88 



THEORY OF THE MICROSCOPE. 



to be deduced are generally accurate. Let P (Fig. 41) be a 

 radiant point situated on the axis, which emits a cone of rays, 

 as shown in the figure, to the anterior limiting surface N Q of the 

 refracting system. The corresponding cone of rays proceeding 

 from the posterior limiting surface N* has now to be constructed. 

 For this purpose we trace only the marginal ray Pg, which limits 

 the angle of aperture. Its prolongation (in this case backwards) 

 meets the anterior principal plane E Q in the point i ; consequently 

 the line drawn parallel to the axis through i cuts the second 

 principal plane E* in the point a, which belongs also to the 

 emergent ray. Further, if we draw through the anterior focus 

 a line parallel to i P to meet the anterior principal plane, and, 



I 



FIG. 41. 



from the point of intersection, r, a second line parallel to the axis, 

 the latter meets the posterior focal plane F* in a point p, which 

 likewise belongs to the emergent ray, and consequently determines 

 its direction. The prolongation backwards of the line a p gives 

 at the same time the position of the virtual image-point in the 

 axis. If we add also the marginal ray which proceeds symmetri- 

 cally on the other side, the emergent cone of rays is evidently com- 

 pletely determined. The question now arises, What is the ratio 

 of the two angles of aperture ? As a glance at the figure shows, 

 the incident cone of rays has an aperture o> = 2 x angle i P k, 



