206 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
[665 
Hence in AH the irrational part is 
2 i a ! + a 2 ~ a (d™ + a l 3uf) 
= ( a i “ a ) {(¿M' 2 ~ aa l (S«) 2 } = 2 ^/ F {(<M 2 - aa l (3w) 2 }- 
But we have 
whence 
and the term thus is 
(Vde) 2 = {abcd^A + aJbjCjdef— 2 VX F}, 
-~7ïr- ~ it (abcdiCifi + aibjCxdef) — \ (Vide) 2 ; 
o~ o' 
i (abcdjejfj + aJ^Cjdef ) — | (Vde) 2 
O' 
{(Sot) 2 — aaj (0ii) 2 {. 
Joining hereto the rational part of ^ AH, and multiplying the whole by 4, we have 
n 
AH = aa 1 M + 
+ 
/ 2X 
X'\ 
2aj X 
l d 3 
dv 
a d 2 
/ 2F 
F'\ 
2a F~ 
l d 3 
d*J 
ai d 2 
(dvr + a !0*i) 2 
(0'oj + a du) 2 
+ 
(abcdjeJj + ajbiCjdef) — (Vde) 2 {(0ct) 2 — aaj (du) 2 }, 
where M has its foregoing value = (Ho' — (Vde) 2 } (du) 2 + 233' du 0tzr + (S' (0trr) 2 . 
Xtrsi siep o/ the reduction. 
Writing bcdef = U, bjCjdjeJ^ U 1 , then X = aU, F=a 1 H 1 , and consequently 
X' = -H + aH', F' = - U. + aJK, 
the accents in regard to U, U 1 denoting differentiations as to x, y respectively: then 
2 U U'\ aJJ 
/ 2X X'\ 2aj X /2aU U - aU'\ 2& l aU 
1 1 - ' 1 ' 
6 s + d 2 
Û2 ^ 
a d- 
[ 03 02 
and similarly 
2 F F'\ 2a F 
T d^ = aai 
H _ ¿A _ 
p d 2 ) 
_ 2F _ F'\ _ 2a F _ 
d 3 d 2 J a, d 2 ” 
/ 2H X HA a H, 
= aa ' - d 3 d 2 J- d 2 •