Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/185

513.] We shall next consider in what way it is mathematically conceivable that the elements $PQ$ and $P^{\prime }Q^{\prime }$ might act on each other, and in doing so we shall not at first assume that their mutual action is necessarily in the line joining them.

We have seen that we may suppose each element resolved into other elements, provided that these components, when combined according to the rule of addition of vectors, produce the original element as their resultant.

We shall therefore consider $ds$ as resolved into $\cos \theta \,ds=\alpha$ in the direction of $r$, and $\sin \theta \,ds=\beta$ in a direction perpendicular to $r$ in the plane $P^{\prime }PQ$.
Fig. 30.

We shall also consider $ds^{\prime }$ as resolved into $\cos \theta ^{\prime }\,ds^{\prime }=\alpha ^{\prime }$ in the direction of $r$ reversed, $\sin \theta ^{\prime }\cos \eta \,ds^{\prime }=\beta$ in a direction parallel to that in which $\beta$ was measured, and $\sin \theta ^{\prime }\sin \eta \,ds^{\prime }=\gamma ^{\prime }$ in a direction perpendicular to $\alpha ^{\prime }$ and $\beta ^{\prime }$.

Let us consider the action between the components $\alpha$ and $\beta$ on the one hand, and $\alpha ^{\prime }$, $\beta ^{\prime }$, $\gamma ^{\prime }$ on the other.

(1) $\alpha$ and $\alpha ^{\prime }$ are in the same straight line. The force between them must therefore be in this line. We shall suppose it to be an attraction

where $A$ is a function of $r$, and $i$, $i^{\prime }$ are the intensities of the currents in $ds$ and $ds^{\prime }$ respectively. This expression satisfies the condition of changing sign with $i$ and with $i^{\prime }$.

(2) $\beta$ and $\beta ^{\prime }$ are parallel to each other and perpendicular to the line joining them. The action between them may be written

This force is evidently in the line joining $\beta$ and $\beta ^{\prime }$, for it must be in the plane in which they both lie, and if we were to measure $\beta$ and $\beta ^{\prime }$ in the reversed direction, the value of this expression would remain the same, which shews that, if it represents a force, that force has no component in the direction of $\beta$, and must therefore be directed along $r$. Let us assume that this expression, when positive, represents an attraction.

(3) $\beta$ and $\gamma ^{\prime }$ are perpendicular to each other and to the line joining them. The only action possible between elements so related is a couple whose axis is parallel to $r$. We are at present engaged with forces, so we shall leave this out of account.

(4) The action of $\alpha$ and $\beta ^{\prime }$, if they act on each other, must be expressed by