A treatise on some new geometrical methods

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A TREATISE ON SOME

NEW GEOMETRICAL METHODS,

CONTAINING ESSAYS ON

THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS,

ROTATORY MOTION,

THE HIGHER GEOMETRY,

AND CONICS DERIVED FROM THE CONE,

WITH

AN APPENDIX TO THE FIRST VOLUME.

Nova methodus, nova seges.

IN TWO VOLUMES.— VOL. II.

BY

JAMES BOOTH, LL.D., P.R.S., tf.R.A.S., &c. &c.,

VICAK OF STONE, BUCKINGHAMSHIRE.

LONDON:
LONGMANS, GREEN, READER, AND DYER,

PATERNOSTER ROW.

MDCCCLXXVII.
[All riffhft reserved.]

T.KRE Y

AT.KRE T FI.AMMAM.

PRINTED BY TAYLOK AND FRANCIS,
RED UON COXTRT, FLEET STREET.

INTRODUCTION

TO THE SECOND VOLUME.

AFTER the lapse of nearly four years, in the face of many hindrances,
untoward events, and difficulties, I have succeeded in bringing
through the press this second and concluding volume of my mathe-
matical and physical researches.

It is proper to mention that the volume will be found to contain
four distinct treatises : — (a) on Elliptic Integrals, ($) on Rotatory
Motion, (y) on the Higher Geometry, and (8) on Conic Sections,
followed by an Appendix to the first volume.

An outline of the following researches on the Geometrical Pro-
perties of Elliptic Integrals was published in the Philosophical
Transactions of the ROYAL SOCIETY for 1852, p. 311, followed by a
Supplement "printed in the volume for 1854, p. 53. Ample time
and unbroken leisure have enabled me to recast and enlarge those
essays. Though the work was onerous, it was also, I may say, a
labour of love, lightened by the discovery, sometimes unexpected,
of new truths of great geometrical beauty.

Amongst these researches not the least important is the discovery
of three curves of double curvature whose rectification may be
effected by elliptic integrals of the first and third orders. These
are the geometrical types of those transcendental expressions due to
Legendre and Lagrange. The algebraical relations discovered by
these illustrious geometers are the exponents of the geometrical
properties of those curves. Those versed in the subject will not
need to be told how the simplicity of these relations contrasts with
the abortive attempts of the most illustrious mathematicians to
devise, on a plane, curves whose quadrature or rectification might
represent those expressions. I do not here propose to give an
analysis of the work ; but, for the sake of the few who care to
inquire into those matters, I would call attention to Chapter VIII.
on conjugate amplitudes, and to Chapter X. on derivative hyper-
conic sections.

IT INTRODUCTION TO THE SECOND VOLUME.

In the course of these investigations this important truth is
clearly established, that the theory of those celebrated functions
constitutes a general trigonometry for those curves in which sur-
faces of the second order intersect. Of this general trigonometry
circular and parabolic trigonometry are the extreme cases. In the
former the modulus is zero, in the latter unity. Thus an unbroken
analogy runs throughout the whole, and the several cases are linked
together under the great mathematical law of continuity.

As a test of the utility of those researches in physics, I have
applied them, in the following essay, to the discussion of the cele-
brated problem, to determine the rotation of a rigid body, in free
motion, round a fixed point j and I have shown how the position
of such a body at the end of any given epoch may be made to
depend on the evaluation of those algebraical expressions or their
equivalents, the arcs of hyperconic sections.

The investigations on rotatory motion given in this volume were
made, the greater portion of them, very many years ago. Some
of them appeared from time to time in those periodical publications
whose pages are open to discussions on subjects of this nature.

In this treatise a complete investigation has been attempted of
the laws of the motion of a rigid body revolving round a fixed point,
and free from the action of accelerating forces — an investigation
based on the properties of surfaces of the second order, of the curves
in which these surfaces intersect, and on the theory of elliptic inte-
grals. The results which have been obtained are exact and not
approximate, general and unrestricted by any imposed hypothesis.

I have carefully abstained from introducing any methods which,
to one moderately versed in the first principles of the integral cal-
culus, might not fairly be assumed as known. There is but one
exception. In a few cases, where the method was peculiarly appli-
cable, I have ventured to make use of tangential coordinates, the
theory of which is fully developed in the first volume of this work.
The reader may, however, if he chooses, omit those applications,
without in any way breaking the continuity of the subject.

I have not been led away by mathematical pedantry to attempt
to render this essay purely algebraical, by rejecting geometrical
conceptions and the aids thence derived to simplicity and clearness,
knowing that, very often, the elegance of the analysis is owing to
the distinctness of the graphical conception, and that, though the
forms of the reasoning may be different, the subject is identically
the same.

The problem of the rotation of a rigid body round a fixed point
is one that has engaged the attention of the most eminent mathe-
maticians of Europe. More than a century has passed away since
D'Alembert first, and Euler soon after, investigated the analytical
conditions of such a motion, and formed those differential equations,

INTRODUCTION TO THE SECOND VOLUME. V

on the integration of which the determination of the motion ulti-
mately depends. In their investigations, which were purely alge-
braical, they used to a great extent the principles of the transfor-
mation of coordinates — a method of research, it must be conceded,
which, though unexceptionable on the ground of mathematical
rigour, is generally found to lead through operose and cumbrous
processes to complicated results.

Some years afterwards, Lagrange took up the subject, and deve-
loped the theory in formulae of great symmetry and generality.
Combining the principle of D'Alembert with that of virtual velo-
cities, he deduced the differential equations of motion, containing
six quantities to be determined. By means of these equations, the
three components of the angular velocities round the principal
axes, which determine the position of the instantaneous axis of
rotation in the body, and the three other angular quantities which
define the position of the body itself in space, at any epoch, may
be expressed in terms of the time. But these quantities, however
they may be determined, furnish us, as it has been justly observed,
with no conception of the motion during during the time. They
prove to us that the body, after the lapse of a certain time, must
be in a certain position ; but we are not shown how it gets there.
We may, by means of calculations, more or less long and compli-
cated, ascertain the bearings of the body at any required instant ;
we cannot, so to speak, accompany it during its motion. It is
determined per saltum, and not continuously ; we are wholly
unable to keep it in view and follow it, as it were, with our eyes
during the whole progress of rotation.

To furnish a clear idea of the rotatory motion of a body round a
fixed point, and free from the action of accelerating or other external
forces, but in motion from the influence of one or more primitive
impulses, was the object of a memoir, presented many years ago
to the Institute, by that eminent mathematician, M. Poinsot. In
this memoir, the motion 'of a body round a fixed point, and free
from the action of accelerating forces, is reduced to the motion of
a certain ellipsoid, whose centre is fixed, and which rolls, without
sliding, on a plane fixed in space'. The axes of this ellipsoid are
assumed proportional to the inverse square roots of the moments
of inertia round the principal axes of the body, passing through
the fixed point, and -coinciding in direction with them. He states
as his final result, that the time and the other ultimate quantities
must be determined by the aid of elliptic integrals. He does not,
however, give any account of the processes by which he arrived at
his results ; and few of the attempts which have since been made to
supply that omission have been very successful.

Some time afterwards the late Professor M'Cullagh, of Dublin,
turned his attention to this problem, which, owing to the mvnt

VI INTRODUCTION TO THE SECOND VOLUME.

researches of Poinsot, had then attracted considerable notice.
He adopted an ellipsoid, the reciprocal of that chosen by the latter
geometer, and deduced those results which had long before been
arrived at by the more operose methods of Euler and Lag-range.
His method of investigation, however, was peculiarly his own ; but,
so far as the author is aware, he never published his method of
research.

The idea of substituting, as a means of investigation, an ideal
ellipsoid, having certain relations with the actually revolving body,
claims as its author the illustrious Legendre. Although he conducts
his own investigations on principles altogether different, he yet
seemed to be, in his Traite des Fonctions Elliptiques, well aware of
the use which might be made of this happy conception.

Several years ago, when engaged in applying the new analytical
method, founded on my peculiar system of tangential coordi-
nates, I was led to views somewhat similar to those of Legendre,
from remarking the close analogy or rather identity which exists
between the formulae for finding the position of the principal axes
of a body, and those for determining the symmetrical diameters of
an ellipsoid. I still further observed, that the expression for the
length of a perpendicular from the centre on a tangent plane to an
ellipsoid, in terms of the cosines of the angles which it makes with
the axes, is precisely the same in form with that which gives the
value of the moment of inertia round a line passing through the
origin. Guided by this analogy, I was led to assume an ellipsoid,
the squares of whose axes should be directly proportional to the
moments of inertia round the coinciding principal axes of the body.

At first sight the inverse ellipsoid, assumed by Poinsot, may
seem to possess some advantages over the direct ellipsoid, at least
so far as such an ellipsoid may be said to approximate in form to
the natural body. For example, if we consider the case of the
rotation of a solid homogeneous ellipsoid round its centre, the ideal
or mathematical ellipsoid of Poinsot will bear a resemblance to the
figure actually in motion. In the direct ellipsoid of moments,
which is made the instrument of investigation in the following
pages, this resemblance does not exist ; for the coinciding axes of
the material and mathematical ellipsoids are such that the sum of
their squares is constant. Should the revolving figure be an oblate
spheroid, its mathematical representative will be a prolate spheroid.
The reader must bear this diversity of figure in mind, in applying
the conclusions of theory to an actually revolving ellipsoid.
Although it may seem a matter of little moment which of the
ellipsoids we choose as the geometrical substitute for the revolving
body, it is not so in reality when we come to treat of the properties
of the integrals which determine the motion. These integrals
depend on the properties of those curves of double flexion in which

INTRODUCTION TO THE SECOND VOLUME. vn

cones of the second degree are intersected by concentric spheres,
liy means of the properties of these curves, a complete solution
may In* obtained, even in the most general case, to which only an
approximation lias hitherto been made. The solution of a mathe-
matical problem may only then be said to be complete, when in the
final result the calculation of the sought quantities may be made
to depend on some known elementary quantity or quantities, such
as a certain straight line, an arc of a circle, &c. So in this problem,
the elliptic transcendents, to the determination of which the calcu-
lation of the motion is ultimately reduced, are shown to represent
arcs of spherical conic sections, whose elements depend on the
nature of the body and on the magnitude and position of the im-
pressed moment. In all the solutions of this problem which have
hitherto appeared, the investigations are brought to a close when
the expressions, either for the time or other sought quantity, are
reduced so as to include the square roots of quadrinomials involving
the independent variable to the fourth power. In this treatise the
investigations are continued beyond that point, and the quadrino-
mials have been reduced, as shown in the preceding treatise, to
arcs of hyperconic sections.

An elliptic integral of the first order being shown to be only a
particular case of elliptic integrals of the third order, as the circle
is a species of ellipse, it follows that the analogies between integrals
of the first and third orders will be more numerous and intimate
than between the second and either of the others. Such is in fact
the case. Elliptic integrals of the first and third orders constantly
occur in combination. In the discussions of the following pages,
for example, integrals of the first and third orders present them-
selves in various combinations, while an integral of the second
order does not once occur in the essay.

The application of the theory of elliptic functions to the discus-
sion of the problem of a rigid body revolving round a fixed point,
has led to the following remarkable theorem :

The length of the spiral between two of its successive apsides,
described in absolute space, on the surface of a fixed concentric
sphere, by the instantaneous axis of rotation, is equal to a quadrant
of the spherical ellipse described by the same axis on an equal
sphere, moving with the body.

The ordinary equations of motion being established, the author
proceeds to show that if the direct ellipsoid of moments be con-
structed, the rotatory motion of a body, acted on solely by primitive
impulses, may be represented by this ellipsoid moving round its
centre, in such a way that its surface shall always pass through a
point fixed in space. This point, so fixed, is the extremity of the
axis of the plane of the impressed couple, or of the plane known to
mathematicians as the invariable plane of the motion.

Vlll INTRODUCTION TO THE SECOND VOLUME.

But a still clearer idea of the motion of such a body may be
formed by the aid of another theorem, which shows that the whole
motion of a revolving body may be represented by a cone which
rolls, without sliding, on a fixed plane passing through its vertex,
while this plane revolves with a uniform motion round its own
axis. This, perhaps, is the simplest conception we can form of a
revolving body. Now the principal axes of this cone, and its focal
lines, depend on the constitution and form of the body, or, in other
words, are functions of the moments of inertia round the principal
axes ; while the initial position of the plane of the impressed couple
in the body will determine the tangent plane to this cone. But
when the two focal lines of a cone, and a tangent plane to it, are
given, the cone may as completely be determined as a conic section
when its foci and a tangent to it are given. Nothing more simple
than this conception : a cone rigidly connected with the body, the
position of whose focal lines, and whose principal vertical angles,
depend on the form and constitution of the body, revolves without
sliding on a plane, while the plane itself revolves uniformly round
its own axis. We may also observe, that when the plane of the
impressed couple passes through one of the focals of the rolling
cone the motion is sui generis ; it no longer may be represented by
a rolling cone. The cone degenerates into a plane segment of a
circle, which swings round one or other of the cyclic axes of the
ellipsoid of moments, these cyclic axes being the boundaries of the
circular segment.

Although it may be, and doubtless is, very satisfactory in this
way to be enabled to place before our eyes, so to speak, the very
actual motion of the revolving body, yet it is not on such grounds
solely that the following essay has been published. Were the theory
of no other use than to give strength and clearness to vague and
obscure notions on this confessedly most difficult subject, enough
had been already accomplished by the celebrated geometer whose
name is so deservedly associated with this subject. It is as a
method of investigation that it must rest its claims to the notice of
mathematicians — as a means of giving simple and elegant inter-
pretations of those definite integrals, on the evaluation of which
the dynamical state of a body at any epoch can alone be ascer-
tained. If the author has to any degree succeeded in accomplish-
ing this, it is because he has drawn largely upon the properties of
lines and surfaces of the second order, and of those curve lines in
which these surfaces intersect. If he has been enabled to advance
any thing new, it is solely owing to the somewhat unfrequented
path he has pursued. That it was antecedently probable such
might lead to undiscovered truths, no one conversant with the
applications of mathematical conceptions to the discussions of those
sciences will deny. To introduce auxiliary surfaces into the dis-

i\ i uoi»i rno\ ro i n i. si:eo\i> \ oi.r M i:. ix

ciissions and investigations of physical science is an idea no less
luminous than it has been successful. The properties of such sur-
faces often aid our conceptions or facilitate our calculations in ;i
very remarkable manner. M. Dupin, for example, reduces the
problem of the equilibrium of a floating body to that of a solid
resting on a horizontal plane, the solid being the envelope of all
the planes which retrench from the floating body a given volume.
We have a still more striking instance in the wave-theory of light.
Therein we find the surface of elasticity the equimomental surface in
the theory of rotation. Few indeed there are among mathema-
ticians who require to be informed how the wave-surface of Fresnel,
and its reciprocal polar, the surface of wave-slowness of Sir William
R. Hamilton, have served to clear our conceptions on a subject as
yet scarcely understood, to realize and embody an indistinct and
shadowy theory. Nay, more, the geometrical properties of the
surface of wave-slowness in the hands of Sir W. Rowan Hamilton
have led to the anticipation of the theory of conical refraction.
They have enabled us to invert the natural order of induction and
to place theory in advance of experiment. Were further illustration
needed, one might refer with confidence to the treatise of Maclauriu
on the figure of the earth, to the researches of Clairaut on the
same subject, and to the investigations of Poisson, C basics, and
Ivory on the attraction of ellipsoids. A theorem in surfaces of the
second order, on which he has bestowed his name, enabled Ivory to
evade the difficulties of the problem on which he was engaged. So
true is the fine anticipation of Bacon : — Ct For as Physicall know-
ledge daily growes up, and new Actioms of nature are disclosed ;
there will be a necessity of new Mathematique inventions"*.

The author has taken occasion, during the progress of the essay,
to derive those partial solutions on particular hypotheses, whieh
are given in the usual text-books on this portion of dynamical
science. To the reader familiar with those solutions it will, doubt-
less, be satisfactory to see tHern follow, as simple conclusions, from
principles more widely general. These partial solutions serve, as
it were, to test the truth and accuracy of the principles on which
the entire theory is based. To those who read the subject as a
portion of academical study, it will, no doubt, prove interesting to
discover an additional link connecting the deductions of pure
thought with the laws of matter and motion. They will not fail
to observe the analogy, that as the properties of the sections of a
cone by a plane have elucidated the motions of translation of the
planets in their orbits, so likewise the theory of the rotation of
those bodies, round their axes, may be founded on the properties
of the sections of a cone by a sphere.

* Of the Advancement of Learning, book iii. chap. 6.
VOL. 11. b

X INTRODUCTION TO THE SECOND VOLUME.

As introductory to the treatise on conies, I have given an essay-
on what may be called the higher geometry on a plane. This
embraces the theory of transversals, invented and developed by
Carnot, and the principles of harmonic and anharmoriic ratio, a
powerful instrument in the able hands of Chasles. The properties
of triangles with reference to inscribed and circumscribed circles,
the properties of orthocentres and of orthocentral triangles, the
remarkable theory of the nine-point circle, and of the excentral
triangles connected with it are also fully developed. In this old
and seemingly worn-out subject the reader will yet find something
new.

The substance of the following essay was read before the Royal
Irish Academy, nearly forty years ago (March 1837)*. It has
lain by me unpublished ever since. 1 have been strongly recom-
mended to add it to this volume by a friend of mathematical
attainments of a very high order to whom I had shown this essay
(Mr. W. J. C. Miller, Mathematical Editor of the ( Educational
Times,' and Registrar of the General Medical Council, to whom I
am much indebted for his judicious advice and suggestions in this
portion of the volume, and also for the care and accuracy which he
has bestowed on the correction of the press). The shortness and
simplicity of the demonstrations encouraged me to submit those
propositions to geometers, few of them requiring any more know-
ledge than that of the simplest propositions of Euclid.

It may be objected to the method developed in the following
pages that all the properties of the conic sections are derived almost
exclusively from those of the right cone. In reply to this objec-
tion, it may be observed that the object is not to investigate the
properties of cones or other surfaces of the second order, but those
only of plane curves ; that the right cone is used as a simpler and
more powerful instrument of discovery than the oblique cone ; and
that any argument for deriving those properties from this latter

* The Secretary communicated the substance of a paper "On the Conic
Sections," by James Booth, Esq.

The methods hitherto adopted in deducing the central and focal properties of
the conic sections from arbitrary definitions having appeared to the author
defective in geometrical elegance, he has endeavoured in this paper to derive
them from a new definition.

If two spheres be inscribed in a right cone touching the plane of a conic
section, the points of contact are called foci.

The property from which the definition of a focus here given is derived,
though known for some time, has not been hitherto applied" further than to
show that this point is identical with the focus as usually defined.

By the help of the above definition, and of the simplest elementary principles,
the central and focal properties already known have been deduced, generally in
one or two steps, and several new theorems have been likewise discovered in the
development of the method.— Extract from the Proceedings of the Royal Irish
Academy, March 16, 1837.

IN I K01)UC'T10.V TO THE SECOND VOLUME. XI

surfacr would be equally applicable in favour of deducing them
from any other suitable surface of the second order. Besides, any
conic section being given on a plane, a right cone of which it may
be considered a section, can always be constructed. The mere
extension to the oblique cone is too trivial, when compared with the
number of other surfaces of the second order having like properties,
to merit any special attention*.

The right cone with a circular base is selected in preference to
any other surface, because the properties of its plane sections, hence
called conic sections, may be derived with more clearness, brevity,
and simplicity, than those of like sections in any other surface. It
must be borne in mind that the surface is used simply as a means
or instrument to obtain the properties of its plane sections ; and
these can be deduced from the right circular cone with greater
facility than from any otherf.

The prolix difFuseness of most of the treatises' on this subject,
the interminable series of proportionals which cumber every page,
and the tcdiousness of the demonstrations follow from the fact that,
as soon as the cone had afforded one or two principal properties
of its sections, these have been selected as definitions of the sections,
and the attempt is made, often with much ingenuity, to base a
wide and general system of these curves on the apex of one narrow
definition J.

* La construction que nous veaons de donner des foyers des coniques, prises
dans le cone oblique, ne se prete pas a la demonstration des proprietes de ces points,
et n'est pas propre meme a iudiquer a priori leur existence dans les coniques. II
reste done a recnercher comment, par la consideration des coniques dans le cone,
on peut etre conduit a la decouverte de leurs foyers. — CHASLES, Aperqu, p. 286.

t Les Anciens avaient considere les sections coniques dans le c6ne, mais seule-
ment pour en concevoir la generation et en demontrer quelques proprietes prin-
cipales, et faire servir ensuite ces proprietes primitives a la recherche, et a la
demonstration de toutes les autres : de sorte qu'ils formaient leur theorie des
coniques sans connaitre la nature ni aucune propriete du cone, et independamment
de celles du cercle qui lui sert de'base. — CHASLES, Apcrpi, p. 119.

\ Nous dirons, en passant, qu'outre la methode des Auciens et celle adoptee
par De la Hire, nous en concevons une troisieme qui n'a point ete employee, et
qui cut ete propre pourtant, si nous ne nous abusons, a sirnplifier extreinement les
demonstrations, et a mettre dans tout leur jour les principes et la veritable origins
des diverses proprietes des couiques : sous ce rapport, on ne pent se dissimuler
que la methode des Anciens n'onrait qu'obscurite.

Cette methode eut consist^ a etudier les proprietes du cone lui-meme, et a les
forniuler, independamment et abstraction faite de celles des coniques ; et cellea-
ci se seraient deduites des premieres avec une facilite et une generalite ravissantes.
On le concevra sans peine, car partout ou les Anciens employaient trois demon-
strations differentes pour demontrer la meme propriete dans les trois sections
coniques, ellipse, hyperbole et parabole, parce qinls s'appuyaient sur les caractercs
particuliers a chacuue de ces courbes, une seule demonstration suffira pour do"-
raontrer, dans le cone meme la proprieie analogue, d'ou celles des trois coniquea
doivent se deduire comme de leur vraie et commune origine.

Pe cette maniere, on erit vu prmdre naissance dans le cone a plusieur? pro-

ill INTRODUCTION TO THE SECOND VOLUME.

Thus if we were to assume the determining ratio, so simply
established in the following treatise, as the basis of a system of
conic sections, we should follow that adopted by Boscovich, Walker,
Sir John Leslie, and others, in their several treatises on this
subject. The numerous books compiled for the use of the Uni-
versities start from the same definition. De la Hire suggested as a
fundamental definition of a system of conies the constancy of the
sum or difference of the focal vectors to any point on the conic.

But a much more fertile property was derived by Dr. Hugh
Hamilton, author of a treatise of conic sections published in 1758,
and very celebrated in its day. He shows that if two fixed lines
be drawn, and two other intersecting lines parallel to them, but
variable in position and cutting the cone, the ratio of the rectangles
under their segments is constant, and independent of their position,
subject only to the condition that they remain parallel to the two
fixed lines given in position. This is perhaps the most general
property of the cone with reference to the properties of its several
plane sections. But Dr. Hamilton's anxiety to abandon the cone
and to arrive as speedily as possible at those theorems which relate
to the foci, directrices, and centres, led him into a course of inves-
tigation but little calculated to exhibit the peculiar advantages of
the basis he had chosen*.

The definition of a focus, on which this treatise chiefly rests, is
derived from a beautiful theorem discovered a few years since by
MM. Quetelet and Dandelin, first published in 1822.

It follows indeed so obviously from prop. 37, lib. ii. of Hamilton's
Conic Sections, that one is at a loss to understand how this acute
and original geometer failed to discover it. The wonder is how he
missed stumbling over it, as it lay so obviously in his way ; and none
of his readers has since supplied the omission.

Although largely to augment the number of general and remark-
able properties of those curves which have been brought to light by
the continuous labours of accomplished geometers in successive
ages maybe considered very arduous, (as I wrote in 1837), yet it is
hoped that several new theorems, especially those on the curvature
of these sections, derived from the properties of the cone, will not
be found elsewhere.

prie"tes des coniques, telles que celle des foyers, qu'il semble qu'Apollonius ait
devinee ; et que ce geometre, ni aucun de ceux qui 1'ont suivi, n'ont rattache'e ni
aux proprietes du cercle, ni a celles du cone ; de sorte que 1'origine premiere de
ces points singuliers, celle qui ne participe que de la nature du cone ou la courbe
prend naissance, est reste"e ignoree. — CHASLES, Aperqu, p. 121.

* Quoniam Apollonius omnia fere conicorum demonstrata conatusestin planum
redigere, antiquioribus insignior : neglecta conorum descriptione, et aliunde
quserens arguments, cogitur perssepe obscurius et indirecte demonstrare id, quod
contemplando solidae figures sectionem apertius et brevius demonstratur. — D.
Francisci Maurolici opera Mathematica, p. 280. See CHASLES, Aperqu, p. 120.

INTRODUCTION TO THE SECOND VOLUME. X1I1

The properties of conic sections may be divided into two distinct
das-rs, the angular and the metrical. The former will be found
chiefly to depend on the focal properties of the sections developed
from the definition of the foci as the points of contact of the plane
of the section with spheres inscribed in the cone, while the latter
will be more easily established by the methods of harmonic lines
and planes. The definition of a centre is founded on the properties
of harmonic pencils. Thus the two classes of properties are quite
distinct. The shortness and simplicity of the demonstrations prove
that these two principles, the definitions of the foci and the centres
of these curves, afford the true key to their investigation.

In most modern treatises on this subject, the three sections are
treated independently, as if they had no common genesis, and the
demonstrations rest, not on geometrical constructions, but on
endless rows of tedious and repulsive proportionals. In the fol-
lowing pages an attempt is made to derive the cardinal properties
of those celebrated curves from their common origin, the cone,
independently of any arbitrary definition. Some of those pro-
perties, and these amongst the most important, which are com-
monly established by the tedious processes of a disguised algebra,
come out at once clear and self-evident from mere inspection.
When those leading theorems are once established for conies in
general, it becomes a matter of the utmost facility to apply them
to the investigation and discussion of theorems and problems of a
less general character on a plane.

There is also to be observed in some of those treatises a puerile
affectation of geometrical rigour, in rejecting the use of such
abbreviations as sin, cos, tan, so generally used in mathematical
works to denote certain constantly occurring ratios. One is at a
loss to understand how the force of a demonstration is augmented
by using instead of sin A the circumlocution " In the right-
angled triangle ABC the ratio of the perpendicular BC to the
hypotenuse BA." This notation, borrowed from trigonometry,
wherever it is adopted, gives a singular clearness and brevity to
the demonstrations. And again, it is difficult to imagine in what
respect it is less rigorous to say a than the straight line AB.

The reader's attention is specially directed to Chapter XXIX.,
in which the radius of curvature of conies is derived directly from
the right cone, without the help either of the Differential Calculus,
or of Infinitesimals or of any other such device. I am not aware
that any attempt has ever been made to obtain the curvature of a
conic directly from the cone whereof it forms a section.

There cannot be a more powerful help to develop that faculty of
the mind which may be called geometrical imagination, that power
to place clearly before the mind's eye the several positions which
planes, lines, and surfaces assume as they intersect in space, than

XIV INTRODUCTION TO THE SECOND VOLUME.

the contemplation of those curves considered as the intersections
of planes and surfaces. In no science is this power of clear and
steady conception so necessary as in Astronomy and Mechanics.

It is worthy of remark that solid geometry as it is called, or a
reference to space of three dimensions, facilitates very often, and
that too in a striking manner, the proofs of theorems concerning
figures on a plane. A signal example of this will be seen in the
simple proofs of the principal properties of conies established by
the help of the right cone.

The object which the author has proposed to himself in the fol-
lowing pages is not so much to use a single method in the solution
of a cloud of problems arid theorems, many of them remarkable
only for their intricacy, but to apply a variety of methods to the
discussion of a class of selected properties, and to show that while
some questions yield with ease to one method, they are almost
insoluble by another.

Thus in some instances several demonstrations will be found for
the same theorem. It is of far greater importance, and will give a
wider grasp of the subject, to contrast and compare different methods
when applied to the investigation of the same theorem. The student
will then perceive that every method has something inherent to
recommend it, and that the method which in one case will give a
simple and easy demonstration, will afford obscure and complicated
results in other cases apparently not more difficult.

For this reason I have been more solicitous to develop a variety
of methods than to follow out some one selected principle into all
its details. It is no doubt a test of ingenuity and mathematical
ability to be able to build up an imposing structure of mathematical
demonstration based upon one fundamental principle alone. But
this apparent simplicity is found often to lead to long calculations and
complicated results in the development of the principle assumed.

To the well-informed reader it will be evident that the modern
methods of geometrical investigation which in recent times have
been applied to the development of geometry have to a great extent
superseded the old. In the geometry of the Greeks, the demon-
strations were partial, often requiring a separate proof for every
modification of figure. Some one property (as in the conic sections
for example) was made the basis of a superstructure erected with
infinite ingenuity and matchless skill, but often tedious, compli-
cated, and involved, owing to the narrowness and remoteness of
the definition.

It has been well observed by a very profound mathematician and
elegant writer, that when a subject is contemplated from a true
point of view it may be explained in a few words to a passenger in
the street*. As disjointed limbs and broken fragments (confused

* Nous ajouterons avec un des g^ometres inodernes qui ont le plus m6dit6 snr

i \ i Koin 'i THIN in mi. SKCOND VOLUME. xv

images) when viewed from the focus of a conical mirror range
themselves in symmetrical order and assume definite forms, so it is
with the truths of science ; confused, isolated, and indistinct they
remain until their true stand-point of view be taken.

The aim and scope of the modern geometry widely transcend
the limits which ancient science imposed on itself, while the tradi-
tional reverence in which those old methods were held was long an
obstacle to the development of physical and mathematical know-
ledge*. AVe have no just reason, however, to be surprised at this
superstitious veneration for the great works and mighty genius of
antiquity. Strange indeed had it been otherwise. It is sometimes
said that we do not retain that traditional reverence for antiquity,
that veneration for great names, which distinguished the pro-
moters of intellectual advancement at the birth of modern civi-
lization— that we no longer feel that exclusive admiration for the
literature and science of Greece and Rome, which, three or four
centuries ago, was a marked characteristic of every one who pro-
fessed to cultivate them. Now this veneration for ancient wisdom
is founded on a fallacious analogy. The young naturally confide
in the experience and knowledge of the old ; and as the old have
preceded them in point of time, we are led by the seeming analogy
to look upon the early life of the world as its old age instead of
its youth. Lord Bacon, in his Advancement of Learning, says,
" certainly our times are the ancient times when the world is
now ancient, and not those which we count ancient, ordine retro-
grade, by a computation backward from our own times." Again,
an exaggerated admiration of antiquity, and a sort of longing
regret for times passed away, are by no means hopeful signs of a
present healthy progress. It has sometimes been remarked of those
who can boast a long line of ancestors, and yet have degenerated

la philosophic des mathe"inatiques, " qu'ou ne peut se flatter d'avoir le dernier
mot d'une the"orie, tant qu'on nia peut pas 1'expliquer en peu de paroles a un
passant dans la rue."

Et en etfet, les ve'rite's grandes et primitives, dont toutes les autres derivent,
et qui sont les vraies hases de la science, ont toujours pour attrihut caracte"risque
la simplicity et 1'intuition. — CHASLES, Aperpi, p. 115.

* Si pre"sentement on me demande mon opinion sur la ge'ome'trie pure, je
demanderai a mon tour de faire une distinction s'agit-il de la ge'ome'trie a'Archi-
mede, d'Euclide, d'Apollouius, et de tous ceux d'entre les modernes qui, comme
Viviani, Halley, Viete et Fermat, ont marche* sur leurs traces? J'avouerai
franchement, quelque opinion que 1'on puisse en prendre de moi, que je n'en suis
pas enthousiaste. Que si, au contraire, on veut parler de cette ge'ome'trie qui,
ne'e, pour ainsi dire, des meditations de 1'illustre Monge, a fait de si immensea
progres entre les mains de ses nombreux disciples, on me trouvera toujours dis-
pose" a lui rendre le plus gclatant homtuage, et a reconnaitre qu'elle nous a fait
de'couvrir en vingt anne"es plus de proprie'te's de l'e"tendue qu on n'en avait pu
de"couvrir dans IPS vingt suNcles qui les avaient pre'ce'de'es. — Annalex <le Matfit-
matiqnp, torn. viii. p. 169.

XVI INTRODUCTION TO THE SECOND VOLUME.

in the descent, that they were satisfied to base their claims to con-
sideration, not on the grounds of personal merit, but on the great-
ness of those who had gone before them. The same is as true of
nations as of individuals. Diodorus and Plutarch, by their extra-
vagant eulogies of the extinct republics and legendary heroes of
antiquity, tried to console themselves for the degeneracy of the
times in which they wrote. By their enthusiastic admiration
of forms of government that had been abolished, they indirectly
censured the enormities of the grinding despotisms under which
they could scarcely call even their lives their own ; and the language
in which they lauded the liberties they had lost was the surest
index of the slavery under which they groaned. The same tone of
saddened retrospection breathes through the fine preface of Livy's
immortal history.

But, independently of these considerations, there is a legitimate
cause and weighty reason for this profound admiration of antiquity.
Let us in imagination go back to the year 1500 of our era, or
thereabouts ; let us imagine a man somewhere in the south of
Europe, or in one of the Greek cities of the lesser Asia, within sight
of that purple sea, beyond whose sunny shores civilization had
never yet been able to advance. Let us further suppose him to be
profoundly versed in all human learning, and acquainted with every
cardinal event in man's history. What are the reflections that
would naturally arise in the mind of so accomplished and philo-
sophical a spectator taking a comprehensive view of the annals of
mankind, and of the progress of civilization from its earliest
recorded dawn down to his own time?

He would have seen all human knowledge either absolutely sta-
tionary or actually retrograding. He would have seen that the
mathematical science of his own day had not made a single step in
advance during the long period of 1700 years, from the state in
which it was left by Archimedes and Euclid and Apollonius ; for
the Roman civilization throughout its long duration never produced
even a fifth-rate mathematician. He would have seen that since
the days of Hippocrates and Galen the science of medicine had dege-
nerated into a mere empirical art ; that there was no body of laws
worthy of the name but the Roman codes ; that alchemy flourished,
for chemistry was not yet ; that astrology had displaced the little
astronomy that was known ; that there was absolutely no such
thing as physical science ; that the multitudinous facts of natural
history had yet to be observed and noted, excepting those only
investigated by Aristotle, that most profound and accurate physicist ;
that in poetry, oratory, architecture, and the kindred arts of painting
and sculpture, the ancients transcended rivalry or even successful
imitation ; in short, that the whole sum of human knowledge, scant
as it was, had continued without augmentation or accession during

INTRODUCTION TO TUB SECOND \ol.l\ll . XVII

lit'tivii long centuries of man's eventful history; that the acutest
\\ii^ and the most subtle intellects were forced to move round and
round in the same dull mill-circle, and thresh the straw that had
bern threshed a thousand times before; that the profoundest
thinkers failed to make even the shallowest discovery either in
science or in art ; that the most learned men occupied themselves,
century after century, in piling up pyramids of commentaries on
those wondrous men Aristotle and Plato, who, like the Pillars of
Hercules in the old mythology, separated the clear, the definite,
the settled, and the known from the dark, the vague, the boundless,
and the obscure, — when, moreover, our supposed inquirer, con-
tinuing his survey, would have learned that whole regions of the
earth's surface were passing clean out of the knowledge of civilized
man, that the ideas which learned professors and adventurous
travellers formed about countries not far remote were vague and
contradictory, that less was known four centuries ago about the
geography of the world and the relative magnitudes and positions of
the several regions thereof than in the times of Scylax, Herodotus,
Strabo, Ptolemy, or even Agatharchides, that the knowledge of
many fine inventions and curious processes in the arts had actually
perished (and has never to this day been rediscovered) — when, in
addition to this, looking to the political aspects of the world, he would
have seen the very fairest and most hallowed regions of the earth's
surface overrun by the wild fanatics of Arabia, or trodden down by
the savage hordes of Turkestan, who with unbroken front were
advancing like the ocean tide rushing up an estuary, to overwhelm
under one undistinguishing flood every monument and every insti-
tution that survived of the ancient civilization (even now who
shall truly say that the liberties of the west and the civilization of
our own time, beginning to show symptoms of early decline and
marks of premature decay, are entirely beyond the reach of the
ever advancing wave of Russian despotism, urged onwards by the
barbarous hordes of the deserts of Eastern Asia ?) — and when, lastly,
to such an ideal spectator, reviewing the history of man's progress
upon earth, that great renovating institution the Church, would
have been presented to his view, not as the living, breathing incar-
nation of the Gospel, giving health and vigour to the nations of
antiquity worn out and effete, but like Niobe of old petrified into
stone, and becoming herself a huge stumblingblock in the way of
progress, a rock of offence to those who saw not that her corrup-
tions and errors were, in some measure at least, due to the evil days
through which she had had to pass.

Nor from such a retrospect could our spectator have drawn, with
regard to the future, other than the most desponding anticipations.
No man could foresee that as the night is darkest before the dawn,
so out of this dense moral night and deep darkness of the human

VOL. II. C

XV111 INTRODUCTION TO THE SECOND VOLUME.

understanding a new order of things was soon to arise, and the light
of a higher and better civilization to gladden mankind. It is no
wonder then that men, looking back through the vista of a length-
ened period of time, and seeing that every thing that was worth pre-
serving in literature, science, and art — whether it be poetry, oratory,
or the drama — whether it be architecture, sculpture, or painting,
was the creation of comparatively a small number of gifted minds
and the birth of a few remote centuries, it is no wonder that men
in those days held the deep conviction that nearly every thing that
could be known was already discovered. In fact they had a special
name for it. They called it the " omne scibile." They called it
not "omnis Scientia/' but "omne scibile," not merely every thing
that was known, but every thing that could be known. It is not
strange, then, that a feeling of admiration apparently akin to hero-
worship should have been felt for those who at a bound had reached
the limits and touched the very outer verge of knowledge attain-
able by man.

It is generally assumed, as an assertion not admitting of dispute,
that the origin of the present methods of physical investigation is
due to Bacon, and that an outline of those methods may be traced
throughout his works, more especially in the ' Novum Organum/
the ' instauratio Magna/ and the ' De Augmentis Scientiarum.'
It requires some hardihood to call in question such an established
opinion ; yet, to one who, free from prejudices and preconceived
notions, shall carefully read those works, it will be abundantly
evident that Bacon's great merit lay in giving form and pressure
to the accepted modes of thought of his own time. His chief object
seems to have been to denounce authority, to set at naught anti-
quity, to undervalue ancient philosophers and their theories, to
prove that 110 natural knowledge could be established by their
methods of procedure, and that the ancient syllogism was an im-
potent instrument of investigation. Now this was the very spirit
of Bacon's age. Human authority had already been denounced in
Ecclesiastical affairs ; and the fruit of this was the Reformation.
The authority of Aristotle and the old Greek philosophers was
questioned ; and a general scepsis identified was the result. In
politics this denial of human supremacy led to the great rebellion
of 1641. Bacon deserves the credit of realizing the spirit of his
own times, which was intensely sceptical. He first snowed that
all advance in the natural sciences must be based on original and
independent inquiry, without reference to the theories of the old
philosophy.

A. very brief examination of Bacon's works would completely
establish this view. In the 84th aphorism of the first book of the
' Novum Organon ' he says " Reverence for antiquity has retarded
mankind, and thrown as it were a spell over them, and the autho-

l\ ruoni < llo\ I.) I III: NK, o.M) you \u . \;\

nty of men who \\erc held to be great in philosophy. It is a mark
of feebleness to yield every thing to ancient authors, and to deny
his supremacy to time; for truth is the daughter of time, not of
authority." He adds that "the present time is to be considered
as the ripe maturity of the world, with all our accumulated facts
and experiences, and not antiquity, which may rather be called the
ehi Id hood of mankind." In fact the whole tone and spirit of the
book is a powerful protest against the influence of authority in
matters of science.

It is often said that Bacon was opposed to the construction of
philosophical hypotheses. This is true in one sense, but not in
another. There are what may be called provisional, as well as
established theories. When Newton saw the historical apple fall
to the ground, and conjectured whether the moon might not itself
be a big apple, he made his calculations, assuming the law of gra-
vitation as his hypothesis. But when he found that, owing to an
erroneous estimate of the mass of the earth, then accepted by
astronomers as correct, his calculations did not confirm his theory,
he abandoned his hypothesis. Now this is an instance of a, provi-
sional hypothesis. When, some years afterwards, Newton obtained
a more correct value of the mass of the earth, he resumed his cal-
culations, established his theory, and thus turned his provisional
into an established hypothesis, which, for countless ages yet to
come, is likely to respond to the mechanism of the heavens.

Bacon agrees with Cousin that the syllogism does not investigate
first principles. This, however, nowise invalidates the use of logic.
It is not the business of logic to investigate first principles. In the
longest and most subtle demonstration there can be found nothing
in the conclusion that was not previously involved in the principles
assumed as the basis of the proof. In most physical inquiries —
it \\c except .Mathematical Astronomy and, perhaps, Optics — there
are but very few steps in the process of physical induction.

Bacon, however, was much more successful in the work of
destruction than in that of reconstruction. He could pull down ;
but he could not build up. The specimens of philosophical induc-
tion which he gives in the second book of the ' Novum Organon '
are most of them puerile, if not silly, and frequently contradict his
own principles. He equally fails in laying down the true goal and
just object to be kept in view in the cultivation of natural knowledge.
He holds up no higher standard than gross material utility. He
proposes to make men comfortable in their persons and dwellings.
This is a low standard ; it falls far below that of the old Greeks.
But some allowance must be made for him. He lived in a cold
ungenial clime, very different from the bright and sunny lands of
Attica. In the great object of his works — the subversion of the

XX INTRODUCTION TO THE SECOND VOLUME.

authority of the ancient philosophers, and the uprooting of all
reverence for antiquity — he has thoroughly succeeded ; and he
succeeded because he embodied the spirit of his age and cleared
the ground for those who were to follow.

The word science has in these latter days been divorced from its
original meaning, geometry and the creations of the pure intellect.
It is now appropriated to observations in natural history and to
experiments in chemistry. These subjects of research are no doubt
very interesting and valuable ; but they are not science in the original
and best sense of the word. Yet without a knowledge of mathe-
matics it is impossible to make any real advance in the discoveries
of physical science. Take the case of that great science Physical As-
tronomy, of which Sir J. Herschel says, " admission to its sanctuary
and to the privileges and feelings of a votary is only to be gained by
one means — sound and sufficient knowledge of mathematics, the
great instrument of all exact inquiry, without which no man can ever
make such advances in this or any other of the higher departments
of science as can entitle him to form an independent opinion on
any subject of discussion within their range/'

But, notwithstanding the concurrent testimony of the greatest
men of every age, it is in the mouths of many a very common
objection which leads them to ask, "What possible use can there be
in mathematics? how few are they to whom they can be of
the least utility in after life \" So it might with equal plausibility
be asked why practise running, leaping, or wrestling ? seeing that
very few become professed athletes. But just as athletic exercises
develop the muscles, improve the health, and invigorate the body,
so severe studies strengthen the understanding, form habits of
thinking, and deepen the grooves of thought, even though the
subjects of those studies be in the course of time wholly forgotten.
Like those old quarries we read of in Pentelicus or Paros, though
the blocks of marble, the material of the breathing bust or god-
like statue have gone, never more to return, yet the ruts of the
wheels that bore them, the grooves in the face of the rock along
which the guiding gear and cordage ran, are as fresh and as sharp
as if they had left off working only yesterday.

And nowhere is this low utilitarian sentiment more loudly
expressed than amongst those who have acquired such attainments
as they possess at our national Universities. Those persons pick up
just as much learning or science as may suit their purpose and help
them forward on the path of life they have selected. In fact,
learning and science are valued just as acquaintance with book-
keeping by double entry is valued, as a means to an end, and that
end by no means the noblest. To secure their approbation,
research must have a bearing on some useful practical money-

INTRODUCTION TO THE SECOND VOLUME. \\1

making object. This is in accord with the spirit of the age, a
spirit of pretence and vanity and sham*.

At this state- of things we ought not to feel any surprise. Our
Universities are no longer calm retreats for the encouragement of
patient and continuous thought expended on the development of
branches of science which do not promise an immediate ready-
money return ; they are now almost wholly engaged in conducting
the elementary education of the upper and middle classes of this
country. And hence it is that some of those who have most widely
extended the boundaries of knowledge are men who early abandoned
their college retreats, or have never been inside the portals of a
University college at all. Men, such as Thomas Simpson, and
Boole, and Davies, and Horner and others, not to speak of those
whom, as still alive, it might be invidious to mention, have had
the genial current of their souls frozen by a chill penury, or were
relegated to a dull oblivion, or at least to a passing obscurity, by
combinations of cliques, nowhere more general or more potent
than in the mathematical world. It would be a curious but
perhaps a bootless inquiry to discuss why, from the days of Apollo-
nius of Perga, called the great geometer, to our own, a characteristic
failing of mathematicians has always been envy.

The education of our own day tends to produce a dead level of
mediocrity. There will be few to note for crass ignorance, and
scarcely any to admire for profound learning. The age is so fast
that it cannot stop to think ; it cannot pause to ponder. Nay, more,
it cannot with common propriety express its own wants and wishes ;
for the " pure well of English undefiled " is rapidly turning into a
puddle of slang. If ridicule be a test of truth, as the author of the
Characteristics asserts it to be, we ought by this time to have reached
the very extreme limit of correct opinion. For every thing, now-
a-days, is treated in a spirit of mockery, levity, or contemptuous
indifference. That this happy result has not yet been obtained is
a proof of the fallacy of LORD SHAFTESBURY'S great discovery in
ethics. There will be, as in all human affairs, a reaction and a
change ; and men will once again follow the more excellent way.

Attempts are perseveringly made to remove the Elements of
Euclid from the high position which it has held for more than
two thousand years, of being unquestionably the best introduction to
geometry. It is assailed on the ground that it is too tedious, too
rigorous in its demonstrations, that it wants order, and is deficient
in symmetry. It is asserted that it is time such old-world notions

* At apud plerosque tantum abest, ut homines id sibi proponant, ut scieu-
tiiirum et artium niassa au<rmeutum obtineat ; ut ex ea, qiue pnesto est, mnssa
nil amplius sumaut aut quaerant, quam quantum ad usum professorium, aut
lucrum, aut existimationem, aut hujusmodi compendia convertere poasiut. —
BACON, Nov. Org. lib. i. Aph. 81.

XX11 INTRODUCTION TO THE SECOND VOLUME.

and methods were exploded, and that what we want now, is some
easy, handy compilation, on a level with the comprehension of most
people, which would commend itself by its practical utility in
meeting the passing needs of daily life ; and if such a short cut to
geometry be not rigorous in its demonstrations, what possible dif-
ference could it make to any one whether the proofs were real or
only seeming ?

But Euclid is not likely to be dethroned for some little time
longer. Not very long ago a Committee was appointed by a new
geometrical Society to draw up a syllabus of the elements of geometry
to supersede the tedious and repulsive work of Euclid. The Com-
mittee, which consisted of six members, was requested to draw up
a joint report 011 the subject. But, Quot homines tot sententia, six
different reports were sent in ! ! no two members so far agreeing
in their views as to unite in drawing up a joint report.

It is also, we are told, likely that the study of Greek in this
country will soon be given up, if not altogether, at least in a great
measure. This is a prospect even still darker ; for it implies a decline
in the cultivation of the finest language that has ever yet been spoken
on the earth, and a consequent degradation of the standard of that
learning by which a nation is ennobled.

It hardly needs to be said that I publish these volumes not only
without the expectation of reimbursement, but with the certainty
of heavy pecuniary loss. I can appeal to no University syndicate
to share my burden. It is perhaps right that for this act of indis-
cretion I should make an apology to the public, whose one sole
test of literary and scientific excellence is Will it pay ? That old-
world notion of working for work's sake is now utterly exploded,
not alone among the ignorant and the vulgar, in whom it might be
forgiven, but even amongst those who stand highest in the ranks
of science in our own day. How often do we hear such researches
stigmatized as unprofitable and vain ! Yet the great masters of
wisdom in every age have otherwise taught ; and I have followed
their teaching, not deterred by the conviction that abstract science
has become obsolete and stale. Many of those discoveries, the
fruit of a long and desultory life, I would not willingly let die.
Popularity as an author or reputation as a discoverer in science is
to me a matter of supreme indifference. Neither is it an object
with me of any importance to make money by the publication of
my discoveries, as I am fortunately placed above those needs which
sometimes press so heavily on many of the most illustrious culti-
vators of literature and science.

J. B.

Stone Vicarage,
New Year's Day, 1877.

TABLE OF CONTENTS.

[The numbers on the left hand denote the sections, the numbers on the
right hand the pages.]

CHAPTER I.

1.] On the general forms of elliptic integrals, table of thirteen distinct forms,
the types of curves, the symmetrical intersections of surfaces of the

second order 5

2.] On the spherical ellipse 8

6.] Rectification of a curve on the surface of a sphere 12

7.] Expression for the arc of a spherical ellipse 13

8.] Expression for the area of a spherical ellipse 15

9.] Relations of supplemental cones 16

10.] Arc of spherical ellipse determined by protective coordinates 18

11. J Another method of rectification 20

13.] Application of this method to the rectification of the spherical ellipse . . 23

15.] Legendre's theorem 26

17.] Another form of rectification 26

CHAPTER II.

20.] On the spherical parabola and its genesis. Properties thereof. An
elliptic integral of the Jlrst order represents an arc of the spherical

parabola 28

21, 22.] Properties of the spherical parabola 30

23.] Another method of rectification for the arc of the spherical parabola. . 33

24.] Lagrange's theorem 33

26.] Legendre's theorem 37

20, 27.] Comparison of formulae of rectification 38

28, 29.] Geometrical interpretation of the transformations of Lagrange .... 42

30.] On imaginary parameters 45

CHAPTER III.

31.] On spherical conic sections with reciprocal parameters 47

32, 33, 34.] Properties of spherical conies with reciprocal parameters 48

XXIV TABLE OF CONTENTS.

CHAPTER IV.

35.] On the logarithmic ellipse and its genesis ; rectification thereof 51

36.] Integration effected 54

37.] Simple expressions for the parameter, modulus, and constants of the

logarithmic ellipse 56

38.] Rectification of the logarithmic ellipse by another method 58

39.] Important property of the paraboloid 61

40.] Rectification continued. Legendre's formula 62

41.] The serniaxes of the elliptic base expressed in terms of the conjugate

parameters 64

42, 43.] Values of the arc in particular cases 65

44.] Particular case of the logarithmic form when the parameters are equal . 70

CHAPTER V.

45.] On the logarithmic hyperbola and its genesis j rectification thereof .... 76

46.] Rectification by another method 79

47.] Values of the semiaxes in terms of the parameters 80

48.] Comparison of expressions found for an arc of a logarithmic hyperbola . . 81
49.] Expression for the difference between the arc of a logarithmic hyperbola

and the corresponding arc of the tangent parabola 86

50.] On the rectification of the curve when the parameters are equal 88

51.] On the rectification of the arc of the logarithmic hyperbola when the

parameter I is infinite 91

52.] On the double rectification of the common hyperbola, analogous to that

of the logarithmic hyperbola 92

CHAPTER VI.

53.] On the values of complete elliptic integrals of the third order. Differ-
entiating under the sign of integration. Coefficients of the complete
elliptic integrals are themselves elliptic integrals. Legendre's formula

of verification, geometrical origin of 95

54.] Properties of inverse spherical ellipses — tests of accuracy 100

65.] On cyclic areas 103

66.] Geometrical representatives of the integrals of the first order in sec. [53] 106
57.] On the value of the complete elliptic integral of the third order and

logarithmic form 107

CHAPTER VII.

58.] On the logarithmic parabola. Genesis of it 110

59.] Different cases Ill

60,] On the curve of symmetrical intersection of an elliptic paraboloid by a

sphere 114

TABLE OF CONTENTS. XXV

CHAPTER VIII.

01.] On conjugate amplitudes and conjugate arcs of hyperconic sections.

The equation of conjugate amplitudes 110

62.] Equation between conjugate amplitudes of the first order . . ; 117

63.] Equation between conjugate amplitudes of the second order 118

64.] Equation between conjugate amplitudes of the third order 119

66.] Determination of the residuals 121

66.] Normal relations between conjugate amplitudes in the three orders . . 122

67.] On conjugate arcs of a spherical parabola 122

68.] Sum of the conjugate arcs of a spherical parabola equal to the sum of

the protangent circular arcs 123

60.] On conjugate arcs of a spherical ellipse 124

70.] Simple relation between the five protangent circular area 125

71.] On conjugate arcs of a logarithmic ellipse 120

CHAPTER IX.

72.] On the maximum protangent arcs of hyperconic sections 181

73.] On the maximum protangent arc in a spherical hyperconic section. . . . 132

74.] Geometrical construction for finding this arc 133

76.] On the maximum protangent arc in a logarithmic ellipse 135

76.] Geometrical proof that an elliptic integral of the third order whose
amplitude has a certain value may be expressed by integrals of the

first and second orders only 137

CHAPTER X.

77.] On derivative hyperconic sections 187

78.] Expressions for successive derivative spherical hyperconic sections . . 140

79.] Relations between the successive moduli and parameters 143

80.] Analogous expressions found for derivative logarithmic ellipses 144

81.] Relations between the successive parameters. Remarkable relation

between elliptic integrals of the third order and their derivatives . . 140
83.] New classification of elliptic integrals. On inverse functions. Recti-
fication of certain plane curves may be effected by elliptic integrals
of the third order. Protective properties of hyperconic sections.
Examples 161

CHAPTER XI.

84.] On the quadrature of the logarithmic ellipse and the logarithmic

hyperbola. To find the area of the logarithmic ellipse 166

85.] To find the area of the logarithmic hyperbola 169

VOL. II. d

XXVI TABLE OP CONTENTS.

CHAPTER XII.

86.] On the rectification of lenmiscates 102

87.] On the hyperbolic lemniscate 164

CHAPTEB, XIII.

88.] Application of the foregoing theory to the problem of the rotation of a
rigid body round a fixed point. General statement, fundamental
formulae. Moments of inertia of a rigid body defined and determined 170

89.] Tangential equation of a surface of the second order 172

90.] Determination by this method of the axis of figure of an ellipsoid . . 173

91.] Definition of instantaneous axis of rotation 175

92.] Determination of the equations of this axis 175

93.] Angular velocity round this axis 170

94.] Formulae for the determination of the velocities of a single particle

parallel to the axes of coordinates 177

95.] These formulae extended to the entire body 178

96.] The tangential coordinates represent the angular velocities round the

axes of any rectangular system. Instantaneous plane of rotation . . 178

97.] Centrifugal forces. Theorems on centrifugal couples 179

98.] Relations between the centrifugal couple and centrifugal triangle ;
simplification of the preceding formulas when the axes of coordinates
coincide with the axes of the ellipsoid ; simple expression for the

centrifugal couple 180

99.] The instantaneous axis of rotation coincides with a perpendicular from

the centre of the ellipsoid on a tangent plane. The angular velocity

round this axis is inversely proportional to this perpendicular .... 182

100.] During the rotation the diameter of the ellipsoid perpendicular to the

plane of the impressed couple is invariable. The surface of the

ellipsoid always passes through a fixed point in space 183

101.] The angular velocity round the instantaneous axis of rotation varies
inversely as the perpendicular from the centre of the ellipsoid on the
instantaneous plane of rotation. The square of the angular velocity
round the instantaneous axis of rotation varies as the area of the
diametral section of the ellipsoid perpendicular to this axis. The
angular velocity round the axis of the impressed couple is constant
during the motion. The centrifugal couple varies as the tangent of
the angle between the instantaneous axis of rotation and the axis of
the impressed couple. The velocity of the vertex of the axis of the
impressed couple along the surface of the ellipsoid varies as the

tangent of this angle 184

102.] Expressions for the values of this velocity resolved parallel to the

principal axes 185

103.] The axis of rotation due to the centrifugal forces lies in the plane of

the impressed couple 186

104.] Instantaneous axis of rotation due to the centrifugal couple 188

105.] Component of the angular velocity due to the centrifugal couple .... 189
106.] Investigation of the lengths of the axis of the centrifugal couple and
of the instantaneous axis of rotation due to that couple. Expression
for the angle between the axes of rotation due to the impressed and
centrifugal couples 189

TABLE OP CONTENTS. XXV11

CHAPTER XIV.

107.] On the cones described by the several axes during the motion of the

1 >.nlv. Investigation of the locus of k the axis of the impressed couple 101

108.] Of the cone described by the instantaneous axis of rotation 102

109.] Of the cone described by the axis w of the centrifugal couple 192

110.] Of the cone described by the axis of rotation due to the centrifugal

couple 103

111.] The plaues of the circular sections of the invariable cone coincide with

tli«' planes of the circular sections of the ellipsoid 104

112, 113.] Some general theorems on rotatory motion 195

CHAPTER XV.

114.] Determination of the time by means of an elliptic integral of the first

order 198

115.] Geometrical interpretation of the modulus of this function 200

116.] Geometrical interpretation of the amplitude 201

117.] Expressions for the coordinates of the vertex of the axis of the impressed

couple in terms of the time, and the constants of the invariable cone 202

118.] Determination of the angular velocity in terms of the time. Deter-
mination of the angles which the instantaneous axis of rotation
makes with the axes of coordinates in terms of the time 203

119.] The angle made by the line of the nodes is determined by an elliptic
integral of the third order and circular form. This integral repre-
sents a spherical ellipse supplemental to the spherical elliptic base
of the invariable cone 205

120.] Hence a method of representing rotatory motion by the motion of a

cone which rolls upon a plane revolving uniformly round its axis . . 208

121.] Determination of the angle between the axis of rotation and the line

of the nodes 209

122.] Determination of the angle between the lino of the nodes and the axis

« of the centrifugal couple 210

CHAPTER XVI.

123.] The body referred to axes fixed in space 211

124.] The area described by the axis of the centrifugal couple on the plane

of the impressed couple varies as the time 212

123.] Determination of the position of the instantaneous axis of rotation in

absolute space at auy epoch 214

120.] Determination of the angle %> which 6 the vector arc, drawn from the

vertex of k, makes with a fixed plane passing through k the axis of

the impressed couple 219

127.] Determination of the relation between the amplitudes 221

128.] The nutation of the instantaneous axis of rotation, and the angular

velocity round it, expressed in terms of the time 221

XXV111 TABLE OF CONTENTS,

CHAPTER XVII.

129.] The spherical spiral described by the pole of the instantaneous axis of
rotation on a fixed concentric sphere. Asymptotic circles of this
spiral 2'23

130.] The length of one undulation of this spiral is equal to a quadrant of

the spherical elliptic base of the cone of rotation . . . . ; 224

181.] Relation between the focal angles of the invariable cone, of the cone

of rotation, and of the cone of nutation 229

132.] When the ellipsoid is very nearly a sphere, the cone of rotation is
indefinitely greater than the cone of nutation. Relation hence
derived between the nutation of the axis of rotation of the earth
and the path of its pole in absolute space 230

133.] On the velocity of the pole of the instantaneous axis of rotation along

the spiral 231

CHAPTER XVIII.

134.] On the spirals described on the surface of an immovable sphere by
the three principal axes of the body during the motion, and on their
asymptotic circles 233

135.] Velocities of the poles of the principal axes 234

136 . . . .139.] The lengths of the spirals described by the greatest and the
least principal axes of the ellipsoid may be expressed by elliptic
integrals of the third order and logarithmic form. Values of the
lengths of the spirals in particular cases 235

140.] The spiral described by the mean principal axis may be rectified by

an elliptic integral of the third order and circular form 241

141.] On the velocities of the poles of the principal axes along their spirals 245

CHAPTER XIX.

142.] Investigation of the motion of rotation when the plane of the impressed
couple is at right angles to the plane of one of the circular sections.

The time may be expressed by a logarithm 245

143.] Determination of the angle between the axis of the impressed couple
and the instantaneous axis of rotation. Determination of the angle

made by the line of the nodes * 248

144.] Determination of the spiral described on an immovable sphere by the
pole of the instantaneous axis of rotation when k=b. Equation of

this spiral 250

145.] This spiral a species of rhumb-line 250

146.] Length of this spiral , 252

147.] Velocity of the pole along this spiral. When the axis of the impressed
couple coincides with the mean axis of the ellipsoid, the lengths of
the spirals described by the greatest and the least principal axes
may be expressed by logarithms 252

TABLE OF CONTENTS.

148.] When the plane of the impressed couple coincides with the plane of
one of the circular sections of the ellipsoid, the elliptic integral may
be reduced from the third order to the first. The elliptic integral
of the first order which determines the position of the axes of the
impressed couple in the body. The two elliptic integrals of the first
order which determine the motion may be expressed by arcs of the
same spherical parabola. The moduli are two successive terms of
Lagrange's modular scale 253

CHAPTER XX.

150.] On transversals 257

164.] Definitions of the orthocentric triangle and the orthocentre. Appli-
cations of the method of transversals . 260

CHAPTER XXI.

100.] On harmonic ratio. Definition of harmonic pencils 265

161.] Applications of the method. 266

166.] On anhannonic ratio 271

167.] Properties of anharmonic ratio ... 278

171. J The theorems of Pascal and Brianchon proved by anharmonic ratio. . 275

CHAPTER XXII.

173.] Definition of poles and polars 277

174.] Applications of the method of poles and polars 280

177.] Newton's theorem 283

180.] Maclauriu's theorem 287

CHAPTER XXIII.

On circles inscribed, exscribed, and circumscribed to a triangle. Defi-
nition of the circles of contact 288

181.] On the properties of these circles 289

185.] On the trigonometrical relations of the angles of a triangle 293

191.] On triangles inscribed in one circle and circumscribed to another. . . . 298

CHAPTER XXIV.

194] On the orthocentric triangle, and the properties thereof 299

196.] New expression for the area of a triangle 301

197.] Relations between a triangle and its orthocentric triangle 301

208.] On the median lines of a triangle, and the properties thereof 306

210.] On the properties of the centroid of a triangle 307

216.] On the properties of excentral triangles 311

XXX TABLE OF CONTENTS.

CHAPTER XXV.

222.] On the nine-point circle and the properties thereof 316

224.] On the triangles whose vertices are, three by three, the four centres of
the three exscribed and the inscribed circle. Definition of the prin-
cipal excentral triangle 318

231.] On the radical circles of a triangle , 321

234.] The nine-point circle touches the inscribed and the three exscribed

circles — several demonstrations of this theorem 323

CHAPTER XXVI.

243.] On some elementary properties of quadrilaterals 334

244.] On quadrilaterals inscribed in one circle and circumscribed about

another, remarkable theorems 335

253.] On the properties of chords drawn from a point in the circumference
of a circle to the angles of an inscribed regular polygon of an odd
number of sides 345

CHAPTER XXVII.

255.] On Conies. Definitions 347

256.] On the focal properties of conies 349

280.] Value of the semiparameter derived from the cone 369

283.] On the conospheroid, and its properties 371

CHAPTER XXVIII.

287.] On the central properties of conies 376

294.] On the hyperbola and its asymptotes. 382

CHAPTER XXIX.

On the curvature of the conic sections derived from the curvature of

the right cone 384

301.] MEUNIEB'S Theorem 385

302.] Values of the radii of curvature of the normal sections of a right cone 386
303.] Radius of curvature of a conic at a given point on the cone, whose

plane passes through a given tangent to the cone at this point .... 386
304.] Centre of the sphere of curvature for all the sections of the cone whose

planes pass through the same tangent to the cone 387

305.] Expression for the radius of curvature of a conic whose plane passes

through a given tangent to the cone 388

306.] Definition of a normal 390

308.] Definition of the central sphere. Theorems 391

CHAPTER XXX.

310.] On the properties of confocal conies derived from the right cone .... 392
312.] Theorems on confbcal conies 394

TABLE OP CONTENTS. XXXI

CHAPTER XXXI.

814.] On similar conic sections 806

810.] Some general theorems 397

CHAPTER XXXII.

817.] On conies in a plane 308

328.] On the eccentric anomaly in an ellipse 406

CHAPTER XXXIII.

880.] On orthogonal projection 408

841.] On divergent projection 418

SUPPLEMENTARY CHAPTER.

Appendix to the first volume, with notes and corrections 420

343.] Theorems of Euler and Cauchy on polyhedrons 420

344.] Pascal's theorem on hexagons inscribed in conies established by the

method of transversals 422

345.] Brianchon's theorem on hexagons circumscribed to conies established

by the method of poles and polars 423

340 . . . .354.] Theorems established by the method of tangential coordi-
nates 424

355.] General tangential equation of a surface of the second order, referred
to three rectangular axes in space, in terms of the coefficients of the
profective equation of this surface referred to the same axes 430

350 300.] Theorems established by the method of tangential coordi-
nates 433

301.] Extension of a principal focal property of conies 439

302.] Theorem in parabolic trigonometry 440

ERRATA IN THE FIRST VOLUME.

Note. — The number denotes the page ; 5 a. and 7 b. that the line in which the
error is found is the fifth line from the top or the seventh line from the bottom ;
and a numeral or letter within brackets, as (13) or (f), denotes a formula.

For Stibstitute

29. 22 a. point points

31. la. a: xt
19b. Place 2 before the square root.

32. la. BO -AC BO -AC,
69. 15 a. x— z =

114. lb. a'f+ftV aV+62£a

115. 2 a. (oy^+C&r)1 (ar)*+(iy)f
(g). a2£2+&V «

154. 9 a. a^+JV

287. (f)

ERRATA IN THE SECOND VOLUME.

For Substitute

14. 5 a. cos2*' cos2/3

37. (70). 1+J (l+j)

40. 5,

41. (90).
41.

43. 10 b. Vl— cos2y sin/x A/1— cos2 y sin2 /

81. (248). [n+m-2wzw2]

96. 27 a. [9] [7]

ON' THK

GEOMETRICAL PROPERTIES

OF

ELLIPTIC INTEGRALS,

INTRODUCTION.

IN publishing the following researches on the geometrical types of
elliptic integrals, I may be permitted briefly to advert to what had
already been effected in this department of geometrical research.
Legendre, to whom this important branch of mathematical science
owes so much, devised a plane curve whose rectification might be
effected by an elliptic integral of the first order. Since that time
many other geometers have followed his example, in contriving
similar curves, to represent, either by their quadrature or rectifi-
cation, elliptic functions. Of those who have been most successful
in devising curves which should possess the required properties,
may be mentioned M. Gudennann, M. Verhulst of Brussels, and
M. Serret of Paris. These geometers, however, have succeeded in
deriving from those curves scarcely any of the properties of elliptic
integrals, even the most elementary. This barrenness in results
was doubtless owing to the very artificial character of the genesis
of those curves, devised, as they were, solely to satisfy one condi-
tion only of the general problem*.

In 1841 a step was taken in the right direction. MM. Catalan
and Gudermann, in the journals of Liouville and Crelle, showed
how the arcs of spherical conic sections might be represented by
elliptic integrals of the third order and circular form. They did
not, however, extend their investigations to the case of elliptic in-

* Legendre a cherche" a repre"senter en ge'ne'ral, la fonction dig. (c, </>) par un
arc de coiirbe ; mais ses tentatives ne nous ont pas sembl£ heureuses, car il n'est
parvenu a r^soudre comple'tenient le probleme, qu'en employant une courbe
transcendante, dans laquelle I'arnplitude <f> et Tares ont entre eux une relation
ge'ome'trique encore plus difficile a saisir que dans la lemniscate. — VEBHULST,
Traitt den Functions Elliptiques, p. 295.

VOL. II. B

2 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

tegrals of the third order and logarithmic form ; nor even to that
of the first order. These cases still remained without any analo-
gous geometrical representative, a hlemish to the theory.

It will be shown in the following pages that the elliptic integral
of the first order, which is merely a particular case of the circular
form of the elliptic integral of the third order, represents a spherical
conic section whose principal arcs have a certain relation to each
other, and that the true geometrical representative of an elliptic
integral of the third order and logarithmic form, is the curve of
intersection of a right elliptic cylinder by a paraboloid of revolu-
tion having its axis coincident with that of the cylinder. The
geometrical representative of the peculiar form when the parameter
is negative and greater than 1, is shown to be a curve which I call
the Logarithmic hyperbola, and which may be thus generated. If
a right cylinder standing on a plane hyperbola as a base, be sub-
stituted for the elliptic cylinder, the curve of intersection may be
named the logarithmic hyperbola. It will have four infinite
branches, whose asymptotes will be the infinite arcs of two equal
plane parabolas. This curve, and not the spherical ellipse, is the
true analogue of the common hyperbola.

The main object of the following treatise is to prove, that Elliptic
Integrals of every order, the parameter taking any value whatever
between positive and negative infinity, represent the intersections of
surfaces of the second order.

To these curves may be given the appropriate name- of Hyper-
conic sections.

These surfaces divide themselves into two classes, of which the
sphere and the paraboloid of revolution are the respective types ;
from the one arise the circular functions, from the other the loga-
rithmic and exponential. The circular integral of the third order
is derived from the sphere, while the logarithmic function of the
same order is founded on the paraboloid of revolution.

Although in the following pages I have, for the sake of simplicity,
derived the properties of those curves, or of the integrals which
represent them, from the intersections of these normal surfaces
(the sphere and the paraboloid) with certain cylindrical surfaces,
yet the intersections so produced may be considered as the inter-
sections of these normal surfaces with various other surfaces of the
second order. Let U=0 be the equation of the sphere or parabo-
loid, and V=0 the equation of the cylinder. The simultaneous
equations U=0, V=0 give the equations of the curve of intersec-
tion. Let / be any abstract number whatever; then U+/V=0
is the equation of another surface of the second order passing
through the curve of intersection. Let U=0 be the equation of
a sphere, for example. Accordingly as we assign suitable values
to the number /, we may make the equation U+/V=0 repre-

ON mi: (.1 DM 1. 1 UK \i. I'KOIM:KTII:S or i.i.urnc INTEGRALS. 3

sent any central surface of the second order. 15tit \vc cannot, by
any snhst itution or rational transformation, make the equation
U+/V=0 represent a non-central surface instead of a central
one, or vice versd.

Although a remarkable relation exists between the areas and
lengths of some of these hyperconics, such as the circle and the
spherical ellipse, yet more distinctly to show the analogy which
pervades all those curves, I have not had recourse in any case to
the method of " elliptic quadratures/' as it is termed*. We can-
not admit such a violation of the law of geometrical continuity as
to suppose that while a function in one state represents a curve
line, in another, immediately succeeding, it must express an area.
Such can only be taken as a conventional explanation, until the real
one, characterized by the simplicity of truth, shall present itself.

In the course of these investigations, it will be shown that the
formulae for the comparison of elliptic integrals, which are given
by Legendre and other writers on this subject, follow simply as
geometrical inferences from the fundamental properties of these
curves, and that the ordinary conic sections are merely particular
cases of those more general curves above referred to under the
name of hyperconic sections.

It will doubtless appear not a little singular that the principal
properties of those functions, their classification, their transforma-
tions, the comparison of integrals of the third order with conju-
gate or reciprocal parameters, were all investigated and developed
before geometers had any idea of the true geometrical origin of
those functions. It is as if the formulae of trigonometry had been
derived from an algebraical definition, before the geometrical con-
ception of the circle had been admitted. As circular trigonometry
may be defined the development of the functions of circular arcs,
whether described on a plane or on the surface of a sphere, and
parabolic trigonometry f as the development of the relations which
exist between the arcs of a p'arabola, so this higher trigonometry,
or the theory of elliptic integrals, may best be interpreted as the
development of the relations which exist between the arcs of hy-
perconic sections.

* En conside"rant lea fonctions elliptiques comnie des secteurs, dont Tangle est
pre'cise'ment 6gal a 1'amplitude rf>, nous avons eu Tavantage de justifier la d«?no-
Miination d'aruplitude appliqutSe a Tangled) ; et meme celle de functions ellipti-
ques, en ge"ne"ral, puisque les courbes aJgeoriques par lesquelles nous avons re-
pre'sente' ces transcendantes, se construisent avec facilite" au moyen desnmms
vecteurs d'une ou de deux ellipses donne"es. — VERHULST, ' Traite des Fonctions
i;//i/>tiques,' p. 295.

M. Verhulst has represented the three kinds of elliptic integrals by means of
sectorial areas of certain curves. It is manifest, however, that it is incomparably
<-;t-i<T to do this than to represent these transcendents by means of the arcs of
curves. — R. L. ELLIS, Riyurt un the recent proyress of Analysis, p. 73.

t See Vol. I. page;',l:;

4 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Indeed it may with truth be asserted that nearly all the princi-
pal functions, on which the resources of analysis have chiefly heen
exhausted, whether they he circular, logarithmic, exponential or
elliptic, arise out of the solution of this one general problem, to
determine the length of an arc of a hyperconic section.

It may be said, we cannot by this method derive any properties
of elliptic integrals which may not algebraically be deduced from
the fundamental expressions appropriately assumed. But surely
no one will assert that the properties of curve lines should be alge-
braically developed without any reference to their geometrical
types.

We might, from algebraical expressions suitably chosen, derive
every known property of curve lines, without having in any in-
stance a conception of the geometrical types which they represent.
The theory of elliptic integrals was developed by a method the in-
verse of that pursued in establishing the formulae of common trigo-
nometry. In the latter case, the geometrical type was given — the
circle — to determine the algebraical relations of its arcs. In the
theory of elliptic integrals, the relations of the arcs of unknown
curves are given, to determine the curves themselves. This is
briefly the object of the present paper.

The true geometrical basis of this theory would doubtless long
since have been developed, had not geometers sought to discover
the types of those functions among plane curves. They were be-
guiled into this course by observing, that in one case — that of the
second order — the representative curve is obviously a plane ellipse.
Hence they were led by a seeming analogy to search for the types
of the other integrals among plane curves also.

I have attempted thus to place on its true geometrical basis a
somewhat abstruse department of analysis, and to clear up the ele-
mentary notions from which it may, with the utmost simplicity, be
developed. It is only in the maturity of a science that the rela-
tions which bind together its cardinal ideas become simplified. An
author, who has himself contributed much to the progress of
mathematical science, well observes, — " qu'il est bien rare qu'une
theorie sorte sous sa forme la plus simple des mains de son premier
auteur. Nous pensons qu'on sert peut-etre plus encore la science
en simplifiant, de la sorte, des theories deja connues, qu'en Fen-
richissant de theories nouvelles, et c'est la un sujet auquel on ne
saurait s'appliquer avec trop de soin." — GERGONNE, ' Annales des
Mathematiques,' torn. xix. p. 338.

It may be asked, of what use is the theory of elliptic integrals ?
This is a very natural inquiry in an age when every intellectual
acquisition, when every exercise of the understanding is tested by
its gross material utility. Yet it may suffice to say in reply, that
this theory will be found of use in many geometrical and physical

• IN Mil (. I oMKTKlCAL PROPERTIKS OF ELLIPTIC INTEGRALS. 5

inquiries. These I'liiietioiis not only exhibit the rectification and
quadrature of conic and hyperconic sections, but they subserve the
theories of the common and conical pendulums and of the elastic
curve. In Astronomy, the elements of the orbits of the planets,
the attraction of ellipsoids, and the problem of the rotation of a
solid body round a fixed point, receive their final and complete
solutions by the help of these integrals. M. Lame has proved how
questions \\hieh involve the distribution of heat and the nature of
isothermal surfaces may be reduced to tbe same functions.

In a subsequent portion of this volume, it will be shown that
the complete mathematical solution of that celebrated problem the
rotation of a solid body, has been for the first time obtained by the
aid of those functions in their state of complete development.

CHAPTER I.

1.] The theory of Elliptic Integrals is founded on the develop-
ment of the quadrinomial integral,

f( t>\ f\ V

VA + Btf + Ca^ + DtfS+tf4'

in which A, B, C, and D are constants, while f (x) denotes a ratio-
nal function of x.

It has been shown by Legendre, and, after him, by Verhulst,
Hymers, and others, that by the help of some ingenious transfor-
mations the above integral may be reduced to one or other of the
following fundamental forms,

dtp c

1 l/»* /I 2 " 2 /»*

•J

and

or, as they have been denoted by Legendre,

Fc(<p), Ec(<p), and Uc(p,(f>).

I have ventured to make some alterations in the established no-
tation of elliptic integrals. I have written i for the modulus, in-
instead of c, and j for its complement instead of b; so that

The symbol c, used by writers on this subject to designate the
modulus, was adopted by analogy from the formula for the recti-
fication of a plane elliptic arc by an integral of the second order.

6 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Although in the circular forms of the third order it still signifies a
certain ellipticity, yet it has no longer the same signification in the
usual form of the first order, or in the logarithmic form of the
third.

Instead of the usual symbol, A= \/l — C2sin2<p= \^l — &2sin2<p,
v/I has been substituted when i is the modulus. Should it be-
come necessary to designate the amplitude cp, the expression may
be written v/I^, while \/I, may denote a function whose modulus
is it.

For the elliptic integrals of the first and second orders,

\—^f and Jd<p v/I have been substituted. Hence J^TT represents

1—7= ==, and fdilr VI, maybe put for fdi/r \/l — i/sin2^.

Ivl— «2snr<p J J

c/

The surface of revolution may be named the generating surface,
while the intersecting surface is always a right cylindrical surface.
The parameter, of which p is the general symbol, we shall suppose
to vary from positive to negative infinity, and to pass through all
intermediate states of magnitude.

The nature of the representative curve will depend on the value
assigned to the parameter p in the expression

[ + p sin2 <p] v 1 — i2 sin2 <p

The modulus i we shall assume to be invariable and less than 1.
In this progress from +00 to — co , the parameter passes through
thirteen distinct values, each of which will cause a variation in the
species or properties of the hyperconic section, the representative
curve of the given elliptic integral.

In the following Table we may observe that the generating sur-
face in passing from a sphere to a paraboloid, in its course of trans-
ition becomes a plane.

It is somewhat remarkable that the common form of the elliptic
integral of the first order does not appear in the Table, although
it is implicitly contained in cases II. and VIII. ; for the circular
form of the third order, when the parameter is equal to the modu-
lus i, may be reduced to the first. The reason why the first form
of elliptic integral does not appear in the Table is this : in the
thirteen cases given, the origin is placed at the centre, or sym-
metrically with respect to the represented curve. When the elliptic
integral of the first order is given in the usual form, without a
parameter, it represents a spherical parabola, but the origin is non-
symmetrical, that is, the origin is placed at a focus.

Instead of p, the general symbol for the parameter, we may sub-

ON rin. <.I:O\II:TKICAL

<>r i I.LIPTIC INTEGRALS.

stitntc for it particular values, such as /, m, or n, as the case may
require. The (juautitios /, m, n, i, and./ are connected by the fol-
low inj; e(iuations : —

i9 -f j*= 1 , I m= i2, and m — n + mw=i2, in the circular form, 1
t2 +jz= 1 , ///=i2, and m + n — mn=i9) in the logarithmic form, J

/// and // may be called conjugate parameters ; while / and m, or
/ and // may be termed reciprocal parameters.

For (1 — wsin9<p) we may put M, and N for (l+wsin2<p).

These thirteen cases are exhibited in the following Table : —

Case.

Sign.

Parameter.

Generating
surface.

Cylindrical
surface.

Hyperconic
section.

I.

+

p = n = cx> .

Sphere.

Elliptic cylinder.

Circular sections of
elliptic cylinder.

II.

+

p=n = i, or
m = n.

Sphere.

Elliptic cylinder.

Spherical parabola.

in.

+

p=n>0.

Sphere.

Elliptic cylinder.

Spherical ellipse.

IV.

±

p = n—().

Plane.

Elliptic cylinder.

Plane ellipse.

V.

—

p = m = \—j,
or m = n.

Paraboloid
indefinitely
attenuated.

Circular cylinder.

Circular logarithmic
ellipse.

VI.

—

p = m, or
_p=n<ta.

Paraboloid.

Elliptic cylinder.

Logarithmic ellipse.

VII.

-

p=m = f.

Plane.

Elliptic cylinder.

Plane ellipse.

vra.

-

p = m = i.

Sphere.

Elliptic cylinder.

Spherical parabola.

IX.

—

p=m>{*
p = m<:l.

Sphere.

Elliptic cylinder.

Spherical ellipse.

X.

—

p=l=l.

Plane.

Hyperbolic
cylinder.

Plane hyperbola.

XI.

—

p=l>\.

Paraboloid.

Hyperbolic
cylinder.

Logarithmic
hyperbola.

XII.

—

P-l-l+j,
or m=n.

Paraboloid.

Hyperbolic
cylinder.

Squiparametral loga-
rithmic hyperbola.

XIII.

—

p = /=oc.

Paraboloid.

Vertical plane.

Parabola.

Cases I., IV., VII., X., XIII. give the formulae for the rectifi-
cation of the ordinary conic sections, the generating surface in
these cases being a plane. When the generating surface is a
sphere, we get the spherical hyperconic sections ; when a parabo-
loid, the logarithmic hyperconic sections result.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

ON THE SPHERICAL ELLIPSE.

2.] A spherical ellipse may be denned as the curve of intersec-
tion of a cone of the second degree with a concentric sphere.

In the spherical ellipse there are two points analogous to the
foci of the plane ellipse, such that the sum of the arcs of 'the great
circles, drawn from these points to any point on the curve, is con-
stant. Let a and /3 be the principal semiangles of the cone ; 2a
and 2/3 are therefore the principal arcs of the spherical ellipse.
Let two straight lines be drawn from the vertex of the cone, in
the plane of the angle of 2a, making with the internal axis of the
cone equal angles e, such that

cos a
cose= -- - ........ (2)

cos/3

These lines are usually called focals, or the focal lines of the cone.
The points in which they meet the surface of the sphere are termed
the foci of the spherical ellipse.

Every umbilical surface of the second order has two concentric
circular sections, whose planes, in the case of cones, pass through
the greater of the external axes. Perpendiculars drawn to the
planes of these sections, passing through the vertex (they may be
called the CYCLIC AXES of the cone], make with the internal axis of
the cone in the plane of 2/3 (the plane passing through the internal
and the lesser external axis] equal angles 97, such that

sin/3

COS V) = -r - ........ (3)

sin a

Let a series of planes be drawn through the vertex, and perpen-
dicular to the successive sides of the cone. This series of planes
will envelop a second cone, which is usually called the supple-
mental cone to the former. The cones are so related, that the
planes of the circular sections of the one are perpendicular to the
focal lines of the other, and conversely.

The equation of the spherical ellipse may be found as follows,
from simple geometrical considerations.

Let 2a and 2/3 be the greatest and least vertical angles of the
cone ; the origin of coordinates being placed at the common centre
of the sphere and cone. Let the internal axis of the cone meet
the surface of the sphere in the point Z, which may be taken as
the pole. Let p be an arc of a great circle drawn from the point
Z to any point Q, on the curve, ty being the angle which the plane
of this circle makes with the plane of 2a. We shall then have for
the polar equation of the spherical ellipse,
1 cos2 sin2 •xr

tan2 p tan2 a tan2 ft'

(»N THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 9

To show this, through the point Z let a tangent plane be drawn to
the spin-re. This plane will intersect the cone in an ellipse. This
ellipse may be called the plane base of the cone, while the portion
of the surface of the sphere within the cone may be termed the
spherical base of the cone. The plane of the great circle passing
through Z and Q will cut the plane base of the cone in the radius
vector R ; and if we write A and B for the semiaxes of this ellipse,
whose plane touches the sphere, we shall have for the common
polar equation of this ellipse, the centre being the pole,

Now, the radius of the sphere being k, and p, a, /3 the angles sub-
tended at the centre by R, A, B, we shall clearly have

R = £ tanp, A=£tana, B = £ tan/9;

1 cos2 -Jr sin2 -*r
whence —5—= * +

—5— * 7 — 5-5 ......

tanap tan2 a tan2/3

We may write this equation in the form

1— sin2p cos2-Jr -oN sin2 -Jr., . 20N

. g r = ^-g-i- (1— sm2a)+-^-2^-(l— sm2/3);
sm2p sm2a v sm2£

1 cos2 -^ , sin2 -\/r
or reducing, = . T + -^-^ ...... (5)

sm2p sm2a sin2/?

This is the equation of the spherical ellipse under another form,
which may be obtained independently by orthogonally project-
ing the spherical ellipse on the plane of the external axes ; or by
taking the spherical ellipse as the symmetrical intersection of a
right elliptic cylinder with the sphere.

3.] If in the major principal arc 2a of a spherical ellipse, we
assume two points equidistant from the centre, the distance e being

determined by the condition cbse= -- -x, as in (2), the sum of the

arcs of the great circles drawn from these points — Fig. 1.
the foci — to any point on the spherical ellipse is con-
stant, and equal to the principal arc 2a.

Let 0 and ff denote the arcs drawn from the
points F, F to a point Q upon the curve, QZ=p,
and the angle QZF=<f, FZ=F'Z=e.

Then, as FZQ, FZQ, are spherical triangles, we

cos 0— cose cosp

get cosilr = -- : — -t , . . . (a)

sin e sin p

COS ff — COS 6 COS p

— COSllr=— — ; — _T . . Hj)

sine smp

VOL. 11. C

10 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

COS a

— -- -- (c), and the equation of the curve given in [(2)]

C/Oo A.J

COS 6

n2'«/r ..... (d)

Between (a), (b), (c), (d), we must eliminate p, ^Jr, and e. Adding
together (a) and (b), also subtracting (b) from (a), we get

cos# + cos0'=2cosp cose; and cos 6 — cos 6' = 2 smp sine cos-^r;
from (d), I = cot2atan2p cos2^ + tan2p cot2£-tan2p cot2/3 cos2^-;

/cos2/3 — cos2a

I

V

\ .
i s
/

cosp
-- --

. 2 , _ , . . „

sm2p cos2i|r=cot2a -- r-o-k : substituting for
sin2 /3

or

sm a sm

sinp cosijr, its value deduced by subtracting (b) from (a), we find
cos2 a (cos 6 — cos ff) 2 + sin2 a (cos 6 + cos ff) 2 = sin2 2a,
or cos2 B -f cos2 & — 2 cos 6 cos & (cos2 a — sin2 a) = 1 — cos2 2a ;
whence cos22a— 2cos# cos & cos2a=l — cos2^ — cos2#'.
Completing the square and reducing, we obtain

cos 2a= cos 6 cos 6' + sin 6 sin 0'= cos (6 ± 0'} or

2a=0±0' ........ (e)

The positive sign to be taken when the curve is the spherical
ellipse.

4.] The product of the sines of the perpendicular arcs let fall from
the foci of a spherical ellipse on the arc of a great circle touching it,
is constant.

Let IB and OT' be the perpendicular arcs let fall from the foci on
the tangent arc of a great circle ; we shall have

sin -BT sin id = sin (a + e) sin (a— e) .

Let OT, tff', fff", be the perpendicular
arcs, let fall from the centre, and the
two foci F and Fy, on the tangent
arc mn. These three arcs will meet
in the point o, the pole of the arc
mn. Let p be the perpendicular
from the centre on the straight line
which touches the plane elliptic base ;
of this straight line, mn is the projec-
tion. We shall therefore have

j92 = A2 cos2 X + B2 sin2 X,
or tan2 -or = tan2a cos8 \ + tan2 /3 sin2 X,
cos2 a

Fig. 2.

whence cos2 IF =

1 — sin2 e sin2 X'

OH I Hi: i;i:<)MKTRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 11

Now FZQ = X, whence iu the spherical trhui^lc FZO,
FO = |-<ZO=£-CT,

sin in1 — cos e sin or

\\r anal! have cosX= = — — .

sin e cos ta-
in the other spherical triangle F'ZO, we shall also have

sin HT" — cos e sin -sr
— cosX=-

sm e cos m

Adding first, and then subtracting these equations, one from the
other, we shall find

sin iff1 + sin tv" =2 cos e sin iff,

sin iff1 — sin iff" =2 sin e cos iff cos X.

Squaring these equations, and subtracting the latter from the
former, we shall obtain

sin a' sin iff" = cos2 e— cos2 iff (1 — sin2 e sin2 \) .
Substituting for cos iff its value given above, and reducing,

sinw' smtsr"=sin (a-f e) sin (a— e) (6)

5.] The area of any portion of a spherical surface bounded by a
closed curve, may be determined by the formula,

r*« fo

area=l <ty I do- [sin o-],

«/0 •'O

where a- is the arc of a great circle intercepted between the fixed
point Z taken within the curve as pole (fig. 3), and any variable
point m assumed within the bounding curve on the surface of the
sphere, p being the spherical radius vector of the curve measured
from the pole Z, and passing through m, while i/r is the angle
which the plane of this gre"at circle, passing through the points
Z, m, makes with the fixed plane of a great circle passing through Z.
Let O be the centre of the sphere, Z the pole, m the assumed
point, ZQ, the great circle passing through
them. Through Z let a great circle OZQ' Fig. 3.

be drawn, indefinitely near to the former,
d^/r being the angle between the planes.
Through m let a plane be drawn perpendi-
cular to the axis OZ, meeting the great circle
OZQ! in m1. Through n, a point on ZQ in-
definitely near to m, a parallel plane being
drawn, it will meet the great circle OZQ' in
a point n', indefinitely near to m1. Now it is
manifest from this construction that the

12 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

whole spherical area to be determined is the sum of all the indefi-
nitely small trapezia, such as mnm1 n1, into which in this manner
it may be divided. To compute the value of this elementary tra-
pezium, we have mm'=sino-d-\Jr, mw=dcr. As the pole Z is
within the curve, the limits of er are 0 and p ; and as the surface
is assumed to extend all round Z, the limits of -ty are 0 and 2vr.

f** ff

Whence area= 1 <tyl dcr[sin<r] ....... (a)

Jo Jo

Integrating this equation between the limits 0 and p, we find

/*217

area = I di|r[l— cos/>] ....... (b)*

Jo

The second integration can be accomplished only when we know
the relation between p and i/r, or the equation of the bounding
curve.

6.] To find an expression for the length of a curve described on
the surface of a sphere, whose radius
is 1.

Let u and u' be two consecutive points
on the curve, ZQ, ZQ' the arcs of two
great circles passing through them in-
clined to each other at the indefinitely
small angle di/r. Through u let a plane
be drawn perpendicular to OZ, and meet-
ing the great circle ZQ' in v.

Then ultimately uvu' may be taken
as a right-angled triangle, whence

Now uu'=d<r, wv=sinp d^} u'v=Ap} whence
do-= [dp8 + smutty2]*.

Integrating this expression between the limits pt and pn, or i/r
and 0, accordingly as we take p or ty for the independent variable,
we get

* Equation (b) may be established by the help of the simplest elementary
principles. We know that the surface of the segment of a sphere comprised
between a tangent plane and a parallel secant plane is equal to the circumference
of a great circle multiplied into the distance between these planes. This* dist-
ance is 1 — cos p ; p being the arc of a great circle, measured from the point of
contact of the tangent plane to the parallel secant plane. If through the dia-
meter perpendicular to these planes we draw two great circles, inclined one to
the other at the angle d\|^, the surface of the sperical wedge thus formed will be

— cosp).

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 13

7.] To apply these expressions to find the length of an arc of a
spherical ellipse.

In this ease it will be found simpler to integrate the differential
expression for an arc of a curve, taking p instead of ty as the in-
dependent variable. We may derive from (5) the following ex-
pressions,

• 2 r _sin2/3 fsin2a— sin2pl ^|
^ siii2p I sin2 a— sin2/3j [

2 sjn^arsin2p-sin2/31 (

T sin2p \ sin2 a— sin2/3J J

Differentiating the former with respect to ty and p, and elimi-
nating sin ty, cos i/r, using for this purpose the relations established
in (a), we shall find

d>/r — sin a sin ft cos p

(b)
r p v sin* p — sin* p

Substituting this value of - in the general expression for the

dp sin p \/sin2 a — sin2 p v/sin2 p — sin'2 ft
ting this value of -p in the general express
arc in the last section, the resulting equation will become

sin p y/ cos2 p— cos2 a cos2 /3
\/ (sin2 a— sin2 p) (sin2 p — sin2 ft)

], . . . (

an elliptic integral which may be reduced to the usual form by the
following transformation : assume

9 _ sin2 a cosa <p 4 sin2 /3 sin2 <p
8 P~22 2/3 sin2<p .....

7T

The limits of integration are 0 and -. Differentiating this ex-

ti/

pression, and introducing inte (c) the relations assumed in (d), we
shall obtain for the arc the following expression : —

tan/3 .

. , "|4/, /8in2a-s

m H V 1 - ( 8in2a

sm2j8

- (8)

Let e be the eccentricity of the plane base of the cone, whose semi-
axes are A and B, as in sec [2] ,

A2-Ba tan2 a- tan2 ft sin2 a- sin2 ft

f/X « ._, — • f •— — — _^^^_^_^^^_^^^ _

A2 tan2 a sin2 a cos2 /3 '

we may derive from (2) and (3)

sin2 a — sin2 ft , . , sin2 a — sin2 /3
sin8 77= -- —s — - and sm2e=— — q-&~ >
sm2 a cos2 ft

14 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

or grouping these results together,

tan2 a — tan2 B sin2 a — sin2 3

* — — '.^—

£>£,—*

_ _

tan2 a sin2 a cos2 /3

. Q sin2 a — sin2/3 ,2 . 2 sin2 a— sin2 /3
sin2 77= -- =-3 -- -=&2, sm2e=— —5—. — =rc. . (9)
sin* a cos"1 a

These quantities m, n, and z2 fulfil the equation of condition
assumed in (1)

i'2 ........ (e)

If we introduce these values into (8) , the transformed equation will

become

_tan/3 . Cr _ df _ -i
~tan a S1 *' J |_[1 - e2 sin2 <p] ^/l -sin2 17 sin2<pj '

an elliptic integral of the third order and circular form, since e2 is
greater than sin2 77, and less than 1 .

This is case IX. in the Table, page 7.

This is the simplest form to which the rectification of an arc of
a spherical ellipse can be reduced. The parameter of the elliptic
integral is the square of the eccentricity of the plane elliptic base ;
and the modulus is the sine of half the angle between the planes of
the circular sections of the cone.

If we write m for e2, i for sin 77, and express the coefficient

- - sin B in terms of m and i, the expression (10) may be trans-
tan a

formed into

/ —
V

"I
j '

[l-msin2<p] Vl^P sin2<p

It is easily shown that the coefficient -^ — sin 8 of the elliptic in-

run a

tegral in (10) or its equivalent I - — J \/mn is the square root of
the criterion of sphericity,

m

For if we substitute in this expression for i its value given in (1)
m — n + mn=i2} we shall find

.- tan/3 .
K=— "sin
tana

a A— m\ /—

5=1- -\vrnn. . . . . (f)
\ m /

As \/ K is manifestly real, the elliptic integral is of the circular
form.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 15

8.] To find the area of a spherical ellipse.

Resuming equations (4) and (5) of the spherical ellipse,

1 cosa>/r sin2-^ , 1 cos2-^- sin2^

tan2p tan2 a tan2/S' ' sin2p sin2 a sin2/3*
dividing the former by the latter, and reducing, we shall find

tan2 a

.
co.p-co.a- ^- - (a)

Substituting this value of cos p in the general expression for the
spherical area (b) sec. [5], we obtain the result

tan2 a

area=-^r — cos a j

(b)

To integrate this equation, let us assume

tan/3

tanyi = — - tan<p : (c)

tana

and we shall find, on making the necessary transformations in the

,. . tan#

preceding expressions, the area=<y< — , cos a x

tan a

J . /tan8 a- tan2 ft\ . 2 1 / /cos2£-cos2a\ .
— 2— -)sm2<p >\/ I — [- -)si

\ tan* a / J V \ cos2p /

(12)

Let A and B be the semiaxes of the plane elliptic base of the
cone, and e its eccentricity, then we shall obviously have

2_A2-B2_tan2a-tan2ff

~A*~ tan'a

and e being the angle between the spherical focus and centre,

cos a . r .. , . _ cos2 ft— cos2 a

cos e = » as m sec 1 21 , whence sin2 e = — — . (e)

cos p cos2 ft

Introducing these relations into (12), we shall obtain the formula

tan/9 fr d<p 1

area=y— — — cosal 71 2 . 2 n — , . . ==f^- . (13)

tana J L[l— e2sm2<p] yl— sm2esin2(pj

This is an elliptic function of the third order and circular form,
since e2 is less than 1 , and greater than sin2 e.

16 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

This seems to be the simplest form that can be found for the
quadrature of the spherical ellipse, the parameter and squared
modulus of the elliptic transcendent being the squares of the ec-
centricities of the plane and spherical ellipses respectively.

We shall show hereafter that there is a class of spherical ellipses
whose quadrature may be effected by elliptic functions of the first
order.

To determine the geometrical signification of the angle of re-

Fig. 5.

duction <p, in the above trans-
formation.

On the major axis of the
plane elliptic base of the cone,
let a semicircle be described.
Let OP be drawn, making the
angle ^ with the major axis
OB. Let the ordinate through
P be produced to meet the
circle in Q, join OQ, ;

tan-Jr PD B tan/3 , , tanilr tan/3 , A,

then- — 7^5=rvis=ir=r- -;but- -*- =— -; see (10)
tanQOB QD A tana tan<p tana

whence Q,OB=<p, or <p is the eccentric anomaly of the point P.

*7T 7T

Now, when -^=0, <p=0, and when ^ = o> <f=2' wnence *P

and ty coincide at these limits. Writing S for the area of the
quadrant of the spherical ellipse ; as the surface evidently consists
of four symmetrical quadrants, the area or length of one quadrant
will manifestly be one fourth of the area or length of the whole ;
whence

tan/3

area=vr — — - cos a
tana

d<p

. — e2 sin2 <p] V 1 — sin2 e sin2

;]• (W)

9.] Let 2a' and 2/3' be the principal arcs of the supplemental cone,
a! being in the plane of /3, and /3' in that of a. Let S' be the length
of a quadrant of the spherical ellipse the intersection of this cone
with the concentric sphere. Then we may deduce from (10)

d<p

. — e'2 sin2 <p} V 1 — sin2 if sin2 <p

Now, as the cones are assumed to be supplemental,

]. . (a)

7T

a + /3' = „> ft + a' = o > whence sin a' = cos /3, sin /3' = cos a,

ON Illi; (iKOMKTKIl'AI. IMUM'KKTIKS OK Kl.UI'TK IM'KI.KALS. 17

. . . tan/3' tan/3 ,2

cos a' = sin B, cos/3' = sm a ; therefore- , = ; — -, t'* = e*, and

tan a' tan a

sin T/ = -si» e (b)

Introducing these transformations into the last formula

v, tan/3 f£r <fy -|

i' = A -cosal TT 5-7-5 /, . 9 . , . (15)

tana , L{1 — e* sm* <pV yl — sure sin2 <pj

c U

Now, if we turn to the expression found for the area of a spheri-
cal ellipse, given in (13), we shall find that it consists of two parts —
a circular arc, and an elliptic integral identically the same with the

IT

one just investigated, when taken between the limits 0 and -. We

SB

thus arrive at the very remarkable result, that the rectification of
a spherical ellipse depends on the quadrature of the supplemental
ellipse, and reciprocally.

If we add together (13) and (15),

S+S'=|; ....... (16)

or taking the whole surface 4S of the spherical conic, and the cir-
cumference 4S' of the supplemental conic, introducing, moreover,
k the radius of the sphere, we obtain the remarkable theorem

4S + 4£2' = 2A:27r (17)

Now 4A:S' is twice the lateral surface of the supplemental cone,
and 4S is the surface of the spherical ellipse. We may therefore
infer that

The spherical base of any cone, together with twice the lateral
surface of the supplemental cone, is equal to the surface of the hemi-
sphere.

Let 4S' denote the spherical base of the supplemental cone, and
L the lateral surface of the original cone : from the preceding
equations we obtain

Adding these equations,

4(S + S') + 2 (L + L') = 4*27r. "I

Subtracting one from the other, . . . . (18)

4(S-S')=2(L-L');

or, if any two cones, supplemental one to the other, are cut by a
concentric sphere,

The sum of their spherical bases, together with twice the sum of
their lateral surfaces, is equal to the surface of the sphere.

And, The difference of their bases is equal to twice the difference
of their lateral surfaces.

VOL. II. D

18 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Again, let a cone whose principal angles are supplemental be
cut by a concentric sphere,

The area of the spherical base, together with twice the lateral sur-
face, is equal to the surface of the hemisphere.

10.] We may, by the method of projective coordinates, derive
an expression for the arc of a spherical ellipse.

In this case we shall consider the spherical ellipse as the curve
of intersection of a right elliptic cylinder by a sphere having its
centre on the axis of the cylinder.

Let

be the equations of the cylinder
and sphere, ABCD and FGCD ;
then, do- being the element of an
arc on the surface of a sphere
whose radius is 1, k&o- will be the
element of the corresponding arc
on the surface of the sphere whose
radius is k.

(19)

Fig. 6.

Hence

£-_-

A.-TT-

dX,

A/TWW

x, y and z being functions of the independent variable X;.
Assume

^2_ ^cos2^ .£•_ 64sin2X;

) l.O\ • O -x

'— 0*) sin2Xy

'' . . (21)

Differentiating these expressions,

and as
d?^2

5£.\ _. <^" \u~ — o~) " sinrA^cps-8 X,

dXy / [a2 cos2 X; + W- sin2 Xy] 3 [a2 (A:2 — a2) cos2 Xy -f W^—W-} sin2xT '
Substituting these expressions in (20), we find

/ //

(23

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 19

The numerator of this expression may be resolved into the factors

•a2) cos2X/+ (£2 — b*} sin8Xy],
and the equation may now be written

-gg) cos2X,+ (k*-t>

dX, A;[a2cos2X/ + 62sin2X/] V«H*S -~«a)cos% +**(**--

(jfZ — b*
70 5
ft* _ /7 *

Hence

(24) may now be transformed into
do-do-dX

d<p; dX7 dtp/ A: [a2 (**- 62) cos2 <py + &2(A2- a2) sin8<pj \/a2 cos2^ + 6

If we imagine a concentric cone to pass through the mutual inter-
section of the cylinder and the sphere, we shall have

, b=k sin/3,

in2 o_

~

tan2«
Whence (26) may be transformed into

tan 0 . n Cr d<f>,

- — " - ¥1

. n ,

0-=- — sin/3 I - 1— (28]

tan a KJ [[1 -e2 sin2 <py] Vl -sin2 17 sin2^J '

an expression identically the same with (10).

The angle <p/ in this expression is identical with <p in (10).

For 2+ g _ ffl4 cos% + ^4 sin2 X, _ a4 + 64 tan2 X,

+ y "22 2~2 2;

eliminating tanX/ by (25),

^ , tfg=^(*a-^t) c««

«2(A2-62) cos2<p/ + 62(^-a2) sin2<p/

No w a2 = *2 sin a, &2 = #2sin2/3, /:2-fl2=^2cos2a, /t2-i2 =
and ,r2 -f y2 = A;2 cos2 p.

Reducing, we get

~>r-ir^r- «Q • J1 (29)

tan* a cos^^ + tan^p sm*<p/

Comparing this expression with (d) sec. [7], it follows that <p = <p;. (30)

20 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

In the foregoing expressions (11) sec. [7] and (28) for the recti-
fication of an arc of a spherical ellipse, the elliptic integrals are of
the third order and circular form, with negative parameters. We
shall now proceed to show that the same arc may be expressed by
an elliptic integral of the third order and circular form, having a
positive parameter.

11.] It is shown in the first volume of this work, at page 184,
that if p, the perpendicular let fall from a fixed point as pole on a
tangent to the curve, makes the angle X with a fixed straight line
drawn through the pole, t being the intercept of the tangent between
the point of contact and the foot of the perpendicular, we shall
have

(31)

Fig. 7.

the upper sign to be taken when the radius of curvature is greater
than /?, the lower sign to be used when it is less than p.

To investigate an analogous formula for the rectification of a
spherical curve, the intersection of a cone of any order with a
concentric sphere.

Let a point Z be assumed on
the surface of the sphere as pole,
and through this point a tangent
plane ZAQB, or (@) , to the sphere
being drawn, the cone whose
vertex is at O, the centre of the
sphere, and which passes through
the given spherical curve, will
cut this tangent plane (©) in a
plane curve AQB, whose rectifi-
cation may be effected, when pos-
sible, by the preceding expression.
Now a tangent plane OOP, or
(T) , may be conceived as drawn
touching the cone, and cutting
the tangent plane (®) in a straight
line QP or t, which will be a tan-
gent to the plane curve in (©).
It will also cut the sphere in an
arc of a great circle (KTS) which will touch the spherical curve in K.
Let the distance QO of the point of contact of the line t with the
plane curve from the centre of the sphere be R. Through the
centre of the sphere let a plane OZP, or (II), be drawn at right
angles to the straight line t. Now this plane, as it is perpendicu-
lar to t, must be perpendicular to the planes (©) and (T) which
pass through t. As the plane (II) is perpendicular to the plane (®),

0\ THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 21

it must pass through (Z) the point of contact of this plane with the
sphere, and cut the plane of the curve AQJB in a straight line ZP,
or p, which passes through the pole, the point of contact of (®)
with the sphere. This line p being in (II), must be perpendicular
to /. The plane (II) will also cut the sphere in an arc of a great
circle Zw = «rJ perpendicular to /ew, the tangent arc to the spherical
curve ; for these arcs must be at right angles to each other, since
the planes in which they lie, (II) and (T), are at right angles. Let
P be the distance OP of the point in which the plane (II) cuts the
straight line t, from the centre of the sphere ; r the distance Q,Z of
the pole of the plane curve to the point in which / touches it, T
being the angle which / subtends at the centre of the sphere, and k
its radius

p = k sin iir, t = P tan T,

T is the angle between OQ and OP.

Let ds be the element of an arc of the plane curve between any
two consecutive positions of R, indefinitely near to each other ;
#d<r the corresponding element of the spherical curve between the
same consecutive positions of R. Then the areas of the element-
ary triangles on the surface of the cone, between these consecutive
positions of R, having their vertices at the centre of the sphere,
and for bases the elements of the arcs of the plane and spherical
curves respectively, are as their bases multiplied by their altitudes.
Let S and S' be these areas ; then

P •

1 dX:

But the areas of triangles are also as the products of their sides
into the sines of the contained angles, i. e. in this case as the squares
of the sides, or

S:ff::R«:*«,. '. (b) «£-£*,. . . (c)

1 ~D f J2 ""^

putting for ds its value given in (31), -r- = ^2< fi$+p f • • (d)

-T. dP dp dp

whence PJT-=#J€> and/ = — •£.

UA, CIA, dA.

Substituting these values in (d),
do-

dX~0'""^R«li dX«"^ ^ f (6)

22 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

We shall now proceed to show that the last term of this equa-
tion is the differential of the arc, with respect to X, subtend*
the centre of the sphere.

t P

This arc being T, tanr=p cosr=^-.

- .dP

Therefore

or as r=— -^, j^ = — ^ ^ -r 3T2~;n dX ( • • • * '»'

HA/ tl/v ti/v ^

Adding this equation to (e), we get for the final result,

cr= fdX sin-cr — T. ]

d» tj ' ' * W

If t=-^-, the formula becomes cr=ldX sm-or + T.
dX J }

12.] This formula serves a twofold purpose ; for it will also enable
us to give the quadrature of the supplemental figure on the
surface of the sphere. Let p1 be that radius vector of the supple-
mental figure on the surface of the sphere which is the prolonga-
tion of -or; p' + -5T=^, and therefore sin -07= cos p'; X remains the
same in both curves ; whence

\ sin'BrdX= \ cosp'dX (h)

But it was shown in (b) sec. [5] that the expression for the area of
a spherical curve is

area=J(l— cosp') dX=X— JsinOTdX. . . . (i)

Thus the proposition established in sec. [9] as to the reciprocal re-
lations between the rectification and quadrature of supplemental
spherical conies of the second order, is shown to hold with respect
to supplemental conies of any order described on the surface of a
sphere.

Throughout these pages, to avoid circumlocution and needless
repetitions, we shall designate as the ^ro-jected tangent, or briefly
as the protangent, that portion of a tangent to a curve, whether it
be a straight line, a circle, or a parabola, between its point of con-
tact, and a perpendicular from a fixed point let fall upon it, whe-
ther this perpendicular be a straight line, or a circular or a para-
bolic arc. This definition is the more necessary, as the protangent
will continually occur in the following investigations. The term
is not inappropriate, as the j»ro-tangent is the projection of the
radius vector on the tangent.

o\ THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 23

Hi.] To apply the formula (33) to the rectification of the sphe-
rical ellipse.

Let, as before, A aiid B be the semiaxes of the plane elliptic
base of the cone, r the central radius vector drawn to the point of
contact of the tangent t, p the perpendicular from the centre on
this tangent, / the intercept of the tangent to the plane ellipse be-
tween the point of contact and the foot of the perpendicular, X the
angle between p and A. Let a, ft, p, tx, r be the angles subtended
at the centre of the sphere, whose radius is 1, by the lines A, B, r,
p, t, we shall consequently have

A=A:tana, B=A:tan/3, r = k tanp, p=k tan -or,

1 • • •

T.J

and /= V£2+./>2 tanr = P tan

Now in the plane ellipse

(A«-B*)»gin*Xcoe«X

sm2X, and /2=^— - :

(34)

P*

therefore in the spherical ellipse

whence sec2 -BT =sec2 a cos2 X + sec2 ft sin2 X.

Dividing the former by the latter,

tan2 a cos2 X + tan2 ft sin2 X

sin2 13-= — -9 — 0_ — <TO • 9 ^ • • • • (36)
sec"1 a cos- X + sec* p sur X

Introducing this value of sin -or into (33), the general form for
spherical rectification, the resulting equation will become

"tan2 a cos2 X + tan2 ft sin2 X~| i
sec2 a cos2 X -I- sec2 ft sin2 X J

14.] To reduce this expression to the usual form of an elliptic
integral.
Assume tan ^= cos e tan X (38)

It must first be shown that this amplitude ^ is equal to the ampli-
tude <p in (d) sec. [7], and therefore to <pt in (25), as was established
in sec. [10].

In an ellipse, if -^r and X are the angles which a central radius
vector, and a perpendicular from the centre, on the tangent drawn
through its extremity, make with the major axis, we know that

B2 tan2 B

tani/r = — ^ tanX = — ^— tanX. Introducing this value of tan -*fr

into (5) sec. [2] and reducing,

Qr tan8 a cos2 X + tan2 # sin2 X

cos2 p=cos- a cos2 ft t — 5 3^ s^ o-^ — 5 — r-~- .

I_tan2a cos2/3 cos2X + tan2/? cos2 a snrAJ

24 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

•

Comparing this value of cos2 p with that assumed for cos2 p in (d)
sec. [7], namely,

sin2 a cos2 <p + sin2 8 sin2 <p
prm* n — _ - _ : - - - —
P

we get, after some reductions, tan <p= cose tan X ..... (39)

But in (38) we assumed tan %= cose tanX. Hence the amplitudes
<p, <p', and % in (d) sec. [7], (25), and (38) are equal. We may ac-
cordingly write <p instead of % or <pr Substituting the value of
tan A, derived from the equation, tan <p= cose tanX, in (38), the in-
tegral in (37) becomes

t.

cos a cos /9 [sin2 a — (sin2 a — sin2 8} sin2 <p] d<p

[cos2 a + (sin2 a —sin2 8) sin2 <p] -v/sin2 a cos2 <p + sin2 8 sin2 <p

cos a , „ sin2a — sin2/? . 0 sin2a — sin2/3

Now cose= r>, tan2e=— — =,— — . sm2?7= r-s— — . (40)

cos p cos2 a sin2 a

Making the substitutions suggested by these relations, and redu-
cing, we shall find

$_ Cr
m aj L

cos a sm aj L [1 + tan2 e sin2 <p] V 1 — sin2 T; sin2
cos a cos 8 C d<p

sin a v/1— sin2^ si

~ P

sn<p

an elliptic integral of the third order, with a positive parameter,
and therefore of the circular form.

This is case IX. in the Table, page 7.

Writing n for tan2 e, i for sin 77, and expressing sin a, cos a, sin/3,
cos/3 in terms of w and z, (41) becomes

fl"

J L

__
V 1 — «2 sin2 <p

If we put A, for the criterion of sphericity, as in sec. [7], with
respect to n the positive parameter, or «,= (!+») (w+-j,

or */*,=(- -) Vww, it may easily be shown that V/^= C°S/?

cos asm a
the coefficient of the preceding integral. Hence also, K K,=J*.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 25

15.] To express the protangent r in terms of \ and <p. We found
in sec. [11]

2 __**__ jV_ _ (A2-B2)2*in2\co82\ _
~P2~Py~ [/fc2 + a2cos2\ + 62sin2\] [a2cos*A. + 62sin2X]'

Now

A2-B2 sin2a-sin2/3

A. = k tana, B = A: tana, e2 = — 7-5 — , and snre= — — ^ — -.

A2 cos2 /3

e2 sin a sin X cos \

whence tanr = — — — — - — — . . . (43)

V 1 — e2 sin* X y 1 —sin2 e sin2 \

To express tan r in terms of the amplitude <p.

Assume the relation established in (d) sec. [7] or (25) or (38) or
(39), tan <p = cos e tan \. Introducing this condition into (43), we
obtain

e tan e sin <r> cos <p

tanr = — == . y . L • ..... (44)
v 1— 8111*77 sm^fp

or as \m = e, v^

. , mnsn<p cos<p

the last equation becomes tanr = - --, >. ~^J . . . . (45)

-v/l-i2sin29

Hence (42) may now be written

_/l+
~\ n

~"

• (46)
sin <p cos <p

A/1— i2 siu2<p \/l— i2sin2<p

Now this formula and (11) represent the same arc of the spherical
ellipse ; they may therefore be equated together. Accordingly

(!±^ft ^-= =J]}

\ n / J L [1 + n sin2 <p] v' 1 —i2 sin2 <pj

/l-m\ f r d(p -I

\~wT/J L[l-msin2<p] \/l-i2 sin2 <pj

d<p 1 _, r \/mn sin <p cos

.g . o + ~7=" tan -===:

— i2sm2<p ymw L \/l — i2sm2<r

This is the well-known theorem established by LEGENDRE, Traite
des Fonctions Elliptiques, torn. i. p. 72, for the comparison of
elliptic integrals of the circular form, with positive and negative

VOL. II. E

26 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Fig. 8.

parameters respectively. These circular forms arise from treating
the element of the spherical conic either as the hypotenuse of an
infinitesimal right-angled triangle, or as an element of a circular
arc, having the same curvature. When we adopt the former prin-
ciple, we obtain for the arc an elliptic integral of the third order,
circular form, and negative parameter. When we select the latter,
we get a circular form of the same order, with a positive parameter.
Equating these expressions for the same arc of the curve, the re-
sulting relation is Legendre's theorem. We thus see how an el-
liptic integral with a positive parameter may be made to depend on
another with a negative parameter less than 1 and greater than i2.

16.] We must not confound
the angle X in the preceding
article with the angle X, in Art.
[10] . We shall investigate the
relation between them. Through
ZO, the axis of the cylinder, let
a plane be drawn making the
angle ty with the plane ZOA«.
Let this plane cut the spherical
ellipse in the point K, and the
plane ellipse the orthogonal pro-
jection of the latter in the point
Q. Through K draw an arc of
a great circle KIT touching the curve, and through Q draw a right
line touching the plane ellipse. From Z let fall the perpendicular
arc ZTT on the tangent arc of the circle, making the angle X with
the arc Za. From O let fall on the tangent to the plane ellipse
at Q, the perpendicular OP making the angle \, with OA.

Then

tan X =

- — g 'a

sin''

tan\|r.

Hence we derive

tanX,
tanX

Consequently tan X . tan Xy = cos2 e tan2 X.

But we have shown in (39) that

tan2 <p — cos2 e tan2 X,
whence tan2<p=tanX tanXy, (48)

on the tangent of the amplitude <p is a mean proportional between
the tangents of the normal angles which a point of contact K on the
spherical ellipse, and its projection Q, on the plane ellipse the base of
the cylinder produce.

17.] We may obtain, tinder another form, the rectification of
the spherical ellipse.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 27

Assume the equations of the right cylinder and generating
sphere as given in (19),

r* + -J*=l, and x* + y* + z* = k*.

• t (/

Make x=asin0, y = /3cos#;

hence z* = k*-a'2 sin20-62 cos20;' ..... (49)

and therefore *d<7/- [a*(k*~^ ™** » + &(&- a*} sin20-U

-U
J ' (

Now

a2 (A:2 - A2) =#» sin2 a cos2 0, 62(/c2 - a2) = /c4 sin2 0 cos2 a,

k*-b* = k* cos2 /3, /c2 - <z2 = £2 cos2 a.
Substituting these values in (50) , and integrating,

,CW ftan2 a cos2 0 + tan2 ff sin2 0"| ^
~J Lsec2a cos20 + sec2/3 sin2 0 J '

If we now compare this formula with (37) and make 0=\, we
shall have </— CT = T .......... (52)

Hence we may represent the difference between two arcs of a sphe-
rical ellipse, measured from the vertices of the major and minor
arcs of the curve, by an arc T of a great circle which touches the
spherical ellipse.

18.] We may thus, by the help of the foregoing theorems, show
that when any elliptic integral of the third order and circular form
is given, whether the parameter be positive or negative, we may
always obtain the elements of the spherical ellipse, of whose arc
the given function is the representative.

Let the parameter be negative.

9 tan2 a— tan2 /3 . . „ sin2 a— sin2 /3

As e2 = — —=m, andsm2«=— — —i 2

tan2 a sm2a

•9 '9

I ll — i - vn —~ £

we shall have tan2 a = ^ri -- r. tan2 6= — ^— . . . . (53)

iz(l— m) t2

In order that these values of tan a, tan /3 may be real, we must
have m>i2 and m<l.

Let the parameter be positive.

3 sin2 a— sin2 /3 , . . sin2 a— sin2 ft

Now tan2e=— —=n. 2

cos* a sin* a

hence . tan«a=|, tan2/3=| ^ l2 ..... (54)

There is in this case no restriction on the magnitude of n.

28 ON THE GEOMETEICAL PROPERTIES OF ELLIPTIC INTEGRALS.

19.] To determine the value of tlie expression

l + n\ /— ff
n ) V/™JL(

n J ^ J L(l+rasin2<p) VI -#
when n is infinite.

As m— n + mn=i*} or (I— m) (l + tt) = l— i2=/2,

when n is infinite, w=l.

Resuming the expression given in (47) ,

/l+n\ fr dtp -|

0"=l- I </mn\ T-. — r~s : 7l -a • g

\ /i / J |_(1 +w srrr <p) v 1 — t sin <pJ

t I (J.(p - 1 I ^y Tflffy S1H ® COS ® I

_ — =- 1 — T . =^ — tan L r r i

\/mn 1 vl— i2 sin2 <p L \/l— i2 sin2(p J )

c/

we find that when n is infinite, a is a right angle.

0 sin2 a — sin2 ft ,, ,, TT

For yz=tan2e= — — = co ; therefore a =„.

cos'' a ^

Now -^ being the angle between the spherical radius vector drawn
to the extremity of the arc, and the major principal arc, we have

tan \!r = — 5— tan X. and tan <p = — -0 tan X,
tan2 a cosp

tan ft sin ft

or tan/w< = — — tan (p.

tan a sin a

Hence -^ is indefinitely less than <p, when w is infinite, or when a
is a right angle. In this case therefore cr=0, and we get, when n
is infinite, and <p not 0,

'i+»V,-fr d$ n_T (55)

We might have derived this theorem directly from (46) , by the trans-
formation

\n sin <p = tan G>.

This is case I. in the Table, p. 7.

CHAPTER II.

ON THE SPHERICAL PARABOLA.

20.] It remains now to exhibit a class of spherical conic sections
whose rectification may be effected by elliptic integrals of the first
order.

ON' THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 29

Tin- curve which is the gnomonic projection of a plane parabola
on the surface of a sphere, the focus being the pole, may be rectified
by an elliptic integral of the first order.

Let a sphere be described touching the plane of the parabola at
its focus. The spherical curve which is the intersection of the sphere
with a cone, whose vertex is at its centre, and whose base is the
parabola, may be called the spherical parabola.

To find the polar equation of this curve.

The polar equation of the parabola, the focus being the pole, is

r=— — , 4g being the parameter of the parabola. Let y be the

angle which g subtends at the centre of the sphere, and p the angle
subtended by r, then

(56)

. . .

1 + COS &)

Let p be the perpendicular from the focus on a tangent to the
parabola, p the angle which this perpendicular makes with the

axis of the parabola; p = - -. Whence in the spherical curve,

-

as j9 = A:tanw, ff = ktan.<y,

tan 7 sin 7

tanCT = — -ij or smCT= , ' . . (57)*

cos//, Vl — cos2 7 sin2/*

* The expression for a perpendicular arc from the focus of any spherical ellipse
on a tangent arc to it may be found as follows : —

The spherical triangle, fig. 2, sec. [4], FOF', in which OFF'=/z, OF=5--or',

'

. TT . sinw" — cos2t sinw'

=-w , grves

from (6) we have sin -or' sinw" = sin (a-f-«) sin (a — t); eliminating sin -or" between
these equations, we obtain, after some reductions,

. 2 _ sin2(2e) cosV+2 sin(a +f) sin(a — e) cos(2e) +sin(2e) cos /* V sin2(2a) — sin2( 2«) sin2/*

felll" "W, SS _ . -n_ TT f\r-i • •> sc\ \ • 5 T •--•--- ' •

2[l-sm2(2f)sm'!/i]
When the curve is the spherical parabola, a-f-f = t-j, a — e=y, and the preceding

expression becomes sinw'= ^ - - or sinw'=l as we take the sign

V l-cos-'-ysm2/*
-or +•

The locus of the foot of this perpendicular is a great circle touching the sphe-
rical parabola at its vertex. Draw the tangent circle at A, and produce the
perpendicular -or' until it meets this tangent circle in D. Write 8 for this pro-
duced perpendicular arc. Hence in the right-angled spherical triangle D 0 A,

cos a = tan y cot 8. or tan 8= — ^. Buttaniir'= — -. Whence -0^=8. The
cos p. cos p.

second value of w', when the circle ie drawn touching the spherical parabola at

the other vertex B, is K, as shown above. This is manifestly the true value of

m

w', since the focus F is the pole of the great circle touching the curve at B.

30 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS,

Introduce this expression into the general form for spherical rec-
tification, G- = Jsin-ard/A + r, given in (31); we use the positive

sign with T, since t = ~- ; and as r, *&, and p are the sides and an

angle of a right-angled spherical triangle, since 2yu, = o>, we get, by
Napier's rules, tan r = sin or tan/*, whence, by substitution,

(58)

=rin f

'J -

. — cos"7sm p
When the sphere becomes indefi-
nitely great, the spherical parabola
approaches in its contour indefi-
nitely near to the plane parabola ;
k being the radius of the sphere,

a
sin 7= tan 7=^-,

since 7 in this case is indefinitely A
small, whence cos2 7 = !. In this
manner, since s = ka, (58) may be
transformed into

_ sin/*

yl— cps*ysui

9

1
ui*/frJ

f

r=<7i

J

COS fJ, COS^jU.

the well-known formula for the rectification of a plane parabola.
When, on the other hand, the sphere becomes indefinitely small
compared with the parabola, 7 approximates to a right angle, and
(58) becomes

s = fj, + tan"1 (tan //,) = 2/i,

as it should be, since 2fj> is the angle which the radius vector p
makes with the axis.

We shall find the notice of these extreme cases useful.

21.] Although we have called this curve the spherical parabola,
as indicating its mode of generation, it is in fact a closed curve,
like all other curves which are the intersections of cones of the
second degree with concentric spheres. It is a spherical ellipse ;
and we shall now proceed to determine its principal arcs.

Let ADG be a parabola, F its focus, O being the centre of the
sphere which touches the plane of the parabola at F, and being also
the vertex of the obtuse-angled cone, of which the parabola ADG is
a section parallel to the side of the cone OB. Let the angle AOF or
the arc Fa be 7, a and /3 being the principal semiangles of the cone ;

whence

.
1 — sin y

ON THE CKUMKTRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 31

Fig. 10.

To determine the angle /3, or the arc Cb. Bisect the vertical angle
AOB of the cone by the line OD, and draw DG an ordinate of the

x'pv/-! v 2

parabola. Then tan2/3=(^r-pr \ . As AOD is an isosceles triangle,

OF

=AO = : and

cosy

OF2

20 F:

sur

We have also, as DG is an ordinate of the parabola,

f\ Tfl f\ Ll2

DG2=4AF x AD=4OF.tan 7 x — =4 —

cosy cos

Hence, substituting, tan2/3=v -i-^-.

1— 81117

-l±™y, tan* £= ,2 si" ? .
1 — sm7' 1 — siny'

"We may therefore announce the following important theorem : —
The spherical ellipse ,whose principal arcs are given by the equations

(59)

7 being any arbitrary angle, may be rectified by an elliptic function
of the first order.

Write x for tan a, y for tan ft, and eliminate sin 7 from the pre-
ceding equations,

tan2a-tan2/3=#2— ya=l, .... (59*)
the equation of an equilateral hyperbola. We thus obtain the fol-
lowing theorem : —

32 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Any spherical conic section, the tangents of whose principal semi-
arcs are the ordinates of an equilateral hyperbola whose transverse
semiaxis is 1, may be rectified by an elliptic function of the first order.
The quadrature of a spherical conic may be effected by an elliptic
function of the first order, when the cotangents of the principal semi-
angles of the cone are the ordinates of an equilateral hyperbola whose
transverse semiaxis is 1.

22.] When we take the complete function, and integrate between

the limits 0 and -^, we get, not the length of a quadrant of the

spherical parabola, as we do when we take the centre as origin, but
the length of two quadrants or half the ellipse. We derive also
this other remarkable result, that when /* is a right angle, the
spherical triangle whose sides are the radius vector, the perpendi-
cular arc on the tangent, and the intercept of the tangent arc be-
tween the point of contact and the foot of the perpendicular, is a

7T

quadrantal equilateral triangle. For when /u-=— ,

TT 7T 7T

p=~2' ' =2' ^ = 2'

It may also easily be shown, that the arc of a great circle which
touches the spherical parabola, intercepted between the perpendi-
cular arcs let fall upon it from the foci, is in every position con-
stant, and equal to a quadrant*.

Hence the spherical parabola is the envelope of a quadrantal arc
of a great circle, which always has its extremities on two fixed great
circles of the sphere, the angle between the planes of these circles

being =— + y.

If we take the spherical conic supplemental to the given sphe-
rical parabola, the foci of this latter are the extremities of the
minor principal arc of the former, and the cyclic arcs of the former
are tangents to the latter at the extremities of its major principal
arc.

Resuming the equations given in (59), which express the tan-
gents of the principal semiarcs of the spherical parabola in terms
of sin y, namely

,
1 — sra.7'

, sin y

As sin •or = . ..-- ' -, and sin w' sin -or" =sin y. see (6). we must have
V 1 - cos2 y sm2 yu,

siniir"= Vl— cos2y sin2 p. Hence, as &'=—— FO, and w"=^— F'O,

2i 2

coa FO . cos FO' = sin y = cos FF : or the angle FOF' is a right angle. (Fig. 2.)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 33

writing i for cos 7, and j for sin 7, we get

whence

TT
Again, since 2e + y=-, sin2e=cos7=z,andcos2e=t/.

Now n=tan2e, 7n=e2: hence n=w=

(CO)

It is proper to remark that, in the case of the spherical parabola,

l — j
i is not the modulus, but ^ — -..

23.] We shall now proceed to the rectification of an arc of the
spherical parabola, the centre being the pole. By this method we
shall obtain certain geometrical results which have hitherto ap-
peared as mere analytical expressions. In (8) or (28) we found
for an arc of a spherical ellipse measured from the major principal
arc, the following expression, the centre being the pole,

>*£.,;- of d(P

a—.

"tan a J (1 — e2 sin2 <p) \/l — sin2 rj sin2 <p'

or, substituting the values of the constants given by the preceding
equations,

** (61)

But when the focus is the pole, we found for the arc the following
expression in (58),

Equating these values of <?; we obtain the resulting equation,

. . (62)

24.] We shall now show that the amplitudes <p and /u, in the pre-
ceding formula are connected by the equation

tan(<p — /it) =j tan /*, (63)

a relation long ago established by Lagrange.

VOL. II. y

34 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Let «r and -or' be the perpendicular arcs from the centre and focus
of the spherical parabola on a tangent arc to the curve. Let X
and //, be the angles which these perpendicular arcs make with the
major principal arc. The distance between the centre and focus of
the spherical parabola, with the complements of these perpen-
diculars, constitute the sides of a spherical triangle. We shall
therefore have

Now sec2«r=sec2a cos2X + sec2/3 sin2X, as in (35) j or writing for
sec a, sec j3 their particular values in the spherical parabola, given

in (59) 2

sec2 -53- = = sin2X (65)

1— sin 7

. tan 7 2 , tan2 7 + cos2 /* .
Again, as tan*r' = ^, CosV

reducing (64), the result is

2^ 2(1+ sin 7) , ,

tanX=(cot/.-sinytan/u)2

In the case of the spherical parabola,

C0s2e_!±^_^ whence (66) becomes

1+siny , tan/i + sinytan^

cosetanX= —. — , or cosetanX= = — — . (67)

cot p,— sin 7 tan p' 1— sin y tan //,. tan p

The second member of this equation is manifestly the expression
for the tangent of the sum of two arcs fj, and v, if we make
tan v = sin y tan /tt.

Hence cos e tan X = tan (/z + v) .

In (25) , or (38) or (39) , we assumed tan <p = cos e tan X.
Hence <p=//, + v, or tan (<p — /i)=tauv = sm7 tan /A.

A simple geometrical interpretation of Lagrange's theorem,

tan (<p — fj,} = sin -jPtan fj,
may be given by the aid of the spherical parabola.

Let DR'B be the great circle, the base of the hemisphere, whose
pole is F (fig. 11). Let BQA be a spherical parabola, touching the
great circle at B, and having one of its foci at F the pole of the
hemisphere whose base is the circle DR'B. Let RQ, be an arc of a
great circle, a tangent to the curve at Q. From F let fall upon it
the perpendicular arc FR. The point R is in the great circle AR
which touches the curve at its vertex A (see note to p. 29) . The

pole of this circle is the second focus F'; for AF' = FB = -. Let
the arcs RF, RF make the angles fi and v with the transverse arc

ON TFIE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

11

AM. Hence AR=v. In the spherical triangle FAR, right-angled
;it A, \\c IKIVI- sin AF = tan v
cot ft. Now, as AF = y, sin A F
r=siny=y; and if <p =
v = tp— ft, or, reducing,

tan (p —/A) ==/ tan fj, ;

whence wr may infer that while
Me original amplitude is the
angle fj, at the focus F, Me rfe-
rived amplitude <p w Me sum
of the angles p and v at the
foci F and F', or the ampli-
tude <p is the sum of the arcs
of two great circles, touching
the spherical parabola at the
extremities of the principal
major arc of the curve, inter-
cepted between those points of contact and the perpendicular arc FR
let fall from the focus F on the tangent arc RQ to the curve.

Hence while the original amplitude p is equal to an arc of the
tangent circle at B, made by RF produced to meet this circle BR',
the derived amplitude <f> is equal to the sum of two arcs of the tan-
gent circles drawn at A and B, and given by the same construction.

When the function is complete, or ^ — -^, R will coincide with
R' the pole of the great circle AB, whence v is also = -; and as

= 7r. This shows that when the function is complete,
or the amplitude is a right angle, the amplitude of the derived
function will be two right angles.

When the spherical parabola approximates to a great circle of
the sphere, the second focus F' will approach to F the immovable
focus. The arc RF' will .therefore approach to coincidence with
the arc RF, or the angle v will approximate to p, so that p = p + v
= 2fj, nearly.

This is the geometrical explanation of the analytical fact observed
in this theory, that when the modulus diminishes, or the spherical
parabola approximates to a great circle of the sphere, the ratio of
any two successive amplitudes approximates to that of two to one.

When the greater principal arc of the spherical parabola is a

right angle and a half, sin 7=—^, and, if C be its circumference,

C=

-f TT. But two quadrants 2s, or the loop

36 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

w

of a lemniscate, are= S%F- / /i~ "' HenC6 25=C~7r'

Or the loop of a lemniscate is equal to the difference between the
circumference of the spherical parabola whose greater principal arc

is -y, and a semicircle.

When a quadrant of the spherical parabola is taken, or when the
point of contact Q coincides with the extremity of the principal

7T

minor arc of the curve, we shall have <p = —y

Sinceinthis caseRQ=PQ,FV=FV; therefore A6=.OFV=OFV,
or RF'V=/i + v. As V is the pole of RP, and F is the pole
of AR, the point R is the pole ,,. , „

of VF. Hence RFV is a
right angle ; but /* + v =RF'V,

whence <p = o- As

tan(<p — /u.) —j tan p,

when 0=77, tan /*= — -r-. If

z VJ

in the expression

;' tan u,
tanr= . J . r==

given in (58), we substitute
this value of tan ft, we shall get

7T

tanr=l, or r=--
4

Since FVF' (fig. 12) is an isosceles spherical triangle, and
cosFF = cos2e=/, and tan2 FFV=tan2/* =i, cos FF tan2 FFV = 1,

or the angle V is a right angle, or PR is a quadrant.

As two quadrants of a spherical parabola are together double of

one, we shall have, writing the integral I — ^ — in the

J vl— «2sin2/A

abbreviated form I — %=,

J VI

7T

- or

r

ON THE GEOMETRICAL PROPERTIES OK ELLIPTIC INTEGRALS. 37

Now, when i is nearly 1,1 -^= = 1- - = log (sec/u, + tan /*).
Taking this expression between the limits /u.=0, and^fc=tan~'(-j ,

\r /

we shall have, since sin /*= , - ., cos u= — .-——, and neglecting

Vl+j Vl+j

j and its powers when added to l,j being very small,

2 f *?y d/a / 2 \

=—-=, whence I -r^ = l°g( — 7=l-

v> Jo VI V vj/

w

Therefore (68) gives f2^?? =log(^)* ....... (69)

Jo VI V'

25.] To show that

ffy 1

vra^-i+

the amplitudes <p and /u, being connected as before, by the equation
tan(<p— /*) =;' tan/A. Since, as in 67,

1 + sin 7

tanp = — - = - —/- -- ,

cot fji — sin 7 tan /M cot /it — ^ tan /*

differentiating this expression with respect to <p and /A,
1+^ d^ = cosV+^sinV^
sin8 ^ d/x, cos2 /^ sin2 /L6 * '

tan* f = (l+
(co

Whence, after some reductions,

We have also tan* f = . {7

(cos2 /A —7 sin2 /A) 2

(7

I _ A 2

Multiplying this expression by ^ — -.} , and reducing,

Multiplying together the left-hand membei*s of the equations (70),

* ...... " resultat fort remarquable, d^j4 signal^ par Legendre ; mais nous

ignorons comment il y est parvenu.' — VEHHULST, Traitt Eltmentaire dcs Fonctions

EUipfiqncs, p. 168.

38 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

(72) and (73), and also the right-hand members together, we shall
get, after some obvious reductions, and integrating,

This is the well-known relation between two elliptic integrals of

1—7

the first order whose moduli are i and — -., or, in the common

.1 l~b
notation, whose moduli are c and ^ ,.

;' tan a

26.1 Let r be the arc whose tangent is — r" .„ .-.,—'

V 1— 1 2sm2/A

then tan2T=sn^c°s^ ~ an . . . (75)

cos4 fji — i* sin4 fj,

and combining (71) and (73), we shall find

^ (1 +j) sin /A cos //, V 1 — i* sin2 ,

— «'\2 . cos4/!.— y2 sin4 fj,

Dividing (75) by (76), the result becomes

2;
•'

tan2r= . ... (77)

We are thus enabled to express T, the portion of the tangent arc
between the point of contact and the foot of the perpendicular arc
on it from the focus, in terms of <p instead of /A.

If we introduce this value of T into (62) and combine with it the
relations established in (74), the resulting equation will become

r /i ,*\ -i /

1 i J \ ' 2 tr, 1 \

/l_/x2 \ /

fl-j\*a. 2

L \l+j)a ll ^J'V

\i+^/ J V

:

\ ^j /

/ /I — 7\2
A / 1 1 «/ \ «in2 ..

V y J[ J_ iy

/

. (78)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 39

Adopting for the moment the ordinary notation of elliptic integrals,

I—/ 2;

m = — c = , — -.. whence 1 + c = =-^-.«
1 +J 1 +J

Introducing this notation, the last formula will become

. . (79)

In the ' Traite des Fonctions Elliptiques/ torn. i. p. 68, we meet
with the formula

v 1 —

Now, when rc= — c, this formula becomes

c)tany

(80)

(81)

whence (79) and (80) are identical.

27.] Let us now proceed to rectify the spherical parabola by the
formula for rectification given in (47), the centre being the pole.
For this purpose, resuming the formula for rectification established
in (41), and deducing the values of the parameter, modulus, and
coefficients in that expression from the given relations,

(82)

we get

1 — sin 7 1— j L— sin 7 1— j

The parameter, tan2e =
The modulus,

The coefficient -.

sm a cos a l+j

The coefficient cosacos/3=J-L/
sin a 1 +j

l-j

sin 77 = :

cos/3 2

and

etane=

i . (83)

Making these substitutions m (41), the resulting equation will become

n - (84)

40 ON THE GEOMETRICAL PEOPERTIES OF ELLIPTIC INTEGRALS.

But from (58), the focus being the pole, we derive

• • • (88)

In (74) we showed that

Introducing this relation into the last formula, and equating
together the equivalent expressions for the arcs in (84) and (85),
we get for the resulting equation,

..(86)

We shall now proceed to show that the common formula for the
comparison of elliptic integrals having the same modulus and am-
plitude but reciprocal parameters is, in this particular case, identical
with the geometrical theorem just established.

The formula is, in the ordinary notation,

. . (87)

We must accordingly show that, c being tan2e, and therefore

-. I sm <p cos <p

tan~

+tan2e)tan<p

• (88)

If we write T, T', and 3 for these angles respectively, we have to
show that

:* (89)

ON 1'Hi: <;I:OMKTKH:AL PROPERTIES OF ELLIPTIC INTEGRALS. 41

r-f- r1 is the arc of the great circle, which touches the spherical
parabola, intercepted between the perpendicular arcs let fall from
the centre and focus upon it.

We must, in the first place, by the help of Lagrange's equation
between the amplitudes, established on geometrical principles in
sec. [24] , reduce these angles to a single variable, fju is taken as
the independent variable instead of <p, as the trigonometrical func-
tion of <p in terms of fi is in the first power only.

We have, therefore,

,

vl—

The equation between the amplitudes <p and /
tan (p — n] —j tan /-t, gives

cos2 /A— ^ sm-'/A

Eliminating p by the help of this equation, from the value of tan t
given in the preceding group,

(1 —7) sin LL cos u, cos2 u, + j sin2 u,

tanr = — ,±L — ^= — x — 9 J. . 9~.

v 1 — i2 sin2 fj, cos2 /* — j sm2 /u,

Using this transformation and reducing,

tan (T + T7) = tan /A V 1 — i2 sin V, .... (92^
a simple expression for the length of the tangent arc to the spherical
parabola between the perpendicular arcs let fall from the centre and
focus upon it.

From the last equation we may derive

'2^ (93

cos p, —j- sn ^
Using the preceding transformations, we may show that

cos A^l ""** 8^n2 /*

cos4/!*— y2 sin4 /A
Hence d=2(T + r'). .......... (94)

VOL. II. O

42 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Therefore (86) becomes

(95)

We have thus shown that in the particular case of the general
formula for comparing elliptic functions of the third order with
reciprocal parameters, when the parameter is positive and equal to
the modulus, the circular arc in the formula of comparison (87) is
equal to twice the arc of the great circle touching the curve and
intercepted between the perpendicular arcs let fall from the centre
and focus upon it.

If we take the parameter with a negative sign, the circular arc r
in (62) will represent the tangent arc between the point of contact
and the foot of the focal perpendicular.

The spherical parabola, like any other spherical ellipse, may be
considered as the intersection of an elliptic cylinder with a sphere
whose centre is on the axis of the cylinder.

Let a and b be the semiaxcs of the base of the cylinder, and k
the radius of the sphere, a and /3 being the principal semiarcs of
the spherical parabola,

but in (59*) we found tan2 a — tan2/3 = !
shall have, i being the eccentricity of
cylinder,

*2 = «s(l+i). t ^ }

28.] The foregoing investiga-
tions furnish us with the geome-
trical interpretation of the trans-
formations of Lagrange. Let the
successive amplitudes <p, -fy, % of
the derived functions be con-
nected by the equations

tan (<p — fi) =j tan //,, •>

tan(^r— <p)==/,tan<p,

tan (x— ^r) =ju tan ty,
&c., &c.,

We may imagine a series of con-
focal parabolas having a common

; hence, substituting, we
the base of the elliptic

(96)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 43

axis, described on a plane in contact with a sphere at their common
focus. These parabolas will generate a series of confocal spherical
parabolas on the surface of the sphere, BCA, BC'A', BC"A", BC'"A'",
which will all mutually touch at the vertex B remote from the
common focus F. Let the distances between the common focus F
and the vertices of the plane parabolas subtend, at the centre of the
sphere, angles 7, «/, 7", &c., whose cosines i, it, ilt, &c. are connected
by the equations

l_-v/fZ]2 i_^/iZT« l-i/fTTa

t.= -- — , z..= - - - '., z,,,= - , - " . . . &c., (yoj

nvi=? i + si-i*' 1-^i+y

it is plain that y = FA, «/=FA', / = FA", 7"' = FA'", &c.

We may repeat this construction successively, until the parameter
of the last of the applied tangent plane parabolas shall become so
indefinitely small, compared with the radius of the sphere, that it
may ultimately be taken to coincide with its projection. We shall
in this way reduce, at least geometrically, the calculation of an
elliptic integral of the first order to the rectification of an arc of a
parabola — that is, to a logarithm, as in sec. [20] . If, on the con-
trary, the moduli i, i,, itl, &c. proceed in a descending series, the
angles y, yt, ylt continually increase, the magnitudes of the con-
focal applied parabolas increase, till at length their parameters
become so large, compared with the radius of the sphere, that their
central projections pass into great circles of the sphere. The eva-
luation of the elliptic integral will therefore ultimately be reduced
to the rectification of a circular arc. These are the well-known
results of the modular transformation of Lagrange.

The formula established in (58) for the rectification of the sphe-
rical parabola, gives

_
v 1 — cos2 7 sin2 /z, L v 1— cos27sin

or, writing i for cosy,,/ for'sin 7, and VTfor ^1 — i2sin2

<7 — T =

cr' and T' being the corresponding quantities for the next derived
spherical parabola,

Now ,>4 and=T}.v M in (98) and (74),
whence 2(<7 — T) = v//(a'~ T') ...... (99)

44 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Thus a simple ratio exists between the arcs, diminished by the
protangents, of two consecutive confocal spherical parabolas.

TT

When the functions are complete, //. is taken between 0 and -^ ;

it

<p therefore, as in sec. [24], must be taken between 0 and TT; but
when the amplitude is taken between 0 and TT the function is
doubled. Moreover, when the functions are complete, the point
Q coincides with B ; so that in this case the complete function
represents not one, but two quadrants of the spherical parabola,

*7T"

the focus being the pole. Hence as T = -^, T^TT. It must be re-

&

membered that <r denotes two quadrants of the spherical parabola
as shown in sec. [24] .

Whence putting C, C', C", C'", &c. for the circumferences of the
successive confocal spherical parabolas, derived by the preceding
law, we may write

C -w=s V; (C, -TT) ^

C, -"=•>; (C,,-9T)

— TT

Multiplying successively by the square roots of j, j,, jn, jni, &c.,
adding, and stopping at the fifth derived parabola,

C -TT= jjJuJMJn &c. (Cy-*r).
Let this coefficient be vTf and we shall have

C— 7T=VJ (CV-7T) ....... (101)

Now we may extend this series, until the last of the derived
spherical parabolas shall differ as little as we please from a great
cirale of the sphere. Let the circumference of this last derived
spherical parabola be Cv. Then Cv=27r, and (101) becomes

C=7T(1+^J) ........ (102)

Hence, calculating the quantity J, we may express the circum-
ference of a spherical parabola by the circumference of a circle.

When all the spherical parabolas are nearly great circles of the
sphere, i, i,, i,,, «w = 0, nearly; and jjjlfjHju,= lt nearly. Whence
J=l, nearly; or

C = 27T ..... ... (103)

When the spherical parabolas are indefinitely diminished,

^ i» *// = l> nearly, and j,jt,jtl,jtll = Q, nearly, therefore J = 0 nearly;

or C = TT ......... (104)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 1 ."»

Hence the circumferences of all spherical parabolas are greater
tl 1:111 two and less than four quadrants of a great circle of the
sphere.

XM).] Denoting the angles at the centre of the sphere, subtended
by the halves of the semiparameters of the applied confocal para-
bolus, by 7, y, y", &c., we shall have cos 7 = ?, cos y =ij, cosy' = iw,
cos /' = iul, and sin 7 =;' , sin y ==;'„ sin y ' =j,,, sin y" ==;„,. We may,

using successively the equation i,= _H — _i, determine in terms

1 + ^1 — i2
of / the successive values of i., i,,, i,,,, and of;., /,,. ;',.,. &c., as follows : —

•/ I' II' III'' */ l' V tl'*' III' *

i*T .• .-rfl±3t^T

-

«-= [ ^

=

I . (105)

Hence we may derive the successive values of jpju,jllt in terms
For

o«,-

(106)

We may express the coefficient J, or the continued product of
J>Ji>Jii>Jm> &c-> in terms ofy> tne complement of the original mo-
dulus. Including in our approximation the fifth derived modulus,
we get

(2)i . (2)'+* . (2)'+*+* . (2)

(107)

As an elliptic integral of the first order may be multiplied, or
divided into any number of equal parts, as shown in every treatise
on this subject, so its representative, an arc of the spherical para-
bola, like that of the circle, may be multiplied, or divided into any
number of equal parts.

30.] It may not be out of place here to show, although the in-
\ cstigation more properly belongs to another part of the subject,
that the arc of a spherical parabola may be represented as the sum

46 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

of two elliptic integrals of the third order, having imaginary para-
meters ; or in other words, that every elliptic integral of the first
order may be exhibited as the sum of two elliptic integrals of the
third order, having imaginary reciprocal parameters.

Parameters, whose product is equal to the square of the modulus,
may be called reciprocal parameters.

Assume the expression given in (58) for an arc of the spherical
parabola, the focus being the pole, and /j, the angle which the per-
pendicular arc from the focus, on the tangent arc of a great circle
to the curve, makes with the principal transverse arc,

Jdit . f sin y tan p

.. r _ + tan-' .{-7-=

*/l-cos2-ysinV (vi— Cos2 y sin

Let cos 7=2, sin 7=7, and, to preserve uniformity in the notation,
write <p for p. Then differentiating the preceding equation, it
becomes after some reductions,

dor _ j[l— P sin2 <P + cos2 <p +J2 sin2 <p]

'

dtp [cos2 <p — iz sin2 <p cos2 <p +j* sin2 <p] </ 1 — issin*p'

Now the numerator is equivalent to 2/(l — i2sin2<p), and the first
factor of the denominator may be written in the form

1 — 2i2sin2<p + i2sin4(p.

But i*=i2(i*-t-j*)) hence this last expression may be put under the
form 1 — 2z2 sin2 <p + i4 sin4 <p + i2/2 sin4 <p. This expression is the sum
of two squares. Resolving this sum into its constituent factors, we
get

_ _ _ b

(i-<;V-l)sin2(p]^l-z2sin2<p'

Now this product may be resolved into the sum of two terms.
Let

do P

. — i(i —jv—l) sin2 <p] v 1 — i2 sin2 <p J
or, reducing these expressions to a common denominator,

sn < 1 - ii- 1 sin2

and comparing this expression with that given in (b), we shall see
that

= 2/, P-Q=0; therefore P=>, Q=/. . . (e)

o\ 1111 ill o\ll 1 KICAL PROPERTIES OF ELLIPTIC INTEGRALS. 47

Integrating (c), we get

- I)sin

d<p

(108)

_ 2 (j —j v/ _ l ) sin2 <p] v 1 — i2 sin2 <p
If we replace i by cos y, and j by sin y, the parameters become
cos 7 (cos 7 + ^ — 1 sin 7) and cos 7 (cos 7 — v' — 1 sin y) , whose pro-
duct is cos2 y, the square of the modulus. They are therefore re-
ciprocal ; and putting m for cos 7 (cos 7 + v' — 1 sin y) and — n for
cos y (cos y — */ — 1 sin 7) , we shall find that these values of m and n
satisfy the equation of circular conjugation, m — n + mn=i?. It
follows therefore that when the rectification of the spherical para-
bola is effected, the centre being the origin, the representative elliptic
integral is of the third order and circular form ; the parameters m
and n are equal to each other, and to the modulus, and are therefore
reciprocal. But when the focus of the spherical parabola is as-
sumed as the origin, the rectification of this curve may be effected
by an elliptic integral of the first order, and this integral may also
be exhibited as an integral of the third order and circular form,
but with imaginary parameters, which are also reciprocal.

CHAPTER III.

ECTIONS WITH REC
X ?/

31.] Let -2 + ^=1 be the equation of an ellipse, the base of an

ON SPHERICAL CONIC SECTIONS WITH RECIPROCAL PARAMETERS.
X9

-2

elliptic cylinder. Let two spheres be described, having their centres
at the centre of this elliptic base, and intersecting the cylinder in
two spherical conic sections. These sections will have reciprocal
parameters, if k, H, the radii of the spheres, are connected by the
equation

(*2-«2)(*,2-a2)=a4*2, ..... (109)

a2 — 62
i2 being, as before, equal to -- ^— .

0

When k and kt are equal, we get k2 = az(l + i). This value of k
agrees with that found for k in (96) ; or, in other words, when the two
spheres coincide, the section of the elliptic cylinder by the sphere is
a spherical parabola. Hence also, a spherical parabola always lies
between two spherical conic sections having reciprocal parameters.

Let e2 and e'2 be the parameters of those sections of the cylinder
made by the spheres. Then, as shown in (9),

_=
sin2 a cos2 0 ~ a2 (/t2 - fc2) A2 - a2 + a9? '

48 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

but the equation of condition (109) gives

In the same manner the spherical conic whose radius is k,' gives

e =

, 2
: therefore e-ef —

or e2 and ef are reciprocal parameters.

To compute in this case the value of the coefficient — - sin
in the expression given in (10) for rectification,

_tan_/3 . C _ dp

~~ tan a Sm P J [1 _ e2 sin2 <p] V 1 - i2 sin2 <p'

Since

tan2/3 .
we obtain by substitution, t-^ sm ^ =

but the equation of condition (109) gives

As this expression is symmetrical, we shall have for the spherical
conic section whose radius is kt,

tan/3, . a W- n^^\

-^sm^yj ....... (Ill)

tan at kk,

tan/3 . tan/3, . . /no\

Hence -sm/3 = - — aMHftj .... (112)

tan a tan at

or the coefficients of the elliptic integrals which determine the arcs
of two spherical conic sections having reciprocal parameters are
equal.

Let K be the criterion of sphericity ; then, as

«=*, ......... (113)

32.] To determine the values of the angles \ and A/ which cor-
respond to the same angle <p in the expressions for the arcs of sphe-
rical conic sections having reciprocal parameters.

cos2 a F-a2 kz-a2

Since cos* e = — =-5 = -5 — ^ = 75 - ~~ — ^o,

cos2/3 k2 — 62 A2— a2 + a2r

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. I'J

introducing the equation of condition (£2 — a2) (k?— a'2)=a4 P,

we get cose=T; but tan <p = cose tan\, as in (39); hence
Kt

k k

tan X = -' tan <p, and tan X, = - tan <p ;

if CL

Fig. 14.

therefore k tanX=^ tanXy, (114)

or the tangent of the angle X which the perpendicular arc from
the centre of the spherical conic, on the arc of a great circle
touching it, makes with the principal major arc, is inversely as
the radius of the sphere.

A simple geometrical construction
will give the magnitude of those
angles X and Xr Let the ellipse
OAB be the base of the cylinder ;
OCC', ODD' being the bases of the
hemispheres whose intersections with
the cylinder give the spherical conic
sections with reciprocal parameters.
Erect the tangents DP, CQ, each

kk
equal to — ' tan <p, and join PO, QO.

The angles AOP, AOQ are X and Xr
\\hen DP = CQ=0, X=Xy = 0; "

whenDP=CQ=ao,X=X,=£. The condition (109) shows that
when k = a, £y = cc . Now as k, tanX,= a tanX is finite always so

*7T

long asXis not absolutely = ~> in order that its equal ^tanXy may

m

be finite also, we must have Xy always equal to 0 for every finite
value of tan X.

33.] The tangent of the principal arc of a spherical parabola is
a mean proportional between the tangents of the principal arcs of
two spherical conies with reciprocal parameters, the three curves
being the sections of the same elliptic cylinder by three concentric

spheres.

Since

Introducing the equation of condition (A:2— a2)(^2 — a2) =«4 t2
(109), we get

tan a tan 0,= . ........ (115)

VOL. n.

50 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Let kn be the radius of the sphere whose intersection with the
cylinder gives the spherical parabola; then £y/2=a2 (l+«). See
(96).-

Hence

2-a*=a2

and tan2 ^=£-2^2 =

tan a tan a^tan2 alt.

(116)

15.

therefore

The altitudes of the vertices
of the three principal major
arcs of the two spherical
conies with reciprocal para-
meters, and of the spherical
parabola, above the plane of
the elliptic base of the cylinder,
are in geometrical progres-
sion. Let AQ, be the alti-
tude of the vertex of the
major arc of the spherical
parabola ; AP, AR the corresponding altitudes of the vertices of the
major arcs of the spherical ellipses which have reciprocal parameters.
Then AP= V k* — a2, AR= V^/2 — a*> AQ,= \/ ktf — a/t=a \/i.
The equation of condition gives, as in (109), APx AR=AQ2.

We shall give, further on, an expression for the sum of the arcs
of two spherical conic sections having the same amplitude, but re-
ciprocal parameters.

34.] The projections of supplemental spherical ellipses on the
plane of XY are confocal plane ellipses.

For sin ?7=sin e', sin 7/=sin e. See sec. [9] .

Hence

a*-b*_a*-b* a?-bf_a*-

This gives as the resulting value,

or

Two cones, supplemental to each other, are cut by a plane at
right angles to their common internal axis. The sections are con-
centric similar ellipses, having the major and the minor axes of
the one coinciding with the minor and major axes of the other.

For

tan2a-tan*£_
tan2 a

9 tan2 a.~ tan2 /9, cot2/3 — cot2 a tan2 a- tan2 /S
and f,—— s -=— -Tg^j- — =— — 2- -> oit'sse.
tan2ay cot2/3 tan2 a

ON THE GEOMETRICAL PROPKKT1 US of ELLIPTIC INTEGRALS. .") 1

CHAPTER IV.

ON THE LOGARITHMIC ELLIPSE.

35.] The logarithmic ellipse may be defined as the curve of
symmetrical intersection of a paraboloid of revolution with an
elliptic cylinder. This section of the cylinder by the paraboloid
is analogous to the section of the cone by the concentric sphere in
[7] ; for this cylinder may be viewed as a cone having: its

sec.

vertex at the centre of the paraboloid, i. e. at an infinite distance.

Let the axes of the paraboloid and cylinder coincide with the
axis of Z, the vertex of the paraboloid being supposed to touch
the plane of XY at the origin O.

It may be proper to note that every tangent plane to the elliptic
cylinder will cut the paraboloid in a parabola, just as every tan-
gent plane to a cone will cut a concentric sphere in a great circle.

Let k be the semiparameter of the paraboloid Oab, and let a and b
be the semiaxes of the base of the elliptic cylinder ACB ; then the
equations of these surfaces, and consequently of the curve in which
they intersect, are

(117)

52 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Let dS be an element of the required curve,

.... (118)

#, y, and 2- being dependent variables on a fourth independent
variable 0.

Assume

x = acos0, y = b sin 9, then o2 cos2 6 + £2 sin2 0 = 2fo. (119)

Differentiating and substituting,

(r\ a — }

j* ) = a2 sin2 0 + &a cos2 0 + (- —gr-1- sin2 0 cos2 0. (120)

To reduce this expression to a form suited for integration, it may
be written,

/t2 + (a2-62)[F + a2-Z>2]sin20-(a2-62)2sin40. (121)

This expression may be reduced as follows :

Let P = 62#2/Q=(a2-62)[F + a2-&2], R=-(a2-62)2; (122)

and the preceding equation will become

in4 "0 ..... (123)

Let this trinomial be put under the form of a product of two
quadratic factors,

(A + B sin20) (C - Bsin20) =AC J- B (C - A) sin2 0 -B2 sin4 0. (124)
Comparing this expression with the preceding in (121), we get

AC=62*2, C-A=/t2 + a2-62, B = a?-£a. . . (125)

To integrate (123) : assume tan2<p = ~ ?tan20. . . (126)

J\.

The limits of integration of the complete functions will continue
as before. Making the substitutions indicated by the preceding
transformations, the integral will now become

. (127)

.... (128)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 53

These values of A, B, C satisfy the equation m + n — mw = i2, as
assumed in (1). As A;2=C— A— B, C>A + B, or n>m, the pre-
ceding expression may now be written

It will presently be shown that A and C must always have the
same sign, whence i^>n.

'+B

As i2= -- -, and as C is always greater than B, i2<l. From

1 + B

(125) we may derive

B) AC

k*~ (C-A-B)2 ' A2-(C-A-B)2

Now, that the values of a and b may be real, we must have OB,
while A and C must be of the same sign ; but as B is essentially
positive, C, and therefore A, must be positive.

B , A + C i2

Since -T — T5=n> an" — n — =~> as m (1^8),

A. ~p -U \*j 71

we may eliminate A, B, C from the values of the semiaxes of the
base of the elliptic cylinder, and express a, b, and k in terms of
i and n. We may thus obtain

a2 _ n( Ij^Kf^- n) b^ _n(^-n)(l-n)z
¥ [2n=P=n*¥' k*~ [2w-i2-»2]2 ;

or more simply in terms of n and m,

a?_mn(l— m) 62_mn(l— n)

In order that these values of a and b may be real, we must have
n positive, i2 > w, and 1 > i2.

This is case VI. in the Table, p. 7.

If we put c for the eccentricity of the plane elliptic base of the
cylinder, we shall have after some obvious reductions,

(!_;*) (I_c2)=(l_w)2, or c2=«. . . . (131)

Now this simple equation between n, m, and c enables us with
great ease to determine the eccentricity c of the base of the elliptic
cylinder whose section with the paraboloid gives the logarithmic
ellipse, when we know the parameters m and n, or the modulus i,
of the given elliptic integral.

54 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

36.] To integrate the expression given in (127), we must assume

_sin<p cos<p

—

f! — — • — 9^T - .....

[1— ft sm2<p]

Differentiate this expression with respect to <p, and we shall have
<M>n_l — 2(l+i2) sin2<p + 3i2sm4<p 2rc(sin2(p— sin4tp) (1 — t* sin2(p)
d<p " [1 — wsin2<p] Vl— z'2sin2<p [1— wsin2<p]2 Vl — ^sin2<p
Let 1 — w sin2 <p = N, 1 — i2 sin2 <p=I, as before.

Separating the numerators of the preceding expression into their
component parts, and attaching to each their respective denomi-
nators, we shall have

1 1

and

2(l+i2)sin2(p_2(l+i2) (l-nsin2(p — l)_2(l+ig)

The next term gives

3^2sin4(p __ 3J2 (1 — rcsin2<p — I)sin2<p _ 3z2 sin2cp 3^sin2<p
NVI= ""« NVI ""» VI —nNVl' '

Now these two terms may be still further resolved ; for

3z2 sin2(p_3 (l-^2 sin2tp-l) _3 VI 3

n VI

3z2 (1— wsin2<p-

-i- •

'

VI n2 NVI w2Vl »aN VI'

whence (d) becomes

3s'2 sin4 <p 3s/ 3 3i2 Si8

" * '

NI n n Vlw2 Vl w2N VI

Combining the expressions in (b), (c), (d) or (e), the first term of
the second member of (a) may be written

[1-2 (1 + 22) sin2 <p + 3P sin4(p] 3 V
[1— w sin2 (p] Vl— z'2sin2^ n

>• (£)

ON THE CKOMKTRICAL PROPERTIES OK ELLIPTIC INTK<!K\I>. .")."»

nn 2n(sin2<p — sin4<p) VI e i \

Lhe second term, - — . 9 \'9 -- , of (a) may be thus dc-

(1— wsm2<p)2

velopi'd,

VI __ 2rca-ft8in2(p-l) Vl_ 21 21

N2 N2 NV"l + N*~Vi;

and these two latter expressions may be written
21 2l-i2sin2< 2

_

N Vl~ N VI N VI ~» ~ N VI

_2J2J_ 2fl 2

"

n VI w N VI N VI
(g) becomes

N2 wv/I

2w sin4 <pl
1 he term -- — _ may be written

2ralsin4(p_ 21 fl — 2/tsin2(p4-^2sm4(p— 2 -I- 2nsin2<p+ 1"]

' N2VI "«l"L N2V^

.. (k)

21 41 21

+

. \/Iw.N VI w.N2\/T
21 2

41 4(1 -i2 sin2 y) 4 4J2 (l-nsin2<p-l)

wN VI" wN VI ~wN VI4 w2

41 4i2 . (i*

=

Combining (k) with (m), we shall have

2rclsm4(p_2 VI 4i2 4 2_ 1 21

2 ;

N2 VI w2 VI N VI «N2 VI

adding (n) to (h),

2w(sin?<p— sin4<p)I 2 VI /4i2 2i2\ 1

>X n / yi

i - (P)

56 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

adding (£) and (p) together, we shall obtain as the final result,

or multiplying by n, putting for i* its value n + m-mn, transposing
and integrating,

. . - • (r)
(l-») "

But we have shown in (129) that

" Id<p

N2 VI
whence

2(n—m) v

v mn k J

\ . . (133)

+ ™n_n)(^+(n-m^l-nlC d(?-

*

!N VI,

Hence an arc of a logarithmic ellipse may be expressed by a
straight line k<&n, and in terms of elliptic integrals of the first,
second, and third orders, the latter being of the logarithmic form.

The expression j — ™ •- may be reduced to

PC dip _»»fl_ xf d<p
^JN VI n "W)JN2 VI J
and therefore combining this expression with (r) ,

1- • (134)
m

37.] When the elliptic cylinder and the paraboloid are given, we
may determine the parameter, modulus, and constants of the func-
tions which represent the curve of intersection of these surfaces, in
the terms of the constants a, b, and k.

The modulus, parameter, coefficients, and criterion of sphericity
may be expressed as linear products of constants haying simple
relations with those of the given surfaces.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 57

Resuming the equations given in (125),

. (135)

we find ( A + C)2= (£2 + a2 - A2)2 +

Assume 4/>*=F +(« + £)*, 408=

we sliall then have the following equations : —

C=

A=(b+p-q)(b + q-

-a •

-b

Substituting these values in (129) , AVC obtain as the resulting expres-
sions

.2 4 (a -f b) (a—b)pq

~(p + q + b)(p + q-bj(a+p-q)(a + q-p)

<—a}(b+p — q}(b + q—p} [ Q37)

n= (a + b}(a-b}
(a+p—q}(a + q—p)'

These values of m, n, and i2 satisfy the second equation of condition
in (1),

m + n— wm = i2;

and if we denote by K the criterion of sphericity,

p + q — aV

1 i I ) • ' • (J-Oo)

p-\-q — b/

we may express the parameters and modulus of the elliptic integral

of the third order and

logarithmic form by a Fig. 17.

geometrical construction c

of remarkable simplicity

when the intersecting

surfaces are given, or

when a, b, and k are

given.

Take BA=«, BD = 6,
and from O the point of
bisection of AD, erect

k B^9L
the perpendicular OC = -.

A

Then (135) gives ;? = BC, q— AC; and putting P and Q, for the angles
BAG and ABC, a + b=2p cosQ, a— 6 = 20 cos P. As p, q, b are the

VOL. II. 1

58 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

sides of the triangle BCD, and the angle BCD = P— Q,
<P-QN

008

Again, as a, p, q are the sides o£ the triangle ABC,

Substituting these values in (137), we get

cos P cos Q cos P cos Q,

\ 2 J

2/P-Q\'
«*(-?-)

_ 2 [cos P cos Q] * , ._cosQ,—cosP

I

- — 7^. — — T
cos Q + cos P

;= - ^ — — ~
cos Q, + cos P

and if c be the eccentricity of the elliptic base of the cylinder,

sm2P.sin2Q

" sin«(P + Qr

These. are expressions remarkable for their simplicity.
We also find for the criterion of sphericity tc,

(P-i-Q) ~

sin1

/P+Q\
— —

k.

COS » — ] COS

L_ \ 6 /

2

(140)

(141)

As — is the altitude of a triangle whose sides are a, p} q.

38.] In the preceding investi-
gations, the element of the curve
has been taken as a side of a
limiting rectilinear polygon in-
scribed within it. We may how-
ever effect the rectification of the
curve, starting from other ele-
mentary principles. Let APB be
the plane base of the elliptic cy-
linder, and let a series of normal
planes PPW wtn-W be drawn to
the cylinder, indefinitely near to
each other, and parallel to its
axis. We may conceive of every
element P-cr of this plane ellipse
between the normal planes as the
projection of the corresponding

O.V THK GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. .V.)

element s-vr1 of the logarithmic ellipse. Let r be the inclination of
the element dS of the logarithmic ellipse to the corresponding
element ds of the plane ellipse. We shall have, dX being the ele-
mentary angle between the planes PPW and •crsrVv',

..... (142)

(1# Q /?

Now (31) gives dx=^ + ^>

c P rd2«

and therefore 2 = 1 -^— d\ + 1 ~-a sec r . dX. (143)

J cos T J dX2

In the plane ellipse p<2=a* cos2 X + i2 sin2 X, whence

d*p = (a9 - b*} (a2 cos4 X - 62 sin4 X)
dX2 ~ a« "~

We have now to express cos T in terms of X.

From (119) combined with (120) we may derive

~

d#2 + dt/2 A2 (a2 sin2 6 + b cos2 9)

7/ fj^> fj if A

Eliminating - between the equations tanX=^ -, and -= tan 6,

X U X X U

we shall have

a,

taiiX= , tan 0.
o

If we eliminate tan 6 by the help of this equation from (145), we
shall obtain

_
~

__
aa-62) [o2-62 - /^ ] sin* X ^(

Substituting this value of COST in (143), and writing P', Q!, R' for
the coefficients of powers of sin X, the resulting equation
become

*2= f dX

J

(a2 b*}( dM«2cos4X-&2sin4X) >' ' '
' J k (a2 cos2 X + 6* sin2 X)» cos rj

As the first of these integrals is precisely similar in form to the
integral in (123), we may in the same manner reduce the expression
into factors. Accordingly let

F -f Q' sin2 X + R' siu4 X= (a + /3 sin2 \) (y - |8 sin- X) . . (149)

60 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Writing a, /3, y instead of A, B, C, and following step by step the
investigation in sec. [35], we shall have, as in (126) and (128), -»/r,
m, and i{ being the amplitude, parameter and modulus,

As «7=a2F, /3=a2— i2, and 7-«=a2-i2-F, . (151)

we shall have the following relations between the constants «, 6, y,
m, it, and A, B, C, w, i, in (150) and (128),

/3 = B, « = C-B, y =

y-/3=A, « + /3 = C, y

• C) .a

. ' — • ? •*

. (152)

/3 B
= &*, or «y=z, »t= fl = r>-

Hence the moduli are the same in the two forms of integration,
and the parameters m and n will be found to be connected by the
equation m + n-mn = i* ; (153)

m and n are therefore conjugate parameters, as they fulfil the con-
dition assumed in (1).

The amplitudes <p and •x/r are equal; for in (126) we assumed

J, and in (150) tan2'^^- tan2X; but

.

tan X=7 tan 6, as in (146), whence tan2 -^=73—- — tan2 <p.
o u (A. + JjJa

In (152) we have found « + /3 = C, and A + B=7, whence
tan2 f =^5 ~ tan2 <p. But AC= W, and «7=a2/t2,

as shown in (125) and (151), whence

^•=<p ......... (154)

We shall now proceed to find the value of the second integral
in (148).

From (147) we may derive

Differentiating this expression, reducing, dividing by cos r, and
integrating, we shall finally obtain

f. dX(tf ea'
J cos T a2 cos2

cos2 X + 42 sin3 X)i

.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. <<1

(148) may now be written

A2=fd\ VI* + Gt' sin2 X+ R

J

-*2 --.

COS3T

(157)

If we measure the arc of the logarithmic ellipse from the minor
principal axis, or from the parabolic arc which is projected into b,
instead of placing the origin at the vertex of the major axis as in
(119), we must put

and following the steps indicated in that article, we shall obtain

. . (159)

VI" -f Q! sin2 S + R'^sm1^. . .

If we now make S=\, and subtract the two latter equations one
from the other, the resulting equation will be

COST

(160)

Fig. 19.

But this integral is, we know, the expression for an arc of a
common parabola whose semiparameter is k, measured from the
vertex of the curve to a point on it where its tangent makes the
angle r with the ordinate.

Thus the difference between two elliptic arcs measured from the
vertices of the curve, which in the plane ellipse may, as we know,
be expressed by a straight line, and in the spherical ellipse by an
arc of a circle (as shown in sec. [15]), will in the logarithmic ellipse
be expressed by an arc of a parabola. As a parabolic arc can be
rectified only by a logarithm, we may hence see the propriety of
the term logarithmic, by which this function is designated.

39. If from the vertex A of a
paraboloid, an arc of a parabola be
drawn, at right angles to a parabolic
section of the paraboloid, it will meet
this parabolic section at its vertex.
Let the arc AQ be drawn at right
angles to the parabolic section Qv
of the paraboloid, the point Q is the
vertex of the parabola Qy.

Draw QT and Q/ tangents to the
arcs QA and Qv. Then QT and Qt
are at right angles, since the arcs
AQ, Q.V are at right angles. As QT
is a tangent to a principal section
passing through the axis of the para-
boloid, it will meet this axis in a
point T ; and as QMs a tangent to
the surface of the paraboloid, it will
be perpendicular to QN the normal to the surface. Now as Q.t is per-

62 ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS.

pendicular to Q,T and to Q,N, it is perpendicular to the plane QTN
which passes through them, and therefore to every line in this
plane, and therefore to the axis AN, or to any line parallel to it,
as the diameter Q,n. Hence, as the tangent Qt to the parabola Q,v
is perpendicular to the diameter Q,n, Q is the vertex of the para-
bola.

Hence, in the logarithmic ellipse, one extremity of the protangent
arc is always the vertex of the parabola which touches the loga-
rithmic ellipse at its other extremity.

This is a very important theorem, as the protangents are arcs of
equal parabolas, all measured from the vertices of the parabolas.
Hence also the length of the protangent arc depends solely on its
normal angle.

As an arc of a circle may be expressed by the notation

5=sin~1(|), y being the ordinate and k the radius, so in like
manner an arc of a parabola may be designated by the form
s = tan~M|), y being the ordinate, and k the semiparameter. To

\/C -

distinguish the parabolic arc from the circular arc, the former may
be written s==rav~} (-,} . Again, as we say, in the case of the

circle, the angle « and the arc kco, co being the angle contained
between the normals to the curve at the extremities of the arc, so
in the parabola, we may write &> for the angle between the normals,
and (Ar.t») for the corresponding parabolic arc. In the case of the
parabola the arc is always supposed to be measured from the vertex ;
in the circle the arc may be measured from any point, as every point
is a vertex.

40.] Resuming the equation (157),

*2-fdX V?' + Q! sin2 X + R' sin4 X- A2

J

we shall now proceed to develop the first integral of the second
side of this equation. As the integral is precisely the same in form
as (123), and the amplitude ^=<p, as also the modulus it=i} we
may substitute «, /3, y for A, B, C, m for n, 3>m for <£>„, retaining
the modulus and amplitude, which continue unchanged, as we have
established in (152) and (154) ; or substituting for a, /3, 7 their
values in m and i, we get

d<p

m . ri-msin2®! Vl-i2sin2

m

v/1 — i2sin2<p J V»z(ia — m)(l— m]

ON THi: (1EOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 63

I f \\ •(• eliminate i from the coefficients of this equation, putting M for
( 1 — m sin2 <p) , and N for (1— wsin2<p), as also VI for Vl — J2sin2<p,
(133) may be written

2(»-m)S_ . (l-itU«-i»)f d? 1

~^S~*~ w » JNVI

. (162)

and (161) will be transformed into

(1— m}(n— m) C dtp

m

\C&? (d /f 2(^~m) r d7
J VI V»»» J cos'

dr

J VI J \™>n jc s"r j

If we compare together (162) and (163), which are expressions
for the same arc of the logarithmic ellipse, and make the obvious

reductions, putting for <!>„ and <£„, their values g]n(P costp y an(j

— , we shall get the following as the resulting equation
of comparison,

d<p

/IN Vi V m /JM Vi

)" • (164)
i2 Td<p 2 C dr sintpcosp VI

~"~ / r ^^ / -••--»•»•

WU VI V^WjCOSaT

From (155) we may deduce

sinT= ^~ "r — r. (165^

we shall therefore have

tan r sec T== V^sinjpcosp ^ (166)

MN

It may easily be shown that tan T sec T represents the portion of a
tangent to a parabola intercepted between the point of contact and
the perpendicular from the focus.

C dr C dr

Hence tanTsecT=2l . — 1 — (167)

J cos3 T J cos T

64 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Combining (164), (166) and (167), and using the ordinary notation
of elliptic integrals,

dr

cos T

d sin T _J_f_dT__
1 — sin2 T' v0*nj cos r

f» d |~ \Jrnn sin <p cos <p~| ,
1 I d® L v/I— z2sin2(p -I

^ sin <P cos <P~] 2

^2sm2> J

we have therefore

rd T Vm^sin^costpl^ I . (169)
dtpL v/l-^sin2^ -I

~

rnn sn <p cos <

;

This is the expression given by LEGENDRE, Traite des Fonctions
Elliptiques, torn. i. p. 68. Written in the notation adopted in this
paper, the formula would be

_ ,

VI

f

41.] We may express a and b, the semiaxes of the elliptic base
of the cylinder, in terms of m and n, the conjugate parameters of
the elliptic integrals in the preceding equations. From the equa-
tion of condition m + n—mn = i'2, and the expressions given in (130),
we may eliminate i2, and obtain

a?_mn(I— m) b<2_mn(\—n)

~ ' ' ' '

Therefore -=
a

ri

(l — m)

m

1 — m

Hence the ratio of the axes of the elliptic base of the cylinder is a
function of the modulus and parameter.

The ratio of the corresponding quantities in the case of the
spherical ellipse may be derived from the equation

—
or -= VI -**=.;.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. G5

This ratio is therefore independent of the parameter. There is,
then, an important difference in the two cases. In the one case,
the ratio of the axes is independent of the parameter, and will con-
tinue invariable while the parameter passes through every stage
of magnitude. But in the logarithmic ellipse the vertical cylinder
will change its base with the change of the parameter. We shall
see the importance of this remark presently.

These ratios are : —

In the sphere, -=j; in the paraboloid, -= jj . . . (172)

42.] Resuming equation (157) and developing it by a process
similar to that applied to (127), we get

= ?? I L* ' "*" TJ"T _J.| "' (173)

Now (151) and (152) give

— m

, __

Making these substitutions, we get

>=a

— C [l-i*sinWf ,C AT

J [1— wasin2<p]2 Vl — i2sin*<p J cos3r'

Now let m=0, then (165) gives r=0, and we shall have

x? C i / T *o • o

2,=«J dip v I— t* sm2<p.

This is the common expression for the rectification of a plane
ellipse whose greater semiaxis is a, and eccentricity i. This is
case IV. of the Table, p. 7.'

We cannot arrive at this limiting expression by making e'2=m=Q
in (53) ; for this supposition would render z=0, which, throughout
these investigations, is assumed to be invariable.

43.] If, as in the case of the spherical parabola, we makew=m,

or n=l— v/1 — iz> the values of r and y become infinite. What,

then, is the meaning of the elliptic integral of the logarithmic

form of the third order when n=m, or n=l— \/l — i2? In the

I—/
circular form of the third order, when m = n, w = r — -., and the

spherical ellipse becomes the spherical parabola, which, as we
know, may be rectified by an elliptic integral of the first order.
VOL. n. K

66 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Not only do the ratios -r, -r become infinite, but they become

K K
1.1 1 n

equal: for -==- = 1 when m = n. What, then, does the in-

a2 1— m

tegral in this case signify ? It does not become imaginary or change
its species.

Resuming the equation established in (133),

If we now introduce the relation given in (130)

a_ \in(i*—ri) (1 — i*)
~k~ Zn-P-ri*

we shall have by substitution

Vl-n a

If we now suppose m = n, or n=l
the last equation will become

. . • (176)

In this case ^= ........ <177)

This is the expression for the length of an arc of a logarithmic
ellipse, the intersection of a cylinder, now become circular, with a
paraboloid whose semiparameter k = Q; therefore, the dimensions
of the paraboloid being indefinitely diminished in magnitude, this
intersection of a finite circular cylinder by a paraboloid indefinitely
attenuated must take place at an infinite altitude. We naturally
should suppose that the section of a cylinder which indefinitely
approaches in its limit to a circular cylinder by a paraboloid of
revolution, would be a circle ; yet the fact is not so. The inter-
section of these surfaces, instead of being a circle, is a logarithmic
ellipse, whose rectification may be. effected by an elliptic integral
of the second order, as we shall now proceed to show.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 67

In the first place let us conceive the paraboloid as of definite
magnitude, and the cylinder to be elliptical, its semiaxes as before
being a and b. Then, as a and b are the ordinates of a parabola,
at the points where the elliptic cylinder meets the paraboloid, at
its greatest and least distances from the axis of the surfaces, we

sha11 have * l * " (178)

Hence a*-b* = 2k (J—z"). Let s? — z" = h, then h is the thickness
or height of that portion of the cylinder within which the loga-
rithmic ellipse is contained.

,1irl. . 2 12 k*mn . „ 0, kmn

Now (171) gives a2— o2= : therefore 2h =•

n— m n—in,

k *Jmn(\ — m) , a ^ inn

and we have also a — — — '-: hence A = 77

n— m 2 y/1— m

Now when n — vn, a=b, k=Q, while we get for h
=°— JL-^g*IL/'

(179)

We thus arrive at this most remarkable result, that though the
cylinder changes from elliptic to circular, while the parameter of
the paraboloid approximates to its limiting value 0, yet the thick-
ness of the zone (that is, h) does not also indefinitely diminish, but
assumes the limiting value given above.

Now if we cut this circular cylinder, the radius of whose base
is a, by a plane making with the plane of the circular section, or

with the plane of XY, an angle whose tangent is , the semiaxes

Ot

£| and 9$ of this plane section will manifestly be

n = a, and ft=Va*+/P or fc»^=L. . . (180)

A \ i ~— 71

If we denote the eccentricity of this plane ellipse by i,,

n 1— Vl— i2 1— j TT • 1— »'/

«.= -; =— =- -.. Hence 7=; r. . (181)

' 2-n i+A/i_i« 1+^ 1+t,

It is shown by Legendre and other writers on this subject that,
if c and c, are two moduli connected by the equation

68 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

and <p and i/r two angles related as in (63), writing <p for /i, so that

tan(i/r— <p)=6tan<p, ...... (183)

we shall have

An independent demonstration of this theorem will be given in
sec. [44].

Now l + '.- ^-^ hence

c - c,

and, using the common notation for the present, (74) gives
6Fc(<p) = - - Fe/(ijr) . Adding these equations, we get

lism^ . . . (185)

or, using the notation adopted in this work,

^Jdf ,/!, + ! sin ^r- [ Jd<p Vl+j =0, . (186)

since n = 1 — b = 1 — 7 .

Substituting the value of the first member of this equation in
(176), the resulting equation will be

sin <p cos <p VI

2 J" '2 co88<p+/smY

Having put for <!>„ its value in this case, namely

- _sin<p cos <p \/I

^« — ~ <2^r~i — • • 9 ->
cos cp -\-j sin <p

we must now combine the last two members of this equation.
Adding, they become

-n 'S sin ilr ^ : — ; — ^ — r» • • • • (1"8)

^ ( cos^ <p +^ sm^ ^> j

From this expression we must eliminate the functions of <p.

Now (73) gives ^1=-^^^^, (189)

V 1
writing <p for /A,

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 69

Substituting this value of ^l in the preceding expression, for
which we put t, we get

2sin<pcos<p

( }

From this equation we must eliminate sin<p, cos<p.
If we solve the preceding equation (189), we shall obtain as the
resulting expressions

2sin2<p = l — v/I/c

2cos2<p = l + y'lyCOS'^r— ^sin2^/

Multiplying these equations together, and recollecting that
J,= 1 — iy2 sin2 ijr, we shall find

4 cos2 <p sin2 <p = sin2 ^ [I, + 2 \/V/ cos ^ + if cos2 i|r] . (1 92)
Now the second member of this equation is a perfect square,
whence 2sin<p cos<p = sin-\/r[ V-l/ + */ cos A/r] . . . . (193)

Substituting this value of 2 sin <p cos<p in (190), we get
n . ~ v + «cos-vK n i. sin ilr cos i/r

---

» = !-
equation (187) may now be written

« a (2— n) a (1+j)
Now, as ^ = -A-^ -- L=._ v \-" an(i

2 vi-w 2 vy

we get ultimately

The second term of the last member of this equation is evidently
the common expression for the protangent to a plane ellipse between
the point of contact and the foot of a perpendicular on it from the
centre; while Hi j d-^r y% or & j"d-\Jr \/l — «,2sm2^r, is the expression
for the arc of a plane ellipse whose semi- transverse axis is H, and
eccentricity i,.

IT

"When the function is complete, # = K and ty—ir. See (183).

70 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

TT
/17T p5

Hence, as I di/r \/Iy=2 1 d>|r x/I,,
Jo J»

7

T

2 = &l

Jo

d^/r VI, (197)

'0

S therefore, in this case, is equal to a quadrant of the plane ellipse
whose principal semiaxis H, and eccentricity it are given by the
equations

, and »,= - -. - - -
P v J — i*-

To distinguish this variety of the curve, we may call it the
circular logarithmic ellipse, as it is a section of a circular cylinder.
Accordingly, in the two forms of the third order, when the con-
jugate parameters are equal, or m=n, the representative curves of
these forms become the spherical parabola and the circular loga-
rithmic ellipse.

This is Case V. in the Table, p. 7. The results of the preceding
investigation will reappear in the demonstration of the theorem,
that quadrants of the spherical or logarithmic ellipse may be ex-
pressed by the help of integrals of the first and second orders.

44.] It is not difficult to show that this particular case of the
logarithmic form, when the parameters m and n are equal, also
represents the curve of intersection of a circular cylinder by a
paraboloid wrhose principal sections are unequal.

Let o?2 + y2=o8 and ^+^,=2z .... (199)

be the equations of the circular cylinder and of the elliptic para-
boloid.

(cos2$ sin2#)
Assume x= a cos 6, y = asin0; then 2z = a2 \ — - --- h — T^— [•, (200)

I * k )

and ^=-«sin0, ^ = acos0,|r|:=a2(p— Jsin0cos0. (201)

Hence =al + a2,-sin20cos20T. . . (202)

Now we may reduce this expression by two different methods
to the form of an elliptic integral.

By the first method, eliminating cos2 6, this expression becomes

' (203)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 71

We may, as in (124), reduce this expression to the form of a
product of two quadratic factors,

(A + B sin2 6} (C - B sin2 0) = AC + B (C - A) sin2 0 - B2 sin4 d. (204)
Comparing this expression with the preceding,

AC-*, B-.-

or C = A + B, and AC=
Let us now, as in (126), assume

..... (206)

A

and, following the steps there indicated, we shall have

S=Afef^\A-f^-^

an expression of the same form as (127).

B B(2A + B) .«

Ut A + B="' TSW*

A A2 ^

therefore 1 — n=-r TJ, and 1 — i2=7T- ^-^ I

A + B (A + JbJ)-4 ( f (209)

Hence 1— n— v'l— i2, or n=m J

If we develop this integral by the method indicated in sec. [36],

£\ •<•> 5 /"• J — .

«C7l — £~ — ^A i Q(Z)

the coefficient - of the integral I

n J(l— nsin2^) Vl-^sin2^)'

in the result, will be 0, and the reduced integral will become, since

B

A+B~~J

= ' and B = a2~ ' '

Let z1 and 2" be the altitudes of the points above the plane of
XY, in which the principal sections of the elliptic paraboloid meet
the circular cylinder. Then 2" — z' is the height or thickness of the
zone of the cylinder on which the curve is traced.

Now a2 = 2)b/, a2=2*'c"; whence 2"-r/ = ~-

72 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Let this altitude or thickness of the zone be put h, and we shall
have

Hence the arc of this species of logarithmic ellipse may be ex-
pressed by integrals of the first and second orders.

It is not a little remarkable that whether the integrals of the
third order be circular or logarithmic, or, looking to their geome-
trical origin, spherical or parabolic, when the conjugate parameters
are equal, or m=w, we may express the arcs of the hyperconic
sections thus represented, in terms of integrals of the first and
second orders only, the integral of the third order being in this
case eliminated.

If we now resume equation (202) and make

(213)

sin 20=2 sin 0 cos 0 — cos ^, and 2d0 = d^. Therefore (202) will
become

§

hence, asA = — 1-^— -I, we shall have

22= V^+T2dx\l -- rin«x. . . (215)

This is the common form for the rectification of a plane ellipse,
whose principal semiaxes are V«2 + ^2 and a. Let i, be the eccen-
tricity of this plane ellipse,

h B n _i_ y/l— jg

~2-n~ ' (2]

and the relation between <p and ^ is given by the equations

7T A

20=2 + X> tan20=£— gtan2«p, or tan0= Vl
Hence

1 + sin ^

=1-nt

, or secx + tanx= V/tanp. (217)

When x=0, tanp=J-; when %=^ <p=£; when ^=-^"^ = 0.

\J A & 2

Hence ^ is measured from the perpendicular on the tangent to the
ellipse, at the point which divides the elliptic quadrant into two

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 73

segments whose difference is equal to a — b, as will be shown further
on ; while p is measured from the semi- transverse axis a. Thus

while x varies from — •» (that is, from the position at right angles
to this perpendicular, and below it) to 0 (that is, to the perpendicular

itself ),<p varies from 0 to tan"1 — --- 1 an(^ while ^ varies from 0 to

V>

7T 7T

^, <t> varies from tan"1 — -= to ^. Thus while y passes over two

VJ *
right angles, <p passes over one right angle.

We may now equate the two expressions (211) and (215) ; and
the resulting equation will be

or

Thus we may express an elliptic integral of the first order by means
of two elliptic integrals of the second order. Hence we obtain the
geometrical origin of the well-known theorem, given in (184) .
When the functions are complete, since

IT IT

fa _ /*2 _

&X Vl— i;2sin2y=2 j dy %/l — i/2sin2y, we get

I— 1

dtp

__ fa C

d*Vl-t,2sin2X=(l+./) d<pVl + (l-n)

[_»/o Jo

,(219)

which agrees with (186).

44*.] From the foregoing investigations it will follow that, if
there are two moduli so related that

'~i+ </!=?- l+j*

and two amplitudes such that

...... (b)

we may express an elliptic integral of the first order by the help
of two elliptic integrals of the second order, whose moduli are
i and i, and whose amplitudes are <p and %.

VOL. II. L

74 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

A like relation is established by Lagrange's theorem (186) in
which the moduli are the same, but the amplitudes are given by
the equation tan (^— <p) = / tanp ....... (c)

Lagrange's theorem as given in (186) is

=0. . (d)

While the theorem established in (218) is

It may be proper to show that these theorems (d) and (e), though
apparently diverse, are identical.

These equations will be identical if we can prove that

Vxi;+ (1 -»*. (f)

To show this, we must eliminate <p between the equations
tan (ty — <p) =j tan <p, and sec % + tan ^ = \J j tan <p.
Eliminating <p and reducing,

tan*Vrtan*%=lii^. -. ..... (g)

Hence sin^— C°SX ...... (h)

[1— if sin8x>

,r,^ , _ ^

and *P= ri — r-s-T- = , , . -. since w=— j :

[1—n sin2 <p] 2'

consequently 2<I>=

/I- A

(

therefore sin ir — 24> =

We have also

— ^- /;\

-2

i)\ THE UEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 7~>

But

n

[1— i/o

,,
'

as will be shown further on ; consequently

Substituting these values in (j), the equations are manifestly
identical.

We may thus by the help of Lagrange's formula, as given in (d) ,
or by the new expression enunciated in (e) , express an elliptic in-
tegral of the first order by the help of two elliptic integrals of the
second order ; but we are unable to reverse the process, and ex-
hibit an elliptic integral of the second order, as a function of two
elliptic integrals of the first order. The problem has been tried,
but in vain.

If we multiply (218) by a, bearing in mind that a2 — 62=a2z2
and b = aj, we shall have, since n = l— J,

-(*-V*'> . . (m)

but when the functions are complete, since

IT

J'

we shall have

-

f

(a -(- b) and 2 \/ab are the semiaxes of the ellipse whose amplitude
is ^ and modulus ir Hence we may derive the following
theorem : —

The difference between the quadrants of two ellipses whose semi-
axes are a, b, and (a + b], 2 \/ab is equal to a complete elliptic in-
tegral of the first order whose modulus is i; or, The difference
between the quadrants of two ellipses whose semiaxes are a, b and
(a + b), 2 \/ab is equal to half the difference between the circum-
ference of a spherical parabola and a semicircle, both described on a
sphere whose radius is a.

76 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

It may be worth while to mention that a + b is the length of the
tangent drawn to the ellipse whose semiaxes are a, b, and inter-
cepted between the axes ; while the point of contact is the critical
point, or the point where, as Fagnano has shown, the constituent
arcs of the quadrant of the ellipse differ by a — b.

\/ab is the perpendicular from the centre on this tangent.

CHAPTER V.

ON THE LOGARITHMIC HYPERBOLA.

45.] The logarithmic hyperbola may be denned as the curve of
symmetrical intersection of a paraboloid of revolution with a right
cylinder standing on a plane hyperbola as a base.

Let Oxx1 be a paraboloid of revolution, whose vertex is at O,
and whose axis is OZ. Let ACB be an hyperbola in the plane of

XY, whose vertex is at A, and whose axis is the straight line OAD.
Let the planes ZOX, ZOD, ZOY cut the paraboloid in the plane

ON TUB GEOMETRICAL PROPERTIES OK ELLIPTIC INTEGRALS. 77

parabolas Ox, Od, Oy, and let cab be the curve on the surface of
the paraboloid whose orthogonal projection on the plane of xy is
the plane hyperbola ABC. Then acb is the logarithmic hyperbola.
Vertical planes erected on the asymptotes of the hyperbola in
the plane of XY will pass through the axis OZ, and will cut the
paraboloid in two parabolas passing through the vertex O, which
will be asymptotic curves to the logarithmic hyperbola. These
curves will be found to have properties analogous to those of the
plane hyperbola and its asymptotes.

Let *~P = 1' anda?2 + y8=2** .... (220)

be the equations of the hyperbolic cylinder and of the paraboloid
of revolution, and consequently of the curve in which they inter-
sect ; let T be an arc of this curve,

T

x, y, z being functions of a fourth independent variable \.

a4cos2X A4 sin2 A,

Assume ar=-~ - ~^ — ,2 . 2., y —-5 - ^ — ,g . a* (222)
a2 cos2 A, — A2 sin2 A. * a*cos*X — £2sm2X v

It is manifest that these assumptions are compatible with the first
of equation (220) ; and the second of that group gives

2 . 2_
+ y ~a2cos2X~

Differentiating (222), we get

(a2 cos2X-A2 sin2X)3'

dX/ " yt2 (a2 cos2X-62 sin2X)4 \

We might, by the help of the imaginary transformation sin 6= V — 1 tan &,
e from the el '

ulting equati

e mg, y

pass at once from the elliptic cylinder to the hyperbolic cylinder. Let tan0'=w,
and the resultin equation will be of the form

dY

an expression which, on trial, it would be found very difficult to reduce. The
difficulty is eluded by making the transformation pointed out and adopted in
the text

78 ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS.

Hence

k dT

(«2cos2X-62sin2X)2
Let this radical be put = \/R,.
Assume B, = ( A + B sin2 X) (C - B sin2 X) )

=AC + B(G-A)sin2X-B2sin4X;j

hence AC=a2£2, B=a2 + 62, C— A=

and therefore k* = A + B — C .
Let us now assume sin <p such that

and «2 cos2X-62 sm2X=a2-

or as

Making

we find =

'j (226)

there resiilts «2cos2X— 62 sin2X=-r — -~r 1 — . , ^sin2^ .

A + Ccos2iL A + C rj

Hence -- ^T_ VAC . [A + C cos^] cos<p

a2*2 dX a4(A + C)[i-/sin2<p]2 '

differentiating the equation sin2 X=-r-^ — " g , . . (230)

dT_dTdX .2_

~' r ~ ' ' ' '

, £ n T i2 (* cos2(pd(p

we get . finally, T= — . - 1- - (233)

* VB(A+C)J[l-/sin2<p]2 Vl-z2sinV

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 79

46.] We may develop another formula for the rectification of
an arc of the logarithmic hyperbola.

Assuming the principles established in sec. [38] , we may put

T=-Jj9secudX-

In this formula p is the perpendicular from the axis of the hyper-
bolic cylinder let fall on a tangent plane to it, passing through the
element of the curve, and v is the angle which a tangent to this
element makes with the plane of the base, v in this equation is
analogous to r in the last section.

In the above expression the negative sign is used, as the curve
and the angle X are incremented in opposite directions.

dz
d\

//d^\2 /dy

V (aO +(-l

Now p* = d2 cos2 X — A2 sin2 \, and tan v=

We must substitute for these differentials, their values given in
(223), and introduce the value of <p assumed in (227), whence

(A + C)2ACcos2<p

o* II — _ - _ ' __ . (OQ^}

~9 cos2?]2 (a2 cos2X-£2 sin2X) '

,

But (231) gives =
d<p

whence

«2/i:cos2<pdip

y — . (237)

We must now determine the value of the second integral in
(234), namely

jo^secvdX.
Since p* = a2 cos2 X - 62 sin2 X,

?£. « udX- -(a' + y)Caa cos'X + ^sin'XJsecudX
dX2 b< ~^s2X^62 8iii«~

80 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Now we may derive from (223)

sin \ cos X
tanv = — * — — i ..... (239)

Differentiating this expression, then multiplying by sec v, and in-
tegrating, we obtain

*42*

Comparing this expression with (238), and introducing into (234)
the values found in (237) and (240), we obtain

?= f dv g* f_ coa'fdp

* J cos3u VB (A + C) J [1-w sin2?]8 Vl-*2 sinaf '

n

Making m

A + B , , C/A + BN i2 C

since /=---, and i2=-, assume w== . (243)

and we shall have m and n connected by the equation of condition,
denned in (1),

m + n— mn = i2.

The three parameters /, m, n, and the modulus i are connected
by the equations

mn = i2 ...... (244)

/ and n are reciprocal parameters, the reader will recollect, while
m and n are conjugate parameters.

By the help of these equations, any one of the quantities /, m,
n, ft may be eliminated, and an equation established between the
three remaining quantities.

47.] It was shown in (226) , that C - A = a2 + i2 - £2, B = a2 + £2,
/fc2=A+B-C, and «2A2=AC, whence

AC_ y_(A + B)(B-C)
C)2' A2~ (A + B-C)

2 '

In order that these values of a and b may be real, we must have
B > C, and A of the same sign with C, both positive ; otherwise

\/R in (225) would be imaginary. As /= , />! ; here the

.A.-)- \j

parameter / is greater than 1, while m and n are each less than 1.
We may express the semiaxes of the hyperbola, the base of the

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 81

hyperbolic cylinder, in terms of the modulus i and the parameter / ;
for by the equations immediately preceding we may eliminate A,
B, and C in (243). We thus find

therefore

_ /(/-I)

We may express the semiaxes in terms of the conjugate para-
meters m and tt

F~ [» + »»-2»m8]
hence

B «2 + 52 m . A;2

nF-

or we may express a and 6 more simply in terms of / and m.
Eliminating n and i2, we get

a*_m(l-m) 62 _/(/-!) ,

jfc«- (/-m)2 ' ^2~(/-m)2'

Let ^ be the eccentricity of the hyperbolic base of the cylinder,
the following equation between c,, i and /_, analogous to (131), will
follow from (246),

(«,«- 1).*/ =(*-«*)* ...... (251)

Hence when i and I are given, ct may easily be found.

48.] If we equate together the values found for T, the arc of the
logarithmic hyperbola, in (233) and (241), we shall have

2T
J

[1 - / sin2<p] 2 Vl - i2 sin2<p

[.(252)
du

_

[1 - m sin2<p]2 Vl -i2 sin2<p " cos3 v

For brevity, put

L = 1 — / sin2<p, M = 1 — m sin2<p, N = 1 — n sin2<p, I = 1 — i2 sin2«p. (253)
The preceding equation may now be written

(254)

VOL. II. M

82 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

or if we substitute for the coefficients of this equation their values
given in (246) , we shall have

1

. . . (255)

cosv,

Let

sin<pcos<pVl
- - --

. . . (256)

Now the process given in sec. [36] will enable us to develop the
integrals

i, as follows : —

-i2) f d<p Vi

and
2^(1-1*;

; • (257)

)Ca9 /^(i-^

J \/I m(i*—m)

d<p

. (258)

The equations of condition ln=i* and m + w — mn — ? give

?£t3_,-i. and g=Q!+?fld5_2=Qf. . (259)

z2— m / m »(«"-l)

We have also, since

Making these substitutions, adding together (257) and (258),
the coefficient of J dip VI vanishes, and we shall have

o(; .ff)a fcos^djp [ gja (1 .g) rcos2(pdj)_/sin(p cos(p VI

L2 VI • 'J M2 VI

A

M VI

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 83

but (255) gives

r

' J

L2 VI

Combining this equation with the preceding,

L VI

m JM

r-i)J VI

/sin<p cos<p Vl_,
"LIT

ELf

/(/-l)JcosDj

. (261)

Now

and as

i'2 >

m (1-1} '

In the last equation, substituting this value of fp and then di-
viding by /, we obtain

(J-1

}

Vl /(/-l) VI

sin<pcos<p Vl_2A / I— ft r dt>
LM V /(/-1)J cos^;

Now

Iv , C dv , 2 LM

T- = tanuseci;4-l andcos^f = — 5-,

cosdu Jcosu cos* f

as may be shown by combining (226) with (235).
Hence sini> =\f -j — ^r *an<p VI,

and therefore tan v sec v = A / \ ~ ' —

v I — i

LM

Substituting this value in the preceding equation, we find
/-J2

(262)
(263)

(264)
(265)

//-

\ i

-i VI

_ /T

V //—

l)

cosu

> . . (266)

84 ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS.

In (170) we showed that, m and n being conjugate parameters
connected by the equation m + n— mn=t2}

l-n

d(p (1-m) £ dtp _ P_ C dtp _ 1 ^ dr

~ '

AT

Now

1(1-1)

Substituting these values in the preceding equation, and dividing

by 75, we obtain
i

fdr

— 1COST

If we add this equation to (266), the coefficient of the integral

I .- will vanish, and the resulting equation will become
J J

.-
JM Y A

f_jg_ , f

JL VI J

,

N VI VI V(J-l)(/-;2)

V"^ r f dt; r dr -[

-l)(/-;2)Ucosv JcosrJ-

We shall now proceed to show that

/» J /*

I -- 1

/I

JCOSV JCOST

/• T t

under the form 1 — ^—.} if we make the assumption
J cos ir

^C

may be put

•*

VI

ief being equal to (l-n) (--l) -

Now
hence

(269)

s

^i2" f dv _ f dp [-[l-i2sin2(p-i2sin2 <p cos2 <p] ~]
-J VI L ~LM~

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 85

But we derive from (165) and (166) the value

dr _T d<p [ncos2<p — n sin2 ft + m2 sin4 ft]

~

or, subtracting,

/T^

V/7=

(l)cosi;

N

,. (272)

These two latter integrals may be combined into the single
integral

'[1 -i2 sin2 ft -n cos2 ft] [1 -i2 sin4 ft] dft

LMNVI

Now, as m + n — mn=fi, the first factor of the numerator becomes
(1 — n) (1 — m sin2 ft) = (1 — n) M ; and therefore

/7-i8 f f du C Ar 1 _ (I- i*\ f [1 -i* sin4 ft]
V/(7=I)LJcosi, JcosrJ-V / /J LN VI

Substituting the first member of this equation for the last term
in (268), we find

j f* j /*n *9 '

U(p I Q(p t [1 — t SI]

5vT J vr~J

Now, since we have assumed in (269)

.
smi/=

VI

LN

; cos2 <f>3

hence
and consequently

dt/
- — .
cos i/

fj?

JL V

,

L Vi N

This formula is usually written

LN VI

= — 7=+ =

VI V*J cos

(276)

(277)

f_
J [l-

f

J [1 — rin

d

. (278)

V. A / ,

We have thus shown that from the comparison of two expres-

86 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

sions for the same arc of the logarithmic hyperbola, we may derive
the well-known equation which connects two elliptic integrals of
the third order, and of the logarithmic form, whose parameters are
reciprocal *.

Hence also it follows that if v, r, and i/ are the angles which
normals to a parabola make with the axis, and if these angles, which
may be called conjugate amplitudes, are connected by the equations
ML

/m

o , =A / -tan<r>

2 V n

cos^ <p
-M aJnT- V^sinpcosp (27g)

i ; VI

LN . , /m n x tan <p

smi/=, ' n — *"N

Vyvyo v — T g ^> •••• " — (\ / — i^- •"/ 7_-

Icos2<p V n VI ^

we shall have

......

J COS V I COS I/ J COS T'

49.] The difference between an arc of a logarithmic hyperbola,
and the corresponding arc of the tangent parabola, may be expressed
by the arcs of a plane, a spherical, and a logarithmic ellipse.
Resuming the equation (241) ,

du _T_ a2 Tcos2<pd<p
us^u~~k~ VB(A-fC)J M2 VI*
and combining (248) with (249), we may easily show that

—2nm
and from (258) we may deduce that

M VI

* We might by the aid of the imaginary transformation sin 0= V —1 tan ^
have passed from this theorem, connecting integrals with reciprocal parameters,
to the corresponding theorem in the circular form. It seems better to give an
independent proof of this theorem by the method of differentiating under the
sign of integration, as we shall do further on. Although these theorems have
algebraically the same form, their geometrical significations are entirely different.
In the logarithmic form, the theorem results from the comparison of two expres-
sions for the same arc of the logarithmic hyperbola. But in the circular form, the
theorem represents the sum of the arcs of two different spherical conic sections
described on the same cylinder by two concentric spheres, or on the same sphere
by two cylinders having their axes coincident.

t These values of v, r, and v' satisfy the equation of condition which connects
the conjugate amplitudes in parabolic trigonometry, tan G> = tan <p sec x +tan x sec <P-
We must replace a>, <p, x by v, v', and T. See vol. i. p. 313, (a).

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 87

Let G=(l-m)+Jd<pVI-m<I>m. . . (282)

Substituting this value of 1 ^~T in the preceding equation, we

^) ivj. \ i

find, after some obvious reductions,

C Av' 2T=

"Jcos3v k ~

\/mnGc _ n(l—m) C d<p
m + n— 2mn \/mn J M VI

Now, a; and bt being the semiaxes of the base of an elliptic cy-
linder whose curve of section with the paraboloid is a logarithmic
ellipse, let, as in (171),

af_mn(\—m) b*_mn(l—n) .

A2~ (n-m)*' A*" (n-m)* '

and if we put 2 for an arc of this logarithmic ellipse, we shall have,
as in (163),

_

k n—m VT/W J M -v/I

cos T

Subtracting this equation from the preceding, we shall finally
obtain

(284)

-- -— .

cos3 u J cos3 T (w — m) (m + w — 2mn)

We may express the arc T by the help of one parabolic arc only,
if we introduce the equation established in (160),

— — , hence

COS3T

i

(285)

— — -

COS3 V '

• /»,
^mn(l-n)mk Vn _ C _d^ ,

(w-m)(w + m-2mw) Lm v ;J VI J

replacing G by its value in (282).

When sin <p= — — , 1^=75, and the arc of the logarithmic hyperbola

v/ *>

becomes infinite, the arc of the parabola also becomes infinite and
an asymptote to the logarithmic hyperbola ; the difference, how-
ever, between these infinite quantities is finite, and equal to

G_2 mtegratea between the limits <p=0,

__

(n — m) (n 4- m — 2mri)
and < = sin~1/~*.

88 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

It is needless here to dwell on the analogy which this property
bears to the finite difference between the infinite arc of the common
hyperbola and its asymptote. When n=m, the above expression
becomes illusory. We shall, however, in the next article find a
remarkable value for the arc of the logarithmic hyperbola when
m=n.

We may express the above formula somewhat more simply.

As in (248) 7=^ *m(l~~n), and

bt_ \/mn(\ — ri) ^_/___L_ *Jmn(\ — ri)m

__
n— m \/m(n— m)(n + m—2mri)

The equation given in (285) now becomes

Q ..... . (286)

COS V i K

The ratio between the axes of the original hyperbolic cylinder
and of the derived elliptic cylinder may easily be determined ; for

y_««(l-m) (} d b*_l-m „.

- a * - b

Let ct be the eccentricity of the hyperbolic base, and c that of the
elliptic base, then

Comparing (a) with (6),

/-«/ /-A-ij 2ro(l-«)

\n = \l~i==*-~\ — i - r~*
a b (n—m)

This equation gives at once the ratio between the axes of the hyper-
bolic and elliptic cylinders.

50.] On the rectification of the logarithmic hyperbola when the
conjugate parameters are equal, or m=n.

We have shown in sec. [43] that, when m = n, the arc of the
logarithmic ellipse is equivalent to an arc of a plane ellipse; so,
when m=w, the arc of a logarithmic hyperbola maybe represented
by a straight line, an arc of a parabola, and an arc of a plane
hyperbola.

In (262), if we make m=n, or l=I+j, n=l—j, we shall have,
writing N for M,

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 89

and in (170), if we make m=n, and M=N,

Adding these equations together, as 1— n—j, we get

sin <p cos <

.N

N LIN uj j

while the arc of the logarithmic hyperbola, as in (233), is
T W-

In this case, the coefficient

= K> as may be shown by
2'

putting, in the general value for this expression given in (249),
m=.n-, hence

2T

T

Now (257) gives

and the general value of / being /2 + z2— 2/e2, as in (256),
/=2/(l-n)2, /=2-w, and /-i2=/(l-ra), since /w=i2.

The last equation may now be written, combining (e) with it,

Adding this equation to (c),

/ ,

M /<!>/ _ (1

JN O W r-

j

VOL. II.

^) sin <p cos <p v/I _ tan <p V'l , tan <p \/i

.^ - ^ - ; - T = - •

jL j L

N

90 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Combining this value of <£/ with the preceding equation, we shall
find

cosi; cosT

^Traces2® 2 cos2® 1

. j ... /T I £._ I- -I

+tan<p yi|^ -^2 -^

7*tan<p \/I
and this latter term, in this case, may be reduced to — - — ~

But, a and b being the semiaxes of the hyperbolic cylinder, (248)
ab mnii . , , . 2 sj ab k

glves^=7^r "L^* or m thls case^ M m=w^ -T7--=;-

Now A /— is the distance from the centre to the focus of an
V ij

1 1

hyperbola the squares of whose semiaxes are - ab and 4 ab ; hence

J l

represents an arc of an hyperbola the squares of whose semiaxes

7 7

are - ab and 4 ai, as will be shown in sec. F521 .
J l

k
Introduce this value of -., and divide by 2,

COS8 V 1 COS3 T

- (290)

Now, when this equation is integrated between the limits <p=0
and <p = sin"1,* / _, or, taking the corresponding values, between

» I/

r=0 and 7=8^-' ( =-r4 )* or Between v = 0 and v=^, T is infinite,
\1 ~rj/ £

J-\
— is also infinite; but
COS3 V

twice the difference A between these infinite quantities is finite.

1 I—/

Let sin2 <p, = -j, sin r, = — -< ; then

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 91

Hence the difference between an infinite arc of the equilateral
logarithmic hyperbola, and the corresponding infinite arc of its
asymptotic parabola, is equal to a straight line + an arc of a plane
jwirabola— an arc of a plane hyperbola.

When the parameters m and n are equal, the logarithmic hyper-
bola may by analogy be called equilateral, seeing that though the
squares of the axes of the hyperbolic base of the cylinder are not
equal, they differ by a constant quantity.

Resuming (250),

b* ll-l

But when m = n, 1=1 +j, m = \—j) substituting these values in
the preceding expressions,

2(b*-a*)=k*.

51.] On the logarithmic hyperbola when /=oo. Case XIII.,
p. 7.

T

3)' °r

Now, as ln=i2, and as i is finite, while /=<x> , n=0.

The equation of condition m + n — mn=i2, gives therefore m = i2.
Equations (248) and (249) give « = 0, 6 = k.

B V»

And as \/B (A -f C) = — -=-, we get;
v m

62 If* A/*M */7

AT V "* V * /T • -o »

= ^-= — ^-_= vt, since m = i^ = n/ ;

T ,- C_ cos2 tpd(p

^=: V*J (-x./gi^^p Vl-^sin2^*

Let /sin2<p=sin2i/r; therefore

V^ cos <pd<p=cos T/rdi/r, [1 — / sin2 <p]2=cos4 i/r,

and cos<=

Making these substitutions in the preceding equation, we get

T vi f d* ? i

—=—!-= I — — . When /=oo, ^=0, n=0;

k /

92 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

hence T=k--, ....... (292)

J cos3 a/f-

or the logarithmic hyperbola in this case becomes a common para-
bola.

As «=0, b=k, the hyperbolic cylinder becomes a vertical plane,
at right angles to the transverse axis.

Hence, comparing this result with that in sec. [19], we find that,
when the parameters are either + oo or — co , the corresponding
hyperconic section is a plane principal section of the generating
surface, i. e. either a circle or a parabola.

52.] By giving a double rectification of the common hyperbola,
we shall the more readily discover the striking analogy which exists
between this curve and the logarithmic hyperbola.

Let Y be an arc of a common hyperbola, whose equation is

A a tt4 COS2 X u am /v

Assume a?2=-^ ~- — e . . r= 9 -9. -,9— -, ,^. . (a)

a2 cos2 X — 62 sin2 X a2 cos2 X — 62 sm2 X

Differentiating these expressions, and substituting, we get

dY_ 62

dX~

2 __

Assume sin2 <t> = =-- sin2 X, and let i2 = O to. . . (b)

-1 z 2

O to.

az + o2

^ ~ '

-wj,. i. f ,, . ,. f dX dY dY dX

finding trom this equation the value of T--. as ^— = 3^-. 3-, we

d<p' d<p dX '

shall finally obtain, since

« « J [1 -sin2 <p] Vl -«2sin <p2'
Sec. [88] gives Y=JjpdX+J^, or Y=-fj»dX-i?. . . (d)

Now,as^=a2cos2X-52sin2X, ^=- J«2 + &2) sin X cos X

MX («2cos2X-62sin2X)^

as sin2<p=-JL*in2X, ...... (e)

CL ~T~ u

dX a cos <p

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 93

hence -=- V«2 + 62tan<p \/l-*2sin2<p ; . . (g)

and as p = a cos <p,

__ «cos __ —

= ~ ~

whence, finally,

. . . . (i)

/¥ * sm <P cos <P
VI- -^j-f. • W

as may be shown by putting

g= sin(pco8(p_
V/l-z^sin2^'
Differentiating this expression and multiplying by i2,

^•2d<£ _ z2 — 2i2 sin2 <p + f4 sin4<p , ,

df ~ [l-i2sin2<p]f

rj _ i2sin2 <pl2 _ fl _ i2}
This expression may be put in the form - - £J - ^ - ^ ,

integrating

r d<p /.

1 71 — -2-2 i" = A
J [1— z2sm2<p]5 J

^ « sn <

- * sn <-

-2-2 i" -

[1— z2sm2<p]5 J Vl-« sm2<p

This is the integral referred to in sec. [44*] .
Adding the integral (k) to,(i),

,- i2sin<pcos<p

Hence, dividing by (1— i2),

3^+Ji VI"^+J VT ' ' ' ' (o)

»v C* i\tf\

and (c) gives — = I ^ — .

a(l— -i2) J fl— sin2<pl Vl — i2sin2<p

94 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Eliminating Y from these equations, we obtain

Jd(p C d<p

[1 — sin2<p] \/l — *2 sm2 <P J [1— «2sin2<p] \/l — i2sina

r J.. (293)
tan<p

I" /^ ~" ^n~~. i 7\ I

The parameters are reciprocal in this equation, being 1 and z2.

Now this is the extreme case of the formula for the comparison
of elliptic integrals of the third order and logarithmic form. We
perceive that this formula results from the comparison of two ex-
pressions for the same arc of a common hyperbola. We may also
see that it is the limiting case of the general formula for the com-
parison of elliptic integrals of the third order having reciprocal
parameters — a formula which in like manner has been deduced
from the comparison of two expressions for the same arc of the

logarithmic hyperbola. It is also evident that^'2 — •— being the

2

sn<z> cos<p •.. . J_. ,.„
difference between tan<p. VI and- — J= — -, it is the difference

between tangents, one drawn to the hyperbola, the other to the
plane ellipse; for tan<p v'l denotes the portion of a tangent to an
hyperbola between the point of contact and the perpendicular on

it from the centre, and - — denotes a similar quantity in

an ellipse. This difference is precisely analogous to the expression

(* i\ f* r\

that occurs in (284) 1 — ^ -- 1 - — -^— , which denotes the difference
J cos3 v J cos3 T

between two parabolic arcs, one drawn a tangent to the logarithmic
hyperbola, the other a tangent to the logarithmic ellipse.

Hence a hyperbolic arc may be expressed by two elliptic arcs.
(Landen's theorem.)

For, eliminating the integral of the first order between (i) and
(218), we get, putting

Y-/tan? Vl=

The difference A between the infinite arc of the hyperbola and its
asymptote is found by integrating the above expressions between

7T

0 and ^. <£ becomes =0; and the difference is given by the equation

ON TUB GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 95

CHAPTER VI.

ON THE VALUES OF COMPLETE ELLIPTIC INTEGRALS OF THE
THIRD ORDER.

53.] We have hitherto investigated the properties and lengths
of elliptic curves, on the supposition that the generating surface,
whether sphere or paraboloid, was invariable, and that the lengths
of the curves were made up by the summation of the elements
produced by the successive values given to the amplitude <p between

certain limits ; 0 and — , suppose, if the arcs are to be quadrants.
A

Thus the length of the quadrant is obtained by adding together
the successive increments that result from the continuous additions,
indefinitely small, which are made to the amplitude. We may,
however, proceed on another principle to effect the rectification of
those curves. If, to fix our ideas, we want to determine the length
of a quadrant of the spherical ellipse, we may imagine the vertical
cylinder, which we shall suppose invariable, to be successively
intersected by a series of all possible concentric spheres. Every
quadrant will differ in length from the one immediately preceding
it in the series, by an infinitesimal quantity ; and if we take the
least of these quadrants, and add to it the successive elements by
which one quadrant differs from the next immediately preceding,
we shall thus obtain the length of a quadrant of the required sphe-
rical ellipse, equal to the least quadrant which can be described on
the elliptic cylinder, plus the sum of all the elements between the
least quadrant and the required one. Thus, for example, the least
'quadrant which can be drawn on an elliptic vertical cylinder, is its
section by a horizontal plane, or a quadrant of the plane ellipse,
whose semiaxes are a and b. In this case the radius of the sphere
is infinite. The least sphere is that whose radius is a, and which
cuts the cylinder in its circular sections. Hence the greatest sphe-
rical elliptic quadrant is the quadrant of the circle whose radius
is a. All the spherical elliptic quadrants will therefore be comprised
between the quadrants of an ellipse, and of a circle whose radius
is a. Any quadrant, therefore A of a given spherical ellipse is equal
to a quadrant of a plane ellipse plus a certain increment, or to a
quadrant of a circle minus a certain decrement. The same rea-
soning will hold as well when we take any other limits besides

7T

0 and -q. These considerations naturally lead to the process of
tii

differentiation under the sign of integration, because we cannot
express, under a finite known form, the arc of a spherical or loga-

96 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

rithmic ellipse, and then differentiate the expression so found,
with respect to a quantity which hitherto has been taken as a
constant.

We may conceive the generation of successive curves of this
kind to take place in another manner. Let us imagine an inva-
riable sphere to be cut by a succession of concentric or coaxal right
cylinders indefinitely near to each other, and generated after a given
law. These cylinders will cut the sphere in a series of spherical
ellipses, any one of which will differ from the one immediately
preceding by an indefinitely small quantity. If we sum all these
indefinitely small quantities, or, in other words, integrate the ex-
pression so found, we shall discover the finite difference between
any two curves of the series separated by a finite interval. One
of the limits being a known curve, the other may thus be deter-
mined.

To apply this reasoning.

In the following investigations we shall assume the generating
sphere to be invariable, and the modulus i with the amplitude <p
to be constant. The intersecting cylinder we shall suppose to vary
from curve to curve on the surface of the sphere. But i is con-

a2 _ £2

stant, and i2= — ^ — > see (<W). Now, a and b being the semiaxes

Gf

of the base of the cylinder, it follows that the bases of all the vary-
ing cylinders are concentric and similar ellipses. Hence in the
elliptic integral of the third order, which represents the spherical
ellipse, the parameter e2 or m and the criterion of sphericity V '*
will vary.

In [9] we found for a quadrant of a spherical conic section, which
we may denote by <7, the expression

'= V«

— e2sin2<p] VI — i2sin2p
Let k be the radius of the sphere.

Since ^—^ — ^-3, e will vary, as being a function of a the
A; — j a

transverse semiaxis of the variable cylinder. We have also

e2*=(l-e2)(e2-i2) (294)

Hence -^— — 2e(l——]:

de \ e4/

and if, as before, we write M for l-msin2(p, or 1— e2sin2<p, we
shall have

TT

/-f5

•= VK\

Jo

(295)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 97

Differentiating this expression on the hypothesis that i and p are
constant, while e is variable, we shall have

ir |"~ ? ^ ~

J w -. - j /»~ j . — /»o j f* o J

cj(T_ . o/c I •* dp \Kn \ dp I •* up
de=2~7ScUUo M Vl+ ~T~ Jo M8 VI Jo M VI j*

Multiplying this equation by ***, and recollecting that

e

= — 2e ( I — —. ), we shall have
V e4/

oV
de

v*?!- A _A

-de V eVJo

,

2

But (134) gives, writing M2 for N2, e2 for m, and z2 for m—n + mn,

9if

&K

e

\

VI

. . (297)

)

Introducing this value into the preceding equation, the coefficient

r

— <P-^ will vanish, and we shall have

M VI

Dividing by -^— , and integrating on the hypothesis that <p and *

e
are constant,

~ _ ~i n - ~i

f2i /T C de C2 d<p Tde(e2— z2) ,

<r= I d<p VI 1 =— /- + constant;

_J« JJe V* L_Jo VIJJ e3 V*

or, as in (294) e •/ K = V' (1 — e2) (e2 — i8) , we shall have

~ «• ~i ^

f"2

o"= I d<p VI
_Jo

~LJo7

4- constant

. . . (299)

VOL. II.

98 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

We must recollect that the complete integrals within the brackets
are functions, not of <p, but of i2, 0, and |. They are therefore

constants.

It is not a little remarkable that the coefficients of the complete
elliptic integrals are themselves also elliptic integrals of the first
and second orders. To show this, assume

e2=cos20 + i2sin20 (300)

Therefore l-e2=/2sin2 6, and e2 — P=j* cos2 0; we have also
ede= — j2 sin 0 cos 0d0.
Hence, if I-/ sin2 0=J,

f d* f _jg^ _=_

J \/(e2— i2)(l — e2) J */l — ^sin20

^'2 sin 0 cos 0
and v/g="~7T_-g • g~a- •

In the same manner we may show that

=? fc_ f _Jg

l-e2e2-"J Vl-/si

(301)

; (303)

_ sin 0 cos 6
— ld<9 \/l— 72sin20— 712— — • (304)

d

(3°5)

Substituting these values in (299), we obtain

(306)

To determine this constant. We must not suppose i = 0, in this
case, as is generally done, to determine the constant. This would
be to violate the supposition on which we have all along proceeded,
namely, that the variable cylinders are all similar, and therefore
that i must be constant. We must determine the constant from
other considerations.

i2/t2

Since e^=-^ — ^-5, when a=0, e2 = z'2. But as in (300)
2 —

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 99

7T

e*=cos*0 + iz sin20, therefore 0=^> As a, the major semiaxis of

the base of the cylinder, is supposed to vanish, the curve diminishes
to a point, and therefore o- = 0.

t7T

When a = k, e2 = l, and 6=0. We have in this case cr=-; for

Z

the sections of a sphere by an elliptic cylinder, whose greater axis
is equal to the diameter of the sphere, are two semicircles of a

great circle of the sphere. Hence, when 6=0, &=T)> sin 0=0,

f C dfl

1 d<s \/J = 0 I — F- = 0; therefore the constant is equal to a- when
J 'J v J

frr

= 0. But when 6=0, (r = -^, or the constant is equal to — .

<* A>

The formula now becomes

7T

' r- ~\t*ia

i a ( A0

dip VI ~~TT
jo jj yJ

Ja d<p f d0 f sin 0

|-7f-ld0VT-hf-
o v* J yJ J V

sn cos

yj

r

(307)

7T

When Q=-x, e = i, and o-=0, as the variable cylinder is in this

• .w

case diminished to a straight line ; therefore the preceding formula
will become

,

f 1 d0 f ? f a d(p f 5

1 •^JU.<WvJJ-U. viJU.

or, using the ordinary notation of elliptic integrals,

;

, (308)

-FjFy (308*)

Hence we obtain the true geometrical meaning of this curious for-
mula of verification discovered by Legendre. In its general form
(307) represents the difference between the quadrants of a great
circle and of a spherical ellipse. When the spherical ellipse va-
nishes to a point, this expression must represent, as in (308) , the
quadrant of a circle.

100 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

54.] If we now apply the preceding investigations to the curve
described on the same sphere by the reciprocal cylinder, or by the
cylinder which gives a function having a reciprocal parameter as
denned in sec. [31], we shall find by substitution in (299)

" ?

-Jo

T

JS

-J»

VI

.f

— e'2 e'2

L
~ + constant

. (309)

But by the conditions of the question, as in (110),

• :9

I • JO

pp' 1 />' *

,c — *, c —

l-/sin20'

C Ae1 _C d0

J \/ (e'2 — z'2) ( 1 — e'2) J V 1 — j* sin2 0'

Jde' /(J2—& C ;'2sin20d
* / ^^ „. • ^
,,/2 % / i 10 ~^~ I /"i "5 ~
C V I — />'* J 4/ I ----r 1* Gl I
^ » JL "~~ C %/ 'V X — y gjjj

. (310)

d0

Vl-/sin20 ^
Substituting these values of the integrals in (309),

We shall now show that the constant C = 0.

When 0=0, e=l, and therefore e' = i. Since e'=i, and cr is a
quadrant of the vanishing spherical ellipse whose principal arcs

a=0, /3=0, we shall have ^=0. Hence also Jd0 \/J=0, I -y =0;

therefore the constant is 0. When 0=^, e' = 1, and (309) becomes

JL

r «•

o VJ.

f

vi

-, \

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 101

7T

or, in the common notation, •gssEiFy+EyFi— FfF/,

it

a formula already established in (308) .

If we add together (307) and (312), we shall have, since
j* sin 0 cos 0

r *

7T

-ir

(313)

. ' /l-m\ ,— C d<p

Now, as m (11) o- = ( - ] iJmn\-^r . 0 _ .— , . a >

\ m / J [1— msm2<p] vl — * sm2<p

, /I — mA f d<p

= l~^T J ^^J [l-m,sin2<p] ^T=?*tfj>'
in which mw' or eV2 = i2.

Whence, as i- -\ \/mn = \- ; — M \Sm.n.= \/K, as we have shown
\ m / \ mt /

in (113),

— wsin8<p] \/l— z2sin2<p

! * (314)

+ V4 V- -?- =Vir

The reader will observe how very different are the geometrical
origins of two algebraical formulae apparently similar. In the
logarithmic form of the elliptic integral, the formula for the com-
parison of elliptic integrals, with reciprocal parameters (one of
which is greater, while the other is less than 1), resulted from
putting in equation two algebraical expressions for the same arc of
the one logarithmic hyperbola. See sec. [48] . In the preceding
case, that of the spherical ellipse, the analogous formula expresses
the sum of the arcs of two inverse spherical ellipses, whose ampli-
tudes are the same.

We shall use the term inverse spherical ellipses to denote curves
whose representative elliptic integrals have reciprocal parameters.
The terms reciprocal and supplemental have long since been appro-
priated to curves otherwise related.

Let a and /3, «y and /3y denote the principal semiarcs of two such
curves. Since the modulus i is the same in both integrals, the
orthogonal projections of these curves, on the base of the hemi-
sphere, are similar ellipses. (9) gives

e2 = i2sec2£, p,2 = i2 sec2 0, ; and we assume e^e^ — i^.

102 ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS.

Hence sec/3 sec@li=l (315)

Again, as tan2 «(1 - e2) = tan2 18 = sec2 18 - 1 ,

and tan2 «,(! -e,2)=tan2 /S^sec2/?,-!,

multiplying these expressions together, and introducing the relation
established in (315),

tan2 a tan2 a.i2 = L555L^!L^ '— __ J_ _ - 1 . (316)

Hence the principal arcs of the inverse spherical ellipses are con-
nected by the symmetrical relations

tan«tana,i=l, and sec/3 secj3y»=l. . . (317)

When the inverse curves coincide, « = «,, /3 = /3;, and the last
equations may be reduced to tan2 a— tan2J3 = l. Now we have
shown in (59) that when the principal arcs of a spherical hyperconic
section are so related, the curve is the spherical parabola, or when
the curve becomes its own inverse it is the spherical parabola.

sin2 a — sin2 /3 sin2/3

We have shown in (9) that z2=— -=1 —

sin1 a

(3) gives cos 17=- — , Zy being the angle between the cyclic arcs

of the spherical ellipse. Hence i=sin 17, but i is constant. There-
fore all inverse spherical ellipses have the same cyclic arcs.

Resuming equation (314), and making the assumption that the
two inverse spherical ellipses coalesce and become identical, the
resulting curve is the spherical parabola. In this case m = n=i,
and (314) may now be written

2V«17^ — . Q n — /, o • =f== V*i — 7^=
J [1— msm2<p] VI — m 2sm2<p J vl—

But as

sm <p

I —

~

1 _ j 2;

and m=--+ see (60), we shall have v^=T~-j and the foregoing

equation becomes

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 103

But (62) gives

' \\/ i-flLjMrin**

J V Vl +jt .. (319)

=/f_=i= ^+tan->-t^L
J v I — 32 sin2/* v 1 — i2 tan2/i

Now it is shown in (68) that when the second side of this equation
is integrated between 0 and filt tan p, being = — -^, the quadrant of

/»-»AV

, . , , , , .4 tanvy/ d/a TT

the spherical parabola becomes j j — — ==^=^— = 4. — since

*} 0 \ X ^~ ? Sill LL

— -2— — — is equal to 1 when tan/i= — ^.

j\ i2sin u *J i

Hence the first side of this equation represents a quadrant of a
spherical parabola, or

JT

j C* d<P 7T

and this expression is identical with (313), since V ' K= r-^r-- when

an expression derived from principles quite remote from those
established in the earlier portions of this book. These coincidences
may be taken as satisfactory tests of the accuracy of some rather
complicated investigations, based on principles both obscure and
remote.

55.] That portion of the surface of a sphere which lies between
the cyclic circles may be called the cyclic area.

The spherical parabola divides the cyclic area into two regions.

104 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

In the one, between the pole and the spherical parabola, lie all the
inverse curves, whose parameters range from z2 to i. In the other,
between the spherical parabola and the cyclic circles, lie all the
conjugate inverse curves, whose parameters range from i to 1.

Let acb, adb be

the cyclic circles, the *5j>- **•

intersection of the
sphere by an elliptic
cylinder whose trans-
verse axis is equal to
the diameter of the
sphere, and whose
minor axis is 2j. Let
a series of concyclic
spherical ellipses be
described within this
cyclic area, whose
semitransverse arcs
are 01,02, 04, 05,
and let 03 be the spherical parabola of the series. For every
curve, 01 or 0 2, inside the spherical parabola, there may be found
another outside it, 05 or 04, such that their principal arcs are
connected by the equations

tana tan «yi=l, sec /3 sec /3; i — 1 .

The algebraic expressions for the arcs of 'these curves having the
same amplitude give

elliptic integrals with Fig. 22.

reciprocal parameters.

The concyclic sphe-
rical ellipses will be or-
thogonally projected on
the base of the hemi-
sphere into as many
concentric and similar
plane ellipses, whose
semiaxes are 01, 02, 04,
05. The cyclic area will
be projected into the
plane ellipse A B C D,
and the spherical para-
bola into the area of the
plane ellipse, whose
transverse semiaxis is

k

Let E be the

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 105

area of the plane ellipse the projection of the cyclic area, and II
the area of the plane ellipse the projection of the spherical para-

bola. Then E = 7r/A:2, and II =

.,

X ~T~ 1

whence

—

or the

ellipse the projection of the spherical parabola divides the area of
the ellipse the projection of the cyclic area into two portions, such
that the outer is to the inner as i : 1. The semiaxes of E are k

and kj, while the semiaxes of II are

and k

—i), where

/ - -

i = sin r), 2rj being as in (9), the cyclic angle.

The importance of this curve, the spherical parabola, in the dis-
cussion of the geometrical theory of elliptic integrals is obvious.

We may determine the principal arcs of two inverse spherical
ellipses by a simple geometrical construction. Let AZB be a ver-

Fig. 23.

T'

tical section of the hemisphere on which the curves are to be de-
scribed. Let F be the focus of the elliptic base of the maximum
cylinder, whose principal transverse axis is accordingly equal to the
diameter of the sphere. ' Join OZ, FZ, and draw ZC at right angles
to ZF, meeting the line AO in C. Produce ZO until OD=AC,
and on OD as diameter describe a circle. We are required, given
one principal arc Za, to determine the corresponding principal arc
Zo' of the inverse hyperconic. Draw the tangent ZG. Through a
draw the line GOu. Through D draw the line DwG7. Join OG7,
it will cut the sphere in a', the vertex of the principal arc 7id .
Let OZ = £, then ZG=£tana; and as CZF is a right-angled tri-

A:2 k

le, CO = ZD= — -=-, k and B being the semiaxes of the

VOL. II.

106 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

maximum cylinder. As all the bases of the cylinders are similar,

Now as ZOG and ZDG' are similar triangles, ZG : ZO : : ZD : ZG',

or /fctana : k : : * : ZG' or ZG'=T- . But ZG' = £tan«,; hence

i i tan a

tan a tan «, i = 1, or the arcs « and a, are connected by the equation
established in (317).

When we require to know which of these successive curves on
this sphere is the spherical parabola, the same construction will
enable us to determine it. Draw ZT, a tangent to the circle on
OD, take ZT' = ZT" = ZT, and join T' and T" with O cutting the
sphere in c and c'. Zc = Zc' is the principal semi- transverse arc of the

spherical parabola; for ZT,2 = &2 tan2 a = OZ .

> = — , or tan2 « = -r.
i i

7T

As ZT'>ZO, cZc'>-^ ; or the principal arc of a spherical para-
bola is always greater than a right angle. Since in the spherical
parabola y + 2e=^, the angle COT'=e, or COT' is equal to half

the distance between the foci of the curve.

56.] It is easy to show that the integrals of the first order in
sec. [53] may be represented by two spherical parabolas having one
common focus at F, the nearer vertex of the one curve coinciding
with the focus of the other.
Thus, let F be the pole of Fig. 24.

the hemisphere ABD. Let D

BC/ and AC,F, denote two
spherical parabolas having
one common focus at F, Fy
and / being the other foci.
Let F/=7, and therefore

FF,=— — -7. Hence the mo- A.

«e

dular angles of the two
curves are 7 and ^—7;
and if we make cos 7 = i,

/7T \

eos(— — 7 )=;.

Thus, while the arc of the one is given by the integral
i 1 — /i J . ==r, the arc of the other depends on the integral

— z"smz

f d?

J VI -/si

sm2<p

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 107

57.] On the value of the complete elliptic integral of the third
order and logarithmic form.

Let

•

Jo [l-n si]

d<p

N

r. . . (320)

sin2 <p J V 1 — z2 sin2 <p

/i2 \
Assume /c the criterion of sphericity= (1 — n) I 1 j , . . (321)

then

dp

d<p

a dp

Multiply by 2/c, then

N

_. (322)

But (134) gives, making the necessary substitutions as in (297),

\

>'; . (323)

and

p_df_ _ [2z_2_2_2^ I f
Jo N Vl~ L»2 » n JJ0

N VI

Introducing the substitutions suggested by the two latter equations
into (322),

Now T- =

— ( -g— 1 ),

\n2 /

. . . (324)

whence

~T

an

VI

d*
dn

. (325)

108 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

If we divide this equation by 2 */K, tne first member will be the

differential of \/K

. Integrating this equation,

, (% d<p 1

— L). viJ

=. (326)

Assume

rc=i2sin20, then /C=

~, dn = 2i2 sin 0 cos 0d0. (327)

L I/
0 i~ 14. a /\ l~^~~ """•£» • S 7\

J n^ v* •/ *

^
J
/^

Jsi

We must now integrate this expression,
dfl

sin20 \/l— *2si]

i2 cos2 0d(9
sin2 0)3

f J2cos2
J (1 -i2 si

= (^--^ f y.

J sin20 \/l — i2sin20 J Vl — *2sin20

" vT^i* Sin2 0 +J (1 _ J2 Sin2 ^f
=J -v/l^FSn2^" * J (l-i2sin20)i

(329)

f

J

adding these equations,
dfl

cot0

tan26> Vl-i2si

_____ ,
(299). J

We have next to compute the value of the integral I - -=.
„ Jn V*

f d0 fdfl

J Vl-i2sin20 J Vl#

flV'*

Substituting these values of the integrals in (326),

d<p

.. . (331)

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 109

If we now substitute this value of

C 2 d<p

I T/7 m the equation given

Jo N

VI

in (175) for a quadrant of the logarithmic ellipse, namely

IT IT TT

^f^i+J>Vl,

2 Vl-

— n

]IV

since

•= (1— z^sin2 0) — cot2 0, we shall obtain the result-

ing equation,

(332)

writing H for

P~2 d<p f - f"2

— -73 J d# V (Ie) — d<p VI
^.Jo V1J i_J0 j

J

(333)

or in the ordinary notation,

When we require to determine the constant, we must not suppose
0=0; for this would render n=Q, and so change the nature of the
curve. Neither should we be justified in making z'=0 (as some
writers do) ; for this would be to violate the original supposition
(and all the conclusions derived from it), namely that i is constant
and less than 1. Moreover, since m + n—mn=i'2=Q, on this hypo-
thesis, m + n=mn ; or m and n would each be greater than 1, which
is inconsistent with the possible values of those quantities.

We have now to determine the value of the constant. In these
investigations we have all along supposed n>m. The least value
n can have is n=m. Were we to suppose n to be less than m, it
would be nothing more than to write m for n, since m and n are
connected by the equation m -f n — mn = i2. Hence, if m is not equal
to n, one of them must be the greater, and this one we agree to call
n, writing m for the lesser. To determine the constant, let us
assume n=m.

Now n=i2sin2#, as in (327), and n, when equal to

cot2 6 =

m

s
and

= 1 — Vl— i2 = 1 —j, (LO) = i — z* smz (/ =/

tan#=/-. | . Hence the coefficient of H in the last equation,

110 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

-x- FT, becomes 0. since in this case cot0= \/j; and

cot0 Vile)7

as n = m, the curve is the circular logarithmic ellipse. See sec. [43] .
The last equation now becomes

'1
•'o

' • (334)

Now, if we turn to (176), we shall find this, without the constant,
to be the expression for the quadrant of a circular logarithmic
ellipse, or the curve in which a circular cylinder, the radius of
whose base is a, intersects at an infinite distance a paraboloid inde-
finitely attenuated. Hence the constant is 0 ; and (332) without
the constant represents a quadrant of the logarithmic ellipse ex-
pressed by elliptic functions of the first and second orders.

CHAPTER VII.

ON THE LOGARITHMIC PARABOLA.

58.] The logarithmic parabola may be defined as the curve of
intersection of a parabolic cylinder and a paraboloid of revolution —
the vertex of this surface being supposed to touch at its focus the
plane of the parabola, the base of the parabolic cylinder.

Let the equation of the paraboloid be

(a)

and y2 = 4A2 + 4A# that of the parabolic base of the cylinder, the
origin being at the focus, k is the semiparameter of the para-
boloid, and h is one fourth of the parameter of the base.

Therefore a?a + y*= (2h + at)* = 2kz; ..... (b)

hence, x being the independent variable,

dg h

therefore

dS (2A +#)[*«+ (h+x) (2A + *)

' '
Now the expression under the radical being a quadrinomial in x,

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. Ill

must be reducible to the usual form of an elliptic integral. We
must choose a suitable transformation. Let

an'r- _

Lull I — i o . i a~~ To " i .... lei

2 - 2 2

deriving this value from (c). , Substituting this value in (d) and
reducing, we obtain the simple expression

d2_2A+a?
da? k sin T'

T is evidently the inclination to the plane of XY, of a tangent drawn
to the curve.

We must now eliminate x. Since

k* tan2 T = 2#> + 3hx + x*,

adding and subtracting 2h?—hx, we shall have
/t2tan2r = (2A + #)2-;

Completing the square by adding — , and taking the square root,

The positive sign only must be taken ; for when x=—h, tan r=0.
Substituting this value of 2h + x in the expression for the arc,

d2 h + <S (4k* tang T + h?)

dx~ 2ksinr ~ ..... W

If now we differentiate (e), we shall obtain
d# 2A:2sinT

dr cos3T\/(4£2tan2T
Multiplying the last equation by this expression,

QZf di ux Ilk k

dr ~~ da; dr = cos3 T v' (4^;2 tan2 T + A2) + cos8 r

dr , f dr

59.] There are now three cases (a), (/3), (y) to be considered :

2k=h, 2k<h, 2k>h.
Case (a). Let h=2k, and the last equation will become

COS3T COS2T

112 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

f dr

Now k tanr is the ordinate of a parabola, and k 1 — g— is the length

of an arc of this parabola from the vertex to a point where a tangent
to it makes the angle r with the ordinate. Hence, if we assume on
the logarithmic parabola a point M, and through this point draw a
plane touching the parabolic cylinder, this plane will be vertical,
and will cut the vertical paraboloid in a parabola whose semipara-
meter will be k. This parabola will touch the logarithmic parabola
at the point M. Hence in this case the length of the logarithmic
parabola to the point M will be equal to the arc of the plane para-
bola from its vertex to the point M, plus the ordinate of this para-
bola at the point M.

Case (/3). Let h>2k.

The general expression may be written

dr

cosr — — T — smT

.jV^z^ri

(c)

and the last equation becomes

dr dr

_,f _ dr f dr

J cos2 T V (1 -i2 sin2 T) + J cos3 r

cos2T-v/(l-i2sin2T)"r'vJcos3T (d)

Now, Y being the arc of an hyperbola, a the transverse axis,
and iz-= 2 ,2, it was shown in (c) sec. [52] that

dr

; . . . . (e)

' cos2 T V (1 — i2 sin2 r)
hence, if k=— - — = — -, we shall have

Logarithmic parabola = plane hyperbola + plane parabola, (f)

The semiaxes a, b of this hyperbola may easily be determined by
the equations

We may eliminate the arc of the hyperbola and introduce instead
elliptic integrals of the first and second orders.
Let 4/1 = 1 — i2 sin2 T, then as in (d)

2 f dr dr

'cos2r\/I cosar'

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 113

and the formula (293), for comparing elliptic integrals with re-
ciprocal parameters, gives

We have also, as in (1) sec. [52]

Jdr — 1 . z sn T cos r

r7i=r^72 drv/I f

Adding and reducing,

Case (7). Let 2k >h.

To integrate in this case, we must transform the second member
of the equation (h) sec. [58] . Assume

2£tanr=A tanv ...... . . (j)

4^2 _ ^2
Then if we make — —rg- =J2> we sna^ nave

\ ?. , *. , da? A sin v

a?)=A + A secu, and J-=TT

dt> 2 cos2 1/
But sin2T=-rT5-7^ .0 . a x .

d2' A V ( 1 — /2 sin2 v) A -/ ( l — ;* sin* „ ,
hence -r- = o — • — ^ - + s -^ ~^~ —• • • (k)

ri»i V. j^r\o* » i *J r>/^o" *i x '

Now, since

2 cos2u 2

dv

_

CO82T 2 COS2l»'

cos v cos

and cos T= - ^

v (1 — <;2 sin2 v) v J

writing J for (I—/2 sin2f), we shall have

— J2 sin2u)

_ cos3r 2J cossy

or

S'=^ f dv

2J -/(I'/sin^

A r___du_ _ *fJi_

2( J ^ J cos* !/•(!-/ sin8 w)+ Jcos3T

Now the second term of the right-hand member of this equa-
VOL. n.

114 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

tion is the expression for an arc of an hyperbola the distance
between whose foci is h. Hence

y=^-f--d;...-0--+Y+n, . . . (m)

II being an arc of the parabola.

We may eliminate the function of the first order and represent
in this case the arc of the logarithmic parabola by the arcs of an
ellipse, an hyperbola, and a parabola.

Let Y be the arc of an hyperbola whose semitransverse axis is

-, and putting E and II for the elliptic and parabolic arcs,

«/

n(T), . . (n)

or, as the equation may be written,

ON THE CURVE OF SYMMETRICAL INTERSECTION OF AN ELLIPTIC
PARABOLOID BY A SPHERE.

60.] The curve of symmetrical intersection of a sphere by a para-
boloid, whose principal sections are unequal, may be rectified by an
elliptic integral of the third order and circular form.

Let a?2 + y8 + a«=2r*and^-+£-=2* . ... (a)

K K-t

be the equations of the sphere and paraboloid, in contact at the
vertex of the latter. Then, finding the values of Ax, dy, and dz,

~z[z-2(r-k)][2(r-kl)-z']'
Assume z=2(r — k) cos2 6+2(r — k1)sinz 0 ..... (c)

Introducing the new variable 6 and its functions,

V(r-A) + (r-*,) tan2 0
Assume k(r — A:1)2tan2^ = A1(r-A)2tan2<p; . . . . (e)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 115

then, introducing the variable <p and its functions,

— k , ) cos2 <p + ^ (r — k) sin2

nd __ <- _ (f)

~ '

Multiplying together the foregoing values of -73 and ^-, and inte-

grating,

d£_

'

if we write m for

Tli andi* for ..... - (h)

J

r—

^ „ sin2 a— sin2 /3 2 sin2 a— sin2 /3 . . .

JNow, as i"«— ^-3— — , and m = ez=-^ - =-£, as m (9),
sm2 a sm2 a cos2 /3

we get from these equations

whence Vr2^/^, sin {>=

'tana

Making these substitutions, (g) will become

,= V^^M, ^^ sm j8 f- -^= (j)

t/ci tan a K J [i _c« sin* <p] v/1 -sin2 77 sin2 <p

Now, as we have shown in (10), this expression denotes an arc
of the spherical ellipse whose principal angles are given by the
equations (i), and whose radius is VY2 — kk}. Hence, if a sphere
be described whose radius is not r, but \/r2— kk}, the length of the
curve, the intersection of the sphere (r) with the paraboloid (kkj
will be equivalent to that of a spherical ellipse described on the
sphere whose radius is *Jr* — kkr

116 ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS.

When r=k, k being greater than klf (d) becomes

-A or s =

Hence s is an arc of a circle. That such ought to be the case is
manifest ; for in this case the sphere intersects the paraboloid in its

circular sections, and A / ~ * is the cosine of the angle which the
V k

plane of the circular section of the paraboloid makes with its axis.
It is obvious that the square of the radius of the sphere must
be greater than the product of the semiparameters of the principal
sections of the paraboloid ; otherwise the surface of the sphere would
fall within that of the paraboloid and their intersections would
become imaginary.

CHAPTER VIII.

ON CONJUGATE AMPLITUDES, AND CONJUGATE ARCS OP
HYPERCONIC SECTIONS.

61.] Conjugate arcs of hyperconic sections may be denned, as
arcs whose amplitudes <p, %, to are connected by the equation

cos 6) = cos (p cos ^ — smpsiny^ yl • — z2sin2o>. . (335)

This is a fundamental theorem in the theory of elliptic integrals,
and may be called the equation of conjugate amplitudes. It holds
equally in the three orders of elliptic integrals.

The angles <p, %, <o may be called conjugate amplitudes.

When the hyperconic section is a circle, i=0, and

cos w = cos <p cos v — sin <p sin v,

/\t /X*

whence t» = (p + ^, or the conjugate amplitudes are <p + %, <p, and ^.
The development of this expression is the foundation of circular
trigonometry.

THT-L T . COS 0 , ^

When CD = — , sin ^ = — r - and

Vl — z2sin2ip

}• (a)

cosy

sm<P= /, .. a-

When the hyperconic section is a parabola, t = l, and (335) may
be reduced to

tan o> = tan <p sec ^ + tan ^ sec <p (b)

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 117

If we make the imaginary transformations

tano> = V — lsino>', tan<p = V — lsin<p', tan%= i/ — Isiny/,
sec <p = cos <p', sec ^ = cos y/,

the preceding formula will become, on substituting these values,
and dividing by V — 1,

sin a)' =sin <p' cos ^ + sin y/ cos <p',

the well-known trigonometrical expression for the sine of the sum
of two circular arcs.

Hence, by the aid of imaginary transformations, we may inter-
changeably permute the formulae of the trigonometry of the circle
with those of the trigonometry of the parabola. In the trigono-
metry of the circle, co = <p + ^ ; and in the trigonometry of the para-
bola co is such a function of the angles <p and ^ as will render
tan [(<p, yj]=tan<p sec^ + tan^ sec<p. We must adopt some appro-
priate notation to represent this function. Let the function (<p, %)
be written (p-1-^, so that tan(<p-i-%) =tan<p sec^ + tan^ sec<p. This
must be taken as the definition of the function <p -1- ^.

The theory of parabolic trigonometry, which more properly
belongs to this part of the subject, has been fully developed in the
first volume of this work (see page 313) .

If we take (335) , square it, and add (cos a cos yj2 to each side to
complete the squares, and reduce, we shall have

cos<p = coso) cos^ + sino) sin^ \/l— i2sin2<p. . . (c)
In like manner

snw l~

since (335) shows that when <p = 0, coso> = cos^;, it follows that in
(c) and (d) the radical must be affected with the positive sign.

62.] Let us assume the equation given in (335) between the
conjugate amplitudes,

cos <o=cosp cos ^ — sin p sin^ \ 1 — i* sm*G>.
Differentiating this equation on the assumption that w is constant,

" d<p

(336)

writing Vl« for Vl— i2sin2w.

_ cos <p cos y — cos to

But \/L = — • ^— -i

sin <f> cos y

118 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

substituting this value of VL in the preceding equation,

[sin <p cos % + cos <p sin % VX] becomes sin co v%>
and

cos <p sin % + sin <p cos ^ VIM becomes sin co v'lp ;

consequently

sinw Vlxdp + sinw V%d%=0; .'. . . (b)

or dividing by sin co Vl<p V^x^ we sna11 °btain

- + -=-0. . ...... (c)

Integrating this expression,

-C. ..... (d)

Now, when % = 0, <p = co. Hence C = 1 -

»J

and the resulting expression becomes

, ,

(337)

This is the fundamental equation that connects conjugate elliptic
integrals of the first order, if their conjugate amplitudes are con-
nected by the algebraical equation

cos co = cos p cos % — sin <p sin % \/l — z2 sin2 co.
63.] The equation (335) between the conjugate amplitudes,
cos co =cos <p cos % — sin <p

which gives the foregoing relation between conjugate elliptic inte-
grals of the first order, naturally leads to the assumption of such a
form as the following,

as equal to some function of <p, p^, and w ; or as co may be assumed
to be independent, and ^ a function of <p by virtue of (335), we may
assume, using the notation adopted in this work

• • (338)
and proceed to determine

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 119

Differentiating this expression,

..... (a)

cos <f> — cos eu cosv
bllt VI.= Sn,riBX *. ...... 00

cos %•— cos o> cos ft ,, .

~~

or, reducing to a common denominator,

d<p(2sin<p cos<p— 2 sin<p coso> cosy,) +d^(2sin^ cosy,— 2 sin y, coso> cos<p)

2sin<psinY,sina>

Now 2 sin <p cos <pd<p = d sin2 <p, and 2 sin ^ cos ^d^ = d sin2 ^,

while

— 2d<psin<pcos^cos6>— 2d^sin^cos<pcoso)=2d. (cos<pcos^coso)).

Hence

,— d fsin2 <p + sin2 v + 2 cos <p cos y cos o>l

d<p Vl« + dY Vlv = -J^- -^- • (c)

^ v x 2sin<psm^smcu

But if w$ square and reduce the conjugate equation (335) we
shall have

sin2<p -f sin2^ + 2 cos<p cos^ cos w = 1 + cos2 to + i2sin2<p sin2^ sin2o) ;
hence

d [sin2 <p + sin2 % + 2 cos <p cos Y, cos w] = iM (sin <p sin^ sin o>)2, (d)
or df(<p) = i2d(sin^) sin^sinw) ...... (e)

Substituting this value of f (<p) and integrating

Jd<p Vlp+Jdy, V'Ix==^ + *2|S^n(P sin ^ sin w.
To determine the constant C. When %=0, <p=o>, and f(<p)=0;
and therefore C=Jdft> \/l»;
Hence finally

Jd<p \/I<p+Jd^ Vlx— j*dft> Vl<o = *2sin<psin^sina). . (339)
64.] To prove that

_C dw _ 1_ *an-ir w *^# sin <p sin y shift)

J (l+wsm2a)) vX> V/AC Ll-j-n — n cos <p cosy cos oJ

or putting U =sin <p sin y sin <u, and V=cos <p cos y cos o», . (a)

120 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

and using the notation hitherto adopted in this book,

3»VC

_ f d% f _tan-.

VJNxVIx J

Differentiating the foregoing expression on the supposition that co
is independent of <p and %, and that ^ is a function of <p, as in (335),
we shall have

=d%), . (342)

assuming as before f(<p) for the unknown function of <p.

d<Z) dy

But as — -== — — 4t, as shown in sec. [621, the last expression

V% Vlx

may be written

d<p

sn<p ~| _.,/./ x

2 2 2 ~

0 Li + n sin2 <p + n sin2 % + w2 sin2 p sin2 ^
or . -^ — ^= |^j

d<p r i^-i,

_

2 + ra sin2 <p + w sin2 X + n* sin2 <p sin2

_

and — -=*•=.— -^, as in sec. [62], substituting in the preceding

\ -10 r

expression

i _ [vi^d(p+ vr;dx] =^f( } (343)

i2 [1 + n sin2 <p + n sin2 ^ + w2 sin2 <p sin2 %]
But it has been shown in sec. [63] that

Vlipd(p+ v/Ix d;j£=i2d (sin^ sin^ sinw) ;

[1+ n sin2 <p + sin2% + w2 sin2<p sin2%] =
For brevity let this denominator be put D, and let
sin <p sin ^ sin co = U, as in (a),

then the preceding expression becomes -tpr- = df((p). . . (345)

We must develop this expression. (335) gives

cos3 <p cos2 % = cos2 co + 2 cos a) sin <p sin ^ VI + sin2 <P sin2
and cos2<p cos2^; = 1 — sin2<p — sin2^; + sin2<p sin2^;.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 121

By the help of this relation we may eliminate sin2<p + si
the denominator D, and we shall get finally,

D=l +wsin2&> — 2wcosG>sin<psiu^ \/Iu> + sm

U U2

from

or =

smw

writing as before U for sin <p sin % sin «.
The equation now becomes

/en sin2<adU

, (346)

(347)

having multiplied this expression by V '*> of which we shall pre-
sently see the need.

65.] Assume the equation,

tan®=^^ (348)

comparing the denominator of this expression with that of the
preceding formula, and also the numerators, so that the coefficients
of U and U2 may be the same in both expressions, we shall then
have

n2&>), CB= — w sin &> cos ft> \/Iw

in which A, B, C are undetermined constants.
Differentiating this expression,

ABdU

and A2 + C2 = n2 + ni2 sin2 &> .

Hence B=sino) yl-f«am*»j C= —

\/mn

n cos co

. /l+w\
and A=nsin&>(-

\ n /

But it has been shown in (42) that

, A n sin w V * -, . -r,

hence A= — -— , and AB=»

(350)

Having thus found for A, B, C values which satisfy the equation
(347), and render the differential expressions (347) and (349)

VOL. II.

122 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

identical, their integrals must be identical. But the integral of

r AU 1
(347) is V*f(?)i and the integral of (349) is 0 = tan-1-

Hence v//gf< = tan-1 ; .... .(351)

or substituting for A, B, C, U their values as given in (350), we shall

cos<p cos y — cos &>
have, putting for y !„, in the constant C its value — s —

as derived from (335),

/-->•« . r n \/K sin (p sin v sin o> ~i

-1— \>

>J

foK(y\
(352)

— n cos <p cos y, cos o>
or finally, putting for A//C its value ( \ \/mn,

r Cl , \/mn sin <p sin y sinw

V«f(<fi)=tan-1

1 — - - cos <p cos v cos a)
l + n

66.] Hence if we assume the conjugate amplitudes p , y, to as
denned by the equation (335), and take the sums of the conjugate
integrals of the first, second, and third orders, we shall find them
connected by the following equations : —

,f dx f d« _0.

J v£ Jyr

\/I

d(P

+ r %_ _ r dm = 1 tan_1 r n V^si

ip JNX Vlx JN^VL V* ' Ll + w— w

wcos(pcos%cose»

When * is negative, f (<p) is no longer a circular function, but a
logarithm, or, in other words, the circular arc becomes the arc of a
parabola, since the elliptic integral of the third order and loga-
rithmic form represents the sections of a paraboloid.

On Conjugate Arcs of a Spherical Parabola.
67.] The well-known relations between elliptic integrals of the
first order, whose amplitudes are conjugate, develop some very
elegant geometrical theorems.

Thus, in fig. 25, since the arc AQ=/ f— JL + QR, and the

T d '

1 — |

arc

T d
=J 1 — = + QR' (see sec. [20]), the arcs

o\ Mil. (,l.o\li:Tltl(\l. I'KOPERTIES OF ELLIPTIC I M l.i - K A I.S.

iiT. 25.

Now AQ + BQ=two quadrants of the spherical parabola, and

tr
QR + QR'= ^ -, whence half the circumference, or

In sec. [22] it has been shown that the complete integral repre
sents the semicircumference, whence

Comparing these equations (a) and (b) together, we get

^ f x

vr/J virj

J VI,

Now, as the triangle RR'P is a quadrantal right-angled triangle,
see sec. [24], the relation between the angles AFR, BER', or <p
and ^, is easily discovered. Since I?PE is a spherical triangle

7T

right-angled at P, and FE = 2e=- — y, we get j tan<p tan^;=l,

, A

since sin 7==;. When AQ=BQ, <p = x> and tan<p = — -=..

The locus of the point P is a spherical ellipse, supplemental to
the former, having the extremities of its principal minor arc in
the foci F, E of the former.

68.] Let <r, <Tt, a-n be three arcs of a spherical parabola, corre-
sponding to the conjugate amplitudes <p, %, to. Then, successively
substituting these amplitudes in (58), the resulting equation
becomes

T— J

124 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

But as the amplitudes <p, %, co are conjugate, the sum of these
integrals of the first order is = 0, whence

(T + a, — (Ttl=T + r — ru ....... (353)

Or, when the amplitudes of three arcs of a spherical parabola are
conjugate amplitudes, the sum of the arcs is equal to the sum of the
protangent circular arcs. The word sum is used in its algebraical
sense.

On Conjugate Arcs of a Spherical Ellipse.

69.] If, in (42), we substitute successively <p, %, w, and add the
resulting equations, we shall have

l+n\ —

^'

rC d<p C dy C d<o ~i
1 — 7=4 1 — — ~l~r^ — T — T/ + T// I

(354)

Now the conjugate relation between <p, %, and a> renders the sum
of the integrals of the first order =0, and the sum of the integrals
of the third order equal to a circular arc ©, which is given by the
following equation, as shown in (352),

n
1 — - - cos <p cos % cos w

Hence cr + a-l — a-ll=® — T—rl + Tll ...... (356)

Or, when the amplitudes are conjugate, the sum of three arcs of a
spherical ellipse may be expressed as the sum of four circular arcs.

When one of the amplitudes eo is a right angle, crlt becomes a
quadrant of the spherical ellipse =0-. ^=0, and © = T=TP as we
shall show presently, whence

(cr — o-;)— O-SST, which agrees with (52).

Or the difference between two arcs of a spherical ellipse, measured
from the vertices of the curve, may be expressed by a circular arc.

T IAK\ f j -L J

In (45) we found tanr = - =^= =•*-, tanr.=

A/i— i2sin2<p ~

Now, when w = ^, (a) sec. [61]

/i

COS(D COSV

gives 9m%=~-~==£==, sm(p= — ,-f^
Vl— « sm2cp vl — * sm2%

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 125

\^mn sin <t> cos <p *Jmn sin v cos y

\\ 1 1 once Vmn sm 9 sin y = — =.— = ,. „ . ^=^>

7T

or @ = T=Ty when r/;=0 or o> = -^-

70.] When we take the negative parameter m instead of the
positive n, (11) gives

Now the sum of these arcs is equal to a circular arc — ®/, which
may be determined by the expression

tfm ft = V^ sin p sin % sin a, (35g)

m

1 + - - cos <p cos v cos cu
1— m *

as in (352) ; whence O- + CT,— cr;/= — ®; ...... (359)

A little consideration will show that ®, must be taken with the
negative sign ; for if we compute the values of tan ® and tan ®/
from (355) and (358), we shall find

tang-tan » = UV V^ («» + »)
1

a symmetrical expression which remains essentially positive, how-
ever we may transpose tan © and tan <*)y.

Hence —tan (3^= — tan( — ©,) =tan ©/} or ®y must be taken with
the negative sign.

If we compare together (356) and (359) , we shall have the fol-
lowing simple relation between the five circular arcs ®, ©y, r, r,, r,,,

<H) + ®/ = T + T/-T/y ....... (360)

We may give an independent proof of this remarkable theorem.
The primary theorem (335) cos co = cos <p cos ^ — sin <p sin ^

sin to cos w sin <p sin y sin o> cos o>
gives — — — — °t — —

\/Iu cos "P cos X ~ cos m

and cos2<p + c

Let snupsinysinwrsU, cos<pcos^;cosa)=V. . (361)

M _ \lmtl sin CD cos o> V^^ U cos2 a>

IN OW tan TII -- ===== -- ~~ - 5 - rr= - ^

Vl — »2sin2a> cos2w— V

V^Ucos^ t|mT= V^Ucos2^.
cos2<p — V cos2^— V

126 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

, x tan T -f- tan T, — tan T,, + tan T tan T, tan TW

and tan (T + r, — T,,) = — -'-'- ,

1 + tan T/y tan r, + tan T tan /; — tan r, tan r

whence tan (T + T, — r;/) =

I _T r COS2(p COS2V COS2ft) m/iU2 COS2<p COS2^ COS2G)

^^Lc^s^^ + cos2X-V + c^2^^V~ (cos2<p-V) (cos2X-V)(cos2a)-V)

o. o __O O* ~,-,'2v*.,-,.^»-.l2_,.

_ TJ2

Cos2% cos2&) cos2o) cos2(p __ cos2cp

"

__
"Y) (cos2o) - V) (cos2o) - V) (cos2,? - V) (cos2<p- V) (

If we reduce this expression, we shall have, on introducing the
relations

, (363)
and cos2<w cos2% + cos2<p cos2a> + cos2% cos2<p

( . ,„„„.

tan (r+r,-r,,) - • (363)

If we now combine the values of tan ® and tan ©,, given in
(355) and (358), we shall have

tan (0 + a) _ (364)

^ /H-(i2 + mw)V-mn(V2+/U2) '

whence O + O^T + T, — r/y,

as is evident from an inspection of the preceding formulae.

On Conjugate Arcs of a Logarithmic Ellipse.
71.] In (162) substitute % and w successively for <p. Let

sin <P cos <p
— ,

(

(365)

1— wsin2^ 1— wsin2o)

we shall have, adding the three resulting equations together, and

,. .,. , n—m

dividing by — -r=,

\mn

TO 0_— n) rf d<p f d^ f dw

~»(»-»)9b>y^5+J^-J
dtp.:: f d% /* dta

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 127

Now, as <p, x> and w are conjugate amplitudes,

fJL

J VI

Vi/J Vix JVC

Vlx — Jda) Vl<o = i2sin<P sin^ sineo. See (339).
Whence

2 PC V V T V»»»

v w X— p - -
K n—m

— nil—

/.
f d

J N

/- .

' V/CU N0 VI Nx VI

We have now to compute the sum of

Since VI-=co8<pco8x-CQ8a,

sin <p sm ^

sn to cos to

UN

if we m ake, as before, cos <p cos ^ cos w = V, and sin <p sin ^ sin G> = U .
Finding like expressions for <I> and X, we shall have

„ n rsin2<pcos2<p sin2 v cos2 v sin2o>cos26)~|
— nfl^^l - £? - ^ + - 4= - *— - - ^ -

UL N^ Nx Nw J

V rn sin2 w w sin2 ^ n sin2 <

~~" ~~ ~

,, n sin* <p cos* <p _ cos'y (1 +n sin2^— 1) _ cos2 <p cos2 <p

TT-VT TT-*. T ^ "W-*^ ^™" '" — " -

nnd cos2(p_l+n-nsin2^-l_ 1 (1-n)

UN^ ~

, nsin2(pcos2(p_ 1 _cos2<p (1-n)

~^^~ ~nU ~U~"^lJ^rj

Vw sin2<p V V

^^ L = Yy— =T=

Finding similar expressions for the functions of o> and ^, and
recollecting that, as in (362),cos2<p + cos2x + cos2o>= 1 +2V— i8U2,

(368)

128 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

we shall have, making W = l — n + riV,

nU(n<5> + nX-nSl)=3-n + nV+ni?Uz-W^ + ^- + ^-

Now fd<p -v/I^+Jd^ VI*— Jd&> VL=*2U; as in (339), whence

<-fda> ^C)] 1

i" i i • '369)

Nx+N;-1]J

We shall find, after some complicated calculations,

»2*U8, ..... (370)
and NxNw + NttN?+N?Nx=W2 + 2W-ra(l-ra)(i2 + m)U2. (371)

Substituting the values hence derived, the whole expression
becomes divisible by rail2, and we shall obtain, finally, the follow-
ing expression,

n—m

, . . (372)

2mn^

+ (n-m)(W2-n2/cU2)

It will be shown that

_ r C dd> I dv C d

/I _ i 1 /v •

•IjN,^ jN.vixlN.^,,^ {373)
=5log

or writing, as before, W for 1— n + wV, and multiplying numerator
and denominator by the numerator,

/-rf ^ r dx _r do, n r

' v" U N? vi* J NX vrx JNU vi J L

When /c becomes negative or V ' K imaginary, we may pass from
the circular to the logarithmic form of the third order by the usual
imaginary transformation. When K is negative (352) gives

____ /n \/ /C\J

= V — 1®, where tan @ = — —

It is a well-known theorem (see vol. i. p. 335) that

i±^3*^|l- • • - (374)
I- V-ltan©J

0\ THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 129

Now instead of \/— Ttan @ we must write sin f = — ^ — , and the
preceding equation becomes

=log

,

=log

,„_>

; (375)

and this logarithm becomes log (sec£-f tan£), which is, we know,

the integral of 1 —-£* We shall therefore have
Jcos£

n v* d df

0
; and as 2

f d£ , f

j ^|=secf tan£+J

the result, dividing by 2, becomes

(n- m) (W2-n2/cU2)

. . (376)

Hence the sum of three arcs of a logarithmic ellipse may be ex-
pressed by an arc of a parabola and a straight line.

When one of the arcs %„ is a quadrant, V=0, and the equation
becomes

(377)

which coincides with (160).

If we apply to (163) the same process, step by step, and make
sin i

sin 5"= — w / , in which W;=l— m + rriV, we shall find

Wy

_? _? 7, f d$. km*n V*fUV

* J cos3^"1" (w-m)(W2-m2A:/U2

, f dr , r dr, _ , C drn

J COSST Jcos3^ Jco&Sru,

(378)

If we subtract this equation from (376), we shall have

J/» 10. /» j /» j /» j
d? , r <*£ =r d>r ! i d<r/ r dT/.
COS3^ J COS8 ^ JCOS3T JCOS3^ JCOS3T

n— m

(379)

VOL. II.

130 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Now this last term is divisible by (n— rri), and may be reduced to
the expression

mn

If in (170), which gives the relation between elliptic integrals of
the third order, we substitute successively the conjugate amplitudes
<p, %, and a), and add the equations thence resulting, we shall have

f «

Jcos£

fJlL-

Jcosry

cosr

(381)

in which

. e_ \Jrnn sin <p sin ^ sin ft>

AJ

COS <p COS y^ COS ft)

i

, .,_ v inn sm <p sin y^ sm o>

Sill L —

w

_

SUIT — —

l—m

smT.=

smr/;=

Vl — i2sin2<p .
\/mn sin eo cos to

(382)

If, in these equations, we change n into — w, and therefore sin£
into V — 1 tan (H), sin £ into V — 1 tan @y,

sinr into V — 1 tanr, sinTy into V — 1 tanry,
and sinr/y into V — 1 taiiT/y,
the preceding equations will become

__

~

' 1T/~

__

, an ;_

\/

wm sn eo cos w

and ® + @/ = T + Ty — ry/, as in (360), values which coincide with
those found in sec. [69] for the circular form. Or we may pass
from the logarithmic to the circular form, or from the paraboloid
to the sphere, or inversely, by the imaginary transformations above
referred to*.

Un examen plus approfondi des fractions de troisieme espece, nous fera con-
naitre que ces deux classes sont essentiellement irre"ductibles entre elles. — VER-
HULST, Traite des Functions Elliptiques, p. 78.

u\ THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 131

CHAPTER IX.

ON THE MAXIMUM PROTANGENT ARCS OP HYPERCONIC SECTIONS.

72.] Since the protangents vanish at the summits of these curves,
there must be some intermediate position at which they attain their
maximum. When the curve has but one summit, as is the case in
the parabola, the hyperbola, the logarithmic parabola, and the
logarithmic hyperbola, there evidently can be no maximum*.

«i2sin<pcos<p T,.
In the plane ellipse, the protangent /=— -= ; . J . It we

Vl — * sm2<p

differentiate this expression with respect to <p, and make the dif-
ferential coefficient 3— =0, we shall get

•4= (384)

yj

Substituting this value of tan <p in the preceding expression,

t=a-b (385)

In this case, the arcs drawn from the vertices of the curve, and
which are compared together, have a common extremity, or they
together constitute the quadrant, as may be thus shown.

The coordinates x, y of the arc measured from the vertex of the

I/ U

minor axis are # = a sin 3, y = b cos •& : therefore - =- cot 3 = / cot 3,

x a

since ja=b. If we now make cot3= y?jy #=$• Again, as

tt u u

tan X =Yg— „ — ,=/2tan\; or making \=^, or tan \= — -p, hence
b x a, , yj

11 if 11

Z-j=j7} or 7j=-. Therefore the arcs have a common extremity.

We have also tan2X=r. This property of the plane ellipse, called

Fagnani's theorem, may be found in any elementary treatise on
elliptic functions.

* The investigation of these particular values of those portions of the tangent
arcs to the curves, which lie between the points of contact and the perpendicular
arcs from the origin upon them — or, as they have been termed in this paper, pro-
tangent arcs — is of importance, because, as we shall show in the next chapter,
in the different series of derived hypercouic sections, the maximum protangeut
arc of any curve in the series becomes a parameter in the integral of the curve
immediately succeeding.

132 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

On the Maximum Protangent Arc in a Spherical Hyperconic Section.

73.] If we assume the expression found for this arc T in (45),
where i represents sin?;, 2?) being the angle between the cyclic
planes of the cone,

VWsin<pcos<p , .

tan r = -^7r_ J!^— r , ..... 386)

y 1 — z2snr<p

T being the angle which the linear protangent t to the elliptic base
subtends at the centre of the sphere. Differentiate this expression,

dx

as in the last article, and make j-=0, we shall find, as before,

d<

tanp= ,_*. (387)

\j V sin /3

If we substitute this value of tan <p in the preceding expression, we
shall obtain

tan T = tan « sec /3 — tan/? sec a, . . . (388)

writing f to denote the maximum protangent.

Now if we turn to sec. [68], we shall there find that this equa-
tion connects the amplitudes of three conjugate arcs of a plane
parabola. -Or if r, ft, and a are made the three normal angles of
a plane parabola, and (k.r), (#./3), (k.a] the three corresponding
arcs of the parabola, we shall have

(k . a) — (k ./3) — (k . T) = k tan a tan j3 tan T.
If in (386) we substitute for sincp and cos<p their values

\/7

and — , J , the expression will become

tanr=. (389)

(1-Kfl

We shall see the importance of this value of T in the next chapter.

I—/
In the spherical parabola, as m=n=i, tan2r=^ — -.=ir

Precisely in the same manner as in the plane ellipse, we may
show that when tan T has the preceding value, the arcs drawn from
the vertices of the curve have a common extremity. This will be
shown by proving that the vector arcs, drawn from the centre of
the curve to the extremities of the compared arcs, have the same
inclination to the principal arc 2«. Now, ->|r and o/r' being these
inclinations, as in sec. [14] , we find

tan4 «

o\ THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 133

;iud (39) shows that tan <p = cos e tan X. Hence, reducing,

tan2 3 sin2 B

Again, (49) shows, when we measure the arc from the -minor

principal arc, that cot#=,-, or cot#=— -r tamlr'. Now, in

b x sm p

order that we may compare these arcs together, we must have
#=X. Hence

, . . tan2 B 1 ,, .

tan2>Jr' = - — -r.-— ....... (b)

tan2 a tau2<p

When we substitute for <p any particular value, (a) and (b) will
give the corresponding values of tan -fy and tan i/r' ; but when we

make tan2<p = - — «=-., the values of
sin B j

and i/r' become equal, or

Fig. 26.

the compared arcs together constitute the quadrant.

74.] To determine the inclination, to the horizontal plane, of
the tangent drawn to any point of the spherical ellipse. The
spherical ellipse being taken as the curve of intersection of a cylinder
by a sphere as in sec. [10] , through a side Rr of the cylinder let
a plane be drawn, it will cut the sphere in a small circle, which will
touch the spherical ellipse in the point r, and will cut the base of
the hemisphere in the straight line HP, which touches the base of
the cylinder at the point R. Let O be the centre of the sphere
and Z the centre of the spherical hyperconic. Through the line
OZ let a plane be drawn at right
angles to the plane of the small
circle Rr-n-P, it will cut the sphere
in the arc of a great circle ZTT at
right angles to the arc rir ; and as
the three planes, namely the hori-
zontal plane, the plane of the small
circle Rr-TrP, and the plane of the
great circle ZOP?r, are mutually at
right angles, the straight lines in
which they intersect PR PTT, PO
are mutually at right angles; there-
fore P is the foot of the perpen-
dicular drawn from the centre O
of the base of the cylinder, to the tangent RP which touches the
curve. P is also the centre of the small circle Arrr, since A13 is a
chord of the sphere. Hence ATT is a quadrant, and therefore rir or
v is the inclination of the element of the spherical ellipse at r to
the base of the hemisphere. Now ZO is the radius of the sphere,
and Pr that of the small circle. RPO is a right angle ; and.thcre-

134 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

fore OR2 = OP2 + PR2. Hence Rr2 = Or2-OR2. Now for the
moment putting A and B for the semiaxes of the base of the
cylinder, OP2=A2cos2X/ + B2sin2X/, and

-- ...(A2-B2)2sin2X/cos2X/ -^ y /

PR2=^-T2 - o^ - .jo' . ^ ; whence OR2=-ro - 2. , r»2 • ^ , (a)
A2cos2X/-t-B2sm2X/ - A2cos2X; + B2sm2X/

T—S — , A4cos2X- + B4sin2X/

and therefore Rr2=Or2— ^2 - 2^ . x>2 • s^ •

A2 cos2 Xy + B2 sin2 X;

Let Or=l, A = sin«, B = sin/3, ..... (b)

RP2 J (sin2 « - sin2 /3)2 sin2 X, cos2 X,

and as tan2i/==H-, tan^v = . — 5 -- -- • — •

sin2a cos2« cos^Xy + sin2^ cos2/3 sin2A,,*

COS OL

t, as in (25), tan Xy= cose tan <p = ^tan<p. Substituting,

we get the expression

sin e sin « sin <p cos <p

tan v= — r . T . (390)

V(l~ sin2 e sin2<p) (1— sin2 97 sin2<p)

In supplemental spherical ellipses, since sin 77 and sin e, see
sec. [9] , are respectively equal to sin e' and sin 77', we infer
therefore that in supplemental spherical ellipses the inclinations
to the plane of XY of the tangents to the curves are the same
when the amplitudes <p are the same.

If we now differentiate this expression, and make -r-=0, we

shall find that tan*<p=- — -„. If we substitute this value of tan<p
in (390), we shall get

tanv=tan(« — /3), or j/=«— /3. . . . (391)

Hence the maximum inclination to the plane of XY of the tangent
to the spherical ellipse is equal to the difference between its prin-
cipal semiarcs. It is remarkable that the point of the curve which
gives the maximum difference between the arcs, which together
constitute the quadrant of the spherical ellipse, is not the point of
greatest inclination; for this latter point is found by making

tan2 <p=i — -p) while the point of maximum difference is obtained
by- putting tan2 <p=— — . This is the more worthy of notice, as we

Sill O

shall find the two points— the point of maximum division, and the

point of greatest inclination — to coincide in the logarithmic ellipse.

If we take the two plane ellipses which are the projections of

the spherical ellipse, one being the perspective, and the other the

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 135

orthogonal projection, and seek on these plane ellipses their points
of maximum division, we shall find that the angles, which tin; per-
pendiculars on the tangents, through these points of maximum
division of those plane curves, make with the principal arc, are the
values which must be assigned to the amplitude <p, to determine
the point where the tangent to the curve has the greatest inclina-
tion to the plane of XY, and the point which divides the quadrant
into two parts such that their difference shall be a maximum.
This is plain ; for the semiaxes of one ellipse are Stance, Artan /9;
while the semiaxes of the other are A: sin a and Arsin/8. And these
angles are given by the equations

tan a sin a

tan2 X= - -- n ; and tan2 X ,=- — ~.

tan/3 sm/3

On the Maximum Protangent Arc in a Logarithmic Ellipse.

75.] We must follow the steps previously indicated, and differ-
entiate the expression found in (165),

njmn sin <z> cos <p / x

SIIIT=— T* ...... (a)

Vl — i?siii*<p

T here denotes the inclination of the element of the curve to its
orthogonal projection on the ellipse, the base of the cylinder, which
intersects the paraboloid in the logarithmic ellipse, see sec. [38] .
T is also the normal angle of the tangent parabolic arc to the loga-
rithmic ellipse, whose plane touches the vertical cylinder. This
expression will be a maximum when the parabolic arc is a maxi-

dr
mum. Put the differential coefficient j-=0. This gives, as before,

tan<p= — -=. Substituting this expression in (a), we get

"We shall find the importance of this expression in the next chapter.

From (392) we derive tan2T=Tj - .-,

(1+.;)

.-,9 - .
2— mn

Now (l+y)2=2 + 2/— ia=2 + 2/— m— n+mn. Hence, as

whence we get tanr = — . Multiply this equation,

yl — m+ yl—n

numerator and denominator, by *J\ — m—V\—n, and the last

136 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

expression will become

\/mn \l\-m \/mn \/l — n
tanr=

n— m n—m

In (171) we found for the semiaxes of the cylinder, whose in-
tersection with the paraboloid is the logarithmic ellipse,

a_\/mn\/l—m b_ *Jmn \l\ —n ^ -_l^^"\ (393)
k~ n—m ' k n—m \ * /

This gives a simple expression for the tangent of the maximum
parabolic arc, analogous to (385) and (391). We have only to
take in the parabola, whose semiparameter is k, an arc whose ordi-
nate is a — b, to determine the maximum protangent parabolic arc.

The value tan<p = — p, which fixes the position and magnitude

VJ
of the maximum protangent arc to the logarithmic ellipse, renders

tan2X= For (150) gives tan2<p = tan2X. But (152) gives

«+/3 C C 1 , tan2\.

— = ^ — =:, and ~ — ^=5^ -- ; hence tan^<p=- -- .
a C— B C— B l—m 1 — m

If we now make

/l—m a

=V i=a=v

as we may infer from (171). Now, substituting this value of
tan2 A, in (155), we shall get

a— b

tanr=— r— .
k

Comparing this expression with (393), we find that the maximum
protangent arc is equal to the maximum inclination.

Again, if we differentiate the values of x, y, z given in (158),
the coordinates of the extremity of the arc measured from the
minor axis, and substitute them in the general expression for the
tangent of the inclination of any curve to the plane of XY, namely

, and make S=X, as before, putting for tan2X=tan2&
2

the value T, we shall get — ^^^ _. =— r— . Hence the arcs

have a common extremity, since they have the same inclination to
the plane of XY. As T=tan2X is the value of tan2X which gives

the maximum protangent =a — b in the plane ellipse the base of
the cylinder, it follows that the point of maximum division on the

ON THE GEOMETRICAL PROPER-DIES OP ELLIPTIC INTEGRALS. 137

logarithmic ellipse is orthogonally projected into the point of
maximum division on the plane ellipse; and the corresponding
protimgent in the latter a — b is the ordinate of the parabolic arc
which expresses the difference between the corresponding arcs of
the former. Thus, while the arcs which together constitute the
quadrant on the plane ellipse differ by the difference of the semi-
axes a— by the corresponding arcs of the logarithmic ellipse will
differ by an arc of a parabola whose ordinate is a — b.

76.] When the amplitude <p is given by the equation tan<p= — j=.,

V>

or when the protangent is a maximum, the corresponding arc of
the spherical ellipse, or of the logarithmic ellipse, may be expressed
by functions of the first and second orders only. This may be

shown as follows. When tan<p = — j=, the arcs cr and <rt of the

V>

spherical ellipse, or the arcs 2 and S, of the logarithmic ellipse,
together make up the quadrants Q, or Q,,; see sections [73] and [75] .
Hence <r + o-,— Q,, or 2-j-S^Q,. But we have also cr,— <T=T,
as in (52), and S,— 2 = r, as in (160). Therefore 2<7=Q,— T,

= — T. Or a- and cr,, or S and

2y may be expressed as simple functions of the quadrant
and T. Now the quadrant, as we have shown in the last
section, may be expressed by functions of the first and second
orders only, while T is an arc either of a circle or of a parabola.
Hence an elliptic integral of the third order, whose amplitude

(p=tan~1( — -=] may be expressed by functions of the first and

second orders only''6'.

CHAPTER X.

ON DERIVATIVE HYPERCONIC SECTIONS.

77.] We shall now proceed to show that, when a hyperconic
section is given, whether it be spherical or paraboloidal, we may
from it derive a series of curves whose moduli and parameters
shall decrease or increase according to a certain law ; so that ulti-

* Tout kappa dont 1'amplitude a pour tangente trigonom&rique — -j-=, [or, as

V b

it is written in this work, I TT^ll peut s'exprimer par des fonctions d'une espece

infMeure. — VEBHULST, TravU des Fonctions Ettiptiques, p. 99.

VOL. II. T

138 ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS.

mately the rectification of these curves may be reduced to the cal-
culation of circular or parabolic arcs, or, in other words, to circular
functions or logarithms. We shall also show that all these derived
curves, together with the original curve, may be traced on the same
generating surface, i. e. on the same sphere or paraboloid.

In sec. [44]* we have shown that the rectification of a plane
ellipse whose semiaxes are a and b, may be reduced to the rectifi-
cation of another plane ellipse whose semiaxes a,, bt are given by
the equations al=a + b,bl=2 \/ab, of which the eccentricity is less
than that of the former, a + b is that portion of the tangent, drawn
through the point of maximum division, which lies between the
axes ; and \/ab is the perpendicular from the centre on it.

We have shown in (63) and (74), that if <p and ty are connected
by the equation tan (ty— <p}=jtfui<p, while i and it are so related
that

we shall have

J \/l — i2sina^) 2 j Vl — z^sin2^ 2

Let us now introduce this transformation into the elliptic inte-
gral of the third order, circular form, and negative parameter. In
(191) we found

2 sin2 <p = 1 -f it sin2 ty — cos ty \^I,.

C dip C d<p

Now IT- — 7^ = 1 r~^ — ^ — . =^=-

JM yl J [1— msm2^)] VI— * sm2<p

Or replacing <p by its equivalent functions in ty, and recollecting
that m — n + mn=iz} since m and n are conjugate parameters of
the circular form, we shall find

jizvr t1-1-^— H^'^cc.t VQ vr; (894)

We may eliminate the radical m cos ty \/Iy from the denominator
of this expression by treating it as the sum of two terms.

Multip^ing and dividing the function by their difference, since

,-.,
l+J

2 (

M VI

mn

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 139

It is truly remarkable that whether the parameter of the original
function we start from be positive or negative, the parameter of the
first derived integral will always be positive. Indeed it is necessary
that this should be the case, because the parameters of the derived
functions, increasing or diminishing as they do, must at length pass
from between the limits 1 and i2. Should they do so, the integral
would be no longer of the circular form, but of the logarithmic.
Now we cannot pass from one of these forms to the other by any
but an imaginary transformation. This objection does not hold
when the parameter is positive, because the limits of the positive
parameter are 0 and co . It is, too, worthy of remark that the first
derived parameter is always the same, whether we transform from
positive or negative parameters. Write

mn /orm\

n.= .. a; (396)

(I+j)z

nt is the first derived parameter.
We may transform (395) into

d(Z> ,, . vf*i , |~rt WMy/i - a . , /T~l

-L-== (l-Mi)i d-ur 2 — m '(1 +w,sm2o!r— 1)— Tncosy VI,

M VI I L n, ^ J

J [i+^sin2^] vr,

Now

mil
~^~

-, and
n

2 m +

mit m + n

. (397)

n,

n

Hence

2^

1 — m)C dip

(m + n) Vwj

J

di/r )

.. (398)

m JM VI

«f7*

*Jmn

(n-

1 + n,

sin2->|r] VI,

2 m

n] A/T

2 v

'n

' tan 1

We shall now show that

cos <

/— . /Qno\

= V^sini/r ..... (399)

Vl— » sm2(p
If we revert to (189), (191), and (193), we there find

2sin<pcos<p=sini/r[ Vl/ + *"/cosi/r],
and 2 VI = (1 +./) [ V% + */ cos ^r] .

, — Vwinsin<pcos<p ,- .

(396) gives ^Jmn— vn,(l +j); therefore - -? v»j»»^.

140 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

If we replace 0 I — ^=- in the preceding equation by its value

3 J VI,
f dp
I — /=, and put Ny for 1 + nt sin2 o/r,

V;

M VI

i2 f d<p 1
1 — ?- ^=tan ]

mnj VI \mn

N,

VI

(400)

Now the common formula for comparing circular integrals with
conjugate parameters is, we know, see (47),

l—m \ f d<p

/l+ft\ f dtp /
V w /J N VI \

VI

, _ tan_ i

l — i2sin2f

Adding these equations we obtain this new formula,

l-m\ ,—

By the help of this important formula we may establish a simple
relation between the sum of the original conjugate functions of the
third order and the first derived function of the same order.

78.] If a- be the arc of a spherical ellipse, it is shown in (46) that

-«sinp

. (401*)

and in (11) that ^=

Adding these equations together, and introducing the relation just
now established,

ty i2 C dp _j r Vwz^sintp

VI< V^wJ VI L Vl-*'2sin2<p

No w, as m — w= i2 — m n, (m -f- w) 2= i4 — 2z2mn + w%2 + 4mn.

mn

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 141

1—7

We have also mn=nl(\+j}'2, it=—-*-, i*=(l — j)(l+j), and

2(2-?) =2(1 +/)=/ • hence

m + n = (l+j)*(l+nt) »Jmlt .... (403)

/m + n\ '••_ /l+w.\ ; _
and therefore V"^~J Vw/=\~^~/ Vw/w/- • • • (404)

It is worthy of especial remark that this coefficient of I — —^

fN'd

in (401) is precisely the same in form as the coefficient of 1 — Z-~.

JN VI
The preceding equation (402) may now be written

Let cry, wy, it, ty be analogous quantities for the derived spherical
ellipse a-, ; substituting their values in (401*) ,

i2 i2 f 2

Let y, gy, y;,, g/w, &c. denote — T==, — ~r=, — ,- — , &c., and put

»•* ^ ^ *•«»» &c- fo

(1 +.;•) (1 +./;) (1 +ju) (1 +y/w) , &c. Let also 4>, ^, ^, ^, &c. denote

the arcs, whose tangents are

Vwiw sin <p cos <p \/mlnl sin i/r cos i|r *Jmtlntl sin ^ cos ^ o
'

Making these substitutions, and writing Q, Qy, Q/p &c. for the
coefficients of

(405) - (406)

142 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Taking the derivatives of these expressions, we may write

Subtract (a,) from (a), (by) from (b), and (c;) from (c), the inte-
grals of the third order disappear, and we shall have

. (407)

If we add these equations together,
(o— <rj= (?«/

-?) - + ^/-*. (408)

If we multiply the first of (407) by 23, the second by 22, the
third by 2, and the fourth by 2°, and add the results,

an integral which enables us to approximate with ease to the value
of the integral of the third order and circular form, in terms of an
integral of the first order.

We have shown in sec. [28] how the integral of the first order
may be reduced.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 143

The above expressions may be reduced to simpler forms when
the functions are complete. In this case <I> = 0, ^=0, ^ = 0,
^=0, &c. ; and when <r is a quadrant, <rt will be two quadrants,
crn will be four quadrants, <rni will be eight quadrants, and so on.
Tin- preceding expression may now be written, denoting a quadrant
by the symbol <7,

-Sq) I

(410)

-=.
In (396) we found for the parameter of the derived integral of

vn/n

the third order the expression nt— .Ng. On referring to the
geometrical representatives of these expressions, we find for the focal
distance e, of this derived curve the expression w.=tan2e,= -; -

(i +;)

but if we turn to (389) we shall see that this is the expression for
the maximum protangent to the original spherical ellipse, which is

TflJl

given by the equation tan2r= 2. We thus arrive at this

curious relation between the curves successively derived, that the
maximum protangent of any one of the spherical ellipses becomes the
focal distance of the one immediately succeeding in the series.

79.] Given m, n, and i, we may determine m,, nt, and it, for

T —• y 7/i7i

i,= T—4, ft/=Tj — ya- Substituting these values of it and nt in the
equation which connects the parameters, ml—n{-

Hence, given m, n, and i} we can easily compute the values of mt,
n,, and i,, and then of mtl),nlt, and ilt, and so on as far as we please.

Given the semiaxes a and b of the elliptic cylinder whose inter-
section with the sphere is the original spherical ellipse, to determine
the semiaxes a/ and bt of the cylinder whose intersection with the
sphere shall be the first derived spherical ellipse.

We may derive from (53) and (54) the values of a and b in terms
of m, n, and i, or, eliminating i, in terms of m and n only ; for

*""• •• — io^~ /i . r» o»*» t*f — TO — —

A:2 w(l+w)' *2 m

_O JL2*«/T \

TT U i Tti Oi 7l,( 1 "~ ?/i, I

Hence -4 = — —-' — . -L — Jl il .

m

144 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Or substituting the values of mt and nt in terms of m and n, and
therefore of a and b,

a + b _2 \/ab

When the radius of the sphere is infinite, or the derived curve is
a plane ellipse, a,=a + b, bt = 2 \/ab, as in sec. [77].

When m=n=i, mt=nl=ii-, or when the given curve is a sphe-
rical parabola, the derived curve will also be a spherical parabola.
Hence all the curves of the series will be spherical parabolas.

If we take the corresponding integral of the third order with a
reciprocal parameter I, such that /m=z'2, and deduce by the fore-
going process the first derived function of the third order, we shall
find the parameter lt of this function to be positive also, and reci-
procal to nt, so that /ywy=zy2.

Hence, if we deduce a series of derived functions from two pri-
mitive functions of the third order and circular form, having either
positive or negative reciprocal parameters, the parameters of all the
derived functions /;, lu, lnl, n,, ntl, nllt, will be positive, and reciprocal
in pairs, so that Ifn^if, ^/w//=z//2, ^y;W;;/=iy//2, &c.

80.] We may apply the same method of proceeding to the loga-
rithmic ellipse, or to the logarithmic integral of the third order,

=^, in which iz > m.

(1 — msin2<p) Vl — «2sin2<p

If on this function we perform the operations effected on the
similar integral in (394), we shall have, after like reductions,

f d<P_- (1+>'<) f

JMVI 4(1-™)J

We must recollect that

T 1 '2 • 2 f mn

I/= -z/2sm2^, and mt=^

We may reduce this expression.

The numerator may be put under the form

m, '• '

Now 2—m — — !=± 1 and — '=-.

m, n m, m

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 145

We have also — :-*= = ..

t !+_;

Hence, making the necessary transformations,
(1 — m)C d<p ^(n—m) VI, C <ty i \fIlC&^_ \/i,Cc<

~^T~JMVI=~"^ TjM^i+^rj VA~~I j

If into this expression we introduce the relation given in (74),

T 2 j vi;

writing <p for /z, and >/r for <p, we shall have

* J M/ Vl/ mwj Vl

, — . V^^sincp cos<p

Now in (399) it has been shown that V^/smi/r= — /^ — -9-^—a ^~>

Vl — 1> sm2<p

and as ^nm= ^m,(l +j], the last term of the preceding equation
may be written

V^ f \lrnn sin <p cos <p~
d<pL v/l-i2sin2<p
. r .

mn sin2 <p cos2 <p
1 — i2 sin2 <p

Substituting this value in the preceding equation, and comparing
it with (169) or (170), we shall find

' V- (416)

__

M VI \ n /J N VI wm i J M/

This equation is analogous to (401) . By the help of it and the last
equation we can always express

Jd<p f d<p . - C d^

— - or \ - in terms 01 I - L-=.
MVI JNVI JM,VI,

Since m{-=—^- — ^ is symmetrical with respect to n and m, we should
have obtained the same value for the derived parameter had it been

deduced from 1 — *-= instead of I — ^. Since «'.=- — -.
JNVI JMVI !+J

m» (1- j)a— mn f Vl— »»— Vl— »12

andm.= „ n/=rr . -\g - = 7 - = - 7r= '

(1+2' 1+;)2— mn Ll-m+l— «J

rr . -\g -
(1+;)2— mn

VOL. II.

146 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

81.] We may express mt and nt simply, in terms of a and b, the
semiaxes of the base of the elliptic cylinder, whose curve of section
with the paraboloid is the logarithmic ellipse.

In (171) we have found the values of m and n in terms of a, b,
and k, namely,

a \Jmn(\ — m) b_ \^mn(l — n)

_

k~ n — m k n — m

a — b \/\ — m— \/l — n • ?

Hence -== — ; -.= , or, assuming the value 01 n, in

a + b Y/i_m+ VI— n

Now

mn (1+/)2— mn (\/l—m+ Vl — nY"

(IT* flsi 1YI — I — 7W,= — nr — —

vyA wj iivi — . -. .\ Q, .*• / / "i i "\ 2 /I i *\ 2

a — 6 v'fH*

and (a) gives — r- = — 7 ^

therefore l-^,= ( Vl-m- yi-

M! mn

Hence, reducing, w'

If we now compare together these expressions for m, and n,,
namely,

m.=

we shall find that n^m^ so long as k >2 \fab, that when ^r=
nt-=ml} and that when &<2 V ' ab, nt<mt.

To determine the axes of the base of the cylinder whose inter-
section with the paraboloid gives the derived logarithmic ellipse.

Since-74-= '' .„ -. -^——^ - -£- as we may infer from

" (ni~mi) " (ni~ mi)
(171), we shall have, substituting the preceding values of mt and nt,

af_

-

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 147

When £=co, or when the paraboloid is a plane, at = (a
b,=2 's/ab, -which are the values of the semiaxes of a plane ellipse

, . . . a — b 1 — \/i— *2
whose eccentricity is - -r=— . as we should have anti-

a + b 1 + Vl-«2

cipated ; for these are the values found in sec. [77] and sec. [79]
for the axes of the derived plane ellipse.

mn /I — ;V . 2
When m = n=l-j, my= 2=- =t*, and n,=0.

Hence, when the original logarithmic ellipse is of the circular form,
the first derived ellipse is a plane ellipse.

When F=4«6, (418) shows that m^n,, or -^=-J—Qct as in

sec. [43] ; but mt-=nt, is equivalent to n = m( V1+./ + \0')2-

Whenever therefore this relation exists between the parameters
and modulus of the original integral, the first derived integral will
represent the circular logarithmic ellipse, which may be integrated
by functions of the first and second orders. Accordingly whenever
the above relation exists between the parameters, the integral of
the third order may be reduced to others of the first and second
orders.

If in the second, third, or any other of the derived logarithmic
ellipses we can make the parameters equal, this derived ellipse will
be of the circular form, and its rectification may be effected by
integrals of the first and second orders only ; accordingly the rec-
tification of all the logarithmic ellipses which precede it in the scale
may be effected by integrals of the first and second orders only.

We may repeat the remark made in sec. [79]. The derived
functions of two integrals of the logarithmic form with reciprocal
parameters, have themselves reciprocal parameters.

82.] If we now add together (162) and (163), we shall have

4(n-m) 2_ , . r»* P__

r (n—m)C dr

+ 2 d<p VI— 2 , — 1 — g-

J r V A/TMOT 1 COS3'

(420)

We must now reduce this equation into functions of ^ instead
of $, >|r and p being connected, as before, by the fundamental
equation

tan (^ — p) =j tan p.

148 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

The elements of these transformations are given in page 69, namely

/r- . *Jmn sin <t> cos <p ,— .

2sm2<p=l + z/sm2i/r — cos>/r Vl» and — j= \ T= vn^my.

V 1 — * sm2<p

From this last equation we may derive

(1 — n sin9<p) (1 — m sin2<p)=I (1 — mt sin2 ^r) .

Now, as 3>n =

1 — n sin2 <p

--l

or, putting for sin2 <p its value,

2 V^ [1— mysi

In the same manner, we may find
<*, _

¥¥l "

. - p-| •

2 Vmw [1— ^s

Adding these equations together, and recollecting that m + n — mn = i2,
we shall get

*?xb , + m$> ^ V CT/Psin ^ + VOT, V^ cos -f sin ^ y/I,

\/»^ [l-m^in2^]

Now, as

«2= (1 +»(! — /), and \/mn= ^

In (186) we found

»«n^ (426)

Subtracting this expression from the preceding, the terms involving
sin T/T will disappear.

We must now compute the sum of the coefficients of 1 — ^= in
(420) and (426). Since

fd<p (l-H',)f <ty ... (l+^f*8 ,*8 on , -xl

1 /Y= — o~^ 1 — -^> this coefficient becomes - ' I — H --- 2(l-f/jJ

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 149

Or as in + n = j2 + mn, this coefficient may be written

2 ri 2 ~~\

Or as mn=m.(l +;)2, it becomes finally, - — r - — 1 I. . (427)

l + ijLm, J

H— E+S-'M^f-S^-ifdS-1^ (428)

,_ (l-n

and

yl ^

(»— m) (ft— w») 1 r d^/r

[1 — mi si

Now, as n + m^=i2 + mn
Hence (n— m)'2=i4

and as »4=(l+./)9(l~.7)2j mn=ml(l +J)12, substituting

therefore w-m2=

4
and as -^ - ^5= (1 +s';)2) the expression will finally become

n-m=

»— m i, /I — m,
hence - =~~

If we now add together (420), (425), (426), (428), and (429|, we

shall have, dividing by (-^^, putting %, for ^^cos^/r VI

V wm (1 — wy sin2 A/T)

42_ _mL*JmL_y. Vmf f, ,f

F= (l-m,)^ +(1-^)VV

- f d^ A; f d^ f dr

M^r+V »J vrr J^^J

Let us now take the logarithmic ellipse whose equation contains
m,, nt, it, >/r instead of m, n, i, and <p, we shall have from (163),

150 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

If we now subtract these equations one from the other, combining
together like integrals, the integral of the third order will vanish
and we shall have

COS8T,

,.(433)

Hence, as we may express an arc of a plane ellipse by an arc of a
derived ellipse, an integral of the first order, and a straight line —
a known theorem — so we may extend this analogy and express an
arc of a logarithmic ellipse by an arc of a derived logarithmic
ellipse, by functions of the first and second orders, by an arc of a
parabola and by a straight line. The relations between the moduli
and amplitudes are the same in both cases,

1—7

it = - — -., and tan (ijr — p) =j tan <p .

Let mn, n,,, in, ty, be derived from mr n , i,, ty, by the same law
as these latter are derived from m, n, z, <p, namely,

I-/

it = - — -., tan (ijr — <p) =j tan <p, m =

mn

.•\«J

'1— w + VI—!

and derive an arc of a third logarithmic ellipse, we shall have,
putting A, B, C, D for the coefficients of the integrals, and II for
the parabolic arc,

*

- c'*'+ D'n"

Multiply the first of these equations by 2 and add them, 2y will be
eliminated. In this way we may successively eliminate 2y, S/p 2yy/,
until ultimately we shall have

&*—>V f^ i i •*-*

_±_2-+i-=

v being the number of operations, and denoting by F and E, the
sum of the integrals of the first and second orders, by W the sum
of the straight lines, and by II the sum of the parabolic arcs.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 151

If in (401) and (416) we substitute the coefficients of the derived
integrals as transformed in (404) and (430), the relation between
the original and the derived integrals of the third order will be, for
the circular form or the spherical ellipse,

q+/A _f dp

(l+n sin2 p) V 1 — *2 sin2 p

\-m ._

dtp

. — »»sin2p) \/l — i2sin2p

(l+n, sin2 i/r) V 1 - «,2 sin2
and for the logarithmic form or logarithmic ellipse,

(434)

— isinp

fc

•', (435)

i j / » / l i

83.] The preceding investigations lead us to consider a new classi-
fication of elliptic integrals, which, in a geometrical point of view,
would seem to be more natural than the one at present in use.
As the first order is merely a particular case of the circular form
of the third, its geometrical type (the spherical parabola) being
a particular species of spherical conic, while the two forms which
are classed under the third order are irreducible one to the other,
representing, as they do, curves of different species, it would seem
a more appropriate division to found their classification on their
geometrical types, the plane, the logarithmic, and the spherical
ellipses, which those integrals represent. Thus that which is now
the second would stand the first, the logarithmic form of the
third order would hold the second place, while the circular form
of the third order, of which the present first order is a particular
case, would occupy the third rank. However, as the present
division has been sanctioned by time, and by the great names of
the founders of this department of mathematical science, Legendre,
Jacobi, Abel, and others, it would be presumptuous to propose to
change it. Besides, in a point of view purely analytical (the view
of the inventors) the present division of these integrals may be
held to be the most appropriate ; for example, it naturally presents
itself in the computation of tables of the numerical values of those
integrals.

152 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Hitherto we have considered the elliptic integral or its equiva-
lent, the arc of the hyperconic section cr, as a function of its
amplitude <p, or assumed as it were, the amplitude <p as the inde-
pendent variable. But we may reverse this course and consider
the amplitude as a function of the arc cr of the hyperconic section.
A notation has been devised by which the amplitude <p may be ex-
pressed as a function of the integral or its equivalent a. When
the modulus of the elliptic integral is 0, the integral becomes

Cx da?

I — ^= or sin"1^. Now this is a function very little used as

Jo Vl-*2

compared with sin a? ; so that sin x is always considered the direct
function, and sirred? or the arc the inverse function. The reason
of this is, as I have elsewhere shown, that our acquaintance with
circular functions is not derived from the integral calculus, while
our knowledge of the properties of the arcs of hyperconic sections
can in no other way be obtained. It will render our language
more precise, if we apply the term elliptic integral to those ex-
pressions in which the amplitude is the independent variable, and
elliptic functions to these expressions in which the arc is the inde-
pendent variable.

In this way, writing sin p = sin amp. a, we might develop a
great system of trigonometry for the hyperconic sections. In this
general system when the modulus i = 0, we pass into circular
trigonometry, and when the modulus z = l, we may develop an
equally extensive system of parabolic trigonometry as given in
the first volume of this work, p. 313. In truth that essay ought
to have been incorporated in this treatise, in which passing over
elliptic functions, we confine our researches to the geometrical
properties of elliptic integrals. To enter on the wide field of
elliptic functions, or as it may be called the trigonometry of the
hyperconic sections, would lead us very far beyond the limits we
have prescribed to ourselves; and it has, moreover, been amply
treated by Legendre, Jacobi, Abel, and other great continental
mathematicians.

There are several plane curves whose lengths we may express
by elliptic integrals of the third order. For example, the length
of the elliptic lemniscate, or the locus of the intersections of
central perpendiculars on tangents to an ellipse, is equal to that
of a spherical ellipse which is supplemental to itself, or the sum of
whose principal arcs is equal to IT, as shown in vol. i. p. 196. We
cannot represent elliptic integrals of the third order generally by
the arcs of curves whose equations in their simplest forms contain
only two constants. Thus let a and b be the constants. We shall
have two equations between the constants, the parameter, and the
modulus of the function, i = i (a, b}, n = ¥,(a) b). Assume a as inva-

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 153

riablc, and eliminate b, we shall have one resulting equation between
i, n, and a, or F(«, i, ri)=Q; or n depends on i.

When there are three independent constants, as in the preceding
investigations, a, b, and k, we shall have z = f(«, b, k), n=i'(a, b, k) .
Eliminating successively b and k, we shall have two resulting equa-
tions, instead of one, F(a, k, i, ri)=Q, and F'(a, b, i, n) = 0; or i
and n depend on two equations, and may therefore be independent.

The general fundamental expressions for the rectification of curve
lines, whether of single or double flexion, show that the arc of a
curve may in general be represented as the sum of two quantities,
an integrated and a non-integrated part ; or, as the proposition may
be more briefly put, an arc of a curve may be expressed as the sum
of an integral and a residual. Thus the arc of a plane ellipse is
equal to an integral and a residual, which latter is a straight line.
An arc of a parabola is the sum of an integral and a residual, which
latter is also a straight line. An arc of a spherical ellipse is the
sum of an integral and a residual, the latter being an arc of a
circle, while an arc of a logarithmic ellipse is made up of two por-
tions, one a sum of integrals, the other (the residual) being an
arc of a common parabola. It appears therefore to be an expendi-
ture of skill in a wrong direction to devise curves whose arcs should
differ from the corresponding arcs of hyperconic sections by the
above-named residuals. Thus geometers have sought to discover
plane curves whose arcs should be represented by elliptic integrals
of the first order, without any residual quantity — the common
lemniscate for example, when the modulus has a particular value.
It is possible that such may be found. In the same way, an expo-
nential curve may be devised whose arc shall be represented by the

f* r\fl

integral k \ —*, instead of taking it with the residual quantity

k tan 6 sec B as the expression for an arc of a common parabola.
Thus geometers have been led to look for the types of elliptic inte-
grals among the higher orders of plane curves, overlooking the
analogy which points to the intersection of surfaces of the second
order as the natural geometrical types of those integrals.

It has thus been shown that the curves of intersection of con-
centric surfaces of the second order may in all cases be rectified by
elliptic integrals. When the intersecting surfaces are not con-
centric, the rectification of the curve of intersection may be reduced
to the integration of an expression which may be called an hyper-
elliptic integral.

The general expression for the length of an arc of this curve
will be an integral of the form

s =

ax4 -f bx3 + ex3 -f ex +f

VOL. II.

154 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

When the surfaces are symmetrically placed and have a common
plane of contact, the above expression may be reduced to

s =

aa? + bx2 + ex + e

This form may be reduced to an elliptic integral.

When, moreover, the surfaces are concentric and symmetrically
placed, the preceding expression may still further be simplified to

r = I dx A /

0.x2- + fix + 7
ax2 -f bos + c

which is the general form for elliptic integrals.

We can perceive therefore that the solution of the general
problem, to determine the length of the curve in which two sur-
faces of the second order may intersect, investigated under its most
general form, far transcends the present powers of analysis. It is
only when one of the surfaces becomes a plane, or when they are
concentric and symmetrically placed, that the problem under these
restricted conditions admits of a complete solution.

We may hence also surmise how vast are the discoveries which
still remain to be explored in the wide regions of the integral cal-
culus. We see how questions which arise from the investigation
of problems, based on the most elementary geometrical forms
(surfaces of the second order) baffle the utmost powers of a refined
analysis, with all the aids of modern improvements. It is not a
little curious, that nearly all the branches of modern analysis, such
as plane and spherical trigonometry, the doctrine of logarithms
and exponentials, with the theory of elliptic integrals, may all be
derived from the investigation of one geometrical problem — to
determine the length of an arc of the intersecting curve of two
surfaces of the second order.

In the logarithmic hyperconic sections, we may develop pro-
perties analogous to those found in the spherical and plane
sections, if we substitute parabolic arcs for arcs of great circles in
the one, and for straight lines in the other. Here follow a few of
those theorems.

1. From any point on a parabolic section of the paraboloid let
two parabolas be drawn touching the logarithmic ellipse or the
logarithmic hyperbola, the parabolic arcs joining the points of
contact will all pass through one point on the surface of the para-
boloid.

2. If a hexagon, whose sides are parabolic arcs, be inscribed in
a logarithmic ellipse or logarithmic hyperbola, the opposite para-
bolic arcs will meet two by two on a parabola.

3. If a hexagon, whose sides are parabolas, be circumscribed to

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 155

a logarithmic ellipse, the parabolic arcs joining the opposite ver-
tices will pass through a fixed point on the surface of the para-
boloid.

4. If through the centre of a logarithmic ellipse or logarithmic
hyperbola two parabolic arcs are drawn at right angles to each
other, meeting the curve in two points, and parabolic arcs be
drawn touching the curve in these points, they will meet on
another logarithmic ellipse or logarithmic hyperbola.

5. If a circle whose radius is a be described on the surface of
the paraboloid, and therefore touching the logarithmic ellipse or
the logarithmic hyperbola at the extremities of its major axis, and
from the extremities of any diameter two parabolic arcs be drawn
to any third point on the circle, if one of these parabolic arcs
touches the logarithmic ellipse or the logarithmic hyperbola, the
other will pass through a fixed point on the surface of the para-
boloid.

6. If on the paraboloid we describe a circle whose radius is
V«2±^ and if from the extremities of any diameter of this circle

we draw parabolic arcs touching the logarithmic ellipse or the
logarithmic hyperbola, these tangent parabolic arcs will meet on
the circle.

These theorems will suffice. There would be little difficulty in
extending the list. In fact nearly all the projective properties of
straight lines and conic sections on a plane may be transformed
into analogous properties of great circles and spherical conic sec-
tions on the surface of a sphere, and of parabolic arcs and loga-
rithmic sections on the surface of a paraboloid.

CHAPTER XI.

ON THE QUADRATURE OF THE LOGARITHMIC ELLIPSE AND OF THE
LOGARITHMIC HYPERBOLA.

84.] The properties of the Logarithmic Ellipse and the Loga-
rithmic Hyperbola have the same analogy to the paraboloid of
revolution that spherical conies have to a sphere, or which conic
sections bear to a plane. To determine the areas of these curves,
or rather the portions of surface of the paraboloid bounded by
them, is a problem not undeserving of investigation.

The Logarithmic ellipse has been defined in Chapter IV. as the
curve of intersection of a paraboloid of revolution with an elliptic
cylinder whose axis coincides with that of the paraboloid.

The Logarithmic hyperbola, in like manner, has been defined in
Chapter V. as the curve of intersection of a paraboloid of revolution

156 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

with a cylinder whose base is an hyperbola, and whose axis coincides
with that of the paraboloid.

Through the vertex Z of the paraboloid let two parabolas be
drawn indefinitely near to each other, ZP, ZQ, and let two planes

indefinitely near to each other at right angles to the axis OZ cut
the parabolas in the points u, u1, v} v1.

The little trapezoid uvu'v' is the element of the surface ; and if
the normal un makes the angle /JL with the axis OZ, d-fy being the
elementary angle between the planes, uu' = ktan.fjidty, k being the
semiparameter of the generating parabola.

Hence the elementary trapezoid

Now uv = ds=

, .
uvu'v' =

cosc

cos4 p.

f f sin/* j
Integrating this expression, area = k2 1 d-^r I — ^— dp ; . (436)

or performing the integration with respect to fi}

K* C
area =-^- 1 dty sec3 p + constant.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 157

Now when the area is 0, sec/u=l, and therefore

k*C
= — — I oS/r.

constant = — — I oSr. Whence

(437)

This is the general expression for the surface of a paraboloid
between two principal planes, and bounded by a curve.

When this curve is the logarithmic ellipse, let the area be put
[LE].

We must now express -fy and /* as functions of another variable, 6.

Let x=aco&0, y = bsin0, the base of the cylinder being the

at* v2
ellipse whose equation is — + j^ = ^> ^ is the angle which

CL 0

V#2+y2 makes with the axis a.

•vr V b . ,.

JNow tan -ur = -:=:- tan 0. (438)

x a

and d^=-g 27. ,2 . ,a (439)

*t- Asxaal H I /, - o i > * £ H

But

therefore secV=VfV "^ >™° v£ v<v ~r" ;ai" v. . (440)

Hence substituting these -values in (437), we get for the area

I -LJ-LJ I -^— ~^" *"»"Q 1 ' -. . • • ^ •* I

v . (441)
d0

Let

t being the modulus and e2 the parameter, as in (15).

158 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

The above expression may be written

3d

a2 cos2 0 + 62 sin 0) V (#2 + «2) - («2 - #*) sin2 0.

.]'

2ab

- «2 - *2 sin2 0

-. (443)

d(

1 +

Therefore, integrating the preceding expression,
r 1= A0

3 C

T^J [1 -e2 si

sin2 0} Vl-i2sin20

— i2 sin2 0

ab

. (444)

Hence the area of the logarithmic ellipse, or rather the area of
the paraboloid bounded by the logarithmic ellipse, may be ex-
pressed as a sum of elliptic integrals of the first, second, and third
orders, with a circular arc.

Since 5— > 2 /2, e2 >i2, or the function of the third order is

a* a2 + /fc2'
of the circular form. Assume a spherical conic section such that

a f> nZ—Kt

tana = 7,

therefore — -cos«=
tan a

bk

a

sin2 e=

Combining the first and last terms of the preceding equation, they
become

-* ftan-t(*
\a

tan 0-

cos a I r

J [1 — e2sii

sn em

in* 01'

Now this is the expression for the surface of a segment of a sphe-
rical ellipse whose principal angles are 2« and 2B, as shown in
sec. [8]. Let this be S.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 159

In the next place, k

is a portion of the elliptic cylinder whose altitude is k, and the
semiaxes of whose base are V^2 4- ^2 and v^2 + k*2. Let this be C,

abk
and

abk C d0

^+1V Vl-^si

is an expression for an arc of the spherical parabola whose focal
distance is one half the focal distance of the former. Let this be
denoted by P.

Hence, if we denote the entire surface round Z by [LE] ,

3[LE]=4AC + 7 gP-4*2S. . . (445)

Or the area of the logarithmic ellipse may be expressed as a sum
of the arcs of a plane ellipse, of a spherical ellipse, and of a sphe-
rical parabola, multiplied by constant linear coefficients.

85. J To find the area of the logarithmic hyperbola.

The general expression for the area, as in (437), is — J (sec3 p— 1) di|r.
Now, the equation of the base of the hyperbolic cylinder being

f*~ ti

*** */ 1 1 i /\ Z.J./1 / A A f*\

g —— I |f*T y ^— ft sf*f* f/ 7/ ~~~ f) L JlTl \j (4<4iO }

a2 A2 '

V b . „

then tan y = ^- = - sin V.

x a

and aT = i cos#d#, cos2

hence

T - , = .

cos2 t/r a a2 + 62 sin2 0

ab cos 0d0

. . ,.
a2 + Zr sm2 0

2 • r2
Since tan-4 /i = =

secs/i=

cosz^

jin2^]i

Let [LY] denote the area of the logarithmic hyperbola, then

-y f

J

160 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Let V be put for *« cos2 0 + a2 + b* sin2 0} .... (448)

and the last equation will become

-

f

+J

VV VV

cos2 e

and this may be written in the form

Let

, — -— g
A:2 + a2

and the preceding equation may be written

3[LY]=-P

dfl

cos20 yl-
a (/r— flz)2r Ad

•Ftan-'f-

Since

. (449)

and as (1 — m) (l + ri)=I — i2, m=~ — — , and (47) gives

CL -J- K

/l + ^\T d^ /I— m\C &6
\ n /JN Vl~^ m /JM VI

- JL r ^ 1 _, / \lrnn sin ^ cos ^\
mnj Vl Vm» V VI '

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 161

hence

l + w\ ,_f d0 fl-m\ , C M

But
Hence

/

V

3[LY]

i2 C A0

I — = -4- tan~

'mnj VI L

_i f Vww sin $ cos 0

. . (451)

T=

VI

' mn=-

+ •

VI

-[-

cos

VI

. (452)

Now, if Y be an arc of the plane hyperbola of which ^k^—b* is the
transverse axis, and i the reciprocal of the eccentricity, we shall
have

ababaz + b* C A0

V«2+>fc2Jcos26' VI'

(453)

And if we take the spherical ellipse whose principal semiangles,
a and /3, are given by the equations

b b

we shall have sin2e=

tana

cos«=

nh

-., also i/r= tan"1/ -sin 0|.

Hence the sum of the first and last terms may be written

r, tan/3 f d0 ~1

^ -- cos a I :

tan a J [l _e2sin2^] Vl -sin2esm2^J

and this expression is S, the value of the area of the spherical
ellipse («/3), as shown in (13) .

Now, let $| be the transverse axis of the auxiliary hyperbola;

VOL. II.

162 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Hence the coefficient of I — — •- may be written -p ~**j> and the equa-

J V 1
tion (452) finally assumes the form

3*[LY] =oft [Y + ^j f~] -A»S +•*» tan" [jV/^sin£cos^J <

Or the area of the logarithmic hyperbola may be expressed as a
sum of the arcs of a common hyperbola, of a spherical ellipse, of a
spherical parabola, and of a circular arc, multiplied by constant
coefficients.

There is one particular case in which the area of the logarithmic
hyperbola may be represented by a very simple expression. Let
k = b ; then, if we turn to (448), V=a2-f W-, and 1 = 1, since i = 0.
Hence (452) may be changed into

3 [LY] = a *JtfTb* tan 0 + & tan-1 ( .-^—- tan 0\

+ 62tan-1( - 4= =a sin 0 cos 0} -b* tan-1 (-sin 0] ;
\aV«2 + ^2 \a '

and this expression may be reduced to

8[LY]=a ' " ' ' ~" ' "'}

Y, (455)

j

a value entirely independent of elliptic integrals, and which may
be represented by a straight line and the difference of two circular
arcs.

CHAPTER XII.

ON THE RECTIFICATION OF LEMNISCATES.

86.] There is a particular class of plane curves, of which the
lemniscate of Bernoulli is an example, to which the principles
established in the foregoing pages may be applied with much
elegance.

Definition. — This entire class of curves may be defined by the
following property. The square of the rectangle under the radii
vectores drawn from the foci to any point on the curve is equal to
a constant, plus or minus the square of the semidiameter passing
through this point multiplied by a constant quantity.

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 163

Let Q, Q! be the foci, and O the centre, p, p, r the lines drawn

Fig. 28.

from these points to any point on the curve. Let OQ,=OQ' = c,
and let /be a variable constant.
Then by the definition

p*p,*=C*±f*iA ......... (a)

But p2p/2 = (*? + y2) 2 + c4 + 2cV - 2c2#2,

and r2=a?2 + y2;

hence (a?2 + y2)2 =(/* + &; Vs + (/2-2c2)y2. . . (456)

This is the general equation of the curve, which assumes different
forms as we assign varying values to / and c. Some examples
may be given.

(«) Let c = 0, or/=oo , the equation is that of a circle.
(/3) Let/2>2c2, and make/2 + 2c2=a2, /2-2c2=62,
the equation will become (x2 -f y2)2 = a2#2 + b'zy2.

This is the equation o£ a curve which may be called the elliptic
lemniscate. It is the locus, as is well known, of the intersection
of central perpendiculars with tangents to an ellipse ; and its recti-
fication has been fully investigated in vol. i. p. 196.

(y) Let /2 = 2c2. The equation becomes (#2+y2)2=4cV, or
the equation is that of two equal circles in external contact.

(8) Let/2<2c2. The equation becomes

(a* + y*)*=(2(?+f*)a?-(2<*-f*)y*, and

(e) Let/2=0. The equation becomes (#2+y2)2=
or the equation is that of the lemniscate of Bernoulli.

(f) Let f2, passing through 0, be taken with a negative sign.
The equation in this case becomes

and

In one case only does the equation of the lemniscate in its general

form coincide with that of Cassini's ellipse, namely when/=0

and h=c, A2 being the product of the radii vectores from the foci.

The definition of Cassini's ellipse being " a curve such that the

164 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

product of the radii vectores drawn from two fixed points, the foci,
to a third point on the curve, shall be constant and equal to ti2"
its equation will obviously be, 2c being the distance between the
foci,

A4-c4=(#2 + y2)2-2c2(#2-#2); .... (b)

when h=c, (#2 + 2T2)2=2c2(#2-y2) ...... (c)

This is the equation of the lemniscate of Bernoulli.

These elliptic lemniscates may also be denned as the orthogonal
projections of the curves of symmetrical intersection of a paraboloid
of revolution with cones of the second degree, having their centres
at the vertex of the paraboloid. Let a and /3 be the principal
semiangles of one of the cones. Its equation is

(d)
es

(e)

Let the equation of the paraboloid be #2 + y2 -f 2kz.

Eliminating z, the equation of the projection of the curve of inter-

section will become

(#2 + y2)2=«2#2 + 6y ...... (457)

When the section is an ellipse, the equation of this curve is, as

2k 2k

Make tana= — , tan/3=y, and the equation of the cone becomes

On the Hyperbolic Lemniscate.
87.] The equation of the lemniscate in this case is

Following the steps indicated in sec. [86] , we shall find

dX2 «2cos2X-62sin2X'

(a)

the limits of X are 0 and tan"1 T.

b

Assume 8in2\=_- _^? - . (b)

a262 -f a4 sin2 <p -f b4 cos2 <p

The limits of <p, corresponding to X=0 and X=tan-1 -?, are

<p = 0, and <p=- ......... (c)

ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 165

Substituting this value of sin2 \ in the preceding equation, we shall
find

dX cos<p
From (b) we may derive

dX a*b

,.

'

Multiplying the two latter equations together and reducing, we get
a3 dip

When a = 6, or when the lemniscate is that of Bernoulli, there
results the well-known expression

When a > b the integral is of the third order and circular form ;
but when a<b the integral is of the third order and logarithmic
form. That it is of the logarithmic form may thus be shown.

Let -_JL?Mj and *2=-H— r»-

a4
Hence i2~m==^(a2 + 62)> ....... (460)

or i2 is greater than m ; but we know that the form is logarithmic
when the square of the modulus is greater than the parameter,
when it is affected with a negative sign.

This is a remarkable result. All analysts know the impossibility
of transforming the circular form into the logarithmic, or vice versa,
by any other than an imaginary transformation ; the utmost efforts
of the most accomplished analysts have been exhausted in the
attempt; yet in this particular case their geometrical connexion
is very close. The modulus and the parameter are connected by
the equation

2j ....... (461)

the upper sign to be taken in the circular form, the lower in the
logarithmic.

There are two distinct cases to be considered — when a is greater
than b, and when a is less than b.

166 ON THE GEOMETEICAL PROPERTIES OF ELLIPTIC INTEGRALS.

Case I. a>b.

Let a plane ellipse be constructed whose principal semiaxes A
and B are given by the equations

!, B2=a2, ...... (f)

and let a sphere be described from the centre of this ellipse with a
radius

B2 = ==K.

Then we can find, as follows, the length of an arc of the spherical
ellipse, the intersection of the sphere whose radius is R with the
cylinder standing on the ellipse whose semiaxes are A and B.

Since

and

therefore

A2

s/ — _
sm _=5_

R2~ a2

-, cos* a = -41

cosx:p = -3j
a2

R cos /3 _
cos a sin a 6 (a9— i

We have also

R cos /3 cos «_ _ atP
sin a ~(a2 — b*) V^

2 _2-22-
tan e —

cos a

sin2 a- sin2 /3

sm a

9 . 70'

a2 + 62

(g)

Substituting these values in (46) the expression for an arc of a
spherical ellipse with a positive parameter, and writing s for the
arc, we get

a8

C

/

J V

dtp

[ . (462)

62

ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 167

Comparing this with (459) , we find

or the difference between an arc of a hyperbolic lemniscate and an
arc of a spherical ellipse may be expressed by an integral of the
first order, together with a circular arc. When a=b, the radius
of the sphere is infinite, the sphere becomes a plane, so that it is
not possible to express an arc of a spherical ellipse by the common
lemniscate.

Case II. Let b>a.

In this case the arc of the hyperbolic lemniscate may be ex-
pressed by an arc of a logarithmic ellipse of a particular species, or
one whose parameter and modulus are connected by the relation
given in (461).

Resuming the expression in (459) for the arc of the hyperbolic
lemniscate,

:2sin2<p

^ S ~ w V T M

22 62 ,

__ =<m. . — «* • 1

b*

(i)

m + n— mn=i2, n=

Let & and & be the semiaxes of the base of the elliptic cylinder,
k the parameter of the paraboloid whose intersection with the
cylinder gives the logarithmic ellipse. Assume for the principal
semi- major axis of the elliptic base

m, n,

In (171) we found the following relations between &,iS, k

nz=mn(l—n) W_mn(l—m}
A2~~ (n-m)2' A2"' (n-m)^'

and as we assume & = V«2 + 62, we get, substituting for m and n
their values in terms of a and b, the semiaxes of the hyperbola

98=—, and &=— - — , ... (k)

« a2 \/62— a2

In (163) we found for the equation of the logarithmic ellipse

168 ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS.

measured from the minor axis, and multiplied by the undetermined
factor Q,

or.v (l-m\ /— 7nf _ d?
20,2,= — I - 1 ymnkQ,]-;-, — — . 2 -, /, .* .

\ m / J [l-wsm2<p] Vl.-«2s

\. . (464)

__ Q-

I£ in this equation we substitute for m, n, and k their values as given

(^ _ ifyi\ __
- } \imnkQ, with the

a3
coefficient — of the expression for the lemniscate in (459),

we shall find

Q=

hence the last equation, substituting this value of Q;, will become

li(465)

abb C d<p fflrA2 — a?\ 2 V«2 + b* ~

— ! ,.. <P

Jd(p ^fl
Vi ~

or the swm of an arc of a hyperbolic lemniscate and of an arc of a
logarithmic ellipse may be expressed as a sum of integrals of the
first and second orders with a circular arc.
When 6= a, the above expression will become

dtp

In this case the parameter of the paraboloid becomes infinite,
and therefore the paraboloid a plane, just as the sphere became a
plane in the last case ; so that we cannot express integrals of the
third order, whether circular or logarithmic, by an arc of a common
lemniscate.

Although the lemniscates may be rectified by elliptic integrals of
the third order, as well circular as logarithmic, yet these curves
cannot be accepted as general representatives of integrals of the
third order, because, in the functions which represent those curves,
the parameters and the moduli are connected by an invariable
relation, as in (461). Thus the elliptic lemniscate, whatever be
the ratio of the axes of the generating plane ellipse, can be repre-
sented only by a particular species of spherical ellipse, that whose
principal arcs are supplemental.

THE THEORY

AND THE

PROPERTIES OF SURFACES OF THE SECOND ORDER,

APPLIED TO THE INVESTIGATION 01'

THE MOTION OF A RIGID BODY

ROUND A FIXED POINT.

" Quant am sciences des phenomenes nnturels, nous nc doutons point que Jes surfaces
du second degr6 ne doivent s'y presenter aussi dans un grand noiubre de questions, et y
joiior un role uussi important que celui des sections coniques dans le systeme planetaire."
— CHASLES, Apcrfu Historiqiic, p. 251.

VOL. II.

170 ON THE MOTION OF A RIGID BODY BOUND A FIXED POINT.

CHAPTER XIII.

88.] We shall now proceed to apply the principles developed in
the foregoing pages to the investigation of a physical problem of
much celebrity and great interest in Astronomy — :the motion of
rotation of a rigid body round a fixed point. The discovery of the
geometrical properties of elliptic integrals may be applied with
singular felicity to the illustration of the complicated motions of
the several axes of this body, the spirals, curves, and cones described
by them during its rotation round the fixed point. Let this point
be taken as the origin of three rectangular coordinates, their direc-
tion being arbitrary as well with respect to the body as to absolute
space. Let us, moreover, make the supposition that the body is
not subject to the action of accelerating forces, but in a state of
motion originated by a single impulse, or by any number of single
impulses, which may be combined into one. This may be consi-
dered as the normal state of the rotation of a body ; because if it
should besides be subjected to accelerating forces, such new forces
will introduce variations into the arbitrary constants of the problem.
It has, moreover, the advantage of admitting a complete solution ;
we are not compelled to have recourse to approximations. It
will be shown that the curves which the final integrals represent
are spherical conic sections — curves which may as easily be deter-
mined, from the principles laid down in the preceding chapters, by
means of the constants which enter into the integrals, and the am-
plitudes of those functions, as the arc of a circle may be ascertained
when we know its radius and the angle which the arc subtends at
the centre. Hitherto there has not been any attempt made, at
least so far as the author is aware, to carry the solution further
than to show that as the final integrals involve the square roots of
quadrinomial expressions with respect to the independent variable,
they might be reduced to the usual forms of elliptic functions.
Bui these integral* have not been interpreted so as to give a
graphic representation of the motion, by means of the properties
of those functions.

Assuming the usual definition of the moment of inertia of a body
with respect to a certain straight line (that it is the sum of all the
constituent elements of the body, each multiplied into the square
of its distance from this axis) , we shall briefly give the usual method
of finding it.

Let the given axis make the angles X, /JL, v, with the axes of
coordinates, R being the distance of one of the elements dm from
the origin, and 0 the angle which this line makes with the axis.
The distance, therefore, of the particle dm from the axis is R sin 6 ;
and the moment of inertia round this axis is the sum or integral of

OX THE MOTION OP A RIGID UODY ROUND A FIXED POINT. 171

all the terms, such as R2 sin2 0dm, which the body affords. Writing
H for the moment of inertia round this axis,

H=Jdm[Rsin0]2, ..... (466)

the integral being extended to the whole mass of the body. H is
therefore a quantity of five dimensions.

To transform this integral into another, which shall contain the
rectangular coordinates xyz of the particle dm. We have

Ecos 6=x cosX+y cos /* +z cos v;

deriving the value of sin d from this expression, and substituting it
in (466), we get

— 2 cos//, cos v §dmyz — 2 cosX cos v §dmxz— 2 cosX cos/it Jdm^ry j

Now these six integrals depend solely on the assumed position of
the coordinate planes with respect to the body, and not on the posi-
tion of the axis of moments, which is determined by the angles
X, p, v. These integrals, referred to the same system of coordi-
nates, will therefore be the same for every assumed axis. Let them
be computed and designated as follows —

=N,|

. j

The value of H may now be written,

H=L cos2X-f M cos2//, + NCOS* i/—2U cos /x cos v) (A,ra\
— 2V cos X cos v — 2W cos X cos p, j

We may reduce this expression to represent a straight line
drawn from the origin to some curved surface, by the following
transformations :

=nA, M=nA;, N=nA//,|
=nB, V=nB,, W=nB//.j

let H=nP2, L=nA, M=nA;, N=nA/
U

Substitute these values, and divide by the cubical constant n,
equation (469) becomes

A cos2X+ A; cos2/* + A/; cos'v— 2B CQSJJ, cos v| ..„ .
— 2By cos X cos v — 2BW cos X cos p = P2 j

Now this, as may easily be shown, is the expression for the
length of a perpendicular let fall from the centre of a surface of
the second order on a tangent plane to this surface. As the coeffi-

172 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

cients L, M, N are necessarily finite and positive, the coefficients
of the surface A, A,, A/;, which have a given ratio to the former,
must also be finite and positive. The surface is therefore an ellip-
soid. That the above expression represents such a perpendicular
may be shown as follows.

89.] The tangential equation of a surface of the second order
(see vol. i. p. 66), the origin being at the centre, is

w + 2B/fC+2B//fi;=l. . (472)

In this equation £, v, % denote the reciprocals of the portions of
the axes of coordinates between the origin and the variable tangent
plane, supposed to envelop the surface in every successive possible
position. The squared reciprocal of the perpendicular from the
centre on the tangent plane is £? + y2 + £2. If X, ft, v denote the
angles which this perpendicular Py makes with the axes of coordi-
nates, cosX=Py£, cos/-i=PyU, cosv=P/£. Substituting these values
of f, v, % in the preceding equation, and multiplying by P,2, we find

A cos2 X + A; cos2 jj, + A/; cos2 v + 2B cos //, cos j>)
+ 2B; cos \ cos v + 2B/; cos \ cos ^ = Py2 j '

an equation which coincides with (471) ; hence Py=P.

If we divide (469) by P2, and introduce the quantities £, v, £ by
the help of the equations cos X= P£, cos fj, = Py, cos v = P£, H = nP2,
we shall find

L^+Mva + N?-2U£i;-2VfS-2W£i;=n. . (474)

It is shown in the first volume of this work, p. 63, that, if x, y, z
denote the projective coordinates of the point of contact of the
tangent plane to the surface,

(475)

Let xlylzl denote the coordinates of the foot of the perpendicular
P on the tangent plane; then as Pcos\=a?;, andP£=cos\,#y=P2|;;
in like manner, y;=P2y, zt—^^ : whence

U£. . . (476)
-*) = (N -nP2)£-Uu - V| )

Now, writing T for the distance measured along the tangent
plane between the foot of the perpendicular upon it from the
centre, and the point of contact of this tangent plane, xt—x, y( — y,
z,—z are the projections of T upon the three coordinate axes. It

ON THE MOTION OP A RIGID BODY ROUND A FIXED POINT. 173

is also evident that (xyf^, (xyz], and (0,0,0) are the projective
coordinates of the three angles of the right-angled triangle whose
vertex is at the centre and whose base is T.

It may easily be shown, and we may therefore assume, that the
orthogonal projections of the area of this triangle upon the coor-
dinate planes of xy, yz, and xz are

\y(x-x)-x(y-y}'\, 0(y,-y)-y (*,-*)],) >

and \x(zt— z}— z(x,— x)~\ )
respectively.

If we substitute in these expressions the values of the projective
coordinates, which may be deduced from (476) , writing A for the
area of this triangle, and A/,, Am;, Aw, for its projections on the
coordinate planes of yx, xz, and xy, (I,, m,, n, being the direction
cosines which a normal to the plane of A makes with the axes of
x, y, z respectively) , we shall have

. (478)
=P2[(L -M)£u-(Vi;-U£)(;-W(t;2-f)]

We shall discover the dynamical illustrations of these expressions
further on.

90.] To determine the axes of figure of the ellipsoid.

It is manifest, whenever the distance T between the foot of the
perpendicular from the centre on the tangent plane, and the point
of contact of this tangent plane with the surface, vanishes, that the
radius through the point of contact becomes also a perpendicular
to the tangent plane, and therefore one of the axes of the surface.
When T=0, its projections on the coordinate axes vanish, or
xl—x=Q, yl—y=6, 2,—z=Q; (476) then becomes, putting n, as
we evidently may do, equal to 1,

(L _pa)f_V$-Wi/=0)

(479)

From these equations eliminating the quantities £, v, £, we get the
following cubic equation in P2,

(L-PKM-P*)(N-P)_U2(L-P2)-V2(M-P*H
-W2(N-P2)-2UVW=OJ

The roots of this equation are the three semiaxes squared of the
ellipsoid.

We need not here stop to show that the three roots of this cubic
equation are real, as the proposition has already been established
in various ways, see vol. i. sec. [84]. The following is a group of

174 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

symmetrical formulae for determining the position of any one of
these axes in space when its magnitude is determined.

Let P,2 be one of the roots of the cubic equation, or the square
of one of the semiaxes ; let L - P 2 = Q, M - P,2 = Qt,, N - P 2 = Qy/ ;
also let \, p, v be the angles which this axis P/2 makes with the axes
of coordinates ; then cosA,= Py£, cosjj,= ¥lv, cos )/ = ?,£.

This equation may also be written

QQ/^-QUS-Q/V^-Q^-SUVWrrrO. . . (481)

Resuming equations (479), and introducing the given value P,2
of P2

(482)
V£=O.

Combining the first of these equations with the second, and
eliminating v,

£_ QQ,-W8

combining the second with the third, and again eliminating u,

multiplying the two latter,

— U*'

In like manner g^-g^!!.

|2 COS2\ QyyQ^— U2

whence, adding,

y— U2

(Qy/Qy-U2) + (QQW- V2) + (QyQ- W2) '

and like expressions for cos2//, and cos2)/ may be found. See vol. i.
p. 73.

We may express these formulae in a more compact notation as
follows :

If we take the first derivative of (480), we shall find it to consist
of three members. Substituting for P2 one of its values, P,2 suppose,
the resulting expression may be written

T -f <0 + O, and the last formula becomes

also QOS^==- cos2v= (484)

u\ THK MOTION OF A RIGID BODY ROUND A FIXED POINT. 175

91.] In every revolving body there exists an instantaneous axis
of rotation, or a line of particles which remain at rest during an
instant. Let C be the position of a point in the revolving body at
any given time, C' the position of the point during the next instant.
Let the arc CC' be ds. At the extremities of this arc d* let normal
planes be drawn to the curve. If these planes are parallel, the
motion is one of rotation round an axis infinitely distant, or the
motion is one of translation. If the planes are not parallel, let
them meet ; the straight line in which they intersect is the axis of
rotation during the indefinitely small time in which the arc CC'
or ds has been described.

This line, the intersection of the normal planes, must pass through
the fixed point, if there be one in the body ; otherwise there would
exist in the body a fixed point and a fixed straight line not passing
through the point, which would retain the body in a state of rest,
contrary to the supposition.

Again, there cannot be, during the same instant, two or more
axes of rotation in the body ; for two fixed lines are equivalent to
three fixed points, which would retain the body in a state of rest.

The same considerations will show that the instantaneous axis of
rotation could not possibly be a curve.

The angular velocity of a body is defined to be the arc of a circle
whose radius is 1, described in the element of the time, and whose
centre is on the axis of rotation.

92.] To determine equations of the instantaneous axis of rota-
tion.

The fixed point being taken as origin, let z'y'z' be the coordinates
of the point C, (acf + do?) , (y1 + dy') , (z1 + dz>) of the point C'. The
equation of the normal plane passing through C is

xdz'+ydyt+zdzl=x'dx'+y'dy' + 2'd2'=0, ... (a)
since the plane must pass through the origin ; hence as
0,'dx1 + y'dy1 + z'dz'= 0,

the point C must move on "the surface of a sphere. The equation
of the normal plane passing through C' is

a?dV + yay + 2rdV=0 ...... (b)

The equation of the osculating plane passing through the arc ds
being

we may determine the constants from the consideration that the
osculating plane is perpendicular to each of the normal planes.
The osculating plane is therefore perpendicular to the intersection
of these planes — that is, to the instantaneous axis of rotation.
Let \, p, v be the angles which this line makes with the axes of

176 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

coordinates, then - =77, — -=T\ ', and the equations of this
cos v C cos v C

straight line become

Az-Cx=0, B,r-Ay=0, Cy-Bz=0 ..... (d)

Let to be the angular velocity round the instantaneous axis of
rotation ;

d*
then «=T7-, R being the radius of curvature.

Make r = a>cosv, and as

cos v =

C

ds

Now R (as is shown in treatises on the geometry of three dimen-

ds3
sions*)isequalto •

whence r=^-%. In like manner, let p = (ocos\, q^cocosft;

A B

. (485)

Substituting in (d) these values of A, B, C, we get
pz—rx—Q, qx—py=0, ry—qz—0.

Fig. 29.

These are the equations of the instantaneous axis of rotation, as
we shall show presently from dynamical considerations.

93.] The angular velocity round the instantaneous axis being &>,
the angular velocity round any other axis which makes the angle
6 with the former is to cos 6.

Let OA be the instantaneous axis of rotation, OB an axis which
makes the angle 6 with the former.
Through O let a plane be drawn
perpendicular to OB. In this plane
assume any point C, with the centre
O and radius OC let a sphere be
described, and through C let a
plane be drawn perpendicular to
OA and meeting this line in Q.
The point C will move, in conse-
quence of this rotation, on the cir-
cumference of the circle the inter-
section of the sphere by this plane, and therefore on the surface of
the sphere itself. Hence the tangent CC' is perpendicular as well
* LKROY, Analyse appliquee a la Geometric des Trois Dimensions, p. 295.

ON THE MOTION OP A RIGID BODY ROUND A FIXED POINT. 177

to the line CO as to CQ. Let the angle CQC'=o>, the angle
COC/ = G/; thenCC'=CQ.a> = OC.o>', and CQ=OCcos0; hence

d =6*00*0 ........ (486)

Now, as the angular velocities of every other element of the body,
round the axes OA, OB, are w and &>' respectively during this
instant, it is plain that the angular velocity of every particle of
the body round these axes is connected by the relation

to' = to cos 0 ;

hence p, q, r in the last section are the angular velocities round
the axes of x, y, z.

94.] Let as before Ox, Oy, Oz be any three rectangular coordi-
nates passing through the fixed point O, and X, Y, Z the velocities of
the particle dm of the body resolved along these axes, x, y, z being
the coordinates of the particle dm. These velocities being trans-
lated to the origin are there equilibrated by the resistance of the
fixed point O; while they generate the moments (Y# — Xy)dm,
(Zy— Y>)dm, (X.z— Z,r)dm in the planes of xy, yz, sx respectively.

We may conventionally assume that the rotations from x to y,
from y to z, and from z to x, shall be taken as positive, and the
rotations in any of the opposite directions as negative. Let a be
the angular velocity round the instantaneous axis of rotation,
X, fi, v the angles this axis makes with the axes of coordinates,
p, q, r the components of the angular velocities atong the axes of
xyz, so that

j9 = wcos\, <7=6>cos/A, r=wcos)/. . . (487)

The velocity of the particle dm parallel to the plane of xy is
r V#* + y2; and this resolved along the axes of x and y is

-r V^T.= and r V*2T.=> or-yr and xr.

in accordance with the conventional agreement as to the signs of
rotation in the coordinate planes ; whence

the velocities parallel to the axes of x and y are — yr and xr,
„ „ „ of y and z are — zp and yp,

„ „ „ of z and x are — xq and zq,

whence X=zq— yr, ^i — xr—zp, Z = yp—xq;

and these velocities translated to the origin generate the moments

~\x—^y=(xr — zp}x—(zq—yr}y, in the plane of xy,}

Zy — Y.Z == (yp—xq}y — (xr— zp}z, in the plane of yzf r . (488)

X^r — 7tX = (zq—yr}z— (yp—xq}x} in the plane of xz. )

VOL. II. 2 A

178 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

"We may determine the position of that group of particles (if any)
in the body which at the given instant are at rest, by making X=0,
Y=0, Z=0. These conditions are satisfied by making xr—zp=Q,
zq—yr = Q, yp—xq = Q.

These, it is hardly necessary to observe, are the equations of a
straight line passing through the origin, equations which we have
already found in (485) from geometrical considerations.

95.] If we extend to the whole mass the velocities found for the
single particle dm in the preceding section, we must integrate the
expressions for these velocities. Introducing the notation adopted
in (468), we find, multiplying the last equation by dm and inte-
grating,

= L -Vr -

(489)
J(Y#-Xy)dm=Nr -Ug -Vp.

Now, as the impressed couple or the resultant of all the impressed
couples must, by the principle of D'Alembert, be equivalent to the
effective moments, if we make this impressed couple K, and /, m, n
the direction-cosines of its axis k,

(490)

When the principal axes are the axes of coordinates, U=0,
V=0, W=0, and we get the well-known equations

K/=Ljo, Km=Mq, Kw=Nr. . . . (491)

Hence the components of the angular velocity round the prin-
cipal axes are equal to the components of the impressed couple
at right angles to these axes, divided by the moments of inertia
about them, or

K/ Km Kw

96.] If we compare together the formulae given in (475) and
(490), we shall make the second members identical by assuming

P~f^> q=fv> r=f£> f being a linear quantity ; . (492)

f2

whence <o2=/2 (£2 + vz + £2) =^; or the angular velocity is inversely

proportional to the perpendicular on the tangent plane, which may
be called the instantaneous plane of rotation.

ON THE MOTION OK A RIGID BODY BOUND A FIXED POINT. 179

Resuming the equations (475) and (490), introducing also the
relations established in (492), we obtain

K/=L/>-Vr-W?=/(L£-V£-Wu)==/n.r, or
K/=/n.r; in like manner Kw=/ny, Kn=fnz, whence

K*=/*n*(a*+y* + z*)=f*D*k* ..... (493)

Now x, y, z are the coordinates of the point of contact of the
tangent plane ; whence we infer that k, the semidiameter drawn
from the centre to the point of contact of the instantaneous plane of
rotation, is constant during the motion.

From the relations of (492), it also follows that if through the
fixed point we draw any three rectangular axes in the body, the
angular velocities round these axes are always inversely propor-
tional to the segments of those axes cut off by the instantaneous
plane of rotation ; or, in other words, the symbols £, v, £, the tan-
gential coordinates of the instantaneous plane of rotation, will
denote the components of angular slowness round those axes.

97.] Resulting from the rotation of the body, there arises a new
class of forces, which in general tend to alter .

the position of the axes of rotation of the &'

body. They are known as the centrifugal
forces. When translated to the origin they
generate a couple, whose magnitude and
position we are now to determine.

Let OQ be the instantaneous axis of rota-

J} Q 7*

tion, — , -, — the cosines of the angles it

Q) 0) 0)

makes with the axes, x, y, z are the coor-
dinates of the particle dm. The centrifugal
force which acts on this particle dm is equal
to the square of the velocity divided by the radius — that is,

2Q — 2

&> um _,M2 Qm. an(j this force, as it acts in the direction of Qm,

may be resolved into the forces o>2(#— x,), o^(y — y^), ^(z—z,),
respectively parallel to the axes of x, y, and z. x{y^ zt are the
coordinates of the point Q. Now

we also have

w
^=^+^±^l^, or

(O O)2

; but a>2x=x(p'2 + q'* + r2), whence
o)2^-^) =q(qx-py] +r(rx— pz) =X', )
a>*(y-y,)=r(ry-gz)+p(Py-qz)=\',[. . . (494)
o>2(2 — 2-,) =p(pz — rat) +q(qz -ry}=7J. }

180 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

From these equations we obtain

Y'a? - X'y =pq (f - #2) + yx(p* - q*} + rz(py- qz] ,
or, extending this expression to the whole mass,

Writing analogous formulae for the other axes, making Glt,
Gm,, Gnt equal respectively to Jdm(Z'y — Y'^r), Jdm(X'j3r— Z'a?),
fdm(Y'# — X'y), and using the notation established in (468), we
get

Gl, = (M-N)?r+(V?-Wr)jB + Ute2-r2),j

->*),[ . (495)
'

When the principal axes coincide with the axes of coordinates,
U = 0, V = 0, W = 0, and the formulae become

(496)

When one of the axes of coordinates, that of z suppose, coincides
with the instantaneous axis of rotation, we havej9 = 0, 5=0, r = <o,
and (495) becomes

(496*)

J) Q

If we multiply the first of (495) by -, the second by — , the third

7*

by -, and add the results, the sum will be zero, or

0; . . - (497)

whence it follows that the plane of the centrifugal couple always
passes through the instantaneous axis of rotation.

Multiply together line byline the groups in (490) and (495), and
add the results ; the sum will be cipher, or

KGj7/y + wm,+nwy]:=0 ..... (498)

Whence we may infer that the planes of the impressed and centri-
fugal couples are always at right angles to each other.

98.] If we compare (478) with (495), we shall find the second
members identical, if we assume, as in (492),

P=tf, q=fr, r=ft; whence /2=P2o>2,
and therefore G=Anw2 ........ (499)

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 181

We may hence infer that the triangle whose sides are the semidia-
meter to the point of contact of the tangent plane, and the perpendi-
cular on this tangent plane from the centre, coincides in position with
the plane of the centrifugal couple. The centrifugal couple is also equal
to the centrifugal triangle multiplied by the mass and the square of
the angular velocity, as shown in the preceding formula.

The reader will not fail to have observed the ease and simplicity
with which the properties of the ellipsoid, treated generally, without
reference to the principal axes, by the method of tangential coor-
dinates, may be used to illustrate and establish the corresponding
states of a body in motion round a fixed point. The subsequent
investigations might in most cases have been discussed with the
same generality and facility; but as the principle of this new
analytical geometry, the method of tangential coordinates, as deve-
loped in the first volume of this work, is probably as yet but little
known, it may be more satisfactory to conduct these investigations
on principles universally admitted. To simplify the results, we
shall adopt a particular system of coordinates which will render
the formulae much more manageable. If we choose the principal
axes of the body as axes of coordinates, U = 0, V = 0, W = 0, and
our investigations will therefore be very much simplified.

Let a > b > c be the three semiaxes of the ellipsoid in the order
of magnitude, L, M, N the moments of inertia about the coinciding
principal axes of the body. We may assume, as in (470), the squares
of the semiaxes of the ellipsoid proportional to the moments of
inertia round these axes, so that

a2n=L, 62n=M, e2n=N, .... (500)

n being a constant depending on the mass and constitution of the
body.

This ellipsoid we shall call the ellipsoid of moments.

Introducing these transformations and simplifications, (469),
(490), and (495) become,

os2>/], ...... (501)

=n62?, Kw=nc2r, ....... (502)

J,=n(A8-c*)gr, Gf»,=n(c*-aa)/w, Gn,=n(a2-Z>4)^. (503)

In formula (501) it is evident that the part within the brackets is
the expression for the square of a perpendicular from the centre on
a tangent plane to the ellipsoid. Let this perpendicular be P, and
(501) will become

H=nP« ........ (504)

Hence it follows that the moment of inertia of any rigid body
round a given axis is the mass of the body multiplied into the square

182 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

of the coinciding perpendicular from the centre on a tangent plane
to the ellipsoid of moments.

Square the terms of (502), add them, and multiply by o>2, we
shall obtain the result

K2<u2 = ii2 [aY + *y + c4/-2] (p* + <?2 + r2) ; .

also, as <o cos \=p, <o cos ^ — q, ft) cos v=r,

H2ft)4 = ii2 [a V + 62?2 + c2r2] 2,

whence we shall obtain

G2 = K2ft)2-H2o)4, ..... (505)

an important formula, which gives the value of the centrifugal
couple in terms of the impressed couple, the moment of inertia, and
the angular velocity round the instantaneous axis of rotation.

99.] Assume the impressed couple K = nfk) k beiug the semi-
diameter of the ellipsoid perpendicular to the plane of K. The
product fk is of course constant ; it will be shown presently that
/ and k are each constant.

As the axes of coordinates are the principal axes,

Kl Km Kn „ //)f,,N

p= i? q=w r=ir See (491)-

Let x, y, z be the coordinates of the vertex of k, then
1=2 m=4, n = j, L = na2, M = n62, N=nc2,>|

K K K

f* fy fr f ' ' (506)

and K=n//t; whence P = ^, 1 = p> r~^}
Squaring these values and adding,

. . . (507)

The cosines of the angles which this perpendicular makes with

P# Py P^

the axes are —5-, -vf . — _-. while the cosines of the angles which the
er b* c2

instantaneous axis of rotation makes with the same axes are

p q r fa f . » Pa? . ., , o Py

-,-,-; but p=J-z and <w = ^, whence -=-o-; similarly - = T| .
to CD CD a2 P a) a2 J &) 62 '

r ¥z

-=—$', we may therefore infer that

The instantaneous axis of rotation coincides with the perpendicular
from the centre on the instantaneous tangent plane drawn through the
vertex of k the axis of the impressed couple. The angular velocity
round this axis is inversely proportional to this perpendicular.

o\ THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 183

100.] During the whole period of rotation, the semidiameter k of
the ellipsoid, perpendicular to the plane of the impressed couple, is
constant.

Through any point Q on the surface of an ellipsoid let a tangent
plane be drawn, and through

the centre a plane parallel to *l£- "•*••

it. Let a concentric sphere be
described through the point Q,,
intersecting the surface of the
ellipsoid in the curve of double
curvature Q*. To this curve,
let a tangent QT be drawn
at the point Q ; and through
this tangent let a diametral
plane be drawn intersecting in the straight line Ob the diametral
plane ROA parallel to the tangent plane through Q.

Hence it follows that QO, Ob are the semiaxes of the plane sec-
tion QO6 of the surface. Let OQ = &, Ob = u. Let fall from O a
perpendicular OP on the tangent plane QPr. This line will also
be perpendicular to the parallel diametral plane O£R, and therefore
to every line in this plane, and therefore to the line Ob. Now the
tangent line QT, as it is on the tangent plane to the ellipsoid, and
passes through the point Q, must be a tangent to the plane section of
the ellipsoid passing through it ; and as it is besides a tangent to a
curve drawn upon the surface of the sphere, it must be at right
angles to the radius of the sphere OQ ; hence OQr is a right angle,
and therefore OQ must be a semiaxis of the section OQr, because,
when a tangent to a conic section is perpendicular to the diameter
passing through the point of contact, this diameter must be an
axis of the section. Now, as the parallel planes QPr, OR6 are
cut by the plane QOr, Ob is parallel to QT and consequently
at right angles to OQ. Hence OQ, Ob are the semiaxes of the
section OQT.

Since Ob is perpendicular to OP as well as to OQ, it is perpen-
dicular to the plane of OPQ, which passes through OP, OQ — that
is, to the plane of the centrifugal couple ; whence we are led to
infer that the semiaxes k and u of the diametral section of the
ellipsoid, whose plane passes through the tangent to the curve of
double curvature in which the ellipsoid and sphere intersect, coin-
cide with the axes of the impressed and centrifugal couples K and
and G respectively.

Assume a point v on the line Ob, so that Ov may be to k as the
centrifugal couple G is to the impressed couple K. The diagonal
OT of this instantaneous rectangle will represent, as well in mag-
nitude as in direction, the axis of the resultant couple at the end
of the first instant. During this instant, accordingly, the vertex of

184 ON THE MOTION OP A RIGID BODY ROUND A FIXKD POINT.

the axis of the impressed couple will have travelled on the surface
of the ellipsoid, as also on the surface of the concentric sphere
whose radius is k. It follows therefore that, at the end of the first
instant, the vertex of the axis of the resultant couple will be found
on the curve of double curvature in which the ellipsoid and sphere
intersect. The same proof will hold for the second and for every
succeeding instant, whence k continues always invariable. Now
the impressed couple K was assumed in sec. [99] equal to n/A: ;
but as n and k are each constant, / must likewise be constant.

If, to fix our ideas, we take the plane of K horizontal, and k
therefore vertical, we may infer that the rotatory motion of the
body will be such that its representative ellipsoid will bring all its
semidiameters which are equal to k successively into a vertical
position, and therefore the surface of the representative ellipsoid
will always pass through a fixed point in space.

Hence the motion of rotation of a rigid body round a fixed
point may easily be conceived by the help of the ellipsoid of
moments.

Let us imagine the centre of this ellipsoid to be fixed, that its
surface always passes through a fixed point in space, and that tangent
planes are always drawn to the ellipsoid through this fixed point . The
perpendiculars from the centre on these successive tangent planes
will represent in magnitude and position the instantaneous axis of
rotation.

101.] It was shown in (507) that the angular velocity a> was

f
equal to ^ ; and as / is constant, the angular velocity round the

instantaneous axis of rotation varies inversely as P (the perpendi-
cular let fall from the centre on the instantaneous plane of rotation) .

Hence it follows that the square of the angular velocity round the
instantaneous axis of rotation is always proportional to the area of
the diametral section of the ellipsoid perpendicular to this axis.

The angular velocity K round the axis of the impressed couple is
constant during the motion.

p

Let 6 be the angle between k and P. Then cos^=^; now

f /P f

K = ca cos 9, as shown in (486), and w = ^, whence /c=;^-=4 ;

Jr Jr iC ic

f

but /and k are each constant, or K — -.= constant. . . . (508)

K

The magnitude of the centrifugal couple G varies as the tangent
of the angle between the axis of the impressed moment and the
instantaneous axis of rotation.

Resume the equation given in (505), G2=K2<w2 — HV. Write
for K, H, and o> their values as given in sec. [99], (501), and

ON THE MOTION OP A RIGID BODY ROUND A FIXED POINT. 185

f

(507)— namely, K=nfk, H = nP2, and «=£. We have also

Jvi _ pa f

tan0=_! _, and *=T, whence

P K

G=K/ttan0 ....... (509)

It will be evident on inspection, that the indefinitely small
portion Ou of the line Ob parallel to the tangent drawn at Q, to
the section of the ellipsoid whose semiaxes are k and u, and which
is equal to QT, may be taken as the element of the arc of the
spherical curve traced out by the vertex of k during the element

d#
of the time dt. Writing -r- for this element Ov, and referring to

Qf

sec. [100], we have the ratio Ou : k : : G : K,

ds Gk
or Ou=-j-=^, but G=K/etan0, andf—ick.

(I/ K

Whence ^=/tan0 ........ (510)

Qf

ds

Now -TT is the velocity with which the curve of double curvature

passes through Q, the fixed point in space. We may thence infer
that the velocity with which the pole of the impressed couple
passes along this curve, or the velocity with which the curve
passes through the fixed pole, varies as the tangent of the angle 6
between the axis of the impressed couple and the instantaneous axis
of rotation.

102.] To find the values of -^-, -^, -,-, or of the velocities of

the pole of the impressed couple in the direction of the principal
axes of the body.
Az

•nr i dz dt , ds ,L n , &Z dz ,

We have j-=j-> and -r-=/tan0, whence •^-=^-fta.nff, and

d7

ds2 cLr2 dy2

^2=l+j-2 + T^2. Now (xyz) is a point on the surface of the

ellipsoid of moments, as also on that of a concentric sphere whose
radius is k. The equations of these surfaces are

o p o

+ i + = l, and .r2 + y2 + *2=**. . . (511)

VOL. II. 2 B

186 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Eliminating y and x successively, and then differentiating, we
find

#c- -

_ 4(62 - c2) 2yV + 64(c« - a2)2^2
hence gp-- (*(a*-b*)*aPy*

and #2=d?2 + y2 + .s2 ; hence

or eliminating x and ?/ by (511),

- -F)^ _ (514)

Making the substitutions suggested by these equations, we
shall obtain

22 *

103.] The axis of rotation due to the centrifugal forces lies in the
plane of the impressed couple.

Let w' be the angular velocity round the axis of rotation due to
the centrifugal couple, and p,, q,, r, its components round the prin-

* When the axis of the impressed moment very nearly coincides with one of
the principal axes (that of c suppose), the differential equations of motion may
easily be deduced.

In this case as x and y are each very small, their product xy may be neglected ;

w^=, ?=/f, r=, and ===-^=0. Hence r is constant,

, ,

equal to n suppose. We also have

whence =~ nq=--nq, or writing A = n«2, B = n&2, C=nc2;

at or be f2 a

Adf-|-(C-B)M?=0. Similarly

These are the equations deduced by Poisson for this particular case. (Traitt
de Mecanique, torn. ii. p. 159.)

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 187

oipal axes. Thru, as the angular velocity round any principal axis
is equal to the couple which produces the motion resolved at right
an»k>s to this axis, and divided by the corresponding moment of

inertia,

dx

Gds
j0/=-y — ; now G = Ktc tan 6, K = n/fc, L = n«2,

dx dx dx

, dx (I/ dt ~dl

and -j- = , --= -jr- M whence p.—f—^.

As ds f tan 6 'a?

dt

Making corresponding substitutions for qt and rt, we shall have
dx dy dz

ft=/|, ?,=/f , r,-/f (516)

Now the cosines of the angles which this axis of rotation makes
with the axes of coordinates are — , —,—; and the cosines of the

X 11 2

angles which the axis k makes with the same axes are -r, ~, 7. If

we denote the angle between the axis k of the impressed couple
and Py the instantaneous axis of rotation due to the centrifugal
couple by #6Py,

Ixdx ydy zdz\

cos ArOPyzz:- — (Pfp + qiy + i'tz} = T~ \ ~<r~l~~r2~ — 2" / ==^ (517)

since the part within the brackets is the differential of. the
equation of the ellipsoid.

We may infer, therefore, that not only is the axis of the centrifugal
couple contained in the plane of the impressed couple, but the axis
round which the centrifugal couple would give the body a tendency to
revolve lies in the same plane also *.

* To determine the angular velocity when L=M, or, using Poisson's notation,
when A=B.

fz dr fdz f(a*-b2)

As r='^2» j7=^^= ay a *#="> since a2 = 63. Hence r is con-
stant = w.

./*» —2 I -.0

; then

/"*&* Ka

a>2=»a-j--— sin2*. We have K = n/A, A=na2; whence w2=Ma4-^1 sin3*.

The expression given by Poisson, Traitt de Mtcaniquc, p. 159.

188 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

104.] Through the vertex of u the axis of the centrifugal couple,
let a tangent plane to the ellipsoid be drawn. The perpendicular
from the centre on this tangent plane, is the instantaneous axis of
rotation due to the centrifugal couple.

Let xt yt zt be the coordinates of the vertex of u ; lt, m,} nt, the
cosines of the angles it makes with the axes; \t, fJ>t, v,, the angles
which Py the instantaneous axis of rotation due to the centrifugal
couple makes with the same axes. Then, as u is perpendicular as
well to k as to Py,

k k k ' 'La* 62 c2~
Eliminating from these equations mt and I, successively,

-..r cosX/

cos v; ±y, or*/ cos v, Q'ZI

£f

and _/==___' —'—It- whence

rc, *, */ n, z,'

u

cos \y c2 /, cos /*. c2 m 0 , . . ,, /, m. , .

=-o — , =T^ — '. Substituting for -*-. — ' their values

cos vt a2 nt cos v, 62 w^ n, n,

given in the preceding equations, and reducing, we find

cos2 v = ^ ^ (518^

CU» y^ . a L2\2~.2,,2 i /J,2 s,2\2,,.g~2 i /'r.S ^,2\2^,2™2* V."-10/

We may find analogous expressions for cosXy and cos/^.

( 1 ' » >

-.-,

( 1 ' » > fi 7/ fi 2T

Introducing the terms -.-, -, -, by the help of (515),

- • - (519)

Now the cosine of the angle which the axis due to G makes

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 189

M

with the axis of z is -*•; writing for rt and to, their values as given

in (516),

r<

'/

Whence, comparing (519) with (520),

^i=cosi'/: in like manner ^=cos\,, ±l=
o>, to, &>,

or, The perpendicular let fall from the centre on the tangent plane,
drawn through the vertex of the axis of the centrifugal couple, coin-
cides with the instantaneous axis of rotation due to this couple.

The perpendicular Py is therefore in the plane of the impressed
couple.

105.] To find the component of the angular velocity toy due to
the centrifugal couple resolved along the instantaneous axis of
rotation.

Let 8 be the angle between the axes of the rotations due to the
impressed and centrifugal couples. Then

or substituting the values of o>, p, q, r, a>,, pt, qt, r,, as given in
(506) and (516), we shall have

dx y dy z dz~|
ff+|af+-4BJ.

Now the part within the brackets is the differential of

whence e^cos 8= — p, d7=/d/(pr ^ut as a)==p

da> .d/l\ , da> ^

=^ ' whence ~=ta' cos S' • •

Or, The increment of the angular velocity round the instantaneous
axis of rotation, is due to the component of the angular velocity
arising from the centrifugal couple, and resolved along the axis.

106.] To investigate expressions for the lengths of u and P;.

As u makes angles with the coordinate axes whose cosines are

1-, a , -r- , since u is parallel to the tangent to the common inter-

190 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

section of the ellipsoid and sphere, and is besides a semidiameter
of the surface,

! AW l(

\ds} \dSj

12 ^ '

•vr dx dx dt dy dy At dz dz At , ds e^ -.

Now 1T=A> & Z~£T& 7T=i7 TJ and -r;=/tan^asm(511).

ds dt ds' ds dt ds' d

Whence ^= ^ 2.2 2/T '9a '-. . . (522)

Mi, ft £ f\f. s*£t * TOTl* H ^ '

U U C J LcLH U

Again, as P/2=«2cos2X; + A2 cos2 ^ + c2 cos2 v,, we shall have,
putting for cosA^, cos/^, cosv; their values as given in (518),

i=

If we combine (511), (522), and (523), we shall find

/ds\2 /d^\2 /dy\« /d^y
\dt) _(dt) \dt) \dt)
~~ ~~ ~~

dy\ /d£

, */ A« • ^i^

but Pl=f-, q{=f±-J-} rt-f±-f as shown m (516).

Whence =' ' (524)

d« Gyfc /"

And as ^=-^- [see (501)] and eo=^, we shall have

(525)

To investigate an expression for the angle p, between the axes of
rotation due to the impressed and centrifugal couples.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 191

The cosines of the angles which the axes of rotation make with
tin- axes of coordinates are

£,1, -,£',-£ A whence co3P=

ft) ft) 0) ft), ft), ft)*

XT *-*

Now - and =

___

•Pxyz(b*-c*\ »

whence pp.=J •'( - - - I. Finding like expressions for
a262c2 V a2 /

a2 - 62 i2 - c2 c2 - a2"!
and rr acos— _- + _-+_ _J ;

/f

but

_

whence -^p^-^^"^. • (526)

The values of &> and o>, are given in (507) and (524).

This formula shows that whenever any two of the axes of the
ellipsoid of moments are equal, or whenever the axis of the impressed
couple happens to lie in one of the principal planes of the ellipsoid,
the angle between the axes of rotation due to the impressed and cen-
trifugal couples is a right angle.

CHAPTER XIV.

ON THE CONES DESCRIBED BY THE SEVERAL AXES DURING THE
MOTION OF THE BODY.

To determine the cones described by the axes of the impressed
and centrifugal couples, as also by the axes of rotation due to those
couples — in other words, to investigate the loci of k, P, u, and P/
referred to the principal axes of the body during the motion, will
be the object of the present chapter.

107.] To find the locus of k, the axis of the impressed couple.

The equation of the cone whose vertex is at the centre, and
which passes through the curve in which the ellipsoid of moments,

192 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

and the invariable sphere whose radius is k, intersect, may easily
be investigated, as k passes through the intersection of the ellipsoid
and sphere —

the equation of a cone of the second degree, whose axes coincide
with those of the ellipsoid.

This cone and the spherical conic section which constitutes its
base will repeatedly present themselves in the course of the fol-
lowing pages ; it may therefore be proper to denote them by some
appropriate name.

As the side of this cone is constant, being the axis of the im-
pressed couple, it may with propriety be named the invariable
cone ; and the spherical conic may be termed the invariable spherical
ellipse.

108.] To investigate the nature of the surface described by P
the instantaneous axis of rotation.

\y jj,, v, being the angles which P makes with the axes,

cos v a*z cos v o*z
We have also the equations of the ellipsoid and sphere,

rp2 yt ^2

~2 + TS + -3= 1, ^2 + «/2 + 2^2 = A:2. Eliminating x, y, z, we get

Let xyz be the coordinates of any point on the surface of

sc

the cone at the distance R from the origin; then cos A, =^7,

it

>y g

=^, cosv=^, and the equation of the cone becomes

the equation of a cone which is also of the second degree.

As this cone too will frequently recur, we may name it the cone
of rotation.

109.] To determine the equation of the cone described by the
axis u of the centrifugal couple.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 193

Let d ]f d be the coordinates of a point on the axis u of the
centrifugal couple; then

dr dy

a/ dJ a2/62-eV V1 5* b*fa*-c*\z

D=j- = -2(-2 — r«)-> n=T-=-9(-a — 15)-- See (512).

r Az c2\a2— 62/# 2' d? c2\a2 — b*)y

<l.v (l.v

From these equations and the equations of the ellipsoid and sphere,
eliminating x, y, z, we find, omitting the traits as no longer neces-
sary, the following equation of the fourth degree*,

110.] To determine the equation of the cone described in the
body by Py the axis of rotation due to the centrifugal couple.

The axis P, makes with the axes of coordinates the angles \t, (it, vr
Let xt y,2t be the coordinates of a point on the surface of this cone;
then

y22 22

z1 cosv, xw—b*/' zt cos v, y\a

* It may not be out of place to show that the equations of the invariable cone,
and of the cone of rotation given in sec. [107J and sec. [108] are equivalent to
the equations of the same cones given by Poisson in his Trait6 de Mfaanique
(torn. ii. pp. 151, 152). To show this, assume the equation of the vis viva given
at page 140 of the same volume, A=A/)a+Bj2+Cr2. Now

A = no", p = &, whence Aj>2 = n/" ^ ;
finding similar values for By2 and O2, we obtain

we also have A = na2, B=n62, C = nc2.

A fi2

k'=K=nfk; hence A;'2=n.n/2A;2=^.A.^, or £-2=

(I K

But the coefficient

I 1\ I/, a2\ n/, AA\ n k'*-Ah

be wntten I-'=I-=

making similar substitutions for the other coefficients and dividing by r^, we get

m

In the same way (628) may be transformed into

(A'3 - AA)*a + (*'a - B%2 + (k« - CA)za = 0.
These are the equations given by Poisson.
VOL. II. 2 C

194 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Eliminating x, y, z from these equations, as also from those of the
ellipsoid and sphere,

2222 2 2-c2)2a2c2*V|

)C2a2b*xY=0)' * '

which is also an equation of the fourth degree.

111.] The circular sections of the invariable cone coincide in
position with the circular sections of the ellipsoid.

It is a property of surfaces of the second order*, that if in two
such surfaces referred to the same or parallel axes the coefficients
of the squares of the corresponding variables differ all by the same
quantity, the circular sections of any two such surfaces are parallel.

Now the coefficients of the squares of the variables in the

equation of the ellipsoid are -^ T^, -%, and the coefficients of

* Let A:r2+Ay+A"z2+2Byz+2B'a:3+2B".ry+2Ca:+2C'?/+2C"z = l

be the equation of a surface of the second degree, referred to rectangular axes.
Let the surface now be referred to a new system of rectangular coordinates, such
that the plane of x'y' shall be parallel to one of the umbilical tangent planes, or
to one system of circular sections of the surface. If in this transformed equation
we make z' = 0, we shall obtain the equation of a circle referred to rectangular
axes, if the roots are real. The equation being that of a circle, we thence derive
two conditions — the equality of the coefficients of the squares of the variables,
and the evanescence of the coefficient of the rectangle x'y' . Let 6 be the angle
between the axes of z and z'. If we take the intersection of the plane of xy with
the plane of one of the circular sections as the axis of x', ty being the angle
between the axis of x and x'} we shall have, by the known transformations of
coordinates, and putting z'=0,

x = cos ^x' + cos 6 sin -^y', y = — sin -fyx' +cos 6 cos tyy', z = — sin Qy '.

Substituting these values of T, y, z in the given equation, the resulting equation in
x1 and y' is that of the conic section in which the plane of x'y' intersects the
given surface. As this section must be a circle, we get the two conditions

[( A — A'')cos24/+ ( A' - A")sin2i// - 2B"sin $ cos i/>] tan2<9+ 2 [B cos $ +Br sin i//]tan 6

= 4B" sin ^ cos ty — (A — A') (cos2 ^ — sin2 i//)
and

, ^_B"(cos2^ — sin2^)+(A — A') sin^cosi/'
B'cosi// — Bsini/r

From these equations eliminating tan 6, we should obtain a resulting equation
of condition in ij/, whose coefficients would be functions of (A — A'), (A— A"),
(A' -A"), B, B;, B".

As the coefficients of the squares of the variables do not enter the coefficients
of the resulting equation, but the differences of those coefficients only, it follows
that two surfaces of the second order whose equations are of the form

/ &c. =
will have the planes of their circular sections parallel.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 195

the equation of the cone are -,— p, i^~"^> ~«~~p> °f w^icu the

1

constant difference is To-
rt2

112.] There are some general properties of rotatory motion,
such as the principles of the conservation of areas, the conservation
of living forces , &c., which may with much simplicity be here estab-
lished.

Resuming the equation (466) and multiplying by eo2, we get

Ha>2=Jdm[Ro>sin0]2,

the integral being extended to the whole mass of the body. Now
R« sin 6 is the velocity of the particle dm. The above integral
therefore denotes the sum of all the elementary particles of the
body multiplied each into the square of its velocity. This is termed

the vis viva of the body.

f
In (504) it was shown that H=nP2, and G>=^; whence

Ha>2=n/2, or the vis viva of the body is constant, since n and/
are constant.

Let the vis viva of the body be denoted by F, we shall have

F= constant ....... (531)

Multiply the tangential equation of the ellipsoid of moments
given in (474) by/2, then

n/2 = L/2!2 + M/V + N/2 ? - 2U/X- 2 V/2££- 2W/2£y .
In (492) it was shown that p=f%, q=fv, r=f£, whence

F=Lj99 + M?2 + Nr2-2IV--2Vpr-2^, . . (532)

which is the equation of the vis viva in its most general form.
When we take the principal axes as axes of coordinates,

U = 0, V=0, W=0, or F=Lp2 + M?2 + Nr2, . (533)

the form in which the equation of the vis viva is usually exhibited.
If we square the equations given in (490), and add the results,

(L2 + V2 -f W>2 + (M2 + W2 + U2) q* + (N2 + U2 + V2)r2 }

. (534)

In this equation is contained the principle of the conservation of
areas; for (Kl=Lp— Vr— Wg-), see (490), is the sum of the areas
described on the plane of yz, multiplied into the particles which
describe those areas. Now these areas are projected on the plane

196 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

of the impressed couple, by multiplying this expression by the
cosine of the angle between the planes — that is, by / or its equal

Lp-Vr-Wq , (Lp-Vr-Wg)2 , ., ,

— - == - * ; and therefore -i-3- - ^ - — denotes the sum ot
&. J\-

the particles of the body multiplied into the areas described by
these particles on the plane of yz, and then projected on the plane
of the impressed couple. Finding analogous expressions for the
two other coordinate planes, we get for the sum of all the particles
of the body multiplied into the areas which they describe on the
plane of the impressed couple,

, ,

~~

but the sum of these expressions must, we know, be equal to K,
whence we obtain the formula given above.

When the axes of coordinates are the principal axes, V=0,
U=0, W=0, and we get the well-known equation,

K2=Ly + My + N2r2 ...... (535)

We may, in a very simple manner, establish the equations which
embody the principles of the vis viva, and the conservation of areas,
without using the method of tangential coordinates, when we restrict
our choice of coordinates to the principal axes of the body ; for

L=na2, j9 = -2, as shown in (500) and (506).

Finding like values for the other analogous quantities and
adding,

2=F. . (536)

Again, Ly +MV + Nr2=n2/2(#2+y2+*2) =n2/2A2=K2. (537)

Let p,, q,, rt, denote the angular velocities round the principal
axes, the components of the angular velocities due to the centri-
fugal couple ; then

. . . (538)

(I)
We have L=na2, p,=f— ^-. Writing similar expressions for

0

the other analogous quantities,

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 197

Now ^=/tan 0, see (501), and £=*, as in (508) ;

at K

whence L2/;,2 + M V + N2ry2 =n2/2/2 tan2 0

=n*f*k*
We may also show that,

. tan8 0 = K2*2 tan9 0.
k*

. . (539)

113.] Using the principles established in the foregoing pages,
the reader will find little difficulty in verifying the following
theorems : —

....... (a)

0, ...... (b)

pi am r ,n 1 d 2 , .

The sum of the squares of the distances of the vertices of the
three semiaxes of the ellipsoid of moments from the plane of the
impressed couple, divided by the corresponding moments of inertia,
is constant during the motion.

Let xt be the distance of the vertex of a from the plane of the

X (/ J'

impressed couple. Then xt=zal, and l=-r; hence ^/=-r- and

K K

nft * -jp2

L=na8, or --—— , whence

The sum of the squares of the distances of the vertices of the
three semiaxes of the ellipsoid from the plane of the impressed
couple, divided by the squares of the corresponding moments of
inertia, is constant during the motion.

a*#2 a?2 1 /a"2\

As before xf= — -, L2=n2a4; therefore =L — --—[ — } whence
A;2 L* n2£2\a2/

Let tangent planes be drawn to the vertices of a, b, c, the three
semiaxes of the ellipsoid, cutting off from the axis of the plane of
the impressed couple three segments. The sum of the squares of
the reciprocals of those segments will be constant during the

198 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

motion. Denoting these reciprocals by f, v, £ we shall have

/y> ty*

=£2, during the motion; for «f =£=-, or k%=-; hence

K K tt

Again, a2!2 + Z>2u2 + c2£2 = (/2 + m? + w2) = 1 .

£, v, £, the reciprocals of the segments cut off from the axis of
the plane of the impressed couple by three tangent planes drawn
through the vertices of the axes of the surface, may be the segments
of the axes of coordinates cut off by any tangent plane to the
ellipsoid.

If through the vertex of k, which is a point fixed in space, a
plane be drawn parallel to the plane of the impressed couple, this
fixed plane will cut off segments from the axes of the ellipsoid
during the motion, the sum of the squares of the reciprocals of
which is constant.

Writing £, v, £ for these reciprocals, we have

=l} kv=m, k$=n; hence ^ + v2 + ^=. . . (g)

CHAPTER XV.

INVESTIGATION OF THE POSITION OF THE BODY AT THE END OF
A GIVEN TIME.

114.] We must now proceed to the investigation of formulae by
whose aid we may be enabled to determine the position of the body
at the end of a given epoch. For this purpose we shall obtain two
distinct classes of formulae, to determine not only relatively to
certain fixed lines within the body (the principal axes suppose)
the position of certain other lines, but also absolutely the position
of these lines themselves in space. This double investigation is
necessary, because the locus of a point will vary accordingly as we
choose the axes of coordinates fixed in space, or varying in position
according to some given law. For example, the instantaneous axis
of rotation describes on a sphere concentric with the body, and
moving along with it, a spherical conic, while it describes on a con-
centric sphere fixed in space a spiral which undulates continually
between two small parallel circles of the sphere.

Again, under certain conditions the same straight line may
describe in the body a plane, or on the moving sphere a great

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 199

circle, while it describes in absolute space a sort of spiral cone, or
on the surface of the fixed sphere a spiral approaching very nearly
to the loxodromic or rumb line.

We have hitherto assumed k as lying between the mean and
least semiaxes of the ellipsoid, or a2>Z»2>A:2>e2. Should we
require to consider the case when k lies between the greatest and
mean semiaxes of the ellipsoid, the formulae will be most easily
modified so as to embrace this hypothesis also, by taking in that
case c as the greatest semiaxis, and b the mean semiaxis as before,
or a2<i*<A:2<c2. While on the former supposition the binomials
a2 — 62, a2 — c2, A2 — c2, a2— A;2, 62— A:2, A;2 — c2, are all positive, on
the latter they will all be negative. Now, in the formulae which
we shall have to deal with in the remaining portion of this subject,
these binomials occur generally in pairs, connected either by mul-
tiplication or division. It will result, therefore, that no effective
change of sign will generally take place, whether we suppose k to
lie between the greatest and mean semiaxes, or between the mean
and the least. The case where k is equal to the mean axis will
require a separate investigation. When the body is a solid of
revolution we cannot take N equal to L or M, or c equal to a or b,
because we suppose c to be the greatest or the least of the three
semiaxes. The only hypothesis, not inconsistent with previous
assumptions, is L = M, or a = b; and this is the assumption gene-
rally made when the case of a solid of revolution is considered.

Resuming one of the equations (515),

-y ........

If we agree to take -r- with the positive sign when a>b, we must

attach the negative sign when a<b.

To integrate this equation, we must express x and y in terms of z.
This we can easily do by eliminating x and y alternately from the
equations of the ellipsoid of moments and the concentric sphere.
We hence find

=a V(6«-cV8 -<?*(&*-*«) _b Vc*(a-*2)- (a*-c2)z8 „

c v^=£2 c v^^F

Making these substitutions in (a), the last equation becomes

. (540)

* If we assume the relations established in the note at page 186,

A = na% B=n6>, C = nc>, A=n/*, A'.n/A, r= =, =,

200 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

To facilitate the integration of this equation, assume

«c [a ~~ K ){o ~~ K ) /*tt\

t, — — — ' -. . (o41)

Substituting the value of z derived from this equation in (540),
\df ) =/2[(a2 - £2) (b* - c2) cos2<p + (62 - A2) (a2 - c2) sin2<p] ' "

or integrating, we obtain the following elliptic integral of the first
order,

f- ±abc f , ^ (542)

-

f ,
-C2)\ / r(a*-^(*2-

JV L(62-c2)(a2-

2S1

115.] 7%e modulus of this function is the sine of the semifocal
angle of the invariable cone.

Resuming the equation of this cone given in (527), and writing
a and /3 for its principal semiangles,

tan2«=-

C*(0* — K*)' C'(a"—K"-)

Now, e being the semifocal angle of this cone, cos e= ^

cos/3

. ,n. . 9 cos2jS— cos2« (a2— 62)(&2— c2)

as in (2), or sm2e= — -5-^ = /r2 aw2 — T^\> • - (D)

cos2 13 (o2 — c2) (a2 — A2)

_ _

hence cos^=F-f,andsec«cose=?=;. (e)

Consequently the coefficient of the elliptic integral in (542),

abc , ... abc2 sec « cose

- ... may now be written - , (d)

2222

/ V (a2-

In (508) it was shown that f=kfc. Introducing this relation
into the preceding coefficient, and making

.* -- .... (543)

abc*

and by the help of these relations eliminate from (540) the quantities a, b, c, /,
z, kt we shall obtain the resulting equation

±\/AB.Cdr

the expression which Poisson arrives at, Traite de Mfaanique, torn. ii. p. 140.

IHE MOTION OF A RIGID BODY ROUND A FIXED POINT. 201

(54-2) may now be written

(544)

l— sm2esm2<p

In (58) it was shown that the arc <r of a spherical parabola whose
principal arcs a and /3 are given by the equations

2 _ 1 4- sin 7 . - 2 sin 7
tan2 * = - -- : — -, tan2 B = — —,
1—81117 1 — siny

may be represented by an integral of the first order, or

C dip sin 7 tan <p 1

o- = smyl ^ + tan-M . r " - ; (e)

J v 1 — cos8 y sm2 <p L y 1— cos2ysm2<p-J

writing 9 for the circular arc, we get the simple formula

jt=a — <; ........ (545)

In this case, tan2 «=- - — =cot2i e, or 2a + e=7r. . . (f)
1— cos e

2« and e are therefore supplemental.

tjf _ ej^

When e vanishes, a = — , /3=-, or the spherical parabola becomes
A ii

a great circle of the sphere.

When the moment N of the body is very nearly equal to L or
M, c2 must very nearly be equal to a2 or b*, and the coefficient j
becomes indefinitely small.

116.] It may easily be shown that the amplitude <p of the elliptic
integral assumed in (541) is the eccentric anomaly of the vertex of k,
the axis of the impressed couple. Let Q and b be the semiaxes of
the plane ellipse, the intersection of the invariable cone with a plane
which touches the sphere whose radius is k, which is drawn at right
angles to the axis c of the ellipsoid, the internal axis of this cone.

Let the plane which passes through the axis c and k cut the
plane of the ellipse in th'e semidiameter R, making the angle ^
with the axis a of the ellipse. Then, as a = £tana, li = A;tanj8,
and, p being the angle which k makes with the axis of z, R =k tan p,
we shall have

1

tan2* tan2/?

cos2p= --, as shown m sec. [81.

*

Let <p' be the eccentric anomaly, then tan p'=j-tan -^r, . (a)

, tan /3 . cos2 a

or tany-=— — tan®', and cos2p = 1 -- — . — .-« ,- . . . (b)
tana 1— sm2e siu2

VOL. II. 2 D

20.2 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

In (541) we assumed

. 2

_ _

and 7-5 — igT-Wo - g,'=sin2 e. Comparing this expression with (b)

(a2 — k*) (o® — c2)

we find <£>=<£>'.

Or <p is the eccentric anomaly of the vertex of k.

117.] Resuming the equation established in (544), we may invert

the formula, ;7=cose 1 — . , and express the ampli-

J Vl-sin2esin2(p

tude <p in terms of the function jt. Accordingly let <p be a function
of jt, or <p = (jt) *, the parenthesis denoting a function of jt. Sub-
stituting this value in the value assumed for z in (541) , we find the
following values of x, y, z —

2 _ «2 (62 - 7c2) (** - c2) sin2 (jt) }

(a*-k*) (62-c2) cos2 (jt) + (b*-k2) («2-c2) sin2 (jt)'

7*> >• (546)

cos + - a-c sn

c2) cos2 (jt) + (62 — A2) (a2 — c2) sin2 (jt) ' )

* That the assumption here made is allowable, may be shown as follows.
Let (1 - i2 sin2 <p)-% be developed in a series of cosines of multiple arcs ; for the
successive integral powers of sin2 Q may be so developed. Accordingly let

J . - = A +2B cos 2<p +-4C cos 4<p +6D cos 6<p &c.

Integrating these equivalent expressions, and putting t for j . — •j-^f, we

j \ L — t sin (p

sin4<p+Dsin6<p .... &c. now

Substituting these values of the sines of the multiple arcs of <p in the preceding
equation,

or, by the inverse method of series,

<p=«M+

or <p may be taken as a function ofj't, or we may put <p=(;Y), as in the text.

o\ I ill MOTION OF \ HKill) UOJ)V HOl'M) A IIM.I) I'olNT.

We may also express x, y, z in terms of the time and of the
constants of the invariable cone. Transforming the expressions
in the preceding formulae, we find

tan*/3sm2Q7)

see2 a cos2 (jt) + see2 ft sin2 (jt) '

tan2 a. cos2 (jt}
sec2 « cos2 (jt) + sec2 /3 sin2 (jt)'

1

(547)

£2 sec2 a cos2 (jY) + sec2 >3 sin2

From either of these groups of equations we may find the coor-
dinates xy z of the vertex of k the axis of the impressed couple, in
terms of the time. We can thus determine the particular diameter
of the ellipsoid which happens to coincide with the axis of the im-
pressed couple at the end of the time t. And if we suppose the ellip-
soid brought into this position, we shall have the inclination of the
equator of the body to the plane of the impressed couple. This,
however, is not sufficient to determine completely the position of
the body. The body might take any position round this line as an
axis, xy z remaining unchanged. We must therefore determine
the position of some other fixed line or plane in the body. One
of the most obvious is the intersection of the plane of the equator
of the body or of the plane of x y with the plane of the impressed
couple. The position of this line being ascertained at any epoch,
the position of the body will be completely determined.

118.] To determine the value of (o the angular velocity at the
end of any given time.

/•a r™ 2 2 2™i

Since to2 = ^ =/2 \—4 + |j + ^ , substituting for x y z their values
given in terms of the time in (547), we find

tan2a ,. ... tan2)3
7+ -- cos2 O/) + —

, ...1
(jt)

(548)

-

sec2 a. cos2 (jt) + sec2 /3 sin2 (jt)

This formula may be simplified as follows.

It was shown in sec. [108] that the instantaneous axis of rotation
describes a cone of the second degree, whose equation is

Let «' and ft be the principal angles of this cone. It may c asily
be shown that

" c2(^t>2)-

204 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

c2 c2
whence tan a' = 7^ tana, tan /3' =-3 tan /8 (b)

Introducing into the value of a> these functions, we get

l'sin2C//)-

c4 Lsec2a cos2 (jt) +sec2/3 sin2 (jt)

* Let the axis of the impressed couple very nearly coincide with one of the
principal axes (that of c suppose) ; then k is very nearly equal to c, or to z, and

the angular velocity round the axis of z, being given by the equation r=-~, as
in (506), r=-, a constant quantity which may be put equal to wt or K = H.

C

In this case the invariable cone becoming indefinitely attenuated, sec«=l,
sin e = 0, and k=c nearly ; so that the formula given in sec. [114]

-xcose
kf

/(q»
\/ ~

f- -*

J Vl-si

niav now be written. nt= - . To use the notation adopted by

A(oa-caX&8-ca)

\/ " ~~^6»~

Poisson in the Traits de Mecaniqne, let A, B, C denote the moments of inertia
round the principal axes ; then A=na2, B=n62, C = nc2,

whence . /(*-*)(»-*)_ /(A-C)(B-C)

V <M* "V AB

or n8t = q>, whence y= ??S.

,,.„, ,. ' «2

In (546) we found ^ = _,,

Since k2 is equal to e2 nearly, let 7v2 = c2+«'2, in which v is a quantity indefi-
nitely small ; the above formula may now be written

,_ _ yV[fe2-c2— i/2]sin2M^ _
~ (a2 - cs)(63 - c2)— v\(W— 62)cos2wS^+(a2- c2) sin2^]'
or, neglecting i/2 when added to finite quantities,

(a? -(*)(&-<?)
Taking the square root and reducing,

vf

Now assume . = a,

2

•v«2&2(a2— c2)(62-c2)
fx _

whence ^ = «V B(B — C) sin(»8£+y). y is added, since a? and ^ may be sup-

fy _

posed not to vanish together. In like manner, 4? = « V A(A — C) cos (n^-f-y).

o\ Illi: MOTION 01' \ HIOIl) HODY KOCM) A FIXED POINT. 205

We may also express the components p, q, r of the angular
velocity in terms of the time —

(550)

= n«QV) 1 '

P = a4 Lsec2 a cos2 (.;/)+ sec2 /3 sin2 (./0-T

|*!r tan2* cos* (jt) . I

64 Lee2 « cos2 ( #) + sec2 B sin2 ( ;7) J '

r2 =

_

c4 sec2 a cos2 (jt) + sec2/3 sin

The angles which the instantaneous axis of rotation makes with
the principal axes, are given by the equations

COS\_C2 X _C'

cos v a*z~a

cos

c c

or, as tana' =7g tana, tan/3' = -2 tan/9, as in (b),

cos\
cos v

— tan /3' sin

= tan«' cos i

tan2 /Q' sin2 (jt)

COS2/X =

sec2 «' cos2 (,/V) + sec2 ft sin2

tan2 a' cos2 (jt)
sec2 a' cos2 (^Y) + sec2 ft sin2 (^7) '

(551)

COS2V =

sec2 « cos2 (y/) +sec2 (3 sin2 (,/V)*

These equations give us the position of the instantaneous axis of
rotation with reference to the principal axes, in terms of the time.

119.] We must now, in order completely to determine the position
of the body at the end of 'the time /, investigate a formula which
will enable us to ascertain the position of some other line in the
body at the end of the given epoch. We may take the straight line

In (606) it was shown that p = ^, q=ji ; whence

= * A/A(A-C)cos(«&+y).

These are the formulae established by Poisson, on this particular hypothesis,
by methods wholly dissimilar. ( Traitt de Mtcanique, torn. ii. p. 154.)

When k is absolutely equal to c, v=Q, and therefore «=0, or p=0, q=0,
whatever be the value of t. Since K =/fcn, F=/2n, we get

•j^a _ pja

=LM(L-N)(M-N)' °r' U8in* Poi8SOn'8 notation' *a=

206 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

in which the equator of the body (the plane of x y suppose) and
the plane of the impressed couple intersect.

The angular velocity of the body round the axis k being uniform
and equal to K, the angle described on the plane of the impressed
moment in the element of the time d/ will be K&t, or the angle tct
in the time t, measured from a given line in this plane, its inter-
section with the plane of the equator of the body, or the plane of
the axes a, b. But this line, which may be called the line of the
nodes, will itself have an angular motion on the plane of the im-
pressed moment during the time ; this angle may be denoted by ty,
whence the whole elementary angle will be

d>Jr , d3 . cty d3

-TT-I-K- Let this angle be -T-, then -~ + /e= -. . (a)

Now this elementary angle is the projection, on the plane of the
impressed moment, of the angle on the plane of a b, over which the
projection of the axis k on the plane of a b passes in the time dt.
Let p be the angle between these planes, or the angle between k and

the axis of z. Then cos/o = -r, and the angle of which -r- is the

-
projection is — =-. Hence the area described on the plane of a b

Add
by the projection of k upon it is £(#2 + y2) --ri* This area may

also be represented by the expression Wy^r,— # ~~Tj\- Equating
these expressions for the same elementary area,

-vr # — cv c—x .

Now At=L~T^' it-"-**? --• " m

Whence

The equations of the ellipsoid and sphere give
62c2«r2 + aVy2 = a262c2 - a2A2*r2, «2£2«/2 + «262#2 = «2*2/t2

Consequently y-^=/^t2. . . . (d)

o\ I UK MOTION' or A Kl(ill) 1UJDV Hol'M) A KIXKI) POINT. .'J07

And as a2 + y2 = A2 — z*, 4 = K, we at length obtain

T

To integrate this equation, assume as in (541)

_ C COS

_ __ • iwi

----62)sin2<p'

and writing for d/ its value as given in (541*), we obtain by inte-
gration the elliptic integral

No\v, e being the eccentricity of the plane base of the cone the
locns of the axis of the impressed couple, (a) sec. [115] gives

2_-_-

tan2* ~"b*(a*-k*y

ac(b*-k*) tan/3

We hnd also ,. ., g ,gv/A«. - 57= +* -- cos «, . . . (i)
/2 — *— c2 -tan

taking the negative sign when A > £.

Introducing these transformations, the last equation (h) becomes

(553)

tan a J [1 -e2 sin* <p] Vl -sin2 e sin2

If we now turn to the formula given in (15), we shall there find
that this elliptic integral is the algebraical expression for an arc
of the spherical ellipse, supplemental to the one whose principal
arcs are « and /3, supplemental in this case, therefore, to the
invariable spherical conic. Writing a for this arc, we get the
simple relation

....... (554)

We may hence infer that the line of the nodes, or the intersection

208 ON THE MOTION OV A RIGID BODY ROUND A FIXED POINT.

of the plane of the equator ab with the plane of the impressed
couple, describes an angle which is made up of two parts : one of
these parts is a circular arc increasing uniformly with the time ;
the other, <r, is an arc of the spherical ellipse which is the base of
the cone supplemental to the invariable cone. Now, as the axis of
the impressed couple is always a side of the invariable cone, the
plane of the impressed couple will always be a tangent plane to the
supplemental cone ; and it may easily be shown that the line of
contact of the plane of the impressed couple with this cone is always
at right angles to the line of the nodes.

It follows, therefore, that the line of the nodes is retrograde, and
in the time t will describe the angle Kt + a-.

The angle — -^ equal to Kt + a, we may imagine to be thus described.
Let this supplemental cone be conceived to roll on the plane of the
impressed couple with such a velocity that the axis of the con-
jugate tangent plane may describe the invariable cone with the
velocity given in (510). Let, moreover, the invariable plane be
conceived to revolve uniformly round its axis. We shall then have
a perfect idea of the rotatory motion of a body revolving round a
fixed point, free from the action of accelerating forces. In this
manner it is shown that the most general motion of a body round
a fixed point may be reduced to that of a cone which rolls without
sliding with a certain variable velocity on a plane whose axis is
fixed, while this plane rotates round its axis with a certain uniform
velocity.

This cone is always given, and may be determined as follows : —
The circular sections of the invariable cone coincide with the
circular sections of the ellipsoid of moments (see sec. [Ill]),
whence the cyclic axes of the ellipsoid, or the diameters perpen-
dicular to the planes of those sections, will be the focal lines of the
supplemental cone. As the invariable plane is always a tangent
plane to this 'cone, we have elements sufficient given to determine
it ; for when the two focals of a cone and a tangent plane to it
are given, we may determine it, just as we may a conic section when
its foci and a tangent to it are given.

120.] From these considerations it follows that we may altogether
dispense with the ellipsoid of moments, and say that if two straight
lines are drawn through the fixed point of the body, in the plane of
the greatest and least moments, making equal angles with the
axis of greatest moment, whose cosines shall be equal to the square

root of the expression ^ ' ' and a cone be conceived having

these lines as focals, and touching, moreover, the plane of the im-
pressed couple, the entire motion of this body will consist in the
rotation of this cone on the invariable plane, with a variable velocity,
while the plane revolves round its own axis with a uniform velocity.

OX Till: MOTION OF A RIGID BODY 1UH \I> A FIXED POINT. 209

Fig. 32.

Let ACB be the mean plane section of the ellipsoid, or that which
passes through the axes 2a, 2c ; ON,
ON' the cyclic axes ; then, if the
plane of the impressed couple coin-
cides with any of the principal
planes, the cones round the cyclic
axes as focals become planes also,
and the axis of rotation coincides
with one of the axes of the figure.

Again, if the plane of the im-
pressed couple intersects the mean plane between N and C, it will
envelope the cone whose focals are ON, ON', and whose internal
axis is therefore OA. But if it intersect between A and N, it will
envelope the cone whose focals are ON, OM, and whose internal
axis is OC. Whence the range in the former case (which may be
taken as the measure of the stability of rotation round the axis
whose moment is the greatest) is to the range in the latter case
(which may also be assumed as the representative of the stability of
rotation round that axis whose moment of inertia is the least) as
the supplement of the angle between the cyclic axes of the ellip-
soid is to the angle between these axes.

It is also evident that the sign of the spherical elliptic arc will
depend on the sign of the binomial (62 — k*) in (j) sec. [119]. The
signs of Kt and <r being contrary when b < k, they will be the same
when b>k. We may therefore infer that the direction in which
the angle <r shall be described will depend upon the position of the
axis k in the body — whether it lies within the region between the
planes of the circular sections of the ellipsoid, or without.

From the theorem established in sec. [4] we may infer that the
product of the sines of the angles, which the cyclic axes of the body
make with the plane of the impressed couple, is constant during
the motion ; for the cyclic axes of the ellipsoid of moments are the
focals of the cone suppleme'ntal to the invariable cone.

121.] To determine the angle between the instantaneous axis of
rotation and the line of the nodes.

Let this angle be Br The cosines of the angles which the axis
of the impressed couple makes with the axes of coordinates being
as before /, m, n, let the cosines of the angles which the line of the
nodes makes with the same axes be lllt mtl, nlt ; X, JJL, v, are the angles
which the instantaneous axis of rotation makes with the same
axes.

Then cos 8, = llt cos X + mu cos p -f nn cos v (a)

As the line of the nodes lies in the plane of the impressed
couple, and is therefore at right angles to its axis k,

(b)

VOL. II.

2 E

210 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

and as it is perpendicular to the axis of Z, see sec. [119],

hence (a) and (b) become

cos S, = ln cos X + mtl cos p, l,,l + mnm — 0 ; and /y/2 + w?v/2 = 1 .

These equations give mn— — -j==

whence

£ _/cos/ti — wcosX ,_x _y ^_^x _^V

I . I '19 i Q ' If' If* /»2 ' ' A2 }

s -

orcosd.= - / a ^ ...... (555)

22

When two of the moments of inertia are equal (L = M, suppose),
a — b, and cos8y=0, or 8y=900. Whence we may infer that when
the body is a solid of revolution, the angle between the instantaneous
axis of rotation and the line of the nodes is always a right angle.

The angle 8t is also a right angle whenever the axis of the im-
pressed couple lies in one of the planes of the principal sections of
the ellipsoid; for then x=Q, or y = 0.

122.] To determine the angle between the line of the nodes and
the axis u of the centrifugal couple.

Let ^ be the angle which the axis u of the centrifugal couple
makes with a fixed line, ty the angle which the line of the nodes
makes with the same fixed line ; then as the line of the nodes and
u are in the plane of the impressed couple, see (498), the angle to
be determined is (%— ^).

Now the cosines of the angles which u makes with the axes are
Ax dy Az , , d# dy , Az

'''' whence cos fc-^Hw+^^

The values of ltl, mlt) nlt were found in the last section to be
m — / _ft ,_x _y

' H"~ ~~' "'

We may hence deduce

y da? x dyl

lai-T^h -.. (a)

but T-

ds at as as at as

-, dx ,(52— c2) dy . (c2 — a2)

ana -j-=j ,9 0 yz, 3^=/- — o— » —xz. as in
cu o^c elf a^c^

ON THE MOTION OF A UK! ID BODY ROUND A FIXED POINT. 211

Whence

d* fz

dtC< =~VF=

_ C

The part within the brackets is - — % — -; and -j-=/tan Q, see (510) ;

z /k2 — c2\
cos(x-^)=— -—-cotO. . . . (b)

p being the angle between the axes c and k, cos p= -. Introducing

K

this value of z into (514) and the trigonometrical functions of a
and /3 the principal semiangles of the invariable cone, as given in
(a), sec. [115], _

tan 0= (*^\ A /cosV-cos2* cos* /3 (c)

V c2 /V sin2 « sin2 /3

whence cos2 (v-^r) = sin2* sin2 /3

sm2p-cos2« cos2|

and tan2 (y- ^) = ^n2«-sin2p)-(sin2p-sin2^)

sin2 « sin2 /9 cos2 p

This formula leads us to infer that when « = /3, %— i|r is always
0, or %=<^1; whence the axis of the centrifugal couple, when
the solid is one of revolution, always coincides with the line of the
nodes.

Again, when p — ot} or p=(3, X=ty'> *^at ^ whenever the axis
of the impressed couple lies in one of the principal planes of the
solid, the axis of the centrifugal couple coincides with the line of the
nodes.

CHAPTER XVI.

123.] In the'preceding sections formulae are given which enable
us to determine the position of the axis of rotation, and of the axis
of the plane of the impressed couple, with reference to fixed lines
taken within the body. It still, however, remains to determine
the positions not only of those lines, but of the fixed lines within
the body, relatively to absolute space. True, we may by trans-
formations of coordinates, and by the choice of other variables,
obtain solutions from the formulae already established, by methods
which, however, are tedious, complex, and not a little obscure. It
will be found not only the most direct, but by far the most elegant

212 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

method of procedure, to conduct the investigation independently,
and start from first principles.

As the body must now be referred to fixed lines in space, it is
no less obvious than natural that we should assume the plane of the
impressed couple as one of the coordinate planes. Let this plane
be taken as that of x y, its axis that of z. Moreover let the plane
of the greatest and least principal axes of the ellipsoid of moments
coincide with the plane of x z} at the beginning of the time t. The
instantaneous axis of rotation will be in the same plane at the same
epoch, and will make with the vertical axis k an angle whose tan-
gent is given by the equation

(557)

This may easily be shown ; for the perpendicular from the centre
on a tangent through the vertex of k, a semidiameter of an ellipse
whose semiaxes are a and c, makes with k an angle whose tangent
is given by the last formula.

In like manner, for the principal section whose semiaxes are
b and c, we get

C2). .... (557*)

(D and O/ are the maximum and minimum values of 0, the angle
between the axis of the impressed couple and the instantaneous
axis of rotation.

124.] We now proceed to establish the following proposi-
tion : —

The area described by the axis u of the centrifugal couple, on the
plane of the impressed couple, varies as the time.

The following relations were established in (524), (510), (507),
(508) —

As

At to, As _,, / /

=> =i™6 "- and "=

whence -'=/<: tan <9 ...... (558)

&) V,u

Let O be the centre of a sphere whose radius is 1, concentric
with the ellipsoid of moments, Z the point in which the axis of the
plane of the impressed couple meets it, and OI the direction of the
instantaneous axis of rotation at the end of time t. Let the plane
which passes through these lines OZ, OI, or the plane of the cen-
trifugal couple coincide with the plane of x z at the same instant.
Then the axis of Y will at that instant be the axis of the centrifugal

OX THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 213

couple ; and the perpendicular from the centre on the tangent
plane to the ellipsoid, at the point where the axis of Y intersects

Fig. 33.

it, will be the axis of rotation
due to the centrifugal couple,
see sec. [104] . Let the direc-
tion of this perpendicular be
OJ. Through OIJ let a plane
be drawn. If, along Of, OJ the
instantaneous axes of rotation,
we assume lengths OI, Or,
proportional to the angular
velocities o>, &>' round these
axes, the diagonal OI', of the
parallelogram constructed with
those lines as sides, will repre-
sent in direction the instanta-
neous axis of rotation at the
end of the time t + At.

Let OI, Or taken in this proportion, be the sides of the paral-
lelogram ; the diagonal OI' will be the contemporaneous position
of this axis of rotation.

Let the angle ZOI = 0, YOJ = 0'; also let B be the angle between
the planes of IOJ and ZOX. Then, as the instantaneous axis of
rotation due to the centrifugal couple lies always in the plane of
the impressed couple, see sec. [103], the line OJ is in the plane of

*7T

xy, and the angle JOX = — — &. Let ^ be the angle which the

<i/

vector arc 0 makes with a fixed great circle of the sphere passing
through Z. The instantaneous axis having moved into the position
OI', the arc ZI will have moved into the position ZI', or through
the angle d^, in the time At. Let Iv be an arc of a great circle
perpendicular to ZI', and as II'u is an infinitesimal right-angled

triangle we shall have II' sin 8 = If = -*~ sin 0. Again, as I JX is a
spherical triangle, right-angled at X ; sin IJ : sin JX : : 1 : sin 8,

or sin IJ =

cosfl'
sinS

We are also given by the construction,

a)' _ sin II' _ II' sin 8 _ d% sin 9
w sin 1 J cos & At cos 6' '

and (525) gives

' ~Pk

-=_T_/e tan v.
P'u

214 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

/

Equating these values of — , and introducing the relations
P= k cos 6, P'=w cos & ', we get

u*^H = Kk* (559)

Now uz ~ is the elementary area described on the plane of the

U.6

impressed moment by the semidiamcter u of the ellipsoid which
coincides with the axis of the centrifugal couple ; whence the area
described by this semidiameter is proportional to the time, or

. v.J=/rfA24- constant (560)

dt

125.] To determine the position of the instantaneous axis of
rotation in absolute space, at the end of any given time.

If along the axes of rotation due to the impressed and centrifugal
couples, we take two lines to represent the angular velocities due
to those couples, the diagonal of the parallelogram, constructed
with these lines as sides, will represent the instantaneous position
of the axis of rotation.

Now, if we turn to the figure at p. 213, we shall see that

sin II' : sinU : : to' : &>, and ultimately TT=sinII'; whence

do- to' . TT /dcr\2 w'2 a)'2

T- = — smIJ: or I -1 =—5- s-cos^IJ. . . (a)

dt a \ dt ) eo2 &)2

The general formula for the element of an arc measured on the
surface of a sphere is

d(9>

We must now reduce this formula.

dy /ck2
In (559) it was shown that -^ = —^-, and in (525) that

(16 tl

at1 /ck*2

— ==pT~ sin 0- Making the substitutions suggested by these

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 215

transformations, we shall find

(C-jf)a=AVsina0r --1~|_^COS2IJ. (c)

\(U / LI U2 W4J G)2

We shall now proceed to reduce the first term of the second
member of this formula. To facilitate the calculations, let

(d)

(ds\4
Y. ) , we shall have

S?1 Q=/tVsin20

/cUT

\&L{*?f-[*i i (e^

LP>2id/J \"l I' ' '

s, it must be borne in mind, is the arc of the invariable conic ; and
zyx are the coordinates of the vertex of k referred to the principal
planes of the ellipsoid.

Now, if we turn to sec. [106] and sec. [1071, we shall there
find

dfV /dyV /d£
t) \(\t) \to

2 1

1

/d^\2 /d^

yd// (at

I

62 C

SV+

Introducing the substitutions suggested by these transformations,
we shall obtain

*4*2 sin2 6

at

(6)

216 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Making the obvious reduction in this equation,

,

We have also, see (515),

_ /dtA2
' (to) =->

n _ U y djA 2 /d£ Y _
V2 #7 W/ \d// =

Finding similar values for the other symmetrical expressions, sub-
stituting, introducing the relation x'2 + y'2 + z'2 = k'*) and writing

ds
for j- its value /tan 6, we shall finally obtain

_ |-
"L

n

O)'2

We have now to compute the term —5- cos2 I J.

<u2

In sec. [106] it was shown that the angle between the axes of
rotation due to the impressed and centrifugal couples, was given by
the formula

whence —k

or \

In (506) and (516) it was shown that

•//y T * ^ A* /^*^ -i/ **

/«/ / i (/ ~~~ C J V^

•*^ /^2 ^ -* ^ "^ ^»2 A2^>2 ^ JrSr / ^"~ 2 iQ '

Finding analogous expressions for gg^ and rr;,

7.2 _ 7,2-1

• • • (J)

<>\ TIII: MOTION OK \ RKJID BODY ROUND A FIXED POINT. 217

f f
Now <u =4=

-c* c2-a2 Q2-6a_(62-c2)(ffl2-c2)(a2-62) ,
2 ~~~ ~~ ***

a>'2 /g*4 cos4 0 (a2 - 62)2 (a2 - c2) 2(62 - c2) 2# V*2

C<

Multiplying this expression, numerator and denominator, by
tan4 6, writing ick for /, and in the expression

substituting for the terms of the second member the values found
in the preceding equations, reducing, and taking the square root,

dfl_ K&? sin 0 cos 6 (a2- 62) (62-c2) (a2- c*}xyz ,.

~ 444 tan2 0

We have now to express x, y, z in terms of 0.

Combining the simultaneous equations of the ellipsoid of moments,
of the concentric sphere, and of the perpendicular from the centre
on the tangent plane to the ellipsoid, namely

we obtain from these equations,

#*_[62c2tan20-(62-

a4'

\ / \ /

. . (562)

0 + (a2 -

Substituting these values of x,y,z in (561), the resulting equation
will become

a'ftV sine sec3 fl

VOL. II. 2 p

218 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

This is an elliptic integral of the first order, which may be reduced
to the usual form by assuming

(564)

Before we proceed further, we shall give the geometrical inter-
pretation of this assumption.

Let a cone be conceived whose internal axis shall coincide with
the axis of the plane of the impressed couple, or with the axis of z,
and whose principal arcs shall be the greatest and least elongations
of the instantaneous axis of rotation from the axis of the impressed
couple. This cone will generate on the surface of the sphere a
spherical conic, the tangents of whose principal arcs (2a", 2/3") are
given as in (557) by the equations,

*)

This cone may be named the cone of nutation.

Now, if from the centre of this curve the vector arc 6 is drawn
to a point on it, X is the angle which the perpendicular arc from
the centre on the tangent arc through the vertex of 6, makes with
the principal arc a".

To simplify the notation, let

Y=(a2-Ar2)(A2-c2)-a2c2tan20, I . . . (565)
Z = a262 tan2 6 + (a? - A2) (62 - A8) , )

and the equation (563) will become

At a?b*c* tan2 6

0 v/X.Y.Z.'

If we differentiate (564), and make the transformations resulting
from that assumption, we shall get the following relations : —

= A2 (a2 - b2) (k* - c2) cos2 X ; j

42 Y = *2(a2 - 62) (F - c2) sin2 X ; and I . (566*)

2-&2) (a2-c2)sin2X.)

By the help of these transformations, equation (566) takes the
form

+ abc f _ dX

2222 2- >

f _ dX

-c2) \ / _ r(«2-62)(A2-
J V L62-c2«2-

S1
which is precisely the same elliptic integral we found in (542),

OK THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

_ from it only in the amplitude X and the sign. When
b>a the positive sign must be taken. We shall show presently
that $ and X have opposite signs.

Tins formula may be thus written, as in (544),

+ abc* sec « cose C dX

t = I • . (OOO)

1.9 - . 1 1 _a -«\ / '<?__ _2\ i »/l _ait ° - -'- -° "*•

When the integrals are complete they are identical, as they
manifestly should be, because the maximum and minimum values
of 6, the greatest and least elongations of the instantaneous axis
of rotation from the axis of the plane of the impressed couple,
should be given by the same formula, whatever system of axes we
choose — since this value must be independent of the position of any
axes chosen at will, being a function of the constitution of the
body, and of the magnitude and position of the impressed couple.

126.] To determine the angle %, which 6 the vector arc, drawn
from the vertex of k, to the pole of the instantaneous axis of rota-
tion, makes with a fixed plane passing through k the axis of the
impressed couple.

dy icl?
Resuming the equation -57 =— g-j established in (560), we have

now to express M2 in terms of X.

If we turn to (522) , we shall there find

d#\2 /d?A2 /Az\*
— ) I -A ) / — )

or

e

™- • . L ,

Eliminating by the relation = + +

as shown in (515).

Having made these substitutions, we shall find

d*

220 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Eliminating #2 and y2 by the equations of the ellipsoid and sphere,

As
introducing also the relations ^-—ftanO and

a*b2c4 tan2 8= (a2 - c2) (b2 - c2) k2z2 - c4 (a2 - k2) (b2 - k9} ,
as given in (514), we get

tan20+(a2-*2)(62-/:2)(c*-*2)

a2b2c2tan28 ' ' ' (*W)

In this equation substituting the value of tan 6, given in terms
of \ in (564), we obtain

!_ (g2- *2) cos2 x + (&2 -

w2~62(a2-F)cos2X + a2(62-F)sin2
Now this may easily be reduced to the form

But it has been already shown in (i) sec. [119] that

e being the eccentricity of the plane elliptic base of the invariable
cone.

w,

Whence

*!=!_(*!_ -)[_ ir-gr-l. .... (572)

rt/z \ /jx / I ^.. ^>*QTn^ A I ^ '

Ui \ U /I_A — c olU /v_l

k2 C k2

Introducing this value of —2 into the equation % = « I -3 d/,

writing for d£ its value as given in (567), and integrating, we shall
obtain the final result,

ac (b2 — k2) C d^

= — Ki + ^f — - 1 . (573)

-bk ^(a*-k2)(b2-c2)J [l-e2sin2\] Vl-s^es^X v

The positive sign to be taken when b > k.

This elliptic integral differs from (553) only in the amplitude.

When the integrals (553) and (573) are complete, the values of
i/r and ^ become identical, as they manifestly ought to be, because
in sec. [122] it was shown that the line of the nodes coincides
with the axis of the centrifugal couple whenever the instantaneous
axis of rotation lies in one of the principal planes of the ellipsoid.

ON TIN: MO i KIN oi' \ KHMD BODY ROUND A FIXED POINT. 221

If we eliminate z and tan# between (511), (541), and
•"'ill), we shall get the following relation between <p and X,

tan<p tanX = sece; ......

dp sin 2<p
hence ^= — r — ^ ; or <p and X have opposite signs.

ClX Sin <wX

But these angles differ in their origin by a right angle, since <p is
measured from the plane of be, while X is measured from that of

TT .

ac ; subtracting <p from — to make their origins coincide, then

«

tan <p= cose tan X;

this formula coincides with that given in (39) .

Now, when the ellipsoid is a figure of revolution (a equal to b,
suppose) , the invariable cone becomes a right cone of revolution,
whence the angles between its focals vanish, or e=0. Therefore
f is always equal to X ; that is, the amplitudes of the functions are
identical throughout their whole extent, as plainly they ought to
be, because in this case the line of the nodes always coincides with
the axis of the centrifugal couple.

when <p=0, X=0; and when <p=^, X=^.

We may repeat here what has been said in sec. [119], that the
expression

ac(b*-k*) C dX

=*)J[l-<

bk V(a2-#2)(62-c2) [l-e2sin2X] Vl-sin2esinsX
may be transformed into this other,
tan/3

cosal

J [1 - e2 sin2 X] V 1 - sin2 e sin2 XJ

which represents, as has' been shown in sec. [8] , an arc of the
spherical conic, supplemental to the invariable spherical ellipse.

The relation between ^ and X is given by the following elliptic
integral,

[(a2-*2)(62-c2)]*v=flC(62~*2) f - ^

M J [l-e2sin2X] Vl-si

_gbcC dX

* J Vl~

128.] We may now determine the angular velocity round the
instantaneous axis of rotation, and the nutation of this axis, in for-
mulse of great simplicity.

222 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Since in (568) the time is given in terms of X, we may reverse
the formula and obtain X a function of t1. (See note, p. 202) .
t1 in this equation is no longer the same numerical quantity as t
in sec. [117] ; for while all the constants in (542) and (568) are
the same, the amplitudes <p and X are different. Accordingly let

j, :j :: t' : t ; hence jtt=jt' ..... (a)
Let X

Then in (186) writing for tan20 its value p2 — 1, we get

1 (fla + c8_ff) (^ + ca_F) (

= -~ --

Let Py and P/; be the greatest and least values of P ; then
1 sin2 \ cos2 X

I f ! I

vu;

or P is a semidiameter of a plane ellipse whose principal semiaxes
are P; and Pw.

If fl and Oy are put for the greatest and least angular velocities,

O — f— fl1 — L- •

** — T) > ^L — -p >

ru ri

we hence get for the angular velocity the very simple expression

m20'/0; • • • (577)

or the angular velocity varies as the perpendicular on a tangent to
a plane ellipse whose principal semiaxes are proportional to H
and fl'.

In the same way writing ® and (H)' for the greatest and least
values of 6, the nutation of the instantaneous axis of rotation from
the axis of the plane of the impressed couple, we obtain

tan2 6 = tan2 fc) cos2 (j, t} + tan2 & sin2 (j, t) . . (578)

This formula may easily be obtained, if we multiply (d) by A2,
subtract 1 from the first number, and cos2\-fsin2X from the
second.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 223

CHAPTER XVII.

ON THE SPIRAL DESCRIBED ON A FIXED CONCENTRIC SPHERE BY THE
INSTANTANEOUS AXIS OF ROTATION OF THE BODY.

129.] If it were possible to eliminate \ from the equations
(5(51) and (576), we should have a direct equation between 6 and
£, the polar spherical coordinates of the curve. We cannot do
this; but still we may perceive that as the equations involve the
angle % simply and no trigonometrical function of it, while 6 is
a periodic function involving sines and cosines of arcs which
increase with the time, the curve must be some sort of spiral
described on the surface of the fixed sphere. But although this
direct elimination is in the general case extremely difficult, perhaps
impossible to effect, we may however be enabled successfully to
investigate some of the more important properties of this spiral in
the general case, and to give its polar equation in a particular case
of rotatory motion.

The spiral, analogous to the herpoloid of Poinsot, has two
asymptotic circles on the surface of the sphere.

The angle r which the vector arc 6 of a spherical curve, drawn
from the origin to any point on the curve, makes with a tangent
at that point, is given by the equation

(579)

This is evident, because the sides of the elementary right-angled
triangle on the surface of the sphere are the element of the arc, the
differential of the vector arc 6, and the distance sin 0dy. at that
point between two consecutive meridians.

We may transform this equation into

. ., dy At
tanr = sm0J.^ ....... (a)

dy k2
Now in (559) it was shown that /r = /e -3, and in (569) that

tit «*62c2tan20

while 563) gives - =_— -- /vvv' whence
Q0 K sin 6 cos 6 V XYZ

_q*62c* tan2 0 + (a2-**) (&«-*«) (c2-*2)

224 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Now,, whatever supposition we make with, respect to the magni-
tude of k, some one of the factors X, Y, Z, in (565), must be
essentially positive, and cannot become cipher. In this case Z is
essentially positive. Making X = 0, and Y = 0, successively, we
get

and

but when X=0, or Y=0, tan T= oo, or r is a right angle; hence,
when 9 has either of these values, the spiral touches one or other
of the circles whose spherical radii are the values of tan 6 given
above.

If we make 9 greater or less than the limiting values just given,
either X or Y will become negative, and the value of tan 9 there-
fore imaginary. We may hence infer that the spiral on the surface
of the sphere is confined between two planes parallel to the plane
of the impressed couple, and that it always undulates between
two parallel small circles of the sphere, having its apsides alter-
nately upon them.

Let Py and Ptl be the greatest and least values of P, the perpen-
dicular from the centre of the ellipsoid of moments on the instan-
taneous tangent plane. The area of the spherical belt or zone,
within which the undulations of the spiral are contained, is equal
to27rA(P,-Py/).

130.] It was shown in sec. [108] that the instantaneous axis of
rotation referred to the principal axes of the body generates a cone
of the second degree. We shall now proceed to establish the fol-
lowing remarkable theorem.

The length of the spiral between any two successive apsides is
constant, and equal to a quadrant of the spherical ellipse generated
by the cone of rotation.

Let <r be the arc of this spiral,

(566) (559) and (569) give us (^) =

_ d/v

also u —£

dt

and

sin2 B

sin2 9[a?b*c'* tan2 9+ (a2

ON THE MOTION OP A RIGID BODY ROUND A FIXED POINT. 225

Making the requisite substitutions in the general formula for the
spherical arc, we shall find

/ d*\ - = K- *in- e ro,- 0 (X-Y-ZV-hK2 sin2 0 f a2AV Uir 6 + (a'-gX**-**)^-**)]'
VW u'i'c-'lan'y

In (5G5) we found

X = /J2c2 tan2 e-(b*- k*) (/c2 - c2) ,
Y = (a2 - /fc2) (/i2 - c2) - <z2c2 tan2 0,
Z = a*W- tan2 0 + (a2 - £2) (b* - A2) .

Substituting these values of X, Y, Z in the preceding formula,
squaring the second member, and adding, we shall find, after some
rather complicated reductions,

We must now reduce this formula to a form suited for integration.
In (564) we made the assumption,

o262c2 tan2 0= (k2 - c2) [bz (a2 - *2) cos2 X + a2 (62 - **) sin2 \] .
Let us continue this assumption : reducing we find

. g0^(*2-c2)[" &2(a2-*2)cos2X + a2(62-£2)sin2X "1

A2 UV + c«^*)~co8« X + a2 (62 -I- c2 - A2) sin2\J ' (<
and

80_ _ «262C2 _

~A2[62(a2 + c2-A2)co2 2222-
Substituting and reducing

__
c2) ~ ~(b*(a* + c2 - A* jTxis12 X + «- (62 + c2 - /c2) sin2\] 2 '

-j- denotes the velocity of the pole of the instantaneous axis of

rotation along the spiral which it describes. We thus have the
velocity of this point given in terms of X. We shall return to this
expression.

To change the independent variable from t to X.

Multiply the last equation by the equivalent expression given in
(567), namely

tc*k* [(a2 - /c2) (A2 - c2) cos2 X + (A2 - A-2) (a2 - c2) sin2 V

VOL. II. 2 G

226 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

and we shall have

d<r

*" A" i" __/____ _J f^ftl^

z2(£2 + <? - #) sinSX]2[(a2 - £2)(62 - c2)cos2 \+ (62 - /42)va2 -c2)sin2A] '

We shall now proceed to show that this expression may be reduced
to an elliptic integral of the third order and circular form. To
simplify the calculations, write

' — (fjli If^ (13. (3\ \

'j'< (582)

(583)

Making these substitutions, dividing by a2£2c2, and taking the square
root, we shall obtain

abc \/£2 — c2

To integrate this equation, assume

Vtan2X=Utan23>

(584)

Introducing the changes arising from this transformation, the
last equation may be reduced to

r (AV-BU) |~ abc\/k*-cz

E> (DU-CV) VAV / rAV-BUVin2^

L'V ^ AV r j

U(AD-CB) I abc>Jk*-c*

C(DU-CV) VAV r /DU-CVN 1 / /AV-BU

W23>

1 ^ QV J.i ^JV - V AV

We have now to compute the values of the coefficients, modulus,
and parameter of this expression.

From the relations established in (582), we get, writing E and F,
for the first and second coefficients,

_ U ( AD - CB) abc VAa -
C(DU-CV)

a (V* - c2) (b* + c2 -

A2)'

(DU-CV) VAV

DU-CV c2(«2-

= a6 / ' (q2-F)(62-F)

" c V (F^2) (a2 - c2) (a2 + c2 - /t2) '

the parameter =

CV

the square ot the modulus =

AV-BU

- -^ —

A V

-g - -g - ^ —

(a2 — c2) (a2 •+ c2 —

(585)

> (586)

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 227

Let us now take the cone described by the instantaneous axis of
rotation, with reference to the principal axes of the body. The
equation is given in (528), namely,

a V - * V + *2 (P ~ *2)2/2 + <?(<? - k*)a* = 0 ;

and we shall find, writing as before a' and (3? for the principal arcs
of the spherical ellipse the intersection of this cone with a concen-
tric sphere, that

C2(F-C2)

-

COS2 «' = --5 - „ ' - ror, COS2 P =

- „ a - ror, 7-5 - vT—5 - - re-.,

— c2) (62 + c2 — A:2) (<r— <r) (a2 + c2— A:2)

• 2 '_ _

~ ' L -

If we write 261 for the angle between the focals of this cone, we
know from (e) sec. [8] that its value, in terms of the principal arcs
of the spherical ellipse, is given by the equation

tan2 e' = COs

COS2 a!

Substituting the particular values of these functions just given,
we obtain

a2 - c2} (b* - A*) («* + c2 - /t2) '

Hence tan2 e' is the parameter.
Let 2r/ be the angle between the circular sections of the same

T, ,. j • /ft\ AT. -9i sin2 «'— sin'2 yS'

cone. It was found m (9) that sin27/=— — —9 . ,

2 '

sin a

or sin rf is the modulus.

Let us compute the value of the first coefficient E.

Making the necessary substitutions, we obtain the resulting
expressions,.

£=

cos/3'

be V (a2 - c2) (62 - A:2) (A2 - c2) (a2 + c2 - A2 ) " cos «' sin a7'
In like manner we find for the second cpeflBcient F,

F=— A/_ ' "(a*-*8)(A2-^) _cosa'_cos/3'

c V a2_«-c2r/2 + c2-/t2~ "sina' '

228 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Making all the substitutions just indicated, (585) may be trans-
formed into

A r • ! cos/3' f
Arc or spiral = — — . I -

cos a. sin a' J

d<3>

[1 + tan2 </ sin2 <X>] VI — sin2 vf sin3*

cosa' cos/3' r d<3;>

sin a'

a^cos/3' f

sin «' J Vl-si

sm2?/sin2(I>

(588)

When the body is one of revolution or a = Z>, a'=/3' and the pre-
ceding expression becomes, Arc of spiral — sin «' <3>, an arc of a
circle, since e'=0 and ?/=0.

It may be shown by comparing (10) with (41) that if there are
two circular elliptic integrals of the third order with positive and
negative parameters, having the same modulus and amplitude, the
parameters being respectively the square of the tangent of the
semi- focal angle, and the square of the eccentricity of the plane
elliptic base of the cone, the expressions are connected by the
following equation : —

cos/3 C d< }

cos « sin a J ~

589)

tan2 e sin2 <p]
cos a cos /3

sn a

3 C d<p

J Y/l— sin2 77 si

tan |3 . , f d<p

= - — - sin 8 \
tana J[l-e2si

sin2 <p

sin2 <p] V 1 — sin2 rj sin2 <p

I tan-iretan6sin(PCOS(P]
L \/l— sin2 17 sin2 <pJ

>l

If now we introduce this relation into the preceding equation
(588), we shall obtain for the final result,

Arc of spiral = ^7 sin B' \

tana' J[i_

6/2 sin2 <£j y ! _ sin2 ^ sin2 <j>

_1 re' tan e1 sin <I> cos 4>~i j
L v/l-s^^'sin^J J

In sec. [7] it was established that the elliptic integral
tan^^of d(P

J n-<

(590)

tana

sin

[1 — e2 sin2 <p] V 1 — sin2 77 sin2 <p

is the value of an arc of the spherical ellipse, the principal angles
of whose generating cone are 2« and 2/3, the angle between whose

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 229

circular sections is 2t], and the eccentricity of whose plane elliptic
is e. And it is shown in (44) that

Fig. 34.

, r e tan e sin <p cos <z> ~i
tan~'

L vl— sm277sm2<pj

is the arc of a great circle touching
the spherical conic, intercepted
between the point of contact and
the foot of the perpendicular arc
from the centre on the tangent
arc.

Make the angle AOD = <p, draw
the arc Dn a secondary to AB, and
through C draw the tangent arc Cr.

The length of the spiral = spherical elliptic arc AC + circular
arc CT.

The length of the spiral between any two successive apsides is

7T

found by taking <E> between the limits 0 and — . At these limits

A

tangent vanishes, and the expression becomes the length of a quadrant
of the ellipse ; hence we obtain this remarkable proposition : —

The length of the spiral, described on a fixed concentric sphere,
between any two of its successive apsides, is equal to a quadrant of
the spherical ellipse, described by the pole of the instantaneous axis
of rotation, on an equal concentric sphere which moves with the
body.

If we turn to the relation assumed in (584) between X and 3>
for the purpose of facilitating the integrations, and substitute for
U and V their values in the equation

Vtan2X=Utan23>,

we shall find tan23> = ^ tan2\, or tan2$=^2 _

or tan <l> = cos e tan X. This result is identical with the expression
found in (39).

But X and the amplitude <p used in the investigations in this and
the foregoing chapter, are connected by the relation established in

(575),

tan <p = cose tan X.

Hence

(591)

131.] Let e, d , e" be the semi- focal angles of the invariable cone,
of the cone of rotation, and of the cone of nutation respectively.

230 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Then

cos*

- g-g 7~2 - 72WA2 -

cos2 p (a2 — A;2) (o2 — c2)

2 , _cos2a' _ 62 (62 - k*) (a? - c2) (a2 + c2 - *2) .
~ cos2 3' ~ a2 a2 - F 62 - c2 62 + c2 - A2 a

cos2e" = — 3-^/7 = i2/ 2 . — 2 — /~2\ frorn (n) sec- [125].
cos2 j8" A2 (a2 + c2 — A;2)

Whence cos e= cose' cose" (592)

Let e" be the eccentricity of the plane base of the cone of nuta-
tion. From (n) sec. [125] we may derive

fan2 «" fan2 /3" Jf^ffi2 7)2\

110 Ldll Ot ^^ Ldll O ft I W ^^ U j

(* z^ — — — ~ — — .

tan2*" £r(«2 — A;2)

But it was shown in (i) sec. [119], that e2=- ^ 2_7-2\ ; whence

e = e", or the plane elliptic base of the cone of nutation is similar
to that of the invariable cone.

132.] When the revolving body is very nearly a sphere, as in the
case of the planetary bodies, a, b, c are very nearly equal. In this
case, the ellipse of rotation is indefinitely greater than the ellipse
of nutation, as may thus be shown :

tan2a' =

, „ - .„ --

tan2«"=v— -4V- — ^ tan^" =-i— ya e * whence

^^ ^^

tana" 62 / (a2 - k9) (62 - £2) tan/3" a2 /(a2-£2)(£2-

tan a'

Now, when a, b, c are very nearly equal, k also must nearly be
equal to one of these quantities ; whence as k approaches in mag-
nitude to one of the axes, the above ratio becomes indefinitely
small.

As the length of one undulation of the spiral depends solely on
the magnitude of the principal arcs of the ellipse of rotation, and
is independent of that of nutation ; it is evident that when the body
approaches in shape to a sphere, several revolutions of the body
must occur between one extreme position of the axis of rotation
and the one immediately following.

When the body is very nearly a sphere, we may approximate to
this number. In this case the ellipses are very nearly circles, and

ON THE MOTION OF A RIGID BODY RODND A FIXED POINT. 231

the number of revolutions n will be the ratio of their circum-
I'rrruces, or

circumference of circle of rotation _ sin «'_tan a'_ N
~ri iv u inference of circle of nutation"" sin a" ~~ tana" L— N '

Q

or. in the usual notation, w= — '—^ nearly, since a = b = k = c

A. — L/

nearly.

133.] On the velocity of the pole of the instantaneous axis of
rotation along the spiral.

The velocity V along the spiral is the value of the expression 3--.
This value has been found, (f) sec. [130], to be, in terms of X,

2

We shall now proceed to find the maximum and minimum values
of V by the ordinary process of differentiation. For this purpose
differentiating equation (c) of sec. [130] and putting the differ-

ential of I -r- ) equal to 0, we shall obtain

0=^.sin0cos0[Q(sin20-cos20)-2Wcos20], . (595)

writing Q for a2

2. r(a2-A2)(62-*2)(c2-/t2n
and W for (a2 + b* + c2 - *2) |1 g«yc« J •

In this equation there are four factors, any one of which, equated
to cipher, would satisfy the equation ; either -r- = 0, sin 6 = 0,

cos 0=0, or Q(sin2 0-cos2 6) -2W cos2 0=0.

We shall now proceed to show that they are all inadmissible
except the first.

We cannot have sin 0=0, or cos 0 = 0; or 0=0, or 0=-;

0

because the magnitude of the angle 0 is confined within certain
limits, given by the equations (557) ; neither can we have
Q(siu20 — cos20) — 2Wcos20 = 0 ; for if we assume the truth of this
supposition, we shall find, writing 0; for 0,

Q-2W 2(Q-W)

tan20/= — ^— , or sec20;=-i-Q -- '-. ... (a)

We must now compute the value of this expression.

232 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Since Q = a2 + 62 + c2 - 2k2, and

we get, after some reductions,

/72/)2^2

ig£(Q- W) = a2£2

- a

; _1_ 9 /72/-2 i O /72A2 1 (h)
-f- <*c* c ~r ^i* w r \u/

Now this expression may be reduced to the symmetrical form

-*2); . . . (c)

^)
~- • (*)

The greatest value of sec Q, which the conditions of the problem
admit, is given by the equation (557),

Let the ratio of these secants be n, we shall find that n is always
greater than 1 : put sec 6f=n sec @,

_ _
sec2®"

or w,^ = 2 —

As the extreme limits of k are a and c, let £2=a2 — a2, £2=e2 + y2,
a and y being positive quantities, which are small when compared
with the axes. This expression may now be written

M2

or w is equal to V2 nearly, since the second term may be neglected.
We have therefore

sec 6 = \/2 sec @,

a value of 6 which cannot be admitted, since © is the maximum
value of B.

1/3

The only remaining factor is ^- • differentiating (564) and

o\ Nil. MoJii'N OK A Kl(ill) BODY HOUND A FIXKD POINT. 233

(I/

= 0, we get — £2(a2 — A2)sin2X=0, an equation \\hirh

is satMied by X=0 or X = ^; but these values of X give 0=&, and

ii

0 = (-)' ; or, the maximum and minimum velocities of the pole of the
mttantaneotu axis of rotation along the spiral are at its greatest or
/<'<ist (/iff lances from the centre of the spiral, as we might indeed
have anticipated.

Taking the second differential of this expression,

-/t2(a2-62)cos2X,

7T

this is negative when X=0, and positive when \ = -~. Therefore

A

IT

the velocity is a maximum whenX=0, and a minimum when X=— .

Z

Or the velocity is least at the inner, and greatest at the outer
apside.

CHAPTER XVIII.

134.] We shall now proceed to determine the curves traced ont
by the poles of the principal axes of the body, during the motion,
on an immovable concentric sphere. We shall first investigate the
curve traced out by the axis c of the ellipsoid, or the C spiral, as
for the sake of brevity it may be named.

Let p be the angle between the pole of the impressed couple and
the pole of the axis c. Then the usual formula gives us

Now, p being the angle between k and the axis c of the ellipsoid,
>=T, sinp= -r , tanp = — — ; hence (y-) =rs

/d<?\2 /«2/t2XY
In (51-0) it was shown that (-rr) = 2,2 4 >

where X=[(i2 — c2)^2 — i
In (r>52) we found -^=

Before \ve proceed further, it is proper to show that the curve

VOL. II. 2 H

234- ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

has two asymptotic circles ; for, r being the inclination of the vector
arc to the curve at the point of contact,

_-<

~~ •••••••

77"

When X=0, or Y=0, we shall have tan r=oo , or T=— . . (c)

A

The radii of the asymptotic circles may be found by making
X=0 and Y=0,

or («2-yt2)c2-(a2-c2)^2=0;') ,„

and (62-c2)*2-(62-A2)c2 = 0.j

Resuming our equations, and making the suggested substitutions
in (a),

a2Z>V / do- V X Y + a262 (c2 - z2)2 .

ic*k* \dt) '" (A2-*2)

This expression, by the help of the preceding relations, becomes

c2(a2+62~A2)~(a2^2"c2)^ • (596)

135.] Let distances a', b1, c? be assumed along the axes of the
ellipsoid a, b, c, and inversely proportional to these axes, so that
aa! =bb' = cc' = hz. Let v, v', v" be the velocities of the extremities

of these lines respectively. Whence ^(-j:) will be the velocity of

the extremity of cj,

It t I U.U V // / ilU V . / 1 1 U \ " C/~~

or v"=d(— ) = _(_); hence U- I =74

Substituting this value in the last equation, and multiplying by
2, we find

<rW (a2 + tf - A2) -f 2a2b*c*z* - (a2 + b* + c2) aWz*.

Writing analogous expressions for the other axes, and introducing
the relations given by the equations of the ellipsoid and sphere, we
shall find, on adding those expressions,

_A2)< ... (a)

ON TIIK MOTION OF A RIGID BODY ROUND A FIXED POINT. 235

\\ r have therefore this theorem : —

If s trail/ lit lines are taken along the three principal axes of the
body from the centre, and inversely proportional to the square roots
of the moments of inertia round these axes, the sum of the squares of
the velocities of their extremities is constant during the motion.

Let segments equal to R measured from the centre be assumed
on the three principal axes of the body, the sum of the areas
described by the projections of these lines on the plane of the
impressed couple varies as the time.

Let Sc be the area described by the projection of a portion of
the axis of c equal to R on the plane of the impressed couple ;
then the projection of R on this plane is R sin p, and the differ-
ential of the area

dSff , -no • o dilr ., N

-j— = £ It* sin2 p -p- (b)

Now sin2/o = — 77T— ,

and J-L=

whence .Jrftl- ......... (c)

Inlikemanner ^=

dS0 dS$
Whence + +

or Sa + SA + Sc=R2/rt + constant ...... (d)

Should the lengths R, instead of being equal, be proportional to
the square roots of the moments of inertia round the corresponding
axes, the sum of the areas described by the projections of those
lines, on the plane of the impressed couple, is still proportional to
the time.

N nc2
Let R2 = ™:=^. W being a constant. Then (b) in the last

article may be changed into the following, ~r-£=/« ^ (c2— z*).
Whence S0 + S6 + SC=^* (a2 + A2 + c2-/t2)f+ constant. . (e)
136.] Let us now resume the general equation, and proceed to

236 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

find the lengths of the spirals traced by the principal axes during
the motion. The equation for the C spiral is, as in (596),

(a2 - k2) (b* -

Assume, as in (541),

(aa-#2)(62-c2) cos2<p+ (62-*2) (a2-c2) sin2<p'
and substitute this value of z in (a) ; we shall then have

~ Y= -

dtJ a2

-Jfcaa«-c* sin«<

and

a

whence

Let

then

,
'

|^!_A

Vd<p/~C

cos2<p + D sin2<p

cos2 9 + B sin
and this expression may be transformed into

/do;\= BC-AD
U<P/~ C(C-D)

A-B

V*2-,

-. (598)

Equations (d) give us

BC-AP= -(aa
C(C-D) (^

A- B (a2 - &2)(a2
A "«2(a2-yt2)

(C-D) VA A /. /A-B\ .
V 1~(~A~/8m

' C-D
C-D a2

_

C ~(A2-c2)(a2-*2)'

ON THE MOTION OF A IUO1D BODY ROUND A FIXED POINT. 2'J7

Nowe/2=— ., ', — -, as in sec. [7] . Substituting the values
of tan a', tan /3' given in (587), we get

e?*=- — — - = — T — ; hence e' is the modulus.

a2 (a2— A2) A

In (b) sec. [115] it was shown that

c« C-D

whence sin2 e is the parameter. Making these substitutions, and
integrating, we obtain the resulting equation,

/c«(A;a-ca) /(a2 + 62-c2) (Aa-*a)\ f
"V aa(aa-Aa)\ (/fc2-c2)(62-c2) /J [l-si

snesin2<p] \/l-e'2sin2<

As sin2e is less than e/2, this elliptic integral is of the third order
and logarithmic form. That it is so, may be shown by constructing

/ *2\
the expression (l+ri) (l-\ — J ; or in this case, in which n— — sin2e

/ e'2 \

a nd i2 = ^, cos2e ( 1 - -=-f- ) = cot2 e (sin2 e - e'2) ;

\ Bra'e/

whence the criterion of sphericity becomes, as in (138),
_ (62 - k*) 2 (a2 - c2) (a2 + 62 - ca)

a2^2_yt2)(/fc2_c2)^__c2) ..... (g)

This is a quantity essentially negative, whatever be the value we
assign to k between its limits a and c. Hence the polar spiral
described during the motion by the least principal axis, may be
rectified by an elliptic integral of the third order and logarithmic
form.

When the ellipsoid is one of revolution, the elliptic integral may
be reduced from the third order to a circular arc. In this case
a = b, since sine=0, e' = 0.

Adding together the coefficients of the integrals, now become
identical, we get

238 ON THE MOTION OF A RIGID BODY HOUND A FIXED POINT.

137.] Multiply equation (599) by the expression

abc

which depends solely on the moments of inertia of the body. Let
» be written for this factor ; then (599) will become

dcp
~

d(P

,. (601)

a26(62-c2) V(«s-*3K**-c8)J [l-sin2esin2<p] V 1 - e'2 sin2 <p

Now e is the focal angle of the invariable cone, and e1 is the
eccentricity of the plane base of the cone of rotation. Let there
be a cone which shall have the same focal lines as the invariable
cone, and a plane elliptic base similar to that of the cone of rota-
tion. Then, ctt and /3; being the principal angles of such a cone, we
shall have, see (19),

tan2«,— tan2/3, ,0 , sin2 a,— sin2/3, . 0

L-^— -?-i = e'2, and- -'-<, 0 — ' = sm2e, . (a)
tan2 a; cos2 /3,

or tan2 «, = — ~ [ tan2 B, = • (b)

whence cos2a/=rirrA2 — 72(7*2 K> . . . . (c)

a8 (a8 -A*) ~(tf

By the help of these relations, if we construct the expression

e'2
- — 3- we shall find it to be equal to the coefficient of the elliptic

integral of the first order in the equation (601). In like manner

e2
if we construct the expression ' cos2 a, we shall obtain the

coefficient of the elliptic integral of the third order in the same
equation. Accordingly (601) may be written,

. «i* r a?

tan/3J VI -e2 si

sn®

. (602)

cos-'

tan/3; M [l-sin2e,sin2<p] vT-

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 239

138.] When the parameter of the elliptic integral of the third
order is negative and less than the square of the modulus, the
function no longer represents any spherical curve of the second
order. It is possible, however, to construct a spherical curve whose
rectification may be effected by an elliptic integral of the third
order, and logarithmic form.

Let us conceive a spherical curve which shall cut all its spherical
vectors in angles whose cosines shall have a given ratio to the sines
of double the angles which the equal central vectors of a certain
spherical ellipse make with the major arc. Let r be this angle, and
p the distance of the point from the centre of the curve. In the
spherical ellipse, of which the principal arcs are a and /3, let this
vector p make with the major arc the angle ^r. Then, by the law
of the generation of the curve,

cos r =j sin >|r cos -fr ....... (a)

Now, as the spherical radii of the ellipse which are equal to a

7T

and ft respectively, make with the major arc angles 0 and ^, at

these distances cos r=0, and the curve has therefore apsides at
these distances from the centre.

To find the length of the curve, we must compare the values of COST.
cos T=^ sin-^r cos ^ (this relation maybe taken as the definition of

the curve) ; and cos T =( ^- ); j '( ,- \ — . g . -- s— . j . . . (b)
\da-J \dp J sm2 ^r cos2 ^

while the equation of the spherical ellipse gives

cot2 p = cot2* cos*^ + cot2/3siu2<\Jr. . . . (c)
Let <p be the eccentric anomaly, as in (c) sec. [8] ; then

tan>/r=- — -tan<p: (d)*

tana

tan2 ft sin2 <p

whence sm8>lr= — 5 — — » ~ . 9 ,

tan* « cos* <p + tan* /o sin* <p

.. . . (e)
tan2" a cos2<p

COS v" ~~ : .

tan2 « cos2 <p + tan2 ft sin2 <p* y
Substituting these values of sin ^r, cos >/r in (b), we find

.g /do-\2 _ [tan2 a cos2 ft 4- tan2 ft sin2 <p2] ....

\dp/ " tan2 a tan* $ sin2 <p cos2 <p

* The eccentric anomaly <p in (c) sec. [8] is not the same angle as 0 in (d)
sec. [7].

240 ON THE MOTION OP A RIGID BODY ROI7XD A FIXED POINT.

Again, as tan2p=tan2« cos2 <p + tan2 /3 sin2<f>, . . . (g)

/dp \2 (tan2 « - tan2 /3)2 sin2 p cos2 <p
differentiating, (^J = __ ___

whence, as7'=y.

•a f d<7 V— (tan* a~ tan2 ff)2 [tan2 g cos2 (p + tan2 ft sin2 <p]
^ Up/ = tan2 « tan2 /3 [sec2 a cos2 <p + sec2 #~sin2 <p] 2'

tan2 «— tan2 /3 0 sec2 a — sec2 /3 . 2 ,.»

l\ow _ -- — f" — sin e •

JL^ wff o - C . Q - Oil! C • • • VII

tan2 a sec2 «

making these substitutions, reducing and taking the square root,
the transformed equation becomes

=
~

tan/3j Vl-e2sin2<p

e2 cos2 a f d(p

tan/3 J [i_ sin2 e sin2 p] \/l — e2sin2<

As e2 > sin9 e, this is an elliptic integral of the third order and
logarithmic form.

Now, if we compare this formula with (602), we shall find them
identical, — whence we may infer that the length of the spiral
described by the pole of the greatest or the least axis of the ellip-
soid on a fixed sphere (the semidiameter k being the next in the
order of magnitude to such greatest or least axis] will be equal to
the length of the curve there defined as generated on the surface
of a sphere according to a given law.

139.] On the spiral described by the pole of the greater principal
axis, or the A spiral.

In the general equation (596) substitute x for z, and interchange
a and c ; we shall then have

'\2

In (546) we found

(a2 - A;2) (62 - c2) cos2 <p + (62 - /fc2) (a2 - c2) sin2 <p'
Substituting this value of x* in the preceding equation, and

o\ mi1: MimoN or A iti<;ii> uonv uorsi) A KIXKD HOINT. X? 1 1

introducing the value of I-,-) given in (511*), we shall obtain the

\<*f/
resulting equation

a a-

This expression may be reduced in the same way as (597),
omitting the steps for the sake of brevity. The resulting equation
will be found as follows : —

a a -

] * f df

^] J vi=s?2M

an elliptic integral which is also of the third order and logarithmic
form.

The parameter is the square of the sine of the semifocal angle of
the invariable cone, while the modulus is the sine of the major
principal arc of the cone of rotation.

\Yhen a = b, sine = 0, and the above expression assumes the
form,

(605)

' I — T — ^ i

J Vl-s^a'sin2^

In (58) it was shown that cos a' I — T — ^ is the alge-

braical representative of 'an arc of the spherical parabola whose
major principal arc a, is given by the equation

. , 1 + COS «' 1 , a' 7T

tan*«,= — ,= ,; whence «+9=-,

1 — cos «' „ «' * 2

tan2 —

or «' and 2«/ are supplemental.

140.] On the spiral described by the mean axis b of the ellipsoid,
or the mean or B spiral.

In the general equation (596), interchanging b and c, also y
and z, we obtain the result

• • (a)
ii. 2 i

242 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

For z/2 substitute its value given in (546) ,

__ _ _

- (a« _ kz) (62 - c2) cos2 <p + (62 - A2) (a2 - c2) sin2 <p'

Introducing the value of f-r-J found in (541*), we shall obtain

.

v/62^2 \ dip / (a2 - F) (62 - c2) cos2<p + (62 - A2) (a2 - c2) sin2<p '

Let A = a2 (a2 - it8) , C = a2 -

B

= a2 (a2 - it8) , C = (a2 - *2) (62 - c2U

= (a2- c2) (a2 + c9 -A2) , D = (F- k*) (a2 -c-2) ,j

and the preceding equation may be written

1 _ idff"\ _ A cos2(p + B si
62 — yt2 \ d<p / ~

d<p / ~ C cos2<p + D sin2<p

^ as B > A, this equation may be transformed into

_
'

b Vi2-**

D(C-D)

(B-A)
C-D

K (606)

If we now compute the value of the coefficients in this equation
by the help of (c), we shall find, 2e being the focal angle of the
invariable cone, as shown in (b) sec. [115],

c2(/t2 _ c2)

^
-k*}~

' & being the lesser PrinciPal

~B~ (a2 - c2) (a2 + c2 - k*)

angle of the cone of rotation as in (587). We have also

and

BC-AD_(«2-yfc2) (flg + c8-.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 243

Making those substitutions, (606) becomes
b (a2 - A2) (g* + c*-b*)

a" =

[l + tan2ecos?p] Vl-s

c*b »JW^P f d<p

.-sm2/3'cos2<p ;

As the parameter tanae is positive, the elliptic integral of the
third order is of the circular form.

When a = b, tane-=0 and the elliptic integral of the third order
in the preceding equation is reduced to the first. Adding the
above expressions together, and reducing,

/a2 + c2-*2\ ,f

a" = I , — Y^- ) cos a! \

\ a2- A2 / ) Vl-si

dip

sin2a' cos2 <p

(0

This expression agrees with the one found for the greater spiral,
differing from it only in the amplitude, which is complementary.

We shall now proceed to eliminate from the preceding equation
the interal of the first order.

_

Multiply this equation by the factor A /(a*~ ^2)(6g-gg).

V 62(a2-fc2 — b2)

Let as before <xt and ft, be the principal semiangles of a cone
whose focals shall coincide with those of the invariable cone, and
the planes of whose circular sections shall make the angles /3' with
the internal axis; then, assuming the equations established in
sec. [8] and (e), we shall have

tan^-tan2/?, (a2-62) (*2-c2)

-L—— —-' = tan2 6,— tan2e= 7-5 - 57775 — 7* >
sec2/^ (a2— c2)(62— A2)

and

sin2 a,— sin2 8, . „ c2(A;2 — c2)

- ?— 5 — -^=:8ln277.=sin8/S'=7-i - ,.\ ., , o — 7s7>
sm2^ (a2 — ^(a^ + c2 — A2)

as in (587) ; whence, making the substitutions indicated,
(a*

tan *'=-

by the help of these equations we may show that

cos/3, _ (ag-A*)y(62-c2Ka* + c*-Aa) ^ ^ „ }

cos «; sinu, v/ (a2 - c2 ) (a2 - b9) (b* - A*) (a2 + c8 - A*) '
and

*

24 i ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

Whence (600) may now be written
aa — 6S)(62 — c8)-|*H CQS£ * d

f_\ /I — -il _— . !__/

— 62) J cos^sinaj [l+tan2ecos2<p] , .

}. . (608)
cos /3; cos «/ /* cli

/l_ sin

sma/ J \71 — sin2 77, cos'

If now we turn to (41) and (47), in which elliptic integrals are
compared, having the same amplitude, but positive and negative
parameters respectively, we shall find them identical with the pre-
ceding equation, which may now therefore be written

c* -i

> ,, tan/3, . [ dtp
<r' = — — sm5,l

tan «, r'J [1 _ e,2cosV! Vl — sin277.cos2<z>

L. / T J ' // /£$r\C\\

}. . (609)
_, ret tan et sin <p cos <p~|

L A/1— ' ' 2 2 -I

If we take the complete function, the circular arc vanishes. We
may therefore conclude that the length of the mean or B spiral, or
of the spiral described by the pole of the mean axis b of the ellipsoid,
between any two of its asymptotic positions, is equal to a quadrant
of a spherical ellipse. The cone of which this spherical ellipse is
the base, may with ease be determined. It must have the same
focal lines as the invariable cone ; and its minor principal arc is
the angle between the cyclic diameters of the ellipsoid ; for the
cyclic semidiameter whose square is a2 + c2— V2 makes with the axis
c an angle the square of whose tangent is

-^ATT; =r . Ill (s) W6 f OUnd

or 2/3, is the angle between the cyclic diameters of the ellipsoid.

We have thus investigated the equations of the spirals described
on a fixed concentric sphere by the three principal axes of a body,
which we have named the greater, mean, and lesser, or the A, B,
and C spirals. It is not a little remarkable that the rectification
of the greater and lesser spirals must be effected by elliptic inte-
grals of the third order and logarithmic form, while the rectification
of the mean spiral depends on an elliptic integral of the third order
and circular form. It will moreover be evident, on referring to
the preceding sections, that the elliptic integrals which express the
lengths of the spirals described by the instantaneous axis of rota-
tion and the mean principal axis of the body have the same ampli-
tude, and are each of the circular form ; while the integrals which
determine the spirals described by the greatest and the least prin-

us i HI: MOTION or A KKJID BODY HOUND A FIXKD POINT.

ripal axes of the body also have the same amplitude, which is com-
plementary to the former, and are of the logarithmic form.

ML] We may determine the maximum and minimum velocities
\\ ith \\ hieh the poles of the priiu-ipal axes of the body describe their
rr^pet -tivc spirals on the fixed concentric sphere. Resuming the
equation of the spirals traced by the principal axes,

differentiating and putting the differential equal to cipher, we get

1 ./*/ 2

It \vas shown in (515) that = — — g

This is =0 whenever the position of the axis k renders # = 0
or y = 0 ; and as k is at its greatest or least distance from the axis
c of the ellipsoid whenever it lies in one of the principal planes,
the velocity of the pole of c on the spiral is the greatest or the
least whenever the axis c is at its greatest or least distance from
the axis k.

The same proof may be applied to determine the extreme velo-
cities of the poles of a and b.

CHAPTER XIX.

142.] There are two particular cases of the general problem
•which require separate investigations — when the plane of the
impressed couple is at right angles to, or coincides with, the plane
of one of the circular sections of the ellipsoid of moments.

We shall first take the case when the plane of the impressed
couple is at right angles to the plane of one of the circular sections
of the ellipsoid, or k = b. If we introduce this value of k into the
equation of the invariable cone in (527), we shall obtain the follow-
ing equation :—

This expression is the equation of the two plane circular sections
of the ellipsoid which intersect in the mean axis b. If, then, to fix
our ideas, we conceive the plane of the impressed couple to be
horizontal, one of the circular sections of the ellipsoid will be ver-
tical during the motion.

To determine in this case the locus of the instantaneous axis of

246 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

rotation in the body. If we write b for k in the equation of the
cone of rotation (528), we get

a2(a2-6V2 + c2(c2-62)22 = 0, .... (b)

the equation of two plane sections of the ellipsoid passing
through the mean axis, and perpendicular to the umbilical dia-
meters of the ellipsoid.

We may perceive therefore that the axis of the impressed couple,
and the instantaneous axis of rotation, describe planes in the body
daring the motion.

To find the greatest elongation of the axis of rotation from the
axis k. This is nothing more than to find the angle which a per-
pendicular from the centre, on a tangent passing through the vertex
of k or b, makes with it, in an ellipse whose semiaxes are a and c.
Now, h being the conjugate diameter to k or b} and P the perpen-
dicular on the tangent,

7i2 -+. #2 _ aa _j_ ca^ an(j p^ _ ac Let this angle be 3.
Then tan*9==(f^!Kt!=£!W. . . . (c)

To determine the time.

In the general equation (540) let k = b, and we shall find

d* fz V62 - c2
Assume (a2-c2)z2 = c2(a2-&2)sin2<p, . . . . (e)

in which <p is the angle between k and the mean axis of the ellip-
soid, measured on a circular section of the surface. By this trans-
formation, equation (a) may be changed into

dt a

-c2)Un<p)

d<p * \/«2-
It was shown in (c) that tan .& is the maximum value of tan 0.

Hence

j=Kw; the preceding equation, when integrated, will become,
putting C for the constant,

(g)
To determine the value of this constant. Let 8 be the initial

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 247

distance of the pole of k from the axis 6, at the beginning of the

t

motion; then 0 = log tan 75 + C. Subtracting we shall have

li

tan

(h)

Let tan-=m, and the last equation may be written

A

sj = KW, tan ^ = we* ', . (610)

IV

e being the base of the Neperian logarithms.

When B is absolutely equal to 0, m also is equal to 0, and <p is 0,
however large the value we may assign to the time t. But when
B is only very small, m will be a very small quantity, and therefore
t must be very large before its magnitude can have any appreciable
effect on the magnitude of <p. Hence the pole of k will diverge
slowly from the mean axis b. When the initial distance B is sup-
posed to be considerable, then m is no longer an indefinitely small
quantity, and a small increase in t will produce a considerable
effect in the magnitude of <p.

Again, let the axis of the impressed couple, by the motion of
the semicircular section passing through it, be approximated to
indefinitely, by the prolongation of the principal axis b, within a
very small angle B'.

Let T be the future time at which the prolongation of the axis b
shall arrive within a certain small angle 8' of k. Then p = TT— B',

and /T = log tan (-9 — 77) +C. As the initial distance of b from k
must be supposed as before to be B,

0=logtan (!)+C, whence -/T = log [t

(S\
2 ) as before ; then

tan tan

rotanf=e-/r ....... (611)

20

In this equation T will be infinite on two suppositions, either

8'
m = 0, or tan iT=0. The former shows that T will be infinite if b

never departs from coincidence with the axis of the impressed

248 ON THE MOTION OF A RIOID BODY ROUND A FIXED POINT.

couple. From the second we may infer that b never can, having
once departed from coincidence with k, again coincide with it.

We may therefore infer that the motion of k in the body will be
as follows. When the coincidence of k with the mean axis is dis-
turbed, and the disturbance takes place along one or other of the
circular sections of the ellipsoid, the axis b at first diverges very
slowly from k, then with greater rapidity until this velocity
reaches a maximum state. The velocity then decreases, so that b,
with a motion continually retarded, approaches indefinitely near
to, without ever absolutely reaching, the axis of the impressed
couple.

143.] To find the value of 6 the angle between the axis of rota-
tion and the axis of the plane of the impressed couple.

In (514) writing b for k, and c2(a2 — 62)sin2<p for (a2 — c2)^2, we
obtain tan 6 = w sin <p . Hence B varies from its inferior limit to •&

as <p varies from 8 to — .

tu

It was shown in (510) that the velocity of the pole of the plane
of the impressed couple along the invariable conic was /tan 6.

Writing V for this velocity, V = btcw sin <p ...... (611*)

/ £2_p2

As tan# = wsin<p, w = ^, tan20 = — ^— , w being the angular

velocity, whence o>2 = «:2[l +w>2sin2<p], or &> = /csec#. . . (612)

To determine the angle ty which the line of the nodes makes
with a fixed line in the plane of the impressed couple.
Resuming the equation (552), putting b for k, and

C2(a2-62)sin2<p for *2(a2-c2), as in (e) sec. [142], we get

2-£2)

'

in2*? c2(«

— a • W Tlting tan'1 77 TOr -3^-75

2 *

TT, - on - —

bz— z* l+tan2?7cos

which represents the tangent of half the dihedral angle between the
circular sections of the ellipsoid, or half the angle between the
cyclic axes. We also have

T-= - = — , as in (f) sec. [142].
d<p KW sin <p

Making these substitutions in the equation (552),

/*2-c2\f *2d*
>/r= -Kt + K—- _, we find

— -^r = Kt + tan"1 [tan 77 ccs <p] + constant. . (613)
To determine this constant.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 2 W

At the beginning of the motion let the axis of the plane of the
impressed couple very nearly coincide with the mean axis of the
ellipsoid. Then p is very small, and cos <p very nearly equal to 1 :
we thus get 0 = tan-'(tan 77) -f C, or C=— 77; hence

cos<p) — 77. . . (614)

The limits of <p are 0 and TT, between which limits the pole of the
impressed couple lies during the motion. Now when <p = 0,
cos<p=l, and tan"1 (tan 77 cos <p)= 77. When <p=7r, cos<p= — 1,
and tan~' {tan 77 x — 1}=— 77. Whence

(614*)

writing T for the period in which the semicircle is described
by k.

Thus we perceive that the infinite angle -fy is made up of two
parts, one of which increases as the time, while the other continually
approximates to a fixed limit 2i), 2rj being the angle between the
cyclic axes of the surface. .

The geometrical interpretation of this formula it is not difficult
to discover. In sec. [119] it was shown that the angle ^ was
made up of two parts, one of which id increases as the time, while
the other may be represented by an arc of the spherical ellipse,
generated by the cone supplemental to the invariable cone. As
the circular sections of this latter coincide in direction with the
circular sections of the ellipsoid, the cyclic axes of this latter surface
will coincide with the focals of the supplemental cone. Hence, as
before mentioned, the whole motion of the body may be repre-
sented by conceiving this supplemental cone to roll without sliding
on the plane of the impressed couple, while this plane revolves
uniformly round its axis. But when the plane, as in this case,
passes through one of thfc cyclic axes of the ellipsoid, this supple-
mental cone degenerates into a plane sector of a circle, the angle
between whose bounding diameters is 2rj. Now, when the plane
of the impressed couple is slightly disturbed from coincidence with
the plane of this circular sector (for when k coincides with b, the
plane of the impressed couple coincides with the principal plane *ac,
which contains the cyclic axes), it will revolve round a straight
line (one of the cyclic axes bounding the circular sector) instead
of rolling upon a conical surface; and this straight line (the cyclic
axis of the ellipsoid, or the focal of the rolling cone) becomes, in
the ultimate state of this cone, the edge of the circular sector.
The plane of an, being disturbed from a state of coincidence with
the plane of the impressed couple, will revolve round one of the
cyclic axes until it approximates indefinitely on its other side to
this plane.

Now if, instead of the cone, we imagine the sector of the circle

VOL. II. 2 K

250 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

to revolve upon the plane, the line of contact with the plane will
no longer advance continuously upon this plane, but per saltum,
starting forward through an angle 2tj at each half-revolution ; so
that if we imagine a number of semirevolutions to occur, the line
of contact of this sector with the plane would advance through the
angles 2tj, 4*?], &c. From the nature of this motion, however, we
can have but half a revolution, and even that only as a limit. It
follows, therefore, that when half the semicircle is completed, or
when the axis of the plane of the impressed couple comes into the
plane of ac, an angle 77 must at once be added to the angle i/r, or
that the line of the nodes starts forward through the angle 77.

144.] We shall now investigate the nature of the spiral described
by the pole of the instantaneous axis of rotation in the case when
k = b.

The spherical polar coordinates of this spiral are 6 and %.

They are connected as follows : —

CAt
In general X=K^\~^> as snown m (560) : put b for k in the

*«J ^
equation (571) which determines u, and we shall have u=b; hence

X = Kt ......... (a)

This equation shows that the motion of the radius vector arc 0
is uniform, being proportional to the time.

It was shown in (610) that ta,u^=mQKWt : writing ^ for ict, we

"A

gettan^=me% and tan0=w>sin<p. "....... (b)

2

These are the equations of the spiral. We must eliminate <p
from these equations.

2 tan |
As sin <p = 2 sin £ cos ^ = 2 tan| cos2^=— — , we get

A Z A & „ <£

l+tan2|
. 2mwew* a 2w

tan e= «** or tan *"-« «-• (615)

a relation between the variables Q and ^, consequently the equation
of the spiral.

145.] The rhumb line may be denned as the curve described on the
surface of a sphere which cuts all the meridians in a given angle.
Let this constant angle be the complement of <&, then its cotangent
is w, <p and % being the polar spherical coordinates of the curve ;
therefore

d<p ,, x

w sm <f> = y1- ........ (a)

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. .'.'."> 1

This is the equation of the rhumb line.

Taking the integral of this equation, log tan ^=1

Let the value of <p be & when %=0. Then log tan - = C,

9

and tan-=m; hence
A

tan

or tan = we1"*-
A

• (616)

This is the usual equation of the rhumb line, and is identical with
(610). Hence the polar spiral is a sort of curtated rhumb line.
If a rhumb line be described on the surface of the sphere, its ordi-

nate angle being (-„— SJ, and if we shorten its spherical central

vectors <p in the constant ratio given by the equation tan 6 = tan •& sin <p,
the extremity of 6 will describe the polar spiral.

Another construction exhibiting the relation between these
spirals may be given.

Let a concentric sphere be described, whose radius OA =tan-& = w.
On this sphere let a rhumb line be constructed, having its pole at A
in the axis of z. Let this rhumb line be orthogonally projected on
the tangent plane to the sphere whose radius is 1, parallel to the
plane of xy. Now, if this plane curve be considered as the gnomonic
projection (i. e. the eye being
supposed at the centre) ' of a
spherical curve described on
the surface of the outer sphere,
this latter curve will be the
polar spiral, or Q and D are
corresponding points.

This we may thus show. In
this construction we always
have tan 0 = tan •& sin <p.

Now CB = Qn, CB = tan0,

and Qw = tan •& sin <p. Q, and D are therefore the corresponding
points of the rhumb line and of the polar spiral, whose vector arcs
are CD = 0, AQ=p.

It is evident that the polar spiral has an asymptotic circle, whose
radius is sin 3. In the vicinity of the pole, the polar spiral approxi-
mates indefinitelv to the rhumb line.

Fig. 35.

252 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

146.] To find the length of this spiral from the pole to the
asymptotic circle.

/do-\2

w ,

dd w cos tp d^ At _

in2 <p d/ d<p K

and sin2# =

; sin2 <p

. •
sin2 <p

Introducing these relations, we get
do

dividing by cos2<p, and integrating, we shall find

<r=tan-1( Vl+^2tan<p) ..... (617)

7T 7T

When <p = 0, <r=0, and when <p=-s, <r = 'o-

~ ^

We thus find that the length of the polar spiral between the pole
and the asymptotic circle is equal to a quadrant of a great circle of
a sphere, — a result in strict accordance with the more general
theorem established in sec. [130].

rJT

-

When \\ + w^t3m<p = l} or tan<p = cos$, o- = tan-1(l) or <r = -7-

4

147.] To determine the velocity of the pole along the spiral.

A w-

~dt) \d</ \dt

, — o .. o — nr= — — dV~ - =K (l+w 2) sm2 ^ cos2 ^ :
{ 1 + w2 sin2 <p}2 sec4 6

~ — • oa -XT- , .

or V= — 1__ - sm2^, or V=i -- ^-, since tt7 =

*

It may be shown that when k coincides with the greatest or
the least principal axes of the body, the spirals described by the
two other axes are equivalent to circular arcs. But when k coin-
cides with b the mean axis, the lengths of the spirals described by
the greatest and the least principal axes are given by logarithms.
Omitting the investigations (which, though somewhat complicated,
the reader, assuming the principles established in the foregoing

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 253

pages, may supply), the final result will be found as follows —

4

pv = q log tan £ + log (1 +gsec^))p and q being constants.

it

148.] When the plane of the impressed moment coincides with
the plane of one of the circular sections of the ellipsoid of moments,
the elliptic integrals which determine the motion may be reduced
from the third order to the first,

In this case 2k is the cyclic axis of the ellipsoid, or the diameter
perpendicular to the plane of one of its circular sections.

Accordingly yo=-p — i5 + -o. Substitute this value of k in (i)
J K* cr o* c2

and (j) sec. [119], and (553). Reducing, we shall have

b*

-1 / cV2-£2

v x -?(?=?

68

This integral, as the parameter is equal to the modulus, may be
reduced to the first order as follows : —

Let 7 as in sec. [20] be the parametral angle of the spherical

1 — siny c2(a2 — 62) , . , ir ,

parabola. Assume ^ — : — -= 0/,0 s(=tan2n, 77 being half the

1+smy cr(o2— c2)

angle between the circular sections of the ellipsoid. Whence

The preceding equation' may now be written

2sm7 f* d<p

1+sinylr /l-sin7\ . Ij^ f. /l-sin7\« .

1— (- — -. — ' Ism2<p \/ 1— I- : — f|tm9

JL Vl+smy/ r J V \l+sm7/

or, as it may be more succinctly written,

= (1 -tan2 17)

f

J [1—

[1 — tan2»;8m2<p] Vl — tan4i;sin2<p

If we compare (619) with (62), we shall find that the second
member is equivalent to the following elliptic integral of the first
order,

d^, _, r sin 7 tan/x

— cds2 7 sin2 fj, L v/l — CO

s

254 ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT.

the amplitudes <f> and fj, being connected by Lagrange's for-
mula, tan (<p — /i)=sin7 tan /A, as in (63), or, as it may in this case
be written,

tan<p =

^5-15 Sm A*

(c)

Should we require to reduce the integrals of the third and first
order of the same amplitude, equation (58) will enable us with
ease to do so, by assuming the theorem established in that equa-
tion,

,_ siny f
1 4- sin y

)\A-(i

-f i tan-1

/2 — siny

L . (620)

\l+siny/ '

Hence ^ depends on an integral of the first order, — the theorem
it was proposed to establish.

Again, if we substitute the foregoing value of k in (542), which
connects the time with the amplitude q>, on which immediately
depends the position of the axis k in the body at the end of the
given time, we shall have

b*

Kt —

, 2 siny \c2

and as ^ r-^-=

1 + sin y

sin 7

dip

,
1 —

1 + sin y

f dp

\ / /l-sin7
J V U + sin7

(621)

.
sm2 <p

But this elliptic integral, as shown in sec. [25], is the expression
for an arc of a spherical parabola whose parametral angle is 7, the
centre being the pole. In this case the two elliptic functions which
determine the motion are represented by arcs of the same spherical
parabola.

ON THE MOTION OF A RIGID BODY ROUND A FIXED POINT. 255

We may eliminate the latter integral by the equation established
in sec. [25], and the last equation will now become

~i !_n

o * i o i o i I >*
fl 6 LT _l 1^

tCt^^dr 7~^ ^— ^ I * •

V1-

The moduli are two successive terms of Lagrange's modular scale.
149.] Thus have we shown in the foregoing investigations how
the properties of elliptic integrals applied to the theory of the
motion of a rigid body round a fixed point have led us to a complete
solution of this celebrated problem, a solution which has enabled us
to place before our eyes, so to speak, the very actual motion of the
revolving body. Yet it is not on such grounds solely that this
treatise has been published. Were the investigations of no other
use than to give strength and clearness to vague and obscure
notions on this confessedly most difficult subject, enough had been
already accomplished by the celebrated geometer whose name is so
deservedly associated with this theory. It is as a method of inves-
tigation that it must rest its claims to the notice of mathematicians,
as a means of giving simple and elegant interpretations of those
definite integrals on the evaluation of which the dynamical state
of a body at any epoch can alone be ascertained. If the author
has to any degree succeeded in accomplishing this, it is because he
has drawn largely upon the properties of lines and surfaces of the
second order, and of those curve lines in which these surfaces
intersect. If he has been enabled to advance any thing new, it is
owing solely to the somewhat unfrequented path he has pursued.
That it was antecedently probable such might lead to undiscovered
truths, no one conversant with the applications of mathematical
conceptions to the discussions of those sciences will deny. To
introduce auxiliary surfaces into the discussions and investigations
of physical science is an idea as luminous as it has been successful.

A TREATISE

o.v

THE HIGHEE GEOMETRY,

AND ON

COMICS

CONSIDERED AS SECTIONS OF A
RIGHT CIRCULAR CONE.

"Les sections coniques offrent une source intarissable de proprietes, et Ton ne pent
dire sans temerit^ que cette matiere est 6puisee."— QUETELET, Correstpondance Mathi-
matique et Physique, torn. i. p. 162.

ON THE HIGHER GEOMETRY. 257

CHAPTER XX.

ON TRANSVERSALS.

150.J If through any point O in the plane of a triangle ABC
transversals are drawn from the vertices A, B, C, and meet the
opposite sides in the points A,, B,, C,, the continued products of the
alternate segments of the sides are equal.

Fig. 1.

G C D

Through one of the vertices C let a parallel DG to the opposite
side AB be drawn, and let the transversals AA,, BB, meet it in the
points D, G. Then, by similar triangles, we have

AC,: BC,=DC : CG,
BA, : CA,= AB : DC,
CB, : AB,= CG : AB.

Compounding these proportions together, we shall have
AC, . BA, . CB,= AB, . BC, . C A,.

This product is a maximum when O is the centroid (that is,
the centre of gravity) of the triangle.

151.] If the sides of a triangle ABC are cut by a transversal
C^B,, it divides the sides into segments such that the continued
product of these alternate segments are equal.

Through one of the vertices B draw the straight line BD parallel
to the opposite side AC. Then, by similar triangles, we have

AB,:AC,= BD:BC,, CA, : BA,=CB,: BD, BC,:CB,=BC,: CB,.

VOL. II. 2 L

258 ON THE HIGHER GEOMETRY.

Compounding these proportions together, we shall have
AB, . BC, . CA,= AC, . BA, . CB;.

These propositions are of very wide and important application.
Thus, from the former proposition (150) it immediately follows (a)
that the bisectors of the angles of a triangle meet in a point, (0)
that the bisectors of the sides of a triangle meet in a point'35', (7)
that the lines drawn from the vertices to the points of contact of
the inscribed circle meet in a point, (8) and that the perpendiculars
from the vertices on the opposite sides meet in a point.

If an odd number of the points A;, B,, C/ (that is, either one or
three) are on the sides between the angles, transversals drawn from
the vertices will meet in a point; but if an even number (that is,
either two or none) are so found, then the three points A, B, C will
range in a straight line.

152.] Through a point O in the plane of a triangle ABC, let
straight lines be drawn from the vertices A, B, C, meeting the opposite
sides in the points A,, By, C, ; then we shall have

AO BO CO OA OB OC

AA + BB+ CC-^ ' AA + BB+ CC-

We have manifestly, see fig. 1,

AO + OA, BO + OB, ^

AA, BB, CC,

But this may be written

BO m OA OB ocy_

AA, BB, CC, AA, BB/ CC,~

* These bisectors (J3) are called by French geometers median lines, a term
which we shall adopt and make use of hereafter.

ON Till: IIICI1KK r.KO.MKTKY.

259

Let A denote the area of the triangle, and 8t, 8tl, Bllt the areas of
the component triangles whose vertices are at O, and whose bases
are BC, AC, and AB. Then A = S/ + S//+S///.

Hut

/~\ A £ /"MJ £ f\C* S:

JAj_c, !3*u '^i-^iii
'A' BB^A' CC;~A'

Hence we have

and therefore

OA OB OC,

"T™ /~t y^« "™ " "™ J- •

AO BO CO

AA, BB, CO,

When the point O is assumed outside the triangle, the above
proportions still hold good, but one of the component triangles
must then be taken with a negative sign.

153.] In any triangle ABC, if lines be drawn from the vertices
through a point O to the opposite sides, making with the sides at the
vertices A, B, C the angles a, a, ; /3; /3, ; y, y,, then the following rela-
tion will hold good : —

sina sin/3 siny=sina/sm/3/ siii7/ (a)

The triangles BAA, CAA, are as their bases BAP CAr But
twice the triangle BAA;=BA . AA;sin«, and twice the triangle

^BA, : CAr

CAA, =
Hence we have

,, or BA . AAysina : CA.

sin_«=CA> BA
sina/ BA CA,

(b)

In like manner

.
BC

, und =•>

CA EC

260 ON THE HIGHER GEOMETRY.

Multiplying these expressions together, we obtain

sin a sin B sin 7 BA, .CB..AC.

— • — -n — = — = TTT-^ — * ' ' = 1 by sec. 150 . (c)
sin «, sm /3, sin 7, CAy . ABy . BC,

154.] From the vertices A, B, C of a triangle, pairs of lines are
drawn, making with the adjacent sides equal angles a, a; ft, (3 ; y, 7.
If the first set of three lines pass through a point O, the second set
will also meet in a point Q.

Let the angles which the lines AO, BO, CO (fig. 4) make at the
vertices A, B, C with the adjacent sides be a, A— a ; ft, B— ft; 7,
C — 7; then the angles which the second set of lines makes at the
same vertices will be A— a, a; B— ft, ft; 0—7,7. Now, since
the first set of lines pass through a fixed point O, we shall have, by
sec. [153],

sinasin/3sin7 _ , ,

sin (A - a) sin (B - 18) sin (C - 7) ~

and we must therefore have

sin(A — «)sin(B — /3)sin(C— y) _, > .,.

sin a sin /3 sin 7

hence the second set of lines must pass through a fixed point Q.

cv

Hence it obviously follows that since the perpendicular drawn
on the opposite side from any vertex of a triangle, and the diameter
of the circumscribing circle passing through this vertex, make equal

ON THK HIOHBH OEOMKTRY.

'Mil

angles with the adjacent sides, if one set of these lines pass through
a point, the other set will also pass through a point. But the dia-
meters of the circumscribing circle drawn through the three angles
of the triangle concur in a point, the centre ; hence also the three
perpendiculars meet in a point. This point, which is of constant
occurrence in the higher geometry, has been called by some geo-
meters the orthocentre. We shall henceforth adopt this term.
The triangle formed by joining the feet of these perpendiculars
may be appropriately called the orthocentric triangle.

155.] Three lines meeting in a point O are drawn from the vertices
of a triangle ABC, and produced to meet the opposite sides in the
points Ay, B/? Cy. The sides of the triangle AyByC, will meet the
sides of the triangle ABC in three points which range in a straight
line.

Through A draw the straight line Aa parallel to the side BC of
the triangle. Let the corresponding sides of the two triangles
ABC, A;B,C, meet in the points L, M, N. Then L, M, N will
range in a straight line.

Fig. 5.

In the two triangles LAa and LBAy, we have

LA : LB = Aa : BA,, and Aa : CA,=AB, : CB,.

Hence LA_CA,.AB,

LB~BArCB;

MB BC..AB. , N CA..BC,

In like manner ==.ff.= .' * and Nfnr=T.A Ar/-

MC ACy.CB, NA BAy.AC,

Multiplying these expressions together, we obtain
LA.MB.NC rCA.AB.BCn •

262

ON THE HIGHER GEOMETRY.

But as the three lines meet in a point, the expression between
., , . . . ., LA.MB.NC .

the brackets is equal to unity; hence i^r> TV/TO ISJA= ' a tnere~

fore, by sec. [151], L, M, N range in a straight line.

When the point in which the three lines concur lies outside the
triangle, a slight modification of the same proof will apply.

156.] If from any point P in the plane of a triangle ABC, per-
pendiculars are drawn to meet the opposite sides in the points
A/} By, C, ; then we shall have

For

+ PA,2 =

Adding these equations together, the squares on the perpendiculars
cancel each other, and we shall have

AB; + CA; + BC;= AC; + BA; + CB;.

Fig. 6.

From this proposition we may
at once infer, («) that the per-
pendiculars through the middle
points of the sides of a triangle
meet in a point, and (ft) that
the perpendiculars from the
angles of a triangle on the
opposite sides meet in a point.

For as the square on each
perpendicular is the difference
between the squares on the ad-
jacent sides of the triangle and A
the squares on the adjacent c/

segments of the opposite side, the proposition becomes manifest.

157.] From the ends A, B of the base of a triangle ABC, lines
AE, BF, of arbitrary equal lengths, are drawn parallel to the oppo-
site sides of the triangle ; and through E, F lines ED, FD are drawn
parallel to the adjacent sides of the triangle and meeting in D.
Then if AF, BE cut the opposite sides in Ay and E,, these lines will
intersect in the line DC.

By similar triangles, we have

AE : CB = AB/ : CB, and AC : BF = CAy : BA/;
but AE = BF ; hence AC : CB=AB/ . CA, : CB, . BA/.

ON THE HIGHER GEOMETRY. 263

Fig. 7.

C, n

But since the line DC bisects the angle C,
we have AC : CB = AC, : BC,,

therefore AB, . CA, . BC,= AC, . CB, . BA, ;

and therefore the three lines AF, BE, CD meet in a point*.

158.] Prom any point O within or without the angle BAG a trans-
versal is drawn cutting the sides of the angle in the points B and C ;
the sum of the reciprocals of the areas of the triangles AOB and
AOC is constant, and independent of the position of the transversal.
For the sum or difference of the triangles, we find

AOC±AOB=ABC;

or dividing by the product of the areas of the triangles AOC and
AOB, we shall have «

1 1 ABC 2 AB. AC. sin A

AOB±AOC~AOB.AOC~(AO.AB.sinBAO)(AO.ACsmCAO)

2 sin A

~~AO2. sin BAO.sinCAO'

* The following extension of this theorem is due to Mr. W. J. C. Miller, Vice-
Principal of Hudderstield College : — From the ends of the base of a triangle
straight lines are drawn — in the same or in a different direction — parallel to the
opposite sides, and proportional in length to the adjacent sides ; then (1) the
straight lines joining the ends of these parallels with the remote ends of the
base, intersect each other on one of two straight lines which pass through the
vertex of the triangle, and divide the base internally and externally in the
duplicate ratio of the adjacent sides; (2) if the vertical angle is a right angle,
the internal locus is perpendicular to the base ; and (3) if the parallels are pro-
portional in length to the opposite sides, the locus of the intersections will be a
line from the vertex bisecting the base, or else parallel to the base.

Proofs of those theorems will be found in pages 18, 19 of vol. xxii. of the
mathematical ' Reprints from the Educational Times,' edited by Mr. Miller.

264

ON THF HIGHER GEOMETRY.

which is a constant quantity independent of the positions of the
points B and C — that is, of the direction of the transversal.

159.] The method of transversals may also be applied to prove
the following theorem : —

At the ends of the base of a triangle perpendiculars to the adjacent
sides are drawn, having the same ratio to these sides ; the lines
joining the ends of these perpendiculars with the opposite corners
of the triangle will meet on the perpendicular drawn from the vertex
to the base.

Fig. 8.
c

C, B

Let AF, BG be perpendicular to AC and BC, having the same
ratio 2m to the sides AC, BC. Join FC and GC. Let CF=2rc. AC,
CG=2ra.BC, the angle ACF=BCG=y. Put AC=6 and BC=«,
then the area of the triangle FCB is nab sin (C + <y) , and the area
of the triangle AFB is mcb cos A. But these areas are in the pro-
portion of CB, to AB, ; hence

CB. wasin(C+y) CA, nbsin(C+j)

-r-=r'=— T-J and similarly TQ-T- — p •

ABy me cos A ' BA, me cos B

Therefore CBJ.BAj_gcos B

AB,.CA,-5^A'

But ACt=bcos A, and BC/=acosB; consequently

CB, . BA; . AC^AB/ . CAy . BC,.
If the ratio be one of equality and the vertical angle C be a right

angle, it follows that the transversals will meet on the perpendicular

in the diagram of Euclid's proof of the Pythagorean theorem (Euc.

I. 47).

In the same way we may establish the following theorems : —
1. If on the sides of a triangle similar rectangles be drawn and

the adjacent extremities of these rectangles be joined, the perpendi-

ON THE HIGHER GEOMETRY.

ciildi-a from the three vertices of the triangle on these lines will meet
in a point.

2. If on the sides of a triangle similar isosceles triangles be drawn,
the lines joining the vertices of these triangles with the opposite ver-
tices of the given triangle will meet in a point.

CHAPTER XXI.

ON HARMONIC AND ANHARMONIC RATIOS.

160.] The principles developed in these methods will be found
of wide application, and most powerful instruments of investigation.
Let any straight line LM (fig. 9) be bisected in N ; and from any
point V let straight lines be drawn through the points L, M, N ;
and let VD be drawn parallel to LM. The four lines VL, VM, VN,
VD form what is called an harmonic pencil.

If any straight line — called a transversal — be drawn across this
pencil, it will be divided so as to have AC : CB=AD : BD.

Draw aCb parallel to LM or VD; then, since aC=C6, we have

VD_AD VD_BD

r*~ A r*> •mil ^,, — ^ »/-<(•

Fig. 9.

i.

But
consequently

VOL. II.

VD_VD

Ca ~ C6 J

= , or AD : BD = AC : HC.

. . (a)

266 ON THE HIGHER GEOMETRY.

The points C and D are called Harmonic points with reference
to the line AB.

When the transversal is drawn parallel to one of the rays of the
harmonic pencil, its segments between the remaining three rays of
the pencil are equal. This is evident from the preceding figure.
On this property may be based the development of the properties
of the centres of the conic sections.

161.] Let the line AB, harmonically divided in C and D, be
bisected in O.

A.

Fig. 10.

O C B

i i i —

D

\

Then we have
but
therefore

AC : AC+CB=AD : AD+DB;
AC + CB=2AO, and AD + DB = 2DO;
AC : AO — AD : DO,

and

AO: DO-BC : BD.

(b)
(c)
If we take the original ratio AC : CB=AD : DB,

and apply the principle of the composition and division of ratios,
we shall have

AC + CB : AC-CB=AD+DB : AD-DB.
But AC + CB = 2AO, AC-CB = 2CO,

AD + DB=2DO, and AD-DB = 2AO,
or AO : CO = DO : AO, or AO2 = CO .DO; . . (d)

CO CO . DO AQ2_ BC2 _ AC2
D02~BD2~AD2

_ . _ _ _ (}

-^" ~2~2~2'

or the ratio of the distances of the middle point of the given line
from the harmonic points of division is equal to the square of the
ratio of the distances of the same middle point to the ends of
the whole line AD — and also equal to the squares of the ratios of the
distances from the ends of the given line A, B to the harmonic points
of division C, D.

162.] Let L, M, N be the points in which a transversal meets
the sides of the triangle ABC ; then the lines drawn from the vertices
of this triangle to the harmonic points of L, M, N will meet in a
point G.

Let H, F, E be the harmonic points of L, M, N ; then we have

MA FA LB_HB EC_NC
FC' LA" HA' EB~NB*

ON TUB HIGHER GEOMETRY.

Fig- 11. Ac

267

Multiplying these equations together, we shall have

MA . LB I EC _ FA . H B . NC
MC . LA . EB~ FC . HA . NB'

But as the points L, M, N are on a transversal, the first side of
the equation is, by sec. [151], equal to unity, and therefore also the
second side ; hence the straight lines AE, BF, CH must pass through
the same point G.

The point G may be called the pole of the transversal LMN with
respect to the triangle ABC.

Let the segments LH, FM, and EN be bisected in the points
X, Y, Z ; then the points X, Y, Z will range in a straight line.

ZBN'

XB LB"

=
"'

MA"' ZC

consequently

XA.ZB.YC (-LA.MC.NBI*

XB . zc . YA~ LLB . MA . NCJ

268 ON THE HIGHER GEOMETRY.

But the latter member of this equation is,, by sec. [151] /equal to
unity; therefore

XA.ZB.YC = XB.ZC.YA;
hence the three points X, Y, Z lie in the same straight line.

163.] In a complete quadrilateral ABOCED any diagonal ED is
divided harmonically by the two other diagonals AO and BC in
the points F and/.

DEFINITION.

A complete quadrilateral is that in which all the sides are pro-
duced to meet two by two, as ABOCED in fig. 12.

Fig. 12.

^D

For as FC is a transversal to the triangle AED, we shall have

FE.DC.AB = FD.CA.BE, see sec. [151], . . (a)
and as O is a point in the triangle AED, through which pass the
three lines A/, CE, BD, we shall have, see sec. [150],

DC.AB.E/=CA.BE.D/; (b)

dividing the preceding equation by this latter, we shall have

^-~} or FE : FD=E/: D/. . . . . (c)

Hence the diagonal DE is harmonically divided in F and /.

The line A/ may be called the harmonic conjugate of the point
F ; and FGr is, similarly, the harmonic conjugate of the point A.

164.] If a quadrilateral ABCD be cut by a transversal in the
points X, fjb, v, -or, the continued product of the alternate segments
will be equal, or

. (d)

f'\ IIIK IIIOHKK OEOMETRV. 2G9

Fig. 13.

By comparing the partial triangles, we have

AX : \Tar=sin'Br : sin A, Cv : /Av=sin/z : sinC,
B/i : X//, = sin X : sin B, DOT : «n/ = sin v : sin D.

Therefore, compounding these proportions, we have

XCT . ay . \LL . nfv . sin X . sin u, . sin v . sin «r

AX.O.B/A .Dor= — . . r— ^ — . „ • r\ —•

sm A . sin B . sin C . sm D

In like manner

,,. Xu, . LLV . -GTJ/ . X-BT. sinX . sin u, . sin v . sin tr

ACT. a\.CfA.Dv=~ —. — r ^R — • r* • T\ —•

sin A . sin B . sin C . sm D

Hence the truth of the proposition, which may be extended to a
polygon of any number of sides as follows : — •

When it is proved that in a triangle cut by a transversal the
products of the alternate* segments of the sides are equal, we may
extend the proposition to the case of the quadrilateral or to any
other linear polygon.

On one of the sides of the given triangle let another triangle be
constructed, whose sides shall be cut by the given transversal.

Let «, $, y be the ratios of the alternate segments of the sides
of the triangle; then ay9y = l. Let 8 and e be the ratios of the

5j

segments of the sides of the newly applied triangle, then — = 1 ;

consequently a.j3Se=l.

In like manner the theorem may be extended to a polygon of
any number of sides.

It is obvious that in going round the triangle ABC we proceed
from A to B, from B to C, and from C to A. In the same way in
going round the quadrilateral we proceed from A to B, from B to
C, from C to D, and from D to A ; while in going round the applied
triangle we proceed from C to D, from D to A, and from A to C.

270

ON THE HIGHER GEOMETRY.

Hence, if the ratio of the segments of CA be 7, the ratio of the

segments of AC will be — .
7

165.] The middle points of the diagonals of a complete quadri-
lateral lie in the same straight line.

Let \, fj,, v be the middle points of these diagonals ; they lie in
a straight line. Since the line BD is harmonically divided in the
points Gr, H, and bisected at A,, we shall have

XGr BG

~~CG2 vl'fp El2

We shall also have

EHS

Multiplying these expressions together, we shall have

BG . CI . EH 2

But the three diagonals constitute a triangle GHI, of which the
line BCE is a transversal. Consequently BG . CI .EH = BH . CG . El,
and therefore \G . /J . j/H = \H . /*G . vl, or \, p, and v range along
a straight line.

ON THE H Hill EH GEOMETRY. 271

ON ANHARMONIC RATIO.

166.] This theory, the invention of M. Chasles (unquestionably
the greatest geometer of this age, and, perhaps, equal to the best
in any age) , is an extension of the principle of harmonic ratio.

The theorem on which this powerful instrument of investigation
is founded maybe traced to the mathematical collections of Pappus*.
This simple relation has been made the basis of a general system of
conies by M. Chasles. Before his day it lay barren of results, until
he developed its properties and gave it the name of anharmonic
ratio, from its analogy to harmonic ratio, a particular case of the
more general relation. There is one signal peculiarity of this
method. If we take any theorem and its dual, as for example
Pascal's and Brianchon's hexagons, the one inscribed in, and the
other circumscribed to, a conic section, or any other like dual
property, and if the one admits of investigation by Cartesian or
protective coordinates, the dual must be treated by tdhgential coor-
dinates, as discussed in the first volume of this work. But the
anharmonic method is alike applicable to each, as we shall show
further on. Another element of the great power of the anhar-
monic method is that its properties are projective.

From a point Viet four fixed lines be drawn, meeting a fifth straight
line variable in position, in the points A, B, C, D. Let these lines
be put VA = a, VB=6, VC = c, VD = d, and let the sines of the
angles between a and c be a, between c and b be 7, between b and
d be fi, and between d and a be 8. Let the sine of the angles
between a and b be (a + y) , and that between c and d be (/3 + y), and
let p be the perpendicular from the point V on the range.

Now twice the area of the triangle AVC is

AC.p = acot; therefore AC=— •
In like manner CB=^ BD=*^, AD=^

p p

Now these six segments of the range may be combined in the
three following distinct groups — and no more — so that the variable
rays and the common perpendicular/? may be eliminated by division.
CA. DA_ca.<fl>«|3

(b)

AC^DC_ac.db a/3
AB '

AB CBa6 .cd

AD ' CD~arf.e&~ 78

* See Commandine's translation, Prop. 129. Lib. vii.

272 ON THE HIGHER GEOMETRY.

Dividing by a, b, c, d, the three anharmonic ratios become

AC DC •«£

• m ^.-

1 '* CB ' DB~8' AB '

and fill) ^_

AD ' CD"

The first of these forms may be easily recollected, as it is the
form of an harmonic pencil. The second has the same arrange-
ment of the rays in the numerator as the first, ca . db, while the
only arrangement possible for the denominator is ab . dc. The
third form is the result of dividing the denominator of the second
by that of the first.

There are in fact six different forms, which may be reduced to
three.

It is not possible to write the four letters a, b, c, d two by two
in more than three ways, namely ab . cd, ac .bd, ad.cb; hence
there can be but three anharmonic arrangements of the segments
of the range.

A peculiar notation may be devised to indicate briefly the several
ratios of the anharmonic range.

Let V be the vertex of the pencil, and A, B, C, D the four points ;
then the ratio

DA •«. v/AVr«

""~DB may wntten v( gJtC-

AC DC

A ir/v.

^jj-^-pB may be written V( ^J(A-f-D),

AB CB ...

^ may be written

The following relations may be easily established.

Hence these six forms may be reduced to three.

If the given pencil of rays be cut by any other transversal, the
ratios of the segments of this latter range will be the same as those
of the former; for the sines of the radial angles remain unchanged.

ON THE HIGHER GEOMETRY. 278

If four points A, B, C, D be taken on a range, and through any
point in space four rays be drawn through these four points, the
inharmonic ratio of this pencil will be the same as that of the four
points on the range.

The aiiharmonic ratio of any four points ranged along a straight
line in one figure is equal to the anharmonic ratio of the corre-
sponding pencil on the reciprocal polar of the original figure.

167.] Should the rays a, b, c, d meet the circumference of a
circle in four fixed points, while the vertex V of the pencil moves
along the circumference, the anharmonic ratios of these successive
pencils will continue unchanged, because the sines of the radial
angles (that is, of the angles between the rays) continue unchanged.

When the pencil is turned through a right angle, the anharmonic
ratios continue unchanged, because the sines of the radial angles
are still the same.

If four fixed tangents drawn to a circle be intersected by a fifth
tangent variable in position, the anharmonic ratio of the segments
of this tangent made by the fixed tangents will be constant and
independent of its position.

It may easily be shown that if two fixed tangents are drawn to
a circle, the segment of a third variable tangent intercepted between
them subtends a constant angle at the centre, equal to half the
external angle of the two fixed tangents. Hence the variable
segments of the tangent range to the circle subtend fixed angles at
the centre ; and consequently their anharmonic ratio is constant.

168.] If two equal anharmonic pencils have a commonray or axis.

the three other pairs of rays will intersect two by two in three points
rani/i' in a straight lint'.

VOL. II. 2 N

274 ON THE HIGHER GEOMETRY.

Let the two equal anharmonic pencils OQ, OA, OB, OC and
QO, QA;, QB;, Q,C/ have a common ray or axis OQ, the remaining
three rays will intersect in three points a, ft, y, which range in a
straight line.

Join a and /3, and produce «/3 to 8 in the common ray OQ, and
let it meet the fourth rays OC, QC, in the points 7 and y, ; then
these points must coincide, since the anharmonic ratio of 8«/3y
is equal to the anharmonic ratio of S«/3yr

When the anharmonic ratios of two straight lines which meet in
a point are equal, the straight lines which join the corresponding
points two hy two will all three meet in a point. Let OABC and
be two equal anharmonic ranges. Join AA,, BBy, and let

them meet in V. Then if VC, be drawn, it will pass through C ;
for if it cut OAy in some other point D, the anharmonic range
OA/Bp, would be equal to OAjB,!).

169.] If two triangles ABC and AfijC, (fig. 17) have their corre-
sponding vertices on three straight lines which meet in a point O, the
corresponding sides will meet two by two in three points a, ft, 7
which range in a straight line.

Join Oy; then since the pencil OyBAC is cut by the trans-
versal DABy, and also by the transversal DyA^y, the anharmonic
ratios of these two straight lines or ranges are equal ; and as the
pencils CD, CA, CB, C<y and C,!),, C;Ap CyBy, C/y have a common
ray CC;, and their anharmonic ratios are equal, the three remaining
pairs of rays CA, C,AI} CB, C^B/, and AB, A;By will meet in the
three points /3, a, y, which range in a straight line.

The triangles ABC, AjBp, are called by PONCELET homologous
triangles ; the common point in which the three directrix lines meet,

ON THE HIGHER GEOMETRY.

Fig. 17.

275

the centre of homology ; and the straight line in which each pair of
sides meet, the homologous axis.

170.] Let two homologous triangles ABC and Afifi, (fig. 17)
have their sides AB, AjB, meeting in y, their sides BC, B/C/ meeting
in at, and their sides AC, A,C, meeting in /3; then, if at, (3, y range along
a straight line, the lines joining the points AA,, BBy, CCt will meet
in a point.

As the pencil OCABy is cut by the two ranges yBAD and
yByAjD,, their anharmonic ratios are equal, and they have besides an
homologous point y, therefore the lines joining the homologous
points AA,, BB,, CC, meet in a point.

It is rather remarkable that when the two triangles are in the
same plane, some such demonstration as that above given is required,
but when the triangles lie in different planes the proposition becomes
self-evident, the triangles constituting the bases of the same pyra-
mid, and their sides will manifestly meet in the line in which the
plane bases intersect — that is, in a straight line.

171.] If the opposite sides of a hexagon inscribed in a circle be
produced, they will meet two by two in the same straight line.

Let B and E be the ends (fig. 18) of one of the diagonals of the
hexagon BAFEDC, A and C the angles adjoining to B, and F and
D the angles adjoining A and C.

Then, as these points lie on the circumference of a circle, the an-

276 ON THE HIGHER GEOMETRY.

harmonic ratios A(BFED),C(BDEF) will be equal. See sec. [167].
And as the pencil A(BFED) is cut by the transversal LGED, and
the pencil C(BDEF) is cut by the transversal MIEF, the anhar-
monic ratios of these two transversals will be equal. Moreover

Fig. 18.

they have a common or homologous point E; hence the lines joining
the other homologous points will all three meet in the same point,
or the lines joining the points L and M, G and F, I and D, will
meet in the same point N. Hence L, M, N are in the same straight
line.

172.] The diagonals of a hexagon circumscribed to a circle meet in
a point.

Since the four tangents CB, AF, FE, ED (fig. 19) meet the two
tangents AB and CD in the points B and C, A and L, I and N, and
in the points M and D, and as the anharmonic ratios of these two
ranges BAIM and CLND are equal, the anharmonic pencils which
pass through them will be equal. Therefore the anharmonic ratio
of the pencil E (BAIM) will be equal to the anharmonic ratio of the
pencil F(CLDN) ; and as these pencils have a common ray EF, the
remaining three rays of each pencil will meet two by two in three
points which range in a straight line : that is, EB and FC will meet
in O, EA and FL will meet in A, while EM and FD will meet in D.
Hence the point O must be on the line AD, or the three diagonals
meet in the same point O.

ON THE HIGHER GEOMETRY.

277

Fig. 19.

CHAPTER XXII.

DEFINITION.

173.] If from the centre of a circle a perpendicular be drawn on
any straight line in its plane, and if in this perpendicular a point
be taken so that the rectangle contained by its distance from the
centre and the perpendicular shall be equal to the square on the
radius, the point so found and the straight line are called pole and
polar with respect to the circle*.

LEMMA I.

If a chord be drawn in a circle, and any point taken in this chord,
the polar of this point will divide the chord into segments which will
have to each other the same ratio as the segments of this chord made
by the pole.

* Like most of our terms in this important branch of geometry, we owe the
very convenient terms pole and polar to French geometers — the former to Serrois,
the latter to Gergonne.

278

ON THE HIGHER GEOMETRY.

Let P be the pole taken in the chord AB, and let this chord be
cut in the point Q by the polar TT, of the point P. Then we shall
have

PA_QA

PB~QB'

Fig. 20.

Since PT is a tangent to the circle, we have

PT2==PA . PB = PQ . PB + AQ . PB;
and as PTT, is an isosceles triangle,

therefore FT3 = QT.QT, + ptf = QA . QB + PQ . QB + PQ . PB,
= PA.QB + PQ.PB.

2

Equating these two values of PT , and taking away the common
rectangle PQ . PB, we shall have QA . PB = PA . QB ;

, PA QA

PB = QB-

Since PA-PQ=QA and PQ - PB = QB, we have

PA-PQ_QA_PA
PQ-PB~QB~PBj

hence PA, PQ, PB are in harmonical proportion, since the first is
to the third as the difference between the first and the second is to
the difference between the second and the third.

ON THE HIGHER GEOMETRY.

LEMMA II.

279

Let a point and a straight line be assumed as pole and polar with
reference to a circle. The polar of any point taken in this straight
line will pass through the point assumed as pole.

Fig. 21.

(a). Let the pole Q be taken within the circle. Join OQ, and
produce it to P, so that OQ.OP=R2; then, by the definition of
pole and polar, the polar- of Q will pass through P and be at right
angles to OQ.

Through Q draw a chord TT,, and tangents TP,, T,P, meeting
in P,; and join PP,. Then, as OQ. OP = OQ .OP, (since each
rectangle is equal to R2), the triangles OQ.Q and OPP, are similar,
and the angle OQ,Q is equal to the angle OPP,; but OQ,Q is a
right angle, therefore OPP, is a right angle, or the line PP, is the
polar of the point Q.

(/3) . Let the polar PP, cut the circle. Then, if OP be the per-
pendicular on PP, the distance of the pole of PP, from the centre

R2
18 OP'

From any point P, in the polar PP, let tangents P;T and P/T, be
drawn to the circle, the line TTP the polar of P,, will pass through
the pole of PPr Let TT, meet the perpendicular OP in the point Q ;
then, as triangles P.OP and QOQ, are similar, P.O . OQ= QO . OP ;

R8
but Pp . OQ, = R2 ; therefore QO . OP = R2 or QO = ; therefore

280

ON THE HIGHER GEOMETRY.

the point Q, in which the secant TT; cuts the perpendicular OP,
coincides with the pole of the polar PPr

Fig. 22.

It is evident that if we substitute a sphere for the circle, and a
plane for the polar straight line, we may infer that if any point be
assumed in the plane, the polar plane of this point, taken with
reference to a sphere, will pass through the pole of the polar
plane.

174.] If the external angles of a triangle be bisected, the bisectors
will meet the opposite sides of the triangle in three points «, /3, y
which range in a straight line.

Kg. 23.

'C

Let a circle be inscribed in the triangle, and let the points of

»\ NIK HIGHER GEOMETRY.

contact A,, B,, Cy be joined. Let the inscribed circle be taken as a
l>ol<iri~in</ circle. Then, as the bisector of the external angle at C
is the polar of the point c (the middle point of the line AjB,), and as
the side AB is the polar of the point C,, the point y, in which the
side AB meets the bisector of the vertical angle at C, is the pole of
the line Ctc. In the same way it may be shown that a and b are the
poles of the two other bisectors, while A,a and B,A are the polars
of the points in which these bisectors meet the opposite sides. But
the lines drawn from the angles of a triangle to the middle points
of the opposite sides meet in a point, the centre of gravity or cen-
troid of the triangle. Consequently the centroid of the triangle
AjBjC, is the pole of the straight line a/3y, and the perpendicular from
the centre of the circle on this line will pass through the centroid.
175.] If the opposite sides of a quadrilateral inscribed in a circle
be produced to meet in V, V; (fig. 24), and the diagonals AD, BC be

Fig. 24.

V

drawn to meet in O, and tangents to the circle be drawn at the point*
A, B, C, D, these tangents will meet two by two on the lines VO, V,O

VOL. II. 2 O

282 ON THE HIGHER, GEOMETRY.

in the points //., v, m, n, so that the points V/; p., O, v and V, m, O, n
will lie on the straight lines VO and V,O.

Since ABCD is a quadrilateral, the line AB is harmonically
divided in c and V, and the line ab in O and V, and CD in d and V.
See sec. [163] . And again, as ABCD is a quadrilateral inscribed
in a circle, the polar of V will divide harmonically the chords AB
and CD in c and d ; therefore the line cd is the polar of V, and
this line will therefore pass through the poles //,, v of AB and CD.
Hence the points V/( p, O, v are in the same straight line.

In the same way it may be shown that the points V, n} O, m are
in the same straight line.

Without using poles and polars the proposition may be proved
as follows by the method of transversals : —

If we can show that the straight lines V;/A, A/A, B/A make angles
with the sides of the triangle V;AB, such that the product of the
sines of the alternate angles may be equal, these lines must meet in
one point //,, see sec. [153] — that is, if

sin /AAV, . sin //,V;B . sin /iBA = sin /*VyA . sin V;B/i . sin BA/tt.

Now sin /zBA = sin BA/A, since A//, and B/A are tangents to the circle;
also sin AV,O : sin BVp = V,B . Ac : V; A . cB ; and as the angle
V,A/A is equal to the angle ABO, and the angle V;B/A equal to BAO,

P P

sin V,A/i : sin V,B/A = sin ABO : sin BAO=~T> ' : ^-. — r- : but

OB . Be OA . Ac

since the angle V/AO = V/BO, — '—^ -- = ; ' - , P and P, being

*jj
the perpendiculars drawn from A and B on the line OV, ;

or sin V .Au, : sin

and sin BV^ : sin AVyc = V^ . Be : V,B . Ac.

Hence sin VyA//, . sin BV^ = sin VyB//, . sin AV^e.

176.] If a quadrilateral be inscribed in a circle, then (a) the square
on the outer diagonal of the complete quadrilateral is equal to the sum
of the squares on the tangents drawn from Us ends to the circle,
(/3) the diagonal itself is equal to the sum of the tangents drawn
from its middle point, and (y] the circle drawn on this diagonal
as diameter will cut the given circle at right angles.

(a) Since P is the pole of EG (fig. 25), the outer diagonal of the
complete quadrilateral, therefore On . OE = Ow . OG = R2.

But EG2=EO2 + GO2-2GO.Om.

, GO2=GK2+R2, and 2GO.Om=2R2.
Hence we have EGa = ELa+ GK2.

ON THE HIGHER GEOMETRY.

(/3) Let (AM be a tangent drawn to the circle from the middle
point Q of EG.

Then,

therefore

, GO2=GKa

E02=EL2

EQ=GQ=QM.

(y) Since QM is a tangent to one circle and a radius of the
other, the circles must cut orthogonally.

It may also be shown that the squares of the inner diagonals are
to each other as the distances of their middle points from the middle
point Q of the outer diagonal.

177.] The line joining the middle points of the diagonals of a
quadrilateral circumscribing a circle passes through the centre*.

Let a and b be the middle points of the diagonals AC, BD of the
quadrilateral ABCD (fig. 26) circumscribing the circle. Through
B and C draw straight lines BG and CH parallel to the diagonals
AC and BD. Through m and n, the points of contact of the qua-
drilateral, draw the chord mn meeting BG in T, and the line Ba in /.

* Of this theorem, — which is duo to Newton, — a proof by the method of tan-
gential coordinates will be found in the firet volume <>f this work. p. 40.

284

ON THE HIGHER GEOMETRY.

Now since BG, BA, Ba, BC is an harmonic pencil, since AC is
bisected in a and is parallel to BG,the transversal mn is harmonically
divided in / and T ; therefore the polar of T passes through t ; and
as moreover T is a point in the chord mn, the polar of T will pass
through B ; therefore Ba is the polar of T. Now producing the
chords pn, qm they will meet in a point Q, the polar of which point,
as it is on the line pn, will pass through C ; and as it is on the line
qm it will pass through A ; therefore AC is the polar of the point Q ;
and as it has been shown that T'is the pole of the line B«, it
will follow that the straight line QT will be the polar of the point
in which AC and B/ intersect — that is, the point a, the middle point
of the diagonal AC. In the same way it maybe shown that PH is
the polar of b. Now as BG is parallel to AP, we have
Tn : wP = Bw : nC, and QM : wH = Bw : nC ;

Fig. 26.

Therefore TV : Qw=wP : wH. Hence the triangles QTw and HnP
are similar; therefore QT is parallel to PH. But these lines are

ON THE HIGHER GEOMETRY.

285

the polars of the points a and b; the point in which they meet will
therefore be the pole of the line ab. But as they meet at infinity,
the line ab must pass through the centre of the circle.

The same proof will hold when the curve is a conic.

178.] If from any point P (fig. 27) perpendiculars PA,, PB,, PC,
are drawn on the sides of a triangle ABC, a circle through the three
points A,, B,, C, will cut the sides of the triangle in three other points
A;/, By/, C0, such that if perpendiculars to the sides of the triangle be
drawn through these points, they will also meet in a point P .

C ,*

c,

Since AC, . AC,, = AB, . AB,,, and AC = AC, + C,C,,, while
AB,=AB,, + B,B,,, we shall have

AC,,a + AC,, . Cpa-lB? + AB, . B,B „
and AC,3 - AC, . C,C,,= AB,a - AB, . B,B, ;

adding these two expressions, we shall have

so also B A,2 + BA,,2 - A,A,,2 = BC,a + BC,,a - C,C,,2,

and CF ~~

adding these equals, the squares on the intervals between the feet

286 ON THE HIGHER GEOMETRY.

of the perpendiculars mutually cancel, and we shall have

AC; + BA + CB

= AB; + BO; + c A; + AB,,2 + BC/ + c A/.

But since PAy, PBy, PCy are perpendiculars to the sides of the tri-
angle, we shall have, see sec. [156], ,

therefore AC/ + B A/ + CByy2 = AByy2 + BCyy2 + C A;/2 j

hence the perpendiculars through the points A/y, B/y, C;/ meet in a
common point Py.

The line drawn from A to P is perpendicular to the line By,Cyy,
which joins the feet of the perpendiculars P,Byy, P,C,y. For since
AC;PBy is a quadrilateral that may be inscribed in a circle, the
angle APCy is equal to the angle AByC,; and as CyByByyCy, is a
quadrilateral inscribed in a circle, the angle AC,,Byy is equal to the
angle AByCy — that is, to the angle APC,. Consequently the angle
ASCy/ is a right angle. Hence, if from the angles of the triangle
ABC lines be drawn to the points P, Py, the lines drawn to P will
be perpendicular to the sides of the triangle A,,By,Cy/, and the lines
drawn to Py will be perpendicular to the sides of the triangle AyB C/t

The lines drawn from any vertex A to the points P, Py will
make equal angles with the sides AB and AC*.

For the angle P,AB/y is equal to the angle PyCy,Byy which is equal
to the angle PByCy, which has been proved equal to the angle PAC.

179.] The foregoing theorem maybe proved in a simpler way by
the help of the property given in sec. [134].

Let the perpendiculars By,Py and C,,P, be erected at B,y and Cyy
to meet in Py. Then, as AByyP,C/y is a quadrilateral that may be
inscribed in a circle, the angle By,APy is equal to the angle By/C,,Py ;
and as CyB,B,yC,, is a like quadrilateral, the angle ACy,Byy is equal

* Hence the points P, P, are the foci of an ellipse inscribed in the triangle
ABC, of which O is the centre, and the major axis the diameter of the circle.

Produce P^,, to Q until A/;Q is equal A/yP. Join PQ cutting the side of the
triangle in «. Join P,P«. Then as P,« is equal to Q« and P« = Pa,
P,a+ P«=PQ = 20AJ/, or P,a+Pa is constant, being equal to the diameter of
the circle.

We may hence infer that if three tangents to an ellipse be given, and one of
its foci, we can at once construct the ellipse. From the focus draw perpen-
diculars on the three tangents, the circle that passes through the feet of the
perpendiculars will cut the tangents in three other points, through which if per-
pendiculars be drawn, they will meet in the second focus. The major axis of
this ellipse will be the diameter 2R of the circle ; and the eccentricity will be
PP,
2R'

ON THE HIGHER GEOMETRY.

287

to the angle ABjC,; but AC^P, and AB;P are right angles; hence
tin; angle B//Cy/P/ is equal to the angle C^P — that is, to the angle
C,AP, sinoe ABjPC is also a quadrilateral that may be inscribed in
a circle. Hence the angle B//AP/ is equal to the angle CAP,. But
in sec. [154] it is shown that if two sets of lines be drawn from
the angles of a triangle making equal angles with the adjacent
sides, and if one set meet in a point, so likewise the other set will
also meet in a point.

180.] If through a given point P (fig. 28) two secants PAB, PCD
be drawn to a circle, the first fixed, the second movable, and if from the
points of intersection of this latter with the circle tangents be drawn
meeting the fixed secant in the points M, N, we shall have *

I I "1 X r ~""~ T"4 4 I T~l T ~» *

PM

Fig. 28.

Through C and D let tangents be drawn meeting in G, and cutting
the fixed secant in the points M, N. Join AD, BC meeting in O,
and AC, BD meeting in V. Then VO will pass through G, the
intersection of the tangents at C and D, and will cut the line AB

» This theorem is taken from Maclaurin's Tractattis de lineanim currantm
proprietatibtts gcneralibus, p. 11, a treatise of rare originality and beauty. The
theorem in the text, which is proved for algebraical curves of all orders by
a simple application of an elementary principle of the differential calculus,
Maclaurin makes the foundation of a system of geometry of curve lines of singular
elegance.

288 ON THE HIGHER GEOMETRY.

in a point Q. Let the line GE be drawn through G parallel to AB
meeting the lines VA, VB in H and F. Then by similar triangles

PM : PC = EG : CE and PC : PA = CE : EH.
Compounding these ratios, PM : PA=EG : EH.

i i TJ^TJ l 1 l^T^

Hence = PA ' EG" In Uke manner WG °Uain PN = PB' EG"

But EH=EG + GH and EF=EG-FG;

1 1 rEG + GHl 1 , ,
therefore = — = 1 +

= PA EG '

1 1 rEG-FGI 1 r_ FG1
PN = PBHsQ— ^PB^-EGJ5

f 1111 JTGH FG1

+ -+ + PB_T

But as the line AB is harmonically divided in P and Q,

PA : PB = AQ : BQ=GH : FG; and therefore p^=|§;
consequently + J_ + J^.

This proof, without any modification, will hold for conies.

CHAPTER XXIII.

ON CIRCLES INSCRIBED, EXSCRIBED, AND CIRCUMSCRIBED
TO A TRIANGLE,

When a triangle is given, sixteen circles may be described in con-
nexion with it : — one circumscribed to the triangle ; one inscribed
in it ; three touching, each a side and the other two sides produced ;
six passing throiigh the centres of the circles of contact and the ver-
tices of the given triangle taken two by two ; four through the
centres of the inscribed and exscribed circles taken three by three ;
and, lastly, a sixteenth circle passing through the feet of the perpen-
diculars drawn from the vertices of the triangle on the opposite sides.
This may be called the orthocentric circle, as it circumscribes the
orthocentric triangle. It is also known as the nine-point circle.
The other circles will be named as definitions are required.

The four circles which touch the sides of this triangle may with
propriety and brevity be named the circles of contact ; and their
centres may be called the centres of contact.

o\ Till: HIOHKR GEOMETRY.

281)

181.] Let r, r,, rn, rnl, 11 be the radii of the inscribed, exscribed*,
and circumscribed circles of the triangle ABC, and let to, fl, £lp £llt, O
be the centres of these circles, while 0 is the centre of the circle,
whose radius is p, inscribed in the orthocentric triangle. Let the
inscribed circle touch the sides of the triangle in the points B/} A,, F,
and the exscribed circle touch the same sides in the points G, H, F/}
and as BG=BF,, and AH=AF,, BG + AH = BA=c, if a, b, c be
the sides of the triangle opposite to the angles A, B, C. Hence
CG + CH is equal to the perimeter of the triangle, or as CG = CH,
CG or CH is half the perimeter of the triangle; let this semiperi-
meter be denoted by s. And as CA^CB,, GA^HB,; and as
GA/=BF/ + BFandHB,=AF + AF/,thereforeBF/+BF=AF + AF,
or BF=AF; hence BA=GA/=HB/=c. Therefore BG=*-a,
BA=s — b, and CA^*— • c.

Fig. 29.

Let A be the area of the triangle, then it is well known that

A- / 7 W —K\( •* — f^£ ( \

* Not ctcribed, as it is usually written, but extcribtd, in accordance with
the analogy of the pronunciation'of other like words, such as r.rscind, exttrtion,
t.rsert, exsiccate, &c.

VOL. II. 2 P

290 ON THE HIGHER GEOMETRY.

We have also r.= . r,,= - -, )',,,= (b)

s— a s—b s—c

Therefore r^^/^/// = ^ w TTT r=A2. . . . (c)

1 "' s(s— a) (s — b) (s— c)

Taking the reciprocals of (b), we shall have

-=- + — + —, (d) whence r= ^J^JU . (e)

T***^***-7^' 7* >* —I- 7* /* —I- 7* 7*

1 1 ' n 'in ' I ' il ~ ' II' III ~ 'I' III

sr , j sr

O —**. ft — • O \\ f I O /I •

«> I* ^— 1 1 1 n i o u ^— •

rirnriu

therefore 28-a-b=c =

(r -f- f \
~ - /

In like manner « = rY^, b=r,,, . . . (g)

and since 4R= — , substituting the foregoing values of a, b, c,

Si

we shall have 4R=^>l, + r,J(r,,, + r,)(r +,-„)_ . . . (h)

But r,r,,r,,,=sV, and s= »Jrlru-\-rllrul +rlllrl, as in (f) ;
hence 4R~ d + ^+

ri rn rn riu """ r/;/ ri
Now if we develop the numerator and add to both sides

V V V

r= — — , as given in (e) , we shall have

r, + rtt + rw, ....... (j)

Thus the sum of the radii of the exscribed circles is equal to the
radius of the inscribed circle together with four times the radius
of the circumscribed circle.

sr , sr

Since s — a — —) and s — 6 = — , we have

sV2
(s — a)(s — b)= - or s2— (a + b)s

rirn ri rn riu

ON THE HIGHER GEOMETRY.

Finding like expressions for the other sides, and adding,

we obtain 3*2— 4s* + bc + ac + ab= — — — — .

ri ru rui

But rtrltrtn=s*r', consequently 6c + ac + «6=s2 + r(r/ + r// + r///).

hence 6e'+ac + ad=s2+4Rr + r2, (k)

and therefore a2 + 62 + c2=2s2— 8Rr-2r2 (1)

These useful theorems may be more simply established by suc-
cessively eliminating (bc + ca + ab} and (a2 + A2 + e2) between the
formulae 2s = a + b + c and sr'2 = (s — a)(s— b)(s— c).

182.] Since 1 + 1 + 1=!, see (d) sec. [181],

ri ru in

squaring

or

_^+ 1_ + J_ = ^_2|~_L + J_ +_L"j ,
r, ru r,n r Ir, rttrllt ry//r,J

2 *~ 2 ~~ 2

r* r* r, r.. r,,. r

• f* • " I It III

Now r. + r,. + r,,.=4R + r and rlr..rlllr=

I If til I It HI

e 1111 2s2-8Rr-2r2

therefore -^-\ — -z+— 3 + ^ = 5-, . . . . (a)

r.

But it has been shown, in (1) in the last section, that

f 1111

and therefore ++_+=_ 1^ - ..... (b)

ri rn riu r ^

Thus the sum of the squares of the reciprocals of the radii of the four
inscribed and exscribed circles to a triangle is equal to the sum of
the squares of the three sides divided by the square of the area of
the triangle.

183.] Let ht, htl, hltl denote the perpendiculars from the vertices
of a triangle on the opposite sides, then ah,, bhn, chtll are each

1 s — a 1 s— b

equal to 2sr; and as — = - , — = - ,
rt sr rlt sr

we have ! + ! = ^-?£_, Qf h /l+i\ 2

ri ru sr hnf \ri rJ

In like manner hi — H -- ) = 2, and /< (-H -- ) = 2;
v// rni' \ri rm'

consequently &±te+&L±te+&l±Mm6. . (a)

292 ON THE HIGHER GEOMETRY.

184.] The sum of the squares of the sides of a triangle is equal
to twice the sum of the products of each height multiplied by the
distance between the corresponding angle and the orthocentre.

Let h be the altitude corresponding to the angle A ; then the
distance from the vertex A to the orthocentre is 2R cos A, and the

cos A-
product by h is 2RAcos A=4AR — — , putting A for the area of

___

the triangle; and this may be written 2AR- -- r -- -• Finding

like expressions for the other angles, and bearing in mind that
«6c=4RA, we get

(a)

In any plane triangle we shall have the relation
a cos A + 6 cosB + ccos C r

2s • R*

For ifp,Pi>Pn denote the perpendiculars from the centre of the cir-
cumscribed circle on the sides of the triangle a, b, c, we have

cosA=4 cosB=§, cosC=§'. (b)

XV xv XV

Hem

The sum of the ratios of each perpendicular from the centre of the
circumscribing circle on a side of the triangle to the perpendicular
from the opposite angle on the same side is equal to unity.

p area COB ,, p p. pn

For \= -- TTTS; therefore £+£/+O=l. . . (d)
h area CAB h h, ht

The sum of the reciprocals of the perpendiculars from the angles
of a triangle on the opposite sides is equal to the reciprocal of the
radius of the inscribed circle.

Let <o be the centre of this circle, then

r_area BwC
h ""area BAG '

finding like expressions for the other terms, and adding, we shall
have

T i "7" ~r T~ == • ....... (®/

h ht htl r
If we turn to fig. 29 (p. 289) it will easily be seen that

flft) O^ft) fiy/0> _ ,

C« B« A^T

ON THE HIGHER GEOMETRY. 293

c n,&> b il.M a

„

and the sum of these ratios is obviously 2.

ON THE TRIGONOMETRICAL RELATIONS OF THE ANGLES OF A TRIANGLE.

185.] In the following propositions the terms sin, cos, tan are
used as brief and familiar expressions to denote certain ratios of
lines connected with a triangle and its inscribed and circumscribed
circles.

A s— a A s

Since cot— = - , and coW=— , ..... (a)

& i m & / i

finding like expressions for the other angles, and adding, we have

,A . B , C s—as—b s—c s
C0t +C0t +COt =

2 2 2

A , , B , C s s s
and cot^-f cotli + cotT5-=-H --- f- — ;

3 ri r,i r,n
hence dividing these equations by s, we obtain

r ri ru r,n
Multiplying together the cotangents in (a),

A ,B ^C (s-a)(s-b)(s-c)
we have cot cot cot - = - a ~>

and

cot -y cot -Q- cot

s3? s(s —

2 r,rurlu'

•) «*r' .

rrtrnrut

sr*

~«r»~?

or

sr = »J

rr r r

(c)

Hence Me *gware root of the continued product of the four radii
of the inscribed and exscribed circles is equal to the area of the
triangle.

To prove the following relations : —

a • T> b
Since sin A = 775, sinB = ;

294 ON THE HIGHER GEOMETRY.

adding these expressions,

Q

sinA + sinB4sinC = ^ ........ (d)

XV

Multiplying these values,

7*S

(/3) sin A sin B sin C = 52 ........ (e)

If we square (a) and subtract the values of the squares of the
sines, we shall have

4$

(y) sin A sin B + sin B sin C + sin A sin C = —

But a2 + 62 + c2 = 2s2 - 2r2 - 8Rr.

s? _i_ ^2 i 4 j^y
Hence sin A sin B + sin B sin C + sin A sin C =

(B)

= A /(s-b}(s-c}
be

. . A . B . C 4?(* — a)(s— b)(s— c) r
Hence 4»n¥.m]s.u,?— -^ 2=fi. . . (g)

w 1 + cosA=Mir£);

finding like expressions for cos B and cos C, adding, we shall have

—
abc

but «2 + 62 + c2=2*2-2r2-8Rr, as in (1) sec. [181] ;

T

hence cos A + eosB+cosC = l+^p ...... (h)

XV

If p, pp ptl denote the perpendiculars from the centre of the cir-
cumscribed circle on the sides,

j0=Il cos A, j»/=R cos B, pn =R cos C.

Hence P+Pi+Pi,=^ + r ....... (i)

186.] To prove that a cot A + b cot B + c cot C = 2(R + r) . (a)

a cos A 2aRcosA 2ap
acotA.=—. — r-= o-p . — T- = — —=2p.
sm A 2R sin A a

Hence a cot A + b cot B + c cot C = 2 ( p +pt +pj .
But p-i-pl+pll='R-^r, as shown in (i), last section.

ON THE HIGHER GEOMETRY. 295

1 1 \ve square the expression (h) in sec. [185] , and put for cos* A its

a2
value 1 — -, and like expressions for cos2 B, cos2 C, we shall have

= l+ -3+

putting for a2 4 V2 + e2 its value, and reducing,

__

cosB cos C + cos A cos C 4- cos A cos B= - 7^73 - . . (h)

4H

187.] To show that

cos2 A 4- cos2 B 4 cos2 C= 1—2 cos A cos B cos C. . (a)
Since A + B-fC=7r, cosC = — cos (A + B);

therefore cos2 C=cos2 A cos2 B — 2 cos A cos B sin A sin B
+ 1 — cos2 A — cos2 B 4- cos2 A cos2 B,

putting for sin2 A sin2 B in the developed form

its equivalent 1 — cos2 A + cos2 B -f cos2 A cos2 B.
Hence the expression cos2 A 4- cos2 B 4- cos2 C now becomes
1+2 cos A cos B (cos A cos B — sin A sin B) = 1 —2 cos A cos B cos C.
We have also, as shown in (e) section [185],

(l4-cosA)(l+cosB)(l+cosC)=^. . . (b)
If we multiply together the expressions

(1 + cosA), (1+cosB), (1 4- cos C), we shall have

(1 4- cos A)(l 4- cos B) (1 + cos C) =1 + cos A 4- cos B 4- cos C

*
4- cos A cos B 4- cos B cos C 4- cos A cos C 4- cos A cos B cos C=<

Substituting for (cos A 4- cos B 4- cos C) and

cos A cos B 4- cos B cos C 4- cos A cos C their values
as given in (h) and (i) in section [185], we shall find

*2-(2R4-r)2
cos A cos B cos C= - - — — ..... (j)

296 ON THE HIGHER GEOMETRY.

Since co4-A/*<£:=3,

* V oc

A B C 4-s.sr s

4003-008^08- = =; . . . . (k)

comparing this expression with (d), sec. [185], we find

ABC

sin A + sin B -f sm 0=4 cos — cos -^ cos — .

id til til

188.] To show that

tan A + tan B + tan C=tan A tan B tan C.

Let a( and alt be the segments of the side a, made by the per-
pendicular h drawn to it from the vertex A, then

„ a,a,. ACT
cot B cot C=4V = -re-*
h* h*

vr being the perpendicular from the orthocentre on the side a.
Hence cot B cot C= T-=T— • Let 8, &,, 8,, be the component tri-

IL fltt

angles of the original triangle, then

vra=28 and Aa=2A ;

g . * _|_g
hence cot B cot C + cot C cot A + cot A cot B = — ~—^ — '-'= 1 .

Multiplying these expressions by tan A tan B tan C,

tan A + tan B -f tan C =tan A tan B tan C. . . (a)

o- * A /(s-b}(s-c]

Since tan ^= A / i M > ,

& V s(s— a)

Again, as

tanftanftan^-^T^iTT^M- • <b>
222 s \/s(s — a)(s— b)(s — c) s

tan-=

Q ""

-v/*(s— a)(s—b) (s-c) ~~
A B C 4R + r

(c)

C C, A^ A, B ,

- tan— +tan — tan— =1 ; . . (d)

ON THK HICIIKR (i KOM KTKY. '.".I?

for tan-= - -., tan^=— -; therefore

2 s—b 2 s—c

tan — tan -= = - . Hence results the theorem.
22s

2A

189.1 Since A« = 2A. h = — , and therefore

a

I 1 1\ A (bc + ac + ab)
- -=2A^- -£=

(a)

- T -- -—

a b c/ abc 2R

$2 -Lr2_4R2

To show that ST + CT, + «•„= — -^ -- . . . . (b)

Let a;, ay/ be the segments of the side a made by the perpendicular
h from the vertex, -sr the corresponding perpendicular from the
orthocentre; then

„ «,«,. A-sr 2A«r
cos B cos C=-7-^=-^ = — T —
oc oc abc

r
(b) sec. [186] gives cos B cosC + cosAcos C -f cosAcos B =

S«_J_?.2_4R2

Hence & + &, + •&,!= -- ^ -- , as above ;

and therefore ( h + h, + h,) — («• + vr, + «rw) = 2 ( R + r) . . . (c)

But this quantity denotes the sum of the lines drawn from the
orthocentre to the vertices of the triangle ; and as it may be shown
that the sum of these distances is equal to twice the sum of the
perpendiculars on the sides of the triangle, these perpendiculars
being written

P>Pi>Pii> we shall have, as in (i) sec. [185],

p+pt+ptl=K + r ....... (d)

190.] In any triangle the sum of the reciprocals of the sides of the
six inscribed and exscribed squares is equal to twice the reciprocal
of the radius of the inscribed circle.

Let a be the base and h the height of the triangle, and x the

side of the square inscribed, then x—— —j .

Let X) be the side of the square exscribed, then a:;=— — .

112
Hence -H — =T. Let y, y., and z, zt be the sides of the squares

/ •'', //

on the other two sides of the triangle, and we shall have

as shown in (e) sec. [184].

VOL. n. 2 Q

298

ON THE HIGHER GEOMETRY.

ON TRIANGLES INSCRIBED IN ONE CIRCLE AND CIRCUMSCRIBED
TO ANOTHER.

191. Let the triangle ABC be inscribed in the circle AEBG and
circumscribed to the circle Ycok ; we proceed to find an expression
for the distance between O and w the centres of these circles. Let D
be this distance, and let R and r be the radii of the circles ; then
manifestly (R + D)(R-D) = Cw . G<o or D2=R2-Co> . Go>.

Through G draw the diameter GOE ; join AG and AE. Since
the triangles Cfoo and AGE are similar, GE . tak = Ca) . GA; but
GE = 2R, and wk = r, while GA = GB = Gw. Consequently
2Rr=Co> . Go>, and therefore

D2=R2-2Rr ........ (a)

The value of D is independent of the sides of the triangle. Hence,
if two circles be described so that the interval between their centres

shall be equal to \/}tf — 2Rr, any triangle inscribed in the one may
be circumscribed to the other *.

* Another proof of this theorem may be given. Let two tangents to the
inscribed circle be drawn from the points A and B meeting in C. If C be on the
circumference the proposition is established. But if not let another circle be
described passing through the points A, B, C, Let R, be the radius of this
circle, its centre will be on the line GE, suppose at O(, and let D( be the distance
from O, to &). Let OO, =/*, and let OT) = k ; then

But Di2 = Ri2— 2R,r ; or, substituting the value of R;,

and D2=R2-2Rr; consequently R + /* = V It2 + \?

which is impossible unless p,=0 ; or the two centres of the circumscribing circles

must coincide ; and as they pass through the same points A and B, they must be

identical.

ON THE HIGHER GEOMETRY. 2'J'J

192.] Let rt be the radius of one of the outer circles of contact;
then, making the necessary transformations, it may be shown that

D2 = R2 + 2Rr, ........ (a)

If we take like expressions for the other two sides we shall have,
adding them together,

But rt + rlt + rtn— r=4R, as shown in (j) sec. [181]; hence

D' + D' + Dj' + D^-iaR*; . . . . (b)

or the sum of the squares of the distances from the centre of the cir-
cumscribed circle to the centres of the four circles of contact is equal
to twelve times the square of the radius of the circumscribed circle.
It may easily be shown that G, the middle point of the arc AB, is
the centre of the circle which passes through A, B, the ends of the
base AB, and through the centres to and fl of the inscribed and
exscribed circles.

193.] If a triangle circumscribe one circle and be inscribed in
another circle, the circles will have a common pole and polar.

Let d be the distance from O the centre of the circumscribing
circle to the common polar, let 8 be the distance between the centre
of the inscribed circle and the common pole, and, as before, let D
be the distance between the centres of tne circles whose radii are
R and r.

Then obviously we shall have

(D + S)</=«R2, and (rf-D)8=r«.

Eliminating 8, we shall find for d, the distance of O from the
common polar,

, (R + r)(B-r)+B(R-2r)±r

CHAPTER XXIV

ON THE ORTHOCENTRIC TRIANGLE.

194.] The orthocentric triangle has been defined in sec. [lot] as
the triangle formed by joining the feet of the perpendiculars drawn
from the vertices of a triangle to the opposite sides ; and these [>er-
pcndiculars, as it has been shown, meet in the orthocentre.

The circle which circumscribes this triangle IIKIV bo called the
orthocentric circle. It has also by PONCELET been named the nun--
point circle, from a property which will be established further on.

300 ON THE HIGHEK GEOMETRY.

Let A, B, C be the angles of the given triangle, a, b, c the opposite
sides, and R the radius of the circumscribing circle.

The sides of the orthocentric triangle are a cos A, b cos B, c cos C.

Let Ap B;, C, be the vertices of the orthocentric triangle opposite
the vertices A, B, C of the given triangle, then the sides of the
triangle A^C, are a cos B, c cos B, and A]Cr Hence

KjC?=a* cos2 B + c2 cos2 B -2ac cos3 B,

or A^2= cos2 B [a2 + c2 - 2ac cos B] .

But the part put within brackets is equal to AC2 or ft* ;

hence A/^2=AC2cos2B, or b,=b cos B ...... - . (a)

We have also 2R = - — r. a well-known theorem.
sin A

But the sides of the orthocentric triangle are a cos A, b cos B,
c cos C ; and if Ay,Bp C; be the angles of the orthocentric triangle
opposite the sides a cos A, b cos B, c cos C, we shall have

A+2A=7r, or sin Aj= sin 2A = 2 sin A cos A. . . (b)

Hence, if Ry be the radius of the circle circumscribing the ortho-
centric triangle, we have

OT? _ a cos A. _ GCosA a

1 sin At ~ 2 sin A cos A 2 sin A'

Hence 2Ry^=R, or the diameter of the circle circumscribing the
original triangle is twice that of the circle circumscribing the
orthocentric triangle.

195.] To determine the area of the orthocentric triangle.

In general the area of a triangle A is determined by the equation
a£c=4RA, A being the area of the triangle.

In the orthocentric triangle the sides are a cos A, #cosB, ccosC,
and2Ry:=R; hence

abc cos A cos B cos C=4R,Ar
But «fo=4RA, and 2R,=R;

hence cos A cos B cos C = — L or -^ =2 cos A cos B cos C. . (c)

.-•£* ZA

If perpendiculars be drawn from the vertices of a triangle to the
sides of its orthocentric triangle, they will pass through the centre
of the circle circumscribing the given triangle.

As the perpendiculars drawn from the vertices of the given tri-
angle ABC on its opposite sides bisect the angles of the ortho-
centric triangle, the perpendiculars drawn from any two vertices
of the given triangle, A and B suppose, to the sides of the ortho-

ON THE HIGHER GEOMETRY. 301

centric triangle will make equal angles with the side C. Hence
by sec. [154] these lines will meet in a point; and as these three
lines are equal, they must meet in the centre of the circle ABC.

Hence, as the perpendiculars drawn from the vertices of the tri-
angle ABC to the opposite sides determine by their intersection
the orthocentre, so the perpendiculars drawn from the same vertices
to the sides of the orthocentric triangle determine by their inter-
section the centre of the circumscribing circle.

196.] Since the perpendiculars drawn from 0, the orthocentre, to
the sides of the original triangle bisect the angles of the ortho-
centric triangle, @ is the centre of the circle inscribed in it.

To find the value of the radius p of the circle inscribed in the
orthocentric triangle.

Let st be half the sum of the sides of the orthocentric triangle
and A/ its area, then Ay=s,p. But, as in sec. [194],

2s, = a cos A -f b cos B + c cos C ; hence 2s, = ^ (ap + bp, + cplt) , (a)

P> Pi> Pn bemg the perpendiculars drawn from the centre O on the
sides of the triangle.

But ap + bpl + cp,,=2&; hence p——1, and s,—^ ; . . (b)

Si Xv

therefore p=. But — i ! =2 cos A cos B cos C, as in (c) sec. [195];

hence p=2Rcos AcosB cosC. (c) We have also = -r-'> (d)

or the areas of the orthocentric and original triangles are to each
other as the radii of the circles inscribed in the former and circum-
scribed to the latter.

Hence, as (b) gives A = R.?/} it follows that the area of a triangle
is equal to the radius of the circumscribed circle, multiplied into the
semiperimeter of its orthocentric triangle.

We have thus an additional theorem for finding the area of a
triangle. This simple expression may be added to those given ill
sec. [181].

197.] To show that

OB _L — 9i? 2

2R+p_2R

The area of the original triangle is the sum of the areas of
the orthocentric triangle and the three component triangles on its
sides ; and twice the area of one of these triangles is be cos2 A sin A.
Hence

be cos2 A sin A + ac cos2 B sin B + ab cos2 C sin C + 2A,=2 A,

302 ON THE HIGHER GEOMETRY.

or be sin A + ac sin B + ab sin C

, fain8 A sin3B sin3Cl OA
— abc\ -- 1 -- - -- 1 -- =2A — 2A,.
La b c J

But be sin A=acsinB=a6 sinC=2A;

hence 4A

or, as RAy=Ap,

t j, A fsin3 A sin3 B sin8 Cl

hence 4A+2A,=4RA - — H -- - -- (-— L

La o c J

sin3C

(a)

198.] Tb determine an expression for the square of the distance
between the centre of the circumscribed circle and the orthocentre, or
an expression for O©2.

If we take the triangle whose vertices are O, ®, and one of the
vertices, A suppose, of the given triangle, the sides of this new tri-
angle will be 60, R, and 2R cos A, while the angle at A will be
C — B. Hence obviously

O@2=R2 + 4R2cos2A-4R2cosAcos(C-B). . . (a)
But A = 7r-(C + B)j hence

Oea=R2[l-4cosA{cos(C + B) + cos(C-B)}],

or O@2=R2[l-8cosAcosBcosC]. . . . (b)

In (c) sec. [196] it was shown that p=2R cos A cos B cos C.

Hence O@2«=R2-4Rp ........ (c)

Since -r*=2 cos A cos B cos C, see (c), sec. [195],

(d)

We have also O@2=9R2-(a2 + 62 + c2) ...... (e)

199.] The sum of the squares of the distances of the vertices of a
triangle to the orthocentre, diminished by the square of the distance
of this point from the centre of the circumscribed circle, is equal
to three times the square of the radius of the circumscribing circle.

Since A©2=4R2cos2A, we shall have

A®2 + B©2 + C@>2 =4R2(cos2 A + cos2 B + cos2 C) .

But cos2A + cos2B + cos2C = l— 2cos AcosBcosC, see sec. [187];

and as O02 = R2 [1 - 8 cos A cos B cos C],

subtracting,

(a)

ON THE HIGHER GEOMETRY. 303

Since p the radius of the circle inscribed in the orthoccntric
triangle is equal to 2R cos A cos B cos C,

Hence, adding twice this expression to the above, we shall have

=5R2-8Rp. . . . (b)

200.] If p, pt, plp pllf denote the radii of the circles inscribed and
Described to the orthocentric triangle, and if 22* = a* + b* + c2, we
shall have

_22-4R2 _22-a2 _22-62 Z2-c2

2R ' p> --- 2R~' P»-~ZEr> P"'=-MT' ' ^

Since p=2R cos A cos B cos C, as in sec. [196] ,

and 2 cos A cos B cos C = 1 — (cos2 A + cos2 B + cos2 C) ,

p= R [sin2 A + sin2 B + sin2 C - 2] .

But sin2 A = ?L ; substituting, p =

be
Again, since ft— ~ cos A sm A,

o

_OAcosA cosA
-2A-

hence

and like expressions for p,. and pui may be found.
If we add these expressions

or P + P/ + Pu + P///= . ..... (b)

V2 _ OTJ2

R

201 .] Since cos2 A -f cos2 B + cos2 C = 1 — 2 cos A cos B cos C and
p the radius of the circle inscribed in the orthocentric triangle

..«
is equal to 211 cos A cos B cos C, see (c) sec. [196] , while cos* AjMfcp

p being the perpendicular from the centre of the circumscribing
circle on one of the sides, then we shall have

202.] Three times the sum of the squares of the distances of the

304 ON THE HIGHER GEOMETRY.

centres of the four circles of contact from the centre of the circum-
scribing circle is equal to four times the sum of the squares of the
sides, and four times the square of the distance of the orthocentre
from the centre of the circumscribing circle.
In sec. [192] it has been shown that

o<»2 + on2 + on,2 + on,,2 = i2R2,

and O®2=R2(1 — 8 cos A cos B cos C), as in sec. [198] ;
but 2 cos A cos B cos C= sin2 A + sin2 B + sin2 C — 2 ;
Hence, reducing, O®2=9R2- («2 + 62 + c2) (a)

Substituting this value of O® the proposition is manifest; that is,

. (b)

203.] The squares of the sides of a triangle added to the squares
of the radii of the four exscribed and inscribed circles is equal to
sixteen times the square of the radius of the circumscribing circle.

In (f) sec. [181] it is shown that

2=4*2=4(r/r,, + r,,ry,/ + r,r,,,), . . (a)

and bc + ca-\-ab — rlru + rllrlll + rlrlll-\-r(4!R + r). . (b)

But 4R + r=r, + r,, + r,,,; ...... (c)

hence, subtracting twice (b) from (a), we get

a2 + 62 + c2 = 2 (r, ru + ru rul + r, rlu) - 2r (r, + ru + rltl) ;
and as 4R = r/ + rw + r///— r, squaring and subtracting,

16R* = r* + rwa + r//y2 + r8 + fl2 + £2 + ca. . . . (d)

204.] The sum of the squares of the sides of a triangle is equal to
twelve times the square of the radius of the circumscribing circle,
diminished by four times the sum of the squares of the perpendiculars
from its centre on the sides.

For a2 + 62 + c2 = 4R2 [sin2 A + sin2 B + sin2 C] ;

hence «2 + b2 + c2 = 12R2 -4R2(cos2 A + cos2 B + cos2 C),

or

205] In any triangle ^+- + -

, n .
the letters having the usual signification.

c. a a(s—a) 2*2 - (a2 + b2 + c2)

Since -=— — '-. the first factor is -- v ^ ^ '.
r, sr sr

ON THE HIGHER GEOMETRY. 305

But 2«a - (a9 + 62 + c2) = 2r(4R + r), see (1) sec. [181] ,
while r, + rw + r///=4R + r, and a-f 6 + c=2*.

Substituting these values we obtain the result.

206.] To determine an expression for ©GJ, the distance between
the centres of the circles inscribed in the original and orthocentric
triangles.

These centres and a vertex A of the original triangle constitute
the vertices of a triangle whose sides are r cosec ^A, 2R cos A, and
Sa>, while the vertical angle of this triangle is £(C — B).

^—2 4RrcosAcosi(C— B) , N

Hence©o>2=4R2cos2A + r2cosecHA -- jf -- -; (a)

sin ., - v

but cos4(C-B)=c-t-6

sin^A a '

1)1 _1_ (A _ fl2

while cos A = - ^^ --

2bc

Substituthig these values in the original equation, we shall have

0^2=4R2-8Rr + «6 + flC + 6c-(a«-f62 + c2). . . (b)
But it has been shown in (k) and (1) [sec. 181] that

be -f- ac + ab — s* + r9 + 4Rr,
and a* -f b9 + c* = 2s9 - 2r9 - 8Rr .

Hence ©w3=4Rr + 4R2 + 3r2— s* ...... (c)

Let fl, np S1H denote the centres of the exscribed circles ; then,
making the necessary substitutions, we shall find

), >. . (d)
a4-(a2 + 62 + c8

and as in the preceding formula (b)

Adding these expressions together, and bearing in mind that

r,+rw+rw/— r=4R, see sec. [181],
we shall have

8). . . (e)

VOL. II. 2 R

306 ON THE HIGHER GEOMETRY.

Let the distances of the orthocentre from the vertices of the tri-
angle be A®, B®, C© ; then we have

A®2=4R2-«2, B@2=4R2-62, C®2

substituting we obtain

©n2+©a2+©n2/+@^2=4(I02+B02+c©2). . . (f)

Since O02 = 9R2-(a2 + 62 + c2), see (e) sec. [198],

and 0^2 + OH2+Oll2 + On2=12R2, see (b) sec. [192],

therefore

=o« +oir+on;+on;;+4oej,

Hence the sum of the squares of the distances of the centres of the four
circles of contact from the orthocentre, exceeds the squares of the dis-
tances of the same points from the centre of the circumscribing circle
by four times the square of the distance between the orthocentre and
the centre of the circumscribing circle.

207.] A perpendicular is drawn from the vertex of a triangle on
the opposite side ; a line is drawn bisecting the vertical angle and
meeting the base ; and a circle is inscribed in the triangle. The dis-
tances from the middle point of the base to the foot of the perpen-
dicular, to the point of contact of the inscribed circle, and to the
point where the bisector meets the base, are in geometrical progression.
For these distances are, as may easily be shown,

c2-62 c-b a(c-b)
~^a~ ~2~~J 2 c + b '

When circles are exscribed to and inscribed in any triangle, each
side, a suppose, is touched in four points — in two, F F,, within the
angle A, and in two external to it. The circles, one inscribed, the
other exscribed, which touch the side a within the angle A are on
opposite sides of it ; and their distance is (c — b) or^(c—b] from M
the middle of a. The side a is touched by the two remaining
exscribed circles on the same side at two points, L and N, outside
the angle A, distant from the angles B, C, by s — a ; and the distance
between these two points L and N is 2(s — a) +a = c + b; and the
distance of L and N from the middle point M of a is ^(c + b)

208.] If a, /3, 7 be the median lines of a triangle whose sides are
a, b, c, we shall have the following relations between these lines : —

(b)

ON THE HIGHER GEOMETRY. 307

By a well known theorem

262 + 2c2-a2 = 4«2 ........ (c)

Finding analogous values for /32 and 72, and adding, we obtain

) ...... (d)

If we square (c) and the other like expressions, and add them,
we shall have

(e)

If we square the expression (d) and subtract (e) from it, we shall
find

16(/3272 + aV + *W = 9(62c2 + «2c2 + a2i2). . . (f)

209.] If through the points of contact A,, Bp C, of the circle in-
scribed in a given triangle perpendiculars are drawn to meet the cor-
responding median lines in the points I, m, n, we shall have

_L. _L J__2
^~

The distance between the middle point of the base and the foot

c2 — b*

of the perpendicular on it from the vertex A is — » --

iia

The distance between the middle point of the base and the point

c—b
of contact of the inscribed circle is — ^— ; and their distances are as

h and A/.
Hence A,l : <

h-C~b

«i a
»/ — b

-~ n ' f \ h •

2

2a
1 c + b c + b.

2A '

consequently _+_+_ = _ = -.

210.] To determine the distance between the centroid K, and the
centre a> of the inscribed circle.

Let us take the triangle of which the vertices are tc, A, a>. Now

the sides of this triangle are w/e, - ^-r and '{a, * being the median

sin 2^*-
line from A to the opposite side.

Let 6 be the angle between the median line a, and the side r.
suppose.

308 ON THE HIGHER GEOMETRY.

Then, as the median line bisects the triangle ABC, we shall have

• a T. • /A a\ o c + bcosA

cam 9=0 Bin (A— 0), or, as 2«=

/

, .
(a)

COS (

b sin A. ,, c+6cosA

Let e be the angle between the median line 2« and the bisector
of the vertical angle A ; then we have

e = (^ A — 6], and 2« cos e = (b + c) cos ^ A.

Now

sm2^A ' sm-^A

_ be + ab + ac (a2 + 62 + c2)

. . . . (b)

But as 2s2 - 2r2 - 8Rr = a2

2 4

Substituting this value of 4Er in the preceding equation, we get

^2 = /¥(a2 + 62-|-c2)-|(6c + «c + a6)+r2. . . (d)
In the same way, making the necessary substitutions, we shall have

+ ^ (ab + bc- ac) + r,*, - • (e)
n^c2 = ^ (a2 + b* + c2) + £ (fa + ca - ai) + rw
Adding these expressions together, we shall have

+r2 + r,2 + r/2 + r///2. (f )

But it has been shown in (d) sec. [203] that

r2 + rf + r/y2 + r^= 16E2 - (a2 + b2 -f c2).
Hence, eliminating, we obtain

|(a24-A2 + c2). . . (g)

211.] To determine an expression for the distance between the
centroid and the centre of the circumscribing circle.

Taking the triangle whose vertices are O, K, and the middle point
of the base, the sides of this triangle are OK, R cos A, and ^«, while

the cosine of the angle opposite to OK is -. Hence we have

ON THE HIGHER GEOMETRY. 309

2 A

... (a)

*J 6*

,. / fft, _i A2 I /»2\

Reducing, we find O/c =R2 — - — — ' (b)

9

Comparing this expression with that found for the distance of the
orthocentre from the centre of the circumscribing circle, we shall
have

O® = 3O* (c)

212.] The sum of the squares of the twelve lines drawn from the
angles of a triangle to the points of contact of the circles of contact
in the opposite sides is equal to five times the sum of the squares of
the sides of the triangle.

Let the side BC of the triangle be produced to L and N, so that
BL = s — a, CN=s — a; then it may easily be shown that L and N
are the external points of contact, while the distance between F and
Fy, the internal points of contact is c — b. B ut a + * — a + s— a= c + b.
Let AM =«, where M is the middle point of the side a (see fig. 29) ;

then AL2 + AN2 = 2a2 + £ (c + 6)2, and AF2 + AF/5 = 2a2 + ± (c - 6)2;
therefore

Making similar constructions on the other sides ; the sum of the
squares of the twelve lines' will be found equal to

But 4(

Hence the sum of the squares of the twelve lines is equal to

213.] The sum of the squares of the twelve lines drawn from the
middle points of the sides of a triangle to the centres of the circles of
contact, together with the sum of the squares of the sides of the tri-
angle, is equal to twelve times the square of the diameter of the cir-
cumscribing circle.

Let the lines drawn from the middle points of the sides a, b, c
to the centre XI of the exscribed circle opposite the angle A, be
«, j3, 7, and that to to the centre of the inscribed circle be S.

Then

2y9 + ic9; . (a)

310 ON THE HIGHER GEOMETRY.

adding these expressions, and dividing by 2, we have

But

and

adding these expressions, we obtain

) „

j

Writing analogous expressions for the two other centres flt and
we shall have

3(r * + ry/2 + rtlf + r2) + 3 (a2 + i2 + c2) = (a2 + /32 + y2 + S2)

But r- 2 + r,,2 + rw/2 + r2 + a2 + bz + c2= 1 6R2, see sec. [203]

Hence, substituting, we find,

214.] The sum of the areas of the four triangles formed by joining
three by three, the points of contact of the circles of contact is constant,
and equal to twice the area of the given triangle.

The area of the triangle formed by joining the three interior
points of contact must be taken with the negative sign.

In the first place let us take the triangle whose vertex is A and
base a, and construct the triangle whose vertices are the points of
contact of the exterior circle of contact with the side a, and b, c
produced. Then twice the area of this triangle is manifestly

s2 sin A— (s — 6)2sinC— (s — c)2sinB — fosin A ;

and if we make like constructions for the other angles B and C of
the given triangle,

s2 sin B— (s— c)2 sin A — (s — a)2 sin C — ac sin B,
s2 sin C — (s— a}* sin B — (*— 6)2 sin A— ab sin C,

will be twice the areas of the two other triangles.

Let us first combine those terms of which sin A is a factor, or

which may be reduced by obvious substitution to

[4Er + r2]sinA.

ON THE HIGHER GEOMETRY. 311

Making like reductions for the other angles B and C, we get for
twice the sum of the areas of the three triangles

(4Rr + r2) (sin A + sin B + sin C) .

o

But sin A + sinB + sinC = — see (d) sec. [185].

Jtv

Hence twice the sum of the areas of the three triangles is

r*s
Now 4rs is twice the area of the given triangle, and -^- is twice the

Di

area of the triangle whose vertices are the points of internal contact.
215.] In a triangle ABC, let the internal bisectors of the angles
A, B, C meet the opposite sides in the points A/} By, Cp and let the
external bisectors of these angles meet the same sides in the points
A//} Bw, C/y; then, if a>b>c, we shall have

BB CC

A, A,, Ipw C,CW~ 8R2(a + b + c)
Now c : 6=BA,: CA,, or

but as the angle AyAAw is a right angle,

A A
cos AA..B =— — J/=sin AA.B.

. A/A«

But sinAA^ : sin£A = c : BA;.

rm. e / • /.\ a sin AA.B

Therefore (c + b}= - : — TT^-^

sin ^ A

or, putting for sin AA;B its value,

(c + b) . . A A,,

i - '- sin i A = -r- r".

a V;/

Finding like expressions for the other two sides, and bearing in
mind that

m

4 sin ^A . sin £B . sin ^C = ^5,

K

we obtain the theorem.

216.] To find expressions for the sides, angles, and areas of the
excentral triangles, £l£lfln, fl(aflt, fl/ofl^, fiy/uft.

Since (fig. 31) BF=(«— a) is the projection of HB, therefore

In like manner we obtain

312

ON THE HIGHER GEOMETRY.

o __ ft _L o ^m, /»

Therefore HB + BH.=flQ.= . ,* , or the side

sm^B

Let R be the radius of the circle circumscribing the triangle ABC ;

b=2R sin B=4E sin £ B cos £ B.
Hence HH,=4Bcos£B ........ (b)

In like manner nn;/=4B cos^A, and H;fl//=4B cos^C.
Hence the semiperimeter S of the excentral triangle is

S = 2B(cos£A + cos^B + cosiC) ...... (c)

The area of this excentral triangle may be found. For this area
is equal to ^illl/ . 1111^ sin ^ (A + B) ; or, substituting for these ex-
pressions their values, we have

Area of excentral triangle = 8R2 cos £ A cos £B cos £C . . (d)

ON THE HIGHER GEOMETRY. 313

Since 4cos£Acos£B.cos£C = sinA + sinB-|- sin C, see sec. [187],
this area is equal to 2R2 (sin A + sin B + sin C) ; but

Hence the area of the excentral triangle is equal to

(e)

This expression coincides with that found for the area of a tri-
angle in sec. [196] ; for ABC is the orthocentric triangle of the
excentral triangle Ofl;flw, the radius of whose circumscribing circle
is2R.

217.] The area of the excentral triangle is

8R2 cos £A cos £B cos £C,

and the side opposite the angle A is 4R cos £A ; hence the perpen-
dicular from the vertex A on the opposite side is

4Rcos£Bcos£C ......... (a)

The radius p of the circle inscribed in the excentral triangle may
be thus found. Since the radius of the inscribed circle is equal to
the area of the triangle divided by its semiperimeter, therefore

1 _ cos ^ A + cos ^B -4- cos ^C „ .

P~4R cos £A cos £B cos£C*

218.] To find the values of flm, fiyo>, flw«. Since the projection
of n&> on the side a is equal to c,

c 2RsinC .,, . ,.~

therefore llw = - -T7= = — -r-^ =4Rsm*C ..... (a)
cos £C cos £C

In like manner n,a> = 4Rsin£A and n//<u=4R sin £B. . (b)
Hence flfy . fl,^ . nnw .Hw .fi/u . n/ya>=64R8a£c. . (c)

To find the area of the triangle flwH^ The area of twice this
triangle is Oa> . tip sin £(A + C). Hence this area is

8R2 sin ^A sin ^C cos ^B, which may be put under the form
8R2 cos A cos B cos C tan

Finding like expressions for the two other component triangles
of the triangle ftn,Q/y we shall have for the sum of the three,

8R2cos £A cos £B cos £C[tan ±B tan ^C + tan

+ tan£

VOL. II. 2 8

314 ON THE HIGHER GEOMETRY.

But the sum of the terms within the brackets is equal to 1, as
shown in (d) sec. [188] .

219.] The square of the distance between the centres of two of the
exscribed circles of a triangle exceeds the square of the sum of their
radii by the square of the opposite side of the triangle.

Let the exscribed circles be taken which are opposite to the angles
A and C of the given triangle,

sr sr

then we have r,= - and r.,=

—

s — a s — c

,, srb

consequently r.+r,,= 7 -- ^7 -- r =

1 " (s—a)(s — c)

or

Let nn,, be the line which joins fl and fl, ; then the projection of
liB on the side c is s — c, and the projection of Bfly on the side a is
(s — a) ; consequently the sum of these projections is b, or

Oil, sin $8 «*.

Hence

220.] The sum of the squares of the tangents drawn from the centres
of the four circles of contact of a triangle, to any circle which passes
through the centre of the circumscribing circle, is equal to three times
the square of the circumscribing diameter.

Let co, fl, £1,, £ly/ be the centres of the four circles of contact,
and O the centre of the circumscribing circle through which the
diameter HD perpendicular to the base BC passes.

Let Q, be the centre of the arbitrary circle passing through O ;
and draw the lines Qo>, Qfl, Qfl/3 QH/p QO, QH, QD, H«o.

Then QfT + Qo>2 = 2QD2 + 2DC2, since DC = Deo,
and

But 2QD2 + 2QH2=4r2 + 4R2,

and 2DC2 + 2HC2 = 8R2 ;

therefore

~"-r2)=12R2.

But these expressions are the squares of the tangents drawn from
the centres of the circles of contact to the circle whose radius is r.

When r — 0, or the arbitrary circle vanishes to a point, we get
the theorem established in sec. [192].

ON THE HIGHER GEOMETRY.
Fig. 32.

315

221.] If the sides of the excentral triangle fmfln be produced,
and circles of contact be drawn touching the sides of this triangle,
and the centres of these new circles of contact be joined so as to form a
new excentral triangle, and if this process of construction be conti-
nued, the successive excentral triangles will approximate to an equi-
lateral triangle.

Let A, B, C be the angles of the given triangle ; A,, B,, C, the
angles of the first derived triangle, A,,, Bw, Cu the angles 01 the
second derived excentral triangle, and so on ; then

Therefore

B,-A,:=±(A-B), C,-B,=*(B-C), C,-A,=i(A-C).

Hence the differences between the angles of the first derived ex-
central triangle are one half those between the angles of the original
triangle.

Again as A^i^ + C,), B, = *(C,+ ,A)> C,,= i(A,+B,),

Hence the difference between the angles Aw and Bw is one fourth
of the difference between the angles A and B. The same is true for
the other angles. Hence the successive excentral triangles approxi-
mate to an equilateral triangle.

316

ON THE HIGHER GEOMETRY.

CHAPTER XXV.

ON THE NINE-POINT CIRCLE.
DEFINITION.

The circle which passes through the feet of the perpendiculars
drawn from the vertices A, B, C of a given triangle to the opposite
sides has been named the Nine-point circle.

The properties of the Nine-point circle are unquestionably the
most remarkable and elegant in the entire range of plane geo-
metry. Some of the leading properties of this circle were discovered
by PONCELET in the early part of the present century. It is a sin-
gular fact that the theory of the Nine-point circle escaped the notice
not only of the ancient geometers but of modern mathematicians
almost to our own time — another proof, were another wanting, how
inexhaustible are the truths of geometry, and how many yet remain
to be brought to light.

222.] The nine-point circle passes through the middle points of the
sides of the triangle ABC.

Fig. 33.

Let the nine-point circle which passes through the points A,, By, C,

ON THE HIGHER GEOMETRY. 317

cut the sides of the given triangle ABC in the points A;/, B/y, Cw.
Join A/yC/r As AyCyC^A^ is a quadrilateral inscribed in the nine-
point circle, the angle BC/;A/y is equal to the angle BAyC,, which
is equal to the angle BAG, since AyCyAC is also a quadrilateral that
may be inscribed in a circle. Hence, as the angle BC/y A/; is equal to
the angle BAG, AyyCyy is parallel to AC a side of the given triangle
ABC. In the same way it may be shown that AWB;/ is parallel to
AB and BWCW parallel to BC. But when a triangle inscribed in
another triangle has its sides parallel to those of the latter, it obvi-
ously follows that the vertices of the former will be on the middle
points of the latter.

This is a particular case of a far more general theorem which
will be given further on.

223.] The distances of the orthocentre ® from the vertices A, B,
C of the given triangle are double the distances of the centre of the
circumscribing circle from the opposite sides a, b, c.

From C draw the diameter COD; then CBD is a right angle.
Join AD, then CAD is a right angle, and therefore AD is parallel
to BBy while A® is parallel to BD, each being perpendicular to
BC. Therefore A®BD is a parallelogram ; and therefore A® = BD.
But BD is equal to 2A/yO, since BC=2CA// ; hence A© is equal to
twice A,yO.

Bisect A® in -or, and join wA/y meeting O® in v; then, as
yy=®w, wv is equal to Ay/v, and Ov is equal to ®K.

Now v will be the centre of the nine-point circle. For v is the
intersection of the perpendiculars drawn through the middle points
of AyAyy, ByByy, CyCyy the choYds of the nine-point circle.

Since A® is equal to twice AWO, A« is equal to twice Ayy/e, or K
is the centroid of the triangle ABC.

Hence the line which joins the centre of the circle circumscribing
the triangle with its orthocentre passes through the centre of the
nine-point circle and the centroid.

Since A«r is equal and parallel to AWO, OA is equal and parallel
to Ayy-sr. But OA is the radius of the circumscribed circle, and Anvr
is the diameter of the nine-point circle ; hence the radius of the
circumscribed circle is equal to the diameter of the nine-point circle.

As the orthocentric or nine-point circle passes through the feet
of the perpendiculars drawn from the vertices of the given triangle
to the opposite sides, through the three middle points of the
sides of this triangle, and through the three middle points of the
lines which join the orthocentre with the opposite vertices A, B, C
of the given triangle, this circle has therefore been called the nine-
point circle.

The angle A of the triangle BAG is equal to the angle BDC ;

"Rf

and BC = CD sin CDB ; hence CD or 2R= -r— • r .

sin A

318

ON THE HIGHER GEOMETRY.

ON THE TRIANGLES WHOSE VERTICES ARE, THREE BY THREE, THE FOUR
CENTRES OF THE THREE EXSCRIBED AND THE INSCRIBED CIRCLE.

224.] («) In the given triangle ABC (fig. 34) let a circle be con-
ceived to be inscribed whose centre is o>.

Let n, Oy, flw be the centres of the circles of contact. Join OB,
y ; then, as B&> bisects the internal angle B, and BO bisects the
external angle B, these bisectors Bw and BH meet at right angles,
and therefore OB and Bf^ are in a straight line.

In the same way it may be shown that Hfiw and £lft,lt are in a
straight line.

This may be called the principal excentral triangle.

ON THE HIGHER GEOMETRY. 319

(|3) There are three other excentral triangles, whose vertices are
il, H,, a), iiy, Q,tl, a>, and XI, Xl/;, o>.

(7) The sides of these three triangles also pass through the ver-
tices of the given triangle ABC.

(8) The circles which circumscribe these four triangles are all
equal.

It is shown in the last section that the diameter of a circle circum-
scribing a triangle is equal to a side of the triangle divided by the
sine of the opposite angle.

But — -_ ''^r-=- — n-'-jT, since Aa>B is the supplement of the
sin XlX^il,, sm Xlo>Xiy/

angle AXiBr

(e) The triangle ABC is the orthocentric triangle of the excen-
tral triangle X2X2 X2M ; and a>, the centre of the circle inscribed in it,
is the orthocentre of the triangle Iiiiyiiw.

This is evident ; for AH, BU/y, CX2; are perpendiculars drawn
from the vertices Xlfl^Xi, of the excentral triangle to the opposite
sides, all passing through the orthocentre o>.

225.] Any one of the four centres of the circles of contact is the
orthocentre of the triangle whose vertices are the other three centres
of the circles of contact.

Thus w is the orthocentre of the triangle XHl^Xl,,, XI is the ortho-
centre of the triangle Q.t<aQ,lt, Xi, is the orthocentre of the triangle
QLtfaQ,, and X2;/ is the orthocentre of the triangle XlcoXl,.

This is evident from an inspection of the figure.

226.] Since the perpendiculars drawn from the vertices of a tri-
angle on the sides of its orthocentric triangle meet in a point (the
centre), it will follow that

If twelve perpendiculars be drawn to the sides of the triangle ABC
from the four centres of the circles of contact, these perpendiculars
will meet three by three in four points, and these four points will be
the centres of the circles which circumscribe the four excentral tri-
angles.

This follows from sec. [195] ; for the perpendiculars on the sides
of the common orthocentric triangle from the four centres of
the circles of contact make equal angles with the sides of the tri-
angles X1X1X1,,, XleoXl,, flpQ,,,, and Xly/toX2, and therefore the product
of their sines taken three by three are equal.

227.] Since the given triangle ABC is the orthocentric triangle
of the triangles XlXl^, XlruXl,, flfoO^, and XlajXl,,, the radius of the
circle which circumscribes ABC is one half the radius of the circle
X2Xi(X2/y, or its equals Xla)Xly, X^aXl,,, and X2y/&)Xl.

228.] The nine-point circle ABC bisects all the vectors drawn from
the orthocentre to the circumferences of the circles which circumscribe
the given triangles nn;O;/, Htufl,, Zlp>flH, and Ho)!^.

Let w (fig. 35) be the orthocentre of the nine-point circle ABC

320 ON THE HIGHER GEOMETRY.

to the triangle &&,&„. Let v be the centre of the nine-point circle ;

Fig. 35.

therefore v is the middle point of the line Oa>, as shown in the pre-
ceding section ; and as the radius of the circle which circumscribes
the triangle Q&flu is twice that of the nine-point circle ABC,
OT is equal to twice vr ; hence OT is parallel to vr ; and therefore
toT=TT; consequently cay =jflt, o>A = A'7r, Q)C = CC,, o)A,=Afl.

If we take the triangle Hcoil, of which Hw is the orthocentre, and
O; the centre of the circle circumscribing it, then, as OyM is equal
to twice vjjb and fl^O, is equal to twice &,,v, the triangles Qltvfj, and
HyyOjM are similar. Hence fi/;//.=M/i. Thus the nine-point circle
bisects all the vectors drawn from O 7 the orthocentre to the circum-
ference of the circle which circumscribes the triangle £l<a£lt.

229.] The lines drawn from the orthocentres of the four excentral
triangles to the centres of the circles which circumscribe these triangles,
all four pass through the centre of the nine-point circle.

This is evident ; for a>O, ilwOp &c. all pass through v.

230.] If from the centres £1, £ll} £lu of the circles of contact
straight lines be drawn to the middle points of the opposite sides of
the triangle ABC, these lines being produced will meet in a point.

ON THE HIGHER GEOMETRY. 321

In fig. 31 let I be the middle point of the side BC. Then the
area of the triangle flBI is equal to that of the triangle HCI.
But twice the area of the triangle fiBI is equal to HI . HB . sin Bill,
and twice the area of the triangle flCI is equal to HI . HC . sin CHI.

Hence HI . OB . sin BOI = OI . OC . sin CHI ; or, dividing by HI,
we shall have

8mBHE_OC_cos/3

sn
Finding like expressions for the centres fl, and £l(l we shall have

sin Bfll . sin CQ^T, . sin AIl/T//_ cos /3 cos y cos «
sin CHI . sin Bn,Iw . sin AO^I," 0037 cos a cos/3 ~

But it has been shown in sec. [153] that when three lines are
drawn from the vertices of a triangle, making with each side pairs
of angles so that the continued product of the three sines of the
angles of one triad is equal to the continued product of the three
sines of the angles of the alternate triad, these lines will meet in a
point.

ON THE RADICAL CIRCLES OF A TRIANGLE.

231.] If on the six lines, as diameters, which join, two by two,
the four centres of the circles of contact of a triangle, namely eofl,
wfl,, a>nw, £l£l,, flfia, nn;/, six circles be described, it may be shown
that the centres of these circles (see fig. 36) range along the circum-
ference of the circle ABC..

Dividing these diameters into two sets, those which end in the ortho-
centre o>, and those which end in the centres H, H,, flw of the external
circles of contact, and which may be called the inner and outer
diameters, the centres of the inner radical circles are on the middle
points N, N,, N/y of the arcs AB, BC, C A, while the centres of the
outer circles are on the points of bisection M, M/f Mw of the supple-
mental arcs of AB, BC, CA ; so that the six centres of the radical
circles are on the circumference of the circle ABC, and on its tluvr
diameters which are perpendicular to the sides of the triangle ABC.

The sides of the triangle are radical axes of each pair of outer and
inner circles, while the orthocentric perpendiculars are radical axes
of each pair of inner circles.

If from any angle ft of the excentric triangle tangents be dra\vn
to the circles HyB A and £1WCA, these tangents will be equal ; for
their squares are manifestly equal to the rectangle Afio>.

It is evident that o> is the radical centre of the three circles.

Since o> is the orthocentre of the triangle ftn,Qw, o>N,=Nnw
and ft)N = NH; therefore N,N is one half of flfl, and parallel to
it. In like manner since flw is the orthocentre of the triangle
,, nMM = MH, and H</M//=nM//; therefore MMM is one half

VOL. II. * T

322

ON THE HIGHER GEOMETRY.

Fig. 36.

of fin; and parallel to it. Hence MMW=NN,. In like manner
NM = N;My/, since each is equal to one half Oy/o>, and NMM^N,
is obviously a rectangle of which the sides are
2RcosiB and 2Rsin^B.
232.] In sec. [216] it has been shown that

and in sec. [217] that

11,6) = 4R sin ^ A, Hw&> = 4Rsin^B, and flea = 4R sin i|C. (b)
If we square these expressions and add them, two by two, we
shall have

and

ON THE HIGHER GEOMETRY. 323

Therefore the square of a side of a triangle, and the square of the
distance of its orthocentre from the opposite vertex are together equal
to the square of the diameter of the circumscribing circle.

In sec. [216] it has been shown that, if S denote the semiperi-
meter of the excentral triangle,

S = 2R (cos £A + cos |B + cos

So also, if S, denote half the sum of the three lines drawn from
the orthocentre &> to the vertices of the excentral triangle,

S,=2R(sm £A + sin iB + sin £C) ;

such are the geometrical interpretations of these trigonometrical
expressions.

If we square the expressions in (a) and (b) and add them, we
shall have

^?— 16R2(cos2 £A + cos2 £B + cos

and this expression becomes by reduction 8R(4R + r).
In like manner we have

n/»2 + JV>2 + Ho? = 8R (4R - r) .

These expressions when added give the result obtained in (c).

Hence the sum of the squares of the sides of the excentral tri-
angle is equal to 8R(4R-f-r), and the sum of the squares of the
lines drawn from these vertices to to is equal to 8R(4R— r).

233.] The radical axes of the circles inscribed and exscribed
to any triangle intersect each other, two by two, at right angles, in
the middle points of the sides of the triangle, and are parallel to
the sides of the principal excentral triangle.

Let ABC be any triangle, o>, fi, H,, fln the centres of the
inscribed and exscribed circles; then the twelve circles described
about the component triangles of the complete quadrilaterals £lflfi)£lu,
flflcoSl,!, and fl^afl will intersect four and four in ABC, and
their centres will lie two and two in six points on the circumscribing
circle.

234.] The nine-point circle of a triangle touches the inscribed and
the three exscribed circles.

Let O (fig. 37) be the centre of the circle circumscribing the
triangle ABC, and let v be the centre of the nine-point circle which
passes through D the middle point of AB, and through «r the middle
point of PC. Then D«r = R the radius of the circumscribed circle.
Let &) be the centre of the inscribed circle whose radius is r, and
which touches the base AB in the point F. Let Q. be the foot of
the perpendicular CP on AB. Join DG>, and let fall on it the per-

324

ON THE HIGHER GEOMETRY.
Fi". 37.

pendicular «ru. Let the distance vco between the centres of the
nine-point circle and the inscribed circle be d, and let e be the

/ T\ g

angle between Dv and Dca ; then, since Do>2= ' — -r— — + r2,

TJ

2

or, putting k for F^ = V (a — 5) 2 + 4r2,

4fi?2=R2+(«-6)2 + 4r2-2

As 2DQ=acosB-6cosA=-^, and DF=,

(a)

-
2c 2

. (b)

ON THE HIGHER GEOMETRY. 325

On Dw let fall the perpendicular FX and produce it to meet PQ
in K. Now Q*=FQtan QF/e. But as in (b)

and tanQF«=-, substituting

.......

2cr

Let the angle KCG=KHC=0,
then KG=KC sin 0, and KC=2R sin 0,

or KG=2Rsin20 ........ (d)

. ra2 -

Sm0=

2Rsin20= (a-b)* (a-b)*(s-c)

2cr

consequently

2cr

Hence, as GU=KG, C/c=KD, and OT*=R.

This is a new as well as an important property of the circle.

As Du is the projection of DOT or R on the line Do>, and as it is
also the projection of VTK of R and DF on the same straight line, we
shall have

Rcose=Rsin & + $(a— b)cosS, (g)

putting 8 for the angle o>DF.

Nowsin8=-r-, cos8=— r— , where k= *J(a — A)2-|-4r8.

Hence 2R/t cos 6=4rR+ (a—b)*.

Substituting this value of 2R cos e in (a) , we shall obtain

Reducing, this becomes e?=£R— r (h)

235.] Let dt, dn, din denote the distances of the centre vof the
nine-point circle from the centres of the exscribed circles ; we shall
then have by making the necessary transformation of the figure,

rf^R+r,, rfw=iR + rw, dw/=*R + rw; ... (a)
adding these results, we shall have

r// + r/w-r. . . . (b)

326 ON THE HIGHER GEOMETRY.

Now it has been shown in sec. [192] that if D, D/3 T>,,, D/;/ denote
the distances of the centre of the circumscribing circle to the same
four points,

consequently

rf//,). . . . (d)

Hence the sum of the squares of the distances of the centre of the
circle circumscribing a triangle to the centres of the inscribed and
exscribed circles divided by the diameter is equal to the sum of the
distances of the centre of the nine-point circle to the same four
points.

Another proof of this important theorem may be given.

236.] Let ABC be the given triangle as before, circumscribed
by the circle whose radius is R, and whose centre is at O. Let F
and F, be the points in which the inscribed and exscribed circles
touch the base AB or c.

Then BF,=,s— a, &F=s—b.

Let v be the centre of the nine-point circle, and eo that of the
inscribed circle ; join CF,. It may easily be shown that this line
CF/ or/, will pass through i the extremity of that diameter of the
inscribed circle which passes through F its point of contact with
AB. Let D-BT be the diameter of the nine-point circle ; then, as
OD is equal and parallel to C«r, OC or R is equal and parallel to
D-BT, and as r : DF=2r : FF,, DOT is parallel to CFy or to/;, writing
ft for CF,. Hence the angle OCF/ is equal to the angle j/Do>. Let
this angle as before be e, and let OF, be u ; then,

since AOB is an isosceles triangle, R2=w*+(s — a}(s— b). . (a)

But w2=R2+//2-2R//cose; ..... (b)
consequently

2/,R cos e=/,2 + (*-«)(« -b) ..... (c)

Let 8 be the angle which CF, or /, makes with AB the base of
the triangle; then, as CF, or/y is parallel to Da>,

. . . . (d)

But FF,=«-£, and FQ=^ _!', see (b) sec. [235] ;

o

therefore/, cos 8=- (a—b], and consequently

C

ON THE HIGHER GEOMETRY. 327

and therefore, substituting for,/] its value in (c),

sk* c
2R£cose= -- !--(*— a}(s— b) ..... (f)

C 8

Now, as before in (a) sec. [234],

4C?2 = R2 -|- (a - 6)2 + 4r2 - 2Ek cos e ;
eliminating cos e between these equations,

But c- (s-a)(s-b}=cs-c(a + b}+—.

8 S

Now =4Rr, and —c(a + b) = c2-2*c;

5

making these substitutions, the equation becomes

4^= (R-2r)2+ (a-i)2-? [(a-&)2 + 4r2] -c*+sc,

c

or 4rf2=(R-2r)2+[(a-£)2-c2]-- [(a-b)*-c*]- — .

c c

Reducing, the final equation becomes as before,

<*=*R-r ......... (g)

237.] The demonstration of the case when the exscribed circle
touches the base of the triangle differs but little from the preceding.

Join O the centre of the circumscribing circle with F the point
in which the inscribed circle touches the base ; then, as before,

R2=w2+(s-fl)0-i).

Now as CI meets the circumference of the exscribed circle in
the point I the extremity of the diameter F.H, and as in = HF,
and F;D = DF, the line Dfl is parallel to CF or to/, and OC is
parallel to ~Dv as before. Let the angle OCF in the triangle OCF
be put e, then the angle ODi/ in the triangle flDv is TT— e, since
the sides of this triangle are parallel to those of the former. Now
in the former triangle, putting u for OF, as in the last section,

M«=R2+/2-2R/cose,
and R2= w2 + (* - a) (s - b] .

But in the triangle £lDv, putting £lv=d,,

(7r-e); (a)

328 ON THE HIGHER GEOMETRY.

or writing k, for [(«— 6)2 + 4r/2]*, and substituting for cose the
value found above, we shall have

. (b)

s — c

•vr i 9 . i j.\9 79 j («—«)(* — b)c abc

Now 4r,2+ (a— 6)2=£/% and i - ^ - t-= -- $<•

5 — —

ile «ic=4Rsr. But sr=(s-— )cr, : he
Introducing these values we shall have

while «ic=4Rsr. But sr=(s-— )cr, : hence - = 4Rr,.

—

or 4d2=(R + 2r,)2--

c

V2.
*

But c*-(a-b}*=(c + a-b}(c + b-a)

A/ \t i\ j fs—c\ . o 4s.s — a.s—b
=4>(s — a)(s—b), and ( - )4r/2=— -;

hence the expression now becomes

4? 4?

— (*-«)(*-£) +>-«) (*-£), . (c)

or ,

The lines /, f, drawn from the vertex C of the triangle to the
points of contact F, F, in which the exscribed and inscribed circles
touch the base c of the triangle are of much importance. It will
be shown further on that these lines also pass through the extre-
mities of the diameters which pass through the points of contact of
the two focal spheres with the plane of the conic section — the foci.
These lines may therefore be called the vertical focals of the conic
section.

Let r and r, be the radii of the inscribed and exscribed circles to
the base c.

Let 4r * + (a - b}* = kf, 4r2 + (a - b) 2 = k2. Then it may easily
be shown that

f,='-k, and/=^*y.

C- I/

If we put h and h, for the distances of the vertex C of the triangle
to the other extremities the diameters of the inscribed and exscribed
circles, we shall have

. (s — c) , - . s ,
h.— - - k, and h=-k..
c c '

Hence also we hsvejO^ssAA,, or the area of the triangle CFF/ is

ON THE HIGHER GEOMETRY.

equal to the area of the triangle iCI, i and I being the other extre-
mities of the diameters of the inscribed and exscribed circles.

These focal lines/ and/,, passing through F and F/, the bisector
of the vertical angle of the triangle, and the perpendicular from the
vertex on the base of the triangle constitute an harmonic pencil.

The distances F* and F^ from the point * the foot of the bisector

8 S ' _ C

of the vertical angle are - (a— b) and - —(a— b). Hence the

C C

bisector of the vertical angle divides the distance between F, F,
the focal points of the triangle in the ratio of *:* — c; that is (as
s : s — c = rt : r), in the ratio of the radii of the exscribed and inscribed
circles.

238.] A trigonometrical proof of this theorem may be given.

As in fig. 37, let O be the centre of the circumscribing circle,
v that of the nine-point circle, and o> the centre of the inscribed
circle; and let va>=d.

Let the angle vDA. be 7, and the angle <0DA be S ; then as DP
is parallel to the diameter 2CO, and the angle COK is equal to the
difference between the angles A and B, we shall have ^TT— y = A — B.
The radius Dv of the nine-point circle is equal to £R ; and

2D<»=[(a-&)a-r4r2]*=A ...... (a)

Let e be the angle between the sides of the triangle Dv and Deo,
then e = y — B, arid

-8). . . (b)

But as^Tr— y=A— B, cos7=sin(A— B), and siny=cos(A— B) ;
hence cos e= cos (A— B) sin 8 + sin (A— B)cos8 ..... (c)

Now cot 8=^-, and cot £B— cot £A= — -= -- j

2r r r r

therefore 2cot8=cot£B — cot^A.

.j 2 sin £A sin £B
: 8ini(A-B) '

Multiplying this expression by 2sin£(A— B), we have
2 sin* £ (A - B) tan 8 =4 sin £ A sin ±B sin i (A - B),
or, reducing,

2 siuH(A- B) tan 8=2 sin B srnHA-2 sin A sin^B. (d)

VOL. II.

330

ON THE HIGHER GEOMETRY.

Substituting for the squares of these sines their values in terms
of the double angles, we have

cos (A— B) sin 8 + sin (A — B) cos 8= sin 8+ (sin A — sinB)cos 8.
Now in (c) substituting this latter value for the first, we obtain
i + (a-6)2-2IU[sin 8+ (sin A-sin B) cos 8].

a — b

. z 2r . . . ,, a — b
But smo = -T-, smA— smB=-^p-,

K f-\\

5.
and cos 6 =

hence, making these substitutions in the preceding equation, we get

d=±R-r (e)

239.] A proof of this theorem founded on other principles may
be appropiately here given.

Four circles whose radii are r, rt, r,,, rtll touch a fifth circle,
whose radius is R, in four points A, B, C, D, all externally or all
internally, or some externally and others internally. To find
a general relation between these five circles and their common
tangents. Let us assume the particular case of one internal and
three external contacts, as in fig. 38. Let O be the centre of the

common circle of contact, and let w, fl, fl,, £lu, be the centres of
the four circles touching the common circle in the points A, B, C, D.

ON THE HIGHER GEOMETRY. 331

Now in any triangle of which the sides are a, b, c, we shall have,
as may easily be shown,

«2-(i-c)« ...... (a)

But in the triangle OtuH we shall have
~A~R3

snce

Let t be the common transverse tangent crcr, to the circles whose
centres are to and fl, then £2=flo>2— (Ao>-f Bfl)2; consequently

-7= (c)

* /

In like manner BC=-^L, CD= ^L., DA=-^L. (d)

Let T and T, be the common tangents to the opposite circles
whose centres are a> and H as also O, and fltl,

AC=— == and BD= — ; — '—.

Now as ABCD is a quadrilateral inscribed in a circle, we have,
by Ptolemy's theorem,

AB.CD + BC. AD-AC. BD=0, .... (e)

substituting the preceding values found for these lines, we obtain
tf,, + ^,/;-TT,=0 (0

Hence we may infer that when this relation holds between the
six common tangents to the four circles, they are all in contact
with a fifth circle.

Now let four circles be inscribed in and exscribed to a triangle.
Then in this case the four circles have three common tangents, the
sides of the triangle, and on each side of the triangle there will be
four points of contact, a point of contact with each of the four
circles, as shown in sec. [207]. The six tangents coalesce two by
two into three tangents. Each side of the triangle will be a direct
tangent to two of the circles and an indirect tangent to the other two.

Let 7, yt, 7;/, yni be the four points of contact of the side c with
the four circles. Then, as

ry///=fl— ** and y/y^AC^ + BCrt— AB=»+«— e=a + £,
it will follow that the product of the two tangents in the base
ABor c, touching the four circles is (a— b) (a -I- b) =a*—b*.

332

ON THE HIGHER GEOMETRY.

Therefore the sum of the products of the three sets of coincident
tangents taken two by two, is

\&/

Since this relation holds, the four circles must touch one common
circle; and this circle maybe easily shown to be the nine-point circle.

240.] As the triangle ABC is the nine-point circle not only to
the principal excentral triangle O H, flw, but also to the other
excentral triangles flwH,, ClfaQ.,,, Oa>ny/, it follows that the nine-
point circle will be in contact with the sixteen circles which are
exscribed to and inscribed in these four triangles. This relation
may be still further extended, as we now proceed to show.

Let ABC be a triangle inscribed in a circle. Let Oa, O/3,
Oy be the perpendiculars drawn from the centre O on the sides
«, b, c, and produced to A^C,, so that O« = aAy, O/3 = /3By, and
Oy=yCy. Through the points A, By Cy let a circle be described,

Fig. 39.

and a triangle A/B/C/ inscribed in it. This circle and this triangle
may be called the derivative circle and the derivative triangle of
the former.

Since a. and /3 are the middle points of CB and CA, «/3 is the
half of AB ; and as a and ft are the middle points of OAy and OB,,
aft is the half of AyBy. Therefore A;B; is equal to AB and is also
parallel to it. In the same way it may be shown that the other

ON THE HIGHER GEOMETRY.

333

sides of the two triangles are equal and parallel. Hence the cir-
cumscribing circles are equal ; and while O is the centre of the given
circle circumscribing the triangle, © its orthocentre is the centre
of the derived circle. Therefore the circles interchange their
centres and orthocentres. The two triangles have the same nine-
point circle, whose centre is at v the middle point of O®.

Hence it follows that this nine-point circle touches the thirty-two
circles which are circuminscribed* to the excentral triangles of the
original triangle and its derivative.

241.] If a quadrilateral be inscribed in a circle, the orthocentres
of its four constituent triangles will range on another circle equal to
the former.

Let ACBD be the quadrilateral, and let ®,®;, ®/p ©;// be the ortho-
centres of the four constituent triangles ABC, DEC, ADB, ACD.

As A® and D@; are parallel and equal, since each is double of
OQ, therefore @®; is equal and parallel to AD. In the same way
it may be shown that ©/©// is equal and parallel to AC ; so is ©//©///
equal and parallel to CB, while ®®/;/ is equal and parallel to BD.

Hence the two quadrilaterals are equal and alike in every respect,
and therefore the circles in which they are inscribed are equal.

Fig. 40.

Since BD is equal and parallel to ©®,/;, and D®, equal and
parallel to A©, therefore B®, is equal and parallel to A©^, and
is a parallelogram whose diagonals A®, and B®^ bisect

* A short term to denote circles one circumscribed and one inscribed in the
same triangle.

334 ON THE HIGHER GEOMETRY.

each other. Hence the lines joining the corresponding points of
the two quadrilaterals all pass through the same point.

242.] Let the derivative circle be taken, and the four derivative
triangles inscribed in it. Since the four original triangles are
inscribed in the same circle, and have four orthocentres, the deri-
vative group will have only one orthocentre for the four derivative
triangles, and these triangles will be circumscribed each by a distinct
circle. There will be four nine-point circles, whose centres will be
the middle points of the lines joining the common orthocentre with
the four centres of the derived circles.

Hence these four nine-point circles will be in contact with the
hundred and twenty -eight circles of contact, and every vector drawn
from this common orthocentre to the circumferences of these one
hundred and twenty-eight circles of contact will be bisected by one
or other of the four nine-point circles.

CHAPTER XXVI.

ON SOME ELEMENTARY PROPERTIES OP QUADRILATERALS.

243.] («) If the middle points of the opposite sides of a quadri-
lateral be joined, their intersection O will lie in the line joining the
middle points of the diagonals, and these three lines will mutually
bisect each other.

Let a, b, c, d be the middle points of the sides of the quadrilateral
ABCD. Then ab is the half of the diagonal AC and parallel to it.
Therefore abed is a parallelogram, and its diagonals ac, bd are

ON THE HIGHER GEOMETRY. 335

therefore bisected in O. Since bft and dS are each the half of CD
and parallel to it, b/3 = d8, and therefore /3O = 8O.

(|8) The sum of the squares of any two opposite sides of a quadri-
lateral, together with twice the square of the line joining their middle
points is constant ;

that is AB2

(y) Hence also

that is, in any tetrahedron the sum of the squares of the six edges is
equal to four times the squares of the lines joining the middle points
of the opposite edges.

(8) We have also AB2 + BC2 + CD2 + DA2 = AC2 + BD2
that is, the sum of the squares of the four sides of a quadrilateral is
equal to the sum of the squares of the two diagonals, together with
four times the square of the line joining the middle points of the two
diagonals.

ON QUADRILATERALS INSCRIBED IN ONE CIRCLE AND CIRCUMSCRIBED

ABOUT ANOTHER.

244.] In that very celebrated and highly original work, the ' Traite
des proprietes projectives' of PoNCELET(pp.260— 283) the very elegant
properties of circles inscribed in and circumscribed to the same
quadrilateral are treated with much originality. In fact the dis-
covery of those elegant properties is due to Poncelet. The methods
of investigation, however, which he has used, have not hitherto been
admitted into elementary geometry. As these properties deserve
to be better known, and admit of rigorous geometrical demon-
stration, they should take their place in every treatise of pure
geometry. We shall first, by way of preface, state some of those
properties of quadrilaterals in connexion with circles which are
elementary and have been long known.

(«) In every quadrilateral inscribed in a circle the sum of the
opposite angles is equal to two right angles.

(/3) In every quadrilateral inscribed in a circle the rectangle under
the segments of one of the diagonals is equal to the rectangle under
the segments of the other.

(y) In every quadrilateral so inscribed the rectangle under the diago-
nals is equal to the sum of the rectangles under the two pairs of opposite
sides, and the diagonals are to each other as the sums of the rectangles
under the sides which terminate in the extremities of these diagonals.

When, moreover, the diagonals of the inscribed quadrilateral are
at right angles we shall have the following properties : —

(8) The sum of the squares of the four sides is double the square
of the diameter.

336 ON THE HIGHER GEOMETRY.

(e) The sum of the squares of the four segments of the diagonals
is equal to the square of the diameter ; and

(£) The sum of the squares of the two diagonals is equal to the
square of the diameter diminished by four times the square of the
distance between the centre and the point in which the diagonals
intersect.

(77) If circles be described on the three diagonals of a complete qua-
drilateral inscribed in a circle, they will have the same radical axis,
and the orthocentres of the four component triangles of the complete
quadrilateral range on the same straight line.

And with respect to quadrilaterals circumscribed about a circle,
it is easy to show that

(6) The sum of two opposite sides is equal to the sum of the two
others.

(t) In any quadrilateral circumscribed to a circle, the sum of any
two opposite angles is equal to twice the external angle of one of
component quadrilaterals into which the given quadrilateral is divided
by the two chords.

When these chords are at right angles the external angles of the
component quadrilaterals are right angles ; therefore the sum of the
opposite angles of the circumscribing quadrilateral is equal to two
right angles, or the quadrilateral circumscribing the circle may also
be inscribed in a circle.

The proof is very simple, and depends on the equality of the
angles which a chord of a circle makes with the tangents at its
extremities.

245.] If two quadrilaterals are the one inscribed and the other
circumscribed to the same circle, so that the vertices of the inscribed
may be on the points of contact of the circumscribed quadrilateral,

(a) The chords which join the points of contact of the circum-
scribed quadrilateral will be at right angles.

(/3) The diagonals of the two quadrilaterals will cut all four in the
same point.

(y] The points of concourse of the opposite sides of the two qua-
drilaterals will range all four on the same straight line; and

(8) The point of intersection of the four diagonals will be the pole
of the straight line which contains the points in which the opposite
sides of the quadrilaterals intersect.

(e) The diagonals EGr and FH of the inscribed quadrilateral
meet the intersection of the lines joining the points of contact of the
circumscribed quadrilateral; and the angles between the former are
bisected by the latter.

We shall now proceed to establish the foregoing theorems,
beginning with the last (e) .

As the angle CBD is equal to the angle CAD, and the angle
BFP equal to AHP, therefore the triangles BFP and APH are

ON THE HIGHER GEOMETRY.

Fig. 42.

337

similar; therefore BF or BE : BP as AH or AE : AP. Conse-
quently BE : AE =BP : AP, or in the triangle APB the angle APB
is bisected by PE. In the same way it may be shown that the
other angles between the diagonals of the inscribed quadrilateral
are so bisected.

Hence also the chords of contact EG and FH are at right angles.
VOL. ii. 2 x

338 ON THE HIGHER GEOMETRY.

246.] The diagonals of the circumscribed quadrilateral will pass
through the pole P.

Through E and G let tangents be drawn intersecting in M, then
M is the pole of EG; in like manner N is the pole of HF; hence
MN is the polar of P. Let EF and GH meet in L ; then, as the
polar of L must pass through P, the point L must be on the line
MN ; and as L is a point in EF, the polar of L must pass through
B ; and as L is a point in GH, the polar of L must pass through D ;
and as L is a point in MN, the polar of L must pass through P.
Hence BPD is a straight line, the diagonal of the circumscribed
quadrilateral ; and it passes through P. In the same way it may be
shown that the other diagonal AC passes through P.

247.] Since the angle EwH is equal to the angle FCG, the half
of EwH is equal to the half of FCG ; hence the triangles AEw and
FCo> are similar. Consequently

AE.FC=FwW2, ....... (a)

if r be the radius of the inscribed circle.

In like manner BF.DH = r2 ........ (a,)

Let AE = «, BF=5, CG = c, DH = d, . . . . (b)

the radius of the circumscribed circle being R, and r that of the
inscribed circle.

Hence ac = bd=r2 ......... (c)

We have also AP : AE = sin AEP : sin 1 BPA,
and CP : CG=sin AEP : sin^ BPA.

Let AP = w . AE, and CP=w . CG, writing n for the quotient of
sin AEP divided by sin £ BPA. . . . (d)

Hence AC=w(AE + CG) or AC=ra(« + c). . . . (e)

In like manner BD = n(b + d) ; and therefore

AC : BD = a + c: b+d ....... (f)

But AC.'BD-

or, by Ptolemy's theorem, AC . BD = 4r2 + (a + c) (b + d) , . . (g)
since ac = bd=r'2.

Multiply this expression by :pjA = 7 , « and we shall have

and

ON THE HIGHER GEOMETRY. 339

Since AV=na, and CP = nc, \ve have

AC a + c , — Q __ o / a \2

AFW

Let O be the centre of the circle circumscribing the quadrilateral
D, <o the centre of the inscribed circle, P the common pole
and let the straight line Oo>P meet the common polar MN in Q •
then we shall have

-_

Oo>
To show this, in the triangle AwP we have

P^2= A^2+ AP-SAw . AP . cos PA<u,
and C^2= CA2 + A^2- 2CA . Ao> cos PAo> ;

hence, eliminating cosPAw, we obtain

_ A p

Po)2=Ao)2 + AP2-AP . AC+ (C^2-A

Now A^i2=a2 + r*, C^2=c2 + r«, AF=a* + 7 — -; hence

4r4
or, reducing, H-P» = ....... (j)

From O the centre of the circumscribing circle draw the perpen-
diculars OTT and Or on the diagonals AC and BD ; then TT and T are
the middle points of AC and BD. Hence, by Newton's theorem given
in page 283, the line TTT passes through o> the centre of the inscribed
circle; and as OP?rT is a quadrilateral that may be inscribed in a circle,

Oo> . PtO = <07T . 0)T ....... (k)

Now as the sum of the squares of the four sides of a quadrilateral
is equal to the sum of the squares of the diagonals and four t

340 ON THE HIGHER GEOMETRY.

the square of the distance between the middle points of these dia-
gonals, as shown in (S) sec. [243] ,

But 47TT2 = 4ft>7T2 + 4<UT2 -f 8o)7T X <0T ,

and SCOTT . o)T = 80a> . Pa>.

Hence (a + 6)2 + (6 + c)2+ (c + c?)2 + (d+a)*

since AC + 4w^=2(A2 +C), and
Now

and as a2=Al?-r2, 62=Ik>2-r2, c2=O?-r2, c?2=D^2-r2,

] + 8Oa> . Pa>,

or (a + c)(6 + c?)=4rs + 40w.Pa> ...... (1)

But in the preceding paragraph it has been shown that

4

-2-, ...... (m)

equating these values of (a + c)(# + d), we obtain the relation

Since r2=Pa> . Q.a>, this expression may be reduced to

1 1 1
Ow Qw~Pft>'

a simple relation between the distances of the centre w of the
inscribed circle from O the centre of the circumscribed circle, and
from P and Q the pole and polar.

248.] To express R the radius of the circumscribing circle in
terms of r the radius of the inscribed circle and p the distance
between the centre of this circle and the common pole.

Let q be the distance between the pole P and the polar MN, and
D the distance between the centres, then we shall have

r*, ....... (a)

ON THE HIGHER GEOMETRY. 341

•4
and R2 = (D +p] (D +p + q}, or, since p + q =. — ,

or

Substituting in this expression the value of D given in (n) , last
section,

r — «
we finally obtain

The least value of R is when the circles are concentric, or p=0.
In this case R= \/2r.

From these expressions, namely

=r-l^p!, . . . . (e)
*z

it follows that when r and ja are given, D and R are completely
determined, or, however the rectangular chords of the inscribed
circle may vary in position, the centre and radius of the circum-
scribed circle are fixed.

If we eliminate p between the preceding expressions,

r*]* ..... (f)

249.] Hence it follows that if through any fixed point in a given
circle two rectangular chords be drawn, and at their extremities four
tangents be drawn constituting a quadrilateral, this quadrilateral
may be inscribed in a circle, and the centre and radius of this circle
will be fixed and independent of the directions in which the rectan-
gular chords may be drawn.

The square of the area of the quadrilateral is equal to

(a + b)(b + c)(c+d)(d+a),

since half the sum of its sides is (a + b + c + d).

Multiplying out this expression, bearing in mind that ac=bd=r*,
and dividing by abed, we obtain the very remarkable symmetrical
expression

In every quadrilateral which may be inscribed in one circle and
circumscribed to another the centres of the two circles and the common
point in which the four diagonals intersect are in a straight line.

342

ON THE HIGHER GEOMETRY.

In every such quadrilateral the distances from the vertices to the
point of intersection of the diagonals are proportional to the tangents
drawn from these vertices and touching the internal circle.

The diagonals are proportional to the sum of the tangents drawn
from their extremities to the interior circle.

The distance PQ, between the common pole P and its polar MN,
multiplied by the distance between the centres of the inscribed and
circumscribing circles, is equal to the square of the radius of the
inscribed circle.

For if q be this distance, it has been shown that p(p-\-q)=r'* or

22 2

<7= — and D = 9 9. Hence D<7=r2.

p r2 — p*

250.] If a quadrilateral be inscribed in a circle, the squares of the
inner diagonals are to each other as the distances of their middle
points from the middle point of the outer diagonal. See sec. [176].

It has been shown in sec. [165] that the middle points of the
three diagonals range in the same straight line.

Fig. 43.

Let ABDC be the inscribed quadrilateral. Let m and n be

ON THE HIGHER GEOMETEY. 343

the middle points of the inner diagonals AD and BC. Let M
be the middle point of the outer diagonal. Then M, m, n are in
a straight line. P the intersection of the inner diagonals is the
pole of the outer diagonal EG and O the centre of the circle. Om
and On are perpendicular to the diagonals AD, BC, and they bisect
them.

Since the line AD is bisected in m, and harmonically divided in
P and F, as shown in (d) sec. [161], we have

Dm2=Pm.Fw ........ (a)

But Pm= PO sin F, and Fm : Mm = sin ¥Mn : sin F,

x, Mm . sin FMn

or Fm= - ; — =^ ---

smF

Therefore Dm2=PO .sinFMm .Mm ...... (b)

In" like manner Cn2=PO . sin FMn . Mn.

Therefore Dm2 : Cn2=Mm : Mn ....... (c)

251.] This property will enable us to give a very simple and
elegant solution of the following celebrated problem : — Given a
circle and the lengths of the three diagonals of a quadrilateral to
be inscribed in it, to construct the quadrilateral.

Let 2G, 2G,, 2GW be the lengths of the three diagonals, 2G
being greater than 2G,, and 2GW the outer diagonal ; let y=mn.

Since Mm : Mn = G*: G,2,

Mm : Mm-Mn=G* : G*-G,«.
But Mm — Mn = # ; therefore

Mw=G?§^ and Mn=
Let e be the angle OMm ; then in the triangle OmM

Om2 = OM3 + Mm2 - 2OM . Mm cos e. )
We have also • • • (e)

Eliminating cos e from these expressions, we get

Mm Mn

Now as the tangent drawn from M to the circle is equal to Glt)
see sec. [176], and R being the radius of the circle,

........ (0

344

ON THE HIGHER GEOMETRY.

But we have found

G2-G2' ~G2-G,2'
substituting these values in the preceding equation, we get

sr

(g)

(h)

This enables us to express the distance (g) between the middle
points of the inner diagonals in terms of the three diagonals.
Hence the three sides of the triangle Omn are given, and this tri-
angle has its vertex at O ; and hence the diagonals may be drawn
and the quadrilateral constructed.

The circles described on the three diagonals G, G,, G/; of the
quadrilateral as diameters intersect, two by two, in the same two
points. Their centres, therefore, range along the same straight
line, and have a common radical axis, the common chord. The
distance d between the common chord of any two of the circles
and the centre of one of them is given by the symmetrical formula
G2G2 + G2G/-G/2G/ ,

GG,GW
If C be the common chord of the three circles,

C2 = 2(G2 + G,2 + G, 2) - G2G,2Gy 2(G-4 + G,-4 + G/r4) . ( j)
252.] Let the sides of a quadrilateral inscribed in a circle be
cut by a transversal, the continued product of the ratios of the
segments of the sides made by the transversal will be equal.

Fig. 44.

Let L, M, N, P be the points in which the transversal is cut by

ON THE HIGHER GEOMETRY.

546

the sides of the quadrilateral ; from the points A, B, C, D let per-
pendiculars to the transversal be drawn. Let these perpendiculars
be put a, b, c, d. Then we have

AL_a BM_6 CN_c BP_rf
6' CM~c' DN~V AP~«*

Hence

AL . BM . CN . DP abed

1. We have also

BL.CM.DN.AP~~6o/a"
AL . BL . BM . CM . CN . DN . DP . AP= [BL . CM . DN . AP]2.

Let I, m, n, p be the tangents from the points L, M, N, P.
Then Imnp = BL . CM . DN . AP, or Imnp = AL . BM . CN . DP.
This property may be extended to inscribed regular polygons
of anv number of sides.

ON THE PROPERTIES OF CHORDS DRAWN FROM A POINT IN THB CIR-
CUMFERENCE OF A CIRCLE TO THE ANGLES OF AN INSCRIBED
REGULAR POLYGON OF AN ODD NUMBER OF SIDES.

253.] When the polygon is an equilateral triangle the properties
are obvious and known.

When the polygon is a pentagon. In general let the side of the
polygon be put * ; let the chord which subtends two adjacent sides
of the polygon be t, that which subtends three consecutive sides
be u, and that which subtends four sides be z, &c.

Let the chords drawn from the point P to the angles A, B, C,
D, E, F, G, &c. be a, b, c, d, e, f, g, &c.

VOL. ii. 2 Y

346 ON THE HIGHER GEOMETRY.

Then in the case of the pentagon we have

at-i-et=cs,

(a)

Adding these expressions together, and dividing by (s + t), we have

We shall have also c*=(a + e)(b + d), (c)

and 5c2= (a -\-b-\-d-\-e)* (d)

254.] When the regular polygon is a heptagon, then we shall
have the following twelve equations : —

as + cs =bt, >
au-]-ffu=ds,
bt +ft = du,
cs +es=dt,

at-\-gs=bs,
fft+as=fs,

bu+fs=cu, \

et + bs=du,

, ,

=eu, ) (a-\ c + e+g)s=du.

Adding these twelve equations, we shall have

(a 4- c + e +ff) (s + 1 + u) •
or, dividing by (s + 1 + u),

(b)

or, in other words, the sura of the odd chords drawn from the point
P to the alternate vertices of the heptagon will be equal to the sum
of the even chords.

We have also ds= (a+g) (b+f)(c + e)', ..... (c)

that is, the cube of the middle chord is equal to the continued product of
the sums of the first and seventh, of the second and sixth, of the third
and fifth.

When the point P is assumed in the middle of the arc AG, then
PD is a diameter 2R, and a=g, b=f, c=e, and therefore

abc = W ......... (d)

These properties thus established may be extended to regular
polygons of (2n + l) sides inscribed in a circle.

Thus let M be the middle chord of a polygon of (2w + l) sides,
and let Cl C2C3 . . . CZn-\, C2n, C2n+i be the chords drawn from the
point P to the angles of the polygon ; then we shall have

^.,) &c.

When P is the middle point of the arc, M is a diameter 2E, and
the preceding expression becomes

A TREATISE

ON

C O N I C S

CHAPTER XXVII.

DEFINITIONS.

255.] Let a straight line be drawn perpendicular to the plane
of a circle through its centre, and a point in it assumed, through
which a straight line of indefinite length passes, always touching
the circumference of the circle ; the surface thus generated is called
a cone, the perpendicular is called the axis of the cone, and the fixed
point the vertex.

The surface thus generated is divided by the vertex of the cone
into two portions, which may be called the upper and lower sheets
of the cone.

n.

If this surface be cut by a plane, the line in which the cone and
the plane intersect is called a conic section, or in short a conic.

in.

If a sphere be inscribed in this cone touching the plane of the
conic section, the point of contact is called & focus of the conic.

As there may be in general two spheres so inscribed, one touching
the plane of the section above, the other below— or one in each sheet
of the cone, both touching the plane of the section on the same
side — there are in general two foci in a couic section.

These spheres may be called focal spheres.

348 ON CONICS.

IV.

The straight line which passes through the foci, and is termi-
nated by the surface of the cone, is called the major axis.

v.

The plane drawn through the vertex of the cone and the major
axis of the section, cuts the surface of the cone in two straight
lines, which together with the major axis constitute a triangle,
which may be called the focal triangle, since its plane passes
through the foci.

VI.

The focal spheres touch the surface of the cone in two circles
which may be called the circles of contact.

The planes of these circles are manifestly parallel, since they are
at right angles to the axis of the cone.

VII.

The straight line in which the plane of a circle of contact cuts
the plane of the section is called a directrix.

As there are in general two circles of contact, there are in
general also two directrices, and they are parallel to each other.

VIII.

A plane drawn through the vertex of the cone parallel to the
plane of the section is called the vertical polar plane; and the straight
line drawn through the vertex of the cone, the polar line of this
vertical plane with respect to this cone, is called the polar axis, and
it meets the plane of the conic section in a point called the centre.

IX.

The straight line in which the vertical polar plane cuts the plane of
the circle of contact is called the dirigent. As there are in general
two circles of contact, there are two dirigents, and they are parallel
to the directrices.

x.

The dirigent is the polar of tfre point in which the polar axis of
the cone meets the plane of the circle of contact with respect to
this circle.

xi.

If a straight line be drawn from the vertex of the cone in the
vertical polar plane, the polar plane of this straight line will pass
through the polar axis of the cone, and is called & polar plane of the
cone.

XII.

When the vertical polar plane lies outsi'de the cone the parallel

ON CONICS. '51'.)

section is called an ellipse ; when it touches the side of the cone the
parallel section is a parabola ; and when the vertical plane cuts the
surface of the cone, the parallel section is an hyperbola.

In this latter case the polar axis will lie outside the cone ; and if
two planes be drawn through this line touching the cone, they will
cut the plane of the hyperbola in two straight lines called asym-
ptotes ; and as the polar axis (the intersection of the tangent planes)
cuts the plane of the conic in its centre (see def. vm.), the asymptotes
will meet in the centre of the hyperbola. Moreover, as the polar axis
touches the surface of the cone when the conic is a parabola, the two
tangent planes drawn through it to the cone coincide and become
parallel to the plane of the parabola ; consequently the asymptotes
of the parabola are two straight lines parallel to the axis of the
parabola but at an infinite distance from this axis.

XIII.

The ordinate drawn through the focus of a conic, at right angles
to the major axis, is called the parameter or latus rectum.

XIV.

The radical plane of the focal spheres cuts the plane of the conic
in a straight line called the minor axis.

xv.

Lines drawn from the vertex of the cone to the extremities of the
diameters of the focal spheres which are perpendicular to the plane
of the conic may be called vertical focals of the conic.

ON THE FOCAL PROPERTIES OP CONICS.

256.] If a sphere be inscribed in a right cone, the curve of contact
is a circle.

Since all tangents drawn from a point to a sphere are equal,
the vertex of the cone may be considered as the centre of a sphere
whose radii are the sides of the cone intercepted between the vertex
and the line of contact with the inscribed sphere. This sphere,
therefore, will intersect the inscribed sphere in the line of contact ;
but two spheres intersect each other in a circle ; hence the line of
contact is a circle.

257.] The plane which passes through the vertex of the cone and
the two foci, passes also through the axis of the cone, and is at right
angles to the plane of the conic.

The radii of the inscribed spheres which pass through the foci
are at right angles to the plane of the conic, since it is a tangent
plane to the focal spheres; but these radii are parallel, sincr tlu-y
are perpendicular to the plane of the conic ; and therefore the plane

350

ON CONICS.

which passes through them will pass through the centres of the
focal spheres, which are manifestly on the axis of the cone ; con-
sequently the plane of the focal triangle (see def . v.) will be per-
pendicular to the plane of the conic.

258.] The directrix is perpendicular to the major axis of the
conic.

Since tne plane of the conic and the plane of the circle of contact
are each perpendicular to the plane of the focal triangle, their inter-
section, the directrix (Euclid, XI.) will be perpendicular to the
same plane, and therefore perpendicular to any straight line on it,
and therefore to the major axis.

259.] The distance of any point of a conic from a focus is to its
perpendicular distance from the corresponding directrix in a constant
ratio — namely, as the distances of the vertex of the cone from the

Fig. 46.

H.

circumference of the circle of contact and from the dirigent the inter-
section of the plane of the latter by the vertical polar plane.

ON CONIC8. 351

The inscribed sphere touches the plane of the conic MDANFn
in the point F ; and the cone touches the sphere along the circle of
contact CGQP.

Draw NM parallel to AF, join NF, NV. Draw the vertical plane
VYZ parallel to the plane of the conic, meeting the plane of the
circle of contact in the straight line, the dirigent, YZ. In this
vertical plane draw the line VY parallel to the major axis AF, to
which MN is parallel. Join YQ, and produce it to meet the direc-
trix RX. It must meet the line MN also in the directrix ; for as
YQ is in the plane of the circle of contact, it can meet the plane
of the conic only in their intersection, the directrix EX ; but as
MN is parallel to VY, a plane may pass through VY, VQN, and
NM ; hence NM must meet YQ ; and as it lies in the plane of the
conic, it can only meet it in the directrix RX.

Now as the triangles M Q N and V Y Q are similar,
NQ : NM= VQ : VY. But NQ=NF, since Q and F are points on
the same sphere; therefore NF : NM = VQ : VY=VC : VY, since
VQ is equal to VC. But VC has a constant ratio to VY inde-
pendently of the position of the point N; therefore NM has a
constant ratio to NF.

This is the theorem which has been made by De la Hire, and by
others since his time, the basis of a system of conies in a plane.

Cor. i. When the vertical plane touches the cone, as when the
conic is a parabola, VC=VY, consequently NF = NM.

Cor. ii. When the conic is an ellipse, VY is greater than VC, or
NM is greater than NF ; when VY is equal to VC, NM is equal
to NF ; when V Y is less than VC, or when the vertical plane VY
falls within the cone, or NM is less than NF, the conic is an
hyperbola.

The ratio of VC the side of the cone between the vertex and the
circle of contact to the perpendicular VY from the vertex of the
cone on the dirigent YZ is called the eccentricity of the conic, and
is usually denoted by e.

260.] If from any point in a conic a line be drawn to the directrix
parallel to the straight line the intersection of the vertical plane with
the cone, it will be equal to the focal distance of the same point.

Let VP be the intersection of the vertical plane with the cone ;
join PQ ; and by the same construction and demonstration as the
preceding, NF=NM, since VC = VP.

Hence, If from a point in a conic a line be drawn to the directrix
parallel to the axis of a parabola, or to one of the asymptotes of an
hyperbola, this straight line will be equal to the focal distance of the
same point.

261.] The major axis of a conic is equal to the segment of a »ide
of the cone intercepted between the circles of contact.

In the focal triangle ABC the base AB (that is, the major axis

352

ON CON1CS.

of the conic) is equal to the segment of the side CB intercepted
between A, and G, the points of contact of the side CB with the
inscribed circles (see fig. 29). But these are great circles of the
focal spheres which touch the plane of the conic in its foci F and Fr

262.] The sum or difference of the focal distances of any point in
a conic — the sum, if an ellipse, the difference, if the conic be an
hyperbola — is constant, and equal to the portion of the side of the cone
intercepted between the circles of contact (that is, to the major axis}.

Let VQNQ/ be a side of the cone touching the focal spheres in
the points Q,, Q, and passing through N a point on the conic.
Then, as Q, and F are points on the same sphere, NF = NO, ; so also
NF,=NQy.

Fig. 47.

Therefore NF + NF, is equal to QGL,, the portion of a side of the
cone intercepted between the circles of contact.

In the last proposition it was shown that this segment of the side
of the cone is equal to the major axis of the conic. Therefore the
sum of the focal vectors of an ellipse is equal to its major axis.

ON CONICS.

353

In the case of the hyperbola (fig. 48), since NF is equal to NQ
and NF, equal to NQ,, therefore NF,— NF is equal to QQ, the
segment of the side VN of the cone intercepted between the circles
of contact.

Fig. 48.

Cor. i.] The distance between the directrices is equal to that
portion of the major or transverse axis intercepted between the planes
of the circles of contact (see fig. 47) .

The ratio of the major axis of the conic to the distance between
the directrices is as e : 1.

For, in fig. 46, NM : NF= VY : VC ; so also with respect to the
other directrix NM, : NF, = VY : VC.

Therefore NM + NM, : NF + NF,= V Y : VC.

But NM + NM, is the distance between the directrices, and
NF + NF, is equal to AB the major axis; therefore the distance
between the directrices is to the major axis as 1 : e.

Cor. ii.] In the same way it maybe shown that the distance between
the dirigents is equal to the distance between the directrices.

VOL. ii. 2 z

354 ON CONICS.

Cor. iii.] The distance between the foci is equal to the difference
between the sides of the cone terminated in the extremities of the
major axis, namely VB and VA.

For VB-VA=BC-AG=BF-AF=FFr
In the hyperbola we have VB-f VA=FF; (see fig. 48).
263.] The rectangle under the radii of the focal spheres is equal to
the rectangle under the focal distances of the vertices of the conic.
Let R and R, be the radii of the inscribed spheres, w and fl their
centres. The triangles B«oF and HBF/ are similar, since the angle
o>BF = BnF. Therefore RR,= BF, . BF ; but BF,= AF.
Therefore RR,=AF . BF (see fig. 29).

264.] Planes which intersect in a tangent to a conic and pass
through the centres of the focal spheres are at right angles.

It is evident that the dihedral angle between the plane of the
conic and the tangent plane to the cone which contains the tangent
to the conic is bisected by the plane passing through the tangent
their intersection and the centre of the sphere which touches
these two planes ; for this point is equidistant from the planes of the
section and tangent plane.

In like manner the supplement of this dihedral angle is bisected
by the plane which passes through this tangent and the centre of
the other focal sphere. Hence planes drawn through the centres of
the focal spheres and a tangent to the conic are at right angles.

265.] If perpendiculars are drawn from the foci of a conic on a
tangent to the curve, the rectangle under these perpendiculars is equal
to the rectangle under the radii of the focal spheres ; that is, PP, = RRr
Let FP (fig. 49) be the perpendicular from the focus F on the
tangent PQ. Join «P, and erect the perpendicular PN to the plane
of the conic ; it is parallel to <oF, and is therefore in the plane PtuF';
and as QP is perpendicular to FP by construction, and to PN, it is
perpendicular to the plane which passes through them — that is, to
the plane PNtuF. Consequently the angle FPta is the measure of
the dihedral angle between the plane of the conic and the plane
which passes through the tangent to it and the centre &> of the focal
sphere. Let this angle be w, we shall have, since wFP is a right angle,
R = Ptan«r; and as the angle between the plane of the conic and the
plane passing through the tangent QP and the centre fl of the
other focal sphere is the complement of the former, we shall have
R, = P, cot «r, or RR,=PP .

ON COXICS.

350

Fig. 49.

266.] The locus of the feet of perpendiculars let fall from the foci
of a conic on a tangent to the curve is a circle (see fig. 50).

Since the angle o>PQ is a right angle, see last section, and
the planes o>PQ, HPQ, are at right angles, wPfl is also a right
angle; therefore the sphere described on G>£}, the line which joins
the centres of the focal spheres, will pass through the points P, p,
and also through the points A and B, since 12 15o> and o)AH arc
right angles. But the points P, p, A, B are also in a plane, namely
that of the conic. Hence they lie in the intersection of a plane
and a sphere — that is, a circle.

Let Am, Bra be the lines in which the plane of the conic intersects
the tangent planes along the sides of the cone VA, VB. Now t hrsr
planes are perpendicular to the focal plane passing through the
axis of the cone, which is also a diametral plane of the sphere whose
diameter is o>ft ; therefore these lines are tangents to the sphere ;
and as they are parallel, the line which joins their points of contact
A, B is a diameter of the circle. The locus of the feet of the per-
pendiculars is therefore a circle whose diameter is the major axis
of the conic.

356

ON CONICS.

Fig. 50.

267.] A tangent to a conic makes equal angles with the focal vector
and the side of the cone passing through thepoint of contact (see fig. 51).

In the tangent mM.n, through which the tangent plane VQnMmQ,
passes, assume any point m ; draw m¥, and mQ, to the point Q where
the side VM of the cone meets the circle of contact CQC/. Then
in the triangles mMQandmMFthe side wF = mQ,; soalsoMF = MQ;
and Mm is common to the two triangles ; hence these triangles are
equal, and therefore the angles FMw and QMm are equal. In the
same manner the angles FjMra and QtMn are equal.

ON CONICS.

Fig. 51.

357

The angles which the focal vectors make with a tangent to the curve
at the point of contact are equal.

The angle FMm is equal to the angle QMw ; and the angle FyMn
is equal to the angle QyMw. But the angle QMm is equal to the
angle Q,Mn as they are vertically opposite angles, the angles which
the side of the cone makes with the tangent to the curve. Hence
the angles FMw and F,Mn are equal.

268.] The directrix is the polar of the focus, or the locus of the
intersection of every pair of tangents whose chord of contact passes
through the focus (see fig. 52).

Through VF, the line which joins the vertex V of the cone with
the focus F, let a plane be drawn, cutting the plane of the conic in
the line BAD and the plane of the circle of contact in the points
C,CD. As D is a point in the plane of the conic and in the plane
of the circle of contact, D must be on the directrix.

358

ON CONICS.

Fig. 52.

Now as the two sides of the cone VA, VB and the line AB in
which this plane cuts the plane of the conic constitute a triangle in
which the circle FCCy a section of the focal sphere is inscribed, the
lines AC,, VF, and BC will meet in a point. As in the triangles
DAC and DBC the angle at D is common, and the angle DCA is
supplemental to the angle DC;B,

DA : AC=DB : BC;; but AC=AF, and BC,=BF;

therefore DA : DB=AF : BF, or VB, VF, VA, VD constitute an
harmonic pencil. And as this proof will hold good for any plane
drawn through the focus and the vertex of the cone, it is clear that
the directrix is the polar of the focus.

Cor. Join Da and produce it to m. Then mn is harmonically
divided in D and a ; and therefore ma : an=mD : T)n. VC and VC,
are tangents to the circle ; therefore Da is also a tangent.

ON CONIC8.

269.] If any chord of a conic be drawn and produced to meet the
directrix, and from the corresponding focus two lines be drawn, one
to the intersection of the tangents drawn to the ends of the chord,
the other to the intersection of this chord with the directrix, these
two lines will be at right angles to each other (see fig. 53) .

Let the chord mn meet the directrix DX in the point Y ; draw
the tangents mT, wT meeting in T. Draw the tangent planes
VmT, VwT whose lines of contact with the cone meet the circle of
contact in the points a, c. Join a, c ; also T F, F Y ; the angle
TFY is a right angle.

Join TY, and let a plane be drawn through TY touching the
focal sphere in the point S. As TY is in the plane of the conic
which touches the focal sphere in F, the line SF is the conjugate

Fig. 53.

360 ON CONICS.

polar of the line TY ; and as Y is a point in the line TY, the polar
plane of the point Y will pass through FS the conjugate polar of
TY ; and as Fe is the conjugate polar of the directrix in which Y is
a point (see fig. 52), the polar plane of Y will also pass through Ye.
Hence the plane which passes through FS andFe is the polar plane
of the point Y. This plane will also pass through the point T ; for
as the point T is in the intersection of the tangent planes to the
cone VmT and VnT, the polar plane of T will pass through the chord
a c ; and as this chord is in the plane of the circle of contact,, it
must meet the directrix which also lies in the plane of the circle of
contact; for otherwise it would be parallel to it, and then the directrix
could never meet the secant plane Vmn, contrary to hypothesis.
Therefore the chord ac of the circle of contact meets the directrix
in the point Y ; therefore the polar plane of T passes through Y ;
therefore T is a point in the plane FSe; and therefore STis a tangent
to the base of the cone whose vertex is at Y, and which touches the
focal sphere in the points FSe. Consequently YST is a right angle.
Now in the triangles YTF and YTS, YF is equal to YS, TF is equal
to TS, and YT is common ; therefore the angle YFT is equal to the
angle YST. But YST is a right angle ; and therefore YFT is a right
angle.

270.] If a line be drawn from the focus to the pole of a focal
chord, it will be at right angles to it.

For, by the preceding proposition, when the chord mn passes
through F the focus, the point T, the intersection of the tangents
mT, wT will be found on the directrix, and the equal angles mFT,
nFT become right angles.

271.] If a focal chord be drawn perpendicular to the axis, and a
tangent to the curve be drawn at its extremity, it will cut off from the
tangent to the vertex of the curve a portion equal to the distance of
the vertex of the curve from the focus (see fig. 54) .

Through F draw the ordinate FP, and through the point P a
tangent meeting in S the vertical tangent Am drawn through A;
AS is equal to AF.

AS : AD = FP : FD ; but FP : FD=VG : VU, as in sec. [259],
and VG : VU = AG : AD. Therefore AS is equal to AG=AF.

Therefore AS : AD=AG : AD ; hence AS=AG=AF.

Letjflbe the semiangle of the cone, and i the inclination of the plane
of the conic to the axis of the cone ; then VCy = VUcosi=VGcos0;

. VG cos i

consequently = ' = '

ON CONICS.

Fig. 54.

272.] If two tangents be drawn to a conic, the line connecting
their point of meeting with a focus bisects the angle contained by the
focal vectors drawn from this focus to the points of contact (see fig. 55) .

Let Tm, Tw be tangents to the conic at the points m and n, and
meeting in T. JoinTF; thenTF bisects the angle wF». Through Tm,
Tn let tangent planes to the cone be drawn touching it in the sides
Vm, Vtt, and therefore touching the sphere in the points Q and Qr
Join Vt, TQ, and TQ;. Then in the triangles TVQ and TVQ,
since VQ=VQ,, TQ = TQ/, and VT common, the angle VQT is
equal to the angle VQ,T, and their supplements arc therefore equal ;
that is, the angle TQw is equal to the angle TQ,n. Now wF is equal
to wQ, TQ is equal to TF, and Tw is common ; therefore in the
triangles TmQ. and TmF the angle TQw is equal to the angle TFm.
In the same way the angle TQ^ may be proved equal to the angle
TFw ; consequently the angles TFw and TFn arc equa'.

VOL. II.

362

ON CONICS.

Fig. 55.

273.] If from the intersections X, Y, of the tangents Tm, Tn with
the directrix YDX, focal chords XF, YF be drawn, they will make
equal angles with the focal chord FT.

Since X is the pole of mF, wzFX is a right angle ; and since Y is
the pole of J?n, wFY is a right angle. But the angle TFw is equal
to the angle TFw ; therefore the angle TFX is equal to the angle
TFY.

OX CONICS.

363

274.] If from the intersection of two tangents to a conic, chords be
drawn to the two foci, they will make equal angles with the tangents.

Let Tm, Tn (fig. 56) be the tangents to the conic, meeting in T.
Let the tangent planes VmT, VnT be drawn, touching the cone
along the sides VQwQ,, Vtmw,, and draw TF, TF,. The angles
FTn and F,Tw are equal. Join Fw, Ypn.

Now as TF, is equal to TQ, since they are tangents to the same
sphere, and mF, for the same reason is equal to wQ/, and mT is
common, the triangles TwF, and TwQ, are equal, and therefore
the angle F/Tw is equal to the angle Q/Tm.

364 ON CONICS.

In the same way, as TF is equal to TQ, Fm equal to mQ,, and
Tw common, the angle TmF is equal to the angle TwQ,.

Hence the angles Q/Tm and mTF are together equal to QTQ/ ;
or, as the angle G/Tm is equal to the angle F/Tm, twipe the angle
F/Tm together with the angle F/TF are equal to the angle G/TQ.
For the same reason twice the angle FTra with the angle F/TF are
equal to the angle uTur But as TQ, is equal to Tu,, TO, equal
to Tu, and QQ, is equal to uuf, the triangles TQ/Q, and Tuut are
equal ; therefore the angle Q.TQ, is equal to the angle uTu,.
Therefore twice the angle FyTm with the angle F/TF are equal to
twice the angle FTw with the angle F/TF ; taking away the common
angle F/TF, the angle FyTm is equal to the angle FTra.

275.] If two tangents be drawn to a conic, and from their inter-
section two lines be drawn to the points where the tangent plane to
the cone drawn through one of the tangents touches the focal spheres,
the angle contained by the two latter lines will be equal to the angle
between the tangents (see fig. 56) *.

The angle 2F/Tm together with the angle F/TF are equal to the
angle QTQr But the angle F/Tm is equal to the angle FTn. Hence
the angles F/Tm + FTw + F/TF are together equal to QTQ,.

But the angle between the tangents is made up of the component
angles F/Tm + FTw + F/TF. Therefore the angle between the tan-
gents to the conic is equal to the angle QTQr

It is a matter of indifference through which of the tangents to
the conic the tangent plane to the cone be drawn ; for the angles
QTQ, and uTu, are equal.

276.] If a tangent plane be drawn to the cone, meeting two parallel
tangents to a section of this cone in the points m and n, and touching
the focal spheres in the points Q and G,, the quadrilateral QimQin
may be inscribed in a circle (see fig. 57).

By the last proposition the angle TmN is equal to the angle
QmQ,, and T;wN is equal to QraQ, ; therefore the angles QmQy and
QnQ,, are together equal to TmN and T^N. But as the tangents
Tm and T,rc are parallel, the sum of the angles TmN and TywN is
equal to two right angles ; therefore the sum of the angles QmQ;
and QwQ/ is equal to two right angles, or the quadrilateral QmQ^
may be inscribed in a circle.

Cor.] Since Nm . ~Nn is equal to NQ . NQy, while NO, is equal to
NF, and NQy equal to NFy, therefore the rectangle under the seg-
ments of a tangent between its point of contact and its intersections
by two parallel tangents is equal to the rectangle under the focal
chords drawn through the point of contact.

* This is perhaps the most important proposition in the theory of conies
derived from the cone.

ON CONIC8.

Fig. 57.

365

277.] If two fixed tangents be drawn to a conic, and a third tangent
variable in position, the segment of this latter tangent between the
two former will subtend angles at the foci whose sum is constant and
equal to the supplement of the angle contained by the two fixed
tangents (see fig. 58).

Let Tw, Tn be the two fixed tangents touching the conic in the
points m, n. Let *Sr be the variable tangent touching the conic in
S and cutting the fixed tangents in / and r. The tangent fr will
subtend at the foci F, F, angles whose sum is constant, and equal
to the supplement of the angle at T.

366

ON CONICS.

Fig. 58.

The vertex of the cone is omitted from tie figure.

Through tsr let a tangent plane VQtSrGt, to the cone be drawn
touching the focal spheres in the points Q,, Qy, cutting the fixed
tangents to the conic Tm, Tn in the points t, r, and touching the
conic in S. Join tQ, rQ,, tQ.,, rQr By sec. [275] the angle mtS
is equal to the angle QtQ.,, and the angle m-S is equal to the angle
QrQv. Now these two external angles of the triangle Ttr together
with the external angle at T are equal to four right angles ; and the
four angles of the quadrilateral Q/Q/r are also equal to four right
angles. But two of the angles of this quadrilateral QrQy and Q7d/
have been shown to be equal to the external angles of the triangle
tTr ', therefore the remaining two tQ,T and /Q;r must be equal to the
external angle at T.

Now, in the triangles tQr and tYr, since tQ, is equal to t~F, and
rQ is equal to rF, since the points Q and F are on the same sphere,
and tr is common, the triangle tQ,r is equal to the triangle t¥r} and
therefore the angle tQ,r is equal to the angle tFr. The same may

ON CONIC8.

367

be shown for the other focus. Hence the angle Q, of the quadri-
lateral is equal to the focal angle at F, and the other angle Q, of
the quadrilateral is equal to the angle at Fy.

Hence the sum of the angles which tr subtends at the foci is
equal to the supplement of the angle T.

Cor.] When the fixed tangents are parallel, we get the theorem
in sec. [276] .

278.] Two tangents are drawn to a conic ; a perpendicular drawn
to the chord of these tangents from their point of meeting will cut
the major axis in a point which with the two foci and the intersection
of the chord with this axis will be four harmonic points.

Through F the focus of the conic let the diameter of the focal
sphere be drawn meeting its surface in G ; the focal vertical VG will
meet the major axis in the other focus ¥,, see def. xv. Draw

Fig. 59.

the plane Vmnc cutting the focal sphere in a circle zpv, and let
this plane cut the focal diameter FG in O. Let T be the inter-

368 ON CONICS.

section of the tangents Tw, Tn. Then U the vertex of the cone
zfiv\] is on VT, since the tangent planes meet in VT. The polar
plane of the point O is a plane drawn through U parallel to the
plane of the conic. Let this plane meet FG- in Q. Join VQ,
meeting the major axis in <BT. Through T-sr let a line be drawn,
meeting the chord mn in u ; this line will be at right angles to the
chord mn. Then, as the plane through U parallel to the plane of the
conic is the polar plane of O, QF : FO = QG : GO. Through O let
a plane be drawn parallel to the plane of the conic and cutting the
sphere in the line Os. This line Os will be parallel to the chord mn.

Let this lesser circle be the base of a cone whose vertex is at Q,
on the plane through U parallel to the plane of the conic.

Now the line UQ, which joins the vertices of the cones, is the
harmonic conjugate of the line O*, in which the bases of the two
cones intersect. Hence UQ is at right angles to O*. But UQ is
parallel to Tu, since they are in parallel planes ; and sO is parallel
to the chord mn. Hence mn is at right angles to Tu. Since
GO :OF=GQ : QF, therefore (as the vertical focal VG passes
through Fy) VF/} Vc, VF, V-GT constitute an harmonic pencil;
therefore F,c : Fc = F/5i :F-sj.

When mn passes through F, c and «r coincide with the chord F,
and TF is perpendicular to the focal chord as shown in sec. [273] .

Hence also it follows that uc bisects the angle F,u¥, which is one
of the most general theorems in conies, and may be given in the
following form : —

If two rectangular axes are drawn in the plane of a conic, so that
the pole of the one may be a point on the other, the lines drawn from
their intersection to the foci will make equal angles with these axes.

279.] If any point be assumed in the plane of a conic, and tangents
be drawn from this point to the curve, the rectangle under the focal
distances of this point is equal to the rectangle under the major axis
and a perpendicular from this point on a focal chord drawn through
a point of contact divided by the sine of the angle between the tangents,

or TF, . TF=-J^5- (see fig. 56).

sinmTn

Since TF=TQ and TF,=TQ,, therefore TF . TF, . sin wTra is the
area of the triangle QTQ, ; but this area is also equal to QQ, ( = 2a)
multiplied by the perpendicular drawn from T to QQy. This perpen-
dicular it may easily be shown is equal to the perpendicular from T
drawn to the focal chord mY.

Cor.] Hence all the perpendiculars let fall from T on the focal
chords are equal. Consequently, if any two points be assumed on a
conic section, and two focal chords be drawn through each, the centre
of the circle described touching these four chords will be on the inter-
section of the tangents touching the curve in the two given points.

ON CONICS.

Fig. 60.

280.] If a sphere be described about that portion of the cone cut
off' by the circle of contact, the semiparameter is a third proportional
to the side of the cone cut off by the sphere, and the tangent from the
focus to this sphere.

Let CwDQQ, be the circle of contact, which is also the common
intersection of the focal sphere with the circumscribing sphere
VPCDT.

Draw the ordinate Fa perpendicular to the axis AB ; through V
and a draw a side of the cone Va meeting the circle of contact in
the point u, touching the focal sphere in n, and meeting the cir-
cumscribed sphere in V and u.

Then Va . au =l?aa + FV . F#, since Fa is perpendicular to F* .
Now Va = au + VM = Fa + VC ; for Fa and au are tangents_to the same
focal sphere. Therefore au = Fa ; hence (Fa + VC) Fa = Fa1 + FV. FJT.

VOL. ii. SB

370 ON CONICS.

But FV.F#=FT2 ; consequently VC . F«=FT2, or the semipara-

FT2
meter F«=^^.

281.] The semiparameter is equal to the perpendicular distance
between the plane of the section and the vertical polar plane, multi-
plied by the tangent of the semiangle of the cone.

Through the focus F let a perpendicular be drawn to the plane
of the conic, meeting the sphere circumscribed to the cone VCD in
the point P. This line passes through S the centre of the focal
sphere ; and as this point S is on the diameter VS of the circum-
scribed sphere, VPS is a right angle, or VP is the intersection of
the vertical polar plane with this sphere. Consequently FP is the
perpendicular distance between the planes. Now FT2 = FS . FP.
But FS —r, the radius of the focal sphere ; and 20 being the vertical

angle of the cone, tan0=; therefore FT2=FP . VC tan 6.

FT2
But in the preceding proposition it was shown that ^L=^T;

therefore £L=Ptan 6, writing P for FP.

62

Cor. i.l Since P tan 6——, and the area of the ellipse is irab, the
a

volume of the cone which stands on the ellipse as base is ^irb3 cot 6.

Cor. ii.] If a sphere be described with the vertex of the cone as
centre, all the plane sections of this cone which touch this sphere
have equal parameters.

282.] Twice the rectangle under the segments of any focal chord
is equal to the rectangle under this focal chord and the semiparameter.

Let the segments of the focal chord mFn (fig. 61) be/ and /;, and
let c be the distance from the vertex of the cone to a point C on the
circle of contact. Through the vertex V of the cone and the focal
chord m¥n or/+/; let the plane VamFweV pass, intersecting the
cone in the triangle Vmw and the focal sphere in the circle aFeQ,, of
which the radius is p. From s the centre of the focal sphere draw
the perpendicular sx on VF. The plane through sx perpendicular
to VF will pass through x the centre of the circle made by the above
secant plane whose radius is p.

Now f=m$ = ma, fl = nY = ne, and VC=VGw=c.

The following are well known expressions for the area of the
triangle Nmn circumscribing the circle Q,GltFa : —

Uf+f, + Wf]*=(f+fi + JP = i(f+ftP, ... (a)
p being the perpendicular from V the vertex of the cone on the
plane of the conic. But, by similar triangles, VF : p^=¥st or p : F#.

F.2?

Hence j»=VF . — , and x is a point on the circumscribing
sphere (see fig. 60).

ON CONICS.

371

Multiplying together the two latter values for the area, and
equating the product with the square of the former, we shall have

Now ja

VF

therefore 2$ = (/+/,) ^.
c

= VF . — , while c = VC ; and the semiparameter

fig

— , as in the last section ; consequently 2fft=: (/+/y) (£L).

Fig. 61.

283.] To find the locus of the intersection of pairs of tangents to
a conic, meeting at a given angle (see fig. 62).

Draw any tangent plane VQmQjE to the cone, and on the line
Q.Q,, equal to the major axis of the conic, and in this tangent plane,
let a segment of a circle be described capable of containing the
given angle. Let a solid be generated by the revolution of this
tangent plane to the cone carrying the circular segment with it as
described in this plane on the chord QQr

372

ON CONICS.

The intersection of this solid (which may be called the cono-
spheroid) with the plane of the conic will be the required locus.

In this curve of intersection assume any point E ; draw the
tangents Em, Ew to the conic. They will contain the angle niEn.
But this angle is equal to the angle Q,EQ/ by the theorem estab-
lished in sec. [275] .

It is evident that this solid will consist of two sheets, the one

ON CONICS. 373

described by that segment of the generating circle which contains
the given angle, and which has QQy for its chord ; the other will be
described by the remaining segment of the circle, which contains
the supplement of the given angle. It is plain that the two sheets
of this conospheroid meet in the two circles of contact of the focal
spheres with the cone.

Every plane section of the conospheroid at right angles to the axis
of the cone is a circle.

From the point E draw Er at right angles to the side of the cone
VQQ;, and draw rs at right angles to the line VQQ/ until it meets
the axis of the cone in s. Then, as Er, rs are each at right angles
to the side VQQ,/ the plane Er* will be perpendicular to the side
of the cone VQ,Qr Therefore the axis of the cone makes a constant
angle with this plane *rE. E is therefore on the surface of a right
cone whose vertex is s and axis *V ; and *E is constant, since
Esa = *r2 + Er2, each of which is constant. Consequently, E being
on the surface of a right cone, and at a constant distance from the
vertex, E must describe a circle at the distance *E from the vertex
of this cone.

The protective equation of the conospheroid may be found from
the genesis of the surface.

Let 6 be the semiangle of the cone, p the perpendicular from the
centre of the generating circle on the chord 2a. Let 2s be the sum
of the radii of the circles of contact. Let the origin of coordinates
be taken on the axis of the cone equidistant from the planes of the
circles of contact ; let the plane perpendicular to this axis be taken
as the plane of xy, and the plane of the focal triangle as the plane
of xz.

Then it will not be difficult to show that the projective equation
of the conospheroid is

#2 + y9 + z*. - 8s + aa + 2p« - 2sz tan 0 ± 2p (a2 +j»8 - sec2 0s*) *. (a)
The volume V of this surface is

V = 2irr [f r* + ** + *r tan 0 +p* ± ±irr cos 0] ;
Vy and VM being the volumes of the two sheets,

an equation of the fourth degree, as it evidently should be ;

r= Va2+J92 is tne radius of the generating circle.

Since the expression for the difference of the volumes of the two
sheets does not contain 2* the sum of the radii of the circles of
contact, it will follow that this difference will depend on the form
but not on the magnitude of the cone.

284.] We shall now proceed to find the algebraical equation

374 ON CONICS.

of the curve which is the locus of the vertex of a constant angle
whose sides always touch a conic.

It will add to the simplicity of the investigation, and not detract
from its generality, if we assume a right circular cylinder instead
of a cone as the dirigent surface.

The equation of the conospheroid as given in (a) is

xz + y* + z*= a2 + s* + 2j»2 - 2sz tan 6 ± 2p (a2 +pz - z* sec2 6} *;

but when the cone becomes a cylinder, 20 its vertical angle becomes
0, and s=b, where b is the radius of the base of the circular cylinder.
The equation of the conospheroid now becomes

^ + y2 + r2=a2 + 62 + 2/>2±2??(«2+j??2-^2)^ . . (b)

Let the axis of the conic make the angle <p with the base of the
cylinder, the axis of Y continuing unchanged. Then we shall have

x =#cos< + ,2' sin< z = xsm< + z cos <.

But as we require only the equation of the curve in which these
surfaces intersect, we must put z(=Q ; and then x=oet cos <p,
z=xt sin (p.

Substituting these values in the preceding equation, bearing in

a a2

mind that cos<p=T, and sin<p=e, we get, since tan2a=— ,,

b p2

4 [a V + Px* ~ «2&2] = l>2 + y* - («2 + *2) ] 2 tan2 «. . (c) *
285.] When the given conic is a parabola, the locus of the vertex

of the constant angle touching the parabola is an hyperbola (fig. 63).
This case may be simply proved by the theorem established in

sec. [279] .

Let p and pt be the focal distances of any point T outside a conic,

2aP
then ppt=~ — , where a is the angle between the tangents drawn to

the conic from the point T.

When the curve is a parabola py=2« = oo .

Hence p=-¥—, ........ (a)

sma

where p is the perpendicular from T on the chord Fm.

From the point T let tangents TOT, Tmt be drawn to the parabola,
containing the angle a.

Let F be the focus of the parabola, and let the angle FmT be
X- Let Tm=t, TP=p, FT=p, the angle AFT=X, and Tc a per-
pendicular to the axis of the parabola. We shall have p2=Ta . Tb.

* In the 8th volume of the Annales de Mathematiques by GEBGONNE the
problem to find the locus of the vertex of a given angle is solved by PONCELET.
The proof he gives by algebra is complicated and tedious.

DE LA HIRE has also given a solution of this problem. See CHASLES, Aperqu,
p. 125.

For

ON CONICS. 375

Ta . T6 = 1 : sin2 % • therefore^ t* sin2 x=T« . T*- But
consequently p* =Ta . Tb =f? -ca8. But Tc =p sin \ •
and ca being an ordinate of the parabola whose parameter is 4/t,
while j»2 = p2 sin2 «, therefore coa = 4A(&— pcosX).

Reducing, we shall find p=- J*k^_*=-^k™°" ,» (b)

cosX — cos a
Fig. 63.

sec a . cosX— 1

If we now compare this expression with the general form of the
focal equation of an hyperbola

ecosX-T
they will be identical if we make e=sec«, and

tan (t sin « tan8 a

(c)

The parabola, and the hyperbola which is the locus of the revolving
angle, have the same directrix. For the distance of the focus of an
hyperbola from its directrix is A(e— c"1) ; putting for A its value
given above, we get for this distance 2k, the same as in the parabola.
« is the angle between the asymptotes of the hyperbola.

286.] When the given angle is a right angle, the generating
segments of the tangent circle become semicircles, the two sheets
of the conospheroid coalesce, and it becomes a sphere of which the
circles of contact are lesser circles.

The radius of this sphere may be thus found. Let U be the
radius of this sphere; since GG^Jia, and nP=£(R+r),

or »* =

376

ON CONICS.

But R — r=2atan0, 20 being the vertical angle of the cone.

Therefore $t2 = a2 sec2 0 + 62.

To find the diameter of the circle AEB, since £1 is the centre of
•the sphere described through the two circles of contact it is the
middle point of caQ)t ; consequently the circle described round the
focal triangle VAB passes through the point fl.

Now 2O/?=R— r=2atan 0.

Hence, if t be the radius of this circle,

Cor.] When the section is a parabola, the second circle of contact
recedes to infinity, the sphere becomes the plane of the circle of
contact, CQ,G therefore the locus becomes the intersection of this
plane with the plane of the section, i. e. the directrix.

CHAPTER XXVIII.

ON THE CENTRAL PROPERTIES OF CONIC SECTIONS.

287.] The rectangle under the distances of the vertex of the cone
from the centres of the focal spheres is equal to the rectangle under
the sides of the cone ending in the vertices of the major axis of the
conic.

Fig. 64.

ON CONICS. 377

In fig. 64, since flBw is a right angle, the quadrilateral AtoBH
may be inscribed in a circle. Hence the angle VflB is equal to the
angle caAB, which is equal to VAo>. Therefore the triangles VflB
and Vo»A are similar ; consequently

Vft . Vo>=VA . VB.

288.] Through C the middle point 0/AB the major axis of the conic,
which is the centre by def. xiv., let a plane be drawn at right angles
to the axis of the cone. This plane will cut the cone in a circle.
The line CD in which the planes of this circle and the conic intersect
will be the minor axis of this section.

The square of the common ordinate CD gives CD*=C/i . Cm.

Now since BC = CA, Cn= Aa= VA sin 6.

In like manner Cm=B6=VB sin 6.

Therefore Cn . Cw=VA . VB sin2 0. But by the preceding
theorem VA . VB= VO . Vw ; and therefore

VA . VB sin2 0=Vn sin 0 . Vo> sin 0=Rr.

But Rr=BF,. AF,, or AF . BF, as in sec. [263].

Therefore thesquare of half the minor axis is equal toAF. BF = Rr,
which is equal to ppt, as shown in sec. [265] .

289.] The parameter (that is, double the ordinate through a focus)
. 2b*

a

Through F, let a plane be drawn at right angles to the axis of
the cone; then, F,G being^half the ordinate, the intersection|of the
planes of the circle and conic, F/G9= Y,v . F^.

But F,v : C»=BF, : BC or ty=^B5=^5.

p A "El

In like manner F^= f. But AF, . BF,= AF . BF.

A F1 RF"
Therefore F,v. F/*=Cn .Cm . t .

Now in the preceding theorem it has been shown that
C».Cm=A«, and AF.BF=6«;

hence Fl? =F,v. F^=S; therefore 2F.G=— .

a* a

290.] The polar axis VO of the cone meets the plane of the conic
in a point C, the centre (see fig. 65) ; and this point bisects all the dia-
meters of the conic.

Let the vertical polar plane, see def. viii., cut the plane of the circle
of contact abmn in the dirigent XY, and let the polar axis VOC of
this plane meet the plane of the circle of contact in the point O,
and the plane of the conic in C. Now as the dirigent XY and the
pole O are polar and pole with respect to the circle of contact
abmn, any plane which passes through the polar axis VO will cut

VOL. II. 3 C

378

ON CONICS.

the vertical polar plane in the line VU and the cone in the sides
Va, Vb, so that VU, V6, VO, Va constitute an harmonic pencil
in the plane VU6O« which cuts the plane of the conic in the dia-
meter BC A ; and as the plane of the conic is parallel to the vertical
polar plane VXY, VU is parallel to AB. Therefore AB is bisected
inC.

In the same way, let any other line VY be drawn in the vertical
polar plane. Through this line and the polar axis VOC let a plane
be drawn cutting the cone in the lines Vw, Vn, and the plane of
the conic in the diameter MCN. Then as VY, Vm, VO, Vrc con-
stitute an harmonic pencil, and as MN is parallel to VY, MC = CN.

291.] In any conic the rectangles under segments of parallel
chords are proportional to each other.

Through the polar axis VOC (fig. 66) let two planes be drawn
cutting the plane of the conic in the diameters AB, MN, and the
vertical polar plane in the lines VU and VY. Through a and b the
points in which the plane VUO cuts the circle of contact draw the
lines av, bu parallel to the polar axis VOC meeting VU in v and u.

Then we have by similar triangles

AC : Vv=VC : va and Vv : aO=Uv : Ua. (a)

ON CONICS.

Fig. 66.

379

Compounding these proportions, we obtain

AC :aO = VC.U« :va.Va (b)

In like manner BC .bO = VC . U« -.ub.Mb, (c)

and therefore

AC.BC :aO.bO = VC'Uv.\Ju:Va.lJb.va.ub. . (d)

* By a suitable alteration in fig. 66 the theorem may as easily be proved when
the point in which the chords meet is outside the cone.

380 ON CONICS.

Uv Vu UV .

Now, by introducing the relations — =~h==\rci> preceding

expression becomes

VC2 VU3
AC.BC=aO.iO.=

V02U«.U6'

Let aO . bO = A:2, since O is a fixed point in the plane of the
circle of contact, and let Ua . \Jb=t'2, t being the tangent drawn
from U to the circle of contact.

The preceding expression now becomes

(ft

Through V let any other straight line VY be drawn in the ver-
tical polar plane. Through this line VY and the polar axis VC
let another plane be drawn cutting the plane of the conic in the
straight line MCN and the plane of the circle of contact in the
secant Xmn. Then in the same way it may be shown that

and therefore, eliminating the common factors, we find
AC.BC TO2*2

MC.NC

(g)

(h)

Through UV let a tangent plane to the cone be drawn cutting
the plane of the circle of contact in the tangent UT, and the plane
of the conic which is parallel to the vertical polar plane in the
tangent to the conic at T. Then the side VT of the cone will make
equal angles with the tangent to the cone at T, and with YU which
is parallel to it. Let this angle be %. Then as a side of the cone
VT is at right angles to the tangent UT, UV sin ^=UT or t. In
like manner YV sin %/=£/•

Making these substitutions in the preceding expressions, we get

AC.BC_sin2y/ ({]

MC.NC ~sm2%'

It has been shown in sec. [267] that the angle which a side of
the cone makes with the tangent to the conic at the point where
the side of the cone meets it is equal to the angle which the focal
vector makes with the tangent at the same point ; and as AC = BC
and MC=NC, since C is the centre, we may infer that any two
diameters of a conic are to each other inversely as the sines of the
angles which parallel tangents to these diameters make with the focal
vectors passing through the points of contact.

292.] If now through any other point Oy, in the plane of the

ON CONIC8. 381

circle of contact which is not the pole of the dirigcnt XY, a
straight line be drawn from the vertex, and meeting the plane of
the conic in the point C;, and if through this line VC; and the
two lines VU, VY in the vertical polar plane be drawn meeting
the plane of the conic in the straight lines A^B, and MjCyN,,
these lines will be parallel to ACB and MCN, since one pair of
planes passes through UV, and the other pair through YV, which
are each parallel to the plane of the conic. The point C/? however,
will not be the middle point of the chords A^B,, M;N/} since
VU, VB/} V0;, VA; do not constitute an harmonic pencil.
Now repeating the same construction as before, we shall have

comparing this expression with (i) in the last section, we see that

A,C, . B,C/= ACa f.

M/VNjC, MC3*

Thus may the well known relation between the rectangles under
the segments of parallel chords be simply derived from the pro-
perties of the right cone, and from this other that if a straight line
be drawn parallel to a plane, all the planes drawn through this
straight line will cut the plane in parallel straight lines.

293.] Let us assume one of the foregoing rectangles or squares
(since AC=BC and MC = NC) as the square of half the major
axis a2, and let the other square be ay2 ; then

a2 sin2*. , ..

-a = --oi as before ...... (b)

a* sin2 ^

Now when the tangent to the conic is drawn parallel to 2a, the

major axis of the conic, sin 2^= -,, and sin2 ^y=^, p, pt being the

a ppt

focal perpendiculars on the tangent whose focal angle is y,, and p, pt
the focal vectors of the point of contact. Now ppt— b\ as shown

b*
in sec. [288]. Hence sin2^y= — . Substituting for

PPi
their values, we get

In sec. [276] it has been shown that the rectangle under the
segments of a tangent to a conic, intercepted between two parallel
tangents to the curve, is equal to the rectangle under the focal
vectors of the point of contact. Hence, by the preceding theorem,
the rectangle under the segments of the tangents is equal to the
square of the parallel semidiamcter*.

* This is the theorem which connects the focal and central properties of the

conic factions.

38.2

ON CONICS.

ON THE HYPERBOLA AND ITS ASYMPTOTES.

294.] In the preceding sections, the vertical polar plane as defined
in def. vni. is drawn outside the cone, while its polar with respect
to this cone, the vertical polar axis, is drawn within the surface of
the cone. We may, however, invert these conditions, and draw
the vertical polar axis OV outside the cone (as in fig. 67).

Fig. 67.

Through this axis let two tangent planes be drawn to the cone
touching it in the sides VD, VE, and cutting the base of the cone
in the line DE. These tangent planes may be called Asymptotic
Planes. The plane of this triangle VDE will be the vertical polar
plane of the axis VO, which meets the tangents DO, EO in the
point O.

Let a plane AGH be drawn parallel to the vertical polar plane.
This plane will cut the cone in an hyperbola ASaBA The polar
axis OV being produced will meet the plane of the hyperbola in a
point C, which, as will be shown, is the centre of the hyperbola ;
and if the asymptotic tangent planes to the cone drawn through the
polar axis OV, and touching the cone along the sides VD, VE, be

ON CONICS. :{s:{

produced, they will cut the plane of the hyperbola in two straight
lines CG, CH meeting in C ; and these lines are called the asym-
ptotes of the hyperbola.

Since the plane of the hyperbola is parallel to the vertical polar
plane VDE, the asymptotic tangent planes to the cone through
VD, VE will cut these planes in parallel straight lines VD, CG and
VE, CH ; or the asymptotes are parallel to the sides of the cone.

Cor. i.] No hyperbola can be cut from a given right cone the
angle between whose asymptotes is greater than the vertical angle
of the cone.

Cor. ii.] All hyperbolas whose planes are parallel will have the
same asymptotic planes ; and therefore the angles between their
several pairs of asymptotes will be equal.

295.] Through the vertical polar axis VO let a plane be drawn
cutting the vertical polar plane in the line VL, the sides of the
cone in the lines YM and VN, and the plane of the hyperbola in
the line ACB. Then as VO, VM, VL, VN is an harmonic pencil,
and the line ACB is parallel to the line YL in the polar plane VDE,
CA=CB, or C the centre bisects all the chords which pass through
it.

Since the asymptotes CG, CH are parallel to the sides of cone
YD, VE, a line TZ drawn from any point T of an asymptote to
the parallel side VZ of the cone and parallel to YC is equal to
VC the distance between the vertex of the cone and the centre of
the hyperbola, since VCTZ is a parallelogram.

290.] Since the plane of the hyperbola is parallel to the vertical
polar plane, the straight lines in which these planes are cut by the
asymptotic tangent planes are parallel. As the distance between
the plane of the hyperbola and the vertical polar plane is constant,
the surface of the cone as it enlarges from the vertex will approach
more and more closely to the asymptotes ; so also, therefore, will
the hyperbola, as it is a curve on the surface of the cone, and whose
plane is at a fixed distance from the side of the cone in which it is
touched by the asymptotic plane.

297.] If a straight line meet the hyperbola and its asymptotes,
the portions of the line between the curve and the asymptotes are equal.

Let the secant meet the hyperbola in the points 11, S, and the
asymptotes in the points Q, P. Through the points <d, P let
tangents QI, PK be drawn to the cone parallel to YC, and touching
the cone in the points I, K on the sides of the cone V K, Y I ). Thru
as VCPK and VCQI are parallelograms, PK is equal to Ql^as each
is equal to VC. But the rectangle QS . QR : 1' K . 1>S = OP : PKa;
but QI = PK, and therefore QS . QR= 1'K . PS or US = PK.

298.] //' a tangent be drawn to an hyperbola, the portion of it
between the asymptotes will IK- hiscctctl at the ji'iint of contact.

Through A let a tangent TX be drawn, AT = AX.

384 ON CONICS.

From X and T let tangents TZ, XY be drawn to the cone parallel
to VC. They are therefore equal, as each is equal VC. But

AT2 : AX2=TZ2 : XT2;

and as TZ = XY, AT = AX.

299.] The rectangle under the segments of a secant between the
asymptotes and a point on the curve is constant, and equal to the square
of the parallel tangent between the point of contact and the asymptote.

Let AT be parallel to the secant QSR ; draw tangents to the
cone TZ and QI from T and Q, parallel to VC. These tangents
are equal, each being equal to VC ; hence the rectangle

QS.QR : AT2 = QI2:TZ2.
But QI = TZ, as each is equal to VC ; therefore QS . QR = AT2.

Hence also, the rectangles under the segments of any parallel secants
between the asymptotes and points on the curve are equal.

300.] While the vertical polar plane and the vertical polar axis
are interchanging their positions, the former becomes a tangent
plane to the cone, while the polar axis becomes that side of the
cone in which it is touched by the vertical polar plane. Hence
the plane of the conic which is always parallel to the vertical polar
plane, now becomes parallel to a side of the cone; that is, the
section is a parabola : and as the centre of the conic is always on
the polar axis (in this case the side of the cone), the centre of the
parabola will be the point in which the side of the cone will meet
the plane of the parabola, to which it is parallel — that is, at infinity.

Again, as the vertical polar axis is the line in which the asym-
ptotic planes intersect, and as these tangent planes merge iuto one
when their line of intersection becomes a side of the cone, the
asymptotic plane spreads out on either side and meets the plane of
the parabola in straight lines parallel to the axis of the conic, but
at an infinite distance from it. Hence the parabola partakes of
the nature of the hyperbola. It has asymptotes; but they are
parallel to its axis at infinity.

CHAPTER XXIX.

ON THE CURVATURE OF THE CONIC SECTIONS DERIVED FROM THE
CURVATURE OF THE RIGHT CONE.

DEFINITION.

The curvature of a surface at a point A, may be defined as the
aggregate of all the curvatures of its sections whose planes pass
through the normal to the surface at the point A.

ON CONICS.

LEMMA.

301.] A tangent being drawn to any curved surface, and 11
being the radius of curvature of a normal section drawn through
this tangent, at the point of contact, the radius of curvature of any
other plane section drawn through this tangent is 11 cos i, i being
the angle between the planes. MEUNIER'S theorem.

Let AB be the tangent at the point A, CAOD the normal section,
in the plane of the paper, suppose ; then the tangent plane to the

Fig. 68.

surface through the tangent AB will be perpendicular to the plane
of the paper, and the curve*d surface on either side of CAD inde-
finitely near to A is perpendicular to the plane of the curve CAOD.
Let 7 AS be a section of the surface made by a plane passing through
AB, inclined at an angle i to the plane CAOD. Through at a point
assumed on AB indefinitely near to A let the plane aru be drawn
perpendicular to A B, meeting the normal section in rand the other
section in v. Then wot is a right angle, and the angle var = i.
Let £ be the radius of curvature of the section yAS. Then
Aa2=2$l . «r, and Aaa=2lT . <xv. But otr=av cos i ; consequently

C=Hcosi (a)

Cor. i.] Hence if through the circle of curvature of the normal
section of the surface, whose plane passes through the tangent Aet, a
sphere be described having its centre coincident with that of the circle
of curvature of the normal section, a plane passing through the tangent
Aa will cut the surface in a curve and the sphere in a lesser circle,
such that the latter will be the circle of curvature of the former at the
point A.

Cor. ii.] If on the normal to a curved surf ace, as diameter, a sphere
be described passing through the given point A, and if the sections of
the surface and the sphere made by a plane passing through the

VOL. II. 3D

386

ON CONICS.

tangent AB have the same curvature, any other plane passing through
AB will cut the surface and the sphere in sections having the same
curvature.

302.] We shall now proceed to apply this theorem to cones and
conies.

If a tangent AB be drawn to a p- gy

right cone at a point A, and AC
be drawn in the tangent plane at
right angles to the side of the cone
AV, the radius of curvature of the
normal section passing through AC
is to the radius of curvature of the
normal section passing through AB
at the point A as sin2 VAB : 1 ;
or if C be the radius of curvature
through AC, i& the radius of cur-
vature through AB, and the angle
VAB be Y,

r=$tsm2X.

(a)

Let a plane AVD be drawn
through the axis of the cone, and
a tangent plane to the cone along
the side AV, and let another plane
A' Vmn be drawn parallel and inde-
finitely near to the former, cutting
the tangent plane AVB in the
straight line A/V, parallel to AV,
the tangents AB, AC in the points
m and n, and the cone in the
hyperbola r, v, U, of which A,V;
is one of the asymptotes. Draw
nr, mv parallel to the normal at
A, and meeting the hyperbola in
T and v; then nr, mv are ultimately equal ; for in the infinitesimal
hyperbola Ut>r, Vyra . mv=Vtn . nr. But ultimately V/m = V,w,
as each is ultimately equal to VA. Therefore mv = nr. Now
Am2 = 2iH .mv , and An2 = 2 C . nr, while Aw2 = Am2 sin2 %; consequently

r=ifcsin2x ........ (b)

Hence the radii of curvature of all the normal sections of a cone
at a given point, and whose planes pass through tangents to the cone
at this point, are to each other inversely as the squares of the sines
of the angles which these tangents make with the side of the cone
passing through the given point.

303.] To find the radius of curvature of a conic section at a given

ON CONIC8. 387

point on the surface of a cone, whose plane passes through a given
tangent to the cone at this point (fig. 69).

Let A be the given point, AB the given tangent, and VAB=^.

Let the tangent AC be drawn at right angles to VA. Then if
a sphere be described on the normal to the cone touching the
tangent plane at A, it will follow from cor. ii. sec. [301] that if any
common section of the cone and sphere passing through the tangent
AC have the same curvature, every other common section of the
sphere and cone passing through the same tangent AC will have
the same curvature. Let the sphere now be supposed to be inscribed
in the cone, touching the tangent plane at A ; it is manifest that
the common sections of the cone and sphere passing through the
tangent AC parallel to the base of the cone will have the same cur-
vature at A, as the sections in this case are one and the same circle,
the " circle of contact " of the sphere with the cone ; consequently
the great circle of this sphere whose plane passes through the
tangent AC is the circle of curvature of the normal section of the
cone at A whose plane passes through the tangent AC.

Let VA = /, and let the semiangle of the cone be 0, while r is the
radius of this sphere inscribed in the cone, then r is manifestly
equal to / tan 6.

Consequently, if 11 be the radius of curvature of the normal

section of the cone through AB, H = _?L- (see sec. [302]), and
r=/tan0. Therefore it i

sm

,
If now a sphere be described touching the tangent plane VACB

at A, its radius being l ta? , every plane passing through the

2

sm

tangent AB will cut the cone in a conic section, and the sphere in
a circle, such that the latter will be the circle of curvature of the
former at the point A.

304.] To find the centre of the sphere of curvature for all the
sections of the cone whose planes pass through the tangent to the cone
AD (fig. 70).

Let A be a point on the surface of the cone through which the
tangent AD is drawn. To the tangent plane VAD draw the normal
AO meeting the axis of the cone in O. Through ADO let a plane
be drawn, and in this plane make the angle DAC equal to the
angle VAD. Through the point O draw the line OC parallel to
AD, and meeting the line AC in C. Through the point C draw
CQ at right angles to AC, meeting the line AO in Q. AQ is the
radius of the sphere of curvature.

Since QCA is a right-angled triangle at C, and OC is at rigl.t
angles to AQ, the angle CQA=OCA = CAD=VAD=X. Therefore

388

ON CONICS.

AC = AQ sin
and

and

AC sin ^ = AO. Therefore AO = AQ sin2 %,

Hence AQ= . a^ .
sln X

305.] To find an expression for the radius of curvature of any
conic section whose plane passes through the tangent Tp (fig. 71).

Let ACB be the conic section, Tp the tangent to the cone at the
point T in the plane of the conic. From the vertex of the cone draw
the perpendicular VP to the plane of the conic, and through VP let
a plane be drawn at right angles to the tangent Tp, meeting this tan-
gent in p. Then Vp, T?p are each at right angles to Tp ; and therefore
the angle V/?P is the inclination of the tangent plane to the cone to
the plane of the conic. Let VT=/, and the angle VTp = %. The sine
of the angle which the plane of the conic makes with the tangent

VP VP

plane is — — = — : - . Hence the cosine of the angle which the
\p I sm %

plane of the conic makes with the normal plane passing through

VP

AB is — : - . But the radius of curvature of this section is equal to
/ sin ^ *

the radius of curvature of the normal section passing through Tp,
multiplied by the cosine of the angle between the planes, by

ON CONIC8.
Fig. 71.

MEUNIER'S theorem ; or thg radius of curvature of the conic section

at the point T is - — . Now in sec. [281] it has been

sin* •% I sin x

shown that VP tan 0 is the semiparameter. Hence the radius of

curvature is equal to .V

sin X

Now as x is also the angle between the tangent and the focal
vector at the point A,

sin x =• - = -• But ppt = b*, and pp, = a*, p and pt being the perprn -
P Pi

diculars from the foci on the tangent: therefore siu3v= _,and

* fl.3

b* a3

4L = — . Therefore the radius of curvature = -'.-. ... (b)
a ab

Cor.] Hence also the radii of curvature of all conic sections
whose planes pass through a given tangent to the cone are, at their
points of contact, as their parameters.

In some treatises on conic sections bt is put for the semidiameter
parallel to the tangent, while at represents the seraidiaroeter through
the point of contact ; here the notation is reversed.

390 ON CONICS.

DEFINITION.

306.] A normal to a conic, at a given point, may be defined as
the projection, on the plane of the section, of the radius of the
sphere inscribed in the cone, touching the conic at this point.

As the centre of the inscribed sphere is always on the axis of the
cone, and as the projection of any point in the axis of the cone on
the conic is always on its major axis, therefore the foot of the
normal will always be found in the major axis of the conic.

Cor.] The normal is always perpendicular to the tangent to the
cone at the given point; for as AO is perpendicular to AB (see
fig. 69), its projection on any plane passing through AB will be also
perpendicular to AB.

To find an expression for the normal N.

Let N be the normal at the point A, VA=/; then the cosine of
the angle between the normal plane to the cone passing through

VP

AB and the plane of the conic is — : - , as shown in the last

sin "£

section ; and the radius of the inscribed sphere is / tan 6 : conse-

VP

quently the normal is /tan Q . j- r — . Now VP tan 6 is the semi-

parameter (^L), as shown in sec. [281] ; therefore the expression
for the normal becomes

ng

307.] In any conic section the normal is to the radius of curvature
at any given point as the radius of the inscribed sphere is to the radius
of the sphere of curvature at that point.

The radius of curvature of the conic at the given point is • . ,. .

sin3 v

(4L)

The normal at the same point is ^1 — -. The radius of the sphere

sm%

of curvature is -^— 5 — . The radius of the inscribed sphere is I tan 9.

sm %
Hence the proposition is manifest*.

* Intelligent students of this subject may have been at a loss to understand
why the radius of curvature of a conic section at any point should vary inversely
as the cube of the sine of the angle between the tangent and focal vector at that
point. These quantities do not appear to have any connexion ; there are other
quantities with which the radius of curvature would seem to be more nearly allied.
But when it is shown that the angle ^ is not only the angle between the tangent
and the focal vector, but that it is also the angle between a side of the cone aqd
the plane of the normal section of curvature whose radius varies inversely as the
square of the sine of this angle, and that the cosine of the angle between the
plane of the conic and the plane of this normal circle of curvature varies also
inversely as sin x> we may thus see how the radius of curvature of the conic section

varies as the product of . • by -, --

sin x * sin X

ON CONICS.

DEFINITION.

391

The sphere described on the portion of the axis of the cone
between the centres of the focal spheres as a diameter, may be
called the central sphere.

308.] The distance between the centre of the conic and the foot of the
normal is (a — p)e,p being the focal vector to the focus F from the point
N to which the normal is drawn, and e the eccentricity of the conic.

Since the centre S of the central sphere is on the axis of the cone,
and the centre v of the normal sphere is also on the same axis, the
projections of these two centres on the major axis of the conic will
give the centre of this conic and the foot of the normal, as shown
in sec. [306] .

Fig. 72.

Thus, in fig. 72, let VC = c be a side of the cone between the
vertex V and the circle of contact ; and as DD,=2a, see sec. [261],

the distance of V to S, the centre of the central sphere, is I 2),

\CO80/

6 being the scmiangle of the cone, and t the angle which the axis
of the cone makes with the plane of the conic. This line projected
on the major axis of the conic, becomes

COS I

i:un * •, r.»- i T

since e— *, as shown in sec. [2/1J.

cosa

392 ON COMICS.

In like manner, c + p being that portion of the side of the cone
to the point N, the distance of the vertex of the cone to the centre

v of the normal sphere will be I -- £ I ; and this line projected on

\COS0/

the major axis of the conic will become

I - £ ) cosi=
\cos#/

Now On is the difference of the projections OP and nP ; hence
OP-nP = On=(c + a)e — (c + p)e=(a- p)e. . . (b)

Cor. i.] The distance between the foot of the normal and the
focus is

ae — (a—p]e = pe ........ (c)

Cor. ii.] The distance of the foot of the normal from the other
focus is pte ; therefore the rectangle under these distances is

a*<* ......... (d)

309.] The rectangle under the perpendiculars, on the major axis,
from the vertex of the cone and the centre of the central sphere is
equal to the square of half the minor axis.

As the major axis (fig. 72) of the conic is a chord of the central
sphere whose radius is a seed, the perpendicular on the major axis
from this centre will be a tan 0 ; and p being the perpendicular from
the vertex of the cone on the major axis, the rectangle is

a tan 6 .p — a .p tan 6.

fo<z
•But p tan 0 = — , as shown in sec. [281] ;

N

therefore the rectangle under the perpendiculars is equal to 62.

CHAPTER XXX.

ON THE PROPERTIES OF CONFOCAL CONICS DERIVED FROM
THE RIGHT CONE.

310.] The consideration of groups of conies that shall have the
same centre and foci may be based on an extension of the properties
of focal spheres.

If we conceive the radii of the focal spheres inscribed in the cone
to be increased in the same ratio, while the points of contact of the
spheres with the plane of the conic continue the same, and if cir-
cumscribing cones be drawn to each pair of spheres, whose radii

ON CONICS.

393

are R, r : nR, nr : nfi, ntr and so on, we shall have as many cones
circumscribing these spheres, which will cut the plane of the original
conic in as many concentric and confocal conies.

DEFINITION.

The point in which the axis of the cone meets the major axis of
the conic may be called the point of axial intersection.
These cones possess some curious properties.

Fig. 73.

VOL. II.

3 E

394 ON CONICS.

(a) The axes of these cones all pass through a fixed point (the
point of axial intersection) on the major axis.

(fi) The vertices of all these cones range along the same perpen-
dicular to the plane of the conic.

(y) The ratio of the distances from the vertex of any one of the
cones to the centres of the inscribed focal spheres is constant.

311.] Let planes be drawn through the axes of these cones, they
will all cut the major axis in the axial point of intersection Q ; and
P being the foot of the perpendicular drawn from the vertices of
all these cones, we shall have

F,Q : QF = F;P : PF;
for VF,, VQ, VF, VP is an harmonic pencil, as shown in sec. [278] .

The angle between the vertical focals VF and VF, may be thus
found.

The tangent of the angle y between the vertical focals may be
found from the expression

tan y=2e tan 0,

0 being the semiangle of the cone, while B, and r are the radii of
the focal spheres. Let these focal vectors make the angles 8, Sf

O6 tt€

with the major axis ; then tan 8=—, and tan 8;= — .

tan 8 — tan 8. ae(R,—r)

or =

But Rr = 62, as in sec. [288], and (R-r)=2atan0.

Therefore tan y = 2e tan 6.

312.] If in sec. [278] the chord mn be supposed to pass through
Q, the point of axial intersection, the perpendicular on mn from
the intersection of the tangents drawn at the extremities of this
chord mn will pass through P the foot of the perpendicular from
one of the vertices of the cones.

Hence, if mn be a segment of a common chord to any number of
confocal conies, the intersections of every pair of tangents whose
common chord is mn will meet in the straight line drawn at right
angles to mn through P the foot of the perpendicular to the plane
of the section, the locus of the vertices of all the confocal cones.

More generally, if any number of confocal conies have a
common chord, and if tangents in pairs be drawn to the conies at
the points in which they are met two by two by the common chord,
these tangents will meet in pairs on the straight line passing
through -or at right angles to the common chord. If q be the inter-
section of the common chord mn with the major axis of the conic,
we shall have

F,g : gF=F,w : *rF.

Hence the position of the point w may be ascertained.

ox CONICS. 395

Should the chord mn become a tangent instead of a secant to
one of the confocal conies, the pair of tangents coalesce into one
tangent meeting on the perpendicular.

313.] Hence we may obtain this other theorem established by a
very different method in the first volume, p. 20 : — If a secant to a
conic be a tangent to another confocal conic, and tangents be drawn
to the outer conic at the ends of this chord meeting in a point, the
line drawn from this point of intersection to the point of contact of
the inner confocal section will be perpendicular to this secant.

CHAPTER XXXI.
ON SIMILAR CONIC SECTIONS.

DEFINITION.

314.] The sections of a cone made by parallel planes may be
called similar conic sections.

Hence similar conies have the same vertical polar plane and the
same polar axis; and therefore all their centres range along the
same straight line, the polar axis.

Therefore all circles, parabolas, and equilateral hyperbolas are
similar figures ; for their vertical polar planes are identical.

Hence all similar hyperbolas have the same asymptotes.

In similar and similarly posited conies all parallel diameters,
and homologous lines generally, are in the same ratio, that of the
parameters of the conies.

Through the axis VOQ of the cone let a plane be drawn cutting
the planes of the parallel conies ABCD and abnm in the lines
QA, Oa, which lines are themselves parallel ; hence (fig. 74)

Oa : QA=VO : VQ=VP : VP,=VP.tan0 : VP,tan0.

But VP tan & and VP( tan Q are the semiparameters of the two
sections, as shown in sec. [281]. In the same way it may be
shown of any two homologous lines in the similar sections.

315.] In two similar concentric and similarly posited conies two
parallel chords of one are drawn cutting the other; the rectangle
under the segments of the one is equal to the rectangle under the
segments of the other.

Through the opposite cone VA^B^C^D^ let a plane AII'BIICUDU
be drawn parallel to the plane ABCD and equidistant from the
vertex V. The section of the cone made by this plane will be in
every respect equal snd similar to the section ABCD. Now if we
conceive a cylinder erected on this base, and having its axis coin-
cident with that of the cone, it will meet the plane of the parallel

396

ON CONICS.

Fig. 74.

section in a section equal, similar, and parallel to the given section ;
hence the cylinder will meet the upper sheet of the cone in the
section A^B^C^D^. Through any point A; in the cylinder let a
plane be drawn parallel to the given plane A BCD ; it will cut the
cylinder in a section A^ByCyDy equal and similar to ABCD, and the
cone in a section abmn parallel to the section ABCD, and therefore
similar to it. Hence the sections A/B^D, and abmn of the cylinder
and the cone are similar and concentric. Through C, and D/? any
two points on the surface of the cylinder and in the plane of the

ON CONIC8. 397

section AjB^D,, let two parallel chords be drawn meeting the
section abmmtnnl in the points m, n and the points m.nr

Through C, and D/ let two sides of the cylinder be drawn meeting
the cone in the points C, C/y and D, D/y. Then, as CC,,, DD;/ and
Cm, D/m/ are parallel secants of the cone,

C,C . C,Ctl : Cltm . Ctn- D,D . D,Dtf : D,w, . D,n, ;

but as the three common secant planes of the cylinder and cone are
parallel, C,C = D/D and C/^DjD,,; therefore C,m . Cfi=Dfn.Dft.
Hence also, if one of the parallel secants of the similar conies
becomes a tangent, this tangent will be bisected at the point of
contact.

It is manifest that the segments of any chord drawn to meet
the similar conies are equal between the sections.

The following properties of right cones and their sections are
worthy of notice.

316.] (a) A tangent to a cone being drawn, there may always be
drawn through it two planes cutting the cone in two sections which
shall have equal parameters.

(/S) The conic of maximum parameter which can be drawn through
a point on the surface of a right cone is that whose plane is at right
angles to the side of the cone passing through the given point, and
having its tangent at this point parallel to the circular base of the
cone.

(y) Through a given point on the surface of a cone there may be
drawn any number of planes cutting the cone in conies having the
same parameter ; and their planes will all touch a right cone, whose
vertex is the given point and whose axis is the side of the original
cone passing through the given point.

(8) The locus of the foci of all the parabolas which can be con-
structed on a given right cone is also a right cone. The locus of the
foci of all equal parabolas on the cone is a circle whose plane is
parallel to that of the base ; and the locus of the foci of all the para-
bolas whose planes are parallel is a straight line passing through the
vertex of the cone. Hence the locus of the foci of all the parabolas
that can be drawn on the cone is the combination of the above named
loci, a cone.

398 ON CONICS.

CHAPTER XXXII.

ON CONICS IN A PLANE.

317.] Having now, by the help of the right circular cone, estab-
lished the principal properties of its sections by short and simple
demonstrations based on the most elementary principles of the
ancient geometry, it is proper to show how these principles may be
applied to the development of the properties of conies viewed as
curves on a plane. When the cardinal properties of conies have
once been established by the help of the right circular cone, there
is but little difficulty in applying them to the investigation on a
plane of the countless theorems relating to those curves. Geo-
meters have in general ignored their real origin, and have founded
their investigations on some arbitrary definition. It is worthy of
notice that some of the theorems which are most easily and briefly
demonstrated when these curves are taken as sections of a right
cone, can be established only by tedious and complicated methods
when they are treated as plane curves. Of this the focal properties
furnish a striking example. There is, however, a very large class
of theorems to whose solutions neither the right cone nor any other
cone affords any help. I refer to the properties of minor foci and
minor directrices, which will be found discussed at some length in
the first volume of this work (see Vol. I. sec. [288]).

The method of reciprocal polars applied to oblate and prolate
spheroids is the source of innumerable novel properties of conies.
One special application of the method of reciprocal polars to the
prolate or elongated ellipsoid of revolution round its major axis
(see Vol. I. p. 218) enables us to develop to a very great extent
and with much simplicity the properties of surfaces of the second
order whose three axes are unequal. In particular it is shown that
every such surface has four foci and four directrix planes. Every
new method may be a fertile field of new truths.

In sec. [265] it is shown that the rectangle under the focal
perpendiculars p, p, on a tangent drawn to a central conic is
equal to 6". Let P be the perpendicular from the centre on the
same tangent, making the angle X with the major axis. Then,
obviously

P+P,=W (a)

Let p, Pf be the focal vectors to the same point.

Then p-\-pl=2a, as shown in sec. [262].

Let ^ be the focal tangential angle, then we have

2P=p +p,= (p + /o,) sin x= 2« sin %.
Hence P = «sin^ . (b)

ON CONIC8.

Since j» = P -f ae cos X, and ^=P — oecosX, . . . (c)

62 =pp, = P2 — «2e2 cos2 X.
Therefore P2=a2(l-e2 sin2X) ....... (d)

Comparing this expression with (b) we find

cos^=esin\, ....... (e)

a simple relation which connects the focal tangential angle % with
the latitude X.

318.] Since p+pt = 2a, squaring,

p2 + 2pp, + p,z= 4a2. Now 2pp, — 2af, as shown in (b) , sec . [293] ,
while p2 + p? = 2b? + 2a2e2.

Hence, substituting, a/2 + i/2=a2 + 62 ....... (a)

Since a?=ppt and P2*=a2sin2^, fl^P2 = fl2p sin ^ . p, sin ^,
or afP*=aippl=dib'i ....... (b)

Hence the areas of parallelograms about the conjugate diameters
of a conic are equal.

Let b* = #2 + y*, then a*b* = a2*2 + «y . But a2y2 = a262 - 6V ;
hence a*b* = a262 + (a* — i2>2, or b* = b9 + e V ; )

and in like manner fl^2 = a2 — e2a;2. j

319.] The following values of the radius of curvature, and chorda
of curvature passing through the foci and centre, may easily be
derived from the expressions in sees. [303], [304], [305], which
have themselves been deduced from the properties of the right cone.

In sec. [305] it has been shown that

2 b*

and N= — : -- , .... (a)

while P=asin^, and sin^=- ...... (b)

From these values we may obtain the following expressions for
the radius of curvature and the normal —

aN=^y, »=, »sin2x=N.

If -fy and x b6 tne angles which a tangent to the curve makes
with the central and focal vectors,

• aasin4Y .,.

(d)

9 . g -- r — « ......

a2 sm2 Y — o2 cos* ^

2a9

Hence C, the chord of curvature through the centre is —^-t (c)

while the chord of curvature through the focus is — L. . . (f )

400

ON CONICS.

320.] If a line be drawn from the focus to the pole of a focal chord,
it will be at right angles to this chord, see sec. [270] .

In the parabola mpn is a right angle.

Since wF=mY, and the angle p¥m is equal to the angle mYp,
both being right angles, the angle Y^?F is bisected by the tangent
pm. In the same way the angle XjoF is bisected by the tangent pn ;
consequently the angle mpn is a right angle.

Cor. i.] We have also Y/?=Xj9=Fp.

Hence, if from the ends of a focal chord of a parabola perpen-
diculars are drawn to the directrix, the pole of the focal chord will
bisect the portion of the directrix between the feet of the perpen-
diculars.

Fig. 75.

A tangent to a parabola makes equal angles with the focal vector
drawn to the point of contact and with the axis of the curve.

This is evident from an inspection of fig. 75.

321 .] The focal vector drawn through the intersection of a pair of
tangents to a parabola divides the angle between these tangents into
two, which are respectively equal to the alternate angles which the

ON COtflCS.

401

tangents make with the focal vectors passing through the points of
contact.

Let the tangents Tm, Tn to the parabola meet in T ; let F be

Fig. 76.

the focus of the parabola ; and let the tangents meet the axis of the
curve in the points C, C,.

Then the angle wFE is equal to twice the angle mCF, and the
angle »FE is equal to twice the angle nC;F ; hence, adding, the
angle mFn is equal to twice the angle mTn, or half the angle mF»
is equal to the angle wT«. Now the line TFG bisects the focal
angle wF»; therefore the angle wFG is equal to the anglr /wT/i.
But, being external, it is also equal to the sum of the angles Fw*T
and FTw. Therefore the sum of the angles FwT and FTw is equal
to the sum of the angles l<Tn and FTw ; consequently the angle
FwT is equal to the angle FTn.

Hence, since the angle TF/n is equal to the angle TFn, the two
triangles TFw and TF/i are similar ; therefore t»F : TF=TF : nF, or

mF.nF=TF2.

Hence, in a parabola, the square of the focal vector drawn to the
intersection of a pair of tangents to the curve is equal to the rectangle
under the focal vectors drawn to the points of contact of these tangents.

322.] The squares of the tangents Tm, Tn (fig. 76) drawn to a
parabola from any point T are in the same ratio as the focal vector*
drawn to the points of contact in, n.

VOL. II. 3 T

402 ON CONICS.

Let FP be a perpendicular drawn from the focus to the tangent
Tm, then the area of the triangle TFm is =^FP . Tm. But if -^
be the angle TFm, the area of this triangle is also £FT . Fm . sin ^ ;
therefore

FP.Tm

"k^FfTI* .......

But the angle TFn is also equal to ty ;

FP. .Tn ,M

therefore Sm = .......

Equating these values of sin >/r, squaring, putting for FP2 and FP,2
their values k . Fm and k . Fn, we get

fma_Fm

W~*v

Hence also the chord mn is divided into segments- by the line TF,
which are to each other in the duplicate ratio of the tangents Tm
and Tn.

323.] If a parabola be inscribed in a triangle, the circle which cir-
cumscribes the triangle passes through the focus of the parabola.

This theorem follows immediately from that established in sec.
[277], in which it is shown that, if a conic be inscribed in a triangle,
the sum of the angles subtended at the foci by the base of the tri-
angle is equal to the external vertical angle of the triangle. Now
when the conic becomes a parabola, the remote focal angle vanishes,
and therefore the angle at the near focus, subtended by the base
of the triangle, is equal to the external vertical angle of the triangle;
and therefore a circle may be drawn through the vertices of the
quadrilateral ACBF*.

Since FC3=FA/.FB/, FB2=FC,.FA,, FA2=FB,.FC,, . (a)
then, as ACBF is a quadrilateral in a circle,

CB . (FB,FC,)* + CA . (FC, . FA,)*=AB . (FA, . FB,)*;
consequently by division we obtain finally
CB CA AB

VFA,

Hence the sum of the sides of the circumscribing triangle, each
divided by the square root of the focal vector drawn to its point of
contact with the parabola, is constant.

If we multiply together the expressions in (a), we shall have

FA.FB.FC^FA^.FB^FC, (c)

Hence, when a triangle circumscribes a parabola, the product of the
focal vectors drawn to the vertices of the triangle is equal to the

* This theorem is otherwise established, and very simply, in the first volume,
see sec. [53], by an application of the method of tangential coordinates.

ON COMICS.

108

Fig. 77.

product of the focal vectors drawn to the points of contact of the
sides of the triangle with the parabola.

324.] Since the sums of the rectangles under the adjacent sides
of a quadrilateral inscribed in a circle are as the diagonals which
join the points in which the sides of the rectangles meet, we have

AB . BC . CA=CA . FC . FA + CB . FC . FB-AB . FA . FB.
But FA2= FB, . FC,, FB3= FA, . FB,, and "FCa= FA,FB(.

Substituting these values in the preceding equation, we get

AB . BC . CA

VFA,-AB .

[FA,.FB,.FC,]r

325.] The directrices of all the parabolas inscribed in a triangle
pass through the orthocentre of this triangle (see fig. 78) .

From the focus F, on the circumference of the circle, draw the
perpendiculars Fm, Fn, Fr on the sides AC, CB, AB of the «ri\ vn
triangle. The points m, r, n range along a straight line, whirh is
a tangent to the parabola at its vertex. Produce Fr until rx is
equal to Fr, and through x draw xTt parallel to mm. xZ is the
directrix. Produce rA to meet the directrix in D. Join DF
meeting the circle in Q. Join CQ, FB. Then the angle CQD = the
angle FBC, since CQFB is a quadrilateral in a circle. The angle
QFr=the angle Dar=the angle wrF=the angle FBC, since Fr/»B
is a quadrilateral that may be inscribed in a circle. Therefore the
angle DQC is equal to the angle DFx, or F* is parallel to C/».

404

ON CONIC*.

Fig. 78.

Therefore CpD is a right angle ; and therefore ps=pQ,} or s (a point
on the directrix) is the orthocentre.

326.] The inscription of the maximum parabola in a triangle
involves the trisection of an angle (see fig. 79) .

Let ABC be the triangle, and let F be the focus of the maximum
parabola. From F draw the perpendiculars FM, FN, FU on the sides
of the triangle AC, AB, BC ; the line MNU is a tangent to the para-
bola at its vertex (see preceding theorems) . To this tangent MNU
draw the perpendicular FP; FP will be one fourth of the parameter
of the maximum parabola inscribed in the triangle.

Assume a point Fy on the circumference of the circle indefinitely
near to F, and from this point draw the perpendiculars FyM,, F;N, to
the sides of the triangle AC, AB. The line M,Ny will be a tangent
to the parabola whose focus is F, ; draw to this tangent the per-
pendicular FyPy. FyPy is one fourth the parameter of the parabola
drawn indefinitely near to the former ; therefore F;Py=FP; and they
are ultimately parallel, therefore FG the tangent to the circle at F
is parallel to MNU. But as FAMN is a quadrilateral in a circle, the
angle FAB is equal to the angle FMN= GFM. Therefore Fra=FB.
Draw OD parallel to AC, cutting the line Fw in I, then FI=wI ;
therefore FI is equal to the half of FB ; and therefore the angle
FOI is one half the angle FOB, or the arc BFD is trisected in F.

This question may be taken as a good illustration of the appli-
cation of the method of infinitesimals to the solution of problems
in geometrical maxima and minima.

When the given triangle ABC is isosceles, the angle to be tri-
sected becomes a right angle.

ON CONIC8.

Fig. 79.

400

327.] By this method of geometrical limits problems which
present great difficulty if treated by algebra or the differential
calculus, may be solved with great simplicity. For example.

To draw the minimum line through a given point within a given
angle (see fig. 80) .

Let BAG be the given angle, O the given point, and BOC the
minimum straight line. Draw the perpendicular AD from A to BC,
and through O draw the line bOc indefinitely near to the line BOC,
meeting the sides of the given angle in the points c, b. Then
as BOC is the minimum line through O, bOc which is indefinitely
near to it, is therefore equal to it. With O as centre draw the
circles whose radii are OC, Ob cutting the lines be and BC in the
points m, n. Then as OC=Om, and Qb=On, cm=Rn. Let a>
be the infinitesimal angle between the minimum lines. Then
Bn=£n cot B, and in = OB . a>. Therefore B/i=OB . o> . cot B.
In like manner cw = OC . <u . cot C. Therefore as Bn=cro,

OCtanB=OBtanC

But

tan

• AD A

i=77fv and

OC tan C

hence 7vu = : 5-

OB tauB

AD

406

ON CONICS.

hence

This may be reduced to

OC_KD
OB~CD*
CO + OB CD + DB

(a)

OB CD

ButCO + OB = CD + DB = BC; therefore OB = CD. . . (b)
Hence the minimum line drawn through a point within an angle
may be denned as the line to which if a perpendicular be drawn
from the vertex, the distance between the foot of the perpendicular
and one end of the base shall be equal to the distance between the
yiven point and the other end.

The point D is the intersection of a semicircle, drawn on the
line AO, with a curve of the fourth order.

ON THE ECCENTRIC ANOMALY IN AN ELLIPSE.

328.] Let a circle be described on the major axis of an ellipse
as diameter, and an ordinate ~

PD to the major axis be pro-
duced through the point P to
meet the circle in the point Q.
The radius of the circle through
this point makes the angle p
with the major axis. The angle
p is called the eccentric ano-
maly; and the angle X which the
perpendicular from the centre
on the tangent through the point P makes with the major axis is
called the latitude.

ON CONIC8. 407

Since #=acos/^ y — b sin /A, the semidiameter

(5P 2* *«=«*«»*/*+**«*'*• (a)

In like manner

ay2=a2sin2/A + &2cosV, and P^a2 cos2X + £2 sin2X. (b)
Thus P and bt reciprocate their forms. Since x=aco»fj,, and

a2£ = #, fl£=cos/*. In like manner bv = sin/A.

Let d be that semidiameter of the ellipse which coincides with the

eccentric radius OQ, and which makes the angle fj, with the major

1 cos2 /A sin2/*
axis, then -^— — g---f g . But P being the perpendicular on

the tangent through the point P,

1 _x* y2_cosV sin2/!, , .

P2~a4"l"64~ a2 62

therefore P=c?, whence this theorem : —

The perpendicular from the centre of the ellipse on the tangent
through the point P is equal to the semidiameter which coincides
with OQ, the eccentric radius of the circle.

329.] To find the relation between the angle of the eccentric
anomaly p, and the focal tangential angle %.

omce

Hence
To fin
Since

p2 — -5 -f 7-4 2 — 1 — 11 — > au<J
tan v — — — ....

. ^, we niiu
fdl

in- sin /.L

. . . . (e)
in*u

a sin IJL-T-O cos /t
d the relation between X and p.

f*OQ // Q'

P2 //2 Pn«s2 X 4- A2 sin2 X ami ^ -4-

fl tan it— 6 tan X: .

tfl

therefore X is greater than /t.

In sec. [305] it has been* shown that if li be the radius of cur-
vature,

b* fa2 sin* it + 62 cos* u] * / A \
» = — r-=— . (c) Hence $1=^ ^— ^ "_. (d)

So also

- . . . ,

[a2 cos2 X + O2 sm8 X] t

Since the normal N is equal to — : . a'N2 = A2 [a2 sin*/* + b* cos2/*] •

a sin %

Comparing this experiment with the preceding, we get H = 7TTT«>

or the radius of curvature is equal to the cube of the normal divided
by the square of the semiparameter.

408 ON CONICS.

CHAPTER XXXIII.

ON ORTHOGONAL PROJECTION.

330.] In orthogonal projection the several points and lines of the
original or projective figure generate another or projected figure on
a plane inclined to the former, the locus of the feet of the perpen-
diculars drawn from every point of the projective to the projected
figure. These terms will be found simple and useful in saving
much circumlocution. The projective figure is cast into its pro-
jected derivative. Thus in a right circular cylinder, the projective
ellipse generates the projected circle on a horizontal plane.

The principles of orthogonal and divergent projection are often
found to be simple yet powerful instruments of investigation,
especially where it may be required to pass from the projective
properties of a circle to those of a conic. Let an ellipse be con-
ceived as an oblique section of a right cylinder standing on a circle
as base. The projective properties of the circle may be at once
transferred without demonstration to the ellipse. For example : —

(a) All the radii of a circle are equal; and therefore all the dia-
meters of an ellipse are bisected in the centre.

(/3) The squares which circumscribe a circle are equal, and the
diameters which join the points of contact of the sides of the square
are parallel to the sides ; hence all parallelograms about conjugate
diameters are equal in area, and the rectangular diameters in a
circle are projected into conjugate diameters in an ellipse.
' (y) The locus of the angles of a square circumscribed to a circle
is a circle whose radius is to that of the former as ^2 : 1 . Hence
the locus of the vertices of parallelograms about the conjugate dia-
meters of an ellipse is an ellipse similar to the original ellipse, whose
axes are in the ratio of \/2 : 1.

(8) Since the locus of the intersection of perpendiculars from
the centre of a circle on the chords joining the extremities of dia-
meters at right angles to each other is also a circle, so in an ellipse
the locus of the intersections of lines drawn from the centre to the
middle points of the chords joining the extremities of conjugate dia-
meters is an ellipse similar to the former, and whose area is to that
of the original ellipse as 1 : \/2.

Hence the area of the original ellipse is a mean proportional
between the areas of these loci.

(e) As the area of a square circumscribing a circle is the least of all
circumscribing quadrilaterals, so the parallelogram about the conju-
gate diameters of an ellipse is the least of all circumscribing quadri-

ON CONICS. 1()<J

laterals. As the square is the greatest quadrilateral that may be
inscribed in a circle, so the area of the parallelogram formed by joint ,,>i
the extremities of conjugate diameters in an ellipse is a maximum. '
(£) As the equilateral triangle is the least triangle that can be
circumscribed to a circle, so the triangle whose sides are bisected at
the points of contact is the least triangle that can be circumscribed
to an ellipse.

(4) As the equilateral triangle is the greatest triangle that may be
inscribed in a circle, so the greatest triangle that may be inscribed in
an ellipse is one whose vertex is at the extremity of one conjugate
diameter, and whose base is an ordinate to this diameter bisecting it
between the centre and the remote vertex.

Hence all such triangles are equal in area, and their centres of
gravity coincide with the centre of the ellipse.

(6) In a circle a chord drawn from a point in which two tangents
intersect is divided harmonically by this point and the chord of con-
tact; so also in a conic.

331.] A perpendicular is drawn from a given point to a given
straight line. The point and line are orthogonally projected on a
given plane into another point and another straight line ; and from
the former a perpendicular is drawn to the latter. The ratio of these
inrpendiculars is independent of the position of the points from which
the perpendiculars are drawn (fig. 82).

Let OA, OB, OC be a set of three rectangular axes in space ;
let BP be the perpendicular from the given point B on the given
line AC ; let this line AC be orthogonally projected into the line
AO inclined to AC by the angle i ; let BO be the perpendicular
on this line : then the ratio of BP to BO is independent of the posi-
tion of B. Let OQ be the perpendicular from O to the plane ABC
inclined to OC by the angle B.

Now the volume of the rectangular pyramid OACB is one sixth
of the volume OA . OB . OC. But it is also one sixth of the volume
of the triangular base ABC multiplied by OQ.

Therefore OA . OB . OC = OQ . AC . BP.

But OQ = OC cos 6, and OA=AC cost; hence we obtain

T-rr= 7, a ratio independent of the position of the point B.

BP cos i '

This is a most important theorem. It enables us to pass from
the properties of perpendiculars about a circle to the analogous
properties of perpendiculars about a conic. By the help of this
relation we may give a very simple proof of the following celebrated
theorem of PAPPUS, " Ad quatuor tineas," as also of many others.

VOL. ii. 3 o

410

ON CONICS.

Fig. 82.

332.] If from any point P, in the circumference of a circle, per-
pendiculars be drawn to the four sides of an inscribed quadrilateral,
the rectangles under each pair of perpendiculars on the opposite sides
will be equal ; that is (see fig. 83) ,

PA,.PD,=PB,.PC,.

From P let the lines PA, PB, PC, PD be drawn to the four
angles of the quadrilateral, and let R be the radius of the circle.
Then (Euclid, Book VI. Prop. C) we have

PD . PC = 2R . PD, and PA . PB=2R . PAy;
therefore PA . PB . PC . PD=4R2PA, . PD,.
In like manner PA . PB . PC . PD=4R2PB,PC;.
Hence PA,. PD,=PB,PC,.

If now we orthogonally project the circle into an ellipse, the
point P will be projected into a point •or on the conic ; the perpen-
diculars PA, -era will have to each other a ratio, the cosine of the
inclination of the side AB to its projection a/3, and so for the other

ON CONIC8.

411

sides. Hence the theorem of the "Ad Quatuor tineas," viz. : —
If from any point vr in the circumference of a conic perpendiculars
b<> drawn to the sides of an inscribed quadrilateral, the rectangles
under each pair of perpendiculars on the opposite sides will i,
constant ratio.

It is evident that the inclination of the planes will not enter into
the constant ratio, as this relation will be eliminated by division.

Fig. 83.

333.] If tangents be drawn at the vertices of a triangle inscribed
in a circle, and if from any point in the circumference of this circle
perpendiculars be drawn to the tangents and to the sides, the product
of the perpendiculars on the tangents will be equal to the product of
the perpendiculars on the sides.

Since PBT and PAQy are similar triangles, we have

PB:PT=PA:PQr

In like manner we have

PA

PT/y=PB

PQ.

PT,=PC : PQ,, and PC
Compounding these proportions, we obtain

PT . PT, . PTW= PQ . PQ, . PQ,r

Hence, if a triangle be inscribed in a conic, and tangents be drawn
to its vertices, and if from any point in the conic perpendiculars be
drawn to the three tangents and to the three sid^s, the product of
the perpendiculars on the tangents will have a constant ratio to the
product of the perpendiculars on the sides.

412

334.] If from any point in the circumference of a circle perpen-
diculars be drawn to a pair of tangents to the circle, the rectangle

ON CONICS.

413

under these perpendiculars will be equal to the square of the perpen-
dicular drawn from this point to the common chord.

Let PA;, PB,, PC, be the three perpendiculars. Then by similar
triangles

PB, : PC = PA/ : PB, and PC, : PB=PA/ : PC.
Hence PB, . PC,=PA,2.

Therefore, if from any point in a conic , perpendiculars be drawn to
a pair of tangents and their chord of contact ; the rectangle under
the perpendiculars on the tangents will have a fixed ratio to the
square of the perpendicular on the chord.

335.] Let a, b, c be the sides of a triangle inscribed in an ellipse
of which the semiaxes are A and B, while the radius of the circle
circumscribing the triangle is R ; let d, d,, dlt be the semidiameters
parallel to the sides of the triangle a, b, c ; then

Fig. 86.

414 ON CONICS.

Let the ellipse be projected into a circle whose radius is B ; let
the triangle in the ellipse whose sides are a, b, c be projected into
another inscribed in the circle whose sides are ex., /3, y ; let the
areas of the projective and projected triangles be S and Sy, then

S=^, and 8,=^*. But S : S, = A :. B. . . (b)

Now as the lengths of any two parallel lines on a plane have the
same ratio to one another as their projections on another plane,
and as d is parallel to a, a : a. = d : B, or

da. r ,., , d./B d,,j

a = -5~. In like manner 6=^-, and c = -^-.
1 > Da

, ctfty.dd.d,, .„ abc ES , S A , >

Hence abc= -L-^'. But ^=M> and g-=g ; . . (c)

hence RAB=e? . dt . dn.

Let /, /,, fn be the three focal chords drawn through any
focus, and parallel to the sides a, b, c of the triangle ABC ;

then from sec. [282] and [291] it follows that c?2=^. Sub-

A

stituting for d, d,, dlt their values, and writing D for 2R and L for
2B2
A'* D2L=//,/,, ........ (d)

336.] If a circle be described cutting a conic in four points, the
vertices of an inscribed quadrilateral, and from a focus six chords
be drawn parallel to the four sides and two diagonals of the inscribed
quadrilateral, we shall have

This follows obviously from the preceding theorem ; for we may
consider the inscribed quadrilateral with its diagonals as equivalent
to four triangles, to which the construction in the foregoing theorem
being applied, we should have twelve focal chords, three for each
triangle. But each focal chord is once repeated ; this reduces the
number to six different chords. Hence the theorem may be enun-
ciated as follows : —

If a circle meet a conic in four points, the vertices of an inscribed
quadrilateral, and from any focus focal chords be drawn parallel
to the four sides and the two diagonals of this quadrilateral, we
shall have the square of the diameter of the circle multiplied by the
parameter of the conic equal to the square root of the product of the
six focal chords.

ON CONICS.

415

In the preceding theorem the products of the six focal chords,
taken two by two, are equal, or

/I -/2=/3 ./4, A -/3=/l «/4, A -/5=/4 '/6, and /2 ./,=/, ./,. (b)

Fig. 87.

337.] Without having recourse to orthogonal projection, it maybe
shown that the product of three focal chords drawn parallel to the
sides of an inscribed triangle is equal to the product of the para-
meter L of the parabola multiplied by the square of the diameter
D of the circumscribing circle, or

Let ABC be the inscribed triangle, V the vertex, and F the focus.
Let CG be drawn parallel to the axis, meeting the side AB in G,
which makes the angle -fy with the axis; and let CP==p be the per-
pendicular on the side AB, and the angle ACG be o>.

Let AG=<
Now sin2 v :
therefore

sin2<»=e2
sin2 v :

CG = c, AB = /,
: a2 and sin2 o>
1=C8L :/,a2.

BC = w, AC=».
= !=!;:/,;

In like manner

sin2/Lt : 1=C2L :/2A2.

But

sin2\ : a

in2>/r=/2c2:n8w

?,

and

sin2 •$> :

l=L:/3;

therefore

sin2X : 1

=L/*c2 : An2/*4

416 ON CONICS.

Hence, multiplying these expressions, we obtain

£ £ f 0 70 4

= f , fa /q asbWn

But D3 = -^ — = - : — , and Dp — mn;

sin A, sin p, sin i/

therefore LVD* =/, /2 /8a*A*jB*.

But j9=csini/r, and «6sin2iJr=Lc.

Making these substitutions, we have finally

M*=/i./a./8 ........ (b)

338.] //"a come described on the surface of a right cone be ortho-
gonally projected on a plane passing through the vertex at right
angles to the axis of the cone, the vertex of the cone will be a focus
of the projected conic.

Let a, b, e be the semiaxes and eccentricity of the conic drawn
on the surface of the right cone ; let 6 be the semiangle of the
cone ; and let /, I, be the lengths of the sides of the cone between
the vertex of the cone and the ends of the major axis of the given
conic.

Then 4a2=/2 + /,2- Sty cos 20; ..... (a)

W-
2ae=(l—lj), see sec. [262], cor. iii. ; — =p tan 0, see sec. [281],

0

where p is the perpendicular from the vertex of the cone on the
plane of the given conic.

»,-,/. T , . i . //,sin20 Z/,sin0cos0

The area of the focal triangle gives p =-L-= - = -' — , or

6*=J/,sin*0 ......... (b)

Then 2a,= (l + lt) sin 0, and bt=b, since b is parallel to the plane
of projection through the vertex. As A2 = II t sin2 0, b? = II \ sin2 0, or
£;2=/sin0 . /, sin 0 = VA/VB,. Therefore V is a focus of the pro-
jected curve.

The semiparameter of the projected curve is

*L= 262 211, .sin0

a~(l + ll)sinO~ l + l, '

and as ef=— — 3-*-, substituting, e,=j-—^ ..... (d)

d i ' ~\~ it

339.] The surface of a right cone bounded by a conic is developed
on a plane passing through the vertex of the cone at right angles to
its axis ; to determine the curve which the conic becomes when the
surface of the cone becomes a plane.

ON CONIC8. 417

Let the focal equation of the projected conic be

a n e*\

Pi= 1 ( e cos , p, being the focal vector. . . (a)

Through the axis of the cone and the focal vector p, let a plane
be drawn ; it will cut the surface of the cone in a side of the cone *
so that p,=s.sin0. Let 2w sin 0=1; then 2n is a constant.
Let dtp and dtp, be the corresponding elementary angles between
two successive values of p, the focal vector of the projected conic,
and * the corresponding vector along the side of the cone, so that
<pt=2n<p; hence the equation of the projected conic

becomes s = (b)

—

But cos 2w<p = cos2 nip— sin2w<p; hence this equation now becomes

1 sin2w<p cos2»;p

~j— —J- — — (c)

This is a species of spiral curve having two apsides at the distances
/ and lt from the vertex of the cone, when w<p = or when n$—ir.

7T 7T

In these cases the vector-angle <p = — or <p =— •

2n n

Hence the curve undulates between two concentric asymptotic
circles whose radii are /and 7y.

When the conic is a parabola, /,= <», and the equation of the

1 cos2 n<p
locus becomes -= — -j— -.

When the conic is an hyperbola the equation of the locus becomes
1 cos2w<p sin2n<p ...

~L— ~*~i 7 w

o I 1 1

If we refer to the ninth section of NEWTON'S Principia, we shall
see that the formula above given is the equation for movable orbits
whose apsides recede.

340.] If secant planes be drawn through a horizontal tangent to a
right circular cylinder, the locus .of the foci of the elliptic sections
will be the logocyclic curve (see fig. 88) .

Let AD be the horizontal tangent to the right circular cylinder
ABBjA,. Let AB be the major axis of the ellipse, and let F, F, be
its foci. Let a be the radius of the circular base ; then it is manifest
that a is half the minor axis of the ellipse. Then, as F is a focus,
we shall have AF. BF=a*.

VOL. II. 3 I!

418

ON CONICS.

Let zx be the ordinates of
the point F, the axes of coor-
dinates being AA;, AC ; and let
i be the inclination of the
secant plane to the circular
base of the cylinder.

Then

Fig. 88.

AB=2«seci, AF= V#2
and

BF=2asecz— V#9
and

sec i —
therefore

X

Substituting these values for
AF and BF, we get

the equation of the logocyclic
curve, substituting y for z, as
shown in sec. [319] of the first B
volume.

'A,

ON DIVERGENT PROJECTION.

341.] In perspective or central or divergent projection (as it may
be called), the projecting lines are no longer parallel as in orthogonal
projection. They radiate from a single point which may be called
the vertex or centre, and so transfer the lines and points of one
surface to those of another. In general, as here, the surfaces are
planes ; one plane figure is projected into another. This sort of
projection has been named central projection by PONCELET, the
great authority on this subject. This is a simple and powerful
method of investigation, so far as the graphical properties of figures
are concerned. It is more general in its application than orthogonal
projection, in which the vertex or centre of projection is at infinity.
For example, in the application of these methods to conies, only the
properties of the ellipse may be derived from those of the circle by

ON CONICS. 419

orthogonal projection, while divergent projection may be applied
to all conies.

This method of projection rests on the following simple
theorem : —

If a straight line be drawn parallel to a given plane, all planes
drawn through this straight line will cut the given plane in parallel
straight lines ; and if a straight line be drawn meeting the given plane
in a point, all planes drawn through this straight line will meet the
given plane in the same point.

Of the several ways in which this method may be applied the
following appears the simplest.

Through the vertex of a right circular cone let a plane be drawn
at right angles to its axis and intersecting one of the plane sections
of this cone in a straight line which may be called the cyclic axis
(while the plane drawn through this axis and the vertex of the cone
may be called the cyclic plane). Any plane drawn through the
axis of the cone will cut the cone in two straight lines, and the
cyclic plane in a straight line ; and these four lines evidently con-
stitute an harmonic pencil.

The figure whose projective properties it is sought to develop
may be drawn on the plane of one of the circular sections of the
cone, the vertex of the cone being the centre of projection.

One or two applications of this method must here suffice.

342.] Let two right cones having the same vertex and axis be
drawn, they will be cut by a plane at right angles to the common
axis in two concentric circles. Let these circles be drawn so that
a square inscribed in the one shall be circumscribed to the other ;
the diagonals of the inscribed square and its chords of contact with
the circle inscribed in it will pass through the common centre of
the two circles ; and if the square be turned about between the two
circles, it is obvious that its angles will remain on the outer circle,
and its sides remain in contact with the inner circle. If we now
project these circles and tjie square, the circles will become conies
and the square a quadrilateral inscribed in one conic and circum-
scribed to the other. As the opposite sides of the square are
parallel, and the chords joining the points of contact are also
parallel, the projections of these eight lines will meet two by two
in four points along the cyclic axis ; and this cyclic axis is the polar
of the point in which the common axis of the two cones meets the
plane of the two conies. It is also obvious that any number of
quadrilaterals may be inscribed in the one conic and circumscribed
to the other.

420

APPENDIX

TO THE FIEST VOLUME,

WITH NOTES AND CORRECTIONS.

343.] At page xii. of the introduction, reference is made to
a theorem of Euler's connecting by a simple and invariable relation
the numbers denoting the solid angles, faces, and edges of any
polyhedron.

A very elegant and simple demonstration of this curious theorem
which had so long baffled that illustrious geometer Euler, will be
found at page 333 of the XlX.th volume of the Annales Mathema-
tiques of GERGONNE, based on the relations of a group of reticulated
polygons. But the following proof, which some years ago I disco-
vered, will be found still simpler, and requires no knowledge, beyond
that of common arithmetical addition, to understand it.

Let the relation s+f—e = 2 be assumed as established for any
one polyhedron, where s denotes the number of solid angles, / the
number of faces, and e the number of edges. From this solid let
a pyramid be removed whose vertex is one of the solid angles of
the polyhedron ; let n be the number of plane angles which
together constitute the solid angle, the vertex of the retrenched
pyramid. Let S, F, and E represent the numbers of the solid
angles, faces, and edges of the new polyhedron made by retrench-
ing the aforesaid pyramid. Now, by the removal of the pyramid
whose vertex is a solid angle of the first polyhedron, we take
away from this figure one solid angle, but we add n solid angles,
the number round the base of the retrenched pyramid ; so that by
the removal of this pyramid we add n — 1 solid angles to the ori-
ginal polyhedron, or

Bast +(»—!).

By this operation we add n new edges, which are the sides of the
polygon that constitute the polygonal base of the pyramid, or

APPENDIX TO TIIK FIRST VOLL'MK. 421

E =e -f n ; and we evidently add one more face to the original poly-
hedron by removing the pyramid, or F=/+ 1 ; consequently

or

or the same relation exists between the numbers which represent
the solid angles, faces, and edges of the original and derived poly-
hedrons.

We may now assume any simple polyhedron, a cube suppose, in
which the relation s+f— e — 2 = 0 is evident, and by the successive
removal of pyramid after pyramid thus increase the numbers that
denote the solid angles, faces, and edges of the derived polyhedrons,
and still find the same invariable relation,

s+f-e=2.

Cauchy's theorem, which is as follows, may be proved with equal
brevity and simplicity. Let m denote a number of polyhedrons,
agglutinated together like a mass of crystals, and let S, F, E denote
the numbers of the solid angles, faces, and edges of this cluster ot
polyhedrons, we shall have

This is Cauchy's theorem. When there is but one polyhedron,
w=l, and we obtain Euler's theorem.

Let *, /, e denote, as before, the numbers of the solid angles,
faces, and edges of any polyhedron ; then by Euler's theorem we
shall have s+f— e = 2. Now if we conceive one of the polygons
which constitute the faces of this polyhedron to have n edges and
n solid angles, the removal of this polygon with its n solid angles
and n edges will make the closed polyhedron an open polyhedron ;
and we shall have the following relation between the numbers that
denote the solid angles, faces, and edges of an open polyhedron,

s+f-e=l.

Let s,, ft, et denote the numbers of the solid angles, faces, and
edges of the open polyhedron thus derived, we shall have s = st + n,
f=f, + 1, and e = e, + n ; substituting these values in Eider's formula,
(*/ + ») + (/,+ !) -(*/ + ») = 2, or 5, +/-<?,= !.

Let us now conceive a closed polyhedron having an open poly-
hedron applied to one of its faces, so as to fit, or, in other words,
so that the projecting edges of the open polyhedron may be applied
to the solid angles of the closed polyhedron; then we shall have, by
Eider's theorem for the closed polyhedron,

422 APPENDIX TO THE FIRST VOLUME.

and for the open polyhedron

as just now shown. But if S,, Fy, and E, denote the numbers of
the solid angles, faces, and edges of the compound polyhedron, we
shall have

consequently

or the difference is one more than in the case of the single poly-
hedron.

Consequently for every additional open polyhedron we attach,
the absolute number is increased by unity ; or if there be m agglu-
tinated polyhedrons,

[Page xiii.]

344.] The opposite sides of a hexagon inscribed in a conic meet,
two by two, in three points which range along a straight line.

Fig. 89.

APPENDIX TO THE FIRST VOLUME. -I'.' 3

Let A, B, C, D, E, F be the vertices of a hexagon inscribed in a
conic, whose opposite sides meet two by two in the three points
G, H, K. These points range along a straight line.

Let the alternate sides of the hexagon be taken as forming a
triangle LMN, whose sides are cut in the points A, B, C, D, and
K, I-1 by a conic and also by the three transversals BC, DE, and FA.

Through a focus of the conic let chords be drawn /, ft, fn parallel
to the sides of the triangle LMN; then, by a well known theorem,

MA.MB:MF.ME=/ :/„ J

NC.ND:NA.NB =/„:/, [ (a)

LE .LF :LD.LC=//://r)

Multiplying these expressions,

MA.MB.NC.ND.LE.LF = MF.ME.NA.NB.LD.LC. (b)
Since the triangle LMN is cut by the transversals HB, GE, and KA,

HL.NC .MB = HM.NB.LC, )
GM.ND.LE=GN .LD.ME, [ . . . (c)
KN.LF .MA=KL .MF.NA. )

Multiplying together these three sets of proportionals, and dividing
the product by the products in (b), we shall have

HL.GM.KN=KL.GN.HM; . . . . (d)

or the three points G, H, K range along the straight line GHK,
which is a transversal to the triangle LMN*.

345.] A hexagon is circumscribed to a conic ; the diagonals which
join the opposite vertices meet in a point.

Through the points A, B, C, D, E, F (see preceding figure) let
tangents be drawn to the conic, meeting in the points a, b, c, d, e, f,
which therefore constitute a hexagon circumscribed to the conic.
Now as a is the pole of the chord AB, the polar of any point in AB
will pass through a. But G is a point on AB ; therefore the polar
of the point G will pass through a. In like manner the polar of
any point in DE will pass through d. But G is also a point on DE ;
therefore the polar of G will pass through d ; therefore ad is the
polar of the point G. So also be is the polar of the point H, and
cf is the polar of the point K. But, as shown above, G, H, K range
along a straight line ; therefore ad, be, cf, the diagonals of the circum-
scribed hexagon, meet in a point, the pole of the straight line GHK.

* This solution was riven in the ' Ladies' Diarv ' for 1842 under the initial*
J. B. B. C.

424 APPENDIX TO THE FIRST VOLUME.

Page 15. SEC. [24].

346.] More generally, let the protective equation of the conic
section, referred to rectangular axes, be

Ax* + A#2 + 2Exy + 2C# + 2C;y -1=0.

Therefore by sec. [22] £=^ „ *„ , v=

1 — Cx— C,y'

Her (A;+C/2)^-(B+CC>-(A,C-BC/)

(A^-B2) + (A^-BCJw+CA^C-BC,

-(B + CC,)g--(AC,-BC)
(AA,-B2) + (AC,-BC> + (A,C- BC,)f

Substituting these values of x and y in the dual equation x^+yv — \,

In the protective equation of the parabola, B2 — AAy=:0 j hence
the tangential equation of the parabola has no absolute term.

Page 21. SEC. [32].

347.] If from any point Q,, in the plane of a rectangular polygon,
perpendiculars are drawn to the sides , if the feet of these perpen-
diculars be joined two by two, so as to constitute another polygon,
and if the area of this latter polygon be constant, the locus of the
point Q will be a conic section.

Let x and y be the projective coordinates of the point Q, and let
£, v and £p vt be the tangential coordinates of two successive sides
of the polygon, and let 6 be the angle between them ; then, P and Py
being the first pair of perpendiculars,

1-fo-uy _\-tp-vty

VFTV2' ' VI,2 W'

and

sin 0=—r-Z'v~*v' o (see p. 4).

V(£2 + "2)(£2 + i,,2) V

Hence the area of the first component triangle is
PP sin 9- (l-fr

But j;v and %jut being constants, we may put

A=

APPENDIX TO THE FIRST VOLUME.

therefore the first component triangle is equal to

In like manner the next component triangle will be equal to

and if the sum of these component triangles be assumed as constant
and equal to C, we shall have a resulting equation of the form

P#2 + Gty2 + ZRzy + 2S# + 2Ty = C,

the project! ve equation of a conic section — P, Q, R, S, T being
functions of the constants %, £,, v, v,, &c.

Page 25. SEC. [36].

348.] To find the equation to the envelope of equal chords of a
given ellipse.

Proposed by Mr. A. MARTIN in the Educational Times, No. 4519.

Let (x, y) be a point on the ellipse, and (£, v) the tangential coor-
dinates of a tangent passing through this point ; then, eliminating
y between the equations

a*y* + b*ar*=aP ... (a) and x£+yv=l, . . . (b)
we shall find (a2^ + £ V) #2 - 2a2|* + «*(!- 6 V)=0. . . (c)
Let xt and xlt be the roots of this quadratic equation, we have

_aa

* ~"

a y-

consequently (*,-*„)*= «« ^~

In like manner (y,-y,,)2=

Let 2c be the chord. Then (^-^)2+(^-yJ2=4c2, . (e)

or a262(r2 + u2)[a2f2 + 6V-l]=c2(a?^ + 62u2)2. . . (f)

Hence the projective equation of the pedal is

- • (g)

Page 40. SEC. [48].

349.] Two parallel tangents are drawn to a conic, and a third
tangent between them, variable in position. This tangent will cut off
segments from the parallel tangents between its intersections with

VOL. ii. •'* '

APPENDIX TO THE FIRST VOLUME.

them and the points of contact, such that the rectangle under these
segments will be constant.

Fig. 90.

Let two tangents to the curve be taken as axes of coordinates,
the axis of Y being one of the fixed tangents to the curve, while
the axis of X is parallel to the diameter conjugate to the two
parallel tangents. Then the tangential equation of the curve
referred to these tangents as axes is, as shown in sec. [48],

l ....... (a)

Let the variable tangent cut off from the axes of coordinates
OB = b, OA=a. Then, as this line is a tangent to the curve,

-, j-, are tangential coordinates, and satisfy the equation (a) . Hence

(/ U

(a) becomes

(b)

and as the axis of X is a tangent to the curve, we shall have
/3f + 7/ = 0, see sec. [19] ; and as in this case OD = £=7, we shall

have

(c)

APPENDIX TO THE FIRST VOLUME. l.'J?

Now BQ=6 — 7y; and ByQ, it may be shown is equal to

it a a

putting for (3 its value — y7/.

Hence BQ . ByQy= — ^ (b— yy)(2y — a) ; or, multiplying,
BQ . ByQy = - 7/ [2/3 + 27/a + 2yb -ab- fly] .

fl

But 2/3 + 27y« + 2y6 — ci = 0, from (b).

Hence BQ . B;Qy=y7/ (d)

Page 41. SEC. [49].

350.] The tangential equation of the parabola may be obtained
from the projective equation of the curve, 2/2 = 4&r-f 4kz, as follows.
The equations of transition give, see sec. [22] ,

.. —1 y

t — — t> =

and xg + yv=1. Eliminating x and y we get k(%*+ v9) -f £=0.

Page 43. SEC. [50].

351.] A parabola is inscribed in a triangle ABC (fig. 91), touching
the sides of the triangle in the points A, C/ Br The rectangle under
the sides CB, CA of the triangle is equal to the rectangle under the
segments of these sides produced until they touch the parabola;

In sec. [55] it is shown that the tangential equation of the para-
bola, the axes of coordinates being the tangents to the curve, is

ffSv + h£ + hf>=Q ....... (a)

Let CB = 6, CA = a, and, as these values must satisfy this equation,
seeing that AB is a tangent, we shall have

ff + hb + h,a=Q ........ (b)

The value of CAy the tangent to the curve is found by making
t = 0 in the given equation, which reduces it to ov + h=0, or

CA,= -j¥-; in like manner CBy=-r^.
•*

Therefore BA

/= - (| + b\ and AB/= - (| + a) .

428

APPENDIX TO THE FIRST VOLUME.

Consequently BA,AB;= ~ (g + hb + hta] -f ab.

/l/li

But the expression between the brackets =0 by the tangential
equation (b) of the parabola; hence CB . CA = BA/ . ABy.

Fig. 91.

If we now take the other two angles successively as the origin
of coordinates we shall have the same property repeated.

Consequently a26V=AB/ . BA, . BC; . CBy. CA, . AC,. . (c)

Page 43. SEC. [50].

352.] A parabola is inscribed in a triangle. The triangle whose
vertices are the three points of contact is twice the area of the given
triangle (see fig. in last section).

Let two of the tangents to the parabola be taken as the axes of
coordinates. The triangle A/C/B,=2 ABC. Let CB = £, CA = «;
since the tangents CA; CBy are axes of coordinates, the tangential
equation of the parabola is

#£u + A£ + A,v = 0=V; (a)

and the protective coordinates of the point C, in which the curve is

APPENDIX TO THE FIRST VOLUME.

touched by the tangent AB is found to be, using the equation* of
transition (as in sec. [22]),

dV dV

dg _ d^

or

(b)

or, putting - for f and -7 for v, as the point C is on the line AB,

9
Let CA=Y; the value of Y is found by putting £=oo in (a).

1

Hence - or Y=-T^, and X = — =^.
v h nt

Now the triangle A^C^CA^-CCjB-CCyA,. . . . (c)

But CAyB^ j-r, putting a. for the sine of the angle of ordination,
/

and

« • •• n \ n i n. •- •• i

* ' ' ^

In like manner «Y# =-r- (ff + 6h). J

Consequently C A,B, - CC,B, - CC A, = Jf- (g + bh -f aAy) + 2oA<r.

/i/iy

But since- and -7 are tangential coordinates, y + bh + ah,=Q;

hence the triangle A/B/^ = 2««6. But «a6 is the area of the triangle
ABC; therefore the triangle A^C, is equal to twice the triangle ABC.
Since

CB_ — bh , AC,_a-z_ —bh

OB AC, ABy -bh bh

it Will follow that TrT~ = TJ7T — TTT^ — = — -•

BA, BC, CA

353.] The tangential equation of a parabola referred to two
tangents as axes of coordinates, see sec. [55] is

g%v-l- h% + ^=0 = V, (a)

to determine the projective equation of the curve referred to the
same axis.

430 APPENDIX TO THE FIRST VOLUME.

dV dV

Since, as in sec. ,j -r-

V . . (b)

v.. dv

(c)

Hence g= ,= . . . . (d)

Substituting these values of £ and v in the tangential equation
of the parabola (a), we get

0. . . (e)

Page 47. SEC. [55].

354.] A quadrilateral is circumscribed to a parabola. Two of its
sides are fixed, while two are variable in position. These latter
intercept, on the former, segments which are always in a constant
ratio.

Let the tangential equation of the parabola referred to the fixed
sides of the quadrilateral as axes of coordinates be

let a and b, at and bt be the tangential coordinates of the two
variable lines ; then we shall have ( since £ = -, v = -; j

g \-hb + hfl = Q and g + hbt-}-hfif=Q.

Subtract these equations one from the other, the result is
b-bl=-^(a,-a).

But h and ht are constant quantities depending on the equation

of the curve : hence - '- is constant.
a,— a

Page 65. SEC. [76].

355.] The projective equation of a surface of the second order,
f(x,y,z)=Q, referred to three rectangular coordinates in space
being

lt
the tangential equation of the same surface referred to the same

APPENDIX TO THE FIRST VOLUME.

431

rectangular axes may be found by eliminating x, y, z between the
following equations, given in this section.

.._
-

.- • (a)

They may be reduced as follows : —

(A
(A,

-v=0, [ . (b)

Let us now assume the three formal linear equations

ty-\-ctlz=dir (c)

Comparing the coefficients of these expressions with those of the
preceding equations (b), we shall have

a =A

b =

/=B +Cuv,

%p-*-C,,l (d)

If we now solve the group of formal equations (c) for x, y, z, we
shall have

/ (*c« - V) + «// (*/c - *c/) '

_

z=

~

> - (e)

substituting for the nine constants a, A, c, a/} A,, cp and o;;, 4;/, ctf
their values as derived from (d), we shall have, putting A for
this common denominator,

(0

A= AB2 + A^,2 + AWBW2 + AA/Aw-2BB/By;

'-AAJC^ (BA-BIBI()CI,+ (B^-BB^CJu
02- AA,)CM-f (B^-BBJC-f (BA - B

432

APPENDIX TO THE FIRST VOLUME.

We have also, multiplying by £,

(C//B-A//C/)pu+ (BCj-AA
- (B2-A,A,,)C£-(C,,B- AflC,)C&;- (BC,-

(g)

If we now add these expressions together we shall have, since
the triple products of %, v, $ vanish,

= [(B«-A,AW) +260^-^0 f- Kp

-f [(A^-BBJ + (A/C//-BC/)C+ (B^-B^C
+ [(A,Aa-B«)C+ (BBj-A^C,* (B^B-
In like manner

= [(B^-AAJ +2BJCC//-AC/-A//C2] v2
[(AB-B^) +AC//-CB,)C/+ (BC-B/!w
[(A/yBw-BBy) + (A;/C-B,CW)C/+ (B^-

and also

= [(BW-AA,) +28^0,0- A^-AC,

Bearing in mind that
A(
we shall have, making the necessary reductions,

APPENDIX TO THE FIRST VOLUME. 433

= [(B2-A,AJ

+ [(B2-AA/;)

+ [(Brt«-AA,)

+ 2[(AB-B,B,,) + (BC-B,,C,,)C+

+ 2[(A,B/-BB,/) + (B,C,-BC)C,+ (A^-

+ 2[(A,,B//-BB,) + (B/^-

+ 2[(A/AH-B2)C+

Let X, Y, Z be the project! ve ordinates of the centre of the
surface ; then, as shown in sec. [75] ,

y_(A<A/,-B2)C4-(BB<-Af<Bf/)C/+(B</B-A<Bj)CH

AB2 + A,B,2 + A;,B,,2- AA/A;/-2BBJB/,
and like values for Y and Z may be found.

When the surface is a paraboloid, as in this case the absolute
term vanishes, we shall have

AB2 + A/B/2 + A//B/-AA;A/,-2BB>Bw=0. . . (k)
When C — 0, 0^=0, Cw=0 or when the origin is at the centre;
the protective equation of the surface becomes

and the tangential equation of the surface referred to the same
axes is

AB2 + A,B,2 + AWBW* - A A, Atf - 2BByBH

. (1)

'Page 133. SEC. [134].

356.] In a system of confocal ellipses the envelope of the normal that
makes with the major axis an angle whose sine is b(a*—b*)-lisafuur-
cusped hypo cycloid with two opposite cusps at the foci of the system.

Proposed by Mr. J. L. McKenzie, in the Educational Time*, No. 4420.
The tangential equation of the evolute of an ellipse (see sec.
[156]), since a2=£2 + A:2, may be transformed into

6*(u* + £«) + yfcV==/t«£V; ..... (a)
but it is assumed that the sine of the angle which the normal makes

VOL. II. 3 K

434 APPENDIX TO THE FIRST VOLUME.

with the axis is y, and as the normal is parallel to the perpendicular

K

b* £2

from the centre on the tangent, we shall have 73=^5 - 5, or

A; 5 •+• V

#2 (|2 + ,,2) = £2£2 . (b) ; eliminating i2 between (a) and (b), we shall
have £2 + v2-/t9£V=0=V, ..... (c)

the tangential equation of the quadrantal hypocycloid, as shown in
sec. [131].

The protective equation of this curve may easily be found by the
help of the formula of transition given in sec. [22].

For 2jT=2(*M-l), ^=2(*2£2-l);

£2^2 _ }

hence x = — ^-5-, or &2(1 — £a?)i»2 = l.

/r£tr

But (c) gives (£2£2 — l)t»2 = £2; eliminating u2 between these
equations,

f3=ro-J hence Esssjltfft. In like manner -5= X:^.
A2# £2 w"

But £2 + -2=£2j hence we have ^+y^=A-ff, the protective equation

of the quadrantal hypocycloid.

357.] Let a, b be two conjugate semidiameters of an ellipse, and
xt, y{ the coordinates referred to them of a variable point in the
curve ; to show that the envelope of a series of ellipses whose semi-
diameters are coincident in direction with a, b, and in magnitude are

mean proportionals between a, xt and b, yt is f-J +(jj =!•

Proposed by Mr. W. J. C. Miller, in the Educational Times, No. 4463 *.

If az, and byt be the squares of the semiaxes of the variable
ellipse, its tangential equation is a«r/P + 6y/t»2 = l, . . (a)

and the equation of the given ellipse is a2y/2 + 62^/2 = a2^»2, . (b)
Eliminating y, between these equations,

To eliminate xt> yt we must manifestly have three equations — (a) ,
(b) , and the differential of F with respect to xt, or

dF A ,,. dF xt «V_

— =0. . . (d) But as -r-=0, -J = r 4c.4 , 74 4T
dxt d^ ' a [a4|4 + o y J

* This solution embodies an important principle. It shows how the tangential
method may he extended to those cases in which the envelope is generated by the
successive intersections of curves whose parameters vary according to a given law.

APPENDIX TO THE FIRST VOLUME. 435

Finding a like expression for yt, |'= F^M^^,,-

Substituting these values of xt and yt in (b) , the resulting ex-
pression is

K)4+(^)4=i ........ (d)

This is the tangential equation of the curve required.
If we require the projective equation of the same curve, we must
put

V = (a£)«+(Ai/)4-lj. . (e) then =4a4£8, =46V. . (f)
dV

cfiF

But x=-Ty -- jy , see sec. [22]. A like value for y is obvious.

Hence -=«3£3, or (-\*=tfP.
a \«/

Finding a like value for y, the projective equation becomes

©'+(*)'-> ........ w

Page 112. SEC. [119].

yt2
358.] //"atf each point of an ellipsoid a distance -p- be measured

along the normal, P being the perpendicular from the centre on the
tangent plane at that point, the locus of the point so defined is another
ellipsoid, the envelope of which for different values of k is the " surface
of centres " of the original ellipsoid.

Proposed by Mr. R. F. Scott, B.A., in the Educational Times, No. 4460.
Let the tangential equation of the ellipsoid be

...... (a)

and = y^ + ^ + f*; hence ^=A« V? + »*

Let x, y, z be the projective coordinates of the point on the
surface to which the normal is drawn; then ar = ag£, and the pro-

iection of the line -^ on the axis of X is — —-^^-- *-=/;*£;

V £* + »* + £*
and if x, be the projection of its extremity, we shall have

X

x^a^ — k^, since (x— a?,) =&2f. Consequently £ = -y^frg- In
like manner

USB and ^= ...... »

436 APPENDIX TO THE FIRST VOLUME.

Substituting these values of £, v, % in (a) we shall have

_ -I

-

3 A2,. 2 ,.2

i . ° y/ - c

"• '-

for the projective equation of this surface.

Hence by the formula of transition, p. 68, the tangential equation
of this same surface will be

dV

"We must now eliminate k between this equation and -^=0.

£2 + t,9_|_£S

This elimination gives k2 =~ - ^ — -% ........ (e)

^ + P + ?
Substituting this value of k in the equation V = 0 we shall have

(f)

the tangential equation of the surface of centres as found in
sec. [119].

359.] A given ellipse F is one of a system of concentric similar
and similarly situated ellipses. A line is drawn touching any other
ellipse H of the system ; and the perpendicular distance of the tangent
from the centre is a mean proportional between the semi- major axi»
of H, and the semi- minor axis of F. To show that the envelope of
the tangent is the first negative pedal of F, but turned round a right
angle about its centre.

Proposed by Mr. J. L. McKenzie, in the Educational Times:, No. 4368.
The tangential equation of the first negative pedal of

«y + 6V-«2Z<2=0 ...... (a)

is aV + 62f = «262(f + u2)2: . . . , (b)

see ((3) sec. [163] , The projective equation of the reciprocal polar
of (a) , a being the radius of the polarizing circle, is

and the tangential equation of its first negative pedal is

«2|2 + 6V = «4(f + v2)2 ...... (c)

Let «2£2 + iV-l=0=V ...... (d)

b.e the tangential equation of the given ellipse, and let

-l=0=W . . . . (e)

APPENDIX TO THE FIRST VOLUME. 437

be the tangential equation of one of the concentric and similar
ellipses. But, by the conditions of the question,

) = l; ........ (f)

eliminating n between this and the preceding equation, we get

a2f + 6V=«2£2(£* + v2)2 ..... (g)

This equation would coincide with (b) were the axes of coordi-
nates turned through a right angle, or if £ and v were changed
into v and £.

If the duplicate ratio of the perpendicular on the tangent to the
linear unit be equal to the ratio of linear similarity of V and W,
the envelope of this tangent is the first negative pedal of the polar
reciprocal of V.

For, by supposition, » = as(£*-f-t;8) ; ...... (h)

eliminating n between (h) and (f) we get

a«$* + aV= «*#($« + i/8)8, ..... (i)
which coincides with (g).
360.] Prove that the ellipses

«y + b*x* = tfb*, a2*?2 sec4 <p -I- b*y* cosec4 <p =aV . (U, V)

are so related that the envelope of (V), for different values of <p, is
the evolute of (U), and that a point of contact of (V) with its
envelope is the centre of curvature at the point of (U) whose
eccentric angle is <p.

Proposed by Mr. R. Tucker, in the Educational Times, No. 4240.

Let
a*y 2 + 6V — o26* = 0 =\J, a2*2 sec2 <p + b*y 2 cosec4 <p — a4e4 = 0 = V .

dV

Find the value of <p from' the equation -,— = 0; substitute this value

in V = 0, and we shall have W— («?)* + (by)*- (a*— 62)*=0; and
this is the protective equation of the evolute of U=0.

(a2-i2)cos3<p (a'-A2) .

Again, assuming x= - , and y—- — r — -unrfj (a)

we shall find that these values of x and y satisfy the equations
V=0, W=0. Hence this point is common to the ellipse V=0 and
its evolute W = 0.

Moreover, if x and y be the coordinates of a point on the ellipse
U=0, of which point <p is the eccentric angle, we shall have

j- = flcos<p, and y = Asin<p; ..... (b)

438 APPENDIX TO THE FIRST VOLUME.

and if we eliminate cos <p and sin <p between (a) and (b) we shah
have

__
~2' y~'

Hence x and y are the projective coordinates of the centre of
curvature of the point (x, y} .

2. The question may be solved as follows by tangential coor-
dinates : —

Let a?£2 + £V-l=0— U', ...... (d)

(/72_A«V2 f/72_ A2\2

and _^COs4<^+^— ^8in4?u2-l=O^V', . (e)

be the tangential equation of the two ellipses. Then, finding the

dV' a^v2

value of -5 — =0. we shall have cos2<p = -a-o ,0..0. Eliminating
d<p a v + 0 g

sin <p, cos <p from V'=0, we shall have

aV + A2^-(a2-i2)2^u2=0=W, . . . (f)

which is the tangential equation of the evolute of U = 0 (see
vol. i. p. 115).

«sec<p ocosec<p
Assume f-^CI^' and VSB(oTI^ fe)

Now, substituting these assumed values of % and y in the equations
V'=0 and W'=0, we shall find that they satisfy these equations ;
consequently the ellipse V'=0, and its evolute W=0, have a
common tangent.

Let £ and £ denote the tangential ordinates along the axis of X,
made by two tangents passing through a point on an ellipse, one
to the ellipse, the other to the evolute, and let <p be the eccentric
angle of U'=0 at this point; then

a£=cos<p, iu=sin<p, and 0££=£ cos<p = -2 — y^, from (a), (h)

G/ ^~ \j

Hence (a2 — i2 )££=!; consequently the common tangent to
"V'=0 and \V'=0 passes through the point onU' = 0, of which the
eccentric angle is <p.

If we substitute the values of x, y, £, v assumed in the equations
(a) and (g), we shall find that they satisfy the dual equation
xg+yv=*l} consequently the common tangent passes through the
common point of the two given ellipses.

APPENDIX TO THE FIRST VOLUME.

439

Page 230. SEC. [254].

361.] From this focal property of a surface of the second order
having three unequal axes may be derived this new theorem : —

Let two equal semidiameters k be drawn in an ellipse whose semi-
axes are a, b. Assume two points C and D on the major axis, such

_
that CO = \/a*-k* and DO = -7-1 ==, O being the centre.

V« — «

Through the point D let two straight lines be drawn parallel to the
equal semidiameters k. From any point Q, on the ellipse let perpen-
diculars P, P, be drawn to these two lines, and a vector R from

P. P 4*a2

Q, to C : we shall have ^.o/=7o/ a — T&: Q> constant ratio.
rlr Ar(flr — o*)

Cor. i.] When k=b, the perpendiculars P, P, coincide and
become equal, and the ratio becomes -§, the common focal property

C

of the ellipse.

Cor. ii.] When k = a, the point D is at infinity, the lines to which
the perpendiculars from a point on the curve are drawn become the
minor directrices, of which the properties are developed in sec. [288] .

Page 329. SEC. [354] .

The following are the numerical values of sec e, tan e, e, and e~l.
sec 6 = 1-5430806348 &c., tan e= 1-175201 1936 &c.,
sec e + tan e=e = 2-7182818284 &c.,
cos e = e-' = 0-3678794411 &c.

Fig. 92.

440 APPENDIX TO THE FIRST VOLUME.

Page 313. SEC. [343].
Proposed by the Rev. W. Roberts, M.A., in the Educational Times, No. 1749.

362.] In a right-angled triangle ABC (fig. 92) a straight line is
drawn from the right angle A. to a point D in the line BC, whose
distance from the middle point of BC is one third of the radius of the
circumscribing circle. The line AD is produced to meet the circle
in Q. Through Q, draw the radius QO meeting the side AB in G.
Let the angle ABC be <p, and the angle OGB be i/r ; then

(sec ^ + tan i/r) = (sectp + tan <p)3 ..... (a)
Let the angle AGIO be 0, AO=r, and QD = nr; hence

3nsinf^--^l = sin(^ + <p),l . . . . (b)

and 9w2= 10 + 6 cos
Eliminating Q, n, and reducing, we find

2 (sec -»/r sec <f> — tan -ty tan <p)2 — (sec >|r sec <p -f tan ijr tan <p) = 1 . (c)
Subtracting from the preceding equation the identical expressions
(sec2-»/r— tan2 -t/r) (sec2 <p — tan2<p) = l, we find

(sec2 T/T + tan2 >/r) (sec2 p + tan2 <p) — 4 sec -^ tan A/T sec <p tan <p ) .,.

= sec i/r sec <p + tan ^r tan <p. j
But in sec. [344], (^), (77), and (7), it is shown that

sec2 T/T -f tan2 i/r=sec(-»/r-l-/\Jr'), 2 sec -^r tan -^r= tan (/^-i-'^r})
and sec-^r secip + tani^ tan<p = sec (^/r -'-(p).

Substituting these values in the preceding equation (d), it becomes
sec (^-L<^) sec(<p-L<p) — tan(-^r-L^) tan(ip J-<p)=sec (<^-Lf). (e)
But this formula may be written, as shown in sec. [344],
sec (>/r -L i|r -r <p -T- <p) =sec (^-^(p),

Or •^r-i-^-r(p-r(p = '^r-i-(p.

Transposing and changing -p into -1-

hence sec i|r +tani^=(sec(p + tan f>)3 ..... (g)

Generally, the following relation exists in parabolic trigono-
metry : —

sec (<p -1- <p . . . to n angles) -f- tan (<p -1- <p . . . to n angles) ±= (sec^p -f tan <p)n,

THE END OF THE SECOND VOLUME.

A Treatise on some New Geometrical Methods, contain iiiy
Essays on Tangential Coordinates, Pedal Coordhmli'*,
Reciprocal Polars, the Trigonometry of the Pa,-<tbol<t,
Ike Geometrical Origin of Logarithms, the Geometrical
Properties of Elliptic Integrals, on Rotatory Motion,
the Higher Geometry, and Conies. By J. Booth, LL. I).,
E.R.S., E.R.A.S., &c., Vicar of Stone, Bucking!,. •
shire. (In Two Volumes.) Vol. I. with Photographic
Portrait of the Author, 416 pp. and 87 Diagrams.
Medium 8vo. Price 18s. (June 1873.)

The following reviews and notices of the first volume have
appeared : —

From the ( Bulletin des Matheinatique.i ,' Paris, June 1873.

" Le developpement du grand principe do la J)ualite geonu'triquo cst
1'idee fondamentale de cet ouvrage. Dans los vingt-deux premiers
chapitres, Fauteur etablit un systeme do coor.lonuees qu'il appelh
• loan A-.s taiKjentielles, le correlatif du syatume Lieu connu des coordonnees
cartesiennos, et qu'il base sur une notation algebriqne particuliorc. II
applique sa methode et a la discussion et a la solution de diffcrcnts theo-
remes et problemes, en I'tablissaut dans chaquc cas la correlation des
figures geometriques. Cette theorie est uppllquuc non-soulcraeut aux
courbes et surfaces courbos du second dcgre, mais a celles dea dcgres
superieurs.

" L'autcur devoloppe la thoorio des polaires reciprocjues par 1'applica-
tion des relations metriques, pt plus partiouEexemeni; il dcduit les pro-
prietes des surfaces du second ordro, ayant trois axes in<$gaux, de cellos
des surfaces de revolution. II continue ensuite a appliquer co priucijje
sans exception de dualite universclle do la Trigonometric, et etablit, pour
la parabole, unc trigonometric analogue a celle du cercle. II denionlro
1'origine geometrique des logarithmes et fait voir que, si les nombres nature-Is
sont representes par les rayons vocteurs d'une courbe qu'il nommo coin-he
Lii/uci/clique, les logarithmos correspond ants seront rcpresentc's par les arcs
do puraboles .correspondantes. Les principes dc la Trigononic-trio parn-
boliquo servent cnsuitc a otablir de nombreusos relations entrc los arcs de
la ])urribolc ; et 1'autour a soin de signaler les relations eemblablos quo
pnW'iitent les arcs do la chainette et, par suite, les rapports de cette courbo
avcc. la traction.

" Ces quolques mots no donnent qu'un resume succinct d'un important
ouvrage qui est, aiuai que lo declare avoc ruison 1'auteur, eutiercment
original."

11

From the ( Standard' of July 21, 1873.

" The mere title of this book will suffice to show that it treats of the
highest and most profound geometrical and mathematical problems, and
that, were we to discuss at length the various abstruse questions with
which Dr. Booth deals, and to follow him through the new methods of
solution of these problems which he proposes, there are but few of our
readers who would care to follow us. "NYe notice the appearance of the
work, however, because, in the first place, it is a very remarkable addi-
tion to mathematical science, and because, in the second place, it suggests
a number of questions of general importance, many of which are touched
upon by the author himself in his introductory remarks. There is a
tendency of the present age to believe that although in the domain of
practical science and invention there is still great progress to be made,
yet that in the region of abstruse scientific problems there is but slight
range open to us, and that, even if there were, it would be altogether
useless to investigate it. Unfortunately, too, the spirit of the age is
entirely utilitarian. In our universities high mathematics are taught
and studied with a view that the learner may obtain high honours, and
so reap the substantial benefits of scholarships and fellowships. Men do
not study these things for their own sake, nor, having once acquired them
for the sake of distinction or pecuniary advantage, do they keep up the
knowledge after leaving the University. It is difficult, however, to say
that any new scientific problems and discussions whatsoever are useless.
The utility may not, indeed, be evident at the time ; but, for example, our
highest astronomical problems could never have been solved had it not
been for the application of mathematical problems hitherto condemned as
useless. The world is, indeed, deeply indebted to men like Dr. Booth —
deep and original thinkers and students, men who make but little stir in
the world, who have nothing in common with the gentlemen who love to
place themselves in the front rank, and to sound their own trumpets
before the world upon all occasions, but who are content to live quiet and
retired, seeking neither fame nor profit, but studying laboriously, and
issuing perhaps but one book, conveying to the world the result of a life-
time of unremitting mental toil."

From the ' Cambridge Chronicle / August 2, 1873.

" It is upwards of thirty years since the Eev. Jas. Booth published his
first essay on Tangential Coordinates, since which time he has set himself
the task of discovering some method of expressing by common algebra the
properties of reciprocal curves and curved surfaces. Having been suc-
c^ssfuF in the discovery of a simple method and compact notation, he now
gives the public the result of his prolonged labours and researches in this
volume of essays on ' Tangential Coordinates, Pedal Coordinates, Reciprocal
Polars, the Trigonometry of the Parabola, the Geometrical origin of Loga-
rithms, the Geometrical properties of Elliptic Integrals", and other kindred
subjects,' first explaining in the introduction the considerations which led

Ill

to the discovery of his method. With the usual modesty of great mind*
the Ilcv. Jus. Booth apologizes for thus making public the medi1
of the ' better part of a lifetime,' during which lie has watched
tion that some accomplished mathematician would take up these BI.
ami expand them, producing a treatise from which any student <>i' mode-
rate ability might glean enough to enable him to extend those researched
still further. Xo such mathematical champion having appeared, our
Iciirned author has compiled this volume, containing at length results of
which he has from time to time frequently given abstracts in the Proceed-
ings of learned societies. It would have been difficult to have found a man
better fitted for the task, or one who would bring to bear on the subject
more ability, more original and deep thought, or more careful and untiring
research ; indeed this work is the fruit of a life of laborious study in thy
deepest and highest branches of mathematical science; and those who
deal in abstruse scientific problems will frequently find their path m;tde
comparatively easy by the arduous labours of their pioneer, the Itev. Jas.
Booth."

From the ' Educational Times' August \, 1873.

" This is by far the most interesting of the mathematical works which
have for a long time been brought under our notice. Here we find
gathered up, and placed before us in a connected form, and with singular
clearness and elegance of exposition, the various contributions which Dr.
Booth has, from time to time, made to our mathematical literature, along
with much now matter, which is both valuable and original. The chief
feature of the work is the development of the method of Tangential
Coordinates, which now, in some form or other, constitutes a recognized
portion of the Modern Geometry. *

" The method of Tangential Coordinates, however, forms but a small
portion of the contents of the elegant volume before us. Indeed, wo do
not remember to have ever met with a mathematical book containing so
great a variety of interesting, novel, and important matter. This will bo
dearly seen from the following brief analysis of the contents of the book.
The first twenty-four chapters of the volume treat of the development of
the principle of duality, as involved in the system of tangential coordi-
nates, applied to space of two and three dimensions. » In the twenty-filth
chapter the principle of duality is established geometrically, and then ap-
plied— in what we consider one of the most remarkable and original
chapters of the book — to the investigation of the properties of surfaces of
the second order having three unequal axes, derived from the corresponding
properties of surfaces of revolution. In chapter xxix. metrical methods
are applied to the discussion of the great principle of duality with
ence to the theory of reciprocal polars. In chapter xxx. the logoc\ die
curve and the geometrical origin of logarithms are discussed ; while in
chapter xxxi. the trigonometry of the parabola is fully investigated, and
the properties of this new branch of mathematical science applied
catenary and tho tractrix. The last chapter is devoted to the discussion of
certain properties of confocal surfaces.

IV

" From this rapid analysis it will he seen that there is much in this
volume that cannot fail to meet the tastes of all geometers. In some
parts of his work, Dr. Booth professes not to he able to find room for
many illustrative examples, as he states that his main object is to lay
down the principles of the various methods discussed, as applied to a few
particular instances, without following out the investigations into all their
details. Yet even in the most sparsely illustrated portions of the work we
find a few judicious examples, most aptly chosen, while in those portions
wherein the author expresses his fears — which we cannot but think alto-
gether groundless — that examples may be thought to bo unduly multi-
plied, the illustrative exercises are in the highest degree valuable. To the
readers of this' journal these examples will be especially interesting, inas-
much as many of them have appeared in our mathematical columns, and
have there received solutions by methods different, for the most part, from
those given by the author in the volume before us. Occasionally a solu-
tion is quoted entire from our own columns, with appropriate acknowledg-
ment— an act of justice to ourselves which, we regret to say, is not always
rendered — as, amongst other instances, in Mr. Spottiswoode's investigation
of the Tangential Equation of the Cardioid (p. 142), and the Editor's
method (p. 126) of deriving the projectivo equations of the bicuspcd and
unicusped hypocycloid from the general tangential equation.

" A noteworthy feature of the volume before us — and it is one which
we cannot praise too highly — is the clear and elegant style in which it is
written. Usually our mathematical books are little more than mere
collections of algebraical symbols, with scarcely two consecutive sentences
of English of any kind beyond what is required to connect them, from one
end to the other. But Dr. Booth possesses a vigorous and forcible style,
and very properly devotes much attention and ample space to the interpre-
tation of the results at which he arrives, and to a lucid exposition of the
principles of the methods of which he treats.

" The work treats of subjects of great interest and importance to mathe-
maticians, develops methods of much power and efficacy in geometrical
research, is written, as wo have already stated, in a remarkably clear and
vigorous style, and — what is not by any means one of its least recom-
mendations— is one of the best-printed mathematical books that has ever
issued from the English press. The woodcuts, eighty-seven in number,
are admirably engraved, and really serve to illustrate the book, a well-
drawn diagram being introduced wherever it would be of use in enabling
us more easily to follow the demonstrations.

" AVe cannot but express a hope that some of our own contributors
will take up Dr. Booth's methods, and develop and apply them in the
mathematical pages of this journal, and its connected volumes of reprints.
And we hope, too, that Dr. Booth will find, in the reception which mathe-
maticians will accord to this volume, sufficient encouragement to induce
him to carry on soon to its completion the promised second volume, wherein
he proposes, ' if declining years and failing strength should permit ' him,
to embody his researches on the geometrical origin and properties of
Elliptic Integrals, and to apply them to the investigation of the free motion
of a rigid body round a fixed point, together with other collateral inquiries.

" In this country we have no ' Minister of Public Instruction,' or
' Keeper of the Seals,' under whose auspices a costly and unremunerative

mathematical work could be brought out without any ex; the

author; and it would be a subject of regret if, when ;ui 1 ina-

ticiun t.1>kes upon himself some of the duties of tin- aho\ e- incut i<. •
tionai-ies, so useful to men of science across the Channel, ami brin^ out. ;:'
<>\vn cost, a work like the one before us, in every way lit to • /'.ace

amongst the best French and <;rnnan treaties, lie *h:>uld. after all hi-
and trouble, be taught by painful experience that, in this country, no
mathematical work has any chance of success unless it belongs to tin- petty
and trivial class of cram-books, drawn up for the use of candidates pre-
paring for some one of the innumerable competitive examinations which
have become the rage of the day. We hope that the volume Dr. Booth
has now given to the world will meet with such a reception as may show
the writer that there are still 'a chosen few' who can appreciate a work
like that before us, of which it is not too much to say, judging from the
instalment we have already received, that it promises to be one of the moat
valuable contributions to mathematical science which has appeared for
many years.

" We have hitherto said nothing about what we regard as one of the
most attractive portions of the book, the excellent Introduction, which
occupies the first twenty-two pages of the volume. The rest of the work
is addressed more exclusively to mathematicians ; but this is a part which
will not bo without interest even to the general reader. We should h
been glad, had our space permitted, to lay this introduction t/» exttnso
before our readers."

From the f Cambridge Express' October 25, 1873.

" The work consists of separate essays on tangential coordinates, pedal
coordinates, reciprocal polars, the trigonometry of the parabola, the geo-
metrical origin of logarithms, geometrical properties of elliptic integrals,
and other kindred subjects. Most of these are old friends that have ap-
peared long since, cither as pamphlets or in mathematical journals ; but
they have all grown in the interval since we last saw them. Thus the
essay on tangential coordinates is known to most mathematicians us a tract
of :?1J pp., published at IXuhlin in 1840, and entitled ' On the A plication of
a New Analytic Method to the Theory of Curves and Curved Surfaces,'
while here it is presented under its now well-known name of ' Tan;.r'
Coordinates,' and occupies, perhaps, over 200 pp. This was one of the
earliest of Dr. Booth's works, and is the one by which ho is best known ; in
fact the method is always associated with his name. In the original tract
of 1840 Dr. .Booth's said that ho feared that ' brevity and compression had
been too much studied in the following essay ;' and here, after an interval
of thirty-three years, we have the essay amplified and expanded to a size
proportional to the value of the method,' and with tho addition of the notes
and examples which have occurred to its author in a period exceedi
average working length of a lifetime. It would not bo easy to give an
idea of the contents of the work without transcribing the titles of the
different chapters, thirty-three in number. The matter in the bo,.;
course, not consecutive," as it is formed by reprinting, with •dditta
Booth's original papers : but there is a ' u-olden thrc:-d ' which runs through
and connects all the subjects di.-cus -cd in tho volume.

VI

" There is prefixed to the volume, by way of introduction, an interest-
ing essay, written in a spirit which here and there recalls Babbage's
' Decline of Science in England.' Dr. Booth laments the utilitarian spirit
of the age in this country, and points out how all knowledge is subordi-
nated to the grand question of money-making. On this point we cannot
refrain from making the following extract : —

" ' Will it pay? is the test of all mental labour. It was very different
in the schools and agorae of that nation we are so prone to hold up for
admiration as exhibiting models of intellectual greatness hitherto un-
equalled. Nor is this exclusive devotion to the adaptation of science to
money-making so universal in other countries as amongst ourselves. Yet
it was not always so. One might appeal to the age of Newton and Locke,
the age of deep thinking and profound learning, in proof of this position.
The causes of this degradation in the objects of intellectual pursuit are
many, and some of them deeply seated. Not the least of these is the
influence which the philosophy of Bacon has exerted on the tone and
tendency of public opinion in this country. No doubt the author of the
' Novum Organon ' conferred great benefits on mankind by laying down so
clearly the true principles of physical investigation. He has marred this
philosophy, however, by the motives he presents to us for its cultivation,
lie who could propound the maxim, worthy of Epicurus, that the true
object of science is to make men comfortable, had no very exalted con-
ception of the dignity of man's understanding.

" ' It is plain from his tone of thought that the philosophical Chancellor
had a very clear promotion, to use his own phraseolegy, of that emphati-
cally English idea, comfort. There is little doubt that he would have
valued more the invention of an efficient kitchen-range, or an ingenious
corkscrew, than the ideas of Plato or the discoveries of Archimedes.'

" What particularly charms us in the above quotation is the estimate of
Bacon's philosophy, which we are afraid is not very far from the truth.
It is becoming more apparent to the present age that Bacon's views are
very different to those of the savant, and that his philosophy is not in all
respects the magnificent structure it was, till recently, heresy to have any
doubt about.

" No one, however, can fail to read with much interest Dr. Booth's
views on" the subject ; and it must be remembered that they come from
him as from one of the most earnest labourers in the field of education.
If all the time that Dr. Booth devoted to the formation and improvement
of the Society of Arts' schemes of education had been given to his own
pursuits, the volume before us would have been a much larger one.

" A mathematician who republishes his scattered writings collected in
a volume, not only thereby secures whatever posthumous fame is his due,
but also confers a benefit on his science. Their accumulation in the same
volume places the whole in a much higher rank than would belong to the
sum of the parts if separate. It is also to be remembered that in many a
country house, cut off from the great journal literature of mathematics,
the appearance of a book containing original work (not written for
teaching-purposes) is hailed with joy."

BY THE SAME AUTHOR.
Examination the Province of the State. Being

an attempt to show the proper function of the State in
Education. 8vo.

" . . . . The first suggestion of this system seems to have been in an able
pamphlet, published by the K> v. Dr. Booth, addressed to the Marqub of

Uffliaowne "— Thonyht* oil National Munition, lij /,„/•,/ I.i/ti.'lt,,,,. p. 1<).

" Dr. Booth, in his pamphlet, ' Kxamination the Province of the State,' pub-
lished some years ago, laid duwn the general outlines of the system of promoting
education by means of examinations, which now nieut.s with Mich .
acceptance." — Daily News,

flow to Learn and What to Learn. Two Lectures

advocating the System of Examinations established by the
Society of Arts, and delivered, the former at Lewes on the
24th of September, and the latter at Hitchin, on the IGth of
October, 1856. Published by the Society of Arts.

" Among the many pamphlets, speeches, and addresses, with which the press
has this year teemed, on the all-engrossing subject of education, these lectures
by Dr. Booth are far the best in our estimation. They are more liberal and
more comprehensive ; they are marked by sounder sense ; and, what will
still more with most men, they are evidently the production of a man who has
thought much and deeply on the subject of which ne speaks, and who brings to
the aid of a mind at once vigorous and rapacious the benefit of an ext.
experience. Dr. Booth is the Treasurer of the Society of Arts, which has done
more than any other body of men to promote the general improvement and
extend education among the yeoman classes of this country, or ratlier anmn^
those who hold a position in society akin to the ancient yeoman
found in town or country. We have no better name by which we can <li-t '. •
them; they are not the very poor; they are not strictly the middle clat^<
they range indefinitely between these two poles of society.

" In the success of so good a cause we feel the deepest sympathy. \\Y i
that these two lectures cannot fail in exciting that sympathy when- it is not now
felt ; and in that persuasion we recommend them to those who are deeply inter-
ested in the cause of education, and who believe, as we do, that it is the great
and absorbing question of the day." — Momimj llrrald.

" Worthy of the high reputation of the author." — Daily New*.

" We should be glad to see these lectures of l>r. Booth MTV exten-ively circu-
lated among the clergy and laity. We agree with much that he says; but what
we especially desire to commend as an example is, the very lucid and spirited
style in which his lectures are written." — Eiiyluh Ckurckma*.

• We recommend to general notice two lectures ],\ l»r. James Booth, entitled
How to Leant and ll'/«it to Isarn, in which the subj'ect here slightly touched on
is fidly and ably treated." — Chambers1 Journal.

On the Female Education of the Industrial Classes.

A Lecture delivered at Wandsworth. 1855.

LONGMANS, GRl.i.N, UKADKK. AM' DYJ

Ill One Volume, crown 8vo, price 5s. cloth,

The Lord's Supper, A Feast after Sacrifice.

With Inquiries into the Doctrine of Transubstantiation, and
the Principles of development as applied to the Interpretation
of the Bible. By JAMES BOOTH, LL.D., F.R.S., F.R.A.S., &c.,
Vicar of Stone, Buckinghamshire.

" This is a careful and scholarly attempt at a via media between the merely comme-
morative theory of the Eucharist and the doctrines of Transubstantiation and Con-
substantiation. Dr. Booth evidently regards the former as bald and defective, and both
of the latter as extravagant and superstitious. The nature of the Holy Rite preferred
by the author is the Epuhim Sacrificiule of Mede and Cudworth, answering to the meal
of the Jews after, and upon parts of, their sacrifice. We commend the treatise as a
valuable contribution to this discussion, which never was more rife amongst polemical
divines than at present, and which may grow in heat and range within a few years."—
English Churchman, June 9, 1870.

" This volume will well repay perusal. It is the work of a clear thinker and well-
informed man. Dr. Booth is well known to mathematicians as one who is at home in
the most abstruse problems. When we state that, our readers will know they are in the,
hands of a man with powers of continuous thought, who is able to trace his way through
all intricacies and obscureness, if a route be possible to human powers. But the ordi-
nary reader (we mean non-mathematical reader) will observe nothing of the mathema-
tician in our author's manner of handling his present subject. His style and inethod
are distinguished solely by their clearness, simplicity, and orderliness. And the book
consists mainly of quotations from able divines of the past. Quotations from such acute
and learned thinkers as Cudworth, and Waterland, and Mede, wit h ot her divinea of lesser
note, form ihe staple of a large portion of the volume. This remark, however, does not
apply to the latter half of the volume, which-consists of two chapters, the one entitled
' On the Principle of Development as applied to the Interpretation of the Bible,' and the
other ' On Trunsubstantiation,' Taken as a whole, the volume brings together much
that is valuable and suggestive, and in the 7iiain thoroughly sound, on the sacraments,
and specially on the Lord's Supper ; and the doctrine of Trnnsubst-nitiation is handled
as might have been expected by so able and profound a mathematician. The history of
the rise and progress and final result of the doctrine is given briefly, yet truly. It ia
traced to a false philosophy long since buried out of sight and forgotten. It, would be
profitable work for some of the author's co-religionists to read, mark, and inwardly
digest the chapter on Transubstantiation, that not cunningly but clumsily devised fable."
— Weekly Iteview, June 18, 1870.

" This is a learned and well-written attempt to establish, in a logical manner, the true
nature of the Lord's Supper, reliance being mainly placed on the brief narratives of the
Gospels and of St. Paul, further elucidated by a reference to the ancient Jewish language,
history, and customs. Dr. Booth's position embraces the view once (he says) almost
universally held in the Church of England, ' That the Lord's Supper is-a Feast upon a
Sacrifice ;' and to set it forth he has combined and expounded the views of such men as
Joseph Mecle, Cudworth, Potter, Warburton, Waterland, Hampden, and others. This
gives to the treatise a somewhat fragmentary air ; but, taken as a whole, it is clearly,
intelligently, and devoutly written, and will doubtless be acceptable to some disciples of
those famous men. On a subject of such subtlety — where the widest diversity of opinion
still fiercely prevails — it cannot hope to please the many, though it is well worthy of
careful examination. Dr. Booth has studied his subject with care, and brought to hia
diflicut task the fruits of extensive reading." — Standard, June 23, 1870.

" Dr. Booth's modest volume is avowedly not so much an original production as an
attempt to recall by selected citations what he thinks the too much neglected learning
of the fathers of thie Church of England. The volume is divided into four chapters, in
the first of which he adduces authorities to prove that the Lord's Supper is not a mere
service of commemoration ; in the second he adduces authorities to prove that it ought
to be regarded as a feast of thanksgiving, implying a preceding sacrifice ; in the third
he treats of the principle of development as applied to the interpretation of the Bible ;
and in the fourth he discusses and dismisses the doctrine of transubstantiation, inci-
dentally treating at some length of the influence of the philosophy of Aristotle. The
most original thoughts and illustrations occur in the third chapter, and the reasoning
seems to us most conclusive in the fourth. The quotations have evidently been selected
with thought and care, and evince much research ; and the author's own writing is finished
and good.. The volume is the careful production of a thoughtful scholar, though it
conveys the impression to us that the mind of the writer has been somewhat overlaid by
scholastic learning, so as to be in an artificial state, and partially disabled from receiving
in their freshness and simplicity the truths which we conceive to be really revealed in
the scriptures to the human heart." — Theological Review, October 1870.
LONGMANS, GREEN, READER, AND DYER.