ANALYTIC CONDITIONS. 155 
no matter to which of the types f belongs, among the coeffi- 
cients of f’ will be found a homogeneous polynomial of degree 
rin the elements of the first column of M’. Let us replace 
@,, 4,4, by %, y, 2, respectively in this function; let us call 
this function f, and use it as the generator of the system /f;. 
Let f, be transformed by 7’; the following statements hold : 
(a) The number of terms in f, and therefore in the trans- 
(r+1) (7+2) 
; : 
(b) The coefficients of the transformed form f/f,’ consti- 
formed form f,’ is 
tute a set of a homogeneous polynomials each of 
degree r in the elements of the columns of M’ and each con- 
: 41) (r+2 
tains a terms. 
(c) The corresponding terms of this set of polynomials all 
have the same coefficient, viz.: the coefficient of the corre- 
sponding term of /,. 
(d) Three of the coefficients of f/ are homogeneous poly- 
nomials in the elements of a single column each of M’. 
(e) Whenr>i, ae of the coefficients of f,’ are ho- 
mogeneous and symmetrical polynomials in the elements of 
pairs of columns of WM’. 
(g) When r>1, r—2 of the coefficients of f,/ are homo- 
geneous and symmetrical polynomials in the elements of the 
three columns of MW’. 
In the set of 7t2"**”) 
efficients of f,’ we replace the a’s by 2, y, z; the b’s by 2’, y’, 
2’; the c’s by «”, y”, 2’; we now have a set of functions f; 
which must be contained within the set /;. 
Let f and f, be homogeneous and symmetrical polynomials 
of degree 7 in two and three sets, respectively, of three varia- 
bles each, and let each of them be linearly transformed by the 
same J’ as above. Precisely the same set of statements hold 
also for each of these polynomials as for the polynomial /,. 
polynomials obtained from the co- 
