of Electrons, and the Spectrum of Canal Rays. 681 



much larger. For stability we must have p = q + iic with k 

 positive for each root. 



§ 21. We shall solve the general case where Z=ae Lpt , the 

 amplitude a being slowly variable, on the assumption that 

 for t=t , a — a , a constant. This makes 



f= 7 aj YT 'JoTt<t , 



terms of orders pfpz 9 pi/pgj.-« being neglected. We find in 

 the usual way to the same approximation, for tj>t , 



t= -^ ■ — \ **■* I aetP-Prf 1 . dt' 



(P-Pl){p-Pz) Pi-Pi <- J t Q 



Integrating repeatedly by parts we find 

 6 ipt 



£= 



r— +— ] 



(P-P*)(P-P2) 



a Up— pi) 2 Cp -pO (p -i ; 2) + (p-p 2 ) 2 J ' " J 



a ^l J 1_ £=S 6lMt -t 0) + ZZ£1 6 ^-*o) i (45) 



(P-Pl)(p—P2) I Pl-^2 ^1~P2 J 



The first two lines obviously represent the forced vibration 

 induced by the force of variable amplitude ; the third 

 represents the free vibration left over after the amplitude 

 first began to vary ; if the constant state be sufficiently far 

 back the free vibration disappears owing to damping. The 

 transformation is obviously legitimate only when the series 

 in the first line is convergent, that is, when a/a... are all 

 small compared with f—px and with^?— p 2 . This condition 

 ceases to hold when there is resonance. For instance, when 

 p=2?i the corresponding terms in the first line must be 



te ip x t rt 



replaced by the term 1 adtf, which represents a 



Pi—pzjtQ 



free vibration of variable amplitude. 



§ 22. When the disturbing force Z is of class h the waves 

 emitted by the disturbance J are : — 



(1) A forced wave of frequency p + lea and class k, of 

 variable amplitude. 



(2) Two free waves of frequencies p 1 + ka, p 2 + kco, 

 which vanish when the constant state is sufficiently 

 far back. 



