SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 53 



13. Next, for < and Q, 



/,\ 7. PQ" bdO _ <TS : , b ^/(rrs-^ds 



~PY'"PQ ~^7' a 



(2) Sl ~* = 



_ 



a s 3 \ 2a / o s :i \ 2a / ' <r * :t a' 



(3) 

 V ; 



(4) $ = 4 



in accordance with the previous expression for E, or B (2/G'), and 



(5) J2 = 2*-- * = 2 T (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 sm/ K' = G z 1 1 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' = n y cn 2 / G/ , or 2 



1+y 



squaring it, and integrating both sides with respect to / 



According as the modulus y or K is employed, connected by the quadric transfor- 

 mation, as in MAXWELL'S ' E. and M.,' 702, we take, to the modulus y, 



(8) p = 4oG = 



r 



M= 



Q(/) = Q = 27r(l-/)-2Gzn2/G'-2Gy'sn2/G / , 



,, n/ 6 MP ,, n , a-A MB 



= = > ~ 



dn 2/G ' = ^ = E = C o S BAP = cos FPi/, y 

 r 3 liiJJ 



I 2 



' = 



