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AN INTRODUCTORY MEMOIR UPON QUANTICS. 
[139 
I say that it is made up of partial systems corresponding to the different facient sets 
respectively; it is clear from the definition that the quantic is reduced to 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 be a covariant of the quantic, and 
in the particular case in which it contains only the elements, an invariant. (It would 
be allowable to define as a covariant quoad any set or sets, a function 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 
[xdy], although represented by means of the same symbols x, y have, as regards the 
different quantics, different meanings, and we may form the sum 2 {<xd y }, where the 
summation refers to the different quantics: we have only to consider in place of the 
system before spoken of, the system 
%{xd y } — xdy, ...; ^{xdy’) —x'd 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 8 {xd y }, where the summation refers to the entire series of cor 
responding sets. We have in place of the system spoken of in the original definition, 
to consider the system 
[xdy] — S (xdy), ... [xdy] — 8 (x'dy-), ... &c. &c., 
or if we are dealing with two or more quantics, then the system 
2 {xdy} — 8 (xdy), ... ; X {¡x'dy'} — 8 (x'dy-), ... &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 homogeneous 
in regard to the coefficients or elements. And the term “ covariant ” includes, as already 
remarked, “ invariant.” 
It is proper to remark, that if the same quantic be represented by means of different 
sets of elements, then the symbols {xd y } which correspond to these different forms