Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/46

The definitions of other quantities, and of the units to which they are referred, will be given when we require them.

In transforming the values of physical quantities determined in terms of one unit, so as to express them in terms of any other unit of the same kind, we have only to remember that every expression for the quantity consists of two factors, the unit and the numerical part which expresses how often the unit is to be taken. Hence the numerical part of the expression varies inversely as the magnitude of the unit, that is, inversely as the various powers of the fundamental units which are indicated by the dimensions of the derived unit.

On Physical Continuity and Discontinuity.

7.] A quantity is said to vary continuously when, if it passes from one value to another, it assumes all the intermediate values.

We may obtain the conception of continuity from a consideration of the continuous existence of a particle of matter in time and space. Such a particle cannot pass from one position to another without describing a continuous line in space, and the coordinates of its position must be continuous functions of the time.

In the so-called ‘equation of continuity,’ as given in treatises on Hydrodynamics, the fact expressed is that matter cannot appear in or disappear from an element of volume without passing in or out through the sides of that element.

A quantity is said to be a continuous function of its variables when, if the variables alter continuously, the quantity itself alters continuously.

Thus, if $u$ is a function of $x$, and if, while $x$ passes continuously from $x_{0}$ to $x_{1}$, $u$ passes continuously from $u_{0}$ to $u_{1}$, but when $x$ passes from $x_{1}$ to $x_{2}$, $u$ passes from $u_{1}'$ to $u_{2}$, $u_{1}'$ being different from $u_{1}$, then $u$ is said to have a discontinuity in its variation with respect to $x$ for the value $x=x_{1}$, because it passes abruptly from $u_{1}$ to $u_{1}'$ while $x$ passes continuously through $x_{1}$.

If we consider the differential coefficient of $u$ with respect to $x$ for the value $x=x_{1}$ as the limit of the fraction

${\frac {u_{2}-u_{0}}{x_{2}-x_{0}}}$,

when $x_{2}$ and $x_{0}$ are both made to approach $x_{1}$ without limit, then, if $x_{0}$ and $x_{2}$ are always on opposite sides of $x_{1}$, the ultimate value of the numerator will be $u_{1}'-u_{1}$ and that of the denominator will be zero. If $u$ is a quantity physically continuous, the discontinuity