104 
MR, A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
number of covariants or invariants will be infinite. The number of invariants is first 
infinite in the case of a quantic of the seventh order, or septic ; the number of cova- 
riants is first infinite in the case of a quantic of the fifth order, or quintic. 
29. Resuming- the theory of binary quantics, I consider the quantic 
[a, b, .. b\ «'X^, I/)™. 
Here writing {yB^}=aB 4 -l- 26 Bc..+m 6 '^„.=X 
any function which is reduced to zero by each of the operations X— Y— is a 
covariant of the quantic. But a covariant will always be considered as a rational 
and integral function separately homogeneous in regard to the facients {x,y) and to 
the coefficients {a, b , .. d). And the words order and degree will be taken to refer 
to the facients and to the coefficients respectively. 
I commence by proving the theorem enunciated. No. 23. It follows at once from 
the definition, that the covariant is reduced to zero by the operation 
X— Y —xb^— Y —xby.'K—yb^, 
which is equivalent to 
X.Y-Y.X-f 3 /B,-a;B,. 
Now 
X. Y=XY-bX(Y) 
Y. X=YX+Y(X), 
where XY and YX are equivalent operations, and 
X(Y) = lmdd^'\-2m— 1 . . -j-m 1 b'b/,- 
Y(X)= mlbbi.. -\-2m—lb''di-\-lma'ba, 
whence 
X(Y) — Y(X)=maB„-|-m — 26 B*..— m— 26 '^* — 
=.k suppose, 
and the covariant is therefore reduced to zero by the operation 
k+^/^y—xb,. 
Now as regards a term a!^¥ .x^y^ , we have 
k—ma,-\-m—2^.., — m — 2|3'— wa 
yby-xb,=j-i ; 
and we see at once that for each term of the covariant we must have 
wia + m — 2(3 , . — m — 2|8' — ma -J-j — * = 0, 
i. e. if (x, y) are considered as being of the weights respectively, and {a, b, ..b\ d) 
as being of the weights —^m, \m respectively, then the weight 
of each term of the covariant is zero. 
