Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/438

Rh satisfy the equations, one positive and the other negative, the positive value being numerically greater than the negative.

818.] We may obtain the equations of motion from a consideration of the potential and kinetic energies of the medium. The potential energy, $$V$$, of the system depends on its configuration, that is, on the relative position of its parts. In so far as it depends on the disturbance due to circularly-polarized light, it must be a function of $$r$$, the amplitude, and $$q$$, the coefficient of torsion, only. It may be different for positive and negative values of $$q$$ of equal numerical value, and it probably is so in the case of media which of themselves rotate the plane of polarization.

The kinetic energy, $$T$$, of the system is a homogeneous function of the second degree of the velocities of the system, the coefficients of the different terms being functions of the coordinates.

819.] Let us consider the dynamical condition that the ray may be of constant intensity, that is, that $$r$$ may be constant.

Lagrange's equation for the force in $$r$$ becomesSince $$r$$ is constant, the first term vanishes. We have therefore the equationin which $$q$$ is supposed to be given, and we are to determine the value of the angular velocity $$\dot{\theta}$$, which we may denote by its actual value, $$n$$.

The kinetic energy, $$T$$, contains one term involving $$n^2$$; other terms may contain products of $$n$$ with other velocities, and the rest of the terms are independent of $$n$$. The potential energy, $$V$$, is entirely independent of $$n$$. The equation is therefore of the formThis being a quadratic equation, gives two values of $$n$$. It appears from experiment that both values are real, that one is positive and the other negative, and that the positive value is numerically the greater. Hence, if $$A$$ is positive, both $$B$$ and $$C$$ are negative, for, if $$n_1$$ and $$n_2$$ are the roots of the equation,The coefficient, $$B$$, therefore, is not zero, at least when magnetic force acts on the medium. We have therefore to consider the expression $$Bn$$, which is the part of the kinetic energy involving the first power of $$n$$, the angular velocity of the disturbance.