490 MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
or 
r> 
which can clearly only happen if 2p-\-l <3 or the density of the sphere less than the 
fluid. 
In general, then, when the body is denser than the fluid it is attracted. If its 
density is less than the fluid there will be a critical point (as mentioned by Sir W. 
Thomson), beyond which there will be repulsion, and within which it is attractive. 
This critical distance is given by 
VI.-V5 
+ 1 
3 
(24) 
in using which it must be remembered that if r comes out nearly equal to b, the 
formula fails to give a correct value, as it was obtained on the supposition that the 
distances were large. It is, however, extremely accurate if we remember that it is 
true up to inverse powers of the twelfth at least. If the density of the sphere be ‘9 
the critical distance would be 7'648 times its radius. It may be noticed that while 
the principal term in the acceleration depends on r~~, if the density be the same as 
the fluid it depends on r~ 9 . 
In the case of a sphere vibrating within another sphere, along the line of centres, 
the effect of the fluid will be represented by supposing the inertia of the sphere 
increased by a mass 
= ^{l + 3Q(q.q 1 ) X mass of fluid displaced by it 
where Q has the value given in § 9: provided it is not close to the boundary of the 
containing sphere, as in that case ~ becomes infinite, and the small motions of the 
sphere will produce great changes in the value of Q. When its mean position is the 
centre, yr = 0 and Q may be considered constant when we neglect in our equations of 
motion cubes of small quantities. The value of Q in this case is, as lias been already 
mentioned, 
1 b z + 2ft 3 
z'tf-a? X 
mass of fluid displaced 
The foregoing serves to solve the problem of a ball pendulum within a spherical 
envelope when it is so suspended that its centre lies in the horizontal line through the 
centre of the envelope. When it oscillates in any other position the value of the 
coefficient of inertia may be approximated to as in §§ 15, 16. 
