On Mechanical Phosphorescence. 353 
frequency in its spring*, while it', on the other hand, the 
motion is initially horizontal, a vertical vibration is forced by 
the isochronous variation in the tension. Some notion of the 
marked nature of this interaction may best be gained from 
experimental observation. 
Now let us suppose that the vertical coordinate is subject 
to kinetic friction, while the other is frictionless. Then, 
whatever the relation between the periods, the pendulum 
motion forces a small oscillation vertically, in general of 
second order amplitude, maintaining it against friction; thus 
the energy is gradually dissipated and the frequency of the 
emission is double that of the pendulum. 
Consider the effect of a periodic disturbance applied radially. 
In general it will merely produce an isochronous forced 
vibration vertically, but if the frequency is approximately 
double that of the pendalutn motion, this oscillation will in 
turn magnify any existing small pendulum swing ; thus the 
system stores energy absorbed from incident disturbance of 
the particular frequency. 
For a given intensity of disturbance the amount of energy 
obtained in this way will depend upon the time of action, but 
evidently it must have some definite upper limit. 
When the exciting force is removed the pendulum motion 
generates a comparatively small forced oscillation vertically 
of double its own frequency, which continues until the energy 
is dissipated against the kinetic friction. Thus the system 
gives out the stored energy in a definite frequency during an 
interval which is comparable with, and may indeed be large 
compared with, the time of action of the incident disturbance. 
That is, the system phosphoresces. 
It is to be noted that in the special case when the frequencies 
of the normal motions vertically and horizontally are in the 
ratio 2/1, the forced oscillation vertically under the exciting 
disturbance, the energy stored, and the subsequent rate of 
energy emission have their greatest values. 
A disturbance acting on a coordinate through periodic 
change of spring has a continually cumulative effect when 
the ratio of its frequency to that of the free motion of the 
coordinate lies anywhere within a certain range. In the 
above, therefore, . energy is stored under incident force of 
any period within a range, but the resulting emission is 
always of one definite frequency, t. e. double that of the free 
pendulum motion. 
* " On a Class of Forced Oscillations," § 2, Quarterly Journal of 
Mathematics, no. 168 (1906). 
