COVARIANTS OF PLANE CURVES. 
1203 
We may noAv form the coefficient of the highest power of y in the equation to 
the reciprocal curve. This coefficient is to be found, as has been shown, from the 
discriminant of the terms of the highest degree in this cubic, regarded as a binary 
cubic, and is therefore 
+ 4 ha.^ + 4 h-^a — Gab h^a 2 — 3 
From this the equation to the reciprocal curve would be developed according to the 
previous rules but with an interchange of the operators 14^ and fia- the quantities 
in the expanded coefficient are, however, solutions of = 0, except tto, and hence 
the degree of the reciprocal curve is indicated by the power of % in the expanded 
expression. 
25. The matrix of the non-singular cubic, and its differential equation. 
The condition of intersecting the standard curve at a number of points coincident 
with the temporary origin is an invariantal relation, and in finding the condition 
may treat the expressions as if they contained only differential invariants in their 
coefficients. 
Thus we write the equation to the cubic 
t/zTr^ — {(f) + u-^xp^) — ufu^ {(f)^ + %%) 
— u^^TT^ + ( 2 / + = 0 , 
and, for the consecutive points on the standard curve, putting 
we have (33) 
$=h, 
hi + hs q_ 'b'Fg 
3LL 
h^ “b &c. 
Equating to zero the coefficients of the several powers of h, on substitution for tt 
and f in the equation to the cubic, we get 
from tbe coefficient of h^, 
— 0 and 
/+ = 0 
?? 
55 
II 
o 
?? 
55 
o 
11 
?) 
55 
II 
? J 
55 
h\ 
l/, = — 
?? 
5 5 
h\ 
u,^ = Uff 
55 
55 
h\ 
Vs/ 
7 0 2 
(46). 
