170 



PROi - OF Till: AMERICAN M'\1H..MV. 



If we use as oormal function the product 



... . ///-./■ . >/->/ 

 . I rm • e • sin • sin — '-, 



1 5 1 



where \ : = -f)(nr + /r) < lin-i and m and n are positive integers, 

 and write 



m = os n = x 



II = > > J mn • e _A -' • sin — • sin — -, 



m=l tu=1 





this expression will satisfy all conditions if J,,,,, be so taken that when 

 t = (», the second number of the equation shall be equal to //„ for all 

 values of x and y within the filament. We have, therefore, the 

 equation 14 



J, 



and 



when m and n are both odd ; 



i _ " ; " 



- 1 m n — ' ', 



when either /// or n is even, so that 



// = 



!'••//„ 



y—OO A-=0O 



,— \"-t 



i\^(2j+ 1 (2 k -• 1) 

 y=o a_o 



\» = ^- a [(2*+l)»+(2i+l) a ]. 



Bin sin 



'' , -- 

 " 



(58) 



From (58) it appears that the whole Hux of magnetic induction 

 through the core at the time / is 



•'■> 



c 



—\n 



or, if 



^^r_\, + 1 .-■(■_•/■ I 

 9 = ~l> l !'•-• 



(59) 



14 Byerly, Treatise on Fourier's Series, etc., § 71. Riemann-Weber, Die par- 

 tiellen Differential-gleichungen der mathematischen Physik, Bd. II, § 99. 



