302 
MATHEMATICS: E. J. WILCZYNSKI 
For the purposes of our general theory, which shall include all of the in- 
stances mentioned as well as infinitely many others, we postulate a class [F] 
of forms, or functions of two general arguments; F{p, x). As the notation indi- 
cates, the arguments of such a form are of two kinds, the coefficients p, and the 
variables x. Both p and x are supposed to be general variables in the sense of 
E. H. Moore, each argument varying on its own range. This range may be 
a continuous or discrete class of numbers, or a class of functions, or any other 
well-defined range; it maybe the same for p&nd x, or different Finally the 
range over which F varies may be different from either or both of the ranges of 
p or x. In the language of ordinary analysis this general formulation includes 
single forms or systems of forms, whose coefficients and variables may be finite 
or infinite in number, and in the latter case denumerably or non-denumerably 
infinite. 
We further assume that the class of forms [F] is well-defined. This means 
that a criterion is given by means of which we may decide whether a given 
form does or does not belong to [F]. This criterion will also enable us to dis- 
tinguish between the coefficients and the variables of the form 
Two forms of the class [F] are said to be identical if their corresponding coef- 
ficients are equal. Equality of the correspondng variables is not required for 
the identity of two forms. For this reason it is frequently convenient to sup- 
press the notation for the variables in the symbol for a form, and to use the 
simpler symbol F = (p) to replace F (p,x). 
We postulate, in the second place, a group of transformations, which operates 
upon the variables of a form F of [F] and transforms every F of [F] into another 
form F of the same class. The coefficients p of F will depend on the coefficients 
p of F and upon elements which occur in the transformation used. We assume 
that we obtain in this way a new group G operating upon the coefficients of the 
form. G is said to be induced by g. 
In this postulate the word group is uded in the usual sense. Thus we call 
a set of operations a group if the identity is included among its operations, if 
the operations possess an associative law of combination, if the product of any 
two operations of the set and also the inverse of every operation of the set 
belong to the set. The two groups, g and G, may however contain a finite 
number of operations or infinitely many; in the latter case they may be dis- 
crete or continuous; if they are continuous, they may be finite or infinite, in 
the sense of Lie. 
By an appropriate modification of the group G, we may include in our theory 
not only the invariants of forms, but also the theory of invariants of equations and 
systems of equations. This has been pointed out already in connection with 
our preliminary discussion of instance (A). 
// there exists at least one transformation in the group G which, when applied 
to a form Fi of [F], transforms Fi into a form F 2 , the two forms Fi ad F 2 are said 
to be equivalent under the group G. 
