302 Dr. L. Natanson on the Dynamical 



A system of hard elastic spheres, which exert upon one 

 another purely collisional forces, has been found to be in many 

 respects an adequate dynamical representation of what is 

 called an ideal gas. The laws of ideal gas-pressure cannot, 

 however, be accounted for, unless the spheres, or molecules, 

 of which the body is supposed to consist are assumed to be 

 infinitely small. 



Now we know that, in order to explain the properties of 

 real or imperfect gases, it is not sufficient to correct the 

 original theory under this head solely, i. e. to consider the 

 molecules as bodies of some small but definite size. Van der 

 Waals was led therefore to investigate the effect of attractive 

 or cohesive forces, which he supposed to operate between 

 molecules independently of impulsive forces, generated on 

 collision. Accordingly in his equation two correctional quan- 

 tities appear, determined respectively by both of these distinct 

 kinds of molecular interference. To my mind such a cumu- 

 lation of essentially heterogeneous assumptions is utterly 

 unsatisfactory ; and it certainly fails to connect the properties 

 to be explained with one dynamical fact only. Hence, I ven- 

 ture to think, we have to take some broader view of the nature 

 of molecular interaction. 



On the other hand, although no serious discordance is likely 

 to arise as to the highly important role of Yan der Waals's 

 theory in modern thermodynamical science, it must never- 

 theless be affirmed that his equation does not apply to known 

 facts with due approximation, so that the kinetic conception 

 on which it is founded may be said to be disproved. 



1. The fundamental assumption I propose to examine is, 

 that the molecules of a gas may be taken as mere material 

 points, which exert upon one another certain forces every 

 time the distance between them becomes equal to, or less than, 

 a given limit, R say. Whenever two molecules approach 

 within distance R, they will be said to undergo encounter ; and 

 the moment the distance becomes again greater than R will 

 be taken as the ending moment of entanglement. When 

 two or more molecules are thus under mutual interaction, 

 I call them a molecular system ; I shall confine myself, 

 however, at first to the consideration of bimolecular systems 

 by supposing the ratio of systems of any higher degree of 

 complexity to be small enough to be neglected. 



Consider a volume Y, in which N points or molecules are 

 constantly moving. Let ~Rf{v) dv denote the number of mo- 

 lecules whose velocities lie between v and v + dv. These we 

 shall call " r-inolecules " ; and similar abbreviations will be 

 used in other cases. Among the members of that class con- 



