MnncJicster Memoirs, Vol. I. (1906), No. 8. 9 



Hence 



"^rA,. = / cos r^^(^ 

 



— / COS r(0 + iijSin0 + ^jSin 2^ 4- ...)r/<^ . (10). 



The series for kx—<^ can also be similarly replaced by 

 a series in terms of sines of multiples of ^, 

 As before we are entitled to assume 



kx - <^ = J/j sin n + A[„ sin 2^t + . . . (11). 



Treating this equation as we have just treated (9), we 

 obtain 



"^rM^ = / — i^-'v - <f) cos rfxdi.1 

 2 J du. 



' 



= / — (-^.v - (h) COS rud(h 

 J df 



== / ^I/„t COS m(p cos r{(p + ^j sin (/< + ^., sin 2f... )d(p 

 / ' . (12). 



Finally the form properly to be assumed for k{y — b') 



is 



k{y - />)= - A^j cos iLi - iV„ cos 2^1- ... (13). 



and, as before, 



— {ky) sin rudu 

 dp. 



= / — {ky) sin rj^idij) 

 J ^0 



= / 277,,, sin w0 sin r(0 + ^j sin(^ + ^., sin 20 + ...)^0... 

 / ' ■ • (14). 



