On the Estimation of Aperture. By the late 0. Eockinjun. 341 



supposed to recede from the instrument while subtending a constant 

 angle at some point near the instrument, the form that the equa- 

 tion ultimately takes is 



n . d . a =■ n' . pq . sin v . . , . A' ; 



or since, in the present case, n = nJ, 



d . a == pq , sin v, 



where d is the distance of any ray incident parallel to the axis from 

 that axis, and a the angle that P Q subtends. 



Fig. 2 represents this case. 



Law A' put in a geometrical form shows that the directions of 

 any incident ray and the corresponding emergent ray cut on the 

 surface of a circle centered at the image. 



In practice the radius of the circle is so large in comparison 

 with the diameter of the objective that v is always a small angle and 

 V, sin 'y, and tan v for our present purpose may be treated as equal. 



In the figure the equidistant parallel lines from a to & repre- 

 sent the direct incident pencil of uniform intensity, and the lines from 

 a! to V converging at C the corresponding emergent pencil. The 

 directions of these rays meet within the objective on the circle c d 

 of large radius. The consecutive lines between a and b' therefore 

 make nearly the same angle with each other, and the light is nearly 

 uniform in the pencil a' C h'. 



The direct pencil only is drawn, but as the object subtends 

 always but a small angle at the instrument, the breadth of all 

 incident pencils may be treated as equal, and the total number of 

 rays incident on the objective is measured on the convention 

 adopted by 



d . a 



where d is the diameter of the objective. 



Counting the rays emergent from the objective, they consist of 



d 

 pencils of angle -^ and converge to a line of length p q ; their 



number on the convention is therefore 



d 



where /is the focal length of the instrument ; or, since 



pq=f .a, 



the number of the rays = d . a, the same expression as before, 

 and d represents the number of rays received from an object of 

 unit angular dimensions and is the recognized measure of the 

 aperture of the instrument. 



