338 Transactio7is of the Society. 



Professor Abbe remarks tbat bis expression for aperture repre- 

 sents the " number of rays," and not only tbe amount or " mere 

 quantity of ligbt " admitted by tbe objective and utilized for tbe 

 formation of tbe image. The " number of rays " and " quantity 

 of light " are distinguished in the course of his demonstration, as 

 I understand it, by the fact that the first is measured in one 

 dimension of space, and the second in two dimensions.* 



Prof Abbe's definition of aperture may, I think, be treated as 

 a strictly photometrical one, and that it expresses the relation of the 

 total amount of light utihzed by an objective of given magnifying 

 power in an axial plane to the total amount of light emitted by an 

 object supposed to be in air and under a fixed illumination. 



The object is also supposed to be a small portion of a plane 

 surface. 



That this is the case appears from the assumption made by 

 Prof Abbe that the pencils of light of small angular aperture, which 

 proceed from the objective to the image, are properly measm^ed by 

 their breadth at a fixed distance from their point of convergence. 

 This assumption involves evidently the condition that the pencils of 

 light in question have the same intensity over their breadth, and 

 this condition is fulfilled approximately only if the object is nearly 

 plane. 



All this will appear, perhaps, more clearly in the course of 

 the following demonstration which, following the lines of Prof. 

 Abbe's proof, is somewhat more detailed, and supplies demonstra- 

 tions of certain of the formulae which Prof. Abbe omits as too well 

 known to require further illustration. 



Lemma. — An instrument collects all the rays that emanate 

 from some one point in its axis in front of the instrument (and 

 which are contained within a certain finite angle), and causes them 

 to converge accurately to a point in its axis at the back of the 

 instrument, and further collects, under similar restrictions, all rays 

 from a second point in front of the instrument, indefinitely near the 

 first and situate in a plane normal to the axis of the instrument and 

 passing through the first point, and causes them also to converge 

 accurately to a second point behind the instrument, lying in a plane 

 normal to the axis, and passing through the point of convergence of 

 rays incident from the first point. 



Then the sine of the angle that any ray incident on the instru- 

 ment from the first point mentioned makes with the axis, bears a 

 constant relation to the sine of the angle that the corresponding 

 emergent ray makes with the axis. 



Let P, Q be the foci of incident and p, q the corresponding foci 



* [See Note by Prof. Abbe, infra, p. 346,— Ed.] 



