THE AID OF THE ACHROMATIC FRINGES. 43 



ure 23, the mirror n being a half -silvered plate, the quantity nw" = 2&a/sin <p 

 must be deducted. This makes the first term negative, so that the equation 

 for reversed rays is, apart from sign, 



i cos (0 a} 

 n\ = ba 

 sm <p 



One may notice that when 7 = these expressions coincide with the equa- 

 tions for the prism-prism methods above, except for the factor 2. 



It is now necessary to apply the correction for the occurrence of a constant 

 radius of rotation, whereby the mirror n is both rotated and displaced. 

 This correction is, for a normal displacement e and an angle of incidence i, 

 2 e cos i. If the distances Pn b and Pn' = b" 



e=(b"-b} sin ( 9 o -(/3- 7 )/2) = (&"-&) cos (pfi/2 



The angle of incidence is now (/3+7)/2 a, so that the path-difference in 



question becomes 



2 (&"-&) cos(/3-7)/2.cos ((,3+7)/2 -a) =(&"-&) (cos(j3-a)+cos (7-")) 



Since b" = b sin 0/sin (/3 a) if a is a small angle, this may be written (6" 

 b) = ab cos /3/sin 5. Hence the correction is 



cos 2 /3+cos cos 7 

 ba - 



sm j8 



Subtracting this from the first equivalent of n\, and reducing, the equation 

 to be used for non-reversed rays becomes \n = ba (sin /3 + sin 7) when a. is small. 

 Except for the factor 2 this result is again identical with the above for two 

 prisms, if 7 = 0. As the angle of incidence at the micrometer at n is i = 

 (/3+7)/2, and if AN is the displacement 



2 AN cos (/3-f-7)/2 =b (sin /3+sin 7) Act 



For reversed rays, e = (b" b} sin (/3 7)/2 at once, and the path-difference 

 is then for small a 



2ba cot j8 sin -cos (90 -- -+a) =bacot /3 (cos (7 a) cos (/3 a)) 

 2 2 



since the angle of incidence is now 90 (/3 + 7) / 2 + a . Subtracting this path- 

 difference from the above equivalent of n\, the practical equation for reversed 

 rays becomes, after reducing, 



n\ = ba (sin /3 sin 7) 

 and since * = o, then 



. ,, ,, . _ . 

 2AA/=o(sin j8 sin 



Both equations contain the distance d of a remote object in sin /3. 



23. Observations. In the first experiments a sharp-angled prism was placed 

 at S, reflecting the two rays d and d r of white light from a collimator beyond. 

 The method of figures 22 and 23 with a micrometer at n was first used, the 

 screw being in the direction of the bisectrix of the angle 18+7. The collimator 

 is convenient, as otherwise the coincident slit-images are liable to separate 



