452 Transactions of the Society. 



intensities of the two luminous centres being supposed equal. 

 Under these conditions it is clear that A and P are not separated 

 in the image. The question is, to what amount must the distance 

 AP be increased in order that the difference of situation may make 

 itself felt in the image. This is necessarily a question of degree ; 

 but it does not require detailed calculations in order to show that 

 the discrepancy first becomes conspicuous when the phases corre- 

 sponding to the various secondary waves which travel from P to B- 

 range over about a complete period. The illumination at B due 

 to P then becomes comparatively small, indeed for some forms of 

 aperture evanescent. The extreme discrepancy is that between the 



L' 



B 



Fig. 118. 



wares which travel through the outermost parts of the object-glass 

 at L and L' ; so that, if we adopt the above standard of resolution,, 

 the question is, where must P be situated in order that the relative 

 retardation of the rays PL and PL' may on their arrival at B- 

 amount to a wave-length (\). In virtue of the general law that 

 the reduced optical path is stationary in value, this retardation 

 may be calculated without allowance for the different paths pursued 

 on the further side of L, L', so that its value is simply PL — PL'. 

 Now since AP is very small, AL' — PL' is equal to AP . sin a„ 

 where a is the semi-angular aperture L'AB. In like manner PL — 

 AL has the same value, so that 



PL- PL' = 2AP.sina. 



According to the standard adopted, the condition of resolution is 

 therefore that AP, or e, should exceed -JX/sin a, as in (1). If e be 

 less than this, the images overlap too much ; while if e greatly 

 exceed the above value the images become unnecessarily separated. 



In the above argument the whole space between the object and 

 the lens is supposed to be occupied by matter of one refractive 

 index, and X represents the wave-length in this medium of the 

 kind of light employed. If the restriction as to uniformity be 

 violated, what we have ultimately to do with is the wave-length 

 in the medium immediately surrounding the object. 



The statement of the law of resolving-power has been made in 

 a form appropriate to the microscope, but it admits also of im- 

 mediate application to the telescope. If 2E be the diameter of 



