Vibrations of a Dielectric Sphere. 443 



part of A,. The period is accordingly 27ra%fdl(pc)%, where 

 p =4*493, and this leads to the value in question. We may 

 now consider two cases : firstly, that in which the mass is 

 entirely of electrical origin ; and, secondly, that in which in 

 and m' have an ordinary finite ratio. The second is the case 

 favoured by Walker from his analysis of the results of 

 Kaufmann's experiments. 



In the first place,, when the mass is wholly electrical, the 

 decrement reduces from (10) merely to k 2 = p 2 c/a/c 2 as in (11 ). 

 With a = 10 -13 , and /e = 10 16 , this gives A 2 = 6.10 -7 , indicating 

 a very permanent vibration. This vibration would inevitably 

 persist throughout the time daring which the equations for 

 the motion of the sphere under an applied force can be re- 

 garded as furnishing good approximations to that motion. 

 An exception is of course presented to this statement when 

 the applied force is of a periodic character, so that the forced 

 motion is vibratory, and the sphere never deviates far from 

 its initial position, and therefore the equations for the dis- 

 placement f of the sphere at any subsequent time, and the 

 function defining the state of things outside, never tend to 

 become less accurately representative. They only do so, for 

 example, in the problems discussed in the earlier papers, of 

 motion under a uniform force, on account of their assumption 

 that the sphere remains approximately at the origin. But in 

 the face of this consideration, the periodic applied force must 

 not be regarded as exceptional. For, the decrement being 

 of order 10" 7 , it is a matter literally of months,and not seconds, 

 before the free vibrations could be neglected, and therefore 

 all kinds of new agencies would have introduced further free 

 vibrations in the meantime. We may conclude, therefore, 

 that it is never possible to regard the free vibrations as in 

 any way less important than the forced motion. 



This persistence is rather more pronounced on Walker's 

 view of the negative electron, for if 7n' and m are of the 

 same order, we may ignore mp^JK, but not ???, in comparison 

 \n ith m\ so that k 2 = p 2 m / JK 2 (in + in'). The effect of including 

 m is therefore to decrease k 2 ln the ratio m'ftm + m'). For 

 example, the actual ratio of m to m derived by Walker is 

 about unity for Kaufmann's second set of experiments, so 

 that k 2 is halved. In other words, the oscillations may be 

 said to be twice as permanent as they would be if the mass 

 were wholly electrical. 



Finally, then, we see that if a free period of the negative 

 electron, regarded as dielectric, is to come within the visible 

 spectrum at all, it is necessary to suppose that its vibrations 

 are extremely permanent, and therefore that a constant force 



2G2 



