154 THEORY OF COLLINEATIONS. 
but this must be identical with e!/™+mAi+"n ; equating par- 
ameters in these two forms we get 
la, +mb,+nce,=l, 
las+mb.+neo=m, 
laz;+mb,+n¢c3=n; 
we do not get by this process the form e/a +mht+na, It is 
easy to see that the only forms of the functions ?; which re- 
produce themselves by this process are complete homogene- 
ous polynomials in the elements of M. 
187. The Polynomials 9;. The homogeneous polynomials 
$; are not the most general form of such polynomials, but are 
restricted to such forms as occur in the coefficients of the 
transformed forms ¢,’.. In order to determine this restriction 
we must examine more closely the phenomena of linear 
transformation of homogeneous forms in three cogredient 
sets of three variables each. 
Let f; be a set of 7 homogeneous polynomials in one, two, or 
three sets of three variables each; and let each polynomial of 
the set be transformed by a linear transformation T with 
matrix M’ in three cogredient sets of variables ; it is required 
that the coefficients of the transformed polynomials shall re- 
produce in the elements of the matrix of T the original set of 
polynomials and no others. 
Our method of procedure is as follows: We choose from 
the set of functions f; any one f of the set and transform this 
by 7; the coefficients of the transformed form f’ are also func- 
tions of the set f, and so we join these to f and call the new 
system thus found f;, We now transform the system f; by 
T and join tof; the new functions appearing among the co- 
efficients of the transformed forms; we continue this process 
until the system closes. We then have the system of func- 
tions f;. 
Let us choose from the set f; any function f of the set; fwill 
be a homogeneous function of some degree r either in one, 
two, or three sets of variables ; for only polynomials of these 
three types occur in f,. Let f be transformed by T into f’; 
