81 
of Edinburgh , Session 1869-70. 
hoffs 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). 
In 1846, Thomson drew attention to the orthogonal systems (3) 
and (4), as an example of Lame’s theorem.* He showed that the 
rings and brushes of biaxal crystals are a special case of these curves. 
They correspond, in fact, as we shall see, to the equipotential lines 
and lines of flow in an infinite plate with two equal sources of 
electricity. 
Maxwell, in 1856, suggested the application to problems of 
electric currents of his beautiful theory of the motion of an imma- 
terial incompressible fluid in a resisting medium, but does not appear 
to have developed the suggestion.! 
The object of this paper is to show that, by regarding, in accor- 
dance with Maxwell’s suggestion, every point of exit or issue as a 
source or sink, spreading or absorbing electricity, independently of 
all other sources, Kirchhoff’s general equations may be deduced by 
easy geometrical processes, and extended to certain cases of flow 
in curved surfaces. We shall, by this method, be naturally led to 
look mainly at the function u , which in the analytical investigation 
is subordinated to v. The equation u = 0 will receive an obvious 
physical interpretation, and we shall then proceed to consider in 
detail the nature of the flow in certain special cases apparently not 
yet examined. 
If a source P, in an infinite uniformly resisting plate, steadily 
give forth a quantity of electricity E per unit of time, the flow per 
second over the whole circumference of all circles with P as centre is 
equal. Hence the rate of flow at each point of the circumference of 
E 
such a circle is inversely as the radius = - — . The potential due 
to P satisfies the equation 
dv 
dr 
A 
2ttt ’ 
or, 
v = 0 — - — log r . 
2i7T 
* Camb. and Dub. Math. Journ. vol. i. p. 124. 
t Cambridge Phil. Trans, vol x. 
