524 Transactions of the Society. 



quired, and a further simplification may be effected by writing k 

 for the reciprocal of X. The telescopic formula — — is therefore 



expressed by its microscopic equivalent (the numerical 



value of k being 45,287). If now c be made equal to unity, for a 

 square aperture, 2 N.A. k is precisely identical with the formula of 

 Professor Abbe, based upon his spectrum theory, the numerical 

 value of which is given in the " Table of Eesolving Limits " at 

 the end of our Journal (see No. 5 in the following table). 



So long as the investigation proceeds upon the lines of the 

 spectrum theory, it matters not an iota whether the objective be' 

 square or round (both in the telescope and microscope with a square 

 aperture the line of separation must be placed parallel to a side of 

 the square), for that table of microscopical limits only holds good 

 when an objective is illuminated by a narrow pencil of extreme 

 obliquity, and only a small portion at the opposite ends of a 

 diameter of an object-glass is utilised — under which conditions the 

 resultant limit must be the same whether an objective is square or 

 round, and to integrate the action of interference over the circular 

 aperture would be labour wasted. 



An astronomer is consistent, for he says that the theoretical 

 limit for his telescope is so much, but that if it had a square aper- 

 ture, theory points out that it would be able to show him still finer 

 detail in an object ; but, for some unexplained reason, his telescope 

 with a circular aperture does in practice show him the detail theory 

 assigns to a square aperture. An Abbe theorist, on the other hand, 

 is wholly inconsistent, for he contends that, although the aperture 

 of his objective is circular, the theoretical limit for its resolving 

 power must be that for a square aperture. 



The experimental results and the theoretical values detailed 

 above are arranged in the following synoptic table, where each 

 telescopic experiment has its microscopic equivalent adjoined, and 

 vice versa. The telescope limit is given in seconds, which must be 

 divided by the aperture of the object-glass in inches. The micro- 

 scope limit is given in the " number of lines to the inch " that 

 an objective of N.A. 1 " will resolve with a full cone, and with 

 white light ; to adapt it to any other objective whose N.A. differs 

 from unity, it must be multiplied by the N.A. of that objective. 



The experiments with artificial double stars did not prove 

 satisfactory until the holes, which were *01 in. in diameter, were 

 removed to at least 200 yards from the telescope, so that the 

 holes themselves should subtend very small angles. The first dark 

 ring No. 2 was measured by removing an artificial double star from 

 a telescope until the dark rings made contact ; the distance of the 

 telescope from the star was then measured, and the separation of the 



