Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/285

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641.]

AMPÈRE'S THEORY OF MAGNETS.

253

By Art. 403,

{\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}=0

(8)

Multiplying this equation, (8), by $\alpha$ , and dividing by $4\pi$ , we may add the result to (7), and we find

X={\frac {1}{4\pi }}\left\{{\frac {d}{dx}}\left[a\alpha -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right]+{\frac {d}{dy}}[b\alpha ]+{\frac {d}{dz}}[c\alpha ]\right\}

,

(9)

also, by (2),

L={\frac {1}{4\pi }}((b-\beta )\gamma -(c-\gamma )\beta )

,

(10)

{}={\frac {1}{4\pi }}(b\gamma -c\beta )

,

(11)

where $X$ is the force referred to unit of volume in the direction of $x$ , and $L$ is the moment of the forces about this axis.

On the Explanation of these Forces by the Hypothesis of a Medium in a State of Stress.

641.] Let us denote a stress of any kind referred to unit of area by a symbol of the form $P_{hk}$ , where the first suffix, ${}_{h}$ , indicates that the normal to the surface on which the stress is supposed to act is parallel to the axis of $h$ , and the second suffix, ${}_{k}$ , indicates that the direction of the stress with which the part of the body on the positive side of the surface acts on the part on the negative side is parallel to the axis of $k$ .

The directions of $h$ and $k$ may be the same, in which case the stress is a normal stress. They may be oblique to each other, in which case the stress is an oblique stress, or they may be perpendicular to each other, in which case the stress is a tangential stress.

The condition that the stresses shall not produce any tendency to rotation in the elementary portions of the body is

P_{hk}=P_{kh}

.

In the case of a magnetized body, however, there is such a tendency to rotation, and therefore this condition, which holds in the ordinary theory of stress, is not fulfilled.

Let us consider the effect of the stresses on the six sides of the elementary portion of the body $dx\,dy\,dz$ , taking the origin of coordinates at its centre of gravity.

On the positive face $dy\,dz$ , for which the value of $x$ is ${\frac {1}{2}}\,dx$ , the forces are—