with Special Reference to the Microscope. 173 



the discrepancy first becomes conspicuous when the phases 

 corresponding to the various secondary waves which travel 

 from P to B range over about a complete period. The illumi- 

 nation at B due to P then becomes comparatively small, in- 

 deed for some forms of aperture evanescent. The extreme 

 discrepancy is that between the waves 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 retarda- 

 tion 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 

 I/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 unneces- 

 sarily 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 

 immediate application to the telescope. If 2R be the diameter 

 of the object-glass, and D the distance of the object, the angle 

 subtended by AP is e/D, and the angular resolving-power is 

 given by 



2D sin a = m' (0) 



the well-known formula. 



This method of derivation makes it obvious that there is no 

 essential difference of principle between the two cases, 

 although the results are conveniently stated in different 

 forms. In the case of the telescope we have to do with a 



Phil. Mag. S. 5. Yol. 42. No. 255. Aug. 1896. 



