248 



THEORY OF MICROSCOPIC OBSERVATION. 



view act alternately as convex and concave lenses our first task 

 is to examine the resulting image from this more general point 

 of view. 



Let a b (Fig. 136) be a surface-element which gives real images, 

 and let us assume the illuminating mirror 

 to be so placed that all incident rays are 

 deflected to the same side of the vertical. 

 Let the minimum of this deviation be 8, and 

 the maximum 8'; the angle of aperture of 

 the objective may be co as. before. On these 

 suppositions the bounding lines of the 

 umbra and penumbra are determined by 

 the corresponding values of the angle of 

 deviation p. Their dependence upon the 

 inclination of the incident rays of light is 

 more clearly exhibited if their distances 

 from the centre of the refracting surface-elements are expressed 

 directly by the focal length (/) of the latter. If F and F* are the 

 two focal planes, s q and s' q' incident rays, with the inclination 8', 

 which are so refracted that they still just reach the objective, then 

 the triangles o p q and o p q' give the trigonometrical relations 



CO . /CO 



op: o q = cos -77- : sin 1-5- -f 



2t * 2i 



FIG. 136. 



sn 



+ S') 



consequently o q = 



oq = f 



or, since o p = 



cos . op 



cos 



2 ' 



cos cos 



m 



=/(tan|- + tan S') ; 



and similarly we get 



oq' =f (tany -tan 8'). 



The bounding line of the umbra moves forward, therefore, on 

 one side towards the optical centre, and coincides with it as soon 



as 8f = -JT- ; it moves further from it on the other side, and soon* 



