230 
AN INTRODUCTORY MEMOIR UPON QUANTICS. 
[1.39 
17. I pass next to the quantic 
(*&»> y) m , 
which I shall in general consider under the form 
(a, b, ...b\ djx, y) m , 
but sometimes under the form 
(a, b, ...b\ d~\x, y) m , 
the former notation denoting, it will be remembered, 
ax m + y bx m ~ l y ... + -y b''xy ,n ~ 1 + dy m , 
and the latter notation 
ax m + bx m ~ l y... + dxy 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? + 3bx 2 y + ocxy 2 4- dy 3 , 
and (a, b, c, d\x, y) 3 will be used to denote ax? + bx-y + cxy 2 + dy 3 , and so in all 
similar cases. 
Applying the general methods to the quantic 
(a, b,...b\ a'Jx, y) m , 
we see that [yd^ = ad b + 2bd c ... + mb'd a ', 
{xdy} = nibd a + (m — 1 cd b ... + a d b ; 
in fact, with these meanings of the symbols the quantic is reduced to zero by each 
of the operations \yd x ) — yd x , {xd y } — xd y ; 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, a). If instead of the single set (x, y) 
the covariant contains the sets (® lt y^), (ar 2 , y 2 ), &c., then it must be reduced to zero 
by each of the operations [yd x ] — Syd x , [xd y \ — Sxd y (where Syd x = y 1 d Xl + y 2 d x , + •••)> 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 1 , U 2 , ... 
be what U becomes when the set (x, y) is successively replaced by the sets {x l} yi), 
(x 2 , y 2 ), &c. Suppose moreover that 12 = d Xi d y2 — d Xt d y> , &c., then the function 
... ujJjJt..., 
in which, after the differentiations, the new sets {x ly y^), (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