330 THE NORTHERN MICROSCOPIST. 

up very wide angles of light, and we have the means of condensing 
it upon the object by suitable lenses placed beneath the stage. If 
we take a very wide-angled homogeneous immersion objective (say 
1.43 N A) ze, one constructed upon a formula which requires the 
immersion medium to be of the same refractive index as the crown 
glass of which the front lens is composed, and measure its aperture 
in air by means of Abbe’s apertometer, we shall find it to be 
represented by 1.0, or the greatest air-angle of 180°. If we now 
place a drop of water in front of the objective, we shall find that 
the aperture has suddenly expanded to 1.33, the numerical equiva- 
lent of 180° in water, and one-third greater than the aperture in air. 
If again, we remove the drop of water and substitute a drop of oil, 
we shall find that the aperture has again extended itself, that the 
full capacity of the objective is now seen to be in excess of 1.33, 
and therefore greater than 180° in water. 
The truth revealed by the apertometer is confirmed in another 
way. ‘The researches of Professor Abbe, on what is known as the 
“law of aplanatic convergence,” have shown that the aperture of 
an objective requires a certain ratio to be observed between its 
focal length and the utilised diameter of its back lens measured at 
the plane of emergence. (See Fig. 34.) This ratio, in a dry 
objective of 180° air-angle, supposing that angle to be possible, 
will be as 1 to 2; or, in other words, the focal length will be equal 
to the semi-diameter, which latter may, therefore, be expressed in 
terms of the focal length by 1.0. In a water-immersion objective 
of 180° water-angle, this ratio must be increased by one-third, and 
the expression for the semi-diameter will be 1.33. Similarly an 
oil-immersion objective of the same angle will require this ratio to 
be expressed by 1.52. In Fig. 39 is shown the relative proportion 
of the diameters of the back lenses of three objectives of the same 
focal length and angle, in air, water and oil respectively. As 
these numbers represent also the refractive indices of the immersion 
media in each case, as well as the numerical apertures of the 
objectives employed, it is very evident that aferture, in the sense 
of opening merely, must be compounded of the sine of the semi- 
angle of aperture with the refractive index of the immersion 
medium, and that angles greater than 180° in air or water are not 
mere theoretical expressions. 
The practical advantage of the new notation over the old may be 
illustrated by a very simple comparison. In comparing two dry 
objectives of 140° and 160° angular aperture respectively, one 
would naturally suppose that the real aperture of the one exceeded 
that of the other by one-seventh. The numerical notation informs 
us that this is not the case, but that their real apertures are as 94 
to 98, and that the excess of the greater aperture is not one- 
seventh but only one-twenty-fourth, Other inconsistencies of the 

