where Pikl are functions of the independent variable x, into another system of the same form is given by the equations Yk = Σ λ=1 αky, (k = 1, 2........,n), (1) where a are arbitrary functions of x and where the determinant ▲ 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 seminvariants of system (A) has been calculated.2 purpose of this paper to obtain the additional semi-covariants necessary for a complete system of semi-covariants. If equations (1) are solved for ya, there results where A, is the algebraic minor of a in A. If the coefficients in (1) are assumed to satisfy the conditions for the transformation of (A) into the semi-canonical form, the successive differentiation of (2) gives 2 The most general form of (1) which leaves the semi-canonical from in the semi-canonical form is given by the equations2 where a are arbitrary constants whose determinant D is not zero. The semi-covariants in their semi-canonical form are obtained by transforming the semi-canonical form of (A) by (5). We shall let Tikl denote the coefficients of the semi-canonical form of (A) which correspond to the T coefficients Pikl of (A). The effect of the transformation (5) upon (T) is given by the equations2 Σ j=1 Tij, m-2, 1-1 (i = 1, 2,...n, l = 1, 2,....n-1), (7) l where joyj, it is easily verified that each of the sets of quantities r(i=1, 2,.... ..,n) is transformed by (5) cogrediently with y; (i=1, 2,...... n). Therefore, the determinant is a set of quantities which are transformed by (5) cogrediently with y; (i = 1, 2,..........,n). We therefore have n 1 additional relative semi-covariants Since the coefficients in (5) are constants, each set y!") (i = 1, 2,...,n) of derivatives of y; are transformed by (5) cogrediently with y, (i = 1,2,...,n). We therefore have mn n relative semi-covariants n A comparison of (3) with the inverse of (5) and of the expressions,2 Tikl, and Tikl, in terms of the coefficients of (A), with (6) shows that the semi-covariants R, S, T, may be expressed as semi-covariants of (A) simply by replacing y" by tir, Tikl by uikl, and Tikl by Vikl, where Uikl and Vikl are functions of the coefficients of (A) and their derivatives which appear in the expressions for Tik and Tikl If the transformation (1) and the corresponding transformations for the derivatives of y; 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 seminvariants. 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-covariants3 for the special case of (A) where m = 2 shows R and S; to be independent. Again, the functional determinant of T, with respect to y") (i = 1,2,...,n) for each value of = 1, 2,......, m-1 shows that Ti, are independent, among themselves and of R, of S, and of the seminvariants. We have now proved the following theorem: All semi-covariants are functions of seminvariants and of R, S, (i = 1 2,..... n−1), T1 (l 0, 1,....... n-1; T 1, 2,..... .m−1). = = 1 Wilczynski, E. J., 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 OLIVER E. GLENN DEPARTMENT of MathemaTICS, UNIVERSITY OF PENNSYLVANIA 1. Comprehensive as the existent theory of differential parameters is, as related to quantics F = (ao, a1, ..., am) (dx1, dx2)" (aj under arbitrary functional transformations (1) X;= X; (V1, V2) (i = 1, 2), Xi = aj (x1, x2)), 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(1,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 differential 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. It leads also to systems of certain new types which I designate as orthogonal1 and the extended orthogonal types of differential parameters, and methods of enumeration relating to these systems, both general and particular. 2 In the paper of which the present article is an abstract, references to the literature are only incidental to the developments but mention may be given to memoirs by Christoffel, Ricci and Levi-Civita,3 to the tract of J. E. Wright, containing bibliography, and to the symbolical theory of Maschke.5 2. The poles of the transformations on the differentials, -1 thus obtaining another transformation upon the differentials of the same degree of arbitrariness as T, since the functional determinant of f+1,ƒ−1 will not vanish when A 0. The quantics df1 are relative differential covariants appertaining to a domain R(1,T,A) whose defining quantities are the y1, y derivatives of x and x2, and the expression A. The covariant relations are are the factors in R(1,T,A) of the determinant D of T. дх1/дуг = α1, dx1/dy2 = α2, dx2d/y1 = ẞo, dx2/dy2 = B1, and write, after Maschke, the quantic F as the m-th power of a symbolical exact differential, Then when F is transformed by the inverse of T, and the result expanded, the coefficients m-2i (i : = 0, ....., m) of the terms in df+1, df_1 are differential parameters of the domain R(1,T,A), forming, with df1, a complete system in this domain. Their explicit form is m ❤m—2;= [(Y1 — A)fı−2ẞoƒ2]TM-[− (Y1+A) ƒ1+2ẞoƒ2] * (−4ßo▲)] ̄TM (i =o,. .,m) 4. By differentiation of the equations (4) relations are obtained as follows: we obtain a general category of relative differential parameters, 2i m-2i p12iD3), (k) -1 k=o then the parameters of the extended orthogonal type are defined to be those which can be generated in totality by forming such rational expressions in - 2i, df±1 as simplify by multiplication into functions appertaining to the domain R(1,T,0), free, that is, from the expression A. The essential forms from which to construct this totality are evidently P+1 ± P-1. When r = 0 the type is called orthogonal.1 Finite complete systems can be derived in this theory. In fact the products P form a Hilbert system of monomials whence it follows that a complete system of concomitants of the extended orthogonal type is furnished by the finite set of irreducible solutions of the diophantine equation Particular systems of the orthogonal type have been constructed by the present writer in this and previous papers for the quantics of orders one to six inclusive, the system of the sextic for example being composed of 31 parameters. For the case r = 1, m = 2 the system comprises eleven concomitants of the extended type as follows: |