COVARIAN'TS OP PLANE CURVES. 
1183 
The capital letters may now be conveniently replaced by the smaller type, and the 
equations of condition written 
(A) 
(”• -p) {(f - + \-"-p)sj^)- ‘I 
= SS [p i) 7f^py/yH-p'^ 
also 
”’-^+ a(|^ = 
9y« 
and 
S/ 
{d +d.)/=(^- p) + Sy, A E5 (>r - P) g 
+ H/ 
(C) 
(ot) 
(^) 
K15), 
of which equations the last two will be identically satisfied by any functions which 
follow the laws of weight and degree which the simple conditions impose upon a 
covariant. 
We may call these the form, weight, and degree conditions, respectively. 
[A precisely analogous process may be applied to the covariants of surfaces, as is in 
this paper used for plane curves. 
If we, in this case, write 77 — to stand for 
^-y)’ 
and Cm n for 
^»i + nzldx”'dy'‘ ml n \, 
the set of conditions corresponding to (15) are 
5/ 
iv 
(tt 
(tt 
• y) 
x) 
p) 
■p) 
d(^ - 
x) 
5 / 
S(v - 
y) 
3 / 
3 (^ - 
x) ~ 
3 / 
s/ 
- S;«S/{ (w “f" 1) + « — _ ’ 
COjfi ji 
0 / 
d(r^- y) 
— { 1 % “J~ 1) Cm — \^7i+\ Pi ^ ’ 
—p 1 n — q ’ 
- ^Cp^q Cf}i — p^)i — q-\-\ 
S^/H. n 
