780 Proceedings of the Royal Society 
alternately come sharply to rest at successive equal intervals of 
time. This arrangement makes an excellent and instructive class 
experiment, and its value may be greatly increased by placing round 
the exterior end of one of the magnets a vertical coil of copper - 
wire connected with a distant galvanometer. The nature of the 
motion of this magnet at any instant is readily deciphered from 
the signals given by the reflected light on the galvanometer scale, 
which is also visible to the whole class. A more complex, hut 
with practice easily intelligible, signal is given by placing the coil 
round the contiguous ends of the magnets. 
The extension of this arrangement to three, four, and more equal 
magnets, all vibrating in one line, and of nearly equal mass, 
magnetic power, and (independent) period is of course obvious, and 
forms a beautiful mechanical illustration of the solution of a differen- 
tial equation. 
In thinking how most simply to explain such results to an 
elementary class, I was led to the following, which can hardly he 
new, though I have never met with it, but which is certainly not 
as well known as it ought to be. Take first the case of the two 
equal magnets. 
Since there are but two moving parts of the system, and each 
has but one degree of freedom, it is obvious that if we can find two 
different forms of motion of the system which, once established, 
will persist for ever, any motion whatever of the system must he a 
mere superposition of these two modes with arbitrary amplitudes 
and epochs. Now, one such mode is obviously the motion of the 
pendulums as one piece at their equilibrium distance from one 
another. As the magnetic force does not vary during this motion, 
the time of vibration is that of either pendulum when left to itself. 
The other fundamental mode is that in which the centre of inertia 
of the two remains fixed, i.e ., the simultaneous displacements of 
the two magnets are equal and in opposite directions. The time 
of small oscillations now will evidently be the same as if one of the 
magnets were held fixed and its magnetic strength doubled. It 
will, therefore, be shorter or longer than the former period, according 
as the poles presented to one another attract or repel, and its 
actual value is easily calculated. Hence, as these small motions 
separately can be represented by expressions such as cos ( mt + c), 
