286 LIPKA. 



3) Optics. The paths of light in an isotropic medium in which the 

 index of refraction v \'aries from point to point, are determined accord- 

 ing to Fermat's principle by 



i 



'(Pi) 



V ds = minimum. 



(Po) 



Here the integral is proportional to the time. 



4) Catenaries. When a homogeneous, flexible, inextensil)le string 

 is acted on liy conservative forces, the forms of equilibriimi (general 

 catenaries) are determined by 



X 



(Pi) 



2 (]]' + h) ds = minimum. 



(Po) 



5) General case; natural jamUies. We note that each one of the 

 above problems leads to a set of curves determined by an expression 

 of the form 



X 



F ds = minimum. 



(Po) 



where F is a function of the coordinates of position only, and ds is the 

 element of length in the space under consideration. The curves 

 defined by such an expression may be called, following Painleve, a 

 natural family. For the dynamical trajectories, brachistochroncs, 

 and catenaries, there are oo^ such families corresponding to all possible 

 values of the constant h ; but in the optical case there is only one such 

 family. 



§2. The general problem of dynamics. Consider a system 

 with n degrees of freedom, i. e. the position of the system at any 

 instant is determined by n independent parameters or coordinates; thus 



X = j\ {Xi, X2, . . . . , 2'„), y = J2 {Xi, X2,. . . ., Xr^, Z = fs {Xi, X2,. . . ., Xn). 



Then 



(Ix ^ dfidxi .dfj^dxn dy^ ^ dfidxi djjdxn 



dt dxi dt dXn dt dt dxi dt dXn dt 



and twice the kinetic energy is 



^^ = "' = (f)+(f)+(IJ=i-ft ('-'-''^ '"■ 



