Mechanical Phosphorescence. 359 
diminish /x 2 ; and if this is appreciable the frequency of 
emission after the removal of the incident disturbance must 
gradually become greater as the energy is given out. 
Again. 
* -1 = <7 2 ^, where<7 2 <3. 
P 
/ ' 
-* 21 ±n 2 -c 2 > 
9 
.'. from (v.) ±b* = f(l-lv 2 )a, (vi.) 
according as n is less or greater than fi, Now if c>2n, a is 
negative when « is positive, and therefore the above equation 
gives b 2 positive if n greater than //,, but negative if n less 
than fjb. It follows that in the latter case there cannot be 
steady motion and the amplitude of the 6 coordinate oscil- 
lates periodically. In the steady motion which holds for the 
upper half of the range it is evident from (vi.) that the 
total store of energy necessary for saturation is proportional 
to the square root of the incident intensity, and gradually 
decreases to zero as the applied frequency is taken greater 
within the range. It is to be noted that in the deduction of 
(vi.) only the terms of the first approximation were retained. 
5. When the exciting disturbance is removed the 6 swing 
directly forces a small oscillation vertically of exactly half 
its own period, the energy being thereby dissipated against 
the kinetic resistance of the vertical coordinate. It is neces- 
sary to obtain the conditions under which the emission is of 
constant frequency throughout its decay. Evidently from 
the preceding A, must be small compared with « 2 ; and on 
substitution for b 2 in the value of A, we find that this is 
brought about by taking c sufficiently close to 2n. Further- 
more, during emission there is motional resistance to the r 
coordinate, and by making this large we bring /3 towards the 
limit 7r/4, and therefore cause the frequencies of and r to 
approach the limits fi and 2//, respectively to any required 
degree of approximation. Thus the variation in the phos- 
phorescence frequency during decay is brought within any 
assigned range however small. 
We now seek to express the rate of emission of energy as 
a function of the time that has elapsed since the removal of 
the applied force. 
The forced motion of r due to 6 is of amplitude propor- 
tional to the square of the 6 amplitude, and the work done 
per oscillation against the motional resistance is therefore 
