LIMITS OF OPTICi^L CAPACITY OF THE MICROSCOPE. 419 



Or 



n' . a' . (3' = n' . a . (3" (5) 



q. e. d. 



From this theorem it follow s — 



(Firstly), that when a ray (B) proceeding from a luminous 

 point has an absolute smaller divergence-angle than the ray A, the 

 divergence-angle of B will, after subsequent refraction, remain 

 always less than that of A, because the product obtained by our 

 theorem for B is from the beginning less than that obtained for A, 

 and for the same reason must continae to be smaller after each 

 refraction. 



(Secondly), when two rays, starting from the same point on the 

 axis, with equal angles of divergence, but following planes which 

 extend in opposite directions through the axis, their divergence- 

 angles continue to be equal after each refraction, a result which 

 appears indeed at once evident from the symmetrical disposition of 

 a lans system round its axis. 



If now we imagine the illuminating rays, on their way 

 to the object, to be circumscribed by interposing a diaphragm 

 pierced with a circular opening whose centre coincides 

 with the axial line, the plane of the diaphragm being at right 

 angles with the optical axis, then those rays which pass through 

 the opening close to its margin have all alike the largest divergence- 

 angle, and retain the same relation after each fresh refraction. 

 These rays obviously occupy the exterior outline of cones having a 

 circular base, and whose axis is the optical axis of the lens system, 

 and they constitute the boundary of the cone of light proceeding 

 from the luminous point. The divergence-angle of these border 

 rays is, in this case, throughout their entire course, the angle which 

 the semi-aperture of the conical surface bounding the illuminating 

 cone, measures. 



From this there follow, (firstly), certain important results in 

 regard to the photometric conditions of the microscope image. 



According to known laws of photometry, we may equate Z the 



