modation requisite for distinct vision, it at the same time diminishes, by 3 metre-angles, the convergence required for binocular vision.

Let, for instance, A (Fig. 124) be an object situated 20 centimetres from each eye; distinct and binocular vision of this will require 5 D and 5 metre-angles. If this same object be looked at through the lens LL' of 3 dioptries, the eyes will need 3 D less of accommodation, i.e., 5-3=2 D, if they are emmetropic. Hence the lens has had, for the accommodation, the same effect as if we had carried the object from to (from 20 to 50 centimetres), from A to B (Fig. 124).

1 m.

1 m.

At the same time, thanks to the prismatic effect of the lens, rays emanating from A will undergo, at O and O', such a deviation that they will seem to come from B. Hence the angle of convergence will be diminished, proportionately with the accommodation, by three units that is to say, by 4 metre-angles.

It will be noticed, moreover, that this is true no matter what the distance between the eyes. The farther they are from each other, the farther the points of the lens, through which they look, are from the centre. The prismatic effect of the lens increases from its centre towards its periphery, proportionately with the base-line, i.e., with the requisite effort of convergence.

In practice it would not do to order a single lens of this kind, to be worn in front of both eyes, and it would hardly be convenient. Two spectacle-glasses, however, may be cut out of it, in such a way that each eye may be furnished with that portion of the lens which is on the eye's axis, when the lens is whole. In Figure 124 this portion, for the right eye, is shaded.

Spectacles made in this way are called orthoscopic. In order to test their accuracy, it is necessary only to place them at such a distance from the wall that the glasses will produce upon it the image of a candle-flame. The images of the two glasses ought to be exactly superposed, if they form the desired segments of the same lens, and if they are separated by the distance, which would be between them in the whole lens.1

The following is the simplest way of representing to one's self this double action of a convex lens :

Let O and O' (Fig. 125) be the two eyes, and M M' the median line; A a point situated on the latter, at a distance A from each eye. In order

1 A

that it be seen distinctly, a refractive power of =a is necessary.

1 Scheffler, Die physiologische Optik., Braunschweig, 1865, vol. ii., p. 95. Compare, also, Bruecke, Arch. f. Ophth., v. 2, p. 180, 1859: dissecting spectacles, formed by two glasses, which are neither more nor less than two halves of a lens whose focal distance is 22 centimetres (4.5 D).

If we wish each eye, when fixing A, to need no more refraction than it would when looking at a more distant point, B, for instance, at a distance

B, which requires only=b of refractive power, we evidently need a convex lens whose refractive power shall be f-a-b, or, expressed in terms of

1 1 1
if F be the focal distance of the lens.
A B'

distance, F

=

Let us mark off, on the median line, the focus F of the lens at the distance F from either eye.

In order to fix, binocularly, the point A, each eye must have a convergence-angle, which is also the reciprocal of the distance A, or = a (see p. 188).

1 Α

1 In order to fix the point B, there must be a convergence of = b. And, B to change convergence toward A into convergence toward B, we evidently

1 1 1 A B F

=

need a prism whose strength is a-b=f, or -

This expression is identical with the preceding one, which ought to be the case, according to what we have shown on page 190.

Hence the angle of deviation of this prism will be the angle O FM. It represents ƒ metre-angles, if F is measured with the metre. To obtain its absolute value, let us call it 8, and designate by D=0 M half the base-line.

Suppose that we have obtained the desired refractive effect by means of a large convex lens, placed before the eyes as is indicated in Fig. 126. Then the left eye, looking through the point O of the lens, will see the point A as clearly as if it were at B.

What will be the prismatic effect of this point O of the lens? To ascertain this, let us first draw a tangent at this point, i.e., a line perpendicular to the radius C O of the lens. Repeating the same thing for the posterior surface, which we suppose to have the same curvature as the anterior, we obtain the prism M L P, which corresponds to the two points, of the respective surfaces of the lens, through which the left eye looks. Now, the angle MLS, or A, is equal to the angle M C O, because their sides are mutually perpendicular.

For the angle MC O, we have the expression:

if we designate by D the distance M O, and by R the radius of curvature of the surfaces of the lens.

The angle M L P of the prism = 2 λ or A is, therefore :

2 D
A
Ꭱ

supposing the tangent and the angle equal.

Now, the angle of deviation of a prism is, according to a well-known law of physics:

▲ ▲ (n − 1).

=

From our formula 20" (p. 55), it results, on the other hand, that

This D is nearly the same as half the base-line. The deviation ▲, produced by the lens at the point O, is, therefore, identical with that, &, which we have found above as being requisite, in order to change convergence

for A into convergence for B. Hence, the lens thus employed fulfils the aim of relieving the accommodation and convergence of the eyes, by identical quantities.

It should not be forgotten that the prismatic action of spherical glasses does not make itself felt solely in the horizontal, where it diminishes or increases convergence, and may bring about a homonymous. or crossed diplopia, but also in the vertical, or in any other direction.

This action manifests itself in a vertical or oblique diplopia, much more troublesome than a similar disturbance of equilibrium in the horizontal, whenever somewhat strong glasses are placed at different heights before the two eyes. Hence, we should be careful that the line, joining the optical centres of the glasses, is parallel with the baseline. It must, moreover, be contained in the plane of fixation passing through the latter and through the lines of fixation, so that objects seen binocularly shall appear to be where they really are.

When one looks through the edge of a convex glass, placed at a certain distance from the eye, and in such a way that its edge is just opposite the corresponding pupillary circumference, the point of fixation disappears and, with it, a more or less extended portion of its surroundings. This gap is greater in proportion as the glass is stronger and farther removed from the eye. This is easy to understand. Among the rays coming from the point of fixation, some pass outside the glass, but do not enter the pupil, since the glass entirely covers this; others are deviated by the prismatic edge represented by the border of the convex glass, so that they fall beyond the opposite margin of the pupil.

If the centre of the lens is placed in front of the pupil, the same result will, evidently, be produced for all objects situated on a line joining any point of the edge of the glass with the corresponding point of the pupillary circumference. The field of vision of an eye furnished with a convex glass will, therefore, present a larger or smaller annular hiatus corresponding to the circumference of the lens.

When one looks through the latter along its axis, this suppression of a part of the visual field is less troublesome in proportion as the glass is weaker and as the gap remains more peripheral. But it may become a serious inconvenience for persons who, like those who have undergone cataract operations, use strong glasses and sometimes look through their edges.

The spherical and chromatic aberrations presented by the convex glasses used in ophthalmology are not noticeable enough to constitute serious defects. They scarcely manifest themselves except in strong glasses, when the line of vision passes through them elsewhere than through their centres. In this case these secondary effects of convex glasses are the stronger in proportion as the part of the glass in front of the pupil is farther from the centre.

It has been attempted, without much success, to remedy these defects by making achromatic and aplanatic lenses. Crown-glass lenses are preferable to those of ordinary glass, because they are much less dispersive. Perfect achromatism is obtained by the combination of a crown-glass lens with one of flint-glass. But these combinations