BEIGHTNESS OF THE FIELD OF THE MICROSCOPE. 89 



and the emergent cone an aperture w = 2 x angle a P* @. Since 

 i k = a y8, and since k P is equal to the anterior, and /3 P* to the 

 posterior focal distance, we obtain for the trigonometrical tangents 

 of the two angles 



taniPk = ^; tanP*/3 = ^f, 

 p p*' 



in which p and p*, as before, denote the two conjugate foci. But 

 the squares of these tangents are proportional to the areas of 

 transverse sections of the two cones at the distance 1 from the 

 apex. For these surfaces we therefore obtain the proportion 



but the same proportion indicates also the areal magnification, 



p* 



because - - is equal to the coefficient of linear amplification. 



Hence we arrive at the conclusion, that the areas of transverse 

 sections of corresponding cones of light at the distance 1 from the 

 apex are proportional to the squares of the coefficients of linear am- 

 plification. If, now, the emergent cone completely fills the aper- 

 ture of the pupil a n, as assumed in the figure, the aperture of the 

 incident cone is fixed thereby ; for as soon as the aperture is 

 further enlarged, as the above relation involves, the transverse 

 section of the emergent cone becomes larger than the aperture of 

 the pupil, and the surplus of light received by the objective there- 

 fore remains functionless. Since the surface of the transverse 

 section of the incident cone is always somewhat larger (though 

 when treating of a small aperture only infinitesimally so) than 

 the corresponding spherical calotte at an equal distance from the 

 vertex, we are led to the conclusion, that a luminous surface ob- 

 served through the Microscope cannot, under any circumstances, 

 appear brighter than ivhen seen with the naked eye. 



In all these calculations of the brightness, it has, of course, 

 been tacitly assumed that the cones of light proceeding from the 

 object reach the eye without loss. The losses caused by reflexion 

 and absorption are therefore not taken into account. Their 

 amount in a given case cannot be readily determined with ac- 

 curacy, but so much is certain, that they form only a small 



