89-90] SPHERICAL BOUNDARY. 129 



the term of zero order being necessarily absent, since we must 

 have 



I 



^cfe = 0. 

 dr 



on account of the constancy of volume of the included mass. 

 The value of < for the internal space is of the form 



n S n + ......... (3), 



for this is finite and continuous, and satisfies V 2 $ = 0, and the 

 constants can be determined so as to make d<f>/dr assume the 

 given surface-value (1); viz. we have nA n a n ~ 1 = I. The required 

 solution is therefore 



The corresponding solution for the external space is found in 

 like manner to be 



(5) ' 



The two solutions, taken together, give the motion produced 

 in an infinite mass of liquid which is divided into two portions 

 by a thin spherical membrane, when a prescribed normal velocity is 

 given to every point of the membrane, subject to the condition (2). 



The value of $ changes from aSSn/n to aZS n /(n -f 1), 

 as we cross the membrane, so that the tangential velocity is now 

 discontinuous. The motion, whether inside or outside, is that 

 due to a double-sheet of density 



, w , 



4?r n(n+ 1) 



See Art. 58. 



The kinetic energy of the internal fluid is given by the 

 formula (4) of Art. 44, viz. 



ch r ............ (6), 



the parts of the integral which involve products of surface - 

 harmonics of different orders disappearing in virtue of the 

 conjugate property of Art. 88. 



L. 9 



