1880.] Mr Hicks, On two pulsating spheres in a fluid. 31 



But by Laplace's equation 



where in -^ , ^-? ... r is put = b, and therefore ~ = u ; 

 or or or 



so that 



/= | J*^ |^26 M + 6 2 |f - 2</>) sin 20 + 2 cos 2 1|| ^ - | [<£ 2 f. 



Now ["" 2 cos 8 .^d9= (V sin 20e*0+ [<ff = £ + [^f, 



Jo . o" Jo 



and r</> ^ sin 26d0 = f U ?j£- if) sin 20eZ0 



whence 



and 



when r — b. 



d 2 C" ' 3 2 



7 = ||2^P-Q + i6 2 ^|, 



8. We must now shew how P, Q may be determined when all 

 the images are known. Still considering the general problem of 

 symmetrical motion, it is known that <f> can be expanded in a 

 series of Zonal Harmonics 



Kb/ * \rJ 

 ^tCfv say. 



The values of A , , B t being supposed known we have at once 



P=2.§.C 1 = 3C 1 . 



