Hitherto plane objects only have been considered. When we deal with objects of other shape it is self-evident that angular aperture as such will have a marked influence on the appearance of the object, and whether that influence is useful or not, it must of course be taken into account in interpreting what is actually seen on viewing an object.

It would seem then that the "angular aperture" of an objective should be stated as well as the "numerical aperture." When it is known that a lens admits a pencil of such and such angular breadth from an object in a medium of given refractive index, the complete description of the lens is given in all qualities except magnifying power, though we still want the standard of comparison afforded by the numerical aperture notation. Take the case referred to by Prof. Abbe, say a cubical crystal of common salt. We do not see clearly at the same time the horizontal face of the crystal and its vertical sides, but by lowering the objective a narrow band on the four vertical sides is fairly focused, and by observing the apparent dimensions of this square band we are able to say that the crystal is a true rectangular parallelopiped at any rate. Moreover the clearness of the image of the band will depend evidently on the angular aperture of the lens.

Again, if oblique illumination is employed, what was at first symmetrical about the axis of the instrument is now symmetrical about an axis forming a definite angle with that axis, and so angular aperture will be important as such.

These considerations are perhaps too evident to require notice, but they appear to me to have some weight.

Lastly, it may be remarked that by law C the "depth of focus" in an immersion lens is greater than that of a dry air lens in the proportion of N: 1, by formula C.

66

Note by Prof. Abbe.

On the preceding paper Prof. Abbe writes as follows:

I agree that the measure of aperture is a photometrical one in principle. But by the expression, "number of rays" as opposed to mere quantity of light," I desire to convey that the bearing of the notion is not confined to the photometrical functions of the lenses. The expression "quantity of light" would imply the intensity of the rays, which must be excluded in the estimation of aperture, because a greater intensity does not compensate for a smaller angle in regard to "aperture"; whilst it does so in regard to quantity of light, and a purely photometrical measure would have to be based on the estimation of the rays in the whole cone, not only in a plane section. In that respect the difference is, in fact,

that the one is measured in one dimension, the other "in two dimensions," as is said by the author. The author makes the same difference (1) by excluding the intensity of the rays, and (2) by introducing the square root of the photometrical equivalents of the angles as the measure of "aperture.'

This being understood, I should agree that it is better to base the definition of numerical aperture upon photometrical principles directly, instead of on an indirect demonstration, by means of the ratio of linear aperture to focal length, which ratio should be considered as a secondary expression of aperture.

IX.-Note on the Proper Definition of the Amplifying Power of a Lens or Lens-system.

By Prof. E. ABBE, Hon. F.R.M.S.†

(Read 12th March, 1884.)

THE generally adopted notion of "linear amplification at a certain distance is in fact a very awkward and irrational way of defining the "amplifying power" of a lens or a lens-system. Unfortunately, there is little hope that a more rational expression will be generally adopted, because it will seem to be "too abstract," but it may, nevertheless, be useful to consider the following:

In the formula N = the " amplification" of one and the same system varies with the length 1, or the "distance of vision," and an arbitrary conventional value of 7 (e. g. 10 in. or 250 mm.) must be introduced, in order to obtain comparable figures. The actual "linear amplification" of a system is, of course, different, in the case of a short-sighted eye, which projects the image at a distance of 100 mm., and a long-sighted one which projects it at 1000 mm. Nevertheless, the "amplifying power" of every system is always the same for both, because the short-sighted and the longsighted observers obtain the image of the same object under the same visual angle, and consequently the same real diameter of the retinal image. That this is so will be seen from fig. 48, where the thick lines show the course of the rays for a short-sighted eye, and the thin lines for a long-sighted one, the eye in each case being supposed at the posterior principal focus of the system.

The semi-visual angle u* under which an object of semidiameter h is seen, is the same for both observers, as the change resulting from the different positions of the object concerns only the degree of divergence of the various pencils from the various points of the object (and the image), and does not alter the refraction of the principal (central) rays from the various points.

This consideration leads to an expression of the "power" which is in conformity to the last-mentioned salient fact. The quotient

tan u*
h

where u is the semi-visual angle corresponding to h the semidiameter of the object, is a constant quantity for every system, not depending on the particular circumstances of the observing eye; and this quotient indicates, obviously, the greater or smaller visual

†The original paper is written by Prof. Abbe in English.

angle, under which a given length h is displayed by the system to every eye. The numerical value of this quotient gives the tangent of the visual angle, under which the unit of length (h = 1) is shown through the system. This quotient, therefore-i. e. the ratio of tan u toh, or the visual angle corresponding to the unit of length, measured by the tangent—is the rational expression of the

FIG. 48.

magnification or "power" of an optical system, because every observer will see every object enlarged through different systems in the exact proportion of the value of that quotient.

From the fundamental definition of the "equivalent focallength" of a lens-system results the identity of the above-mentioned quotient with the reciprocal value of ƒ (focal length): we have

Consequently the reciprocal of the focal length of a system is, by itself, the proper expression of its amplifying power, because this reciprocal expresses numerically the visual angle (measured by the tangent) under which the unit of length appears through the system.

This very simple expression of the amplifying power of a lenssystem is a strict one, it is true, under the supposition only which was mentioned above, that the place of the eye be at the posterior principal focus of the system, or, what is the same thing, that the principal rays from the various points of the object cross at that focus.

As far as the compound Microscope is in question, no other

case needs to be considered. For the "eye-point" of the Microscope coincides, practically, with the posterior principal focus of the total system. With a hand-magnifier, however, the eye may change its place to some extent, and the crossing-point of the principal rays will therefore be subjected to deviations from the posterior focus of the system. In regard to this more general case the exact formula for determining the power of a system in the manner indicated above is

in which denotes once more the distance of distinct vision of the observing eye, and d the deviation of the said crossing point, or the place of the eye, from the posterior focus. (d must be introduced with positive sign if the focus is behind the eye, and with negative sign if it is in front.)

According to this general formula the exact ratio of the visual angle to the linear magnitude of the object is expressed by a principal term which is independent of all particular cir

cumstances, and an additional term

d

ī

position and the accommodation of the observer's eye. As in all practical cases will be a small fraction, the additional term indicates merely a small correction, and this correction alone depends on the distance of vision and the place of the eye. The simple reciprocal of the focal length will therefore afford in all cases a proper measure of the amplifying power of a lens-system, because it expresses that component of the amplifying power which is inherent in the system itself, and independent of the variable circumstances under which it may be used.

The other generally adopted expression of the power by N = may be put on a somewhat more rational basis than is generally done, by defining the length 7 (10 in.) not as "distance of distinct vision," but rather as "distance of projection of the image." As far as "distinct vision" is assumed for determining the amplification, the value of N has no real signification at all in regard to an observer who obtains distinct vision at 50 in. instead of 10 in., and in fact many microscopists declare the ordinary figures of amplification to be useless for them, because they cannot observe the image at the supposed distance. It appears as if-and many have this opinionthe performance of the Microscope in regard to magnification depended essentially on the accommodation of the observer's eye.