Pulsating Spheres in a Liquid. 117 



Let (/> A be the velocity-potential in the liquid when A pulsates 

 and B does not, </> B the velocity-potential when B pulsates and 

 A does not. Then </> A + <£ B is the velocity-potential when both 

 pulsate. 



For if n h n 2 are normals to X, Y, drawn into the liquid, 



d<pjdn 1 = d, d(j) B /dn 2 = b, d^Jd?i 2 = 0, d^> B /d7i l = 0, (1) 



Therefore, if <p A -\-cf> B = </?, 



d$\dn x = a and d<f>jdn 2 = b. 



Also $ satisfies Laplace's equation, and vanishes at infinity. 

 If T be the kinetic energy of the liquid, p its density, 



the volume-integration extending through the liquid, and the 

 surface-integrals being taken over S 1? S 2 , the surfaces of X 

 and Y. 



But by the surface conditions (1) and by Green's theorem, 



JK4/ s -JM 2 ^ 



(3) 



Substituting <j> A -\-(f) B for $ in (2), and employing (1) and 

 (3), we have 



2T/ / o=-ajj0 A rfS 1 -26JJ^ A dS 8 -6jJ^ B rfS a . . (4) 



Let P (fig. 3) be the inverse of A in the sphere Y; Q, C 

 the inverses of P, B in X ; R, D the inverses of Q, C in Y. 



If Y were absent, the pulsations of X would 2'ive the same 

 velocity- potential as a source a 2 a at A. 



The image of this in Y is a line of doublets BP ; that of 

 BP in X is a line of doublets QC ; that of QC in Y is a line 

 of doublets RD ; and so on. 



Now the flow across the surface X due to BP, QC ; RD, . . . 

 taken in successive pairs is zero ; and the flow across Y due to 

 A, BP; QC, RD; . . . taken in pairs is also zero. 



</) A is therefore the velocity-potential due to these systems. 



It is simpler to transform each line of doublets into the 

 equivalent source and line-sink. 



Let AB = c, and denote a 2 d by ft. 



