174 
Proceedings of the Royal Society 
been given as an example in Kinetics. And from the physical 
point of view it presents a very beautiful example of excessively 
slow, but continued, transformation of mixed potential and kinetic 
energy into kinetic energy alone. 
If r and 0 denote the polar coordinates of the disturbed mass, 
we have (supposing the curvature of the pulley to be large) by 
Lagrange’s method— 
to , the radius vector. 
[The work done by or against this system, along any arc of a 
curve, is the difference between the values of rO 2 at its ends.] 
Changing to rectangular coordinates ( x vertical), and maintaining 
the same degree of approximation as before, we have — 
The first suffices, without farther analysis, to show that the 
vertical acceleration of the disturbed mass is persistently downwards. 
Hence, the result of the disturbance must be the continuous trans- 
formation of the mixed potential and kinetic energy, of the 
vibration originally given to the disturbed mass, into kinetic energy 
of translation of the whole system. 
The equation of energy is easily seen to be— 
2 r - rQ 2 = — \gQ 2 , 
^(r 2 6)= -grO. 
Writing \gr for r, and QJ 2 for 0, these become— 
r - r6 2 = - 0 2 , 
~(r 2 0) = - 2 rQ . 
dt 
Hence, the motio?i of the disturbed mass is the same as that of a 
particle of unit mass under forces - 6 2 along, and - 20 perpendicular 
maxima* 
