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% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2000.

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\centerline{\Largebf RESEARCHES RESPECTING QUATERNIONS:}

\vskip12pt

\centerline{\Largebf FIRST SERIES}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Transactions of the Royal Irish Academy,
vol.~21, part~1 (1848), pp. 199--296.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

This edition is based on the original text in volume~21 of the
{\it Transactions of the Royal Irish Academy}.

The following errors in the original text have been corrected:---

\smallskip

\item{}
in article~18, equation (157), ${\sc r}_{s'}$ was printed as
${\sc r}_s$ in the original text;

\smallskip

\item{}
in article~42, equation (412), the surd on the right hand side
of the equation was missing in the original text.

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

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\line{\hfil Dublin, March 2000}

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{\largeit\noindent
Researches respecting Quaternions. First Series.  By}
{\largerm Sir} {\largesc William Rowan Hamilton},
{\largerm LL.D., V.P.R.I.A.,}
{\largeit Fellow of the American Society of Arts and Sciences; of
the Society of Arts for Scotland; of the Royal Astronomical
Society of London; and of the Royal Northern Society of
Antiquaries at Copenhagen; Corresponding Member of the Institute
of France; Honorary or Corresponding Member of the Royal or
Imperial Academies of St.~Petersburgh, Berlin, and Turin; of the
Royal Societies of Edinburgh and Dublin; of the Cambridge
Philosophical Society; the New York Historical Society; the
Society of Sciences at Lausanne; and of other scientific
Societies in British and foreign Countries; Andrews' Professor of
Astronomy in the University of Dublin; and Royal Astronomer of
Ireland.}

\bigbreak

\centerline{Read November 13, 1843.}

\bigbreak

\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~21 (1848), pp. 199--296.]}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

The researches respecting Quaternions, of the first series of
which an account is submitted in the following pages, are to be
considered as being, at least in their first aspect and
conception, a continuation of those speculations concerning
algebraic Couples, and respecting Algebra itself, regarded as the
science of Pure Time, which were first communicated to the Royal
Irish Academy in November, 1833, and were published in that year
1835 in the seventeenth Volume of its Transactions.  The author
has thus endeavoured to fulfil, at least in part, the intention
which he expressed in the concluding sentence of his former
Essay, in the volume just referred to, of publishing, at a time
then future, some applications of the same view of algebra to a
theory of {\it sets\/} of moments, steps and numbers, which
should include that former theory of {\it couples}.  Some general
remarks on this whole train of speculation, and on its
application to geometrical and physical questions, will be
offered at the end of this paper.  And the author indulges a hope
that the papers containing an account of those subsequent
investigations respecting Quaternions, which he has made, and (in
part) communicated to the Academy, since the date prefixed to
this First Series of Researches, will tend to place the subject
in a still clearer point of view: and, by exhibiting more fully
to mathematicians its interest and its importance, increase the
likelihood of their contributing their aid to its development.

\nobreak\bigskip

{\it Observatory of Trinity College, Dublin, May\/}~3, 1847.

\vfill\eject

\centerline{\it General Conception and Notation of a System or
Set of Moments.}

\nobreak\bigskip

1.
When we have in any manner been led to form successively the
separate conceptions of any number of moments of time, we may
afterwards form the {\it new\/} conception of a {\it system}, or
{\sc momental set}, to which all these separate moments belong;
and may say that this set is of the second, third, fourth, or
$n^{\rm th}$ {\it order\/}, according as the number of moments
which compose it is $2$, $3$, $4$, or $n$: we may also call those
moments the {\it constituent moments\/} of the set.  A
{\it symbol\/} for such a set may be formed by enclosing in
parentheses, with commas interposed between them, the separate
symbols of the moments which compose the set; thus the symbol of
a {\it momental quaternion}, or set of the fourth order, will be
of the form
$$({\sc a}_0, {\sc a}_1, {\sc a}_2, {\sc a}_3),$$
if $A_0$, $A_1$, $A_2$, $A_3$ be employed as symbols to denote
the four separate moments of the quaternion.  If we employ any
other symbol, such as the letter~${\sc q}$, to denote the same
quaternion, or set, we may then write an {\it equation\/} between
the two equisignificant symbols, as follows:
$${\sc q} = ( {\sc a}_0, {\sc a}_1, {\sc a}_2, {\sc a}_3 );
   \eqno (1)$$
and, in like manner, if ${\sc q}'$ denote another quaternion, of
which the four separate moments are denoted by
${\sc a}_0'$, ${\sc a}_1'$, ${\sc a}_2'$, ${\sc a}_3'$,
we shall have this other similar equation,
$${\sc q}' = ( {\sc a}_0', {\sc a}_1', {\sc a}_2', {\sc a}_3' ).
   \eqno (2)$$
An equation of this sort, between two symbols of equinumerous
momental sets, is to be understood as expressing that the
{\it several\/} moments of the one set coincide respectively with
the {\it homologous\/} moments of the other set, primary with
primary, secondary with secondary, and so on: thus if, with the
recent significations of the symbols, we write the
{\it quaternion equation},
$${\sc q}' = {\sc q},
   \eqno (3)$$
or more fully,
$$( {\sc a}_0', {\sc a}_1', {\sc a}_2', {\sc a}_3' )
   =  ( {\sc a}_0, {\sc a}_1, {\sc a}_2, {\sc a}_3 ),
   \eqno (4)$$
we indicate concisely, thereby, the system of the {\it four\/}
following {\it momental equations}, or expressions of four
coincidences between moments of time denoted by different
symbols:
$${\sc a}_0' = {\sc a}_0,\quad
  {\sc a}_1' = {\sc a}_1,\quad
  {\sc a}_2' = {\sc a}_2,\quad
  {\sc a}_3' = {\sc a}_3.
   \eqno (5)$$
The same complex equation, or system of equations, may also be
thus written:
$$( {\sc a}_0', {\sc a}_1', {\sc a}_2', {\sc a}_3' )
   \, ( =, =, =,  = ) \,
      ( {\sc a}_0, {\sc a}_1, {\sc a}_2, {\sc a}_3 );
   \eqno (6)$$
or more concisely thus:
$${\sc q}' \, ( =, =, =,  = ) \, {\sc q}.
   \eqno (7)$$

\bigbreak

\centerline{\it
Characteristics of momental Separation, Recombination, and
Transposition.}

\nobreak\bigskip

2.
In the foregoing article, {\it parentheses\/} have been used as
{\it characteristics of systematic combination}, in order to
combine the symbols of separate moments into the symbol of a
common set.  If we now agree to prefix, conversely,
{\it characteristics of momental separation}, such as
${\sc m}_0, {\sc m}_1,\ldots$ to the symbol of a momental set, in
order to form separate symbols for the {\it separate moments\/}
of that set, we may resolve the equation~(1) into the four
following:
$${\sc m}_0 {\sc q} = {\sc a}_0;\quad
  {\sc m}_1 {\sc q} = {\sc a}_1;\quad
  {\sc m}_2 {\sc q} = {\sc a}_2;\quad
  {\sc m}_3 {\sc q} = {\sc a}_3;
   \eqno (8)$$
and an equation, such as (3), between two momental quaternions or
other sets, ${\sc q}$ and ${\sc q}'$, may, in like manner, be
resolved into equations between moments as follows:
$${\sc m}_0 {\sc q}' = {\sc m}_0 {\sc q};\quad
  {\sc m}_1 {\sc q}' = {\sc m}_1 {\sc q};\quad
   \hbox{\&c.}
   \eqno (9)$$

With these characteristics of combination and separation of
moments, we may write, {\it for any four moments},
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,
the {\it identical\/} equations,
$${\sc a} = {\sc m}_0 ({\sc a}, {\sc b}, {\sc c}, {\sc d});\quad
  {\sc b} = {\sc m}_1 ({\sc a}, {\sc b}, {\sc c}, {\sc d});\quad
   \hbox{\&c.}
   \eqno (10)$$
and for {\it any momental quaternion\/}~${\sc q}$, the identity,
$${\sc q}  = (
      {\sc m}_0 {\sc q},
      {\sc m}_1 {\sc q},
      {\sc m}_2 {\sc q},
      {\sc m}_3 {\sc q} );
   \eqno (11)$$
with other similar expressions for other sets of moments.

The identical expression~(11) may also conveniently be written
thus:
$$1 {\sc q}
   =  ({\sc m}_0, {\sc m}_1, {\sc m}_2, {\sc m}_3) {\sc q}
   =  {\sc m}_{0,1,2,3} {\sc q};
   \eqno (12)$$
$1 {\sc q}$ being regarded as a symbol equivalent to ${\sc q}$,
and the third member of the formula being an abridgment of the
second; and then, by omitting the symbol~${\sc q}$ of that
quaternion of moments which is here the {\it common operand}, we
may write, more concisely,
$$1 = ( {\sc m}_0, {\sc m}_1, {\sc m}_2, {\sc m}_3 )
    = {\sc m}_{0,1,2,3};
   \eqno (13)$$
and may call the second or the third member of this last
symbolical equation a {\it characteristic of recombination\/} (of
a momental set).  The same analogy of notation enables us easily
to form {\it characteristics of momental transposition}, which
shall serve to express the effect of changing the places or
ranks, as primary, secondary, \&c., of the moments of any set,
with reference merely to that conceived and written arrangement
on which the set itself depends for its subjective or symbolic
existence, and without any regard being {\it here\/} had to the
objective or phenomenal succession of the moments in the actual
progression of time.  Thus, from the proposed or assumed
quaternion~(1), we may, in general, derive twenty-three other
quaternions, which shall be all different from it, and from each
other, in consequence of their involving different mental and
symbolic arrangements of the same four moments of time; and these
new quaternions may be denoted by the following expressions:
$$\left. \eqalign{
({\sc a}_0, {\sc a}_1, {\sc a}_3, {\sc a}_2)
   &= {\sc m}_{0,1,3,2} {\sc q};\cr
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
({\sc a}_3, {\sc a}_2, {\sc a}_1, {\sc a}_0)
   &= {\sc m}_{3,2,1,0} {\sc q}.\cr}
   \right\}
   \eqno (14)$$
In this notation we may write the symbolical equations,
$${\sc m}_{3,0,1,2}^4 = 1;\quad
  {\sc m}_{3,0,1,2}   = 1^{1 \over 4};
   \eqno (15)$$
to imply that {\it four successive transpositions}, which are each
of the kind directed by the characteristic~${\sc m}_{3,0,1,2}$,
will {\it reproduce\/} any proposed momental quaternion
$({\sc a}, {\sc b}, {\sc c}, {\sc d})$, as the last of the four
successive results:
$$({\sc d}, {\sc a}, {\sc b}, {\sc c}),\quad
  ({\sc c}, {\sc d}, {\sc a}, {\sc b}),\quad
  ({\sc b}, {\sc c}, {\sc d}, {\sc a}),\quad
  ({\sc a}, {\sc b}, {\sc c}, {\sc d}).
   \eqno (16)$$
And generally, for any set of moments, we may write, by an
analogous use of exponents, the formula
$${\sc m}_{n-1,0,1,\ldots \, n-2}^n = 1;
   \eqno (17)$$
which allows us to establish also this other symbolical equation:
$${\sc m}_{sn-1,0,1,\ldots \, sn-2}^{rn} = 1^{r \over s}.
   \eqno (18)$$
For example, if we take, in this last expression, the values
$n = 4$, $r = 1$, $s = 2$, we are conducted to the following
characteristic of a certain transposition of the moments of an
{\it octad}, which transposition, if it be once repeated, will
{\it restore\/} those eight moments to their original
arrangement, and which is therefore to be regarded as being a
{\it symbolical square root of unity\/}; namely,
$$\omega = 1^{1 \over 2},
   \eqno (19)$$
if
$$\omega = {\sc m}_{4,5,6,7,0,1,2,3}.
   \eqno (20)$$

It may also be here observed, as another example of the notation
of the present article, that if, in addition to this last
characteristic~$\omega$, we introduce three other signs of the
same sort, which we shall call (for a reason that will afterwards
appear) {\it three coordinate characteristics of octadic
transposition}, and shall define as follows:
$$\left. \eqalign{
\omega_1 &= {\sc m}_{5,0,7,2,1,4,3,6};\cr
\omega_2 &= {\sc m}_{6,3,0,5,2,7,4,1};\cr
\omega_3 &= {\sc m}_{7,6,1,0,3,2,5,4};\cr}
   \right\}
   \eqno (21)$$
then these four symbols $\omega$, $\omega_1$, $\omega_2$,
$\omega_3$, will be found to be connected by the relations,
$$\omega_1^2 = \omega_2^2 = \omega_3^2
   =  \omega_1 \omega_2 \omega_3 = \omega;
   \eqno (22)$$
$$\omega \omega_1 = \omega_1 \omega;\quad
  \omega \omega_2 = \omega_2 \omega;\quad
  \omega \omega_3 = \omega_3 \omega;
   \eqno (23)$$
from which, when combined with the equation
$$\omega^2 = 1,
   \eqno (24)$$
these other symbolic equations may be deduced:
$$\left. \multieqalign{
\omega_1 \omega_2 &= \omega_3; &
\omega_2 \omega_3 &= \omega_1; &
\omega_3 \omega_1 &= \omega_2; \cr
\omega_2 \omega_1 &= \omega \omega_3; &
\omega_3 \omega_2 &= \omega \omega_1; &
\omega_1 \omega_3 &= \omega \omega_2; \cr}
   \right\}
   \eqno (25)$$
$$\left. \eqalign{
      \omega_1 \omega_2 \omega_3
   =  \omega_2 \omega_3 \omega_1
   =  \omega_3 \omega_1 \omega_2
   =  \omega;\cr
      \omega_3 \omega_2 \omega_1
   =  \omega_1 \omega_3 \omega_2
   =  \omega_2 \omega_1 \omega_3
   =  1;\cr}
   \right\}
   \eqno (26)$$
$$\left. \eqalign{
      (\omega \omega_1)^2
   =  (\omega \omega_2)^2
   =  (\omega \omega_3)^2
   =  \omega;\cr
      (\omega \omega_1)^4
   =  (\omega \omega_2)^4
   =  (\omega \omega_3)^4
   =  1;\cr
      \omega_1^4
   =  \omega_2^4
   =  \omega_3^4
   =  1.\cr}
   \right\}
   \eqno (27)$$

\bigbreak

\centerline{\it
Forms of ordinal Relations between Moments, or Sets of Moments;
and Comparisons of}

\nobreak\vskip 3pt

\centerline{\it
Pairs of Moments, or Pairs of Sets, with respect to Analogy or
Non-analogy.}

\nobreak\bigskip

3.
If the moment denoted by the symbol~$A'$ be supposed to be
essentially, as well as symbolically, distinct from the moment
denoted by ${\sc a}$, so that these two symbols denote two
{\it different moments\/} in the progression of time, and that
therefore the momental equation ${\sc a}' = {\sc a}$ does
{\it not\/} hold good; then it is an immediate and necessary
result of our notion or intuition of {\it time}, that the
moment~${\sc a}'$, since it {\it is not coincident
with\/}~${\sc a}$, must be {\it either later or earlier\/} than
it.  Using, therefore, as in a former Essay,\footnote*{On Algebra
as the Science of Pure Time.---Transactions of the Royal Irish
Academy, vol.~xvii.  Dublin, 1835.}
the signs $>$~$<$, which are commonly employed as marks of
inequality of magnitude, to denote these {\it two modes of
ordinal diversity}, and thus employing the formula
$${\sc a}' > {\sc a},
   \eqno (28)$$
to express, {\it without any reference to magnitude}, that the
moment~${\sc a}'$ is {\it later\/} than ${\sc a}$; but, on the
contrary, using this other formula, in like manner without
reference to magnitude,
$${\sc a}' < {\sc a},
   \eqno (29)$$
to express that ${\sc a}'$ is {\it earlier\/} than ${\sc a}$; so
that the character~$>$ is here used as a {\it sign of
subsequence}, whereas the mark~$<$ is, on the contrary, in this
notation, a {\it sign of precedence\/}; while the formula, or
equation,
$${\sc a}' = {\sc a},
   \eqno (30)$$
still expresses that the moment~${\sc a}'$ is {\it coincident\/}
or (simultaneous) with ${\sc a}$, so that the mark~$=$ is at once
an expression of {\it symbolic equivalence\/} and also a
{\it sign of simultaneity\/}; we see that the comparison of any
{\it sought\/} moment~${\sc a}'$, regarded as an {\it ordinand},
with any {\it given\/} moment~${\sc a}$ regarded as an
{\it ordinator}, must conduct to {\it one or other\/} of these
{\it three forms of ordinal relation}, (28), (29), (30); and that
no such comparison of two moments can conduct to two of these
three forms, or modes of relation, at once.  In like manner, if
we compare any {\it set\/} of $n$ moments
$({\sc a}_0', {\sc a}_1',\ldots \, {\sc a}_{n-1}')$,
regarded as an {\it ordinand set}, with any other equinumerous
momental set
$({\sc a}_0, {\sc a}_1,\ldots \, {\sc a}_{n-1})$,
regarded as an {\it ordinator set}, by comparing {\it each\/}
moment of the one set with the homologous moment of the other
set, primary with primary, secondary with secondary, and so
forth, we shall obtain in general $n$ different ordinal
relations, which may, however, be combined, in thought and in
expression, into {\it one system}, or {\sc ordinal set\/}; and
this set, which may be said to be of the $n^{\rm th}$
{\it order}, will admit of $3^n$ different {\it forms}, obtained
by attributing separately to each of its $n$ {\it constituent
ordinal relations\/} each of the three forms $>$~$<$~$=$.  For
example, the complex ordinal relation which a sought momental
quaternion~${\sc q}'$, regarded as an ordinand, bears to a given
momental quaternion~${\sc q}$, regarded as an ordinator, is
composed of four ordinal relations between the homologous moments
of these two momental sets, of which four relations each
separately may be one of subsequence ($>$), or of precedence
($<$), or of simultaneity ($=$): and hence this complex ordinal
relation of ${\sc q}'$ to ${\sc q}$ may receive any one of
$3^4 = 81$ different forms, of which one, namely, the case of
{\it quadruple momental coincidence}, has been considered in the
first article, and of which the others may be denoted on a
similar plan.  Thus to write the formula
$${\sc q}' \, ( >, =, <, = ) \, {\sc q},
   \eqno (31)$$
if ${\sc q}'$ and ${\sc q}$ denote the quaternions (1) and (2),
may be regarded as a mode of concisely expressing the following
system of four separate ordinal relations between moments,
$${\sc a}_0' > {\sc a}_0;\quad
  {\sc a}_1' = {\sc a}_1;\quad
  {\sc a}_2' < {\sc a}_2;\quad
  {\sc a}_3' = {\sc a}_3;
   \eqno (32)$$
or, in the notation of the second article,
$${\sc m}_0 {\sc q}' > {\sc m}_0 {\sc q};\quad
  {\sc m}_1 {\sc q}' = {\sc m}_1 {\sc q};\quad
  {\sc m}_2 {\sc q}' < {\sc m}_2 {\sc q};\quad
  {\sc m}_3 {\sc q}' = {\sc m}_3 {\sc q};
   \eqno (33)$$
and similarly in other cases.

\bigbreak

4.
Again, as we have compared two moments, or two sets of moments,
or have conceived them to be compared with each other, with a
view to discover the (simple or complex) {\it ordinal
relations\/} existing between them, so we may now compare, or
conceive to be compared, {\it two pairs\/} of moments, or of
momental sets, with respect to their (simple or complex)
{\it analogy\/} or {\it non-analogy\/}; that is, with respect to
the {\it similarity or dissimilarity of the two simple or complex
ordinal relations}, which are discovered by the two separate
comparisons of the moments or sets belonging to each separate
pair.  Representing (as in the former Essay) by the notation
$${\sc d} - {\sc c} = {\sc b} - {\sc a},
   \eqno (34)$$
the existence of an {\it analogy\/} of this sort between the two
pairs of moments, ${\sc a}$,~${\sc b}$, and ${\sc c}$,~${\sc d}$,
or the supposition of an {\it exact similarity\/} between the two
ordinal relations of ${\sc d}$ to ${\sc c}$, and of ${\sc b}$ to
${\sc a}$; we may, in like manner, denote by the formula,
$${\sc q}''' - {\sc q}'' = {\sc q}' - {\sc q},
   \eqno (35)$$
the {\it complex analogy\/} which may be conceived to exist
between the two pairs of quaternions, or other momental sets,
${\sc q}$,~${\sc q}'$, and ${\sc q}''$,~${\sc q}'''$, belonging
all to any one determined order~$n$, that is, containing each $n$
moments.  This analogy (35) requires, for its existence, in the
view here taken, that the $n$ constituent ordinal relations
between moments which compose, by their mental and symbolic
combination into one system, the complex ordinal relation of the
set~${\sc q}'''$ to the set~${\sc q}''$, should,
{\it separately\/} and respectively, be exactly similar to those
$n$ other constituent ordinal relations between moments, which
collectively compose the other complex ordinal relation of the
set~${\sc q}'$ to the set~${\sc q}$; for then, but not otherwise,
do we regard the one {\it complex\/} ordinal relation as being in
{\it all\/} respects similar to the other.  In symbolical
language, the complex {\it set-analogy\/} (or analogy between
pairs of sets) {\it of the $n^{\rm th}$ order\/} (35) may be
resolved into $n$ {\it momental analogies\/} (or analogies
between pairs of moments), namely, the following:
$$\left. \eqalign{
   &  {\sc m}_0 {\sc q}''' - {\sc m}_0 {\sc q}''
    = {\sc m}_0 {\sc q}' - {\sc m}_0 {\sc q};\cr
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
   &  {\sc m}_{n-1} {\sc q}''' - {\sc m}_{n-1} {\sc q}''
    = {\sc m}_{n-1} {\sc q}' - {\sc m}_{n-1} {\sc q};\cr}
   \right\}
   \eqno (36)$$
of which each separately is to be interpreted on the same plan as
the analogy~(34).  The two formul{\ae} of {\it momental
non-analogies}, or of {\it dissimilar ordinal relations\/}
between pairs of moments,
$$\left. \eqalign{
{\sc d} - {\sc c} &> {\sc b} - {\sc a},\cr
{\sc d} - {\sc c} &< {\sc b} - {\sc a},\cr}
   \right\}
   \eqno (37)$$
may still be interpreted as in the former Essay; the first formula
(37) denoting that the relstion of the moment~${\sc d}$ to
${\sc c}$ is, as compared with the relation of ${\sc b}$ to
${\sc a}$, a relation of {\it comparative lateness\/}; and the
second formula~(37) denoting, on the contrary, that the former
ordinal relation, as compared with the latter, is one of
{\it comparative earliness\/}: and because, in the first case the
moment~${\sc d}$ is {\it too late}, while in the second case this
moment is {\it too early}, to satisfy the analogy~(34), we may
still call the first formula~(37) a momental {\it non-analogy of
subsequence}, and may call the second formula~(37) a
{\it non-analogy of precedence}.  By compounding several such
momental non-analogies, or even one such, with any number of
momental analogies, into one system, we shall compose a
{\it complex non-analogy\/} between two pairs of momental sets,
which may easily be denoted on the plan of recent notations;
thus, if we make, for abridgment,
$$\left. \eqalign{
{\sc q}''
   &= ({\sc a}_0'', {\sc a}_1'',
            {\sc a}_2'', {\sc a}_3''),\cr
{\sc q}'''
   &= ({\sc a}_0''', {\sc a}_1''',
            {\sc a}_2''', {\sc a}_3'''),\cr}
   \right\}
   \eqno (38)$$
retaining for ${\sc q}$ and ${\sc q}'$ the same meanings as in
the equations (1), (2), and then write the formula
$${\sc q}''' - {\sc q}'' \, ( >, =, <, = ) \, {\sc q}' - {\sc q},
   \eqno (39)$$
we are to be considered as expressing concisely hereby a complex
non-analogy between two pairs of momental quaternions,
${\sc q}$,~${\sc q}'$, and ${\sc q}''$,~${\sc q}'''$,
which may be resolved into the following system of mixed
analogies and non-analogies between four pairs of moments:
$$\left. \multieqalign{
   {\sc m}_0 {\sc q}''' - {\sc m}_0 {\sc q}''
      &> {\sc m}_0 {\sc q}' - {\sc m}_0 {\sc q};
         \quad\hbox{or,} &
   {\sc a}_0''' - {\sc a}_0'' &> {\sc a}_0' - {\sc a}_0;\cr
   {\sc m}_1 {\sc q}''' - {\sc m}_1 {\sc q}''
      &= {\sc m}_1 {\sc q}' - {\sc m}_1 {\sc q}; &
   {\sc a}_1''' - {\sc a}_1'' &= {\sc a}_1' - {\sc a}_1;\cr
   {\sc m}_2 {\sc q}''' - {\sc m}_2 {\sc q}''
      &< {\sc m}_2 {\sc q}' - {\sc m}_2 {\sc q}; &
   {\sc a}_2''' - {\sc a}_2'' &< {\sc a}_2' - {\sc a}_2;\cr
   {\sc m}_3 {\sc q}''' - {\sc m}_3 {\sc q}''
      &= {\sc m}_3 {\sc q}' - {\sc m}_3 {\sc q}; &
   {\sc a}_3''' - {\sc a}_3'' &= {\sc a}_3' - {\sc a}_3.\cr}
   \right\}
   \eqno (40)$$

A little consideration suffices to show, by the aid of the
fundamental notion of {\sc time}, which enters essentially into
this whole theory (at least as the subject is here viewed), that
every simple or complex analogy or non-analogy of the kind
considered in the present article admits of {\it alternation\/};
that is to say, if we call the moments ${\sc b}$ and ${\sc c}$,
or the sets ${\sc q}'$ and ${\sc q}''$, the {\it means}, and call
the moments ${\sc a}$ and ${\sc d}$, or the sets ${\sc q}$ and
${\sc q}'''$, the {\it extremes}, of the analogy or
non-analogy, it is allowed to {\it interchange the means\/} or to
{\it interchange the extremes\/} among themselves, without
destroying the truth or changing the character of the formula.
For example, under the conditions (40), we may write, instead of
(39), either of the two following forms:
$$\left. \eqalign{
{\sc q}''' - {\sc q}'
   \, ( >, =, <, = ) \, {\sc q}'' - {\sc q};\cr
{\sc q} - {\sc q}''
   \, ( >, =, <, = ) \, {\sc q}' - {\sc q}'''.\cr}
   \right\}
   \eqno (41)$$
We may also employ {\it inversion}, that is, we may substitute
extremes for means, and means for extremes, provided that we, at
the same time, change each of the two signs of ordinal diversity
between moments, and every complex sign of ordinal non-analogy
between momental pairs, to the contrary or {\it opposite sign},
by changing $>$ to $<$, and $<$ to $>$; thus we may write the
complex non-analogy (39) under this other or {\it inverse\/}
form:
$${\sc q}'' - {\sc q}''' \, ( <, =, >, = ) \, {\sc q} - {\sc q}'.
   \eqno (42)$$
And with the same conceptions, and the same plan of notation, we
are led to regard the following formula of {\it quadruple
momental analogy},
$${\sc q}''' - {\sc q}'' \, ( =, =, =, = ) \, {\sc q}' - {\sc q},
   \eqno (43)$$
as being only a fuller expression of that complex analogy between
the two pairs of quaternions ${\sc q}$,~${\sc q}'$, and
${\sc q}''$,~${\sc q}'''$, which is more briefly denoted by the
formula (35).

\bigbreak

5.
Consistently with the same modes of interpreting formul{\ae} for
the expression of any simple or complex analogy or non-analogy
between pairs of moments or of sets, or of any similarity or
dissimilarity between simple or complex ordinal relations, if we
agree that the symbol~$0$, {\it when it occurs as one member of
any such formula}, shall be regarded as a {\it symbol of the
relation of ordinal identity}, writing thus for any two identical
moments, or identical sets,
$${\sc a} - {\sc a} = 0,\quad
  {\sc q} - {\sc q} = 0;
   \eqno (44)$$
we may then not only write
$${\sc a}' - {\sc a} = 0,\quad
  {\sc q}' - {\sc q} = 0,
   \eqno (45)$$
as transformations of the equations (30) and (3); but also
$${\sc a}' - {\sc a} > 0,\quad
  {\sc a}' - {\sc a} < 0,
   \eqno (46)$$
as transformations respectively of the two formul{\ae} of ordinal
diversity, (28) and (29); and may write
$${\sc q}' - {\sc q} \, ( >, =, <, = ) \, 0,
   \eqno (47)$$
instead of the formula (31).  And if we employ small Roman
letters, with or without accents or indices, such as
${\sr a}$,~${\sr a}_0$, \&c., to denote generally {\it any ordinal
relations\/} between moments, which may or may not be relations of
identity, and which may otherwise be denoted by such symbols as
${\sc b} - {\sc a}$, ${\sc a}_0' - {\sc a}_0$, \&c., which have
been already used as members of formul{\ae} expressing analogies
or non-analogies; writing, for example,
$$\left. \multieqalign{
{\sc a}_0' - {\sc a}_0 &= {\sr a}_0, &
{\sc a}_1' - {\sc a}_1 &= {\sr a}_1, \, \ldots \cr
{\sc a}_0''' - {\sc a}_0'' &= {\sr a}_0', &
{\sc a}_1''' - {\sc a}_1'' &= {\sr a}_1', \, \ldots \cr}
   \right\}
   \eqno (48)$$
and extending this notation so as to introduce the corresponding
abridgments,
$${\sc q}' - {\sc q} = {\sr q},\quad
  {\sc q}''' - {\sc q}'' = {\sr q}';
   \eqno (49)$$
then we may not only transform the formula~(31), or the system of
the formul{\ae}~(32), by writing
$${\sr q} \, ( >, =, <, = ) \, 0;
   \eqno (50)$$
but also, on the same plan, may substitute for the expression of
the complex non-analogy~(39) this more concise expression,
$${\sr q}' \, ( >, =, <, = ) \, {\sr q}.
   \eqno (51)$$
For in this notation (as in that of the former Essay), the first,
second, and third of the three formul{\ae},
$${\sr a} > 0,\quad {\sr a} < 0,\quad {\sr a} = 0,
   \eqno (52)$$
express, respectively, that the ordinal relation between moments,
denoted by the letter~${\sr a}$, is one of lateness, or of
earliness, or of simultaneity; and in like manner, the three
written assertions,
$${\sr b} > {\sr a},\quad
  {\sr b} < {\sr a},\quad
  {\sr b} = {\sr a},
   \eqno (53)$$
express, respectively, that the ordinal relation between the two
moments of one pair, denoted by ${\sr b}$, as {\it compared with
the relation\/} between the two moments of another pair, denoted
by ${\sr a}$, is one of {\it comparative\/} lateness, comparative
earliness, or comparative coincidence, that is, analogy.  And to
mark generally the {\it unity of the conception of an ordinal
set}, or system of ordinal relations, such as was considered in
the foregoing article, we may agree to denote such a system or
set of relations by writing in parentheses, with commas
interposed, the symbols of those separate relations; and thus may
write the formula,
$${\sc q}' - {\sc q}
   =  ( {\sc a}_0' - {\sc a}_0,
        {\sc a}_1' - {\sc a}_1,\ldots \,
        {\sc a}_{n-1}' - {\sc a}_{n-1} );
   \eqno (54)$$
or, more concisely, by the abridgments (48) and (49), if we
confine ourselves to the case of an ordinal quaternion,
$${\sr q} = ({\sr a}_0, {\sr a}_1, {\sr a}_2, {\sr a}_3).
   \eqno (55)$$

\bigbreak

\centerline{\it
Operations on an Ordinal Set; Coordinate Characteristics of
Quaternion-Derivation.}

\nobreak\bigskip

6.
We may now treat this last expression for an ordinal quaternion
in the same way as the expression for a momental quaternion was
treated in the second article.  Let ${\sc r}_0$,~${\sc r}_1$,
\&c., be {\it characteristics of ordinal separation}, analogous
to the characteristics of momental separation,
${\sc m}_0$, ${\sc m}_1$, \&c.; we may then, with their help,
decompose the equation (55) into four others, as follows:
$${\sc r}_0 {\sr q} = {\sr a}_0;\quad
  {\sc r}_1 {\sr q} = {\sr a}_1;\quad
  {\sc r}_2 {\sr q} = {\sr a}_2;\quad
  {\sc r}_3 {\sr q} = {\sr a}_3;
   \eqno (56)$$
we may therefore write, {\it for any four ordinal relations},
${\sr a}$,~${\sr b}$,~${\sr c}$,~${\sr d}$, between moments, the
identical equations,
$${\sr a} = {\sc r}_0 ({\sr a}, {\sr b}, {\sr c}, {\sr d});\quad
  {\sr b} = {\sc r}_1 ({\sr a}, {\sr b}, {\sr c}, {\sr d});
   \quad\hbox{\&c.};
   \eqno (57)$$
and, for {\it any ordinal quaternion}, we may write the
corresponding identity,
$${\sr q} = (
      {\sc r}_0 {\sr q},
      {\sc r}_1 {\sr q},
      {\sc r}_2 {\sr q},
      {\sc r}_3 {\sr q} );
   \eqno (58)$$
or more concisely, by abridgments analogous to those marked (13),
$$1 = ( {\sc r}_0, {\sc r}_1, {\sc r}_2, {\sc r}_3 )
   =  {\sc r}_{0,1,2,3};
   \eqno (59)$$
with formul{\ae} of the same kind for ordinal sets of higher
orders.  {\it Characteristics of ordinal transposition\/} are
easily formed on the same plan; and we may write, for example, as
the expression of one such transposition performed on the ordinal
quaternion (55),
$${\sc r}_{3,0,1,2} {\sr q}
   =  ({\sr a}_3, {\sr a}_0, {\sr a}_1, {\sr a}_2);
   \eqno (60)$$
and hence may deduce this symbolic equation, analogous to (15),
$${\sc r}_{3,0,1,2}^4 = 1.
   \eqno (61)$$

If, instead of thus {\it transposing the ordinal relations}, we
transpose, in the expression of any one relation, {\it the two
related moments}, or momental sets, we then obtain, in general, a
new ordinal relation, which is the {\it inverse\/} or
{\it opposite\/} of the old relation, or is that old one with its
{\it sign\/} ({\it or signs\/}) {\it changed}, each constituent
relation of {\it earliness\/} being altered to a relation of
{\it lateness\/} (in the same degree) and {\it vice vers\^{a}\/}:
a change which may be expressed, according to known analogies of
notation, by prefixing the sign~$-$ to the symbol of the simple
or complex relation which has thus been altered: for example, the
equations (48), (49) give, by this change of signs,
$${\sc a}_0 - {\sc a}_0' = - {\sr a}_0,\quad
  {\sc a}_1 - {\sc a}_1' = - {\sr a}_1,
   \quad\hbox{\&c.};
   \eqno (62)$$
and
$${\sc q} - {\sc q}' = - {\sr q},
   \quad\hbox{\&c.}
   \eqno (63)$$
Hence we may write, as a consequence of the formula (55), the
following:
$$- {\sr q}
   = ( - {\sr a}_0, - {\sr a}_1, - {\sr a}_2, - {\sr a}_3 );
   \eqno (64)$$
that is, for any ordinal quaternion, we have
$$-1 = ( - {\sc r}_0, - {\sc r}_1, - {\sc r}_2, - {\sc r}_3 ),
   \eqno (65)$$
with similar results for other ordinal sets.  The notation may be
abridged if we agree to write, for the present, such formul{\ae}
as the following:
$$\left. \eqalign{
  - {\sc r}_0 = {\sc r}_{-0};\quad
  - {\sc r}_1 = {\sc r}_{-1};\quad\ldots \cr
  ( {\sc r}_{-1}, {\sc r}_{-0},\ldots )
   = {\sc r}_{-1,-0,\ldots};\quad\hbox{\&c.}\cr}
   \right\}
   \eqno (66)$$
for then we can not only express the symbolical equation (65)
under the shorter form,
$$-1 = {\sc r}_{-0,-1,-2,-3},
   \eqno (67)$$
but can compose, generally, {\it characteristics of ordinal
derivation}, which shall express the joint or combined
performance of several simultaneous or successive acts of
separation, invertion, transposition, and recombination of the
constituent relations of any ordinal set.  Thus if we operate
twice successively on an {\it ordinal couple\/}
$({\sr a}_0, {\sr a}_1)$, by the characteristic of derivation
${\sc r}_{-1,0}$, we obtain thereby the two new or derived
couples:
$$\left. \eqalign{
{\sc r}_{-1,0} ({\sr a}_0, {\sr a}_1)
   &= (- {\sr a}_1, {\sr a}_0);\cr
{\sc r}_{-1,0}^2 ({\sr a}_0, {\sr a}_1)
   &= {\sc r}_{-1,0} (- {\sr a}_1, {\sr a}_0) \cr
   &= (- {\sr a}_0, - {\sr a}_1)
    = - ({\sr a}_0, {\sr a}_1);\cr}
   \right\}
   \eqno (68)$$
of which the last is merely the original couple
$({\sr a}_0, {\sr a}_1)$ with its sign changed; so that we have
the symbolic equation,
$${\sc r}_{-1,0}^2 = -1.
   \eqno (69)$$
This symbolic result, presented under a slightly different form,
was made the foundation of the theory of algebraic couples, and
of the use of the symbol $\surd -1$ in algebra, proposed by the
present writer, in that Essay, already several times referred to,
which was published in a former volume of the
Transactions of this Academy; for the symbolic equation
(vol.~xvii, page~417, equation 157)
$$\surd (-1) = (0,1),$$
was there given, in which the essential character of the
{\it number-couple\/}  $(0,1)$ was that, when used as a
multiplier, it transformed one {\it step-couple\/}
$({\sr a}_1, {\sr a}_2)$, that is to say, one couple of steps,
${\sr a}_1$,~${\sr a}_2$, in the progression of time, or one
couple of ordinal relations between moments, into another couple
of steps or of relations in the same progression of time,
according to the law,
$$(0,1) ({\sr a}_1, {\sr a}_2)
   =  (- {\sr a}_2, {\sr a}_1);$$
which agrees with the process directed by the recent
characteristic of derivation,~${\sc r}_{-1,0}$, and was included
in the equation (37), page~401, of the volume lately cited.
Again, if we now regard $i$,~$j$,~$k$ as three characteristics of
operation on an ordinal quaternion, defined as follows:
$$\left. \eqalign{
i &= {\sc r}_{-1,0,-3,2};\cr
j &= {\sc r}_{-2,3,0,-1};\cr
k &= {\sc r}_{-3,-2,1,0};\cr}
   \right\}
   \eqno (70)$$
we shall have the four following symbolic equations, which will
be found to be of essential importance in the present theory of
quaternions:
$$\left. \eqalign{
i^2 &= -1;\cr
j^2 &= -1;\cr
k^2 &= -1;\cr
ijk &= -1;\cr}
   \right\}
   \eqno (71)$$
and which may be concisely expressed under the form of a single
but continued equation, as follows:
$$i^2 = j^2 = k^2 = ijk = -1.
   \eqno (72) = ({\sc a})$$

\bigbreak

7.
To leave no doubt respecting the truth or meaning of these
important symbolical relations (72) or ({\sc a}), between the
{\it three coordinate characteristics of quaternion-derivation},
$i$,~$j$,~$k$, defined by the equations (70), we shall here
exhibit distinctly the successive steps or stages of the
transformations which are indicated by those characteristics.
Suppose then that any ordinal quaternion~${\sr q}$, or any set of
four ordinal relations,
${\sr a}$,~${\sr b}$,~${\sr c}$,~${\sr d}$,
between moments of time, is proposed as the subject of the
operations.

For the purpose of operating on this quaternion by the
characteristic of derivation~$i$, we may first write the
following definitional equation between its two symbols,
$${\sr q} = ({\sr a}, {\sr b}, {\sr c}, {\sr d}),
   \eqno (73)$$
and then resolve this complex equation into its four components,
or constituents, with the help of the signs of ordinal
separation, ${\sc r}_0$, \&c., as follows:
$${\sc r}_0 {\sr q} = {\sr a};\quad
  {\sc r}_1 {\sr q} = {\sr b};\quad
  {\sc r}_2 {\sr q} = {\sr c};\quad
  {\sc r}_3 {\sr q} = {\sr d}.
   \eqno (74)$$

In the next place, the definition (70) of $i$, combined with the
notation~(66), directs us to change the signs of the second and
fourth of these equations (74), and then to make the first and
second equations change places with each other, interchanging
also, at the same time, the places of the third and fourth, so as
to form this new system of four equations:
$${\sc r}_{-1} {\sr q} = - {\sr b};\quad
  {\sc r}_0    {\sr q} =   {\sr a};\quad
  {\sc r}_{-3} {\sr q} = - {\sr d};\quad
  {\sc r}_2    {\sr q} =   {\sr c}.
   \eqno (75)$$
We are then to combine these four constituent ordinal relations,
thus partially inverted and transposed, namely,
$- {\sr b}$,~${\sr a}$,~$- {\sr d}$, and ${\sr c}$, into a new
ordinal quaternion; and this will be, by definition, the
{\it first coordinate derivative},~$i {\sr q}$, of the proposed
quaternion~${\sr q}$; so that we may now write, as derived from
the equation~(73), by the {\it first coordinate mode of
quaternion derivation}, the equation,
$$i {\sr q} = ( - {\sr b}, {\sr a}, - {\sr d}, {\sr c}).
   \eqno (76)$$
If now we repeat this process of derivation, we get successively
the two following systems of four equations:
$${\sc r}_0    \mathbin{.} i {\sr q} = - {\sr b};\quad
  {\sc r}_1    \mathbin{.} i {\sr q} =   {\sr a};\quad
  {\sc r}_2    \mathbin{.} i {\sr q} = - {\sr d};\quad
  {\sc r}_3    \mathbin{.} i {\sr q} =   {\sr c};
   \eqno (77)$$
$${\sc r}_{-1} \mathbin{.} i {\sr q} = - {\sr a};\quad
  {\sc r}_0    \mathbin{.} i {\sr q} = - {\sr b};\quad
  {\sc r}_{-3} \mathbin{.} i {\sr q} = - {\sr c};\quad
  {\sc r}_2    \mathbin{.} i {\sr q} = - {\sr d};
   \eqno (78)$$
and, finally, by a new combination of those four last ordinal
relations into one ordinal quaternion, which is {\it the
derivative of the derivative of ${\sr q}$ in the first coordinate
mode}, we find
$$i^2 {\sr q} = i \mathbin{.} i {\sr q}
   =  ( - {\sr a}, - {\sr b}, - {\sr c}, - {\sr d} )
   = - {\sr q};
   \eqno (79)$$
so that this {\it repeated process of derivation\/} by the
characteristic~$i$ has {\it changed the sign of the
quaternion},~${\sr q}$, by changing the sign of {\it each\/} of
its four constituent ordinal relations,
${\sr a}$,~${\sr b}$,~${\sr c}$,~${\sr d}$;
which is the property expressed by the first equation~(71),
namely, by the formula,
$$i^2 = -1.
   \eqno (71,1)$$

By exactly similar operations, except so far as the second
symbolic equation~(70) differs from the first, we find, for the
{\it second coordinate derivative},~$j {\sr q}$, of the same
proposed quaternion,~${\sr q}$, the expression,
$$j {\sr q} = (- {\sr c}, {\sr d}, {\sr a}, - {\sr b});
   \eqno (80)$$
and for the {\it derivative of the derivative in the second
mode},
$$j^2 {\sr q}
   =  j \mathbin{.} j {\sr q}
   =  (- {\sr a}, - {\sr b}, - {\sr c}, - {\sr d})
   =  - {\sr q}
   =  - 1 {\sr q};
   \eqno (81)$$
the symbols~$1 {\sr q}$ and ${\sr q}$ (like $1 {\sc q}$ and
${\sc q}$) being regarded as equivalent: which result (81)
justifies the second equation~(71), by giving the symbolic
equation,
$$j^2 = -1.
   \eqno (71,2)$$
And in like manner the {\it third coordinate
derivative},~$k {\sr q}$, is, by the third equation~(70),
expressed as follows:
$$k {\sr q} = (- {\sr d}, - {\sr c}, {\sr b}, {\sr a});
   \eqno (82)$$
so that, by repeating this process of derivation, we find that
the {\it derivative of the second order, in the third mode}, as
well as in each of the two other modes, is {\it the original
quaternion with its sign changed},
$$k^2 {\sr q} = k \mathbin{.} k {\sr q}
   =  (- {\sr a}, - {\sr b}, - {\sr c}, - {\sr d})
   =  -1 {\sr q};
   \eqno (83)$$
or, by detaching the symbols of operation from those of the
common operand,
$$k^2 = -1.
   \eqno (71,3)$$

Finally, if we operate on the expression (82) for $k {\sr q}$, by
the characteristic~$j$, we find
$$\eqalignno{
j \mathbin{.} k {\sr q}
   &= {\sc r}_{-2,3,0,-1}
         (- {\sr d}, - {\sr c}, {\sr b}, {\sr a}) \cr
   &= (- {\sr b}, {\sr a}, - {\sr d}, {\sr c})
    = i {\sr q};
   &(84)\cr}$$
and, therefore, operating on this result by $i$, we obtain,
$$i \mathbin{.} j \mathbin{.} k {\sr q}
   = i \mathbin{.} i {\sr q}
   =  -1 {\sr q},
   \eqno (85)$$
that is,
$$ijk = -1;
   \eqno (71,4)$$
so that {\it the first coordinate derivative, of the second
coordinate derivative, of the third coordinate derivative of any
ordinal quaternion, is equal to that quaternion with its sign
changed\/}; and all the parts of the compound assertion~(72),
or~({\sc a}), are justified.

\bigbreak

8.
We see, at the same time, by (84), that
$$jk = i;
   \eqno (86)$$
or that a derivation in the third mode, followed by a derivation
in the second mode, is equivalent to a derivation in the first
mode.  If, on the contrary, we had effected the two successive
derivations in the {\it opposite order}, operating first in the
second mode, and afterwards in the third mode, we should have
obtained an {\it opposite result}, that is, a result which might
be formed from the previous result by changing the sign of the
final ordinal quaternion: for if we operate on the
expression~(80) by $k$, we get
$$kj {\sr q} = ({\sr b}, - {\sr a}, {\sr d}, - {\sr c})
   =  - i {\sr q},
   \eqno (87)$$
giving the symbolic equation,
$$kj = -i,
   \eqno (88)$$
of which the contrast to the equation~(86) is highly worthy of
attention.  Another contrast of the same sort presents itself,
between the results of operating on the expression~(80) by the
characteristic~$i$, and on the expression~(76) by the
characteristic~$j$; for these two processes give,
$$\left. \eqalign{
ij {\sr q} = (- {\sr d}, - {\sr c}, {\sr b}, {\sr a})
   &= k {\sr q};\cr
ji {\sr q} = ({\sr d}, {\sr c}, - {\sr b}, - {\sr a})
   &= -k {\sr q};\cr}
   \right\}
   \eqno (89)$$
or, more concisely,
$$ij = k;\quad ji = -k.
   \eqno (90)$$
And, finally, we find, in like manner, by operating on (76) by
$k$, and on (82) by $i$, the two contrasted results,
$$\left. \eqalign{
ki {\sr q} = (- {\sr c}, {\sr d}, {\sr a}, - {\sr b})
   &= j {\sr q};\cr
ik {\sr q} = ({\sr c}, - {\sr d}, - {\sr a}, {\sr b})
   &= -j {\sr q};\cr}
   \right\}
   \eqno (91)$$
giving
$$ki = j;\quad
  ik = -j.
   \eqno (92)$$
The importance and singularity of these results (86) (88) (90)
(92) induce us to collect them here into one view, as follows:
$$\left. \multieqalign{
ij &=   k;& ji &= - k;\cr
jk &=   i;& kj &= - i;\cr
ki &=   j;& ik &= - j.\cr}
   \right\}
   \eqno (93) = ({\sc b})$$

\bigbreak

9.
It ought, however, to be observed, that when once the fundamental
formula, or continued equation~({\sc a}), has been established,
no new operations of {\it actual\/} derivation of quaternions, by
inversions and transpositions of ordinal relations between
moments, such as have been performed in the foregoing article,
are {\it necessary}, for the deduction of these
equations~({\sc b}).  Thus if we knew, by any process
independent of the actual derivations (84), that
$i^2 = ijk = -1$, or that
$i^2 {\sr q} = ijk {\sr q} = - {\sr q}$,
whatever ordinal quaternion~${\sr q}$ may be, we could infer
immediately that
$$jk {\sr q}
   =  - i^2 \mathbin{.} jk {\sr q}
   =  - i \mathbin{.} ijk {\sr q}
   =  - i (- {\sr q})
   =    i {\sr q},
   \eqno (94)$$
and thus could return to the symbolic equation~(86), or to the
essential part of the relation (84), from the
equations~({\sc a}).  Again, from those equations~({\sc a}) we
can infer that
$$ij \mathbin{.} k {\sr q} = ijk {\sr q}
   =  - {\sr q} = k^2 {\sr q} = k \mathbin{.} k {\sr q},
   \eqno (95)$$
and, therefore, suppressing the symbol~$k {\sr q}$ of the common
operand, which may represent any ordinal quaternion, we obtain
the first equation~(90), namely $ij = k$.  Operating on this by
$i$, and changing $i^2$ to $-1$, we find the second
equation~(92), $ik = -j$.  Operating with this on $-k {\sr q}$,
we obtain again $i = jk$. Operating on this by $j$, we get
$ji = -k$; that is, we are conducted to the second equation~(90).
Operating with this on $- i {\sr q}$, we find the first
equation~(92), namely, $ki = j$.  And, finally, operating on this
equation by $k$, we are brought to the equation~(88), namely,
$kj = -i$, which completes the symbolic deduction of ({\sc b})
from ({\sc a}).

Either by a deduction of this sort, or by actually performing the
operations indicated, we find also that
$$kji = 1;
   \eqno (96)$$
that is to say, if we operate successively on any ordinal
quaternion~${\sr q}$ by the three modes of coordinate derivation,
$i$,~$j$,~$k$, in their order (first by $i$, then by $j$, and
finally by $k$), the result will be the original quaternion
itself.  And if we make, for abridgment, in the notation of the
sixth article,
$$\left. \eqalign{
i' &= {\sc r}_{1,-0,3,-2};\cr
j' &= {\sc r}_{2,-3,-0,1};\cr
k' &= {\sc r}_{3,2,-1,-0};\cr}
   \right\}
   \eqno (97)$$
so that the results of the operation of these three new
characteristics, $i'$, $j'$, $k'$, on the quaternion~(73), are,
respectively,
$$\left. \eqalign{
i' {\sr q} &= ({\sr b}, - {\sr a}, {\sr d}, - {\sr c});\cr
j' {\sr q} &= ({\sr c}, - {\sr d}, - {\sr a}, {\sr b});\cr
k' {\sr q} &= ({\sr d}, {\sr c}, - {\sr b}, - {\sr a});\cr}
   \right\}
   \eqno (98)$$
we shall then have not only the relations,
$$i' = - i,\quad
  j' = - j,\quad
  k' = - k,
   \eqno (99)$$
but also these others,
$$\left. \eqalign{
i' i = i i' &= 1;\cr
j' j = j j' &= 1;\cr
k' k = k k' &= 1;\cr}
   \right\}
   \eqno (100)$$
on which account we may call these three new signs,
$i'$,~$j'$,~$k'$, as compared with the signs $i$,~$j$,~$k$,
{\it coordinate characteristics of contra-derivation}, performed
on an ordinal quaternion.

\bigbreak

\centerline{\it
Connexions between the coordinate Characteristics of
Quaternion-Derivation and those of}

\nobreak\vskip 3pt

\centerline{\it
Octadic Transposition, introduced in the foregoing
Articles.}

\nobreak\bigskip

10.
It may serve to throw some additional light on the
foregoing relations between the coordinate characteristics,
$i$,~$j$,~$k$, of quaternion-derivation, if we point out a
connexion which exists between (1st) the system of these three
signs and the sign~$-$, which enters with them into the
formula~({\sc a}), on the one hand, and (2nd) the system of the
four characteristics of octadic transposition,
$\omega_1$,~$\omega_2$,~$\omega_3$, and $\omega$, which were
considered in the second article, on the other hand.  In general,
an {\it ordinal set\/} of the $n^{\rm th}$ order, since it
involves $n$ constituent ordinal relations, which are each
between two moments, or because it is a complex ordinal relation
between two momental sets, which are each of the $n^{\rm th}$
order, may be regarded as containing, in its first conception, a
reference to $2n$ moments; and these moments may always be
supposed to be collected, in thought and in expression, into a
{\it new momental set, of twice as high an order\/} as the
ordinal set which was proposed.  In symbols, the ordinal
set~(54), which may be thus denoted:
$${\sc q}' - {\sc q}
   =  ({\sc a}_0', {\sc a}_1',\ldots \, {\sc a}_{n-1}')
    - ({\sc a}_0, {\sc a}_1,\ldots \, {\sc a}_{n-1}),
   \eqno (101)$$
may naturally suggest the consideration of the following momental
set, with which it is {\it connected\/}:
$$({\sc a}_0', {\sc a}_1',\ldots \, {\sc a}_{n-1}',
   {\sc a}_0,  {\sc a}_1,\ldots \,  {\sc a}_{n-1} );
   \eqno (102)$$
and if the latter set be given, the former can be deduced from
it.  Hence every operation of transposition performed on the $2n$
moments of the set (102), is connected with, and determines, a
certain {\it corresponding\/} change of the $n$ ordinal relations
of the set~(101).  For example, if in the formula of momental
transposition~(18) we make $s = 2$, $r = 1$, then, with reference
to a certain operation on the momental set (102), which consists
here in exchanging the places of each moment~${\sc a}$ with the
corresponding moment~${\sc a}'$, we obtain the symbolic equation,
$${\sc m}_{2n-1,0,1,\ldots \, 2n-2}^n = 1^{1 \over 2};
   \eqno (103)$$
which implies that a repetition of this process of transposition
would restore the set~(102) to its original state.  But the same
operation on this momental set corresponds to, and determines, a
certain other operation, performed on the ordinal set~(101),
which consists in changing the sign of each constituent ordinal
relation, and in therefore changing, by the sixth article, the
sign of the ordinal set itself, or in operating on that ordinal
set by the characteristic~$-$, or $-1$; we might therefore, in
this way, be conducted to the known result, or principle, that
the sign~$-$, or the coefficient~$-1$, is a symbolic square root
of unity.  And we might be led to express in words the
corresponding conception, by saying that as two successive
interchanges of the places of two moments, or of two momental
sets, regarded respectively as ordinand and as ordinator, do not
finally affect their ordinal relation to each other; the
{\it second transposition\/} of these two moments or sets having
{\it destroyed the effect of the first\/}: so too, and for a
similar reason, the character (as well as the degree) of an
ordinal {\it relation\/} is not changed, or is {\it restored},
when it undergoes {\it two successive inversions\/}: the
{\it opposite of the opposite\/} of a relation being the same
with that {\it original relation\/} itself.  Thus, in particular,
for the case $n = 4$, the characteristic of octadic
transposition,~$\omega$, of which the symbolic square was unity,
is connected with the sign~$-$, or $-1$, prefixed, as a
characteristic of inversion, to the symbol of an ordinal
quaternion.

\bigbreak

11.
Again, with respect to the {\it sign of\/}~{\sc semi-inversion},
$\surd (-1)$, we may observe that {\it if the exponent~$n$ of the
order of the ordinal set be an even number}, $= 2m$, then we
shall have in general, as a symbolic {\it fourth root of unity},
the following characteristic of momental transposition, which may
be changed by changing $r$ to $1$, $s$ to $4$, and $n$ to $m$, in
the formula~(18):
$${\sc m}_{4m-1,0,1,\ldots \, 4m-2}^m = 1^{1 \over 4};
   \eqno (104)$$
and which takes the particular form (15), when $m$ is changed to
$1$.  And because the symbolic square of the first member of
(104) acquires the form (103) by restoring $n$ in the place of
$2m$, we see that {\it an ordinal set, if it be of an even order},
such as is an ordinal couple or quaternion, may always be
{\it semi-inverted, and therefore operated on by the
sign\/}~$\surd (-1)$, in, {\it at least, one way}, through the
medium of that momental transposition, performed {\it on a
momental set of an evenly even order}, which is indicated by this
first member.  For example, when we operate on a {\it momental
quaternion\/}
$({\sc a}_0', {\sc a}_1', {\sc a}_0, {\sc a}_1)$ by the
characteristic~${\sc m}_{3,0,1,2}$ we obtain the new momental
quaternion,
$$({\sc a}_1, {\sc a}_0', {\sc a}_1', {\sc a}_0)
   =  {\sc m}_{3,0,1,2}
      ({\sc a}_0', {\sc a}_1', {\sc a}_0, {\sc a}_1);
   \eqno (105)$$
and it is evident that, as was remarked in the second article,
and as is included in the more general assertion (104), four
successive transpositions of this sort {\it reproduce\/} the
momental quaternion which was originally proposed to be operated
on.  But we now see, further, that if, on the plan of the article
immediately preceding the present, we {\it connect}, in thought,
this {\it momental quaternion\/} with the {\it ordinal couple},
$$({\sc a}_0', {\sc a}_1') - ({\sc a}_0, {\sc a}_1)
   =  ({\sc a}_0' - {\sc a}_0, {\sc a}_1' - {\sc a}_1),
   \eqno (106)$$
we shall thereby {\it connect\/} the foregoing operation of
{\it momental transposition\/} with an operation of
{\it ordinal derivation}, which must admit of being symbolically
represented by the sign $\surd (-1)$, and which here consists
in passing from the couple (106) to this other ordinal couple:
$$({\sc a}_1, {\sc a}_0') - ({\sc a}_1', {\sc a}_0)
   =  ({\sc a}_1 - {\sc a}_1', {\sc a}_0' - {\sc a}_0).
   \eqno (107)$$

In fact, if we examine the changes of ordinal relation which have
been made, in passing from the form (106) to the form (107), we
shall perceive that they may be said to consist in first
inverting the second constituent relation of the couple, namely,
${\sc a}_1' - {\sc a}_1$, which thus becomes
${\sc a}_1 - {\sc a}_1'$, and in then transposing the two
constituent relations.  But this is precisely the process of
ordinal derivation which was indicated in the sixth article by
the characteristic ${\sc r}_{-1,0}$, and which we saw to be a
symbolic square root of $-1$.  Indeed, as was noticed in that
sixth article, it was on this property of this mode of
derivation, that the present writer proposed, in a former Essay,
to found a theory of algebraic couples, and of the use of the
symbol $\surd (-1)$ in algebra.

\bigbreak

12.
Proceeding on a similar plan, through not precisely by the
formula~(104), to illustrate those {\it new symbolic fourth roots
of unity\/} which enter into the present theory of algebraic
quaternions, by regarding those roots as certain characteristics
of ordinal derivation, which are connected with certain other
characteristics of momental transposition, we are now to consider
a {\it momental octad}, which we shall denote as follows:
$$\Omega
   =  ( {\sc a}_0', {\sc a}_1', {\sc a}_2', {\sc a}_3',
        {\sc a}_0, {\sc a}_1, {\sc a}_2, {\sc a}_3 );
   \eqno (108)$$
and shall regard as being {\it connected}, on the plan of the
tenth article, with the {\it ordinal quaternion},
$${\sr q}
   =  ( {\sc a}_0', {\sc a}_1', {\sc a}_2', {\sc a}_3' )
    - ( {\sc a}_0, {\sc a}_1, {\sc a}_2, {\sc a}_3 );
   \eqno (109)$$
that is, by (48) and (49), with the ordinal quaternion (55).  If
we operate on the octad~$\Omega$ by the characteristic of
transposition~$\omega$, defined by the symbolic equation~(20) of
the second article, then, according to a remark lately  made, the
resulting octad $\omega \Omega$ {\it corresponds\/} to, or is (on
the present plan) connected with, the quaternion $- {\sr q}$; and
thus the two signs~$\omega$ and $-$, as here used, have a certain
correspondence, or connexion, though not an identity, with each
other.  Again, if we operate on the same octad~$\Omega$ by the
three coordinate characteristics of transposition
$\omega_1$,~$\omega_2$,~$\omega_3$,
defined by the equations (21), we obtain these three new octads:
$$\left. \eqalign{
\omega_1 \Omega
   &= ( {\sc a}_1,  {\sc a}_0', {\sc a}_3,  {\sc a}_2',
        {\sc a}_1', {\sc a}_0,  {\sc a}_3', {\sc a}_2 );\cr
\omega_2 \Omega
   &= ( {\sc a}_2,  {\sc a}_3', {\sc a}_0', {\sc a}_1,
        {\sc a}_2', {\sc a}_3,  {\sc a}_0,  {\sc a}_1' );\cr
\omega_3 \Omega
   &= ( {\sc a}_3,  {\sc a}_2,  {\sc a}_1', {\sc a}_0',
        {\sc a}_3', {\sc a}_2', {\sc a}_1,  {\sc a}_0 );\cr}
   \right\}
   \eqno (110)$$
to which {\it correspond\/} these three derived quaternions:
$$\left. \eqalign{
i {\sr q}
   &=  ({\sc a}_1 - {\sc a}_1', {\sc a}_0' - {\sc a}_0,
            {\sc a}_3 - {\sc a}_3', {\sc a}_2' - {\sc a}_2);\cr
j {\sr q}
   &=  ({\sc a}_2 - {\sc a}_2', {\sc a}_3' - {\sc a}_3,
            {\sc a}_0' - {\sc a}_0, {\sc a}_1 - {\sc a}_1');\cr
k {\sr q}
   &=  ({\sc a}_3 - {\sc a}_3', {\sc a}_2 - {\sc a}_2',
            {\sc a}_1' - {\sc a}_1, {\sc a}_0' - {\sc a}_0);\cr}
   \right\}
   \eqno (111)$$
the characteristics of derivation $i \, j \, k$ being easily seen
to have the same effect and significance here as in the recent
articles.  Thus the three coordinate characteristics of
quaternion-derivation, $i$,~$j$,~$k$, {\it correspond\/}
respectively to the three coordinate characteristics of octadic
transposition, $\omega_1$,~$\omega_2$,~$\omega_3$; and since the
sign~$-$ has been seen to correspond in like manner, as a sign of
ordinal inversion performed on the quaternion~${\sr q}$, to the
other octadic characteristic~$\omega$, we see that a
correspondence is at once established between the symbolic
equations (22), respecting transpositions of the moments of an
octad, and the formul{\ae} (72) or ({\sc a}), respecting
derivations of an ordinal quaternion.  The equations (25)
correspond in like manner to the formul{\ae} (93) or ({\sc b});
the octadic characteristics,
$\omega \omega_1$, $\omega \omega_2$, $\omega \omega_3$
correspond to the characteristics of contraderivation of a
quaternion, $i'$,~$j'$,~$k'$; the equation (27) might remind us
that $i$,~$j$,~$k$, $i'$,~$j'$,~$k'$ are, all of them, symbolic
fourth roots of unity; and, finally, the equations (26) show, by
the same kind of correspondence of relations, that we may write
the following formul{\ae}, which include the results (71,4) and
(96):
$$\left. \eqalign{
ijk = jki = kij &= -1;\cr
kji = ikj = jik &= 1.\cr}
   \right\}
   \eqno (112)$$

\bigbreak

\centerline{\it
Addition and Subtraction, or Composition and Decomposition of
Ordinal Relations between}

\nobreak\vskip 3pt

\centerline{\it
any Sets of Moments.}

\nobreak\bigskip

13.
The usual correlation between the signs $+$ and~$-$ may be
extended by definition to expressions involving those signs in
conjunction with symbols for momental and ordinal sets; and thus,
by the use already mentioned of {\it zero}, the following
equations,
$$\left. \eqalign{
({\sc q}' - {\sc q}) + {\sc q} &= {\sc q}',\cr
({\sc q}'' - {\sc q}') + ({\sc q}' - {\sc q})
   &= {\sc q}'' - {\sc q},\cr
0 + {\sc q} &= {\sc q},\cr}
   \right\}
   \eqno (113)$$
together with those others which are formed from them by changing
each ${\sc q}$ to ${\sr q}$, may here, as elsewhere, be regarded
as {\it identically\/} true.  At the same time, the two symbols
$0 - {\sr q}$ and $- {\sr q}$ will thus be equisignificant, each
denoting the inverse or {\it opposite\/} of that complex ordinal
relation between two sets of moments, which is denoted by the
symbol~${\sr q}$; because the symbol~$- {\sr q}$ has been already
defined to denote that inverse relation, and therefore we have
now the two equations, $(- {\sr q}) + {\sr q} = 0$,
$(0 - {\sr q}) + {\sr q} = 0$; and the other isolated, but
{\it affected\/} symbol, $+ {\sr q}$, may in like manner be
interpreted as being equivalent in signification to
$0 + {\sr q}$, and therefore to ${\sr q}$.  With the conceptions
of {\it addition\/} and {\it subtraction}, or of
{\it composition and decomposition of ordinal relations}, which
correspond to these notations, we may write:
$$( {\sr a}', {\sr b}',\ldots ) \pm ( {\sr a}, {\sr b}, \ldots )
   =  ( {\sr a}' \pm {\sr a},
        {\sr b}' \pm {\sr b},\ldots );
   \eqno (114)$$
$$\left. \eqalign{
{\sc r}_0 ({\sr q}' \pm {\sr q})
   &= {\sc r}_0 {\sr q}' \pm {\sc r}_0 {\sr q};\cr
{\sc r}_1 ({\sr q}' \pm {\sr q})
   &= {\sc r}_1 {\sr q}' \pm {\sc r}_1 {\sr q};\ldots \cr}
   \right\}
   \eqno (115)$$
or, using $\Sigma$ and $\Delta$ as characteristics of {\it sum\/}
and {\it difference}, we may establish the important identities:
$${\sc r}_m \Sigma {\sr q}
   =  \Sigma {\sc r}_m {\sr q};\quad
  {\sc r}_m \Delta {\sr q}
   =  \Delta {\sc r}_m {\sr q}.
   \eqno (116)$$
{\it Addition of ordinal sets\/} is a {\it commutative\/} and
also an {\it associative\/} operation; that is, we have the
formul{\ae},
$${\sr q}' + {\sr q} = {\sr q} + {\sr q}';
   \eqno (117)$$
$$({\sr q}'' + {\sr q}') + {\sr q}
   =  {\sr q}'' + ({\sr q}' + {\sr q});
   \eqno (118)$$
the former of these two properties of addition being connected
with the principle of {\it alternation of an analogy}, which was
mentioned in the fourth article.  An ordinal set, of any
order~$n$, may always be regarded as the {\it sum\/} of $n$ other
sets of the same order, in each of which only {\it one\/}
constituent ordinal relation (at most) shall be a relation of
diversity; for we may write, generally,
$${\sr q}
   =  ({\sc r}_0 {\sr q}, 0,\ldots )
    + (0, {\sc r}_1 {\sr q},\ldots )
    + \&c.
   \eqno (119)$$
Thus, for example, the ordinal quaternion~(73) may be expressed
as the {\it sum of four others}, which may be called respectively
a {\it pure primary\/} (ordinal quaternion), a {\it pure
secondary}, {\it pure tertiary}, and {\it pure quaternary}, as
follows:
$$({\sr a}, {\sr b}, {\sr c}, {\sr d})
   =  ({\sr a}, 0, 0, 0)
    + (0, {\sr b}, 0, 0)
    + (0, 0, {\sr c}, 0)
    + (0, 0, 0, {\sr d}).
   \eqno (120)$$

\bigbreak

\centerline{\it
Multiplication of an ordinal Set by a Number.}

\nobreak\bigskip

14.
With these preparations it is easy to attach a perfectly clear
conception to the act or process of {\it multiplying\/} any
single ordinal relation,~${\sr a}$, or any ordinal
set,~${\sr q}$, {\it by any positive or negative number},~$m$.
For having already agreed to regard $1 {\sr q}$ and ${\sr q}$, as
well as $1 {\sr a}$ and ${\sr a}$, as being symbols equivalent to
each other, so that we have identically, or by definition,
$${\sr a} = 1 {\sr a},\quad
  {\sr q} = 1 {\sr q};
   \eqno (121)$$
and adopting also from common Arithmetic, which may itself be
regarded as a branch of the {\it Science of Pure Time}, since it
involves the conception of {\it succession\/} between things or
thoughts as {\it counted}, the abbreviations $2$,~$3$, \&c., for
the symbols $1 + 1$, $1 + 1 + 1$, \&c., we shall have an
analogous {\it system of abbreviated symbols\/} to denote the
{\it composition of\/} any number of {\it similar ordinal
relations}, whether those components be simple, as~${\sr a}$,
or complex, as~${\sr q}$; namely, the following:
$$\left. \multieqalign{
{\sr a} + {\sr a} &= 2 {\sr a}, &
{\sr a} + {\sr a} + {\sr a} &= 3 {\sr a}, \quad\hbox{\&c.};\cr
{\sr q} + {\sr q} &= 2 {\sr q}, &
{\sr q} + {\sr q} + {\sr q} &= 3 {\sr q}, \quad\hbox{\&c.} \cr}
   \right\}
   \eqno (122)$$
We may also agree to write, at pleasure, $2 \times {\sr a}$,
$3 \times {\sr q}$, \&c., instead of $2 {\sr a}$, $3 {\sr q}$,
\&c.; and with this use of elementary notations, the
{\it distributive\/} and {\it associative\/} properties of
multiplication offer themselves in the present theory, under the
well-known and elementary forms,
$$m ({\sr a}' \pm {\sr a}) = m {\sr a}' \pm m {\sr a};\quad
  (m' \pm m) {\sr a} = m' {\sr a} \pm m {\sr a};
   \eqno (123)$$
$$(m' m) \times {\sr a} = m' \times (m {\sr a});\quad
  (m' \div m) \times m {\sr a} = m' {\sr a};
   \eqno (124)$$
in each of which each symbol~${\sr a}$ or ${\sr a}'$ of a simple
ordinal relation may be changed to the corresponding symbol
${\sr q}$ or ${\sr q}'$ of an ordinal set, and in which we may,
{\it at first}, suppose that $m$,~$m'$, $m' - m$, and
$m' \div m$, denote positive whole numbers.  The writing (as
usual),
$$0 \times {\sr a} = 0,\quad 0 \times {\sr q} = 0,
   \eqno (125)$$
we shall be able, with the help of the interpretations in the
last article, to {\it remove\/} the last mentioned restriction,
and to suppose that $m$, $m'$, $m' + m$, $m' - m$,
$m' \times m \, ( = m' m)$, and
$\displaystyle m' \div m \, \left( = {m' \over m} \right)$,
denote {\it any\/} numbers, whole or fractional, and positive or
negative, or null, from $-\infty$ to $+\infty$, without violating
any of the usual rules for operating on such numbers, by
addition, subtraction, multiplication, and division; or rather we
might {\it deduce anew\/} all those  known rules for those
fundamental operations on what are usually called {\it real\/}
numbers, as consequences of the foregoing formul{\ae}, or as
necessary conditions for their generalization; observing, indeed,
that for the case of {\it incommensurable\/} (but still real)
multipliers, whether operating on a simple ordinal
relation~${\sr a}$, or on an ordinal set~${\sr q}$, we are to use
also an equation of {\it limits}, of the form,
$$(\lim m) \times {\sr a} = \lim (m \times {\sr a}).
   \eqno (126)$$
It is a consequence of these conceptions and notations that
{\it an ordinal set\/}~${\sr q}$ {\it is multiplied by a
number~$m$, when each of its constituent ordinal relations,
${\sc r}_0 {\sr q}$, ${\sc r}_1 {\sr q}$, \&c., is separately
multiplied thereby\/}; so that we may establish the formula,
$$m ({\sr a}, {\sr b}, {\sr c},\ldots )
   =  (m {\sr a}, m {\sr b}, m {\sr c},\ldots );
   \eqno (127)$$
and therefore also,
$${\sc r}_0 \mathbin{.} m {\sr q}
   = m \, {\sc r}_0 {\sr q};\quad
  {\sc r}_1 \mathbin{.} m {\sr q}
   = m \, {\sc r}_1 {\sr q};
   \quad\hbox{\&c.}
   \eqno (128)$$
And any ordinal relations, such as $m {\sr a}$,~$m {\sr b}$,
\&c., or any ordinal sets, such as $m {\sr q}$,~$m {\sr q}'$,
\&c., which are thus obtained from others such as
${\sr a}$,~${\sr b}$, \&c., or ${\sr q}$,~${\sr q}'$, \&c., by
multiplying them respectively by any common number~$m$, may be
said to be {\it proportional\/} to those others.

We may also say that any ordinal relations, such as
$m {\sr a}$,~$m' {\sr a}$, \&c., and that any ordinal sets, such
as $m {\sr q}$,~$m' {\sr q}$, \&c., are {\it proportional to the
multiplying numbers\/} $m$,~$m'$, \&c., by which they are
generated from any common relation~${\sr a}$, or set~${\sr q}$,
as from a common multiplicand, when such generation is possible.

\bigbreak

\centerline{\it
Case of Existence of a simple numeral Quotient, obtained by a
particular Division of one}

\nobreak\vskip 3pt

\centerline{\it
Ordinal Set by another.}

\nobreak\bigskip

15.
The recent theory of the {\it multiplication\/} of an ordinal set
by a number, enables us to assign, in one extensive case, an
expression for the result of the {\it division\/} of one ordinal
set by another; for if we regard the equations
$$({\sr a}' \div {\sr a}) \times {\sr a} = {\sr a}',\quad
  ({\sr q}' \div {\sr q}) \times {\sr q} = {\sr q}',
   \eqno (129)$$
as being identically or definitionally true by the general
symbolical correlation of the signs $\times$ and~$\div$, we may
then write, in virtue of the formula (127), this other and
correlative formula,
$$({\sr a}', {\sr b}', {\sr c}',\ldots )
      \div ({\sr a}, {\sr b}, {\sr c},\ldots )
   =  m,
   \eqno (130)$$
whenever the following conditions are satisfied:
$${\sr a}' \div {\sr a}
   =  {\sr b}' \div {\sr b}
   =  {\sr c}' \div {\sr c}
   =  \ldots  = m.
   \eqno (131)$$
In other words, we know how to {\it interpret the quotient\/}
${\sr q}' \div {\sr q}$, of {\it one ordinal set~${\sr q}'$
divided by another\/}~${\sr q}$, namely, as being another
expression for a simple or single number~$m$, in the case when
the $n$ constituent ordinal relations of the one set are
{\it proportional\/} (in the sense lately defined) to the
$n$ {\it homologous constituents\/} of the other set; and we
have, {\it in that case}, the continued equation,
$${\sr q}' \div {\sr q}
   =  {\sc r}_0 {\sr q}' \div {\sc r}_0 {\sr q}
   =  {\sc r}_1 {\sr q}' \div {\sc r}_1 {\sr q}
   =  \hbox{\&c.}
   \eqno (132)$$
But in the infinitely many {\it other\/} cases in which this
condition of proportionality is {\it not\/} satisfied, the $n$
numerical quotients,
${\sc r}_0 {\sr q}' \div {\sc r}_0 {\sr q}$,
${\sc r}_1 {\sr q}' \div {\sc r}_1 {\sr q}$, \&c., being at least
partially different among themselves, and therefore not being
each equal to one common number~$m$ (whether commensurable or
incommensurable, and whether positive or negative or null), it
is, for the same reason, {\it impossible to find any\/} {\sc one}
{\it such number,~$m$, which shall be correctly equated to the
quotient ${\sr q}' \div {\sr q}$ of the two proposed ordinal
sets}, in consistency with the foregoing principles.  It is,
however, {\it not impossible to find a\/} {\sc system}
{\it of numbers}, which may, consistently with those principles,
be regarded as representing this {\it quotient of the direction
of one ordinal set by another\/}; and we proceed to give an
outline of a process by which such a numeral system, or
{\it complex quotient}, may be found.

\bigbreak

\centerline{\it
Investigation of a complex numeral Quotient, resulting from the
general symbolical Division}

\nobreak\vskip 3pt

\centerline{\it
of one ordinal Set by another.}

\nobreak\bigskip

16.
Conceive that from any proposed expression of the form,
$${\sr q}
   =  ({\sr a}_0, {\sr a}_1,\ldots \,
          {\sr a}_t, \ldots \, {\sr a}_{n-1}),
   \eqno (133)$$
for an ordinal set~${\sr q}$ of the $n^{\rm th}$ order, we form
$n$ other expressions of {\it coordinate derivative sets},
${\sr q}_0, {\sr q}_1,\ldots \, {\sr q}_{n-1}$, according to the
type,
$$1 \times_r {\sr q} = \times_r {\sr q} = {\sr q}_r
   =  ({\sr a}_{r,0}, {\sr a}_{r,1},\ldots \,
          {\sr a}_{r,s},\ldots \, {\sr a}_{r,n-1});
   \eqno (134)$$
in which it is supposed that the constituent ordinal
relation~${\sr a}_{r,s}$, of the derivative set~${\sr q}_r$, has
a determinate and known dependence on the $n$ constituents, such
as ${\sr a}_t$, of the proposed set~${\sr q}$; and let us
conceive that this dependence is expressed by a formula such as
the following:
$${\sr a}_{r,s}
   =  c_{r,s,0} {\sr a}_0 + \ldots
          + c_{r,s,t} {\sr a}_t + \ldots
          + c_{r,s,n-1} {\sr a}_{n-1};
   \eqno (135)$$
the $n^3$ {\it coefficients of coordinate
derivation},~$c_{r,s,t}$, being all regarded as constant and
known numbers, whether positive or negative or null.  It will
then be possible, without altering the constant numerical values
thus supposed to belong to those $n^3$ coefficients,~$c_{r,s,t}$,
to form a {\it complex\/} and variable
{\it derivative\/}~${\sr q}'$ of the set~${\sr q}$, by
multiplying each of the $n$ {\it simple\/} or {\it elementary
derivatives\/} already obtained, such as ${\sr q}_r$, by a
{\it variable number\/}~$m_r$, and adding the $n$ products
together; and the resulting set may be denoted thus:
$$\eqalignno{
( m_0 \times_0 + m_1 \times_1 + \ldots
          + m_r \times_r + \ldots
          + m_{n-1} \times_{n-1} ) {\sr q} \cr
   =  m_0 {\sr q}_0 + m_1 {\sr q}_1 + \ldots
          + m_r {\sr q}_r + \ldots
          + m_{n-1} {\sr q}_{n-1}
   &= {\sr q}';
   &(136)\cr}$$
where we shall have
$${\sr q}'
   = ({\sr a}_0', {\sr a}_1',\ldots \, {\sr a}_s',\ldots \,
            {\sr a}_{n-1}'),
   \eqno (137)$$
if we make, for abridgment,
$${\sr a}_s'
   =  m_0 {\sr a}_{0,s} + m_1 {\sr a}_{1,s} + \ldots
          + m_r {\sr a}_{r,s} + \ldots
          + m_{n-1} {\sr a}_{n-1,s};
   \eqno (138)$$
and the entire collection of signs of operation,
$m_0 \times_0 +$~\&c., which is prefixed between parentheses to
the symbol~${\sr q}$ in the first line of the formula (136), may
be said to be a {\it characteristic of complex derivation}, or a
{\it complex symbolic multiplier}.  But instead of thus
conceiving the set~${\sr q}'$ to be {\it deduced\/} from
${\sr q}$ by this mode of complex derivation, or {\it symbolical
multiplication\/}~(136), with the assistance of the constant
coefficients of derivation~$c$, and of $n$ {\it given values\/}
for the variable multiplying numbers~$m$, we may enquire,
conversely, {\it what system of numerical multipliers},
$m_0,\ldots \, m_r,\ldots \, m_{n-1}$,
{\it must be assumed}, in order to produce or generate a
{\it given ordinal set\/}~${\sr q}'$, as the {\it symbolical
product\/} of this sort of multiplication; the {\it multiplicand
set\/}~${\sr q}$, and the {\it constant
coefficients\/}~$c$, being still supposed to be
{\it given}.  This inverse or reciprocal process may be called
the {\it symbolical division of one ordinal set by another},
namely, of the set~${\sr q}'$ by the set~${\sr q}$; and it may be
denoted by the following formula, which is the reciprocal or
inverse of the formula (136):
$${\sr q}' \div {\sr q}
   = m_0 \times_0 + m_1 \times_1 + \ldots + m_{n-1} \times_{n-1}.
   \eqno (139)$$

To describe more fully the process which is thus briefly
indicated, we may observe that, besides the $n^3$ constant
coefficients~$c$, there are now given, or supposed to be known,
$2n$ ordinal relations of the forms ${\sr a}_t$ or ${\sr a}_s'$
(or numbers proportional to these $2n$ relations), as the
constituents of the two given ordinal sets of the $n^{\rm th}$
order, ${\sr q}$ and ${\sr q}'$; which sets are here regarded as
the {\it divisor set\/} and the {\it dividend set\/}
respectively.  Thus the $n^2$ ordinal relations of the form
${\sr a}_{r,s}$ are conceived to be known, as depending in a
known manner on the $n$ given relations~${\sr a}_t$, by the $n^2$
expressions of the form (135); and on substituting for these
$n^2$ ordinal relations, and for the $n$ other given relations of
the form~${\sr a}_s'$, in the $n$ formul{\ae} (138), any system of
numerical values which shall be (in the sense of the 14th article)
{\it proportional\/} to these different ordinal relations, we
shall thereby obtain $n$ {\it linear equations}, of an ordinary
algebraical kind, between the $n$ sought {\it numbers\/},~$m_r$:
from which these latter numbers may then in general be deduced, by
any of the usual processes of solution of such ordinary and linear
equations.

For example, after fixing upon any standard ordinal relation, or
relation between two selected moments of time, and calling
it~${\sr a}$, we may first prepare the equation (138) by putting
it under the form,
$${\sr a}_s' \div {\sr a}
   =  \Sigma_r \mathbin{.}
         m_r ({\sr a}_{r,s} \div {\sr a});
   \eqno (140)$$
in which $\Sigma_r$ is the characteristic of a
summation performed with respect to~$r$, and the quotients in
both members are numerical.  And then, by suitable combinations
of the numerical quotients in the second member of this last
equation, which combinations are determined by the given
expressions (135), we may find a system of $n^2$ numerical
{\it coefficients of elimination\/},~$l_{r,s}$, of which the
values depend on the constant coefficients~$c$, and on the $n$
given numerical quotients of the form ${\sr a}_t \div {\sr a}$,
but are independent of the $n$ other quotients
${\sr a}_s' \div {\sr a}$, and satisfy the $n^2$ conditions
included in the formula,
$$\Sigma_s l_{r,s}
   ({\sr a}_{r',s} \div {\sr a}) = 0,
   \quad\hbox{or}\quad = l,
   \quad\hbox{according as}\quad
   r' \gtlt \hbox{ or } = r;
   \eqno (141)$$
$l$ being here another number, namely, the {\it common
denominator\/} of the elimination.  For in this manner we shall
have $n$ final expressions of the form,
$$m_r = l^{-1} \Sigma_s \mathbin{.}
            l_{r,s} ({\sr a}_{r,s}' \div {\sr a});
   \eqno (142)$$
by which the $n$ sought {\it coefficients of the symbolical
quotient\/} (139) can be, in general, determined.

\bigbreak

\centerline{\it
Successive complex Derivation: Conception of a numeral Set.}

\nobreak\bigskip

17.
Suppose that, after deducing ${\sr q}'$ from ${\sr q}$, by the
complex derivation or symbolical multiplication (136), we again
derive another ordinal set ${\sr q}''$ from ${\sr q}'$ by another
multiplication of the same sort, with the {\it same constant
coefficients of derivation},~$c$, but with a {\it new system of
variable numerical multipliers},~$m$; which supposition we shall,
on the same plan as before, express as follows:
$$( m_0' \times_0 + \ldots + m_{r'}' \times_{r'} + \ldots
          + m_{n-1}' \times_{n-1} ) {\sr q}'
   =  {\sr q}''.
   \eqno (143)$$
Making now, in imitation of the expression (137),
$${\sr q}''
   =  ({\sr a}_0'', \ldots \,
         {\sr a}_{s'}'',\ldots {\sr a}_{n-1}''),
   \eqno (144)$$
we shall have, as expressions analogous to (138) and (135), the
following:
$${\sr a}_{s'}''
   =  \Sigma_{r'} \mathbin{.}
         m_{r'}' {\sr a}_{r',s'}';
   \eqno (145)$$
$${\sr a}_{r',s'}'
   =  \Sigma_s \mathbin{.}
         c_{r',s',s} {\sr a}_s';
   \eqno (146)$$
and thus the result of this {\it successive multiplication\/}
will be a determined and known set~${\sr q}''$.  In the next
place, let this resulting set, or {\it successive symbolical
product},~${\sr q}''$, be {\it divided by the original
set\/}~${\sr q}$, which was at first proposed as a multiplicand;
we shall then obtain, by the method described in the foregoing
article, a symbolical quotient of the form,
$${\sr q}'' \div {\sr q}
   =  m_0'' \times_0 + \ldots
          + m_{r''}'' \times_{r''} + \ldots
          + m_{n-1}'' \times_{n-1};
   \eqno (147)$$
in which, on the same plan as in the formula (142), and with the
{\it same system of eliminational coefficients\/} of the
form~$l$, determined by (141), we have,
$$m_{r''}'' = l^{-1} \Sigma_{s'} \mathbin{.}
         l_{r'',s'} ({\sr a}_{s'}'' \div {\sr a}).
   \eqno (148)$$
Substituting for ${\sr a}_{s'}''$ its value, given by (145),
(146), and by (138) or (140), and eliminating the numerical
denominator~$l$ by (141), we find that we may write:
$$m_{r''}''
   =  \Sigma_{r,r'} \mathbin{.}
            m_r m_{r'}' n_{r,r',r''};
   \eqno (149)$$
if we establish, for conciseness, the following formula,
including $n^3$ separate expressions for so many separate
numbers:
$$n_{r,r',r''}
   =  \left(
         \Sigma_{s,s'} \mathbin{.}
         l_{r'',s'} c_{r',s',s} {\sr a}_{r,s}
      \right)
      \div
      \left(
         \Sigma_s \mathbin{.}
         l_{r,s} {\sr a}_{r,s}
      \right);
   \eqno (150)$$
in which it is to be observed that the sum which enters as a
divisor is the same for all the $n^3$ quotients.  The value of
each of these numerical quotients (150) will, {\it in general},
depend on the $n - 1$ ratios of the constituents
${\sr a}_0, {\sr a}_1,\ldots \, {\sr a}_{n-1}$
of the first proposed ordinal set~${\sr q}$, or the ratios of the
numbers to which these $n$ ordinal constituents are proportional;
but it may be possible to {\it assign\/} (at the outset)
{\it such values to the constant but arbitrary coefficients of
derivation\/}~$c$, or to subject those $n^3$ coefficients
to such restrictions, that {\it these $n - 1$ arbitrary ratios of
the $n$ constituents\/}~${\sr a}_t$, in the expression (133),
{\it shall have no influence on the value of any one of the $n^3$
numbers\/} included in the expression (150).  When this last
condition, or system of conditions, is satisfied, we are allowed
to {\it detach the characteristics of the successive symbolical
multiplications of an ordinal set from the symbol of the original
multiplicand\/}; and as the result of the comparison of the
formul{\ae} (136) and (143), and of (147) under the form,
$${\sr q}''
   =  (m_0'' \times_0 + \ldots + m_{n-1}'' \times_{n-1})
         {\sr q},
   \eqno (151)$$
we may write,
$$m_0'' \times_0 + \ldots + m_{n-1}'' \times_{n-1}
   =  (m_0' \times_0 + \ldots + m_{n-1}' \times_{n-1})
      (m_0 \times_0 + \ldots + m_{n-1} \times_{n-1});
   \eqno (152)$$
which will denote the {\it reduction of a system of two
successive and complex derivations}, or symbolic multiplications
of the kind (136), {\it to one complex derivation of the same
kind}.  Under the same conditions, the {\it successive
performance of two simple or elementary derivations}, of the kind
(134), will be equivalent to the performance of {\it one complex
derivation}, of the kind (136), {\it with numerical coefficients
independent of the original derivand}, as follows:
$$\times_{r'} \times_r
   =  \Sigma_{r''} \mathbin{.}
         n_{r,r',r''} \times_{r''}.
   \eqno (153)$$
We may also regard the $n$ variable numerical coefficients~$m_r$,
in the quotient (139), obtained by the symbolical division of one
ordinal set by another, as composing, under the same conditions,
a {\sc numeral set}; and this new sort of {\it set\/} may be
{\it detached}, in thought and in expression, from the two
ordinal sets which have served, by their mutual comparison, to
{\it suggest\/} it.  The quotient (139), when thus regarded as a
{\it numeral set}, may be denoted as follows:
$${\sr q}' \div {\sr q} = q = (m_0, m_1,\ldots \, m_{n-1});
   \eqno (154)$$
the letter~$q$, when used as a symbol of {\it such\/} a set,
being written in the Italic character: and then the $n$ numerical
relations, which are included in the formula (149), may be
supposed to be otherwise summed up in the one equation:
$$(m_0'',\ldots \, m_{r''}'',\ldots \, m_{n-1}'')
   =  (m_0',\ldots \, m_{r'}',\ldots \, m_{n-1}')
      (m_0,\ldots \, m_r,\ldots \, m_{n-1}).
   \eqno (155)$$
And conversely, this last equation, which asserts that {\it the
numeral set in its first member is equal to the symbolical
product of the two numeral sets in its second member}, may be
considered to receive its {\it interpretation\/} from the formula
(149); in which the $n^3$ numbers $n_{r,r',r''}$ may be called
the {\it coefficients of multiplication of a numeral set}.  But
it is necessary to consider more closely what are the
{\it forms\/} of those {\it conditions of detachment\/} which
have been above alluded to, and which (according to the view here
taken) are required for the (separate) {\it existence\/} of such
a numeral set; it will also be proper to give, at least, some
examples of the possibility of satisfying the conditions thus
determined.

\bigbreak

\centerline{\it
Conditions of Detachment.}

\nobreak\bigskip

18.
The following appears to be a sufficiently simple mode of
discovering the {\it conditions of detachment}, under which the
values of the numerical coefficients, $n_{r,r',r''}$, in (149) or
(150), shall be independent of the ratios of the ordinal
constituents of the set~${\sr q}$, which is originally operated
upon.  Employing the characteristics of ordinal separation, as
explained in a former article, we may now regard it as being the
{\it definition\/} of the sign of derivation~$\times_r$, that
this sign satisfies the symbolic equation,
$${\sc r}_s \times_r
   =  \Sigma_t \mathbin{.} c_{r,s,t} {\sc r}_t;
   \eqno (156)$$
which gives
$$\eqalignno{
{\sc r}_{s'} \times_{r'} \times_r
   &= \Sigma_s \mathbin{.}
         c_{r',s',s} {\sc r}_s \times_r \cr
   &= \Sigma_{s,t} \mathbin{.}
         c_{r',s',s} c_{r,s,t} {\sc r}_t.
   &(157)\cr}$$
On the other hand, the equation (153), when operated on by the
characteristic of separation~${\sc r}_{s'}$, gives, by
changing $r''$ to $s$, and by afterwards changing $r$,~$s$ in
(156) to $s$,~$s'$:
$$\eqalignno{
{\sc r}_{s'} \times_{r'} \times_r
   &= \Sigma_s \mathbin{.}
         n_{r,r',s} {\sc r}_{s'} \times_s \cr
   &= \Sigma_{s,t} \mathbin{.}
         n_{r,r',s} c_{s,s',t} {\sc r}_t.
   &(158)\cr}$$
We are then to satisfy the equation,
$$\eqalignno{
0  &= \Sigma_s
         (  n_{r,r',s} {\sc r}_{s'} \times_s
          - c_{r',s',s} {\sc r}_s   \times_r ) \cr
   &= \Sigma_{s,t}
         (  n_{r,r',s} c_{s,s',t}
          - c_{r',s',s} c_{r,s,t} ) {\sc r}_t;
   &(159)\cr}$$
and because we are to do this independently of the ratios of the
$n$ constituent ordinal relations~$a_t$, which are obtained from
the ordinal set~${\sr q}$ by the $n$ operations of
separation~${\sc r}_t$, we must endeavour to satisfy all the
numerical conditions which are included in the form,
$$0 = \Sigma_s
         ( n_{r,r',s} c_{s,s',t}
          - c_{r',s',s} c_{r,s,t} ).
   \eqno (160)$$
The {\it number\/} of these {\it conditions of detachment\/}
(160) is $n^4$, because each of the four indices,
$r$,~$r'$,~$s'$,~$t$, may receive any one of the $n$ values
$0,1,\ldots \, n-1$; and they involve only $2n^3$ numerical
coefficients, or rather their ratios, which are fewer by one, to
be determined; from which it may at first sight seem to be
impossible to satisfy all these conditions of detachment, except
by making all the coefficients of derivation vanish.  Yet we
shall see that when $n = 2$, namely, for the case of
{\it numeral couples}, the conditions admit of an
{\it indeterminate\/} form of solution: and for the case $n = 4$,
it will be shown that they can also be satisfied by that system
of coefficients on which is founded our theory of {\it numeral
quaternions}, and even by a system of coefficients somewhat more
general.  A more complete discussion of the important formula
(160) will not be needed for the purposes of the present Essay.

\bigbreak

\centerline{\it
Case of Couples.}

\nobreak\bigskip

19.
If we suppose $n = 2$, then the index~$s$, with respect to which
the summation is to be performed, can be only $0$ or $1$; the
formula (160) becomes, therefore, in this case,
$$n_{r,r',0} c_{0,s',t} + n_{r,r',1} c_{1,s',t}
   =  c_{r',s',0} c_{r,0,t} + c_{r',s',1} c_{r,1,t}.
   \eqno (161)$$

If we suppose also that the two simple or elementary derivations
of one ordinal couple from another are denoted thus:
$$\left. \eqalign{
\times_0 ({\sr a}_0, {\sr a}_1)
   &= ({\sr a}_{0,0}, {\sr a}_{0,1})
    = (a {\sr a}_0 + a' {\sr a}_1,
            b {\sr a}_0 + b' {\sr a}_1);\cr
\times_1 ({\sr a}_0, {\sr a}_1)
   &= ({\sr a}_{1,0}, {\sr a}_{1,1})
    = (c {\sr a}_0 + c' {\sr a}_1,
            d {\sr a}_0 + d' {\sr a}_1);\cr}
   \right\}
   \eqno (162)$$
we shall have, by (135), for the $2^3 = 8$ coefficients of
derivation of the form $c_{r,s,t}$, the abridged symbols:
$$\left. \multieqalign{
c_{0,0,0} &= a;  &
c_{0,0,1} &= a'; &
c_{0,1,0} &= b;  &
c_{0,1,1} &= b'; \cr
c_{1,0,0} &= c;  &
c_{1,0,1} &= c'; &
c_{1,1,0} &= d;  &
c_{1,1,1} &= d'. \cr}
   \right\}
   \eqno (163)$$
And if we employ in like manner these other temporary
abridgments, for the eight coefficients of multiplication of one
numeral couple by another,
$$\left. \multieqalign{
n_{0,0,0} &= e;  &
n_{0,0,1} &= e'; &
n_{0,1,0} &= f;  &
n_{0,1,1} &= f'; \cr
n_{1,0,0} &= g;  &
n_{1,0,1} &= g'; &
n_{1,1,0} &= h;  &
n_{1,1,1} &= h'; \cr}
   \right\}
   \eqno (164)$$
the {\it equations of detachment}, included in the general
formula (160), will then, by (161), be the sixteen following:
$$\eqalignno{
(t = 0) \hskip 7.1em (t = 1) \hskip 11.9em\cr
\left. \multieqalign{
   & (s' = 0) & e  a  + e' c  &= a  a  + a' b ; &
                e  a' + e' c' &= a  a' + a' b'; \cr
   & (s' = 1) & e  b  + e' d  &= b  a  + b' b ; &
                e  b' + e' d' &= b  a' + b' b'; \cr}
   \right\}
   \quad (r = 0,\, r' = 0) & &(165)\cr
\left. \multieqalign{
   & (s' = 0) & f  a  + f' c  &= c  a  + c' b ; &
                f  a' + f' c' &= c  a' + c' b'; \cr
   & (s' = 1) & f  b  + f' d  &= d  a  + d' b ; &
                f  b' + f' d' &= d  a' + d' b'; \cr}
   \right\}
   \quad (r = 0,\, r' = 1) & &(166)\cr
\left. \multieqalign{
   & (s' = 0) & g  a  + g' c  &= a  c  + a' d ; &
                g  a' + g' c' &= a  c' + a' d'; \cr
   & (s' = 1) & g  b  + g' d  &= b  c  + b' d ; &
                g  b' + g' d' &= b  c' + b' d'; \cr}
   \right\}
   \quad (r = 1,\, r' = 0) & &(167)\cr
\left. \multieqalign{
   & (s' = 0) & h  a  + h' c  &= c  c  + c' d ; &
                h  a' + h' c' &= c  c' + c' d'; \cr
   & (s' = 1) & h  b  + h' d  &= d  c  + d' d ; &
                h  b' + h' d' &= d  c' + d' d'. \cr}
   \right\}
   \quad (r = 1,\, r' = 1) & &(168)\cr}$$

Now the twelve equations (165) (166) (167) are all satisfied,
independently of $c$, $c'$, $d$, $d'$, if we suppose
$$a = b' = e = f' = g';\quad
  a' = b = e' = f = g = 0;
   \eqno (169)$$
and then the four remaining equations (168) take the forms,
$$\left. \multieqalign{
ha + (h' - c) c &= c' d; & (h' - c - d') c'   &= 0; \cr
(h' - c - d') d &= 0;    & ha + (h' - d') d' &= c' d;\cr}
   \right\}
   \eqno (170)$$
which are satisfied by supposing
$$h' = c + d';\quad ha = c' d - c d'.
   \eqno (171)$$
Accordingly, with the values (169), the sign of
derivation~$\times_0$ reduces itself to the ordinary numeric
multiplier~$a$, so that we may write simply,
$$\times_0 = a;
   \eqno (172)$$
and while the other sign of linear derivation~$\times_1$ retains
its greatest degree of generality, consistent with the
{\it order\/} of the sets, namely, {\it couples}, which are at
present under consideration, so that the four numerical constants
$c$~$c'$~$d$~$d'$ remain entirely unrestricted, the symbolic
equations of the form (153) become now, by (164), (169), and
(171):
$$\left. \eqalign{
\times_0 \times_0
  &= e \times_0 + e' \times_1 = a \times_0;\cr
\times_1 \times_0 &= f \times_0 + f' \times_1 = a \times_1;\cr
\times_0 \times_1 &= g \times_0 + g' \times_1 = a \times_1;\cr
\times_1 \times_1 &= h \times_0 + h' \times_1
   = a^{-1} (c' d - c d') \times_0 + (c + d') \times_1;\cr}
   \right\}
   \eqno (173)$$
and these equations are, as we aimed that they should be,
independent of the original derivand, that is, here, of the
ordinal couple $({\sr a}_0, {\sr a}_1)$.  In fact, the three
first equations (173) are evidently true, by (172), whatever the
constant coefficients of derivation included in the
sign~$\times_1$ may be; and if, by the definition (162) of that
sign of derivation, we form the {\it successive derivative},
$$\eqalignno{
\times_1 \times_1 ({\sr a}_0, {\sr a}_1)
   &= \times_1 ({\sr a}_{1,0}, {\sr a}_{1,1}) \cr
   &= ( c {\sr a}_{1,0} + c' {\sr a}_{1,1},
        d {\sr a}_{1,0} + d' {\sr a}_{1,1} ) \cr
   &= ( c ( c {\sr a}_0 + c' {\sr a}_1)
            + c' ( d {\sr a}_0 + d' {\sr a}_1),
        d ( c {\sr a}_0 + c' {\sr a}_1)
            + d' ( d {\sr a}_0 + d' {\sr a}_1) ),
   &(174)\cr}$$
we are conducted, whatever the two original constituent ordinal
relations ${\sr a}_0$ and ${\sr a}_1$ may be, to the same final
ordinal couple, as if we add together the two partial results,
which are obtained by the two derivations represented by the two
terms of the last member of the fourth equation (173), namely,
the two following couples:
$$\left. \eqalign{
a^{-1} (c' d -c d') \times_0 ({\sr a}_0, {\sr a}_1)
   &= ( (c' d - c d') {\sr a}_0, (c' d - c d') {\sr a}_1 ); \cr
(c + d') \times_1 ({\sr a}_0, {\sr a}_1)
   &= ( (c + d') (c {\sr a}_0 + c' {\sr a}_1),
        (c + d') (d {\sr a}_0 + d' {\sr a}_1) ).\cr}
   \right\}
   \eqno (175)$$
We may therefore express the result of two successive and complex
derivations of this sort, performed on an ordinal couple
$({\sr a}_0, {\sr a}_1)$, by a {\it symbolical equation
independent of that original derivand}, or operand couple,
namely, by the following:
$$(m_0' \times_0 + m_1' \times_1) (m_0 \times_0 + m_1 \times_1)
   =  m_0'' \times_0 + m_1'' \times_1,
   \eqno (176)$$
which is included in the form (152), and in which we have now
these two relations, of the form (149), between the numerical
coefficients:
$$\left. \eqalign{
m_0'' &= a m_0' m_0 + a^{-1} (c' d - c d') m_1' m_1;\cr
m_1'' &= a m_1' m_0 + a m_0' m_1 + (c + d') m_1' m_1.\cr}
   \right\}
   \eqno (177)$$
Under the same conditions we may also write, more briefly,
$$(m_0'', m_1'') = (m_0', m_1') (m_0, m_1),
   \eqno (178)$$
as in the general form (155); and may regard the one {\it numeral
couple\/} $(m_0'', m_1'')$ as the {\it symbolical product\/} of
the other two.  If we simplify the formul{\ae} by assuming the
five constant coefficients of derivation which still remain
disposable, namely $a$, $c$, $c'$, $d$, $d'$, as follows:
$$a = 1,\quad c = 0,\quad c' = -1,\quad d = 1,\quad d' = 0,
   \eqno (179)$$
we shall then have
$$\times_0 ({\sr a}_0, {\sr a}_1) = ({\sr a}_0, {\sr a}_1);\quad
  \times_1 ({\sr a}_0, {\sr a}_1) = (-{\sr a}_1, {\sr a}_0);
   \eqno (180)$$
or more concisely,
$$\times_0 = 1;\quad \times_1 = {\sc r}_{-1,0};
   \eqno (181)$$
this last symbol being here the same characteristic of derivation
of an ordinal couple which was considered in former articles of
this paper.  And the eqaution for the {\it multiplication of two
numeral couples\/} will then reduce itself to the following
form:
$$(m_0', m_1') (m_0, m_1)
   =  (m_0' m_0 - m_1' m_1, m_1' m_0 + m_0' m_1);
   \eqno (182)$$
which agrees with that assigned in the earlier Essay.  With the
same values of the coefficients of derivation, and consequently
with the same values of the coefficients of multiplication
likewise, we may write also, as in that Essay (compare the page
just cited), a {\it formula for the division of one numeral
couple by another}, namely:
$${(m_0'', m_1'') \over (m_0, m_1)}
   =  (m_0', m_1')
   =  \left(
         {m_0 m_0'' + m_1 m_1'' \over m_0^2 + m_1^2},
         {m_0 m_1'' - m_1 m_0'' \over m_0^2 + m_1^2}
      \right).
   \eqno (183)$$

It is not necessary, and it would detain us too long from the
main subject of this memoir, to consider here any other and less
simple formul{\ae} of the same sort, which may be obtained for the
same case of couples, by any other systems of coefficients of
derivation and multiplication, which satisfy the same conditions
of detachment, assigned in the present article.

\bigbreak

20.
It may be instructive, however, to consider here the same case of
couples, as an exemplification of some other general formul{\ae}
which have been already given in this Essay.  Writing, for
abridgment,
$${\sr a}_t \div {\sr a} = a_t;\quad
  {\sr a}_s' \div {\sr a} = a_s';\quad
  {\sr a}_{s'}'' \div {\sr a} = a_{s'}'';
   \eqno (184)$$
and in like manner,
$${\sr a}_{r,s} \div {\sr a} = a_{r,s};\quad
  {\sr a}_{r',s'}' \div {\sr a} = a_{r',s'}';
   \eqno (185)$$
the quotients thus denoted being numerical; we have, by article
16, for the case $n = 2$, the commas in the compound indices
being here omitted for the sake of conciseness:
$$\left. \multieqalign{
a_{00} &= c_{000} a_0 + c_{001} a_1; &
a_{01} &= c_{010} a_0 + c_{011} a_1; \cr
a_{10} &= c_{100} a_0 + c_{101} a_1; &
a_{11} &= c_{110} a_0 + c_{111} a_1; \cr}
   \right\}
   \eqno (186)$$
$$a_0' = m_0 a_{00} + m_1 a_{10};\quad
  a_1' = m_0 a_{01} + m_1 a_{11};
   \eqno (187)$$
$$\left. \eqalign{
l   = l_{00} a_{00} + l_{01} a_{01}
   &= l_{10} a_{10} + l_{11} a_{11};\cr
0   = l_{00} a_{10} + l_{01} a_{11}
   &= l_{10} a_{00} + l_{11} a_{01};\cr}
   \right\}
   \eqno (188)$$
and, consequently,
$$\left. \eqalign{
  l m_0 &= l_{00} a_0' + l_{01} a_1';\cr
  l m_1 &= l_{10} a_0' + l_{11} a_1'.\cr}
   \right\}
   \eqno (189)$$
Again, by article~17, for the same case $n = 2$, we have the
analogous formul{\ae}:
$$\left. \multieqalign{
a_{00}' &= c_{000} a_0' + c_{001} a_1'; &
a_{01}' &= c_{010} a_0' + c_{011} a_1'; \cr
a_{10}' &= c_{100} a_0' + c_{101} a_1'; &
a_{11}' &= c_{110} a_0' + c_{111} a_1'; \cr}
   \right\}
   \eqno (190)$$
$$a_0'' = m_0' a_{00}' + m_1' a_{10}';\quad
  a_1'' = m_0' a_{01}' + m_1' a_{11}';
   \eqno (191)$$
and then, assuming these other expressions,
$$a_0'' = m_0'' a_{00} + m_1'' a_{10};\quad
  a_1'' = m_0'' a_{01} + m_1'' a_{11},
   \eqno (192)$$
we find, by (188), two equations of the same forms as (189),
namely,
$$\left. \eqalign{
l m_0'' &= l_{00} a_0'' + l_{01} a_1'';\cr
l m_1'' &= l_{10} a_0'' + l_{11} a_1''.\cr}
   \right\}
   \eqno (193)$$
Making, therefore, according to the general rule contained in the
formula (150),
$$\eqalignno{
l n_{r r' r''}
   &= \Sigma_{s,s'} \mathbin{.}
         l_{r'' s'} c_{r' s' s} a_{r s} \cr
   &=    (l_{r'' 0} c_{r' 0 0} + l_{r'' 1} c_{r' 1 0}) a_{r0}
       + (l_{r'' 0} c_{r' 0 1} + l_{r'' 1} c_{r' 1 1}) a_{r1},
   &(194)\cr}$$
we have results included in the formula (149), namely,
$$m_0'' = \Sigma_{r,r'} \mathbin{.}
            m_r m_{r'}' n_{r r' 0};\quad
  m_1'' = \Sigma_{r,r'} \mathbin{.}
            m_r m_{r'}' n_{r r' 1};
   \eqno (195)$$
that is, more fully,
$$\left. \eqalign{
m_0''
   &=    m_0 m_0' n_{000} + m_0 m_1' n_{010}
       + m_1 m_0' n_{100} + m_1 m_1' n_{110};\cr
m_1''
   &=    m_0 m_0' n_{001} + m_0 m_1' n_{011}
       + m_1 m_0' n_{101} + m_1 m_1' n_{111}.\cr}
   \right\}
   \eqno (196)$$
Thus, in particular, the coefficient of the product $m_0 m_0'$,
in the expression thus obtained for $m_0''$ is,
$$n_{000}
   = l^{-1} l_{00} ( c_{000} a_{00} + c_{001} a_{01} )
       + l^{-1} l_{01} ( c_{010} a_{00} + c_{011} a_{01} ).
   \eqno (197)$$
The equations (188) permit us to write
$$l_{00} =   a_{11};\quad
  l_{01} = - a_{10};\quad
  l_{10} = - a_{01};\quad
  l_{11} =   a_{00};
   \eqno (198)$$
provided that we assign to $l$ the value
$$l = a_{00} a_{11} - a_{10} a_{01}.
   \eqno (199)$$
Hence
$$n_{000}
   =  {  a_{11} ( c_{000} a_{00} + c_{001} a_{01} )
       - a_{10} ( c_{010} a_{00} + c_{011} a_{01} )
         \over a_{00} a_{11} - a_{10} a_{01} }.
   \eqno (200)$$
If we substitute, in this expression for $n_{000}$, the values
(186) for $a_{00}$, $a_{01}$, $a_{10}$, $a_{11}$, we shall
thereby obtain, in general, a certain function of $a_0$, $a_1$,
which will be homogeneous of the dimension zero, because it will
present itself under the form of a fraction, of which the
numerator and the denominator will be homogeneous and quadratic
functions of the same $a_0$,~$a_1$.  In order that this quotient
of two quadratic functions of the number expressing the ratio of
$a_1$ to $a_0$, or of ${\sr a}_1$ to ${\sr a}_0$, may be itself
independent of that ratio, we must have certain relations between
the coefficients $c_{000}$, \&c., and the fraction itself must
take a particular value connected with those coefficients; which
relations and value may be determined by the three equations:
$$\eqalignno{
n_{000} ( c_{000} c_{110} - c_{100} c_{010} )
   &=    c_{110} ( c_{000}^2 + c_{001} c_{010} )
       - c_{100} c_{010} ( c_{000} + c_{011} );
   &(201)\cr
n_{000} ( c_{000} c_{111} - c_{100} c_{011}
        + c_{001} c_{110} - c_{101} c_{010} )
      \hskip -9.2em \cr
   &=    c_{111} ( c_{000}^2 + c_{001} c_{010} )
       - c_{101} c_{010} ( c_{000} + c_{011} ) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + c_{110} c_{001} ( c_{000} + c_{011} )
       + c_{100} ( c_{010} c_{001} + c_{011}^2 );
   &(202)\cr
n_{000} ( c_{001} c_{111} - c_{101} c_{011} )
   &=    c_{111} c_{001} ( c_{000} + c_{011} )
       - c_{101} ( c_{010} c_{001} + c_{011}^2 ).
   &(203)\cr}$$

In like manner, each of the seven other coefficients,
$n_{010}$,~\&c. in the expressions (196), will furnish three
other equations of condition, which must all be satisfied, in
order that the values of these coefficients of multiplication of
couples may be independent of the original ratio of $a_1$ to
$a_0$, or of ${\sr a}_1$ to ${\sr a}_0$; and each of the
twenty-four equations thus furnished, of which the equations
(201), (202), (203), are three, is an equation of the third
dimension, with respect to the coefficients of derivation and
multiplication, $c_{000}$,~\&c., $n_{000}$,~\&c.  We should,
therefore, by this method, have obtained equations more numerous
and less simple than those which were given by the method of the
eighteenth article: which method there is, therefore, an
advantage in introducing, even for the case of couples, and much
more for the case of quaternions, or other ordinal and numeral
sets; although the method above exemplified appears to offer
itself more immediately from the principles of the seventeenth
article.

But to exhibit by an example the agreement of the two methods in
their results, let the symbols defined by the equations (163),
(164), be employed to abridge the expression of the equations
(201), (202), (203); the latter will then become:
$$\left. \eqalign{
e (ad - cb)
   &= d (a^2 + a' b) - c b (a + b');\cr
e (ad' - cb' + a'd - c'b)
   &= d' (a^2 + a'b) - c'b (a + b')
       + da' (a + b') - c (ba' + b'^2);\cr
e (a'd' - c'b')
   &= d'a' (a + b') - c' (ba' + b'^2);\cr}
   \right\}
   \eqno (204)$$
and it is evident, upon inspection, that these three equations
(204) may be deduced by elimination of $e'$ from the four
equations of detachment (165), which were obtained by the
simplified method; and which, in that method, formed part of a
system of only sixteen (instead of twenty-four) equations, each
rising no higher than the second (instead of the third)
dimension.

\bigbreak

\centerline{\it
Associative Principle of the Multiplication of numeral Sets:
Characteristics of numeral}

\nobreak\vskip 3pt

\centerline{\it
Separation.}

\nobreak\bigskip

21.
Whenever, for any value of the exponent~$n$ of the order of a
set, we have succeeded in satisfying the $n^4$ simplified
{\it equations of detachment}, included in the formula (160) of
the eighteenth article, and have thereby found a system of $n^3$
coefficients of derivation, and a connected system of $n^3$
coefficients of multiplication, with reference to which two
systems of coefficients an equation, or rather a system of
equations, of the form (153) can be established, independently of
the $n - 1$ ratios of the constituents of that ordinal
set~${\sr q}$, on which the two successive derivations are
performed; it is evident that we can then proceed, in like
manner, to perform on the resulting set a third successive
derivation; and that, with respect to such successive operations
of derivation, the following simple but important formula holds
good:
$$\times_{r'} \mathbin{.} \times_r \times_t
   =  \times_{r'} \times_r \mathbin{.} \times_t.
   \eqno (205)$$

To develope this symbolical equation, which may be said to
contain the {\it associative principle of the multiplication of
numeral sets}, we may conveniently employ a {\it characteristic
of numeral separation},~${\sc n}$, analogous to those two
characteristics, ${\sc m}$ and ${\sc r}$, which we have already
introduced in this paper, for the purpose of expressing
separately the different moments of a momental set, and of
separating, in like manner, those constituent ordinal relations
between moments which compose an ordinal set.  Let us, therefore,
agree to regard the $n$ equations,
$$m_0 = {\sc n}_0 q;\quad
  m_1 = {\sc n}_1 q;\quad\ldots \,\quad
  m_{n-1} = {\sc n}_{n-1} q,
   \eqno (206)$$
as jointly equivalent to the one complex equation or expression
(154), for a numeral set~$q$, of any proposed order~$n$; in such
a manner that we shall have, identically, for numeral
coefficients and numeral sets, the equations
$$\left. \eqalign{
m_0 &= {\sc n}_0 (m_0, m_1,\ldots \, m_{n-1}),\cr
m_1 &= {\sc n}_1 (m_0, m_1,\ldots \, m_{n-1}),\ldots\cr}
   \right\}
   \eqno (207)$$
and
$$q = ({\sc n}_0 q, {\sc n}_1 q,\ldots \, {\sc n}_{n-1} q);
   \eqno (208)$$
which are analogous to those marked (10) and (11), for moments
and momental sets, and also to the formul{\ae} (57), (58), for
constituent ordinal relations, and for the ordinal sets to which
they belong.  We may then substitute for the formula (153) of
symbolic multiplication, or of successive derivation, the
following:
$${\sc n}_s \mathbin{.} \times_{r'} \times_r = n_{r,r',s};
   \eqno (209)$$
which will give, also, by suitably changing the letters,
$${\sc n}_{s'} \mathbin{.} \times_s \times_t = n_{t,s,s'};
   \eqno (210)$$
the commas in the indices being here, for the sake of greater
clearness, restored.  In this manner we find that
$${\sc n}_{s'} (\times_{r'} \times_r \mathbin{.} \times_t)
   =  \Sigma_s \mathbin{.} n_{r,r',s'} n_{t,s,s'}.
   \eqno (211)$$
But, also,
$${\sc n}_s \mathbin{.} \times_r \times_t = n_{t,r,s};\quad
  {\sc n}_{s'} \mathbin{.} \times_{r'} \times_s = n_{s,r',s'};
   \eqno (212)$$
and, therefore,
$${\sc n}_{s'} (\times_{r'} \mathbin{.} \times_r \times_t)
   =  \Sigma_s \mathbin{.} n_{s,r',s'} n_{t,r,s};
   \eqno (213)$$
consequently, by operating with the characteristic~${\sc n}_{s'}$
on the symbolical equation (205), we obtain this other form for
the expression of the associative principle, considered as
establishing a certain system of relations between the
coefficients of multiplication:
$$0 = \Sigma_s
         (  n_{r,r',s} n_{t,s,s'} - n_{s,r',s'} n_{t,r,s} ).
   \eqno (214)$$
We are, therefore, entitled to regard this last formula, or the
system of numerical equations of condition which it includes, as
being a consequence of the analogous system of conditions
included in the formula (160), because the associative property
of multiplication is a consequence of the principle of
detachment.  And on comparing the two formul{\ae}, we perceive
that as soon as the one last deduced, namely (214), has been
satisfied by a suitable system of coefficients of multiplication,
then the one previously established, namely, (160), can be
immediately satisfied also, by connecting with this latter system
a system of coefficients of derivation, according to the rule
expressed by the following very simple equation:
$$c_{r,s,t} = n_{t,r,s}.
   \eqno (215)$$

For example, in the case of couples, with the abridged symbols
(163), (164), for the two systems of coefficients, this rule
(215) would have shewn that if we had in any manner succeeded in
satisfying the sixteen equations of detachment
(165)~$\ldots$~(168) between $a \, b \, c \, d$
$a' \, b' \, c' \, d'$ and $e \, f \, g \, h$
$e' \, f' \, g' \, h'$, we could then satisfy the same equations
of detachment with the same values of the eight latter symbols,
and with the following values for the eight former:
$$\left. \multieqalign{
a  &= e; & b  &= e'; & c  &= f; & d  &= f';\cr
a' &= g; & b' &= g'; & c' &= h; & d' &= h';\cr}
   \right\}
   \eqno (216)$$
which, in fact, will be found to agree with the values of the
nineteenth article.

\bigbreak

\centerline{\it
Connexion between the Coefficients of Derivation and of
Multiplication; simplified}

\nobreak\vskip 3pt

\centerline{\it
Conception of a numeral Set, regarded as expressing the complex
Ratio of an ordinal Set}

\nobreak\vskip 3pt

\centerline{\it
to a single ordinal Relation.}

\nobreak\bigskip

22.
The rule (215), for connecting together the two systems of
coefficients, of derivation and of multiplication, admits of
being interpreted or accounted for in a very simple manner.

The coefficient~$c_{r,s,t}$, introduced in the sixteenth article,
may be regarded as having been generated, or, at least, brought
under our view as follows.  We first supposed an ordinal
set,~${\sr q}$, to be operated on by the elementary
characteristics of derivation~$\times_r$, so as to produce
thereby a derivative set,~${\sr q}_r$.  We then operated on this
derived set, in a way which may be indicated by the
characteristic of ordinal separation,~${\sc r}_s$, and so
obtained a result of the form
$${\sc r}_s \times_r {\sr q} = {\sr a}_{r,s}.
   \eqno (217)$$
And, lastly, we analyzed this result, so as to find the part of
it which depended on, and arose from, the constituent~${\sr a}_t$
or ${\sc r}_t {\sr q}$ of the original operand set; and the
coefficient of this constituent~${\sr a}_t$, in the part obtained
by this analysis, was denoted by $c_{r,s,t}$, and was regarded as
a coefficient of derivation.  On the other hand, the coefficient
of multiplication,~$n_{t,r,s}$, may be said to arise thus: an
elementary derivation, denoted by $\times_t$, is succeeded by
another, denoted by $\times_r$; the compound operation,
$\times_r \times_t$, is detached from the operand, and regarded
as equivalent to a single complex derivation, of which the
characteristic may be symbolically equated to a certain numeral
set; this last set is subjected to the characteristic of numeral
separation,~${\sc n}_s$, or to an analysis equivalent thereto;
and the result is, by (212), the coefficient of multiplication in
question.

Now the agreement of the results of the two processes, which is
expressed by the equation (215), becomes quite intelligible and
natural, if we conceive that the constituent~${\sr a}_t$ of the
operand set~${\sr q}$, on which constituent alone we really
operate in the former process, the others being, in fact, set
aside, as contributing nothing to the result here sought for, has
been {\it itself produced\/} or generated {\it by an earlier
operation of the form\/}~$a_t \times_t$ (where $a_t$ has the same
signification as in (184)), from some one primary or original
ordinal relation, such as that which was denoted in some recent
articles by the letter~${\sr a}$.  In this manner we may be led
to look upon {\it any ordinal set}, such as the set~${\sr q}$, in
the equation (133), as being {\it generated by a certain complex
derivation}, which is expressed by a certain numeral
set~$q$, {\it from a single standard ordinal
relation},~${\sr a}$, or from the relation between some two
standard or selected moments of time, according to either of the
two reciprocal formul{\ae}:
$${\sr q} = q {\sr a}
   = \Sigma_t \mathbin{.} a_t \times_t {\sr a};
   \quad\hbox{or,}\quad
  q = {\sr q} \div {\sr a}
   = \Sigma_t \mathbin{.} a_t \times_t;
   \eqno (218)$$
in which last equation the members are symbols for a numeral set.
And thus a {\it numeral set\/} $(q)$ may come to be conceived as
being a {\it system or set of numbers, serving to mark or express
the complex ratio which an ordinal set $({\sr q})$ bears to a
simple or single ordinal relation\/}~$({\sr a})$, regarded as a
standard of comparison.

\bigbreak

\centerline{\it
Case of Quaternions; Coefficients of Multiplication.}

\nobreak\bigskip

23.
In the case of quaternions, the formula (214) gives a system of
$4^4 = 256$ equations of condition, included in the following
type (in which $u$ has been written instead of $r'$, and the
accent common to all the indices~$s'$ has been omitted as
unnecessary in the result):
$$\eqalignno{
      n_{r,u,0} n_{t,0,s}
    + n_{r,u,1} n_{t,1,s}
    + n_{r,u,2} n_{t,2,s}
    + n_{r,u,3} n_{t,3,s} \hskip-12em \cr
   &= n_{0,u,s} n_{t,r,0}
    + n_{1,u,s} n_{t,r,1}
    + n_{2,u,s} n_{t,r,2}
    + n_{3,u,s} n_{t,r,3};
   &(219)\cr}$$
each of the four indices $r$, $s$, $t$, $u$, in this last
formula, being allowed to receive any one of the four values
$0$,~$1$,~$2$,~$3$.  And all these two hundred and fifty-six
equations are satisfied when we establish the following system of
numerical values of the sixty-four coefficients of multiplication
(in which the commas between the indices are again omitted for
conciseness):
$$\left. \multieqalign{
n_{000} &=  1; &
n_{001} &=  0; &
n_{002} &=  0; &
n_{003} &=  0; \cr
n_{010} &=  0; &
n_{011} &=  1; &
n_{012} &=  0; &
n_{013} &=  0; \cr
n_{020} &=  0; &
n_{021} &=  0; &
n_{022} &=  1; &
n_{023} &=  0; \cr
n_{030} &=  0; &
n_{031} &=  0; &
n_{032} &=  0; &
n_{033} &=  1; \cr}
   \right\}
   \eqno (220)$$
$$\left. \multieqalign{
n_{100} &=  0; &
n_{101} &=  1; &
n_{102} &=  0; &
n_{103} &=  0; \cr
n_{110} &= -1; &
n_{111} &=  0; &
n_{112} &=  0; &
n_{113} &=  0; \cr
n_{120} &=  0; &
n_{121} &=  0; &
n_{122} &=  0; &
n_{123} &= -1; \cr
n_{130} &=  0; &
n_{131} &=  0; &
n_{132} &=  1; &
n_{133} &=  0; \cr}
   \right\}
   \eqno (221)$$
$$\left. \multieqalign{
n_{200} &=  0; &
n_{201} &=  0; &
n_{202} &=  1; &
n_{203} &=  0; \cr
n_{210} &=  0; &
n_{211} &=  0; &
n_{212} &=  0; &
n_{213} &=  1; \cr
n_{220} &= -1; &
n_{221} &=  0; &
n_{222} &=  0; &
n_{223} &=  0; \cr
n_{230} &=  0; &
n_{231} &= -1; &
n_{232} &=  0; &
n_{233} &=  0; \cr}
   \right\}
   \eqno (222)$$
$$\left. \multieqalign{
n_{300} &=  0; &
n_{301} &=  0; &
n_{302} &=  0; &
n_{303} &=  1; \cr
n_{310} &=  0; &
n_{311} &=  0; &
n_{312} &= -1; &
n_{313} &=  0; \cr
n_{320} &=  0; &
n_{321} &=  1; &
n_{322} &=  0; &
n_{323} &=  0; \cr
n_{330} &= -1; &
n_{331} &=  0; &
n_{332} &=  0; &
n_{333} &=  0. \cr}
   \right\}
   \eqno (223)$$

We might content ourselves with proving the truth of this
assertion by actual arithmetical substitution of these
sixty-four values in the two hundred and fifty-six equations; but
the following method, if less elementary, will probably be
considered to be more elegant, or less tedious.  It will have,
also, the advantage of conducting to a somewhat more general
system of expressions, by which the same equations can be
satisfied; and will serve to exemplify the application of the
fundamental relations, ({\sc a}),~({\sc b}), which were assigned
in the sixth and eighth articles, between the important symbols
$i$~$j$~$k$, and on which the present Theory of Quaternions may
be regarded as essentially depending.

\bigbreak

24.
Let us, then, first form, from the type (219), by changing the
index~$r$ to the value~$0$, the following less general type,
which, however, contains under it sixty-four out of the two
hundred and fifty-six equations of condition to be satisfied:
$$       n_{0u0} n_{t0s}
       + n_{0u1} n_{t1s}
       + n_{0u2} n_{t2s}
       + n_{0u3} n_{t3s}
   =     n_{0us} n_{t00}
       + n_{1us} n_{t01}
       + n_{2us} n_{t02}
       + n_{3us} n_{t03}.
   \eqno (224)$$
Make, for abridgment,
$$q_{tu}
   =  n_{tu0} + i n_{tu1} + j n_{tu2} + k n_{tu3};
   \eqno (225)$$
$i$~$j$~$k$ being the three symbols just now referred to; we may
then substitute for (224) the following formula, deduced from it,
but not involving the index~$s$:
$$       n_{0u0} q_{t0}
       + n_{0u1} q_{t1}
       + n_{0u2} q_{t2}
       + n_{0u3} q_{t3}
   =     q_{0u} n_{t00}
       + q_{1u} n_{t01}
       + q_{2u} n_{t02}
       + q_{3u} n_{t03}.
   \eqno (226)$$
This, again, will reduce itself, by the same definition (225) of
the symbol~$q_{tu}$, to the identity,
$$q_{0u} q_{t0} = q_{0u} q_{t0},
   \eqno (227)$$
and therefore will be satisfied, if we satisfy the six conditions:
$$\left. \multieqalign{
q_{t1} &= i q_{t0}; &
q_{t2} &= j q_{t0}; &
q_{t3} &= k q_{t0}; \cr
q_{1u} &= q_{0u} i; &
q_{2u} &= q_{0u} j; &
q_{3u} &= q_{0u} k. \cr}
   \right\}
   \eqno (228)$$
If, instead of making $r = 0$, we make $r = 1$, in (219), we then
obtain, instead of (224), the formula:
$$       n_{1u0} n_{t0s}
       + n_{1u1} n_{t1s}
       + n_{1u2} n_{t2s}
       + n_{1u3} n_{t3s}
   =     n_{0us} n_{t10}
       + n_{1us} n_{t11}
       + n_{2us} n_{t12}
       + n_{3us} n_{t13};
   \eqno (229)$$
and the symbolic equation (226) is replaced by the following:
$$       n_{1u0} q_{t0}
       + n_{1u1} q_{t1}
       + n_{1u2} q_{t2}
       + n_{1u3} q_{t3}
   =     q_{0u} n_{t10}
       + q_{1u} n_{t11}
       + q_{2u} n_{t12}
       + q_{3u} n_{t13};
   \eqno (230)$$
which, under the conditions (228), becomes first, by the
definition (225),
$$q_{1u} q_{t0} = q_{0u} q_{t1};
   \eqno (231)$$
and then is seen to be satisfied, in virtue of the same
conditions.

In like manner by making $r = 2$, in (219), we find
$$       n_{2u0} n_{t0s}
       + n_{2u1} n_{t1s}
       + n_{2u2} n_{t2s}
       + n_{2u3} n_{t3s}
   =     n_{0us} n_{t20}
       + n_{1us} n_{t21}
       + n_{2us} n_{t22}
       + n_{3us} n_{t23};
   \eqno (232)$$
and this, under the form
$$       n_{2u0} q_{t0}
       + n_{2u1} q_{t1}
       + n_{2u2} q_{t2}
       + n_{2u3} q_{t3}
   =     q_{0u} n_{t20}
       + q_{1u} n_{t21}
       + q_{2u} n_{t22}
       + q_{3u} n_{t23},
   \eqno (233)$$
is satisfied by the same conditions (228), since they give
$$q_{2u} t_{t0} = q_{0u} q_{t2}.
   \eqno (234)$$

Finally, the formula obtained from (219) by making $r = 3$,
namely,
$$       n_{3u0} n_{t0s}
       + n_{3u1} n_{t1s}
       + n_{3u2} n_{t2s}
       + n_{3u3} n_{t3s}
   =     n_{0us} n_{t30}
       + n_{1us} n_{t31}
       + n_{2us} n_{t32}
       + n_{3us} n_{t33},
   \eqno (235)$$
or this other, deduced from it by the help of (225),
$$       n_{3u0} q_{t0}
       + n_{3u1} q_{t1}
       + n_{3u2} q_{t2}
       + n_{3u3} q_{t3}
   =     q_{0u} n_{t30}
       + q_{1u} n_{t31}
       + q_{2u} n_{t32}
       + q_{3u} n_{t33},
   \eqno (236)$$
is satisfied by the same conditions (228), which give
$$q_{3u} q_{t0} = q_{0u} q_{t3}.
   \eqno (237)$$
We shall therefore satisfy not only the sixty-four arithmetical
conditions included in the type (224), but also the sixty-four
others included in the type (229), sixty-four included in (232),
and the sixty-four included in (235); that is to say, we shall
satisfy the whole system of the {\it two hundred and fifty-six
arithmetical\/} (or ordinary algebraical) {\it conditions\/}
included in the formula (219), if we satisfy the system of the
{\it six symbolical equations\/} (228), which involve the three
symbols $i$~$j$~$k$ in their composition; provided that we do so
{\it without establishing any linear relation between those three
symbols and unity}.  This last restriction is necessary, in order
that each of the four symbolical formul{\ae}, (226), (230), (233),
(236), not involving the index~$s$, may be, as we have supposed,
equivalent to the corresponding one of the four arithmetical
formul{\ae}, (224), (229), (232), (235), in which that index~$s$,
occurs, and is permitted to receive any one of the four values,
$0$,~$1$,~$2$,~$3$.

\bigbreak

25.
If we write, for conciseness,
$$q_0 = n_{000} + i n_{001} + j n_{002} + k n_{003},
   \eqno (238)$$
the conditions of the preceding article give the sixteen
symbolical equations:
$$\left. \multieqalign{
q_{00} &=   q_0; &
q_{01} &= i q_0; &
q_{02} &= j q_0; &
q_{03} &= k q_0; \cr
q_{10} &=   q_0 i; &
q_{11} &= i q_0 i; &
q_{12} &= j q_0 i; &
q_{13} &= k q_0 i; \cr
q_{20} &=   q_0 j; &
q_{21} &= i q_0 j; &
q_{22} &= j q_0 j; &
q_{23} &= k q_0 j; \cr
q_{30} &=   q_0 k; &
q_{31} &= i q_0 k; &
q_{32} &= j q_0 k; &
q_{33} &= k q_0 k; \cr}
   \right\}
   \eqno (239)$$
in which, while still retaining the {\it linear independence\/}
lately assumed to exist between $i$,~$j$,~$k$, and $1$, we may
now suppose that the {\it squares and products\/} of the three
symbols, $i$,~$j$,~$k$, are determined, or eliminated, by the
help of the fundamental formula~({\sc a}), assigned in the sixth
article, namely,
$$i^2 = j^2 = k^2 = ijk = -1;
   \eqno ({\sc a})$$
together with those others which this may be considered as
including, especially the following:
$$ij = k,\quad ji = -k;\quad
  jk = i,\quad kj = -i;\quad
  ki = j,\quad ik = -j.
   \eqno ({\sc b})$$
In this manner, by (225) and (238), while the first of the
sixteen symbolical equations (239) is identically satisfied, each
of the other fifteen will resolve itself into four ordinary
equations, independent of the three symbols $i$,~$j$,~$k$; and
thus, if we denote, for conciseness, four of the numerical
coefficients of quaternion multiplication as follows,
$$n_{000} = a,\quad
  n_{001} = b,\quad
  n_{002} = c,\quad
  n_{003} = d,
   \eqno (240)$$
the other sixty coefficients of such multiplication may be
expressed in terms of these; and the values so obtained will
satisfy the two hundred and fifty-six conditions included in the
formula (219); whatever four numbers may be chosen for
$a$,~$b$,~$c$,~$d$.

And if we farther simplify the formul{\ae} by supposing
$$a = 1,\quad
  b = 0,\quad
  c = 0,\quad
  d = 0,
   \eqno (241)$$
which will be found in the applications to involve no
{\it essential\/} loss of generality, we then obtain, from this
last-mentioned system of expressions, that system of sixty-four
numerical values for the sixty-four coefficients of
multiplication of quaternions, which was assigned in the
equations (220)~$\ldots$~(223), of the twenty-third article.

\bigbreak

\centerline{\it
Coefficients of Quaternion-Derivation; Comparison of
Characteristics.}

\nobreak\bigskip

26.
Adopting, then, those values, (220)~$\ldots$~(223), for the
sixty-four coefficients of multiplication, let us, at the same
time, in accordance with the rule~(215), adopt also such a
connected system of values for the sixty-four connected
coefficients of derivation,~$c_{r,s,t}$, as shall give the
continued equation,
$$\eqalignno{
1  &= c_{000} = c_{011} = c_{022} = c_{033}
    = - c_{101} = c_{110} = - c_{123} = c_{132} \cr
   &= - c_{202} = c_{213} = c_{220} = - c_{231}
    = - c_{303} = - c_{312} = c_{321} = c_{330};
   &(242)\cr}$$
ten of these coefficients~$c$ being thus equal to $+1$, and six
other being each equal to $-1$, while the other forty-eight
coefficients of derivation shall, by the same rule, vanish.

The formula (135) will thus give the sixteen following equations:
$$\left. \multieqalign{
{\sr a}_{00} &= \mathbin{\phantom{-}} {\sr a}_0; &
{\sr a}_{01} &= \mathbin{\phantom{-}} {\sr a}_1; &
{\sr a}_{02} &= \mathbin{\phantom{-}} {\sr a}_2; &
{\sr a}_{03} &= \mathbin{\phantom{-}} {\sr a}_3; \cr
{\sr a}_{10} &=                     - {\sr a}_1; &
{\sr a}_{11} &= \mathbin{\phantom{-}} {\sr a}_0; &
{\sr a}_{12} &=                     - {\sr a}_3; &
{\sr a}_{13} &= \mathbin{\phantom{-}} {\sr a}_2; \cr
{\sr a}_{20} &=                     - {\sr a}_2; &
{\sr a}_{21} &= \mathbin{\phantom{-}} {\sr a}_3; &
{\sr a}_{22} &= \mathbin{\phantom{-}} {\sr a}_0; &
{\sr a}_{23} &=                     - {\sr a}_1; \cr
{\sr a}_{30} &=                     - {\sr a}_3; &
{\sr a}_{31} &=                     - {\sr a}_2; &
{\sr a}_{32} &= \mathbin{\phantom{-}} {\sr a}_1; &
{\sr a}_{33} &= \mathbin{\phantom{-}} {\sr a}_0; \cr}
   \right\}
   \eqno (243)$$
and therefore, by comparing the definitions (134) and (70), we
shall have the four expressions:
$$\left. \eqalign{
\times_0 {\sr q}
    = (  \mathbin{\phantom{-}} {\sr a}_0,
         \mathbin{\phantom{-}} {\sr a}_1,
         \mathbin{\phantom{-}} {\sr a}_2,
         \mathbin{\phantom{-}} {\sr a}_3 )
   &= 1 {\sr q};\cr
\times_1 {\sr q}
    = (  - {\sr a}_1,
         \mathbin{\phantom{-}} {\sr a}_0,
         - {\sr a}_3,
         \mathbin{\phantom{-}} {\sr a}_2 )
   &= i {\sr q};\cr
\times_2 {\sr q}
    = (  - {\sr a}_2,
         \mathbin{\phantom{-}} {\sr a}_3,
         \mathbin{\phantom{-}} {\sr a}_0,
         - {\sr a}_1 )
   &= j {\sr q};\cr
\times_3 {\sr q}
    = (  - {\sr a}_3,
         - {\sr a}_2,
         \mathbin{\phantom{-}} {\sr a}_1,
         \mathbin{\phantom{-}} {\sr a}_0 )
   &= k {\sr q};\cr}
   \right\}
   \eqno (244)$$
for the results of operating, by the four elementary
characteristics of derivation,
$\times_0$, $\times_1$, $\times_2$, $\times_3$,
which are thus seen to be equivalent to $1$, $i$, $j$, $k$, on
the ordinal quaternion,
$${\sr q} = ({\sr a}_0, {\sr a}_1, {\sr a}_2, {\sr a}_3).
   \eqno (55)$$
Whatever the constituents of this original operand may be, since
the equations of detachment have been satisfied by the choice of
the constant coefficients, we shall have, by the formula (153),
and by the values (220)~$\ldots$~(223), sixteen expressions for
the symbolic squares and products of these elementary
characteristics of derivation, which are independent of the
quaternion first operated on; namely, the sixteen expressions
following:
$$\left. \multieqalign{
\times_0 \times_0 &=                       \times_0; &
\times_1 \times_0 &= \mathbin{\phantom{-}} \times_1; &
\times_2 \times_0 &= \mathbin{\phantom{-}} \times_2; &
\times_3 \times_0 &= \mathbin{\phantom{-}} \times_3; \cr
\times_0 \times_1 &=                       \times_1; &
\times_1 \times_1 &=                     - \times_0; &
\times_2 \times_1 &=                     - \times_3; &
\times_3 \times_1 &= \mathbin{\phantom{-}} \times_2; \cr
\times_0 \times_2 &=                       \times_2; &
\times_1 \times_2 &= \mathbin{\phantom{-}} \times_3; &
\times_2 \times_2 &=                     - \times_0; &
\times_3 \times_2 &=                     - \times_1; \cr
\times_0 \times_3 &=                       \times_3; &
\times_1 \times_3 &=                     - \times_2; &
\times_2 \times_3 &= \mathbin{\phantom{-}} \times_1; &
\times_3 \times_3 &=                     - \times_0; \cr}
   \right\}
   \eqno (245)$$
which might also be deduced from the equations,
$$\times_0 = 1;\quad
  \times_1 = i;\quad
  \times_2 = j;\quad
  \times_3 = k.
   \eqno (246)$$

\bigbreak

\centerline{\it
Product and Quotient of two numeral Quaternions; Law of the
Modulus.}

\nobreak\bigskip

27.
We may also write, by (155),
$$(m_0'', m_1'', m_2'', m_3'')
   =  (m_0', m_1', m_2', m_3') (m_0, m_1, m_2, m_3),
   \eqno (247)$$
and may say that the {\it numeral
quaternion\/}~$(m_0'', m_1'', m_2'', m_3'')$ is equal to the
{\it product\/} obtained when the numeral quaternion
$(m_0, m_1, m_2, m_3)$ is {\it multiplied} as a
{\it multiplicand}, by the numeral quaternion
$(m_0', m_1', m_2', m_3')$ as a {\it multiplier\/}; provided
that, by the formula (149), with the same values of the
coefficients of multiplication, we establish the four following
equations between the twelve numerical constituents of these
three numeral quaternions:
$$\left. \eqalign{
m_0'' &= m_0' m_0 - m_1' m_1 - m_2' m_2 - m_3' m_3;\cr
m_1'' &= m_0' m_1 + m_1' m_0 + m_2' m_3 - m_3' m_2;\cr
m_2'' &= m_0' m_2 - m_1' m_3 + m_2' m_0 + m_3' m_1;\cr
m_3'' &= m_0' m_3 + m_1' m_2 - m_2' m_1 + m_3' m_0.\cr}
   \right\}
   \eqno (248)$$

Under the same conditions we may say that the {\it multiplier\/}
quaternion (or the {\it left-hand factor\/} in the expression for
a product) is the {\it quotient\/} obtained by {\it dividing\/}
the product by the multiplicand; and may write the formula,
$$(m_0', m_1', m_2', m_3')
   =  {(m_0'', m_1'', m_2'', m_3'')
         \over (m_0, m_1, m_2, m_3)}.
   \eqno (249)$$
It is easy to see that if we make, for abridgment,
$$\left. \eqalign{
\mu^2   &= m_0^2   + m_1^2   + m_2^2   + m_3^2,\cr
\mu'^2  &= m_0'^2  + m_1'^2  + m_2'^2  + m_3'^2,\cr
\mu''^2 &= m_0''^2 + m_1''^2 + m_2''^2 + m_3''^2,\cr}
   \right\}
   \eqno (250)$$
and regard $\mu$,~$\mu'$,~$\mu''$ as positive (or absolute)
numbers, the equations (248) give the following very simple but
important relation:
$$\mu'' = \mu' \mu.
   \eqno (251)$$
If then we give the name of {\it modulus\/} to the (positive or
absolute) square-root of the sum of the squares of the four
(positive or negative or null) numbers, which enter as
{\it constituents\/} into the expression of a numeral quaternion,
we see that it is allowed to say, for such quaternions (as well
as for couples and their analogous moduli), that {\it the modulus
of the product is equal to the product of the moduli}.  The
equations (248) give also, for the numerical constituents of the
quotient (249), the expressions:
$$\left. \eqalign{
m_0' &= \mu^{-2}
      ( + m_0'' m_0 + m_1'' m_1 + m_2'' m_2 + m_3'' m_3 );\cr
m_1' &= \mu^{-2}
      ( - m_0'' m_1 + m_1'' m_0 - m_2'' m_3 + m_3'' m_2 );\cr
m_2' &= \mu^{-2}
      ( - m_0'' m_2 + m_1'' m_3 + m_2'' m_0 - m_3'' m_1 );\cr
m_3' &= \mu^{-2}
      ( - m_0'' m_3 - m_1'' m_2 + m_2'' m_1 + m_3'' m_0 );\cr}
   \right\}
   \eqno (252)$$
which may be compared with the expression (183) for the quotient
that results from the division of one couple by another.  As a
verification, we may observe that they give, as it is not
difficult to see that they ought to do,
$${(m_0, m_1, m_2, m_3) \over (m_0, m_1, m_2, m_3)}
   = (1, 0, 0, 0).
   \eqno (253)$$
And these results respecting products and quotients of two
numeral quaternions may easily be remembered, or reproduced, if
we observe that we have the following {\it general expression for
a numeral quaternion\/}:
$$q = (m_0, m_1, m_2, m_3) = m_0 + i m_1 + j m_2 + k m_3;
   \eqno (254) = ({\sc c})$$
where $i$,~$j$,~$k$ are still those three coordinate symbols, or
new fourth roots of unity, already introduced in this Essay, of
which the squares and products are subject to the fundamental
formula:
$$i^2 = j^2 = k^2 = ijk = -1;
   \eqno ({\sc a})$$
and to the relations which are consequences of this formula,
especially the following:
$$ij = -ji = k;\quad
  jk = -kj = i;\quad
  ki = -ik = j.
   \eqno ({\sc b})$$
These equations, ({\sc a}) and ({\sc b}), had indeed occurred
before in this paper; but on account of their great importance in
the present theory, they have been written once more in this
place, in connexion with the general expression~({\sc c}), which
may represent any numeral quaternion.

\bigbreak

\centerline{\it
On the more general System of Coefficients, obtained by a recent
Investigation.}

\nobreak\bigskip

28.
If we had not adopted the particular numerical values (241), but
had allowed the four letters $a$,~$b$,~$c$,~$d$, in the equation
(240), to denote any four constant numbers, which numbers, or
their symbols, should thus enter as {\it arbitrary constants\/}
into the expressions for the coefficients of multiplication, and
into those for the connected coefficients of derivation of
quaternions; then it is not difficult to see that, with the same
fundamental system of expressions for the squares and products of
$i$,~$j$,~$k$, contained in the formula~({\sc a}), the results of
the investigation in the twenty-fourth and twenty-fifth articles
might be concisely presented as follows:
$$m_0 \times_0 + m_1 \times_1 + m_2 \times_2 + m_3 \times_3
   = (m_0 + m_1 i + m_2 j + m_3 k) (a + bi + cj + dk).
   \eqno (255)$$
And then the formula of symbolic multiplication of one numeral
quaternion by another, which is included in (152), namely,
$$\eqalignno{
m_0'' \times_0 + m_1'' \times_1 + m_2'' \times_2 + m_3'' \times_3
   \hskip - 12em \cr
   &= (m_0' \times_0 + m_1' \times_1
            + m_2' \times_2 + m_3' \times_3)
      (m_0 \times_0 + m_1 \times_1
            + m_2 \times_2 + m_3 \times_3),
   &(256)\cr}$$
would become, with the same system of non-linear relations
between the same three symbols $i$,~$j$,~$k$:
$$\eqalignno{
m_0'' + m_1'' i + m_2'' j + m_3'' k
   \hskip - 9em \cr
   &= (m_0' + m_1' i + m_2' j + m_3' k)
      (a + bi + cj + dk)
      (m_0 + m_1 i + m_2 j + m_3 k).
   &(257)\cr}$$
This formula resolves itself, by those relations, and by the
linear independence of $i$,~$j$,~$k$, and~$1$, into four separate
equations, which may be obtained from the four equations (248),
by changing $m_0$,~$m_1$,~$m_2$,~$m_3$, respectively to,
$$\left. \eqalign{
m_0^\backprime
   &= a m_0 - b m_1 - c m_2 - d m_3;\cr
m_1^\backprime
   &= a m_1 + b m_0 + c m_3 - d m_2;\cr
m_2^\backprime
   &= a m_2 - b m_3 + c m_0 + d m_1;\cr
m_3^\backprime
   &= a m_3 + b m_2 - c m_1 + d m_0;\cr}
   \right\}
   \eqno (258)$$
so that, with these abridgments, the four equations included in
the formula (257) may be thus written:
$$\left. \eqalign{
m_0'' &=    m_0' m_0^\backprime
          - m_1' m_1^\backprime
          - m_2' m_2^\backprime
          - m_3' m_3^\backprime;\cr
m_1'' &=    m_0' m_1^\backprime
          + m_1' m_0^\backprime
          + m_2' m_3^\backprime
          - m_3' m_2^\backprime;\cr
m_2'' &=    m_0' m_2^\backprime
          - m_1' m_3^\backprime
          + m_2' m_0^\backprime
          + m_3' m_1^\backprime;\cr
m_3'' &=    m_0' m_3^\backprime
          + m_1' m_2^\backprime
          - m_2' m_1^\backprime
          + m_3' m_0^\backprime.\cr}
   \right\}
   \eqno (259)$$
In this manner we should obtain the four expressions:
$$\left. \eqalign{
m_0'' &= a A_0 + b B_0 + c C_0 + d D_0; \cr
m_1'' &= a A_1 + b B_1 + c C_1 + d D_1; \cr
m_2'' &= a A_2 + b B_2 + c C_2 + d D_2; \cr
m_3'' &= a A_3 + b B_3 + c C_3 + d D_3; \cr}
   \right\}
   \eqno (260)$$
where
$$\left. \eqalign{
A_0 &= m_0' m_0 - m_1' m_1 - m_2' m_2 - m_3' m_3;\cr
A_1 &= m_0' m_1 + m_1' m_0 + m_2' m_3 - m_3' m_2;\cr
A_2 &= m_0' m_2 - m_1' m_3 + m_2' m_0 + m_3' m_1;\cr
A_3 &= m_0' m_3 + m_1' m_2 - m_2' m_1 + m_3' m_0;\cr}
   \right\}
   \eqno (261)$$
$$\left. \eqalign{
B_0 &= - m_0' m_1 - m_1' m_0 + m_2' m_3 - m_3' m_2;\cr
B_1 &= + m_0' m_0 - m_1' m_1 + m_2' m_2 + m_3' m_3;\cr
B_2 &= - m_0' m_3 - m_1' m_2 - m_2' m_1 + m_3' m_0;\cr
B_3 &= + m_0' m_2 - m_1' m_3 - m_2' m_0 - m_3' m_1;\cr}
   \right\}
   \eqno (262)$$
$$\left. \eqalign{
C_0 &= - m_0' m_2 - m_1' m_3 - m_2' m_0 + m_3' m_1;\cr
C_1 &= + m_0' m_3 - m_1' m_2 - m_2' m_1 - m_3' m_0;\cr
C_2 &= + m_0' m_0 + m_1' m_1 - m_2' m_2 + m_3' m_3;\cr
C_3 &= - m_0' m_1 + m_1' m_0 - m_2' m_3 - m_3' m_2;\cr}
   \right\}
   \eqno (263)$$
$$\left. \eqalign{
D_0 &= - m_0' m_3 + m_1' m_2 - m_2' m_1 - m_3' m_0;\cr
D_1 &= - m_0' m_2 - m_1' m_3 + m_2' m_0 - m_3' m_1;\cr
D_2 &= + m_0' m_1 - m_1' m_0 - m_2' m_3 - m_3' m_2;\cr
D_3 &= + m_0' m_0 + m_1' m_1 + m_2' m_2 - m_3' m_3.\cr}
   \right\}
   \eqno (264)$$

And thus may the problem of the multiplication of numeral
quaternions be resolved, without any restriction being laid on
the numerical values of the four arbitrary constants,
$a$,~$b$,~$c$,~$d$.  The {\it modular equation\/} (251), namely
$\mu'' = \mu' \mu$, will extend to this more general system, if
we define the modulus~$\mu$ of the quaternion
$(m_0, m_1, m_2, m_3)$ by the formula:
$$\mu^2
   =  (a^2 + b^2 + c^2 + d^2) (m_0^2 + m_1^2 + m_2^2 + m_3^2).
   \eqno (265)$$
Thus, with the recently established forms (261)~$\ldots$~(264),
of the sixteen functions $A_0 \, \ldots \, D_3$, we must have, as
an {\it identity}, independent of the values of the
{\it twelve\/} numbers denoted by the symbols
$a$~$b$~$c$~$d$ $m_0$~$m_1$~$m_2$~$m_3$
$m_0'$~$m_1'$~$m_2'$~$m_3'$,
the following equation:
$$\eqalignno{
      (a A_0 + b B_0 + c C_0 + d D_0)^2
    + (a A_1 + b B_1 + c C_1 + d D_1)^2
   \hskip -21em \cr
   &\mathrel{\phantom{=}} \mathord{}
    + (a A_2 + b B_2 + c C_2 + d D_2)^2
    + (a A_3 + b B_3 + c C_3 + d D_3)^2
   \hskip -12em \cr
   &= (a^2 + b^2 + c^2 + d^2)
      (m_0'^2 + m_1'^2 + m_2'^2 + m_3'^2)
      (m_0^2 + m_1^2 + m_2^2 + m_3^2);
   &(266)\cr}$$
and therefore, independently of the values of the {\it eight\/}
numbers $m_0 \, \ldots \, m_3'$, we must have these {\it ten\/}
other equations:
$$\left. \eqalign{
(m_0'^2 + m_1'^2 + m_2'^2 + m_3'^2)
      (m_0^2 + m_1^2 + m_2^2 + m_3^2)
   \hskip -12em \cr
   &= A_0^2 + A_1^2 + A_2^2 + A_3^2
    = B_0^2 + B_1^2 + B_2^2 + B_3^2 \cr
   &= C_0^2 + C_1^2 + C_2^2 + C_3^2
    = D_0^2 + D_1^2 + D_2^2 + D_3^2;\cr}
   \right\}
   \eqno (267)$$
$$\left. \multieqalign{
0  &= A_0 B_0 + A_1 B_1 + A_2 B_2 + A_3 B_3; &
0  &= A_0 C_0 + A_1 C_1 + A_2 C_2 + A_3 C_3; \cr
0  &= A_0 D_0 + A_1 D_1 + A_2 D_2 + A_3 D_3; &
0  &= B_0 C_0 + B_1 C_1 + B_2 C_2 + B_3 C_3; \cr
0  &= B_0 D_0 + B_1 D_1 + B_2 D_2 + B_3 D_3; &
0  &= C_0 D_0 + C_1 D_1 + C_2 D_2 + C_3 D_3. \cr}
   \right\}
   \eqno (268)$$

Although these identities admit of being established in a more
elementary way, yet it has been thought worth while to point out
the foregoing method of arriving at them, because that method
follows easily from the principles of the present theory.

\bigbreak

\centerline{\it
On the Extension of the Theory of Multiplication of Quaternions
to other numeral Sets.}

\nobreak\bigskip

29.
This seems to be a proper place for offering a few remarks on the
treatment of the general equation (214), which may assist in the
future extension of the present theory of multiplication of
quaternions to other numeral sets; and may serve, in the
meanwhile, to throw some fresh light on the process which has
been employed in the twenty-fourth and twenty-fifth articles, for
discovering a mode of satisfying that general equation, in the
case when the exponent~$n$ of the order of the set is~$4$.

Let $i_0, i_1,\ldots \, i_{n-1}$ be a system of $n$ symbolical
multipliers, which we shall assume to be unconnected with each
other by any {\it linear\/} relation; and let us establish the
following formula, analogous to (225),
$$q_{t,u} = i_0 n_{t,u,0} + \ldots + i_{n-1} n_{t,u,n-1}
          = \Sigma_v \mathbin{.}i_v n_{t,u,v}.
   \eqno (269)$$
Then, operating by the characteristic
$\Sigma \mathbin{.} i_{s'}$
on the equation (214), we shall transform that equation into the
following:
$$0 = \Sigma_s
      (n_{r,r',s} q_{t,s} - q_{s,r'} n_{t,r,s});
   \eqno (270)$$
and may satisfy it by supposing
$$q_{t,u} = i_u q_0 i_t;\quad
  q_0 = i_0^{-1} q_{0,0} i_0^{-1};
   \eqno (271)$$
for we shall then have
$$\Sigma_s \mathbin{.} n_{r,r',s} q_{t,s}
   =  q_{r,r'} q_0 i_t = i_{r'} q_0 i_r q_0 i_t
   =  i_{r'} q_0 q_{t,r}
   =  \Sigma_s \mathbin{.} q_{s,r'} n_{t,r,s}.
   \eqno (272)$$
We are therefore to endeavour to satisfy the symbolical
condition,
$$\Sigma_v \mathbin{.} i_u^{-1} i_v i_t^{-1} n_{t,u,v}
   =  \hbox{const.} = q_0;
   \eqno (273)$$
this constant~$q_0$ being independent of $t$ and $u$, and the $n$
symbols $i_0$,~$i_1$,~\&c., being still unconnected by any linear
relation.  When this shall have been accomplished, we may then
employ the formula,
$$\times_t = i_t q_0;
   \eqno (274)$$
which will give
$$\times_u \times_t = i_u q_0 i_t q_0
   =  q_{t,u} q_0
   =  \Sigma_v \mathbin{.} n_{t,u,v} \times_v;
   \eqno (275)$$
and therefore will agree with the formula (153).  And thus the
equations of detachment will have been satisfied, and a numeral
{\it set}, of the kind above supposed, will be found under the
form,
$$q = \Sigma_t \mathbin{.} m_t \times_t
    = \Sigma_t \mathbin{.} m_t i_t q_0.
   \eqno (276)$$
For the case of {\it couples}, we may make
$$i_0 = 1;\quad i_1 = \surd (-1);\quad q_0 = 1;
   \eqno (277)$$
and then the condition (273) will be satisfied by the values of
the coefficients of multiplication assigned in the nineteenth
article; and the numeral couple will present itself under the
well-known form, $m_0 + m_1 \surd (-1)$.

For the case of {\it quaternions}, if we suppose
$$i_0 = 1,\quad i_1 = i;\quad i_2 = j;\quad i_3 = k;
   \eqno (278)$$
the symbols $i$,~$j$,~$k$ being still connected by the
fundamental relations~({\sc a}); the six symbolical equations
(228), and the sixteen symbolical equations (239), will then be
included, by (269), in the formula (273), in which we may write,
by (240), and by (271), or (238),
$$q_0 = a + bi + cj + dk;
   \eqno (279)$$
and the expression (255) will be included in the more general
expression (276).  And if we farther particularize, and at the
same time simplify, by adopting, as we propose henceforth to do,
the values (241), which reduce $q_0$ to $1$, we shall then obtain
from (276), by (278), the same expression (254), or ({\sc c}),
which has already been assigned in the twenty-seventh article, as
the representation of a numeral quaternion.

\bigbreak

\centerline{\it
Successive Multiplication of Quaternions: Application of the
associative Principle.}

\nobreak\bigskip

30.
It has been stated that we design to adopt, in our present theory
of numeral quaternions, the simplifications contained in the
equations (241).  We shall therefore regard, henceforth, the
constituents of any product of {\it two\/} numeral quaternions as
being given by the simpler formul{\ae} (248), and not by the more
complex formul{\ae} (260), in which $A_0 \, \ldots \, D_3$ are
abridged representations of the sixteen quadrinomials
(261)~$\ldots$~(264).  Yet the trouble of investigating these
latter expressions will not have been thrown away: for we may
see, by (257), that they will serve, hereafter to express
the result of a successive multiplication, or the {\it continued
product of three numeral quaternions}.  And by applying the
{\it associative principle}, already considered in the
twenty-first article, to such {\it successive multiplication}, we
see that, instead of developing the formula (257) by a process
which was equivalent to the development of the system of the two
equations,
$$    m_0^\backprime
    + m_1^\backprime i
    + m_2^\backprime j
    + m_3^\backprime k
   =  (a + bi + cj + dk) (m_0 + m_1 i + m_2 j + m_3 k),
   \eqno (280)$$
and
$$m_0'' + m_1'' i + m_2'' j + m_3'' k
   =  (m_0' + m_1' i + m_2' j + m_3' k)
      (   m_0^\backprime
        + m_1^\backprime i
        + m_2^\backprime j
        + m_3^\backprime k ),
   \eqno (281)$$
we might have developed the same formula (257) by a different,
but analogous process, founded on a different mode of grouping or
{\it associating\/} the three quaternions which enter as symbolic
factors.  For we might have introduced this other quaternion,
$$    m_0^{\backprime\backprime}
    + m_1^{\backprime\backprime} i
    + m_2^{\backprime\backprime} j
    + m_3^{\backprime\backprime} k
   =  (m_0' + m_1' i + m_2' j + m_3' k)(a + bi + cj + dk);
   \eqno (282)$$
which would have given the expression,
$$m_0'' + m_1'' i + m_2'' j + m_3'' k
   =  (   m_0^{\backprime\backprime}
        + m_1^{\backprime\backprime} i
        + m_2^{\backprime\backprime} j
        + m_3^{\backprime\backprime} k )
      (m_0 + m_1 i + m_2 j + m_3 k);
   \eqno (283)$$
and then the four values (260), for the four constituents of the
final product of the three quaternion factors which enter into
the second member of the formula (257), would have presented
themselves as the result of the elimination of the four
constituents of the intermediate quaternion product (282),
between the eight following equations:
$$\left. \eqalign{
m_0^{\backprime\backprime}
   &= m_0' a - m_1' b - m_2' c - m_3' d;\cr
m_1^{\backprime\backprime}
   &= m_0' b + m_1' a + m_2' d - m_3' c;\cr
m_2^{\backprime\backprime}
   &= m_0' c - m_1' d + m_2' a + m_3' b;\cr
m_3^{\backprime\backprime}
   &= m_0' d + m_1' c - m_2' b + m_3' a;\cr}
   \right\}
   \eqno (284)$$
$$\left. \eqalign{
m_0'' &=    m_0^{\backprime\backprime} m_0
          - m_1^{\backprime\backprime} m_1
          - m_2^{\backprime\backprime} m_2
          - m_3^{\backprime\backprime} m_3;\cr
m_1'' &=    m_0^{\backprime\backprime} m_1
          + m_1^{\backprime\backprime} m_0
          + m_2^{\backprime\backprime} m_3
          - m_3^{\backprime\backprime} m_2;\cr
m_2'' &=    m_0^{\backprime\backprime} m_2
          - m_1^{\backprime\backprime} m_3
          + m_2^{\backprime\backprime} m_0
          + m_3^{\backprime\backprime} m_1;\cr
m_3'' &=    m_0^{\backprime\backprime} m_3
          + m_1^{\backprime\backprime} m_2
          - m_2^{\backprime\backprime} m_1
          + m_3^{\backprime\backprime} m_0.\cr}
   \right\}
   \eqno (285)$$
And accordingly, on comparing these eight equations with the four
expressions (260), we arrive at the same quadrinomial values for
the sixteen coefficients $A_0 \, \ldots \, D_3$, which have been
already given in the equations (261)~$\ldots$~(264).  We may
perceive that they would conduct also to the relations (267),
(268) between those coefficients, and to the formula (266) for
the {\it decomposition of a product of three sums, containing
each four squares}, by eliminating the
modulus~$\mu^{\backprime\backprime}$ of the quaternion (282)
between two equations analogous to (251), namely, the two
following:
$$\mu^{\backprime\backprime} = \mu' e,\quad
  \mu'' = \mu^{\backprime\backprime} \mu;
   \eqno (286)$$
where $\mu$, $\mu'$, $\mu''$ have the significations (250), and
where
$$\mu^{\backprime\backprime 2}
   =     m_0^{\backprime\backprime 2}
       + m_1^{\backprime\backprime 2}
       + m_2^{\backprime\backprime 2}
       + m_3^{\backprime\backprime 2};\quad
  e^2 = a^2 + b^2 + c^2 + d^2.
   \eqno (287)$$

\bigbreak

\centerline{\it
Addition and Subtraction of Numeral Sets; Non-commutative
Character of}

\nobreak\vskip 3pt

\centerline{\it
Quaternion Multiplication.}

\nobreak\bigskip

31.
Any two numeral sets may be {\it added\/} to each other, by
adding their respective constituent numbers, primary to primary,
secondary to secondary, and so forth; and on a similar plan may
{\it subtraction\/} of such sets be performed; thus, for any two
numeral quaternions we may write,
$$(m_0', m_1', m_2', m_3') \pm (m_0, m_1, m_2, m_3)
   =  (m_0' \pm m_0, m_1' \pm m_1, m_2' \pm m_2, m_3' \pm m_3);
   \eqno (288)$$
and generally, by using $\Sigma$ and $\Delta$ as the
characteristics of sum and difference, and employing those signs
of numeral separation which were proposed in the twenty-first
article, we may write formul{\ae} for sums and differences of
numeral sets, which are analogous to, and may be considered as
depending upon those marked (116), for the addition and
subtraction of ordinal sets; namely, the following:
$${\sc n}_r \Sigma q = \Sigma {\sc n}_r q;\quad
  {\sc n}_r \Delta q = \Delta {\sc n}_r q.
   \eqno (289)$$

For the {\it multiplication\/} of numeral sets, we have already
established principles and formul{\ae} which involve, generally,
the {\it distributive\/} and the {\it associative\/} properties of
the operation of the same name, as performed on single numbers;
but which {\it do not retain}, in general, {\it the commutative
property\/} of that ordinary operation upon numbers.  Thus we may
write,
$$\Sigma q' \times \Sigma q = \Sigma (q' \times q),
   \eqno (290)$$
and also,
$$q'' \times q' q = q'' q' \times q = q'' q' q,
   \eqno (291)$$
the mark of multiplication being allowed to be omitted, because
its place is unimportant to the result, in the successive
multiplication of any three or more numeral sets.  But we are
{\it not\/} at liberty to write, {\it generally}, for
{\it any two\/} such sets, as factors, the commutative formula,
$$q' q = q q';$$
since, although, by the equation (182), this last formula of
commutation of factors holds good, not only for single
{\it numbers}, but also when the factors are numeral
{\it couples}, of the kind considered in the nineteenth article
of the present paper, and in the earlier Essay there referred to,
yet, for the case of numeral quaternions, the relations~({\sc b})
between the products of the symbols $i$,~$j$,~$k$, give results
opposed to the commutative formula, namely, the following:
$$ij = - ji,\quad
  jk = - kj,\quad
  ki = - ik.$$

In fact, by (149), or by (209), to justify generally this
commutative formula of multiplication, as applied to numeral sets
of the order~$n$, it would be necessary that the $n^3$
coefficients of multiplication should be connected with each
other by the relations included in the type,
$$n_{r,r',s} = n_{r',r,s}.
   \eqno (292)$$
Now these relations have, indeed, been established in our theory
of numeral couples, since, in the abridged notation of the
nineteenth article, and with the values there adopted, we have
the equations,
$$f = g,\quad f' = g';
   \quad\hbox{or}\quad
  n_{010} = n_{100};\quad n_{011} = n_{101};
   \eqno (293)$$
but they do {\it not\/} hold good in our theory of numeral
quaternions, since we have been led to adopt values for the
coefficients of multiplication, which give, on the contrary,
$$n_{123} = - n_{213};\quad
  n_{231} = - n_{321};\quad
  n_{312} = - n_{132}.
   \eqno (294)$$
Thus, if we still adopt the system of values of the coefficients
of quaternion multiplication assigned in the twenty-third
article, we must {\it reject\/} the commutative property; and may
establish a formula which is {\it opposite\/} in its character to
the equation (292), namely, the following:
$$n_{r,r',s} = - n_{r',r,s},
   \quad\hbox{if}\quad
   r \gtlt r,\quad r > 0,\quad r' > 0.
   \eqno (295)$$

\bigbreak

\centerline{\it
General Division of one numeral Set by another: Combination of
the Operations of}

\nobreak\vskip 3pt

\centerline{\it
Division and Multiplication of Quaternions.}

\nobreak\bigskip

32.
The general {\it division\/} of one numeral set by another, if
regarded as the operation of {\it returning to the multiplier},
from the product and the multiplicand, involves no theoretical
difficulty, since it depends on the solution, by elimination or
otherwise, of a finite system of ordinary equations of the first
degree, between the sought numerical constituents of the
quotient; and it has been already exemplified, for couples and
quaternions, in the nineteenth and twenty-seventh articles.  But
it is of essential importance to observe that, if division of
numeral sets be thus defined by the formula,
$$(q'' \div q) \times q = q'',
   \eqno (296)$$
in which, as in all other cases, we conceive {\it the symbol of
the multiplier to be placed at the left hand}, and which is
analogous to (129), we shall then {\it not have, generally}, for
numeral sets, as for numbers, this other usual relation:
$$q \times (q'' \div q) = q''.$$
In fact, if we were to assume, for example, that this latter and
usual equation, though true for numbers and for numeral couples,
was generally true for numeral quaternions also, we should then,
in consequence of the definitional formula (296), which fixes the
correlation of the signs $\times$ and $\div$, with respect to the
numerical sets, be virtually assuming, also, that equation of
commutative multiplication, $q' q = q q'$, which, for the case of
quaternions at least, we have already seen reason to
{\it reject}.  Hence follows the important consequence that, in
this case of quaternions, the first member,
$q \times (q'' \div q)$, of the lately rejected equation, is the
{\it symbol of a new quaternion, distinct in general from the
operand quaternion,~$q''$, which has been first divided and
afterwards multiplied by one common operator quaternion,~$q$;
these two operations, thus performed, having not generally
neutralized each other, on account of the generally
noncommutative character of the multiplication of numeral
quaternions}.  It is, therefore, already an object of interest in
this theory, and will be found to be a problem of which the
geometrical and physical applications are in a high degree
important, {\it to determine the constituents of that new
quaternion},~$q_{\prime\prime}$, distinct from $q''$, which is
thus represented by the symbol $q \times (q'' \div q)$, or which
satisfies the equation
$$q \times (q'' \div q) = q_{\prime\prime}.
   \eqno (297)$$
To express the same problem otherwise, with the help of the
definition of division, (296), we have now the system of the two
equations,
$$q'' = q' q;\quad q_{\prime\prime} = q q';
   \eqno (298)$$
$q''$ and $q_{\prime\prime}$ being those two distinct quaternion
products which arise from the multiplication of the same two
quaternion factors, $q$ and $q'$, with two different arrangements
of those factors; and we are to eliminate the four constituents
of one of those two quaternion factors, namely, the constituents
of the factor~$q'$, between the eight separate and ordinary
equations into which the two quaternion equations (298) resolve
themselves.  If we write, for this purpose,
$$\left. \eqalign{
q &= w + ix + jy + kz,\cr
q' &= w' + ix' + jy' + kz',\cr
q'' &= w'' + ix'' + jy'' + kz'',\cr
q_{\prime\prime} &= w_{\prime\prime} + ix_{\prime\prime}
       + jy_{\prime\prime} + kz_{\prime\prime},\cr}
   \right\}
   \eqno (299)$$
we shall then have the four equations,
$$\left. \eqalign{
w'' &= w' w - x' x - y' y - z' z;\cr
x'' &= w' x + x' w + y' z - z' y;\cr
y'' &= w' y - x' z + y' w + z' x;\cr
z'' &= w' z + x' y - y' x + z' w;\cr}
   \right\}
   \eqno (300)$$
together with the four others which result from these by
interchanging, in the right-hand members, the accented with the
unaccented letters, and by changing in the left-hand members
upper to lower accents; namely, the four following:
$$\left. \eqalign{
w_{\prime\prime} &= w w' - x x' - y y' - z z';\cr
x_{\prime\prime} &= w x' + x w' + y z' - z y';\cr
y_{\prime\prime} &= w y' - x z' + y w' + z x';\cr
z_{\prime\prime} &= w z' + x y' - y x' + z w'.\cr}
   \right\}
   \eqno (301)$$
It thus appears immediately that
$$w_{\prime\prime} = w'';
   \eqno (302)$$
and the elimination, above directed, of the four numbers
$w'$,~$x'$,~$y'$,~$z'$, that is, of the constituents of the
numeral quaternion~$q'$, between the eight equations (300),
(301), gives these three other equations, which complete the
solution of the problem, so far as it depends on the
above-mentioned elimination:
$$\left. \eqalign{
w x_{\prime\prime} + z y_{\prime\prime} - y z_{\prime\prime}
   &= w x'' + y z'' - z y'';\cr
w y_{\prime\prime} + x z_{\prime\prime} - z x_{\prime\prime}
   &= w y'' + z x'' - x z'';\cr
w z_{\prime\prime} + y x_{\prime\prime} - x y_{\prime\prime}
   &= w z'' + x y'' - y x''.\cr}
   \right\}
   \eqno (303)$$
These equations conduct to the relations,
$$x x_{\prime\prime} + y y_{\prime\prime} + z z_{\prime\prime}
   =  x x'' + y y'' + z z'',
   \eqno (304)$$
and
$$x_{\prime\prime}^2 + y_{\prime\prime}^2 + z_{\prime\prime}^2
   =  x''^2 + y''^2 + z''^2;
   \eqno (305)$$
which, as it is easy to foresee, will be found to have extensive
applications, and which may also be easily obtained, by observing
that, before the elimination of $w'$, $x'$, $y'$, $z'$, the
equations (300), (301) give
$$\left. \multieqalign{
x_{\prime\prime} + x''
   &= 2 (wx' + w'x); &
x_{\prime\prime} - x''
   &= 2 (yz' - zy'); \cr
y_{\prime\prime} + y''
   &= 2 (wy' + w'y); &
y_{\prime\prime} - y''
   &= 2 (zx' - xz'); \cr
z_{\prime\prime} + z''
   &= 2 (wz' + w'z); &
z_{\prime\prime} - z''
   &= 2 (xy' - yx'). \cr}
   \right\}
   \eqno (306)$$

\bigbreak

33.
Although these latter combinations (306), of those equations
(300), (301), conduct without difficulty to the equations (303),
(304), (305), yet it is still more easy, when once the principles
of the present theory have been distinctly comprehended, to
deduce the last-mentioned equations, by treating in the following
way the problem of the foregoing article.

Instead of {\it resolving\/} the numeral quaternion~$q'$ into the
four separate terms, $w'$, $ix'$, $jy'$, $kz'$, as is done in the
second of the four expressions (299), and then eliminating the
four constituent numbers $w'$, $x'$, $y'$, $z'$ between the eight
ordinary equations into which the two quaternion equations (298)
resolve themselves, we may {\it eliminate the quaternion\/}~$q'$
itself between those two equations (298), and so obtain
immediately, without any labour of calculation, this new
quaternion equation,
$$q_{\prime\prime} q = q q'';
   \eqno (307)$$
which, by the three remaining expressions (299), and by the
equality (302), becomes:
$$(i x_{\prime\prime} + j y_{\prime\prime} + k z_{\prime\prime})
         (w + i x + j y + k z)
   =  (w + i x + j y + k z) (i x'' + j y'' + k z'').
   \eqno (308)$$

If now we perform the multiplications here indicated, attending
to the fundamental expressions ({\sc a}) ({\sc b}), for the
squares and products of the three symbols, $i$,~$j$,~$k$, and to
the linear independence, already supposed to exist, between the
four symbols, $i$,~$j$,~$k$ and $1$, we find that the {\it one\/}
quaternion formula (308) resolves itself into the {\it four\/}
equations, (303) and (304).  And either from the four equations
thus obtained, or by an application of the law of the modulus to
the quaternion equation~(308), the relation (305) may be
obtained.  It is worth while observing that we may also write the
quaternion formula,
$$(w^2 + x^2 + y^2 + z^2) q_{\prime\prime}
   =  (w + ix + jy + kz) (w'' + i x'' + j y'' + k z'')
         (w - ix - jy - kz);
   \eqno (309)$$
or, more fully,
$$\eqalignno{
(w^2 + x^2 + y^2 + z^2) (w_{\prime\prime} - w''
          + i x_{\prime\prime}
          + j y_{\prime\prime}
          + k z_{\prime\prime})
   \hskip -18em \cr
   &= (w^2 - x^2 - y^2 - z^2) (i x'' + j y'' + k z'')
       + 2 (x x'' + y y'' + z z'') (ix + jy + kz) \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 2 w \{ i (y z'' - z y'') + j (z x'' - x z'')
                  + k (x y'' - y x'') \};
   &(310)\cr}$$
by resolving which {\it one\/} formula, the same separate values
for $w_{\prime\prime}$, $x_{\prime\prime}$, $y_{\prime\prime}$,
$z_{\prime\prime}$ may be obtained, as from the system of the
{\it four\/} ordinary equations (302), (303).

\bigbreak

\centerline{\it
On the Operation of pre-multiplying one numeral Set by another,
and on fractional}

\nobreak\vskip 3pt

\centerline{\it
Symbols for Sets.}

\nobreak\bigskip

34.
Since we have seen that we are not at liberty to assume
generally, for {\it all\/} numeral sets, that the commutative
formula of multiplication holds good, we must (in general)
distinguish between {\it two modes\/} of combination of two such
sets with each other, {\it as factors}, in some such way as the
following.  We saw reason, in the twenty-second article, to
regard an ordinal set,~${\sr q}$, as having been generated by a
certain symbolical multiplication, or complex derivation, from a
single standard ordinal relation,~${\sr a}$, as from an original
operand, or derivand; the operator, or symbolical multiplier,
having been a numeral set,~$q$.  If such an ordinal
set,~${\sr q}$, or $q \times {\sr a}$, be again operated on by
the new numeral set,~$q'$, as by a new symbolical multiplier, the
result will be a new ordinal set, $q' \times (q \times {\sr a})$,
which, in this theory, admits of being denoted also by
$(q' \times q) \times {\sr a}$; and generally, in the same
theory, the conditions of detachment entitle us to write the
formula
$$q' \times (q \times {\sr q}^\backprime)
   = (q'\times q) \times {\sr q}^\backprime,
   \eqno (311)$$
whatever operand set (of the same order) may here be denoted by
the symbol~${\sr q}^\backprime$.  Thus, to multiply the numeral
set~$q$, as a multiplicand, by the numeral set~$q'$, as a
multiplier, comes to be regarded as being equivalent to the
operations of multiplying some single standard ordinal
relations,~${\sr a}$, or some ordinal set,~${\sr q}^\backprime$,
{\it first by the given multiplicand set},~$q$, and
{\it afterwards by the given multiplier set},~$q'$; and of then
finding that  third set,~$q''$, namely, the
{\it product\/}~$q' \times q$, or $q' q$, which, acting {\it as a
single multiplier}, would produce the {\it same final result},
and would, therefore, serve, by its single operation, to replace
this twofold process.  In this view of the multiplication of one
numeral set by another, the set proposed as a
{\it multiplicand\/} is itself a {\it previous\/} multiplier, and
may, therefore, by called a {\it premultiplicator}, or, more
familiarly, a {\it premultiplier}.  And thus, instead of saying
that the product $q' \times q$, or $q' q$, is obtained by
multiplying $q$ by $q'$, we may be permitted occasionally to say
that the same product results from {\it premultiplying\/} $q'$
{\it by\/} $q$; the symbol of the {\it pre\/}multiplier being
placed towards the {\it right hand}, as that of the multiplier is
placed towards the left.

With this phraseology, and with the definitional formula (296),
which easily gives also this other connected formula,
$$(q' \times q) \div q = q',
   \eqno (312)$$
{\it division and premultiplication are mutually inverse
operations\/}; that is to say, a numeral set,~$q'$, remains, upon
the whole, unchanged, when it is {\it both\/} divided and
premultiplied, or both premultiplied and divided, by any other
numeral set,~$q$ (of the same order).  We may also agree to
express the same results by symbols of {\it fractional forms}, a
{\it fraction\/} being defined to be the {\it quotient\/} which
is obtained when the numerator is divided by the denominator, so
that we shall adopt here, as a definition, the formula
$${q' \over q} = q' \div q;
   \eqno (313)$$
for then we may say that {\it a fraction gives its numerator as
the product, when it is premultiplied by its denominator\/};
though it does {\it not\/} always, at least for the case of
quaternions, produce that numerator when it is {\it multiplied\/}
by that denominator (the order of the factors being then
different).  In symbols, the equations
$${q'' \over q} q = q'',\quad {q' q \over q} = q',
   \eqno (314)$$
are here regarded as {\it identical\/}; whereas these other usual
equations,
$$q {q'' \over q} = q'',\quad
  {q' q \over q'} = q,$$
of which the first is only an abridged way of writing a formula
already rejected, while the second is connected therewith, are
{\it not generally true\/} (or, at least, not universally so)
{\it for numeral sets\/}; because the {\it order of the
factors\/} in multiplication is, in the present theory of such
{\it sets, not generally unimportant to the result}.  We have
seen, for example, in the foregoing article, that the quaternion
which may now be denoted by the symbol
$\displaystyle q {q'' \over q}$,
or by this other symbol,
$\displaystyle {q q'' \over q}$,
or by $q q'' \div q$, instead of being generally equal to the
quaternion~$q''$, is equal, in general, to {\it another
quaternion},~$q_{\prime\prime}$, distinct from the former, though
having several simple relations thereto, which will be found to
be connected, in their geometrical and physical applications,
with questions respecting the transformation of rectangular
coordinates in space, and the rotation of a solid body.  It may,
therefore, be not useless to remark expressly here, that the
following usual equations {\it continue true\/} in the present
theory of numeral sets, as well as in common algebra:
$$q \times {q'' \over q} = {q q'' \over q};\quad
  {q_2 q \over q_1 q} = {q_2 \over q_1}
   =  {q_2 q \div q \over q_1 q \div q};
   \eqno (315)$$
or, in other words, that a {\it fraction is multiplied\/} by a
numeral set when its {\it numerator\/} is multiplied thereby; and
that the {\it value\/} of a fraction, regarded as representing a
numeral set, remains {\it unchanged}, or represents the same set
as before, when its numerator and denominator are {\it both
premultiplied}, or {\it both divided, by any common set\/} (of
the same order); both which results depend on the associative
property of multiplication, and on the principle that two numeral
sets cannot {\it generally\/} give equal products, when operating
as multipliers on one common multiplicand (different from zero),
unless they be themselves equal sets.  These general remarks will
become more clear by their future applications; meanwhile, we may
here agree to use occasionally, for convenience and variety,
another form of expression, consistent with the foregoing
principles, and to say that, in the product $q' q$, the left-hand
factor,~$q'$, is multiplied {\it into\/} the right-hand
factor,~$q$, as the latter has been said to be multiplied
{\it by\/} the former, and as that former factor again has been
said to be {\it pre\/}multiplied by the latter.

\bigbreak

\centerline{\it
On the Operations of submultiplying, and of taking the Reciprocal
of a numeral Set.}

\nobreak\bigskip

35.
As it has been found necessary to distinguish, in general,
between {\it two modes of multiplication\/} of one numeral set by
another, with different arrangements of the factors, so it is
also necessary in this theory to distinuish generally between
{\it two inverse operations}, namely, between the operation of
{\it division}, and another closely connected operation, which
may be called {\it sub-multiplication}.  For if this last-named
operation be now defined to the the {\it returning to the
multiplicand}, when the product and the multiplier are given, it
will then be evidently distinct, in general, or, at least, for
the case of quaternions, from the operation of {\it division},
which has been already defined to be the {\it returning to the
multiplier}, when the multiplicand and product are given; because
these two factors, the multiplier and the multiplicand, when
regarded as numeral sets (at least if those sets be quaternions),
cannot generally change places with each other, without altering
the value of the product.  To denote conveniently this new
operation of {\it sub\/}multiplication, or of returning from the
set~$q' q$ to the set~$q$, when the set~$q'$ is given, we shall
now introduce the conception of a {\it reciprocal set}, which may
be denoted by any one of the three symbols,
$$1 \div q = {1 \over q} = q^{-1};
   \eqno (316)$$
and of which the characteristic property is, that it satisfies
generally the two reciprocal conditions,
$$q^{-1} \times q {\sr q} = {\sr q},\quad
  q \times q^{-1} {\sr q}' = {\sr q}',
   \eqno (317)$$
of which the second follows from the first, and which may be more
concisely written thus:
$$q^{-1} q = q q^{-1} = 1.
   \eqno (318)$$
Thus, whether a numeral set~$q$ be multiplied or premultiplied by
its reciprocal set~$q'$, the product in each case is unity; and
when these two reciprocal sets are employed to operate, as
successive multipliers, on any ordinal or numeral set as a
multiplicand, they {\it neutralize\/} the effects of each other.
It follows hence, that {\it to submultiply by any numeral set is
equivalent to multiplying by the reciprocal of that set\/}; so
that we may write generally, for such sets, the {\it formula of
submultiplication\/} (as in ordinary algebra) thus:
$${1 \over q'} \mathbin{.} q' q = q'^{-1} \mathbin{.} q' q = q.
   \eqno (319)$$
It is evident from what has been said, that the {\it reciprocal
of the reciprocal\/} of a numeral set is equal to that {\it set
itself\/}; and that to {\it divide by\/} such a set is to
{\it premultiply by\/} (or to {\it multiply into\/}) its
{\it reciprocal}; thus generally,
$$q' q \div q = q' q \times {1 \over q}
   =  q' q \mathbin{.} q^{-1} = q'.
   \eqno (320)$$
The {\it reciprocal of a quaternion\/} is given by the formula,
$$(w + ix + jy + kz)^{-1}
   =  (w^2 + x^2 + y^2 + z^2)^{-1} (w - ix - jy - kz).
   \eqno (321)$$

In general, the {\it reciprocal of the product\/} of any number
of sets is equal to the {\it product of the reciprocals\/} of
those sets, arranged in the {\it contrary order\/}: thus we may
write,
$$( \ldots \, q_2 q_1 q_0 )^{-1}
   =  q_0^{-1} q_1^{-1} q_2^{-1} \, \ldots
   \eqno (322)$$

\bigbreak

\centerline{\it
On Powers of a Numeral Set, with whole or fractional Exponents;
Square and Square Root}

\nobreak\vskip 3pt

\centerline{\it
of a Quaternion; Indeterminate Expressions, by Quaternions, for
the Square Roots of}

\nobreak\vskip 3pt

\centerline{\it
Negative Numbers.}

\nobreak\bigskip

36.
The symbol~$q^{-1}$, for the reciprocal of a numeral set, is only
one of a system of symbols of the same sort, which may easily be
formed by an adaptation of received algebraic notation.  For with
the notions given already, respecting multiplication and division
of sets, there is no difficulty in interpreting now, in an
extended sense, adapted to the present theory, the following
usual system of equations,
$$\left. \eqalign{
q^0 = 1,\quad
q^1 = q,\quad
q^2 = q \times q^1,\quad
q^3 = q \times q^2,\, \ldots \cr
q^{-1} = {1 \over q},\quad
q^{-2} = {1 \over q} \times q^{-1},\quad
q^{-3} = {1 \over q} \times q^{-2},\, \ldots \cr}
   \right\}
   \eqno (323)$$
and then the well-known {\it equation of the exponential law},
$$q^s \times q^r = q^r \times q^s = q^{r+s},
   \eqno (324)$$
will hold good, as in ordinary algebra, the {\it exponents\/} $r$
and $s$ being here supposed to denote any two positive or
negative whole numbers, or zero.

These two other usual equations,
$$(q^r)^s = q^{sr},\quad (q^{t \over s})^s = q^t,
   \eqno (325)$$
will then also hold good for numeral sets, at least when $r$,
$s$, $t$ and $\displaystyle {t \over s}$, denote whole numbers;
and the latter of these two formul{\ae} may be employed as a
{\it definition\/} to {\it interpret\/} the symbol
$q^{t \over s}$, when the exponent is a numeral fraction; thus,
$q^{1 \over 2}$ will denote that numeral set, {\it or any one of
those numeral sets}, which satisfy, or are roots of, the
equation,
$$(q^{1 \over 2})^2 = q^1 = q.
   \eqno (326)$$
For example, it results from what has been already shown, that if
$q$ denotes the first numeral quaternion (299), then its
{\it symbolic square}, or {\it second power}, is another
quaternion,~$q_2$, given by the formula
$$q_2 = q^2 = (w + ix + jy + kz)^2
   =  w_2 + i x_2 + j y_2 + k z_2,
   \eqno (327)$$
where
$$\left. \eqalign{
w_2 &= w^2 - x^2 - y^2 - z^2;\cr
x_2 &= 2wx;\quad y_2 = 2wy;\quad z_2 = 2 wz.\cr}
   \right\}
   \eqno (328)$$
And hence, conversely, the symbolic {\it square root of the
quaternion},~${\sr q}_2$, or its power with the
exponent~${1 \over 2}$, is to be regarded as being equivalent to
this other numeral quaternion,
$$q =  q_2^{1 \over 2}
    = (w_2 + i x_2 + j y_2 + k z_2)^{1 \over 2}
    =  w + ix + jy + kz;
   \eqno (329)$$
where the constituents, $w$,~$x$,~$y$,~$z$, are {\it any four
numbers\/} (positive, negative, or zero), {\it which satisfy the
system of the four equations\/}~(328).  Those equations give the
relation
$$w_2^2 + x_2^2 + y_2^2 + z_2^2
   =  (w^2 + x^2 + y^2 + z^2)^2,
   \eqno (330)$$
which is included in the more general result (251), respecting
the multiplication of any two quaternions; therefore, conversely,
$$w^2 + x^2 + y^2 + z^2 = \surd ( w_2^2 + x_2^2 + y_2^2 + z_2^2);
   \eqno (331)$$
and, consequently, by the first of the four equations (328),
$$2 w^2 = w_2 + \surd (w_2^2 + x_2^2 + y_2^2 + z_2^2),
   \eqno (332)$$
where the radical in the second member of (331) is to be
considered as a positive number: and, therefore, the first
constituent,~$w$, of the sought quaternion~$q$, or of the square
root of the given quaternion~$q_2$, is itself given, generally,
by (332), as either the positive or the negative square root of
another given positive number.  And after choosing either of
these two values (the positive or the negative) for $w$, the
other three constituents, $x$,~$y$,~$z$, of the sought
quaternion~$q$, become, {\it in general}, entirely determined by
the three last equations (328).  There are, therefore, {\it in
general, two, and only two, different square roots of any
proposed numeral quaternion\/}; and they differ only in their
signs.  But there is one very important {\sc case of
indeterminateness}, in which an {\it infinite variety\/} of roots
takes the place of that {\it finite ambiguity}, which has thus
been seen to exist generally in the expression for the square
root of a quaternion, namely, the {\it case where the proposed
square is equal to a negative number}, presented under the
{\it form\/} of a quaternion, of which the first constituent is
negative, while the three last separately vanish.  For, if we
suppose the data to be such that
$$w_2 = - r^2,\quad x_2 = 0,\quad y_2 = 0,\quad z_2 = 0,
   \eqno (333)$$
$r$ being some positive or negative number, then the positive
radical in (331) becomes
$$\surd (w_2^2 + x_2^2 + y_2^2 + z_2^2) = r^2 = - w_2,
   \eqno (334)$$
and the equation (332) reduces itself to the following:
$$w = 0.
   \eqno (335)$$
And while the three last of the four equations (328) are then
satisfied, independently of the three remaining constituents,
$x$,~$y$,~$z$, the first of those four equations gives this
{\it one\/} relation, between those three constituents of the
sought quaternion~$q$,
$$x^2 + y^2 + z^2 = r^2,
   \eqno (336)$$
which is the {\it only condition\/} that they must satisfy.  And
since we may satisfy this condition by assuming
$$\left. \eqalign{
x &= {lr \over h},\quad
y  = {mr \over h},\quad
z  = {nr \over h},\cr
h &= \surd (l^2 + m^2 + n^2),\cr}
   \right\}
   \eqno (337)$$
without any restriction being imposed on the three (positive, or
negative, or null) numbers, $l$,~$m$,~$n$, we see that, in our
theory of quaternions, the {\it square root of a negative number
is a partially indeterminate quaternion}, belonging, however, to
a certain peculiar {\it class}, and admitting of being thus
denoted:
$$(-r^2)^{1 \over 2}
   =  {(il + jm + kn) r \over \surd (l^2 + m^2 + n^2)}.
   \eqno (338)$$
In fact, if we square the second member of this last formula,
attending to the fundamental expressions, ({\sc a}), ({\sc b}),
for the squares and products of the three symbols, $i$,~$j$,~$k$,
we find, as the result of this operation, the negative
number~$- r^2$, which is the square of the first member; for
those fundamental expressions give, generally, this very simple
and remarkable equation,
$$(ix + jy + kz)^2 = - (x^2 + y^2 + z^2).
   \eqno (339) = ({\sc d})$$
For example, in this theory, the square root of $-1$ itself is
represented by a partially indeterminate symbol of the foregoing
class, and we may write
$$(-1)^{1 \over 2}
   =  {ix \over r} + {jy \over r} + {kz \over r},
   \quad\hbox{where}\quad
  r^2 = x^2 + y^2 + z^2.
   \eqno (340)$$
That is to say, whatever three positive, or negative, or null
numbers may be denoted by $x$,~$y$,~$z$, provided that they do
not all together vanish, we are allowed in this theory to
establish the following {\it general expression for any one of
the infinitely many square roots of negative unity\/}:
$$(-1)^{1 \over 2}
   =  {ix + jy + kz \over \surd (x^2 + y^2 + z^2)}.
   \eqno (341) = ({\sc e})$$
Or, with the recent meaning of $r$, and with a notation which
more immediately suggests the conception of a numeral set, we may
establish the formula,
$$(-1,0,0,0)^{1 \over r}
   =  \left( 0, {x \over r}, {y \over r}, {z \over r} \right).
   \eqno (342)$$

\bigbreak

\centerline{\it
Cubes and Cube Roots of Quaternions; partially indeterminate
Expressions by}

\nobreak\vskip 3pt

\centerline{\it
Quaternions for the Cube Roots of positive and negative Numbers.}

\nobreak\bigskip

37.
With the same condition of abridgment, (336), we may write
generally, for any numeral quaternion, this expression
$$q = w + (-1)^{1 \over 2} r;
   \eqno (343)$$
or still more briefly and, at the same time, more determinately,
$$q = w + \iota r,
   \quad\hbox{where}\quad
  \iota^2 = -1,
   \eqno (344)$$
and where $\iota$ may be conceived to be in general determined
when $q$ is determined, since
$$\iota = {i x \over r} + {j y \over r} + {kz \over r},\quad
   r = \surd (x^2 + y^2 + z^2).
   \eqno (345)$$
The cube of this expression (344) for $q$ is
$$q^3 = w^3 - 3 w r^2 + \iota (3 w^2 - r^2) r;
   \eqno (346)$$
and this {\it cube}, or {\it third power of a quaternion}, may be
equated to a new quaternion, denoted as follows:
$$q_3 = q^3 = w_3 + i x_3 + j y_3 + k z_3
   =  w_3 + \iota_3 r_3,
   \eqno (347)$$
where
$$r_3^2 = x_3^2 + y_3^2 + z_3^2,\quad \iota_3^2 = -1;
   \eqno (348)$$
provided that we satisfy the two conditions,
$$w_3 = w^3 - 3 w r^2,\quad \iota_3 r_3 = \iota r (3 w^2 - r^2),
   \eqno (349)$$
of which the second again resolves itself into three others, on
account of the mutual linear independence of the three symbols,
$i$,~$j$,~$k$.  These last equations give
$${x_3 \over x} = {y_3 \over y} = {z_3 \over z}
   = 3 w^2 - r^2;
   \eqno (350)$$
and, therefore, it is allowed to write
$$\iota_3 = \iota,\quad r_3 = r (3 w^2 - r^2);
   \eqno (351)$$
provided that, if we still choose to consider the radical~$r$ as
positive, we regard the other radical,~$r_3$ as varying its sign,
according to the law
$$r_3 \gtlt 0,
   \quad\hbox{according as}\quad
  3 w^2 \gtlt r^2.
   \eqno (352)$$
If, now, it be required to find conversely the {\it cube
root\/}~$q$, or the power with exponent ${1 \over 3}$ of a given
quaternion~$q_3$, we shall have, first, the two equations
$$w^2 + r^2 = (w_3^2 + r_3^2)^{1 \over 3};\quad
   {r \over w} \left( 3 - \left( {r \over w} \right)^2 \right)
         \left( 1 - 3 \left( {r \over w} \right)^2 \right)^{-1}
   = {r_3 \over w_3};
   \eqno (353)$$
of which the second may be written more concisely thus:
$$3 t - t^3 = (1 - 3 t^2) t_3,
   \quad\hbox{if}\quad
  r = t w,\quad r_3 = t_3 w_3;
   \eqno (354)$$
so that
$$t_3^2 = w_2^{-2} (x_3^2 + y_3^2 + z_3^2).
   \eqno (355)$$
The value of this positive number,~$t_3^2$, is known, because the
four constituents of the quaternion~$q_3$ are now supposed to be
given; hence, three different positive values for $t^2$ can, in
general, be deduced from the square of the first equation (354),
which is a well-known cubic; for each such value of $t^2$, the
sign of $t_3$, and therefore, also (by the same cubic equation),
the sign of $t$ may be determined by the condition that $r_3$ or
$t_3 w_3$ is, by (352), to receive the same sign as $3 - t^2$;
but $r$ is supposed positive, therefore $w$ has the same sign as
$t$; and
$$w^2 (1 + t^2) = w_3^{2 \over 3} (1 + t_3^2)^{1 \over 3},
   \eqno (356)$$
so that the constituent~$w$ is entirely determined: therefore $r$
(being $= tw$) is known, and then the three remaining
constituents, $x$,~$y$,~$z$, of the sought quaternion,~$q$, are
given by (350).  Thus, the sought cube root,~$q$, of the proposed
numeral quaternion~$q_3$, is, in general, determined; or, at
least, is restricted to a finite and {\it triple variety},
answering to the {\it three\/} (real, numerical, and)
{\it unequal roots of the known cubic equation\/} (354); which
roots can always be found by the help of a table of trigonometric
tangents.  We see, then, by the foregoing process, which will
soon be replaced by one more simple and more powerful, that there
are, {\it in general, three, and only three, distinct cube roots
of any proposed numeral quaternion}.  But when it is required to
find, on the same plan, under the form of a quaternion, the cube
root of a positive or negative number,~$w_3$, regarded as an
abridged expression for the quaternion $(w_3,0,0,0)$, then
$x_3$, $y_3$, $z_3$, and $r_3$ all vanish; and while the ratios
of $x$,~$y$,~$z$ remain entirely arbitrary, the numbers $w$ and
$r$ are to be determined so as to satisfy the two equations,
$$w_3 = w^3 - 3 w r^2;\quad 0 = r(3w^2 - r^2);
   \eqno (357)$$
which require that we should suppose either
$$r = 0,\quad w^3 = w_3,
   \eqno (358)$$
or else,
$$r^2 = 3 w^2,\quad w^3 = - {\textstyle {1 \over 8}} w_3.
   \eqno (359)$$
For example, if we seek the quaternion cube roots of positive
unity, regarded as equivalent to the quaternion $(1,0,0,0)$, we
find not only unity itself, under the form of the same
quaternion, but also this other, and partially indeterminate
expression,
$$1^{1 \over 3} = (1,0,0,0)^{1 \over 3}
   = (- {\textstyle {1 \over 2}}, x, y, z);
   \eqno (360)$$
where the three positive or negative numbers, $x$,~$y$,~$z$, are
only obliged to satisfy the condition
$$x^2 + y^2 + z^2 = {\textstyle {3 \over 4}}.
   \eqno (361)$$
And, in like manner, besides negative unity itself, there are
infinitely many quaternion cube roots of negative unity, included
in the expression
$$(-1)^{1 \over 3} = (-1, 0, 0, 0)^{1 \over 3}
   =  ( + {\textstyle {1 \over 2}}, x, y, z),
   \eqno (362)$$
under the same condition (361) respecting the sum of the squares
of the constituents $x$,~$y$,~$z$.  The values of this last
expression (362), as well as the values of the expression (360),
are, therefore, included among those quaternions which are (in
this theory) {\it sixth roots of unity}, or are among the values
of the symbol~$1^{1 \over 6}$.  As one other example, it
may be remarked that, by the rule (359), the number negative
eight has, for one of its cube roots, the quaternion of which
each of the four constituents is equal to positive unity; thus
{\it one value\/} of the symbol
$$(-8, 0, 0, 0),
   \quad\hbox{is}\quad
  (1, 1, 1, 1);
   \eqno (363)$$
and, accordingly, we shall find that
$$(1 + i + j + k)^3 = - 8,
   \eqno (364)$$
if we develop the first member of this last equation, employing
the distributive property of multiplication, but {\it not\/} the
commutative property, and reducing by the values of the symbolic
squares and products of $i$,~$j$,~$k$, which have been already
assigned.  It may be noted here that, in the more general problem
of finding the cube root,~$q$, of a quaternion,~$q_3$, of which
the three last constituents, $x_3$,~$y_3$,~$z_3$, do not all
vanish, so that $r_3$ is different from~$0$, we might have
eliminated $r^2$ between the first equation (349) and the first
equation (353), and so have obtained an ordinary cubic equation
in $w$, which, as well as the equation in $t$, can be resolved by
the trigonometrical tables, namely, the cubic:
$$4 w_3 - 3w (w_3^2 + r_3^2)^{1 \over 3} = w_3.
   \eqno (365)$$

\bigbreak

\centerline{\it
Connexion of Quaternions with Couples, and with Quadratic
Equations.}

\nobreak\bigskip

38.
In general, if a numeral quaternion~$q$ be required to satisfy
any ordinary numerical equation (with real coefficients) of the
form
$$0 = a_0 + a_1 q + a_2 q^2 + a_3 q^3 + \hbox{\&c.},
   \eqno (366)$$
we may first substitute for $q$ the expression (344), namely,
$w + \iota r$, where $\iota^2 = -1$.  Then, after finding any one
of those systems of values of the two (real) numbers $w$  and
$r$, which satisfy the system of the two equations, obtained by
the foregoing substitution, and by equating separately to zero
the sums of the terms containing respectively the even and odd
powers of $\iota$, namely, the equations
$$\left. \eqalign{
0  &= a_0 + a_1 w + a_2 (w^2 - r^2) + a_3 (w^3 - 3 w r^2)
       + \hbox{\&c.},\cr
0  &= a_1 r + a_2 (2wr) + a_3 (3 w^2 r - r^3)
       + \hbox{\&c.};\cr}
   \right\}
   \eqno (367)$$
we shall only have to change $\iota r$, in the expression for
$q$, to $ix + jy + kz$, and to suppose, as before, that
$x^2 + y^2 + z^2 = r^2$.  But the process by which the two
numbers $w$ and $r$ are thus supposed to be discovered, is
precisely the process by which a numerical couple $(w, r)$, of
the kind considered in the nineteenth article of this paper, and
in the earlier Essay there referred to, would be determined, so
as to satisfy the couple equation,
$$0 = a_0 + a_1 (w, r) + a_2 (w, r)^2 + \hbox{\&c.}
   \eqno (368)$$
The calculations required for finding a {\it couple\/} $(w, r)$
which shall satisfy this equation (368), are therefore the same
as those required for finding a {\it quaternion\/}
$(w, x, y, z)$, which shall satisfy the equation
$$0 = a_0 + a_1 (w, x, y, z) + a_2 (w, x, y, z)^2 + \hbox{\&c.};
   \eqno (369)$$
provided that we suppose the constituents of these two numeral
sets to be connected with each other by the relation already
assigned, namely,
$$x^2 + y^2 + z^2 = r^2.
   \eqno (336)$$
Thus, in particular, if it be proposed to satisfy, by a
quaternion~$q$, the quadratic equation,
$$0 = a_0 + a_1 q + a_2 q^2,
   \eqno (370)$$
which we may put under the form
$$q^2 - 2 a q + b = 0,
   \eqno (371)$$
we may first change $q$ to the couple $(w, r)$, and so obtain the
{\it two\/} separate equations,
$$w^2 - r^2 - 2aw + b = 0,\quad 3wr - 2 ar = 0;
   \eqno (372)$$
of which the latter requires us to suppose, either,
$$\hbox{1st,}\quad r = 0;
   \quad\hbox{or,\quad 2nd,}\quad w = a.
   \eqno (373)$$
The first equation conducts to a quadratic equation in $w$,
namely,
$$w^2 - 2 a w + b = 0,
   \eqno (374)$$
which is precisely the proposed equation (371), with the
symbol~$q$ of the sought quaternion changed to the symbol~$w$ of
a sought number; and reciprocally if it be possible to find a
real number~$w$, or rather (in general) two such numbers, which
shall satisfy the quadratic (374), that is to say, if (the
equation have {\it real roots}, or if) the condition
$$a^2 > b,
   \quad\hbox{or}\quad
  a^2 = b + c^2,
   \eqno (375)$$
be satisfied, where $c$ is a positive or negative number, then
the equation (371) will be satisfied by either of the two
quaternions which are included in the following expression, and
by no other quaternion,
$$q = (w, 0, 0, 0) = (a \pm \surd (a^2 - b), 0, 0, 0).
   \eqno (376)$$
The same expression holds good, giving one solution of the
equation (371), for the case $a^2 = b$.  But in the remaining
case, where
$$a^2 < b,\quad a^2 = b - c^2,
   \eqno (377)$$
$c$ being still a positive or negative number, we are to adopt
the remaining alternative (373), namely $w = a$; and instead of
supposing $r = 0$, we are now, by the first equation (372), and
by (377), to suppose
$$r^2 = w^2 - 2aw + b = b - a^2 = c^2;
   \eqno (378)$$
and the solution of the quadratic equation (371) is now expressed
by the partially indeterminate quaternion, connected with the two
couple-solutions $(a, \pm c)$,
$$q = (a, x, y, z),
   \quad\hbox{where}\quad
  x^2 + y^2 + z^2 = b - a^2.
   \eqno (379)$$
And thus we may perceive that, if we denote by $\mu$ the modulus
of the first numeral quaternion (299), which may represent any
such quaternion, then {\it this quaternion,~$q$, is a root of a
quadratic equation}, with real coefficients, namely, the
following:
$$q^2 - 2 w q + \mu^2 = 0.
   \eqno (380)$$

\bigbreak

\centerline{\it
Exponential and Imponential of a numeral Set; general Expression
for a Power, when}

\nobreak\vskip 3pt

\centerline{\it
both the Base and the Exponent are such Sets.}

\nobreak\bigskip

39.
The investigations, in some recent articles, respecting certain
powers and roots of a quaternion, may be made at once more simple
and more general by the introduction of a well-known exponential
series.  We shall, therefore, write
$${\sc p} (q)
   =  1 + {q^1 \over 1}
       + {q^2 \over 1 \mathbin{.} 2}
       + {q^3 \over 1 \mathbin{.} 2 \mathbin{.} 3}
       + \hbox{\&c.}
   \eqno (381)$$
and shall call this series the {\it exponential function}, or
simply, {\it the exponential of the numeral set\/}~$q$, with
respect to which the operations are performed; we shall also
denote this exponential still more concisely by writing simply
${\sc p} q$ instead of ${\sc p} (q)$, where no confusion seems
likely to arise from this abbreviation.  The {\it inverse
function}, which may be conceived to express reciprocally~$q$, by
means of ${\sc p} q$, may be called by contrast the
{\it imponential\/} function, and denoted by the
characteristic~${\sc p}^{-1}$; thus, we shall suppose
${\sc p}^{-1} q$ to be such that
$${\sc p} {\sc p}^{-1} q = q.
   \eqno (382)$$
or that, more fully,
$$q = 1 + {\sc p}^{-1} q
       + {1 \over 2} ({\sc p}^{-1} q)^2
       + {1 \over 2 \mathbin{.} 3} ({\sc p}^{-1} q)^3
       + \hbox{\&c.}
   \eqno (383)$$
Then, because the function~${\sc p}$ is such that
$${\sc p} q' \times {\sc p} q = {\sc p} (q' + q),
   \quad\hbox{if}\quad
  q' q = q q';
   \eqno (384)$$
and because, by the {\it associative\/} principle of
multiplication, {\it any two whole powers of the same numeral
set,~$q$, are commutative as factors}, that is to say, may change
their places with each other, without altering the value of the
product; we shall have, generally,
$${\sc p} f'(q) \times {\sc p} f(q)
   =  {\sc p} (f'(q) + f(q)),
   \eqno (385)$$
because we shall have
$$f'(q) \times f(q) = f(q) \times f'(q),
   \eqno (386)$$
if the symbols $f(q)$ and $f'(q)$ denote here any combinations of
whole powers of {\it one common numeral set},~$q$, and of any
given numerical coefficients.  For example, if $a$ denote a
{\it number}, we shall have
$${\sc p} a \times {\sc p} q = {\sc p} (a + q).
   \eqno (387)$$
We may also deduce, from the formula (385), this other important
corollary, which is general for numeral sets, and in which the
symbol ${\sc p} \mathbin{.} sq$ represents the same function as
${\sc p}(sq)$, while $s$ may, at first, be supposed to denote a
whole number:
$$({\sc p} q)^s = {\sc p}(sq) = {\sc p} \mathbin{.} sq.
   \eqno (388)$$
We have, therefore, for {\it any two whole numbers}, $s$ and $t$,
the relation
$$({\sc p} \mathbin{.} sq)^t
   =  ({\sc p} \mathbin{.} tq)^s;
   \eqno (389)$$
and, therefore, as an equation of which the second member is, at
least, {\it one of the values\/} of the first, we have
$$({\sc p} \mathbin{.} sq)^{t \over s}
   =  {\sc p} \mathbin{.} tq.
   \eqno (390)$$
We are thus led to write, as an equation of the same sort, giving
an expression for, at least, one value of any {\it fractional
power of a set}, whenever the {\it imponential\/} of that set can
be discovered,
$$q^{t \over s}
   =  {\sc p} \left( {t \over s} {\sc p}^{-1} q \right).
   \eqno (391)$$
The simplicity of this equation may now induce us to
{\it extend\/} it, as we propose to do, by {\it definition}, to
the cases where the {\it exponent of the power}, instead of being
a numeral fraction, is an incommensurable number, {\it or even a
numeral set}.  We shall therefore, write generally
$$q^{q'} = {\sc p} ( q'{\sc p}^{-1} q);
   \eqno (392)$$
and thus we shall have a {\it general expression for any power of
a numeral set}, through the help of the characteristics of the
exponential and imponential thereof.

\bigbreak

\centerline{\it
Application to Quaternions; Amplitude and Vector Unit;
Coordinates, Radius,}

\nobreak\vskip 3pt

\centerline{\it
and Representative Point.}

\nobreak\bigskip

40.
On applying these general principles to the case of a quaternion,
we have first, by (387),
$${\sc p} q = {\sc p} (w + ix + jy + kz)
   =  {\sc p}w \mathbin{.} {\sc p} (ix + jy + kz);
   \eqno (393)$$
and then, if we use the notations (345), and attend to the
connexion already established between quaternions and couples, we
find that
$${\sc p} (ix +jy + kz)
   = {\sc p} (\iota r) = \cos r + \iota \sin r;\quad
   \iota^2 = -1;
   \eqno (394)$$
where $\cos r$ and $\sin r$ denote, as usual, the cosine and sine
of $r$, so that, in the theory of couples, the following equation
holds good:
$$P(0,r) = (\cos r, \sin r).
   \eqno (395)$$
(Compare the earlier Essay, where the functional sign~${\sc f}$
was used instead of ${\sc p}$).  Thus the {\it exponential of a
quaternion\/}~$q$ is expressed generally, with these notations,
by the formula,
$${\sc p}q = {\sc p} w \mathbin{.} (\cos r + \iota \sin r).
   \eqno (396)$$
Reciprocally the {\it imponential\/} ${\sc p}^{-1} q'$, of any
other quaternion,~$q'$, is to be found by comparing this formula
(396) with the expression of that quaternion~$q'$, when put under
the form,
$$q' = w' + \iota' r'
   =  \mu' (\cos \theta' + \iota' \sin \theta'),
   \eqno (397)$$
where
$$\mu' = \surd (w'^2 + r'^2),\quad
  \tan \theta' = {r' \over w'}.
   \eqno (398)$$
We find, in this manner, that we may suppose
$$q' = {\sc p} q,\quad q = {\sc p}^{-1} q',
   \eqno (399)$$
provided that we make
$${\sc p} w = \mu';\quad
  r = \theta' + 2 n' \pi;\quad
  \iota = \iota';
   \eqno (400)$$
where $n'$ is any whole number, and $\pi$ is, as usual, the least
positive root of the numerical equation,
$$\pi^{-1} \sin \pi = 0.$$
Hence, the sought {\it imponential of the quaternion\/}~$q'$ is
$${\sc p}^{-1} q'
   =  {\sc p}^{-1} \mu' + \iota' (\theta' + 2 n' \pi);
   \eqno (401)$$
and, in like manner, by suppressing the accents, the imponential
of $q$ is found to be
$${\sc p}^{-1}q = {\sc p}^{-1} \mu + \iota (\theta + 2 n \pi),
   \eqno (402)$$
where $\theta$ may be said to be the {\sc amplitude}, and $\mu$
is what we have already called the {\sc modulus} of $q$.

\bigbreak

41.
We may also say that $\iota$ is the {\it imaginary unit}, or
perhaps, more expressively, that it is the {\sc vector unit}, of
the same quaternion~$q$.  For in the applications of this theory
to geometrical questions, this imaginary or vector unit~$\iota$
may be regarded as having in general a given {\it direction in
space\/} when $q$ is a given quaternion; and if we denote its
{\it direction cosines\/} by $\alpha$,~$\beta$,~$\gamma$, so that
$$\alpha = {x \over r},\quad
  \beta  = {y \over r},\quad
  \gamma = {z \over r},\quad
  \alpha^2 + \beta^2 + \gamma^2 = 1,
   \eqno (403)$$
we may write, generally, by (345),
$$\iota = i \alpha + j \beta + k \gamma,\quad \iota^2 = -1.
   \eqno (404)$$
{\it This power of representing any\/} {\sc direction in
tridimensional space}, {\it by one of the quaternion forms of
$\surd (-1)$, is one of the chief peculiarities of the
present theory\/}; and will be found to be one of the chief
causes of its power, when employed as an instrument in researches
of a geometrical kind.  If $\alpha$,~$\beta$,~$\gamma$ be
conceived to be the three rectangular coordinates of a
point~${\sc r}$ upon a spheric surface, with radius unity,
described about the origin of coordinates as centre, we may also
write, more concisely and, at the same time, not less
expressively,
$$\iota = i_{\sc r};\quad i_{\sc r}^2 = -1.
   \eqno (405)$$
A numeral quaternion~$q$ may therefore, in general, be thus
expressed:
$$q = \mu (\cos \theta + i_{\sc r} \sin \theta);
   \eqno (406)$$
where
$$\mu = \surd (w^2 + x^2 + y^2 + z^2),\quad
  i_{\sc r} = i \alpha + j \beta + k \gamma = \surd (-1).
   \eqno (407)$$
Its imponential, by (402), will then take the form
$${\sc p}^{-1} q = \log \mu + i_{\sc r} (\theta + 2 n \pi),
   \eqno (408)$$
$n$ denoting here any positive or negative whole number, or zero;
and $\log \mu$ denoting the (real and) natural or Napierian
logarithm of the positive (or absolute) number~$\mu$; or in other
words, that determined (real) number, whether positive or
negative or null, which satisfies the equation
$$\mu = {\sc p} (\log \mu).
   \eqno (409)$$

\bigbreak

42.
Substituting this expression (408) for the imponential of a
quaternion in the general expression (392) for a power of a set,
we find, for a power of a quaternion~$q$, with another
quaternion~$q'$ as the exponent of that power, the expression,
$$q^{q'}
   =  {\sc p} \{ q' \log \mu
          + q' i_{\sc r} (\theta + 2 n \pi) \};
   \eqno (410)$$
which, however, it is not {\it generally\/} allowed to resolve
into the two factors, ${\sc p} (q' \log \mu)$ and
${\sc p} \{ q' i_{\sc r} (\theta +2 n \pi) \}$
because $q'$ and $q' \iota_{\sc r}$ are not, in general,
{\it condirectional quaternions\/}; if this latter name be given
to quaternions which have vector units equal or opposite, so that
in each case they are commutative with each other, as factors in
multiplication.  But if we change the exponent~$q'$, in (410), to
any numerical fraction,
$\displaystyle {t \over s}$,
where $s$ and $t$ denote whole numbers, then this resolution into
factors is allowed, and the formula becomes
$$\eqalignno{
q^{t \over s}
   &= {\sc p}
         \left\{
            {t \over s} \log \mu
          + {t \over s} i_{\sc r} (\theta + 2 n \pi)
         \right\} \cr
   &= {\sc p} \left( {t \over s} \log \mu \right) \mathbin{.}
         {\sc p}
         \left\{
            i_{\sc r}
               \left(
                  {t \theta \over s} + {2 t n \pi \over s}
               \right)
         \right\} \cr
   &= \mu^{t \over s} (\cos + i_{\sc r} \sin)
               \left(
                  {t \theta \over s} + {2 t n \pi \over s}
               \right);
   &(411)\cr}$$
and thus it will be found that the chief results of the
thirty-sixth and thirty-seventh articles, respecting certain
powers and roots of a quaternion, are reproduced under a
simpler and more general aspect; for instance, the square root of
a quaternion is now given under the form
$$q^{1 \over 2}
   =  \mu^{1 \over 2} (\cos + i_{\sc r} \sin)
         \left( {\theta \over 2} + n \pi \right)
   =  \pm \surd \mu
         \left(
            \cos {\theta \over 2}
          + i_{\sc r} \sin {\theta \over 2}
         \right).
   \eqno (412)$$
But in the particular case where the original quaternion,~$q$,
reduces itself to a negative number, $q = w = - \mu$, so that its
amplitude,~$\theta$, is some odd multiple of $\pi$, while the
direction of its vector unit is indeterminate or unknown, the
formula (412) for a square root becomes simply
$$(-\mu)^{1 \over 2} = \mu^{1 \over 2} i_{\sc r};
   \eqno (413)$$
the position of the point~${\sc r}$ upon the {\it unit sphere\/}
being now likewise indeterminate or unknown, which agrees with
our former results respecting the indeterminate quaternion forms
for the square roots of negative numbers.  In like manner, the
quaternions, distinct from unity itself, which are {\it cube
roots of unity}, are now included in the expression
$$1^{1 \over 3}
   =  \cos {2 n \pi \over 3} + i_{\sc r} \sin {2 n \pi \over 3};
   \eqno (414)$$
where the {\it direction\/} of $i_{\sc r}$ remains entirely
undetermined.  But, {\it in general, the power, $q^{t \over s}$,
of a quaternion,~$q$, admits of $s$, and only $s$, distinct
quaternion values}, if the exponent,
$\displaystyle {t \over s}$,
be an arithmetical fraction in its lowest terms, so that the
numerator and the denominator of this fractional exponent are
whole numbers prime to each other; and if the proposed
quaternion~$q$ does not reduce itself to a number~$w$, by the
three last constituents, $x$,~$y$,~$z$, all separately vanishing
in its expression.  As an example of the operation of
{\it raising a quaternion to a power of which the exponent is
distinct from all positive and negative numbers, and from zero},
we may remark that the formula (410) gives, generally, for the
powers of an imaginary unit, such as $i_{\sc r}$ (for which we
have $\mu = 1$,
$\displaystyle \theta = {\pi \over 2}$), the expression
$$i_{\sc r}^{q'}
   =  {\sc p}
         \left\{
            q' i_{\sc r} \left( {\pi \over 2} + 2 n \pi \right)
         \right\};
   \eqno (415)$$
making then, in particular, $i_{\sc r} = i$, and $q' = k$, we
find, by ({\sc b}),
$$i^k
   =  {\sc p}
         \left\{
            ki \left( {\pi \over 2} + 2 n \pi \right)
         \right\}
   =  {\sc p}
         \left\{
            j  \left( {\pi \over 2} + 2 n \pi \right)
         \right\}
   =  {\sc p} \left( {j \pi \over 2} \right) = j;
   \eqno (416)$$
and by a similar process we find, more generally,
$$i_{\sc r}^{i_{{\sc r}'}} = i_{{\sc r}'} i_{\sc r},
   \eqno (417)$$
whenever $i_{{\sc r}'}$ and $i_{\sc r}$ denote {\it two
rectangular imaginary units}, so that the points ${\sc r}$ and
${\sc r}'$, which mark their directions, are distant from each
other by a quadrant on the sphere.  We may here introduce a few
slight additions to the nomenclature already established in this
paper, and may say that, in the general expression
$q = w + ix + jy + kz$, the three coefficients, $x$,~$y$,~$z$,
which multiply respectively the three coordinate characteristics,
$i$,~$j$,~$k$, are {\it the three\/} {\sc coordinates} {\it of
the quaternion}, and that the square root~$r$ of the sum of their
squares is the {\sc radius} of the same quaternion.  We shall
also say that the point~${\sc r}$, on the surface of the unit
sphere, which constructs or represents the direction of the
vector unit in its expression, is at once the {\sc representative
point} {\it of that vector unit},~$i_{\sc r}$, and also (in a
similar sense) {\it the representative point of the
quaternion\/}~$q$ itself.

\bigbreak

\centerline{\it
On the general Logarithms of a Set, and especially on those of a
Quaternion.}

\nobreak\bigskip

43.
Though we cannot enter here at any length into the theory of
{\it logarithms of sets}, yet it is obvious that if we make
$$q'' = q^{q'},
   \eqno (418)$$
the general expression (392) for a power of a set gives this
inverse expression for the exponent~$q'$:
$$\log_q \mathord{} \mathbin{.} q''
   = q' = {{\sc p}^{-1} q'' \over {\sc p}^{-1} q};
   \eqno (419)$$
in which expression, however, for a {\it logarithm of a set},
under the form of a fraction, the numerator and the denominator
are to be regarded as {\it separately subject to that
indeterminateness}, whatever it may be, {\it which arises in the
return from the exponential of a set to the set itself}, or in
the passage from a set~$q$ to its {\it imponential\/}
${\sc p}^{-1} q$.  Thus in the case of quaternions, the
{\it general logarithm of the quaternion~$q''$, to the base~$q$},
may, by (419) and (408), be written thus:
$$\log_q \mathord{} \mathbin{.} q''
   =  {  \log \mu'' + i_{\sc r} (\theta'' + 2 n'' \pi)
         \over
         \log \mu   + i_{\sc r} (\theta   + 2 n   \pi) }.
   \eqno (420)$$
It involves, therefore, {\it two arbitrary and independent whole
numbers}, $n''$ and $n$, in its expression, as happens in the
theories of John T. Graves, Esq., Professor Ohm, and others,
respecting the general logarithms of ordinary imaginary
quantities to ordinary imaginary bases; and also in that theory
of the general logarithms of {\it numeral couples}, with other
numeral couples for their bases, which was published by the
present author (as part of the Essay already several times cited,
on Conjugate Functions and Algebraic Couples, and on Algebra as
the Science of Pure Time), in the seventeenth volume of the
Transactions of this Academy.

\bigbreak

\centerline{\it
Connexion of Quaternions with Spherical Geometry.}

\nobreak\bigskip

44.
Let ${\sc r}, {\sc r}', {\sc r}'',\ldots \, {\sc r}^{(n-1)}$ be
any $n$ points upon the surface of the unit sphere, so that they
may be generally regarded as the corners of a spherical polygon
upon that surface; and let them be regarded also as the
determining or representative points (in the sense of the
forty-second article) of the same number of vector units,
$i_{\sc r}$,~$i_{{\sc r}'}$,~\&c.  Then the associative property
of multiplication will give, on the one hand, the equation
$$    i_{\sc r} i_{{\sc r}'} \mathbin{.}
      i_{{\sc r}'} i_{{\sc r}''} \mathbin{.}
      i_{{\sc r}''} i_{{\sc r}'''} \, \ldots \,
      i_{{\sc r}^{(n-1)}} i_{\sc r}
   =  (-1)^n;
   \eqno (421)$$
because
$$i_{\sc r}^2
   =  i_{{\sc r}'}^2 =  i_{{\sc r}''}^2 =  \ldots =  -1;
   \eqno (422)$$
and, on the other hand, on substituting the expressions for these
vector units, involving their respective direction-cosines and
the three fundamental units $i$,~$j$,~$k$, which expressions are
of the forms
$$i_{\sc r} = i \alpha + j \beta + k \gamma,\quad
  i_{{\sc r}'} = i \alpha' + j \beta' + k \gamma',\ldots
   \eqno (423)$$
we shall have, for the product of the two first, by the
fundamental relations ({\sc b}), the expression
$$\eqalignno{
i_{\sc r} i_{{\sc r}'}
   &= ( i \alpha  + j \beta  + k \gamma  )
      ( i \alpha' + j \beta' + k \gamma' ) \cr
   &=  - ( \alpha \alpha' + \beta \beta' + \gamma \gamma' )
       + i ( \beta  \gamma' - \gamma \beta'  )
       + j ( \gamma \alpha' - \alpha \gamma' )
       + k ( \alpha \beta'  - \beta  \alpha' ),
   &(424)\cr}$$
that is,
$$i_{\sc r} i_{{\sc r}'}
   =  - \cos {\sc r} {\sc r}'
         + i_{{\sc p}''} \sin {\sc r} {\sc r}',
   \eqno (425)$$
if ${\sc r} {\sc r}'$ denote the arc of rotation in a great
circle, round a positive pole~${\sc p}''$, from the
point~${\sc r}$ to the point~${\sc r}'$ upon the sphere, with
other similar transformations for the other binary products.  By
combining these two principles, (421), (425), it is not difficult
to see that, {\it for any spherical polygon}, regarded as having
its corners ${\sc r}, {\sc r}',\ldots$ at the positive poles
of the sides of another polygon, the following formula holds
good:
$$    ( \cos {\sc r} + i_{\sc r} \sin {\sc r} )
      ( \cos {\sc r}' + i_{{\sc r}'} \sin {\sc r}' )
      \ldots
      ( \cos {\sc r}^{(n-1)}
         + i_{{\sc r}^{(n-1)}} \sin {\sc r}^{(n-1)} )
   =  (-1)^n;
   \eqno (426)$$
in which the symbols ${\sc r}, {\sc r}',\ldots$ under the
characteristics $\cos$ and $\sin$, denote the (suitably measured)
successive angles at the corners ${\sc r}, {\sc r}',\ldots$.  In
particular, for the case of a {\it spherical triangle},
${\sc r} {\sc r}' {\sc r}''$, the formula (426) gives this less
general formula, which however, may be considered as
{\it including spherical trigonometry\/}:
$$    ( \cos {\sc r} + i_{\sc r} \sin {\sc r} )
      ( \cos {\sc r}' + i_{{\sc r}'} \sin {\sc r}' )
      ( \cos {\sc r}'' + i_{{\sc r}''} \sin {\sc r}'' )
   =  -1.
   \eqno (427)$$

\bigbreak

45.
Multiplying both members of this formula (427) into
$\cos {\sc r}'' - i_{{\sc r}''} \sin {\sc r}''$,
we put it under the less symmetric but sometimes more convenient
form,
$$    ( \cos {\sc r} + i_{\sc r} \sin {\sc r} )
      ( \cos {\sc r}' + i_{{\sc r}'} \sin {\sc r}' )
   =  - \cos {\sc r}'' + i_{{\sc r}''} \sin {\sc r}''.
   \eqno (428)$$
Developing the first member of this last equation, and
substituting, for the product of the two vector units, its value
(425), we find that it resolves itself into the two following
formul{\ae}:
$$\cos {\sc r} \cos {\sc r}'
       - \cos {\sc r} {\sc r}' \sin {\sc r} \sin {\sc r}'
   =   - \cos {\sc r}'';
   \eqno (429)$$
$$    i_{\sc r}     \sin {\sc r}  \cos {\sc r}'
    + i_{{\sc r}'}  \sin {\sc r}' \cos {\sc r}
    + i_{{\sc p}''} \sin {\sc r} \sin {\sc r}'
         \sin {\sc r} {\sc r}'
   =  i_{{\sc r}''} \sin {\sc r}''.
   \eqno (430)$$
Of these two equations, the first agrees with the known
expression for the cosine of a side ${\sc r} {\sc r}'$ of a
spherical triangle ${\sc r} {\sc r}' {\sc r}''$, regarded as a
function of the three angles ${\sc r}$,~${\sc r}'$,~${\sc r}''$;
and the second expresses a theorem, which can easily be verified
by known methods, namely, that if a force $= \sin {\sc r}''$ be
directed from the centre of the sphere to the point~${\sc r}''$,
that is, to one corner of any such spherical triangle
${\sc r} {\sc r}' {\sc r}''$,
this force is statically equivalent to the system of three other
forces, one directed to ${\sc r}$, and equal to
$\sin {\sc r} \cos {\sc r}'$; another directed to ${\sc r}'$, and
equal to $\sin {\sc r}' \cos {\sc r}$; and the third equal to
$\sin {\sc r} \sin {\sc r}' \sin {\sc r} {\sc r}'$, and directed
towards that point~${\sc p}''$ of the arc ${\sc r} {\sc r}'$,
which lies at the same side of this arc as does the
corner~${\sc r}''$.

\bigbreak

46.
In this, or in other ways, we may be led to establish, as a
consequence from the principles which have been already stated,
the following general {\it formula for the multiplication of any
two numeral quaternions\/}:
$$\eqalignno{
q \times q'
   &= \mu  ( \cos {\sc r}  + i_{\sc r}    \sin {\sc r}  )
      \times
      \mu' ( \cos {\sc r}' + i_{{\sc r}'} \sin {\sc r}' ) \cr
   &= \mu \mu' \{ \cos (\pi - {\sc r}'')
         + i_{{\sc r}''} \sin (\pi - {\sc r}'') \};
   &(431)\cr}$$
and to interpret it as being equivalent to the system of the
three following rules or theorems.  First, that (as was seen in
the twenty-seventh article), the {\it modulus~$\mu''$ of the
product\/} is equal to the {\it product~$\mu \mu'$ of the
moduli\/} of the factors.  Second, that if a {\it spherical
triangle\/} ${\sc r} {\sc r}' {\sc r}''$ be constructed with the
representative points of the factors and product for its three
corners, the {\it angles of this triangle\/} will be respectively
equal to the {\it amplitudes of the two factors}, and to the
{\it supplement of the amplitude of the product\/}; the
amplitude~${\sc r}$ of the multiplier quaternion~$q$, for
example, being equal to the spherical angle at the
corner~${\sc r}$ of the triangle just described.  And third, that
the {\it rotation round the product point,~${\sc r}''$, from the
multiplier point}, which is here denoted by ${\sc r}$, {\it to
the multiplicand point}, denoted here by ${\sc r}'$, is
{\it positive\/}; or, in other words, this rotation is in the
{\it same direction\/} (towards the right hand, or towards the
left), as the rotation round the positive semiaxis of $z$ or of
$k$ ($= ij$), from that of $x$ or of $i$, to that of $y$ or of
$j$.  The same third rule may also be expressed by saying that
the rotation of a great semicircle {\it round\/} the multiplier
point,~${\sc r}$, {\it from\/} the multiplicand
point,~${\sc r}'$, {\it towards\/} the product point~${\sc r}''$,
is positive; whereas the rotation to the same product point,
{\it from\/} the multiplier point, {\it round\/} the multiplicand
point, is, on the contrary, negative.  (Compare the remarks in
Note~A, printed at the end of the present series.)

\bigbreak

47.
The associative character of multiplication shows that if we
assume any three quaternions $q$,~$q'$,~$q''$, and derive two
others $q_\prime$,~$q_{\prime\prime}$ from them, by the equations
$$q q' = q_\prime,\quad q' q'' = q_{\prime\prime},
   \eqno (432)$$
we shall have also the equations
$$q_\prime q'' = q q_{\prime\prime} = q''',
   \eqno (433)$$
$q'''$ being a third derived quaternion, namely, the ternary
product $q q' q''$.  Let
${\sc r}$~${\sc r}'$ ${\sc r}''$ ${\sc r}_\prime$
${\sc r}_{\prime\prime}$~${\sc r}'''$
be the six representative points of these six quaternions, on the
same spheric surface as before; then, by the general construction
of a product assigned in the foregoing article, we shall have the
following expressions for the six amplitudes of the same six
quaternions:
$$\left. \multieqalign{
\theta
   &= {\sc r}' {\sc r} {\sc r}_\prime
    = {\sc r}_{\prime\prime} {\sc r} {\sc r}'''; &
\theta_\prime
   &= {\sc r}'' {\sc r}_\prime {\sc r}'''
    = \pi - {\sc r} {\sc r}_\prime {\sc r}'; \cr
\theta'
   &= {\sc r}_\prime {\sc r}' {\sc r}
    = {\sc r}'' {\sc r}' {\sc r}_{\prime\prime}; &
\theta_{\prime\prime}
   &= {\sc r}''' {\sc r}_{\prime\prime} {\sc r}
    = \pi - {\sc r}' {\sc r}_{\prime\prime} {\sc r}''; \cr
\theta''
   &= {\sc r}_{\prime\prime} {\sc r}'' {\sc r}'
    = {\sc r}''' {\sc r}'' {\sc r}_\prime; &
\theta'''
   &= \pi - {\sc r}_\prime {\sc r}''' {\sc r}''
    = \pi - {\sc r} {\sc r}''' {\sc r}_{\prime\prime}; \cr}
   \right\}
   \eqno (434)$$
${\sc r}' {\sc r} {\sc r}_\prime$ being the spherical angle at
${\sc r}$, measured from ${\sc r} {\sc r}'$ to
${\sc r} {\sc r}_\prime$, and similarly in other cases.  But
these equations between the spherical angles of the figure are
precisely those which are requisite, in order that the two points
${\sc r}_\prime$ and ${\sc r}_{\prime\prime}$ should be the two
{\it foci of a spherical conic inscribed in the spherical
quadrilateral\/} ${\sc r} {\sc r}' {\sc r}'' {\sc r}'''$, or
touched by the four great circles of which the arcs
${\sc r} {\sc r}'$, ${\sc r}' {\sc r}''$, ${\sc r}'' {\sc r}'''$,
${\sc r}''' {\sc r}$, are parts; this geometrical relation
between the six representative points
${\sc r}$~${\sc r}'$ ${\sc r}''$ ${\sc r}_\prime$
${\sc r}_{\prime\prime}$~${\sc r}'''$
of the six quaternions, $q$, $q'$, $q''$, $q q'$, $q' q''$,
$q q' q''$, which may conveniently be thus denoted,
$${\sc r}_\prime {\sc r}_{\prime\prime} (..)
      {\sc r} {\sc r}' {\sc r}'' {\sc r}''',
   \eqno (435)$$
is, therefore, a consequence, and may be considered as an
interpretation of the very simple algebraical formula for
associating three quaternion factors,
$$q q' \mathbin{.} q'' = q \mathbin{.} q' q''.$$
It follows, at the same time, from the theory of cones and
conics, that {\it the two straight lines}, or {\it radii
vectores}, which are drawn from the origin of coordinates to the
points ${\sc r}_\prime$,~${\sc r}_{\prime\prime}$, and {\it which
construct the imaginary parts of the two binary quaternion
products, $q q'$,~$q' q''$, are the two focal lines of a cone of
the second degree, inscribed in the tetrahedral angle, which has
for four conterminous edges the four radii which construct the
imaginary parts of the three quaternion factors $q$,~$q'$,~$q''$,
and of their continued or ternary product\/}~$q q' q''$.

\bigbreak

48.
We have also, by the same associative character of
multiplication, an analogous formula for the product of any
{\it four\/} quaternion factors,
$q$,~$q'$,~$q''$,~$q'''$, namely,
$$q \mathbin{.} q' q'' q'''
   =  q q' \mathbin{.} q'' q'''
   =  q q' q'' \mathbin{.} q'''
   =  q^{IV},
   \eqno (436)$$
if we denote this continued product by $q^{IV}$; and if we
make
$$q q' = q_\prime,\quad
  q' q'' = q_\prime',\quad
  q'' q''' = q_\prime'',\quad
  q q' q'' = q_\prime''',\quad
  q' q'' q''' = q_\prime^{IV},
   \eqno (437)$$
and observe that whenever ${\sc e}$ and ${\sc f}$ are foci of a
spherical conic inscribed in a spherical quadrilateral ${\sc a}
{\sc b} {\sc c} {\sc d}$, so that, in the notation recently
proposed,
$${\sc e}{\sc f} (..) {\sc a} {\sc b} {\sc c} {\sc d},
   \eqno (438)$$
then also we may write
$${\sc f}{\sc e} (..) {\sc a} {\sc b} {\sc c} {\sc d},
   \quad\hbox{and}\quad
  {\sc e}{\sc f} (..) {\sc b} {\sc c} {\sc d} {\sc a},
   \eqno (439)$$
we shall find, without difficulty, by the help of the formula
(435), the five following geometrical relations, in which
each ${\sc r}$ is the representative point of the corresponding
quaternion~$q$:
$$\left. \eqalign{
   &{\sc r}_\prime {\sc r}_\prime' (..)
         {\sc r} {\sc r}' {\sc r}'' {\sc r}_\prime'''; \cr
   &{\sc r}_\prime' {\sc r}_\prime'' (..)
         {\sc r}' {\sc r}'' {\sc r}''' {\sc r}_\prime^{IV}; \cr
   &{\sc r}_\prime'' {\sc r}_\prime''' (..)
         {\sc r}'' {\sc r}''' {\sc r}^{IV} {\sc r}_\prime; \cr
   &{\sc r}_\prime''' {\sc r}_\prime^{IV} (..)
         {\sc r}''' {\sc r}^{IV} {\sc r} {\sc r}_\prime'; \cr
   &{\sc r}_\prime^{IV} {\sc r}_\prime (..)
         {\sc r}^{IV} {\sc r} {\sc r}' {\sc r}_\prime''. \cr}
   \right\}
   \eqno (440)$$

These five formul{\ae} establish a remarkable {\it connexion
between one spherical pentagon and another\/} (when constructed
according to the foregoing rules), through the medium of {\it five
spherical conics\/}; of which five conics each touches two sides
of one pentagon, and has its foci at two corners of the other.
If we suppose, for simplicity, that each of the ten moduli is
$= 1$, the dependence of the six quaternions by multiplication on
four (as their three binary, two ternary, and one quaternary
product, all taken without altering the order of succession of
the factors) will give eighteen distinct equations between the
ten amplitudes and the twenty polar coordinates of the ten
quaternions here considered; it is therefore in general permitted
to assume at pleasure twelve of these coordinates, or to choose
six of the ten points upon the sphere.  Not only, therefore, may
we in general take {\it one of the two pentagons arbitrarily},
but also, at the same time, may {\it assume one corner of the
other pentagon\/} (subject, of course, to exceptional cases);
and, after a suitable choice of the ten amplitudes and four other
corners, the five relations (440), between the two pentagons and
the five conics, will still hold good.

A very particular (or rather limiting) yet not inelegant case of
this theorem is furnished by the consideration of the plane and
regular pentagon of elementary geometry, as compared with that
other and interior pentagon which is determined by the
intersections of its five diagonals.  Denoting by
${\sc r}_\prime$ that corner of the interior pentagon which is
nearest to the side ${\sc r} {\sc r}'$ of the exterior one; by
${\sc r}_\prime'$, that corner which is nearest to
${\sc r}' {\sc r}''$, and so on to ${\sc r}_\prime^{IV}$; the
relations (440) are satisfied, the symbol $(..)$ now denoting
that the two points written before it are foci of an ordinary (or
plane) ellipse, inscribed in the plane quadrilateral, whose
corners are the four points written after it.  We may add, that
(in this particular case) two points of contact for each of the
five quadrilaterals are corners of the interior pentagon; and
that the axis major of each of the inscribed ellipses is equal to
a side of the exterior figure.

\bigbreak

49.
By combining the principles of the forty-seventh with the
calculations of the twenty-eighth and thirtieth articles, we see
that, with the relations, (258), (259), (284), from which the
relations (285) have been already seen to follow, we may regard
$m_1^\backprime$,~$m_2^\backprime$,~$m_3^\backprime$
as the rectangular coordinates of a point on one focal line, and
$m_1^{\backprime\backprime}$,~$m_2^{\backprime
   \backprime}$,~$m_3^{\backprime\backprime}$
as the rectangular coordinates of a point on the other focal line
of a certain cone of the second degree, having its vertex at the
origin of those coordinates, and having, on the successive
intersections of four of its tangent planes, four points, of
which the coordinates are respectively
$m_1$,~$m_2$,~$m_3$; $b$,~$c$,~$d$; $m_1'$,~$m_2'$,~$m_3'$; and
$m_1''$,~$m_2''$,~$m_3''$.  Hence, with the same relations
between the symbols, the known theory of reciprocal or
supplementary cones enables us to infer that the two equations
$$\left. \eqalign{
x m_1^\backprime + y m_2^\backprime + z m_3^\backprime
   &= 0,\cr
x m_1^{\backprime\backprime}
       + y m_2^{\backprime\backprime}
       + z m_3^{\backprime\backprime}
   &= 0,\cr}
  \right\}
   \eqno (441)$$
represent two cyclic planes of a certain other cone of the second
degree, which has its vertex at the origin, and contains upon its
surface the four points which are determined by the twelve
following rectangular coordinates:
$$\left. \multieqalign{
     & m_2 d - m_3 c,
   & & m_3 b - m_1 d,
   & & m_1 c - m_2 b; \cr
     & c m_3' - d m_2',
   & & d m_1' - b m_3',
   & & b m_2' - c m_1'; \cr
     & m_2' m_3'' - m_3' m_2'',
   & & m_3' m_1'' - m_1' m_3'',
   & & m_1' m_2'' - m_2' m_1''; \cr
     & m_2'' m_3 - m_3'' m_2,
   & & m_3'' m_1 - m_1'' m_3,
   & & m_1'' m_2 - m_2'' m_1. \cr}
   \right\}
   \eqno (442)$$
It would have been easy to have given a little more symmetry to
these last expressions, if we had not wished to present them in a
form in which they might be easily combined with some that had
been already investigated, for a different purpose, in this
paper.

\bigbreak

50.
If we denote by the symbol~$i_{{\sc r} {\sc r}'}$ that vector unit
which is directed towards the positive pole of the arc
${\sc r} {\sc r}'$ ({\it from\/} the point~${\sc r}$ {\it to\/}
the point~${\sc r}'$ on the unit sphere), then the general
formula (425) for the {\it product of any two vector units},
$i_{\sc r}$ and $i_{{\sc r}'}$, becomes
$$i_{\sc r} i_{\sc r}'
   =  (\cos + i_{{\sc r} {\sc r}'} \sin)
      (\pi - {\mathop{{\sc r} {\sc r}'}\limits^\frown});
   \eqno (443)$$
and because the positive pole of the arc~${\sc r} {\sc r}'$ is
the negative pole of the reversed arc~${\sc r}' {\sc r}$, so that
in this reversal the change of sign may be conceived to fall upon
the vector unit,
$$i_{{\sc r}' {\sc r}} = - i_{{\sc r} {\sc r}'},
   \eqno (444)$$
while the arc itself may thus be regarded as not having changed
its sign, but only its pole, we may also write, generally, in
this notation, for the {\it quotient of any two vector units},
the expression
$$i_{\sc r} i_{{\sc r}'}^{-1}
   =  - i_{\sc r} i_{{\sc r}'}
   =  ( \cos + i_{{\sc r}' {\sc r}} \sin ) \mathbin{.}
         {\mathop{{\sc r}' {\sc r}}\limits^\frown}.
   \eqno (445)$$
Hence the associative principle of multiplication gives this
other property of any spherical polygon,
${\sc r} {\sc r}' {\sc r}'' \,\ldots$,
which may be regarded as a sort of {\it polar conjugate\/} to the
property (426), as depending on the consideration of the polar
polygon, or {\it polygon of poles}, namely, the following:
$$( \cos + i_{{\sc r}' {\sc r}} \sin )
        {\mathop{{\sc r}' {\sc r}}\limits^\frown} \mathbin{.}
     ( \cos + i_{{\sc r}'' {\sc r}'} \sin )
        {\mathop{{\sc r}'' {\sc r}'}\limits^\frown} \, \ldots \,
     ( \cos + i_{{\sc r} {\sc r}^{(n-1)}} \sin )
        {\mathop{{\sc r} {\sc r}^{(n-1)}}\limits^\frown}
   = 1.
   \eqno (446)$$
Thus, in particular, for any spherical triangle, of which the
three sides may be briefly denoted thus,
$${\mathop{{\sc r}' {\sc r}}\limits^\frown} = \theta'';\quad
  {\mathop{{\sc r}'' {\sc r}'}\limits^\frown} = \theta;\quad
  {\mathop{{\sc r} {\sc r}''}\limits^\frown} = \theta';\quad
   \eqno (447)$$
while the three corresponding vector units, directed to the
positive poles of these three arcs, may be thus denoted,
$$i_{{\sc r}' {\sc r}} = \iota'';\quad
  i_{{\sc r}'' {\sc r}'} = \iota;\quad
  i_{{\sc r} {\sc r}''} = \iota';
   \eqno (448)$$
the following equation holds good, and may be employed, instead
of (427), as a formula for spherical trigonometry:
$$    ( \cos \theta'' + \iota'' \sin \theta'' )
      ( \cos \theta   + \iota   \sin \theta   )
      ( \cos \theta'  + \iota'  \sin \theta'  )
   =  1.
   \eqno (449)$$
Hence also may be derived this other and not less general
equation, analogous to (431), and serving in a new way to express
the result of the multiplication of any two numeral quaternions,
in connexion with a spherical triangle:
$$    \mu   ( \cos \theta   + \iota   \sin \theta   ) \times
      \mu'  ( \cos \theta'  + \iota'  \sin \theta'  )
   =  \mu \mu'
            ( \cos \theta'' - \iota'' \sin \theta'' ).
   \eqno (450)$$

The sides of the triangle here considered are $\theta$,
$\theta'$, $\theta''$, that is, they are the amplitudes of the
two factors and of the product; and the angles respectively
opposite to those three sides are the supplements of the mutual
inclinations of the three pairs of vector units,
$\iota'$,~$\iota''$; $\iota''$,~$\iota$; $\iota$,~$\iota'$;
they are therefore, respectively, the inclinations of the two
vector units $\iota'$ and $\iota$ to $- \iota''$, and the
supplement of their inclination to each other.  But, in the
multiplication (450), $\iota$,~$\iota'$, and $- \iota''$ are
respectively the vector units of the multiplier, the
multiplicand, and the product; if then we agree to speak of the
mutual inclination of the vector units of any two quaternions as
being also the mutual {\it inclination\/} of those two
quaternions themselves, we may enunciate the following Theorem,
with which we shall conclude the account of this First Series of
Researches: {\it If, with the amplitudes of any two quaternion
factors, and of their product, as sides, a spherical triangle be
constructed, the angle of this triangle, which is opposite to the
side which represents the amplitude of either factor, will be
equal to the inclination of the remaining factor to the product;
and the angle opposite to that other side which represents the
amplitude of the product, will be equal to the supplement of the
inclination of the same two factors to each other}.

\vfill\eject

\centerline{NOTE A}

\nobreak\bigskip

\centerline{\it
Extract from a Letter of Sir William R. Hamilton to John T.
Graves, Esq.}

\nobreak\bigskip

\line{\hfil ``{\it Observatory of Trinity College, Dublin},
24{\it th October}, 1843.}

\nobreak\bigskip

---``The Germans often put $i$ for $\surd - 1$, and therefore
denote an ordinary imaginary quantity by $x + iy$.  I assume
{\it three\/} imaginary characteristics or units, $i$,~$j$,~$k$,
such that {\it each\/} shall have its square $= - 1$, without any
one being the equal or the negative of any other;
$$i^2 = j^2 = k^2 = -1.
   \eqno (1)$$
And I assume (for reasons explained in my first letter) the
relations
$$ij = k;\quad jk = i;\quad ki = j;
   \eqno (2)$$
$$ji = - k;\quad kj = - i;\quad ik = - j;
   \eqno (3)$$
each imaginary unit being thus the product of the two which
precede it in the cyclical order $i$~$j$~$k$, but the negative of
the product of the two which follow it in that order.  Such being
my fundamental assumptions, which include (as you perceive) the
somewhat strange one that {\it the order of multiplication of
quaternions is not, in general, indifferent}, I have at once the
theorem that
$$(w + i x + j y + k z) (w' + i x' + j y' + k z')
   =  w'' + i x'' + j y'' + k z'',
   \eqno (4)$$
if the following relations hold good:
$$w''  = w w' - x x' - y y' - z z';
   \eqno (5)$$
$$\left. \eqalign{
x'' &= w x' + x w' + y z' - z y';\cr
y'' &= w y' + y w' + z x' - x z';\cr
z'' &= w z' + z w' + x y' - y x';\cr}
   \right\}
   \eqno (6)$$
and reciprocally that these four relations (5) and (6) are
{\it necessary\/} (on account of the mutual independence of the
three imaginary units, $i$,~$j$,~$k$, except so far as they are
connected by the conditions above assigned), in order that the
quaternion $w'' + i x'' + j y'' + k z''$ may result as a product
from the multiplication of $w' + i x' + j y' + k z'$, as a
multiplicand, by $w + i x + j y + k z$ as a multiplier.

``Making, for abridgment,
$$x_\prime'' = w x' + x w';\quad
  y_\prime'' = w y' + y w';\quad
  z_\prime'' = w z' + z w';
   \eqno (7)$$
$$x_{\prime\prime}'' = y z' - z y';\quad
  y_{\prime\prime}'' = z x' - x z';\quad
  z_{\prime\prime}'' = x y' - y x';
   \eqno (8)$$
and observing that
$$x x_{\prime\prime}''
          + y y_{\prime\prime}''
          + z z_{\prime\prime}''
   = 0;\quad
  x' x_{\prime\prime}''
          + y' y_{\prime\prime}''
          + z' z_{\prime\prime}''
   = 0;
   \eqno (9)$$
we see easily that
$$    x_\prime '' x_{\prime\prime}''
    + y_\prime '' y_{\prime\prime}''
    + z_\prime '' z_{\prime\prime}''
   = 0;
   \eqno (10)$$
therefore, since
$$x'' = x_\prime'' + x_{\prime\prime}'',\quad
  y'' = y_\prime'' + y_{\prime\prime}'',\quad
  z'' = z_\prime'' + z_{\prime\prime}'',
   \eqno (11)$$
we have
$$x''^2 + y''^2 + z''^2
   =  x_\prime''^2 + y_\prime''^2 + z_\prime''^2
          + x_{\prime\prime}''^2
          + y_{\prime\prime}''^2
          + z_{\prime\prime}''^2.
   \eqno (12)$$
Again,
$$(x x' + y y' + z z')^2
       + x_{\prime\prime}''^2
       + y_{\prime\prime}''^2
       + z_{\prime\prime}''^2
   =  (x^2 + y^2 + z^2) (x'^2 + y'^2 + z'^2),
   \eqno (13)$$
$$-2 w w' (x x' + y y' + z z')
       + x_\prime''^2
       + y_\prime''^2
       + z_\prime''^2
   =  w^2 (x'^2 + y'^2 + z'^2) + w'^2 (x^2 + y^2 + z^2);
   \eqno (14)$$
therefore,
$$w''^2 + x''^2 + y''^2 + z''^2
   =  (w^2 + x^2 + y^2 + z^2) (w'^2 + x'^2 + y'^2 + z'^2).
   \eqno (15)$$
Let
$$\left. \multieqalign{
w   &= \mu   \cos \theta;   &
x   &= \mu   \sin \theta   \cos \phi;   &
y   &= \mu   \sin \theta   \sin \phi   \cos \psi;   \cr
   &  &  &  &
z   &= \mu   \sin \theta   \sin \phi   \sin \psi;   \cr
w'  &= \mu'  \cos \theta';  &
x'  &= \mu'  \sin \theta'  \cos \phi';  &
y'  &= \mu'  \sin \theta'  \sin \phi'  \cos \psi';  \cr
   &  &  &  &
z'  &= \mu'  \sin \theta'  \sin \phi'  \sin \psi';  \cr
w'' &= \mu'' \cos \theta''; &
x'' &= \mu'' \sin \theta'' \cos \phi''; &
y'' &= \mu'' \sin \theta'' \sin \phi'' \cos \psi''; \cr
   &  &  &  &
z'' &= \mu'' \sin \theta'' \sin \phi'' \sin \psi''; \cr}
   \right\}
   \eqno (16)$$
and let $\mu$, $\sin \theta$, and $\sin \phi$, be treated as
positive (or, at least, not negative) quantities; we shall then
have
$$\mu'' = \mu \mu';
   \eqno (17)$$
which may be enunciated by saying that {\it the modulus of the
product of two quaternions is the product of the moduli of those
two factors}.

``At the same time we shall have
$$r = \mu \sin \theta,
   \quad\hbox{if we make}\quad r = \surd (x^2 + y^2 + z^2);
   \eqno (18)$$
and may call this quantity,~$r$, the {\it modulus of the pure
imaginary triplet}, $ix + jy + kz$.  We may also call it the
{\it radius\/} of the imaginary part of the quaternion
$w + ix + jy + kz$, or even the radius of the quaternion itself;
and may speak of the {\it inclination\/} of one such radius to
another, the cosine of this inclination being
$$\cos \mathbin{.} r r'
   =  \cos \phi \cos \phi'
    + \sin \phi \sin \phi' \cos (\psi' - \psi).
   \eqno (19)$$
The angle~$\phi$ may be called the {\it colatitude}, and $\phi$
the {\it longitude}, of the radius, or triplet, or quaternion.
And $\theta$ may be called the {\it amplitude\/} of the
quaternion; so that the real part multiplied by the tangent of
the amplitude, produces the radius of the quaternion, or of its
imaginary part,
$$w \tan \theta = r.
   \eqno (20)$$
The amplitude,~$\theta$, may be supposed to range only from $0$
to $\pi$.  It vanishes for a pure, real, positive quantity, and
becomes
$\displaystyle = {\pi \over 2}$
for a pure imaginary; it is $= \pi$ for a pure real negative.

``The equation (5), combined with (16) and (17), gives
$$\cos \theta''
   =  \cos \theta \cos \theta'
    - \sin \theta \sin \theta'
         \{
            \cos \phi \cos \phi'
          + \sin \phi \sin \phi' \cos (\psi' - \psi)
         \};
   \eqno (21)$$
if, therefore, we construct a spherical triangle, of which one
side is the inclination of the factors, while the two adjacent
angles are the amplitudes of those factors, the remaining angle
will be the supplement of the amplitude of the product.

``Combining (5) with (6), we find that
$$\left. \eqalign{
w w'' + x x'' + y y'' + z z''
   &= (w^2 + x^2 + y^2 + z^2) w';\cr
w' w'' + x' x'' + y' y'' + z' z''
   &= (w'^2 + x'^2 + y'^2 + z'^2) w;\cr}
   \right\}
   \eqno (22)$$
therefore, by (16) and (17),
$$\left. \eqalign{
\cos \theta'
   &= \cos \theta'' \cos \theta
    + \sin \theta'' \sin \theta
         \{
            \cos \phi'' \cos \phi
          + \sin \phi'' \sin \phi  \cos (\psi  - \psi'')
         \};\cr
\cos \theta
   &= \cos \theta'' \cos \theta'
    + \sin \theta'' \sin \theta'
         \{
            \cos \phi'' \cos \phi'
          + \sin \phi'' \sin \phi' \cos (\psi' - \psi'')
         \};\cr}
   \right\}
   \eqno (23)$$
so that in the spherical triangle lately mentioned, the two
remaining sides are the inclinations of the two factors to their
product.  This spherical triangle may, therefore, be constructed
by merely joining the points ${\sc r}$,~${\sc r}'$,~${\sc r}''$,
where the sphere, with radius unity, and with centre at the
origin of $x$,~$y$,~$z$, is met by the directions of the radii,
$r$,~$r'$,~$r''$, of the two factors and the product.  The
spherical coordinates of these three points are
$\phi$,~$\psi$; $\phi'$,~$\psi'$; $\phi''$,~$\psi''$;
the spherical angles at the same points are $\theta$, $\theta'$,
$\pi - \theta''$.  In the solid corner, at the origin, made by
the three radii, $r$,~$r'$,~$r''$, whatever the lengths of these
radii may be, the three dihedral angles are
$$r'' r r' = \theta;\quad
  r r' r'' = \theta';\quad
  r' r'' r = \pi - \theta'';
   \eqno (24)$$
that is, they are the amplitudes of the factors, and the
supplement of the amplitude of the product.

``Though this theorem of the spherical triangle,
${\sc r}$,~${\sc r}'$,~${\sc r}''$,
or solid corner, $r$,~$r'$,~$r''$,
when combined with the {\it law of the moduli\/}
($\mu'' = \mu \mu'$), reproduces four relations between the four
{\it constituents}, $w''$, $x''$, $y''$, $z''$, of the quaternion
product, and the eight constituents of the two quaternion
factors, namely, $w$,~$x$,~$y$,~$z$, and $w'$,~$x'$,~$y'$,~$z'$,
that is to say, the two relations (5) and (15), and the two
relations (22); yet it leaves still something undetermined, with
respect to the direction of the product, which requires to be
more closely considered.  In fact, we can thus fix not only the
modulus,~$\mu''$, and the amplitude,~$\theta''$, of the product,
but also the inclinations of its radius,~$r''$, to the two radii,
$r$ and $r'$; but the construction, so far, fails to determine
{\it on which side of the plane\/}~$rr'$ of the radii of the
factors does the radius of the product lie.  In other words, when
we deduced the relations (15) and (22), we may be considered as
having employed rather the equations (9) and (13), which were
derived from (8), than the equations (8) themselves; the three
quantities, $x_{\prime\prime}''$, $y_{\prime\prime}''$,
$z_{\prime\prime}''$, might, therefore, all change signs
together, without affecting the law of the moduli, or the theorem
of the spherical triangle.  And the additional condition, which
is to decide between the one and the other set of signs of these
three quantities, or between the one and the other set of signs
in the expressions
$$x'' = x_\prime'' \pm x_{\prime\prime}'';\quad
  y'' = y_\prime'' \pm y_{\prime\prime}'';\quad
  z'' = z_\prime'' \pm z_{\prime\prime}'';
   \eqno (25)$$
is easily seen, on reverting to first principles, to be the
choice of the cyclical order $i$~$j$~$k$, rather than
$i$~$k$~$j$, or the choice of the upper rather than the lower
signs in the assumptions
$$ij = -ji = \pm k,\quad
  jk = -kj = \pm i,\quad
  ki = -ik = \pm j.\quad
   \eqno (26)$$
This gives a clue, which may be thus pursued.  Let
$$\left. \multieqalign{
x_\prime''
   &= r_\prime'' \cos \phi_\prime'', &
y_\prime''
   &= r_\prime'' \sin \phi_\prime''
         \cos \psi_\prime'', &
z_\prime''
   &= r_\prime'' \sin \phi_\prime''
         \sin \psi_\prime''; \cr
x_{\prime\prime}''
   &= r_{\prime\prime}'' \cos \phi_{\prime\prime}'', &
y_{\prime\prime}''
   &= r_{\prime\prime}'' \sin \phi_{\prime\prime}''
         \cos \psi_{\prime\prime}'', &
z_{\prime\prime}''
   &= r_{\prime\prime}'' \sin \phi_{\prime\prime}''
         \sin \psi_{\prime\prime}''; \cr}
   \right\}
   \eqno (27)$$
then, by (12) and (16), and by the meaning which we have assigned
to $r''$, we have
$$r''^2 = r_\prime''^2 + r_{\prime\prime}''^2,\quad
  \mu''^2 = w''^2 + r''^2.
   \eqno (28)$$

``By (9), $r_{\prime\prime}''$ is perpendicular to the plane of
$r r'$; and therefore, by (10), $r''$ is in that plane, being, in
fact, the projection of $r''$ thereupon.  This projection is
entirely fixed by the construction already given; and it remains
only to determine the direction of the
perpendicular,~$r_{\prime\prime}''$, as distinguished from the
opposite of that direction.  And a rule which shall fix the sign
of any one of the coordinates,
$x_{\prime\prime}''$,~$y_{\prime\prime}''$,~$z_{\prime\prime}''$,
will be sufficient for this purpose.  It will be sufficient,
therefore, to study any one of the equations (8), for instance
the first, namely,
$$x_{\prime\prime}'' = y z' - z y',$$
and to draw from it such a rule.

``Substituting for $y$, $z$, $y'$, $z'$, their values (16), we
find
$$x_{\prime\prime}''
   =  \mu \mu' \sin \theta \sin \theta'
         \sin \phi \sin \phi' \sin (\psi' - \psi);
   \eqno (29)$$
so that (the other factors having been already supposed positive)
$x_{\prime\prime}''$ has the same sign as the sine of the excess
of the longitude $\psi'$ of $r'$ over the longitude~$\psi$ of
$r$.  But these longitudes are determined by the rotation of the
plane of $xr$ round the positive semiaxis of $x$, from the
position of $xy$ towards the position of $xz$, or from the
positive semiaxis of $y$ towards that of $z$; which direction of
rotation is here to be considered as the positive one.
Consequently, $x_{\prime\prime}''$ is positive or negative,
according as the least rotation from $+ x$, from $r$ to $r'$, is
itself positive or negative; in each case, therefore, the
rotation round $x_{\prime\prime}''$, and, consequently, round
$r_{\prime\prime}''$, or finally round $r''$, from $r$ to $r'$,
is positive.  {\it The rotation round the product line, from the
multiplier to the multiplicand, is constantly right-handed or
constantly left-handed}, according as the rotation round $+i$
from $+j$ to $+k$ is itself right-handed or left-handed.  Hence,
also, to express the same rule otherwise, {\it the rotation round
the multiplier, from the multiplicand to the product}, is (in the
same sense) {\it constantly positive}.  In short, the cyclical
order is multiplier, multiplicand, product; just as, and
precisely because, we took the order $i$~$j$~$k$ for that in
which the rotation round any one, from the next to the one after
it, should be accounted positive, and chose that $ij$ should be
$= k$, not $= - k$.  {\it The law of the moduli, the theorem of
the spherical triangle, and the rule of rotation, suffice to
determine entirely the product of any two quaternions.}

``In my former letter I gave a theorem equivalent to that which I
have here given as the theorem of the spherical triangle,
answering, in fact, very nearly to the polar triangle, conjugate
therewith, but, as I think, much less geometrically simple,
because the three corners had no obvious geometrical meanings,
whereas now the corners, ${\sc r}$,~${\sc r}'$,~${\sc r}''$ mark
the directions of the factors and product respectively.  In the
new triangle, if we let fall a perpendicular from the
extremity~${\sc r}''$ of that radius of the sphere which
coincides in direction with $r''$, on the arc ${\sc r} {\sc r}'$,
which represents the inclination of the factors to each other,
and call the foot of this perpendicular ${\sc r}_\prime''$, we
shall have
$$r_\prime'' = r'' \cos {\sc r}'' {\sc r}_\prime'',\quad
  r_{\prime\prime}'' = r'' \sin {\sc r}'' {\sc r}_\prime'';
   \eqno (30)$$
also the spherical coordinates of ${\sc r}_\prime''$ will be
$\phi_\prime''$, $\psi_\prime''$, and
$\phi_{\prime\prime}''$, $\psi_{\prime\prime}''$, in (27), will
be the spherical coordinates of a point
${\sc r}_{\prime\prime}''$ which will be one pole of the arc
${\sc r} {\sc r}'$, and will be distinguished from the other pole
by the rule of rotation already assigned; it might, perhaps, be
called the {\it positive pole\/} of ${\sc r} {\sc r}'$, though it
ought then to be considered as the negative pole of
${\sc r}' {\sc r}$.

``We saw that $r_\prime''$ was in the plane of $r$ and $r'$, and
this is now constructed by ${\sc r}_\prime''$ being on the great
circle ${\sc r} {\sc r}'$.

``There seem to be some advantages in considering the quaternion
$$w + i x_\prime'' + j y_\prime'' + k z_\prime''
   \eqno (31)$$
as the {\it reduced product\/} of the two factors already often
mentioned in this letter, it is the {\it part\/} of their
complete product (4) which is {\it independent of their order};
and its radius~$r_\prime''$, is, as we have seen, the
{\it projection\/} of the radius~$r''$ of the complete product on
the plane of the two factors $r r'$.  We now see that
$$\tan \theta \sin r r_\prime''
   =  \tan \theta' \sin r' r_\prime''
   =  \tan r'' r_\prime'';
   \eqno (32)$$
the radius~$r_\prime''$ of the reduced product divides the angle
between the radii $r$,~$r'$, of the factors, into parts, of which
the sines are inversely as the tangents of the amplitudes,
$\theta$,~$\theta'$.  Indeed this radius~$r_\prime''$, is the
statical resultant, or {\it algebraical sum}, of two lines which
coincide in direction with $r$ and $r'$ respectively, if $w'$ and
$w$ be positive, but have their lengths equal to the products
$w' r$ and $w r'$, or
$\mu \mu' \sin \theta \cos \theta'$ and
$\mu \mu' \sin \theta' \cos \theta$, or
$w w' \tan \theta$ and $w w' \tan \theta'$;
as appears (among other ways) from the equations (7).  For the
same reason, or by a combination of the equations (7), (16),
(27), we have
$$r_\prime''^2 \mu^{-2} \mu'^{-2}
   =  \cos \theta^2 \sin \theta'^2
    + \cos \theta'^2 \sin \theta^2
    + 2 \sin \theta \cos \theta
            \sin \theta' \cos \theta' \cos r r';
   \eqno (33)$$
and because, by (21),
$$\cos \theta''
   =  \cos \theta \cos \theta'
    - \sin \theta \sin \theta' \cos r r',
   \eqno (34)$$
we arrive at the following pretty simple expression for the
radius of the reduced product,
$$r_\prime''
   =  \mu \mu' \surd ( \cos \theta^2 + \cos \theta'^2
       - 2 \cos \theta \cos \theta' \cos \theta'' ).
   \eqno (35)$$
But also, by the general analogy of the present notation, if we
denote by $\mu_\prime''$ and $\theta_\prime''$ the modulus and
amplitude of the same reduced product (31), we shall have
$$\mu_\prime'' \cos \theta_\prime''
   =  w'' = \mu \mu' \cos \theta'',\quad
  \mu_\prime'' \sin \theta_\prime'' = r_\prime'';
   \eqno (36)$$
therefore,
$$\mu_\prime''
   =  \mu \mu' \surd (
         \cos \theta^2 + \cos \theta'^2 + \cos \theta''^2
       - 2 \cos \theta \cos \theta' \cos \theta'' );
   \eqno (37)$$
and
$$\cos \theta_\prime''
   =  { \cos \theta''
         \over \surd (
         \cos \theta^2 + \cos \theta'^2 + \cos \theta''^2
       - 2 \cos \theta \cos \theta' \cos \theta'' ) }.
   \eqno (38)$$
Again, by (17), (28), (34), (36), (37),
$$\left. \eqalign{
r_{\prime\prime}''
    = \surd (\mu''^2 - \mu_\prime''^2)
   &= \mu \mu' \surd (
         1 + 2 \cos \theta \cos \theta' \cos \theta''
       - \cos \theta^2 - \cos \theta'^2 - \cos \theta''^2 ) \cr
   &= \mu \mu' \sin \theta \sin \theta' \sin r r' \cr}
   \right\}
   \eqno (39)$$
an expression for the radius of the pure imaginary triplet,
$$i x_{\prime\prime}''
       + j y_{\prime\prime}''
       + k z_{\prime\prime}'',
   \eqno (40)$$
that is, of the complete product (4) minus the reduced product
(31), which agrees with the second equation (30), because, by
spherical trigonometry,
$$\sin \theta \sin \theta' \sin r r'
   =  \sin \theta'' \sin r'' r_\prime'';
   \eqno (41)$$
and which gives
$$\mu_\prime''
   =  \mu \mu' \surd
         ( 1 - (\sin \theta \sin \theta' \sin rr')^2 ).
   \eqno (42)$$
We might call the triplet (40), (which remains when we subtract
the reduced product from the complete product), the {\it residual
triplet}, or simply, the {\it residual}, of the product of the
two proposed quaternions (4).  And we see that this {\it residual
is always perpendicular to the reduced product}, when it exists
at all; for we shall find that it may sometimes vanish.  It is
the part of the complete product which changes sign when the
order of the factors is changed.

``These remarks on the geometrical construction of the
{\it equations of multiplication\/} (5) and (6) have, perhaps,
been tedious; they certainly are nothing more than deductions
from those equations, and, consequently, from the fundamental
assumptions (1), (2), (3).  Yet it may not be altogether useless,
in the way of illustration, to draw some corollaries from them,
by the consideration of particular cases.

``{\it Multiplication of two Reals.}---It is evident from the
figure that, as [the two internal angles] $\theta$ and $\theta'$
tend to $0$, [the external angle] $\theta''$ tends to $0$
likewise; and that the same thing happens with respect to
$\theta''$, when $\theta$ and $\theta'$ both tend to $\pi$.
Hence the product of two positive or two negative real quantities
is a real positive quantity.  But when one of the two amplitudes
of the factors, $\theta$ or $\theta'$, tends to $0$, and the
other to $\pi$, then $\theta''$ also tends to $\pi$; the product
of two reals is, therefore, real and negative, if one of the two
factors is positive and the other negative.

``{\it Multiplication by a Real.}---If $\theta$ tend to $0$,
$\theta''$ tends to become $= \theta'$, and ${\sc r}''$ tends to
coincide with ${\sc r}'$; also $\mu$ tends to become $= w$.  If,
therefore, a quaternion be multiplied by a positive real
quantity, $w = \mu$, the effect is only to multiply its modulus
by that quantity, without changing the amplitude or direction.
But if $\theta$ tend to $\pi$, then $\mu$ tends to $-w$;
${\sc r}''$ tends to become diametrically opposite to ${\sc r}'$;
and $\theta''$ tends to become supplementary to $\theta'$.  If a
quaternion be multiplied by a real negative, $w = - \mu$, the
effect is to multiply the modulus,~$\mu'$, by the real positive,
$-w = \mu$; to change the amplitude~$\theta'$ to $\pi - \theta'$;
the colatitude,~$\phi'$, to $\pi - \phi'$; and the
longitude,~$\psi'$, to $\pi + \psi'$.  Accordingly, by inspection
of the second line of the expressions marked (16), we see that
these changes are equivalent to multiplying each of the four
constituents, $w'$, $x'$, $y'$, $z'$, of the proposed quaternion,
by $- \mu$.  In each of these two cases of multiplication by a
real, the {\it residual\/} triplet disappears by (39), because
$\sin \theta$ vanishes.

``{\it Multiplication of a Real by a Quaternion.}---We have only
to suppose that $\theta'$ tends to $0$ or to $\pi$.  The residual
vanishes, and the order of multiplication is indifferent.

``{\it Multiplication of two pure Imaginaries.}---Here
$\displaystyle \theta = \theta' = {\pi \over 2}$,
$\mu = r$, $\mu' = r'$; ${\sc r}''$ coincides with
${\sc r}_{\prime\prime}''$, that is, with the positive pole of
${\sc r} {\sc r}'$; the direction of the product is perpendicular
to the plane of the factors; and the amplitude of the product is
the supplement of the inclination of those two factors to each
other.  Introducing the consideration of the reduced product and
residual, since
${\sc r}'' {\sc r}_\prime'' = {\displaystyle {\pi \over 2}}$,
we have, by (30), $r_\prime'' = 0$, $r_{\prime\prime}'' = r''$;
the reduced product is a pure real, namely, the real part of the
complete product; and the residual is equal to the imaginary
part.  The amplitude of the reduced product is $= \pi$, or $= 0$,
according as the inclination of the factors is less or greater
than
$\displaystyle {\pi \over 2}$;
such, then, is the condition which decides whether the real part
of the product of two pure imaginaries, taken in either order,
shall be negative or positive.  The real part itself
$= \mu \mu' \cos \theta'' = - r r' \cos r r' =$ the product of
the radii of the factors multiplied by the cosine of the
supplement of their mutual inclination.  The radius of the
residual $= r r' \sin r r' =$ the product of the same radii of
the factors multiplied by the sine of their inclination to each
other.  The product is a pure imaginary, if the factors be
mutually rectangular; but a pure real negative, if the factors
coincide in direction; and a pure real positive, if their
directions be exactly opposite.

``{\it Squaring of a Quaternion.}---As ${\sc r}'$ tends to
coincide with ${\sc r}$, and $\theta'$ to become equal to
$\theta$, ${\sc r}''$ tends to coincide likewise with ${\sc r}$,
and $\theta''$ to become double of $\theta$, at least if $\theta$
be less than
$\displaystyle {\pi \over 2}$.
But if $\theta$ be greater than
$\displaystyle {\pi \over 2}$,
then ${\sc r}''$ tends to coincide with the point diametrically
opposite to ${\sc r}$, and $\theta''$ tends to become equal to
the double of the supplement of $\theta$.  If
$\displaystyle \theta = {\pi \over 2}$,
then ${\sc r}''$ tends to become distant by
$\displaystyle {\pi \over 2}$
from ${\sc r}$, but in an indeterminate direction, which is,
however, unimportant, because $\theta''$ tends to become $= \pi$,
and the square (of a pure imaginary triplet) is thus found to be
{\it a pure real negative\/}; which agrees with the recent result
respecting the product of two pure imaginaries, coincident in
direction with each other.  In general, {\it the square of a
quaternion may be obtained by squaring the modulus and doubling
the amplitude\/}; that is, the square of
$$\mu \cos \theta + \mu \sin \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ),
   \eqno (43)$$
may {\it always\/} be thus expressed:
$$\mu^2 \cos 2 \theta + \mu^2 \sin 2 \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi );
   \eqno (44)$$
for instance,
$$(         i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi )^2
   =  -1;
   \eqno (45)$$
although, when
$\displaystyle \theta < {\pi \over 2}$, $\theta < \pi$,
it is supposed, in the {\it construction}, that we treat
$\cos 2\theta$ as $= \cos (2\pi - 2\theta)$;
$\sin 2\theta \cos \phi$ as
$= \sin (2\pi - 2\theta) \* \cos (\pi - \phi)$;
$\sin 2\theta \sin \phi \cos \psi$ as
$= \sin (2\pi - 2\theta) \* \sin (\pi - \phi)
         \* \cos (\pi + \psi)$;
and
$\sin 2\theta \sin \phi \sin \psi$ as
$= \sin (2\pi - 2\theta) \* \sin (\pi - \phi)
         \* \sin (\pi + \psi)$;
all which is evidently allowed.

``{\it Cubing a Quaternion.}---The cube may always be found by
cubing the modulus, and tripling the amplitude.

``{\it Raising to any whole Power.}---The $n^{\rm th}$ power of
the quaternion (43) is the following, if $n$ be a positive whole
number:
$$\mu^n \cos n\theta + \mu^n \sin n\theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ).
   \eqno (46)$$

``{\it Extracting a Root.}---The $n^{\rm th}$ root has, in
general, $n$ and only $n$ values, included under the form
$$    \mu^{1 \over n} \cos {\theta + 2 p \pi \over n}
    + \mu^{1 \over n} \sin {\theta + 2 p \pi \over n}
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ).
   \eqno (47)$$

``{\it Roots of Reals.}---If $\theta = 0$, so that we have to
extract the $n^{\rm th}$ root of a positive real quantity,~$w$,
considered as the quaternion
$$w + i0 + j0 + k0 = w,
   \eqno (48)$$
$\phi$ and $\psi$ remain entirely undetermined, in the formula
$$(\mu + i0 + j0 + k0)^{1 \over n}
   =  \mu^{1 \over n} \cos {2p \pi \over n}
    + \mu^{1 \over n} \sin {2p \pi \over n}
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ).
   \eqno (49)$$
For example, unity, considered as $1 + i0 + j0 + k0$, has not
only itself as a cube root, but also every possible quaternion
which has its modulus $= 1$, and its amplitude
$\displaystyle = {2\pi \over 3}$.
(The amplitude
$\displaystyle = {4\pi \over 3}$
corresponds merely to quaternions which directions opposite to
those with the amplitude
$\displaystyle = {2\pi \over 3}$,
and direction is here indifferent.)  But unity has only two
square roots, $\pm 1 + i 0 + j 0 + k 0$.

``If $\theta = \pi$, so that we have to extract the $n^{\rm th}$
root of the quaternion (48), when $w = - \mu$, we have still
$\phi$ and $\psi$ left undetermined, but the formula is now
$$(-\mu + i0 + j0 + k0)^{1 \over n}
   =  \mu^{1 \over n} \cos {(2p + 1) \pi \over n}
    + \mu^{1 \over n} \sin {(2p + 1) \pi \over n}
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ).
   \eqno (50)$$
For example, the square root of $-1$ may have any arbitrary
direction, provided that it is a pure imaginary with modulus
$= 1$;
$$(-1 + i0 + j0 + k0)^{1 \over 2}
   =        i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi.
   \eqno (51)$$

``{\it Exponent any positive quantity.}---The power is
$$    \mu^{m \over n} \cos
         \left( {m \over n} \overline{\theta + 2 p \pi} \right)
    + \mu^{m \over n} \sin
         \left( {m \over n} \overline{\theta + 2 p \pi} \right)
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ),
   \eqno (52)$$
if $\displaystyle {m \over n}$ be any positive fraction; and it
is natural to define that the power with incommensurable exponent
$$ \{ \mu \cos \theta + \mu \sin \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi )
   \}^\nu
   \eqno (53)$$
is the limit of the power with exponent
$\displaystyle {m \over n}$,
if $\nu$ be the limit of
$\displaystyle {m \over n}$;
hence generally, the power (53) is
$$    \mu^\nu \cos (\nu \theta + 2 \nu p \pi)
    + \mu^\nu \sin (\nu \theta + 2 \nu p \pi)
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi );
   \eqno (54)$$
at least, if $\nu$ be positive.  The reason for this last
restriction is, that we have not yet considered {\it division},
at least in the present letter, which I am aiming to make
complete in itself, so far as it goes.

``{\it Multiplication of codirectional Quaternions.}---If, in
fig.~1, we conceive ${\sc r}'$ to approach to ${\sc r}$, then, in
general, ${\sc r}''$ will approach either to ${\sc r}$ of to the
point diametrically opposite; and, in the first case, $\theta''$
will tend to become the sum of $\theta$ and $\theta'$; but, in
the second case, the sum of their supplements.  In each case we
may treat $\theta''$ as $= \theta + \theta'$, if we treat
${\sc r}''$ as coinciding with ${\sc r}$, or $\phi''$ and
$\psi''$ as equal to $\phi$ and $\psi$.  Thus, generally,
$$\eqalignno{
\{ \mu \cos \theta + \mu \sin \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ) \}
   \hskip -21em \cr
   &\mathrel{\phantom{=}} \mathord{}
   \times
   \{ \mu' \cos \theta' + \mu' \sin \theta'
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ) \} \cr
   &= \mu \mu' \cos (\theta + \theta')
    + \mu \mu' \sin (\theta + \theta')
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi );
   &(55)\cr}$$
which accordingly agrees with the equations of multiplication (5)
and (6), whatever $\mu$, $\mu'$, $\theta$, $\theta'$, $\phi$,
$\psi$ may be.  (Indeed, if $\theta' + \theta = \pi$, the
position of ${\sc r}''$ is undetermined; but this is indifferent,
because its amplitude is now $=\pi$, and the product is a pure
real negative.)  For example, by making $\phi = 0$, we fall back
on the old and well-known theorem of ordinary imaginaries, that
$$(\mu \cos \theta + i \mu \sin \theta)
      (\mu' \cos \theta' + i \mu' \sin \theta')
   =       \mu \mu' \cos (\theta + \theta')
       + i \mu \mu' \sin (\theta + \theta').
   \eqno (56)$$

``{\it Division\/} [{\it Submultiplication\/}].---By (55),
$$\eqalignno{
\{ \mu \cos \theta + \mu \sin \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ) \}
   \hskip -21em \cr
   &\mathrel{\phantom{=}} \mathord{}
   \times
   \{ \mu^{-1} \cos \theta - \mu^{-1} \sin \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ) \} \cr
   &= 1.
   &(57)\cr}$$

``The {\it reciprocal\/} of a quaternion may be found by changing
the modulus to its reciprocal, and then either changing the
amplitude to its negative, or else the direction to its opposite;
this latter change (of direction rather than amplitude), agreeing
better than the former with the construction in fig.~1.
Accordingly, in that figure or in this, in which ${\sc r}$
represents the direction of multiplier, and may be called the
multiplier-point, ${\sc r}'$ multiplicand point, and ${\sc r}''$
product point, if we prolong ${\sc r} {\sc r}'$ and
${\sc r} {\sc r}''$ till then meet in ${\sc r}^\backprime$, the
point diametrically opposite to ${\sc r}$; then, in the triangle
${\sc r}^\backprime {\sc r}'' {\sc r}'$, the point~${\sc r}'$,
with amplitude~$\theta'$, will be equal to the product of
${\sc r}^\backprime$ as multiplier, with amplitude~$\theta$, and
${\sc r}''$ as multiplicand, with amplitude~$\theta''$, by the
theorems already established.  {\it We may therefore, return from
product to multiplicand, by multiplying by reciprocal of
multiplier.}  But it is natural to call this return
{\it division\/} [{\it submultiplication\/}].  To {\it divide\/}
[or rather to {\it submultiply\/}] is, therefore, to multiply by
the reciprocal of the proposed divisor, if this reciprocal be
determined by the rule assigned above.  These definitions and
theorems respecting division of quaternions lead us to put the
equation~(4) under the form
$$w' + i x' + j y' + k z'
   =  \ldots
   =  {w - ix - jy - kz \over w^2 + x^2 + y^2 + z^2}
         (w'' + i x'' + j y'' + k z'');
   \eqno (58)$$
and so conduct us not only to the relation
$w' = (w^2 + x^2 + y^2 + z^2)^{-1}
          (w w'' + x x'' + y y'' + z z'')$,
which we had already, but also to these others, which can
likewise be deduced easily from the equations of multiplication,
(5) and (6),
$$\left. \eqalign{
x' &= (w^2 + x^2 + y^2 + z^2)^{-1}
         ( w x'' - x w'' + z y'' - y z'' );\cr
y' &= (w^2 + x^2 + y^2 + z^2)^{-1}
         ( w y'' - y w'' + x z'' - z x'' );\cr
z' &= (w^2 + x^2 + y^2 + z^2)^{-1}
         ( w z'' - z w'' + y x'' - x y'' ).\cr}
   \right\}
   \eqno (59)$$
The modulus of the quotient is the quotient of the moduli.
$$\left. \eqalign{
{\mu'' \cos \theta'' + \mu'' \sin \theta''
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi )
   \over \mu \cos \theta + \mu \sin \theta
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi )}
   \hskip -21em \cr
   &= {\mu'' \over \mu} \cos (\theta'' - \theta)
    + {\mu'' \over \mu} \sin (\theta'' - \theta)
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi ).\cr}
   \right\}
   \eqno (60)$$

``Codirectional quaternions may be divided by each other, by
division of moduli and subtraction of amplitudes; and
diametrically opposite quaternions may be treated as
codirectional by changing an amplitude to its negative.  A
quaternion divided by itself gives unity, under the form
$1 + i0 + j0 + k0$.

``{\it Raising to any Real Power.}---The transformation (54) of
the $\nu^{\rm th}$ power of a quaternion is now seen to hold
good, if the exponent~$\nu$ be any real quantity.

``{\it Napierian Exponential.}---If
$$f(t) = 1 + {t \over 1}
       + {t^2 \over 1 \mathbin{.} 2}
       + \hbox{\&c.},
   \eqno (61)$$
then, $r$ being $= \surd (x^2 + y^2 + z^2)$, \&c.,
$$f(ix + jy + kz)
   =  \cos r + \sin r
      (     i \cos \phi
          + j \sin \phi \cos \psi
          + k \sin \phi \sin \psi );
   \eqno (62)$$
{\it the modulus of the function~$f$ of a pure imaginary is
unity.}''

The foregoing is an extract from a letter, hitherto unpublished,
which was addressed by the author to his friend, Mr.~Graves, at
the time specified in the date.  Two figures have been
suppressed, as it was thought that the reader would find no
difficulty in constructing them from the indications given.  A
fractional symbol in the formula (58) has also been suppressed,
as not entirely harmonizing, under the circumstances in which it
occurs, with a notation subsequently adopted.  And the reader is
reminded by the words ``submultiplication'' and ``submultiply,''
inserted within square brackets, that these words have since come
to be preferred by the author to the words ``division'' and
``divide,'' when it is required to mark the return from the
product to the multiplicand, in cases when the order of the
factors is not indifferent to the result: {\it division\/} being
(in the text of the present paper) defined to be, in such cases,
the return from the product to the multiplier.  With these slight
changes, it may be interesting to some readers to see how nearly
the author's present system, although it has been, since the date
of the foregoing letter, in some respects, simplified and
extended, besides being applied to a great variety of questions
in geometry and physics, agrees with the formul{\ae} and
constructions for quaternions, which were employed by the writer
in October,~1843; and were in that month exhibited by this
letter to a scientific correspondent, and also soon afterwards to
a brother of that gentleman, the Rev.\ Charles Graves, before the
Meeting of the Academy at which the first public communication on
the subject was made, and of which the date (November 13th, 1843)
is prefixed to the present series.  As that public communication
of November,~1843 was in great part oral, and as a considerable
interval has since elapsed, the author thinks it may be not
irrelevant to mention expressly here that not only were the
fundamental formul{\ae} (1) (2) (3) of the foregoing letter
exhibited to the Academy at the date so prefixed, and a general
sketch given of their relation to spherical trigonometry, but
also the theorems respecting the connexion established through
quaternions between certain spherical quadrilaterals, pentagons,
and conics, which form the subject of the forty-seventh and
forty-eighth articles of this paper, were then communicated, and
illustrated by diagrams.  Those theorems have since been printed
in the Number of the ``London, Edinburgh, and Dublin Philosophical
Magazine'' for March, 1845.  The fundamental equations between
$i$,~$j$,~$k$ received their first {\it printed\/} publication in
the Number of that Magazine for July,~1844; and other articles on
Quaternions, by the present writer, which will probably be
continued, have appeared in the Numbers of that Magazine for
October, 1844; July, August, and October, 1846; and in that for
the present month, June, 1847, in which these last sheets of the
present paper are now passing through the printers' hands.  The
articles on Symbolical Geometry, in the ``Cambridge and Dublin
Mathematical Journal,'' are also designed to have a certain
degree of connexion with this subject.

The ``first letter'' to Mr.~Graves, referred to in the one here
printed, was written on 17th of October, 1843, and has been
printed in the Supplementary Number of the same Philosophical
Magazine for December, 1844.  It contained a sketch of the
process by which the writer had succeeded in combining, through
Quaternions, his old conception of {\it sets of numbers}, derived
from the conception of {\it sets of moments of time}, with the
notion of {\it tridimensional space}.  The former conception had
been familiar to him since the year 1834, about the end of which
year, and the beginning of the following one, he tried a variety
of triplet systems, and obtained several geometrical
constructions, but was not sufficiently satisfied with any of
them to give them publicity; attaching, perhaps, too much weight
to the objection or difficulty, that in every such system of
{\it pure triplets}, the product was found liable to vanish,
while the factors were still different from zero.  It should be
here observed that the {\it triplets\/} described in the author's
two letters of October, 1843 are really {\it imperfect
quaternions\/}; they are, therefore, strictly speaking, {\it not
proper triplets}, such as he had once sought for (and in some
degree found); and they cannot be regarded as having at all
anticipated the independent discoveries since made by Professor
de Morgan, nor those made subsequently by John T. Graves,
Esq.\ and the Rev.\ Charles Graves, in 1844, respecting certain
remarkable systems of {\it such pure and proper Triplets, with
products of a triplet form}, connected with imaginary cube roots
of negative or positive unity.

The writer hopes that a very interesting theory of {\it octaves},
including an extension of Euler's theorem respecting products of
sums of squares from four to eight, which was communicated to him
as an extension of his quaternions, about the end of 1843 and
beginning of 1844, in letters from his friend, Mr.~John Graves,
will yet be published by that gentleman, who has also contributed
to the ``Philosophical Magazine'' for April, 1845, a remarkable
paper on Couples.  Some valuable papers on Couples, Quaternions,
and Octaves, have also been communicated to the same magazine,
since the commencement of 1845, by Arthur Cayley, Esq.,
especially an application of quaternions (which appeared in the
February of that year) to the representation of the rotation of a
solid body.  That important application of the author's
principles had indeed occurred to himself previously; but he was
happy to see it handled by one so well versed as Mr.~Cayley is in
the theory of such rotation, and possessing such entire command
of the resources of algebra and of geometry.  Any further remarks
which the writer has to offer on the nature and history of this
whole train of inquiry, must be reserved to accompany the account
of his Second Series of Researches respecting Quaternions.

\bye