THE DYADICS IN THREE DIMENSIONS. 395 



then A, B,C lie on a line as can easily be verified by expressing the 

 matrix in terms of coordinates. 



3. Regressive Matrices and products. We can consider space 

 as generated by planes as well as bj' points. If its coordinates are 

 a,-, a plane a is represented by the matrix 



a — II ai tt'j as (X4 || . 



The same plane may be represented by the matrix [ABC] of any three 

 noncollinear points lying in it. If ai is equal to the coefficient of x,- 

 in the expansion of the determinant (ABCX), we shall write 



a = [ABC]. 



The line of intersection of two planes a and /3 can be represented by 

 the matrix 



[a/S] = \\qzi 942 923 Qu qzi qu\\, (13) 



where 



Qik = 



I3i 13k 



If the same line is the join of two points A and B, we know from line 

 geometry that the coordinates qik are proportional to the coefficients 



of the minors * * in the determinant (ABxy). If qnc is equal to 



\ Ifi Vk 



the coefficient oi \ Xi yk\in that determinant we shall write 



M] = [AB]. 



Three planes a, (3, 7 intersect in a point A. The coordinates a^ of 

 this point are proportional to the coefficients of ^i in the determinant ^ 

 (^a|37). In particular, if a^ is equal to the coefficient of ^» in that 

 determinant, we write 



A = [a/37]. 



There is a determinant (al3j8) of four planes just as of four points. 

 The matrices [a/3], [ai37], [ajSyd] can be regarded as products. They 



8 It is to be observed that the variable ^ is put in the first row of this determi- 

 nant while in §1 we wrote for points (ABCX). This change is made in order 

 to make the reduction formulas agree in sign with the ones Grassmann gave. 



