Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/385

one place and appear in the other, but it must travel along a continuous path, so that if a closed surface be drawn, including the one place and excluding the other, a material substance in passing from the one place to the other must go through the closed surface. The most general form of the equation in hydrodynamics is

${\frac {d(\rho u)}{dx}}+{\frac {d(\rho v)}{dy}}+{\frac {d(\rho w)}{dz}}+{\frac {d\rho }{dt}}=0;$

(18)

where $\rho$ signifies the ratio of the quantity of the substance to the volume it occupies, that volume being in this case the differential element of volume, and $(\rho u),\,(\rho v),$ and $(\rho w)$ signify the ratio of the quantity of the substance which crosses an element of area in unit of time to that area, these areas being normal to the axes of $x,\,y,$ and $z$ respectively. Thus understood, the equation is applicable to any material substance, solid or fluid, whether the motion be continuous or discontinuous, provided the existence of the parts of that substance is continuous. If anything, though not a substance, is subject to the condition of continuous existence in time and space, the equation will express this condition. In other parts of Physical Science, as, for instance, in the theory of electric and magnetic quantities, equations of a similar form occur. We shall call such equations 'equations of continuity' to indicate their form, though we may not attribute to these quantities the properties of matter, or even continuous existence in time and space.

The equation (17), which we have arrived at in the case of electric currents, is identical with (18) if we make $\rho =1,$ that is, if we suppose the substance homogeneous and incompressible. The equation, in the case of fluids, may also be established by either of the modes of proof given in treatises on Hydrodynamics. In one of these we trace the course and the deformation of a certain element of the fluid as it moves along. In the other, we fix our attention on an element of space, and take account of all that enters or leaves it. The former of these methods cannot be applied to electric currents, as we do not know the velocity with which the electricity passes through the body, or even whether it moves in the positive or the negative direction of the current. All that we know is the algebraical value of the quantity which crosses unit of area in unit of time, a quantity corresponding to $(\rho u)$ in the equation (18). We have no means of ascertaining the value of either of the factors $\rho$ or $u,$ and therefore we cannot follow a particular portion of electricity in its course through the body. The other method of investigation, in which we consider what passes