A Theory of the Widening of Lines in Spectra. 319 



where e is the electrostatic measure of the charge on a corpuscle, 

 r the distance between the corpuscles and 9 the angle the line 

 joining the corpuscles makes with the axis of x. If p 1 and p % are 

 now the times of vibration of the system of two corpuscles, we 

 see from equation (1) that p 1 and p 2 are the roots of the equation 



(fjb — a — mp 2 )* = a 2 



where a is written for 



^(l-3cos'0); 



thus fx — a — mp~ = ± a. 



If p is the undisturbed period /u, — mp 2 — 0, then 



Pi 2 = p 2 , 



. , 2 « 

 P* =Po . 



One period is thus the same as the undisturbed period and if 

 Bp is the small change in the other period we have 



Sp = - — = — , (1 - 3 cos 2 6). 



mp mp ir 



Thus if 3cos 2 is less than unity the period will be diminished, 

 i.e. the line will be displaced towards the red end of the spectrum, 

 while if 3 cos 2 is greater than unity the period will be increased, 

 i.e. the line will be displaced towards the violet end. The greatest 

 negative value of 1 — 3 cos 2 6 is — 2, the greatest positive value 1, 

 thus the maximum increase in the period is twice the maximum 

 decrease. 



Thus if we have a crowd of vibrating corpuscles symmetrically 

 distributed round one vibrating w T ith the same period, the sharp 

 line corresponding to this period will be broadened out into a 

 band, the maximum displacement towards the violet end will be 

 twice that towards the red end, but inasmuch as X(l— 3cos 2 #) 

 vanishes for a symmetrical distribution, the amount of light in the 

 band on the red side of the original line will be the same as that 

 on the blue side. 



We see that the maximum amount of widening varies as l/? ,s , 

 i.e. is proportional to the density of the synchronously vibrating 

 corpuscles ; when the spectrum is produced by the electric dis- 

 charge the number of these corpuscles will increase with the 

 current density, hence we see the reason for the well-known 

 widening of the lines when the current density is increased. 

 A large number of luminous particles would not give widened 

 lines unless the particles were vibrating synchronously, it is only 

 the density of the synchronous particles that affects the widening 



