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

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513.]
153
GEOMETRICAL RELATIONS OF TWO ELEMENTS.

513.] We shall next consider in what way it is mathematically conceivable that the elements and 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 as resolved into in the direction of , and in a direction perpendicular to in the plane .

We shall also consider as resolved into in the direction of reversed, in a direction parallel to that in which was measured, and in a direction perpendicular to and .

Let us consider the action between the components and on the one hand, and , , on the other.

(1) and 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 is a function of , and , are the intensities of the currents in and respectively. This expression satisfies the condition of changing sign with and with .


(2) and 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 and , for it must be in the plane in which they both lie, and if we were to measure and 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 , and must therefore be directed along . Let us assume that this expression, when positive, represents an attraction.

(3) and 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 . We are at present engaged with forces, so we shall leave this out of account.

(4) The action of and , if they act on each other, must be expressed by

.