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\centerline{\Largebf ON QUATERNIONS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   3 (1847), pp.\ 1--16.)}

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

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\centerline{\largerm 1999}

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\noindent
\centerline{{\largeit On Quaternions.  By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

\bigskip

\centerline{Read November~11, 1844.}

\bigskip

\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), pp.\ 1--16.]}

\bigskip

In the theory which Sir William Hamilton submitted to the Academy in
November, 1843, the name {\it quaternion\/} was employed to denote a
certain quadrinomial expression, of which one term was called (by
analogy to the language of ordinary algebra) the {\it real part},
while the three other terms made up together a trinomial, which
(by the same analogy) was called the {\it imaginary part\/} of the
quaternion: the square of the former part (or term) being always a
positive, but the square of the latter part (or trinomial) being
always a negative quantity.  More particularly, this imaginary
trinomial was of the form $ix + jy + kz$, in which $x$, $y$, $z$ were
three real and independent coefficients, or {\it constituents\/}, and
were, in several applications of the theory, constructed or
represented by three rectangular coordinates; while $i$, $j$, $k$ were
certain {\it imaginary units}, or symbols, subject to the following
{\it laws of combination\/} as regards their {\it squares and
products},
$$i^2 = j^2 = k^2 = -1,  \eqno {\rm (A)}$$
$$ij = k,\quad jk = i,\quad ki = j, \eqno {\rm (B)}$$
$$ji = -k,\quad kj = -i,\quad ik = -j, \eqno {\rm (C)}$$
but were entirely {\it free from any linear relation\/} among
themselves; in such a manner, that to establish an equation between two
such imaginary trinomials was to equate {\it each\/} of the three
constituents, $xyz$, of the one to the corresponding constituent of
the other; and to equate two quaternions was (in general) to establish
{\sc four} separate and distinct equations between real quantities.
{\it Operations\/} on such quaternions were performed, as far as
possible, according to the analogies of ordinary algebra; the
{\it distributive\/} property of multiplication, and another,
which may be called the {\it associative\/} property of that
operation, being, for example, retained: with one important departure,
however, from the received rules of calculation, arising from the
abandonment of the {\it commutative\/} property of multiplication, as
{\it not\/} in general holding good for the {\it mixture\/} of the new
imaginaries; since the product $ji$ (for example) has, by its
definition, a different sign from $ij$.  And several constructions and
conclusions, especially as respected the geometry of the sphere, were
drawn from these principles, of which some have since been printed
among the Proceedings of the Academy for the date already referred to.

The author has not seen cause, in his subsequent reflections on the
subject, to abandon any of the principles which have been thus briefly
recapitulated; but he conceives that he has been enabled to present
some of them in a clearer view, as regards their bearings on
geometrical questions; and also to improve the algebraical method of
applying them, or what may be called the {\sc calculus of quaternions}.

Thus he has found it useful, in many applications, to dismiss the
{\it separate} consideration of the three real constituents, $x$, $y$,
$z$, of the imaginary trinomial $ix + jy + kz$, and to denote that
trinomial by some {\it single\/} letter (taken often from the Greek
alphabet).  And on account of the facility with which this so called
{\it imaginary\/} expression, or square root of a negative quantity,
is constructed by a {\it right line having direction in space}, and
having $x$, $y$, $z$ for its three rectangular components, or
projections on three rectangular axes, he has been induced to call the
trinomial expression itself, as well as the line which it represents,
a {\sc vector}.  A {\it quaternion\/} may thus be said to consist
generally of a {\it real\/} part and a {\it vector}.  The fixing a
special attention on this part, or element, of a quaternion, by giving
it a special name, and denoting it in many calculations by a single
and special sign, appears to the author to have been an improvement in
his method of dealing with the subject: although the general notion of
treating the constituents of the imaginary part as coordinates had
occurred to him in his first researches.

Regarded from a geometrical point of view, this algebraically
imaginary part of a quaternion has thus so natural and simple a
signification or representation in space, that the difficulty is
transferred to the algebraically real part; and we are tempted to ask
what this last can denote in geometry, or what in space might have
suggested it.

By the fundamental equations of definition for the squares and
products of the symbols $i$, $j$, $k$, it is easy to see that any
(so-called) real and positive quantity is to any vector whatever, as
that vector is to a certain real and negative quantity; this being
indeed only another mode of saying that, in this theory, {\it every
vector has a negative square}.  Again, the product of any two
rectangular vectors is a third vector at right angles to both the
factors (but having one or other of two opposite directions, according
to the order in which those factors are taken); a relation which may
be expressed by saying, that the fourth proportional to the real unit
and to any two rectangular vectors is a third vector rectangular to
both; or, conversely, that the {\it fourth proportional to any three
rectangular vectors\/} is a quantity {\it distinct from every vector},
and of the kind called {\it real\/} in this theory, as contrasted with
the kind called {\it imaginary}.

Now, in fact, what originally led the author of the present communication
to conceive (in 1843) his theory of quaternions (though he had, at a
date earlier by several years, speculated on {\it triplets\/} and
{\it sets}\footnote*{See Transactions of the Royal Irish Academy,
vol.~xvii. p.~422.  Dublin, 1835.}
of numbers, as an extension of the theory of {\it couples}, or of the
ordinary imaginaries of algebra, and also as an additional
illustration of his view respecting the Science of Pure Time), was a
desire to form to himself a distinct conception, and to find a
manageable algebraical expression, of a {\it fourth proportional to
three rectangular lines}, when the {\sc directions} of those lines
were taken into account; as Mr.\ Warren\footnote\dag{Treatise on the
Geometrical Representations of the Square Roots of Negative
Quantities, by the Rev. John Warren.  Cambridge, 1828.}
and Mr.\ Peacock\footnote\ddag{Treatise on Algebra, by the Rev.
George Peacock.  Cambridge, 1830.}
had shewn how to conceive and express the fourth proportional to any
three lines having direction, but situated {\it in one common plane}.
And it has since appeared to Sir William Hamilton that the subject of
quaternions may be illustrated by considering more closely, though
briefly, this question of the determination of a fourth proportional
to three rectangular directions in space, rather in a geometrical than
in an algebraical point of view.

Adopting the known results above referred to, for proportions between
lines having direction in a single plane (though varying a little the
known manner of speaking on the subject), it may be said that, in the
horizontal plane, ``West is to South as South is to East,'' and
generally as any direction is to one less advanced than itself in
azimuth by ninety degrees.  Let it be now assumed, as an extension of
this view, that in {\it some analogous sense\/} there exists a fourth
proportional to the three rectangular directions, West, South, and Up;
and let this be called, provisionally, {\it Forward}, by contrast to
the opposite direction, {\it Backward}, which must be assumed to be
(in the same general sense) a fourth proportional to the directions of
West, South and Down.  We shall then have, inversely, Forward to Up as
South to West, and therefore, as West to North: if we admit, as it
seems natural and almost necessary to do, that (for directions, as for
lengths) the inverses of equal ratios are equal; and that ratios equal
to the same ratio are equal to each other.  But again, Up is to South
as South to Down, and also as North to Up: and we can scarcely avoid
admitting, or defining, that (in the present comparison of directions)
ratios similarly compounded of equal ratios are to be considered as
being themselves equal ratios.  Compounding, therefore, on the one
hand, the ratios of Forward to Up, and of Up to South; and on the
other hand the respectively equal (or similar) ratios of West to
North, and of North to Up, we are conducted to admit that Forward is
to South as West to Up.  By a reasoning exactly similar, we find that
Forward is to West as Up to South; and generally that if $X$, $Y$, $Z$
denote any three rectangular directions such that $A:X::Y:Z$, $A$ here
denoting what we have expressed by the word Forward, then also
$A:Y::Z:X$ (and of course, for the same reason $A:Z::X:Y$); so that
the {\it three\/} directions $X\,Y\,Z$ may be {\it all changed
together\/} by advancing them in a {\it ternary cycle}, according to
the formula just written, without disturbing the proportionality
assumed.  But also, by the principle respecting proportions of
directions in one plane, we may cause {\it any two\/} of the three
rectangular directions $XYZ$ to revolve together round the third, as
round an axis, without altering their ratio to each other.  And by
combining these two principles, it is not difficult to see that
{\it because\/} Forward has been supposed to be to Up as South to
West, {\it therefore\/} the same (as yet unknown) direction
``Forward'' must be supposed to be {\it to any direction $X$
whatever}, as {\it any\/} direction $Y$, perpendicular to $X$,
is to {\it that third\/} direction~$Z$ which is perpendicular to
both $X$ and $Y$, and which is obtained from $Y$ by a right-handed
(and not by a left-handed) rotation, through a right angle, round $X$;
in the same manner as (and because) the direction West was so chosen
as to be to the right of South, with reference to Up as an axis of
rotation.  Conversely we must suppose that if {\it any three
rectangular directions}, $X\,Y\,Z$, be arranged, as to order of
rotation, in the manner just now stated, then $Z:Y::X:A$; or in other
words, we must admit, if we reason in this way at all, that the
direction called already Forward, will be the fourth proportional to
$Z\,Y\,X$.  And if we vary the order, so as to have $Z$ to the left,
and not to the right of $Y$, with reference to $X$, then will the
fourth proportional to $Z\,Y\,X$ become the direction which we have
lately called Backward, as being the opposite to that named Forward.

Again, since Forward is to Up as South to West, that is in a ratio
compounded of the ratios of South to East and of East to West, or in
one compounded of the ratios of West to South, and of any direction to
its own opposite; or, finally, in a ratio compounded of the ratios of
Up to Forward and of Forward to Backward, that is, in the ratio of Up
to Backward, we see that the third proportional to the directions
Forward and Up is in the direction Backward: and by an exactly
similar reasoning, with the help of the conclusions recently obtained,
we see that if $X$ be {\sc any} {\it direction in tridimensional
space}, then $A:X::X:B$; $B$ here denoting, for shortness, the
direction which has been above called Backward.

The geometrical study of the relations between directions in space,
combined with a few very simple and guiding principles respecting the
composition of relations generally, might therefore have led to the
conception or assumption of a certain {\it pair of contrasted
directions}, namely, those which we have called Forward and Backward,
and denoted by the letters $A$ and $B$.  And these are such that if we
conceive a quantitative element to be combined with each, and give the
name of {\sc positive unity} to the unit of magnitude measured in the
direction of {\it Forward}, but that of {\it Negative Unity\/} to the
same magnitude measured backward; and if we extend to {\it this\/}
positive unity and to lines having direction in space the received
definitions of multiplication, that ``Positive Unity is to Multiplier
as Multiplicand is to Product,'' and that ``the product of two equal
factors is the square of either;'' we may then consistently and
naturally be led to assert the same results as those already
enunciated from the theory of quaternions respecting the product of
two vectors, in the two principal cases, first, where those two
vectors are rectangular, and second, where they are coincident with
each other.  And thus may we justify, or at least interpret and
explain, the fundamental definitions (A) (B) (C) of this theory, by
regarding the symbols $i\,j\,k$ as denoting three vector-units having
three rectangular directions in space.

But, farther, we derive from this view of the whole subject an
illustration (if not a confirmation) of the remarkable conclusion that
the so-called {\it real and positive unit\/} $+1$ is not (in this
theory) to be confounded with {\it any\/} vector unit whatever, but is
to be regarded as of a kind essentially distinct from every vector.
For this positive unit $+1$ is in the direction above called Forward,
and denoted by $A$.  Now if this could coincide with a direction $X$
in tridimensional space, then, whatever this latter direction might be
supposed to be, we could always, by the general formula $A:X::Y:Z$
(where $X$ is arbitrary), deduce the inadmissible proportion
$X:X::Y:Z$, in which the two directions in one ratio are identical,
but those in the other are rectangular to each other.  If then we
resolve to retain the assumption of the existence of a fourth
proportional $A$ to three rectangular directions in space, as subject
to be reasoned on at all in the way already described, and as
determined in direction by its contrast to its own opposite $B$
(corresponding to an opposite order of rotation in the system
$X\,Y\,Z$), we must think of these two opposite directions $A$ and $B$
as merely {\it laid down upon a scale}, but must abstain from
attributing to this {\sc scale} any one direction rather than another
in tridimensional space, as having such or such a zenith distance, or
such or such an azimuth, rather than such or such another.  And the
progression {\it on this scale\/} from negative to positive infinity,
obtained by combining a quantitative element with the contrast between
two opposite directions, corresponds less to the conception of {\it
space\/} itself (though we have seen that considerations of space
might have suggested it) than to the conception of {\it time}; the
variety which it admits is not {\it tri-\/} by {\it uni-\/}
dimensional; and it would, in the language of some philosophical
systems, be said to appertain rather to the notion of {\it
intensive\/} than of {\it extensive\/} magnitude.  Though answering
precisely to the progression of the quantities called {\it real\/} in
algebra, it has, when viewed from the geometrical side, somewhat the
same sort of {\it imaginariness}, and yet (it is believed) of {\it
utility}, as compared with lines in space, which the square root of an
ordinary negative has, when compared with positive and negative
quantities.  This analogy becomes still more complete when we observe
that (in this theory) the fourth proportional to any direction~$X$ in
space, and either of the two directions $A$ or $B$ upon the scale, is
the direction opposite to $X$; so that, {\it if a vector-unit\/} in
any determined direction $X$ {\it had been taken for positive unity},
then each of the two {\it scalar\/} units in the directions $A$ and
$B$ (in common, it is true, with every vector-unit perpendicular to
$X$) {\it might have been called}, by the general nomenclature of
multiplication, {\it a square root of negative one}.

It is, however, a peculiarity of the calculus of quaternions, at least
as lately modified by the author, and one which seems to him
important, that it select {\it no one direction in space as eminent\/}
above another, but treats them as all equally related to that
{\it extra-spatial}, or simply {\sc scalar} direction, which has been
recently called ``Forward.''  In this respect it differs in its
processes from the Cartesian method of coordinates, and seems often to
admit of being more simply and directly applied to the treatment of
geometrical problems, because it requires no previous selection of
axes, rectangular or other.  The author is, indeed, aware that the
cooperation of other and better analysts will be necessary in order to
bring the method of quaternions to anything approaching to perfection.
But he hopes that an instance or two of the facility with which
{\it some\/} questions at least allow themselves to be treated by this
method, even in its present state, may not be without interest to the
Academy.  And he conceives that two examples in particular, one
relating to the composition of translations, and the other to the
composition of rotations in space, may usefully be selected for
statement on the present occasion.

As preliminary illustrations of the operations employed, it may be
remarked that for any system of lines having direction in space, it is
required by many analogies (and is, for lines in one plane included
among the definitions or results of the theories of Mr.\ Warren and
Mr.\ Peacock), that the {\it sum\/} should be regarded as being equal
to that one line which constructs or represents the {\it total
effect\/} of all the different rectilinear motions which are expressed
by the different summands.  {\it Vectors\/} are therefore to be
{\it added\/} to each other by a certain geometrical composition,
exactly analogous to the composition of motions, or of forces, and
following the same known rules.  {\it Scalars}, on the other hand
(that is to say, the so-called real parts of any proposed
quaternions), admitting only of a progression in quantity, and of a
change of sign, without any other changes of direction, are to be
added among themselves by the known rules of algebra, for the addition
of positive and negative numbers.  The addition of a scalar and a
vector to each other can be no otherwise performed, or rather
indicated, than by writing their symbols with the $+$ sign
interposed; each being, as we have seen, in some sense, imaginary
with respect to the other.  These operations of addition are all of
the commutative, and also of the associative kind; that is to say, the
order of all the summands may be changed, and any group of them may be
collected or associated into one partial sum.

{\it Scalars\/} are {\it multiplied}, as well as added, by the known
rules of ordinary algebra, for the multiplication of real numbers,
positive or negative; because the positive unity of the system has
been assumed to be itself a scalar, and not a vector unit.

For the same reason, {\it to multiply any vector by any scalar\/}~$a$,
is in general to change its length in a known ratio, and to preserve
or reverse its direction, according as $a$ is $>$ or $< 0$; the
product is therefore a new vector, which may be denoted by $a \alpha$.
The same new vector is obtained, under the form $\alpha a$, when we
multiply the scalar~$a$ by the vector $\alpha$.  If $a + \alpha$ and
$b + \beta$ be two quaternion factors, of which $a$ and $b$ are the
scalar parts, and $\alpha$, $\beta$ the vectors, then with a view to
preserving the distributive character of multiplication, it is natural
to define that the product may be distributed into the four following
parts:
$$(a + \alpha)(b + \beta) = ab + a\beta + \alpha b + \alpha \beta.$$
And if the multiplicand vector $\beta$ be decomposed into two parts,
or summands, one $= \beta_1$ and in the direction of the multiplier
$\alpha$, or in the direction exactly opposite thereto, and the other
$= \beta_2$, and in a direction perpendicular to the former (so that
$\beta_1$ and $\beta_2$ are the projections of $\beta$ on $\alpha$
itself, and on the plane perpendicular to $\alpha$), then it may be
farther defined that the {\it multiplication of any one
vector\/}~$\beta$ {\it by any other vector\/}~$\alpha$ may be
accomplished by the formula
$$\alpha \beta = \alpha (\beta_1 + \beta_2)
   = \alpha \beta_1 + \alpha \beta_2;$$
in which, by what has been shewn, the partial product
$\alpha \beta_1$ is to be considered as equal to a scalar, namely, the
product of the lengths of $\alpha$ and $\beta_1$, taken with the sign
$-$ or $+$, according as the direction of $\beta_1$ coincides with, or
is opposite to that of $\alpha$; while the other partial product
$\alpha \beta_2$ is a vector, of which the length is the product of
the lengths of $\alpha$ and $\beta_2$, while its direction is
perpendicular to both of their's, being obtained from that of
$\beta_2$, by making it revolve right-handedly through a right angle
round $\alpha$ as an axis.  These definitions, which are compatible
with the formul{\ae} (A), (B), (C), and may serve to replace them,
will be found sufficient to prove generally, and perhaps with somewhat
greater geometrical clearness than those formul{\ae}, the distributive
and associative properties of quaternion multiplication, which have
already been stated to exist.  They give easily the following
corollaries, which are of very frequent use in this calculus:
$$\alpha \beta + \beta \alpha = 2 \alpha \beta_1
   = 2 A B \cos (A,B);
   \eqno {\rm (a)}$$
$$\alpha \beta - \beta \alpha = 2 \alpha \beta_2
   = 2 \gamma A B \sin (A,B);
   \eqno {\rm (b)}$$
$A$ and $B$ denoting here the {\it lengths\/} of the lines $\alpha$
and $\beta$, and $(A,B)$ the {\it angle\/} between them; while
$\gamma$ is a vector-unit perpendicular to their plane, and such that
a right-handed rotation, equal to the angle $(A,B)$, performed round
$\gamma$, would bring the direction of $\alpha$ to coincide with that
of $\beta$.  For example, when $\beta = \alpha$, then $B = A$,
$(A,B) = 0$, and
$$\alpha \beta = \beta \alpha = \alpha^2 = - A^2,$$
so that the length~$A$ of any vector~$\alpha$, in this theory, may be
expressed under the form
$$A = \sqrt{-\alpha^2}.
   \eqno {\rm (c)}$$
More generally we have the equation
$$\alpha \beta - \beta \alpha = 0,
   \eqno {\rm (d)}$$
when the lines $\alpha$ and $\beta$ are coincident or opposite in
direction; while, on the contrary, the condition for their being at
right angles to each other is expressed by the formula
$$\alpha \beta + \beta \alpha = 0.
   \eqno {\rm (e)}$$

These simple principles suffice to give, in a new way, algebraical
solutions of many geometrical problems, of various degrees of
difficulty and importance.  Thus, if it be required, as an easy
instance, to determine the length of the resultant of several
successive rectilinear motions, or the magnitude of the statical sum
of several forces acting together at one point, as a function of the
amounts of those successive motions, or of those component forces, and
of their inclinations to each other, we have only to denote the
components by the vectors $\alpha_1, \alpha_2,\ldots \alpha_n$, and
their sum by $\alpha$, the corresponding magnitudes being
$A_1, A_2,\ldots A_n$, and $A$; and the equation
$$\alpha = \alpha_1 + \alpha_2 + \cdots + \alpha_n$$
will give, by being squared,
$$\eqalign{\alpha_2
   &= \alpha_1^2 + \alpha_2^2 + \cdots + \alpha_n^2 \cr
   &\quad + \alpha_1 \alpha_2 + \alpha_2 \alpha_1 + \cdots
          + \alpha_1 \alpha_n + \alpha_n \alpha_1 + \cdots;\cr}$$
that is, by the foregoing principles (after changing all the signs),
$$\eqalign{A^2
   &= A_1^2 + A_2^2 + \cdots + A_n^2 \cr
   &\quad + 2 A_1 A_2 \cos (A_1, A_2) + \cdots
          + 2 A_1 A_n \cos (A_1, A_n) + \cdots;\cr}$$
a known result, it is true, but one which can scarcely be derived in
any other way by so very short a process of {\it calculation}.  For it
is not {\it quite\/} so easy, on the {\it algebraical\/} side of the
question, to see that
$$({\textstyle \sum} x)^2
      + ({\textstyle \sum} y)^2
      + ({\textstyle \sum} z)^2
   = {\textstyle\sum} (x^2 + y^2 + z^2)
      + 2 {\textstyle \sum} (xx' + yy' + zz'),$$
however easy this may be, as it is to see that
$$({\textstyle \sum} \alpha)^2
   = {\textstyle \sum} (\alpha^2)
      + {\textstyle \sum} (\alpha \alpha' + \alpha' \alpha):
   \eqno {\rm (f)}$$
although the {\it geometrical interpretation\/} of the first of these
two formul{\ae} is of course more obvious than that of the latter, to
those who are familiar with the method of coordinates, and not with
the method of quaternions.

Again, let us consider the more difficult problem of the composition
of any number os successive rotations of a body, or, at first, of any
one line thereof, round several successive axes, through any angles,
small or large.  Let the axis of the first of these rotations have the
direction of the vector-unit $\alpha$, ($\alpha^2 = -1$), and let the
amount of the positive rotation round this axis be denoted by $a$,
which letter here represents still a scalar or real number.  Let
$\beta$ be the revolving line, considered in its original position;
$\beta'$ the same line, after it has revolved through the angle~$a$
round the axis~$\alpha$.  The part, or component, of $\beta$, which is
in the direction of this axis, is that which was denoted lately by
$\beta_1$; and the formula (a), when multiplied by
$-{1 \over 2} \alpha$, gives, as an expression for this part,
$$\beta_1 = {\textstyle{1 \over 2}} (\beta - \alpha \beta \alpha),
   \eqno {\rm (g)}$$
because it has been supposed that $\alpha^2 = -1$.  This part of
$\beta$ remains unaltered by the rotation.  The other part, or
component of $\beta$, is, in like manner, by (b),
$$\beta_2 = {\textstyle{1 \over 2}} (\beta + \alpha \beta \alpha);
   \eqno {\rm (h)}$$
and this part is to be multiplied by $\cos a$, in order to find the
part of $\beta'$, which is perpendicular to $\alpha$, but in the plane
of $\alpha$ and $\beta$.  Again, multiplying by $\alpha$, we cause
$\beta_2$ to turn through a right angle in the positive direction
round~$\alpha$, and obtain, for the result of {\it this\/} rotation,
$$\alpha \beta_2
   = {\textstyle{1 \over 2}} (\alpha \beta - \beta \alpha);$$
an expression which is the half of that marked (b), and which is to be
multiplied by $\sin a$, in order to arrive at the remaining part of
the sought line $\beta'$, namely, the part which is perpendicular to
the plane of $\alpha$ and $\beta$.  Collecting, therefore, the three
parts, or terms, which have been thus separately obtained, we find,
$$\eqalign{\beta'
   &= \beta_1 + (\cos a + \alpha \sin a) \beta_2 \cr
   &= {\textstyle{1 \over 2}} (\beta - \alpha \beta \alpha)
      + {\textstyle{1 \over 2}} \cos a (\beta + \alpha \beta \alpha)
      + {\textstyle{1 \over 2}} \sin a (\alpha \beta - \beta \alpha) \cr
   &= \left( \cos {a \over 2} \right)^2. \beta
      - \left( \sin {a \over 2} \right)^2. \alpha\beta\alpha
      + \cos {a \over 2} \sin {a \over 2}.(\alpha \beta - \beta \alpha);\cr}$$
that is,
$$\beta' = \left( \cos {a \over 2} + \alpha \sin {a \over 2} \right)
           \beta
           \left( \cos {a \over 2} - \alpha \sin {a \over 2} \right);
   \eqno {\rm (i)\hbox{*}}$$
the\footnote\null{*[{\it Note added during
printing.\/}]---The printing of this abstract having been delayed, the
Author desires to be permitted to append the following remarks:

If we should make, for abridgment
$$\alpha \tan {a \over 2} = - \gamma,$$
the formula (i) for any single rotation might be thus written,
$$\beta' = (1 + \gamma)^{-1} \beta (1 + \gamma).
   \eqno {\rm (i')}$$
And if we then made
$$\beta = ix + jy + kz,\quad
  \beta' = ix' + jy' + kz',\quad
  \gamma = i\lambda + j\mu + k\nu,$$
$i$, $j$, $k$, being the same three rectangular vectors, or imaginary
units, as in the formul{\ae} (A) (B) (C), but $x$, $y$, $z$, $x'$,
$y'$, $z'$, $\lambda$, $\mu$, $\nu$, being nine real or scalar
quantities, we should obtain the same general formula for the
transformation of rectangular coordinates (with the same geometrical
meanings of the coefficients $\lambda$, $\mu$, $\nu$,) as that which
Mr.\ Cayley has deduced, with a similar view, but by a different
process, and has published, with other ``Results respecting
Quaternions,'' in the Philosophical Magazine for February, 1845.

The present writer desires to return his sincere acknowledgments to
Mr.\ Cayley for the attention which he has given to the Papers on
Quaternions, published in the above-mentioned Magazine: and gladly
recognizes his priority, as respects the printing of the formula just
now referred to.  But while he conceives it to be very likely that
Mr.\ Cayley, who had previously published in the Cambridge
Mathematical Journal some elegant researches on the rotation of
bodies, may have perceived, not only independently, but at an earlier
date than he did himself, the manner of applying quaternions to
represent such a rotation; he yet hopes that he may be allowed to
mention, that a formula differing only slightly in its notation from
the formula (i) of the present abstract, with the corollaries there
drawn respecting the composition of successive finite rotations, had
been exhibited to his friend and brother Professor, the Rev.\ Charles
Graves, of Trinity College, Dublin, in an early part of the month
(October, 1844), which preceded that communication to the Academy, of
which an account is given above.}
operations here indicated being thus sure to make no change in the
part $\beta_1$, which is in the direction of the axis of rotation, but
to cause the other part $\beta_2$ to revolve round that axis~$\alpha$
through an angle $= a$.  Again, let the same line $\beta'$ revolve
round a new axis of rotation denoted by a new vector unit $\alpha'$,
through a new angle $a'$, into a new position $\beta''$; we shall have,
in like manner,
$$\beta'' = \left( \cos {a' \over 2} + \alpha' \sin {a' \over 2} \right)
           \beta'
           \left( \cos {a' \over 2} - \alpha' \sin {a' \over 2} \right);
   \eqno {\rm (j)}$$
and so on, for any number~$n$ of rotations.  Let the last position of
$\beta$ be denoted by $\beta_n$; and since it can easily be proved, by
the theory of multiplication of quaternions, that the continued
products which present themselves admit of being thus transformed:
$$\left.  \eqalign{
\biggl( \cos {a^{(n-1)} \over 2}
      + \alpha^{(n-1)} \sin {a^{(n-1)} \over 2} \biggr) \cdots
   \biggl( \cos {a' \over 2} + \alpha' \sin {a' \over 2} \biggr)
      \hskip-72pt\cr
   \biggl( \cos {a \over 2} + \alpha \sin {a \over 2} \biggr)
   &= \cos {a_n \over 2} + \alpha_n \sin {a_n \over 2};\cr
      \noalign{\vskip 3pt}
   \biggl( \cos {a \over 2} - \alpha \sin {a \over 2} \biggr)
   \biggl( \cos {a' \over 2} - \alpha' \sin {a' \over 2} \biggr) \cdots \cr
\biggl( \cos {a^{(n-1)} \over 2}
      - \alpha^{(n-1)} \sin {a^{(n-1)} \over 2} \biggr)
   &= \cos {a_n \over 2} - \alpha_n \sin {a_n \over 2};\cr}
      \right\} \eqno {\rm (k)}$$
in which $\alpha_n$ is a new vector unit, and $a_n$ a new real angle,
we find that the result of all the $n$ rotations is of the form
$$\beta_n = \left( \cos {a_n \over 2} + \alpha_n \sin {a_n \over 2} \right)
           \beta
           \left( \cos {a_n \over 2} - \alpha_n \sin {a_n \over 2} \right).
   \eqno {\rm (l)}$$
It conducts, therefore to the same final position which would have
been attained from the initial position~$\beta$, by a {\it single
rotation\/} $= a_n$, round the {\it single axis\/}~$\alpha_n$; the
amount and axis of this resultant rotation being determined by either
of the two equations of transformation (k), and being independent of
the direction of the line~$\beta$ which was operated on, so that they
are the same for all lines of the body.

If the present results be combined with the theorem marked (R), in the
account, printed in the Proceedings of the Academy, of the remarks
made by the Author in November, 1843, it will at once be seen that if
the several axes of rotation be considered as terminating in the
points of a spherical polygon, and if the angles of rotation be equal
respectively to the doubles of the angles of this polygon (and be
taken with proper signs or directions, determined by those angles),
then the total effects of all these rotations will vanish; or, in
other words, the body will at last be brought back to the position
from which it set out.

Finally, it may be mentioned that the author is in possession of a
general method for expressing by quaternions the tangent planes and
normals to curved surfaces, and that in applying this method to find
the cone of tangents enveloping a given sphere, and drawn from a given
point, the geometrical impossibility of the problem, when the point is
an internal one, is expressed by the square of a vector becoming in
this case positive.

\bye