1068 Mr. C. F. Bickerdike on the 



first have a conception o£ how the change is being made when 

 we imagine N to be increased ad infinitum. If we simply 

 add to the number o£ air-molecules, with the average kinetic 

 energy per molecule unaltered (except so far as may be the 

 result of abstracting energy from the heavy particle) we are 

 packing our finite space with an infinity of mass and of energy. 

 No one can conceive what this would mean — whether the 

 gaseous molecules have any room to move at all, in fact. 



Let us, first of all, therefore, conceive the air-molecules as 

 diminished in mass just in proportion to the increase in 

 their numbers. We can imagine each molecule — supposed 

 originally as uniformly of mass m— to be divided into x parts, 

 and x can be made as large as we like. The original mass and 

 density of the gas are then unaltered, and its total energy is 

 unaltered excepting for transference from the heavy 

 particles *. 



The accepted laws of distribution of energy are arrived at 

 on the supposition that the actual time of any collision is so 

 small in comparison with the time of free motion of each 

 molecule that it can be neglected, and the heavy particle is 

 conceived as colliding successively with individual molecules 

 of the gas. 



If the mean length of free path is / originally and the 

 radius of the molecules is r, I is assumed to be so much 

 larger than r that r is negligible"!". 



When we suppose each molecule to divide into x parts, 

 I and r are diminished indefinitely, but remain of the same 

 order of magnitude relatively to one another. 



On this supposition it is possible that the time required 



* Tt is clear that on this hypothesis the formula of Jeans (Report, 

 p. 6) for total energy in terms of energy of waves would not have the 

 meaning which he attributes to it. Am is the limiting wave-length, 

 which may be made as small as we like provided it is not comparable 

 with the average distance between molecules. The total energy is 



4R1 



-s t — 3. Jeans says this gives infinite energy when Am is infinitely small. 



This can only be on the assumption that RT is not similarly reduced in 

 magnitude and must imply that M is increased ad infinitum by mere 

 addition of molecules with the same mass and energy as 1he original 

 ones, i. e. packing the finite space with infinite mass and energy. Of 

 course, if that could be conceived, the energy of the heavy particles 

 would be small in comparison. If we multiply N, however, by the 

 method proposed above, the total energy is unchanged and RT must be 

 reduced as much as A' . This consideration alone, however, does not 



suffice to dispose of the argument that the energy of the heavy particles 

 would all go into the gas. 



t Vide Edgeworth, "On the Application of Probabilities to the 

 Movement of Gas Molecules," Phil. Mag. Sept. 1920, pp. 249, 250 and 

 footnote. 



