LIMIT OF ACCURACY IN THE MIRROR METHOD. II 



which gives 



D 



zd 2 \ d 



In ordinary conditions in which d= D, we get = /3; the delicacy, 

 whatever be the distance of the scale and the telescope, is exactly the 

 same as if the telescope were mounted directly on the movable part. 

 The apparent advantage obtained by doubling the displacement by 

 reflection is compensated by the double distance of the image of the 

 scale. The sensitiveness increases when the scale is placed further 



than the telescope; it would even be doubled if the ratio were 



d 



very small, but in that case it would be necessary to have scales of 

 great dimensions, for which it would be difficult to get a good 

 illumination. 



This reasoning always pre-supposes that the optical power of the ' 

 telescope is entirely utilised that is to say, that the pencil of rays 

 which proceed from a point of the scale covers the whole of the 

 object-glass, or, at any rate, the horizontal diameter at right angles 

 to the lines. If A is the diameter of the object-glass, a the hori- 

 zontal dimensions of the mirror, we must have 



The least size of the mirror is thus 



Ad 



U + d' 



V 



it is half the diameter of the object-glass for the ordinary arrange- 

 ment, and is equal to this diameter itself when the scale is at a 

 great distance. 



It must be added that the construction of plane surfaces presents 

 the greatest difficulties ; all defects in the mirror injure the purity 

 of images, and contribute to diminish the accuracy of the obser- 

 vations. 



When it is merely proposed to observe the divisions of a scale, 

 it is sufficient if the mirror has the dimensions defined by the 

 preceding equation, in the direction, at right angles to the lines. 

 The sharpness of the image may, moreover, be greatly improved 

 by covering the object-glass with a diaphragm, in which is a 



