400 MATHEMATICAL APPENDICES 22* 



highest value which the energy of affinity can have ; in other cases 

 the stability of the molecule is destroyed. 



The value of the constant S3 can therefore be determined by 

 our noting that the sum of the probabilities of all possible cases 

 must be certainty, and that therefore 



when the limits of the integrations are values connected by the 

 equation 



x = m(ll 2 + tt 2 + tt) 2 ) + 20 = 2$. 



Without knowing the value of the function we can carry the 

 integration only so far as to express S3 by the transcendental 

 equation 



where 



x = mg 2 + 2^, 



and the limiting value g is given by 



mg 2 + 2fy = 2*, 



<f> q and </>g being the mean values of </> corresponding to the 

 velocities q and g. 



We meet the same difficulty in attempting to find that part 

 of the energy which we have called the atomic energy ; for, in 

 taking the sum of the kinetic and potential energies within the 

 molecule we obtain on the average 



where the limits of the integrations are again conditioned by 

 i x = im(u 2 + tt 2 + n>' 2 ) + ^ = *. 



Without a knowledge of the function the integration can be 

 carried no further than as is given above for S3. 



23*. Rotations of the Molecules 



It can still be doubted whether the formulae for the individual 

 motions of the atoms, as they have so far been developed, really 

 contain everything of importance for the nature of the case. If 

 we consider that composite molecules are doubtless thrown into 

 rotation on collision, we must consider our procedure open to 



A 



