126 COEFFICIENTS OF INDUCTION. 



In order to abbreviate the calculations, we shall put, for example, 



We have then, for the first terms, 

 M 3* 



_ =I+27 _ 3/J __ 



+ 0-3750001 [1 + 127-15/5] 



+ o-23437a 2 [i + 307 -350] 



(18) + o-i7o9oa 3 [i + 567 - 63/2] 



+ o-i3458a 4 [i + 907 - 990] 



i + 1327- 1430] 

 i + 1827- 1950]. 



An analogous expression appended to expression (41) of (741) 

 will give the coefficient of mutual induction of two very distant coils ; 

 with the expression (44) of (742) we shall in like manner obtain the 

 coefficient of induction of a long coil on another coil in the interior. 



When the convergence of the expressions thus obtained is not 

 sufficiently rapid, recourse should be had to the use of elliptic 

 integrals, of which they are indeed only the development in series. 



763. DETERMINATION OF M BY ELLIPTIC INTEGRALS. If r be 

 the distance of the two elements ds and ds' of the contours, e the 

 angle which these two elements make between them, we might 

 calculate the coefficient of mutual induction by Neumann's formula 



M = 



If < and <' are the angles which the elements ds and ds' make 

 respectively with a fixed plane passing through the axis, the distance 

 r is given by equation. 



r 2 = x 2 + a? + a' 2 - 2aa'cos (</> - <') . 

 Since we have 



