252 
MR. J. LARMOR OX A DTXAMICAL THEORY OF 
and L = By — C/3. Under the usual circumstances, in which the oethereal displace¬ 
ment-current can be neglected, these expressions are identical with the ones given 
without valid demonstration in Maxwell’s ‘Treatise.’*" The remarkable property 
is there established {loc. cit., § 643) that, independently of the form of the relation 
between magnetic induction and magnetic force in the medium and whether there is 
permanent magnetism or not, this bodily forcive (with the last terms neglected) can 
be formally represented in explicit terras as equivalent to an imposed stress : viz. ^ 
denoting magnetic force and magnetic induction, the bodily forcive is the same as 
would arise from (i) a hydrostatic pressure (il) a tension along the bisector of 
the angle e between ^ and 33, equal to Ilf 33 cos ~e/47r, (iii) a pressure along the 
bisector of the supplementary angle, equal to |^33 sin ~e/47r, together with an out¬ 
standing bodily torque turning from 33 towards |lf and equal to ||33 sin 2e/47r. 
When 33 and |lf are in the same direction, the torque vanishes, and a pure stress 
remains in the form of a tension /47r along the lines of force and a 
pressure in all directions at right angles to them. There is no warrant for 
taking this stress to be other than a mere geometrical representation of the bodily 
forcive. It is however a convenient one for some purposes.! Thus the traction acting 
on the layer of transition between two media, in which (a, y) changes very rapidly, 
which might be directly deduced in the same manner as the electric traction above 
(§ 35), may also be expressed directly as the resultant of these Maxwellian tractions 
* Vol. II., § 640. It will be observed that the force acting on the moving electrons which constitute 
the true current is here taken to be (w '7 — w'ji, In the investigation of Part II., § 15, 
which determines the motional force on a single electron, the expression for T represents the kinetic 
energy of the cetlier; it is transformed so as to be expressed in terms of the electric displacement of the 
a 3 ther and the electrons of the materials by introducing (F, G, H) whose curl gives the actual velocity 
of the oether near the electron ; and finally, after the forces acting on the electrons and on the tethereal 
displacement have thus been separated out, (F, G, H) is eliminated by the same relation. Thus the 
force acting on the single moving electron comes out as e — 27 — F...), where (f, is 
the velocity of the medium ; and the average force acting on the electrons in the element of volume, that 
is, the induced electric force causing electric separation in the element, is e (yc — zb — -,-), as 
there given. But in computing, as in Part II., § 23, the electromagnetic force on an element of volume 
carrying a current, it must be borne in mind that part of the above force on the single electron arises 
from the magnetism in this element of volume itself; and the principle of energy forbids that any part 
of the forcive on the mechanical element of volume of the material can arise from mutual actions inside 
the element, so that this part must be compensated by a reciprocal action of the moving electrons which 
constitute the current on those which constitute the magnetism, in a manner which might be expressed 
if necessary. Hence, when this local part is omitted in accordance with the general principle, the 
transmitted electromagnetic force is (r '7 — u''/3, as above, not (v'c — ^c'h, as 
previously stated in closer accordance with the AMP^:RE-M.\XWELL formula. Cf. § 44 rn/ru. [Observe, 
however, that in quoting Part II., § 15 (f, >], f) must now represent the velocity of the (ether multiplied 
by the square root of 4>7r times its very high coefficient of inertia; the unit of time was there tacitly 
chosen so that this factor should be unity.] 
[t For example, the repulsion exerted by alternating currents on pieces of copper or other con¬ 
ducting masses may thus most conveniently be represented.] 
