42 SIR G. GEEENHILL ON ELECTROMAGNETIC INTEGRALS. 



as in (14), 3 ; so calling it 2B, as a standard type of the III. E.I., it is proved by 

 the lemma (l) that 



in which the variables are separated, t and T. Then with 



to the modulus y = \f -, and co- modulus y = \f -^^ , an d w ^h e and /to 



V tity v EI 1 3 



denote fractions, such that 



(6) eG , f V 



J 



f 



V 1 



(8) 2/G' = m - , = c,r' , = dn- 



tg 



t-t* dt 



= 2/G' f (I -dn 2 eG) deG - 2/G' (G-H), 



Jo 

 where H denotes E (y), the complete II. E.I. to modulus y ; 



(10) {{(T t} dT dt - G " T ~ t 



^ " 



.= G dn 2 2/G' c22/G' = G (2/H' + zn 2/G') 



Jo 



with H' = E (y') ; and then 



(11) B = G(2/H' + zn2/G')-2/G'(G-H) 



= 2/(GH'+G'H-GG') + Gzn2/G / 

 = 7r/+Gzn2/G', 



by LEGENDEE'S relation, and this is the equivalent statement of his equation (m'}, 

 expressed in the notation of JACOBI. 



Then L is given by the three E.I.'s in the form 



(12) L = 



a, 

 and so may be said to be expressed in finite terms, that is, by tabulated functions. 



