GEOMETRIC INVESTIGATIONS ON DYNAMICS. 301 



fi and all partial derivatives of all orders of /, with respect to the w's, 

 shall take on arhiirary numerical values, or so that the quantities /f, P„ 



-^, — ', shall take on arbitrary numerical values, subject only to the 

 dUr dur 



relations (33) and (34). Before applying these conditions in a eu- 

 clidean space of 7i dimensions, it will be of advantage to apply them 

 to a euclidean space of four dimensions. 



§9. Proof of the converse theorem in a euclidean space of 4 

 dimensions. For a space of four dimensions we may write the 

 equations of the base hypersurface (32) in the form 



(32') Xi=fi(u,v,iv), (i=l,2,3,4), 



and the six conditions (33) and (34) as 



(330 Pif + P2f?+P3^+P4^=0, (u^.->..)^^ 

 oil ou ou oil 



(34') /^^_^^^ + ^^^_^2^/2^_^/aP3^3_^3^3^ 



\dv du du dv J \ dv du du dc J \dv du du dv J 



+ =— = 0, (w ^ v — > to) '■^ 



^ = 0, {u ^v 



\ dv du du dv ) 



The three necessar;^ conditions (41) become 



(41') '•2'^/^^•_^^^^'•v'^i/^'^_^^^ = 0, 



a dxi \dv du du di J a dpi \ dv du du dv J 



(ti -^v -^ w) . 



We may evidently place each of the two summations in the left 

 member of this equation equal to zero. 



Expanding the first summation in (41') we may evidently group the 

 terms into the following eighteen: 



dj\df3_dfidf, 

 dv du du dv 



df,df,_df^df, 

 dv du du dv 



(42) (^Il-^](^^- ^^) ^(^-^ 

 \dxz dxij \dvdu du dv J \c).ri dx^ 



,fdF\_dF,\ fdf2dfi_df2dfj\ /d_F\_df\\ /( 

 \c).r2 5a'i / \dv du du dv J V^-''i •^•^■■i / v 



12 We shall use the symbol {u -*v —^ w) to mean that-similar relations may 

 be written by a cyclical interchange of m, v, w. 



