30 DISPLACEMENT INTERFEROMETRY BY 



not been disturbed. This is the interesting feature of the method, for the 

 angle a between the two positions of the rigid beam will also be the angle 

 between all corresponding normals of the mirrors, as indicated in the diagram. 

 If we take the case on the left, the angle between incident and reflected ray 

 at P will therefore be ^2+40: for the original mirror at P and TT/ 2 -\-4a- 2 

 a = 7r/2 + 2Q! for the new position at PI. But the angle between the rays re- 

 flected at m and m' respectively as 2 a. Hence if T\P\ is prolonged backwards 

 it must intersect the line mn at the original angle \ and thus P\Ti is paral- 

 lel to PT, Similar reasoning applies on the other side for PiTi and will still 

 hold if the direction of the ray Sn prolonged is reversed. Finally, ir/2 may 

 be any reasonable angle. 



It will contribute to a more adaptable design of the apparatus for general 

 interferometry if the ray Sn' may also be reversed by reflection (fig. 15, 

 mirror n") in parallel to itself, allowing a small lateral offset, similar on both 

 sides for clearance of the mirrors. Reflection between fixed parallel mirrors 

 on the left in d and between mirrors set at a reentrant right angle on the 

 right, say at n" ', would accomplish this at corresponding distances for the 

 transverse rays. Again, half -silvers may be used at m and n for reflection, 

 which method is probably best. These details will here be disregarded. 

 If small angles are to be measured the direct method is enormously more 

 sensitive. 



16. Estimate. The full expression for the path-difference corresponding 

 to the rotation of rail a will be complicated and of no interest here. It is not 

 sufficient to regard the intercepts y and x as solely contributing to the path- 

 difference, which would therefore be x-\-y for the direct case and x y for 

 the case when the ray d' is reversed somewhere at n" (fig. 15) and returned 

 parallel to itself. It may be shown that for small angles a, if /3 is the angle 

 between incident and reflected rays originally at m and n and b the distance 

 mP = Pn, d the distance Sm = Sn, 



are sufficiently approximate equations up to the squares of small quantities 

 to meet the interference for the direct and the reversed cases respectively. 

 Hence, if for instance a=i =0.0175; b=i meter = io 2 cm.; d= i kilom. = io 5 

 cm; X = 6X io~ 6 cm., the number of fringes corresponding to each of the terms 

 may be computed as 



(Direct) w = 6Xio 4 io 3 +6o (Reversed) w = io 3 60 



In the first case over 61,000 fringes pass per degree of rotation, a= i. This 

 makes about 2. 9 Xio~ 7 or about 0.06 second of arc per fringe. But the 

 method is insensitive as regards distances d, unless the first two terms can be 

 compensated. In the second or reversed-ray case, the method would be 

 relatively much more sensitive as regards d if the first term 2ba z could be 

 compensated. The difficulty lies in the occurrence of 2 in the term, 

 whereas most compensators would act as the first power of a. 



