136 PROBLEMS IN THREE DIMENSIONS [CHAP. V 



Hence in the case of irrotational motion we have 



smM0~ dr dr~ M 



Thus if $ = r n S n .............................. (9), 



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

 have, putting //, = cos 6, 



d * _ nr n+i S d _- r n Q _. ^ d8 



d^~ * n dr~ ^ } d^ 



The latter equation gives 



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 * = r P B Gt), * =- &quot;(l -tf ...... (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 formula? there given with 

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

 sphere is 



^ = -i U -sin 2 ........................ (1). 



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

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

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



