65 
665] 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
205 
-his 
We 
e f) 
Expression for A d 2 A — (dA ) 2 ; several subheadings. 
Writing for shortness A d 2 A - (dA) 2 = AA, as before, and so in other cases: then 
in general APQ = P 2 AQ + Q 2 AP, and thence AP 2 = 2P 2 AP or A \/P = ~ AP. Hence 
starting from A = il fa = il aa,, we have 
A A = AO Vaa, = aa, Ail + 2 — (a 2 Aa, + a, 2 Aa), 
where 
Aa = a 3 2 a — (3a) 2 = — a d' 2 x — (3x) 2 , Aa, = — a 3 2 y — (3y) 2 . 
Hence writing 
we have 
Aii = I Mil 2 , 
\ 0 A A = I aa, M — (a, 3 2 ^ + a d 2 x,) — | i— (dx) 2 + - (3 y) 2 
( ct cil 
fl 2 
But we have 
dx = (dv — y du), dy = — (dv — x du) ; 
squaring the first of these and differentiating, we find 
2dx d 2 x = 
2X X'\ 0 2X ' 
W + f dx + W dÿ 
X 
(dv — y du) 2 — 2dy du ^ (dv — y du) 
where as regards X the accent denotes differentiation as to x (and further on, as 
regards Y, it denotes differentiation to y), viz. this is 
2X X'\ a ,21 ' 
w + w dy 
V 
(dv — y du) 2 — 2d y du ^ (dv — y du), 
= ( - dx (dv - y du) 2 + (dv - y du-d du) (dv - y du) dy, 
where the second term is 
which is 
2X 
d 3 
(dv — x du) (dv — y du) dy, 
2 fXY 
(dv — x du) 2 dx : 
hence dividing by 2dx, the equation is 
and similarly 
/ Y Y'\ VZF 
d 2 x =(-f 3 + ^d 2 ) ( dv ~ V dU)2 & r ~ ^ 8U ) 2 ’ 
a/yv /Y Y'\ 
&y = - f (dv - y duf + ( ÿ w J (dv-xduf-, 
and we may in these values in place of dv — y du and dv x du write 3ar + a, 3u and 
3sr + a du respectively.