Page:MillerTheory.djvu/5

 which interests us. If we could measure the perpendicular distance between these wave-fronts at a sufficient distance from B, we should know the angle between them. But h" is only a fictitious line. What we cannot measure between h and h" we can measure between a and a, provided we can determine the point of intersection between a and a, and provided this be found in a convenient position. We have therefore to determine the point of intersection of a and a', knowing that of h and h".

The observing telescope is shown at T, fig. 5. Its axis is parallel to B II. We will show that the phase-difference of a and a' is constant at all points on any line parallel to the line B II, or to the axis of the telescope.

If we write, ', not for wave-lengths, but for the perpendicular distance between consecutive wave-fronts of the same phase, and , ' for the total aberration of the wave-fronts of the two systems, we have to show that $$\frac{\lambda'}{\cos\delta'}-\frac{\lambda}{\cos\delta}$$ is identically zero for eight specified equidistant azimuths, and is not greater than $$0.3\frac{v^{4}}{\mathrm{V}^{4}}$$ for other azimuths. Each of these quantities is determined by a complicated expression; and the equality specified can be most readily determined by trigonometric computation.

To prove the proposition, therefore, we will take that azimuth where, according to Dr. Hicks, the shifting of the intersection is a maximum, and we will assume the extreme case where the velocity of the apparatus is half that of light.

In fig. 6, the mirrors D, I, and II are accordingly supposed to move in the direction of the arrow. Let be the period of the waves of light incident on D; according to the previous specification, the angle between these wave-fronts and the plane of II is $$\sin^{-1}\frac{v}{\mathrm{V}}\cos\alpha$$; that is, they are parallel to II. Lay off on c d, the line of motion of a certain point of the mirror D, the positions of this point at the times 0,, 2, &c. Positions of D and of I and II at certain times are also noted in the same way: all numerical subscripts denote times. The source moves with the apparatus, and therefore, with the assumed ratio of velocities, the apparent wave-length of the light incident at D is half the wave-length in the case of rest, and is half the distance described by a wave-front in the unit of time. Let the initial position be one in which a wave-front passes through the given point in the mirror and through the point O in the line of motion. At the time $$t=\tau$$ the mirror is at 1, and the wave-front in