SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 45 



As the result is independent of the size of the segment, it holds true when the 

 segment is made small, and this can be proved in a couple of lines of elementary 

 geometry, as given in the 'American Journal of Mathematics' (A.J.M.), October, 

 1917, p. 237. Thence, by summation, the result for P holds in the same form when 

 the spherical segment has any arbitrary boundary not restricted to be circular. 



For the analytical expression of 12 the complete elliptic integral of the third kind 

 (E.I. III.) is required. This is not attempted by MAXWELL, and he leaves 12 in the 

 form of a series of zonal harmonics obtained and written down from the axial 

 expansion. 



But the chief difficulty in the theory of the ampere balance is the reduction and 

 manipulation of 12, a multiple-valued function with a cyclic constant 4-rr for a magnetic 

 circuit through the circle on AB, say round the circle on ED, linked with the electric 

 circuit round AB. 



8. The III. E.I. required for 12 will be of the same nature as B which occurs in 

 L (6), 2, (14), 3, and to obtain the relation, take MAXWELL'S M and differentiate 

 with respect to A, then 



A dQ _ 1 cZM f a cos 6 dd _ \ A cos 0(A+acos 0) 

 I 1 / 



db 2,r dK " J PQ J PQ 3 



Making use of the lemma, 



( 9 \ f db 1 b 



J PQ 3 MQ 2 ' PQ ' 



12 is obtained by an integration with respect to b, 



, Q x AT f i b la f Aacose(A + acos0) bde 



AQ = Jaco80th-'pQ^-j- -^QT- -p- 



Integrating the first of these integrals by parts, 



/ . b V* f - n Aasin0 bde f Aacos 6(A + a cos (?) Id 6 



th PQJo - J a 8m * "MO 5 " ' PQ ' T^T 'PQ 



rAct(Acos(9 + Qi) bdO 

 " J " MQ 2 ' PQ ' 



the * marking the place of a term which vanishes at both limits, 



bdQ 



(5) Q = constant 



= constant 12 (MQ), 

 suppose, where 



(6) 12(MQ) = f " AacOS + a2 bd6 



MQ 2 PQ 



H 2 



