ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 367 



Hence, for the nth. image to be excluded, nX/e must be greater than 

 sin a, but according to Abbe, for resolution the first diffracted image 

 must appear, and hence resolution is just possible if \/e is equal to sin 6. 



It has been assumed that air is the medium on either side of the 

 object glass ; if on the object side we have a medium of refractive index 

 fi, then it is easy to show that we must replace sin by /x sin 0, and the 

 condition of resolution is that e should be equal to A///, sin 0, or intro- 

 ducing the term numerical aperture for the quantity /x sin 0, we have 

 the result that a grating is resolvable if the space between the lines is 

 not less than the result found by dividing the wave-length of light by 

 the numerical aperature. 



Now, while the truth of this result can in certain cases be estab- 

 lished, the reasoning given in the books under consideration is 

 insufficient to prove it. 



In order to decide if the grating can be resolved we must establish 

 the law of variation of intensity in the view plane, and then consider 

 whether these variations are such that they can be detected by the eye. 

 This has been done by Lord Rayleigh. The images formed in a Micro- 

 scope are, like all other images, produced by interference ; in considering 

 resolving power we have to consider diffraction effects, it is true, but the 

 diffraction which concerns us mainly is that due to the aperture of the 

 object glass, and only indirectly that due to the object viewed. 



Neither is it necessary, if we know completely the distribution of the 

 light over the stage, to go back to the source in our consideration of the 

 problem ; having given the distribution over the stage both in amplitude 

 and phase, we are potentially able to determine that in the view plane 

 without reference to the source. Difficulties of calculation may stop us, 

 it is true, but that is another matter. 



Let us take, again, the case of a grating illuminated by plane waves, 

 their plane being parallel to that of the grating ; we have to consider 

 the effect due to a series of equidistant lines of light ; these differ, 

 however, from a series of independent equidistant linear sources in that, 

 with the grating, the phases of the various sources are the same ; we 

 have therefore to remember that interference will take place between 

 the light from the different lines, while with a series of independent 

 lines there is no relation between the phases ; we can calculate the 

 intensity due to each source separately, and superpose the whole. 



Lord Eayleigh's solution of the problem, which is presented when 

 a narrow double line in a spectrum is viewed through a telescope, or 

 when the attempt is made to resolve two close double stars, is better 

 known than his equally valid solution of the grating problem, and as it 

 is simpler it will be useful to indicate it first. 



The intensity in the view plane for a single linear source, assuming 

 for the moment that we are dealing with a telescope with a rectangular 

 aperture, is given by a certain curve. If we assume a second independent 

 source parallel to the first we get a similar curve alongside the first. The 

 resultant intensity is found by adding the corresponding ordinates of 

 the two curves, and the lines will appear as double when the drop in the 

 resultant intensity curve is sufficient to be detected by the eye. 



2 c 2 



