SCIENCE. 



289 



the image are admitted from a field of — mm. in the 



N 



first case and of — — mm. in the second. Inasmuch as 

 2 N 



the " opening " of the objective is estimated by the diam- 

 eter (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 

 leDgths. 



In the case cf the single lens, therefore, its aperture 

 must be determined by the ratio between the clear open- 

 ing and the focal length, in order to define the same 

 thing as is denoted in the telescope by the absolute 

 opening. 



Dealing with a compound objective, the same consid- 

 erations obviously apply, substituting, however, for the 

 clear opening of the single lens, the diameter of the pen- 

 cil at its emergence from the black lens of the objective 

 —that is, its clear effective diameter. 



All equally holds good, whether the medium in which 

 the objective is placed is the same in the case of the two 

 objectives or different, as an alteration of the medium 

 makes no difference in the power. 



Thus we arrive at the general proposition for all kinds 

 of objectives. 1st. When the power is the same, the 

 admission of iays varies with the diameter of the pencil 

 at its emergence. 2nd. When the powers are different 

 the same admission requires different openings in the 

 proportion of the focal lengths, or, conversely, with the 

 same opetting the admission is in inverse proportion to 

 the focal length — that is, the objective which has the 

 wider pencil relatively to its focal length has the larger 

 ' aperture. 



Thus we see that, just as in the telescope, the absolute 

 diameter of the object-glass defines the aperture, so in 

 the microscope, the ratio between the utilized diameter 

 of the back lens and the focal length of the objective de- 

 fines its aperture. 



This definition is clearly a definition of aperture in its 

 primary and only legitimate meaning as "opening" — that 

 is, the capacity of the objective for admitting rays from 

 the object and transmitting them to the image ; and it at 

 once solves the difficulty which has always been involved 

 in the consideration of the apertures ot immersion ob- 

 jectives. 



So long as the angles were taken as the proper expres- 

 sion of aperture, it was difficult for those who were not 

 well versed ; n optical matters to avoid regarding an angle 

 of 180° in air as the maximum aperture that any objec- 

 tive could attain. Hence water-immersion objectives of 

 96 and oil-immersion objectives of 82 were looked 

 upon as being of much less aperture than a dry objective 

 ot 180 , whilst, in fact, they are all equal — that is, they 

 all transmit the same rays from the object to the image. 

 Therefore, f8o° in water and 180 in oil are unequal, " 

 and both are much larger apertures than the 180 which 

 is the maximum that the air objective can transmit. 



If we compare a series of dry and oil-immersion ob- 

 jectives, and, commencing witli veiy small air-angles, 

 progress up to 180 air-angle, then taking an oil-immer- 

 sion of 82° and progressing again to 180 oil-angle, the 

 ratio of opening to power progresses continually also, 

 and attains its maximum, not in the case of the air-angle 

 of i8o Q (when it is exactly equivalent to the oil-angle of 

 82 ), but is greatest at the oil-angle of 180 . 



If we assume the objectives to have the same power 

 throughout, we get rid of one of the factors of the ratio, 

 and we have only to compare the diameters of the emer- 

 gent beams, and can represent their relations by dia- 

 grams. Our figure (which is taken from Mr. Crisp's 

 paper) illustrates five cases of d fferent apertures of ^ in. 

 objectives — viz., those of dry objectives of 6o°, 97° and 



180 air-angle, a water-immersion of 180 water-angle, 

 and an oil-immersion of 180 oil-angle. The inner dotted 

 circles in the two latter cases are of the same size as that 

 corresponding to the 180° air angle.* 



RELATIVE DIAMETERS OF THE (UTILIZED) BACK-LENSES OF 

 VARIOUS DRY AND IMMERSION OBJECTIVES OF THE SAME 

 POWER (%) FROM AN AIR-ANGLE OF 6d q TO AN OIL-ANGLE 

 OF 180°. 



Numetical Aperture 



1-52 



= 180 oil-angle. 



o 



Numerical Aperture 

 1-33 



— 180 2 water-angle. 



Numerical Aperture 

 1.00 



= 180° air-angle. 

 = 96° water-angle. 

 = 82° oil-angle. 



Numerical Aperture 



'•75 



= 97 air-angle. 



Numerical Aperture 

 •5° 



= 6o° air-angle. 



A dry objective of the full maximum air-angle of 180 

 is only able (whether the first surface is plane or concave) 

 to utilise a diameter of back lens equal to twice the focal 

 length, while an immersion lens of even only ioo" (in 

 glass) requires and utilises a larger diameter, i. e., it is 

 able to transmit more rays from the object to the image 

 than any dry objective is capable of transmitting. When- 

 ever the angle of an immersion lens exceeds twice the 

 critical angle for the immersion-fluid, i. e., 96 for water 

 or 82 for oil, its aperture is in excess of that of a dry 

 objective of 180 . 



Having settled the principle, it was still necessary, 

 however, to find a proper notation for comparing aper- 

 tures. The astronomer can compare the apertures of his 

 various telescopes by simply expressing them in inches ; 

 but this is obviously not available to the microscopist, 

 who has to deal with the ratio of two varying quantities. 



Prof. Abbe here again conferred a boon upon micro- 

 scopists by his discovery (in 1873, independently con- 

 firmed by Prof. Helmholtz shortly afterwards) that a 

 general relation existed between the pencil admitted into 

 the front of the objective and that emerging from the 

 back of the objective, so that the ratio of the semi-diam- 

 eter of the emergent pencil to the focal length of the ob- 

 jective could be expressed by the sine of half the angle 

 of aperture («) multiplied by the refractive index of the 

 medium («) in front of the objective, or n sin. u (n being 

 1.0 for air, 1.33 for water, and 1.5 for oil or balsam). 



When, then, the values in auy given cases of the ex- 

 pression n sin. u (which is known as the " numerical 

 aperture") has been ascertained, the objectives are in- 

 stantly compared as regards their aperture, and, more- 



* The explanation of the mistaken supposition that the emergent beam 

 is wider in the case of the immersion objectives because the immersion- 

 fluid abolishes the refractive action of the first plane surface of the ob- 

 jective (which, in air, reduces all pencils to 80° within the glass), belongs 

 rather to the controversial branch of the matter. It is, however, fully 

 dealt with in the papers referred to. 



