Middle of tke Nineteenth Century. 4 25l 



in electrostatics was not yet correctly known. Then, within 

 the substance of any homogeneous conductor, the function V 

 must satisfy Laplace's equation V- V= ; 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 

 Jed VfoN shall be continuous, where k denotes the ohmic specific 

 conductivity of the conductor, and 3/3^ 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 them- 

 selves 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 



a /ar\ a 



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 Kudolph 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 



*Ann. d. Phys. Ixxv (1848), p. 220. 



