THEORY OF ELECTROMAGNETISM. 
749 
Taking for simplicity the case where there is no external force (F), or force potential 
(W), we have 
<£'A'= 0, [<£'UV] a + 6 = 0. 
Substituting now from equation (27), § 55 for we see that this means that there 
is equilibrium owing to the simultaneous existence of three stresses : (1) the 
qrdinary elasticity theory stress, owing to terms which onty involve ^ ; (2) the stress 
which is independent of l Q and, therefore, depends only on electromagnetic quantities; 
(3) a stress due to terms in l 0 , which involve both 'F and electromagnetic quantities. 
When these last are linear in NF, the resulting stress will depend, like the second, upon 
electromagnetic quantities only. If not linear, they will depend both upon T and the 
electromagnetic quantities. It is quite possible that there should be no strain at all, 
and yet a very sensible stress due to electromagnetic actions. 
In fact, in solving the elasticity problem—having given the distribution throughout 
the field, of dielectric displacements, of currents, and of magnetisation, required the strain 
at any point—the only wa,y in which the electromagnetic data can be used is, by 
finding the force per unit volume and surface respectively due to them, and then 
treating these forces as external. That is, the knowledge of the stress which produces 
the mechanical effects of electromagnetism is of no use in discovering the strain 
actually resulting; all the knowledge we can thus utilise is that of the forces (per 
unit volume and surface) due to such stresses. This shows that, to find the true 
expression for l it is not sufficient to investigate experimentally what strain accom¬ 
panies a given displacement, or current, or magnetisation at a point. 4 ' The problem 
is much more complicated. The shapes of all the bodies present must be assumed of 
quite as great importance as the electromagnetic quantities in deciding the form of l 
from such experiments. 
76. These remarks may be illustrated by considering the effect of Maxwell’s stress 
in two different cases. Choosing one shape of soft-iron body it will be found that the 
magnetisation will, according to Maxwell’s stress, compress the body ; choosing 
another shape, expansion results. 
Suppose we have (l) an anchor ring of soft iron, (2) surrounding this a layer of air 
of uniform thickness, (3) surrounding this n coils of insulated uniformly distributed 
wire carrying a current c. Take columnar coordinates r, ■'), z, the axis of z being the 
axis of the anchor ring, and let i, j, h be unit vectors (functions of the position of a 
point) in the directions of dr, dS, dz respectively. At any point inside the coil 
we have H' = 2ncjjr. Assuming f to be a constant scalar I' = (f —• 1) JL'/in. 
Hence, from equation (27), § 67, 
{f} A' = - A'SI'H'/Z = - (ft - 1) Vj'SH'HV/ 47r = - n 2 c° (f - 1) i/irr 3 . 
* This seems to be the meaning of the third sentence of the small print on p. 269 of vol. 15., 
‘ Encyc. Brit.,’ 9th ed. 
