detect Aberrations of the Second Degree 677 



angle of aberration of the second degree, which it is the 

 object lo 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 con- 

 idered. We use wave-lengths to measure a length of less 

 than O0002 mm., to determine the angle h B It". 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 equa- 

 tion (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 O9899o05 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 

 a- 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 aberra- 

 tion produced by reflexions from the mirrors of the apparatus. 

 These can be developed in a series of powers of the velocity 

 ratio, and of .-ines and cosines of the azimuth and of its 

 multiples. But numerical estimates seem desirable, and the 

 formulas 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 10 2 L; another, for the ratio 

 Inn. may have the length 100" L ; and the third, for the 

 ratio 1000, may have the length 1000 2 L. What we can 

 readily learn for the ratio 10, with <^\en place logarithms, 

 will apply to the ratio 100, except for the circumstance that 

 angles are nut -o 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 

 11. for certain azimuths with the velocity-ratios 10 and 1000, 

 and for 18 azimuth- of the apparatus with the velocitv-ratio 

 Inn. From th<-se aberration- wo subtract that part of the 

 aberration which is annulled by the motion of the telescope, 



