DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 333 
and that e denotes the longitudinal extension; then if in the expression for the kinetic 
energy there is a term of the form 
wf .< 75 > 
where 6 denotes the temperature, y the quantity of electricity which has crossed any 
section of the wire, and /(e) denotes any function of e, Lagrange’s equation shows 
that the E.M.F. round the circuit equals 
.< 7s > 
this vanishes when 6 is constant and also when e is constant, so that the theory does 
not indicate the existence of a current when experiment shows that there is none. 
If we suppose that the circuit consists of two pieces of the same kind of wire, in one 
of which the longitudinal strain is constant and equal to e while the other piece is 
unstrained, and that one of the junctions of the strained and unstrained pieces is at a 
temperature 9 1 and the other at a lower temperature 0, 2 , then the E.M.F. round the 
circuit tending to make the current flow from the strained to the unstrained wire 
across the hot junction is 
.(U) 
where f(o) denotes the value of /(e) when (e) is zero. von Titnzelmann (Phil. 
Mag. [5], 5, p. 339) has proved that the current is reversed when the strain exceeds 
a certain limit, this shows that/(e) cannot be a linear function of e, but must involve 
powers of e above the first. 
Since the term 
(a y$£f(e)ds 
involves the coordinates which fix the state of strain of the wire, it indicates the 
existence of certain stresses in it. Let a be the displacement of a particle parallel to 
ds an element of the arc of the wire, then if a becomes a-f-Sa, e is increased by 
d.hajds, when this change occurs in <?, the change in the term 
(a ,jej/( e )ds 
! if we integrate this by parts we see that the coefficient of §a is 
|A y {^/(e)-|(r(4)}* 
2 x 
MDCCCLXXXV. 
