274 
MATHEMATICS: E. B. STOUFFER 
Proc. N. A. S. 
where p iki are functions of the independent variable x, into another sys- 
tem of the same form is given by the equations 
n 
yk = 2 ak ^ y ^ 2 ----> w )» (!) 
X = l 
where are arbitrary functions of x and where the determinant A 
of the transformation does not vanish identically. 
A function of the coefficients of (A) and their derivatives and of the 
dependent variables and their derivatives which has the same value for 
(A) as for any system derived from (A) by the transformation (1) is 
called a semi-covariant. If a semi-covariant does not contain the dependent 
variables or their derivatives, it is called a seminvariant. A complete 
system of semin variants of system (A) has been calculated. 2 It is the 
purpose of this paper to obtain the additional semi-covariants necessary 
for a complete system of semi-covariants. 
If equations (1) are solved for y x , there results 
n 
A >x = 2 A * y 3> (2) 
where A, x is the algebraic minor of a jx in A. If the coefficients in (1) 
are assumed to satisfy the conditions for the transformation of (A) into 
the semi-canonical form, 2 the successive differentiation of (2) gives 
n 
where 
n 
The most general form of (1) which leaves the semi-canonical from in 
the semi-canonical form is given by the equations 2 
n 
X=l 
where a kx are arbitrary constants whose determinant D is not zero. The 
semi-covariants in their semi-canonical form are obtained by transform- 
ing the semi-canonical form of (A) by (5). We shall let ir ik i denote the 
coefficients of the semi-canonical form of (A) which correspond to the 
