386 Messrs. G. C. Foster and O. J. Lodge on the Flow of 



experimentally the theoretical conclusions, together with some 

 results obtained by these methods. 



The general subject treated of in this paper has attracted the 

 attention of a considerable number of mathematicians and phy- 

 sicists. The earliest published investigation relating to it was 

 contained in a remarkable memoir by Kirchhoff, which appeared 

 in PoggendorfPs Annalen in 1845 (vol. lxiv. p. 497). In this 

 paper Kirchhoff established the general mathematical theory of 

 the flow of electricity in an unlimited uniformly conducting- 

 sheetj and in a limited sheet with a circular boundary, with so 

 much completeness as to leave little for others to do beyond 

 working out the application to special cases of the general prin- 

 ciples he laid down, or finding other methods of establishing the 

 conclusions he deduced from them. 



We cannot better indicate the general plan of Kirchhoff' s in- 

 vestigation than by quoting the following account of it from a 

 paper by Professor W. Robertson Smith*, to which we shall 

 have to make further reference immediately : — " By an applica- 

 tion of Ohm's law, he [Kirchhoff] expressed analytically the 

 condition to be satisfied by v [the potential] . When the elec- 

 tricity enters and issues by a number of individual points, he 

 found (apparently by trial) that an integral of the form % (a log r) f 

 where r l} r 2 , &c. are the distances of the point (x, y) from the 

 successive points of entrance and issue, satisfied the conditions 

 when the plate is infinite. For a finite plate, it is necessary that 

 the boundary of the plate should be orthogonal to the curves 



2 (a log r) = const. * . (3) 



He was thus led to form the orthogonal curves whose equation 

 he gives in the form 



2(«[r,R]) = const., . . (4) 



where [r, R] is the angle between r and a fixed line R. These 

 equations he applies to the case of a circular plate, completely 

 determining the curves when there is one exit and one entrance 

 point in the circumference, and showing that in any case a proper 

 number of subsidiary points would make the equipotential lines 

 determined by (3) cut the circumference at right angles. Kirch- 

 hoff's paper is throughout properly busied with the function v, 

 and the stream-lines are only dealt with incidentally. There is no 

 attempt to give a physical meaning to the equation (4)." To 

 this we have only to add that Kirchhoff proved the accuracy of 

 his theoretical deductions by determining experimentally the 

 form and distribution of the equipotential lines on a circular 

 disk with two electrodes on the edge, as well as (Pogg. Ann. 



* Proc. Edinb. Roy. Soc. 1869-/0, pp. 79-99. 



