404 MATHEMATICAL APPENDICES 24* 



24*. Molecular and Atomic Energy 



With respect to the formula 



thus proved for the whole mean value of the kinetic and potential 

 energy within a molecule, we have already remarked, at the end 

 of 22*, that the integrations cannot be strictly carried out without 

 a knowledge of the function ^ and of the limits of the integrals 

 which depend on its values. 



We may, however, approximately evaluate the triple integral 

 by considering that the functions 



c ~* x and 



change value like each other. The values of the integrals of the 

 two functions, if taken between equally wide limits, will therefore 

 be of the same order of magnitude. But the limits extend to 

 infinity in that integral only by means of which the molecular 

 energy E is calculated from the velocity w, and not in (, in which 

 the variables u, t>, tt) are limited by the finite magnitude $ in 

 accordance with the equation 



i x = i m (u 2 + tt 2 + tD 2 ) + = <*>. 

 We may therefore conjecture that integration of the function 



will give a value which is less than E. If this conjecture is 

 correct, we may conclude, with due regard to the summation over 

 all the atoms in the molecules indicated in the value of G by the 

 sign S, that 



if n denotes the number of atoms combined in the molecule. 



If we divide this internal energy of the whole complex of 

 atoms in equal shares among the n atoms, each share being 



e = (S/tt, 



we may express this conclusion thus, that the mean energy e of 

 an atom is less than the energy E of the motion of the centroid of 

 the molecule, or 



C < E. 



