Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/183

509.] It may be observed with reference to these experiments that every electric current forms a closed circuit. The currents used by Ampère, being produced by the voltaic battery, were of course in closed circuits. It might be supposed that in the case of the current of discharge of a conductor by a spark we might have a current forming an open finite line, but according to the views of this book even this case is that of a closed circuit. No experiments on the mutual action of unclosed currents have been made. Hence no statement about the mutual action of two elements of circuits can be said to rest on purely experimental grounds. It is true we may render a portion of a circuit moveable, so as to ascertain the action of the other currents upon it, but these currents, together with that in the moveable portion, necessarily form closed circuits, so that the ultimate result of the experiment is the action of one or more closed currents upon the whole or a part of a closed current.

510.] In the analysis of the phenomena, however, we may regard the action of a closed circuit on an element of itself or of another circuit as the resultant of a number of separate forces, depending on the separate parts into which the first circuit may be conceived, for mathematical purposes, to be divided.

This is a merely mathematical analysis of the action, and is therefore perfectly legitimate, whether these forces can really act separately or not.

511.] We shall begin by considering the purely geometrical relations between two lines in space representing the circuits, and between elementary portions of these lines.

Fig. 29. Let there be two curves in space in each of which a fixed point is taken from which the arcs are measured in a defined direction along the curve. Let $A$, $A^{\prime }$ be these points. Let $PQ$ and $P^{\prime }Q^{\prime }$ be elements of the two curves.

Let	$AP=s$,	$A^{\prime }P^{\prime }=s^{\prime }$		(1)
	$PQ=ds$,	$P^{\prime }Q^{\prime }=ds^{\prime }$,

and let the distance $PP^{\prime }$ be denoted by $r$. Let the angle $P^{\prime }PQ$ be denoted by $\theta$, and $PP^{\prime }Q^{\prime }$ by $\theta ^{\prime }$, and let the angle between the planes of these angles be denoted by $\eta$.

The relative position of the two elements is sufficiently defined by their distance $r$ and the three angles $\theta$, $\theta ^{\prime }$, and $\eta$, for if these be