10 MEASUREMENT OF ANGLES. 



reason we are often led to increase the distance from the mirror 

 to the scale to an inconvenient extent 



666. These difficulties may be lessened, and at the same time 

 the errors, due to small displacements of the axis during the experi- 

 ments, may be eliminated by using* two mirrors, directly opposite, 

 and two parallel scales. If p and </, p and d ', are the distances from 

 the axis of each of the mirrors and of the corresponding scale, and 

 if two observers note the deflections x and x' produced by the same 

 rotation, we shall have 



tan 2 = 



d- P 



The sum d+d' is the distance of the two scales which are fixed ; 

 p + p' is the distance of the two mirrors which are connected with 

 each other. This latter distance may be reduced to the thickness of 

 a glass plate silvered on both sides. 



667. LIMIT OF ACCURACY IN THE MIRROR METHOD. The 

 accuracy obtainable in the method of reflection depends simply on 

 the telescope and on the dimensions of the mirror, and it is often 

 wrongly thought that the accuracy may be improved by unduly 

 increasing the distance from the scale to the mirror. The advantage 

 of a great distance with a plane mirror and a rectilinear scale consists 

 especially in this, that the telescope can observe the ends of the 

 scale as well as the centre without change of sighting. On the other 

 hand, if the telescope has been constructed for long distances, a 

 small displacement of the eye-piece will allow the scale to be read. 

 These are minor points ; the principal question is the smallest angle 

 of rotation of the mirror which the telescope can estimate. 



Suppose the mirror at a distance d from the scale, and at a 

 distance D from the object glass of the telescope, and let /? be the 

 smallest angle which the telescope can distinguish. The image of 

 the scale being at a distance D + d from the object-glass, this limiting 

 angle corresponds to a length of scale e such that 



the corresponding deflection in the mirror is 



* W. WEBER and ZOLLNER. Berichte der K. S. Gesellschaft. Leipzig, 1880. 



