﻿Fixed- Arm Spectroscopes. 347 



2sin0/2 r . ,,/n . 7i ^+acosa 



jr- |asin 0/2 +t> 



cos 6 L cos a 



11 = ~ tan « H- tan ~ ' 



_ 2 sin 0/2 cos a [a sin 0/2 4- &] — (d 4 a cos a) cos 



sin (a 4 0) _ ' 



rt . «, ft - . a /ft . 7-1, a 7 4 a cos a, 



2 sin 0/2 [a sin 0/2 4 />J tan a ■ tan 



/ 



COS a 



^ tan a 4 tan 



2 sin 0/2 sin a [a sin 0/2 + 6] — (d + a cos a) sin 

 = _ sin (a+01 - """ 



The equation of the reflected ray CD will be 

 ?y ~y = tan (04 2a)(a:—a?), 



and the length of the perpendicular let fall from the origin 

 will be 



tan (0 + 2ay-y = 

 P Vl + tan 2 (0 4-2«) * V ' * V ;? 



Substituting the values of a/ and ?/, reducing, and introducing 

 the relations 



+ 2 a = £ = constant, 



+ « = (£-«), 

 we finally obtain 



p = 2a cos 2 /3/2 + 26 sin 0/2 + 2d cos (« + 0). 



The term 2a cos 2 /3/2 is a constant, but the last two terms are 

 variable. It will be seen at once that we may make p a 

 constant, viz., prevent any lateral shifting of the ray, by 

 simply making both b and d equal to zero. In other words, if 

 we simply fulfil the condition that the axis of rotation of the 

 system shall be at the intersection of the plane bisecting the 

 refracting angle of the prism with the plane of the reflecting 

 mirror, there will be no lateral displacement of the ray which 

 passes through the system at minimum deviation, for different 

 values of 6, viz. for different wave-lengths. There is one 

 case in which this does not hold, viz., when these two planes 

 are parallel. Then we have 



a = 9O°-0/2, ora + 0=18O°-«. 



Hence 



6 sin 0/2 + d cos (a + 0) = (b—d) cos a ; 



and therefore we may in this case make p constant by making 

 b — d : in other words, by making the two planes coincident. 

 Then the second half of the prism becomes useless, and we 



