116 Mr. W. Sutherland on the 



Let us obtain general expressions for the potential energy 

 which the molecules in unit mass of a single chemical sub- 

 stance possess by virtue of their mutual attractions, and also for 

 the virial of these attractions. Let mm'(f>(r) be the potential 

 energy of two molecules, then d<j>(r)=—f(r)dr. For the 

 potential energy of the molecules in unit mass and for the 

 virial of their attractions, we shall have the expressions 

 2 22> 2( A W an( l i • i 22 m VW> tne double 2 relating to the 

 two distinct summations that have to be performed, and the 

 \ being introduced as a factor to neutralize the effect of 

 counting twice each pair of molecules as acting on one 

 another, in the process of summation. 



To obtain values for these sums in the form of integrals we 

 have to imagine the matter of the molecules as uniformly 

 spread throughout the space in which they are distributed. 



If there are n molecules in a space v, then - is the volume of 



a space which we will call the domain of a molecule, to dis- 

 tinguish it from the spatial extension of the molecule, and 

 from what is often called its sphere of action. To find the 

 boundaries of a domain, let us imagine at any moment the 

 distribution of the molecules to be uniform, and that the 

 matter of each expands out in the form of a sphere, the rate 

 of expansion being the same for all. Let them finally press 

 upon one another like plastic expanding bodies, until no 

 portion of the space is unoccupied by matter ; then the whole 

 space is divided into n polyhedra, of which each is the domain 

 of a molecule. Imagine the matter of one polyhedron 

 gathered into its centre, then, by drawing matter uniformly 

 from the continuous mass, and building it up round the faces 

 of the polyhedron so as to form a spherical cavity, or, if neces- 

 sary, by cutting matter away from the faces of the polyhedron 

 and distributing it uniformly throughout the continuous mass, 

 we can finally, with only an infinitesimal alteration of the 

 density of the continuous mass, form a continuous mass round 

 any molecule gathered into the centre of its domain ; so that 

 the potential energy of the molecule and of the continuous 

 mass so formed is the same as that of the molecule and all 

 the other original molecules. The radius a of the spherical 

 domain thus defined is then the lower limit to be used in the 

 desired integrals. The upper limit is the length L already 

 defined. 



With a molecule m as centre describe a spherical shell of 

 radius r and thickness dr; let p be the density of the continu- 

 ously distributed matter, which, of course, is the same as that 

 of the original molecular distribution, then the potential 



