1879.] Mr Hicks, On two pulsating spheres in a fluid. 277 



masses of the spheres, neglecting fifth and higher powers of the 

 ratio of the radii to the central distance. The object of the present 

 communication is twofold, first to demonstrate a remarkable re- 

 lation between the successive "mass-images" of a source in the 

 centre of one sphere which enables us to determine rigorously the 

 action between two spheres ; and secondly, to show how this action 

 may be applied to explain gravitation, and especially the gravita- 

 tion of the vortex atoms of Sir William Thomson. 



2. It is known* that if a source of fluid P exist in an infinite 

 incompressible fluid in which a sphere is at rest, that the motion 

 of the fluid is compounded of that due to the source and a certain 

 arrangement of a source and sinks within the sphere. The image 



consists of a source -j-p at the inverse point of P, and a line sink, 



whose line density is — - , thence to the centre of the sphere, 



where /x is the magnitude of the source at P and a the radius of 

 the sphere. If now we have in a fluid two spheres A, B, oi which 

 A is performing pulsations, then at any time, the motion of the 

 fluid due to A alone would be that of a source at the centre A of 



magnitude /u. = — a^ -j- „ But on account of the presence of B we 



must suppose an image of this, as above defined, within B, and 

 again an image of this within A, and so on indefinitely. It might 

 seem that these images would soon become extremely complicated, 

 but a remarkable relation holds between the successive images, 

 which enables us to determine them all, and which we proceed to 

 demonstrate. 



It will be seen that in the image the amount of the whole 

 line sink is equal and opposite to the source, as clearly ought to 

 be the case, since no fluid must on the whole be generated or 

 destroyed within the sphere. Suppose now in a fluid in which 

 the sphere A is at rest, that there exist a source of fluid (/*) at P, 

 and a line sink thence to Q, FQ being in a line with the centre 

 oi A, and P being nearer the sphere than Q. Suppose also that 

 the whole amount of the line sink is equal to the source — in other 



words, that the line density is - p^. Then, if P', Q' be the 



inverse points of P, Q with respect to the sphere, the image of the 



above arrangement will consist of (1) a source at P' = ^j^ =/"-'; 



and (2) a constant line sink from P' to Q\ whose line density is 



* Proceedings, Royal Society, No. 197, 1879. 



21—2 



