42 DISPLACEMENT INTERFEROMETRY BY 



If the angle SnP is /3 and Pnp is 7, the values of the branch paths may be 

 found to be (since nP = mP = b), if /3 a = d and 7 a = r, 



wp = w = 6/cos 7 



wn' = b sin a/sin 5 



nq =b sin /3/ sin 5 sin T 



_ ^ f sin a sin /3 sin r+sin 2 7 sin 6 cos r 1 



sin o cos T cos 7 [ sin /3 sin 7 sin r cos r j 



Hence the path-difference is equivalent (after some reduction) to the equation 



% _ & [sin a (cos 7 cos r + sin fi sin r ) + sin /3 cos 7 



sin 5 cos T cos 7 j cos 2 7 cos r sin 5 sin /3 sin 7 sin r cos rj 



If a = o, then /3 = 5, 7 = ^, and the right-hand member becomes zero, as it 

 should. I have not succeeded in putting this equation in a much more con- 

 venient form. 



If a is very small, so that differential expressions may be introduced, the 

 rigorous equation, to an approximation of the second order in a, may be 



reduced to / f r,\\ to \ 



cos 7(i-f-cos (p 7); i+cos (p y) 

 n\ = ba - = ba 



sin 5 cos r sin j8 



If /3 is nearly 90 and if the distance Pp is p, then 



n\ = ba (i-j-sin 7) =ba-\-pa cos 7 



The same expression may be obtained geometrically by prolonging n'p and 

 T'q to m'. The triangle n'qm' is isosceles. Hence if we draw the wave- 

 front Pq', normal to pT, we may deduct the common length pq' from both 

 rays, or add it to both. Hence the path on the right will be x+b' cos (y a) 

 -\-p sin 7, where b f is the line Pn' and on the left as before, 6/cos 7. Hence 

 (if b = p/tan 7) the path-difference is 



/sin a b' i \ 



b I ; h~cos (7 a)+tan 7 sin 7 I 



\ sin 5 b cos 7/ 



which, as above, also reduces to 



ba. (i+cos (18-7)) / sin /3 



There is a source of discrepancy which enters when the face PP' does not 

 pass through the axis P, but is eccentric. In such a case, if e is the distance 

 of the plate from the axis, a correction equivalent to e (i cos a) cos 7 = 

 e 2 7, nearly, will have to be supplied. Again, if there is lack of symmetry 

 from such a cause, the base-lines will be b and b' and the angles 7 and 7', 

 so that a modified equation is suggested. Finally, from all these expressions 

 the changes of glass path on the left with a, if not compensated, must be 

 deducted. As the method admits of a good achromatic phenomenon of 

 reversed slit-images, it is theoretically interesting and I have given it some 

 study. 



For the case where the ray Sn prolonged returns on itself, as from n" in fig- 



