GEOMETRIC INVESTIGATIONS ON DYNAMICS. 



319 



Now (37) may be written 



(92) Fiiuu U2,...., «„-l) = Fi ifuh, ....Jn; Pl,P2, ....,Pn) 



= F'i {flyfi, ■ ■ ■ -ffn', Q], Qi,--, Qn), 



where Fi represents the form which the function Fi takes when the 

 P's are expressed in terms of the Q's, by (87). Then 



(93) 



2 ^^ +v ^^ 

 I dxi dur I dqi dur 



duk I dxi duk I dqi duu 



dF\ 



d II r 



with similar expressions for the partial derivatives of Ox^, and (91) 

 then becomes 



(94) 2 



dia^.Fi) fdfi df, dfi df. 



+ 



dxi \durdui; dukdur 

 dia^.Fi) fdQidf, dQidf, 



dqi 



durduk dukdu 



+ 2 

 p 



dQ^dP^ dQ^dP, 



dUr dvk duk du, 

 (r, i- = 1, 2 . . 



= 0, 



,n-l). 



These (n — 1) (w — 2)/2 conditions are therefore necessary conditions 

 for a normal hypercongruence. Our problem is to find the forms of 

 the functions F'i or Fi in order that this condition of orthogonality 

 should hold for every hypersurface taken as a base. We may simplify 

 our problem by the following considerations. Since the geodesies in 

 our space form a normal hypercongruence for every hypersurface as 

 base, the equations of the geodesies 



(18) 



or 



(95) 



must satisfy conditions (94). Substituting 6'x for F\ in (94), sub- 

 tracting the result from (94) and writing 



