COFFIN. — THE SELF-INDUCTANCE OF COILS. 795 



sinh x = x + —. + —. + . . .; sin x = x — — + — 

 3 ! o ! o\ o 



x- x* , X' x* 



cosh a: =1 + 2] + 4n+ " ' * ; °° SX = * ~ 2! + T\~ 



Equation (10) may then be written 



L — Z 2 d /3 sinh cr — sin x\ 2 c? _. 



3 r^ V a: cosh x — cos x J 3 ?'i 



(20) 



(21) 



By this equation we see that the fractional change of self-inductance in 

 any coil may be calculated if the properties of the function 



3 sinh x — sin x -_ g . 



x cosh x — cos x 



be known for all values of a; from zero to infinity. 



A formula for small frequencies may be derived by expanding the 

 hyperbolic and circular functions and retaining only low powers of the 

 variable x. Q then becomes 



1 x* x s 

 1 1 1- • • • 



9 = 3 3! 7! H! 



1 X 4 x s 



1 1 h . . . 



2! 6! 10! 



which, retaining only terms in a: 4 is 



1 3x 4 6 a; 4 



_ 2 + TT _ + TV Gx 4 _ 14 x 4 _ _ 8 x" _ x*_ 



■ l 6! ~ 2x' 4_ + TT - TT~~ ~7T~ "630* 



2 z 4 ^ 6 ! 



This equation is applicable for small frequencies only, and (21) may be 

 written, since 64 tt 2 = 630 very approximately. 



L »- Lco =\- {I -<»*#**). (23) 



IiX) O Vi 



It agrees as to its form and dimensions with that of Max Wien.7 When 

 w = 0, it reduces to 2/3 d/r 1 . As a verification of the term 2/3 d/r 1} 

 the approximation formula for Bessel's formula with small argument 



II = n J r -i TinJL j»t s (24) 



V r r. 2 — r x 

 7 Loc. cit., p. 16. 8 Sorumerfeld, loc. cit, p. 679. 



