SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
53 
13. Next, for $ and 
7 , _ PQ^ hdO_rr — S:^ h — 
PY"^'p(y “ ys 
( 2 ) 
Si-a- ^ / n + r^ _ ^ 
0 -—S 3 \ 2a / 
(T —S 3 \ 2a / 
\/{Si — (T ■ q- —S2) _ & 
(T—S 3 a 
( 3 ) 
( 4 ) 
do 
ds 
^-s yS’ 
0 = 4 
ds = 27 r/+ 4 Kzn/K', 
S3 iT 
-s ys 
in accordance with the previous expression for B, or B {2fG'), and 
( 5 ) Q = 27r-0 = 27r(l-/)—4Kzn/K'. 
Comparing this with the previous expression for Q in (ll), § 8,we have the theorem 
of the quadric transformation of the zeta function 
(6) 2K zn/K' = G zn 2/G' + Gy' sn 2/G'' 
This is obtained by taking the quadric transformation formula, obtained from the 
geometry of fig. 2, 
(7) dn/K' = 2/G' + y'cn 2fG' ^ 2K dn/K'= G dn 2/G' + Gy'cn 2/G', 
1 +y 
squaring it, and integrating both sides with respect to / 
According as the modulus y or /c is employed, connected by the quadric transfor¬ 
mation, as in Maxwell’s ‘ E. and M.,’ § 702, we take, to the modulus y. 
(8) p^^^4aGv/y'^ = 4Gy'sn2/G', 
'^’.3 \/ ('^V’.s) 
M = -27r7’3[(2-y')G-2H], 
L2 (/) = Q = 27 r (1 -/)-2G zn 2/G'-2Gy' sn 2/G', 
S2 (/) + Q(l-/) = 2x-Pb 
811 2/G' = 
MP 
PB ’ 
cn 2/G' 
a - A _ MB 
^"PB’ 
dn 2/G' = = ^ = cos BAP = cos FPp', 
Vg cjU 
I 2 
/ 
y 
PB 
PA’ 
