Page:MillerTheory.djvu/6

 question cuts the line of motion in 2, and intersects the mirror in e. The wave-front reflected from D at $$t=0$$ will have reached the point f1, and the tangent ef1 establishes the reflected wave-front. At the times 2, 3, &c., this particular disturbance will be found at f2, f3, f4, &c.

When D is at a position 2, a new disturbance will have been established at g, which, at the time $$t=5\tau$$, will be found at g5. In the same way, h5, i5, k5 will have arranged themselves in the line f5 k5. At the time $$t=10\tau$$ the six wave-fronts will have been reflected from I, and will be placed along the line o10 f10. The angle f10 O I o is equal to the aberration of the wave-fronts after reflexion from D. As, at this azimuth, the angles of incidence and of reflexion at I are equal, this angle is also the aberration of the emergent rays.

Part of the wave-front f, indicated by f, will be transmitted through the mirror D. It will overtake the mirror II at the time $$t=8\tau$$, when II will have position marked II8, fig. 7. Returning after reflexion, it will take the position noted for the times $$t=9\tau$$, $$t=10\tau$$; and meeting the mirror D at $$t=10\tfrac{2}{3}\tau$$, it will be reflected as shown f11, at an angle whose tangent is given by the formula below. The following wave-front g' will be reflected one period later, at II9, and it is shown in several positions. The wave-front f, belonging to the other system, having passed through the mirror, and having reached the line, Sd, at $$t=10\tau$$, is shown at f11.

In fig. 8 is shown the position of the wave-fronts below the mirror D for the time $$t=15\tau$$. f15, and f'15 have moved along the paths indicated, while the other wave-fronts have moved in a corresponding manner, their position at the time $$t=15\tau$$ being as shown in the figure. The wave-fronts of the unaccented system are placed on the line op; the aberration of the system is equal to the angle. The wave-fronts of the accented system are placed on the line qr; the corresponding aberration is the angle '; the line T15q being the position of the axis of the observing telescope at the time $$t=15\tau$$. Produce the planes of the wave-fronts, draw line l, l', parallel to qT15, the axis of the telescope, each terminated by the planes of consecutive wave-fronts. Their lengths are $$l=\frac{\lambda}{\cos\delta},\ l'=\frac{\lambda'}{\cos\delta'}$$. It is to be proved that $$l=l'$$.

Putting for the angle of incidence and ' for the angle of reflexion, we have for both reflexions at D, =45°. For this azimuth there is no change of angles at I and II. The