64 Prof. A. Gray on the Calculation 



distance from the origin ; «, £ the corresponding quantities for 

 the other, so that r = \/x l + a?, p = \Z^ + oc 2 ; and let dx, d% 

 be the axial lengths of the elements. Then if unit current 

 flow in each turn of the coils the currents in the elements 

 are n das, nd%. Writing down, then, by (1) the expression 

 for the energy of the two elements, and integrating from 

 x = x x to x=x 2 in the one case, and from S — Si to £=£2 ni 

 the other, we get for the mutual electrokinetic energy of two 

 coils of lengths # 2 — # 13 f 2 — fi, and carrying unit currents, 

 the expression 



I.4AW2^ ) Z l {ff){J"^f&} { jV'W)rff} • ( 2 > 



7i( '(<//) is found by differentiation with respect to cos (// 

 from the well-known expression, and f/p then written for 

 cos ft converts the result into a fraction the numerator of 

 which is a rational integral function of f only, and the deno- 

 minator of which is p 1-1 . This denominator is cancelled 

 by the multiplier p f_1 , and the second integral can thus be 

 found at once in all cases without any difficulty. Z/(<£) can 

 also be found by differentiation in the same manner, and the 

 integral then found by direct integration for each value of i; 

 but the following process, which gives by successive differen- 

 tiation of a simple function at once the indefinite integrals 

 required for the evaluation of the first definite integral, and 

 the values of Z/ ((//), is much more convenient. 



The solid angle subtended by one of the circles, say that, of 

 radius a and axial distance x, at a point distant p from the 

 origin is given if p < r by the equation 



■ = 2tt 1 1 - cos <$> + sin 2 <f> 2 j Z/((£) Z, (0) (£ Y 1 , 



(3) 



where 6 is the angle between the axis of the circle and the 

 line from the origin to the point in question. 



Now let the angle 6 be 90°, and a be zero, so that p is the 

 distance, y, of the point considered from the axis ; then all the 

 harmonics Z f (0) of odd order vanish, and the general expres- 

 sion for the harmonic of even order 2i is 



y 1.3...(2»-l) 



1 ; 2.4 ... 2z ' 



Hence 



2^ [l -*-«>{ J $Z,'(*)- 2 LJ^Z^) + ...}]. (4) 



