388 Messrs. G. C. Foster and 0. J. Lodge on the Flow of 



the mode of treatment adopted in it has enabled us to establish 

 by strictly elementary mathematical methods, and so to render 

 accessible to a greater number of readers, many of the conclu- 

 sions which previous writers had arrived at by more abstruse 

 processes. 



2. In order to avoid ambiguity, it may be well to begin by 

 giving a few general explanations and definitions of terms that 

 will be frequently used in what follows. It will be assumed that, 

 in all cases considered, the flow of electricity has attained a sta- 

 tionary condition — that is to say, that on the whole there is 

 neither gain nor loss of electricity at any part of the sheet, but 

 that at every instant the quantity that enters any part and the 

 quantity that leaves it are equal. Positive electrodes, or such as 

 supply positive electricity to the sheet, will be spoken of as 

 "sources," while negative electrodes will be termed "sinks." 

 The rate at which a source or a sink supplies or removes electri- 

 city will be spoken of as the " strength " of the source or sink. 

 Through every point of the sheet a straight line can be drawn 

 such that no electricity crosses it at that point ; the direction of 

 a line so drawn through a given point is the " direction of the 

 current" at that point. A "line of flow" or " stream-line " is 

 a line drawn so that no electricity crosses it anywhere; or, in 

 other words, it is a line whose direction at all points coincides 

 with the direction of the current. The quantity of electricity 

 flowing between two consecutive lines of flow is everywhere the 

 same; for since no electricity crosses either of them, whatever 

 quantity starts between them must remain between them through- 

 out their whole course. It may be useful to observe that flow- 

 lines are not loci of constant strength of flow, they are lines 

 of direction simply. 



The " strength of the current " at any point is given by the 

 quantity of electricity which crosses, in unit time, a short line 

 normal to the flow-line through the point, divided by the length 

 of the line. 



If a line be drawn so as to intersect the lines of flow every- 

 where at right angles, there will be no flow of electricity along 

 it ; but since the sheet on which the line is drawn is a conductor, 

 this implies that all points of the line are at the same potential, 

 or that the line is an " equipotential line." 



3. It is obvious that a line of flow and an equipotential line 

 might be drawn through every point of a conducting sheet tra- 

 versed by a current, so that, if it were possible to draw them all, 

 the sheet would be completely covered by two sets of lines cut- 

 ting each other at right angles. The general distribution, how- 

 ever, of the lines of each set can be correctly shown without 

 drawing more than a limited number, if those that are drawn 



