95-97] RANKINE'S METHOD. 139 



97. The motion of a liquid bounded by two spherical surfaces 

 can be found by successive approximations in certain cases. For 

 two solid spheres moving in the line of centres the solution is 

 greatly facilitated by the result given at the end of Art. 95, as to 

 the ' image ' of a double-source in a fixed sphere. 



Let A, B be the centres, and let u be the velocity of A towards B, u' that 

 of B towards A. Also, P being any point, let AP=r, BP = r', PAB=B, 

 PEA = B'. The velocity-potential will be of the form 



(i), 



where the functions < and $' are to be determined by the conditions that 

 V 2 </> = 0, V 2 <'=0 .............................. (ii), 



throughout the fluid ; that their space-derivatives vanish at infinity ; 

 and that 



dd> d<f>' 



-^r = -cos0, -j- =0 



dr dr 



over the surface of A, whilst 



-o =-*' 



over the surface of B. It is evident that is the value of the velocity- 

 potential when A moves with unit velocity towards B, while B is at rest ; and 

 similarly for 0'. 



To find 0, we remark that if B were absent the motion of the fluid would 

 be that due to a certain double-source at A having its axis in the direction 

 AS. The theorem of Art. 95 shews that we may satisfy the condition of zero 

 normal velocity over the surface of B by introducing a double-source, viz. the 

 'image' of that at A in the sphere B. This image is at H lt the inverse 

 point of A with respect to the sphere B ; its axis coincides with AB, and its 

 strength ^ is given by 



where /u, =|a 3 , is that of the original source at A. The resultant motion due 

 to the two sources at A and H l will however violate the condition to be 



