MATHEMATICS: E. J. WILCZYNSKI 
303 
Let us now define a proper sub-class [$>], of [F], by imposing some property 
upon the coefficients p which is not satisfied by all forms of [F]. It is assumed 
that this property is well defined, so that we may be able to decide whether a 
given form F, of [F], does or does not belong to the sub-class [<£]. Let us 
assume that, for every form F of [F], there exists in G at least one transforma- 
tion S which transforms F into an equivalent form of the sub-class [<£]. We 
shall then say that <£ = S[F] is a canonical form of F. It may happen that such 
a canonical form equivalent to F under the group G, and belonging to the 
sub-class [<f>] exists merely for those forms of F which do not belong to a well- 
defined sub-class \&] of [F]. We shall speak of the forms of the sub-class [&] 
as exceptional forms, and call all the other forrns of [F] generic forms. Of 
course the term, general form of [F], includes both the exceptional and the 
generic forms. Under these circumstances we shall still speak of $ as a canoni- 
cal form of Fj but we shall add the qualifying phrase for the generic case when- 
ever the distinction becomes necessary. 
In general there will be many transformations of G which transform every 
generic F into a canonical form of the sub-class [<£]. If all of these transforma- 
tions transform every generic F into the same form of the sub-class [$>], we 
shall say that the canonical form <£ is unique. 
It remains to define the term invariant. A function /, of the coefficients 
p of a form F, is an absolute invariant under the broup G, if it is equal to the 
same function of the corresponding coefficients of any form F which is equiva- 
lent to F, by means of a transformation of the group G. 
This notion may be regarded as including also the notion covariant. For, 
we may replace the given form F by another form, or system of forms, who 
coefficients depend also upon the variables of F. 
We are now ready to prove the following theorem. 
Let [F] be a well-defined class of forms, and let [<£] be a proper sub-class of [F]. 
Let G be a group of transformations which transforms every form of [F] into a 
form of the same class. By a generic form of [F] we mean one which is equiva'ent 
to a form of [<i>] under G. Then, there exists, for every generic form of [F] at 
least one transformation in G which transforms F into a canonical form of the 
sub-class [<£]. If this canonical form is unique, its coefficients are one-valued 
absolute invariants of the form F for the group G. 
Proof. — Let F be a generic form of [F], and let <E> be its canonical form. Let 
5 be the most general operation of G which transforms F into 3>. We shall 
have symbolically 
S(F)=$. (1) 
The operation S will depend, in general, upon the coefficients p, of F, and 
may contain, besides, arbitrary elements in great number. The canonical 
form <£ belongs to the sub-class [<£] of [F]. Since this canonical form is, by hy- 
pothesis, unique, its coefficients it are independent of the arbitrary elements 
which may occur in S, and they are one-valued functions of the coefficients 
