ME. A. CAYLEY ON TSCHIENHAITSEN’S TEANSEOEMATION. 
5G7 
which is equal to 
I [(IU'+6J'0T+I(I^-27P)0'^] ; 
so that the condition in order that this invariant may be equal to zero is 
IU'+[6J+2^/-i(P-27P)]0=O, 
which agrees with a result of M, Heemite’s. 
There should, I think, be an identical equation of the form 
which would serve to express the square of the invariant cp' in terms of the other 
invariants U', H', 0', I, J ; but assuming that such an equation exists, the form of the 
factor M remains to be ascertained. The invariant J (cubinvariant) of the form 
(1, 0, C, 13, 1)^ contains and it would be necessary to make use of the iden- 
tical equation just referred to in order to reduce it to its simplest form ; and (this being 
so) I have not sought for the expression of the cubinvariant of (1, 0, C, l)h 
For the quintic we have the equations 
(a, b, c, d, e,fX^\ 1)'=0, 
{ax -\-byQ 
-\-{ax^-{-hhx -f-'lcjC 
+ («P 5 + 1 0 -h 6 
-f- (aP+ 55P-fl0cP-{- lOf/a-h 4^>)E, 
and this leads to the system of equations 
-JB-4cC-6dD- 
■4eE, 
-aB-5^C-10cD- 
lOtZE, 
P 
o 
\ 
P 
1 
Q 
1 
/E, 
y— iB — 4cC— 6c?D-j- 
eB, 
-«B-55C-10i?D 
/D 
? 
5eD-{- 
/E, . 
y-hB-icC-B 4fZDH- eB, 
/c 
5 
5eC+ fB 
lOdC-j- 5eD-b /E, 
/B 
9 
SeB-f- fC 
lOiZB-j-h^G-h /D 
1 
P 
1 
— «E TTl, X, X-, x\ P)=0, 
— 
aC 
-5bB 
— 
aB 1 
— aB — 
bhC 
5 
— aC 
1 
y — 6cC-)-4fZD-j- eE, — 
lOdjB-j-lOfZC-}- 5eD-j- jfE, y 
and the transformed equation is obtained by equating to zero the determinant formed 
out of the matrix contained in this equation. 
MDCCCLXir. 4 H 
