MATHEMATICS: E. J. WILCZYNSKI 301 
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whose coefficients p k depend upon po,p h . . . , p n and X. The corresponding 
invariants are of considerable importance. Here also we must distinguish 
between the theory of invariants of forms, and the theory of invariants of 
equations. 
As a third instance, let us consider the class of analytic functions 
/(*) = p Q +pix + p 2 x 2 + . . . + p n x n +...., (C) 
where the coefficients po, pi ... ,p n ,---, and the variable x are complex num- 
bers. We introduce transformations of the form 
aX+(3 
x = , ao — py q= 0, 
yx-\- 5 
where a, ft 7, 5 are arbitrary constants, and attempt to find invariants of 
f(x) under all such transformations. In this case we are dealing with a form 
which has infinitely many coefficients, forming a denumerable set. » 
We shall list one further instance of our general theory. Let K (x, £) be 
a real continuous function of a: and £ in the region 0^x^l,0^£^l, 
and let <p(£) be a real continuous function of £ in the interval 0 ^ £ ^ 1 . 
Then 
I(x) = f (D) 
may be regarded as a form whose value, as function of x, depends upon the 
choice of the K and <p functions. We may think of the functions of x to which 
K (x, £) reduces, for all special values of £ between 0 and 1, as the coefficients of 
the form. Thus the number of these coefficients is non-denumerable, even 
if the function K(x, £) be regarded as given. As to the variables of the form 
I(x) we still have two choices. We may think of the function <p(£) as ranging 
over the class of all continuous functions and regard <p(£) as trie only variable 
of the form; or else we may consider the functional values of <p(if) as the varia- 
bles. In the latter case, the range of the variables would be the class of real 
numbers, and the number of variables would be continuously infinite. 
We may transform (D) by putting 
<p($ = ^ (a 
where is an arbitrary continuous function of £. This will transform I(x) 
into I(x), where 
7 (x) = £ K (x, 0 v (f) dt, K (x, 0=K (x, f) X (£). 
In particular it will be possible to choose X(£) in such a way as to make 
*»;«) = i 
except for those values of £ for which K(\, £) 
be said to be in its canonical form. 
= 0; the resulting integral may 
