Voi,. 6, 1920 
MATHEMATICS: E. B. STOUFFER 
645 
SEMINVARIANTS OF A GENERAL SYSTEM OF LINEAR HOMO- 
GENEOUS DIFFERENTIAL EQUATIONS 
By E. B. Stouffkr 
Department of Mathematics, University of Kansas 
Communicated by B. J. Wilczynski, September 27, 1920 
The system of linear homogeneous differential equations 
m — In 
where 
ax 
and where pi^i are functions of the independent variable x, is evidently 
converted into another system of the same form by the transformation 
n 
x=i 
where a^x are arbitrary functions of x, and where the determinant \ \ = A 
does not vanish identically. Furthermore, it is known'^ that (1) is the 
most general transformation of the dependent variables which leaves 
(A) unchanged in form. 
A function of the coefficients of (A) and their derivatives which has 
the same value for (A) as for every system derived from (A) by a trans- 
formation of the form (1) is called a seminvariant. The seminvariants 
of (A) have been calculated for the case^ n = 1, m = any positive integer, 
and for the case^ m = 2, n = any positive integer. It is the purpose of 
the present paper to obtain them for the general case of (A). 
The calculations are considerably simplified by first obtaining the 
seminvariants in a so-called semi-canonical form and then expressing them 
in terms of the coefficients of (A). The possibility of a simplification of 
this general nature was first suggested by Green. ^ 
The transformation (1) converts (A) into a new system (B) in which 
the coefficients of the derivatives of order w — 1 of the dependent variables 
are zero provided that a^- is so selected that 
-2 
Pi,k,m—\o^kj = ^ qikiOLkj, (hi = 1,2, ,n). (2) 
^=1 k=i 
Such a selection is always possible. The system {B) is the semicanonical 
form of (A). 
