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 dty/dc, where < has either of the above forms ; so that 

 the required potential is, for r < c, 



9 P *} P f(\\ 



Zij. , o.t o \ O /. 



f& c^ c 



and for r > c, 



1 ^.2 ' 2 |*3 ' * \ /' 



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

 can be put into the more compact form* 



where 



2w-5 



7 - P 



"H ; -L n-i 



1 . n O ^7fc L) 



This function Q n (/A) is sometimes called the zonal harmonic ' of 

 the second kind.' 



Thus 



(M) = i (5^ - 8/i) log J-+ - {/ + f. 



* This is equivalent to Art. 84 (4) with, for n even, A = 0, B = ( - )i t - ' ;" 7 \ x 



1 . o...(/i 1) 



whilst for n odd we have A =*(-)*+*> ' .'", , ^ = 0. See Heine, t. i. 



3 .O...7t 



pp. 141, 147. 



