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M S34

DUBLIN UNIVERSITY PRESS SERIES.

THE COLLECTED WORKS

OF

JAMES MAC CULLAGH, LLD.,

FELLOW OF TRINITY COLLEGE,
AND

PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF DUBLIN.

EDITED BY

JOHN H. JELLETT, B.D.,

AND

SAMUEL HAUGHTON, CLK., M.D.

DUBLIN: HODGES, FIGGIS, & CO., GRAFTON-STREET.
LONDON: LONGMANS, GREEN, & CO., PATERNOSTER-ROW.

1880.

“hi o
als a

ie

ek

PRESS. ‘. #

‘

PRINTED AT THE UI

PREFACE.

———

THE present volume contains a complete collection of
the scientific works of the late Professor Mac Cullagh.
They have been reprinted for the most part from the
Proceedings and Transactions of the Royal Irish Aca-
demy, in which they originally appeared. Some few
have been taken from the Philosophical Magazine.

Prof. Mac Cullagh’s most important contributions
to Science were made in the departments of Physical
Optics and Geometry—the first class being very much
the larger. The discrepancy is not, however, as great
as might at first sight appear. A considerable part
of his Optical researches, more especially those of
earlier date, really belong to the domain of Geometry;
and, if they were so classed, the inequality between
the classes would be much reduced. Such a classifica-
tion, however, involving the separation of the purely
geometrical propositions from their physical applica-
tions, would be exceedingly- inconvenient, and these
propositions have been allowed to remain in the con-

nexion in which the author placed them.
b

iv Preface.

In his earlier Optical Memoirs, Prof. Mac Cullagh
aimed chiefly at elucidating, by means of geometrical
theorems, the physical theory of Fresnel. This is the
principal object of the Memoirs I-IV. in the present
volume. In V. occurs the first notice of a problem
which subsequently occupied so large a space in Prof.
Mac Cullagh’s researches, namely, ‘the investigation
of the laws according to which polarized light is re-
flected and refracted at the surface of a crystalline
medium. This problem is discussed at length in XI.
and XIV. In the former of these memoirs he deduces
a solution of the problem from certain assumed phy-
sical principles. In the second he seeks to establish
the theory upon a strictly mechanical basis by means of
the general dynamical equation of Lagrange. These,
which.are the principal memoirs treating of the general
question, are supplemented by Memoirs XVI.—XIX.,
in which the same problem is discussed. Two other
important questions, namely, metallic reflexion and the
double refraction of quartz, which required a peculiar
mode of treatment, are considered in Memoirs VL,
VII, XV., XVII., XXI.

Prof. Mac Cullagh’s contributions to pure Geo-
metry, excluding, as has been said, all those theorems
which have been introduced by the author as auxiliary
to his Optical researches, form the second Part of the
present volume. The first of these is a Memoir on

Preface. Vv

the Rectification of the Conic Sections, which (with
his first Optical Memoir) was communicated to the
Royal Irish Academy shortly after he obtained his
B. A. degree. This was followed, many years after-
wards, by an elaborate memoir, which may, indeed, be
fairly called a treatise: ‘‘On Surfaces of the Second
Order.’’ In this memoir a new definition is given for
this class of surfaces analogous to the well-known
mode of defining curves of the second order by means
of a focus and directrix.

_ The articles contained in the third and fourth parts
of the present volume were not published during the .
lifetime of the author. They are records of Courses
of Lectures on the subjects of Rotation and Attrac-
tion, given by Prof. Mac Cullagh. These records
were preserved by Professors Haughton and Allman,
and communicated by them to the Royal Irish Aca-
demy.

Two short Papers on Egyptian Chronology, which,
like most of Prof. Mac Cullagh’s writings, were origi-
nally communicated to the Royal Irish Academy, have
been printed at the end of this volume.

CONTENTS.

PART I.-

PHYSICAL OPTICS.

PAGE

I. On the Double Refraction of Light in a Crystallized Medium,
according to the principles of Fresnel, : eS oe
II. On the Intensity of Light when the Vibrations are Bliptial, . 14
III. Note on the subject of Conical Refraction, ‘ 17

IV. Geometrical Propositions applied to the Wave Theory of Light, 20
V. A Short Account of some Recent Investigations Concerning the
Laws of Reflexion and Refraction at the Surface of Crystals, 55

VI. Laws of Reflexion from Metals, : ‘ ; . 58

VII. On the Laws of the Double Refraction of Quartz, ; ; . 63

VIII. On the Laws of Reflexion from Crystallized Surfaces, ° 75
TX. On the Probable Nature of the Light Transmitted by the Dia-

mond and by Gold Leaf, . ; : ‘ ‘ ; . 82

X. On the Laws of Crystalline Reflexion, . : . 83

XI. On the Laws of Crystalline Reflexion and Tatenntian, ‘ 37

XII. On a new Optical Instrument, intended chiefly for the purpose
of making Experiments on the Light Reflected from Metals, 138

XIII. Laws of Crystalline Reflexion.—Question of priority, ‘ . 140
XIV. An Essay towards a Dynamical Theory of Reflexion and Refrac-
: tion, . J 5 ; . .

XV. On the Optical Laws of Rock. ents: ‘ : ; ; . 185

XVI. On a Dynamical Theory of Crystalline Reflexion and Refraction :
(Supplement), . ‘ : ; ; ; ’ ; . 194

ars (4%

a

Viii Contents.

XVII. Notes on Some Points in the Theory of Light,
XVIII. On the Problem of Total Reflexion,
XIX. On the Dispersion of the Optic Axes, and of itis Axes of
Elasticity in Biaxal Crystals, . . . «. .
XX. On the Law of Double Refraction, . ;
XXI. On the Laws of Metallic Reflexion, and on the Mode of
making Experiments upon Elliptic Polarization,

XXII. On the attempt lately made by M. Laurent to explain, on
mechanical principles, the Phenomenon of Circular Pola-
rization in Liquids, . ‘ ; ‘

XXIII. On Total Reflexion, .

PART II.
GEOMETRY.

I. Geometrical Theorems on the Rectification of the Conic Sec-

tions, . 5 r

II. On the Surfaces of the Second Gris. $ .

III. Note relative to the comparison of the Arcs of Corser tari
larly of Plane and Spherical Conies, ° ‘

IV. Note on Surfaces of the Second Order,

PART ITI.
ROTATION.

I. On the Rotation of a Solid Body round a Fixed Point; being an
account of the late Professor Mac Cullagh’s Lectures on that
subject. Compiled by the Rey. Samuel sea He Fellow of
Trinity College, Dublin, ; ‘ ; ;

PAGE

187
218

221
227

230

249
250

. 255
. 260

. 318

. 321

. 329

Contents. ix

PART IV.
ATTRACTION.
PAGE
I. On a Difficulty in the Theory of the Attraction of Spheroids, . 349

II. On the Attraction of Ellipsoids, with a new demonstration of
Clairaut’s Theorem, being an account of the late Professor
Mae Cullagh’s Lectures on those subjects. Compiled by George
Johnston Allman, LL.D., of Trinity College, Dublin, . . 362

SUPPLEMENT.
EGYPTIAN CHRONOLOGY.
I. On the Chronology of Egypt, . . 373

II. On the Catalogue of Egyptian Kings, viioh is ‘malts es by
the name of the Laterculum of Eratosthenes, . ; . 376.

I.—ON THE DOUBLE REFRACTION OF LIGHT IN A CRYS-
TALLIZED MEDIUM, ACCORDING TO THE PRINCIPLES
OF FRESNEL. .

[Transactions of the Royal Irish Academy, Vou. xv1.—Read June 21, 1830.]

Tue mathematical difficulties under which the beautiful and
interesting theory of Fresnel has hitherto laboured are well
known, and have been regarded as almost insuperable. He tells
us, in his Memoir (see the Memoirs of the Royal Academy of
Sciences of Paris, tom. vii. p. 136), that the calculations, by
which he assured himself of the truth of his construction for
finding the surface of the wave, were so tedious and embarrass-
ing, that he was obliged to omit them altogether. A direct de-
monstration has since been supplied by M. Ampere (Annales de
Chimie et de Physique, tom. xxxix. p. 113) ; but his solution is
excessively complicated and difficult.

Judging from the simplicity and elegance of the results that
there must be some simple method of arriving at them, I have
been led to consider the subject with the attention which it
merits, and have succeeded in discovering a method by which
the whole may be explained with that simplicity which is cha-
racteristic of every theory that is founded in nature.

In the following Paper I shall give a brief view of this
method, sufficient to enable those who are acquainted with the
mechanical principles laid down in the original memoir of
Fresnel, to trace, at a glance, the connexion between the several

B

2 The Double Refraction of Light in a Crystallized

parts of his theory. For this purpose it will be convenient to
premise the following Geometrical Lemmas :—

1. If a, b,c, be the semiaxes of an ellipsoid, and a, (3, y,
the angles which they make with a perpendicular from the centre
on a tangent plane, the square of the perpendicular will be
equal to

a’ cos’ a + b* cos* B + ¢ cos’ y.

Let a plane through the point of contact Q, and one of the
semiaxes OA, intersect the ellipsoid in
the ellipse AON, and the tangent plane

in the tangent QZ, and draw QW per- m Q
pendicular to OA; then OA is a semi- rae

axis of the ellipse 4OJ, and therefore is 0
a mean ole between OW and a
OL; whence OL =—, denoting OW by Fig. 1.

x. But if p denote “the length of the perpendicular from the
centre O on the tangent plane at Q, the cosine of the angle

which it makes with 0.4 will be equal to

La

Re
OL? and therefore

we
cos a ==, and @ cos a=

Similarly,

Hence
x y 2
a cos’ a + b? cos* 3 + & cos’ y = p? (5+ Jo. 5) oe

Cor.—Since 2, y, 2, are as the cosines of the angles which OQ
makes with the semiaxes, it appears from the demonstration that
the cosines of the angles which the perpendicular to a tangent
plane makes with the semiaxes are, with respect to each other,
directly as the cosines of the angles which the semidiameter
through the point of contact makes with the semiaxes, and in-
versely as the squares of the semiaxes themselves.

2. If the semiaxes a, 6, c, and a’, b’, ¢, of two concentric

Medium, according to the Principles of Fresnel. 3

ellipsoids coincide in direction, and be reciprocally proportional,
so that ada’ = bb’ = cc’ = k*; andif a semidiameter OR of the one
be cut perpendicularly in P by a plane which touches the other,
then will OR be inversely as OP, so that OP x OR will be
always equal to 4’.

Let OR be a semidiameter of the ellipsoid whose semiaxes
are a’, b’, ¢; and let a, 3, y, be the angles which it makes with
them ; then if 2, y, s, be the co-ordinates of R, we have

2 2 2
x z
Feast 1 ie R P

im sala P
and therefore :
Q
1 cos’ a cos’ 3 i. cos? y

OR? = a? + b? a =

Fig. 2.
= = (@ cos’ a + BD? cos® B + ¢ cos? .)
But by the preceding lemma, since OP is perpendicular to the
tangent plane at Q, we have

OP? = a cos’ a + 8 cos’ B + ¢ cos? y.

Hence
1 OP
OR Kk’?
and therefore
OP x OR = k’.

3. If through the point of contact Q the straight line OQ be
drawn to meet in VV the tangent plane at R, it will meet it at
right angles.

For the cosines of the angles made by OP with the semi-
axes are directly as the cosines of the angles made by OQ with
them, and inversely as the squares of a, b, ¢ (Lem. 1, Cor.) ; and
the cosines of the angles made with the semiaxes, by a perpen-
dicular to the tangent plane at R, are directly as the cosines of
the angles made with them by OP, and inversely as the squares
of a’, 0’, c, or as the same cosines and the squares of a, b, ¢, di-
rectly ; that is, simply, as the cosines of the angles made by OQ

B2

4 The Double Refraction of Light in a Crystallized

with the semiaxes. Hence OQ coincides with the perpendicular
to the tangent plane at R. :

4. If a perpendicular at O to the plane POQ meet the sur-
face of the ellipsoid abe in g, then will OQ and Og be the semi-
axes of the section QOg made by a plane passing through
them.

For the tangent plane at Q and the plane QOg are perpen-
dicular to POQ, and therefore the intersection of the two former,
which is a tangent to the ellipse QOq at Q, is perpendicular to
OQ; whence OQ is one semiaxis, and Og the other.

If the perpendicular Og meet the other ellipsoid in r, then
OR and Or will be the semiaxes of the section ROr made by a
plane passing through them; for (by lem. 3), the straight line
OQN is perpendicular to the tangent plane at 2.

5. In a straight line, at right angles to any diametral section
QOq of the ellipsoid abc, let OT and OV be taken equal to OQ
and Og, the semiaxes of the sectica, and imagine the double sur-.
face which is the locus of all the points Zand V; then if OS
be perpendicular to the plane which touches this surface in 7,
and OP to that which touches the ellipsoid in Q, the lines OP
and OS will be equal and perpendicular to each other, and the
four, OP, OQ, OS, OT, will lie in the same plane, which will be
at right angles to Og.

By the preceding lemma it is evident that Og is perpendicular
to the plane POQ; and since OT is perpen-
dicular to the plane QOq, it follows that OP, T P
OQ, OT, are in the same plane at right
angles to Og. In the surface which is the
locus of 7, and in the plane 70g, let a point
T, be taken indefinitely near to 7'; then the
plane of the section at right angles to O7, [ °
will pass through OQ, and will have one of
its semiaxes (that to which OT, is equal)
indefinitely near to OQ, and therefore differ-
ing from OQ by an indefinitely small quan-
tity of the second order, adopting, for brevity, the language of

Fig. 3.

Medium, according to the Principles of Fresnel. 5

infinitesimals. ‘To see this, it is only necessary to recollect that,
by the property of maxima and minima, the semiaxis differs
from any semidiameter indefinitely near to it, such as OQ, by an
indefinitely small quantity of the second order. Hence, since
OT is equal to OQ, and OT, to the above-mentioned semiaxis,
it follows that OT and OT, differ by an indefinitely small quan-
tity of the second order, and that therefore the angle OTT’ is
- ultimately a right angle: consequently the tangent to the curve
in which the plane TOq intersects the locus of 7’ is perpendi-
cular to the plane PQOT. But the tangent plane at 7' passes
through this tangent, and therefore the perpendicular OS to the
tangent plane must lie in the plane PQOT.

Again, let a point 7” indefinitely near to 7, and in the plane
PQOT, be taken in the surface which is the locus of 7, and let
the plane of the section which is perpendicular to OT” intersect
the plane PQOT in the straight line OQ which meets the ellip-
soid in Q’. Then that semiaxis of the section to which O7” is:
equal will be indefinitely near to OQ’, and will therefore differ
from it by an indefinitely small quantity of the second order.
Hence, since OT is equal to OQ, the angle OTT’ will be ulti-
mately equal to OMQ’; and therefore, 7'S and QP being tan-
gents, the angles OTS and OQP are equal. But O7'= OQ, and
the angles P and S are right; therefore OS = OP, and the angle
SOT = POQ; whence SOP = TOQ = a right angle.

Similarly, if one perpendicular be let fall from O on a plane
touching the locus in V, and another on the plane touching the
ellipsoid in g, it may be proved that the two perpendiculars are

equal and at right angles to each other, and that, with the lines
OV and Og, they lie in a plane at right angles to OQ.

6. An ellipsoid being cut by any plane through its centre,
the difference between the squares of the reciprocals of the semi-
axes of the section is proportional to the rectangle under the
sines of the angles which the plane of the section makes with
the planes of the two circular sections of the ellipsoid.—(See
Fresnel’s Memoir, p. 150.)

6 The Double Refraction of Light in a Crystallized

Let the plane which cuts the ellipsoid intersect its circular
sections in the lines OR’, OS’, and
let the principal section AOC of the Py
ellipsoid cut the circular sections in
OR and OS; then OR, OS, OR, ®
OS’, will be all equal to the mean
semiaxis OB, and hence the semiaxes 8
OA’, OC’ of the section A’OC’ will c
bisect the acute and obtuse angles . % ;
made by OF’ and OS’. Leta plane «
through OB and OA’ intersect the
principal plane ACO in the line OT. “
Then, by the nature of the ellipse, we Fig. 4.
have, in the ellipse A’OC’,

1 1 1 Pe ee
OR* OA? aa a 5a) Baas ORs

and in the ellipse BOT,

POE SO - Gp) sin’ BOA

Hence, observing that OB and OR’ are equal, we have

1 % 1 “(5 - 1 \sin? BOA’
OC? OA?® \OB OT?) sin? AOR”

But the ellipse OAC gives
Meas ace ay, Caf) YokT” taal
OR? OT* \OC? OA

) (sin? AOR - sin? AQT)

1 1 j j
; (oo= - Gar) * 82 ROT sin SOT.
Therefore

Looe died Abie dol se BOs
200% OA® NOO* OAs) ane a Ole
because OB and OR are equal.

sin ROT sin SOT,

Medium, according to the Principles of Fresnel. 7

Imagine a sphere described from the centre O with a radius
equal to OB, and passing through # and S’, and cutting A’O
in P. The sides BP and P&’ of the spherical triangle BPR’
subtend the angles BOA’ and A’OR’ at the centre; and its
angle PBR’ is equal to the angle ROT; whence the sines of the
sides being proportional to the sines of the opposite spherical
angles, it follows that

ee sin ROT
is equal to the sine of the spherical angle BR’P, which is the
angle made by the section A’OC’ with the plane of the circular
section BOR. Similarly, by means of the spherical triangle
BPS’, it may be shown that

sin BOA’

sin A’OS’

is equal to the sine of the angle made by the plane 4’OC’ with

the plane of the circular section BOS. Therefore, since the
angles A’OR’ and A’OS’ are equal, it follows that

ic ene ee ee
00% 0A» °" OC? OA

multiplied by the product of the sines of the angles which the
plane A’OC’ makes with the planes of the two circular sections.

I shall now demonstrate a geometrical construction for find-
ing the magnitude and direction of the elastic force arising from
a displacement in any direction—a construction which, with the
help of the preceding lemmas, will lead us immediately to all
the conclusions established by Fresnel.

Let O be the position of a point in the medium when
quiescent, and let three rectangular axes passing through it, and
fixed in space, be taken for the axes of co-ordinates. For a small
displacement in the direction of x, let the elastic forces, excited
in the directions of a, y, z, be a, b, c; for an equal displacement
in the direction of y let the forces be a’, ', ¢ ; and for the same

sin SOT

8 The Double Refraction of Light in a Crystallized.

in the direction of < let them be @’, 0”, c’. Then, if a point
receive an equal displacement in a direction OZ making with
OX, OY, OZ, the angles a, B, y, the forces in the direction of
#, y, (denoting then by X, Y, Z, respectively), will be

X=acosa+d cos 3 +a” cos y,
Y =bcosa+ b’ cos B + 0” cos y,
Z=ccosa+c cos B+ c cos 7,

as follows from considering (see Fresnel’s Memoir, p. 82) that
the force arising from a displacement in any direction is the
resultant of the forces arising from the three displacements in
the directions of x, y, s, which are the statical components of
that displacement. But since (p. 90) the force in the direction
of one of the axes, arising from a displacement in the direction
of another, is equal to the force in the direction of the latter,
arising from an equal displacement in the direction of the former,
it follows that a = 6, a’ =, b” = ¢; and hence

X=acosa+ bcos 3 + ¢ cos y,
Y =) cosa+ b cos3+e¢ cosy,
Z=ccosat+¢ cos +c" cosy.

Let aa* + by? + c’s* + 2c¢ys + 2czu + Way = 1 be the equation
of a surface of the second order. Let OJ intersect it in J, the
co-ordinates of J being 2’, 7/, x’; then the equation of the tan-
gent plane at J will be, by the known formule

(ax! + by’ + cx’) a + (ba + Wy + ee) y + (ce + ly + e#) 2 =1.

Put OF=~r, and let the tangent plane intersect OX, OY, OZ,
in the points P, Q, R; then from this equation we have

Apne! + by +08 = 9 (a cos a + 6 cos B + ¢ cos y) = 7X,
Gan thy ted =r (6 cos a+b’ cos B + ¢ cos y) =n,
1

OR * ca + ¢y +é'% =r (ec cos ate cos B +e” cos y) = 7Z,

Medium, according to the Principles of Fresnel. 9

* Now if p be the length of the perpendicular let fall from O
on the tangent plane, the cosines of the angles which this per-
_pendicular makes with the axes of co-ordinates will be equal to

ea ee

OP? OD OR
respectively, that is, to prX, pr Y, prZ; and since the sum of the
squares of these cosines is equal to unity, we have

pr f/X?+Y?+ Z*=1,

Hence it appears that the perpendicular let fall from O on
the tangent plane is parallel to the direction of the resultant
elastic force, and that the magnitude of the resultant is ex-

| 1
pressed. by ee

From this conclusion we may, with the greatest facility,
deduce several corollaries.

1°. Since the elastic force is supposed finite, whatever be the
direction of the displacement, it is manifest that the above-men-
tioned surface of the second order must be an ellipsoid, and that
when the displacement is in the direction of any of the three
axes of the ellipsoid, the elastic force excited will be in the
direction of the same axis, because the tangent plane will be
perpendicular to it. Hence the remarkable consequence, that
there are always three axes of elasticity at right angles to each
other.—(Memoir, p. 93.) Also the elasticities arising from equal
displacements in the directions of the three axes are inversely
as the squares of the axes; and hence, the positions of the axes
and the elasticities in their respective directions being given, the
ellipsoid may be constructed.

2°. The ellipsoid being thus constructed, the direction of the
elastic force, arising from a displacement in the direction of any
of its semidiameters, will be parallel to the normal at the extre-
mity of that semidiameter; and for equal displacements the
magnitude of the force will be inversely as the rectangle under
the semidiameter and the perpendicular from the centre on the
tangent plane at its extremity. If the displacements are pro-

10 The Double Refraction of Light in a Crystallized

portional to the semidiameters, the elastic forces will be both
parallel and proportional to the normals; for the normal, ter-
minated by any of the principal planes, is inversely as the per-
pendicular on the tangent plane.

3°. If the elastic force be resolved into two, one parallel and
the other perpendicular to the direction of the displacement, the
former will be inversely as the square of the semidiameter i in
the direction of the displacement. .

4°. If the ellipsoid be cut by a plane through its centre, and
if the elastic force arising from a displacement in the direction
of either axis of the section be resolved parallel and perpendi-
cular to that axis, the part perpendicular to the axis will also be
perpendicular to the plane of the section. For (by Lem. 4) the
plane passing through one axis and the perpendicular to the
tangent plane at its extremity, is perpendicular to the other axis,
and therefore to the plane of the section. But if the displace-
ment be in the direction of any other diameter of the section, |
the elastic force, resolved perpendicularly to that diameter, will
be oblique to the plane of the section.

To apply these things to the double refraction of light
in a crystallized medium, imagine the ellipsoid to be described
as above, and let it be cut through its centre by a plane
parallel to that of a plane wave of light incident on the crystal ;
then if the vibrations of the light be parallel to either of the
axes of the section, the plane containing the direction of the
vibrations and that of the elastic force arising from them will
be perpendicular to the plane of the wave (No. 4, preceding) ;
and therefore, according to Fresnel’s theory, the direction of the
vibrations will remain parallel to itself, while the wave is pro-
pagated. But if the light be common light, or if it be polarized,
and the plane of polarization be not perpendicular to either of
the axes of the section, the wave will be divided into two others
having the directions of their vibrations parallel to the semiaxes
of the elliptic section, and their planes of polarization perpendi-
cular to them: their velocities of propagation—measured in a
direction perpendicular to their plane—will be different, and

Medium, according to the Principles of Fresnel. 11

will be in the sub-duplicate ratio of the elasticities in the direc-
tion of their respective vibrations, and therefore (by No. 3) in-
versely as the semiaxes of the section to which those vibrations
are parallel.

If a wave be propagated in ali directions from an origin O
within a crystal, its surface will at each instant be touched by
the simultaneous position of a plane wave which passed through
O at the instant when the former began to be propagated
(Memoir, p. 127). Hence, to find the surface of the double
wave in a crystal, let the above-mentioned ellipsoid be cut by
any plane through its centre O, and imagine two other planes
parallel to this section, and at distances from it which are third
proportionals to its semiaxes, OR, Or, and any given line /: the
double surface which touches these planes in all their positions
will be the surface of the wave.

Now conceive another concentric ellipsoid, having the direc-
tions of its semiaxes the same, but their lengths a, b, c, inversely
proportional to those of the former, so that the rectangle under
any coinciding pair of semiaxes is equal to /*: then if a plane,
touching this second ellipsoid in Q, cut OR perpendicularly in
P, the line OP will be a third pro-
portional to ORand & (Lem. 2) ; and T s
if Or intersect the second ellipsoid ¥
in g, the semiaxes of the section

QOq will be OQ and Og (Lem. 4). fis
is

Draw OT perpendicular to QOq
and equal to OQ, and conceive the
surface which is the locus of the . 8

point J to be described, and a tan- Fig. 6.

gent plane, to which the line OS is drawn perpendicular, to be
applied at the point Z. Then OS will be perpendicular to the
plane ROr and equal to OP (Lem. 5): and hence the point 7
always lies in the surface of the wave. Similar things may be
proved with respect to the other semiaxis Or of the ellipse ROr.
Hence we deduce the following construction for the surface of
the wave :—

12 The Double Refraction of Light in a Crystallized

“ Describe an ellipsoid whose axes are in the directions of the
axes of elasticity, the squares of their lengths being directly as
the elasticities in their respective directions ; cut the ellipsoid by
a plane through its centre, as QOg, and in a perpendicular to
that plane take two portions O7' and OV equal to the semiaxes
OQ and Og of the section. The double surface which is the locus
of the points 7’ and V is the surface of the wave.”

As to the planes of polarization of the rays belonging to the
two parts of the wave, Fresnel has shown how to find them by
means of a surface which he calls the surface of elasticity. But
it is desirable to be able to find them by means of the same
ellipsoid which serves to find the surface of the wave. Now 7'S,
parallel to OR, is the direction of the vibrations of the ray O7,
and the tangent plane at Q is perpendicular to OR, and there-
fore parallel to the plane of polarization of the ray O7. In like
manner the plane of polarization of the ray OV is parallel to
the tangent plane at g. Hence the planes of polarization of two
different rays, having a common direction, are parallel to the
planes which touch the ellipsoid at the extremities of the semi-
axes of the diametral section perpendicular to their common
direction.

When two of the axes are equal, the ellipsoid becomes a
spheroid, and the crystal is said to be uniaxal, the double refrac-
tion being regulated by the third axis which
is perpendicular to their plane. Let 4OB
be a section of the spheroid through the
third axis O.A, which is its axis of revolu-
tion; take OB’ = OB, and OA’ = OA, and
let the ellipse A’OP and the circle BOB’
revolve about OB as an axis; they will
describe a surface compounded of a sphe- Fig. 6.
roid and sphere, which will in this case be the surface of the
double wave. For if OW be the direction of a ray, and if a
plane perpendicular to OM cut the ellipse AOB in OQ, and the
equator of the spheroid in Og, the lines OQ and Og, of which
the latter is equal to OB, will be the semiaxes of the section

Medium, according to the Principles of Fresnel. 13

QOq. Taking, therefore, O7 and OV equal to OQ and OB, the
locus of V will evidently be the circle BOB’; and the locus of
T will be the ellipse A’OB’, since the ae BOT’ is equal to
the angle BOQ.

In the general case, O7' and OV, which are equal to OQ and
Oq, represent the velocities of the two different sorts of rays
having a common direction in the crystal; and two right lines,
which lie in the plane of the greatest and least axes of the ellip-
soid, and are perpendicular to its two circular sections, are
called the optic axes. It appears, therefore, by the 6th Lemma,
that the difference of the squares of the reciprocals of the velo-
cities of the two rays, having a common direction in the crystal,
_ is proportional to the product of the sines of the angles which
that direction makes with the optic axes. This is the celebrated
law of M. Biot, to which he was led by analogy, and which he
afterwards found to agree with that previously laid down by
Dr. Brewster.

Such, in their simplest form, are the principal features of the
Mechanical Theory of Double Refraction, invented by the late
M. Fresnel—a theory which would do honour to the sagacity of
Newton, and which gives us ample reason to regret that the life
. of its author was not longer spared, to enrich with further dis-
coveries his favourite science. Of him it may be said, as New-
ton said of Cotes—and apparently with much greater reason—
that if he had lived longer, we should have known something at
last of the laws of nature.

II.—ON THE INTENSITY OF LIGHT WHEN THE VIBRA-
TIONS ARE ELLIPTICAL.

[Edinburgh Journal of Science, April, 1831.]

Accorprnc to the opinions commonly received, the intensity of
light, in the undulatory hypothesis, is proportional to the vis
viva, which again is proportional to the square of the greatest
velocity. Now the greatest velocity will be the same in an
ellipse and a right line which have the same period, if the
greater axis of the former be equal to the whole extent of the
latter ; so that in elliptic vibrations the intensity would be in-
dependent. of the minor axis, which is far from being true. I
would propose the integral Sv’dt—so remarkable for its mecha-
nical properties—as the measure of the intensity, the integral
being extended to the whole time of a vibration. This gives
precision to the notion of vis viva, and leads, moreover, to an
elegant result ; for if a and } denote the semiaxes of the ellipse,
and 7' the time of vibration, the integral, by an easy calculation,

will be found equal to = (a? + b’), showing that for the same

colour the intensity is proportional to the sum of the squares of
the semiaxes, and that for different colours it increases with the
rapidity of the vibrations, as it would be natural to suppose
d priori.

This theorem assigns very simply the reason why two por-
tions of light polarized at right angles do not interfere; but to

On the Intensity of Light, &c. 15

show this it will be necessary to lay down the following general
rule for compounding rectilinear vibrations having the same
period, whatever be the difference of their origin and direc-
tion :—

Let AA’ and BB (Fig. 7), bisecting each other at O, represent
in extent and direction the vibrations
to be compounded, and suppose C
and D to be two simultaneous posi-
tions of the moving molecule, which
it would have in virtue of each vi-
bration singly. Complete the paral-
lelograms OF and OP, and through
P describe an ellipse having O for
its centre, and touching the sidés of Fig. 7.
the parallelogram OF; this ellipse will represent the resulting
vibration; it will have the same period as the compound one,
and equal areas will be described in equal times about its centre.

To apply this construction to the case proposed, it is neces-
sary to show that when OA and OB are constant and at right
angles to each other, the intensity of the elliptic vibration, or
the sum of the squares of the semiaxes, is independent of the
difference of origin, or of the position of the points C and D.

Now in an ellipse, when a perpendicular from the centre ona
tangent makes an angle ¢ with the major axis, the square of its
length is equal to a? cos* ¢ +0? sin’ ¢. If @ be the angle which
the major axis makes with OA, it will make its complement
with OB, and we shall have

OA? = a’ cos’ > + 0’ sin’ ¢,

OB = a’ sin’ } + 0° cos? ¢,
and therefore
OA? + OB =a’ + 0.

Hence the intensity is independent of the difference of sa
and therefore the rays do not interfere.
This remarkable circumstance is commonly accounted for by

16 On the Intensity of Light, &c.

observing, that since the component velocities are at right angles
to each other, the square of the actual velocity must be the sum
of their squares ; but this proves the proposition only when the
greatest velocities are simultaneous, which happens only in the
eases of a complete accordance or of a difference of a semi-
undulation in the interfering portions.

It may be observed also, that the method usually given for
finding the intensity of the vibration resulting from two or more
rectilinear vibrations proceeds upon no certain grounds. All the
‘vibrations may be reduced to two in rectangular directions, the
expression for the square of the velocity in each of their diree-
tions consisting of two parts, one of which is constant, and the
other depends on the cosine of an are increasing proportionally
to the time: the square of the resultunt velocity, therefore, con-
sists also of a constant part, and a part depending similarly on
the time ; and it is assumed that the intensity of the resulting
vibration is proportional to the former, which is true only in
the very particular cases just mentioned, if the intensity be
measured by the square of the greatest velocity.

This assumption, however, gives a correct result ; and the
reason that it does so is obvious from the principle laid down in
the commencement ; for if the expression for the square of the —
velocity be multiplied by the differential of the time and inte-
grated, the variable parts will vanish when the aa is ex-
tended to the whole time of a vibration.

Ill.—_NOTE ON THE SUBJECT OF CONICAL REFRACTION.

[From the Philosophical Magazine, Vou. u1., 1833.]

WueEn Professor Hamilton announced his discovery of Conical
Refraction, he did not seem to have been aware that it is an ob-
vious and immediate consequence of the theorems published by
me, three years ago, in the Transactions of the Royal Irish Aca-
demy, vol. xvi., pt.ii., p. 65, &c. The indeterminate cases of my
own theorems, which, optically interpreted, mean conical refrac-
tion, of course occurred to me at the time; but they had nothing

to do with the subject of that Paper; and the full examination
of them, along with the experiments they might suggest, was
reserved for a subsequent essay, which I expressed my intention
of writing. Business of a different nature, however, prevented
me from following up the inquiry.

I shall suppose the reader to have studied the passage in
pp. 75, 76, of the volume referred to. He will see that when
the section of either of the two ellipsoids employed there is a
circle, the semiaxes—answering to OR, Or, and to OQ, Og, in
the general statement*—are infinite in number, giving of course
an infinite number of corresponding rays. And this is conical
refraction. I shall add a few words on the two cases :—

1. When ROr is a circle, any two of its rectangular radii
may be taken for OR and Or. The line OS and the tangent
plane perpendicular to it at S are fixed; but the point of con-

* The right line Ogr is perpendicular to the plane of the figure, and intersects
the two ellipsoids in g and r.

a

Cc

18 Note on the Subject of Conical Refraction.

tact 7’ is variable, for the plane ROS in which it lies changes
with OR. Thus we get a curve
of contact on the tangent plane
of the wave surface, and a cone v
of rays OT derived from the same
incident ray. The vibrations of =}.
any ray OT are in the line 7S \ ’
passing through the fixed point S,
as follows from a general remark
in the place referred to.

The three right lines OQ, Or, OT, are at right angles to each
other, and a geometer will observe that the first two of them are
confined to given planes. For Or is always in the plane of the
circle ROr ; and the point Q must be in a given plane, because
the line OP, perpendicular to the plane that touches the ellip-
soid in Q, is in a given plane ROr.

2. When QOq is a circle, the points 7’ and V coincide in a
nodal point », where the two sheets of the wave surface cross
each other. At this point there are an infinite number of tan-
gent planes, for OQ and Og are now indeterminate. The same
refracted ray On may therefore be derived from any one of an
infinite number of incident rays, and its polarization will differ
accordingly ; for the vibrations are in the line »S drawn from
the node to the foot of the perpendicular OS on the tangent
plane. The ray On, however, is always accompanied by another,
but variable, refracted ray.

The lines, OP, Og, OS, are at right angles to each other, and
the first two of them are confined, as before, to given planes.
For 07 is in the plane of the circle QOq; and OP, being perpen-
dicular to the tangent plane at Q, must lie in a given plane.
These given planes are parallel to two principal tangent planes
passing through n, and touching the circle and ellipse that com-
pose the wave section in the plane of the nodes: whence it is
easy to see that every nodal tangent plane intersects the two
principal tangent planes in lines that are constantly at right
angles; for these lines are parallel to OP and Og.

iy s

Fig. 8.

Note on the Subject of Conical Refraction. 19

The examination of both cases is completed by the following
theorem :—

When three right lines at right angles to each other pass
through a fixed point, in such a manner that two of them are
confined to given planes, the plane of these two, in all its posi-
tions, touches the surface of a cone-whose sections parallel to the
given planes are parabolas ; while the third right line describes
another cone, whose sections parallel to the same planes are
circles.

The application is obvious. We see that the curve of contact
in the first case is a circle. The points S in the second case are
also in a circle.

Nore ON THE ABOVE, ADDRESSED BY Proresson Mac CuLtaGH 10 THE
Epitors oF tHe Purtosopnican Macazine.—Vo.. ur. 1838.

Tue introductory part of my Note which appeared in your last Number
was written in haste, and I have reason to think it may not be
rightly understood. You will therefore allow me to add a few obser-
vations that seem to be wanting.

The principal thing pointed out in the Paper published some time
ago in the Zransactions of the Royal Irish Academy is a very simple
relation between the tangent planes of Fresnel’s Wave Surface and the
sections of two reciprocal ellipsoids. Now this relation depends upon
the axes of the sections, and therefore naturally suggested to me the
peculiar cases of circular sections in which every diameter is an axis.
Thus a new inquiry was opened to my mind. And accordingly, with-
out caring just then to obtain final results, which seemed to be an
_ easy matter at any time, I expressed in conversation my intention of
returning to the subject of Fresnel’s theory in a supplementary Paper.
The design was interrupted, and I was prevented from attending to it
again, until I was told that Professor Hamilton had discovered cusps
and circles of contact on the wave surface. This reminded me of the
eases of circular section, and the details given in my last note were
immediately deduced.

c2

IV.—GEOMETRICAL PROPOSITIONS APPLIED TO THE
WAVE THEORY OF LIGHT.

[ Transactions of the Royal Irish Academy, Vou, xvu.—Read June 24, 1833.]

Parr I.—GerometricaL Propositions.

1. Turorem I.—Conceive a curved surface B to be generated
from a given curved surface 4 in the following manner: having
assumed a fixed origin O, apply a tangent plane at any point Q
of the given surface, and perpendicular to this plane draw a
right line OPR cutting the plane in P,

and terminated in R, so that OP and OR ae
may be reciprocally proportional to each x 7
other, their rectangle being equal to a Fig. 9.

constant quantity 4’, and let all the points R taken according
to this law generate the second surface B. Then the relation
between these two surfaces, and between the points Q and R, will be
reciprocal ; that is to say, if a tangent plane be applied at the
point F& of the second surface, a perpendicular OY to this plane
will pass through the point Q of the first surface, and ON and
OQ will be reciprocally proportional to each si the rect-
angle under them being also equal 4’.

2. To prove this theorem, take a point q, in the tangent plane
of the surface A, and near the point of contact Q (Fig. 9).
Through q let several other planes be drawn touching the sur-
face A in points Q’, Q”, Q”, &c., and draw the perpendiculars
OP R’, OP’ R’, OP’ R”, &e., according to the same law as
OPR. The points R, Rf’, Rk’, R”, &e., will thus be upon the
second surface B, and they will moreover be all in the same —

Geometrical Propositions. : 21

plane ; for from any one of them #’ let R’n be drawn perpendi-
cular to the right line Og and meeting Og in 7; then, on account
of the similar right-angled triangles OP’g and Onk’, the rect-
angle nOq will be equal to the rectangle R’OP’, or to the
constant quantity /*, so that the point ”, or the foot of the per-
pendicular let fall upon Og, will be the same for all the points
R, RF’, Rk’, R”, &e., and consequently all these points will lie in
a plane cutting the right line Ogv perpendicularly in n, so as to
make the rectangle nOqg equal to #*. Now, while the point Q
remains fixed, let the point g approach to it without limit in the
tangent plane at Q; and the points 2’, R’, R”, &e., will in like
manner approach without limit to the fixed point R; the plane
which contains all those neighbouring points having for its limit-
ing position the tangent plane at R. Also the point » will
ultimately coincide with VV. It follows, therefore, that the tan-
gent plane at & cuts the right line OQ perpendicularly in J, so
as to make the rectangle VOQ equal to A”.

3. Corollary.—lf any point Q upon the surface A should be
a point of intersection, where the surface admits an infinite
number of tangent planes, the perpendiculars from O upon these
planes will form a conical surface having O for its vertex. In
OQ take, as before, a point NV, so that ON x OQ =k’, and let a
plane passing through WV at right angles to OQ cut the conical
surface. The intersection will be a certain curve. From the
preceding demonstration it is evident that every point of this
curve belongs to the surface B, and that the plane which touches
this surface at any point of the curve cuts OQ perpendicularly
in V; or, in other words, that the same plane touches the surface
B through the whole eatent of the curve.

4. Two surfaces related to each other like A and B in the
preceding theorem may be called reciprocal surfaces, and points
like Q and R reciprocal points ; the radii OQ and OR may like-
wise be termed reciprocal. A familiar example of such surfaces
is afforded, as I have shown on a former occasion,” by two ellip-

* Transactions of the Royal Irish Academy, Vol. xvt., pt. ii., pp. 67, 68.—Supra, p. 3.

22 Geometrical Propositions applied

soids having a common centre at the point O, and their semi-
axes coincident in direction, and connected by the relation
ad = bl’ = cc =k? ; where a, b, c, are semiaxes of one ellipsoid in
the order of their magnitude, a being the greatest ; and a, b’, ¢,
those of the other ellipsoid, a being the least. The mean semi-
axes b and J’ coincide, and the circular sections of both ellipsoids
pass through the common direction of } and 0’.

5. It has also been shown with regard to those ellipsoids,
that if Q and & be reciprocal points on the surfaces of abe and
dv’¢ respectively, and if a right line Ogr, perpendicular to the
plane QOR, cut the first ellipsoid in g and the second in ”, the
lines OQ and Og will be the semiaxes of the section made in the
ellipsoid abe by a plane passing through them ; and the lines
OR and Or, in like manner, will be the semiaxes of the section
made in the other ellipsoid a b’ c’ by the plane in which they lie.

6. It may further be remarked, that if the radius OQ in one
of the reciprocal ellipsoids describe a plane, the corresponding
radius OR will describe another plane. For the planes touching
the ellipsoid abc in the points Q will all be parallel to a certain -
right line, and therefore the perpendiculars OR to these tangent
planes will all lie in a plane perpendicular to that right line.
These two planes, containing the reciprocal radii, may, for
brevity, be called reciprocal planes.

When two reciprocal radii lie in a principal plane, at rip
angles to a semiaxis of the ellipsoids, it is evident that two planes
intersecting in this semiaxis, and passing through the i es
radii, are reciprocal planes,

7. Turorem IT.—If three right lines at right angles to each
other pass through a fixed point O, so that two of them are con-
fined to given planes, the plane of these two, in all its positions,
touches the surface of a cone whose sections parallel to the given
planes are parabolas ; while the third right line describes ano-
ther cone, whose sections parallel to the given planes are circles.

Let the plane of the figure (Fig. 10), supposed parallel to one
of the given planes, be intersected by the other given plane in
the right line MN; and let OQ be perpendicular to the latter

to the Wave Theory of Light. 23

plane, while OP is perpendicular to the former and to the plane
of the figure, so that PQ being joined will meet IZW at right
angles inR. Let OA, OB, OC, be the three perpendicular lines,
of which OA is parallel to the plane
of the figure ; this plane will be inter-
sected by the plane of OA and OB in
a right line BT parallel to OA, and
therefore perpendicular to both OB
and OP, and to the plane BOP, and
to the line BP. Thus the angle PBT
is always a right angle, and therefore
BT always touches the parabola whose
focus is P and vertex R; or, which
comes to the same thing, the plane AOBT always touches the
cone which has O for its vertex, and the parabola for its section.

Again, since OB, OP, OC, are all at right angles to OA,
they are in the same plane, and therefore the points B, P, C,
are in the same straight line; and as BOC is a right angle, the
rectangle under BP and PC is equal to the square of the per-
pendicular OP ; but QOR is also a right angle, and therefore
QP x PR=OP*; whence BP x PO = QP x PR, and therefore
the points B, Rk, C, Q, are in the circumference of a circle, so
that the angle at Cis a right angle, being in the same segment
with the angle at R. Thus the point C describes the circle whose
diameter is PQ, and OC describes the cone of which this circle
is the section. ;

8. Of the two right lines OP and OQ perpendicular to the
given planes, one is also perpendicular to the plane of the section.
That one is OP. Its extremity P is the focus of the parabola.
The extremities of both are the extremities of the diameter PQ
of the circle. The vertex of the parabola is the point R, where
the diameter of the circle intersects that given plane to which
the plane of section is not parallel.

9. Turorem III.—In a straight line at right angles to any
diametral section QOq of an ellipsoid abc whose centre is O, let
OT and OV be taken respectively equal to OQ and Og, the semi-
axes of the section, and imagine the double surface which is the

N =

Fig. 10.

24 Geometrical Propositions applied

locus of all the points 7’ and V; then if OS be perpendicular to
the plane which touches the surface in 7, and OP to the plane
which touches the ellipsoid in Q, the lines OP and OS will be
equal and perpendicular to each other, and the four straight
lines OP, OQ, OS, OT, will lie in the same plane at right
angles to Og

10. This theorem is taken from a former communication to
the Academy.* The surface to which it relates, being the wave
surface of FRESNEL, is one of frequent occurrence in optical in-
quiries, and it is therefore desirable to give it a distinctive name
not derived from any physical hypothesis. I shall call it a
biaxal surface, from the circumstance implied in its construction,
and adopted as the definition on which the preceding theorem is
founded—namely, that any pair of its coincident diameters are
equal to the two axes of a central section made in the generating
ellipsoid abc, by a plane perpendicular to the common direction
of the two diameters. The name, perhaps, may appear the more
appropriate, as it reminds us of the place which the surface holds
in the optical theory of biaxal crystals.

11. Turorem IV.—The biaxal surfaces generated by two
reciprocal ellipsoids are themselves reciprocal.

For if Q and R (Fig. 11) be reciprocal points on the two
ellipsoids, abe and a’d’c’, a tangent plane at Q will cut OR pen-
pendicularly in P; a tangent plane at R
will cut OQ perpendicularly in WV; and “7
the rectangles ROP and NVOQ will be
equal to each other and to #’ (Art. 4).
Also if the straight line Ogr, at right angles
to the plane of the figure, cut the first ellip-
soid in g and the second in 7, then (5) the * e
elliptic section QOq will have OQ and Og ‘8
for its semiaxes, and the lines OR and Or Fig. 11.
will be the semiaxes of the other section ROr. Draw, therefore,
in the plane of the figure, the right lines OTZ and OSM pen-
pendicular to the right lines OQN and OPR, making OT, OL,

T 8

* Transactions of the Royal Irish Academy, Vol. xv1., pt. ii., pp. 67, 68.—Supra, p.4.

to the Wave Theory of Light. ee

OS, OM, equal to OQ, ON, OP, OR, respectively ; the angles.
at S and Z being of course right angles. Then it is evident
that the point 7' is on the biaxal surface generated by the ellip-
soid abe, bécause OT is perpendicular to the plane of the ellipse
QOq and equal to the semiaxis OQ; and by Theorem III. it
appears that OS is perpendicular to the tangent plane at 7. In
like manner, the point I/ is on the biaxal surface generated by
the other ellipsoid ad’c’, and OL is perpendicular to the tangent
plane at WZ. Moreover, the rectangles MOS and LOT, being
- equal to the rectangles ROP and NOQ, are each equal to /’.
Hence the proposition.is manifest.

12. As the ellipsoid whose semiaxes are a, b, c, may be called
the ellipsoid abc, so the biaxal surface generated by this ellipsoid
may be called the biaxal abc; and that which is generated by the
ellipsoid a‘d’c’ may be called the biaxal 0’?

13. Proposrrion V.—To find what properties of biaxal sur-
faces are indicated by the cases wherein one of the two sections
Q0q, ROr, in the preceding theorem, is a circle.

Case 1.—When QOg is a circular section of the ellipsoid abe,
the points 7’ and V (9), in the description of the biaxal surface
abe, coincide in a single point x. At this point there are an in-
finite number of tangent planes; because the semiaxes of the
circular section QOq being indeterminate, any two perpendicular
radii of the circle may take the place of OQ, Og, in the general
construction. The point x is therefore a point of intersection (3),
where the two biaxal sheets cross each other, and it may be called.
a nodal point, or simply a node. As OQ always lies in the plane
of the circle QOq, the. line OR, which is reciprocal to OQ, must
lie (6) in a given plane reciprocal to the plane of the circle.
And as 0¢ lies in the plane of the circle, we have three right
lines OR, Og, OS, which are at right angles to each other, and
of which the first two are confined to given planes. Therefore
by Theorem II. the third line OS describes a cone whose sections
parallel to the given planes are circles. Now, Z7S—or in the
present case »S—is parallel to the fixed plane which contains
OR, and therefore the point S describes a circle; or, in other

26 Geometrical Propositions applied

words, the feet of the perpendiculars OS, let fall from O on the
nodal tangent planes, occupy the circumference of a circle pass-
ing (8) through the nodal point.

14. Parallel to the plane of the circle and to its reciprocal
plane, conceive two planes passing through the node, and call
them the principal tangent planes at n. The plane of the circle
and its reciprocal plane are intersected in the right lines Og, OR,
by the plane gOR, which is parallel to a tangent plane at n.
Consequently this tangent plane at » intersects the two principal
tangent planes in lines that are parallel to Og, OR; and as Og,
OR are perpendicular to each other, it follows that every nodal
tangent plane intersects the two principal tangent planes in lines
that are at right angles. .

Hence again, the nodal tangent planes touch (7) the surface
of a cone whose sections, parallel to the principal tangent planes,
are parabolas. As this cone touches the biaxal surface all round
the point », it may be called the xodal tangent cone.

15. Case 2.—When Or is a circular section of the ellipsoid
dl’¢, any two perpendicular radii of the circle may be taken for
OR, Or: and because OR = UW’, and OR x OP =k? = bb’, we have
OP or OS equal to 4, the mean semiaxis of the ellipsoid ade.
Hence OS is given both in position and length ; for it is perpen-
dicular to the fixed plane ROr, and it is equal to d. Now, a
plane cutting OS perpendicularly at S is a tangent plane to the
biaxal abe; and we have just seen that this tangent plane re-
mains the same, whatever pair of rectangular radii are taken for
OR, Or. But the point of contact 7'is variable, for the plane
ROS in which it lies changes with OR. Therefore as OR re-
volves, the point 7’ describes a curve of contact on the tangent
plane of the biaxal abe.

The lines OR, Or, are in the fixed plane ROr; and as OQ
is reciprocal to OR, it lies in a fixed plane reciprocal to the plane
Ror (6). Therefore the first two of the three perpendicular right
lines Or, OQ, OT, are confined to fixed planes. Hencé the third
line OT describes a cone, whose sections parallel to these planes
are circles. But the tangent plane is parallel to the fixed plane

to the Wave Theory of Light. 27

ROr, and its intersection with OT describes the curve of contact.
Therefore the curve of contact is a circle passing (8) through the
point S.

16. We have examined the two cases of circular section with
reference only to the biaxal abe. If we examine the same cases
with regard to the second biaxal a’b’c’, we shall find that their
indications are reversed; the supposition which gives a node
upon one biaxal, giving a circle of contact on the other: and
that the node and the circle, thus corresponding, are so related,
that a line drawn from 0 to the node passes through the circum-
ference of the circle, cutting the plane of the circle perpendicu-
larly ; whilst every line drawn from O through the circumference
of the circle is perpendicular to some nodal tangent plane.

These things are evident on looking at the figure. For when —
ROr is a circle, it is plain that the point UY is a node of the
biaxal a’b’c’, since OI is perpendicular to the plane of the circle
ROr and equal to its radius OR. But we have already seen (15)
that when ROr is a circle, the other biaxal abc has a circle of
contact, whose plane is perpendicular to OY at the point S of its
circumference. The line O7'TL is perpendicular, in general (11),
to a tangent.plane at MW, and therefore perpendicular, in the
present case, to a nodal tangent plane ; whilst the point 7,
through which it passes, is on the circle of contact. It is also
evident that OT x OL = k’.

We have here an example of the general remark in the corol-
lary of Theorem I.

17. The section made in the biaxal surface abe, by any of
the principal planes of its generating ellipsoid, consists of an
ellipse and a circle.

For, let the plane QOq pass through one of the semiaxes, a,
and let it revolve round this semiaxis, while the right ine O7V
(9), perpendicular’ to the plane QOgq, revolves about O in the
plane of the semiaxes }, c. Then the semiaxis a of the ellopsoid
- will always be one of the semiaxes of the ellipse QOq; and if
OT be equal to this semiaxis, the point 7’ will describe a circle
with the radius a about the centre O. The other semiaxis of the

28 Geometrical Propositions applied

ellipse QOq is that semidiameter of the principal ellipse be which
lies in the intersection of the plane dc with the plane QOq; and
as OV is equal and perpendicular to this semidiameter, the point
V describes an ellipse equal to.bdc, but turned round through a
right angle, so that the greater axis of the ellipse described by V
coincides in’ direction with the less axis of the ellipse bc. As the
radius a of the circle is greater (4) than both the semiaxes 4, ¢,
of the ellipse, the circle will lie wholly without the ellipse.

In like manner, the section made in the biaxal surface by
the plane ab consists of a circle with the radius ¢, and an ellipse
with the semiaxes a, 6; and as the radius of the circle is less than
both the semiaxes of the ellipse, the circle lies wholly within the
ellipse. ,

18. But when the section lies in the plane of the greatest
and least semiaxes a, c, the circle and ellipse, of which it is com-
posed, intersect each other. ‘For the radius b of the circle is less
than one semiaxis of the ellipse ae and greater than the other.
Leaving the ellipse ac in the position which it has as a section of
the ellipsoid abc, if we describe the circle b with the centre O and
radius 6, the ellipse and the circle will cut each other in four
points at the extremities of two diameters; and planes, passing
through these diameters and through the semiaxis b of the ellip-
soid, will evidently be the planes of the two circular sections of
the ellipsoid. Now, turning the ellipse ac round through a right
angle (17), the circle and the ellipse in its new position will con-
stitute the section of the biaxal sur-
face, and will cut each other (Fig. 12)
in four points 7 at the extremities of
two diameters nOn, nOn, which are
perpendicular to the two former dia-
meters, and therefore perpendicular
to the planes of the two circular
sections. Consequently, the biaxal
surface has four nodes at the four
points ». These nodes, it is manifest, are alike in all their pro-
perties ; and they are the only points common to the two biaxal

Fig. 12.

to the Wave Theory of Light. 29

sheets, since the points Z'and V (9), in the description of the
biaxal surface, cannot coincide unless the section QOg, perpen-
‘dicular to OT7'V, be a circle.

19. The plane of the greatest and least semiaxes, a, c, of the
generating ellipsoid, may be called the plane of the nodes; and
the two diameters nOn, nOn, passing peveeh the nodes, may
be called the nodal diameters.

At one of the nodes » (Fig. 12) draw tangents nf, nk, to the
ellipse and the circle that compose the biaxal section; and
through O draw Op perpendicular to On, cutting the circle in p.
Then as On is perpendicular to the plane of a circular section of
the ellipsoid abe, this circular section will have Op for its radius,
and its circumference will cross that of the ellipse ac (belonging

to the ellipsoid) in the point py. A line touching the ellipse ae
_ at p will be parallel to every plane that touches the ellipsoid in a
point of the circular section, and will therefore (6) be perpendi-
cular to the plane which is reciprocal to the plane of the circular
section. But the tangent at pis perpendicular to the tangent nf,
since the two tangents would coincide if the ellipse ac were turned
round (18) through a right angle, the point p then falling upon
nm. Hence the circular section and its reciprocal plane are
parallel to the tangents nk, nf; and therefore two planes perpen-
dicular to the plane of the figure, and passing through these tan-
gents, are the planes that we have called (14) the principal
tangent planes at n.

20. Produce Op to meet nf in v, and conceive a parabola
having its focus at O, its vertex at v (8), and its plane perpendi-
cular to the plane of the figure. A cone, with its vertex at
and this parabola for its section, is (14) the nodal tangent cone.

Draw Of perpendicular to nf at f, and meeting nk in k. The
perpendiculars let fall from O upon the nodal tangent planes
form a cone, of which the circles described in planes perpendi-
cular to the figure upon the diameters nf, nk, are sections (8).
On the other biaxal surface a’d’c’ there is (16) a circle of contact
whose plane is perpendicular to On. This circle of contact is
(16) another section of the cone last mentioned.

30 Geometrical Propositions applied

21. To the circle 6 and to the principal section ac of the
ellipsoid abe conceive a common tangent di’ to be drawn, in a
quadrant adjacent to that which contains the node », and let it
touch the circle in d@’ and the ellipse ac in 7’. A radius Od’,
drawn through the point @’ to meet the ellipsoid a’J’c’ in the
point d”, will be reciprocal to the radius 01’, because it is perpen-
dicular to a tangent at 7’, and it will be equal in length to J’, be-
cause Od” x Od’ =k? = bb’, and Od’ =b; whence Od” =b. There-
fore Od” is in a circular section of the ellipsoid ad’c’. Two planes
perpendicular to the plane of the figure, and passing through
the reciprocal radii Od”, Oi’, are (6) reciprocal planes, and we
have seen that the first of them makes a circular section in the
ellipsoid @d’c’. They are therefore (15) the fixed planes in the
second case of Prop. V.

22. Now draw di a common tangent to the circle d and
ellipse ae composing the biaxal section, and let it touch the circle
in d and the ellipse in¢. The lines Od, Oi, are of course perpen-
dicular to the lines Od’, Oi’, and therefore perpendicular to the
fixed planes just mentioned. Hence the line Od and the point
d are the same as the fixed line OS and the point S in the second
case of Prop. V. The plane of the circle of contact is therefore
perpendicular to Od at the point d (15) ; and the points d and 4,
where its plane intersects the right lines Od, Oi, perpendicular
to the fixed planes, are (8) the extremities of a diameter.

These things agree with the obvious remark, that the points
of contact d and i must be points of the circle of contact; and
that di must be a diameter, because the plane of the circle is per-
pendicular to the plane of the figure, and this latter plane divides
the biaxal surface symmetrically.

As the circle and ellipse may have a common tangent oppo-
site to each node, there are four circles of contact in planes per-
pendicular to the plane of the nodes.*

23, The biaxal surface belongs to a class that may be called

* The curves of contact on biaxal surfaces, and the conical intersections or nodes,

were lately discovered by Professor Hamilton, who deduced from these properties
a theory of conical refraction, which has been confirmed by the experiments of

’

to the Wave Theory of Light. 31

apsidal surfaces, from the manner in which they are conceived to
be generated.

Let G be a given surface, and O a fixed origin or pole. If
a plane passing through O cut the surface G, the curve of inter-
section will in general have several apsides A, A’, A”, &c., where
the lines OA, OA’, OA”, &c., are perpendicular to the curve.
Through the point O conceive a right line perpendicular to the
plane of the curve, and on this perpendicular take from O the
distances Oa, Oa’, Oa’, &c., respectively equal to the apsidal dis-
tances OA, OA’, OA”, &. Imagine a similar construction to be
made in every possible position of the intersecting plane passing
through O, and the points a, a’, a’, &c., will describe the dif-
ferent sheets of an apsidal surface.

The apsidal surface has a centre at the point O, because the
lengths Oa, Oa’, Oa’, &c., may be measured on the perpendicular
at either side of the intersecting plane.

Referring* to the demonstration of Theorem III., it will be
seen to depend only on the supposition that the point Q is an
apsis of the section made by the plane QOq; or, which is the ©
same thing, that OQ is a position wherein the radius vector from
O to the curve of section is a maximum or a minimum. Hence
we have the following general theorem :—

24. Prop. VI. TxHreorem.—lf tangent planes be applied at
corresponding points A, a, on the surface G and the apsidal sur-
face which it generates, these tangent planes will be perpendi-
cular to each other and to the plane of the points O, A, a.

This is equivalent to saying that perpendiculars from O on
the tangent planes are equal to each other, and lie in the plane
of the lines OA, Oa.

25. If Qand RZ be reciprocal points on two reciprocal surfaces,
of which Q is the fixed origin or pole, the tangent plane at Q

Professor Lloyd. See Transactions of the Royal Irish Academy, Vol. xvu., part i.,
pp. 182, 145; and the present Paper, Art. 55-58.

The indeterminate cases of circular section—at least the case of the nodes—had
occurred to me long ago; but having neglected to examine the matter attentively,
I did not perceive the properties involved in it (13).

* Transactions of the Royal Irish Academy, Vol. xyt., part ii., p. 68.

32 Geometrical Propositions applied

will be (1) perpendicular to OR and to the plane QOR. Let a
plane also perpendicular to the plane QOR pass through OQ,
cutting the surface to which the point Q belongs in a certain
curve, and the tangent plane at Q in a tangent to this curve.
The tangent is evidently perpendicular to OQ, and therefore the
point Q is an apsis of the curve.

In like manner, the point £# is an apsis of the section mas
in the other surface by a plane passing through OR and perpen-
dicular to the plane QOR.

26. From these observations, and from Prop. VI., it appears
that if the points Q, R, in the figure of Theorem I'V., be reciprocal
points on any two reciprocal surfaces, and if the same construe-
tion be supposed to remain, the points 7 and I will be points
on the apsidal surfaces generated by these reciprocal surfaces, and
the tangent planes at 7’ and Iwill be perpendicular to the lines
OM and OT respectively. Also the rectangles LOT and MOS
will be equal to k*. Hence we have another general theorem :—

Prorv. VII. Txurorrem.—The apsidal surfaces generated by
two reciprocal surfaces are themselves reciprocal.

27. A very simple example of apsidal surfaces, with nodes
and circles of contact, may be had by supposing the generatrix
G to be asphere, and the pole O to be within the sphere, between
the surface and the centre C.

It is evident that the apsidal surface in this case will be one
of revolution round the right line OC
as an axis. Therefore taking for the
plane of the figure (Fig. 13) a plane
passing through OC and cutting the
- sphere in a great circle of which the
radius is CS, let a plane at right
angles to the figure revolve about 0,
cutting the circle CS in the points
A, A’. The section of the sphere
made by the revolving plane will have
only two apsides -A, A’, with respect Fig. 13.
to the point O, except when the plane is perpendicular to OC.

to the Wave Theory of Light. 33

Hence, if we draw the right line Oaa’ perpendicular to 404’,
taking Oa, Oa’, always equal to OA, OA’, the points a, a’, will
describe a section of the apsidal surface. This section will evi-
dently consist of two circles C'S’, C’S”, equal to the circle CS,
and having their centres C’, C”, on the opposite sides of O in a
right line C’OO” perpendicular to OC; the distances OC, OC’,
OC” being equal. The circles C’S’, C”’S”, intersect in two
points 7, n’, on the line OC, and have two common tangents di,
dv’, which are bisected at right angles by OC in the points ¢, c’.

28. Now let the circles C’S’, C’S”, with their common tan-
gents, or only one of the circles with the half tangents, revolve
about the axis OC, and we shall have the apsidal surface with
nodes at n, n’, and with circles of contact described by the
radii cd, cd’.

The section of the sphere, by a plane passing through O at
right angles to On, is a circle of which O is the centre. If there-
fore we suppose that the point n answers to a in Prop. VL., the
_ apsis A corresponding to ” will be indeterminate, and the posi-
tion of the tangent plane at ” will also be indeterminate, which
ought to be the case at a node.

The surface reciprocal to the sphere, the pole being at O, is
evidently a surface of revolution about the axis OC (it is easily
shown tobe a spheroid having a focus at O) ; and the section of
this reciprocal surface, by a plane perpendicular to the axis at O,
is a circle of which O is the centre. This circumstance indicates
(15) that on the apsidal surface there is a curve of contact, whose
plane is parallel to. the plane of circular section; which agrees
with what we have already seen. ‘

29. When the point O is without the sphere, the axis OC
will pass between the circles C’S’, C’S”, without intersecting
either of them. The apsidal surface, described by the revolution
of one of these circles about OC, will be a circular ring. The
nodes have disappeared; but the circles of contact still exist, as
is evident.

dD

34 Geometrical Propositions applied

Part II.—On toe Wave Tueory or Licur.

30. Some of the foregoing propositions lead to a simple trans-
formation of the theory of light.

In this theory, the swrface of waves, or the wave surface, is a
geometrical surface used to determine the directions and veloci-
ties of refracted or reflected rays, being the surface of a sphere
in a singly refracting medium ; a double surface, or a surface of
two sheets, in a doubly refracting medium; a surface of three
sheets on the supposition of triple refraction ; and having always
a centre O round which it is symmetrical. The radii of the wave
surface, drawn from its centre O in different directions, repre-
sent the velocities of rays to which they are parallel.

31. We shall consider particularly the case of a doubly re-
fracting crystal, with two plane faces parallel to each other, and
surrounded by a medium of the common kind wherein the con-
stant velocity is V: supposing, for the sake of clearness, that
the crystal refracts more powerfully than the surrounding
medium, so that the velocities in the crystal are less than the
velocity V.

A ray S’O, falling on the first surface of the crystal at the point
O, is partly reflected according to the common law of reflection,
and partly refracted. The two refracted rays pass on to the
second surface, where each of them is divided by internal re-
flection into a pair, the two reflected pairs being parallel to each
other ; while the two emergent rays—one from each refracted
ray—are parallel to each other and to the incident ray S’O.
The directions of the rays within the crystals are usually found
by the following construction :— ;

32. Describe a wave surface of the crystal, having its centre
at O the point of incidence. By the nature of the wave surface,
a right line OTU, drawn from the point O, will in general cut
this surface in two points 7, U, on the same side of 0; and a
ray passing through the crystal in a direction parallel to OTU
will have one of the two velocities represented by the radii O7,

to the Wave Theory of Light. 35

OU, taking a line of a certain length & to represent the uniform
velocity V in the external medium. With the centre O and a
radius OS equal to this line & describe a sphere. As the veloci-
ties in the crystal are supposed to be

less than V, the wave surface will lie e
wholly within this sphere. Let the i

lane of the figure (Fig. 14) be the
P the figure (Fig. 14) tes

plane of incidence, perpendicular to
the parallel faces of the crystal, and
intersecting the first face in the right
line FA. Through the point 8,
where the incident ray S’O, produced
through the crystal, cuts the surface
of the sphere, draw SJ at right angles
to OS and meeting FA in the point J. A right line perpendi-
cular to the plane of the figure, and passing through this point J,
we shall call the right line J.

33. Through the right line J draw two planes touching the
two sheets of the wave surface, on the side remote from the inci-
dent light, in the points 7, 7’, which will lie within the sphere
- (32) ; then the incident plane wave, perpendicular to OS, will
be refracted into two plane waves parallel to these two tangent
planes; and the lines OT, OT’, will be the directions of the re-
fracted rays along which the refracted waves are propagated.
The lengths OT, OT’, represent the velocities with which the
light moves along the rays; and of course the normal velocities,
which are the velocities of the refracted waves, are represented
by the perpendiculars OG, OH, let fall from O on the two tan-
gent planes at 7, 7’. These two perpendiculars OG, OH, evi-
dently lie in the plane of the figure; but the points 7, 7’, in
general, do not lie in this plane.

34. Again, through the right line J draw two other planes
_ touching the wave surface, at the side of the incident light, in
the points ¢, #. The rays O7, OZ", arriving at the second sur-
face of the crystal, will each be divided by internal reflection
into two rays parallel to Of, Ot’; and these four reflected rays,

D2

Fig. 14.

36 Geometrical Propositions applied

arriving at the first surface, will each be divided, by a new re-
flection, into two rays parallel to OT, OZ’; and so on, for any
number of reflections. Any of the rays emerging at the tirst
surface, after internal reflections, is parallel to the ray Os pro-
duced by ordinary reflection at the point of incidence; and any
ray emerging at the second surface is parallel to the incident
ray S’OS.

35. This construction may be changed into another that will
be found more convenient both in theory and practice.

Through S draw SR perpendicular to OJ, and meeting OG,
OH, produced, in the points P, MZ. Then as the angles at G
and # are right angles, the points J, R, G, P, are in the cireum-
ference of a circle, and therefore OP x OG = OI x OR = OS* =i;
and similarly, OM x OH =k’. If then we take O for the fixed
origin, or pole, and /* for the constant rectangle (‘Theorem I.),
and describe the surface which is reciprocal to the wave surface,
it is evident that the poimts P and & will be points of the sur-
face so described, and that OT, OT’, will coincide in direction
with perpendiculars let fall from O on planes touching the sur-
face at P and JY, and will be inversely proportional to these
perpendiculars. It follows in the very same manner, that if per-
pendiculars Og, Oh, let fall from O on the tangent planes at ¢, /,
be produced to meet SR in the points p, m, these points will also
be on the surface reciprocal to the wave surface.

In the present case, it is manifest that this reciprocal surface
lies wholly without the sphere OS.

36. The surface reciprocal to the wave surface, the pole
being at O, we shall call the surface of refraction.

It is hardly necessary to observe that the surface of refrac-
tion has a centre at the point O, round which it is symmetrical ;
that it is a sphere in a singly refracting medium, a double sur-
‘ace in a doubly refracting medium, and a surface of three sheets
if we suppose a case of triple refraction.

37. In the case that we are considering, let the figure
(Fig. 15) represent a section made in the double surface of re-
‘raction and its attendant sphere by the plane of incidence.

to the Wave Theory of Light. 37

Through the point S, where the incident ray 8’O, prolonged,
euts the circular section of the sphere, draw SR perpendicular to
the face of the crystal, or to #A; and let SR, produced, cut the
circle again in the point s. Then Os is the direction of the ray
given by ordinary reflection at the first surface of the crystal.

Fig. 15.

Produce the right line SRs both ways, to cut the surface of re-
fraction in the points P, VW, behind the crystal, and in the points ~
Pp, m, before it; and conceive planes to touch the surface of re-
fraction at the points P, WU, p,m. Suppose also that perpendi-
culars OP’, OM’, Op’, Om’, are let fall from O upon these tangent
planes, and that they intersect the planes in the points P’, /’,
p’, m’, respectively.

Then from the preceding observations (33, 34, 35), it is
manifest that OP’, OM’, are the directions of the rays into
which S’O is divided by refraction ; that each of these refracted
rays, on arriving at the second surface of the crystal, is divided
by internal reflection into two rays parallel to Op’, Om’; and that
each of the four reflected rays, on arriving at the first surface, is
again divided by reflection into two rays parallel to OP’, OM’,

38 Geometrical Propositions applied

and soon. In general, every ray going into the crystal from
the: first surface, whether after refraction or after any even
number of internal reflections, is parallel either to OP’ or to OW’;
and every ray returning from the second surface of the crystal
after any odd number of internal reflections, is parallel either to
Op’ or to Om’. Thus the direction of every ray in the interior
of the crystal is the same as the direction of some one of the four
lines OP’, OM’, Op’, Om’; and the velocity of the ray is in-
versely as the length of this line ; so that the velocity of the ray
OM’, for example, or of any ray parallel to OW’, is to the velo-
city Vas OSisto OM’. The little plane waves that, keeping
always parallel to themselves, move along these rays, are respec-
tively perpendicular to the lines OP, OM, Op, Om; and the
lengths of these lines are inversely as the velocities of the waves
estimated in directions perpendicular to their planes ; so that the
velocity of the wave which moves along the ray OW’, or along
any parallel ray, is to the velocity V as OS is to OW.

38. The ray OP’, and all the rays parallel to it, are perpen-
dicular to the plane which. touches at P the surface of refraction ;
and the waves which move along these rays are perpendicular to
the right line OP. Any ray of this set may be called a ray P,
and any of the waves a wave P. In like manner, the rays WM, p,m,
are rays that are perpendicular to the tangent planes at the
points WW, p, m, respectively; and the waves I, p, m, are the
waves that belong to these rays, and that have their planes re-
spectively perpendicular to the right lines OW, Op, Om. The
rays P, W/, all come from the first surface of the crystal ; the
rays p, m, from the second.

As the ordinates RP, Rp, are greater than the ordinates ©
RM, Rm, so the rays P, p, are more refracted or more reflected
than the rays M, m. The former rays may therefore be said to
be plus refracted, or plus reflected, and the latter to be minus re-
Sracted, or minus reflected. Or—tfor the convenience of naming—
the rays P, p, may be called plus rays; and the rays MU, m,
minus rays. The waves P, p, in like manner, may be termed
plus waves, and the waves WM, m, minus waves.

to the Wave Theory of Light. 39

For a medium of the common kind, or a singly refracting
medium, we may use the letters S and s. Thus the incident ray
S’OS, or any ray emerging parallel to OS from the second sur-
face of the crystal, may be marked by the letter S; while the
ray Os produced by common reflection, or any ray emerging pa-
rallel to Os from the first surface, may be denoted by the letter s.

39. The course of a ray through the crystal may now be
easily expressed. A ray SWps, for example, is a ray (S) inci-
dent on the crystal, undergoing minus refraction (J/) at the
first surface, plus reflection (p) at the second, and emerging (s)
from the first surface in a direction parallel to Os. - Of this ray
the part within the crystal is Mp. A ray SPS is a ray plus re-
fracted, and then emerging in a direction parallel to that of
incidence. A ray SPpMS is a ray plus refracted at the first
surface, then plus reflected at the second surface, then minus
reflected at the first surface, and finally emerging from the second
surface in a direction parallel to that of incidence. Its path
within the crystal is Ppl.

These examples indicate the general method of expressing
the path of a ray. ;

Suppose light to be moving in the same direction and with
* the same velocity along two proximate parallel rays, so that it
is at the point A in one ray when it is at the point Bin the
other ; and through the points 4d and B conceive two planes
perpendicular to the common direction of the rays. These planes
are either coincident, or maintain a constant distance. In the
first case, the rays are said to be in complete accordance. In the
second case, the constant distance between the planes is called
the interval between the portions of light composing the rays, or
the interval between the waves that move along the rays.

We proceed to find the lengths of these intervals in the case
of rays emerging parallel to each other, at either side of the
crystal that we have been hitherto considering.

41. Let the tangent planes at P, MW, p, m, intersect the plane
of the figure (Fig. 15) in the right lines PP, MM, pp, mm,
which of course are tangents to the section of the surface of re-

40 Geometrical Propositions applied

fraction represented in the figure ; let a perpendicnlar at O to the
face of the crystal cut these tangents in the points P, MW, p, m,;
and let the lines OP”, OM”, Op”, Om”, respectively parallel to
PP, MM, pp,, mm, cut the line SRs in the points P”, W”’, p”, m”.

The length of the path which a ray P describes within the
crystal is equal to the thickness © of the crystal divided by
the cosine of the angle P’OP, which the path of the ray makes
with a perpendicular to the faces of the crystal ; and the velocity

oF P as pqual’so Px mse (87): dividing therefore 4he lemme

of the path by the velocity, we find that the time in which a ray
P crosses the crystal is equal to
6 x OF”
V x OS x cos POP,

But as OP’ is perpendicular to the tangent plane at P, we have

OP :
eEPOL i te
Therefore the time is equal to ss . Similarly, the times in

which rays W, p, m, pass from one surface of the crystal to the
other are equal to

ex MM” Oxpp” Ox mn’
VxOS’ Vx0OS’ Vx OS

42. Now suppose the path of a ray P to be projected per-
pendicularly on a right line having any proposed direction in
space. Through O conceive a right line OL parallel to the pro-
posed direction, and meeting in Z the tangent plane at P. The
length of the projection is equal to the length of the path multi-
plied by the cosine of the angle P’OL which the ray P makes

; : Se are cos POL
with OL; that is, the projection is equal to 0 sos POP’ But

, respectively.

because OP” is perpendicular to the tangent plane at P, we have

OPE OE.

; OP” y 2 ,
cos POL = oF, ang con FOF" Dp: ope

to the Wave Theory of Light. 41

therefore
cos P’ OL 3 Pp”
cs POP, OL °
PP”

Hence the Sécieekinie is equal to 6 ~— OL”
If the path of a ray P be projected on the incident ray OS,

then producing OS to meet PP, in /, we see, by what has just

been proved, that the length of the projection is equal to

POOR ya =n OB?
by similar triangles. In like manner, the projections of the
paths of rays M/, p, m, on the direction of the incident ray OS,
are equal to aie aii ett
sp m
"os 90s’. Os”

respectively. .

48. Let each rectilinear path be measured in the direction
in which the light moves along it; and according as the
- direction so measured makes an acute or an obtuse angle with
the direction OS, measured from O to S, let the projection of
the path on OS be reckoned positive or negative. Then if
SPmMpMS be any ray entering the crystal at O, and emerging
from its second surface at #, and if a perpendicular ETI be let
fall from F# upon OS, meeting OS in J, the distance OJ, from
O to the foot of this perpendicular, will evidently be equal to
the algebraic sum of the projections of the paths P, m, WV, p, UV,
contained within the crystal, taking each projection with its
proper sign. It is obvious that the projections of the P and
rays are always positive. And as the lines Op’, Om’—the
directions of the rays p, m=lie in planes which are respectively
perpendicular to pp, mm, or to Op”, Om”, it is easy to see that
these directions make acute or obtuse angles with OS, according
as the points p”, m”, lie below the point S or above it; that is,
the projections are positive or negative according as the points
p’, m’, lie without the circle OS towards P, M, or within the

42 Geometrical Propositions applied

circle. -Therefore the distance OJ, in the case of the figure, is
equal to

oH (SP” — Sn” + SM” — Sp’ + SM”).

44. If the paths of rays P, M, p, m, be projected on the
direction Os of the ordinarily reflected ray, the lengths of their
projections will be

or si” sp” sm”
os’ 20s? 20s ° os

respectively. The projections upon Os of the rays p, m, will be
always positive; and the projections of the rays P, WU, will be
positive or negative according as the points P”, WM”, lie above
the point s or below it; that is, according as the points P”, WZ”,
lie without the circle OS towards p and m, or within the circle.
So that if SPmMps be a ray entering the crystal at.O and
emerging from the first surface at e, and if a perpendicular
et be let fall from e upon Os, the distance Oi, from the point O
to the foot of this perpendicular, or the algebraic sum of the
projections of the paths P, m, M, p, contained within the
erystal, will be equal to ;

8

an (-sP” + sm” -sM” + sp”),
in the case of the figure.

45. Let us imagine that the light in the incident ray S’O,
instead of being interrupted at O by the crystal, had continued
to move with the same velocity V in the same right line OS,
leaving the point O at the moment when the refracted light
enters the crystal at O. Comparing the light in this imaginary
ray with that in a ray emerging parallel to it from the second
surface of the crystal, after an even number of internal reflec-
tions, we shall find that the emergent is behind the imaginary
ray, and that the interval between them (40)—or the retar-
dation of the former—may be derived very easily from the
letters that designate that ray. Let SPmIMpMS be any such

to the Wave Theory of Light. 43

ray. The sum of the distances of the point S from each of the
points marked by the letters (PmMpM) that denote (89) the
part of the ray contained within the erystal, is proportional to
the interval of retardation, that interval being equal to

ay (SP + Sm + 8M + Sp + SM).

For if from the point EZ, where the last internal ray WU
emerges from the second surface of the crystal, a perpendicular
ET be let fall upon OS, meeting OS in J, the time of pen
OL with the velocity V would (43) be

tS)
VxO

But (41) the actual time of describing the broken path PmMpM
is

og (SP’ - Sm” + SU” - Sp” + SM’).

os = ag (PR + mm” + UM" + pp" + MM’) ;

and, on inspecting the figure, this time is seen to be greater
than the time of describing OJ, by

ee OE ih
V x OS

or by the time in which the line

(SP + Sm + SM+ Sp + SN),

ay (SP + Sm + SM + Sp + SM)

would be described with the velocity V. Consequently, at the
moment when the light in the ray SPmMpMS emerges at the
point E from the second surface of the crystal, the light in the
imaginary uninterrupted ray OS will have passed the point I
by an interval equal to the line just mentioned ; and as the two
rays afterwards have the same velocity and nari directions,
this interval is the retardation of the emergent ray.

46. The rays emerging from the first surface after any odd
number of internal reflections are to be compared with the

44 Geometrical Propositions applied

ordinarily reflected ray Os to which they are parallel, the light _
in Os, which moves with the velocity V, being supposed to leave
O at the moment when the refracted light enters the crystal at
O. The mode of proceeding in this case is exactly similar to
that in the last, and the interval is determined in the same way,
using s in place of S; the retardation of the ray SPmMps, for
example, of which the part PmMp is contained within the
crystal, being equal to

oo

a (sP + sm + si + sp).*

47. It isremarkable that the preceding demonstration in no-
wise depends upon the supposition that the planes perpendicular
to the rays P, WM, p,m, are tangent planes to the surface of
refraction at the points P, M, p,m. If we had supposed any
planes—different from the plane of the figure—to pass through
the points P, WW, p, m, and the rays to coincide in the direction
with perpendiculars let fall from O upon these planes, and to
have velocities inversely proportional to the lengths of the
perpendiculars, the intervals of retardation would have remained
unchanged. Hence the retardations are the same as if the lines
OP, OM, Op, Om, were the directions of the rays in passing
through the crystal, as will appear by conceiving the planes
that we have spoken of to be perpendicular to these lines.

If the incident ray S’O were refracted in the ordinary way

with an index equal to 2 it would take the direction OP; if

it were refracted, in like manner, with the index aa it would

take the direction OM; and if the two rays, thus ordinarily
. refracted, were to emerge from the second surface of the crystal
in directions parallel to OS, it is evident, from what has been
said, that they would be in complete accordance, respectively,
with the rays SPS and SUS. —

* The change of phase, which may take place at a surface of the crystal, is not
here considered as affecting the intervals.

to the Wave Theory of Light. 45

If the surface of refraction should happen to have a node
NV, which is a point of intersection where it admits an infinite
number of tangent planes (3), let the direction of the incident
ray S’OS be chosen, so that the right line RS perpendicular
- to the face of the crystal, being produced below S, may pass
through JV, and we shall have a cone of refracted rays formed
by the perpendiculars let fall from O upon the tangent planes
at V; all of which rays, on emerging parallel to OS from the
second surface of the crystal, will be in complete accordance
with one another. For we have just seen that if the ray S’OS
were supposed to emerge after being refracted-in the ordinary

way with an index equal to on it would be in complete

0.
accordance with any ray of the cone.

48. The interval between any two rays emerging at the
same side of the crystal is the difference of their retardations.
In taking the difference, the letters that are common to the
names of the two rays may be left out. Thus the ray SPmMS
is behind the ray SPS by the interval

9
ay (sm + SM) = OS —~ Mm.

The line ne Pp is the interval between the rays SMS and
SMpPS, or between the reflected ray Os and the ray SPps,
and so on.

49. The retardations of the two refracted rays SPS and

SMS, emerging without internal reflection, are me SP and

~ —— SM respectively. The difference of these is © vil Con-
sequently, when the two refracted rays have emerged from the
second surface in directions parallel to the incident ray, the
light in the plus emergent ray is behind the light in the minus
0x PM
OSs
words, the incident plane wave, perpendicular to OS, produces

emergent ray by an interval equal to Or, in other

46 Geometrical Propositions applied

two emergent waves parallel to each other and to the incident
wave, moving along the emergent rays with equal velocities V,
and preserving the distance eee between their planes, the
minus wave being foremost. If OS, the radius of the sphere, be
taken for unity, PI will be a number—generally a very small
fraction—and the interval will be the thickness of the crystal
multiplied by this number.

50. Suppose the right line PIR, remaining always perpen-
dicular to the face of the crystal, to describe a cylindrical sur-
face, with the condition that the part PM, intercepted between
the two sheets of the surface of refraction, shall remain of a con-
stant length; the point R will then describe, on the surface of
the crystal, a curve whose radii OR are the sines (to the radius
OS) of the angles of incidence of a cone of rays; and every ray
S’O of this cone, when refracted by the crystal, will afford two.
emergent rays, or two waves, having the same given interval
between them. Lines drawn from the eye parallel to the sides
of this cone are the emergent rays belonging to a ring, when
rings are made to appear, in any of the usual ways, on trans-
mitting polarized light through the plate of crystal. In nominal
conformity to this, we see that the line PY describes a ring of
constant breadth between the two sheets of the surface of refrac-
tion. The ring described by supposing pm to remain constant
corresponds to the interval between two rays p and m reflected
at the same point of the second surface of the crystal, and then
emerging at the first. The other intercepts Pp, Mm, Pm, Mp,
are proportional (48) to intervals like those in Newton’s rings—
to the intervals, namely, between the reflected ray Os and the
rays SPps, SMms, SPms, SMps, emerging at the first surface
after one reflection within the crystal ; or to the intervals between
rays that are twice reflected in the crystal and the rays trans-
mitted without reflection.

51. The general investigation of the figure of a geometrical
ring does not distinguish between the different intercepts, and
will therefore include all the rings PM, pm, Pp, Mm, Pm, Mp ;

to the Wave Theory of Light. 47

so that it will be sufficient to contemplate any one of them, as
PM, of which the breadth PY is equal to.a given line J.

The points P and UM describe, in general, similar and equal
curves of double curvature, which may be called ring-edges, as
being the edges of the ring; and if we imagine the surface of
refraction, carrying these curves .along with it, to be shifted
either way, in a direction parallel to PY, through a distance equal
to J, it is clear that the new position of one of the ring-edges
will exactly coincide with the first position of the other, and that
therefore the curve of the latter ring-edge will be given by the
intersection of the two’equal surfaces in these two positions.
Let U = 0—where UV is a function of «, y, 3, and given quanti-
ties—be the equation of the surface of refraction in its original
position ; and, the axes of co-ordinates being fixed, suppose that
by the shifting of the surface the co-ordinates of a point assumed
on it are diminished by the given lines f, g, h, which are the
projections of the given line J on the axes of x, y, 2, respectively.
Then the equation of the surface in its new position will be had by
substituting «+f, y+g, 2+h, for 2, y, z, in the equation U=0,
which will thus become U + V= 0, where V is the increment of
U produced by the substitution. These two equations com-
bined are equivalent to the equations U = 0, V = 0, which are
therefore the equations of one of the ring-edges. If the surface
had been shifted the opposite way, in a direction parallel to PY,
the intersection would have been the other ring-edge, whose
equations are therefore deducible from those already found, by
changing the signs of f, g, h.

52. If the equation of the surface of refraction be trans-
formed, so that the plane of zy may coincide with the face of the
crystal, and the axis of s be perpendicular to it, the origin of
co-ordinates being at the centre O, no change will be produced
in # or in y by the motion of the surface, because PY, the direc-
tion of the motion, is now parallel to the axis of s ; but s will be
diminished or increased by J; and, accordingly, if U’=0 be the
equation of the surface in its first position, when the centre
is at O, and if U’ become U’ + V’ when z becomes z + J, the

48 Geometrical Propositions applied

equation of the surface in its second position, when the centre
has moved through a distance equal to J along the axis of z, will
be U’ + V’=0; and these two equations combined will give
U’ = 0, V’ = 0, for the equations of one of the ring-edges. The
equations of the other ring-edge are deduced from these by
changing the sign of J.

The projection of each of the ring-edges on the plane zy is
the curve traced by the point R on the surface of the crystal
(50). This curve may be called a ring-trace. Its equation is
obtained by eliminating s between the equations of a ring-edge; -
and as the result must be the same whether J be taken posi-
tive or negative, the equation of the ring-trace, when found by
this general method, will contain only even powers of J. The
radii drawn from O to the points R of the ring-trace are (50)
the sines (to the radius OS) of the angles of incidence or emer-
gence of the rays that form an optical ring, the rays that come
from this ring to the eye being parallel to the sides of the cone
described. by the right line S’OS, while the point R describes the
ring-trace.

53. It is evident that tangents to the ring-edges, at the
points P and HY, are parallel to each other, and therefore parallel
to the intersection of two planes touching the surface of refrac-
tion at P and MV, because these tangent planes pass through the
tangents. But the directions OP’, OM’, are perpendicular to
the tangent planes, and therefore the plane P’ OW’, containing the
two rays, is perpendicular to the intersection of the tangent planes,
and of course perpendicular to the parallel tangents. Hence
the plane P’OW’ intersects the face of the crystal in a right line
perpendicular to the projection of the parallel tangents on the
face of the crystal. As this projection is a tangent to the curve
described by R, it follows that the normal to the ring-trace at
the point # is parallel to the line joining the points in which
the two refracted rays cut the second surface of the crystal.

In like manner, taking any two consecutive rays (P and m),
having a common extremity on one surface of the crystal, the
line joining the points where these rays cut the other surface is

to the Wave Theory of Light. 49

parallel to the normal at the point & of the ring-trace which is
described when the intercept (Pm) between the letters that mark
the rays is supposed to remain constant.

54. In all that precedes we have made no supposition about
the surface of refraction except that it is a surface of two sheets ;
and if we supposed it to have three sheets, the conclusions would
be easily extended to this hypothesis.

In the theory of Fresnet, the wave surface is* a biaxal
whose generating ellipsoid has its centre at the point O, and its
semiaxes parallel to the three principal directions of the crystal,
the length of each semiaxis being equal to OS divided by one
of the principal indices of refraction. The surface of refraction
is reciprocal to the wave surface, and is (11) therefore another
biaxal generated by an ellipsoid reciprocal to the former, having
its centre at the same point O, and the directions of its semiaxes
the same as before, the rectangle under each coincident pair of
semiaxes being equal to 4? or OS*. Hence the semiaxes of the
ellipsoid which generates the biaxal surface of refraction are
equal in length to OS multiplied by each of the three principal
indices. This biaxal surface is of course to be substituted for
the surface of refraction in the preceding observations.

55. When the line RS, produced below S, passes through a
node WV of the biaxal surface of refraction, the points P, I,
eoincide in the point JV, and the interval PM vanishes. At the
point VV there are an infinite number of tangent planes, and the
perpendiculars from O on these tangent planes give a cone of
refracted rays whose sections we have already shown how to
determine (20), All the rays in this cone, on arriving at the
second surface of the crystal, emerge parallel to the incident ray
OS; and if the rays in the emergent cylinder be cut by a plane
perpendicular to their common direction, they will all arrive at
this plane at the same instant, because the interval PZ vanishes,
See Art. 47.

56. Suppose fig. 12 to be a section of the wave surface. The

* Transactions of the Royal Irish Academy, Vou. xv1., p. 76 (supra, p. 11).
~E

50 Geometrical Propositions applied

right line Od will pass through V; and the circle of contact,
described on the diameter di in a plane perpendicular to the
right line OdN, will be a section of the refracted cone. Now it
will be recollected* that, in general, the vibrations of a ray O7,
which goes to any point 7' of the wave surface, are parallel to
the line which joins the point 7 with the foot of the perpendi-
cular let fall from O on the tangent plane at 7. In the present
case, the perpendicular is the same for all the rays of the re-
fracted cone, and its extremity coincides with the point d: so
that the line d7, drawn from d to any point 7 of the circle of
contact, is parallel to the vibrations of the ray OT which passes
through 7. Conceive, therefore, a plane perpendicular to OV
at the nodal point V. This plane will cut the refracted cone in
a circle whose circumference will pass through V; and a line
NT’, drawn from the node to any other point 7” of the circum-
ference, will be the direction of the vibrations in a ray OT" which
crosses the circle at this point. The plane of polarization is
perpendicular to the direction of the vibrations.

57. The transverse section of the emergent cylinder is always
a very small ellipse, affording a hollow pencil of parallel raysin ~
complete accordance (55). If the crystal be thin, this: ellipse
will be of evanescent magnitude. Hence the line OS will be
the direction of a line drawn from the eye to the centre of the
rings commonly observed (50) with polarized light; or it will
be what is called the apparent direction of one of the optic axes.
The diameter passing through WV will be the direction of the
optic axis within the crystal. There are therefore two optic
axes, parallel to the two nodal diameters (19) of the surface of
refraction.

As OW is equal to the mean semiaxis of the generating
ellipsoid, or to the mean index of refraction, when OS is unity,
it follows that the apparent direction of an optic axis is the
direction of an incident ray, which, if refracted in the ordinary
way, with an index equal to the mean index of refraction, would
pass along a nodal diameter of the surface of refraction.

* Transactions of the Royal Irish Academy, Vou. xvi., p. 76 (supra, p. 12).

to the Wave Theory of Light. 51

58. We have seen (15) that there is a’circle of contact on
the biaxal surface of refraction. If an incident ray S’OS be
taken, cutting the sphere in .S, so that the line RS produced
may pass through the circumference of this circle, it is manifest
that the direction of the refracted ray will be the same through
whatever point II of the circumference the line RS may pass,
because that direction is perpendicular to the tangent plane at
Il, which is in fact the plane of the circle itself. If, therefore,
the line RS move parallel to itself along the circumference: of
the circle, cutting the sphere in a series of points S, every inci-
dent ray S’OS which passes through a point S so determined
will be refracted into two rays, of which one will have a fixed
direction in the crystal, being perpendicular to the plane of the
circle of contact, and therefore coinciding (16) with On, one of
the nodal diameters of the wave surface. But though the direc-
tion On of the refracted ray is fixed, its polarization changes
with the incident ray from which it is derived ; for if II be the
point in which the line RS, corresponding to any position of
the incident ray, crosses the circle of contact, the vibrations of |
the refracted ray On will be contained in the plane of the lines
On, Of], and will be perpendicular to OI]. Conceive a circle
described on the diameter nf in a plane perpendicular to the
figure (Fig. 12). This circle, and the circle of contact on the
surface of refraction, are (20) sections of the same cone. Let II’
therefore be the point at which OI], in any position of the inci-
dent ray, crosses the circumference of the circle nf’; and the line
IIn, drawn to the node of the wave surface, will be the corre-
sponding direction of the vibrations in the ray On.

59. With regard to the general law of polarization in the
theory of Fresnzx, it may be observed, that if the ellipsoid abe
which generates the biaxal surface of refraction be cut by a
plane perpendicular to OP, the vibrations of the ray P will be
parallel to the greater axis of the section, and therefore the plane
of polarization will pass through OP and the less axis ; whence
it is easy to show that the plane of polarization of a ray P bisects
one of the angles made by two planes intersecting in OP and

E 2

52 Geometrical Propositions applied

passing through the nodal diameters of the surface of refraction ;
the bisected angle being that which contains the least semiaxis
e of the generating ellipsoid. The plane of polarization of the
ray p is found in like manner. But for the rays WU, m, the
angle to be bisected is that which contains within it the greatest
semiaxis a.

If OP’ be perpendicular to a tangent plane at P, the vibra-
tions of the ray P will be perpendicular to OP, and will lie in
the plane POP’. A similar remark applies to the rays HZ, p,m.

60. When two semiaxes a, b, of the ellipsoid abe become
equal, it changes into a spheroid aac described by the revolution
of the ellipse ac about the semiaxis c; and the biaxal aac, gene-
rated by this spheroid, is* composed of a sphere whose radiusis a,
and a concentric spheroid ace described by the revolution of the —
ellipse ac about the semiaxis a; so that, the diameter of the
sphere being equal to the axis of revolution of the spheroid, the
two surfaces touch at the extremities of the axis. This combina-
tion of a sphere and a spheroid is the surface of refraction for
unjaxal crystals. In these crystals, therefore, the refracted ray
whose direction is determined by the intersection of the right line
RS with the surface of the sphere follows the ordinary law of a
constant ratio of the sines, and is called the ordinary ray ; whilst
the other, whose variable refraction is regulated by the intersec-
tion of RS with the spheroid, is called the extraordinary ray. And
hence uniaxal crystals are usually divided into the two classes
of positive and negative, according to the character of the extra-
ordinary ray ; being called positive when it is the plus ray, and
negative when it is the minus ray. The first case evidently
happens when the spheroid is oblate, and therefore lies without
the sphere described on its axis; the second, when the spheroid
is prolate, and therefore lies within the sphere. The second case
(which is that of Iceland spar) may be supposed to be repre-
sented in the figure (Fig. 15), where the elliptic section of the
spheroid, made by a plane of incidence oblique to the axis, lies -

* Transactions of the Royal Irish Academy, Vou. xvt1., p. 77 (supra, p. 12).

to the Wave Theory of Light. 53

within the circular section of the sphere, and the minus ray is
of course the extraordinary one.

61. Let PY, preserving a constant length J, move parallel
to itself between the surfaces of the uniaxal sphere and spheroid,
so as to form a ring (50). Then supposing the spheroid, with
the ring-edge described on it by the point I/, to remain fixed,
imagine the sphere, carrying the ring-edge P along with it, to —
move parallel to PM, from P towards UW, through a distance
equal to J, and the two ring-edges will exactly coincide.

Hence the uniaxal ring-edge is the intersection of a sphere
and a spheroid, the diameter of the sphere being equal to the
axis of revolution of the spheroid, and the line joining their
centres being perpendicular to the faces of the crystal and equal
to the breadth J of the ring. And the projection of this inter-—
section, on a plane perpendicular to the line joining the centres
of the sphere and the spheroid, is the wniaxal ring-trace.

62. The biawal ring-edge is (51) the intersection of two equal
biaxal surfaces similarly posited, the line joining their centres
being perpendicular to the faces of the crystal and equal to the
breadth of the rmg. And the projection of this intersection, on
a plane perpendicular to the line joining the centres of the sur-
faces, is the biaxal ring-trace.*

* In applying the general theory (51, 52) to biaxal rings, it is necessary to
know the equation of a biaxal surface, which may be found in the following
manner :—Let 7, 7’, 7”, be three rectangular radii of the generating ellipsoid abc,
the two latter being the semiaxes of the section made by a plane passing through
them; so that if from the centre O two distances OT, OV, equal to 7’, 7”, be taken
on the direction of 7, the points Z' and V will belong (9) to the biaxal surface; and
let a plane parallel to the plane of 7’, 7”, and touching the ellipsoid, cut the direc-

tion of r at the distance y from the centre. Then ifr make the angles a, B, y,
with the semiaxes a, 5, c, we shall have, by the nature of the ellipsoid,

cos* a cos*B cos? y

1
a ae eee ea a

?

p? = a? cos? a + b? cos? b + & cos? y.

Now, since the sum of the squares of the reciprocals of three rectangular radii of

54 Geometrical Propositions, &c.

an ellipsoid is constant, as well as the parallelopiped described on three conjugate
semidiameters, we have the equations

1 1 1 1 1 1

yet pt ty — tt

pr? of =a bc;
or,

1 Rea Rr | cos* a . cos*B cos* ¥
ye at ( a yw te ) mts
1 a* cos? a + 62 cos B + c* cos?

= 7.
"ak le a® b? ¢ ip

Whence it appears that »”, 7”, are the values of p in the equation

Ti Soe

in which p denotes indifferently either semidiameter, O7' or OV, of the biaxal sur-
face. Therefore putting for Mand N their values, and writing a * .. instead of
p

cos a, cos B, cos y, and 2” + y® + 2? instead of p*, we obtain, for the equation of the
biaxal surface,

(a? + y? +2) (a2a+ bey? + 022) — a2 (B? + 0%) a? 3? (a? + 7) y*
— 02 (a2 402) 2-4 a2 = 0.

This is the equation of the surface of refraction for a biaxal crystal in which
a, b, e, are (54) the three principal indices of refraction, taking OS the radius of ~
the sphere to be unity. The left-hand member of the equation is therefore the ex-
pression supplied by Fresnet for the function VJ in Art. 51.

When the faces of the crystal are parallel to any of the principal planes of the
ellipsoid—to the plane of zy for example—the nature of the ring-trace may be
found very easily. For if the difference of the two values of z, deduced from the
preceding equation of the surface of refraction, be put equal to a constant quantity —
J, the result, when cleared of radicals, will be an equation of the fourth degree in
z and y, which will be the equation of the corresponding ring-trace. This is a case
that occurs frequently in practice ; the crystal being often cut with its faces per-
pendicular to the axis of x or of z, because these lines bisect the angles made by the
optic axes.

V.—A SHORT ACCOUNT OF SOME RECENT INVESTIGA-
TIONS CONCERNING THE LAWS OF REFLEXION AND
REFRACTION AT THE SURFACE OF CRYSTALS.

[Fifth Report of the British Association, 1835. ]

To understand the nature of the general problem which a com-
plete theory of double refraction requires to be solved, let it be
supposed that a ray of light is reflected and refracted at the
separating surface of an ordinary medium and a doubly refract-
ing crystal, the light passing out of the former medium into the
latter. This limited view of the subject is taken merely for the
sake of clearness of conception ; since we might suppose that both
media are crystallized, without increasing the difficulty of the
problem. The question, it is obvious, naturally divides into
two distinct heads. The first relates to the laws of the propa-
gation of light in the interior of either of the two media, before
or after it has passed their separating surface; and this part of
the subject has been fully treated, according to their different
methods, by MM. Fresnel and Cauchy. The second division of
the subject had been left completely untouched. It relates to
the more complex consideration of what takes place at the sepa-
rating surface of the media, the laws according to which the
light is there divided between the reflected and refracted rays,
including a determination of the attendant circumstances indi-
cated by the wave theory, with regard to the vibrations in the
reflected and refracted rays. In the case above mentioned, when
the incident light is polarized, there are four things to be deter-

56 Laws of Reflexion and Refraction

mined, namely, the magnitude and direction of the reflected vi-
bration, with the magnitudes of the two refracted vibrations. The
four conditions necessary for this determination are furnished
by two new laws, which could not be easily stated without en-
tering too much into detail. The results applied to determine
the polarizing angle of a crystal, in different azimuths of the
plane of reflexion, agree very closely with the.admirable experi-
ments of Sir David Brewster on Iceland spar. In the course of
these experiments it was observed that the polarizing angle re-
mained the same when the crystal was turned half round (through
an angle of 180°) ; although the inclination of the refracted _
rays to the axis of the crystal was thereby greatly changed.
This remarkable fact is a consequence of theory. After some
complicated substitutions in the primary equations, the value of
the polarizing angle is found to contain only even powers of the -
sine or cosine of the azimuth of the plane of reflexion, and there-
fore a change of 180° in the azimuth produces no change in the
polarizing angle.

The two new laws above mentioned, on which the theory
depends, occurred to the author in the beginning of last Decem-
ber; but, owing to an oversight in forming one of the equations,
they were not fully verified until the beginning of June.

In this theory it is supposed that the vibrations are parallel
to the plane of polarization, according to the opinion of M. Cau-
chy. This is contrary to the views of Fresnel, whose theory
of double refraction obliged him to adopt the hypothesis that
the vibrations are perpendicular to the plane of polarization. It
is further supposed that the density of the vibrating ether is
the same in both media; and the hypothesis of a constant den-
sity in different media, which was found necessary for the
theory, seems to accord, better than the supposition of a varying
density, with the phenomena of astronomical aberration.

If we conceive the three principal indices of refraction for
the crystal to become equal, we shall obtain the solution of a
very simple case of the general problem with which we have
been occupied—the case of an ordinary refracting medium such

at the Surface of Crystais. 57

as glass. This simple case, it is well known, was solved by
Fresnel. The foregoing theory leads to a simple law, expressing
all the particulars of the case, but differing with regard to the
magnitude of the refracted vibration from the formule of
Fresnel. The law may be stated by saying that the refracted
vibration is the resultant of the incident and reflected vibrations ;
the first vibration being the diagonal of a parallelogram, of
which the other two vibrations are the sides, just as in the com-
position of forces. The plane of the parallelogram is the plane
of polarization of the refracted ray. It is to be remembered,
that the vibrations in each ray are perpendicular to the ray itself,
and parallel to its plane of polarization.

This simple case has been considered by M. Cauchy in a
short Paper inserted in the Budletin Universel, tom. xiv. ; but it
does not seem to have been observed by anyone that his solu-
tion is erroneous. His formula for light polarized parallel to
the plane of reflexion is that which belongs to light polarized
perpendicular to the plane of reflexion, and vice versd.

VI.—LAWS OF REFLEXION FROM METALS.

[ Proceedings of the Royal Irish Academy, Vou. 1., p. 2.—Read Oct. 24, 1836.]

THe author observes that the theory of the action of metals -
upon light is among the desiderata of physical optics, whatever
information we possess upon this subject being derived from’
the experiments of Sir David Brewster. But, in the absence of
a real theory, it is important that we should be able to represent
the phenomena by means of empirical formule; and, accord-
ingly, the author has endeavoured to obtain such formule by
a method analogous to that which Fresnel employed in the case
of total reflexion at the surface of a rarer medium, and which,
as is well known, depends on a peculiar interpretation of the
sign f -1. For the case of metallic reflexion, the author
assumes that the velocity of propagation in the metal, or the re-
ciprocal of the refractive index, is of the form

m (cos x + 4/ - 1 sin x);

without attaching to this form any physical signification, but
using it rather as a means of introducing two constants (for
there must be two constants, m and y, for each metal) into
Fresnel’s formule for ordinary reflexion, which contain only
one constant, namely, the refractive index.

Then if 7 be the angle of incidence on the metal, and «’ the
angle of refraction, we have

sin” = m (cos x + / —1sin x) sin ¢, (1)

Laws of Reflexion from Metals. &e

and therefore we may put

cos i” = m’ (cos x’ — f — 1 sin x’) cos, - (2)

if
m’* cost = 1 — 2m* cos 2x sin*i + m* sin *?, (3)

and

m? sin 2y sin*?

wneXS 1 - m’ cos 2x sin?’ (4)

Now, first, if the incident light be polarized in the plane of
reflexion, and if the preceding values of sin 7’, cos 7’, be substi-
tuted in Fresnel’s expression

sin (i - 7)
anise) "

for the amplitude of the reflected vibration, the result may be
reduced to the form

7

a (cos 8 — of —1 sind),
if we put

tan p = — (6)

tan 6 = tan 2p sin (y + x’), (7)

, 1 —sin 2y cos (x + x’) 3
“ “1+ sin 2y cos (x + x)’ (8)

Then, according to the interpretation, before alluded to, of
/ —1, the angle 8 will denote the change of phase, or the retar-
dation of the reflected light ; and a will be the amplitude of the
reflected vibration, that of the incident vibration being unity.
The values of m’, x’, for any angle of incidence, are found by
formule (3), (4), the quantities m, x, being given for each metal.
The angle x’ is very small, and may in general be neglected.

60 Laws of Reflexion from Metals.

Secondly, when the incident light is polarized perpendicu-
larly to the plane of reflexion, the expression

tan (7 — 7’)
tan (¢ + 7)’

treated in the same manner, will become

a’ (cos 0 — .f - 1 sin 8), (9)

if we make
tan Y = mm’, (10)
tan & = tan 2/’ sin (y — y’), (11)

_ 1-sin 2y' cos (x - x’) , (12)
1 + sin 2y’ cos (x - x’)’

12

and here, as before, & will be the retardation of -the reflected
light, and a’ the amplitude of its vibration.

The number If = = may be called the modulus, and the

angle x the characteristic of the metal. The modulus is some-
thing less than the tangent of the angle which Sir David
Brewster has called the maximum polarizing angle. After two
reflexions at this angle a ray originally polarized in a plane in- »
clined 45° to that of reflexion will again be plane-polarized in a
plane inclined at a certain angle ¢ (which is 17° for steel) to the
plane of reflexion ; and we must have

12

tan ¢ = 5 (13)

Also, at the maximum polarizing angle we must have
o - 3 =90°. (14)
And these two conditions will enable us to determine the con-

stants Mf and y for any metal, when we know its maximum
polarizing angle and the value of ¢; both of which have been

Laws of Reflexion from Metals. 61

found for a great number of metals by Sir David Brewster.
The following Table is computed for steel, taking WM = 33,
x = 54°.

a 5 a a a’? % (a? + a?)
0 | 27° 27° 0-526 0526 _ | 07526
30 23 31 0-575 0-475 0°525
45 19 38 0°638 0-407 0-522
60 13 54 0-729 0-308 0-518
75 7 98 0-850 0-240 0-545
_| 85 2 152 0-947 0-491 0-719
90 0 180 1-000 1-000 1:000

The most remarkable thing in this Table is the last column,
which gives the intensity of the light reflected when. common
light is incident. The intensity decreases very slowly up to a
large angle of incidence (less than 75°), and then increases up to
90°, where there is total reflexion. This singular fact, that the
intensity decreases with the obliquity of incidence, was dis-
covered by Mr. Potter, whose experiments extend as far as an
incidence of 70°. Whether the subsequent increase which ap-
pears from the Table indicates a real phenomenon, or arises from
an error in the empirical formule, cannot be determined without
more experiments. It should be observed, however, that in these
very oblique incidences Fresnel’s formule for transparent media
do not represent the actual phenomena for such media, a great
quantity of the light being stopped, when the formule give a
reflexion very nearly total.

The value & — 8, or the difference of phase, increases from
0° to 180°. When a plane-polarized ray is twice reflected from
a metal, it will still be plane-polarized if the sum of the values
of & — 6 for the two angles of incidence be equal to 180°.

It appears from the formule that when the characteristic x
is very small, the value of 6’ will continue very small up to the

62 Laws of Reflexion from Metals.

neighbourhood of the polarizing angle. It will pass through
90°, when mm’ = 1; after which the change will be very rapid,
and the value of & will soon rise to nearly 180°. This is exactly
the phenomenon which Mr. Airy observed in the diamond.
Another set of phenomena to which the author has applied
his formule are those of the coloured rings formed between
a glass lens and a metallic reflector ; and he has thus been
enabled to account for the singular appearances described by
M. Arago in the Memoires d’ Arcueil, tom. iii., particularly the
succession of changes which are observed when common light
is incident, the intrusion of a new ring, &c. But there is one
curious appearance which he does not find described by any
former author. It is this. Through the last twenty or thirty
degrees of incidence the first dark ring, surrounding the central
spot, which is comparatively bright, remains constantly of the
same magnitude; although the other rings, like Newton’s rings
formed between two glass lenses, dilate greatly with the obli-
quity of incidence. This appearance was observed at the same
time by Professor Lloyd. The explanation is easy. It depends
simply on this circumstance (which is evident from the Table),
that the angle 180°-- 0’, at these oblique incidences, is nearly
proportional to cos 7. |
As to the index of refraction in metals, the author conjec-

tures that it is equal to :
cos x

(oy68-)

VII.—ON THE LAWS OF THE DOUBLE REFRACTION OF
QUARTZ.

[ Transactions of the Royal Irish Academy, Vou. xvu1.—Read Feb. 22, 1836.]

Tue singular laws of the double refraction of quartz, which have
been discovered by the successive researches of Arago, Biot,
Fresnel, and Airy, are known merely as so many independent
facts; they have not been connected by a theory of any kind.
I propose, therefore, to show how these laws may be explained
hypothetically, by introducing differential coefficients of the
third order into the equations of vibratory motion.

Suppose a plane wave of light to be propagated within a crys-
tal of quartz. Let the co-ordinates x, y, s, of a vibrating mole-
cule be rectangular, and take the axis of s perpendicular to the
plane of the wave, and the axis of y perpendicular to the axis of
the crystal. Let us admit that the vibrations are accurately in
the plane of the wave, and of course parallel to the plane of zy.
Then, using & and » to denote, at any time ¢, the displacements
parallel to the axes of x and y respectively, we shall assume the
two following equations for explaining the laws of quartz :—

ae ast * O ae ()
dn d’n BE
ae Pa Oe | (2)

The peculiar properties of this crystal depend on the con-
stant C. When C = 0, the third differentials disappear, and

64 On the Laws of the Double Refraction of Quartz.

the equations are reduced to the ordinary form, in which state
they ought to agree with the common equations for uniaxal
crystals. Hence, putting a for the reciprocal of the ordinary
index, } for the reciprocal of the extraordinary, and ¢ for the angle
made by the axis of s with the axis of the crystal, we must have

A=a@, B=a-(a@-V) sin’ 9, (3)

supposing the velocity of propagation in air to be unity.
Now, from the nature of equations (1) and (2), the vibra-
tions must be elliptical. In fact, if we put

E =p 00s = (st- 2), n=q sin ee (st -2)| (4)

where p, q, 8, are constant quantities, the differential equations
will be satisfied by assigning proper values to s and to the ratio

z For, after substituting in equations (1) and (2) the values

of the partial differential coefficients obtained by differentiat-
ing formule (4), we shall find that every term of each equa-
tion will have the same sine or cosine for a factor! omitting

therefore, the common factors, and making ; =k, we shall get

the following equations of condition :—

s=A- T Ck, (5)
A 247 C
v= B- we a (6)
Subtracting these, we have
A-B+7E (7-2) =0, (7)

which, by formule (3), becomes
1 ;
bas a Le ae oe ee 2 mar
k ao @ b*) sin’? ¢.k =1 (8)

Let us now interpret these results. It is obvious, from for-

On the Laws of the Double Refraction of Quartz. 65

mule (4), that s is the velocity of propagation fora wave whose
length is 7, and that each vibrating molecule describes a little
ellipse whose semiaxes p and q are parallel to the directions of x
andy. But the number /, expressing the ratio of the semiaxes,
has two values, one of which is the negative reciprocal of the
other, as appears by equation (8) ; and each value of / has a cor-
responding value of s determined by equation (5) or (6). Hence
there will be two waves elliptically polarized, and moving with
different velocities, the ratio of the axes being the same in both
ellipses; but the greater axis of the one will coincide with the
less axis of the other. The difference of sign in the two values —
of k shows that if the vibration be from left to right in one wave,
it will be from right to left in the other. These laws were dis-
covered by Mr. Airy.

The law by which the ellipticity of the vibrations depends on
the inclination ¢, and on the colour of the light, is contained in
equation (8). The value of the constant C will be determined
presently. In the mean time we may observe, that C denotes a
line, whose length is very small, compared with the length of a
wave.

- When ¢ = 0, the light passes along the axis of the crystal.
In this case we have kh? =1, and k=+1; which shows that there
are two rays, circularly polarized in opposite directions. The
value of s for each ray may be had from equation (5) or (6), by
putting + 1 and - 1 successively for &. Calling these values s’
and s”, we find

4 C r ar

=a — ar, ¢-a(1-%), (9)
4 C a” rO0
stad +ir—, fo a(1 +55) (10)

Suppose a plate of quartz to have two parallel faces perpen-
dicular to the axis, and conceive a ray of light, polarized in a
given plane, to fall perpendicularly on it. The incident rectilinear
vibration may be resolved into two opposite circular vibrations,

EF

66 On the Laws of the Double Refraction of Quartz.

which will pass through the crystal with different velocities ; and
which, after their emergence into air, will again compound a
rectilinear vibration, whose direction will make a certain angle p
with that of the incident vibration : so that the plane of polariza-
tion will appear to have been turned round through an angle
equal to p, called the angle of rotation. This angle may be de-
termined by means of the preceding formule, Putting @ for the
thickness of the crystalline plate, the circularly polarized wave
whose velocity is s’ will pass through it in the time

0 “( 7)
Peed Pa
Fe al

and the wave whose velocity is s” in the time

9, 20)
8” a a
Therefore, if 8 be the difference of the times, we have

_ 20700
Tat 5 geek

") (11)

But, since the velocity of propagation in air is supposed to
be unity, the time and the space described are represented by the
same quantity; and therefore 6, which is evidently a line, de-
notes the distance between the fronts of the two circularly pola-
rized waves, when they emerge into air. The waves being at
this distance from each other, if we conceive, at the same depth
in each of them, a molecule performing its circular vibration,
and carrying a radius of its circle along with it, the two radii
will revolve in contrary directions, and will always cross each -
other in a position parallel to the incident rectilinear vibration.
Now let two series of such waves be superposed, so as to agitate
every molecule by their compound effect, and it is evident that,
when the radius vector of one component vibration attains the
position just mentioned, the radius vector of the other will be se-

parated from it by an angle equal to ans where A is the length of

On the Laws of the Double Refraction of Quartz. 67

a wave in air. The resultant rectilinear vibration will bisect this
angle; and therefore p, the angle of rotation, will be equal to

a Hence, substituting for 3 its value, and observing that /,

r
the length of a wave in quartz, is equal to aX, we find

2

p= (12)
which gives the experimental law of M. Biot, that the angle of
rotation is directly as the thickness of the crystal, and inversely
as the square of the length of a wave for any particular colour.
By changing the sign of C, we should have an equal rotation in
the opposite direction. And here we may remark, that C may
_be made negative in all the preceding equations, its magnitude
remaining. There are two kinds of quartz, the right-handed
and left-handed, distinguished by the sign of C.

The angle of rotation, ‘for a given colour and thickness, is
known from M. Biot’s experiments. We can therefore find the
value of C by means of the last formula; and substituting this
value in equation (8), we shall be able to compute * when ¢ and
Zare given. Now it happens that Mr. Airy,* by a very inge-
nious method of observation, has determined the values of & in
red light for two different values of ¢; and of course we must
compare these observed values of / with the independent results
of theory. As Mr. Airy’s experiments were made upon red
light, we shall select, for the object of our calculations, the red
ray which is marked by the letter C in the spectrum of Fraun-
hofer. For this ray, Fraunhofer has given the length A, which,
expressed in parts of an English inch, is equal to (0000258 ; and
M. Rudberg has found a = *64859, } = 64481. Moreover, from
the experiments of M. Biot, we may collect that the are of rota-
tion, produced by the thickness of a millimetre, is something
more than 19 degrees for the ray we have chosen; so that the
fraction } may be taken to-express nearly the length of that

* Transactions of the Cambridge Philosophical Society, Vol. 1v., p. 205.
F 2

68 On the Laws of the Double Refraction of Quartz.

are in a circle whose radius is unity. We have, therefore,
6 = 03937 inch, and p = 333. Substituting these values in the
formula

som tet ee
27C a®Ap
derived from (12), we find
l
a0 52710 ;

from which it appears that C is about twenty thousand times less
than the millionth part of an inch.
Again, since a’ — 0° = 00489, we have

Y 2 ae
| on (a = b’) = 258,
so that equation (8) becomes
k? — 258 sin? o.k = 1. (13)

The results of this formula are compared with Mr. Airy’s
experiments in the following Table, in which the less root is
taken for k, and its sign is neglected.

Values of k.
Values of .
Observed. Calculated.
6% 15’ tan 16° 38’ = °2897 *2980
8° 54’ tan 8° 56’ = +1582 *1579

The angles ¢, in the first column, are deduced from the ob-
served inclinations of the rays in air to the axis of the crystal ;
and as k was observed to be somewhat different for the ordinary
and extraordinary rays, its mean values are given in the second
column. The exact coincidence between these and the calculated
values is, perhaps, in some degree accidental ; but a less perfect
agreement would be sufficient to confirm the theory.

On the Laws of the Double Refraction of Quartz. 69

The magnitude of & varies considerably with the colour of
the light, increasing from the red to the violet, while the coeffi-
cient of sin*#./, in formula (13) diminishes. If we take the
violet ray H, for example, this coefficient will be 159. But it
would be useless to make any more calculations, as we have no
experiments with which they might be compared.

The figure of the wave surface yet remains to be examined.

Eliminating / between formule (5) and (6), we obtain the
equation

G 2

(s° — A) (8° - B) = 40° =

?? > (14)

which expresses the nature of the surface, s being a perpendi-
cular from the origin on a tangent plane. From this equation
it follows that the two values of s can never become equal in
quartz, as they do in other crystals; for ifwe solve the equation
for s’, and put the radical equal to zero, we shall get the condi-
tion
Co 2
(A - B)? + 167? Fae 0,

which cannot be fulfilled, since the quantity which ought to
vanish is the sum of two squares. The two sheets, or nappes, of
the wave surface are therefore absolutely separated.

It is commonly assumed that one of the rays is refracted
according to the ordinary law; but this is not the case, since
neither of the values of s is constant. However, the ray which
has the greater velocity (a being greater than }) may still, for
convenience, be called the ordinary ray. Of the two roots of
equation (8), the one, /,, whose numerical value (supposing ¢ not
to vanish) is less than unity, corresponds to this ray. When C
is positive, k, is negative ; and when C is negative, 4, is positive :
therefore in both kinds of quartz, by formule (5) and (6), we
have s,? > A, and s,2< B; denoting by s, and s, the respective
velocities of propagation of the ordinary and extraordinary waves.
Hence, if we conceive a sphere of the radius a, with its centre at
the origin, and a concentric prolate spheroid, whose semiaxis of

70 On the Laws of the Double Refraction of Quartz.

revolution is also equal to a, and parallel to the axis of the
crystal, while the radius of its equator is equal to b, the ordi-
nary nappe of the wave surface will fall entirely without the
sphere, and the extraordinary nappe entirely within the spheroid,
whether the crystal be right-handed or left-handed. With re-
spect to the little ellipse in which the vibrations are performed,
and of which the semiaxes parallel to w and y are represented by
pand gq respectively, it is evident that p > g for the ordinary
wave, since k,< 1; and that p< q for the extraordinary wave.
When C vanishes, the minor axis of each ellipse also vanishes,
and the rays become plane-polarized, the ordinary vibrations —
being then parallel to the direction of x, and the extraordinary -
parallel to that of y. This is exactly what ought to happen on
the supposition that the vibrations of a plane-polarized ray* are
parallel to its plane of polarization—a supposition which was
kept in view in framing the fundamental equations (1) and (2).

To show, with precision, how the two kinds of quartz are to
be distinguished by the sign of C, we must give definite direc-
tions to the axes of co-ordinates. To this end, let us imagine the
plane of zy to be horizontal, and a circle to be described in it
with the origin O for its centre; and let the north, east, and
south points of this circle be marked respectively with the letters
N, E,8. Let the direction of + w be eastward, from O to E;
that of + y northward, from O to V; and that of + s vertically
downwards ; the progress of the light through the crystal being

* On this point there are two very different opinions. Fresnel supposed, as is
well known, that the vibrations of a plane-polarized ray are perpendicular to its
plane of polarization; whereas, according to M. Cauchy, whom I have followed,
they are parallel to that plane. I am induced to adopt the latter supposition, be-
cause I have succeeded, by means of hypotheses which are grounded on it, in dis-
covering the laws of reflexion from crystallized surfaces ; laws which include, as a
particular case, those discovered by Fresnel for ordinary media. The hypotheses
alluded to, along with-some of their results, are published in the London and Edin-
burgh Philosophical Magazine, Vol. viu., p. 103, in a letter to Sir David Brewster
(supra, pp. 75, et seg.) See also Vol. vu, p. 295, of the same Journal (supra,
pp- 55, et seg.) I hope soon to offer the Academy a detailed account of my researches
on this subject. ep

On the Laws of the Double Refraction of Quartz. 71

also downwards, and the plane of the wave moving parallel, as
before, to the plane of wy. ‘Then the crystal will be right-handed
or left-handed, according as C is positive or negative. For, if C
be positive, /, will be negative, and formule (4) will become, by
exhibiting the sign of /;,,

— =p cos Fey}, n=-—k, p sin HF Gt - 2), (15)

for the ordinary vibration ; and
E=k,q cos 5 (ta), n= q sin IF or —=){, (16)

for the extraordinary vibration. Now if we suppose the are
= (s¢ - s) either to vanish, or to be a multiple of the circum-
ference, the molecule will be at the east point of its vibration ;_
and upon increasing the time a little, the value of » will become
negative in (15), and positive in (16), so that the movement will
be towards the south in the first case, and towards the north in
the second. Therefore, when C is positive, the ordinary vibra-
tion takes place in the direction VES, or from left to right, and
the extraordinary in the direction SEN, or from right to left,
supposing a spectator to look in the direction of the progress of
the light. It may be shown, in like manner, that when C is ne-
gative, the ordinary and extraordinary vibrations are in the
directions SEN and NES, or from right to left and from left to
right respectively. _ Now if a plane-polarized ray be transmitted
along the axis of the crystal, the plane of polarization will be
turned in the direction of the ordinary vibration, because this
vibration, being propagated more quickly, will be in advance of
the other, upon emerging from the crystal. Hence, the rotation
is from left to right when C is positive, and from right to left
when C is negative; and the crystal is called right-handed i in
the first case, and left-handed in the second.

We have all along supposed that C is a constant quantity,
and the agreement of our results with experiment proves that

72 On the Laws of the Double Refraction of Quartz.

this supposition is at least very nearly true in the neighbourhood
of the axis. It is probable, however, not only that C varies with
¢, but that it becomes different in equations (1) and (2) ; that
is to say, it is probable that the following equations

(17)

in which C’ is a little different from C, would be more correct
than those which we have assumed. Indeed Mr. Airy’s experi-
ments seem to indicate that C’ is greater than C; for he found,
as we have already said, that the ratio of the axes of the little
ellipse described by a vibrating molecule is somewhat different
for the two rays, being more nearly a ratio of equality for the
ordinary than for the extraordinary ray. Now if we set out
from equations (17), instead of (1) and (2), and proceed in all
respects as before, we shall arrive at the formula

,

ie (at — 8) sin $k = A (18)

instead of formula (8). The quantity _ will be greater than

unity, if C’ be greater than C, and the value of &, will be greater
than before. This seems to be the explanation of the difference
between the ratios observed by Mr. Airy.

It may be proper to state briefly the considerations which led
to the foregoing theory. Beginning with the simple case of a
ray passing along the axis, the first thing to be explained was
the law of M. Biot, that the angle of rotation varies inversely as
the square of / or of A. Now it was remarked by Fresnel, who
first resolved the phenomena of rotation into the interference of
two circularly polarized waves, that the interval 6 between these
waves, at their emergence from the crystal, must be inversely as
1, if the angle of rotation be inversely as the square of 7, This re-

eet

On the Laws of the Double Refraction of Quartz. 73

mark suggested* to me the idea of adding, to the equations of the
common theory, terms containing the third differential coefficients
of the displacements; for it was evident that such additional
terms would give, in the value of s’, a part inversely proportional
to 7. It was also evident that the third differential coefficient
of — should be combined with the second differential coefficients
of n, and the third of n with the second of &, in order that, after
substitutions such as we have indicated in deducing formule (5)
and (6), the sines or cosines might disappear by division, and
that thus the value of s? might be independent of the time, as
it ought to be. This kind of reasoning led me to assume the
equations 2 Be 2
S = 7+ Os, (19)
—=¢@—+D (20)

for the case of a ray passing along the axis of quartz; and then,
substituting in these equations the values of the differential co-
efficients obtained by differentiating the formule

E— =p cos °F (et — 5) , n=+tpsin IF oes),

which express a circular vibration (from right to left, or from
left to right, according to the sign of the second p), the result
was

from (19), and

* «The singular relation between the interval of retardation [8] and the length
of the wave [7] seems to afford the only clue to the unravelling of this difficulty.””—
* Report on Physical Optics,’’ by Professor Lloyd (‘‘ Fourth Report of the British
Association,” p. 409). It was in reading this Report that Fresnel's remark, about
the relation between 6 and /, first came to my knowledge.

74. On the Laws of the Double Refraction of Quartz.

from (20) ; which showed that D =- C, since the values of s,
corresponding to the same circular vibration, ought to be equal.
The transition from this simple case to that of a ray inclined at
a given angle @ to the axis was easily made, by taking into ac-
count the doubly refracting structure of the crystal. This was
done by supposing € and » parallel to the principal directions in
the plane of the wave, and by changing a’, in equation (20), into
a — (a?-0*) sin’; and thus the fundamental equations (1) and
(2) were obtained.

ee
7 3

C7520)

VIII.—ON THE LAWS OF REFLEXION FROM CRYSTAL-
LIZED SURFACES.

[From the Philosophical Magazine, Vou. v1u1., 1835: ]

To Str Davin Brewster.

Dear Str—I have great pleasure in sending you an account
of the laws by which I conceive that the vibrations of light are
regulated when a ray is reflected and refracted at the separating
surface of two media; especially as the only guide which I had,
in my inquiry after these laws, was your Paper on the action of
crystallized surfaces upon light, published in the Philosophical ~
Transactions for the year 1819. The observation which I found
there, that the polarizing angle was the same for a given plane
of incidence, “ whether the obtuse angle of the rhomb [ of Iceland: :
spar | was nearest or furthest from the eye, or whether it was to
the right or left hand of the observer,” disappointed me at first,
being contrary to what I had anticipated from principles ana-
logous to those which had been employed by Fresnel in the-
problem of reflexion from ordinary media. I then sought other
principles, and the observation is now a result of theory.
Assuming, as a basis for calculation, that Fresnel’s law of
double refraction is rigorously true, I have been obliged to
make an essential change in his physical ideas. Conceive an
ellipsoid whose semiaxes are parallel to the three principal direc-
tions of the crystal, and equal respectively to its three principal
indices of refraction, and let a central section of the ellipsoid
be made by a plane parallel to the plane of a wave passing

76 On the Laws of Reflexion

through the crystal. The section will be an ellipse, and the
wave will be polarized by the crystal in a plane parallel to either
semiaxis of this ellipse, the index of refraction for the wave
being equal to the other semiaxis. This is Fresnel’s law of

double refraction; and the theory which led him to it makes it

‘necessary to waive that the vibrations of the wave are perpendi-
cular to its plane of polarization; whereas, according to the
views which I have adopted, the vibrations of the wave are pa-
rallel to its plane of polarization, and to one semiaxis of the
elliptic section, while its index of refraction is equal to the other

semiaxis. These views nearly agree with the theory of M. Cauchy,

according to whom the vibrations of polarized light are parallel

to its plane of polarization, but inclined at small angles to the
plane of the wave in crystallized media, instead of being exactly
parallel to the latter plane, as I have supposed them to be. Be-
sides, the theory of M. Cauchy, founded on the six equations of
pressure in a crystallized medium, implies the existence of a
third ray of feeble intensity, and for the other two rays gives a
law somewhat different from that of Fresnel. Being obliged, in
order to account for your experiments, to abandon the physical
ideas of Fresnel, and to approximate towards those of M. Cauchy,
I was embarraégsed by this third ray ; and wishing to get rid of
it, as well as of the slight deviations from the symmetrical law
of Fresnel, I adopted the expedient of altering the equations of
pressure, in such a way as to make them afford only two rays,
and give a law of refraction exactly the same as Fresnel’s. The

]

equations which I found to answer this purpose are the follow- _

ing :—

A=- 24024 +02) V* p,
dy

Jrom Crystallized Surfaces. 77

di dé\
R= v (Es ae

a, dn
free & dy +z V?

In these equations, the axes of co-ordinates are perpendicular
to each other, and parallel to the principal directions of the erys-
tal; x, y, s are the co-ordinates of a vibrating molecule at the
time ¢; &, », ¢ are the components of the displacement of the
same molecule at the same time; 4, b, ¢ are the three principal
indices of refraction out of the crystal into an ordinary medium
in which the velocity of light is equal to V; and p is the density
of the ether, which density I suppose to be the same in all media.
The quantities A, F, E are the components, parallel to the axes
of x, y, s, respectively, of the pressure upon a plane perpendi-
cular to the axis of x; F, B, D are the components of the pres-
sure upon a plane perpendicular to the axis of y; and FZ, D, C
the components of the pressure upon a plane perpendicular to
the axis of s. The values of D, H, Fare the same as those given
by M. Cauchy; but the values of A, B, C are different from
his, and much simpler. By introducing into the equations of
M. Cauchy the condition that the vibrations shall be performed
without any change of density, the resulting values of A, B, C
might be shown to agree nearly with those given above. “The
six pressures, A, B, C, D, E, F, being known, it is easy to find
the pressure upon a plane making any given angles with the
axes of co-ordinates.

These things being premised, it is time to mention the laws,
or rather hypotheses, which I have imagined for discovering the
relations that exist, as to direction and magnitude, among the
vibrations in each ray, when reflexion and refraction take place at
the separating surface of two media, whether crystallized or not.
In stating the two very simple laws that have occurred to me

78 On the Laws of Reflexion

for this purpose, it will be convenient, when the first medium is
an ordinary one, to suppose that the incident light is polarized.
Then, by the first law, the vibrations in one medium are equivalent
to those in the other ; that is to say, if the incident and reflected
vibrations be compounded, like forces acting at a point, their
resultant will be the same, both in length and direction, as the
resultant of the refracted vibrations similarly compounded. By
the second law, the lateral pressure upon the separating surface is
the same in both media ; the lateral pressure being understood to
mean the pressure in a direction perpendicular to the plane of
incidence. .

As it would engage us too long to follow these laws into
detail, I shall merely state some of the results which I have ob-
tained from them, for the. case of a uniaxal crystal into which
the light passes out of an ordinary medium.

Imagine the surface of the crystal to be horizontal, and call
the point of incidence J. With the centre J and any radius, con-
ceive a sphere to be described, cutting in the point Z a vertical
line JZ drawn through the centre, z
and let a radius IP, parallel to the
axis of the crystal, meet the surface

of the sphere in P. Let the great ‘i

circle ZOE be the plane of incidence, EB

containing both the direction JO of

the ordinary refracted ray produced P
backwards, and the direction LH of rr

a normal to the extraordinary wave ;
and draw the great circles PZ, PO, PE. The angle Zwill be the
azimuth of the plane of incidence. Let Z0= 9, ZH =9', PO=y,
PE=YwW, the angle ZOP = 0, and the angle ZEP = 0’. Call the
angle of incidence 7, and suppose 0 to be the reciprocal of the ordi-
nary refractive index and a the reciprocal of the extraordinary.
Each of the refracted rays, in turn, may be made to disappear,
by polarizing the incident ray in a certain direction assigned by
theory. When the extraordinary ray disappears, the reflected
ray is polarized in a plane inclined to the plane of incidence at

Jrom Crystallized Surfaces. 79

an angle 6 determined by the formula
tan 3 = cos (7+ p) tan 0+ 2 (a® - 0”) sn sin cosy mt (2)

When the ordinary ray disappears, the plane of polarization
of the reflected ray is inclined to the plane of incidence at an
angle (3 determined by the formula :

- tan 3’ = cos (7+ ¢’) cotan

‘cos 2° sin? 7
(a tar,

- ha mY sin { cos y/

And when the angles (, 3’, become equal, the plane of polariza-
tion of the reflected ray becomes independent of the plane of po-
larization of the incident ray; and the angle of incidence #, at
which this equality takes place, is the polarizing angle of the
crystal. Hence we have the equation of condition

cos (7+ @) tan 8 + 2 (a* — ”) sin @ sin y cos cee |
* y=0

a acigh , 9\ COS 20 , sin? 7 |
+ cos(7 +9’) cotan 0 + (a - = OY a gy sin cosy’ suG=4)
to be fulfilled at the polarizing angle.
Since 7 + ¢, in this equation, is nearly equal to a right angle,

put i+ = ‘ + 6, and 6 will be a small quantity. Draw PR

an are of a great circle perpendicular to ZOZ, and let ZR = p,

PR-=gq. Then we shall find from equation (4), after various

substitutions and reductions,

(a - B) (1+28*) :
Ba-5° ©

8 = K cos’ g (cos* ¢ — cos’ p) ; where K =

In deducing this value of 6, the approximations were made
with a tacit reference to the case of reflexion in air from a com-
mon rhomb of Iceland spar. The coefficient X, in this case, is
equal to about nine degrees, and the resulting numerical values of

80 On the Laws of Reflexion

the polarizing angles in various azimuths agree very well with
your experiments. You will perceive that the value of 6 is the
same in supplementary azimuths, which explains the observation,
cited in the beginning of my letter, relative to the equality of
the polarizing angles at opposite sides of the perpendicular [7
in a given plane of incidence.

When the point & falls upon O, we have 6=0, andi +@
equal to a right angle. Hence, when the cotangent of ZR is
equal to the ordinary index, the tangent of the polarizing angle
is equal to the same index. This theorem, though deduced from
an approximate equation, might be shown to be exact.

When the axis of the crystal lies in the plane of incidence, we
may obtain an exact expression for the polarizing angle. The
condition of polarization then becomes

sin’ 7

thie (i+ #’) = (a a b) sin yp cos y inGs@) me

0; (6)

from which, by the proper substitutions, we obtain the following

expression :—
.,. L-a@ cos’ - 0b’ sin’ A
sin’ 7= es a i? ] (7)

where ) denotes the complement of ZP, or the inclination of the
axis to the face of the crystal, and ¢ is the polarizing angle.
This formula, in a shaz‘e somewhat different, was communicated,
above a year ago, ti/ Professor Lloyd, who has noticed it, in con-
nexion with your "ap, in his “ Report on Physical Optics.”
When a and 0 becoinjiefes al the formulk:gives your law of the
tangent for ordinary’ incisy > Ta

The foregoing %/P :. show that, when a ray is Polarized by
reflexion from a cry 4n¢, the plane of polarization deviates from
the plane of inciden'p,, except when the axis lies in the latter
plane ; and that ies deviation may be made very great by
psi: Mw cTYS/41 in contact with a medium whose refractive
power is nearly €4 ual to that of the crystal itself ; for when ¢ is

nearly equal to p 0. 4, g’, the divisor sin (7 — ) or sin (i- 9’) is

sé — lad

Laws of Reflexion from Crystallized Surfaces. 81

very small, and therefore tan or tan f’ is very great. But
this remark is of no value whatever in explaining the very sin-
gular phenomena which you have observed in the extreme case
just mentioned ; nor can I imagine any reason why there should
be a deviation, as there was in some of your experiments, when
the axis lies in the plane of incidence, since everything is then
alike on both sides of this plane. Indeed the whole of this
subject, which occupies the latter part of your Paper of 1819, is
very extraordinary and interesting ; and I was glad to hear that
you had resumed the investigation of it, and made many experi-
ments which have not been published.

I wish you would publish them. They seem to be of ee
importance in the present state of optical science.

I am, dear Sir, ever truly yours,
J. Mac Cuttacu.

Trin. Cotu., Dusuin, Dec. 22, 1835.

IX.—ON THE PROBABLE NATURE OF THE LIGHT TRANS.-
MITTED BY THE DIAMOND AND BY GOLD LEAF.

[ Proceedings of the Royal Irish Academy, Vou. 1. p. 27.—Read Jan. 9, 1887.]

Proressor Mac CuLiacu made a verbal communication on the
probable nature of the light transmitted by the diamond and by
gold leaf. He conceives that as there is a change of phase caused
by reflexion from these bodies, so there is also a change of phase
produced by refraction ; the change being different according as
the incident light is polarized in the plane of incidence, or in the
perpendicular plane. Consequently, if the incident ray be po-
larized in any intermediate plane, the refracted ray should be
elliptically polarized ; and on examining the light transmitted
by gold leaf, this was found to be the case. Of course the same
thing is true of the light which enters the other metals, and
which # subsequently absorbed. The same remark explains the
appearance of double refraction in specimens of the diamond
which give only a single image; and it is likely that other
precious stones will be found to possess similar properties. Mr.
Mac Cullagh has obtained a general formula for the difference
of phase between the two component portions of the refracted
light—one polarized in the plane of incidence, and the other
perpendicular to it. He finds from this formula, that the dif-
ference ‘of phase, which is nothing at a perpendicular incidence,
increases until it becomes equal to the characteristic at an inci-
dence of 90°; and when the light emerges into air, the difference
of phase is doubled. The formula has not yet been submitted
to the test of experiment.

(cx835-)

X.—ON THE LAWS OF CRYSTALLINE REFLEXION.

[ From the Philosophical Magazine, Vou. x., 1837.]

In a Number of Poggendorff’s Annalen (No. 6, for 1836), which
reached Dublin late in November, there are some remarks by
M. Seebeck on a Paper of mine which appeared in the last
February Number of this Journal (vol. viii. p. 103). That
Paper contains a general theory of reflexion at the surfaces of
crystallized media; and M. Seebeck, in comparing the results
with his own experiments, has fully confirmed some of my
formule, while he has shown that others are defective. I have
therefore been obliged to revise my theory, and I have ascer-
tained that it was vitiated by the introduction of a certain re-
lation among the quantities. denominated pressures, which,
following the example of M. Cauchy, I had supposed to be
concerned in the problem. This relation I had observed to
hold in the case of singly refracting media, and I concluded,
without any other reason, that it would hold good generally.
But though it led to the correct formula for the polarizing
angles in different azimuths, it was nevertheless arbitrary and
unfounded; and therefore it is now banished entirely from the
investigation, the place which it occupied being supplied by the
natural and simple law of the preservation of vis viva, while
everything else remains as before. I hope the imperfection of
my first essay will be excused, when it is considered that the
erroneous proposition bears but a small proportion to the whole
theory; and, moreover, that the general problem, which I under-
took to resolve, is one that has not been attempted by any other
G 2

84 On the Laws of Crystalline Reflexion.

person, although the want of a solution has long been felt. The
difficulties which we have to deal with, in entering upon this
problem, are not mere mathematical difficulties, but difficulties
arising from the want of first principles; and, in physical
questions of this kind, where we must, at the outset, have re-
course to conjecture, in order to supply the very principles of
our reasoning, it can hardly be expected that the whole truth
should be divined at once. I think, however, that I have now
obtained a true mechanical theory; and if so, it will help to-
decide, not only the question immediately before us, but also
the other much-disputed, though more elementary, questions
concerning the density of the ether in different media, and
the direction of the vibrations in polarized light. In fact, a
particular supposition respecting each of the latter questions
is included in my theory, the several principles of which,
making the single alteration that has been mentioned, I shall
here enumerate :—

1. The density of the ether is the same in all media.

2. The vibrations of plane-polarized light are parallel to the
plane of polarization.

3. The vis viva is preserved.

4. The vibrations are equivalent at the common surface of
two media.

To these may be added the definition of the polarizing
angle of a crystal; namely, the angle of incidence at which
the plane of polarization of the reflected ray becomes indepen-
dent of the plane of polarization of the incident ray. At the
polarizing angle, the former plane does not, in general, coincide
with the plane of reflexion, but makes with it a small angle
which may be called the deviation.

It is curious that, about a year and a half ago, I employed
these four principles, precisely as I have now enumerated them,
in deducing Fresnel’s well-known laws of reflexion for ordinary
media; but I did not then apply the law of vis viva to crystals,
because my mind was preoccupied by the notion that there ex-
isted some relation among the pressures. This notion I had

On the Laws of Crystalline Reflexion. 85

taken up from reading a little Paper, by M. Cauchy, in the
Bulletin des Sciences Mathématiques for July 1830; and by
combining such a relation with the three conditions afforded
by my own law of equivalent vibrations, I had actually obtained,
for the polarizing angles in different azimuths, a formula (that
marked (5) in my former Paper), which I found to agree very
well with Sir David Brewster’s experiments, and which M.
Seebeck has found to agree still better with his own.

The formula for the polarizing angle is obtained by equat-
ing two values of the deviation; and it is remarkable that the
very same formula comes out in my present theory, although
the values of the deviation are entirely different. Referring,
for brevity, to the notation of my former Paper, I find, for the
ease of a uniaxal crystal,

ees G08 45--+ ) TAN Gc ee ge ee wees (a)
sin p’ cosy’ sin® i W)

sin 0’ sin (i- ¢’)

These equations (a) and (0) are to be substituted for equa-
tions (2) and (3),* which are the equations that M. Seebeck found
to be at variance with his experiments.

By means of formula (5), equation (a) becomes

— tan 3’ = cos-(i + ¢’) cotan 6’ + (a* -— 0°)

B= 9 Sin 2g sin i seese, Pipe dee aia Sane seas Tt (c)

from which the deviation in any azimuth may be readily calcu-
lated. The azimuth (as M. Seebeck reckons it) begins when 0
= 0, and p is then positive. This formula (c) perfectly represents
the experiments of M. Seebeck on Iceland spar. The corre-
sponding expressions for biaxal crystals may be easily deduced,
and will be given in a Paper which I am preparing to lay
before the Royal Irish Academy.

At the time of my last communication I was not aware that
the case in which the plane of incidence is a principal section of
the crystal (or the azimuth = 0) had been solved by M. Seebeck,
and that formula (7),f which I regarded as my own, had been
obtained by him long before.

* Supra, p. 79. + Supra, p. 80.

86 On the Laws of Crystalline Reflexion.

It remains to say a word respecting the new principle of
equivalent vibrations, the most important, perhaps, of all, as it
is certainly the simplest that can be imagned. If we conceive
an ethereal molecule situated at the common surface of two
media, it would seem that its motion ought to be the same,
whether we regard the molecule as belonging to the first me-
dium or to the second. Now the incident and réflected vibra-
tions are superposed in the first medium, and the refracted
vibrations in the second; and therefore we may infer (when
the phase is not changed by reflexion or refraction), that if the
incident and reflected vibrations be compounded, like forces acting at
a point, their resultant will be the same, both in length and direc-
tion, as the resultant of the refracted vibrations similarly com-
pounded. This is the law of equivalent vibrations, and it
gives, at once, three equations. A fourth equation is afforded
by Fresnel’s law of the vis viva; and thus we have the four
conditions necessary for a general solution of the problem.
From the principle of equivalent vibrations, as we have
stated it, it follows that the vibrations resolved parallel to the
separating surface are equivalent in the two media; and, in
fact, this part of the general principle was assumed by Fresnel ;
but the other part, namely, that the vibrations perpendicular
to the separating surface are equivalent, was not assumed by
him, nor is it by any means true in his theory. It appears
then that three conditions only are afforded by the hypotheses
which Fresnel successfully employed in solving the problem of
reflexion from ordinary media. These hypotheses, therefore,
are not sufficient when applied to crystals; except, indeed, in
the case before alluded to, where the azimuth = 0, which has
been solved by M. Seebeck. It should be observed, that though
the reasons which I have assigned for the principle of equivalent
vibrations are extremely simple, yet it was not by such simple
reasoning that I was led to it originally.

Trinity CoLttecE, Dusiriy, December 13, 1836.

287.)

XI.—ON THE LAWS OF CRYSTALLINE REFLEXION AND
REFRACTION.

[ Transactions of the Royal Irish Academy, Vou. xvu1.—Read Jan. 9, 1837. ]

Wuen a ray of light, which has been polarized in a given plane,
suffers reflexion and refraction at the surface of a transparent
medium, the rays into which it is divided are found to be po-
larized in certain other planes; and it becomes a question to
determine the positions of these planes, as well as the relative
intensities of the different rays; or, in theoretical language, to
find the direction and magnitude of the reflected and refracted
vibrations, supposing those of the incident vibration to be given.
The transparent medium may be either a singly-refracting sub-
stance, such as glass, or a doubly-refracting crystal, like Iceland
spar. When the medium is of the first kind, the problem is
comparatively simple, being, in fact, nothing more than a par-
ticular case of the problem which we have to consider when the
medium is supposed to be of the second kind. In the progress
_of knowledge it was natural that the simpler question should be
first attended to; and accordingly Fresnel, during his brief and
brilliant career, found time to solve it. But the general problem,
relative to doubly-refracting media, had not been attempted by
anyone, when, in the year 1834, my thoughts were turned to
the subject. I then recollected a conclusion to which I had
been led some years before, and which, on this occasion, proved
of essential service to me. Being fond of geometrical construc-

88 On the Laws of Crystalline

tions, I amused myself, when I first became acquainted with
Fresnel’s theories, by throwing his algebraical expressions, when-
ever I could, into a geometrical form ; and treating in this way
the well-known formule in which he has embodied his solution
of the problem just alluded to, I obtained a remarkable result,
which gave me the first view of the principle that I have since
employed under the name of the principle of the equivalence of
vibrations. In order to state this result briefly, I will take leave
to introduce a new term for expressing a right line drawn
parallel to the plane of polarization of a ray, and perpendicular
to the direction of the ray itself. Calling such a right line the
transversal of the polarized ray, I found, from the formule
of Fresnel, that when polarized light falls upon a singly-
refracting medium, the transversals of the incident, of the re-
flected, and of the refracted rays are all parallel to the same
plane, which is the plane of polarization of the refracted ray ;
and that the magnitudes of the vibrations, or the greatest ex-
cursions of the ethereal molecules, in the incident and the
reflected rays, are to each other inversely as the sines of the
angles which the respective transversals of those rays make with
the transversal of the refracted ray. I was struck by the strong
analogy which these relations among the transversals bore to
the composition of forces or of small vibrations in mechanics ;
but it happened unfortunately that, in the theory of Fresnel,
the vibrations of light were supposed to take place, not in the
direction of the transversals, but perpendicular to them, so that
there was no physical circumstance to support the analogy, there
being no motion in the direction of the transversals ; while, on
the other hand, no such analogy existed among the vibrations
themselves in the directions which Fresnel had assigned to them.
It was therefore with some interest that I afterwards learned,
upon the publication of the tenth volume of the Memoirs of the
Institute, that M. Cauchy* had actually inferred, from mecha-
nical principles, that the vibrations of polarized light are in the

* Mémoires de l’ Institut, tome x. p. 304.

Reflexion and Refraction. ‘89

direction of the transversals; but this inference was to be re-
ceived with caution, as being contrary to the hypothesis of
Fresnel; and besides, I had in the meantime contrived a way
of adapting my analogy, in some degree, to that hypothesis, by
supposing areas to be compounded instead of vibrations ; so that
I hesitated which of the two opinions to prefer. Taking, how-
ever, the opinion of M. Cauchy as that which fell in more na-
turally with the aforesaid analogy, I was led to the conclusion,
that the vibration in the refracted ray is probably the resultant
of the incident and reflected vibrations; and I saw that if this
principle were true for singly-refracting media, it should also,
from its very nature, be true, when properly generalized, for
doubly-refracting crystals ; so that in such crystals the resultant
of the two refracted vibrations would be the same, both in length
and direction, as the resultant of the incident and reflected
vibrations.

This was the principle of equivalent vibrations. But I had
no sooner begun to regard it as probable, than an objection
started up against it. In the case of a ray ordinarily refracted
out of a rarer into a denser medium, the magnitude of the
refracted vibration, as deduced from this principle, was greater
than that which came out from the theory of Fresnel, in the
proportion of the sine of the angle of incidence to the sine of
the angle of refraction. Consequently, assuming with Fresnel
that the ether is more dense in the denser medium, the law of
the preservation of vis vida was violated.

There was another embarrassment which I felt in my early
efforts to find out the laws of crystalline reflexion. Taking
for granted the hypothesis of Fresnel, that the density of the
ether in an ordinary medium is inversely as the square of its
refractive index, I was at a loss what hypothesis to make, in
this respect, for doubly-refracting crystals, wherein the refrac-
tive index changes with the direction of the ray. For the
density, being independent of direction, could not be con-
ceived to vary with the refractive index. About two years
ago I got over this difficulty, by supposing the density of the

go On the Laws of Crystalline

ether to be the same in all media.* At the same time I was
compelled to employ the principle of equivalent vibrations, in
order to have a sufficient number of conditions, though for a
while I overlooked the perfect agreement which now subsisted
between this principle and the law of vis viva: it happened, in
fact, that the new hypothesis of a constant density made the
vis viva of the refracted ray exactly the same as in the theory
of Fresnel.t

But to see why it was necessary to assume the principle of
equivalent vibrations, we must observe, that when a polarized
ray is incident on a crystal there are four things to be deter-
mined, namely, the direction and magnitude of the reflected

vibration, and the magnitudes of the two refracted vibrations. .

Hence we must have four conditions, or we must have relations
affording so many equations. But the hypotheses of Fresnel, -
by which he solved the problem of reflexion for ordinary media,
afford only three conditions. We will state his hypotheses at
length :—

1st. The vibrations of polarized light are in the plane of the
wave, and perpendicular to the plane of polarization.

2nd. The density of the ether is inversely as the square of
the refractive index of the medium.

3rd. The vis viva is preserved.

4th. The vibrations parallel to the separating surface of two
media are equivalent; that is, the refracted vibration parallel to
the surface is the resultant of the incident and reflected vibra-
tions parallel to the same.

We see that the fourth hypothesis gives two conditions, and
the law of vis viva gives a third.

Let us now take the more general principle of equivalent
vibrations, in place of the fourth hypothesis of Fresnel, altering

* This hypothesis is maintained by Mr. Challis; and certainly it falls in ex-
tremely well with the astronomical phenomenon of the aberration of light.—See,
on this subject, Professor Lloyd’s Report on Physical Optics, ‘‘Fourth Report of the
British Association for the Advancement of Science,’’ pp. 311, 313.

t Supra, p. 100, note.

Reflexion and Refraction. gl

the first hypothesis in the way that we have shown to be neces-
sary in order to suit that principle, and making the ethereal
density constant. Then, if we retain the law of vis viva, our
new hypotheses will be these :—

Ist. The vibrations of polarized light are in the plane of the
wave, and parallel to the plane of polarization; which may be
_ expressed in a word, by saying that the vibrations are transver-
sal, according to the peculiar sense in which I use the term.
2nd. The density of the ether is the same in all bodies as in
_ vacuo.

3rd. The vis viva is preserved.

4th. The vibrations in two contiguous media are equivalent;
that is, the resultant of the incident and reflected vibrations is

the same, both in length and direction, as the resultant of the
refracted vibrations.

It is evident that the last hypothesis affords three equations,
by resolving the vibrations parallel to three axes of co-ordinates;
‘and the law of vis viva supplies a fourth equation. Thus we

have the requisite number of conditions.

The hypotheses that we have last enumerated are those
which will be employed in the present Paper. They have been
made to include the law of vis viva, because I lately found that
this law must necessarily accompany the rest; but at first I
neglected it, and even made considerable progress without it;
for, by the help of another hypothesis, I obtained formule
which represented such experiments as I was aware of at the
_ time. This other hypothesis I took up from reading an article
by M. Cauchy in the Bulletin des Sciences Mathématiques,* in
which he arrives, by a peculiar process, at the formule of
Fresnel for the case of ordinary reflexion. The hypotheses

* «Sur la Réfraction et la Réflexion de la Lumiére,’’ Bulletin des Sei. Math.,
Juillet, 1830. “In this Paper the vibrations of polarized light must be supposed
perpendicular to the plane of polarization, though the Paper was published im-
mediately after the author had promulged the contrary opinion. The latter
opinion, which I adopted from him because it harmonized with my analogy

before mentioned, he has formally renounced of late, and has returned to the

92 On the Laws of Crystalline

which he chiefly employs are relations among certain quanti-
ties called pressures; and it was such a relation that I adopted
instead of the law of vis viva. I supposed that, at the confines
of two media, the pressure on the separating surface, in a direc-
tion perpendicular to the plane of incidence, ought to be the
same, whether it be considered as resulting from the vibrations
in the first medium or in the second. This hypothesis I con-
ceived to be true in general, because I found it to be true for
ordinary media; but I could never assign any better reason for
it. Combining it, however, with the principle of equivalent
vibrations, I deduced several expressions for uniaxal crystals,
and among others a formula for the polarizing angles in diffe-
rent azimuths of the plane of reflexion. When this formula was
compared with the experiments of Sir David Brewster* on the
polarizing angles of Iceland spar, the accordance was so satis-
factory as to leave no doubt upon my mind that I had arrived
at the true formula for these angles; and though the truth of
the conclusion did not allow me to argue that the premises

hypothesis of Fresnel. M. Cauchy supposed too, in the above Paper, that the
ethereal density is the same in different media; but he has found cause to abandon
this hypothesis also. See his notes addressed to M. Libri, in the Comptes rendus
des Stances de V Académie des Sciences, Séance du 4 Avril, 1836, where he gives the
reasons for his present opinions, He says, ‘‘Ainsi Fresnel a eu raison de dire,
non-seulement que les vibrations des molécules éthérées sont généralement com-
prises dans les plans des ondes, mais encore que les plans de polarisation sont
perpendiculaires aux directions des vitesses ou des déplacements moléculaires.
J’arrive au reste & cette derniére conclusion d’une autre maniére, en établissant
les lois de la réflexion et de la réfraction a l’aide d’une nouvelle méthode qui
sera développée dans mon mémorie...... [cette méthode] ne m’oblige plus a
supposer, comme je l’avais fait dans un article du Bulletin des Sciences, que la
densité de l’éther est la méme dans tous les milieux. Mes nouvelles recherches
donnent lieu de croire que cette densité varie en général quand on passe d’un
milieu 4 un autre.’’ More lately, in his Nouveaux Exercices de Mathématiques,
7¢ Livraison, M. Cauchy states positively that his principles do not permit him
to adopt the hypothesis that the density of the ether is the same in all media. He
also gives the differential equations which, as he has found by his new method,
ought to subsist at the separating surface of two media, and from which he has
obtained the formule of Fresnel for ordinary reflexion. But these equations do
not include the laws of crystalline reflexion.
* Phil. Trans., 1819, p. 150.

-

Reflexion and Refraction. 93

were true, yet the presumption in their favour was very strong,
insomuch that, upon remarking, as I did soon after, that the
law of vis viva harmonized with my other hypotheses, I did not
think it worth while* to try what would be the consequence of
using this law, instead of the relation which I had put in its
place. In this state of my theory, I gave an account of it at
the meeting of the British Associationt in Dublin, in August,
1835 ; and the leading steps and results were atterwards pub-
lished j in a letter to Sir David Brewster.

Now we are to observe, that when common light is polarized -
by reflexion at the surface of a doubly-refracting crystal, the
polarization does not, in general, coincide with the plane of
reflexion, as in the case of ordinary media, but is inclined
to it at a certain angle, which may be called the deviation; and
it was by equating two values of the deviation that I obtained
the formula above mentioned for the polarizing angle. This
formula, as we have seen, was correct; but it happened, singu-
larly enough, that the expressions for the deviation, which were

* T had, besides, an objection to the law of vis viva, on the ground that it would
give an equation of the second degree; and I wished to have all my equations
linear, lest, in the seemingly complicated question of crystalline reflexion, they
should give two answers when the nature of the question required but one. This
has actually happened, since the present Paper was read, in applying my hypothe-
ses to the case of internal reflexion at the second surface of a uniaxal crystal.
Supposing an ordinary ray to emerge after double reflexion, and putting @ for the
angle which the emergent transversal makes with the plane of incidence, I found,
for determining @, an equation of the form

A+ Btané+ Ctan’@ = 0,

wherein 4 is very small, but does not vanish; so that the equation gives two roots,
one very small, the other about the proper value. It is clear, therefore, that there
is a want of adjustment somewhere: but I am now inclined to think that the fault
is not in the principle of vis viva. Possibly our laws of the propagation of light in
doubly refracting media are not quite accurate. Whatever supplementary law
shall be found to remedy this untoward result will probably, at the same time,
account for the extraordinary phenomena observed by Brewster, in reflexion at
the jirst surface when the crystal is in contact with a medium of nearly equal
refractive power.

+ London and Edinburgh Philosophical Magazine, Vol. v1. p. 295.

Ibid., Vol. vit. p. 103; February, 1836. (Supra, p. 75.)

94 On the Laws of Crystalline

used in obtaining the formula, were erroneous. It is to M. Seebeck
that I am obliged for pointing out this curious circumstance.
In Poggendorff’s Annals,* he gave an abstract of my letter to
Sir David Brewster, and compared my results with his own
numerous and accurate experiments, both on the polarizing
angles of Iceland spar and on the angles of deviation. He
found that my formula represented the former class of ex-
periments as well as could be wished; but the theoretical —
values of the deviations did not at all agree with his experi-
mental measures. These measures of the deviation he pub-
lished on this occasion; and, with their assistance, I traced
the error to its source, which was the relation among the press-
ures. The principle of vis vica was therefore introduced, instead °
of that relation, and the theory became much simpler by the
change. I now obtained, for the deviation, a new expression,
which agreed with the experiments of M. Seebeck; but the
formula for the polarizing angle came out the very same as
before. This correction was made on the 6th of December,
and was published in the Philosophical Magazinet on the first
of the present month.

In the interval I have arrived at very elegant geometrical
laws, which can be easily remembered, and which embrace the
whole theory of crystalline reflexion. In enunciating these, it
will be convenient to draw our transversals always through the
same origin O, which we shall suppose to be the point of inci-
dence, as this point is common to all the rays, whether incident,
reflected, or refracted; and we may imagine wave planes to be
drawn through the origin, parallel to the plane of each wave,
so that every transversal will lie in its own wave plane. The
incident and reflected wave planes will be perpendicular to the
incident and reflected rays, but the two refracted wave planes
will in general be oblique to their respective rays. In the latter
case, a right line drawn through the origin perpendicular to

* Annalen der Physik und Chemie, Vol. xxxvut. p..276. _
t+ London and Edinburgh Philosophical Magazine, Vol. x. p. 43. (Supra, p. 84.)

Reflexion and Refraction. 69.

the wave plane is called the wave normal. It is scarcely
necessary to remark, that all the four wave planes intersect
the surface of the crystal in the same right line which is per-
pendicular to the plane of incidence; and_that the angles of
refraction are the angles which the refracted wave normals
make with a perpendicular to that surface. The index of
_ refraction is the ratio of the sine of the angle of incidence to
the sine of the angle of refraction, just as in ordinary media;
but here it is a variable ratio, and has different values for the
same angle of incidence. I have elsewhere* shown how to find
the refracted rays and waves when the incident ray is given.

As we suppose the ethereal molecules to vibrate parallel to
the transversals, we may take the lengths of the transversals
proportional to the magnitudes or amplitudes of the vibrations ;
these lengths being always measured from the common origin
O. Then, in virtue of our fourth hypothesis, the transversals
will be compounded and resolved exactly by the same rules as
if they were forces acting at the point O.

We must now conceive a wave surface of the crystal, with
its centre at O, the point of incidence. As the veloci-
ties of rays which traverse the crystal in directions

parallel to the radii of its wave surface are repre- 9
sented by those radii, so let a concentric sphere be =
described with a radius OS, which shall represent,

er

on the same scale, the constant velocity of light in
the medium external to the crystal. At any point
T on the wave surface apply a tangent plane, .on
which let fall, from O, a perpendicular OG, meeting 4

the plane in G. On this perpendicular take thelength — Fig. 17.
OP from towards G, so that OP shall be a third proportional
to OG and the constant line OS. Then, while the point 7’
describes the wave surface, the point P will describe another
surface reciprocal? to the wave surface. This other surface may

* Trish Academy Transactions, Vol. xvi. p. 252.
t For the general theory of reciprocal surfaces, see Irish Academy Transactions,
Vol. xvit. p. 241.

96 On the Laws of Crystalline

very properly be called the index surface,* because its radius
OP is the refractive index of the ray whose velocity is OT, or
rather of the wave 7G, which belongs to that ray; for if we
conceive an incident wave, touching the sphere, to be refracted
into the wave 7G, touching the wave surface in 7, the sine
of the angle of incidence will be to the sine of the angle of
refraction as OS to OG, or as OP to OS; so that, taking the
constant OS for unity, the index of refraction will be repre-
sented by OP. The wave surface and the index surface will
thus be reciprocal to each other, every point 7 on the one
having a point P reciprocally corresponding to it on the other.

It is remarkable that the transversal of the ray OT is per-
pendicular to the plane OPT; for in the theory of Fresnel, as
I formerly proved,t the direction of the vibrations is the right
line 7G; and as I suppose the transversal to be perpendicular
to the vibrations of that theory, and to be, at the same time, in
the wave plane, which is perpendicular to OP, it follows that
the transversal must be perpendicular to both the right lines
TG and OP, and therefore perpendicular to their plane OPT.
Therefore conceiving the transversal to be drawn through O
at right angles to the plane OPT, the plane of polarization
of the ray OZ must needs pass through it. But there is
nothing else to fix the position of this last plane. We may
make it pass through the ray itself O7, as an ordinary media,
or we may draw it through the wave normal OP with Fresnel.
Or, instead of drawing it through either of these two sides of
the triangle OPT, we may make it parallel to the third side
PT. The last is what I should prefer, because the plane so
determined possesses important properties. I shall call it, how-
ever, the polar plane, because the name, plane of polarization, is
along one; and the signification of the latter may, if any one

* This is the surface which I formerly called (Zrans., p. 252) the surface of
refraction; a name not sufficiently descriptive. Sir W. Hamilton has called it
the surface of wave slowness, and sometimes the surface of components. But the
name index surface seems to recommend itself, as both short and expressive.

+ Ibid. Vol. xvi. p. 76. (Supra, p. 12.)

Reflexion and Refraction. 97

chooses, be kept distinct, though in an ordinary medium both
terms must mean the same thing. ‘The polar plane then of the
ray OT is a plane passing through its transversal and parallel
to the right line PT’; so that if OX be drawn parallel to PT,
the polar plane will pass through OX. In general, to find the
transversals and the polar plane of any ray, we take the point
where the ray meets its own nappe of the wave surface, and join
it with the corresponding point on the index surface, drawing a
plane through the origin and the joining line. Then a right
line perpendicular to this plane at the origin will be the trans-
versal, and a plane drawn through the transversal parallel to
_ the joining line will be the polar plane.

Now let a polarized ray be incident at O upon the crystal.
It will in general be divided into two rays. But each of these
rays in turn may be made to disappear by polarizing the inci-
dent ray in a certain plane. Let us suppose then that there is
only one refracted ray OT. In what direction must the incident
ray be polarised, or, in other words, what must be the position
of its transversal, in order that this may be the case? and what
will be the corresponding transversal of the reflected ray? The
answer is simple—both transversals will lie in the polar plane of
the refracted ray. Let us pursue this remark a little.

The refracted ray O7' being given, we can find its polar
plane, and thence the intersections of this plane with the inci-
dent and reflected wave planes. ‘These intersections will be the
positions..of. the incident and reflected transversals when OT is
the sole refracted ray. The refracted transversal lies: also in
the polar plane; and this transversal is, by our fourth hypo-
thesis, the diagonal of a parallelogram, whose sides are the
other two transversals, which determines the relative lengths
of the three transversals, or the relative amplitudes of the
vibrations. The intensities of the reflected and incident rays
are, of course, proportional to the squares of their transversals.
When the ray OT dissappears, we must take the polar plane of
the other ray, and proceed as before.

Thus there are, in the incident wave een two transversal

H

98 On the Laws of Crystalline

directions which give only a single refracted ray. These, as well
as the corresponding ones in the reflected wave plane, may be
called uniradial transversals. They are the intersections of the
two refracted polar planes with the incident and reflected wave
planes.

When the incident transversal does not coincide with either
of the uniradial directions, it is to be resolved parallel to them,
and then each component transversal will supply a refracted
ray, according to the foregoing rules. The reflected transver-
sals, arising from the component incident ones, are to be found
separately by the same rules, and then to be compounded.

In ordinary reflexion, if the incident transversal be in the
plane of incidence, or perpendicular to it, the reflected trans-
versal will be so likewise. But this does not hold in crystalline
reflexion. The general method just given will, however, enable
us to determine the positions and magnitudes of the reflected
transversals in these two remarkable cases; and then, if we
choose, we can reduce any other case to these two, by resolving
the incident transversal in directions parallel and perpendicular
to the plane of incidence.

If we conceive a pair of incident transversals, at right angles
to each other, to revolve about the origin, it is evident that there
will be a position in which the reflected transversals correspond-
ing to them will also be at right angles to each other. There
is no difficulty in finding this position, and there will be an
advantage in using it when common unpolarized light is in-
cident on the crystal. For, the incident transversals being
rectangular, we may suppose the light to be equally divided
between them, and then the intensities of the corresponding
reflected portions can be found by the preceding rules. As
the reflected transversals are also rectangular, the sum of these
intensities will be the whole intensity of the reflected light, and
their difference will be the intensity of the polarized part of
it. This part will be polarized in a plane passing through the .
greater of the two reflected transversals.

Common light will be completely polarized by reflexion when

Reflexion and Refraction. 99

the two uniradial directions in the reflected wave plane coincide
with each other; that is, when this plane and the two refracted
polar planes have a common intersection. For then, if the inei-
dent light be polarized, it is manifest that the reflected transver-
sal will lie in that intersection, whatever be the position of the
incident transversal; and therefore if common light be incident,
with its transversals in every possible direction, the reflected
transversals will have but one direction. Thus the reflected light
will be completely polarized in a plane passing through the above
intersection.

Hence, as the reflected ray is perpendicular to its wave plane,
it follows that, at the polarizing angle of a crystal, the reflected ray
ts perpendicular to the intersection of the polar planes of the two re-
Sracted rays. The reflected transversal, as we have seen, is this
very intersection. This transversal is inclined, in general, to the
plane of incidence, and we have had occassion to speak of its in-
clination under the name of the deviation. If we now suppose the
double refraction to diminish until it disappears, the intersection
of the polar planes will at last coincide* with the refracted ray.
There will then be no deviation, and the reflected and refracted
rays will be at right angles to each other, agreeably to the law
of Brewster, which prevails at the polarizing angle of an ordi-
nary medium. .

There is a case in which the construction that we have given
for determining the polar plane of a ray becomes useless. It is
when the ray O7'is a normal to the wave surface; for then OP
coincides with OZ, and we cannot fix the transversal by our con-
struction. But it is precisely in such a case that the polar plane
is most easily ascertained, for it is then nothing more than the
plane of polarization of the common theory. For example, if we
take the ordinary ray of a uniaxal crystal, its polar plane will
pass through the ray itself. and the axis of the crystal. Of course
in an ordinary medium the polar plane and _ plane of polari-
zation are synonymous.

* For the polar planes will become two planes of polarization at right angles to
each other.
H2

100 On the Laws of Crystalline

It may not be amiss to apply our general rules to the case
of ordinary reflexion and refraction. Suppose then a polarized
ray to fall on the surface of an ordinary medium. Draw a plane
through the incident transversal and the refracted ray; this will
be the plane of polarization of the refracted ray, and it will
intersect the reflected wave plane in the reflected transversal.
The refracted transversal will be the diagonal of a parallelo-
gram, whose sides are the other two transversals; hence we
have the relative lengths of the transversals, and thus every-
thing is determined.*

_ * This construction was mentioned at the meeting of the British Association in
Dublin.—<Sce the Reports of the Association, or London and Edinburgh Phil. Mag.
' vol. vii. p. 295. The following is an extract from the Paper which I read at that
meeting :—

‘‘ The formule given by Fresnel for the same purpose will be found to agree
exactly with this rule, in determining the positions of the planes of polarization ;
and his expression for the amplitude of the reflected vibration is also in accordance
with our construction. But the coincidence does not hold with regard to the am-
plitude of the refracted vibration, though the vis viva of the refracted ray is the
same in both theories.

“ Now it is very remarkable that if we alter the hypotheses of Fresnel where
they are at variance with the preceding principles, we shall, from his own equa-
tions of condition, deduce formule agreeing in every respect, even as to the ampli-
tude of the refracted wave, with the construction which we have accounted for in
a different way (i. e. by using the relation among the pressures instead of the law
of vis viva). The requisite alterations are two in number. First, the vibrations
are to be supposed parallel to the plane of polarization, and not perpendicular to it,
as Fresnel conceived; and secondly, the density of the ether is to be considered the
same in both media, from which it follows, that the corresponding ethereal masses,
imagined by Fresnel, are to each other as the sine of twice the angle of incidence
to the sine of twice the angle of refraction. Substituting in Fresnel’s equations of
condition this value of the ratio of the masses, we obtain the formule which I am
inclined to regard as correct.’

The equations spoken of in this extract are those which arise from the prin-
ciple of vis viva, and from the equivalence of vibrations parallel to the separating
surface of the two media. - But it is worth while to observe, that when the vibra-
tions are all in the same direction, that is, when the light is polarized perpendicular
to the plane of incidence, the very same formule will come out from Young’s re-
markable analogy of the two elastic balls, one of which impinges directly on the
-other, supposed previously at rest, the masses of the balls being to each other in the
ratio of the ethereal masses mentioned above, And, perhaps, this consideration
affords the simplest possible explanation of Brewster’s law relative to the pola-

Reflexion and Refraction. 101

The reason of this construction will be evident, if we con-
sider that, in an ordinary medium, the polar plane is the same
as the plane of polarization; and that, when there is only one
refracted ray, the three transversals lie in the polar plane of
that ray, according to the general remark with which we set
out. We now proceed to show that the theorem asserted in
this remark is a consequence of our hypotheses, and we shall
afterwards deduce a few results which may be readily compared
with experiments.

Let us suppose then that the direction of the incident trans-

rizing angle ; for, as there is no reflected motion when the balls are equal, the whole
velocity of impact being communicated to the ball that was at first quiescent, so
there is no reflected vibration when the ethereal masses are equal; that is, when
the sine of twice the angle of incidence is equal to the sine of twice the angle of
refraction, or when the angles of incidence and refraction are together equal to a
right angle. The whole of the incident vibration then passes into the refracted
ray. In general, if ¢,, 1, denote the angles of incidence and refraction, the masses
of the imaginary balls will be as sin 2:,, to sin 2:,; and, if the velocity of the ori-
ginal impact be taken for unity, the common theory of the collision of elastic bodies
will give

sin 21, - sin 22, tan (1, — ¢,)

sin 21, +sin 21, * tan (1, +4)’

(1)

for the velocity retained by the impinging ball after the impact; and
2sin 2c, ag sin 21,
sin 21, + sin 21; sin («, +4,) Cos (¢, —2,)’

(11.)

for the velocity communicated to the other ball. . These expressions (1.) and (11.),
are the same as the values of r, and r,, which we should deduce from equations (1)
and (2), on the next page, by supposing 7, to be unity, and the angles 0), 2, 63 to
be right angles. The general construction given in the text will lead to the same
results, if we find from it the limiting ratios of the transversals, on the supposition
that their directions approach each other indefinitely, and ultimately coincide in a
right line perpendicular to the plane of incidence.

When the transyersals are all in the plane of incidence, or when the light is
polarized in that plane, the incident, the reflected, and the refracted transversals
are to each other as sin («,+1,), sin (t,—1,), and sin 2:, respectively ; because each
transyersal is proportional to the sine of the angle between the other two; and, in
the present case, the angle between any two transversals is equal to the angle
between the corresponding rays. Hence, taking the incident transversal for unity,
the reflected transversal is

sin (¢’—1,) |
sin (¢, +1)’

(111,)

iof * On the Laws of Crystalline

versal is such that there is only one refracted ray. It is evi-
dent that, in this case, the three transversals must lie in the
same plane, since, by the fourth hypothesis, the refracted vibra-
tion is the resultant of the other two vibrations; and, therefore,
we have only to prove that the plane of the transversals is the
same as the polar plane of the refracted ray. Let 71, 72, 73 be
the respective lengths of the incident, refracted, and reflected
transversals; let 6,, ., 0; be the angles which they make with
the plane of incidence, the angle 0, being known from the
theory of Fresnel; put u, &, « for the angles made by the
respective wave planes with the surface of the crystal, and
M1, 1%, ms for the relative quantities of ether set in motion»
by each wave. ‘Then our hypotheses will give us the four
following equations :—
My, Ty” = Mz 72" + M373" (1)

a sin 6, + T3 sin 6, =T2 sin 0, (2)
71 00S 0; cos 4 + 73 COS 8; cos t3= 72008 8,008, _ (8)
7,008 9; sin 4 +73 Cos 0; sin t3 = 72 C08 8, SiN t2. (4)

The first equation is manifestly the translation of the law of ©

and the refracted transversal is
sin 2:, anid

sin («, +4) P
It has been already observed that our theory differs from that of Fresnel with
regard to the magnitude of the refracted transversals. The expressions (11.) and (rv.)
sins

must, in fact, be multiplied each by = in order to produce the corresponding ex-

sin t,
pressions which result from Fresnel’s hypotheses. But the two theories also differ
as to the relative directions of the incident and reflected transyersals. For, suppos-
ing the light to fall upon the denser medium, or 1, to be greater than 1,, our con-
struction indicates that these transversals, when the angle of incidence is small,
point in the same direction; whereas Fresnel concludes the contrary to be the case.
However, the disagreement in this respect ceases as we approach the limiting inci-
dence of 90°; for then, according to both theories, the incident and reflected trans-
versals point in opposite directions. This last conclusion is conformable to the

, inference which Professor Lloyd has drawn from his experiments on the interfer-
ence of direct light with light reflected at a very oblique incidence.—See Irish
Acad. Transactions, vou. xvi. p. 176.

Reflexion and Refraction. 103

the preservation of vis viva ; the other three are obtained from
the principle of equivalent vibrations, by resolving the vibra-
tions, or transversals, in three rectangular directions. In the
second equation, the transversals are resolved perpendicular to
the plane of incidence; in the fourth, perpendicular to the sur-
face of the crystal; and in the third equation they are resolved
parallel to the intersection of these two planes. When the angles
0,, @., 8; begin, the transversals are in the plane of incidence
in such a relative position, that, if they were turned round
together in that plane through a right angle, they would point
each in the direction of its own wave’s progress. These angles
increase on the same side of the plane of incidence, and range
through the whole circumference. The angles 1, &, ts are those
of incidence, refraction, and reflexion; but, for the sake of sym-
metry, they are taken to be the angles which the wave normals,
drawn from the origin in the direction of each wave’s motion,
make with the perpendicular to the surface, this perpendicular
being directed towards the interior of the crystal. Thus it
happens that ., is the supplement of 4. Attending to this
circumstance, equations (3) and (4) give us

Cost
7, COS 0, — T3 COS 0; =T2 COS 6, =, )
COS U1
(5)
sin t
7, 008 0, +7;0080;=7,00 &%—=—}
sin
and, by adding and subtracting these, we find
cos #, sin (4 + &) 4
TT, =Te2 >
cos@, sin 2z,
(6)

cos 6, sin (ts — 4)
cos@, sin2i, ”’

T3= Te
which values if we substitute in equations (1) and (2), observing
that m; = m,, as is evident, we shall get

sin’? (4 + tr) sin? (a — ts) Me sin? Qu,
cos? 0, cos? 0, m, cos? @,”

(7)

104 On the Laws of Crystalline

sin (t:+ ¢) tan 0, — sin (;—«) tan 0; = sin 2x, tan 0). (8
Subtracting from (7) the identity .

sin? (¢; + te) — sin? (4 — 2) = sin 2y sin 2ra,
there remains
sin’ (4; + ¢) tan? 0, — sin’ (4 — &) tan? 0;
am
= a & — ee cos* 0 : (9)

and this, by making

Mz sin 2+ 2hsin? 0,
Ney sin 21

(10
becomes
sin? (¢; +4) tan’ @,— sin? (4. —) tan* 6,= sin 2. (sin 2+ 2/) tan’, (11)
which is divisible by equation (8), the quotient being

sin (4 + &) tan 6, + sin (4 - &) tan 0; = (sin r+ 2h) tan @,. (123

Then, by adding and subtracting equations (8) and (12) we —

obtain
Atan 0,

sin (1 of to)

htan 0, i (18)

tan 9,=— cos (¢: +t) tan 0,.+ ————..
sin (t = 2)

tan 0, = cos (t:— tz) tan 0,+

These equations give the positions of the incident and reflected
transversals when / is known.

Now let the. directions in which the transversals have been
resolved in equations (2), (3), (4), be taken for the axes of 2,2, y
respectively ; so that, the origin being at O, the plane of ay may
be the plane of incidence, and the axis of # may lie in the sur-
face of the crystal. And, the reflected ray being conceived to lie
within the angle made by the positive directions of # and y, let
the initial condition that we have assumed for the angles 0,, @:,
@; be satisfied by supposing that, when these angles begin, the

Reflexion and Refraction. 105

transversals 7,, 7, lie between the negative directions of 2 and
y, and the transversal 7; between the directions of + # and —y.
Then if @,, 0:,; 8; be reckoned towards the positive axis of z, so
that each angle may be 90° when the corresponding transversal
points in the direction of s positive, the equations of the trans-
versal 7, will be

6-3 Bee ¥ 4
tan 0, COS 4 sin ty ue)

and those of r; will be
3 wv % Y (15)

tan@, cos, sina’
Let
s+ Ax+By=0 (16)

be the equation of a plane passing through the directions of 7,
tT, and7;. To determine A and B, let the variables be eliminated
from this equation by means of (14) and (15) successively, and
we shall get the two equations of condition,

tan 0,-— Acosu—- Bsin «= 0, (17)
tan 0,+ A cos, — Bsins,=0;
which, by addition and subtraction, give

tan. 6, + tan 0,
2 sin ty

B=

tan 6, — tan 0; | 78)

2008 4

A=

substituting, in these values, the expressions (13) for tan @,,
tan @;, we have

B= tan 0,(sinn + ana) =
sins, — sin*t, |
~ Psi Gm)
A = tan 0, (cos. pd a ‘ )
81N"t, — 81N“te
whence, by making
tan k = . (20)

sin*i, = sin*:,”

106 On the Laws of Crystalline

we find
B tanw + tank
A 1-tany tank | ee Gn
But if s = 0 in (16), we have
Av + By =0, (22)

for the equation of the right line in which the plane of the trans-
versals intersects the plane of incidence. This right line, lying,
like the refracted wave normal, between the directions of + «
and — y, makes with the direction of - y an angle v which ob-

viously has = for its tangent; and therefore, by (21),

A
V=EHt+K3 (28)

which shows that the intersection of the two planes is inclined
to the refracted wave normal at an angle equal to x. (as

We must now find the value of 4, which depends on the rela-
tive ethereal,masses put in motion by the incident and refracted
waves. Coriceiving the incident and refracted rays to be cylin-
drical pencils, having of course a common section in the plane
of xz, which is the surface of the crystal, let each pencil be cut
by a pair of planes parallel to its wave plane, and distant a
wave’s length from each other; then the cylindrical volumes so
cut out will represent the corresponding masses, since, by our
second hypothesis, the densities are equal. These volumes are
to each other in the compound ratio of their altitudes, whieh are
the wave lengths, and of the areas of their bases. The altitudes
are evidently as sinu to siny. The first base is a perpendicular
section of the incident pencil ; the second base an oblique section
of the refracted one, the obliquity being equal to the angle « at
which the wave normal is inclined to the ray. The perpendi-
cular sections are to each other as the cosines of the angles which
they make with the common section of the cylinders, or as cos 4
to COS U2); putting ¢) forthe angle which the refracted ray makes
with the negative direction of y. The second base is greafer
than the perpendicular section of the refracted pencil in the pro-

Reflexion and Refraction. 107

portion of unity to cose. Therefore, compounding all these
ratios, we find
Me SIN tz COS t(2)
m, Silt COS COSE

(24)

The same result may be otherwise obtained. by observing that,
in a system of waves, the corresponding masses are proportional
to the ordinates , y of the points where the rays meet their wave
surfaces. By a system of waves, I mean an incident wave with
all that are derived from it by reflexion or refraction at the same
surface of the crystal, or at parallel surfaces. If, at the point
where the incident ray intersects its spherical wave surface, we
apply a tangent plane intersecting the plane of zz in a right
line parallel to z, through which right line other planes are
drawn touching the wave surface of the crystal in four points,
these tangent planes will be the waves derived from the inci-
dent wave which touches the sphere ; and the points of contact,
including that on the sphere, will be the points where the rays
meet the wave surfaces. Then the corresponding masses will
be represented by prisms having a common rectangular base in
the plane zz, one side of this rectangle being the distance, on
the axis of x, between the origin and the common intersec-
tion of the tangent planes; and the triangular face of- each
prism having the same distance for one side, and a point of con-
tact for the opposite angle. These prisms, as they have a com-
mon base, will be proportional to their altitudes, which are the
ordinates y of the points of contact, ‘The expression (24) may
be easily deduced from this relation.

Let OT, OP, and the negative direction of y meet the sur-
face of the wave sphere (described: with the Li;
radius OS) in the points 7, P,, Y,; and let NA,
the right line, in which the plane of the trans-
versals intersects the plane of incidence, meet
the sphere in Z, Then the points Y,, P,, LZ, Fig. 18.
being all in the plane of incidence, will be on the same great
circle Y,P,L,; and drawing the great circles 7’P, Y,7, we

108 On the Laws of Crystalline

shall have YP, =u, Y,T, = T.P,=«, YL, =v=un+k, by
(23) ; whence PL, = x.

As the transversal 7, is perpendicular to the plane O7P, or to
the plane of the great circle 7’\P,, the cosine of the spherical
angle 7'P_Y, is the sine of 0,; and therefore, from the triangle
T, PY, we have

rr)

COS t(2) = COS tz COSE + SIN & Sine sin O,, (25)
which being substituted in (24), gives

M, sin 2u+2sin*%, sin 6, tans

m, sin 24 . Ge)
and comparing this result with (10), we find
sin*,, tan e_
“ans? (27)
whence, and from (20), it follows that
m2
Mee a sin’, tan € (28)

(sin*, — sin*t,) sin 6,”

Draw the great circle LK, at right angles to 7/P,, and meet-

ing it in ,; then the plane of Z,K, will be the plane of the ~

transversals, since the latter plane passes through Z,, and is per-
pendicular to 7,P,. But the tangent of PK, is equal to the
tangent of PL, multiplied by the cosine of the angle P, or by
the sine of 0,; therefore, denoting P,K, by a, and recollecting
that PL, = «, we find, by (28),
tan & ox is a, (29)
tane sin*,—sin*.,
Now we have seen that the ratio of OP to OS, or OS to OG
(Fig. 17), is the index of refraction ; so that sin’, is to sin’, as
OP to OG. Therefore, by (29),

tena OG OG.
tane OP-OG GP’

(30)

Reflexion and Refraction. 109

but OG is to GP as the tangent of the angle GPT is to the tan-
gent of the angle GOT; and since < is the angle GOT, it follows
that s is equal to the angle GPT or KOP. - Consequently, OK
will meet the surface of the sphere in the point XK, Thus we
have proved our assertion, that, when there is only one refracted
ray, the plane of the transversals is the polar plane of that ray.
The sign of the quantity / is always the same as that of the
cosine of the spherical angle 7.P,Y;. But to remove all ambi-
guity respecting signs, we must make a few additional conven-
tions. Supposing, as we have hitherto done, that the refracted
light moves from O to 7, and conceiving a right line to be
drawn from the origin parallel to G7, and directed from G
towards 7’, let the angle $., which this right line makes with the
plane of incidence, be reckoned, like 6,, 02, from an initial posi-
tion comprised between the negative directions of 2 and y; and
let $2, like the angles 6, 9., @;, increase on the side of s positive,
and range from 0° to 360°. Then $, will always be equal either
to the angle P, of the spherical triangle 7'PY,, or to the re-
entrant angle, which is the difference between P, and 360°. In
either case, the cosine of $, will be the same, both in magnitude
and sign, as the cosine of the angle 7'P,Y,. Consequently, if,
instead of (25), we use the direct trigonometrical formula

COS (2) = COS t2 COS € + SID tz SiN € COS Sp, (31)
we shall find
sin’, tane cos S.
Rie eee Rs (32)
sin’@, —

showing that the sign / is always the same as the sign of cos 3;.
Now as 0, differs from $, by aright angle, we will suppose

0. = S + 90°, (38)

and then we shall have sin @, = cos &, algebraically as well as
numerically. Thus we see that, by adopting these conventions,
the value of / in (27) will have the proper sign. Therefore,
substituting this value of / in formule (13), we obtain

1IO On the Laws of Crystalline

sin*, tan « )
cos @, sin (t; + ta)’

tan 8, = cos (t; — w) tan @ +
(34)
sin’, tane
cos #2 sin (t:~ ta)

tan 0; = — cos (u +t) tan 0, +

These formule give the uniradial directions, or the positions
of the incident and reflected transversals, when the sole refracted
ray is that with which we have been occupied. The like direc-
tions, when the other ray exists alone, will be given by the
formulze

sin*:’, tan & )
cos 0’, sin (11 Pty):

tan 0’, = cos (1 —0’2) tan 0’, +

(35)
rat ‘ : sin*’,tane’
tan = — cos («1 + t's) tan 0, + —— 0, 8in (4-0) J

where all the quantities, except 4, which remains the same, are
marked with accents, to show that they belong to the second
refracted ray.

The uniradial directions having been found by these equa-
tions, the relative magnitudes of the uniradial transversals are
determined by equations (6). When the incident transversal
is not uniradial, it is evident, as we said before, that it may be
resolved* in the two uniradial directions; that each component

* That, if an incident transversal be resolved in any two directions, the reflected
and refracted transversals arising from it will be the resultants of those which would
arise from each of its components separately, is a principle which appears very
evident, insomuch that we can hardly suppose it to be untrue, without doing
violence to our physical conceptions. Nevertheless, it is necessary to prove that
this principle is not contrary to the law of vis viva ; for though the vis viva may be
preserved by each set of components (as it is when these are uniradial), yet we
cannot therefore conclude that it will be preserved by their resultants. Here then
is a test of the consistency of our theory; for we are bound to show that the law of
vis viva is not infringed by the adoption of the principle in question. Now it is
easy to see that, whatever be the two directions in which the incident transversal
is resolved, the final results will always be the same; because, taking the compo-
nent in each of these directions separately, the reflected and refracted transversals
belonging to it must be obtained, in the first place, by the help of a resolution per-

Refiexton and Refraction. III

transversal, as if the other component did not exist, will furnish
a refracted ray and a partial reflected transversal uniradial in its
direction; and that the total (or actual) reflected transversal will
be the resultant of the two partial ones.

When 0; = @;, the partial reflected transversals will coin-
cide, and their resultant will have a fixed direction, independent
of the direction of the incident transversal. The angle of inci-
dence at which this takes place is the polarizing angle, and the
common value of @; and 0’; is the deviation. If, at the polarizing
angle, the partial reflected transversals be equal in magnitude,
and opposite in direction, their resultant will vanish, and the
reflected ray will disappear. ‘This will happen when the inci-
dent transversal is in the plane of the two refracted transversals,
and therefore in the intersection of this plane with the incident
wave plane; for, when there is no reflected ray, the incident
transversal alone must be equivalent to the two refracted trans-
versals.

Since the reflected transversal can be made to vanish at the
polarizing angle, this angle might be found directly by putting
the vis viva of the incident ray equal to the sum of the vires vive
of the two refracted rays, and by making the incident trans-
versal the resultant of the two refracted transversals. Resolv-
ing the transversals parallel to the axes of co-ordinates, these
conditions would give four equations, from which we could

formed in the uniradial directions. We need not, therefore, consider any case but
that in which the resolution is uniradial throughout.

The incident transversal being denoted by 7}, let 73 be the reflected transversal
determined by the rules given in the text; and let the uniradial components of the
former be 71, 7'1, while those of the latter are 73, 7’3. Then will

T;? r—] T? ob a’; tL Qr17') cos (01 —= 6'1),
Ts? = 73" + 1's? + 2737's cos (03 — 6’3) ;
where the signification of 61, 6/1, 03, 6’3 is the same as in the text. The vis viva of

one refracted ray is m, (71? — r3?), and that of the other is m (7/1 — 7’s*) ;_ there-
fore the vis viva of both refracted rays is

m, (71? + 1/12 — 13? — 7’3?),

112 On the Laws of Crystalline

eliminate the two ratios of the three transversals, together with
the angle at which the incident transversal is inclined to the
plane of incidence. In the equation produced by this elimina-
tion, the angle of incidence would be the polarizing angle, and
the other quantities would be known functions of that angle ; ;
whence the angle itself would be known.

a quantity which ought to be equal to
m, (Ty? -- 3?) ;
and consequently the equation
717’ COS (81 — 6’1) = T37’3 cos (63 — 6’3) (v.)

ought to be true, This equation, by help of the expressions (6) for 71, 73, and the
like expressions for 71, 7’s, becomes

in (1, + ¢,) sin (, + /,) (1 +tan @, tan @’,)
=sin (1, —1,) sin (4, —/’,) (1+ tan@, tan 6’,) ; (vr.)

which again, by substituting the values (13) and the other similar values, is
changed into

in (1, + /,) {cos (1, — “,) + cotan 6, cotan #,} +h+ h’=0, (vi1.)

where /’ denotes for one refracted ray what / denotes for the other, the value of
being given by formula (27), and that of ’ by the same formula with accented

letters. The angle of incidence, wé may observe, has disappeared from the

equation.

If, therefore, the laws of reflexion, which we have endeavoured to establish, are
consistent with cach other, this last equation must be satisfied by means of the rela-
tions which the laws of propagation afford; or rather, the equation must express a
property of the wave surface of the crystal, however strange it may be thought that
such a property should be derived from the laws of reflexion—laws which would
seem, at first sight, to have no connexion at all with the form of the wave surface.
Now I have found that the equation (vir.) really does express a rigorous pro-
perty of the biaxal wave surface of Fresnel; a very curious fact, which not only
shows that the laws of reflexion and the laws of propagation are perfectly adapted
to each other, but also indicates that both sets of laws have a common source in
other and more intimate laws not yet discovered. Indeed the laws of reflexion are
not independent even among themselves; for the expressions (111.) and (ry.) in the
note on ordinary reflexion (page 101) have been deduced solely from the principle of
equivalent vibrations, and yet they satisfy the law of vis viva. Perhaps the next step
in physical optics will lead us to those higher and more elementary principles by
which the laws of reflexion and the laws of propagation are linked together as parts
of the same system.

oe oe? eed

Reflexion and Refraction. 113

It deserves to be remarked, that, at any angle of incidence,
if the incident and reflected wave planes be intersected by a
plane drawn through the two refracted transversals, the inter-
sections will be corresponding transversal directions; that is to
say, if the incident transversal coincide with one intersection,
the reflected transversal will coincide with the other. For it is
evident, from our fourth hypothesis, that if three of the trans-
versals be in one plane, the fourth transversal must be in the
same plane.

We come now to apply our theory to the case of uniaxal
erystals; and, in doing so, we shall take the crystal to be of the
negative kind, like Iceland spar, so that the ordinary refraction
will be more powerful than the extraordinary.. On the sphere
described with the centre O and radius OS, let XY be a great
circle in the plane of incidence, the radii OX, OY being the po-
sitive directions of the co-ordinate axes of z and y. Suppose the
right lines ‘O and 07’, intersecting the sphere in ¢ and 7’, to be
the incident and reflected rays; let the ordinary refracted ray and
the extraordinary wave normal be produced backwards from O to
meet the sphere, at the side of the incident light, in the points 0
and e respectively ; let the right line OA, cutting the sphere in
A, be the direction of the axis of the crystal ; and draw the great
circles do, Ae, AY. The points ¢, e, 0, 7’ are all on the circle
XY. The point Z, where the extraordinary ray OF produced
backwards meets the sphere, will be on the circle Ae; and if, as
in the figure, the are Ae be less
than a quadrant, the point e will
lie between A and #. The polar
plane of the ordinary ray is ob-
viously the plane of the circle
Ao; but the polar plane of the
other ray must be found by. a Fig. 19.
construction. On the are AcH
take the portion ef, so that the point e may lie between the
points HZ and /, and so that the tangent of ef may be to the tan-
gent of He as the square of the sine of the are eY is to the dif-

I

-

114 On the Laws of Crystalline

ference between the squares of the sines of iY andeY. Through
J draw the great circle /# perpendicular to the circle AcH; and
it is manifest from (29) that the plane of /¢ is the polar plane of
the extraordinary ray. On each circumference Ao and ft, the
points which are distant 90° from # and 7’, the distances being
measured by ares of great circles, are the points where the unira-

dial transversals, prolonged from the centre, intersect the sphere. .

Let Ao and /¢ intersect each other in ¢, and let ¢’ be an are of a
great circle connecting the point ¢ with the point 7’. When the
connecting are ¢i’ is a quadrant, the two uniradial transversals,
belonging to the reflected ray, coincide with each other and with
the right line O¢; the angle of incidence is then the polarizing
angle; the plane of #7’ is the plane of polarization of the reflected
ray; and the angle #’ Y is the deviation.

To find the equations appropriate to uniaxal crystals, wemay ~

suppose formule (34) to belong to the ordinary, and formulze
(35) to the extraordinary ray. Then will « = 0, and & = the are
Ee. Putting 0 and 0’ for the spherical angles Aoi and Ae, we
shall easily see that 0, = 6 + 180°, and 0’, = 0 + 90°, if we con-
ceive the point A and the positive axis of s to be both on the
upper side of the plane XOY. And if w’ denote the are Ae,
while 5 and a respectively express the reciprocals of the prin-
cipal indices, ordinary and extraordinary, the law of Huyghens,
for the double refraction of uniaxal crystals, will give us

ab.
tan ¢= —— sin w’ COS w’, (36)
where
sin’ he °
ha yaaa: (a? — B*) sin? w’. (37) >
ty

Observing these relations, we have, from (34),

tan 0, = 008 (1; ~ ts) tan 8, (38)

tan 0; =— cos («, +) tan 8,

for the ordinary ray; and from (35) we get

Reflexion and Refraction. 115

Sin w COS w’ sin? 4
sin 0 sin (u + 2) 5

tan 9; =— 008 (4 - ¢’) cotan 6” — (a? — b°)
| (39)

Sin w’ COS w’ sin’,
sin. y sin (a = U2) 4

tan 6’; = cos (¢: +42) cotan @ — (a? — 0”)

for the extraordinary ray.
The four preceding equations determine the uniradial Sia
tions; and the following equation,
sin w cosw’sin’t
sin #’ sin (4-2)
obtained by putting tan 0,=tan 0’;, is that which determines the
polarizing angle.

In making use of this last equation to deduce the law of the
polarizing angles in various positions of the axis of the crystal,
we shall confine ourselves to the case in which the reflexion from
the crystal takes place in air, because the angle « — «’, will then
be considerable, and the quantities cos (t: + 2) and cos (4 + ¢2)
will be small, so that it will be easy to arrive at approximate
results. For we shall have, in the first place,

608 (1, + «) tan 9+c08(u +12) cotan 6 — (a? -0?)

=0, (40)

cos (1. + t's) = C08 (4. + 2) — (2 -w2), (41)

nearly, since 4 +¢ will not differ much from a right angle; and
because
sin 2=Osin ny, sin ’2=ssiN 4, (42)

we shall also have, rigorously,

sin? c’, — sin? t, = (s* — 6’) sin? 4 = (a? — 6”) sin? w’ sin’ 4, (43)
or
: sin? w’ sin? ty

OY a uh

fhe (44)
_ which may be written

sin? w sin? ¢
i. -&k= (a? af De) ;
81n Que

(45)

with sufficient accuracy. This value of ’,—« having been sub-
stituted in (41), the resulting expression for cos («,+¢2) must be
12

ee ——

116 On the Laws of Crystalline

substituted in equation (40), which will then become

cos (1; +4) (tan 6+ cotan 8) — (a* — 2°) sin? « sin (= cotan @
S1N Qt.
COS w
sin @ cos 5-) = 0, i

if, denoting the are Ao by w, we confound w’ with w, # with 0,
and write cos 2y. instead of sin (4-¢2). Multiplying all the
terms of (46) by sin @ cos 0, we find

sinwcos@ cos 5) (47)

cos (t; + &) = (a? — 0”) sin *c; sin w cos 8 . +
(1 + «2) = ( ) : sin 2t Cos 2u

From A draw the arc AR meeting the are iY at right angles
in the point R, and put RY=p, ARk=q. Then by means of the -

values
COS w = COS J COS (P—b), (48)
sin w cos @=cosq sin (p—w),
afforded by the right-angled triangle 4Ro, the equation (47)
will take the form

(a? - B) sin’,

* 2 . ‘3 . 7
COS (¢ + te) FO RE aah be (p-) sin (pte), (49)
or
cos («+ 2) = K cos’g (sin*p — sin’), (50)
where ;
ee HM), (51)

21-8)?

this value of K being found by assuming tan » = cotan u = 3,
which is accurate enough for the purpose.

Thus we have obtained 4. + , or the sum of the polarizing
angle and the angle of ordinary refraction. The former angle

itself may be inferred from formula (50) by help of the relation

sint,=6siny. In this way, if we use a, instead of « to dis-
tinguish the polarizing angle from other angles of incidence,

and if we put
K ap

‘= TTR 7 Md 8)

(62)

Reflexion and Refraction. 117

we shall find
@, = a — k cos’q(sin’p — sin’), (53)

in which a is the angle whose cotangent is equal to d; in other
words, @ is the polarizing angle of an ordinary medium whose
refractive index is equal to the ordinary index of the crystal.
This result accounts for a remarkable fact observed by Sir
David Brewster, who, in the year 1819, led the way in the ex-
perimental investigation of the laws of crystalline reflexion. He
found that the polarizing angle remains the same when the crystal
is turned round through 180°, though one of the angles of refrac-
tion is changed, and though the situation of the refracted rays,
with respect to the axis of the crystal, becomes quite different
from what it was. ~This circumstance, which surprised me when
I first met with it, is an immediate consequence of formula (53);
for the effect of a semi-revolution of the crystal is to change the
signs of p and g; but the nature of the formula is such that
these changes of sign do not alter the value of a. Neither is
that value altered by turning the crystal until the azimuth, as
the spherical angle A Yi is usually called, is changed into its
supplement ; for then the sign of p alone is affected.

Another remark, made by the same distinguished observer,

is also a consequence of formula (53). From his experiments
it appears that, on a given surface of the érystal, the polarizing
angle differs from a constant angle by a quantity proportional
to the square of the sine of the azimuth AYi. Now, calling
this azimuth a, and putting A for the acute angle at which the
axis of the crystal is inclined to its surface, so that A may be the
complement of the arc A Y, we have

sin g = cosX sina, tan p = cotan A cosa ; (54)

and by making these substitutions in formula (53), after having
changed sin 1, into cos w, that formula becomes

@, = @ — k (sin*a — sin’) + * sin’a cos*A sin’a, (55)

which agrees with the remark of Brewster.

118 On the Laws of Crystalline

The deviation 0; or 6’; is found from the second of equations
‘ tan g hee
(38), by putting TEES for tan 0, and by substituting for

cos (t; + &) the value (49) or (50) which it has at the polarizing
angle. The result is

0, = 0’, = - % sin 2q sin (p + 0), (56)

since the small are 0, may be taken for its tangent. This result

is easily transformed into
0,=6',=-Ksing cosg, (57)

where ¢ denotes the are A, or the angle which the incident ray
makes with the axis of the crystal; and this last expression is
equivalent to the following,

6; = 0, = — K cosX sina (sinA cosa + cosA sina cosa), (58)

which gives the deviation in terms of \ and a.

As an example of the application of our formule, we shall
make some computations relative to Iceland spar. According
to M. Rudberg, the ordinary index of that crystal, for a ray
situated in the brightest part of the spectrum, at the boundary
of the orange and yellow, is 1:66; and the least extraordinary
index for the same ray is 1:487. Dividing unity by each of
these numbers, we get a ="6725, b="6024; whence a = 58° 56’ ;
k ='1164 = 6° 40’; K='1587=9° 5’. Having thus determined
the constants, we can readily calculate the polarizing angle and
the deviation, for any given values of \ and a. ;

First, let us see how the polarizing angle varies on different
faces of the crystal.

1. When A = 90°, the face of the crystal is perpendicular to
its axis, and «, is independent ofa. In this case the formula
(55) gives

-m=a+k cos? w=60° 42,

which is the maximun value of the polarizing angle.

Reflexion and Refraction. _ 119

2. When \=0, the axis lies in the face of the erystal, and
formula (55) becomes

@,=a-ksin’a-cos’a,
showing that «,=a, when a is either 90° or 270°. But when a
is 0 or 180°, we have
@,=a—ksin’a = 54° 2,
which is the minimum value of the polarizing angle.

3. For the natural fracture-faces of the crystal the value of
A is 45° 23’. Hence, when a=0 or 180°,

@, =a -k (sin?a —sin’d) =57° 26;
and when a=90° or 270°,
@,=a+kcos’a sin? \ = 59° 50’.

These values of the polarizing angles agree very well with thie
experiments of Sir David Brewster, and still better with those
of M. Seebeck.

If we wish to know in what azimuths a, is equal to a, on a
given surface of the crystal, it is obvious from (55) that we
must make

sin’@ — sin’ = sin?a@ cos’ sin’*a,

whence we have, simply,
tan »

cos a=+
tan a

(59)

which shows that the thing is impossible when A is greater
than @; and that, when A is less than a, there are four such
azimuths; as indeed there are, generally speaking, four values
of a corresponding to any other particular value of the polariz-
ingangle. Ifa’ be the least of these azimuths, the others will
be 180°-a’, 180°+a’, and 360°-a’. On a natural face of the
erystal, the value of a’, answering to the supposition a,=2, is
found to be 52° 22’.

Next, let us trace the changes which the deviation under-
goes in some remarkable cases.

120 On the Laws of Crystalline

1. When the face of the crystal is perpendicular to its axis
there is evidently no deviation.

2. When the axis lies in the face of the crystal the devia-
tion vanishes in the azimuths 0, 90°, 180°, 270°. In the inter-
mediate azimuths, differing 45° from each of these, the deviation

is a maximum; for if we put A=0 in formula (55) the result will
be |

¢,--4 sin a sin 2a;

and this quantity (neglecting its sign) is a maximum when sin
2a=+1. The coefficient of sin 2a is equal to 3° 54’, which is
consequently the greatest value of the deviation. According to
the experiments of M. Seebeck, the value is 3° 57’.

8. On the fracture-faces of the crystal the deviation va-
nishes in the azimuths 0 and 180°, as also in two other azi-
muths for which .

cos a=-

tan a’

and in which, therefore, a, is equal to. In the azimuth 45°
the deviation is — 3° 35’; in the azimuth 90° it is —2° 32’; and
in the azimuth 127° 38’ it vanishes; after which it attains a
small maximum with a positive sign, and vanishes again in
azimuth 180°. The calculated values of the deviation agree
pretty well with the values observed by M. Seebeck.

The sign of the deviation shows at what side of the plane
of incidence the plane of polarization lies. But the position of
the latter plane is best indicated by that of the transversal of
the reflected ray. If this transversal and the axis of the erys-
tal be produced from the origin, towards the same side of the
plane of #z, until they intersect the sphere in the points ¢ and
A respectively, these points will be on the same side of the
great circle XY when the deviation and the sine of the azi-
muth have unlike algebraic signs; and they will be on opposite
sides of that circle when those quantities have like signs. ‘There-
fore if the crystal be supposed to revolve in its own plane, be-

Reflexion and Refraction. 121

ginning at the azimuth 0, the points ¢ and 4 will lie on the same
side of XY until A reaches the position. 4’, where the angle
A’Y?i is equal to 127° 38’; the point ¢ will then pass over to the
side opposite .A, at which side it will remain until A arrives at
the azimuth 232° 22’. Thenceforward, to the end of the revo-
lution, both points will be found on the same side of the circle
Say

We have seen that the deviation always vanishes when the
axis of the crystal lies in the plane of incidence. The reason is,
because the crystal is then symmetrical on opposite sides of that
plane. In this case the problem of reflexion offers peculiar faci-
lities for solution, since the uniradial directions are obviously
parallel and perpendicular to the plane of incidence. Let us,
therefore, consider the case at length.

1. In the first place, when the only refracted ray is the
ordinary one, the three transversals are in the plane of incidence,
and the transversal of each ray is proportional to the sine of the
angle between the other two rays. Hence the proportions are

TI “ T2 a T3
sin (a += tz) 7 sin Qu 5 sin (u = to)

(60)

the same as in ordinary media.

2. In the second place, when the sole refracted ray is the
extraordinary one, the three transversals are perpendicular to
the plane of incidence; and, if we use accents to mark the quan-
tities connected with this ray, we have the equations

Tit73=7.
1 bed SPF
72 , 73 72 (61)
MT Y=M2T2+MT 35
which give the proportions
r r. r’
1 2 3
= ght, (62)
M+m, 2m, ™m-mM,
wherein
m, sin 2’,+2sin*’, tan &’ (63)

Mm, sin 2¢,

by (26) ; the upper or lower sign being taken, in the numera-

122 On the Laws of Crystalline

tor of (63), according as the refracted ray or its wave normal
makes the smaller angle with a perpendicular to the face of
the crystal.

To find the polarizing angle, we have only to make m= m’s,
for then r’s will vanish by (62); and therefore, if common light
be incident, the whole reflected pencil will be polarized in the
plane of incidence. Supposing the crystal to be a negative one,
let us conceive the refracted ray to lie within the acute angle
made by the axis of the crystal with a perpendicular to its
surface. We shall then have to take the positive sign in the
numerator of (63), and the polarizing angle will be given by
. the condition
sin 21, = sin 21’, + 2 sin)’, tan «’. (64)

But from (36) we have, in general,
sin*’, tan ¢ = (a’?— 0’) sin w’ cos w’ sins, (65)
and in the present instance it is evident that

w =90°-A-v2,

where A denotes, as before, the angle which the axis of the

crystal makes with its surface. Substituting these values in
(64), and multiplying all the terms by tan «2, we get

sin’, = sin 4 cos 4 tan ’: — (a’ — b*) sin (A + 2) cos (A + ¢2) tan vy sin*y.
Again, from (37) we have |

sin’:’, = b? sin*:, + (a? — b*) cos’ (A + (2) sin’); (66)
and by equating these two expressions for sin*’,, we find

a’ cos® X + b? sin’? X
cotan 1, + (a* — 6) sin A cos A

tan ¢’, = (67)
Then, if this value of tan’, be substituted in equation (66), after
all its terms have been divided by cos’’,, we shall obtain the
simple and rigorous formula

1—a’ cos? \ — 0’ sin? A

pope
ae 1—a’ }°

= sin’a,, (68)

;
wa
§
|

are a ae ee Ae

Reflexion and Refraction. 123

for determining the polarizing angle a, when the axis of the
crystal lies in the plane of incidence. It is manifest, from the
nature of the formula, that this angle is the same, whether the
azimuth is 0 or 180°; that is, whether the light is incident at
the right or left side of the perpendicular to the surface of the
erystal.

This formula might be deduced more briefly by recollecting
what we have already proved, that the corresponding masses ™,
and m’, are proportional to the ordinates y of the points where
the incident ray and the extraordinary refracted ray meet their
respective wave surfaces; whence it follows that these ordinates
must be equal at the polarizing angle; and thus the question is
reduced at once to a geometrical problem. For as both rays are
in the plane of incidence, the axis of # will be intersected in one
and the same point by right lines touching the wave surfaces,
or their sections, at the extremities of the ordinates. Now the
sections in the plane of zy are a circle and ellipse with their
common centre at the origin, the radius of the circle being unity,
and the semiaxes of the ellipse being a and d, of which 0 is in-
clined at the angle to the axis of «; and therefore it is re-
quired to draw, parallel to the axis of z, a right line intersecting
the circle and ellipse, so that if tangents be applied to them at
two points of intersection which lie on the same side of the axis
of y, these tangents, when produced, may cut each other on the
axis of z The angle which the tangent to the circle makes with
the axis of x is then the polarizing angle a,; and the solution
of the problem just stated leads directly and easily to the for-
mula (68). From this way of viewing the matter we see the
reason why the polarizing angle is the same in the azimuths 0
and 180°; for if tangents be applied at the two remaining
points where the parallel that we have spoken of intersects the
circle and ellipse, it is evident that these tangents also will cut
each other on the axis of x; since tangents drawn at the ex-
tremities of any chord, either of a circle or an ellipse, intersect
the parallel diameter at equal distances from the centre.

- Let the reflecting surface of the crystal be in contact with

124 On the Laws of Crystalline

a fluid medium whose index of refraction out of vacuo is repre-
sented by WV, and let B and A respectively denote the ordinary
and the principal extraordinary indices of refraction out of vacuo

into the erystal. Then putting a for a, and . for b, in the pre-
ceding formula, and making

TL? = A’ sin? \ + B’ cos’ A,
we readily deduce
A* B- [? N*
N* (L? - N*)

tan’ a, = (69)

Hence we perceive that if Z’ = 4B, that is, if
B
tan A = VA a

(in which case A will never be much above or below 45°), the
value of a, will be always possible; for then we shall have

AB
NY.

But if A be different from this, and of course L’ not equal to
A B, the value of a, may become impossible for certain values
of V. For it is clear that if WV lie between the limits Z and

cen the numerator and denominator of the fraction (69) will

4

tan? a, =

(70)

have unlike signs, and the tangent of a, will be the square root ~

of a negative quantity. In this case, therefore, if common light
be incident, it will “refuse to be polarized,” as Brewster ex-
presses it; in other words, it will be impossible to find an angle
of incidence at which the reflected pencil will cease to contain
light polarized perpendicularly to the plane of incidence, or at
which the reflected transversal 7’; will vanish. With all values
of NV, except those which are included between the narrow limits

ZL and = the polarizing angle is possible. It is zero at the

latter limit, and 90° at the former. Outside these limits it

Reflexion and Refraction. 125

changes rapidly at first, until VV has passed either of them by
a quantity considerable in proportion to the interval between
them.

From (68) we find a,=A, when a=1, or N=; and also
a,=90°—A, when 6=1, or N=B. In the latter case it is re-
markable that no light is reflected when. common light is inci-
dent at the angle 90°—A. For then we have 7’,=0; and because
t:=t, we have likewise r,=0. Therefore no light can enter the
reflected pencil. But this case deserves that we should consider
it more at large, without restricting ourselves to the supposition
that the axis of the crystal lies in the plane of incidence.

Assuming then that V=B, or that the refractive index of
the fluid, which covers the reflecting surface, is equal to the
ordinary index of the crystal itself, we may observe that, in
this case, every angle of incidence, in every azimuth, has a
right to be regarded as a polarizing angle. In fact, common
light cannot suffer reflexion at the separating surface of the
erystal and the fluid, without becoming completely polarized.
For if polarized light be incident, and if 7; and 7’; be the uni-
radial reflected transversals, respectively belonging to the ordi-
nary and to the extraordinary ray, the former transversal must
necessarily vanish, for the same reason that no reflexion can
take place at the separating surface of two ordinary media
whose refractive indices are equal; and thus the actual re-
flected transversal will always coincide in direction with 7’,
whatever be the direction of the incident transversal. Conse-
quently, if common light be incident, the whole reflected pencil
will be polarized in a plane passing through 7’;, and making
with the plane of incidence an angle 6’, determined by the
second of formule (39). By putting .=4 in that formula,
and employing the expression (44), we first obtain

_ 608 (+ ¢’2) cos 0’ tan w’ + sin (4. + «’2)
sin 0 tan w’

tan &,

_ C08 (tr +02) tan (pez) +8im (44/2) |
7 sin 0’ tan w’ ;

126 On the Laws of Crystalline

and thence .
nie sin (p+u) _ sin (p +4) cosa’ |

sin 0’ tan w’ cos(p-v’) sing cos (p—¢2)’

and finally,
tan ’;=sin (p+) cotang; (71)

a result which shows that the plane of polarization of the re-
flected ray is perpendicular to a plane drawn through the ray
itself and the axis of the crystal.

Moreover, we find, from the first of formule (39), by
proceeding as above,

tan 0’, =— sin (p — 1) cotan g=—cotan 6; (72)

and from (88) it is evident that 0:=0. Therefore all that re-
lates to the case under our consideration may be summed up in
the following statement :

When V=8, and the incident light is polarized in a plane
passing through the axis, the course of the light is unaltered,
and there is neither reflexion nor refraction. When it is pola-
rized in the perpendicular plane, all the light which enters the
crystal undergoes extraordinary refraction. Whatever light is
reflected is always polarized in a plane at right angles to that
which passes through the reflected ray and the axis of the
crystal; and this is true, whether the incident light is pola-
rized or not.

Here, for the present, we must terminate our deductions
from the general theory propounded in this Paper. Several
other questions remain to be discussed, such as the reflexion of
common light* at the first surface, and the internalt reflexion

*The mode of treating the case in which common light is incident has been
pointed out at the bottom of p. 100.

+ I have since found that the problem of reflexion at the second surface may be
reduced to that of reflexion at the first surface by means of a very simple rule.
Let us suppose the two surfaces of the crystal to be parallel; and let a ray A,
uniradially polarized, and incident on the first surface, give the ray Rs by reflexion,
and the single ray Rz by refraction. Let R2 be the ray which suffers internal reflex-
ion at the second surface, thereby giving the two reflected rays R,,, R’,,, and the

Reflexion and Refraction. 127

at the second surface of a crystal; but these must be reserved
for a future communication. It would be easy, indeed, to write
down the algebraical solutions resulting from our theory; but
this we are not content to do, because the expressions are rather
complicated, and, when rightly treated, will probably contract
themselves into a simpler form. It is the character of all true

single refracted ray Ri) emerging from the crystal in a direction parallel to R,. Put
* 1, T3, T2, and 7,,, 7’,,, 7(1) for the transversals of the rays in the order in which they
have been named. As the transversal r2 is supposed to be given in magnitude, the
lengths as well as the directions of 7; and r3 can be found by the construction in
page 97.

Now, the direction of 73 being changed, and its magnitude retained, let the ray
R3 be turned directly back, so as to be incident again on the crystal, and to suffer
reflexion and refraction at the first surface. Then the two refracted rays which it
gives will be parallel to 2,,, R’,,, and their transversals will be equal and parallel
to, 7’, The reflected ray which it gives will coincide with 21; and the re-
flected transversal, when compounded with 7, will furnish a resultant equal and
parallel to the emergent transversal 7,1).

Thus the constructions, -yhich have been given for the first surface, may be
made available for the second surface, and every question relative to crystalline
reflexion may be solved geometrically by means of the polar planes.

The foregoing rule was not, properly speaking, deduced from theory. I first
formed a clear conception of what the rule ought to be, and then verified it for the
simple case of singly-refracting media, and finally proved it for doubly-refracting
crystals. The truth of the rule, in crystals, depends upon the truth of the three
following equations :—

sin (¢,,+ «’,,) {cos (t,,—«’,,) + cotan @,,cotan 6’,,} +4,,+h',,=0,
sin («,—1,,) {cos (¢;++,,) —cotan 62 cotan @,,} +/he-h,,=0, (vrz1.)
sin (t.~-1',,) {cos («, +¢’,,) — cotan #2 cotan 6’,,} +h2—h',, =0,

in which the notation is intelligible without any explanation. The first equation
is the same as equation (vi1.) already noticed; and the other two differ from it
only in appearance, the change in the signs being occasioned by a change in the
relative position of the rays.

When the reflexion is total, I suppose we may follow the example which
Fresnel has set us in the case of ordinary media. The general algebraic expres-
sion for each reflected transversal will then become imaginary ; and by putting it
under the form

T (cos +\/ —1sin$),

we shall have 7 for the reflected transversal, and # for the change of phase.

128 On the Laws of Crystalline

theories that the more they are studied, the more simple they
appear to be. And we may add, that a close examination of
such theories always meets with its reward, in the unexpected*
consequences which present themselves to view. Nothing can be
simpler than the laws of double refraction, as they were deli-
vered by Fresnel; yet the properties of his wave surface still
continue to furnish the geometer with beautiful and curious re-

lations. So we may hope that a little more time, devoted to

the laws of reflexion, will not be spent in vain. They promise
to supply many other theorems, not undeserving of attention,
though perhaps not as simple and comprehensive as those that
have already: been made known.

From the nature of the rules which we have given for treating the question of
reflexion at either surface of the crystal, it follows that the final equation, for
determining the position of a transversal, is always linear, though the equation
of vis viva is of the second degree. This result very strongly confirms the theory ;
but it shows, at the same time, that the law of the preservation of vis viva is not to
be regarded as an ultimate principle, but rather as a ‘consequence of some elemen-
tary law not yet discovered.

It now appears that the conjectures put forward in the note, p. 93, were hasty,
and that there was some mistake in the calculations which gave rise to them. It
38 scarcely necessary to mention, that the sheet in which that note is found was
printed off before I had obtained the result’ announced in the subsequent note,
p- 111. Various delays occurred while my Paper was going through the press;
and I took advantage of them to increase its value, by appending notes on some
of the questions which I had overlooked or omitted in the first consideration of
the subject.

* As an instance of this, it may be mentioned, that the conclusion arrived at
in the note,’p. 111, was wholly unexpected. And in verifying the equation (vm),
an unexpected and useful theorem was obtained; for it became necessary to find
a manageable expression for the tangent of the angle « which the wave normal
makes with the ray. This expression is wanted in applying the formule (34) and
(35) to biaxal crystals, and therefore I shall make no apology for introducing it
here.

Having described a sphere concentric with the wave surface, let the wave
normal OP and the two optic axes (which are the nodal diameters of the index
surface) be produced from the centre O to meet the sphere in the points P,, 4, 4,,
respectively, thus marking out the angles of a spherical triangle PAA, The same
wave normal may belong to two different rays; and if we select one of these
rays, its transversal must lie in a plane drawn through the wave normal, and
bisecting either the internal angle 4P A, of the spherical triangle, or the ex-

Reflexion and Refraction, 129

If we are asked-what reasons can be assigned for the hypo-
theses on which the preceding theory is founded, we are far
from being able to give a satisfactory answer. We are obliged
to confess that, with the exception of the law of vis viva, the
hypotheses are nothing more than fortunate conjectures. These
conjectures are very probably right, since they have led to
elegant laws which are fully borne out by experiments; but
this is all that we can assert respecting them. We cannot
attempt to deduce them from first principles; because, in the
theory of light, such principles are still to be sought for. It

ternal supplementary angle. By producing the optic axes in the proper direc-
tions, we may always make the above plane (which Fresnel calls the plane of
polarization) bisect the internal angle. Supposing this to have been done for the
ray which was selected, put w and w, for the sides PA and P_A, of the spherical
triangle, and y for the contained angle 4P,A, Let s be the length of the wave
normal from the centre 0 to the point where it intersects the tangent plane applied
at the extremity of the ray, that is, applied at the point where the ray meets its
own nappe of the wave surface; and let @ and ¢ be the greatest and least semiaxes
of the ellipsoid which generates the wave surface. Then we shall have

ac

tan e= 3 sin (w —w,) sin} yp. (Ix.)

&

And it is now manifest that if «, be the angle which the other ray makes with the
same waye normal, and s, the length of the wave normal intercepted between the
centre and the tangent plane at the extremity of this ray, we shall also have

sin (w + w,) cos} v. (x.)

a

- tan €,= 282
If a ray is given in direction it will have two wave normals; and then the
angles «, ¢,, which it makes with each normal, may be found from the formule

tane= 5 (5-‘) sin (w —w,) sin} y,

(xI.)

es MG SP
tane,="5-(1 - zs) sin (w+,) cosy,

where r and r, are the two radii of the wave surface which are in the direction of
the ray ; the spherical triangle P4.4,, of which the sides and contained angle are
expressed by the same letters as before, being now formed by producing the ray
and the two nodal diameters of the wave surface, until they intersect the sphere in
the points P, 4, A.

K

130 On the Laws of Crystalline

is certain, indeed, that light is produced by undulations, pro-
pagated, with transversal vibrations, through a highly elastic
ether; but the constitution of this ether, and the laws of its
connexion (if it has any connexion) with the particles of bodies,
are utterly unknown. The peculiar mechanism of light is a
secret that we have not yet been able to penetrate. As a proof
of this, we might observe, that some of the simplest and most
familar phenomena have never been explained. Not to mention
dispersion, about which so much has been fruitlessly written,
we may remark, that the very cause of ordinary refraction, or
of the retardation which light undergoes upon entering a trans-
parent medium, is not at all understood. Much less can it be
said that double refraction has been rigorously explained ; its
laws alone have been clearly developed by Fresnel. In short,
the whole amount of our knowledge, with regard to the propa-
gation of light, is confined to the /aws of phenomena: scarcely
any approach has been made to a mechanical theory of those
laws. And if the case of uninterrupted propagation through
a continuous medium presents such difficulties, it would be use-
less to think of accounting for the laws which subsist at the
confines of two media, where the continuity is broken.

But perhaps something might be done by pursuing a con-
trary course; by taking those laws for granted, and endeavour-
ing to proceed upwards from them to higher principles. In
this point of view, our second law, or hypothesis, is extremely
remarkable; for it seems to be opposed, in some degree, to the
notion that the ethereal molecules are strongly attracted or
repelled by the particles of bodies. However that may be, it
would appear that a true theory must be in accordance with
this hypothesis; and that any mechanical ideas, which would
make the mean density of the ether vary from one medium to
another,* cannot be admitted to represent the real state of

* Those who maintain that the density of the ether is different in different media,
ought to consider the following question :—What function of the three principal
indices of a doubly-refracting crystal represents the density of the ether within
the crystal ?

Reflexion and Refraction. 131

things in nature. It is no objection to the hypothesis in
question, to say that it increases the difficulty of accounting
for refraction ; for, as there is positive evidence in favour of
the hypothesis, we ought rather to conclude that the common
opinion, which attributes refraction to a change of density in
the ether, is altogether erroneous.

In the next place we may remark, that our first hypothesis,*
concerning the direction of vibrations in polarized light, will be
useful in testing any proposed theory; for as it now seems to be
certain that the vibrations are parallel to the plane of polariza- -
tion, and not perpendicular to it, as Fresnel supposed, such a
direction of the vibrations ought to be a consequence of the
theory which we adopt.

The third hypothesis, or the principle of the preservation of
vis viva, is the most natural that can be imagined, inasmuch as
it implies only this, that the incident light is equal to the sum
of the reflected and refracted lights. Yet it is probable that
even this principle, like the law of vis viva in ordinary mechanics,
is a result of simpler laws, and will be shown to be so as soon
as the true mechanism of light shall be discovered.

The fourth hypothesis is a very important one, because the
whole theory turns upon it; and therefore, in the beginning of
this Paper, a particular account has been given of the manner in
which it was originally suggested. If we wish to give a reason
for this hypothesis, we might say that the motion of a particle
of ether, at the common surface of two media, ought to be the
same, to whichsoever medium the particle is conceived to
belong ; and as the incident and reflected vibrations are super-
posed in one medium, and the refracted vibrations in the other,
we might infer that the resultant of the former vibrations ought
to be the same, both in length and direction, as the resultant of
the latter. At first sight this reasoning appears sufficiently
plausible; but it will not bear a close examination. For as
the argument is general, it would prove that the principle of

* This hypothesis properly belongs to the laws of propagation, as it relates only
to what passes within a given medium.

K 2

132 — On the Laws of Crystalline

the equivalence of vibrations is true for metals,* as well as for
crystals, which it certainly is not. :It is not easy to see why
the principle should hold in the one case and not in the other;
but it is probably prevented from holding, in the case of metals,

* A few days after this Paper was read, I found reason to persuade myself
that in metals the vibrations parallel to the surface are equivalent, but not those
perpendicular to it; and that in metals, as well as in crystals, the vis viva is
preserved. This persuasion was founded on a system of formule which I had
invented for expressing the laws of metallic. reflexion and refraction; and which
seem to represent very satisfactorily the experiments of Brewster, Phil. Trans.,
1830. As metallic and crystalline reflexion are kindred subjects, and will one
day be brought under the same theory, however distinct they may now appear,
it will not be out of place to insert the formule for metals here. These formule
are not proposed as true, but as likely to be true; and they will be found to
express, at least with general correctness, all the circumstances that have hitherto
been regarded as anomalies in the action of metals upon light.

I suppose that for every metal there are two constants, M and x, of which
the first is a number greater than unity, and the second is an angle included
between 0 and 90°. The number MI call the modulus, and the angle x the
characteristic of the metal. Both M and x vary with the colour of the light,

and the ratio ee
periments it appears that I diminishes from the red to the violet; and therefore
I should suppose that cos x diminishes in a greater ratio, in order that the index
of refraction may increase as in transparent substances.

Put 1, for the angle of incidence, and 1, for the angle of refraction, so that

is probably the index of refraction. From Brewster’s ex-

sats od : (x11)

sini, cosx
and let » be a variable determined by the condition

COS t,
=—+, : XIII.
Me cos fe ( )

These two relations combined will give

1 cos?
Fhe 1+ ( -) tan*,, (xTv.)

which shows that w is equal to unity at a perpendicular incidence, and that it
vanishes at an incidence 90°, decreasing always during the interval.

Now if plane polarized light be incident on the metal, we must distinguish two
principal cases, according as the light is polarized in the plane of incidence, or in
the perpendicular plane. In the first case, denoting the reflected and refracted
transversals by 73 and rz respectively, let us put A3 for the change of phase in

Reflexion and Refraction. 133

by the same cause, whatever it is, which produces a change of
phase in metallic reflexion.

It will be proper to conclude this Essay with a brief sketch
of the researches of Sir David Brewster and M. Seebeck, the

the reflected ray, and Az for the change of phase in the refracted ray. Let the
same symbols, marked with accents, be used in the second case with similar signi-
fications. Then if the incident transversal be taken for unity, we shall have the
following formule :

1. When the incident transversal is in the plane of incidence,

ree + - 2M pw cos x )
OM? + pe + 2M pcos x’
4M? p22
2 —
7S +e + 2M pcos x’ > ( xv.)
2M p sin x pw sin x
tan A3 = ———_-_=- tan Ao.
nM ge? mB Oe" Me woos’ |

2. When the incident transversal is perpendicular to the plane of incidence,

_1+ M* w?- 2M yp cos x 7
1+ M? pw? + 2M yu cos x’

4M* p?
72 = res > MAEVE}
1+ WM? w+ 2M p cos

2M pw sin x Rei ates sin x

axe Be Bn By
peel M? p2—-—1’ Myu+cosx J

When x = 0, there is no change of phase, and the formule become identical
with those given in the note, p.101. When x = 90°, there is total reflexion at
all incidences. The case of pure silver approximates to this. For good speculum
metal, x is about 70°. The value of M ranges from 2} to 6 in different metals.

When the incident transversal is inclined to the plane of incidence, its compo-
nents, parallel and perpendicular to that plane, will give two reflected transversals
with a difference of phase equal to A’3- A3. The reflected vibration will then be
performed in an ellipse; and the position and magnitude of the axes of the ellipse
may be deduced from the preceding formule. The consequences of these formule
are very simple and elegant, but I cannot dwell upon them here. Suffice it to
observe, that every angle of incidence has another angle corresponding to it, which
I call its conjugate angle of incidence; and that the value of A’3 — A3 gis one of

these angles is the supplement of its value at the other, while the ratio “is the

same at both angles ; whence it follows that, ceteris paribus, the elliptic ibcatioes.
reflected at conjugate angles, are similar to each other, and have their homologous

134 On the Laws of Crystalline

only other writers who have treated of the subject of crystalline
reflexion.

So early as the year 1819, Sir David Brewster published,
in the Philosophical Transactions, a Paper “ On the Action of
Crystallized Surfaces upon Light.” * In this Paper the Author
details a great variety of experiments on the polarizing effects
of Iceland spar. He gives the measures of the polarizing angles
in different azimuths, when the reflexion takes place in air; but
he does not notice the accompanying deviations, which were pro-
bably too small to attract his attention. In another instance,
however, he obtained very large deviations. He conceived the
idea of pushing his experiments into an extreme case, by mask-
ing, as it were, the ordinary reflecting action of the crystal, and
leaving the extraordinary energy at full liberty to display itself.
This was done by dropping on the reflecting surface a little oil
of cassia, a fluid whose refractive index is nearly equal to the
ordinary index of Iceland spar. When common light, incident
at 45°, was reflected at the separating surface of the oil and the
spar, the reflected pencil was found to be partially, and some-
times completely, polarized in planes variously inclined to the
plane of incidence, the inclination going through all magnitudes
from 0 to 180°, as the crystal was turned round in azimuth.
This general result is no more than what theory would lead us
to expect, when the angle of incidence is nearly equal to one of
the angles of refraction; but to institute a minute comparison
of theory with experiment would require troublesome caleula-

axes equally inclined to the plane of incidence, but on sabe sides of it. When
A's — A3 = 90°, the conjugate incidences are equal, the ratio ~ = is a minimum, and

the axes of the elliptic vibration are parallel and perpensieular to the plane of
incidence. When A’; = 90°, or My = 1, the value of 7’3 is a minimum, and equal
to tan 4 x.

The foregoing formule differ slightly from those which I have given in No. I.
of the Proceedings of the Royal Irish Academy. The small quantity x’, which
occurs in the latter, has been purposely neglected, as its presence interferes with
the simplicity of the expressions.

* Phil. Trans. 1819, p. 146.

Reflexion and Refraction. | 135

tions, which I have not had time to make. With the view,
however, of showing clearly, from theory, that the range of
the deviation is unlimited, I have considered the simple case
in which WV = B, or in which the refractive index of the fluid
is ewactly equal to the ordinary index of the crystal. This case,
moreover, is remarkable on its own account; and it might be
worth while to try whether it could not be verified by direct
experiment. If a fluid could-be procured whose refractive
index, for some definite ray of the spectrum, should be equal
to the ordinary index of the crystal for the same ray, and if
common light, incident at any angle and in any azimuth, were
reflected at the confines of the fluid and the crystal, then,
supposing the theory to be exact, the definite ray aforesaid
would, as we have seen, be completely polarized by reflexion,
and the plane of polarization would always be perpendicular
to a plane drawn through the direction of the reflected ray
and the axis of the crystal. This experiment would be an
elegant test of the theory in its application to these extreme
and trying cases; and if it were successful, no doubt could
be entertained* as to the rigorous accuracy of the geome-
trical laws of reflexion.

*T was at this time in doubt whether the phenomena observed with oil of
cassia could be reconciled to theory; and when the note in page 93 was written,
- I was almost certain that they could not. But I have since, I think, found out
the cause of this perplexity. Some of Brewster’s experiments were made with
natural surfaces of Iceland spar; others with surfaces artificially polished. I
believe (though I have made very few calculations relative to the point) that the
former class of experiments will be perfectly explained by the theory ; the latter
I am certain cannot be so explained, nor ought we to expect that they should.
For the process of artificial polishing must necessarily occasion small inequalities,
by exposing little elementary rhombs with their faces inclined to the general
surface; and the action of these faces may produce the unsymmetrical effects
which Brewster notices as so extraordinary (Sixth Report of the British Asso-
ciation, Transactions of the Sections, p. 16). If this will not account for such
effects, I do not know what will. From an old observation of Brewster (Phi/.
Trans., 1819), it would appear that imperfect polish does actually produce a want
of symmetry in the phenomena; for when common light was reflected between
oil of cassia and a badly polished surface perpendicular to the axis, he found that
the reflected ray was polarized neither in the plane of incidence, nor perpendicular

136 On the Laws of Crystalline

The experiments with oil of cassia must be very difficult
on account of the great feebleness of the reflected light. Sir
David Brewster, however, resumed them at different times;
and he laid an extensive series of his results before the Physi-
eal Section of the British Association at its late meeting in
Bristol.

It was not until the latter end of November, 1836, that I
became acquainted with the investigation of M. Seebeck, who

has contributed greatly to the advancement of the subject. He.

made very acurate experiments on the light reflected in air from
Iceland spar. He detected the deviation, notwithstanding its
smallness, and measured it with great care. He also made the
first step in the theory of crystalline reflexion; and the remark-
able formula (68), which gives the polarizing angle when the
axis lies in the plane of incidence, is due to him. The hypo-
theses which he employed were similar to those of Fresnel,
and they enabled him to solve the problem of reflexion in the
case just mentioned, but not to attempt it generally. The date

of his first Papers* is the year 1831; but he did not publish his

experiments on the deviation until a recent occasion, when he
was led to compare them t+ with the theory which I had origi-
nally given in my letter to Sir David Brewster. I have already
stated the correction which the theory underwent in conse-

to it, but 75° out of it. The same surface, when the light was reflected in air,
gave the polarizing angle more than two degrees below its proper value.

To show that, in other respects, the general character of the phenomena is in
accordance with theory, we may observe that when NV = B, and A=0 or 90°, if
common light be incident at 45° in the plane of the principal section of the crystal,
the whole of the reflected light will be polarized perpendicularly to that plane; and
therefore if V be nearly equal to B, while every thing else remains the same, the
reflected pencil will contain some vnpolarized light, and will be only partially pola-
rized in a plane perpendicular tu the plane of incidence; so that (as Brewster has
found by experiment) the crystal\will then produce by reflexion the same effect
which is produced by ordinary rviraction. This (as he also found) will not happen
when A and the angle of incidence are each equal to 45°, because the light is then
incident at the polarizing angie.

* Poggendorff’s Annals, Vols xxi. p. 290; Vol. xx1. p. 126.

t Ibid., Vol. xxxvut. p. 280.\

a.
eh

Reflexion and Refraction. 137

quence of those experiments, and by which it was brought to
its present simple form.*

* Two or three months after this correction had been published in the Philoso-
phical Magazine, a notice of it was inserted in Poggendorff’s” Annals, vol. xl.
p. 462. Up to that time, I believe, nothing had been published in Germany on
the general theory of crystalline reflexion; at least the writer of the notice (whom
I take to be M. Seebeck) does not seem to have heard of any other theory, or any
other principles than mine. But in the next number of Poggendorff, vol. xl.
p- 497, there appeared a letter from M. Neumann, in which the writer speaks of
a theory of his own, founded on principles exactly the same as those which I had
already announced, and refers to a Paper which he had communicated on the
_ subject to the Academy of Berlin. The Paper has been printed in the Transactions
of that Academy for the year 18353; and through the kindness of the author I have
received a copy of it, just in time to acknowledge it here. On casting my eye over
it, 1 recognize several equations which are familiar to me—in particular, the equa-
tions (v1l.), (vri1.), (Ix.), (x.), which I discovered independently in November last.
M. Neumann’s Paper is very elaborate, and supersedes, in a great measure, the
design which I had formed of treating the subject more fully at my leisure; nor
can I do better than recommend it to those who wish to pursue the investigations
through all their details.

Trinity Cottece, Dustin, March, 1838.

(7838)

XII.—ON A NEW OPTICAL INSTRUMENT, INTENDED
CHIEFLY FOR THE PURPOSE OF MAKING EXPERI-
MENTS ON THE LIGHT REFLECTED FROM METALS.

[ Proceedings of the Royal Irish Academy, April 9, 1838.]

‘Tue instrument consists of two hollow arms or tubes, moveable
about the centre, and in the plane, of a large divided circle, each
arm being provided with a Nicol’s eye-piece, or some equivalent
contrivance for polarizing light in a single plane; while in one
arm, which is of course crooked, a Fresnel’s rhomb is interposed
between the eye-piece and the centre of the circle. At this centre
is placed a stage for carrying the reflector, with its plane perpen-
dicular to the plane of the circle, and having a motion to and fro
for adjustment. Hach eye-piece, as well as the Fresnel’s rhomb,
turns freely about the axis of the arm to which it belongs, and
is provided with a small circle for measuring its angle of rota-
tion. When the two arms are set at equal angles with the re-
flector, and the observer looks through the crooked arm, he will
see a light admitted through the straight one; and then, by
turning the Fresnel’s rhomb, and the eye-piece next his eye, he
will be able, by means of their combined movements, to find a
position in which the light will entirely disappear. An obser-
vation will then have been made; for the light, before its inci-
dence on the metal, is polarized in a given plane by the first
eye-piece; but after reflexion from the metal (as we know from
Sir David Brewster’s experiments) it is elliptically polarized ;
and our object is to determine the position and species of the

On a New Optical Instrument, &c. 139

little ellipse in which the reflected vibration is supposed to be
performed. Now, the axes of this ellipse are parallel and per-
pendicular to the principal plane of the rhomb, when it is in
the situation above described, where the light completely dis-
appears; and the ratio of the axes is the tangent of the angle
which that plane makes with the principal section of the eye-
piece next the eye. The angles are read off from the divided
circles; and thus, for any angle of incidence, and any plane
of primitive polarization, we can at once ascertain the nature
of the reflected elliptic vibration. Professor Mac Cullagh men-
tioned that the instrument was made last year with the view
‘of testing certain formule which he has proposed for the case
of metallic reflexion, and which have been printed in Vol. xv.
pp- 70, 71, of the Transactions of the Academy*; but that he
had not yet found leisure to make the various adjustments
which are necessary in order to obtain satisfactory results with
it. The instrument is beautifully executed by Mr. Grubb, who
himself contrived the subordinate mechanism, by which the
requisite movements are effected with perfect ease to the ob-
server.

* Supra, pp. 132, 133.

( 140 )

XIII.—LAWS OF CRYSTALLINE REFLEXION.—QUESTION
OF PRIORITY.

[ Proceedings of the Royal Irish Academy, Nov. 30, 1838,]

Tue President, Sir William R. Hamilton, read the following
letter which had been addressed to him by M. Neumann of
K6nigsberg, on some points connected with the history of the
Laws of Crystalline Reflexion :—

Monsieur, ;
Le haut prix que j’attache 4 votre suffrage et a celui de l’illustre Académie,
a laquelle vous présidez, et l’honorable mention, que vous avez voulu faire de mon
mémoire sur la théorie de la lumiére dans la séance de cette Académie du 25 Juin,
m’engagent & vous dresser la lettre suivante. Vous avez donné dans cette séance
un jugement dans la question de priorité, qui pouvait s’élever entre Mr. Mac
Cullagh et moi par rapport 4 la découverte des lois suivant lesquelles la lumiére
est reflechie et refractée par des milieux crystallins—j’ai l’honneur de vous com-
muniquer dans ce qui suit quelques faits et quelques reflexions fondées sur ces
faits, et qui auraient été peut-étre de quelque influence sur ce jugement.

Au commencement de l'année 1833 j’ai communiqué a M. Seebeck de Berlin
non seulement T'ensemble des principes de ma théorie tels qu’ils se trouvent
imprimés dans le § 2 de mon mémoire, mais j’avais illustré encore ces principes
par leur application aux milieux non crystallins. En méme tems j’ai annoneé 4
M. Seebeck, que les résultats tirés de ces principes par rapport aux milieux
erystallins étaient parfaitement d’accord avec ses observations sur l’angle de
polarisation du kalkspath, et je lui fis part de la formule méme, qui exprime
V’inclinaison du plan de polarisation du rayon polarisé par réflexion vers le plan
de reflexion. Sous la date du 11 Mai, 1833, M. Seebeck m’écrivit, que cette
formule aussi s’accordait parfaitement avec ses observations, qu'il n’avait pas
encore publiées et qu’ il avait la complaisance de me communiquer en manuscrit.
Dans le printems de 1834 le manuscrit de mon mémoire tel qu’il a paru depuis
allait étre achevé; mais un voyage que je fis dans ce tems et qui m’éloigna

t a

Question of Priority. 141

assez long-tems de Kénigsberg, m’empécha de la publier incessamment. Cepen-
dant j’avais pris soin d’en faire un abrégé dans lequel je développai complétemen;
les principes de ma théorie et les résultats auxquels elle m’ayait conduit par
rapport aux crystaux.4 un axe.

J’envoyai cet extrait en Mai ou Juin, 1834, par la librairie de M. Schropp
de Berlin 4 M. Arago, en le priant de le faire imprimer dans les Annales de
Chimie et de Physique, ce savant ayant dans une note publiée dans ce tems
marqué un grand intérét pour l’investigation des lois des intensités du rayon
ordinaire et extraordinaire, lois qui se trouvaient parmi les résultats mentionnés,
Il n’y a pas de doute que cet extrait ne soit parvenu dans les mains de M. Arago, .
entre lesquelles il doit se trouver encore 4 présent. Du reste, M. Jacobi en avait
pris une connaissance détaillée, et 4 Berlin il a été entre les mains de MM. Weiss
et Poggendorf.

En passant par Viemnne dans 1’été de 1834, j’avais le plaisir d’entretenir de mes
résultats et de ma méthode M. Ettinghausen, savant trés distingué et trés versé
dans les parties les plus épineuses de l’optique. Antérieurement j’avais enseigné
mes doctrines 4 M. Senff, maintenant professeur 4 l’ Université de Dorpat, pendant
le séjour que fit 4 Koenigsberg ce jeune et habile physicien, qui vient de publier
un excellent travail sur les propriétés optiques et crystallographiques du fer sulfaté.

Il suit de tout ce qui précéde, que déja en 1834, mes resultats trouvés par
rapport aux lois de reflexion et de refraction des crystaux n’étaient guére
inconnus aux physiciens de |’ Allemagne, qui s’occupent de l’ optique, et si dés lors
ils n’ont pas regu une plus grande publicité, vous voyez, Monsieur, cela tenait
aux Annales de Chimie. La publication de mon mémoire a été rétardée par
Vespoir que j’avais congu de pouvoir lui ajouter une partie expérimentale.
Mais l’exécution des appareils me faisant attendre trop long tems, j’ai présenté
vers la fin de 1835 4 l’Académie de Berlin mon ouvrage tel qu'il a été imprime
depuis parmi les mémoires de cette Académie. La partie expérimentale a été
publiée en 1837 dans le volume 42 des Annales de M. Poggendorf.

Je vois du discours que vous avez tenu, Monsieur, dans la Séance de votre
Académie du 25 Juin passé, et qui vient de m’étre communiqué, que c’est déja
en Aoiit, 1835, que Mr. Mac Cullagh a fait 4 l’Association Britannique une
communication sur les lois de reflexion et refraction par les crystaux, et qui a
été imprimée dans le Lond. and Edinb. Phil. Mag., Février, 1836. Je crois trés
yolontiers, que Mr. Mac Cullagh est parvenu aux résultats qui se trouvent dans
cette publication, par ses propres efforts et sans avoir eu connaisance de mes
travaux sur ce méme sujet. Toutefois ce ne sont pas ces résultats qui, pourraient
@tre Vobjet d’une question de priorité. En effet dans une note publiée dans les
Annales de M. Poggendorf (vol. xxxviii. 1836), M. Seebeck a montré que les
formules auxquelles est parvenu Mr. Mac Cullagh ne sont pas justes, et qu’elles
ne representent pas les lois de reflexion et de refraction par les crystaux. Dans
la méme note M. Seebeck a exposé, comment les lois de reflexion et de refraction
des milieux non erystallins conformes 4 cette definition du plan de polarisation, a
laquelle on est conduit dans la théorie dela double refraction, peuvent étre déduites.
des suppositions faites par Fresnel, avec la seule modification de l’homogénéité

142 Laws of Crystalline Reflexion.

de l’éther dans tous les milieux. © Mais les suppositions de Fresnel ainsi modifiées
forment la base principale de ma méthode, dont j’avais déja fait part 4 M. Seebeck
depuis plusieurs années. II est vrai, que dans les deux milieux Fresnel ne suppose
que l’égalité de deux composantes paralléles au plan de séparation, mais Végalité
de la troisiéme n’ est qu’une simple conséquence de celle des deux autres et des
autres suppositions. Ce sont les suppositions de Fresnel modifiées de la dite
maniére, qu’a adoptées Mr. Mac Cullagh, aprés s’étre convaincu par la note de
M. Seebeck de la faussete des résultats qu’il avait jusque-la obtenus, conviction
qui l’engagea a rejeter tout ce qui n’était pas conforme a ces suppositions, et dés
lors seulement en 1837, dans le Lond. and Edinb. Phil. Mag., Mr. Mac Cullagh
est parvenu aux mémes lois de reflexion et de refraction que j’avais eues l’honneur
de présenter 4 |’ Academie des Sciences de Berlin en 1835.

Vous voyez par tout ceci, Monsieur, que dés 1833 j’ai été en pleine possession
de la méthode, et que dés le commencement de 1834 j’ai été en pleine possession
des résultats qu’elle fournit, que dans ce méme tems j’ai envoyé un abrégé con-
tenant ces résultats et lu en manuscrit par plusieurs savans bien connus 4 M.
le redacteur des Annales de Physique et Chimie pour le publier dans ce recueil,
et qu’d la fin de 1835, j’ai présenté l’ouvrage complet 4 present imprimé &
Y Académie de Berlin; vous voyez en méme tems, que Mr. Mac Cullagh ayant
communiqué a l’Association Britanique en 1835 des lois de réflexion et de refrac-
tion crystallin, ces lois ont été demontrées étre fautives par M. Seebeck in 1836,
et que Mr. Mac Cullagh n’est parvenu en 1837 aux vraies lois qu’ aprés avoir
pris connaissance du fondement de ma méthode, et s’en étre servi.

De tout cela résulte, Monsieur, que la priorité de la découverte des lois de
réflexion et réfraction par des crystaux n’est pas douteuse, et’qu’il n’y a pas de
simultanéité entre mes travaux et ceux de Mr. Mac Cullagh, dont du reste
personne ne peut estimer plus que moi le talent distingué.

Daignez, Monsieur, agréer les assurances de la plus haute considération avec

laquelle je suis, &c.,
F. E. NEUMANN.

Kénicssere, 5 Octobre, 1838.

When this letter was read, Professor Mac Cullagh requested
permission to make a few remarks. After expressing much
regret that his researches in the theory of light should have |
clashed with those of any other person (though in the present
state of science such collisions were perhaps inevitable), he
proceeded to say, that he did not think it necessary to detain
the Academy with a formal reply to the communication which
had just been read; it would be sufficient for him to observe,
in general, that the facts brought forward by the writer, with
reference to the history of his own investigations, were all,
without exception, of a private nature, not one of them being

Question of Priority. 143

taken from any published document; that the first document
of the kind, which professed to give any account of M. Neu-
mann’s “method,” or any statement of the principles employed
in it, appeared in the Annals of Poggendorf (Vol. xx. p. 497),
some months after Mr. Mac Cullagh had published his Jast
Paper on the subject in the Philosophical Magazine (Vol. x.
p. 43), and even after that Paper had been noticed in the
aforesaid Annals (Vol. xu. p. 462) ; that M. Neumann’s Memoir
in the Berlin TZransactions was not published until a later
period; that, therefore, there could be no question about prio-
rity of publication; and that, consequently, if it were to be
imagined for a moment that either author had borrowed from
the other, the presumption must necessarily be against M. Neu-
mann. With respect to M. Seebeck’s note, it would be enough
to state, that M. Neumann is not mentioned there at all; that
the principles there given by M. Seebeck are not adequate to
the general solution of the problem; and that such of them
as differ from those of Fresnel had been previously published
by Mr. Mac Cullagh. It was clear, therefore, that Mr. Mac
Cullagh owed nothing on the score of theory to anyone but
Fresnel. He had, indeed, made one alteration in his theory
as it originally stood; for he had at first rejected Fresnel’s law
of the vis viva, and had been obliged to restore it afterwards,
in order to account for certain experiments of M. Seebeck,
which M. Seebeck himself, from want of sufficient principles,
had not attempted to account for; but the real service which
M. Seebeck had rendered him, and for which he had frequently
acknowledged his obligations, was the communication of these
experiments, and not any suggestion of the law of vis viva,
which he knew well enough before. In all this, however, it
was plain that M. Neumann had no concern, unless he chose
to say that he had appropriated to himself Fresnel’s law of
the vis viva, that he had determined to regard it as the foun-
dation of his method (le fondement de sa méthode), and that
thenceforward no one else (however ignorant of such appro-
priation) could have any right to use it.

144°: Laws of Crystalline Reflexion.

Having thus endeavoured to prove his claim to priority of
publication, and to establish the independence of his own re-
searches, which was all that was necessary for self defence,
Mr. Mac Cullagh concluded by saying, that he would there
drop the argument, without discussing his claim to priority in
the abstract, as he had an objection to disputes of such a kind,
and did not wish to pursue them any farther than he was
compelled to do. But if anyone thought it worth while to
examine the merits of this second question, he would find
the circumstances relating to it very fully and clearly stated in
the last number of the Proceedings of the Academy,* and
-would thence be enabled to form a judgment for himself.

* Vol. 1. p. 217.

| aA RS)

XIV.—AN ESSAY TOWARDS A DYNAMICAL THEORY OF
CRYSTALLINE REFLEXION AND REFRACTION.

[ Transactions of the Royal Irish Academy, Vou. XxIe—Read December 9, 1839.]

Sxcr. I.—Intropucrory OBsERVATIONS.— EQUATION OF MoTION.

Nearty three years ago I communicated to this Academy* the
laws by which the vibrations of light appear to be governed in
their reflexion and refraction at the surfaces of crystals. These
laws—remarkable for their simplicity and elegance, as well as
for their agreement with exact experiments—I obtained from a
system of hypotheses which were opposed, in some respects, to
notions previously received, and were not bound together by any
known principles of mechanics, the only evidence of their truth
being the truth of the results to which they led. On that occa-
sion, however, I observed that the hypotheses were not indepen-
dent of each other; and soon afterwards I proved that the laws
of reflexion at the surface of a crystal are connected, in a very
singular way, with the laws of double refraction, or of propaga-
tion in its interior ; from which I was led to infer that ‘all these
laws and hypotheses have a common source in other and more
intimate laws which remain to be discovered ;” and that “the
next step in physical optics would probably lead to those higher
and more elementary principles by which the laws of reflexion

* In a Paper ‘‘On the Laws of Crystalline Reflexion and Refraction.’’ —TZrans-
actions of the Royal-Irish Academy, Vou. xvi. p. 31. (Supra, p. 87.)

L

146 On a Dynamical Lheory of

and the laws of propagation are linked together as parts of the
same system.’”* This step has since been made, and these anti-
cipations have been realised. In the present Paper I propose to
supply the link between the two sets of laws by means of a very
simple theory, depending on certain special assumptions, and
employing the usual methods of analytical dynamics.

In this theory, the two kinds of laws, being traced from a
common origin, are at once connected with each other and
severally explained ; and it may be observed, that the explana-
tion of each, as well as the source of their connexion, is now
made known for the first time. For though the laws of erys-
talline propagation have attracted much attention during the
period which has elapsed since they were discovered by Fresnel,t+
they have hitherto resisted every attempt that has been made to
account for them by dynamical reasonings; and the laws of re-
flexion, when recently discovered, were apparently still more
difficult to reach by such considerations. Nothing can be easier,
however, than the process by which both systems of laws are
now deduced from the same principles.

The assumptions on which the theory rests are these :—First,
that the density of the luminiferous ether is a constant quantity;
in which it is implied that this density is unchanged either by
the motions which produce light or by the presence of material
particles, so that it is the same within all bodies as in free
space, and remains the same during the most intense vibrations.
Second, that the vibrations in a plane-wave are rectilinear, and
that, while the plane of the wave moves parallel to itself, the
vibrations continue parallel to a fixed right line, the direction of
this right line and the direction of a normal to the wave being
functions of each other. This supposition holds in all known
crystals, except quartz, in which the vibrations are elliptical.

Concerning the peculiar constitution of the ether we know

* Ibid, p. 53, note. (Supra, p. 112.) The note here referred to was added
some time after the Paper itself was read.

+ These laws were published in his Memoir on Double Refraction—Mémoires
de V Institut, tom. vii. p. 46.

Crystalline Reflexion and Refraction. 147

nothing, and shall suppose nothing, except what is involved in
the foregoing assumptions. But with respect to its physical
condition generally, we shall admit, as is most natural, that a
vast number of ethereal particles are contained -in the differen-
tial element of volume; and, for the present, we shall consider
the mutual action of these particles to be sensible only at dis-
tances which are insensible when compared with the length of a
wave.

By putting together the assumptions we have made, it will
appear that when a system of plane waves disturbs the ether, the
vibrations are transversal, or parallel to the plane of the waves.
For all the particles situated in a plane parallel to the waves are
displaced, from their positions of rest, through equal spaces in
parallel directions ; and therefore if we conceive a closed surface
of any form, including any volume great or small, to be de-
scribed in the quiescent ether, and then all its points to partake
of the motion imparted by the waves, any slice cut out of that
volume, by a pair of planes parallel to the wave-plane and inde-
finitely near each other, can have nothing but its thickness
altered by the displacements; and since the assumed preserva-
tion of density requires that the volume of the slice should not
be altered, nor consequently its thickness, it follows that the
displacements must be in the plane of the slice, that is to say,
they must be parallel to the wave-plane. And conversely, when
this condition is fulfilled, it is obvious that the entire volume,
bounded by the arbitrary surface above described, will remain
constant during the motion, while the surface itself will always
contain within it the very same ethereal particles which it en-
closed in the state of rest; and all this will be accurately true,
no matter how great may be the magnitude of the displace-
ments.

Let x, y, = be the rectangular co-ordinates of a particle before
it is disturbed, and 7+, y+», 2+ its co-ordinates at the time ¢,
the displacements &, n, being functions of 2, y,z and ¢. Let
the ethereal density, which is the same in all media, be regarded
as unity, so that dxdydz may, at any instant, represent indif-

L2

148 On a Dynamical Theory of

ferently either the element of volume or of mass. Then the
equation of motion will be of the form

{if dadyds (= o& + 2 on + 2 a) = {\f dadydz8V, (1)

where V is some function depending on the mutual actions of
the particles. The integrals are to be extended over the whole
volume of the vibrating medium, or over all the media, if there
be more than one.

Setting out from this equation, which is the general formula
of dynamics applied to the case that we are considering, we per-
ceive that our chief difficulty will consist in the right determina-
tion of the function V ; for if that function were known, little
more would be necessary, in order to arrive at all the laws which
we are in search of, than to follow the rules of analytical me-
chanics, as they have been given by Lagrange. The determina-
tion of V will, of course, depend on the assumptions above stated
respecting the nature of the ethereal vibrations ; but, before we
proceed further, it seems advisable to introduce certain lemmas,

for the purpose of abridging this and the subsequent investiga-
tions.

Secr. IJ.—Lemmas.

Lemma I.—Let a right line making with three rectangular
axes the angles a, (3, y, be perpendicular to two other right lines
which make with the same axes the angles a’, 9’, y’ and a”, 3”, 7”
réspectively, and which are inclined to each other at an angle

denoted by 8; then it is easy to prove that
sin 8 cos a = cos 23’ cos y” — cos 3” cos y’,
sin 0 cos 3 = cos y’ cos a” — cos y” cosa’, (A)
sin 9 cos y = cos a’ cos (3” — cos a” cos 3’ ;

supposing the first right line to be prolonged in the proper direc-
tion from the origin, in order that the opposite members of any

Crystalline Reflexion and Refraction. 149

one of these equations may have the same sign, as well as the
same magnitude.

If the last two right lines be perpendicular to each other,
we have sin @ = 1, and the formule become

cos a = cos 3’ cos y” — cos 3” cos 7’,
cos 9 = cos y’ cos a” — cos y” cos a’, (B)

cos y = cos a’ cos 3” — cos a” cos f’;

but in this case the three right lines are perpendicular to each
other, and therefore we have, in like manner,

cos a’ = cos3” cos y — cos 3 cos y”,
cos 9 = cos y” cos a — cos y cos a”, (B’)

cos y’ = cos a” cos — cos a cos 3”;
and also
cos a” = cos 3 cos y’ — cos 3’ cos y,

cos 3” = cos y cos a’ — cos 7’ COs a, (B”)

cos y” = cos a cos 3’ — cosa’ cos f3.

The last three groups of formule will still be true, if we
suppose the first right line to make with the axes the angles
a,a,a’, the second the angles 3, 9’, 3”, and the third the
angles y, y’, y”

Lemma II.—Let &, n, ¢ denote, as before, the displacements
of a particle whose initial co-ordinates are x, y, ; and after
putting

hh hye Ry i
de dy ~~" da de’ =~ dy de

suppose the axes of co-ordinates, still remaining rectangular, to
have their directions changed in space, whereby the quantities
X, Y, Z will be changed into X’, Y’, Z’, answering to the new
co-ordinates 2’, y’,z’, and to the new displacements &’, n’, 2’ ; then
will the quantities X’, Y’, Z’ be connected with X, Y, Z by

150 On a Dynamical Theory of

the very same relations which connect the co-ordinates 2’, y/, ¢ .
with 2, y, s, or the displacements &, n’, 2’ with &, n, ¢.

That is to say, if the axes of z, y, s make with the axis of a
the angles a, 3, y, with the axis of x the angles a’, 9’, y’, and

with the axis of s’ the angles a”, 6”, y” respectively, we shall

have
X= X’ cosa+ Y’ cosa’ +Z’ cosa’,
Y=X’ cosB + Y’ cos’ + Z’ cos 3”, ; - (d)
£=X’' cosy + Y’ cosy’ + Z cosy’,
and

xX’ = X cosa + Y cos B + Z cos y,
Y’= X cosa’ + Y cos’ + Zeosy’, (D’)
Z = X cosa’ + Y cos” + Z cosy” ; é
just as we have, for example,
E=f cosa+y cosa +Z cosa’,
n = & cos 3+ 7 cos P+ Z cos”, (d)
=F cosy+n cosy +f cosy’,
and
a =2 cosa +y cos B +2 COS y,
y = cosa’ +y cos PB’ + 8 cosy’, (d’)
x =a cosa’ + y cos 3” + 8 cos y”.

For, the change of the independent variables z, y, s into
2, y, % gives us the equations

dy _dn det dn dy, dn de
dz da ds di ds. ds’ ds’
ae _ de de’ dh ay | de ie
dy da dy dy dy dz dy’

in the right-hand members of which we have to substitute the

Crystalline Reflexion and Refraction. 151

values of the differential coefficients obtained from (d) and (d’).
Thus we get

dy dz’ dr’ , ag’
== (F< os B + —, cos B ae ) 008 y

is (= cos 3 + cos a+” ° cos ”) cos y

+ Gs cos (3 + = cos (3 + = cos B”) cos y”,
Sale cos 7 + 2%, 008 + F008 7” ) 00s

+ a cos y + 06 7 +F cos ") 908

dé’ def dz’ 1
+ (Ge 008 9 + Sir 008 + a7 008 7" ") 008

and when we subtract these equations, attending to the formule
in Lemma I., we find

oF (2S Dr fu Sees
Cn af Ce Mae a dab a

i (= dn’
de’ ay) cos a” 9

or simply,
X =X’ cosa+ Y’ cosa’ +Z cosa’,

which is the first of formule (p). And in like manner the others
may be proved.

The same things will obviously hold with Poigoct to quanti-
ties derived from X, Y, Z in the same way that these are derived
from &, n, . That is, if we put

Re ag ey ign ae
‘ds ay’ ‘da dz’ «dy da’

and then suppose the axes of co-ordinates to be changed, the

152 On a Dynamical Theory of

formule for the transformation of the quantities X, Y, Z, will
be similar to those for the transformation of the co-ordinates
themselves. The like will be true of the quantities X,, Y,, Z,,
if we put

@Y/ ak ae a ae ae
OG dy 9 dan et dy ae

and so on successively.

It is to be observed that, in this Lemma, the displacement
is not limited by any restriction whatever. Hach of its com-
ponents may be any function of the co-ordinates. But the
displacements produced by a system of plane waves are re-
stricted by our definition of such waves; they must be the
same for all particles situated in the same wave plane. If
the waves be parallel, for instance, to the plane of «’, 4’, the
quantities &’, n’, 2’ will be independent of the co-ordinates 2’, y/,
and will be functions of z’ only. This consideration reduces
formule (p) to the following :

dy dé’

X=77 COS a — 7g 084s

Y= = cos 3 - = —, cos 9’, (EB)
dr’ dé’

Z == cosy — 77 cos 7’;

"in which it is remarkable that the normal displacement 2’ does
not appear. If é’ = 0, these formule become

, mf n

x= cosa, oS 7 082: Z=% 0087; (F).

or if »’ = 0, then we have

=— -& cosa’ v--5, cos, Z =~ 008 7. (F’)
Lemma III.—If, in an ellipsoid whose semiaxes are equal
to a, b, c, there be two rectangular diameters, one making with

Crystalline Reflexion and Refraction. 153

the semiaxes the angles a, 8, y, and the other the angles
a’, 3’, y’, such as to satisfy the condition

= A sB’ cosy cosy’
selene a Red sp CORR ROE Y) «I Hy, (a)

’
a” C

these diameters will be the axes of the ellipse in which their
plane intersects the ellipsoid.

For, the above condition expresses that either diameter is
parallel to the tangent plane at the extremity of the other ;
they are therefore conjugate diameters of the elliptic section ;
and hence, as they are at right angles to each other, they must
be its axes.

If the semiaxes of the ellipsoid be represented by - _ -
the equation of condition will become
@ cosa cosa’ + b* cos B cos 9’ + ¢ cos y cos y’ = 0. (c’)

Lemma IV.—Let s, s’ be the lengths of perpendiculars let
fall from the centre of an ellipsoid upon any two tangent
planes, and 7, 7’ the lengths of radii drawn to the respective
points of contact. Then putting w for the angle between the
directions of 7 and s’, and w’ for the angle between the directions
of 7’ and s, we shall have

rs8 COSw = 7's’ COS w’.

For if the semiaxes of the ellipsoid, having their lengths
denoted by a, 0, c, make with the direction of s the angles
a, B, y, and with that of s’ the angles a’, 9’, y’; with the
direction of 7 the angles a, (3, yo, and with that of » the
angles a, 61, y:, there will exist the relations

acosa=178 COSam, Ob cosB=rs cos, ¢ Cosy = 718008 yo,
@ cosa’ =1"s cosa, 6’ cos B’ = 7's’ cos, © cos y’ =7"s' cosy,
by one set of which the quantity

-@ cosa cosa’ + 0’ cos3 cos 3’ + & cosy cos y’

154 On a Dynamical Theory of
will be converted into |

78 (COS ap COS a’ + COS (3p cos SB’ + COS yo COs y’) = 18 COS w,

and by the other set into

7” s’ (cOS a, COS a + Cos (3; cos 3 + Cos y: Cos y) = 7” 8 COS ww’;

so that we shall get

78 COS w = 1" 8’ cos w = a’ cos a cosa’ + LU? cos B cos

+ecosycosy’. (H)
Corollary.— When the condition

a cos a cos a’ + 0° cos B cos (’ + ¢ cos y cos 7’ = 0 (1)

is satisfied, each of the angles w, w’ is a right angle. Let us
suppose, at the same time, that the direction of s is perpendi-
cular to that of s’. Then will the directions of s and r’ coin-
cide with the axes of the ellipse in which their plane intersects
the ellipsoid; for s is perpendicular to 7’ and parallel to the
tangent plane at its extremity. The directions of s’ and r, in
the same manner, will coincide with the axes of another elliptic
section.

Secor. IJI.—DerrerminatTIon oF THE FUNCTION ON WHICH THE
Morton DEPENDS. Princrpat Axzs or A OrysTAL.

We come now to investigate the particular form which
must be assigned to the function V, in order that the formula
(L) may represent the motions of the ethereal medium. For
this purpose conceive the plane of 2 y to be parallel to a
system of plane waves whose vibrations are entirely transversal
and parallel to the axis of y’, so that &’=0,2’=0. Imagine
an elementary parallelepiped dz’ dy’ dz’, having its edges parallel
to the axes of 2’, 7’, z’, to be described in the ether when at rest,
and then all its points to move according to the same law as the
ethereal particles which compose it. The faces of the parallele-
piped which are perpendicular to the edge ds’ will be shifted,
each in its own plane, in a direction parallel to the axis of y’;
but their displacements will be unequal, and will differ by dn’,

Crystalline Reflexion and Refraction. 155

so that the edges connecting their corresponding angles will no
longer be parallel to the axis - s’, but will be inclined to it at

an angle x whose tangent is —; os 7°
Now the function V can only depend upon the directions
of the axes of 2’, 7’, 3’ with respect to fixed lines in the crystal,
and upon the angle «, which measures the change of form pro-
duced in the parallelepiped by vibration. This is the most
general supposition which can be made concerning it. Since,
however, by our second assumption, any one of these directions,
suppose that of 2’, determines the other two, we may regard V
as depending on the angle « and on the direction of the axis of
z alone. But from the equations (F) it is manifest that the
angle « and the angles which the axis of « makes with the
fixed axes of x, y, s are all known when the quantities X, Y, Z
are known. Consequently V is a function of X, Y, Z.
Supposing the angle « to be very small, the quantities
X, Y, Z will also be very small; and if V be expanded
according to the powers of these quantities, we shall have

V=K+AX+BY+ 07+ DX*+ EY?+ F??
+ GYZ+ HXZ+IXY+ &e.,

the ae K, A, B, C, D, &e., being constant. But in the
state of equilibrium the value of 6 V ought to be nothing, in
whatever way the position of the system be varied; that is to
say, when the displacements &, yn, 2, and consequently the
quantities X, Y, Z, are supposed to vanish, the quantity

OV = AOX + BOY + C8Z + 2A2DXSX + &e.,

ought also to vanish independently of the variations 8£, Sn, 8Z,
or, which comes to the same thing, independently of 8X, SY,
6Z. Hence* we must have 4 = 0, B= 0, C= 0; and therefore,
if we neglect terms of the third and higher dimensions, we may
conclude that the variable part of V is a homogeneous function

*See the reasoning of Lagrange in an analogous case, Mécanique Analytique,
tom. I. p. 68.

156 On a Dynamical Theory of

of the second degree, containing, in its general form, the squares —
and products of X, Y, Z, with six constant coefficients.

Of these coefficients, the three which multiply the products
of the variables may always be made to vanish by changing the
directions of the axes of x, y,s. For this is a known property
of functions of the second degree, when the co-ordinates are the
variables; and we have shown, in Lemma II., that the quan-
tities X, Y, Z are transformed by the very same relations as
the co-ordinates themselves. Therefore, in every crystal there
exist three rectangular axes, with respect to which the function
V contains only the squares of X, Y, Z; and as it will presently
appear that the coefficients of the squares must all be negative,
in order that the velocity of propagation may never become ima-
ginary, we may consequently write, with reference to these axes,

V=-4(@7X*+ PY’ +2’), (2)
omitting the constant X as having no effect upon the motion.

The axes of co-ordinates, in this position, are the principal axes
of the crystal, and are commonly known by the name of awes of
elasticity. Thus the existence of these axes is proved without
any hypothesis respecting the arrangement of the particles of
the medium. The constants a, b, c are the three principal velo-
cities of propagation, as we shall see in. the next section.

Having arrived at the value of V, we may now take it for
the starting point of our theory, and dismiss the assumptions by
which we were conducted to it. Supposing, therefore, in the
first place, that a plane wave passes through a crystal, we shall
seek the laws of its motion from equations (1) and (2), which con-
tain everything that is necessary for the solution of the problem.
The laws of propagation, as they are called, will in this way be
deduced, and they will be found to agree exactly, so far as mag-
nitudes are concerned, with those discovered by Fresnel; but the
direction of the vibrations in a polarized ray will be different
from that assigned by him. In the second place, we shall in-
vestigate the conditions which are fulfilled when light passes
out of one medium into another, and we shall thus obtain the
laws of reflexion and refraction at the surface of a crystal.

Crystalline Reflexion and Refraction. 157

Secr. [IV.—Propacation oF Licut 1n A CrystatiizeD ME-
piuM.—Laws oF FRESNEL.—ALTERATION REQUIRED TO BE
MADE IN THEM.— WAvE-SURFACE, INDEX-SURFACE, AND
THEIR PROPERTIES.

The principal axes of the crystal being the axes of 2, y, ,
we have, by equation (2),

~ §V = @ X8X + BYSY + ZZ; (3)

or, by taking the variations from formule (c), and interchanging
the characteristics d and 6,

ay (Bn _ BS) ey (AE _ BE) 4 (dE _ Bn.
pore er 3) OY eae) 22 (ay ae

and if we substitute this value in equation (1), and then inte-
grate by parts the right-hand member, in order to get rid of the
differential coefficients of the variations, we shall obtain

a
{if dedyds (Fe oF +5 2 a+ 5 - 4)

= ff dyda (?Zén - 0? Yor) + ff dads (a? X8Z - &°Z8E)
+ [[ dady (0 YO — a X8n) (4)

+ [ dedyds (2 ak a) +(e 0X | oD
Ee

But as the variations 6&, sn, 8¢ are arbitrary and independent,
this equation cannot hold unless the double integrals, which re-
late to the limits of the system, reduce themselves to zero, leav-
ing the equality to subsist, independently of the variations,
between the triple integrals alone. Equating, therefore, the

158 On a Dynamical Theory of

coefficients of the corresponding variations in the triple integrals,
we get

@E dz ,,a¥
Gan ape ae
@y dX ,aZ
dt ed ds” de’ : (5)
2 _,dY_ dX
d@. de” dy’

which are the equations of propagation, giving the expression
for the accelerating force parallel to each axis of co-ordinates.
When there is a single medium extending indefinitely on all
sides, the conditions relative to the limits are of no importance,
and we have only to consider the equations (5), from which we
shall now deduce the laws by which a system of plane waves is
propagated. .
Supposing the waves to be parallel to the plane of a’ #/, the
displacements will be functions of x’ only; and if y~ be any
function of the displacements, we shall have, by formule (d’),

ay ded’ a, db a , bw,
de de ded OP dg ae OP a

so that the equations (5) may be written

PE dZ

oe = 008 8p” - By cos

an ., ,dZ

FT sig ay 008” cy 008 a” ;

fy en | , aX R
ap! Gg C8 * a’ 7 008 B :

and when we combine these with the following,

7 a? 2 2
- oF ae oe vos Bee

dé de dé ae °° »

Crystalline Reflexion and Refraction. 159

es a? Ne :
< = FF cos a + Gq 0088 + o* 008 7
Go ae » an » 46 ”
a i + Fa 008 B +e cosy ;

attending to the relations (8), (B’), we find

d. , adY , az ,
eG + & i cos 3’ + ¢ 7 8)

ot 2 cos a+ UT cos +o? S008 7,
ae
ae 7°

from which it appears that there is no accelerating force in the
direction of a normal* to the wave, and consequently no vibra-
tion in that direction. Introducing now the values of X, Y, Z
from formule (£), the first two of these equations become

ve 2 27 2 2 (Q/ 2 2-7 PE’
TE = (@ cos’ a’ + 6° cos’ (’ + ¢* cos Y) GA
— (a’ cos a cos a’ + 8 cos B cos fy’ + c* cos y cos vy ahs
(6)
Pf 2 age? 2 ang? ree As
ap 7 4 cos’ a + b? cos’ 8 + ¢ cos’y) Fas

7

&
— (a cos a cos a’ + 0’ cos 6 cos 3’ + c* cos y cos y’) 6S
B

But as the axes of 2’, 7/ are arbitrarily taken in the plane of 2’ 7/
we may subject their directions to the condition

a cosa cos a’ + 5? cos B cos 8’ + & cos y cos y’ = 0; = (7)

* In the ingenious, but altogether unsatisfactory theory, by which Fresnel has
endeavoured to account for his beautiful laws, the direction of the elastic force
brought ixzp.play by the displacement of the ethereal molecules is, in general, in-
clined to jective? of the wave. He supposes, however, that the force normal to
that pl. - t produce any appreciable effect, by reason of the great resistance

whic’ plane ffers to compression.—Mémoires de l’ Institut, tom. vii. p. 78.
lines

een

160 On a Dynamical Theory of

and then, if we put

8° = a’ cos’ a + 0? cos* 3 + c* cos y,

/ , b Al (8)
s?= a cos’ a’ + b?.cos* B’'+ c* cos’ y’,
the equations (6) will be reduced to the well-known form
Pe! ae dy’ dy
ae? ag de 7” ag (9)

This result shows that, when the directions of 2’ and »/’ fulfil
the condition (7), the vibrations & and 7’ are propagated inde-
pendently of each other, the former with the velocity of s’, the
latter with the velocity s. The vibrations must therefore be
parallel exclusively to one or other of these directions, else the
system of waves will split into two systems, one vibrating pa-
rallel to a, the other parallel to 7/.

When the plane of the wave is parallel to one of the prin-
cipal axes, it is easy to infer that the vibrations must be either
parallel or perpendicular to that axis; and that, in the latter
case, the velocity of propagation is constant, being equal to
a, b, or c, according as the wave is parallel to the axis of 2, y,
or s. These constants are therefore called the principal velo-
cities of propagation ; and we now perceive the reason of the
negative sign in equation (2); for if any of the terms in the
right-hand member of that equation were positive, the corre-
sponding velocity would be imaginary.

According to Fresnel, the wave which is propagated with
the velocity a has its vibrations not perpendicular to the axis
of z, but parallel to it; and itis to be observed that a difference
of the same character distinguishes his views, throughout, from
the results of the present theory. It will appear in fact, by
what immediately follows, that the equations (7), (8), (9), ex-
press exactly the laws of Fresnel, provided the quantities &’ and
7, in the equations (9), be interchanged. ‘To make these laws
agree with our theory, it is therefore necessary to ‘them in
one particular, and in one only ; it is necessary ti, ‘se that

Crystalline Reflexton and Refraction. 161

the direction of the vibrations is always perpendicular to that
assigned by Fresnel. And since, in order to make his views
agree with the phenomena, Fresnel was obliged to s8y that, in
tain plane are perpendicular to that plane, it is clear that, on the
present principles, we must come to a different conclusion, &0

say that the vibrations of a polarized ray are parallel to its plane

Conceive 22 ellipsoid with its centre at O, the common origia
of the co-ordinates ”; Ys %» a, Yr%3 and let its gemiaxes be pa-
rallel to % Y> * their lengths being equal M

ee
a

coincide with the axes of the ellipse in

lipsoid 5 and if the right line OR, meet- Fig. 20.
ing the ellipsoid in R, be the direction of gf, we have

_ @ costa + & cos’ B + & cosy»

or, by (8)s

so that OR is the reciprocal of the velocity with which the vi-
prations pat Ylel to yf are propagated. Thus we see that the
vibrations parallel to either gemiaxis of the elliptic section are
propagated with velocity which is measured by the reciprocal

of the other semiaxis-
Again, conceive a0 ellipsoid with its centre at O, and its

plane which cuts OR perpendicularly +n P, and draw the right
lines OP, PQ. Then the condition (7) is ‘dentical with that
M

162 On a Dynamical Theory of

marked (1) in the corollary to Lemma [V., it follows that Oy/
(if we so call the direction of y’) is perpendicular to OQ, and
also that Oy/ and OQ coincide with the axes of the elliptic sec-
tion made in this ellipsoid by the plane QOy’, just as Oy’ and
OR coincide with the axes of the section ROy’ in the first ellip-
soid. The plane QOR is therefore perpendicular to Ov/ and to
the plane of the wave. Moreover, we have

(OP)? = a cos’a + b cos’B + & cos*y = 8°,

so that OP is the reciprocal of OR, and is equal to the velocity
s with which the wave is propagated when its vibrations are
parallel to Oy’.

Now let the figure 7OSM be equal in all respects to QOPR,
but in a position perpendicular to it, so that if QOPR were
turned round in its own plane through a right angle, the point
O being fixed, the points Q, P, R would fall upon 7, S, If re-
spectively ; and supposing the wave-plane ROv’ to take various
positions passing through O, imagine a construction similar to
the preceding one to be always made by means of the two ellip-
soids. Then while the points R and Q describe the ellipsoids,
the points M and 7 describe two biaxal* surfaces reciprocal to
each other, the latter surface being touchedt in the-point 7’ by
a plane which cuts OM perpendicularly in S. But this plane
is parallel to the central wave-plane ROy’, and distant from it
by an interval OS (= OP) which represents the velocity of the
wave; and as the surface whose tangent planes possess this
property is, by definition, the wave-surface of the crystal, it is
obvious that the point 7’ describes the wave-surface. The radius
OT, drawn to the point of contact, is then, by the theory of
waves, the direction of the ray which belongs to the wave ROv/,
and the length O7' represents the velocity of light along the
ray. As to the surface described by the point WU, it is that

* See Transactions of the Royal Irish Academy, Vou. xvit. p. 244 (supra, p. 24).
+ Ibid. Vou. xvi. p. 6 (supra, p. 4).

Crystalline Reflexion and Refraction. 163

which I have called the swrface of indices, or the index-surface,*
because its radius OW, being the reciprocal of OS, represents
the index of refraction, or the ratio of the sine of the angle of
incidence to the sine of the angle of refraction, when the wave
ROy’, to which OW is perpendicular, is supposed to have passed
into the crystal out of an ordinary medium in which the velocity
of propagation is unity. The angles of incidence and refrac-
tion are understood to be the angles which the incident and
refracted waves respectively make with the refracting surface
of the crystal.

The wave-surface and the index-surface have the same geo-
metrical properties since they are both biaxal surfaces. Let us
consider the former, which is generated by the ellipsoid whose
semiaxes are a, b, c; and let us conceive this ellipsoid to be in-
tersected by a concentric sphere of which the radius is 7. Then
the equations of the ellipsoid and the sphere being respectively

a Owe ©

1 P+y+2
"geil go a Sea il Rage

y*

1,

we get, by subtracting the one from the other,

ees ee ah eed
e(e-z)t¢ eos) G-z)-% ao

for the equation of the cone A which has its vertex at O, and
passes through the curve of intersection. If OQ be equal to 7,
it will be a side of this cone; and a plane touching the cone
along OQ will make in the ellipsoid a section of which OQ will
be a semiaxis; so that O7' will be perpendicular to that plane,
and equal in length tor. Therefore, as OQ describes the cone
A, the right line OT describes another cone B reciprocal to A,
and the point 7’ describes the curve in which the wave-surface is
intersected by the sphere above mentioned ; this curve being a
spherical ellipse, reciprocal to that which the point Q describes on

* See Transactions of the Royal Irish Academy, Vou. xvr1. p. 38 (supra, p. 96).
I had previously called it the surface of refraction, Vou. xvi. p.-252 (supra, p. 36).
M2

164 On a Dynamical Theory of

the surface of the ellipsoid. The equation of the cone B is
found from that of A, by changing the coefficients of the squares
of the variables into their reciprocals, and is therefore

ax? by? cs?
P-e@ Pp-BP P-e

= 0, (11)

which, of course, is also the equation of the wave-surface, if 7 be

supposed to be the radius drawn from O to the point whose

co-ordinates are x, y,s. Combining this equation with that of

the sphere, we have
x y° 3°

ra a r — Bb? ¥ P—e

= I, (12)

which represents a hyperboloid passing through the common
intersection of the sphere, the cone B, and the wave-surface.

Since the differences between the coefficients of the squares
of the variables in the equation (10) are the same as the corre-
sponding differences in the equation of the ellipsoid, the cone A
has its planes of circular section coincident with those of the
ellipsoid. The cone B, being reciprocal to A, has therefore its”
focal lines perpendicular to the circular sections of the ellipsoid.
These focal lines are consequently the nodal diameters* of the
wave-surface, that is, the diameters which pass through the
points where the two sheets of that surface intersect each
other.

If the direction of OT cut the other sheet of the wave-
surface in 7”, and if two radii of constant lengths, equal to OT
and OT” respectively, revolve within the surface, the cones B
and B’ described by these radii will intersect each other at right
angles, since they have the same focal lines. And supposing
the axis of y to be the mean axis of the ellipsoid, so that the
nodal diameters lie in the plane of wz, the axis of x will lie
within one of the cones, as B, and the axis of s within the other
cone B’. Now the angle contained by the two sides of either

* See Transactions of the Royal Irish Academy, Vou. xvut. p. 247 (supra, p. 29).

Crystalline Reflexion and Refraction. 165

cone, which lie in the plane of xz, is given by the angles @ and
& which the direction of the right line O77’ makes with the
nodal diameters; because the angles which any side of a cone
makes with its focal lines have a constant sum, or a constant
difference, according to the way in which they are reckoned.
But if the angles @ and @’ be reckoned (as they may be) so that
their sum shall be equal to the angle contained by the two sides
of the cone B which are in the plane of wz, their difference will
be equal to the angle contained by the two sides of the cone B’
which are in the same plane’; the contained angle, in each case,
being that which is bisected by the axis of z. Therefore, the
lengths OT and OT’, which we denote by r and 7’, are equal to
two radii of the ellipse whose equation is
2 2
3 = att
these radii making with the axis of z the angles $(@ + 0’) and
4(0 — 0’) respectively. Hence

1 _.sin?} (0+) cos (0+0) , (- ¥ =)
ie par bs Be ae
¢

+
a a? c

a et
(18)
1 sin®} (0-0) cos*h- (9-6) (2 =)
fe ae 2 + 2 =2z\ ata) -
r a c e ¢
Pe ised
-+(& - 5) cos (0-0

These formule give the two velocities of propagation along a
ray which makes the angles 0, 6’ with the nodal diameters.
Subtracting them, we have

an yan (gz) sin O sing. (14)

rf a

All the preceding equations, relative to the wayve-surface,

166 On a Dynamical Theory of

may be transferred to the index-surface, by changing the quan-
tities a, b, c into their reciprocals. For example, if the normal
to a wave make the angles @, 0, with the nodal diameters of
the index-surface, the formule (13) give

s=1(a +c’) -1(a@ -c’) cos (0+ A),
(15)
s?=1(a +) —4(a@ - c) cos (0 — 4);
observing that s and s’, the two normal velocities of propaga-
tion, are the reciprocals of the radii of this surface which coin-
cide with the wave-normal. Subtracting these expressions, we
get
8° — §? = (a — c*) sin ® sin hy. (16)

As the position of the tangent plane, at any point 7 of a
biaxal surface, depends on the position of the axes of the section
Q0,/ made in the generating ellipsoid by a plane perpendicular
to OT, it is obvious that when this section is a circle, that is,
when the point 7’is a node of the surface, the position of the
tangent plane is indeterminate, like that of the axes of the
section ; and it is easy to show that the cone which that plane
touches in all its positions is of the second order. Again, when
the section RO, of the reciprocal ellipsoid is a circle, the right
line OS is given both in position and length; and the tangent
plane, which cuts OS in S, is fixed; but the point of contact 7
is not fixed, since the semiaxis OR, to which the right line ST
is parallel, may be any radius of the circle ROy’. In this case,
the point 7 describes a curve ,in the tangent plane, and this
curve is found to be a circle. But both these cases have been
fully discussed elsewhere.*

* See Zransactions of the Royal Irish Academy, Vou. xvi. pp. 245, 260 (supra,
pp. 26-7, 49-51).

Crystalline Reflexion and Refraction, 167

Srcr. V.—ConDITIONS TO BE SATISFIED WHEN LIGHT PASSES
OUT OF ONE MEDIUM INTO ANOTHER.—REMARKABLE Crrcum-
STANCES CONNECTED WITH THEM.—RELATIONS AMONG THE
TRANSVERSALS OF THE INCIDENT, REFLECTED, AND RE-
FRACTED Rays.

Now let light pass out of one medium into another—sup-
pose out of an ordinary into a doubly-refracting medium ; and
taking the origin of rectangular co-ordinates x, yo, % at a point
O on the surface which separates the two media, let this surface
be the plane of ay. Then if the components of the displace-
ment of a particle whose initial co-ordinates are x, %, % be
denoted by &’o, n’o, 2’) when the particle is in the first medium,
and by &”, mo’, Go” when it is in the second, *he equation (1),
adapted to the present case, will be

227
{hf da dy d% Ce 0&9 + = 8n/o + Ae 82" )
de” dno

“A ” ad’ “9 a"
+ edge diy (“Fe 86" +H? By’, + SE 8)

— = [If dao dyo dz. dV’ + Sif dary dyo dz SV” ; (17)

wherein oV’ and dV” are the respective values of 8V for the two
. media, which are conceived to extend indefinitely on each side
of the plane of x, y,; that plane being an upper limit of the
integrations relative to one medium, and a lower limit of the
integrations relative to the other. Each medium is conceived
to be oceupied by systems of plane waves—the first by incident
and reflected waves, the second by refracted waves; and, except’
where they are bounded by the plane of a %, these waves are

regarded as unlimited in extent.
For the ordinary medium, if we put
ae dn’ de’ / ag’ dé’, » dé, dy’, :
= a se Dagens A

ds, dy,’ © de, di,” sd aly ld

168 On a Dynamical Theory of

and suppose the velocity of propagation to be unity, we have*

_ gy yr (Bn'o ASE\ «, (d82', ddEs\ . , (ddE AdBn'0\.
av -x4(e- Es), yr, (Se Te) +20 (G-Se)

For the crystallized medium, if its principal axes be those of
2, y, &, the value of 6V” will be the same as that of SV in for-
mula (3); but instead of the variations of &, n, Z, we must use
those of &”,, 1”, 2”. Denoting the cosines of the angles which
the principal axes respectively make with the axis of x, by J, m,n;
with the axis of y, by 7’, m’, n’; with the axis of z, by 7”, m”, n”;
and putting

ne dR > ig © dat) = AB’ ° dy day’
gf A BE
o¥"o= dit, d&”

ddy’, x ds2”,
dzo dy.’

- 08") dy"
aa oe 5 Yi Bs ;
: dyo dite

we have
OX = 6X", + VEY", + O20

oY - Moxy = ms Be + mL" oy
6Z =n 8X" + WEY") + n’OZ".

These expressions for 6X, dY, dZ having been written in
formula (3), the resulting value of 8 V”, as well as the above value
of dV’, is to be substituted in the equation (17), and then the
right-hand member of that equation is to be integrated by parts,
in order to get rid of the differential coefficients of the varia-

tions. When this operation is performed, the triple integrals -

_ on one side of the equation will be equal to those on the other ;
and by equating the coefficients of the corresponding variations

* It is assumed here, and in what follows, that when there are two or more
coexisting waves in a given medium, the form of the function V is the same as for
a single wave, provided the displacements which enter into the function be the re-
sultants of the displacements due to each wave separately. This, however, ought
evidently to be the case, in order that the principle of the superposition of vibrations
may hold good.

Crystalline Reflexion and Refraction. 169

in each medium, we should get the laws of propagation in each,
But we are not now considering these laws, and we need only
attend to the double integrals produced by the operation afore-
said. The double integrals are together equal to zero; but we
are concerned only with that part of them which relates to the
common limit of the media, the plane of x y,; and this part
must be separately equal to zero, since the conditions to be ful-
filled at the plane of x, y, are independent of anything that
might take place at other limiting surfaces, if such were sup-
posed to exist. Collecting therefore the terms produced by in-
tegrating with respect to z,, and observing that a negative sign
must be interposed between those which belong to different
media, we get

[de dy, (Y 0069 — Xs dn’) — ff dx, dy, (Q0&”, - P8y",) = 9, (18)
where
P=@1X+0mV+enZ, Q=AVX+RmV+en'Z, (19)

In each of these equations it is understood that z,=0. But
when z, = 0, we have obviously

e5 = <a No = es (20)
oe’, = 08 on’, = én”, ;

and therefore

so that the equation (18) becomes
ff da, dy, {(Y", — Q) d&, — (X', - P) dn’,} = 9,

which, as the variations d&’, and dy’, are arbitrary and indepen-
dent, is equivalent to the two equations

X= ?P. . =.0, (21)

Thus, to find the relations which subsist among the vibra-
tions incident, reflected, and refracted, at the common surface
of two media, we have four conditions, expressed by the equa-
tions (20) and (21); and these conditions are sufficient to deter-
mine the reflected and refracted vibrations, when the incident

170 On a Dynamical Theory of

vibration is given. But though, by the nature of the question,
four conditions only are required for its solution, there remains
another condition which ought to be satisfied; for we ought
evidently to have

2,=2", when g, = 0. (22)

This condition is apparently independent of the rest; but it
cannot really be so, if the preceding theory is consistent with
itself. We shall accordingly see, in what follows, that the last
condition is included in the other four; which is a remarkable
circumstance, and a singular confirmation of the theory.*

As the incident and reflected waves coexist in the first me-
dium, and two sets of refracted waves in the second, the resolved
displacements, and all the quantities which depend upon them,
are composed of two parts, due to the coexisting waves. Let
the point O be, for each set of waves, the origin of a system of
rectangular co-ordinates, which we shall call ~, y:, %, for the in- |
cident, and #;, 7/1, 2: for the reflected wave, the axes of s, and z
being perpendicular to the respective waves, and their positive
directions being those of propagation. Let the displacements in
these waves be parallel to y,, 7/:, and be denoted by m, 1, re-
spectively. Then if the axes of a, y,, s, make with the axis of
a, the angles a, (1, y:, and with the axis of x’, the angles
ay, 31, y's, we have, by the formule (F),

dy; a
oosay; 7 Y").= —

dn’ :
cos (3; + a cos 3';. (23)
Again, let the co-ordinates #2, #2, #. have reference to one set of
refracted waves, and 2’, y's, 3, to the other, the axes of x
and s’, being perpendicular to the respective waves, and their
positive directions being those in which the waves are propa-

dn,
Sed cosa, + 77

* In considering the question of reflexion at the common surface of two ordi-
nary media (Mémoires de V Institut, tom. ix. p. 396), Fresnel assumes the conditions
(20) ; but his other suppositions violate the condition (22). In fact, this last con-
dition is inconsistent with the supposition that, in a polarized ray, the direction of
the vibrations is perpendicular to the plane of polarization. See the Transactions of
the Royal Irish Academy, Vou. xvimt. p. 32 (supra, p. 88).

Crystalline Reflexion and Refraction. 171

gated. Suppose the displacements to be parallel to y., y’2, and
to be denoted by n., n’, respectively. Then if the axes of z, y, z
make with the axis of x, the angles ay), By), ya), and with the
axis of wv’, the angles =~ 2’ ()) y@)) we have, by the formule(/’),

,

d
x. * C08 aa) + — ih,

cos
we dz's a @)»

dng dy’,
Fe ms cos (3 2) vey cos 3’ w),

dn. dys
Z= 7 008 ya) + COS y’ (2) 3

and thence, by the relations (19),

d.
P= a (a7 co8 a) + 6’m cos Bi) + cn COS y(2))
2

,

dis
+ i, = (al? cos a’) + Bm cos Bia) + en C08 y’@)),
dn. oy 2,,/ ; 2,0
Q= 2a (a°V’ cos ay) + b’m’ cos By + cn’ cos y~))
dh'zy 27 2
tal ((a°’ cos a’ (2) + bm’ cos (2) + ¢?n’ cos y’@)).

Suppose the ellipsoid which generates the wave-surface of
the second medium to have its centre at O, and to be touched in
the points Q and Q’ by two planes which cut the axes of 2, and
a, perpendicularly in the points P and P’; the lengths OP and
OP’ being expressed by s, s’, and the lengths OQ and OQ’ by
r, 7. Let the axes of 2, y, % make with the direction of OQ
the angles az, 92, y2, and with the direction of OQ’ the angles
a’, 3x, y’2 Then, from the equation (H) in;Lemma IV., it is
manifest that

a’l C08 aw) + b’m cos Be) + €’n COS ya) = 7°8 COS az,
al cos a’ (2) + b’m cos (3’2) + cn COS y’~) = 7's" COs a’2,
a’l’ cos aw) + b’m’ cos By) + cn’ COs ya) = 78 C08 Ba,

al’ cos a’ (2) + b’m’ cos 9’) + en’ COB y’ (a) = 2"s" cos (3's.

172 On a Dynamical Theory of

Hence,
dnz dy.
P=— 5 18 COS az + WH, r’s’ cosa’s,
(24)
ad. 7,
Q-& = 78 008 + rs cos (3's.

The quantities s, s’ are, as appears by the last section, the
normal velocities with which the two sets of refracted waves are
propagated. The velocity with which the incident and reflected
waves are propagated is taken as unity. Therefore, if 7, 7/1 be
the transversals, or amplitudes of vibration in the incident and
reflected waves, and r, 7’, the transversals of the refracted waves,
the lengths of the latter waves being denoted by Az, A’2, and the
length of an incident or reflected wave by A,, and if we put

, ee eet
= ost v4),

i= (t-mty), — $

Qa Qa
2 = —— (st - 2+), dr= > (st-#2+v2),
rz r2
where 1;, v’;, v2, v2 are constants, and 7 is the ratio of the cir-
cumference to the diameter of a circle, we may write
Mm =7 COS pis ns = v1 cos $1, N2 = Tz COS 2s N2 = rT’. cos re (26)

By means of these values the formule (23) and (24) become

ae, ; :
X= (7, cos a, SiN g + 7; COSa’, Sing’),
1

Se = (7; cos 3, sin p: + 7; cos 9% sin 9’),
(27)

2

Y be A
rs rs
P=27 (Fz T2 COS a: sin 2 + XY, t's 008 a sin g's),

7s

Q = 27r & 72 C08 [33 SiN pd, + v7? cos 8’, sin 9’ »)

The angles ¢:, $1, ¢2, ¢'2 are the ee of vibration in the

Crystalline Reflexion and Refraction. 173

different waves at the time ¢. To see how they depend on the
co-ordinates x, %, %, conceive the axis of z, to be directed from
O towards the interior of the second medium, and the axis of z,
to lie in the plane of incidence, so that the positive directions of
%1, 82 2 may lie within the angle made by the positive direc-
tions of z, and z,, while the positive direction of sj lies within
the angle made by the positive direction of « and the negative
direction of z,. Let 7, be the angle of incidence, and %,, 7’, the
angles of refraction; then

E31 > Xo sin a, + Zo cos hy zy = Hoy sin dy a Zo cos lhy
(28)

2 = % sin to + Z COS tay Bo = sin t's + 2 COS gts

These values are to be written in the expressions (25). They
show that the phases, and therefore the displacements, are in-
dependent of %.

Since the conditions relative to the plane of x y) must hold
at every instant of time, and for every point of that plane, the
co-efficients of ¢, as well as those of 2, in the values of the
different phases, must be identical; so that we must have

, eee MAS sing; sing sind’,
Ai a de el “Ka ae. Az ie nN (29)
Therefore, when x = 0, the supposition
Lt Nome vy; pei fe vs (30)

renders the phases identical, independently of ¢ and a. And,
from the form of the equations of condition, it is easy to see
that this supposition is necessary; because the equations (20),
when the values (26) are substituted in them, contain only the
cosines of the phases; and the equations (21), when the values
(27) are substituted in them, contain only the sines of the

174 On a Dynamical Theory of

phases. Making the latter substitution, and attending to the
relations just mentioned, we find

7, COS a; + 7’; COS a), = 1'T2 COS ag + 1°72 COS a2,
(31)

7, cos 3; + 71 cos 3’; = rr, COS B2 + 77's cos 32.

In these equations, the angles by whose cosines each trans-
versal is multiplied are the angles which a plane, passing
through the directions of that transversal and of the corre-
sponding ray, makes with the planes of y % and % %. This
is evident with regard to the incident and reflected rays. And
if we refer to the diagram in the preceding section, it will also
be evident with regard to the refracted rays; for OQ is perpen-
dicular to the transversal 7., and to the right line O7, which is
the direction of the corresponding ray.

Taking O for the point of incidence, let right lines proceed-
ing from it represent the different rays; and let the length of
each ray, measured from O in the direction of propagation, be
assumed proportional to the velocity with which the light is
propagated along it. Through the extremity of each ray con-
ceive its transversal to be drawn, and let the transversals so
drawn have their moments taken, with respect to the point O,
as if they represented forces applied to a rigid body. The
length of the incident or reflected ray being considered as
unity, the lengths of the refracted rays (as appears by the
last Section) are r and 7’ respectively. Hence, as each trans-
versal is perpendicular to its ray, the moments of the incident
and reflected transversals are proportional to 7, 71, and the
moments of the refracted transversals to rr, r’r’, respectively.
The equations (31) therefore signify that when the moments
are projected, either upon the plane of y %, or upon the plane
of a %, the total projected moments are the same for the two
media; or that, if the transversals themselves be projected on
either of these planes, the moments of the projections of the
incident and reflected transversals are together equal to the
moments of the projections of the refracted transversals.

Crystalline Reflexion and Refraction. 175

But the second of the equations (31) has another signifi-
cation. For if the transversals applied at the extremities of
the refracted ray8 be projected on the plane of a %, which is
the plane of incidence, and contains the axes of z, and z’,, the
projections will be perpendicular to these axes, since the trans-
versals themselves are perpendicular to them; and the distances
of the projections from the point O will be proportional to s
and s’, or, by the relations (29), to sin ¢ and sin 7’,; so that
if 0, and 6’, be the angles which the transversals make with
the plane of incidence, the moments of the projections will
be represented by 72 cos 0, sin 7, and 7’, cos 0, sin 7. At the
same time, if 0, and 6’, be the angles which the incident and re-
flected transversals make with the plane of incidence, the mo-
ments of the corresponding projections of these transversals will
evidently be represented by 7; cos 6; sin ¢,, and — 7’; cos 0, sin 4 ;
the latter quantity being taken with a negative sign, because
the extremity of the reflected ray, where the transversal 7’;
is applied, lies in the first medium, while the extremities of
the incident and refracted rays lie in the second, and it is
supposed that when any of the angles 0, 61, 02, 0’, is zero,
the direction of the corresponding transversal makes an acute
- angle with the axis of ~. Hence we have

7, 008 9; sin 2; — 7’; cos 0; sin 7; = 72 cos O, sin 7, + 7’, cos M2 sin 72;

an equation which expresses that if each transversal be pro- |
jected upon the axis of x, the sum of the projections of the
incident and reflected transversals will be equal to the sum
of the projections of the refracted transversals. Therefore,
since the phases of the different vibrations are identical when
% = 0, the condition (22) is fulfilled, as it ought to be.

On account of this identity of phases, it follows from the
conditions (20) and (22), that if the transversals be drawn
through the point O, and those which belong to each medium
be compounded like forces acting at a point, their resultants
will be the same; that is, the resultant of the incident and

176 On a Dynamical Theory of

reflected transversals will be the same as the resultant of the
refracted transversals. *

Hence, recollecting what has been proved respecting the
moments of the transversals applied at the extremities of the
rays, we have the following theorem:

Supposing the length of each ray, measured from the point
of incidence and in the direction of propagation, to be taken
proportional to the velocity with which the light is propagated
along it, and its transversal to be drawn through the extremity
of this length, the incident and reflected transversals having
their proper directions, but the refracted transversals having
their directions reversed; if all the transversals so drawn be
compounded like forces applied to a rigid body, their resultant
will be a couple, lying in a plane parallel to the plane which
separates the two media.

This theorem affords a complete solution of the question
of reflexion and refraction.* Expressed analytically it gives
five equations, of which four are independent.

To apply the preceding results to a simple case, suppose
the second medium, as well as the first, to be an ordinary one.
We have then only one refracted ray, and one refracted trans-
versal 72.

1°. When the incident ray is polarized in the plane of inci-
dence, the transversals are all in that plane; and as they are
perpendicular to the rays, and the refracted transversal is the
resultant of the other two, we have evidently

, sin (i; — #2) © sin 2i;
rae sin (+ 2)” "oT gin (i) +)

(32)

2°. When the incident ray is polarized perpendicularly to

* The same theorem applies to the other case of reflexion and refraction, when
a ray which has entered the crystal emerges from it into an ordinary medium,
undergoing double reflexion at the surface where it emerges. In fact, the con-
ditions (20) and (21) hold good whether the ordinary medium is the first or the
second; and in the latter case, as well as in the former, it may be shown that
the condition (22) is fulfilled, and that the theorem above mentioned is true.

Cryst Reflexion and Refraction. 177

have, by Lepcidence, the transversals are all perpendicular

s Taking 2sin 4, to represent the length of the

©08 a:= reflected ray, the proportional length of the re-

ay is 2sin %, and the projections of these lengths on

5 plane of y% are 2 sin i, cos i, and 2 sin #, cos #2, or sin 2i,
sin 27,. The transversals applied at the extremities of the
rays are not altered by being projected on the plane of % =;
therefore the moments of the incident, reflected, and refracted
_ transversals, projected on this plane, are represented by the
quantities r, sin 27, — 7, sin 2%, and r. sin 2i respectively.
_ Egquating the last moment to the sum of the other two, and
the refracted transversal to the sum of the other two trans-

versals, we get

(rn = 71) sin 2%, = T2 sin Ding 1s 4G Tv; = T2;
and thence
"a tan (t, = iz) oe sin 2%,
ee Te tan (i, + i,)’ a gf BR (i, +4) cos (i, — i) (33)

This case has been considered by Fresnel. The relative
magnitudes of the incident and refiected transversals, as given
by him, are in accordance* with the formule (32) and (33);
but ,with respect to the refracted transversals, his results do
not agree with the formule.

Secr. VI.—PreservaTion oF Vis Viva—THEOREM OF THE
Potar Pranze—Concivsion.

Returning to the general question, if we resolve the trans-
versals parallel to the axes of ~, ¥, %, and equate the sums
of the parallel components in one medium to the corresponding

* There is, however, a difference as to the relative directions of the incident
and reflected transversals. When the second medium is the denser, and the inci-
dence is perpendicular, these transversals, according to the present theory, have
the same direction, but according to Fresnel they have opposite directions.

N

\
\

178 On a Dynamical T; heorye, |

\
sums in the other, we get the three conditions tant of the
: \

(r, cos 0, + 7’, cos 0’) cos 7; = 72 cos A, cos 7, +72 COS Orn 5 te

Ti sin 0, + ry sin 0, =T2 sin 6, + T's sin %., > the

(7, cos 0, —7’, cos 6’;) sin 7, =72 cos 0, sin 7, + 7’, cos 0’, sin 72.

A fourth condition is supplied by the first of the equations (31),
in which equation we have to write

Cos a; = sin 4, cos 4, cos a’; = — sin 6; cos %,

and to substitute similar expressions for cos a2, COS a’s.

The right line OQ is perpendicular to the transversal 7, and
to the ray OT. The cosines of the angles’a:, (2, y, may therefore
be found by means of the cosines of the angles which the trans-
versal and the ray make with the axes of 2%, Yo, &o.

The cosines of the angles which the transversal r, makes
with these axes are respectively

cos 8, cos 2, sin @., — cos @, sin %.

As the plane which passes through the ray and the wave-
normal OS is perpendicular to the transversal 7, this plane
makes with the plane of incidence an angle equal to 90° + @, or
90° — @,. Let a sphere, having its centre at O, be intersected in
the points S,, Z, by the right lines OS, OT, and in the points
Xo, Yo, Z by the axes of x, yo, 2; and conceive the points Z)
and Y, to be at the same side of the plane ~, z,, the spherical
angle 7,S,X, being 90° + @,, and the spherical angle 7,S,Z,
being 90° -— 0,. Let <« be the angle which the ray makes with
the wave-normal. Then, the angles which the ray makes with
the axes of co-ordinates being measured by the ares T,X,, T, Yo,
T,Z,, the cosines of these angles respectively are a

sin 7, cos e — sin 9, cos % sin ¢, cos @, sin ¢,
COS 72 COS € + Sin 9, sin % sin «.

Hence, as the transversal is at right angles to the ray, we

Crystalline Reflexion and Refraction. 179

have, by Lemma I.,
COS a2 = sin 7, sin e+ sin 8, cos 7%, cose, cos (3, =— cos 8, cose,

COS ‘Y2 = COS #, sin ¢ — sin ; sin #, Cos «. (35)
In like manner, putting <’ for the angle which the other re-

fracted ray makes with its wave-normal, we have

COs a’, = sin ?’, sin « +sin #’,cos?’,cos«, cos 9’, =— cos 0’, cos &,

cos y’s = cos 7”, sin « — sin 9’, sin 72 cos &’. (36)
If we substitute, in the first of the equations (31), the values

just given for cos a2, cos a’, along with the above values of
COS a;, COS a’,, and attend to the relations
Y COS € = 8. r cose = 8,
(37)
sin 7, = $ sin %, sin 7’, = s sin 4,
we find, after multiplying all the terms of the equation by
sin 7, :
(71 sin 6, = rs sin %,) sin ty cos On = T2 (sin 0. sin tz COS te (
38
+ sin’7, tan «) + 7’, (sin @, sin 7’, cos 7, + sin??’, tan ¢).

Joining this equation to the equations (34) we have all the con-
ditions that are necessary for the solution of the question.
Multiplying the first of the equations (34) by the third, also
the second of these equations by the equation (38), adding the
products together, and then dividing by sin 7, we obtain

pa(t? — 71?) = pote + wer’? + Mrr’r, (39)
where we have put
fi = COS%, pe, =8 (cos % + sin @, sin ¢, tan «),
b's = 8 (cos, + sin @, sin 7, tan &),
Hf sin 7; = sin (2 + #’2) {cos 0, cos 6’, + sin 0, sin 0’, cos (i, — 7’) }

+ sin &, sin??, tan e + sin 0, sins’, tan ¢’.
N2

180 On a Dynamical Theory of

The last expression may be put under the form y
MM sin , sin (% — 72)

= sin*?,{cos 0, cos 8’, + sin @, sin 9’, cos (%, — 2) (40)

+ sin 0, sin (ts me i) tan &}

—sin’?’,{cos @, cos 0’, + sin 0, sin 4’, cos (t; — 72)

~ sin 0, sin (is aes i's) tan e'}.
Let the axes of x, 4%, %) make with the direction of OP the
angles a,, B,, y,, and with the direction of OP’ the angles —
a’, 3, y¥, The cosines of these angles may be found from

the expressions (35) and (36) by supposing « and «’ to vanish.

Therefore

cosa,,=sin @, cost, cos,,=—cos 6, cosy, =-—sin @, sin i, (41)
11

cos a’,,=sin 0’, cos 72’, cos3’,,=—cos 0, cosy’,,=—sin #,sin?’.

If w be the angle which OQ makes with OP’, and w’ the
angle which OQ’ makes with OP, so that

COS w = COS az COS a’, + CoS P, cos P’,, + COS 2 COS Y’,,5
COS w’ = COS a’; COS a,, + COS [3’2 Cos [3,, + COS y’2 COS Y,,5
we find, by the formule (35), (36), (41),
cos w = 008 €{cos 8, cos 6’, + sin 0, sin 6’, cos (i, — #2)
: + sin 0, sin (t, = to) tan e},
cos w’ = 008 «’ {cos 8, cos 0’, + sin 8, sin 6’, cos (i, — 72)
— sin 0, sin (¢, — ¢’,) tan «’}.

Hence, observing the relations (37), we see that the right-hand
member of the equation (40) is equal to the quantity |

sin’? (1s cos w — 7's’ cos w’).

Crystalline Reflexion and Refraction. 181

But, by the property of the ellipsoid (Lemma IV.), this quantity
_is zero;* therefore M = 0, and the equation (39) becomes

fat” = pa vy? + pate + wets. (42)

On each ray let a length, representing the velocity with
which the light is propagated along it, be measured, as before,
from the point O. The distances of the plane of x y from the
extremities of these lengths will be proportional to the coeffi-
cients of the squares of the transversals in the preceding equa-
tion. For if we take, on the incident or reflected ray, a length
equal to unity, its projection on the axis of 2 will be cos 4; or pn;
and if, on the refracted ray OT, we take a length equal to 7, its
projection on the same axis will be

r (cos #2 cos ¢ + sin @, sin % sin ¢),

which is equal to w,. Similarly, the length 7’, assumed on the
other refracted ray, will have its projection equal to nw’. The
quantities by which the squares of the transversals are multi-
plied, in the equation (42), are therefore the corresponding ethe-
real volumes} which we may conceive to be put in motion by the
different waves ; and as we suppose the density of the ether to
be the same in both media, the equation expresses a principle
analogous to that of the preservation of vis viva.t

By giving a certain direction to the incident transversal,
that is, by polarizing the incident ray in a certain plane, we may
make one of the refracted rays disappear. If O7 be the ray
which remains, we have 7’, = 0, and the equations (34) and (38)

become

* The equation M = 0 is the same as the equation (vi1.) in my former Paper.—
Transactions, R.I.A., Vou. xvi. p. 52 (supra, p. 112).

t Ibid., p. 48 (supra, p. 106).

t A similar equation of vis viva holds when the light passes out of a crystal into
an ordinary medium.

182 On a Dynamical Theory of

(7, cos 0, + 7’; cos 0’;) cos %; = rz Cos 8; cos én,
7 sin 6, + 7’, sin 0’, = r, sin 0,
(7, cos 0; — 7’, cos @;) sin 7, = 7, cos @, sin %, (43)
(7, sin 6, - 7’; sin @,) sin 4 cos ¢,
= 7, (sin 0, sin 7, cos 7, + sin??, tan ).
In this case, the three transversals are in the same plane, the

refracted transversal being the resultant .of the other two.
Therefore if we find this plane, everything will be determined.

The axes of x, Y%, % make, with the incident ieqereet ?

angles whose cosines are
cos 0, cos 1, sin 0,, — cos 4, sin 4,

and, with the reflected transversal, angles whose cosines are
cos 8, cos 4, sin 0’,, cos 6, sin 4 ;

therefore, by Lemma I., the cosines of the angles which these
axes make with a right line perpendicular to the plane of the
transversals are proportional to the quantities

sin (0, + 0’:)sin7,, —cos 0, cos 6’, sin 24, sin (0: —6;) cost).

Now from the product of the first and second of the equations
(43), combined with the product of the third and fourth, we
find, by the help of the relations (37),

2Qr, ry sin (0, + ;) sin a = v2" tan a, cos 6, {sin 6, cos Io

— s’(sin 0, cos % + sin % tan «)}.

From the squares of the first and third of those equations we ©

find
— 2r17’, cos O, cos 0’, sin 2i, = 7,’ tan i, cos*@, (s* — 1),
and from the product of the first and fourth, combined with the
product of the second and third,
2717, sin (0; — 6,) cos 7, =r," tan? cos 0,{— sin 0, sin %,
+s’ (sin 8, sin 7, — cos i, tan e)}.

er:

Crystalline Reflexion and Refraction. 183

In the three equations just found, the left-hand members are
proportional, as we have seen, to the cosines of the angles which
a right line perpendicular to the plane of the transversals makes
with the axes of co-ordinates; and the right-hand members, as
appears by the formule (85) and (41), are proportional to the
quantities

cos a cos 8

_cos
——H_ COS a2, tp cos P2, ketal
8 8 oa

= 7 cos 29

which are obviously the differences between the corresponding
co-ordinates of the points R and Q. The plane of the transver-
sals is therefore perpendicular to the right line QR, which joins
those points.

A plane parallel to the right line 7, and passing through
the transversal of the ray OT, is that which I have called the
polar plane of the ray ;* and this plane is perpendicular to QR.
Therefore, when there is only one refracted ray, the incident
and reflected transversals lie in the polar plane of that ray ; and
their directions being thus determined, the relative magnitudes
of the three transversals are known. In this case the incident
and reflected transversals are called uniradial ; and as each re-
fracted ray in turn may be ‘made to disappear, there are two
uniradial directions in the plane of the incident wave, and two
in that of the reflected wave.

When the incident transversal is not uniradial, it may be
considered as the resultant of two uniradial transversals, each of
which will supply a refracted ray, and will produce a uniradial
component of the reflected transversal.

It is needless to extend these deductions further. They
have been carried far enough to show that the results of the
foregoing theory are in perfect accordance with the laws estab-
lished in my former Paper on the subject of crystalline reflexion.
The theory itself suggests much matter for consideration; but
at present we shall confine ourselves to one remark, which may

* Transactions, Royal Irish Academy, Vou. xviir. p. 39 (supra, p. 96).

184 On Crystalline Reflexion and Refraction.

be necessary to prevent any misconception as to the nature of
the foundation on which it stands. In this theory, everything
depends on the form of the function V; and we have seen that,
when that form is properly assigned, the laws by which crystals
act upon light are included in the general equation of dynamics.
This fact is fully proved by the preceding investigations. But
the reasoning which has been used to account for the form of
~ the function* is indirect, and cannot be regarded as sufficient, in
a mechanical point of view. It is, however, the only kind of
reasoning which we are able to employ, as the constitution of
the luminiferous medium is entirely unknown.

* Since this Paper was read to the Academy, I have found that the form of the
function V is more general than it would seem to be from the mode in which it is
here deduced ; and I have obtained from it a theory of the Total Reflexion of Light.
For a sketch of this theory, see the Proceedings of the Royal Irish Academy,
Vou. 1. p. 96 (supra, p. 187).

(* 865A)

XV.—ON THE OPTICAL LAWS OF ROCK-CRYSTALS.

[ Proceedings of the Royal Irish Academy, Vou. 1. p. 385.—Read Jan. 18, 1840.]

Proressor Mac CutitacH made a communication respecting
the optical Laws of Rock-crystal (Quartz).

In a Paper read to the Academy in February 1836, and
published in the Transactions (Vou. xvu. p. 461),* he had shown
how the peculiar properties of that crystal might be explained,
by adding to the usual equations of vibratory motion certain
terms depending on differential co-efficients of the third order,
and containing only one new constant C. This hypothesis,
which was very simple in itself, not only involved as conse-
quences all the laws that were previously known, but led to
the discovery of a new one—the law, namely, by which the
ellipticity of the vibrations depends on the direction of the ray
within the crystal. He was not able, however, to account for
his hypothesis, nor has it since been accounted for by anyone.

But the theory developed in the Paper which he read at
the last meeting of the Academy now enables him to assign,
with a high degree of probability, the origin of the additional
terms above mentioned, and, if not to account for them mecha-
nically, at least to advance a step higher in the inquiry. In
that theory it was supposed (and the supposition holds good
in all known crystals, except quartz), that tie molecules of the
ether vibrate in right lines, the displacements remaining always
parallel to each other as the wave is propagated; and it was
shown that the function V, by which the motion is determined,

* Supra, p. 63.

186 On the Optical Laws of Rock-Crystals.

then depends only on the reative displacements of the molecules.
But when this is not the case—when, as in quartz, each mole-
cule is supposed to vibrate in a curve—then it is natural to con-
ceive that the function V may depend, not only on the relative
displacements, but also on the relative areas which each mole-
cule describes about every other more or less advanced in its
vibration. This idea, analytically expressed, introduces a new
term v into the value of the function 2V; and, if the plane of
the wave be taken for the plane of zy, it is easy to show that
-o(2 VE dé a}

dz dz’ dz dz’
Now if we integrate by parts the expression
[lj dedydz80,

so as to get rid of the variations of differential co-efficients, the
reduced form of the triple integral will be

20 {il dedyds & SE - ae bn)

from which it appears that the quantities

d'n PE
C eae C 7
are to be added to the usual expressions for the force in the
directions of x and y respectively. These are the very terms
in the addition of which the hypothesis before alluded to

consists.

C582)

XVI.—ON A DYNAMICAL THEORY OF CRYSTALLINE
REFLEXION AND REFRACTION.

[Proceedings of the Royal Irish Academy, Vou. 11. p. 96.—Read May 24, 1841.]

Proressor Mac CuttacH read a supplement to his Paper
“On a Dynamical Theory of Crystalline Reflexion and Refrac-
tion.”

In his former Paper on that subject,* the author had given the
general principles for solving all questions relative to the propa-
gation of light in a given medium, or its reflexion and refrac-
tion at the separating surface of two media; but he had applied
them only to the common case of waves, which suffer no dimi-
nution of intensity in their progress, and in which the vibration
may be represented by the sine or cosine of an are multiplied
by a constant quantity. Some months after that Paper was read,
it occurred to him that he might obtain new and important
results by substituting in his differential equations of motion a
more general expression for the integral, that is (as usual in
such problems), by making the displacements proportional to
the sine or cosine of an are, multiplied by a negative exponen-
tial, of which the exponent should be a linear function of the
co-ordinates. Such vibrations would become very rapidly insen-
sible, and would, therefore, be fitted to represent the disturbance
which, in the case of total reflexion, takes place immediately
behind the reflecting surface; and, the laws of this disturbance
being thus discovered, the laws of polarization in the totally

* Soe Proceedings, 9th December, 1839 (supra, p. 145).

~ eee

188 On a Dynamical Theory of

reflected light would also become known, by means of the —

. general formule which the author had established for all cases

of reflexion at the common surface of two media.

The present supplement is the fruit of these considerations.
It contains the complete theory of the new kind of vibrations,
not only in ordinary media, but in doubly-refracting crystals ;
and also the complete discussion of the laws of total reflexion at
the first or second surface of a crystal, including, as a particular
case, the well-known empirical formule of Fresnel for total
reflexion at the surface of an ordinary medium.

The existence of vibrations represented by an expression con-
taining a negative exponential as a factor had been recognised
by other writers, and was indeed sufficiently indicated by the
phenomenon of total reflexion ; but it was impossible to obtain
the laws of such vibrations, so op as the general i bs for
the propagation of light were unknown.

The method of deducing these equations was given in the
abstract of the author’s former Paper ;* but as they were not there
stated, it may be well to transcribe them here. If then we put

dn a ade dé de dy
"fe dy do de ay ae
and suppose the axes of co-ordinates to be the principal axes of
the crystal, the equations in question may be thus written :

vt dZ_ya¥
de ° dy dz’
Cn _ dX ? dZ 2)

dP” de” dy’
and if we further put

dm dé, dé, dé, = dé, dm
me ont 7 Mae ae omar ee (3)

* See Proceedings, 9th December, 1839 (supra, p. 157).

Crystalline Reflexion and Refraction. —189

they will take the following simple form :

eS a oh 2-#Y,

ALi ee
in which it is remarkable that the auxiliary quantities &, m, 21,
are exactly, for an ordinary medium, the components of the dis-
placement in the theory of Fresnel. In a doubly-refracting
crystal, the resultant of 1, »,, 2: is perpendicular to the ray, and
comprised in a plane passing through the ray and the wave-nor-
mal. Its amplitude, or greatest magnitude, is proportional to
the amplitude of the vibration itself, multiplied by the velocity
of the ray.

The conditions to be fulfilled at the separating surface of two
media were given in the abstract already referred to, From
these it follows,.that the resultant of the quantities &, m, G1,
projected on that surface, is the same in both media; but the
part perpendicular to the surface is not the same; whereas the
quantities €, n, 2, are identical in both. These assertions, analy- .
tically expressed, would give five equations, though four are
sufficient ; but it can be shown that any one of the equations is
implied in the other four, not only in the case of common, but
of total reflexion ; which is a very remarkable circumstance,
and a very strong confirmation of the theory.

The laws of double refraction, discovered by Fresnel, but not
legitimately deduced from a consistent hypothesis, either by him-
self or any intermediate writer, may be very easily obtained, as
the author has already shown, from equations (2), by assuming

E=pcosa sing, n=pcosPsing, J=p cosy sing, (5)
icone = oF (le + my + n& ~ st);

but the new laws, which are the object of the present supple-
ment, are to be obtained from the same equations by making
E=<(p cosa sin ¢ + g cosa’ cos@),
n = €(p cos sin @ + ¢ cos)’ cos ¢), (6)
f =«(p cosy sin ¢ + g cosX’ cos g),

190 On a Dynamical Theory of
where ¢ has the same signification as before, and

sre Tilia wy He),
the vibrations being now elliptical, whereas in the former case
they were rectilinear. In these elliptic vibrations the motion
depends not only on the distance of the vibrating particle from
the plane whose equation is

le+my+ns=0, — (7).
but also on its distance from the plane expressed by the equation
fe+gy+he=0; (8)

and if the constants in the equation of each plane denote the
cosines of the angles which it makes with the co-ordinate planes,
we shall have A for the length of the wave, and s for the velocity
of propagation ; while the rapidity with which the motion is ex-
tinguished, in receding from the second plane, will depend upon
the constant 7. The constants p and g may be any two conju-
gate semi-diameters of the ellipse in which the vibration is per-
formed ; the former making, with the axes of co-ordinates, the
angles a, (3, y, the latter the angles a’, (’, y’.

As vibrations of this kind cannot exist in any medium, unless
they are maintained by total reflexion at its surface, we shall
suppose, in order to contemplate their laws in their utmost
generality, that a crystal is in contact with a fluid of greater re-
fractive power than itself, and that a ray is incident at their
common surface, at such an angle as to produce total reflexion.
The question then is, the angle of incidence being given, to de-
termine the laws of the disturbance within the crystal.

The author finds that the refraction is still double, and that
two distinct and separable systems of vibration aré transmitted
into the crystal. He shows that the surface of the crystal itself
(the origin of co-ordinates being upon it at the point of inci-
dence) must coincide with the plane expressed by equation (8),
a circumstance which determines the three constants f, g, .
The plane expressed by (7) is parallel to the plane of the re-.

Crystalline Reflexion and Refraction. IgI

fracted wave; and a normal, drawn to it through the origin,
lies in the plane of incidence, making with a perpendicular to
the face of the crystal an angle w, which may be called the angle
of refraction ; so that, if 7 be the angle of incidence, we have
sin w = 8 sin?,

the velocity of propagation in the fluid being regarded as unity.

To each refracted wave, or system of vibration, corresponds
a particular system of values for 7,s,w. These the author shows
how to determine by means of the index-surface (the reciprocal of
Fresnel’s wave-surface), which he has employed on other occa-
sions,* and the rule which he gives for this purpose affords a
remarkable example of the use of the imaginary roots of equa-
tions, without the theory of which, indeed, it would have been
difficult to prove, in the present instance, that there are two, and
only two, refracted waves. Taking a new system of co-ordi-
nates 2’, 7/, x, of which 2 is perpendicular to the surface of the
crystal, and 7/ to the plane of incidence, while # lies in the in-
tersection of these two planes; put 7/ = 0 in the equation of the
index-surface referred to those co-ordinates, the origin being at
its centre ; we shall then have an equation of the fourth degree
between «’ and z’, which will be the equation of the section made
in the index-surface by the plane of incidence. In this equation
put 2 = sin?, and then solve it for z’. When 7 exceeds a certain
angle 7’, the four values of z’ will be imaginary ; and if they be

denoted by
utov/-l, utv /-l,

each pair will correspond to a refracted system, and we shall
have, for the first,

_ gin? sin w
tanw =—_, &= ni? r= 8v; (9)
and for the second,
. . s ,
, sint , silnw NBad >.
tan w a ad os ass r= sv. (10)

* Transactions of the Academy, Vos. xvi. and xvuit. (supra, pp. 36, 96).

=

192 On a Dynamical Theory of

When # lies between 7’ and a certain smaller angle i”, two of the
roots will be real, and two imaginary. The real roots correspond
to waves which follow the law of Fresnel; the imaginary roots
give a single wave, following the other laws just mentioned.

Lastly, when 7 is less than 7”, all the roots are real, the re-
fraction is entirely regulated by Fresnel’s law, and the reflexion
by the laws already discovered and published by the author.

If the crystal be uniaxal, and all the values of * imaginary,
the ordinary wave-normal will coincide with the axis of #’;
whilst the extraordinary wave-normal and the axis of <’ will be
conjugate diameters of the ellipse in which the index surface is
cut by the plane of incidence.

When a= =c, the crystal becomes an ordinary medium ;
there is then only single refraction, and the refracted wave is
always perpendicular to the axis of w’.

‘With regard to the ellipse in which the vibrations are per-
formed, it may be worth while to observe, that if it be projected
perpendicularly on the plane of incidence, the projected diameters
which are parallel to the surface of the crystal and to the wave-
plane will, in all cases, be conjugate to each other, and their re-
spective lengths will be in the proportion of r to unity. The
vibrations, it is obvious, are not performed in the plane of the
wave, though they take place without changing the density of
the ether.

The new laws here announced are, properly speaking, laws
of double refraction, and are necessary to complete our know-
ledge of that subject. Between them and the laws of Fresnel
a curious analogy exists, founded on the change of real into
imaginary constants.

The laws of the total reflexion, which accompanies the new
kind of refraction, need not be dwelt upon in this abstract, as
nothing is now more easy than to form the equations which con-
tain them. In fact, the difficulties which formerly surrounded
the problem of reflexion, even in the simplest cases, have com-
pletely disappeared, since the author made known the conditions
which must be fulfilled at the separating surface of two media.

Crystaliine Reflexion and Refraction. 193

In what precedes, it has been supposed that the reflexion and
refraction take place at the first surface of the crystal, because
this is the more difficult and complicated of the two cases into
which the question resolves itself. But it will usually happen
in practice that a ray which has entered the crystal will suffer
total reflexion at the second surface, while the new kind of vibra-
tion is propagated into the air without. The refracted wave will
then be always perpendicular to the axis of x’; the two reflected
rays, within the crystal, will be plane-polarized, according to the
common law, but they will each undergo a change of phase;
and the vis viva of the two rays together will be equal to that of
the incident ray, the vis viva being measured by the square of
the amplitude multiplied by the proportional mass.

In conclusion, the author states a mathematical hypothesis
by which both the laws of dispersion, and those of the elliptic
polarization of rock crystal, may be connected with the laws
already developed.

( 194°)

XVII—NOTES ON SOME POINTS IN THE THEORY OF
LIGHT.

[Proceedings of the Royal Irish Academy, Vou. u. p. 189.—Read Noy. 8, 1841.]

15

On a Mechanical Theory which has been proposed for the Explana-
tion of the Phenomena of Circular Polarization in Liquids, and
of Circular and Elliptic Polarization in Quarts or Rock-crystal;
with Remarks on the corresponding Theory of Rectilinear Pola-
rization. . 3

Tuer theory of elliptic polarization, which I feel myself
called upon to notice, was first stated by M. Cauchy, and has
been made the subject of elaborate investigation by other
writers. That celebrated analyst, conceiving (though without
sufficient reason, as will presently appear) that he had fully
explained the known laws of the propagation of rectilinear
vibrations by the hypothesis that the luminiferous ether, in
media transmitting such vibrations, consists of separate mole-
cules symmetrically arranged with respect to each of three
rectangular planes, and acting on each other by forces which
are some function of the distance, was led very naturally to
imagine that he would find the laws of circular and elliptic
vibrations, in other media, to be included in the more general
hypothesis of an unsymmetrical arrangement. Accordingly,
in a letter read to the French Academy on the 22nd of
February, 1836—a letter to which he attached so much im-
portance that he desired it might not only be published in

Notes on some Points in the Theory of Light. ?°%

the Proceedings, but also “deposited in the Archives” of that
body*—he gave a precise statement of his more extended views,
informing the Academy that he had submitted his new theory
to calculation, and that, among other remarkable results, he had
obtained (with a slight variation or correction) the laws of cir-
cular polarization, discovered by Arago, Biot, and Fresnel. Re-
ferring to his Memoir on Dispersion, published at Prague, under
the title of Nouveaua Exercices de Mathématiques, he observes, that
the results therein contained may be generalized, by “ceasing to
neglect” in the equations of motion [the equations marked (24)
in § 2 of that memoir] certain terms which vanish in the case of
* a symmetrical distribution of the ether. He then goes on to
say—

“Nos formules ainsi généralisées représentent les phé-
noménes de l’absorption de la lumiére ou de certains rayons,
produite par les verres colorés, la tourmaline, &ec., le phe-
noméne de la polarisation circulaire produite par le cristal
de roche, Vhuile de térébenthine, &c. (Voir les expériences
de MM. Arago, Biot, Fresnel). Hlles servent méme a dé-
terminer les conditions et les lois de ces phénoménes;’ elles
montrent que généralement, dans un rayon de lumiére po-
lariseé, une molecule d’éther décrit une ellipse. Mais dans
certains cas particuliers, cette ellipse se change en une droite,
et alors on obtient la polarisation rectiligne.” ‘ Enfin le cal-
cul prouve que, dans le cristal de roche, Vhuile de térében-
thine, &c., la polarisation des rayons transmis parallélement
a Vaxe (s'il s’agit du cristal de roche) n’est pas rigoureusement
circulaire, mais qu’alors l’ellipse différe trés peu du cercle.”

Thus, to say nothing for the present of the questions of
dispersion and absorption, it appears that M. Cauchy conceived
he had completely accounted for the facts of circular and
elliptic polarization, and that he had deduced the formulas
“which serve to determine the conditions and laws of these
phenomena.” But neither in this letter, nor in any subse-

See the Comptes Rendus des Séances de V’ Académie des Sciences, tom. ii. p. 182.
02

yO LVotes on some Points in the Theory of Light.

quent version* of his theory, has he given the formulas them-
selves. Nor has he told us the nature of the calculations by
which he was enabled to correct the received opinion, and to
prove that the vibrations in a ray transmitted along the axis
of quartz, or through oil of turpentine, are not rigorously
circular, as Fresnel and others have supposed, but slightly
elliptical. Now—to take the case of quartz—if we consider
that the vibrations of a ray passing along the axis are in a
plane perpendicular to it, and if we admit, as M. Cauchy
always does in the case of other uniaxal crystals, that there
is a perfect optical symmetry all round the axis, we shall find
it hard to conceive on what grounds he could have come to
the conclusion that the vibrations of such a ray are performed
in an ellipse. For if all planes passing through the axis of
the crystal be alike in their optical properties, there will be
absolutely nothing to determine the position and ratio of the
axes of the ellipse; there will be no reason why its major axis,
for example, should lie in one of these planes, rather than in
any other. But, whatever may be thought of this case inde-
pendently of observation, it is manifestly absurd to suppose
that the vibrations are elliptical in the case of a ray passing
through oil of turpentine, or any other /iguid possessing the
property of rotatory polarization; for, in a liquid, all planes
drawn through the ray itself are circumstanced alike. From
these simple considerations it is evident that the theory of
M. Cauchy is unsound; but a closer examination will show
that it is entirely without foundation, and that it is directly
opposed to the very phenomena which it professes to explain.
To make this appear, however, in the easiest way that the
abstruseness of the subject will allow, it will be necessary to

* From some statements that have been made within the last few days by
Professor Powell (Phil. Mag. Vou. xrx. p. 374), at the request of M. Cauchy
himself, it appears that the latter republished his views about circular and
elliptic polarization, in a lithographed memoir of the date of August, 1836;
but I do not find that he published, either then or since, the detailed calcula-
tions which he seems to have made.

Notes on some Points in the Theory of Light. 20%

advert to some former researches of my own, which have a
direct bearing on the question.

The same day on which M. Cauchy’s letter was read to
the French Academy, I had the honour of reading to the
Royal Irish Academy a Paper “On the Laws of Double
Refraction in Quartz”* wherein I showed that everything
which we know respecting the action of that crystal upon
light is comprised mathematically in the following equations:—

aE BE dn

ae = aae t CF

1

ae Paige ae?

which differ from the common equations of vibratory motion
by the two additional terms containing third differential co-
efficients multiplied by the same constant C, this constant
having opposite signs in the two equations. The quantities
E and »n are, at any time ¢, the displacements parallel to the
axes of x and y, which are supposed to be the principal di-
rections in the plane of the wave, one of them being there-
fore perpendicular to the axis of the crystal. The constants
A and B are given by the expressions

A=a@, B=a@-(d- 0) sin’y,

where a and 6 are the principal velocities of propagation,
ordinary and extraordinary, and y is the angle made by the
waye-normal (or the direction of s) with the axis of the crys-
tal. The only new constant introduced is C, which, though
the peculiar phenomena of quartz depend entirely on its ex-
istence, is almost inconceivably small: its value is determined
in the Paper just referred to. The equations are there proved
to afford a strict geometrical representation of the facts; not
only connecting together all the laws discovered by the dis-
tinguished observers to whom M. Cauchy refers, and includ-

* See Transactions, R. I. A., Vou. xvu. p. 461 (supra, p. 63).

78 Notes on some Points in the Theory of Light.

ing the subsequent additions for which we are indebted to
Mr. Airy, but leading to new results, one of which establishes
a relation between two different classes of phenomena, and
is verified by the experiments of M. Biot and Mr. Airy.
Having, therefore, such conclusive proofs of the truth of these
equations, we are entitled to assume them as a standard
whereby to judge of any theory; so that any mechanical
hypothesis which leads to results inconsistent with them may
be at once rejected.

Now I assert that the mechanical hypothesis of M. Cauchy
contradicts these equations, and therefore contradicts all the
phenomena and experiments which he supposed it to repre-
sent. But before we proceed to the proof of this assertion,
it may perhaps be proper to remark, that previously to the
date of M. Cauchy’s communication, and of my own Paper, I
had actually tried and rejected this identical hypothesis, and
had even gone so far as to reject along with it the whole of
M. Cauchy’s views about the mechanism of light. For though,
in my Paper, I have said nothing of any mechanical investiga-
tions, yet, as a matter of course, before it was read to the Aca-
demy, I made every effort to connect my equations in some
way with mechanical principles; and it was because I had
failed in doing so to my own satisfaction, that I chose to
publish the equations without comment,* as bare geometrical
assumptions, and contented myself with stating orally to the
Academy, as I did some months after to the Physical Section
of the British Association in Bristol? that a mechanical account
of the phenomena still remained a desideratum which no attempts
of mine had been able to supply. I am not sure that on the first
occasion I stated the precise nature of these attempts, though I

* The circumstances here related will account for what Mr. Whewell (History
of the Inductive Sciences, You. 11. p. 449) calls the “ obscure and oracular form ”’ in
which those equations were published. Having, at that time, no good explanation
of them to give, I thought it better to attempt none. But in the general view which
I have since taken (see p. 224 of this volume), they do not offer any peculiar diffi-

culty.
+ See ‘‘ Transactions of the Sections,’”’ p. 18.

Notes on some Points in the Theory of Light. 201

incline to think I did; but I have a distinct recollection of having*

done so on the second occasion, in reply to questions that were —
_ asked me by some Members of the Association.* Now, my first
attempt to explain those equations, which was made almost as soon
as I discovered them, actually turned upon the very idea which
about the same time found entrance into the mind of M. Cauchy—
I mean the idea of an unsymmetrical arrangement of the ether.
For as it was generally believed, at that period, that the hypothe-
sis of ethereal molecules symmetrically distributed had led, in the
hands of M. Cauchy, to a complete theory of rectilinear polari-
zation in crystals,+ the notion of endeavouring to account for the
phenomena of elliptic polarization, by freeing the hypothesis from
any restriction as to the distribution of the ether, would natu-
rally occur to anyone who was thinking on the subject, no less
than to M. Cauchy himself. And though, for my own part,
I never was satisfied with that theory, which seemed to me to
- possess no other merit than that of following out in detail the
extremely curious, but (as I thought) very imperfect, analogy
which had been perceived to exist between the vibrations of the
luminiferous medium and those of a common elastic} solid (for

* At the period of this meeting, M. Cauchy’s letter on Elliptic Polarization had
been published for some months; but I was not then aware of its existence. Indeed
the letter appears not to have attracted any general notice; for the theory which it
contains was afterwards advanced in England as a new one, and M, Cauchy has
been lately obliged to assert his prior claim to it, through the medium of Professor
Powell.—<See notes, pp. 196, 202-3.

t See his Exercices de Mathématiques, Cinquitme Année, Paris, 1830, and the
Mémoires de ? Institut, tom. x. p. 293.

{ The analogy was suggested by the hypothesis of transversal vibrations, which,
when viewed in its physical bearing, was considered by Dr. Young to be ‘‘perfectly
appalling in its consequences,’’ as it was only to solids that a “ lateral resistance”’
tending to produce such vibrations had ever been attributed. (Supplement to the
Encyclopedia Britannica, Vou. vt. p. 862, Edinburgh, 1824.) He admits, how-
ever, that the question, whether fluids may not “transmit impressions by lateral
adhesion, remains completely open for discussion, notwithstanding the apparent
difficulties attending it.’? As far as I am aware, Fresnel always regarded the
ether as a fluid. M. Poisson affirms that it must be so regarded, and attributes its
apparent peculiarities to the immense rapidity of its vibrations, which does not

$6 Votes on some Points in the Theory of Light.

.. is usual to regard such a solid as a rigid system of attracting
or repelling molecules, and M. Cauchy has really done nothing
more than transfer to the luminiferous ether both the constitu-
tion of the solid and differential formulas of its vibration), still
I should have been glad, in the absence of anything better, to
find my equations supported by a similar theory, and their form
at least countenanced by the like mechanical analogy. Besides,
I recollected that Fresnel himself, in his Memoir on Double Re-
fraction, had indicated a “‘helicoidal arrangement,” or something
of that sort, as a probable cause of circular polarization* and as
this was an hypothesis of the same kind as the other, only not so
general, I was prepared to find that the supposition of an arbitrary
arrangement, whatever might be thought of its physical reality,
would lead to equations of the same form as those which I had
assumed. Upon trial, however, the very contrary proved to be
the case; for though it was possible to obtain additional terms, con-
taining differential co-efficients of the third order, multiplied by
the same constant C, yet this constant always came out with
the same sign in both equations, whereas a difference of sign
was essential for the expression of the phenomena. I had no
sooner arrived at this result than I perceived it to be fatal to
the theory of M. Cauchy, and to afford a demonstration of its
insufficiency, not only in the particular application which I had
made of it, but in all its applications. For the hypothesis

allow the law of equal pressure to hold good in the state of motion (Annales de
Chimie, tom. xliv. p. 482). M. Cauchy calls the ether a fluid, though he treats it
as a solid. My own impression is, that the ether is a medium of a peculiar kind,
differing from all ponderable bodies, whether solid or fluid, in this respect, that it
absolutely refuses, in any case, to change its density, and therefore propagates to
a distance transversal vibrations only ; while ordinary elastic fluids transmit only
normal vibrations, and ordinary solids admit vibrations of both kinds. This hypo-
thesis also includes the supposition that the density of the ether is unchanged by
the presence of ponderable matter. As to M. Cauchy’s third ray, with vibrations
nearly normal to the wave, there is no reason to believe that it has even the
faintest existence; but it is necessarily introduced by his identification of the
vibrations of light with those of an indefinitely extended elastic solid.
* Mémoires de V Institut, tom. vii. p. 73.

Notes on some Points in the Theory of Light. 201

which I used was, in fact, identical with that theory, in the
most general form of which it is susceptible, when unrestricted
by any particular supposition as to the arrangement of the
ethereal molecules; and therefore the fundamental conception
of the theory could not be true, as it not merely failed to ex-
plain a large and most remarkable class of phenomena—those
of circular and elliptical polarization—but absolutely excluded
them, and left no room for their existence. It followed from
this, that the mechanical explanation, which the same theory
was supposed to have given, of the phenomena of rectilinear
polarization and double refraction in crystals, could not be well
founded: indeed, as I have said, I had always distrusted it, and
that for various reasons, of which one has been already men-
tioned, and another was suggested by the forced relations which
M. Cauchy had found it necessary to establish among the con- |
stants of his theory, and by which he had compelled, as it were,
his complicated formulas to assume the appearance of an agree-
ment (though, after all, a very imperfect one) with the simple
laws of Fresnel.
Such were the conclusions at which I arrived, and the re-
flections which they forced upon me, nearly six years ago.
They have been frequently mentioned in conversation to those
who took an interest in such matters, and their general tenor
may be gathered from what I have elsewhere written ;* but I
did not think it worth while to publish them in detail, because
it seemed probable that juster notions would prevail in the
course of a few years, and that the ingenious speculations to
which I have alluded would gradually come to be estimated at
their proper value. But from whatever cause it has arisen—
whether from the real difficulties of the subject, or the extreme
vagueness of the ideas that most persons are content to form of
it, or from deference to the authority of a distinguished mathe-
matician—certain it is that the doctrines in question have not
only been received without any expression of dissent, but
have been eagerly adopted, both in this country and abroad, by

* Transactions, R. I, A., Vou. xvi. p. 68 (supra, p. 129).

202 Votes on some Points in the Theory of Light.

a host of followers; and even the extraordinary error, which it
is my more immediate object to expose, has been continually
gaining ground up to the very moment at which I write, and
has at last begun to be ranked among the elementary truths of
the undulatory theory of light. Notwithstanding my un-
willingness, therefore, to be at all concerned in such discus-
sions, I do not think myself at liberty to remain silent any
longer. There are occasions on which every consideration of
this kind must give way to a regard for the interests of science.

To show that the principles of M. Cauchy contradict, in-
stead of explaining, the phenomenon of elliptic polarization,
let us take the axes of co-ordinates as before; and let us sup-
pose, for the sake of simplicity, and to avoid his third ray,
that the normal displacements vanish. Then his fundamental
equations take the form

PE

Ta = Sfaé = =hAn,

2)
r (
a = SgAn + ShAE,

where f, g, 2 are quantities depending on the law of force
and the mutual distances of the molecules.* If, therefore,

*T have not thought it necessary to transcribe the original equations of M.
Cauchy, which are rather long. He has presented them in different forms; but
the system marked (16) at the end of § 1 of his Memoir on Dispersion, already
quoted, is the most convenient, and it is the one which I haye here used. The
directions of the co-ordinates being arbitrary, I have supposed the axis of z to be
perpendicular to the wave-plane. Then, on putting ¢= 0, A¢= 0, in order to get
rid of the normal vibration, the last equation of the system becomes useless, and
the other two are reduced to the equations (2), given above; the letters f, g, A,
being written in place of certain functions depending on the mutual actions of the
molecules. It will be proved, further on, that this simplification does not at all
affect the argument. As the directions of x and y still remain arbitrary, I have
made them parallel to the axes of the supposed elliptic vibration.

It may be right to observe, for the sake of clearness, that, when the medium is
arranged symmetrically, it is always possible to take the directions of x and y such
that the two sums depending on the quantity 4 may disappear from the equations
(2), and then the vibrations are rectilinear. But when the arrangement is unsym-
metrical, this is no longer possible.

Notes on some Points in the Theory of Light. 203

we assume that each molecule describes an ellipse, the axes of
which are parallel to those of # and y; that is to say, if we
make

E=pcos¢, n=¢sin ¢,
(3)

and consequently,
AE = p (sin 20 sin ¢ — 2 sin? 8 cos 4),
An = — q (sin 20 cos ¢ + 2 sin’ sin 9),

where 6 = a, we shall find, by substituting these values in the
equations (2), which must hold good independently of ¢,

8 =A +0, &=B - we (4)

Sf sin 20 — 2k Sh sin’O = 0,

Sg sin 20 + S Sh sin’O = 0,

The equations (2) are precisely the same as those which have been employed
by Mr. Tovey and by Professor Powell, the latter of whom, in his lately published
. work, entitled, ‘‘.4 General and Elementary View of the Undulatory Theory, as ap-
plied to the Dispersion of Light, and other Subjects,” has dwelt at great length on
the theory of elliptic polarization which they have been supposed to afford, and
which he regards as a most important accession to the Science of Light. Professor
Powell has also made some communications on the subject to the British Asso-
ciation, and has written two Papers about it in the Philosophical Transactions
(1838, p. 253: and 1840, p. 157), besides several others in the Philosophical Ma-
gazine. He, however, always attributed this theory of elliptic polarization to
~ Mr. Tovey, until his attention was directed, by a letter from M. Cauchy, to some
investigations of the latter which he had not previously seen (Phil. Mag. Vou. xx.
p. 374). Mr. Tovey set out with the principles of M. Cauchy, and therefore
naturally struck into the same track, in pursuit of the same object, apparently
quite unconscious that anyone had preceded him. It was, indeed, an obvious
reflection, that these principles, when generalized to the utmost, ought to include,
not only the laws of elliptic polarization, but (as really has been thought by M,
Cauchy and his followers) of dispersion and absorption, and, in short, of all the
phenomena of optics.

204. Votes on some Points in the Theory of Light.

wherein & = £ expresses the ratio of the semiaxes of the elliptic

vibration, and
Sem : ? ;
A’ = oF Sfsin*#, B= gn3 29 sin’6,
2
O = te =; 2A sin 20.

Equating the two values of s*, we get, ane the determination of
k, the following quadratic :—

k+1=0. (5)

Now making the substitutions (3) in equations (1), page 197,
we have

e=-A4-> O, #=B- >, a
and thence
r
sy (A B)k-1=0, (7)

a result which is perfectly inconsistent with the former, since
the two roots of (5) have the same sign, if they are not imagi-
nary, while those of (7) have opposite signs, and cannot be
imaginary. If, therefore, one equation agrees with the phe-
nomena, the other must contradict them. The last equation
indicates that, in the double refraction of quartz, the two
elliptic vibrations are always possible, and performed in oppo-
. site directions, which is in accordance with the facts; whereas
the equation (5), deduced from M. Cauchy’s theory, would
inform us that the vibrations of the two rays are either im-
possible or in the same direction.*

To apply the results to a particular instance, let us con-
ceive a circularly polarized ray passing along the axis of
quartz, or through one of the rotatory liquids, such as oil of

* This conclusion, which shows that M. Cauchy’s Theory is in direct opposition
to the phenomena, might have been obtained without any reference to the equa-
tions (1). But these equations are necessary in what follows.

Notes on some Points in the Theory of Light. 205

turpentine; the position of the co-ordinates # and y, in the
plane of the wave, being now, of course, arbitrary. In each
of these cases we have k=+1, and A= B=<a’, so that the
value of s* in equation (6)-is expressed by the constant a’,
plus or minus a term which is inversely proportional to the
wave-length \; the sign of this term depending on the direc-
tion of the circular vibration. Now it will not be possible
to obtain a similar value of s* from the formulas (4), unless
we suppose A’ = B’ = a*, since it is only in the expansion. of
O’ that a term inversely proportional to A can be found; but
on this supposition the formulas are inconsistent with each
other, nor can they be reconciled by any value of &. Indeed,
when A’= B’, the equation (5) give k=+./-—1. Thus it
appears that circular vibrations, such as are known to be propa-
gated along the axis of quartz, and through certain fluids, can-
not possibly exist on the hypothesis of M. Cauchy. It was
probably some partial perception of this fact that caused M.
Cauchy to assert that the vibrations, in these cases, are not ex-
actly circular, but in some degree elliptical—a supposition which,
if it were at all conceivable, which we have seen it is not (p. 196),
would be at once set aside by what has just been proved; for no
assumed value of 4, whether small or great, will in any way hate
to remove the difficulty.

But this is not all. Rectilinear vibrations are excluded as
well as circular; for we cannot suppose / = 0 in the equations
(4), so long as the quantity C’, resulting from the hypothesis of
unsymmetrical arrangement, has any existence. Thus the in-
consistency of that hypothesis is complete, and the equations to
which it leads are utterly devoid of meaning.

The foregoing investigation does not differ materially from
that which I had recourse to in the beginning of the year 1836.
To render the proof more easily intelligible, and to get rid of
M. Cauchy’s “third ray,”’ which has no existence in the nature
of things, I have suppressed the normal vibrations; a procedure
which is not, in general, allowable on the principles of M.
Cauchy. It will readily appear, however, that this simplifica-

206 Votes on some Points in the Theory of Light.

tion still leaves the demonstration perfectly rigorous in the case
of circular vibrations, and does not affect its force when the
vibrations are elliptical. For in the rotatory fluids it is obvious
that the normal vibrations, supposing such to exist, must, by
reason of the symmetry which the fluid constitution requires, be
independent of the transversal vibrations, and separable from
them, so that the one kind of vibrations may be supposed to
vanish when we wish merely to determine the laws of the other.
The equations (2) are, therefore, quite exact in this case; and
they are also exact in the case of a ray passing along the axis’
of quartz, since such a ray is not experimentally distinguishable
from one transmitted by a rotatory fluid, and its vibrations must
consequently be subject to the same kind of symmetry. In these
two cases, therefore, it is sigorously proved that the values of f,
which ought to be equal to plus and minus unity, are imaginary,
and equal to+.,/-—1. And if we now take the most general
case with regard to quartz, and suppose that the ray, which was
at first coincident with the axis of the crystal, becomes gradually
inclined to it, the values of / must evidently continue to be ima-
ginary, until such an inclination has been attained that the two
roots of equation (5) become possible and equal, in consequence
of the increased magnitude of the co-efficient of the second term.
Supposing the last term of that equation to remain unchanged,
this would take place when the co-efficient of / (without regard-
ing its sign) became equal to the number 2, and the values of &
each equal to unity, both values being positive or both negative.
The vibrations which before were impossible would, at this in-
clination, suddenly become possible; they would be circular,
‘ which is the exclusive property of vibrations transmitted along
the axis; and they would have the same direction in both rays,
which is not a property of any vibrations that are known to
exist. At greater inclinations the vibrations would be ellip-
tical, but they would still have the same direction in the two
rays. These results would not be sensibly altered by regard- —
ing the equation (5) as only approximate in the case of rays
inclined to the axis; for the last term of that equation, if it

LVotes on some Points in the Theory of Light. 207

does not remain the same, can never differ much from unity ;
since it must become exactly equal to unity, whatever be the
direction of the ray, when the crystalline structure is sup-
posed to disappear, and the medium to become a rotatory
fluid.

That a theory involving so many inconsistencies should
have been advanced by a person of M. Cauchy’s reputation
would, perhaps, appear very extraordinary, if we did not re-
collect that it was unavoidably suggested by the general prin-
ciples which he had previously adopted, and which were
supposed, not merely by himself, but by the scientific world
generally, to have already afforded the only satisfactory ex-
planation of the laws of double refraction in the common
and well-known case where the vibrations are rectilinear.
This supposed explanation was obtained, as has been said,
by restricting the application of M. Cauchy’s principles to
the hypothesis of a vibrating medium arranged symmetrically,
in which case it was shown that the vibrations were neces-
sarily rectilinear; and of course the removal of this restric-
tion was the only way in which it was possible, on those
principles, to account for the existence of circular and ellip-
tical vibrations. Accordingly, when M. Cauchy perceived
that, on the hypothesis of unsymmetrical arrangement, the
existence of rectilinear vibrations became impossible, and that
of elliptic vibrations, generally speaking, possible, he found
it very easy to persuade himself that he had obtained a new
proof of the correctness of his views, and a new and most im-
portant application of the fundamental equations by which
his general principles were analytically expressed. To have
supposed otherwise would have been to admit that his general
principles were false. If the elliptical or quwasi-circular vibra-
tions which he was now contemplating were not capable of
being identified with those which had been recognized in the
phenomena presented by quartz and the rotatory fluids—if
their laws were essentially or very considerably different—his
theory would be inconsistent with a wide range of well-known

208 Votes on some Points in the Theory of Light.

facts, and, notwithstanding its so-called explanations of other
laws, should be finally abandoned. Under these circumstances,
therefore, he very naturally supposed that his new results
must be in complete harmony with the phenomena discovered
by M. Arago, and analyzed so successfully by MM. Biot and
Fresnel; although, had he taken the precaution of acquiring
such a clear notion of the phenomena as would have enabled
him to translate them into analytical language, he must have
perceived that they were entirely opposite to his results, and
that this opposition furnished an argument which swept away
the very foundations of his theory. For, if the constitution
of the luminiferous medium were such as M. Cauchy sup-
poses, the well-known phenomena of circular and elliptic
polarization would, as we have seen, be absolutely impossible.

Thus the argument which overturns the particular theory
of elliptical polarization destroys at the same time all the
other optical theories of M. Cauchy, because they are all
built on the principles which we have now demonstrated to be
false. But though the principles of M. Cauchy are now, for
the first time, formally refuted, they were objected to, on
general grounds, so long ago as the year 1830, by a person
whose opinion, on a question of mechanics, ought to have had
considerable weight. This was M. Poisson, who, having de-
duced from the equations of motion of an elastic solid the con-
sequence that such a body admitted vibrations perpendicular
to the direction of their propagation, thought it right to re-
mark that this conclusion could not be supposed to account
for transversal vibrations in the theory of light, because (as
he expressed himself) ‘‘the same equations of motion could
not possibly apply to two systems [of molecules] so essen-
tially different from each other” as the ethereal fluid and
an elastic solid.*—(See the Annales de Chimie, tom. xliv.

* As the theory of M. Cauchy (Mem. de I’ Institut, tom. x.) had been communi-
cated to the Academy of Sciences some months before the period (October, 1830) at
which M. Poisson wrote, there can be no doubt that M. Poisson’s remark was di-
rected against that theory, though he did not expressly mention it. ;

Notes on some Points in the Theory of Light. 209

p. 432). The remark, however, did not meet with much at-
tention from mathematicians, who were, perhaps, not dis-
posed to scrutinize too closely any hypothesis which gave
transversal vibrations as a result. Besides, the hypothesis
appeared to go much further, as it offered primd facie expla-
nations of a great variety of phenomena; it was one to which
calculation could be readily applied, and therefore it naturally
found favour with the calculator; and as to M. Poisson’s objec-
tion, it was easily removed by a change of terms, for when the
elastic solid was called an “ elastic system,” there was no longer
anything startling in the announcement that the motions of the
ether are those of such a system. The hypothesis was there-
fore embraced by a great number of writers in every part of
Europe, who reproduced, each in his own way, the results of
M. Cauchy, though sometimes with considerable modifications.
Every day saw some new investigation purely analytical—some
new mathematical research uncontrolled by a single physical
conception—put forward as a “mechanical theory” of double
refraction, of circular polarization, of dispersion, of absorption ;
until at length the Journals of Science and Transactions of
Societies were filled with a great mass of unmeaning formulas.
This state of things was partly occasioned by the great number
of “ disposable” constants entering into the differential equa-
tions of M. Cauchy and their integrals; for it was easy to in-
troduce, among the constants, such relations as would lead to
any desired conclusion; and this method was frequently
adopted by M. Cauchy himself. Thus, in his theory of double
(or rather triple) refraction, given in the works already cited
(p. 145), he supposes three out of his nine constants to vanish,
and assumes, among the other six, three very strange and im-
probable relations, by means of which each of the principal
sections of his wave-surface (considering only two out of its
three sheets) is reduced to the circle and ellipse of Fresnel’s
law; and the ti.ree principal sections being thus forced to coin-
cide, it would not be very surprising if the two sheets were
found to coincide in every part with the wave-surface of Fres-
P

210 Votes on some Points in the Theory of Light.

nel. The coincidence, however, is only approximate; but
M. Cauchy is so far from being embarrassed by this cireum-
stance, that he does not hesitate to regard his own theory as
rigorously true, and that of Fresnel as bearing to it, in point of
accuracy, the same relation which the elliptical theory of the
planets, in the system of the world, bears to that of gravita-
tion.* Nor is he at all embarrassed by the supernumerary ray
belonging to the third sheet of his wave-surface; he assumes
at once that such a ray exists, though it was never seen, and
promises, for the satisfaction of philosophers, to make known
the means of ascertaining its existence.t But he afterwards
contented himself with observing that as its vibrations are in
the direction of propagation they probably make no impression
on the eye, and he then gave it the name of the “invisible
ray.” +

In these investigations, the suppositions which M. Cauchy
had made respecting the constants led to the result that the
vibrations of a polarized ray are parallel to its plane of pola-
rization ; but in the year 1836 he changed his opinion on this
point, and then, by reinstating the constants that he had be-
fore supposed to vanish, and establishing proper relations
amongst them and the rest, he arrived at the conclusion that
* the vibrations are perpendicular to the plane of polarization.§
All his other results, of course, underwent some correspond-
ing change; and it is this new theory which must now be
regarded as rigorous, while that of Fresnel is to be looked
on as approximate. But it is needless to say, that if the
accuracy of Fresnel’s law of double refraction. is to be disputed,
it must be on much better grounds than these; and the results
of M. Cauchy are certainly too far removed from that law to
have any chance of being consonant with truth. Although, for
example, his new views respecting the direction of the vibra-

* Mémoires de U’ Institut, tom. x. p. 313.
t Ibid. p. 305.

{ Nowveaux Exercices, p. 40.

§ Comptes Rendus, tom. ii. p. 342.

Notes on some Points in the Theory of Light. 211

tions agree, in a general way, with those of Fresnel, there is
yet, in one particular, an important difference between them ;
for, according to Fresnel, the vibrations are always exactly in
the surface of the wave, while, according to M. Cauchy (in his
old theory as well as the new), they are only so in ordinary
media. In a biaxal crystal he finds—and this is one of the
ways in which the “invisible ray” manifests its influence—that
the direction of vibration, in each of the two rays that are
visible, is inclined at a certain angle to the wave-plane; but
this angle, though small, is by no means inconsiderable, as
M. Cauchy seems to intimate, overlookiug the fact, which ap-
pears from his own equations, that it is of the same order of
magnitude as the quantities on which the double refraction
depends. It is true, the deviation measured by this angle can-
not, if it exists, be directly observed in the refracted light; but
its indirect effects on reflected light ought to be very great,
since the action of the crystal on a ray reflected at its surface
differs from that of an ordinary medium by a quantity of the
same order merely as the aforesaid angle; and as the problem
of crystalline reflexion has been already solved* on the suppo-
sition (which is an essential one in the solution) that the
vibrations are eaactly in the plane of the wave, it is highly
improbable, considering the complex nature of the question,
that it will be solved, in any satisfactory way, on a supposition
so different as that which is required by the theory of M. Cauchy.
However, as the laws of such reflexion are now well known, by
means of the solution alluded to, it is possible that M. Cauchy
may, as in the case of double refraction, succeed in deducing the
same laws, or, if not the same, what may seem to be more exact
~ laws, from certain principles + of his own, helped out, if need be,

* Transactions, Royal Irish Academy, Vou. xvyutt., p. 31 (supra, p. 87).

+ In applying these principles to the question of reflexion and refraction at the
surface of an ordinary medium (Comptes Rendus, tom. ii. p. 348), M. Cauchy has
arrived at the singular conclusion, that light may be greatly increased by refraction
through a prism, at the same time that it is almost totally reflected within it. Sup-
posing the refracting angle of the prism to be very little less than the angle of

P 2

212 Votes on some Points in the Theory of Light.

by proper relations among his constants ; especially if, to allow
greater scope for such relations, the number of constants be in-
creased by the hypothesis of two coexisting systems of mole-
cules, an hypothesis which M. Cauchy has already considered
with his usual generality, but without making any precise ap-
plication of it.*

Perhaps one cause why M. Cauchy’s views on the subject
of double refraction have met with such general acceptance
may be found in the fact, that a theory setting out from the —

total reflexion for the substance of which it is composed, a ray incident perpendi-
cularly on one of the faces will emerge, making a very small angle with the other
face; and as the reflexion at the latter face is nearly total, it is self-evident that
the intensity of the emergent light, as compared with that of the incident, must be
very small. M. Cauchy, however, finds by an elaborate analysis that a prodigious
multiplication of light [‘‘ wne prodigieuse multiplication de a lumiére’’] takes place,
the emergent ray being nearly six times more intense than the incident when the
prism is made of glass, and nearly nine times when the prism isof diamond. This
result was, in a general way, actually verified experimentally by himself and ano-
ther person; so easy it is, in some cases, to see anything that we expect to see.
Had the result been true, it would have been a very brilliant discovery indeed ; for
then we should have been able, by a simple series of refractions, to convert the
feeblest light into one of any intensity we pleased; but the very absurdity of such
a supposition should have taught M. Cauchy to distrust both his theory and his ex-
periment. Far from doing so, however, he considers the fact to be perfectly esta-
blished, and to afford a new argument against the system of emission. “ Ici,’ says
he, ‘un rayon, réfléchi en totalité, est de plus transmis avec accroissement de
lumiére; ce qui est un nouvel argument contre le systéme d’émission.”” The sys-
tem of emission has at least this advantage, that by no possible error could such a
conclusion be deduced from it. For if all the particles of light be reflected, cer-
tainly none of them can be refracted.

The truth is, that M. Cauchy mistook the measure of intensity in the hypothe-
sis of undulations, supposing it to be proportional simply to the square of the
amplitude of vibration ; whereas it is really measured by the vis viva, or by that
square multiplied by the quantity of ether put in motion, a quantity which in the
present case is evanescent, since the corresponding volumes of ether, moved by the
ray within in the prism and by the emergent ray, are to each other as the sine of
twice the angle of the prism to the sine of twice the very small angle which the
emergent ray makes with the second face of the prism. The intensity of the emer-
gent light is therefore very small, as it ought to be, though the amplitude of its
vibrations is considerable.

* Exercices d’ Analyse et de Physique Mathématique, tom. i. p. 33.

Notes on some Points in the Theory of Light. 213

same principles, and leading, by the same relations among con-
stants, to formulas identical in every respect with his earlier
— results, was advanced independently, and nearly at the same
time, by M. Neumann of Konigsberg.* A coincidence so re-
markable would be looked upon, not unreasonably, as a strong
argument in favour of the theory; though it must be allowed
that, in the effort to extend fhe knowledge of any subject,
there is a tendency in different minds to adopt the same errors
respecting it, as well as the same truths; a fact of which we
have seen other examples in the course of the present article.

According to M. Neumann,f the “third ray,” not being
perceived as light, must manifest its existence as radiant heat,
or as a chemical power, or as some other agent [ ‘als strahlende
Warme, oder chemisch wirkend, oder als irgend ein anderes Agens’’ |
and he thinks that the nature of this ray will be more easily
investigated, if the laws of reflexion shall be deduced from the
aforesaid theory. But we have seen that the laws of reflexion
are, to all appearance, at variance with the theory, and they
take no account whatever of the third ray. Besides, the dis-
coveries which have been made of late years respecting the po-
larization of radiant heat, and the strong analogies that have
been traced between it and light, amount to a demonstration
that its vibrations are transversal, and of course essentially dif-
ferent from those of the supposed third ray, which are normal,
or nearly so. There is every reason to believe that the vibrations
of the chemical rays are also transversal; and we may confi-
dently assert, that the three species of rays—those of light and
heat, and the chemical rays—are produced not only by vibra-
tions of the same medium, but by the same kind of vibrations,
propagated with nearly the same velocities. If, therefore, the
third ray of MM. Cauchy and Neumann has any existence, it
must be referred to ‘some other agent,” the nature of which it
is impossible to conjecture.

Enough has now been said to show that the optical theory

* Poggendorff’s Annals, Vol. xxy. p. 418.
+ Ibid. p. 454.

214 Votes on some Points in the Theory of Light.

which we have examined, and which has passed current in the
scientific world for a considerable period, is quite inadequate to
explain the leading phenomena of light, and that it is based
upon principles which are altogether inapplicable to the subject.
M. Cauchy states, in the memoir so often quoted,* that the first
application which he had made of his principles was to the
theory of sound, and that the formulas which he had deduced
from them agreed remarkably well with the experiments of
Savart and others on the vibrations of elastic solids. As I have
already intimated, it is in the solution of such questions (which,
however, have long been familiar to mathematicians) that the
fundamental equations of M. Cauchy may be most advanta-
geously employed ; and had he pursued his researches in this
direction, his labours would doubtless have been attended with
more success, and with greater benefit to science.

Il.—On Fresnel’s Formula for the Intensity of Reflected Light,
with Remarks on Metallic Reflexion.

When Mr. Potter discovered, by experiment, that more
light is reflected by a metal at a perpendicular incidence than
at any oblique incidence (at least as far as 70°), the fact was
looked upon, by himself and others, as contrary to all received
theories; and certainly the universal opinion, up to that time,
was, that the intensity of reflexion always increases with the
incidence. It may therefore be worth while to remark, that the
formula given by Fresnel for reflexion at the surface of a trans-
parent body, though not of course applicable, except in a very
rude way, to the case of metals, would yet lead us to expect, for
highly refracting bodies, as the metals are supposed to be, pre-
cisely such a result.as that obtained by Mr. Potter. For when
the index of refraction exceeds the number 2 + V 3, or the tan-
gent of 75°; the expression for the intensity of reflected light
will be found to have a minimum value at a certain angle of in-

* Mem, del’ Institut, tom. x. p. 294.

_LVotes on some Points in the Theory of Light. 215

cidence ; while for all less values of the refractive index the in-
tensity will be least at the perpendicular incidence.

Let 7 and @’ be the angles of incidence and refraction, and
po =

sin 7 _ COS 7 |
sin?” cosa”?

then if I be the intensity of the reflected light, when common
light is incident, Fresnel’s expression

oe ae pa

2 (sin (@4+7) tan? (i+7)S’

in which the intensity of the incident light is taken for unity,
may be put under the form

the value of I being in that case

a 1
os -x) -4 (ag) 8
Oia Maca naeuele e
(Ag) cA Bh ag)
and the geentapanding angle of incidence being given by the
formula

VUVF=1 1
in i= aang hh h = M+ =~),
sin 7 cars a pg where « 2( 77)

Since p + ; cannot be less than 2, it is easy to see that, when

216 Notes on some Points in the Theory of Light.

there is a minimum, If + s cannot be less than 4, and therefore

M cannot be less than 2 + 3, or 3°732.
As an example, let M+ = =6. Then, at a perpendicular

incidence, one-half the incident light will be reflected. The
miniritum will be when 7 = 65° 36’, and at this angle only 7% of
the incident light will be reflected. The value here assumed
for the refractive index is that which Sir J. Herschel* assigns
to mercury ; but if my ideas be correct, it is far too low for that
metal.

The only person who supposes that the refractive: index of a
metal is not a large number is M. Cauchy, It has always
been held as a maxim in optics, that the higher the reflective
power of any substance, the higher also is its refractive index.
But M. Cauchy completely reverses this maxim; for, as I have
elsewhere shown,t it follows from his theory that the most re-
flective metals are the least refractive, and even that the index
of refraction, which for transparent bodies is always greater
than unity, may for metals descend far below unity. Thus,
according to his formula, the index of refraction for pure silver
is the fraction 4, so that the dense body of the silver actually
plays the part of a Very rare medium with respect to a vacuum.
It appears to me that such a result as this is quite sufficient to
overturn the theory from which it is derived. The formulas,
however, which he gives for the intensity of the reflected light,
are identical with the empirical expressions which I had given
long before, and are at least approximately true.

In framing my own empirical theory,t two suppositions re-
lative to the value of the refractive index presented themselves.
Putting U/ for the modulus, and x for the characteristic, I had to

choose between the values / cos x and my The latter value
is that which I adopted; the former, which is M. Cauchy’s, was

* Treatise on Light, Art. 594.
+ Comptes Rendus, tom. viii. p. 964; vid. note at the end of this volume.
t See Proceedings, Vou. 1. p. 2 (supra, p. 58).

Notes on some Points in the Theory of Light. 217

rejected, because I saw that it would lead to the result above
mentioned.

Another result of M. Cauchy’s, which he has given twice in
the Comptes Rendus,* requires to be noticed. When a polarized
ray is reflected by a metal, the phase of its vibration is altered ;
and if the incidence be oblique, the change of phase is different,
according as the light is polarized in the plane of incidence, or
in the perpendicular plane. But when the ray is reflected at a
perpendicular incidence, it is manifest that the change is a con-
stant yuantity, whatever be the plane of polarization. In fact,
the distinction between the plane of incidence and the perpen-
dicular plane no longer exists, and the phenomena must be the
same in all planes passing through the ray. Yet M. Cauchy,
in the two places above quoted, asserts it to be a consequence of
his theory, that in this case the alterations of phase are different
for two planes of polarization at right angles to each other, and
that the difference of the alterations amounts to half an undula-
tion. ‘The same singular hypothesis had been previously made
by M. Neumann,t whom M. Cauchy appears to have followed;
but M. Neumann has since admitted it to be erroneous.t

* Tom. ii. p. 428, and tom. viii. p. 965.
+ Poggendorfi’s Annals, Vol. xxv1. p. 90.
t Ibid. Vou. xu. p. 513.

( 218 )

XVIII.—ON THE PROBLEM OF TOTAL REFLEXICN.

[Proceedings of the Royal Irish Academy, Vou. u.—Read November 30, 1841.] _

Proressor Mac CutiacH communicated to the Academy a
very simple geometrical rule, which gives the solution jof the
problem of total reflexion, for ordinary media, or for aaa
crystals.

First, let the total reflexion take place at the pntaith sur-
face of two ordinary media, as between glass and air, and let it
be proposed to determine the incident and reflected vibrations,
when the refracted vibration is known. It is to be observed,
that the refracted vibration (which is in general elliptical) can-
not be arbitrarily assumed; for, as may be inferred from what
has been already stated,* it must be always similar to the sec-
tion of a certain cylinder, the sides of which are perpendicular
to the plane of incidence, and the base of which is an ellipse
lying in that plane, and having its major axis perpendicular to
the reflecting surface, the ratio of the major to the minor axis
being that of unity to the constant vr. The value of 7, as deter-
mined by the general rule in p. 191, is

2
r= |1-——.
| n* sin? 7’

where # is the angle of incidence, and m the index of refraction
out of the rarer into the denser medium. The ellipse is greatest

* Proceedings of the Academy, Vou. 11. p. 102 (supra, p. 192).

On the Problem of Total Reflexion. 219

for a particle at the common surface of the media; and for a
particle situated in the rarer medium, at the distance s from that

surface, its linear dimensions are proportional to the quantity
Qrrz =

e * ; so that for a very small value of s the refracted vibra-
tion becomes insensible.

Now, taking any plane section of the aforesaid cylinder to
represent the refracted vibration for a particle situated at the
common surface of the two media, let OP and OQ be the semi-
axes of the section, and let them be drawn, with their proper
lengths and directions, from the point of incidence 0; through
which point also let two planes be drawn to represent the inci-
dent and reflected waves. ‘Then conceive a plane passing
through the semiaxis OP, and intersecting the two wave-
planes, to revolve until it comes into the position where the
semiaxis makes equal angles with the two intersections; and in
this position let the intersections be made the sides of a parallel-
ogram, of which the semiaxis OP is the diagonal. Let OA
and OA’, which are of course equal in length, denote these two
sides. Make a similar construction for the other semiaxis OQ,
and let OB, OB’, which are also equal, denote the two sides of
the corresponding parallelogram. Then will the incident vi-
bration be represented by the ellipse of which OA and OB are
conjugate semidiameters, and the reflected vibration by the
ellipse of which OA’ and OB are conjugate semidiameters.
And the correspondence of phase in describing the three ellipses
will be such that the points 4, A’, P will be simultaneous posi-
tions, as also the points B, B’, Q.

The same construction precisely will answer for the case of
total reflexion at the surface of a uniaxal crystal, which is
covered with a fluid of greater refractive power than itself. It
is to be applied successively to the ordinary and extraordinary
refracted vibrations, and we thus get the wniradial incident and
reflected vibrations, or rather the ellipses which are similar to
them. And as any incident vibration may be resolved into two
which shall be similar to the uniradial ones, we can find the re-

220 On the Problem of Total Reflexion.

flected vibration which corresponds to it, by compounding the
uniradial reflected vibrations.

It may be well to mention that, in a uniaxal crystal, the
plane of the extraordinary refracted vibration is always perpen-
dicular to the axis, and therefore the ellipse in which the vibra-
tion is performed may be easily determined by the remark in
p- 192. The plane of the ordinary vibration has no fixed po-
sition in the crystal; but if we conceive the auxiliary quantities
E1, my 2: (p. 188), to be compounded into an ellipse (as if they
were displacements), the plane of this auxiliary ellipse will be
perpendicular to the axis of the crystal.

Whether the preceding very simple construction, for finding
the incident and reflected vibrations by means of the refracted
vibration, extends also to the case of diaxal crystals, is a point
which has not yet been determined, on account of the compli-
cated operations to which the investigation leads, at least when
attempted in any way that obviously suggests itself.

Gices20')

XIX.—ON THE DISPERSION OF THE OPTIC AXES, AND OF
THE AXES OF ELASTICITY, IN BIAXAL CRYSTALS.

[From the Philosophical Magazine, Vou. xx1., October, 1842.]

In the last number of the Philosophical Magazine (p. 228),
there appeared an extract from the Proceedings of the Royal
Irish Academy, containing a notice of a memoir which I had
the honour of reading to that body on the 24th of May, 1841;
and in the concluding paragraph of the notice a brief allusion
is made to a “mathematical hypothesis” by which I had con-
nected the laws of dispersion and those of the elliptic polari-
zation of rock-crystal with the other laws that were there
announced. My present object is to indicate the development
of that hypothesis, with reference more particularly to the sub-
ject of dispersion in crystals, and to communicate a very simple
result which I have lately had occasion to obtain from it. The
result is remarkable as embracing and explaining a class of in-
tricate phenomena which hitherto have not been connected with
any theory, or rather have stood in opposition to all theories ;
I mean the phenomena of the dispersion of the optic axes, and
of the axes of elasticity (as they are called) in biaxal crystals.
The name of axes of elasticity was given by Fresnel to
three rectangular directions, which, according to his theory,
exist in every crystallized medium, and which are distinguished
by the property, that if a particle of the medium be slightly
displaced in the direction of any one of them, the elastic force
thereby called into play will act precisely in the line of the dis-

——

222 On the Dispersion of the Optic Axes, and

placement. These directions coincide with the axes of the
ellipsoid by which he constructs his wave-surface; and the po-
sition of the axes being thus fixed, it is only their /engths that
can be supposed to vary for the differently coloured rays.
Such is the view taken by Fresnel with regard to crystalline
dispersion, and it is obviously the only view that his theory
admits. Succeeding theorists, in their numerous attempts to
deduce Fresnel’s beautiful laws from dynamical principles, have
always been obliged to assume that the medium is symmetri-
cally arranged with respect to three rectangular planes; and
as, in this hypothesis, the axes of elasticity, or of optical sym-
metry, necessarily coincide with those of symmetrical arrange-
ment, their directions are fixed, as before, independently of
colour.

From these principles it follows that the optic axes for dif-
ferent colours all lie in the same plane, namely, the plane of
the greatest and least axes of the ellipsoid, and that they are
equally inclined to each of the latter axes, so that the angle
made by any pair, to whatever colour they belong, is always
bisected by the same right line. This was accordingly, for a
long time, believed to be the case; and the earlier experiments
of Sir J. Herschel,* which are appealed to by Fresnel, as well
as the observations of Sir David Brewster, seemed to establish
it as a general law. But it was afterwards discovered by Sir J.
Herschel that, in borax, the optic axes for different colours lie
in different planes inclined at very sensible angles to each
other; and the same discovery was made about the same time
(1832) by M. Norrenberg. The latter observer further ascer-
tained, that even when the optic axes all lie in the same plane
there are cases, as in sulphate of lime, wherein their angles are
not bisected by the same right line. These facts, and others of
a like nature that have been since observed, show the falsehood
of the supposition that the lines called the axes of elasticity
have always the same directions whatever be the colour of the

* Phil. Trans. 1820.

of the Axes of Elasticity, in Biaxal Crystals. 223

light ; they are inconsistent with all received notions, and con-
tradict every theory that has been hitherto proposed. No
person, as far as I am aware, has even attempted to explain
them.

But in the theory which I have constructed to represent the
laws of the action of crystallized bodies upon light, and which
has already brought so much within its grasp, the phenomena
in question do not offer any difficulty whatever: on the con-
trary, they are of a kind that would naturally be looked for,
antecedently to experiment. For, in this theory, I make no
hypothesis as to the constitution of the ether, or the arrange-
ment of its molecules; nor any hypothesis, like’that of Fresnel,
respecting the mechanical signification of the axes of elasticity.
The existence of three rectangular axes possessing peculiar pro-
perties is not a principle, but a result, of theory: their direc-
tions are determined by conditions -perfectly analogous to those
which determine the principal axes of an ellipsoid from its ge-
neral equation; and these directions are functions of certain
quantities which are constant when differentials of the second
and subsequent orders are neglected, but which vary when these
are taken into account. The differentials of higher orders in-
troduce terms depending on the wave-length; and thus the
directions, as well as the lengths, of the principal lines depend on
the colour of the light, or, to speak more accurately, on the
length of the wave.

All this will be easily understood if we recur to the first
principles of the theory. According to these, everything de-
pends on the form assigned to the function V in the general
dynamical equation

2
{il dady dz (3 o€ + 3 on + = a) = [if dedydz8V,
from which the motion of the ether is deduced. In my first
memoir on the subject (read to the Academy on the 9th of De-
cember, 1839), I showed that when differentials of the first
order only are preserved, the function V—which may perhaps

224 On the Dispersion of the Optic Axes, and

with propriety be called the potential, since the motion of the
system is potentially, or virtually, included in it—is a function
of the second degree, composed of the three quantities X, Y, Z,
which are connected with the displacements &, n, Z, by the fol-
lowing relations :— |

dn daz
x = dz ra dy’
To show this, I make use simply of the consideration that the
motion must be such as to satisfy the condition
dé dn dz

tee apt Fer

y% dé Z d— dy

de ds’ ~ dy de’

which seems to be characteristic of the vibrations of light. But.
the same condition allows us to suppose that the potential con-
tains not only the quantities X, Y, Z, but their differential co-
efficients of any order with respect to the co-ordinates. This
supposition, however, is too general, and requires to be limited
by other considerations. Now the most natural restriction
which can be imposed consists in the assumption that the quanti-
ties of all orders are formed on the same type, those of any
order being derived from the preceding in the same way that
the quantities X, Y, Z are derived from &, n, €: there are par-
ticular reasons also which go to strengthen this hypothesis, and
have led me to adopt it. Putting therefore

dY dZ dZ dX aX dY
a iy ae tel a
dY, dd, dz, aX, dX, a¥,

Ge ay? de de? dy de

and so on, I suppose the potential to be a function of the second
degree, composed of all the quantities X, Y, Z, X, Vi, Z, Xy
Y,, Z,, &c.; and this is the ‘‘ mathematical hypothesis” alluded
to in the beginning of this article. The hypothesis occcurred to
me more than three years ago (June, 1839), but I did not ven-

of the Axes of Elasticity, in Biaxal Crystals, 225

ture to communicate it to the Academy until the date of my
second memoir. (May, 1841); and even then I had not studied
it with the attention which I now conceive it merits. It was
only very lately, in fact, in some conversations which I had
with M. Babinet during a short visit to Paris, that my attention
was strongly drawn to the subject of dispersion in crystals, par-
_ ticularly the dispersion of the axes of elasticity. My thoughts
then naturally reverted to the hypothesis which I have men-
tioned, and since my return I have found that it affords a com-
plete explanation of all the phenomena.*

T have also found that it gives the general law, extended to
biaxal crystals, of that elliptic and circular polarization which
has hitherto been detected only in quartz and in certain fluids ;
_ while for the case of rectilinear polarization it gives a law (very
possibly a true one) more general than that of Fresnel, but
quite as elegant, and differing very slightly from it. The hy-
pothesis, therefore, is still too general for our present purpose.
To make it include only those crystals to which the law of
Fresnel is rigorously applicable, the alternate derivatives X,,
Y,, Z, X;,-Y3, Z, &c., must be-supposed to vanish in the func-
tion which represents the potential. Then, the axes of co-
ordinates having any fixed directions within the crystal, the
axes of elasticity will be the principal axes of an ellipsoid re-
presented by an equation of the form |

Ax’ + By? + Cz? + 2Dyz + 2Exz + 2Fry = 1,

in which each of the six coefficients—the first, for example—ex-

presses a series of the form
Ba Sy + &0,

where A denotes the wave-length, and all the other quantities

* I am indebted, for my information on the subject, to a short article, drawn
up by MM. Quetelet and Babinet, in the Bulletin of the Royal Academy of
Brussels, Vou. 1. p. 150; as also to Poggendorff’s Annals, Vou. xxvi. p. 309;
Vou. xxxv. p. 81.

Q

226 On the Dispersion of the Optic Axes, &e.

are constant. The ellipsoid itself is the reciprocal of that ellip-
soid by which the wave-surface is constructed, and its semiaxes

are the three principal indices of refraction. As A is supposed
to vary, not only the lengths but the directions of the principal
axes vary, and thus we have a different. wave-surface for every
different wave-length within the crystal.

The optic axes are perpendicular to the circular sections of
the above ellipsoid, and describe, in general, two fragments of a
cone, the equation of which may be found by supposing A to be
variable in the equation of the ellipsoid. But only very par-
ticular cases have been hitherto observed, and I shall not stop
to discuss them. -

J. Mac CuLnacu.

Trin. Cotu., Dusiin, September, 1842.

ieee s yee

XX.—ON THE LAW OF DOUBLE REFRACTION.

[From the Philosophical Magazine, Vou. xx1., 1842].

Havine mentioned, in an article * which I sent a few days ago
for insertion in the Philosophical Magazine, that I had been led,
in following out an hypothesis, to a law of double refraction
more general than that of Fresnel, I think it may be well to
state very briefly the nature of that law, and to point out the
difference between it and the law of Fresnel, especially as I
have since observed that the difference is one of a very extra-
ordinary kind, and one which, if it has a real existence (a
question which experiment only can decide), may serve to
account for phenomena that have seemed hitherto inexpli-
cable.

I have said, in the article referred to, that when the poten-
tial V, which is a function of the second degree, is supposed
to contain only the squares and products of the derivatives
X, Y, Z, X2, V2, Z, Xs, &e., we get the law of Fresnel, as well
as the law of crystalline dispersion; but if we make the more
general, and apparently the more natural supposition, that it
contains also the squares and products of the alternate deriv-
atives X1, Yi, 7, X;, Y3, Z, &c., then we get, of course, a dif-
ferent law. Now I find that there will still be two optic axes
for éach colour, and that the two directions of vibration in a
given wave-plane will have the same relation to them as be-

* ** On the Dispersion of the Optic Axes, and of the Axes of Elasticity, in Biaxal
Crystals ’’ (supra, p. 221).
Q2

228 = On the Law of Double Refraction.

fore; while the difference of the squares of the two velocities of
propagation will continue proportional to the product of the
sines of the angles which the wave normal makes with the optic
axes; but the swm of the squares of these velocities will be in-
creased or diminished by a quantity proportional to the square
of a perpendicular let fall from the centre on the tangent plane
of a certain very small ellipsoid, this tangent plane being sup-
posed parallel to the wave. Such is the general result for biaxal
crystals; but its bearing will be best perceived by taking the
case of a uniaxal crystal, wherein the law of Fresnel reduces
itself to that of Huyghens. .

In this case the wave-surface will, instead of the sphere and
spheroid of Huyghens, consist of two ellipsoids touching each
other at the extremities of a common diameter, which coincides
with the axis of the crystal; one ellipsoid differing slightly
from a sphere, the other slightly from a spheroid. Neither of
the rays will be refracted according to the ordinary law, nor
will the wave-surface be symmetrical round the axis. As the
law of refraction is unsymmetrical, that of reflexion will be so
likewise, and thus we may perhaps obtain an explanation of
the extraordinary phenomena observed by Sir David Brewster
in reflexion at the common surface of oil of cassia and Iceland
spar.

It will no doubt appear strange to call in question the ac-
curacy of the Huyghenian law, which is generally considered
to be established beyond dispute by the experiments of Wol-
laston and Malus. But the fact is, that no exact experiments
have ever been made on the refraction of the ordinary ray.
Neither of those philosophers seems to have entertained any
suspicion that the ordinary law might be inapplicable to it;
they both took for granted that it followed the law of Snellius.
But their results seem to be quite consistent with the suppo-
sition that the ordinary index, for rays passing in different
directions through Iceland spar, may vary in the third place
of decimals, perhaps even in the second. The experiments of
Rudberg throw no light upon the question, for it happens,

On the Law of Double Refraction. 229

oddly enough, that though he had two prisms in every other
case, he used only one of Iceland spar: he could not there-
fore compare the velocities of rays passing in different direc-
tions. On comparing his numbers, however, with those of
Wollaston and Malus, there is, as Sir David Brewster has
observed,* a “surprising discrepancy,” so great indeed as to
be quite “alarming.” After remarking the difficulty of find-
ing any explanation of it, Sir David concludes that it must
arise fronrthe different refractive powers possessed by different
specimens. But though this cause must operate in some
degree, we cannot tell to what extent it is effective, and the
discrepancy may notwithstanding be occasioned, in a great
measure, by a deviation from the Huyghenian law. The whole
question must therefore be reopened, and the ordinary in-
dices for the fixed lines of the spectrum must be determined by
means of different prisms cut out of the same piece of Iceland
spar.
Whatever the result may be—whether it shall confirm the
law of Huyghens, or show that another must be substituted for
it—it will at least be useful for science, by removing the uncer-
tainty in which the subject is at present involved.

* Phil. Mag., 8. 3, Vou. 1. p. 8.

Tain. Coxt., Dusxin, Sept. 24, 1842.

(230°)

XXI.—ON THE LAWS OF METALLIC REFLEXION, AND
ON THE MODE OF MAKING EXPERIMENTS UPON
ELLIPTIC POLARIZATION.

[Proceedings of the Royal Irish Academy, Vou. 1. p. 376.—Read, May 8, 1843.]

SEVERAL years ago, as the Academy are aware, I made an at-
tempt to investigate the laws according to which light is re-
flected at the surface of metals, and I then proposed certain
formule which represented, with sufficient accuracy, all the
facts and experiments which I was able to collect upon the sub-
ject.* But in order to test these formule satisfactorily, it was
necessary to obtain measurements far more exact than any that
had previously been made; and for this end I devised an in-
strument, which was constructed for me by Mr. Grubb, and of
which a brief description has been given in the Proceedings.t I
regret to say, however, that nothing of much consequence has
yet been done with the instrument. Some preliminary trials of
its performance were indeed made in the summer of 1837, and
the results of one of these shall presently be given; but an ac-
cidental strain which it suffered, while I was preparing to un-
dertake a series of experiments, caused me to discontinue the
observations at the time; and being then obliged to superintend
the printing of my essay on the “ Laws of Crystalline Reflexion
and Refraction,”+ my attention was drawn afresh to this latter

* See the Proceedings of the Academy, Vou. 1. p. 2, October, 1886 (supra, p. 58) ;
Transactions, Vou. xvi. p- 71, note (supra, p. 133).

+ Vou. 1. p. 159 (supra, p. 138).
{ Transactions R. I. A. (supra, p. 87).

Laws of Metalhe Reflexion. 231

subject, respecting which some new questions suggested them-
selves, which I thought it right to discuss in notes appended to
the essay. I was not afterwards at leisure to take up the ex-
perimental inquiry, until the beginning of the year 1839, when
I began to think of putting the instrument in order for that
purpose. The strain which it had suffered rendered some slight
alterations necessary; and I now resolved to make additions to
it also, with the view of operating upon the fixed lines of the
spectrum, as a few trials had convinced me that measures suffi-
ciently precise could not be obtained without employing light
of definite refrangibility. I wished, moreover, to take the oppor-
tunity which the nature of the proposed experiments presented,
of verifying the theory of Fresnel’s rhomb, or rather of verify-
ing, by means of the rhomb, the formule which Fresnel has
given for computing the effects of total reflexion, when it takes
place at the common surface of two ordinary media. I wrote
therefore to Munich for several articles which I wanted; among
others, for a set of rhombs cut at different angles, out of dif-
ferent kinds of glass. But while I was waiting for these some
months elapsed, and in the meantime I got sight of a new
theory, which, from its connexion with my former researches,
possessed more immediate interest, and the pursuit of which, in
conjunction with other studies and various engagements, caused.
me again to suspend the inquiry respecting the laws of metallic
reflexion. I allude to the “ Dynamical Theory of Crystalline
Reflexion and Refraction,’ communicated to the Academy in
December, 1839.* This was followed soon after by a general
“ Theory of Total Reflexion,”+ founded on the same principles.
The latter theory, forming a new department of physical optics,
and involving the solution of questions not previously attempted,
was analytically complete when it was communicated to the
Academy in May, 1841; but its geometrical development has _
since required my attention from time to time, and has not yet

been brought to that degree of simplicity of which it appears to

* Proceedings, Vou. 1. p. 374, (supra, p. 145).
{ Ibid. Vou. 11. p. 96 (supra, p. 187).

232 Laws of Metallic Reflexion, and Mode of

be susceptible.* Indeed I have found that, in this instance,
the geometrical laws of the phenomena are by no means obvious
interpretations of the equations resulting from the analytical
solution of the problem; and in endeavouring to verify such
supposed laws I have often been led to algebraical calculations
of so complicated a nature that it has been impracticable to
bring them to any conclusion, and I have been obliged, from
mere weariness, to abandon them altogether. On returning,
however, to the investigation, after perhaps a long interval of
time, I have usually perceived some mode of eluding the calcu-
lations, or of directly deducing the geometrical law; and, when
the theory comes to be published in its final form, no trace of
these difficulties will probably appear.

From the causes above mentioned, combined with frequent
absence from Dublin, the researches which I had entered upon,
respecting the action of metals upon light, have been hitherto
interrupted ; and as it may still be. some time before they are
resumed, I venture, in the meanwhile, to submit to the Aca-
demy the results already spoken of, which were obtained on the
first trial of the instrument, and which afford the best data
that can yet be had for comparison with theory.

The results, it must be confessed, are those of very rough
experiments, made one evening (about the month of July, 1837)
in company with Mr. Grubb, before I had received the instru-
ment from his hands, and merely with the view of showing him,
when it was finished, the kind of phenomena that I proposed to
observe with it, and the mode of observing them. But the in-
strument was so far superior (in workmanship at least) to any
apparatus previously employed for this sort of experiments,
that it was impossible, without great negligence in using it, not
to obtain measures of considerable accuracy. I did not, how-
ever, at the time, set much value on these measures, because I
expected shortly to possess a series of observations made with
every possible precaution ; but having chanced to preserve the
paper on which they were noted down, I was tempted, a few

* Proceedings, You. 11. p. 174 (supra, p- 218).

making Experiments upon Elliptic Polarization. 233

Jays ago, to try how far they agreed with my formule; and the
agreement turns out to be so close, that I think myself justified
in publishing them. Besides, it will be curious hereafter to
compare them with more careful measurements.

Before we proceed, however, to the details of the experi-
ments, it may be well to give the formule in a state fitted for
immediate application. The light incident on the metal being
polarized in a certain plane, let a denote the azimuth of this
plane, or the angle which it makes with the plane of incidence ;
and as the reflected light will be elliptically polarized, or, in
other words, will perform its vibrations in ellipses all similar
and equal to each other, as well as similarly placed, put 0 for
the angle which either axis of any one of these ellipses makes
with the plane of incidence, and let 8 be another angle, such
that its tangent may represent the ratio of one axis of the
ellipse to the other. Then when the optical constants I and x
(of which I suppose the first to be a number greater than unity,
and the second an angle less than 90°) are known for the par-
ticular metal, the angles @ and $B may be computed for any
value of a, at any given angle of incidence, by the following
formule :

(v’ — v) sin 2a
2f + (v’ + v) cos 2a’

tan 20 =
(a)

2g sin 2a
v +v+2f cos 2a’

sin 23 =

in which fand g are constant quantities given by the expres-
sions

f= (zp) 008 x, y= (at+ 5) sin x, (B)

and v, v’ are quantities depending on the angle of incidence #,
in the following way. ‘Let «’ be an angle such that
sins M .
- (c)

sin 7’ cos x’

234 Laws of Metallic Reflexion, and Mode of

and put
cos 7
cos (0)
then will
vate Fish. ae
pee lig = (x)

The angles 0 and 6 are given by immediate observation with
the instrument; and from their values at any incidence, and
for any azimuth a of the plane of primitive polarization, we can
find the constants WJ and y, which we may afterwards use to
calculate the values of @ and 3 for all other incidences and
azimuths, in order to compare them with the observed values.
It is indifferent, in the formule, whether @ be referred to the
major or the minor axis of the elliptic vibration, as also whether
tan 3 be the ratio of the minor to the major axis, or the reci-
procal of that ratio; but in what follows we shall suppose 0 to
be the inclination of the plane of incidence to that axis, which,
when ais 45° or less, is always the major axis; and ( shall be
supposed less than 45°, in order that its tangent may represent
the ratio of the minor axis to the major.

When the azimuth a is equal to 45°, the formule (4) become

he es, Se st
miei oe sin 28 = ———; (F)
from which we may deduce the remarkable relation
tan 28g
cos 20 f’ (<)

showing that, in the case supposed, the ratio of tan 26 to
cos 20 is independent of the angle of incidence. In the experi-
ments which I made with Mr. Grubb this azimuth was always
45°; and the following Table contains the results of observation
compared with those obtained by calculation from formule (F).
The experiments were made upon a small disk of speculum
metal ; and in the calculations I have taken

M= 2-94, y = 64° 28’.

making Experiments upon Elliptic Polarization 235

Value of 6. Value of B.
Angle of
Incidence.
Observed. Calculated. Observed. Calculated.
65° 27° 65! 27°53’ | 28° 0! 28° 0/
70 15 41 15 44 30 7 33 (1
75 — 8 46 —- 9 16 34 10 34 6
80 - 30 15 — 29 25 27 0 26 53
84 — 37 22 — 37 25 16 47 17 17

The light used in these observations was that of a candle
placed at a short distance, and was admitted through small
apertures at the ends of the tubes.* The Nicol’s prism in the
first tube having been secured in a position in which its princi-
pal plane was inclined 45° to the plane of incidence, and the
two arms having been set at the proper angle with the surface
of the metal, the Fresnel’s rhomb and the Nicol’s prism in the
second tube were moved simultaneously, until the image of the
candle became as faint as possible. Had light perfectly homo-
geneous been employed, the image could have been made to
vanish altogether; but instead of vanishing, it became highly
coloured; and our rule in observing was to make the blue at
one side of it, and the red at the other, equally vivid, so as to get
results which should belong, as nearly as possible, to the mean
ray of the spectrum. When this was done, the angles 6 and
(subject, however, to certain corrections which will be hereafter
explained) were respectively read off from the divided circles
belonging to the rhomb and the prism. The observations were
made at large incidences, because it is within the last thirty de-
grees of incidence that the phenomena go through their most
rapid changes.

If we now cast our eyes on the above Table, making due
allowance for the uncertainty arising from the dispersion of the
metal, we shall be struck with the agreement between the cal-
culated and observed numbers. The differences are greatest in

* See the description of the instrument in the Proceedings, Vou. 1. p. 159 (supra,
p- 188).

236 Laws of Metallic Reflexion, and Mode of

the last two observations, which, however, were really the first ;
for the observations were made in the inverse order of the inci-
dences, and their accuracy may have improved as they went on.
However that may be, the differences are quite within the
limits of the errors of observation; and they are actually less
than those which Fresnel found to exist between calculation
and experiment in the much simpler case of reflexion at the
surface of a transparent ordinary medium, when he proceeded
to verify the formula which he had discovered for computing
the effect of such reflexion.*

It may seem extraordinary that these experiments should
have been in my possession for nearly six years before I became
aware of their close agreement with my formule; but the fact
is, that I did not regard them with much interest, because,
from the circumstances in which they were made, I did not ex-
pect more than a general accordance with theory. And even
now I am in no haste to infer the absolute exactitude of the
formule, though they are found to represent the phenomena
so well. It was far more allowable to infer that the formula of
Fresnel was exact in the case just mentioned, though it appeared
to represent the phenomena less perfectly. For, to say nothing
of the small number of our experiments, the present is a much
more complicated case, and the phenomena depend on two con-
stants instead of one, so that the formule might be slightly
altered, and yet perhaps continue to agree very well with rough
experiments. Where there is only one constant this is not so
probable. Again, there is one of the quantities in the preceding
formule which may be greatly altered without producing more
than a slight effect on the values of 0 and 8. This quantity is
the ratio of sin ¢ to sin 7’, which, according to the value in for-
mula (c), is a number so large as to make the angle 7’ always
small, so that its cosine never differs much from unity; and
therefore if the above ratio were taken equal to any other large
number, the value of u in formula (p) would remain nearly the

* See the Table which he has given in the Annales de Chimie, tom. xviii.
p. 314. :

making Experiments upon Elliptic Polarization. 237

same, and consequently the values of @ and 6 would be but
slightly changed.

It is with regard to the value of mu as a function of the inci-
dence that I entertain the greatest doubts, and if any defect
shall be found in the formule I think it will be here. The re-
lations (c) and (p), from which » may be deduced in terms of 7,
were not indeed adopted without strong reasons; but I am not
entirely satisfied with them ; because, when we reverse the prob-
lem, and seek to determine the constants Wand y from the ob-
served values of 8 and 8 at a given incidence, the results are
rather complicated and involved, though the approximate deter-
mination is easy enough. As the formule are in a great mea-
sure built upon conjecture, we must not be disposed to receive
them without the strongest experimental proofs; and it will
certainly require experiments of no ordinary accuracy to decide
some of the questions which may be raised respecting them.

When plane-polarized light is incident on a metal, if its vi-
brations be resolved in directions parallel and perpendicular to
the plane of incidence, the effect of the reflexion is to change
unequally the phases of the resolved vibrations ; and it may be
useful to have the formule which express the difference of
phase after reflexion, and the ratio of the amplitudes of vibra-
tion. Put @ for the difference of phase; and supposing, for
simplicity, the incident light to be polarized in an azimuth of
45°, let o be an angle less than 45°, such that tan o may represent
the ratio of the reflected amplitudes respectively perpendicular
and parallel to the plane of incidence ; then we shall have

2g af
Re oy 008 2o = 5 (11)
from which we may infer that
sin @ tan 20 = g (1)

F

or that the product on the left side of the last equation is inde-
pendent of the angle of incidence. It is to be observed that the

238 «Laws of Metallic Reflexion, and Mode of

relations (G) and (1) are independent of the value of u, and may
hold good though that value should require to be changed.

All the preceding formule are merely mathematical conse-
quences of those which I published long ago in the Zransactions*
of the Academy. The formule which I had previously given
in the Proceedings,t are slightly different, and, I think, less
likely to be exact, because they are less simple, and do not lead
to any of the remarkable relations which may be deduced from
the others.

Having had occasion, in the course of the few experiments
which I made with the instrument before mentioned, to study
the nature of Fresnel’s rhomb, which constitutes an important
part of it, I shall here describe the method which must be fol-
lowed in order to obtain true results, when the rhomb is em-
ployed in observations on light elliptically polarized. A ray in
which the vibrations are supposed to be elliptical is given, and
what we want is to determine the ratio of the axes of the
elliptic vibration, and their directions with respect to a fixed
plane passing through the ray; in other words, to determine
the angles which we have denoted by 6 and @ in the case of a
ray reflected from a metal. For this purpose the ray is ad-
mitted perpendicularly to the surface at one end of the rhomb,
and after having suffered two total reflexions within, passes out
perpendicularly to the surface at the other end. Then causing
the rhomb to revolve about the ray, we shall find two positions
of it in which the emergent light will be the plane-polarized,
these positions being readily indicated by a Nicol’s prism inter-
posed between the rhomb and the eye; for such a prism, by
being turned round the ray, can make the light totally disap-
pear when it is plane-polarized, but not otherwise. These two
positions of the rhomb will be exactly 90° from each other; in
one of them the principal plane of the rhomb (the plane of re-
flexion within it) will be parallel to the major axis of the elliptic
vibration, and the angle which it makes with the plane of inci-

*Vou. xvii. p. 71 (supra, p. 183). ¢t Vou. 1. p. 2 (supra,"p. 58).

making Experiments upon Elliptic Polarization. 239

dence on the metal will be equal to 6: while in the same posi-
tion the angle which the principal plane makes with the plane
of polarization of the emergent ray (as given by the Nicol’s
prism) will be equal to 3. In the other position, the principal
plane will be parallel to the minor axis of the elliptic vibration,
and the corresponding angles will be equal to 90° - @ and
90° — 8 respectively. This, however, proceeds on the supposi-
tion that the rhomb is exact. When it is not so, which is of
course the proper supposition, and a very necessary one in the
experiments with which we are concerned, there will still be,
generally speaking, two positions of it in which the emergent
ray will be plane-polarized, or in which a disappearance of the
light may be produced by the Nicol’s prism ; but these positions
will no longer be 90° from each other, nor will the principal —
plane, in either of them, coincide with an axis of the elliptic
vibration. If we now measure the angles between the different
planes as before, and denote them by 0’, (3’ in the first position,
and by 90° — 0”, 90° — 3” in the second, we shall find that 0’
and 9” are unequal, but we shall have 3’ equal to 8”. The va-
lues of @ and ( will then be given by the formule

+ 0” _ 608 2’ ;
er ee cos 23 = cos ( — 0”): (x)

The error of the rhomb may easily be found. Supposing
the vibrations to be resolved in directions parallel and perpen-
dicular to its principal plane, the rhomb is intended to produce
a difference of 90° between the phases of the resolved vibrations,
or to alter by that amount the difference of phase which may
already exist; but the effect really produced is usually different
from 90°, and this difference, which I call ¢, is the error of the
rhomb. The value of « is given by the formula

_ sin (0 - 6”).
tan ¢ = fan 2p ; (1)

and as the error of the rhomb is a constant quantity, we have

240 Laws of Metallic Reflexion, and Mode of

thus an equation of condition which must always subsist between
the angles 6’- 0” and 3. For any given rhomb the sine of the
first of these angles is proportional to the tangent of twice the
second, and therefore 6’ — 0” constantly increases as (3 increases
towards 45°, that is, as the axes of the elliptic vibration approach
to equality. When # is equal to 45°-4<, we have 0’- 6” =90°;
and for values of (3 still nearer to 45°, the value of sin (6’ — 6”)
becomes greater than unity, that is to say, it becomes impos-
sible, by means of the rhomb, to reduce the light to the state of
plane-polarization. This is a case that may easily happen with
an ordinary rhomb in making experiments on the light re-
flected from metals; because at a certain incidence, and for a
certain azimuth of the plane of primitive polarization, the re-
flected light will be circularly polarized.

The rhomb which I used in the experiments tabulated above
was made by Mr. Dollond, and was perhaps as accurate as
rhombs usually are; it was cut at an angle of 544°, as pre-
scribed by Fresnel. Its error was about 3°, and the value of
6 — 0’, at the incidence of 75°, was about 7°. But in another
rhomb, also procured from Mr. Dollond, and cut at the same
angle, the value of 0’ — 0”, under the same circumstances, was
about 20°, and the value of « was therefore about 8°. The
angle given by Fresnel was calculated for glass of which the
refractive index is 1:51; and the errors of the rhombs are to be
attributed to differences in the refractive powers of the glass. I
was not at all prepared to expect errors so large as these when
I began to work with the rhomb, and they perplexed me a good
deal at first, until I found the means of taking them into ac-
count, and of making the rhomb itself serve to measure and to
eliminate them. ‘The value of the rhomb as an instrument of
research is much increased by the circumstance that it can thus
determine its own effect, and that it is not at all necessary to
adapt its angle exactly to the refractive index of the glass. It
may also be remarked, that this circumstance affords a method
of directly and accurately testing the truth of the formule
which Fresnel has given for the case of total reflexion at the se-

making Experiments upon Elliptic Polarization. 241

parating surface of two ordinary media; for we have only to
measure the angle of the rhomb and the refractive index of the
glass, and to compute, by Fresnel’s formula, the alteration
which the rhomb ought to produce in the difference between the
phases of the resolved vibrations; which alteration of phase
we may then compare with that deduced, by means of the for-
mule (K) and (x), from direct experiment.

If, in each position of the rhomb, we measure the angle
. which the plane of polarization of the emergent ray makes with
the plane of incidence on the metal, and call the two angles re-
spectively y’, y’, we shall have

y’ 7 = B, y” = 0” as Bp, (m)
and therefore |
oy ty’ =0' +0" =20, 28’ =y" - 7 + 0'- 0"; (x)

from which it appears that if the rhomb were perfectly exact,
that is, if 0’ and 6” were equal to each other, the angle @ would
be half the sum of y’, y”’, and the angle 3 half their difference.
It would then be sufficient to measure the angles y’ and y”, in
order to get 9 and B accurately. And if the rhomb were erro-
neous, the true value of @ would still be half the sum of y’,
7’; but the true value of $8 would not be discoverable without
measuring the angles 0’, 0”, by the help of which it can be de-
duced from the second of formule (N), combined with the
second of formule (Kk). Nor can we discover whether the
rhomb is erroneous or not, without measuring the angles @’,
6” ; and therefore as these angles must be measured in any case,
the former method of determining 0 and (3 is to be preferred.
In making experiments on elliptically polarized light, a
plate of mica, or any other doubly-refracting crystal, placed per-
pendicular to the ray, may be used instead of Fresnel’s rhomb.
If the thickness of the crystalline plate be such that the interval
between the two rays which emerge from it is equal to the
fourth part of the length of a wave, for light of a given refran-
gibility, the plate will, for such light, perform all the functions

R

242 Laws of Metallic Reflexion, and Mode of

of the rhomb; the principal plane of the rhomb being repre-
sented by the plane of polarization of one of the emergent rays.
But unless the light be perfectly homogeneous, this method is
liable to great inaccuracy in practice, since the effect of the plate
in producing or altering the difference of phase between the two
rays which interfere on their emergence from it is inversely
proportional to the length of a wave, and will therefore be ex-
tremely different for light of different colours, and will change
yery perceptibly even within the limits of the same colour. It
is true, the effect of the rhomb always varies with the colour of
the light: but this variation is trifling compared with that which
exists in the other case. It was for this reason that Iemployed ~
the rhomb in my experiments, instead of a crystalline plate.
The apparatus, however, is much simplified by using such a
plate ; and if any one chooses to do so, and to work with homo-
geneous light, he must take care to follow, in every respect, the
directions which I have given for conducting experiments with
the rhomb. The two cases are precisely similar; and if it be
necessary not to neglect the errors of the rhomb, it is certainly
not less necessary to take into account those which may arise
from a want of accuracy in the thickness of the plate, consider-
ing how difficult it is to make the thickness correspond exactly
to the particular ray which we wish to observe.

I have been induced to enter into these particulars, respect-
ing the mode of making experiments on elliptic polarization,
because the subject is one which has not hitherto been studied ;
nor does it seem to have occurred to anyone that any precaution
was requisite beyond that of getting the rhomb cut as nearly as
possible at the proper angle, or the crystalline plate made as nearly
as possible of the proper thickness. This, indeed, was quite suf-
ficient for ordinary purposes. For example, light polarized ina
plane inclined 45° to the principal plane of the rhomb or of the
plate, would, as far as the eye could judge, be circularly pola-
rized after passing through either of them. Notwithstanding a
certain error in the angle of the one, or in the thickness of the
other, such light would, when analyzed by a rhomboid of Iceland-

making Experiments upon Elliptic Polarization. 243

spar, give two images always sensibly equal in intensity. But
an error which could not be at all detected in this way might
produce a very great effect in such experiments as those upon
the metals, and, for the purpose of comparison with theory,
might render them entirely useless, if in the first method of
observing we relied upon one set of observations, taking (sup-
pose) the values of 0 and 9’ for the true values of 0 and 6; or
if, in the second method, we contented ourselves with merely
measuring the angles y’ and y”.

The necessity of attending to the foregoing rules and re-
marks will appear from an examination of the experiments of
M. de Senarmont, published in the Annales de Chimie.* In
these very elaborate experiments, which were made upon light
reflected at various incidences from steel and speculum metal,
the author followed a plan similar to that which I have adopted,
and which, in a general way, I had previously sketched in the
Proceedings + of the Academy. There was this difference, how-
ever, that he used a plate of mica instead of Fresnel’s rhomb.
Now as he worked with common white light, the use of the
mica plate must have rendered two kinds of errors unavoidable.
In the first place, it would be impossible always to take the ob-
servations for the same ray of the spectrum; and next, as a
consequence of this, the thickness of the plate would be gene-
rally inexact for the particular ray to which the observations
happened to correspond. If the thickness of the plate were
exact for a certain ray, it would be very sensibly inexact even
for the neighbouring parts of the spectrum; and as the part of
the spectrum to which the observations belonged was continually
changing, the results obtained for different incidences and azi-
muths would not be comparable with each other, even though,
in each separate case, the error of the plate were allowed for
and eliminated. The values of 0, however, as determined by
M. de Senarmont, would be correct, so far as this error is con-
cerned; those of (§ alone would be erroneous. For the values
of 0 were determined in two ways: by measuring the angles

_* Tom, Lxxiii, pp. 351-358. + Vo, 1. p. 159.
R2

244 Laws of Metallic Reflexion, and Mode of

0, 6’, and taking their sum for 20; also by measuring the
angles y’, y”’, and taking their sum for the same quantity.
Now each of these methods gives a true value of 0, because by
the preceding formule we have 20=0'+@0’=y'+y"; and
this accounts for the agreement, shown by the tables of M. de
Senarmont, between the values* of 20 obtained by these different
methods. But the values of were deduced from the angles
Y; vy’; by simply making their difference equal to 23; and we
see by the second of formule (n) that, when the plate is not of
the proper thickness, this value of 26 is erroneous by the whole
amount of the angle 6’ - 0”, the difference between (’ and B
being supposed so small that it may be neglected. As M. de
Senarmont proceeded on the common assumption that when the
thickness of the plate has been adjusted to that part of the
spectrum to which the observations are intended to refer, it
may afterwards, through the whole series of experiments, be re-
garded as exact, he necessarily conceived 6’ and 6” to be the
same angle; and it was on the principle of taking an average
between two measures of the same quantity that he made the
supposition 20 = 0’ + 6”, which happened to be correct. When
therefore he found @ and 0” to be different, he of course looked
upon the difference as merely an error of observation, which it
would be superfluous to tabulate. Not having the values of
this difference, therefore, we have not the means of immediately
correcting the values of 28. But as observations were made
for several azimuths at each angle of incidence, we may use the
values of @ to determine those of 8; for when at any inci-
dence (except that of maximum polarization, where @ = 0 for all
azimuths) the values of @ are known for two given values of a,
we can deduce the corresponding values of 6, without any other
theory than that of the composition of vibrations. The values

* Or rather the values of 180° + 26; because the angle w, the double of which
appears in the tables of M. de Senarmont, is equal to 90°+ 6. The angles which
he calls +1 and y2 are equal to 90° + y” and 90°+ yy’ respectively. It therefore
comes to the same thing, whether the one set of angles or the other is supposed to
be measured. The letter 6 has the same signification in both notations.

making Experiments upon Elliptic Polarization. 245

of B so deduced must indeed be expected to be very inaccurate,
partly because of errors in the observed values of 0, partly be-
cause the observations in different azimuths do not answer to
the same ray of the spectrum; but they will be accurate enough
to show the great amount of the error committed by neglecting
the difference 0-6”. For example, putting @, and (3, for the
values of @ and 8 when a = 45°, M. de Senarmont gives, at the
incidence of 60° upon steel, 20, = 64° 15’ (taking the mean of
his two determinations), and for the azimuths 55°, 30°, 25°, he
gives 20 equal to 88° 5’, 37° 2’, and 29° 36’ respectively. Com-
bining these values of 20 in succession with that of 20, we get
for 2 the series of values 32° 38’, 33° 28’, 34° 30’; the dif-
ferences between which are to be attributed to the causes above
stated. The mean value of 28, thus found is 33° 32’; while
its value, as given by M. de Senarmont, is only 28° 41’. The
difference 4° 51’ is the value of 0’- 6’, which, divided by the
tangent of 2(6,, gives 7°19’ for the mean value of «, the error of
the mica plate corresponding to that part of the spectrum which
was observed at the incidence of 60°.

At incidences nearer the angle of maximum polarization
the errors are probably much greater. Beyond that angle
they again diminish, and in some cases they almost vanish.
Thus, at the incidence of 85° upon steel, with the value of 20,
and the value of 20 corresponding to a = 20°, we get, by com
putation, a value of 2/3, which differs only by a few minutes
from that given by M. de Senarmont. Nearly the same thing
happens at the same incidence when we take a = 25°. In these
cases, therefore, the results belong to that particular ray for
which the thickness of the plate was exact.

The observations of M. de Senarmont on speculum metal
were not carried beyond the incidence of 60°. He states that
he was unable to observe at higher incidences, on account of the
uncertainty arising from the dispersion of the metal; but though
this cause operated in some degree, his embarrassment must
have been really occasioned by the increasing magnitude of the
difference 6’ - 0”, as he approached the angle of maximum po-

246 Laws of Metallic Reflexion, and Mode of

larization ; that difference being perhaps twice as great as in
the case of steel. My own experiments on speculum metal were
all made, as has been seen, at incidences greater than 60°.

The experiments of M. de Senarmont do not at all agree
with the formule; and therefore I have been obliged to analyze
his method of observation, and to show that it could not lead to.
correct results. It is to be regretted that his method was de-
fective, as the zeal and assiduity which he has displayed in the
inquiry would otherwise have put us in possession of a large
collection of valuable data.

I shall conclude by saying a few words respecting the in-
tensity of the light reflected by metals. The formule for com-
puting this intensity have been given in the Transactions of the
Academy, in the place already referred to; but they may be
here stated in a form better suited for calculation. If we sup-
pose ¥ and y’ to be two angles, such that

cotan y = 7 cotan = My, (0)

and then take two other angles w, w’, such that
cosw=sin 2p cos x, cosw =sin 2y~’ cos x, (R)
we shall have
r=tan}w, 7 =taniw, (Q)

where r is the amplitude of the reflected rectilinear vibration,
when the incident light is polarized in the plane of incidence,
and +r’ is the amplitude of the reflected vibration when the
incident light is polarized perpendicularly to that plane; the
amplitude of the incident vibration being in each case supposed
_ to be unity. Hence when common light is incident, if its in-
tensity be taken for unity, the intensity J of the reflected light
will be given by the formula

I=} (tan’4w + tan?4w’). (R)

If with the values of If and x determined by my experi-

making Experiments upon Elliptic Polarization. 244

ments we compute, by the last formula, the intensity of re-
flexion for speculum metal at a perpendicular incidence, in
which case w= 1, we shall find J= +583. This #s considerably
lower than the estimate of Sir William Herschel, who, in the
Philosophical Transactions for 1800 (p. 65), gives *673 as the
measure of the reflective power of his specula. The same num-
ber, very nearly, results from taking the mean of Mr. Potter’s
observations.* It might seem therefore that the formula is in
fault ; but I am inclined to think that the metal which I em-
ployed had really a low reflective power. Its angle of maxi-
mum polarization was certainly much less than that of the specu-
lum metal used by Sir David Brewster,t who states the angle
to be 76°, whereas in my experiments it was only about 734° ;
and ‘any increase in this angle, by increasing the value of /,
raises the reflective power. On the other hand, the maximum
value of 8 (when a = 45°) was greater than that given by Sir
David Brewster, namely, 32°; and any increase in 8 tends also
to increase the reflective power. Now it is not unreasonable to
suppose that the highest values of both angles may be most
nearly those which belong to the best specula ; and accordingly,
if we take 76° for the incidence of maximum polarization, and
retain the maximum value of 3, namely, 34° 37’, which results
from my experiments, we shall get I = 3°68, y = 66° 16’, and the
value of I at the perpendicular incidence will come out equal to
662, which scarcely differs from the number given by Herschel.

It is clear from what precedes that the optical constants are
different for different specimens of speculum metal, and this is
no more than we should expect, from the circumstance that the
metal is a compound, and therefore liable to vary in its optical
properties from variations in the proportion of its constituents ;
but I am disposed to believe that the same thing is generally
true, though of course in a less degree, of the simple metals ;
so that in order to render the comparison satisfactory, the
measures of intensity should always be made on the same spe-

* Edinburgh Journal of Science, New Series, Vou. 11. p. 280.
+ Philosophical Transactions, 1830, p. 324.

248 Laws of Metallic Reflexion, &c.

cimen which has furnished the values of Mand y. There is
one metal, however, with respect to which there can be no
doubt that the experiments of different observers are strictly
comparable, when it is pure, and at ordinary temperatures—I
mean mercury. For this metal Sir David Brewster states the
angle of maximum polarization to be 78° 27’, and the maximum
value of 3, when a= 45°, to be 35°; from which I find UW = 4°616,
x = 68°13’, and, at the perpendicular incidence, J = °734.
Now Bouguer observed the quantity of light reflected by mer-
cury, but not at a perpendicular incidence. His measures were
taken at the incidences of 69° and 784°, for the first of which
he gives, by two different observations, ‘637 and ‘666; for the
second, by two observations, ‘754 and ‘703, as the intensity of
reflexion.* If we make the computation from the formula,
with the above values of Mand x, we find the quantities of
light reflected at these two incidences to be, as nearly as pos-
sible, equal to each other, and to seven-tenths of the incident
light, the intensity of reflexion being’a minimum at an inter-
mediate incidence; and if we suppose these quantities to be
really equal at the incidences observed by Bouguer, we must
take the mean of all his numbers, which is 69, as the most pro-
bable result of observation. This result differs but little from
one of the two numbers given by him at each incidence, and
scarcely at all from the result of calculation. |

The angle at which the intensity of reflexion is a minimum,
when common light is incident, may be found from the formula

(ar + 7) (u+ 2) = (W- F)V FFA - 4 008, (s)

which gives the values of u, and thence that of ¢. This inci-
dence for mercury is, by calculation, 75° 15’, and the minimum
value of J is ‘698, which is less than its value at a perpendicular
incidence by about one-eighteenth of the latter. According
to the formule, the reflexion is always total at an incidence
of 90°. | ;

* See his Traité d Optique sur la Gradation de la Lumiére: Paris, 1760; pp. 124,
126.

( 249 )

XXIJ.—ON THE ATTEMPT LATELY MADE BY M. LAURENT
TO EXPLAIN, ON MECHANICAL PRINCIPLES, THE
PHENOMENON OF CIRCULAR .POLARIZATION IN
_ LIQUIDS. ©

[Abstract of a Communication addressed to the BritTisH ASSOCIATION,
in the year 1843. Report, p. 7.]

Tue Author showed that this attempt had not succeeded. M.
Laurent supposes the particles of the luminiferous ether not
to be simply material points, but to have dimensions which
are not insensible when compared with their distances; and on
this hypothesis he deduces a system of differential equations,
the integrals of which he conceives to represent the phenome-
non in question. The integrals given by M. Laurent are,
however, altogether erroneous, though this circumstance was
not noticed by M. Cauchy in the remarks and comments
which he made on M. Laurent’s Memoir. The true integrals
of these equations (supposing the equations to be correctly
deduced) were shown by Professor MacCullagh to indicate
motions of the ether which do not correspond to the observed
phenomenon. The account of M. Laurent’s theory, with M.
Cauchy’s remarks upon it, will be found in the eighteenth
volume of the Comptes Rendus of the Academy of Sciences
of Paris.

( 250 )

XXIIT.—ON TOTAL REFLEXION.

Proceedings of the Royal Irish Academy, Vou, ut. p. 49.—Read, January 13, 1845.

Proressor Mac OvtiacH made a communication on the subject
of Total Reflexion.

In the case of total reflexion the vibrations which take place
in the rarer medium are in general elliptical, and when this
medium is a crystal, the equations by which the ellipse of vi-
bration is determined are very complicated. The projection
of this ellipse upon the plane of incidence may, however, be
easily found by the remark in p. 12 of the present volume;
the projecting cylinder is therefore known, and as the ellipse of
vibration is a section of this cylinder, the question of deter-
mining the ellipse is reduced to that of determining its plane.
For this purpose Mr. Mac Cullagh gave the following rule.
Having constructed the ellipsoid of indices (that whose axes
are parallel to the axes of elasticity, and inversely proportional
to the three principal velocities of propagation in the erystal),
let its two planes of circular section intersect the aforesaid
cylinder. The curves of intersection will be ellipses, which
shall be supposed to have a common centre O in the axis of
the cylinder. Let OP, OP’ be the greater semiaxes of these
ellipses, and OQ, OQ the less semiaxes; the lengths of the
two former being denoted by p, py’, and the lengths of the two
latter by g, g’. Join the extremities P, P’ of the greater
semiaxes, and the extremities Q, Q’ of the less semiaxes; and

On Total Reflexion. | 251

divide each of the right lines PP’, QQ’, in the ratio of

/p-¢ to /p?-¢*. Then a plane drawn through the
centre O and the two points of division will be the plane of
vibration. In the application of this rule some precautions
are to be observed, but they need not here be insisted on.

The foregoing rule was deduced (in the year 1843) from
the general equations by a peculiar use of imaginary quanti-
ties, after the author had several times tried in vain to obtain a
geometrical interpretation of those equations by considerations
of a more obvious and ordinary kind. This use of imaginaries
is founded on a remarkable theorem relative to the ellipse, by
which it appears, that the plane of an ellipse and its species
(that is, the directions and the ratio of its axes) may be ex-
pressed by two imaginary constants, just as the direction of a
right line in space is expressed by two real constants. By
means of this theorem—which it is unnecessary to repeat, as
it has been published in the University Calendar*—we may find
such properties of elliptical vibrations as are analogous to those
of rectilinear vibrations; and it was in this way that the above
rule was discovered. It is analogous (though it scarcely appears
so at first sight) to the rule by which, in the theory of Fresnel,
the direction of rectilinear vibrations is determined, when the
plane of the wave is given.

* “Examination Papers” of the year 1842, p. lxxxiy. [The theorem is as
follows :—

Given an ellipse in space, the origin of co-ordinates being taken at the centre.
Let the points x, y, 2, 2’, y’, 2’, be the extremities of conjugate diameter. Then
the imaginary quantities

ytyf-1 2+2f-1
ciel feat

are constant for every system of conjugate diameters. ]

¢

bo
a:
Hook
ae
oe
Bre,
o

I.—GEOMETRICAL THEOREMS ON THE RECTIFICATION
OF THE CONIC SECTIONS.

[ Transactions of the Royal Irish Academy, Vou. xvi. p. 79.—Read, June 21, 1830.]

Lemma 1. Let 7 and ¢ be two points indefinitely near each
other on any given curve AT, and let tangents at Zand ¢ meet
in the points P and p any
other given line WN,
straight or curved, and
draw Pq perpendicular to
ty; then the difference
between the are AT and
the tangent TP will ex-
ceed or fall short of the ak
difference between the arc At and the tangent ¢p by a quantity
which is ultimately to pq in a ratio of equality.

For the increment of 7'P, or the difference of TP and 7p, is
ultimately equal to the sum of pg, VT, and Vt (V being the
intersection of pt and PTZ’ produced); and 7Z¢, or the increment
of the arc AT, is ultimately equal to the sum of VZ' and Vt;
therefore the difference of the increments is ultimately equal to
pq. Whence the proposition is manifest.

PrositeM.—To find the Length of the Are of a Parabola.

Let F' be the focus and A the vertex of a parabola AT:
draw AK perpendicular to AF, and let a tangent at the point
T meet it in P; take p indefinitely near to P, and let Fp

256 Geometrical Theorems on the

intersect PT’ in g: then since YP and Fp are perpendicular to
the tangents P7' and pt, it follows (by the preceding lemma) that
Pq is ultimately equal to the increment
of the difference between the are AT’
and the tangent 7'P. With the centre
F and semiaxis 7A describe the equi-
lateral hyperbola.G.4H, and _bisect
the angles AFP, Ap by the straight
lines FL, F/. Then (since the square
of the radius vector of an equilateral
hyperbola is inversely as the cosine of
twice the angle which it makes with
the axis) the square of LF will be
equal to Af’ x PF; and because the
angle LF/ is half the angle PJ, there- Mig.

fore the area LF7 will be equal to one-fourth of the rectangle
under AFand Py.

Hence the hyperbolic sector AFL is equal to one-fourth of
the rectangle under AF and the difference between the para-
bolic are AT and the tangent 7'P.

Lemma 2. If on either axis of an ellipse a semicircle be
described, of which CD and CH
are two radii at right angles to
each other, and if DN and HM
be drawn perpendicular to the
axis aA, and meeting the ellipse
in Hand Z; then CE and CZ Fig. 3.
will be conjugate semidiameters.

For tangents at H and LZ will meet in a point 7 in Aa pro-
duced; the triangles HMT and DNC will be similar, and DV
and HWM will be similarly divided in Z and LZ; therefore COE —
and TZ will be parallel, and consequently CH and OL will be
conjugate semidiameters.

Lemma 3. Take in the ellipse a point / indefinitely near to
ZL, and draw through it mh parallel to ZH; join Ch, and with
the centre CO and a radius equal to CH describe the are Kh

Rectification of the Conic Sections. 257

meeting CH and Chin K andk: then Kk will be ultimately
equal to Li.

For Li: Hh::L7:HT::CE: CD::CK: CH:: Kk: Hh.
Therefore Li = Kh.

Lemma4. If the semiaxes AC and BC of an ellipse be equal
to the sum and difference of the sides PQ, QR, of a triangle PQR,
and if the angle BCD be

equal to half the contained -~ aus
angle PQR (ADa being the B ae
semicircle on Aa); then, *
DEN being drawn perpn- *~ “~“ ° _ ss
dicular to Aa, CE will be ahs Bi

equal to the base PR. Take QU and QS equal to QR; then
PS and PU will be equal to AC and CB, and the angle CDN
to TSP; therefore drawing PT perpendicular to 7S, DN and
NC will be equalto TSand 7P. But URS being a right angle,
UR is parallel to PT, and therefore TS: TR:: PS: PU:: AC
>: BC:: DN: EN; but TS= DN, therefore TR = EN; and
since PT = CN, it follows that PR = CE.

THEOREM.

Let AT and A’7” be an ellipse and hyperbola, the semiaxis
(CA or C’A’) of either being equal
to (C’F’ or CF) the distance between
the focus and centre of the other; and
let tangents at the points 7’ and 7”
meet in P and PF” the circles described
on the axes, so that FP = F’P’: let Fig. 5.
also a” B”’ A” be another ellipse whose semiaxes (4” C” and
B” C”) are equal to aF and FA,
and take in its circumference a T
point Z so that the semidiameter
conjugate to that passing through
I may be equal to FP or FP’; rae ee Gres area
then will the excess of the ellip- Fig. 6.
tic are AT' above its tangent 7P be greater than the excess of

s

258 Geometrical Theorems on the

the hyperbolic are A’Z” above its tangent 7’P’, by twice the
elliptic are A” LZ.

Take the point Z so that when the ordinate MZH is drawn
to meet in H the semicircle described on the axis, the angle
HC” M may be equal to D
half the angle PCF (or
P’C'F’, for the triangles : k
PCF and P’C’F’ have H
all their sides and angles E B"
equal); draw C”’D per-
pendicular to C’H, and @” o” 7M A"
DE to A’ a’, meeting the Fig. 7.
ellipse in #; then, by lem. 2, C’”E will be conjugate to OC” L,
and by dem. 4 it will be equal to FP, since the angle B’C”’D is
equal to HC” M, and is therefore half of PCF, whilst the semi- ~
axes C’” A”, C” B”’, are the sum and difference of PC and CF.
Hence the point Z thus found is that required by the enuncia-
tion. Take p, yp’, h, indefinitely near to P, P’, H, and similarly
related to each other; let Fp and Fp’ intersect. 7P and 7’P’ in
q and ¢, and with the centre C” and a radius equal to C”# de-
scribe the evanescent arc Ak. Then FP, F’P’ are always per-
pendicular to 7P, T’P’, and therefore Py is ultimately the
increment of the difference between the are A7' and the tangent
TP (lem. 1.) and P’¢ the increment of the difference between
the are A’Z” and the tangent Z7’P’; also by lem. 3. Kk is ulti-
mately equal to Z/, the increment of the are A” LZ. ‘Now the
angles CFP and Cp are equal to C’P’F” and C’p’F’; therefore
PFp is equal to the sum of P’F’p’ and P’C’p, or to P’ Fp’ with
twice HO”h: but FP, F’P’, and C’ K are all equal, and there-
fore Pq is equal to P’g’ and twice Xk, or to P’¢’ and twice L/.
Whence the proposition is manifest.

Schol. The angle B’C” H is half the angle SCP or f’C’P’,
and therefore by /em. 4, the semidiameter C’Z is equal to the
straight line P or /’P’. This gives another and easier method
of finding the point L.

Hence every are of a hyperbola may be found by means of

Rectification of the Conic Sections. 259

two elliptic ares. This beautiful theorem was discovered by
Landen, and is now for the first time demonstrated geometri-
cally; but the manner in which it is stated by him and by suc-
ceeding writers differs from the above, and is much more
complicated. The different forms under which they have pre-
sented it may be easily deduced from the preceding, by means
of the theorem of Fagnani, of which a geometrical demonstra-
tion was first given in vol. ix. of the Transactions of the Royal
Irish Academy, by the Lord Bishop of Cloyne.

December 23; 1829.

1d (260)

IIl.—ON THE SURFACES OF THE SECOND ORDER.

[Proceedings of the Royal Irish Academy, Vou. 1. p. 446.—Read Nov. 80, 1843.]

THERE is hardly any geometrical theory which more requires to
be studied, or which promisés to reward better whatever thought
may be bestowed upon it, than that of the surfaces of the second
order. My attention was drawn to it, many years ago, by the
consideration of mechanical and physical questions. In the
dynamical problem of the Rotation of a Solid body, and in the
investigation of the properties of the Wave-Surface of Fresnel,
I found, so long since as the year 1829, that the ellipsoid could
be employed with very great advantage; while the discussion of
these questions, but especially of the former,* suggested proper-
ties of the ellipsoid and its kindred surfaces which I might not
otherwise have perceived. In this manner I was led to consider
systems of confocal surfaces, and thence to notice the focal
curves, which I discovered to be analogous, in the theory of the
surfaces of the second order, to the foci in that of the plane
conic sections. That theory now began to interest me on its
own account, and, guided by analogy, I struck out the leading
properties possessed by the surfaces in relation to their focal
curves; but the interference of other matters prevented me from
continuing the inquiry. I had done enough, however, in this

* The Theory of Rotation, here spoken of, was completed in the year 1831 ; but,
from causes which need not be mentioned at present, it was not published. The
investigations relative to Fresnel’s Wave-Surface will be found in the Transactions
of the Royal Irish Academy, Vou. xvi. p. 65; Vou. xvi. p. 241. See also Vou.
Xx. p. 82, of the same Transactions.

On the Surfaces of the Second Order. 261

and other parts of the theory, to open new views respecting it ;
and the results at which I had arrived seemed so fitted for in-
struction, that, when I was appointed Professor of Mathematics
in the University, I made them the subject of the first lectures
which I gave in that capacity, in the beginning of the year 1836.
Next year the heads of these lectures were communicated to this
Academy, in a Paper of which a very short abstract appeared
in the Proceedings.* The subject soon became a favourite one
among the more advanced students in the University, who are,
for the most part, excellent geometers, and in the present Article
very little will be found which is not well known amongst them ;
very little, indeed, which was not communicated to the Academy
on the occasion just mentioned, or which may not be gathered,
in the shape of detached questions, out of the Examination
Papers published in the University Calendar. But as nothing
has yet been published on the subject in a connected form,
except the brief notice in the Proceedings of the Academy, and
as mathematicians in other countries attach some importance to
researches of this kind, and appear to be in quest of certain
principles which are familiar to us here, it seems proper to col-
lect together the chief results that have already been obtained,
in order that persons wishing to pursue these speculations may
be better able to judge where their inquiries should begin, and
in what direction further progress is most likely to, be made.

Part I.—GEnNERATION OF SURFACES OF THE SECOND ORDER.

§ 1. The different species of surfaces of the second order are
obtained, as is usually shown in elementary treatises, by the
discussion of the general equation of the second degree among
three co-ordinates ; but it is necessary that we should also be
able to derive these surfaces from a common geometrical origin,
if we would bring them completely within the grasp of geo-
metry. Now as the different conic sections may (with the

* Proceedings of the Royal Irish Academy, Vou. 1. p. 89.

262 On the Surfaces of the Second Order.

exception of the circle), be described in plano by the motion of a
point whose distance from a given point bears a constant ratio
to its distance from a given right line,* it is natural to suppose
that there must be some analogous method by which the sur-
faces of the second order may be generated in space. Accord-
ingly I have sought for such a method, and I have found that
(with certain analogous exceptions) every surface of the second
order may be regarded as the locus of a point whose distance
from a given point bears a constant ratio to its distance from a
given right line, provided the latter distance be measured pa-
_ rallel to a given plane; this plane being, in general, oblique to
the right line. The given point I call, from analogy, a focus,
and the given right line a directrix; the given plane may be
called a directive plane, and the constant ratio may be termed
the modulus.

To find the equation of the surface so defined, let the axis
of s be parallel to the directrix; let the plane of xy pass through
the focus, and cut the directrix perpendicularly in A, the co-
ordinates being rectangular, and their origin arbitrarily assumed.
in that plane; and let the axis of y be parallel to the intersec-
tion of the plane wy with the directive plane, the angle between
the two planes being denoted by ¢. ‘Then if we put x, y, for
the co-ordinates of the focus, and z., y, for those of the point A,
while the co-ordinates of a point S upon the surface are denoted
by 2, y, 2, the distance of this last point from the focus will be
the square root of the quantity

(e - a)? + (y- my +2;

and if a plane drawn through S, parallel to the directive plane,
be conceived to cut the directrix in D, the distance SD will be
the square root of the quantity

(w — a2)’ seo*p + (y — ya)? 5
* This method of describing the conic sections is due to the Greek geometers.

It is given by Pappus at the end of the Seventh Book of his Mathematical Collec-
tions.

On the Surfaces of the Second Order. 263

- so that, m being the modulus, the locus of the point S will be a
surface of the second order, represented by the equation

(e-m)F+ (yy) +2 = mt (o—2,)* 8009 + (y-y)}y (1)
which, by making
A=1- mi’ sec’¢, B=1-m’,.
Game, se? ¢-n, H=my-y, (2)

K = m’ (x,? sec? + y.*) — a," - y,

may be put under the form

Av + By + 3+ 2G + 2Hy = K, (3)

showing that the plane of xy is one of the principal planes of
the surface, and that the planes of vz and yz are parallel to
principal planes.

Before we proceed to discuss this equation it may be well to
observe, that as it remains the same when ¢ is changed into - 4,
or into 180° - ¢, the directive plane may have two positions
equally inclined to the plane of wy, and therefore equally in-
clined to the directrix. Indeed it is obvious that, if through
the point S we draw two planes making equal angles with the
directrix, and cutting it in the points D and D’ respectively, the |
distances SD and SD’ will be equal. Every surface described in
this way has consequently two directive planes; and as each of
these planes is parallel to the axis of y, their intersection is
always parallel to one of the axes of the surface. This axis
may therefore be called the directive axis. The directive planes
have a remarkable relation to the surface, as may be shown
in the following manner :—

Suppose a section of the surface to be made by a plane which
is parallel to one of the directive planes, and which cuts the
directrix in D; then the distance of any point S of the section
from the focus F' will have a constant ratio to its distance SD

264 On the Surfaces of the Second Order.

from the point D; and, as the locus of a point whose distances
from the two points /' and D are in a constant ratio to each
other is a plane or a sphere, according as the ratio is one of
equality or not, it follows that the section aforesaid will be a
right line in the one case, and a circle in the other. Hence
it appears that all directive sections, that is, all sections made in
the surface by planes parallel to either of the directive planes,
are right lines when the modulus is unity, and circles when the
modulus is different from unity.

Since the equation (8) is not altered by changing the sign of
¢, or by changing ¢ into its supplement, we may suppose this
angle (when it is not zero) to be always positive and less than
90°; for the supposition ¢ = 90° is to be excluded, as it would
make the secant of ¢ infinite, and the directive planes parallel
to the directrix. In the discussion of the equation there are two
leading cases to be considered, answering to two classes of sur-
faces. The first case, when neither A nor B vanishes, gives the
ellipsoid, the two hyperboloids, and the cone; the second, when
either or each of these quantities is zero, includes the two para-
boloids and the different kinds of cylinders.

§ 2. First Class of Surfaces.—W hen neither A nor B vanishes,
we may make both G and H vanish, by properly assuming the
origin of co-ordinates. Supposing this done, we have

a, = mt, Sec? p, yi = MYr, (4)

the equation of the surface being then
A?’ + By +2=K, (5)

in which the axes of co-ordinates are of course the axes of the
surface. When £ is not zero, the surface is an ellipsoid or hy-
perboloid, having its centre at the origin of co-ordinates ; when
K = 0, the surface is a cone having its vertex at the origin.
Eliminating «., y, from the value of X, by means of the re-
lations (4), we get
K=

A 2 B 2.
Bae Bae Bea ©)

On the Surfaces of the Second Order. 265

and eliminating ,, y, in like manner, we get
K=A(1-A)#?+B(1-B) y’; (7)

from which expressions it appears that, everything else remain-
ing, the focus and directrix may be changed without changing
_ the surface described. or in order that the surface may re-
main unchanged, it is only necessary that H should remain con-
stant, since A and B are supposed constant. This condition
being fulfilled, the focus may be any point F whose co-ordinates
%, y Satisfy the equation (6), and A (the foot of the directrix)
may be any point whose co-ordinates 7,, y2 satisfy the equation
(7) ; it being understood, however, that when one oMbhese points
is chosen, the other is determined. The locus of F (supposing
& not to vanish) is therefore an ellipse or a hyperbola,* which
may be called the focal curve, or the focal line; and the locus of
A is another ellipse or hyperbola, which may be called the diri-
gent curve or line: the centre of each curve is the centre of the
surface, and its axes coincide with the axes of the surface which
lie in the plane of zy. Moreover, as the quantities 1- A and
1-B are essentially positive, the two curves are always of the
same kind, that is, both ellipses, or both hyperbolas ; and when
they are hyperbolas, their real axes have the same direction.
The directrix, remaining always parallel to the axis of s, de-
scribes a cylinder which may be called the dirigent cylinder.

Since, by the relations (4), the corresponding co-ordinates
of F and A have always the same sign, these points either lie
within the same right angle made by the axes of # and y, or lie
on the same axis, at the same side of the centre. And as these
relations give

A : B
eta: ay GRY 7 Ly Ya- N= TRY (8)
it is easy to see that the right line AF is a normal to the focal

* In the Proceedings of the Academy, Vou. 1. p. 90, it was stated inadvertently
that “if we confine ourselves to the central surfaces, the locus of the foci will be
an ellipse.’’

266 On the Surfaces of the Second Order.

curve ; for the quantities 2, - 2, and y, —- y are proportional to
the cosines of the angles which that right line makes with the
axes of w and y respectively, while the values just given for
these quantities are, in virtue of the equation (6), proportional
to the cosines of the angles which the normal to the focal curve
at the point F makes with the same axes.

It may also be shown, that if the directrix prolonged through
A intersects a directive plane in a certain point, and if a right
line drawn through F, parallel to the directrix, intersect the
same plane in another point, the right line joining those points
will be a normal to the curve described in that plane by the first
point.

§ 3. To find in what way the focal and dirigent curves are
connected with the surface, let the equations (5), (6), (7) (when
K does not vanish), be put under the forms

&
P + Q + R = i; (9)
a? Yr XP Ys
— — = — —_ = Z; 10
P, : Q, 4 P, 53 Q: ( )

so that the quantities P, Q, R may represent the squares of the
semiaxes of the surface, and P,, Q:, P2, Q, the squares of the
semiaxes of the curves, these quantities being positive or ne-
gative, according as the corresponding semiaxes are Teal or
imaginary. ‘Then we have

K K
P="; Q= > R= kK,
P,=P(1-A), Q=Q(1-3B), (11)
P Tee
P,=7 > = TR

whence it follows that
P, P, = P”, Q: 2. = Q", (12)
P,=P-R, Q=Q-R. (13)

and also that

On the Surfaces of the Second Order. 267

From equations (12) we see that P, and P, have always the same
sign, as also Q, and Q,; and that, neglecting signs, the semi-
axes of the surface are mean proportionals between the corre-
sponding semiaxes of the focal and dirigent curves. These curves
are therefore reciprocal polars with respect to the section made
in the surface by the plane of wy; and it would be easy to show
that the points F and A are reciprocal points, or that a tangent
applied at one of them to the curve which is its locus has the
other for its pole.

The focal curve, when we know in which of the principal
planes it lies, is determined by the conditions (13); and as it
depends on the relative magnitudes of the quantities P, Q, R,
it will be convenient to distinguish the axes of the surface, with
relation to these magnitudes. Supposing, therefore, the quan-
tities P, Q, R to be taken with their proper signs, as they are
in the equation (9), that axis to which the greatest of them
(which is always positive) refers shall be called the primary
axis ; and that to which the quantity algebraically least has re-
ference shall be termed the secondary axis; while the quantity
which has an intermediate algebraic value shall mark the middie
or mean axis. Then, since P, and Q, will be negative, if R be
the greatest of the quantities aforesaid, the focal curve cannot
lie in the plane of the mean and secondary axes. Its plane
must therefore pass through the primary axis: it will be the
plane of the primary and mean axes, if R be the least of the
three quantities; but the plane of the primary and secondary
axes, if be the intermediate quantity. In the former case the
curve will be an ellipse, in the latter a hyperbola ; and we shall
extend the name of focal curves to both the curves so deter-
mined, though it may happen that only one of them can be
used in the generation of the surface by the modular method, as
the method of which we are treating may be called, from its
employment of the modulus. A focal curve which can be so
used shall be distinguished as a modular focal; but each focal,
whether modular or not, shall be supposed to have a dirigent

curve and a dirigent cylinder connected with it by the relations
already laid down.

268 On the Surfaces of the Second Order.

Since P, - Q, = P — Q, the foci of a focal curve are the same
as those of the principal section in the plane of which it lies,
and they are therefore on the primary axis of the surface. It
will sometimes contribute to brevity of expression, if we also
give the name of primary to the major axis of an ellipse and
to the real axis of a hyperbola. We may then say that the
primary axes of the surface and of its two focal curves are
coincident in direction ; and that (as is evident) the foci of
either curve are the extremities of the primary axis of the
other. |

If K be supposed to approach gradually to zero, while A and
B remain constant, the focal and dirigent ellipses will gradually
contract, and the focal and dirigent hyperbolas will approach
to their asymptotes, which remain fixed. When £ actually
vanishes, the surface becomes a cone; the two ellipses are each
reduced to a point coinciding with the vertex of the cone, and
each hyperbola is reduced to the pair of right lines which were
previously the asymptotes. The dirigent cylinder, in the one
case, is narrowed into a right line; in the other case it is con-
verted into a pair of planes, which we may call the dirigent
planes of the curve.

§ 4. We have now to show how the different kinds of sur-
faces belonging to the first class are produced, according to the
different values of the modulus and other constants concerned
in their generation.

I. When m is less than cos ¢, the quantities A,B, K,P,Q,R
are all positive, and Q is intermediate in value between P and
R. The surface is therefore an ellipsoid, and its mean axis is
the directive. As the quantities 1- A and 1-8 are always
positive, the focal and dirigent curves are ellipses.

_ Here we cannot suppose & to vanish, as the surface would
then be reduced to a point.

When ¢ = 0, that is, when the directive planes coincide with
each other, and therefore with a plane perpendicular to the
directrix, so that SD is the shortest distance of the point 8 from
the directrix, the surface is a spheroid produced by the revo-

On the Surfaces of the Second Order. 269

lution of an ellipse round its minor axis, and the focal and diri-
gent curves are circles.

II. When m is greater than unity; A and B are negative ;
and if K be finite, it is also negative; whence P and Q are po-
sitive, and K is negative. Also, supposing ¢ not.to vanish, Q
is greater than P. The surface is therefore a hyperboloid of
one sheet, with its real axes in the plane of ay; and the direc-
tive axis is the primary. The focal and dirigent curves are
ellipses. But when ¢ = 0, the surface is that produced by the
revolution of a hyperbola round its imaginary axis, and the
focal and dirigent* are circles.

If K = 0, which implies, since A and B have the same sign,
that x, 71, %2, y, are each zero, the surface is a cone having the
axis of s for its internal axis; and the focal and dirigent are
each reduced to a point. The focus and directrix are conse-
quently unique; the focus can only be the vertex of the cone;
the directrix can only be the internal axis; and the directrix
therefore passes through the focus. The directive axis, which
coincides with the axis of y, is one of the external axes; that
one, namely, which is parallel to the greater axes of the elliptic
sections made in the cone by planes perpendicular to its internal
axis. This is on the supposition that is finite; for, when
» = 0, the cone becomes one of revolution round the axis of z.

III. When m is greater than cos ¢, but less than unity, we
have A positive and B negative, and the species of the surface
depends on #. It is inconsistent with these conditions to sup-
pose » = 0, and therefore the surface cannot, in this case, be one
of revolution. The value of HK may be supposed to be given
by the formula .

1-A 1-B
K=*F (a, - 2) +S w-y, (14)

which contains only the relative co-ordinates of the focus and

the foot of the directrix, and is a consequence of the equations
(6) and (7).

* When the term dirigent stands alone, it is understood to mean a dirigent Jine.

270 On the Surfaces of the Second Order.

1°. If His a positive quantity, the surface is a hyperboloid
of one sheet, with its secondary axis in the direction of #; the
primary axis, as before, in the directive, but the focal and diri-
gent are now hyperbolas.

2°. If Kis a negative quantity, the surface is a hyperboloid
of two sheets, having its primary axis coincident with that of 2.
The secondary axis is the directive; the focal and dirigent are
hyperbolas.

3°. If K = 0, the surface is a cone, having the axis of x for
its internal axis; the directive axis being, as before, that exter-
nal axis to which the greater axes of the elliptic sections, made
by planes perpendicular to the internal axis, are parallel. The
axis of s is the other external axis, which may be called the
mean axis of the cone, because it coincides with the mean axis of
any hyperboloid to which the cone is asymptotic. As A and B
have different signs, it is evident, from the equations (6) and
(7), that the focal and dirigent are each a pair of right lines
passing through the vertex, each pair making equal angles with
the internal axis. Two planes, each of which is drawn through
the mean axis and a dirigent line, are the dirigent planes of
the cone.

The corresponding focal and dirigent lines are those which
lie within the same right angle made by the internal and direc- ©
tive axes ; and since by the equations (6) and (8) the value of K
may be written

K = 2% (x. = 21) + (Y2 = Yr);

we see that, as K now vanishes, the right line joining corre-
sponding points F and A upon these lines is perpendicular to
the focal line. Of the two sides of the cone which are in the
plane wy, one lies between each focal and its dirigent; and ‘it
may be inferred from the equations, that the tangents of the
_ angles which the internal axis makes with a focal line, with one
of these sides of the cone, and with a dirigent line, are in con-
tinued proportion, the proportion being that of the cosine of
tounity. And hence it follows, that these two sides of the cone,

On the Surfaces of the Second Order. 271

with a focal line and its dirigent, cut harmonically any right
line which crosses them.

§ 5. From this discussion it appears, that the ellipsoid and
the hyperboloid of two sheets can be generated modularly, each
in one way only, the modular focal being the ellipse for the
former, and the hyperbola for the latter; but that the hyperbo-
loid of one sheet can be generated in two ways, each of its focals
being modular, and each focal having its proper modulus. The
cone also admits two modes of generation,* in one of which,
however, the focus is limited to the vertex of the cone, and the
directrix to its internal axis. But when the hyperboloid of one
sheet, or the cone, is a surface of revolution, it has only one
mode of modular generation. In cases of double generation,
the directive planes of course remain the same, as they have a
fixed relation to the surface. A modular focal, it may be ob-
served (and the remark applies equally to surfaces of the second
class), is distinguished by the circumstance that it does not in-
tersect the surface. The only exception to this rule are the
focal lines of the cone, which pass through its vertex. A focal
which is not modular may be called wmbilicar, because it inter-
sects the surface in the umbilics; an umbilic being a point on
the surface where the tangent plane is parallel to a directive
plane. Thus the focal hyperbola of the ellipsoid, and the focal
ellipse of the hyperboloid of two sheets, are umbilicar focals,
and pass through the umbilics of these surfaces ; but the hyper-

* The double generation of the cone, when its vertex is the focus, may be proved
synthetically by the method indicated in the Examination Papers of the year 1838,
p- xlvi. (published in the University Calendar for 1839). Supposing the cone to
stand on a circular base (one of its directive sections), and to be circumscribed by a
sphere, the right lines joining its vertex with the two points where a diameter per-
pendicular to the plane of the base intersects the sphere, will be its internal and
mean axes. Thenif P be either of these points, V the vertex, ( the point where
the axis PV cuts the plane of the base, and B any point in the circumference of
the base, the triangles PV B and PBC will be similar, since the angles at V and B
are equal, and the angle at P is common to both triangles ; therefore BV will be to
BC as PV to PB, that is, in a constant ratio. It is not difficult to complete the
demonstration, when the focus is supposed to be any point on one of the focal
lines.

272 On the Surfaces of the Second Order.

boloid of one sheet has no umbilics, and accordingly both its ”
focals are modular, and neither of them intersects the surface.
The umbilicar focals and dirigents have properties which shall
be mentioned hereafter. |

An umbilicar focal and the principal section whose plane
coincides with that of the focal are curves of different kinds,
the one being an ellipse when the other is a hyperbola; but a
modular focal is always of the same kind with the coincident
section of the surface, being an ellipse, a hyperbola, or a pair of ©
right lines, according as the section is an ellipse, a hyperbola,
or a pair of right lines; and when the section is reduced to a
point, so likewise is the modular focal.

The plane of a modular focal always passes through the
directive axis. When the directive axis is the primary, asin ~
the hyperboloid of one sheet, both focals are modular. But in
the ellipsoid and the hyperboloid of two sheets, where the pri-
mary axis is not directive, only one of the focals can be modular.
The plane of an umbilicar focal is always perpendicular to the
directive axis; and therefore, when that axis is the primary,
there is no umbilicar focal.*

When the surface is doubly modular, the two moduli m, m’
are connected by the relation

2 nN2
cos’ mts Be (16)

m m

* Tf the first of the equations (10), when P; and Q; are both negative, be sup-
posed to express an imaginary focal, there will, ina central surface, be three focals,
- two modular and one umbilicar; the two modular focals being in the principal
planes which pass through the directive axis, and the umbilicar focal in the remain-
ing principal plane. Then, when we know which of the axes is the directive axis,
we know which of the three focals is imaginary, because the plane of the imagi-
nary focal is perpendicular to the primary axis. A modular focal may be imagi-
nary, and yet have a real modulus; this occurs in the hyperboloid of two sheets.
In the ellipsoid, the imaginary focal has an imaginary modulus. In all cases the
two moduli are connected by the relation (16).

It will appear hereafter that the vertex of the cone is an umbilicar focus. The
cone has therefore three focals, none of which is imaginary ; but two of them are
single points coinciding with the vertex.

On the Surfaces of the Second Order. 273

where ¢ is the angle made by a directive plane with the plane
of the focal to which the modulus m belongs. One modulus is
greater than unity; the other is less than unity, but greater
than the cosine of the angle which the plane of the correspond-
ing focal makes with a directive plane. In the hyperboloid of
one sheet, the less modulus is that which belongs to the focal
hyperbola. In the cone, the less modulus belongs to the focal
lines. Of the two moduli of a cone, that which belongs to the
focal lines may be termed the Jinear modulus ; and the other, to
which only a single focus corresponds, may be called the singu-
lar modulus.

§ 6. Second Class of Surfaces.—In this class of surfaces, one
of the quantities A, B vanishes, or both of them vanish.

I. When m = cos ¢, and ¢ is not zero, A vanishes, but B
does not; and the surface is either a paraboloid or a cylinder.

1°. Ifthe surface is a paraboloid, we may suppose the origin
of co-ordinates to be at its vertex, in which case both H and K
vanish, and we have the relations

G =%- MN, Yi = Y2 COS’ d,
(17)
te? + yi? cos’ - a -y =0;
the equation of the surface being
y sin’d + 3? + 2Gz = 0, (18)

which shows that the paraboloid is elliptic, having its axis in the
direction of x, and the plane of zy for that of its greater prin-
cipal section. From the relations (17) we obtain the following :

yi: tan’ + 2Ga, + G’ =0,
ae 2 ue)
yz sin’ ¢ cos’ + 2Ga, —- G?=0;

from which we see that the focal and dirigent curves are para-

bolas, having their axes the same as that of the surface; and

their vertices equidistant from the vertex of the surface, but at

opposite sides of it. The cavity of each curve is turned in the
T

274 On the Surfaces of the Second Order.

same direction as that of the section vy. The focus of the focal
parabola is the focus of the section wy, and its vertex is the
focus of the section «z of the surface; its parameter being the
difference of the parameters of these two sections. The para-
meter of the section wy is a mean proportional between the para-
meters of the focal and dirigent parabolas.

2°. If the surface is a cylinder, we may make G and H
vanish, by taking the origin on its axis. "We then have

ty = his is = Yo COS 9,
(20)
K = y;* tan’ = y? sin’ cos’¢;
the equation of the cylinder, which is elliptic, being
y sin’¢ + 3° = K. (21)

Here the focal and dirigent are each a pair of right lines —
parallel to the axis of the cylinder, and passing through the foci
and directrices of a section perpendicular to the axis. The cor-
responding focal and dirigent lines lie at the same sides of the
axis.

IT. When m = 1, and ¢ is not zero, B vanishes, but 4 does
not.

1°. If the surface is a paraboloid, and the origin of co-ordi-
nates at its vertex, the quantities G and # vanish; and the
equation of the surface becomes

a tan’ — 3 = 2Hy, (22)
and we have the relations

H= Y2 _ Yiy Xv, = Le seo’,
(23)
HF sec’ + y.? — 2? - y, = 0.

The paraboloid is therefore hyperbolic, its axis being that of y,
which is also the directive axis; and as the tangent of ¢ may
have any finite value, the plane of wy, which is that of the focal —

On the Surfaces of the Second Order. 275

curve, may be either of the principal planes passing through the
axis of the surface. The relations (23) give

x? sin’ = 2Hy, = a? = 0,
(24)
a,° tan’ sec’ — 2Hy, + H*? = 0,

for the équations of the focal and dirigent, which are therefore
parabolas, having their axes the same as those of the surface,
and their concavities turned in the same direction as that of the
section wy; their vertices being equidistant from the vertex of the
surface, and at opposite sides of it. The focus of the focal para-
bola is the focus of the section ry, and its vertex is the focus of
the section yz, its parameter being the sum of the parameters of
these two sections. The parameter of the section vy is a mean
proportional between the parameters of the focal and dirigent
parabolas.

2°. If the surface is a cylinder, and the origin on its axis,
G and H vanish, and we have

a = Xe sec’ d, W=Yx
(25)
- K =x; sin’¢ = x tan’¢ sec’¢ ;
the equation of the cylinder, which is hyperbolic, being
etan’g-=-K. . (26)

The focal and dirigent are each a pair of right lines parallel
to the axis of the cylinder; the corresponding lines passing
through a focus and the adjacent directrix of any section per-
pendicular to the axis. The directive planes are parallel to the
asymptotic planes of the cylinder.

In this case, if K = 0, the surface is reduced to two directive
planes, and the focal and dirigent to the intersection of these
planes.

III. When m = 1, and ¢ = 0, both A and B vanish, and the
surface is the parabolic cylinder. If, as is allowable, we sup-

T2

276 On the Surfaces of the Second Order.

pose G and £ to vanish, the equation of the cylinder becomes

3° + 2Hy = 0, (27)
and we have
H=y2- Yr H = He,
(28)
ty! + Ys — @Y — yr = 0;
whence
"=-tH, y=4H. (29)

The focal and dirigent are each a right line parallel to the axis
of x, the former passing through the focus, the latter meeting
the directrix of the parabolic section made by the plane of ys.
The plane of xy is the directive plane.

§ 7. We learn from this discussion that, among the surfaces of
the second class, the hyperbolic paraboloid is the only one which
admits a twofold modular generation ; the modulus, however, _
being the same for both its focals. In the elliptic paraboloid
the modular focal is restricted to the plane of that principal
section which has the greater parameter ; we shall therefore
suppose a parabola to be described in the plane of the other
principal section, according to the law of the modular focals ;
the law being, that the focus of the parabola shall be the focus
of the principal section in the plane of which the parabola lies,
and its vertex the focus of the principal section in the per-
pendicular plane. The parabola so described will have its
concavity opposed to that of the surface; it will cut the sur-
face in the umbilics, and will be its umbilicar focal, the only
such focal to be found among the surfaces of the second
class. We shall of course suppose further, that this focal has
a dirigent parabola connected with it by the same law as in
the other cases, the vertices of the focal and dirigent being equi-
distant from that of the surface and at opposite sides of it, while
the parameter of the dirigent is a third proportional to the para-
meters of the focal and of the principal section in the plane of
which the curve lies. The two focals of a paraboloid are so re-
lated, that the focus of the one is the vertex of the other. The
cylinders have no other focals than those which occur above.

On the Surfaces of the Second Order. 277

§ 8. In this, as in the first class of surfaces, the right line
FA, joining a focus F with the foot of its corresponding direc-
trix, is perpendicular to the focal line; and the focal and diri-
gent are reciprocal polars with respect to the section wy of the
surface. These properties are easily inferred from the preced-
ing results; but, as they are general, it may be well to prove
them generally for both classes of surfaces. Supposing, there-
fore, the origin of co-ordinates to be anywhere in the plane of
wy, and writing the equation of the surface in the form

(@@-m)'+ y-y)+s=L(e-a,)'+U(y-y)’, — (80)
which, when identified with (3), gives the relations
A=1-TL, B=1-UH,
G=In-«, H=My-yY, (31)
K = La; + My? - 2? - y?,
we find, by differentiating the values of the constants Gi;

and K,
Leda, = dix, May, = dy,

Lx, dx. + My, dy —- dx, = Vy dy, = 0. (32)
Hence we obtain
(a2 _ #;) dat, + (y2 = 1) dy, = 0 ; (33)

an equation which expresses that the right. line joining the
points F and A is perpendicular to the i which is the locus
of the point F.

Again, the equation of the section xy of the surface being

Ag? + By? + 2Gx + 2Hy = K, (34)

the equation of the right line which is, with respect to this
section, the polar of a point A whose co-ordinates are «, y2, is
(Ax, + G)x+ (By + H)y = K- Ga. - Hy; (35)
but the relations (31) give
Ax, + G=%—- tH, By, + H = 2 - Wy

(36)
K -— Ga, - Hy, = % (a. - %) + n(y2-—) 3

278 On the Surfaces of the Second Order.

and hence the equation (35) becomes
(#2 — @) (@ — 1) + (y2-%) (Y - %) = 9, (37)
which, as is evident from (83), is the equation of a tangent
applied to the focal at the point F corresponding to A. This
shows that the focal and dirigent are reciprocal polars with
respect to the section vy, and that in this relation, as well as in
the other, the points F and A are corresponding points.
Supposing FE’ and A’ to be two other corresponding points
on the focal and dirigent, if tangents applied to the focal at F
and F’ intersect each other in T, the point T will be the pole of
the right line AA’ with respect to the section vy, as well as the
pole of the right line FF’ with respect to the focal; and hence
if any right line be drawn through T, and if P be the pole of
this right line with respect to the section, and N its pole with
respect to the focal, the points P and N will be on the right
lines AA’ and FF’ respectively. Now it is useful to observe
that the distances AA’ and FE” are always similarly divided
(both of them internally or both of them externally) by the
points P and N, so that we have AP to A’P as FN to F'N.
This property may be proved directly by means of the fore-
going equations; or it may be regarded as a consequence of the
following theorem :—If through a fixed point in the plane of
two given conics having the same centre, or of two given para-
bolas having their axes parallel, any pair of right lines be
drawn, and their poles be taken with respect to each curve,
the distance between the poles relative to one curve will be in
a constant ratio to the distance between the poles relative to
the other curve.* In fact, the poles of the right lines TF, TE’,
with respect to the focal, are F, F’; and their poles with respect
to the section ay are A, A’; therefore, since the focal and the
section zy may be taken for the given curves, and the point T

* There is an analogous theorem for two surfaces of the second order which
~have the same centre, or two paraboloids which have their axes parallel. If
through a fixed right line any two planes be drawn, and their poles be taken with
respect to each surface, the distance between the poles relative to the one surface
will be in a constant ratio to the distance between the poles relative to the other.

On the Surfaces of the Second Order. 279

for the fixed point, the ratio of FF’ to AA’ is the same as the
ratio of FN to AP, or of F’N to A’P; and consequently the
distances FF’ and AA’ are similarly divided in the points N
and P. =

§9. In the equation (80), considered as equivalent to the
equation (1), the constants Z and YW are both positive; but the
properties which have been deduced from the former equation
are independent of this circumstance, and equally subsist when
one of these constants is supposed to be negative (for they can-
not both be negative). This leads us to inquire what surfaces
the equation (30) is capable of representing when the constants
LI and MW have different signs; as also, for a given surface, what
lines are traced in the plane of zy by points F and A, of which
£15 Yr) AN a, y, are the respective co-ordinates. After the ex-
amples already given, this question is easily discussed, and the
result is, that the only surfaces which can be so represented are
the ellipsoid, the hyperboloid of two sheets, the cone, and the
elliptic paraboloid—that is to say, the umbilicar surfaces to-
gether with the cone; and that, for an umbilicar surface, the
locus of F is the umbilicar focal, and therefore the locus of A is
the corresponding dirigent; while for the cone the points F and
A are unique, coinciding with each other and with the vertex of
the cone. A geometrical interpretation of this case is readily
found; for as Z and MU have different signs, the right-hand
member of the equation (30), if IZ be the negative grey
the product of two factors of the form

S(e-%)+9(y-y), S(@-%)-g(y -%);

in which f and g are constant; and these factors are evidently
proportional to the distances of a point whose co-ordinates are
2, ¥, , from two planes whose equations are

S(@-m)+9(y-y)=9, f(e-m)-g(y-) =9,

which planes always pass through a directrix, and are inclined
at equal and constant angles to the axis of x or of y. There-
fore, if F be the focus which belongs to this directrix, the square

280 On the Surfaces of the Second Order.

of the distance of F from any point of this surface is in a con-
stant ratio to the rectangle under the distances of the latter
point from the two planes. And these planes are directive
planes; because, if a section parallel to one of them be made
in the surface, the distance of any point of the section from the
other plane will be proportional to the square of the distance of
the same point from the focus; and, as the locus of a point,
whose distance from a given plane is proportional to the square
of its distance from a given point, is obviously a sphere, it
follows that the section aforesaid is the section of a sphere, and
consequently a circle; which shows that the plane to which the
section is parallel is a directive plane. Thus,* the square of the

© *In attempting to find a geometrical generation for the surfaces of the second
order, one of the first things which I thought of, before I fell upon the modular
method, was to try the locus of a point such that the square of its distance from a
given point should be in a constant ratio to the rectangle under its distances from
two given planes; but when I saw that this locus would not represent all the
species of surfaces, I laid aside the discussion of it. Some time since, however,
Mr. Salmon, Fellow of Trinity College, was led independently, in studying the
modular method, to consider the same locus; and he remarked to me, what I had
not previously observed, that it offers a property supplementary, in a certain sense,
to the modular property; that when the surface is an ellipsoid, for example, the
given point or focus is on the focal hyperbola, which the modular property leaves
empty. This remark of Mr. Salmon served to complete the theory of the focals,
by indicating a simple geometrical relation between a non-modular focal and any
point on the surface to which it belongs.

In a memoir “ On a new method of Generation and Discussion of the Surfaces
of the second Order,’’ presented by M. Amyot to the Academy of Sciences of Paris,
on the 26th December, 1842, the author investigates this same locus, conceiving it
to involve that property in surfaces which is analogous to the property of the focus
and directrix in the conic sections ; and the importance attached to the discoyery of
such analogous properties induced M. Cauchy to write a very detailed report on
M. Amyot’s memoir, accompanied with notes and additions of his own (Comptes
rendus des Séances de l’ Académie des Sciences, tom. xvi. pp. 783-828, 885-890 ;
April, 1848); and also occasioned several discussions, principally between M.
Poncelet and M. Chasles, relative to that Memoir (Comptes rendus, tom. xvi.
pp. 829, 938, 947, 1105, 1110). But the property involved in this locus cannot
be said to afford a method of generation of the surfaces of. the second order, since
it applies only to some of the surfaces, and gives an ambiguous result even where
it does apply. It is therefore not at all analogous to the aforesaid general property
of the conic sections, and moreover it was not new when M. Amyot brought it

-

On the Surfaces of the Second Order. 281

distance of any point of the surface from an umbilicar focus
bears a constant ratio to the rectangle under the perpendicular
distances of the same point from two directive planes drawn
through the directrix corresponding to that fogus; and it is
easy to see that this ratio, the square root of which we shall
denote by u, is equal to L-—W, or, neglecting signs, to the
sum of the numerical values of Z and M. Of course, if the
distances from the directive planes, instead of being perpendi-
cular, be measured parallel to any fixed right line, the ratio
will still be constant, though different. For example, if the
fixed right line for each plane be that which joins the corre-
sponding umbilic with either focus of the section wy, the ratio

forward. Mr. Salmon had in fact proposed it for investigation to the students of
the University of Dublin, at the ordinary Examinations in October, 1842; and it
was published, towards the end of that year, in the University Calendar for 1848,
some months before the date of M. Cauchy’s report, by which the contents of
M. Amyot’s memoir were first made known. The parallelism of the two given
planes to the circular sections of the surface is also stated in the Calendar ; but this
remarkable relation is not noticed by M. Amyot, nor by M. Cauchy (see the ‘‘ Exa-
mination Papers’ of the year 1842, p. xlv., quest. 17, 18; in the Calendar for 18438).
It is scarcely necessary to add, that the analogue which M. Amyot and other
mathematicians have been seeking for, and which was long felt to be wanting in
the theory of surfaces of the second order, is no other than the modular property
of these surfaces, which appears to be not yet known abroad. M. Poncelet insists
much on the importance of extending the signification of the terms focus and direc-
triz, so as to make them applicable to surfaces; and he supposes this to have been
effected, for the first time, by M. Amyot. These terms, however, applied in their
true general sense to surfaces, had been in use, several years before, among the
mathematical students of Dublin, as may be seen by referring to the Calendar
(‘‘ Examination Papers”’ of the year 1838, p. c. 1839, p. xxxi.).

The locus above mentioned, being co-extensive with the umbilicar property,
does not represent any surface which can be generated by the right line, except
the cone. To remedy this want of generality, M. Cauchy proposes to consider a
surface of the second order as described by a point, the square of whose distance
from a given point bears a constant ratio either to the rectangle under its distances
from two given planes, or to the sum of the squares of these distances. This enun-
ciation, no doubt, takes in both kinds of focals, and all the species of surfaces; but
the additional conception is not of the kind required by the analogy in question,
nor has it any of the characters of an elementary principle. For the given planes,
according to M. Cauchy’s idea, do not stand in any simple or natural relation to
the surface; and besides there is no reason why, instead of the sum of the squares

&

282 On the Surfaces of the Second Order.

of the square to the rectangle will be the square of the number
m sec ¢, where m is the modulus, and ¢ the angle which the
primary axis makes with a directive plane.

When the umbilicar property is applied to the cone, the
vertex of which is, as we have seen, to be regarded as an umbi-
licar focus, having the directive axis for its directrix, it indi-
cates that the product of the sines of the angles which any side
of the cone makes with its two directive planes is a constant
quantity.

It is remarkable that the vertex of the ‘cone affords the only
instance of a focal point which is at once modular and umbi-
licar, as well as the only instance of a focal point which is
doubly modular. This union of properties it may be con-
ceived to owe to the circumstance that the cone is the asymp-
totic limit of the two kinds of hyperboloids. For if a series
of hyperboloids have the same asymptotic cone, and their
primary axes be indefinitely diminished, they will approach
indefinitely to the cone; and, in the limit, the focal ellipse
and hyperbola of the hyperboloid of one sheet will pass into
the vertex and the focal lines of the cone, thus making the
vertex doubly modular; while the focal ellipse of the hyperbo-
loid of two sheets will also be contracted into the vertex, and
will make that point umbilicar.
of the distances from the given planes, we should not take the sum after multi-
plying the one square by any given positive number, and the other square by
another given positive number; nor is there any reason why we should not take
other homogeneous functions of these distances. This conception would therefore
be found of little use in geometrical applications; while the modular principle, on
the contrary, by employing a simple ratio between two right lines, both of which
have a natural connexion with the surface, lends itself with the greatest ease to the
reasonings of geometry. Indeed the whole difficulty, in extending the property of
the directrix to surfaces of the second order, consisted in the discovery of such a
ratio inherent in all of them—a ratio haying nothing arbitrary in its nature, and
for which no other of equal simplicity can be substituted.

It may be proper to mention that the term modulus, which I have used for the
first time in the present Paper, with reference to surfaces of the second order, has
been borrowed from M. Cauchy, by whom it is employed, however, in a significa-
tion entirely different. Several other new terms are also now introduced, from the
necessity of the case.

On the Surfaces of the Second Order. 283

When the two directive planes coincide, and become one
directive plane, the umbilicar property is reduced to this, that
the distances of any point in the surface from the point F and
from the directive plane are in a constant ratio to each other ;
and therefore the surface becomes one of revolution round an
axis passing through F at right angles to that plane; the point
F being a focus of the meridional section, or the vertex if the
surface be a cone. When the directive planes are supposed to
be parallel, but separated by a finite interval, we get the same
class of surfaces of revolution, with the addition of the surface
produced by the revolution of an ellipse round its minor axis;
the point F being still on the axis of revolution, but not having
any fixed relation to the surface.

§ 10. If in the equation (30) we supposed the right-hand -
member to have an additional term containing the product of
the quantities « - 2, and y - y, with a constant coefficient, all
the foregoing conclusions regarding the geometrical meaning
of that equation would remain unchanged, because the addi-
tional term could always be taken away by assigning proper
directions to the axes of z and y. If, after the removal of this
term, the coefficients of the squares of the aforesaid quantities
were both positive, the locus of F would be a modular focal of
the surface expressed by the equation ; but if one coefficient
were positive and the other negative, the locus of F would be ©
an umbilicar focal. The equation in its more general form is
evidently that which we should obtain for the locus of a point
S, such that the square of its distance SF from a given point F -
should be a given homogeneous function of the second degree
of its distances from two given planes; the plane of zy being
drawn through F perpendicular to the intersection of these
planes, and 22, y, being the co-ordinates of any point on this
intersection, while 7, y, are the co-ordinates of F. The point
F might be any point on one of the focals of the surface de-
scribed by S; the intersection of the two planes (supposing
them always parallel to fixed planes) being the corresponding
directrix.

284 On the Surfaces of the Second Order.

These considerations may be further generalized, if we
remark that the equation of any given-surface of the second
order may be put under the form

(w—ay)*+(y-yi)? + (8-21)? =L (w-a,)? + (y-y2)? +N (2 -%)?
+L (y-y) (s-#) + (e-«,) (s-m) +N" (@=an) (y-y2), (88)

where L, WU, N, L’, WM’, N’ are constants, and a, 7,2; are con-
ceived to be the co-ordinates of a certain point F, and a, ys, %
the co-ordinates of another point A. The constants L’, M’, NV’
may, if we please, be made to vanish by changing the directions
of the axes of co-ordinates; and when this is done, the new co-
ordinate planes will be parallel to the principal planes of the
surface. Then, by proceeding as before, it may be shown that,
without changing the surface, we are at liberty, under certain
conditions, to make the points F and A move in space. The
conditions are expressed geometrically by saying that the two
surfaces, upon which these points must be always found, are
reciprocal polars with respect to the given surface, the points F
and A being, in this polar relation, corresponding points; and
that the surface which is the locus of F is a surface of the
second order, confocal with the given one, it being understood
that confocal surfaces are those which have the same focal
lines. The surface on which A lies is therefore also of the
second order, and the right line AF is a normal at F to the
surface which is the locus of this point. Moreover, if through
the point A three or more planes be drawn parallel to fixed
planes, and perpendiculars be dropped upon them from any
point S whose co-ordinates are «, y, z, the right-hand member
of the equation (38) may be conceived to represent a given
homogeneous function of the second degree of these perpendi-
culars; and the given surface may therefore be regarded as the
locus of a point 8, such that the square of the distance SF is
always equal to that function.

§ 11. In the enumeration of the surfaces capable of hess
generated by the modular method, we miss the five following

On the Surfaces of the Second Order. 285

varieties, which are contained in the general equation of the
second degree, but are excluded from that method of genera-
tion by reason of the simplicity of their forms—namely, the
sphere, the right cylinder on a circular base, and the three
surfaces which may be produced by the revolution of a conic
section (not a circle) round its primary axis.* These three
surfaces are the prolate spheroid, the hyperboloid of two sheets,
and the paraboloid of revolution; and the circumstance, that
the foci of the generating curves are also foci of the surfaces,
renders it easy to investigate their focal properties.t In point
of simplicity, the excepted surfaces are to the other surfaces of
the second order what the circle is to the other conic sections,
the circle being, in like manner, excepted from the curves
which can, be generated by the analogous method in plano ;
and the geometry of the five excepted surfaces may therefore
be regarded as comparatively elementary. These five surfaces
were, in fact, studied by the Greek geometers,} and, along with
the oblate spheroid and the cone, they make up all the surfaces
of the second order with which the ancients were acquainted.
Except the cone, the surfaces considered by them are all of
revolution; and there is only one surface of revolution, the
hyperboloid of one sheet, which was not noticed until modern
times. This surface is mentioned (under the name of the
hyperbolic cylindroid) by Wren,§ who remarks that it can
be generated by the revolution of a right line round another
right line not in the same plane. As to the general conception
of surfaces of the second order, the suggestion of it was reserved
for the algebraic geometry of Descartes. In that geometry the

* The case of two parallel planes is also excluded, but it is not here taken into
account. The case of two parallel right lines is in like manner excluded from the
corresponding generation of lines of the second order.

_ + A Paper by M. Chasles, on these surfaces of revolution, will be found in the
‘* Memoirs,” of the Academy of Brussels, tom. y. (An. 1829).

{ The hyperboloid of two sheets, and the paraboloid of revolution, were known
by the name of conoids. Archimedes has left a treatise on Conoids and Spheroids,
as well as a treatise on the Sphere and Cylinder.

§ In the Philosophical Transactions for the year 1669, p. 961.

286 On the Surfaces of the Second Order.

curves previously known as sections of the cone are all expressed ~
by the general equation of the second degree between two co-
ordinates; and hence it occurred to Euler* about a century ago,
to examine and classify the different kinds of surfaces comprised
in the general equation of the second degree among three co-
ordinates. The new and more general forms thus brought to
light have since engaged a large share of the attention of geo-
meters; but the want of some other than an algebraic principle
_ of connexion has prevented any great progress from being made
in the investigation of such of their properties as do not im-
mediately depend on transformations of co-ordinates. This
want the modular method of generation perfectly supplies, by
evolving the different forms from a simple geometrical concep-
tion, at the same time that it brings them within the range of
ideas familiar to the ancient geometry, and places théir relation
to the conic sections in a striking point of view.

It may be well to remark that the excepted surfaces are
. limits of surfaces which can be generated modularly, as the
circle is the limit of the ellipse in the analogous generation of
the conic sections. Thus the sphere is the limit of an oblate
spheroid, one of whose axes remains constant, while its focal
circle is indefinitely diminished ; and the right circular cylinder
is the limit of an elliptic cylinder, whose focal lines are con-
ceived to approach indefinitely to coincidence with each other
and with the axis of the cylinder, while one of the axes of the
principal elliptic section remains constant. In these cases the
dirigent lines, along with the directrices, move off to infinity.
The other three excepted surfaces correspond to the supposition
= 90°, which was excluded in the discussion of the general
equation (1). For if we make m sec ¢ = n, the quantity which
constitutes the right-hand member of that equation may be
written

n? (x — a2)? + n®(y — Y2)* cos? o;

and if we suppose ” to remain finite and constant, while @ ©

* See his Introductio in Analysin Infinitorum, p. 3738. Lausanne, 1748.

On the Surfaces of the Second Order. 287

approaches to 90°, and m indefinitely diminishes, this quantity
will approach indefinitely to n’ (#—- 2)’, which will be its limit-
ing value when ¢=90°. But #-® is the distance of the point
S from a fixed plane intersecting the axis of x perpendicularly
-at the distance x, from the origin of co-ordinates; and there-
fore, in the limit, the equation expresses that the distances of
any point S of the surface, from the focus F and from this fixed
plane, are to each other as n to unity, that is, in a constant
ratio, which is a common property of the three surfaces in
question. This property also belongs to the right cone, but the
right cone does not rank among the excepted surfaces.

§ 12. We have seen that, when the modulus is unity, any
plane parallel to either of the directive planes intersects the
surface in a right line; whence it follows, that through any
point on the surface of a hyperbolic paraboloid two right lines
may be drawn which shall lie entirely in the surface. The
plane of these right lines is of course the tangent plane at that
point, and therefore every tangent plane intersects the surface
in two right lines, This is otherwise evident from considering
that the sections parallel to a given tangent plane are similar
hyperbolas, whose centres are ranged on a diameter passing
through the point of contact, and whose asymptotes, having
_ always the same directions, are parallel to two fixed right lines
which we may suppose to be drawn through that point. For
as the distance between the plane of section and the tangent
plane diminishes, the axes of the hyperbola diminish; and they
vanish when that distance vanishes, the hyperbola being then
reduced to its asymptotes. The tangent plane therefore inter-
sects the surface in the two fixed right lines aforesaid. The
same reasoning, it is manifest, will apply to any other surface
of the second order which has hyperbolic sections parallel to its
tangent planes; and therefore the hyperboloid of one sheet,
which is the only other such surface,* is also intersected in two

* The double generation of these two surfaces by the motion of a right line has
been long known. It appears to have been discovered and fully discussed by some
of the first pupils of the Polytechnic School of Paris. This mode of generation had,

288 On the Surfaces of the Seconda Order.

right lines by any of its tangent planes. These right lines are
usually called the generatrices of the surface.

From what has been said, it appears that the generatrices of
the hyperbolic paraboloid, and the asymptotes of its sections (all
its sections, except those made by planes parallel to the axis,
being hyperbolas), are parallel to the directive planes. The
generatrices of the hyperboloid of one sheet, and the asymp-
totes of its hyperbolic sections, are parallel to the sides of the
asymptotic cone; because any section of the hyperboloid is
similar to a parallel section of the asymptotic cone; and when
the latter section is a hyperbola its asymptotes are parallel to
two sides of the cone.

Part I].—Prorerties oF SurFAcES oF THE SECOND
ORDER.

§ 1. In the preceding part of this Paper it has been neces-
sary to enter into details for the purpose of communicating fun-
damental notions clearly. In the following part, which will
contain certain properties of surfaces of the second order, we
shall be as brief as possible; giving demonstrations of the more
elementary theorems, but confining ourselves to a short state-
ment of the rest. |

Many consequences follow from the principles already laid
down.

Through any directrix of a surface of the second order let a
fixed plane be drawn cutting the surface, and let S be any point
of the section. If the directrix and its foeus F be modular, and
if a plane always parallel to the same directive plane be con-
ceived to pass through § and to cut the directrix in D, the direc-
tive distance SD will be always parallel to a given right line,
and will therefore be in a constant ratio to the perpendicular dis-
tance of 8 from the directrix. This perpendicular distance will

however, been remarked by Wren, with regard to the hyperboloid of revolution.
It does not seem to have been observed, that the existence of rectilinear genera-
trices is included in the idea of hyperbolic sections parallel to a tangent plane.

On the Surfaces of the Second Order. 289

consequently bear a given ratio to SF, the distance of the point
S from the focus. And the same thing will be true when the
directrix and focus are umbilicar, because the perpendicular dis-
tance of the point 8 from the directrix will be in a constant ratio
to its distance from each directive plane drawn etenven the
directrix.

The fixed plane of section will in general contain another
directrix parallel to the former, and belonging to the same
focal ; and it is evident that the perpendicular distance of S from
this other directrix will be in a given ratio to its distance SE”
from the corresponding focus EF’, the ratio being the same as in
the former case. Hence, according as the point S lies between
the two directrices, or at the same side of both, the sum or dif-
ference of the distances SF and SF’ will be constant.

If the plane of section pass through either of the foci, as F,
this focus and its directrix will manifestly be the focus and di-
rectrix of the section. In this case the plane of section will be
perpendicular to the focal at F. And if the surface be a cone,
the point F being anywhere on one of its focal lines, the distance
of the point S from the directrix will be in a constant ratio to
its perpendicular distance from the dirigent plane which con-
tains the directrix, and therefore this perpendicular distance will
be in a given ratio to the distance SF. Now, calling V the
vertex of the cone, and taking SV for radius, the perpendicular
distance aforesaid is the sine of the angle which the side SV of
the cone makes with the dirigent plane; and SF, which is per-
pendicular to VF, is the sine of the angle SVF. Consequently
the sines of the angles which any side of a cone makes with a
dirigent plane and the corresponding focal line are in a given
ratio to each other.

§ 2. Conceive a surface of the second order to be intersected
in two points 8, 8’ by a right line which cuts two parallel direc-
trices in the points E, H’, and let F, F” be the foci correspond-
ing respectively to these directrices. The perpendicular dis-
tances of the points 8, 8’ from the first directrix and from the
second are to each other as the lengths SE, S’E, SE’, 8’ H’

U

290 On the Surfaces of the Second Order.

respectively, and therefore the ratios of FS to SE, of FS’ to YH,
of F’S to SE’, and of E'S’ to S’H, are all equal.

Hence, the right line FE bisects one of the angles made by
the right lines F'S and FS’; and the right line F’E’ bisects one
of the angles made by F’S and FS’.

When the points 8, 8’ are at the same side of EH, the angle 7
supplemental to SF’ is that which is bisected by the right line
FE. Now if the point S be fixed, and 8’ approach to it inde-
finitely, the angle SFE will approach indefinitely to a right
angle. Therefore if a right line touching the surface meet a
directrix in a certain point, the distance between this point and
the point of contact will subtend a right angle at the focus
which corresponds to the directrix. And if a cone circumscrib-
ing the surface have its vertex in a directrix, the curve of contact
will be in a plane drawn through the corresponding focus at right
angles to the right line which joins that focus with the vertex.

When the surface intersected by the right line SS’ is a cone,
suppose this line to lie in the plane of the focus F and its direc-
trix, that is, in the plane which is perpendicular at F to the focal
line VF (the vertex of the cone being denoted, as before, by V) ;
the angles made by the right lines FE, FS, FS’, are then the
same as the angles made by planes drawn through VF and each
of the right lines VE, VS, VS’; and the last three right lines are
the intersections of a plane VSS’ with the dirigent plane on which
the point E lies, and with the surface of the cone. Therefore if
a plane passing through the vertex of a cone intersect its sur-
face in two right lines, and one of its dirigent planes in another
right line, and if a plane be drawn through each of these right
lines respectively and the focal line which belongs to the dirigent
plane, the last of the three planes so drawn will bisect one of
the angles made by the other two. And hence, if a plane touch-
ing a cone along one of its sides intersect a dirigent plane in a
certain right line, and if through this right line and the side of
contact, respectively, two planes be drawn intersecting each other
in the focal line which corresponds to the dirigent plane, the two
planes so drawn will be at right angles to each other.

On the Surfaces of the Second Order. 291

Let a right line touching a surface of the second order in S
meet two parallel directrices in the points E, HE’, and let F, F”
be the corresponding foci. Then the triangles FSE and F’SE’
are similar, because the angles at F and FE’ are right angles, and
the ratio of FS to SE is the-same as the ratio of FS to SH’.
Therefore the tangent EE’ makes equal angles with the right
lines drawn from the point of contact 8 to the foci F, F’. When
the surface is a cone, let the tangent be perpendicular to the
side VS which passes through the point of contact; the angles

.FSE and F’SE’ are then the angles which the tangent plane
VEE’ makes with the planes VSF and VSE’, because the right
line FE is perpendicular to the plane VSF, and the right line
F’E’ is perpendicular to the plane VSE’. Therefore the tangent
plane of a cone makes equal angles with the planes drawn
through the side of contact and each of the focal lines.

Supposing a section to be made in a surface of the second
order by a plane which cuts any directrix in the point EH, if the
focus F belonging to this directrix be the vertex of a cone having
the section for its base, the right line FE will be an axis of the
cone. For if through FE any plane be drawn cutting the base
of the cone in the points 8, 8’, one of the angles made by the
sides E'S, FS’ which pass through these points will always be
bisected by the right line FE; and this is the characteristic
property of an axis.

§ 3. Two surfaces of the second order being supposed to
have the same focus, directrix, and directive planes, so that they
differ only in the value of the modulus m, or of the umbilicar
ratio u (see Part I. § 9): let a right line passing through any
point E of the directrix cut one surface in the points 8, 8’, and
the other in the points &, 8, and conceive right lines to be
drawn from all these points to the common focus F'. Since, if
ratios be expressed by numbers, the ratio of FS to SE (or of
FV’ to SE) is to the ratio of FS, to 8. (or of FS, to 8,E) as
the value of m for the one surface is to its value for the other,
when the focus is modular, or as the value of » for the one sur-
face is to its value for the other when the focus is umbilicar, the

u2

292 On the Surfaces of the Second Order.

sines of the angles EF'S, and EFS (or of the angles EFS, and
EFS’) are in a constant proportion to each other, because these
sines are proportional to those ratios. And since the right line
FE bisects the angles SFS’ and S,F8,, both internally or both
externally, in which case the angles SFS, and S’F8, are equal,
or else one internally and the other externally, in which case the
angles SFS, and 8’FS, are supplemental, it is easy to infer,
from the constant ratio of the aforesaid sines, that in the first
case the product, in the second case the ratio of the tangents of
the halves of the angles SFS, and S’FS, (or of the halves of the
angles SF'S; and S’F'8,) is a consequent quantity.

If the point 8’ approximate indefinitely to 8, the right line
passing through these points will approach indefinitely to a tan-
gent. Therefore when two surfaces are related as above, if a
right line passing through any point E of their common direc-
trix intersect one surface in the points S,, S,, and touch the other
in the point S, the chord 8,8, will subtend a constant angle at
the common focus F', and this angle will be bisected, either in-
ternally or externally, by the right line FS drawn from the focus
_to the point of contact. And the angle EFS being then a right
angle, the cosine of the angle SFS, or SFS, will be equal to the
ratio of the less value of m or u to the greater.*

§ 4. Among the surfaces of the second order, the only one
which has a point upon itself for a modular focus is the cone,
the vertex of which is such a focus, related either to the internal
or to the mean axis as directrix. In the latter relation the
vertex belongs to the series of foci which are ranged on the focal
lines. To see the consequence of this, let V be the vertex of
the cone, and VW its mean axis perpendicular to the plane of
the focal lines. On one of the focal lines and its dirigent assume
any corresponding points F and A, and let AD be the directrix
passing through A. Then if a directive plane, drawn through
any point S of the surface, cut this directrix in D and the mean

* See Exam. Papers, An. 1839, p. xxxi., questions 9,10. These and some of
the preceding theorems were originally stated with reference to modular foci only.
They are now extended to umbilicar foci.

On the Surfaces of the Second Order. 293

axis in W, the ratio of SF to SD will be expressed by the linear
modulus, as will also the ratio of VF to WD, since V is a point
of the surface, and WD is equal to the directive distance of V
from AD. But since V is a focus to which the mean axis is
directrix, the ratio of SV to SW is expressed by the same mo-
dulus. Thus the triangles SVF and SWD are similar, the sides
of the one being proportional to those of the other. Therefore
the angle SVF is equal to the angle SWD ; that is to say, the
angle which the side VS of the cone makes with the focal line
VF is equal to the angle contained by two right lines WD and
WS, of which one is the intersection of the directive plane with
the dirigent plane VWD corresponding to VF, and the other is
the intersection of the directive plane with the plane VWS
passing through the mean axis and the side VS of the cone.

Hence it appears that the sum of the angles (properly
reckoned) which any side of the cone makes with its two focal
lines is constant. For if EF’ be a point on the other focal line,
and D’ the point where the directrix corresponding to F’ is in-
tersected by the same directive plane SWD, it may be shown, as
above, that the angle SVE" is equal to the angle SWD’, that is,
to the angle made by the right line WS with the right line
WD’, in which the directive plane intersects the dirigent plane
corresponding to VE”. Conceiving therefore the points F, F’, 8,
and with them the points D, D’, to lie all on the samé side of
the principal plane which is perpendicular to the internal axis,
the right line WS will lie between the right lines WD and WD’,
and the sum of the angles SVF and SVE" will be equal to the
angle DWD’, which is a constant angle, being contained by
the right lines in which a directive plane intersects the two diri-
gent planes of the cone. This constant angle will be found to be
equal, as it ought to be, to one of the angles made by the two
sides of the cone which are in the plane of the focal lines, namely,
to the angle within which the internal axis lies.

If we conceive the cone to have its vertex at the centre of a
sphere, and the points F’, F’, S$ to be on the surface of this sphere,
the ares of great circles connecting the point S with each of the

294 On the Surfaces of the Second Order.

fixed points F, F’ will have a constant sum. The curve formed
by the intersection of the sphere and the cone may therefore,
from analogy, be called a spherical ellipse, or, more generally, a
spherical conic, because, by removing one of its foci F', F’ to the
opposite extremity of the diameter of the sphere, the difference
of the arcs SF and SF’ will be constant, which shows that the
spherical curve is analogous to the hyperbola as well as to the
ellipse. Hither of these plane curves may, in fact, be obtained
as a limit of the spherical curve when the sphere is indefinitely
enlarged, according as the diameter along which the enlarge-
ment takes place, and of which one extremity may be conceived
to be fixed while the other recedes indefinitely, coincides with
the internal or with the directive axis of the cone. The fixed
extremity becomes the centre of the limiting curve, which is an
ellipse in the first case, and a hyperbola in the second.

The great circle touching a spherical conic at any point
makes equal angles with the two ares of great circles which join
that point with the foci, because the sum of these ares is con-
stant. This is identical with a property already demonstrated
relative to the tangent planes of the cone. Indeed it is obvious
that the properties of the cone may also be stated as properties
of the spherical conic, and this is frequently the more convenient
way of stating them.

§ 5. If the sides of one cone be perpendicular to the tan-
gent planes of another, the tangent planes of the former will be
perpendicular to the sides of the latter. For the plane of two
sides of the first cone is perpendicular to the intersection of the
two corresponding tangent planes of the second cone; and as
these two sides approach indefinitely to each other, their plane
approaches to a tangent plane, while the intersection of the two
corresponding tangent planes of the second cone approaches in-
definitely to a side of the cone. Thus any given side of the one
cone corresponds to a certain side of the other; and any side of
_ either cone is perpendicular to the plane which touches the other
along the corresponding side. This reasoning applies to cones
of any kind.

On the Surfaces of the Second Order. 295

Two cones so related may be called reciprocal cones. When
one is of the second order, it will be found that the other is also
of the second order, and that, in their equations relative to their
axes, which are obviously parallel or coincident, the coefficients
of the squares of the corresponding variables are reciprocally
proportional, so that the equations

2

Pé+Q+R2=0, %+%+2-0 1
2+ Qy’ + Kz’ =0, Piat = 0, (1)

=<]

express two such cones which have a common vertex. These
cones have the same internal axis, but the directive axis of the one
coincides with the mean axis of the other, and it may be shown
from the equations that the directive planes of the one are per-
pendicular to the focal lines of the other. The two curves in
which these cones are intersected by a sphere, having its centre
at their common vertex, are reciprocal spherical conics. In
general, two curves traced on the surface of a sphere may be said
to be reciprocal to each other, when the cones passing through
them, and having a common vertex at the centre of the sphere,
are reciprocal cones. Any given point of the one curve corre-
sponds to a certain point of the other, and the great circle which
touches either curve at any point is distant by a quadrant from
the corresponding point of the other curve.

By means of these relations any property of a cone of the .
second order, or of a spherical conic, may be made to produce a
reciprocal property. Thus, we have seen that the tangent plane
of a cone makes equal angles with two planes passing through
the side of contact and through each of the focal lines; there-
fore, drawing right lines perpendicular to the planes, and planes
perpendicular to the right lines here mentioned, we have, in the
reciprocal cone, a side making equal angles with the right lines
in which the directive planes of this cone are intersected by a
plane touching it along that side. It is therefore a property of
the cone, that the intersections of a tangent plane with the two
directive planes make equal angles with the side of contact; a

296 On the Surfaces of the Second Order.

property which it is easy to prove without the aid of the reci-
procal cone.

The two directive sections drawn through any point S of
a given surface of the second order may, when they are circles,
be made the directive sections of a cone, and this may obviously
be done in two ways. Each of the two cones so determined will
be touched by the plane which touches the given surface at the
point S, because the right lines which are tangents to the two —
circular sections at that point are tangents to each cone as well
as to the given surface; therefore the side of contact of each
cone bisects one of the angles made by these two tangents ; and
hence the two sides of contact are the principal directions in the
tangent plane at the point 8, that is, they are the directions of
the greatest and least curvature of the given surface at that
‘point; for these directions are parallel to the axes of a section
made in the surface by a plane parallel to the tangent plane,
and the axes of any section bisect the angles contained by the
right lines in which the plane of section cuts the two directive
planes.

§ 6. It has been shown that the sum of the angles which any
side of a cone makes with its focal lines is constant. Hence
we obtain the reciprocal property,* that the sum of the angles

- (properly reckoned) which any tangent plane of a cone makes
with its two directive planes is constant. This property may
be otherwise proved as follows :—

Through a point assumed anywhere in the side of contact

* This property, and that to which it is reciprocal, as well as some other pro-
perties of the cone, were, together with the idea of reciprocal cones and of spherical
conics, suggested by my earliest researches connected with the mechanical theory
of rotation and the laws of double refraction. I was not then aware that the focal
lines of the cone had been previously discovered, nor that the spherical conic had
been introduced into geometry. Indeed all the properties of the cone which are
given in this Paper were first presented to me in my own investigations. Its double
modular property, related to the vertex as focus, was one of the propositions in the
theory of the rotation of a solid body, and was used in finding the position of the axis
of rotation within the body at a given time. But the modular property common to
all the surfaces of the second order was not discovered until some years later.

On the Surfaces of the Second Order. 297

let two directive planes be drawn. As the circles in which the
cone is cut by these planes have a common chord, they are circles
of the same sphere ; and a tangent plane applied to this sphere,
at the aforesaid point, coincides with the tangent plane of the
cone, because each tangent plane contains the tangents drawn to
_ the two circles at that point. The common chord of the circles is
bisected at right angles by the principal plane which is perpen-
dicular to the directive axis, and therefore that principal plane
contains the centres of the two circles and the centre of the
sphere. Now the acute angle made by a tangent plane of a
sphere with the plane of any small circle passing through the
point of contact is evidently half the angle subtended at the
centre of the sphere by a diameter of that circle; therefore the
acute angles, which the common tangent plane of the cone and
of the sphere above mentioned makes with the planes of the
directive sections, are the halves of the angles subtended at the
centre of the sphere by the diameters of the sections. But the
- diameters which lie in the principal plane already spoken of,
and are terminated by two sides of the cone, are chords of the
great circle in which that plane intersects the sphere; and the
halves of the angles which they subtend at its centre are equal
to the angles in the greater segments of which they are the
chords, and consequently equal to the two adjacent acute angles
of the quadrilateral which has these chords for its diagonals.
Hence, as two opposite angles of the quadrilateral are together
equal to two right angles, it follows that the four angles of the
quadrilateral represent the four angles, the obtuse as well as the
acute angles, which the tangent plane of the cone makes with the
planes of the directive sections ; the two angles of the quadrila-
teral which lie opposite to the same diagonal being equal to the
acute and obtuse angles made by the tangent plane with the
plane of the section of which that diagonal is the diameter.
Thus any two adjacent angles of the quadrilateral may be
taken for the angles which the tangent plane of the cone makes
with the directive planes. If we take the two adjacent angles
which lie in the same triangle with the angle « contained by the

4

298 On the Surfaces of the Second Order.

two sides of the cone that help to form the quadrilateral, the
sum of these two angles will be equal to two right angles
diminished by «; and if we take the two remaining angles of
the quadrilateral, their sum will be equal to two right angles
increased by x; both which sums are constant. But if we take
either of the other pairs of adjacent angles, the difference of the
pair will be constant, and equal to x.

The same conclusion may be deduced as a property of the
spherical conic. Let a great circle touching this curve be inter-
sected in two points, one on each side of the point of contact, by
the two directive circles, that is, by two great circles whose
planes are directive planes of the cone which passes through the
conic and has its vertex at the centre of the sphere. Since the
right lines in which the tangent plane of a cone intersects the
directive planes are equally inclined to the side of contact, the
are intercepted between the points where the tangent circle of
the conic intersects the directive circles is bisected in the point of
contact ; therefore, either of the spherical triangles whose base is
the tangent arc so intercepted, and whose other two sides are the
directive circles, has a constant area; because, if we suppose the
tangent are to change its position through an indefinitely small
angle, and to be always terminated by the directive circles, the
two little triangles bounded by its two positions and by the two
indefinitely small directive arcs which lie between these posi-
tions, will have their nascent ratio one of equality, so that the
area of either of the spherical triangles mentioned above will
not be changed by the change in the position of its base. But
in each of these triangles the angle opposite the base is constant ;
therefore the sum of the angles at the base is constant.

From this reasoning it appears that if a spherical triangle
have a given area, and two of its sides be fixed, the third side
will always touch a spherical conic having the fixed sides for its _
directive arcs, and will be always bisected in the point of contact.

_ § 7. The intersection of any given central surface of the
second order with a concentric sphere is a spherical conic, since
the cone which passes through the curve of intersection, and has

On the Surfaces of the Second Order. 299°

its vertex at the common centre, is of the second order. The
cylinder also, which passes through the same curve and has its
side parallel to any of the axes of the given surface, is of the
second order; and the cone, the cylinder, and the given sur-
face are condirective, that is, the directive planes of one of them
are also the directive planes of each of the other two. This may
be seen from the equations of the different surfaces; for, in ge-
neral, two surfaces whose principal planes are parallel will be
condirective, if, when their equations are expressed by co-ordi-
nates perpendicular to these planes, the differences of the coeffi-
cients of the squares of the variables in the equation of the one
be proportional to the corresponding differences in the equation
of the other.

If any given surface of the second order be intersected by a
sphere whose centre is any point in one of the principal planes,
the cylinder passing through the curve of intersection, and hay-
ing its side perpendicular to that principal plane, will be of the
second order, and will be condirective with the given surface.
This cylinder, when its side is parallel to the directive axis, is
hyperbolic ; otherwise it is elliptic. Ifa paraboloid be cut by
any plane, the cylinder which passes through the curve of sec-
tion, and has its side parallel to the axis of the paraboloid, will
be condirective with that surface ; and it will be elliptic or hy-
perbolic, according as the paraboloid is elliptic or hyperbolic.*

If two concentric surfaces of the second order be reciprocal
polars with respect to a concentric sphere, the directive axis of
the one surface will coincide with the mean axis of the other,
and the directive planes of the one will be perpendicular to the
asymptotes of the focal hyperbola of the other. "When one of
the surfaces is a hyperboloid, the other isa hyperboloid of the
same kind; the asymptotes of the focal hyperbola of each sur-

* T have introduced the terms directive and condirective, as more general than
the terms cyclic and biconeyclic employed by M. Chasles. The latter terms suggest
the idea of circular sections, and therefore could not properly be used with refe-
rence to the hyperbolic paraboloid, or to the hyperbolic or parabolic cylinder, in each
of which surfaces a directive section is a right line.

300 On the Surfaces of the Second Order.

face are the focal lines of its asymptotic cone; and the two
asymptotic cones are reciprocal.

When any number of central surfaces of the second onde
are confocal, or, more generally, when their focal hyperbolas
have the same asymptotes, it is obvious that their reciprocal sur-
faces, taken with respect to any sphere concentric with them, are
all condirective.

§ 8. If a diameter of constant length, revolving within a
given central surface, describe a cone having its vertex at the
centre, the extremities of the diameter will lie in a spherical —
conic. And if the cone be touched by any plane, the side of
contact will evidently be normal to the section which that plane
makes in the given surface, and will therefore be an axis of the
section. As the axes of a section always bisect the angles made
by the two right lines in which its plane intersects the directive
planes of the surface, and as the cone aforesaid has the same
directive planes with the given surface, it follows that the right
lines in which a tangent plane of a cone cuts its directive planes
are equally inclined to the side of contact—a theorem which has
been already obtained in another way.

If a section be made in a given central surface by any plane
passing through the centre, the cone described by a constant
semidiameter equal to either semiaxis of the section will touch
the plane of section ; for if it could cut that plane, a semiaxis
would be equal to another radius of the section. Denoting by
r, ” the semiaxes of the section, conceive two cones to be de-
scribed by the revolution of two constant semidiameters equal
to r and »” respectively. These cones are condirective with the
given surface, and have the plane of section for their common
tangent plane. Supposing that surface to be expressed by the
equation

2 2 2

P oR (2)

and the directive axis to be that of y, the axis of w will be the
internal axis of one cone, say of that described by 7, and the axis

On the Surfaces of the Second Order. 301

of s will be the internal axis of the other cone. Let « be the angle
made by the two sides of the first cone which lie in the plane
az, and «’ the angle made by the two sides of the second cone
which lie in the same plane; the former angle being taken so
as to contain the axis of x within it, and the latter so as to con- ©
tain within it the axis of s. Then, considering r, 7’ as radii of
the section wz of the surface, we have obviously

1 cos*g« , sin’ gn 4 1ii 4 2 peat
2 Pp Ri AAP eR Pap) 2
(3)

a. cos’ 3K’ | sin’ aK 11\_ 1 Ls ok m
ri R Po OB BL We BY?

observing that when these formule give a negative value for 7°
or x’, in which case the surface expressed by the equation (2)
must be a hyperboloid, the direction of r or r’ meets, not that
surface, but the surface of the conjugate hyperboloid expressed
by the equation

Ss

2 2 2

Sieg tee | (4)
Now calling @ and @ the angles made by the tangent plane of
the cones with the directive planes of the given surface, which
are also the directive planes of each cone, the angles x, x’ depend
on the sum or difference of 0 and @’. If the latter angles be
taken so that their sum may be equal to the supplement of x,
their difference will be equal to x’, and the formule (8) will
become

a74(p+5)- i(3- 3) cos (8 + 0’)

n-3(p* 3B) i(5- 7) 08 0- 0’),

by which the semiaxes of any central section are expressed in
terms of the non-directive semiaxes of the surface, and of the

(5)

302 On the Surfaces of the Second Order.

angles which the plane of section makes with the directive
planes.*

§ 9. From the centre O of the surface expressed by equa-
tion (2) let a right line OS be drawn cutting perpendicularly in
> the plane which touches the surface at 8. Let o denote the
length of the perpendicular OS, and a, (3, y the angles which it
makes with z, y,z. Then

o = P cos’a + Q cos’ B +R cos’ y. (6)

From this formula it is manifest that, if three planes touching —
the surface be at right angles to each other, the sum of the
squares of their perpendicular distances from the centre will be
equal to the constant quantity P + Q +R, and therefore the
point of intersection of the planes will lie in the surface of a
given sphere. If another surface represented by the equation

be touched by a plane cutting OS perpendicularly in 3), and if
oy be the length of OX, then

a? = P, cos’ a + Q cos® B + Ry cos y 5
and therefore when the two surfaces are confocal, that is, when
P-P,=Q-Q=R-R =k,

we have o*® — o, = k, which is a constant quantity. Hence, if
three confocal surfaces be touched by three rectangular planes,
the sum of the squares of the perpendiculars dropped on these
planes from the centre will be constant, and the locus of the
intersection of the planes will be a sphere.

The focal curves of a given surface are the limits of surfaces
confocal with it,t when these surfaces are conceived, by the pro-

* See Transactions of the Royal Irish Academy, Vou. xx1., as before cited.
The formule (5) were first given, for the case of the ellipsoid, by Fresnel, in his
Theory of Double Refraction, Mémoires de I’ Institut, tom. vii., p. 155.

+ It was by this consideration, arising out of the theorems given in this and the
next section about confocal surfaces, that I was led to perceive the nature of the

On the Surfaces of the Second Order. 303

gressive diminution of their mean or secondary axes, to become
flattened, and to approach more and more nearly to a plane
passing through the primary axis. And it will appear here-
after, that if a bifocal right line, that is, a right line passing
through both focal curves, be the intersection of two planes
touching these curves, those two planes will be at right angles
to each other. Therefore the locus of the point where a tangent
plane of a given central surface is intersected perpendicularly
by a bifocal right line is a sphere. The primary axis of the sur-
face is evidently the diameter of this sphere.

Hence we conclude that the locus of the point where a tan-
gent plane of a paraboloid is intersected perpendicularly by a
bifocal right line is a plane touching the paraboloid at its vertex.
For a paraboloid is the limit of a central surface whose primary
axis is prolonged indefinitely in one direction, and a plane is
the corresponding limit of the sphere described on that axis as
diameter. As this consideration is frequently of use in deduc-
ing properties of paraboloids from those of central surfaces, it
may be well to state it more particularly. It is to be observed,
then, that the indefinite extension of the primary axis at one
_ extremity may take place according to any law which leaves the ~
other extremity always at a finite distance from a given point,
and} gives a finite limiting parameter to each of the principal
sections of the surface which pass through that axis. The
simplest supposition is, that one extremity of the axis and the
adjacent foci of those two principal sections remain fixed, while
the other extremity and the other foci move off, with the centre,
to distances which are conceived to increase without limit. Then,
at any finite distances from the fixed points, the focal curves
approach indefinitely to parabolas, as do also all sections of the
surface which pass through the primary axis, while the surface
itself approaches indefinitely to a paraboloid; so that the limit

focal curves, and the analogy between their points and the foci of conics. And I
regarded that analogy as fully established when I found (in March or April, 1832)
that the normal at any point of a surface of the second order is an axis of the cone
which has that point for its vertex and a focal for its base.

304 On the Surfaces of the Second Order.

of the central surface is a ‘paraboloid having parabolas for its
focal curves. The limit of an ellipsoid, or of a hyperboloid of
two sheets, is an elliptic paraboloid, having one of its focals
modular and the other umbilicar, like each of the central sur-
faces from which it may be derived ; and the limit of a hyper-
.boloid of one sheet is a hyperbolic paraboloid, having, like that
hyperboloid, both its focals modular.

§ 10. Let the plane touching at S the surface expr by
equation (2) intersect the axis of x in the point X, and let the .
normal applied at S intersect the planes yz, ws, vy, in the points
L, M, N respectively. Since the section made in the surface by
a plane passing through OX and the point S has one of its axes
in the direction of OX, it appears, by an elementary property of
conics, that the rectangle under OX and the co-ordinate « of
the point 8 is equal to the quantity P; but that co-ordinate is
to LS as OS or o is to OX, and therefore the rectangle under o
and LS is equal to P. Similarly the rectangle under o and MS
is equal to Q, and the rectangle under o and NS is equal to R.
Thus the parts of the normal intercepted between the point 8
and each of the principal planes are to each other as the squares
of the semiaxes respectively perpendicular to these planes; the
square of an imaginary semiaxis being regarded as negative,
and the corresponding intercept being measured from 8 in a
direction opposite to that which corresponds to a real semiaxis.

The rectangle under o and the part of the normal inter-
cepted between two principal planes is equal to the difference
of the squares of the semiaxes which are perpendicular to these
planes. This rectangle is therefore constant, not only for a given
surface, but for all surfaces which are confocal with it. .

Hence the part of the normal intercepted between two prin-
cipal planes bears a given ratio to the part of it intercepted
between one of these and the third principal plane, whether the
normal be applied at any point of a given suai or at any
point of a surface confocal with it.

If therefore normals to a series of confocal surfaces be all
parallel to a given right line, they must all lie in the same plane

On the Surfaces of the Second Order. 305

passing through the common centre of the surfaces, because
otherwise the parts of any such normal, which are intercepted
between each pair of principal planes, would not be in a constant
ratio to each other.

The point S being the point at which any of these parallel
normals is applied, the plane touching the surface at S is parallel
to a given plane; the perpendicular OS dropped upon it from
the centre has a given direction, the plane OS® is fixed, and the
directions of the lines OL, OM, ON, in which the plane intersects
the principal planes, are also fixed. And as the angle OXS is
always a right angle, and the normal at S is always parallel to
GX, the distance S bears a given ratio to each of the distances
OL, OM, ON, and therefore also to each of the intercepts MN.
Hence, since the rectangle under OS and any one of these in-
tercepts is constant, the rectangle under OS and S> is constant.

Therefore if a series of confocal central surfaces be touched
by parallel planes, the points of contact will all lie in one plane,
and their locus, in that plane, will be an equilateral hyperbola,
having its centre at the centre of the surfaces, and having one
of its asymptotes perpendicular to the tangent planes. This
hyperbola evidently passes through two points on each of the
focal curves, namely the points where the tangent to each curve
is parallel to the tangent planes.

If a series of confocal paraboloids be touched by parallel
planes, it will be found that the points of contact all lie ina
bifocal right line, and that the normals at these points lie in a
plane parallel to the axis of the surfaces; so that the part of
any normal which is intercepted by the two principal planes is
constant. This theorem may be proved from the two following
properties of the paraboloid :—1. A normal being applied to the
surface at the point 8, the segments of the normal, measured
from § to the points where it intersects the planes of the two
principal sections, are to each other inversely as the parameters
of these sections. 2. Supposing the axis « to be that of the sur-
face, the difference between the co-ordinates ~ of the point S and
of the point where the normal meets the plane of the principal

: x

306 On the Surfaces of the Second Order.

sections, is equal to the semiparameter of the other principal
section.

§ 11. Let a tangent plane, applied at any point S of a surface
of the second order, intersect the plane of one of its focals in the
right line ©, and let P be the foot of the perpendicular dropped
from S upon the latter plane. The pole of the right line ©, with
respect to the principal section lying in this plane, is the point
P. Let N be its pole with respect to the focal. Then if T be
any point of the right line ©, the polar of this point with respect
to the section will pass through P, and its polar with respect to
the focal will pass through N ; and if the former polar intersect
the dirigent.curve in A, A’, and the latter intersect the focal in
F, F’, the points F, F’ will correspond respectively to the points
A, 4’, and the distances AA’ and FF’ will be similarly di-
vided by the points P and N (see Part I. § 8). But since the
point S is in the plane of the two directrices which pass through
A and A’, the lengths AP and A’P, which are the perpendicular
distances of S from the directrices, are proportional to the lengths
FS and F’S. Therefore FN is to F’N as FS is to E'S, and the
right line NS bisects one of the angles made by the right lines
FS and F’S, And as this holds wherever the point T is taken
on the right line ©, that is, in whatever direction the right line
FF’ passes through the point N, it follows that the right line
NS.is an axis of the cone which has the point § for its vertex
and the focal for its base. Further, if FF’ intersect © in the
point Q, we have FN to F’N as FQ is to F’Q, because N is the
pole of © with respect to the focal; therefore FQ is to F’Q as_
FS is to F’8, and hence the right line QS also bisects one of the.
angles made by FS and F’S. The right lines NS and QS are
therefore at right angles to each other; and as the latter always
lies in the tangent plane, the former must. be perpendicular to
that plane. |

Consequently the normal at any point of a surface of the
second order is an axis of the cone which has that point for its
vertex and either of the focals for its base.

It is known that when two confocal surfaces intersect each

On the Surfaces of the Second Order. 307

other, they intersect everywhere at right angles; and that
through any given point three surfaces may in general be de-
scribed, which shall have the same focal curves. If three con-
focal surfaces pass through the point S, the normal to each of
them at § is an axis of each of the cones which stand on the
focals and have S for their common vertex. The normals to the
three surfaces are therefore the three axes of each cone.

If the points at which a series of confocal surfaces are touched
by parallel planes be the vertices of cones having one of the
focals for their common base, each of these cones will have one
of its axes perpendicular to the tangent planes. Therefore when
an axis of a cone which stands on a given base is always parallel
to a given right line, the locus of the vertex is an equilateral
hyperbola or a right line, according as the base is a central conic
or a parabola.

§ 12. A system of three confocal surfaces intersecting each
other consists of an ellipsoid, a hyperboloid of one sheet, and a
hyperboloid of two sheets, if the focals be central conics; but it
consists of two elliptic paraboloids and a hyperbolic paraboloid,
if the focals be parabolas. In the central system, the ellipsoid
has the greatest primary axis, and the hyperboloid of two sheets
the least ; and the focal which is modular in one of these sur-
faces is umbilicar in the other. The asymptotic cones of the
hyperboloids are confocal, the focal lines of each cone being the
asymptotes of the focal hyperbola. In the system of parabo-
loids, the two elliptic paraboloids are distinguished by the cir-
cumstance that the modular focal of the one is the umbilicar
focal of the other.

The curve in which two confocal surfaces intersect each other
is a line of curvature of each, as is well known ;* and a series of
lines of curvature on a given surface are found by making a
series of confocal surfaces intersect it.

Now if a series of the lines of curvature of a given surface
be projected on one of its directive planes by right lines parallel

* See Dupin’s Développements de Géomeétrie.

x2

308 On the Surfaces of the Second Order.

to either of its non-directive axes, the projections will be a series
of confocal conics; and when the surface is umbilicar, the foci
of all these conics will be the corresponding projections of the
umbilics.* When the surface is not umbilicar, its directive axis
will be parallel to the primary axis of the projections.

The same line of curvature has two projections, according as
it is projected by right lines parallel to the one or to the other _
non-directive axis. In the ellipsoid these projections are always
curves of different kinds, the one being an ellipse when the other
is a hyperbola; but in a hyperboloid the projections are either
both ellipses or both hyperbolas. In the hyperbolic paraboloid
the projections are parabolas. In the elliptic paraboloid one —
of the projections is always a parabola, and the other is either
an ellipse or a hyperbola.

The corresponding projections of two lines of curvature which
pass through a given point of the surface are confocal conics in-
tersecting each other in the projection of that point, and of
course intersecting at right angles.

§ 13. A difocal chord is a bifocal right line terminated both
ways by the surface.t In a central surface, the length of a
bifocal chord is proportional to the square of the diameter which
is parallel to it; the square of the diameter being equal to the
rectangle under the chord and the primary axis.

More generally, if a chord of a given central surface touch
two other given surfaces confocal with it, the length of the chord
will be proportional to the square of the parallel diameter of the
first surface, the square of the diameter being equal to the rect-
angle under the chord and a certain right line 2/, determined by
the formula

L or

°= (BP) (P- PY!’ @)

wherein it is supposed that the equation (2) represents the first

* “Exam. Papers,’’ An. 1838, p. xlvi., question 4; p. xcix., question 70.
+ The theorems in § 13 are now stated for the first time.

On the Surfaces of the Second Order. 309

surface, and that P’, P” are the quantities corresponding to P
in the equations of the other two surfaces.

In any surface of the second order, the lengths of two bifocal
chords are proportional to the rectangles under the segments of
any two intersecting chords to which they are parallel.

In the paraboloid expressed by the equation

y* + x = L,

P ¢
if y be the length of a bifocal chord making the angles B and y
with the axes of y and z respectively, we have

1 _ cos’p ¥ cos’ y
aes gi;

(8)

§ 14. At the point S on a given central surface expressed by
the equation (2), let a tangent plane be applied, and let 4, x’
be the squares of the semiaxes of a central section made in the
surface by a plane parallel to the tangent plane; each of the
quantities k, k’ being positive or negative, according as the
corresponding semiaxis of the section is real or imaginary, that
is, according as it meets the given surface or not. Then the
equations* of two other surfaces confocal with the given one,
and passing through the point 8, are

x y° 2? ted y 2?

Rept ost Bok Ploy t.O2et BY

=1. (9)

The given surface is intersected by these two surfaces respec-
tively in the two lines of curvature which pass through the
point S; the tangent drawn to the first line of curvature at S is
parallel to the second semiaxis of the section, and the tangent
drawn to the second line of curvature at 8 is parallel to the first
semiaxis of the section.

When two confocal surfaces intersect, the normal applied to

* ‘Exam, Papers,’’ An, 1837, p.c., questions 4, 5,6; An.1838, p. c., questions 71, 72,

310 On the Surfaces of the Second Order.

one of them at any point § of the line of curvature formed by
their intersection lies in the tangent plane of the other, and is
parallel to an axis of any section made in the latter by a plane
parallel to the tangent plane. Supposing the surfaces to be cen-
tral, if two normals be applied at the point S, and a diameter of
each ‘surface be drawn parallel to the normal of the other, the
two diameters so drawn will be equal and of a constant length,
wherever the point 8 is taken on the line of curvature; the
square of that length being equal to the difference of the squares
of the primary axes of the surfaces, and the diameter of the sur-
face which has the greater primary axis being real, while that of
the other surface is imaginary. As the point S moves along the
line of curvature, each constant diameter describes a cone condi-
rective with the surface to which it belongs; the two cones so
described are reciprocal, and the focal lines of the cone which
belongs to one surface are perpendicular to the directive planes
of the other surface. :

When two confocal paraboloids intersett, if normals be ap-
plied to them at any point S of their intersection, and a bifocal.
chord of each surface be drawn parallel to the normal of the
other, the two chords so drawn will be equal and of a constant
length, wherever the point S is taken in the line of intersection
of the surfaces ; that constant length being equal to the differ-
ence between the parameters of either pair of coincident principal
sections.

§ 15. The point 8 being the common intersection of a given
system of confocal surfaces, of which the equations are

Sis ey oe le a
Bt ot Rh Pig R=)
(10)
a” y° 3? ’
Pg * Ro)

suppose that another surface A confocal with these, and ex-
pressed by the equation

y* 3?

B Gite (11)

On the Surfaces of the Second Order. 311

is circumscribed by a cone having its vertex at 8. If the nor-
mals applied at S to the given surfaces, taken in the order of the
equation (10), be the axes of new rectangular co-ordinates &, n, Z,
the equation of the cone, referred to these co-ordinates, will be*
2 2 2
é fr a eis & &
P= Py OP Py FFP
The surfaces of the given system, in the order of their
equations, may be supposed to be an ellipsoid, a hyperboloid
of one sheet, and a hyperboloid of two sheets; the axes of
2, y, = being respectively the primary, the mean, and the
secondary axes of each surface. Then P is greater than P,
and P’ greater than P”. |
The normals to the given surfaces are the axes of the cone
-expressed by the equation (12); and if the surface A be
changed, but still remain confocal with the given system, it
is obvious from that equation that the focal lines of the cir-
cumscribing cone will remain unchanged, since the differences
of the quantities by which the squares of &, n, @ are divided
are independent of the surface A. As P” is intermediate in
value between P and P”, the normal to the hyperboloid of
one sheet is always the mean axis of the cone; the focal lines
lie in the plane &Z, and their equation is
&? oe
a
P=aP =P"
* The equation (12) was obtained in the year 1832, and was given at my
Lectures in Hilary Term, 1836. The most remarkable properties of cones circum-
scribing confocal surfaces are immediate consequences of this equation. That
such cones, when they have a common vertex, are confocal, their focal lines
being the generatrices of the hyperboloid of one sheet passing through the
vertex, was first stated by Professor C. G. J. Jacobi, of Kénigsberg, in 1834
(see Crelle’s Jowrnal, Vou. x11. p. 137). See also the excellent work of M. Chasles,
published in 1837, and entitled Apergu historique sur T’ Origine et le Développe-
ment des Méthodes en Géométrie, p. 387. The analogy which exists between the
focals of surfaces and the foci of curves of the second order was supposed by
M. Chasles to have been pointed out in that work for the first time (Comptes

Rendus, tom. xvi., pp. 883, 1106); but that analogy had been previously taught and
developed in the Lectures just alluded to. a

0. (12)

= 0, (13)

312 On the Surfaces of the Second Order.

which shows that they are parallel to the asymptotes of a
central section made in the hyperboloid of one sheet by a
plane parallel to the plane &Z, since the quantities P’ - P and
P’-P” are (including the proper signs) the squares of the
semiaxes of the section which are parallel to € and Z re-
spectively. The focal lines are therefore the generatrices of
that hyperboloid at the point S.
When & = 0, the equation (12) becomes
ae bates a

— oe — = {
R 7 R’ + R’ 0, (14)

which is that of the cone standing on the focal ellipse and
having its vertex at 8S. When Q,=0, the same equation
becomes

= 0, (15)

which is that of the cone standing on the focal hyperbola,
and having its vertex at 8. The normal to the hyperboloid
of one sheet at the point S is the mean axis of both cones;
the normal to the ellipsoid is the internal axis of the first
cone and the directive axis of the second, while the normal
to the hyperboloid of two sheets is the directive axis of the
first and the internal axis of the second.
The three surfaces expressed by the equations

ee ar hg

Pig Se Ns 4 Bed
P Pp 7 1, Q QQ Ue ~
eof a
2 n 2
a? Wo Rt

are a confocal system, having their centres at 8, and being re-
spectively an ellipsoid, a hyperboloid of one sheet, and a hyper-
boloid of two sheets. They intersect each other in the centre
of the system expressed by the equations (10), and their normals
at that point are the axes of w, y, s respectively. The relations

On the Surfaces of the Second Order. 3 13

between the two systems of surfaces are therefore perfectly reci-
procal. From the equations (14) and (15) it is manifest that
the asymptotic cones of the hyperboloids of one system pass
through the focals of the other.

§16. The point S being the intersection of a given system
of confocal paraboloids whose equations are

(17)
2 2
r + we =a+h”,
tee
where p-p’=q-(7=4(h-/’), and p-p”=q-q7’=4(h-h"):
suppose that another paraboloid A confocal with these, and
expressed by the equation

2 2
Yat hy (18)

is circumscribed by a cone having its vertex at S. Then if the
normals applied at 8 to the given system of surfaces, taken in
the order of their equations, be the axes of the co-ordinates

E, n, € respectively, the equation of the circumscribing cone will
be

é + , sl + ih Ce

Pk ei aaa a I
showing that those normals are the axes of the cone, and that
the focal lines of the cone are independent of the surface A,
provided it be confocal with the given surfaces. If the hyper-
bolic paraboloid be the second surface of the given system, the
parameter p’ will be intermediate in value between p and p”,

and the equation of the focal lines of the cone will be

= + = Tim

Yl Se ‘ead
which is the equation of a pair of right lines parallel to the
asymptotes of a section made in the hyperbolic paraboloid by

OS ens’ (19)

0, (20)

314 On the Surfaces of the Second Order.

a plane parallel to the plane &Z, since the quantities p’—p and
p’-p” are proportional to the squares of the semiaxes of the
section which are parallel to € and Z respectively. The focal
lines are therefore the generatrices of the hyperbolic paraboloid
at the point S.

Putting p, and q alternately equal to zero in the equation
(19), we get

2 2 2

Se a Ge 2 Sp ee (21)

eee ae 33
the equations of two cones which have a common vertex at §,
the first of them standing on the focal which lies in the plane
az, the second on the focal which lies in the plane zy. The
mean axis of each of these cones is the normal at S to the
hyperbolic paraboloid; the internal axis of either cone is the
normal to the elliptic paraboloid which has the base of that
cone for its modular focal.

As the cones which have a common vertex, and stand on
the focals of any surface of the second order, are confocal, they
intersect at right angles. Therefore when two planes passing
through a bifocal right line touch the focals, these planes are
at right angles to each other. And as cones which have a
common vertex, and circumscribe confocal surfaces, are confo-
cal, two such cones, when they intersect each other, intersect at
right angles. Therefore when a right line touches two confo-
cal surfaces, the tangent planes passing through this right line
are at right angles to each other. :

§ 17. When two surfaces are reciprocal polars* with respect
to any sphere, and one of them is of the second order, the other
is also of the second order. ‘Let the surface B be reciprocal to
the surface A before mentioned, with respect to a sphere of
which the centre is S; and suppose R’ and R to be any cor-
responding points on these surfaces. Then the plane which

* Transactions of the Royal Irish Academy, Vou. xvir. p. 241; “Examination —
Papers,’’ An. 1841, p. cxxvi., question 4.

On the Surfaces of the Second Order. 315

touches the surface A at the point R intersects the right line
SR’ perpendicularly in a point K, such that the rectangle under
SR’ and SK is constant, being equal to the square of the radius
of the sphere. Now if the point K approach indefinitely to §,
the distance SR’ will increase without limit, the surface B being
of course a hyperboloid; and if through S any plane be drawn
touching the surface A, a right line perpendicular to this plane
will evidently be parallel to a side of the asymptotic cone of the
hyperboloid. The asymptotic cone of B is therefore reciprocal
to the cone which, having its vertex at S, circumscribes the
surface A. Hence, as the directive planes of a hyperboloid
are the same as those of its asymptotic cone, it follows that the
directive planes of the surface B are perpendicular to the gene-
ratrices of the hyperboloid of one sheet, or the hyperbolic para-
boloid, which passes through S, and is confocal with the surface
A. And this relation between two reciprocal surfaces ought to
be general, whatever be the position of the point 8 with respect
to them ;* for though it has been deduced by the aid of the
circumscribing cone aforesaid, it does not, in its enunciation,
imply the existence of such a cone. This conclusion may be
verified by investigating the equation of the surface B in terms
of the co-ordinates &, n, Z. Suppose p to be the radius of the
sphere with respect to which the surfaces A and B are recipro-
cal. Then if A be a central surface expressed by. the equation
(11), and having &, m, 2 for the co-ordinates of its centre, the
surface B will be represented by the equation

(P - P,) £*+(P’- P,) x? +(P”- P,)2*

= 2p? (EE + non + SoS) — p4

but if A be a paraboloid expressed by the equation (18), the
equation of B will be

(p — po) * + (p’ — po) n° + (p” — po) 2?

(22)

(23)
= p4’(E cosa +n cos 9 + cos y),

* This relation was first noticed by Mr. Salmon.

316 On the Surfaces of the Second Order.

where a, 3, y are the angles which the axis of « makes with the
axes of &, n, Z respectively. In the first case, the equation (22)
shows that the directive planes of B are perpendicular to the
right lines expressed by the equation (13); in the second case,
the equation (23) shows that the directive planes of B are per-
pendicular to the right lines expressed by the equation (20).
When the surface A is a paraboloid, and the distance of the
point R from its vertex is indefinitely increased, the plane
touching the surface at R approaches indefinitely to parallel-
ism with its axis, and the right line SK, perpendicular to that
plane, increases without limit. Therefore the surface B passes
through the point 8, and is touched in that point by a plane
perpendicular to the axis of A, | .
When the point 8 lies upon the surface A, the co-efficient
of the square of one of the variables, in the equation (22) or
(23), is reduced to zero, and the surface B is a paraboloid
having its axis parallel to the normal applied at 8 to the
surface A. This also appears from considering that when § is
a point of the surface A, the normal at that point is the only
right line passing through 8, which meets the surface B at an
infinite distance.
If a series of surfaces be confocal, their reciprocal surfaces,
taken with respect to any given sphere, will be condirective.
When the equations of any two condirective surfaces are ex-
pressed by co-ordinates perpendicular to their principal planes,
the constants in the equations may be always so taken that the
differences of the co-efficients of the squares of the variables in
one equation shall be equal to the corresponding differences in
the other. Then, by subtracting the one equation from the
other, we get the equation of a sphere. Therefore when two
condirective surfaces intersect each other, their intersection is,
in general, a spherical curve. But when the surfaces are two
paraboloids of. the same species, their intersection is a plane
curve. :
§18. Through any point S of a given surface four bifocal
right lines may in general be drawn. Supposing the surface

On the Surfaces of the Second Order. 317

to be central, let a plane drawn through the centre, parallel to
the plane which touches the surface at S, intersect any one of
these right lines. Then the distance of the point of intersec-
tion from the point 8 will always be equal to the primary semi-
axis of the surface.*

If through any point S of a given central surface a right
line be drawn touching two other given surfaces confocal with
it, and if this right line be intersected by a plane drawn
through the centre parallel to the plane which touches the
first surface at 8, the distance of the point of intersection from
the point S will be constant, wherever the point 8 is taken on
the first surface. If this constant distance be called /, and the
other denominations be the same as in the formula (7), the
value of / will be given by that formula. t

* ** Examination Papers,’ An. 1838, p. xlvii., question 9.

¢ In the notes to the last-mentioned work of M. Chasles, on the History of
Methods in Geometry, will be found many theorems relative to surfaces of the
second order. Among them are some of the theorems which are given in the
present Paper; but it is needless to specify these, as M. Chasles’s work is so well
known.

(338: ,)

III.—NOTE RELATIVE TO THE COMPARISON OF ARCS OF
CURVES, PARTICULARLY OF PLANE AND SPHERI-
CAL CONICS.

[Proceedings of the Royal Irish Academy, Vou. 11. p. 446.—Read Noy. 30, 1843.]

Tue first Lemma given in my Paper on the rectification of the
conic sections* is obviously true for curves described on any
given surface, provided the tangents drawn to these curves be
shortest lines on the surface. The demonstration remains exactly
the same; and the Lemma, in this general form, may be stated
as follows :—

Understanding a tangent to be a shortest line, and sup-
posing two given curves E and F to be described on a given
surface, let tangents drawn to the first curve at two points T, ¢,
indefinitely near each other, meet the second curve in the points
P,p. Then taking a fixed point A on the curve H, if we put s
to denote (according to the position of this point with respect
to T) the sum (or difference) of the arc AT and the tangent
TP, and s + ds to denote the sum (or difference) of the are A¢
and the tangent ty, we shall have ds equal to the projection of
the infinitesimal arc Pg upon the tangent; that is, if a be the
angle which the tangent TP makes with the curve F at the
point P, we shall have ds equal to Py multiplied by the cosine
of a.

Now through the points P, p conceive other tangents T’P,

* Transactions of the Royal Irish Academy, Vou. xvi. p. 79 (supra, p. 20).

Note relative to the comparison of Arcs of Curves. 319

tp to be drawn, touching the curve E in the points T’, ¢’; and
let x and ds’ have for these tangents the same signification which
s and ds have for the former tangents. Supposing the nature of
the curve F to be such that it always bisects, either internally or
externally, the angle made at the point P by the tangents TP
and T’P, it is evident that ds = + ds’, and therefore either s +
ors—s isaconstant quantity.

A simple example of this theorem is afforded by the plane
and spherical conics. If the curves E and F be two confocal
conics, either plane or spherical, and tangents TP, T’P be
drawn to F from any point P of E (the tangents being of
course right lines when the curves are plane, and ares of great
circles when they are spherical; in both cases shortest lines),
it is well known that the angle TPT’ made by the tangents is
always bisected by the conic E. The angle is bisected inter-
nally or externally according as the conics intersect or not.
Hence we have the two following properties* of confocal
conics :—

1. When two confocal conics do not intersect, if one of
them be touched in the points T, T’ by tangents drawn from
any point P of the other, the sum of the tangents TP, T’P will
exceed the convex are T'l’ lying between the points of contact,
by a constant quantity.

2. When two focal conics intersect in the point A, if one
of them be touched in the points T, T’ by tangents drawn from
any point P of the other, the difference between the tangents

* The first of these properties was originally given for spherical conics by the
Rey. Charles Graves, Fellow of Trinity College, in the ‘‘ notes and additions’’ to
his translation of M. Chasles’s Memoirs on Cones and Spherical Conics, p. 77
(Dublin, 1841). Mr. Graves obtained it as the reciprocal of the proposition, that
when two spherical conics have the same directive circles, any tangent arc of the
inner conic divides the outer one into two segments, each of which has a constant
area. Both properties, with the general theorem relative to curves described on
any surface and touched by shortest lines, were afterwards given in the University
Calendar. See ‘‘ Examination Papers,’’ An. 1841, p. xli., questions 3-6; An. 1842,
p. lxxxiii., questions 30-34, These two properties of conics were communicated, in
October, 1848, to the Academy of Sciences of Paris, by M. Chasles, who supposed
them to be new. See the Comptes Rendus, tom. xvii. p. 838,

320 LVote relative to the comparison of Arcs of Curves.

TP, T’P will be equal to the difference between the ares AT,
AY’.

These properties give the readiest and most elegant solution
of problems concerning the comparison of different arcs of a
plane or spherical conic. Any are being given on a conic, we
may find another are beginning from a given point, which shall
differ from the given arc by a right line if the conic be plane,
or by a circular are if the conic be spherical.

- *
ats "

eae,

IV.—NOTE ON SURFACES OF THE SECOND ORDER.

[Proceedings of the Royal Irish Academy, Vow. 11. p. 429.—Read April 12, 1847.}

Let a surface A of the second order be represented by the
equation

its primary axis being that of 2 Through a given point S
whose co-ordinates are 2’, 7’, x’, conceive three surfaces confocal
with A to be described, and let P, P’, P’, be the squares of
their primary semiaxes. Then if normals drawn to these sur-
faces respectively. at the point S be the axes of a new system of
co-ordinates &, n, 2, and if we put

/2 72 12
P-P,=k, P-P,=%, P’-P.=k’, = pt et® tog
the equation of the surface A, referred to the new co-ordinates,
will be
& us . 505 non CoG .
Tt+pt f= (f- 1) (#4 mr + 1), (a)

where £&, m, 2) are the co-ordinates of its centre.
From the form of this equation it is evident that, if the sur-
face be intersected by the plane whose equation is
EE non , 606
Pes er aa (0)
Y

322 Note on Surfaces of the Second Order.

it will be touched along the curve of intersection by the cone
whose equation is
2 2 2
eee (c)
This mode of deducing, in its simplest form, the equation of a ©
cone circumscribing a surface of the second order, is much easier
than the direct investigation by which the equation (c) was
originally obtained.

Let a right line passing through S intersect the plane ex-
pressed by the equation (b), in a point whose distance from S is
equal to «, while it intersects the surface A in two points, P and
P’, the distance of either of which from S is denoted by p. Let
the surface B, represented by the equation

£? 2 a
Set eet als (d)

be intersected by the same right line in a point whose distance
from S is equal to 7, the distance r being, of course, a semidia-
meter of this surface. Then it is obvious that the equation (a)
may be written

so that, if p and p’ represent the distances SP and SP’ respec-
tively, we have

(¢)
and therefore

(f)

This result is useful in questions relating to attraction. For
if A be an ellipsoid, every point of which attracts an external
point S with a force varying inversely as the fourth power of
the distance, and if the point S be the vertex of a pyramid, one

Note on Surfaces of the Second Order. 323

of whose sides is the right line SPP’, and whose transverse sec-
tion, at the distance unity from its vertex, is the indefinitely
small area w, the portion PP’ of the pyramid will attract the
point S, in the direction of its length, with a force expressed by
the quantity

and putting @ for the angle which the right line SP makes with
the axis of &, the attraction in the direction of & will be

2w cos 6
te

(9)

Now, supposing the axis of — to be normal to the confocal ellip-
soid described through S, it will be the primary axis of the
surface B, which will be a hyperboloid of two sheets; and the
surface being symmetrical round this axis, it is easy to see, from
the expression for the elementary attraction, that the whole
attraction of the ellipsoid will be in the direction of &. There-
fore when the force is inversely as the fourth power of the dis-
tance, the: attraction of an ellipsoid on an external point is
normal to the confocal ellipsoid passing through that point.

Hence we infer, that if U be the sum of the quotients found
by dividing every element of the volume of an ellipsoid by the
cube of its distance from an external point, the value of U will
remain the same, wherever that point is taken on the surface of
an ellipsoid confocal with the given one.

The question of the attraction of an ellipsoid, when the law
of force is that of the inverse square of the distance, has been
treated by Poisson, in an elegant but very elaborate memoir,
presented to the Academy of Sciences in 1833.* In the preced-
ing year I had obtained the theorems just mentioned, by con-
sidering the law of the inverse fourth power; and, as well as I
remember, they were deduced exactly as above, by setting out
from the equation (2). But I did not then succeed in applying

* Mémoires de T Institut, tom. xiii.

¥2

324 Note on Surfaces of the Second Order.

the same method to the case where the law of force is that of
nature, probably from not perceiving that, in this case, the ellip-
soid ought to be divided (as Poisson has divided it) into concen-
tric and similar shells. This application requires the following
theorem, which is easily proved :—

Supposing JA’ to be another ellipsoid, concentric, similar, and
similarly placed with 4; let the right line SPP’ intersect
it in the points py and p’, respectively, adjacent to P and P’;
then if the direction of that right line be conceived to vary, the
rectangle under Pp and P’p (or under Pp’ and P’p’) will be to
the rectangle under SP and SP’ in a constant ratio.

Denoting the constant ratio by m, and combining this theo-
rem with the formula (f), we have

Ppx Pp mr
PP 2° (1)

Now let the two surfaces A and A’ be supposed to approach in-
definitely near each other, so as to form a very thin shell, then
ultimately P’p will be equal to PP’, and we shall have

Pp = Py’ =>

where m is indefinitely small. Therefore if the point S, external
to the shell, be the vertex of a pyramid whose side is the right
line SP, and whose section, at the unit of distance from the
vertex, is w, the attraction of the two portions Pp and P’p’ of
this pyramid, which form part of the shell, will be equal to mrw.
Hence it appears, as before, on account of the symmetry of the
surface B round the axis of &, that the whole attraction of the
shell on the point S is in the direction of that axis, and conse-
quently (as was found by Poisson) in the direction of the inter-
nal axis of the cone whose vertex is S, and which circumscribes

the shell.
To find the whole attraction of the shell, the expression

mrw cos 0 (¢)

Note on Surfaces of the Second Order. 3.25

must be integrated. Let @ be the angle which a plane, passing
through SP and the axis of &, makes with the plane &n, then

= w = sin 0d0d¢
1 (— sin’@ cos’ — sin? me 1
Be. Nee + 7 rf 7 ——:
r ee a a a

When these values are substituted in (7), that expression may be
readily integrated, first with respect to 0, and then with respect
to ¢.

It is evident that, by the same substitutions, the expression
(g) may be twice integrated.

An investigation similar to the preceding has been given by
M. Chasles, for the case in which the force varies inversely as
the square of the distance.* He uses a theorem equivalent to
the formula (7), but deduces it in a different way.

From what has been proved it follows that, if V be the sum
of the quotients found by dividing every element of the shell by
its distance from an external point S, the value of V will be the
same wherever that point is taken on the surface & of an ellip-
soid confocal with the surface 4 of the shell.

Let > be another ellipsoid confocal with A, and indefinitely
near the surface =. The normal interval between the two sur-
faces = and &’, at any point S on the former, will be inversely
as the perpendicular dropped from the common centre of the
ellipsoids on the plane which touches 3 at S. Hence, supposing
the point S to move over the surface =, that perpendicular will
vary as the attraction exerted by the shell on the point S, when
the force is inversely as the square of the distance, or as the at-
traction exerted by the whole ellipsoid 4 on the point S, when
the force is inversely as the fourth power of the distance.

When the point S is on the focal hyperbola, the integrations,
by which the actual attraction is found in either case, are sim-
plified, for the surface B is then one of revolution round the
axis of &, and its semidiameter r is independent of the angle ¢.

* Mémoires des Savants Etrangers, tom. ix.

326 Vote on Surfaces of the Second Order.

From the expression for the attraction of a shell we can find,
by another integration, the attraction of the entire ellipsoid when _
the law of force is that of nature. And thus the well-known
problem of the integral calculus, in which it is proposed to deter-
mine directly the attraction of an ellipsoid on an external point,
without employing the theorem of Ivory to evade the difficulty,
is solved in what appears to be the simplest manner. =

PAR TORT,

Oel ATT.EON.

.

ig:
UP Fey ie

Fy

feo

—_
-

ry, %

z= we

as

tee

IL—ON THE ROTATION OF A SOLID BODY ROUND
A FIXED POINT; BEING AN ACCOUNT OF THE
LATE PROFESSOR MACCULLAGH’S LECTURES ON
THAT SUBJECT.* COMPILED BY THE REV. SAMUEL
HAUGHTON, FELLOW OF TRINITY COLLEGE, DUBLIN.

[ Transactions of the Royal Irish Academy, Vou. xxu. p. 139.—Read April 23, 1849. ]

I.—Composition oF Rorarions.

Ler O be the intersection of two axes of rotation, OR, OR’;
and let the magnitudes of the rotations be represented by w, w’;
then the motion impressed upon the body by these two rotations
will be the same as the motion produced by a single rotation
round an axis, which is represented in magnitude and position
by the diagonal of the parallelogram formed by w, w’. For,
draw through any point I of the body a plane perpendicular to
the line OI, and project upon this plane the parallelogram
formed by w, w’; the sides of this parallelogram will be w sin
ROI and o’ sin R’OI. Now the velocities impressed upon the

* This Essay—‘‘ On the Rotation of a Solid Body round a Fixed Point,”’—
has been compiled from my notes of Professor Mac Cullagh’s Lectures, delivered
in the Hilary Term of the year 1844, in Trinity College. A short account of some
of the results contained in it was published by Professor Mac Cullagh himself, in
the Proceedings of the Royal Irish Academy.t As it has appeared to many of Mr.
Mac Cullagh’s friends desirable that a somewhat more detailed account of his re-
searches in this subject should be published, I have, in accordance with this desire,
drawn up and presented to the Academy an account of his Lectures on
Rotation. I haye endeavoured to arrange the subject in a systematic order, and
to give the results proved by him during the course of the Lectures, carefully ex-
cluding all theorems and proofs of theorems, which were not originally given by
him, as here stated. —Samvet Haveuron,

+ Vol, 1. pp. 520, 542.

330 Rotation of a Solid Body round a Fixed Point.

point I by the rotations w and w’ are Ol.w sin ROI and
’ Ol.’ sin ROI; and the directions of these velocities are per-
pendicular to the sides of the projected parallelogram. Hence, :
if this parallelogram be turned in its plane through 90°, its
sides will represent in magnitude and direction the actual
velocities: the resultant of these velocities is perpendicular to
the projection of the diagonal of the parallelogram (w, w’):
this projection, turned round through 90°, will represent the
actual velocity, which is therefore the same in magnitude and
direction as would be produced by a single rotation represented
by the diagonal of (w, w’). Hence rotations may be resolved
along three rectangular axes by the same laws as couples, and
they must be counted positive when the motion produced is
from z to z, x to y, y to z, and vice versa.

II.—LinEArR VELOCITIES PRODUCED BY A GIVEN RoTaTION.

Let the origin of co-ordinates be assumed on the axis of
rotation, and let the magnitude of the rotation and of its com-
ponents be represented by (w, p, g, 7): the velocity of any
point (#, y, s) is in a direction perpendicular to the plane con-
taining the axis of rotation and the point (a, y, s); and its
magnitude is represented by the area of the triangle whose
angles are situated at the origin, the point (7, y, s), and the
point (p,q, 7). Hence, the components of the linear velocity
are represented by the projections of this triangle on the co-
ordinate planes. These projections are

w= qE— ry;
V=12 — ps; (1)
w = py — qe.

Rotation of a Solid Body round a Fixed Point. 331

TII.—To REPRESENT GEOMETRICALLY THE Moments oF INERTIA
OF A Bopy WITH RESPECT TO AXES DRAWN THROUGH A
Fixep Pornt.

The moment of inertia of a body with respect to any axis

(a, B, y) is

M = A’ cos’a + B’ cos’ B + C’ cos*y - 2L’ cos B cos y
— 2M’ cos a cos y — 2’ cos a cos DP;
where
; A’ = f(y’ + 8’) dm, DT’ = fyzdm;
B = {(# + 8’) dm, M’ = frzdm ;
C’ = {(a? + y’) dm, N’ = faydm.

Assume If = at u being the mass of the body, and r a distance

measured on the line (a, (3, y), and construct the ellipsoid whose
equation is

A’e? + By? + C's? — 2L’ys — 2Mxz - 2N'ry = pb; (2)

then it is evident that the moments of inertia of the body with
respect to axes passing through the fixed point are represented
by the squares of the reciprocals of the radii vectores of this
ellipsoid. Assume A = ya’, B= yb*, C= yc’, and let the axes of
co-ordinates be the axes of the ellipsoid; its equation will thus
become

Cr+ Pyticv=1; (3)

and the equation of the reciprocal ellipsoid will be

a 2 2?
ter ae. (4)

This latter ellipsoid may be called the ellipsoid of gyration, as
the perpendiculars on its tangent planes represent the radii of

332 Rotation of a Solid Body round a Fixed Point.

gyration ; this is evident from the consideration, that these per-
pendiculars are reciprocal to the radii vectores of the ellipsoid
(3). In fact, the moment of inertia with respect to any axis
will be represented by the formula

Uf =(a’ cos’ a + b*-cos? 3 + ¢ cos? Y) w= wP* = Fa; (5)

(R, P) denoting the radius vector and perpendicular on tangent
plane of the ellipsoid (4); and (R’,P’) the corresponding lines in
the ellipsoid (3).

IV.—To Finp THE Maenirupe, Posrrion, AnD DrrEcrion oF
THE StTaTicAL CouPLE PRODUCED BY THE CENTRIFUGAL
Forces.

If from any point (x, y, s) of the body, a perpendicular be
let fall on the axis of rotation (a, (3, y), the centrifugal force
will be represented by the product of the square of the angular
velocity and this perpendicular: the corresponding elementary
statical couple will be found by multiplying the centrifugal
force by the distance from the foot of the perpendicular to the
origin, which is represented by the quantity (# cos a + y cos 6
+2 cosy). The components of the elementary couple will be
proportional to the projections of the triangle formed by the
lines before mentioned. The components of the elementary
couples must be integrated for the entire extent of the body,
and the integrals thus found will be the components of the
couple produced by centrifugal force: the expressions gre as
follows :

w” (a cos a + y cos 3 + 8 cos y)(% cos B — y cos y) dm;
w’ (x cos a + ¥ cos, [3 + 8 Cos y)(# cos y — & cos a) dm;

w” (# cos a + ¥ Cos [3 + 8 cos y)(y cos a — x cos 3) dm.

If the axes of co-ordinates be principal axes, these expressions,
when integrated, will become

w* cos 8 cos y (B- C) =qr(B-C);
w’ cos a cos y (C —- A) =pr (C'— A) ; (6)
w’ cos a cos 3 (A — B)=pq(A-B):

Rotation of a Solid Body round a Fixed Point. 333

P; q, 7 being the components of the angular velocity w. The
position of the resultant couple may be expressed by means of
the ellipsoid (4). Ifa tangent plane be drawn to this ellipsoid
at the point (a, y, s), and perpendicular to the line (a, B, y), it
’ may be easily shown that the projections of the triangle formed
by the radius vector and perpendicular are represented by the
quantities

cos 3 cosy (b°-c*), cosacosy (c’-a’), cosa cos 3 (a? — b*) :

these three expressions multiplied by pw? will produce the quan-
tities used in (6). Hence it appears that the couple produced
by the centrifugal forces lies in the plane of the radius vector
and perpendicular to a tangent plane of the ellipsoid (4) ; the
tangent plane being perpendicular to the axis of rotation. Also,
the magnitude of the resultant couple is proportional to the
triangle formed by the radius vector and perpendicular.

The differential equations of motion commonly used in the
solution of this problem may be deduced immediately from
equations (6). In fact, as the axes of co-ordinates are axes of
permanent rotation, the increment of angular velocity round
each axis will be equal to the statical couple of the applied forces
(including centrifugal forces), divided by the moment of inertia
round that axis. The statement of this fact, in analytical lan-
guage, will give the equations of motion :

dt
dq
Ba, =(C-A) pr+ MU; (7)
Om = (4=B) pg N
dé pat

(LZ, M, N) being the components of the applied statical couple.

The position and magnitude of the couple produced by the
centrifugal forces are easily found by the method which has been
just given; but the direction will be found more readily by

334 Rotation of a Solid Body round a Fixed Point.

taking more particular axes of co-ordinates. Let the axis of
rotation be the axis OZ, and the plane of radius vector and
perpendicular be the co-ordinate plane Zz
XOZ. In the accompanying figure OR’

and OP” are the radius vector and per- RB
pendicular of the ellipsoid (2), and OR,. “4 P’
OP the radius vector and perpendicular jy,
of the ellipsoid (4), which is reciprocal~
to the former; the rotation is positive,
in the direction indicated by the arrow. 06 >
As the rotation is round the axis of z, it is Fig. 1.
easy to see that the statical couple produced by centrifugal force
will have for components, round the axes of w and y respectively,
the quantities w°{yzdm, w*{xzdm taken with their proper sign ;
i.e. the components are + w*L’, + w° I’; L’, M’ being coefficients
in the equation of the ellipsoid

=

x

A’? + By? + O's? — 20 ys - 2Mas — QN’ ry = p.

The tangent plane to this ellipsoid, applied at the point (x, y, s),
will be

(A’e- M's - N’y) a + (B’y - N’a— L's) y+ (C’s- L’'y- M2) # =p.

At the point R’ the tangent plane will be perpendicular to the
plane XOZ, and will be found by making wz = 0, y = 0, and de-
stroying the coefficient of y/ in the preceding equation. These —
conditions give us L’ = 0, which proves that the statical couple
produced by centrifugal force lies altogether in the plane XOZ.
The equation of the tangent plane is the same as the equation
of the line R’P’, and is

C's’ —- M / Sat:

Hence we obtain

The value of the centrifugal couple is w*J/’, which is found

Rotation of a Solid Body round a Fixed Point. 335

from the preceding equation by replacing O’ and tan ¢ by their
values uP’, and = Q being the line RP.

We thus obtain finally the centrifugal couple lying in the
plane XOZ, and expressed by the equation

w{azdm =— pw’ PQ. (8)
It thus appears that the centrifugal couple lies in the plane of
radius vector and perpendicular, is proportional to the area of ~

the triangle ROP, and has a direction opposite to the direction
of rotation.

V.—To FIND THE RELATION BETWEEN THE PLANE OF PRINCIPAL
MoMEntTs AND THE Axis oF Roration at Any Iysrant.

The motion of the body at any instant consists of a rotation
of a certain magnitude round a certain axis; this rotation might
be produced by an impulsive couple of a determinate magnitude
and direction. The statical impulsive couple thus conceived is
the couple of principal moments. Let this couple be represented
by G, and act round the axis OR (fig. 1, p. 334); then the cor-
responding axis of rotation will be the perpendicular OP, and
the relation between G and w may be thus found :—Let the axes
of co-ordinates be the axes of the ellipsoid (4), the radius vector
being determined by the angles (A, u, v), and the axes of rota-
tion by the angles (a, 8, y). From mechanical considerations
we obtain the equations

G cos A = Ap = pw @ cosa;

G cos p = Bg = pw BD’ cosB;

G cos v = Cr = pw © cosy.
Hence we obtain

cosX a@’cosa cosn 0’ cos
cosy ¢ cosy’ cosy c* cosy’

_ G _ Geos
Pie. per

W

336 Rotation of a Solid Body round a Fixed Point,

The first two of these equations prove that the axis of rota-
tion is the perpendicular on tangent plane of the ellipsoid, and
the last equation gives the magnitude of the rotation in terms
of the impressed couple and quantities determined by the nature
of the body itself. Equations (9) are true, whatever be the
forces acting on the body; if no forces act, G will be fixed in
magnitude and position in space, by the principle of conserva-
tion of areas, but will change its position in the body, the axis
of rotation accompanying it, and changing its position both in
the body and in space.

VI.—Roration propuceD By CENTRIFUGAL FoRCE; PARTICU-
LAR PROPERTIES OF THE Morion WHEN No Forces Act.

The axis of rotation produced by the centrifugal couple always

lies in the plane of principal moments. This theorem may be.

thus proved: Let the radius vector and perpendicular be drawn,
which coincide with the axis of principal moment and axis of ro-
tation at any instant; a line perpendicular to the plane of radius
vector and perpendicular is the axis of centrifugal couple ;
this line and the original radius vector are axes of the section of
the ellipsoid made by their plane: at the point where the axis

of the centrifugal couple pierces the ellipsoid let a tangent plane ~

be applied ; the perpendicular let fall on this tangent plane is
the axis of rotation produced by centrifugal forces. From the
construction it is evident that the plane of the second radius
vector and perpendicular is perpendicular to the axis of G ; hence
the axis of the centrifugal couple and the axis of rotation pro-
duced by it always lie in the plane of principal moment. Two
important corollaries follow from the theorem just demonstrated,
in the case where no forces act :—First, the component of an-
gular velocity round the axis of primitive impulse is constant
during the motion. Secondly, the radius vector which coincides
with the axis of G is of constant length during the motion. The
first theorem is obvious ; for as the axis of rotation produced by

centrifugal force is always perpendicular to the axis of G, it

Rotation of a Solid Body round a Fixed Point. 337

cannot alter the rotation round that axis. The second theorem
follows from equation (9), from which we deduce

w COS @ = (10)

G
Re
The left-hand member of this equation is constant bythe pre-
ceding theorem; and G is constant, since there is no external
force; therefore # is constant.

As the axis of G is fixed in space, and the line # is constant,
it is evident that the axis of G will describe in the body the cone
of the second degree, determined by the intersection of the ellip-
soid (4) with the sphere whose radius is R. The equation of
this cone is q

s— e+ a y+ 2 3° = 0. (11)

As the axis of principal moments describes this cone in the body,
it is accompanied by the axis of rotation, which is always the
corresponding perpendicular on tangent plane of the ellipsoid.
The cone described by the axis of rotation might be found thus.
Let tangent planes be applied to the ellipsoid along the sphe-
rical conic in which the cone (11) cuts the ellipsoid. From the
centre let fall perpendiculars on these tangent planes ; the locus
of these perpendiculars is the required cone.

ViIl.—Tue Axis or princrpAL Moments Is FIXED IN
SPAcE.

This is evident from D’Alembert’s principle, but may be
shown by geometrical considerations in the particular case under
consideration. ‘The axis varies in position in the body, in con-
sequence of the centrifugal couple, which must be compounded
with the impressed couple at each instant. Referring to equa-
tion (8), the value of the centrifugal couple is — ww*PQdt, the
principal moment being G = pwPR (vid. (9)). Hence the angle

Z

338 Rotation of a Solid Body round a Fixed Point.

through which the axis of principal moments shifts in an element
wQdt
R ‘
vector, will give the elementary motion on the spherical conic
traced by the axis of principal moment on the surface of the
ellipsoid : this motion is therefore - wQd¢; but in the same time
the point of the body which coincides with the point where the
axis of moments pierces the spherical conic will describe the

of time is — : this angle, multiplied by the constant radius

angle + wQdt in consequence of the angular rotation. Hence

the axis of moments will remain fixed in space, and will move
in the body with a velocity proportional to the tangent of the
angle between the radius vector and perpendicular, the motion
being in a direction opposite to the direction of the rotation.
This is evident from the consideration that Qw = Pw tan 9, Pw

being constant and equal to a (wid. (9)).

VIII.—To Finp THE MoTION oF THE PRINCIPAL AXIS IN THE
Bopy.

First Method.

The point of the principal axis of moments, which is situated
at the distance R from the centre, moves on the spherical conic
which has been determined. Let this point be projected on the
three co-ordinate planes; then, since the spherical conic is pro-
jected into a conic section, the movement of the axis of moments
is reduced to the movement of a point on a conic section, accord-
ing to a law which must be determined. The radius vector
describes an elementary triangle in the surface of the cone (11):
let the projections of this triangle on the co-ordinate planes be
(dA,, dA», dA;) ; we obtain easily

dA, dz dy dA, de ds dA; _ dy ya
is “99s

Had” BP dk Oe ae

Substituting in these equations the values of the velocities given
by (1), we obtain

Rotation of a Solid Body round a Fixed Point. 339

a. 229
tA Py = (R= a’) p;
dA, R- wv ered
aq -Fe py = eas (12)
ee ee:
he Seats s=(R-c*)r

Ms Se
These equations prove that the areolar velocity of the projection
on a co-ordinate plane varies as the ordinate to that plane. By
means of the method of quadratures, we may determine from
equations (12) the position of the projections of the principal
axis at any instant, and hence deduce the position of the axis
itself.

Second Method.

If the spherical conic be projected on a cyclic plane of the
ellipsoid of gyration, by lines parallel to v and z, the projections
will be two concentric circles, and the v
corresponding projections will lie on @ ook Shae,
the same ordinate SII’ (fig. 2). The ;
inner circle will belong to the projec-
tion parallel to x, if R be greater than
b, and will belong to the projection
parallel to z if R be less than 0; and
if R be equal to b, the two circles will
coincide with each other and with the
spherical conic, which in this case becomes the circular section of
the ellipsoid. The projected point will revolve round the cir-
cumference of the inner circle, and will vibrate on the circum-
ference of the outer circle between the dotted lines. It is evi-
dent that the mean axis of the ellipsoid OY lies in the plane of
the figure. Let SI and SI’ be equal to p, p’, and let C, C’ denote
the radii of the two circles: the velocities of the projections in
the circles will evidently be

24 pi lw.
~ p dt? pdt?
Zz?

340 Rotation o a Solid Body round a Fixed Point.
C and OC’ having the values

ee (ye | eS
O-0 (S) - 0 aoe)
The value of + deduced from (1) is

me; w as sv = Pw a pp sin @ cos 0.
a

tod

6 being the angle made by the plane of the circular section with
the plane (x, y),

: e |/a-0 a |(bP-@
sind =§ (<=) co 0-5 |=)

Introducing these values of = sin 8 and cos@, and for Pw its

value s we obtain finally for the velocities
Mu

rll Beem
rf Ni GB ome

The velocity of each projection, therefore, varies as the ordinate
of the other. This theorem enables us to find a simple expression
for the time. Using the angle (#) marked in fig. 2, we obtain

C09 = /(C" ~ C* sin’)
(¢, C, K) belonging to the projection parallel to axis of x. If

(~, C’, K’) be the corresponding quantities for the other projec-
tion, we obtain also

— = I’ ./(C? - CO” sin’ y) ;

Rotation of a Solid Body round a Fixed Point. 341

or, since it is easily seen that sia “a we obtain finally

K’
K'dt = ot :
1 - —, sin’ ¢
Bhi
(14)
Kat = a

acer)

The motion of the principal axis of moments is, therefore, ex-
pressed by an elliptic function of the first kind.

The motion of the axis of moments is determined by the
magnitude of the radius vector of the ellipsoid, which is the axis
of the original couple impressed upon the body ; if this radius
vector be greater than the mean axis of the ellipsoid, the cor-
responding spherical conic will have the axis of x for its internal
axis; and if the radius be less than the mean axis, the axis of
% will be the internal axis of the conic; in no case will the mean
axis be the internal axis of the spherical conic. If the radius R
be nearly equal to either the greatest or least semi-axis, the ex-
pression (14) for the time may be integrated. Let & be nearly
equal to the greatest semi-axis. The first of the equations (14)
belongs to the interior circle, which is of small dimensions in
the case supposed ; the second equation expresses the vibratory
motion of the projection, through a small are of the outer circle,
which will have a radius much greater than the inner circle;
we may, therefore, suppose the angle ~ to be equal to its sine.

Multiplying both sides of the equation by . we obtain.

nts
A AO ea
oe RB)
Ce.
Hence
CF sin (K't + A). - (15)

If T, denote the time of a complete oscillation or revolution of

342 Rotation of a Solid Body round a Fixed Point.

the axis of moments about the axis of 2, and 7’, the time of a re-
volution of the body round the axis of x, the following relation
between these two periods may be readily deduced from (15) :

_T be

“/ ((@ — b) (a - oe)’
If the axis of moments, and consequently the axis of revolution,
be situated near the axis of greatest or least inertia, it will
always continue near this axis; if, however, it be situated near
the mean axis, the movement of the body will be determined by
the following construction :—Let the two cyclic planes of the
ellipsoid be drawn through the mean axis; they will divide the
ellipsoid into two regions, in one of which is situated the axis of
maximum inertia, and in the other the axis of minimum inertia.
The spherical conic described by the axis of principal moments
will have the first or second of these axes for its internal axis,
according as # is greater or less than the mean axis. If the
axis of principal moments lie in one of the cyclic planes, the
spherical conic becomes a circle, and its two projections become
identical with itself (fig. 2, p. 3389); the expressions (14) are
reduced to the form

?, (16)

fit =
cos

which when integrated gives

" eens
Kt + A= log oot (7 5)

or, :
cot G - 5) = cot ¢ - *) ats (17)

¢o being the value of @ corresponding to ¢ = 0, and K being ex-
pressed by the following quantity :

oie - ¥) &= 6)
in Bac :

It is evident from the equation (17) that the axis of moments
will coincide with the mean axis of inertia at the end of an in-
finite time.

Rotation of a Solid Body round a Fixed Point. 343

TX.—To FIND THE PostTION oF THE Bopy IN SPACE AT THE
END oF ANY GIVEN TIME.

First Method.

The radius vector of the ellipsoid of gyration, which is per-
pendicular to the plane containing the axes of principal moment
and of rotation, always lies in the plane of principal moment,
and describes in that plane areas proportional to the time.

_ Let OG, OQ be the axes of principal moment and of rota-
tion ; OR’, OQ’, the axes of centrifugal
couple and of corresponding rotation ;
the plane QOQ’ will contain the two
successive positions of the axis of rota-
tion. Let OI be the position of the

_ axis of rotation at the end of the time
_6¢; then dw will be equal to the angle
described in the fixed plane by the
line OR’. Let R’ and P” be the radius
vector and perpendicular correspond- /)’
ing to the centrifugal couple and its Fig. 3.

axis of rotation. The following relations are evident from the

figure :

sin Q/O1 _ cosg¢’ |
smQOI singdu’

*
a
@

because

cos

sin pou.
sin®’”

sin O’OI = eins Bee

sin QOI =

but, from mechanical considerations,

oO” ES ee es ,__ Gw sin pot
w PRwsin pot’ it uPR’ Se 3
Hence, by equating the geometrical and mechanical expression,
we obtain
— R°Su = wPR8t -< of. (18)

344 Rotation of a Solid Body round a Fixed Point.

The position of the body in space is thus reduced to quadra-
tures; but the problem may be solved more readily in the fol-
lowing manner.

Second Method.

The axis of principal moments, appearing to move in a
direction opposite to the rotation, describes in the body the cone
whose equation has been given (11). If the cone reciprocal to
this cone be described, one of its sides will lie in the fixed
plane, and the whole motion of the body in space will be the
same as the motion of this cone, which partly slides and partly
rolls on the fixed plane, the sliding motion being uniform. This
theorem is evident by resolving the angular velocity w into two
components, one round the axis of principal moments, and the
other in a direction perpendicular to this, round the side of the
reciprocal cone, which is in contact with the fixed plane. These
components are w cos @ and w sing; w cos ¢ being constant and
producing the sliding motion, while w sin ¢ represents the an-
gular velocity round the side of the cone in contact with the
fixed plane. The angle described by the side of the reciprocal
cone in the fixed plane at the end of a given time is, therefore,
the algebraic sum of two angles, one of which is proportional to
the time, and the other is the angle described in the cone in con-
sequence of the rotation w sin ¢, and is, therefore, measured by
the are of a spherical conic. The position of the body at the
end of the time ¢ is thus found :—determine by equation (14)
the position of the axis of principal moments in the cone (11) ;
the corresponding position of the component axis of rotation in
the reciprocal cone is therefore known. Hence the angle de-—
scribed in the time ¢ in the fixed plane is

ds 8
0 = fw cos pdt + |F=o cos p.t +5. (19)

The equation of the reciprocal cone is

au? . by? Z es?
Ro RF. Hae

=0. (20)

Rotation of a Solid Body round a Fixed Point. 345

In (19) the positive or negative sign must be used according
as R is less or greater than the mean axis of the ellipsoid ; this
__ is evident from the composition of rotations, and from the con-
sideration that in the former case the axis of rotation falls inside
the cone (11), while in the latter case it falls outside.

X.—To Finp a Port, rF any, 1n 4 Given Axis or Rotation,
WHICH BEING FIXED, THE AXIS WILL BE PERMANENT.

Let R’R” (fig. 4) be the given axis, round which the body
revolves with a rotation expressed by w; describe the ellipsoid
of gyration round the centre of gra-
vity O, and draw OF” parallel to R’R”.
The centrifugal force wrdm at any
point (7, y, s) may be resolved into two
components, w*pdm and w’. R’P’.dm;
rand p denoting the distances of the
point from the axes R’R” and OP’ re-
spectively ; the effect of the rotation
round R’R” is therefore the same as an
equal rotation round OP’, together with
a number of parallel and equal forces applied to each point of
the body. The rotation round OP’ produces a centrifugal couple
represented by —pw*.OP. PR (vid. (8)); or, determining the point
Ty by the condition OP. PR = OP’. P’R’, the centrifugal couple is
— pw’.OP’. PR’. The resultant of the parallel forces is a force
applied at the centre of gravity, acting in the direction parallel
to RP’, and equal to uo’. R’P’. Comparing this with the cen-
trifugal couple, it is evident that the forces at O destroy each
other, and, therefore, the total result of the rotation round R’R”
is to produce a force acting at the point R’, which has been just
determined. If this point be fixed, the axis R’R” will be a per-
manent axis of rotation. The condition by which the point R’
is found is, that the triangle OR’P” is equal to and in the same
plane with the triangle ORP: hence, if an ellipsoid confocal to

Fig. 4.

346 Rotation of a Solid Body round a Fixed Point.

the ellipsoid of gyration be described through the point R’, it
-will be perpendicular to the line R’R”. The general construc-
tion for permanent axes is, therefore, the following:—Let the
ellipsoid of gyration be described, and confocal ellipsoids: any
line which pierces one of these ellipsoids at right angles is a per-
manent axis of rotation for the point of intersection.

ATTRACTION.

a i

ea

hang49: )

I.—ON A DIFFICULTY IN THE THEORY OF THE ATTRAC-
TION OF SPHEROIDS.

[ Transactions of the Royal Irish Academy, Vou. xvi. p. 237.—Read May 28, 1832.]

An approximate theorem, discovered by Laplace, and relating
to the attraction of a solid slightly differing from a sphere, on
a point placed at its surface, has given rise to many disputes
among mathematicians.* I hope the question will be set in a
clear light by the following remarks.

Let us consider the function which expresses the sum of
every element of a solid divided by its distance from a fixed
point, and let us denote it, as Laplace has done, by the letter V.
It is necessary to find the value of V for a pyramid of inde-
finitely small angle, the fixed point being at its vertex. Calling
¢ the small solid angle of the pyramid (or the area which it
intercepts on the surface of a sphere whose radius is unity and
centre at the vertex), it is manifest that the element of the
pyramid at the distance r from the vertex is gr’dr; dividing
therefore by 7, and integrating, we have 497’, or @ multiplied
into half the square of the length, for the value of V.

Again, supposing the force to vary inversely as the square
of the distance—the only hypothesis that can be of use in the
present inquiry—the attraction of the same pyramid on a point
at its vertex, and in the direction of its length, is manifestly
equal to or.

Let us now consider a solid of any shape, regular or irregu-
lar, terminated at one end by a plane to which the straight
line PQ is perpendicular at the point P; and let there be a
sphere of any magnitude, whose diameter P’Q’ is parallel to
PQ. Let P” be a fixed point, and from the points P, P’, P”,
draw three parallel straight lines Pp, P’p’, P’p”, the first two

* See Pontécoulant, Théorie analytique du systéme du monde, tome ii. p. 380; with
the references there given.

350 Difficulty in Theory of A traction of Spheroids.

terminated by the surfaces of the solid and of the sphere, the
third, P’p”, in the same direction with them and equal to their
difference, without regarding which of them is the greater, and

BR

yeaa fa), Ke
NS ae

suppose all the points yp”, taken according to the same law, to
trace the surface of a third solid. Let Pp, P’p’, Pp”, be edges
of three small pyramids with their other edges proceeding from
P, P’, P”, parallel, and having of course the same solid angle,
which we shall call ¢, and denote by 7, 7’, 7”, their respective
lengths, and by V, V’, V”, the values of the function V for each
of them. Drawing pR perpendicular to PQ, the attraction of the
pyramid Pp in the direction of PQ will be equal to ¢ x PR.
Call this attraction .A, and let a be the radius of the sphere.

Since y” is the difference of 7 and 7’, we have 7” + 7? — +”
= 2rr =2PRx PQ’, and multiplying by 1¢ we find $¢r*
+ tor? — tor? = 2ap x PR, that is V+ V’- V’=2adA. The
same thing is true for any other three pyramids similarly related
to each other, throughout the whole extent of the three solids
which are exhausted by them at the same time; and hence, if
we now denote by V, V’, V’”, the whole values of the function V
for the three solids, and by A the whole attraction of the first of
them parallel to PQ on a point at P, we shall still have V+ V’
— V" =2aA.

To express this general theorem in the notation of Laplace,
we have merely to observe that the attraction A is synonymous

with -(F) and that the quantity V’ for the sphere is equal

to * at Substituting these values, we find
dV 45% mg
V+ 2a(7-)- ~3 ra + V 3
an exact equation, differing from the approximate one of Laplace
only in containing the quantity V”, and totally independent of

Difficulty in Theory of Attraction of Spheroids. 351

the nature of the surface or of the magnitude of the sphere; the
only things supposed being that all the lines drawn from P meet
the surface again but once, and that no part of it passes beyond
a plane through P at right angles to PQ.

With respect to the limit of the quantity V”, it is obvious
that if a hemisphere be described from P” as a centre, with a
radius equal to the greatest difference 5 between the lines Pp,
P’p’, the solid P”p” will lie wholly within this hemisphere, and
consequently V” will be less than the value of V for the hemi-
sphere, that is, less than 70’; for here all the little pyramids
from the centre have the same length 8, and their bases are
spread over the hemispherical surface; wherefore V” = 27
x 10? =76*. All this is independent of anything but the suppo-
sition just mentioned. -

If now PQ be supposed to be a spheroid of any sort,
slightly differing from the sphere P’Q’, and such that the
line PQ, perpendicular to the surface at P, passes nearly
through the centre, then all the differences, of which 6 is the
greatest, being of the first order, the quantity V”, which is less
than 7d’, will be of the second order ; and therefore neglecting,
as Laplace has done, the quantities of that order, we get the
theorem in question.

It may be well to apply the general theorem to the simple
ease in which the first solid is a sphere of the radius a’, because
both Lagrange and Ivory have used this case to show that the
reasonings of Laplace are incorrect. In this instance, then, the
surface described by the point p” is that of a sphere whose
radius is the difference between a and a’; and the values of

V, V’, V’, and A, are ma ora’ om (a —a)* and snd respec-

tively.
Substituting these values in the equation V+ V’-— V” =2aA,

and omitting the common factor om, the resulting equation

a? +a’ — (a’ - a)? = 2ad
ought to be identical; and so it manifestly is.
November, 1831.

(= 3a§2y »)

II.—ON THE ATTRACTION OF ELLIPSOIDS, WITH A NEW
DEMONSTRATION OF CLAIRAUT’S THEOREM,
BEING AN ACCOUNT OF THE LATE PROFESSOR
MAC CULLAGH’S LECTURES ON THOSE SUBJECTS.
COMPILED BY GEORGE JOHNSTON ALLMAN, LL.D.,
OF TRINITY COLLEGE, DUBLIN.

i

[ Transactions of the Royal Irish Academy, Vou. xxu. p. 879.—Read June 13, 1853.]

Proposition I.

If P be any point on the surface of an ellipsoid, and PC, be
drawn perpendicular to an axis OC, and an ellipsoid be described
through C, concentric, similar, and similarly placed to the given
ellipsoid ; then the component of the attraction of the given ellipsoid
on P in a direction parallel to OC is equal to the attraction of the
inner ellipsoid on the point C;,.

This theorem is an extension of that given by Mac Laurin*
relating to the attraction of a spheroid on a point placed on its
surface. It may, moreover, be established by means of the same °
geometrical proposition from which MacLaurin deduced his
theorem.

Through the point P let a chord PP’ of the given ellipsoid
be drawn parallel to the axis OC. Now, suppose both ellipsoids
to be divided into wedges by planes parallel to each other, and
passing respectively through this chord and the parallel axis of
the inner; and suppose the wedges to be divided into pyramids,
the common vertex of one set being at P, and of the other at C,.

* De caus. Phys. Flux. et Refl. Maris, sect. 3; or see Airy’s Tract on the
Figure of the Earth, Prop. 8.

On the Attraction of Ellipsoids. 353

Observing that any two of these parallel planes cut the two
surfaces in similar ellipses, such that the semi-axis of one is
equal to the parallel ordinate of the other, it is easy to see that

Fig. 1.

the reasoning employed by Mac Laurin may be used to esta-
blish the truth of the theorem stated above.

Proposition II.

To calculate the Attraction of an Ellipsoid on a Point placed at
the extremity of an Axis.*

Let the semi-axes of the ellipsoid be a, 6, c, where a>b>e,
and let the point on which it is required to find the attraction
be C (Fig. 1), the extremity of the least axis.

Suppose the ellipsoid to be divided by a series of cones of revo-
lution which have a common vertex C and a common axis CC’, 0’
being the vertex of the ellipsoid opposite to C; it will be sufficient
to find an expression for the attraction of the part of the ellipsoid
contained between two consecutive conical surfaces, whose semi-
angles are @ and 0 + d@ respectively. Suppose now the part of
the ellipsoid between two consecutive cones to be divided into

* Proceedings of the Royal Irish Academy, Vou. ut. p. 367.
2A

354 On the Attraction of Ellipsoids.

elementary pyramids with a common vertex OC. Let CP be one
of these elementary pyramids, whose solid angle is w; let PQ
be drawn perpendicular to CC’ ; from the centre O draw a radius

vector OR parallel to CP, and from the extremity R let fall a
perpendicular RS on the axis CC’.

Now the attraction of the elementary pyramid CP on the
material point », placed at its vertex = ufpw.CP; and the com-
ponent of this attraction in the direction of the axis is

ufpw.CQ = 2ufpw. oe

Now suppose the radius vector OR to revolve around the axis
OC’, then the attraction on the point C of the portion of the
ellipsoid bounded by the two cones of revolution, whose semi-

On the Attraction of Ellipsoids. 355

angles are @ and @ + d@ respectively, since it is made up of the
components in the direction CC’ of the attractions of all the
elementary pyramids CP, is

PHP ost. & (OR w) = We

P cos’ d0. = (OR dg),
dp being the angle between two consecutive sides of the cone
generated by the revolution of OR.

But S(OR?d@) is equal to twice the superficial area of the
part of this cone which is enclosed within the ellipsoid. More-
over, the projection on the plane ad of this portion of the sur-
face of the cone is an ellipse, whose semi-axes are 7’, sin 0, 7, sin 0,
and whose area is 777; sin’ #, 7, and r, being the maximum and
minimum values of OR: the superficial area of the portion of
the cone within the ellipsoid is therefore 77,7, sin 0.

Hence it follows that

> (OR? do) = Qrr V2 sin 6.
The attraction on the point C of the part of the ellipsoid con-
tained between the two cones of revolution, whose common
vertex is at C, and whose semi-angles are 6 and 0 + d@ respec-
spectively, is therefore

— cos’ 4d0.7,72 sin 6,
where |

1 | /eos’@ sin’ =) 1 | eS 0 sin? ")
== ( + —} and.—-= sot I.
r; c a 12 c b

On substituting these values, the expression given above
becomes

4 abe cos’@ sin 0d0
mHIe J (a cos’ + c* sin’@) ,/ (0° cos’@ + ¢* sin*)"

Hence the attraction of the solid ellipsoid on the point C at the
extremity of the least axis is

TT

Anu 2 abe cos’@ sin 0d0
mh » |. / (a* cos’@ + c* sin’) ,/ (0° cos’ @ + c? sin*@)’
2a2

356 On the Attraction of Ellipsoids.

Let cos @ = u, this expression becomes

abe u? du (1)

Anufo | J{e+w(@—e)) J {e+ wv (e—c)}’

In the same way it may be shown that the attraction of the
ellipsoid on a point u placed at the extremity of the mean
axis is

abe u? du

snutp| Jere e-)) Jere @-By?

and on a point at the extremity of the greatest axis,

‘ abe u? du

4nufo | ama (P-a)} f{ae+u (c? — a*)}"

It will be seen in a subsequent proposition, that these three

expressions are not independent of each other, the values of the _

three attractions in question being connected by an equation.

Proposition III.

To give Geometrical Representations of the Attraction of an
Ellipsoid on Points placed at the extremities of its least and mean
Axes.* “2h

On the greater axis OA, of the focal ellipse assume a point —

K, such that OK, = : OA,; from the point K, draw a tangent

* Proceedings of the Royal Irish Academy, Vot. 11. p. 367.

On the Attraction of Eltipsoids. 357

K,Q, to the focal ellipse, and let 7’ = tan K,Q, - are A,Q,; then
the attraction of the ellipsoid on the particle « placed at the ex-
tremity C of the least axis is
4a uf p abc?
@-) @-2)
For let a point K (see fig. 3) be assumed on the greater axis OA,
of the focal ellipse, such that

oK = te + wu? (Bb? -c’)};

F (2)

from K let a tangent KQ be drawn to the focal ellipse, and let
OP be the perpendicular let fall from O and KQ; then y denot-
ing the angle A,OP,

OK?. cos? a) =
Moreover,

OK?. cos’) = OP? = (@ - c’) cos? yp + (B? —c*) sin’ y.

ne {c? + u? (b? — c*)}. cos? y.

Equating these values, and solving for sin’ y, we get

(?-¢) uv

paca e+ u* (a —c)
Now
d. (tan KQ - are A,Q) = sinyd.OK*
_(@-¢) @-2) udu

C V/{e+wv (?-e)} f{e+wv (Bb-e)

By comparing this expression with (1), given in the last propo-
sition, it appears that the attraction on the point C of the portion
of the ellipsoid contained between the two conical surfaces whose
semi-angles are 0 and 0 + d@, respectively, is

4rpfp abc’
@—c) (Fe) d.(tan KQ — are A,Q).

* Transactions of the Royal Irish Academy, Vou. xvi. p. 79. Proceedings of
the Royal Irish Academy, Vou. 11. p. 507 (supra, p. 255).

358 On the Attraction of Eltipsords.

Now, in order to obtain the attraction of the whole ellipsoid
on the point C, we have to integrate the expression given above
between the limits w= 0 and w=1, or OK =OA, and OK =OK,;
from which it appears that its value is

4mufp abe*

(a° -—c’) (b’-c’)
It is easy to see that the attraction of the part of the ellipsoid
contained within the conical surface, whose semi-angle @ is equal
to the angle cos u, is

ye

abe’

4rufp (@ o e) G = @) (7 t), (3)

where ¢ = tan KQ ~ are A,Q.
To represent the attraction on a point u placed at the extre-
mity of the mean axis, assume on the transverse OA, of the focal

hyperbola a point K, such that OK, = OA, and from K,

draw a tangent K,Q, to the hyperbola, and let 7'= tan K,Q,;
—are A,Q,; then the attraction of the ellipsoid on the point p is

abe
= 4ufp (a a¥ °) Gi ac 2?)

P. (4)

To prove this, assume a point K such that

OK = Oe 18 +u? (e - 8)};
from K draw a tangent KQ to the hyperbola, and from O let

fall a perpendicular OP on this tangent; then if ~ = angle

A,OP,
(a — 0°) w?

sin p “= P+ we (a = b)’

Hence, by following a method similar to that used in finding the
representation of the attraction on a point at the extremity of
the least axis, the expression given above may be easily obtained.

The attractions O, B of the ellipsoid on points placed at the
extremity of the least and mean axes are thus represented by

On the Attraction of Ellipsords. 359

means of arcs of the focal ellipse and hyperbola respectively. In
‘consequence of the third focal conic of the ellipsoid being ima-
ginary, no direct geometrical representation can be given for the
attraction A on a point placed at the extremity of its greatest
axis. It will, however, be found, as was intimated above, that
a simple relation exists between the three attractions, which
enables us to represent this last by x means of ares of both focal

conics.
The relation alluded to is

= + ee + g = 4rpfp.* (5)

This can be easily proved by ihe help of the following -geo-
metrical theorem :—

If from the extremities A, B, C of the three axes of an ellip-
soid three parallel chords Ap, Bg, Cr, be drawn, and if these ©
chords be projected each on the axis from whose extremity it is
drawn, then the sum of these three projections, Aa, BB, Cy,
divided respectively by the lengths of the axes AA’, BB’, CC,
on which they are measured, will be equal to unity.

Now conceive three chords Ap, Ap’, Ap”, to be drawn from
A, making each with the other two very small angles, and so
forming a pyramid with a very small vertical solid angle w ;
and from B and C let two systems of chords Bg, By’, Bg’, and
Cr, Cr’, Cr’, be drawn, each system forming a very small pyra-
mid whose three edges are parallel to the three edges Ap, Ap’,
Ap”, of the pyramid which has its vertex at A.

The attractions of the three pyramids, reduced each to the
direction of the axis passing through its vertex, will be equal to
ufpw.Aa;, ufpw. BB, ufpw.Cy respectively ; and, therefore, the
sum of those attractions divided respectively by the lengths of
the axes will be

Aa BB.
wow (57+ BB’ + ot)" Hip.

* Proceedings of the Royal Irish Academy, Vou. uy, p. 526,

360 | On the Attraction of Ellipsotas.

Let pyramids thus related be indefinitely multiplied, and
the ellipsoid will be simultaneously exhausted from the three
points A, B, C.

Hence the sum of the whole attractions at A, B, C, divided |

respectively by the lengths of the corresponding axes, will be

2rufp, or
= ie Pie = 4rpfp.

Proposition LY.

To find an expression for the potential V of a system of par-
ticles at a point M, whose distance from the centre of gravity of the
system is very great compared with the mutual distances of the
particles.

It is proved by Poisson,* that if the origin of co-ordinates
be at the centre of gravity of the system ee

> (aa! + yy’ + 22’) dm - bs (a? + y? + 2°) dm,

3
As ao ars

27"

a, y’, # being the co-ordinates of the distant point, and 7” its
distance from the origin. Let now the principal axes at that
centre be taken as axes of co-ordinates; then, since

Stydm=0, Tazdm=0, Syzsdm=0;

V=5+

2r

4 , 1
Dy’ D(a +y°y 2 + 32 *) dm -S% > (a? x y + 2) dm.

Hence, if A, B, C be the three principal moments of inertia,
and J the moment of inertia of the system round OM,

M1
Ste hg

* Mecanique, tom i. p. 178.

9,3 (4+B+ C- 31). (6) 3

On the Attraction of Ellipsoids. 361

~ Proposition V.

A system of material particles attracts a point M, whose distance
JSrom the centre of gravity O of the attracting system is very great
compared with the mutual distances of the particles ; then if a tan-

_ gent plane be drawn to the “ ellipsoid of gyration,”* perpendicular
to OM, the whole attraction lies in the plane OST, where S is the
point in which this tangent plane intersects OM, and T its point
of contact with the ellipsoid.

Let a, 8, y be the direction angles of OT; a’, 9’, y’ of OM;
and a, (1, y: of TS; and a, Bo, yo of the normal to the

\m
soene \

Fig. 4.

plane OST; and let OS, OT and the angle SOT, be denoted by
p, r, and @ respectively. It will be sufficient to prove that the
component of the attraction in the direction of the normal to
the plane OST is zero.

We shall first find the components X, Y, Z, of the attraction
in the directions of the axes, and thence deduce the value of Q.

* The centre of this ellipsoid is at the centre of gravity ; its axes are in the
directions of the principal axes, and their lengths are determined by the equations

M@= A, MP?=B, Me=C. .
This ellipsoid is used by Professor Mac Cullagh in his ‘‘ Theory of Rotation’’: see

Rey. 8S. Haughton’s Account of Professor Mac Cullagh’s Lectures on that subject,
Transactions R. I, A., You. xxu. p. 139 (supra, p. 329).

Now,
heap Seem: a aE
X= - 77 = 7 008a +o (4+ B+ C-SI) cosa + 975 a?
dh 8dr ~
Ys oe Fe ;(4+B+ C-38T) cos’ + 37 ar ay”
Z=- 57 = 5 008y' + 5; (4+ B+C- 31) cosy’ +=; Baw?
but, e
aI _2(A-T)cosa’, dT _2(B-1) cos, al _4(0- T) cosy”
i , ’ dy yr’ ; dz’ y

Hence we have

X =F, 008 a! + or (4+B+0- 5I) cos a a

Y- 7, 008 8+ a7 (se B ae 51) 00s p’ + Soe (7)
Z = Tr 008 9 + oy ; (4+ B+ C-SI) cosy’ Path
Now,
Q = X cosa) + Y cos B, + Z cos yp ;
but,

sin @ COS ay = cos 3 cosy’ — cosy cos,
sin p COs 3) = Cos y COS a’ — COS a COS y’,
sin @ COS yo = cosa cos 9’ — cos 3 cosa’ ;
The following relations, moreover, exist :
a’ cosa’ =rp cosa, B° cos 3’ =rp cosB, c* cosy =rp cosy: (8)

hence, by substitution, we have

v-¢ ea’
—— cos 3’ cosy’, cos Bo = :
pr sin p pr sin

2 2

pr sin

cosy’ cosa’,

COS ap =

COS Yo = cos a’ cos (9.

On the Attraction of Ellipsoids. 363

Substituting these values for cos ao, cos (39, COS yo, in the expres-
sion for Q, and observing that
COS a’ COS ay + 008 [3’ Cos (3, + COS y’ COS yo = 0,
we get
Q- a’ (b?-c*) +B? (e’-a’) +e (a’—b")

ne / fox 3 9
Bias cosa’ cos 3’ cosy’=0. (9)

Proposition VI.

The same things being supposed, to find the other Components
of the Attraction, namely R in the direction of the centre of gravity
MO, and P in the transverse direction TS.

To find R;
R =X cosa’ + Y¥ cos PB’ + Zcosy ;
Me 8 al
f=, +o (d+ B+ C- dL) + 2?
Mee, ha=5,+ =, (A+ Bx OW 8D. (10)
To find P ;

P = X cosa, + YoosfB, + Z cosy;
but,
sin COS a; = COSa’ COS @ — COSa,
sin @ cos}, = cos 3’ cos g — cos 3,

sin p COS y1 = COS’ COS — COS y.

Substituting for cos a, cos (3, cosy their values from (8), we get

“o — p hb? — yp?
o08a,=—--—— cosa’, 0086, =-——~ os B,
pr sin pr sin ¢
2 2
Cc _
cos yi = — cos y’.
pr sin

Substituting these values of cosa, cos 3, cos yi, and observing

that
/
cosa’ COS a; + cos [3’ cos 3; + cosy’ cosy, = 0,

364 On the Attraction of Ellipsords.

we have
3H p*(r*— p’)

pa ag: 7 Gee MRS Beret ok

* pr sin yr’ Te ets
or, =

x ST. (11)

The negative sign indicates* that the force P acts in the direction
TS, ¢.e. from the radius vector towards the perpendicular of the
ellipsoid of gyration. If the force P be resolved into three
others in the direction of the axes, it is evident from the values
given in Proposition V. for X, Y, Z, that these components are

ao) coma oats 8’, a ae. #) cosy’.t (12)

* The direction of the force P, which Professor Mac Cullagh determines by the
interpretation of the negative sign, may be very clearly seen from the following
considerations. This force exists in every case where the three principal moments
of inertia of the system at O are not all equal, that is, when the ellipsoid of gyra-
tion is not a sphere. The greatest axis of that ellipsoid is manifestly towards that
part of the body in which there is a deficiency of attracting matter. If we now con-
sider the position of a perpendicular on a tangent plane of an ellipsoid with rela-
tion to the corresponding radius vector, we shall find that it always lies away
from the greatest axis. But the transverse force has been shown to be in the
plane of radius vector and perpendicular. Therefore, the direction of the transverse
force, being towards the preponderating matter, must be from T to 8.

+ The results given by Professor Mac Cullagh in Propositions V. and VI. may
be otherwise obtained, and, perhaps, with greater facility, by introducing the con-
sideration of the statical moment of the attracting force. *

If the three principal moments of inertia were equal to each other, then the
whole attraction would be in the direction of the centre of gravity, and its mag-

nitude would be
M

re
In general, however, the attracting mass will be of an irregular shape ; there will
exist then, in addition to the principal part of the attraction, which will be central,
a transverse force which will tend to cause a motion of rotation about the centre of
gravity.

The components of the moment of this transverse force in the three principal
planes are

wY-yX, YZ-<?Y, ?’X-2Z;

* See Rev. R. Townsend, in the ‘‘ Dublin University Examination Papers,” 1849, p. 51.

On the Attraction of Ellipsords. 365

Proposition VII.

An Ellipsoid is composed of ellipsoidal strata of different densities
and of variable but small ellipticities ; to find the Components, cen-
tral and transverse, of its Attraction on an external point,

The values found in the last Proposition for the components
of the attraction of any mass on.a very distant point will be
found to hold in the present case, whatever be the position of
the attracted point. In order to show this, we shall first prove
it for a homogeneous ellipsoid of small ellipticities. Such an
ellipsoid being given, another confocal with it can be con-
structed so small, that the distance to the attracted point may be

but from (7), ;
Bae
vY-yX=- = cos a’ cos B’ = — ed (a? — b*) cosa’ cos fp’,
B- 3M
y Z- pee aad: cos B’ cosy’ = — = 75 (8? — ¢*) cos B’ cosy’,
2X-aZ=—- sc cos 7’ cosa’ = — = (c? — a?) cosy’ cosa’.

Now it is well known, that
4 (a — 6”) cosa’ cosp’, 4 (b?—c*) cosB’ cosy’, 4} (c* — a?) cosy’ cosa’,

are the areas of the projections of the triangle OST on the principal planes. Hence
it follows that the resultant moment lies in the plane of the radius vector OT and
the perpendicular OS to the corresopnding tangent plane of the ellipsoid of gyra-
tion ; the tangent plane being perpendicular to OM. It appears, also, that the
magnitude of the resultant moment is

- + Mog x ST,
and therefore that the transverse component of the attraction
Pat O80 BF
yt

Or, the values of the central force and the moment of the transverse force may
be obtained directly from the expression (6) for the potential .V. This function is
of such a nature, that its differential coefficient with relation to any line (the sign
being changed) is equal to the resolved part of the attraction in that direction ; and
the differential coefficient with relation to any angle (the sign being changed as

366 On the Attraction of Ellipsoids.

regarded as very great, compared with the axes of this ellipsoid.
The components of the attraction of this small ellipsoid on the
distant point are given by the expressions (10) and (11): now as
the attractions of two confocal ellipsoids on an external point are
in the same direction, and proportional to their masses: the
components of the attraction of the proposed ellipsoid will, —

therefore, be

M78? GE
R=7,+ on gp (Avt Bt G- 3th),

p-- 278, x O81, ;

the letters with suffixes referring to the small ellipsoid.
The attracting ellipsoids being confocal, their ellipsoids of
gyration are confocal also; hence it follows that

7 (4: + B+ 0,-8h)=4+ B+ 0-3,
é ;

and

OS, x 8,T, = Os x ST.

before) gives the component in the plane of that angle of the moment of the
attractive force. .
Hence,

since

dr’

but a

d
( aoe erat) z) F (x'* + y’*) = 0 where F is any function ;
3

iat Sah pe ee hn ee d\ (Ax? + By’? + C2?)
° Oi dy’ y dx’ a Or'3 dy’ -y' dz’ aD ) :

3 (4 — B)

. =
.N=-—5

(cos a’ cos B’).

The two other components of the moment may be similarly obtained. The
remainder of the proof is the same as in the former part of this note.

On the Attraction of Eliipsords. 367

It appears, from this, that the central and transverse components
of the attraction of a solid ellipsoid of uniform density, and
whose ellipticities are small, on any external point whatever,
are given by the same formule as the corresponding components
of the action of any mass on a distant point.

Now it is a property of moments of inertia, that they are
subtractive, that is, the difference of the moments of inertia of
two masses with relation to any axis is equal to the moment of
inertia of the difference of those masses with relation’to the same
axis. And the values at which we have arrived for the central
- force, and for the three components of the transverse force, con-
tain in each term either a mass or a moment of inertia in the
first power, and therefore, these values also are subtractive.
Hence the two components of the attraction of a homogeneous
mass contained between two concentric and coaxal ellipsoids of
small ellipticities, are given by formule (10) and (11). Now
suppose an ellipsoidal mass to be composed of strata bounded
by ellipsoids of different but small ellipticities, each stratum
being homogeneous throughout its extent, while the density
varies from one stratum to another according to any law; then,
since those formule hold for the action of each stratum sepa-
_ rately, and since the terms of which they are made up are in
their nature additive, they hold for the entire mass.*

Proposition VIII.

An oblate Spheroid is composed of spheroidal strata of different
densities and of variable but small ellipticities ; to find the Com-
ponents of its Attraction on any external point.

The expressions given in the last Proposition for R and P
become simplified in this case. Let OZ be the axis of revolu-
tion, and let X denote the angle which OM makes with the plane

* See Professor Mac Cullagh, in the ‘‘ Dublin University Examination Papers,”
1833, p. 268.

368 On the Attraction of Eliipsoids..

XY; then since 4 and B are equal, we have

I= A cos’ + Csin?A;
and therefore

A+B+C-38I=(C- A) (1-38 sin’A);
MOS x OT=(A-C)sindoosr.

_ Substituting these values in the expressions for R and P, we
have

also

M 30-4 nk
R=3+5—~ (L-ssin’d), (13)
P=3 Oe cos A sin A. (14)

The direction of the force P is towards the plane of the
equator ; this appears from the shape of the “ ellipsoid of gyra-
tion,”’ which in this case is a prolate surface of revolution.

Proposition [X.—CuarraAut’s THEOREM.

Whatever be the law of variation of the Earth’s density at
different distances from the centre, if the ellipticity of the surface be
added to the ratio which the excess of the polar above the equatorial
gravity bears to the equatorial gravity, their sum will be : q, where
q is the ratio of the centrifugal force at Equator to equatorial gravity.

_ For suppose the attracted point M to be on the surface of
the earth, which is known to be an oblate spheroid of small
ellipticity. Then, from the principles of Hydrostatics, since the
tangential force is zero, we have
R cos 0 — P sin # — w*r cos cos (8 — A) = 0, (15)

where w denotes the angular velocity, and @ the angle which
the tangent to the meridian through the attracted point makes
with the radius vector. Developing cos (6 — A) and arranging,

we obtain |
(R - w*r cos’) cos 0 = (P + w’r cosA sin A) sin 0. (16)

On the Attraction of Ellipsoids. 369

But, from the property of the elliptic section made by the plane

of the meridian, we have
e* sinA cos

cot 9 = Sa. ey ia 2esinA cosA, 4. p.,

where e¢ is the excentricity and « the ellipticity of this ellipse.
Substituting in (16) this value of cot 0, and the values of R
and P from (18) and (14), the equation of equilibrium becomes

\: —

+5 (1-3 sin’A) -o”* *rcostn| 2 sind cosA

e 2

= sh oo +w ' sin X cos A,
r
or, approximately,
ae a4 3 sin’ A) —w a cos*A| 26=3
_ lee aan

If we neglect quantities of the second order, this equation
becomes

+ a.

Sal Nace wd. (17)
a a

We have thus arrived at a relation which enables us to ex-
press the unknown quantity C —-A, in terms of quantities which
are all known, and, therefore, to eliminate the former from any
other equation in which it may occur.

Now let &, and &, denote the equatorial and polar attrac-
tions respectively ; we have from the general value of & (13),

M 3C-A
eee ag oa
wal aA,

but
Bre 1 wert |
e=a(l-«) “17g = a (1+ 2e) and (= 7 (1 + 4e)
nyo 2M 0-4
a a a‘

2B

A a >
. : 2 ~ a) & a
“ee

370 On the Attraction of Ellipsoids.

r

G, = R, and G. = Rp - wa; aa a

ee gee ome |
“. Gy~ Ge= —G ~ 9 ev Sahat

= Se

Eliminating oe by means of equation (17), we get

> ome

'
‘
F #
= i

.
-
*
.
+
-
-

- SUPPLEMENT.

a
©
J]
1S)
Zi
O
aay
a
oO
4

-

i is

——————

C2395.)

I.—ON THE CHRONOLOGY OF EGYPT.

' [Proceedings of the Royal Irish Academy, Vou. t., p. 66.—Read April 24, 1837.]

In this Paper the author endeavours to ascertain the names
of the Egyptian sovereigns who were contemporary with Moses,
For this purpose he finds it necessary to determine the interval
between two celebrated epochs—the reign of Menes and the
Exodus of the Israelites. He conceives that the former epoch
is fixed by the “old chronicle” at the distance of 443 years
from the beginning of a cynic (or canicular) cycle; and he thinks
it strange that this simple meaning should not have occurred to
chronologists, who have universally supposed the “ cynic cycle”
of the old chronicle to be a series of demi-god kings who derived
that appellation from the dog-headed Anubis. The canicular
eycle is a well-known period of 1460 years, which the Egyptians
seemed to have used for computing time, as we sometimes use
the Julian period. One of these cycles commenced in the year
2782 before the Christian era; and if we reckon 443 years in
advance, we shall have the year B. c. 2339 for the commence-
ment of the reign of Menes. This date agrees well with the com-
putation of Josephus, who says that the interval from Menes .
to Solomon was upwards of 1300 years. Again, we are told by
Clemens of Alexandria, that the Exodus of the Israelites took
place 345 years before the beginning of a canicular cycle. This
is evidently the cycle which commenced Bs. c. 1822; and hence
we have B. c. 1667 for the date of the Exodus. The mterval
between Menes and the Exodus was, therefore, about 670 years.

.

374 On the Chronology of Egypt.

If, now, we take the catalogue of Eratosthenes, which com-
mences with Menes, we shall find, at the distance of 670 years
from Menes, a king named Achescus Ocaras, who reigned only
one year; preceded by a king named Apappus, who reigned a
hundred years, and succeeded by queen Nitocris, who reigned
six years. Mr. Mac Cullagh thinks that Apappus is the king
in whose reign Moses was born ; that Ocaras is he who pursued
the Israelites to the Red Sea; and that Nitocris is the famous
queen mentioned by Herodotus. It may be objected that
Eratosthenes gives us the succession of Theban kings, whereas
the Pharaohs of the Mosaic history reigned in Lower Egypt;
but it is remarkable that the three sovereigns mentioned above
are found in Manetho’s dynasties among those who reigned at
Memphis ; and it is singular that these are the only sovereigns
(except Menes and his immediate successor) in which the dynas-
ties of Manetho and the catalogue of Eratosthenes agree. All
the other names are different. Of course the predecessors of
Apappus, at Thebes and at Memphis, were different; and thus
we can easily understand how there arose up at Memphis “a
new king who knew not Joseph.” It would appear, in fact,
that Apappus was of a Theban family, and that he succeeded, for

some reason or another, to the throne of Lower Egypt. He was
- only six years old (as we learn from Manetho) when he came to

the throne ; and it is natural to suppose that his chief advisers, —

as he grew up, were the courtiers who accompanied the young
king from his own country to Memphis, and who knew nothing
of Joseph, and cared nothing for his people. Accordingly, when

Apappus arrived at manhood he issued an order that every —

male child of the Hebrews should be destroyed, lest they should
grow too numerous for the Egyptians ; and, under these cireum-
stances, Moses was born in the twenty-first year of his reign,
and was saved by the king’s young daughter, a girl about ten
years old. About the. sixtieth year of Apappus, Moses was
obliged to fly to the land of Midian, for having killed an
Egyptian ; and when at length the king of Egypt died—
“after many days,”’ as it is in the original—Moses returned in

a

On the Chronology of Egypt. 375

the beginning of the reign of Ocaras, before whom were per-
formed those signs and wonders which prepared the way for the
departure of the Israelites. On the night of the Passover, the
king lost his first-born, perhaps his only son; and this may be
the reason that he was succeeded by his sister Nitocris. The
short reign of Ocaras (a single year) might be explained by
supposing he was drowned in the Red Sea; but as there is
nothing in the sacred narrative which obliges us to admit that
the king-perished in this manner, we may adopt the account of
Herodotus, that he was murdered by his subjects. We may
imagine that some of his nobles remained with Pharaoh on the
shore ; and that when they saw the sea return and swallow up all
that had gone in after the Israelites, they murdered the king,
whose obstinacy had brought such calamities on his people, and
then placed his sister Nitocris on thethrone. As Nitocris was the
daughter of Apappus, there is nothing to prevent us from sup-
posing that the queen, now ninety years old, was the princess who
had saved the infant Moses. Weary of her life, she lived only
to avenge her brother. For this purpose, says Herodotus, she
constructed a large subterranean chamber, to which, when it was
finished, she invited the principal agents in her brother’s death;
and there, by the waters of the Nile admitted through a secret
canal, they were drowned in the midst of the banquet. The
queen then threw herself into a room filled with ashes, where
she perished.

(<376")

IIl.—ON THE CATALOGUE OF EGYPTIAN KINGS, WHICH
IS USUALLY KNOWN BY THE NAME OF THE
LATERCULUM OF ERATOSTHENES.

[ Proceedings of the Royal Irish Academy, Vov. u. p. 366.—Read January 9, 1843.]

Tis Catalogue, which the distinguished mathematician and
philosopher whose name it bears drew up by command of
Ptolemy Euergetes, contains a long series of kings who reigned
at Thebes in Upper Egypt, and has been preserved to us in
the Chronographia of Georgius Syncellus, a Greek monk of the
eighth century. It is a document which has been made much
use of by chronologers ; by some of whom (as by Sir John Mar-
sham for example, who calls it “ venerandissimum antiquitatis
monumentum’’), it has been reckoned of the very highest autho-
rity ; but it is extremely corrupt in the latter part, owing to the
carelessness with which it was transcribed either by Syncellus
himself or his immediate copyists. The writers on Egyptian
antiquities have in consequence been much perplexed in settling
the chronology of the reigns in which the errors exist, and the
attempts that have been made to remove the confusion have
only served to increase it. It was the object of the author to
restore the document to its original state, and he showed that
this might be effected, with complete certainty, by a proper at-
tention to the manuscripts of Syncellus. Of these only two are

On the Catalogue of Egyptian Kings. 377

known: one has been used by Father Goar, the first editor of
the Chronographia (Paris, 1652) ; the other, which is a much
better one, has been collated by Dindorf, the second and latest
editor. Dindorf’s edition was published at Bonn, in the year
1829, as part of the Corpus Scriptorum Historie Byzantine, and
on its first appearance Mr. Mac Cullagh had satisfied himself as
to the original readings of the Catalogue, and had seen how to
account for the errors which, probably from Syncellus’s own
negligence, had crept into it; but he did not publish his conclu-
sions at the time, thinking that similar considerations could not
fail to occur to some of the numerous writers who were then
giving their especial attention to such subjects. This, however,
has not been the case. Chronologers have continued to follow in
the footsteps of Goar, a man of little learning, and of no critical
sagacity, who corrected the Catalogue most injudiciously, and
whose corrections, strange to say, are left without any remark
by Dindorf. Thus Mr. Cory, in his Ancient Fragments, a work
much referred to, merely transcribes Goar’s list ; and Mr. Culli-
more, in attempting to reconcile ancient authors with each other
and with the monuments, has adopted an hypothesis respecting
the identity of two sovereigns which is not tenable when the
true version of the Catalogue is known. Even in Goar’s edition,
however, there was quite enough to have led a person of ordi-
nary judgment to the correct readings of the Catalogue, though
perhaps they could not be said to be absolutely certain without
the additional light obtained from that of Dindorf.

The Catalogue in question professes to contain the names of
thirty-eight sovereigns, with the years of their reigns ; the whole
succession occupying, as is stated, a period of 1076 years; but
it is only in the last eight reigns that the errors and inconsist-
encies occur. The thirty-second prince is called Stamenemes ,
that is, Stamenemes the Second, though there is, at present, no
other of that name in the list ; and the beginning of his reign—
as appears from the years of the world, which Syncellus has
annexed according to the Constantinopolitan reckoning—follows

378 On the Catalogue of Egyptian Kings.

the termination of the preceding one by an interval of twenty-
six years. Jackson, in his Chronological Antiquities, is positive that
this prince is called the Second by a mistake, and adds the years
that are wanting to the reign of his predecessor, as Goar had
previously done. In the first part of this view all authors, with-
out exception, are agreed, though they do not explain how a
mistake, so very odd, could have originated; but the learned
Marsham—who, having adopted the short chronology of the
Hebrew Bible, is so hard pressed to find room for the Egyptian
dynasties that he is obliged to begin the reign of Menes the
very year after the Deluge—is glad to omit the twenty-six
years altogether, thus reducing the sum of all the reigns to
1050 years, contrary to what is expressly stated by Syncellus.
The natural inference from the state of the MSS. is, however,
simply this: that the thirty-second king was Stamenemes L.,
that he reigned twenty-six years, and was succeeded by Stame-
nemes II. We may easily conceive that the eye of the tran-
scriber, deceived by the identity of names, passed over the first,
and rested on the second, thus occasioning the error. Indeed
there can now be no doubt that this was the fact; because, in
the MS. marked (B) by Dindorf, the next king is numbered as
the thirty-fourth, the next but one as the thirty-fifth, and so on;
which shows that a name had dropped out, and this name could
be no other than that of Stamenemes I., who must have filled
the vacant interval, and must consequently have reigned the
number of years that has been assigned to him.

As neither Goar nor any other writer perceived this omission,
the successor of Stamenemes II. has always been reckoned as
the thirty-third in the list, and the next following as the thirty-
fourth, &c. But as one error begets another, the omission was
compensated by the insertion of an anonymous king, who is
placed thirty-sixth in the list, with a reign of fourteen years ;
the insertion being necessary to complete the number (thirty-
eight) which the Catalogue ought to contain. And, by a fur-
ther error, these fourteen years are taken out of the reign of the

On the Catalogue of Egyptian Kings. 379

thirty-seventh sovereign, who ought to have nineteen years
instead of the five that have been hitherto assigned to him. This
last error was occasioned by an ignorant correction of a mistake
which is found in both the MSS., and which therefore probably
arose from the carelessness of Syncellus himself. The thirty-
seventh king and his predecessor are stated to have begun to
reign in the same year of the world, and to have reigned the
same number of years (five). Now from what goes before it is
plain that both these numbers belong to the thirty-sixth king ;
and from the year of the world in which the thirty-eighth and last
king began to reign, it is clear that the thirty-seventh reigned
nineteen years. ‘The mistake in the MSS. is one which might
easily be made by a thoughtless writer; for the Catalogue is
given in detached portions—a few reigns at a time—separated
by a great quantity of other matter, and the name of the thirty-
sixth king ends one of these portions, while that of the thirty-
seventh begins another ; so that, not having both before his eyes
at the same moment, a person so careless as Syncellus might,
without being conscious of it, attach the same reign and date to
the two names, by transcribing twice over the same line of num-
bers in the Catalogue which he was copying ; the whole of which
Catalogue, in all likelihood, he had previously drawn up in a
tabular form, with the years of the world annexed according to
his own chronology, that it might be ready, as any portion of it
was wanted, for immediate transference to his pages. Such
seems to be the natural account of the matter; but, as usual, it
does not occur to Goar, who takes the opportunity, which the
confusion affords him, of foisting in his supplementary king
between the two last mentioned, giving each of these five years,
as in the MS., by which means he obtains room for him; while
on the other hand he alters the year of the world attached to the
thirty-seventh king, so as to make it suit his hypothesis.

The following is a view of the last eight reigns, as they ap-
pear to have stood in the original document, compared with the
erroneous list of Goar. The years of the world are omitted, as

380 On the Catalogue of Egyptian Kings.

being of no importance, except so far as they are useful in the
preceding argument.

I. Goar’s List. II. Correctep List.

Years. Years
31. Peteathyres reigned 42 | 81. Peteathyres reigned 16
32. Stamenemes of 23 | 382. Stamenemes I. ,, 26
33. Sistosichermes ,, 55 | 33. Stamenemes II. ,, 23
34. Maris - 43 34. Sistosichermes ,, 55
35. Stphoas a 5 35. Maris ss 43
36. Anonymous s 14 36. Siphoas 5 5
37. Phruoro a 5 | 37. Phruoro FN 19
38. Amutharteus ,, 63 | 388. Amutharteus ,, 63

The interval of time which has been shown to belong to the
first Stamenenes, and which was added by Goar to the reign of
Peteathyres, is differently disposed of by Mr. Cullimore, in a
chronological table which he has given in the second volume of
the Transactions of the Royal Society of Literature. His object
being to compare the lists of Eratosthenes, Manetho, &c., with the
supposed hieroglyphical series, he makes Saophis, the fifteenth
in Eratosthenes’ Catalogue, the same as a king whose name
is read Phrathek Osirtesen ; but the forty-third year of the
latter is mentioned on the monuments, whereas Saophis has
only twenty-nine years in the Catalogue. To escape from this
difficulty, therefore, Mr. Cullimore adds the unappropriated in-
terval to the reign of Saophis, thus giving him fifty-five years
instead of twenty-nine. But it now appears that such a suppo-
sition is altogether inadmissible, and consequently the two per-
sonages in question cannot be identified—a circumstance which
proves that there is some fault in Mr. Cullimore’s assumptions,
and that his other conclusions, at least in this part of his Table,
cannot be relied on.

The corrections here given do not interfere with the infe-
rences drawn by Professor Mac Cullagh from the Catalogue of
Eratosthenes in a former Paper on Egyptian Chronology (Pro-
ceedings of the Royal Irish Academy, vol. i., p. 66),* because the

* Supra, p. 373.

On the Catalogue of Egyptian Kings. 381

portion of the Catalogue with which he was there concerned ter-
minates with the reign of Queen Nitocris, the twenty-second in
the list. The corrections, indeed, though not hitherto published,
were made long before the date (April, 1837) of that Paper,
but not before he had adopted the hypothesis therein proposed,
as an answer to the old and ever-recurring question— Who were
the Egyptian sovereigns that were contemporary with Moses ?
For it was in consequence of this hypothesis, which had sug-
gested itself to him at a very early period, that he was led to
examine the Catalogue minutely, in order to discover whether
his chronology was affected by its errors.

Having been led to refer to his hypothesis, Mr. Mac Cullagh
took occasion to observe that, in the interval which had elapsed
since it was published, he had not met with any facts that were
opposed to it: on the contrary, the more he considered it, the
more he was inclined to believe in its reality; though it was
entirely different from every other that had been proposed,
either by modern chronologers or by the early Fathers of the
Church, in their manifold attempts to connect the narrative of
Moses with the remaining fragments of Egyptian history. The
hypothesis, indeed, is the on/y one which, while it gives a pro-
bable date for the Exodus, also satisfies what Mr. Mac Cullagh
conceives to be the necessary conditions of the question ; namely,
a very long reign—of at least eighty years—during which the
Israelites were persecuted, succeeded by a very short one—
apparently not more than a year—during which their deliver-
ance was wrought ; and it is interesting in itself, on account of
the remarkable connexion which it establishes between sacred
and profane history, and the highly dramatic character of the
events which are thus, for the first time, brought into view.

THE END.

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gow. Fourth Edition, enlarged. $vo,
price 10s, 6d,

INDEX.

A bbey & Overton's English Church Life

"s Photography .......4. eavbsategevsbien
Acton’s Modern Cookery ......s.e«ss ash cigaceauk
Alpine Club Map of Lhe oe beataeces +e
Alpine Guide (The) .. Rosescucegragcnaies

Amos's urisprudence .

mer of the Constitution......... oe

Fifty Years of the English Con-
BEOURION cconseassouse Rosecsunsaartes Reapslessanse?
Anderson's Strength of Materials .........00
Armstrong's Organic Chemistry ....... eves
Arnold's by ) Lectures on Modern History
Miscellaneous Works ...++-

SETMONDS 55. ssoniosasegecnses vsee

——(T.) English Literature ............
Arnott's Elements Of PhySics......c00e eseeees
Atelier (The) du Lys. .........s000« caceceesocece
Atherstone PYiory........cseccccsnseseee auch oonst

Autumn Holidays of a Country Parson ...
Ayre's Treasury of Bible Knowledge ......

Bacon's a by Whately .....+. secdonee
ife and Letters, by ‘Spedding . cos
Ws Ss, caves sie ceateccss Sevbeostectvads 2
Bagehot's Economic Studies ...ccscrsceceeeeee
—_—_ Literary Studies ...c.ceocesseseecesce
Bailey's Festus, a Poem .......068 ebdapuievengen
Bain's Mental and Moral Science.......++.0+
on the Senses and Intellect .........
Emotions and Will...... RAsveacieseenne

Baker's Two Works on Ceylon.......scessesee
Fold Alpine Guides
a:

SOOO ee emer ee eereeeoeneee

on Railway Appliances

eee eeeeenraceee

17
A

Beaconsfield” s (Lord) Novels and Tales 38 & %
8

Becker's Charicles and Gallus........000. bekene
Beesly's Gracchi, Marius, and Sulla....
Black's Treatise on Brewing ........+000 Sir
Blackiley's German-English Dictionary......
larne's Rural Sports ccidcsvveceveryonsvessstec
LORIE S WEEOIS | acsedncussocecetccsdagancstensad
Bolland and Lang's Aristotle's Politics...
Boultbee on 39 Articles.........sssssseveseccevees
-'s History of the English Church...
Bourne's Works on the Steam Engine......
Bowdler's Family Shakespeare sissverseesees
Bramley-Moore's Six Sisters of the Valleys .
Brande's Dictionary of Science, Literature,
and Art ..ccccoses svececsnesveceacssccscocsss cosas
Brassey s Sunshine and Storm in ‘the East.
Voyage of the Sunbeam.....-...0+
Browne's Exposition of the 39 Ria cater
Ptah oda BS Modern England .
Buckle's

sence

istory of Civilisation .sscssssesseese
Posthumous Remains ...... Fidsdence
Buckton’s Food and Home @obkery svete
—— Health in the House ..........00000
——— Town and Window Gardening...
Bull's Hints to Mothers .........sseneeeees sepee
Maternal Management of eae sta ‘
Burgomaster’s Family (The) ......csecsseeeees
Burke's Vicissitudes of Families...

Cabinet TAwyertsccstboeccuscbsscvcssepcestoverantavs
Capes’s Age of the Antonines.scscscssssssseses
Early Roman Empire ...scccssecseee
Cayley's Wiad of Homer .....ccsscsecessceecees
Cetshwayo’s Dutchman, translated by
DISROP COLEHSO .scsedestescscucsecses svecaccene’

3

Changed cee of Sagres. Truths...
Chesney's Indian Polity ...... severdesseseseeese
———— Waterloo Campaign .s....s0ccerese
Church's Beginning of the Middle Ages...

Colenso on Moabite Stone KC... nesanseseuasaee
”s Pentateuch and Book of Sher
Commonplace Philosopher...
Comte's Positive Polity ......cesseccecsessessncs
Conder’s Handbook to the Bible sess
Congreve's Politics of Aristotle .......ss.s+e0
Conington's Translation of Virgil’s Zneid
Miscellaneous WritingS......++.

Contanseau’s Two French Dictionaries ....

Conybeare and Howson's St. Patil ...c..sseeee
Cooper's Tales from Euripides «...... svees
Coney s Sia against Absolute Mon-

ALCHY i.) 5 c0cd>saanasoyesaqaunyees jiapebswceaqaenen
Cotta on Rocks, by Lawrence ssscirsaseavene
Counsel and Comfort from a City Pulpit...
Cox's (G. W.) Athenian Empire ..

Crusades ......+. ocenceucevestess
Greeks and. Persians...+.+.++
Creighton's Age of Elizabeth ....... wegitied seed

England a Continental Power .

Shilling History of England...
——_——— Tudors and the Reformation
Cresy's Encyclopzedia of Civil, Engineering
Critical Essays of a Country Parson......+..
Crookes’s Anthracen ...cesccrececescessescevesaces
Chemical Analyses ....e+0. cosseseee
Dyeing and Calico-printing Seebes
Culley's Handbook of ae eee
Curteis's Macedonian Empire ......+..
De Caisne and Le Maout's Botany .....
De Tocqueville's Democracy in America...

Dixon's Rural Bird Life ..... scoetvsvveaweae shee
Dobson on the OX .....0000- Siixehaean ee
Dove's Law of Storms ....... scguiacduboeesaahale
Doyle's (R.) Fairyland ........000+ Laets osaeuente
Drummond's Jewish Messiah ....s+seeseeseeee

Eastlake’'s Hints on Household Taste ssisee
Edwards's Nile.. “a
Ellicoté’'s Scripture Commentaries senseneses
Lectures on Life of Christ
Elsa and her Vulture

eeeeeere See eeweewennerneee

Epochs of Ancient History....cccccccsecseseeee ©

—— English History ....ccscscssesere
Modern History ....cscccsceeccoee
Ewald's History of Israel .......cesssenceeseee
Antiquities of Israel...
Fairbairn’s Applications of Iron ......++0008
Information for Engineers......
Mills and Millwork .....s..sseesse
Farrar’s Language and Languages .....0++
Francis's Fishing Book .....ssecssssereseceeeee
Frobisher’ s Life Dy Foes.scecsecsvererecanesvese
Froude’s Cesar sees. Wsvensetaiee
———— English in Ireland

seeeeeeeecereee

Cent eeteeeeeenee

see eeeee esse eereee

———History of England ..... eeesescevense a

Lectures on South Africa ....s.0.
——Short Studies......
Gairdner’s Houses of Lancaster and York

Richard III. & Perkin Warbeck
Ganot's Elementary Physics ...cccesreseeeeees
Natural Philosophy ...... aunveauebane
Gardiner's Buckingham and Charles «+...

see eeeeeeseeseneetene

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NAWWWWWWWYNW ©

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HH
NOONW OO

oT

Lessons of Miadie Age CONE R ERO T RETF ET eee ee ee

WORKS published by LONGMANS & CO. 23
Gardiner’s Personal Government of Charles I. 2 Lewes's Biographical History of Philosophy 3
— First Two Stuarts. ...s.ccccscreee 3 Lewis on Authority aigitddhnck sOipteeakisninakee.n CO
— Thirty Yeats’ War severe 3 | Leddelland Scott's Greek-English Lexicons 8
German Home Life.......sccssssseeeesssseesveee 7 ~+| Lindley and Moore's Treasury metrtitars sas 7 2E
Goodeve’s Mechanics....sccccccccsserccecceccecees IL Lioyd’s Magnetism ......... sbheessevanes net
IMechanisii=. cesses cevevcstysovccvsee’ VIE Wave-Theory of Light.. atisbant content
Gore's Art of Scientific Discovery............ 14 | Zongman’s (F. W.) Chess Openings... Soseusss QE
Electro-Metallurgy ......scecceseewees II German Dictionary... 8
- Gospel (The) for the Nineteenth Century . 6 (W.) Edward the Third......... 2
Grant's Ethics of Aristotle ..........seeeee0 ser oS Lectureson Historyof England 2
Graver Thoughts of a Country Parson.. 7 Old and New St. Paul's 13
Greville's Journal .... se seveseseseaseeeeseeees I | Loudon’s Encyclopzedia of Agriculture ... 15
Griffin's Algebra and Trigonometry. veeee « IF Gardening...... 15
Griffith's A B C of Philosophy ............ oe 5 Plants....ccccveee 12
Grove on Correlation of Physical Forces... 10 | Lwddock's Origin of Civilisation ........e0008 I2
Gwilt's Encyclopzedia.of Architecture....... 14 | Zudlow's American War of Independence 3
Hazle's Fall of the Stuarts.........seee0e vintwyes 3 Lyyra’ Germanica, .cresccvcssesstsuves tt ee
Hartwig's Works on Natural ory and Macalister's Vertebrate Animals seghevaecces EE
Popular Science Westheevavtsserssevermad, | Mfacaulay's: (Lintd):ESSAYS’ scticccesséhsaasensoenne
Hassall’s Climate of San Remo.. sre k7 History of England . aa
Haughton's Animal Mechanics «...++0++00008 10 Lays, Illustrated.......++
Hayward's Selected Essays wseecccsee 6 Cheap Edition... 19
Heer's Primeval World of Switzerland...... 12 Life and Letters......... 4
Heine's Life and Works, by eee covers 4 Miscellaneous Writings 7
Helitholtz or Tone ssrcccressseseccsseveseeseees IO SPCECHES ist. cw sandeonteees eee
Helmholta's Scientific Lectures s.ssssese Io Works ........ Sescesnasocee,
Herschel's Outlines of Astronomy ........00. 9 Writings, Selections from 7
Hillebrand's LecturesonGerman Thought 6 | AZcCulloch’s Dictionary of Commerce .... 8
Hobson's Amateur Mechanic ..........0:e00006 14 | AZacfarren on Musical Harmony .......0008 13
Hophins's Christ the Consoler .....s04004. 17 | Macleod's Economical Philosophy......s+s. 5
Hoskold’s Engineer's Valuing Assistant cos 34 Economics for Beginners 2r
Hullah's History of Modern Music ...... 12 Theory and Practice of Banking 2r
——— Transition Period .......cscseceee0e 2 — Elements of Banking............006. 22
IMIDE SULSCAYS Autacscs scsetssvenspetstcssescdesess © 6 prea = regalia Rematcinarscs ie
Treatise on Human Nature........ 6 India sssccssesee coeevecsseesossvececeeesscseess IG
Thne's Rome to its Capture by the Gauls... 3 | Mademoiselle Mori ...........+000 sce XO
History Of ROME weccccsssssessesessneee 3 | AZahafy's Classical Greek Literature ...... 3
Lngelow's Poems ss+.seseseveseerees sesessveseeses IQ | AZalezt's Annals of the Road ........... tessese IQ
Sameson's Sacred and Legendary Art... 13 | Zanning's Mission of the Holy esi svsese 17
—_ Memoirs by Macpherson......... 4 | MMarshman's Life of Havelock wesc 4
enkin's Electricity and Magnetism......... 11 | Martineau’'s Christian Life.......ccccccsceeseee 7
errold’s Life of Napoleon........ SET CCrSS abe Hours of Thought.....ssscsscose 17
ohnson's Normans in Europe sssesseseseeeee 3 | A AY MMS. eeeeeecccceceeeeneceveneene IF
— — Patentee’s Manual .......csccccecese 2I Maunder's Popular Px cent al alemeaie 20
Fohnston’ 's Geographical Dictionary......... 8 | Maxwell's Theory of Heat .....ccccscssseseveee IZ
ukes's Types Of Genesis .......ceceecsseeeese 16 | AZay’s History of Democracy ...cccorccsoeee 2
ukes ON Second Death cisocccsedevecvsseese 16 History of England ........+00 Goel a nae
Kalisch's Bible Studies .....sssccccccsesesseseee 16 | AZelville's (Whyte) Novels and Tales ., 19
—- Commentary on the Bible......... 16 | AZendelssohn's Letters wrrccccrsssrsesssresseeness
— Path and Goal........seccccseeseess 5 | AMerivale's Early Church History stacveveree IS
Keller's Lake Dwellings of Switzerland.... 12 | ————-— Fall of the Roman Republic... 2
Kerl's Metallurgy, by Crookes and Rohrig. 15 General History of Rome ...... 2
Kingzett's Alkali Trade ........scsccssssssocese, 13 Roman TriumvirateS.e.eceeee 3
—— Animal Chemistry .... sseeeesesseaeee 13 | —————— Romans under the Empire...... 2
Kirby and Spence's Entomology ........ we* 12 | Merrifield Arithmetic and Mensuration... xx
Klein's Pastor's Narrative .....-.sscecssesevees 7 | Miles on Horse's Foot and Horse Shoeing 20
Knatchbull-Hugessen's Fairy-Land ......+ -- 18 on Horse’s Teeth and Stables......... 20
Higgledy-Piggledy 18 | A¢¢iZ(J.) on the Mind..... WNvaselsunseas 5
Landscapes, Churches, &c......... tssseersseree 7 | Adil’ (J. S.) Autobiography ..... Rie 4
Latham’'s English Dictionaries ........0..++. 8 Dissertations & Discussions 5
Handbook of English Language 8 —— Essays.on Religion ......000008 16
Lecky's History of England...... a HE Sy ee | —— Hamilton's Philosophy ...... 5
European Morals.... 3 —— Liberty ....secceresseccvvceesesess | -'S
Rationalism ....ccccrseees 63 Political Economy «e008 5
Leaders of Public Opinion............ 4 | — Representative Government 5
Leisure Hours in TOWN ..scccccsssscsveseseses 7\|-—_— Subjection of Women......... 5
Leslie's Essays in Political and Moral —— System of Logic ...... vssseneee 5S
Philoso phy TOOT ROOT O Ree R ERE HERE A EEO R EERE ER® 6 “ext Unsettled Questions seeteenee 5
7. onnees: Utilitarianism Seer ewer eter etree 5

WORKS published by LONGMANS & CO.

24
Miller's Elements of Chemistry. «0.00.06. 13 °| Sewell’ s Passing Thoughts on Religion ...
Inorganic Chemistry...sreseceserseee IT - Preparation for Communion ...... _
——-— Wintering in the Riviera.. 17 Stories.and ‘Tales *v.cciscconsacvene chon
Minto (Lord) :in India... ..cc0ssescccssesedecose 2-2 Thoughts for the Age ....cccccesreee |
Mitchell's Manual of Assaying .. srsssseeseesere, 15 | Shelley's Workshop Appliances ..ss.sseseeeeee
Modern Novelist's Library .......ssssee0s 48 & 19 Short's Church History
Monsell's Spiritual Songs.......ssereeseess 17 | Smith's (Sydney) Wit and Wisdom .........
Moore's Irish Melodies, eTiiustrated Edition 13 (Dr. R. A.) Air and Rain ..........65
Lalla Rookh, Illustrated Edition... -13 (R. B,)Carthage & the Carthaginians
Moreil's Philosophical Fragments...... asics —S Southey's Poetical Works........0+0 veveneeseees
Morris's Age of Anne .......+++ Litutpinsctectiee 2S Stanley's History of British. Birds: veravesies
Mozart's Life, by NOkl .ccccrssceesscevseoees 4 | Stephen's Ecclesiastical Biography..... ...++«
Miiller’s Chips from a German Workshop. 7 Stonehenge, Dog and Greyhound ...... s+.
>———Hibbert Lectures‘on Religion ... .16 | Sfomey on Strains ..s.secsccessecsececvanas¥aswese
Science of Language ...... beiadete eer, Stubbs's Early Plantagenets .......csse00s ste
Science of Religion ........csseveeeee 16 | Sunday Afternoons, by A. K, H.B. .........
Nelson onthe Moon.....cccccccccscsccssesese Fine cag Supernatural Religion .. re i
Nevile's Horses and Riding .......cccceceere . 20 | Swinbourne's Picture Logic eedonsbereseseshen
Newman's Apologia pro Vita Sua........0.. + 4 | TZancock's England during ~ the Wars,
Nicols's Puzzle of Life ....ccs.c.ssesseveveseee I2 1778-1820. ss setoarvass cvabonadpnet eames te sai5oue
Noiré’s Miiller & Philosophy of Language 7 Taylor's History of India ........000.. <osaia eet
Northcott's Lathes & Turning ....sccccccee 4 ———— Ancient and Modern History ...
Ormsby's Poem of the Cid sceccrescessssssevees 19 (Feremy) Works, des by fie
Owen's Comparative Anatomy and Phy- Text-Books Of Science...ss.ssescseeseseceensenes
siology of Vertebrate Animals .+.......... IZ Thomé's Botany .. evewesunace
Pache's Guide to the Pyrenees .recssesresevee 18 Thomson's Laws of Thought .. way ben yecanarenee
Pattison’s Casaubon........00068 Gavesctvers Secu LA Thorpe's Quantitative Analysis. ....sscsseseses
Payen's Industrial Chemistry....... hg oaecea heed Thorpe and Muir's Qualitative Analysis ...
Pewtner s Comprehensive Specifier ..ssccoee 2E Thudichum's Annals of Chemical Medicine
Phillips's Civil War in Wales .....s040 one ae Tilden's Chemical Philosophy «....ccescosseee
Piesse's Art of Perfumery .....++ evansestevopee Say — Practical-Chemistry ......sssseseseees
Pole’'s Game OF Whist .....ccccecccssssecseseoeee 2I Todd on Parliamentary Government.....+++
Powell's Early England ...0....ssecseoeeee atte Trench's Realities of Irish Life .......sees000e
Preece & Sioowright s Telegraphy... Prey It Trollope's Warden and Barchester Towers —
Present-Day Thoughts..........0+ Se petreey erly Twiss's Law of Nations ....... a3 «aah cusgieenan
Proctor's Astronomical Works ......ssseesee 9 Tyndall s (Professor) Scientific Works ...
Scientific Essays (Two Series)... 11 NID AWAXES cei cineys savancersnsases ccnvcusetheceoeeane
Prothero’s Life of Simon de Montfort ...... 4 Unwin's Machine Design .. vovesukesdecaeraan
Public Schools Atlases ........ Sroka heepisdste =e Ge Ure's Arts, Manufactures, and Mines ...60
Rawlinson's Parthia.......cc.scccsccscres ssdevaes’ Q\|° Venn's Life, by. Aight \si.cscccnaneenveameeeaee
Sassanians .......s0 seveseseteean 03 Ville on Artificial Manures.. seeeseeeevens
Recreations of a Country Parson... = 7 Walker on Whist.......05+ fis dndceteegextga oRaUEn
Reynardson's Down the Road sseersreeeereee 19 Walpole's History of England .....scccsceees
Rich's Dictionary of Antiquities .......006. pee Warburton’s Edward the Third ........0008
Rivers s Orchard House «....0.seeeee. constraint a Watson's Geometery ...ssccscoessseeesscesegece
Rose Amateur's Guide... sectss 12 Watts's Dictionary of Chemistry s...++s0see
Rogers's Eclipse of Faiths scassasecvecdec bbasdss MO Wede's Civil War in Herefordshire .........
Defence of Eclipse of Faith ...... 16 Weinhold's Experimental Physics.....+.0+0«»
Roget's English Thesaurus .......0+-e+e00 sia 8 Wellington's Life, ky Gletg ccccccssscesseeese
Ronalds’ Fly-Fisher’s Entomology .s..000. 20 Whately’'s English Mesa depiies *
. Rowley's Rise of the People ....... beaestendéan oo LOBIC “css sicsosachacsacuvekeuoaenenneee
Settlement of the spiegatiensr see 3 | m=ApAp-p- RhCAOTIC 2... .,srereccevscveeeeroeceees
Russia and England .cecccsscceseees eieke oe White's Four Gospels in Greek.......ceeeeees
Russia Before and After the War.. vegtaxsnoostieo a and Riddles Latin Dictionaries ...
Rutley's Study of Rocks...... savehsben chek cas ES oh Wilcocks’s Sea-Fisherman. ....sccsesessscesvees
Sandars's Justinian’s Institutes secs 6 Williams's Aristotle’s Ethics.....sececoosssese
' Sankey's Sparta and Thebes ..... ebecaenstidesn eg Wilson's Resources of Modern Countries...
Savile on Apparitions .......se00008 soscesesscone = 77 Wood's (J. G.) Popular Works on Natural
Schellen's Spectrum Naciaa can Fievebatesinaticge a 0O History ....... shes cust gaevgnsoeannnley scene
Seaside Musings ......+0000 sdawnraeesen Woodward's Geology .. vusebateesasasacene
Scott's Farm Valuer ..... atewebhpdubens tetestvvaans ‘ited Yonge's English-Greek Lexicons sccsesedsbus
Rents and Purchases Segteatsal etaisuse; ad FIOTACE:..ccsasccsvngtasveoWeuveuniedeehenes
Seebohm’s Oxford Reformers of 1498......... 2 | Youatt on the Dog we vee
ee Protestant Revolution ....06010008 3 Gn the Forse: oi. cdasecabasteis stvecsacne
Sewell’s History of France sesccsssersseseoese 2 | Zeller’'s Greek Philosophy sserssesscorsersseoes
f ‘ Spottisqoode & Co, Printers, New-street Sguare, London,

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