Mechanism of Radiation. 427 



Instead o£ supposing the complete potential-function to be 



we are compelled to suppose it to be of the form 



eeftyiir) + mm ! yjr 2 (r) , 



where ee' are the charges, and mm! the masses of the two 

 ions*. When r = 0, we must have 



ijr 2 (r) = cc, and ^{^I^Sf) — ^ • 



When r is very great we must have, in the limit, 



*i(r) = r~\ f 2 (r)=0f. 



This complete potential-function must necessarily contain 

 a reference to some fixed length or lengths J, and this will 

 determine the scale of the atom. Let us suppose the potential- 

 function to be such that some stable configurations of ions exist, 

 and examine the consequences of regarding such a configuration 

 of ions as our picture of an atom. Seeing that there are several 

 hundreds of ions in an atom §, we should naturally expect 

 the configuration of stable equilibrium to be such that the 

 shape of the atom is very approximately spherical, and this 

 conclusion agrees with the result already reached in § 7. 



The Equations of Small Displacements of the Ideal Atom. 



§ 12. To determine the positions of equilibrium and the 

 vibrations about these positions, for a dynamical system con- 

 sisting of a large number of ions, is obviously impracticable. 

 It is, however, known that the number of ions in an ate 22 is 

 very large, so that we shall attack the problem in the simple 

 case of an " ideal " atom in which the ions are supposed to 

 be so small that the number in an atom may be regarded as 

 infinite. After the investigation it will be shown how it is 

 possible to obtain an approximate knowledge of the spectrum 



* For ions such as we are now considering e=+e' and m=m\ T 

 have, however, stated the law in general terms. 



f In an exhaustive treatment of rether-force, ^ 2 (>') would probably be 

 identical with the gravitational potential, but for our present purpose 

 gravitational forces may be disregarded. 



X E. g. we might have as the potential-function 



r J' 



and a might be referred to as the " size " of the ion. 



§ Roughly, 700 in the hydrogen atom, and for ether elements in pro- 

 portion to the atomic weights of the elements. We here suppose the mass 

 of the atom to consist solely of the masses of ions of which it is composed. 

 See J. J. Thomson, Phil. Mag. vol. xlviii. p. 547, and earlier papers. 



