THE ELECTB.IC AND LUMINIFEROUS MEDIUM. 
817 
(P, Q, K) = 
/(IF (IG 
{ d/ dx ’ dt dy ’ dt dz / 
when 'F is some function of position as yet undetermined, whose value is to be 
adjusted to satisfy the restriction to circuital flow which the present analysis for 
conduction and convection currents involves. The electrodynamic forcive acting on 
the conductors carrying the currents is 
(X, Y, Z) = 
, dF , dG , dH 
dF , dCr . dH 
dy dy dy 
dF . dG , dH\ 
hut this involves, in addition to the usually recognized foreives of Ampere’s law and 
Faraday’s rule, a forcive in the direction ds of the resultant current F and equal to 
— F dN/cZs, where N is the component of the vector-potential in the direction of ds, 
This additional forcive may be represented as balanced by a tension ZN, in each 
filament or tube of flow carrying a current i, just as above. The existence of this 
tension seems to admit of easy test by a suitable modification of Ampere’s third 
crucial experiment. 
It is now a simple matter to complete this theory, which at present applies to 
circuital convection and conduction currents, so as to include the effect of convection 
without this restriction. It will suffice to consider a uniform current i' flowing in an 
open path, thus accumulating electrification at one end and removing it from the 
other end. The second term in M when integrated with respect to ds yields 
ids. ^ % 
dr 
ds 
; thus in the energy of the element oiids there is a term ids.\\^^ cosdeZr, 
where 6 is the angle between ds and the distance r of dr from it, and dpjdl is the rate 
of increase of the density of electrification at the element dr. Thus there is an addi¬ 
tional electric force — 4-^ ff—, 
^ dt ]\r r 
-j- z^ + -f 2/~\ dp 
^ dr, and an additional electromagnetic 
force 
j dt 
dr, where {x, y, z) have reference to the element dr 
as origin. These expressions are appropriate where, in place of following the convec¬ 
tion of single electrons, we contemplate the change of electric density at a point in 
space; they suffer from an apparent want of convergency, which would be real were 
it not that |p dr is null. 
121. It may be observed finally, that the question as to how far it is permissible to 
entertain the view that the non-electrlc properties of matter may also be deducible 
from a simple theory of free electrons in a rotationally fluid aether, has hardly here been 
touched upon. The original vortex-atom theory of matter has scarcely had a begin¬ 
ning made of its development, except in von Helmholtz’s fundamental discovery of 
the permanence of vortices, and the subsequent mathematical discussions respecting 
their stability. How far a theory like tbe present can take the place of or supplement 
MDCCCxciv. —A. 5 m: 
