of Electrons, and the Spectrum of Canal Rays. 679 



Let (f, 77, f) be the displacement of the iih. electron from 

 its position in steady motion ; f , 77, f may be expressed as 



sums of terms of the type A exp. i(pt — k z — )... where p is 



generally complex, of the form q + iic ; ^ is the frequency 

 relative to the ring, tc measures the damping, k is an integer 



between +3. The frequency to an outside observer is 



q + hco ; for the vibration has k waves, which are carried 

 round by the ring in its rotation. 



Changing the notation let u, r, w } X, Y, Z, L 5 ]\I, X, F, G, H 

 now refer to the new axes of f, 77, f ; denote the disturbing 

 force by F, and its components by JT, Y, Z. We have 



H=0, 



du du_C*/3* dw~ 



dt ~ ■ dt~ p ' ^ " ' 



-rsmaf^-sm(/)-f —cos 9 J + -^sma(-Xcos^+I sine/)) 



+ -f^d0 smaSin ^ 

 ^ = -^ s in a (||sm^+^cos^)+^(Xcosa + Zsina S in^). 



In the last three equations X, F,... are all independent 



of t ; in the functions ^^... f, 77, f must be put equal to 



of 

 zero o/ier differentiation. Thus we see that in the disturbing 

 force terms of two classes only occur : 



(a) Class : the term -—Xcosa in Z, a constant. 



(IS) Class —1 : the remaining terms, all of which involve 



'2tti 

 cos cf> or sin (j> linearly, where $ = cot + B + — — . 



The corresponding relative frequencies are 0, co ; the 

 ■absolute frequencies, being q + kco, are both zero. 



Hence neither set of terms gives rise to periodic waves in 

 the surrounding medium. 



Y= 



