AND DOUBLE THETA-FUNCTIONS. 
927 
_ A (u ) , , A (u—u) , , bc\/X [(d—x)dx 
log 77- 7 = const. + — , -- -777-, 
A(%) °A (u + u) ^/af(a—x)J Vi\/X 
showing how the integrals of the third kind 
f(d—x)dx f(d—x)dx f(d — x)dx f {d—x)dx 
J Via/X ’ J V3\/ X ’ J V3\/ x ’ J (£C — £C') v/X 
depend on the theta-functions. 
If, instead, we work with the original equation 
p., 0 (u + u')G(io—u') _ D ~u ~Dhd 
°~° C 2 «.CV “ i ~c hi GV’ 
we find in the same way 
C'CQ CJm - A) _ C(^ + A) 
^ C (A) C (it—A) C (u + A) 
2t 2 KsnKv/cnKAdnKAsn 3 K?i.. 
1 — /c 2 sn 2 KAs:u 2 Bfit 
<Z . / D 2 ztD 2 A\ 
“ du ' l0g [ _ C%C%7 
= —log ( 1 — IdsirKu sn 3 K A), 
or multiplying by \du and integrating 
C'(A) , . . C (u-ii) 
“c(U +i 
which is in fact Jacobi’s equation 
7o 3 snKAcnKAdiiKAsn 2 K'a.K(di 
1 — & 2 sn 2 KAsn 2 Kht 
©A , - ~ 
U fan 2 (HI 
. , %(u — a) f ,sn a cn a dn a sn% du 
1 i™ A - 2 - ———;-7-’ =n {u, a). 
J 1 — A,sira sirii ' ’ 
®(« + a) 
I do not effect the operation but consider the forms first obtained, 
A(u-\-u)A.(u — u') = — Vi, &c., 
as the equivalent of Jacobi’s last-mentioned equation. 
Additioi i-fon i mice . 
56. The addition-theorem for the quotient-functions is of course given by means of 
the theorem for the elliptic functions : but more elegantly by the formulae relating to 
