Vol.. 7, 1921 
MATHEMATICS: E. B. STOUFFER 
275 
coefficients pm of (A). The effect of the transformation (5) upon w\ 7 k ] 
is given by the equations 2 
n n 
= 22 A * tx *< a ** ( ' =0, 1 m-2) (6) ■ 
X=l m=1 
where is the algebraic minor of a X ; in Z). 
If we put 
n 
U = 2 2 fyj-i (* = 2,... », / = 1, 2, (7) 
i=l 
where = it is easily verified that each of the sets of quantities 
ru(i=l, 2, ,n) is transformed by (5) cogrediently with ji (*=■!, 
2, , n). Therefore, the determinant 
R 
yu y* y n 
nu rn r n i 
ri, n -i T2, n -1 *w,n-l 
is a relative semi-covariant. 
Again, it is evident from (6) that 
h = 2 ^3^2 yj, (*'=1, 2,....n), 
is a set of quantities which are transformed by (5) cogrediently with 
ji (* = 1, 2, ,n). We therefore have n — 1 additional relative 
semi-covariants 
s> - = 2^-^r- - (*= 1.2,.. 
i=i 
Since the coefficients in (5) are constants, each set yj T) = 2,. 
of derivatives of are transformed by (5) cogrediently with = 1,2, . . . ,n). 
We therefore have — n relative semi-covariants 
r = 2^ |^>y = 0,l,....n-l;r = l,2,....,m-l) 
i=i 
A comparison of (3) with the inverse of (5) and of the expressions, 2 
Tiki, and iTjki, in terms of the coefficients of 04), with (6) shows that the 
