50 PROCEEDINGS OF THE AMERICAN ACADEMY. 



coordinates a^ are proportional to the coefficients of .Vj in the determi- 

 nant \ABCX\.^ 

 If ai is equal to the coefficients of Xi in that determinant we shall write 



a=[AB C]. 



Thus a three-rowed matrix is for our purpose equivalent to a one-rowed 

 matrix in contragredient variables. 



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

 by a matrix 



0-1 0.2 0.3 Cl'4 

 ^1 ISo ^3 ^4 



The coordinates of the line are here 



[a^] = 



Qik 



Oi aic 

 I3i fik 



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

 tical geometry that the coordinates qtk are proportional to the co- 



'i a-fc' 



efficients of the minors 



If gii is ec 

 shall write 



in the determinant \ A B X Y \. 



Vi Vkl ' ' 



If qik is equal to the coefficient of | Xi yk \ in the determinant we 



[a/3] = \AB]. 



This amounts to saying that in the determinant [A B a ^ ], each 

 minor in the first two rows is then equal to its algebraic compliment 

 (coefficient in the expansion of the determinant). 



Similarly we represent the point of intersection of three planes by a 

 matrix [a jS 7]. The coordinates a^ of this point A are proportional 

 to the coefficients of ^i in the determinant [| a [] 7]. In particular 

 if tti is equal to the coefficient of ^j in that determinant we write 



A = [al3y]. 



In this case each term of the first row in the determinant (A a ^ y) 

 is equal to its algebraic compliment. 



There is a determinant [a /3 7 S] of four planes just as of four 

 points. These quantities [a /3], [a (3 7], (a /3 7 S) can be regarded 

 as products formed according to the same laws as the products of 

 points. These products of matrices expressed in plane coordinates 

 we shall call regressive. 



3 It is to be observed that here X is written last. If we take the coefficients 

 of Xi in the determinant \X A B C\ they will have different signs from the 

 coefficients used here. j. 



