80 Proceedings of the Roycd Society 
the nature of the distribution will be indicated to the eye by form¬ 
ing the equipotential curves 
v = const. .... (1). 
From the second point of view, we should endeavour to deter¬ 
mine the lines of flow by equations of the form 
u = const. .... (2). 
The curves determined by equations (1) and (2) are obviously 
orthogonal, and since 
d?v drv _ ^ 
dx l dy 2 ; 
we know, by a theorem of Lame and Stokes,* that 
dx 1 dy 2 
Kirchhoff, in the year 1845, took up the problem for plane surfacesf 
in the first of the two ways we have indicated. By an application 
of Ohm’s law, he expressed analytically the conditions to be satis¬ 
fied by v. When the electricity enters and issues by a number of 
individual points, he found (apparently by trial) that an integral 
of the form 2}(a log r ), where r 1 r 2 , &c., are the distances of the 
point (x, y) from the successive points of entrance and issue, satis¬ 
fies these 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(a [r,R]) = const. . . . (4), 
where [r, B] is the angle between r and a fixed line E. 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- 
* See Thomson and 'bait’s Natural Philosophy, i. 542. 
t Poggendorffs Annalen, Bd. lxiv. 
