346 Prof. J. J. Thomson on 
The terms in J cancel on integration, and we have 
a2 ff'oee-orteans 
Pe ret bble Vi (aG—yF dex dy dz 
=e value of (xG—yF) at the origin 
— 
Hence the moment of the momentum about any axis through 
the electrified point vanishes, so that the resultant momentum 
passes through the point. 
The magnetic force due to a moving particle is the same 
as that due to an element of current at the particle. The 
direction of the current being parallel to the direction of 
motion of the particle, and the intensity « of the current 
being determined by the relation .ds=e (velocity of particle), 
ds is the length of the element of current. Hence we 
may apply the preceding results to find the momentum in 
the field due to the mutual action of any system of currents 
and a fixed electrified point; and hence it follows that the 
resultant momentum passes through the electrified point. 
We have previously seen that when the magnetic field is 
due to a system of magnetic poles ina non-magnetic medium, 
the momentum in the field due to the action of the magnetic 
field on a fixed electrified point, has zero resultant but finite 
moment of momentum, and hence can be represented by a 
momentum passing through the point and an equal and 
opposite momentum through some point of the system pro- 
ducing the magnetic force. In the case when the magnetic 
field is due to currents the second momentum, 2. e. that 
passing through the system producing the magnetic field, is 
absent. This result seems at first sight not in accordance 
with the principle that we can replace a magnet, as in 
Ampére’s hypothesis, by a system of electric currents ; for in 
this case there would be a finite resultant momentum, while 
the moment of momentum about any line passing through 
the fixed electrified point would vanish. The explanation of 
the discrepancy is that in calculating the momentum due 
to the magnetic poles, we have supposed that the poles 
were surrounded by a medium which was non-magnetic, 
i. é. that the magnetic induction was equal to the magnetic 
force throughout the field. If the magnetic induction is not 
equal to the magnetic force, the results of our investigation 
would have to be modified, for the momentum is proportional 
to the magnetic induction and not to the magnetic force. 
When we allow for this difference, we see that in the 
case of the magnets, as well as in that of the currents, the 
