58 



THEORY OF THE MICROSCOPE. 



The refracted ray is then to be regarded as incident on the second 

 refracting surface the surface of contact of the' two lenses ; the 

 angle a lt which it makes with a straight line drawn from the 

 centre of curvature through the point of incidence, is the angle 

 of incidence. Its magnitude is given by the trigonometrical 

 ratio 



a^c : ci = sin 04 : sin fa, 



or, denoting by r the radius of curvature, and by d the thickness 

 of the Hint-lens, 



(/! 4- d + r) : r = sin a x : sin fa ; 

 hence 



sin a, = 





 sin fa . 



The resulting angle of refraction, which may be called a/, is ob- 

 tained from the known refractive ratios of the flint- and crown- 

 glass. In this case the direction of the ray after the second 

 refraction, together with the angle <f> 2 which it makes with the 

 axis, are to be regarded as known. In the triangle ^ i a. 2 the sum 

 of the angles a^ and i is equal to the exterior angle < 2 , or, since 

 the angle i = a/ a 1? 



02 = 01 + a l ~ a l' 



Hence, for the distance / 2 of the point a. 2 from the second 

 refracting surface, we get 



f 2 = a. 2 c r, or, since a 2 c : ci = sin a/ : sin fa , 



sin a/ 



A = r - - r - 



sin 2 



Similarly may be calculated the angle of incidence a 2 for the last 

 refracting surface of the double-lens, and from it the angle of 



FIG. 20. 



