<■//. /.] MICROSCOPE A ND ACCESSORIES. 17 



In general, the angle increases with the size of the lenses forming the objective 

 and the shortness of the equivalent focal distance ($ [3). If all objectives were 

 dry or all water or all homogeneous immersion a comparison of the angular aper- 

 ture would give one a good idea of the relative number of image forming rays 



Fig. 26. Diagram illustrating the angular aperture of a 



microscopic objective. Only the front tens of the objective is 

 shown. 



Axis, the principal optic a.vis of the objective. 



If / , />' C. the most divergent rays that can enter the objective, 

 they mark the angular aperture. A B D or C B D hat f the 

 angular aperture. This is designated by u in making Nu- 

 merical Aperture computations. See the table, \ 30. 



transmitted by different objectives; but as some are dry, 

 others water and still others homogeneous immersion, one 

 can see at a glance that, other things being equal, the 

 dry objective (Fig. 27) receive- less light than the water immersion, and the 

 water immersion (Fig. 28) less than the homogeneous immersion (Fig. 29). 

 In order to render comparison accurate between different kinds of objectives, 

 Professor Abbe takes into consideration the rays actually passing from the back 

 combination of the objective to form the real image ; he thus takes into account 

 the medium in front of the objective as well as the angular aperture. The term 

 " Numerical Aperture," {N. A.) was introduced by Abbe to indicate the capacity 

 of an optical instrument "for receiving rays from the object and transmitting 

 them to the image. 



i< 29. Numerical Aperture (abbreviated X. A.), as now employed for micro- 

 scope objectives, is the ratio of the semi-diameter of the emergent pencil to the 

 focal length of the lens. Or as the factors are more readily obtainable it is 

 simpler to utilize the relationship shown in the La Grange-Helmholtz formula, 

 and indicate the aperture by the expression : X. A. = n sin it. In this formula ;/ 

 is the index of refraction of the medium in front of the objective i air, water or 

 homogeneous liquid 1, and sin it is the sine of half the angle of aperture ( Fig. 26, 

 D B A . For the mathematical discussion showing that the expressions 



semi-diameter of emergent pencil . , . . ., 



tt r^ r- ,-■ 1 — = n sm u, the student is referred to the Journal 



tocal length of the lens 



of the Royal Microscopical Society, 1S81, pp. 392-395. 



For example, take three objectives each of 3 mm. equivalent focus, one being a 

 dry, one a water immersion, and one a homogeneous immersion. Suppose that 

 the dry objective has an angular aperture of 106 , the water immersion of 94 and 

 the homogeneous immersion of 90 . Simply compared as to their angular aper- 

 ture, without regard to the medium in front of the objective, it would look as if 

 the dry objective would actually take in and transmit a wider pencil of light than 

 either of the others. However, if the medium in front of the objective is con- 

 sidered, that is to say, if the numerical instead of the angular apertures are com- 

 pared, the results would be as follows : Numerical Aperture of a dry objective of 

 106 , X. A. =;/ sin it. In the case of dry objectives the medium in front of the 

 objective being air, the index of refraction is unity, whence n = 1. Half the 

 angular aperture is ' '1*-° = 53 . By consulting a table of natural sines it will be 

 found that the sine of 53 is 0.799, whence X. A. =n or 1 X sin u or 0.799 = 0.799.* 



* I 29a. Interpolation. In practice, as in solving problems similar to those on 

 the following pages and those in refraction on p. 50, if one cannot find a sine exactly 



