394 MATHEMATICAL APPENDICES 21* 



new motion of the atoms which now comes in is not free ; it takes 

 place under the influence of forces which maintain the combination 

 of the atoms into a molecule, and whose joint effect we call affinity. 

 In addition, then, to the kinetic energy, which is made up of the 

 sum of the kinetic energies of the atoms and molecules, we must 

 bring the potential energy also into the theorem, which, therefore, 

 takes the form 



N(E + <) = |S. {m(u* + v 2 + w*) + S.m(u 2 + to 2 + tt> 2 )} + SS.a. 



To the mean energy E of the to-and-fro motion of a molecule, 

 which I will call the molecular energy, there is here added on the 

 left-hand side the atomic energy (5 which is present inside the 

 molecule, in the form partly of heat-motions of atoms, and partly 

 of chemical affinity, while on the right-hand side these magnitudes 

 are taken into consideration as the kinetic energy of the molecules 

 and atoms, and as chemical work performed by heat. The last 

 magnitude is introduced by a function which represents the part 

 of the work which is done on a single atom contained in the 

 complex of the molecule. So that S.0 expresses the amount of 

 chemical work in the molecule, and SS.^ the whole amount of 

 chemical work in the medium. 



The value of this chemical energy we may easily express by 



/he attractive forces between the atoms if these forces belong to 



/. the class named__by Helmholtz central forces. Thus, if one 



atom exerts on another distant by r from it a force f(r), the work 



required to increase this distance by dl is 



fW; 



to overcome the affinity, therefore, in the infinitely small dis- 

 placement of an atom, an amount of energy or of heat 



is used up, so that we find 



as the value of the energy spent, the constant 6 here denoting the 

 smallest distance from the atom in question to which the attracted 

 atom can come. 



The new more general formulae, like the earlier simpler ones, 

 hold good for all possible states of motion, and may therefore 

 undergo variation in the same way as the others. But in this 



