Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/383

291.] If $dS$ denotes the section of a tube of flow by a plane normal to $x,$ we have by the theory of the change of the independent variables,

$\delta \lambda \cdot \delta \lambda '=\delta S\left({\frac {d\lambda }{dy}}{\frac {d\lambda '}{dz}}-{\frac {d\lambda }{dz}}{\frac {d\lambda '}{dy}}\right),$

(13)

and by the definition of the components of the current

$u\,\delta S=L\,\delta \lambda \cdot \delta \lambda '.$

(14)

$\left.{\begin{matrix}{\mbox{Hence}}&&u=L\left({\frac {d\lambda }{dy}}{\frac {d\lambda '}{dz}}-{\frac {d\lambda }{dz}}{\frac {d\lambda '}{dy}}\right).\\{\mbox{Similarly}}&&v=L\left({\frac {d\lambda }{dz}}{\frac {d\lambda '}{dx}}-{\frac {d\lambda }{dx}}{\frac {d\lambda '}{dz}}\right),\\&&w=L\left({\frac {d\lambda }{dx}}{\frac {d\lambda '}{dy}}-{\frac {d\lambda }{dy}}{\frac {d\lambda '}{dx}}\right).\end{matrix}}\right\}$

(15)

292.] It is always possible when one of the functions $\lambda$ or $\lambda$ is known, to determine the other so that $L$ may be equal to unity. For instance, let us take the plane of $yz,$ and draw upon it a series of equidistant lines parallel to $y,$ to represent the sections of the family $\lambda '$ by this plane. In other words, let the function $\lambda '$ be determined by the condition that when $x=0\;\lambda '=z.$ If we then make $L=1,$ and therefore (when $x=0$)

$\lambda =\int u\,dy;$

then in the plane $(x=0)$ the amount of electricity which passes through any portion will be

$\iint u\,dy\,dz=\iint d\lambda \,d\lambda '.$

(16)

Having determined the nature of the sections of the surfaces of flow by the plane of $yz,$ the form of the surfaces elsewhere is determined by the conditions (8) and (9). The two functions $\lambda$ and $\lambda '$ thus determined are sufficient to determine the current at every point by equations (15), unity being substituted for $L.$

On Lines of Flow.

293.] Let a series of values of $\lambda$ and of $\lambda ;$ be chosen, the successive differences in each series being unity. The two series of surfaces defined by these values will divide space into a system of quadrilateral tubes through each of which there will be a unit current. By assuming the unit sufficiently small, the details of the current may be expressed by these tubes with any desired amount of minuteness. Then if any surface be drawn cutting the