Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/397

alterations may be performed independently, reversibly, and isothermally, and that the state of the large electrode H₀, is not altered thereby. Let de denote the quantity of electricity which passes through the cell from H₀, to H, when the state of the system is thus varied: then if E denote the available energy of the system, and γ the surface-tension at H, we have

$dE=\gamma dS+Vde$,

γ being measured by the work required to increase the surface when no electricity flows through the circuit.

In order that equilibrium may be re-established between the electrode and the solution when the fall of potential at the cathode is altered, it will be necessary not only that some hydrogen cations should come out of the solution and be deposited on the electrode, yielding up their charges, but also that there should be changes in the clustering of the charged ions of hydrogen, mercury, and sulphion in the layer of the solution immediately adjacent to the electrode. Each of these circumstances necessitates a flow of electricity in the outer circuit: in the one case to neutralize the charges of the cations deposited, and in the other case to increase the surface-density of electric charge on the electrode, which forms the opposite sheet of the quasi-condenser. Let Sf (V) denote the total quantity of electricity which has thus flowed in the circuit when the external electromotive force has attained the value V. Then evidently

$de=d\{Sf(V)\}$;

so

$dE=\{\gamma +Vf(V)\}dS+VSf^{\prime }(V)dV$.

Since this expression must be an exact differential, we have

${\frac {d\gamma }{dV}}+f(V)=0$;

so that -dy/dV is equal to that flux of electricity per unit of new surface formed, which will maintain the surface in a