588 Mr. J. H. Jeans on the Conditions 



may include the potential energy of the system in a perma- 

 nent field of force, if such exists, and it may include electro- 

 static or electrodynamic energies, or any other energies which 

 are such that the equations of motion may be derived from 

 the function E in the Hamiltonian manner. The equations 

 of the system are 2n in number, being of the forms 



bqr dE 



a* - 



~6pr' 



~bPr 



dE 



~dt 



~d<lr ' 



('•=■1,2, 

 ('•=1,2, 



(y) 



The elimination of t from these equations will give the 

 equations of the stream-lines which determine the paths of 

 the representative points in our generalized space. These 

 equations will be 2n — l in number, and will be capable of 

 expression in the forms 



i|r A . = constant(.§=l, 2, . . . 2n — 1), . . (10) 



where yjr s is a definite function of the 2n coordinates*. 



There is one further equation which can be derived from 

 equations (9), and this may be expressed in the form 



^ 2n = constant + /(/), .... (11) 



where ^ 2n is a function of the 2n coordinates. This last 

 equation determines the motion of the particles along the 

 stream-lines. The 2n equations (10) and (11) are the exact 

 equivalents of the 2n equations (9). 



Now let us transform coordinates in our generalized space, 

 from the coordinates of scheme (3) to the generalized co- 

 ordinates 



^1, 'f 2, • • • • fan-l, fan- • • • (12) 



Transformed into these coordinates equation (8) becomes 



g =l bf s bt V ' 



For the first (2n — l) values of s (s = l,2,... 2n— 1) we have 



dt 



* From another point of view equations (10) may be regarded as first 

 integrals of the equations of motion. The whole question turns on the 

 fact that the equations are 2n—l in number. That this is so is evident 

 from the fact that the path of every point must be definitely and 

 uniquely determined by them. 



