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

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\centerline{\Largebf ON QUATERNIONS AND THE ROTATION}

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\centerline{\Largebf OF A SOLID BODY}

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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,
   4 (1850), pp.\ 38--56.)}

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

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\centerline{\largerm 2000}

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\centerline{{\largeit On Quaternions and the Rotation of a Solid Body.}}

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

\bigskip

\centerline{Communicated January~10, 1848.}

\bigskip

\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~4 (1850), pp.\ 38--56.]}

\bigskip

Sir William Rowan Hamilton gave an account of some applications
of Quaternions to questions connected with the Rotation of a
Solid Body.

I.
It was shown to the Academy in 1845, among other applications of
the Calculus of Quaternions to the fundamental problems of
Mechanics, that the composition of statical couples, of the kind
considered by Poinsot, as well as that of ordinary forces, admits
of being expressed with great facility and simplicity by the
general methods of this Calculus.  Thus, the general conditions
of the equilibrium of a rigid system are included in the following
formula, which will be found numbered as equation (20) of the
abstract of the Author's communication of December~8, 1845, in
the Proceedings of the Academy for that date:
$${\textstyle\sum} \mathbin{.} \alpha \beta = -c.
   \eqno (1)$$

In the formula thus cited, $\alpha$ is the {\it vector of
application\/} of a force denoted by the other vector~$\beta$;
and the scalar sysmbol, $-c$, which is equated to the sum
$\alpha \beta + \alpha' \beta' + \cdots$ of all the quaternion
products $\alpha \beta, \alpha' \beta',\ldots$ of all such pairs
of vectors, or directed lines $\alpha$ and $\beta$, is, in the
case of equilibrium, independent of the position of the point
from which all the vectors $\alpha, \alpha',\ldots$ are drawn, as
from a common origin, to the points of application of the various
forces, $\beta, \beta',\ldots$.  This requires that  the two
following conditions should be separately satisfied:
$${\textstyle\sum} \beta = 0;\quad
  {\textstyle\sum} V \mathbin{.} \alpha \beta = 0;
   \eqno (2)$$
which accordingly coincide with the two equations marked (18) of
the abstract just referred to.  The former of these two
equations, $\sum \beta = 0$, expresses that the applied {\it
forces\/} would balance each other, if they were all transported,
without any changes in their intensities or directions, so as to
act at any one common point, such as the origin of the
vectors~$\alpha$; and the latter equation,
$\sum V \mathbin{.} \alpha \beta = 0$, expresses that all the
{\it couples\/} arising from such transport of the forces, or
from the introduction of a system of new and opposite forces,
$- \beta$, all acting at the same common origin, would also
balance each other: the {\it axis\/} of any one such couple being
denoted, in magnitude and in direction, by a symbol of the form
$V \mathbin{.} \alpha \beta$.  When either of these two
vector-sums, $\sum \beta$, $\sum V \mathbin{.} \alpha \beta$, is
different from zero, the system cannot be in equilibrium, at
least if there be no fixed point nor axis; and in this case, the
quaternion quotient which is obtained, by dividing the latter of
these two vector-sums by the former, has a remarkable and simple
signification.  For, if this division be effected by the general
rules of this calculus, in such a manner as to give a quotient
expressed under the original and standard form of a
{\it quaternion}, as assigned by Sir William R. Hamilton in his
communication of the 13th of November, 1843; that is to say, if
the quotient of the two vectors lately mentioned be reduced by
those general rules to the fundamental {\it quadrinomial form},
$${\sum V \mathbin{.} \alpha \beta \over \sum \beta}
   = w + ix + jy + kz,
   \eqno (3)$$
where $i$,~$j$,~$k$ are the Author's {\it three co-ordinate
imaginaries}, or {\it rectangular vector-units}, namely, symbols
satisfying the equations,
$$i^2 = j^2 = k^2 = ijk = - 1,
   \eqno (4)$$
which have already been often adduced and exemplified by him, in
connexion with other geometrical and physical researches; then
the {\it four constituent numbers}, $w$, $x$, $y$, $z$, of this
quaternion (3), will have, in the present question, the meanings
which we are about to state.  The algebraically real or {\it
scalar\/} part of the quaternion (3), namely, the number
$$w = S ( {\textstyle \sum} V \mathbin{.} \alpha \beta
            \div {\textstyle \sum} \beta ),
   \eqno (5)$$
which is independent of the imaginary or symbolic coefficients
$i,j,k$, will denote the (real) quotient which might be otherwise
obtained by {\it dividing the moment of the principal resultant
couple by the intensity of the resultant force\/}; with the known
direction of which force the axis of this {\it principal\/} (and
known) couple coincides, being the line which is known by the
name of the {\it central axis\/} of the system.  And the three
other numerical constituents of the same quaternion (3), namely,
the three real numbers $x,y,z$, which are multiplied respectively
by those symbolic coefficients $i,j,k$, in the algebraically
imaginary or {\it vector\/} part of that quaternion, namely, in
the part
$$ix + jy + kz
   =  V ( {\textstyle \sum} V \mathbin{.} \alpha \beta
            \div {\textstyle \sum} \beta ),
   \eqno (6)$$
are the three real and rectangular {\it co-ordinates of the foot
of the perpendicular let fall from the assumed origin\/} (of
vectors or of co-ordinates) {\it on the central axis of the
system}.  These co-ordinates vanish, if the origin be taken on
that axis; and then the direction of the resultant force
coincides with that of the axis of the resultant couple: a
coincidence of which the condition may accordingly be expressed,
in the notation of this Calculus, by the formula
$$0 = V ( {\textstyle \sum} V \mathbin{.} \alpha \beta
            \div {\textstyle \sum} \beta );
   \eqno (7)$$
whereas the second member of this formula (7) is in general a
vector-symbol, which denotes, in length and in direction, the
perpendicular let fall as above.  In the case where it is
possible to reduce the system of forces to a single resultant
force, unaccompanied by any couple, the scalar part of the same
quaternion (3) vanishes; so that we may write for this case the
equation,
$$0 = S ( {\textstyle \sum} V \mathbin{.} \alpha \beta
            \div {\textstyle \sum} \beta );
   \eqno (8)$$
which agrees with the equation (19) of the abstract of December,
1845, and in which the second member is in general a scalar
symbol, denoted lately by $w$, and having the signification
already assigned.  When the resultant force vanishes, without the
resultant couple vanishing, then the denominator or divisor
$\sum \beta$ becomes null, in the fraction or quotient (3), while
the numerator or dividend, $\sum V \mathbin{.} \alpha \beta$,
continues different from zero; and when both force and couple
vanish, we fall back on the equations (18) of the former abstract
just cited, or on those marked (2) in the present communication,
as the conditions of equilibrium of a free but rigid system.
Finally, the scalar symbol
$$c = - {\textstyle \sum} S \mathbin{.} \alpha \beta,
   \eqno (9)$$
which enters with its sign changed into the second member of the
formula (1), and which, when the resultant $\sum \beta$ of the
forces~$\beta$ vanishes, receives a value independent of the
assumed origin of the vectors~$\alpha$, has also a simple
signification; for (according to a remark which was made on a
former occasion), there appears to be a propriety in regarding
this scalar symbol~$c$, or the negative of the sum of the scalar
parts of all the quaternion products of the form $\alpha \beta$,
as an expression which denotes the total {\it tension\/} of the
system.  In the foregoing formul{\ae} the letters $S$ and $V$are
used as characteristics of the operations of taking respectively
the {\it scalar\/} and the {\it vector}, considered as the two
{\it parts\/} of any quaternion expression; which parts may still
be sometimes called the (algebraically) {\it real\/} and
(algebraically) {\it imaginary\/} parts of that expression, but
of which {\it both\/} are {\it always}, in this theory, entirely
and easily {\it interpretable\/}: and in like manner, in the
remainder of this Abstract, the letters $T$ and $U$ shall
indicate, where they occur, the operations of taking separately
the {\it tensor\/} and the {\it versor}, regarded as the two
principal {\it factors\/} of any such quaternion.

\bigbreak

II.
To apply to problems of {\it dynamics\/} the foregoing
{\it statical\/} formul{\ae}, we have only to introduce, in
conformity with a well-known principle of mechanics, the
consideration of the equilibrium which must subsist between the
forces lost and gained.  That is, we are to substitute for the
symbol~$\beta$, in the equations (1) or (2), the expression
$$\beta = m \left( \phi - {{\rm d}^2 \alpha \over {\rm d} t^2} \right);
   \eqno (10)$$
where $m$ denotes the mass of that part or element of the system
which, at the time~$t$, has $\alpha$ for its vector of position,
and consequently
$\displaystyle {{\rm d}^2 \alpha \over {\rm d} t^2}$
for its vector of acceleration; while the new
vector-symbol~$\phi$ denotes the accelerating force, or $m \phi$
denotes the moving force applied, direction as well as intensity
being attended to.  Thus, instead of the two statical equations
(2), we have now the two following dynamical equations, for the
motion of a free but rigid system:
$$\eqalignno{
{\textstyle \sum} \mathbin{.} m {{\rm d}^2 \alpha \over {\rm d} t^2}
   &= {\textstyle \sum} \mathbin{.} m \phi;
   &(11)\cr
{\textstyle \sum} \mathbin{.} m V \mathbin{.} \alpha
         {{\rm d}^2 \alpha \over {\rm d} t^2}
   &= {\textstyle \sum} \mathbin{.} m V \mathbin{.} \alpha \phi;
   &(12)\cr}$$
of which the former contains the law of motion of the centre of
gravity, and the latter contains the law of the description of
areas.  If the rigid system have one point fixed, we may place
at this point the origin of the vectors~$\alpha$; and in this
case the equation (11) disapears from the statement of the
question, but the equation (12) still remains: while the
condition that the various points of the system are to preserve
unaltered their distances from each other, and from the fixed
point, is expressed by the formula
$${{\rm d} \alpha \over {\rm d} t} = V \mathbin{.} \iota \alpha,
   \eqno (13)$$
where the vector-symbol~$\iota$ denotes a straight line drawn in
the direction of the axis of momentary rotation, and having a
length which represents the angular velocity of the system; so
that this vector~$\iota$ is generally a function of the time~$t$,
but is always, at any one instant, the same for all the points
of the body, or of the rigid system here considered.  The
equation (12) thus gives, by an immediate integration, the
following expression for the law of areas:
$${\textstyle \sum} \mathbin{.} m \alpha V \mathbin{.} \iota \alpha
   =  \gamma + {\textstyle \sum} \mathbin{.} m V
         \int \alpha \phi \, {\rm d} t;
   \eqno (14)$$
where $\gamma$ is a constant vector; and if we operate on the
same equation (12) by the characteristic
$2 S \int \iota \, {\rm d} t$,
we obtain an expression for the law of living forces, under the
form:
$${\textstyle \sum} \mathbin{.} m (V \mathbin{.} \iota \alpha)^2
   = - h^2 + 2 \sum \mathbin{.} m S \int \iota \alpha \phi \, {\rm d} t;
   \eqno (15)$$
where $h$ is a constant scalar.  The integrals with respect to
the time may be conceived to begin with $t = 0$; and then the
vector~$\gamma$ will represent the {\it axis of the primitive
couple}, or of the couple resulting from all the moving forces
due to the initial velocities of the various points of the body;
and the scalar~$h$ will represent the {\it square root of the
primitive living force\/} of the system, or the square root of
the sum of all the living forces obtained by multiplying each
mass into the square of its own initial velocity.  Again, the
equation (13) gives by differentiation,
$${{\rm d}^2 \alpha \over {\rm d} t^2}
   =  V \mathbin{.} \iota {{\rm d} \alpha \over {\rm d} t}
       + V \mathbin{.} {{\rm d} \iota \over {\rm d} t} \alpha
   =  \iota V \mathbin{.} \iota \alpha
      - V \mathbin{.} \alpha {{\rm d} \iota \over {\rm d} t};
   \eqno (16)$$
and for any two vectors $\alpha$ and $\iota$, we have, by the
general rules of this Calculus, the transformations,
$$V \mathbin{.} \alpha ( \iota V \mathbin{.} \iota \alpha )
   =  V \mathbin{.} \iota ( \alpha V \mathbin{.} \iota \alpha )
   =  {\textstyle {1 \over 2}} V \mathbin{.} (\iota \alpha)^2
   =  S \mathbin{.} \iota \alpha \mathbin{.} V \mathbin{.} \iota \alpha
   =  {\textstyle {1 \over 2}} V \mathbin{.} \iota (\alpha \iota \alpha)
   =  - {\textstyle {1 \over 2}} V \mathbin{.} \alpha (\iota \alpha \iota);
   \eqno (17)$$
therefore, by (12) and (14),
$${\textstyle \sum} \mathbin{.} m \alpha V \mathbin{.}
         \alpha {{\rm d} \iota \over {\rm d} t}
       + {\textstyle \sum} \mathbin{.} m V \mathbin{.} \alpha \phi
   =  V \mathbin{.} \iota {\textstyle \sum} \mathbin{.}
            m \alpha V \mathbin{.} \iota \alpha
   =  V \mathbin{.} \iota \gamma
       + {\textstyle\sum} \mathbin{.} m V \mathbin{.}
            \iota V \int \alpha \phi \, {\rm d} t.
   \eqno (18)$$

Hence also the {\it time\/}~$t$, elapsed between any two
successive stages of the rotation of the body, may in various way
be expressed by a definite integral; we may, for example, write
generally,
$$t = \int {2 \sum \mathbin{.} m \alpha V \mathbin{.} \alpha \, {\rm d} \iota
         \over \sum V \mathbin{.} m ((\iota \alpha)^2 + 2 \phi \alpha)};
   \eqno (19)$$
the scalar element ${\rm d} t$ of this integral being thus expressed as
the quotient of a vector element, divided by another vector;
before finding an available expression for which scalar quotient
it will, however, be in general necessary to find previously the
geometrical manner of motion of the body, or the law of the
succession of the positions of that body or system in {\it
space}.  It may also be noticed here, that the comparison of the
integrals (14) and (15) gives generally the relation:
$$S \mathbin{.} \iota \gamma + h^2
   =  {\textstyle \sum} \mathbin{.}
         m \, S \int \iota \alpha \phi \, {\rm d} t.
   \eqno (20)$$

\bigbreak

III.
When no accelerating forces are applied, or when such forces
balance each other, we may treat the vector~$\phi$ as vanishing,
in the equations of the last section of this abstract; which thus
become, for the {\it unaccelerated rotation of a solid body about
a fixed point}, the following:
$$\eqalignno{
{\textstyle\sum} \mathbin{.} m \alpha V \mathbin{.} \iota \alpha
   &= \gamma;
   &(21)\cr
{\textstyle\sum} \mathbin{.} m (V \mathbin{.} \iota \alpha)^2
   &= -h^2;
   &(22)\cr
{\textstyle\sum} \mathbin{.} m \alpha V \mathbin{.}
         \alpha \, {\rm d} \iota
   &= V \mathbin{.} \iota \gamma \, {\rm d} t;
   &(23)\cr}$$
which result from (14) (15) (18), by supposing $\phi = 0$, or,
more generally
$${\textstyle\sum} \mathbin{.} m V \mathbin{.} \alpha \phi = 0,
   \eqno (24)$$
that is, by reducing the differential equation (12) of the second
order, for the motion of the rigid system, to the form
$${\textstyle\sum} \mathbin{.} m V \mathbin{.}
      \alpha {{\rm d}^2 \alpha \over {\rm d} t^2}
   =  0.
   \eqno (25)$$
At the same time the general relation (20) reduces itself to the
following:
$$S \mathbin{.} \iota \gamma + h^2 = 0;
   \eqno (26)$$
which may accordingly be obtained by a combination of the
integrals (21) and (22); and the vector part of the
quaternion~$\iota \gamma$, of which the scalar part is thus
$= -h^2$, may be expressed by means of the formula:
$$2 V \mathbin{.} \iota \gamma
   =  V {\textstyle\sum} \mathbin{.} m (\iota \alpha)^2
   =  V \mathbin{.} \iota {\textstyle\sum} \mathbin{.}
         m \alpha \iota \alpha;
   \eqno (27)$$
which gives, by one of the transformations (17),
$$V \mathbin{.} \iota \gamma
   =  V \mathbin{.} \iota {\textstyle\sum} \mathbin{.}
      m \alpha S \mathbin{.} \alpha \iota;
   \eqno (28)$$
so that we have, by (13) and (23),
$${\textstyle\sum} \mathbin{.} m \alpha V \mathbin{.}
      \alpha \, {\rm d} \iota
   =  {\textstyle\sum} \mathbin{.} m \, {\rm d} \alpha \,
      S \mathbin{.} \alpha \iota.
   \eqno (29)$$
But also, by (21), because
$S \mathbin{.} \iota \, {\rm d} \alpha = 0$,
we have
$${\textstyle\sum} \mathbin{.} m \alpha V \mathbin{.}
      \alpha \, {\rm d} \iota
   =  {\textstyle\sum} \mathbin{.} m \, {\rm d} \alpha \, V \mathbin{.}
      \alpha \iota
      + {\textstyle\sum} \mathbin{.} m \alpha \iota \, {\rm d} \alpha;$$
we ought, therefore, to find that
$${\textstyle\sum} \mathbin{.} m
      ({\rm d} \alpha \mathbin{.} \alpha \iota
         - \alpha \iota \mathbin{.} {\rm d} \alpha)
   = 0,$$
or that
$$0 = V {\textstyle\sum} \mathbin{.}
         m (V \mathbin{.} \iota \alpha \mathbin{.} {\rm d} \alpha);
   \eqno (30)$$
which accordingly is true, by (13), and may serve as a
verification of the consistency of the foregoing calculations.

\bigbreak

IV.
We propose now briefly to point out a few of the
{\it geometrical\/} consequences of the formula{\ae} in the
foregoing section, and thereby to deduce, in a new way, some of
the known properties of the rotation to which they relate; and
especially to arrive anew at some of the theorems of Poinsot and
Mac Cullagh.  And first, it is evident on inspection that the
equation (22) expresses that {\it the axis~$\iota$ of
instantaneous rotation is a semidiameter of a certain ellipsoid,
fixed in the body, but moveable with it\/}; and having this
property, that if the constant living force~$h^2$ be divided by
the square of the length of any such semidiameter~$\iota$, the
quotient is the {\it moment of inertia of the body with respect
to that semidiameter as an axis\/}: since the general rules of
this calculus, when applied to the formula (22), give for this
quotient the expression,
$${\textstyle\sum} \mathbin{.} m (T V \mathbin{.} \alpha U \iota)^2
   =  - h^2 \iota^{-2} = h^2 T \iota^{-2};
   \eqno (31)$$
where $TV \mathbin{.} \alpha U \iota$
denotes the length of the perpendicular let fall, on the
axis~$\iota$, from the extremity of the vector~$\alpha$, that is,
from the point or element of the body of which the mass is $m$.
In the next place, the equation (26), which is of the first
degree in $\iota$, may be regarded as representing the
{\it tangent plane\/} to the ellipsoid (22), at the extremity of
the semidiameter~$\iota$; because this equation is satisfied by
that semidiameter or vector~$\iota$, when we attribute to it the
same value (in length and in direction) as before; and because if
we change this vector~$\iota$ to any infinitely near vector
$\iota + \delta \iota$, consistent with the equation (22) of the
ellipsoid, this near value of the vector will also be compatible
with the equation (26) of the plane; for when the variation of
the equation (22) is thus taken (by the rules of the present
calculus), and is combined with the equation (21), it agrees with
the equation (26) in giving
$$S \mathbin{.} \gamma \, \delta \iota = 0.
   \eqno (32)$$

But the plane (26) is {\it fixed in space}, on account of the
constant vector~$\gamma$ and the constant scalar~$h$, which were
introduced by integration as above; consequently {\it the
ellipsoid\/} (22) {\it rolls\/} (without gliding) {\it on the
fixed plane\/} (26), {\it carrying with it the body in its
motion}, and having its centre fixed at the fixed point of that
body, or system, while the {\it semidiameter of
contact\/}~$\iota$ represents, in length and in direction, the
{\it axis of a momentary rotation}.  This is only a slightly
varied form of a theorem discovered by Poinsot, which is one of
the most beautiful of the results wherewith science has been
enriched by that geometer: for the ellipsoid (22), which has here
presented itself as a mode of {\it constructing the integral
equation which expresses the law of living force of the system},
and which might for that reason be called the {\it ellipsoid of
living force}, is easily seen to be concentric with, and similar
to, the {\it central ellipsoid\/} of Poinsot, and to be similarly
situated in the body.  It may, however, be regarded as a somewhat
remarkable circumstance, and one characteristic of the present
method of calculation, that {\it it has not been necessary, in
the foregoing process, to make any use of the three axes of
inertia}, nor even to assume any knowledge of the
{\it existence\/} of those three important axes; nor to make any
other reference to any {\it axes of co-ordinates\/} whatsoever.
The result of the calculation might be expressed by saying that
``the ellipsoid of living force rolls on a plane parallel to the
plane of areas;'' and nothing farther, at this stage, might be
supposed known respecting that ellipsoid (22), or respecting any
other ellipsoid, than that it is a closed surface represented by
an equation of the second degree.  With respect to the {\it path
of the axis of momentary rotation~$\iota$, within the body}, it
is evident, from the equations (21), (22), that this path, or
locus, is a {\it cone of the second degree}, which has for its
equation the following:
$$\gamma^2 {\textstyle\sum} \mathbin{.} m (V \mathbin{.} \iota \alpha)^2
   = - h^2 ( {\textstyle\sum} \mathbin{.}
              m \alpha V \mathbin{.} \iota \alpha )^2;
   \eqno (33)$$
where the symbol $\gamma^2$, by one of the fundamental principles
of the present calculus, is a certain negative scalar, namely, the
negative of the square of the number which expresses the length
of the vector~$\gamma$, and which (in the present question) is
constant by the law of the areas.  Thus, according to another of
Poinsot's modes of presenting to the mind a sensible image of the
motion of the body, that motion of rotation may be conceived as
the {\it rolling of a cone}, namely, of this cone (33), which is
fixed in the body, but moveable therewith, on a certain other
cone, which is the fixed locus in space of the instantaneous
axis~$\iota$.

\bigbreak

V.
But we might also inquire, what is the {\it relative locus}, or
what is the path {\it within\/} the body, of the vector~$\gamma$,
which has, by the law of areas, {\it a fixed direction}, as well
as a {\it fixed length in space\/}: and thus we should be led to
reproduce some of the theorems discovered by Mac Cullagh, in
connexion with this celebrated problem of the rotation of a solid
body.  The equations (26) and (32) would give this other formula,
$$S \mathbin{.} \iota \, \delta \gamma = 0;
   \eqno (34)$$
and thus would shew that the vector~$\gamma$ is (in the body) a
variable semidiameter of an ellipsoid {\it reciprocal\/} to that
ellipsoid (22) of which the vector~$\iota$ has been seen to be a
semidiameter; and that these two vectors $\gamma$ and $\iota$ are
{\it corresponding semidiameters\/} of those two ellipsoids.  The
tangent plane to the new ellipsoid, at the extremity of the
semidiameter~$\gamma$ (which extremity is fixed in space, but
moveable within the body), is perpendicular to the axis~$\iota$
of instantaneous rotation, and intercepts upon that axis a
portion (measured from the centre) which has its length expressed
by $h^2 T \iota^{-1}$, and which is, therefore, inversely
proportional to the momentary and angular velocity (denoted here
by $T \iota$), as it was found by Mac Cullagh to be.  To find the
{\it equation\/} of this reciprocal ellipsoid we have only to
deduce, by the processes of this calculus, from the linear
equation (21), an expression for the vector~$\gamma$ in terms of
the vector~$\iota$, and then to substitute this expression in the
equation (26).  Making, for abridgment,
$$\left. \eqalign{
n^2   &= - {\textstyle\sum} \mathbin{.} m \alpha^2;\cr
n'^2  &= - {\textstyle\sum} \mathbin{.} m m'
            (V \mathbin{.} \alpha \alpha')^2;\cr
n''^2 &= + {\textstyle\sum} \mathbin{.} m m' m''
            (S \mathbin{.} \alpha \alpha' \alpha'')^2;\cr}
   \right\}
   \eqno (35)$$
so that $n$,~$n'$,~$n''$, are real or scalar quantities, because
the square of a vector is negative, and introducing a
characteristic of operation~$\sigma$, defined by the symbolic
equation,
$$\sigma = {\textstyle\sum} \mathbin{.}
         m \alpha S \mathbin{.} \alpha,
   \quad\hbox{or}\quad
  \sigma \iota = {\textstyle\sum} \mathbin{.}
         m \alpha S \mathbin{.} \alpha \iota;
   \eqno (36)$$
it is not difficult to show, first, that
$$(\sigma^2 + n^2 \sigma + n'^2) \iota
   = - {\textstyle\sum} \mathbin{.} m m' V \mathbin{.}
         \alpha \alpha' \mathbin{.} S \mathbin{.} \alpha \alpha' \iota;
   \eqno (37)$$
and then that the {\it symbol~$\sigma$ is a root of the symbolic
and cubic equation},
$$\sigma^3 + n^2 \sigma^2 + n'^2 \sigma + n''^2 = 0;
   \eqno (38)$$
in the sense that {\it the operation denoted by the first member
of this symbolic equation\/} (38) {\it reduces every
vector~$\iota$, on which it is performed, to zero}.  But the
linear equation (21) may be thus written:
$$(\sigma + n^2) \iota = \gamma;
   \eqno (39)$$
it gives, therefore, by (38),
$$(n^2 n'^2 - n''^2) \iota = (\sigma^2 + n'^2) \gamma;
   \eqno (40)$$
that is, by (37) and (36),
$$(n''^2 - n^2 n'^2) \iota
   =  n^2 {\textstyle\sum} \mathbin{.} m \alpha
         S \mathbin{.} \alpha \gamma
       + {\textstyle\sum} \mathbin{.} m m'
         V \mathbin{.} \alpha \alpha'
         S \mathbin{.} \alpha \alpha' \gamma.
   \eqno (41)$$
Such being, then, the solution of this linear equation (21) or
(39), the sought equation of Mac Cullagh's ellipsoid becomes, by
(26),
$$(n^2 n'^2 - n''^2) h^2
   = n^2 {\textstyle\sum} \mathbin{.} m (S \mathbin{.} \alpha \gamma)^2
       + {\textstyle\sum} \mathbin{.} m m'
         (S \mathbin{.} \alpha \alpha' \gamma)^2;
   \eqno (42)$$
and we see that the following inequality must hold good:
$$n^2 n'^2 - n''^2 > 0.
   \eqno (43)$$
If then a new and constant scalar~$g$ be determined by the
condition,
$$(n^2 n'^2 - n''^2) h^2 + g^2 \gamma^2 = 0,
   \eqno (44)$$
(where $\gamma^2$ is still equal to the same constant and
negative scalar as before), we may represent the {\it internal
conical path}, or relative locus, of the vector~$\gamma$ in the
body, by the equation:
$$0 = g^2 \gamma^2 + n^2 {\textstyle\sum} \mathbin{.} m
         (S \mathbin{.} \alpha \gamma)^2
       + {\textstyle\sum} \mathbin{.} m m'
         (S \mathbin{.} \alpha \alpha' \gamma)^2.
   \eqno (45)$$
We see then, by this analysis, that {\it the straight
line~$\gamma$ which is drawn through the fixed centre of
rotation, perpendicular to the plane of areas, describes within
the body another cone of the second degree\/}: while the {\it
extremity\/} of the same vector~$\gamma$, which is a {\it fixed
point in space, describes, by its relative motion, a spherical
conic in the body}, namely, the curve of intersection of the cone
(45) and the sphere (44): which agrees with Mac Cullagh's
discoveries.  Again the normal to the cone (45), which
corresponds to the side~$\gamma$, has the direction of the vector
determined by the following expression:
$$\theta = \iota + h^2 \gamma^{-1};
   \eqno (46)$$
and this new vector~$\theta$ is always situated in the plane of
areas, and is the side of contact of that plane with another cone
of the second degree in the body, which is {\it reciprocal\/} to
the cone (45), and was studied by both Poinsot and Mac Cullagh.
But it would far exceed the limits of the present communication,
if the author were to attempt here to call into review the
labours of all the eminent men who, since the time of Euler, have
treated, in their several ways, of the rotation of a solid body.
He desires, however, before he concludes this sketch, to show how
his own methods may be employed to assign the values of the three
principal moments, and the positions of the three principal axes
of inertia; although it has not been necessary for him, so far,
on the plan which he has pursued, to make any use of those axes.

\bigbreak

VI.
Let us, then inquire under what conditions the body can continue
to revolve, with a constant velocity, round a permanent axis of
rotation.  The condition of such a {\it double\/} permanence, of
both the direction and the velocity of rotation, is completely
expressed, on the present plan, by the one differential equation,
$${{\rm d} \iota \over {\rm d} t} = 0;
   \eqno (47)$$
that is, in virtue of the formula (23), by
$$V \mathbin{.} \iota \gamma = 0;
   \eqno (48)$$
or, on account of (28) and (36), by this other equation,
$$(\sigma + s) \iota = 0,
   \eqno (49)$$
where $\sigma$ is the characteristic of operation lately
employed, and $s$ is a scalar coefficient, which must, if
possible, be so determined as to allow the following symbolic
expression for the sought permanent axis of rotation, namely,
$$\iota = (\sigma + s)^{-1} 0,
   \eqno (50)$$
to give a value different from zero,or to represent an actual
vector~$\iota$, and not a null one.  Now if we assumed any actual
vector~$\kappa$, such that
$$(\sigma + s) \iota = \kappa,
   \eqno (51)$$
we should find, by the foregoing Section of this Abstract, and
especially by the equations (37) and (38), a result of the form,
$$(s^3 - n^2 s^2 + n'^2 s + n''^2) \iota = \sigma' \kappa,
   \eqno (52)$$
where $\sigma'$ is a new characteristic of operation, such that
$$\sigma' = \sigma^2 - s \sigma + s^2 + n^2 (\sigma - s) + n'^2,
   \eqno (53)$$
and that, therefore,
$$\sigma' \kappa
   =  s^2 \kappa
       + s {\textstyle\sum} \mathbin{.} m \alpha
         V \mathbin{.} \alpha \kappa
       - {\textstyle\sum} \mathbin{.} m m'
         V \mathbin{.} \alpha \alpha'
         S \mathbin{.} \alpha \alpha' \kappa;
   \eqno (54)$$
so that the solution (41) of the linear equation (39) is included
in this more general result, which gives, for any arbitrary value
of the number~$s$ the symbolic expression:
$$(\sigma + s)^{-1}
   = (s^3 - n^2 s^2 + n'^2 s + n''^2)^{-1} \sigma'.
   \eqno (55)$$
Hence the condition for the non-evanescence of the expression
(50), or the distinctive character of the permanent axes of
rotation, is expressed by the cubic equation,
$$s^3 - n^2 s^2 + n'^2 s - n''^2 = 0.
   \eqno (56)$$
The inequality (43) shows immediately that this equation (56) is
satisfied by at least {\it one\/} real value of $s$, between the
limits $0$ and $n^2$; and an attentive examination of the
composition (35) of the coefficients of the same cubic equation
in $s$, would prove that this cubic has in general {\it three\/}
real and unequal roots, between the same two limits; which roots
we may denote by $s_1$,~$s_2$,~$s_3$.  Assuming next any {\it
arbitrary\/} vector~$\kappa$, and deriving from it two other
vectors, $\kappa'$ and $\kappa''$, by the formul{\ae}
$${\textstyle\sum} \mathbin{.} m \alpha V \mathbin{.} \alpha \kappa
   =  \kappa';\quad
  - {\textstyle\sum} \mathbin{.} m m'
         V \mathbin{.} \alpha \alpha'
         S \mathbin{.} \alpha \alpha' \kappa
   =  \kappa'';
   \eqno (57)$$
making also
$$\left. \eqalign{
\iota_1 &= s_1^2 \kappa + s_1 \kappa' + \kappa'',\cr
\iota_2 &= s_2^2 \kappa + s_2 \kappa' + \kappa'',\cr
\iota_3 &= s_3^2 \kappa + s_3 \kappa' + \kappa'';\cr}
   \right\}
   \eqno (58)$$
we shall thus have, in general, a system of three rectangular
vectors, $\iota_1$,~$\iota_2$,~$\iota_3$, in the directions of
the {\it three principal axes}.  For first they will be, by (54),
the three results of the form $\sigma' \kappa$, obtained by
changing $s$, successively and separately, to the three roots of
the ordinary cubic (56); but by the manner of dependence (53) or
the characteristic $\sigma'$ on $\sigma$ and $s$, and by the
symbolic equation of cubic form (38) in $\sigma$, we have, if $s$
be any one of those three roots of (56), the relation
$$(\sigma + s) \sigma' \kappa = 0;
   \eqno (59)$$
consequently the three vectors (58) are such that
$$0 = (\sigma + s_1) \iota_1
    = (\sigma + s_2) \iota_2
    = (\sigma + s_3) \iota_3.
   \eqno (60)$$
Each of the vectors $\iota_1$,~$\iota_2$,~$\iota_3$, is
therefore, by (49), adapted to become a permanent axis of
rotation of the body; while the foregoing analysis shows that in
general no other vector~$\iota$, which has not the direction of
one of those three vectors (58), or an exactly opposite
direction, is fitted to become an axis of such permanent
rotation.  And to prove that these three axes are in general at
right angles to each other, or that they satisfy in general the
three following equations of perpendicularity,
$$0 = S \mathbin{.} \iota_1 \iota_2
    = S \mathbin{.} \iota_2 \iota_3
    = S \mathbin{.} \iota_3 \iota_1,
   \eqno (61)$$
we may observe that, for any two vectors $\iota$, $\kappa$, the
form (36) of the characteristic~$\sigma$ gives,
$$S \mathbin{.} \kappa \sigma \iota
   =  {\textstyle\sum} \mathbin{.} m
      S \mathbin{.} \kappa \alpha
      S \mathbin{.} \alpha \iota
   =  S \mathbin{.} \iota \sigma \kappa,
   \eqno (62)$$
and therefore, for any scalar~$s$,
$$S \mathbin{.} \kappa (\sigma + s) \iota
   =  S \mathbin{.} \iota (\sigma + s) \kappa;
   \eqno (63)$$
consequently the two first of the equations (60) give (by
changing $\iota$,~$\kappa$,~$s$ to $\iota_2$,~$\iota_1$,~$s_1$),
$$(s_1 - s_2) S \mathbin{.} \iota_1 \iota_2 = 0;
   \eqno (64)$$
and therefore they conduct to the first equation of
perpendicularity (61), or serve to show that the two axes,
$\iota_1$ and $\iota_2$, are mutually rectangular, at least in
the general case, when the two corresponding roots, $s_1$ and
$s_2$, of the equation (56), are unequal.  The equations (48) and
(32), namely,
$V \mathbin{.} \iota \gamma = 0$,
$S \mathbin{.} \gamma \, \delta \iota = 0$,
show also that these three rectangular axes of inertia are in the
directions of the {\it axes of the ellipsoid\/}~(22), which has
presented itself as a sort of construction of the law of living
force of the system; and a {\it common property\/} of these three
rectangular directions, which in general belongs {\it
exclusively\/} to them, and to their respectively opposite
directions, may be expressed by the rules of this calculus under
the very simple form,
$$0 = V {\textstyle\sum} \mathbin{.} m (\iota \alpha)^2;
   \eqno (65)$$
or under the following, which is equivalent thereto,
$${\textstyle\sum} \mathbin{.} m (\iota \alpha)^2
   =  {\textstyle\sum} \mathbin{.} m (\alpha \iota)^2.
   \eqno (66)$$

With respect to the geometrical and physical significations of
the three values of the positive scalar~$s$, the equation (49)
gives
$$s \iota^2 + S \mathbin{.} \iota \sigma \iota = 0;
   \eqno (67)$$
and consequently by (36), and by the general rules of this
calculus,
$$s = {\textstyle\sum} \mathbin{.} m
         (S \mathbin{.} \alpha \, U \iota)^2
   =  {\textstyle\sum} \mathbin{.} m x^2,
   \eqno (68)$$
if $x$ denote the perpendicular distance of the mass $m$ from the
plane drawn through the fixed point of the body, in a direction
perpendicular to the axis~$\iota$.  We may therefore write the
follow expressions for the three roots of the cubic~(56):
$$s_1 = {\textstyle\sum} \mathbin{.} m x^2;\quad
  s_2 = {\textstyle\sum} \mathbin{.} m y^2;\quad
  s_3 = {\textstyle\sum} \mathbin{.} m z^2;
   \eqno (69)$$
if $xyz$ denote (as usual) three rectangular coordinates, of
which the axes here coincide respectively with the directions of
$\iota_1$,~$\iota_2$,~$\iota_3$; and we see that the {\it three
principal moments\/} of inertia, or those relative to these three
axes, are the three sums,
$$s_2 + s_3,\quad s_3 + s_1,\quad s_1 + s_2,
   \eqno (70)$$
of pairs of roots of the cubic equation which has been employed
in the present method.  At the same time, the conditions above
assigned for the directions of those three axes take easily the
well-known forms,
$$0 = {\textstyle\sum} \mathbin{.} m x y
    = {\textstyle\sum} \mathbin{.} m y z
    = {\textstyle\sum} \mathbin{.} m z x,
   \eqno (71)$$
if (for the sake of comparison with known results) we change the
vectors $\alpha, \alpha',\ldots$ of the masses $m, m',\ldots$ to
the expressions
$$\alpha = ix + jy + kz,\quad
  \alpha' = ix' + jy' + kz',\ldots
   \eqno (72)$$
where $xyz$ are the rectangular co-ordinates of $m$, and $ijk$
are the three original and fundamental symbols of the present
Calculus, denoting generally three rectangular vector-units, and
subject to the laws of symbolical combination which were
communicated to the Academy by the author in 1843, and are
included in the formula (4) of the present Abstract.  And then,
by (35), the coefficients of the cubic equation~(56) will take
the following forms, which easily admit of being interpreted, or
of being translated into geometrical enunciations:
$$\left. \eqalign{
n^2   &= {\textstyle\sum} \mathbin{.}
            m (x^2 + y^2 + z^2);\cr
n'^2  &= {\textstyle\sum} \mathbin{.}
            m m'
            \{
               (y z' - z y')^2
             + (z x' - x z')^2
             + (x y' - y x')^2
            \};\cr
n''^2 &= {\textstyle\sum} \mathbin{.}
            m m' m''
            \{
               (y z' - z y') x''
             + (z x' - x z') y''
             + (x y' - y x') z''
            \}^2.\cr}
   \right\}
   \eqno (73)$$
In fact, the first of these three expressions is evidently the
sum of the three quantities (69); and it is not difficult to
prove that, under the conditions (71), the second expression (73)
is equal to the sum of the three binary products of those three
quantities; and that the third expression (73) is equal to their
continued or ternary product: in such manner as to give
$$\left. \eqalign{
s_1 + s_2 + s_3 &= n^2;\cr
s_1 s_2 + s_2 s_3 + s_3 s_1 &= n'^2;\cr
s_1 s_2 s_3 &= n''^2.\cr}
   \right\}
   \eqno (74)$$
Perhaps, however, it may not have been noticed before, that
expressions possessing so {\it internal\/} a character as do
these three expressions (73), and admitting of such simple {\it
interpretations\/} as they do, without any {\it previous\/}
reference to the axes of inertia, or indeed to {\it any axes\/}
(since all is seen to depend on the {\it masses and mutual
distances\/} of the several points or elements of the system),
are the coefficients of a cubic equation which has the well-known
sums,
$\sum \mathbin{.} m x^2$,
$\sum \mathbin{.} m y^2$,
$\sum \mathbin{.} m z^2$,
referred to the three principal planes, for its three roots.  In
the method of the present communication, those expressions (73),
or rather the more concise but equivalent expressions (35), have
been seen to offer themselves as coefficients of a {\it symbolic
equation of the third degree\/} (38), which is satisfied by a
certain {\it characteristic of operation\/}~$\sigma$, connected
with the solution of a certain other symbolic but {\it linear\/}
equation: and the Author may be permitted to mention that this is
only a particular (though an important) application of a general
method, which he has for a considerable time past possessed, for
the solution of those linear equations to which the Calculus of
Quaternions conducts.  To those who have perused the foregoing
sections of this Abstract, and who have also read with attention
the Abstract of his communication of July, 1846, published in the
Proceedings of that date, he conceives that it will be evident
that {\it for any fixed point~$A$ of any solid body\/} (or rigid
system), {\it there can be found\/} (indeed in more ways than
one) {\it a pair of other points $B$ and $C$, which are likewise
fixed in the body, and are such that the square-root of the
moment of inertia round any axis~$AD$ is geometrically
constructed or represented by the line~$BD$, if the points $A$
and $D$ be at equal distances from $C$}.

\bigbreak

VII.
Finally, he desires to mention here one other theorem respecting
rotation, which is indeed more of a geometrical than of a
physical character, and to which his own methods have led him.
By employing certain general principles, respecting powers and
roots, and respecting differentials and integrals of Quaternions,
he finds that for any system or set of diverging vectors,
$\alpha, \beta, \gamma,\ldots \, \kappa, \lambda$,
the continued product of the square roots of their successive
quotients may be expressed under the following form:
$$    \left( {\alpha  \over \beta}   \right)^{1 \over 2}
      \left( {\beta   \over \gamma}  \right)^{1 \over 2}
      \, \cdots \,
      \left( {\kappa  \over \lambda} \right)^{1 \over 2}
      \left( {\lambda \over \alpha} \right)^{1 \over 2}
   = (\cos \pm U \alpha \sin) {s \over 2};
   \eqno (75)$$
where $s$ is a scalar which represents the
{\it spherical excess\/} of the pyramidal angle formed by the
diverging vectors; or the {\it spherical opening\/} of that
pyramid; or the {\it area\/} of the spherical polygon, of which
the corners are the points where the vectors
$\alpha, \beta, \gamma,\ldots, \, \kappa, \lambda$,
meet the spheric surface described about their common origin with
a radius equal to unity.  And by combining this result with the
general method stated to the Academy by the Author\footnote*{The
same connexion between the Author's principles, and geometrical
and algebraical questions, respecting the rotation of a solid
body, or respecting the closely connected subject of the
transformation of rectangular coordinates, was independently
perceived by Mr.~Cayley; who inserted a communication on the
subject in the Philosophical Magazine for February, 1845, under
the title, ``Results respecting Quaternions.''  It is impossible
for the Author, in the present sketch, to do more than refer here
to Mr.~Cayley's important researches respecting the Dynamics of
Rotation, published in the Cambridge and Dublin Mathematical
Journal.  An account of the speculations and results of the late
Professor Mac Cullagh on this subject may be found in
part~viii.\ of the Proceedings of the Royal Irish Academy; and a
summary of the views and discoveries of Poinsot has been given by
that able author in his very interesting tract, entitled,
{\it Th\'{e}orie Nouvelle de la Rotation des Corps}, Paris,
1834.}
in November, 1844, for connecting quaternions with rotations, it
is easy to conclude that if a solid body be made to revolve in
succession round any number of different axes, all passing
through one fixed point so as first to bring a line~$\alpha$ into
coincidence with a line~$\beta$, by a rotation round an axis
perpendicular to both; secondly, to bring the line~$\beta$ into
coincidence with a line~$\gamma$, by turning round an axis to
which both $\beta$ and $\gamma$ are perpendicular; and so on,
till, after bringing the line~$\kappa$ to the position~$\lambda$,
the line~$\lambda$ is brought to the position~$\alpha$ with which
we began; then {\it the body will be brought, by this succession
of rotations, into the same final position as if it had revolved
round the first or last position of the line~$\alpha$, as an
axis, through an angle of finite rotation, which has the same
numerical measure as the spherical opening of the pyramid
($\alpha, \beta, \gamma,\ldots \, \kappa, \lambda$)
whose edges are the successive positions of that line}.

\bye