e Tracts in Mathematics
and Mathematical -fthy sic s
J. G. LEATHEM, M.A.
E. T. WHITTAKER, M.A., F.R.S.
THE ELEMENTARY THEORY
J. G. LEATHEM, M.A.
Fellow and Lecturer of St John's College
and University Lecturer in Mathematics
Cambridge University Press Warehouse
C. F. CLAY, Manager
London : Fetter Lane, E.G.
Edinburgh : 100, Princes Street
Price 2s. 6d. net
Cambridge Tracts in Mathematics
and Mathematical Physics
J. G. LEATHEM, M.A.
E. T. WHITTAKER, M.A., F.R.S.
The Elementary Theory
Symmetrical Optical Instrument
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
C. F. CLAY, MANAGER.
ILonfcon: FETTER LANE, E.G.
OFUinimrgl) : 100, PRINCES STREET.
F. A. BROCKHAUS.
#cto gork: G. P. PUTNAM'S SONS.
Bombag anb Calcutta: MACMILLAN AND CO., LTD.
[All rights reserved]
THE ELEMENTARY THEORY
J. G. LEATHEM, M.A.
Fellow and Lecturer of St John's College
and University Lecturer in Mathematics
at the University Press
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS.
TN Gauss's Dioptrische Untersuchungen there is little trouble with
sign conventions, and continued fractions are not employed.
These are, however, prominent features in more recent presentations
of the first-order theory of the optical instrument, and render the
subject somewhat difficult to the beginner.
It is the aim of the present Tract to eliminate all unnecessary
difficulties and to give a quite elementary account of the theory ; and,
to this end, it has seemed desirable to follow (in Sections I IV) the
general lines of Gauss's memoir. I am indebted to a suggestion of
Mr T. J. I'A. Bromwich for a feature of the present scheme which
seems to me to mark a great advance in simplification, namely the
postponing of the study of the functional form of the constants of the
instrument till after its general optical properties have been established,
and the employing of an elementary theorem in algebraic linear trans-
formations to obtain the fundamental equations and the relation
between the constants.
Limits of space have prevented any close examination of the
application of the theory to particular instruments, but one or two
questions in connection with the equivalent thin lens and the adjust-
ment of field-glasses, not usually treated in text-books, have been
discussed ; and a few pages have been devoted to bringing reflecting
instruments within the scope of the theory.
In Section IX a brief and, I hope, easy account is given of the
I am deeply indebted to Mr Bromwich for reading the manuscript,
for his assistance in drawing up the syllabus which constitutes
Section X, and for other most valuable suggestions. My thanks are
due to Mr W. M. Page, Fellow of King's College, for reading the
proofs, and to Mr S. D. Chalmers, of the Northampton Institute, for
giving me the benefit of his knowledge of technical optics.
J. G. L.
ST JOHN'S COLLEGE,
1 May 1908.
I. Approximate formulae for a succession of refractions at nearly
normal incidence 1
II. The mathematical solution of the refraction problem for a
symmetrical instrument ....... 8
III. The optical properties of a symmetrical instrument . . 17
IV. The manner in which the optical properties of an instrument
depend on its constitution ...... 26
V. The equivalent thin lens 36
VI. Reflecting instruments . . . . . . . . 39
VII. Entrance and exit pupils 43
VIII. Chromatic defects of the image 44
IX. The aberrations of the third order .... 58
X. Syllabus of propositions concerning the characteristic function
and the focal lines of a pencil of rays of light ... 66
THE ELEMENTAEY THEORY OF THE
SYMMETRICAL OPTICAL * INSTRUMENT.
I. APPROXIMATE FORMULAE FOR A SUCCESSION OF
REFRACTIONS AT NEARLY NORMAL INCIDENCE.
1. Analytical formulae expressing the laws of refraction.
These laws are : (i) The incident ray, the refracted ray, and the
normal to the refracting surface at the point of incidence are in one
plane, (ii) The ratio of the sines of the angles of incidence and
refraction is a constant, depending only on the nature of the media
in which the light is propagated.
It is important to express these laws in terms of the direction
cosines of the lines involved. Let /x be the index of refraction of the
medium in which is the incident ray, // the index of the medium in
which is the refracted ray; let the cosines of the incident ray be
(/, m, n), those of the refracted ray (/', m', n'\ those of the normal
to the refracting surface at the point of incidence, drawn towards
the medium /x', (L, M, N) ; let the angles of incidence and refraction
be < and <' respectively.
In the diagram AO is the incident ray, OC the refracted ray, and
OK the normal. On the incident ray produced a point B is taken so
that OB is /A units of length, and on the refracted ray a point C is
taken so that OC is // units of length. BJ, OK are perpendiculars
drawn to the normal. Then the angles BOJ, COK are </>, <' respec-
tively, and the lengths of JB and KG are /* sin < units and i*! sin <f>'
The first law tells us that JB and KG are in the same plane
through JK, and therefore parallel to one another. The second law
tells us that JB and KG are of the same length. The two laws are
expressed in the statement that JB and KG have equal projections on
LAWS OF REFRACTION
each of the coordinate axes. Now the projection of JB is the excess
of the projection of OB over that of OJ; and OB is of length p. and
has 'cosines' 1 $ to/ w}, 1 white *.0.T is of length /xcos< and has cosines
(L, M, N). Likewise the projection of KG is the difference of the
projections of OC and OK. So the equalities of the projections of
JB and KG on the three axes of coordinates are expressed by
equations of the type
fjil-fjLCOs<j)L = fji'l' ft' cos $ L.
Rearranging, we get the following equations to express the laws of
pl'-nl = (X cos <' - //. cos <) L \
p'm -fJLm = (p cos<j>' -pcos<f>) M j- ............... (1).
p'ri ILU= (ft cos </>' - /A cos <) N }
The three equations are not all independent, as is readily seen by
multiplying them by L, M, N respectively and adding, it being
remembered that ^Ll' = cos<', and
2. Approximate formulae for nearly normal incidence.
When the incident ray is very nearly normal, it is readily seen
that the refracted ray is also very nearly normal. It is therefore
possible so to choose the axis of z that it shall be nearly parallel to the
incident ray, the refracted ray, and the normal at the point of in-
cidence. When the axis of z has been so chosen, /, m, I', m, L, M,
are all small. We shall obtain approximate formulae on the hypothesis
that these quantities are so small that their squares and products may
be neglected ; this is equivalent to the supposition that aberration is
to be neglected.
On this hypothesis n, which is equal to
differs from unity by small quantities of the second order; so also
1-3] APPROXIMATE B^ORMULAE 3
ri and N. Hence we may replace all three by unity. Also < and <'
are small of the same order as /, etc. ; so cos <f>, cos <f>' differ from unity
by small quantities of the second order, and may be replaced by unity.
Thus the third of equations (1) becomes, on the basis of the
proposed approximation, an identity; and the other two take the
= (n f -fji) M )"
3. Expression of L, M in terms of the coordinates of
the point of incidence.
The most usual application of the formulae (2) is to the case in
which a pencil of rays pass nearly normally through a comparatively
small portion of the refracting surface, so that the rays and the
normals at all the points of incidence are very nearly parallel to one
another. It is then advantageous to take as axis of z the normal at
some one of the points of incidence, say at a point so centrally
situated that it may be called the centre of the portion of the surface
through which refraction takes place ; the ray incident at this point
may be called the central ray of the pencil ; it need not be more
Unless the point at which the axis of z is normal, say (0, 0, c), is
a singular point on the refracting surface, the part of the surface in
the neighbourhood of this point can be approximately represented by
an equation of the type
2 (z-c) + az* + 2hxy + by 2 = .................. (3),
the approximation neglecting terms of the third order in x and y.
The constants a, h, b are of course known when the shape of the
surface is known.
At the point (#, y, z) the cosines of the normal are approximately
(ax + hy\(hx + by\ 1,
when the squares and products of #, y are neglected. The first two of
these may be substituted for Z, M in the formulae (2), and so we get
the refraction of the ray incident at (x, y, z) determined by the
' - pm = (X - //.) (hx + by) )
It is, of course, possible to choose the coordinate planes of x and y
4 SERIES OF REFRACTING SURFACES [l
so that the h of formula (3) shall be zero. If this were done the
approximate equation of the surface would take the form
2( z - c ^~+^-=0 ..................... (5),
and PI, p 2 would be the principal radii of curvature of the surface at
the point (0, 0, c), reckoned positive when the corresponding con-
vexities of the surface are towards the medium /A'. The formulae
which would then take the place of (4) are
4. A series of refracting surfaces having a common
When a ray traverses a succession of different media arranged in
such a way that the refracting surfaces have a common normal with
which the ray is always nearly coincident, it is interesting to see how
the equations of the previous Article enable us to derive from a know-
ledge of the position of the ray before its first incidence a complete
specification of the ray after its final emergence.
The incident ray (say in a medium /x ) is known when we know its
cosines 1 , m , and the coordinates (a- 1} y lt z^ of the point where it
meets the first surface,
2 (z - Ci) + a^ + ^xy + b 1 f = 0.
Clearly ^ differs from c a only by quantities of the second order, so we
may replace Zi by c l5 and regard the incident ray as specified by the
four quantities / , m<>, #1, y\.
After the first refraction (into a medium ^ the cosines of the ray
are changed to l ly m l , given by the equations
Equations of this type, which determine the change of direction due to
refraction, may be called " optical " equations.
The coordinates (# 2 , y 2 , c 2 ) of the point where the refracted ray
meets the second refracting surface
2 (z c 2 ) + a z a? + 2/i 2 ay + b 2 y 2 = 0,
can be obtained by putting z = c 2 (a sufficient approximation) in the
3-5] SERIES OF REFRACTING SURFACES 5
equations of the ray, i.e. of the line which proceeds from (x lt y lt c x ) in
the direction (h, m lt 1). Now the equations of this line are
(x - #0/4 = (y- yO/wij = z-c lt
and so x. 2 , y*. are determined by the equations
Equations of this type, which determine the coordinates of a point
of refraction in terms of those of the previous point of refraction, may
be called " geometrical " equations.
Having found # 2 , 3/2 by the geometrical equations, we are in a
position to use the optical equations corresponding to the next
equations which we could not use till we knew # 2 > 3/2? but which now
give us the values of 4 ^2-
Eliminating l lt m l from the six equations we are left with four
equations which give us explicit formulae for # 2 , 3/2, 4, ^2 in terms of
/oi Wo* #1, yi ; that is, the quantities specifying the ray in the medium
to in terms of the quantities that specify the incident ray.
Thus the problem of refraction is solved for the case of two
surfaces. If there are n surfaces and n + I media, we get in a similar
manner 2n optical equations and 2n - 2 geometrical equations. From
these we can eliminate successively the 4w - 6 quantities
/!, m l , Z 2 , #2, 4, ^2, #3) 3/3, #_!, 3/tt-l, 4-1, W n _!,
and obtain finally four equations expressing x n , y n , l n , m n , the quan-
tities specifying the emergent ray, in terms of 1 , m , x lt y lt the
quantities that specify the incident ray*.
5. Case in which all the refracting surfaces are sym-
metrical about the same two planes through the axis.
By suitable choice of the planes of x and y it is always possible to
make the coefficient of xy in the equation of one of the refracting
surfaces vanish ; but in general this choice would leave the corre-
sponding coefficients for all the other surfaces different from zero. But
if the surfaces are such that their indicatrices at the points where they
are met by the common normal (the axis of z) all have their principal
* Cf. Prof. R. A. Sampson on Gauss's Dioptrische Untersuchungen, Proc.
London Math. Soc. xxxix. 1898, p. 33.
6 EQUATIONS FOR GENERAL INSTRUMENT [l
axes in the same two directions, then the same choice of coordinate
planes will make all the ^'s vanish simultaneously.
In this case the equations of the preceding Article are greatly
simplified, for when all the A's are zero, the equations divide them-
selves into two sets, one set involving only #'s and /'s, the other set
involving only #'s and m's. To solve the problem of n surfaces we now
have only to eliminate the 2n - 3 quantities l^ # 2 , ts - -#n-i 4-i> from
the n optical and the n 1 geometrical equations of the first set. The
result, with suitable change of symbols, serves also for the corresponding
equations of the second set ; so that in this particular case the labour
of elimination is much less than half that required in the general case.
6. Form of the results in the general case.
Taking the #'s, ys, /'s, m's in the order which presents itself
naturally as one follows the course of a ray, we see that each of
these quantities is a homogeneous linear function of those that precede
it. Consequently when the elimination has been performed we get
%n, y n > ln> m n as homogeneous linear functions of /, m Q , x^, y^. A
more useful result is arrived at if we specify the positions of the
entering and the emergent rays, not by the coordinates (a?,, ^),
(#n, yn) of the points where they meet the planes z = c l9 z = c n , but
by the coordinates ( , ^ )> (, */) of the points where they respectively
meet twe other arbitrarily selected planes z = c l -p, z = c n + q, which
we call planes of reference. This implies the introduction of four
more geometrical equations, namely two of the type
and two of the type = x n + ql,
and the addition of x, y^ x n , y n to the quantities to be eliminated.
If we denote by a's with double suffix the coefficients in the
expressions for /, m, x n , y n in terms of x lt y lt 1 , m^, so that, for
example, l = a n a?i + a lz y } +^3/0 + ^4^0, it is readily verified that the
suggested elimination leads to
+ (a 4S +pa 4l + qa<% +pqa 2l ) 1 + (a u +pa& + qa^+ pqa 22 ) m
and two others which we may regard as contained in these two, since
they may be derived by differentiating with respect to q and remem-
tt j ^
= L = m.
5-7] OPTICAL INTERPRETATION OF EQUATIONS 7
Thus the whole theory of the set of n surfaces is contained (to
the degree of approximation postulated at the outset) in two linear
equations involving a rather large array of constants*.
7. Optical interpretation of the equations for the general
Without going into detail, it may be useful to indicate how these
equations may be interpreted optically, in other words to shew how
they give us information as to the image formed by the rays of light
that proceeded originally from a given bright point. We regard
(o> i/o i Cip) as the coordinates of the bright point, and suppose a
white screen to be placed in the plane z = c n + q. If we fix attention
on the ray which sets out from the source in a given direction,
specified by given values of / and m , the equations tell us the co-
ordinates of the point in which the ray after refraction strikes the
screen. If the pencil of refracted rays form a point image, it must
be possible, namely by placing the screen where it will receive the
image, to make all the refracted rays strike the screen in the same
point ; in fact, it must be possible to give such a value to q that and
?) shall be independent of / and m . This means that the giving of a
suitable value to q makes four coefficients vanish. Though these four
conditions are not all independent, they are in general equivalent to
three independent conditions, and so cannot be simultaneously satisfied
except in special cases.
Failing to obtain a point image, we next try to find the positions
of the focal lines of the pencil of emergent rays. For the points in
which the rays strike the screen to lie in a straight line, the necessary
and sufficient condition is that there should be a linear relation
between and ^ with coefficients independent of 4 and m . This
is the case if the same value of A makes the coefficients both of / and
of m Q vanish in the expression for + Ar/. The condition is therefore
the vanishing of
There are two values of q which satisfy this condition, and these
define the positions of the two focal lines of the emergent pencil.
* Amongst the 16 constants there are 6 relations, which are the conditions for
the existence of a Characteristic Function. For the form of the relations see
Sampson, I.e. pp. 38, 39 ; for the connexion with the Characteristic Function see
Bromwich, Proc. London Math. Soc. xxxl. 1899, p. 8.
8 POWER OF A SPHERICAL SURFACE [l, II
8. Case of two planes of symmetry.
When the surfaces are as described in Article 5, the mathematical
solution of the refraction problem is much simpler. The equations
n ) /
n ) / ) ,^
. 22 ) m )
and the optical interpretation is easier than in the general case.
II. THE MATHEMATICAL SOLUTION OF THE REFRACTION
PROBLEM FOR A SYMMETRICAL INSTRUMENT.
9. The symmetrical optical instrument.
We proceed now to the detailed discussion of a very particular case
of the general arrangement discussed in Article 4, namely the case in
which all the refracting surfaces are symmetrical with respect, not
merely to two planes, but to every plane through the axis of z. The
refracting surfaces are then necessarily surfaces of revolution having
the axis of z as common axis of revolution ; they are generally spheres,
but are sufficiently well represented by approximate equations of the
2(z-c-)+^^- = (10).
Such an arrangement is called a symmetrical optical instrument, and
is the kind of instrument most frequently employed in every-day life.
The approximate theory here developed applies to any symmetrical
instrument which is used in such a way that the rays which it trans-
mits are very nearly parallel to the axis, for example the telescope
and opera-glass ; it is practically useless in the case of wide-angle
instruments such as the microscope or portrait-camera.
10. Power of a single spherical refracting surface.
When refraction takes place from a medium of index /x to a
medium of index /*' through a surface represented approximately
by equation (10), the optical equations are equations (6) of Article 3,
simplified by the equality of both p x and p 2 to p. They are, in fact,
pm [j-m =
8-10] POWER OF A SPHERICAL SURFACE 9
Thus it appears that, to the approximation contemplated, the optical
properties of the spherical refracting surface are all contained in the
This constant is called the "diverging power" or simply the " power"
of the surface, and is usually denoted by K.
It is most important that the definition of the power of a surface
should be understood precisely, without confusion on account of the
conventions of sign which are implicitly involved in the above formula.
These conventions are ultimately two, namely (i) that the light passes
from the medium //. into the medium //, and (ii) that p is positive
when the convexity of the refracting surface is towards the second
Suppose the direction of the ray of light to be reversed. Then /*'
becomes the index of the first medium, and //, that of the second, so
that fj* p must take the place of // /x in formula (12); but con-
vexity towards the medium // is concavity towards the medium /A,
which is now the second medium, so that the case required by
definition is that in which p is negative, and p must take the
place of p in the denominator. Thus the power of the surface for
a ray going from // to //. is
that is, the same as before. Thus it appears that the power of a
surface is not altered by reversing the ray, and its definition has
nothing to do with the sense in which the ray passes.
The second of the conventions is thus the only one that affects the
definition of the power, and its bearing on the definition is clearly
summed up in the following rule :
The power of a refracting surface is to be regarded as positive
when the medium on the convex side o/ the surface has a greater
refractive index than tJie medium on the concave side; the power is
negative when the medium of greater index is on the concave side.
This is only another way of saying that K is positive if // > n and
P > 0, or if p < //- and p < 0.
For a plane refracting surface p is infinite, and K is zero.
The optical equations for a surface of power K are
nm = Ky (13).
10 KEDUCED PROJECTED INCLINATIONS [ll
11. Reduced projected inclinations of a ray.
The equations of the ray in the medium /x being of the form
x -a y ft
7- - - = *-%
it follows that the equation of the projection of the ray on the plane of
xz is of the form
x- a. = l(z-y}.
Hence / is the tangent of the angle that this projection of the ray
makes with the axis of z, or, to our degree of approximation, the angle
itself. In fact / and m are the inclinations to the axis of z of the
projections of the ray on the coordinate planes through that axis. We
call / and m the " projected inclinations " of the ray.
Putting 1*1 = 8, pm = , we may call S and e the " reduced projected
inclinations" of the ray, the word "reduced" in this connexion
meaning that the projected inclinations are multiplied by the index
of the medium in which the ray is passing.
In terms of this notation, the optical equations for a single surface
take the form
V-B = KX, f -* = K y... (14).
12. Divergence produced by a refracting surface.
If the effect of refraction were an increase in the projected inclina-
tions of each ray, a pencil of rays would, after passing through the
surface, be more divergent or less convergent than before ; the surface
would literally produce divergence. But it is to be noticed that the
optical equations give information, not as to the changes produced by
the refraction in the projected inclinations, but as to the changes
produced in the reduced projected inclinations ; and the power of the
surface measures the degree to which, for a given point of incidence,
it is capable of increasing the reduced projected inclinations. It is
therefore convenient to abandon the literal meaning of the word
" divergence," and to apply the term to what is really quite different,
namely increase of the reduced projected inclinations. With this
understanding, a diverging surface is one whose power is positive, a
converging surface is one whose power is negative. A plane surface
produces neither convergence nor divergence.
In the case of an instrument consisting of several surfaces, if the
first and last media are the same there is no difference between the
literal and the special meanings of divergence. Thus a double convex
11-14] REDUCED DISTANCES 11
lens is a converging instrument, a double concave lens is a diverging
A thin double concave lens of glass, whose two surfaces have the
same curvature, produces divergence ; and as this results from the
successive refractions at two refracting surfaces of equal power, it
may be assumed that each surface separately produces divergence.
In fact a surface with air on the concave side and glass on the convex
side is a diverging surface or surface of positive power. This fact,
which is easy to remember, helps one also to remember the rule given