MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANT1CS. 
255 
gives rise to a fresh series of results. We have in this case 
{•zBj,} = — (a+f3..)aB a +« 2 B a +/3 2 B /3 + .. ; 
in fact with these meanings of the symbols the quantic is reduced to zero by each 
of the operations xd y , {yd x } — yd x , and we have consequently the definition of 
the covariant of a quantic considered as expressed in the form a(x—uy)(x—(3y).. 
And it will be remembered that these and the former values of the symbols and 
{ 3 /B,} are, when the same quantic is considered as represented under the two forms 
(a, b, ,b\ a"$jx,y) m and a{x— ay)(oc— identical. 
19. Consider now the expression 
a e (x—ay) j (x—(3y) k ..(tt,—(3) p ...), 
where the sum of the indices j, p... of all the simple factors which contain a, the sum 
of the indices k,p... of all the simple factors which contain (3, &c. are respectively 
equal to the index 0 of the coefficient a. The index 0 and the indices j 0 , &c. may be 
considered as arbitrary, nevertheless within such limits as will give positive values 
(0 inclusive) for the indices^, k, ... 
The expression in question is reduced to zero by each of the operations xb s , 
{yd x } — yb x ; and this is of course also the case with the expressions obtained by inter- 
changing in any manner the roots a, (3, y and therefore with the expression 
a 9 '2(x—ccyy(x—(3y) k ..(cc—(3) p ...), 
where 2 denotes a summation with respect to all the different permutations of the 
roots a, j 8 ... 
The function so obtained (which is of course a rational function of a, b..b\ a') will 
be a covariant, and if we suppose ^—m0 — 2Sp, where S p denotes the sum of all the 
indices p of the different terms ( a—(3) p , &c., then the covariant will be of the order 
(i. e. of the degree p in the facients x, y), and of the degree 6 in the coefficients. 
20. In connexion with this covariant 
a 9 1(x—ay) p (x—(3y) k ..(ci—(3) p ..) 
of the order ^ and of the degree 0 in the coefficients, of the q uantic U= 
consider the covariant 
of a quantic V= 
a(x—uy)(x—(3y).., 
2 (l 2 p ..)V 1 V 2 ..V m 
(*!*> y)*> 
in which, after the differentiations, the sets {x„ y x ), (x 2 , y^).. are replaced by the 
original set {x,y). The last-mentioned covariant will be of the order m(<p—0)-\-pj, 
and will be of the degree m in the coefficients ; and in particular if <p=0, i. e. if V be 
a quantic of the order 0, then the covariant will be of the order y, and of the degree 
