1902.] A Portable Telemeter, or Rctnge-fincler. 353 



From (3) we have 



cos (9 + 89) - cos = - sin cos <j> . i 

 so that 86 = i . cos <j> (4). 



Now the first prism deflects the line (9 h fa) or (#/, fa') the direc- 

 tion of the object, to (9 2i fa) or (9 2 , fa') the direction of the image, 

 where 9% = 9 L , and the second prism deflects this line back to (9$, fa'), 

 where 8 S ' = 8 2 . Also, since the angle between reflecting surfaces is 

 equal in the pair of prisms, and generally taken equal to we 

 have 



fa' ~ fa' = fa- fa = \* (5). 



Now the deflection of the image vertically, in the plane through 

 the axis of the second prism, after reflection in the two prisms, is 



9 B ' - 9{ = 9-1 - 9' = (9 2 ' - 9 Y ) - (9{ - 9{) 

 = (9 2 '-9 2 )^(9 1 '-8 1 ) = 89 2 -89 1 . 



Therefore vertical displacement = i (cos fa - cos fa) (6). 



Also from (2) 



. / , f,> • , , sin 9 . , sin 9 . . 



sm (<£ + 6 fa = sm^> = - — -Q- . sm = - . a . sin <p 



sin 9' sm 9 + 89 cos 9 



= (1-89 cos 9) sin fa 



and therefore by (4) 8<f> = -50 cos tan = - i cos 9 sin <£. 



The error of observation is the angular deflection of the image 

 horizontally, supposed to be in a plane perpendicular to the axis of the 

 second prism, after reflection in the two prisms, and is 



fa' - fa' = [by (5)] (fa' -fa + fa) - fa' 



= 8 fa - 8 fa = - i cos 9 (sin fa - sin fa) 



or error = - i . a (sin fa - sin fa) (7). 



Now, translating there, we see from (6) that the vertical displace- 

 ment of the image by one-half of the base is 



i (cos fa - cos fa), 



which is clearly a maximum for a fixed value of i, if fa- fa = Jtt> 

 when 



<£ 2 = f 7T, and fa = \tt. 



This occurs when the plane cutting one prism symmetrically 

 through its axis is parallel also to the axis of the other prism ; and, if 

 I be the greatest possible inclination of the axes of the two prisms to 

 each other, through bad adjustment, then the vertical displacement 

 cannot exceed 



J2 . I for one half-base and 2^/2.1 for the whole base. 



