in Article 10, that a surface has positive power when the medium of
greater index is on the convex side.
13. Reduced distances.
When reduced projected inclinations are substituted for projected
inclinations in the geometrical equations, of the type of equations (7)
(the equations of a ray in medium of index y^), the forms assumed are
These will be simplified if we introduce new symbols to represent
fe ~ Ci)//*i and similar expressions which occur in other geometrical
equations. Now c 2 - c t is a distance measured along the axis of the
instrument ; if we define 0,1 by the relation
/ \ I / + f\
(.c 2 c l )/fi l = ai v L ;>
we may call x the corresponding "reduced distance." Reduced
distance means distance parallel to the axis or along the ray, divided
by the index of the medium in which the ray is passing. In other
parts of Optics the phrase " reduced distance " is used in a different
sense, so care must be taken not to extend the use of the definition
here given beyond its present application.
With the notation of (15) the geometrical equations take the
x 2 = \ + ct>i$i )
#2 = */l +1 1 J
14. Refraction at a single spherical surface.
Let the power of the surface be *, let the first medium have index
fjL, the second index /w/. Let it be agreed to specify the incident ray by
12 REFRACTION AT A SINGLE SURFACE [ll
means of its reduced projected inclinations 8, e, and by the coordinates
&, y, of the point where it meets a certain plane of reference ; and let
the emergent ray be defined by its reduced projected inclinations 8', e',
and the coordinates #', y', of the point where it meets a second plane
of reference. The planes of reference are perpendicular to the axis ;
the first is at a reduced distance u in front of the refracting surface,
the second is at a reduced distance u' behind the refracting surface.
The corresponding true distances are accordingly pu and pu. The
reduced distance u is to be measured positively from the surface into
the first medium, while u' is to be measured positively from the
surface into the second medium. This kind of convention will be
adhered to throughout. Later on we shall associate u with the
position of an object, u (or a corresponding symbol) with the position
of the image, and it is convenient to remember that u is positive for
a real object, and u' positive for a real image.
If the coordinates of the point of incidence be , ^ we have three
pairs of equations at our disposal. The first are the geometrical
equations for the incident ray, typified by
= x + u$.
The second are the optical equations for the refraction, typified by
The third are the geometrical equations for the refracted ray,
of = + u'%. ^
Eliminating from these we get two equations, namely,
8'=*tf + (+l)8 \
X' = (KU + 1) X + (KUU' + U + u') & j '
which express 8' and x in terms of x and 8, that is, the quantities
specifying the refracted ray in terms of those specifying the incident
ray. The corresponding equations expressing e' and / in terms of
y and need not be written down, for it is clear that they are of
the same form and involve precisely the same constants. The four
* The order in which the symbols #, 3, 5', x' appear in these equations and in
many later ones may, at first, be thought rather unnatural, but it has been
deliberately chosen as representing in a sense the historic or chronological order.
In the case of the initial ray we know first the point from which it sets out, and
second the direction in which it goes ; hence x, y precede 5, e. In the case of the
emergent ray we learn first (from the optical equations) its direction, and afterwards
(from the geometrical equations) the coordinates of its point of arrival ; hence 5', e'
precede *', y'.
14, 15] THEOREM ON LINEAR SUBSTITUTIONS 13
equations constitute a complete solution of the refraction problem for
a single spherical surface.
The form of equations (17) must be carefully noted. They are of
8' = ax +
where a, b, c, d are constants depending partly on the power of the
surface, and partly on the positions of the planes of reference. And
(KU +1) (KU' + 1) - K (KUU' + U + *') = 1,
the constants of (18) are subject to the condition
bc-ad=l .......................... (19).
It will be shewn that the theory of any symmetrical optical instru-
ment, no matter of how many surfaces it is composed, is contained in
equations of the same type as (18), with the relation (19) between the
15. Algebraical lemma on linear substitutions.
Let two variables, 19 ^ be homogeneous linear functions of two
other variables 2 , <fes, determined by the relations
and let 0*, < 2 be similar functions of another pair of variables 3 ,
From these equations it follows, by the elimination of 2 and < 2 , that
0! and fa are expressible as homogeneous linear functions of 3 , </> 3 ,
where, in fact,
p"=pp' + qr f , q"=pq +qs',
r" - rp + sr, s" = rq' + ss f .
Now it is clear from these formulae that p", q", r", s" are the elements,
formed by the ordinary rule for multiplying determinants, of the
determinant which is the product of
14 SYMMETRICAL INSTRUMENT OF n SURFACES [ll
That is to say, O l and ^ are homogeneous linear functions of 3 and < 3>
and the determinant formed by the coefficients of these functions is
equal to the product of the determinants formed respectively by the
coefficients of the expressions for O lt ^ in terms of # 2 , $25 and the
expressions for 2 > <t>2 in terms of 3 , < 3 .
The theorem is true equally for three sets of n variables, whatever
integral value n may have ; but for our purpose the case of n = 2
If now we consider not merely three, but any number of sets of two
variables, the sets being taken in a definite order, and those of each set
being homogeneous linear functions of those of the set next in order,
we find by repeated application of the theorem just proved that the
variables of the first set can be expressed as homogeneous linear
functions of those of the last set, and that the determinant of the
corresponding coefficients is the product of all the determinants of
the coefficients of the relations between intermediate pairs of con-
16. Symmetrical instrument consisting of any number
n of refracting surfaces.
In the case of an instrument consisting of any number of refracting
surfaces, let the indices of the first and last media be /^ and //. re-
spectively, and let the powers of the surfaces be K lt K 2 ,...*. Let the
entering ray be specified by its reduced projected inclinations 8 o>
and by the coordinates # 05 3/o of the point where it strikes a plane of
reference at reduced distance u in front of the first surface of the
instrument. Let the emerging ray be specified by similar quantities
8, e, a?, y, and a plane of reference at reduced distance v behind the last
surface. Let the ray in any intermediate medium (/^.) be specified
by 8 r , e,,, # r , y r , and a plane of reference chosen anywhere in the
Then, by Article 14, it is seen that the S's and #'s of consecutive
* The theorem is a particular case of a well known theorem concerning
Jacobians, namely that if 6 and are functions of u and v, and if u and v
are functions of x and y, then
6 (0, 0) _ d (6, 0) 9 (u, v)
d (x, y)~ d (u, v) d (x, y) '
15-17] CONSTANTS OF THE INSTRUMENT 15
rays are connected by equations of the same type as (18), subject to a
condition of the type of (19). In fact
subject to GU di _
So also # 2 G^XI + d 2 $i }
And there is a whole series of such relations, ending with expressions
for x, 8, in terms of x n - lt 8 n _!.
Hence, by the lemma of Article 15, it appears that there are
relations, got by eliminating x^ 81, ... #-i, 8 n -i, of the form
where the determinant of the coefficients C', D', A', B' equals the
product of n determinants each of which is unity, and therefore is
itself unity; in fact,
B'C'-A'D' = l ........................ (21).
These coefficients A', B', C', D' must of course depend on the
values of u and v ; they are not therefore constants depending merely
on the nature of the instrument, but are constants for the instrument
so long as we keep to the same first and last planes of reference.
It is obvious that the relations between the entering and emerging
rays must be quite independent of the particular method which we
adopted of specifying the intermediate rays within the instrument,
and therefore the constants do not depend on the particular choice
of the intermediate planes of reference.
Of course y and e are expressible in terms of y Q and e by equations
involving the same four constants.
17. Dependence of A', B', C", D' on. u and v.
In order to interpret optically the equations (20), it is necessary to
ascertain in what way the constants depend on u and v. With this
in view we introduce a new set of constants, A, B, C, D, which are the
values that A', B', C', D', respectively, would have if u and v were both
zero. These are genuinely constants of the instrument, and correspond
16 FUNDAMENTAL EQUATIONS [ll, III
to planes of reference taken close up to the first and last refracting
surfaces respectively. They are, of course, subject to the relation
BC-AD=l ........................... (22).
If , ty) and , y are the coordinates of the points where the ray
meets the first and last surfaces of the instrument respectively, these
coordinates must take the place of # > # > x -> V i n equations (20) when
these surfaces and the corresponding planes of reference coincide. So
we have the relations
, 3 ,
The geometrical equations of the entering and emerging rays give
relations between # and > an d between and #, namely
The elimination of and between (23) and (24) leads to
= AxQ + (Au + B) 8 I ^^
These must be identical with (20), and so we have the relations
D' = Auv + v + Cu + D
The equations (25) are the most useful form of the equations of the
instrument. The second alone involves the whole theory and may be
called the fundamental equation. The first is really contained in the
second, for it may be derived from it by differentiation with respect to
v t it being seen from the geometrical equation that
It appears therefore that the whole of the theory of the instrument,
when aberration is neglected, depends on four "constants of the
instrument," which are equivalent to only three independent constants
on account of the relation (22).
The constant A is called the "power" of the instrument.
17, 18] CONJUGATE FOCAL PLANES 17
III. THE OPTICAL PROPERTIES OF A SYMMETRICAL
18. Conjugate focal planes.
In proceeding to deduce from equations (25) some of the optical
properties of the instrument, it is advantageous to think of the point
(#01 #o) in the first plane of reference as the seat of a point source of light,
and to regard the last plane of reference as occupied by a white screen.
The fundamental equation shews that the value of x in general
depends on that of 8 , so that rays setting out from the luminous point
in different directions and passing through the instrument will strike
the screen in different points. Thus there is a bright patch on the
screen, but not an image. It is, however, possible, when the position
of the bright point is prescribed, to choose such a value of v, and
therefore such a position of the screen, that x shall be independent of
S ; this is effected by choosing v so that the coefficient of 8 in the
fundamental equation vanishes. The coefficient of e in the expression
for y, being the same coefficient, vanishes for the same choice of v.
And so the point in which the ray meets the screen is the same
whatever be the direction in which it originally set out. Thus all the
rays come together to form a point image at the point (a?, y) in the
final plane of reference.
The relation between the reduced distances u, v, of the bright
point and its image, from the ends of the instrument is
Auv + Sv+Cu + D = ..................... (28).
The value of v thus defined does not depend on X Q or y , but only on
u, and so a number of bright points having the same u give images
having the same v. Consequently a small plane object placed on the
axis in the plane defined by u has a plane image at the position defined
by v. The relation between object and image is, in a sense, a reciprocal
one, since light, proceeding from an object in the plane v and traversing
the instrument in the reverse direction, would form an image in the
Planes which correspond to one another in this fashion are called
" conjugate focal planes."
When A is not zero, the relation between conjugate focal planes
may be put in another form if we multiply (28) through by A, and
make use of (22). We get
= A*w + ABv + ACu + BC- 1,
whence (Au + B)(Av+C) = l ..................... (29).
18 LINEAR MAGNIFICATION [ill
19. Conjugate foci.
A bright point and its image point are called "conjugate foci."
The expression of the coordinates of the image point in terms of
those of the object point to which it is conjugate is contained in the
#o tyo Ou + D
x = Au + S' '-A***' V = ~A^S
which are immediate consequences of (25) and (28). Here u and v are
coordinates, for origins in the first and second planes of reference
respectively, in the special case in which /x and /x are both unity ; in
the more general case u and v are z coordinates divided respectively
by /MO and /*.
20. Linear Magnification.
The ratio of the linear dimensions of the image to those of the
object is called the "linear magnification," and may be denoted by m.
We may measure the object from the point on the axis to some other
point (# , y ) ; and then, when v has been adjusted so as to satisfy (28),
the image is measured from that point of its plane which is on the
axis, to the point (x, y). So far as regards dimensions parallel to the
axis of x, the linear magnification is the ratio of x to x , and this is
given by the second of equations (25). So we have
m = Av+C ........................... (31).
Combining this with (29) we get
= Au + B ........................... (32).
These are the two magnification formulae ; the symmetry of the
instrument about the axis ensures that these formulae, proved in
the first instance only for dimensions parallel to the axis of x, shall be
true for dimensions in all directions perpendicular to the axis of the
A negative value of m indicates inversion of the image as com-
pared with the object.
* The correspondence between a point (x', y', z'), referred to one set of axes,
and a point (#, y, z), referred possibly to another set of axes, which is deter-
mined by the relations
px + qy + rz + s
is called a homographic correspondence. It is a one-to-one correspondence in
which straight lines correspond to straight lines. Clearly the correspondence
between conjugate foci is of this type.
19-22] UNIT PLANES AND PRINCIPAL FOCI 19
Since m is the same for dimensions in all directions, the image is
of the same shape as the object ; for example the image of a circular
object is circular. If we had precise image formation with an asym-
metrical instrument, the kind referred to in Article 8, the linear
magnification would be different for different directions and the image
would be of a different shape from the object, i.e. distorted ; a circular
object, for example, would have an elliptic image.
21. Unit Planes or Principal Planes.
It is possible so to choose u and v that the linear magnification
shall be unity. Denoting the special values by u and v lt we obtain,
by putting m \ in (31) and (32), the relations
l=Av l + C, l = Au 1 + (33).
The conjugate focal planes defined by u and Vi are called the "planes
of unit magnification," or more briefly "unit planes" or "principal
planes." The points where these planes meet the axis are called the
"unit points" of the instrument. The characteristic property of the
unit planes may be expressed by the statement that the (#, y) co-
ordinates of the point where any entering ray crosses the first unit
plane are the same as those of the point where the corresponding
emergent ray crosses the second unit plane.
22. Principal Focal Planes.
If in the relation (28) we put u GO , the corresponding value, F,
of v is given by
if we put v = so , the corresponding value U of u is given by
AU+B = (35).
The planes specified by U and V are called the "principal focal
planes " of the instrument, and their intersections with the axis are
called the " principal foci." The second principal focal plane, that
given by F, is the place where a screen should be placed in order to
receive the image of an infinitely distant object ; in other words it is
the locus of the points of concurrence of emergent rays corresponding
to incident pencils of parallel rays. The first principal focal plane,
that given by [7, is such that if a point source of light be placed at
any point of it (near the axis), the rays from it will, after traversing
the instrument, emerge as a parallel beam. If the direction of passage
through the instrument were reversed, the rdles of these two planes
would be interchanged.
20 FOCAL LENGTHS OF AN INSTRUMENT [ill
23. Focal Lengths.
The distances of the principal focal planes beyond the corre-
sponding unit planes are called the "focal lengths" of the instrument.
The word "beyond" is used to indicate that distances are to be
measured from the respective ends of the instrument, away from the
instrument ; i.e., opposite to the direction of the light for the end at
which incidence takes place, and in the same direction as the light at
the end where the light emerges.
The reduced distances of the principal focal planes beyond the
corresponding unit planes are respectively
UU-L and YVI,
each of which equals - 1 /A. The actual distances, or focal lengths,
are therefore F^ F 2 , where
F! = -IJ^A and Fs = -n/A (36).
These lengths are negative if the instrument has a positive power,
and vice versa. They are equal if the first and last media have the
same index; and in the particular case of /x = l, /* = !, which is
practically true for an instrument in air, the focal length is
24. Relation between the distances of conjugate foci
from the principal focal planes.
When distances are measured from the principal focal planes
instead of from the ends of the instrument the magnification for-
mulae and the condition for conjugacy assume specially simple forms.
Let u U=p, v- V=q t so that p, q are the reduced distances of the
planes of reference beyond the principal focal planes. Substituting
U+p, V+q, for u y v, in formulae (2.6), and remembering (34) and
(35), we obtain
so that the standard formulae (20) assume the form
From these it appears that the condition that x should be inde-
pendent of 8 , that is, the relation between conjugate focal planes, is
or pq = (l/AY ........................... (39).
23-25] NODAL POINTS 21
The associated magnification formulae are
^ = Ap, m = Aq ........................ (40).
These formulae are, of course, equivalent to
where p', q are the true distances corresponding to p, q. In this form
they can be strikingly illustrated by a diagram in which the axes of
the instrument, the unit planes, and the principal foci, S lt B 2 , are first
drawn. From a point P of the object two rays are drawn, the first
parallel to the axis and meeting the first unit plane in X, the second
through BI meeting the first unit plane in Y. The corresponding
emergent rays are a ray which leaves the second unit plane at X' and
passes through S 2j and a ray which leaves the second unit plane at Y'
and proceeds parallel to the axis ; the point Q of intersection of these
two marks the position of the image of P. The lines XX' and YY'
are parallel to the axis, in virtue of the property of unit planes. All
the lengths named in (41) are easily identified in the diagram, and the
formulae are seen to be the expressions of very simple geometrical
25. Nodal Points.
If an incident ray cuts the axis of the instrument at the point
whose reduced distance from the first refracting surface is ^J, then for
.this ray # = 0, and the first of equations (25) becomes
8 = (4ti + )8 .
Now if u have the value u given by the relation
Au 2 + = n/n ........................... (42),
we have S//x, = 8 //u, ,
so that the projected inclinations of the entering and the emergent
rays are equal. The emergent ray must pass through the image point
of the point on the axis determined by u z , namely a point on the axis
at reduced distance v 2 beyond the end of the instrument from which
rays emerge, v 2 being given by
Av*+C=pJiL ........................... (43).
These two conjugate points on the axis are called the "nodal
points " of the instrument. They have the property that if an enter-
ing ray passes through the first nodal point the emergent ray passes
through the second nodal point and is parallel to the entering ray.
22 EQUIVALENT SINGLE SURFACE [ill
If H = P O , it is seen by comparison of the formulae (33), (42), and
(43) that U! = u 2 and v 1 = v 2 . In other words, if the first and last
media have the same index the nodal points are the points where the
unit planes cut the axis.
Suppose the instrument to be directed towards a very distant
bright point, and the image caught on a properly placed screen. If
now the instrument be rotated about a point on its axis through a
small angle, the image will still be (very approximately) on the screen,
but not in general at the same point of the screen. One ray suffices
to determine the position of the image, and we shall take that ray to
be the ray through the nodal points. The object being very distant
the direction of the entering ray is not altered by the rotation of the
instrument, and so the direction of the ray that emerges and passes
through the second nodal point is unaltered. Thus the image is
shifted on the screen by an amount equal to the distance moved
through by the second nodal point on account of the rotation of the
instrument. If the point about which the rotation takes place is the
second nodal point itself, that point does not move, and so the image
does not move on the screen. This fact is taken advantage of in
practice, to determine experimentally the nodal points of an optical
instrument. The image is studied through a microscope while the
instrument to be tested is turned successively about different points
on the axis ; when a point has been found such that a small rotation
about it does not move the image it is known to be the nodal point.
26. Equivalent Single Refracting Surface.
The equations (26) shew how the constants, A', B', C', D', depend
on the positions of the planes of reference. If we take the unit planes
as planes of reference we must substitute the values of HI and Vi for
u and v in these equations, and the corresponding constants are
A' = A, B'=l, C' = l, D' = Q ............ (44);
so the equations (20) assume the form
Let us compare these with the corresponding equations for a single
refracting surface of power K, when the planes of reference both coin-
cide with the surface itself. The coincidence of the planes of reference
ensures the equality # = # , and we have already seen (Article 11,
eqn. 14), that
8 = r + 8 ........................... (46).
25-27] APPARENT DISTANCE 23
Thus it is seen that, if K = A t the equations for the single surface are
identical with those of the instrument ; and, if the initial and final
media have respectively the same indices in the two cases, the inter-
pretations of corresponding symbols are identical. To a given entering
ray, therefore, there correspond in the two cases emergent rays having
the same direction and proceeding from points having the same (a?, y)
coordinates. The only difference is in the z coordinates of the points
of departure from the second unit planes ; these necessarily differ by
the distance from the first to the second unit plane of the original
instrument. So, for a given object at a given distance from the first
unit plane, the single refracting surface of power A produces an
image which differs only in position from the image produced by the