Pulsating Spheres in a Liquid. 121 



When the axes of the doublet and of the derived sources are 



directed towards B, this becomes ( — ) n ~ l h ■ 



J 



Now let the spheres X, Y vibrate so that the normal 

 velocities are Zonal Harmonics uV n , vP m , having AB, BA as 

 their respective axes. 



Then, since 



*df n \r) r n+l 



X, if it vibrated by itself, would behave like a source of the 



a n+2 u 

 nth. order of strength — — f , with its axis along AB. 



Let </> A be the potential due to this source and its images. 

 This is found by calculating successive images of the complex 

 source H in Y and X, and making /= c after all operations 

 are completed. 



If Y vibrated by itself it would be equivalent to a source 



b m+2 v 

 of the mth order, f ; and with its images it would give 



a potential cf) B . 



Then, as before, if both spheres vibrate, 



2T/ P = -u^^ A PJS 1 -2v^<f> A P m dS 2 -v^<i> B P m dfi i . (10) 

 To evaluate J f $ A P m dS 2 , we proceed as follows:— 



a n+2 u 

 where ty is the potential due to a doublet — (with axis 



along AB) distant/ from B, and to the images of the doublet; 

 the subscript c denoting that after all operations are completed 

 c is to be written for/. 



Now, if v be a source of order zero distant f from B, <§> v its 

 velocity-potential, 



according as £> or <h. 



