52 Transactions of the Society, 



3. It was not always easy to ascertain the virtual diameter of 

 the pencil admitted from the object-glass or a ; if the limiting 

 diaphragm or " stop " is in front, as is sometimes the case, it would 

 give a measure very much too small ; if placed at some distance 

 behind, an addition is required to be made to its diameter in the 

 ratio of its distance from the object-glass to its distance from the 

 back conjugate focus, and with glasses composed of several separate 

 lenses, unless they are of very short focus ; a further addition is 

 requisite for the distance within the compound glass at which the 

 rays cross. 



4. The apertures in most of the observations in the third table 

 were taken by measurement, with such corrections when neces- 

 sary ; but from other measurements and trials, wdiich it would be. 

 tedious to give, I found also that, for determining the defining- 

 power, their diameter might be considered to be equal to the chord 

 of the angle of the pencil admitted from a point in focus, the focal 

 distance d being radius.* This angle can be taken by placing a 

 candle at, say, 10 ft. distant, and turning the Microscope on its 

 focal point as a centre till the edge of the light bisects the field 

 on each side; but to the chord so obtained about 0'006 in. has to 

 be added, and more if the aperture is much less than ^V ^^^ ^^ 

 correct an effect of the dispersion at the edge, which under the 

 circumstances causes the angle to appear too small. 



It may be inferred from the above that so long as we can increase 

 the angle of the pencil received and bring to focus the rays ad- 

 mitted, the intrinsic defining-power will continue to increase in 



a s 

 the ratio of the chord to radius, so that taking -j- as before (p. 45) 



at 0-000021 in., a being the aperture of any object-glass of a 

 Microscope and d its front focal length, s, or the smallest separa- 

 tion visible with that glass, will be ; or if the angle 



of the pencil is known, calling d unity, s will be = 0-000021 in., 

 divided by tlie chord of the pencil ; the ultimate limit then will 

 be theoretically when the aperture is twice the focal distance! or 

 - 000021 in., which would bring the smallest separation that can be 

 imagined visible in ordinary light of a series of lines or squares to 

 be little more than 0-00001 in. And although any near approach 



* Note by A. E. C. — The chord of an angle being twice the sine of half the 

 angle, this is the first proof that J. J. Lister knew in 1832 that the resolving-power 

 of Microscope objectives grows proportionately, not with the angle itself, as was 

 thought for many years after that date, bnt with the chord — in other words, with 

 what, since Abbe, is called numerical aperture. 



t Note by A. E. C. — It will be seen that this is a direct reference to N.A I. as 

 the limit of a dry objective. There is a slip of paper amongst the miscellaneous 

 notes and papers, not published, with the modern formula in terms of the sine of 

 half the angle, which J. J. Lister evidently used for calculating resolving-powers. 



