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 •