The equation (2) is itself one of Euler's equations; to obtain the others, arrange the equation (1) in the form

- sino cos4 (d.Дw, – Bw ̧w,) + sino sing (

dt

2

(d.Bw2+ Aww, Aw ̧w,)

dt

..(5).

dt

Taking L, M, N, as the moments of forces about the principal axes, and resolving these moments about the axes C, F, and Z respectively, where F is the intersection of AB and XY,

dU

аф

dU -N, de

=

= L sin + M cos,

=

L sino cos+M sin 0 sin + N cos 0.

From these equations we find that L sin0, and M sin are respectively equal to the right-hand members of equations (6) and (7), and hence Euler's equations follow at once.

It is instructive to notice the interpretation of the equations (1), (2), and (3); they represent the equalities of the rates of change of the effective angular momenta about the axes of C, F, and Z, and of the moments of forces about those axes.

Thus the angular momentum about Z, is

- Aw, sin cos + Bw, sin 0 sin4+ Cw, cos 0,

2

from which the equation (3) follows at once, the axis Z being
dU
fixed, and the moment of forces about it.

df

For the equation (1), observe that in the fig. 30, K is the intersection of ZC with XY, and that F, the intersection

df of AB and XY, is the pole of ZC; and hence that is the dt rate of rotation of the axes K and F.

Let P and Q be the angular momenta about these axes,

then

and

P=Aw, cose cosp - Bw, cose sin + Co, sin 0,

2

3

but the rate of change of the angular momentum about the moving axis F, is

Lastly, the equation (2) follows from the expression for the rate of change of angular momentum about C, i.e.

THE SMALL OSCILLATIONS OF A PARTICLE AND OF A RIGID BODY.*

By ROBERT STAWELL BALL, A.M, Professor of Applied Mathematics and Mechanism, Royal College of Science for Ireland.

I.

Introduction.

LAPLACE investigated the small oscillations of a particle on a sphere. Poisson solved a special case of the same problem on the ellipsoid. Lagrange discovered the general laws of small oscillations, and his methods have been improved by Messrs. Thomson and Tait. The results, of which the following is an abstract, have been obtained by an union of the method of Lagrange in its improved form, with some elegant theorems of kinematics discovered by M. Chasles. Demonstrations of some of the theorems here enunciated will be found in two papers written by the author.

"On the small oscillations of a particle on a surface under the action of any forces." Quarterly Journal of Mathematics, No. XXXIX. 1869.

"On the small oscillations of a rigid body about a fixed point under the action of any forces, and more particularly when gravity is the only force acting." Transactions of the Royal Irish Academy, Vol. XXIV. Science, Part XVI.

II.

A Particle.

1. There are in general three lines called normal lines, such that whatever be the small oscillations of a particle free in space, the movement is compounded of simple harmonic vibrations along the normal lines.

2. When the forces have a potential, a constant small quantity of energy would draw the particle along any radius vector from its position of rest to the surface of a certain ellipsoid, the normal lines are in the principal directions of this ellipsoid, and the lengths of the isochronous simple pendulum are proportional to the squares of its principal axes.

pool.

Abstract of a paper read before the British Association at Liver

3. When the particle is constrained to a surface, the motion is compounded of vibrations in two directions on the surface, and when the forces have a potential, the tangent lines to these directions are at right angles.

III.

A free Rigid Body.

4. A free rigid body may receive any displacement by being screwed along an axis in space, the distance it travels along the axis when rotated through the angular unit being termed the pitch of the screw.

5. The movement of a free rigid body when making small oscillations is compounded of six normal movements, cach consisting of a to and fro vibration about a normal screw, the position, pitch, and period of which depends upon the forces.

6. Whatever be the initial motion of the body, it may be distributed uniquely among the six normal screws, and thus the entire motion is determined.

IV.

A constrained Rigid Body.

7. If a rigid body have & degrees of freedom, its motion is compounded of vibrations about k normal screws.

8. A body capable of turning around a fixed axis, and sliding along it, has two degrees of freedom, its motion is compounded of that about two normal screws whose pitch is different, but both of which lie in the fixed axis.

9. A body, three points of which are limited to a plane, has three degrees of freedom. Its motion is compounded of vibrations about three normal screws whose pitch is zero, and the directions of which are perpendicular to the plane.

10. A body rotating about a fixed point has three degrees of freedom. Its motion is compounded of vibrations about three normal screws, whose pitch is zero, and whose directions pass through the point.

N.B. The screws in this case may be conveniently called the normal axes.

V.

A rigid body rotating about a point, the forces having a

potential.

11. The body may be moved from one position to any other position by rotation through a certain angle, about a certain axis passing through the point; this angle and axis are called the axis of displacement and the angle of displacement respectively.

12. On any axis through the point take a radius vector proportional to the small angular velocity, which a small quantity of energy would be able to communicate to the body about the axis. The quantity of energy being constant, the locus of this point on different axes is the wellknown momental ellipsoid.

13. On any axis through the point take a radius vector proportional to the small angle, through which a small quantity of energy would be able to rotate the body about the axis from its position of equilibrium against the forces. The quantity of energy being constant, the locus of this point on different axes may be called the ellipsoid of equal

energy.

14. The three common conjugate diameters of the momental ellipsoid and the ellipsoid of equal energy are the normal axes; the body would vibrate about either of these axes, as about a fixed axis, and its motion is always compounded of vibrations about these axes.

15. The length of the simple pendulum isochronous with the vibration about each normal axis is proportional to the square of the ratio of the corresponding diameter in the ellipsoid of equal energy to that of the momental ellipsoid.

16. The body is slightly disturbed from its position of rest by rotation about an axis of displacement, through an angle of displacement, and also by receiving a small angular velocity about an initial instantaneous axis. This displacement and velocity may be uniquely resolved into corresponding displacements and angular velocities about the normal axes, and thus the motion of the body is completely determined.