﻿Fixed-Arm Spectroscopes. 343 



parts is essentially that employed in the first case described, 

 except that a concave mirror is now used instead of the lens. 

 The calculation of the lateral displacement, although not quite 

 so simple as in the first case, is easy. 

 In fig. 2, 



let d = distance o a from axis of rotation to the face of 

 the concave mirror ; 

 r = perpendicular distance o b from the axis to the 



plane of the reflecting-mirror ; 

 <w = angle oac ; 

 6 and a have the same meanings as before. 



When in adjustment the line of collimation passes through 

 the axis of the rotation, and the deflected ray o a will fall on 

 the centre of the concave mirror. Then we have from ana- 

 lytical geometry the following equations : — 

 For the line a c, 



y = tan (0-f ft)) [%— dcosd] +dsin# 

 = — tan2a[# — d cos 0] + d sin 0, ... (2) 

 since d + co = 180°-2*; 



and for the line d c, 



?/=- tan *(%)+—. (3) 



J w cos a v ' 



The lateral displacement (Ay) of the ray will be the ordi- 

 nate of the point of intersection of these two lines. Solving 

 (2) and (3) for y we have 



Ay = d sin (2a + 0) — 2r cos a. 



If the system be turned through a small angle e, 6 becomes 

 + e, and « becomes a — e/2. Hence + 2a= const., and 



Ay' = d sin {6 + tot) - 2r cos (a - e/2) , 



Ay — Ay' = 2r [cos a — cos (a — e/2)] s r cos (6 + to) sin e ; 



or the lateral displacement is directly proportional to r, to the 

 cosine of the angle of deviation, and to the angle of dis- 

 placement. Since e is fixed by the prism used, and co must 

 be small in order to secure good definition, the only way in 

 which the error of displacement may be reduced is by redu- 

 cing r ; hence the object in placing the mirror just as close 

 to the prism as possible. In the actual case, the values of r, 

 0, and ft) were 



rSlO centim., 0s5O° (for D lines), and ft)s3°. 



Hence for e = 3° we have Ay' = 3 millim., and the angular 



error is, as before, 3/R= n^K =3' nearly. 



