ANALYTIC CONDITIONS. 153 
three sets of three variables each, we see that our functions 
@; may contain homogeneous polynomials in one, two, or three 
sets of three variables each. For example, these may be homo- 
geneous linear, quadratic, cubic, etc., forms in one set of 
variables ; bilinear, quadrato-linear, etc., in two sets; or tri- 
linear, ete., in three sets of variables. From the theory of 
linear transformation we are led to the following general 
statement: A necessary and sufficient condition that a set of 
functions ¢; shall be invariant in form under a linear trans- 
formation T in three cogredient sets of three variables each is 
that each function of the set be a function only of complete* 
homogeneous polynomials in one or more of these sets of 
variables. 
186. The Second Condition. The second condition, that 
the constants of the transformed functions shall reproduce 
the original functions, gives us at once the specific form of 
the functions ~;. They can be none other than complete ho- 
mogeneous polynomials in the elements of the rows or col- 
umns of M; for the constants in the transformed functions 
are the coefficients of the powers and products of «,, 3,, etc., 
in the transformed polynomials entering into the functions, 
and these coefficients are complete homogeneous polynomials 
in the elements of the rows or columns of M. An illustration 
will make this clearer. Suppose, for example, that @ is an 
exponential function of a linear homogeneous polynomial in 
the elements of one row of M,; thus e /4:+m&+n% | We 
then have the three forms e/“tmbtna, g@latmiatnn , 
elAi+mBi+nC: | Making the substitutions U, in the last form 
we get A 
e (la, +mb,+ne)) 44+ (las+mbo+ ner) i + (Las + mb; +Nes)7rs 
* By a complete homogeneous polynomial in a given number of variables we mean 
one that contains all the terms consistent with the number of variables and the de- 
gree of the polynomial. Such and only such polynomials are invariant in form under 
a general linear transformation in all the variables of the set. 
—10* 
