1880.] Mr Hicks, On two pulsating spheres in a fluid. 33 



cfQ 

 We require the value of b 2 —■ — 2Q, when r = b. 



Now O.-A.ftf+B.®'"; 



whence it may be shewn that 



(*£-») cla 



= 2«(2,»-3) J 4„_ 1 J„+2m(2m + 3)B,„. 1 i?„+^.-,B„-2.4,A,-,. 

 which determines X in terms of the A, B. 



9. It remains to shew how <£ can be determined in a series of 

 Zonal Harmonics. In doing this we will consider the fluid-motion 

 due to any number of spheres, lying in a line, either at rest, or 

 pulsating, or moving along the line in any manner. This is a 

 slightly more restricted case than is considered above, but the 

 method will apply to that as well. 



Take any portion of an image within the sphere B under con- 

 sideration, say /n n at a distance p n from the centre of B. The 

 potential due to this is 



p n \r \rj J 



therefore the part of B i due to this is — p> n p^lb m and 



1 



B, = - 



gm 2 (PuPj), 



the summation extending to all the sources and sinks within B. 

 So also 



A — Tt'V Vn 

 A-. — — 2* —^ , 



the summation extending to all without B. 



In the particular case we are considering, the sources and sinks 

 will be arranged in systems of mass-images, each consisting of a 

 source, and a line sink of constant density, and of magnitude 

 equal to the source. The part of B. depending on the mass-image 

 whose source is p, n is 



_ LL„. e^_ *£zfll 



*>"'r p " p. -p." ''+i 



VOL. IV. PT. I. 



