PRESENT PROBLEMS OF GEOMETRY 571 



number of integrals of certain types. The chief problem for the 

 geometer, however, is the discovery of the precise relations between 

 the connectivities of the Riemann manifold and the various genera 

 of the algebraic surface. That relations do exist between such di- 

 verse geometries the one operating with all continuous, the other 

 with the algebraic, one-to-one correspondence is one of the most 

 striking results of recent mathematics. 



Geometry of Multiple Forms 



For some time after its origin, the linear invariant theory of 

 Boole, Cayley, and Sylvester confined itself to forms containing a 

 single set of variables. The needs of both analysis and geometry, 

 however, have emphasized the importance and the necessity of 

 further development of the theory of forms containing two or more 

 sets of variables (of the same or different type), so-called multiple 

 forms. 



In the plane we have both point coordinates (x) and line coor- 

 dinates (u). A form in x corresponds to a point curve (locus), a 

 form in u to a line curve (envelope), and a form involving both x 

 and u to a connex. The latter was introduced into geometry, some 

 thirty years ago, by Clebsch, the suggestion coming from the fact 

 that, even in the study of a simple form in x, co variants in x and u 

 present themselves, so that it seemed desirable to deal with such 

 forms ab initio. 



Passing to space, we meet three simple elements, the point (x~), 

 the plane (u), and the line (p). Forms in a single set of variables 

 represent, respectively, a surface as point locus, a surface as plane 

 envelope, and a complex of lines. The compound elements composed 

 of two simple elements are the point-plane, the point-line, and the 

 plane-line. The first type, leading to point-plane connexes, has been 

 studied extensively during the past few years; the second to a more 

 limited degree; the third is merely the dual of the second. To com- 

 plete the series, the case of the point-line-plane as element, or forms 

 involving x, u, and p, requires investigation. 



In the corresponding n-dimensional theory it is necessary to take 

 account of n simple elements and the various compound elements 

 formed by their combinations. 



The importance of such work is twofold: First, on account of 

 connection with the algebra of invariants. A fundamental theorem 

 of Clebsch states that, in the investigation of complete systems of 

 comitants, it is sufficient to consider forms involving not more than 

 one set of variables of each type : if in the given forms the types are 

 involved in any manner, it is possible to find an equivalent reduced 

 system of the kind described. On the other hand, it is impossible 

 to reduce the system further, so that the introduction of the n types 



