10 DISPLACEMENT INTERFEROMETRY BY 



To obtain g it is sufficient to treat the similar triangles (3, 8, 9') and (9, 8', 

 9'), where ^ = (9,4), ^' = (3,8), =(9,9'), =(9,8') may be found in succession, 

 as the normal distance between the mirrors M and M' is R \/2 , so that finally 



g=(h-l) sin (45 2 a) q = (h-l) cos (45 2a) 



If these quantities are introduced into the above equation for n\, we may 

 obtain, after some reduction, 



wX = 4-R sin a (cos a sin a) 



Since n\ = 2 AN cos i, AN being the normal displacement of the mirror 

 M' and 4 = 45, the corresponding equation to the second order of small quan- 

 tities, a, is 



AN zR 



~= ~ r (cos ex sin a) = z\f zR(i a c? z) 

 Aa COS I 



If a. is sufficiently small, the coefficient is simply 2.R/cos i as used above. 

 There remain the glass paths which for the rays d and d' are compensated. 

 Additionally the upper ray has a glass path (3) displaced to (4'), The lower 

 ray has the fixed path at (i), and this is equal to the other at (i), since the 

 angles are 45. Thus the variable part of the glass paths at (3) to (4') is 

 uncompensated and the angle of incidence changes from 45 to 45 2 a. The 

 reflecting sides of the plates are silvered. Hence 



e (sin i cos * tan r)z^a = -\/z (i tan r)e 



must be added to the equation. 



(b) The second case, figure 3, in which the auxiliary mirror of the preceding 

 apparatus is omitted, is curiously enough inherently simpler. MM', NN' 

 are mirrors (half -silvered at (i) and (3), and the two latter on a verti- 

 cal axis a, and rigidly joined by the rail (2, 3). The mirrors ^/ Jf # 

 being preferably at 45, the component rays are i, 2, 3, T // i 

 and i, 5, 3, T, the mirror M' being on a micrometer with 

 the screw normal to the face. The ray parallelogram is made 

 up as before of (i, 2) = b= (3, 5) and (i, 5) = 2^ = (2, 3). 

 When the rail (2, 3) is rotated over an angle a, the mirrors 

 take the position A/i and N'i, at an angle a to their prior 

 position, and the angle of incidence is now 45 a. The new paths, if (4, 6) 

 is the final wave-front, are thus (i, 2, 2', 6, T 2 ) and (i, 5, 4, 7\). The rays 

 Ti and TZ are parallel and interfere in the telescope. Hence the path- 

 difference introduced by rotation is (n being the order of interference) 



zR 



w\ = 6-f-.Rtana+ - -cos a (zR+b R tan a) 

 cos a 



or 



n\ = zR tan a 



for the triangle (a, 7, 2') is isosceles and its acute angles each a. 



The rays T\ and T% have now separated and the amount (4, 6) is also 

 zR tan a. When this exceeds a few millimeters the interferences vanish. 



