On the Estimation of Aperture. By Prof. E. Alibe. 393 



obliquity u* of any emergent ray from the obliquity u of the same 

 ray at its entrance, by means of the equation 



sin u* = c sin u ; (2) 



and if this equation is applied to the ray of utmost obliquity which 

 is transmitted through the system, u* will express the semi-angle 

 of the emergent pencil, whilst u is the semi-angle of the admitted 

 cone of light or the semi-angle of a2}erture. 



The linear opening of the system, or the diameter of the 

 delineating pencil at the plane of its emergence, is readily calculated 

 by means of the angle u* and the distance at which the image is 

 projected. If J is the plane of emergence (the plane of the back 

 surface of the system) and I the distance of the image from J, the 

 linear seww'-diameter p of the pencil is, obviously, 



p = I tan M*, 



for which may be substituted the identical equation 



, sin M* 

 p = I 



cos u'^ 



In the case of microscope-objectives, the distance I (the length 

 of the microscope-tube) is always many times greater than p, and 

 accordingly the angle of convergence u* is always very small, 

 never exceeding a few degrees. The cosine of such an angle may 

 be put = 1 without appreciable error ; and taking now the value of 

 sin u* from the equation (2) we obtain 



p = c I sin u, (3) 



which expresses the linear semi-aperture of the system by the semi- 

 angle of aperture. 



The question will now arise: how is the value of c for every 

 particular case to be obtained ? 



This is established by a dioptrical proposition of older date, 

 which is known as the Lagrange-Helmholtz law of convergence of 

 infinitesimally narrow pencils. If and 0* denote conjugate 

 foci, h the diameter of an object at 0, and h* the diameter of its 

 image at 0*, n and n* the refractive indices of the media in front 

 and at the back of the system, whilst v and v* are the angles of 

 obliquity of any ray traversing the system close to the axis, then 

 we have always 



V* n h n \ . 



V 11* h* ' n* N ' 



where N denotes the linear amplification of the system for that pair 

 of conjugate foci ; and this holds good for every composition of the 

 system and for every position of the conjugate foci. According 



