318 Prof. Thomson, 



A Theory of the Widening of Lines in Spectra. By J. J 

 Thomson, M.A., F.R.S., Cavendish Professor of Experimental 

 Physics. 



[Read 12 March 1906.] 



The well-known widening which occurs under suitable 

 conditions in the lines of the spectra of many elements is 

 regarded in this paper as due to the effect of resonance between 

 systems which if free from each other's influence would vibrate 

 in the same period. If two systems whose free periods are p 1} p 2 

 are brought near together so as to exert small forces on each 

 other, the actual periods of the compound system thus formed 

 will in general differ from p 1} p 2 , the periods of the constituent 

 systems. If p x is not equal to p 2 this difference however will be 

 very small as the changes in the periods will be proportional to 

 the squares of the small forces which the one system exerts on 

 the other. If, however, p x and p 2 are equal the changes in the 

 periods will be much larger as they will be proportional to the 

 first power instead of to the square of the forces between the 

 systems. We shall confine our attention to the influence on each 

 other of systems whose free periods are the same ; unless this 

 influence is very large that of systems with unequal periods will 

 be quite inappreciable. 



We shall assume that the lines in the spectra are due to the 

 vibrations of negatively electrified corpuscles: the forces exerted 

 by one vibrating corpuscle on another are of two kinds, (1) the 

 electrostatic repulsion between the corpuscles, (2) those due to 

 electromagnetic induction between the moving corpuscles. We 

 shall consider these separately, beginning with the electrostatic 

 forces. Consider two corpuscles (1) and (2) vibrating parallel to 

 the axis of x; let x 1} x 2 be the displacements from equilibrium of 

 these corpuscles respectively, m the mass of either corpuscle, 

 /x the force of restitution for unit displacement, then if the 

 corpuscles were vibrating freely and not influencing each other 

 we should have 



d 2 x, . d?x 2 



m -=— + fxx 1 = 0, m —r- + fue 2 = 0. 



When we take the electrostatic forces between the two corpuscles 

 into account these equations become 

 d 2 x { 



m -~ + fix 1 = ~(l - 3 cos 2 6) (#! — Xo) 



CvL i 



m —j-^ + fxx 2 = — (1 — 3 cos 2 6) (x 2 - x x ) 



■in 



