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XII. Molecular Dynamics of a Crystal. By Lord Kelvin *. 



§ 1. FT1HE object of this communication is to partially 

 JL realise the hope expressed at the end of my 

 paper of July 1 and July 15, 1889, on the " Molecular 

 Constitution of Matter j ": — " The mathematical investigation 

 must be deferred for a future communication, when I hope 

 to give it with some further developments." The italics are 

 of present date. 



Following the ideas and principles suggested in §§ 14-20 

 of that paper (referred to henceforth for brevity as M. C. M.), 

 let us first find the work required to separate all the atoms of 

 a homoo-eneous assemblage of a great number n of molecules 

 to infinite distances from one another. Each molecule may 

 be a single atom, or it may be a group of i atoms (similar to 

 one another or dissimilar, as the case may be) which makes 

 the whole assemblage a group of i assemblages, each of n 

 single atoms. 



§ 2. Remove now one molecule from its place in the 

 assemblage to an infinite distance, keeping unchanged the 

 configuration of its constituent atoms, and keeping unmoved 

 every atom remaining in the assemblage. Let W be the 

 work required to do so. This is the same for all the molecules 

 within the assemblage, except the negligible number of those 

 (§ 30 . below) which are within influential distance of the 

 surface. Hence itiW is the total work required to separate 

 all the n molecules of the assemblage to infinite distances 

 from one another. Add to this n times the work required 

 to separate the i atoms of one of the molecules to infinite 

 distances from one another, and we have the whole w r ork 

 required to separate all the in atoms of the given assemblage. 



Another procedure, sometimes more convenient, is as 

 follows : — Remove any one atom from the assemblage, keeping- 

 all the others unmoved. Let w be the work required to do 

 so, and let %w denote the sum of the amounts of work 

 required to do this for every atom separately of the whole 

 assemblage. The total amount of work required to separate 

 all the atoms to infinite distances from one another is |2t£. 

 This (not subject to any limitation such as that stated for the 

 former procedure) is rigorously true for any assemblage 

 whatever of any number of atoms, small or large. It is, in 

 fact, the well-known theorem of potential energy in the 



* From the Proceedings of the Royal Society of Edinburgh for 

 1901-2, communicated by the Author. 



t Proc. Roy. Soc. Edin., and vol. iii. of Mathematical and Physical 

 Papers, art. xcvii. 



