the Magnetic Influence on Spectra. 511 



exists a steady motion in which the cord whirls round bodily, 

 and which will be generated when the velocity of the motion 

 imposed on the end is gradually increased from a very small 

 initial value to its final amount. 



The difficulty is, however, not thus surmounted ; for this 

 steady motion which does not involve radiation is really a 

 state of stationary undulation arising from the superposition 

 of a wave-train travelling outwards on another travelling 

 inwards, and the genesis of the latter one would have to be 

 accounted for. We might assume that these non-radiating 

 vibrations consisted of stationary waves reflected backwards 

 and forwards between two vibrating molecules, or between 

 two ions in the same molecule ; but even that would not be 

 satisfactory. As a matter of fact, however, no explanation of 

 this kind is needed. The effective electric inertia of an ion e 

 by itself is |^ 2 a -1 *, where a is the radius of its nucleus 

 supposed spherical : the rate at which it loses energy by 

 radiation is proportional to e 2 , and involves its motion, but 

 does not depend on a at all. The kinetic reaction to change 

 of its velocity which is connected with loss of energy by 

 radiation can thus be made negligible in comparison with 

 the kinetic reaction arising from its inertia. In fact, the 

 energy of the asthereal motion carried along by the moving 

 ion depends on the first term in H, involving r~ 2 , and the 

 radiated energy depends on the second term, involving r~\ 

 But in types of oscillation in which there are crowds of ions 

 moving close together in step, loss of energy by radiation is 

 an important feature in the dynamics of free vibrations. 



These considerations can be developed by aid of the analysis 

 of § 9 above. In consequence of the stream-function property 

 of Hp, the components of d/dt of the electric force, along Br 

 and along r89, are respectively 



c 2 dBp . c? clRp 



~- and — - 



p rdu p dr ' 



p being r sin 6; thus they are 



and 



-«» sin 6 { *'-/■/'> + £^p± + ft-*) } ; 

 ( r 6 cr err J 



and the electric force is obtained by integrating with respect tot. 

 * Phil. Trans. A, 1894, p. 812. This inertia is no longer quite constant 

 when the velocity of the ion is considerable compared wilh that of 

 radiation. In that case also the simple computation of the radiation here 

 given would not be exactly applicable ; and the problem would have to 

 be treated by continuous differential anatysis after the manner of Phil. 

 Trans. A, 1895, p. 718. 



