208 THEORY OF MICROSCOPIC OBSERVATION. 



the point in which the line of direction of the ray passing through 

 unrefracted cuts the plane of adjustment drawn through the centre. 



To determine this point, let S T (Fig. 113) be a ray incident 

 vertically from below, which is reflected at H and refracted for the 

 second time in M ; further, let a and a be the angles of incidence 

 and refraction, r the short and R the long radius ; and let p 

 = L MKNuml 77 = L MHK: then the triangle MHK gives 



p = 180 - [(a - a') + 77] ; 

 consequently 



sin p = sin (a a + 77) = sin (a a) cos 77 + cos (a a) sin 77. 

 In the triangle C H M we get the proportion C M : C H = sin 77 : 



sin a : and hence sin 77 = 7 ^ T sin a sin a. The above formula 



therefore passes into the form 



sin p = sin (a a) |/ 1 ^ sin a'\ 



T> 



+ cos (a a) sin a'. 



By the help of this equation the direction of the emergent ray 

 may be determined for any given angle of incidence. In the case 

 where p = 90, 77 is the complementary angle a a. We get 

 therefore 



7) 



cos (a a') = sin 77 = sin a, 



and hence 



r sin a 



R cos (a a) ' 



From the last expression we find that with any angle of in- 

 cidence a proportion between the radii may be imagined that will 

 give to the emergent ray the direction of the incident one. Since, 

 then, the line of direction of the ray emerging without deviation, 

 and with it the virtual focus, moves the further inwards the smaller 

 a is, it is not without practical interest to compare the position of 

 the focus, which appears in the microscopic image as a bright line 

 with the ratio between R and r, and to investigate the general 

 relations which here exist. We have collected below a few values 

 of r : R with the corresponding distances of the virtual focus from 

 the centre (denoted by F in the table), and have added the 

 corresponding angles of incidence and refraction. The refractive 



