294 LIPKA. 



transformed into F ds, the hypersurfaces which are the loci of equal 

 arc lengths are transformed into the hypersurfaces which are the 

 loci of equal actions. 



We shall now give a more direct proof of the Lipschitz theorem with- 

 out using the Beltrami Theorem as a basis. 



Consider the equations of the natural family given by 



(10) — SFff;;— = — S T^T' (^= 1.2, ....,w). 



ds i ds It ik ox I ds ds 



Let the parameter curves Xi be curves of the natural type, and let the 

 parameter .I'l represent the action along these curves measured from 

 the corresponding intersection with a definite orthogonal parameter 

 hypersurface of the system xi = constant, e. g. xi — 0. Thus 



a-i = / Fi dsi = j FiV an dxi, 

 since, along the Xi parameter curves, we have by (2) 



dsi = V an dxi, 

 so that 



FiV^i = 1. . 



Applying equations (10) to our .ri parameter curves (along which 

 X2, xs, . . . ., Xn are all constants), we have 



(19) A/M=o, {1 = 2,-3,.... ,71). 



dxi \au/ 



Since the parameter curves .Ti are normal to the hypersurface xi = 0, 

 we have from (6) for the angle between the parameter curve xi and 

 any other parameter curve xi, 



cos cou = — -^= = 0, for .I'l = 



Van Vail 



or 



^=0, (/ = 2, 3,...., n), for .1-1 = 0. 



an 

 But as shown by (19), au/un is independent of Xx, so that 



^1=0, (/ = 2, 3, . . . . , 7i), for all values of .ri 

 On 



