of Methods for Observing Small Rotations. 91 



about 20 for each inch of aperture of the object-glass or 

 mirror"*. The attainable accuracy in this case, therefore, 

 depends very largely on the brightness of the object whose 

 position is to be determined, and in the great majority of 

 cases this is not sufficiently great to enable the limit indicated 

 above to be reached with either the microscope or mirror and 

 telescopet- Under the most favourable conditions as regards 

 illumination, &c, the practical limit of accuracy seems to be 

 about the same in this case as in the preceding, i. c, about 

 A the limit of resolution for the microscope, and j 1 ^ to -^ 

 that limit for the mirror. 



From a review of the preceding cases, which cover all 

 those usually met with in practice, we see that the mirror 

 has a somewhat greater efficiency with respect to the micro- 

 scope than is indicated by Rayleigh's comparison. The limit 

 of accuracy in the case of the microscope is at most 1 

 the limit of theoretical resolution given by (2) ; in the case 

 of the mirror it is (in the last two cases) from A to A, the 

 limit given by (1). This higher efficiency is largely due to 

 the fact that the most favourable conditions are far more 

 easily attained with the mirror and telescope than with the 

 microscope. Indeed, the limit of resolution indicated by (2) 

 for the latter supposes that the angular aperture of the micro- 

 scope objective is 180°, a condition never fulfilled in practice. 

 If we assume the greatest practicable aperture that can be 

 used for the purpose under consideration to be 90°, the re- 

 solution will be less than the limit indicated in the proportion 

 of 7 to 10. Again, it is far more difficult to secure the 

 requisite degree of illumination with the microscope than 

 with the mirror ; and this will still further decrease its 

 measuring power in comparison with the latter. 



In general, I think it would be safe to assume from both 

 the preceding theoretical considerations and from the results 

 of experiment, that in order to make the microscope and 

 pointer method as accurate as the mirror method for observ- 

 ing small angular movements, the length of the pointer would 

 have to be at least two to three times the diameter of the 

 mirror. The greater convenience and efficiency in other 

 directions of the first method makes it, as stated in the first 

 part of the paper, far more generally useful. 



To these two methods of measuring angular motions we 



* Phil. Mag. Oct. 1879, p. 263. 



t It is, however, readily reached in the case of the interferometer. 

 because with this instrument the width of the fringes may ba ma le as 

 great as we please without diminishing their brightness. 



