448 Transactions of the Society. 



The limit can be depressed only by a diminution in X , such as 

 photography makes possible, or by an increase in fi, the refractive 

 index of the medium in which the object is situated. 



This method, in which the object is considered point by point, 

 seems the most straight-forward, and to a great extent it solves 

 the problem without more ado. When the representative disks 

 are thoroughly clear of one another, the two points in which they 

 originate are resolved, and on the other hand, when the disks 

 overlap the points are not distinctly separated. Open questions 

 can relate only to intermediate cases of partial overlapping and 

 various degrees of resolution. In these cases (as has been insisted 

 upon by Dr. Stoney) we have to consider the relative phases of the 

 overlapping lights before we can arrive at a complete conclusion. 



If the various points of the object are self-luminous, there is 

 no permanent phase-relation between the lights of the overlapping 

 disks, and the resultant illumination is arrived at by simple 

 addition of separate intensities. This is the situation of affairs in 

 the ordinary use of a telescope, whether the object be a double 

 star, the disk of the sun, the disk of the moon, or a terrestrial 

 body. The distribution of light in the image of a double point, or 

 of a double line, was especially considered in a former paper,* and 

 we shall turn to the subject later. 



When, as sometimes happens in the use of the telescope, and 

 more frequently in the use of the microscope, the overlapping 

 lights have permanent phase-relations, these intermediate cases 

 require a further treatment ; and this is a matter of some import- 

 ance as involving the behaviour of the instrument in respect to 

 the finest detail which it is capable of rendering. We shall see that 

 the image of a double point under various conditions can be 

 delineated without difficulty. 



In the earliest paper by Prof. Abbe f which somewhat pre- 

 ceded that of Helmholtz, similar conclusions were reached ; but 

 the demonstrations were deferred, and, indeed, they do not appear 

 ever to have been set forth in a systematic manner. Although 

 some of the positions then taken up, as for example that the larger 

 features and the finer structure of a microscopic object are de- 

 lineated by different processes, have since had to be abandoned,! 

 the publication of this paper marks a great advance, and has con- 

 tributed powerfully to the modern development of the microscope. § 



* 'Investigations in Optics, with special reference to the Spectroscope,' Phil. 

 Mag., vol. viii. p. 266 (1879). t Arch. f. Mikr. Anat., vol. ix. p. 413 (1873). 



X Dallinger's edition of Carpenter's ' Microscope,' p. 64, 1891. 



§ It would seem that the present subject, like many others, has suffered from 

 over specialisation, much that is familiar to the microscopist being almost unknown 

 to physicists, and vice versa. For myself I must confess that it is only recently, in 

 consequence of a discussion between Mr. L. Wright and Dr. G. J. Stoney in the 

 'English Mechanic' (Sept., Oct., Nov., 1894; Nov. 8, Dec. 13, 1895; Jan. 17, 1896), 

 that I have become acquainted with the distinguishing features of Prof. Abbe's 

 work, and have learned that it was conducted upon different lines to that of 

 Helmholtz. I am also indebted to Dr. Stoney for a demonstration of some of Abbe's 

 experiments. 



