or A CUEVE or the eoueth oedee. 
361 
tution, Hi=H, but and thus 
^a!>x^h!'y^fz, Wx^Wy^fz, /^+/V+c";5)=(2B.H, 2B,H, 2B.H); 
and similarly, on making the substitution, 
{d\ h\ d\f, /, h!'Xx, y, ;s)^=6.5H-3.3.2H,=(30-18)H=12H. 
Hence writing therein U=0, the foregoing equation becomes 
-9 (a", b", c", f", g", h"XB.U, B,U, 
= 12H.(a, B, c, F, G, hXB, ,By ,B^)®(H-3H0 
— 4 (a, b, c, f, g, hXB,U, B^U, B^U)", 
which may also be written 
-9(a", b", c", f", g", h"XB,U, B,U, B,U)^ 
= 12H.(a, B, C, F, G, hXB, , Bj, , B 
-36H.(a, b, c, f, g, hJB^ , B^ , BJ^H, 
— 4 (a, b, c, f, g, hX^^U, Bj,U, B^U)", 
where (Wi, z,) are ultimately to be replaced by {w, y, z). The second line in fact 
vanishes, which 1 show as follows : — 
Demonstration of my Theorem for a Cubic Curve. 
Let U=(*X'^ 5 3^? 3, cubic function; it may by a linear transformation of the 
coordinates be reduced to the canonical form 3d-\-y^-\-z^-\-^lxyz, and we then have 
(a, b, c, f, g, hX^,, Bj,, BJH-^6^ 
= {yz— ^ V) . — 6 l^x 
+ {zx—l‘^y^). — U^y 
+ {xy—Pz‘). — %l^z 
■^2{l‘^yz-lcd).{l+2?)x 
+ 2{Vzx-hf).{l+2l^)y 
-{■2{l‘^xy-lz^).{l-\-2V')z 
= — IWxyz +^1^ {x^-\-y'^-\-z^) 
-\-QlHl-\-2P)xyz—2lil-\-2l^){cd-\-y^-\-z^) 
= {-\2l^.^^m^)xyz-\-{-2l^2l%x^-\-y^-irz^) 
= 2{ — l-yl^)[cd-Yf-\-z^-\-Qlxyz). 
Or since — Z+^Ms equal to the quartinvariant S, and the equation is an invariantive one, 
we have for any cubic function whatever 
(a, b, c, f, g, hXB„ Bj,, BJ"H-^6®=2S. U, 
which is the theorem in question. There is a difference of notation, and consequently a 
