Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/238

CHAPTER VII.

THEORY OF ELECTRIC CIRCUITS.

578.1 We may now confine our attention to that part of the kinetic energy of the system which depends on squares and products of the strengths of the electric currents. We may call this the Electrokinetic Energy of the system. The part depending on the motion of the conductors belongs to ordinary dynamics, and we have shewn that the part depending on products of velocities and currents does not exist.

Let $A_{1}$, $A_{2}$, &c. denote the different conducting circuits. Let their form and relative position be expressed in terms of the variables $x_{1}$, $x_{2}$, &c., the number of which is equal to the number of degrees of freedom of the mechanical system. We shall call these the Geometrical Variables.

Let $y_{1}$, denote the quantity of electricity which has crossed a given section of the conductor $A_{1}$, since the beginning of the time t. The strength of the current will be denoted by ${\dot {y}}_{1}$, the fluxion of this quantity.

We shall call ${\dot {y}}_{1}$ the actual current, and $y_{1}$ the integral current. There is one variable of this kind for each circuit in the system.

Let $T$ denote the electrokinetic energy of the system. It is a homogeneous function of the second degree with respect to the strengths of the currents, and is of the form

$T={\frac {1}{2}}L_{1}{\dot {y}}_{1}^{2}+{\frac {1}{2}}L_{2}{\dot {y}}_{2}^{2}+\And c.+M_{12}{\dot {y}}_{1}{\dot {y}}_{2}+\And c.,$

(1)

where the coefficients $L$, $M$, &c. are functions of the geometrical variables $x_{1}$, $x_{2}$, &c. The electrical variables $y_{1}$, $y_{2}$, &c., do not enter into the expression.

We may call $L_{1}$, $L_{2}$, &c., the electric moments of inertia of the circuits $A_{1}$, $A_{2}$, &c., and $M_{12}$ the electric product of inertia of the two circuits $A_{1}$ and $A_{2}$. When we wish to avoid the language of