Page:A Dynamical Theory of the Electromagnetic Field.pdf/10

where the summation is extended to all the bodies with their proper values of $C$,$p$,$a$, and $q$. Then the momentum of the system referred to $A$ is

$Lu+Mv,$

And referred to $B$,

$Mu+Nv,$

And we shall have

$\left.{\begin{array}{l}X={\frac {d}{dt}}(Lu+Mv)\\Y={\frac {d}{dt}}(Mu+Nv)\\\end{array}}\right\}$

(2)

Where $X$ and $Y$ are the external forces acting on $A$ and $B$.

(25) To make the illustration more complete we have only to suppose that the motion of $A$ is resisted by a force proportional to its velocity, which we may call $Ru$, and that of $B$ by a similar force, which we may call $Sv$, $R$ and $S$ being coefficients of resistance. Then if $\xi$ and $\eta$ are the forces on $A$ and $B$

$\left.{\begin{array}{l}\xi =X+Ru=Ru+{\frac {d}{dt}}(Lu+Mv)\\\eta =Y+Sv=Sv+{\frac {d}{dt}}(Mu+Nv)\\\end{array}}\right\},$

(3)

If the velocity of $A$ be increased at the rate ${\tfrac {du}{dt}}$, then in order to prevent $B$ from moving a force, $\eta ={\tfrac {d}{dt}}(Mu)$ must be applied to it.

This effect on $B$, due to an increase of the velocity of $A$, corresponds to the electromotive force on one circuit arising from an increase in the strength of a neighbouring circuit.

This dynamical illustration is to be considered merely as assisting the reader to understand what is meant in mechanics by Reduced Momentum. The facts of the induction of currents as depending on the variations of the quantity called Electromagnetic Momentum, or Electrotonic State, rest on the experiments of Faraday^[1], Felici^[2], &c.

Coefficients of Induction for Two Circuits

(26) In the electromagnetic field the values of $L$, $M$, $N$ depend on the distribution of the magnetic effects due to the two circuits, and this distribution depends only on the form and relative position of the circuits. Hence $L$, $M$, $N$ are quantities depending on the form and relative position of the circuits, and are subject to variation with the motion of the conductors. It will be presently seen that $L$, $M$, $N$ are geometrical quantities of the nature of lines, that is, of one dimension in space; $L$ depends on the form of the first conductor, which we shall call $A$, $N$ on that of the second, which we call $B$, and $M$ on the relative position of $A$ and $B$.

(27) Let $\xi$ be the electromotive force acting on $A$, $x$ the strength of the current, and

1. Experimental Researches, Series I., IX
2. Annales de Chimie, ser, 3, XXXIV. (1852), p. 64.