HOW ' APERTURE ' IS OBTAINED 45 



the pencil emergent from the objective \\'ith the focal length of that 

 objective. 



It will be desirable to explain somewhat more in detail how 

 this conclusion is arrived at, as given in Professor Abbe's 

 papers. 



Taking in the first case a single-\&as microscope, the number of 

 rays admitted within one meridional plane of the lens evidently in- 

 creases as the diameter of the lens (all other circumstances remaining 

 the same), for in the microscope we have at the back of the lens the 

 same circumstances as are in front in the case of the telescope. The 

 larger or smaller number of emergent rays will therefore be properly 

 measured by the clear diameter; and, as no rays can emerge that 

 have not first been admitted, this must also give the measure of the 

 admitted rays. 



Suppose now that the focal lengths of the lenses compared are 

 not the same what, then, is the proper measure of the rays 

 admitted ? 



If the two lenses have equal openings but different focal lengths, 

 they transmit the same number of rays to equal areas of an image 

 at a definite distance, because they would admit the same number if 

 an object were substituted for the image that is, if the lens were 

 used as a telescope-objective. But as the focal lengths are different. 

 the amplification of the images is different also, and equal areas of 

 these images correspond to different areas of the object from which 

 the rays are collected. Therefore the higher-power lens, with the 

 same opening as the lower power, will admit a greater number of 

 rays in all from the same object, because it admits the same number 

 as the latter from a smU<>r portion of the object. Thus, if the focal 

 lengths of two lenses are as 2:1, and the first amplifies X diameters, 

 the second will amplify 2 N with the same distance of the image, so 

 that the rays which are collected to a given field of 1 mm. diameter 



of the image are admitted from a field of ^- mm. in the first ca.se 



and of -v nim. i 11 the second. Inasmuch as the ' opening ' of the 



A J^l 



objective is estimated by the diameter (and not by the area), the 

 higher-power lens admits twice as many rays as the lower power, 

 because it admits the same number from a field of half the diameter, 

 and in general the admission of rays with the same opening 

 but different powers must be in the inverse ratio of the focal 

 lengths. 



In the case of the single lens, therefore, its aperture must be 

 determined by the ratio between the clear opening and the focal 

 length, in order to define the same thing as is denoted in the telescope 

 by the absolute opening. 



Consider now the compound objective the most important case 

 in the microscope. What is the opening of this composite system ? 

 We must adhere to the diameter of the admitted cone at that plane 

 where it has its ultimate maximum value, which is obviously the 

 diameter of the pencil at its emergence, from the system, or, practi- 



