Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/437

 

817.] The disturbance, being a vector, can be resolved into components parallel to $x$ and $y$, the axis of $z$ being parallel to the direction of the ray. Let $\xi$ and $\eta$ be these components, then, in the case of a ray of homogeneous circularly-polarized light,

$\xi =r\cos \theta$,⁠$\eta =r\sin \theta$,

(1)

where

$\theta =nt-qz+\alpha$.

(2)

In these expressions, $r$ denotes the magnitude of the vector, and $\theta$ the angle which it makes with the direction of the axis of $x$.

The periodic time, $\tau$, of the disturbance is such that

$n\tau =2\pi$.

(3)

The wave-length, $\lambda$, of the disturbance is such that

$q\lambda =2\pi$.

(4)

The velocity of propagation is ${\frac {n}{q}}$.

The phase of the disturbance when $t$ and $z$ are both zero is $\alpha$.

The circularly-polarized light is right-handed or left-handed according as $q$ is negative or positive.

Its vibrations are in the positive or the negative direction of rotation in the plane of ($x$, $y$), according as $n$ is positive or negative.

The light is propagated in the positive or the negative direction of the axis of $z$, according as $n$ and $q$ are of the same or of opposite signs.

In all media $n$ varies when $q$ varies, and ${\frac {dn}{dq}}$ is always of the same sign with ${\frac {n}{q}}$.

Hence, if for a given numerical value of $n$ the value of ${\frac {n}{q}}$ is greater when $n$ is positive than when $n$ is negative, it follows that for a value of $q$, given both in magnitude and sign, the positive value of $n$ will be greater than the negative value.

Now this is what is observed in a diamagnetic medium, acted on by a magnetic force, $\gamma$, in the direction of $z$. Of the two circularly-polarized rays of a given period, that is accelerated of which the direction of rotation in the plane of ($x$, $y$) is positive. Hence, of two circularly-polarized rays, both left-handed, whose wave-length within the medium is the same, that has the shortest period whose direction of rotation in the plane of $xy$ is positive, that is, the ray which is propagated in the positive direction of $z$ from south to north. We have therefore to account for the fact, that when in the equations of the system $q$ and $r$ are given, two values of $n$ will