Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/384

system of tubes, the quantity of the current which passes through this surface will be expressed by the number of tubes which cut it, since each tube carries unity of current.

The actual intersections of the surfaces may be called Lines of Flow. When the unit is taken sufficiently small, the number of lines of flow which cut a surface is approximately equal to the number of tubes of flow which cut it, so that we may consider the lines of flow as expressing not only the direction of the current but its strength, since each line of flow through a given section corresponds to a unit current.

On Current-Sheets and Current-Functions.

294.] A stratum of a conductor contained between two consecutive surfaces of flow of one system, say that of $\lambda ',$ is called a Current-Sheet. The tubes of flow within this sheet are determined by the function $\lambda .$ If $\lambda _{A}$ and $\lambda _{P}$ denote the values of $\lambda$ at the points $A$ and $P$ respectively, then the current from right to left across any line drawn on the sheet from $A$ to $P$ is $\lambda _{P}-\lambda _{A}.$ If $AP$ be an element, $ds,$ of a curve drawn on the sheet, the current which crosses this element from right to left is

${\frac {d\lambda }{ds}}ds.$

This function $\lambda ,$ from which the distribution of the current in the sheet can be completely determined, is called the Current-Function.

Any thin sheet of metal or conducting matter bounded on both sides by air or some other non-conducting medium may be treated as a current-sheet, in which the distribution of the current may be expressed by means of a current-function. See Art. 647.

Equation of 'Continuity.'

295.] If we differentiate the three equations (15) with respect to $x,y,z$ respectively, remembering that $L$ is a function of $\lambda$ and $\lambda ',$ we find

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$

(17)

The corresponding equation in Hydrodynamics is called the Equation of 'Continuity.' The continuity which it expresses is the continuity of existence, that is, the fact that a material substance cannot leave one part of space and arrive at another, without going through the space between. It cannot simply vanish in the