358 Mr. A. Stephenson on 
The effect of the r motion on 6 is then merely to keep the 
latter in a forced vibration of constant amplitude, and the 
system is therefore in the steady state, no further energy 
being absorbed. It may be noted that when n = fi for the 
steady motion « = 0: in this case of exactly double frequency 
it is evident without analysis that at a certain amplitude of 
swing, if the phase is properly adjusted, the periodic varia- 
tion in tension is exactly balanced by the radial incident 
disturbance, so that r remains constant and all the incident 
energy is reflected. 
It is of interest to inquire how the amount of energy 
required for saturation depends upon the intensity of the 
disturbance and its frequency within the necessary range. 
When the system is saturated, 
and /3 = 90° or 0° respectively, so that 
The signs and the phase of 6 depend upon the sign of a 0r 
the initial value of a. There is no loss of generality in the 
choice of a so that « is positive. Then as the 6 swing 
increases a is gradually diminished from a to the steady 
value, which is therefore positive also. Hence in the above 
equations for a//, k, and 6 the upper or lower values are to- 
be taken according as n 2 is less or greater than jjl 2 . 
By substitution in (ii.) 
-tc 2 \ + ^1 ^ — 1 ) /cos 2 nt + \[ =an 2 cos 2nt r 
where X is the change in the mean value of r due to the 
/ n 2 \ 2 
6 motion. The terms in (— 2 — 1) being of the second 
order are neglected. We have then 
\ = ±1b 2 n 2 /e\ 
and ± , V= l{ 2a? ^| (f2 _ 4 „ 2) 0;_ 1 ) / }. . (v .) 
The effect of the increase, \, in the mean value of r is to- 
