792 PROCEEDINGS OP THE AMERICAN ACADEMY. 



H= V - ". u " V rr-^—, ; # «*"* (8) 



Now, Sommerfeld 6 has shown that the expression 

 gifcfa-r) _ e — a&v-r) 



r e **(»V— rj _ e — tfc(r,— ri) 



is a very good approximation formula for the case in which k?\ and kr 2 

 are large, and he finds also that it is valid even when these quantities are 

 small, this being an unexpected result. 

 Since 



1 3r 



A7rq = ik V-^77 , .,, -,Ke tut (9) 





where in differentiating, on account of the largeness of h r only the ex- 

 ponential terms were considered variable. 



For the total current /per unit of length we obtain the value 



I-\, fqdr = - — ^ f^dr = — ^- T \H 7 . - H r A 



(10) 



• r_ H * J»t 



where .AT" is the number of turns per unit length of the coil. 



The electro-kinetic energy of a system is given by 



T=\LI* (11) 



where L is its self-inductance, and I is the total current. In the case 

 of alternating currents, the mean square values are meant, or 



t=\lt--. 



Hence we may write 



u 



o 



p i r 



= 2 4= (12) 



dt 



where t is the period. 



6 Loc. cit, p. 678. 



