when the aperture is diminished; in practice, this generally
occurs. This ray, named by Abbe a ``principal ray'' (not to be
confused with the ``principal rays'' of the Gaussian theory),
passes through the centre of the enttance pupil before the first
refraction, and the centre of the exit pupil after the last
refraction. From this it follows that correctness of drawing
depends solely upon the principal rays; and is independent
of the sharpness or curvature of the image field. Referring
to fig. 8, we have O'Q'/OQ = a' tan w'/a tan w = 1/N,
where N is the ``scale'' or magnification of the image. For
N to be constant for all values of w, a' tan w'/a tan
w must also be constant. If the ratio a'/a be sufficiently
constant, as is often the case, the above relation reduces
to the ``condition of Airy,'' i.e. tan w'/ tan w= a
constant. This simple relation (see Camb. Phil. Trans.,
1830, 3, p. 1) is fulfilled in all systems which are symmetrical
with respect to their diaphragm (briefly named ``symmetrical
or holosymmetrical objectives''), or which consist of two like,
but different-sized, components, placed from the diaphragm
in the ratio of their size, and presenting the same curvature
to it (hemisymmetrical objectives); in these systems tan
w' / tan w = 1. The constancy of a'/a necessary for
this relation to hold was pointed out by R. H. Bow (Brit.
Journ. Photog., 1861), and Thomas Sutton (Photographic
Notes, 1862); it has been treated by O. Lummer and by M. von
Rohr (Zeit. f. Instrumentenk., 1897, 17, and 1898, 18, p. 4).
It requires the middle of the aperture stop to be reproduced in
the centres of the entrance and exit pupils without spherical
aberration. M. von Rohr showed that for systems fulfilling
neither the Airy nor the Bow-Sutton condition, the ratio a'
tan w'/a tan w will be constant for one distance of the
object. This combined condition is exactly fulfilled
by holosymmetrical objectives reproducing with the scale
1, and by hemisymmetrical, if the scale of reproduction
be equal to the ratio of the sizes of the two components.
Analytic Treatment of Aberrations.---The preceding review
of the several errors of reproduction belongs to the ``Abbe
theory of aberrations,'' in which definite aberrations are
discussed separately; it is well suited to practical needs, for
in the construction of an optical instrument certain errors are
sought to be eliminated, the selection of which is justified by
experience. In the mathematical sense, however, this selection
is arbitrary; the reproduction of a finite object with a finite
aperture entails, in all probability, an infinite number of
aberrations. This number is only finite if the object and
aperture are assumed to be ``infinitely small of a certain
order''; and with each order of infinite smallness, i.e. with
each degree of approximation to reality (to finite objects and
apertures), a certain number of aberrations is associated. This
connexion is only supplied by theories which treat aberrations
generally and analytically by means of indefinite series.
A ray proceeding from an object point O (fig. 9) can be defined
by the co-ordinates (x, e). Of this point O in an object
plane I, at right angles to the axis, and two other co-ordinates
(x, y), the point in which the ray intersects the entrance
pupil, i.e. the plane II. Similarly the corresponding image
ray may be defined by the points (x', e'), and (x',
y'), in the planes I' and II'. The origins of these four
plane co-ordinate systems may be collinear with the axis
of the optical system; and the corresponding axes may be
parallel. Each of the four co-ordinates x', e', x', y'
are functions of x, e, x, y; and if it be assumed that the
field of view and the aperture be infinitely small, then x,
e, x, y are of the same order of infinitesimals; consequently
by expanding x', e', x', y' in ascending powers of x,
e, x, y, series are obtained in which it is only necessary
to consider the lowest powers. It is readily seen that if the
optical system be symmetrical, the orqins of the co-ordinate
systems collinear with the optical axis and the corresponding
axes parallel, then by changing the signs of x, e, x,
y, the values x', e', x', y' must likewise change their
sign, but retain their arithmetical values; this means that the
series are restricted to odd powers of the unmarked variables.
The nature of the reproduction consists in the rays proceeding
from a point O being united in another point O'; in general,
this will not be the case, for x', e' vary if x, e be
constant, but x, y variable. It may be assumed that the
planes I' and II' are drawn where the images of the planes
I and II are formed by rays near the axis by the ordinary
Gaussian rules; and by an extension of these rules, not,
however, corresponding to reality, the Gauss image point
O'0, with co-ordinates x'0, e'0, of the point O at
some distance from the axis could be constructed. Writing
Dx'=x'-x'0 and De'=e'-e'0, then Dx' and
De' are the aberrations belonging to x, e and x, y,
and are functions of these magnitudes which, when expanded in
series, contain only odd powers, for the same reasons as given
above. On account of the aberrations of all rays which
pass through O, a patch of light, depending in size on
the lowest powers of x, e, x, y which the aberrations
contain, will be formed in the plane I'. These degrees,
named by (J. Petzval (Bericht uber die Ergebnisse einiger
dioptrischer Untersuchnungen, Buda Pesth, 1843; Akad.
Sitzber., Wien, 1857, vols. xxiv. xxvi.) ``the numerical
orders of the image,'' are consequently only odd powers; the
condition for the formation of an image of the mth order
is that in the series for Dx' and De' the coefficients
of the powers of the 3rd, 5th . . . (m-2)th degrees must
vanish. The images of the Gauss theory being of the third
order, the next problem is to obtain an image of 5th order,
or to make the coefficients of the powers of 3rd degree
zero. This necessitates the satisfying of five equations;
in other words, there are five alterations of the 3rd order,
the vanishing of which produces an image of the 5th order.
The expression for these coefficients in terms of the constants
of the optical system, i.e. the radii, thicknesses, refractive
indices and distances between the lenses, was solved by L.
Seidel (Astr. Nach., 1856, p. 289); in 1840, J. Petzval
constructed his portrait objective, unexcelled even at the present
day, from similar calculations, which have never been published
(see M. von Rohr, Theorie und Geschichte des photographischen
Objectivs, Berlin, 1899, p. 248). The theory was elaborated
by S. Finterswalder (Munchen. Acad. Abhandl., 1891,
17, p. 519), who also published a posthumous paper of Seidel
containing a short view of his work (Munchen. Akad.
Sitrber., 1898, 28, p. 395); a simpler form was given by A.
Kerber (Beitrage zur Dioptrik, Leipzig, 1895-6-7-8-9). A.
Konig and M. von Rohr (see M. von Rohr, Die Bilderzeugung
in optischen Instrumenten, pp. 317-323) have represented
Kerber's method, and have deduced the Seidel formulae from
geometrical considerations based on the Abbe method, and have
interpreted the analytical results geometrically (pp. 212-316).
The aberrations can also be expressed by means of the
"characteristic function'' of the system and its differential
coefficients, instead of by the radii, &c., of the lenses;
these formulae are not immediately applicable, but give,
however, the relation between the number of aberrations and the
order. Sir William Rowan Hamilton (British Assoc. Report,
1833, p. 360) thus derived the aberrations of the third order;
and in later times the method was pursued by Clerk Maxwell
(Proc. London Math. Soc., 1874--1875; (see also the treatises
of R. S. Heath and L. A. Herman), M. Thiesen (Berlin. Akad.
Sitzber., 1890, 35, p. 804), H. Bruns (Leipzig. Math.
Phys. Ber., 1895, 21, p. 410), and particularly successfully
by K. Schwartzschild (Gottingen. Akad. Abhandl., 1905,
4, No. 1), who thus discovered the aberrations of the 5th
order (of which there are nine), and possibly the shortest
proof of the practical (Seidel) formulae. A. Gullstrand (vide
supra, and Ann. d. Phys., 1905, 18, p. 941) founded his
theory of aberrations on the differential geometry of surfaces.
The aberrations of the third order are: (1) aberration of the
axis point; (2) aberration of points whose distance from the
axis is very small, less than of the third order---the deviation
from the sine condition and coma here fall together in one class;
(3) astigmatism; (4) curvature of the field; (5) distortion.
(1) Aberration of the third order of axis points is dealt with
in all text-books on optics. It is important for telescope
objectives, since their apertures are so small as to permit
higher orders to be neglected. For a single lens of very
small thickness and given power, the aberration depends upon
the ratio of the radii r:r', and is a minimum (but never
zero) for a certain value of this ratio; it varies inversely
with the refractive index (the power of the lens remaining
constant). The total aberration of two or more very thin lenses
in contact, being the sum of the individual aberrations, can be
zero. This is also possible if the lenses have the same algebraic
sign. Of thin positive lenses with n=1.5, four are necessary
to correct spherical aberration of the third order. These
systems, however, are not of great practical importance. In
most cases, two thin lenses are combined, one of which has
just so strong a positive aberration (``under-correction,''
vide supra) as the other a negative; the first must be a
positive lens and the second a negative lens; the powers,
however: may differ, so that the desired effect of the lens is
maintained. It is generally an advantage to secure a great
refractive effect by several weaker than by one high-power
lens. By one, and likewise by several, and even by an
infinite number of thin lenses in contact, no more than two
axis points can be reproduced without aberration of the third
order. Freedom from aberration for two axis points, one of which
is infinitely distant, is known as ``Herschel's condition.''
All these rules are valid, inasmuch as the thicknesses and
distances of the lenses are not to be taken into account.
(2) The condition for freedom from coma in the third order
is also of importance for telescope objectives; it is
known as ``Fraunhofer's condition.'' (4) After eliminating
the aberration On the axis, coma and astigmatism, the
relation for the flatness of the field in the third order is
expressed by the ``Petzval equation,'' S1/r(n'-n) =
0, where r is the radius of a refracting surface, n
and n' the refractive indices of the neighbouring media,
and S the sign of summation for all refracting surfaces.
Practical Elimination of Aberrations.---The existence of
an optical system, which reproduces absolutely a finite plane
on another with pencils of finite aperture, is doubtful; but
practical systems solve this problem with an accuracy which
mostly suffices for the special purpose of each species of
instrument. The problem of finding a system which reproduces
a given object upon a given plane with given magnification
(in so far as aberrations must be taken into account) could
be dealt with by means of the approximation theory; in most
cases, however, the analytical difficulties are too groat.
Solutions, however, have been obtained in special cases (see
A. Konig in M. von Rohr's Die Bilderzeugung, p. 373; K.
Schwarzschild, Gottingen. Akad. Abhandl., 1905, 4, Nos.
