PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 77 



where r = 0, 1, 2, ... m. Equations (30), (31), (32), (33) show 

 that 



a,- = Sn-i ..... (36). 



25. Distance and angle. The distance between two points 

 A, B is 



-— _ (qAB) 



(</> A) (0 B) 



Similarly we define the angle between any two spaces R, T of the 

 same order r by the equation 



iq- ffr-i R-<rr-\ T) 



RT = 



(^r R) {<Tr T) 



(Tr-\ being the complex which multiplied by R and T respectively give 

 points. This expression can be put into two other forms which we 

 shall now obtain. We consider three cases depending on the form 



of Cr+l. 



(1) If <^r.X=^,, 



we have 



Then 



(2). If 



then 



(p-(Tr-lR-<7r-l T) = (ff^+l i? • (7r_l T) . 



[Fp'] 



k+1 R] = 



k\ ' 

 (/i?)F [p-Fp^-^R\ 



k\ i]c-l)\ 



Since in this case (Tr-\ also contains F, we have 



{p -(Jr-.R- Or^^ T) = — ((7,^1 R ■ (r,_i T) . 



In both of the preceding cases 



(/^•a,_i/?-(7,_i T) =(- iy^'(a,^,R-ar^, T). 



(3) If a^+i is of the form -- or ~— by the dual of the preced- 



A;! k\ 



