GEOMETRIC INVESTIGATIONS ON DYNAMICS. 321 



Letting K stand for the eqnal ratios in (100), we may write these 



Ji ^ (t>i + Qi K, (t = 1, 2, . . . , ?0 

 or, by (98) 



2 flxi II\ = (f)i -j- Qi K, (i = 1, 2, . . . n). 



X 



To solve these for II { , we multiply by A u and sum with respect to i, 

 then 



H'i = i:Aii<i>i + K^Aiiqi. 



i i 



Hence by (87) 



Hi = I, Aiicl)i + Kpi. 



i 



To find the value of K, we now multiply by a\i p),, sum with respect to 

 X and /, and use the relation (88), 



i: oxipxHi = I, pi(pi-\- K'L axipxpi. 



U i \l 



t 



Now since 



'^ttxiPxpl = 1, 



then, by total differentiation with respect to s, we may show that 



2 oxi px F2 + I S —^ Pi pxpi = 



\l iU OX'i 



or "SauPxPi -'EaxiP\Gi = = XaxiPxHi. 



X2 X/ \l 



Hence, 



and 



K = —'2pi(})i 



Hi = "2 An 4>i — Pill, Pi 0,-. 



Finally, replacing Hi by Fi— Gi, Gi by its value from (IS), and 0,- 

 by its value from (101), we get 



(102) x'i-\-i:Vl\x'ixl = ^^ {Aii-x'ix'i), (1=1,2,. ..,n) 

 ik '-'' > i dXi 



