394 Transactions of the Society. 



to this proposition, tlie ratio of infinitesimal angles v and v* (per- 

 taining to one and tlie same ray at its entrance and emergence) 

 may be completely determined by the refractive indices of the 

 media at the radiant and at the image and by the linear amplifica- 

 tion of the image, ivithout regard to the elements of the optical 

 system or to the position of the foci. 



This important theorem was established by Professor Helmholtz 

 in 1866.t Its earliest origin must be traced back to Lagrange, 

 who pomted out a similar proposition, confined however to less 

 general conditions, in 18034 



The way in which equation (-1) leads to a general determina- 

 tion of the constant c, which appears in the law of convergence 

 of w/fZe-angled pencils, will be readily understood. Any wide- 

 angled cone of rays admitted to an aplanatic system, embraces 

 axial rays of infinitesimal obliquities v and v*, and as in the case 

 of very small angles the ratio of the sines becomes identical with 

 the ratio of the angles, the value of e in equation (1) must, as far 



V* 



as it pertains to those axial rays, coincide with the value of — as 



determined by the Lagrange-Helmholtz formula. But as the con- 



dition of aplanatism requires the same value of the quotient —. 



sm w 



for all rays of the wide-angled pencil, we must have for all cases 



n 1 



Introducing this expression of c into equation (3) and taking 

 into account that in the case of the Microscope the medium at the 

 back of the system is always air {n* =1), the linear semi-opening 

 of an objective is 



I I 



p = — . n mn u ; or p = — . a, (6) 



N N 



a being put for n sin u, and therefore denoting the product of the 

 sine of the semi-angle of aperture and the refractive index of the 

 medium to which this anojle belongs. 



When an objective has a focal length = / and an image is pro- 

 jected at a distance = I from the lens, the amplification N of this 

 image will be, very approximately, 



whence it follows 



t ' Physiologisclie Optik,' 186G, p. .50. % Mem. Aca.l. Beiliu, 1803. 



