PHILLIPS AND MOORE. — LIXEAR DISTANCE AND ANGLE. 79 



last term cannot appear more than once as a factor of any term of the 

 resnlt. Hence 



Ih' = (a FY 1/ - r (a Fy'^ [a •/;• F- //"']. 



= {aFy-'{{aF)i/ -r[a-p-F-i/-']\. 



= {aFr'[a-Fi/], 



provided that 2 r + 1 ^ n. IMultiplying the first of these values of 

 py by Fi = [a J)], since [a j)] is ah-eady a factor of the second term, 

 we get 



(aFYia-ir^] 



[F,iH^] = (aFriap-lf] 



r+ 1 



provided that 2 r -\- 2 ^ n. Dividing the above expressions for pY 



and [jPi py] by rl, we get 



p/ {aFr-'[a-Fp^] ] 



rl r\ 



[F^pY] (a F)^ [a- 7/^1] 



(37) 



rl O'+l)! J 



these expressions being valid if the order of the left side is equal to 

 or less than that of a hyperplane. 



We next find the intersection of a second hyperplane /3 with the 

 system of complexes p^, [FijiY]- Let 



^2= L^i^i] = I3a-Fp] 



' (aF) 21 



By the same argument as before we get 



p/_ (j3F,r'[i^-F,pn 



rl (aFYrl 



[F2P2']_ Q^F.YW-pr'] 



r! (aF)'-(r+l)! 



Using the vajues of pi and [Fi p/] in (37) we haA'e 



p./ _ (,3apY-'h3a-p^^^] 



rl (/•+!) 



rl rl 



