FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
473 
and -gp 2 ,Tt 3 2 = To in the second, parallel to R L and R 2 respectively, each combined 
with an equal lateral pressure, so that the tensor of the stress vector is constant. 
But, so far as the attraction is concerned, we may ignore the stress in the second 
medium altogether, and consider it as the 2P>, of the stress-vector in the first medium 
over the surface of the second medium. The tangential component summed has zero 
resultant; the attraction is therefore the sum of the normal components, or 2 r J\ cos 2 0 lf 
where 6 X is the angle between R t and the normal. This is the same as 21^ (R N 2 — R/), 
if R N and R x are the normal and tangential components of R, ; or 
2 Trrdr\fi^ £ 
-/L 
\4t TfXy^ fly + fX„J 
i mz I. 
fly'll* (fly + fly)/ J ’ 
(209) 
which on evaluation gives the required result (203). 
But this method does not give the true distribution of translational force due to the 
stresses. In the first medium there is no translational force, except on the magnet. 
Nor is there any translational force in the second p 2 medium. But at the interface, 
where p changes, there is the force — i|R 2 Vp per unit volume, and this is represented 
by the stress-difference at the interface. It is easily seen that the tangential stress- 
difference is zero, because 
T sin 20 = yuR N R T ,.("210) 
and both the normal induction and the tangential magnetic force are continuous. 
The real force is, therefore, the difference of the normal components of the stress- 
vectors, and is, therefore, normal to the interface. This we could conclude from the 
expression — 4rR 2 Vp. But since the resultant of the interfacial stress in the second 
medium is zero, we need not reckon it, so far as the attraction of the pole is concerned. 
The normal traction on the interface, due to both stresses, is of amount 
Wl" /.l.i 
87rV 6 (fly + fly) 2 
t 2 + a 2 ^ d l j 
( 211 ) 
per unit area. Summed up, it gives (203) again. 
That (211) properly represents the force — ^R' 3 Vp when p is discontinuous, we may 
also verify by supposing p to vary continuously in a very thin layer, and then proceed 
to the limit. 
The change from an attraction to a repulsion as p 3 changes from being greater 
to being less than p 1; depends upon the relative importance of the tensions parallel to 
the magnetic force and the lateral pressures operative at different parts of the 
interface. In the extreme case of p 3 = 0, we have R L tangential, with, therefore, a 
pressure everywhere. For the other extreme, R! is normal, and there is a pull on the 
second medium everywhere. When p 3 is finite there is a certain circular area on the 
interface within which the translational force due to the stress in the medium 
containing the pole m is towards that medium, whilst outside it the force is the other 
MDCCCXCII.—A. 3 P 
