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V. On the Probabilities of Molecular Configurations. By 

 Ladislas Natanson, Phys. Z>., Lecturer on Natural 

 Philosophy ■, University of Cracow*. 



WITH reference to Sir William Thomson's recently re- 

 published kinetic explanation of dissipation of energy, 

 I should like to lay some few remarks upon the subject of the 

 probabilities of molecular configurations before the readers of 

 the Philosophical Magazine. 



Let us imagine a volume V divided into n equal parts or 

 " elements," and suppose in that volume N points to be con- 

 tained. These points may represent ordinary material mole- 

 cules for the sake of investigating the distribution of density 

 in a fluid medium, phenomena of diffusion, and some further 

 cases of material equilibrium; in other problems, however, 

 the points may be taken to represent anything else. They 

 may mean elementary chemical atoms in discussing dissocia- 

 tion and other cases of chemical equilibrium ; or ends of lines 

 drawn from a fixed point so as to represent the speeds of 

 various molecules at a given time (in order to investigate the 

 distribution of velocities in a crowd of molecules) ; or, again, 

 in molecular theory of capillary action, they may be regarded 

 as ends of lines constituting a (similar) space-diagram intended 

 to represent in direction and magnitude all the forces ex- 

 perienced at a given time by a molecule of a liquid. 



To take the simplest case, let us find the probability of the 

 following arrangement of N molecules in n elements of volume 

 Suppose the first element contains N x molecules, the second 

 N 2 * molecules, . . ., the nth contains N w molecules. This 

 particular arrangment may be symbolized thus : — 



(N^N,,...,^,); (1) 



its probability will be denoted by Q. It will be admitted at 

 first that the chance of any given molecule being in a given 

 element does not depend in any way upon the simultaneous 

 presence, in that element, of any number of other molecules. 

 Could we endow with one, and only one, molecule every 

 element of volume, we should be able to realize the corre- 

 ponding arrangement of molecules in 



N(N-l)... 3.2.1 or N! . . . \ (2) 



distinct manners. Now in the case before us the first element 

 contains by supposition not one but N x molecules ; and 



* Communicated by the Author. 

 E2 



