GEOMETRIC INVESTIGATIONS ON DYNAMICS. 315 



pt Fi — Pi Fk = Pk (f>i — Pi 4>k (:r,k= 1,2, . . . , n) 

 or 



Fi — 01 Fo — 4>i Fi — 4>i Fk — 4>k 



(SO) 



Pi Pi Pi Pk 



Fn — 4>n 



■Pn 



Letting K stand for these equal ratios, (SO) may be written 

 (81) Fi=cl>i-\-piK, {i= l,2,...,n). 



Multiplying by pi and summing with respect to i, we get 



S Pi Fi = I, pi(f)i-{- Ki:, p^, 



i i i 



and remembering that 



S p,^ = S {x'if =1; S Pi Fi = 2 x'i x"i = 0, 



■t" i i 



we have 



t 



K = — 2 pi(f)i, 



i 



and (81) becomes 



(82) Fi = (t>i- pi^pk 4>k (1= 1,2, ..., n). 



k 



Now, we must still satisfy equations (69), and this will introduce 

 conditions on the forms of the 0's. Substituting the values of the F's 

 as given by (81) into (69), these reduce to 



\dxi dxij \dXm dxij \dx,n dxj 



{i, I, m = 1, 2, . ..,n). 



Since these equations are to hold for abitrary values of the j^'s (the 

 only condition being pi^ -{- pi^ -\- . . . -}- Pn = 1), and since the expres- 

 sions in parentheses are independent of the p's, we must have 



(84) ^ = '^. {U=l,2,...,») 



OXi OXi 



