GEOMETRIC INVESTIGATIONS ON DYNAMICS. 



291 



de = F'-ds'' = S (FoiA-) dxi dxk, 



ik 



2 dt 8 (dt) = i:^-^P^d.v. dxk 8x1 + 2 S (Fhiu) dxi 8 (dxd 

 iki ox I il 



iW dxi dt dt il dt 



Integrating both sides, and noting that, by partial integration, 



(\ (F'-au) '^di8xt) = S {F'au) ^'Sx, [ - /" ^2 (F^m) ^dt 5.r, 

 J/o il dt il at Ifo *Jio at il ai 



and that the first term of the right member vanishes since 8xi is zero 

 at the fixed end points Po and Pi, we finally get 



Since 5^^ is arbitrary, this expression will vanish if 



:) 



dXi 



dt 



dt 8xi. 



d dxi 



0/ i dt ik 



diFhiik) dxidxu 



dxi dt dt 

 Replacing dt by F ds, this becomes 



(/- 1,2, ....,7l). 



^ d ^ dxi 1 d (Fhiik) dxidxk ,, ^ -, n 



ds i ds 2 F ik dxi ds ds 



For a euclidean space, we may use the relations (7), and (10) becomes 

 d / dxi\ 1 d F /dxiV „ , , , 



or, expanding and solving for 



d^ 

 ds^ ' 



d?xi _ 1 

 ds"- ~ F 



dF_ ^ /dxi' 

 dxi'^i \ds J 



dxi dF dxk 



ds k dxk ds 



, (/= 1,2, ....,7t). 



7 ^ W. 



If we let L = log F, write^ xi = -^, x" = ~, and choose the arc 



ds ds- 



length 6- as the parameter along the curves, so that 



8 Throughout this paper, primes refer to total derivatives with respect to s. 



