48 SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 



In the reduction of this form we employ a new substitution, putting 



(5) cos 



where s is the new variable, and <r a constant ; and take s = s :t at A where QPY = 0, 

 cos'QPY = 1 = m*(<r-s.,), 



lc\ EX? - * QY 2 s-s, A a sin 2 6 = rf-i* . r*-r a * 



P(^~ <r -S 3 i PQ^'oSa r 2 4aV 



so that s = co at r = 0, oo. Then take s = s 2> s l for r 2 = +r.,r- A ; this makes 

 s a -* 8 ^r.-r.V / 2 A \ a Sl -s s /r,+r,V _ 2 A 



(7) 



>-3-^ 



With the 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 with 

 the elliptic integral 



(o\ K '" - " 



(9) S = 4 . *! s. *, s.s s :j , 2; = 4. Sj o-.o * 2 .o 



(10) > 1 >r >,>>,>-', 



(11) ca , az 



and with fractions e and /, such that 



(12) 2eK = f V^i-^Ms y K / = f" 



J3 \/S J(T 



(13) 2e K = sn- A/^^ - en- A/^ = dn-' J *-*= 



V s 2 -s 3 V s 2 -s 3 V Sj- 



(14) 



S 3 



