300 LIPKA. 



dt i dur duk \i durduk idukdur/ 



dt i dUk dUr \i dUkdUr idUrdUk 



Substituting in (26') we shall get our condition in the form 



(38) ao + ait+aof -{-asf -\- =0. 



Since this must vanish independent of t, every coefficient, a, must 

 vanish. We find that ao vanishes identically, which is as it should be 

 since the hypercongruence is normal to the initial surface i = by 

 hypothesis. Let us first discuss the condition that ai vanish. This 

 gives 



(39) 2-—- S-— ^=0, (r, A:=l,2, ,n-l). 



i dUrOUk X OUkOUr 



Now by differentiating (37), we get 



dUr I dxi dUr I dpi dUr 



(40) !Zi=2^^+2^^' 



duk I d.i'i dUk I dpi dUk 



and (39) becomes 



(41) v^(^^^_^^\ + v^/^^^_^^fi\^0, 



ildXi\dUrdUr dUkdUr) ildpi\dUrdUk dUkdUr) 



{r,k= 1,2, ....,n-l). 

 These conditions arc therefore necessary conditions in 



order that the oo"-i curves belonging to the system (27) and orthogonal to 

 the hypersurface (32) shall form a normal hypercongruence. 



Our problem is to find the forms of the functions Fi in order that 

 this condition of orthogonality should hold for every hypersurface (32) 

 taken as a base. Ecjuations (41) must then necessarily hold for n 

 arbitrary functions fi(i = 1,2,...., 7i). These functions may be so 

 chosen that for any assigned values of Ui, u-i,. . . . , Wn-i, the quantities 



