56 
SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
Expanded in a series 
(2) 
til 
-1 _ v 
PQ 2n + iVPQ 
1 / h ^ 1 ^6 I 
2n+l\rJ (Ao) 
where, as befoi’e, in (3), § 3, 
(3) 
0 = 2 
O), 
r~ = cos^ n) + ’i '2 sin^ w = (w, y), y' = — > 
^3 
(4) 
G,h- 4- . adfl = 2 ^ (A)’”' r 2^5^ =4a2 -L (A)'** p, („), 
where P„('?() is the toroidal function, introduced by W. M. Hicks, ‘Phil. Trans.,’ 
1881-4, defined by 
(5) 
p.(«) 
do 
0 (ch 11 +sh u cos O)"'*'" Jo \ EQ 
" /EA. EB'\”+= 
= 2yyr('+hT*, y' = e-. 
0 \ y 
given by the substitution 
( 6 ) 
ch u + sli u cos 0 = 
diPn ^ EQ^ ^ EA.EB 
y EA.EB Eg 
2 ’ 
|0 = am {v, y); 
and P„ satisfies the differential equation 
( 7 ) 
d P 11 dP d i r\^ -1 \ dP / 2 1 \ T) 
^ “^ = 3c dO = 
with C = ch u, and the sequence difference equation 
(8) (2ii+])P„^i-4nCP„+(2n-l)P„_i = 0. 
Expressed otherwise, with u = eG, v = G + 2y’GT, 
(9) 
sn V 
0!’ + A 
cn V = 
ih 
Cl +A 
dn V = 
g —A 
CL + A 
0 = 2 am u, 
(10) th-' A = t'l-' = »' tan-' 
r Q sn V dn u sn v dn w 
= [am (u—v) — am (11 +v)]. 
(G) 
I = 2ai [am {u — v) — am {u + v)] d am u. 
Jo 
