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

748 Definition of Ray.

23. In § 13 we defined a ray as the path of a labelled disturbance, 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.|undefined

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.|undefined

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 $= \int_{\mathrm{A}}^{\mathrm{B}} \frac{\cos\epsilon}{1 - \alpha^2} \cdot \frac{ds}{\mathrm{V}} - \frac{\phi_\Beta - \phi_\Lambda}{\mathrm{V}^3 \left(1 - \alpha^2 \right)} =$ minimum.|undefined

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