100 Lord Rayleigh on the Laic of 



lie on the same path, i. e. are attained by the system in its 

 free motion sooner or later. This fundamental assumption, 

 though certainly untrue in special cases, would appear to 

 apply in Lord Kelvin's problem ; and, if so. Maxwell's 

 argument requires the equality of kinetic energies for A and 

 C in the time-averages of a single system. 



In view of this contradiction we may infer that there must 

 be a weak place in one or other argument ; and I think I can 

 show that Lord Kelvin's conclusion above that the average 

 of the sum of the potential and kinetic energies of A is equal 

 to the average kinetic energy of C, is not generally true. In 

 order to see this let us suppose the repulsive force F to be 

 limited to a very thin stratum at H, so that A after penetrating 

 this stratum is subject to no further force until it reaches the 

 barrier K; and let us compare two cases,, the whole energy 

 being the same in both. 



In case (i.) F is so powerful that with whatever velocity 

 (within the possible limits) A can approach, it is reflected 

 at H, which then behaves like a fixed barrier. In case (ii.) F 

 is still powerful enough to produce this result, except when A 

 approaches it with a kinetic energy nearly equal to the whole 

 energy of the system. A then penetrates beyond H, moving 

 slowly from H to K and back again from K to H, thus 

 remaining for a relativelv lono- time bevond H. Lord Kelvin's 

 statement requires that the average total energy of A should 

 be the same in the two cases ; but this it cannot be. For 

 during the occasional penetrations beyond H in case (ii.) A 

 has nearly the whole energy of the system ; and its enjoyment 

 of this is prolonged by the penetration. Hence in case (ii.) A 

 has a higher average total energy than in case (i.) ; and a 

 margin is provided which may allow the average kinetic 

 energies to be equal. I believe that the consideration here 

 advanced goes to the root of the matter, and shows why it is 

 that the possession of potential energy may involve no 

 deduction from the full share of kinetic energy. 



Lord Kelvin's " decisive test-case " is entirely covered by 

 Maxwell's reasoning — a reasoning in my view substantially 

 correct. It would be possible, therefore, to take this case as a 

 typical example in illustration of the general argument ; but 

 I prefer for this purpose, as somewhat simpler, another test- 

 case, also proposed by Lord Kelvin, This is simply that of a 

 particle moving in two dimensions; and it may be symbolized 

 by the motion of the ball upon a billiard-table. If there is to 

 be potential energy, the table may be supposed to be out of 

 level. The reconsideration of this problem may perhaps be 

 thought superfluous, seeing that it has been ably treated 



