572 GEOMETRY 



of variables is necessary for the algebraically complete discussion. 

 Geometry must accordingly extend itself to accommodate the 

 configurations defined by the new elements. 



Second, on account of connection with the theory of differential 

 equations. The ordinary plane connex in x, u, assigns to each point 

 of the plane a certain number of directions (represented by the 

 tangents drawn to the corresponding curve), and thus gives rise to 

 an (algebraic) differential equation of the first order in two variables; 

 the point-plane connex in space, associating with each point a single 

 infinity of incident planes, defines a partial differential equation 

 of the first order; the point-line connex yields a Monge equation. 

 The point-line-plane case has not yet been interpreted from this 

 point of view. 



One special problem in this field deserves mention, on account of its 

 many applications. This is the study of the system composed of a 

 quadric form in any number of variables and a bilinear form in con- 

 tragredient variables, that is, a quadric manifold and an arbitrary 

 (not merely automorphic) collineation in n-space. For n = 6, for 

 example, this corresponds to the general linear transformation of 

 line or sphere coordinates. 



In addition to forms containing variables of different types, the 

 forms involving several sets of variables of the same type require 

 consideration. Forms in two sets of line coordinates present them- 

 selves in connection with the pfaffian problem of differential systems. 

 The main interest attaches, however, to forms in sets of point coor- 

 dinates, since it is these which occur in the theory of contact trans- 

 formations and of multiple correspondences. For example, while 

 the ordinary homography on a line is represented by a bilinear form 

 in binary variables, the trilinear form in similar variables gives rise 

 to a new geometric variety, the so-called homography of the second 

 class (associating with any two points a unique third point), which 

 has applications to the generation of cubic surfaces and to the con- 

 structions at the basis of photogrammetry. The theory of multilinear 

 forms in general deserves more attention than it has yet received. 



Other important problems, connected with the geometric phases of 

 linear invariant theory, can merely be mentioned: (1) The general 

 geometric interpretation of what appears algebraically as the sim- 

 plest protective relation, namely, apolarity. (2) The invariant dis- 

 cussion of the simpler discontinuous varieties, for example, the poly- 

 gon considered as n-point or as n-line. 1 (3) The establishment of a 

 system of forms corresponding to the general space curve. (4) The 

 study of the properties and the groups of the configurations cor- 



1 Cf. F. Morley "On the geometry whose element is the 3-point of a plane," 

 Trans. Amer. Math. Soc., voL v (1904). E. Study in his Geometric der Dynamen 

 develops a new foundation for kinematics by employing as element the Soma or 

 trirectangular trihedron. 



