422 LIlillTS OP OPTICAL CAPACITY OF THE MICROSCOPE. 



angles of a pencil of rays whose section is a circle, then 



X = JdS I 2 TT. cos a. sin a. da = tt JdS. sin ^a. 

 J 



If after a series of refractions the surface d S^ is completely and 



accurately imaged in d Si with the brightness -^ Ji and a^ of the 



respectively appertaining divergence-angles ; then the amount of 

 light must be 



L =■- TT J ■^. d iSi . sin^a.. 

 n^ 



as now, ^*S^ : dSi = /3^ : (d^ there follows from these equations, 

 n . (d . sioia == ni . /3i . sin ui (7) 



which renders this formular of equation (5) valid for larger angles 

 of divergence, assuming that (3 and /3i are two images exactly re- 

 producing each other, and whose surfaces are perpendicular to the 

 axis. 



Brightness of Image. When the pupil of the observer's eye is 

 fully immersed in the pencil of rays proceeding from any point of 

 an image, the observer will see the image illuminated as brightly 

 as the object. This result was already announced by Lagrange. 

 Unfortunately he had not investigated a second case, which happens 

 to be more common just when high powers are used, namely, when 

 the pencil of rays does not entirely occupy the pupil of the eye. 



If a pencil of light having only small divergence-angle ay does 

 not entirely fill the pupil when the image /3iis situate at the proper 

 distance of distinct vision, then the brightness R of the retinal 

 image in that eye will be less than that entering the free eye J5^, 

 whose pupil is entirely filled with light. 



Let s indicate the distance of vision, p the radius of the pupil, 

 then the area of its surface will be tt ^^, the cross section of the 

 pencil of light tt s". sin^^i and the general relation will be 



H: E^=s-. sin^a, ! P' 



