434 Prof. A. W. Porter and Mr. R. E. Gibbs on 



change of phase. Attention may therefore at first be con- 

 centrated on the latter change alone. 



Differentiating R^+L' 2 /? 2 with regard to .r, one obtains 

 .finally 



2^,27 =a 1 -t-« 2 + J? 77 " 



.as the condition for maxima and minima : the maxima are 

 separated by half a wave-length and the minima are midway 

 •between them. 



Fig. 1. 



The above result can be shown diagrammatical!}' as a 

 -vector diagram (fig. 1). The triangle A represents the 



operation L -j +Ri, the triangle B the operation L 2 -r +R2* 



whilst C is an allowance for the mutual action. The 

 maxima an 1 minima of Y occur when M. 2 p 2 (which may 

 first be considered constant) is in line with X, i. e. when 

 2g r # = a 1 4 a 2 + P7r. If M is not constant, then with centre 

 :a helix (shown dotted) is constructed such that its radius 

 vector represents W 2 jr for each value of x. Y is a 

 maximum now, for those values of x for which it cuts the 

 helix perpendicularly. 



(ii.) Simple Dynamical case illustrating maintenance of 

 vibrations. 



Whereas in the previous case the frequency was fixed, in 

 that of the reacting telephones it is not so : the problem, 

 in fact, becomes one of the stability of chance vibrations 

 that may be excited. Such problems have been discussed 

 by the late Lord Rayleigh who, taking an ordinary dynamic 

 equation of the second order 



d 2 y dy •„ _ 



inquired the effect of disturbing terms arising in phase 



with either 11 or -f . In order that a small disturbance if 



,J at 



