Page:MillerTheory.djvu/9

 angle of aberration of the second degree, which it is the object to detect.

We have shown that the wave-lengths of the two rays, when resolved in the direction in which alone they are used, are equal. One other point as to wave-lengths must be considered. We use wave-lengths to measure a length of less than 0.0002 mm., to determine the angle h B h". Is the scale of variable value? The light from a source moving with the apparatus has its wave-length modified by the motion. Dr. Hicks gives the formula for this effect in equation (4), page 17. If with this and the equation (2) we compute the wave-length resolved in the axis, at the azimuths where the effect is a maximum, and for the velocity ratio 100, the two minima are 0.9899995 L and 0.9899505 L, while the two maxima are both 0.9999500 L, where L is the wave-length in the case of no motion. For the ratio 10,000 these quantities differ from unity by about a hundredth part as much, and the inequality is negligible, even if we had to multiply this unit by a large number. But we have to do with only a fraction of the unit.

We next inquire as to the amount and the laws of aberration produced by reflexions from the mirrors of the apparatus. These can be developed in a series of powers of the velocity ratio, and of sines and cosines of the azimuth and of its multiples. But numerical estimates seem desirable, and the formulæ are such that these can more easily be obtained from trigonometrical computation. For the actual velocity ratio the computation is not easy, because trigonometrical tables of fifteen decimal places are not available. Imagine, then, three different apparatus, each of the dimensions proper to the special value of the velocity ratio for which it is specially designed. One apparatus, for the ratio 10, may have the length B II, fig. 3, equal to 10² L; another, for the ratio 100, may have the length 100² L; and the third, for the ratio 1000, may have the length 1000² L. What we can readily learn for the ratio 10, with seven place logarithms, will apply to the ratio 100, except for the circumstance that angles are not so small that sines and arcs are identical in value. What we compute for the ratio 100 with ten place logarithms tells us everything we desire to know for the ratio 1000 and for the actual ratio.

We have computed the aberrations of the two rays I and II, for certain azimuths with the velocity-ratios 10 and 1000, and for 18 azimuths of the apparatus with the velocity-ratio 100. From these aberrations we subtract that part of the aberration which is annulled by the motion of the telescope,