Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/382

Hence, if $\gamma$ is the component of the current normal to the surface,

$\gamma =N\left\{u{\frac {d\lambda }{dx}}+v{\frac {d\lambda }{dy}}+w{\frac {d\lambda }{dz}}\right\}.$

(7)

If $\gamma =0$ there will be no current through the surface, and the surface may be called a Surface of Flow, because the lines of motion are in the surface.

288.] The equation of a surface of flow is therefore

$u{\frac {d\lambda }{dx}}+v{\frac {d\lambda }{dy}}+w{\frac {d\lambda }{dz}}=0$

(8)

If this equation is true for all values of $\lambda ,$ all the surfaces of the family will be surfaces of flow.

289.] Let there be another family of surfaces, whose parameter is $\lambda ',$ then, if these are also surfaces of flow, we shall have

$u{\frac {d\lambda '}{dx}}+v{\frac {d\lambda '}{dy}}+w{\frac {d\lambda '}{dz}}=0$

(9)

If there is a third family of surfaces of flow, whose parameter is $\lambda '',$ then

$u{\frac {d\lambda ''}{dx}}+v{\frac {d\lambda ''}{dy}}+w{\frac {d\lambda ''}{dz}}=0$

(10)

Eliminating between these three equations, $u,v,$ and $w$ disappear together, and we find

${\begin{vmatrix}{\frac {d\lambda }{dx}},&{\frac {d\lambda }{dy}},&{\frac {d\lambda }{dz}}\\{\frac {d\lambda '}{dx}},&{\frac {d\lambda '}{dy}},&{\frac {d\lambda '}{dz}}\\{\frac {d\lambda ''}{dx}},&{\frac {d\lambda ''}{dy}},&{\frac {d\lambda ''}{dz}}\end{vmatrix}}=0;$

(11)

$\mathrm {or} \quad \quad \lambda ''=\phi (\lambda ,\lambda ');$

(12)

that is, $\lambda ''$ is some function of $\lambda$ and $\lambda '.$

290.] Now consider the four surfaces whose parameters are $\lambda ,\lambda +\delta \lambda ,\lambda ',$ and $\lambda '+\delta \lambda '.$ These four surfaces enclose a quadrilateral tube, which we may call the tube $\delta \lambda \cdot \delta \lambda '.$ Since this tube is bounded by surfaces across which there is no flow, we may call it a Tube of Flow. If we take any two sections across the tube, the quantity which enters the tube at one section must be equal to the quantity which leaves it at the other, and since this quantity is therefore the same for every section of the tube, let us call it $L\delta \lambda \cdot \delta \lambda '$ where $L$ is a function of $\lambda$ and $\lambda ',$ the parameters which determine the particular tube.