Partition of Kinetic Energy. 



105 



see that x cannot enter, or that / is a function of E only. 

 The only permanent distributions accordingly are those 

 included under the form 



f(W)dxdu, (18) 



where E is given by (17), and /is an arbitrary function. 



It is especially to be noticed that the limitation to the 

 form (18) holds only for phases lying upon the same pnth. 

 If two phases have different euergies, they do not lie upon 

 the same path, but in this case the independence of the 

 distributions in the two phases is already guaranteed by the 

 form of (18). The question is whether all phases of given 

 energy lie upon the same path. It is easy to invent cases 

 for which the answer will be in the negative. Suppose, for 

 example, that there are two centres of force 0, 0' on the line 

 of motion which attract with a force at first proportional to 

 distance but vanishing when the distance exceeds a certain 

 value less than the interval 00'. A particle may then vibrate 

 with the same (small) energy either round or round 0' ; 

 but the phases of the two motions do not lie upon the same 

 path. Consequently / is not limited by the condition of 

 steadiness to be the same in the two groups of phases. In 

 all cases steadiness is ensured by the form (18) ; and if all 

 phases of equal energy lie upon the same path, this form is 

 necessary as well as sufficient. 



All the essential difficulties of the theory appear to be 

 raised by the particular case just discussed, and the reader to 

 whom the subject is new is recommended to give it his 

 careful attention. 



In the more general problem of motion in two dimensions 

 the discussion follows a parallel course. In order to find the 

 distribution at time t ! corresponding to (9) at time t, we have 

 to transform the multiple differential, regarding x, y, u, v as 



Here again we take the 

 y, a/, y' as an intermediate 



known functions of x' ', y f 

 initial and final coordinates 

 set of variables. Thus 



u\ v 



dx' dy' du' dv r = dx' dy' dx dy x 



dx dy du dv — dx dy dx 'dy' x 



du' dv' 



dx ' dx 



du' dv r 



dy ' dy 



du dv 



dx" dx 1 



du dv 



dy' ' dy' 



. (19) 



. (20) 



