19 



S with respect to the corresponding q's respectively, and consider Ihe partial 

 differential equation: 



„/ ds es\ 



E(q„...qs,^^^^,...^J =«„ (17) 



obtained by putting this expression equal to an arbitrary constant a,. If then 



S = F (^1, ... 7s, «1, . . . as) + C, 



where a^, . . . as and C are arbitrary constants like «j, is a total integral of (17), we 

 get, as shown by Hamilton and Jacobi, the general solution of the equations of 

 motion (4) by putting 



II -^+ A, g -A. {k^2,...s) (18) 



and 



|| = p;„ (k = 1,.. s) (19) 



where t is the time and ß^, . . . ßs a new set of arbitrary constants. By means of 

 (18) the q's are given as functions of the time t and the 2s constants «j, ... «$, 

 ßj^, . . . ßs which may be determined for instance from the values of the q's and 

 q's at a given moment. 



Now the class of systems, referred to, is that for which, for a suitable choice 

 of orthogonal coordinates, it is possible to find a total integral of (17) of the form 



s 



s =^S„{qk,a„...as), (20) 



1 



where Sk is a function of the s constants a^, . . . «s and of qi; only. In this case, in 

 which the equation (17) allows of what is called "separation of variables", we get 

 from (19) that every /; is a function of the «'s and of the corresponding q only. If 

 during the motion the coordinates do not become infinite in the course of time 

 or converge to fixed limits, every q will, just as for systems of one degree of freedom, 

 oscillate between two fixed values, different for the different q's and depending on 

 the «'s. Like in the case of a system of one degree of freedom, p^ will become 

 zero and change its sign whenever qk passes through one of these limits. Apart 

 from special cases, the system will during the motion never pass twice through 

 a configuration corresponding to the same set of values for the g's and p's, but it 

 will in the course of time pass within any given, however small, distance from any 

 configuration corresponding to a given set of values q{, ... qs, representing a point 

 within a certain closed s-dimensional extension limited by s pairs of (s — l)-dimensional 

 surfaces corresponding to constant values of the q's equal to the above mentioned 

 limits of oscillation. A motion of this kind is called "conditionally periodic". It 

 will be seen that the character of the motion will depend only on the «'s and not 

 on the ß's, which latter constants serve only to fix the exact configuration of the 



3* 



