COYARTANTS OF PLANE CURVES. 
1199 
and expressing it in terms of powers of ^ and y], we get 
Irf 
- 3 {hij^ - b,) rf^ 
+ 3 — ilhVi + ttg) 
— ^ {h{y — y^x) + h^x — h^] yf kc. = 0 \ 
the coefficients being easily written down. 
Now, whatever be the point {x.y) taken upon the curve, we obtain the equation to 
the curve in an identical form, and therefore the ratios of corresponding coefficients 
in any two such equations are equivalent. 
This being the case, we obtain nine first integrals of the differential equation to the 
cubic, to which we may give the forms 
K 
= constant, 
h 
2/1 — 2 - + y = constant, 
y — y^x X — = constant, 
&c. 
On account of the obvious character of these expressions, I do not write them 
down, nor shall I perform the simple processes whereby they are simplified, but 
merely write down the final forms in which I propose to deal with them. 
For this purpose, write 
~b 
m 
1 ) 
^ 1 . 
¥ ■ 
¥ 
■ ^2 
a 
b 
b 
3 JO + ^ j:3 
O W _ 7 . 
¥ ^ ¥ ¥ “^3 
(37). 
