Theory of the Magnetic Influence on Spectra. 503 



Relative linear velocity of surfaces. . . 3040 cm. per sec. 



Distance between cylinders - 38 cm. 



Least relative retardation which could 



have been detected 00005 X. 



Assuming the speed and sensibility attained by Kundt in 

 this case the same as that in case of water — it could not have 

 been greater — the limits of the experiment with solution of 

 sodium hyposulphite are extended by a factor of 100. 



LXIII. On the Theory of the Magnetic Influence on Spectra; 

 and on the Radiation from moving Ions. By J. Larmor, 

 D.Sc, F.R.S* 



A THEORETICAL analysis of somewhat general cha- 

 racter can be developed in connexion with Zeeman's 

 phenomenon f, which may help to throw light on the nature 

 of the electric vibrations in the molecule. It will be con- 

 venient to begin with a simple case. 



1. Consider a single ion e, of effective mass M, describing 

 an elliptic orbit under an attraction to a fixed centre pro- 

 portional to the distance therefrom. The equations of motion 

 will be (x,'y, z) = —a 2 (x,y,z) ; and the frequency of oscil- 

 lation in any direction will be o/27r. Now suppose that a 

 uniform magnetic field H, in a direction (l,m,n), is intro- 

 duced : the equations of motion will become 



x = — a 2 cc + k (ny — mi ) 



y=- — a 2 y + /c(lz—nx;) 



z = — a 2 z -f ic(rnx — l'y) 



where K — eH/Mc 2 , in which c is the velocity of radiation f. 

 To obtain the frequencies (p/27r) of the oscillations thus 

 modified, we make as usual (x,y, z) proportional to e ipt . 

 This gives, after easy reduction, an equation for p, 



{p 2 — ct 2 y — K 2 j) 2 (p 2 —a 2 ) = 0. 



Thus, corresponding to each original period represented by 

 p=a, there are three modified ones represented hj p = a and 

 p 2 ±Kp — a 2 =0 ; when, as in practice, k is very small the two 



* Communicated by the Author. 



t Zeeman, Phil. Mag. March and July 1897 ; Michelson, Phil. Mag. 

 May 1897 ; Lodge, Proc. Roy. Soc. Feb. and June, 1897. 



\ Cf. Phil. Trans. A, 1895, p. 718. These equations only apply strictly 

 [infra, § 10) when the velocity of the ion is small compared with c. 



