41 
sm G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
( 10 ) 
( 11 ) 
( 12 ) 
(13) 
a ^— A ? _ \/(t— 
MQ" “ T-t 
_ y(-U) dt 
MQ" PQ T-t VT’ 
— U = 4 . —T . T —^2 • T —^3, T > 
(14) 
'■^^cd-AHdd ^ p 2 y(-U) dt 
0 MQ^ PQ }ts r-t ■ v/T ’ 
in a standard Weierstrassian form of the III, E.I, 
The expression of this TIL E.I. when complete, by means of the E.I., I. and II., • 
complete and incomplete, was given by Legendre, ‘ Fonctions Elliptiqnes,’ chap. 23, 
and (14) falls under his class (m'). 
4. But we shall avoid the Legendrian form, and start by making use of the 
lemma 
( 1 ) 
v/T 
dt T — t 
dr 1 — t 
proved immediately by effecting the differentiation. 
Integrating, in either order, with respect to the differential elements 
dt 1 dr 
v/T y-jj 
and between the limits ^2 > ^ > ts, and t^, t of r. 
( 2 ) 
(3) 
dr p- dt /rp d/|-v/T\ 
J.y-UJ^ 3 v/T ^ dt\T-t I 
d(i^\ 
dt\ T— t ) 
dt 
f dr 
J7^ 
. T- 1 )u 
- 0 , 
dt dr / T'\ d (2 \/ U\ 
TtJ. drX T-t ] 
dt [ d 
v/T I di 
U 
■-t 
d. 
_ { dt 
ix/-U 
J v/T V T-t 
_d^ j- yZ-U ^ 
Jts y/T T-t 
