As the result of this erroneous supposition, it followed that an immersion objective could have no advantage over a dry objective, except in the case of the latter being used upon a balsam-mounted object, its aperture then being (as was supposed) “cut down.” The error lies simply in overlooking the fact that the rays which are reflected back when the object is mounted in balsam (Fig. 37_a_) are not rays which are found when the object is in air (Fig. 37), but are _additional and different_ rays which do not exist in air, as they cannot be emitted in a substance of so low a refractive index.
Lastly, it should also be noted that it is numerical and not angular aperture which measures the quantity of light admitted to the objective by different pencils.
First take the case of the medium being the same. The popular notion of a pencil of light may be illustrated by Fig. 38, which assumes that there is equal intensity of emission in all directions, so that the quantity of light contained in any given pencils may be compared by simply comparing the contents of the solid cones. The Bouguer-Lambert law, however, shows that the quantity of light emitted by any bright point varies with the obliquity of the direction of emission, being _greater_ in a perpendicular than in an oblique direction. The rays are less intense in proportion as they are more inclined to the surface which emits them, so that a pencil is not correctly represented by Fig. 38, but by Fig. 38_a_, the density of the rays decreasing continuously from the vertical to the horizontal, and the squares of the sines of the semi-angles (_i.e._, of the numerical aperture) constituting the true measure of the quantity of light contained in any solid pencil.
If, again, the media are of different refractive indices, as air (1·0), water (1·33), and oil (1·52), the total amount of light emitted over the whole 180° from radiant points in these media under a given illumination is not the same, but is _greater_ in the case of the media of greater refractive indices in the ratio of the squares of those indices (_i.e._, as 1·0, 1·77 and 2·25). The quantity of light in pencils of different angle and in different media must therefore be compared by squaring the product of the sines and the refractive indices, _i.e._ (_n_ Sin _u_^2), for the square of the numerical aperture.