Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/271

 in electrostatics was not yet correctly known. Then, within the substance of any homogeneous conductor, the function V must satisfy Laplace's equation &nabla;2V = 0; while at the air-surface of each conductor, the derivate of V taken along the normal must vanish. At the interface between two conductors formed of different materials, the function V has a discontinuity, which is measured by the value of Volta's contact force for the two conductors; and, moreover, the condition that the current shall be continuous across such an interface requires that k&part;V/&part;N shall be continuous, where k denotes the ohmic specific conductivity of the conductor, and &part;/&part;N denotes differentiation along the normal to the interface. The equations which have now been mentioned suffice to determine the flow of electricity in the system.

Kirchhoff also showed that the currents distribute themselves in the conductors in such a way as to generate the least possible amount of Joulian heat; as is easily seen, since the quantity of Joulian heat generated in unit time is

where k, as before, denotes the specific conductivity, and this integral has a stationary value when V satisfies the equation

Kirchhoff next applied himself to establish harmony between electrostatical conceptions and the theory of Ohm. That theory had now been before the world for twenty years, and had been verified by numerous experimental researches; in particular, a careful investigation was made at this time (1848) by Rudolph Kohlrausch (b. 1809, d. 1858), who showed that the difference of the electric "tensions" at the extremities of a voltaic cell, measured electrostatically with the circuit open, was for different cells proportional to the electromotive force