794 PROCEEDINGS OF THE AMERICAN ACADEMY, 



and h(r 2 — r 1 )=y 1 , 



we obtain for Hand its conjugate H, the expressions : 



l r Jy P v _ p—iy p—ii 

 ylr, _^j !__!_ ff e ^t 



V r pVJr e Vi _ p-nj! e ~Vi 



and (18) 



e iy e~ y 

 e l1Jl e~^ 



/,. p—iy p>j _ pW p—y 

 ff= V- 1 ±-± 6 —^— H e~^ 



A conjugate expression is obtained by putting — i for i everywhere in 

 an equation. 



Their product becomes 



or 



zi if ui r i cosh 2y — cos 2y 

 JtlH = Ji Q ' — — — . 



r cosh 2 y x — cos 2 y x 



Multiplying this expression by r dr and integrating from r x to r 2 gives 



cosh r dr = sinh r 



/« 



putting x = 2 h (r. 2 — i\) = 2 h d 



H 2 r x d sinh x — sin x 

 8 x cosh x — cos x ' 

 We may now write 



2 A72 2 [~i 2 d /3 sinh a? — sin a; \ ~j , 



>co 3 i\ \x cosh a; — cos ay J 



where a; = 2 c? V2 tt w o- = (4 7r rf \/<t) y/n where a> '= 2-rrn 



Now, 4 tt 2 N 2 i\ 2 is the self-inductance per unit length of a coil with a 



mean radius r lt and is therefore the self-inductance for infinite frequency, 



according to the remarks of Wien. This is also shown by equation (19), 



sinh x 

 as the second term in parenthesis is zero when x = co for — - — — 1, 

 1 cosh x 



and sinh x and cosh x increase without limit as x increases without limit, 



as may be seen by their expansions 



