122 On two Pulsating Spheres in a Liquid. 



Whence, for a doublet v at the same point, with axis 

 along AB, 



JJ m v " 2 (2m+l)r +2 (2m+l)& m_1 ' 



^ being the velocity-potential due to v. 



But taking lj>b, the image of the doublet v is — vb s /^ 3 , 

 distant b 2 /£ from B. 



The two doublets together give the term 



A ^ 



and it remains to evaluate 



t-r^S). 



d/r L \ r 



a w+2 w 



where the summation extends to the doublet — r (or a) 



n + 1 I ' 



and its images outside Y. For this doublet 



* = /*; ?=/• 



For Q, the next image outside B, 



- a%z z- °lL 

 V ~ P {ef-b 2 ) 3 ' *- G c f_ b r 



The succeeding image introduces a term of order c -12 com- 

 pared with that due to /it ; we shall neglect this. 

 Making /== c after all operations are over, we have 



"*JJ P.*A^,= cm+n+1 ( m+1 , n+1 , 



+ M 1 + ? + ?j + •■••/' 



where 



/? = (m + 2) (m + 3) « 4 + 2(m + 2) (n + 2)a 2 6 2 + (n + 2) (n + 3)b\ 

 It remains to find j* f^P^Si, from which Jf0 B P TO ^S 2 



can be derived by symmetry. 



The source //, at A contributes a constant, which we need 



not write. 



The images P and Q contribute a term found by making 



v = ^ a lf, e = o-v/f. 



The image H and its successor contribute only a small 

 term ; here f can be regarded as constant, since / only 

 occurs in a term of order c~ 5 ; and v is approximately 

 pa s b 6 c- 6 f-\ 



