THE DYADICS IN THREE DIMENSIONS. 391 



elements are equal. The matrix will be said to be zero only when all 

 its elements are zero. 



If two points A and B have coordinates a,- and hi respectively 

 and Oj = k hi, we shall write 



A = kB. 



Geometrically A and B are the same point. But in Grassmannian 

 analysis a point has magnitude as well as position. The magnitude 

 of ^ is A; times that oi B. In this paper we have no need to define 

 unit magnitudes and therefore have not done so. 



A linear function of two or more points A, B, C etc. is defined by the 

 matrix 



\A -\- ixB + vC -\- . . . = II Xa, • + fJ.bi + vc, ■+ . . . II (2) 



If the matrix does not vanish identically, it represents a point in the 

 space determined by A, B,C, etc. Conversely, any point in that 

 space can be represented in that way. For example any point on a 

 line can be represented as a linear function of two, any point in a plane 

 as a linear function of three not on a line, and any point in space as a 

 linear function of four not in a plane. If the matrix vanishes identi- 

 cally, and the multipliers X, jx, v, etc. are not all zero, those points lie 

 in a space of lower dimensions than that determined by a like number 

 of independent points. 



The Pliicker coordinates ^ of a line are certain two rowed determi- 

 nants 



Pik = 

 in the matrLx 



Oi ak 

 bibk 



Oi 02 as ai 

 hi hi hz hi 



(3) 



where ai and hi are the coordinates of any two distinct points A and B 

 on the line. We shall represent the line AB by the one-rowed matrix 



[AB] = II Pl2 Pl3 PU P23 P2i PSill (4) 



As above the matrLx will be considered zero if all its elements are zero, 



7 The facts concerning Pliicker coordinates and line geometry in general 

 which are here assumed are all to be found in Jessop's Line Complex, Cam- 

 bridge, 1902. 



