THE AID OF THE ACHROMATIC FRINGES. 33 



On inserting the values of the quantities as given above and reducing, an 

 equation identical with the above appears, which for small a. is 



n\ = 2ba (cosec /3 a+cot 0) 



If the prism has its nose at P (nearly), or in the axis of rotation, a small 

 correction is to be added to the preceding expression. The path-difference 

 on the right is increased by 



, i + cos (90 20) t 2 sin a 



cos a (cos a+sin a) cos a sin a 



and increased on the left by 



z 'I (cos a sin a) cos a 



Hence the correction is on reduction 



2b sin 2 a sin 0/sin 5 = 2&a 2 , nearly. 

 This merely wipes out the small middle term, a, of the above equation, 



leaving n\ = (2ba/sin 0) (i +cos 0) 



When the prism is reduced to reflectors in its plane of symmetry, as treated 

 at the beginning of this paragraph, the equation loses the terms w v and 

 reduces to 



n\ = x J t-y-\-z-\-z'-\-c' c, or to n\ = 2ba (i/sin /3-f-i) 



In the practical apparatus the mirrors m and n rotate on a fixed radius 

 b, whereas b in the diagram elongates on the right and contracts on the left 

 respectively to 



b" = b sin 0f sin 8 b' = b sin j3/sin a 



Hence the mirrors in the apparatus are displaced normally on the right and 

 left by 



e(b" b) cos j8/2, inward, and e' = (b b') cos 0/2, outward. 



The path-difference thus introduced is the sum of the decrease on the right 

 and increase on the left and its value is 2e cos i, when i is the angle of inci- 

 dence in question. Thus the correction is (after reduction) 



(b" -b} (cos a+cos (0-a)) + (6-b') (cos a+cos (0-f-a)) 

 The expression may be further reduced to 



260: cos sin . 



-(cos 0+cos a) 

 sin a sin 5 



when a is a small angle, or to 



260: (i/sin sin /3-f-cot 0) 



If this quantity is deducted from the above equation for path-difference and 

 direct rays there remains simply n\ = zba sin 0. The latter, therefore, is 

 the equation to be used in interpreting the observation. So that generally 

 when i = 0/2 for the micrometer at n 



2 cos i A./V/Aa: = 26 sin 



In the case of reversed rays the conditions on the left remain the same 

 as before. But on the right the mirror n is set at an angle 0/2 to the rail 

 3 



