451 



Transactions of the Society. 



dispensed with. By this means an image of a distant source 

 • composed of independently radiating parts, such as a lamp-flame, 

 may be thrown upon the object, and it might at first sight be 

 supposed that the problem under consideration was thus completely 

 solved in all cases, inasmuch as the two apertures correspond to 

 different parts of the flame. But we have to remember here and 

 everywhere that optical images are not perfect, and that to a point 

 of the flame corresponds 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. 118 and the former argument apply almost without modifica- 

 tion, and they show that the radius AP of the disk has the value 

 iX / 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 independent 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 [especially that the optical distance 

 measured along a ray is a minimum] already applied. As before, 

 AB (fig. 119) represents the axis of the instrument, A and B being 



A 



Fig. 119 



■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 dis- 

 tance between them is the same for all [ray-]paths, e.g. for A E S B 

 and A L M B. [For the same reason the optical distance from P to 

 Q is the same along the various rays, one of which lies infinitely 



* [1902. In the original diagram Q was shown upon the wrong side of B. I owe 

 the correction to a correspondence with Prof. Everett.] 



