254 
MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
and the latter notation 
ax m -j- bx m ~ x y . . -f- b x xy m ~ 1 + dy m . 
But in particular cases the coefficients will be represented all of them by unaccen- 
tuated letters, thus {a, b, c, d\x,y ) 3 will be used to denote ax 3 -\- 3 bx 2 y -\- 3 cxy 2 -\- dy 3 , 
and (a, b, c, d^fx, y ) 3 Mull be used to denote ax 3 -\-bx 2 y cxy 2 dy 3 , and so in all 
similar cases. 
Applying the general methods to the quantic 
(a, b..b\ a'Xx,y) m , 
we see that 2Z>B c ...+?w6'd a . 
{x‘b y }—mbb a -\-{m— lcB 4 ..+a'^; 
in fact, with these meanings of the symbols the quantic is reduced to zero by each of 
the operations {3/d.J — y~d x , ; hence according to the definition any function 
which is reduced to zero by each of the last-mentioned operations is a covariant of 
the quantic. But in accordance with a preceding remark, the covariant may be con- 
sidered as a rational and integral function, separately homogeneous in regard to the 
facients (x, y) and the coefficients (a, b..b\ a'). If instead of the single set (x, y) 
the covariant contains the sets (x I5 yd), {x 2 ,y 2 ), & c., then it must be reduced to zero 
by each of the operations {?/<),.} — ^yb x , {xS^}— Sxc^ (where 
but I shall principally attend to the case in which the covariant contains only the 
set (x,y). 
Suppose, for shortness, that the quantic is represented by U, and let U 15 U 2 .. be 
what U becomes when the set (x, y) is successively replaced by the sets (x„ y x ), 
(■*25 y?), &c. Suppose moreover that 12=B^ 2 — &c., then the function 
Ii p T3 ? 23 r ..U 1 U 2 U 3 .., 
in which, after the differentiations, the new sets (x 15 */,), (x 2 , y 2 ).. may be replaced by 
the original set (x,y)—, will be a covariant of the quantic U. And if the number 
of differentiations be such as to make the facients disappear, i. e. if the sum of all the 
indices p, q.. of the terms 12, &c. which contain the symbolic number 1, the sum of 
all the indices p, r, &c. of the terms which contain the symbolic number 2, and so 
on, be severally equal to the degree of the quantic, we have an invariant. The 
operative quantic 0 becomes in the case under consideration 
— V..+d a Xx,#) m , 
the signs being alternately positive and negative ; in fact it is easy to verify that this 
expression gives identically 0U=O, and any covariant of 0 operating on a covariant 
of U gives rise to a covariant of U. 
18. But the quantic 
(a, b..b\ a'Jx, y) m , 
considered as decomposable into linear factors, i. e. as expressible in the form 
a(x—cty)(x—( 3 y).., 
