with Special Reference to the Microscope. 



175 



in the image, not a point, but a disk of finite magnitude. 

 When this consideration is taken into account, the same 

 limitation as before is encountered. 



For what is the smallest disk into which the condenser is 

 capable of concentrating the light received from a distant 

 point ? Fig. 2 and the former argument apply almost 

 without modification, and they show that the radius A P of 

 the disk has the value ^X/sin a ; where a is the semi-angular 

 aperture of the condenser. Accordingly the diameter of the 

 disk cannot be reduced below \ ; and if e be less than \ the 

 radiations from the two apertures are only partially inde- 

 pendent of one another. 



It seems fair to conclude that the function of the condenser 

 in microscopic practice is to cause the object to behave, at any 

 rate in some degree, as if it were self-luminous, and thus to 

 obviate the sharply-marked interference-bands which arise 

 when permanent and definite phase-relations are permitted to 

 exist between the radiations which issue from various points 

 of the object. 



As we shall have occasion later to employ Lagrange's 

 theorem, it may be well to point out how an instantaneous 

 proof of it may be given upon the principles already applied. 

 As before, A B (fig. 3) represents the axis of the instrument, 



Fig. 3. 



Q- 



A and B being conjugate points. P is a point near A in the 

 plane through A perpendicular to the axis, and Q is its image 

 in the perpendicular plane through B. Since A and B are 

 conjugate, the optical distance between them is the same for 

 all paths, e. g. for AESB and A L M B. And, since A P, 

 B Q are perpendicular to the axis, the optical distance from 

 P to Q is the same (to the first order of small quantities) as 

 from A to B. Consequently the optical distance P Jx 8 Q is 

 the same as AESB. Thus, if //,, yJ be the refractive indices 

 in the neighbourhood of A and B respectively, « and /3 the 

 divergence-angles E A L, SBM for a given ray, we have 



fi . AP . sin a = /u, / . BQ . sin /3, 

 2 



(6) 



