48 SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
Tn the rerhiction of this form we employ a new substitution, putting 
(5) cosmPY = pL^ = m“(—s), 
where s is the new variable, and a- a constant; and take s = at A where QPY = 0, 
QY^ _ s — s^ _ sin^ d _ r^—r\r'^ — 
W ~ 7^, ~ 4^^ ’ 
Then take s = s.^, for — ±r.jr^; this makes 
53 /C3—>’2^ 
<1 
1 
^ Si S3 _ 
(r^ + r.^^ _ 2A 
cr —S3 2a y 
+ r^) 
<T — S3 
( 2a / 
cosH^PY = 1 = ‘nf{rr — s^), 
( 6 ) 
PY^ 
PQ^ 
(T — S 
(T — So 
so that s = cc at r = 0, oo. 
With tlie variable s we are employing the elliptic function has a new modulus k, 
obtained by a quadric transformation of the former modulus y, and associated wdth 
the elliptic integral 
( 8 ) 
(9) 
( 10 ) 
x/{s^-s^) ds 
K' 
v/(gi-g 3 ) dry 
S 4 . S . So S . S S3, 2 — 4 . Sj a. (T S2. (T S3, 
00 > Si > (T > S 2 > S > S 3 > — 00, 
( 11 ) 
^ 
\7’3 + 7’s 
K 
1 -y 
l+y'’ 
and with fractions e and f, such that 
( 14 ) 
