of Methods for Observing Small Rotations. 87 



in every respect to the first save that it is broadened in the 

 inverse ratio of the aperture of the eye to the aperture of the 

 mirror or objective ; the only effect of interposing the latter* 

 being to change the apparent magnitude of an object of finite 

 dimensions or the apparent brightness of point sources (such 

 as stars). The retinal image of the reference-line or cross- 

 wire will also be a diffraction pattern which will be similar to 

 that of the line under examination. These two patterns will 

 be of the same width when the angular width of the line itself 

 as viewed from the objective (or mirror) is the same as the 

 angular diameter of the cross-wire as viewed from the eye- 

 lens, and when the aperture of this latter is just sufficient to 

 admit the full cone of light from the first aperture. Under 

 these circumstances the conditions are exactly the same as 

 they would be if we were looking through any instrument at 

 two parallel lines of the same width, except that when the 

 observing instrument is moved only one of the images at the 

 focal plane changes its position with reference to the axis of 

 the instrument. In considering the limiting degree of 

 accuracy with which the position of the (apparently) moving 

 image can be determined with reference to the other we must 

 distinguish between at least three cases. 



I. Suppose both reference-line and object are of the same 

 intensity (both light or both dark) and are initially super- 

 posed. It is obvious that no motion of one image with 

 respect to the other will be evident until the separation is at 

 least as great as that required for the " resolution - " of a double 

 line. For lines of negligible width viewed through a rect- 

 angular aperture this separation (angular) is, as is well 

 known f , -, 



b 



In this case therefore the smallest angular movements ob- 

 servable with a mirror and with a microscope and pointer 

 are given by expressions (1) and (3) as assumed by Rayleigh. 

 For lines of finite width the smallest angular separation at 

 which resolution occurs is given by the expression $ 



.9. 



% = <7 + 



2cr + a' 



* Provided only the angular aperture of the eyepiece lens (aud the 

 pupil of the eye) is the same viewed from the focal plane as that of 

 the objective or image-forming lens ; and provided also the magnifying 

 power of the eyepiece be not too high (see paper by Helmholtz already 

 referred to, Pogg. Ann. 1874). 



t See Rayleigh, " Wave Theory," Enc. Brit. vol. xxiv. 



% See paper " On the Resolving Power of Telescopes and Spectro- 

 scopes for Lines of Finite Width," Phil. Mag. vol. xliii. p. 317, May 1897. 



