surfaces of monochromatic or light of single wave length.
(a) Monochromatic Aberration. The elementary theory of optical
systems leads to the theorem; Rays of light proceeding from
any ``object point,' unite in an ``image point''; and therefore
an ``object space'' is reproduced in an ``image space.'' The
introduction of simple auxiliary terms, due to C. F. Gauss
(Dioptrische Untersuchungen, Gottingen, 1841), named the
focal lengths and focal planes, permits the determination
of the image of any object for any system (see LENS). The
Gaussian theory, however, is only true so long as the angles
made by all rays with the optical axis (the symmetrical axis
of the system) are infinitely small, i.e. with infinitesimal
objects, images and lenses; in practice these conditions are
not realized, and the images projected by uncorrected systems
are, in general, ill defined and often completely blurred,
if the aperture or field of view exceeds certain limits.
The investigations of James Clerk Maxwell (Phil.Mag., 1856;
Quart. Journ. Math., 1858, and Ernst Abbe1) showed that
the properties of these reproductions, i.e. the relative
position .and magnitude of the images, are not special
properties of optical systems, but necessary consequences of
the supposition (in Abbe) of the reproduction of all points
of a space in image points (Maxwell assumes a less general
hypothesis), and are independent of the manner in which the
reproduction is effected. These authors proved, however, that
no optical system can justify these suppositions, since they
are contradictory to the fundamental laws of reflexion and
refraction. Consequently the Gaussian theory only supplies
a convenient method of approximating to reality; and no
constructor would attempt to realize this unattainable ideal.
All that at present can be attempted is, to reproduce a single
plane in another plane; but even this has not been altogether
satisfactorily accomplished, aberrations always occur, and
it is improbable that these will ever be entirely corrected.
This, and related general questions, have been treated--besides
the above-mentioned authors--by M. Thiesen (Berlin.
Akad. Sitzber., 1890, xxxv. 799; Berlin.Phys.Ges.
Verb., 1892) and H. Bruns (Leipzig. Math. Phys.
Ber., 1895, xxi. 325) by means of Sir W. R. Hamilton's
``characteristic function'' (Irish Acad. Trans., ``Theory
of Systems of Rays,,' 1828, et seq.). Reference may also
be made to the treatise of Czapski-Eppenstein, pp. 155-161.
A review of the simplest cases of aberration will now be
given. (1) Aberration of axial points (Spherical aberration
in the restricted sense). If S (fig.5) be any optical
system, rays proceeding from an axis point O under an angle
u1 will unite in the axis point O'1; and those under an
angle u2 in the axis point O'2. If there be refraction
at a collective spherical surface, or through a thin positive
lens, O'2 will lie in front of O'1 so long as the angle
u2 is greater than u1 (``under correction''); and
conversely with a dispersive surface or lenses (``over
correction''). The caustic, in the first case, resembles
the sign > (greater than); in the second K (less than). If
the angle u1 be very small, O'1 is the Gaussian image;
and O'1 O'2 is termed the ``longitudinal aberration,''
and O'1R the ``lateral aberration'' of the pencils with
aperture u2. If the pencil with the angle u2 be that
of the maximum aberration of all the pencils transmitted,
then in a plane perpendicular to the axis at O'1 there is
a circular ``disk of confusion'' of radius O'1R, and in a
parallel plane at O'2 another one of radius O'2R2; between
these two is situated the ``disk of least confusion.''
The largest opening of the pencils, which take part in the
reproduction of O, i.e. the angle u, is generally determined
by the margin of one of the lenses or by a hole in a thin
plate placed between, before, or behind the lenses of the
system. This hole is termed the ``stop'' or ``diaphragm'';
Abbe used the term ``aperture stop'' for both the hole and
the limiting margin of the lens. The component S1 of the
system, situated between the aperture stop and the object
O, projects an image of the diaphragm, termed by Abbe the
``entrance pupil''; the ``exit pupil'' is the image formed
by the component S2, which is placed behind the aperture
stop. All rays which issue from O and pass through the aperture
stop also pass through the entrance and exit pupils, since these
are images of the aperture stop. Since the maximum aperture
of the pencils issuing from O is the angle u subtended by the
entrance pupil at this point, the magnitude of the aberration
will be determined by the position and diameter of the entrance
pupil. If the system be entirely behind the aperture stop,
then this is itself the entrance pupil (``front stop'');
if entirely in front, it is the exit pupil (``back stop'').
If the object point be infinitely distant, all rays received
by the first member of the system are parallel, and their
intersections, after traversing the system, vary according
to their ``perpendicular height of incidence,'' i.e. their
distance from the axis. This distance replaces the angle
u in the preceding considerations; and the aperture, i.e.
the radius of the entrance pupil, is its maximum value.
(2) Aberration of elements, i.e. smallest objects at right
angles to the axis.--If rays issuing from O (fig. 5) be
concurrent, it does not follow that points in a portion
of a plane perpendicular at O to the axis will be also
concurrent, even if the part of the plane be very small.
With a considerable aperture, the neighbouring point N will
be reproduced, but attended by aberrations comparable in
magnitude to ON. These aberrations are avoided if, according to
Abbe, the ``sine condition,'' sin u'1/sin u1=sin u'2jsin
u2, holds for all rays reproducing the point O. If the
object point O be infinitely distant, u1 and u2 are
to be replaced by pi and h2, the perpendicular heights of
incidence; the ``sine condition', then becomes sin u,1jh1
sin u'2/h2. A system fulfilling this condition and free
from spherical aberration is called ``aplanatic'' (Greek
a-, privative, plann, a wandering). This word was
first used by Robert Blair (d. 1828), professor of practical
astronomy at Edinburgh University, to characterize a superior
achromatism, and, subsequently, by many writers to denote
freedom from spherical aberration. Both the aberration of axis
points, and the deviation from the sine condition, rapidly
increase in most (uncorrected) systems with the aperture.
(3) Aberration of lateral object points (points beyond the
axis) with narrow pencils. Astigmatism.---A point O (fig.
6) at a finite distance from the, axis (or with an infinitely
distant object, a point which subtends a finite angle at the
system) is, in general, even then not sharply reproduced, if
the pencil of rays issuing from it and traversing the system
is made infinitely narrow by reducing the aperture stop; such
a pencil consists of the rays which can pass from the object
point through the now infinitely small entrance pupil. It
is seen (ignoring exceptional cases) that the pencil does
not meet he refracting or reflecting surface at right angles;
therefore it is astigmatic (Gr. a-, privative, stigmia, a
point). Naming the central ray passing through the entrance
pupil the ``axis of the pencil,' or ``principal ray,'' we
can say: the rays of the pencil intersect, not in one point,
but in two focal lines, which we can assume to be at right
angles to the principal ray; of these, one lies in the plane
containing the principal ray and the axis of the system,
i.e. in the ``first principal section'' or ``meridional
section,', and the other at right angles to it, i.e. in the
second principal section or sagittal section. We receive,
therefore, in no single intercepting plane behind the system,
as, for example, a focussing screen, an image of the object
point; on the other hand, in each of two planes lines O' and
O" are separately formed (in neighbouring planes ellipses are
formed), and in a plane between O' and O" a circle of least
confusion. The interval O'O", termed the astigmatic difference,
increases, in general, with the angle W made by the principal
ray OP with the axis of the system, i.e. with the field of
view. Two ``astigmatic image surfaces'' correspond to one
object plane; and these are in contact at the axis point; on
the one lie the focal lines of the first kind, on the other
those of the second. Systems in which the two astigmatic
surfaces coincide are termed anastigmatic or stigmatic.
Sir Isaac Newron was probably the discoverer of astigmation;
the position of the astigmatic image lines was determined by
Thomas Young (A Course of Lectures on Natural Philosophy,
1807); and the theory has been recently developed by A.
Gullstrand (Skand. Arch. f. physiol., 1890, 2, p. 269;
Allgemeine Theorie der monochromat. Aberrationen, etc.,
Upsala, 1900; Arch. f. Ophth., 1901, 53, pp. 2, 185). A
bibliography by P. Culmann is given in M. von Rohr's Die
Bilderzeugung in opitschen Instrumenten (Berlin, 1904).
(4) Aberration of lateral object points with broad pencils.
Coma. ---By opening the stop wider, similar deviations arise
for lateral points as have been already discussed for axial
points; but in this case they are much more complicated.
The course of the rays in the meridional section is no longer
symmetrical to the principal ray of the pencil; and on an
intercepting plane there appears, instead of a luminous
point, a patch of light, not symmetrical about a point, and
often exhibiting a resemblance to a comet having its tail
directed towards or away from the axis. From this appearance
it takes its name. The unsymmetrical form of the meridional
pencil--formerly the only one considered--is coma in the
narrower sense only; other errors of coma have been treated by
A. Konig and M. von Rohr (op. cit.), and more recently by
A. Gullstrand (op. cit.; Ann. d. Phys., 1905, 18, p. 941).
(5) Curvature of the field of the image.---If the above errors
be eliminated, the two astigmatic surfaces united, and a sharp
image obtained with a wide aperture--there remains the necessity
to correct the curvature of the image surface, especially when
the image is to be received upon a plane surface, e.g. in
photography. In most cases the surface is concave towards the system.
(6) Distortion of the image.--If now the image be sufficiently
sharp, inasmuch as the rays proceeding from every object point
meet in an image point of satisfactory exactitude, it may happen
that the image is distorted, i.e. not sufficiently like the
object. This error consists in the different parts of the
object being reproduced with different magnifications; for
instance, the inner parts may differ in greater magnification
than the outer (``barrel-shaped distortion''), or conversely
(``cushion-shaped distortion'') (see fig. 7). Systems free
of this aberration are called ``orthoscopic'' (orthos ,
right, skopein to look). This aberration is quite distinct
from that of the sharpness of reproduction; in unsharp,
reproduction, the question of distortion arises if only parts of
the object can be recognized in the figure. If, in an unsharp
image, a patch of light corresponds to an object point, the
``centre of gravity'' of the patch may be regarded as the image
point, this being the point where the plane receiving the
image, e.g. a focussing screen, intersects the ray passing
through the middle of the stop. This assumption is justified
if a poor image on the focussing screen remains stationary
