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to A^tti but in magnitude such that A/t^^xA.a,.

Similarly let BJ), be the image of BJj„ and let BM, = yBA- Also let
A,B, = c, and A.X^ = c^.

Then since a, and h„ are the images of a, and \, the line F^aK will be
the direction of the ray after emergence, cutting the axis in F^, (unless x = y.


when a.})^ becomes parallel to the axis). The point F._ may be found, by
remembering that A^a, = B^b^, Ajii = xAfL^, B]j. = yDJj^. We find —

■ " 'y-x

Let g^ be the point at which the emergent ray is at the same distance
from the axis as the incident ray, draw gfi^ perpendicular to the axis, then
we have

' y-x

Similarly, if aSiF^ be a ray, which, after emergence, becomes parallel to
the axis ; and gfi^ a line perpendicular to the axis, equal to the distance of
the parallel emergent ray, then

A,F, = c,-y~, F,G,^^^^ .
x—y ^—y


I. The point F^, the focus of incident rays when the emergent rays are
parallel to the axis, is called the Jirst jprincii^al focus of the instrument.

II. The plane G^^ at which incident rays through F^ are at the same
distance from the axis as they are after emergence, is called the first princi-
pal plane of the instrument. F^G^ is called the first focal length.

III. The point F^, the focus of emergent rays when the incident rays
are parallel, is called the second principal focus.

IV. The plane G,^., at which the emergent rays are at the same distance
from the axis, as before incidence, is called the second principal plane, and
Ffi^ is called the second focal length.

When x = y, the ray is parallel to the axis, both at incidence and emerg-
ence, and there are no such points as F and G. The instrument is then
called a telescope. x( = y) is called the linear ina^nifying power and is denoted

by I, and the ratio - is denoted by n, and may be called the elongation.

In the more general case, in which x and y are different, the principal
foci and principal planes afford the readiest means of finding the position of


Prop. IV. Given the principal foci and principal planes of an instrument,
to find the relations of the foci of the incident and emergent pencils.

Let F„ F„ (fig. 3) be the principal foci, G^, G., the principal planes, Q^
the focus of incident light, Q^P^ perpendicular to the axis.

Through ^1 draw the ray Q^g^F^. Since this ray passes through F^ it
emerges parallel to the axis, and at a distance from it equal to G^g^. Its
direction after emergence is therefore Q.,g^ where G^g„ = G^g^. Through Q^ draw
Q{Yi parallel to the axis. The corresponding emergent ray wiD pass through
F^^, and will cut the second principal plane at a distance G^y^_= G-^y^, so that
jP„y, is the direction of this ray after emergence.

Since both rays pass through the focus of the emergent pencil, Q^, the
point of intersection, is that focus. Draw Q^P^ perpendicular to the axis.
Then PxQi = G{Y^ = G^y., and G,g, = G^g^ = P,Q.,. By similar triangles F,P,Q, and


P,F, : F,G, :: P,Q, : {G,g, = ) P,Q,.
And by similar triangles F^P^Q^ and F^G^y^

Pm = Gry^) ■ P^Q^ ■■■■ ^^. ■■ F^P^ -
We may put these relations into the concise form


F^r p.Qr F,p,'

and the values of F„P^ and PJ^^ are


F..P.= '^'pf^" - and P.Q. = ^'P.Q,.

These expressions give the distance of the image from F^ measured along the
axis, and also the perpendicular distance from the axis, so that they serve to
determine completely the position of the image of any point, when the princi-
pal foci and principal planes are known.

Prop. V. To find the focus of emergent rays, when the instrument is a

Let ^1 (fig. 4) be the focus of incident rays, and let Q^aJ)^ be a ray
parallel to the axis ; then, since the instrument is telescopic, the emergent
ray Q^aM^ will be parallel to the axis, and Q^P^^l. Q^P^.


Let QiOiB^ be a ray through ^,, the emergent ray will be Q,a,J5,, and

AM, ~ A,a,~ I. A,a, " A.a, " A,B, '

so that -FT^ = -4 r>' = n, a constant ratio.

P^B, A,B^

Cor. If a point C be taken on the axis of the instrument so that

^^^ = A,B,-A^, ^'^' = T:^ ^^^"

then CP, = n.CP,.

Def. The point C is called the centre of the telescope.

It appears, therefore, that the image of an object in a telescope has its
dimensions perpendicular to the axis equal to I times the corresponding dimen-
sions of the object, and the distance of any part from the plane through C
equal to n times the distance of the corresponding part of the object. Of
course all longitudinal distances among objects must be multipUed by n to
obtain those of their images, and the tangent of the angular magnitude of an

object as seen from a given point in the axis must be multipHed by - to

obtain that of the image of the object as seen from the image of the given

point. The quantity - is therefore called the angular magnifying power, and

is denoted by m.

Prop. VI. To find the principal foci and principal planes of a combina-
tion of two instruments having a common axis.

Let /, /' (fig. 5) be the two instruments, G^F^Ffi, the principal foci and
planes of the first, G^F^F^G^ those of the second, V^<^^^S, those of the com-
bination. Let the ray g^jJj'g^ pass through both instruments, and let it be
parallel to the axis before entering the fii'st instrument. It will therefore pass
through F„ the second principal focus of the first instrument, and through g.
so that G^^ = (xi(7i.

On emergence from the second instrument it will pass through ^^ the
focus conjugate to F,, and through g^ in the second principal plane, so that


(r.'g' = G^g^. (f>i is by definition the second principal focus of the combination
of instruments, and if T^y^ be the second principal plane, then r„y, = G^g^

We have now to find the positions of <f>, and Tj.

By Prop. IV., we have

^^^== — F:Fr~ •

Or, tlie distance of the principal focus of the combination, from that of the
second instrument, is equal to the product of the focal lengths of the second
instrument, divided by the distance of the second principal focus of the first
instrument from the first of the second. From this we get

r"jp' jp'A ^"'^^ {FjF^ — F^G()
Ctj i^j - -t^2 9a = jrpT ,

oi G,<f>, = jrp7 .

Now, by the pairs of similar triangles ^G^g^, (jtV^y, and FJjr(g', F^G^^,

T,<j>, _ r,y, ^ %, _ F„G,

~g:4>. Gig. G:g( g;f,-

Multiplying the two sides of the former equation respectively by the first and
last of these equal quantities, we get

, Gr^ , . GiF„'

Or, the second focal distance of a combination is the product of the second
focal lengths of its two components, divided by the distance of their consecutive
principal foci.

If we call the focal distances of the first instrument f^ and /,, those of
the second // and //, and those of the combination J\, /j, and put FJF^=d,
then the positions of the principal foci are found fi:om the values

and the focal lengths of the combination from

'~ d ' J'~ d '


When d = 0, all these values become infinite, and the compound instruiaent
becomes a telescope.

Prop. VII. To find the linear magnifying power, the elongation, and the
centre of the instrument, when the combination becomes a telescope.

Here (fig. 6) the second principal focus of the first instrument coincides at J'
with the first of the second. (In the figure, the focal distances of both instru-
ments are taken in the opposite direction from that formerly assumed. They are
therefore to be regarded as negative.)

In the first place, F,' is conjugate to F^, for a pencil whose focus before
incidence is F^ will be parallel to the axis between the instruments, and will
converge to i^/ after emergence.

Also if G^g^ be an object in the first principal plane, G,g„ will be its first
image, equal to itself, and if Hh be its final image

^^^- Gjr-~- f:^

Now the linear magnifying power is 7,- , and the elongation is .' .
because F.' and H are the images of F.^ and G^ respectively ; therefore

l=-4^ and n=££-.
The angular magnifying power = in = -= — 4-7 •
The centre of the telescope is at the point C, such that

When n becomes 1 the telescope has no centre. The efiect of the Instruineni
is then simply to alter the position of an object by a certain distance measured
along the axis, as in the case of refraction through a plate of glass bounded bv
parallel planes. In certain cases this constant distance itself disappears, as in
the case of a combination of three convex lenses of which the focal lengths arr


4, 1, 4 and the distances 4 and 4. This combination simply inverts every object
without altering its magnitude or distance along the axis.

The preceding theory of perfect instruments is quite independent of the
mode in which the course of the rays is changed within the instrument, as
we are supposed to know only that the path of every ray is straight before
it enters, and after it emerges from the instrument. We have now to con-
sider, how far these results can be applied to actual instruments, in which
the course of the rays is changed by reflexion or refraction. "We know that
such instruments may be made so as to fulfil approximately the conditions of
a perfect instrument, but that absolute perfection has not yet been obtained.
Let us inquire whether any additional general law of optical instruments can
be deduced from the laws of reflexion and refraction, and whether the imper-
fection of instruments is necessary or removeable.

The following theorem is a necessary consequence of the known laws of
reflexion and refraction, whatever theory we adopt.

If we multiply the length of the parts of a ray which are in diflerent
media by the indices of refraction of those media, and call the sum of these
products the reduced path of the ray, then :

I. The extremities of all rays from a given origin, which have the same
reduced path, lie in a surface normal to those rays.

II. When a pencil of rays is brought to a focus, the reduced path from
the origin to the focus is the same for every ray of the pencil.

In the undulatory theory, the " reduced path " of a ray is the distance
through which light would travel in space, during the time which the ray
takes to traverse the various media, and the surface of equal " reduced paths "
is the wave-surface. In extraordinary refraction the wave-surface is not always
normal to the ray, but the other parts of the proposition are true in this and all
other cases.

From this general theorem in optics we may deduce the following propo-
sitions, true for all instruments depending on refraction and reflexion.

Prop. VIII. In any optical instrument depending on refraction or reflex-
ion, if ajtti, />i^i (fig. 7) be two objects and a.a.^, h.fi^ their images, A^B^ the
distance of the objects, AM. that of the images, ^i^ the index of refraction of


the medium in which the objects are, /a, that of the medium in which tlie
images are, then

«,a, X /^,y8, _ a,a, x h.fi.,
^' A A ~^' A,B., ''

approximately, when the objects are small.

Since a, is the image of a^, the reduced path of the ray a,6,a,, will be
equal to that of a^^a„_, and the reduced paths of the rays a^/3,cu and a,/Aa, will
be equal.

Also because l)^^ and h.^„ are conjugate foci, the reduced paths of the
rays b^ajj, and h^aj),, and of ^ia,,/8j and ^,a.,/3, will be equal. So that the
reduced paths

afi, + h,a^ = a^ySj + ^.a^

aJ3, + I3,0L, = tti^i + b.cL,

feiOj + Oj^j = b^a^ + alt.,

these being still the reduced paths of the rays, that is, the length of each
ray multiplied by the index of refraction of the medium.

If the figure is symmetrical about the axis, we may write the equation

Fi (aA - «i^i) = /^2 (aA - ci-A),
where aJS^, &c. are now the ax^tual lengths of the rays so named.

Now aA' = A,B;' + 1 (a,a, + b^.f,

so that a^i — aj)^ = OiC^ x 6^8, ,

a.a, X 61)8,

and ft, (a^ - aj),) = fi^

a A + aj)^

Similarly /x, (a^ - a,&,) = fi, ^^^^^j ^'

So that the equation /x, ^ , "T' = /x^ — ^— — , ,


is true accurately, and since when the objects are small, the denominators are
nearly 2A,B^ and 2A^„ the proposition is proved approximately true.

Using the expressions of Prop. III., this equation becomes

1 xy

Now by Prop. III., when x and y are different, the focal lengths /, and /,


. xy ^ 1

^1 'x-y ^ y — ^

therefore -^ = -^ = - by the present theorem.

So that in any instrument, not a telescope, the focal lengths are directly as
the indices of refraction of the media to which they belong. If, as in most
cases, these media are the same, then the two focal distances are eqiial

When x = y, the instrument becomes a telescope, and we have, by Prop. V.,

l = x and n=-; and therefore by this theorem

m n'

We may find I experimentally by measuring the actual diameter of the
image of a known near object, such as the aperture of the object glass. If be
the diameter of the aperture and o that of the circle of light at the eye-hole
(which is its image), then

From this we find the elongation and the angular magnifying power

n = ^'l\ and m = ^'y.

When ix, = fi„ as in ordinary cases, m = y = -, which is Gauss' rule for deter-
mining the magnifying power of a telescope.


Prop. IX. It is impossible, bj means of any combination of reflexions
and refractions, to produce a perfect image of an object at two different distances,
unless the instrument be a telescope, and

l = n=-, m=l.

It appears from the investigation of Prop. VIII. that the results there
obtained, if true when the objects are very small, will be incorrect when the
objects are large, unless

ajSi + tti^i : a^^ + a,h :: A^B^ : A^^,

and it is easy to prove that this cannot be, unless all the Hnes in the one figure
are proportional to the corresponding lines in the other.

In this way we might show that we cannot in general have an astigmatic,
plane, undistorted image of a plane object. But we can prove that we cannot
get perfectly focussed images of an object in two positions, even at the expense
of curvature and distortion.

We shall first prove that if two objects have perfect images, the reduced
path of the ray joining any given points of the two objects is equal to that
of the ray joining the corresponding points of the images.

Let tto (fig. 8) be the perfect image of a^ and yS^ of /B^. Let

Ajai = a^, BJ3, = b„ Ajx^ = a^, B.J3., = b., A^B^ = c^, A^^ = c^.

Draw a^D^ parallel to the axis to meet the plane B^y and aJD, to the plane
of A.

Since everything is symmetrical about the axis of the instrument we shall
have the angles D^Bfi^ = D.M.fi, = d, then in either figure, omitting the sufl&xes,

= c' + a' + b'-2ahcose.

It has been shown in Prop. VIII. that the difference of the reduced paths
of the rays aj)^, afi^ in the object must be equal to the difference of the reduced
paths of a^^j, a^^ in the image. Therefore, since we may assume any value for 6

/^i J{(^x + &i' + Ci* - lajb, cos 6) - fi, J{a^ + h^ + c^ - 2a,h cos 6)


13 constant for all values of 6. This can be only when

and fi, J{aJ),) =fi,J (aM,),

which shows that the constant must vanish, and that the lengths of lines
joining corresponding points of the objects and of the images must be inversely
as the indices of refraction before incidence and after emergence.

Next let ABC, DEF (fig. 9) represent three points in the one object
and three points in the other object, the figure being drawn to a scale so that
all the lines in the figure are the actual lines multiplied by /Xj. The lines of
the figure represent the reduced paths of the rays between the corresponding
points of the objects.

Now it may be shown that the form of this figure cannot be altered with-
out altering the length of one or more of the nine lines joining the points ABC
to DEF. Therefore since the reduced paths of the rays in the image are equal
to those in the object, the figure must represent the image on a scale of /n,
to 1, and therefore the instrument must magnify every part of the object alike
and elongate the distances parallel to the axis in the same proportion. It is
therefore a telescope, and m=l.

If iJi, = ix,, the image is exactly equal to the object, which is the case in
reflexion in a plane mirror, which we know to be a perfect instrument for all

The only case in which by refraction at a single surface we can get a
perfect image of more than one point of the object, is when the refracting
surface is a sphere, radius r, index /x, and when the two objects are spherical

surfaces, concentric with the sphere, their radii being - , and r ; and the two

images also concentric spheres, radii /ar, and r.

In this latter case the image is perfect, only at these particular distances
and not generally.

I am not aware of any other case in which a perfect image of an object
can be formed, the rays being straight before they enter, and after they emerge
from the instrument. The only case in which perfect astigmatism for all pencils
has hitherto been proved to exist, was suggested to me by the consideration





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of the structure of the crystalline lens in fish, and was published in one of
the problem-papers of the Camhiidge and Dublin Mathematical Journal. My
own method of treating that problem is to be found in that Journal, for
February, 1854. The case is that of a medium whose index of refraction varies
with the distance from a centre, so that if fi, be its value at the centre, a
a given line, and r the distance of any point where the index is /x, then

/^ = /Ao

a' + r''

The path of every ray within this medium is a circle in a plane passing through
the centre of the medium.

Every ray from a point in the medium, distant b from the centre, will

converge to a point on the opposite side of the centre and distant from it ^ .

It will be observed that both the object and the image are included in
the variable medium, otherwise the images would not be perfect. This case
therefore forms no exception to the result of Prop. IX., in which the object and
image are supposed to be outside tho instrument.

Aberdeen, 12th Jan., 1858.

[From the Proceedings of the Royal Society of Edinburgh, Vol. rv.]

XYIII. On Theories of the Constitution of Saturn's Rings.

The planet Saturn is surrounded by several concentric flattened rings, which
appear to be quite free from any connection with each other, or with the planet,
except that due to gravitation.

The exterior diameter of the whole system of rings is estimated at about
176,000 miles, the breadth from outer to inner edge of the entire system,
36,000 miles, and the thickness not more than 100 miles.

It is evident that a system of this kind, so broad and so thin, must
depend for its stability upon the dynamical equihbrium between the motions of
each part of the system, and the attractions which act on it, and that the
cohesion of the parts of so large a body can have no effect whatever on its
motions, though it were made of the most rigid material known on earth. It
is therefore necessary, in order to satisfy the demands of physical astronomy,
to explain how a material system, presenting the appearance of Saturn's Kings,
can be maintained in permanent motion consistently with the laws of gravitation.
The principal hypotheses which present themselves are these —
I. The rings are solid bodies, regular or irregular.
II. The rings are fluid bodies, liquid or gaseous.
in. The rings are composed of loose materials.

The results of mathematical investigation appHed to the first case are, —

1st. That a uniform ring cannot have a permanent motion.

2nd. That it is possible, by loading one side of the ring, to produce
stability of motion, but that this loading must be very great compared with
the whole mass of the rest of the ring, being as 82 to 18.


3rd. That this loading must not only be very great, but very nicely
adjusted; because, if it were less than '81, or more than 83 of the whole,
the motion would be unstable.

The mode in which such a system would be destroyed would be by the
collision between the planet and the inside of the ring.

And it is evident that as no loading so enormous in comparison with the
ring actually exists, we are forced to consider the rings as fluid, or at least
not solid ; and we find that, in the case of a fluid ring, waves would be gene-
rated, which would break it up into portions, the number of which would
depend on the mass of Saturn directly, and on that of the ring inversely.

It appears, therefore, that the only constitution possible for such a ring is
a series of disconnected masses, which may be fluid or solid, and need not be
equal. The \iomplicated internal motions of such a ring have been investigated,
and found to consist of four series of waves, which, when combined together,
will reproduce any form of original disturbance with all its consequences. The
motion of one of these waves was exhibited to the Society by means of a small
mechanical model made by Ramage of Aberdeen.

This theory of the rings, being indicated by the mechanical theory as the
only one consistent with permanent motion, is further confirmed by recent obser-
vations on the inner obscure ring of Saturn. The limb of the planet is seen
through the substance of this ring, not refracted, as it would be through a
gas or fluid, but in its true position, as would be the case if the light passed
through interstices between the separate particles composing the ring.

As the whole investigations are shortly to be published in a separate form,
the mathematical methods employed were not laid before the Society.

XIX. On the Stability of the motion of Saturn's Rings.

[An Essay, which obtained the Adams Prize for the year 1856, in the University

of Cambridge.]


The Subject of the Prize was announced in the following terms ; —

The University having accepted a fimd, raised by several members of St John's Collegp,
for the purpose of foun ding a Prize to be called the Adams Prize, for the best Essay
on some subject of Pure Mathematics, Astronomy, or other branch of Natural Pliilosophy,
the Prize to be given once in two years, and to be open to tlhe competition of all persons
who have at any time been admitted to a degree in this University: —

The Examiners give Notice, that the following is the subject for the Prize to be adjudged
in 1857:—

The Motions of iSaturn's Rings.

*** The problem may be treated on the supposition that the system of Rings is exactly or
very approximately concentric with Saturn and symmetrically disposed about the plane of his Equator,
and different hypotheses may be made respecting the physical constitution of the Rings. It may
be supposed (1) that they are rigid: (2) that they ai-e fluid, or in part aeriform: (3) that they
consist of masses of matter not mutually coherent. The question will be considered to be answered
by ascertaining on tliese hypotheses severally, whether the conditions of mechanical stability are
satisfied by the mutual attractions and motions of the Planet and the Rings.

It is desirable that an attempt should also be made to determine on which of the above

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