Sm G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
We may drop G' without ambiguity, and then 
(16) 
(17) 
(18) 
V 6n f sn f - 
cn/, dn/ = . 
h ^'2 • h 
cn/, dn/ = ^ 
2ah 
h "^2 
[a. + /(A^ + 6^)]^’ 
(19) sn (/,-/,) = gP -//■ = Bn 2./G- 
I—y sn/] sn/, 
r. 
/ 2-/1 = 2/ cn 2/G' = 
a —A 
ro 
dn 2/G' = 
d + A 
r. 
(20) sn (/+/) = 
ah 
a 
,-V(A' + «>’) “ v/(A4-?.“) 
OP' 
OB 
sin BOP = sin OBP', 
OBP' = am (/ 2 +/ 1 ) G' = am 2/'G', suppose. 
When OP is produced to cut the circle on AB in R, and the circle on ED again in 
P', PP' will touch the co-axial circle in R; and by the poristic property of these 
circles with the elliptic function interpretation, 
(21) OBPi = am/G', OBP, = am/G', 
if the tangent at the lowest point e, e' of the R circle, and of the other co-axial circle 
touching P'P", where EP" = EP, cuts the circle on ED in Pj and P,. 
22. Treating Q (PR) of (3), § 9, in the same way 
( 1 ) 
Q (PR) = 
i ^3 ~h ii, 
^ b do 
J PQ ’ 
0 if \/{a^ + h^)- 
-A h do ( 
» _ 1 1 \/{a^ + h^) +A 
b do 
^ J /(d^-t/P) -\-a cos 6 PQ 
5 
* ^ [a^ +1/) — a cos 
0 PQ ’ 
and 
a similar reduction will give 
(3) 
II 
1 y-u., 
1 T 3 — t 
dt 
^/T“ 
■^/i + 2G zn/G', 
(4) 
—--i 
II 
II 
7ry4 4-2G zs/G', 
cc > G> T, > G. > ^ > G > T4 > - 0°, 
