Spectra and Planck's Law. 431 



where ds is an element of a line of electric force, ay the cross- 

 section of a tube of force, m the mass of an electron, h Planck's 



constant, and p determined by the equation ~ = . 



When the force is due to a single positive charge we may 

 put p = l/r, co = r 2 , where r is the distance from the charge. 



We notice that when the law of force reduces to the 

 inverse square law, a)R is constant and B vanishes. 



Integrating by parts we have 



B= 2 p[,,«R + fRrf,] (1) 



Hence, if Bj, B 2 be the values of B at two places of 

 equilibrium where R vanishes, 



B 1 .-B f =^f Rds, 



or if w be the work done on an electron in moving from one 

 place to another, 



X5 l —B. 7 = — —w. 



eli 



If n l; n 2 are the frequencies of the vibrations of an electron 

 at the two places 



1 e „ 1 



V.rrr in 



Hence 



h ' 



P 



'2tt m 



w 

 ni — n 2 = 



Thus, if an electron falls from a place where the magnetic 

 force vanishes to another position of equilibrium, the fre- 

 -quency of the vibration is equal to w/li t where iv is the 

 energy converted into radiation. Thus the transference from 

 potential energy to energy of radiation is in accordance with 

 Planck's law. 



Assuming the relation between the electric and magnetic 

 force given in the preceding investigations, it is easy to find 

 laws of electric force such that the frequencies of the vibra- 

 tions of electrons would be connected by a relation similar to 

 that expressed by a liydberg series. 



Thus, for example, suppose that the force R exerted by 



