464 
MR, O. HEAVISIDE OH THE FORCES, STRESSES, AND 
about the stress in the interior of a magnet, there can be no question as to the 
possible validity of this stress in the air outside our magnet, for we know that the 
force R is then a polar force, and that is all that is wanted, m and h being merely 
auxiliaries, derived from R. 
Now consider a region A, containing magnets of this kind, enclosed in B, the rest 
of space, also containing magnets. The mutual force between the two regions is 
expressed by 2P N over the interface, which we may exchange for 2Em through 
either region A or B, still on the assumption that R remains polar. 
But if we remove this restriction upon the nature of R, and allow it to be arbitrary, 
say in region B or in any portion thereof, we find 
NF = div P N = RN aiv R + NV curl R . R; 
or 
F = Rm + A JR, 
if J = curl R. This gives us, from a knowledge of the external magnetic field of polar 
magnets only, the mechanical force exerted by a magnet on a region containing J. 
whatever that may be, provided it be measurable as above ; and without any experi¬ 
mental knowledge of electric currents, we could now predict their mechanical effects in 
every respect by the principle of the equality of action and reaction, not merely as 
regards the mutual influence of a magnet and a closed current, but as regards the 
mutual influence of the closed currents themselves; the magnetic force of a closed current, 
for instance, being the force on unit of m, is equivalently the force exerted by m on the 
closed current, which, by the above, we know. Also, we see that according to this 
magnetic notion of electric current, it is necessarily circuital. 
At the same time, it is to be remarked that our real knowledge must cease at the 
boundary of the region containing electric current, a metallic conductor for instance; 
the surface over which P N is reckoned, on one side of which is the magnet, on the 
other side electric current, can only be pushed up as far as the conductor. The stress 
P N may therefore cease altogether on reaching the conductor, where it forms a distri¬ 
bution of surface force fully representing the action of the magnet on the conductor. 
Similarly, we need not continue the stress into the interior of the magnet. Then, so 
far as the resultant force on the magnet as a whole, in translating or rotating it, and, 
similarly, so far as the action on the conductor, is concerned, the simple stress P N of 
constant tensor -|R 3 , varying from a tension parallel to R to an equal pressure laterally, 
acting in the medium between the magnet and conductor, accounts, by its terminal 
pulls or pushes, for the mechanical forces on them. The lateral pressure is especially 
prominent in the case of conductors, whilst the tension goes more or less out of sight, 
as the immediate cause of motion. Thus, when parallel currents appear to attract one 
another, the conductors are really pushed together by the lateral pressure on each 
conductor being greater on the side remote from the other than on the near side : 
whilst if the currents are oppositely directed, the pressure on the near sides is greater 
than on the remote sides, and they appear to repel one another, 
