228 THEORY OF MICROSCOPIC OBSERVATION. 



t /' (Fig. 125) be the principal focal planes of the objective 

 (through want of space the focal lengths are represented much too 

 short in proportion to the aperture a, b) ; then, as the construction 

 shows, the dark line will appear at q, and the bright line at 7;. 

 Similarly, rays deflected towards the opposite side will form a dark 

 line at q'. The distance p q = p q is given in the figure by the 

 trigonometrical tangent of the angle a = 30, drawn to a radius 

 equal to the focal length ; with a focal length say of 3 mm., this 



would be = - x 3 = 1'732 mm. In reality, however, 



these distances are proportional to the sine of the angle of deflexion, 

 as has been shown by Abbe (sec page 25). 



If we increase the width of the opening a b from 1 mic. to 

 1 " mic., then, ccvteris paribus, a i = '75 mic. = 3 half wave-lengths. 

 The deflected pencils may therefore be divided into three parts 

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

 Each pair will therefore annul each other, as in the preceding 

 case; the third ray, however, will remain unaffected, and will 

 produce at q a briyht line instead of a dark one. This line is the 

 first bright line which in the diffraction image follows the focal 

 line of the direct rays. Between this line and the point q the 

 dark line already mentioned appears, for which a i is equal to a 

 whole wave-length ; for, obviously, a smaller inclination of the de- 

 flected rays corresponds to this value for the supposed larger aper- 



ture. We get sin a = = f rom w hi c h a = 19 2 8'. 



(10 O 



The change just mentioned takes place, of course, on the other 

 side of the optic axis ; a bright line appears at q', and a dark 

 one between p and q. 



If the angle of aperture of the objective is large enough to 



F) 



admit other pencils, for which a i = - wave-lengths, then another 



bright line will appear below q (and, similarly, above <?'), which 

 again will be separated from the preceding ones by a dark one. 



5 wave-length 5 



lor this new line we get sin a = ~- ~ - = -; conse- 



^2 a o o 



quently, a = 56 26', which supposes an angle of aperture 2 a 

 = 112 52'. Any further deflected pencils cannot be admitted 

 under the given conditions. 1 



1 If X represents the wave-length, and b the width of the opening, then the 



