PROCEEDINGS OF THE SOCIETY. 159 



" I adopt both of these suggestions for my illustration. 

 " Draw B C through the centre O parallel to H L, and D q, 'E q, 

 through the points B and C respectively, and join B Q and C Q, 

 B Q C will be the ' angular aperture ' of the hemispherical lens, 

 which will be about 113° or 114°, and D g E the angular pencil, after 

 refraction, about 66°, the spaces between the radiant point Q and 

 the front of the lens being occupied by media of about the same 

 refractive index as the glass. 



" Suppose, now, that we draw the ray Q G at an angle of 40° 49' 

 with the axis (the critical angle), and P q through the point G, it be- 

 comes possible to arrive at some faint idea of the probable way in 

 which the fallacy of the ' supposed limit ' may have given rise to 

 the notion of an immersion lens being able to refract rays that a 

 dry one could not ; for if the lens BAG were in air, it is manifest 

 that the ray F G Q would, on reaching the surface B 0, be refracted 

 to O, and all the rays between F q and D q would suffer internal 

 reflection, never getting out of the lens at all, and erroneously 

 conclude that they must therefore necessarily be lost by a dry 

 lens. 



" But those who argue thus overlook the simple fact that it is by 

 no means necessary to let the lens intercept the pencil DqE at the 

 points B C if it be intended for use as a dry lens ; nor, in fact, is it 

 necessary to employ so large a portion of the sphere as a hemisphere, 

 nor even to use a lens of the same radius of curvature as is suitable 

 for the immersion lens. 



" However, to show what can be done with a lens of the same 

 radius of curvature, draw the pencils KqM and D 5 E, Fig. 33, exactly 

 similar to those in Fig, 32, and let the lens b' a' c' of the same radius 

 of curvature cut the said pencil at the points h' c', which are situated 

 so as to form an angle of 81° 58' with the centre 0' of curvature of 

 the lens. The lines D q,'Eq will be refracted at b' and c', and falling 

 on the flat front surface at x and y — at a smaller angle of inclination 

 with the normal than the critical angle — will pass out of the lens and 

 come to a focus at 2, the angle xz y being a larger angle than B Q C , 

 that of the immersion lens. 



" The thickness of the lens b' a' c' for a dry objective can be varied 

 to a great extent to suit the exigencies of construction without loss of 

 aperture, because the front surface performs a very large part of the 

 refraction, while in the immersion lens the back surface of the front 

 lens alone is effective ; but in any case there must be more ' work- 

 ing distance ' available with an immersion lens than with a dry one, 

 hence one of its most important advantages. 



"I have now demonstrated beyond dispute the following facts, 

 viz. : — 



" That a dry lens can have as large an ' angular aperture ' as an 

 immersion one, and that the assumed difference of aperture between 

 dry and immersion lenses does not exist ; 



" That no lens can have an ' aperture ' of any kind which exceeds 

 that of 180° angular in air ; and 



" That consequently the table of ' numerical apertures ' published 



