128 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



This is equivalent to prescribing an arbitrary value of < over the 

 surface ; the value of <f> in the interior is thence determinate, 

 by Art. 40. To find it, we may suppose the given surface value 

 to be expanded, in accordance with the theorem quoted in Art. 88, 

 in a series of surface-harmonics of integral order, thus 



<j ) = S + S 1 + S,+...+ S n + .................. (1). 



The required value is then 



4> = 8 a+ r -S l + r +...+ r 8 n + ............ (2), 



for this satisfies V 2 <=0, and assumes the prescribed form (1) 

 when r = a, the radius of the sphere. 



The corresponding solution for the case of a prescribed value 

 of <f> over the surface of a spherical cavity in an infinite mass of 

 liquid initially at rest is evidently 



(3). 



Combining these two results we get the case of an infinite 

 mass of fluid whose continuity is interrupted by an infinitely thin 

 vacuous stratum, of spherical form, within which an arbitrary 

 impulsive pressure is applied. The values (2) and (3) of < are of 

 course continuous at the stratum, but the values of the normal 

 velocity are discontinuous, viz. we have, for the internal fluid, 



and for the external fluid 



g = - 2 (n + l)8 n /a. 



The motion, whether internal or external, is therefore that due 

 to a distribution of simple sources with surface-density 



over the sphere. See Art. 58. 



90. Let us next suppose that, instead of the impulsive 

 pressure, it is the normal velocity which is prescribed over the 

 spherical surface; thus 



............... (1), 



