DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
521 
is rj, is established, and then breaks down by reason of the re-arrangement of the 
dielectric, the whole effect is the same as if a quantity of electricity equal to y passed 
through each unit area of the dielectric. 
Let us suppose that in the expression for the electrical part of T — Y for unit 
volume of a conductor through which the electrical displacements are f g, h, in 
addition to the usual term 
1 
2 
f+ M f 
+ N 
where L, M, N, are the components of the electrical momentum parallel to the axes 
of x, y, z, respectively, there is the term 
- *Q (/ 3 + r + w ); 
then Lagrange’s equations give 
§ + Q/= external electromotive force tending to increase f 
CtL 
= X say. 
Now, in the case of a conductor, f is changing very rapidly, while the other terms in 
this equation only change gradually. Let us then take the mean over unit time of 
each side of the equation, then we have 
ft + Q \'/ dt = Y .< 79 > 
where a bar placed over a symbol denotes the mean value of that symbol, and we 
suppose that Q does not change quickly with the time. 
Now let us suppose, as before, that the polarisation is continually being broken down 
and renewed ; let it break down n times a second, and let the mean value of f over 
one of these intervals of l/n of a second be f and the maximum value be af: the 
breaking down of the field each time is equivalent to the passage of af units of elec¬ 
tricity across unit area at right angles to the axis of x. Since the field breaks down n 
times a second, it is equivalent to the passage of naf across this unit section in unit 
time, or, if u be the component of currents parallel to the axis of x, 
u naf 
Now 
i fdt = n \'/ dt=n fj=f 
3 X 
MDCCCLXXXVII.—A. 
