Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/53

point of view, is not so necessary to be observed for the sake of the mathematical methods. This is the distinction between longitudinal and rotational properties.

The direction and magnitude of a quantity may depend upon some action or effect which takes place entirely along a certain line, or it may depend upon something of the nature of rotation about that line as an axis. The laws of combination of directed quantities are the same whether they are longitudinal or rotational, so that there is no difference in the mathematical treatment of the two classes, but there may be physical circumstances which indicate to which class we must refer a particular phenomenon. Thus, electrolysis consists of the transfer of certain substances along a line in one direction, and of certain other substances in the opposite direction, which is evidently a longitudinal phenomenon, and there is no evidence of any rotational effect about the direction of the force. Hence we infer that the electric current which causes or accompanies electrolysis is a longitudinal, and not a rotational phenomenon.

On the other hand, the north and south poles of a magnet do not differ as oxygen and hydrogen do, which appear at opposite places during electrolysis, so that we have no evidence that magnetism is a longitudinal phenomenon, while the effect of magnetism in rotating the plane of polarized light distinctly shews that magnetism is a rotational phenomenon.

On Line-integrals.

16.] The operation of integration of the resolved part of a vector quantity along a line is important in physical science generally, and should be clearly understood.

Let $x$, $y$, $z$ be the coordinates of a point $P$ on a line whose length, measured from a certain point $A$, is $s$. These coordinates will be functions of a single variable $s$.

Let $R$ be the value of the vector quantity at $P$, and let the tangent to the curve at $P$ make with the direction of $R$ the angle $\epsilon$, then $R\cos \epsilon$ is the resolved part of $R$ along the line, and the integral

$L=\int _{0}^{s}R\cos \epsilon ds$

is called the line-integral of $R$ along the line $s$.

We may write this expression

$L=\int _{0}^{s}{\Big (}X{\frac {dx}{ds}}+Y{\frac {dy}{ds}}+Z{\frac {dz}{ds}}{\Big )}ds$,