CONDITIONS UNDER WHICH Hp = 1 293 



Let Rm be the greatest distance from the point .r, y to any point on the 

 geometrical boundary as in Fig. VII. 13. With few exceptions, in phase 

 microscopy \he^^ — l| < 1. Furthermore, lJi(27rpir)| < 0.52. There- 

 fore 



\Hs{x, y)\ < 2TrpiiRm - R)0.52. (15.17) 



The condition which pi must satisfy in order that Hs(x,y) shall be 

 negligibly small as compared with Hr {x, y) of Eq. 15.12 is 



7rpi(/?,„ - R)^l. (15.18) 



Larger values of pi are of course tolerable, but the largest tolerable 

 values are difficult to estimate. 



In conclusion, a point x, y belongs to the region A provided that it is 

 possible to draw about the point .r, y (Fig. VH.IS) an inscribed circle 

 whose radius R is so large that 



Jo(27rp,„/?)-^0, 



provided also that the conjugate area is chosen so small that 



Jo(27rpi/?)^l, 



and provided that pi is so small that 



TTPxiRrn - R) <^ I 



where R^ is the greatest distance in wavelengths from the point .r, y to 

 any point on the boundary of the geometrical image of the particle. 



|ilf |p^ is the usual N.A. of the objective. As the numerical aperture 

 of the objective is increased, the region A creeps toward the edge of the 

 geometrical image of the particle. |M|pi is the numerical aperture of 

 the conjugate area with respect to the object space of the objective. 

 As pi is decreased, the region A spreads throughout the interior of the 

 geometrical image of increasingly larger particles. For points x, y of 

 the region A the function Hp{x, y) of Eq. 15.6 is practically unity and 

 the total energy density Ga is given by Eq. 14.8 of the elementary 

 theory of phase microscopy. 



When the required hole in the condenser diaphragm assumes pinhole 

 dimensions, it is preferable to discard the lamp and substage condensers 

 and to illuminate the object directlj^ by means of a distant zirconium 

 arc, etc. The question as to the self-luminous character of a \'irtvial 

 image is thus avoided. 



Osterberg and Pride have performed experiments with polanret 

 systems involving narrow-coned axial illumination and ha\'e found that 

 the above predicted A region exists. In one of the polanret systems 



