with similar expressions for the remaining components of the impulse. We have here written for shortness

It is plain that §。, no so, λo, Mo, vo, are the components of the impulse of the cyclic fluid motion which remains when the solid is (by forces applied to it alone) brought to rest.

122. As a simple example we may take the case treated by Thomson*; viz. where the solid is a circular ring (of any form of section), and has therefore only one aperture. If we take the axis of the ring as axis of x, we see by the same reasoning as in Art. 116 that if the situation of the origin in this axis be properly chosen we may write

dp

=0, or p= const. as is

dt

The fourth of equations (20) then gives obviously the case. Let us suppose that = p 0, and that the ring is slightly disturbed from a state of steady motion parallel to its axis. In the beginning of the disturbed motion v, w, q, r are small quantities whose squares and products we may neglect. The first

of (20) then gives

du

dt

=0, or u const., and the remaining equations

=-(Au+§)r, Qd2 = - {(4 − B)u + §}w,

dq

dt

dt

right-hand side of (48) should be negative; and the time of a small oscillation, in the case of disturbed stable motion, is

123. The general equations of motion of the ring are also

satisfied by ξ, η, ζ, λ, μ = 0, and v constant. We have then

The motion of the ring is then one of uniform rotation about an axis in the plane yz parallel to that of y, and at a distance

r

from it.

Case of two or more moving solids.

124. The foregoing methods fail when we have two or more moving solids, or when the fluid does not extend in all directions to infinity, being bounded externally by fixed rigid walls. In such cases we may suppose the position at the time t of each moving solid to be defined by means of six 'co-ordinates,' in the manner explained in treatises on Kinematics. It is easy to see that must be a linear function of the rates of variation of these coordinates (in other words, of the 'generalized velocity-components' of the system), and thence that the kinetic energy of the system is, as in Art. 110, a homogeneous quadratic function of these generalized velocities, with however the important change that the coefficients in this function are not constants, but themselves functions of the co-ordinates of the system. The equations of motion are then most conveniently formed by Lagrange's method*, the applicability of which to systems of the peculiar kind here considered requires however to be in the first place established†.

The accompanying references will be of service to the reader who wishes to pursue the study of the general problem in the manner indicated. We content ourselves here with the discussion of a very simple case in which the forces acting on the solids can be readily calculated by the direct method.

125. Let us suppose that we have two spheres in motion in the line joining their centres A, B. Let u be the velocity of the first in the direction AB, v that of the second in the direction BA. Further, P being any point of the fluid, let

also let a, b be the radii of the spheres and c the distance AB of their centres. If the sphere B were absent, and its place occupied by fluid, the velocity-potential 1 due to the sphere A alone would be, by Art. 102,

3

cos θ.

To find the value of 6, in the neighbourhood of B we have

S

2

=(1+2cosx+33 cosx-1+ &c.) *.

C

COS

This gives at the surface of B,

αφ

ds

1

=

8

2

- (2cosx+2.3.0.3cosx-1+ &c.).

The relation which actually holds at the surface of B, viz.

cos

' zonal harmonics' of orders 1, 2, &c. respectively. In fact, remembering that γά is the potential at the point P due to a small magnet of unit moment placed at A with its axis pointing in the direction AB, we readily find from the definition of the aforesaid zonal harmonics Q1, Q2, &c., that

is however no longer satisfied; but it is plain from the course of the above investigation that the error in the normal velocity there

ab3

will be of the order and that if this be rectified by the addi

,

tion of a properly chosen term, to the above value of $, the effect of this at the surface of B will be of the order

In the

particular cases examined below we shall suppose a and b both small in comparison with c, and shall not take into account small quantities of so high an order as that last written. We have then at the surface of B

as

Зи

a

=-10(+3) cosx-b2 u. 3 cos2x-1&c... (49),

2

2

The total effect of the fluid pressure on the sphere B evidently reduces to a force in the direction AB, the amount of which is

0

p. 2πb sin x. bdx. cos x............(50),

dφ where p is to be found from (18). In calculating at we must remember, as in Art. 102, that the origin B of the polar co-ordinates s, x is itself in motion with velocity v in the direction BA. The rates at which the values of s, x for a fixed point are increas