80 



PROCEEDINGS OF THE AMERICAN ACADEMY, 



these expressions being valid if the order of the left member is equal 

 to or less than that of [/9 a]. Similarly we obtain the intersection of 

 this system with a hyperplane 7, etc. We thus get finally. 



Px = [R Sx+2]. 



as point and complex in a space R of order n — \. For these we 

 have the equations 



R9r = 



^2r+l — 



r! 



= (RS^y-'lR-Sor.x] 



= {R S},y[R-S2r+\+l] 



(38) 



these expressions being valid for values of r such that the orders of 

 the left members are equal to or less than n — X. 



li {R Sx) ¥" we choose the magnitude of R such that 



{R Sx) = 1. 

 Then the above equations become 



We consequently have 



\R,_x-rAB\R [RSn-r-AB]R 



[Rn-x-i A] [Rn-x^i B] [R S„_i • A] [R S„_i • B] 

 [A BR S„_2l R R{AB 8^-2) R 



(39) 



[A • R Sn-i] [B ■ R S„_i] R {A S„_i) R {B 5„_i) 

 provided that A B is contained in R}^ 



Hence we have 



AB = 



(Sn-rAB) 



f-R„-x-2 A B] R 



(Sn-l A) (Sn-1 B) [i?,_x_i A] [iJ„_x-l B] 



We may consider R as the unit quantity in the space R. Then the 

 right side of the above equation is the expression for distance relative 

 to the system of complexes in R. Thus whether we take distance in 

 R relative to the fundamental system of complexes Si or relative to 

 the sections Ri in R, the result is the same. Similar relations of the 

 angles between other spaces in R relative to Sj and i?j can be shown. 



Massachusetts Institute of Technology. 



12 Cf. Grassmann, Gesammelte Werke, Vol. I, theil 2, page 91. 



