Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/230

We have therefore

$T=T_{m}+T_{e}+T_{m}e,\,$

where

${\begin{aligned}T_{m}&={\frac {1}{2}}(x_{1}x_{1}){\dot {x}}_{1}^{2}\,+\And c.+(x_{1}x_{2}){\dot {x}}_{1}{\dot {x}}_{2}\,+\And c.,\\T_{e}&={\frac {1}{2}}(y_{1}y_{1}){\dot {y}}_{1}^{2}\,+\And c.+(y_{1}y_{2}){\dot {y}}_{1}{\dot {y}}_{2}\,+\And c.,\\T_{me}&=(x_{1}y_{1}){\dot {x}}_{1}{\dot {y}}_{1}\,+\And c.\end{aligned}}$

572.] In the general dynamical theory, the coefficients of every term may be functions of all the coordinates, both x and y. In the case of electric currents, however, it is easy to see that the coordinates of the class y do not enter into the coefficients.

For, if all the electric currents are maintained constant, and the conductors at rest, the whole state of the field will remain constant. But in this case the coordinates y are variable, though the velocities ${\dot {y}}$ are constant. Hence the coordinates y cannot enter into the expression for T, or into any other expression of what actually takes place.

Besides this, in virtue of the equation of continuity, if the conductors are of the nature of linear circuits, only one variable is required to express the strength of the current in each conductor. Let the velocities ${\dot {y}}_{1}$, ${\dot {y}}_{2}$, &c. represent the strengths of the currents in the several conductors.

All this would be true, if, instead of electric currents, we had currents of an incompressible fluid running in flexible tubes. In this case the velocities of these currents would enter into the expression for T, but the coefficients would depend only on the variables x, which determine the form and position of the tubes.

In the case of the fluid, the motion of the fluid in one tube does not directly affect that of any other tube, or of the fluid in it. Hence, in the value of T_e, only the squares of the velocities ${\dot {y}}$, and not their products, occur, and in T_me any velocity ${\dot {y}}$ is associated only with those velocities of the form ${\dot {x}}$ which belong to its own tube.

In the case of electrical currents we know that this restriction does not hold, for the currents in different circuits act on each other. Hence we must admit the existence of terms involving products of the form ${\dot {y}}_{1}{\dot {y}}_{2}$, and this involves the existence of something in motion, whose motion depends on the strength of both electric currents ${\dot {y}}_{1}$ and ${\dot {y}}_{2}$. This moving matter, whatever it is, is not confined to the interior of the conductors carrying the two currents, but probably extends throughout the whole space surrounding them.

573.] Let us next consider the form which Lagrange's equations of motion assume in this case. Let X' be the impressed force