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MA THEM A TICS: 0. E. GLENN Proc. N. A. S. 
semi-covariants R, S if T lr may be expressed as semi-covariants of (A) 
simply by replacing y\ r) by t iT , ir m by uikly and T ikl by v m> where u m 
and v ik i are functions of the coefficients of (A) and their derivatives 
.which appear in the expressions for Tr ik i and t^i- 
If the transformation (l) and the corresponding transformations for 
the derivatives of y t are made infinitesimal, and the resulting system of 
partial differential equations for the semi-covariants is set up, it is found 
that there are exactly mn relative semi-covariants which are not semin- 
variants. We thus have the proper number of semi-covariants, but it 
remains to show that they are independent. 
A comparison of R and S, with the corresponding semi-covariants 3 
for the special case of (A) where m — 2 shows R and S* to be independent. 
Again, the functional determinant of T ir with respect to y\ T ^ (i = 1,2, . . . ,n) 
for each value of r = 1, 2, , m — 1 shows 3 that T tr are independent, 
among themselves and of R, of 5 t - and of the semin variants. 
We have now proved the following theorem: 
All semi-covariants are Junctions of seminvariants and of R, S t (i = 1 
2, ,n-l), T lT (l = 0, 1, ,n-l;r = 1, 2,... m-l). 
1 Wilczynski, E- Projective Differential Geometry of Curves and Ruled Surfaces, 
Teubner, Leipzig, Chap. I. 
2 Stouffer, these Proceedings, 6, 1920(645-8). 
3 Stouffer, London, Proc. Math. Soc, (Ser. 2), 17, 1919 (337-52). 
AN ALGORISM FOR DIFFERENTIAL INVARIANT THEORY 
By Ouver B. Glknn 
Department of Mathematics, University of Pennsylvania 
Communicated by I„ B. Dickson, April 16, 1921 
1. Comprehensive as the existent theory of differential parameters 
is, as related to quantics 
F = (a Q , a h . . a m ) {dx h dx 2 ) m (aj = aj(x h x 2 )), 
under arbitrary functional transformations 
(1) %i = oc { (y h y 2 ) (i = 1, 2), 
developments of novelty relating to the foundations result when emphasis 
is placed upon the domains within which concomitants of such classes may 
be reducible, particularly a certain domain R(i,T,A) defined in part by 
certain irrational expressions in the derivatives of the arbitrary functions 
occurring in the transformations. For a given set of forms F all differen- 
tial parameters previously known are functions in R of certain elementary 
invariants, which we designate as invariant elements, and their derivatives. 
The theory of invariant elements serves, therefore, to unify known theories 
and, for the various categories of parameters, gives a means of classification. 
