2 and 3). At the present time constructors almost always employ
the inverse method: they compose a system from certain, often
quite personal experiences, and test, by the trigonometrical
calculation of the paths of several rays, whether the system
gives the desired reproduction (examples are given in A.
Gleichen, Lehrbuch der geometrischen Optik, Leipzig and Berlin,
1902). The radii, thicknesses and distances are continually
altered until the errors of the image become sufficiently
small. By this method only certain errors of reproduction are
investigated, especially individual members, or all, of those named
above. The analytical approximation theory is often employed
provisionally, since its accuracy does not generally suffice.
In order to render spherical aberration and the deviation from
the sine condition small throughout the whole aperture, there
is given to a ray with a finite angle of aperture u* (width
infinitely distant objects: with a finite height of incidence
h*) the same distance of intersection, and the same sine
ratio as to one neighbouring the axis (u* or h* may not be
much smaller than the largest aperture U or H to be used in the
system). The rays with an angle of aperture smaller than
u* would not have the same distance of intersection and
the same sine ratio; these deviations are called ``zones,''
and the constructor endeavours to reduce these to a minimum.
The same holds for the errors depending upon the angle of
the field of view, w: astigmatism, curvature of field
and distortion are eliminated for a definite value, w*,
``zones of astigmatism, curvature of field and distortion,'
attend smaller values of w. The practical optician names
such systems: ``corrected for the angle of aperture u*
(the height of incidence h*) or the angle of field of
view w*.'' Spherical aberration and changes of the sine
ratios are often represented graphically as functions of the
aperture, in the same way as the deviations of two astigmatic
image surfaces of the image plane of the axis point are
represented as functions of the angles of the field of view.
The final form of a practical system consequently rests
on compromise; enlargement of the aperture results in
a diminution of the available field of view, and vice
versa. The following may be regarded as typical:--(1)
Largest aperture; necessary corrections are--for the axis
point, and sine condition; errors of the field of view are
almost disregarded; example-- high-power microscope objectives.
(2) Largest field of view; necessary corrections are--for
astigmatism, curvature of field and distortion; errors of
the aperture only slightly regarded; examples--photographic
widest angle objectives and oculars. Between these extreme
examples stands the ordinary photographic objective: the
portrait objective is corrected more with regard to aperture;
objectives for groups more with regard to the field of
view. (3) Telescope objectives have usually not very large
apertures, and small fields of view; they should, however,
possess zones as small as possible, and be built in the simplest
manner. They are the best for analytical computation.
(b) Chromatic or Colour Aberration. In optical systems
composed of lenses, the position, magnitude and errors
of the image depend upon the refractive indices of the
glass employed (see LENS, and above, ``Monochromatic
Aberration''). Since the index of refraction varies with
the colour or wave length of the light (see DISPERSION),
it follows that a system of lenses (uncorrected) projects
images of different colours in somewhat different places
and sizes and with different aberrations; i.e. there are
``chromatic differences'' of the distances of intersection,
of magnifications, and of monochromatic aberrations. If
mixed light be employed (e.g. white light) all these images
are formed; and since they are ail ultimately intercepted
by a plane (the retina of the eye, a focussing screen of a
camera, &c.), they cause a confusion, named chromatic
aberration; for instance, instead of a white margin on a dark
background, there is perceived a coloured margin, or narrow
spectrum. The absence of this error is termed achromatism,
and an optical system so corrected is termed achromatic.
A system is said to be ``chromatically under-corrected''
when it shows the same kind of chromatic error as a thin
positive lens, otherwise it is said to be ``over-corrected.''
If, in the first place, monochromatic aberrations be neglected
---in other words, the Gaussian theory be accepted---then
every reproduction is determined by the positions of the focal
planes, and the magnitude of the focal lengths, or if the focal
lengths, as ordinarily happens, be equal, by three constants of
reproduction. These constants are determined by the data
of the system (radii, thicknesses, distances, indices, &c.,
of the lenses); therefore their dependence on the refractive
index, and consequently on the colour, are calculable (the
formulae are given in Czapski-Eppenstein, Grundzuge der
Theorie der optischen Instrumente (1903, p. 166). The
refractive indices for different wave lengths must be known
for each kind of glass made use of. In this manner the
conditions are maintained that any one constant of reproduction
is equal for two different colours, i.e. this constant is
achromatized. For example, it is possible, with one thick
lens in air, to achromatize the position of a focal plane of
the magnitude of the focal length. If all three constants
of reproduction be achromatized, then the Gaussian image for
all distances of objects is the same for the two colours,
and the system is said to be in ``stable achromatism.''
In practice it is more advantageous (after Abbe) to determine
the chromatic aberration (for instance, that of the distance
of intersection) for a fixed position of the object, and
express it by a sum in which each component conlins the amount
due to each refracting surface (see Czapski-Eppenstein, op.
cit. p. 170; A. Konig in M. v. Rohr's collection, Die
Bilderzeugung, p. 340). In a plane containing the image point
of one colour, another colour produces a disk of confusion;
this is similar to the confusion caused by two ``zones'' in
spherical aberration. For infinitely distant objects the
radius Of the chromatic disk of confusion is proportional to
the linear aperture, and independent of the focal length (vide
supra, ``Monochromatic Aberration of the Axis Point''); and
since this disk becomes the less harmful with au increasing
image of a given object, or with increasing focal length,
it follows that the deterioration of the image is propor-,
tional to the ratio of the aperture to the focal length,
i.e. the ``relative aperture.'' (This explains the gigantic
focal lengths in vogue before the discovery of achromatism.)
Examples.--(a) In a very thin lens, in air, only one constant
of reproduction is to be observed, since the focal length and
the distance of the focal point are equal. If the refractive
index for one colour be n, and for another n+dn, and the
powers, or reciprocals of the focal lengths, be f and f + d
f, then (1) df/f = dn/(n-1) = 1/n; dn is called
the dispersion, and n the dispersive power of the glass.
(b) Two thin lenses in contact: let f1 and f2 be
the powers corresponding to the lenses of refractive indices
n1 and n2 and radii r'1, r"1, and r'2,
r"2 respectively; let f denote the total power, and d
f, dn1, dn2 the changes of f, n1, and n2
with the colour. Then the following relations hold:--
(2) f = f1-f2== (n1 - 1)(1/r'1-1/r''1) +(n2-1)(1/
r'2 - 1/r''2) = (n1 - 1)k1 + (n2 - 1)k2; and
(3) df = k1dn1 + k2dn2.
For achromatism df = 0, hence, from (3),
(4) k1/k2 = -dn2 / dn1, or f1/f2 = -n1/
n2. Therefore f1 and f2 must have different algebraic
signs, or the system must be composed of a collective and a
dispersive lens. Consequently the powers of the two must be
different (in order that f be not zero (equation 2)), and
the dispersive powers must also be different (according to 4).
Newton failed to perceive the existence of media of
different dispersive powers required by achromatism;
consequently he constructed large reflectors instead of
refractors. James Gregory and Leonhard Euler arrived at the
correct view from a false conception of the achromatism of
the eye; this was determined by Chester More Hall in 1728,
Klingenstierna in 1754 and by Dollond in 1757, who constructed
the celebrated achromatic telescopes. (See TELESCOPE.)
Glass with weaker dispersive power (greater v) is named
``crown glass''; that with greater dispersive power, ``flint
glass.'' For the construction of an achromatic collective lens
(f positive) it follows, by means of equation (4), that a
collective lens I. of crown glass and a dispersive lens II. of
flint glass must be chosen; the latter, although the weaker,
corrects the other chromatically by its greater dispersive
power. For an achromatic dispersive lens the converse must be
adopted. This is, at the present day, the ordinary type,
e.g., of telescope objective (fig. 10); the values of
the four radii must satisfy the equations (2) and (4). Two
other conditions may also be postulated: one is always the
elimination of the aberration on the axis; the second either
the ``Herschel'' or ``Fraunhofer Condition,'' the latter being
the best vide supra, ``Monochromatic Aberration''). In
practice, however, it is often more useful to avoid the second
condition by making the lenses have contact, i.e. equal
radii. According to P. Rudolph (Eder's Jahrb. f. Photog.,
1891, 5, p. 225; 1893, 7, p. 221), cemented objectives of
thin lenses permit the elimination of spherical aberration
on the axis, if, as above, the collective lens has a smaller
refractive index; on the other hand, they permit the elimination
of astigmatism and curvature of the field, if the collective
lens has a greater refractive index (this follows from the
Petzval equation; see L. Seidel, Astr. Nachr., 1856, p.
289). Should the cemented system be positive, then the more
powerful lens must be positive; and, according to (4), to the
greater power belongs the weaker dispersive power (greater
v), that is to say, clown glass; consequently the crown
glass must have the greater refractive index for astigmatic
and plane images. In all earlidr kinds of glass, however,
the dispersive power increased with the refractive index;
that is, v decreased as n increased; but some of the
Jena glasses by E. Abbe and O. Schott were crown glasses of
high refractive index, and achromatic systems from such crown
glasses, with flint glasses of lower refractive index, are
called the ``new achromats,'' and were employed by P. Rudolph
in the first ``anastigmats'' (photographic objectives).
Instead of making df vanish, a certain value can be assigned
to it which will produce, by the addition of the two lenses,
any desired chromatic deviation, e.g. sufficient to eliminate
one present in other parts of the system. If the lenses I.
and II. be cemented and have the same refractive index for one
colour, then its effect for that one colour is that of a lens
of one piece; by such decomposition of a lens it can be made
chromatic or achromatic at will, without altering its spherical
effect. If its chromatic effect (df/f) be greater than
that of the same lens, this being made of the more dispersive
of the two glasses employed, it is termed ``hyper-chromatic.''
For two thin lenses separated by a distance D the condition
for achromatism is D = v1f1+v2f2; if v1=v2
(e.g. if the lenses be made of the same glass), this reduces
