Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/48

tinuous. Let the values of the variables for which the discontinuity of the derivatives occurs be connected by the equation

$\phi =\phi (x,y,z\ldots )=0,$

and let $F_{1}$ and $F_{2}$ be expressed in terms of $\phi$ and $n-1$ other variables, say $(y,z...)$.

Then, when $\phi$ is negative, $F_{1}$ is to be taken, and when $\phi$ is positive $F_{2}$ is to be taken, and, since $F$ is itself continuous, when $\phi$ is zero, $F_{1}=F_{2}$.

Hence, when $\phi$ is zero, the derivatives ${\frac {dF_{1}}{d\phi }}$ and ${\frac {dF_{2}}{d\phi }}$ may be different, but the derivatives with respect to any of the other variables, such as ${\frac {dF_{1}}{dy}}$ and ${\frac {dF_{2}}{dy}}$ must be the same. The discontinuity is therefore confined to the derivative with respect to $\phi$, all the other derivatives being continuous.

Periodic and Multiple Functions.

9.] If $u$ is a function of $x$ such that its value is the same for $x$, $x+a$, $x+na$, and all values of $x$ differing by $a$, $u$ is called a periodic function of $x$, and $a$ is called its period.

If $x$ is considered as a function of $u$, then, for a given value of $u$, there must be an infinite series of values of $x$ differing by multiples of $a$. In this case $x$ is called a multiple function of $u$, and $a$ is called its cyclic constant.

The differential coefficient ${\frac {dx}{du}}$ has only a finite number of values corresponding to a given value of $u$.

On the Relation of Physical Quantities to Directions in Space.

10.] In distinguishing the kinds of physical quantities, it is of great importance to know how they are related to the directions of those coordinate axes which we usually employ in defining the positions of things. The introduction of coordinate axes into geometry by Des Cartes was one of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations performed on numerical quantities. The position of a point is made to depend on the length of three lines which are always drawn in determinate directions, and the line joining two points is in like manner considered as the resultant of three lines.

But for many purposes in physical reasoning, as distinguished