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A MEMOIR ON THE ABELIAN AND THETA FUNCTIONS. 
165 
in regard to the variations of 3, and in regard to the variations of 4 : we must 
therefore have 
12 
dw ;i 
312 
d 3 134 d 3 234 
134 + 234 
and the like equation obtained herefrom by the interchange of the numbers 3 and 4. 
95. The equation just written down relates to any four points 1, 2, 3, 4 of the 
conic; and if for 3, 4 we write 0, 3 respectively, it becomes 
d®__d.031 d. 023 
Z 012 ~ 031 + " 023”’ 
which relates to the points 0, 1, 2, 3 of the conic: writing, as before, 023, 031, 012 = p, a, t, 
this equation is 
i o _ da dp 
I z — 1 , 
t a p 
which may be verified as follows : the equation of the conic is f— 23.<rr + 31. rp + 12.pa, = 0 : 
we have dco = where — 23. a + 31p, = — , that is, dco = — (— — + -£), 
dj dT r t t { a p) 
dr 
the equation in question. 
96. We have, as a property of any four points 0, 1, 2, 3 of a conic, 
23 
-01 
or 
23 -01 . . 23 o- 01 
123.023 ~ 012.031 ’ ° r Say A . p~ ar ’ that 1S) A p ~ T ’ 
hence considering 0 as a variable point, and differentiating the logarithms, 
7 . 01 da dp 
-d log—= + —, 
T cr p 
and the foregoing equation 12 — = — — +— thus becomes 12— = — dlog —, 
restoring for r its value 012, 
i ^ dco 01 
12 012 ~ f/ ° § 012' 
Taking now ax + /3y + r yz = 0 for the equation of the line 012; this meets the conic 
in the points 1, 2, coordinates (x 1 , y ly zi) and (x 2 , y 2 , z 2 ) respectively: and we have 
a, ¡3, 7 = yiz 2 -y 2 z 1 , z 1 w a -z a as 1 , x 2 y 1 -x 1 y 2 , 
12 = y u z^x 2 , y 2 , z 2 ), 
and from this last value 
12 8 = {(a, y u z$x 2 , y 2 , z 2 )} 2 -(a, yi, z^f. (a, ...][x 2> y 2 , z 2 f 
(the second term being of course =0), viz. this is 
12 2 = -(6c-/ 2 , ...\yiZ 2 -y 2 z^ Z x X 2 -Z 2 X ly x 2 y 2 -x 2 y^f 
— - (be —/ 2 , ...#a, /3, y) 2 , 
to 
¡, 4 
bely