32 DISPLACEMENT INTERFEROMETRY BY 



is now n'rs and from m', m'P\ before they meet in the common wave-front 

 P\qs. Hence the intercepts 



rs = v = (2+2') (cos a sin a)/(sin a-f cos a) 

 rP 2 = w = (0-fV)/ cos a (sin +cos a) 



will enter in treating the path-differences. On the left the rays have not 

 been distributed. 



If we take the direct case first the original path-difference SnP and SmP 

 may be regarded zero or n and m in the same phase. On rotation, therefore 

 (angle a), the path-difference is increased on the right by x-\-c f w-\-v and 

 increased on the left by y+c, so that the total path-difference is equiva- 

 lent to the equation 



riK = c' c(w v)-\-x+y 



If the above equivalents are inserted, this equation may be reduced to 



sin /3 cos /3/cos a sin 2 /3 sin a + sin /3 cos a 

 n\ = 2b sin a- 



sm (p-\-a) sin (pa) 



in which the three terms in the numerator correspond to the respective 

 intercepts c' c, w v, x-\-y. 



Since a and /3 are small angles, we may write sin a = a, cos a=i a 2 / 2 . 

 and cos = b/d. Therefore the equation would, for practical purposes, 

 become 



n\ = 2ba 2ba z -\-2b z a/d 



the three terms corresponding to the xy, wv, and cc' effect. 



In the case of reversed ray (fig. 2) we may consider the points m' and 

 n' in the same phase. Hence the original path-difference ( = o) is x y. 

 The path-difference after rotation c' w-\-v c. The total change of path- 

 difference due to rotation is thus given by 



n\ = c' c (w v) x-\-y 



This differs from the preceding by the deduction of 2*. The rays again ter- 

 minate in the common wave-front P\qs to enter the telescope. Hence after 

 reduction 



sin /3 cos /3/cos a sin 2 8 sin a sin a cos /3 

 n\ = 20 sin cc - 



sin (j3-f a) sin (/3 a) 



the terms showing the cc', wv, and xy effects. The approximate form of this 

 equation is thus practically 



The wv effect predominates, the cc' effect is intermediate, and the xy effect 

 very small if d is large, as already instanced. 



The preceding equations may also be obtained geometrically by letting 

 fall the normal from n (fig. 14) to the prism-mirror and prolonging the ray 

 at s backward. In the isosceles triangle so formed the angle at the base is 

 45 a. Hence in the above notation the path-difference takes the form 



x+2 (c'w)cos 2 (45 a) (2' 2) (c y) 



