168 Lord Rayleigh on the Theory of Optical Images, 



fi being the refractive index of the medium ; and thus 



6 = |X //* sin a (3) 



The denominator /a sin a is the quantity now well known 

 (after Abbe) as the " numerical aperture." 



The extreme value possible for a is a right angle, so that 

 for the microscopic limit we have 



«=iV/* (4) 



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

 as photography makes possible, or by an increase in /*, 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 repre- 

 sentative 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 fas 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 return to the 

 subject later. 



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

 and more frequently in the use of the microscope, the over- 

 lapping lights have permanent phase-relations, these inter- 

 mediate cases require a further treatment ; and this is a 

 matter of some importance 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 t> which somewhat 



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

 Phil. Mag. vol. viii. p. 266 (1879). 



t Archiv.f. Mikr. Anat. vol. ix. p. 413 (1873). 



