MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 185 



The relation between the parts of x and depending on a free period is 



so that 



Thus the approximate dissipation function is 



m' 



and this is necessarily positive. Tf, however, the exciting period exactly coincides 

 with a free period, the approximation is invalid, and we cannot draw this conclusion. 



Turning next to the case of negatively charged particles, these carry the same 

 electric charge as the positively charged particles, biit have a much smaller true 

 mass. As a consequence m'fm for these is no longer small, but of finite order. Our 

 deduction from KAUFMANN'S experiments gave m' of about the same order ot 

 magnitude as m. This gives a value for a = 1'Gx 10~ 13 which is much less than the 

 atomic radius. 



Consequently a value of K = 10 16 would be required to give a free period in the 

 visible spectrum. 



When m'fm is finite, the roots of 



// ? 

 ( 1 + 1' 2 a 2 



make ka finite and do not further concern us. 

 The roots of 



m 



also, in general, make ka finite. But if >n'f)ii is just less than 2, there is a possible 

 root which makes ka small, given approximately by 



_-l-^L 1 = 2/1 m 



l l\ I A 



2m 

 The rate of radiation is approximately 



Hence, if m'fm is greater than 2, there is emission from infinite wave-length to 

 very far out in the ultra-violet. 



If m'fm is less than 2, there is absorption from infinite wave-length to a certain 

 VOL. ccx. A. 2 B 



