42 
SIR G. GREENHILL 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 
W Vtv/-U’ 
* 
in which the variables are separated, t and r. Then with 
( 5 ) 
G 
_ v/(h ^.s) Q.' _ r 
J u \/"T J U 
/ \/(h ^ 3 ) 
'-U 
to the modulus y = aX~— co-modulus y' — ^^ with e and y to 
denote fractions, such that 
( 6 ) 
( 7 ) 
(») 
( 9 ) 
n/ f' \/(h ^3) 
y-u 
eG = ^ 2/G' = £ 
,G = sn - Vyf, = dn- V 
2/6' = s,.- y = cn- y = d„- y ^ 
/ + \ dt dr _ o yp/ r t — dt 
' •' yx y-u " J yx 
= 2yG' [ (I —dn^cG) deG = 2yG'(G—H), 
Jo 
Id 
^3 
where H denotes E (y), the complete II. E.I. to modulus y ; 
dT 
( 10 ) 
/ , \ dT dt _ r-\ 
(r-ts) yx “ ^ 
•2/ 
T — tg 
\/(h t'?) \/ U 
= G r dn^2/G'd2/G' = G(2/H' + zn2/G') 
J 0 
with H' = E (y') ; and then 
( 11 ) B = G( 2 /H'-fzn 2 /’G')- 2 /G'(G-H) 
= 2/(GH' + G'H-GG') + Gzn2/G' 
= Tzf + G zn 2y G^, 
by Legendre’s relation, and this is the equivalent statement of his equation (-^^i'), 
expressed in the notation of Jacobi. 
Then L is given by the three E.I.’s in the form 
( 12 ) 
L = ixP(a+ — )i + JM6-2xB(a’‘-A''), 
a 
and so may be said to be expressed in finite terms, that is, by tabulated functions. 
