F. J. CHESHIRE ON A METHOD OF USING ABBE's APERTOMETER. 351 



Since the angle ft never exceeds a few degrees cos /? = 1 to a 



sin 



first approximation, and from (1) sin (i = — ^- -, thus finally : — 



N.A.i - N.A. + N.A. i^^ 



— "N" A -I. f^ ^ ^^^ "'■ ^^^ ^ ('\\ 



The determined N.A. is thus always in excess of the true 

 N.A. by the quantity represented by the last term of equation 

 (3). This quantity for a given lamp distance becomes a maximum 

 when a = 45°, that is for N.A's just over unity. In the limiting 

 case, when D is as small as possible (a fixed indicator on the stage, 

 say), and a also is small, the error may amount to more than 

 10 per cent. With a lamp a foot away the error in general is 

 not more than from 1 to 2 per cent., but it increases rapidly as 

 D is diminished, and diminishes slowly as D is increased. 



The various attempts that have been made from time to time 

 to improve upon Abbe's apertometer, or his method of using it, 

 have not met with such success as to encourage further eflbrts in 

 the same direction, but it may be pointed out that the auxiliary 

 objective and stop may be dispensed with by dropping a stop of 

 about 3 mm. diameter into a low-power eyepiece — the lower the 

 better — and examining the image of the circular edge of the 

 apertometer disc in the eye-ring by means of a powerful hand 

 magnifier. This method has the further advantage that it does 

 away with the necessity for withdrawing the draw-tube to screw 

 in the objective after focussing. Further, an un graduated 

 apertometer disc * may be substituted for the usual graduated 

 one by mounting the two indicators, 1,1, to slide on a scale St 

 and using it as shown in broken lines to measure the chord of 

 the maximum aperture angle. Now the N.A. to be determined 

 is equal to /x sin a, where a is half the angle subtended by the 

 chord. Therefore sin a equals the choi-d divided by twice the 

 radius of the disc, thus the 



N.A. = Chord X ^— -^ 



2 X radius of disc 

 = Chord X a Constant. 



* Zeiss, cost 25*. 



J Brown & Sharpe, 100 mm. divided into fifths, cost U. lOd. 



