222 
MR. HEARN ON DISCREPANCIES OBSERVED 
We are sure that some degree of magnetism must be induced in the masses and 
balls by terrestrial magnetic influence, and then the magnetic systems may be capable 
of sensibly influencing each other. Now when balls are magnetized, their action is 
the same as if there were a small magnet concentric with each ball, and therefore the 
mutual action of balls on each other is reduced to that of small magnets, the distance 
of whose centres is great compared with their lengths. In the case of the earth being 
one of the magnetic bodies, the distance of the centres may be well considered infi- 
nite in comparison with the lengths of the supposed magnets, and therefore magnetic 
action of the earth upon any body at its surface will not tend to produce any motion 
of translation, or to increase the weight of such body. But when two balls at a 
moderate distance from each other are magnetized, the force producing motion of 
translation does not vanish, except for certain relative positions of the magnetic axes, 
and is sometimes positive and sometimes negative ; that is, sometimes attractive and 
sometimes repulsive. Imagine then that the masses have remained in the positive 
position for some hours ( e . g. during the night) ; the mutual influence of the masses 
and balls combined with terrestrial influence must have induced a certain very 
minutely magnetic state in those bodies, and the resulting attraction of the masses 
and balls is mixed up with the attraction of their mutual gravitation. Now reverse 
the masses by turning the plank round the centre of motion through nearly 180 °, the 
action arising from magnetism, if we suppose the masses capable of preserving their 
minutely magnetic state, will now probably be one of repulsion, though perhaps not 
precisely equal in amount to the magnetic attraction in the positive position. This 
repulsion will also be mixed up with the action of mutual gravitation. 
Now, when we reflect that the attraction arising from gravitation between a mass and 
one of the balls is exceedingly minute (about the tenth of a grain), it is clear that an 
almost inconceivably feeble magnetic state may be the cause of great perturbations. 
I now proceed to apply mathematics to these views. The term depending on the 
. „ . . , . £E . . , , .. . . . &M . , 
action of gravitation being -|r i.<px, let that depending on magnetic action be -|r i.yx, 
then the differential equation of motion will be 
d?x dx „ , kEi kM.i . 
and if we suppose <px and x expanded to the first power of x, we may write 
<px= 1 -{-ax, \J/x= 1 -f- cx ; 
and making 
m 2 — 
kEia 
~T~ 
kM.ic 0 
we have 
d^x dx 0 
dF + Sdi+l* x '- 
97 i ki(E + M) ^ 
dt Tr ^ j, \ •) 
Let P be the value of x for the resting point or position of equilibrium, then 
^P=m^+ M(E t t M) » (*•) 
