45 
SIR G. GREENHILL OX ELECTROMAGNETIC INTEGRALS. 
As the result is independent of the size of the segment, it holds true wlien 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 ii the complete elliptic integral of the third kind 
(E.I. III.) is required. This is not attempted by Maxwell, and he leaves 0 in the 
form of a series of zonal harmonics obtained and written down from the axial 
expansion. 
But the chief dithculty in the theory of the ampere balance is the reduction and 
manipulation of il, a multiple-valued function with .a cyclic constant Itt 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 Q 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 
( 1 ) 
. dQ _ 1 cZM 
dh ~ 2,r dl 
a cos Qd9 f An cos 0 (A + a cos B) dO 
PQ J PQ^ 
Making use of the lemma, 
(2) 
dh 
PQ3 MQ^ PQ 
II is obtained by an integration with respect to h, 
( 3 ) 
AH = 1 a cos 6 th ^ dO — 
Aa cos 0 (A + « cos 6) h dO 
Integrating the first of these integrals by parts, 
Aa sin 9 h d9 
MQ^‘ W 
(4) AH = (a sin 9 th ^ ~ \ ^ 
^ PQ/o 
An cos 0 (A +a cos 9) hd9 
MQ^ 
I An (A cos 9 +a) h d9 
MQ^ PQ 
the * marking the place of a term which vanishes at both limits, 
( 5 ) 
[ Aa cos 0 + cP b d9 
H = constant— - 
suppose, where 
( 6 ) 
MQ^ 
constant —H (MQ), 
PQ 
H(MQ) = 
Aa cos 0 + cP b d9 
0 MQ^ 
H 2 
PQ 
