390 MOORE AND PHILLIPS. 



in three dimensions and apply the theory to the study of the complex 

 line. The dyadies are of four main types : (a) Those which transform 

 points into points or planes into planes, (b) those which transform 

 points into planes or planes into points, (c) those which transform 

 lines or complexes into lines or complexes, (d) those which transform 

 points or planes into lines or complexes and those which transform 

 lines or complexes into points or planes. The first class represent 

 the collineations and to this type belong most of the dyadies hitherto 

 discussed. The second class represent the correlations. The last 

 two types so far as we know have not been discussed before. These 

 are not in general contact transformations. 



By means of double multiplication of two dyadies (one of which 

 may represent an identical transformation) we determine many 

 invariants and covariants. From the geometrical interpretation of 

 this double product we obtain a series of descriptive theorems analo- 

 gous to the Pappus theorem for the hexagon inscribed in a plane two- 

 line. 



INTRODUCTION. 

 I. Matrices and Outer Products. 



1. Progressive Matrices. In a former paper ^ we gave an inter- 

 pretation of the products of Grassmann ® in which we represented 

 points, lines, etc. as rectangular matrices and expressed the products 

 as operations performed on those matrices. As those products form 

 a fundamental part of the present paper, we shall here briefly outline 

 the method there used. 



Our space is a projective point space of three dimensions and so we 

 represent a point A by the matrix 



^= II ai 02 as a4 II = II tti 11 (1) 



where ai, Os, as, at are the homogeneous coordinates of the point. 

 Two matrices of this kind will be called equal when their corresponding 



5 A theory of linear distance and angle, These Proceedings, 48, 1912. 



6 Expositions of Grassmann's product theory can be found in the following 

 places: 



Ausdehnungslehre, 1862. 



Whitehead's Universal Algebra, Chapter I, Book IV. 



Encyclopedia, French edition, Complex Number, tome 1, 1. 



The treatment here given is somewhat different from any of the above. 



