PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QU ANTICS. 
a bipartite cubic function l) 3 (x, y) 3 ; and the determinant formed out of the matrix 
is at once seen to be an invariant of this bipartite cubic function. 
Assume now that we have identically 
(a, b, c, djx, y) 3 =« b', d, dX&+iy), 
viz. this equation written under the equivalent form 
(«', b\ d , d'XX, Y ) 3 =(a, b, c, <ZJX+ Y, z(X-Y)) 3 , 
determines (a 1 , b', d, d') as linear functions of (a, b, c, d), it in fact gives 
a'=(a, b, c, dX 1, — *) 3 =<z — 3bi-{-3ci 2 — di 3 , 
V-=.(a , b , c, dXXi — &) 2 (1> i) = a— bi— ci 2 -\-di 3 , 
</ = («, &, c, —i) (1, i) 2 =a+ bi- cv-di\ 
d'=(a, b , c, dXh if =a-\- 3fo’+ 3ce 2 + c?i 3 , 
then observing that ky -\-lx±i(ly — kx)={x+iy)(^ik-\-l), we have 
{a, b, c, dXfy+lx, ly—Jcx 3 =(a', b', d , d'X^r-Hy)(— ik+l), \{x—iy){ik+l))\ 
and if in the expression on the right-hand side we make the linear transformations 
x+iy= x'j2, —ik+l— Id J 2, 
X—iy=.—iy'*J 2, ik-\-l=—illj2, 
which are respectively of the determinant +1, the transformed function is 
=(a',b,d, d'Xk'x', - I’yJ , 
that is, we have 
(a, b, c, dXky + lx, ly—kxf=(a!, b', d, d'Xk'x 1 , —I'y'Y- 
The last-mentioned function is 
k' 3 k'H' k’l' 2 l' 3 
. -W 
+3c' . 
and (from the invariantive property of the determinant) the original determinant is equal 
to the determinant of this new form, viz. we have 
4 F 
MDCCCLXVII. 
