﻿of a Loaded Spiral Spring. 389 



respond to Ba = 0, and those of i & , V period to 8<f> = 0, 



If, however, -j— is finite and equal to 2\, we have, when 



approximately 

 If, however, 

 the system is vibrating in one of its normal modes, either 



or 



(\-V\ 2 + l)Bx=Bk<j> 



throughout the motion, the periods corresponding to these 

 modes being nearly equal. In this case, if the system 

 receives a displacement not represented by either of these 

 equations, the subsequent motion will be compounded of two 

 vibrations, one of which slowly gains upon the other, and will 

 thus exhibit phenomena of intermittence. 



For example, if the displacement (&t* = X, 6\/> = 0) be given, 

 this may be resolved into 



and 



2*/\ 2 + l ^ 2^X 2 + 1 5 



8*=X ^' + l + \ W =-X 



2^X 2 +1 ' r " 2* / X 2 + l' 



and therefore, when the vibrations of one normal mode have 

 gained half a period on those of the other, the half-amplitude 



of the ^-vibration will have decreased from X to X — . a 



and a ^-vibration of half-amplitude X ,— — = will have 



appeared, while when another half-period is gained the initial 

 conditions will be restored. Thus, while at first the system 

 moves simply with an ^-vibration, this gradually diminishes 

 to a minimum value, and at the same time a c/>-vib ration is 

 gradually set up and grows to a maximum ; the latter vibra- 

 tion then decreases and finally vanishes, while the former 

 increases until it reaches its initial value, and then the phe- 

 nomena recur. 



It is easy to see that a similar intermittence will be 

 exhibited if the system is started with a 0- vibration only. 



The above results may readily be verified experimentally 

 by employing as the mass M a body of adjustable moment of 

 inertia. The most interesting case is that in which k is 

 adjusted so that \ is rendered very small, when the energy 



