248 MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
zero by each of the operations of the entire system. Now, besides the quantic 
itself, there are a variety of other functions which are reduced to zero by each of the 
operations of the entire system ; any such function is said to he a covariant of the 
quantic, and in the particular case in which it contains only the elements, an inva- 
riant. (It would be allowable to define as a co variant quoad any set or sets , a func- 
tion which is reduced to zero by each of the operations of the corresponding partial 
system or systems, but this is a point upon which it is not at present necessary to 
dwell.) 
7. The definition of a covariant may however be generalized in two directions : 
we may instead of a single quantic consider two or more quantics ; the operations 
{.rB^}, although represented by means of the same symbols x, y, have, as regards the 
different quantics, different meanings, and we may form the sum where the 
summation refers to the different quantics: we have only to consider in place of the 
system before spoken of, the system 
— xb y , .. .. &c. &c., 
and we obtain the definition of a covariant of two or more quantics. 
Again, we may consider in connexion with each set of facients any number of 
new sets, the facients in any one of these new sets corresponding each to each with 
those of the original set; and we may admit these new sets into the covariant. 
This gives rise to a sum S(#c^), where the summation refers to the entire series of 
corresponding sets. We have in place of the system spoken of in the original defi- 
nition, to consider the system 
{.:rc^} — S(xBJ, .. {x'By} — S(x'By-), .. & c. &c,, 
or if we are dealing with two or more quantics, then the system 
— S(«B y ), .. 2{.r'B y } — S(x'B y ), .. &c. &c., 
and we obtain the generalized definition of a covariant. 
8. A covariant has been defined simply as a function reduced to zero by each of 
the operations of the entire system. But in dealing with given quantics, we may 
without loss of generality consider the covariant as a function of the like form with 
the quantic, i. e. as being a rational and integral function homogeneous in regard to 
the different sets separately, and as being also a rational and integral function of the 
elements. In particular in the case where the coefficients are mere numerical multi- 
ples of the elements, the covariant is to be considered as a rational and integral 
function homogeneous in regard to the different sets separately, and also homoge- 
neous in regard to the coefficients or elements. And the term £ covariant’ includes, 
as already remarked, 4 invariant.’ 
It is proper to remark, that if the same quantic be represented by means of differ- 
ent sets of elements, then the symbols {.zBj,} which correspond to these different forms 
of the same quantic are mere transformations of each other, i. e. they become in 
virtue of the relations between the different sets of elements identical. 
