76 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Hence q^ is not zero. Now q"^ is a point, and since q contains F 

 as a factor, must be the point F. Also 



Consequently 



again from (32) we have 



1 = [F4>]. 



= F 



ml 



[0 q'"~'] 

 (m - 1) 



= P 



{C) 



{B') 



The equations {A), (B), (C) and {A'), {B'), {C) show that F, p are 

 related to 0, q in the same way that the latter are to the former. 



Hence 



where r = 0, 1, 2 



1 

 rl 



[p"'-'--F] 

 (m — r) ! 



(33) 



m. 



The formulae (30), (31), (32), (33) show that the system of quanti- 

 ties />'", F p'' is generated in the same way from F, p or from </>, q. 

 If the product of two of these quantities is progressive the factors can 

 be associated and the result is either zero or equal to a third. If the 

 product is regressive we replace p^ and F p^ by their expressions in 

 terms of </> and q. The product in this form is represented by a sum 

 of matrices (in hyperplane coordinates) having a smaller number of 

 rows than columns. The factors can therefore be commuted and 

 associated giving a result which is number times a quantity of the form 

 q^ or 4> q^. Hence the product of any two quantities of the funda- 

 mental system is a numerical multiple of a third. 



Let 



S2r 



r! 

 Fp' 



(34) 



J 



where r = 0, 1, 2, ... m. Dually we have the quantities ctj such 

 that, 



q' 



r! 



0'2r+l — 



rl J. 



(35) 



m 



