7 MICROSCOPE AND ACCESSORIES. 



According to some other authors the angle of aperture is the angle between the 

 extreme rays from the focal point which can be transmitted through the entire 

 objective. This would give a somewhat greater angle than by the first method as 

 the focal point of the objective is nearer to it than the axial point of the object 

 (PI. I, Fig. i, 3 and 5). 



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

 and the shortness of the equivalent focal distance (\ 7). If all objectives were dry 

 or all water or homogeneous immersion a comparison of the angular aperture 

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

 mitted by different objectives ; but as some are dry, others water and still others 

 homogeneous immersion, one can see at a glance (see PI. V, Fig. 42, 43, 44) that 

 other things being equal, the dry objective (Fig. 42) receives less light than the 

 water immersion, and the water immersion (Fig. 43) less than the homogeneous 

 immersion (Fig. 44). In order to render comparison accurate between different 

 kinds of objectives, Professor Abbe takes into consideration the rajs actually pass- 

 ing 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" was introduced by Abbe to indicate 

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

 transmitting them to the image, and the aperture of a microscopic objective is 

 therefore determine J by the ratio between its focal length and the diameter of the 

 emergent pencil at the point of its emergence, that is the utilized diameter of a 

 single-lens objective or of the back lens of a compound objective." 



Numerical Aperture (abbreviated N.A.) is then the ratio of the diameter of the 

 emergent pencil to the focal length of the lens, or as usually expressed, the factors 

 being more readily obtainable, it is the index of refraction of the medium in front 

 of the objective (i.e., air for dry, and water or homogeneous fluid for immersion 

 objectives) multiplied by the sine of half the angle of aperture. The usual form- 

 ula is N. A. = a sin u ; N. A. representing numerical aperture, n the index of refrac- 

 tion of the substance in front of the objective, and u the semi-angle of aperture. 



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 , N.A. = « sin u. In the case of dry objectives the medium in front of the 

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

 aperture is •Mp-° = 53°. By consulting a table of natural sines it will be found 

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



With the water immersion objective in the same way N. A. =u sin u. In this 

 case the medium in front of the objective is water, and its index of refraction is 

 1.33, whence 7/ = 1.33. Half the angular aperture is - 9 Z 40 ==47 , and by consult- 

 ing a table of natural sines, the sine of 47 is found to be 0.731 i.e. sin u = 0.731, 

 whence N.A. =n or 1.33 X sin u or 0.731 =0.972. 



With the oil immersion in the same way N.A. =n sin u ; n or the index of refrac- 

 tion of the homogeneous fluid in front of the objective is 1.52, and the semi-angle 



