GEOMETRIC INVESTIGATIONS ON DYNAMICS. 287 



where the o's are functions of the n parameters Xi, Xo,. . . ., Xn- If 

 the acting forces are conservative, there will exist a work function 

 ir(.ri, a-2, . . . . , .r„) independent of the time, and a constant of energy h 

 so that T — ]V = h. The trajectories are then determined by 



X 



(^') / / . . 



V 2 (H + h) V 2 ciik dxi dxk = minimum. 



(Po) k 



Employing geometric language, we may regard Xi, xo,. . . . , Xn as 

 the coordinates of a point in a space of 7i dimensions, and Po and Pi as 

 two points in such a space. As the element of arc length in such a 

 space is given by . 



ds^ = 2 (liK-dxidxic, 



ik 



we are led to the problem of minimizing I F ds, where P is a function 



of the coordinates Xi, X2, . . . . .t„ and ds is the element of arc in a space 

 of n dimensions. The space 'may be euclidean (of zero curvature), 

 or non-euclidean, of constant or variable curvature. The natural 

 family consists of co2(i-i) curves, one through each point in each 

 direction. 



If the work function W is zero, then P is a constant, and our prob- 

 lem reduces to finding the curves for which 



J-'CPi) 

 ds = 

 (Po) 



minirnum; 



'(Po) 



this evidently defines the geodesies in our space. 



§3. Geometric Properties of a dynamical system. In 1869, 

 Beltrami ^ proved a remarkable theorem concerning the geodesies in 

 a space of 71 dimensions, !'„. This theorem may be stated as follows: 



The 00 "-1 geodesies which pass out normally from points of any hypcr- 

 surfaee, F„_i, form a normal hypcrcongruencc, i. e. admits 00 1 normal 

 hyper surf aces, which arc the loci of equal arc lengths measured from any 

 one of the system of hyper surf aces. 



■ The theorem is a generalization of Gauss's theorem for the geo- 

 desies on an ordinary surface, V2. 



In 1871, Lipschitz ^ announced the corresponding theorem for the 



1 E. Beltrami, Sulla teorica generale dei parametri differenziali. Memorie 

 deir Academia delle Scienze dell' Institute di Bologna, series 2a, vol. 8, p. 549. 



2 R. Lipschitz, Untersuchung cines Problems der V ariationsrechnung in- 

 welchem das Problem der Mcchanik enthalten ist. Crelle's Journal, vol. 7-4. 



