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PROCEEDINGS OF THE AMERICAN ACADEMY. 



ing reasoning we have, since the sign must be positi\'e in the first 

 and negative in the second case 



{ar^i /?-(r,_i T) = (- l)"--! ((7,^1 R-q-a^^, T) 

 For every case the following equation holds: 



{p ■ (Tr-l R ■ (Jr-l T) _ {q-a r+lR ' Cr.+i T) _ ^ +j (o-r+1 R " (Tr-l T) 



RT = 



{cr R) {a, T) {ar R) (cr, T) 



It is evident from the definition that 



= (-1)' 



(^r R) i<rr T) ' 



RT = 



TR. 



This together with the linearity of the expression, the factored form 

 and symmetry of the denominator, shows that three spaces R, R', R" 

 of a pencil determine angles such that 



RR' + R' R" -\- R" R --= 0. 

 To prove this directly it is only necessary to place 



R" = \ R + fx R' 



in the expression for the above sum and clear of fractions. 



26. Distance and angle in a section of hyperspace. A space 

 R of our space of order n intersects the complexes »Si of the funda- 

 mental system in a set of complexes. For spaces contained in R we 

 can define distance and angle relative to these last complexes. We 

 wish now to show the relation between those invariants and tlie cor- 

 responding invariants relative to the complexes Sj. 



First consider the section made by a hyperplane a. This deter- 

 mines with the complex j^ a point 



i^i = [a 2J], 

 and with the complex [F 2>], a complex 



2h = [a-F pi 

 We can write this last expression in the form 



Pi = (o-F) p - [a-p-F]. 

 If we multiply this by itself r times, since the last term is a line, this 



