GEOMETRIC INVESTIGATIONS ON DYNAMICS. 293 



(10) Jmi = 0, if /± 7»; Amm=i, (/, m = 1, 2, ,n). 



Thus any natural famih/ of curves is represented by differential equations 

 of the form (13) for euclidean space and of the form (15) for any space. 

 Here L is an arbitrary point function. From these equations it is 

 evident that, given a point Xi of I'„ and a direction ^t through this 

 point, one and only one curve of a given natifral family of curves is 

 determined. There are thus oo"-i curves passing through a point and 

 a totality of oo2("-i) curves composing the family. 



If the function F is constant, our problem reduces to finding the 

 curves for which 



(17) j ds = minimum, 



and equations (15) become 



(18) x'^ + ^Y^\x'ixi = Q, {m=\,2,....,n) 



ik ^ ' 



the well-known equations of the geodesies in any space. -^^ 



§6. Proof of the direct or Lipschitz theorem. Darboux^^ 

 proves this by noting that the determination of the geodesies in any 

 space F„, where 



ds~ = S F^aik dxi dxk, 



ik 



leads to a minimizing of 



/V S F^Uik dxi dxk = / F V 2 "ik dxi dxk . 

 ik %J ik 



If, now, the geodesies in the space F„ are conformally represented in a 

 space V'n in which 



ds- = S aifc dxi dxk, 



ik 



the representing curves will evidently form a natural family in !'„. 

 Since conformal representation leaves angles unchanged, the orthog- 

 onal property expressed by Beltrami's theorem for the geodesies 

 will be immediately carried over into the orthogonal property expressed 

 by Lipschitz 's theorem for a natural family. Further, since ds is 



10 Bianchi, ibid., vol. 1, p. 334. 



11 Darboux, ibid., vol. 2, p. 510. 



