FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
435 
then have a distribution of forces and torques in the surface-layer which equilibrate 
the internal forces and torques. 
To illustrate; we know that Maxwell arrived at a peculiar stress, compounded of 
a tension parallel to a certain direction, and an equal lateral pressure, which would 
account for the mechanical actions apparent between electrified bodies, and endea¬ 
voured similarly to determine the stress in the interior of a magnetised body to 
harmonise with the similar external magnetic stress of the simple type mentioned. 
This stress in a magnetised body I believe to be thoroughly erroneous; nevertheless, 
so far as accounting for the forcive on a magnetised body is concerned, it will, when 
propei'ly carried out with due attention to surface-discontinuity, answer perfectly 
well, not because it is the stress, but because any stress would do the same, the only 
essential feature concerned being the external stress in the air. 
Here we may also note the very powerful nature of the stress-function, considered 
merely as a mathematical engine, apart from physical reality. For example, we may 
account for the forcive on a magnet in many ways, of which the two most prominent 
are by means of forces on imaginary magnetic matter, and by forces on imaginary 
electric currents, in the magnet and on its surface. To prove the equivalence of 
these two methods (and the many others) involves very complex surface- and volume- 
integrations and transformations in the general case, which may be all avoided by the 
use of the stress-function instead of the forces. 
§ 11. Next as regards the activity of the stress P N and the equivalent translational, 
distortional, and rotational activities. The activity of P N is P N q per unit area, if q 
be the velocity. Here 
PnQ. gq.NQi ~t 3 r 2 , ®'®2 "h .(45) 
bv (42); or, re-arranging, 
Pn* 1 — 1 X -f- g^CL + "— N2 Qq, 
= N 2 a 2 ,.(46) 
where Q ? is the conjugate stress on the q plane. That is, qQ ? or 2Q q is the 
negative of the vector flux of energy expressing the stress-activity. For we choose 
P NN so as to mean a pull when it is positive, and when the stress P N works in the 
same sense with q energy is transferred against the motion, to the matter which is 
pulled. 
The convergence of the energy-flux, or the divergence of qQ q , is therefore the 
activity per unit volume. Thus 
div + Q, 2 gr 2 + 0,3^3) = q (i div Q, L + j div + k div Ct 3 ) + (d 1 Vg 1 + Q , 2 Vq% + Q3V23) . ( 47 ) 
- q (VjPi + V„P 2 + V3P3) + PjVjq + P,V 2 q + P 3 V 3 q .... ( 48 ) 
where the first form (47) is generally most useful. Or 
3 k 2 
