On the Estimation of Ajperture. By the late C. Hockin, jun. SiS 



and the number of rays in any medium is counted by tbe 

 length of the object x angular breadth of the small pencils coming 

 from it X refractive index of the medium under the conditions 

 implied in what goes before. 



To return to the Microscope : Law A applied and interpreted 

 geometrically shows that the direction of any incident ray cuts the 

 directions of the emergent ray on the circumference of a small 

 circle centered a little behind the object, but so near it, that for 

 purposes of illustration it may be supposed to be centered at the 

 object.* Fig. 4 represents this case. 



P Q being a plane, the intensity of the light emergent from it at 

 any angle varies as the cosine of the angle that the ray makes with 

 the axis. If then the diameter A B is divided into equal parts, and 

 lines through the subdivisions are drawn parallel to the axis 

 cutting the circle in the points between a and h, the unevenly dis- 

 tributed lines in the pencil aVh represent the distribution of light 

 in the incident pencil, and the nearly evenly distributed lines in the 

 pencil a! p h', the distribution of the light in the emergent pencil, 

 which is seen to be nearly uniform. 



The number of rays forming the half-image is therefore 

 measured hj pg x /." a pV 



= |N.PQ. Z-^a'ph' 

 = i 2 N" . P Q . sin ^ a> 6' 



1 n 



= N . PQ.^« -, . siuaPi) 



or since n' = 1 

 = w . sin ^ a P 6 

 = n sin u in the notation of law A. 



This is Prof Abbe's formula for aperture, and the proof of it 

 which has been given will not, I think, be found to differ essentially 

 from Prof. Abbe's proof. 



The only difference is that the form " ratio of clear opening to 

 focal length is omitted." This definition seems, at first sight, 

 somewhat arbitrary, but when we see that this expression, inter- 

 preted as it is in Prof. Abbe's demonstration, expresses the square 

 root of the light admitted from a standard object under fixed 



* If Ms the distance between the object and the image, the centre of the 

 circle is situated at a distance = ; ; behind the image, and its 



diameter = 2 / x ; . 



{-ly- 



