COVARIANTS OF PLANE CURVES. 
1201 
Hence the functions involved in the first integrals are the solutions of a set of 
differential equations analogous to (40), and it is the common solutions of these two 
sets of equations treated as simultaneous that we are seeking. It is easy to prove 
that the number of such simultaneous solutions is four, and they can easily be 
constructed as follows :— 
Pg — (^3^4 d" “h h.2 {kJiz — d" (^ 1^3 d“ 
Pg = 
P^ —^ ^ 3 ^ 3 )^ *d” ^^2 ^^2^3 ^^3^3^ (^^3^1 ^^^3^3 
d“ Ag (AgA| — ~~ ^^ 1 ^ 3 )*^ 
H42). 
!hh) 
Of these P^ can be shown to be Matrix of Hessian/(Matrix of General Cublc)^. 
23. To find the first integral of the differential equation to the general cubic ivhich 
is a differential invariant. 
For this purpose we are to find a function of the P’s which is a solution of kLff= 0. 
With rather more difficulty than in the case of fig, we find 
Clfi^ = Aq — 4 
= Aq _ 3 ^ ^ 
HiAg = Ag 2 — Ag 2 — Ao, 
HiAq = — 6 A/ — 6 ^ Aq, 
• J) J) 
nff = - ^ 1 
HiAg = 2hfi^ + Ahf — 4 ^ Ag + 2 Aq, 
P^A^ = GAgAg ~ 3 ^ Aq — 3 Aq. 
The problem is now to find functions of the P’s, which satisfy simultaneously 
and 
dh^ dli^ dJi^ dk-^ dk^ cWq dk^ 
Aj Aq — Ag — 6/q2 — G/qA^ ^/qA^ + 4:hf GA^Ag 
f?A| dlv^ dk-^ dk^ dA'g dk^ 
4/q SAj 2 A 3 GA], 5Aq 4Aq 3Aj. 
the third equation being satisfied by each of the six first integrals. 
MDCCCXCIII.—A. 
7 o 
