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MATHEMATICS: E. J. WILCZYNSKI 
INVARIANTS AND CANONICAL FORMS 
By E. J. WlLCZYNSKI 
Department of Mathematics, University of Chicago 
Communicated by E. H. Moore, August 7, 1918 
Every student of the theory of invariants has observed the fact that the 
coefficients of a unique canonical form are invariants. But a general a priori 
proof of this fact, sufficiently general to cover all of the cases needed in the 
applications, seems to be lacking. It is the purpose of this paper to furnish 
such a proof, making use only of the abstract principles which are common to 
all known invariant theories. 
We begin by giving a brief outline of some of these invariant theories, so 
that we may have these instances in mind when we formulate our general 
theory. Consider first a binary n-ic, 
n 
(p) = 2 A***?"' (A > 
2'= 0 
This binary form is a function of po,pi,. . , p n and of Xi, x 2 . The p's are 
called the coefficients, and the x's the variables of the form. We introduce 
new variables by putting 
Xi = an Xi + oi i2 x 2 ,(i = 1> 2), 
where the quantities are arbitrary constants with a non-vanishing determi- 
nant, thus transforming the form (p) into a new form (p). Those combina- 
tions of the coefficients p k which are equal to the same functions of the co- 
efficients pk are called invariants of the form. In the classical theory of in- 
variants it is really the equation (p) = 0 which is the object of study rather 
than the form or function (p). Consequently the additional transformations, 
operating upon the coefficients only, pi = \p i} (i = 0, 1, 2, . . , n), where X 
is an arbitrary constant, are introduced. An invariant of the equation must 
remain unaltered by these transformations also. 
A second invariant theory is concerned with the class of linear differential 
expressions or forms 
where p 0 ,pi . . . , p n , and y are functions of x. The coefficients p k in this case are 
functions of x, whereas in instance (A) they are numbers, real or complex. 
In instance (A) the variables Xi,x 2 are also numbers; in (B) the variables 
y,dy/dx,. . ., d n y/dx n are functions. In this case we may even think of y 
as the only variable, since the variables dy/dx, etc., are determined when y 
is given. 
Let us now transform the variable y by putting y = \(x)y, where X(x) is 
a n arbitrary function. Then (B) goes over into a new differential form (B) 
