GEOMETRIC INVESTIGATIONS ON DYNAMICS. 317 



(27) x"i = Fi (xi, X2, , .T„; .xi', xi, , Xn), a = 1,2,..., ii), 



and passing out orthogonally to an arbitrary hypersurface 



(32) Xi = fi (Ui, Uo, , Un-\), (l = 1, 2, , ?0- 



This hypercongruence may be written 



(36) Xi = fi-\-Pit + \Fit^ + iMi^-\- ...., (i= l,2,...,n), 

 where 



(37) Fi (wi, W2, . . . . , w„_i) = Fi (/i, /2, ....,/„; Pi, P2. ..., Pn) 



and the P's are connected by identities similar to (33) and (34) but 

 of a more general form, viz., 



5/ 

 ; ^ a\^ r\ Tf, = i] z. ax^ r\ 



(34 



(33') Sax^PxP^=l; i:ay.,Py^4^ = 0, (r = 1, 2,. . . ., 7i - 1); 



\n \n OUt 



n 2P, f2ax,^+Px^') =0, (r=l,2,....n-l); 



Xm \ OUr OUr J 



\^ du^\ duk dukj \^ duk\ '^ dUr dur J' 



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



Before applying the conditions of orthogonality to the curves (36) 

 we shall introduce considerable simplification in the work by intro- 

 ducing a set of new quantities, Qi, defined by the linear transforma- 

 tion 



(87) Qm = 2 ax^ Px, and hence, Pi, = ^ A^^Qx 



X X 



where ^Ix^ is the minor of a\^ in the determinant a — \a\^\ divided by a 

 itself, so. that 



(88) S ai^ A\^ = 0, if i ±X ; S a\^ A\^ = 1. 

 Now, the identities (33') and (34') take the much simpler form 



(89) S(?,P, = 1; 2Q,^=0, (r= 1,2,... .,71-1); 



n u OUr 



