THE AID OF THE ACHROMATIC FRINGES. 9 



so that the sensitiveness is from (6), 



f x 



(7) w = (AAO = 



(a) It will next be desirable to deduce the above fundamental equations 

 more rigorously than has thus far been done. Figure 2 is supplied for this 

 purpose, and represents the more sensitive case where, in addition to the 

 mirrors MM,' NN' (all but M being necessarily half-silvers), there is an 

 auxiliary mirror, mm, capable of rotation (angle a) about a vertical axis a. 

 The mirrors, M ---- N', in their original position, are 

 conveniently at 45 to the rays of light, while mm is <m'm 

 normal to them. Light arriving at L is thus separated 

 by the half-silver Af at i, into the two components 

 i, 2, i, 9, 3, T and i, 6, 7, 6, 3, T, interfering in the 

 telescope at T. 



When mm is rotated over a small angle a, these paths 

 are modified to i, 2, 2', 4, 4', 5, T 2 and i, 6, 7, 8', 2 C" 4~ 

 TI and T 2 enter the telescope in parallel and produce 



interferences visible in the principal focal plane, provided the rays TI 

 and T 2 are not too far apart, in practice not more than i or 2 mm. Inter- 

 ference fringes therefore will always disappear if the angle a is excessive, but 

 the limits are adequately wide for all purposes. The essential constants 

 of the apparatus are to be: 



(9, i) = (6, 3 )=& (i, 2) = (6, 7 )=C (9, 3) = (i, 6) = (2, 7)-* * 



.R being the radius of rotation. 



Where the mirror mm is rotated to m'm' over the angle a the new upper 

 path will be: 



C+R tan 



where (2' 4)=^, (4, 4')=^, (4/5)=g, the plane (8, 5) =5 normal to 7"i, and 

 TZ being the final wave-front. The lower path is similarly 



2 R + (C-R tan a)+d' 



to the same wave-front (8, 5), where (?', 8)=d'. Hence (apart from glass 

 paths, which are preferably treated separately) the path-difference n\ (n 

 being the order of interference) should be 



n\ = 2 R (tan a i)-\-d-d'-\-e +g 

 The figure, in view of the laws of reflection, then gives us in succession 



d =(b-\-c-\-R tan a) /(cos 2 a+sin 2 a) 



d' (b+cR tan a) /(cos 2 a+sin 2 a) 



e =2 R /(cos 2 a+sin 2 a) 



g =2 7? sin 2a (i+ tan a) (cos 2 a sin 2 a)/(cos 2 a + sin 2 a) 



q =2 R sin 2 a (i+tan a) 



