Introduction The beautiful symbolic method of Maschket for the study of differential invariants has been applied to a certain extent to the study of geometry. Maschke himself has made application to the theory of curvature of hyperspace and also to the study of directional relations.! The more important formulas of ordinary differential geometry were developed by Smith § in terms of the symbolic notations, and finally, Bates]] has used the method in a further study of curvature. Of these

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... re. Of these applications, the most complete and satisfactory is that of Maschke in his paper Differential parameters of the first order. This paper contains a complete discussion of the relations between the tangent vectors to the subspaces Bk of the space Sn under consideration. The development of those properties of hyperspace which depend upon invariants and differential parameters of the second order is by no means as complete. Several of the papers mentioned above are, to be sure, devoted to special problems involving second order properties, but no systematic study of these properties by means of the symbolic method has been attempted. There is, of course, a very extensive development of this subject by means of unsymbolic methods. A quite complete treatment of the geometry of two-dimensional surfaces has been given by Wilson and Mooref and