THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
to Xn/x. For the case of a mixture X is the same for both constituents ; a result 
which may or may not hold good for a solution or a chemical compound. 
19. The value of X, namely fv, which has here been assumed, is not merely 
dictated by the form desired for the final result. That value has in fact already 
been specified as the first approximation in quite another connexion.* As this is 
the crucial point of uhe theory, it may be allowable to present the argument in 
detail. The total electric force acting on a single molecule is derived from the 
aggregate potential V = S d/dx d/d^ + d/dz) r-\ where p,, are 
the components of the moment /x of a polarized molecule. This potential, when the 
point considered is inside the polarized medium, involves the actual distribution of 
the suirounding molecules; and thus the force derived from it changes rapidly at 
any instant of time, in the interstices between the molecules. But when the point 
cmisidered is outside the polarized medium, or inside a cavity formed in it whose 
dimensions are considerable compared with molecular distances, the summation in 
the expression for V may be replaced by continuous integration ; so that, ( f', g', h') 
being the intensity of polarization in the molecules of the dielectric, 
^ — |(y d/dx + g' djdy -{- Id djdz) dr ; 
and the force thence derived is perfectly regular and continuous. This expression 
may be integrated by parts, since, the origin being outside the region of the integral, 
no infinities of the function to be integrated occur in that region. Thus 
^ + 'nig' -f nh') r-i dS - \{df’/dx + dg'ldij + dh'jdz) r"! dr ; 
that IS, the potential at points in free mther is due to Poisson’s ideal volume 
density p, equal to — {df'/dx -{- dg'jdy + dh'ldz), and surface density cr, equal to 
If + mg' + nh'. When the point considered is in an interior cavity, this surface 
density is extended over the surfiice of the cavity as well as over tlie outer boundary. 
Now when it is borne in mind that, at any rate in a fluid, the polar molecules are 
in rapid movement, and not in fixed positions which would imply a sort of crystalline 
structure, it follows that the electric force on a molecule in the interior of the 
material medium, with which we are concerned, is an average force involving the 
average distribution of these polar molecules, and is therefore properly due to an 
ideal continuous density like Poisson’s, even as regards elements of volume which 
are very close up to the point considered. To compute the average force which 
causes the polarization of a given molecule we have thus to consider that molecule 
as situated in the centre of a spherical cavity whose radius is of the order of 
molecular distances; and we have to take account of the eftbct of a Poisson 
distribution cn the surface of this cavity, or more precisely of the result of an 
averaged continuous local polarization, surrounding the molecule, whose intensity 
increases from nothing at a certain distance from the centre up to the full amount i' 
* “ On the Theory of Electrodynamiof^,” ‘ Proc. Roy. Soc.,’ ,52 (1892), p. fit 
VOL. CXC.—A. 9 TT 
