COVARIANTS OF PLANE CURVES. 
1175 
geometrical condition, such as that of moving along an arc, free from singularities, of 
a curve of degree not lower than the 
The equation to the resulting curve, which will osculate the assumed curve at the 
chosen point may now be deduced. Most easily, write 
x = h, y]i-y = • 
For further simj)llcity, write tt — y) for 17 — y — (£ — a:), so that 
y% 
and transform the determinant so that, in the first row y — y may be replaced by tt — p. 
To fill up the several columns in the determinant we write under — xY (tt -pY 
the coefficients of 1?, , . . in the expansion of /i" (tt —We thus obtain 
/(a;,y,. . .) = 
{^—xY, (I — xY 1 (tt —p), 
0 , 0 , 
{^—x){tt—p), tt — p 
0 , 
h 
2 ! 
0 , 
0 , 
2/3 
2 t 
Is 
2 ! 
X h 
d {d + 1) {d + 2) (d + 3) d (d + 3) 
8 2 
The transformed equation is affected in the same way, so that 
f{x, y,. . = + {M+ By + C)-''(AX + BY + + 
^ JJd(d + l)(d + 2)W + 3)/8-cnrf + 3)/2 —X)'^, (|—Xy^~^ (77 —P), 
0 , 0 , 
0 , 0 , 
Now write the determinants shortly 
/(I — X, TT —p,yz . . .) and /(^ - X, tt - P, Yg . . .). 
Bjh zzr dX/c/x = 
TT — P 
0 , 
21 
111 
2 ! ’ 
2 ! 
Also 
where 
AX + BY + C = )a. 1 
(AX + BY + C) (Ai + BJY - (A,X + B,Y + C^) (A + BY^) = X ' 
Also write ^ 
A.^ -f- By -[- C = J 
I- ( 2 )- 
