J4 ON INTERFERBNCl IN THE MICROSCOPE. 



If we suppose fee'/' (fig. *) to be the focal and principal planes 

 of an objective (in the figure they are drawn for want of 

 space disproportionately close together in comparison with the 

 size of the diffracting opening 'or transparent space, ab) then, 

 as the construction shews, there will appear a dark band at q, 

 and a light band at p. On the other side of p the diffraction 

 pencil would form a similar dark band at q'. The distances 

 fp q, p q'J are given in the figure as calculated with the trigono- 

 metric tangent of the angle of divergence (in this case =30^) and 

 radius having the same value as the focal length. For instance, with 



tang. 30 



a focal length of 3 mm., the distance = . 3 = 1.732 mm. 



Strictly, however, these distances are, as Professor Abbe has 



shown, proportional to the sine of the diffraction angle. 



When the width {db) of the opening is increased from i to i .^ 



mik.. fl« becomes, cceteris paribus ^ 0.7^ mik.=3 half-wave lengths. 



The diffracted pencils can then be separated into three parts 



whose marginal rays will be displaced by a half wave length. 



Two of these annul each other, as in the foregoing example ; but 



the jthird remains unaffected, and, therefore, produces a bright 



instead of a dark line at q. This is the first bright diffraction line 



which follows the direct bright ray. Between this latter and the 



point q comes now the dark band before described, where ai = a 



whole wave length ; for it self-evident that, corresponding with the 



new value of wave length for the supposed wider opening there 



must be a slighter deflection of the diffracted rays. Thus we get sine 



, wave-length . . ,^, n^, 



divergence angle a= , =^, whence 0=19° 28. The 



same change occurs, of course, on the other side of the optic axis, 



and there appears at q' a bright, between^ and q' a dark band. If 



the aperture of the objective is wide enough to take in another 



pencil, for which a c=§ of wave length, there will appear beyond 



q and q' another bright line, which again would be separated from 



the last by another dark interspace. For this new line 



wave lensfth 

 sine a=^ . , =|, and consequently a=^6^ 26' which 



