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Cambridge Tracts in Mathematics
and Mathematical Physics

GENERAL EDITORS

J. G. LEATHEM, M.A.

E. T. WHITTAKER, M.A., F.R.S.

No. 7

THE THEORY

OPTICAL INSTRUMENTS

by '
E. T. WHITTAKER, M.A., F.R.S.

Hon. Sc.D. (Dubl.) ; Royal Astronomer of Ireland.

Cambridge University Press Warehouse

C. F. CLAY, Manager
London : Fetter Lane, E.G.
Glasgow: 50, Wellington Street

1907

Price 2s. 6d. net

Cambridge Tracts in Mathematics
and Mathematical Physics

GENERAL EDITORS

J. G. LEATHEM, M.A.
E. T. WHITTAKER, M.A., F.R.S.

No. 7

The Theory
of Optical Instruments.

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

C. F. CLAY, MANAGER.
EonlJon: FETTER LANE, E.G.

©laagoto: 50, WELLINGTON STREET.

Ertpjtg: F. A. BROCKHAUS.

#rb) gorfc: G. P. PUTNAM'S SONS.

ttombag anU Calcutta: MACMILLAN AND CO., LTD.

THE THEORY

OPTICAL INSTRUMENTS

by
E. T. WHITTAKER, M.A., F.R.S.

Hon. Sc.D. (Dubl.) ; Royal Astronomer of Ireland.

THE

UNIVERSITY

£*LIF

CAMBRIDGE:

at the University Press

1907

(Eambrtoge :

PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS.

PREFACE.

of Astronomy, Photography, and Spectroscopy, have
frequently expressed the desire for a simple theoretical account of
those defects of performance of optical instruments to which the names
coma, curvature of field, astigmatism, distortion, secondary spectrum,
want of resolving power , etc., are given : it is hoped that the need will
to some extent be met by this little work, in which the endeavour is
made to lead up directly from the first elements of Optics to those parts
of the subject which are of greatest importance to workers with optical
instruments. A short account of the principal instruments has been
added.

While the tract is primarily written with this practical aim, the
writer ventures to hope that it may be useful in drawing the attention
of Pure Mathematicians to some attractive theorems : of special
interest is Klein's application of the imaginary circle at infinity to
establish the result (§ 30) that no optical instrument can possibly
be constructed, other than the plane mirror, so as to be capable of
transforming all the points of the object-space into points of the
image-space.

The writer moreover believes that the customary course of
Geometrical Optics presented to mathematical students in Universities
might with advantage be modified : and offers the present tract as a
suggestion to this end.

E. T. W.

DUNSINK OBSERVATORY, Co. DUBLIN,
November 1907.

180806

CONTENTS.

CHAPTER I. THE POSITION AND SIZE OF THE IMAGE.

PAGE

Sect. 1. Rays and waves of light 1

2. Reflexion 3

3. Refraction : Format's principle . . . . 3

4. Object and image 5

5. Image-formation by direct refraction at the spherical

interface between two media 6

6. Image-formation by direct refraction at any number of

spherical surfaces on the same axis ... 7

7. The Helmholtz-Clausius equation .... 9

8. The transformation of the object-space into the image-

space ...*..... 10

9. The measure of convergence of a pencil . . . 12

10. The lens 12

11. The thin lens 15

12. The spherical mirror 16

13. Astigmatism 17

14. Primary and secondary foci 19

15. Oblique refraction of a thin pencil at a single spherical

surface ......... 20

16. The entrance-pupil and the field of view ... 22

17. The magnifying power of a visual instrument . . 23

CHAPTER II. THE DEFECTS OF THE IMAGE.

Sect. 18. The removal of astigmatism from an optical instrument

with a narrow stop 24

19. The removal of astigmatism from an optical instrument

used at full aperture 26

20. Seidel's first condition : the removal of spherical

aberration 28

21. Evaluation of the spherical aberration in uncorrected

instruments ... .29

Vlll CONTENTS

PAGE

Sect. 22. Coma and its removal : the Fraunhofer condition . 32

23. The sine condition . 34

24. Aplanatism 35

25. Derivation of the Fraunhofer condition from the sine-

condition ......... 37

26. Astigmatism and Seidel's third condition ... 39

27. Petzval's condition for flatness of field ... 39

28. The condition for absence of distortion . . 41

29. Herschel's condition 45

30. The impossibility of a perfect optical instrument . . 47

31. Removal of the primary spectrum .... 48

32. Achromatism of the focal length . . . . 50

33. The higher chromatic corrections .... 52

34. The resolving power of a telescope objective . . 52

35. The resolving power of spectroscopes .... 54

CHAPTER III. SKETCH OF THE CHIEF OPTICAL INSTRUMENTS.

Sect. 36. The photographic objective 56

37. Telephotography 58

38. The telescope objective 58

39. Magnifying glasses and eyepieces .... 61

40. The visual astronomical refractor .... 63

41. The astronomical reflector 64

42. Field, marine, and opera glasses 65

43. The Microscope 67

44. The Prism Spectroscope .... ^ . 70

OF THE

UNIVERSITY

CHAPTER I

THE POSITION AND SIZE OF THE IMAGE.

1. Rays and waves of light.

The existence of " shadows," which is constantly observed in every-
day life, is most simply explained by the supposition that the influence
to which our eyes are sensitive, and which we call light, travels (at any
rate in air) in straight lines issuing in all directions from the "luminous "
bodies with which it originates, and that it can be stopped by certain
obstacles which are called opaque. This supposition of the rectilinear
propagation of light is not exactly confirmed by more precise observa-
tions : light does in fact bend round the corners of opaque bodies to
a certain very small extent. But the supposition is so close an
approximation to the truth that it may be taken as exact without
sensibly invalidating the discussion and explanation of many of the
most noteworthy phenomena of light.

If an opaque screen, pierced by a small hole, be placed at some
distance from a small source of light, the light transmitted through
the hole will therefore travel approximately in the prolongation of the
straight line joining the source to the hole. Light which is isolated in
this way, so as to have approximately a common direction of propaga-
tion, is called a pencil : and a luminous body is to be regarded as
sending out pencils of light in all directions. As there is a certain
amount of vagueness in this statement, owing to the absence of any
definite understanding as to what the cross-section of a pencil is to be,
it is customary to make use of that principle of idealisation which is of
such constant occurrence in mathematics : we introduce the term ray
to denote a pencil whose cross-section is infinitesimally small, so that
the light can be regarded as confined to a straight line : and then the
above idea can be expressed by the statement that a luminous body
sends out rays of light in all directions.

A more intimate study of the physical properties of light tends to
the conviction that light consists in a disturbance of a medium which

2 THE NATURE OF LIGHT [CH. I

fills all space, interpenetrating material bodies : to this medium the name
aether is given. A luminous point is then to be regarded as sending
out waves of disturbance into the surrounding aether, in much the
same fashion as a stone dropped into a pond sends out waves of
disturbance in the water of the pond. In the latter case, we can
distinguish between the crests of the waves, where the water is heaped
up, and the troughs, where the surface is depressed below the normal
level : these crests and troughs form a system of circles having for
centre the point where the stone struck the water : we can speak of
any crest or trough, or indeed any circle which has this point for
centre, as a wave-front, meaning thereby that at all points of such
a circle the water is at any instant in the same phase of disturbance.
Similarly in the case of the waves emitted by a luminous point in any
medium which is homogeneous (i. e. has the same properties at all its
points) and isotropic (i. e. has the same properties with respect to all
directions), the aether is in the same phase of disturbance at any
instant at all points of a sphere having the luminous point as centre :
and these surfaces of equal phase are called wave-fronts. It is evident
that the rays of light proceeding from the point are simply the normals
to the wave-fronts.

The luminous disturbances with which we are familiar in nature
are generally of a very complicated character, but can be regarded as
formed by the coexistence of a number of disturbances of simpler type,
in which those wave-fronts which have the same phase (e.g the
" crests ") follow each other at regular intervals of distance. This
distance is called the wave-length of the simple disturbance : and the
time taken by one crest to move over one wave-length, i.e. to replace
the crest in front of it, is called the period. Differences of wave-length
or period affect the eye as differences of colour.

The wave-fronts are propagated outwards from a luminous point, in
the same way as the water-waves on the pond : the velocity with which
a wave-front moves along its own normal depends on the material
medium (e.g. air or glass) in which the propagation is taking place.
The ratio of the velocity of light in vacuo to the velocity in any given
medium is called the index of refraction of the medium : it is
proportional to the time light takes to travel 1 cm. in the medium.
The refractive index depends to some extent on the colour of the light
considered : we shall suppose for the present that we are dealing with
light of some definite period, so that the index of refraction has a
definite value for every medium considered.

1-3] REFLEXION 3

2. Reflexion.

It is a familiar fact that light is to some extent thrown back or
reflected from the surfaces of most bodies on which it is incident. In
most cases the incident wave-front is so broken up by the small
irregularities of surface of the reflecting body, that any regularity
which it may have possessed before reflexion is destroyed : but if the
reflecting body is capable of being used as a mirror, i.e. if its surface is
optically smooth, reflexion has a regular character which we shall now
investigate.

Let the plane of the diagram be perpendicular to the reflecting
surface and the incident wave-front,
and let AC, A B be the traces of the
reflecting surface and the incident
wave-front respectively. Let DC
be the trace of the wave-front after

r\ \s

reflexion, and let #(7 and AD be per-
pendicular to the respective wave-fronts, so that they are respectively
parallel to the incident and reflected beams of light.

Then the time taken by the wave-front to travel from one position to
the other is proportional to either EG (which represents the time taken
by B to move to its new position (7) or to AD (which represents the
time taken by A to move to its new position D) : we have therefore

BC = AD, or BCA=DAC.

The angle between the incident ray BC and the normal to the
surface is called the angle of incidence : the angle between the
emergent ray AD and the normal is called the angle of reflexion.
The last equation may be expressed by the statement that the
reflected ray is in the same plane as the incident ray and the normal
to the reflecting surface, and the angle of reflexion is equal to the angle
of incidence. This is the law of reflexion.

3. Refraction : Fermat's principle.

If a thick piece of glass or any other transparent substance be
interposed in air between a luminous body and the eye, the luminous
source will in general still be seen, but will appear distorted or
displaced in some manner. From this it is evident that while the rays
from the luminous body which strike the glass are in part reflected at
the surface of the glass, they are also partly transmitted through the
glass, and at the same time experience a certain amount of deflexion

1—2

4 KEFRACTION [CH. I

from their original course. It can easily be shewn experimentally that
this deflexion, to which the name refraction is given, takes place at
the entry of the ray into the glass, and again at its emergence from the
glass : there is no change of direction of the ray during its passage
through the glass, if the latter be homogeneous.

If a ray of light passes from one medium into another, the acute
angle between the incident ray and the normal to the interface between
the media is called the angle of incidence, and the acute angle between
the refracted ray and the normal is called the angle of refraction.

Refraction is easily explained as a consequence of the difference of
velocity of propagation of light in different media. Let A C be the
trace of a small part of the refracting
surface : let AB be the trace of the
incident wave-front, so that its normal
BC is parallel to the incident beam : let
DC be the trace of the wave-front after

refraction, and AD its normal : and let , ,

ft and /A' denote the refractive indices of
the media.

Then the time taken by the wave-front to travel from one position to
the other is proportional to /A . BC (which represents the time taken by
B in travelling to (T) or to /*' . AD (which represents the time taken by
A in travelling to D). We have therefore,

p.BC=p.AD, or fjismBAC^f^'smACD.

Thus the law of refraction is that the sines of the angles of incidence
and refraction are in the ratio /A'//*- This is readily seen to be
equivalent to the statement that the cosines of the angles made by the
incident and refracted rays with any line in the tangent-plane to the
interface are in the ratio /A'//U,.

Media for which the index of refraction has comparatively large
or small values are spoken of as optically dense or optically light
respectively.

When the refraction takes place from a dense into a light medium,
so that n > /A', the law of refraction gives a real value for the angle of
refraction only when the angle of incidence is less than sin"1 (/A'//A).
This value of the angle of incidence is called the critical angle :
when the angle of incidence is greater than the critical angle, refraction
does not take place, all the light being reflected. This phenomenon is
known as total internal reflexion.

The laws of reflexion and refraction can be comprehended in

3, 4] PRINCIPLE OF LEAST TIME 5

a single statement known as Fermat's principle, which may be thus
stated : The path which is actually described by a ray of light between
two points is such that the time taken by light in travelling from one
point to the other is stationary (i.e. is a maximum or minimum) for that
path as compared with adjacent paths connecting the same terminal points :
the velocity of the light being everywhere proportional inversely to the
refractive index. In the case of reflexion the condition must of course
be added that the path of the ray is to meet the reflecting surface.

To shew that Fermat's principle is equivalent to the ordinary law of
refraction, let OA be an incident ray
in a medium of index /u, A I the
refracted ray in a medium of index ///,
B any point near to A on the refract-
ing surface AB. The excess of length
of OB over OA is evidently AB cos
OB A, and the excess of length of AI

over BI is ABcosBAl: so the

difference between the times of propa-

gation of luminous disturbance along the two paths OBI and OA I is

proportional to

/a . AB cos OB A -fJi'.AB cos BAI,

which vanishes in consequence of the law of refraction : this establishes
the stationary property which is enunciated in Fermat's principle.
Fermat's principle is analytically expressed by the statement that

(where /x denotes the refractive index for the element ds of the path)
has a stationary value, when the integration is taken along the actual
path of a ray between two given terminals, as compared with adjacent
curves connecting the same terminals.

4. Object and image.

In the preceding discussion of reflexion and refraction we have
considered only the direction of the tangent-plane to a wave-front at
some particular point : we must now proceed to consider the curvature
of the wave-front, which of course depends on its distance from the
luminous point from which it is diverging. The same idea can be
otherwise expressed by the statement that we have hitherto treated
only single rays, but are now about to study pencils.

Consider a luminous point which is emitting waves in air ; we shall

6 . OBJECT AND IMAGE [CH. I

call this the object-point. Suppose that the light, after proceeding
some distance from the object-point, is incident almost perpendicularly
on a convex lens (i.e. a piece of glass bounded by two spherical faces
and thickest in the middle). The waves before incidence on the lens
are convex in front, so that the part of the wave-front which strikes the
centre of the lens is originally a little ahead of the parts of the wave-
front which strike the rim of the lens : but as the luminous disturbance
travels more slowly in the glass than in air, that part of the wave
which passes through the centre of the lens, and therefore has the
greatest thickness of glass to traverse, will be retarded relatively to the
outer parts of the wave in passing through the lens ; and it may
happen that this takes place to such an extent as to make the outer
portions of the wave-front ahead of the central portion when the wave
emerges from the lens, so that the wave is now concave in front. This
concave wave will propagate itself onwards, in the direction of its own
normal at every point, and thus its radius of curvature will gradually
decrease until the wave finally converges to a point. This point, to
which the luminous disturbance issuing from the object-point and
caught by the lens is now ingathered, is said to be a real image of the
original object-point.

In any case the centre of curvature of the wave-fronts after
emergence from the lens is said to be an image of the object-point, the
image being called virtual if the luminous disturbance does not
actually pass through it.

5. Image-formation by direct refraction at the spherical
interface between two media.

The fundamental case of image-formation is that in which the light
issuing from an object is refracted at a spherical interface between two
media. Let the refractive indices of the first and second media be ju, and
ft' respectively, and let r be the radius of curvature of the interface,
counted positively when the surface is convex to the incident light.
Let 0 be the object-point, A the vertex or foot of the normal from 0
to the interface, P a point on the interface near A, PN perpendicular
to the axis or central line OA. We
shall consider the formation of an
image by a luminous disturbance
which is propagated approximately I

along the axis. o~ AVN u

A spherical wave-front originating

4-6] IMAGE-FORMATION BY SPHERICAL REFRACTING SURFACES 7

from 0 would, but for its encounter with the second medium, occupy at
some time a position represented by the trace P U, where U is a point
on the axis such that OU= OP. But owing to the fact that the dis-
turbance does not travel with the same velocity in the two media, the
disturbance along the axis will have reached only to a point V, where

or (p'-fiAN-iL. NU= n' . VN.

But by a well-known property of circles, we have

PN* = NU(ON+OU\ and PN* = NA(2r - NA\
and the equation can therefore be written in the form

2r-NA

which when P approaches indefinitely near to A takes the form

**' ^ _ JL. = 2//' VN
r OA~ PN* '

shewing that V and P lie on a sphere of centre /, where

r OA AI'

This sphere evidently represents the wave-front after refraction, and
its centre /, determined by the last equation, is the image-point corre-
sponding to the object-point 0. This equation shews that the range
formed by any number of object-points on the line OAI is, in the lan-
guage of geometry, homographic with the range formed by the
corresponding image-points.

6. Image-formation by direct refraction at any number
of spherical surfaces on the same axis.

We shall consider next the successive refraction of a pencil of light
at any number of spherical refracting surfaces whose centres of
curvature are on the same line or axis : the object- point will be
supposed for the present to be also situated on this axis, and the
pencil of light to be directed approximately along the axis.

Let x denote the abscissa of the object-point, measured (positively
in the direction of propagation of the light) from any fixed origin on the
axis : and let the abscissae of the successive images be alt #2, ..., x .

Then the homographic property found in § 5 shews that #x is given
in terms of x by an equation which can be written in the general form

*=£-:£.

where (aa, ft, y1} 8X) are constants which depend on the position and

8 POSITION OF THE IMAGE [CH. I

curvature of the first refracting surface and on the refractive indices of
the first and second media.

Similarly the positions of the successive images are given by
equations which may be written in the form

_ o^ + ft _ a3#2 + ft

~ ~

Combining these so as to eliminate the intermediate images, we see
that the position x of the final image-point is determined in terms of
the position x of the original object-point by an equation which can also
be written in the form

, ax + (3

X = - 5-

yx + 8

where (a, ft y, 8) are constants depending on the system of refracting
surfaces, but not depending on the position of the object-point.

If y is zero, the system is said to be a telescopic system : the
equation which determines x in terms a/ then becomes

which by change of origin can be written

x' — kx,
where k is a constant.

If y is not zero (which is the more general case), we can evidently
without loss of generality take y to be unity : the equation can then
be written

xx1 + 8af - ax - J3 = 0 ;

so if we now measure x from a point at a distance 4- S from the original
origin, and also measure x' from a point at a distance -a from the
original origin, the equation will take the form

where C is a constant. This equation determines the position x of the
final image. The origin from which x is now measured is called the
First Principal Focus of the optical system : it is evidently the
position in which the object must be placed in order that the image
may be at an infinite distance, i.e. in order that the emergent
wave-fronts may be plane. Similarly the origin from which x is
measured is called the Second Principal Focus : it is the position taken
by the image-point when the object-point is at an infinite distance,
e.g. a star. In the accustomed language of geometry, the Principal
Foci are the " vanishing points " of the homographic ranges formed by
any set of object-points and the corresponding image-points.

6, 7] HELMHOLTZ'S EQUATION 9

7. The Helmholtz-Clausius equation.
The equation xx = C

determines the position x of the image formed by a given optical
system, in terms of the position x of the object : we shall next shew
how to determine the size of the image in terms of the size and position
of the object, when the latter is supposed to be no longer a point but
a body of finite (though small) dimensions.

Let AB be an object, perpendicular to the axis AA' of the instru-
ment, and let A'B' be its image ; we can regard AB and A'B' as two

B D

B' D'

positions of a wave-front, when small quantities of the second order are
neglected (the ratio of the height AB to the dimensions of the instru-
ment being taken as a small quantity of the first order). Let AD,
A'D', be the corresponding two positions of another wave-front (pro-
ceeding of course from another source) slightly inclined to the first.
Then the time taken by the luminous disturbance to travel from B to B'

= „ „ „ „ „ „ ,,DtoD'.

It follows that the time taken by the light to travel the distance
BD in the initial medium is equal to the time taken to travel B'D' in
the final medium : or

where /* and // are the refractive indices of the initial and final media.
If then we denote the heights AB, A'B' of the object and image by
y and y respectively, and the initial and final angles BAD, B'A'D
between slightly inclined wave-fronts by a, a', respectively we have

This is known as Helmholtzs equation : it gives the linear magnifi-
cation y jy in terms of the angular magnification a! I a.

It is obvious that the above reasoning does not depend essentially
on the circumstance that the optical instrument has been supposed to
be symmetrical about an axis : we can therefore abandon this suppo-

;

10 OBJECT-SPACE AND IMAGE-SPACE [CH. I

sition, and state the theorem in a more general form due to Clausius*.
Suppose that a small line-element / in a medium of index /* has for
image a small line-element /' in a medium of index /*', and that a pencil
of light which has a small angular aperture a when it issues from
a point of / has an aperture a when it converges to the corresponding
image-point on /' : and let $ and \j/r be the angles made by I and /'
respectively with the normals to the pencil in its plane at the two ends.
Then / cos ^ will correspond to the y of Helmholtz's equation, and
/' cos \f/' to y : so we obtain Clausius' equation

fJ.la COS \l/ = l^l'a! COS ^'.

8. The transformation of the object-space into the image-
space.

We are now in a position to obtain formulae which completely
determine the manner in which an optical instrument forms an image
of a small object situated on its axis of symmetry.

The position of any point of a possible object, or any point of tlie
object-space as it is generally called, will be specified by its abscissa x
measured along the axis (positively in the direction of propagation of
the light) from the First Principal Focus of the instrument, and its
ordinate y drawn perpendicularly to the axis : and similarly the
position of a point in the image-space will be specified by coordinates
(#'» y'\ °f which x is measured from the Second Principal Focus of the
instrument.

Suppose that two objects, of heights yl9 y^ respectively, are at the
points whose abscissae are #1} #2 : let their images be of heights #/, #2',
respectively. Then the equation of § 6 gives

aw'=C,
so we have

Distance between images = --- = --- x Distance between objects.

If therefore a denote the inclination to the axis of the ray from the
axial point of the first object to the topmost point of the second object,
and if a! denote the inclination of this ray to the axis after passing
through the instrument, we have

a' 3/2' Distance between objects

i/2 Distance between images

i J c\^~

* Ann. der Phys. cxxi. (1864), 1.

7, 8] THE OPTICAL COLLINEATION 11

Now if /A and /*' denote the refractive indices of the initial and final
media, we have by Helmholtz's equation

and therefore, substituting the value just found for a'/a, we have

We now suppose that the two objects approach each other so as
ultimately to coincide in position : thus (omitting the suffixes) we have

The equation which determines the height y of an image in terms
of the height and position of the corresponding object is therefore

where /is a constant connected with the constant G by the equation

Thus the optical instrument transforms points (x, y) of the object-
space near the axis into points (#', y') of the image-space, in a manner
defined by the equations of transformation.

> fy
y = —-

9 X

This transformation is of the kind called in Geometry a collineation,
a name which is given to those transformations of space which
transform points into points and also transform straight lines into
straight lines.

When the initial and final media have the same refractive index, as
in the case of an optical instrument in air, the above equations become

*'=-£ y-^.

a? ' y x

The constant / is called the focal length of the instrument. If the
object is at an infinite distance (e.g. a pair of stars) and subtends
an angle a at the instrument, it is evident from the last equation that
the length of the image will be fa. Thus the focal length of a
photographic telescope determines the scale on which the heavens will
be depicted in the photographs taken with the instrument.

12 MEASURE OF CONVERGENCE [CH. 1

The object-point and image-point for which the linear magnification
y'ly is unity are sometimes called the Principal Points of the system.
The preceding equations give for the coordinates of these points

SO X =/, X = -/.

The principal points are therefore at distances from the principal foci
equal to the focal length.

9. The measure of convergence of a pencil.

"When light-waves are propagated outwards from a point 0 in
a homogeneous isotropic medium, the product of the refractive index /A
and the curvature of the wave-fronts at any point P is called the
divergence of the system of waves at the point P : the divergence
is therefore measured by the quantity pf OP.

Similarly if the luminous disturbance is converging to an image-
point 0, the quantity p/PO is called the convergence at P. Con-
vergence is evidently equivalent to a divergence equal in magnitude
but opposite in sign.

The theorem of § 5 can thus be expressed by the statement that
the effect of direct refraction at the spherical interface (radius r)
between two media p and /*' is to increase the convergence (or diminish
the divergence) of the incident pencil, by an amount (/*' - /*)/r. This
mode of stating the formula makes it easier to form a mental picture of
the effect of a direct refraction on a pencil.

10. The lens.

We shall now discuss the formation of images by lenses. A lens
consists of a slab of glass, or some other transparent substance, whose
faces are polished, and generally spherical. The line passing through the
centres of curvature of the faces is called the axis of the lens. We
shall denote the refractive index of the material of the lens by p, and

shall suppose that the lens is placed in a medium of index unity. The
points A, B, in which the axis meets the faces, are called the vertices,

8-10] THE LENS 13

and the distance AB between them is called the thickness of the lens,
and will be denoted by t : the radii of curvature of the faces (counted
positive when convex to the incident light) will be denoted by r, s ; so
that refraction at the first face increases the convergence of a pencil by
an amount (/x - l)/r, which we shall write fcl9 and refraction at the second
face increases the convergence by (!—/*)/«, which we shall write k2.

Suppose that a ray OP issuing from an object-point 0 on the axis,
and inclined at a small angle a to the axis, meets the first face of the
lens at P and is refracted into the direction PQ, making an angle
04 with the axis ; and is afterwards refracted at the second face of the
lens into the direction QR, making an angle a' with the axis. Let I±
and I be the points in which the ray meets the axis after its first
and second refraction respectively, so that II is the place of the
intermediate image of 0 and / is the place of the final image.

The formula of § 5, applied to the second refraction, is

or a! = - fa . BQ + /* . a,

&yj = Af + t . <*!.

But the formula of § 5, applied to the first refraction, is
_/*__ 7 J^_

"~ ~T" 7" — *kl ~s\ T 9

/! A OA

or //a! =-&! . 4P + a,

so substituting for aA in the preceding equation, and writing a for £//*,

we have

or -8. + l

since AP = a. OA ;

hence a /a = — K . OA + 1 — ak2,

where K is written for the quantity ^ + Jc^-

Now Helmholtz's equation shews that y'jy = a/a', where y'jy is the
ratio of the height of the final image at / to that of the object at 0.
Thus we have

y -. + l-a2

But it was shewn in § 8 that when an image is formed by direct
refraction through any optical system symmetrical about an axis, the

14 IMAGES FOKMED BY A LENS [CH. I

ratio of the heights of the image and object is given by an equation of
the form,

2/-/

jr *'

where / is the focal length of the system and x = FA - OA is the
distance of the object-point from the first principal focus F, measured
positively in the direction of propagation of the light. Comparing
these two equations, we have

1 l-akz

= FA --'

These equations determine the focal length of the lens and the position
of its first principal focus ; the position of the second principal focus F'
is similarly given by the equation

The position and size of the image are therefore given by the
equations

1
x —— T^2 , where F I— x ,

and y. — -j^- ,

which completely determine the image-forming action of the lens.

The distance of the vertex A from the first principal point H
is (§ 8)

IT A •— T? A /" — L"~ 2 ^2

HA-1A-J-- _r_-__-_,

and the distance of the second principal point from the vertex B,
measured outwards from the lens, is similarly - akJK. The distance
between the principal points is therefore t — a (kl + k^)jK ; or if t be
small compared with the focal length, it is approximately (/x - 1) t\i*..

It is easily seen from the above formulae that generally speaking
the effect of the thickness in a double-convex lens is to decrease the
converging power of the lens, while in a double-concave lens the thick-
ness increases the diverging power. When one surface of the lens is
plane, the thickness has no effect on the power.

A single thick lens possesses what is known as an optical centre,
characterised by the following property : any ray whose direction in the

10, 11] THIN LENSES 15

glass (i.e. between the two refractions) passes through the optical centre
will emerge from the second face parallel to its direction at incidence on
the first face. For if the incident ray passes through the first principal
point of the lens, the emergent ray will pass through the second
principal point, and by Helmholtz's equation their inclinations to the
axis will be equal : so the optical centre is the image of either principal
point in the corresponding face.

In the case of a lens of which one face is plane, the optical centre
and one of the principal points coincide at the vertex of the curved face.
In the case of a deep meniscus, i.e. a concavo-convex lens of great
curvature, the optical centre may be at a considerable distance from
the lens.

11. The thin lens.

When the lens is so thin that its thickness is negligible in comparison
with its focal length, the vertices may be regarded as coincident in one
point A, and the general formulae become

The principal points are now coincident at A : and the effect of the
lens is simply to increase the convergence of an incident pencil by an

amount (/*—!)(- —)'• This is called the converging power of the

\r s /

lens ; if the lens is thicker in the middle than at the rim, it is said to be
convergent and the focal length / is positive : if the lens is thinner in the

middle than at the rim, it is said to be divergent, and (/x, - 1) ( - - J

\S T /

can then be called its diverging power : the focal length is in this case
negative. Convergent lenses form real images of objects which are
situated so that the first principal focus is between the object and
the lens : for the divergence of a pencil on its arrival at the lens from
such an object is smaller than the converging power of the lens, so the
emergent pencil converges. Divergent lenses, when used alone, cannot
form real images of real objects.

When the object and image are both real (case of the convergent
lens, object in front of first principal focus), or both virtual (case of the
divergent lens, virtual object behind the first principal focus), the
object and image are on opposite sides of the lens and the image is

16 THE SPHERICAL MIRROR [CH. I

consequently inverted, as can be seen by observing that the straight
lines joining corresponding points of the object and image cross the
axis at the optical centre : in other cases the image is upright relatively
to the object.

If several thin lenses are placed in contact, each lens will exercise
its own converging power, and therefore the converging power of the
whole is the sum of the converging powers of the separate lenses : that
is, the reciprocal of the focal length of the system is the sum of the
reciprocals of the focal lengths of the individual lenses.

If two thin lenses of focal lengths /j and /2 are separated by an
interval a, each lens will resemble a single spherical surface in con-
verging power, and we can therefore deduce the formulae for the optical
behaviour of the system from the formulae of a single thick lens, by
replacing (/*-!)//* by 1//1} (1 -/*)/,? by l//2, and t{^ by a. Thus the
focal length of the system is

1 _ /!/,

1+ !_ JL fi+f*-a>

/I /, /l/2

and the distance from the second lens to the second principal focus is

12. The spherical mirror.

The reflexion of a pencil of light at the spherical interface between
two media can be treated in the same way as refraction. Let 0 be an
object-point, A the foot of the normal
from 0 to the interface, P a point on
the interface near A, PN the perpen-
dicular from P on the axis OA.

A wave-front propagated from 0,

which on arrival at P would have occupied the position PU if there
had been no reflexion, will actually occupy the position PVt where
VA = AU, owing to the reversal of direction of the central part of
the wave by the reflexion.

Let r denote the radius of curvature of the reflecting surface,
counted positively when the surface is convex to the incident light ;
then we have

VA=AUt or VN-AN=AN+NU.

11-13] ASTIGMATISM 17

Now PN2 = 2r.AJST,

since r is the radius of curvature of PA ; and

since 0 A is (in the limit) the radius of curvature of P U ; and
PN* = 2AI. VN,

since A I is (in the limit) the radius of curvature of P V, where /
denotes the image-point of 0.

Thus the preceding equation becomes

J__l 1 J_
~AI~ r~ r + OA1

121

or -rj.= -+ 77-7.

AI r OA

The divergence of the wave-front is therefore increased by 2/r as a
result of the reflexion, and the wave is at the same time reversed in
direction of propagation. The quantity 2/r is called the diverging
power of the mirror.

It is easily seen from this equation that a mirror has optical pro-
perties similar to those already found for the instruments which refract
light : its principal foci are coincident at the middle point of A C, where
C is the centre of curvature of the mirror, and its focal length is \r.

13. Astigmatism.

The wave-fronts which diverge from a luminous point in a homo-
geneous isotropic medium are spherical. If one of these spherical wave-
fronts is incident directly on an optical instrument symmetric about
an axis, so that the axis of the instrument points exactly toward the
luminous point, it is obvious from symmetry that the wave-front at
emergence will still be symmetric about the axis, and the part of it in
the immediate neighbourhood of the axis can therefore be regarded as
a portion of a sphere : this is generally expressed by the statement that
the emergent pencil of light is homocentric, a name implying that the
luminous disturbance is converging to (or diverging from) a single
point, namely the centre of this sphere (which of course will be the
image-point of the original luminous point). If for definiteness we
suppose that the pencil at emergence is converging to form a real image-
point, its cross-section will gradually diminish after leaving the
instrument, until at the place of the image the cross-section of the
w. 2

18 ASTIGMATISM [CH. I

pencil reduces to a point : after this the cross-section will again increase
in area ; thus :

O ° ° • ° ° O

When however a thin pencil of light is incident obliquely on a
refracting surface, the wave-front at emergence cannot in general be
regarded as a portion of a sphere, for its curvature will be different in
different directions along its surface : and the cross-section of the
emergent pencil of light will never reduce to a single point at any
distance from the instrument, but will present in succession the
following forms :

It will be observed that the cross-section reduces first to a short
segment of a straight line, and subsequently to a short segment of a
straight line in a direction at right angles to the first segment. These
segments are called the focal lines of the pencil : their origin may be
explained in the following way.

Let AP be the emergent wave-front, and A the point in which it is
met by some ray AR^R^, which we select
as the central or chief ray of the pencil :
this chief ray will of course be the normal
to the wave-front at A.

It is a well-known geometrical theorem

that all the normals to a surface touch the two caustic surfaces which
are the loci of its centres of principal curvature. Let us apply this
theorem to the surface AP. Let J?x and R% be the centres of principal
curvature of the wave-front at A ; we can suppose that the plane of the
diagram is the plane of the principal section for which Rl is the centre
of curvature. Then any ray of the pencil which meets the wave-front
at a point P near A touches the caustic surface through Rl at some
point near R^ and therefore its shortest distance from the line through
Rl perpendicular to the plane of the diagram is a small quantity of at
least the order AP^jARl. Similarly the distance of the ray through
P from the line drawn through R2 in the plane of the diagram perpen-
dicular to ARZ is a small quantity of at least the order AP2/Afi2.
These lines through Rl and R2 are evidently the focal lines, whose
existence was indicated above ; Rl and E2 are called the foci of the

(U
\

V/VERHS(TY

r
13, 14] ASTIGMATISM 19

thin pencil. We thus see that every ray of the pencil approximately
intersects the two focal lines.

The position of the focal lines is evidently not dependent on the
particular wave-front used to obtain them, since so long as the luminous
disturbance remains in the same medium its wave-fronts are a family
of parallel surfaces and have therefore the same caustic surfaces.

In the case of the homocentric pencils which have been considered
in the theory of direct image-formation, and which are symmetrical about
an axis, one caustic surface reduces to the axis itself, and the other
caustic surface has near the axis the form of a surface produced by the
revolution of a plane curve about a cuspidal tangent ; the foci Rl and
R2 in this case coincide at the cusp.

A thin pencil which is not homocentric, but diverges from two
focal lines, is said to be astigmatic. If the pencil originally issued
from a luminov0 point before the refractions, the image of this point
on a screen placed at either of the foci will be a short segment of a
straight line. If the screen is placed at (say) the focus R^ , the image
of a line will therefore be quite fine and sharp if it has the same direc-
tion as the focal line at Rlt since then the short segments of lines
which are the images of its individual points will overlie each other
lengthways : but otherwise the image will be blurred and broad, since
then the short segments which are the images of the individual points
of the original line will stand out more or less perpendicularly to the
general direction of the image of that line, and so will communicate
breadth to it.

The theory of focal lines is really part of the general theory of congruences :
a congruence is a set of oo 2 lines, just as a surface is a set of oo 2 points, and a
ruled surface is a set of QO l lines. Every ray of a congruence is intersected
by two adjacent rays ; these intersections are called the foci of the ray, and
the two planes passing through the ray and either of its two intersecting rays
are called focal planes. The loci of the foci are called the focal surfaces
of the congruence : every ray of a congruence touches the focal surfaces at its
focal points, and the tangent-planes are the focal planes.

If the focal planes are at right angles to each other for every ray of a
congruence (as is the case in the optical application of the theory), the
congruence consists of the set of normals to some surface (in the optical case,
this surface is the wave-front), and is called a normal congruence.

14. Primary and secondary foci.

The general case of the refraction of a thin pencil of light (either
homocentric or already rendered astigmatic by previous refractions)

2—2

20 THE ASTIGMATIC FOCI [CH. I

which is obliquely incident on a refracting surface of any curvature, is
a somewhat complicated subject of investigation: we shall consider
only the case which is of practical importance, namely the refraction of
a thin pencil through an optical instrument consisting of a series of
spherical refracting surfaces symmetrical about an axis, when it is
assumed that the chief ray of the pencil is initially in one plane with
the axis (and inclined at a finite angle to the axis), so that by symmetry
the chief ray never leaves this plane in the course of the subsequent
refractions. This plane through the axis and the chief ray will be
called the meridian plane of the pencil. By symmetry it follows that
the principal sections of the pencil are that by the meridian plane,
which is called the meridian or primary section, and that by the plane
at right angles to this, which is called the sagittal or secondary section ;
the corresponding foci of the pencil, which are the centres of curvature
of the meridian and sagittal sections of the wave-front respectively, are
called the meridian or primary focus and the sagittal or secondary
focus. Either the meridian or the sagittal focus or any point between
them, where the cross-section of the pencil is very small, may be
regarded as in some sense an image of the object-point from which the
thin pencil originally issued ; but as was explained in the last article,
the images thus obtained will be more or less blurred.

It is evident from symmetry that the rays which are at any time in
the meridian plane of the pencil always remain in the meridian plane
after any number of refractions, and that the same is true of the rays
in the sagittal plane.

15. Oblique refraction of a thin pencil at a single spherical
surface.

The analytical formulae for the case of a single refraction are
obtained in the following way.

Let a pencil whose meridian focus is Ol and chief ray O^A be
refracted from a medium of index /A
into a medium of index // at a
spherical interface whose centre of
curvature is C and radius r, counted
positive when the surface is convex
to the incident light. Let AI± be
the refracted chief ray, and let O^PI^
be the path of an adjacent ray in the
meridian section of the pencil, so that when P is indefinitely near to A,

14, 15]

THE ASTIGMATIC FOCI

/i tends to a limiting position, which is that of the meridian focus of
the pencil after refraction.

Let i, i' be the angles of incidence and refraction for the chief ray.
Then the equation

j* sin »*=/*' sin t'
when differentiated gives

ft cos i . di = ft' cos i' . di'

ft cos i (A01P + ACP)=t*'cos i' . (-PI.A + AGP)
./APcosi AP

= M cost -

AP\

T )

ft COS2

ft cos i _ ft cos i — ft cos ^

This is the equation connecting consecutive primary foci. It may easily
be interpreted geometrically as implying that the line GI/I passes
through a fixed point : and when i is replaced by zero it evidently
reduces to the ordinary equation (§ 5) for the direct refraction of
a pencil at a spherical surface'.

Next, let 02 be the sagittal
focus of the incident pencil. The
sagittal focus 72 of the refracted
pencil is, by symmetry, at the
intersection of the chief ray AIZ
of the refracted pencil with the
line of sagittal symmetry 02C.

The law of refraction gives

ft sin 0<>AC= ft' sin GAL.

But

and

Thus we have

02A AI2

C02 cos A C02 = 02A cos i + r,
(7/2 cos A C02 = A 72 cos i ' - r.

02A cos i + r , AI2 cos i' - r

02A

AI.

ft' cos^' — ft cos i

AI2 0,A r

This is the equation connecting consecutive secondary foci ; like the

22 THE ENTRANCE-PUPIL [CH. I

equation for primary foci, when i is replaced by zero it reduces to the
equation for direct refraction.

The union of rays at the sagittal foci is evidently, on account of the
symmetry, one order higher than the union at the primary foci.

Example. A small homocentric pencil of light is incident on and reflected
by a spherical surface of radius r ; shew that the reflected pencil is usually
astigmatic, and that the distance between the focal lines is equal to v^~v2,
where

I _ l - 2 I _ 1 _ 2 cos i

v^ u r cos i ' v.2 u r

i being the angle of incidence and u the distance of the origin of light from
the point of incidence.

16. The entrance-pupil and the field of view.

If an object is placed in front of a single convex lens, and a real
image is formed behind the lens, it is obvious that of all the rays of
light emitted by the object, the only ones which contribute to the
formation of the image are those which pass through the lens ; in other
words, the cross-section of the image-forming pencils is limited by the
rim of the lens. In most optical instruments the cross-sections of the
image-forming pencils are limited not only by the riins of the lenses,
but also by diaphragms or stops, which are generally openings in the
form of circles, whose centres are on the axis of the instrument and
whose planes are perpendicular to the axis ; a stop evidently obstructs
all those marginal rays which are at too great a distance from the axis
to pass through the opening. The rims of the lenses must of course be
included in an enumeration of the stops of an instrument, as also must
the edge of the iris, limiting the pupil of the eye, if the instrument is
used visually.

As will appear later, a judicious selection of the image-forming
pencils by a suitably placed stop of small aperture may effect a great
improvement in the optical performance of an instrument.

In order to find which one of the various stops in a given instrument
is effective in determining the cross-section of the image-forming
pencils, we consider the image of each stop formed by that part of the
optical system which precedes it, and from these images we select that
one which subtends the smallest angle at the axial point of the object
(which may be either in front of or behind it) ; this image is called the
entrance-pupil. It is evident that the cone of rays from the axial
point of the object to the entrance-pupil will be able to pass through
the instrument, but that a larger cone would have its marginal rays
cut off by that stop of which the entrance-pupil is an image.

15-17] THE FIELD OF VIEW 23

The angle subtended at the axial point of the object by the entrance-
pupil is called the angular aperture of the system ; the rays which
proceed from the various points of the object to the axial point of the
entrance-pupil are called the chief rays of the pencils which take part
in the representation.

The image of the entrance-pupil in the entire instrument is called
the exit-pupil : in those instruments which are intended for visual
observations, the entrance-pupil of the eye should be placed at the
exit-pupil of the instrument, when this is physically possible.

The stops also determine the extent of the field of view of the
instrument. In order to find which one of the stops is effective in
limiting the field of view, we consider the image of each stop formed
by that part of the system which precedes it, and from these images we
select the one which subtends the smallest angle at the axial point of
the entrance-pupil : this image has been called the entrance-window by
M. von Rohr, and evidently determines the extreme points of the object
which will be represented by pencils containing chief rays; its image
in the entire system is called the exit-window, and the angle subtended
by the entrance-window at the axial point of the entrance-pupil is
called the angular field of mew of the instrument. If the entrance-
window is not in the plane of the object, part of the object will be seen
only by partial pencils.

17. The magnifying power of a visual instrument.

We define the magnifying power of a visual instrument employed
to examine near objects as the ratio of the angle subtended by the
image of an object at the eye, when the object is so placed that the
image is at a standard distance (generally taken to be 25 cm.) from
the eye, to the angle subtended by the object when viewed directly
with the eye at the standard distance.

The magnifying power is therefore equal to the ratio of the heights
of the image and object respectively when the image is situated at the
standard distance in front of the exit-pupil of the instrument, i.e. it is
equal to the linear magnification when the image is in this position.

When a visual instrument is used for the examination of objects at
infinity, as in the case of the astronomical telescope, it is natural to
define the magnifying power as the angular magnification at the pupils :
this by Helmholtz's theorem (§ 7) is equal to the reciprocal of the linear
magnification at the pupils, so the magnifying power is equal to the
ratio of the radius of the entrance- pupil to the radius of the exit-pupil.

CHAPTER II.

THE DEFECTS OF THE IMAGE.

18. The removal of astigmatism from an optical instru-
ment with a narrow stop.

We now proceed to consider the conditions which must be satisfied
in order that an optical instrument may, as accurately as possible,
transform pencils issuing from the various points of the object into
homocentric pencils in the image-space, so that the image may be
a point-for-point representation of the object without blurring : and
moreover, that the image so formed may be geometrically similar to the
object.

It will be supposed throughout that we are dealing with an object
at some definite distance from the instrument, and that we wish to
eliminate errors in the image for an object in this position alone :
if the object is moved to some other position, errors will of course
reappear in the image. It will therefore be assumed that a plane object
is placed at right angles to the axis of the instrument : and we shall
suppose at first that a diaphragm of very small aperture is placed
at some point on the axis, so that the pencils of light which pass
through it, and by which alone the image is formed, are of very small
cross-section. Under these assumptions we shall find the condition
which must be satisfied in order that these pencils when they finally
emerge into the image-space may be homocentric, i.e. that the image
may be free from astigmatism. The treatment will necessarily be
approximate, the linear dimensions of the object and of the lens-
apertures being supposed as in Chapter I to be small compared with
the radii of curvature of the refracting surfaces ; but the approximation
is now to be carried to a higher order than in Chapter I.

18] INSTRUMENT WITH NARROW STOP 25

Let the ^th refracting surface be taken to separate a medium of
index HI-I from a medium of index Hh and to have a radius of
curvature rt, measured positively when the surface is convex to the
incident light ; let It-, denote the height of the intermediate image
of the object before refraction at this surface, and l{ the height of the
intermediate image after this refraction : let Xi and xi be the distances
of the intermediate images of the diaphragm from this refracting
surface before and after this refraction respectively (distances being
measured positively in the direction of propagation of the light),
and Si and si the distances of the intermediate images of the object
from the surface before and after this refraction ; and let i and i' be the
angles of incidence and refraction at this surface for the chief ray of
the pencil proceeding from the topmost
point of the object.

Then if 0, and 02 are the primary
and secondary foci of this pencil before
its refraction at the rth surface at P,
and I, , 72 are the primary and secondary
foci after this refraction, we have (§ 15)

Hi cos2 i' _ Hi-\ cos2 i __ Hi cos i' - HI-\ cos i _ _HJ
PI, P0l rt ~PI

Since, to our degree of approximation, we have
cos2 i=l— i2, cos2 i' = l- i'*,
these equations give

Hi HJJ'* Hi-i Hi-ii* JH_ _ Hi. Ill* _ Hi-i Hi-i- P

_ _
PI, PI, PO, PO, ~ PI, PI,2 PO, PO,2

st'* s^ s-

Now if yi denote the distance of P from the axis, we have

i^_yi i'-y±_yi -/, ju

~ > ~ >

and we have (§ 5)

an(1

26 REMOVAL OF ASTIGMATISM [CH. II

so our equation can be written

^./!/2 *-1.010a ,/ 1 1

But by similar triangles we have

Xi - Si

and we have

Thus the equation becomes

a* v/ i _j_\

ci- ft;/ \/Mi tH-\*J'

Since by Helmholtz's theorem we have

R«t^ = f*<_i«i'/«

this can be written

Now add together the equations of this type for all the refracting
surfaces in the instrument. The only terms surviving on the left-hand
side will be one involving the Oi$2 of the original object and one
involving the I1I2 of the final image : but the former of these
vanishes, since the pencils issuing from the object are originally
homocentric : and the latter term must vanish if the pencils converging
to the final image are also to be homocentric. Thus we have the
theorem that the condition for absence of astigmatism in the final
image is

Qxi-Q

where the summation is taken over all the refracting surfaces. This is
known as Zinken-Sommer's condition.

19. The removal of astigmatism from an optical instrument
used at full aperture.

If an optical instrument can be constructed so as to give emergent
pencils which are free from astigmatism even when a narrow diaphragm
is not inserted, i.e. when the full aperture of the lenses is filled by the
pencils, it is evident that the emergent wave-fronts will have their

18, 19] INSTRUMENT AT FULL APERTURE 27

principal radii of curvature equal at every point, and will therefore be
spherical : that is, the emergent wave-fronts will converge to points,
and the instrument will furnish an image which corresponds point for
point with the object. Clearly if this absence of astigmatism for
full pencils is to be attained, the condition found in the last article
must be satisfied independently of the diaphragm : in other words, the
last equation must be true whatever value MI may have. We shall now
find the conditions which must be satisfied in order that this may be
the case.

If hj, denotes the height at which a paraxial ray (i.e. a ray whose
path lies indefinitely close to the axis), passing through the axial points
of the intermediate images, meets the ^th refracting surface, and if d^
denotes the distance between the i— 1th and zth refracting surfaces, we
evidently have (the other symbols being defined as in the last article)

hi (Qxi - Q»i) t?i-\ (Qx, i-\ - Q8, i-0

Adding together equations of this type, we have

vwu-w

Now the condition found in the last article for absence of astigmatism
with a narrow diaphragm at xl is

and by use of the preceding equation this can be written

2 l + Q*V V -- + , ., <~" — , - -1- = 0 ...(A).

28 SEIDEL'S FIRST CONDITION [CH. H

The only quantity in this equation which involves the position of the
diaphragm is the quantity Qxl ; so the equation will be satisfied for all
positions of the diaphragm, provided the coefficients of the various

powers of ^ -- 77- are separately zero ; that is, the optical instrument

fyccl — tysl

will give point-images when used at full aperture, provided it satisfies
the three conditions

These are known as Seidel's first, second, and third conditions, re-
spectively*. We shall now proceed to interpret them.

20. Seidel's first condition: the removal of spherical
aberration.

We shall first interpret Seidel's condition (I).

By comparing condition (I) with equation (A) of the last article, it
is evident that condition (I) taken alone represents the condition that
the instrument shall give point-images by all pencils which can pass
through a diaphragm specified by the condition Qxl - Qsl = 0, i.e. subject
to the presence of a narrow7 stop placed at the axial point of the object.
But a narrow stop placed at the axial point of the object would allow
the passage of a full pencil from this axial point, while it would not
allow any light whatever to reach the instrument from the other points
of the object. Condition (/) therefore implies that all rays proceeding
from the axial point of the object are accurately united into the axial
point of the image. This is usually expressed by the statement that
the optical instrument has no spherical aberration.

When condition (I) is not satisfied, the rays proceeding from
the axial point of the object do not reunite to form a single image-
point ; the marginal rays, or rays which pass through the outer zones t
of the lenses, do not meet the axis in the same point as the paraxial
rays. When the instrument forms a real image, if the image as formed

* Seidel, Astr. Nach. XLIII. col. 289.

t The term zone is used to denote a ring-shaped region of one of the refracting
surfaces, bounded by two circles whose centres are on the axis.

19-21]

SPHERICAL ABERRATION

by the marginal rays is nearer to the instrument than that formed by
the paraxial rays, the instrument is said to be under-corrected for
spherical aberration : in the opposite case, it is said to be over- corrected.

(a) (b)

The figures (a) and (b) respectively represent an under-corrected and
an over-corrected pencil.

The curve drawn in the figures, touched by the rays of the pencil, is
the caustic (§ 13) : it is the evolute of the wave-front. If the light is
received on a screen placed nearer to the instrument than the focus of
an under-corrected pencil, the image will evidently be surrounded by a
hard edge (where the caustic meets the screen) : but if the screen
is placed beyond the focus, the image will be surrounded by scattered
light.

Spherical aberration will evidently become more and more noticeable
as the cross-section of the pencil increases, i.e. as the aperture of the
optical system increases.

21. Evaluation of the spherical aberration in uncorrected
instruments.

When the spherical aberration is not eliminated in an optical
instrument, its amount can be determined in the following way.

Let A be the vertex of the zth refracting surface AP, and let 0 be
the intersection of the axis with the prolongation of an image-forming

ray KP in the (•*' - l)th medium, while / is the intersection of the axis
with the same ray PI'm the £th medium. Denote AO by st + A^, and

30 SPHERICAL ABERRATION [CH. II

A I by s^ + Af, so that Ac measures the spherical aberration in the
^th medium for a ray which meets the ith refracting surface at a
height PN=h{.

Then if C be the centre and rt the radius of the refracting surface,
we have

" NO

r< -

to our degree of approximation ; whence we readily have

and

Similarly

i A

Ti W{

Now by the law of refraction we have

/*,_! sin (.4 CP - A OP) = & sin (A CP - AIP).

Substituting for the sines and cosines in this equation their values just
found, we have

1 1 , A,_A , K-M (I i\2

-^ - j aw

2 j

21] SPHERICAL ABERRATION 31

Now if $i denotes the inclination of the ray to the axis in the
^th medium, we have

Qi-\8i = QiSi =hi,
so the equation becomes

Adding together the equations of this type which refer to the successive
refracting surfaces, we have

-i'ifov/-^- --),

p = l \Pp-iSp PpSp /

, = ^ s ev v- (— --- M •

BpfVyl \l*9-i*p P-pSp/

l*9-i*p

This equation gives the spherical aberration of the image at any stage.

If we apply the formula to the case of image-formation by a single
thin lens, of refractive index /*, radii r and s, and focal length /,
so that

we have for the spherical aberration, along the axis, of a ray incident
at height h and proceeding from an object at infinity (e.g. a star)
the expression

and substituting these values we have

By applying this equation to particular cases it will be found, for
example, that a plano-convex lens is strongly under-corrected when the
plane face is turned towards an object at infinity, but only feebly
under-corrected when the convex face is turned towards the object.

The spherical aberration of a lens can however be completely
changed, and brought to any desired value, if in the process of
polishing the faces of the lens are made to depart from the exact
spherical form. If for example we consider a telescope objective which
is affected by spherical aberration, so that the longitudinal aberration
of a ray at distance h from the axis (the object being supposed at
infinity) is (3h?, it can without difficulty be shewn that this aberration

32 SEIDEL'S SECOND CONDITION [CH. n

can be completely removed by figuring the inner face so as to remove
a film of glass whose thickness at the point h is a constant

H — f-£- — r-^, where f is the focal length of the objective and ft the

4 (^/A — i)J

index of the glass on the inner side.

22. Coma and its removal : the Fraunhofer condition.

We next proceed to the interpretation of Seidel's second condition.
If we write equation (A) of § 19 in the form

X+Y(Qxl-QsJ + Z(Qxl-Qsir~ = 0 (A),

the three Seidel conditions are respectively

X=0, F=0, Z=0.

Now equation (A) represents the condition which must be satisfied in
order that astigmatism may be absent for that position of the stop
which corresponds to the value of (Q^ - Qsl} in the equation : and we
can regard the above form of the equation as a Taylor series developing
the condition in ascending powers of (Qxl - Qsl). The vanishing of X
implies (§ 20) the absence of astigmatism when (Q^ - Qsl) is zero, i.e.
when the stop is exactly at the axial point of the object : similarly the
vanishing of X and Y together implies that the astigmatism is not
only zero when Qxi — Qsi is zero, but that its rate of increase is zero
when Qxi — Qsi is made slightly different from zero, i.e. when the stop is
placed slightly in front of the object but very near to it. But when
the stop is in this position, it will permit the passage of practically
the full pencil from any point of the object which is very near the axis :
and hence the full pencil from such a point will be free from astigmatism
on emergence from the instrument. In this way we see that wJien
Seidel's condition (II) is satisfied in addition to condition (/), there is
a point-for-point representation not only of the axial point of the object,
but also of points of the object which are infinitesimally near the axis.

The defect of the image which is thus removed may be further
elucidated in the following way.

Suppose that the instrument does not satisfy condition (II), and
consider the full meridian pencil from an object-point 0 situated just
off the axis. The rays on emergence from the instrument will touch
a caustic ABC (Fig. a). If the light be received on a screen BK
at right angles to the axis at the place of the image, it is evident that
no light will reach the screen above the point B, where the caustic
meets the screen. The rays which have passed through the central

21, 22]

COMA

zone of the instrument will meet the screen at B in a bright point
(B in Figs, a and b). The rays which have passed through a zone
of the instrument somewhat further from the centre will (as is evident

(a)

from Fig. a) meet the screen lower down than B (at H in Fig. a) in
a circular section (LMN in Fig. 6) : and the rays which have passed
through the outermost zones of the instrument will meet the screen
still lower down, in a still larger circle (F\n Fig. a, PQE in Fig. 6).
In this way we see that the total effect on the screen is a balloon-
shaped flare of light, bright at the tip B and growing fainter as it
expands downwards*. This defect is known as coma (KO^O/, the
hair) : it is of great importance, as e.g. the definition in the outer
parts of the field of an astronomical telescope (assuming good definition
at the centre of the field) depends chiefly on the removal of coma. It
is perhaps more difficult to grasp than any of the other defects, owing
probably to the bewildering variety of (at first sight) unrelated ways in
which it may be described : from one point of view we may regard it as
spherical aberration (of the primary focus) for object-points just off the
axis : from another point of view we may regard it as implying that the
linear magnification of a very small object, situated on the axis of the
instrument, is different when different zones of the instrument are used
to form the image. To our order of approximation, and on the
assumption that there is no spherical aberration for the axial point of
the image, these two statements are evidently equivalent.

The condition (II) for the removal of coma was called by Seidel
Fraunhofers condition, because it was found to be almost exactly

* It is to be observed that each point of a circle such as PQR in the coma
corresponds to two diametrically opposite points of the zone which gives rise to the
circle, e.g. it is evident from Fig. a that the two extreme marginal rays will meet
the screen in the same point F : one-half of a zone gives a whole comatic circle.

w. 3

34 THE SINE CONDITION [CH. II

satisfied by the Konigsberg Heliometer objective, which had been
constructed by Fraunhofer many years before the discovery of the con-
dition, and which was celebrated for the excellence of its definition.

23. The sine condition.

We have seen that Seidel's equation (II) expresses the condition
that the linear magnification of a small object on the axis of the
instrument shall be the same whatever zone of the lenses is used
in forming the image. In all our work hitherto, however, it has been
assumed that the fourth power (and higher powers) of the angular
aperture can be neglected : and we shall now shew that the condition
just stated can be expressed analytically in a form which is rigorous
however large the aperture may be.

Suppose then that the lenses of an optical instrument are of any
size ; and let 0 be a small object situated on the axis in a medium of
index /*, its height / being at right angles to the plane of the diagram.

Let the instrument form an image / of 0, in a medium of index //, by
a thin sagittal pencil whose plane is at right angles to the plane of the
diagram, and whose chief ray OPQI makes an angle 0 with the
axis initially, and 0' finally. Let a denote the angle between the
extreme rays of the pencil initially, and let a' be the final value of
this angle : and suppose that d^ is the angle between the meridian
planes which pass through the extreme rays of the pencil, so

a = sin0.d<£, a' = sin 9'. d$.
Clausius' equation (§ 7) gives at once

so the linear magnification of a small object, when the image is formed

by rays which pass through this zone on the refracting surfaces, is

22-24] APLANATISM 35

This result is true for all optical instruments, independently of whether
they are affected with spherical aberration or not.

Suppose now that the instrument is corrected for spherical
aberration, so that the images of 0 formed by different zones are
situated at the same point of the axis. In order that the images
of a small object at 0 may be in all respects identical, they must be of
the same size ; and therefore the equation

, . -a-, ,
fji sm0

where m is the linear magnification for the image formed by the
paraxial rays, must be satisfied by every ray which issues from the
axial point 0. This equation is called the sine-condition.

As might be expected, the sine-condition also ensures that the
images formed by meridian pencils have the same magnification,
whatever be the zones through which the pencils pass. For again
applying Clausius' equation (§7)

JJL COS i/' . la = fJL COS \j/' . /'a',

we have in this case (the object and image being taken in the plane of
the diagram, perpendicular to the axis)

^ = 0, y' = 0', a = dO, a' = dO',
so the equation becomes

But by differentiating the sine-condition we have

P cos 6dO = mp cosO'dO',
so I'll = m,

i.e. the magnification is m whatever zone of the lenses is employed.

The honour of discovering the sine-condition must be shared

between Seidel*, who first gave that approximate form of it which he

called Fraunhofer's condition, and Clausius t, who first obtained the

\ rigorous form. It remained unnoticed however until in 1873 it was

rediscovered by Abbe and Helmholtz.

24. Aplanatism.

If an optical instrument is free from spherical aberration, and also
satisfies the sine-condition, for a certain position of the object, it is said
to be aplanatic for the object in question.

* Astr. Nach. XLIII. (1856), 289.
t Pogg. Ann. cxxi (1864), 1.

3—2

36 THE APLANATIC POINTS OF A SPHERE [CH. II

In the construction of microscope objectives, use is made of
the fact that there is one position of the object for which a single
spherical refracting surface is aplanatic : a result which we shall
now proceed to establish.

Let C be the centre of a sphere of glass of radius r and of index /x,
situated in a medium of index
unity : suppose that an object 0
is embedded in the glass at a
distance CO equal to r/n from
the centre ; and let / be the
point on CO at a distance pr
from C.

Then if P be any point on
the spherical surface, we have

so the triangles OOP, PCI, are similar : and therefore we have

sin

IPC sinPOC PC

sin OPC sin OPC OC

This shews that a ray proceeding from 0 in the direction OP will
\)G refracted at the surface exactly into the direction IP, whether P is
near the axis IOC or not : in other words, there is no spherical
aberration for the positions 0 and / of the object and image.

But it is also true that the sine-condition is satisfied for this position
of the object : for we have

PIC " smOPC ~" OC

shewing that the linear magnification is independent of the zone of the
spherical surface at which the refraction takes place, and is equal to v?.
The spherical surface is therefore aplanatic for an object in the position
0. The application of this principle to microscopes will be discussed
later.

There is another well-known case in which spherical aberration
is perfectly corrected for pencils of any aperture, namely that in which
the rays of light from a star are received on a concave reflecting
surface having the form of a paraboloid of revolution whose axis is
directed toward the star. In this case, as is obvious from the geometry
of the paraboloid, the rays are accurately united into an image at the

24, 25] THE FRAUNHOFER AND SINE CONDITIONS 37

focus of the paraboloid : but it can readily be verified that in this case
the sine-condition is not satisfied, so the surface is not aplanatic. It is
this want of aplanatism which causes the deterioration of definition in
the outer parts of the field of a reflecting telescope.

25. Derivation of the Fraunhofer condition from the sine-
condition.

We shall now shew analytically (what has already become obvious
from general reasoning) that the Fraunhofer condition for absence
of coma is simply the approximate form of the sine -condition, when
the fourth and higher powers of the angular aperture are neglected.

The sine-condition is (§ 23)

sin0'_ _/x_
sin 6 ~ p!m '

where m is the linear magnification for the paraxial rays.

Now considering separately the refraction at the ith refracting

surface, and using notation similar to that which has been frequently
used before, we have

sin 0/ PO N° +

Consequently we have
r- = the product of the values of -r—- for the separate refractions

= the product of the quantities ^ . ( 4? . ^ . - ^? *

*/ \AI sj ,h^(l 1\

2s- (r, .Si')

38 THE FRAUNHOFER AND SINE CONDITIONS [CH. II

It will be observed that A 0 differs from st by the spherical aberration
A,-,.

Now the product of the quantities -4 is —,— : so if the sine-
condition is satisfied, we must have

f .... i i

product 01 quantities — ; — — -- , 2 - r— = 1,
St 2s/ \Ti si

W } = Q
2/M/J

or _ , , ,

Si zm_i4 2/*i

where the summation is taken over the various refracting surfaces.
Substituting for A^ and Af from § 21, this becomes

i _-. y\o T A I -*•

1_ ^ ^2 ^4 / 1_ + _J_\ + ^ y^2/J L.M =0

/I 1 \

wnere -Ai =: TTS / ^~^ ^s / /rs > • • •

1 1_

.fj-ikiki+i f*i/t
The sine-condition thus becomes

Her*-)-

But Seidel's condition (I), which is supposed to be satisfied, is (§ 19)
SVtktf-l-. >-—) = <>.

i \PiSi Pi-iSi/

Multiplying the latter equation by

and adding it to the former, we have

M —. ~ -J~ = 0,

and this is no other than the Fraunhofer condition already found
in §§ 19, 22.

25-27J CURVATURE OF FIELD 39

26. Astigmatism and Seidel's third condition.

Of Seidel's three conditions (§ 19), only the third now remains for
interpretation. Since the three conditions together ensure freedom
from astigmatism over the whole field, and the two first conditions have
been shewn to relate specially to the central parts of the field, it is
evident that when Seidel's two first equations are satisfied, the third
equation may be regarded as representing the condition for removal of
astigmatism from the outer parts of the field.

27. Petzval's condition for flatness of field.

We have seen that the wave-fronts which issue from points of the
object will, after passage through an optical instrument, converge again
to points forming an image, provided that, in instruments with very
narrow diaphragms, the Zinken-Sommer condition (§ 18) is satisfied;
or, in instruments for which the diaphragm is not narrow, provided
Seidel's equations (I), (II), (III) (§ 19) are satisfied. It remains
to consider whether this image is a faithful copy of the object.

A condition which must obviously be satisfied if this is to be
the case is that if the object is plane and at right angles to the axis,
the image shall also be plane ; by symmetry, it' will also be at right
angles to the axis. We shall now find the analytical equation which
must be satisfied by the lenses of the instrument in order that a plane
object may give a plane image ; it is usually referred to as the condition
for flatness of field.

Let AP be the ^th refracting surface, 000 the intermediate image
before refraction at this surface, /0/
the image after refraction at this
surface, PO and PI the directions of
the chief ray (§ 16) of the pencil by
which the image-points 0 and / are
formed; and let X and X' be the °° x '<> x'

intermediate images of the diaphragm.

Let the radii of curvature of 0,0 and IJ respectively be p^ and
Pi ; and let the notation in other respects be the same as in previous
articles.

Then the coordinates of 0 referred to the vertex A are

( s. +Z!izl f i\ . those of / are (s- + f , /*) , and those of P

\ 2pi_i / \ *Pi '

40 PETZVAL'S CONDITION [CH. n

We have therefore

J_ Mi JL. yf (t-y*)*]

P/ s-V 2Pis-+2riS^ 2s-2 J
to our approximation, and similarly

PO SiV 2p;_^

Thus the equation (§ 15)

Pi COS %' - /tx^j COS i _ J^_ _ /AT-I

~^~ ~PI~PO

becomes

— l— = —, \ 1 - ^~- , + 5^—j -

?*t 2ri s^ I 2p^s/ 2^s/ 2s/2 )

O .1
Since i' =

this equation becomes

e-V'i i\ (fe-fe)2
~

Adding together the various equations of this type which refer to the
various refracting surfaces, we see that if the original object and final
image are each plane we must have

i \xi- si

The first sum is however known to be zero, since the instrument satisfies
Zinken-Sommer's condition (§ 18) : and hence we see that the condition
for flatness of field is

= 0.

This condition was first given by Petzval, and is known by his name.

27, 28]

DISTORTION

If the instrument consists of a number of thin lenses in air,
the refractive index and focal length of the £th lens being pk and/fc
respectively, the condition obviously becomes

k P-kfk

It is interesting to observe that the Petzval condition does not depend
in any way on the distance of the object from the instrument, or on
the separation of its component lenses.

28. The condition for absence of distortion.

Having now secured flatness of field, it remains to ensure that the
object (supposed to be a plane figure at right angles to the axis of the
instrument) shall give rise to an image which is geometrically similar
to itself. When this is not the case, the image is said to be affected by
distortion.

Distortion, in an optical instrument symmetrical about an axis,
simply means that the magnification of the image is not the same
in the outer parts of the field as at the centre. When the magnification
is greatest at the centre, a straight line in the outer part of the object-
field will evidently give rise to an image-line which is curved, with its
concavity turned towards the centre of the field : this is known as
"barrel" distortion. If on the other hand the magnification is greatest
at the margin of the field, a straight line in the outer part of the object-
field will give rise to a curved line in the image-field, with its con-
vexity turned towards the centre of the field : this is known as " pin-
cushion " distortion. All single lenses, whether consisting of one lens
or of several lenses cemented together, produce distortion : it is there-
fore necessary for most purposes in Photography to use objectives
in which there are one or more intervals between the lenses.

We shall use the same notation as in the preceding articles ; and

shall in addition denote by <fo the angle which the chief ray of the
image-forming pencil of a point 7 of the intermediate image makes

42 DISTOETION WITH NARROW STOP [CH. II

with the axis of the instrument in the medium /*»• ; and we shall denote
by xl + EI the distance from the ^th refracting surface AP of the
intermediate image X' of the diaphragm, formed by this pencil in the
medium fti} so that Et really represents the spherical aberration of this
image of the diaphragm.

The distance of the intermediate image 0 from the axis before this
refraction is OK (where OK is the perpendicular to the axis from 0),
or JO^tan^-j, or (^ + Ei-i — s^ tan <£i_j ; and its height after this
refraction is IL = (xl + Et-Si) tan<£t-. If there is no distortion, the
product of the ratios ILJOK at the various refracting surfaces must be
independent of the position of the point 0 in the object; so the
product

i 0* + Ei-l- s^ tan <£,_!

must be independent of the height y{ at which the chief ray PI meets
the ^th refracting surface.

Nnw

1.1 \J W

so the product in question is

Neglecting factors which do not depend on y^ this product is

as this reduces to unity for paraxial rays, it must be always unity : we
must therefore have

, •&- yf E^ Ej | yf \=p
^ 2n^ ^f-Si a?/ 2r^//

__ Mi-i^-i . y*V i ni o

But applying to Et the formula (§ 21) for the spherical aberration
of the intermediate image of an object, we have

28] DISTORTION WITH NARROW STOP 43

The condition for absence of distortion is therefore

Now by Helmholtz's theorem, /^/f _!#<._! is a constant for all the
images ; but 04_! = A,-/s<, so

Pt-ili-i _ constant
~~*T~ ~~^~
and since (§ 18) we have

constant
we see that yt = -™ -

The condition for absence of distortion may therefore be written

ri *i

Writing

I-%fa,i, and i-^forl,
r< /Af_i #; r< /x< a?i

this becomes

C4^+

Q* f Q*? ( i JLV+1^1 -l-NUo

« ^ required condition for absence of distortion, when the
diaphragm is at the position x ; it being assumed that the instrument
already satisfies the Zinken-Sommer condition and the Petzval
condition.

If it is required that distortion should be absent when the image is
formed by pencils filling the whole aperture of the optical instrument,
we must find the condition in order that the last equation may be

44 DISTORTION WITH FULL APERTURE [CH. II

satisfied whatever value x may have ; it being now assumed that the
instrument already satisfies the three Seidel conditions of § 19, and the

Petzval condition (§ 27). For this purpose we substitute for ~ *

«KM ~~ tysi

its value (§ 19)

1 + ht*Q« f 2 -A- + , * a N} 5
(p=i^php hp+i *i ( (fa - f&JJ

making this substitution, and omitting terms which vanish in con-
sequence of the conditions already satisfied, the condition for absence
of distortion becomes

This does not involve the position of the diaphragm, so is the
required condition for absence of distortion with full pencils.
If we denote

we see, on collating the results of the preceding articles, that the
condition for absence of

spherical aberration is 2®i = 0,

coma ,, 2®iUi=Q,

astigmatism „ S®<£7ai = 0,

curvature of field „ 2 - ( --- ) = 0,

rt \fr fr-J

distortion „ ^ 1®^^ +-(--—} uA = 0.

rAf< K-i/

In each case it is assumed that the conditions occurring previously in
the list are fulfilled.

It is however to be remembered that all these conditions have been
derived on the supposition that terms of orders higher than the third
in the angular aperture and angular field of view can be neglected :
when the field of view is large, as in the case of photographic objectives,
or when the angular aperture of the pencils is large, as in the case of
microscope objectives, terms of higher order must be taken into account.

28, 29] HERSCHEL'S CONDITION 45

29. Herschel's condition.

Sir John Herschel formulated the condition which must be satisfied
in order that an instrument, which is free from spherical aberration for
the standard position of the object, may also be free from spherical
aberration for positions of the object indefinitely near to this, i.e. that
a slight displacement of the object along the axis may not introduce
spherical aberration.

It was shewn by Abbe that this condition can be expressed in
a form which is applicable to instruments of any aperture however
large, just as the Fraunhofer condition for absence of coma can be
extended in the form of the sine-condition. We shall first establish
Abbe's condition, and then deduce Herschel's condition by supposing
that the fifth power of the angular aperture can be neglected.

The condition in question, viz. that spherical aberration shall vanish
for a second position of the object, adjacent to the one for which it is
already known to vanish, is evidently equivalent to the condition that
the magnification of a small segment of the axis, situated at the position
of the object, may be the same whatever zone of the refracting surfaces
is used to form the image. Let / be the length of this segment, I' the
length of its image, //, and p! the refractive indices of the initial and
final media. Suppose that the image is formed by a thin meridian
pencil whose chief ray makes an angle 0 with the axis in the initial
medium, and makes an angle 6' with the axis in the final medium.
Applying Clausius' theorem (§ 7), we have

fi fi'

Integrating this, pi sin2 - = ^7' sin2 — ,

2 2

the constant of integration vanishing since 0 and 0' vanish together.

Now the general equations of image-formation by paraxial pencils,
namely (§ 8),

*=-*'/', y'=A,

It. a;' J x'
give &

so if m denotes the linear magnification of a small object at right angles
to the axis, we have

I' u' 2

7 = — m\
I /A

46 HERSCHEL'S CONDITION

Substituting in the preceding equation, we have
/x'2m2 sin2 \6' = /x2 sin2 \6
sinAfl' /A

[CH. II

sin

This is Abbe's condition : it is obviously impossible to satisfy it and the
sine-condition simultaneously, save in exceptional cases.

We shall now proceed to derive Herschel's condition from this.

At the refraction at the ^th surface,
we have p/

sm"

-JM

1 +

NO*
\PN* 3 PN*

and

IPN

/
V1

\
8 NO*) '

in^ NO

or, in our usual notation,

sin |0t-
sin J#i-i

V 1 3 k2
^^-R1 8?

, A h^ 3V
*/ + *<--- i--_i

If Abbe's condition is satisfied, we must therefore have

? ( I,-' */ 2riSi + 2r4V 8 ** + 8 if) ~ °'

.A^.j A4 Vn A 1 _1\\ iv^/1 1
or S -1 -7 +• -S" fti I —; rJj - g ?^ l^a ~ ^

since hi/si=0i_1.

29, 30] KLEIN'S THEOREM 47

The summation occurring here is the same as that occurring in the
derivation of Fraunhofer's condition from the sine-condition : so the
equation can at once be written in the form

2 fji + <M2 21 -YT-} QM (—, — — Y] - \ (V - V) = o.

i L( p=iPPhphp+l) \fiiSi Hi-iSi/J 4V

This is HerscheCs condition. It is evidently compatible with the
Fraunhofer condition only when 00 = + 6n ; this happens either when
the object is at a point for which the angular magnification is ± 1, or
when 00 and 6n are both zero, i.e. when the system is telescopic and
the object at infinity.

30. The impossibility of a perfect optical instrument.

Although it is possible to construct lens-systems satisfying the
conditions which have been found, and therefore giving a satisfactory
image for some definite position of the object when the aperture and
field of view are not too large, we shall now shew that it is theoretically
impossible to construct a really perfect optical instrument, i.e. one
which will transform all points of the object-space into points of the
image-space with some degree of magnification or minimisation. The
proof is due to Klein*.

Suppose for the moment that such a perfect instrument exists.
Since not only are points transformed into points, but lines (rays of
light) are transformed into lines, the transformation of the object-space
effected by the instrument is a collineation.

Now it is known that all the spheres of space have in common an
imaginary circle at infinity, which contains the cyclic points of all the
planes of the space t; an (imaginary) straight line which meets the
circle at infinity is called a minimal line. Suppose then that the ray
incident on one of the refracting surfaces of the instrument is a minimal
line : the sine of the angle formed with the normal to the surface is
infinitely great, and as conversely a minimal line is characterised by
this infinitely large sine, it follows from the law of refraction that the
refracted ray is also a minimal line.

This applies to each refraction; and therefore the collineation
transforms each minimal line in the object-space into a minimal line in
the image-space ; so that the circle at infinity in the object-space is
transformed into the circle at infinity in the image-space.

* Zeitschriftfiir Math. u. Phys. XLVI. (1901), 376.

f For two similar and similarly situated quadrics intersect in one plane curve at
a finite distance and one at infinity : and spheres are similar quadrics.

48 KLEIN'S THEOREM [CH. n

From this it follows at once that the collineation is merely
a similitude : it may be either direct or inverse (i.e. one which
interchanges right and left).

In order to find the ratio of the similitude, suppose that c, c'
denote the velocity of light in the object-space and image-space
respectively. We can suppose that the similitude is direct, as if
inverse it can be changed into a direct similitude by the addition of
a plane mirror to the instrument.

Let the time taken by the light to travel from a point (x, y, z) to
its image-point (x, y z) be denoted by X(x, y, z). Let (#,, ylt z^) be
a point on one of the rays from (X y, z) to (of, y', z'\ at a distance r
from (x, y, z); and let (#2, y^ #2) be a point on another ray from
(X y, z) to (V, y', z\ also at a distance r from (X y, z). Then if A.
denote the ratio of similitude of the image-space and object-space, the
distances of the image-points (#/, #/, ^/) and (#2', y.2', z2') from (x, y', z'}
are each \r, and (since the similitude is direct) they are each behind
(i.e. beyond) (a?', y, z). The time from (a^, ylt Zi) to its image is
therefore

•p-/ ^ r \r

x(*,y>*)--c + j\

and the time from (,r2, 3/2? ^2) to its image is the same. So the times
from (X, i/!, ^) and (#2> 3/25 #2) to their respective images are the
same : but these are really arbitrary points in the object-space, so the
time from any point in the object-space to its image is the same for all
object-points. Hence we have

X(x, y,z)-- + 4- = X(x, y, z)

or = -,

so the dimensions of the object-space are to those of the image-space
as c to c. Thus when the instrument works in air, so that c = c, the
image is merely a life-size copy of either the object, or of the image
obtained from the object by reflexion in a plane mirror.

31. Removal of the primary spectrum.

As already explained, the refractive index of a substance depends
on the colour, i.e. the wave-length, of the light used in its determination.
The behaviour of an optical system, which has been calculated in terms
of the refractive indices, is therefore different for light of different

30, 31] PRIMARY AND SECONDARY SPECTRUM 49

colours : the position of the principal foci, the focal lengths, and the
aberrations, will in general vary when the wave-length of the light is
varied. As ordinary white light contains rays of all colours, there will
therefore be a certain degree of confusion in the images formed by
the optical instrument with white light: to this the name chromatic
aberration is given. With a simple uncorrected lens of tolerably small
aperture, the chromatic aberration is much more serious than the
spherical aberration ; with a convex lens of crown glass, if the red rays
from a star are brought to a focus at a point H, the violet rays will
intersect the plane through R perpendicular to the axis in a circle
whose radius is about -^ that of the lens, whatever be the focal
length.

An optical system which is so contrived as to have the same
behaviour for two standard wave-lengths is said to be achromatic. In
order to achieve this, we must evidently secure that the row of images
of the same object in light of different colours shall be doubled on
itself, so that the images shall coincide in pairs : thus in an ordinary
achromatic lens which is intended for visual observations, the yellow
image is united with the dark green image, the orange-red with the
blue, and the red with the indigo. Obviously at one end of this
doubled row there must be two coincident images which differ
infinitesimably in wave-length, i.e. there will be an image for which the
rate of change of position with change of wave-length is zero : thus in
the achromatic lens just mentioned, the images formed by the
yellowish-green rays are closely united and focussed at minimum
distance from the lens.

This pairing of images does not ensure an entire absence of
chromatic aberration, since the images in three different colours will
not coincide : but other terms, which will be mentioned later, are
employed to denote a more complete freedom from colour troubles.
The coloured fringes due to this outstanding colour-aberration are
generally referred to as the secondary spectrum ; a simple method (due
to Sir G. Stokes) of observing the secondary spectrum of a lens is
the following. Focus the lens on a vertical white line on a dark
ground, and cover half the lens by a screen whose edge is vertical.
Then evidently the yellow and green rays, which form an image nearer
the lens than the mean image, will (coming from the uncovered half
of the lens only) pass the mean image on one side of it, namely the
side on which the screen is : while the red and blue rays, which form
an image beyond the mean image, will pass on the other side of the
w. 4

50 THE ACHROMATIC LENS [CH. II

mean image. The image will therefore have a citron-coloured margin
on one side and a purple margin on the other.

32. Achromatism of the focal length.

The variation of behaviour of a transparent substance for light of
different wave-lengths is usually measured by its dispersion or dispersive
power,

where /x is its refractive index for some standard wave-length and
/x + c?/x is its index for some other standard wave-length not far removed
from this.

Consider now the colour-variation of focal length of a single thin
lens, for which we have (§11)

Differentiating this equation logarithmically, we have

Af
7--m.

The focal length of a compound lens consisting of two thin lenses
in contact, of focal lengths/! and/2, is the reciprocal of

l//i+l//2:
so if the compound lens is to be achromatic, we must have

or + o,

/I /2

where ta^ and or2 denote the dispersive powers. This equation repre-
sents the condition that the focal length, and consequently also in this
case the position of the principal foci, may be the same for the two
standard colours. The combination is therefore achromatic for all
distances of the object.

The above equation shews that one of the lenses (say (1)) must be
convergent and the other (say (2)) divergent : if the focal length of the
whole is to be positive, we must have/i < — fz, and consequently •&1<<sFZt
so the divergent lens must have the greater dispersion. As flint glass
has a greater dispersion than crown, the convergent lens is taken to be
a crown and the divergent lens a flint. Roughly speaking, a flint

31, 32] ACHROMATISM OF THE FOCAL LENGTH 51

whose diverging power is 2, will achromatise a crown whose converging
power is 3, leaving a converging power of 1 for the compound lens.
The Petzval condition for flatness of field (§ 27),

/*! /I + /*2/2 = 0,

requires however that ju,2 should be less than ^ ; so the convergent
lens should have the higher refractive index, though having the
smaller dispersive power, a condition which it was impossible to fulfil
until the Jena glasses were introduced.

Consider next a system consisting of two thin lenses separated by
an interval a. The focal length of the combination is (§11) the

reciprocal of

1 1 a

/1+/2"A72)

so if the focal length is to be achromatised we must have
_ dfi df2 Q*dfi cudfi

~ fz ~~ 1*2! ~*~ flf "*~ f /2>
/I /2 /]/2 /1/2

torj ora a (X
0 = - + -

i \

/i/i

In such a system the two lenses would usually be of the same kind
of glass, in order that whatever degree of achromatism is attained for
two colours may as far as possible be attained for all colours : taking
therefore ^ equal to zsra, we have

11 2a

77 ~7~7>

/i /a /I/a

or a = J (/ +/2),

so the distance between the lenses must be half the sum of their focal
lengths. This condition is applied in the construction of eyepieces.

It is to be observed that the positions of the principal foci of the
combination have not been achromatised, so that we have achromatised
the size but not the position of the image. It is in fact impossible to
achromatise a system of two non-achromatic lenses separated by
a finite interval for both the size and position of the image : for if it
were possible, the intermediate image, which is at the point where the
line joining the object-point (supposed slightly off the axis) to the
centre of the first lens intersects the line joining the image-point to
the centre of the second lens, would be the same for every colour, and
therefore each lens separately would be achromatic.

4—2

52 APOCHROMATISM [CH. II

33. The higher chromatic corrections.

It is possible to remove the secondary spectrum, or more strictly
speaking to replace it by a " tertiary " spectrum, by uniting the images
in three instead of two colours. This can be effected for a combination
of two lenses provided the Jena glasses are available : with the older
glasses three lenses are required.

The variation of spherical aberration with the colour must also be
taken into account. In the ordinary telescope objective, the citron
image is corrected for spherical aberration, so the red image is under-
corrected and the blue image is over-corrected : this defect is, in the
case of the visual telescope, masked by the secondary spectrum : but
with objectives of large angular aperture and short focal length, e.g.
high-power microscope objectives, the correction of chromatic difference
of spherical aberration is of greater importance than the elimination of
the secondary spectrum.

Optical systems in which the spherical aberration is corrected for
more than one colour, but in which the secondary spectrum is not
removed, are called semi-apochromatic ; while systems which have no
secondary spectrum and are aplanatic (§ 24) for more than one colour
are called apochromatic.

34. The resolving power of a telescope objective.

Nothing in our investigations hitherto has suggested the existence
of any limit to the magnification attainable by means of an optical
instrument; and it might therefore appear as if it were possible to
construct a telescope of moderate dimensions which should reveal the
minutest details of structure on the heavenly bodies. As a matter
of fact, it is not possible, or at all events not profitable, to apply
a magnifying power greater than a certain amount to a telescope with
a given objective : and the reason for this is to be found in the
circumstance that the wave-front by which the image of a star is formed
is not a complete sphere, but is merely that fragment of a spherical
wave which has been able to pass through the rim of the objective.
This mutilated wave-front does not converge exactly to a point, as
a full spherical wave would do, but forms a diffraction pattern in
the focal plane of the objective, consisting of a bright disc whose
centre is the image-point of the star as found by the preceding theory,
surrounded by a number of dark and bright rings concentric with it

33, 34] RESOLVING POWER 53

In order to determine the dimensions of this pattern, let a denote
the diameter AB of the telescope objective,
and S the centre of the diffraction pattern.
The disturbance which is brought to a focus A-
at a point T in the focal plane is the
disturbance which at some preceding instant
occupied the plane COD, perpendicular to
the line OT which joins T to the centre 0

of the object-glass. Let SOT =0, and let
(p, </>) denote the polar coordinates of a
point in the plane COD referred to 0 as
origin and the line of greatest slope to the
plane A OB as initial line.

The disturbance at the point (p, <£) is proportional to

pdpd<f> . sin 2?r ( — — J ,

where t denotes the time, T the period of the light, X its wave-length,
and z the perpendicular distance of the point (p, </>) from the plane
A OB.

But z = pO cos <£,

0 being regarded as a small quantity.

The total disturbance at T is therefore

* J JJL • n ft p0COS<f>\

pdpd<f>.sm2Tr(- — — - J,

\T >V /

integrated over the circle COD,

, . 2-n-pO COS <t>

or sm — I pdpdd> . cos —

/Y
JJ

. 2lT* ft" P"

sm — / I

r Jo Jo

since the elements of the integral involving sin — — r cancel each

other in pairs.

Expanding the cosine in ascending powers of its argument, and
integrating term by term, this becomes

{ L

where m denotes -rrOa/2\*.

* The series in brackets is a well-known Bessel-function expansion, being in
fact ^/, (2m). Of. Whittaker, Modern Analysis, p. 267.

54 DAWES' KULE [CH. II

The first dark ring in the diffraction pattern will occur at the first
point T for which the disturbance vanishes, i.e. it will correspond to the
lowest value of m which makes the series in brackets to vanish : this is
found by successive approximation to be

f» = l-92,

giving 0=T22-.

The radius of the central diffraction-disc of a star (measured to the
first dark ring) formed in the focal plane of a telescope objective of
aperture a and focal length f is therefore l'22\f/a.

A telescope is usually estimated to succeed in dividing a close
double star when the centre of the diffraction-pattern of one 'star falls
on the first dark ring of the diffraction -pattern of the other star : when
this is the case, it follows from the preceding equation that the angular
distance between the stars in seconds of arc is 1*22 x 206265 x A/a.
If we express a in inches, and take A =1/50,000, this gives for the
angular distance between the stars,

5"
aperture in inches "

This is known as Dawes rule for the resolving power of a telescope
objective.

35. The resolving power of spectroscopes.

The power of spectroscopic apparatus (prisms or gratings) to
separate close spectral lines involves the same principles as the power
of telescope objectives to separate the components of double stars.
Each spectral line is really a diffraction-pattern, consisting of a narrow
bright band at the place of the geometrical image of the line, flanked
by alternate bright and dark bands : and the spectroscope is said to
resolve two lines of adjacent wave-lengths when the centre of the
central bright band arising from one wave-length falls on the first dark
band of the pattern arising from the other wave-length.

The difference between the telescopic and the spectroscopic cases is
that in the telescope we are dealing with the circular diffraction-pattern
of a point-source, formed by a circular beam, while in the spectroscope
we are dealing with the banded pattern of a line-source, formed by
a beam of rectangular cross-section. The latter case is analytically the
simpler of the two, since all sections of the beam at right angles to the
spectral lines are similar to each other, and the problem can therefore
be treated as a two-dimensional one.

34, 35] RESOLVING POWER OF SPECTROSCOPES 55

Let AB be the wave-front, limited by an aperture AB of breadth a,
of a pencil of parallel light of wave-length X,
representing one of two vibrations which are
just resolved. Draw BC at right angles to
AB, and take BC equal to X. Then the first
dark band of the diffraction-pattern corre-

sponding to the disturbance AB will fall at the place to which a
wave-front occupying the position A C is brought to focus : for the
phase of the ^^-disturbance at C differs by a whole wave-length from
the phase at B, i.e. from the phase at A, and consequently every point
in A C will have a corresponding point in the other half of A C which
is in exactly the opposite phase, and so will interfere with it to produce
total darkness.

Thus the disturbances represented by AB and AC will be just
resolved : so if W denote the angle between two wave-fronts of approxi-
mate wave-length X, the disturbances will be just resolvable when the
beams are of breadth a, provided that

As the product of the inclination of two plane wave-fronts and their
diameter is, by Helmholtz's theorem, unaltered by passage through any
system of lenses, it is evident that the resolvability of two adjacent
disturbances is not altered by passage through any lens-system which
.does not introduce new diaphragm limitations, and so depends solely
on the prisms or grating.

The resolving power of a spectroscope is defined by Lord Rayleigh
to be X/SX, when two spectral lines of wave-lengths X and X + SX
respectively can just be resolved, the slit of the spectroscope being
infinitely narrow. But the result obtained above gives

Thus the resolving power of any grating or train of prisms is
measured by the product of the breadth of the emergent beam of parallel
light and the dispersion ; the dispersion being defined as the rate of
change of deviation with wave-length*.

* The resolving power can be regarded from a different point of view as equal
to the number of separate pulses into which a single incident pulse of light is
broken up by the spectroscope. For references to this aspect of the theory, of. a
paper by the author in Monthly Notices of the Royal Astron. Society, LXVII. p. 88.

CHAPTER III.

SKETCH OF THE CHIEF OPTICAL INSTRUMENTS.

36. The photographic objective.

The simplest form of photographic objective is a single convergent
lens*; the light from an object at some distance is rendered convergent
by the lens, and the real image thus formed is received on a gelatine
film containing emulsified bromide of silver : this salt is acted on by
light, and after undergoing the processes of development and fixation
yields a permanent image in metallic silver.

The rapidity of action of the lens depends only on its aperture-
ratio, which is the ratio of its focal length / to its diameter : if the
diameter be f/n, the time of exposure required is proportional to n2 ;
for the exposure is inversely proportional to the light falling on unit
area of the image, and is therefore proportional to the area of the
image divided by the total light received by the lens from the object:
but the area of the image is proportional to /*, and the total light
received is proportional to the area of the lens, i.e. to (fjnf: so the
time of exposure is proportional to n2.

This theorem applies equally to objectives which are not constituted
of a single lens, provided that instead of the diameter of the lens we
take the diameter of the entrance-pupil. For an average photographic
objective, n is about 7 when the full aperture is used: for portrait
lenses, which are very rapid, n may be as low as 3.

* If a single non-achromatic convergent lens were used, it would be best to
select a deep meniscus with its concavity turned towards the object (this secures
considerable depth of focus and a large field of fair definition) and to use a narrow
stop in front (which reduces the spherical aberration and curvature of image) : when
the focus is being obtained, a weak convex lens must be inserted, so as to reduce
the visual focal plane to the place which the focal plane of the actinic rays occupies
when this lens is absent.

36] THE PHOTOGRAPHIC OBJECTIVE 57

The single convergent lens is practically useless, on account of the
defects which have been discussed in the preceding chapters ; and it is
necessary to design objectives formed of more than one lens, with a
view to the special requirements of terrestrial photography, which are
the following :

1. The system must be achromatised in such a way that the visual
image, which is used in finding the focus, may coincide with the
actinic image which acts on the sensitive plate : the D line of radiation
of sodium is generally united with the blue Hp radiation.

2. The definition should be such that points of the object are
represented by dots of (say) not more than T^ of an inch diameter,
over a field of (say) 50° square : in the case of portrait lenses, this
requirement is sacrificed in order to obtain the greatest possible
rapidity: a portrait lens will not usually cover a greater field than
about 25° square. In any case, the standard of definition is much
lower than is demanded of telescope objectives, but the field is much
wider. The definition is usually improved by stopping down,
i.e. narrowing the aperture of the diaphragm : but this involves a
loss of rapidity.

3. Distortion must as far as possible be eliminated : objectives
consisting of lenses cemented together, with the stop in front, always
give barrel distortion (§ 28), while if the stop is between the lens and
the image there is pincushion distortion. If we combine these two
systems into a doublet, i.e. a system of two compound lenses separated
by an interval in which the stop is placed, the two opposite distortions
neutralise each other and we obtain an objective which is rectilinear,
i.e. free from distortion.

4. The objective must have a certain amount of depth of focus,
i.e. must be able to give fairly sharp images of objects which are in
front of or behind that object-plane which is accurately focussed.
Depth of focus is usually measured by the range of object-distance for
which the pencil meets the sensitive plate in a disc of less than a
certain diameter : it depends on the object-distance, focal length, and
aperture, but does not vary much with the type of lens. The depth of
focus is obviously increased by stopping down, since then all pencils
become narrower : with equal aperture-ratios, the depth is greater for
small focal lengths than for large ones.

5. Among minor requirements may be mentioned freedom from
flare, i.e. from light which has been reflected at some of the refracting

surfaces, and which on reaching the sensitive plate interferes with the
brilliancy of the image.

58 TELEPHOTOGRAPHY [CH. Ill

It is of course not possible here to enter into details regarding the
construction of the various types of photographic objective which are
at present on the market.

37. Telephotography.

When an object at a great distance is photographed with an
ordinary photographic objective, the image is inconveniently small and
the details difficult to distinguish. A more convenient image can be
obtained by making an enlargement from this photograph : but owing
to the grain of the sensitive plate, and the insufficient definition of the
primary image, it is not practicable to enlarge many diameters. It is
therefore desirable to obtain a primary image as large as possible.
Now in order to obtain a large-scale image, the camera must have an
objective of great focal length : and as with most objectives the length
of the camera is nearly equal to the focal length, this requires an
inconvenient or impossible extension of the camera. The difficulty is
surmounted by removing the principal point of the system (which is at
a distance from the principal focus equal to the focal length, and is
generally near the objective) to a considerable distance in front of the
objective, so that although the focal length is great, the distance from
the objective to the sensitive plate is comparatively small. This is
effected in telephotography, in which a divergent lens is introduced
between the convergent objective and the sensitive plate : this divergent
lens diminishes the convergence of the pencils which fall on it from
the convergent combination, so that they become practically the same
as the pencils which would have proceeded from a convergent lens of
great focal length, placed at a considerable distance in front of the
actual position of the objective.

38. The telescope objective.

The conditions which must be satisfied by the objectives used in
astronomical telescopes, whether visual or photographic, differ greatly
from the conditions which must be satisfied by the objectives used in
terrestrial photography. In the latter, definition which will bear a
feeble magnification is required over a field of (say) 50° square : in the
former, definition which will bear a much higher magnification is
required, but over a much smaller field : the field seen at one time in
a large visual telescope is only about J° in diameter, and the region
depicted on the sensitive plate of a photographic telescope is usually
only of about the order of magnitude of a square degree. Consequently

36-38] TELESCOPE OBJECTIVES 59

the defects of astigmatism, curvature of field, and distortion, which
come into prominence at the outer parts of a wide field, are much less
important in celestial than in terrestrial work : while on the other
hand the defects of spherical aberration and coma, which affect the
central parts of the field, must be more carefully eliminated in the
astronomical objective than in the ordinary photographic lens. More-
over, since any diminution in light-gathering power is to be avoided at
all costs in astronomy, it is not permissible to correct errors by means
of diaphragm effects. For these reasons the doublet, which is pre-
dominant in terrestrial photography, is abandoned by astronomers in
favour of an objective consisting of two or three lenses fairly close
together, designed to make the corrections for spherical aberration and
coma as perfect as possible.

The colour corrections also differ in the two cases. In the
terrestrial lens, the actinic image must be made to coincide in position
with the visual image which is used in focussing: but as in astro-
photographic work the focus is found by taking trial plates, there is no
need to trouble about the visual rays, and consequently the colour
correction can be devoted wholly to the improvement of the actinic
image, the blue Up radiation being generally united with a violet
radiation emitted by mercury. In the visual telescope there is no need
to take account of the actinic image, and the yellowish-green rays are
brought to the minimum focus.

We shall now shew how the equations found in Chapter II can be
applied to design what may be called a Fraunhofer objective : this will
be defined as a telescope objective consisting of two lenses whose
thickness will be neglected, in contact at their vertices, and having their
four radii of curvature chosen to satisfy the following conditions

(i) Given focal length F for the objective as a whole,
(ii) Achromatism,

(iii) Absence of spherical aberration for an object at infinity,
neglecting the 5th power of the aperture,

(iv) The sine-condition for an object at infinity, neglecting the
5th power of the aperture.

Let rlt r.2, r3, r4 denote the radii of the refracting surfaces in order
(all taken positively when convex to the incident light), /*, ^ the
refractive indices of the lenses, wj, vr.2 their dispersions for the radiation
which it is desired to have at minimum focus, /! and /2 their focal
lengths.

60 THE FRAUNHOFER OBJECTIVE [CH. Ill

Conditions (i) and (ii) may be written

I-L+I ^ + ^!-o

PVl /*' /I /•"

Thus if K denote Wi/wa, we have

1 1 1 K

and therefore

, __

' 7-3 r4~ -F

These equations are satisfied identically if we write

where /?i and j92 are now to be determined from conditions (iii) and (iv).

These conditions (iii) and (iv) are
(Spherical aberration condition, § 20)

(Sine-condition, § 25)

Q4u4 = 0,

11 /i i\

where Ql = --- =ft( --- ),

n ak r\r, W

and similar equations hold for Qs and Q4: a?j, a?a, #3, ^4, ^5, being the
distances of the object and its successive images from the objective;
and where

11 11 11

U2 = --- , WQ = — ; --- , W4 =

XZ /^2' ft^4 #3 ^5

Now since the object is at infinity, we have

and consequently

38, 39] EYEPIECES 61

Substituting in terms of PI and jt?2, we have (neglecting common
factors)

CM— 1)», ft' — 1 ft' — 1

= *— I^3, «a=l-«i, U* = ^rP*-^T> Ut=-

Substituting in the conditions (iii) and (iv) above, we have

M + 1 //.' 4- 1 /A ft'**

and -/?! - K ^— r- jt?2- ^— - + K--4-— = 0.

ft //. /X-l /* -1

The second equation gives j»2 as a linear function of pl ; substituting
in the preceding equation, we have a quadratic for plt which can be
solved : the radii of curvature of the surfaces of the objective are thus
determined.

39. Magnifying glasses and eyepieces.

For the rough examination of small objects, the magnifying glass
is used. This in its simplest form consists of a single convergent lens,
held between the eye and the object, at a distance from the latter
somewhat less than its own focal length : an enlarged virtual image is
thus formed at some distance behind the object, and this is examined
by the eye. The pupil of the eye is the diaphragm effective in limiting
the aperture of the image-forming pencils, and the rim of the lens
(supposing it to be of greater diameter than the pupil) is the diaphragm
effective in limiting the field of view.

Closely allied to the magnifying glasses are the eyepieces which are
used to examine the images formed by the objectives of visual
telescopes and microscopes. These consist usually of two lenses
separated by an interval : the lens which is nearest the eye is called
the eye-lens, and the other the field-lens.

In Huyghens eyepiece the lenses are placed at a distance apart
equal to half the sum of their focal lengths, in order to satisfy the
condition of achromatism found in § 32. The focal length of the
field-lens is usually three times that of the eye-lens, but in some

62 EYEPIECES [CH. Ill

modern eyepieces, especially those used for low-power magnification
with the microscope, the ratio of the focal lengths is smaller than this.
The lenses used are plano-convex, with the convex sides turned
towards the image to be examined.

The first principal focus of Huyghens' eyepiece falls between the
lenses, and consequently the image to be examined (which must be
placed at this point in order that the emergent wave-fronts may be
plane) can only be a virtual image : in other words, a Huyghens'
eyepiece, when used with a telescope objective, must be pushed in nearer
to the objective than the place at which the objective would form
a real image of the object. On this account the Huyghens construction
cannot be used in micrometer eyepieces, in which it is desired to place
a framework of spider-lines in the plane of the image formed by the
objective, and to examine the spider-lines and the image together by
the eyepiece.

The image formed by high-power apochromatic microscope objectives
is usually examined by a compensating eyepiece, which is specially
corrected chromatically in order to neutralise the chromatic difference
of magnification due to the objective.

In Ramsderis construction, which is always used in micrometer
eyepieces, the first focal plane of the combination does not fall between
the lenses, and the eyepiece can consequently be used in order to
simultaneously examine the image (formed by the objective of the
telescope or microscope) and also a reticle of spider-lines, placed in its
plane with a view to micrometric measurements. In this construction
the two lenses are usually plano-convex with the convex sides turned
towards each other : they are taken to be of the same focal length, and
therefore if the condition of achromatism were satisfied the interval
between the lenses would be exactly equal to this focal length : with
this arrangement however the field lens would be exactly in the focus
of the eye-lens, which is undesirable ; and the interval is consequently
taken to be shorter than the focal length, the resulting chromatic
error being (in the best eyepieces) removed by substituting achromatic
combinations for the simple field-lens and eye-lens.

The field-lens is so near to the real image examined, that its
principal effect is to deflect the chief rays of the pencils towards the
axis of the instrument, without greatly altering the inclination of the
other rays of the pencils to the chief rays : the function of magnifying
is therefore performed almost wholly by the eye-lens.

39, 40] THE VISUAL ASTRONOMICAL REFRACTOR 63

40. The visual astronomical refractor.

The astronomical refracting telescope, as used visually, is formed
by the combination of an astronomical objective (§ 38) with an eyepiece
(§ 39) which is used to examine the image formed by it. In the typical
normal case the eyepiece is so placed that its first focal plane coincides
with the second focal plane of the objective : under these circumstances
the parallel pencil of light from a star is made to converge to an image
situated in this focal plane, and is re-converted into a parallel pencil
by the eyepiece. Short-sighted observers find it convenient to push
the eyepiece nearer to the objective, so that the emergent pencils are
divergent.

The diaphragm effective in limiting the apertures of the pencils is
the rim of the objective : this is therefore the entrance-pupil (§ 16).
The exit-pupil, which is the image of the objective formed by the
eyepiece, is outside the instrument and behind it, and the eye is placed
there. The field of view is generally limited by a diaphragm placed in
the focal plane of the objective : if this were not present, the field
would be limited by the rim of one of the lenses of the eyepiece, and
there would be a " ragged edge" of the field seen only by partial
pencils. The field of view is of course the angle subtended at the
centre of the objective by this diaphragm.

The magnifying power (§ 17) is readily seen to be the ratio of the
focal lengths of the objective and eyepiece. A telescope is usually
furnished with a battery of eyepieces, giving various magnifications.

When the eyepiece is of such short focal length that the magnifying
power of the telescope is greater than a number which may be roundly
stated as equal to the diameter of the objective in millimetres, the
definition is spoiled by the diffraction effects discussed in § 34 : from
this to one-half of it may be regarded as the useful range of magnifying
power, since below this limit the capabilities of the objective are not
being used to their full extent. This corresponds to an exit-pupil of
1 to 2 mm., which is much smaller than the pupil of the eye.

If the object viewed is a star, which may be regarded as a
mathematical point, the brilliancy varies directly as the light gathered
by the objective, i.e. as the square of the aperture, and is independent
of the focal length. The same consideration applies to the rapidity of
an astro-photographic objective.

The aperture-ratio (§ 36) of a telescope objective is usually about
15 : but for small telescopes it is frequently smaller. In the old
telescopes, which were constructed before the discovery of achromatic

THE ASTRONOMICAL REFLECTOR

[CH. Ill

combinations, the aperture-ratio was very large : this was in order to
take advantage of the fact that the influence of chromatic aberration
on the distinctness of an object is inversely proportional to the
aperture-ratio.

41. The astronomical reflector.

In the astronomical reflecting telescope, the light from a celestial
object is received on a concave mirror, which serves the same purpose
as the objective of a refracting telescope, namely to form a real image
of the object in its own focal plane. This image can either be allowed
to impress itself directly on a sensitive plate, or may be examined by
an eyepiece. In the latter case, it is necessary to insert a small plane

Fig. a.

Fig. &.

mirror obliquely in the path of the rays after leaving the large mirror,
in order to divert them to the side of the telescope, where the image is
formed and examined : otherwise the head of the observer would obstruct
the passage of the incident light to the large mirror. This con-
struction is known as the Newtonian reflector (Fig. a) : the path of the
rays from a star P to its real image p will be obvious from the diagram,
Q being the large mirror and R the flat. The magnifying power, as
in the case of the refractor, is the ratio of the focal lengths of the
objective and eyepiece.

In certain cases, e.g. the photography of planets, it is desirable to
obtain on the sensitive plate an image on a larger scale than would be
furnished directly by the concave mirror : this is achieved by making
use of what is essentially the same principle as that on which
telephotography (§ 37) is based, namely receiving the rays from the
large mirror on a small divergent (i.e. convex) mirror before allowing
them to form a real image. This is known as Cassegrain's con-

40-42]

THE FIELD GLASS

structiou (Fig. 6). The path of the rays from the star P to its real
image p, after reflexion at the large mirror Q, the convex mirror R, and
the flat S, will be obvious from the diagram.

The diaphragm effective in limiting the aperture of the image-
forming pencils of a reflector is the rim of the large mirror. The field
of view of a visual reflector is limited by the rim of one of the eyepiece
lenses, or by a diaphragm placed in the plane of the real image in
order to exclude the part of the image formed by partial pencils.

The correction for spherical aberration of the large mirror is effected
by figuring it to a paraboloidal form : as we have seen however (§ 24)
this does not remove coma, which is accordingly an outstanding defect
in all reflecting telescopes.

The reflector is of course perfectly free from chromatic aberration,
and this involves a further advantage over the refractor in that it
permits the construction of reflectors having a much smaller aperture-
ratio than refractors, and consequently much greater rapidity for
objects with an extended area.

The aperture-ratio of the large mirror of a modern reflector is
usually about 5 : the addition of a convex mirror, which usually gives
about a threefold magnification, raises the aperture-ratio to about 15
in Cassegrain's construction.

For the above reasons, and also because it is easier to construct a
mirror than an objective of the same diameter, and therefore easier to
secure light-gathering power, the reflector is specially suited for the
photography of nebulae.

42. Field, Marine, and Opera Glasses.

The visual astronomical telescope cannot be applied to terrestrial
uses without modification, since the image which is formed by the
objective and examined by the eyepiece is inverted. It is possible
to surmount this difficulty by the use of an erecting eyepiece,
which is in principle similar to the microscope (§ 43), forming
a second (erect) real image in its
interior : but the instrument so con-
stituted is of considerable length and
cannot be supported steadily in the
hands without difficulty. Accordingly
field-glasses were until recent years
always formed of a convergent ob-
jective combined with a divergent

THE PRISM BINOCULAR

[CH. Ill

eyepiece : the rays after leaving the objective and before reaching the
plane of the real image were intercepted by the eyepiece, which
destroyed their convergence and rendered them parallel at emergence.
The path of the rays will be evident from the diagram.

Since no real image is formed in this construction, which is known
as the Galilean telescope, there is no inversion of the object. The
diaphragm effective in limiting the field of view is the rim of the
object-glass, and the diaphragm effective in limiting the aperture of the
pencils is the pupil of the eye. The magnifying power, as in other
telescopes, is the ratio of the focal lengths of the objective and
eyepiece.

The Galilean telescope has a much smaller field of view than
an astronomical telescope of the same magnifying power ; on this
account the best modern field-glasses have reverted to the astronomical
type of telescope, with a device suggested originally by Porro for
re-erecting the object and shortening the
tube-length of the telescope. This device,
which is represented in the annexed
diagram, is to interpose a prism in the
path of the light when it has travelled
some distance from the objective : the rays
fall normally on the hypotenuse face of
the prism, and after passing through the
glass to one of the other faces are totally
reflected, passing thence to the third face
where they are again totally reflected :
after this they travel through the glass
to the hypotenuse face again and emerge
normally from the prism. The effect of
the two total reflexions has been to reverse
the direction of the beam, so that the rays
are now travelling back towards the ob-
jective : after proceeding some distance in
this direction they are again intercepted by
a double-total-reflexion prism, with its
principal section at right angles to that of
the first prism : this once more reverses the
direction of the beam and sends it on to
the eyepiece, whence it passes into the eye.
A field-glass, formed of two telescopes of

42, 43] THE MICROSCOPE 67

this construction (one for each eye) is called a Prismatic Binocular :
the folding up of the path of the rays by the two reversals greatly
reduces the length of the instrument, and the total reflexions perform
the other necessary function of erecting the image. The magnifying
power of a Prism Binocular usually ranges from 6 to 12, and the field
ranges from 3° to 8° in diameter.

43. The Microscope.

The simple magnifying glass (§39) cannot advantageously be
constructed to give magnification above a certain limit, owing in part
to the excessive smallness of the lens which would be required for
a high magnification. In order to pass beyond this limit, we can
conceive an astronomical telescope placed immediately behind the
magnifying glass, so that the pencil from a point of the object off the
axis, after being converted by the magnifying glass into a pencil
of nearly parallel rays, passes through the telescope and thereby
increases its angle of divergence from the axis of the instrument. In
this way we attain a magnifying power which is roughly the product of
the magnifying powers of the magnifying glass and the telescope.

This arrangement is essentially a microscope, the combination of the
magnifying glass and telescope objective being called the objective
of the microscope, and the telescope eyepiece being the eyepiece of the
microscope. The object to be viewed is placed in front of the
microscope objective (which always consists of a combination of several
lenses, and has a very short focal length) at a distance from it slightly
greater than the focal length : a real enlarged image is consequently
formed by the objective and examined by an eyepiece.

The magnifying power of the entire instrument, which we have
defined in § 17 as the ratio of the linear dimensions of image and
object when the image is at the standard distance of distinct vision, is
readily found to be approximately equal to

Length of tube x Conventional distance of distinct vision
Focal length of objective x Focal length of eyepiece

A microscope objective must be designed to give the best possible
definition when a small field of view is seen by pencils of very wide
angular aperture : the incident cones of light have apertures as great as
150°. Consequently of the aberrations discussed in Chapter II, the
most important in the construction of microscope objectives are,
spherical aberration, coma (the sine-condition), and chromatic aberration.

68 THE MICROSCOPE [CH. Ill

The pencils are of such wide angle that spherical aberration must
be much more completely removed than would be the case by the
satisfaction of the approximate condition found in § 20 ; this further
spherical correction is usually known as " spherical zones." Moreover
the same circumstance — the wide angle of the pencils — causes the
chromatic variation of the spherical aberration (§ 33) to assume
serious proportions, and in all good objectives it is specially corrected.
In the best or apockromatic objectives (§ 33), the secondary spectrum
is also removed.

In high-power objectives, advantage is taken of the property of the
aplanatic points of the sphere discussed in § 24 ; the front lens of the
objective is a hemisphere with its plane face turned towards the object :
below this is a film of cedar-wood oil (whose refractive index, T51, is
practically the same as that of the hemisphere), separating the
objective from a cover-glass, usually 018 mm. thick, which protects the
object. In this way the object is virtually within a sphere whose
refractive index is that of the glass, and in fact is situated at the
internal aplanatic point of the sphere, a magnified image being formed
at the external aplanatic point.

An immersion objective (i.e. one in which the oil is used) collects
a wider cone of light from the object than a dry objective would do :
for if the cone of light on emerging from the cover-glass passes into air
(as happens with dry objectives), its rays are bent outwards by the
refraction, and consequently the outermost rays of the cone will pass
outside the rim of the objective ; in the immersion objective they are
not refracted on emergence from the cover-glass, and so pass on into
the objective.

We must now discuss the resolving power of the microscope. The
object will first be treated as if it were self-luminous, ignoring the
fact that it is actually seen by light directed on it from another
source.

Let the semi-vertical angle of the cone of light issuing from the
object to the objective be 0, and let the semi- vertical angle of the cone
forming the image be #': let /x denote the refractive index of the
cedar- wood oil, ^ being replaced by unity in the case of dry objectives.
The quantity ^ sin 0 is called the numerical aperture of the objective,
and is generally denoted by the letters N.A.*

The wave-front from the object, being limited by the rim of the

* It is approximately equal to the ratio of the radius of the back lens of the
objective to the focal length of the objective.

43] THE MICROSCOPE 69

objective, will form a diffraction-pattern at the image ; regarding the
objective as compounded of a magnifying-glass and a telescope-objective
in juxtaposition, we can apply the theorem of § 34, which at once shews
that the radius of the central diffraction-disc at the image is

1-22 A

2tan0"

where X is the wave-length of the light. If m denote the magnification,
it follows that the centre of the image of one object will fall exactly
on the first dark ring of the diffraction-pattern of a second object,
provided the distance apart of the objects is

0-61 X

Now the sine-condition gives the equation

**. =m. or N.A. =msmO',
sm0

and as sin 6' and tan 6' are practically equal (0' being a small angle), we
see that the distance apart of tivo objects which can just be resolved is

0-61 X

N.A. '

The best immersion objectives have a numerical aperture of 1*4 :
taking X='0005 mm., we see that two objects which can just be
resolved with these objectives will be approximately at a distance apart
equal to

0-61 x -0005

— — mm., or '00022 mm.

In this discussion we have however neglected one fact of importance,
namely that the object studied by the microscope is not truly self-
luminous, but is illuminated by another source of light. The importance
of this distinction was first shewn by Abbe, who observed that the
light incident from the source is diffracted by the object, and that in
order to obtain an image correctly representing the structure of the
object it is essential that the objective should receive the whole of
this diffraction-pattern. If this condition is not satisfied, the image
obtained will represent a fictitious object, such as would give rise to a
diffraction-pattern consisting of those parts of the actual diffraction-
pattern which are transmitted by the objective.

70 THE PRISM SPECTROSCOPE [CH. Ill

44. The Prism Spectroscope.

A spectroscope is an instrument designed for the work of analysing
any given composite radiation into its constituent simple radiations,
each with its own wave-length. In the prism spectroscope, this is done
by taking advantage of the fact that the refractive index of glass for any
kind of light depends on the wave-length of the light, and that conse-
quently radiations of different wave-lengths can be separated from each
other by causing them to pass through a glass prism, i.e. a piece of
glass bounded by two optically-plane faces inclined to each other.

If for example the light which it is desired to analyse is that
produced by the flame of a Bunsen burner, in which a salt of sodium is
volatilised, the usual practice is to throw an image of the flame (by
means of a convergent lens) on a narrow slit between two jaws of metal,
so that the opening of the slit is strongly illuminated by the yellow
light. This slit is placed in the focal plane of a telescope objective, so
that the sodium light which is able to pass between the jaws of the
slit travels on to the objective and is there converted into a parallel
beam. In this condition it is received on one face of a prism, and
passes through the glass and out at the other face ; the beam is then
received normally on another telescope objective, in the focal plane of
which two images of the slit are formed close together ; these images
correspond to two kinds of yellow radiation emitted by the sodium
flame, which have followed slightly different paths in the prism and
have thus become separated. Each kind of radiation emitted by the
original source of light will give rise in this way to a distinct final image
of the slit : these slit-images are called spectral lines, and collectively
form the spectrum of the source of light : they may be allowed to
impress themselves on a sensitive plate, or may be examined visually
with an eyepiece.

The slit and the first telescope objective are together called the
collimator : and the collimator, prisms (the light may pass through
more than one prism successively), and final telescope, constitute a
prism spectroscope.

We shall first find the dispersion produced by the train of prisms,
i.e. the differential effect of the prisms on two radiations of slightly
different wave-lengths. Suppose that the light consists of two kinds
of radiation, for one of which the refractive index is typified by p and
for the other by /* + 8/x : and let BO denote the angle between the two
emergent beams corresponding to these two kinds of radiation : we
shall now find 80*.

* The method is due to Lord Rayleigh.

44]

THE PRISM SPECTROSCOPE

Let PQ be a wave-front at incidence on the prism-train, P'Q' the
corresponding piece of a wave-front for
the radiation ^ after emergence from the
prisms into air. PP' and QQ' the paths
of the rays from P to P' and from Q to
Q' for this radiation, RQ' and SP' the
paths from the wave-front PQ to Q' and
P' for the light p + 8/z, and T the point
in which the path SP' meets the wave-
front of the light /A + 8/x through Q'.

Then

Difference of values of

I fa 4- S//,) e?s

taken along the paths /STand /SP', since along P'T7 we have /* + 8/x = 1.
Thus P'Q' . 3(9 = /" 0 + 8/x) ^ - [ (> + 3/x) <fe,

j .RQ' J SP'

the integral having the same values along the paths ST and RQ\ since
it is proportional to the time of propagation of the (/* + S/A) wave.

Now pds= I tids,

J E<? JQQ'

by the stationary property of fads (§ 3)

since the time of propagation of the /x wave is the same from any point
on PQ to the corresponding point on P'Q',

=/«."*

by the stationary property of fads.
Thus we have

P'Q'.W=\ Sp.ds-f *fy.ds

J RQ' JPP'

= I SfjL.ds— I fy.ds,

J QQ' J PP'

to our degree of approximation.

If the prisms are all formed of the same variety of glass, this
becomes

ds,

72 THE PRISM SPECTROSCOPE [CH. Ill

where the integration is now to be taken only over those portions of
the path which are inside the prisms, omitting the parts which are in
air. Thus if t denote the difference of the lengths of path travelled in
glass by the two sides of the beam, and if a denotes the breadth of the
emergent beam, the last equation can be written

Now if X and A. + 8/x denote the wave-lengths of the two radiations
//. and fji + Sfi, the resolving power of the spectroscope is (§ 35) a&0/$\.
Thus we have the result that the resolving power of a prism spectroscope
is

•*.

where t denotes the difference of tlw lengths oj path travelled in the glass
of the prisms by the two sides of the beam, and d^dX is the rate of change
of refractive index with wave-length.

In the most usual case, one side of the beam passes through the
refracting edges of the prisms, i.e. it does not travel any distance at all
in the glass : and t then denotes practically the total length of those
sides of the prisms which are opposite the refracting edges. Roughly
speaking, one cm. of glass is required in order to resolve the yellow
light of sodium into its two component radiations. It must, however,
be remembered that the formula has been derived on the assumption
that the slit is infinitely narrow : the small though measurable breadth
of the slit diminishes the power of resolution.

In the early prism spectroscopes it was customary to use a large
number of small prisms — often 12 or more — in order to obtain a high
resolving power. The same end is now better attained by using a
smaller number of prisms — generally not more than four — of much
larger size. In the older arrangement the large dispersion caused a great
separation of the different coloured beams even before their passage
through the last prisms of the train, and consequently made it impos-
sible for them to pass all together through the last prism : the full
resolving power of the instrument was therefore only displayed over a
very narrow range of the spectrum at once. This, though not a matter
of much consequence in visual spectroscopes, where different parts of
the spectrum can readily be brought to the centre of the field in turn,
would be a serious defect if it were desired to photograph the spectrum.
The loss of light by reflexion at the faces of the prisms was also much
greater in the old than in the new type of spectroscope.

CAMBRIDGE : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.

GENERAL LIBRARY
UNIVERSITY OF CALIFORNIA— BERKELEY

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