whence 



FAR INTERIOR POINTS OF A LARGE PARTICLE 287 



M^Foix, y, po, qo) = ^(f^'f^) = ^e''; (12.5) 



M'^Gix, y) = constant = h^ II l(po, qo) dpo dqo. (12.6) 



Thus 



h^ 

 M^ 



1 1 ^ ipo, qo) dpo (iqo U2.7) 



is physically the energy density in the image of the siirroimd at points 

 well removed from the nearest geometrical image of a particle. For 

 brevity we shall call Gg the energy density of the surround. Gg is 

 proportional to the energy transmission }r of the conjugate area and 

 vanishes, as in the Schlieren method, when the conjugate area is opaque. 

 In conclusion, the energy density at any point x, y in the image of the 

 surround approaches Gg when the nearest geometrical image of any one 

 of the particles is located 5 or more Airy units r^ from the point x, y. 

 As a corollary, one particle will not disturb appreciably the image of 

 another provided that the edges of their geometrical images are sepa- 

 rated by 5 or more Airy units. Accordingly, particles whose geometrical 

 images are separated by at least 5 Airy units are said to be isolated. 

 It follows from Eq. 12.3 that particles are more likely to be isolated with 

 objectives having the higher numerical aperture. 



13. THE ENERGY DENSITY IN THE FAR INTERIOR OF A LARGE PAR- 

 TICLE 



Suppose first that the particle is very large and is located in a much 

 larger field of view. For points x, y located well in the interior of the 

 geometrical image of the particle the argument leading to Eq. 11.5 may 

 be repeated with Fa of Eq. 11.3 to show that 



FAX, y, po, qo) = -^ e^-2.iiMHpo.+,,y)p (^^ /^^ (13.1) 



Then from Eqs. 11.1, 11.4, 11.5, and 13.1 we obtain almost directly the 

 result 



(2TrilM)ipnx+Qoy) / \ 



Foix, y, Po, qo) = j^, ge'^'P ( ^ ' ^) ' (13.2) 



