COVARIANTS OF PLANE CURVES. 
1179 
SO that the multiplying factor 
= 1 + + + 
+ {‘^iv - d,) (AX + BY) 
— d (A^ + Bt]). 
4. Determination ofY,^. 
The dependent variable is supposed to be connected with the independent by some 
relation, say 
2/ = ^ (x), 
so that ynjn ! is the coefficient of /t” in the expansion of (/> (cc -j- h). 
The homographic relations may for our present purpose be written 
ir = X + BiY - X (AX + BY) 
2 / = Y - Y (AX + BY), 
whence 
Y = (^ {X + BiY - X (AX + BY)} + Y (AX + BY)... 
where on the right-hand side Y never appears except when affected by a small 
coefficient. 
Now write X + /i in place of X, and consequently 
Y + Y^h + &c. + -f &c. (= Y + t//) for Y, 
the coefficient of /i” in the resulting expression on the right is then 
Y„/w!. 
But this expression may be written 
(f) [x h^-\- B^xjj — (2AX + BY) h — BXi/; — Aid — Bh\p} 
+ Y (AX -f BY) + A (Xi// + Y/^ + hxfj) + B (2Yi// + t//”). 
Hence 
Y, = y,{l+ BiYi - BXYi - 2AX - BY} + A (XY^ + Y) -f 2BYYi, 
and generally Y„/w 1 = the coefficient of from 
7 L 2 
