enter or leave the liquids. Along the line cd the surface contracts and does work at the constant increased difference of potential. The tendency of the difference of potential along ed is to diminish, and to keep it constant electrical energy, E2, must enter the liquids.

Along da the surface contracts and the difference of potential and superficial tension diminish to their original values.

The liquids, after this process, are in every respect in their primitive condition; and the difference of potential across the surface being the same, the total quantity of electricity upon the surface must be the same as at the beginning of the process.

It is obvious from Lippmann's results that the cycle is also completely reversible. Since during the cycle there is a gain of work abcd, the electrical energy which enters the liquids must be greater than that which leaves them, and we have

As the liquids are conductors, the only place at which a difference of potential exists, in the whole body, is at the surface of separation. The quantities of electricity entering and leaving the body, do so at this difference of potential.

Let P, and P, represent the differences of potential for the lines ab and cd respectively.

E1 is proportional to the quantity of electricity Q, which leaves the

liquids along a b.

Ea

P2

c d.

is proportional to the quantity Q, which enters the liquids along

2

Since the total quantity of electricity upon the surface is the same at the end of the process as at the beginning,

As any cycle may be decomposed into special cycles like the one that has been considered, we have for the most general case,

The electrical cycle that has been considered would probably be found useful in the investigation of many similar problems, where electrical energy is transformed into mechanical work.

The analogy between the equations V. and VI. and those derived from the two laws of thermodynamics is very close.

The equations might have been obtained by accepting immediately the theory of the conservation of electricity.*

Since

dE= Pdm= edm,

equation IV. may be written.

fedm=-feds;

f(e dm + kd S) = 0.

Substituting the value of dm,

Se Yde +eXd S+kd S) = O.

As the expression under the integral sign is an exact differential,

X obviously represents the electrical capacity of unit surface at constant potential.

Y is the electrical capacity of the whole surface, as the potential varies. If C is the capacity for unit surface

Y = CS.

The last equation reduces to

From equation VI.,

de

fdm=0,
f(CSde+Xd S)=0.

This expression being also an exact differential,

e C = d (e X + k).

(d.)

C=

dX
de

* Lippmann, Journal de Physique, tom. x.

(e.)

These equations were obtained by Lippmann. They show the close relation existing between contact electricity and superficial tension. The equations contain all the results of Lippmann's experiments; and though these experiments were made only upon special substances, yet it seems probable that the conclusions can be extended to all sub

stances.

If the electrical charge for any transformation is kept constant,

tain value of e for mercury.

de d S

As the curve representing the change of d2k k with e is concave toward e, is minus. Hence is plus, or bed e2 low a certain limit the difference of potential increases as the area of the surface increases.

Substances which do not act chemically on each other follow Volta's tension law. In any closed circuit at uniform temperature the sum of the differences of potential between the elements of the circuit is null. From II. of Lippmann's experiments is obtained a simple method of finding the superficial tension between two liquids, which do not act chemically upon each other, when we know the superficial tension between each of these liquids and a third liquid.

Let the contact electromotive forces between AB, BC, CA, be ee, eg, respectively. Let k, kg, ką be the superficial tensions corresponding to these differences of potential.

Suppose we have a vertical glass tube, the lower half containing mercury and the upper half dilute sulphuric acid. Suppose the + pole of a battery, the electromotive force of which does not exceed 0.9 Daniell, is connected with the acid, and the pole with the mercury. The superficial tension is increased and there is a contraction of the surface. From equation (c) a quantity of heat Q is evolved:

If the poles of the battery are reversed there will be an absorption of heat of the same amount. If the difference of potential exceeds 0.9 Daniell the effects will be reversed.

This is completely analogous to the Peltier phenomenon on each side of a neutral point.

If the surface of separation between mercury at constant potential and dilute sulphuric acid at constant higher potential is increased by the action of any force, work is done against superficial tension, and the energy of electrical separation appears. Positive electricity is separated across the surface from mercury to acid; and the tendency of the electrical action is to oppose the force. Conversely if the energy of electrical separation disappears by the passage of positive electricity from the acid to the mercury, mechanical work is done against superficial tension by dilatation of the surface.

Suppose we have a bent glass tube of uniform bore, containing dilute sulphuric acid in the bent part, and a column of mercury in each arm A and B. Let the temperature of the meniscus at A be T, and that of the meniscus at B be T-d T. Let the potential of the sulphuric acid be maintained constant, by any means, and equal to V. Let the mercury at B be put in contact with a conductor of constant potential V1, and infinite capacity. Let the mercury at A be put in contact with a similar conductor at potential V.

Let these potentials be so arranged that the change in the difference of potential at B is equal and opposite to that at A.

If the difference of potential at B is increased an electrical separation takes place so as to produce a contraction of the surface. At A

the separation takes place so as to produce a dilatation of the surface. The electrical change causes an evolution of heat at B, and an absorption of heat at A.

For the present it will be assumed that the superficial tension is a function of the temperature.

The electrical works at A and B are equal and opposite.

Since, however, the superficial tensions at the two surfaces are different, the changes in the areas of these surfaces, though opposite, are not equal.

Let d S equal the increment of area at A. Let d S equal the increment of area at B.

The quantity of heat absorbed during the expansion d S1 is

dk

A

d SA

d T

A

The quantity of heat given out during the contraction d S is

QB
Q2 = (T — d 11) [dz] d S2,

T)

As long as

dk
is minus, the right-hand member of this equation is
dT

essentially minus, and there is consequently more heat absorbed than evolved.