125.] TWO SPHERES. 145 



where M is the mass of fluid displaced by the sphere. Hence 

 the principal effect of the plane boundary is to increase the inertia 



3a 3 



of the sphere in the ratio 1 + -3- : 1, c denoting double the 



c 



distance of the centre of the sphere from the plane. 



(6) Let us suppose each sphere constrained to move with 

 constant velocity. The force which must be applied to B in order 



to maintain this motion is C- u* approximately, and is in the 



c 



direction BA. The spheres therefore appear to repel one another. 

 The forces to be applied to the two spheres are not equal and 

 opposite except when v = u. 



(c) Let us suppose that each sphere makes small periodic 

 oscillations about a mean position, the period -being the same 

 for each. The average value of the first term of (52) is then 

 zero, and the mutual action of the two spheres is equivalent to 



a force ^ uv, urging them together, \\here uv denotes the 



C 



mean value of uv. If u, v differ in phase by, less than a quarter- 

 period this force is one of attraction, if by more than a quarter- 

 period it is one of repulsion. 



(d) Let A perform small periodic oscillations while B .is 

 held at rest. The mean force on B is now .zero to our order of 

 approximation. To carry the approximation further, we remark 



that the mean value of -f- at the surface of B is necessarily zero, 

 ctt 



and that the next important term in the value (51) of the semi- 



j|ML 



square of the velocity is, when &amp;gt;v = 0, ^ -y u~ sin 2 ^ cos ^, and 



the resulting term in (50) is. found on integration to be - 5 2 , 



c 



where ?7 2 denotes the average value of the square of the velocity 

 of A 



This result comes under a general principle enunciated by 

 Thomson. If we have two bodies immersed in a fluid, one of 

 which A performs small vibrations while the other B is held 

 at rest, the fluid velocity at the surface of B will on the whole 



L. 10 



