354 Mr. A. Stephenson on 
The class of phosphorescent substances exhibits exactly 
similar phenomena. 
The qualitative agreement being thus far established, we 
proceed to examine the quantitative properties of our typical 
system*. 
3. If r is the length of the string and the inclination to the 
vertical at time t, the equations of motion of our system under 
the radial disturbance are 
r0+2'r 0+gO=O (i.) 
r+2fc'r — r 6 2 + c 2 (r— 1) = an 2 cos 2nt . . . (ii.) 
In the case of a system which receives energy from, or 
gives up energy to, a surrounding medium the ' frictional ' 
coefficient, k, may be large during emission and negligible 
during absorption. Thus in the present case if the suspended 
body exposes a large surface the effect of periodic disturbance 
communicated through the air is accounted for by the term 
on the right of (ii.)? an d K represents merely the internal 
frictional force which may be small : when, on the other 
hand, the body is giving up energy by the generation of 
periodic motion in the air, k is large. We shall therefore 
assume that k is zero while the^ system is storing energy, 
but is large during- emission. 
Assuming the amplitude of u to be small initially, we have 
r = l — 2a cos 2nt. 
where I is the length of the string in equilibrium, and 
JL 
— 2 
Putting = (f)/r we have from (i.) 
r<f> +((!— r ) <£ = 0, 
/l ^ cos 2?iH <£ + (// y .4:n 2 cos 2nt)(l)~0, (iii.) 
where /M 2 =g/L 
* The results of the analysis in § 3-§ 5 are summarised in § 6. 
