GEOMETRIC INVESTIGATIONS ON DYNAMICS. 311 



(C8) (^-^)(-,).,,,., + (f^-f") (-,)<«-.„+ - 

 \d.vi dxij \d.v,n oxij 



\dxm dxj 

 Substituting 



?™= (-ir+^;jn,; q,= (-ly^'pi; qi= (-1)'+V-. 

 we get, finally, 



(C9) (fl - m ,..- (p - m ,n+ (f' - 'M ,.= 



\dxi dxiJ \dxm oXiJ \dx„i dxij 



where i, I, m may take all values from ] to n. 



Let us now consider the vanishing of the 2d summation in (41), viz., 



(70) s^_^Y^^|^-^^)=0, (./.■= 1.2....,,,-,) 



a dpi \d U r aUk O Uk du r) 



We must introduce here the identities 

 (33) 2 Fi^ = 



i \dUkOUr dUrOUkJ 



Expanding (70) we get 



(71) ^^f^-^-^^^Ws^ ("— '-^-— '-^V••• 

 _^2aFY^i^^i_^P:«^^_^ ^0,(r,A-=l,2,...,n-l), 



dPm\dUrdUk dukduJ 



and we may reduce the number of terms by eliminating the coefficients 



of ^,^,^2^ . .,^^\ . . . ., from the sums in (71). Thus to 

 d])! dpi dps dpm 



eliminate 



^dPodfi dP2dfi \ 



dUrdUk dUkdUrJ 



