﻿754 Dr. A. C. Crehore on the Construction of 



alone is very large, and were it not for the fact that the 

 definite integrals of so many of them are zero it M T ould be 

 impracticable to employ the process indicated at all. As it 

 is, the resulting equations which have been derived are 

 rather too long to publish here. 



Average Translational Force of Atom on Atom. 



The next process is to use these integrated equations to 

 derive the force that one atom exerts upon another, each 

 consisting of a single ring of electrons and a positive charge 

 equal and opposite in value to the sum of the charges of all 

 the electrons for a neutral atom, the centre of mass of the 

 positive charge being at the centre of the orbit of the electrons. 

 In so doing we may use the same equations for determining 

 the force that an electron e of the one atom exerts upon the 

 positive charge of the other atom by simply changing the 

 sign of the force because the product ( — e) x ( + «') becomes 

 negative, whereas it was positive for two electrons, and bv 

 making the radius of the orbit n of the positive charge equal 

 to zero. 



We have also to consider the force that each electron e 

 in the second atom exerts upon the positive charge of the 

 first atom by making the radius m equal to zero. Fortu- 

 nately, when the three sets of forces so obtained, first, the 

 electrons on electrons; second, the electrons of the one atom 

 on the positive charge of the other atom and the electrons 

 of the other atom upon the positive charge of the one 

 atom; and third, the positive charge of the one atom upon 

 the positive charge of the other, are added together, all the 

 terms involving the even powers of the radii m and n 

 cancel out. This drops a large number of terms and leaves only 

 those containing the product m 2 n z , and gives the final result 

 which applies to any two atoms with certain reservations. 



These special cases to which the general results do not 

 apply are those in which each atom has one single electron 

 or two electrons revolving at the same angular velocitv in 

 each atom ; for the integral equations differ when the 

 angular velocities are equal. Then the phase angle between 

 the two electrons comes into the account. This limitation, 

 however, is restricted to the case where the rings contain 

 one or two electrons. If there are three or more the phase 

 angle disappears in all terms up to and including the sixth 

 pow r er of the distance, and we obtain precisely the same 

 equations for the force of atom on atom by integrating for 

 synchronous revolution as we get for incommensurable 

 velocities when the number per ring is three or more. 



