the Goldschmidt Dynamo Equations. 791 



This solution is excellent, given the legitimacy of a Fourier 

 expansion ; but my brother distrusts this as manifestly divergent, 

 and considers that, apart from physical considerations and elec- 

 trical devices, the strict solution is an exponential one. His own 

 treatment is as follows : — 



Given nx= -j^(y cos 0) (1) 



dd 



d L 

 dd 



n'(y-b)= ^.(x cos d\ (2) 



nn 



' ; ^~ 5)= ^{ COS0 j0 (yCOS ^}' ' * ^ 

 Let y cos d = z, tan = sinh u ; and call wn' = N a . 

 Whence sec — cosh u = du/dd ; 



dz d d dz d 9 z 



also £ =«••£<* «• •); and ^ • Ju = sec fl^,. 



Therefore the differential equation becomes 



cl 2 z 



— 2= N 2 (i/-b)cose; (4) 



i. e., %1 -N 2 2 = - N 2 6 sech u=~N a & . 



= -2jS T2 6(e-«-e- 3il + e- 5w + . . .). 



2N 2 6 

 r = A cosh Nw + BsinhNw4-^— 727^7i{ <? " M -^ 3, ' + ^" 5tt - + ...}. (5) 



Now TTT^ — ^^ !f = 





c £-u p-zu e~~^ w ~\ 



?/ cos = z= A cosh Nit 4- B sinh Nm + 2N 2 6 -J ^- 2 __ 12 — ^ 2 _g, + ^ 2 _ 52 . . . j- 



8= * = N(Asmh N !( +BeoshN«)-2N s 6| j^-p- j^. . . J , 



nx cos 



where cosh u = sec = sec;:>£, and N 2 = BB//jp 2 M 2 , as stated 

 above. 



That this solution satisfies the equations (1) and (2) can be 

 verified : but the series of circular functions is more practically 

 useful, since the higher frequencies which go to build up the 

 above total result are barred by the conditions of the actual 

 problem, 



3B2 



