CHAPTER II.

AMPÈRE'S INVESTIGATION OF THE MUTUAL ACTION OF ELECTRIC CURRENTS.

502.] We have considered in the last chapter the nature of the magnetic field produced by an electric current, and the mechanical action on a conductor carrying an electric current placed in a magnetic field. From this we went on to consider the action of one electric circuit upon another, by determining the action on the first due to the magnetic field produced by the second. But the action of one circuit upon another was originally investigated in a direct manner by Ampère almost immediately after the publication of Örsted's discovery. We shall therefore give an outline of Ampère's method, resuming the method of this treatise in the next chapter.

The ideas which guided Ampère belong to the system which admits direct action at a distance, and we shall find that a remarkable course of speculation and investigation founded on these ideas has been carried on by Gauss, Weber, J. Neumann, Riemann, Betti, C. Neumann, Lorenz, and others, with very remarkable results both in the discovery of new facts and in the formation of a theory of electricity. See Arts. 846–866.

The ideas which I have attempted to follow out are those of action through a medium from one portion to the contiguous portion. These ideas were much employed by Faraday, and the development of them in a mathematical form, and the comparison of the results with known facts, have been my aim in several published papers. The comparison, from a philosophical point of view, of the results of two methods so completely opposed in their first principles must lead to valuable data for the study of the conditions of scientific speculation.

503.] Ampère's theory of the mutual action of electric currents is founded on four experimental facts and one assumption.

Ampère's fundamental experiments are all of them examples of what has been called the null method of comparing forces. See Art. 214. Instead of measuring the force by the dynamical effect of communicating motion to a body, or the statical method of placing it in equilibrium with the weight of a body or the elasticity of a fibre, in the null method two forces, due to the same source, are made to act simultaneously on a body already in equilibrium, and no effect is produced, which shews that these forces are themselves in equilibrium. This method is peculiarly valuable for comparing the effects of the electric current when it passes through circuits of different forms. By connecting all the conductors in one continuous series, we ensure that the strength of the current is the same at every point of its course, and since the current begins everywhere throughout its course almost at the same instant, we may prove that the forces due to its action on a suspended body are in equilibrium by observing that the body is not at all affected by the starting or the stopping of the current.

504.] Ampère's balance consists of a light frame capable of revolving about a vertical axis, and carrying a wire which forms two circuits of equal area, in the same plane or in parallel planes, in which the current flows in opposite directions. The object of this arrangement is to get rid of the effects of terrestrial magnetism on the conducting wire. When an electric circuit is free to move it tends to place itself so as to embrace the largest possible number of the lines of induction. If these lines are due to terrestrial magnetism, this position, for a circuit in a vertical plane, will be when the plane of the circuit is east and west, and when the direction of the current is opposed to the apparent course of the sun.

By rigidly connecting two circuits of equal area in parallel planes, in which equal currents run in opposite directions, a combination is formed which is unaffected by terrestrial magnetism, and is therefore called an Astatic Combination, see Fig. 26. It is acted on, however, by forces arising from currents or magnets which are so near it that they act differently on the two circuits.

505.] Ampère's first experiment is on the effect of two equal currents close together in opposite directions. A wire covered with insulating material is doubled on itself, and placed near one of the circuits of the astatic balance. When a current is made to pass through the wire and the balance, the equilibrium of the balance remains undisturbed, shewing that two equal currents close together in opposite directions neutralize each other. If, instead of two wires side by side, a wire be insulated in the middle of a metal

tube, and if the current pass through the wire and back by the tube, the action outside the tube is not only approximately but accurately null. This principle is of great importance in the construction of electric apparatus, as it affords the means of conveying the current to and from any galvanometer or other instrument in such a way that no electromagnetic effect is produced by the current on its passage to and from the instrument. In practice it is generally sufficient to bind the wires together, care being taken that they are kept perfectly insulated from each other, but where they must pass near any sensitive part of the apparatus it is better to make one of the conductors a tube and the other a wire inside it. See Art. 683.

506.] In Ampère's second experiment one of the wires is bent and crooked with a number of small sinuosities, but so that in every part of its course it remains very near the straight wire. A current, flowing through the crooked wire and back again through the straight wire, is found to be without influence on the astatic balance. This proves that the effect of the current running through any crooked part of the wire is equivalent to the same current running in the straight line joining its extremities, provided the crooked line is in no part of its course far from the straight one. Hence any small element of a circuit is equivalent to two or more component elements, the relation between the component elements and the resultant element being the same as that between component and resultant displacements or velocities.

507.] In the third experiment a conductor capable of moving only in the direction of its length is substituted for the astatic balance, the current enters the conductor and leaves it at fixed points of space, and it is found that no closed circuit placed in the neighbourhood is able to move the conductor.

The conductor in this experiment is a wire in the form of a circular arc suspended on a frame which is capable of rotation about a vertical axis. The circular arc is horizontal, and its centre coincides with the vertical axis. Two small troughs are filled with mercury till the convex surface of the mercury rises above the level of the troughs. The troughs are placed under the circular arc and adjusted till the mercury touches the wire, which is of copper well amalgamated. The current is made to enter one of these troughs, to traverse the part of the circular arc between the troughs, and to escape by the other trough. Thus part of the circular arc is traversed by the current, and the arc is at the same time capable of moving with considerable freedom in the direction of its length. Any closed currents or magnets may now be made to approach the moveable conductor without producing the slightest tendency to move it in the direction of its length.

508.] In the fourth experiment with the astatic balance two circuits are employed, each similar to one of those in the balance, but one of them, ${\displaystyle C}$, having dimensions ${\displaystyle n}$ times greater, and the other, ${\displaystyle A}$, ${\displaystyle n}$ times less. These are placed on opposite sides of the circuit of the balance, which we shall call ${\displaystyle B}$, so that they are similarly placed with respect to it, the distance of ${\displaystyle C}$ from ${\displaystyle B}$ being ${\displaystyle n}$ times greater than the distance of ${\displaystyle B}$ from ${\displaystyle A}$. The direction and </noinclude>strength of the current is the same in ${\displaystyle A}$ and ${\displaystyle C}$. Its direction in ${\displaystyle B}$ may be the same or opposite. Under these circumstances it is found that ${\displaystyle B}$ is in equilibrium under the action of ${\displaystyle A}$ and ${\displaystyle C}$, whatever be the forms and distances of the three circuits, provided they have the relations given above.

Since the actions between the complete circuits may be considered to be due to actions between the elements of the circuits, we may use the following method of determining the law of these actions.

Let ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, ${\displaystyle C_{1}}$, Fig. 28, be corresponding elements of the three circuits, and let ${\displaystyle A_{2}}$, ${\displaystyle B_{2}}$, ${\displaystyle C_{2}}$ be also corresponding elements in another part of the circuits. Then the situation of ${\displaystyle B_{1}}$ with respect to ${\displaystyle A_{2}}$ is similar to the situation of ${\displaystyle C_{1}}$ with respect to ${\displaystyle B_{2}}$, but the distance and dimensions of ${\displaystyle C_{1}}$ and ${\displaystyle B_{2}}$ are ${\displaystyle n}$ times the distance and dimensions of ${\displaystyle B_{1}}$ and ${\displaystyle A_{2}}$, respectively. If the law of electromagnetic action is a function of the distance, then the action, whatever be its form or quality, between ${\displaystyle B_{1}}$ and ${\displaystyle A_{2}}$, may be written
${\displaystyle F=B_{1}.A_{2}f({\overline {B_{1}A_{2}}})\,ab}$,
and that between ${\displaystyle C_{1}}$ and ${\displaystyle B_{2}}$
${\displaystyle F^{\prime }=C_{1}.B_{2}f({\overline {C_{1}B_{1}}})\,bc}$,
where ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ are the strengths of the currents in ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$. But ${\displaystyle nB_{1}=C_{1}}$, ${\displaystyle nA_{2}=B_{2}}$, ${\displaystyle n{\overline {B_{1}A_{2}}}={\overline {C_{1}B_{2}}}}$, and ${\displaystyle a=c}$. Hence
${\displaystyle F^{\prime }=n^{2}B_{1}.A_{2}f(n{\overline {B_{1}A_{2}}})\,ab}$,
and this is equal to ${\displaystyle F}$ by experiment, so that we have
${\displaystyle n^{2}f(n{\overline {A_{2}B_{1}}})=f({\overline {A_{2}B_{1}}})}$;
or, the force varies inversely as the square of the distance.

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.

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 ${\displaystyle A}$, ${\displaystyle A^{\prime }}$ be these points. Let ${\displaystyle PQ}$ and ${\displaystyle P^{\prime }Q^{\prime }}$ be elements of the two curves.

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

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

The relative position of the two elements is sufficiently defined by their distance ${\displaystyle r}$ and the three angles ${\displaystyle \theta }$, ${\displaystyle \theta ^{\prime }}$, and ${\displaystyle \eta }$, for if these be given their relative position is as completely determined as if they formed part of the same rigid body.

512.] If we use rectangular coordinates and make ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ the coordinates of ${\displaystyle P}$, and ${\displaystyle x^{\prime }}$, ${\displaystyle y^{\prime }}$, ${\displaystyle z^{\prime }}$ those of ${\displaystyle P^{\prime }}$, and if we denote by ${\displaystyle l}$, ${\displaystyle m}$, ${\displaystyle n}$ and by ${\displaystyle l^{\prime }}$, ${\displaystyle m^{\prime }}$, ${\displaystyle n^{\prime }}$ the direction-cosines of ${\displaystyle PQ}$, and of ${\displaystyle P^{\prime }Q^{\prime }}$ respectively, then

 ${\displaystyle {\frac {dx}{ds}}=l}$, ⁠ ${\displaystyle {\frac {dy}{ds}}=m}$, ⁠ ${\displaystyle {\frac {dz}{ds}}=n}$, ${\displaystyle {\frac {dx^{\prime }}{ds^{\prime }}}=l^{\prime }}$, ${\displaystyle {\frac {dy^{\prime }}{ds^{\prime }}}=m^{\prime }}$, ${\displaystyle {\frac {dz^{\prime }}{ds^{\prime }}}=n^{\prime }}$,
(2)
and
 ${\displaystyle l(x^{\prime }-x)+m(y^{\prime }-y)+n(z^{\prime }-z)={}}$ ${\displaystyle r\cos \theta }$, ${\displaystyle l^{\prime }(x^{\prime }-z)+m^{\prime }(y^{\prime }-y)+n^{\prime }(z^{\prime }-z)={}}$ ${\displaystyle -r\cos \theta ^{\prime }}$, ${\displaystyle ll^{\prime }+mm^{\prime }+nn^{\prime }=\cos \epsilon }$,
(3)
where ${\displaystyle \epsilon }$ is the angle between the directions of the elements themselves, and
(4)
${\displaystyle \cos \epsilon =-\cos \theta \cos \theta ^{\prime }+\sin \theta \sin \theta ^{\prime }\cos \eta }$.
Again
(5)
${\displaystyle r^{2}=(x^{\prime }-x)^{2}+(y^{\prime }-y)^{2}+(z^{\prime }-z)^{2}}$,
whence
 ${\displaystyle r{\frac {dr}{ds}}}$ ${\displaystyle {}=-(x^{\prime }-x){\frac {dx}{ds}}-(y^{\prime }-y){\frac {dy}{ds}}-(z^{\prime }-z){\frac {dz}{ds}}}$, ${\displaystyle {}=-r\cos \theta }$.
(6)
Similarly
 ${\displaystyle r{\frac {dr}{ds^{\prime }}}}$ ${\displaystyle {}=(x^{\prime }-x){\frac {dx^{\prime }}{ds^{\prime }}}+(y^{\prime }-y){\frac {dy^{\prime }}{ds^{\prime }}}+(z^{\prime }-z){\frac {dz^{\prime }}{ds^{\prime }}}}$, ${\displaystyle {}=-r\cos \theta ^{\prime }}$;

and differentiating ${\displaystyle r{\frac {dr}{ds}}}$ with respect to ${\displaystyle s^{\prime }}$,

 ${\displaystyle r{\frac {d^{2}r}{ds\,ds^{\prime }}}+{\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}}$ ${\displaystyle {}=-{\frac {dx}{ds}}{\frac {dx^{\prime }}{ds^{\prime }}}-{\frac {dy}{ds}}{\frac {dy^{\prime }}{ds^{\prime }}}-{\frac {dz}{ds}}{\frac {dz^{\prime }}{ds^{\prime }}}}$ ${\displaystyle {}=-(ll^{\prime }+mm^{\prime }+nn^{\prime })}$ ${\displaystyle {}=-\cos \epsilon }$.
(7)

We can therefore express the three angles ${\displaystyle \theta }$, ${\displaystyle \theta ^{\prime }}$, and ${\displaystyle \eta }$, and the auxiliary angle ${\displaystyle \epsilon }$ in terms of the differential coefficients of ${\displaystyle r}$ with respect to ${\displaystyle s}$ and ${\displaystyle s^{\prime }}$ as follows,

 ${\displaystyle \cos \theta }$ ${\displaystyle {}=-{\frac {dr}{ds}}}$, ${\displaystyle \cos \theta ^{\prime }}$ ${\displaystyle {}=-{\frac {dr}{ds^{\prime }}}}$, ${\displaystyle \cos \epsilon }$ ${\displaystyle {}=-r{\frac {d^{2}r}{ds\,ds^{\prime }}}-{\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}}$, ${\displaystyle \sin \theta \sin \theta ^{\prime }\cos \eta }$ ${\displaystyle {}=-r{\frac {d^{2}r}{ds\,ds^{\prime }}}}$.
(8)

513.] We shall next consider in what way it is mathematically conceivable that the elements ${\displaystyle PQ}$ and ${\displaystyle P^{\prime }Q^{\prime }}$ 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 ${\displaystyle ds}$ as resolved into ${\displaystyle \cos \theta \,ds=\alpha }$ in the direction of ${\displaystyle r}$, and ${\displaystyle \sin \theta \,ds=\beta }$ in a direction perpendicular to ${\displaystyle r}$ in the plane ${\displaystyle P^{\prime }PQ}$.

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

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

(1) ${\displaystyle \alpha }$ and ${\displaystyle \alpha ^{\prime }}$ are in the same straight line. The force between them must therefore be in this line. We shall suppose it to be an attraction
${\displaystyle {}=A\alpha \alpha ^{\prime }ii^{\prime }}$,
where ${\displaystyle A}$ is a function of ${\displaystyle r}$, and ${\displaystyle i}$, ${\displaystyle i^{\prime }}$ are the intensities of the currents in ${\displaystyle ds}$ and ${\displaystyle ds^{\prime }}$ respectively. This expression satisfies the condition of changing sign with ${\displaystyle i}$ and with ${\displaystyle i^{\prime }}$.

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

This force is evidently in the line joining ${\displaystyle \beta }$ and ${\displaystyle \beta ^{\prime }}$, for it must be in the plane in which they both lie, and if we were to measure ${\displaystyle \beta }$ and ${\displaystyle \beta ^{\prime }}$ 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 ${\displaystyle \beta }$, and must therefore be directed along ${\displaystyle r}$. Let us assume that this expression, when positive, represents an attraction.

(3) ${\displaystyle \beta }$ and ${\displaystyle \gamma ^{\prime }}$ 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 ${\displaystyle r}$. We are at present engaged with forces, so we shall leave this out of account.

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

The sign of this expression is reversed if we reverse the direction in which we measure ${\displaystyle \beta ^{\prime }}$. It must therefore represent either a force in the direction of ${\displaystyle \beta ^{\prime }}$, or a couple in the plane of ${\displaystyle \alpha }$ and ${\displaystyle \beta ^{\prime }}$. As we are not investigating couples, we shall take it as a force acting on ${\displaystyle \alpha }$ in the direction of ${\displaystyle \beta ^{\prime }}$.

There is of course an equal force acting on ${\displaystyle \beta ^{\prime }}$ in the opposite direction.

We have for the same reason a force
${\displaystyle C\alpha \gamma ^{\prime }ii^{\prime }}$
acting on ${\displaystyle \alpha }$ in the direction of ${\displaystyle \gamma ^{\prime }}$, and a force
${\displaystyle C\beta \alpha ^{\prime }ii^{\prime }}$
acting on ${\displaystyle \beta }$ in the opposite direction.

514.] Collecting our results, we find that the action on ${\displaystyle ds}$ is compounded of the following forces,

and
 ${\displaystyle X=(A\alpha \alpha ^{\prime }+B\beta \beta ^{\prime })ii^{\prime }}$ in the direction of ${\displaystyle r}$, ${\displaystyle Y=C(\alpha \beta ^{\prime }-\alpha ^{\prime }\beta )ii^{\prime }}$ in the direction of ${\displaystyle \beta }$, ${\displaystyle Z=C\alpha \gamma ^{\prime }ii^{\prime }}$ in the direction of ${\displaystyle \gamma ^{\prime }}$.
(9)

Let us suppose that this action on ${\displaystyle ds}$ is the resultant of three forces, ${\displaystyle Rii^{\prime }\,ds\,ds^{\prime }}$ acting in the direction of ${\displaystyle r}$, ${\displaystyle Sii^{\prime }\,ds\,ds^{\prime }}$ acting in the direction of ${\displaystyle ds}$, and ${\displaystyle S^{\prime }ii^{\prime }\,ds\,ds^{\prime }}$ acting in the direction of ${\displaystyle ds^{\prime }}$, then in terms of ${\displaystyle \theta }$, ${\displaystyle \theta ^{\prime }}$ , and ${\displaystyle \eta }$,

 ${\displaystyle R=A\cos \theta \cos \theta ^{\prime }+B\sin \theta \sin \theta ^{\prime }\cos \eta }$, ${\displaystyle S=-C\cos \theta ^{\prime }}$, ⁠ ${\displaystyle S^{\prime }=C\cos \theta }$.
(10)

In terms of the differential coefficients of ${\displaystyle r}$

 ${\displaystyle R=A{\frac {dr}{ds}}{dr}{ds^{\prime }}-Br{\frac {d^{2}r}{ds\,ds^{\prime }}}}$, ${\displaystyle S=+C{\frac {dr}{ds^{\prime }}}}$,⁠ ${\displaystyle S^{\prime }=-C{\frac {dr}{ds}}}$,
(11)

In terms of ${\displaystyle l}$, ${\displaystyle m}$, ${\displaystyle n}$, and ${\displaystyle l^{\prime }}$, ${\displaystyle m^{\prime }}$, ${\displaystyle n^{\prime }}$,

 ${\displaystyle R=-(A+B){\frac {1}{r^{2}}}(l\xi +m\eta +n\zeta )(l^{\prime }\xi +m^{\prime }\eta +n^{\prime }\zeta )+B(ll^{\prime }+mm^{\prime }+nn^{\prime })}$, ${\displaystyle S=C{\frac {1}{r}}(l^{\prime }\xi +m^{\prime }n+n^{\prime }\zeta )}$, ⁠ ${\displaystyle S^{\prime }=C{\frac {1}{r}}(l\xi +m\eta +n\zeta )}$,
(12)

where ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \zeta }$ are written for ${\displaystyle x^{\prime }-x}$, ${\displaystyle y^{\prime }-y}$, and ${\displaystyle z^{\prime }-z}$ respectively.

515.] We have next to calculate the force with which the finite current ${\displaystyle s^{\prime }}$ acts on the finite current ${\displaystyle s}$. The current ${\displaystyle s}$ extends from ${\displaystyle A}$ where ${\displaystyle s=0}$, to ${\displaystyle P}$, where it has the value ${\displaystyle s}$. The current ${\displaystyle s^{\prime }}$ extends from ${\displaystyle A^{\prime }}$, where ${\displaystyle s^{\prime }=0}$, to ${\displaystyle P^{\prime }}$, where it has the value ${\displaystyle s^{\prime }}$. The coordinates of points on either current are functions of ${\displaystyle s}$ or of ${\displaystyle s^{\prime }}$.

If ${\displaystyle F}$ is any function of the position of a point, then we shall use the subscript ${\displaystyle {}_{(s,0)}}$ to denote the excess of its value at ${\displaystyle P}$ over that at ${\displaystyle A}$, thus
${\displaystyle F_{(s,0)}=F_{P}-F_{A}}$.

Such functions necessarily disappear when the circuit is closed.

Let the components of the total force with which ${\displaystyle A^{\prime }P^{\prime }}$ acts on ${\displaystyle AA}$ be ${\displaystyle ii^{\prime }X}$, ${\displaystyle ii^{\prime }Y}$, and ${\displaystyle ii^{\prime }Z}$. Then the component parallel to ${\displaystyle X}$ of the force with which ${\displaystyle ds^{\prime }}$ acts on ${\displaystyle ds}$ will be ${\displaystyle ii^{\prime }{\frac {d^{2}X}{ds\,ds^{\prime }}}ds\,ds^{\prime }}$.

Hence
(13)
${\displaystyle {\frac {d^{2}X}{dsds^{\prime }}}=R{\frac {\xi }{r}}+Sl+S^{\prime }l^{\prime }}$.
Substituting the values of ${\displaystyle R}$, ${\displaystyle S}$, and ${\displaystyle S^{\prime }}$ from (12), remembering that
(14)
${\displaystyle l^{\prime }\xi +m^{\prime }\eta +n^{\prime }\zeta =r{\frac {dr}{ds^{\prime }}}}$,
and arranging the terms with respect to ${\displaystyle l}$, ${\displaystyle m}$, ${\displaystyle n}$, we find
 ${\displaystyle {\frac {d^{2}X}{ds\,ds^{\prime }}}}$ ${\displaystyle {}=l\left\{-(A+B){\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}\xi ^{2}+C{\frac {dr}{ds^{\prime }}}+(B+C){\frac {l^{\prime }\xi }{r}}\right\}}$, ${\displaystyle {}+m\left\{-(A+B){\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}\xi \eta +C{\frac {l^{\prime }\eta }{r}}+B{\frac {m^{\prime }\xi }{r}}\right\}}$, ${\displaystyle {}+n\left\{-(A+B){\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}\xi \zeta +C{\frac {l^{\prime }\zeta }{r}}+B{\frac {n^{\prime }\xi }{r}}\right\}}$.
(15)
Since ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ are functions of ${\displaystyle r}$, we may write
(16)
${\displaystyle P=\int _{r}^{\infty }(A+B){\frac {1}{r^{2}}}\,dr}$, ${\displaystyle Q=\int _{r}^{\infty }C\,dr}$,
the integration being taken between ${\displaystyle r}$ and ${\displaystyle \infty }$ because ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$ vanish when ${\displaystyle r=\infty }$.
Hence
(17)
${\displaystyle (A+B){\frac {1}{r^{2}}}=-{\frac {dP}{dr}}}$, and ${\displaystyle C=-{\frac {dQ}{dr}}}$.
516.] Now we know, by Ampère's third case of equilibrium, that when ${\displaystyle s^{\prime }}$ is a closed circuit, the force acting on ${\displaystyle ds}$ is perpendicular to the direction of ${\displaystyle ds}$, or, in other words, the component of the force in the direction of ${\displaystyle ds}$ itself is zero. Let us therefore assume the direction of the axis of ${\displaystyle x}$ so as to be parallel to ${\displaystyle ds}$ by making ${\displaystyle l=1}$, ${\displaystyle m=0}$, ${\displaystyle n=0}$. Equation (15) then becomes
(18)
${\displaystyle {\frac {d^{2}X}{ds\,ds^{\prime }}}={\frac {dP}{ds^{\prime }}}\xi ^{2}-{\frac {dQ}{ds^{\prime }}}+(B+C){\frac {l^{\prime }\xi }{r}}}$.

To find ${\displaystyle {\frac {dX}{ds}}}$, the force on ${\displaystyle ds}$ referred to unit of length, we must integrate this expression with respect to ${\displaystyle s^{\prime }}$. Integrating the first term by parts, we find

(19)
${\displaystyle {\frac {dX}{ds}}=(P\xi ^{2}-Q)_{(s^{\prime },0)}-\int _{0}^{s^{\prime }}(2Pr-B-C){\frac {l^{\prime }\xi }{r}}\,ds^{\prime }}$.
When ${\displaystyle s^{\prime }}$ is a closed circuit this expression must be zero. The first term will disappear of itself. The second term, however, will not in general disappear in the case of a closed circuit unless the quantity under the sign of integration is always zero. Hence, to satisfy Ampère's condition,
(20)
${\displaystyle P={\frac {1}{2r}}(B+C)}$.

517.] We can now eliminate ${\displaystyle P}$, and find the general value of ${\displaystyle {\frac {dX}{ds}}}$,

 ${\displaystyle {\frac {dX}{ds}}=\left\{{\frac {B+C}{2}}{\frac {\xi }{r}}(l\xi +m\eta n\zeta )+Q\right\}_{(s^{\prime },0)}}$ ${\displaystyle {}+m\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {m^{\prime }\xi -l^{\prime }\eta }{r}}\,ds^{\prime }-n\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {l^{\prime }\zeta -n^{\prime }\xi }{r}}\,ds^{\prime }}$.
(21)

When ${\displaystyle s^{\prime }}$ is a closed circuit the first term of this expression vanishes, and if we make

 ${\displaystyle \alpha ^{\prime }=\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {n^{\prime }\eta -m^{\prime }\zeta }{r}}\;ds^{\prime }}$, ${\displaystyle \beta ^{\prime }=\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {l^{\prime }\zeta -n^{\prime }\xi }{r}}\,ds^{\prime }}$, ${\displaystyle \gamma ^{\prime }=\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {m^{\prime }\xi -l^{\prime }\eta }{r}}\,ds^{\prime }}$,
(22)

where the integration is extended round the closed circuit ${\displaystyle s^{\prime }}$, we may write

Similarly
 ${\displaystyle {\frac {dX}{ds}}=m\gamma ^{\prime }-n\beta ^{\prime }}$. ${\displaystyle {\frac {dY}{ds}}=n\alpha ^{\prime }-l\gamma ^{\prime }}$, ${\displaystyle {\frac {dZ}{ds}}=l\beta ^{\prime }-m\alpha ^{\prime }}$.
(23)

The quantities ${\displaystyle \alpha ^{\prime }}$, ${\displaystyle \beta ^{\prime }}$, ${\displaystyle \gamma ^{\prime }}$ are sometimes called the determinants of the circuit ${\displaystyle s^{\prime }}$ referred to the point ${\displaystyle P}$. Their resultant is called by Ampère the directrix of the electrodynamic action.

It is evident from the equation, that the force whose components are ${\displaystyle {\frac {dX}{ds}}}$, ${\displaystyle {\frac {dY}{ds}}}$, and ${\displaystyle {\frac {dZ}{ds}}}$ is perpendicular both to ${\displaystyle ds}$ and to this directrix, and is represented numerically by the area of the parallelogram whose sides are ${\displaystyle ds}$ and the directrix.

In the language of quaternions, the resultant force on ${\displaystyle ds}$ is the vector part of the product of the directrix multiplied by ${\displaystyle ds}$.

Since we already know that the directrix is the same thing as the magnetic force due to a unit current in the circuit ${\displaystyle s^{\prime }}$, we shall henceforth speak of the directrix as the magnetic force due to the circuit.

518.] We shall now complete the calculation of the components of the force acting between two finite currents, whether closed or open.

Let ${\displaystyle \rho }$ be a new function of ${\displaystyle r}$, such that
(24)
${\displaystyle \rho ={\frac {1}{2}}\int _{r}^{\infty }(B-C)\,dr}$,
then by (17) and (20)
(25)
${\displaystyle A+B=r{\frac {d^{2}}{dr^{2}}}(Q+\rho )-{\frac {d}{dr}}(Q+\rho )}$,
and equations (11) become
 ${\displaystyle R=-{\frac {d\rho }{dr}}\cos \epsilon +r{\frac {d^{2}}{ds\,ds^{\prime }}}(Q+\rho )}$, ${\displaystyle S=-{\frac {dQ}{ds^{\prime }}}}$, ⁠ ${\displaystyle S^{\prime }={\frac {dQ}{ds}}}$.
(26)

With these values of the component forces, equation (13) becomes

 ${\displaystyle {\frac {d^{2}X}{ds\,ds^{\prime }}}}$ ${\displaystyle {}=-{}}$ ${\displaystyle \cos \epsilon {\frac {d\rho }{dr}}{\frac {\xi }{r}}+\xi {\frac {d^{2}}{ds\,ds^{\prime }}}(Q+\rho )-l{\frac {dQ}{ds^{\prime }}}+l^{\prime }{\frac {dQ}{ds}}}$, ${\displaystyle {}={}}$ ${\displaystyle \cos \epsilon {\frac {d\rho }{dx}}+{\frac {d^{2}(Q+\rho )\xi }{ds\,ds^{\prime }}}+l{\frac {d\rho }{ds^{\prime }}}-l^{\prime }{\frac {d\rho }{ds}}}$.
(27)

519.] Let

 ⁠ ${\displaystyle F=\int _{0}^{s}l\rho \,ds}$, ${\displaystyle G=\int _{0}^{s}m\rho \,ds}$, ${\displaystyle H=\int _{0}^{s}n\rho \,ds}$, (28) ⁠ ${\displaystyle F^{\prime }=\int _{0}^{s^{\prime }}l^{\prime }\rho \,ds^{\prime }}$, ${\displaystyle G^{\prime }=\int _{0}^{s^{\prime }}m^{\prime }\rho \,ds^{\prime }}$, ${\displaystyle H^{\prime }=\int _{0}^{s^{\prime }}n^{\prime }\rho \,ds^{\prime }}$. (29)

These quantities have definite values for any given point of space. When the circuits are closed, they correspond to the components of the vector-potentials of the circuits.

Let ${\displaystyle L}$ be a new function of ${\displaystyle r}$, such that
(30)
${\displaystyle L=\int _{0}^{\infty }r(Q+\rho )\,dr}$,
and let ${\displaystyle M}$ be the double integral
(31)
${\displaystyle M=\int _{0}^{s^{\prime }}\int _{0}^{s}\rho \cos \epsilon \,ds\,ds^{\prime }}$,
which, when the circuits are closed, becomes their mutual potential, then (27) may be written
(32)
${\displaystyle {\frac {d^{2}X}{ds\,ds^{\prime }}}={\frac {d^{2}}{ds\,ds^{\prime }}}\left\{{\frac {dM}{dx}}-{\frac {dL}{dx}}+F^{\prime }-F\right\}}$.

520.] Integrating, with respect to ${\displaystyle s}$ and ${\displaystyle s^{\prime }}$, between the given limits, we find

 ${\displaystyle X={\frac {dM}{dx}}}$ ${\displaystyle {}-{\frac {d}{dx}}(L_{PP^{\prime }}-L_{AP^{\prime }}-L_{A^{\prime }P}+L_{AA^{\prime }})}$, ${\displaystyle {}+F_{P}^{\prime }-F_{A}^{\prime }-F_{P^{\prime }}+F_{A^{\prime }}}$,
(33)

where the subscripts of ${\displaystyle L}$ indicate the distance, ${\displaystyle r}$, of which the quantity ${\displaystyle L}$ is a function, and the subscripts of ${\displaystyle F}$ and ${\displaystyle F^{\prime }}$ indicate the points at which their values are to be taken.

The expressions for ${\displaystyle T}$ and ${\displaystyle Z}$ may be written down from this. Multiplying the three components by ${\displaystyle dx}$, ${\displaystyle dy}$, and ${\displaystyle dz}$ respectively, we obtain

 ${\displaystyle X\,dx+Y\,dy+Z\,dz={}}$ ${\displaystyle DM-D(L_{PP\prime }-L_{AP\prime }-L_{A^{\prime }P}+L_{AA^{\prime }})}$, ${\displaystyle {}+(F^{\prime }dx+G^{\prime }dy+H^{\prime }dx)_{(P-A)}}$, ${\displaystyle {}-(F\,dx+G\,dy+H\,dz)_{(P^{\prime }-A)^{\prime }}}$,
(34)

where ${\displaystyle D}$ is the symbol of a complete differential.

Since ${\displaystyle F\,dx+G\,dy+H\,dz}$ is not in general a complete differential of a function of ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$, ${\displaystyle X\,dx+Y\,dy+Z\,dz}$ is not a complete differential for currents either of which is not closed.

521.] If, however, both currents are closed, the terms in ${\displaystyle L}$, ${\displaystyle F}$, ${\displaystyle G}$, ${\displaystyle H}$, ${\displaystyle F^{\prime }}$, ${\displaystyle G^{\prime }}$, ${\displaystyle H^{\prime }}$ disappear, and
(35)
${\displaystyle X\,dx+Y\,dy+Z\,dz=DM}$,
where ${\displaystyle M}$ is the mutual potential of two closed circuits carrying unit currents. The quantity ${\displaystyle M}$ expresses the work done by the electromagnetic forces on either conducting circuit when it is moved parallel to itself from an infinite distance to its actual position. Any alteration of its position, by which ${\displaystyle M}$ is increased, will be assisted by the electromagnetic forces.

It may be shewn, as in Arts. 490, 596, that when the motion of the circuit is not parallel to itself the forces acting on it are still determined by the variation of ${\displaystyle M}$, the potential of the one circuit on the other.

522.] The only experimental fact which we have made use of in this investigation is the fact established by Ampère that the action of a closed current on any portion of another current is perpendicular to the direction of the latter. Every other part of the investigation depends on purely mathematical considerations depending on the properties of lines in space. The reasoning therefore may be presented in a much more condensed and appropriate form by the use of the ideas and language of the mathematical method specially adapted to the expression of such geometrical relations—the Quaternions of Hamilton.

This has been done by Professor Tait in the Quarterly Mathematical Journal, 1866, and in his treatise on Quaternions, § 399, for Ampère's original investigation, and the student can easily adapt the same method to the somewhat more general investigation given here.

523.] Hitherto we have made no assumption with respect to the quantities ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, except that they are functions of ${\displaystyle r}$, the distance between the elements. We have next to ascertain the form of these functions, and for this purpose we make use of Ampère's fourth case of equilibrium. Art. 508, in which it is shewn that if all the linear dimensions and distances of a system of two circuits be altered in the same proportion, the currents remaining the same, the force between the two circuits will remain the same.

Now the force between the circuits for unit currents is ${\displaystyle {\frac {dM}{dx}}}$, and since this is independent of the dimensions of the system, it must be a numerical quantity. Hence ${\displaystyle M}$ itself, the coefficient of the mutual potential of the circuits, must be a quantity of the dimensions of a line. It follows, from equation (31), that ${\displaystyle \rho }$ must be the reciprocal of a line, and therefore by (24), ${\displaystyle B-C}$ must be the inverse square of a line. But since ${\displaystyle B}$ and ${\displaystyle C}$ are both functions of ${\displaystyle r}$, ${\displaystyle B-C}$ must be the inverse square of r or some numerical multiple of it.

524.] The multiple we adopt depends on our system of measurement. If we adopt the electromagnetic system, so called because it agrees with the system already established for magnetic measurements, the value of ${\displaystyle M}$ ought to coincide with that of the potential of two magnetic shells of strength unity whose boundaries are the two circuits respectively. The value of ${\displaystyle M}$ in that case is, by Art. 423,
(36)
${\displaystyle M=\iint {\frac {\cos \epsilon }{r}}\,ds\,ds^{\prime }}$,
the integration being performed round both circuits in the positive direction. Adopting this as the numerical value of ${\displaystyle M}$, and comparing with (31), we find
(37)
${\displaystyle \rho ={\frac {1}{r}}}$and${\displaystyle B-C={\frac {2}{r^{2}}}}$.

525.] We may now express the components of the force on ${\displaystyle ds}$ arising from the action of ${\displaystyle ds^{\prime }}$ in the most general form consistent with experimental facts.

The force on ${\displaystyle ds}$ is compounded of an attraction

 ${\displaystyle R}$ ${\displaystyle {}={\frac {1}{r^{2}}}\left({\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}-2r{\frac {d^{2}r}{ds\,ds^{\prime }}}\right)ii^{\prime }\,ds\,ds^{\prime }+r{\frac {d^{2}Q}{ds\,ds^{\prime }}}ii^{\prime }\,ds\,ds^{\prime }}$ in the direction of ${\displaystyle r}$, ${\displaystyle S}$ ${\displaystyle {}=-{\frac {dQ}{ds^{\prime }}}ii^{\prime }\,ds\,ds^{\prime }}$ in the direction of ${\displaystyle ds}$, and ${\displaystyle S^{\prime }={\frac {dQ}{ds}}ii^{\prime }\,ds\,ds^{\prime }}$ in the direction of ${\displaystyle ds^{\prime }}$,
(38)

where ${\displaystyle Q=\int _{r}^{\infty }C\,dr}$, and since ${\displaystyle C}$ is an unknown function of ${\displaystyle r}$, we know only that ${\displaystyle Q}$ is some function of ${\displaystyle r}$.

526.] The quantity ${\displaystyle Q}$ cannot be determined, without assumptions of some kind, from experiments in which the active current forms a closed circuit. If we suppose with Ampère that the action between the elements ${\displaystyle ds}$ and ${\displaystyle ds^{\prime }}$ is in the line joining them, then ${\displaystyle S}$ and ${\displaystyle S^{\prime }}$ must disappear, and ${\displaystyle Q}$ must be constant, or zero. The force is then reduced to an attraction whose value is
(39)
${\displaystyle R={\frac {1}{r^{2}}}\left({\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}-2r{\frac {d^{2}r}{ds\,ds^{\prime }}}\right)ii^{\prime }\,ds\,ds^{\prime }}$.
Ampère, who made this investigation long before the magnetic system of units had been established, uses a formula having a numerical value half of this, namely
(40)
${\displaystyle R={\frac {1}{r^{2}}}\left({\frac {1}{2}}{\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}-r{\frac {dr}{ds\,ds^{\prime }}}\right)jj^{\prime }\,ds\,ds^{\prime }}$.
Here the strength of the current is measured in what is called electrodynamic measure. If ${\displaystyle i}$, ${\displaystyle i^{\prime }}$ are the strength of the currents in electromagnetic measure, and ${\displaystyle j}$, ${\displaystyle j^{\prime }}$ the same in electrodynamic measure, then it is plain that
(41)
${\displaystyle jj^{\prime }=2ii^{\prime }}$,or${\displaystyle j={\sqrt {2}}i}$.

Hence the unit current adopted in electromagnetic measure is greater than that adopted in electrodynamic measure in the ratio of ${\displaystyle {\sqrt {2}}}$ to 1.

The only title of the electrodynamic unit to consideration is that it was originally adopted by Ampère, the discoverer of the law of action between currents. The continual recurrence of ${\displaystyle {\sqrt {2}}}$ in calculations founded on it is inconvenient, and the electromagnetic system has the great advantage of coinciding numerically with all our magnetic formulae. As it is difficult for the student to bear in mind whether he is to multiply or to divide by ${\displaystyle {\sqrt {2}}}$, we shall henceforth use only the electromagnetic system, as adopted by Weber and most other writers.

Since the form and value of ${\displaystyle Q}$ have no effect on any of the experiments hitherto made, in which the active current at least is always a closed one, we may, if we please, adopt any value of ${\displaystyle Q}$ which appears to us to simplify the formulae.

Thus Ampère assumes that the force between two elements is in the line joining them. This gives ${\displaystyle Q=0}$,

(42)
${\displaystyle R={\frac {1}{r^{2}}}\left({\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}-24{\frac {d^{2}r}{ds\,ds^{\prime }}}\right)\,ii\,ds\,ds^{\prime }}$, ${\displaystyle S=0}$,${\displaystyle S^{\prime }=0}$.

Grassmann[1] assumes that two elements in the same straight line have no mutual action. This gives

(43)
${\displaystyle Q=-{\frac {1}{2r}}}$,${\displaystyle R=-{\frac {3}{2r}}{\frac {d^{2}r}{ds\,ds^{\prime }}}}$,${\displaystyle S=-{\frac {1}{2r^{2}}}{\frac {dr}{ds^{\prime }}}}$, ${\displaystyle S^{\prime }={\frac {1}{2r^{2}}}{\frac {dr}{ds}}}$.

We might, if we pleased, assume that the attraction between two elements at a given distance is proportional to the cosine of the angle between them. In this case

(44)
${\displaystyle Q=-{\frac {1}{r}}}$, ${\displaystyle R={\frac {1}{r^{2}}}\cos \epsilon }$, ${\displaystyle S=-{\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}}$, ${\displaystyle S^{\prime }={\frac {1}{r^{2}}}{\frac {dr}{ds}}}$.

Finally, we might assume that the attraction and the oblique forces depend only on the angles which the elements make with the line joining them, and then we should have

(45)
${\displaystyle Q=-{\frac {2}{r}}}$, ${\displaystyle R=-3{\frac {1}{r^{2}}}{\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}}$, ${\displaystyle S=-{\frac {2}{r^{2}}}{\frac {dr}{ds^{\prime }}}}$, ${\displaystyle S^{\prime }={\frac {2}{r^{2}}}{\frac {dr}{ds}}}$.

527.] Of these four different assumptions that of Ampère is undoubtedly the best, since it is the only one which makes the forces on the two elements not only equal and opposite but in the straight line which joins them.

1. Pogg., Ann. lxiv. p. 1 (1845).