DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
329 
The phenomena of thermo-electricity show that { u, v} must be a function of the 
electrical coordinates y, but it is difficult to determine the form of this function with 
certainty. The two most striking phenomena in thermo-electricity are (l) the Peltier 
effect which shows that electromotive forces exist at the junction of different metals 
in the thermoelectric circuit, and (2) the Thomson effect which shows that electro¬ 
motive forces exist throughout an unequally heated conductor. We shall try to find 
a term whose presence in the expression for the kinetic energy would correspond to 
these effects. Let us consider a circuit formed by two metallic wires each parallel 
to the axis of x, \et p, q, r denote the components of the electric displacement parallel 
to the axes of x, y, z respectively, let us suppose that we have in the expression for 
the kinetic energy per unit of volume the term 
Jij_.it ,dr\ @ 
yte dy dzj 
(59) 
where k is a quantity which depends upon the material of which the wire is made. 
Thus the expression for the energy of the wires will contain the term 
(GO) 
In order to avoid considerations about discontinuity let us suppose that k instead 
of changing abruptly as we go from one wire to another changes very rapidly but 
continuously throughout a small region enclosing the junction of the wires, when we 
wish to pass to the actual case we shall suppose the rate of variation to increase and 
the size of the region to diminish indefinitely. 
Now 
,dp do , dr \_ 7 , , 
= — J|kG)(/ p+mg+«?')(/,S—j'|[( Pj(k<S> )+ qj( kH)+ rjy<~d))dxdt/dz . (Gl) 
where d S is an element of the surface bounding the region through which we 
integrate, and l, m, n are the direction cosines of the normal to this surface drawn 
inwards. The case we are considering is that of two wires parallel to the axis of x, 
for by far the greater part of their length the bending which is necessary to make 
their ends join being supposed to occupy an indefinitely small space. Let us suppose 
that the electric displacement is parallel to the axis of x, then the surface integral 
vanishes and the term under consideration 
=-.fff(4(' t ®)+4( K ®)+i<' <e ))^ & 
(62) 
