136 PROBLEMS IN THREE DIMENSIONS [CHAP. V 



Hence in the case of irrotational motion we have 



- n 

 -f , -f- = sm0 ,J- ............ (8). 



dr' dr d6 



Thus if 0=r*S n .............................. (9), 



where 8 n is any surface-harmonic symmetrical about the axis, we 

 have, putting p = cos 0, 



n , 



dr 



The latter equation gives 



^' ............... do), 



which must necessarily also satisfy the former; this is readily 

 verified by means of Art. 85 (1). 



Thus in the case of the zonal harmonic P n , we have as 

 corresponding values 



and <f> = r~ n ~ l P n (/A), i|r = r~ n (1 /^ 2 ) , (12), 



of which the latter must be equivalent to (5) and (6). The same 

 relations hold of course with regard to the zonal harmonic of the 

 second kind, Q n . 



95. We saw in Art. 91 that the motion produced by a solid 

 sphere in an infinite mass of liquid was that due to a double- 

 source at the centre. Comparing the formulae there given with 

 Art. 94 (4), it appears that the stream-function due to the 

 sphere is 



* 2 ,v, " V /* 



The forms of the stream-lines corresponding to a number of equidistant 

 values of ^ are shewn on the opposite page. The stream-lines relative to the 

 sphere are figured in the diagram near the end of Chapter vu. 



