GEOMETRIC INVESTIGATIONS ON DYNAMICS. 



307 



we have 



K = — 2 Pi (t)i, 



i 



and (52) becomes 



(53) Fi=4>i- Pi 2 Pk <t>h (i = 1 , 2, 3, 4). 



k 



We still have to satisfy equations (44). Substituting the vahjes of 

 the i^'s as given by (52) into (44), these reduce to^ 



(54) 



Pi 



d(f)2 d4>z 





dxs dxnj 



dx, 



dx4 



\dX3 dx4 



\dxi dxsj \dxi dxj \dxz dxj 



fd4>i d4>2\ . fd<j)i 

 \dx2 dxJ \dxi 



dxi 



dipi dcf): 



8x2 dx, 



\dxs dxij \dxi dxsj \dx2 dxJ 



Since these equations are to hold for arbitrary values of the ^^'s, (the 

 only condition being p^-\- P'^-\- p^-\- p^= 1), and since the ex- 

 pressions in parentheses are independent of the ^^'s, we must have 



(55) 



(901 502 502 503 503 504 



5x2 5x1' 5X3 dX2 dXi 5X3 



504 _ 501 501 _ 503 502 504 



5xi 5x4' 5x3 5xi' 5x4 5x2' 



Hence, the 0's are partial derivatives of a single function L(xi, X2, xs, X4), 

 i. e. 



(56) 



dL dL dL _ dL 



01 = - — ; 02 = - — ; 03 — ~ — ; 04 - 7—. 

 5xi 5x2 0x3 5x4 



Introducing these values into (53), we fiinally get 



dL 



(57) 



x'; = p _^.,'v|^xL (i= 1,2,3,4), 

 5xt k oxk 



