Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/231

corresponding to the coordinate x, one of those which determine the form and position of the conducting circuits. This is a force in the ordinary sense, a tendency towards change of position. It is given by the equation

$X'={\frac {d}{dt}}{\frac {dT}{d{\dot {x}}}}-{\frac {dT}{dx}}.$

We may consider this force as the sum of three parts, corresponding to the three parts into which we divided the kinetic energy of the system, and we may distinguish them by the same suffixes. Thus

$X'=X'_{m}+X'_{e}+X'_{me}.\,$

The part X'_m is that which depends on ordinary dynamical considerations, and we need not attend to it.

Since T_e does not contain ${\dot {x}}$, the first term of the expression for X'_e is zero, and its value is reduced to

$X'_{e}=-{\frac {dT_{e}}{dx}}.$

This is the expression for the mechanical force which must be applied to a conductor to balance the electromagnetic force, and it asserts that it is measured by the rate of diminution of the purely electrokinetic energy due to the variation of the coordinate x. The electromagnetic force, X_e, which brings this external mechanical force into play, is equal and opposite to it, and is therefore measured by the rate of increase of the electrokinetic energy corresponding to an increase of the coordinate x. The value of X_e, since it depends on squares and products of the currents, remains the same if we reverse the directions of all the currents.

The third part of X' is

$X'_{me}={\frac {d}{dt}}{\frac {dT_{me}}{d{\dot {x}}}}-{\frac {dT_{me}}{dx}}.$

The quantity T_me contains only products of the form ${\dot {x}}{\dot {y}}$, so that ${\frac {dT_{me}}{d{\dot {x}}}}$ is a linear function of the strengths of the currents ${\dot {y}}$. The first term, therefore, depends on the rate of variation of the strengths of the currents, and indicates a mechanical force on the conductor, which is zero when the currents are constant, and which is positive or negative according as the currents are increasing or decreasing in strength.

The second term depends, not on the variation of the currents, but on their actual strength. As it is a linear function with respect to these currents, it changes sign when the currents change