

124 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



We can hence deduce expressions, which will be useful to us 

 later, for the velocity-potential due to a double-source of unit 

 strength, situate on the axis of a? at a distance c from the origin, 

 and having its axis pointing from the origin. This is evidently 

 equal to dcfr/dc, where &amp;lt;/&amp;gt; has either of the above forms; so that 

 the required potential is, for r &amp;lt; c, 



and for r &amp;gt; c, 



The remaining solution of Art. 85 (1), in the case of n integral, 

 can be put into the more compact form* 



where 



This function Q n (yu,) is sometimes called the zonal harmonic of 

 the second kind. 



Thus 



- 3/t) log } + - 



2 4 n 



* This is equivalent to Art. 84 (4) with, for n even, A = 0, B = ( - )i .&quot; ; 



1 . o. ..(/I 1; 



whilst for n odd we have ^^(-)^(+D 2 - tl &quot;^ n ~ 1 - , B = 0. See Heine, t. i., 

 pp. 141, 147. 



