SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
39 
2. An integration of M with respect to h will lead immediately to V. Jones’s 
expression for the mutual induction, L, between the circular current PP' on the 
diameter 2A, and a uniform current sheet flowing round the cylinder on the diameter 
2a, stretching from the circle AB a length h, up to the circle PP'; drawing out the 
circle AQB axially, like a concertina. 
The current sheet is taken as the equivalent of the close helical winding in the 
ampere balance of the wire on the C 5 dinder carrying the electric current and forming 
a solenoidal magnet, of which a constant L gives a line of magnetic force, the one 
passing through P, these lines circulating through the solenoidal tube and closing 
again outside. 
. In the hydrodyhamical analogue L would be the stream function (S.F.) of liquid 
circulating through the tube. 
Employing the lemma of the integral calculus, for the line potential of MP at Q, 
(2) L = [ M = [ [ 27rAa cos 0 = [ ^irKa cos d th~^ dB, 
Jo JoJo Jo 
and integrating by parts, with the lemma of the differential calculus 
(3) 
(4) 
d h _ Ka sin 6 h 
de ^ “ MQ^ '¥q 
L = 27rAa (sin 0 th"^ — [ 27rAa sin 0 ^ 
= * 
I 
iir 
PQ/o 
A^a^ sin^ B h dB 
MQ^ ■ PQ 
MQ^ 
hdB 
PQ 
the * marking a term which vanishes at the limits, and with 
(5) 4AW sin^ B = 4AW— (MQ^—A^)^ 
= -MQ^ + 2(a^ + A^)MQ^-(rt^A^)^ 
( 6 ) 
L 
= ixj 
= h\ 
-MQ’ + 2(a“ + V)-<^^^^ 
a^ + A^ — 2Aa cos B— ^ 
h dB 
b dB 
PQ ’ 
introducing the complete elliptic integral, I., II., III. ; and this is the expression 
employed by V. Jones, but obtained by a complicated dissection. 
