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SCIENCE. 



[N. S. Vol. XXIII. No. 579. 



sary that a sufficient number of collisions 

 take place to reduce this difference to zero. 



The defect of this demonstration lies in 

 the assumption that the velocities in the 

 three rectangular directions can be con- 

 sidered as independent. As they do not 

 enter independently in tlie^ equations of 

 collision between the molecules, we might 

 fairly expect them to be related to each 

 other, until they are proved to be inde- 

 pendent. 



I have dwelt on this first method of Max- 

 well's, because of its historic importance, 

 and because it illustrates the difficulty of 

 deciding by inspection on the conditions 

 which may legitimately be assumed for the 

 application of the calculiis of probabilities. 

 In the elaborate method subsequently de- 

 veloped by Maxwell and by Boltzmann, the 

 same difficulty is met with in another form. 

 In this method the molecules are considered 

 as spheres, freely moving about within a 

 vessel and colliding with each other. The 

 effort is made to determine the character- 

 istics of the motion of the assemblage of 

 molecules which must obtain if the condi- 

 tion of the assemblage is to be, to outside 

 inspection, uniform. To do this, we con- 

 sider a number of molecules belonging to a 

 certain class characterized by possessing 

 certain component velocities before colli- 

 sion, and certain other component velocities 

 after collision, and a number of molecules 

 belonging to another class characterized by 

 possessing the same component velocities in 

 the reverse order, and, considering the 

 probable number in each of these classes as 

 a function of the component velocities, we 

 write down the expression for the probable 

 number of molecules of these two classes 

 which occupy the same element of volume 

 and so are in collision. "With this expres- 

 sion we can obtain an expression for the 

 average increase in the number of mole- 

 cules in one of the classes, due to collisions, 

 and this ought to be zero if the condition 



of the assemblage is to be uniform. From 

 the discussion of this last expression follow 

 Boltzmann 's H-theorem, the formula for 

 distribution and the theorem of equipar- 

 tition. 



When the mode is analyzed in which the 

 expressions giving the probable numbers 

 of molecules in the two classes are com- 

 bined, it appears that an assumption is in- 

 volved in it which is not evident on inspec- 

 tion: namely, that the probability of the 

 presence of a molecule of one class in an 

 element of volume is independent of the 

 coordinates of that element, or, what 

 amounts to the same thing, of the proba- 

 bility of the presence, in the same element 

 of volume, of a molecule of the other class. 

 The gas to which this assumption applies 

 is called by Boltzmann unordered with 

 respect to the distribution of the molecules 

 (molekulm'-ungeordnet) . Jeans uses for 

 this condition of a gas the very convenient 

 phrase 'molecular chaos.' 



Now we evidently can not assert off-hand 

 that this chaotic condition will obtain in 

 all systems of particles which represent 

 real gases. When Boltzmann tries to de- 

 fine it he does so by negative instances, 

 showing what the condition of the system 

 might be which would be 'ordered' with 

 respect to the distribution of the molecules ; 

 and then leaves us to infer that the vast 

 majority of distributions do not possess 

 any peculiarities which would put them in 

 the 'ordered' class. But this inference is 

 not readily drawn. It is evident that any 

 condition of the system is ordered, in the 

 sense that the successive conditions follow 

 on mechanical principles from the initial 

 condition. All that we can do by inspec- 

 tion is to cherish the hope that, while all 

 systems are 'ordered' in this sense, yet a 

 vast number of them — an infinite number 

 of them, it may be, in comparison with 

 those 'ordered' in Boltzmann 's sense— are 

 still 'unordered' in such a sense that the 



