89-90] SPHERICAL BOUNDARY. 129 



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

 have 



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

 The value of &amp;lt;/&amp;gt; for the internal space is of the form 



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



for this is finite and continuous, and satisfies V 2 &amp;lt; = 0, and the 

 constants can be determined so as to make d(f&amp;gt;/dr assume the 

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

 solution is therefore 



The corresponding solution for the external space is found in 

 like manner to be 



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 cf&amp;gt; changes from aS n /n to aZS n /(n +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 



See Art. 58. 



The kinetic energy of the internal fluid is given by the 

 formula (4) of Art. 44, viz. 



*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 



