170 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If we use as normal function the product 



Amn ■ e-^^' ■ sin — ^ • sin — -, (54) 



c c 



where A^ = Trp(m^ + ??^) (4 f^c') and m and n are positive integers, 

 and write 



W!=oon:=oo 



H = > ^Amn e ^ ' • sm — ^ ■ sm — -, (55) 



mrrl n^l 



this expression will satisfy all conditions if A^n he so taken that when 

 t = 0, the second number of the equation shall be equal to Hq for all 

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

 equation i* 



A^n = i^ Cdx fsin^.sin^-^^ (56) 



when either m or m is even, so that 



^' = rT5[(2 i- + 1)-^ + (-V + I)']- (58) 



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

 through the core at the time t is 



i=l A-=l 



or, if g = 7rp/4 /^c^ 



^* Byerly, Treatise on Fourier's Series, etc., § 71. Kiemann-Weber, Die par- 

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



