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I
OBSERVATORY LlBRASf
ASTRON.
OBS.
CONTEIBUTIONS
TO
PHOTOGRAPHIC OPTICS
CONTEIBUTIONS
TO
PHOTOGRAPHIC OPTICS
BY
OTTO SUMMER
DOCTOR OP PHILOSOPHY
PROFESSOR ASSISTANT IN THE REICHSANSTALT, BERLIN
TRANSLATED AND AUGMENTED
BY
SILVANUS P. THOMPSON
DOCTOR OP SCIENCE, FELLOW OF THE ROYAL SOCIETY, PRINCIPAL OF THE
CITY AND GUILDS TECHNICAL COLLEGE, FINSBURY, LONDON
J J
* J
MACMILLAN AND CO., Limited
NEW YORK : THE MACMILLAN COMPANY
1900
AU rights reserved
K-.
7
PREFACE BY THE TRANSLATOE
Three articles which appeared in the autumn of 1897 in
the Zeitschrift fur Iristrumentenhiinde, under the title of
"Contributions to Photographic Optics," from the pen of Pro-
fessor Otto Lummer of Berlin, attracted the attention of the
present writer. They were found to give in concise form
information not to be found elsewhere; and they presented
that information in a manner so logical and so direct as to
be of immediate value in scientific optics. Further, they
contained an exposition of the remarkable theories of Professor
L. von Seidel of Munich, whose work in the domain of
geometrical optics, particularly in relation to the aberrations
of lenses, was far too little known to optical writers. To
those whose knowledge of theoretical optics has been gained
from elementary text-books, and who may be dimly aware
of something called spherical aberration, and of something
else called chromatic aberration, it may come as a surprise
to find that for the purposes of constructing good photo-
graphic lenses there are no fewer than five different kinds
of aberration of sphericity and two difierent kinds of aberra-
tion of colour to be taken into account, as well as the
chromatic differences of the spherical aberrations.
It may be claimed for Professor Lummer that he has,
following von Seidel's mathematical theories, in these articles
succeeded in making clear not only what these several aberra-
tions are, but how they are combated and overcome in the
construction of the modern photographic objective. One all-
vi LUMMER'S PHOTOGRAPHIC OPTICS
important feature of these modem types of lens is that which
has followed the introduction by Abbe and Schott of the
new kinds of glass known as Jena glass, by the use of which
certain advantages can be attained which are physically un-
attainable with any of the optical glasses previously known.
This is not a question of any imaginary superiority of German
glass over that of English or French manufacture ; it is the
discovery of glass- having new physical properties, namely
of new kinds of crown glass which, while having a lower
dispersion than flint glass, have a higher instead of a lower
refractivity. The discovery of this new optical property was
followed by the discovery by Dr. P. Eudolph of a new
principle of construction, which lies at the root of the recent
improvements in camera lenses. The writer knows of no
British text-book of optics in which Seidel's theory of the
five aberrations is even mentioned. He knows of only one
British text-book of optics in which Eudolph's principle is
stated — and there it is stated incorrectly.
It therefore appeared well worth while to prepare an
English translation of Professor Lummer's articles ; and with
his kind consent and willing co-operation the present version
has been prepared. The translation does not profess to be
merely a reproduction of the original. The text has been
freely paraphrased, and elaborated in many places where the
very conciseness of the original made some amplification
desirable. No attempt is made to distinguish between the
original text and the portions added by the translator; but
the translator alone is responsible for Chapters XII. and XIII.,
which are additional. He is also responsible for Appendix I.,
in which is given a r4sum4 of von Seidel's original mathe-
matical investigation, and a brief notice of the subsequent
work of Finsterwalder and others. Appendix II. is adopted
almost piecemeal from Professor Lummer's edition of the
Optics of Miiller-Pouillet, as is also much of Appendix III:,
of which not the least valuable part is the example of the
way of computing lenses actually used in practice.
PREFACE BY THE TRANSLATOR vii
Mention is made in a footnote to p. 6 of two works on
optics which ought to be the familiar possession of every
good student, namely Czapski's Theory of Optical Instruments
(published in 1893 by Trewendt of Breslau), and the volume
on Optics in the ninth (1895) edition of Mtiller-Pouillet's
Physics (published by Vieweg of Brunswick), this latter being
edited by Professor Lummer. Both these works are in
German, and most unfortunately no translation of either has
appeared — most unfortunately, for there is no English work
in optics that is at all comparable to either of these. I say
so deliberately, in spite of the admirable article by Lord Eay-
leigh on "Optics" in the Encyclopaedia Britannica, in spite of the
existence of those excellent treatises, Heath's Geometrical Optics
and Preston's Theory of Light, No doubt such books as
Heath's Geometrical Optics and Parkinson's Optics are good
in their way. They serve admirably to get up the subject
for the Tripos ; but they are far too academic, and too remote
from the actual modem applications. In fact, the science
of the best optical instrument -makers is far ahead of the
science of the text-books. The article of Sir John Herschel
" On Light " in the Uncyclopaedia Metropolitana of 1840 marks
the culminating point of English writers on optics.
The simple reason of the badness of almost all recent
British text-books of optics is that, with the exception of
one or two works on photographic optics, they are written
from a totally false standpoint. They are written, not to
teach the reader real optics, but to enahle him to pass
examinations set by non-optical examiners. The examina-
tion-curse lies over them all. Probably the reason why no
English publisher has yet been found courageous enough to
bring out translations of Czapski's Optical Instruments or
Miiller-Pouillet's Optics is that, even if translated, they would
not command a large sale, because it would be useless for
any student to cram himself up on them for an examination.
The optical books which sell in England to-day are cram-
books for university examinations. And so there is, to
via LUMMER'S PHOTOGRArHIC OPTICS
those who know, little inducement to write treatises upon
real optics.
The present treatise at least is not open to this unreality.
It is for the scientific readers amongst the public to decide
whether it succeeds in giving them something not to be
found elsewhere, and something worthy of being known and
studied.
The thanks of the translator are due to the various optical
firms who have kindly furnished him with information of a
valuable character. He also gratefully acknowledges the able
assistance of Mr. J. Dennis Coales, who has helped in the work
of preparing the translation.
CONTENTS
PAQB
Preface by the Translator v
Introductory, by Professor Lummer
CHAPTER I
Attainment of a Perfectly Sharp Image
CHAPTER II
Seidel's Theory of the Five Aberrations
CHAPTER III
Formation of Images by means of a Small Aperture. Pin-hole
Camera 14
CHAPTER IV
Formation of the Image by a Simple Converging Lens . . 19
CHAPTER V
Influence of the Position of the Stop upon the Flatness of
THE Field 24
X lummee's photographic optics
CHAPTEK VI
PAGE
The Cause of Distortion — Conditions necessary for Distor-
tionless Pictures .29
CHAPTER VII
Systems corrected for Colour and Sphericity, consisting of
TWO Associated Lenses — Old Achromats ... 40
CHAPTER VIII
New Achromats 47
CHAPTER IX
Separation op the Lenses as a means of producing Artificial
Flattening op the Image 57
CHAPTER X
XJNSrMMETRICAL OBJECTIVES CONSISTING OF TWO MEMBERS . . 59
CHAPTER XI
Double -OBJECTIVES consisting of Two Symmetrical Members
WITH THE Stop between them 68
CHAPTER XII
Some Recent British Objectives 85
CHAPTER XIII
Tele-photographic Lenses 94
J
CONTENTS xi
APPENDIX I
PAOB
Sbidel's Theory of the Five Aberrations .103
APPENDIX II
On THE SmE-CoNDmoN 116
APPENDIX III
Computation of Lenses 122
INDEX 131
\V
INTKODUCTOKY
BY PROFESSOR OTTO LUMMER
In working up the subject of Photographic Optics for the ninth
edition of the well-known text-book of physics of Mtiller-
Pouillet, I looked about fruitlessly for a guide. In the
literature pertaining to the topic I sought in vain for one
which did not confuse the reader by an enumeration of the
countless names given by makers to their lenses, or for one
which, on the other hand, should lead him by logical and
convincing reasoning to understand the chief advances of
recent times, and the value of the several types of objectives.
The practical performance of a lens-system, which is the only
aspect needing consideration, is in the last resort only to be
determined in the experimental way. Yet such an experi-
mental test, if it stand alone, would be quite inadequate to
explain the method and means by virtue of which the actual
performance has been attained. It therefore appeared to me
desirable to acquire a means of drawing a judgment, at least
about the more important features in the possible performance
of an objective, by applying theoretical considerations to the
design, and to the various given items of construction, the
kinds of glass, and the number of refracting surfaces employed.
In general, the more numerous the elements that are at
one's disposal in the calculation — radii of curvature of the
refracting surfaces, distances between components, kinds of
glass, and the like — the more may the lens attain. If now
one applies analysis to discover what are the conditions that
may be satisfied when one has at one's disposal a given
number of elements, and what are the conditions that must
B
2 LUMMER'S PHOTOGRAPHIC OPTICS
be complied with in order to produce an image fulfilUng the
prescribed requirements of performance, one is henceforth in a
position to draw a judgment as to the range of possibility in
the performance of any given objective.
In endeavouring to sketch, within narrow compass, the
whole of photographic optics, and to arrange the various kinds
of objectives in typical groups, I am fully aware that such a
beginning cannot succeed at once in being complete. There
is still needed much computative work, and above all much
experimental testing, in order to fill up the spaces left vacant,
and to afiFord a substantial foundation to the systematic
treatment of the subject.
CHAPTEE I
ATTAINMENT OF A PERFECTLY SHARP IMAGE
In order to understand better the objective and its aberra-
tions, we must first briefly consider the whole subject of the
formation of images. So far as this aim of the optical
system is concerned, it may be stated in mathematical
language as follows : The two regions of space in front of the
lens and behind it must be in colliTiear relation — that is to say,
all rays proceeding from a point in one region must unite
again in a corresponding point in the other region, in such
a way, in fact, that any extended geometrical form in the
object -space in front is precisely correlated to a similar
geometrical form in the image-space behind the lens. Or, in
other words, to every point in the object there shall correspond
a conjugate point in the image, and vice versa. Of all optical
systems there is only one that literally fulfils this condition,
namely, the plane mirror ; for only in the case of reflexion at
a plane surface are these conditions of a point-to-point
correspondence between the object-region and the image-
region accurately fulfilled. But since this reflexion produces
only a change of position, without magnification, and more-
over only gives a virtual image, it is of no significance in
photographic optics. In photography, optical systems are
required which cast real images of objects, and which more-
over cast them upon a flat surface, namely the photographic
plate. One must therefore turn one's attention to curved
reflecting or refracting surfaces. But of these it may be
shown in general that they do not even produce a small real
image, by means of wide-angled pencils (as is necessary for
bright images), with accurate correspondence point for point
4 LUMMER'S PHOTOGKAPHIC OPTICS chap.
and accurately similar to the object. But rather, they pro-
duce a truly collinear image of extended flat objects, only if the
delineating pencils are very narrow. Never, except in the
special case — which, however, is of no importance in practical
use — where the effective rays make indefinitely small angles
with the principal axis of the system, that is to say, when both
the visible field and the angular aperture of the system are
small, does the formation of a truly collinear image occur.
Gauss, in treating the equations which express quite rigor-
ously the elements of the refracted ray in terms of the elements
of the incident ray and those of the refracting surfaces, developed
the trigonometrical functions of the angle between the delineat-
ing rays and the axis in series of ascending powers -^ of the arc
subtended. In so doing, and neglecting the third and higher
powers, as being small relatively to the first power, he obtained
simple equations for the production of a stigmatic^ image
in accordance with the well-known laws of geometrical optics,
but accurate to a first approximation only. The formation of
an exact image, according to these expressions of Gauss, is
therefore only realised for rays which fall within an indefinitely
narrow cylindrical space around the principal axis of a centred
system of refracting or reflecting spherical surfaces.
Since the formation of images under such a limitation
^ As is well knowD, the sines and cosines of angles may be expressed in terms
of the corresponding angle (in radians) as follows : —
a* a*
cosa = l-p2 + r2^-«^^-
Since for all angles less than 1 radian ( = 57** 17' 44") the value of a is a proper
fraction, the values of a*, a', a*, a^, etc., are still smaller fractions, so that the
higher terms of these series diminish rapidly. Hence it is that for most purposes,
even for angles over 30**, it suffices to neglect all terms after the second one in
the developed series. For small angles all terms beyond the first may be
neglected.
2 The adjective stigmatic (Greek (rrLyfjuty a point), as used in optics, refers to
the accurate bringing of the rays of a pencil to focus at a point, in contradistinc-
tion to the inaccurate focussing of the rays which would cause them to meet in
focal lines, as in the defect called astigmcUism, Without the stigmatic reunion
of the rays of a pencil, no lens can render as a point the image of a point-object.
A stigmatie lens is one which will perform this. A stigmatic pencil is one which
focusses or of which the rays meet accurately in a point. The adjective anastig-
matie means devoid of astigmatism, and therefore really has the same significa-
tion as stigmatic.
I ATTAINMENT OF A PERFECTLY SHARP IMAGE 5
*
(which virtually means always stopping - down the lens to
the smallest aperture) is of no value for practical purposes,
the endeavour was soon made to widen the limitations that
beset the production of exact images, by resorting to the
principle of the division of work. In other words, the duty
of refracting the rays must be distributed over a number of
separate surfaces, and the effects of their different curvatures
and of the distances between them upon rays at different
obliquities must be ascertained in order to decide what the
various curvatures of the surfaces must be, and what must be
their distances apart. Further, it was necessary to consider
each combination with reference to the special duty required
to be performed by it.
In Microscope objectives (and to a less degree in the objectives
of highly magnifying Telescopes), one is dealing with the forma-
tion of images in relatively small fields of vision, but by the
use of relatively wide -angled pencils. But if this be the
limitation in this case, it is the endeavour, on the other hand,
in the case of Magnifying glasses, Eyepieces, etc., to make the field
of vision as large as possible, while sacrificing the angular
width of the pencils. Upon the combination in one whole
of two systems so constructed, depend, as is known, the great
possibilities of the compound instruments — the Microscope
and the Telescope.
Midway between these two special systems there lies a
third, the Camera lens or Photographic objective, properly so
called, which must of necessity both possess a large field of
vision and form its images by means of wide-angled pencils.
Naturally, to attain both the extension of field and the width
of angular aperture (upon which latter, for a lens of given
magnification, the brightness of the image and therefore the
rapidity of the photographic action depends) one must
sacrifice something; and in this case one deliberately
renounces the production of the precise stigmatic reunion of
the rays that is required in the microscope, in the telescope
objective, and in the magnifying lens. Also, the design of
the photographic system is modified in adaptation to different
purposes (landscape lenses, portrait lenses, lenses for interiors,
telephotographic lenses, etc.), according as great angular width
of the aperture, or great extent of the field may be required.
6 LUMMER'S PHOTOGRAPHIC OPTICS chap, i
In other words, the camera objective is constructed upon some
pre-determined type, according as width of pencil or size of
field is of the more importance. The conditions which must
be satisfied -^ in their construction diflfer correspondingly in the
diflferent types of lens-system.
We are concerned here with photographic lens-systems only.
^ The conditions which must be fulfilled by the Microscope objective, the
Magnifying glass, the eyepiece, etc., are thoroughly treated in a German work
by Dr. Siegfried Czapski, Theory of Optical Instrumenis according to Professor
E, Abbe (published by £. Trewendt of Breslau, 1898). This work forms one
section of Winkelmann*s Hamdbook of Physics, but can be purchased separately.
A less exhaustive modem treatise on this branch of optics will be found in the
new (ninth) edition of Miiller-Pouillet's Physics, of which vol. ii., edited by
Professor 0. Lummer, is devoted to Optics. It is a great pity that neither of
these works has yet been translated into English.
CHAPTEE II
seidel's theory of the five aberrations
In order to formulate, at least to a first approximation, the
conditions to be fulfilled in the construction of these types
of lens-systems, we will resort to the theory of formation of
images as treated by L. von Seidel, whose works on this
topic date back to the years 1855 and 1856.-^ SeideFs theory,
as we may name it, takes into consideration all those rays
which cross the principal axis at angles so great that the
third powers, in the developed series of the sines and cosines
of these angles, must be included in the calculation, while the
fifth and higher powers may still be neglected as not materially
influencing the result. L. von Seidel developed his theory
so far that one can deduce the influence both of the angular
aperture and of the width of the field of vision upon the
perfection of the image, from the relation found for conjugate
rays before and after refraction.
By selecting appropriately the terms in the calculations
for conjugate rays he obtains formulae for the correcting terms,
which have to be added to Gauss's terms in those cases where
third powers as well as first powers must be taken into
account — that is to say, cases where, beside the axial rays or
the paraxial rays, oblique rays also contribute to the formation
^ L. yon Seidel's writings on geometrical optics appear to be quite unknown
in England. The principal of them were published in the Astronomischen
Nachric?Ue7i (Altona), in Nos. 835, 871, 1027-1029 of that publication. Von
Seidel also gave a non-mathematical exposition of the Theory of Aberrations
and the mathematical conditions for their elimination, in voL L of the Ahharvd'
lungen der NcUurunsserischaftlich-technischen Commission hei der koniglicTien
hayerischen Akademie der Wissenschaften in Munchen (Miinchen, 1857). See
also Appendix II.
8 LUMMER'S PHOTOGRAPHIC OPTICS chap.
of the image. [Axial rays here mean those that are close to the
principal axis ; and paraxial those which, though not near the
principal axis, are nearly parallel to it.] In the formulae for
these correcting terms there occur only five non-identical sums,
which sums are to be multiplied into the terms that are
dependent upon the co-ordinates of the incident rays. In
order to annul in the plane of the image all aberrations of the
third order, for all combinations of the co-ordinates of the
incident rays, one has therefore five equations at one's disposal.
If we denote these five sums by the symbols S^, Sg, Sg, S^, and
Sg, we may state the main proposition of the theory as follows :
Given an object in a plane perpendicular to the axis, its image
(produced by von Seidel's rays) will be sharply defined, flat,
and undistorted (and identical with the hypothetical image of
Gauss's theory) if, and only if, all the sums S^ to S^ are
severally nul.
In correspondence with the five sums S^ to S^ we may
distinguish flve aberrations. Each kind of aberration will
disappear if the construction of the lens is such that the
corresponding " sum " or coefficient of the correcting terms is
zero. Suppose the construction of the lens-system is such
that Sj = ; in that case there will be no spherical aherration
in the axis, as ordinarily understood. The amount of axial
spherical aberration is (as is well known) proportional to the
third power of the linear aperture. Everybody knows that
a lens that has spherical aberration will not give a sharp
image of a bright point, and that the definition is improved
by introducing a " stop " to cut off all light except that which
passes through the central region of the lens. But if the lens
is so designed as to fulfil the condition S^ = 0, no such stop
will be necessary to secure sharp definition at the centre of
the plate — and, as the full aperture is thus available, the
time of exposure is greatly shortened. A lens, however, may
be constructed to fulfil this condition without being by any
means perfect. Accurate definition at the middle of the
picture is desirable, but other errors may still be present.
For example, there may be coma ^ at all the other parts of the
image ; worst, of course, at the margins. If the construction
1 Coma is a pear-shaped or comma-shaped blur of light extending from, and
partly surrounding, the image of a bright point.
II SEIDEL'S THEORY OF THE FIVE ABERRATIONS 9
of the lens is such that not only Sj = 0, but 83 = also, then
the defect called coma subsides into the less objectionable
defect which may be called radial astigmatism} To every
point lying outside the axis there are found to correspond
two short focal lines, situated at different distances behind the
lens, and occupying positions at right angles to one another.
For instance, if the object is a bright point considerably lelovx
the level of the principal axis, such as would form an image
at some point on the focussing screen above the middle of the
picture, it will be found that if the screen is pushed too near
in, the image will be distorted into a short bright line in a
horizontal position ; while if the screen is drawn further back,
the image will be a short bright line in a vertical position.
Between these two positions the image will be a luminous
patch of intermediate shape, but will not be an exact point at
any distance of the screen. It will further be found that if the
point that is acting as object is moved to greater and greater
^ The term astigmatism is applied to the property possessed by cylindrical
lenses, and combinations of cylindrical with spherical lenses, of bringing a beam
of light to A focal line instead of a. focal point. The eyes of many persons, owing
to the curvature of the cornea being unequal in different meridians about the axis,
possess this defect. It can be remedied by applying cylindrical lenses having an
equal and opposite amount of astigmatism. The adjective a^igmatic is rightly
applied to cylindrical lenses, since its etymological meaning is "not bringing to
a point, " which is the correlative to stigmatiCy which means ** bringing to a point."
Any person can readily imitate for himself the defect of true astigmatism by
putting in front of his eye a thin cylindrical lens, either positive or negative,
having a power of say + 1 dioptrie or - 1 dioptric. An eye thus rendered
astigmatic, or a naturally astigmatic eye, when directed to an object having
lines in different directions upon it, will see some of these in focus, and others
not in focus. For example, if a normal eye is covered by a positive cylindrical
lens with its axis vertical, and is directed toward a window, the horizontal
window-bars may appear quite sharp, but the vertical bars blurred and out of
focus.
No camera lens ever has astigmatism in this sense. The sense in which some
writers on camera lenses use the term is quite diflferent. Oblique rays going
through the lens may fail to be brought stigmatically to an exact point ; they
may, according to circumstances, give a blur or coma, or they may give focal
lines at different distances : a short tangential line nearer in toward the lens, and
a short radial line further out, as explained in the text lower down. It is to
this that the term radial astigmatism applies. A camera lens having this defect
produces at the margins of the picture a streaky effect for objects which (for
example, the foliage of trees) have a multitude of small points. There will be
a kind of concentric streakiness if the plate is too near in, and a kind of radiating
streakiness if the plate is further out — the central part being all the time fairly
well defined.
10 LUMMER'S PHOTOGRAPHIC OPTICS chap.
distances away from the principal axis, the distance which
separates these two focal lines (as measured along the chief
ray of the pencil, or oblique secondary axis) increases by a
disproportionately great amount. It becomes more than twice
as great, when the lateral distance of the point-object from
the axis is doubled. The aberration Sg, to which coma is due,
is proportional to the square of the aperture and to the simple
distance of the point-object from the axis. Moreover, to the
various point - objects in a plane perpendicular to the axis,
there correspond two sets of images which lie in two separate
curved surfaces — one surface containing all the little tangential
line-images, the other surface containing all the little radial
line-images. These two curved surfaces touch each other at
the point where they cross the principal axis, namely, at
that point where the axis meets the theoretical image-plane
which (on Gauss's theory) is the conjugate plane corresponding
to the object-plane. Now, if we could get rid of this radial
astigmatism, the two little focal lines or line-images would
retreat toward one another, and merge into one sharp point-
image; and at the same time the two curved surfaces just
spoken of would merge into a single focal surface. In fact,
to remove radial astigmatism, and produce on a single focal
surface stigmatically sharp images, we must so construct the
lens as to fulfil the new condition S^ = 0. But the focal
surface, though now united into one, is still curved ; and, as
we cannot use curved or dished photographic plates in the
camera, there will still be bad definition either at the mai^ns
or at the centre of the flat screen. To remove this aberration
of mirvature of focal surface we have to design the lens-system
so as to fulfil the fourth condition, namely, so that S^ = 0.
Then, and then only, when all these four conditions are fulfilled
in the construction of the lens-system, shall we obtain a sharp,
stigmatic, flat image, with equally good definition all over the
plate, the image then really occupying the focal plane assumed
in the theory of Gauss. Both the aberrations corresponding
to Sg and S^ are proportional to the linear aperture, but also
proportional to the square of the size or lateral extension ; or,
strictly, to the square of the tangent of the angle subtended
by the object. There is left only one single aberration of the
third order which may still affect the image, namely distortion
II SEIDEL'S THEORY OF THE FIVE ABERRATIONS 11
of the marginal parts. To remove this the construction must
be such as to fulfil the fifth condition, namely S^ = 0. This
aberration is proportional to the third power of the distance
of the object.
The condition S2 = is identical with the so-called " sine-
condition"-^ for small apertures, which may be expressed
verbally by saying that it requires that all zones of an objec-
tive should possess equal focal lengths ; or, in other terms, that
every ray proceeding from an element of a surface should be
brought by the system to a conjugate element of a surface.
Since Frauenhofer fulfilled the condition Sg = in his cele-
brated heliometer objective, von Seidel calls this Frauenhofer's
condition. It may be equally expressed by saying that if the
spherical aberration for rays parallel to the axis has been
removed (Sg = 0), it has also been eliminated for obliqiie pencils
of the same cross-section as the axial pencil. If Sg is not zero,
then the oblique pencil exhibits the one-sided blur or patch
of light known as coma.
If one wished to investigate the formation of images up to
the seventh, ninth, or higher powers of the angle, then some
further equations of condition,^ which must be fulfilled in order
that a plane object should yield a sharp, flat, and undistorted
image, must be obtained.
Let us pause on von SeideFs theory of the formation of
images, and assume forthwith that there exists a lens-system
free from the five possible aberrations of the third order with
which it deals. Then such a lens-system does produce a fault-
less image of an object situated in a plane at a certain distance
from the system ; but on the other hand, if the same lens-system
is set to produce an image of any other plane lying nearer or
further, that image will again be subject to aberrations of the
third order. In other words, the lens so corrected is truly
accurate only for one particular distance. If it is required to
form images of objects at all distances, taking into considera-
tion aberrations of the fifth order, then there arise in addition
to von Seidel's five equations other new ones, of which one in
^ See Appendix I.
' Compare the memoir of M. Thiesen entitled ** Contributions to Dioptrics "
in the Sitzungsberic?Ue der Berliner Akademie, 1890, p. 799. An abstract of
Thiesen's theory, which includes that of Seidel, has been given by Professor
Lummer in Miiller-Pouillet's Physics (9th edition), vol. ii. (on Optics), pp. 522-25.
12 LUMMER'S PHOTOGRAPHIC OPTICS chap.
particular is known as Herschers equation.^ This condition
is in curious contradiction to the second of von SeideFs con-
ditions (82 = 0), being related to it in such a way that in
general, if the construction is such that one of the two is
fulfilled, the other is not, and vice versa. This implies that if
we would have a lens free from distortion, and giving a flat
field that can be used for ail distances, we shall be com-
pelled to sacrifice definition a little, allowing a slight coma
at every distance; or, if we insist on absolute precision
of definition, we shall not be able to use the lens for all
distances.
But, though nothing has yet been said as to the points in
construction implied in each of these five conditions, it must
be understood that this production of an image free from
aberrations of the third order, and in a flat plane, can only be
attained by combining in the lens-system a sufficient number
of separated surfaces. Each new condition to be fulfilled
practically means an additional refracting surface to carry out
the correction imported. If one were to set down the
distances of the various refracting surfaces from one another as
all equal to zero, then, assuming a like refractivity for the
first and last media (for example, air in front of and behind
the lens), the conditions Si to S5 can only be fulfilled by
taking the focal length of the combination as infinitely great —
in fact, either a plane mirror or a flat sheet of glass without
thickness. The distance between the two adjacent refracting
surfaces {i.e. the thickness of the lenses), and likewise that
between the separate components (as when the camera objec-
tive is made of two separated members), are therefore essential
factors in the attainment of precision of the highest order in
the formation of images.
In addition to these aberrations, all of which might occur
even when monochromatic light is used, there must be con-
sidered those which, when white light is used, arise as a result of
dispersion. Of these the most important are the chromatism
of the positions of the images, and the chromatism of the
true focal lengths. The former has the result of causing the
images for different colours to occupy different places upon
^ Von Seidel himself draws attention to this in the Astronomischen NacJiricJUen,
No. 1029, p. 326, 1856.
II SEIDEL'S THEORY OF THE FIVE ABERRATIONS 13
the axis ; it is inseparable from Gauss's theory of formation of
images. The chromatism of the focal lengths produces a
different size of the image for the different colours^ The
commonly-used term " chromatic aberration " includes both of
these really different aberrations.
CHAPTER III
FOKMATION OF IMAGES BY MEANS OF A SMALL APEKTURE.
PIN-HOLE CAMERA
That we may the better appreciate the performance of
appliances devised by human ingenuity, let us very briefly
consider the simplest method of forming an image, a method
which Nature, as it were, offers us of herself — the formation of
an imojge hy means of a small hole.
Pin-hole images are essentially a consequence of the
rectiliTiear propagation of light, which doctrine finds its expres-
Fio. 1. — Formation of an Image in the Pin-hole Camera.
sion in the saying that light is propagated in the form of rays.
Assuming this as a general principle, the process of the
formation by straight rays is shown diagrammatically in Fig. 1.
Here P represents an opaque screen with the aperture a b for
forming an image, L I the object, and TT the intercepting
screen on which the image is received. light from any single
point of the object falls on the screen through the aperture
only. Consequently, corresponding to each point of the
CHAP. Ill PIN-HOLE IMAGES 15
object there is produced a bright spot, which is situated on
the line drawn from the point on the object to the middle of the
aperture, and which is similar in its shape to the shape of the
aperture. A round aperture naturally produces round spots.
The farther the object is moved away from the aperture, the
smaller is the spot of light on the screen corresponding to
each point of the object; since for an object at an infinite
distance the size of the spot would become equal to that of the
aperture. Thus there arises, to a certain extent, a point-for-
point formation of image, in which to each point of the object
there coresponds, as an image, a small bright spot or disc of at
least the same size as the aperture. So far, then, the image
produced in a pin-hole camera resembles the more or less
badly focussed image due to a converging lens.
The greater the distance of the screen from the aperture,
the less do the discs corresponding to the various points of
the object overlap each other, and the clearer are the details
of the image, because, at least in the case of an object at a
great distance, the diameter of the spots varies only slightly
with the distance of the screen. Here we have the reason
why the pin-hole camera delineates objects at widely different
distances with equal clearness ; it possesses, as photographers
would say, great "depth" of focus. Consequently the
definition of the image does not depend on the flatness of the
screen; and however much the screen is curved, one can
always obtain an equally sharp image. But, of course, the
geometrical similarity of the image with the object is depen-
dent on the form of the screen, and will be altered if the
ficreen is bent. Only when the screen is flaty and its surface
parallel to the plane of the object, is an image obtained
which is perfectly similar to the object, and which is
free from distortion up to the extreme margin of the
field. It is just because of these desirable properties of the
pin-hole camera that " photography without a lens " has been
made use of, until quite recently, in order to take pictures
of architectural buildings, churches with high towers, and the
like ; for in these instances it is necessary that the picture
should be in correct perspective, angle -tms, with a field of
view of wide extent, and free from distortion. For such
objects the very prolonged time of exposure of the plate is
16 LUMMER'S PHOTOGRAPHIC OPTICS chap.
relatively unimportant compared with width of angle and
undistorted perspective.
Besides its capability of giving angle -true or orthoscopic
delineation, and its extraordinary depth of focus, the pin-hole
camera possesses, as just remarked, a very wide field of view.
If in spite of its simplicity it has henceforth to give place to
the new wide-angle systems of lenses of complicated and
costly structure, the reason must be sought in the far inferior
brightness and in the poor definition of the images it yields.
If the theory of the rectilinear propagation of light were
strictly true, then the sharpness of definition of pin-hole
pictures would increase directly with the diminution of the
size of the pin-hole; though the amount of light admitted
would of course be proportionately diminished.
But in reality light consists of waves, not of rays ; and its
propagation only appears to be rectilinear, when taken in the
gross, because the waves themselves are of such minute
dimensions. Whenever one begins to deal with small aper-
tures, pin-holes, or narrow slits, one at once discovers that
though some of the light does indeed travel in straight
lines, some of its waves also spread laterally, giving rise to
diffraction " fringes " and other phenomena of " interference "
characteristic of wave propagation. The only strictly tenable
definition of a "ray" of light is that it is the path along
which waves are marching. Kays of light in the old physical
sense do not exist. Diffraction, or the apparent spreading of
the waves of light into the margins of the geometrical body,
or bending round into the shadows of narrow objects such as
pins or hairs, is an absolute disproof of the "ray" theory.
And the existence of diffraction is a matter which, in the
theory of the microscope, and in the theory of the resolving
power of the telescope, it is as necessary to take into account
as either refraction or dispersion.^
With the use of a relatively small aperture the effects of
this diffraction begin to assert themselves. The smaller the
aperture the more evident becomes the lateral spreading of the
^ See a remarkable article on the Diffraction Theory in Geometrical Optics,
by Dr. K. Strehl, in the Zeitschrift filr InstrumerUenkundef December 1899, p.
364. See also the articles on Optics, and on Wave- Theory, by Lord Kayleigh in
the Encyclopaedia Britannica, Lord Rayleigh's articles in the Philosophical
Magazine for 1879, 1880, and 1886 should also be consulted.
Ill PIN-HOLE IMAGES 17
light-waves ; so that if in the vain attempt to isolate a single
" ray " of light we make the pin-hole smaller and smaller, the
little disc of light which is cast on the screen begins to
appear surrounded with diffraction halos and faint fringes of
colour ; so that the attempt to reduce the size of the pin-hole
below a certain limit defeats itself. For while, down to a
certain degree, diminishing the size of the pin-hole sharpens
the image, after that limit has been attained, any further
decrease in size of the pin-hole will reduce the sharpness of
the definition, until finally, when the size of the aperture^
becomes of the order of magnitude of one wave-length, the
sharpness of the image is quite lost.
The assumption of a rectilinear propagation of light is in
fact an abstraction, which only holds good in the case of
undisturbed propagation in one and the same medium. But
even then, both in the case of the formation of the image by
^ It may be convenient to mention the sizes of some sewing needles of a
standard firm of manufacturers, Messrs. Milward of Redditch. According to
Mr. Dallmeyer these have the following diameters, in mils. (1 mil. =Y^JVTr of an
inch) : —
No.
No. 1
46 mils.
2
42 „
3
38 ,,
4
36 „
32 ,,
6
29 „
7
26 mils.
8
23 „
9
20 ,,
10
18 ,,
11
It) ,,
12
14 ,.
The size of a wave-length of light varies from 32 millionths of an inch for
the extremest red, down to 14*4 millionths for the extremest violet visible.
One may take 16 millionths as about the size of the wave-length of blue light to
which the photographic film is most sensitive. The distance of the plate from
the pin-hole, to give the best concentration of light in the diffraction disc that
is the image of a point, can be calculated by dividing the square of the diameter
of the hole by four times the wave-length. Thus, if a No. 9 needle were used to
make the pin-hole, the hole being 20 mils, in diameter, the calculation would
be—
020X 0-020
best distance of screen =
4x0-000016
0-000400
0-000064
= 12?
64
= 6 inches, approximately.
The reader should also consult a paper on Pin-hole Thotography by Captain
Sir William Abney, F.R.S., in the Catnera Club Jmimal, May 1890; also Mr.
Dallmeyer's Telephotography ^ p. 15.
C
18 LUMMER'S PHOTOGRAPHIC OPTICS chap, hi
a pin-hole camera, and in that by a lens-system, a complete
solution of the problem on the principle of the diffraction theory
is only possible by the aid of the higher mathematics. By its
aid it is possible, in the case of the pin-hole camera, to find a
formula for the particular numerical relation between the size
of the aperture employed, the distance of the point-object from
it, and the size of the resulting discs of light. So also one
obtains the formula for calculating the best size of pin-hole
to use for a given length of camera -body. Little practical
importance, however, attaches to the elementary method of
calculation hitherto in vogue, even if it sufiBces to predetermine
the changes in appearance that occur when the size of the
aperture is changed. The elementary formula, published
indeed long ago by Petzval, shows that the distance of the
screen suitable to give the sharpest image with an aperture of
0*3 millimetre should be 50 millimetres; whilst, according to
the actual experiments of A. Wagner, the best distance for
this same aperture amounts to about 100 millimetres. Ad-
mitting, however, the validity of these approximate figures,
then a simple calculation shows that a Petzval portrait-objec«
tive which, with an aperture of 8 centimetres and a focal
length of 30 centimetres, permits of a tenfold magnification
in the image, surpasses the pin-hole camera, with respect to
the brightness of the image, about 18,000 times, and surpasses
it also in respect of sharpness of definition about 180 times;
the distance of the image from the aperture being the same.
In the modern portrait -objectives which produce brightly-
illuminated pictures, and which, for a focal length of 30
centimetres, can be used with an aperture up to 12 centi-
metres {ix. with //2'5), the brightness of the image is approxi-
mately 40,000 times as great as that of the pin-hole camera,
that is to say, such an objective is 40,000 times more rapid.
CHAPTEE IV
FORMATION OF THE IMAGE BY A SIMPLE CONVERGING LENS
By placing a simple magnifying lens behind the aperture of a
camera, Giambattista della Porta led the way towards the photo-
graphic optics of ^to-day. Imperfect as is the image produced
by a single simple lens, yet at that date its introduction
signified a forward step. For the lens at once lessened the
two chief defects of the pin-hole camera, since it produced an
image which was relatively sharp, and above all bright.
Both these advantages follow from the property possessed by
refracting spherical surfaces, of causing all rays proceeding
from a point to intersect again, approximately, in another point ;
that is to say, such surfaces convert homocentric (divergent)
pencils of rays into other homocentric (convergent) pencils.
Let it be assumed that a single lens may actually bring
about such a stigmatic reuniting of the rays from a luminous
point serving as object. Yet even then, the image is not
itself really a point, but is a more or less extended small
bright disc, A lens-system which, according to its geometrical
construction, would, on the ray theory, bring a homocentric
pencil to reunite in one definite point will, according to the
wave theory, accomplish nothing more than convert the spheri-
cal wave-surfaces of rays that emanate from a self-luminous
point into other wave-surfaces, also spherical and apparently
emanating from or advancing toward another centre. Now
the theory of diffraction shows, in accordance with the principle
of the interference of small wave-elements proceeding from the
effective portion of the wave-surface, that a spherical wave-
surface produces in the plane of its vertex a small disc of light
surrounded by alternate bright and dark rings. Every astron-
20 LUMMER S PHOTOGRAPHIC OPTICS chap.
omical observer is familiar with the spurious discs shown by
even the best telescopes when accurately focussed upon a star*
These are examples of the action of diffraction in preventing
any accurate point -image from being formed at the focus.
Such diffraction discs decrease in brightness from their middle
to their edge. Their diameter depends upon the ratio of the
aperture to the focal length of the lens. The greater the ratio,
the more nearly does the dififraction disc shrink down to a
point-image. As understood in the wave theory, the point-
image is merely the limit toward which the distribution of the
light in the plane of the vertex (of the wave-surfaces emerging
from the optical system) approximates as the operative portion
of the emergent wave-surface is increased in area.
Physical optics recognises no other meaning than this to
the term a " point-image." ^
To endeavour to separate diffraction from the formation of
images would be to separate the effect from the cause. Yet
in spite of this one frequently meets with erroneous statements
upon the influence of diffraction, as if it were a kind of inter-
loper which under certain circumstances might be avoided, or
which was only produced primarily by the light grazing against
the edge of the stop.
If the supposition that the optical system, speaking accord-
ing, to the language of geometrical optics, causes homocentric
pencils to reunite in one point, does not prove to be correct
in fact (that is to say, if there in fact is spherical aberration),
then it follows that the actual operative wave-surface which
emerges from the lens possesses a form not truly spherical ;
and the consequence of this is that the lens-system produces a
faultyformation of images,such faults being what we term aherra-
tions. In order to acquire an understanding of the formation
of the image in this case, one must ascertain the form of the
wave-surface, so as to calculate the diffraction effect produced by
the operative part of the wave -surface. In particular, what is
wanted to be known is the diffraction effect at the place where,
according to the elementary theory of Gauss, the simple point-
image ought to be produced. This calculation is not simple ;
nor indeed is it possible for every form of the wave-surface.^
* K. Strehl has recently undertaken a detailed study of the question what
sort of a distribution of the light is produced in the focal plane at the place of
IV FORMATION OF- IMAGE BY SIMPLE CONVERGING LENS 21
So far as a simple converging lens can succeed in effecting
a precise reuniting of the rays in a point, the reduction to a
minimum of the aberrations due to sphericity is attainable
only by the use of highly refractive materials, and by the
choice of an appropriate form for the lens.
In consequence of the chromatic dispersion of the light,
there always occurs, even with as small an aperture as one
may select, some want of definition ; and the aberrations due
to sphericity, when the aperture is relatively small and the
light employed is monochromatic, are negligibly small in com-
parison with the chromatic aberration. The circle of aberration
due to the chromatic dispersion is in fact of a diameter about
equal to one-thirty-third ^ of that of the aperture of the lens.
In consequence of this chromatic aberration, the image
formed by a single glass lens is not much superior to that
formed by the pin-hole camera. If one takes into account
only the diffraction effect (upon the assumption of a refraction
that is stigmatically accurate) and the circle of aberration
due to dispersion, an approximate calculation shows that the
badness of the definition ^ of a single lens is a minimum when
with an aperture of 3 millimetres there is used an aperture
ratio of //1 00. This minimum of definition gives for this
aperture a disc of about 0*244 millimetres in diameter as the
image of a point.
the theoretical point-image of Gauss, or in the neighbourhood of the focus, by a
ncni-spherical wave-surface. See K. Strehl, Theory of the Telescope on the Basis
of Dijfraxiixm (Leipzig, 1894), and abstracts of his work published in the
ZeUschrift filr Jnstrumentenkunde.
^ This was the reason why, at the time when achromatism appeared to be
unattainable, lenses of enormously long focal length were employed, and why
also people abandoned refracting systems and turned to reflecting telescopes.
These so-called aberratiooiless surfaces have, however, at the best found no appli-
cation except in certain very special cases — for example, in search-light mirrors to
which a paraboloidal instead of a spherical form is given. That they have found
so little other use is due to the circumstance that they are not aplanatic in Abbe's
sense of the term, and do not give images free from aberration except at the one
predetermined point on the axis, not even at points a little on one side in the
focal plane. This can only be attained, according to von Seidel's theory, if beside
Si being =0, Sg is also = 0. In other words, they are not truly aplanatic unless
they also fulfil Frauenhofer's condition — that is to say, unless the sine-conditio-n
(see p. 2 above) is fulfilled for pencils of all and every width.
2 The author here takes as a measure of the badness of definition the diameter
of the small spot or disc of light which is formed, instead of a true pointj as the
image of a luminous point-object.
22 LUMMER'S PHOTOGRAPHIC OPTICS chap.
The image formed by a single glass lens is therefore superior
to that of a pin-hole, about twenty-fourfold in brightness, and
about fivefold in definition.^
By the substitution of a simple converging lens instead of
a mere hole, two of the most important properties — brilliancy
and definition of the image — have been then somewhat
enhanced. But this augmentation is not such as to be of very
direct importance, and it has been bought dearly enough, since
there are brought in certain fresh disadvantages inseparable
from the introduction of the lens.
In the first place, the sharpest image of an object is formed
only at a quite definite position, the location of which varies
with the distance of the object ; while in the case of the pin-hole
camera — at least for all objects that are a moderate distance
away — the image is equally sharp for all distances of the plate
from the pin-hole. The " depth " of image in the case of the
lens is very limited. Any slight displacement of the focussing
screen toward or from the lens spoils the sharpness of the
picture.
Secondly, as a further consequence of dispersion, the single
lens possesses a so-called chemical focus. It is well known
that the chemical actions of light, in which all photography
consists, are not produced equally by rays of all the different
colours. While the eye is sensitive to all the colours from
red to violet of the visible spectrum, it is most sensitive to
those of yellow and yellowish green in the mid-region of the
spectrum. On the other hand, the silver-salts used in the
preparation of photographic plates and films are hardly sensi-
tive at all to red or orange ; their range usually extends
from yellow, through green, blue, and violet, right into the
ultra-violet region of the spectrum, and therefore includes
certain kinds of light to which the human eye is insensitive,
and which are therefore invisible. These " ultra-violet " or
" chemical " or " actinic " rays are of shorter wave-length than
any that the eye can see, consequently they are more refracted
^ A comparison such as this between the performance of the pin-hole camera,
that of a single lens, and that of a double achromatic lens, with respect to bright-
ness and definition of images, has already been made by Petzval. See his
Report on the Results of Certain Dioptric Investigations^ published at Pesth
in 1843. See also an excellent pamphlet entitled The Principles of a Photographic
Lens simply eocplained, by Mr. Conrad Beck (1899).
■^^
IV FORMATION OF IMAGE BY SIMPLE CONVERGING LENS 23
by glass than even the most refrangible (violet) of the visible
rays. Hence a simple glass lens, of which the focal length for
violet light is shorter than that for red light, has a still shorter
focal length for these ultra-violet or chemical fays.
This fault makes itself evident in the following way : — If
the focussing of the picture on the screen has been made as
sharp as possible to the eye, and the sensitive plate is then
substituted at exactly the same position, the picture so taken
will seem badly focussed, because the chemical focus lies a
little nearer to the lens than the visual focus.
Both these defects, the want of focal "depth" and the
aberration due to chromatic differences in the focal length,
are lessened by diminishing the aperture of the lens. And
since already on account of spherical aberration no great
aperture can be permitted, it follows that these aberrations
will also be of relatively small importance in comparison with
those that occur* with oblique pencils. Such pencils must
necessarily be used in taking photographs of extended objects
that occupy a wide field of view, since the rays from
the lateral parts of such objects must enter the lens obliquely.
In the formation of extended images by a simple glass lens
with a small aperture, there come in therefore the special
aberrations due to obliquity, of which the three chief ones are
Radial Astigmatism (see p. 9 above), Curvature of the plane
of the Image, and Distortion, all of which occur the more
markedly the greater the obliquity of the pencils — that is to
sav, the wider the field of view.
For the elimination of these aberrations one cannot do
much, at least by any method of compensation. Yet a
diminution of them may be obtained by choosing a suitable
form (meniscus) for the lens, and by adopting an appropriate
method of " stopping down " the aperture (front stop).
While a meniscus lens (a positive meniscus with the
concave side outwards), by virtue of its form, yields a sharp
image over a wider field than does the ordinary bi-convex
form of lens, the stop set in front of the lens at a suitable
distance causes the best image that the lens can give —
imperfect though that be — to come to focus in one plane as a
flat picture.
CHAPTEE V
INFLUENCE OF THE POSITION OF THE STOP UPON THE
FLATNESS OF THE FIELD
In order to comprehend the influence of the position of the
stop upon the situation of those points of the image which
are not on the axis,
one must recollect
the property called
radial astigmatism
(see footnote to p. 9
above). In virtue
of this property, a
pencil of Kght pro-
ceeding from some
lateral point of an
object as its source,
and traversing the
lens obliquely, pro-
duces instead of a
point-image two focal
lines, m^, m, (Fig. 2),
separated from each
other by a short
distance called " the astigmatic difference." The focal lines m^^
and m^ corresponding to the various points of the object always
lie on curved surfaces K^ and K^, which touch each other at
their common point of intersection E with the axis. These focal
lines show themselves sharply on the nearer surface K^ as bits
of concentric circles or tangential line-elemeTvts, and on the
further surface Kg as bits of radii or radial line-elements. No
Fio. 2. — Radial Astigmatism of Oblique Pencil,
causiug two Curved Focal Surfaces.
CHAP. V
INFLUENCE OF THE POSITION OF THE STOP
25
sharp image is formed of either the tangential or the radial
focal lines on a plane photographic plate inserted at the theor-
etical focus E ; but instead, each oblique pencil produces
on it an oval luminous patch corresponding to the section, by
that plane of the astigmatic pencil. Somewhere along the
pencil between the (horizontal) focal line at m^ and the
(vertical) focal line at m^ the section of the pencil contracts ;
all the rays here — at the place marked s in Fig. 2 — being
concentrated within a round patch called the circle of least
confusion, which is the nearest approach to a well-defined
image of the point-source. When there is little astigmatism,
and when the stop is a hole of circular form, this smallest
cross-section of the oblique pencil is likewise circular. The
circles of least coAfusion which correspond to various points of
the object are in general also situated on a curved surfojce Kg,
which likewise cuts the axis in the focal point E. The exist-
ence of these curved, focal surfaces has long been known. They
are discussed, for example, in Coddington's Treatise on the
Reflexion and Refraction of Light (1829), p. 199, where the
condition for flattening the surface Kg is laid down mathe-
matically.^
If one cannot abolish the radial astigmatism, it is at least
some gain if one can shift the positions of all the circles of
Fig. 3. — Use of Front Stop in Rectification of the Image.
least confusion so that they shall all lie in one plane ; if, in
other words, one could flatten the curved surface K3, and
^ See also R. H. Bow, in the remarkable papers contributed by him to the
British Journal of Photography in the years 1861 and 1863.
26 LUMMER'S PHOTOGRAPHIC OPTICS chap.
bring it into coincidence with the plane 'Em passing through
the focus E. This can, in fact, be accomplished by the use of
a Front stop P (Fig. 3), placed at a proper distance before
the lens.
As the figure shows without need of further explanations,
the stop cuts out from each oblique pencil a partial pencil,
which alone becomes operative, the other rays being intercepted.
Moreover, while if the stop were close to the lens all these
partial pencils would traverse one and the same part of the
lens (namely its central part), when the stop is removed to
some little distance in front the operative partial pencils will
traverse different regions of the lens, those partial pencils
which are most oblique traversing zones of the lens nearest
its periphery, while those pencils that are less oblique will
traverse the lens nearer its middle.
Experience shows that for any given meniscus lens there is
a particular distance of the stop which will bring the circles
of least confusion of all the oblique pencils almost exactly into
the plane Em, which is the plane where the axial rays come
to their focus. This is a very different matter from that
which we have previously called flattening of the image (which
can theoretically be accomplished optically by lens combina-
tions without the use of a stop), so that we may fairly describe
the effect here produced by the use of a front stop as an
artificial rectification of the image. What is understood by
curvature of the image, in the sense of von SeideFs five aberra-
tions (see p. 10, ante), is strictly the curvature of an image
possessing in other respects stigmatic accuracy. One can only
talk of an actual elimination of this aberration if with the
removal of the curvature, and the consequent attainment of
flattening, there remains also attained the condition that the
rays are reunited in stigmatic correspondence ; and this is
not so in the use of a single meniscus lens with a front stop
to trim down the circles of least confusion into approximate
compliance with fair definition in one plane. We shall return
to this matter in considering achromatic double-objectives.
Here, however, we are dealing only with a shifting of the
smallest cross-section of the oblique pencil between m^ and m^,
a shifting accomplished by methods which, so far from annulling
the astigmatic difference m^, m^, even increase it. And indeed
INFLUENCE OF THE POSITION OF THE STOP
27
this artificial rectification of the curved image is only possible
when there is present a sufficiently considerable loant of accwracy
in the convergence of the pencils of rays.^ Only in such
cases can the point of reunion of a partial pencil, cut from a
full pencil by means of a stop, be changed by making a
corresponding change in the position of the stop. In Fig. 4
this operation is exhibited. The full oblique pencil, which
fills the aperture uv of the lens S, after emergence does not
meet in one point, but (so far as those rays are concerned
whose meridian is the plane of the paper) cut one another in
such a way as to form a caustic curve (i\Jyf. It will be
noticed that this caustic curve is not symmetrical with respect
to the axis L^oSL' of the oblique pencil. In fact, this is the
Fig. 4. — Operation of Stop iji selecting Partial Pencils and so shifting Position of Image.
cause of the defect called Coma (see p. 8). Only those rays
intersect each other which lie in neighbouring positions in
the pencil; and they intersect more or less in a point or in
a small focal line. By applying the stop P with the small
aperture ab, all the rays of the entire oblique pencil uv are
cut oft*, except only those of the partial pencil cd, which is
cross-hatched ^ in the Figure 4. Consequently, in place of the
caustic curve QL'W there now appears at ^ a fairly-defined
image of the point-object which is situated far away along Loo.
The nearer the stop P is pushed in toward the lens S, the
^ Nevertheless the Jsnowledge of this simple expedient leads to important
conclusions with respect to the symmetrical double-objectives to be presently
described.
^ The reader is advised that Fig. 4 is purely diagrammatic, and exaggerates
the defect. There is no attempt to represent the accurate refraction of the
individual rays.
28 LUMMER'S PHOTOGRAPHIC OPTICS chap, v
nearer does the region erf, where the incident pencil meets the
lens, shift toward the middle region of the lens, and the point-
image g toward L'. As soon as the stop comes into contact
with the lens, all the operative partial pencils traverse the
middle of the lens, and the tip U of the caustic curves becomes
the position of the point-image for all rays in the vertical
meridian. When, however, the distance of the stop is
sufficiently great, the various partial pencils pass through
different zones of the lens, according to their various obliquities,
and of each caustic there comes into operation only one spot g.
The greater the obliquity of the pencil uv, to w^hich the partial
pencil ah belongs, the nearer does the operating zone lie to the
periphery of the lens, and the further from U is the effective
spot g of the caustic.
After the same fashion one may also produce a similar
series of changes in the position of the focus of a pencil by
pushing the stop P right up to the lens, and displacing it in
a vertical direction along its surface. But only a displacement
of the stop along the axis can simultaneously cut out from the
various pencils appropriate partial pencils ^ that lie in their
several meridians. Here, then, we find the rationale why in
the use of landscape lenses, and other single-component systems
used in photography, the stop is always placed in front.
Hand in hand, however, with the increase of the distance
of the stop, and with the rectification of the image thereby
effected, there enter in other detrimental conditions. In the
first place may be mentioned the rather insignificant fault
that both the size of the field and that of the evenly-illuminated
part of the image are diminished. A far more grievous
fault is, however, the distortion of the picture.
^ In many cases the central ray of a pencil may be taken as representative of
the rest of the rays of that pencil, and may be regarded as its axis, even though
it does not pass through the optical centre of the lens, and though it is itself
refracted in traversing the lens. Such rays are sometimes called chief rays,
because of their representative character.
CHAPTER VI
THE CAUSE OF DISTORTION CONDITIONS NECESSARY FOR
DISTORTIONLESS PICTURES
Suppose that the first four of von Seidel's conditions (S^ to S^,
see p. 8) have been complied with: then the lens -system
will project a stigmatically sharp image, of an object-
plane^ perpendicular to the axis, upon a second plane — the
image-plane — which is also perpendicular to the axis. This
image will be similar to the object and without distortion,
provided von Seidel's fifth condition (S^ = 0) is also fulfilled.
An image which thus is free from distortion is sometimes called
" angle-true," or " orthoscopic," or " true in perspective " ; and a
lens which will perform this, giving, for the full field, images of
straight lines as straight lines, not curved nor sloping at
incorrect angles, is spoken of as a " rectilinear " lens.
This condition of freedom from distortion of the picture has
reference of necessity to the path followed by the chief rays of
each pencil, and may be deduced from simple considerations.
First, it is clear that, particularly where, as in the pin-hole
camera (Fig. 1), the chief rays proceed icnhvJcen from object to
image, point to point, orthoscopic similarity is realised of
itself. The centre of the aperture, where the chief rays
intersect one another, is then the centre of projection, and the
chief rays of all the pencils, when considered geometrically, are
straight lines that intersect two planes that are parallel to
one another, therefore giving geometrically similar figures for
image and object. Since in the pin-hole camera any flat
object produces an image which, whatever its defect of defini-
^ Meaoing of an object all points of which are situated on one plane — as, for
example, a picture or a wall — or, in other words, a flat object.
30
LUMMER'S PHOTOGRAPHIC OPTICS
CHAP.
tion, is likewise flat, it is evident that the pin-hole camera
must project an image which is orthoscopically similar right
up to the margin of the field of view, because of the recti-
linear course of the chief rays.
Entirely similar is the course of the chief rays in the case
of a sphere- lens (Fig. 5) having a central stop ab with a
small aperture. Here also the chief rays proceed in straight
lines from points on the object to the conjugate points on
the image. Here also, if the image is to be similar to the
object, parallel planes, perpendicular to the axis, must be
conjugated together — that is to say, must correspond point-for
point. But as Fig. 5 shows, the sphere-lens produces, from
a Jlat object, an image which, though sharp (at least with a
Fig. 5. — Production of Curved Image by a Sphere-lens having a Central Stop.
relatively small stop), is curved. Assuredly the chief rays will
project on a flat photographic plate an image similar to the
object ; but this image will be sharp only in the middle, the
definition rapidly falling off from the middle towards the
margins. The image Q'L' received upon a suitably curved
plate will be sharp up to the margin ; but, on the other hand,
it is not similar to the distant flat object, but is much distorted.
If, as in most objectives, the chief rays suffer a. deviation
in passing through the system, then their course must con-
form to certain ascertainable laws if orthoscopic conditions are
stm to be fulfilled.
We have mentioned how a simple lens furnished with a
front stop produces in the image a distortion which is the
greater the further the stop is removed from the lens. Let us
next investigate, in this simple case, what the course of the
VI THE CAUSE OF DISTORTION 31
chief rays must be, in order that an orthoBccpically similar
picture shall result. For this the assumption will be made
that the system S is such that it forms a sharp im^ of a
flat object that is perpendicular to the axis, and that the image
is itself also flat, and in a plane perpendicular to the axis. If,
as shown in Fig. 6, tlie stop V lies in/rcmt of the lens-system,
then the chief rays, while still in the region in front of the lens,
all intersect the axis in one and the same point, namely the
point m at the centre of the aperture of the stop. They then
go on to the lens, and are refracted according to the laws of
geometrical optics. The course of these refracted chief raya
varies according to the correction of the lens-system. If the
Bjatem be a simple converging lens, then the marginal rays
after refraction cut the axis nearer to the lens than do the
axial rays. Let m', m", m'" be the points of intersection of
the axis for the chief rays ma, mh, im in question. Further-
more, let a (which is identical with a), ^, 7, etc., be the points
of mutual intersection where the chief rays before and after
refraction meet if produced : these points may be called the
chief poitUs ' of the oblique pencils, or of the chief rays.
' In rendering inf« Engliali bj "chief rays" and "cliief points " tLe words
HaupUiTahUn and Eauptpunkte of the oiigiiial German, the use of the terms
"priudpal raja" and "principal poiuts " has been pnrposely avoided. As the
term " ptiucipitl points " is already recognised in optics to denote the two points
on the axis which in Gauas'a treatment of thick lenses constitute the ]»ir of
optical centres of the lens, there would be some coofiision if the same term were
here used for points not on the axis, and which ate not the optical centres. The
term "chief ray" (as explained in the footnote to p. 28) is nsed for the central
ray of any oblique pencil. It will he noted that tlie poiuCa a, p, 7, at
32 LUMMER'S PHOTOGRAPHIC 0PTIC5S chap.
Now it was assumed that the lens-system S is such as
to give stigmatically a flat image (at L') of a flat object
situated at L. Accordingly the points x, y, z of an object
situated in a plane perpendicular to the axis have their respec-
tive images a/, 1/ , J at the places where the corresponding chief
rays, after refraction through the lens, intersect the conjugate
plane, which is likewise perpendicular to the axis. Orthoscopy,
i.e. the formation of geometrically undistorted images, requires
therefore that all pairs {i.e, rays before refraction and after
refraction) of chief rays should trace similar figures on the
conjugate pairs of planes. But this is the case only if (1) the
chief rays which intersect one another in the " object-region "
(i.e. in front of the lens) also intersect one another at a single
point in the "image region" {i.e. behind the lens), and if (2)
the " chief points " a, fi, 7, etc., lie in a plane that is perpen-
dicular to the axis. In such a case the chief rays would belong
to pencils of rays that are orthoscopically similar, and they
would always trace out geometrically similar figures upon any
plane that might be drawn across the axis perpendicular to
this axis. Provided that both the conditions stated are ful-
filled at once, then the lens-system will be orthoscopic for all
distances of the object.
If then the chief points a, ^8, 7, etc., all lie upon a plane
perpendicular to the axis, there remains to be satisfied as the
one necessary and sufficient condition for distortionless per-
formance the requirement that all the chief rays shxill after
refraction he reunited to O'iu point. But this is equivalent to
saying that the lens shall reunite in one single aberrationless
point all the rays going out from m : or, iu other words, that
the lens shall be ^ free from spherical aberration with respect to
present being dealt with, have certain properties in common with the so-called
''principal points" of Gauss. In fact, on the assumption that the lens is thin,
so that its two "principal planes" are coincident, these points a, jS, 7, here
called "chief points," are points on the "principal plane."
^ In the Aplanatic type of lens, and in Double-objectives of symmetrical con-
struction with the stop in the middle, the position of the "chief points" is of
no particular importance. For since, in consequence of the symmetrical path
of the rays, the conjugate chief rays run parallel to one another, the condition
of accurate reuniting (real or virtual) of the incident and emergent chief rays is
satisfied, as we shall show later. In other words, t?ie spherical correctio7i of the
systein relatively to the ^^ entraiice-pupU'* and tlie **exU-pupil'* suffices for the
production of complete orthoscopy.
VI THE CAUSE OF DISTORTION 33
the point m, which is the centre of the stop P, and with respect
also to its conjugate image m'.
If in any lens-system the " spherical aberration of the chief
rays" has been eliminated, then the questions whether the
system produces distortion, and what is the nature of that
distortion, are determined by the positions of the " chief
points." Now this further condition, that all " chief points "
shall lie upon a plane perpendicular to the axis, is identical
with the fulfilling of the principle known as the tangent-
condition ; which, expressed in words, is that the ratio of the
tangents of the angles between the ray and the axis shall he
constant for all conjugate rays.^ But in any case the require-
ment that the entire object should be delineated without dis-
tortion in the image requires, in addition to the fulfilment of
the condition of equality of ratio of the tangents, that there
should be spherical correction for the position of the stop and its
image, and, indeed, for the aperture-ratio (ag/mS) actually used
for the chief rays.
In the case of a thin bi-convex lens, in which the " chief
points " lie close to its mid-plane ag, the course followed by
the chief rays suffices to affi)rd information about the nature
of the distortion, its amount, and its alteration, when any
alteration is made in the position of the object of the stop, or
of the lens. Firstly, it is known that a simple positive {ie.
convex) lens refracts the marginal rays more strongly than the
axial rays. If, therefore, as in Fig. 6, real images are formed
of the object and of the front stop, it follows immediately from
their mutual positions that spaces of equal size on the object
(Lx = X7/ = yz, etc.) will suffer (see lfocf>x'y^>y^2f) a minifica-
tion toward the margin of the field. If a network of
straight lines at right angles, like Fig. 7, a, is used as object,
there will be produced a distorted image, like Fig. 7, h. This
kind of distortion we will call a negative one. It is sometimes
called a " barrel-shaped " distortion.
According as the stop is situated at a distance BS (Fig. 6)
within the focal length, or at a distance LB between the
^ Compare Czapski's* Theory of Optical Instruments, pp. 110-13. This
tangent-condition appears to have been first formulated by Lagrange. See also
Helmholtz's Physiologische Optik (edition of 1896), p. 70 ; or Pendlebury's
Lenses and Systems of Lenses (1884), p. 26 ; or Heath's Treatise on Ge&tnetrical
Optics (second edition, 1895), p. 57.
D
34
LUMMER'S PHOTOGRAPHIC OPTICS
CHAP.
focal plane and the object, there is produced a real or a virtual
image of it. Whichever of the two it be, the distortion is,
however, negative so long as the stop is situated in front of the
lens-system. In the particular case where the stop coincides
with the front focus at B, its image is at an infinite distance,
the chief rays must therefore go on parallel to the axis,^ and
orthoscopy will be realised. We now can clearly see that the
a
■
Fig. 7. — a, Reticulated Object ; h, Barrel-shaped (negative) Distortion ;
c, Pincushion-shaped (positive) Distortion.
condition of orthoscopy, so far as it relates to the accurate
reuniting of the chief rays, is in contradiction to the require-
ment of great intensity of illumination; for, in general, a
system which has been made aberrationless for the focus and
for its infinitely distant image will not also accurately reunite,
stigmatically, the pencils of rays originating at points on the
objecty at least not with a wide aperture to the system.^
* Abbe calla such a system **telecentric on the side of the image." It is
specially used for micrometric measurements, since the size of the image is
independent of smaU adjustments. See Abbe in the SitzungsherichU der
Jenaer Oesellschaft fur Medizin und Natunmssenscha/tenf 1878 ; see also
Czapski's Theory of Optical InstrumentSf p. 165.
In Czapski's Theory of Optical Instruments, p. Ill, there is given the follow-
ing rule for attaining freedom from distortion : — ** The ratio of the trigonometrical
tangents of the angles which the corresponding chief rays in image and object
make with the axis, must be constant.'* This rule has been the subject of recent
discussions. See Kaempfer, in Eder's Jdhrbudiy xi. 1897, p. 247 ; and M. von
Rohr, in the Zeitschrift fUr Instrumentenkunde, September 1897, p. 271. The
latter writer has shown that the constancy of the tangent-ratio is the necessary
and sufficient condition for freedom from distortion only when the system is itself
such as to yield on both sides a distortionless image of the stop. According to a
statement of Abbe, the degree of distortion, even in the case of symmetrical
objectives, depends in general on the distance of the object ; freedom from dis-
tortion being only attained when the magnification is equal to unity.
VI DISTORTIONLESS PICTURES 35
Bi-convex lenses can be spherically corrected either for the
position of the stop, or for the position of the object, by
choosing the curvatures of the two faces not equal, but such as
will make the amount of refractive work performed by the
two surfaces respectively equal to one another. If the size of
the stop is properly proportioned for reuniting the refracted
pencils, then the positions, on the one hand of the object, on
the other hand of the stop, which will lead to good definition,
are determinate.
As one may find out by shifting the stop along the axis,
any movement of the stop nearer toward the lens increases the
size of the visible field ; but, for equal angular width of field,
the aperture-ratio (ag/mS) used for the outermost chief rays is
diminished, and is diminished, indeed, in a greater proportion
than the distance of the stop. Along with this there is
attained a more accurate convergence of the effective chief rays,
in consequence of which again the distortion will be less.
But even when the stop touches the lens, there must still, at
least with a lens of appreciable thickness, be some distortion
present.^
If the stop is introduced on the far side of the system,
between lens and image, so that it becomes a hind stop, then
the distortion changes its sign and becomes positive. The
magnification increases toward the margin of the field, and the
crossed network of lines (Fig. 7, a) assumes in the image the
form shown in Fig. 7, c, which is known as a pincushion-shaped
distortion.
If in this case the system is to become orthoscopic, the
operative chief rays must either all aim for the point that is
the image of the centre of the stop, or else they must all
appear to come from it, according as whether the image of the
stop lies behind the object or in front of it. It all comes in
general to the same purport, whether the system be furnished
with a front stop or a hind stop : in order that a system may
be orthoscopic and form an undistorted image of any flat
object, it must primarily be free from spherical aberration with
respect to the points where the stop and its image are respec-
* If the *' chief" points do not lie on a plane perpendicular to the axis, one
can learn only by working through the calculations how far the spherical aberra-
tion of the "chief " rays compensates for the former defect.
36 LUMMEKS PHOTOGRAPHIC OPTICS ch*p.
tivelj situated ; and secondarily, the chief points for all the
effective chief rays must lie on a plane perpendicular to the
axis ; or, what amounts to the same, the tangent-condition
must be fulfilled.
Let us here draw a distinction between simple and com-
pound systems of lenses. In the former the stop lies out-
side the system. Compound lens-systems consist mostly of
two component systems, I and II in Fig. 8, between which
Fio. 8. — Doable-objective consisting of Two ComjKiDeiits with Slop between them.
the stop is placed. According to whether the compound
systems are made up of two lite or two unlike com-
ponents, they are denominated " symmetrical " or " ansym-
metrieal." Those which are symmetrical with respect to
the stop in the middle, such as the Aplanats, we will call
" Double-objectives."
Let us consider any compound lens -system. "We shall
assume that, whatever its species, it is such as to give, from a
Jlat object perpendicular to the axis, a corresponding Jlat
image also perpendicular to the axis; and then, proceeding
in a manner analogous to that followed for simple systems
treated above, we will investigate the conditions which it
must fulfil in order that the picture shall be angle-true and
distortionless. First let us consider the course of the chief
rays for a given position of the actual stop P.
To this end we must find by graphic construction, accord-
ing to Abbe's method, the virtual stops, or "pupils," which
limit the field in the object-region and the image-region
respectively. Wherever the actual stop is inserted in the
optical system, it can always be shown that the operative
pencils of rays in the object-region are limited, as it were, by a
VI DISTORTIONLESS PICTURES 37
stop, the image of which, relatively to the system itself, limits
the pencils of rays in the image -region. In other words,
regarded from the front, through the front component of the
lens, the actual stop acts on the incident rays as a virtual
stop of a different size and position ; and when regarded from
behind, through the hinder component, the actual stop acts as
toward the emergent rays as a virtual stop of again a different
size and position. Abbe gives the name of " entrance-pupil "
to the virtual aperture in the front aspect, and that of " exit-
pupil " to the virtual aperture in the hinder aspect. These
two pupils ^ are conjugate one to the othe^, each being the
image of the other so far as the whole lens-system is concerned.
If, as in the compound objective under consideration, the actual
stop P lies between two components I and II, then necessarily
the (virtual) image Pj, which component I gives of the stop
P, acts as erUrance-pupil, while as exit-pupil there is the (virtual)
image Pg, which component II gives of the same. The aper-
tures Pj and Pg, which are conjugate with respect to the whole
system I + II, are accordingly the defining limits of angular
width for the admission of rays. They fully replace, in their
action on the light, the actual stop P. If in any instance
Pj and Pg were "real" images of P, we might remove the
latter and replace both Pj and Pg by bodily stops. But as
P is in almost every case nearer to the component lens
than the principal focus of the latter, the images P^ and Pg
are virtual.
Por the sake of greater intelligibility, let us here consider
the stop as being itself very small. Then the pencils are
reduced almost to their chief rays, and these all intersect one
another at one point, the mid-point m of the stop P.
If the compound system I -}- II is to be orthoscopic and to
produce a similar image of any plane object, then in this case
also the condition is that the pencils of chief rays in the object-
space and in the image-space respectively must trace similar
figures where they are intercepted by the pairs of conjugate
planes perpendicular to the axis. This occurs in any given
compound objective only under the conditions that (1) the
chief rays both before and after refraction pass through one
^ For an excellent and simple account of Abbe*s theory of the entrance- and
exit-pupils of an optical system, see Dallmeyer's Telephotography ^ pp. 94-101.
38 LUMMER'S PHOTOGRAPHIC OPTICS chap.
common point on the axis, or when sufficiently prolonged meet
in one single point ; and that (2), on the other hand, the " chief
points" of the oblique pencils, relatively both to the front
component I and the hinder component II, lie upon a plane
perpendicular to the axis (compare Fig. 8). If the latter con-
dition is fulfilled of itself, then the sole requirement for the
production of an orthoscopically similar image is — The combina-
tion must he free from spherical aberration with respect to the
entrance- and exit-pupils.
The Double -objectives are distinguished from all the
unsymmetrical compound systems by the circumstance that in
them the path of the chief rays through the middle stop is
absolutely symmetrical.
Howsoever oblique may be the course of the chief incident
rays before the front component, those only being operative
which actually cross one another at the aperture of the mid-
stop, the emergent chief rays which are conjugate to them will
emerge from the hinder component parallel to their several
directions before incidence. This relation exists provided the
incident chief rays all aim for one and the same point (namely,
the mid-point m^ of the entrance-pupil P^, which is conjugate,
with respect to the front component, to the real mid-point m) ;
or, in other words, provided the front component is free from
spherical aberration with respect to the position of the stop
and to that of its image. By reason of the symmetry, all the
chief rays emerging from the hinder component will also appear
to come from one single point, viz. from the image of m with
respect to that component. But if the incident chief rays
intersect one another in a single point, the emergent ones also
in a single point, and if the conjugate chief rays run parallel
to one another, then the latter will trace out similar figures
when they cross all planes that are perpendicular to the axis.
However the chief points may be situated, the double-objective
always gives angle-true and rectilinear pictures, provided it is
corrected spherically with respect to the places of the entrance-
and exit-pupils. In the double-objective the removal of the
spherical aberration of the system, with respect to the chief
rays that are operative for a given position of the stop, is the
sole condition for orthoscopy — that is to say, for giving pictures
that are free from distortion. In other words, the system must
VI DISTORTIONLESS PICTURES 39
be spherically corrected with respect to the entrance- and exit-
pwpils}
Without going now any further into the consequences
which might be deduced from the known path of the rays in
the simpler forms of double -objective, such as the Periscope,
etc., by tracing their connection with the principle thus
established, sufl&ce it to say that as in the simple system so
also in the compound system, the condition for attaining
orthoscopy stands in direct contravention of the requirements
for great intensity of light. In general it would be difficult,
to say the least, to procure a lens-system which should be free
from spherical aberration, not only with respect to its entrance-
and exit-pupils, but also at the same time with respect to the
relatively distant object and its image. At least it would be
difficult for a lens having a large aperture-ratio.^
^ Early in the year 1896, when Professor Lummer arrived at the establishment
of this condition for the orthoscopic formation of the image, he asked his friend,
Dr. P. Rudolph of Jena, to be so good as to calculate out how far the objectives
in commerce which were commonly described as orthoscopic were spherically
corrected with respect to the entrance- and exit-pupils. Some calculations
worked out with this purpose showed that most compound lens-systems do not
comply with this condition. Professor Lummer therefore followed out no further
the consequences of this condition, and only approached the matter again when,
shortly afterwards, Dr. Rudolph wrote that the so-called "notoriously distortion-
free " objectives, both symmetrical and unsymmetrical, were far from being free
from distortion, at least not free for all different distances of objects. Professor
Lummer takes the opportunity here of expressing his warmest thanks to
Dr. Rudolph for the advice and information which he has many a time im-
parted to him.
^ Yon Seidel discusses the question as to when a system is spherically corrected
simultaneously for various distances of the object — that is to say, when it
satisfies the so-called Herschel's condition. He finds that the latter contradicts
Frauenhofer's condition. Only in certain quite special cases can both conditions
be fulfilled at once. The telescope, used as a whole, is an apparatus so designed
that both the conditions named will be attained if one of them is realised.
See AstroTvomische Ncu^richteUj xliii. p. 326, 1856.
CHAPTEE VII
SYSTEMS CORRECTED FOR COLOUR AND SPHERICITY, CONSISTING
OF TWO ASSOCIATED LENSES — OLD ACHROMATS
An important advance in the domain of practical optics was
made in the year 1752 by DoUond, when he succeeded, by
combining two lenses of diflferent kinds of glass, in eliminating
the chromatic dispersion without destroying the power of the
lens to refract the rays to a focus. So far as concerns the
first of the five aberrations in von Seidel's list, namely the failure
of the lens to give a sharp image in the middle of the field,
the removal of which is the first term in the correction for
spherical aberration, and so far as concerns the first term ^ in
the corrections for chromatic dispersion, namely the correction
for the focussing of different colours at diflferent distances from
the lens, DoUond's principle affords a satisfactory solution.
For by suitably combining two lenses of diflferent materials a
complete elimination of these two defects can be attained.
The popular method of describing Dollond's invention is to
say that he obtained an achromatic lens by associating together
a lens of crown glass and another of flint glass, one being a
positive, the other a negative lens, and so made one correct the
chromatic aberration of the other. This mode of statement is
^ In other words, this means that the middle part of the objective can be made
to bring to accurate convergence at cme point two pencils of rays of different
colours. In order that the higher members also in the series of chromatic
aberrations, or, as Abbe calls them, the ** chromatic differences of the spherical
aberration," should be eliminated, it is necessary that aZl zones of the objective,
and not its central region only, should be corrected, so as to bring the two
colours from these parts of the lens also to focus in the same point. With two
lenses (flint and crown) only two colours can be accurately brought to coincidence ;
the coincidence for the remainder of the colours is only approximate, because
of the irrationality of dispersion.
CHAP. VII SYSTEMS CORRECTED FOR COLOUR AND SPHERICITY 41
not only loose, but is partly misleading. No lens made of two
kinds of glass only can be achromatic for all diflferent colours
from different parts of the spectrum. It can be designed to
bring together red and violet rays, but in that case will not
accurately focus yellow, green, or blue to the same point. Or
it can be designed to bring orange and blue together, but will
fail in accuracy with respect to red, yellow, green, and violet.
There always remains a residual colour error uncorrected, this
being a secondary chromatic aberration, or, as it is usually
termed, a "secondary spectrum" or a "residual dispersion."
Again, the so-called achromatic lens may bring together the
two colours to one principal focus, and yet not produce images
of the same size for the two colours, because the true focal
length (on which the magnification depends) is the length
from the principal focus back to the optical centre (or " principal
point " of Gauss), and the position of the principal point may
not be the same for the two colours. Again, the so-called
achromatic lens, though it may bring axial pencils of two
colours to meet accurately at one focus, will not be even in
this limited sense achromatic either for wide pencils parallel
to the axis, or for oblique pencils. In fact, just as the errors
due to sphericity were shown by von Seidel to be numerous,
so the errors due to chromatic dispersion are also numerous.
The ordinary so-called achromatic lens of Dollond can be made
to correct the first term of the series of spherical aberrations
and the first term of the series of chromatic aberrations ; but
by putting together two lenses, one flint, one crown, not more
than these two first terms can be corrected, and corrected only
for two colours of the spectrum.
These two -lens combinations corrected in this sense
achromatically and spherically we shall henceforth call
achroToats, The chromatic aberration is removed by selecting
as materials two glasses having for equal amounts of dispersion
unequal amounts of refraction, while the removal of the
spherical aberration depends on selecting the appropriate form
for the lens. The solution of the problem how to make an
achrorruit depends on the application of the principle of com-
pensation, which we shall many times over come to recognise
as the main means for producing the best images through
compound lens-systems. If one chooses two lenses of proper
42 LUMMER'S PHOTOGRAPHIC OPTICS chap.
form, made of appropriate materials, one of which makes a
parallel beam convergent, and the other such as to make
a parallel beam divergent, by suitable choice the positive
chromatic and spherical abeiTations of the first lens can be
compensated by the negative chromatic and spherical aberra-
tions of the second lens, biU withovi entirely removing the con-
vergence of the pencil, as would be the case if both lenses were
made out of the same kind of glass. In this way, therefore,
one obtains an achromat in which both the spherical and the
chromatic aberrations are annulled, but having a definite focal
length.
For the compensation of the spherical and the chromatic
aberrations with the provision of a prescribed focal length —
that is to say, to satisfy three prescribed conditions — there must
be three variable elements at our disposal. These we have in
the circumstance that in an achromat made of two lenses, one
flint, one crown, there are four curvatures which we can choose
at any values we please. To comply with the three conditions
mentioned above we can vary three of these radii. But since
with four available curvatures we might satisfy four conditions,
it is usually preferred so to shape the curvatures of the lenses
that the two inner faces shall have equal radii of curvature
(one convex, the other concave), in order that the two lenses
may be cemented together. The thickness of the glasses is here
left at any convenient amount; but it must be relatively
small.^
For the compensation of the spherical and chromatic
aberrations of the first order, and the production of a definite
focal length, the only requirement, therefore, is two thin lenses
cemented together.
Such an achromat accordingly projects a colourless point
as the image of a white point serving as object; or, more
accurately expressed, it projects a small colourless diffraction'
disc of such a magnitude as corresponds to the aperture-ratio
employed.
We will assume with Petzval that the achromat may be,
as in Daguerre's time, when used with the aperture //1 6, so
^ In the case of great thicknesses of lenses, it is possible even with two lenses
of the saTM kind of glass to produce achromatism either of the focal points or of
the ** principal " points. See F. Eessler, Schldmilch*s Zeitschrift, xix. p. 1, 1884.
JL I -J^ »^B^^^^— ^i^W^P^
VII SYSTEMS CORRECTED FOR COLOUR AND SPHERICITY 48
well corrected spherically that the image may bear a three-
fold enlargement. Such a cemented achromat is inferior to
a PetzvaFs portrait-objective in defining power threefold only,
while it is its inferior in intensity of light nineteenfold. Yet
it surpasses the single glass convex lens by about three times
in definition and about four times in intensity. To this
superiority there must be added the further advantage that it
is free from the so-called " chemical focus.''
If instead of the cemented lenses there are used two
separated lenses, as these four different radii of curvature
at the four surfaces, there arises the possibility of choosing
the curves so as to satisfy a fourth condition in addition to
the three enumerated above. As such a fourth condition the
important one to adopt,^ at least for lenses to embrace a
wide field of view, is Frauenhofer's condition, which is fulfilled
in the objective of his celebrated heliometer. Steinheil pushed
the investigation further, endeavouring in his telescope objec-
tives to comply with the second chromatic condition, which
requires the magnification (and therefore the true focal length,
or length measured back from the principal focus to the
" principal point ") to be of equal value ^ for two colours.
In general, when using ordinary kinds of flint and crown
glass, such as were available in Frauenhofer*s time, solution
of the equations results in two different typical forms ^ of
cemented achromats; that is to say, there are two general
forms of cemented achromat for which the spherical and the
chromatic aberration is annulled, and only these two forms
if the old kinds of glass are used. The first typical form,
Fig. 9, is that commonly used for telescope objectives, since
^ As already mentioned, Frauenhofer*s condition, which is identical with
Seidel's condition that 83=0, is identical with the sine-relation for relatively
small angles of aperture of the emergent [pencils. In the case of telescope
objectives this sine-condition assumes a simple form. It is satisfied if the
** chief ^points '* for the various rays parallel to the axis lie upon a drde having
its centre at the principal focus and the true focal length as its radius.
(Compare Steinheil and Volt's Handbook of Practiced Optics, Leipzig, 1891,
p. 57.) See also Appendix III., p. 122.
2 Compare A. Steinheil's memoir ** On the Orientation of Objectives consisting
of Two Lenses, and on their Aberrations," Astronomische Nachrichten, cix. p. 216,
1884.
' On this point the reader should consult an admirable paper by Mr. Conrad
Beck ' * On the Construction of Photographic Lenses, " in the Journal of the Society
of Arts, 1st February 1889.
44 LUMMER'S PHOTOGRAPHIC OPTICS chap.
even with a relatively large aperture -ratio it gives sharp
images ; it has the convex side (of the crown glass) turned
outwards toward the light. If it is reversed, so that the
concave side is turned outwards toward the light, it gives a
definition that is less sharp at the centre of the field, but
gives more widdy extended images of moderate sharpness.
Fig. 9. Fio. 10.
The two Typical Forms of Old Achromats.
Daguerre took his first photographs in the year 1839 by the
help of such a meniscus lens.^ The second typical form, Fig.
10, which is a meniscus shape, consisting of a positive meniscus
of crown combined with a negative meniscus of flint, if used
as an objective with the concave side outwards, has sharp
definition ovei: a larger region of the field.
As in the case of the simple glass convex lens, achromats
are used in photography with a front stop, in order to shift
into the focal plane the circles of least confusion formed by
the astigmatic oblique pencils — that is to say, in order
artificially to straighten the image. Eecently some advantage
has been found to accrue in using as landscape lenses, instead
of these simple achromats, so-called anastigmatic objectives,
consisting of three or of four lenses cemented together ; and
of these we shall speak in detail later. For the purpose of
understanding aright these anastigmatic multi-lens objectives
with flat fleld, we must examine more closely into the achromat,
and particularly with respect to the kinds of glass which are
used in its construction.
1 So, at least, it is sometimes said. The form Fig. 10 was, however, only
invented in 1854, by the late T. Grubb. The meniscus used by Daguerre more
nearly resembled Fig. 9, the flint being a bi-concave lens, cemented to a bi-
convex crown, the flint being the outward lens as used.
VII SYSTEMS CORRECTED FOR COLOUR AND SPHERICITY 45
Two epochs are to be distinguished with respect to the
modem use of glass in the construction of fine lenses — the
epoch of Frauenhofer and the epoch of Abbe. Before Abbe
and Schott, working in Jena, had completed their epoch-
making researches for the production of new kinds of optical
glass, there existed as available for optical calculations only
certain kinds of glass in which the amount of the disper-
sion went on increasing with the increase of the refractive
index. The higher the refractive index of the glass, the
greater was its dispersive power; not in strict proportion,
indeed, otherwise achromatic combinations would have been
impossible. But there was no known kind of glass which,
with a higher refractive index, had a lesser dispersion. The
flint, which had a greater dispersion than the crown, always
had also a higher refractive index. One has only to look
through the lists of the optical glasses used by Frauenhofer,
or those manufactured by the great houses of Chance and of
Feil, and compare their refractive and dispersive indices, to
see that this was so. The admirably careful measurements
made by Frauenhofer,^ and those subsequently made by
Steinheil,^ by Bailie,® and by Hopkinson * on the values of the
refractive indices for different parts of the spectrum afford no
exception. This property of all glasses known during the
Frauenhofer period necessarily involves as a consequence that
the converging lens (marked 1 in Fig 9) of the achromat is
made out of a glass having both a lesser dispersion and a lower
refractivity than those of the glass which is used for the
diverging lens (marked 2 in Fig 9).
If we denote by/^ the focal length of the lens 1, by /g that
of the lens 2, and by F the resultant focal length of the
achromat, then, as is well known, the relation between them
is expressed as —
1_ 1 1
Since the lens 2 is a diverging or concave lens, we must
^ SUzuTigsbeHckte der k'oniglichen hayerischen Akademie der Wissenschaflen zu
Miinchen, vol. v., or Gilbert's AnnaleUj Ivi. p. 292 (1817).
2 Steinheil and Voit, Handbuch der Angevoandten Optik (1891), pp. 12-33.
^ Annales du Bv/reau des LongUvdes, cxciii. p. 620.
* Proceedings of the Royal Society ^ xxvi. p. 290, June 1877.
46 LUMMER'S PHOTOGRAPHIC OPTICS chap, vii
take /g as being negative, and then the relation may be
written
J 2 ""/i
Now, in order to produce real images, the achromat is to
be itself a positive lens, and F must be positive, and this
obviously cannot be the case unless f^ is greater than f^. In
order that any lens shall have a high power (and therefore
a short focus), it must either have a great curvature, if its
refractive index is moderate, or it must be made of glass of a
high refractive index with a moderate curvature. A converging
lens of short focal length, combined with a diverging lens
of longer focal length, never gives, however, images which are
sharp and free from colour defects, unless the first of its two com-
ponents, in spite of its stronger curvatures or of its higher
refractive index, produces spherical and chromatic aberrations
only just as great as those of the second of the two com-
ponents. If in order to procure compensation of the chromatic
dispersion one must perforce employ one of the lenses with
deeper curves than the other, it is clear that the very deepness
of the curvature will, unless the right selection is found by
calculation, cause trouble by producing too great a spherical
aberration to be compensated by that of the lens with less
steep curves. Now, since the size of the circle of chromatic
dispersion (see p. 23) is inversely proportional to the focal
length, it necessarily follows that the converging lens must be
made out of a glass of lesser dispersion than that used in the
diverging lens, and therefore— so long as these glasses only are
procurable in which the dispersion increases with the refrac-
tivity — it must needs be made out of a glass with a lower
refractive index.
Achromats made of two cemented lenses constructed of
these older kinds of glass of the Frauenhofer period we will
denote by the name Old Achromats, in contradistinction to
the new types of achromat, which can only be constructed
with the use of certain of the modern Jena glasses, and which
may be called New Achromais,
CHAPTEE VIII
NEW ACHROMATS
We will assume that a two-lens objective may be so corrected
that it reunites the rays stigmatically — both for points on the
axis and for others aside of it. In such a case the first three
of von SeideFs aberrations have been eliminated (or in symbols,
Sj = 0, Sg = 0, S3 = 0) ; and while a point-object gives a point-
image to any object, there will correspond an accurately
defined, though in general curved and distorted image. If no
rays are present except such as comply with these conditions of
von Seidel, then one would observe the image of a flat object
as though it were a spherically curved surface whose vertex
only touched the place where a flat image ought to be formed
(compare p. 24).
The condition that this spherical surface should change
into a plane is accordingly set down as von Seidel's fourth con-
dition, namely S^ = ; or, if we introduce symbols for the
quantities, the summation of which is signified in S^, the
condition is
N"
2- = 0; [1]
r
where r is the radius of curvature of any of the surfaces, N
the difference of the reciprocals of the refractive indices of the
two media bounded by that surface, and the summation to be
taken for all the surfaces. Let us denote by the letter 7 this
reciprocal, and by /a a refractive index ; also let us distinguish
the media by giving them odd numbers, and the surfaces by
giving them even numbers as suffixes.
Then von Seidel's condition for a two-lens combination with
48 LUMMER'S PHOTOGRAPHIC OPTICS chap.
four refracting surfaces and media would be, if written in
fuU—
7i-78 I 78-75 | 75^^7 _t /y7""^9 ^0
^'O ^2 ^4 ^6
Or, for two lenses respectively of indices fi^ and fi^ surrounded
with and separated by air of refractive index = 1 —
i-i i-i i-i i-i
=0. . . [2]
^0 »-2 '•4 »•«
or, as it may be written —
As this formula shows, the condition for the flattening of
the image succinctly depends upon the indices of refraction and
upon the radii of curvature, but in nowise upon the distances
of the various refracting surfaces, nor upon their order, nor yet
upon the distance of the object. But there is reason to think
that if any change were made in these matters that do not
enter into present consideration, then the first three of von
Seidel's conditions would in general no longer be fulfilled, and
with them would vanish the formation of stigmatically sharp
images. But if precision of focus no longer existed, it is useless
to speak of prescribing the curvature of the surface in which the
image is formed. Only under the provision that a stigmatic
focussing up to the fifth order of precision is attained (i,e. that
S^ = Sg = S3 = 0), does the condition S^ = for the flattening
of the image become valid and unambiguous.
Now the focal length / of a thin lens, whose surfaces have
radii of curvature r^ and r^y is, as is well known,
[3]
or in words, the reciprocal of the focal length is equal to the
refractivity of the material of the lens (air being taken as
unity) multiplied by the algebraic sum of the curvatures of its
two faces.
Accordingly, if we treat the thickness of both the lenses of
VIII NEW ACHROMATS 49
our objective as negligibly small, the condition for flattening
the image assumes the following simple form ^ : —
/*i/i M2/2
or
Mi/i= -^lU .[*^]
This formula tells us, firstly, that /^ and f^ must have
opposite signs, and secondly, that the lens of shorter focal
length must be made out of glass of a higher refractive index.
The value of /g in terms of /^ may be stated at once as
/,= -^^-/x [5]
Now the value of the resultant focal length F of the
compound objective, whose component lenses have focal lengths
/j and/2 respectively, is
p / l^/2 .
and if in this we substitute for f^ its value in terms of f^,
from equation [5], we obtain —
F=-^-, [6]
the significance of which relation we will discuss more
narrowly.
First it is evident that if fj^i = [^2 ^^^ iocal length F at
once becomes infinite. Two lenses, therefore, if of the same
glass, give a fiat image only when they act together^ as a
piece of parallel flat glass. Equation [4a] shows that if fi^ = fi^
then also /^ = -f^,
^ In this form Petzval, in the year 1843, expressed the condition for flattening
the image, without, however, giving the necessary general proof of its validity, the
demonstration of which is due to von Seidel. Applied to a single lens formula [4]
has no meaning. Considering the small attention paid, down to the most recent
time, to von Seidel's theory of the formation of images, it cannot be wondered at
that the correctness of Petzval's formula has been doubted, and its significance
under- valued. The general principle of equation [4] was discovered by Airy, and
is to be found in Coddington*s Treatise on the Beflcddon and Refraction of Light
(1829), pp. 197-200.
'^ Astronomische Nachrichten, xliii. p. 323, 1856.
E
50 LUMMER'S PHOTOGRAPHIC OPTICS chap.
The discussion is of more interest for the cases where /a^
is either greater or less than fi,^ — that is to say, where the two
lenses are made out of substances of different refractivity, as is
the case in achromats.
In the objective depicted in Fig. 9, p. 64, /^ is positive and
/g negative. If such a lens is to be made achromatic by the
employment of glasses of the Frauenhofer epoch, and also
have a real focal length, then necessarily /2>/j, and consequently
also /irg must be greater than fi^. The condition for obtaining
a flat image, expressed in formula [4a], requires that if f^ is
going to be greater than Z^, then of necessity must /Hj be
greater than fi^ ; for a reference to equation [6] shows us that
if it is not greater, the resulting focal length -Pwill be negative.
Then it is clear that, so long as the old kinds only of glass are
available, on£ cannot possibly make, by combining together two lenses,
an achromat that has ajlat field ; for the condition of achromat-
ism requires the more powerful of the two lenses to be made
of crown and the less powerful of flint, while the condition for
getting a flat field exactly contradicts this, and requires that
the more powerful lens should be of flint and the less powerful
of crown. If, on the other hand, one takes for the conveiging
lens/j a glass of higher refractivity than that used for the diverg-
ing lens f^, then one can only attain to achromatism if one can
find glasses such that the glass of higher refractivity shall have
a less dispersion than the glass of lower refractivity. In Frauen-
hofer's time no such glasses existed.
The flattening of the image required at the same time as
achromatism in the two-lens objective involves then a further
condition as to the sorts of glass to be used ; for these must be
such that high refractivity with a lower dispersion is paired oflT
against a lower refractivity with a high dispersion. Such glasses
were first put at the disposal of practical optics as a con-
sequence of the foundation of the Glass-technical Laboratory
of Schott and Co. at Jena. In the older kinds of glass of
the Frauenhofer period, higher refractivity had always been
associated with higher dispersion. For that reason, in the
two-lens objectives made out of these older kinds of glass, the
production of a flat field was impossible because it contravened
the much more important condition of achromatism.
On the other hand, amongst the many kinds of glass
/
i
YIII
NEW ACHROMATS
61
made in the Jena factory, while there are numbers closely
resembling the old sorts, there are some that differ widely
in their properties.
As there is much misunderstanding about the Jena glass,
it is not inappropriate that something should be said here
about it. Messrs. Schott and Co. have, during the dozen years
of the operation of their factory, put out on the market some
hundreds of kinds of glass, some of which have since been
withdrawn, not being found of permanent value. Their
present catalogue enumerates some seventy-five different kinds,
ranging from a very light boro-silicate crown of index 1*4967
to a densest silicate flint of index 1*9626. A table containing
a selected few from their current list is here appended.
• Table of a Few of the Jena Glasses
Factory
Number.
225
s
30
802
40
138
20
1209
381
726
376
230
118
41
s
67
Description.
Light Phosphate Crown .
Dense Barium Phosphate Crown
Boro-silicate Crown .
Silicate Crown ....
Silicate Crown of high refractivity
Silicate Crown of low refractivity
Densest Baryta Crown
Crown of high dispersion .
Extra Light Flint
Ordinary Light Flint
Silicate Flint of high refractivity
Ordinary Silicate Flint
Dense Silicate Flint .
Densest Silicate Flint
Refractive
Mean
Md-1
Index
Dispersion
Mf-Mc
Z^-
Mf-Mc-
= v.
1-5159
0-00737
70
1-5760
0-00884
65-2
1-4967
0-00765
64-9
1-5166
0-00849
60-9
1 -5285
0-00872
60-2
1-6019
0-00842
59-6
1-6112
0-01068
57-2
1-5262
0-01026
51-3
1-5398
0-01142
47-3
1 -5660
0-01319
42-9
1-6014
0-01415
42-6
1-6129
0-01660
36-9
1-7174
0-02434
29-5
1-9626
04882
19-7
To these may be added for comparison a few other sub-
stances: —
Fluor-Spar
Canada Balsam
Diamond .
Aniline .
Water
Cinnamic Ether
Piperine .
Silver Iodide .
MD.
Mf-Mc-
1 -4338
0-00446
1-526
0-0227
2-4173
0-0251
1-5863
0248
1 -3337
0-006
1-5607
0-0508
1-681
0-069
2-1816
0-123
62 LUMMER'S PHOTOGRAPHIC OPTICS chap*
It will be noted that in this list the glasses selected are
arranged not in the order of their refractive indices (though as
a matter of fact the glass at the head of the list has the lowest,
and that at the bottom of the list the highest refractivity), nor
are they arranged in the order of their dispersivity. The order
chosen is that of the amounts of their mean refractivity for
equal amounts of dispersion. This is best explained by a
little circumlocution. The mean refractive index given for
each particular glass is its index for the yellow light of the
sodium-flame, i.e, for the D-line of the spectrum. It is denoted
by the symbol fij^; and as in lens-formulae, such as [3] on
p. 48, it is the difference between this index and that of air
which constitutes the effective refractivity, we take /^ — 1 as
the mean refractivity of the material. The dispersion is the
diflference between the refractive indices for two rays of different
colour, and may be expressed either for the whole range of
colours in the spectrum, or only for a part of that range. We
might compare the dispersions of two kinds of glass, for
example, over the region of the spectrum that lies between the
A-line at the extreme end of the red, and the D-line in the
yellow, and in that case fij^ — fij^ would be the partial-dispersion
over that region. It is, for purposes of comparison, useful to
know these partial-dispersion values, since if we can find two
kinds of glass, equally satisfactory in other respects, for which
the respective partial-dispersions are nearly proportional to
their dispersions as a whole, then such a pair of glasses will,
if made up into an achromatic combination, have less residual
colour-error — less " secondary spectrum " — than would be the
case if their partial-dispersions were not so proportional. For
ordinary optical purposes, and to bring the focus for red rays to
coincide with the focus for blue rays, it is usual to measure the
dispersion from the C-line that lies at the orange end of the
red region to the F-line in the blue region.^ That is to say,
^ For purely actinic purposes it is necessary to design the lens so as to reunite
all those rays that produce actinic effects, that is to say, from the blue-green of
the spectrum to a point in the ultra-violet, disregarding the red, orange, and
yellow parts entirely. For this purpose the mean refractivity might be taken
as Atfl- 1, and the dispersion as that from the F-line to the bright line in the
violet afforded by an electric spark from a mercury electrode. But for pJioto-
graphic purposes it is desirable to reunite the "chemical" focus with the " visual"
focus ; so /An - 1 is taken as the mean refractivity, while the dispersion is reckoned
VIII NEW ACHROMATS 5S
we measure /a, — fi^y and use this value in our calculations as
a measure of the mean dispersion of the material. But in
lens designing it is still more important to know what propor-
tion the mean refractivity bears to this mean dispersion.
Accordingly, if we divide one by the other we obtain the
quantity which, expressed in symbols, is
for which it is more convenient to use the single symbol v.
We shall call it the refractivity for equal mean dispersion, or the
achromatic refractivity. In the table the order of the various
glasses is that of their values for v. That at the top of the
list — the lightest crown — has the greatest refractivity for a
given amount of dispersion, while that at the bottom of the
list — the densest flint — has the least refractivity for the
given amount of dispersion. The importance of knowing these
values lies in this : that if we know these values of v we can
at once state what the relative powers of two lenses must be
that they may achromatise one another. Suppose, for example,
we have to make an achromatic pair, using for the positive
lens the silicate crown glass called " 40," and for the negative
lens the ordinary light flint "0 376," we see that the former
has value i/=60*9, and the latter i; = 42*9. These are two
old-fashioned glasses of normal sorts. If we take the two
lenses having their powers respectively proportional to these
values, they will have equal dispersions that exactly compensate,
and the resulting lens will have a power proportional to the
simple difference — ^in this case 18*0. For example, to use the
language of the ophthalmic opticians, if we wanted to make an
achromatic combination having a power of 12 dioptrics, we
60'9
must take a "silicate crown" lens of +12 x = +40*6
18
dioptrics, and combine it with an " ordinary light flint " lens
42*9
of —12 X = — 28*6 dioptrics. Put these together. If
from the D-line to the G-line (or from the bright hydrogen line near it), instead
of from C to F. In that case the refractivity for equal mean dispersion will be
denoted for distinction as ?. It equals ^-^- — .
64 LUMMER'S PHOTOGRAPHIC OPTICS chap.
made each as piano-lenses, they will when cemented resemble
Fig. 11, p. 56, and will have a net power of + 12 dioptrics.
If thus taken without reference to the curvature of the cemented
surface, it may indeed be achromatic, but will not be free from
spherical aberration. It will not be a perfect "achromat."
Neither will it have a flat field, since in this case we have
taken a pair of glasses, of which the one with lower dispersion
has also got the lower refractive index. This pair of materials
could make only an " old achromat " at the best.^
Now consider the case of two glasses such as the follow-
ing: —
AAd.
Mf-/*c.
v.
Densest Baryta Crowu .
1-6112
0-01068
57-2
Soft Silicate Crown
1-5151
0-00910
56-6
This barium glass is one of a new species of glasses in which,
by the admixture of barium salts, low dispersive powers have
been obtained along with high refractivity. Out of the above
two glasses an achromat with flat field might be constructed,
since here
whence it possesses a positive focal length. Turning again
to the table of Jena glasses, suppose we selected as crown
the brand called "S 30" — a barium phosphate crown —
and as flint the brand "0 726" — the extra light flint;
we should have
i^=-^'l^ = 43-5/,.
M1-M2
This would give a very long-focus combination, but the field
would be flat if the curves were chosen so as to correct for
spherical aberration.
The higher, moreover, the refractive index fi^ of the material
selected for the positive lens (provided one does not go so far
as to destroy the possibility of achromatising it), so much the
^ See some examples of calculation of doublets and triplets by Mr. E. M.
Nelson, in his Presidential Address to the Royal Microscopical Society, 1898,
Joum. R, M. Society, pp. 156-169. The question of the modern design of
triplets is treated by M. von Rohr in his recent work, Theorie und Geschichte des
Photographischen Objektivs, pp.. 363-387.
VIII NEW ACHROMATS 55
shorter may be the positive focal length of the combination,
along with complete flatness of the image. For diamond (if
one could use diamond as a material), which has a refractive
index of fi^ = 2*4, combined with ordinary flint, fi^ = 1'6, would
give a resulting focal length only about 3 times f^.
Long ago Petzval, in the discussion about the possible
flattening of the image, drew attention to the image-flattening
property of the diamond, without, however, expressly removing
the difficulties which would militate against the achromatising
of a diamond positive lens by a negative lens of other material.
On the other hand, von Seidel raised the objection that the
requirements of achromatism contravene that of procuring a
flat image.
Now that by means of the Jena glasses there are available
the anomalous pairs of glasses which are needful for flattening
the field, achromats have been produced in which the positive
lens has a higher refractive index and a lesser dispersion than
the negative lens.
As suggested above, we describe achromats made out of
anomalous pairs of Jena glass as New achromats, to distinguish
them from those made of the old kinds of glass. The
cemented two-lens new achromat cannot, however, be spheri-
cally corrected so well as the old achromat. Its chief import-
ance, on the contrary, appears, as we shall show, in its use in
combination with the old achromat. In any case, however, we
shall understand by the term New achromat an objective in
which the disposable elements other than those needed for the
attainment of the prescribed focal length are unreservedly
devoted to the best possible annulment of chromatic and
spherical aberrations. It will also be maintained that the
distinction between the n£>w and the old types consists, not
in the order in which the glasses follow one another, but
exclusively in the question whether the positive member
consists of glass of higher index with lower dispersion than
the negative lens, or of glass which has a lower index as well
as a lower dispersion. In any and every case the correcting
lens must have a higher dispersion than that of the lens which
it is to correct.
The first two-lens objective that was made out of anomalous
pairs of glass is that represented in Fig. 11, which is one
56
LUMMER'S PHOTOGRAPHIC OPTICS
CHAP, vm
component of the Eoss's Concentric lens designed by Dr.
Schroder. This, taken by itself,
has, in consequence of its shape, a
considerable uncorrected, spherical
aberration, which, indeed, is not
completely removed in the new
achromats. Also in the Grroup-
aniiplanet lens of C. A. Steinheil,
to be described later, one member
is made out of anomalous pairs of
glass ; but this lens is hyperchro-
Toatic} and expressly so in order to
afford to the two other members
of the aniiplanet very strong aberrations of opposite kinds.
^ More chromatic, therefore, than the equivalent single glass lens.
FiQ. 11. — Achromat used in
Concentric Lens.
CHAPTEE IX
SEPARATION OF THE LENSES AS A MEANS OF PRODUCING
ARTIFICIAL FLATTENING OF THE IMAGE
The Seidel-Petzval formula for the radius of curvature at
the vertex of the image is, as we have already mentioned,
unambiguous only when the formation of the image is stigmati-
cally accurate up to terms of the fifth power. In order to
establish the correctness of this view, one need not apply it to
a system which realises to the highest degree the theoretically
perfect production of focus.^
The magnitudes not included in the Seidel-Petzval equation,
for example, the distance of the object from the system, and
the distance between the lenses, will have a large influence on
the curvature of the image, since when they are altered the
convergence of the rays is also changed.
As an example let us, following Schroder's lead, select the
case in which two plano-convex lenses 1 and 2 (Fig. 12) are
used as a system — first (a), very near together ; and secondly
(6), separated widely from each other. If the image is in the
first case strongly curved, it will always become " flatter " the
further the two lenses are separated from each other.
That the image is curved when the lenses are in contact
is not to be wondered at, for then the two lenses act together
as an individual lens of equivalent focal length, although the
various faults are less than in the simple equivalent strongly-
curved lens, because the work of refracting the rays is now
shared between the two lenses. Here also to each point there
corresponds a caustic curve, so that by suitable stopping ofif
^ See Dr. Schroder's Elements of Photographic Optics (Berlin, 1891).
58
LUMMER'S PHOTOGRAPHIC OPTICS
CHAP. IX
the curved image may be artificially straightened, as explained
in Chapter V.
Now this " stopping " may be effected by the separation of
the lenses themselves (Fig. 12, 6). Here the first lens acts
similarly to a stop as regards the oblique pencils, so that only
a part of them comes into action. Moreover, in consequence
of this, each operative partial pencil passes through the two
lenses in a reversed manner, inasmuch as it traverses the
opposite sides of the two lenses. For example, the pencil which
a
Fia. 12.
traverses the lower part of the first lens traverses the upper
part of the second lens, and vice versa. The rectification
of the image caused by separating the lenses is not to be
confounded with the true formation of flat images, attained
by the choice of suitable kinds of glass, as in the "new
achromats." The latter process is a true correction compatible
with the use of the full aperture, or full at least in comparison
with an " old achromat " of equal power.
The artificial flattening of the image by the use of stops is
attained cU the expense of irdensity of the light, especially of
that of the oblique pencils ; for it is brought about, not by any
appropriate change in the ray-path of each pencil, but only by
exercising a suitable selection among its many partial pencils.
We are now in a position to comprehend, in the case of the
unsymmetrical double-objectives, the correction of aberrations
by methods depending upon the same principle.
CHAPTEE X
UNSYMMETRICAL OBJECTIVES CONSISTING OF TWO MEMBERS
The Petzvai Portrait-Objective, by Voigtldnder
The portrait-objective calculated out by Petzvai in 1840 is
still to-day used in almost the same form as he originally gave
to it, and as such has scarcely been surpassed. This circum-
stance shows very clearly that in the optical art, more than in
all others, theory correctly applied leads to the desired goal.
At the time photography came into existence, when Draper
of New York obtained, in 1840, the first portrait of a living
person with an exposure of two to twenty minutes duration, it
was the keen desire of all concerned to possess an objective
which transmitted more light, such as would shorten the time
of exposure. Petzvai of Vienna and Chevalier of Paris sought,
independently of oie another, to attain this end, and in so
doing they designel lens combinations of several associated
members.
Already in the year
1841 Voigtlander of
Vienna put on the
market the first objec-
tive made according to
Petzval's calculations,
and by this materially
contributed towards
making photography Kxo. 13._Petzva.-s Portrait-Objective.
popular. In this ob-
jective, depicted in Fig. 13, everything else was sacrificed to
the aim of obtaining, with a great aperture-ratio of almost //3,
60 LUMMER'S PHOTOGEAPHIC OPTICS chap.
an image of a point on the axis free from colour defects, and,
above all, free from spherical dberration of the higher order.
In spite of the great intensity of the light transmitted, the
centre of the image should be well enough defined to permit
of being enlarged many times.
If it is necessary to have one radius at one's disposal for
the elimination of the first term of the series of spherical aber-
rations, so for the elimination of five terms, as in Petzval's
objective, there are five conditions to be fulfilled.
Should one wish to satisfy these conditions, by putting
together several suitable lenses without distances between them,
spherical aberration up to a high order might doubtless be got
rid of; but other faults of the first order would reappear which
would only be caused to disappear if the system were such as
to act merely like a plate with parallel sides (see p. 12 above).
Accordingly, the lenses must be separated from each other, and
in consequence the separated parts must be rendered achro-
matic each for itself, in order to obtain stable achromatism.-^
To the five spherical and the two achromatic conditions is
added that of obtaining the prescribed focal length. These
eight conditions were in Petzval's portrait -objective satisfied
by the seven radii of curvature of the lens surfaces and one
distance. In order to obtain this result all tentative guesses
are useless, one must go to work by systematic calculation, as
Petzval did. The Petzval objective carried out by Voigtlander
would under favourable conditions produce pictures capable of
being enlarged ten times ; and it transmitted sixteen times as
much light as the single achromat used by Daguerre. This
great advance was, however, paid for by corresponding sacrifices
which made this objective, so suitable for taking portraits,
yet so very unsuitable for the taking of groups and land-
scapes.^ All the efforts had been directed to the correction
of the centre of the field alone, as a consequence of which the
image outside the central region showed aberrations due to the
^ By this term is to be understood achromatism such that for the given two
colours that are brought to reunion, there shaU be achromatism not only in the
sense that the coloured images shall be found in the same focal plane, but that
they shall be of the same size. In other words, there shall be achromatism of
the principal points as well as of the principal focal lengths.
^ These defects are even purposely exaggerated in the Dallmeyer-Bergheim
portrait lens.
X UNSYMMETEICAL OBJECTIVES 81
oblique pencils. This uae of two widely separated members
involves, on the other hand, a limited field of view, with an
illumination diminishing up to the margin of the field, whilst,
in consequence of there being six air-glass surfaces, a great
number of reflected images are formed, so that the brightness
of the image is not so great as with landscape achromats.
The Aniiplanet of A. Steinheil '
There was, however, a serious objection which clung more
or less to the older systems working with lai^e apertures ;
namely, that in consequence
of radial astigmatism the de-
finition of the imt^e dimin-
ished rapidly from the middle
to the margin. With the
object of getting rid of these
evils, Steinheil in the year
1881 constructed bis Anti-
planet (Fig. 14).
Upon the basis of most
comprehensive calculations.
Dr. Adolph Steinheil came to
the conclusion that the image „.„ ,. „. . . .,, ^ ,. , ,
° FlQ. 14. — steinheil s Anlipland.
IS the more uniform in sharp-
ness the more iinequally the whole performaiice of the objective
is divided between its two members. Accordingly, both the
members^ I and II possess aberrations of opposite hinds
of intentionally large magnitudes, and whilst the focal length
of I is positive, but smaller than the focal length of the
combination, II possesses a sufficiently large negative focal
length. The first member I is subject to the faults of a
simple positive lens, and the second member II posaesses
the faults of a simple negative lens. In this way radial
astigmatism, as well as image curvature, is diminished over
a certain field, but beyond this field the want of definition
' German Patent No. 16,35i of jear 1881,
' From this point ouwarda it will be convenient, for all objectives that consist
of two separated membera, to denote the front member as I and the hinder
member as II.
62 LUMMER'S PHOTOGRAPHIC OPTICS chap.
rapidly increases, so that outside certain limits, even when
well stopped down, there is no sharp definition. The objec-
tives considered in the following sections, which yield an
image free from radial astigmatism, and also flat, could not
possibly have been constructed prior to the invention of the
new Jena glasses.
Zeiss Anastigmat, designed by P. Rudolph
The anastigmatic flattening of the field aimed at by anti-
planets was finally attained in the two members of a composite
unsymmetrical objective by means of the principle of the opposed
gradation of the refractive indices enunciated by Dr. P. Eudolph
of Jena. Steinheil had already obtained a reduction of the
anomalies of the oblique pencils by the device of preserving
in the two members of the combination intentionally high but
opposed aberrations. But Eudolph was able actually to get
rid of them by combining a spherically and chromatically
corrected member made out of a pair of ordinary glasses
with an approximately spherically and chromatically corrected
member made out of an anomalous pair of glasses ; or, as we
may more simply say, according to our newly-founded definition,
by combining a new achromat with an old achromat. Eadial
astigmatism can, however, only be got rid of when combining
two achromats, each of which is approximately chromatically
and spherically corrected, provided the astigmatic aberration
produced by one achromat is of opposite sign to that introduced
by the other. It may now be pointed out that a new achromat,
in consequence of its cemented surface being a positive or
convergence-producing one, does, in fact, bring in an astigmatic
aberration of opposite sign to that brought in by the old
achromat with its negative or divergence-producing cemented
surface. In this opposition of function of the two cemented
surfaces lies the importance of the opposed gradation of the
refractive indices in the two lenses of Zeiss*s anastigmat, as
enabling the elimination of radial astigmatism to be eflected.
The new achromat at the same time offers, as we have seen, a
means of correcting curvature of image. If, as in Zeiss's
anastigmat, Fig. 15, a new achromat I is combined with an
old achromat II of suitable construction, an approximate
I UH8YMMETKICAL OBJECTIVES 63
elimination of the astigmatism of the oblique pencils may be
attained without prejudice to the flattenii^ of a lai^e field.
Also, in Zeiss's aruuiigmat, with a very considerable apertore-
ratio an umisual uniformity of defiuition is obtained over a
large angular width of field.
Each member of the anastigmat per se ia only approximately
achtomatised ; it is a good thing, however, if the combined
system is free both from " chemical focus " and also from
chromatic differences in the size of images.
The Anastigmat depicted in Fig. 1 5 is a wide-angled objective
giving great illumination, and having a maximum aperture //9.
That in Fig. 16 serves as an instantaneous lens of great
intensity. It consiets of a double front-lens and a triple back-
lens, of course preserving opposite gradation of refractive indices
in the two members. The fifth lens only serves to render
possible the elimination of spherical aberration of higher orders,
while employing a large aperture-ratio. There is a third series
of anaatigmats representing special wide-angled lenses.
Later on we shall speak of the attempts made by H. Schroder
and A. Miethe, prior to those of P. Kudolph, to construct
anastigmata by the use of anomalous glasses, which in these
researches both inventors apply to double objectives, disposed
symmetrkally as regards the stop, and therefore consisting of
two identical new achromats.
61 LUMMEK'S PHOTOGEAPHIC OPTICS chap.
The CemejUed Simple Objective, with Anustigmatic Image-
fiaiteniii^, composed of Three or Four Lenses.
The principle laid down and demonstrated by Dr. Eudolph,
according to which an objective, in order that it may yield
a flat and stigmatic image, must be so constructed, is,
according to our nomenclature, that it should be composed of
a new and an old achromat, in one of which the cemented
surface should possess a converging effect and in the other a
diverging effect. This principle once enunciated, it became a
simple matter to construct a single cemented objective with
anastigmatically flat field.
First consider the Zeiss anastigmat of Fig. 15 (so con-
structed) — how in it the outer surface of the second member
(a new achromat) has the same absolute
curvature as the outer surface of the first
{an old achromat). Then if one were to
reverse one of these members and cement
them together, one would so obtain the
anastigraatic simple objective deaigned
by Eudolph in 1894,^ and placed on the
market by C. Zeiss under the name
"anastigmat-lens//12-5," Fig. 17. This
lens, according to Kudolph's statement,
possesses, along with greater illuminating
power and better definition, " a hitherto
unattained perfection of the anastigmatic
^°' !L7seri^Vlr"*' flattening of the field." Since this
objective does not consist of separated
members, it is not necessary that each of the latter should be
in itself achromatic ; it is rather preferable to admit lai^
opposed aberrations in both members, in order to obtain other
advantages. In this form the objective in a certain sense
unites the antiplanet principle of Steinheil with the anastig-
matic principle of Eudolph. At all events the more important
conditions of construction depend only upon the special
purposes of the objective, and on the kinds of glass available
' British Patent No. 19,509 of 1S91 : improvements in and relating to
photogi'aphic objectives. See also British Journal of Photography, 1894, p. 829 ;
or Eder'B Johrbuch <Ur Photographie, 1895, p. 283.
X UNSYMMETRICAL OBJECTIVES 65
in manufacture. Consequently the cemented members may
possess the most diverse characters ; they may be both positive,
or one may be positive while the other is negative or even
neutral, provided only that the system as a whole remains
chromatically and spherically corrected. Also, the order of
the lenses is a secondary matter, if only the type is so
preserved that two of the lenses together form a new
achromat with a convergence-producing cemented surface, and
the two others an old achromat with a divergence-producing
cemented surface ; so that the Eudolph principle of opposite
gradation of the refractive indices may be realised and the
radial astigmatism compensated.
Suppose that in the quadruple anastigmatic simple objective
the two middle lenses are replaced by a single lens whose
refractive index lies between the indices of the two outer lenses,
then we have a triple or three-lens objective (Fig. 18), which
exhibits the above-defined opposed gradations of the refractive
2
a b
Fig. 18. — Triple cemented Anastigmatic Lenses.
indices, and also belongs to the anastigmatic type with two
diflerently acting cemented surfaces (s = converging, z = diverg-
ing). Thus the Eudolph principle is preserved both in the
case where the middle lens and the outer lenses have negative
focal lengths (Fig. 18, a), and in the case where, conversely,
the middle lens is a diverging lens, and the outer lenses have
positive focal lengths (Fig. 18, 6).^
An achromatic single objective of this kind, which consists
of three lenses cemented together, and by which the image
^ See the examples given on pp. 368-376 of the recent work on photographic
lenses by Dr. M. von Bohr.
F
«6 LUMMEKS PHOTOGBAPHIC OPTICS chap.
is rendered anastigmatically flat, besides beii^ spherically
corrected for pointa both on and off the axis, was con-
structed, even lefore the single objective, at the end of
1891, from the calculations of Dr. £udolph, in the work-
shops of C. Zeiss. But it was first put on the market'
by this firm ia 1893 under the name Anastigmat-saizlinse,
Series VI.
Independently of this, but also calculated out in the most
simple form, an anastigmatically aplanatic single lens was
des^ned by von Ho^h. A lens in accordance with von
Hoegh's calculations was protected by patent^ in December
1892 by the firm of Goerz, the patent covering particularly
the combination of two such triple cemented lenses combined as
a 8ymmetrical double-objective, which was placed on the
market under the name DovMe Anastigmat.
The firm C. Zeiss also combined two of its anastigmats,
already corrected as simple objectives, to form its Satz-ana-
stigmat. Series Via.
\ The manufacture of
three-lens simple ob-
jectives has been
nevertheless given up
by this firm, since
they possesa in the
cemented four - lens
system a simple ob-
jective which surpasses
the triple system in
FiQ. 19.— Set o( Convertible Anaetkrmata. , . %. ''
being of greater aper-
ture, more fully atigmatically aplanatic, and better corrected
with respect to the chromatic difference of the sizes of the images
produced by it. The term Satz-aiuxstigmat means " adaptable
anastigmat " or " convertible anastigmat," and under the latter
name they have been put in the market in England by the firm
of Ross, Limited, who are licensees under the Rudolph patents
(see Chapter XII.). Fig. 19 depicts a set of such convertible
1 Britiah Patent No. 4692 of 1893. See alao British Journal of Photography,
1893, p. S31.
» British Patent No. 23,878 of 1802 ; Gennan Patent No. 71,437. See also
British Journal of Photography, 1893, p. 185 ; or PhotographiKhe MUlheiliaigm,
(Berlin, 1893).
X UNSYMMETRICAL OBJECTIVES 67
)
anastigmats. When put together they make a wide-angled
portrait lens ; but either half can (with reduced aperture) be
used singly as a landscape lens, three dififerent focal lengths
I being thus available for use. The Satz-anastigmat objective is
also known as Zeiss's Protar,
CHAPTEE XI
DOUBLE-OBJECTIVES CONSISTING OF TWO SYMMETRICAL
MEMBERS WITH THE STOP BETWEEN THEM
General Properties of the Double-objective
There was a time when, beside the ordinary achromatic cemented-
lens landscape-objective, and the Petzval-Voigtlander portrait-
objective, no lens existed which with a relatively large
aperture gave wide-angled pictures that were sharp and free
from distortion. At that time the resources of photographic
optics were enriched by A Steinheil by his Aplanat, which
belongs to the type of double-objectives, and thus at a stroke
an end was put to the want that had been felt. Before the
aplanats were introduced, symmetrical systems had indeed been
constructed, which, thanks to their symmetry with respect
to the central stop, gave pictures free from distortion. But
the necessary illuminating power, which was possessed by the
aplanat of Steinheil, was absent from these systems. To the
latter circumstance the aplanat, and with it the type of double-
objective, owe directly their rapidly acquired popularity and
extensive use.
Before considering the various kinds of double-objective, we
will briefly describe those advantages which belong to all
systems built up symmetrically with respect to the central
stop, and which, simply in consequence of this disposition,
are moreover independent of the qualities of the single
components.
Let us to this end consider the formation of the image of
an object situated at a distance equal to twice the focal
length — the formation, in fact, of the image which is likewise,
CHAP. XI DOUBLE-OBJECTIVES 69
as is already known, also situated at a distance of double the
focal length at the other side of the lens, and is equal in size
to the object. By the use of a double-objective, even when
each member of the double - objective is merely a simple
lens, the image will be endowed with three advantageous
properties : —
(1) It is free from distortion and is perfectly sunilar to
the object.
(2) It is of equal size for all the various colours, and is
therefore free from chromatic differences of magnification. '
(3) It is free from the defect of coma ; that is to say, the
one-sided residuum of the spherical aberrations of the
oblique pencils is eliminated.
j In Chapter VI. of this treatise the advantage of the ortho-
scopy of symmetrical double-objectives has been thoroughly
investigated. With reference to the elimination of coma,
it may be shown that every double -objective brings to one
point the oblique pencils whose paths lie in a meridional
plane, with the same accuracy and sharpness as it does the
axial pencils.^
But this does not mean that the other rays of the oblique
pencils also meet in the same point in which the meridional
ones are brought to intersection. Further, the " astigmatic
difference" of the oblique pencils remains, in spite of the
symmetrical disposition of the two members, just as a bright
point seen through a prism does not appear as a point when
seen through a second reversed prism. A symmetrical
double-objective is in general subject to a pure radial astigma-
tism, but without coma ; at least so far as concerns images in
the symmetric planes (situated at double focal length) for
which the magnification is equal to minus unity.
If one regards as the image (conjugate to a point-object)
the smallest (and in this case circular) cross-section situated
between the focal lines, then these circles of least confusion
lie in general on a curved surface. As the magnitude of the
" astigmatic difference " depends upon the construction of the
individual members, so the curvature of the image depends
upon the distance by which they are separated. It is usually
^ See Czapski, Theory of Optical InstrumerUSf pp. 201 and 209 ; or Miiller-
Pouillet's Optics (9th edit.), pp. 774-76.
70 LUMMER'S PHOTOGRAPHIC OPTICS chap.
the case that, with a smaller distance between the members,
the astigmatic difference decreases and the curvature of the
image increases ; while conversely, with an increasing separa-
tion of the members, the image becomes more and more
flattened, and the astigmatic difference is increased.
So far as concerns the elimination of chromatism of focal
lengths or of the chromatic differences in magnification for
different colours, it must be remembered that an oblique ray
incident upon a plane parallel glass plate emerges from the
same as a parallel beam of variously coloured rays, whose
directions are parallel to that of the incident rays.^ If, in
spite of this, a glass plate held obliquely to the direction of
vision allows the object to appear without coloured edges, the
explanation is that that which is united in one point in the
image is not a single ray, but a bundle of rays emanating
from a point-object. If one regards the rays of the bundles
as rays parallel to one another, then in each emerging ray are
a large number of differently coloured rays, each of which
belongs to a different ray of the incident pencil. Since all
these rays are reunited by the eye, they produce the sensation
of white light.
Upon similar principles depends the action of the double-
objective in forming from a white object images that are of
equal size for all the constituent colours.
If one considers the zones of the two members of the
double-objective, that are intersected by one chief ray, as
replaced by the prism equivalent to them of equal refracting
power, then it is seen that the same act together as a mere
parallel plate of glass which is intersected obliquely by the
rays ; since both the substituted prisms have equally great
refracting angles, and their respective surfaces are parallel each
to each.
The white pencil of rays proceeding from a point-object
we can further consider as an infinite number of variously
coloured pencils. In consequence of dispersion, tlic chief
ray belonging to each coloured pencil (which principal ray
therefore intersects the axis in the middle of the stop or in
the point of symmetry of the system) has a somewhat different
direction of incidence.
^ Compare MiiUer-PouiUet's Optics (9th edit.), p. 265.
XI DOUBLE-OBJECTIVES 71
Our assumption was that these various coloured rays of a
pencil came from a point-object situated at a distance of twice
the focal length ; since all of these pass through the point of
symmetry, and emerge each parallel to its direction of
incidence, the emerging coloured chief rays of necessity cut
each other in the point conjugate with the point from which
they came, viz. the image point, which according to theory is
likewise situated at a distance of double the focal length, and
is at the same distance from the axis as the point-object.
The chief rays of the variously coloured pencils, into which
one may consider each white pencil of rays to be resolved, all
cut each other, therefore, in a point.
If this point of intersection is identical with the focus or
the circle of least confusion, for example, of the yellow pencil,
then the effective centre of the circle of confusion of the red, blue,
and other pencils will also lie in the same place, whilst the focus
of these colours is situated in the same line, but a little nearer
or more distant. They will all become coincident in one point
if each individual member of the double-objective is achromatic.
VARIOUS KINDS OF DOUBLE -OBJECTIVES
In the double-ohjedive, as the name already chosen by us
should signify, the same individual member appears twice
over. The path of the rays would remain geometrically
exactly the same, if instead of a second member, the hole in
the stop were reflecting. It follows immediately from this,
that by duplicating one of the members it is impossible to
eliminate those aberrations which are only to be got rid of by
compensation of the oppositely acting factors ; such, for example,
are chromatism of the focal lengths, central spherical aberration,
and radial astigmatism.
All these last-named aberrations must first be obviated in
the indimdvAil members of the double-objective if they are not
to render homocentric focussing of the axial and the oblique
pencils illusory. Accordingly, the development of the sym-
metrical double-objective depends upon the improvement of
the single objective, so far as the latter is applicable for
combination in symmetrical pairs.
Certainly nothing stands in the way of applying any objec-
72 LUMMER'S PHOTOGEAPHIC OPTICS chap.
tive, be it the portrait-objective of Petzval, the antiplanet of
Steinheil, or the anastigmat of Zeiss, as a member of a double-
objective. Only there arises a second important question,
whether it pays to bear the expense which the duplication of
a simple member entails, in order to win the advantages
associated with every double -objective. If, for instance, one
arranged two Zeiss anastigmats symmetrically with respect to
the centre of the stop, in order to add aplanatic advantages to
the anastigmatic ones, there would be flare-spots due to the
repeated internal reflexions, and a considerable consequent
diminution of light. There would also be a very small field of
view, on account of the length of the system, a field in which,
moreover, the brightness would diminish rapidly from the
middle to the edge.
In practice hitherto only simple objectives made out of
cemented lenses have been used as members of a double-
objective. However many simple objectives we possess, so
many kinds of double-objectives can exist and actually do exist.
As simple objectives we have recognised the following types: —
(1) Simple converging lens.
(2) Two-lens old achromat.
(3) Two -lens new achromat.
(4) Three -lens cemented objective with anastigmatically
flattened field.
(5) Four -lens cemented objective with anastigmatically
flattened field.
DOUBLE-OBJECTIVE TYPE NO. 1
The complete sphere (Fig. 5, p. 30) with a small central stop
may be considered as the simplest representative of No. 1 double-
objective. One may think of the same as composed of two
hemispheres I and II, which are cemented together at their
middle parts (ah) and stopped off up to this region. In this
case neither refraction nor dispersion of the chief rays takes
place.
Next to the complete sphere-objective comes the panoramic
lens of Sutton (1859). In this the interior space of the
hollow sphere, which constitutes the simple lens, is filled with
water.
XI
DOUBLE-OBJECTI VES
73
The best No. 1 type double-objective made up of simple
lenses is the Periscope of Steinheil (1865),
Fig. 20. On account of its relatively great
illuminating power, along with its " artistic "
fuzzy definition, on account also of its cheap-
ness and the great brilliancy of its . pictures,
the Periscope has since 1890 enjoyed great
popularity, and its manufacture has recently
been taken up again.
FiQ. 20.— Steinheil's
Periscope,
DOUBLE-OBJECTIVE TYPE NO. 2
Harrison's Spfiericai-ohfective, Busch's Pantoscope, as well
as Steinheil's Aplanat (Fig. 21), belong to Type No. 2 of
double -objectives, made with
two old achromats as individual
members.
With the exception of
Steinheil's Aplanat, the above-
named double -objectives can
only be used with a very small
aperture-ratio, since, when used
with larger stops, they render
even points on the axis in-
distinct.
If one would further com-
bine with the advantages of the
double-objective naturally con-
sequent upon the symmetrical
disposition of its members with respect to the central stop,
those of high illuminating power, one must not renounce the
use of achromatic lenses. Also any very steep curves, such as
those of Harrison's spherical-objective and of the Pantoscope
of Busch, must be abandoned ; so we must turn to the use of
slightly curved menisci, if spherical aberration is to become of
small amount with a large aperture, and the image as flat as
possible. When in our design we have satisfied the several
conditions for the elimination of spherical and chromatic
aberration, and for obtaining a given focal length, all the
variable elements at our disposal in the construction of a
FiQ. 21. — Steinheirs Aplanat
74 LUMMER'S PHOTOGEAPHIC OPTICS chap.
cemented achromat are exhausted; and since in the sym-
metrical double-objective the two members are exactly alike,
then the only new elements that come in are the distance
between the individual members and the choice of the kinds
of glass.
Should an improvement upon the above-named objectives
be sought after, then all care must be devoted to the individual
members, and they must be constructed out of such glasses,
that while the whole system is corrected for the greatest
possible aperture-ratio, it shall also transmit with the smallest
aberrations oblique pencils of the widest cross-section. The
credit of having fulfilled these conditions, as completely as the
glasses of that time allowed, is due to Adolph Steinheil as
early as the year 1866.
The achromat used by him consisted of two Frauenhofer
flint glasses, and possessed the form of a meniscus. Obviously,
one may to-day apply the new Jena glasses to the construction
of the individual members of the aplanat in the same way.
Indeed, the only distinction between the various types of
aplanats lies in the kinds of glass selected, and the consequent
modifications in the form of the achromats. In the group-
aplanats of high illuminating power the achromat is spherically
corrected for a relatively great aperture, and the members are
set at moderately great distance apart ; on the other hand, in
the vnde-angle aplanat, while the members are of lower illumin-
ating power, but better adapted for oblique pencils, the distance
between the components is chosen as small as permissible.
As might be expected in view of the small number of
applied elements and the nature of the achromat, the aplanat
cannot be corrected either in respect of the astigmatism or
of the curvature of the field. But the image can be improved,
either with regard to the flattening of the field or to astig-
matism, by separating the components. In order to diminish
the curvature of the image, when the aperture-ratio is large, a
large distance between the two components may be chosen
(see p. 58); whilst, when using a small stop and wide field of
view, one endeavours to render astigmatism as small as possible
by lessening the distance between the members.
Ever since the year 1886 Steinheil has constructed aplanats
with a variable distance between the two components, which,
XI DOUBLE-OBJECTIVES 75
with full aperture and small distance, act as group-objectives
giving large illumination, and on the other hand, with small
aperture and great distance between the components, act as
wide-angle objectives, and consequently within a certain range
unite in themselves the diverse types of aplanats.
The fact of such a change in the distance separating the
two components being successful in practice is clear evidence
that even in the best case perfect stigmatic reunion of the
rays cannot be obtained by means of the aplanat type. For
the process here is similar to that in the case of a simple
achromatic landscape lens with an anterior stop, in which, by a
displacement of the stop, the position of the image and a con-
sequent artificial flattening of the image can be obtained, simply
because the rays do not come strictly to point-foci (see p. 27).
At all events, the aplanat which combined correct delinea-
tion and a wide angle with great intensity of illumination
marked a great improvement upon the objectives existing
prior to that time.
Of the remaining objectives which belong to the type of
aplanats may be mentioned the Euryscope of Voigtlander,
the Lynkeimype of Goerz, the ParUoscope of Hartnack, and
the Rectilinear of Dallmeyer. For the numerous names under
which other examples of the aplanat have been brought, the
work of J. M. Eder, Die photographischen Objektive, ihre EigeU"
schaften und ihre Prilfung (Halle a.S., Verlag von Wilhelm
Knapp, 1891, S. 104), may be consulted.
DOUBLE-OBJECTIVE TYPE NO. 3
A special place among aplanats composed of two double-
lens components is taken by Schroder's concentric lens} and
also by Miethe's anastigmat,^ They represent the double-
^ Ross-Concentrio Lens of Dr. Schroder, British Patent No. 5194 of the
year 1888 ; Photog, Neus, 1889, S. 816. The objective first came into the
market in 1892. See Brit, Joum, of Photography, No. 1669, 30th April, 1892.
3 A. Miethe, Der AnastigmcU; Vogel's Photog. Mitth, 25, S. 123 and 173
to 174. Miethe's first anastigmat was calculated out in 1888, using a highly re-
fracting phosphate crown, and a very light flint glass, and was constructed by
Hartnack of Potsdam. But the phosphate crown does not stand atmospheric
exposure weU, and this was found to be an objection. This lens might be
described as an aplanat made of two equal new achromats symmetrically
arranged.
76 LUMMER'S PHOTOGRAPHIC OPTICS chap.
objective No. 3, which consists of two new eichromats. Since
the lenses are cemented, only three radii stand at one's
disposal in this case, as in the old achromat. Of course, with
new achromats flattening of the field is at once approximately
obtained by the choice of suitable glass (see p. 50). For the
removal of astigmatism there is, however, in this case no
fourth variable element at one's disposal; for by the
duplication of the new achromat to form a double-objective
one does not obtain both a positive and negative cemented
refracting surface, as in the case of the combination of a new
achromat with an old achromat in the Zeiss-Budolph anastigmat
(see p. 62). Experience shows, and it is also deducible from
our systematic treatment of the subject, that these double-objec-
tives cannot, in consequence of the aberrations of the oppositely
dcting cemented refracting surfaces,^ produce chromatically and
stigmatically corrected images, which are at the same time
spherically corrected. By choosii^ the concentric form
Schroder obtained very good flattening of the field and little
radial astigmatism, but this form of lens is at a disadvantage
with respect to the spherical correction, upon which so much
the more stress must be laid, since the new achromat is not
susceptible of being spherically corrected so well as the old
achromat.
Only quite recently, since the date when it became possible
to obtain a suflBciently anastigmatic flattening of the field by
means of triple or quadruple cemented lenses (so following out
Eudolph's principle appUed in Zeiss's anastigmat), have the
obstacles been removed which stood in the way of constructing
"anastigmatic aplanats." By this term is meant a system
which, by its symmetrical construction, combines the advan-
tages peculiar to the aplanat with those that are special to
Zeiss's anastigmat.
^ After Professor Miethe had pointed out, on p. 87 of his book on Photographic
Optica^ that in the objectives denominated by us ''new achromat" it is also
possible to reduce to within narrow limits the axial aberrations, and to construct
with them aplanats of sufficient intensity, he continues thus : '* The lens-systems
carried out according to these principles, and called AnastigmcUSf are still subject
to certain aberrations, both of a mechanical and of an optical nature. Firstly,
the distances of the lenses must be made relatively very great ; secondly, the
removal of astigmatism over the whole field is not possible ; and, thirdly, the
crown glass used is not sufficiently proof against climatic deterioration to be
good for photographic purposes."
DOUBLE-OBJECrrrVBS
DOUBLE-OBJECTIVE TYPE NO. 4
la this type each component consists of an objectiTe of
three lenses cemented t<^ether, and which is ab'eady more or
less spherically and chromatically corrected, and yields an
anast^^atically flattened imi^.
1. Dffuble Anastigmat of C. P. Goers. — The f/rst objective
of thia kind was brought into the market in the year 1893
by C. P. Goerz, under the name Double
Anastigmat, and was made according to
the computation of HeiT von Hoegh,
It is depicted in F^. 22. According
to the data of Goerz's catalogue, the
double anastigmat has, with an aperture-
ratio fj1-1, an angle of 70° (degrees),
and with a smaller aperture-ratio a field
of as much aa 90°. The typical form
of triple cemented component has been
described on p. 65, to which description
we may refer so fer as relates to the
single components. With respect to
its performance as a whole, the double
anastigmat marked a distinct advance
over the ordinary apUnats. How far its performance may be
appraised when compared with the anastigmat of Zeiss, consist-
ing of five lenses only, there is not yet any decided concensus
of opinion.
In the patent specification of Goerz there is stated as a
second claim the use of one component of the double anastig-
mat as a single objective, yet the double anastigmat has mainly
acquired its reputation as a compound system, and might well
be designated, when it appeared, as the best symmetrical double-
oly'ective.
Goerz has also produced ' a form of Double Anastigmat
(Fig. 22a) in which the symmetrical components each consist
of five lenses cemented together. There are thus six different
radii of curvature, and at least four different kinds of glass
are used, the refractive indices of the five lenses, beginning
with the convex outermost lens, being as follows: 161, 1'54,
' British Patent No. 2864 of 1899.
78 LDMMEK'S PHOTOGEAPHIC OPTICS chap,
1-52, 1'6X, 1'51. In order to attain the greatest intensity,
the first or outermost lena must have the highest refractivity,
and the last or innermost (concave)
must have the lowest possible re-
fractivity. Spherical aberration is
corrected by the second surface,
which is a negative or divei^ing
one ; and the difference of the re-
fractivities of the two media which
it limits must be small — not more
than O'Ol- — because it is necessarily
of a deep curvature, its depth being
determined by the condition that the
chief rays of oblique pencils must
meet it at as small a refracting
angle as possible, otherwise there
would be accumulated distorting effects that could not be com-
pensated by the subsequent refractions. To fulfil the anastig-
matic condition are provided the fourth and fifth surfaces, one
convex, the other concave, and as each of these is to act as a
collecting surface, the medium between must have a higher
refractive index than either of those that adjoin it. The fourth
surface serves to neutralise distortion for oblique pencils, while
the fifth must be as flat as possible, to prevent curvature of the
imt^e. Hence the last lens must have a very low refractivity,
and the last but one a very high re-
fractivity. Thus it becomes needful
to insert between the first and third
lenses a positive lens of intermediate
refractivity, its second surface being
either slightly concave toward the
Kght, or slightly convex, as may be
required to correct for chromatic
differences of the spherical aberra-
tion. In other respects this surface
effects little, because the mean re-
fractivities of the materials on the
two sides of it are nearly alike. In
order to secure a good and unalter-
able centering, the three negative lenses are made so that
XI DOUBLE-OBJECTIVES 78
they project over and completely enclose the two positive
lenseB.
The Goerz anastigmats are also made up as unsymmetrical
compoundB, with a smaller size of component for the second
member, as in Fig. 22b. The individual components can be
used singly as landscape lenses, the combination being thus
convertible. The aperture-ratio of the dovhle anastigmat is
stated asf/6-5, and that of the single component as//ll.
2. Convertible Anastigmat, Series Via., of Carl Zeiss. — The
triple anastigmat of the firm Carl Zeiss, mentioned on pp. 64
and 65, admits, with great advantage, of being duplicated to
form a symmetrical objective. The eonvertiUe anastigmai (or
" Satz-anastigmat "), constructed of two three-lens components
of equal focal length, ia expressly within the type of double
anastigmats. Since the components are separately corrected
as well as possible, it makes no practical difference in the
performance of the double-objective, with respect to sliarpness
and anastigmatic flattening of the field,
whether the two components of which it
is composed be of equal or unequal focal
length. The form with unequal com-
ponents, depicted in Fig. 23, belongs to
the class denoted Satz-anastigmat, Series
Via, In consequence of this circum-
stance one may combine in pairs any of
a series of two or three different sizes of
the single anastigmat of Series VI., and
make of them very good anastigmatic com-
pffiind lemes. For instance, the two com-
ponents of Fig, 23 might each be used fi". 23.— Zeiea's Ati^ig-
^ , , , , mai, Series Via. (or Con-
separately for landscape or group purposes, veriibie ATtastig-mtU).
thus affording in one lens three different
possibilities. Because these combinations are possible, the name
Saiz-anastigmats, meaning adaptive or convertible anastigmats,
is given to this series.
3. Collinear of Voigtlander und Sokn. — The collinear
shown in Fig, 24, which was computed by Dr. Kaempfer,^ is
similar to the double anastigmat composed of two simUar
triple components. Each individual member consists of a
' Pholog. ICorr. 1894, S. 495 ; aee also Catslogue of Voig tender und Sohn.
80 LUMMEE'a PHOTOGEAPHIO OPTICS chap.
middle conveigiug meniscus lens of lower re&active index
cemented to two lensea of higher index, of which that facing
the atop ia hi-concave and the other bi-convex. In this wise
there are brought into existence both a convei^ing and a
diverging cemented surface, as is required for producing the
anastigmatic flattening of the field. In this case also the
component is subordinated to the performance of the system
as a whole. As a douhle-objective the coHinear, according to
Fio. 24.— VcJgUttnder'B CWitmsor. Fia. 25.— Steinheil's Orthottiffmal.
the inventor, is well corrected spherically for an aperture-ratio
of about //7, and possesses also good anaatigmatic flattening
with a wide extent of field. These lenses are now manufactured
in England under the Yoigtlander patents by Messrs H. and
J. Beck.
4. Orthostigmai Type II. of C. A. Steinlml Sohne. — The
orthostigmat Type II,, shown in Fig. 25, has been brought
out eomraercially quite recently ^ by C. A. Steinheil and Sons.
Belonging to the same type as the collinear, it consists like-
wise of two components, each of three lenses cemented
together, of which the middle one has a lower index of
* With regard to the date of ita appearance, the most recent catalogue of
C. A. Steinheil Sbhne gives infonnation.
XL DOUBLE-OBJECTIVES gl
refraction than the outer. The details of construction were
published in the British Journal of PhMography, 1896, p.
489.
DOUBLI-OBJECTIVE TVPE No. 5
This double-objective is represented by the convertible
anastigmat. Series Vila., of Carl Zeiss,
which consistsof two quadruple cemented
components of equal or unequal focal
length, and in its later form is shown in
Fig. 26. Since in quadruple components
the theoretical possibility of obtaining
anaatigmatie flattening is more perfectly
realised than in the case of triple com-
ponents, the convertible anastigmats of
Series Vila, should possess theoretically |
a still higher efficiency. With respect to
the correction of the components of Series
Vila,, what was previously said of the Yio.2i.—z&\ss sAnastigimt
convertible anastigmats of Series Via. Series vilo. (convert,
applies directly to this case also — viz.
that excellent anastigmatfl can be made by combining various
simple objectives of Series VII. ; on account of which they
have rapidly come into acceptance.
Zeiss's Plahae and Unar
With the eight-lens convertible anastigmats. Series Vila.,
the improvement of photographic objectives appears to have
attained a certain limit in one direction, namely, that in
which the aim was the utilisation of the new Jena glasses (by
the application of Rudolph's principle of correction), whether in
the single objective, the double objective, or in the convertible
objective. Nevertheless, Dr. Kudolph has designed for Messrs,
Zeiss, under the name of Plartar, a symmetrical lens having
certain advantages over the double anastigmat In this lens
Rudolph starts from the principle of the telescope objective of
Gauss. It is well known that Gauss had shown that if an
82 LUMMEE'S PHOTOGRAPHIC OPTICS fUAP.
achromatic objective is made of the form shown in Fig, 27,
instead of the ordinary cemented form (such as Figs. 9 or 10,
p. 44), it is possible, since there is one more radius available, to
which there can be assigned any desired value, not only to make
the combination achromatic,
but to make it such that it
corrects the spherical aberration
■ for two different parts of the
spectrum, and so gets rid of the
chromatic differences of the
spherical aberration. To adapt
F la. 27.— GilUSSS AchTomutic Obtectivf, , , . , ^ ,.
such a lens to photographic
work, it must be modified so as to give it the additional
property of anastlgmatically flattening the image. This
depends upon finding suitable sorts of glass. In llie modified
lens-system, either one lens or the other, or both, is made up
of a cemented pair chosen so that both the kinds of glass used
have the same, or nearly the same, mean refractive index,
while possessing very different dispersing power. Any
cemented pair, so constructed, will act, eo far as mere refi-ac-
tion is concerned, simply as a homogenous single lens, while,
so far as its dispersive power is concerned, it may be achro-
matic, or under-corrected or over-
corrected for colour, according to
the curvature chosen for the cement-
ing surface. Hence the outer curva-
tures and thicknesses of the lenses
may be predetermined so as to
correct for spherical aberration,
coma, and curvature of field,leaving
to subsequent independent calcula-
tion the choice of the curvature of
the internal cemented surface upon
which the colour -correction de-
pends. Obviously success in using
this principle depends upon having
a sufficiently large selection of Fk. 2S.-P/awr objective
glasses from which to select those
suited for the purpose. A slight departure from exact agree-
ment in the mean refractivity is quite admissible, and indeed
SI DOUBLE-OBJECTIVES 83
has the advantage of enabling the lens belter to approximate
toward fulfilling the sine-condition for eliminatiou of coma.
The Planar lens depicted in Fig, 28 is that manufactured from
^Rudolph's specification by Messrs. Hoss of London. It has a
view-ai^le of from 62° to 72°, according as its aperture-ratio
is adjusted from //3-8 to //6 ; and is
therefore a very rapid wide-angle lens,
well adapted for copying processes of
all kinds and for instantaneous taking
of groups and portraits. They are,
however, inferior for architectural
work to the anastigmats.
The very latest lens of Messrs.
Zeiss, constructed from the computa-
tions of Dr. Rudolph, is denominated
the Unar}
This lens (Fig. 29) is not symmet-
rical, and therefore strictly belongs to
the class described in the preceding
chapter. Its front member consists
of two separated lenses, with an air-
space between them resembling a ^la. 29.— Zwss'a Unar.
positive meniscus, while the hinder
member also consists of two separated lenses, the air-space
between them having the form of a negative meniscus. The
hinder member is therefore like a Gauss objective, while the
front member recalls the back part of a Petzval lens. But
neither part is by itself corrected for colour. Only two kinds
of glass are employed, the two outer lenses (both positive) being
of a dense baryta crown, having a mean refractive index of about
1'61, while the two inner lenses are of an ordinary light
flint of about I'S"?. It might be thought that, as only two
kinds of glass are used, the system would not fulfil Eudolph's
anastigmatic principle of opposed gradation. But a httle
consideration will show that the convex air-meniscus in the
front component operates like a negative lens, while the con-
cave air-meniscus acts like a positive lens. Hence the former
acts like the z surface of Fig. 18, a, p. 65, whilst the latter
acts like the s surface of that figure. In its properties the
' British Patant Specification No. 2489 of 1899.
84 LUMMER'S PHOTOGRAPHIC OPTICS ohap. xi
Unar is intermediate between the Anastigmats and the Planar.
The 6-lens Planar, with an aperture-ratio of f/S'Q, covers a
field of about 65°; while an 8 -lens Aruistigmat, with aperture-
ratio ranging between //6*3 and //8, covers a field of 80°.
The 4-lens Uhar, with an aperture-ratio of //4'5, covers an
angular field of about 70°. It is therefore admirably adapted
for the general purposes of the amateur, and has the merit
of exceedingly simple construction.
CHAPTEE XII
SOME RECENT BRITISH OBJECTIVES
No account would be complete that dealt only with objectiveB
manufactured by the great German firme, and accordingly
some correspouding information is here added respecting some
recent British lenses.
Ross's Lenses
The firm of Ross had already, under the technical advice of
Dr. Schroder, produced the Concentric lens of
which mention was made on p. 56; and Rosa's
concentricB are well known for their excellent
qualities as to covering power with small aper-
tures. These were the first camera objectives in
which use was made of the new Jena glasses
having relatively h^h refraction with small dis-
persion.
Ross's concentric lenses have, however, been B'la. so— EosB'a
for some years largely superseded by J^^"'"' *""
the more modern anastigmata, which
are manufactured under licence under the Rudolph
patents, and double anaat^mats under the Goerz
patents (p. 77).
Fig, 30 shows the construction of the Ross con-
centric lens, used for landscape and copying. With
Fio. 31.— aperture-ratios of //16 to //45, it gives excellent
KoBs'sWide- definition over an angular field of about 75°, but
mrtric Lens. ^ ^ot rapid enough for many purposes. Fig. 3 1
depicts the Ross wide-angle symmetrical lens, used
for views, architectural work, and the like, requiring a field of
86 LUMMEE'S PHOTOGRAPHIC OPTICS chaf.
90°. With aperture-ratios //16 to//64 it gives good defini-
tion and practical freedom from distortion right up to the margin.
In this lena the components are
simply cemented achromats.
More recently, besides adopting
the Zeiss anastigmats and convert-
ible anastigmats described in the
previous chapter, the firm of Eoss,
Limited, has put on the market a
form of very rapid lens known as
the Universal Symmetric Aiiastig-
mat (" new extra rapid series "),
having an aperture-ratio of //5-6.
These lenses surpass the older sym-
metrical lenses in definition, and
are excellent for animal studies and
Fio. 32.— Ross's Unicsraal Sjpit- street scenes, as well as for groups
nc igmat. ^^^ portrait work. With the
aperture- ratio given, they cover a view-angle exceeding 65°.
Each component consists of a triple -cemented lens in which
three kinds of glass are used. It therefore has a certain
resemblance with the Collinear of Voigtlander. As it is
not patented, no data of its radii of curvature have been
published.
Dallmeyeb's Lenses
The firm of Dallmeyer {now J. H. Dallmeyer, Limited)
has long enjoyed a high reputation for its Triple Achromatic,
Wide-angle Rectilinear, and Rapid Rectilinear lenses, the suc-
cessive introductions of the late Mr. J. H. Dallmeyer. The
labours of Mr. T. R Dallmeyer and of Mr. Hugh L. Aldis
have resulted in various new developments, including the
Telephotographic objectives described in Chapter XIIL, and
the new Stigmatic lens now to be described.
In designing the StigmMic lenses, which are double-objec-
tives, the symmetrical form has been abandoned in order to
obtain a new means of eliminating astigmatism and spherical
aberration. They consist of two components, each approxi-
mately corrected for chromatism. As originally designed and
XTi SOME RECENT BRITISH OBJECTIVES 87
described in the patent specification, the front component con-
sisted of a positive meniscus system, possessing strong positive
spherical aberration, made up of two (or of three) lenses
cemented together, the negative lens having the higher disper-
sive power. The back component admitted of several varieties,
but essentially it consisted of an inner positive meniscus, separated
by an air-space from a hinder stronger negative meniscus, one
or both of these menisci being made as a cemented pair, so as
to secure achromatism for the back component, so operating
together that the whole back component is a weak negative
lens, having a negative spherical aberration sufficiently great
to compensate the positive spherical aberration of the front
component. This design, substantially that shown in Fig.
33, has more recently been reversed, back for front, as in
Fig. 34. In symmetrical double -objectives, as has been
previously pointed out, each of the component systems must
be spherically corrected, and the duplication merely enables
distortion of the image to be eliminated. But in order to
correct a compound lens for spherical aberration, its positive
component must, under most conditions favourable to the
construction of photographic lenses, have a lower refractive
index than the negative component (in other words, it must
be an old achromat, see p. 46), and this condition renders
correction for chromatism and for radial astigmatism less easy.
In order to escape this difficulty, the designers of the Stigmatic
lenses reverted to the earlier unsymmetrical form of objective,
and obtained correction, just as did Steinheil, by causing the
faults of the two components to neutralise each other. As
already stated, the back component consists of two parts, the
first a converging meniscus, and the second a stronger diverging
meniscus. It is an essential part of the design that the last
surface of the former has a flatter curvature than the first
surface of the latter, so that they enclose an air-space of the
form of a positive meniscus. This meniscus air-lens, bounded
by glass, acts therefore as a diverging lens. By this device
the back component is caused to have a great negative spherical
aberration, and yet the converging glasses may be of high
refractive index, and the diverging ones of a relatively lower
index. In this way they are enabled to fulfil also the fourth
of Seidel's conditions (p. 47), which secures flatness of field.
88 LUMMEE3 PHOTOOBAPHIC OPTICS chap.
Mr. Aldis has given' the following atatement of the point.
Using the aymbola ft, ft', etc., for the indices of refraction, and
fj, T-j and /j, /g for the respective radii, the condition for secur-
ing flatness of Geld is that the sum of all the terms, such as
+ etc.,
should he zero. This was, indeed, first pointed out by Petzval,
and is practically identical with Seidel's fourth condition.
Aldis then goes on to observe that in order to realise this
condition as far as possible, three conditions have to be
obser\'ed : —
(1) The converging lenses should be of glass of high refrac-
tive index, and the diverging lenses of low refractive
index.
(2) Diverging components should be separated by a con-
siderable interval from convei^ing components.
(3) Thick meniscus lenses should be used.
In order to reconcile the first of these conditions with the
condition of achromatism, it was necessary to have recourse
to one of the new Jena glasses having high refractivity and
Fio. 34,— DaJImeyer's SUgmaiic
low dispersion — in fact, a dense baryta crown — for the converg-
ing lens. In the patent specification three numerical examples
are given. They differ mainly in r^ard to the back component.
In the first form the positive meniscus is a single crown lens
' See specifieatiou of Patent No. 16,640 of 1895.
Ill SOME EECENT BBITI8H OBJECTIVES 88
of h^h refractivity, while the negative meniscua ia a cemented
lens. In the second form, which resembles that used in
Fig. 33, both parts of the back component are cemented lenses.
As described in the specification, all the converging lenses are
of dense baryta crown, while both the diverging lenses uaed in
the back component are of a light silicate crown. In the
third form, which resembles Fig, 34, but reversed in direction
with respect to the light, the first part of the back component
is a cemented lens, while the second part la a simple negative
meniscua of Kght silicate crown glass.
F^. 33 represents StigT^iatic lens (Series I.) of apertnre
fjA:, which is a portrait lens. The following are the data,
kindly furnished by Mr. T. R Dallmeyer, as applied to a lens
of 10 inches equivalent focal length. The glasses used are of
■ the following kinds : —
Lenses L^L^L^ /*= 1-5726, /. = 67-5 (0 211).
LensLj, /i= 1-5738, /. = 41-4 (0 569).
Lenses L^Lj, /t= 1-6151, i-= 56-6 (0 114).
The several radii of curvature are as follows : — r^ = — 2-74 ;
rg=+3-92; r^=-4-07; r^=+4-39; r^=+l-46; r^ =
+ 2-89; r,= +1-67; r^= -5-65; r^= +2-74.
The several thicknesses are as follows: — rfj = 0-67; d^
= 0-42; <£g = 0-43; d^ = 012; dj = 0-10,
but is slightly adjusted in different cases ;
d^ = 0-n-, rf^ = 0-50; rfg = 3-65.
These lenses, like the convertible
anastigmats of Zeiss, are capable of being
separated and the components used as
independent lenses. Fig. 31 above re-
presented a Stigmatic lens (Series II.)
of aperture //6, capable of use aa a
universal lens with a view-angle of
nearly 70°; but if stopped down to /y'16,
it has a view-angle of 85°. Fig. 35 re- Fio. 35.— Front Component
presents the front component as used
alone, and Fig. 36 the back component
as used alone, for landscape purposes. The former has a focal
length about two times, the latter a focal length about one and a
half times, as great as that of the combined system. Fig. 37
90 LUMMER'S PHOTOQEAPHIC OPTICS chap.
depicts a non-coiivertible form of Stigmatic lens (Series III.),
having aperture-ratio //7'7. In this form the front component
Fio. 36.— Back Component of the FlO. 37.— Dallnieyei uon-convert-
Psllmeyei' SUgmatic used as a ibla Stit/maiic Objective (Seriea
single lens. III.)
is that containing a positive air- meniscus, while the back
component is cemented.
The (jOOke-Tayloe Lenses
Mr. H. Dennis Taylor, of the well-known firm of optical
engineers, T. Cooke and Sons, of York, has demised an objective
which is of special interest from its extreme simplicity, in
spite of which it gives, within the range of its capabilities, a
precision of performance probably unsurpassed by any more
complex lens. These Cooke lenses are placed on the market
by Messrs. Taylor, Taylor, and Hobaon, of Leicester ; and
they are also manufactured in Germany by the Voigtljinder
establishment under the name of Triple Anastigmais.
The Cooke lens consists of three parts, the front and
back components being positive lenses, whilst between them,
adjusted carefully to an intermediate position, is a strong
negative lens. Mr. Dennis Taylor's original idea appears to
have been ^ to make each of the positive lenses of a cemented
pair, each pair corrected for colour and for central aberration.
See also specification of
XII SOME RECENT BRITISH OBJECTIVES 91
and to throw upon the intermediate negative lens the whole
burden of the work of 'lengthening out the oblique pencils so
as to correct for coma, astigmatism, and curvature of field.
He also put forward from the beginning the idea that the
negative power of this intermediate lens should be approxi-
mately equal to the sum of the powers of the two outer
positive lenses, the resultant power being not zero, but
positive because of the separation between the lenses. The
conception underlying this feature of the design is apparently
derived from the intention approximately to fulfil von Seidel's
fourth condition (see pp. 47 and 49), the physical meaning of
which is to the effect that, if in the constrioction of the separate
lenses the glass used were all of one kind, so far as • mean re-
fracting power is concerned, and the separate lenses were all
pushed up close together, it would act like a plane thick sheet
of glass. Seidel showed that the fulfilment of this condition
suffices to give a stigmatically flat plane to the image. Mr.
Dennis Taylor's principle is that the separate lenses, if all
pushed up close together, should act like a plane thick sheet
of glass, whatever the refractive indices of the glass. Hence
if, as in fact is the case, glasses of different mean refractivities
are employed for the different lenses, the adoption of the
principle of making the power of the negative lens equal to
the sum of the powers of the two positive lenses can satisfy
von SeideFs fourth condition only approximately.
It then occurred to Mr. Dennis Taylor that it was not
necessary for all the lenses to be made of achromatic cemented
pairs. He made the two positive outer lenses simple crown
glass lenses (approximately of the form of " crossed " lenses
with the greater curvature outwards), and placed between
them an over-corrected powerful concave cemented lens to
compensate for their aberrations. It was indeed no novelty,
per se, to place a concave lens between two convex outer
lenses. That had been done years before by Sutton, by
Dallmeyer, and also by Steinheil in his portrait Antiplanat.
Neither was it novel, per se, to use a central over-achroma-
tised lens to correct the chromatic aberrations of the outer
members. That had been done before by Abbe and Eudolph,
who, however, applied a central triple -cemented lens of
nearly zero magnifying power, which could therefore have no
92 LUMMER'S PHOTOGRAPHIC OPTICS CHiP.
sensible effect on the equivalent focal length or on the
curvature of the field. But it does appear to have been
novel to make the power of this negative central lens
approximately equal to the sum of the powers of all the
positive lenses: In brief, Mr. Dennis Taylor intended hia
central correcting lens to perform the triple function of (1)
correcting for chromatic aberration ; (2) correcting both
axial and oblique pencils for spherical aberration; (3) correct-
ing the combination as regards flatness of field and mai^inal
astigmatism. To secure the firat point requires proper choice
of glass as respects dispersion. To secure the second involves
adoption of proper curves and distances. To secure the last
requires the fulfilment (at least approximately) of von Seidel's
fourth condition. In order, as he supposed, to give the over-
achromatising power to this central negative lens, it was at
first made of an extremely steep bi-concave of a light silicate
flint cemented to a meniscus of baryta crown. But, surprising
as it may seem in view of all that is required of this central
lens, Mr. Dennis Taylor then found that adequate corrections
could be obtained by the use of a
single bi-concave of hght flint glass,
a diaphragm being placed immedi-
ately behind it. Fig. 38 shows the
form adopted for the Cooke lens,
<— Series III,, with aperture-ratio //6'5.
The final corrections in these lenses
are made by adjustment of the separ-
ation between the components. ■
In Mr. Dennis Taylor's second
„„„„,,„,, specification. No. 15,107 of 1895,
FiQ. 38.— Cooka-Taylor Lena ,*^ , ., , .
(Series III.). he describes several series ot lenses.
In these he takes advantage of the
new Jena glass, employing for the positive lenses the densest
baryta crown (0 1209), having a mean refractive index
of 1-6114, while the negative lenses are made of a light
silicate or boro-silicate flint, having a mean index of 1*5482
or 1"5679. The use of the high-refraetivity crown enables
the positive lenses to be made with less steep curves, and
the use of the low - refraetivity flint enables the correcting
negative lens to be made with very steep curves, so compensat-
xit SOME RECENT BEITISH OBJECTIVES 93
ii^ for the other aberratioua Numerical examples^ are
given in the specification,
Fig, 9 of which approxi-
mately corresponds to the
present form of medium
wide-angle lens.
Fig. 39 depicts the
Cook portrait lens, having ,^_
aperture-ratio of //4-5 ;
the angular field being
over 45°. The back glass
is adjustable, so a^ to per-
mit the operator to work
either with
full defini- Fig. 38.— Cooke-Taylor Portrait I^ns.
tion up to
the mai^ins of the plate or to " soften " the
detail by re -introducing spherical aberration.
By removing the back lens and substitnting
another of lower power, known as an " exten-
sion lens" (Fig. 40), the entire focal length of
Fid. 40.— Cooke- the combination may be lengthened without
Sna." ^""°'' sacrificing definition. For process work the
Cooke lenses are much prized, on account of
their freedom from distortion, as well as for their excellent
marginal definition.
' lu the recent treatise of Herr von Bohr on Photographic Objectirea, he
givea aberration curves for many actual lenses of different makers. The curves
given for the Cooke lenses are not, however, taken from an actual lens, but from
the data of the patent specification only. This, we are informed, is also the case
with some of the other lenses there described, which is to be regretted, as, for
obvious reasons, patent specifications are seldom accurate in detail.
CHAPTEE XIII
TELEPHOTOGRAPHIC LENSES
In recent years a type of lens has been developed for the
express purpose of taking photographs of very distant objects.
Supposd, for example, there is a widely-extended landscape
which includes a castle standing on a hillside five or six miles
away. An ordinary landscape lens, even if suitable for
making a whole-plate picture 9 inches wide by 7 inches high,
though it could take into its field of view a wide stretch of
country, could not, if directed towards this castle, produce a
picture of it on any but an exceedingly small scale. It must
be remembered that the size of the images on a plate is
governed by the strict rule of optics that the relative sizes of
image and object are in the same proportion as their relative
distances from the lens.^ Now, let us suppose that the land-
scape lens is one with a focal length of 1 inches : let us see
what size it will give to the image of the castle. Suppose the
latter to-be 100 feet high and 5 miles away. Then the
height of the image of the castle on the plate will bear the
same proportion to 100 feet as 10 inches bears to 5 miles.
It will, in fact, be about -^-^ of an inch high ! The only way
to get a large image of that castle from a distance of 5 miles
is to employ a lens of longer focal length than 10 inches, or
to use something which will optically act as such. Let us
apply the same rule to ascertain what focal length would be
needed in order to produce an image 3 inches high. The
focal length would have to be such that it bears to 5 miles
^ Or, strictly speaking, are in the same proportion as their distances from
the "principal points" of the lens — of which "principal points" more is said
later. See p. 128. See also Harris's Optics^ 1776.
CHAP. XIII TELEPHOTOGRAPHIC LENSES 95
the same proportion as 3 inches does to 100 feet. In fact, it
would have to be 64 feet long. Imagine a camera-body
64 feet long !
Let it be remembered that when we are dealing with objects
many feet away from the camera, the rule that governs the
action of lenses in the case of magnifying glasses and micro-
scopes works in the reverse way. The more powerful the
lens — that is, the shorter its focal length — the less does it
magnify. To produce larger pictures of distant objects we
must use a weaker lens — that is to say, one of longer focus.
Further, when one is using a weak lens of long focus, or its
equivalent, the angular width of the field of view will be
proportionately contracted.
Fig. 41. — Diagram illustrating Relation between Focal Length and Size of Image.
Suppose a distant object AB (Fig. 41) is being photographed
through a lens L which has a short focal length, requiring the
plate to be put at P^. Then by drawing the lines B6 and Aa
through the optical centre of the lens we have marked at ab
the size of its image on the plate. Now suppose we substitute
for the lens another having three times as long a focal length.
We must draw back the plate to Pg, three times as far away
from L, to get a well-defined image, and it will now be of the
size a^b\ three times as large as before. Suppose that the size
ab is the size of our plate, then when we draw it back to the
position cd it will not be large enough to take in the whole
image of AB, but will only take in a part CD one-third as large.
In fact, the longer the focal length of our lens, the narrower
is, as said above, the angular width of the field of view.
In telephotographic work we are content to deal with fields
96 LUMMER'S PHOTOGRAPHIC OPTICS chap.
of view of very narrow angular width; but to make the
images of the distant objects large enough to be of service we
must employ a lens-combination such as to act as a very
weak long-focus lens.
The problem, therefore, of telephotography consisted in
inventing some optical combination which would act as a
long-focus lens, and yet not require an impracticably long body
to the camera. Not very much invention was required,
because the requisite optical system already existed in theory
in the telescope itself. Generally, when people use telescopes
they use them subjectively, that is to say, put them to their
eyes so that they receive the image personally. But there is
another way of using a telescope, namely, to let the light that
comes out through the eye-piece fall upon a white screen, where
it makes a real image that can be seen objectively by many
people at once. Of course, this requires a darkened room,
unless the object is itself very bright. This is — even without
a dark room — an excellent way of observing the sun-spots.
The writer, when a boy at school more than thirty years ago,
used this method to photograph the spots of the sun.
Let us now see how telescope principles may be applied to
make a telephotographic lens-system.
Consider a common plano-convex lens A (Fig. 42) capable
of bringing a parallel beam to converge to a focus at F. If
now a negative lens B of somewhat greater power is introduced
Fig. 42. — Focal Length of Positive Lens.
between the lens A and its focus, as in Fig. 43, it will reduce
the amount of convergence, and as in the lower figure, and
cause the rays to meet at F' further away. If these converg-
ing rays are produced backwards (as shown by the dotted
lines) till they meet the original parallel rays, it will be seen
XIII
TELEPHOTOGRAPHIC LENSES
97
that the effect of the combination is the same as if, instead of
the lenses A and B, there had been used a lens C of less power
than A, and as if it had been placed much further away in the
! !B
True focal length
Fig. 43.
-EflFect of adding a Negative Lens in lengthening the eflfective Focal Length,
and virtually shifting the Lens forward.
direction from which the light enters. All telephotographic
combinations are founded on this principle. The simple
formula for finding the focal length of the equivalent lens is
this : —
Let /i be the focal length of the first (positive) lens.
Let /g be the focal length of the second (negative) lens.
Let w be the width between the lenses.
Let F be the true focal length of the equivalent lens.
Then
A glance at Fig. 43 will show that the distance from the
focus F measured to the second lens B is much less than the
true focal length. This back distance, sometimes, but incor-
rectly, called the " back focus " of the combination, is given by
the formula
BF ^ Wi-^^0
From the first of these formulae it is clear that the focal
length, and therefore the magnification, depends upon the
width by which the lenses are separated. If they were
separated by a width w exactly equal to the difference between
their focal lengths /^ — /g, then the equivalent focal length
would become infinitely great, the rays emerging parallel. In
H
98 LUMMER'S PHOTOGRAPHIC OPTICS chap.
such a cfitse the telescope so adjusted would suit only an eye
adjusted for parallel vision. For distinct vision with a normal
eye adjusted to see distinctly an object or an image situated at
say 12 inches distance, it would be necessary to alter w
slightly, so as to make the image a virtual one at that distance
from the eye, the telescope being drawn out a little, so that w
is slightly greater than /^ — /g. If, however, it is desired that
this telescopic arrangement shall project a real image on a
screen, the telescope must be shortened a little, so that w is
less than /^ — /,. Let us denote this displacement in or out
by the symbol d, and let it be reckoned negative when w
exceeds /^ — /g. Then we have
and F ='
//2
d'
Suppose, for example, /^ = 8 inches, and f^ (the negative
lens) is 4 inches. If we put them 4 inches apart, f^ — /^ = 4 ;
/j — /g — -m; = rf as 0. In this position F = infinity. Now let
the back lens be pushed in 1 inch, or w = 3, and d=l ; then
Fa=32 inches. Or let the back lens be pushed in 2 inches,
then F = 1 6 inches. In the former case the distance of the
camera back from the back lens is 28 inches, in the latter
case the back distance is 12 inches. In any case this back
distance is considerably less than the true focal length, because
the " equivalent planes " of the combination are always dis-
placed toward, or even beyond the front (positive) component.
This displacement is more marked when the two lenses are
of very unequal power. For example, let /^ = 8 inches and
/^ = 2, and let them be put 5^ inches apart Here d =f^ —
/2~ir = ^ inch, and therefore F=32 inches, and the Iwick
distance works out at 10 inches only. Here is an example
of a camera lens, the back focal length of which is only 10
inches, and which is itself only 5^ inches long, but which acts
as a lens (so far as magnifying power is concerned) having an
equivalent focal length of 32 inches. It will be noted that
the focal length can be altered by changing the distance
between the two components.
The application of this principle to telescopes, to shorten
Fio. 45.— View o
XIII
TELEPHOTOGRAPHIC LENSES
99
their length and give them a variable magnifying power, appears
to have been suggested first by Wolf early last century ; but to
Barlow^ is due the realisation of this idea by the employment
Fig. 44. — Barlow Telescope with Extension Lens in Middle.
of a negative dchromatic lens to extend the equivalent focal
length of the objective. Fig. 44 depicts a modern form of the
Barlow telescope on this principle.
Telephotographic lenses were first brought out about the
year 1891 by Mr. T. R Dallmeyer and M. A. Duboscq of
Paris independently, and a little later in the same year by
Professor A. Miethe, now of Charlottenburg. Suggestions of
a more or less definite nature, to lengthen the focus of a
positive lens by adding behind it a negative lens, had been
made years before by Barlow and by Porro ; while Mr. Traill
Taylor had suggested the use of an opera-glass (which has a
negative eye-piece) as an enlarging lens. But for telephoto-
graphic work the opera-glass is not adapted. It does not
give a good flat field, and the negative lenses are not of
sufficiently large aperture to be effective. Mr. T. R Dall-
meyer has recently published an extensive work ^ on telephoto-
graphy, dealing with the whole subject, and particularly the
use of these lenses in portraiture. For photographing distant
objects the telephotographic objective has the great advantage,
over the equivalent lens of ordinary construction, that one
may produce large images without having the very gre^t and
unwieldy camera length that these would require. Fig. 45
shows a view of Miinchen taken by Professor Miethe at a
1 See DoUond and Barlow, Fhil. Trans, 1834 ; also Dawes, Astron, Soc. Notices,
vol. X. p. 175.
^ Telephotography^ by T. R. Dallmeyer : London, W. Heinemann, 1899. To
the courtesy of the author of this book is due the permission to reproduce Figs. 43
and 47. Mr. Dallmeyer had previously read papers on the same subject at the
Camera Club, London, on 10th December 1891 and 10th March 1892. Amongst
other literature on this topic may be mentioned a monograph on the use of
photographic tele-objectives by Dr. P. Rudolph, issued by the firm of Zeiss of
Jena in 1896.
100 LUMMEE'S PHOTOGRAPHIC OPTICS chap.
distance of 2800 metres (over 1^ mile) with an ordinary
objective, a Steioheil's Group-Antiplanet, of 10-inch focal
length ; and Fig. 46 is a view, taken at the same distance with
a tele-objective, of part of Fig. 45. The camera length was
,2 feet 4 inches ; but the equivalent focal length was 8 feet !
The exposure was 3 seconds. For mountain photc^raphy, and
for landscapea taken from balloons for topographical purposes,
the tele-objective is peculiarly adapted.
Mr. Dallmeyer's first construction consisted of a single
cemented positive of wide aperture-ratio combined with a
single cemented negative lens, with a diaphragm between.
But, owing to the presence of a slight distortion, he modified
the construction, and now uses as the front positive component
portrait lens of aperture-ratio //4 or //6,
FlQ. 47.— D»llmejer"3 Tels-objective.
to a tube with rack-work, and as a negative back
component a double combination negative element This is
illustrated in Fig. 47. At first, for the sake of getting high
magnification (which depends on the ratio of/j to/), it was
thought advisable to have the negative lens of focal length /
several times shorter than f^, that of the positive component.
But now the usual practice is to make /^ about half f^.
When viewing very distant objects it is necessary to rack
forward the front lens so that d may be very small ; and to
correct for the altered spherical aberration, it is also advisable
slightly to unscrew the hinder element of the portrait lens.
Professor Miethe employs usually a ColliTiear positive
combined with a triple-cemented negative.
Steinheil also uses a triple-cemented negative in combina-
tion with a Group-Antiplanet (p. 61).
XIII TELEPHOTOGRAPHIC LENSES 101
Messrs. Zeiss use for portraiture a single positive component
made up of four glasses cemented together, and having an
aperture-ratio of about// 3. For all cases in which distortion
is inadmissible they substitute a double anastigmat. In
either case the negative component is a cemented triple with
one face flat It is provided with a rack and an engraved
scale, with an index, in order to read off the amount of the
optical displacement d, in order to calculate the equivalent
focal length.
The second or negative lens used in the tele-objective should
be itself achromatic, and should be preferably also corrected for
central spherical aberration. It should therefore consist of
two cemented lenses ; the principal or negative one being
that with lower refractive index and higher v, the positive
correcting part being of a higher refractive index and lower v*
The difference of the two refractivities affords the means for
correcting spherical aberration, the difference of the two
jz-values the means for achromatising. This can be done with
old kinds of glass, but only by use of relatively steep curves.
Eudolph has, however, shown ^ that, by the application of
anomalous pairs of glass having very nearly equal refractivity,
more favourable curvatures can be used. Accordingly, in such
lenses Zeiss uses a dense baryta crown for the negative glass,
and a silicate crown of high dispersion for the positive correcting
part that is cemented to it to form an achromatic negative
lens, the spherical aberration being under-corrected.
The advantage of the telephotographic lens in portrait
work appears to be that, with a telephotographic lens of
the same equivalent focus, the camera may be placed further
off, has a better perspective effect, and enjoys ^a greater " depth "
of focus, while producing pictures of the same size. In practice
any non-distorting doublet lens, stigmatic, anastigmatic, or
rapid rectilinear, may be used as the positive component of a
tele-objective.
Following out the plan of the telephotographic lens, Mr.
Dallmeyer has, in conjunction with Mr. Bergheim, produced an
exceedingly interesting portrait lens consisting of a single
uncorrected positive lens combined with a single uncorrected
^ British Patent No. 10,000 of 1893; ox British Journal of Photography y xl.
p. 659 (1893).
102 LUMMEE'S PHOTOGRAPHIC OPTICS chap, xiii
negative lens of larger diameter placed some distance behind
it. A stop is employed in front of the front lens. The
positive lens has a high aperture-ratio, and the combination is
free from distortion ; but the spherical aberration that is
present prevents fine definition in the picture, and gives
images with softened outlines, having certain artistic qualities
that are not unpleasing.
APPENDIX I
SEIDEL'S THEORY OF THE FIVE ABERRATIONS
So much is said in Chapter II. about Seidel's theory, and so little beyond
a general outline is actually given, that for the benefit of readers who
may wish to go more deeply into the subject some further account of this
theory is here appended.
Ludwig von Seidel, who was Professor of Mathematics in Munich
(died 1896), contributed to the Astronomische Nachrichten a number of
mathematical papers on the theory of lenses, the chief of them appearing
in Nos. 835, 871, 1027, 1028, and 1029 of that journal The paper
which deals with aberrations and their annulment is to be found in Nos.
1027 to 1029, published April 1865. Its title is " Zur Dioptrik," with a
second title, " On the Development of the Members of the third Order
which determine the path through a system of refracting media of a ray
of light lying out of the plane of the axis." The paper is long and
intricate, the mathematical expressions obtained being for the most part
very complicated. As stated in Chapter IL, the method of procedure is
to obtain trigonometrical expressions for the path of the rays which
traverse the optical system at different angles, then to develop these
trigonometrical expressions in series of ascending powers, and then,
neglecting all powers above the third order for the sake of simplicity, to
deduce from the expressions the conditions which will lead to the annul-
ment of the several aberrations. As explained in the text in Chapter II.,
these conditions are found to be expressed as five different sums, which
are, in fact, the coefficients of the various terms in the equations, each
sum needing to be reduced in turn to zero if the corresponding aberration
is to be eliminated. Thus are obtained the five equations of condition
enumerated in Chapter II.
Yon Seidel has himself given a non-mathematical account of the
matter in voL i. of the Reports of the Scientific Techniccd Commission of the
Royal Bavarian Academy of Sciences, p. 227, 1866. More recently
S. Finsterwalder has furnished a resuws of von Seidel's equations, and
has drawn cet'tain further consequences fxom them. From these two
sources, and from Professor Lummer's edition of Miiller-Pouillet's Optics,
the following account of von Seidel's theory has been compiled.
In von Seidel's investigation he adopted a notation which, if convenient,
is also unusual. Every centred optical system consists of a number of
104
LUMMEE'S PHOTOGRAPHIC OPTICS
APPENDIX
spherical surfaces which are the boundaries betweeil media of different
refractivities. Seidel uses even suffixes to denote quantities relating to the
refracting surfaces, and odd suffixes to denote quantities relating to the
intervening media. His zero is reckoned at the first refracting surface,
the radii of curvatures of the successive surfaces being called pm fht Pa > » >
etc, the last one being called /Og^, where i denotes half the total number
of such surfaces. Similarly, the set of successive refractive indices are
w_i, rij, n_3, Wg, etc., up to 71^+1. Distances, real or virtual, of points
of intersection of rays with the axis, being reckoned from the respective
surfaces, will be denoted with even suffixes ; while quantities that belong
to the intervening media, such as their thicknesses, and the inclinations
of rays traversing them, will be denoted with odd suffixes. The order
followed in the notation is that followed physically by the incident rays,
and most conveniently taken from left to right Badii and distances of
intersections of rays, both of which are measured from the vertices of the
corresponding refracting surfaces, are considered positive when measured
in this direction from left to right.
Any centred dioptric system will then be characterised by its ordinary
data — p the radii of curvature, d the thicknesses of the respective media,
and n their refractive indices ; so that, adopting the suffix notation just
explained, the whole of the given elements of the system are situated
in the following order : —
n_i, p^ Wi, c?i, p^ Wg, (^3, /)4 . . . n^i-i, d2i-i^ p2i, Wai+i.
For these ordinary data von Seidel now substitutes with great success
certain new ones relating to what he calls " normal rays," meaning by the
term "normal ray" one whi,ch, starting from a point in the optical
axis, and continually making indefinitely small angles therewith, passes
through the system.
These new data are the lateral distances h^c from the axis at which the
normal ray intersects the respective refracting surfaces, and the angles
^2i+i which it makes with the axis in traversing the successive individual
media, and any finite multiples of these quantities, since both are small.
Fig. 48. — Path of a Ray through Optical System.
Let us consider the accompanying Fig. 48, in which the path of a
normal ray through a system of four refracting surfaces, the angles being
exaggerated, however, for clearness. Then, since for small angles the
tangent may be taken as equal to the arc, the quotient ^0/^-1 g^ves the
I SEIDEL'S THEORY OF THE FIVE ABERRATIONS 105
distance of the starting-point from the vertex of the first refracting
surface. Further, instead of writing the refractive indices w^i+i, it is
more convenient to introduce their respective reciprocals ^ with the symbol
^2t+i' So then the magnitudes h, a-, v will serve to determine the optical
fiystem just as well as the original /o, rf, and n. The following set of
equations will serve to convert from one set of symbols to the other : —
1 _ ""d ~ ""21 + 2 ,
"21+1 >
0"2i+l
1
^21 + 1 =
^2i+i
y . . in
Pa = -
l'2*-lO-2l + l-^2l + l<^2i-l ,
Or to convert back from /a, d, w, to h, o-, v, the following algorithm given
by von Siedel may be used : —
" First form the constants ao, ai, og, etc., according to the equations
_ ^21-1 — ''^21 + 1 .
d— r
«12l= = +W2i_i7l.2i+i
N2^
«2l+l— - y
P-d p-d
then choose Iiq and o-_i so that h^jo-^i is equal to the distance of the
starting-point of the normal ray from the vertex of the first refracting
surface ; then take K_i = n_iO-_i, Kq — Jiq, and calculate with these initifd
values all the later /c, according to the equation Kfn+\ = <hn.'^m+*^m-i9
then one has in general
^d = i^d
'21 + 1 —
__ '^21 + 1 »
^'2t+l
The effect of introducing into the calculations these successive " deter-
mining quantities " K and o-, instead of following the plan of reckoning
by intermediate virtual focal lengths, is to free the calculations from the
unmanageable continued fractions which would otherwise occur.
Next, in order to determine the position of a ray before refraction,
von Seidel chooses the two pairs of co-ordinates i/_i f_i, i/'_i f'_i of the
points in which the ray meets two fixed planes A_i and B_i perpen-
dicular to the axis of the system, and in a similar fashion the refracted
ray is referred to co-ordinates in two planes Ai and Bi in the second
medium, and so forth to the two final planes ^<d+\ ^nd ^'+1 ^^ ^^
(2t-|- l)th medium. The planes Ai Bi . . . A2i.f.i Bji+i should, however,
be dependent in some known way upon the planes of origin A_i B_i ;
they should, in fact, be situated where, according to the approximation
^ It will be noted here that von Seidel uses the symbol y in a different
sense from that assigned to the symbol v (the achromatic refractivity) in
Chapter VIII.
106 LUMMER'S PHOTOGRAPHIC OPTICS appendix
formulae of Gauss, the images, real or virtual, of the planes A_i and B_i
are situated in the corresponding media.
The purpose for which two sets of planes are chosen is the following :
The A set are supposed to represent the focal planes for the object and
its successive images after the several successive refractions. The B set
are the diaphragmatic planes — that is to say, the planes of the diaphragm
and its several successive images. [In Abbe's method of treating entrance-
and exit-pupils, the positions of these and of the diaphragm would lie in
some of the planes of the B series.]
Let us consider a normal ray,^ defined by its determining values ^, o*,
which passes through the point where the axis intersects the plane A_i ;
then, as already mentioned, — is the distance of the plane A_i from
the vertex of the first refracting surface ; — the distance of the plane Ai
from the same surface ; --? the distance of the plane A^ from the second
refracting surface ; and so on. If now one introduces the consideration
of a second normal ray which passes through the intersection of the plane
B_i with the axis, and which is specified by the "determining quantities *'
h\ (r\ then the distances of the B series of planes may be expressed in a
similar way. The distance between the planes A^+i and B^i+i is then
21 + 2 _ ^''21+2
^2i + l ^21 + 1 ^21 + 1 ^21 + 1
The quantities h, a- and h\ o-' are, of course, not independent of each
other, for they both originated from the original p and d. Whence there
results equality between the following expressions : —
V-i Vi ~ I'l ~ Vs
Let us now write, for brevity, some additional conventions : —
ho
i p=i
2-2
P
—J h^^Ji^p
Then by the introduction of the quantity T above, and these new
conventions, one may express the dependence of h' and o-' on h smd a- in
the following formulae : —
^ By the term *^ normal ray " must be understood one that actually intersects
the axis at some point, so that its path lies wholly in a meridional plane drawn
through the axis similar to the ray drawn in the plane of the paper in Fig. 47
supra.
SEIDEL'S THEORY OF THE FIVE ABERRATIONS
107
o-'iji-i - o-'ui+i = (a-2t_i - (r2i+i)(x - T2) +
TN,
2»
i
2TN2t/i2f
/U:
J • *
21
m
V2*-iO"'2»+i - V2i+i(r'2i-i = (vai-iO-ai+i - Vai+iO-jji-iXx - T2) ;
In the case when i = o the sums 2 on the right vanish (see Astrono-
mische Nachrichten, No. 1028, p. 316).
In selecting the co-ordinates of which one makes use in fixing (in the
various planes of the A and B series) the points in which these planes are
intersected by the rays, one may either choose the rectangular co-ordinates
V2i+ii Cat+i) ^'»+i> C'a'+ij which are parallel to one another and have
their origins in the points of intersection of the several planes with the
optic axis, or else one may choose the polar co-ordinates 7*21+ 1, Vai+ij
^'ai+ij v'2i+i» whose poles lie on the optic axis, and whose angles are
reckoned by parallel straight lines. Then the requirement of collinear
formation of images, in accordance with the usual dioptric formulae of
approximation {i.e. Gauss's theory) — that is to say, that the areas mapped
out in the planes of the A series by themselves, or in the planes of the B
series by themselves, by the intersection of any rays traversing the optic
system, shall be similar to one another — is expressed simply by the propor-
tionality of the linear co-ordinates. The letters iq f, 1/' f ', r v, r' v' may be
used to denote the values, resulting from the dioptric approximation
formulae, of the co-ordinates of the traces of a ray intersecting the A and
B planes ; their proportionality can then be expressed according to the
known relation between the size of the image and the convergence of the
rays in the following formulae : —
"^ -'/-i = %i = "-'/3= • . • H
^-1, -1,-3 ,
O" -1 / 0-, , (To ,
H'
v_
(r'_^v _o-\« a-
v_
1 , --s
Z'
8
0--1 _^l^ _<r. _
V_
lr_j= — Vj = -2r3= . . . R
'8
^-1 ^1 ^3
8
R'
[4a].
The magnitudes H, Z, R, and H', Z', R', may be termed the reduced
co-ordinates of the points of intersection of the rays with the planes.
Further, one may consider the co-ordinates rj, f, r and rj\ f, r as being
measured in their several planes by units of measurement conveniently
chosen, so that in particular the same numerical measures for the co-
108
LUMMER'S PHOTOGRAPHIC OPTICS
APPENDIX
ordinates of the traces of a given ray may be found, on the one hand, for
the A planes, and, on the other, for the B planes. The departures of the
values of the co-ordinates of tlie actual points of intersection from those
thus approximately arrived at, departures which constitute aberrations
of the rays, may be expressed by
^'/a'+i, ^Ca+i» ^^2t+i, ^«^2»+i, 1
^'j'a+i, ^Ca+i, Ar 2f+i, Av'ai+i ; j
and these measured in terms of the reduced units — so far as they are
linear — by
ARat-
AR'
AH'oij.1, AZ'fl
2t+l»
2t + l>
^21+1' J
Then we have, for example,
V2i + \
(H H-AHai+i) = ^a+i + ^'/'ii+i
0-,
V2f-H
^21+1
(R +AR2i+i) =r2i+i +Ar2i+i
[45].
and so forth.
These aberrations AHa+i . . . may be regarded as correcting
terms relatively to the values of the determining quantities H, Z, H', and
Z' of a ray that follows the ideal path in conformity with the approxima-
tion formulae. In proceeding with the calculation of these correcting
terms one at once discovers a great advantage which lies in the ingenious
choice made by von Seidel of these four determining quantities — namely,
that the expression for the correcting term of any one of them, so far as
it relates to any single refraction, does not depend upon the four correct-
ing terms of the previous refraction, but only upon one of them, and
contains only the approximate values of the remaining members.
To simplify the notation, all the magnitudes relating to the ray and
to the medium prior to 2ith refraction, and which should strictly be
distinguished by the index 2i - 1, may be marked by minus signs placed
under the corresponding letters, and in a similar way those magnitudes
as altered after undergoing the 2ith refraction may be indicated by plus
signs written under them in lieu of the index 2^ + 1. Then the reduced
polar co-ordinates of the traces of the ray in the A and B planes
become : —
In the planes
A
B
before the refraction
R + AR, V + Av
R' + AR', v' + Av'
and in the planes
after the refraction
A
+
R + AR, V + Av
+ +
B
+
R' -t- AR', v' + At?'
+ +
SEIDEL'S theory of the five aberrations 109
The difference AR - AR or Av — Av of the correcting terms is to be I
+ - + -
added, before or after the refraction as the case may be, to the constant ]
reduced co-ordinates R, v.
So far all has been preparatory, explaining the notations, abbreviations,
and conventions by which von Seidel was enabled to handle the highly
complicated relations between the various quantities. We shall now see
how he applied them in the trigonometrical calculations of the aberrations. !
The differences AR - AR, and Av - Av of the correcting terms, which
+ - + -
are, in fact, the co-ordinate elements that go to make up the aberrations
of the individual rays, are then expressed by Seidel in the following
formulae, which he develops at length from the ordinary trigonometrical
expressions, neglecting the higher terms. They are accurate up to terms
of the fifth order of the co-ordinates, supposed to be small quantities of
the first order : — *
Order of the co-ordinates supposed infinitely small of the first order —
2T3(AR - AR) = R^3 cos (v' - v)h[ - + Vf v<T-v(r) ... I.
+ - \ N / -- ++
(o- - &){(t' - o-')
- R^2R(-1 + 2 cos W - v) A - + - + (v(t-v<t) . II.
NN -- ++
-I- R'R2 COS {v' - v)
2h{z + (vcr-vo-) . . Ilia.
(o-.crXcr'-cr')
^h - + - + (vfT'^Vfr') . im.
-) (o- — o-X V o-' — V o-') , . . IIIc.
N - + -+ +-
-R3 h(z + )(v(r'-vo-')H--^(o-'-o-0(v(r'-vo-') IV.
L \ N / --. ++ J^ -+-++- J
2T3R (Av - Av) = R' sin {v - v) X all the following :—
R'2;i Z_^ (VO--VO-) . . . . V.
\ N / -- ++
(O- - or)(o-' - O-')
- 2R^R COS (r^ - 1;) ;i " + " '^ (va-^vcr) ... VI.
M iVJ
(O- - or)(o-' - O-')
NN -- ++
(<r-cr)((r'-cr')
h - + z +.(vo-'-vo-')+ ^(o- - cr)( V 0-' - V O VII.
L NN .--++ N_+_++_J
The corresponding formulae for AR' — AR' and Av — A^ are deduced
+ - + -
from the above by making the following substitutions : —
110 LUMMER'S PHOTOGRAPHIC OPTICS appendix
K and AR' for R and AR
V and Av for v and Av
h' for h
a-' and o-' for a* and o*
+ « + _
- T for T.
If we assume now that the dioptric system has k+l refracting surfaces
numbered 0, 2, 4, . . . 2^, we can write the above formulae 2A; + 1 times
after each other, and each time put, in place of the plus and minus signs
under the letters, the indices 2^+1, 2i- 1 of the media following and
preceding the 2tth refraction, and further provide the unmarked letters
h and N (T, R, R', v, and v' remain constant through all the refractions)
with the appropriate indices of the refracting surfaces. Further, if the
object-points are supposed to lie in the first plane of the A series, we can
take AR At? equal to zero. If we then add the right and left sides of
-1 -1
the k + l equations together, there remains on the right-hand side merely
2T^AKik+i in the one case and 2T^RAv2ifc+i in the other ; that is, there re-
main only the reduced aberrations of the co-ordinates of the ideal approxi-
mate co-ordinates of the trace of the ray where it intersects the last plane
of the series. The actual longitudinal aberrations are obtained from the
reduced aberrations by multiplication with — ^. On the left-hand
side, after the addition, we find again the common factors of the simple
formulae built up from the reduced approximate co-ordinates. We
then find that, in place of the parts that vary from refraction to re-
fraction, there enter only sums of 2^ + 1 terms, the common terms of
which are easily formed from the expressions I. to VII., indicated above,
by simply replacing v, v, o-, o-, o-', o-', h, N by Vat-i, Vgi+i, cr^.i ,o-2£+i,
_ + -+_ +
Although the formulae so obtained are already very suitable for
the calculation of the modus operandi of a given optical system, yet the
circumstance that the original determining quantities — the p and d—of the
system to be investigated are contained both in the quantities denomin-
ated by h and a- and in those denominated by h' and a-' creates some
difl&culty in answering the question as to the designing of a system of
prescribed performance. By means of the relation already obtained in
equation (3) between h <r and h' a-', the latter can, however, be eliminated,
with the result that the performance of a dioptric system then appears
actually expressed in a single series of determining quardities. In this way
von Seidel obtained the following system of formulae : —
• Write first, for brevity,
■rj Nsi "^ v^p-id^-i
p=% ^ ^
we may collect into the five following sums those expressions which
recur in the final formulae, and which in reality govern the several
different features of the general aberrations due to form ; —
SEIDEL'S THEORY OF THE FIVE ABERRATIONS
111
S(l
S(2
S(3;
S(4
S(5
= 2(l) = 2^.i(^^^^^^
+i>
i=o
1 =
i—k
=2(2)=2^i)u»i
i=o
i = k
1 =
i=k
=2(3)=2(2)Uai
1 =
i = k
1 =
i=k
->-2(«-S)
1 =
i=k
t=o
i=k
=2(^)=2^4)u^
i=o
1 =
■ [5].
These are von Seidel's famous five sums so frequently referred to, and
explained generally in Chapter II. As just mentioned, they recur in the
final fonnulae, which, as given by Finsterwalder, are again five in number,
as follows : —
S(l)
A =
B =
C=
D =
E =
2T^
XS(1) + TS(2)
3x^S(l) + 6xTS(2) + 2T2S(3) + T2S(4)
2T3
X^S(l) + 3x^TS(2) + 2xT2S(3) + xT^S(4) + T3S(5)
2T3
X^S(l) + 2xTS(2) + T2S(4)
2T3
And, by the aid of these, the reduced aberrations of the polar co-ordinates
of the traces of the ray in the last plane may be written : —
ARafc+i = AR'Scos {v- v) -BR'2R[1 + 2 cos2(v - v)] + CR'R2cos (v- v) ~DR8 ;
RAvafc+i = R' sin {v' - v){ AR'2 - 2BRR' cos (v' - v) + ER2}.
The plane A_i in the first medium was by hypothesis coincident
with the object. The plane B_i in the first medium may be considered
as situated at the place where the front stop (if such exists) is set to
limit the incident rays, or where the front mounting of the lens acts as
a stop. Or if that which limits the working aperture is a stop in one of
the other media, then the plane B_i must be taken at that place where
(by Gauss's theory) the image of the real stop would be found in the
first medium. Then, on the one hand, the magnitude R depends upon
the distance of a point-object from the optic axis, or in other words, upon
the width of the field of view coming into action. On the other hand,
the magnitude R' depends upon the place where the incident ray in
traversing the system meets the plane of the stop ; and therefore, if one
112 LUMMER'S PHOTOGRAPHIC OPTICS appendix
considers the extreme rays which are admitted by the aperture of the
diaphragm, R' depends upon the amount of the effective aperture of the
system. These Seidel formulae are therefore competent to deal with any
given centred optical system as to its performance in any cases that
may be presented of prescribed width of field or size of aperture.
As was pointed out in Chapter II., the five sums have the following
physical properties : —
If S(1) = tJiere will he no spherical aberration at the centre of the
field. It is equivalent to satisfying Euler's condition for the removal of
central aberration.
If this is done, and S(2) = 0, then there will he no coma. The fulfil-
ment of this second reduction to zero is equivalent to satisfying
Frauenhofer's condition or Abbe's sine-condition.
If both these are done, and further, S(3) = 0, then there mil he no
cutigmcUism of oblique pencils. The fulfilment of this third reduction to
zero still leaves the image-surface curved.
If, the first three conditions being achieved, we make also S(4) = 0,
then the7'e will he no curvature of the plane of the image. The fulfilment of
the fourth reduction to zero is equivalent to satisfying Petzval's condition.
It effects the anastigmatic flattening of the image, which, however, may
still suffer from unequal magnification toward the margins ; or in other
words, there may still be distortion.
If, the first four conditions being realised, we have also S(5) = 0, then
there mil be no distortion. So that if all five conditions are fulfilled the
optical system will give a perfectly defined, stigmatically perfect, flat,
distortionless, truly collinear image of a flat object.
In the memoir of Finsterwalder chiefly used in preparing this
summary are given the actual numerical values of the five sums for the
case of the celebrated Heliometer objective of Frauenhofer, which will
serve as a simple example of the theory. This lens is an uncemented
achromatic system of one flint and one crown glass, supposed to be
perfectly corrected for spherical aberration in the axis, for yellow light.
The radii of curvature of its surfaces, and the thicknesses of its successive
parts, are as follows : —
/3y= +838-164
P2= -333-768
p^= - 340-326
Pg= -1168-926
d^ = 6-0
(^3 = 00
(^5 = 4-0
Diameter of aperture is 70-2. The distance of the principal focus
from the vertex of the last surface is 1126*70. The true focal length is
1 131'45. These values are in old Bavarian " lines '' ; and as the Bavarian
foot (of 144 lines) is equal to 0*292 metre, it follows that the true focal
length is 2286 millimetres.
The values of the reciprocals of the refractive indices (for yellow light)
are : —
I SEIDEL'S THEORY OF THE FIVE ABERRATIONS 113
v_i = l-0
vi = 0-653967
Vg = 1-0
V5 = 0-610083
Vy=l-0
From these may be deduced the following values for the quantities h
and (T, which are the ** determining quantities " (see p. 105 above) :
A,(j= 100-00
^2 = 99*7523
;i4 = 99-7523
/ig = 99-6696
o-^j = 0-0
0-1 = 0-041285
0-8 = 0-221270
(7^5 = 0-020675
0-^ = 0-088382
From these there may be deduced the values of the five sums, the
separate totals of the positive and negative terms being given, as well as
their net totals : —
S(l)= -5-50853 +5-50874 =+0-00021
S(2)= -0-11198 +0-108383 =-0-003597
S(3)= -0-00288 +0-0020082 =-0-0008718
S(4)= -0-0046632 +0*0931744 =-0*0014888
S(5)= -0-00011635 + 0-00011783= +0-00000148
From these figures it appears that in this lens the compensation for
central aberration, the compensation of the positive term by the negative,
is correct to within 4 per cent of the value of the former ; while the
residual errors of S(2) and S(5) are an even smaller percentage. The
smallness of S(5) is presumably due to the small thickness of the com-
ponent lenses. Frauenhofer had purposely designed the lens to correct
for coma, and the smallness of S(2) is the measure of his success. This
lens had a very narrow field of view, its angular semi-width being
only 48'.
In a later paper by von Seidel, written in 1881, but published
posthumously in 1898,^ he reviews the equations of the earlier theory ; he
gives additional expressions for the radial and tangential aberrations in
the image-plane ; and also, re-states some of the equations, using rectilinear
co-ordinates x and y in place of the polar co-ordinates of the earlier
paper, x standing for R' cos (v' - v) and y for R' sin (v' — v). He also
shows that Frauenhofer's condition for simultaneous removed of central
aberration and of coma may be more simply written
B = xS(l) + TS(2) = 0.
^ **Ueber die Bedingangen mbglichst pracizer Abbildung durch einen
dioptrischen Apparaf (edited by S. Finsterwalder), Sitzutigsbcrichte der k.
bayr, Akademie, 1898, p. 396.
I
114 LUMMEE'S PHOTOGRAPHIC OPTICS appendix
Finsterwalder, who in 1892 published a remarkable memoir ^ upon the
images produced by optical systems of large aperture, has been the first
to recognise the extraordinary merit of von Seidel's investigations, and to
pursue them further. He has worked out the expressions for the form
of the focal surface in general cases for oblique rays, and in particular
for the special forms which that surface assumes when the Euler con-
dition and the Frauenhofer condition are fulfilled. He also investigated
the distribution of the light in the coma, and its changes of shape when
the position and size of the stop are changed. Finsterwalder further
shows that if, for a given dioptric system and a given object-plane, the
condition S(1)S(3)- [S(2)]2 = is fulfilled, then the focal surface (that
is to say, the surface containing the apices of all the individual focal
surfaces for the separate points of the object) will be a spherical surface,
the curvature of which is
S(l)S(4)-[S(2)f
S(l) vafc+i '
whence it follows that, if S(l) is not zero, the image surface will be flat
if the condition S(1)S(4)-[(S)]2 = is fulfilled. Further, the curvatures
of the two spherical surfaces which contain the tangential and radial
focal lines of the oblique astigmatic pencils are respectively
2_S(3|+_S(4) ^^^ S(4)^ ^
If S(3) = 0, the first curvature becomes equal to the second ; and if
S(4) likewise = 0, the curvature of the focal plane vanishes. Finster-
walder's memoir contains a most elegant investigation of the phenomenon
of coma, and is illustrated by a number of plates to elucidate the singular
shapes thrown upon a screen through an uncorrected lens by oblique
pencils proceeding from a non-axial luminous point. The condition
that S(4) = is equivalent to the proposition of Petzval, that to flatten
the image it was necessary to fulfil the condition 2t- = 0, where / is the
focal length of any of the component lenses and n its index of refraction.
But von Seidel justly remarks that this condition is of itself of no
significance : its significance begins when, as a preliminary, S(l) = S(2)
= S(3) = 0. He also most acutely points out that this condition,
necessary to the flattening of the image, could not possibly be fulfilled so
long as one has to deal with those kinds of glass in which the dispersion
and the refractivity increase or decrease together.
Remarkable as these researches of von Seidel are, it is of interest to
note that an even more general method of investigation into lens
aberrations had been previously propounded. This is the fragmentary
1 *< Die von optischen Systemen grosserer Oeffnang und grosseren Gesichts-
feldes erzeugten Bilder, auf Grand der Seidelschen Formeln untersucht/' von S.
Finsterwalder, Abh. d, IL Classe d. k, Akad, d. Wisamschaften in Munchen,
Bd. iii. p. 519.
I SEIDEL'S THEORY OF THE FIVE ABERRATIONS 115
paper 1 of Sir W. Rowan Hamilton, introducing into optics the idea of a
"characteristic function," namely, the time taken by the light to pass
from one point to another of its path. True, he did not work out the
relations between the constants of his formulae and the data of the optical
system. Yet the method, as a mathematical method of investigation, is
unquestionably more powerful. It has recently, and independently, been
revived by Thiesen,^ whose equations include those of von Seidel.
The latest development of advanced geometrical optics is due to
Professor H. Bruns, who has shown ^ that in general the formulae that
govern the formation of images can be deduced from an originating
function of the co-ordinates of the rays — a function termed by him the
eikonal — by differentiating the same, just as in theoretical mechanics the
components of the forces can be deduced by differentiation from the
potential function. Bruns's work is based upon the theory of contact-
transformations of Sophus Lie. But as yet neither the formulae of Bruns
nor those of Thiesen have been reduced to such shape as to be available
for service in the numerical computation of optical systems.
^ On some Results of the View of a Characteristic Function in Optics," B.A.
Beptyrt for 1833, p. 360.
2 ''BeitragezurDioptrik," Berl Berichte, 1890.
3 "Das Eikonal,'* Abhandlwigen der mcvth,-phy, Claase der k, sdchsischen
Akad, d, Wissenschaften^ Bd. xxi., Leipzig, 1895.
APPENDIX II
ON THE SINE-CONDITION
In the foregoing pages such frequent mention is made of the " sine-con-
dition " to be fulfilled by optical systems, that no excuse is necessary for
adding a short explanatory notice, based upon the paragraphs about this
matter by Professor Lummer in his edition of the Optics of Miiller-Pouillet
Let it be granted at the outset that we know that it is possible to
calculate the form of a lens which shall have no central spherical aberra-
tion — that is to say, one which forms an accurately-focussed image of a
point situated on the axis — and that this can be done even for a wide-
angled pencil travelling along the principal axis. This granted, let us
see what are the conditions to be observed in order that, with equally
wide-angled pencils, such a system may be made also to give well-focussed
images of points that lie, not on the axis, but near to it. As this re-
quirement was fulfilled in optical instruments of small aperture, it was for
a long time supposed it might therefore also be attained without further
conditions in optical systems with a wide aperture. But this is not so
by any means. Even in those cases where the most complete removal of
spherical aberration at the central point of the field has been attained,
those points of the image that lie immediately at one side of the axis are
in general so indistinct that the size of their circle of aberration may be
regarded as comparable with the distance that the object-point is situated
laterally from the axis.
According to Abbe this want of definition for points aside of the axis
originates in the circumstance that, for an indefinitely small element of
the surface of the object, the different zones of the spherically corrected
lens project images having different linear magnifications.
This property is illustrated in Fig. 49. Let the lens-system S be so
corrected that it focusses at the point Q all the rays that go out from the
point-object at P. That is to say, both the central rays A and the
marginal rays M are refracted accurately to meet at Q. But for rays
that emanate not from P, but from a point p a little to one side, it is
quite otherwise. The axial pencil a emerging from this point produces
the image q\ which is quite easily found by the rule for finding images
by any rays in a meridional plane containing the axis, whilst the extreme
pencil m is refracted to some point q\ where it produces a more or less
4
APPENDIX II
ON THE SINE-CONDITION
117
well-defined image of p. Consequently there are formed images of the
small object Pp, such as Qq and Qq\ of sizes that differ according to
whether the part of the lens used in their formation is the middle part
FiQ. 49. — Diagram illustrating Formation of Image by Central and by Marginal Bays.
or a marginal zone. If all zones are acting together, then all these
differently-sized images are formed simultaneously on the top of each
other, their centres coinciding, but not their edges. These differences
between the magnifying powers of the middle and edge of an objective
may in the case of a microscope objective amount to 50 per cent or more.
Such want of definition was for a long time falsely assigned to the
inappropriate designation of ** curvature of image " or " want of flatness "
of the field. The phenomenon of comay or lob-sided deformation of the
image of a bright point situated away from the axis, was indeed recognised ;
but it was not known before the time of Abbe that all the real faults of
curvature of field and radial astigmatism of oblique pencils were masked
by the more important errors due to the inequality of the magnifying
powers of the different zones of the lens, in any lens that is merely
spherically corrected for the centre of the field. Coma is indeed a
manifestation of this same error, as may readily be demonstrated by
placing against a lens an annulus of paper, of such a size as to allow a
central portion and a marginal zone to be used. If a bright point is
caused to cast on a white screen by means of oblique pencils through this
lens, the pear-shaped coma will be seen to be divided into an inner
smaller pear-shaped patch with a bright tip, and an outer ovate and
much more distorted margin.
But if a lens-system is to be truly aplanatic — in the sense in which
Abbe uses that term — that is to say^ if it is to reproduce as a plane
element in the image a plane elemerU of surface of an object, it must,
beside being spherically corrected for a point on the axis, have the same
magnifying power for all its zones. The necessary and sufficient condition
that all the zones of the system S should produce equal-sized images Qg'
of the object Pp' is the following : —
The ratio of the sines of the angles made with the axis hy any and every
ray proceeding from the axial point P, and refracted to the image point Q,
must be constant y or
sinu
siniA
= constant.
118 LUMMER'S PHOTOGRAPHIC OPTICS appendix
This is the Bine-condition which is of so vast an importance in the pro-
duction of correct images. When the sine-condition is not fulfilled, and
only axial spherical aberration has been corrected, then the image of a
small flat object will appear like the tip of a cone viewed from above.
Henceforth, therefore, no lens ought to be termed aplanatic ^ unless it is
so constructed that, while its central spherical aberration is annulled for
the particular focal distance at which it is intended to be used, it shall
also fulfil the sine-condition. The two conjugate points on the axis for
which it is thus doubly corrected, so that a flat element of luminous
surface placed at one is accurately imaged as a flat element of sur£eu^ at
the other, are properly termed the aplancUic points of that lens-system.
Abbe's test for the true aplanatism of a lens consists in viewing through
the lens a system of distorted hyperbolae resembling Fig. 7, c, p. 34,
which, when placed at the proper distance from the aplanatic point of
the lens, yields an image of undistorted straight lines. By means of this
criterion Abbe had come to the conclusion that optical practice had satis-
fied theoretical requirements long '^efore the importance of the sine-law
was known, and even before the publication in 1873 of the sine-condition.
As a matter of fact, all the older microscopic objectives that are truly
aplanatic do also satisfy the sine-condition. The older microscope makers,
while seeking in a purely empirical way to find such combinations of
various lenses as should satisfy the eye by giving the best definition when
applied to test objects, unconsciously varied the combination of lenses of
the objective until unknovdngly they attained not only spherical correction,
but also the fulfilment of the sine-condition. This is but another instance
of the artist, in the practice of his art, outstripping the science of his time.
In order to ascertain the constant of the equation -^ = constant, one
sinu
may proceed in several different ways. Abbe ^ deduced the sine-condition
and the value of the ratio from the requirement that two conjugate
elements of surface should be delineated by all partial pencils with an
equal magnification, or at least provided the departures from equality
are negligibly small compared with the size of the elements in question.
At the same epoch von Helmholtz ^ demonstrated the constancy of the
ratio of the sines of conjugate axis-angles under the condition that all the
light emanating from an element of surface and traversing the system
should actually be reunited in the image which that system, supposed
aberration-free, should cast according to the ordinary rules of geometrical
optics, as taught by Gauss's theory. He therefore thus applied the law of
the conservation of energy to the radiation of light.
In a much more general way, and even before Abbe or von Helmholtz,
^ This is a narrower definition than that usually found in optical treatises.
For example, 'i{eTSGhel{Encijclop. Metrop., a,vt, "Light," p. 389) defines an aplanatic
lens as "one which shall refract all rays, for a given refractive index, and con-
verging to or diverging from any one given point, to or from any other."
^ Archiv fur mikroscopische Anatomie, ix. 40, 1873 ; and Carl's Bepertorium
der Physik, xvi. 303, 1881.
3 ** Ueber die Grenze der Leistungsfahigkeit der Mikroskope," Pogg. Annalen,
Jubelband, 1874, p. 557 ; and JFissenschaftliche Ahhandlungen^ ii. p. 185.
II
ON THE SINE-CONDITION
119
Clausius^ deduced from the second law of thermodynamics the relation
to be satisfied in order that the whole of the energy from a small element
of surface within a cone of indefinitely small solid-angle should be transmitted
to a second element If one applies the equation of Clausius to the
formation of the image of an element of surface hy means of wide-angled
pencils of rays traversing an optical system, one obtains the sine-condition^
and for its constant the same value as Abbe and von Helmholtz have
assigned to it
The simplest and most elementary method of deducing the sine-
condition is that given by Mr. John Hockin.^ He proceeds from the
assumption that in an aplanatic system S (Fig. 50), which forms the image
Fig. 50.
QB of the object PA by means of suitably wide-angled pencils, the
" optical lengths " ^ between the conjugate pairs of points are in every
possible way equal to one another, apart in this case also from small
differences of a n^ligibly small order of magnitude. Hockin's process
consists in taking into consideration only the narrow pencils proceeding
from A and P parallel to each other, and of which the axial pairs of rays
cut each other in (say) N, and the parallel pairs of marginal rays in (say) M.
Then the perpendicular PA represents the wave surface of the rays
which intersect in N, and the perpendicular PD from P on to the back-
ward produced ray At is the wave surface of the rays which cut each other in
M. Consequently the " optical length " PMQ = PNQ ;
but since AMB = ANB
AMB - PMQ = ANB - PNQ
= AN-hNB-PN-NQ,
or since AN = PN
(AM -I- MB) - (PM -I- MQ) = (NB - NQ),
and consequently
(AM - PM) -I- (MB - MQ) = (NB - NQ).
The difference on the right side of the equation vanishes when the object
becomes infinitely smalL
Let us call the divergence-angles of conjugate rays u and u' ; the
linear dimensions of the infinitely small object and image dy and dy' ; and
^ Mechanische Wdmutheorie (3rd edition, 1887), i. 315 ; or English translation
by Browne, p. 321,
2 J(mmaZ of the Royal Microscopical Society, iv. 337, 1884.
^ "Optical length '*= distance traversed by light in vaciw during the time
occupied in traversing the ly&th considered =2 (actual pathx/^i).
120
LUMMER'S PHOTOGRAPHIC OPTICS
APPENDIX
the wave length, or the refractive index, in the object-space X or fx respec-
tively ; in the image-space A' or ft. Then if one equates the ** optical
lengths" to the distances reduced to their equivalents in empty space,
there is obtained : —
. _^ . . ( For X is inversely proportional to /a,
A A
■Dir /^■»# . BC sin u'dif'
BM-QM= +-.-== + — ^
A A
and the reduced length of AD=
ATI ^^
fJ.AD = K—r- •
A
If C is the foot of the perpendicular from Q on to MB, and if the distance
BQ is a small magnitude, we may write MC = MQ. We consequently
obtain for infinitely small elements delineated with wide-angle pencils
the condition
_ sin udy sin udy' _
XT' X"^" '
sin u' _dy X' __dy fx
sin u dy ' \ dy ' fx
then
[!]•
Since this condition must be true for all pairs of values of u and u'y
it must also be true for infinitely small values. In this case, however,
dy'
■J- — /^o» *^*^ ^» ^* ^ equal to the linear magnification for meridional rays.
We therefore obtain
sin u _^ fi 1
sin u fi^ /3q
[2].
This equation assumes a still simpler form for the special case when the
object is moved to infinity, as is approximately the case in the telescope.
7 = const; h being the axial distance
h
Instead of -; — „ we then have
sm u sin u
of the incident ray, as in Fig. 51. But since for meridional rays
/t
I
I
V
w
Fig. 61.
Q
we must put -: — ; =^0, and since further from the triangle EQH it follows
smu
h h
that sin u' = — -, or -: — ; = EQ, our condition simply is
£Q ' sinu
EQ = const. =^0 ;
but that is, the points of intersection of the ^parallel incident rays mth their
II ON THE SINE-CONDITION 121
conjugate etnergent parts lie on a circle whose radius is equal to the focal
length for meridional rays}
If the sine-condition is fulfilled, then an element of surface is distinctly
delineated by pencils of any angular width, but an extended surface is
not necessarily so, nor would this be so either for several elements situated
behind one another. There is consequently only one pair of aplanatic
points, so that an objective must be once for all computed for that pair of
points for which it is to be used. So soon as the object is moved from
the aplanatic point, to another position on the axis, its image, which moves
to a new conjugate point, is now aplanatic no longer.
But even the modest requirement of only two diflferent points on
the axis being rendered aberration -free, cannot be fulfilled, if the
aplanatic delineation of an element of surface has been accomplished.
For in order to satisfy that requirement, as Czapski ^ has proved by a
method analogous to that of Hockin, the condition
. u
2 _ u
-, — const. = p« — . .... [31.
u ' ^ a *• ■•
sin-
must be satisfied ; which stands in contravention of the sine-condition for
sinw u!
aplanatic systems, -; — 7- = Pq— .
From the two conditions it follows that, with all the aid of practical
optics, one can approximate very closely to the attainment of the following
theoretical goal — namely, to form a perfectly sharp inuige, hy means of
wide-angled pencils of any vndth, either of an indefinitely small elemevd of
surface perpendicular to the a^is, or else of an indefinitely short piece of the
axis itself On the other hand, it remains practically impossible to form a
perfectly sharp image of a small finite axially situated element of space.
The conditions that are required to give perfect definition to its element
of longitudinal dimension contravene those required to give perfect
definition of its elements of lateral extension. In the language of the
photographer, a perfect wide-angled rapid lens which will be suitable for
copying a flat picture, with precise definition right up to the extreme
margins, will have little or no " depth of focus," while a rapid lens
which has great depth of focus will be incapable of giving sharp images
right up to the margin of a wide field.
^ The point of intersection E is therefore called the * 'chief point," and the
distance EQ the focal length of the ray associated with it.
2 Theory of Optical InstrumentSf pp. 103-105.
APPENDIX III
COMPUTATION OF LENSES. TRIGONOMETRICAL FORMULAE
OF VON SEIDEL
However useful may be all the approximate formulae which are based,
for the sake of simplicity, upon neglecting the small quantities of the
higher orders in the series of terms, they can only serve to indicate the
approximate form of any desired optical system. They are simple because
they neglect the details which are concerned in the various aberrations,
and they are only approximately fulfilled by pencils of rays of small
angular value. For accurate reckoning of the aberrations of small pencils
they are useless, and are equally useless foi even the rough calculation of
wide-angled lenses. For example, in the design of a microscope objective
which is intended to focus accurately and stigmatically a cone of rays of
180** angle, all mere approximation-formulae aflFord no help. Even if one
introduces into them the higher terms that are usually neglected, they
are still useless, because then they lose their simplicity for computational
purposes. Petzval, in 1867, attempted to develop the series of terms up
to those of the ninth order, and found the task hopeless.
Hence, failing general formulae that combine the two incompatible
conditions of being at once simple and accurate, one is compelled to have
recourse to another method of attacking the calculations — and that
method, though of perfect accuracy, an empirical one. One assumes (on a
basis of experience and guess-work) a tentative optical system, and then
one tests it, whether on paper, or by actually constructing it. The test
oh paper consists in computing accurately the course through the system
of a few typical rays, and so one judges of the perforipance of the system.
The result of the computation suggests a possible modification — ^involving
a re-computation ; and so the work of designing proceeds tentatively.
Sometimes one arrives at a point where it is worth while to grind the
lenses and build up the system, and thus test it optically, when experi-
mental adjustments may aid toward a further perfection.
But even thus on paper one cannot compute accurately the path
followed by even a few selected rays without having formulae by which to
compute. And if one would save time, one would wish to have some
theoretical guidance toward selecting these rays.
The experimental process may follow various lines. The experimenter
tries the optical effect of modifying parts of the system, changing the
APPENDIX III COMPUTATION OF LENSES 123
individual lenses, trying lenses of other kinds of glass, altering the
distances between them, or stopping them down until the image of the
test-object is seen distinctly and free from colour-defects. This procedure
is really a fine art, rather than a science ; and in the hands of a true
artist, such as Hartnack of Potsdam, or as Powell of the firm of Powell
and Leland, it has yielded excellent results.
For large telescope objectives the empirical process — always a fine art
— takes a different course. The curves of the lenses are first calculated
approximately, by the aid of the rough formulae of approximation of the
ordinary text-book, and the glasses are then ground and polished. Then,
directing the telescope upon a fixed star (or upon an " artificial star," to
serve as a luminous point), one observes the images formed in different
parts of the field, aiding the eye by means of a high-power magnifying
glass. Then the objective, or its individual surfaces, are ground or polished
by hand, zone by zone, or bit by bit of the surface, until each zone and
every part of each zone gives a sharp and colourless image in one and the
same plane. This method of local retouching, which was used for
reflecting telescopes by Foucault, has been used for object glasses by
many makers, notably by T. Cooke of York, and by none with more
striking success than by the late Mr. Alvan Clarke.
The process of empirical computation is in any case tedious also. But
it brings with it other possibilities, enabling the computer to estimate
the various individual aberrations as to their several relative values, and
to eliminate one or other of them, according to the ultimate purpose for
which the lens is destined. Moreover, it leads to more general results,
and gives clear indications for such further modifications of the system as
will improve it, so that the desired end may be reached, step by step,
indeed, but by steps the effect of which will he thus known beforehand from
the calculations.
Naturally, then, one starts, in this case also, by means of the formulae
of approximation as already known for treating central spherical aberration
and chromatic aberration, and so calculates roughly a system which shall
have the required focal length, etc., fulfilling the prescribed conditions,
provided only smaW-angled pencils near the axis are used. This is, of
course, exceedingly simple. Then begins the operation of testing. One
must compute the exact path, right through all the successive surfaces, of
a certain number of individual incident rays, and see where they intersect
the focal plane that has been drawn through the principal focus of the
central axial rays (or " null-rays "). For this purpose one must make use
of a rigid trigonometrical computation. The importance of this process is
such that it must be described in full for the special case of rays which
actually intersect the axis of the system, and which therefore lie in some
one meridional plane with the axis. These, which we may call main
rays^ are simpler to calculate than others, because the whole course of such
a ray, before and after each successive refraction, will lie in the same
plane. After we have considered such simple cases, we shall be better
able to appreciate the labours by which L. von Seidel extended the
method of computation by giving exact trigonometrical formulae for those
rays which lie out of any such meridional plane, and which never
intersect the axis.
124
LUMMER'S PHOTOGRAPHIC OPTICS
APPENDIX
Computation of Main Rats
To follow the course of a main ray, we may consider first the simple
case of a spherical refracting surface bounding the junction of two refracting
media. Following von Seidel's notation (p. 104, ante\ we will use odd
suffixes for the media, and even suffixes to denote the surfaces. If this first
surface is numbered zero, then the refractive indices of the anterior
and posterior media will be denoted by /a_j and fi^^ respectively. The
radius of curvature of the surface will be Tq ; its vertex may be denoted
by Sq, and its centre of curvature as M^. Then Fig. 52 will serve to
Fig. 52.
demonstrate the geometrical relations between the incident and refracted
rays. Let the luminous point be considered as situated at L, at an
infinite distance to the left along the axis S^M^, so that its rays are a
beam parallel to this axis. Then any single ray of this beam will be
characterised by specifying the point Pq at which it intersects the
refracting surface, the point P^ lying at a distance h^ from the axis.
Let the refracted ray PqLq, which is conjugate to the incident ray, meet
the axis at Lq, at the distance SQLQ = aQ from the vertex. The radius
Tq — SqM(j = PqMq will be reckoned positive if it lies to the right of the
refracting surface, or negative if it lies to the left The angles
of incidence and refraction are called respectively <^q and yp^, the angle
of deviation Sq, and the angle at which the refracted ray meets the axis
Aq ; these latter being equal to one another, since the incident ray is
parallel to the axis. Then, since PqMq is normal to the surface, we have
sin <^Q = — ^
±^0
sin \j/^ = tirl sin </)y
in
[2].
[3].
And by the fundamental principle of triangles, as applied to the triangle
PqL^Mq, we have
Ill
COMPUTATION OF LENSES
126
^o^^o-S + ^'o^sini/'o
PqMo
±^0
sin
w
'0
This gives the value of cLq + Tq, from whence the value of a^ can be
immediately written as
_ sin ^^ + sin Aq
t*A — » A : . • ...
"O^'O-
sin A,
[5].
If a plane be considered as drawn through the centre of curvature
M, transverse to the axis, the incident and refracted rays will meet it in
the points Q^ and Q'q respectively ; the incident and refracted rays both
lie in the plane triangle PqQqM^, and we have the relation
^^sm^o^^oQo
. [6].
Next let us consider the case of two successive refracting surfaces, a
second spherical surface, with its vertex at S2 and its centre of curvature
at Mg, being the boundary between the medium of refractive index /a^.^
and the third medium of index /tA^g. The radius rg of this second
surface is SgMg or PgMg, where Pg is the point where the ray E^L^ meets
it, and is refracted along PgLg. The angle of incidence is LQPgMj or <^2>
and that of refraction LgPgMg or i//^- The angle of deviation is LoPaL^
or Sg ; and the angle at which the refracted ray meets the axis is PgLgMg
Fig. 53.
or Ag. Then, considering the relations between the angles and sides^of
the triangle PgMgL^, we have
«^^*2 = T7irrsinAo . . . . [7];
P2M2
or, calling the distance between the two vertices Sj and Sg by the symbol
"V
Also, from the law of refraction,
. , gp + r^ - d^ ,
sm ©o = — — r^ sm A,
- ^2
sin \^2 = r- si» ^2 •
r+3
[8].
[9].
126 LUMMER'S PHOTOGRAPHIC OPTICS appendix
Further
^2 = ^2-^2 • • • • [10].
But, in contradistinction to the former case,
^o = '^2-^2 .... [11].
Because (as shown in Fig. 53) /i^^ is less than /Lt^j, and (jy^ is less than
^2> it follows that 8^ is negative. From the triangle LgPoMg it
follows that
MgLg __ a2 + r^ ^ sin ^^
P2M2 ±r^ BiiKf, ' ' • [12J;
whence
" " ^2
'^"•■^ sL-^ .... [13].
If there follow several more refracting surfaces, then one calculates out
for each of them, exactly as for this second surface, each of the five
corresponding quantities <^, ^, S, A, and a. The value of a in the last
medium gives the apparent focal length (or back focal length) of the lens-
system for the ray corresponding to the zonal radius h^, which has been
taken for the incident ray in the calculation ; and if the system had no
spherical aberration, a would come out the same for all values of h^. If,
then, when the computation is made for several values of h^, that is, for
several zones of the lens, the differences give the values of the longitudinal
aberration ; and from these one could calculate the size of the circle of
confusion in any given plane near the focus, and also, approximately, the
distribution of the light within such circle of confusion. .
To compute the axial values for these rays that go through the middle
of the lens, and for which h^ = 0, we have recourse to an artifice, because
if we took s^ = 0, then i/' = and A = 0, and a would become indeterminate.
So we must turn back to formula [2] of p. 124, which, when the angles
are indefinitely small, may be written strictly correctly as
^o^^'^o .... [14].
r'+l
But also, under these conditions
or
"0 ^0 ^^ -^o^^--o^^^_^_^ . . [17].
But there is another way to find the values for axial rays. One may,
after calculating down (as in the example given below) for any particular
zonal radius Hq, simply repeat for the axial ray the same values as far as
[15];
[16].
«
Ill
COMPUTATION OF LENSES
127
to the item log sin \j/ ; and then, following on, write for the axial rays,
instead of the angular values of <^q, \p^ and S^, the values of their natural
sineSy and operate with these instead of the angles. In other words,
instead of forming the item <^q — ip^ substitute the item sin <^q — sin xpQ.
It is easily shown that this process is legitimate. Equation [16] is
rigidly true for main rays. Multiplying both numerator and denominator
by sin ^, we have
sin \// sin xp
at^ + r,
"0
±^0
fi,. . , . , sin <f) — sin i/'*
C-Zi sin ^ — sm \f/ ^ '
that is to say, we may use for axial rays the same formula as in equation
[4] we use for zonal rays, except that for Xq, or <^ - 1/', we write sines
instead of angles. And this saves time, because in computing the zonal
rays we have already had to compute log sin \{/,
Example of Computation of a Parallel Beam throuqh a Simple Lens
Let the following be the data of the simple lens (Fig. 54) : —
/*-! =
/*+! =
d, =
"•o =
'•2 =
: 1-00000
: 1-52964
= 1 00000
8 millimetres
: + 69-250 millimetres
: -216-195 millimetres
1
y\
•
1
1
1
. Y ,
"^'~,
^"*"» «.
s.
'K' —
M.
+3
FiQ. 64.
It is required to find the principal focal length for rays which meet
the lens at a zone of radius /Iq=15 millimetres, for a zone of radius
Aq= 10 millimetres, and for the axial or "null" rays.
128
LUMMER'S PHOTOGRAPHIC OPTICS
APPENDIX
log (I: +r^
log sin <I>Q
log(/x_i:/A+i)
log sin \I/q
^0 = ^0- ^0 = -^©
log sin Xq
log (1 ; sin \q)
log sin x^Q
log±»-o_
log^A^^ + ro)
Ao + ^o
±^0 + ^2-^1
^0 ± »*2 - ^1
log (Aq ± r2 - (^i)
log sin Aq
log(l: ±r2)
log sin <I>Q
log (M+1 : Z^+s)
log sin i/'2
<^2
-'A2
^2 = ^2-^2
^2 = '^0 + ^2
log sin Ag
log (1 : sin Ag)
log sin ^2
log±r2_
logjCAg + rg)
Ag + ^g
±^2
Aq=15.
M7609
8-15958
9-33567
9-81541
9-15108
12° 30' 35"
-8" 8' 27"
4" 22' 8"
8-88183
111817
9-15108
1-84042
210967
128-727
277-447
406174
2-60871
8-88183
-7-66515
-9-15569
0-18459
- 9-34028
-8" 13' 42"
12° 38' 44"
4*^ 25' 2"
4" 22' 8"
8° 47' 10"
9-18397
0-81603
- 9-34028
- 2-33485
2-49116
309-866
-216-197
93-659
Ao=10.
1-00000
8-15958
9-15958
9-81541
8-97499
8" 18' 10"
5' 25' 1-2"
2° 53' 8-8"
8-70196
1-29804
8-97499
1-84042
211345
129-852
277-447
407-299
2-60991
8-70196
-7-66515
-8-97702
0-18459
-916161
- 6° 26' 32-9"
8° 20' 31-1"
2° 53' 58-2"
2° 53' 8-8"
5° 47' 7"
9-00346
0-99654
-9-16161
- 2-33485
2-49300
311-172
-216-197
+ 94-975
Axial Ray.
1-17609
8-15958
9-33567
9-81541
9-15108
0-21661
0-14161
007500
8-87506
1-12494
9-15108
1-84042
2-11644
130-75
277-447
408-197
2-61087
8-87506
-7-66515
-9-15108
0-18459
- 9-33667
-014161
+ 0-21661
0-07500
0-07500
0-15000
9-17609
0-82391
-9-33567
- 2-33485
2-49443
312-20
-216-197
+ 96003
By the above computation we have found the angle of inclination A^
(Fig. 53) of the individual rays, with respect to the axis in the image-
region, and therefore the distance SgLg of their points of intersection
from the lens. This intersection-distance (which photographers inaccur-
ately call the " back focus ") must not be confused with the real focal
length. To find the latter, we must — following out the construction of
Fig. 55 — produce the emerging ray PgLg backwards, until it meets the
prolonged incident ray LPq at C. Then a perpendicular dropped from
C to E on the axis will give at E the " principal point " or " equivalent
point " (" Haupt-punkt " of Gauss), from which the true focal length CLg
Ill
COMPUTATION OF LENSES
12»
is to be measured. For axial rays CL^ is equal to ELg, because of the
smaUness of the angle X^ For the rays of greater zonal distance OL2 is
00
r=*r
Fig. 66.
greater than ELg. The next stage in the computation is to reckon out
these two magnitudes for the three rays chosen to be computed.
From the triangle CELg it follows that
tan Ag
smAg
Let EL2 be called F (the true focal length), and CL2 be called G (the
focal length reckoned obliquely) : then the computation will proceed as
follows : —
Ao = 15.
^0=10.
Axial Rays.
log^o
1-17609
1 -00000
1-17609
log (1 : tan Xg)
0-81090
0-99432
0-82391
logF
1-98699
1-99432
2-00000
F
97-05
98-70
100-00
log^o
1-17609
1-00000
1-17609
log (1 ;sin Ag)
0-81603
0-99654
0-82391
logG
1-99212
1-99654
2-00000
G
98-20
99-21
100-00
One sees that, for the axial rays, the value of G coincides with that of
F. What the point E at the foot of the perpendicular CE is for the
axial (null) rays, the point C is for the other rays that have a finite
zonal distance from the axis. We may accordingly call the point C a
" chief " 1 point of these rays, the distance CLg being the focal length
* Such points must not be confounded with the "principal points" of Gauss,
sometimes called in English the " equivalent points," or the ** optical centres " ;
for these, unlike the "chief" points, are always situated on the axis.
K
130 LUMMER'S PHOTOGRAPHIC OPTICS appendix hi
corresponding to the particular zonal distance h. The focal length EL2
for the axial rays is for distinction called the true focal length. In order
that the sme-condition (see Appendix II.) should be fulfilled by any lens,
it is requisite that the locus of all <* chief points such as C should be at
equal distances from the principal focus L^, as, indeed, is shown by the
dotted circle through C in Fig. 55.
In Lummer's edition of MuUer-Pouillef s Optics, p. 573, there is given
a complete computation of a Qauss's telescope objective, an achromat made
of two non-cemented lenses of Jena glass. Many others are to be found
in the Handbook of Applied Optics of Steinheil and Yoit. Another
example of the complete computation of a two-lens objective — an aplanat
of 43 inches focal length — by Dr. Harting, is given in the Zeitschrift fwr
Instrumentenkundey vol. xix. p. 269 (1899) ; see also vol. xviii. p. 357
(1898).
Computation op Rats which do not intbrsbct the Axis
Rays which do not cut the optic axis, but pass it at some distance later-
ally, are much more difficult of calculation than the main rays considered
above, because their path does not lie in any one plane, but changes
from refraction to refraction at the successive surfaces of the system,
making the formulae for precise computation more complex. For unless
the best selection is made of those co-ordinates or parameters which are
suited to the problem, the mathematical complications would frighten off
even a practised computer. For example, one might at each successive
deviation find it necessary to solve a new and awkward spherical triangle,
or else give up a rigid solution and fall back upon successive approxima-
tions. And if, in lenses of great angular aperture, rays that do not lie
in meridional planes must needs be taken into account, any plan that
will simplify such computations is of real service.
At the instigation of the late Dr. Steinheil, Professor von Seidel under-
took the investigation which led to the enunciation of exact trigonometrical
formulae for this case. These were used first in the establishment of
Steinheil at Miinchen, and, after a few months of successful use, they were
in 1866 published in the Abhandlungen of the Bavarian Academy of
Sciences.
These formulae, as used in Steinheil's establishment, are reprinted in
the Handbook of Applied Optics of Steinheil and Voit, where it is stated
that a computation, by their means, of a ray that does not cut the axis,
takes only about twice as long as that of a ray that is in a meridional
plane. Nevertheless, they are not now used in practical work. For it
has been found sufficient for all practical purposes to compute only — as
in the example given — a few rays which lie in a meridional plane and
meet the lens at different distances from the middle point The calcula-
tion of these is quite laborious enough. The work of Steinheil and Yoit
above referred to gives a large number of cases of the computations for
lenses consisting of two members.
GENERAL INDEX
(Numbers refer to pages ; those in italics to footnotes.)
Abbe, Professor E., researches of, 45
lens-pupils, 37, 37
theory and writings, 6, 34, 36, 40, 118
use of negative correcting lens, 91
Aberration, axial, 8
central (see axial)
chromatic, 11, 21, 23, 40
of focal lengths, 12, 21, 23
of position of images, 12
circle of, 21
spherical, 8
at image of stop, 31, 32
chromatic differences of, 4^
oblique (see Coma), 7, 8
Aberrations, theory of, 7
Aberrationless surfaces, 21
Abney, SirW., pin-hole photography, 17
Achrmnat, 41
" old " and " new " defined, 46
relative focal lengths of components
of, 45, 49, 54
simple rule for construction of, 53
two forms of old, 43
Achromatic lens, 41
components ol^ ^5
simple rule for, 53
Achromatism in double-objectives, 70
Airy, Sir G., curvature of image, 4^
Aldis, H. F., 86, 88
Anastigmat, Zeiss-Rudolph, 62-64, 76
double, convertible, definition of, 66
Collinear, 79
convertible Goerz, 78
convertible Zeiss, 66, 79, 81
Goerz-Hoegh, 66, 77
Miethe, 75
Planar, 81
Universal Symmetrical (Ross), 86
Protar, 67
Triple (Cooke-Taylor), 90
Unar, 83
Anastigpiatic triple-cemented lens, 65
Angle-true delineation, 16, 29
ArUiplanet of Steinheil, 56, 61, 64
Aperture-ratio, influence of stop on, 35
Aplanat, definition of, 36
SteinheU's, 68, 78
Aplanatic combinations, 69
Astigmatic difference, 24
Astigmatism, nature of, 9
radial, 9, 24
Axial rays, 8
Baille, J. B., on indices of glass, 45
Barlow telescope, 99
Barrel-shaped distortion, 33
Beck, Conrad, writings of, ^2, 43
Beck, R. and J., Voigtlander CoUinear,
80
Bebgheim lens (see Dallmeyer)
Bow, R. H., 25
Bruns, H., writings referred to, 115
BuscH, Pantoscqpe, 73
Camera, pin-hole, 14, 17, 29
Caustic curve, 27
Chance Brothers' glass, 45
Chemical focus, 22
Chevalier, time of exposure, 59
Chief rays, 29
points, 31
Chromatic aberration, 11, 21, 23, 40
aberration, correction of, 41
differences of focal lengths, 12, 21, 23
of position of images, 12
of spherical aberration, 4^
dispersion, 40
elimination of, 40
Circle of least confusion, 25
of least chromatic dispersion, 21
Clausixjs and sine-condition, 119
CoDDiNQTON, Treatise on Optics, 25, 49
Collinear (Voigtlander -Kaempfer lens)
79
132
LUMMER'S PHOTOGRAPHIC OPTICS
Collinear, relation between object and
image, 3
Coma, the aberration called, 8, 11, 27
Compensation, principle of, 41
Componnd lenses, distortion in, 36
distortion, condition of elimination of,
88
Concentric (Ross-Schroder lens), 56, 75,
85
Confusion, circle of least, 25
Convertible Anastigmaty 66
Cooke lenses, 90-93
Curvature of focal surface, 10
of plane of image, 10, 23
CzAPSKi, writings referred to, 6, SS, 34,
69, 121
Dagitebre, 42
lens, 44, 44, 60
Dallmbter on Abbe lens-pupils, S7
details of lens, 89
on size of needles, 17
pin-hole photography, 17
Rapid Rectilinear f 86
Re<Uilinear, 75
Stigmatic lenses, 86-90
telephotographic lens, 86, 99
triple achromatic, 86
use of negative correcting lens, 91
Wide-angle Rectilinear^ 86
Dallmbter-Bbrghbim portrait lens, 60,
101
Definition, badness of, 21
Depth of focus, 15, 22
Diamond, image-flattening property of,
55
Difference, astigmatic, 24
Diffraction, 16, 19
Dispersion, chromatic, 12
chromatic, elimination of, 40
equivalents, table of, 51
residual (secondary spectrum), 41
circle of aberration due to, 21
Dispersive qualities of glass, 45
Distortion, 11, 23, 29
barrel-shaped, 33
causes of, 29
freedom from, 15
freedom from, condition for, 29, 32, 33
influence of stop on, 34
in compound lenses, 36, 38
in modem objectives, 39
pincushion-shaped, 35
DoLLOND, achromatic lenses of, 40, 41
Draper, first portrait photographed by, 59
Double-objectives, achromatism ot 70
astigmatism in, 69
general properties of, 68
image by, in symmetric plane, 68
various kinds of, 72
(see Anaatigmat)
DuBOSCQ, A., telephotographic lens, 99
Edbb, text-book of, 75
Eikonal, the, 115
Equivalents, refraction and dispersion
table, 51
Eulbr'b condition, 112
Efu/ry scope (VoigtlSnder lens), 75
Feil^ glass of, 45
Field, flattening of, 47
flattening of, influence of stop on
24
flatness of, 24
influence of stop on size of, 35
FiNSTBRWALDER, writings referred to,
103
Focal lengths, chromatic difference of,
12, 21, 23
lines, 9, 24
surface, curvature of, 10
surfaces, 24
Focus, chemical, 22
depth o^ 15, 22
Frauenhofer, condition, 11, 21, 39, 43,
43, 112, 116
epoch of, 45, 46, 50
glass, used by, 45
objective, 11, 112
Fuzziness, 73
Gauss, achromatic objective, 82, 130
equations of, 4
equations of, Seidel correcting terms
for, 7
reference to theory of, 10, 13, 20, 31,
41
Glass, Jena, factory for, 50
Jena, table of, 51
new and old kinds considered, 45
refractive and dispersive qualities, 45
GoERZ, Anastigmatic Aplanat, 66
Double Anastigmat, 66, 77
Double Anagtigmat, Series Ha, 77
Convertible Anastigmat, Series H., 78
Lynkeiscope, 75
Grubb, lens, 44
Hamilton, Sir W. R., writings referred
to, 115
Harris, writings referred to, 94
Harrison, spherical objective of, 73
Harting, writings referred to, 130
Hartnack, Miethe's Anastigmat, 75
Pantoscope, 75
Hauptpunkte and Hauptstrahlen, 31
Heath, Professor, on Geometrical Optics
33
Heliometer objective, 11, 112
Helmholtz, writings referred to, 33
and sine-condition, 118
Herschbl, Sir J., equation and condition,
12,39
GENERAL INDEX
133
HocKiN, J., and sine-condition, 119
HoEGH, E. von, Anastigmatic A;pla7iati 66
Double Anastiginati 77
HOPKINSON, Dr. J., on glass, 45
Illnmination, intensity of, and orthoscopy,
34
Image, artificial rectification of, 26
collinear, with plane mirror, 3
curvature of plane of, 10, 23
flattening, 26, 47,
flattening by separating two com-
ponents, 57
formation by pin-hole camera, 14
formation by simple lens, 19
point-, 20
production of orthoscopic, 31
production of orthoscopic, condition for,
32
" telecentric " on side of, 3 4
Jena, factory at, 50
glass, optical properties of, 50
glass, table of, 51
Kaempfeb, writings referred to,
GoUineary 79
Kessler, writings referred to, 4^
Lens, achromatic, 41
achromatic, relative focal lengths of
components, 45, 49, 54
anastigmatic, triple-cemented, 65
compound, distortion in, 36, 38
effect of zones of, 28
meniscus, properties of, 23
pupils, 37, 37
separation of^ to flatten image, 57
simple, image formed by, 19
sphere-, 30
use of diamond in, 55
Lie, Sophus, writings referred to, 115
LuMMEB, Professor 0., edition of Mtiller-
Pouillet's Optics, 6, 103, 116
on Orthoscopy, 39
Lynkeiscope (Goerz lens), 75
Meniscus lens, properties of, 23
MiETHE, Professor A., writings referred to,
76
anastigmat, 63, 75
telephotographic lens, 99
MuLLER-PouiLLET, text-book of Optics,
16, 69, 70, 103
Needles, sizes of, 17
Nelson, E. M., on Doublets, 64
New achromat defined, 46
Objectives, achromatism of, 69
astigmatism in, 69
Objectives, distortion in modem, 39
double, 36
double, various kinds {^oitAnastigTimts),
71
performance shared unequally by com-
ponents, 61
Eudolph's principle in, 62
symmetrical and unsymmetrical, 36
unsymmetrical, 59
Antiplanet, 56, 61, 64
Aplanat, 68, 73
Daouerre's, 44, 60
Ettryscope, 75
Fbauenhofeb's, 11, 43
Gauss's, 82,
Gbubb's, 44
Lynkeiscope, 75
Orthostigmat, 80
Panoramic, 72
Pantoscope, 73, 75
Periscope, 73
Petzval's, 18, 43, 59, 60, 68, 72
Spherical, 73
telephotographic, 86
telescope (Steinheil), 43
(see also Anastigmat)
Oblique pencils, 26
Old achromat defined, 46
two forms of, 41
Orthoscopic (or angle-true), 16, 29
image production, 31
Orthoscopy, 31
and intensity of illumination, 34
conditions for, 32
in compound lens, 36
in compound lens, sole condition for,
38
Orthostigmat, 80
Panoramic lens, 72
Pantoscope, 73, 75
Paraxial rays, 8
Pencils, oblique, 26
Pendlebubt, C, on lenses, 33
Periscope, 73
Petzval, Jos., writings referred to, 42
49, 122
brightness of images, 22
curvature of image, 57, 88
exposure, 59
objective, 18, 43, 59, 60, 68, 72
objective as double anastigmat, 72
pin-hole camera formula, 18
use of diamond as lens, 55
Pincushion distortion, 35
Pin-hole camera, 14, 17, 29
needles for, 17
sizes of, 17
Planar, 81
Point-image, 20
Porta, Giambattista della, early camera,
19
13i
LUMMER'S PHOTOGRAPHIC OPTICS
Portrait objective of Petzval, 61
first, 59
ProtaTy 67
Pnpila, lens, 37, 37
Quadruple objectives, 65
Radial astigmatism, 9, 24
focal line, 24
Ratleioh, Lord, writings referred to, 16
Rays, chie^ 29
Rectification of image, 26
Rectilinear (Dallmeyer's), 75
Rapid, 86
Rectilinear, meaning of term, 29
Refraction, table of indices of, 51
Refractive qualities of glass, 45
Residual dispersion (see Secondary spec-
trum), 41
Retouching lenses, 123
ROHB, M. von, writings referring to, S^t
54, 65, 93
on triplets, 54
examples of triplets, 65
Ross, Concentric lens, 56, 75, 86
Satz-aTiastigmat, 66
Universal Symmetrical, 86
Wide-angle Symmetrical, 85
Rudolph, P., Anastigmat, 62, 63, 76
orthoscopy, 39
Planar, 81
principle, 62, 64, 65
Unar, 83
use of negative correcting lens, 91
Saiz-anastigmat, 66
Goerz's, 78
Zeiss's, 66, 79, 81
ScHOTT, 45, 50, 51
ScHBODEB, writings referred to, 57
anastigmats, 63, 85
Concentric lens, 56, 75, 76, 85
use of two separated lenses, 57
Secondary spectrum, 41, 52
Seidel, L. von, writings referred to, 7, IS,
39, 41, 49, 103, 107
achromatism and flat images, 55
conditions and theory referred to, 7
11, 12, 21, 26, 29, 40, 43, 47, 48,
49, 87, 88, 103
fourth, considered, 47
fourth, considered for two-lens com-
bination, 47
fourth, old glasses cannot satisfy, 49
curvature of image, 57
trigonometrical formulae of, 122
Sine-condition (Frauenhofer), 11, 21, 39,
43, 43, 112, 116
Sphere-lens, 30
Spherical aberration, 7, 8, 31, 32, 40
chromatic diflferences of, 40
Spherical aberration, correction of, for
centre of stop, 31, 32
oblique (coma), 7, 8
SphericcU ibjective, 73
Steinheil, writings referred to, 43, 130
Aplanat, 68, 73
Aplanat as double-objective, 72
on glass (ref.)) 45
opposed aberrations, 62, 87
OrthostigTYuU, 80
Periscope, 73
principle, 61
telescope objective, 43
use of negative correcting lens, 91
wide angle and large aperture-ratio, 74
Stigmatic, meaning of term, 4
Stigmatic lens (Dallmeyer), 86, 88, 89, 90
Stop, correction for centre of, 31, 32
use of front, 26
use of hind, 35
influence of, on aperture-ratio, 35
on flatness of image, 24
on size of field, 35
Stbehl, on Diffraction Theory in Geo-
metrical Optics, 16, 30
Surface, aberrationless, 21
convergence and divergence-producing,
64,65
curvature of focal, 10
Sutton, Panoramic lens, 72
use of negative lens in triplet, 91
Symmetricid objective, definition of, 36
(see O^ectives, Anastigmat)
Table of refraction and dispersion equiva-
lents, 51
Tangent condition, 33, 35
Tangential focal line, 24
Taylor, objective, 90
method of correcting with negative
lens, 90-93
opera-glass as enlarging lens, 99
Tdephotographic lenses, 94
advantages of, 101
Thiesen, writings referred to, 11, 115
Trigonometrical formulae of Seidel, 122
Triple achromatic objective, 86
Triple-cemented lenses, 65
Unar, 83
Unsymmetrical objective defined, 36
of two members, 59 (see Objectives,
Anastigmat)-
VoiQTLANDEE, OolUnear, 79, 86
JShiryscope, 75
P6/2wi^ objective, 59, 68
VoiT, writings referred to, 45, 130
Wagner, pin-hole camera experiments, 18
GENERAL INDEX
135
Wide-angle Rectilinear objective, 86
Wide-angle Symmetrical objective, 87
Work, principle of division of, 6, 36, 61
Zeiss, Anastigmat, 62, 63, 64, 76
Anastigmaty as double-objective, 72
Zeiss, Anastigmaty convertible, 66, 79,
81
Planar, 81
Protar, 67
teleobjective, 101
t/war, 83
Zones of lens, effect of various, 28
THE END
Printed by R. & R. Clark, Limited, Edinburgh.
WORKS BY PROR SILVANUS THOMPSON, D.8c.
PRINCIPAL AND PROFESSOR OP PHYSICS IN THE
TECHNICAL COLLEGE, FINSBURY
LIGHT, VISIBLE AND INVISIBLE
Illustrated, Grown 8vo. 6s. net
LIGHT. — "A successful attempt to bring a very recondite subject into the
open. . . . The book is excellently produced, and is alive with helpful diagrams
and clever portraits."
ELECTRICAL ENGINEER.^'' We can recommend this little volume both
to the general reader and to the student who wishes to get a true grasp of the
ascertained facts about light. "
ELEMENTARY LESSONS
IN
ELECTRICITY AND MAGNETISM
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