PHYSICS: C. BARUS 
15 
To obtain g it is sufficient to treat the similar triangles (3,8,9') and (9,8',9') 
where h = (9,4), h' = (3,8), k = (9,9'), I = (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 - I) sin (45° - 2a), q = (h - I) cos (45° - 2a). 
If these quantities are introduced into the above equation for n\ we may 
obtain after some reduction 
nk = 4 R sin a (cos a — sin a). 
Since = 2AN cos i, AA being the normal displacement of the mirror M f , 
and i = 45°, the corresponding equation to the second order of small quanti- 
ties a is 
AN/Aa = 2 R (cos a - sin o:)/cos i = lyjl R (1 - a - a 2 /2). 
If a is sufficiently small, the coefficient is simply 2 R/cos i as used heretofore. 
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 1, and this is equal to the other at 1, 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° — 2a. The reflecting 
sides of the plates are silvered. Hence e (sin i — cos i tan r) 2Aa must be 
added to the equation. 
2. Rotating Doublet. — The second case, figure 2, in which the auxiliary mir- 
ror of the preceding apparatus is omitted is, curiously enough, inherently sim- 
pler. M,M', N,N', are mirrors, half silvered at (1) and (3) and the two latter 
on a vertical axis A and rigidly joined by the rail (2,3). The mirrors being 
preferably at 45°, the component rays are 1,2,3,T and 1,5,3, T, the mirror M' 
being on a micrometer with the screw normal to the face. The ray parallelo- 
gram is made up as before of (1,2) = b = (3,5) and (1,5) = 2 R = (2,3). 
When the rail (2,3) is rotated over an angle a, the mirrors take the position 
Ni and Ni 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 
(1,2,2', 6,T 2 ) and (1,5,4,7^). The rays T\ and T% are parallel and interfere 
in the telescope. Hence the path difference introduced by rotation is (n 
being the order of interference) 
rik = b + R tan a + (2 R/cos a) cos a - (2 R + b - R tan a) = 2 R tan a, 
for the triangle (a,7,2') is isosceles and its acute angles each a. 
The rays Ti and T 2 have now separated and the amount (4,6) is also 
2 R tan a. When this exceeds a few millimeters the interferences vanish. 
A correction must however be applied, since in the practical apparatus the 
mirrors rotate at a fixed distance apart. Hence the mirror iVi must be dis- 
placed toward the right (shortening the path) by the normal distance 
e (R/cos a - R) cos 45° 
and the mirror AY toward the left by the same amount. The path difference 
