Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/844

Definition of Ray.

23. In § 13 we defined a ray as the path of a labelled disturbance,^[1] for it is that which enables an eye to fix direction, it is that which determines the line of collimation of a telescope. Now in order that a disturbance from $\mathrm {A}$ may reach $\mathrm {B}$, it is necessary that adjacent elements of a wave front at $\mathrm {A}$ shall arrive at $\mathrm {B}$ in the same phase; hence the path by which a disturbance travels must satisfy this condition from point to point, viz., that disturbances arriving at any point from a preceding point of a ray agree in phase. This condition will be satisfied if the time of journey down a ray and down all infinitesimally differing paths is the same.

The equation to a ray is therefore contained in the statement that the time taken by light to traverse it is a minimum; or

$\int _{\mathrm {A} }^{\mathrm {B} }{\frac {ds}{\mathrm {V} }}=$ minimum.

If the medium, instead of being stationary, is drifting with the velocity $v$, at angle $\theta$ to the ray, we must substitute for $\mathrm {V}$ the modified velocity $\mathrm {V} \cos \epsilon +v\cos \theta$, and so the function that has to be a minimum in order to give the path of a ray in a moving medium is

$\int _{\mathrm {A} }^{\mathrm {B} }{\frac {ds}{\mathrm {V} \left(\cos \epsilon +\alpha \cos \theta \right)}}=\int _{\mathrm {A} }^{\mathrm {B} }{\frac {\mathrm {V} \cos \epsilon -v\cos \theta }{\mathrm {V} ^{2}\left(1-\alpha ^{2}\right)}}ds=$ minimum.

Path of Ray, and Time of Journey, through an Irrotationally Moving Medium.

24. Writing a velocity-potential $\phi$ in the above equation to a ray, that is putting

$v\cos \theta ={\frac {\partial \phi }{\partial s}}$,

and ignoring possible variations in the minute correction factor $1-\alpha ^{2}$, between the points $\mathrm {A}$ and $\mathrm {B}$, it becomes

Time of journey $=\int _{\mathrm {A} }^{\mathrm {B} }{\frac {\cos \epsilon }{1-\alpha ^{2}}}\cdot {\frac {ds}{\mathrm {V} }}-{\frac {\phi _{\mathrm {B} }-\phi _{\Lambda }}{\mathrm {V} ^{3}\left(1-\alpha ^{2}\right)}}=$ minimum.

Now the second term depends only on end points, and therefore has no effect on path. The first term contains only the second power of aberration magnitude; and hence it has much the same value as if everything were stationary. A ray that was

1. [It has been objected that a bit of wave-front cannot be labelled, because of diffraction effects. This seems to me only a practical difficulty, and a more practical definition based upon preserved phase-con nexion follows a few lines later in the text; but the meaning conveyed by the convenient phrase “labelled disturbance” can equally well and I think unobjectionably be expressed by calling a ray the path of a definite, or identical, portion of energy—the direction of energy-flux.—July, 1893.]