340 
PROFESSOR A. R. FORSYTH ON 
s 
*1 Vi *1 1 =°- 
x. 2 y 2 z. 2 1 
^3 y-i h 1 
Vi- 1 
If we clioose A = 0 and one of the two, B and C, to be unity and the other zero, 
X = 0 reduces to 
x 4 = 0 
and we may therefore take zero as the lower limits of all our integrals. 
Let 
«/i= + (l— as 2 )* 
% x — -f- (1 — ic^x~f j 
then ■± i y l are the roots of F»„ dz z i those of F*. We have 
J = 4y l z l = 4 (1 — x 2 . 1 — k 2 x 2 ) i 
and therefore by our formula (iv) 
f*=4r 
U=1J 
U dx 
o f(x)y(i-x*)(i-kwy 
:© 
1 
IT 
in which the 2 on the right-hand side implies summation for the expressions obtained 
by the substitutions 
y— y L and 2 = z i3 
v— V\ u z — z i> 
(i.) Let f(pc) = 1, U = 1 ; then the right-hand side vanishes and we have 
% + W3+ 
x=smL 
where 
Thus the preceding detenninantal relation will give sn^-ptto+t^)) which is —x 4 , 
in terms of the elliptic functions of u x , u 2 , w 3 . 
(li.) Let/(a?) = lj U = s 2 =l — k 2 x^-, then we have 
E(wi)+E(w 2 )+E(w 3 )d-E(H 4 ) 
_ A r — n,/ — CaI 
— Ci £ j- log (1 — Ax —By—Cz) j 
fh 1 , / I — Ax—C&j —By A , y 1 / l- Aai-hCgt + B,?/! 
— }h [ °g [l-A x -Cz l +Bi/J+y l - g U-Aai + C^-B^ 
