r She Sclty tihravu. 1
PRESENTED TO
THE ( ORWELL UNIVERSITY, 1870,
BY
The Hon. William Kelly
k-
U> A
P*|T
18^4
gklHEMATlCS
11*0OUTLINES
OF
A NEW THEORY
OF
ROTATORY MOTION,
TRANSLATED FROM THE FRENCH OF POINSOT.
WITH
EXPLANATORY NOTES,
BY
CHARLES WHITLEY, M.A., FR. Ast. S
FELLOW OF ST. JOHN’S COLLEGE, CAMBRIDGE, AND
READER IN NATURAL PHILOSOPHY IN THE UNIVERSITY OF DURHAM.
CAMBRIDGE:
PRINTED AT THE PITT PRESS, BY JOHN SMITH,
PRINTER TO TOR JJNIFRRSITT.
AND SOLD BY R. NEWBY, CAMBRIDGE;
AND SIMPKIN A MARSHALL, LONDON.
M.DCCCJLXXIV./cornellX
UNSVEKSH'Y;ADVERTISEMENT.
The following pages contain a version of
the Extract which M. Poinsot has published
of the Memoir presented by him to the French
Institute on the 19th of May 1834.
The object of the Translator is to call the
attention of those engaged in Mathematical pur-
suits in this country to a subject which has
acquired so great an interest for the Mathe-
maticians of the Continent from the animated
discussions respecting it which have arisen out
of the above Memoir.
With a view to facilitate the progress of
the Reader, but more especially in the hope of
rendering this publication useful to Students in
the University, demonstrations of the funda-
mental propositions have been subjoined in the
form of notes, and an Appendix containing
demonstrations of the leading principles assumed,
viz. the existence of an Instantaneous Axis, and
of three Principal Axes, and the Conservation of
Couples, has been added.IV
The Translator has been informed by a dis-
tinguished Member of the Institute that the
Memoir will appear entire in the fourth of a
succession of volumes to be published by that
Society in the course of next year.
	ERRATA.	
PAGE 2 4	LINE 5 from bottom "l last line	J	for “rotation” read “rotatory motion.”
14	Note (6)	after “bb'” insert “ (fig. 7.)”.
20	18 from top	after “ moveable cone” insert “ whose vertex is 0.
24	22	far “ same/nmne” read “ equal/urrows.”THEORY
OF
ROTATORY MOTION.
INTRODUCTORY REMARKS.
The following enquiry in Dynamical Science
is one which has most frequently occupied my
attention, and forms one of the subjects which,
if I may so say, I have been most anxious
thoroughly to understand.
Every one can form for himself a clear idea
of the motion of a point, that is to say, of a
corpuscle supposed to be infinitely small, which
gives us in some degree the notion of a mathema-
tical point. For we have only to figure to ourselves
the line, straight or curved, which the point may
describe, and the velocity with which it moves
in this line. But if we have to consider the
motion of a body of sensible magnitude and defi-
nite shape, it must be allowed that the idea which
we form of it is very obscure.
At first indeed, the idea appears to become
clearer by resolving itself into two others. For
A2
if we confine ourselves to the consideration of
one particular point in the body, we may on the
one hand follow the motion of this point, which
can only describe a certain line in space, and on
the other the motion of the body, which can
only turn at the same time on this point as about
a fixed axis. But this second motion, namely,
that of a body moveable about a point, round
which it is at liberty to turn in every direction,
is one of which we have but a confused notion.
Not but that, by referring the different points
in the body to planes or objects fixed in space,
we can find differential equations, as they are
called, of this motion, which, in the simple case
of a body acted on by no external force, we have
even been able to integrate, or at least to reduce
to quadratures. Euler and d’Alembert (nearly
at the same time and by different methods) were
the first to solve this important and difficult
problem: and afterwards, as is known, the illus-
trious Lagrange undertook to investigate anew
this famous question, and to develop it in his
own manner; I mean, by a series of analytical
formulae and transformations, remarkable for their
symmetry and elegance. But it must be allowed
that in all these solutions we see nothing but
calculations, without having any clear idea of the
rotation of the body. We may be able, by means
of calculations, more or less long and complicated,
to determine the place of the body at the end
of a given time; but we do not see at all how
it arrives there. We are totally unable to keep3
it in view and to follow it, as we might wish,
with our eyes, during the whole course of its
rotation.
Therefore to furnish a clear idea of this Rota-
tory Motion, hitherto unrepresented, has been the
object of my endeavours*.
The result is an entirely new solution of the
problem of the rotatory motion of a body, acted on
by no force, whether it turns freely on its centre
of gravity, or on any other fixed point about
which it is constrained to move: a genuine solu-
tion, inasmuch as it is palpable, and enables us
to follow the motion of the body as clearly as
the motion of a point. And if we would pass
from this geometrical representation to calculation,
in order to measure all the different properties
or affections of this motion, the formulae requisite
for the purpose are direct and simple, each of
* It has always appeared to the Translator, that Geometrical illus-
trations, whenever they can he obtained, are of very great use in
enabling us to form clear and correct ideas of the meaning of
analytical formulae and operations. It may he sufficient to advert
to the connexion between the singular solution of a differential
equation, and the envelope of series of curves described after a
given law, or between the first integrals of a partial differential
equation of the second order, and the characteristics of the curved
surface to which it belongs. In quadratures particularly, a large
class of integrals may he made to depend upon elliptic arcs; and
it may he remarked here, that a distinguishing feature of
M. Poinsot's theory is the explanation of all the complications of
rotatory motion by a reference to the properties of an ellipsoid,
with which Analytical Geometry has made us familiar: while the
calculations for determining the actual position of the body resolve
themselves at once into the form of the elliptic transcendents here
mentioned.4
them expressing a dynamical theorem of which
we have a clear id<^, and which proceeds at
once to its object. My analysis of the question
therefore offers in addition this advantage, that
every thing therein is expressed and developed
in terms of the immediate conditions of the
problem, without the intervention of those co-
ordinates and angles which are foreign to the
question, and which take their rise only in the
indirect method employed to discuss it. For we
may remark generally of our mathematical re-
searches, that these auxiliary quantities, these long
and difficult calculations into which we are often
drawn, are almost always proofs that we have
not in the beginning considered the objects them-
selves so thoroughly and directly as their nature
requires, since all is abridged and simplified, as
soon as we place ourselves in a right point of
view.
I thought then that a solution so simple,
and so well calculated to throw new light on the
most difficult questions of Dynamics, might fur-
ther the real advancement of science, and was
therefore worthy the attention of Geometers : and
this consideration led me to compose the Memoir,
which I had the honour this day to present to
the Institute.
I divide it into three principal sections. In
the first I consider the actual motion of the body,
and the forces which would be capable of pro-
ducing it. In the second, I give the solution of
the problem of the rotatory motion of a free body;5
and in the third, I develope the calculations which
relate to this solution.
But to give a more precise idea of this work,
I shall briefly lay down the first principles of
the new theory, and afterwards go hastily over
the principal theorems which are the object and
result of it.SECTION I.
ON THE ACTUAL MOTION OF BODIES, AND ON THE FORCES
CAPABLE OF PRODUCING ANY GIVEN MOTION.
Signification of the terms, Simple Rotatory Motion,
and Angular Velocity.
The only rotatory motion of which we have
a clear idea, is that of a body which turns on an
immoveable axis. For we see plainly all the cir-
cumferences of the circles which the different
points of the body describe about this axis, and
which they can really describe at the same time,
without changing in any respect their relative
position, or what we may denominate the form
of the body.
We have an equally clear idea of the quantity
or measure of this rotatory motion; for since all
the points in it describe similar arcs in the same
time, the ratio of the velocity of a point to the
radius of the circle which it describes is the same
for all points, and it is this constant ratio which
forms the measure, or, as it is called, the angular
velocity of rotation. (1)7
(l) Let OPp (fig. 1) be the axis of rotation, R> r,
any two points describing the circles RR', rr\ which
must lie in planes perpendicular to Op, and therefore
parallel to each other.
Let Rif, rr\ be arcs described uniformly in the
same time (0.
Draw PS parallel to pr, and when pr comes into
the position pr, let PS' be the corresponding position
of PS.
Then z SPS* - z rpr'. (Euc. xi. 10.)
But since the body is rigid,
SR = S'R',
and therefore, z RPRr = z SPS' = /.rp r'.
Hence every point in the body describes round Op in
the time (0 an angle = RPRf.
Let (w) be the angle described in l" by every point,
then z RPR* = Zto,
RRf
and the actual velocity (i?) of R =
PR. z RPR'
t
PR.at^
and	which is the same for every point, is
called the Angular Velocity of the body, or the Velocity
of Rotation.
Composition of Rotatory Motion.
From these simple notions, and from the pri-
mary elements of Geometry, we may conclude that
from the influence of any two separate causes,
a body tended to turn at the same time round8
the two sides of a parallelogram, with two angular
velocities respectively proportional to the lengths
of these sides, the body would turn round the
diagonal, with an angular velocity proportional
to the length of this diagonal. (2)
(2) Let the straight lines Oa, Ob, (fig. 2.) lying in the
plane of the paper intersect each other in O, and suppose
two impulses to act simultaneously upon a body, one
of which would cause it to turn about Oa with an an-
gular velocity = e. OA, and the other to turn about Ob
with an angular velocity = e. OB. Draw JP, BP,
parallel to OB, OA, and PM, PN perpendicular to
OA, OB; and let QPQ' be the circle which the point
P would describe, if the first impulse wrere communicated
singly, about Oa,
RPR...................................
..........................second......................
.............Ob.
Then the planes in which these circles lie will both of
them be perpendicular to the plane bOa passing through
the axes, and therefore their intersection pPp will be
perpendicular to this plane, and therefore to each of the
lines PM, PN, which are the radii of the circles, and
will therefore be a tangent to both circles at P.
The first impulse therefore, would make P tend to
move in the direction Pp with a velocity = MP x (angular
velocity round Oa)
— e. OA. MP = e. OA . OB sin BOA,
and the second in the direction Ppf with a velocity
= NP x (angular velocity round Ob)
= e. OB. NP = e. OB. OA sin BOA,
that is, in a direction exactly opposite, with the same
velocity. If therefore these impulses were communicated9
together, the point P would remain at rest; and the
same may evidently be proved of every other point
in the line OP, which would therefore remain entirely
at rest in consequence of the two impulses.
Draw AK, AL perpendicular to OB, OP- Then
it is evident that the first impulse would not cause any
tendency to motion in the point A, and the second would
cause it to move in the direction of the tangent to a
circle whose radius is AK, and whose plane is perpen-
dicular to 06, that is, in a direction perpendicular to the
plane bOa, with a velocity = e - OB. AK- But in con-
sequence of the two impulses, the line OP remains fixed
in space, and since the body is rigid, the distance AL is
invariable, and the direction of -4’s motion coincides with
the tangent to a circle whose radius is AL, and whose
plane is perpendicular to OP. The point A will there-
fore describe about L the circle SAS', with an angular
velocity = e	—— = e. OP, (by similar triangles
OPN, PAL-) And in the same manner, it may be shewn
that every point in 0A will describe a circle, in a plane
perpendicular to OP, with an angular velocity = e - OP.
Hence the whole line OA, and consequently the rigid
body in which it lies, will turn about OP with an angular
velocity = e. OP.
The same reasoning clearly holds if an interval
occur between the communication of the impulses.
The velocities of the highest points and R of the
circles described by a point lying within the Z 60a, are
here supposed to be in the same direction, when resolved
perpendicularly to a plane bisecting z 60a; and the
rotatory motions are consequently said to be in the same
direction.
If they are in opposite directions, we must take
OB' = OB (fig. 3.) on the other side of OA and complete
B10
the parallelogram, when OP may be shewn as before
to be the new axis, and e. OP the angular velocity of the
body about it.
Whence it follows that the rotatory motions
about different axes which intersect in any point
are compounded in precisely the same manner as
simple forces applied at the point.
And this similarity of composition is not con-
fined to rotatory motions about axes which inter-
sect; but what is very remarkable, it extends to
rotatory motions about axes situated any how in
space.
Thus rotatory motions about two parallel axes
are compounded into a single one equal to their
sum, about an axis parallel to them, which divides
their distance in the inverse ratio of the component
rotatory motions. (3)
(3) Let the axes a a. bb' (fig. 4.) be as before in the
plane of the paper. Draw AB perpendicular to them,
and let
angular velocity round a a = e. BP,
................... bb’ = e.AP,
then as before,
Velocity of P upwards, in a direction perpendicular
to the plane of the paper, due to the rotatory motion
round ad, = AP x (angular velocity round aa')
= e .BP. AP.
Velocity downwards, due to the rotatory motion round bb',
= e ■ AP. BP.
And P, and similarly every point in pp, remains at rest,
and therefore pp' becomes the new axis of rotation.11
And the velocity of -4 perpendicular to AP, which is
that due to the rotatory motion round bti, = e. AP • AB;
therefore velocity of A round P, or velocity of the
body’s rotation round pp',
= e-J^B = e.AB = e.(BP+AP),
= (velocity of rotation round a o')
+ (velocity of rotation round 66')-
What is meant by saying that the rotatory motions are in
the same direction is evident from the last note : that is,
that the motions of Q' and R are in the same direction.
If these are in opposite directions, the resultant
rotatory motion is equal to their difference, and
the position of the axes is determined by the same
laws as that of the resultant of two parallel forces
acting in opposite directions. (4)
(4) If the rotatory motions are in opposite directions,
the motion of r', (fig. 5.) the highest point of the circle
described by a point P round bbr, will be in an opposite
direction to that of O', which we may suppose to be
the same as in the last note.
Suppose the angular velocities to be unequal, and
let that round a a be the greater. In BA produced
take a point P, such that
angular vel. round aa, which we may call <wo, = e. PP,
bb,	= e. Pjfj
then it is evident that the motion of Rf will be parallel
to that of /, and the motion of P round bb’ will be
in the direction RPRr, while the motion round aa is
in the direction QPQ; and therefore as before,
velocity of P downwards =	. PA = e. PB. PJ,
......... upwards = &b. PB = e. PA, PB\12
therefore pp is the new axis of rotation, and the velocity
of rotation round pp, or
e.PB.AB
PB
= e. AB = ePB — ePA = wa
(t)
We may therefore conclude that the resultant of two
rotatory motions a?a, <w6, about parallel axes aa , bb' is a
rotatory motion about an axis parallel to them in the
same plane, which =	=*= wb, according as the compo-
nent rotatory motions are in the same or opposite direc-
tions; the distance of the axis from bb' being equal
to BP
= BA.
e.BP
eBP ± eAP
= BA.
<*>b

BA being measured in a positive direction.
If the motions are in opposite directions and > wa,
BP will be negative, and P will lie on the other side
of B.
If <aa =	, BP will be infinite, and every point will
describe a circle of infinite radius. This case is con-
sidered in the next note. (See Pritchard's Couples,
Appendix, Prop. C.)
If these two parallel and opposite rotatory
motions are equal, they can never be reduced to
a single one. They form in that case what may
be called a Couple of Rotatory Motions: a rotatory
motion sui generis, and which can never be reduced
to a simple rotatory motion about any axis what-
ever. And in fact it is easy to see that the result
of such a couple would be to give the body a simple
motion of translation in space, in a direction per-
pendicular to the plane of the couple, and measured
by its moment, that is, by the product of one of13
the rotatory motions and the distance between
the parallel axes. (5)
(5) Let the plane of the paper be perpendicular to
the parallel axes ad, bbf, (fig. 6.) which meet it in
the points A and B; and suppose two impulses to be
communicated to the body which, separately, would
generate angular velocities, each = <yo, round them in
opposite directions. Take AB = a. Then it is clear
that if for l" the motion round ad were alone com-
municated, B would describe the arc BB’ = a wa, and
if the motion round ad were then to cease and that
round bb' to commence, A would describe an arc A A"
also = a. At this time therefore the position of the
distance between the axes (A" B") is parallel to the
initial position (AB) and the points of intersection (A)
and (B) of the axes with the paper have moved in the
direction BA through A’A" = a. versin wa. If we sup-
pose the motions to be communicated in reversed order
we shall have the new position of the distance (A" B"')
still parallel to AB, while the points of intersection will
have moved in the direction AB through A' A" = a.ver-
sin o)a. And it is clear that the effects will be similar
however small the intervals may be taken. If therefore
the impulses be communicated together, the direction
of the distance at any instant will be parallel to AB,
and consequently the motion round bb', which tends to
make A move in the direction AB, will tend (since AB
may be considered as an incompressible rod) to make
B move in the same direction, while an equal and con-
trary effect will be produced directly on B by the motion
round ad. Hence every point in AB and consequently
every point in the body will move in a direction per-
pendicular to AB.
And if P be any point in AB,
P’s velocity = AP.wa + BP. wa
= a .wa* which is the moment of the couple14
A Couple of Rotatory Motions is therefore
equivalent to a single force, applied at the centre
of gravity of the body, in a direction perpendicular
to the plane of the couple, and equal to the product
of its moment by the mass of the body.
These couples may be transformed into others
equivalent to them, may be turned about or re-
moved at pleasure in their own planes or into planes
parallel to their own, without any change being
produced in the motion of the body. Their com-
position and decomposition follow exactly the same
law with those of ordinary couples, and we may
apply to them without exception the corresponding
theorems. (6) K *
(6) If aa'f ii, in the plane of the paper be the
axes of a couple whose moment is a.wa, it is evident
that the same motion will be produced in the point
P, perpendicular to the plane, by a couple of rotatory
motions round cc, dd', whose distance CD — b, with
angular velocities each = if a . = b. wb. (See
Pritchard, Prop, iv.)
Also if the axes be turned in the same plane into
the positions ffparallel to each other the effect on
P is the same. (v. Pritchard, Prop, hi.)
And what is here true of P is true of every point
in the body.
The propositions corresponding to (Pritchard,
Prop. ii. and Cor.) are self-evident.
The effect of any number of couples in the same
or parallel planes is clearly equal to their sum, the
angular velocities being taken with their proper signs.
If two couples lie in planes inclined to each other
they may be transformed and removed, until their15
distances are equal and coincide with the intersection
of the planes and with each other, (v. Pritchard,
Prop, vi.) when the motions of translation which they
produce in the point P, taken as before, may be com-
pounded as in Elementary Dynamics. And the same
will apply to the parallel motions of every other point
in the body.
From the parallelogram of Simple Hotatory
Motions, and the parallelogram of Couples of
Hotatory Motions results the composition of any
number whatever of rotatory motions, situated
any how in space, and this general composition
perfectly resembles the general composition of
forces. (7)
(7) If z bOa = 90° (fig. 8.), we have from note (2)
the rotatory motion round Op with an angular velocity
(<yp) = e. OP, equivalent to
rotatory motion round Oa with angular vel. (cnrt) = e. Oa,
.....................Ob ................. (<y6) = e. Ob;
••• w = w +
and m =
<*>a = cos a,
where a — pOa = tan 1—,
sin a,	<wa
and similarly if Op be the diagonal of a parallelopiped,
and a, /3, y, the angles which it makes with the three
edges passing through O, the rotatory motion about it
may be resolved into three rotatory motions about these
axes with angular velocities,
tap cos a, &p cos ($,	<#p cos y, respectively.
We may therefore resolve any number of rotatory mo-
tions about axes passing through a point, into others
about three rectangular axes passing through that point,16
which we may call the origin, and taking the sums about
each axis, compound them again for a general resultant.
If any one of the axes do not pass through the
origin we must suppose two opposite rotatory motions,
each equal to the motion about it, to be impressed upon
the body, about an axis parallel to it through the origin:
we shall then have a motion about the latter axis, and
also a couple of rotatory motions to consider, in addition
to any supposed to exist previously in the body. And
any number of couples of rotatory motions impressed on
the body may be resolved into equivalents in the three
co-ordinate planes, and a resultant obtained as for the
simple rotatory motions.
Thus, in the same way that any number of
forces whatever may always be reduced to a single
one passing through a given point and a single
couple, any number of rotatory motions about
different axes situated any how in space, may
always be reduced to a single rotatory motion about
an axis passing through a point, selected at plea-
sure, and a single couple of two equal and opposite
rotatory motions about axes parallel to one another.
And if, as in the case of forces, we wish to reduce
them so as to leave nothing arbitrary, we may
always assign a position for the point above-men-
tioned, such that the plane of the couple shall
be perpendicular to the axis of the resultant
rotatory motion, which may in this case be deno-
minated the Central Axis of the couples of rotatory
motions*.
Since a couple of rotatory motions is equivalent
to a simple motion of translation of the body in
* See Pritchard, Prop. vn.17
the direction of the axis of the couple, the
above analysis ultimately reduces the whole motion
to a rotatory motion about a determinate axis,
and a simultaneous motion of translation in the
direction of this axis. Which is, as we shall
see farther on, the most general motion that a
body can have in absolute space.
Rotatory Motion about a Point.
We now proceed to give an idea of the mo-
tion of a body about a point on which it appears
to turn in every direction.
The motion of a body which turns on an
immoveable axis being the only one of which
we have a clear idea, it is to our notions thereof
that we must endeavour to reduce the motion of
a body turning any how about a fixed point.
Now it is shewn that this motion, whatever
it may be, if considered only for a single instant,
is none other than a simple rotatory motion about
a certain axis passing through the fixed point,
whose direction remains immoveable during this
instant*.
Hence it is concluded that in the following
instant it is likewise a simple motion of rotation,
but about another axis; and so on, from one in-
stant to another, in such wise that the motion
of the body may be considered as a succession
of simple rotatory motions, of each of which the
* See Appendix.
C18
mind has a distinct idea. It is in the same way
that, in order to form a notion of the motion
of a point in a curved line, we represent this
point as describing successively the sides of an
infinitesimal polygon inscribed in this curve. And
the same is true of the instantaneous axis of
rotation in the motion of a body, as of the tan-
gent to a curve in the motion of a point which
describes it.
Sensible Illustration of this Rotatory Motion.
Though the preceding analysis is exact, and
I have laboured to render it clear, it seems to me
that our idea of a body, turning about an axis
which is perpetually changing, is still rather
obscure. It is therefore desirable to render it
clearer, and to present to the mind a distinct and
sensible image of it.
Now I demonstrate in the most simple manner
that, the rotatory motion of a body about an axis
which incessantly varies its position round a fixed
point, is identical with the motion of a certain
cone, whose vertex coincides with this point, and
which rolls, without sliding, on the surface of a
fixed cone having the same vertex.
I mean to say that the moveable cone, sup-
posed to have a rigid connection with the body,
which it carries along with it, if made to roll
on the other cone, which is fixed in absolute
space, will cause the body to perform the precise
motion with which it is supposed to be endued.19
The line of contact of the two cones will be at
every instant the axis about which the body turns
for this instant, or as it is called, the instantaneous
axis; whence we perceive that this axis is move-
able at the same time in the body and in absolute
space, describing in space the surface of the fixed
cone, and in the interior of the body the surface
of the moveable cone of which we have just
spoken. (8)
(8) The most general motion of which a rigid
body is capable about a fixed point O (fig. 9.) is
manifestly such, that any point P in it, taken at
pleasure, may be made to describe any given curve of
double curvature lying on the surface of a sphere, of
which O is the centre, and OP the radius, with a
velocity varying as any function of its position.
The normal planes at successive points of such a
curve of double curvature intersect in a conical surface*,
whose vertex is O. Suppose this surface to be dis-
connected from the body; and let the body be rigidly
connected with a plane which rolls upon the cone, both
of them being in the interior of the body and perfectly
rough. Let OR be the line of contact and Z POR=a>
which will be the inclination of the osculating plane-f
at P to the rolling plane.
If now an angular velocity (o>) be communicated
to the body about OR it may be resolved into two,
one about an axis perpendicular to OP in the plane
POR, which will cause P to move in the osculating
plane, with an angular velocity = w sin a, and the other
about OP, which being resolved again, part of it about
an axis OQ, perpendicular to the rolling plane and
therefore in the normal plane POIl^ will be destroyed
* Hymers’ Analytical Geometry, Art. 64.	t Ibid. Art. 63.20
by friction, leaving finally an angular velocity about
OR = a* cos a sin a.
Let t be the small time which elapses before the
plane comes into contact with the next axis of rotation,
which will manifestly depend on the form of the cone;
and suppose the whole system to turn about OR in an
opposite direction with an angular velocity /3. Then
/.(a) sin a cos a — /3) will be the angle described by the
rolling plane in space, and sin a cos a in the system.
Now the inclination of the new position of the oscu-
lating plane to the rolling plane, that is, the new value
of a, depends on the latter only, while the new position
of the osculating plane in space depends on the former.
Hence tfie relation between the alteration of P’s velocity
in the osculating plane and the change in the position
of this plane, may be varied arbitrarily by varying /3.
But we may evidently give what value we please to |3
at any point, by introducing a moveable cone, instead
of a moveable plane, to which the body is rigidly at-
tached, and supposing the system absolutely at rest;
for the rolling plane revolving backwards in space
would constantly touch such a cone. So that, con-
versely, we may assign such forms to the cones, and
such a function pf the position for the velocity of the
moveable cone^ as to make P describe any curve what-
ever; with a velocity depending in any manner on the
position.
And this, I believe, is the greatest degree of
clearness of which an idea so complicated and
obscure, as that of the motion of a body which
turns any how about a fixed centre, is susceptible.
There is no motion of this nature which cannot
be produced exactly, by making a certain cone
roll on a fixed cone having the same vertex; so
that if we figure to ourselves all the possible cones21
which can he made to roll in this manner upon
one another, we have a faithful illustration of all
the possible motions of a body round a point, on
which it is at liberty to turn in every direction.
Also, if the rotatory motion to be considered
were discontinuous, that is, if the axis of rotation,
instead of changing its position by insensible
degrees, were to leap abruptly from one position
to the next through a finite angle, we could imitate
equally well the motion of the body, by taking,
instead of two cones, two pyramids having the
same vertex, and causing one to roll on the other,
so that the moveable pyramid turning on their
common edge should bring into contact all its
different faces, one after the other, with the faces
respectively equal to them of the fixed pyramid.
If the motion of the body is given, it is clear
that the two cones or pyramids are also given,
as well as the velocity of rotation round the line
of contact, and consequently the velocity with
which the instantaneous axis traces at the same
time the two surfaces. And reciprocally, if of these
different quantities, which come under our con-
sideration in discussing the motion, any three are
given, we may say that the fourth is also, and
that the motion of the body is thoroughly deter-
mined.
Thus the Earth turns in one day on its axis,
while this axis describes in an opposite direction
a right cone* about the axis of the ecliptic, with
* See Appendix.22
a velocity measured by the retrograde motion
(Precession} of the Equinoxes, and which amounts
to about 50" in a year; we may therefore deter-
mine in the case of the Earth the cone which,
rolling on the former in the interior of the globe,
would generate in the Earth a motion correspond-
ing precisely to that which we observe in it. And
it is easy to see that on the Earth the circum-
ference of the circle which is the base of this
little moveable cone, is to that of the base of the
fixed cone, as a day to the period of a complete
revolution cf the equinoxes: which gives scarcely
six feet for the little circumference which the
instantaneous pole of rotation of the Earth de-
scribes every day on its surface. (This is on the
hypothesis of an uniform diurnal Precession.) (9)
(9) Let P (fig. 10.) be the pole of the Earth,
K the point where a line perpendicular to the plane
of the ecliptic at the Earth’s centre meets the surface.
Then we may consider the centre of gravity as a fixed
centre of rotation; and consequently PQR, the fixed
circle in space which the pole describes, will have its
circumference equal to Sir x radius of Earth x sin (PK
~ Earth’s obliquity), and the little circle which rolls
upon it, and which is the path of the instantaneous
pole on the Earth’s surface, will have its circumference
equal to the portion of PQJ? which is described in one
day, that is
1
365
360 x 60 x 60
50
x radius of 0 x sin 23°. 28'
= 5^ feet nearly.23
And eveTy point in this little circle becomes successively
in the course of a day the instantaneous pole of rotation.
Hence we may find at any instant the actual position,
both in space and on the Earth itself, of the instan-
taneous pole.
The most general Motion that a Body can have in
Absolute Space considered.
From the simple idea of a mere motion of
translation, which carries forward at every instant
all the equal molecules of the body through small
equal and parallel lines in space, and from the
simple idea of the rotation of the body about an
axis, which remains immoveable during this in-
stant, results the complex idea of the most general
motion of which a body is capable in absolute
space. Nothing is more clear than this resolution
of any kind of motion into two others which we
can conceive perfectly, and which we may consider
separately, since they are such that, if at every
instant they were executed one after the other,
every point in the body would be brought to the
same place at which it arrives, by its natural
motion, at the end of the instant of which we
are speaking.
But we may have a curiosity to form for our-
selves a notion of the real and single motion with
which the body is endued, for the purpose of seeing,
in some degree, the nature of the simultaneous curves
which the different points describe, and which24
they can really describe at the same time without
causing any change in the form of the body.
Now since a motion of translation may always
be considered as a Couple of equal and opposite
rotatory motions, it follows that the motion of
a body, whatever it may be, can always be reduced
to a simple rotatory motion about an axis passing
through a point selected arbitrarily in space, and
a certain couple of rotatory motions, whose plane
will be in general inclined to this axis. But
instead of taking a point at pleasure, we may
always assign for it such a position, that the
plane of the couple shall be perpendicular to the
axis of simple rotation; and then the whole mo-
tion is reduced to a rotatory motion about a
determinate axis, and a motion of translation in
the direction of this axis. Whence it results that
the motion is identical with that of an external
screw which turns in the corresponding internal
screw. All the points in the body therefore de-
scribe on concentric cylinders small arcs of helices
which have all the same furrow*. In the next
instant it is a different screw, with another axis
and a different furrow: and so on, whence we
see the way in which are formed the simultaneous
curves which all the points describe in space, and
in which they move as in curvilinear canals,
wherein we may suppose them to be inclosed.
Sometimes the furrow of this screw vanishes,
and then the whole motion is reduced to a simple
* “ Pas," the distance between two contiguous threads.25
rotatory motion about the axis of the screw, which
becomes what is called the spontaneous axis of
rotation.
But in general the furrow of the screw does
not vanish, and there is no spontaneous axis pro-
perly so called: that is, there is no straight line in
the body, all the points in which remain immove-
able for an instant. But there is always what
we may call a sliding spontaneous axis, that is,
a succession of points, forming a straight line,
which have no motion other than a simple one
of translation, in the direction of this line.
Such are the simplest notions and the clearest
illustrations that we can form for ourselves of
the motion of bodies. The mere motion of trans-
lation and the mere rotatory motion require no
explanation to enable us to conceive them. Any
motion whatever may always be reduced, and that
in an infinite number of ways, to two such motions.
And among this infinite number of reductions
there is always one which furnishes3 the axis of
rotation in'the actual direction of the translation:
^so that, the most general motion of which a body
is capable is, as we before observed, that of a
certain external screw which turns in the cor-
responding internal screw.
After considering the actual motion of bodies
in a point of view purely geometrical, I proceed
to enquire into the forces capable of producing it,
in order to determine conversely the motion due
to any given forces; which is the natural object
of the science of Dynamics.
D26
Forces capable of producing a given Motion.
Whatever be the motion of a body there always
exist forces capable of producing it. For at any
instant during the motion of the body we may
consider every molecule as if it were at rest, but
acted on by a force capable of giving it the actual
velocity which we suppose it to possess. Therefore
an infinity of similar forces, applied individually
to all the equal molecules of the body, are capable
of producing in it the actual motion which we
observe; and this too spontaneously, I mean to
say even if these molecules have no mutual con-
nexion, and consequently without causing any
violent strain, which would tend to destroy their
actual connection.
Such are the elementary forces capable of pro-
ducing a given motion in a system of molecules,
either free, or arbitrarily connected with each
other.
But if these molecules constitute, as I here sup-
pose, a system of invariable form, we may compound
together all these elementary forces, and so replace
them by a single force and a single couple, which
will be equally capable of producing a given
motion in the solid body under consideration. (10)
(lO) This follows immediately from the theory of
couples,* and includes d’Alembert’s principle, which is,
that “ if any number of forces act upon a rigid system
to produce motion, the elementary forces which indi-
* Pritchard, Prop. vu.27
vidually propel every molecule of the system, and which
are called the effective forces, are statically equivalent
to these impressed forces.1’ Since the system is rigid,
we may reduce each set of forces to a single force
applied at a given point, and a couple whose plane
passes through this point. Suppose a set of forces,
respectively equal and opposite to each of the impressed
forces, to be applied at the same points; then these may
be reduced to a single force, applied at the same point
as before, and a couple whose plane also passes through
this point. Now the force will clearly be equal to the
resultant of the impressed forces, and in an opposite
direction, and the couple will be equivalent to and in
the same plane with the resultant couple of the im-
pressed forces, and tend in an opposite direction. But
such a set of forces would entirely prevent motion;
therefore their resultant force and resultant couple must
exactly balance the resultant force and resultant couple
of the effective forces. Hence the impressed and effective
forces are statically equivalent.
The impressed force at any one of the points of
application may be more than adequate to produce the
actual motion of the molecule on which it immediately
acts, and hence we might imagine a part of it to be
lost. On the other hand at some points, the impressed
forces may be inadequate to produce the actual motion,
and at others there may be no impressed forces at all;
whence arises an idea of forces being gained at these
points. It follows immediately from what has been said
above that “the forces lost and gained are statically
equivalent,” and this is the original form in which
the principle was stated.
Let us enquire therefore what are respectively
the force and the couple which together correspond
to a given motion.28
And first, if the body has only a mere motion
of translation in space, so that all the equal mo-
lecules have velocities equal, parallel and in the
same direction, it is manifest that all the ele-
mentary forces capable of producing these velocities
are also equal, parallel, and in the same direction,
and consequently reducible to a single force, pa-
rallel and in the same direction, and equal to
their sum, applied at the centre of gravity of the
body. Hence we see conversely, that the effect
ef any force, applied arbitrarily at the centre <f
gravity of a body, is to transport all the particles
thereof in its own direction, with a velocity mea-
sured by the magnitude of the force divided by
the mass ef the body; which is, so to speak, self-
evident.
In the second place, if the body has only
a mere rotatory motion about any axis, it is
evident that the velocities, and consequently the
forces by which the individual molecules are im-
pelled, are all proportional to the distances of
the molecules from this axis, and in directions
always perpendicular to these distances, and to
the axis of which we speak. Now these elementary
forces are always reducible to a single force and
a couple. But if the axis passes through the centre
of gravity of the body the force vanishes, and
the whole is reduced to a couple, whose plane is
in general inclined to this axis. If in addition
the axis is one of the three rectangular ones which
exist in all bodies, and which are called the three
principal axes, we find that the couple is per-29
pendicular to tins axis, and that it is measured
by the product of the angular velocity by the
moment of inertia of the body about this principal
axis. Whence we conclude conversely, that the
effect of a couple acting on a body in a plane
perpendicular to one of its three principal axes
is to make the body turn about this axis* with
an angular velocity equal to the moment of the
couple divided by the moment of inertia of the
body about this axis. (11)
(ll) Let a rigid system turn about an axis OG
(fig. 11.) with an angular velocity m, and let any point
P in it, whose co-ordinates are ON=x, NP = y, OG — %,
be the position of a molecule whose mass is m, PT
being a tangent at P to the circle which it describes
about OG.
Then m is impelled by a moving force equal to
the product of its mass and velocity = m. m . OP, applied
at the point P in the direction TP', which we may
resolve into two others at the same point, one
= mm. OP. cos TPN = mm. OP. sin OPN = m&x
in the direction NP or Gy, and another = maiy in the
direction NO, or — — m&y in the direction Gx.
Now we may suppose the former force to act
on the body at any point N in the line of its
direction. Let two opposite forces, each equal and
parallel to it be applied at 0; then we shall have a
force at 0 in the direction Gy = mmzr, and a couple
whose moment is mma?2 in the plane NOP', and the
former will give us in the same manner a force at G
in the direction Gy = mmzr, and a couple in the plane
%y whose moment is umxz, Proceeding in the same
way with the other force, we obtain,30
A force at G in	the	direction Gx = — wmy,	a couple
in	the plane NOP	which tends in the same	direction
with the former and whose moment is = <omy2,
And a couple	in	the plane %% which	tends in
an	opposite direction*	to that in the plane	ssy, and
whose moment = — wmyz when estimated in the same
direction. And the couples in the plane NOP may
be removed into the plane xy. Hence the resolved
parts of all the elementary forces may be summed up,
and reduced to,
(i)	Two forces applied at G; viz.
2 (ftjrnzr) = a?2(m<r) in the direction Gy,
and — co2(my)................... Gx.
(ii)	Two couples ; viz.
ft)2(mzr^) in the plane %y
— coUfmyx) .............. %x,
tending to make the body turn about some axis in
the plane of xy.
(iii)	Two couples; viz :
w(2m?) and to (2ray2) in the plane xy.
If G be the centre of gravity
2w# = 0, 2my = 0,
therefore the two forces vanish.
If	= and 2my^ = 0,
the resultant couple lies wholly in the plane of #y,
and is equal to
w {(Swia;2) + (Smjr)} = w	+ 3Z2)}
=	OP2),
* See Note (2).31
which is called the moment of inertia of the body about
the axis GO.
For an investigation of the number and directions
of the axes for which, when they are severally taken
as the axis of z, the quantities 2mm and 'S.myss sepa-
rately vanish, and for the method of determining the
moments of inertia of bodies, see Appendix.
Now this one theorem gives us immediately
the effect of a couple acting on a body in any
plane whatsoever. For whatever be the couple,
we can always decompose it into three others,
respectively perpendicular to the three principal
axes of the body: and dividing each of them by
the moment of inertia relative to its axis, we shall
have the three angular velocities with which these
three couples would tend to make the body turn
about their respective axes. If then from the
centre as origin, we take three lines in these di-
rections to represent at once the axes and velocities
of rotation, and complete the parallelopiped, we
shall have in the diagonal the axis and velocity
of the rotatory motion to which the proposed
couple gives rise at the first instant*. The sim-
plicity of this operation is evident.
Motion of a Body arising from any given Forces.
Whatever be these forces they may always
he reduced to a single one, passing through the
centre of gravity of the body, and a single couple.
* See Note (7).32
Now the effect of the force is a mere motion of
translation in the direction of this force, and the
effect of the couple is to impress on the body
a rotatory motion about a certain axis passing
through the centre of gravity, whose direction is
determined by what we have just observed.
But it will be seen from what immediately
follows that we can express much more clearly
the direction of the instantaneous axis, relative
to the plane of the couple which gives birth to it.
The Central Ellipsoid of a Solid Body.
Since the three components of the proposed
couple give three rotatory motions which are re-
spectively proportional to these couples, and in-
versely as the moments of inertia about the three
axes, we perceive from geometrical considerations
that the axis of rotation is in the direction of the
diameter conjugate to the plane of the couple, in
an ellipsoid whose axes are reciprocally propor-
tional to the square roots of the moments of
inertia about these axes. (12)
(12) Let A, B, C, be the moments of inertia about the
principal axes, and let the centre of gravity G (fig- 11.)
be taken as origin, and these three rectangular axes
as the axes of a?, y, and % respectively. Take in them
the lines
1 , 1 „ 1
=	y	-J	U -- C ------------ J------ J
\/n.A	y/n.B	y/n.C
and about these three lines as axes describe an ellipsoid.33
Let a, /3, 7, be the angles which the axis of a
couple, whose plane passes through the origin, makes
with the co-ordinate axes; and let M be its moment.
Then the equation to the plane of the couple is*
x cosa + y cos + *cos^ =0........(i),
and the resolved parts of it respectively perpendicular
to the axes are
M cos a, M cos /3, M cos y ;
and by the theorem laid down in page 29, if <ya be
the velocity generated by the couple M cos a about the
principal axis O,r,

M cosa
Mna2 cos a;
and similarly, wh = Mnb2 cos /3, wc = Mnc* cos y;
and if GP = p be the resultant axis of rotation, meeting
the surface of the ellipsoid in the point P, whose co-
ordinates are a?', y-> and inclined at angles a, 7'
to the axes,
,	f	+	f	t	<»c
x = p cosa= p — , y -p —,	% = p —,
and the equation to the tangent plane at P, which is
xxf yy
a*- + V
SM
"I--o' — b
becomes x cos a + cos 8 + # cos 7 =	—....(ii.),
* Mnp	7
wherefore the planes (i.) and (ii.) are parallel; that
* Hymers, Art. 3. Cor. 1.
t Note (7).
E34
is, GP is the diameter conjugate to the plane of the
couple*.
I suppose therefore an ellipsoid to be construct-
ed about the centre of gravity of the body, having
its three principal axes in the directions of the
principal axes of the body, the squares of their
lengths being reciprocally proportional to the mo-
ments of inertia of the body about them: and
I may here remark that this ellipsoid will pos-
sess the remarkable property, that the moment of
inertia of the body about any one of its diameters
will be inversely as the square of the length of
this diameter, (is)
(IS) The equation to a plane perpendicular to
GP (v. last note) is
S = cos a! + y cos « cos y ;
and if Q be any point in this plane, the distance of
Q from GP = \/ GQ2 — c* ; therefore moment (P) of
the body round GP, which = 2 mass of a particle x
(distance from GP)S,
= 2m (a;2 + y- + a? - $2).
And since the co-ordinate axes are by hypothesis prin-
cipal axes,
Imyz = 0, Imxx = 0, 2ma?y = 0;
.-. P =	sin8 a + (2my2) sin2 /3' + (2m#2) sin2 y',
or since sin2 a — 1 — cos2 a = cos2 /3' + cos2 y',
P - cos2 a . 2 m (y2 + #*) + cos2 /3'. 2m (<v2 + #2)
+ cos2 y 2 m (ar + y2)
* Ilvmers, Art. 24. Cor. 1.35
= A cos2 a' + B cos2 (3' + C cos2 7'
-A cos2 a H-----— cos2 ft! + — cos2 7'
na2	ntf ' nr? r
1	m!'i 1	y'1	1	a/2
BO*	p~	+ w62	pl	+ BCS	p2
1	.	a?'2	w'2	a'2
= —j,	since—+—+—= 1.
np	a2 0 c
Now whatever be the form and constitution
of a body, it has always a centre of gravity and
three principal axes, as distinctly determinate as
those of any homogeneous regular solid*. We
may therefore always conceive such an ellipsoid
as the above to he constructed in it, which has
the triple advantage, of placing before our eyes
the centre of gravity and principal axes, of giving
us all the moments of inertia which we may have
to take into consideration (since the moment
about any diameter is expressed in a simple func-
tion of its length), and of offering us an easy
method of determining the position of the axis
of instantaneous rotation, relative to the plane
of the impressed couple. Now it is this ellip-
soid, the consideration of which will throw great
light on the theory of rotatory motion, that I shall
for the future denominate the central Ellipsoid.
See Appendix.36
The Motion of a Body arising from any given Couple
clearly expressed.
Suppose that a body at rest is struck by a
couple in any plane drawn through the centre of
gravity, which may therefore be considered a
diametral plane of the central ellipsoid: we have
just seen that the instantaneous axis of rotation
to which the couple gives rise is the diameter
conjugate to the plane of the couple: and the
angular velocity will evidently be measured by
the moment of the couple, resolved perpendicularly
to this diameter, and divided by the moment of
inertia of the body about the same diameter. (14)
(l4) This may be proved strictly from the pre-
ceding notes. For if 0 be the angle between the
instantaneous axis and the axis of the couple,
cos 0 = cos a • cos a + cos /3 . cos /3' 4- cos y cos yr
y_
Mna2 p Mnb- p Mncrp
<*>P /£2
Mnp2 \dr
(t)p = Mnp2 cos 0
Jfcos 0
P
It is manifest also from note (10) that the impressed
couple M is equal to, and in the same plane with,
the resultant couple of the effective forces; that is?
the resultant of the two sets of couples (ii) and (iii) in
note (ll).VI
And resolving M parallel and perpendicular to the
plane of the couples (iii), we shall have
M cos 0 = resultant of (iii)
the same equation as before.
Likewise, M sin 0 = resultant of the couples (ii).
Hence the axis of an impressed couple can never
be the corresponding axis of instantaneous rotation,
unless it coincides with a principal axis of the body.
For if 0 = 0, no part of M can be employed to produce
the couples (ii), which form a necessary part of the
effective forces, requisite about any axis not a principal
one to give to each molecule its proper motion. The
theorem (page 29) expressed by M = P. g>p only holds
therefore for a principal axis.
Since a couple may always be removed into
any plane parallel to its own, without causing
any change in its effect on the body, we may
always suppose the plane of the impressed couple
instead of being drawn through the centre, to
touch the surface of the ellipsoid: and then we
may say, that
If a body is struck by a couple situated in
any plane which touches the central ellipsoid of
the body, the instantaneous pole of the rotatory
'motion to which the couple gives birth is the
point of contact. And conversely that
If a body turns on any diameter of its central
ellipsoid, the couple actually impressed on it is
in the tangent plane at the pole.38
Which appears to be one of the simplest
theorems in Dynamical Science, on the difficult
and obscure theory of rotatory motion.
We proceed to enquire how the rotatory
motion changes from one instant to another, and
to trace throughout its whole course the motion
of the body.SECTION II.
SOLUTION OF THE PBOBLEM OF THE ROTATION OF FREE
BODIES.
It is evident that the axis of rotation which
we denominate the instantaneous axis remains
immoveable only for a single instant. For from
the rotatory motion itself there arise, for each
of the equal molecules of the body, centrifugal
forces respectively proportional to, and in the
direction of, the radii of the circles which these
molecules tend to describe. Now the axis of
which we speak being, by hypothesis, not a
principal axis, these centrifugal forces will not
balance each other. When removed parallel to
themselves to the centre of gravity, they give
it is true a resultant which equals nothing, but
the resultant couple does not vanish. There arises
therefore from the rotatory motion itself an ac-
celerating couple, the action of which impresses
on the body at each instant an infinitely small
rotatory motion, which is compounded with the
actual rotatory motion of the body, and causes
both the magnitude and the axis of it to vary. (15)
(15) The whole effect of the couple being, as we
saw in the last note, to produce a rotatory motion about
.	,	.	Jfcosd
GP, with an angular velocity i»p = ———, we may
consider the body in motion as acted on by no external
force. And in such case the tension of the rod (QJ?),40
connecting the molecule m at Q with the axis GP, which
is for the instant immoveable,
(velocity of Q)2
= /ffi.--------------
QP
= m. QR. (a)?)2
applied at R in the direction PQ ; which is equivalent to
a force = (o>p)2.?ra QP at G, perpendicular to GP,
and a couple whose moment is (w^)2. mQP. GP, and
whose plane passes through GP.
And it is clear from the reasoning in note (11) that
the resultant of all these centrifugal forces at G will
vanish, by the property of the centre of gravity.
And the resultant of all the couples will be a couple,
in a plane passing through the axis, equal to
&p x (resultant of all the couples wp. m QR. GP)
= wp x {resultant of the couples (ii) for the axis GP|
which does not vanish; since the axis is not a principal
one.
In order to investigate the motion of the body,
it is necessary therefore to commence by finding
this accelerating couple which arises from the
centrifugal forces. Now it is very easy to see
directly, or to conclude from the principle of the
conservation of couples*, that if we take two lines
to represent respectively, the axis and magnitude
of the impressed couple, and the axis and mag-
nitude of the instantaneous rotatory motion, the
accelerating couple due to the centrifugal forces
is always represented in magnitude and position
* See Appendix.41
by the surface of the parallelogram constructed
on these two lines: a simple and remarkable
theorem, which includes the whole theory of ro-
tatory motion, and which when interpreted ana-
lytically gives immediately those three elegant
equations which we owe to Euler, hut which are
ordinarily obtained by long and circuitous opera-
tions only. (16)
(16) It is clear that since each centrifugal force
is proportional, and in a direction perpendicular, to the
corresponding effective force which must have originally
acted on the particle m to produce its motion, the
resultant couple will be in a plane perpendicular to that
of the resultant couple of the effective forces, and there-
fore, during the first instant, perpendicular to that of
the impressed couple M.
The plane of the accelerating couple will therefore,
during the first instant, pass through the axis of M as
well as through the axis of rotation.
And if (Z) be a linear unit, we may take GP (fig. 12.)
M
and GM in the axis of the couple = -- .
I
Then the area of the parallelogram Gm, which is
manifestly in the plane of the accelerating couple,
= GP.GJf.sin PGM
=	sin 0 ;
which from notes (l4) and (l 5) we know to he the mag-
nitude of the couple, during the first instant.
And since the body is acted on by no external force,
whatever be the position into which it is brought by
the action of the centrifugal forces, we may assume that
F42
the general resultant of the elementary effective forces,
which produce its motion round the instantaneous axis,
will always be the same; that is, that it will always
be a couple equivalent to, and in the same plane with M.
If therefore ff be the angle which the instantaneous axis
makes with the axis of the couple at any time, M cos ff
and M sin ff will still be the respective resultants of
the couples (iii) and (ii) for that axis. Hence the
theorem holds generally.
For the equations of Euler see Appendix.
The accelerating couple arising from the cen-
trifugal forces being always situated therefore in
the plane drawn through the axis of the im-
pressed couple and the instantaneous axis, the
straight line on which it tends to make the body
turn is the diameter conjugate to the plane of
these two axes, and consequently is the diameter
which is at the same time conjugate to the in-
stantaneous axis audits projection on the plane
of tEe^feoup^e. Whence I conclude in the first
place, that the axis of the rotatory motion caused
by the centrifugal forces always lies in the plane
of the impressed couple, {yi)
(17) The instantaneous axis being conjugate to the
plane of the impressed couple, is conjugate to every
pair of conjugate diameters in the section of the ellipsoid
made by that plane, and therefore to the pair, one of
which is GU (fig. 13.) its projection on that plane; i. e.
the intersection of the plane of the resultant couple of
the centrifugal forces, which is perpendicular to the
plane of the impressed couple, with that plane. There-
fore the axis of instantaneous rotation and its projec-
tion on the plane of the impressed couple are conjugate
axes of MPU the section of the ellipsoid made by the43
plane of the centrifugal couple, and therefore the axis
GQ about which this couple tends to make the body
turn is conjugate to this pair. It must therefore be
the diameter conjugate to GU in UGN, the section
made by the plane of the impressed couple.
If then we take two lines, one of which repre-
sents this infinitely small motion, and the other
the actual motion, and complete the parallelogram,
in order to obtaiu, in the diagonal, the line which
represents the rotatory motion at the end of an
instant, it is clear that the extremity of this
diagonal remains always at the same height above
the plane of the couple. Whence we deduce im-
mediately these two corollaries; the first, that
Throughout the whole course of the motion, the
angular velocity is proportional to the length of
the diameter passing through the pole of instan-
taneous rotation on the surface of the central ellip-
soid: th^ second, that
The plane of the couple, considered always as
a, tangent at the pole, remains constantly at the
same distance from the centre of this ellipsoid. (18)
(18) Let GQ (fig. 18.) be the axis about which the
centrifugal forces tend to make the body turn, PRQ
the section of the ellipsoid made by the plane contain-
ing the instantaneous axis GP, and GQ. Then by the
last note GP, GQ, are conjugate axes of this ellipse;
and therefore the tangent at P, and for a small distance
the arc PP, is parallel to GQ.
GP
Take the very small line Gq =---(vel. round GQ)
and complete the parallelogram GP. Then R will be a
point in the curve, and therefore the next pole of in-44
stantaneous rotation; and the height of it above GQ is
manifestly the same as that of P.
GR
Also velocity round GR = a>p ,
we see moreover from the reasoning in note (16) and
from the theorem in page 37, that the body will come
into such a position that the tangent plane at the pole R
shall be parallel to the plane of the couple originally
impressed: which in the original position of the body
we may suppose to be a tangent at P.
But the centre is fixed in absolute space, and
the plane of the couple always parallel to itself;
therefore this plane, which is always a tangent
at the instantaneous pole of rotation, is an in-
variable plane, fixed in absolute space.
Therefore the motion of the body, or what
is the same thing the motion of the central el-
lipsoid, is such, that this ellipsoid remains in
contact with a plane fixed in absolute space: that
is, it turns at every instant about the radius vector
at the point of contact, with an angular velocity
proportional to the length of this radius.
This ellipsoid therefore rolls without sliding
on the fixed plane abovementioned; for since
all its motion consists in turning for an instant
on the line drawn from the centre to the point
of contact, the ellipsoid brings at the end of this
instant a new point into contact with the plane;
and this new point, which becomes the pole of
rotation for the instant following, remains in its
turn immoveable for this instant, and so on to
infinity; whence it is manifest that none of the45
points at which the ellipsoid comes into contact
with the fixed plane can ever slide on this plane.
We have therefore a clear idea of the com-
plicated motion of a body of any shape whatever,
turning freely either on its centre of gravity or
on any other fixed point, due to the action of
a couple whose impulsion it originally received
in any given plane. For about the center of
gravity, or if the body is not free, the fixed centre
of rotation, and the principal axes of the body
at that point, we may describe the Central El-
lipsoid, to which we may confine our attention
and neglect entirely the shape of the body. We
may then suppose this ellipsoid whose centre is
fixed, to roll without sliding on a fixed plane
touching it, which gives us an exact representation
of the geometrical motion of the body: and since
the angular velocity with which it turns about
the radius at the point of contact is proportional
to the length of this radius, we obtain also the
dynamical motion of the body; that is, we have
before us the complete succession of the positions
which the body occupies, and the time which it
takes to pass from any one to any other; which
completely determines the motion.SECTION III.
DEVELOPEMENT OF THE SOLUTION.
The two Curves described by the Instantaneous Pole
of Rotation.
The above illustration of the rotatory motion
of a body leads us at once, and as it were by the
hand, to the calculations necessary to measure all
the different affections of this motion.
And first this succession of points, at which
the central ellipsoid comes into contact with the
fixed plane of the impressed couple, traces on
the surface of the ellipsoid the path of the in-
stantaneous pole in the interior of the body, and
the corresponding succession of points on the
fixed plane traces its path in absolute space. We
can therefore determine immediately these two
curved lines, and consider them as the bases of
two conical surfaces having the same vertex, one
of which, moving with the body, would by rolling
on the other, which is fixed in absolute space,
cause in the body the precise motion with which
it is endued.47
To find the first curve we have only to de-
termine th« succession of points in which the
ellipsoid is touched by a plane which is always
at the same distance from its centre; or what
is the same thing, which touches a concentric
sphere whose radius is equal to the given dis-
tance.
While this plane traces on the ellipsoid the
path of the instantaneous pole, we may remark
that it traces on the sphere the path that the
pole of the couple, which is fixed in space, would
appear to describe in the interior of the moveable
body; a curve of the same nature which we shall
have also occasion to consider.
But to speak of the first only: we see that
it is a re-entering curve of double curvature,
having like the ellipse four principal vertices,
at which it is divided into four equal and sym-
metrical parts; a species of elliptical wheel, whose
is always either the greatest or least radius
of the central ellipsoid, according as the radius
of the sphere is given greater or less than the
mean radius of the ellipsoid. This curve of
double curvature is projected in a complete ellipse
on the plane perpendicular to the axis which
forms its axle, in an elliptic arc on the other
plane, and always in an hyperbolic arc on the
plane perpendicular to the mean radius. (19)
(19) Let Pg (fig. 14.) be a perpendicular section
of the tangent plane touching the ellipsoid in P and
a concentric sphere whose radius Gg = r in g.48
Let y, be the co-ordinates of P,
................................... of g,
x\yf,zf,...................... of any point in
the tangent plane.
Then we have two equations to this plane; viz :
yy zz
a*
c2
r tt	r tr	r tr
x x yy z z
and _ +	1;
•gf X
which must coincide; therefore — = —
r2 a2
and 7“?’
2T =
r* b*’
z‘
z*
c4
_	.	1 XT if z2
Therefore 5 = 1 + 1: + “?
r2 a4 b4 c4
ai	y*
Also ~ +
b2 5 c2
-; = i;
which are the equations to the curve of double curvature.
Let a be the greatest and c the least semi-axis. Then
if t > b9 the curve can never meet the plane perpendicular
to the major axis;
1 1
~ b2
for if x = 0, s2 = c2.	5
1 1
~ b2
a negative quantity. The curve therefore lies in this49
case wholly about the major axis. And vice versa if
r < b. Also the equation to the projection on the
principal plane perpendicular to c is
or 1
a2
semi-axes are
b*+c	(1	a?2	A
	Vi* " a4	ftj ’
<?		c2
	y2	'ft2
c2 1	+ ft2'	—“5 — 1? c
	1 -	~V2
the equation		to an ellipse
c2		/I
' r2	and ft ?	
c2 ’		V 1
a2		ft2
the first of which is always < a ; and if r < 6, the second
is < b9 or the projection is a complete ellipse, concentric
to the principal section of the ellipsoid; but if r > b the
minor semi-axis is >b, and the projection is only part
of an ellipse.
The equation to the projection perpendicular to the
mean semi-axis is
b2	~b^
a2 & 1 ~	_
a2 V + c2	’’
1 “	1 “i5
which is manifestly the equation to an hyperbola, the
coefficients of a? and z1 having necessarily different
signs.
G50
The four vertices of this curve are the points
where the radius vector, and consequently the
velocity of rotation, attain their maximum and
minimum, values; and we may remark that the
maximum always occurs when the instantaneous
pole passes through the two vertices which lie
in the mean principal plane of the ellipsoid, and
the minimum when it passes through the other
two vertices. (20)
(20) Since GP =	+ j/2 + ar,
0 = dJ.GP = GP.dTGP
= x+ ydTy + zd,tz,
,	x y ,	* j
also 0 = - + - dxy + - dxz,
a fea	0

y	% 1
whence we derive separately y = Q, % = 0, which give
possible values for GP when r > 6, the former satisfying
the conditions for a maximum, and the latter for a
minimum; the value of the maximum radius (P) being
- c2 + cr - r2
r \/ a~ — <?
and of the minimum (p),
\/#2 + «Z2 =
a2 y/r2 — 62 4-	\/a2 — r1
r y/ a2 — 62
The second curve, being traced by that which
rolls about the centre on the fixed plane of the
couple, is therefore a plane curve which encircles51
the projection of the centre, forming equal and
regular undulations corresponding to the equal
and symmetrical arcs of the rolling orbit which
produces it: it is a species of circular curve whose
radius varies periodically, and which winds for
ever between two concentric circles whose circum-
ferences it touches alternately. (21)
(21) The projection (gp) of the radius vector GP
on the fixed plane of the couple will evidently arrive
at its maximum and minimum values cotemporaneously
with GP; for when GP is greatest its inclination to this
plane is least.
With centre g, (fig. 15.) the projection of G, and
radii p and P, describe two circles; then the curve being
perpendicular to the radius vector gp at the points
where it attains its maximum and minimum values,
will manifestly touch these circles at those points; that
is, will touch them alternately, since the curve passes
alternately the mean and major principal planes.
The consequences deduced in this and the last note
for r > b are easily adapted to the case when r < h.
If the angle at the centre which corresponds
to two consecutive vertices of these equidistant
undulations is commensurable with four right
angles, the curve re-enters itself after a certain
number of revolutions; and the instantaneous
pole which describes it returns at once to the
same position, both in the body and in space.
But in the contrary case the curve never re-enterk
itself, and the pole, which always returns peri-
odically to the same place in the body, can never52
return at the same time to the same point in
space.
Such are the two curves described by the in-
stantaneous pole, the one in the interior of the
body, and the other in absolute space. And al-
though these curves are of such different forms,
yet since it is one and the same point which
describes them both, the equations to them, be-
tween the radius vector and the arc, exactly
coincide. (22)
(22) To find this equation to the two curves,
we have
p2 = or + y~ + #2,
1	<r2 y2
r2	a4 + b4 + c4
Substituting in the second and third the values of
sr derived from the first, multiplying the former by
-4 +	, and subtracting, we have
b c
+ £_	*
W e2/ \62	6V	fc8	c2 r2’

(a9-fc*)(o8-c4)	’
or x = a ■ 's/a2 + p2, suppose;53
dpx =	~.
\/a? + j?
b'p
Similarly, =
+ p‘
7a + p2

c p
•. dps, which = \/{dpxf + (d?p)2 + (dp!gf

+
+
a2 + p2 /32 + p2 7" + p2
The rolling cone of which the first curve
forms the base is simply a right cone of the
second degree; but the fixed cone on which it
rolls is a transcendent cone, whose surface undu-
lates for ever about the fixed axis of the couple:
it is a species of right circular cone, whose sur-
face however is fluted according to the regular
undulations of the curve which forms its base.
Proposed Names for the two Curves.
We know that a heavy body projected any
how in space turns on its centre of gravity, ex-
actly as if it were free from the action of gravity.
The two remarkable curves therefore above de-
scribed are presented constantly to our notice in
the motion of projectiles, and merit names as
much as the path of the centre of gravity which
is called a parabola.
I propose therefore to give them the names
of relative and absolute Poloids; or rather, in
order to distinguish them by their respective forms,54
to call the first simply the poloid, and the second
the serpoloid.
It is evident that the forms of the curves will
depend entirely on.'four given quantities, viz.'jthe
three semi-axes of the central ellipsoid which are
always given by the nature of the body, and the
height of the centre above the tangent plane of
the couple, which is given by the direction of
the impressed couple.
Particular Case in which the Poloid becomes an Ellipse,
and the Serpoloid a Spiral.
In the particular case in which the height
above the plane of the couple is exactly equal to
the mean radius of the ellipsoid, the poloid be-
comes an ellipse, whose plane passes through the
mean radius, and the serpoloid a spiral, which
when examined throughout its whole extent ap-
pears to be a sort of double spiral. I mean that
it throws out on opposite sides of a vertex two
equal branches whose generating points revolve in
opposite directions about a fixed centre, which
they continually approach, but which is a species
of asymptotic point that they never attain. In
this case therefore the instantaneous pole is always
a new point both in the body and in absolute
space, although the length of the spiral described
is finite, and moreover equal to the semi-circum-
ference of the rolling ellipse which produces it. (23)55
(23) When r = b, we have
a? V f z- bs
a2 + ft5 + ?
<r2 y2
and + 7-, + -7 = b
whence x = —
c2

and the curve lies wholly in a plane passing through
the axis of y. It is therefore an ellipse whose axes are
a2b2 + Ifc2 — a*
a2 — b2
= a and b.
Now the latter is evidently the value of Gg (fig. 14.)
in this case. And as the value of GP diminishes from
the former, the angle PGg, which is the angle (0) be-
tween the instantaneous axis and the fixed axis of the
couple, perpetually diminishes, and the accelerating
couple arising from the centrifugal forces (a?p. M sin 0)
becomes less than any assignable quantity; that is, no
finite couple M can make the pole coincide with g
(fig. 16.) though it continually approaches it in moving
either way from AT, which lies in the circumference of
the circle described with radius GK = a on the plane of
the couple.
Particular Cases in which they are reduced to a Point.
If the distance of the centre from the tangent
plane is given equal to one of the extreme radii
of the ellipsoid, then since this can only happen at
a single point in the surface, the poloid and ser-56
poloid are both reduced to a single point, and
the instantaneous pole remains immoveable both
in the body and in space during the whole motion;
and the same thing will happen in a single variety
of the particular case before considered, namely,
when the plane of the couple touches the central
ellipsoid at its mean pole. (24)
(24) In all these cases the axis of the impressed
couple coincides with a principal axis, and therefore the
theorem of page 29 is applicable, and the axis of
the couple being the instantaneous axis, 0=0 and the
centrifugal couple vanishes, that is, the elementary cen-
trifugal couples mutually destroy each other.
Particular Case, arising from the Constitution of the
Body, in which the two Curves become Circles.
Lastly, if the body is one of those which have
two of their principal moments of inertia equal
to one another, in which case the ellipsoid becomes
a solid of revolution, the poloid becomes a circle
about the axis of this spheroid, and the serpoloid
another circle about the fixed axis of the couple.
In all bodies of this kind the movement is that
of a right cone whose base is a circle rolling
uniformly on a fixed cone of the same kind. It
is one of the simplest cases of rotatory motion,
but we must remark that if, as is usually the
case, the circumferences of these circles are in-
commensurable, the instantaneous pole can never
return at the same time to the same point in
the body and in absolute space. (25)57
(25) If a = b, the equations to the poloid become
2	9	5
a	c~
and
a?2 + y* zr l
+ c4 r2 ’
whence we have
s2 / 1	1\ l 1
c2 \c2	a2)	a2 ’
»nd (4-^-4--^.
a \e a2/ er r
the equations to a circle in a plane perpendicular to the
axis of %.
Hence GP, and therefore the angular velocity, is
constant.
And the serpoloid being traced out by P (fig. 14.),
which always remains at the same distance \/GP2 — r2
from g^ will also be a circle.
The uniformity of the motions of the pole in the
body and in space will be shewn hereafter.
We need not examine the simplest case of
all, in which the central ellipsoid is a perfect
sphere: for in whatever manner the couple is
applied, the axis of rotation and the axis of the
couple coincide, and the instantaneous pole re-
mains immoveable in the body and in absolute
space.
H58
Velocities, with which the Pole describes the two Curves,
with which it approaches and recedes from the
Centre, with which it revolves about the Jived Avis
of the Couple,
After having examined the nature of the two
curves described by the pole, which have the same
differential equation between the radius vector
drawn from the centre of the ellipsoid and the
arc, we may consider the velocity with which the
pole describes them both; that with which it
recedes from or approaches the centre, which is
the increment or fluxion of the radius vector and
consequently of the angular velocity; and the
angular motion of the pole about the fixed axis
of the impressed couple. It is easy to deter-
mine the remarkable points where these different
velocities have their maximum and minimum
values, which occur at the alternate vertices of
the waves of the serpoloid. But we may make
a curious remark on the velocity of the pole along
this curve; viz. that it may have, in certain cases,
a minimum value at the superior vertex of the
wave, and likewise a minimum at the inferior
vertex, and consequently a maximum at a third
point intermediate to them; which will not hold
for the other velocities, , whose maximum and
minimum values occur only at the vertices of the
curve described by the pole. (26)
(26) The velocity in the poloid (d,s) = PP (fig. 13.)
= — (velocity round GQ) ;59
and since the plane of the centrifugal couple passes
through the axis of the impressed couple, the axis GN of
the former lies in the plane of the latter :
and coq
(moment of centrifugal couple) cos NGQ
moment of inertia round GQ
M sin Q • cos NGQ,
~Q
dts = Mnpq2 sin Q cos NGQ.
But by Napier s rules,
cos PGQ = cos PGU. cos QGU,
and cos PGU = sin 0 =
\/p2 — r2
P
cos PGQ = - dsp *,
whence cos QGN =
and (pq)2 sin2 PGQ + (^w)2 sin2 UGQ + (pu)2 sin2PGG
= («c)2 + (a&)2 + (6c)%
also p2 + <f + u2 = dr + 62 + c2,
whence we may obtain q in terms of p and dsp; and
substituting, we have dts in terms of the same quantities.
But from note (22) we have dsp = —, and we
pF(p)
can therefore determine dts in terms of p alone.
And since <ap = Mnp2 cos Q = Mnpr,
dtujp = Mnr. dtp = Mnrdts. dsp,
* Miller s Differential Calculus, Art. 9160
which is reducible to an elliptic function of the kind
treated by Legendre, Chap. xxxm.
If p be the radius vector Gg of the serpoloid,
P = V/jJ+7,
and dps = dps.dpp = —p?-- „. V^ + p2 • F(i* + p2)
V r + p
= pF(r2 + p2),
dts = p/^r2 + p2) dtp,
and at the vertices dtp = 0, dps = oo .
We may afterwards simplify the equation to
the serpoloid by referring it to the radius vector
drawn from its own centre in the plane in which
it lies, and the angle described by this radius
about the centre. And any such expression may
be obtained without difficulty. (27)
(27) If 0 be the angular distance of the radius
vector from a given position,
p	p
which is reducible as before to an elliptic function.
Determination of the Position of the Body at the end
of a given Time.
Lastly, in order to find the formulae which will
enable us to calculate the position of the body
at the end of a given time, we must begin by
determining the velocity of rotation in terms of61
the time, which will be done by a single inte-
gration, and will give the place of the instan-
taneous pole on the surface of the central ellipsoid.
Afterwards we must integrate the above equation
to the serpoloid, which will give the place of the
pole on the fixed plane of the couple. And by
these two quadratures, which naturally belong to
the class of elliptic transcendents, newy say that
the proposed problem is entirely solved; I mean
that we are enabled by means of them to deter-
mine the actual position in space in which a body
is found at the end of a given time. For we
have only to suppose the ellipsoid placed in con-
tact with the fixed plane, so that they touch at
the points which we have just determined in the
ellipsoid and plane respectively; when the central
ellipsoid, and consequently the body, will have the
exact position in space at which it arrives by its
natural motion at the end of the given time.
We may vary these determinations in different
ways, by taking other unknown quantities relative
to the position of the body; but whatever be the
co-ordinates which we employ, the expression of
these quantities in terms of the time will always
require two integrations, which necessarily belong
to the class of elliptic transcendents.
In the particular case in which the height
of the centre above the fixed plane is equal to
the mean radius of the ellipsoid, when the poloid
becomes a simple ellipse and the serpoloid a spiral,
the difficulty is diminished, and the integrals be-
come logarithmic or exponential. (28)62
(28)	In this case dps is reducible at once to an
elliptic function; and the value of q2 deduced from
note (26) is greatly simplified; and thus the values of
dtu)p and d$p are reduced as here stated.
Lastly for bodies whose central ellipsoid is one
of revolution, when the motion becomes wholly
uniform and circular, no integration is required
to determine the position of the body at the end
of a given time. (29)
(29)	In this case all the axes in the equatoreal
plane AGB (fig. 17.) of the spheroid are principal axes.
Let AGC be the plane passing through the axis of the
spheroid and the axis of the couple, GB the intersection
of the equatoreal plane with the plane of the couple, which
will be manifestly perpendicular to AGC- Then the
resolved part of M perpendicular to the principal axis
GB = 0, and therefore the axis of instantaneous rotation
lies in the plane AGC.
Therefore AGC is the plane of the centrifugal couple,
and GB, which is the diameter conjugate to it, the axis
about which this couple tends to make the body turn.
And from note (25) GP (=jo) is constant, as indeed
appears also from the consideration that GB and there-
the small arc PR is perpendicular to GP; substituting
therefore in the equation obtained in note (26) and
observing that NGQ = Q, we have
P
a constant quantity.
For the application to the theory of Precession, see
Appendix.63
Properties of the three principal Axes of the Body,
relative to the Stability of the Rotatory Motion.
When the plane of the impressed couple is so
situated that the instantaneous pole falls on one
of the principal poles of the central ellipsoid, it
always remains there; so that the instantaneous
axis is the immoveable straight line in which the
axes of the body and the couple coincide through-
out the whole motion. Each of the three principal
axes is therefore a permanent axis of rotation. But
there exists between them, as is known, a remark-
able difference with respect to the stability of the
rotatory motion about each.
If the instantaneous pole falls on the greater
or lesser pole of the ellipsoid, and happens by
the impulsion of any small disturbing couple to
be drawn aside to a little distance therefrom, it
will not recede farther, but will describe its poloid
about this particular pole of the ellipsoid. But
this is not the case when the instantaneous pole
falls on the mean pole of the ellipsoid; for on any
the slightest displacement it will recede farther and
farther, and proceed to describe its poloid about the
greater or lesser pole, according as the accidental
disturbance causes the distance of the tangent
plane of the couple from the centre of the ellipsoid
to increase or diminish. And if the disturbance
is such that this distance is not altered, which
happens in the directions of two particular ellipses
which cross each other at the mean pole, the in-64
stantaneous pole will proceed to describe the par-
ticular ellipse along which it has been started, or
rather the half of this ellipse, till it arrives at the
opposite mean pole; which is the greatest dis-
turbance that the body can suffer: whilst, if it
had been started along the other half of the ellipse
it would have returned immediately to the same
mean pole; which is the least possible disturbance.
There is therefore a single case in which the in-
stantaneous axis being drawn aside from the mean
axis, with which it coincided at first, not only does
not recede farther from it, but even returns towards
it immediately, until its distance is less than any
assignable quantity. But in all other cases it
proceeds to describe an elliptical cone about the
major or minor axis, or to trace the plane of one
or other of the ellipses which I have mentioned:
and we may say that the rotatory motion about
the mean axis has no stability. (30)
(30) The rotatory motions about GP, GQ (fig. 13.)
are supposed to be in the same direction ; and in that
case P moves in the direction MPU. Suppose the
direction of the motion which the impressed couple tends
to produce about its own axis to be reversed; then the
motion about the instantaneous axis will be reversed, the
motion which the centrifugal couple tends to produce
about GQ being unchanged. The motions about GP,
GQ will therefore be in opposite directions, and in order
to find R the new pole we must take q on the other side
of G. The pole will therefore move in the direction
UPM.
Hence the direction of the motion of the pole in the
poloid depends on the direction of the motion which the65
impressed couple' tends to produce about its axis. If
therefore, for a given direction of this motion, the instan-
taneous pole, on being started in one direction from
the mean pole along the above-mentioned ellipse, tends
to describe the whole semi-ellipse, it will on being started
along the other half of this ellipse still tend to move
in the same direction as before, i. e. will return towards
the mean pole; which from note (24) it will approach
nearer than by any assignable quantity.
The only axes of stable rotatory motion there-
fore are those about which the moments of inertia
of the body are respectively the greatest and least;
but we must not conclude that it is equally stable
about these two axes: for if one of them differs
little from the mean axis, the stability of the
motion about it will not be greater than of that
about the mean axis, as we shall presently see.
Measure of the Stability about each of the two extreme
Axes.
In order to form a distinct idea of this
stability, and of that which forma in some
degree the measure of it for each of these two
axes, imagine the surface of the ellipsoid to be
cut into four elliptical sections by the two ellipses
above-mentioned, whose planes intersect in the
mean axis. The mean pole is therefore at the
intersection of these two ellipses; the greater pole
is in the centre of one of a pair of sections, and>
the lesser pole in the centre of the supplementary
section.
I66
Now, in the first place, if the instantaneous
pole of rotation falls on the mean pole of the
ellipsoid, it is clear that if disturbed ever so little
it will be thrown into one or other of these two
sections, and describe its poloid about one or other
of the principal poles of the ellipsoid: or else if
the disturbance is in the direction of one of the
two ellipses it will either describe the half of this
ellipse, or return immediately to its former po-
sition ; this being the only case of stability about
the mean axis.
Again, if the instantaneous pole falls on the
greater pole of the ellipsoid it may be removed
at pleasure to any point in the surrounding section
without ceasing to describe its poloid about the
same pole: and if it is in this that we make the
stability about the major axis to consist, we may
say that the magnitude of the section is in some
degree the measure of it. Similarly we perceive
that the supplementary section is the measure of
the stability of the rotatory motion about the
minor axis. Now if one of these two axes differ
little from the mean axis, the corresponding section
is very small, and the supplementary section very
great. The axis which differs little from the mean
axis affords therefore very little stability, and the
other axis very much. It is not therefore correct
to say, as people usually do, that if the instan-
taneous axis is drawn a little aside from the prin-
cipal axis which corresponds to the greatest or least
moment of inertia of the body, it recedes very little
from it, and makes only small oscillations during67
the whole period of the motion: for if the moment
of inertia relative to this axis differs little from
the mean moment, the instantaneous pole may be
thrown by a small disturbance out of the little
section in which it lies into the neighbouring
section, and proceed to describe therein its poloid
about the other axis; or else, if it is only removed
to another point in this little section, it may de-
scribe therein a narrow and elongated poloid, and
consequently make considerable oscillations about
the principal pole from which it has been drawn
aside. (31)
(si) If r > ft, the semi-axes of the ellipse which is
the projection of the poloid on the plane perpendicular
to the axis of a? are
the latter of which, when 6 and consequently r very
nearly equals a, becomes

very nearly,
which may be made to approach b as nearly as we please
by diminishing the difference between t and 6, that is by
throwing the pole very near to the boundary of the
elliptic section mentioned in the text.
The other semi-axis will be increased by this means;
but very slightly, as	is supposed to be very
small in comparison with a2 — c2.
* Note (19). The equation to the projection on yz is obtained
from that on xy by writing ~ for r, a for c, and c for a.68
In bodies where one of the extreme moments
of inertia differs little from the mean moment, and
■where consequently the central ellipsoid is nearly
a solid of revolution about the other axis, the
stability of the rotatory motion is only absolute
for this axis. This is the case in the Earth,
whose motion is stable about its present axis,
but would be very much otherwise about the third
axis, which differs, as we know, very little from
the mean axis.
Motion of the principal Axes of the Body in absolute
Space.
We have considered the motion of the instan-
taneous pole of rotation both in the body and in
space: but it may be asked what are the motions
of the poles of the central ellipsoid themselves;
the velocities with which they revolve about the
fixed axis of the impressed couple, and with which
they approach .or recede from the plane perpen-
dicular to this axis which gives us their motions
of Precession and Nutation: we may examine
into the nature of the three curves or serpoloids
which the projections of these three principal poles
trace at the same time on a fixed plane, &c. and
we shall find in the easiest manner many curious
properties of the motion of a body. For example,
The sum of the areas swept out, during the
motion of the body, by the projections on the plane
of the couple of equal portions of each of the three69
principal axes, measured from the centre, is pro-
portional to the time.
If these three lines, instead of being equal, are
proportional to the square roots of the moments of
inertia, or to what I call the arms of inertia of
the body about the same axes, the sum of the
areas is also proportional to the time.
These are simple, and in some degree geo-
metrical theorems, which must however be dis-
tinguished from the dynamical theorem relative
to the areas traced by all the radii vectores drawn
to the several-molecules of the body, though it is
easy to reduce these expressions, drawn from the
same principle, to one and the same.
Analogous theorems may be proved relative
to the nutations of the three principal axes of
the body towards the fixed plane of the couple
of impulsion. For the sum of the squares of the
distances of the three principal poles of the central
ellipsoid from the axis of the couple is a constant
quantity: and,
The sum of the squares respectively multiplied
by the moments of inertia of the body is constant
throughout the whole motion.
Lastly, if we consider the curves described by
the projections of these poles on the same fixed
plane, we shall find that they are of the same
nature with the serpoloid described by the instan-
taneous pole of rotation.
In general the pole, whether it be the greater
or lesser, which forms the centre of the poloid,
describes* a curve having like the serpoloid equal70
and regular undulations about the same centre;
the superior vertices of the one corresponding to
the superior vertices of the other, and the inferior
to the inferior. During the same time the other
two poles describe also curves undulating regularly:
but when one of them passes the superior vertices
of its path the other is passing the inferior vertices
of its similar path; the poles being at an angular
distance of 90° from one another.
In the particular cases in which the polmd is
an ellipse and the serpoUnd a spiral, the mean
pole of the body describes also a spiral which
approaches the centre continually, and nearer than
by any assignable limit, without ever reaching it:
the two other poles also describe spirals, of a
species in some degree resembling the former;
for each of them recedes continually from a certain
minimum distance from the centre to a certain
maximum which it never reaches; so as to ap-
proach continually the circumference of an asymp-
totic circle. We may make another curious remark
on this particular case of the motion of bodies;
viz. that there exists in the mean plane of the
central ellipsoid a certain diameter which has the
remarkable property of remaining always perpen-
dicular to the fixed axis of the impressed couple,
and therefore of describing the plane of this couple,
and that too with an uniform motion. So that
the whole motion of the body consists in turning
on this particular diameter with a variable velocity,
while this diameter uniformly describes a circle in
space.71
When the central ellipsoid is one of revolution
the pole of the figure describes a circle as well as
the instantaneous pole. In this case there is no
other pole properly speaking than the extremity
of the axis of the spheroid: but if we chose to fix
arbitrarily on two other points in the equator at
an angular distance of 90° from each other and to
examine the two curves which their projections
describe, we should have two perfectly equal but
not circular curves: which would be two equal
serpoloids about the same centre, the superior
vertices of the one being always at an angular
distance of 90° from the corresponding inferior
vertices of the other.
We shall have yet more new properties and
new illustrations of rotatory motion to present.
For instance, it is easy to see that the section of
the central ellipsoid made by the fixed plane of
the couple is an ellipse whose area is constant
throughout the motion.
So that if we consider the central ellipsoid as
plunged into a non-resisting fluid, of which the
fixed plane of the couple forms the level, we may
say that the area of the plane of floatation is
constant.
We may pass from hence to a new illustration
of the motion and represent it by that of an el-
liptical cone which rolls on the plane of the couple
with a variable velocity, and slides with an uniform
velocity. All which properties will be developed
in the Memoir.72
We see how much these illustrations enlighten
and correct our ideas of even the most elementary
portions of the theory of rotatory motion. Those
who cultivate the geometrical properties of surfaces
of the second order will draw from them without
difficulty a great number of curious theorems re-
lative to this kind of motion: for each proposition
in Geometry gives a corresponding one in Dy-
namics. But the great advantage of our mode
of treating the subject consists in the easy demon-
strations which it affords of the motions of pre-
cession and nutation of the equators of the heavenly
bodies, and of the nodes of their orbits: of thus
simplifying and sometimes correcting these difficult
theories, of which we have already seen an example
in the determination of the single and invariable
plane of areas, which I have denominated the
Equator of the system of the Universe.APPENDIX.
I. The Axis of Instantaneous Rotation.
When a rigid body is in motion, it is turning,
during every separate instant, about some straight line
or other, (v. page 17.)
For an analytical proof of the existence of this line
see WhewelPs Dynamics. (Art. 120.)
The following is extracted, by permission of the
Author, from Earnshaw’s Statics. (Art. 10p.)
“ Let P, Q, (fig. 18.) be any two particles of a rigid
body; PP', QQ', the paths which they describe during
the same instant; A, P, the centres of curvatures of
these paths; then the line joining A9 B, will be the axis
about which the whole body turns during this instant?’
“ For the lines which join the successive contempo
raneous positions of P and Q, while they are respectively
passing to P' and Q', will form a species of conical
surface; and since, by reason of the rigidity of the
body, they are all of the same length, the planes PAP\
QBQ', in which the curves (PP, QQ) formed by their
extremities lie, must be parallel. Now since P describes
round A the angle PAP' in the same time that Q
describes round B the angle Q2?Q', the trapezium PABQ.
turns in the same time round AB and comes into the
position PABQ! (for AP = AP, and 2? Q = 2?Q', since
A, B, are the centres of curvature of PP, QQ'). Con-
K74
sequently the motion of every particle of the body
situated in PQ takes place about the axis AB''
“ Hence the motions of the points P and Q take
place about AB, and therefore AB must be perpendi-
cular to PAP, QBQ,', the planes of these motions.
In like manner if the motion of any other particle R
take place about a point C, AC must be perpendicular
to the planes of motion PAP, RCR'; hence both AB
and AC are perpendicular to PAP, which is impossible,
(Euc. xi. 13.) unless they coincide; in which case C is
a point in AB, and the motion of R takes place about
AB; and since R is any particle, the motion of every
particle takes place about AB, that is, the whole body
turns during the instant about the straight line AB^
If any point in the body be fixed the axis must pass
through this point.
For let O be the point and join AO; then we may
consider PAO as a crooked but rigid rod moveable about
the fixed point A, and it is manifest that while one
extremity P moves through PP, the other cannot remain
at rest unless it lies in AB.
In this case the motion of any one point P deter-
mines the motion of every other point.
For if PO be joined, the rod PO considered as a
rigid body must be turning during every separate instant
about some axis passing through 0; and it is shewn in
the course of the above demonstration that the plane
of the motion of any other point Q in the rigid body
is parallel to the plane of the motion of any point in OP.
Therefore the motion of Q is about the same axis as that
of OP; and it is clear from note (l) that the angular
velocities are the same.
If the body is perfectly free, it has during every
instant a simple rotatory motion about some axis pass-75
ing through the centre of gravity; except in the case
when all the particles move in equal and parallel straight
lines, that is, when the body has a mere motion of
translation.
To prove this it is necessary to establish the follow-
ing Dynamical property of the centre of gravity.
If a motion of translation be communicated to a
body which has a simple rotatory motion about any axis
passing through its centre of gravity, the motions will
subsist together, and each will continue to affect the
body precisely as it would have done if the other had
never existed.
Suppose a velocity -u in the direction of a line which
makes angles a, )3, 7, with the co-ordinate axes to be
communicated to every particle of the rigid system in
note (11).
Then the resolved parts of the effective forces which
act on a particle M situated at the point P are
+ mt? cos a, in the direction Ga?,
m + v cos /3).................... Gy,
mv cos 7 .................. G#.
And the resolved parts of all the elementary effective
forces are reducible to
(i.) Three forces applied at G; viz.
- o)2 (my) + v cos a 2 (m),
which by the property of the centre of gravity (if jut = 2 (m)
the mass of the system)
= jULV cos a, in the direction Ga?,
a>2(ma?) + t'cos/32(m) = mdcos/3................ Gy.
■v cos 7 2 (m) = f/. v cos 7....... Gs*.76
(ii.) Two couples; viz.
(- toy + v cos a) -	cos y
— — itil&myz in the plane z&\
'Zmzfax + v cos ($) — 'Zmyv cos y
— a>lLm<vz...............zy.
(iii.) Two couples ; viz :
(tox + v cos /3) and 2my (toy — v cos a),
which together = wSm (zr2 + y2) in the plane wy.
The whole elementary effective forces are therefore
equivalent to a force equal to the resultant of the
forces (i) = \/fj?i? (cos2 a -h cos2 fi + cos4 y) =/j.v applied
at the centre of gravity, in a direction making angles
a, 7? with the axes, and to the resultant of the two
sets of couples (ii) and (iii); and it is manifest that
if the rotatory motion were suppressed, the resultant
force would still be and the resultant couple would
vanish; and therefore conversely that the consequence of
applying a force = yv at the centre of gravity of the
body is to give every particle of it a mere motion of
translation with a velocity t?, in the direction of the force,
whether the body has or has not a rotatory motion; and
that on the other hand the couples (ii) and (iii), which
are necessary and sufficient to produce a simple rotatory
motion about an axis through G, remain the same, what-
ever be the value of t?, and therefore of /uv.
Suppose now that AB (fig. 19.) the line containing
the centres of curvature of the paths of every particle
of the body in absolute space, does not pass through the
centre of gravity G of the body, and that the angular
velocity of the body about it is <0. Then if about Gg a
line through G parallel to AB, we suppose two angularT7
velocities each = w to be communicated to the body in
opposite directions, we shall have a simple rotatory
motion with an angular velocity co about Gg and a
motion of translation perpendicular to the plane GgAB;
and since Gg passes through the centre of gravity these
motions are independent, that is, the body has during
the instant, a simple rotatory motion about an axis
through G.
Hence also it is impossible for a free rigid body to
have a simple rotatory motion about any axis which
does not pass through the centre of gravity.
For the rotatory motion which it has for an instant
about any such axis can always be resolved into a simple
one about an axis through the centre and a motion of
translation. Hence the axis of the screw spoken of in
pp. 24, 25, always passes through the centre of gravity
of the body.
We may also illustrate the above principles by a
reference to the motions of the Earth and Moon in
absolute space.
If the Earth had no rotatory motion about its own
axis we might naturally consider SK (fig. 20.) perpen-
dicular to the plane of the ecliptic as the instantaneous
axis (AB figs. 18 and lp.). The motion about it would
be equivalent at every instant to a motion of translation
in the direction of a tangent to the Earth’s orbit, and a
simple rotatory motion about an axis through the centre
of gravity of the Earth parallel to SK. In the course
of a year the Earth would turn once round on this
axis in the direction WLE, and to a spectator on the
Earth’s surface the Sun would appear to describe in the
same direction a circle round the centre of gravity of
the Earth in the plane of the ecliptic, and the fixed
stars parallel circles in the opposite direction. And
this motion would be totally independent of the motion78
of translation, of which the spectator would only be
aware from the consideration that if it were suppressed
the Sun’s apparent motion would be in the opposite
direction.
But no such apparent annual motion of the fixed
stars is observed. We may therefore conclude that the
Earth has no such simple rotatory motion, and that
its motion round the Sun is a simple motion of trans-
lation of the centre of gravity; which would also appear
from Dynamical considerations. It is observed that the
Moon presents always the same face towards the Earth.
Hence all the particles in it describe similar curves about
a line through the centre of gravity of the Earth perpen-
dicular to the Moon’s orbit. This axis is therefore
always an instantaneous axis of the Moon. Hence the
Moon has a simple rotatory motion in the same di-
rection with its motion of translation about an axis
through its centre of gravity parallel to the instan-
taneous axis, and turns once round on this axis during
a revolution of its centre of gravity round that of the
Earth.
If now we suppose one or more rotatory motions
to be communicated about other axes through the centre
of gravity, the motion of translation at every instant will
not be affected, and the simple rotatory motion about
the centre of gravity at every instant, upon which the
appearances of the heavens depend, will be compounded
of these.
II. Principal Axes and Moments of Inertia,
Through every point in a material system at least
three straight lines may be drawn, in directions mutually
at right angles, for which, when they are severally taken
as the axis of each of the quantities 2 (mtw), 2 (my%)
vanishes.79
In a rigid body or system these lines are called
principal axes. (v. page 28.)
If GP (fig. 11.) make angles /3, 7, with the
co-ordinate axes, and the co-ordinates of Q, the position
of any particle m, be a?, y, %; as in note (13).
Moment round GP (P) = 2m (QP)2
= 2 ma?2 + 2m?/2 + 2m^2
-	2ma?2 cos2 a — 2 m?/2 cos2 [i — Imz2 cos2 y
—	22ma?y cos a cos /3 — 2 2m a?# cos a cos y
— 22mt/# cos /3 cos y.
Let A = 2 my2 -t- 2m«2, which is manifestly the
moment round Ga?,
Af = 2m^#, and so on for the other axes.
Then P = A cos2 a + B cos2 ft + C cos2 y
— A' cos /3 cos y — B cos a cos y — C' cos a cos /3.
And if we take any point a?, y, % in GP at a distance
p from G, we shall have
Pp2 = Ax2 + By2 + C%2 — Afy% — Bf — C'xy.
Now the value of P evidently depends upon the values
of a? 7, that is upon the position of GP; and p being
arbitrary we may take it equal to any function of the
same quantities;
let it =
/y = -
n80
and Ax2 + By2 + C~2 - A'yz - B’xz - Cay = — = e,
n
is the equation to the locus of the extremity of p. We
see at once that it is a surface of the second order
whose centre is G; and since P can never vanish, except
in the particular case when all the particles lie in the
straight line GP, p is always finite, and the surface is an
ellipsoid.
If now the axes of the co-ordinates be transformed
so as to coincide with the Geometrical Axes of the
ellipsoid, the form of the equation becomes
Aar + By2 + Czr = e;
For these axes therefore the quantities A'9 B\ Cf9
respectively = 0; and therefore each of them is a principal
axis of the body or system.
Now a geometrical axis of an ellipsoid coincides with
the normal at the point where it meets the surface; at
this point therefore we have
Ax — Cry — Bf% By — A’% — C’x C% — Bx — A'y
x	y	%	’
A cos a - Cf cos fi — B! cos y
	cos a	
B cos (i -	- Af cos 7 -	- Cf cos a
	cos /3	
C cos 7 -	- Bf cos a -	- Afcos /3
cos y
whence we obtain
(P - A) cos a + C' cos + B' cos 7=0,
C' cos a + (P - P) cos 4- J' cos 7 = 0,
B' cos a + A' cos (i + (P - C) cos 7 = 0.81
and by elimination,
(P-J)(P-P)(P-O-J'2(P-J)-P'2(P-P)_C"2(P~C)
+2j'p'rr=o,
in which the roots of P are of course the moments of
inertia about the geometrical axes of the ellipsoid.
To shew that the roots of this equation are real,
assume A' = 0, B' = 0; then the factor P — C disap-
pears and the equation becomes a quadratic, and if this
has for its roots Pn P2, which we find to be possible
quantities, and we substitute successively in the original
equation
- x , Pj P2, and + oo ,
we obtain results alternately positive and negative,
whence we conclude that there are three real roots be-
tween these limits, (vide Cauchy, Exercices, Vol. in. p. 5.)
The above ellipsoid is manifestly the same with that
which Poinsot calls the central ellipsoid.
The moments of inertia about the principal axes
passing through the centre of gravity are called the
principal moments of inertia of the body.
The moment about the major axis of the ellipsoid is
clearly less, and that about the minor axis greater, than
that about any other axis whatever.
If two of these are equal the ellipsoid whose equa-
tion we have found above becomes a spheroid, and
every diameter in the equatoreal plane is a principal
axis, or the number of principal axes is infinite.
If the three principal moments are equal, the ellip-
soid becomes a sphere, and every diameter is a principal
axis.
L82
When the co-ordinate axes are the principal axes
through the centre of gravity, the value of P is re-
duced to
A cos2 a + B cos2 + C cos2 7.
From this we can deduce the moment about any axis
which does not pass through the centre.
For if a rigid body or system revolve about a fixed
axis AB (fig. 19.) with an angular velocity to, the re-
sultant effective couple in the plane Pag perpendicular
to AB = <w x (moment of inertia (-4) round AB).
But if the axis were not fixed the motion might
be resolved as before into a rotatory motion round Gg,
and a motion of translation; the former of which gives
us a couple = co x (moment (G) round Gg), and the
latter a force applied at G
= yu (velocity of translation)
= /a (o>. a) if GO = a, see note (5),
which is equivalent to a force /a to a applied at 0 and
a couple = /two. a in a plane perpendicular to GOA.
And the force is destroyed by the resistance of the
fixed axis;
wA = toG + tojuflT,
A = G + Ma2.
The principal moments of a homogeneous solid body
are readily determined by integration.
For if nt = 2 (m) be a continuous function of <2? and y,
the value of an individual elementary portion A p. = m
it must entirely depend on the values of x and y
at the point where that portion is situated.83
Therefore the corresponding elementary portion A C
of the moment, which = Aja . (#* + «/2), must depend
entirely on the values of a? and y5 and must therefore
be a function of m-
But — = ®* +
Am
taking the limits, d^C = a?2 4- y2.
Similarly dfl A = y2 +
d^B = tT2 + %2.
If the density of the body vary according to a given
law of the position of a particle,
dx dy dxfi = p	y,z);
■■■	= pfx{yf(®, y, ss),
and dxC = pfxfy (a2 + J/2) x/(a, y, ss)
and similarly for A and J5.
If the body is homogeneous,
dxdydxfl = p; dxfi = pfxfyl,
••• dxC = p fxfy (^ + ys) = p fxfya? + p fxfyy\
and similarly for A and B.
For a plane surface % = 0; and dxdyfjL = p;
C = pf4v^ + pf4yy^
■d = pfi/vV'y	B = p f* fy M2'■>
. C - A + B.84
If AB (fig. 21.) be an uniform physical line whose
middle point is G, which is clearly the centre of gravity,
and a line Gg perpendicular to it be taken for the axis
of (#), we shall evidently have % = 0 for every point
in this line:
= 0, = 0,
and every line perpendicular to AB is a principal axis
thereof.
To find the principal moment, take GA for the axis
of <r, then y = 0 and dx/m = p;

.	a	a
which from -v =-----to os = + -
2 2

= M • —•
12
The moment about an axis through A parallel to Gg
cr	ar
= M	m . —
12	4
= M.--
The moment about an axis Gh inclined at an angle
(a) to Gg
= m . — cos’ a.
1285
If ABCD (fig. 22.) be a rectangular parallelogram,
any axis perpendicular to its plane will be a principal
axis; and lines through G parallel to the sides will
evidently be principal axes at G, since for every product
+ mxy we shall have a corresponding product - vnxy,
and	= and if = AB = b, GN = y,
dy/jL = p • MM* = p . AD;
moment round Gx(A) = f9y*. pa
b3
b2
= pa — = /u, —.
r 12 12
B = pb — = u — .
r 12 *12
Hence in this case
C =
a2 + 62
12
and if we suppose this axis to pass through the centre
of gravity of any number of rectangles of the same
size and density, and exactly parallel to ABCD, the
moment of the system about this axis will evidently
be obtained by multiplying C by the number of these
planes. Hence the principal moments of a rectangular
parallelopiped whose base is ABCD will be
C = pabc,
r* a2 4- c2
a2 +	<£ + b2
12 12 ’
62 + c2
and similarly,
12
1286
Whenever a homogeneous solid body can be divided
by three planes, passing through the centre of gravity at
right angles to each other, into perfectly symmetrical
portions, the intersections of these planes are principal
axes. This follows readily from what has just been said
of the principal axes of a rectangle ; it appears also from
the consideration that each plane will in that case contain
two of the geometrical axes of the central ellipsoid.
If APB be an ellipse (fig. 23.), the principal axes of
G are GJ, GB, and a line perpendicular to the plane
APB,
and B = p	PP, since dxfj, - PP,
= 2p^ |	1 (a2 - tf2)*. x +	(a2 - a?2)5},
| B = 2p {J	a2-a?} ;
a
and A = /m — ;	C = jul
4
487
The principal moments for a circle are immediately
deducible by making b = a, as are those for an elliptic or
circular cylinder by multiplying by the height.
We may however obtain C more readily for a circle.
For if GR (fig. 24.) = r,
r2 = a? + y~ = and drft = 27rr . p ;
„	ttQ^ cl2
C = 27rp Jrr3 = p m > if GA = a.
2 2
Every radius GA is a principal axis;
For a solid of revolution about the axis of z,
dzfi = v'P.p = ir(f%)2.p;
• •• C = ,rpf,(f%y.
And A = B ■,
dp A = A (d^A + dflB) = ^	+ y2) +
= 1^ + ^;
••• A = ^£(f*y + vpf,(f*y.!?.
If G be the centre of an ellipsoid (fig. 25.) the axes
GJ, GB, GC are principal axes;
and dxA = pfyfT (y2 + s2).
But if PNP' be a section at the distance GM = x,
the equation to it is
y2 1	1
b2 ’	a?2 + e2 ’ a?
1 - 1 ~88
MN=b \/l-C, MP = c \A--L
a~	ar
and pjyfz(y2 + between the limits
y = - MN,	% = - MP,
y = + MN,	z = 4- MP,
is manifestly the moment of inertia of the plane ellipse
PNP about GA;
'• J"T-i,,!(6’ + c')W$ + 5?} + e’
which from <v = — a to a? = -j- a
*irp l. /12	<>\ ft2 + c
= —- abc. \b£ + c ) = m	.
15	'	f	5
In all the above cases the moment of inertia is of the
form where A is a constant quantity. And this will
be the case in any system whatever, since the moment is
made up of positive products m (a?2 + ^2) each of which
is of that form.
The line k is called the radius of gyration.
If G (fig. 26.) be the centre of gravity of a free
rigid system, k the radius of gyration at G, and Gg
in the plane of the paper a principal axis perpendicular
to GO, an impulsive force *5 applied in a direction per-
pendicular to the plane of the paper at a point G whose
distance from G is s, is equivalent to a force /S' applied at
G and a couple .S's in a plane perpendicular to Gg.89
The effect of the former is to produce a motion of
translation in the direction of *5. Let v be the velocity
generated, then S = pv.
The effect of the latter is to produce a rotatory
motion round GC. Let co be the velocity of rotation,
then S. s = w/mk'.
Let C be a point such that GC.w = c, then C and
every point in a line through C parallel to Gg will
remain at rest; therefore this line will be the axis about
which the body or system will actually turn during the
first instant in absolute space.
The line thus obtained by reversing the process in
pp. 76, 77> and compounding the motion of translation
with the rotatory motion is called the Spontaneous Axis
of Rotation (v. p. 25.), which is therefore a principal
axis at a point C which is sometimes called the Centre
of Spontaneous Rotation^ and whose position in OG
produced is determined by the consideration that
GC.
S.s
S
s
If the axis Cc were fixed, the shock of any force S on
O would evidently produce no pressure whatever on Cc.
Hence O is called the Centre of Percussion correspond-
ing to the axis Cc. Its position is determined from the
equation
If a rigid body or system turn about any fixed axis
Cc with an angular velocity <y, the motion at any instant
is equivalent to a simple rotatory motion about an axis
Gg through the centre of gravity parallel to Cc, and
a couple of rotatory motions whose moment is CG.cw,
that is, a motion of translation with a velocity = CG •	;
for these are the motions into which, if the axis were
set free, it would immediately be resolved.
M90
Now the motion of G is unaffected by the former,
and the latter would be generated by a force £ applied
at G = fJL. CG. co.
Therefore the resultant couple of the effective forces
in a plane perpendicular to Cc
= co. (moment of inertia round Cc)
=	+	(PaSe83-)
which, if O be a point such that GO = —— , = S. CO.
CG
But a force S applied at O is equivalent to a force
iS at C, which would be destroyed by the reaction of
Cc9 and a couple whose moment is S.CO.
Hence a rotatory motion about a fixed axis Cc with
a velocity w would be produced by a force S = co ./jl. CG
applied at O.
But since the axis is fixed, the same effect would
be produced by a force R applied at G
CO
= S.^-=a>.jUL.CO.
CG
Now if the whole mass were collected at 0 and connected
with C by an imponderable rigid rod, the force which
must be applied at 0, which would then become the
centre of gravity, to cause /jl to revolve about C
would be
M x (linear velocity) = p. . co CO9 as before.
It is evident that the velocity of the particle m at O
is the same in either case.
Any point O lying in a cylindrical surface at a
fc9
distance CG + from Cc is called a centre of oscilla-
CG
tion corresponding to the axis Cc. x91
The angular velocity due to a force R at G=----777. ,
m . CO
and the circumstance of the axis being fixed enables us
to replace any force T acting at Q perpendicular to the
CQ
plane of the paper by a force R = T.	at G, parallel
to r.	CG
The same reasoning holds for a succession of im-
pulsive forces, and thus for a continued force, such as
gravity.
If CJ (fig. 27.) be a vertical, JCG = 0, the im-
pulsive force at G, which acts at every successive
instant during the short time A t to produce an angular
velocity Aw,
= p.g sin 0.
And each successive increment of velocity being
independent of the former, the whole increment Aw in
the time A 2
_ A/ '“g SiH 0 ■
•	M.C0 ’
Aw gsin0
’ * a? = co ;
taking the limits dtw =
gsin 0
~C0'
But A0, the angle described in any small time A£
during which w may be considered uniform,
= w. A£;
d|0 = w;
gsin 0
dl20«d<w = ‘
CO
The same expression for determining t which would
have arisen in considering the motion of any mass sus-
pended from C by an immaterial thread CO.92
III. Conservation of Couples. Equations of Euler.
The following statement and demonstration of the
principle of the Conservation of Couples are translated
from the Memoir presented by M. Poinsot to the Insti-
tute in May 1804.
Let any number of perfectly free bodies, unconnected
with each other, describe uniformly straight lines in
space; then the forces by which they are respectively
impelled remain always the same and in the same
direction.
Therefore the resultant of all these individual forces
passing through any fixed point, and the resultant couple,
are the same at every instant during the whole motion.
Now if we suppose these bodies to be suddenly
connected together so as to act one upon another by
virtue of any reciprocal forces whatever, that is to say,
such that between any two bodies the action and re-
action are perfectly equal and opposite, which com-
prehends all forces of this nature, the individual motions
of the several bodies will be changed, and the forces
which respectively impel them will vary in magnitude
and direction during every instant of the motion. But
the resultant of these forces passing through any fixed
point and the resultant couple will remain the same
as before, and would still be the same if the bodies
were suddenly set free, and each were to fly off in a
straight line with the new velocity by which it is
actually impelled.
This principle, which follows from the Differential
Equations of Motion, may also be demonstrated in the
following manner.93
Since each body by reason of its connection with
the others is unable any longer to obey fully the
impulse which it has received, the force which acts
on it is decomposed into two others, one of which is
destroyed, while the other is that which the body
actually obeys. The circumstances are identical with
those of the case in which a body in motion meets with
an insurmountable obstacle, except that the component
part of the force which would then be annihilated, pro-
ceeds to act upon the other bodies of the system, and
is destroyed by the united action of the similar com-
ponents which each of them contributes.
Now it is evident that the resultant force passing
through any fixed point, and the resultant couple, of the
impulsive forces are respectively equivalent to the re-
sultant forces and resultant couples of the two sets of
forces into which they are decomposed. But the first set
being in equilibrio their resultant force and resultant
couple vanish by the laws of Statics: therefore the ori-
ginal resultant force and resultant couple are identical
with those which the body actually obeys. With respect
to any other forces, such as mutual attractions, which
may exist in the system, since they are reciprocal, that
is to say, distributed in pairs of equal and opposite forces,
they cannot in any way affect the forces above mentioned.
We see therefore that in a system of bodies which
have received any primitive impulses and which act
mutually upon each other in any manner, the resultant
of all the forces which impel them passing through any
fixed point, and the resultant couple, remain always the
same whatever variations the respective moving forces
of individual bodies may experience, whether these varia-
tions take place by insensible degrees, or abruptly, from
any change in the reciprocal actions of the bodies, or
from the sudden introduction of any new connecting
forces among them.94
Hence in note (l6) the resultant of all the effective
couples at any instant during the motion = M.
The centrifugal couple is of course included. It
is the portion of the resultant M sin Q' of the couples
whose planes pass through GP, which lies in a plane
passing also through GM.
The projections of GM = M (fig. 12.) on three fixed
lines Ga?, Gy, G% in space, will represent in magnitude
and direction the axes of the resultant couples of the
effective forces in planes perpendicular to these lines.
Let o, 6, c, be the cosines of the angles which the
axes of the ellipsoid make with the line Ga?, then
Mcos MGx = Jf cos a. a + cos .6 + cos . c
= A (/)a. a + B (t)b . b + C<tfc . c.
Similarly
Mcos MGy = Aua. a! +	6' + Cwc . c,
M cos MG% =	. a" + Bub. 6" + C wc. c".
(v. Poisson, M&canique, Art. 409.)
It must be observed that the portion of the centri-
fugal couple, as obtained in note (l5), which lies in the
plane perpendicular to the axis of instantaneous rotation
is omitted in deducing the final equation, as having
no tendency to change the position of the axis.
IV. Application to the Precession of the Equinoxes.
Let G (fig. 28.) be the Earth’s centre, GC its geome-
trical axis; 5 the Sun ; S'L W the equator, the Earth
being supposed in the position which it has at the95
summer solstice. Then the action of the Sun on S'
is greater, and on W less than the action on G- There-
fore in addition to the force on G which produces the
motion of translation there is a couple of forces in
opposite directions which produces a rotatory motion
round the line L perpendicular to the plane KGS-
Also in winter the opposite face being presented to S
the couple tends to produce motion in the same direc-
tion. If therefore the Earth when originally projected
had a rotatory motion of its own about GC, which would
be the case if the primitive impulse did not pass through
G but through some other point in the equatoreal plane
in a direction perpendicular to a plane passing through
GC, these two rotatory motions would be compounded,
and the pole of rotation on the central ellipsoid would be
drawn aside to a short distance from the geometrical
pole. It would therefore describe as its poloid a circle
at this distance from the pole C. The serpoloid would
also be a circle about GK. For the plane of the re-
sultant couple would be parallel to the plane of the
ecliptic.
The ellipticity of the Earth being very small the
effect of the centrifugal forces is not perceptible.
When the Earth is in any other position the effect of
the Sun’s attraction is at some times to increase and at
others to diminish the obliquity, which however it does
not permanently alter.
To determine the velocity of the Pole we must know
the amount of the effect of the disturbing couple at S'
and IF, which depends on the Sun’s attraction at those
points, (v. Airy, Precession, Art. 21.)
If a be the angular velocity generated by the couple
at S' and W, w the angular velocity of the Earth’s rota-96
tion round the instantaneous axis which is always the
same, p the radius of the poloid,
27tp = (velocity of the Pole) x (one day).
But by note (26.), vel. of the Pole = (radius of ®) — ,
O)
and length of a day = —;
O)
p = (radius of (g) .	.	(v. Airy, Art. 13.)
The value of p is obtained by observation in note (9).

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