GEOMETRIC INVESTIGATIONS ON DYNAMICS. 289 



three dimensional spaces of constant curvature by Kasner.^ In the 

 following sections the author proves this theorem for a euclidean space 

 of four dimensions (§9), then for a euclidean space of n dimensions, 

 (§10), and finally for a general curved space of n dimensions, (§11). 

 §5 contains a derivation of the differential equations of the natural 

 family of curves by the methods of the calculus of variation. In §6 

 the author gives a very simple and direct proof of the Lipschitz theorem. 

 It will be noted that although the proof of the direct theorem is very 

 brief, the proof of the converse theorem is quite long. 



The methods adopted for proving the converse theorems are similar 

 to those used by Kasner for the three dimensional case, except that 

 the use of the arc length as the parameter along a trajectory intro- 

 duces a symmetry without which the rather long and complex algebraic 

 expressions could hardly be handled.^ 



§4. Distance and angle. The totality of points determined by 

 assigning all possible values to n variables or coordinates Xi, Xo,. . . ., x„ 

 are said to constitute a space of n dimensions, F„. If the x's are 

 expressed as functions of m variables, wi, U2, . . . . , Ur 



^m) 



Xi = fi (Wl, Uo, , Urn), (i = 1,2, , 7l) 



the totality of points thus determined are said to form a space of m 

 dimensions, Vm, contained in T^„. In particular, if m — n — 1, we 

 have a^^ype^surface of ?i — 1 dimensions; if m — 1, we have a curve. 

 The element of arc length or distance between two points Xi and 

 Xi-\- dxi in Vn is defuied by a quadratic differential form 



(1) d^ = X ttik dxi dxk,"^ 



ik 



where the a's are functions of the a-'s. The parameter curve Xi is such 

 that along it only the coordinate Xi varies, the other coordinates 



5 E. Kasner, The theorem of Thomson and Tail and natural families of trajec- 

 tories. Trans. Am. Math. Soc, vol. 11 (1910), pp. 121-140. 



6 Natural families of curves have been characterized geometrically in other 

 ways, for euclidean spaces by E. Kasner: Natural families of trajectories: 

 conservative fields offeree; Trans. Am. Math. Soc, vol. 10 (1909), pp. 201-219; 

 and for general curved spaces by the present writer: Natural families of curves 

 in a general curved space of n-dimensions; Trans. Am. Math. Soc, vol. 13 (1912), 

 pp. 77-95. 



7 We write only S and understand that the summation is to be carried 



ik 



out from 1 to n for each of the indicated subscripts, unless otherwise specified. 



