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III. — Note on Reduced Apertures. 

 By Eev. S. Leslie Brakey, M.A. 



When an object is immersed in any liquid it has been shown, with 

 superabundant proof, that the aperture of the object-glass cannot 

 exceed a certain determinate limit, which limit varies with the 

 nature of the medium. But the limit so fixed is only a maximum, 

 which cannot be exceeded ; that is, it is not to be understood that 

 it will in general be reached, or even nearly approached. It is, in 

 fact, the angle which corresponds to the theoretical limit of 180° 

 of aperture in air. As in air every glass falls short, more or 

 less, of this angle, which is only theoretically possible, so with pre- 

 servative media the actual angle will always be less than the 

 maximum referred to. Its absolute magnitude depends upon the 

 absolute magnitude of the angle in air, increasing and diminishing 

 along with it. To determine its amount for any glass, we may of 

 course, if we like, proceed simply by experiment, as exemplified in 

 the case of the glass examined and reported upon in the January 

 number of this Journal, at p. 29. But the kind of experimenting 

 required for such cases is troublesome ; and costly as involving a 

 special apparatus ; and with some fluids dangerous, since the obliquity 

 of the position required in immersing the whole front will in general 

 necessitate the immersion of the working parts (the screw-collar 

 and screw of the nose-piece). It is very unlikely, therefore, that 

 any microscopist will be found so zealous as to work this experi- 

 ment himself for his own glasses. Fortunately, this is unnecessary. 

 In a letter inserted in the November number of last year, at p. 247, 

 I observed that the actual angle may always be found by calculation, 

 without experiment. The calculation is easy, and its rationale not 

 difficult to follow ; and this it is the purpose of the present note to 

 point out. 



(Fig. 1) is the front of the object-glass, F the focus for air, 

 and F the extreme ray, refracted to X. Then a will be the 

 semi-aperture, and of course at the same time the angle of incidence. 

 F' being the focus for the new medium, the extreme ray F' O will 

 also be refracted into X, so that /3 is the common angle of re- 

 fraction. Now the sine of a' is to the sine of /3 as the index of the 

 medium to the index of the glass, inversely ; and the sine of /3 is 

 to the sine of a as the index of the glass to the index of the air, 

 inversely. Compounding these ratios, the sine of a' is to the sine 

 of a as the index of air to the index of the medium ; that is (since 

 the index for air is unity), the sine of a is equal to the sine of a 

 divided by the index of the medium. 



Therefore to find the aperture for any preservative medium : — 

 Find by experiment the semi-aperture in air, divide its sine by the 



