Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at |http: //books .google .com/I
'P^^(^3 54<S. Vc^.a
HARVARD COLLEGE
LIBRARY
FROM THE
FARRAR FUND
The bequeH of Mrs, Elvui Farrar in
memory of her husband, John Farrar,
HoUis Prcfessor of Mathematics,
Astronomy and Natural Philosophy,
1807-1836
THE
GEOMETRY OF CYCLOIDS
LONDON : PRINTED BY
SPOTTISWOODB AND CO., NfiW-STRBBT SQUARE
AND PARLIAMENT STREET
1
1
4#
m
vH
y
A TREATISE ON
THE CYCLOID
AND ALL FORMS OF
CYOLOIDAL CURVES
and on the Use of such Curyes in dealing with the
MOTIONS OE PLANETS, COMETS, &c.
AND OF
MATTER PROJECTED FROM THE SUN
RICHARD A. 5£pCT0R, B.A.
SCHOLAR OF ST JOHN'S COLLEGE, CAMBRIDaB
MATHBUATIOAL SCHOLAR AND HON. FELLOW OF KING'S COLLBOB, LONDON
AUTHOR OF < SATURN AND ITS STSTBM ' < THE SUN ' ' THE MOON ' * TRANSITS OF VENUS '
<THB UNIVBBSE OF STARS' 'B8SATS ON ASTRONOMT' <THE OKOMONIO
STAR ATLAS' < LIBRARY STAR ATLAS' ETC.
WITH 161 ILLUSTRATIONS AND MANY EXAMPLES
FOR the USE nf STUDENT& in UNIVERSITIES ttc.
' " LONDON
LONGMANS, GREEN, AND CO.
1878
All rights reserved
Tkij^ vi%n 8.
3
HARyAiOCOLLEmitM.IIARY
PEEFACE.
This work deals primarily with the geometry of
cycloidsy curves traced out by a point in a circle roll-
ing on a straight line, or on or within another circle,
and trochoids (or hoop-curves), curves traced out by a
point within or without a circle so rolling.
Although the invention of the cycloid is attributed
to Galileo, it is certain that the family of curves to
which the cycloid belongs had been known, and some
of the properties of such curves investigated, nearly
two thousand years before Galileo's time, if not earlier.
For ancient astronomers explained the motion of the
planets by supposing that each planet travels uniformly
round a circle whose centre travels uniformly round
another circle. By suitably selecting radii for such
circles, and velocities for the uniform motions in them,
every form of epicyclic curve can be obtained, including
the epicycloid and the hypocycloid. When the radius
of the fixed circle is indefinitely enlarged, or, in other
words, when the centre of the moving circle advances
vi PREFACE,
uniformly in a straight line, the curve traced out by
the moving point becomes a trochoid, and may either
be a prolate, a right, or a curtate cycloid, according as
the velocity of the moving centre is greater, equal, or
less than the velocity of the point around that centre.
Lastly, if the radius of the moving circle is indefinitely
enlarged, so that a straight line is carried uniformly
round a centre while a point travels uniformly along
the line, the curve traced out becomes a spiral of the
family to which belong the spiral of Archimedes and
the involute of the circle.
It is of these curves, which are all included under
the general name epicyclical curves, that I treat in
the present volume, though the cycloid, epicycloid,
hypocycloid, and trochoid are more fully dealt with,
in their geometrical aspect, than the epitrochoidal and
spiral members of the epicyclic family.
Ancient geometers were not very successful in
their attempts to investigate any of these curves. It
is strange indeed to find a mathematician even of
Galileo's force so far foiled by the common cycloid as
to be reduced to the necessity of weighing paper
figures of the curve in order to determine its area,
Pascal dealt more successfully with this and other
problems. Yet he seems to have regarded their rela-
tions as of guflScient difficulty to be selected for his
PREFACE. vii
famous challenge to mathematicians, to try whether a
priest who had long given up the study of mathematics
was not a match for mathematicians at their own
weapons. The argument, in so far as it was intended
to prove the soundness of Pascal's faith, was fee.ble
enough. But the failure, or partial failure, of many
who attacked his problems, is noteworthy. We
find, for instance, that Roberval laboured for six
years over the quadrature of the cycloid, and only
succeeded at last in solving it by the comparatively
clumsy method indicated at p. 199, inventing a new
curve for the purpose. It will be seen that in the
present work this famous problem comes very early
(Prop. III., pp. 5, 6), and is made to depend on the
fundamental (and obvious) relation of the cycloidal
ordinates. The method — which so far as I know is a
new one — is extended to the epicycloid, hjrpocycloid,
trochoid, epitrochoid, and hypotrochoid. It will be
found that, in all, thirteen distinct methods of solving
the problem geometrically are either given in full or
indicated (seven of these methods being new so far as
I know), while seven independent methods are indi-
cated for determining the area of the epicycloid and
hypocycloid (of which five are new), besides one
method (see footnote, p. 50) derived from the properties
of the cycloid. After the first demonstration of the
yiu PREFACE.
area^ however, those methods only are given in full
which involve other useful relations.
The position of the centre of gravity of the
cycloidal arc, and of the cycloidal area, has been fully
dealt with geometrically in Section I. (so far as I know,
for the first time). It seems to me that the treatment
of such problems by geometrical methods usefully in-
troduces the student to the use of analytical methods.
For instance. Prop. XIV. is a geometrical illustration
— ^in reality, so far as my own mathematical studies
were concerned, a geometrical anticipation* — of the
familiar relation
/dv , P du J
ax J ax
of the Integral Calculus.
Most of the propositions in the first three sections
were established in the same manner as in this volume,
in notebooks which I drew up when at Cambridge ;
* I may mention, as a circumstance in which some may perhaps
find encouragement and others a warning, that (owing chiefly to
my liking for geometrical studies) I knew very little of the Diffe-
rential Calculus, and scarcely anything of Astronomy, when I took my
degree. Possibly I owe to this circumstance no small share of the
pleasure derived from the study of these and other mathematical
subjects since. The hurried rush made at our xmiversities over the
domain of mathematics has always seemed to me little calculated
to develope a taste for mathematics, though it may not invariably
destroy it when it already exists. The withdrawal of the mind
during three years from other subjects of greater importance, —
general literature, history, physical science, and so forth, — is still
more pernicious : yet it is practically forced on those who wish for
university distinctions, fellowships, and so forth.
PREFACE, ix
but the proofs have been simplified and their arrange-
ment altogether modified more than once since then«
In fact anyone who compares the first two sections with
recent papers of mine on the Cycloid, Epicycloid, and
Hypocycloid, in the English Mechanic^ will perceive
even that in the interval since those papers were written
the subject-matter has been entirely rearranged.
In defining epicycloids and hypocycloids I have
made a change by which an anomaly existing in the
former treatment of these curves has been removed.
The definitions hitherto used run as follows : —
l^he \ J: ^ 1 'j\ is the curve traced out by a
y hypocycloid J ^
point on the circumference of a circle which rolls with--
out sliding on a fixed circle in the same plane ^ the two
. , , . . f external 1 . .
circles beinq m \ , , , }• contact.
^ y internal J
For this I substitute : —
The \ -u 1 ^j\ is the curve traced out bu a
[ hypocycloid J ^
point on the circumference of a circle which rolls with-
out sliding on a fixed circle in the same planer the rolling
circle touching the \ - -j \ of the fixed circle.
That the latter is the more correct definition is
proved by the fact that, while the former leads to an
altogether unsymmetrical classification of the resulting
X PREFACE,
curves, the latter leads to a classification perfectly
symmetrical. According to the former every epicy-
cloid is a hypocycloid, but only some hypocycloids are
epicycloids ; according to the latter no epicycloid is a
hypocycloid, and no hypocycloid is an epicycloid.
In the fourth section on motion in cycloidal curves
I have adopted a somewhat new method of arranging
the demonstrations to include cycloids, epicycloids,
and hypocycloids. The proof that the cycloid is the
path of quickest descent is a geometrical presentation
of Bemouilli's analytical demonstration.
The section on Epicyclics was nearly complete
when my attention was directed to De Morgan's fine
article on Trocholdal Curves in the Penny Ct/clopcedia,
the only complete investigation of any part of my
subject (except a paper by Purkiss on the Cardioid)
of which I have thought it desirable to avail mysel£
I rewrote portions of the section for the benefit of
those who may already have studied De Morgan's
essay, deeming it well in such cases to aim at
uniformity of definition, and, as far as possible, of treat-
ment. It will be observed, however, by those who
compare Section V. with De Morgan's essay, that
my treatment of the subject of epicyclics remains
entirely original, and that in some places I do not
adopt his views. For instance, I cannot agree with
PREFACE. xi
him in regarding the angle of descent as negative under
any circumstances consistent with the definition of
the epicyclic itself. The radius vector indeed ad-
vances and retreats in certain cases; but in every
case it advances on the whole between any apocentre
and the next pericentre. De Morgan has also misin-
terpreted the figures on p. 187, as explained, p. 186.
In two respects this treatise has gained from
my study of De Morgan's essay. In the first place,
I had not originally intended to devote a section
to the equations of cycloidal curves. Secondly, and
chiefly, I was led, by the study of the very valuable
illustrations engraved by Mr. Henry Perigal for Prof.
De Morgan's article, to cancel all the drawings which
I had constructed to illustrate Section V., and to
apply to Mr. Perigal for permission to use his me-
chanically traced curves. A study of Plates II., III.,
and IV., and of other figures illustrating Section V., will
show how much the work has gained by the change.
For figs. 119 to 122, and two of those of Plate IV.,
also mechanically drawn, I am indebted to Mr. Boord.
I may add, to show the value of these illustrations,
that Prof. De Morgan, in his * Budget of Paradoxes,'
says that without Mr. Perigal's * diagrams direct from
the lathe,' his article on Trochoidal Curves * could not
have been made intelligible.' Yet even those cuts.
xii PREFACE,
and many others added to them in this volume, will
give the reader but inadequate ideas of the immense
number, variety, and beauty of the sets of diagrams
published by Mr. Perigal himself, in his * Contributions
to Kinematics.' In these the curves are shown white on
a black, background, and hundreds of varieties at once
instructive and ornamental are presented for study
and comparison. Even for the mere patterns thus
formed, and apart from their mathematical interest,
these sets of diagrams possess great value. (See
further the note, pp. 1193-195.)
The portions of Section V. relating to planetary
motions, and the concluding section relating to the
graphical use of cycloidal curves for determining the
motion of bodies in elliptical orbits under gravity and
of matter projected from the sun, will be useful, I
trust, to students of astronomy. In some respects
cycloidal curves are even more closely related to
astronomy than the conic sections. If planets and
comets travel approximately in ellipses about the sun,
and moons in ellip:3es about their primaries, the planets'
paths, relatively to our earth regarded as at rest, are
epicyclic curves ; while the cycloid and its companion
curves supply an effective construction for dealing
with Kepler's famous problem relating to the motion
of a body in an ellipse round an orb in the focus
attracting according to the law of gravity.
PREFACE. xiii
A treatise such as this is rather intended to afford
the means of solving such problems as may be suggested
to the student than of supplying examples. I have,
however^ added a collection of about 150 examples.
All except those to which a name is appended are
original. They are, in fact, a selection from among
those which occurred to me as the work proceeded*
Many which I had intended to present as riders have
ultimately been worked into the text among the co-
rollaries and scholia. If these had been included as
examples, the total number would have amounted to
about 300 ; but it seemed to me better in their case
to indicate the nature of the proof.
EIOII. A. PROCTOR
LoiSTDOiJ : December^ 1877.
P.S. — As the last sheets are receiving their latest
corrections for press, I receive, through Mr. Boord's
kindness, the eight figures on p. 256. Of these, figs.
154, 158 represent orthoidal, figs. 155, 159 cuspidate,
and figs. 156, 160 centric epicyclics; while fig. 157 is
a transcentric, and fig. 161 a loop-touching epicyclic.
Errata,
On p. 69, line 11, for < Area ABD,' read * Area OBD.*
„ 129, „ 17, „ * D,' read < 0.'
. 73
11. r -^
CONTENTS.
»o»
SECTION I.
PAOB
The Right Cycloid ..... 1
SECTION II.
The Epicycloid and Hypocycloid . . 40
Appendix to Section II,
The Straight Hypocycloid ....
Useful General Proposition • . , .
The Four-pointed Hypocycloid
The Oardioid ......
The Bicuspid Epicycloid ....
The Involute of the Circle ....
Centre of Gravity of Epicycloidal and Hypocycloidal Arcs
and Areas . . . . . .85
66
68
72
73
79
80
SECTION in.
Trochoids . . . . . . 92
Appendix to Section III,
Elliptical Hypotrochoids . . . . . 124
The Trisectrix . . . . . .126
The Spiral of Archimedes . . . . . 127
Flanef s Shadow in Space shown to be spiral . . 133
xvi CONTENTS.
SECTION TV.
PAOK
Motion in Cycloidal Curves. . . . 135
SECTION V.
Epicycucs ...... 148
Appendix to Section V.
Bight Trochoids regarded as Epicyclics . . 167
Spiral Epicyclics . . . . . . 168
Planetary and Lunar Epicyclics . . . . . 169
Forms of Epicydic Curves ..... 182
Forms of Right Trochoids . . . . . 195
The Companion to the Cycloid .... 197
SECTION YI.
Equations to Cycloidal Curves . . . 201
SECTION yn.
Graphical Use of Cycloidal Curves to determine
(i) the Motion of Planets and Comets . . . 209
(n) the Motion of Matter projected from the Sun 216
Examples. . . . 234
PLATES.
Plate I. . . . . frontiepiece
Plates II. and HI. to face each other between pp. 182, 183
Plates IV. and V. „ „ „ 192, 193
Plate VI. .... tofacep,2X^
THE
GEOMETEY OF CYCLOIDS.
Section I.
THE RIGHT CYCLOID,
Note. — Any curve traced by a point on the circumfer-
ence of a circle which rolls without sliding upon either
a straight line or a circle in the same plane is called a
cycloidy but the term is usually limited to the right
cycloid^ and will be so employed throughout this work.
definitions.
The right cycloid is the curve traced by a point
on the circumference of a circle which rolls without
sliding upon a fixed straight line in the same plane.
The rolling circle is called the generating circle \
the point on the circumference the tracing point.
Similar terms are employed for all the curves dealt
with in this work.
Let AQB (fig. 1, Plate I.) be the rolling circle,
KL the fixed straight line. Let the centre of the
THE
GEOMETEY OF CYCLOIDS.
Section I.
THE RIGHT CYCLOID,
Note. — Any curve traced by a point on the circumfer-
ence of a circle which rolls without sliding upon either
a straight line or a circle in the same plane is called a
cycloid, but the term is usually limited to the right
cycloid, and will be so employed throughout this work.
DEFINITIONS.
The right cycloid is the curve traced by a point
on the circumference of a circle which rolls without
sliding upon a fixed straight line in the same plane.
The rolling circle is called the generating circle \
the point on the circumference the tracing point.
Similar terms are employed for all the curves dealt
with in this work.
Let AQB (fig. 1, Plate I.) be the rolling circle,
KL the fixed straight line. Let the centre of the
2 GEOMETRY OF CYCLOIDS.
rolling circle move along the line c' C c parallel to
KL through C the centre of AQB, in the direction
shown by the arrow. Then it is manifest that at
regular intervals the tracing point will (i.) coincide with
the line KL, as at D' and D (EyD' and E y rf being
corresponding positions of the generating circle), and
(iL) will be at its greatest distance from KL, as at A
( AQB being the corresponding position of the genera-
ting circle), this distance being the diameter of AQB,
so that ACB the diameter through the tracing point is
at right angles to KL. It is clear also from the way
in which the curve is traced out that the parts AP'D'
and APD are similar and equal. Therefore ACB is
called the axis of the cycloidal curve ; IKD is the
hase\ and A the vertex. The points D' and D are
<3alled the cusps. The radius CA drawn to the tracing
point is called the tracing radivsy the diameter through
the tracing point the tracing diameter. The radius of
the generating circle may be conveniently represented
by the symbol K. Where the tracing diameter coin-
cides with the axis, the generating circle is said to be
central^ and AQB so placed is called the central gene^
rating circle. A diameter to the generating circle
parallel to ACB, that is perpendicular to IKD, is said
to be diametral. The line c' C c is called the line of
centres.
The complete cycloid consists of an infinite number
of equal cycloidal arcs ; but it is often convenient to
speak of the cycloidal arc D' AD as the cycloid.
It is clear that if D'E' and DE be drawn perpen-
THE RIGHT CYCLOID, 3
dicular to D'D, the semi-cycloidal arcs on either side
of D'E' and DE are symmetrical with respect to
these lines. Therefore D'E' and DE may conveniently
be called secondary axes.
A straight line E'AE through A parallel to D'D
manifestly touches the cycloid at A ; for there is one
position^ and one only, of the generating circle (be-
tween D^'E' and DE) which brings the tracing point to
the distance AB from D'D. E'AE is called the
tangent at the vertex.
PROPOSITIONS.
^ Prop. I. — The base of the cycloid is equal to the
circumference of the generating circle.
This is manifest from the way in which the curve
is traced out ; for every point of the generating circle
AQB (fig. 1) is brought successively into rolling con-
tact with the base D'D ; so that necessarily
D'D = circumference of the circle AQB.
Cor. 1. Biy = BD= semicircular arc AQB.
Cor. 2. Drawing D'E' and DE square to D'D and
(/ Cc parallel to D'D,
Area E'D = 2 area AD = 4 area CD
=s 4 rect. under CB, BD
= 4 rect. under CB, arc AQB
s 4 times area of generating circle AQB.
B 2
GEOMETRY OF CYCLOIDS.
Prop. IL — If through P, a point on the cycloidal arc
APD {fig. 2), the straight line PQM be drawn
parallel to the base BDy cutting the central generating
circle in Q and meeting the axis AB in M ; then
QP = arc A Q.
Let A^'PB' be the position of the generating circle
when the tracing point is at P, C'' its centre, A'C^'
diametral, cutting MP in M^ Draw the tracing dia-
meter PC>. Then MQ = MT ; MM' =QP ; and
arc AQ = arc A'P. Now, since VCp is the tracing
diameter, p is the point which had been at B when the
Fig. 2.
A
A'
1
^
^"'"S^ J
\
M
\q
M^*^
"-^p
1
C
\
^W.' 1
J^
^
7
\
'C^
J
/
\
(\
^
/•
\
B
1
**
c
tracing point was at A; hence the arc j?B' = BB',
for every point of p B' has been in rolling contact
with BB'. But
Arc/?B' = arc ATsrarc AQ; and BB'=MM' = QP.
Wherefore, QP=arc AQ.
Cor. 1. PM = arc AQ+MQ.
Cor. 2. Since BD = arc AQB = arc AQ + arc QB,
BD >PM : wherefore the whole arc APD lies on the
left of DE, perpendicular to BD.
THE RIGHT CYCLOID] 6
Cor. 3. Let MP produced meet DE in m. Then
Pm=Mwi-PM=arc AQB-arc AQ-MQ
= arcQB-MQ.
Cor. 4. Arc AT=BB' ; and arc PB'=B'D.
Cor. 5. If through P', a point on the arc PD,
P'yQ' be drawn parallel to BD, meeting AQB in Q'
and cutting ATB' in q\ then Q'P'' = arc A!q^ and
QP=arc A'P; wherefore
?F( = Q'F-Q'g=QT'-QP) = arcA'y-arcAT;
that is, q P^ = arc P q.
Cor. 6. If through R, a point on the arc AP, 5 R S
parallel to BD meet the arcs AQB, A'PB' in S and «,
then
S5 = QP = AT; and SR= arc A'^;
wherefore R « = arc * P = arc SQ.
•^ Prop. 111.— The area DAD {fig. 1, Plate I.) between
the cycloid and its base is equal to three times the area
of the generating circle.
A, B, D, E, C, &c. (fig. 3), representing the same
points as in the preceding proposition ; take CL = CL'
on AB, and draw LP Z, LTY parallel to BD, cutting
the cycloid in P and P^ and the central generating
circle in Q andQ', respectively. Complete the ele-
mentary rectangles PN, P'N% L A, of equal width,
(PM=FMO. Then
QP = arc AQ, and QT'= arc AQ' = arc BQ ;
therefore QP 4- QT' = semicircle AQB = LZ; and
the two rectangles NP and NT' are together equal
6
GEOMETRY OF CYCLOIDS,
to the rectangle L A. Taking all such pairs of rectan-
gular elements as NP and N'P^, it follows that in the
limit area AQBDP = rectangle CE = circle AQB.
(Prop. I. Cor. 2.)
Hence the area between the cycloid and its base
(= 2AQBDP + circle AQB) = three times the area
of the generating circle. Q.E.D.
Another proof. — Let AP'^D be a cycloidal arc
having A as cusp^ D as vertex^ and DE as axis. Let
Fio. 3.
H
.f-
^■;^ "ffS^^^
„^'
A»
K (n~
r*^
fH"
K
c
L
-\ N\
"\
\
X
V
\
1/ ^^^ ^
\j
V
^'^
^ — ■!
It
I L cut AP'^D in V and be produced to meet the
circle AQB in Q''. Then
LP = arc AQ + LQ ; and LP" = arc AQ - LQ
(Prop. II. Cor. 2). Wherefore
FT = LP-LP'' = 2LQ=Q''Q; and the elementary
area Pm=the elementary area Q'^N.
Taking all such elementary rectangles^ we have in
the limit area AP''DP= circle AQB = rectangle CE.
Hence> taking these equals from the rectangle BE^ it
follows that the equal areas ABDP'^ and APDE are
together equal to the rectangle CD, that is, to the
THE RIGHT CYCLOID. 7
circle AQB. Therefore AP^'DB = the semicircle
AQB ; APDB = three times the semicircle AQB ;
and the area between the cycloid and the base = three
times the generating circle.
Cor. 1. Rectangle AZ= area AQP + area BQ'FIX
Cor. 2. Rectangle C/=area QPP'Q'.
Cor. 3. If AE and BD be bisected in H and I^
and HI cut PQ and P'Q' in h and i ; then if, as in«
the figure, P and P' are on the same side of HI,
P A + F2 = P yi + F'A = FT = Q'^Q = 2LQ.
If P falls between AB and HI, as at />, then, com^
pleting the construction indicated by the dotted lines,
p'H-p U^fh'-p h'^p''p=g 9=2 qj.
That is, if two points are taken on the cycloidal arc-
equidistant from Cc, the sum or difference of the per-
pendiculars from these points upon HI will be equal
to the chord of the generating circle formed by either
perpendicular produced, according as the points on the
cycloid are on the same or on opj)osite sides of HI.
This relation will be found useful hereafter in detei-
mining the centre of gravity of the cycloidal area.
Cor. 4. When the tracing point is at P, the gene-
rating circle passes through P'' ; for its chord through
P parallel to AE = QQ''=PF^
Cor. 5. Area AQ'^Q = area AP^'P; and area
AQ''F'=area AQP. The latter relation, established
independently (by showing that QP = Q'^F^), leads
to a third demonstration of the area.
8 GEOMETRY OF CYCLOIDS,
^ Prop. IV. — If P{Jig, 4) is a point on the cycloidal
arc APD^ APBf the generating circle when the
tracing point is at P, A^ C Bf diametral^ then PB' is
the normal and AP is the tangent to the cycloid at
the point P.
Since, when the tracing point is at P, the generating
circle A'PB^ is turning round the point B^ the direc-
tion of the motion of the tracing point at P must be
Fig. 4.
:^
m ^"^^^^
3
'^'^aSiP'
c
v7\
B E
V D
at right angles, to B'P ; wherefore PB^ is the normal
and A^P is the tangent at the point P.
Another demonstration. — The objection may be
raised against the preceding proof y that^ by the same
reasoning, B^ would be proved to be the centre of curva^
ture at Py which is not the case. Although the objection
is not really validy an independent proof may conve^
niently be added.
Take P' a point near to P, and draw PQM, P'Q'N
parallel to BD, cutting AQB in Q and Q', and P'Q'N
cutting ATB' in q. Join PC. Then y P'= arc Py
(Prop. II. Cor. 4), and ultimately PyP^ is an isosceles
TEE RIGET CYCLOID. 9
triangle, whose equal sides Vq and qV are respectively
perp. to the equal sides C'P and C^B' of the isosceles
triangle PC'B' ; wherefore the third side PP' is perp.
to the third side PB' * That is, PB' is the normal at
P, and therefore PA' the perp. to PB' is the tangent
at P.
Cor. 1. If Pw be drawn perp. to P'N, then the
figure PP'n is in the limit similar to the triangles
A'B'P, A'Pm, PB'tw (m being the point in which
A'B' and PM intersect).
Cor. 2. If BT cut FN in /, the triangle /FP is
similar to the four triangles named in Cor. 1.
Cor. 3. Triangles P y /, P y P' are similar respec-
tively to triangles PC'A' and PC'B' ; and Z^ = y P^
Cor. 4. AQ is parallel to the tangent at P.
Cor. 5. If AQ prod, meet FN in r, QQ' ulti-
mately = Q' r.
ScHOL. — A tangent may be drawn to the cycloid
from any point on the curve. For if we draw PQ
parallel to BD, the tangent PA' is parallel to AQ.
To draw a tangent from any point A' on the tangent
at vertex, we draw A'B' perp. to base, and the semi-
circle A'PB' on ADB' intersects APD in the point
P such that A'P is tangent to APD.
* Thus, let the triangle P g' F be turned in its own plane round
the point P till P q coincides with PC — that is, through one right
angle; the other sides qV and PF will also have been turned
through a right angle, therefore q P' will be parallel to C B', and
q P' being equal to g'P, F will fall on B'P (for any parallel to C'B'
will cut off aa isosceles triangle from B'PC) ; hence B'PP' is the
angle through which PP' has been turned, and is therefore a right
angle.
GEOMETEY OF CYCLOIDS.
Prop. Y.—If PQ {Jig. 5), parallel to the base of cy-
cloid APD, and above the line of centres C c, meets
the central generating circle in Q, and QJV, PM are
perpendicular to Cc,
Area Ah QP+ reel. QM=rect. CF
{F being the point in which NQ produced meets the
tangent at the vertex A T).
If P^ be a parallel to the base below the line of centres
Q'i) PM , perpendicular to C c.
Area AhQ'P' - rect. (^M= red. CF
(F being the point in which LQ^ produced meets the
base BD).
Take p a point near to P, and let pn perp. to QN
cut arc AQQ' in q; join AQ and produce to meetpn
-yi^
in r; draw/y L, r'K.,pm perp. to Cc, and join Cq.
Then in passing from P to ^, area A A QP + rect. QM
is increased by PpmM and diminished by Q5LN, orin
the limit, increased by rect. M^ or Kr (sinceQr/iP
THE RIGHT CYCLOID, 11
is a parallelogram^ Prop. IV. Cor. 4) and diminished by
rect. Ny ; wherefore total increase = rect. Lr. But
nq : yQ ( = 9r. Prop. IV. Cor. 5) :: yL : Cy(=NF),
.'. rect. under nq, NF = rect. under qr, qJj\
that is, rect. N/= rect. Lr,
or inert, of rect. CF= inert, of (area AAQP 4 rect. QM).
But these areas start together from nothing, at A,
.-. rect. AAQP -h rect. QM = rect.' CF.
Cor.l. Area AQC'KP= square CT = square CT',
TCT being the tangent to AC'B at C on the line of
centres.
Again, making a similar construction for the second
case (for convenience in figure Q' ^ is so taken that
Q ^ and q Q' are perp. to C c\ we have ultimately
decrement of area (A A QT' - Q^M^ = L y' + F w'
= rect. Ly' + rect. n' K' (ultimately) = rect. N/.
But since wY •' Q'?' (= /^) :: ?'N : C/ (= N/),
rect. under n' ^ , N/' = rect. under ffr' , / N ;
that is, rect. NF' = rect. N/, or
decrt. of rect. CF' = decrt. of area (AAQT'-Q'M').
But these areas begin together from the equal areas
AQC'R and square CT,
.-. area AAQ'F - rect. Q'M' = rect. CF'.
Cor. 2. Area AC'BDR=rect. CBD c= generating
circle, so that we have here a new demonstration of
the area.
12
GEOMETRY OF CYCLOIDS,
Pkop. VI. — If from P a point on the cycloid APD
(Jig- 6) PQ drawn parallel to the base^ meets the
generating circle in Q,
arc AP = 2 chord A Q,
With the same construction as in Prop. IV., join
AQ and Wq\ produce B'jr to meet PP' in A; and
draw C^K perpendicular to B^P. Then ultimately.
Fig. 6. (Join A'j.)
d I
qk\& perpendicular to PP', and the triangle P^P' is
isosceles ;
.-. PP' = 2AP ultimately.
But PP^ is ultimately the increment of the cycloidal
arc AP ; and P A is ultimately the increment of the
chord AT (for A!q ^ A!k ultimately). Hence the
increment of the cycloidal arc AP = twice the incre-
ment of the chord AT or of the chord AQ. There-
fore, since the arc and chord begin together at A,
Arc AP = 2 chord AQ.
Cor. 1. Arc APD = 2 AB = 4R, and the entire
cycloidal arc from cusp to cusp = 4AB = 8R.
THE RIGHT CYCLOID. 13
Cor. 2. SiDce the square on AQ = rect. AB . AM,
sq. on st line equal to arc AP = 4 rect. AB. AM,
and we have,
Arc AP = 2v/2R . v/AM,
that Is, Arc AP (xr\/KM.
Cor. 3. Arc AP : arc PD :: AL : LB.
^ Prop. VII. Prob. — To divide the arc of a cycloid
into parts which shall he in any given ratio.
Let a straight line ab (fig. 6) be divided into any
parts in the points c and d\ it is required to divide the
arc APRD in the same ratio.
Divide AB in L and / so that
AL : L / : /B :: ac : cd : db.
With centre A and radius AL and A/, describe circular
arcs LQ, / r, meeting the semicircle AQB in Q and r.
Through Q, r, draw QP, rR, parallel to BD. Then
Arc AP = 2AQ = 2AL ; and arc AR = 2AZ.
Therefore
Arc PR = 2 L / ; and similarly arc RD = 2 IB.
Therefore
Arc AP : arc PR :: arc RD :: AL: LZ : ZB :: ac : cd :db;
or the arc APD has been divided in the points P and
R in the required ratio.
Similarly may the arc APD be divided into any
number of parts, bearing to each other any given ratios.
14 QEOMETRY OF CYCLOWS.
Prop. VIII.— Wi(A the construcHon of Prop. IV.
AreaAPB'B: tectorial area A' ^ Pk
:: area PB" D : tegment PES' :: 3 : 1.
Let aVb (fig. 7) be the position of the tracing
circle when the tracing point is at P' near to P, on the
FiQ. 7. (JoinaF.)
side remote from A; acb diametral. Join bV, draw
P'y I parallel to BD meeting A'PB' in q and PB' in /,
join q B'j which is parallel to b P', because y P' =
P J = B'fi. Then ultimately V q = ql (Prop. IV.
Cor, 3), wherefore parallelogram qb = twice the tri-
angle IqW and trapezium /P'6B' = 3 times the
triangle Iq^': that is, ultimately (when the triangle
/PP* vanishes compared with /P'ftB'),the elementary
B'PP' 6 = 3 times the elementary area PB'y
= 3 (area A'B' y A - area A'B'P h)
= 3 (area a SF - area A'B'P).
Thus the increment of the area ABB'F = 3 times the
THE RIGHT CYCLOID. 15
increment of the area A^B'P, and the decrement of
area PB'D = 3 times the decrement of the area PFB'.
But the areas ABB'P and A^B'P commence together,
and the areas PB'D and PFB^ end together, as P
passes from A to D. Hence
ABBT = 3 times sectorial area A'BT h.
Area PB'D = 3 times the segment PFB'
and
Area APB'B : sectorial area A'BT h
:: area PB'D : segment PFB' :: 3:1.
Cor. 1. Area PFB'D =: 2 segment FFB\ This
is easily proved independently. For any elementary
parallelograms jQT and FF^ (having sides parallel to
BD), are manifestly equal ; wherefore area g' F ^ F'
=? parallelogram qb = twice triangle B' y / = (ulti-
mately) twice the decrement of segment ^F'P^
Cor. 2. Area AQBBT (BQ straight) = 2 sec-
torial area AQB.
Cor. 3. Area Q^BDP = 2 seg. QsB + par. PB
= 2 seg. Q^B + rect. BM'.
ScHOL. — Prop. VIII. affords another proof of the
relation established in Prop. III. The first corollary,
established independently, gives another proof.
16 GEOMETRY OF CYCLOIDS.
Prop. IX. — With the same construction as in the pre
ceding propositions.
Area APA = segment A'hP.
Join PA', q A^ and V'a. Then ATF is ulti-
mately a diameter of the parallelogram A^aP^qy and
the ultimate triangle A'PP'a is equal to the triangle
A'PP'y, or in the limit to the triangle A'Py. But
A'PP'a is the increment of the area APA', and A'Py
is the increment of the segment A'AP. Since these
areas then begin together and have constantly equal
increments, they are constantly equal. Therefore
Area APA' = segment A' A P.
Cor. 1. Draw PL, PM'M perp. to AE, AB respec-
tively, PM intersecting AB in M'. To each of the equal
areas APA' and A'AP add the equal triangles A'PL
and A'MP. Then the area APL = area A'A PM' =
area AQM. This may be proved independently. For
drawing P'K, P'N' perp. to AE, A'B', we see that
A'PP' is ultimately a diameter of the rectangle N'K,
and therefore the rectangles PK and PN', being com-
plements to rectangles about the diameter, are equal :
or ultimately the increment of the area APL = incre-
ment of the area A'A PM' ; wherefore, since these areas
begin together, area APL = area A'A PM' = area AQM.
Cor. 2. Area AQP = rect. ML— 2 area AMQ.
Cor. 3. Area Q5BDP = circ. AQB -area AQP
= circle AQB — rect. ML + 2 area AMQ
= 2 (semicircle AQB -h area AMQ) — rect. ML.
THE RIGHT CYCLOID.
17
Cor. 4, Area AA'AP = 2 area AA^P=2 segment
A'AP. This may be proved independently, in the
same way as Cor. 1, Prop. VIII. Area A!aV^qhy
ultimately equal to the area A'aP^PA, is shown to be.
equal to the area of the parallelogram A'aP'y, that is,
to twice the area A'PP'a or A'PP^jr (the ultimate in-
crements of AA'P, A^ A P, respectively).
ScHOL, — Prop. IX. and Cor. 1 and 4 (established
independently) afford three new demonstrations of the
area of the cycloid. For they severally show that
area APDE = semicircle DQ^E, on DE as diameter ;
and since BE = twice the generating circle, the area
APDB = 3 times the semicircle AQB.
It will be noticed that the area AEQ'DP = area
A^BDP. This, which may easily be proved inde-
pendently, affords yet another proof of the area of the
cycloid. Thus let APD, AP'D (fig. 8) be cycloidal
Fig. 8.
arcs, placed as in Prop. III.; A'PB'P^ and apbj/
adjacent positions of the tracing circle. Then, Prop.
III. Cor. 4, P^P and p^p are both parallel to BD.
Hence ultimately area A'a^P = area A!ap^V^ ; but
c
18
GEOMETRY OF CYCLOIDS,
these are the increments of the areas AA'P, and AA'P%
which commence together. Hence area AA'P = area
AA'P', wherever P and V may be. Wherefore
^taking P to D) area AEQ'DP = area XEqBV =
area AQBDP. Therefore the arc APD dividea the
area AEQ'DBQ into two equal parts. But area
AEQ'DBQ = area AEDB = twice the generating
circle. Hence ai-ea AQBDP = area APDQ'E = the
generating circle ; area APDB = 3 the semicircle
AQB ; and area AEDP = semicircle AQB.
Prop. X. — The radius of curvature at P (fig. 9) ij?
equal to twice the normal PB\
With so much of the construction of fig. 7 as is con-
»
Fig. 9. (Fop 0', O uead o, o ; and join o a\)
A a' e
.<
B^
ft
1 J^ "^
m'^^y
tained in fig. 9, produce P'i, which is parallel to y B',
THE RIGHT CYCLOID. 19
to meet PB' produced in o\ Then since ultimately
ZP' = 2 7y; lo' ultimately = 2 ZB^ So that if the
normals at the adjacent points P and P', intersect ulti-
mately (when P' moves up to P) in o (which, there-
fore, is the centre of curvature at P),
Rad. of curvature P o = 2 normal PB^
Cor. The radius of curvature diminishes from the
vertex, where it has its maximum length, to the cusp,
where the radius vanishes or the curvature becomes
infinite.
Prop. XI. — The evolute of the cycloid APD {^fig* 9)
is an equal cycloid D ody having its vertex at jD, and
its cusp d on AB produced to d so that Bd =i AB,
Complete the rectangle DBrfe, produce A^B^ to
a', and join o ol. Then in the triangles A'B^P and
d^'o the sides A^B^ B'P, are equal to the sides a'B^,
B^o, each to each, and enclose equal angles ; therefore,
the triangles are equal in all respects, and the angle
alo B' (= the angle B^PA^) is a right angle. Hence a
circle described on B^a^ as diameter will pass through
0. Again, in the equal circles A'B'P and a'B'^, the
ancrles A^B'P and a'Wo at the circumference are
equal. Therefore the arc o a' = the arc P A^ = BB'
(Prop. II. Cor. 4) = rf a!. Wherefore o is a point on
a cycloid having d e for base, a cusp at rf, and B^o a' as
tracing circle. Since de = BD = arc B'oa^ D e is ,
the axis and D is the vertex of the evolute cycloid.
Cor. oY = 2 oB' = arc o D (Prop. VI.) ; so that;^
c 3
20 GEOMETRY OF CYCLOIDS.
if a string coinciding with the arc doJi and fastened
at d be unwrapped from this arc, its extremity will
always lie on the cycloid APD, which may, therefore, be
traced out in this way as the involute of the arc doT>,
Prop. XII. — If APD {fig. 9) be a semi-cycloidal arcy
do D its evolute, avd o B'P the radius of curvature
at any point P on APD, cutting the base BD in
S , then the area APB'B = three times the area
d BB'o.
If P'o' be a contiguous radius of curvature cutting
BD in by and P7 parallel to BD meet PB' in /; then
in the limit o / = 2 o B', and therefore the area of the
ultimate triangle o / P^ = 4 times the area of the ulti-
mate triangle o Wb ; or ultimately the area B7 P'A = 3
times the area o B^b. But these areas are the element-
ary increments of the areas APB'B and rfBB'o, which
begin together from AB d. Wherefore the area APB'B
= 3 times the area d BB^'o.
Cor. 1. Area ABD = 3 times area d BD = 3 times
area AED=| rect. BE = 3 times the generating circle.
We have here another demonstration of the area.
Cor. 2. Area o B'D = \ area B'DP = segm. P y B'
(Prop. VIII. )• This may be proved independently ;
for triangle oW b = triangle W Iq = (ultimately) tri-
angle B^P q ; but triangles o W b, WP q, are decre-
ments of area oWD and segment P q W, which end
together at D ; ,', o B^D = seg. P q B'.
Hence, dWD = ^ generating circle. We have
here, then, yet another demonstration of the area.
THE RIGHT CYCLOID.
21
Pkop. XIIL — If G {fig. 10) is the centre of gravity
of the cycloidal arc APDy then GK, perp. to AE
{the tangent at the vertex ^) = J AB.
Let PP' be an element of the arc APD and let
PM, P'N perp. to AB intersect the semicircle AQB
in Q and Q'. Join AQ^ cutting MQ in n. Then
ultimately PP' is parallel and equal to n Q' (Prop. IV. ).
Now, representing the mass of element PP^ by its
lengthy the moment of PP' about AE ultimately
= PF. AN = n Q' . AN
= MN.AQ'
(since n Q' : MN :: AQ^ : AN)
and may be represented therefore by the elementary
rectangle MN /»», of which the side N/ = AQ^
Thus the moment of the arc APD about AE may
be represented by the area A q'b B obtained by draw-
ing the curve A ^b through all the points obtained as
^ was. But since square of Nj^ = square of AQ^
= rect. under AB, AM ; A y'& is part of a parabola
23
GEOMETRY OF CYCLOIDS.
having A as vertex, AB as axis and parameter (focus
at S, such that AS = ^ AB). Therefore area AB b
= f AB . B5 ; and moment of arc APD about AE
2 AB.BZ^ arc APD. B 6
^^ •
3
(=arcAPD.KG) =
3
or
KG = iBJ= JAB.
Cor. 1. Moment of PP' about AE = MN. AQ'.
Cor. 2. Still representing the mass of arc by its
length, that is, taking for unit of mass the mass of one
unit of length of the arc.
Moment of arc APD about AE = f (AB)^
Cor. 3. Momt. of AP about AE is represented by
area AM y = f AM v/AB"7AM = f AM*. AB* .
Prop. XIV. — If G (^fig, W) is the centre of gravity of
the cycloidal arc APDy then GL perp. to the axis
AB =z BD-i AB.
With same construction as in Prop. XII.,
Fig. 11. (AQ' and NQ intersect in n.)
momt. of PP' abt. AB = PN . PP'=2PN. inct. of AQ
THE RIGHT CYCLOID. 23
(Prop. VI.). Draw P a, P^a' parallel to AB and equals
respectively, to AQ, AQ'; complete the rectangles N a,
M c^ ; and produce a P to meet MP' in A. Also join
BQ' and let NP, MP' prod, meet a cycloidal arc BE
having B as vertex and E as cusp m p and p'. Then,
rect. M of ultimately exceeds rect. N a by
rect. under PN,(P'a'— Pa) -h rect. under aT'. k PV
That is,
inct. of rect. N a = PN. inct of AQ -!- AQ'. A P'
= i momt. of PP about AB + BQ' . MN
(since AQ':BQ'::*P: AF)
= \ momt. of PP' about AB + momt. oipp' about BD
(Prop. XIII. Cor. 1).
Wherefore, taking all increments from A, where rect^
N a has no area, to D, where N a = rect. AD, we have
2 rect AD = momt. of arc APD about AB
+ 2 momt. of B jo E about BD ;
that is,
DE
arc APD. GL = 2 AB . BD - 2 arcB joE . -y- ?
or 2 AB . GL = 2AB . BD - | AB . DE ;
.-. GL = BD-.f AB.
Cor. Draw GH perp. to DE. Then GL + f AB
= BD =r GL + GH. Therefore GH = f AB.
24
GEOMETRY OF CYCLOIDS.
Pbop. XV. — If G (Jig. \\) is the centre of gravity of
the cycloidal arc APD^ and GH^ GJ be drawn perp,
to DE and BDy JH is a square^ whose sides are
each equal to ^ AB,
From Prop. XIII. EH = ^ AB ; .• . DH = f AB.
From Prop. XIV. Cor., GH = f AB. Therefore,
the rectangle JG is a square having each of its sides
= fAB.
Pbop. XVI. — If G' {fig. 12) is the centre of gravity
4)fthe area APDEy then G'Kperp. to AE=iAB.
Take PP'' an element of the arc APD ; draw P' n
perp. to AE, and PQM, P'Q'N perp. to AB, inter-
FiG. 12.
meeting AQB in Q and Q'. Complete rectangles P w,
QN. Th«n from Prop. IX. Cor. 1,
rect. P w = rect. QN.
Now momt. of element P n about AE, ultimately
= i P' « . rect. P n
= i AN . rect. NQ
= i momt. of NQ about AE.
THE RIGHT CYCLOID,
26
Taking all such elements^ we have
Momt. of area APDE about AE = \ momt. of area
AQB about AE.
That is, G'K . area APDE = i AC . area AQB.
But, area APDE = area AQB ;
.•.G'K = iAC = iAB.
Pbop. XVII. — If G' {fig, 13) is the centre of gravity
of the area APDEy HI parallel to AB through II
the bisection of AEy and G'L perp. to Hly then
G'L : AB :: AB : SBI, or G'L = ^. AB.
Take elements MN and M'N' equal to each other
and equidistant from A and B respectively ; draw
Fio. 13.
n' t
MQP, NP^ N'E, and Wq Bf parallel to BD, meet-
ing APD in P, P', R and R' (Q and q being points on
circle AQB). Draw F^n and B/ n' perp. to AE, and
complete the elementary rectangles P w, R w', QN and
q N'. These four rectangles are equal. Now, sum of
26 GEOMETRY OF CYCLOIDS.
moments of P w, R 71' about HI
= H w . rect. P 71 + H 7i' rect. R n'
= (H 71 + H 7i') rect. QN
= 2QM . rect. QN (Prop. III. Cor. 3)
= 2 moment of rect. QN about AB.
[This relation holds whether P n and R n' lie on
the same side as in fig. 13 or on opposite sides of HI;
for in the latter case^ the moments being in opposite
directions^ their difference is the effective moment, and
instead of (H ti' + H n) rect. QN, we get (H ti' — H ti)
rect. QN ; but when n! and n are on opposite sides of
HI, H 72' - H 72 = 2QM. Prop. III. Cor. 3.]
Wherefore taking all the elements such as MN,
M'N', from A and B to the centre C, we get
Momt. of area APDE about HI = 2 momt. of semi-
circle AQB about AB ;
that is, LG'. area APDE = 2 C ^ . area AQB
{g being the centre of gravity of the semicircle AQB
and G g perp. to AB). And since area APDE = area
AQB, LG' = 2C^.
But we know that
C ^ : AB :: AB : 3 arc AQB* (= 3BD) ;
, ^ 2AB\
(orC^=-3^):
wherefore LG' : AB : 2 AB : 3 BD :: AB : 3BI;
* If the reader is unfamiliar with this property, he may esta-
blish it thus : — First show that projection of any element of semi-
circle on tangent at the middle point of the arc has a moment about
THE RIGHT CYCLOID.
27
Prop. XYlll.— If G and G' {fig, 14) are the centres
of gravity of the areas APDB and APDE re-
spectively^ O the centre of gravity of the rectangle
BE {that is the point in which HI^ drawn as in last
proposition, and COy the line of centres y bisect each
other)y and GKy G'L are drawn perp, to HI, then
OK=-^AB=iA C; andGK=^LG'=z ~.AB=:^f~.
Since O is the centre of gravity of the rectangle
BE, that is, of the area APDB + the area APDE, the
P B
moments of APDB and APDE about COC are
equal ; that is,
diameter equal to the moment of the element ; therefore moment
of semicircular arc, or ir rad. x dist. of C.G from diameter = diameter
X rad. ; that is distance of C.G from diam. = diameter — ir. Now a
semicircular area may be supposed divided into an infinite number of
equal small triangles having centre for apex, and each triangle may
be supposed collected at its C.G. at a distance from centre = f rad.
Hence C.G. of semicircular area lies at a dist. from diameter =
2 diameter
g . That is to say C^ : 4r : : 1 : 3ir : : r : 3ht, or
Cff : 2r :: 2r : 3 arc of semicircle.
28 GEOMETRY OF CYCLOIDS,
3 area APDE . OK = area APDE . OL ;
or OK = iOL = iV AB = ^ AC.
Similarly,
GK = iLG'= ^ . AB = i4P.
Cor. 1. Since LG' : AB:: AB : 3BI
(Prop. XVII.),
GK : AB::AB : 9 BI.
Cor. 2. G, O, and G' lie in a straight line, and
0G'=30G.
Cor. 3. Since moment of area AQBD about BD
= (moment of ABDP— moment of AQB) about BD
= (i . 3 AC - AC) ""^^^ = ?A?. T . AC ; it follows
that the C.G. of area AQBD lies at a distance = | AC
from BD.
ScHOL. — The position of G may be thus ob-
tained: —
Take OK = ^ AC. Also, take BM = ^ AB ;
join MI, and let MF perp. to MI intersect DB pro-
duced in F : draw KG perp. to 01 and equal to BF.
Then G is the centre of gravity of the area APDB.
For OK = ^AB ; and
KG( = FB) :BM::BM : BI;
that is, KG : ^AB :: ^AB : BI :: AB : 3 BI,
or KG: AB::AB: 9 BI.
THE RIGHT CYCLOID.
Prop. XIX.— ;//rom G {Jig. 15), the centre of gravity
of semi-cycloidal arc APD, GL be drawn perp, to
AB, and- G I making with AB produced the angle
GIA = the angle ADB ; then the surface gene-
rated by the revolution of the arc APD about the
axis AB is equal to eight times the rectangle having
sides equal to AB and LI.
By Guldinus's First Property (see note following
this Proposition), the surface generated by the revolu-
Flo. 15.
■"^^C^
/
^
\.
/
\
tlon of APD about AB = rect. under straight lines
equal to APD and circumference of circle of radius
LG. But APD = 2 AB, and since GL / is similar to
ABD, and BD = J the circumference of circle of
radius AB, it follows that L / = ^ circumference of
circle of radius LG. Hence the surface produced by
the revolution of APD about AB
= rect. under 2 AB and 4 L /
= 8 times the rectangle under AB and L /.
30 GEOMETRY OF CYCLOIDS.
Cor. 1. In revolving round AB through half a right
angle, APD generates a surface equal to rectangle
under AB and L /.
Cor. 2. Since. GL = BD-AAB (Prop. XIV.),
L/= (BD — ^AB) - ; and the surface generated byre-
volution of APD about AB = 4AB (BD - f AB)^ .
= SAC (^. AC- JAG) TT = ^ (^— t) (AC)\
= 8 (tt — ^) generating circle.
Note. — Guldinus's properties, usually demonstrated by the in-
tegral calculus, are essentially geometrical. His First Property
m&y be stated and established as follows : —
If a plane curve revolve through any angle a about an axis in its
own plane, the curve lying entirely on one side of the aansy tlie a/rea
generated hy the curve is equal to a rectangle liaving its adjacent sides
equal in length to the curve and to the arc described by tJie centre of
gravity of tlie curve, in revolving about tlie axis through the angle a.
Let APB (fig. 16) be a curve lying in the same plane as OX, and
entirely on one side of OX, and let it revolve around OX through
an angle a to the position apb> Then PF, an element of the arc
APB, generates a cotical shred of constant breadth PF and of area
ultimately = PP'. arc Pj? = PP'. PM . a = a . moment of PF about
OX. Taking all the elementary arcs of APB in this way, the sur-
face generated by the arc APB = a . moment of arc APB about OX
= a . GN . arc APB ; (G being the centre of gravity of the arc APB,
and GN perp. to OX).
Or, if length of curve APB « L, GN == 5, and the area of the
surface generated = A* then
A « L . <2 .a
THE RIGHT CYCLOID, 81
If the axis intersect the curve, then the two portions of the
curve lying, on either side of the axis must be separately dealt with.
It is easily seen that if the curve APB is not plane, or if (whether
plane or not) it is not in the same plane as OX, a similar property
may be established. Let the curve be carried once round OX, and
let a plane through OX intersect the surface thus generated in a
curve A'P'B' (any parts of A'P'B' through which more than one part
of APB may have passed being counted twice or thrice or so many
times as they may have been traversed in one circuit of APB). Let
L' be the length of A'FB' (thus estimated) ; G' its centre of gravity
(correspondingly estimating the weight of its various parts), and d'
the distance of G-' from OX. Then the surface generated by the
revolution of APB round OX through the angle o «- L'. d\ a (jamy
part of the generated surface traversed more than once by the
generating curve being counted as often as it has been so traversed).
Again, if APB so move as to generate a cylindrical surface either
right or oblique, and two planes through OX intersect the surface
thus generated, the portion of this surface intercepted between
'those planes may be thus obtained : — through OX take a plane perp.
to the axis of the cylindrical surface and intersecting that surface
in a curve AT'B' of length L' and having centre of gravity G-' at
distance df from OX ; let the portion of a straight line through G'
parallel to the axis of the cylindrical surface, intercepted between
the boundary planes s= h ; then the surface intercepted = L'. d\ h.
The proofs of this and the preceding extensions of Guldinus's
first property depend on the same principle as the proof of the pro-
perty itself given above. In fact, the student who has grasped the
principle of that proof will perceive the extensions to be little more
than corollaries.
It may be of use to note that the two extensions require two
lemmas. The first requires this lemma : — If an element of arc PP'
be projected orthogonally on a plane through OX and P into the
elementary arc P^, then PP' and P^ in rotating through any angle
round OX generate equal surfaces. This is obvious, since they
generate equal elementary surfaces in rotating through an elemen-
tary angle round OX. The second extension requires this lemma : —
If two planes through OX cut two parallel lines Pj?, P'y in P, F
and PfP', the lines PP' and j?p' being elementary, then two other
planes through OX near to these last cutting Fj) and F'jp' in R, R'
and r, r', such that PR =p r, intercept equal areas PRR'F and ^r ?''^'.
These areas are in fact ultimately parallelograms on equal bases and
between the same parallels.
32 GEOMETRY OF CYCLOIDS.
Prop. XX. — If from G {fig* \5\ the centre of gravity
of the semi-cj/cloidal arc APDf GHbe drawn perp. to
EDy and G h making with ED produced the angle
Gh H = angle ABD^ then the surface generated by
the revolution of the arc APD about ED as an
axis is equal to eight times the rectangle under AB
and Hh,
The demonstration is in all respects similar to
that of Prop. XIX.
Cor. 1. In revolving through half a right angle^
APD generates a surface equal to the rectangle under
AB and H A.
Cor. 2. Since GH = | AB (Prop. XIV. Cor.),
H A = - - ; and the surface generated by the revo-
lution of APD about ED=8. AB .^^ *= ^ (AB)«
= - - (AC)^ = Y • generating circle.
Prop. XXI. — If from G{fig. 15), the centre of gravity
of the semi'Cycloidal arc APD^ GK be drawn perp,
to AEy and G k parallel to AD meet AE in A, then
the surface generated by the revolution of the arc
APD about AE as axis ^=- eight times the rectangle
under AB andKk.
The demonstration is similar to that of Prop. XIX.
Cor. 1. In revolving through half a right angle
THE RIGHT CYCLOID, 33
APD generates a surface equal to the rectangle under
AB and E k.
Cor. 2. Since GK = ^AB (Prop. XIV.), K A =
-^ AB ; and the surface generated by the revolution of
APD about AE = 8^AB.A^=^J'AB^=^^'' (AC)^
= ~o • generating circle.
Pkop. XXIII. — If from G {fig» 15), the centre of
gravity of semi-cycloidal arc APD^ GJ he drawn
perp. to BD^ and Gj parallel to AD to meet BD
produced in J, then the surface produced by the revo-
lution of the arc APD about BD as axis weight times
the rectangle under AB andJj,
The demonstration is similar to that of Prop. XIX.
Cor. 1. In revolving through half a right angle
APD generates a surface equal to the rectangle under
AB and Jj.
Cor. 2. Since GJ = f AB (Prop. XV.), 3j =
- AB ; and the surface generated by the revolution of
APD about BD = ®- (AB)« = -^^ (AC)».
= --. generating circle.
D
u
GEOMETRY OF CYCLOIDS.
Prop. XXIV.— 7//rom G {fig. 17), the centre of
gravity of the cycloidal area APDB^ GL be drawn
perp, to ABy and G I making with AB produced the
angle G I A ^ angle ADBy then the volume gene-
rated by the revolution of the are& APDB around
the axis AB is equal to six times the volume of a
cylinder having the generating circle AQB for base
and height equal to LL
By Guldinus's Second Property (see note following
this proposition) the volume generated by the revolu-*
tion of surface APD around AB = volume of a right
cylinder having APDB as base and height = circum-
ference of circle of radius LG. But area APDB =
4 generating circle ; and, as in Prop. XIX., L Z=
i circumference of circle with radius LG. Hence the
volume generated by the revolution of area APD
around AB is equal to (^ x 4 times, or) six times the
volume of a cylinder having circle AQB as base and
height = L /.
Cor. 1. The volume generated by the revolution of
THE RIGHT CYCLOID, 86
APDB through one-third of two right angles about
AB is equal to a cylinder having circle AQB as base
and height = L Z,
Cor. 2. Since LG = OC -^- (Prop. XVIII.)
= -. AC— -— — , L / = ( ^. AC — ^—1^; andthesur-
2 97r U 97r y '
face generated by the revolution of APDB about AB
= 6. (AO^gAC - l^C) . = (^-^-^;) (AC)3.
Cor. 3. Since the rectangle BE in revolving
around AB generates a cylinder whose volume
= AB . 9r . (BD)«=2AC . -T (tAC)^=2^3 , {KC)\ it
follows from Cor. 2 that the volume generated by
APDE in revolving around AB
=V(AC)3-(^'-|-) (AC)3=('-V_^) (AC)3.
NOTB.— -Guldinus's Second Property may be thus stated and es-
tablished : —
If a plane figv/re revolve through an angle a about an axis in ita
own plane (thefigwe lying entirelg on one side oftlie axis), the volume
of the solid generated by tJie figure is equal to that of a cylinder Jiaving
the figure for base and its height equal to tJie arc described by tite
centre of gravity oftlie swrface in reviving through the angle a.
Let AQB (fig. 18) be a plane figure, and let it revolve through
an angle a about an axis OX in the same plane (AQB lying en-
tirely on one side of OX) to the position of a qb. Then PP', an ele-
ment of the figure's area, generates a ring of constant cross section
PP* and of volume ultimately = PP'. Pp = PF. PM . o = a . moment
of PF about OX. Taking all the elements of area of AQB in this
way, the volume generated by the surface AQB = a . moment of the
area AQB about OX = o . G-N . area AQB, G being the centre of
gravity of the figure AQB, and G-N perp. to OX.
Or if area of AQB = A, GN = d, and the volume of the solid
generated = V,
V= A. a, a,
D 2
3C GEOMETRY OF CYCLOIDS.
Prop. XXV.— 7/" from G {Jig. 17), the centre of
gravity of the cycloidal area APDB^ GH be drawn
perp. to BD and G h parallel to AD to meet BD in
hy then the volume generated by the revolution of the
area APDB about BD as axis is equal to six times
the volume of a cylinder having the generating circle
AQB for base and height equal to Hh.
The demonstration is in all respects as in Prop.
XXIV.
Cor. 1. The volume generated by the revolution
of APDB through one-third of two right angles about
It is easily seen that if the figure AQB is not plane, or if,
whether plane or not, it is not in the same plane as OX, a similar
Fig. 18.
property may be established. Let the figure AQB be carried
once round OX, and let a plane through OX intersect the surface
thus generated in a curve A'Q'B' (any parts of the plane figure
A'Q'B' through which more than one part of AQB may have passed
being counted twice or thrice, or so many times as they may have
been traversed in one circuit of AQB). Let A' be the area of
A'Q'B' (thus estimated), G' its centre of gravity (correspondingly
estimating the weight of its various parts), and d' the distance of
G' from OX. Then the volume generated by the revolution of AQB
round OX through the angle o = A', a!, a (any part of the volume
generated which is traversed more than once by the generating
curve being counted as often as it is so traversed).
THE RIGHT CYCLOID, 37
BD is equal to a cylinder having the circle AQB as
base and height = H A.
Cor. 2. Since GH=f AC (Prop. XVIIL), HA=
AC ; and the volume generated by the revolution of
APDB about AB=^.(AC)2.|7rAC=|7r« (AC)^
Cor. 3. Since the rectangle BE in revolving
around BD generates a cylinder whose volume =
BD.^ (AB)2=irAC.4^ (AC)2=4^2 (AC)S it follows
from Cor. 2* that the volume generated by APDE in
revolving around BD
= 4^^ ( AC)3 -|t2 {ACf = 1^2 (AC)^
Again, if AQB so move as to generate a cylindrical surface either
right or oblique, and two planes through OX intersect the surface
thus generated, the portion of the volume of this cylinder inter-
cepted between these planes may be thus obtained : —Through OX
take a plane perp. to the axis of the cylindrical surface, and inter-
secting that surface in a curve A'Q'B', enclosing a figure of area A',
and having its centre of gravity G' at a distance d' from OX ; let
the portion of a straight line through Gr' parallel to the axis of the
cylindrical surface intercepted between these bounding planes = h ;
then the volume intercepted = A'. &'. h.
The proof of this and the preceding extension of Guldinus's
second property will be found to require the two following lemmas :
First, if an element of area PP' be projected orthogonally on a
plane through OX and P into the elementary area P^', then PP'
and Vp' in rotating through any angle around OX generate equal
elementary solids. This is obvious, since they generate equal ele-
mentary solids in rotating through an elementary angle around OX.
Secondly, if two planes through OX cut a parallelopipedbn of ele-
mentary cross section in the parallelograms PF and pp'^ Fp and
P'p' being two opposite edges of the parallelopipedon, then two
other planes through OX near to these last, cutting Fp and F'p' in
K, R', and r, r', such that PR = ^ r, intercept equal elementary solids,
PRR'F SLndprr'p'i These solids are, in fact, ultimately parallelo-
pipedons on equal bases and between the same parallel planes.
88 GEOMETRY OF CYCLOIDS,
Prop. X.'XNl.—If from G' {fig. 17), the centre of
gravity of the cycloidal area APDE^ G'K he drawn
perp, to AE and G'k parallel to AD to meet AE in A,
then the volume generated by the revolution of the area
APDE about AE as axis is equal to twice the volume
oj a cylinder having the generating circle AQB for
base and height equal to Kk,
The demonstration is as in Prop. XXIV., except
that the area APDE = a third only of the area
APDB.
Cor. 1. The volume generated by the revolution
of APDE through two right angles about AE = a
cylinder having circle AQB as base, and height equal
toKA.
Cor. 2. Since G'K=iAC (Prop. XVI.), KA =
-AC ; and the volume generated by the revolution of
APDE about AE = ^r (AC/. ^AC='^'(AC)''.
Cor. 3. Since the volume generated by the revolu-
tion of rectangle BE around AE=47r2 (AC)* (see
Prop. XXV. Cor. 3), it follows from Cor. 2 that the
volume generated by APDB in revolving around AE
=4^«(AC)3-'^'(AC)3= 'i^(AC)^
THE RIGHT CYCLOID, 39
Pkop. XXVII.— ^/rom G' {fg. 17), the centre of
gravity of the cycloidal area APDE^ QJ he drawn
perp. to DE and G'j parallel to AD to meet DE
in js then the volume generated by the revolution of
the area APDE around DE as axis is equal to twice
the volume of a cylinder having the generating circle
AQB for bane and height equal to Jj.
The demonstration is as in Prop. XXIV., modified
as in Prop. XXVI.
Cor. 1. The volume generated by the revolution
of APDE through two right angles about AE = a
cylinder having circle AQB as base, and height equal
to Jy.
Cor. 2. Since G'J = ^^^— _^ AC (Prop. XVII.)
^ Off
= /'-—-) AC, J;= I -—-I AC ; and the volume
generated by the revolution of APDE around DE
=u(AC)« g!-|) AC= {^^-^D (AC)3.
Cor. 3. Since the volume generated by the revo-
lution of the rectangle BE around DE = 2^ (AC)^
(Prop. XXIV. Cor. 3), it follows from Cor. 2 that the
volume generated by APDB in revolving around DE
= 2,3 (AC)3 - Q'-«J) (AC)3= (y + ^3') (AC)'.
40 GEOMETRY OF CYCLOIDS,
SECTION II.
THE EPICYCLOID AND HYPOCYCLOID.
DEFINITIONS.
The Epicycloid is the curve (as D AD^jig. 19, Plate I.)
traced out by a point on the circumference of a circle
{as AQB) which rolls without sliding on a fixed
circle {as BDH) in the same plane ^ the rolling circle
touching the outside of the fixed circle.
The Hypocycloid is the curve {as l/ADy fig. 20, Plate
/.) traced out by a point on the circumference of a
circle {as AQB) which rolls without sliding on a
fixed circle {as BDB^) in the same plane ^ the rolling
circle touching the inside of the fixed circle.
What follows applies to both figures unless special reference is
made to one only, and in every demonstration in this section two
figures are given, one illustrating a property of the epicycloid, the
other illustrating the same property of the hypocycloid, but the de-
monstration applying equally to either figure, unless special refer-
ence is made to one only. The student will do well to read each
proof twice, using first one figure, then the other. For convenience
the word * cycloidal ' throughout this section is to be understood to
signify either epicycloidal or hypocycloidal according to the figure
followed.
[Note. — It will be shown in Prop. I. of the pre-
THE EPICYCLOID AND HYPOCYCLOID, 41
sent section that if two circles AQB and AQ'B%
touching at B, touch a fixed circle BDB' at the ex-
tremities of a diameter BOB', then the same curve is
traced out by the point A on the circle AQB rolling
in contact with the circle BDB', as by the point A on
the circle AQ'B' rolling in contact with the same circle
BDB". We may therefore, in what follows, limit our
attention to cases in which the centre lies outside
the rolling circle. According to the definitions given
above, the curve traced out by A, fig. 19, is an epi-
cycloid whether AQB or AQ'B' is the rolling circle.
It may be well to mention that it has hitherto been
customary to regard the curve traced out by A on
AQB, fig, 19, as an epicycloid^ and the same curve
traced out by A on AQ'B^as an external hypoci/cloid.
Instead of defining the hypocycloid as the curve ob-
tained when the rolling circle touches the outside of
the fixed circle, it has hitherto been usual to define it
as the curve obtained when either the convexity of
the rolling circle touches the concavity of the fixed
circle, or the concavity of the rolling circle touches the
convexity of the fixed circle. There is a manifest
want of symmetry in the resulting classification, see-
ing that while every epicycloid is thus regarded as an
external hypocycloid, no hypocycloid can be regarded
as an internal epicycloid. Moreover, an external hypo-
cycloid is in reality an anomaly, for the prefix ^ hypo '
used in relation to a closed figure like the fixed circle
implies interiorness.]
Let BDB' (radius F) be the fixed circle, AQB
42 GEOMETRY OF CYCLOIDS.
■
(radius R) the rolling circle. If the centre of the
latter circle move in the direction shown by the arrow,
it is manifest that at regular inter\'als the tracing
point will coincide with the circumference BDB', as
at D% D, &c. (E'/D' and EyD being the correspond-
ing positions of the rolling circle), while midway be-
tween two such coincidences the tracing point will be
at its greatest diametral distance from D^BD as at A
(AQB being the corresponding position of the rolling
circle), ACB the diameter through the tracing point
passing when produced through O, the diameter of the
fixed circle. It is clear also from the way in which
the curve is traced out that the parts AP'D and APD
are similar and equal. Wherefore AB is called the
axis of the cycloidal arc D'AD. The circular arc
D'BD is the hase^ A the vertex, and the points D'
and D are the cusps. It is convenient to call the
radius to the tracing point the tracing radius, and the
diameter through the tracing point the tracing diameter.
The tracing circle in the position AQB is called the
central generating circle ] and straight lines passing
through the centres of both the fixed and rolling circles
are said to be diametral. The arc Cc is called the arc
of centres, BXkd the circle of which it is part the circle
of centres.
Let a circle E'AE be described with centre O and
radius OA, and let OD' and OD (produced if neces-
sary) meet this circle in E and E^ ; then it is clear that
D'rf' and Drf, the parts of the cycloidal curve on either
side of D'E' and DE, are symmetrical with regard to
THE EPICYCLOID AND HYPOCYCLOID, 43
these lines respectively, which are therefore secondary
axes. Also E'AE touches the curve IK AD in A.
The complete curve, either of an epicycloid or of
a hypocycloid, consists of an infinite number of equal
cyoloidal arcs, but when the radii F and R are com-
mensurable in length, the curve is re-entering, and
may be described as consisting of a finite number of
arcs.* Thus if R = F the rolling circle will make one
complete circuit of the fixed circle between each suc-
cessive coincidence of the tracing point with the fixed
circle ; hence D and D' will coincide, and there will be
but one cusp. (No hjpocycloid can be traced with
these radii.) If R = ^F, each base as DD' will be
equal to half the circumference of the fixed circle, and
there will be but two cusps. Similarly if R = ^F, ^F,
|F, &c., there will be 3, 4, 5, &c., cusps, respectively.
In these cases the complete cycloidal arc will consist of
a number of equal arcs, standing on equal parts of one
circuit of the fixed circle's circumference. Again, if
wR = nF, where n and m are integers prime to each
other, then m circumferences of the smaller circle
will be equal to n circumferences of the larger. Con-
sequently there will be m cusps in the complete cy-
cloidal curve, and the base of each cycloidal arc will
be equal to one mth part of n circumferences of the
71
fixed circle, that is to the th part of the circumfer-
* Theoretically it consists in that case of an infinite number of
arcs, occupying a finite number of positions, and consequently eacli
arc coinciding with an infinite number of other arcs belonging to
the curve.
44
GEOMETRY OF CYCLOIDS.
ence of this circle. Wherefore if n > m, the base is
greater than the circumference of the fixed circle,
but \{ n<m the base is less than this circumference.
If m = unity, that is R = ^i F, then the base of each
cycloidal arc = n times the circumference of the fixed
circle.
PROPOSITIONS.
Prop. I. — If a circle q'Dq {figs. 21 and 22), having
Fig. 21.
radius Dcy roll in contact with a circle KDb, having
radius ODy c and O lying on the same side of Z>,
then the point D on q^ D q will trace out the same
curve as the point D ona circle Q^DQ having radiuM
D C equal to c O [measured in direction c O), rolling
in contact with the circle K D h.
Let b be the point in which the rolling circle (jfD q
THE EPICYCLOID AND HYPOCYCLOID,
45
touches KD i, when the tracing point is at P, c' being
the centre of y'D q {c\ O, and b lying in the same
straight line). Through O draw OC^ equal and
parallel to c' P, meeting KD ^ in B' ; and join PC^
Then PC'O e?' is a parallelogram ; PC'=e'0 = DC;
also, since OC' = cT = c'^»^ and OB' = 06, C'W = Oc'
= DC. Hence a circle equal to QDQ^ touching KDi
in B' (on the same side as QDQO, has its centre at C
Fio. 22.
and passes through P.
Moreover, since arc P 6 = arc D by
Z.Pc'i(=/ieO*=Z.PC'B0: LJ>0b::0b: c'b,
...ZPC^B': Z.DOB'::.Oc' : 0*::/f)C : OD.
Therefore arc PB'=arc DB', and P is a point on the
curve traced out by D on the circle QDQ' rolling in
contact with the circle KD b,
ScHOL. — It is manifest that when P arrives at the
vertex of the curve the rolling circles are placed (re-
latively to each other) as in figs. 19 and 20.
40 GEOMETRY OF CYCLOIDS.
Prop. II. — The base of the epicycloid or hypocycloid
is equal to the circumjerence of the generating circle.
This, as in the case of the cycloid, needs no demon-
stration.
Cor. 1. Arc D'B (figs. 19 and 20) = arc BD
= half the circumference of the generating circle.
Cor. 2. Arc C c : arc BD :: CO : BO.
Or for the epicycloid,
arc C c = ^ i^. arc BD = 1^ arc AQB,
xC xC
and for the hypocycloid,
arc C c = — "" — . arc BD = T" . arc AQB.
sx xC
Cor.3.AreaE'AEDBD'=2 AED'B
= 4 rect. under AC, C c *
= 4 ^^. rect under AC, BD
= ^^J^. circle AQB
for the epicycloid = 4 \ — ^ circle AQB
IX
for the hypocycloid =4 '.-""-^ circle AQB.
xi
* The relation here employed is almost self-evident. It may be
thiis demonstrated: Divide the area AEDB into a series of ele-
mentary areas by drawing radial lines from : each element is in
the limit a trapezium whose area = rectangle under AB and half
the sum of those elementary arcs of AE and BD which form (in
the limit) the parallel sides of the trapezium. Therefore the area
AEDB = rectangle under AB and half the sum of the arcs AE,BD
= rectangle under AB and the arc C c.
THE EPICYCLOID AA'D BYTOCYCLOID.
47
Prop. III. — If through P, a point on the epicychidal
or hypocycloidal arc APD(^jigs. 23 and 24), the arc
PMbe drawn concentric with the hate BD, cjttttng the
central generating circle in Q and meeting the axis
AB in M, then arc QP: arc AQ:: OM I OB.
Let A'PB' be the position of the generating circle
when the tracing point is at P; C its centre; A'C'B'O
diametral, cutting PM in M'. Draw the tracing dia-
Fio. 23. F:o. 24.
meter FC'6. Then it is mutifest that arc QM = arc
MT ; arc MM' = arc QP ; and arc AQ = arc A'P.
Now b ia the point which was at B when the tracing
point was at A ; and since every point of the arc b B'
has heen in rolling contact with BB', the arc bW= the
arc BB'. But arc b B'= arc A'P = arc AQ ; and
arcMM'(=arcQP) :arcBB'::OM : OB;
.-. arc QP : arc AQ:: OM : OB.
48 GEOMETRY OF CYCLOIDS.
Cor. 1. Arc MP = rr^ . arc AQ+arc MQ.
Cor. 2.
Let arc MQ prod, meet OE(drawn as in figs. 19,20)in m
.U TIT OM ^^ OM ^^„
then arc M m = -_ - arc BD = Qg- . arc AQB.
[But arc BQ > QK perp. to AB; . •. -^ - . arc BQ >
ML, perp. to AB and meeting OQ produced in L (for
OM : OB > OM : OK). But ML > arc MQ. A
fortiori, then, ^^^ . arc BQ>arc MQ.] *
^^ OM ,^ OM
.•. smce arc Mm = (yn^ arc AQ + q^t arc BQ,
while arc MP = -fz^ • arc AQ + arc MQ,
arc M m > arc MP, and P falls between OA and OE ;
that is, the whole arc APD lies between OA and OE.
Cor. 3. The arc P m = arc M tw — arc MP
OM .^ OM .^ ^^
= Qg- arc AQ — vyjv arc AQ — arc MQ
= Qjs arc BQ — arc MQ.
Cor. 4. If through P', a point near P, arc V^p Q'
be drawn concentric with the base BD, meeting
AQB in Q' and cutting A'PB^ in y, then in the limit
* The part in [] fails for hjrpocycloid. Substitute the follow-
ing : — Let OQ pr iduced meet arc BD in H, draw BF perp. to OH and
describe J BFO. Then, arc BH = arc BF (of half rad. and double
Z at centre) ; but arc BF < arc BQ, •/ chd. BF < chd. BQ (BFQ
being a rt. angle) while seg. BF contains a larger angle than seg.
BQ'Q. Hence arc BQ > arc BH > ~ . arc MQ ; i.e. ^ . arc BQ
> arc MQ.
THE EPICYCLOID AND HYPOCYCLOID. 49
(when P' is very near to P), arc P'Q' = T^iT- arc A!q ;
and arc PQ = rv^* arc A'P ; therefore,
arcP'Q'-arcPQ (=yF)= ^^ (arc A'y-arc AT)
""TTr ^^ 5' ' ^^* ^^ limit,
yF : arcPy::OM : OB.
Prop. IV.— ^, B, C, D, E, Sfc. {figs. 25 and 26, p. 51)
representing the same points as in the preceding propo-
sition, the area APDBQ = half the area ABDE ;
or area APDB Q : generating circle : ; OC : OB,
Take CL=CL',on AB ; and LK, L'K' equal ele-
ments of AB, both towards C. Draw LQ, Ky, K^y',
and L'Q' at right angles to AB to meet AQB ; and
about O as centre describe arcs QP, yjo, q'p\ and
QT, meeting APD. Let O y, produced if necessary,
meet QP in n; draw Q A perpendicular to Ky;join
C y, and draw C m perpendicular to O y, produced if
necessary. Then ultimately the triangles Q A y and
y KC are similar, as are the triangles Q y tz and y C m
(for Q y C being ultimately a right angle, Q y w is ulti-
mately the complement of C y wi and therefore equal
to y C m). Hence the quadrilateral Q n y A is similar
to the quadrilat^al ym C A, and
qn : QA(=LK)::Cwi : Ky ::C0 : yO
(triangles CO m and y OK being similar). Hence
E
50 GEOMETRY OF CYCLOIDS.
Area QPjoy (ult.=rect. n y, QP) : rect. LK, QP
::C0 :yO;
but, rect. LK, QP : rect. LK, Ay :: QP : Ay
::yO : BO (Prop. 11.) ;
.•• ex (Bq. area QPjoy : rect. LK, Ay :: CO : BO
::Cc : BD;
similarly, area QT>Y : rect. L'K% A / (or LK, By)
::Cc : BD;
. • . QP/? y + Q'Py y' : rect. LK, Ay + By (or LK, BD)
::Cc : BD;
wherefore QP jo y + QT> Y = rect. LK, C c.
.*. summing all such elements between AE and BD,
Area APDBQ=rect under AC, C c=i area ABDE.
or, area APDBQ : gen. :: OC : OB.
Cor. 1. Since, for epicycloid, C c = — pr-« AQB,
* Tfc-r-w-r^^ F + Iv 4^ A y-^T^ F + R
area APDBQ = — |r- . AC . AQB= -^— . gen.
and the area between epicycloidal arc and base
/ F+R \ ^ 3F+2R
= (2 . — F"" + 1 j g«"' ® = F sen. .
"P T>
For the hypocycloid, area APDBQ = — — — . gen. Q ;
and the area between hypocycloidal arc and base
3F-2R
= Y • ^^^* ® •
Cor. 2. If AB is the axis of a cycloid (A the ver-
tex) and LQ produced meet this cycloid in E., then
Area AQP : area AQR :: OC : 0B.»
* This relation, which follows directly from the proportion on
the fifth line of this page, might have been employed to establish
the main proposition. I preferred, however, to give an independent
proof.
THE EPICYCLOID 4ND HYPOCYCLOID, 61
Cor. 3. Epicyc. area APDE = APDBQ - AQB
/F + R \ F + 2R
= ^HF" "" V ^^°' ® ~ "TF" sen. .
F— 2 R
Hypocycloidal area APDE = — ^~p" gen. .
Fig. 25.
Fig. 26.
Cor. 4. Area AQP + area BQT'D=rect. AL, C c ;
«and, area QQ'PT = rect. under LC, C c.
Prop. V. — If P is a point on the epicycloidai or hy^
pocydoidal arc APD (Jigs. 27 and 28) A'PB' the
generating circle when the tracing point is at P,
A CH diametraly then PB' is the normal and A P
is the tangent at the point P,
Since, when the tracing point is at P, the generat-
ing circle A'PB' is turning round the point B', the
direction of the motion of the tracing point at P must
53 OEOMBTSY OF CYCLOIDS.
be at right angles to PB' ;— wherefore PB' is the
normal and AP is the tangent at P.
Another Demonstration. (See p. 8.)
Take P' a point near to P and draw PQM, P'Q'
concentric with BD ; PQM meeting AB in M and
cutting AQB in Q ; and P'Q'N cutting AQB and
Fia. 27. (Join PC, AQ.) Fig. 2S. (Join AQ.)
A'PB' in Q' and q. Join PC, PO, and let C'e pa-
rallel to PO meet PB' (produced in case of epicycloid)
in s. Then (Prop. HI. Cor. 4)
arc yF :arc Py ::P0 : B'O;: C* : C'B' (=CP);
or the BideE about the angles P ? P', PC< are propor-'
tional ; but these angles are ultimately equal, for P j
is ultimately perp. to C'P, and P'y to PO, that is to C».
Therefore the triangles PqV and PC's are ultimately
similar ; and the third side PP' of one is perp, to the
THE EPICYCLOID AND HYPOCYCLOID. 53
third side P « of the other. That is PB' is the normal
at P, and therefore PA' perp. to PB' is the tangent at P.
Cor. 1. If PB' intersect Q'F in /, and s C pro-
duced meet PA' in A, the triangle PP7 is ultimately
similar to the triangle sVk.
Cor. 2. If B'y be joined and produced to meet PP'
in riy then y w is ultimately perp. to PP' ; wherefore if
C'N be drawn perp. to B'P, the figure V^q P'w / is ulti-
mately similar to the figure PC'^ N k ; whence
PF :P«::P* : PN.
ScHOL. — As in Schol. p. 9 (obviously modified), a
tangent may be drawn to APD from any point on
APD or AA'E.
Prop. VL — With the same construction as in Prop. V,,
Arc AP : chord AQ::2C0 : BO.
Since qn ia ultimately perpendicular to PP', P n
is ultimately equal to the excess of chord A'y over
chord A'P. Now from Cor. 2, Prop. V.,
PF : Pn :: ^P : NP :: 2*P : B'P
:: 2 CO : BO:: 2 CO : BO,
or, inct. of AP : inct of ch. AT (or AQ) :: 2 CO : BO.
But arc AP and chord AQ begin together, wherefore
Arc AP : chord AQ :: 2 CO : BO.
Cor. 1. Arc APD : AB :: 2 CO : BO.
Cor. 2. For the epicycloid,
2(F-fR) _4R(F + R)
Arc APD = AB .
F F
64 GEOMETRY OF CYCLOIDS.
For the hypocycloid.
Arc APD = AB . « ZjD = tB(F=K).
Cor. 3. PF : Pn::2C0 : BO.
Cor. 4. PP' : 71 F :: 2 CO : 2 CO-BO
::2C0 : AO.
Cor. 5. Pw : 72F::BO : AO.
Prop. VII. — Prob. To divide the arc of an epicycloid
or a hypocycloid into parts which shall be in any
given ratio to each other.
Let a straight line a h (figs. 27 and 28) be divided
into any parts in the points c and rf : it is required to
divide the arc APD in the same ratio.
Divide AB in L and K, so that
AL : LK : KB::ac : cd : db;
with centre A and radii AL and AK, describe circular
arcs LQ, K r, cutting the semicircle AQB in Q and r ;
through which points draw the arcs QP, r P, concen-
tric with BD. Then
Arc AP : chord AQ (= AL) :: 2C0 : BO.
Similarly Arc AR : AK :: 2 CO : BO ;
Therefore Arc PR : LK :: 2 CO : BO.
Similarly Arc RD : KB :: 2 CO : BO,
therefore
Arc AP : arc PR : arc RD :: AL : LK : KB
:: ac : cd : db ;
THE EPICYCLOID AND HYPOCYCLOID, 65
or, the arc APD is divided into the points P and R in
the required manner.
Similarly may the arc APD be divided into four,
five, or any number of parts, bearing to each other any
given ratios.
Prop. VIII. — With same construction as in Prop. V,,
Area ABRP{figs. 27 and 28) : sectorial area AB'P
:: area RPD : segm. PFR :: 2 CO ^ BO: BO.
Let b be the point of contact of tracing and fixed
circles, when tracing point is at P' ; join b P', BQ, and
BQ' ; and draw b i perpendicular to P s. Then triangle
b B'l is similar to BX'N, therefore to PC'N, and
therefore (Prop. V., Cor. 2) ioV qn\ and B' ^ = P 5' :
therefore V qn and b Wi are equal in all respects ; and
P w = i2. Now elementary area PP'^B" is ultimately
equal to trapezium Fib P,
=half rect. under P i and (PP' + b i)
= half rect. under PB' and (PP' + P w) ultimately
and elementary area QBQ' is ultimately equal to tri-
angle PB' q
= half rect. under PB' and P w, ultimately.
.-. area PF b B' : area QBQ' ::FF' + Fn: Fn
:: 2 CO + BO : BO (Cor. 3, Prop. VI.).
Thus the increment of area ABB^P, or the decrement
of area B'PD, bears to the increment of area A'B'P,
or the decrement of area PFB', the constant ratio
W GEOMETRY OF CYCLOIDS.
(2 CO + BO) : BO. But the areas ABBT and BTD
commeDce together, and the areas A'B'P, PFB' end
together, as F passes from A to D ; hence
Area ABBT : sectorial area A'BT
:: area BTD : segment PFB' :: 2 CO + BO : BO.
Cor. 1. V n=: bi\
and PF : bi :: PF : P;t :: 2 CO : BO.
Cor. 2. Area BTPD : seg. BTP :: 2 CO : BO.
This can be proved independently, in the same maimer
as the corresponding relation for the cycloid, Cor. 1,
Prop. VIII., Cycloid.^
ScHOL. — The above affords a new demonstration
of the property proved in Prop. IV. Cor. 2 also, if
independently established, gives another proof of the
area.
* The proof may be effected in two ways, both analogous to the
proof for cycloid, — viz., either by making the sides of elements
such as/y* and FF' concentric with BD,or by making them perpen-
dicular to A'B'. In the f oi mer c:ise we find the decrement of space
PFB'D = Fq B' J, that is (ultimately) = F w Wh and the rest of the
proof is like the above. In the latter case we find the decrement of
PFB'D = arect under C'c' (o' centre of JF'F) and projection of
B'q on A'B' ; and decrement PFB' = triangle VB'h = } rect. under
B'& and projection of B'q on A'B' ; therefore
decrement of PFB'D : decrement of PFB' : : 2Q' o' : B' h ;
that is, area PFB'D : area PFB' : : 2C'c' : B' J : : 2C0 : BO.
THE EPICYCLOID AND BYPOCYCLOID. 67
Prop. 1X.~I/ F (Jigs. 28 and 29) be a point on the
epicycloidal or hypocydoidal arc APD, and OP, OA,
OD be joined, and PM be drawn perp. to A'R, the dia-
metral of the generating circle A'PB" through P, then
AreaAPO: rect. OC (arc A'P+PM) :: OA : 2 BO.
The area APO = sector QBE' + AOB'P ± area
ABB'P (taking the upper sign for the epicycloid, and
the lower aign for the hypocycloid, throughout) ;
therefore,
2 area APO = OB . arc BB' + OB' . PM
= OB arc A'F + OB . PM
- bo -(ACarcA-P + ACPM);
= (OB ± AC) «rc A'P + (OB ± AC) PM
58 GEOMETRY OF CYCLOIDS.
^ 2 CO. AC ,,.^^ 2 CO. AC ^^
± — gQ — arc AT ± — g^ . PM ;
= (CO±?^g^)arcA'P
±(C0±^g,^)PM;
^^ /BO ± 2 AC>\ , . ,^ T., .,
= CO (^ jgQ j (arc AT -!- PM) ;
= ^ . CO . (arc AT + PM) ; therefore,
area APO : rect. OC (arc AT + PM) :: AO : 2 BO.
Cor. 1. Area APDO : cect. OC, BD :: AO : 2 BO.
Cor. 2. Area DPO : rect. OC (arc BT - PM)
:: AO: 2 BO.
Cor. 3. APDO : sect. OBD :: AO . CO : {B0)\
Cor. 4. APDO : sect. OC c (figs. 25 and 26)
:: sect. OA a : APDO :: AO : CO.
Note. — The above demonstration might have been readily made
geometrical in form as it is in substance ; but it would have been
more cumbrous and not so easily followed. The student should,
however, note the following independent demonstration (which
occurred to me after the above had been corrected for press) : —
In figs. 27, 28, p. 62, let OP intersect Vlinh\ draw PH perp.
to s k and PM' perp. to A'B'. Then the ultimate increment of area
APO = i rect. OP, h V ; while the corresponding increment of rect.
OC (arc A'P + PM') = rect. OC, inct. of (arc A'P + PM'). Therefore,
former inct. : latter inct. : ; ^ OP . A F : OC, inct. of (arc A'P + PM').
Now, AF:P^::«H: CP
and P q : inct. (arc A'P + PM') : : CP : B'M'
.'.ex ieq.y hV : inct. (arc A'P + PM') : : « H : B'M'
But OP: OB' ::«c':C'B'
OP.AF: OBMnct. (arcA'P + PM')::«H.«C': B'M'.C'B'
: :« P . « N : B'P . B'N (since C, F, H, P, N, lie on a 0).
THE EPICYCLOID AND HYPOCYCLOID, 69
Wherefore, increasing OB' in 2nd term to OC, and B'P in 4th to « P
(or both in the same ratio, since triangles 8 B'C, PB'O are similar),
OP . AF : OC . inct. (arc AT + PM') : :«P . «N : *P . B'N
::*N: b'n :: C'B'+iB'O : ^B'O
::A0: BO;
or, inct. area APO : inct. rect. OC (arc A'P + PM') : : AO : 2 BO
Area APO : rect. OC (arc AT + PM') : : AO : 2 BO.
Cors. 1, 2, 3, and 4, follow as before.
SCHOL. — We have here an independent demonstration of the
area of the epicycloid and hypocycloid, since
Area APDO = area^BD ± area APDB.
Prop. X. — With the same construction as in former
Propositions {Jiffs. 31 and 32),
Area APA : segment A hP w AO : BO,
Let a P^B be the position of the tracing circle when
tracing point is at P' near to P ; acbO diametral.
Draw q P' concentric with BD and AE, join A' q, a P^
A'P'; also producing A'a to T and 5'P to ^, draw P'T
and A' t perp. to A'T and P t respectively.
Then A'PP'a, the increment of AA'P = ^ rect.
under A'a, P'T ultimately ; and A'P q, the increment
of segment A'AP=^rect. under V q, A7 ultimately.
But ultimately the right-angled triangles A^tq and
P'T a are equal in all respects (since A^q = a P', and
angle A'y t = angle at circumference on segment A'y
= angle at circumference on segment a P' = angle
V'a t) therefore F't = A'T, and
increment of AA'P : increment of segment A'A P
:: A'a: Pj(=B'i) :: AO:BO;
80 GEOMETRY OF CYCLOIDS.
or since these axta» begin together,
area AA'P : segmeut A'AP :; AO : BO.
Cot. Area AA'A P : seg. A'AP :: 2 CO : BO
( :: AO + BO : BO). This may readily be established
.independeDtly — by showing that ultimately
area A'aA'FPA : AA'Py :: 2 C'c: B'i:: 2 CO : BO.'
ScHOL. — Since it follows that
area APDE : \ gen. © :; AO : BO,
* A line from K, peip. to A'B', to meet X'kS= Co; ftod a line
Crom F, perp. to A'B', to meet A'9 = P j — B'i.
THE EPICYCLOID AND HYPOCYCLOID. 61
we have here another demonstration of the area of
APDE. Further, since
\ gen. © : area ABDE :: i CB, BD : 2 . CB, arc C c
:: BD:4Cc' :: BO : 4 CO,
it follows, ex (Bqualiy that
area APDE : area ABDE :: AO : 4 CO.
Yet again, from the corollary we see that
Area APDQ'E : i generating circle :: 2 CO : BO
:: \ area ABDE : ^ generating circle,
.-. area APDQ'E = \ area ABDE,
which is the relation established in Prop. IV. If
established independently, as explained above, this
leads to another demonstration of the area.
Note.— Arc APD divides the area AQBDQ'E
into two equal areas.
Prop. XL— 7f PRo {figs. 33, 34) is the radius of
cvrvature at -P, and PB' the normal^ then
Po.PBf :: 2 CO : AO.
With so much of the construction of figs. 27, 28
as is shown in figs. 33, 34, produce P^ b to meet PB'
produced in o', then o is the limiting position of o' as P'
moves up to P. Now since PP' is ultimately parallel
to b I, therefore ultimately
P o' : B'o' :: PP' : b i :: 2 CO : BO
(Prop. VIII., Cor. 1), wherefore
P ^ : PB^ :: 2 CO :: 2 CO - BO : : 2 CO : CO + AC,
or ultimately ' P o : PB' :: 2 CO : AO.
02
GEOMETRY OF CYCLOIDS.
Cor. 1. For the epicycloid,
2 (F + R)
radius of curvature = y x ovt • i^ormal ;
and for the hypocycloid,
2 (F — R)
radius of curvature = -y .^o "R * ^^ormal.
Fm. 33.
Fig. 34.
Cor. 2. VW.Wo :: 2 CO - BO: BO :: AO : BO
:. F + 2 R : F for the epicycloid;
: : F — 2 R : F for the hy pocycloid.
SciiOL. — We see from Cor. 1 that when F = 2 R
the radius of curvature of the hypocycloid is infinite,
or the hypocycloid degenerates into a straight line.
See further the Appendix to this section, pp. 66 to 68.
THE EPICYCLOID AND HYPOCYCLOID. 63
Prop. XII. — The evolute of the epicycloid or hypocy-
chid APD (^figs. 33 and 34) is a similar epicycloid or
hypocycloidy doDy having its vertex at D^ and its
cusp d so placed on OA (produced if necessary), that
dB: BA :: OB i OA;
or, which is the same thing , O d : OB :: OB : OA.
Join OD and describe the arc da'e with O as
centre and radius O d. Produce A!Bf to O, cutting (fig.
33) or meeting (fig. 34) d e in a', and join o a\ Then
B'A' : a:W :: BA : dB :: AO : BO :: PB^ : Wo
(Prop. XL, Cor. 2);
that is, the sides about the equal angles o H'a'y PB' A'
are proportionals ; therefore the triangles o'Wa\
PB' A'' are similar, and the angle a'o B' ( = the angle
B'PA') is a right angle. Hence a circle described
on Wa' as diameter will pass through o. Again the
angles A'B'P and a^JVo at the circumferences of the
circles A'B^P and a^Wo being equal,
arc a' : arc PA ( =arc BB^ :: a'W : WA' :: OA : OB
:: OB : O d :: SiTQ d a^ : arc BB^
Therefore, arc o a' = arc da\ and o is a point on an
epicycloid (fig. 33) or hypocycloid (fig. 34) having de
for base, its cusp at d and Woa^ as tracing circle.
Since rfe: BT>::od: OB::Brf: AB
;: arc B'o a' : arc ATB' (= BD) ;
therefore de = arc Wo a! ;
so that g D is the axis and D the vertex of the epi-
cycloid or hypocycloid rfo D.
64 GEOMETRY OF CYCLOIDS.
Cor. If c is the bisection of e D,
oV : oW :: 2C0 : AO :: 2cO : DO;
therefore (Prop. VI.),
o P = arc o D.
If^ then^ a string coinciding with the arc doT) and
fastened at rf, be unwrapped from this arc, its extremity-
will always lie on the arc APB, which may thus be
traced out as the involute of the arc do J).
ScHOL. — A convenient construction for finding the
base^ &c.> of the evolute doTUvA indicated by the dotted
lines in the figures : thus, join AD, then B e parallel
to AD gives e (on OE, produced if necessary), the
radius of the base e d.
Prop. XIII.— If do D (Jigs. 33, 34) be tlie evolute of
the epicycloid or hypocycloid APD^ and o B^Py the
radius of curvature at any point P on APDy cut the
base BD in Bf ^ then
area APRB : area d BBo
:: rect. under AO {AO + 2 BO) : square on BO.
If F'o' be a contiguous radius of curvature cutting
BD in b, and b i is drawn perp. to o B'P, then in the
limit
oP: oi ::2C0: BO;
therefore
ult. area PoP' : ult. areaot^ :: 4(00)^ : (BO)S
whence, ultimately
area PB'bF : area oB'b :: 4(C0)^ - (B0)« ; (B0)«
:: rect. (2C0 - BO) (2 CO + BO) : sq. on BO
:: rect. AO (AO + 2B0) : sq. on BO.
THE EPICYCLOID AND HYPOCYCLOID. G6
But the areas VWb V and o Wb are the elementary
increments of the areas APB^B and rfBB'c;, which
begin together. Therefore,
area APB'B : area d BB'o
:: rect under OA (AO + 2B0) : sq. on BO.
Cor. 1 . Area APDB : area d o DB
:: area PB'D : area oB'D
:: rect. under OA (AO + 2 BO) : sq. on BO.
Cor. 2. Since
area rfoDB : area APDE :: (BO)^ : (AO)S
it follows {ex (Bq,) that
area APDB : area APDE :: AO (A0 + 2B0) : (A0)«
:: AO 4- 2B0 : AO
:: (3P + 2R) : (F + 2R) for the epicycloid
:: (3F - 2R) : (F - 2R) for the hypocycloid.
SCHOL. — It follows from Cor. 2 that
Area APDE : area ABDE :: AO : 2(A0 + BO)
:: AO : 4C0,
which is one of the relations established in the scho-
lium on Prop. X. Hence we have in Prop. XIII.
another method of demonstrating the area of the epi-
cycloid and the hypocycloid.
66 GEOMETRY OF CYCLOIDS.
Appendix to Second Section.
There are many forms, both of the epicycloid and
of the hypocycloid, which possess interesting proper-
ties. For the most part the general properties esta-
blished in the preceding section will suffice to enable the
student to deduce the properties of special forms of
these curves. For this reason, and also because of the
requirements of space, I shall only touch briefly here
on a few points in connection with the forms assumed
by epicycloids and hypocycloids for certain values ot
the radii of the fixed and rolling circles. I do not
make set propositions of these points, but present them
in such sequence as appears most convenient and suit-
able.
THE STEAIGHT HYPOCYCLOID.
The hypocycloid becomes a straight line when the
diameter of the rolling circle is equal to the radius of
the fixed circle.
This in reality has been already demonstrated, be-
cause we have seen in the scholium to Prop. XI. that
the radius of curvature of the hypocycloid becomes
infinite when F = 2R. Also the relation is involved
in the demonstration of Prop. I. For when the two roll-
ing circles (figs. 2 1 and 22) are equal, each having its dia-
meter equal to the radius of the fixed circle, the curve
THE EPICYCLOID AND HYPOCYCLOID.
67
traced out by each must be a straight Hoe. Thus, —
let BOB' (fig. 35) be the diameter of the fixed circle,
and its halves BO, OB', the diameters of the two equal
rolling circles ; then by what is shown in Prop. I. of
this section the point O on BQO will trace oat the
same curve as the point O on B'Q'O, but since the
circles BQO and B'Q'O are equal, this curve, i
as traced out by O on BQO, must bear the same re-
lation in all respects to the axis OB that the same
curve regarded as traced out by O on B'Q'O bears to
the axis OB', and the only line which can possibly
fulfil this condition is the diameter I^OD at right
angles to BOB'. This then must be the path traced
out by the point O in each case.
Let us proceed, however, to an independent de
monstration.
68 GEOMETRY OF CYCLOIDS.
When the circle OQB has rolled to the position
Op ^ (O c J its diameter), let p be the point which had
been at B, so that drawing the diameter /> c P^ P is the
position of the tracing point. Then the arc pb\& equal
to the arc B b ; and therefore, since F = 2R, the angle
B0£ is equal to half the angle 6cj9, that is to the
angle bVp : but BOjo and O ^ P are alternate angles ;
wherefore &P is parallel to BO ; and OP, which (OPi
being a semicircle) is perpendicular to i P, is perpen-
dicular to BO. P therefore lies on the diameter D'OD
at right angles to BOB'; which was to be shown.
Cor. The point p lies on OB (the angles cOp and
c OB being each equal to half the angle b cp),
USEFUL GENERAL PROPOSITION.
The following property is worth noticing. It is
true of course for the cycloid also.
A diameter of the generating circle of an epicycloid
or hypocycloid constantly touches the epicycloid or
hypocycloid which would be generated by a circle of
half the diameter, alternate cusps of this epicycloid or
hypocycloid falling on successive cusps of the former.
It will suffice to demonstrate the property for the
epicycloid.
Let AQB (fig. 36) be the generating circle of an
epicycloid when the tracing point is at A, the vertex
of the epicycloid. When the circle has rolled to posi-
tion aV byXeX, pcV be the position of the diameter
which had originally been in position ACB. Draw
b P' perpendicular to p P, and on c & describe the semi-
circle cVb^ having (/ as its centre and passing through
THE EPICYCLOID AND HYPOCYCLOID, 69
P' because c Vb is a right angle. Then because the
angle PVi = twice the angle P^c i, and c'b = half
c by the arc P' i = arc pb = arc B b. Wherefore P'
is a point on an epicycloid traced out by the rolling of
c Fi on BD, B being a cusp. D is the next cusp, be-
cause the base of the smaller epicyloid being equal to
the circumference of generating circle c P'ft = circum-
ference of semicircle AQB = BD. Also p P'c P is the
tangent at P^ by what has been already shown respect-
ing the tangent to an epicycloidal arc.
The student will find it a useful exercise to prove
the property established in Prop. I. of the present
Fig. 36. (Draw in epicycloid on base BD, toaching cp in Y.)
A
section in thei manner illustrated by figs. 37 and 38,
where APB is the arc traced out by point A on each
of the circles AQB, AQ'B'. The construction and
proof for the epicycloid (fig. 37) run as follows: —
ABOB' being a common diameter of all three circles
at the beginning of the rolling motion, let P be the
position of the tracing point of the smaller rolling
circle when its centre is at c. Draw the diametral line
aeb O/, and the diameter P cp. Join P b and pro-
70
GEOMETRY OF CYCLOIDS.
duce to meet the circle BDB' in b', produce b'O to c',
taking O c' = R, so that 6V = F + R, aod join P <<;
then since
AP : bb' :: ab i bf:: B : F :: Oc" : O J'
P c' is parallel to O b, and the triangle A'c'P, like tri-
angle b'Ob, is isosceles (c'b' = (/P), With centre </
and radius cT or t/b" ( = F + R) describe the circle
6'P a' ; produce P c' to meet this circle In j/'. Now,
arc 6 & = arc b p ;
,-. angle /)ci : angle BO* :: F : R;
but angle pcb = 2 angle c S P = 2 angle Obb'
= angle i'Oy
. ■. angle b'Of (= angle 6V/) : angle BO i :: F : R ;
and Z. AV;j' : z b'OB' :: F : F + R :: B'O : 6V,
THE EPICYCLOID AND HYPOCYCLOIB,
71
Whence It follows that arc b'p' = arc b'W ; and P is,
therefore, a point on the curve traced out by A (on
the circle AQ'B''), rolling so that its inside touches the
outside of the fixed circle BDB', ABOB' being ori-
ginally diametral. The same curve APB is traced
out, then, by the point A on each of the circles AQB
and AQ'B^
Fio. 38.
Cor. If we produce VO to meet the circle ft'P a' in
«', and join P a', then a P and P c^ are in the same
straight line.
The construction and proof for the hypocycloid
(fig. 38) are similar, writing only — R for + R.
The curve enveloped by a diameter of the gene-
rating circle of an epicycloid produced by the rolling
72 GEOMETRY OF CYCLOIDS.
of a circle larger than the fixed circle, and touching
this circle internally, will be an epicycloid if the radius
of the rolling circle exceeds the diameter of the fixed
circle ; but if the rolling circle has a radius less than
the diameter of the fixed circle, the curve enveloped
by a diameter of the rolling circle will be a hypocycloid.
The proof for both cases is easily derived from the
demonstration in pp. 68, 69, the dotted line and circle
of fig. 37 showing the nature of the construction.
The curve enveloped by a diameter of the gene-
rating circle of a hypocycloid is shown by reasoning
similar to that in pp. 68, 69, to be the hypocycloid
traced out by a generating circle of half the diameter,
alternate cusps of the smaller hypocycloid agreeing with
successive cusps of the larger. The dotted line and
circle in fig. 38 indicate the requisite construction when
the rolling circle has a diameter greater than F.
THE FOUR-POINTED HYPOCYCLOID.
It follows from the property indicated in the preced-
ing paragraph that the diameter OB of the rolling circle
BQO (fig. 35) constantly touches a hypocycloid having
four cusps, at B, D, B', and D'. As the extremities
p and P of the diameter lie always on BB^ and IKD
respectively, we have in this result the solution of the
problem ^ to determine the envelope of a finite straight
line pc^y whose extremities slide along the fixed straight
lines BOB^ and DOD' at right angles to each other. '*
The direct proof is simple, however. Thus let j9 P be
THE EPICYCLOID AND HYPOCYCLOID. 73
the straight line in any position. Complete the rect-
angle O/? 6 P, whose diagonals O h and p P are equal
and bisect each other in c. With centre O and radius
O i, describe the circle B JDB^, and draw b P' perpen-
dicular to /^P. Then a circle on e&, as diameter, passes
through P'. Let c' be the centre of this circle ; then
(fb^^Obi but Z. 6c'P' = 2 L ftcP' = 4z. *0B;
therefore arc iP' = arc/?B. Hence P'' is a point on
the hypocycloid traced out by circle ft P' e rolling on
the inside of the circle BDB', the cusps lying at B,
D, B', and D'.»
THE CARDIOID.
The cardioid^ or epicycloid traced by a point on
the circumference of a circle rolling on an equal circle,
has some interesting properties. Here, however, space
cannot be found for more than a few words about the
chief characteristics which distinguish this curve.
Let AQB (fig. 39) be the rolling circle, B J S the
fixed circle, A the tracing point when at the vertex,
so that ACBOS is diametral. Now let aPft be another
* The four-pointed hypocycloid BDB'D' is interesting in many
respects. It bears the same relation to the evolute of the ellipse
that the circle bears to the ellipse. Its equation may readily be
obtained. Thus, let DOD' be axis of a?, 30B' axis of y, and a-, y
co-ordinates of F ; put BO J = 6 ; OB = a ; then,
a? = ^ F sin B^ph sin * 6 = a sin ■ 6 ;
y = PFcose = Jto cos* = acos»6; p
, , «/ +y5_^2^ the required equation. y j
/a?2 y* \
The equation to the evolute of the ellipse ( -^ + v^ = 1 1 is
(f )l + mi = 1 ; ^here a! = a- *', and J' = f' - 6.
b
74 GEOMETRY OF CYCLOIDS.
position of the rolling circle, acbO diametral. Draw
the common tangent b m, meeting ABS in m ; draw
also m/>c P through c, the centre of circleaPft; join
PS, cutting mb in r ; bk perpendicular to AS ; and
join 6 P, J S. Then, since cb = bO, and i m is perp.
to c O, triangle cbm=^ triangle Obm in all respects ;
and arc bp = arc b B, Wherefore, P is the position
of the tracing point ; P a is the tangent to the cardioid
Fio. 39. (Produce Pi to meet 04BS in jr;joi'»P*.*'P'-)
at P, P 6 is the normal. The curve will manifestly
have the shape indicated in the figure, the only cusp
being at S, and the tracing point retumiug to A after
tracing the other half SP'A. AS divides the curve
symmetrically.
Note first that P n = n S ; or the cardioid is similar
to the curve ohtained by drawing perpendiculars from
THE EPICYCLOID AND HYPOCYCLOID, 75
S (as S n) to tangents at all points of a circle B J S.
We might then obtain the cardioid P'RAPS, by draw-
ing a circle on AS as diameter, and from S letting
perpendiculars fall on tangents to this circle. This
property is expressed by saying that the pedal of a
circle with respect to a fixed point on its circumference
is a cardioid.
Secondly, L nVb — alt. L PJcz: L bV m ^
/ i S m ; hence S n= S A. So that if we draw any line
S n from S, and from i, in which the bisector of BS n
meets the circle on SB as diameter, draw bn per-
pendicular to S n, the locus of /i is a cardioid. [The
larger cardioid, P^APS, would be similarly described
by producing Sw and Si, and from point in which
S6 meets circle on AS as diameter, letting fall perpen-
dicular on Sti (meeting Sn in P).]
Or, thirdly, we may obtain a cardioid by taking
any finite line as SB, drawing B b square to bisector
of any angle BS n, and from b drawing b n square to
S n : the locus of n will be a cardioid.
Fourthly, draw circle OGD about S as centre cut-
ting S 72 in ^, and draw e I perp. to SB ; then S n =
S A = SO + O A = SD + S Z (because S e is parallel and
equal to O i) = D /. Thus the cardioid may be obtained
by drawing radii as S ^ to a fixed circle OGD, and on
8 e, produced if necessary, taking n so that S w = D /.
This is the usual definition of the cardioid.
Fifthly, let P w S cut circle B * S in/. Then pro-
ducing Pi to meet circle J BS in ^, we have i P = ft^,
and rectangle P i . P^ (= 2P i^) = rectangle P/, PS
76 GEOMETRY OF CYCLOIDS.
=:2rect. P/. Pn. Hence Pi« = Pn . P/,andP A/
is a right angle. Wherefore pbf is a straight line,
and (P b bisecting angle p P/) P/= P/? = SB. Hence
the cardioid P'APS may be obtained by drawing
straight lines as Sf to circumference of circle By S,
and taking on S/ produced P/= BS. (The cardioid
is therefore a limagon.)*
Cor. If we draw s'Ss tangent to circle By*S at S,
and take S 5 = S/ = BS, then «, / are points on the car-
dioid. We see that s^s = S A ; and it is easily seen that if
P'SP is a straight line through S, PP' = SA. For,
according to the definition just obtained, we should
have P'' on a point on the curve if /SP' = BS =/P ;
therefore P'SP = SA. It may be well, however, to
show how this can be directly proved when the cardioid
is regarded as an epicycloid. For this purpose we have
only to notice that if on a ft O produced we set centre
of generating circle as at c\ then i'P', the arc of the
generating drcle to tracing point P', must equal i'S,
wherefore P^'S is parallel to e'O, and in same straight
line with PS. But since PSP' is parallel to c enjoining
centres of equal circles a P 6, VY'a\ « P is parallel to
VY\ and therefore PP' = ^ a = 2 ft a = SA. This pro-
perty gives a method of tracing out the cardioid me-
chanically. For if there be a circular groove as B/S,
and we take a ruler of length SA (twice diameter of
groove), having a vertical pencil point at each extremity
* The limagon is the curve obtained by drawing radii vectores to
a circle from a point on its circumference, and producing and re-
ducing all of them by a constant length.
THE EPICYCLOID AND HYPOCYCLOID, 77
and a point at its middle point moving in the groove,
while the rod itself always passes through S (either
through a small ring there or by having a projecting
point at S and a groove along the rod), the pencils at
the extremities of the rod will trace out the cardioid.
While one pencil moves over AP* the other will move
over SP''/, and while the former passes on from s to S,
the latter passes on from / to A, completing the tracing
of the curve.
The evolute of the cardioid A 5 S / is a cardioid
5 r O, having its vertex at S, cusp at dy on OB, such
that 0(/=-^0B, and linear dimensions equal to one-
third those of the cardioid A s s\
S, the cusp of the cardioid, is also called the focus.
Since P i is the normal at P and angle SP b = angle
6 P m, we perceive that if S be a point of light, and
the arc of the cardioid reflect the rays, P m will be the
course of the ray reflected from P. Hence the caustic
or envelope of the reflected rays will be the curve
constantly touched by the diameter P/? in the tracing
out of the cardioid. This curve, as shown at pp. 68,
69, will be the epicycloid traced out by a circle whose
diameter = CB, and which has S as one of its cusps.
The other cusp will be at B, and the curve will have
the position shown by the dotted curve BRS and its
companion lobe in fig. 39.
Let us now determine how far the cardioid ranges
in distance from the diameter AS, and beyond ss\
We note that (i.) when P (fig. 39) is at the greatest
possible distance from AS, the tangent P a must be
78
GEOMETRY OF CYCLOIDS.
parallel to AS; and (ii.) when P is at its greatest
distance from / S j^^ the tangent at P must be parallel
to « /, and therefore W^ the normal, must be parallel
to SA. Wherefore, since W has been shown to be
parallel to P a, we see that when P is at its greatest
distance from SA, P^ is at its greatest distance from
8 /. Now, when P a is parallel to AS, so also is p bfy
and as the arc &/= arc B &, the position of bf is at
once assigned : for if a chord bf (fig. 40) is parallel to
BS, arc Bft = arc S/, and since arc b/=2LVC B^ =
S/, we have B i = Jrd the semi-circumference B J S,
and the angle BS/= two-thirds of a right angle.
S/= SO = SF ; and SP = 3S0. Also,
Tn = Sbn = ^-^SOi andF7i = §5.
071= -^f^; andSm = ^SO.
It follows from the parallelism of the tangent Puz
and the normal P'A^ that when the cardioid is beins:
THE EPICYCLOID AIW HYPO CYCLOID, 79
described by the oontinuous motion above indicated, one
end of the rod is always moving in a direction at right
angles to that of the other end of the rod. Thus the
tangents and normals at P and P'' (fig. 39) intersect
on the circle which has PP' for its diameter. The
normals also intersect on the circle Bffb' (at^), and the
tangents on the circle having centre O and radius OA.
Cor. The curve cuts s/ at equal angles, each equal
to half a right angle.
THE BICUSPID EPICYCLOID.
The epicycloid with two cusps (the dotted curve of
fig. 39, which, from its shape, we may call the nephroid)
presents also many interesting relations. I merely
indicate, however, in a few words the chief points
to be noticed at the outset of an inquiry into the re-
lations of the bicuspid epicycloid.
Let P (fig. 41) be a point on the epicycloidal arc
traced by the rolling of AQB on the circle DBD',
whose radius BO = AB.*
Let a P ft be position of rolling circle through P.
Draw common tangent b t, meeting OA in t ; and join
t a, cutting « P A in p. Then, since Ob=ba, angle
fOi = angle taO, and arc/? ft = arc Bft; wherefore
;7eP is a diameter of circle APft. Angle cflP =
compt. of c fl JO = compt. of ^ O ft = angle ft OD'. Hence
TO a is isosceles, and f ft T is a straight line. Draw
♦ The curve has been omitted from fig. 41. The student should
trace it in pencil from the cusp D through A and P (touching PT)
to ly — forming a branch like either half of the dotted curve of fig. 39.
80
GEOMETRY OF CYCLOIDS.
b n perpendicular to OT, and join n P, ft P, bp; then
triangle J) ft P = triangle Onb in all respect*, iP=6n,
and V m = mn. Wherefore the hicuBpid epicycloid
ra&j be described thus : draw from any point b oa
Fio. 41. (JoinSy.)
[ ^
fiL-
a
/^
W\
D
/N\
circle DBD', bn perpendicular to fixed diameter,
DOD', and n m perpendicular to tangent at b ; then
if n m is produced to P so that m P = /n h the locus
of P is a bicuspid epicycloid.
THE INVOLUTE OF THE CIRCLE
REGARDED AS AN EPICYCLOID.
The curve traced by a point on a straight line which
rolls on a circle iu the same plane may be regarded as
an epicycloid whose generating circle has an infinite
radius. The curve is the involute of the circle. Thus,
let DQB (fig. 42} be a circle, T'DT a tangent at D,
and let this tangent roll without sliding over the circle
DQB (DOB a diameter), the point D tracing out the
curve I)P. Then when the tangent has the positJon
PB'p, having rolled over the arc DQB' once only, B'P
THE EPICYCLOID AND HYPOCYCLOID. 81
haying been in contact with every point of the arc
B^'QD is equal in length to this arc. Therefore the
point P lies on that involute of the circle DQB^ which
commences at the point D. But T^DT may be re-
garded as part of a circle of infinite radius touching
the circle DQB'' in D, and the arc DPR therefore as
an epicycloid. In fact this arc is the extreme case of
the epicycloid when the radius of the rolling circle is
indefinitely enlarged, precisely as the right cycloid is
the extreme case when the radius of the fixed circle is
indefinitely enlarged. The part of the curve near to
DQB manifestly has the shape shown in the figure, D
being the cusp. The branches of the curve extend
without limit outwards. It is obvious that if the line
B'P be produced to meet the next whorl of DPR (not
the curve D p R), the portion of this line intercepted
between P and that whorl will be equal to the circum-
ference of the circle DQB. Again, if PB'' produced
meet the branch D /? R in ;?, VWp is also equal to the
circumference of DQB' ; for B'P = arc B'QD, and
B> = arc B'B'^D. The straight line r DR, perp. to
T'DT, passes through all the points of intersection of
the two branches, for the curve must necessarily be
symmetrical on either side of OD from the way in
which it is traced out. Q t, the tangent parallel to
OD, and equal to the quadrant QD, determines the
greatest range of the branch D ^P above DT, for the
curve is perp. to Q ^ at ^ ; also, if Q ^ be produced both
ways indefinitely, its intersections with the prolongation
of D^P above DT determine the greatest range of
G
82
GEOMETRY OF CYCLOIDS.
each successive whorl of that branch above DT, while
its intersections with the branch D/7R below DT
determine the greatest range of each whorl of that
branch below DT. Similarly of the tangent to DQB
parallel to Q^^ and of the tangents perp. to DOB.
Many other relations of a similar kind exist which the
student will have no diflSculty in discovering for him-
self. Both branches manifestly approach more and
Fig. 42.
more nearly to the circular form as their distance from
the centre increases; for from the manner of generation
the normals to the curve touch the circle DQB^ and
for branches at an indefinitely great distance the di-
mensions of DQB are relatively evanescent^ wherefore
the normal at any remote point of the curve is inclined
at an evanescent angle to the line joining that point
with O. Ovy a whorl of the spiral may be regarded as
dhanging its distance from the fixed point O during one
THE EPICYCLOID AND HTPOCYCLOID, 83
complete circuit by a distance^ as ^R^ p"^'\ &c.
(these lines being diametral), equal to the circumfer-
ence of DQBj and this distance vanishes compared with
the radius vector of the spiral in its remote parts, so
that the radii vectores of a single whorl, though differ-
ing by a finite quantity and therefore not absolutely
equal, are yet in a ratio of equality ; and in that sense
the whorl corresponds with the definition of a circle.
The circle DQB is the evolute of the curve RjoDPR,
&e. ; but we have seen (second section. Prop. XII.)
that the evolute of an epicycloid is a similar epicycloid :
hence we must regard the circle DQB as consisting of
an infinite number of infinitely close whorls, similar to
the remote whorls of the curve RjoDPR.
The rectification and quadrature of the epicycloid
in the preceding section manifestly fail for the involute
of the circle regarded as an epicycloid. But it is easy,
a^ follows, to compare the length of any arc D^P with
the corresponding arc DQB' of the fixed circle, and
the area D^PB^'Q with the area of the sector DQB'O.
ARC OF THE INVOLUTE OF THE CmCLE.
Let PP (fig. 42) be an elementary arc, PB^ P'B'^
the corresponding positions of the tracing tangent, then
since OB' is perp. to BT and OB'' to B"P', the angle
B'OB" = the angle PB"F, in the limit. Hence
Arc PP' : arc 9'B" :: BT : OB' :: arc DQB' : DO.
Now in Dr take Dd = OD ; and in DT take DM =
o 2
84 GEOMETRY OF CYCLOIDS.
arc DQB', and MM' = arc B'B''. Complete the rect-
angles Dd NM, NM^ Also draw MK = DM, perp.
to DM, and complete the rectangle KM^ Then if
we represent the arc B'B'' by the area NM', the arc
PP' will be represented by the area KM', for
KM : NM' :: P'P : B'B".
But since KM = DM, K lies on a straight line, DK,
bisecting the angle rDT ; and every element of arc as
PP' has a corresponding representative element of
area, as KM', in the space KDM. Therefore the
length of the arc D^P is represented ultimately by
the area DMK ; or
ArcD^P : arc DQB' :: area DMK : area rfM
:: iDM.KM: DM.OD
:: i DM : OD (since DM = KM)
:: iarcDQB':OD
:: arc DQB' : BD.
That is, the arc D^P is a third proportional to BD
and the arc DQB'.
This is the relation required. It may conveniently
be replaced by the following : —
Cor. Rect. under arc D ^P and BD = square on B'P,
(B'P)«
or. Arc D^P =
BD
AREA BETWEEN CIRCLE, ITS INVOLUTE, AND
THE NORMAL TO INVOLUTE.
Take Dti = i OD and complete the rectangle nM.
Draw ML perp. to DM, cutting nW parallel to DM
in N', and take L so that
THE EPICYCLOID AND HYPOCYCLOID. 85
ML : MN'(= Dn) :: (PBO^(= DM«) : (OBO^
Complete the rectangle LM'. Then by construction
Area N'M' = triangle OB'B'' ultunately ;
and ultimately
A BTF : A OB'B'' :: (PB^^ : (OB^"
:: rect. LM' : rect. NM'.
Therefore Rect. LM' = triangle BTP'.
Now from the construction L is a point on a parabola
D/L, having D as vertex and n as focus, or BD as
parameter. Hence, every elementary triangle as B'PP'
has a corresponding representative elementary rect-
angle LM^ Therefore
Area D^PB'Q = parabolic area D/,LM
= J rect. imder DM . LM.
Now DM=arcDQB';
and by property of parabola,
.-. LM . BD=(DM)«=(PB7;
or LM is a third proportional to BD and PB',
and therefore, as shown in last page,
LM=arc D^P,
.-. area D^PB'Q= J . rect. under arcs DQB' and D^P.
Cor. Area D^PB'Q = J-^qD"
CENTRE OP GRAVITY OF EPICYCLOIDAL AND
HYPOCYCLOIDAL ARCS AND AREAS.
There is no simple geometrical method for de-
termining the position of the centre of gravity of an
88 QEOMETSY OF CYCLOIDS.
epicycloidal or hypocycloidal arc or area ; and the
fore, strictly speaking, these problems do not belong
my subject. Bat it may be as well to indicate i
analytical method of solving them, which has i
hitherto, so &r as I know, been discuraed in a
mathematical treatise. I shall consider the case
the epicycloid only. The scdution for the hypocycl
is similar, and the result only diSers in the sign of
the radios of the rolling circle.
First, then, to determine the ordinates
THE EPICYCLOID AND HYPOCYCLOID. 87
centre of gravity of the arc APD, fig. 43 (fig. 44 for
the hypocycloid), O being taken as origin, OX perp.
to O A as axis of or, and O A as axis of Y.
Let L A'CT = « ; Z. PC'y = rf fl. Then,
^„, 2E(F + E) 9 ,.
arc PP' = ^^ ^ ^ cos .^.d fl.
Also, if P « is perp. to O A, then ultimately,
moment of arc PP' about O A = P w . PP'
== { (F + 2 R) sin I e + 2 R sin I cos ^^^^^ }
2R(F4-R) « ./,
X — 5l_ ^'cos- d«
2R(F + R) ^,
= ^p majrffl, say;
si,nd similarly,
moment of are PP' about OX=i O n . PP'
= I (F + 2R) cos |fl + 2 R sin i sin ^"^^^ j
2R(F4-R) 9 ..
X ''-^ ^.cos -dfl
2R(F4-R) ,.
= \^ 1 my rf9, say.
We have to integrate these two expressions between
the limits 9 = 0, and 9 =7, to obtain the moments of the
arc APD around the axes O A and OX.
\r r ^A /'rF + 2R . F + 2R.
Now Jm^dd =y L_^_ sm -^^ 9
- g+gsin^7t^9+g sin ?^j!^9ld9,
2 2 F 2 2F J
88 GEOMETRY OF CYCLOIDS.
■/:
4F
2F(F + R) . ,F-2R
F-2R ^'° ~irr''
. 2FR . »3F + 2R ^^
Similarly / Wy rf fl = F sin — ^ t
, F(F + R) . F-2R
+ T=2R-^"^-TF-''
FR . 3F + 2R ,,
+ 3-FT2R ""^ ^2F- ' = ^'' "^y-
. 2R(F+R) M, _„.
• F • arc APD ~ "
and similarly y = My.
To determine X and Y, the coordinates of the centre
of gravity of the area APDE, we haye, —
Area of element A'P'a = ^tf ^ R« sin« | rf « ;
i^ 2
and if g be the C. G. of this element^ ultimately a
triangle^ A'^= J A'P' = — - sin - , ultimately.
Alsoif^TTiisperp. to OA,
moment of element A'P'a about OA^gm . area A'P'a,
(F + 2R)sm^ fl + — sm- cos ■ ^^ 9 j
F + 2R -oo • 9 ^ JA
X — 1- — . K^ sm^ - a 6
^P 2
= (£+jK)R«.„,dfl,
^ — , ,-ij, ^ -, say ;
THE EPICYCLOID AND HYPOCYCLOID. 89
and similarly,
moment of element A'F'a about OX=Om . area A'P'a,
= J(F + 2R)coSj,« + -3-8m-sm -g^ « |
x^tf^.R'.sin'Jrffl
F 2
(F + 2R)R« ..
= ^ ^ — Oy d fl, say.
Now
/Z ji. /•r2F-3R . R. F + R . F + R .
^3F + .5R . F-R. R . 2F + R.-| .,
+ — g sm_^fl--sm_^^fljd9;
.•jjClrffl = (2F-3R) ^ sin"^ x-Fsin'^^ »
, (3F + 5R)F . .F-R
+ 3 (F-R) "° -2F-'
FR . » 2 F + R .
sm« — -.=^— w = A, , say.
3(2F+R) 2F
Similarly
^>9=(2F-3K)^^sin ^. -
.(3F + 5E)F . F~R
^ 6 (F-R) ''^-F~^
FR . 2F+R .
sm — = — 9r = Ay, say.
F . F + R
^ sm — ,,— T
2
6(2F-l-R) F
. s^.(F + 2R)t>2 a. _2^A,
••^"" F • area APDE ~ ^^
since area APDE = ?^±J^ ^ R^ ;
2F
and similarly Y = — ^»
90 GEOMETRY OF CYCLOIDS,
It is easy to obtain in a similar manner X^ and Y\
the coordinates of the centre of gravity of the area
APDB, though the expressions are rather more cum-
brous. We take such elementary areas as Wb B' in
fig. 27 (fig. 28 for hypocycloid), and find,
r . R ^
Moment of element about OA = I (3 F 4- 2 R) sin ^ ^
. 5F + 4R T5 fl . F + 2R,nTjo -9 ,,
+ J cos R - sm — ^-jr- « J R^ cos«- rff.
r R «
Moment of element about 0X= (3 F + 2 R) cos ^ »
,5F+4Rtj fl F + 2Rn TJ2 *^A
+ ^ R cos - cos ' -, d\ R^ cos - a 6.
R 2 2 i^ J 2
These expressions can be easily integrated. It will,
however, be more convenient to proceed as follows :
Moment of area ABDE about O A
= f [(F + 2 Wf - F3] sin' ^ T = B,», say.
Moment of area ABDE about OX
= J [(F + 2 R/-F»] sin ^= B,», say.
Moment of APDB about O A = B», - ^"tf^ R« A, .
Moment of APDB about OX =B''y - ^— R« A,,
... X' = (B3, - l^. R«A,) ^il±|^ . R«.
Y' = (B3, - ^-^ RK A,)* '-1^ . R«.
SCHOL. — It should be noted that these solutions
might be presented geometrically, if it were worth
THE EPICYCLOID AND HYPOCYCLOID, 91
while ; but only at great length and with complicated
diagrams. The student will observe that all the rea-
soning in each demonstration, up to the point where the
integral calculus is employed, is manifestly capable of
being presented geometrically, the ratios dealt with
(including the trigonometrical ones) being those of
lines to lines, areas to areas, or solids to solids (in deal-
ing with moments of areas). Again, the only relations
derived from the integral calculus, are these —
ysin a fl rffl = — (1 —cos a) = 2 sin^-^
a a ^
J.
»0 \
COS a J rf fl = - sin «.
X a
These (which are in effect one) are both capable of
easy geometrical demonstration, and are in fact de-
monstrated further on in the quadrature of the * com-
panion to the cycloid.'
The student not familiar with the integral calculus,
will find no difficulty in proving by trigonometrical
series,* that the sum of the series whose general term is
- sin — (r taking all integral values from to n), is
n n
2 sin^ — when n is indefinitely increased ; and that the
sum of the series whose general term is - cos — , is sin a.
These summations (or such as these) suffice for sum-
ming the elements dealt with in the above demon-
stration.
* See the chapter on the Summation of Trigonometrical Series
in Todhunter's * Plane Trigonometry.'
92 GEOMETRY OF CYCLOIDS.
Section III.
TROCHOIDS.
Note. — Any curve traced by a pointy within or without
the circumference of a circle, which rolls without
sliding upon a straight line or circle in the same
plane, is a trochoid; but the term is usually limited
to the right trochoid, and will be so employed through--
out this section.
DEFINITIONS.
The right trochoid is the curve traced out by a
point either within or without the circumference of a
circle^ which rolls without sliding upon a fixed straight
line in the same plane.
If the tracing point is within the circle, the trochoid
is called a prolate or in/lected cycloid. The shape of
such a trochoid is shown in fig. 45, Plate I.
If the tracing point is outside the circle, the trochoid
is called a curtate or looped cycloid. The shape of
such a trochoid is shown in fig. 46, Plate I.
An epitrochoid is the curve traced out by a point
either within or without the circumference of a circle
which rolls without sliding on a fixed circle in the same
TROCHOIDS, 93
plane^ the rolling circle touching the outside of the
fixed circle.
A liypotrochoid is the curve traced out by a point
either within or without the circumference of a circle
which rolls without sliding on a fixed circle in the
same plane, the rolling circle touching the inside of
the fixed circle.
It may readily be shown that every epitrochoid
can be traced out in two ways — viz., either by a point
within or without a circle which rolls in external con-
tact with a fixed circle, or by a point without or within
a circle which rolls in internal contact with a fixed
circle of smaller radius. Also every hypotrochoid can
be traced out either by a point within or without a
circle which rolls in internal contact with a fixed
circle of radius larger than rolling circle's diameter, or
by a point without or within a circle which rolls in
internal contact with a larger fixed circle, but of radius
not larger than rolling circle's diameter. Instead,
however, of giving a demonstration of these relations,
after the manner of Prop. I., Section II., I leave the
point for more general demonstration in Section V.
In what follows, reference is made to right trochoids,
unless special mention is made of epitrochoids and
hypotrochoids. Either fig. 45 or fig. 46 may be fol-
lowed. The reader is recommended to read the follow-
ing remarks twice over — once with each figure, and to
adopt the same plan with the demonstration of each of
the following propositions.
Let AQB (radius R) be the rolling circle, KL
94 GEOMETRY OF CYCLOIDS.
the fixed straight line. Let the distance of the tracing
point from the centre be r, so that the tracing point
lies on the circumference of the circle aqb, of radius
r, and concentric with AQB. This circle, aqb, ia
called the tracing circle. Let IXD be the fixed straight
line, touching the circle AQB in B. Let the centre
of the rolling circle move along a line c' C c, parallel
to D'D through C, the centre of AQB, in the direction
shown by the arrow. Draw ef e and d! d parallel to
cf C Cy and touching the tracing circle aqb. Then it
is manifest that at regular intervals the tracing point
will fall upon the straight lines ef e and d d. When
at a on the straight line ff e, the tracing point is turn-
ing around the centre of the rolling circle in the direc-
tion in which this centre is advancing, and .is at its
greatest distance from the fixed straight line. When
at df and cf, the tracing point is turning round the
centre of the rolling circle in the opposite direction,
and is at its greatest distance from c^ c on the side
towards which lies the fixed straight line KL. The
curve will manifestly be symmetrical on either side of
the diameter aC&, perp. to KL. Therefore a 6 is
called the axis of the trochoidal curve : d dis the base ;
and a the vertex. The radius C a, drawn to the
tracing point, may conveniently be called the tracing
radius. D'AD is called the generating base. The
rolling circle AQB is called the generating circle,
and when in the position AQB, is called the central
generating circle. The circle aqbis called the tracing
circle, and when in the position aqb, is called the
TROCHOIDS. 95
central tracing circle. The complete trochoid consists
of an infinite number of equal trochoidal arcs^ but it is
often convenient to speak of a single trochoidal arc^
da dy as the trochoid.
It is clear that if D V E', D c E, be drawn perp.
to the fixed straight line through df and dy and inter-
secting ffae m e^ and e, respectively, the parts of the
trochoid on either side of d!e' and de are symmetrical
with respect to these lines. Therefore def and de may
conveniently be called secondary axes.
The straight lines efa e and d bd are tangents to
the trochoid at a, and at d and d, respectively.
PKOPOSITIONS.
Pkop. I. — The base of the trochoid is equal to the
circumference of the generating circle {Jigs. 45, 46).
For d bd^ D'BD = circumference of the circle
AQB.
Cor. 1. d b = V d ^ half the circumference of the
generating circle.
Cor. 2. Area e dd ef =^2 rect. a d = 4 rect. C d
= 4 ^ rect. CD = 4 ^ circle AQB.
xC xC
Cor. 3. The base d bd : circumference of the trac-
ing circle aqb :: circumference AQB : circumference
aqb :: 'R, : r.
Cor. 4. Area edd ef = 4 rect. under Cb, b d
= 4 rect. under C &, — . arc a q 6=4 — . circle aqb.
r r
GEOMETRY OF CYCLOIDS.
Prop. II. — If through p, a point on the trochoidal arc
apd (Jiffs. 47, 48), the straight line pqM be
drawn parallel to the bate bdj cutting the central
tracing circle in q, and meeting the axis AB in M;
then.
qp = ~ arc a q.
Let A'PB', (^ p b' be the position of the generating
and tracing circles when the tracing point is at p.
'^^
^-C>s
^\."
..XA.
•v~4
M^
'M:,
UJ I
,.<i>^j
'•> \
■J
C their common centre, A'CB' diametral cutting
p M in M'. Draw the diameter T pC ^. Then it i»
TROCHOIDS, 97
manifest that M y = W p ; MM' = qp ; and arc 09 =
arc a! p* Now j3 is the point which was at B when the
tracing point was at at, and since every point of the
arc j3 B' has been in rolling contact with BB', the arc
i3B' = BB'.
But arc jS B' = arc A'P = arc c^p = — arca^;
R
and BB'= MM' = 9/?; wherefore qp = arc aq.
Cor. 1. M/? = — arc aq + My.
R
Cor. 2. Since & d = AQB =: - aqb
R , ,v
= — (arc aq -{■ arc q 6),
it follows that in the case of the prolate cycloid, where
H > r, and therefore - . arc q h necessarily > M y,
bd>Mp, and the whole arc apd lies on the same
side of de eis ab.
But in the case of the curtate cycloid (fig. 48),
^here R < r, there must be a point y'' on aqb where
- arc b(f' =^^ q" (drawn perp. to AB),
and if p" be the point in which N q" produced meets
the trochoid, then will /?'' fall on e rf, for
p''N= -arcay^'H- Ny''
R
= - (arc a(^' •\- arc b (/') = bd.
r
The part of the trochoid lying between p'^ and d mani-
H
08 GEOMETRY OF CYCLOIDS.
festly falls on the side oi ed remote from a b ; and as
the complete curve is symmetrical with respect to ed^
it follows that the curtate cycloid has a loop of the
form y r d /. It is also clear that the point j/' lies
between D and e, since if L be the point in which BD
cuts the arc aqby and CL cuts AQ6 in /, the arc B/
is less than BL. The point y may lie nearer to e
than E does, however, and the arc d / /?'' may intersect
a b. It is easily seen either from the mode of genera-
tion or from Cor. 1, that if the ratio r : R be small,
the curve may cut ed a great number of times before
the tracing circle has been carried entirely past ed.
Observe that if C 5^' cuts AQ'B in point Q'
arc BQ' = N q'\
Cor. 3. Let Mp produced (if nece^i?ary, in the
case of curtate cycloid) meet ed in m; then
= - arc aqb — — arc aq — S/Lq
= — arc b q — Mq.
For points of the ^ro. p" rd (fig. 48) this relation still
holds, regarding lines drawn perp. U) ed from the right
as negative.
T
Cor. 4. Arc a p = =p . A^', and
T
arc b' p := ^ . b' d.
Cor. 5. If from p' on p d^ p' q' be drawn parallel to
b dto meet a' p V in q\
q' p* : B,rc pq' :: R : r.
TROCHOIDS, SO
The proof of this is similar to that of Prop. II., sec. 1,
cor. 5.
ScHOL. — The reader will find no difficulty in
making the necessary modifications for the epitrochoid
and hypotrochoid, deducing properties bearing to
those established above the same relation which those
established in Prop. III., section 2, bear to the pro-
perties established in Prop. II., section 1.
Prop. III. — The area d! ad {figs. 45, 46) between
the trochoid and its base : area of the generating circle
:: {bC-\' bA): b C :: 2 R -{- r : r.
This may be proved in either of two ways corre-
sponding in all respects with the two proofs of Prop.
III., section I. In the first proof, we show that ele-
mentary rectangles q'p^q' p' (figs. 49, 50) are equal
to elementary rectangle L I ; whence areas aqp^ q'b'dp\
together, are equal to rectangle LZ; and the area
R
aqb dp to the rectangle C e = - circle aqb. Whence
2 R
area d'ad (figs. 45, 46)= © aqb-\-— ^ aqb^
or
area d' adi © aqb :: 2^ -{■ r i r :: be -{■ b A: bC
In the second proof, having drawn the inverted
trochoid ap" d^ with ae as half base, and c^e as axis,
we show that the elementary rectangles p"p and q"q
are equal, whence
area (]['a q = Bxeap'^a p ; and area a p"dp = circle aqb.
H 2
100
OEOMETRY OF CYCLOIDS,
The equal areas ap"db Budiapde are, therefore, each
= \ (rect. be — circle abq)
= \^~ - i) circle abq;
therefore
R
and
as before.
the area apdb = T- + i) circle abq;
d'ad = f j circle a i y,
Fio. 49.
«r
Fig. 60.
ScHOL. — The reader will find no difficulty in deal-
ing in like manner, so far as first proof is concerned,
with the area between the epitrochoid or hypotrochoid
and the base. The demonstration bears precisely the
same relation to that of Prop. IV., section 2, which the
above first proof bears to the first proof of Prop. III.,
section 1. We thus show that the area between the
\
TROCHOIDS. 101
generating semicircle aqh, the arc base b d (radius F)
and the trochoidal arc «/?(/: generating circle aqh
:: 2 CO (i C + i A) : ^ O . ^ C, that is, in the ratio
compounded of the ratios 2 CO : & O and {bC + ^ A)
: hC.
In all cases, — for cycloid, epicycloid, hypocycloid,
trochoid, epitrochoid, and hypotrochoid, —
area aqb dp in the trochoidal figures = J area aide.
Prop. IV. — If the cycloid^ a PD^ and the trochoid^
ap d {figs. 49 and 50), have a common axis a i,
area aqb dp : area aqb DP :: R : r.
From Prop. II., section 1,
qV ^ arc a q ;
but from Prop. II., of the present section,
R
qp = j ^TQaq
5'/? : 5'P :: R : r,
and elem. rectangle ;? y : elem. rectangle qV ;: R : r;
whence area aqp \ area « y P
: : area aqb dp : area aqb DP : : R : r.
Cor. Area aqp : area a^'P :: R : r.
ScHOL. — A similar property can be readily esta-
blished for epicycloids and epitrochoids, or for hypo-
cycloids and hypotrochoids, having a common axis.
In this case, qp, qVy and b dy are concentric arcs, and
in place of elementary rectangles we have elementary
to;
GEOMETRY OF CYCLOIDS.
areas like Q;>, ?'P' of figs. 26 and 27; but tlie
ratios are the same, and we therefore bUII find
area aqp i area aqV :: area aqbdp : area a jiDP
:: R : r.
Pkop. V. — Ifp (Jiffs. 51, 52) is a point in a trochoidal
arc, a' pb', the tracinff circle when the tracing point
is at p, a' C b' diametral, meeting the generating
base in B',then B'p is the normal at p ; and if
Ta' t is the tangent to the tracing circle at a',
Tp,tp, tangents to the trochoid and tracing circle
respectively at p, then
Tl : a't :: R \ r.
Fio. 52.
Since, when the tracing point is at p, the generating
circle is turning around the point B', the direction of
the tracing point's motion at p must be at right
angles to B'/*, which is, therefore, the normal at p.
The tangent p T at p is therefore perp. to p JS'. Also,
TROCHOIDS. 103
Since pa! \B perp. to p i', and p t to C'/?, triangle
pT a! v& similar to p B'i', and p a!t to /? J' C ; there-
fore
Tf : a'^::B'C' : C'&::K : r.
Another Demonstration,
From p and p' (near /?), on the trochoidal arc, draw
p M, p'W perp. to a'i^ ^M' cutting a'p 1/ irxq. Then
qp' :pq::C'W ' CT( = C»,
Prop. II., cor. 5, and since ultimately the sides qp\
qp are perp. to the sides 0/ B', C'p,
angle jo 5' P'= angle /? C'B'.
Hence the triangle p qp' is ultimately similar to the
triangle p C'B', and p p\ the third side of one, is ulti-
mately perp. to joB', the third side of the other.
Wherefore p B' is the normal at p. And, as in the
preceding proof, T^ : a't:\ C'B' : G'V :: K : r.
Cor. 1. Triangle p qp' being similar to triangle
P C'B',
pp' : pq::Wp : C'p.
Cor. 2. It pm be perp. to /?'M', and p T cut or
meet a'b' in K, theii pp^ m is in the limit similar to
triangles p B'M, Kp M, KB'/?.
Cor. 3. If Wp cut jo' M' in Z, the triangles Z/? m
and /jo'jo are similar to the four triangles named in cor.
1. Also, Ip q is similar to K/? C, BXidq pp' to C'/> B'.
Wherefore
Iq : y/ :: KC : C'B' ::p N : NB'.
104 GEOMETRY OF CYCLOIDS.
Cor. 4. If p b' cut jo' M'in A, Upp' is similar to
a'/? B', and kp q \o a! p C^ Wherefore
pq^q k, and k q : jjo' '-jP? • 9P^ :: r : R.
Cor. 5. If in the case of the prolate cycloid, illus-
trated in fig. 51, the tracing point is at r, where the
tangent from B' meets the tracing circle a q b\ then
the normal B'r has its greatest inclination to a'B',
and its least inclination to the base. It is manifest,
therefore, that r is a point of inflection. At the point
of the prolate cycloid corresponding to /, in which
B' p cuts the tracing circle, the tangent is parallel to
the tangent at p.
Cor. 6. If in the case of the curtate cycloid, illus-
trated in fig. 52, the tracing point is at r on the generat-
ing base, the normal B'r coincides with the generating
base. Therefore the curtate cycloid cuts the generat-
ing base at right angles.
Cor. 7. W q produced to meet pp' inn is ultimately
perp. to JO/?', and if C'N is drawn perp. to p B', p qn
is similar to p C'N, and p^q n to B'C'N ; and
p p^ : pn::pW : p'N.
ScHOL. — It is easy to prove that/? B' is the normal
in the case of epitrochoid or hypotrocfcoid. We have
only to draw C'^ parallel to the line joining p with the
centre of the fixed circle, to meet p B',* and proceed
as in Prop. V., section 2. (In both figs. C'« is drawn
for the case of the epitrochoid ; C'/, for the case of
* The reader will note that, in fig. 51, CV does not extend far
enough. It should be produced to meet jf B\
TROCHOIDS. 106
the hypotrochoid). If, In the former case, the straight
line joining p with O, the centre of the fixed circle, be
perp. to B'/?, which can only happen when r > R (fig.
51), the tangent at jo passes through O. This deter-
mines the position of the tangent from the centre to
the curtate epicycloid corresponding to the direction of
the stationary point in the looped epitrochoid, regarded
as a planetary curve. It is well to note the construction
for determining this point. Produce C'i' (fig. 51) to
0, the centre of the fixed circle, and on B'O describe
a semicircle cutting a'p V in r' ; then B V is perp. to
r'O, and therefore a circle described about O as centre,
with radius O /, intersects the curtate epicycloid in
the point where the tangent passes through O. This
relation is demonstrated and dealt with under Prop. X.
Cor. 8. In the case of epitrochoids and hypotro-
choids the triangle p qp' is similar — not to p CW —
but to p Cs (the s accented throughout for hypotro-
choid) ;
pp' :pq::ps:pC\
and pp^ : np::ps : PN.
Since then Njo and np are the same for the epi-
trochoid or hypotrochoid as for the right trochoid, with
the same generating and tracing circles (and, of course,
the same angle, p C'a', between tracing radius and
diametral), while
pW : B'5::F : R,
and therefore /? B' : p s:: B'O : CO (see figs. 28 and
29), it follows that pp' ^ regarded as an arc of an epitro-
100 GEOMETRY OF CYCLOIDS,
choid or hypotrochoid, bears to p /?', regarded as an
arc of a trochoid (/? q being the same for both), the ratio
sp I pB', or CO : B'O, or F±R : R (the upper sign
for epitrochoid, the lower for hypotroohoid).
The student will find it a useful exercise to com-
plete the construction indicated in the scholium, noting
that the figs. 51 and 52 are correct for the cases there
considered, as well as for the case considered in the
text, except only that the lines p M and p^q M' must
be concentric with the generating base through B' —
that is, must have for centre the point mentioned
in the scholium.
Prop. VI. — From a point p {Jigs. 53 and 54), on the
trochoid ap d, above the line of centres c</ Cy let q p
he draicn parallel to c C to meet the central tracing
circle a c'h in q, and qn^ p m^ perp, to c C ; then, if
the rectangle a c nf be completed^
area a h q p-trect. p n : rect, cf:: R : r.
And if from p' on ap d below c C, p'q' parallel to c C
meet ac'b in q' ; q'n', p'm' are drawn perp, to c C ;
and rect, n c bf is completed, then
area a h c' q' p' —rect, p'n : rect, cf ',: R : r.
Let a PD be a semi-cycloid having a i as axis ;
then it is easily seen that every element of either area
a h qp-bp n or ah q'p' --p'n parallel to c C, bears to
the corresponding element for the case of cycloid a PD,
the ratio R : r ; and therefore the sum of all such ele-
TROCHOIDS.
107
ments of either area in case of trochoid : sum of all
such elements of either area in case of cycloid {i.e.^
Fio. 63.
cf or cf, as shown in Prop. V. sec. 1) :: R : r.
That is,
area aAg'/^ + rect. q m : rect. cf)
area a h q'v> — rect. q'm^ : rect. c'f I "
R
Cor. Area ac'h dr -=. rect. cd •=. — . circle aqh
r
(Prop. I., cor. 4). Thus we have here another de-
monstration of the area of trochoid.
GEOMETRY OF CYCLOIDS.
Prop. VII. — Let a {Jit). 55) be the vertex of the
trochoidal arc ap, a'p b' the tracing circle through p,
^Cb' diametral, A'CfB" the correnponding diameter
of generating circle. Describe the quadrant A' PA"
having b' as centre and b'A' as radius; produce b'p to
meet A'PA" In P; and draw PI perp. to b'A".
Then, if b'B" = UR, and B"PA", an elliptic
quadrant having b'B" and b'A" as semi-axes, inter-
sect PI in P,
arc ap = twice the elliptic arc B"P.
Let/)'" be a poiot on the trochoid near p, and let
p'q parallel to the base meet a'p b' in q. Produce b'q
FiQ. 63.
to meet A'PA" in Q; draw QL perp. to b'A", cut-
ting B"Q'A" in Q', Join a'p, Wp, and draw b'n
parallel to Wp (dividing a'p in n, so that a'n ; np
y.a'b' : fi'B'::A'B" : 3"^). Join C>,PQ, and P'Q'.
The secants PQ, P'Q' being ultimately tangents at
*y does not lie on Pi:
TROCHOIDS, 109
P and P', meet ultimately when produced on h'A!' ; let
them thus meet in T.
Then P'Q' : PQ :: P'T : PT :: b'n : b'a' (since tri-
angle a'b'p is similar to PT Z, and a'p and P I are
similarly divided in n and P' respectively) :: Wp : B'«'.
Also, TQipq:: A'i' ( = a'B') : a'i'
(because PJ'Q is an angle at centre of quadrant A'P A"
and at circumference of semicircle a'p ft').* Where-
fore, ex CBqiiali,
P'Q :pq::B'p : a'h'. But
pp' ipq :: B> : Q'p (Prop. V., cor.l) :: 2 B> : a'h' ;
therefore, p;?' = 2 FQ'.
But pp' and 2P'Q 'are increments of arc ap and arc
B"P' respectively, which arcs begin together.
Therefore, arc a;? = 2 arc B"P'.
Cor, The arcs apd (figs. 45 and 46) = elliptic
arc B" A"B', and arc d'a d = circumference of an
ellipse having semi-axes ft A, ft B, that is, R + r and
Pkop. VIIL— ijr a'pV {Jigs. 56 and 57) is the
position of the tracing circle through /?, a'h' diame-
tral^ a b the axis, and p h' he joined^ then
area apb'b : sect, area abq(or a'h'p A)|
^ xc "ir r m r.
\-
area p Vd : segment b s q {or b' fp)
Let a PD be a cycloid, having a ft as axis, and let
P/? be parallel to ft rf; then area aqb B'P = 2 sec-
* -5^ = circ. meas. oipVq^^^ circ. meas. of ^ Cg' = i ^^f =^£ .
no GEOMETRY OF CYCLOIDS.
torial area A'B'F. But every elemeDt of the area
aqb b'p parallel to base Id {si&ia Prop, III.) : corre-
spondiDg element in caseof cycloH::!^ : r. Wherefore
area aqb b'p : sectorial area ab ^::2 R : r, and area
acbb' : gectorial area aft^::2R + r : t. Similarly
area j) h'd : segment bs q::2^ + r : r.
Cor. 1. Are&p/b'd
Cor. 2. Area aqbh'
Cor. 3. If ^ q produced meet a £ in
2 R
area qsb dp = rect. bm, qp -i —
ScHOL, — Two independent methoda of demonstra-
ting tlie area of trochoids can be derived from the above
proposition, as in the case of cycloid. For, carrying p
to d, we have area ap d b : ^ circle fly6"2R+r: r,
as in Prop. III.
The proof may be extended to epitrochoids ani)
hypotrochoids, and the following proportion esta-
blished : —
TROCHOIDS. Ill
Area abb' p : sectorial area a' bp
:: area b'p d : seg. b sq
:: (2 CO + BO) (2 R + r) : BO . r, where BO is the
radius of the base^ and CO is the radius of the arc of
centres^ or
:: (3F±2R)(2R + r) : F. r
(where F is the radius of fixed circle), the upper sign
for epitrochoid, the lower for hypotrochoid.
Prop. IX. — To determine the area of the loop of the
curtate cycloid apd^ fig, 48.
By cor. 3, Prop. VIII., area q"p"r d i, fig. 48,
(= rect. N rf+^ loop r'r — area N b q")
= rect. b N, q"p" + — seg. y"L b ;
.'. i loop r'r = area N b q" ^ rect. under & N, N q"
2 R
H seg. q"\j b
r
2 R + r
r
seg. q"lj J — triangle ft N y" ;
, , 4R + 2r „^ . ^ ^^
.*. loop rr=^ seg. q 1j o— rect. N n.
Prop. X. — With the same construction as in Proposi-
tion VIII. y area ap ha' : segment a'hp : : 2 J? : r.
Since area a q p : area AQP :: R : r :: area
aqp ha' I aq PH A' (PH A' being the arc of tracing
© A'PB^ for cycloid, not wholly shown in the figure) ;
112 GEOMETRY OF CYCLOIDS,
it follows that area apha' : area aPHA'::R : r.
But area a PH A' = 2 segment A'HP or 2 segt. a'h p ;
. • . area apha' : segt a' hp::2 R : r.
Cor. 1. Area apd^e : \ circle eq'd::2 R : r.
Since ap dq'e =• apd e-\-^ circle « y J,
and rect. be : ^ circle e q'd :: 4 R : r, it follows that
rect. be ' area ap de + ji circle a^i :: 2 : 1
as in schol. to Prop. III., so that we have here a new
demonstration of the area.
Cor. 2. In the case of the prolate cycloid, fig. 57,
in which p a' does not intersect the arc a p,
area ap a' : segment a'hp :: 2 R— r : r.
Cor. 3. Proceeding to (/, area ap de : j^ circles q'd
:: 2 R — r : r, in case of prolate cycloid.
Cor. 4. In the ciase of the curtate cycloid, fig. 56,
p a' cuts the curve in some point A, between p and a\
Here then
area fl A a' — area A JO : segment fl'Aj9 :: 2 R—r : r,
or passing to d,
area a r e — semi-loop rp'a : ^ circle eq'd:: 2 R — r : r.
Schol. — Another independent demonstration of
the area of trochoids is worthy of notice. Let us suppose
that the circle aqb, figs. 49 and 50, slides uniformly
between a e and b dto the position e Qd (^ c/ diametral).
Let p"a'p be the position of the upper segment when
the circle passes through p"p {=:q"q^ so that the circle
reaches p" and v simultaneously), and let a closely
TROCHOIDS. lis
adjacent segment, as in the figure, give the elementary
areas a'p and a'p'\ These are ultimately in a ratio of
equality, but they are the respective increments of the
areas ap a', ap"a' (or as actually drawn in the figure,
they are the elementary increments next before the
attainment of these areas ap a'^ ap"a'\ and these
areas begin together. Hence
area ap a' ^=^ area a p"a' ;
and carrying the moving circle to its final position,
area apdQ,e = area ap^'d Q'e = area ap dbq',
whence the result of Prop. III. follows at once.
Prop. XL — Let p o (Jiffs, 58-62) be the radius of
curvature at p, on the trochoid; a'pb' the tracing
circle through p. Then, if a' C'b' meet the generating
base in JS', and C'N be drawn perp, to p B'y
po : pB' ::p B' : pN.
With so much of the construction of Prop. V. as
is indicated in fig. 58 (illustrating the prolate cycloid),
let jo'L be the normal at p' (near p). Then
q p^ = - arc p q (Prop. II., Cor. 5) = B' L.
Join q B'. Now ;?'L, being parallel to q B', is not
parallel to p B', unless the point q falls on jo B' ; that
is, unless the tangent to the circle a'q b' passes through
B', the case illustrated by fig. 60. In this case the
radius of curvature is infinite, or jo is a point of inflec-
114
GEOMETRY OF CYCLOIDS.
tioiL In all other cases, p W and p'Jj meet when pro-
duced, — towards B'L, when p'q has to be produced to
meet p B' (in /), and towards p p' when p B' intersects
Fig. 68.
Fio. 59.
Fio. 61.
p'q (in /) between p' and y, fig. 59. Let them meet
in 0. Then in the limit
lo : ZB':: Ip' \ Iq :: pW : p^ (Prop. V., cor. 3).
That is, ultimately,
op : pB' ::pB' : pN.
Cor. Rect. under op^p N = square on p B'.
ScHOL. — The following construction is indicated for
determining the centre of curvature. On B'/?, pro-
duced if p is beyond N, otherwise not, take/? H =7? N,
nd on the tangent /? KT at/? take/?T=/?B'; then
TROCHOIDS, 116
T o perp. to HT will meet p B' produced in o, the
centre of curvature at p. For
op,pn = (p T)\
that is, opyp'N= {p B')*.
The student will find no diflSculty in dealing with
the corresponding demonstration for the curtate cy-
cloid. Fig. 61 gives the construction for one general
case, p above the base ; and for the case of a point on
the generating base where B' becomes the centre of
curvature (for the latter case r and r' are put for p
and p'y while the letters H, T, and N are accented).
Fig. 62 gives the construction for a general case,/?
below the base.
For the vertex, N coincides with C',/? N = a'C^ = r,
and j» B' = a'B' = R + r. Therefore,
radius of curvature at a = ^^ —.
r
both for prolate and curtate cycloids.
For the point d, N also coincides with C',/? N = r
in absolute length, and must be regarded as negative
in case of prolate cycloid, because N falls outside p B'
beyond /?, whereas in case of curtate cycloid N falls
on the same side of p as B', though beyond B'. Also
± /? B' = (R — r). Therefore, rad. of curvature at d
^ (R - r)
r
for curtate cycloid.
But it is to be noticed that in considering the
curvature in the case of the curtate cycloid as constantly
positive, regard is had to the .intrinsic nature of the
curve. If the curvature is considered with reference
i2
2
, negative for prolate cycloid, and positive
116 GEOMETRY OF CYCLOIDS,
to the base, there is a change of sign at the moment
when N passes the point B', or where the curve cuts
the generating base — viz., at r.
At this point r,
(r B'Y
radius of curvature = — ^7- = r B' ; or
r x5
square on rad. = (r B')' = (C' rf - (CB')^ = r^ - ^\
PbOP. XII. — Letpo {Jigs. 63, 64) be the radius of cur-
vature at the point p of an epitrochoid or hypotrochoid ;
a'ph' the tracing circle through p; and a'h' O dia-
metral, cutting generating base in B\ Draw C'N
per p. to p B' ; and (7 s parallel to p O meeting p B'
(^produced if necessary) in s. Then
p o : pW ::p s : ps — NB'.
[Two illustrative cases only are dealt with (one of
a prolate epicycloid, one of a prolate hypocycloid). The
student will find no difficulty in modifying the demon-
stration and figure for other cases.]
Ltetp' be a point near j9 ; p'h the normal at p^ ; p'g
concentric with generating base B'L, meeting a'p V in
q. Draw qn perp. \^pp'\ qi\\\ direction perp. to
a'h' to meet p p' in z, and L h perp. to B'/?. Then, as
in case of right trochoid, q i= — avcp q = WIj^
and triangle B'L h is equal in all respects to triangle
qin. Also triangles p qn,pqi,pqp' are similar to
triangles p C'N,p C'B'^p Cs. (See Prop. V., Cors.
and Schol.) Now L A is parallel to p'p; wherefore,
po : hoiipp' : hit {= ni) ::ps : NB',
or ultimately po : pB' ::p's : (ps — NB').
TROCHOIDS.
117
Cor. Since p s : CO ::/? B' : B'O, we see that
poi C'0::{pWf : (j9^~NB') B'O
:: {p BJ : p W. C'O-NB^ B'O.
See p. 166. At vertex, and at pt. on base, rad. of cur-
mature - (R + O'CF + R ) and -(R -0'(F + R)
respectively, R being regarded as negative for hypo-
cycloid.
Fio. 63.
Fig. 64.
z — y
ScHOL. — A construction similar to that for the
radius of curvature at points on right trochoids can
readily be obtained. Thus produce B'^ to H (as in fig.
58), taking pH^p 5— NB' ; on the tangent p K take
^ T, a mean proportional between p W and p s ; then
T perp. to TH will intersect p B' produced, in o, the
centre of curvature at p. For by the construction
poips- NB') = (pTy=pB'.ps
.'. po : pB' ::ps : (ps — NB').
At a point of inflection the radius of curvature
becomes infinite. Now pB' is always finite, and
118
OEOMETET OF CYCLOIDS.
smcept : pB'::C'0 : WO, pg is also necessarily
finite. Wherefore, the radius of curvature can only
become infinite by the vanishing of ^s— NB', that ia,
when NB' = p g, or N p = B'»,
or ;:' must have such a position as is shown in figs. 65
and 66, for the epitrochoid and hypotrochoid respec-
tively. "Wherefore,
NB' :pB'::ps:pB'::C'0 : B'0::F±R : F
(upper sign for epitrochoid, lower for hypotrochoid).
or, drawing p I parallel to NC — that is, perp. to B'Jf
— to meet CO in I,
CB' : B'l :: CO : B'O :: F ± R : F.
Wherefore, the construction for determining points
of inflection is as follows: — Take I in CO (figs. 65
and 66), so that
CB' : BT :: C'C : B'O :: F ± R : F
CB'.B'0_ R^
tR*
CO
or B'l =
TROCHOIDS, 119
Then if the circle on IB' as diameter cuts the tracing
circle, as at /?, a circle about centre O with radius Op
cuts the epitrochoid or hypotrochoid in its points of
inflection. If the circle on IB^ as diameter does not cut
the tracing circle, there are no points of inflection.
Cor. C'B' : CI:: CO : C'B',
and (C'BO'=C'I . CO ; that is, CI = p^?^-
If, in case of epitrochoid, I falls at b\ — that is, if
C'B' : WV :: CO : B'O :: P + R : F,
the radius is infinite at the point d ; but there is no
change of curvature : two points of inflection coincide,
and the curvature has the same sign on both sides of
the double point of inflection. In this case,
Cy : CB' :: CB' : CO :: R : F + R
or r : R::R : F + R.
This indicates the relation between r, R, and F, when
in the case of epitrochoid the curve just fails, at rf, of
becoming concave towards the centre.
If, in case of hypotrochoid, I falls at a', that is, if
CB' ; BV :: CO : B'O :: F - R : F,
the radius is infinite at the vertex a. Two points of
inflection coincide, the curvature having the same sign
on both sides of the double point of inflection. In
this case
CV : CB' :: CB' : CO :: R : F - R
orr: R::R: F- R.
This indicates the relation between r, R, and F, when.
120 GEOMETRY OF CYCLOIDS.
in the case of the hypotrochoid) the curve just fails at
a of becoming concave towards the centre.
Prop. ^111.— If p (Jigs. 65 and ^^) be a point of in-
flection of an epitrochoid or hypotroclioidy a'qp the
corresponding position of the generating circle;
a' O O diametral^ meeting the generating base in
B' ; p z perp. to B' C ; and k the centre of semi-
circle B'p I; then will
rect. CB'. CI ± sq. on Cp = 2 rect Ck, Cz
{the upper sign for epitrochoid, the lower for hypo^
trochoid).
We have
(C>y = {C'zy + {p zf = [C'zy + {k I)«-(A z)\
and for epitrochoid
CB', C'I = (C'A)2-(AI)2
.-. C'B' . CI + {C'pY = {C'zy + {G'kf - {k zy
= 2 C'A . Cz.
For hypotrochoid
C'B'.C'I = (AI)2-(C'A)2
.•: C'B' . CI - (Cpy = (kzy - (Cz)^ - (Cky
= 2 CA . C'z.
ScHOL. — This prop, may also be treated in the
manner adopted for the next — z.e., starting from the
relation (IpY + (B'p)^ = (I B')^, and taking triangles
I Cp and B'C>.
TROCHOIDS,
121
Prop. XIV. — Let p {figs. 67, 68) be the point of the
loop of an epitrochoid or hypotrochoid where the
tangent to the curve passes through the centre of the
fixed circle ; o'p V the corresponding position of the
tracing circle ; and a' OB' diametral, meeting the gene-
rating circle in A' and B'; then, if p K is drawii perp,
to OCy
Rect. OAy CK = sq. on CV + rect. O C", CB'y
for epitrochoidy and
=irect. OC.CB'sq. on Ch',
for hypotrochoid.
Since p B' is the normal at jo, H'p O is a right
angle, and sq. on B'/? -f sq. on /> O = sq. on B'O.
Fig. 67.
Fig. 68.
Now (B»2 = {C'pY + (C'B')' - 2 C'B' . CK
and [OpY = (C»2 + (C'Oy :;: 2 CO . CK
(lower sign for hypotrochoid)
.-. (B»2 + {Op)^ = 2 (C»^ + (CB')' + (C0)2
~2(CB'±C0)CK;
that is, (B'O)^ = 2 {C'pf + (CB')2 + (CO)^
T 2 OA' . CK.
122 GEOMETRY OF CYCLOIDS.
Or, for epitrochoid,
2 OA'. C'K = 2 (C'iO' + (C'0)» + (C'B')'-(B'0)«;
ue. (Euc. II., 7) OA' . C'K = {C'bJ + OC . C'B'.
For hypocycloid,
20A' • CK'=(B'0)2-2(C'y)«-(C'0)»-(C'B')» ;
i.e. (Euc. II.,4)0A'. C'K' = OC'. C'B'-(C'i')'-
ScHOL. — This prop, may also be treated in the
manner adopted for the preceding, bisecting K O in n,
and noting that rectOC . C'B'=± [(C'n)»-(nB')«],
upper sign for epitrochoid, lower for hypotrochoid.
Observe that C'K (regarded as positive or negative,
according as K lies on CO, or C on KO)
_ r»±(F±R)R _ r« + R^ ± FR
F±2R ^^" Fdi2R '
the upper sign for epicycloid, the lower for hypocy-
cloid.
This is the relation existing at a stationary point
in an epicycloidal planetary orbit.
Prop. XV. — If G {figs. 47 and 48) is the centre of
gravity of the trochoidal area d'a dy
^G: 3R + 2 r::r: 2(2R+ r).
Since every elementary rectangle of the part of area
d'a d outside circle aqby taken parallel to base : corre-
sponding element of part of cycloid having a 6 as axis
lying outside same circle a ^ & : : R : r^ it follows that
the distance of CG- of former areas from bd (alon^;
TROCHOIDS. 123
h C, evidently) = distance of CG of latter areas from
h (along i C) = I i C (Prop. XVIII., sec. 1st, cor; 3).
. • . Mom. of d'a d about b d
R . 3r .
= 2— circleao'i .— i--H circle ^fl^i . r
r ^ 4 ^
3R + 2r . ,
= • circle aqb
and area d'a d = circle aqh
r ^
.^2R + r., , 3R + 2r., .
.• . ^(jr . circle aqb = r circle aqb
r ^ 2 ^
,^ 2R + r 3R+ 2r
*G^-— F-= 2
or J G : 3 R -H 2 r : : r : 2 (2 R + r).
3R + 2r r
Cor. iG =
2R + r '2
Prop. X VI. — The volume generated by the revolution
of a trochoid about its base is equal to that of a
cylinder having the circle aqb for base and height
equal to the circumference of a circle of radius
-| R + r ; that isy this volume=r\SE + 2 r) t^
By Guldinus' 2nd prop., vol. = (area d'ad)2'rrbG
_ ,2R + r 3R + 2r _ ,.^t3 ^,
= Qaqb ^^-j^-X__y.= 0a^J(3R + 2r)7r
= vol. of cylinder having circle a y i for base, and
height equal to circumference of a circle of radius
i R + r ; or, vol. = r^ (3 R + 2 r)ir\
OEOMBTRY OF CYCLOIDS.
Appekdix to Section III.
ELLIPTICAL IIYPOTROCROID&
The hypiitrochoid becomes an ellipse when the
ilUnff circle is equal to the radius qftbe^fixed
of the J
circle.
Let BB'D (6g. 69) be the fixed circle, BQO the
rolling circle, when tracing point a is od the radina
Fro. Cn. (Sole that two lowsr as are Orerk.)
JJCO. We have already seen (p. 68) that when the
circle has rolled to position B'A'O, the tracing radius
has itB extremity A' on OD perp. to OB, and B'A' is
pcrp, to OD (OC'B' being diametral). Take Ca' on
,C'A', equal to C a, then a' is the tracing point Taking
C A = Ca, describe arc b I'd about O as centre, cutting
OB' in 6'. Then C'i' = Ca', and .-. b'a' is paraUel to
B'A' and perp. to OD, which let it meet in M, and
draw C'N perp. to B'A', bisecting b'a' in n. Then
TROCHOIDS, 126
a'M : a'n :: a' A! : a'C :: aO i aC
.-. a'M: ^'M(=a'M + 2a'w)::aO : O ^.
Wherefore a' is a point on an ellipse having O a
as semi-minor axis, and bb'd as auxiliary circle, —
i.e., having O d and O a (or R + r and R — r) as semi-
axes.
If r > R, or the tracing point is in CO produced,
as at a, it may be shown in like manner that when the
tracing radius has any other position C'A'a', the
tracing point a' lies on an ellipse having O S (D S = O a)
and O a as semi- axes, that is, having semi-axes equal
to r 4- R and r — R, respectively.
ScHOL. — An ellipse with given semi-axes, a and i,
can be traced out equally by taking the radius of the
fixed circle equal to ^(a + b) or ^(a — b). In the former
case, the tracing radius = ^(a + ^)— -^ = ^(a — ^); in
the latter the tracing radius =^(a — Z>) -f ^= ^ (a-f Z>).
THE TRISECTRIX.
When the radius of the rolling circle of an epitro-
choid is equal to that of the fixed circle, and r = 2 R,
the curve is called the trisectrix. The property of
trisecting angles from which it derives its name may
be thus established.
Let BDB' (fig. 70), centre O, be the fixed circle;
EQD, centre C, the rolling circle (ECDO diametral),
when the tracing radius is in the position CDO, or
(since CD = DO = R = ir) the tracing point is at O.
When the rolling circle is in position B'QA', A'C'B'O
126
QEOMETRT OF CYCLOIDS.
diametral, let C'Pp be the tracing radius, cutting
B'QA' in P. Then arc PE' = arc B'D; .-. angle
OCp = angle C'OC ; and aince C'p = OC, the tri-
angles OCp and C'OC are equal in all respects.
Thus,
angle p OC = angle CC'O
and angle COG' = angle p CO ;
.-.angle pOC = angle ;)CC = angle OC'C — p CO
= right angle — \ angle COC — angle p CO
= right angle — f angle p CO
= right angle — ^ angle OC p.
Fra. 70.
Wherefore, if O^ produced meet in B a circle de-
scribed about C as centre, through O,
angle ROC + angle CRO = 2 angle/) OC
= 2 right angles — 3 angle OCp;
TROCHOIDS. 127
but angle ROC + angle CRO
— 2 right angles — angle RCO ;
.'. angle RCO = 3 angle OC/?.
Hence the trisectrix affords the following construction
for trisecting any given angle RCO. With centre C
and radius CO describe arc OR, cutting CR in R.
Join OR, cutting the loop OBC in p ; then angle
RCO = 3 angle p CO, or C/? trisects the angle RCO.
ScHOL, — Both the tricuspid epicycloid and the
tricuspid hypocycloid are trisectrices. See Exs. 91, 92.
THE SPIRAL OP ARCHIMEDES REGARDED AS AN
EPITROCHOID.
The curve traced out by a point retaining a fixed
position with respect to a straight line which rolls
without sliding on a circle, in the same plane as line
and point, may be regarded as an epitrochoid, whose
generating circle has an infinite radius.
Supposing the tracing point on R r, fig. 71, T'DT
the rolling straight line, it will easily be seen that if
this point is near D, the curve will resemble DPR,
only instead of a cusp near D there will be simply
strong curvature convex towards O, and two points of
inflexion, one on each side of R r. When the point is
remote from D, the curve will be concave towards O
throughout. It is easily seen from the formula at page
119 (or it can be readily proved independently^) that
♦ For the independent geometrical proof, it is only necessary
to show that the tracing point recedes from B r initially at the
128
GEOMETRY OF CYCLOIDS,
if the tracing point lies at d such that D rf = DO, the
radius of curvature will be infinite at rf, the two points
of inflexion coinciding there, for from the proportion
r : K::R: F + R,
R-r : R::F : F + R.
we have
Fig. 71.
Wherefore, since the ratio R : F + R is one of
equality when R is infinite,
R-r= F; thati8,r/D = DO.
"When the tracing point is on DR there will be a loop.
We need not consider the various curves traced out
according to the varying position of the point rf, either
same rate at which the point of c6ntact between the generating line
and the fixed circle recedes from R r ; which is obvious, since D d
as it moves with the rolling tangent is constantly parallel to the
radius from O to the point of contact just named, and in its initial
motion the point D moves in direction D r.
TROCHOIDS. 129
on D r or on DK. There is^ however^ one case
which is historically interesting^ and may therefore be
considered here^ though briefly.
When the tracing point is at O, the curve traced
out becomes the spiral of Archimedes^ a curve so called
because^ though invented by Conon, it was first inves-
tigated by his friend Archimedes. It was defined as
the curve traversed by a point moving uniformly along a
straight line, which revolves uniformly aroimd a centre.
So traced it is only perfect as a spiral when the moving
point is supposed first to approach the centre from an
infinite distance, and after reaching the centre to recede
along the prolongation of its former course to an in-
finite distance. Begarded as a trochoid, the complete
spiral (or rather the part near the centre) will be traced
out by supposing TDT' to roll first in one direction
from the position where the tracing point is at D^ and
afterwards in the other direction.
The identity of this epitrochoid with the spiral
of Archimedes is easily demonstrated. Thus, let p
(fig. 72) be a point on the curve, BT the corre-
sponding position of the rolling tangent, P p being the
position of the line which had been coincident with
OD, so that Pj9 is perp. to B'P, and B'P equal in
length to the arc DQB'. Then, since OB' is perp.
to BT and equal to Pj9, Oje? = BT. And O/? is
parallel to BT, the rolling line, whose direction has
changed through an angle measured by the arc DQB',
which is equal to BT or Op. Hence the distance of
p from O is proportional to the angle through which
n
130 QEOMBTRY OF CYCLOIDS.
Op has revolved from its initial direction OQ' (parallel
to DT'). Therefore ^ is a point on a spiral of Archi-
medes.
AEEA OF THE SPIRAL OF ARCHIMEDBS.
The area of the curve is thus determined ; — Let pp'
be neighbouring positions of the tracing point ; B'P^,
' corresponding positions of the rolling tangent
Fio. 72.
B'T'b
with its perp. Then Op\i equal and parallel to B'P;
Oy to B"P'. Wherefore, in the limit, area /»0/»'=
area PB"P'. Hence, increment of area O k rp=: incre-
ment of area DfPB'Q; and t^iese areas begin together:
they are therefore equal. But PB' and P'B" are nor-
mals to D ( P, the involute of the circle DQB ;
TROCHOIDS. 131
therefore, area D t PB'Q = \ ^ ^ ; (see p. 85)
that is, area O rjo = 3 -^|=x— •
ARC OF THE SPIRAL OF ARCHIMEDES.
The arc of this spiral may he thus determined.
Drawing DK (fig. 72), as in fig. 71, and representing
element of arc PP' by an element of area KM' (KM
= DM = B'P), let LM he so taken that element of
area LM' represents the increment of arc pp\ Now
the tangent at p is perp. to B'/?, so that in the limit
(angle pOp' being equal to angle PB'P'),
pp' : PF::B'j9 : B'P;
••• {PPJ • (PPO' - CB»' •• (BT)»
::(PO)2 + (OB0' : (BT)^
or (LM)2 : (KM)^ :: (DM)2 + (OD)2 : (DM)^
::(KM)2 + (OD)2 : (KM)«
.•.(LM2 = (KM)^ + (DO)2
or (LM)2-(KM)2 = (0D)«
Wherefore L is a point on rectangular hyperbola d j'L,
having D d = OD as semi-axis, D as centre, and DK
as an asymptote ; and
arcOrjE? : arc DfP ::hyperb. areaDdLM : aDKM.
:: recti DL + sq. on ODHog, — jr^^ — j : sq. on DM.
or, since arc D<P = ^^ (p. 84) = ^-^ ;
V n rect. DL , f, DM + ML\
X 2
132 GEOMETRY OF CYCLOIDS.
Cor. The loop cuts the axial line BO dmz. point
r, such that r = Q f (the tangent drawn to DQB,
parallel to OD, meeting involute D f R in ^)=arc DQ.
ScHOL, — The curve, as it recedes from O, ap-
proaches more and more closely to the involute of the
circle Q'DQ, the curves being asymptotic. All that has
been said about the figure of the involute of the circle
at a great distance from O (pp. 82, 83), applies there-
fore to the spiral of Archimedes.
We have seen that the epicycloid, traced by the
point O, fig. 72, carried along with T^DT, as it rolls on
the fixed circle Q'DQ, is a spiral of Archimedes. To
prove the converse of this ^ —
Let a point start from O in direction OQ', tra-
velling uniformly with velocity v along radius OQ',
while this radius turns uniformly with angular velocity
ft) around O in direction Q'DQ. After a time ty let
the point be at /? ; then Op = vt and
angle Q'O q (greater than 2 rt. angles) = co ^.
Now if, with radius OQ' = F, we describe a circle
Q'DQB about O as centre, intersecting O/? in y, then
arc Q'D q =z F oot; and if F be such that F co = r
(in other words, if F be such that motion in a circle of
radius F, with angular velocity o) round the centre,
gives linear velocity v), then arc Q'Dy {= F to t)
= v t = Op. Wherefore, drawing OB' perp. to Op,
and completing the rectangle OB'P p,
BT =0p = arc Q'D q = arc B'QD ;
and Fp = OB' = F.
TROCHOIDS. 138
.•. P IS the position of the point D on tangent DT
after rolling round arc DQB' to tangent at B', and P p
is the position then taken up by DO. Hence as TD T
rolls on the circle Q'DQ^ the point O regarded as rigidly
attached to T OT^the tangent to circle Q'DQ of radius
Fy at jD, will trace out a spiral of Archimedes in which
the linear velocity of the moving point along the revolv^
ing radius is equal to F . angular velocity of the latter.
Prop, — The axis of a planefs shadow in space is a
spiral of Archimedes,
The spiral of Archimedes is interesting as the path
along which the centre of a planet's shadow (the
earth's for example) may be regarded as constantly
travelling outwards with the velocity of light.
This is easily seen if we suppose the earth and its
shadow momentarily reduced to rest, and, with the sun
as pole, imagine a radius vector carried from an initial
position coinciding with the earth and retrograding
through the various portions of the shadow. Let V be
the velocity of the earth in her orbit, D her distance from
V
the sun, and therefore yc her angular velocity about the
sun. Also let L be the velocity of light. Then if our
radius vector, carried back through an angle 0, corre-
sponding to the earth's motion in time f, is equal to r, we
have jrt=:^6y or f==.^; and r = L f = -^ 6,
Wherefore, since the radius vector varies as the vecto-
rial angle, the corresponding point of the shadow's axis
134 GEOMETRY OF CYCLOIDS,
(which was at the earth at time t before the epoch we
are considering) lies on a spiral of Archimedes. We
have in fact L, the velocity of light, for the velocity
along the radius vector (r in the preceding demonstra-
tion), when the angular velocity about the sun is taken
V
equal to the earth's angular velocity in her orbit, or ^i
(corresponding to a> in preceding demonstration).
The radius F of the fixed circle by which this
tremendous spiral could be traced out, would therefore
V L
be such that F-pr = L, orF = ^D = the radius of
the earth's orbit increased in the ratio in which the
velocity of light exceeds the velocity of the earth in
her orbit. Thus
F= 92,000,000 miles x ^^^ (roughly)
= 5,000,000 X 187,000 miles
= 935,000,000,000 miles.
[It is convenient to remember that the sun's dis-
tance is nearly equal to five million times the mean
distance traversed by the earth in one second.]
Note. — The student will find further information
respecting spiral epitrochoids in the examples on pp,
254-256. The solution of these examples presents no
difiiculty.
136
Section IV.
MOTION IN CYCLOIDAL CURVES,
Lemma. — When a body at rest at A {fig, 7S) is acted on by
an attractive force residing at C, and varying as the dis-
tance from the centre, the body wUl travel to C in the same
time whatever the distance CA; and if \i , CA is the measure
of the accderatin/g force at A, time of faU to A ==
TT
2'vV
Let AB, perp. to CA, represent the accelerating force at
A ; join CB, then M jt? perp. to CA, meeting CB in p, repre-
Fig.
73.
1
/ ^
/
/
v
f c
/
;»
<^
Jtt
sents the accelerating force at M ; (vel.^ at M) is represented
by 2 .M^BA*=2CAB(^^,5M).'=rect. 6A.(|^y,
(AQA' being a circle about C as centre). That is,
* Any elementary rectangle p m represents M m . accelerating
force at M ; or since the force may be considered miif orm throughout
the space Mm, pM. represents half the increase of the square of the
velocity (by well-known relation in case of uniform force). Hence,
area ^ A represents J (vel.« at M— vel.* at A) = i (vel.* at M).
136 GEOMETRY OF CYCLOIDS.
Vel. at M is represented by ^-^ . rect. h A ;
OQ
or, ' Vel. atM = 9^. V,
where V is the velocity with which a particle would reach
C after traversing distance AC under the force at A con-
tinued constant.
But if Q g is a small element of arc at Q and q m perp.
to CA, then, ultimately,
Therefore time of traversing m M = — — , = %^ ; or,
^ vel. at M V ' '
inert, of time from beginning = — . the inert, of arc AQ.
Hence, time of fall from A to M = — ^ .
But if /I . CA is the measure of the accelerating effect of
the foi-ce at A, V«=2 /x CA . ^= /i (CA)2
or V =>/jli. CA;
Thus, vel. at M= n/ /i . QM ; and time of fall from A to M
= — J ^ = ^-= . circular measure of /i CQA.
Thus, time of fall to C=: —=- . circ. meas. of rt. angles -—=- ;
and is therefore independent of the original distance C A.
ScHOL. — The general relation of this lemma may be re-
garded as obvious, seeing that a force varying as the dis-
tance from the centre is in this case a force varying as the
distance remaining to be traversed ; and this relation holding
from the beginning, it follows that whether such distance be
MOTION IN CYCLOID AL CURVES, 137
large or small, it will be traversed in the same time. The
general relation may be considered, in this aspect, as
follows : —
Let C, fig. 74, be the centre of force, and let one particle
start from A, another from a, in the same straight lin'* CA.
Divide C A and C a each into the same number of equal
elements, and let ^, m, n, and L, M, N be the points of divi-
Fio. 74.
Tttli-
J^JL.
MMLA
sion nearest to a and A, respectively. Then the force on the
particles starting from A and a may be regarded as severally
uniform while these particles traverse the spaces AL, a
respectively ; hence these spaces being proportional to AC,
a C, that is to the uniform forces under which they are tra-
versed, will be traversed in equal times ; and velocities pro-
portional to the forces, that is to ML and I m respectively,
will be generated in those times. Again, since the forces
acting on the particles at L and I are proportional to the
spaces LM, I m, and the velocities with which the particles
b^in to traverse these spaces also proportional to LM, I m,
it follows that the times in LM, I m, will be equal ; and the
total velocity acquired at the end of those times will still be
proportional to ML and mZ, or to MN and m^i, the spaces
next to be traversed. And so on continually. Hence the
particles will arrive at C simultaneously ; and the velocities
with which they reach C will be proportional to AC and a c.
It is manifest, also, that if the particles during their
progress to C be resisted in a degree constantly proportional
to the velocity, the times of reaching C will still be equal.
138
GEOMETRY OF CYCLOIDS.
PROPOSITIONS.
Prop. I. — If A {fig, 75) he the vertex, AB the aoda of an in
verted cycloid DP Ay a particle let fall from a point F on
the arc APD {supposed perfectly smooth) will reach A in the
same tim^e wherever F may be.
Let P be a point on the arc AF; draw PM perp. to AB
cutting the generating circle in Q and join AQ. Bepreeent
Fig. 76. (Join AQ.)
the accelerating force of gravity by g. Then since the tan-
gent at P is parallel to AQ,
Acc«. force at P along PP' :g :: AM : AQ : : AQ : AB ;
or, the accelerating force at P in direction of motion
AQ arc AP
"■^AB""^* 2AB
Hence if the straight line ad=s arc AD, and we take
af:=axc AF, and a j9=arc AP, the acceleration of the particle
at P is the same as that of a particle moving from/ to a under
the action of a force varying as the distance from a, and
equal at j9 to^ . ~^, or at c? to ^ . --—= =:g. The time of fidl,
2AB 2A^
then, (by lemma, p. 135) is independent of the position of F.
MOTION IN CYCLOID AL CURVES, 139
Since in this case the accelerating force at D = ^ =
- ^ . arc APD, the /x of lemma=^, and time of fell from any
point of arc APD to A = . /^ '1='^ \ /- •
\/ g 2, \/ g
The time of oscillation from rest to rest on either side
of A = 27r. /I.
ScHOL. — This proposition is easily established indepen-
dently. Thus take an elementary arc PP' ; draw ordinates
FHK, PQM, and P Q' (Q,Q', on BQA); arcs Qn, (^'n' about
A as centre, to AH ; and n q, n'q' pei*p. to AH, meeting
quadrantal arc Hg'N on AH in q, q'. Then, (vel.)^ along
PP = 2^.KM
= 2g (AK- AM) = -^ (AH«- AQ^) ^^.{nq^; or,
/. vel. aitF=:./^9 (nq); & PP =2(AQ-AQ )=2nn';
time along PF = 2 ww' -- . /_?£.(n^)= . /?A? . ^ ;
= A /fr^.circ. meas. ofqAq';
and time along FPA
Pkop. II. — A particle toiU pass in the same time to A along
a smooth epicydoidal arc APD {A the vertex, APB dia-
metral,) under the action of a repulsive force at varying
directly as the distance, from whatever point on APD the
particle starts.
Let the particle start from F. At P on the arc FA>
draw the tangent ATT, and the normal PB'; then OB A' cuts
140 OEOMETRY OF CYCLOIDS.
the generating circle through P diametrally in B'A', {B' on
the base BD); and OT perp. to A'T is parallel to B P. De-
scribe arc PQ about as centre, to meet central generating
circle AQB, and join OP, AQ.
Then if the meaaure of aocelerating force at A = ^ , OA,
accelerating force at F ^ fi . OP ; and the accelerating
force in direction PF = /;i OP. p^ = fi PT
= //AP
. Arc APD (Prop. VI., aec. 2).
OP"
OB' ^ 2AQ.0C (OB)'
B A' ^ '' GB~ ■ 2BA . OC
4R(r+E) ■
Fio. 76. (Accent upper n and q.)
Therefore (applying lemma, p. 135,
cycloid) the time in which particle reaches A
^ ^/ JB(r+R) 5^5 / B{F+R)
The time of oscillation from rest to rest on either side of
A is twice this.
ScHOL. — This proposition may be proved independently
of the lemma, by a demonstration similar to that used for the
cycloid. The figure indicates the construction. We begin
MOTION IN CYCLOID AL CURVES. 141
by showing that (Vel.)^ at M = // [(OP)^ - (OF)^]
= fi [(OM)H (MQ)2- (OK)2-(KH)2]
=/ii[MK (OM+OK)-f AM (MK+KB)-KB(MK+AM)]
= y[x.2MK(0A+0B)
= ^ [(AH)«- (AQ)»)] ?^
The rest follows directly, as in case of cycloid.
Prop. III. — A particle vdll pass in the same time to the ver-
tex of a smooth hypocydoidal arc under the action of an
attractive force at the centre varying directly as the distance,
from whatever point on the arc the particle starts.
The construction and demonstration are in all respects
similar to those in the case of epicycloid, Prop. II.
Time of motion from F to A =^ . / ^ (^-^) .
FV ^
and (Vel.)« at P = ytx {nqf (^^\
SOHOL. — ^The time of oscillation in the epicycloid under
force above considered : time of an oscillation in cycloid
under gravity (the radii of generating circles being equal)
:: Vg (F+B) : F^/ji
This follows directly from the values above determined
for the times of motion to A.
That the times of oscillation may be equal, we must have
F + R
Since this gives /n F = — = — g, it follows that the accele-
142 GEOMETRY OF CYCLOIDS,
rating force at A in the epicycloid must exceed the force of gra-
vity in the ratio OC : OB, in order that the oscillations may be
performed in the same time as in a cycloid of equal generating
circle, under gravity. The force in the epicycloid will equal
gravity at a distance from = - — — = OK', obtained as in
F+R
fig. 76 by drawing BK' perp. to 00 to meet semicircle on
00 as diameter in K'.
If we take /x F = ^r, a cycloid in which the oscillations
under gravity will be the same as the oscillations in the epi-
cycloid must have a generating circle whose radius
(F+B)Il 00. OB (OKO^ J,. , ^ . ,, ,
=^ Y^— = — Qg— = 05 = S ^ obtained by draw-
ing semicircle B A; 0, taking B A; = OK', and drawing k h
perp. to BO.
Oorresponding considerations and constructions apply in
the case of hypocycloid.
It is manifest (see scholium to lemma) that if the par-
ticle in its passage along , the epicycloidal, hypocycloidal, or
cycloidal arc, be resisted in a degree constantly proportional
to the velocity, the periods of oscillation will still be isochro-
nous; the arc of oscillation, however, will no longer be sym-
metrical on either side of the axis, but will continually be
reduced, each complete arc of oscillation being less than the
arc last described.
A weight may be caused to oscillate in the arc of an
inverted cycloid in the manner indicated in ^g, 77. Here
a A is a string swinging between two cycloidal cheeks apDy
ap'jy\ a being a cusp, and DD', the common tangent at the
vertices D, D , being horizontal. The length of the string
a A being equal to twice the axis of apD, or to the arc
apjy, the weight swings in the cycloidal arc DAD ' (Prop. XI.
section 1). Such a pendulum would vibrate isochronously,
MOTION IN CYCLOID AL CURVES,
143
if there were no friction and the string were weightless ; but
in practice the cycloidal pendulum does not vibrate with
perfect isochronism.
An approach to isochronism is secured in the case of an
ordinary pendulum by having the arc of vibration small
compared with the length of the pendulum. In this case
the small circular arc described by the bob may be regarded
as coincident with a small portion of the cycloidal arc DAD'
(fig. 75) near to A, and the isochronism thence inferred. But
in reality the approach to isochronism in the case of a long
pendulum oscillating in a small arc, is best proved as a direct
consequence of the relation established in the lemma.
Thus, let ACA' (fig. 73) be the arc of osciUation of a pen-
dulum, whose length Z is so great, compared with AA', that
ACA' may be regarded as straight. Then the accelerating
force in the direction of the bob's motion when at M
= g . sin. deflection from the vertical ^^ . — p very nearly,
or varies as CM. Hence the time of oscillation is very
nearly constant, whatever the range on either side of C, so
only that the arc of oscillation continues very small com-
pared with L
The accelerating force towards at M being ^ . CM,
V
144
GEOMETRY OF CYCLOIDS.
the time of an oscillatioii from rest to rest
and the Vel. at M=:QMa/| ^ a/I (^^*-"^^*)-
A pendulum may be made to swing in an epic^cloidal
arc in the way shown in fig. 78, or in a hypocydoidal arc in
Fig. 78.
the way shown in ^, 79 (Prop. XII. sect. 2) ; but of course
the oscillations will not be isochronous under gravity. In the
Fig. 79.
case of the hypocycloid, if the plane of fig. 79 be supposed hori-
OJP MOTION IN CYCLOID AL CURVES. 146
zontal, P a smooth ring running on the arc DAD', and this
ling be connected with the centre of the fixed ciitsle by an
exceedingly elastic string, very much stretched, the oscilla>
tions of the ring will be very nearly isochronous. For the
tension of a stretched elastic string is proportional to the
extension, and if when the ring is at A the stidng is
stretched to- many times its original length, the extension
when the ring is at different parts of the arc DAD' is very
nearly proportional to the extended length. Suppose, for
instance, that when at A the string were extended to 100
times its original length,, then the extension would only be
less than the actual length by one 100th part.
If the circular arc DD' represent part of a gi'eat circle
of the earth's surface, DAD' a hypocycloidal tunnelling hav-
ing DD' as base, then, since the attraction at points below
the surface of the earth varies directly as the distance from
the centre, a body would oscillate in DAD in equal periods.
It would not, however, be possible to construct such a tun-
nelling, or to make its surface perfectly smooth.
Prop. IV. — The path of quickest descent from D to any
poiTit F not vertically below B, is a cycloidal arc through
F, having its cusp at D and its axis vertical.
The following is a modification of Bernouilli's original
demonstration.
The path of descent will necessarily be in the vertical
plane through D and F. Let it be DPF, and let PP' P" be
a small portion of this path, represented on a much enlarged
scale in fig. 80a.
Let jo be a point on a horizontal line through P', and close
to P . Then since DPF is the path of quickest descent, the
L
146
GEOMETRY OF CYCLOIDS.
time of descent down the arc PPT" is a minimum, and
from the nature of maxima and minima it follows that the
change in the time of fall resulting from altering the arc
pp/p// jjj^ ^g Q^Q VpV" is evanescent, compared with the
total time of fall down PPT'^ If this time were increased in
an appreciable ratio by passing from P' to a point p on one
side, it would be appreciably diminished by passing from P'
to a point on the other side of P', which is contrary to the
supposition that DPF is the arc of quickest descent. Now
regarding PP' and P' P" as straight lines, draw p'l perp. to
PP' and Ym perp. to P" jt?', so that ultimately P^= P/>', and
Fig. 80a.
P"7?i=:P'T'. Therefore, if we suppose PP' and Vp' traversed
with the imiform velocity V, then -^ represents the ex-
cess of time in PP' over time in Pjo' ; and if we suppose
P' P" and p P' traversed with the uniform velocity V,
then ^7 represents the defect of time in P' P" fi*om
time in pl^". Therefore since time along PP P" = time
P7 pm V P7
^ or — ss —
V p'm
cos PP'»'
— —pT-TpT/ • That is, the velocity at different points along
the arc of descent varies as the cosine of the angle at
which the arc is inclined to the horizon at these points. But
along Py P", we must have ^- = \T^ ®^ =" ~
OF MOTION IN CYCLOIBAL CURVES. 147
t^ is a property of motion in an inverted cycloid. For if
DPFAiy is a cycloidal arc having D and D as cusps, AB
as axis, and AB vertical, and PL is drawn perp. to AB,
cutting central generating circle in Q, then
(VeL)»atP = 2(7.BL = 2(7^ = 4(7R.(|^)'
Le. VeLat P = 2 v/^. cos ABQ = 2 ^/^R . cos AQL,
"the required relation, since AQ is parallel to the tangent
»t P.
Hence DPF is part of a cycloid having its cusp at D and
it» axis verticaL
To describe the required arc, draw any cycloid T)f{V
tkSL-^jmg D as cusp, its base D d' horizontal, and cutting DF
ixx J\ then D' so taken that
DD' : J^d :: df : D/
is 'C^he base of the required cycloid through F. The axis BA,
l>isecting DD' at right angles, bears to 6 a, the axis of D a d\
tlx« ratio DF : D/.
ScHOL. — The arc is not necessarily one of descent
^l^^^'oiighout. If F' be the point to be reached, and the angle
^^ inclination of D/' to the horizon is less than tlie angle
* ^^ a, the path from D to F' will include the vertex A, and
*"^ particle will be ascending from A to F'.
^The cycloid DAIV is the path of quickest motion from
^^ D' at the same horizontal level as D.
l2
OEOMETRY OF CYCLOIDS.
Sectiok V.
EFICYCLICS.
DiV.— 1/ a poiiU travels unifortaly round the dreumferenee
of a circ^, wltoM eenire travels uniformly round the cir-
cumference of a Jixed cirde in the same plane, the curve
traced out by tJie moving point is called an epieydic.
Let AQB (£g. 81) be the circle round which the tracing
point travels, CC'K the circle in which the centre C of the
moving circle AQB is carried, the centre of the fixed circle
CC'K. Then the circle CC'K, is called the deferent, AQB the
Fio. 81. (Join Cp.)
epicycle, O the centre, C the nwon point, P the tracing point.
At the beginning of the motion let the tnU'ing point be at A
in OC produced, or at its greatest possible distance from O.
When the centre is at C ' let the tracing point be at P. Draw
the epicyclic radius C'a parallel to CA, and let OC produced
SPICTCUCS. 149
meet ihe ^icyde in A': ilIso kt OA and OA' cut the
epicycle respectiTriT in B and B*. Tbcn C'a is the position
to which C A has been carried bv the motion of the epicTcle,
and a A'P is the arc over which tiie tracing point has tra>
veiled, in the same time. The angle PC a is caUed the epi-
cyclic angle J and the angle C'OC the deferent iai angie. Both
motions being uniform, the deferential ai^le bears a constant
ratio to the ^cyclic ai^le. Call this ratio 1 : n ; so that
1 : 97 is the ratio of the angolar velocities of mean point
round centre, and of tracing point round mean point. If we
^represent the radius of the deferent by D, and the radius of
the epicycle by £, the linear velocities of the motions just
)nentioned are in the ratio D I n E.
The deferential motion may be conveniently supposed to
take place in all cases in the same direction around O, — that
indicated by the arrow on CC. Such motion is called direct.
Angular motion in the reverse direction is caUed retrograde.
When the motion of the tracing point round the mean point is
direct, n is positive ; we may for convenience say in this case
that the epicycle is direct, or that the curve is a direct epicy^
die. When the motion of the tracing point round the meiEui
point is retrograde (as, for instance, if the tracing point had
moved over arc a^' P' while mean point moved over arc
CC), n is n^ative; and we say the epicycle is retrograde, or
that the curve is a retrograde epicyclic.
The straight line joining the centre and the tiucing point
in any position is called the radius vector, A point sucli as
A, where the tracing point is at its greatest distance (D + E)
from O, is called an apocentre, A point where the tracing
point is at its least distance (D -^ E) from the centre is
called a pericentre. Taking an apocentre as A for starting
point, OA is called the initial line, and the angle between the
160 GEOMETRY OF CYCLOIDS.
radius vector and the initial line is called the vectorial angle.
This angle is estimated always in the same direction as the
deferential angle : so that if at the beginning the motion of
the tracing point round O was retrograde, the vectorial angle
would at first be negative.
Whatever value n may have, save 1 (in which case
the tracing point will manifestly move in the circle AA'),
the tracing point will pass alternately from apooentre on the
circle AA' to pericentre on the circle BB', thence to apocentre
on the circle AA', and so on continually. The angle between
an apocentral radius vector and the next pericentral radius
vector is called the angle of descent. It is manifestly equal to
the angle between a pericentral radius vector and the next
apocentral radius vector, called the angle of ascent.
PROPOSITIONS.
Prop. I. — TJie angle of descent ; two right angles :: ni^/l I 1.
When n is positive and greater than 1^ the epicyclic
angle PC a {^, 81) exceeds the deferential angle COC, or
A'C'a, by PC A, or angle PCA':t=(w-l) deferential angle.
But, at the first pericentre, angle PCA'=:2 right angles, and
the deferential angle is the angle of descent. Hence,
2 right angles = (w — 1) angle of descent,
or the angle of descent : two right angles II w — 1 : 1.
When n is positive and less than 1, A'Ca exceeds the
epicyclic angle p Q'a by p CA', or angle p CA' = (1 — w)
deferential angle ; and proceeding as in the last case, we find
the angle of descent : two right angles : : 1 — n : 1.
When n is negative, we have the epicyclic angle a Q'V
+ angle A'Ca = angle P'CA', or (taking the absolute value
EPICYCLICS, Iftl
of n without regard to sign) angle P'C'A'= (w + 1) deferen-
tial angle. Wherefore (proceeding as before),
the angle of descent : two right angles X (n + 1) ' 1.
But n being n^ative, the sum of the absolute values of 1
and n is the difference of their algebraic values, or n *^ 1.
Hence for all three cases,
angle of descent : two right angles \ln^\ : 1.
ScHOL. — The angle of descent is always positive. See
note, p. 185.
Prop. II. — The epicycle traced with deferential and epicyclic
radii D and E^ respectively^ and epicyclic vel, : deferen-
tial vel, II n I 1, can also he traced with deferential and
epicyclic radii E and D respectively^ and epicyclic vel, 1 (fe-
ferential vel, ^ • 1 : w.
In fig. 81, complete the parallelogram PC'Oc'. Then
O (/ = C P = E and c'P = OC = D. Moreover ^ c'OQ
= C PC'a, and c'P is parallel to OC. Wherefore we see that
while the epicyclic curve is traced out by the motion already
described, the point c' travels in a circle of radius E about O
as centre, with the same velocity as P round C ; while P
travels uniformly in a circle of radius E roimd c,' and with
the same velocity as C roimd O.
Therefore the same epicyclic curve is traced out with
deferent and epicycle of radii D, E, respectively, having
angular velocities as ti : 1, or by deferent and epicycle of
radii E, D, respectively, having angular velocities as 1 ; n.
ScHOL. — Thus the deferential and epicyclic radii, D and
E, can always be so taken that D is not less than E. When
D =: E, the curve can still be regarded as traced in either of
two ways, viz., with epicyclic vel. to deferential vel. \\n \ 1
or ; ; 1 ; n. In this case all the pericentres fall at the centre.
152 GEOMETRY OF CYCLOIDS.
Prop. III. — Every epitrochoid is a direct ejncydxc ; and every
hypofrochoid is a retrograde epicycHc.
Let be the centre of a fixed circle BB'D (fig. 82) on
which rolls the circle AQB ; and let the tracing point be at
r on CA.* Let the circle AQB roll uniformly to the position
A'Q'B', G'p P being the position of the gen«<ating radius, p
the tracing point. Draw C'Q' parallel to OC. Then thfe
centre C of the rolling circle has travelled uniformly in
circle CC about O as centre. Again ^. Q'C> = Z Q'C'A'
+ Z A'C>=COcYl+ ?) (since arc. AT = arc B'B).
'Wherefore p is a, point on an epicyclicJ arq, whose defer-
ent and epicycle have radii OC and C r, or (R + F) and r
respectively, and whose epicycHc angle I deferential angle
:: R+F : B. Or, by preceding proposition, we may have
r and R + F f or radii of deferent and epicycle respectively,
having R ^ B+F for ratio of epicycUc and deferential angles.
In this case n is greater than 1 and positive.
Next, fig. 83, let the circle AQB roll around instead of on
the circle BB'D. Then the above proof holds in all respects,
save that the angle QVp now = Z Q'C'A' — /. A'G'p, and
radius OC = R — F instead of R + F. Thus in this case, the
epitrochoid gives an epicyclic curve having for deferential
and epicyclic radii (R— F) and r, respectively, and deferen-
tial angle : epicyclic angle :i R— F : R; or else, deferential
and epicyclic radii r and (R— F) and ratio of deferential and
epicyclic angles as R : R— F.
In this case n is less than 1 and positive.
Next let be the centre of a fixed circle BB'D, inside
^hich, figs. 84 and 85, rolls the circle AQB ; and let the
* Or at /, on CA produced, in which case read p' torp through-
put the demonstration, for all fo.ur casesi. :
EPICYCLIC&
163
iii-ftcing point be al r. Then following the words of proof
for the case of epitrochoid with modifications corresponding
to the two figs. 84 and 85, the student will have no difficulty
in showing that the hypotrochoid, in the case illustrated by
each of these figures, may either have deferential and epi-
cyclic radii (F— R) and r, and deferential angle : epicyclic
Migle : : F — R r E ; or epicyelic and deferential mdii r
Fio. 84. Fio. 85.
and (F — R), and deferential velocity ; epicyclic velocity
::R:F-E.
Since F has moved round C in a direction contrary to
tliat in which C has moved round 0, n is negative in both
cases. IfF-R>RorF>2R,reiK >1; this is the
case illustrated by fig. 84, K F - R < E or F < 2 R, the
caae illustrated by fig. 85, m ia < 1.
IM GEOMETRY OF CYCLOIDS.
ScHOL. — We may find in this proposition another reawm
for regarding the curve traced out by a point on, or within,
or without a circle which roUa outaide a fixed circle, but is
touched by that circle internally, as an epitrocboid, not as a
hypotrochoid, for this definition leads again {while the-ethw
does not) to a Bymmetrical classification, giving epitrochoids
■as direct epicyclic curves, and hypotrochoids as retrc^isde
epicyclio curves.
Prop. IV. — Every direct epioydie it an epUrochoid ; and
every retrograde epieydic it a hypotrochoid.
Let ^ be a point on an epicyclic curve pp', OC ( n: D)
the radius of deferent, Cp ( ^ E) the radios of epicycle ;,
Fid. 8S.
n positiveand > 1. Then the motion of ^ may be resolved
into two, one perp, to CO, the other perp. to C ^, Eepre-
sent these by the straight lines pS, pii, taking p'M. = pC
_co.
and therefore /> N =
j then the diameter ^ T of the pa-
rallelogram NpMT represente the motion of p in direction
and magnitude. Complete the pamllelogram ^COc; take
PN'=pN; and draw N'B parallel to cO to meet OC in B.
EPICYCLICS. 1S5
Suppose the parallelogram NM turned (in its own plane)
roond the point p through one right angle in the direction
shown by the curved arrow, making p M coincide with p C
and the parall. KM with parall. WC Then p B, the dia-
meter of the parallelogram N'C, is the normal at p.
Now, by the preceding proposition, if a circle DBB , having
centre at C and radius CB, roll on the fixed circle KBL having
centre at and radius OB, the epitrochoid traced out by p,
at distance Op from C, will be the epicyclic having Gp as
radius of epicycle, CO as radius of deferent, and epicyclic
ang. veL : deferential ang. vel. ;; 00 : OB ;; n ; 1. It will
therefore be the epicyclic pp\
Fio. 87. Fio.,88.
Thus the epicyclicpp' is an epitrochoid having
r = BO = D Tl-^); E = CB = 5;ftnd7-=E.
We get precisely the same construction for the position
of the normal pB by interchanging the radii and the
angular velocities of deferent and epicycle, that is, taking
O c as radius of deferent ftnd c p as radius of epicycle. Let
p B and c 0, produced (if necessary) intersect in 'b'. Then
166 GEOMETRY OF CYCLOIDS.
b'O I fi'c;:OB ',cp','.n—l : n; and by the preceding propo-
sition, if a circle dh b", with centre at c and radiua c6', roll
outside but in internal contact with the circle k b'l having
centre at and radius O b', the epitrocboid traced out by p at
diatance cp from c will be the epicyclic having ep as radius
of epicycle, cO as radiits of deferent, and epicyclic ang. vel. :
deferential ang. vel. :: cO ; cb' ;; \ ; n. It will therefoitt
be the epicyclic jj^'. Therefore p/>' is an epitnx^oid having
F = 6'0 = D (n - 1) ; R = c 6' = D . « ; and r = E.
It will be found that the demonstration applies equallv
to the case of tlje direct epicyclic where n < 1 , illustrated in
fig. 87, only that N' lies on pe produced. The two corre-
sponding epitrochoids have
(1) F = BO = D Tl - '"] J E = CB = B ; and r =E.
(2) F = yO = D (1 - n) ; R = c J =D M ; and r =E.
Moreover, it will be found that the demonstration
applies 'With slight (and obvious) alterations to the case of
the retrograde epicyclic illustrated in fig. 88. (In the case
illustrated, n > 1 1 it is not necessary to illustrjite sepa-
rately the case in which m < 1 ), We obtain for ^he two
corresponding hypotrocholds, —
EPICYCLICS. 157
(1) P = BO == D fl + IV R = CB = 5 ; and r = E.
(2)F = 50 = b (1 +w) ; R = c6'=Dw; andr = E.
ScHOL. — A number of cases resulting from .varieties in
the position of p are illustrated by the dotted constructions,
and in figs. 89 and 90 (cases in which there is retrogression
about O, h lying between O and B). The reader will have
no difficulty either in understanding these, or in illustrat-
ing many other cases resulting from variations in the values
of D, E, and n.
Prop. V. — The normal at any point p of an epitrochoid or
hypotrochoid passes through the point of coniact B of the
fixed circle with the rolling circle when the tracing point is
at p.
The demonstration of the preceding proposition includes,
the proof of this general proposition. The motion of jo being
at the instant precisely the same as though the circle B were
rolling on the tangent to the fixed circle at B^ 'it follows that
if N^ (= CB) represent the linear velocity of jt? in direc-
tion perp. to CO due to the advance of centre C of rolling
circle D BB , jo M = jt? C represents on the same scale the
linear velocity of jo in direction perp. to C jt? ; wherefore p T,
the diameter of the parallelogram NM, represents the re-
sultant linear velocity of p ; and as in the demonstration of
preceding proposition, if the parallelogram KM be rotated
round jo in its own plane, through a right angle, in the direc-
tion indicated by the curved arrow, pT is brought to coin-
cidence with p B, which is therefore the normal at p.
' V •■*.. . . .
158 GEOMETRY OF CYCLOIDS.
■
Prop. VI. — To determine the apocentral and pericentral
velocities in epicydic curves.
From Prop. IV. fig. 86, we see that if the linear velocity of
p around C is represented by p C, that is, by E, the linear velo-
city of pis represented by p T, perp. to ^ B, in direction, and
by ^ T in magnitude, where CB ( = — ) represents the linear
velocity of C about O.
Hence the velocity at an apocentre is represented on the
same scale by B a, and the velocity at a pericentre by O ft, a
and b being the points in which OC, produced if necessary,
meets the circle pp^ jOg, a the remoter. That is, the linear
velocity at apocentre = — + E. On the same scale the linear
velocity of the mean centre = - ; and
lin. vel. at apocen. : lin. vel. of mean cen. : lin. vel. at pericen.
::?.+ E : ^ : --E
n n n
:: D +wE: D : D - nE;
n being positive in case of direct epicyclic and negative in
case of retrograde epicyclic.
Thus in the case of the direct epicyclic the motion at an
apocentre is always direct ; while the motion at a pericentre is
direct, retrograde, or negative, according as D < or > w E, or
afiCB,fig 86,( =— jVor<C6. In the case of the retrograde
epicyclic the motion at an apocentre is direct or retrograde
according as D> or <n E, or as CB r=: — j > or < Co,
fig. 88 ; while the motion at a pericentre is always direct.
ScHOL. — If D=wE, there is a cusp at pericen. or apocen.
EPICYCLICS,
150
Prop. VII.— 2\) determine the position of the points, if any,
where the motion of the radivs vector becomes retrograde*
It is manifest that if , as in the cases illustrated by figs.
86, 87, and 88, the point B lies outside the circle pp\ p^y or
D > n E, the motion, direct both at apocentres and pericen-
tres, is direct throughout. For the motion to be retrograde
in part of the epicyclic, then, we require that D be < n E, or
CB < a. Since the direction &t pia perp. toBp, the mo-
Fio. 91.
Fio. 92.
tion will be directly towards or from centre if Bp is at right
angles to Op, for then Op will be the tangent at p» We
have then the relations presented in fig. 91 for direct epi-
cyclic, and in ftg. 92 for retrograde epicyclic.
Op is the distance from O at which the epicyclic becomes
retrograde (for all smaller distances in case of direct epicyclic,
and for all greater distances in case of retrograde epicyclic).
Manifestly the distance Op is determined by describing a
semicircle on OB intersecting a'p h' in p. Now the angle
pQ'a' ^{n — I) deferential angle (measured from apocen-
tral initial radius vector), say Z. jt? C'a' = (w — 1) 0, and we
might proceed by the epicyclic method of treatment to
160 GEOMETRY OF CYCLOIDS.
determine geometrically. We have, however, already thfi.
means of doing this, in the result of Prop. XIV., Sect. Ill,
Thus, draw p K perp. to C O ; then
Cos «C«= — — =— ^^ ^ = — ^ L-L ,^.
(O A of Prop. XIV. sec. 3=—')
T) * ^
E2 + D . -
Cos (w — 1) ^ = — —
eCd ,5) (1+«)DE'
71 being negative in case of retrograde epicyclic.
Cor. If ^i be the value of ^ determined from this equation,
the motion is retrograde from ^ ^ i to ^ = — 0i.
n—\
ScHOL. — The angle ^i is of course the angle which OC
makes with the initial line, and does not directly indicate
the arc of retrogradation, which is twice, the angle p O d.
This, however, may be readily deduced in any given case. For
tan « O 6 = ^ -- = — ^ ^ "" '* \ — is known, and there-
^ KG D + Ecos(7i— l)(Pi
180'
fore, «Oc? = 0i+«O6' — — =- is also known.
n-\
It can easily be shown that
^ ' (1 + w) I>E
and tanj»0 6=i./?lE5'
EPICYCLICS. 161
Pbop. VIII. — To determine the tangential, transverse, and
radial velocities, and the angular velocity around the
centre at any point of an epicydic curve.
Let pi (figs. 86, 87, 88) be the position of the point on
the epicycle apih. Join Opi and draw B A perp. to Op^,
Then when Cpi (= E) represents the linear velocity in the
epicycle, Opi represents the linear vel. at j^i in magnitude,
but is at right angles to the direction of motion at joj.
Hence pi h represents the linear velocity perp. to the radius
vector, and B h represents the linear velocity in the direction
of the radius vector, the direction of the motion in either
Case being determined by conceiving pi C turned around pi,
carrying with it jt?i B and jOj A, in the plane of the figure,
through a right angle, to coincidence with the direction of
Pi8 motion in the circle a p b. This includes all cases geo-
metrically, and the student will have no difficulty in effect-
ing the construction and deducing the proper directions for
the tangential, transverse, and radial velocities, for any given
values of D, E, and n, and for any given position of the
moving point. The angular velocities are determined by
the same construction. Thus in the case illustrated by fig. 86 :
The tangential velocity of pi is represented hy p^ B in
magnitude and is in advancing direction shown by arrow
at pi.
The transverse velocity of jo, is represented hy pi h in
magnitude, and in direction by B h.
The radial velocity of jOj is represented by B A in magni-
tude, and in direction by pi h.
The angular velocity oip^ about : uniform angular velo-
city of ;?! about C :: ^ : Pl^wpji : Op^.
Opx P\^
And similarly for all other cases.
M
162 GEOMETRY OF CYCLOIDS.
It is more convenient, however, where so many cases
arise, to obtain the analytical expressions for these quanti-
ties ; for we know that by rightly considering the signs of the
values used and obtained, the same expression will be correct
for all possible cases. Let then the angle p^Qa (iig. 86)
= (n— 1) ; that is, let the deferential angle = ^ ; let the
linear velocity of the mean point (C) be V, wherefore the
E
linear velocity of the moving point in the epicycle = nY .
This is what we have represented linearly by jt?i C in figs.
86, 87, and 88, so that since jt?i = E, we have to affect all
the above linear representations of velocity with the co-effi-
. ,nY
cient - — :
Therefore, the tangential vel.
nY ^ wV
= -^ . ;?i B = -^^ {PiC)2 + (CB)2+2joiC.CBcosjt>Ca.
^^ . / Ti^2^ I>^ . 2DE , Tv^
= j^ >/ D2 + 7i2 E2 + 2 71 DEcos {n-\)(p.
nY
The transverse vel. = ^ . jOi B . cos B jOj O :
now,cosB;,.O=(l£l)l±%0!)^(M'; .-.p.B.cosB^.O
2jO| B. jOi O ^
E2+J?.%25?cos(;i-l)</) + E2+D2+2DEcos(M-m-/'D -.?V
_ n^ n \ n/
and transverse vel. (direct)
^V D^ + nW +{n-h l)DEcos(n~l)j>
^ * >/ r)2qrE2"+"2DE cos (n- 1) <^
Y
The radial vel. = :^ PiB . sin B;?iO :
EPICYCLICS. 103
sin B «i BO
now — ; ^-i— - = — - :
sinjOiOB jOiB'
therefore,
;>,B8inB;>,0=BOsin;>iOB=/'D-^V ^^(^~^)*;
and radial vel. (towards centre)
=(^-l)V. E8in(n~l)^
^/D2 + E*^ + 2DE'cos(n-l)9
The angular velocity about O
_ transv. vel. _ Y D^ + ^^E^-f (^^-fl) DE co s ( n -!)(/ >
rad. vect. "" D * D2 + E2 + 2DEcos(7i-^l)~^
The transverse vel. and the angular vel. about vanish, if
D2 + nE2+ {nJf 1)DE cos (71-1)^ = 0,
the condition already obtained.
If V is the velocity in epicycle, v =: V ^ , or V=v —
XJ 9i E
which value substituted for V in the above formulae gives
formulae enabling us to compare the various velocities with
the velocity in the epicycle.
ScHOL. — We see from the geometrical construction that
the radial velocity has its maximum value towards or from
the centre, when the moving point is at pi orp^ (figs. 86, &c.),
where a tangent from meets the circle apib; for then B h
or B h has its gi'eatest value. This also may be thus seen :
— Since the deferential motion gives no radial velocity, the
radial velocity will have a maximum value when the epicyclic
motion is directly towards or from the fixed centre, — that is,
at the points where a tangent fr*om the fixed centre to the
epicycle meets this circle.
Cor. The angular vel. at apocentre > = or < angu-
lar vel. at pericentre, according as
aB > bB aB > aO
^■^ ^=s - .— . « or as 4 — =L. ^=s rz — ,_^
aO<6 6B^60
m2
104
GEOMETRY OF CYCLOIDS.
Piiop. IK. — To ditermine whzn epicydic loops touch.
For this we mist have /LpOd (figs. 93, 94) = angle of
ddscent ; that is, see Schol. to Prop. VII.,
Fia. 93.
Fio. 94.
— -cos-M —
n— 1 L
w. _L AA/ 180° 180°
(l+n)DE
!j UV D2-eO
or
360°
w-1
Prop. X. — To determine the position of poinds of vnfleadon.
If j9, figs. 95, 96, as in Prop. XIII., sec. 3, is a point of
inflexion, we have as in that proposition
2 Q'k . 0'« = C'B' . C I ± (0»«
(lower sign for retrograde epicyclic)
or (C'B It CI) Q'z = C'B' . CI ± (C'j9)«
• ^=-cos(r.-lU-92?::-51±j(2W
- Qi^- ^^V^ ^)^- (C'B'± CI) C'j.
Now by Cor. to Prop. XIL, Sect. III., CI= f 5 V^ D = 5.
EPICYCLICS
Iflfi
to be r^arded as n^ative for retrograde epicyclie. Ht-nce
FiQ. 06.
n being negative in case of retrograde epicyclie.
Cor. If f 2 be the value of f deteimiseii from this equa-
tion, there is a point of contrary flexure when fl- = flpj and
360^
another when ^ =
ScHOL.^The angular range round of the arc Letveen
the points of flexure can he determined, as in case of arc of
retrogiadation, see scholium to Prop. VII. We have
tajipOb- {%». 95 and 96) = ^-1 _ Eam(«-l)y,_
wherefore, if d be the pericentre
p d= ■ — -— pj — p 6 , is also knowit.
It is easily shown tliat
■ D+Eoo.(m-i)f,
KiO GEOMETRY OF CYCLOIDS.
and tan ;, O 6 = j, (^» +^^i)— „» e«-
For the critical case where the points of inflexion coincide,
we have, from Cor. 1, cos (w— 1) 02 = — 1 ^
that is D« +n3E2 = w(l + w)DE
(the same condition, both for direct and retrograde epicyclic,
due account being taken of the sign of n) ;
or n (w^ E - D) E = {n^ E - D) D
or (w E - D)(w2 E - D) = 0,
which is satisfied, (i), if n = — , the condition (Schol. p. 158)
E
for a cusp (at pericentre in case of direct epicyclic, and at apo-
oentre in case of retrograde epicyclic), and (ii), if n^=~, cor-
E
responding to the case when this curve becomes straight at
pericentre both for direct and retrograde epicyclic. Com-
pare scholium to Prop. XII., Section III., from which the
relation between w^, D, and E, can be directly obtained.
Prop. XI. — To deterrtihie the radius of curvature ^ p, at a
point on epicyclic where deferential angle = f.
From Cor. p. 117, noting value of j^B' (as in p. 162) ;
that CO == w . C'B' ; and that B N= C'B cos jo B'C ; while
p B cos jt? B C =B'C' +p C cos {n— 1) 0, it is easily shown
that
_ [B^+n^ E^-f 2 n DE cos (n-l) (pf
^ I)2+7i3E2 + yi (n+l) DEcos(n-l)0
at apocentre, p==i^±^^ ; at pericentre p =L^^ J .
EPICYCLICS. 167
Appendix to Section V.
RIGHT TROCHOIDS REGARDED AS EPICYCLICS.
It is often convenient to regard right trochoids as
epicyclics. The radius of the deferent is in their case
infinite, the centre of the epicycle moving in a straight
line. Tt is necessary to substitute linear for angular velo-
cities, the value of n becoming infinite when the deferent
becomes a straight line. It is manifest that if the centre of
the rolling circle of a right trochoid moves with velocity v in
the line of centres, the tracing point moves with ve-
locity — V around the tracing circle ; and conversely, it is
manifest that if a point moves with velocity m v round the
circumference of a circle of radius E, whose centre moves with
velocity v in a straight line in its own plane, the point will
trace out a right trochoid, having a tracing circle of radius E
and a generating circle of radius m E. We may put v= 1,
in which case m represents the velocity of the tracing point
round the circimiference of the moving cu*cle ( m = — j . It
is obvious also that if m > 1 there is a loop ; if m=l, a cusp ;
if 7/i < 1 the curve is inflected. These cases correspond to
those of right trochoids in which r > R, r = R, and r < R.
Since right trochoids may be regarded as special cases of
epicyclic curves, it is not necessary to discuss them further
in their epicyclic character. It will be found easy to deduce
any required relation for right trochoids from the relations
above established for epicyclics, combined with the considera-
tions noted in the preceding paragraph, A single illustra-
tion will suffice to show how this may be effected.
108 GEOMETRY OF CYCLOIDS.
Suppose we wish to determine when the tracing point
ceases to advance in the looped trochoid. "We have, from
Prop. YII., in case of epicyclic,
cos (H — 1) 0, = — !— r-^-^
^ '^^ (1 + n) DE
and if m represents the ratio of linear velocities in epicycle and
deferent, n = m — . Also ?i ^ is the angle swept out in
epicycle, and when D becomes infinite is the same as (w — \)<p,
so that the angle ^^ (the angle aCL of fig. 48) is deter-
mined by the equation
cos 01 = — - ^ — ! — ^T--r^- = — — when D is infinite.
^* (E + mD)D m
The student will, however, find it a useful exercise to go
independently through the various propositions relating to
epicyclics, for the case in which the deferent is a straight
line. The relations involved are simpler than those dealt
with in the present section. It is to be noticed that m
may always be regarded as positive, the same curve being
obtained for a negative value of m as for the same positive
value, if r remains unaltered.
SPIRAL EPICYCLICS,
When the radii of epicycle and deferent are both infinite
but (D— E) finite, the epicyclic becomes one of the system of
spirals of which the involute of the circle and the spiral of
Archimedes are special cases. We must of course suppose the
curve traced out on either side of the pericentre, since the
remoter parts of the curve pass off" on each side to infinity.
Instead, however, of imagining a deferent of infinite radius
carrying an epicycle also of infinite radius, it is more con-
venient, in independent researches into these spirals by
epicyclic methods, to consider a deferent radius as revolving
EPICYCLICS. 169
uniformly round a fixed point, this radius bearing at its
extremity a straight line perp. to it in the plane of its own
motion, along which line a point moves with uniform
velocity. Let the length of the revolving radius ^ dy
velocity of its extremity 1, and velocity of moving point m.
Then if m ^ 1, the curve is the involute of the circle traced
out by the end of the revolving radius ; if m > or < 1, the
curve is one of the system of spirals bearing the same relation
to the involute of the circle which the curtate and prolate epi-
cycloid respectively bear to the right epicycloid. If c?= 0, the
infinite straight line revolves about a point in its own centre ;
and the curve traced out by the moving point is the spiral
of Ai'chimedes, whatever the imiform angular velocity of the
revolving line, and whatever the uniform velocity of the
tracing point along the line. See also examples 131-133.
PLANETARY AND LUNAR EPICYCLES.
The ancient astronomera discovered that the paths in
which the planets travel with reference to the earth are
approximately epicyclic. It is easily shown that this follows
from the fact that the planets, as well as our earth, travel in
nearly circular paths about the sun as centre.
The general property is as follows : —
Peop. I. — Regarding the planets as travelling uniformly in
circles about the sun as centre, and in the same plane, the
path of any planet P (Jig, 97) with reference to any other
planet y p, regarded as at rest, is the same as the path of p
with reference to P regarded as at rest, the corresponding
radii vectores lying in opposite directions ; and each such
path is a direct epicyclic.
Let S be the sun, p and P two planets {p being the
inferior planet, and P the superior), in conjunction on the line
170
GEOMETRY OF CYCLOIDS.
Sjo P. Let the planet p move to p', while P moves to P'.
Draw p Q and P q parallel and equal to p' P'. Then, with
reference to the planet p, regai'ded as at rest, the planet P
has moved as if from P to Q ; while considered with refer-
ence to P, regarded as at rest, the planet p has moved as if
from pio q: and since jt? Q is equal and parallel to P g, the
path of the outer planet with reference to the inner, regarded
Fig. 97.
^-- — -"■
w
as at rest, is the same as the path of the inner planet with
reference to the outer regarded as at rest, — each path being,
however, turned round through 180° with regard to the
other.
Join p' q, PT, p' p, and P'Q. Draw S s' parallel to
p'q, and SS' parallel to P'Q, and join s'q, «T, S'/?, and
S'Q. Also draw am and S M parallel to SP, and complete
the parallelograms PMS'S, and p m s'S.
Then, by construction, the figiires S'j9, ^'Q, S'P", 85^,
q P', and a' P', are parallelograms. Wherefore p S'= jo'S =
SjE?; andzSjE?S'==Z.Jt?S/; S'M==SP= SF=S'Q and
Z. MS'Q = Z. PSP' ; so that the relative motion of the outer
planet from P to Q around p may be regarded as effected
by the uniform motion of S to W in a circle about p as centre
EPICYCLICS. 171
(corresponding to the real motion of jo to jt?' around S
as centre), accompanied by the uniform motion of P (which,
if at rest, would have been carried to M), in a circle around
the moving S as centre to Q, — that is, through the arc M Q
= P P'. Hence the motion of P with reference to /? is that
of a direct epicyclic having D = S jt?, E = S P, and
Ang. vel. of P round S
A Tig, vel. of p round S
Similarly the relative motion of the inner planet from p
to q, around P, may be regarded as effected by the uniform
motion of S to a around P as centre (corresponding to the
real motion of P to P' around S as centre), accompanied by
the uniform motion oi p (which, if at rest, would have been
carried to m) in a circle around the moving S as centre to
q, — that is, through the arc mq=^ pp » Hence the motion
of p with reference to P is that of a direct epicyclic having
D = SP, E = Sjo, and
Ang. vel. of jo round S
Ang. vel. of P round S
ScHOL. — If the distances of the planets p and P from the
sun are r and R respectively, the epicyclic of either planet
about the other has D = R, E = r, and
n
= ?)' ,
for the angular velocities of planets round the sun vary
inversely as the periods — that is, as the sesquiplicate power
of the mean distance.
Since (?)*>pO->»
the motion of one planet with reference to another is always
retrograde when the planets are nearest to each other;
therefore every planetary epicyclic is looped.
172 GEOMETRY OF CYCLOIDS
The arc of retrogradation of one planet with reference
to the other may be obtained as explained in scholiuni
to Prop. VII. of this section. The din^ation of the retrogra-
dation follows directly from the form\ila for determining
cos (ti — 1) ^1 as in that proposition ; for 0| is the angle
swept out by the superior planet around the sun between the
time of inferior conjunction and first station. This formula,
with the values above given for D, E, and w, becomes
or, putting P, p, for the respective periods of the planets,
— — ^ 01 = - T> ^i . ~
cos
p ^"^ Rrt + Rf r R* + rt
Ri r\ V'Rr
R - Rl ri + r VRr - (R + r) '
and
p-/>
sin 01= V\ — cos (w— 1) 01 v^l -f cos (w — l)0i
_ s/ ( R -f~r) ( R "" 2 Ri ri'HM' )
R - Rt r* + r
_ ( Ri - ri ) a/R + r
R - Ri rt + r '
Wherefore tan /? O 6' (see fig. 91, and schol. p. 160)
r(R^ - t\ ) a/R + T
R (R - Ri r^ + r) - Ri r*
r (RV - H)A/R-f-r
" R(R 4" r) -Ri r* (R + r) "" Ri. VR -f- r
The arc of retrogradation, —
= 20, + 2^O6'-36O°(^),_
can be readily determined. Thus, the arc of retrogradation
EPICYCLICS. 173
= 2tan-* ^
RVR + r
- ^ /18O-+COS-1 ^g. ) (I)
■
This formula gives the arc of retrogradation. The angle
between pericentral and stationary radii vectored is half the
arc of retrogradation.
Thus the epicyclic path of a superior planet (period P)
with respect to an inferior planet (period jt?), or of latter
planet with respect to former, will have —
Apocentral distance = R + r ;
Pericentral distance = R — r ;
Angle of descent = p _^ • 180°.
The arc of retrogradation is determined by formula (1 ) above.
All the tables of planetary elements give R, r, P and p.
If one of the planets is the earth, the calculation is simpli-
fied, because the tables of elements give the distances of other
planets with the earth's mean distance as unity.
If a satellite be regarded as travelling uniformly in a
circle around its primary, while the primary travels uni-
formly in a circle in the same plane around the sun, the
path of the satellite is an epicyclic about the sun as fixed
centre.
All the satellites travel in the same direction round their
primaries as the primaries round the sun, except the satel-
lites of Uranus, whose inclination is so great that their
motion does not approach the epicyclic character. The
174 GEOMETRY OF CYCLOIDS.
direction of the motion of Neptune's satellite, sometimes
given in tables of astronomical elements as retrograde, can-
not yet be regarded as determined. The inclination of
Saturn's satellites, seven of which travel nearly in the same
plane as the rings, is considerable ; but these bodies may be
regarded as having paths of an epicyclic character. Our own
moon's path is but little inclined to the ecliptic, and the
paths of Jupiter's moons are still nearer the plane of their
planet's motion. The discussion of the actual motions of
these bodies belongs rather to astronomy than to our present
subject. We need consider here only some general relations.*
Prop. 11. — To determine under wJiat conditions a sutellitey
travelling in a direct epicycle about the sun, will have its
motion {referred to the sun) looped, cusped, or direct
throughout, or partly convex towards the sun, or just fail-
ing of becoming convex at perihelion, or partly conca^ve
towards the sun.
Let M be the sun's mass, m the primary's, R the dis-
tance of primary from the sun, r the distance of satellite
from primary ; also (though these values are only for con-
venience) let P be the primary's period, p the satellite's, and
assume that m is so small compared with M, and the satel-
lite's mass so small compared with m, that both the ratios
(M + m) : M,and (m + satellite's mass) : m may be regarded
throughout this inquiry as equal to unity.
We have first to obtain the means of comparing the
velocities in the primary and secondary orbits under any
♦ In a work on the * Principles of Astronomy,' which I am at
present writing, the nature of the planetary and lunar epicycles
will be found fully treated of.
EPICrCLICS,
176
given conditions. The most convenient way of doing this is
perhaps as follows : — Let V, v , be the respective velocities
of bodies moving in circles around the sun, and round the
primary, at the same distance, R ; and let v be the velocity
of the satellite at distance r. Then we know that
R '
g :: M:m,
or
Y:v' :
: >/M : \/m,
and
v' iv :
: v^r : VK
.\Y:v :
:: A/Mr : a/wiR,
and
V V
R • r '
: a/mh : ^/mR
This is the ratio of the angular velocities of primary and
satellite in their respective orbits. It gives us
n: 1 {iiT :p):: ^/mR^ : a/mh.
The path of the satellite will therefore be looped, cusped,
or direct throughout, according as
'm R3 > R
or as
/vi R^ > j
V Mr3 < r
mR^Mr; or^ >4.
< M < R
And the path of the satellite will be partly convex towards
the sun, or just fail of becoming convex at perihelion, or be
partly concave towards the sun, according as
mR» > R
/
Mr3
< r
or as
m R2 i M r2 ; or - « .r^
< M < R*
-; or
V M>
r
R
The student will find no difficulty in obtaining formulro
for the range of the arc of retrogradation, if any, or of the
170 GEOMETRY OF CYCLOIDS.
arc of convexity towards the sun, if any, following the course
pursued at pp. 172, 173 (using in the latter case the formula
of p. 165), remembering that in this case D = R and E = r
p
and 71 = — , as in the case of planetary motion, but that in
P
reducing the formula he must employ the relation
»tR3
I have not thought it necessary to occupy space here
with the reduction of these formulae, because they are of no
special use. The path of our own moon has no points of
retrogradation or of flex\u«, and the position of such points
on the paths of Jupiter's moons, or Saturn's, is not a matter
of much moment.
We may pause a moment, however, to inquire into the
Hmits of distance at which, in the case of these planets and
our earth, convexity towards the sun, or retrogradation,
would occur.
M
In the case of our earth, — = 322,700 = (568)« about ;
and R x= 92,000,000. Therefore a moon would travel in a
cusped epicycle, or come exactly to rest at perihelion, if (the
earth's whole mass being supposed collected at her centre)
the moon's distance from the earth's centre were qoo 700
miles, or about 285 nules. That a moon should travel in a
path convex to the sun in perihelion, the distance should not
exceed — ^~Kak > or about 162,000 miles. Thus the
moon's actual distance being 238,828 miles, her path is
entirely concave towards the sun.
M
In the case of Jupiter, — = 1,046 = (32^)^ about; and
EPICYCLICS, 177
R = 478,660,000 miles. < Therefore a moon would travel in
a cusped epicycle, or come exactly to rest in perihelion, if its
478,660,000
distance from Jupiter s centre were f"()Jfi — ' ^^ about
457,600 miles. Thus the two inner moons, whose distances
are 259,300 and 412,000 miles, have loops of retrogradation ;
whereas the two outermost, whose distances are 658,000 and
1,155,800 miles, have paths wholly direct. But all the
moons travel on paths convex towards the sun for a con-
siderable arc on either side of perihelion ; since for the path
of a Jovian moon to just escape convexity towards the sun at
perihelion, its distance from Jupiter should be ^^ ooX
miles, or about 14,804,000 miles; which far exceeds the
distance even of the outermost moon.
M
In the case of Satiu-n — = 3,510 = (59)^ about, and
R =t 877,570,000 miles. Hence a moon would travel in a
cusped epicycle if its distance from Saturn were — q~51
or about 250,700 miles. This is rather less than the distance
of his fourth satellite, Dione, 253,442 miles ; and, owing to
the eccentricity of Saturn's orbit, it must at times happen
that Dione comes almost exactly to rest for an instant at a
cusp in epicyclic perihelion, or only has a motion perpendicular
for the moment to the path of Saturn. The three satellites
nearer to Saturn ti-avelling at distances of 124,500, of 159,700,
and of 197,855 miles, have loops of retrogradation, as have all
the satellites composing the ring system. The other satellites,
having distances of 353,647, of 620,543, of 992,280, and of
2,384,253 miles respectively, have no loops ; but their paths
are convex towards the sun for a considerable arc on either
178 GEOMETRY OF CYCLOIDS,
side of epicyclic perihelion ; since, for a satellite's path just io
escape convexity towards the sun, the satellite's distance
877,570,000
should be gg miles, or about 14,874,000 miles.
Prop. III. — Regarding the planets as moving uniformly in
circles round the sun in the invariable pkme, the j^rcjec-
tions of the paths of the planets in space upon a Jlxed
plane parallel to the invariable plane of the solar system
are right trochoids.
This follows directly from the fact that the sun is
advancing in a right line (appreciably, so far as ordinary
time-measures are concerned), with a velocity comparable
with the orbital velocities of the planets. His course being
inclined to the invariable plane, the actual path of each
planet is a skew helix, as shown in the last chapter of my
treatise on the sun.
Prop. IV. — To determine the tangential, transversey and
radial velocities {linear) of a planet in its orbit relatively to
another planet, and its angular velocity about this planet.
Let R be the distance, P the jieiiod, V the velocity of the
planet which is regarded as the centre of motion; r the
distance, p the period, v the velocity of the other planet.
Then, in the formulae for the tangential transverse, and
radial velocities in epicyclics, we have to put
D = R;E = r;and7i = y =-;
but it will be convenient to retain n, remembering its value.
We may also conveniently write — = jo, so that n ^ p-i
BPICYCLICS. 170
Moreover, -with the units of distance and time in whioh B, r,
P, and p are expressed.
Also is the angle swept out around the sun by the planet
of reference since the last conjunction of the sun and the
other planet, the conjunction being superior in the case of
an inferior planet.*
Thus the tangential velocity is equal to
= V >/l + p-i + 2p* cos (w — 1) 9 .
The formula can obviously assume many forms, but per •
haps this, which enables us at once to compare the tangential
velocity with Y , the velocity of the planet of reference in
its orbit, is the most convenient.
The transverse velocity (direct)
^/R2 +r« + 2Rrcos(n- l)p
_ y 1 + p^ + (p-^ + p) cos (n — 1)
A/r+^^'+2 p cos (t* — 1)^
The radial velocity (towards centre)
= (p^|-l)V. r sin (n- 1)
^R2 + r2 + 2 Rr cos (w - 1) -^
_ y (p~^ - p) sin {n — 1)0
Vl -h p2 + 2p cos (w - 1)^'
* The conjnnction most be such that the sun is between the two
planets. It is a convenient aid to the memory, in distinguishing
between the superior and inferior conjunctions of inferior planets,
to notice that inferior conjunction is that kind of conjunction with
the sun which only inferior planets can enter into.
k2
180 GEOMETRY OF CYCLOIDS.
The lingular velocity of the planet about the planet of
reference
__ V i-^r^± R^. + (p"* +1) R r cos (n - 1)
— R ' R^-fr'» + 2rcos(7i- 1)0
_ pi 4 1 + (p^ + p) cos (n — 1)
- ^ • 1+ p» + 2p cos (w ^ 1) '
V
putting w = w = angular velocity of the planet of reference
in its orbit.
Cor. 1. In conjunction (superior if moving planet is in-
ferior) = 0;
.'. Angular velocity in superior conjunction
pi + 1 + p-i + p
= <i>
1 +f)» + 2p
(1 + P) X (1 + P* )
^2
= - (^i^)-
Cor. 2. Similarly since in opposition if the moving planet
is superior, or in inferior conjunction if the moving planet is
inferior, (w— 1) = 180°, angular velocity of a planet in op-
position or inferior conjunction
pi 4- 1 — p~i — P
= " l + P^-2p
(l-p)-p-S(l-p) / l-p-^ \
-^ (1 - pf ^ =n"T^=7^;
— W 1 — pa W
A p 1 — P 'v/p + p
ScHOL. — All the above formulae are susceptible of many
modifications depending on the relations subsisting between
the periods, distances, real velocities, and angular velocities
of the planets in their orbits. From Kepler's third law all .
such modifications may be directly deduced.
EPICYCLICS. 181
Pbop. V. — A planet transits the sun! 8 disc at such a rate
that the sUrHs diafneter S would be traversed in time t ;
assuming circular orbits and uniform motion, determine
the planet* s distance from, the sun,*
Let the planet's distance = p, earth's distance being unity,
and let w be the earth's angular vel. about the sun = sun's
angular vel. about earth. Then, if t' be the time in which
the sun in his annual course moves through a distance equal
to his own apparent diameter, w <' = S, and the planet's
angular velocity about the earth when in inferior conjunction
Wherefore, the planet's retrograde gain on the sun (which
advances with angular velocity w)
Vp '\- ii
\ \^p + p J t t '
t_
t' -t
a qiiadratic giving
or p + >v/|o == jr—^ '
v,-=-i±^£ij±i:=i(±^^i+i:_i),
or p-:l(* +< . ^^t + f
The lower sign must be taken, the upper giving a value of
p greater than unity.
Cor. Let us take the supposed case of Vulcan, whose
* This was the problem Lescarbault had to deal with in the case
of the supposed intra-Merciirial planet Vulcan. He failed for want
of such formulae as aie here given.
182 GEOMETRY OF CYCLOIDS.
rate of transit was such that the sun's diameter would Lave
been traversed in rather more than four hours. Sinoe in
March (the time of the supposed discovery) the sun traversed
by his annual motion a space equal to his own apparent
diameter in rather more than 12 hours, we may say that
(with as near an approximation as an observation of this
kind — ^inexact at the best — merits) ^' = 3 ^ Thus
P = i(2->/3)
= i (2 - 1-732) = i (0-268) = 0-134.
This is very near the estimated value of the imagined planet's
distance,
FORMS OF EPICTCLIC CURVES.
The relations discussed in the propositions of this section
enable us to determine the shape and general features of
ej^trochoids or direct epicyclics and of hypotrochoids or re-
trograde epicyclics, for various values of D, E, and n. I
propose to consider these features, but briefly only, because
in reality their consideration belongs rather to the analytical
than to the geometrical discussion of our subject.
In the first place, since we obtain the same curve by
interchanging deferent and epicycle, and at the same time
interchanging the relative angular velocities of the motions
in these circles, we shall obtain all possible varieties of epi-
cyclic curves by taking D as not less than E, so long as we
give to n all possible values from positive to negative in-
finity.
The whole curve lies, in every case, between circles of
radii D + E and D — E, the apocentres falling on the former
circle, the pericentres on the latter. When D = E, the whole
curve lies within the apocentral circle; and all the pericentres
lie at the fixed centre.
184
GEOMETRY OF CYCLOIDS.
side ; so that in this case, as in that of direct epicyclic, we
liave when w' = - two points of inflexion coinciding at
the pericentres. These two cases are illustrated in figs. 114
and 115. The former is a direct epicyclic; 7» = 5; and
D : E : ; 25 : 1 ; (apocentral dist. : pericentral dist. : : D-f-E
: D— E : : 13 : 12. The latter is a retrograde epicyclic;
w= — 3 ; and D : E : : 9 : 1 ; (apocentral dist. : pericentral
dist. : : D + E : D-E : : 5 : 4). Compare figs. 118, 121, 154,
158.
As n continues to decrease from the value
^/l
the
angle of descent continually increases if n is positive and we
have curves of the form shown in ^, 108.
Fio. 114. Fio. 116.
r
I
^^
In diminishing from the value ^^-n passe 3 through
the value unity. When w = + 1 the curve is a circle hav-
ing the fixed point as centre, and having for radius whatever
distance the tracing point may have from that centre ini-
tially; the radius vector therefore always lies in value
between D + E and D — E.
As n continuing positive diminishes in absolute value from
1 to 0, the angle of descent which had become infinite dimi-
nishes, remaining positive.* The curve continues concave
* De Morgan says, * becomes very great and negative.' This is
correct on his assumption that the angle of descent is to be re-
EPICYCLICS. 186
towards the centre, resembling the appearance it had had
before n reached the value unity. As n approaches the value
0, however, the angle of descent becomes less and less, until
when 71=0 it becomes 180°j the curve being now a circle hav-
ing radius D and centre at distance E from the fixed centre.
Thus, if the tracing point is initially at A, fig. 81, p. 148,
the centre is at c, but if the tracing point is initially at P,
the centre is at c', (0 c being parallel to C P).
As n diminishes in absolute value from— v / to — 1,
\ E
the angle of descent increases till it is equal to 90°, the
curve, always concave towards the fixed centre, forming a
series of arcs more and more approaching the elliptical form,
as in fig. 109, till when w = — 1 the ciu've is the elliptical
hypocycloid, see p. 124. We see that the equality of the
diameters of the fixed and rolling circles is equivalent to the
condition ti = — 1 for retrograde epicyclic. The semi-axes
are(D-|.E)and(D-E).
Lastly as n, still negative, diminishes from — 1 towards
0, the curve at first resembles in appearance that obtained
before n reached the value —1, but the angle of descent
gradually increases, until at length, when ti = 0, it is 180°
and the curve becomes the circle already described.
garded as positive when the radius of the epicycle gains in direc-
tion on the radius of the deferent, and negative when the radius of
the deferent gains in direction on the radius of the epicycle. There
is no occasion, however, to make this assumption, which is alto-
gether arbitrary. If we consider the actual motion of the tracing
point coming alternately at apocentre and at pericentre upon the
deferential radius, which oonxtantly advatices whatever the value of n
positive or negative (except + 1 only), we must consider the angle
of descent as always positive. We arrive at the same conclusion
also if we consider that the radius vector advances on the whole be-
tween apocentre and following pericentre, for all epicyclics, direct
or retrograde.
186 GEOMETRY OF CYCLOIDS.
The varieties of form assumed by epicyclics aooording to
the varying values of n, D, and E, are practically infinite.
It will be noticed that in all the illustrative figures, n is a
commensurable number, so that the curve re-enters into itself.
Of course, no complete figure of an epicycle in which n is not
a commensurable number could be drawn.
Certain special cases may here be touched on briefly.
When D = E, the direct epicyclic assumes such forms as
are shown in figs. 110, 112, the retrograde epicyclic such
forms as are shown in figs. Ill and 113. The distinction
between the two classes of epicyclics in these cases is re-
cognised by noting that the angle of descent, which must be
positive, can only be made so by tracing the curves in figs.
110 and 112 the direct way, and by tracing those in figs.
111 and 113 the reverse way.
A distinction must be noted between direct and retrograde
epicyclics, when D is nearly equal to E, and n approaches the
value -, which is nearly equal to unity. For the direct epi-
E
cyclic, the angle of descent, 180° -f- (w— 1), becomes very
great, and we have a curve which passes from apocentre to
pericentre through a number of revolutions, before beginning
to ascend again by as many revolutions to the next peri-
centre.* On the other hand, in the case of the retrograde
epicyclic, when D is very nearly equal to E, the angle of
descent 180° -r- (w -|- 1) approaches in value to 90°, or the
angle between successive apocentres approaches in value to
two right angles, so that the curve has such a form as is
shown farther on in fig. 119.
We have followed the effects of changes in the value of
* Prof. De Morgan strangely enough takes figs. 116 and 117 as
illustrating this case. But in both these figs. »:a^g^j in fig. 117,
D = V E* ^ neither is E very nearly equal to D.
EPICYCLIC8.
187
n, where D and E are supposed to remain unchanged through-
out. The number of apooentres and perioentres depends, as
we have ah'eady seen, on the value of n. It will be a useful
exercise for the student to examine the effect of varying the
value of E, keeping D and n constant, or (which amounts
Fio. 116.
really to the same thing) to examine the effect of varying the
E
value of — , keeping n constant. Since the angle of descent
is equal to 180° -^-(n— 1) ifnis positive, and to 180° -?-
Fio. 117.
E
(w+ 1 ) if w is negative, changing the value of ^ will not give
all the curves having any given number m of apocentres or
pericentres (for each revolution of the deferent). For this
purpose it is necessary to assume first w = (m + 1), giving
all the direct epicyclics having m apocentres and w peri-
ls« OEOMETRY OF CTCLtilDS.
centri*. and wcondl y n = ~{m—\) giving all the retrogradft
epicyplics Imving m tvpaeentres nnd m peric^itreE, for each
■■ovohition of the deferent. (Of eourBe, m is not necewariljr
a whii!p minil»r.)
(Suppose wo take ii^y, so that thp angle of descent
(=180° -^g) is eqiml to gths of two right angles. Then if
K> j\ D we have such a curve as is shown in fig. 116. As
E diminishes until K^yV-'-'' the loops tnm into ciisps ai
shown in fig. 117; as E diminishes still further i
^-^ D (that is9i'^=u the cui've aesiimeetheortJioidal
form shown in fig. 118. Again, take n= —^. Then
EPICYCLICS. 1813
when E U neai-ly etjunl to D the curve has Muoh it form
as is Hhown in fi^, lliJ, merging into the cuaijithite foi'iii
aain %. 120, when E = f D; and into the oi-thoidiil (m*
stnvijjhteneil) form, as in tig. 121, when E =: i^^ D (or
eJ
For fiu-ther illustrations «
If we compare tig. 98 with fig. 122, we perceive that i:
the former the loop between two successive whorls overhv]]
two pi-et^ding loo]>s, while in the latter each loop overlaps
but one pi-eceding loop. A number of varieties arise in thin
way. The determination of the condition under which any
given pi'Bceding looji may he just touched is not difficult;
190
GEOMETRY OF CYCLOIDS,
but in no case does the condition lead to a formula giving n
directly in terms of D and K The simplest of these eases is
dealt with in Prop. IX. of this section. (See fig. 160, p. 256.)
Figs. 123 and 124 illustrate eight-looped epicycHcs direct
and retrograde. By noting the different proportions between
Fio. 1 22,
their respective loops, and by comparing fig. 123 with fig.
100, a ten-looped direct epicyclic, and fig. 124 with fig. 101, a
ten-looped retrograde epicyclic, the student will recognise the
effect of varying conditions on the figures of epicyclics. (In
Fio. 123.
fig. 100, n := 11 ) in fig 101, w t= ^ 9 ; in fig. 123, n =: 9,
and in fig. 124, w as — 7).
It is a useful exercise to take a series of epicyclics and
determine the value of B, E, and n,. from the figure of the
curve. Suppose, for instance, the curve shown in fig. 125,
EPICYCLICS.
191
is given for examination. This closely resembles ^, 108 in
appearance; but in reality fig. 125 is a retrograde, whereas
^g, 108 is a direct epicycHc. The character of the curve in
this respect is determined by tracing it directly from any
apooentre and noting that the next apocentre falls behind
Fio. 124,
the one from which we started. The valuas of D and E are
determined at once from the dimensions of the ring within
which the curve lies, — its outer radius being D + E, its
inner D — E. The value of n is conveniently determined
Fio. 125.
by noting the angle between two neighbouring apocentrea
(indicated best by the uitersections of the curve next within
the apocentres, for from the symmetry of the curve all inter-
sections lie of necessity either on apocentral radii vectoi*es
or on these produced). This angle s one-tenth of 360°, so
192
GEOMETRY OF CYCLOIDS.
that the angle of descent is y^ths of 180° ; or w + 1 = y*.
Thus in absolute value n := ^, but n is negative.
In like manner we find that in fig. 126, w = — ^.
In each of the figs. 127, 128, and 129, n = 2, since
there is only one apooentre. In fig. 127, the trisectrix,
Fio. 126.
Fio. 127.
D = E ; in fig. 128, the cardioid, D = 2 E ; in fig. 129,
D = 3E.
Figs. 1 30 and 131, Plate IV., illustrate some of the pleasing
combinations of curves which may be obtained by the use of
the geometric chuck, the instrument with which all the curves
of the present part of this section have been drawn. In
Fin. 28.
Fio. 129.
tig. 130 we have two direct epicyclics, (D — E) of the outer
being equal to (D + E) of the inner. It will be found that
for the outer w = 7, while for the inner w = 15. In ^, 131
we have four direct epicycles, having (D + E) constant, but
ratio D : E difierent in each. It will be found that there
EPICYCLICS. 193
are 5^ apocentres in each circuit ; whence (w — 1) s=
-^ . 360 = 67^, and n = 68^. The inner part of the figure
is a retrograde epicyclic having 5^ apocentral distances in
each circuit ; whence in absolute value (n + 1) = 67^, and
n = - 66^.
Figs. 132, 133, Plate v., are further examples for the
student.
The remaining eight figures of Plates IV. and V., for
which I am indebted to Mr. Perigal, present the approxi-
mate figures of the epicyclics traversed by the planets, with
reference to the earth regarded as fixed. Of course the real
curves of the planetary orbits with reference to the earth
do not retiu-n into themselves as these do, the value of n not
being in any case represented by a commensurable ratio.
Moreover, the orbits of the earth and planets around the
sun are not in reality circles described with uniform velocity,
but ellipses around the sun as a focus of each and described
according to the law of areas called Kepler's second law.
Therefore figs. 134 — 141 must be regarded only as repre-
sentative types of the various epicyclics to which the plane-
tary geocentric paths approximate more or less closely. In
the case of Mars, I may remark that either of the ratios
15 • 8 or 32 \ 17 would have given a more satisfactory
approximation to the planet's epicyclic path around the
earth. It so chances that I have taken occasion during the
opposition-approach of Mars in 1877 to draw the true geo-
centric path of Mars around the earth for the last forty
years and for the next fifty, taking into account the eccen-
tricity and ellipticity of the paths, and the varying motion
of the earth and Mars in their real orbits around the sun.
The resulting curve, though presenting the epicyclic cha-
racter, yet falls far short of any of the curves of Plates IV.
194 GEOMETRY OF CYCLOIDS.
and y . in symmetry of appearance. The loops are markedly
unequal, a relation corresponding of course to the observed
inequality of the arcs of retrogradation traversed by Mars
at different oppositions.
Note. — Mr. H. Perigal, to whom I am indebted for all the illus-
trations of this part of the present work (except figs. 118-121, 132,
133, and 154-161, engraved by Mr. L. W. Boord, with a similar -
instrument), gives the following account of the geometric chuck: —
*■ The geometric chuck, a modification of Suardi's geometric pen^
was constructed by J. H. Ibbetson, more than half a century ago, as
an adjunct to the amateur's turning-lathe. It is admirably adapted
for the purposes of ornamental turning ; but is still more valuable
as a means of investigating the curves produced by compound cir-
cular motion. In its simplest form it generates bicirdoid curves,
so called from their being the resultants of two circular movements.
This is effected by a stop- wheel at the back of the instrument giving
motion to a chuck in front, which rotates on its centre, while that
centre is carried round with the rest of the instrument and the train
of wheels which imparts the required ratio of angular velocity to the
two movements. A sliding piece gives the radial adjustment, which
determines the phases of the curve dependent upon the radial-ratio.
< By the simple geometric chuck a double motion is given to a
plane on which the resultant curve is delineated by a fixed point ;
but it may act as a geometric pen when it is made to carry the
tracing point with a double circular motion, so as to delineate the
curve on a fixed plane surface. The curves thus produced being
reciprocals, all the curves generated by the geometric chuck may be
produced by the geometric pen, and vice versd, by Tna.ViTig' the angu-
lar velocity of the one reciprocal to that of the other. For instance,
the ellipse may be generated by the geometric chuck with velocity-
ratio 8 1 : 2 ' (see, however, remarks following this extract), < and
by the geometric pen with velocity-ratio ■» 2 : 1, the movements
of both being inverse, that is, in contrary directions.
* The accompanying curves were turned in the lathe with the geo-
metric chuck (by myself, many years ago), of sufficient depth to
enable casts to be taken from them in type metal, so as to print the
curves as black lines on a white ground. These curves are therefore
veritable autotypes of motion.'
Mr. Perigal has invented, also, an ingenious instrument, called
the kinescope (sold by Messrs. R. & J. Beck, of Comhill), by which
all forms of epicyclics can be ocularly illustrated. A bright bead
EPICYCLIC8. 196
is set revolving with great rapidity about a centre, itself revolving
rapidly about a fixed centre, and by simple adjustment, any velo-
city-ratio can be given to the two motions, and thus any epicyclic
traced out. The motions are so rapid that, owing to the persist-
ence of luminous images on the retina, the whole curve is visible as
if formed of bright wire.
He has also turned hundreds of epicyclics (or bicircloids, as he
prefers to call them) with the geometric chuck. There is one point
to be noticed, however, in his published figures of these curves. The
velocity-ratio mentioned beside the figures is not the ratio n : 1 of
this section, but (n— 1) : 1, i.e., he signifies by the velocity-ratio, not
the ratio of the actual angular velocity of the tracing radius in the
epicycle to the angular velocity of the deferent radius, but the ratio
of the angular gain of the tracing radius /rtwi the deferent to the an-
gular velocity of the deferent. This may be called the mechanical
ratio, as distinguished from the mathematical ratio ; for a mecha-
nician would naturally regard the radius C'A' of the epicycle PAT'
(fig. 81) as at rest, and therefore measure the motion of the tracing
radius C'F from C'A', whereas in the mathematical way of viewing
the motions, Q'a is regarded as the radius at rest, and the motion of
C'P is therefore measured from Ca. The point is not one of any im-
portance, because no question of facts turns upon it ; but it is neces-
sary to note it, as the student who has become accustomed to regard
the velocity-ratios as they are dealt with in the present section (and
usually in mathematical treatises on epicyclic motion), might other-
wise be perplexed by the numerical values appended to Mr. Perigal's
diagrams. These values, be it noticed, are those actually required in
using the geometric chuck or the kinescope ; for in all adjustments
the epicycle is in mechanical connection with the deferent.
FORMS OF RIGHT TROCHOIDS.
Right trochoids may be regiarded as epicyclics having the
radius of deferent infinite, the centre of the epicycle travel-
ling in a straight line. A good idea of the form of trochoids
may be obtained by regarding them as pictures of screw-
shaped wires (like fine corkscrews), viewed in particular
directions. This may be shown as follows : —
If a point move uniformly round a circle whose centre
advances uniformly in a straight line perpendicular to the
o 2
396 GEOMETRY OF CYCLOIDS,
plane of the circle, the point will describe a right helix, the
convolutions of which will lie closer together, relatively to
the span of each, as the motion of the point in the circle is
more rapid relatively to the motion of the circle's centre.
Now if any plane figm-e be projected on a plane at right
angles to its own, by parallel lines inclined half a right
angle to each plane (or perpendicular to one of the two planes
bisecting the plane angle between them), the projection of
the figure is manifestly similar and equal to the figure itselfl
Therefore if the circle and the point tracing out the helix just
described be projected on a plane parallel to the axis of the
helix, by lines making with this plane and the plane of
the circle an angle equal to half a right angle, the circle will
be projected into a circle whose centre advances uniformly
in the plane of projection in a right line. The projection of
the tracing point will be a point ti-avelling uniformly round
this circle ; and therefore the projection of the helix will be a
right trochoid. We may say then that every helix viewed
at an angle of 45° to its axis is seen as a trochoid, — or rather
that portion of the helix which is so viewed from a distant
point appears as a trochoid. When the tracing point of a
helix moves at the same rate as the centre of the circle, the
helix viewed at an angle of 45° to its axis appeai-s as a
right cycloid. Thus a helicoid or corkscrew wii-e having a
slant of 45° and viewed from a great distance at the same
slant (so that the line of sight coincides with the direction
of the helix where touched, at one side, by a plane through
the remote point of vipw), appears as a cycloid.
The helix is projected into other cm-ves if the line of sight
is inclined to the axis at an angle less or greater than 45°.
In this case the projected curve is that generated by a point
travelling round an ellipse in such a way that the eccen-
tric angle increases uniformly while the centre of the ellipse
EPICYCLICS. 197
advances uniformly, — in the direction of the minor axis if the
angle of inclination exceeds half a right angle, and of the
major axis if the angle of inclination is less than half a
right angle.
A set of such curves, obtained from a helix of inclination
45°, are shown in fig. 144, plate VI., Abjo T' being a semi-
cycloid, and AbgT, AbgT', (kc, other projections of the
same portion of the helix by lines inclined to the plane of
projection at an angle exceeding a right angle, A b T' being
the orthogonal projection of this portion of the helix.
Such ciu^es, and varieties of them resulting when the
helix is skewed (the centre of the circle advancing in a
direction not perpendicular to the plane of the circle), possess
interesting properties ; but they do not belong to our subject,
not being trochoidal. Moreover, for their thorough investi-
gation much more space would be required than can here be
spared. But one of these curves, the orthogonal projection
AbT' (fig. 144, Plate VI.) of a helix of inclination 45°,
must be briefly mentioned here, because associated histori-
cally as well as geometrically with the right cycloid.
THE COMPANION TO THE CICLOID.
This ciu-ve, called also * Roberval's Curve of Sines,* may
be obtained as follows : —
Let AB (fig. 142) be a fixed diameter of a circle AQB,
and through any point Q on AQB draw MQ j9 perp. to ACB
and equal to the arc AQ ; the locus of this point p is the
companion to the cycloid APD having AB as axis.
If COc, the line of centres of semicycloid APD, be
bisected in 0, the curve passes through O, because CO ^
quadrant AQC.
Drawing p m, Q, n, perp. to CO c, we have tti =
198
GEOMETRY OF CYCLOIDS.
CO-Cm=AC'-AQ=arcQC';;)m:Om :: Qw rarcQC
\ \ sin QCC ; circ. meas. of QCC. Hence the part A jo O of
the companion to the cycloid is a curve of sines.
Produce Qn to meet AC'B in Q', draw MQ'jt)' paralle
Fio. 142.
to BD to meet the curve ApJ^mp' and AB in M', and draw
pm' perp. to CO c. Then
C m' = M>' = AC Q', and OC = AC
,-. Om' = arc C'Q' = arc C'Q = O m ;
And p m' = w Q' = n Q = jt) m.
Therefore the part Op'T> of the curve bears precisely the
same relation to the line c, which the part A jo O bears to
OC. Thus the entire curve is a curve of sines.
Area Ajt? OC = areaO jo' 1>C] wherefore, adding CODB,
area AODB = rect. CD = ^ rect. BE = circle AQB.
It is also obvious that the same curve D^' ^ A will be ob-
tained by taking E o' D as the generating semicircle^ and
drawing m' q jt?=arc q'Y>^mqp'=^ arc qq'T>\ so that the
figure ED p Op A is in all respects equal to the figure
BApOp D.
EPICTCLIC8. 199
Since MQP = arc AQ + MQ ; and Mp = arc AQ,
MQ =pP;
so that an elementary rectangle QN = elementary rectangle
p L of same breadth ; whence it follows that area Ap D P
= semicircle AQB : for we may regard pJj and NQ as
elementary rectangles of these areas respectively, and the
equality of every such pair of elements involves the equality
of the areas. Since
area AODB=circle AQB ; and area A.p DP=i circle AQB ;
/. Area APDB = f circle AQB ;
and 2 area APDB = 3 circle AQB :
this is Eoberval's demonstration of the area of the cycloid.
Draw ar parallel and near to Qj9, and hah, C T, rl
perp. to OC ; then
ZC = A«; mC= AQ ; .*. m Z = Q « ; and
ml: nh:: Q«: wA::CQ(=AA;) : Qn(ult. = rZ)
.*. rect. ml,r Z=rect. nh.hk; that is, rect. r m=rect. n k ;
or inct. of area Apm C=>inct. of rect. A n. But these areas
begin together. Hence area A je> m C = rect. A n ; also
Area AOC = rect. CT ; and area pmO == rect. n T.
Kepresenting angles by their circular measure : —
. QC .Om J xm •/'i Om\
. QU' . Urn J X m « A *
pm^=rem^ — srsin ; and rect. 7iT=r' ( 1— cos
r T \
)
f
therefore, the proof that area pmO ^s rect. n T, may be re-
garded as a geometrical demonstration of the relation
sin a; cZ a; = 1 — cos x ;
J
X
AQ Cm
And aimilarly, since pm^r cos —55 =s r cos , the proof
r r
200 GEOMETRY OF CYCLOIDS.
that area A jt? m C = rect. A n may be regarded as a geome*
trical demonstration of the relation
J
cos xdxr=. sin x,
X
It will easily be seen that for points on 0^'D,
Area AO j^' M' — rect. M' m' = rect, A n, or B w,
leading again to the relation
area AODB = rect. B c.
201
Section VI.
EQUATIONS TO CYCLOID AL CURVES.
Although, properly speaking, the discussion of the equa-
tions to cycloidal curves belongs to the analytical treatment
of om* subject, it may be well, for convenience of reference,
to indicate here the equations to trochoids (including the cy-
cloid), epicyclics, and the system of spirals which may be re-
garded as epitrochoidal {see p. 127, et seq.). For the sake of
convenience and brevity I follow the epicyclic method of
considering all these curves.
Let the centre of a circle aqb (figs. 45, 46, Plate I.), of
radius 6, travel with velocity 1 along a straight line C c in its
own plane, while a point travels with velocity m round the
circumference of the circle. Take the straight line C c for
axis o£x, Ca for axis of y, and let the point start from a, in
direction aqb. When it has described an angle m ^ about C,
the centre has advanced a distance e along C c, and there-
fore, if X and y are the coordinates of the tracing point,
X = e <!> + e sin m <l>, y = e cos m 0. (1)
If we remove the origin to 6, the centre of the base, taking
h d 2A axis of x and 5 a as axis of y, the equations are,
a: = 6 + e sin m 0, y = 6 + 6 cos m 0. (2)
If we remove the origin to a, the vertex, taking a e as
axis of X and a & as axis of y, the equations are
;b = 6 ^ -f 6 sin m 0, y = e — e cos m 0. (3) ,
V . .
202 GEOMETRY OF CYCLOIDS.
If we remove the origin to c', taking c' C aa axis of x,
and t! d' sa axis of y, tbe- tracing point starting from d in
the same direction as before, the equations are
a: = e^-«Bin^, y = ecosm^ (4)
If in this case ve remove the origin to e', taking e'e as
axis of X and e' d' as axis of y, the equations are
je = e ^ — e sin ^, y = e + e cos m ^. (6)
And lastly, if we remove the origin to d', taking d'd as
axis of X and </' e' as axis of j/, we have the equations
x^ti^ — e sin ^ ^ = e — e cos m ^. (6)
Fia. 143. (Join C'p.)
If )/i = I, these equations represent the right cycloid ; if
m < 1, they represent the prolate cycloid ; and if m > 1, they
represent the ciutate cycloid.
For epicyclics, take O (fig. 143), the centre of fixed circle
as origin, OA through an apocentre A as axis of x, axid a
perp. to OA through as axis of y. Put OC, radius of defe-
rent ^rf; CA, radius of epicycle ^e (using italics as more
convenient in equations than capitab) ; £ COC ^ 0, and
angle a C P ^ n ^ . Then, if x and y are the co-ordinates of P
x=d{X)8 ^+ccos n ^, y=rfsin^+eBiii np. (7)
EQUATIONS TO CYCLOID AL CURVES. 203
If OC, instead of passing through an apocentre when pro-
duced, intersects the curve in a pericentre at B, the equations
are
x=c? cos — e cos n ^, y^=d sin ^ — 6 sin n 0. (8)
For a retrograde epicyclic, angle aC*P=n^, and the
equations (A being an apocentre ) are
a5=cZ cos ^ + e cos n ^, y=^ sin — e sin n 0. (9)
If B is a pericentre of retrograde epicyclic, the equations
are
aj=rcZ COS — 6 COS n <p, y==d sin + e sin n 0. (10)
But all these equations are derivable fi*om form (7) ; — (8)
by rotating the axis through the angle of descent,— —-j ; and
(9) and (10) from (7) and (8) respectively by changing the
sign of n. So that equations (7) may be used as the equa-
tions for the epicyclic in rectangular coordinates, without
loss of generality.
When, in (7) and (10), w = -, the equations are those of
the epicycloid and hypocycloid respectively, when an axis coin-
cides with the axis a;; if, in equations (8) and (9), ti=-, the
equations are those of the epicycloid and hypocycloid, respec-
tively, when a cusp falls on axis of x, it will be remembered
that if F is radius of fixed circle and R radius of rolling circle,
c^ = Il-I-F, and e = R ; R being regarded as negative in case
of hypocycloid.
From (7) we get
a;2 + y2 ^ r2 = f^a + e> -F 2 rf 6COs (n - 1)0, (11)
- ^ dooa d) + e cos n<b .,^.
andtane= , . P^ . Z; (12)
which are the polar equations to the curve, being the pole
204 GEOMETRY OF CYCLOIDS,
and OA, though an apocentre, the initial line. [Equation
(11) is obviously derivable at once from the triangle OCP.]
For the epicyclic spirals, suppose OC, %. 143=-/, and that
a tangent at C to circle CK, carrying with it the perp. BCA,
rolls over the arc OR, uniformly, till it is in contact at C ,
the angle C'OC being 0. Then if AC = g, and x and y are
the rectangular coordinates of the point to which A has
been carried, it is obvious (since CA in its new position is
parallel to OC ) that (taking projections on axes of x and y)
a;=(/*+5r)cos^-f/0sin0; y=(/+5r) sin 0-/0 cos ^ ; (13)
the equations to the epicyclic spiral traced by A. The sjiiral
traced by B obviously has for its equations
«=(/— (7)cos04-/^sin0; y=(/— ^r) sin 0-/0 cos ^. (14)
From (13) we get
a;»+ya = r* = (/+(7)2 4-/»0«; or
/•. /■«— TTrr-ax ^ a (/+.7)s in<^-/0co8»
/0= v/r«-(/«+(7«); tan 0= (j:^^)- s^+Z^sin ^ <^^)
the polar equations to these spirals. See also Ex. 133, p. 253.
If (7=0, or the tracing point is on the tangent, equations
(13) become
a; — /cos0 +/0sin0, y =/sin — /0 cos ^ ; (16)
the equations to the involute of a circle. The polar equation
to this curve is (from 15),
/tan J^^z!^^ - ^/W^::j^
taiie= , { V, (17)
/tan ^^ "'^+ s/1^^^
1£ g = — / equations (13) become
a;=/0sin0; y=— /0cos0j
giving x^ -\- y^ =-P (l>^ ; orr=/0;
IT ^
and tan 6 = — cot ; or 6 = — s ^
EQUATIONS TO CYCLOID AL CURVES. 205
whence r =:/d +/^ ; (18)
the polar equation to the spiral of Archimedes, with OD, fig.
72, p. 130, as initial line. If OQ be taken as initial line, the
equation is
r=fd. (19)
All the pairs of equations in rectangular coordinates can
readily, by eliminating ^, be reduced to a single equation
between x and y. Thus (1) becomes
e
X = — COS"
■'g)+ ^e*-2/*; (20)
the. general equation to the right trochoid.
From equation (11)
1 _i aj* + 2^* - d'^ - e^
6 = f COS * ^r-1 :
which combined with either of equations (7) gives the general
equation to the epicycUc in rectangular coordinates. To
obtain this general equation in a symmetrical form, note that
from (7)
y cos — a sin = e sin {n — 1)0. (21)
However, in nearly all analytical investigations of the
properties of these curves, it is more convenient to use the
pair of equations (1) for trochoids, (7) for epicyclics, and
(13) for epicyclic spirals, or the polar equations (11) and (12)
for epicyclics, and (15) for epicyclic spirals.
The only use I propose to make, here, of the equations to
these curves, is to obtain the general equations to the evo-
lutes of trochoids, epicyclics, and epicyclic spirals. These
general equations, though they may be deduced from rela-
tions established geometrically in the text, are more con-
veniently dealt with analytically.
We have in equations (1), (7), and (13), x and y expressed
as functions of a third variable ; wherefore
206 GEOMETRY OF CYCLOIDS.
d}y dx dPxdy '
d^^ d(p'~ d<f^ d<p
and the equation to the evolute is derived from the two
equations
d<l^} \dq»J '^\d(pJ J
^•"^ d'^y dxcf^x dy '
d ut^ d <l> diff^ d<i>
_ . dip] {d(i>j ^\d<t>j r .
d<j>^ d<p dfft^ dfji
where £ and rf are coordinates of the point in the evolute
corresponding to the point x, y, on the curve.
In the case of trochoids, we obtain from (1)
d X d u
7— =e+mecosma): -7-^ = — W6 sinm ^ :
d^y d^x
Also, ^-^=:— m^ecosm^j ^j = — mesinm^;
d^V dx d^x dy ««/ ^, \
• Z—£. zJi _ — - -^ — — e' m' (cos m + w) :
••(^92 (f0 difl^ d(l> ^ ^ ''
(1 4- 2 m cos m 9 + 7?i^)t
wherefore p =: — e ^ « / ^^^ . . ^^ — \
^ ni* (cos m + ^'*)
(1 + 2mcosm0 + ^2)
and if we put — -^ s-7 ; ^ = «^> *^® eqiiations
^ m^ (cos mifi -\- m) *
to the evolute are
^ = e^ + e(l +A;m)sinm0,
j; = e A; + 6(1 + A;m)cosm0, ^*'"'^
EQUATIONS TO CTCLOIDAL CURVES. 207
If we put \ _Ji — =0' and m (l+k m)=zm\ these equations
may be written
f = « (1 + A; m) 0' + e (1 + A; m) sin m'^',
and iy = eA; + c(lH-A;m) cos m'^' ;
from which we see that the evolute of the trochoid may be
regarded as traced by an epicycle of variable radius e ( 1 + A; m),
in which the tracing point moves with velocity bearing the
variable ratio m' to the velocity of the epicycle's centre,
while the deferent straight line shifts parallel to the axis of
X so that its distance from this axis is constantly equal to
6 A; on the negative side of the axis of y.
If m = 1 (or curve (1) becomes the cycloid), A; = — 2,
and equations (22) become
£r=e0 — 6sin^; ij= — 2e — e cos m ; (23)
showing that the evolute is an equal and similar cycloid,
with parallel base, removed a distance 2 e, or one diameter of
the tracing circle, from the base of the involute cycloid
towards the negative side of the axis of y (that is from the
concavity of the involute), and having vertices coincident
with the cusps of the involute cycloid.
From equations (7) we obtain
d sc • • d u
__=: —cf sin 0—71 e sin 710; ^=(f cos0 + 7iecos7i0;
a a
1— -- = — a sm 6 — n*e sm n <b,
a 0^ ^ ^
-; — g = — c^cos — 71^ e cos n <b :
a <i»^ a <i> d u.^ a If) ^
208 GEOMETRY OF CYCLOIDS.
__ [ d^ + n^e^ -^ 2ncgecos(n-l)0j ^
wheretore,p - ^, ^ n^e^J^(n^ + n) <£e cos (ti-I)^ ^
, c?*4-n*e^ + 2nc?6 cos(n— 1)
and If we put j^2^^z^%^ (n^ + n) t£ e cos (n-1) 0=^'
we obtain for the equations to the evolute
l=id cos + e cos w — A; (cZ cos + w e cos w 0),
and 71 = d sin + e sin n ^ — A; (cZ sin + n e sin n 0) ;
or £ = <Z(1 — A;)co8 + 6(1— wA;) cos n0
and i; ^ c?(l— A;) sin0 + e(l— 7iA;)sin w^
whence we see that the evolute may be regarded as traced
by an epicycle of variable radius e (l—nk) carried on a de-
ferent also of variable i-adius d (1—k),
It is easily seen (see p. 117, and figs. 63, 64), that
,_C-B7 pa \
CO' V;>«-NB7*
When d -= ne, 80 that the involute epicyclic is the epi-
cycloid or the hypocycloid (according as 7i is positive or ne-
2
gative), k reduces to , and the equations of the evolute
become
}; (24)
4 = . d cos 0—
w+1 ^ w+l
w— 1 , . ^ w— 1
n =r . a sm0— -
(25)
which (we see from 8) are the equations of an epicycloid or
hypocycloid (according as n is positive or negative), whose
deferential and epicyclic radii (and in fact whose linear pro-
portions) bear to those of the involute the ratio (n— 1) ;
(w-f-l), and whose vertices touch the cusps of the involute
epicycloid or hypocycloid. If w is positive the ratio (w— 1)
: (n f 1) is the same as (d—e) : (c?+e), or F ; (F+ 2 R), as
in Section II. If w is negative the ratio (w— 1) : (^i-f- 1) is
the same as (d+e) : (<£— e),orF : (F— 2R),as in Section II.
209
Section VII.
GRAPHICAL USE OF CYCLOIDAL CURVES.
Graphical Use of the Cycloid and its Companion to
Determine the Motion of Planets and Comets.
[From the Monthly Notices of the Astronomical Society for April
1873.]
The student of astronomy often has occasion to deter-
mine approximately the motion of bodies, as double stars,
comets, meteor systems, and so on, — in orbits of considerable
eccentricity. The following graphical method for solving
such problems in a simple yet accurate manner is, so far as I
know, anew one.* By its means a diagram such as fig. 144,
Plate VI., having, once for all, been carefully inked in on
good drawing card, the motion of a body in an orbit of any
eccentricity can be determined by a pencilled construction of
great simplicity, which can be completed (including the
construction of the ellipse) in a second or two.
Let APA', ^. 145, be an elliptical orbit of which ACA'
is the major axis, C the centre, S being the centre of force,
so that A is the aphelion, and A' the perihelion. Let H be
* New as a method of construction, though the principle on
which it depends is of course not new. The curve ApT' (fig. 146), for
instance, is an orthogonal projection of a particular prolate cycloid
which, as Newton long since showed, if accurately drawn, gives the
means of determining the motion in the ellipse APA'. But, as he
remarks, this prolate cycloid cannot readily be drawn ; whereas the
curve ApT' can be very readily drawn.
P
^M ■ L'lO GEOMETRY OF CYCLOIDS. ]
^H Imlf the iiei-iodic time, and T the time in which the bj
^H moves from A to P.
^1 On AA' desciilie the auKiliary semicircle A 6 A'.
■ The..
H T : H :: ai-«i asp ; area ABA'
^L :: (ACQ +SCQ) rareaAJA'
^^^H :: AC . AQ + CS . QM ; AO . AQA'
^^^P ::aq+^.QM:aqa'
Now if A III T' be a cj'cloid hariog AA' as its diamd
(Ordinate M in = AQ + QM.
Fm, US.
1
H
^
tho
QJ
Aud il' «-,; uki^ Jll 'J = AQ, %ve haw f/ u jioiiit oiiAg
lamijuuiou to tlie cjcloiil, Tlie line q m is then etiual
1 ; iuid if we take a point j) on m Q such that
h&ye
GHAPHICAZ USE OF CYCLOID AL CURVES, 211
SO
M|? = AQ + ^. QM; and AT' = AQA';
wherefore
T: H::M|? : a'T'
Thus we may represent the time in traversing the arc AP
by the ordinate Mj^to a curve A.p T', obtained by dividing
all such lines as ^ 7?i (joining the cycloid and its companion^
and parallel to A'T') so that qp I qm 2l&^Q \ AC.
Accordingly, if we construct such a diagram as is shown
in fig. 144, plate VI., in which AT' is a semi-cycloidal arc
and AbT' its companion, while intermediate curves are
drawn dividing all such lines as b b|Q into ten or any other
convenient number of equal parts, the curves through the
successive points b, b|, b2, &c., to bi©, give us the time-ordi-
nates for bodies moving in ellipses having A and A as apses,
and their centres of force i*espectively at C, Si, 83,83, . . .
S9, and A'.
In the plate the semi-ellipses corresponding to these posi-
tions of the centre of force are drawn in, and it will be
manifest that any ellipse intermediate to those shown can be
pencilled in at once, with sufficient accuracy. Ellipses within
AB9A' have their focus of force between Sg and A', and are
exceptionally eccentric* It is easy to construct such an
ellipse, however, in the manner indicated for the semi-ellipge
AB9A'. For the radial lines and the parallels to AT
thi*ough their extremities are supposed to be inked in ; and
(taking the case of ellipse ABgA') we have only to draw the
semicircle a B9 a', and parallels to AA' through the points
where the radial lines intersect this semicircle, to obtain by
* It is manifest that when the centre of force is at A' we have
the case of a body projected directly from a centre of force, and the
time-curve becomes the cycloid A bj^T'. Thus the above lines give a
geometrical demonstration of the relation established analjrtically
in the paper which follows.
p 2
212 GEOMETRY OJb CYCLOIDS^
the intersections of these parallels with the parallels to AT
a sufficient number of points on the semi-ellipse.
The illustrative diagram has been specially constructed
for the use of those who may have occasion to employ the
method, and will be found sufficiently accurate for all ordi-
nary purposes. Before proceeding, however, to show how
the method is applied in special cases, I shall describe how
such a diagram should be constructed : —
First the semicircle ABA' must be drawn, and the lines
AT, A'T' perp. to AA'. Then CA' must be divided into
ten equal parts (and when the %ure is large, a plotting scale
for hundredths, &c., should be drawn). Next A'T and AT
must be each taken equal to 3*1416 where CA' is the unit.
Join TT'. Now AT and AT' represent, as time-ordinates,
the half-period of any body moving in an ellipse having AA'
as major axis. Each must now be divided into the skme
number of equal parts, and it is convenient to have eighteen
such parts. (So that in the illustrative case of our Earth,
three divisions represent a month.) Next the semicircle
ABA' must be divided into eighteen equal parts. Through
the points of division on the semicircle, parallels to AT and
A'T' are to be drawn,* and the points of division along AT and
A'T' are to be joined by parallels to AA' and TT'. Then the
curve A b T', the * companion to the cycloid,' runs through
the points of intersection of the first parallel to AT and the
first to AA', the second parallel to AT and the second to
AA', the thii'd parallel to these lines, the foiuiih, and so on.
We have now only to take b biQ equal to CB ; q, pi equal to
Ml Pj ; q2 P2 equal to Mg Pg ; and so on, to obtain the re-
quired points on the cycloid A bjoT'; and the equidivision
♦ Practically it is convenient to draw another semicircle on TT,
divide its circumference into eighteen parts, and join the correspond-
ing points of division on the two semicircles.
GRAPHICAL USE OF CYCLOIDAL CURVES. 213
of all suoh lines as b b,o, qi Pi, q2 P2 (into ten parts in the
illustrative diagram) gives us the required points on the
intermediate curves.
Next let us take some instances of the application of the
diagram.
I. Suppose we wish to divide a semi-e'lipse of given
eccentricity into any given number of parts traversed in
equal times, and let the eccentricity be \, and 18 the given
number of parts ♦ : —
Then S.^ is the centre of force ; AB5A' the semi-ellipse ;
and AbgT' the time-curve. The dots along AbgT' give
the intersection of the time-curve with the time-ordinates
parallel to AA'; and therefore parallels to AT, though these
dots (not drawn in the figure, to avoid confusion) indi-
cate by their intersection with the semi-ellipse AB5A1 the
points of division required.
II. Suppose we wish to know how &r the November
meteors travel from perihelion in the course of one quarter
of their period, that is, one half the time from perihelion to
aphelion : —
The curve ABgAj is almost exactly of the same eccen-
tricity as the orbit of the November meteors. To avoid
additional lines and curves, let us take it as exactly right.
Then AbciT' is the time- curve. For the quarter period
fr^m the perihelion (or aphelion), we take of course the
middle vertical line, which intersects AbgT in Cg. This
point by a coincidence is almost exactly on a parallel to AT,
and this parallel meets the semi-ellipse ABg A' in n, the re-
quired point on the orbit. In other words, the journey of
the November meteors from A to n occupies the same time
as their journey from n to A', Sg being the position of the
* This selection is made solely to avoid the addition of lines and
carves not necessary to the completeness of the diagram.
214 GEOMETRY OF CYCLOIDS,
Sun, and the Earth's distance from the Sun approximately
equal to A'Sg.
III. Suppose we require, in like manner, the quai-ter-
period positions in different orbits, all having AA as major
axis, but their centres of force variously placed along CA'.
We get any number of points, n, 1, k, precisely as n was
obtained ; m, of course, is on the parallel through C|o ; and
we obtain, in fine, the curve m n 1 k B, which resembles, but
is not, an elliptic quadrant.
IV. Suppose we require to know in what time the half
orbit from aphelion or perihelion is described in orbits of
different eccentricity. The requii*ed information is manifestly
indicated by the intersection of CC with the time-curves, in
b, bi, bg, <&c. Thus in the circle, A£ is described in the time
represented by C b ; in the semi-ellipse AB3 A', AB3 is
described in the time represented by C bs, and B3A' in the
time represented by bgC; and so on for the other semi-
ellipses?.
V. Suppose we require to determine approximately the
* equation of the centre ' for a body when at any given point
of its orbit of known eccentricity. Take the case of Mars,
whose eccentricity being nearly -j^, his path is feirly repre-
sented by the ellipse next within ABA', and his time-curve
by Ab] T'. Then the equation of the centre, when Mars is
at his mean distance, is represented by bbj ; when Mars is
at Pi (not on the circle, but on the curve just within), the
equation of his centre is represented by qi r^ ; and so on.
Many other uses and interpretations of the time-curves
will suggest themselves readily to those who are likely to use
the diiigram.
After the above method had been briefly described, Pro-
fessor Adams, who was in the chair, mentioned a method
I
GltAPHICAL USE OF CYCLOIDAL CURVES.
(devised by himseff miiny yea:* since) hy which the same
reaults can be obtained from the ' companion to the cycloid '
or ' curve of sines,' Profensor Adams's method may be thus
exhibited ; — Let abn' he the y-positive half of one wave of
the ' curve of sinee,' 6C its diameter ; A6A', a semicircle with
radius li C. Let ABA', fig. 146, be a half-ellipRO having' its
foctia at 8. Then tbe time in any arc AP of this ellipse may
he thus determined. Join b S, produce the ordinate PM to Q
on circle ABA', <h^w Q q parallel to n n', and qp parallel to
h S ; then ap represents the time in traversing AP. where
a a' represents the half period. Andvieever»A. if we require
the iiosition of the moving- body after any time fi-oni tlie ap.se,
say aplie'ion, then ttike n ji to iflpi-eseiit the time, where a a'
is the half [teriod, AOA' the major axis, S the centre of foit-e ;
join S 6, dmw p q pamllel to S ft, 7 Q parallel to AA', and
QP perpendicular to AA' gives P the point i-equii-ed.
It will be manifest that in principle my method ia iden-
tical with this, for in my figure the time is I'cpresented by
iVI JI, whei* M q {fig. 1 45) is equal to the arc AQ, and q p is
equal to QM reduced in the ratio of CS to t!A. Now « ^ in
fig. 146 is the projection of « 5 and qp; and the projection of
o 5 is equal to the arc AQ {see p. 300), while the projection
2118 GEOMETRY OF CYCLOIDS.
of qp is equal to QM reduced in the proportion of CS
to AC.
Althoup^h Professor Adams's consti-uction has the advan-
tage of requiring but a single curve, yet for the particular
purpose described my construction is more convenient. We
see from the fig. 146 that to give the relation between the
times and positions in the case of the ellipse ApA\ we
require a series of parallels to bC, a a' and 68; and the
parallels to 6 S only serve for this one case. Therefore we
could not constiTict a I'eference figure for many cases, without
having many series of parallels and a very confusing result.
In my construction we have, instead, many curves, but a
result which is not confusing because each curve is distinct
from the rest.
Graphical Use of the Cycloid to Measure the Motion
OF Matter projected from the Sun.
IFrom the Monthly Notices of tlie Astronomical Society for
December 1871.]
Whatever opinion we may form as to the way in which
the matter of certain solar prominences is propelled from
beneath the photosphere, there can be little question that
such propulsion i*eally takes place. It seems clear indeed
that some prominences, more especially those seen in the
Sun's polar and equatorial regions, ai*e formed — or rather
make their appearance — in the upper regions of the so!ar
atmosphere, and even assume the appearance of eruption-
prominences by an extension downwards^ somewhat as a
waterspout simulates the appearance of an uprushing column
of water though really formed by a descending movement.
But it is certain that other prominences are really phenomena
of eruption.
GRAPHICAL USE OF CYCLOID AL CURVES, 2\7
' In the case of any matter thus erupted, we shall clearly
obtain an inferior limit for the value of the initial velocity
of outrush, if we assume that the apparent height reached by
the matter is the real limit of its upward motion (that is,
that there is no foreshortening), and that the solar atmosphere
exercises no appreciable influence in retarding the motion.
The latter supposition is, however, wholly untenable under
the circumstances, while the former must in nearly all cases
be erroneous ; and I only make these suppositions in order
to simplify the subject, noting that their effect is to reduce
the estimated velocity of outrush to its lowest limiting
value.
W« are to deal then, for the present, with the case of
matter flung vertically upwards from the sim*s sinface and
subject only to the influence of solar gravity ; J propose to
consider the time of flight between certain observed levels,
not the mere vertical distance attained by the erupted
matter; and (as I wish to deal with cases where a great
distance from the sun. has been attained) it will be necessary
to take into account the different actions of the solar gravity
at different distances. Zollner, in dealing with prominences
of moderate height, has regarded the solar gravity as con-
stant ; but this is evidently not admissible when we come to
deal with matter hm-led to a height of 200,000 miles, since
at that height solar gravity is reduced to less than one-half
the value it has at the surface of the sun.
It is easy to obtain the required formula ; and though it
is doubtless contained in all treatises on Dynamics, it will
be as well to run through the work in this place. In re-
ducing the formula I have noticed a neat geometrical illus-
tration (and a partial proof) which I do not remember to
' have seen in that form in any book. It not only presents
in a striking manner the varying rate at which a body
218
GEOMETRY OF CYCLOIDS.
falls towards a centre attracting according to the law of
natui-e, but it supplies a means whereby the time of flight
between any given distances may be readily obtained from a
simple consti-uction.
Let C, fig. 147, be the centre of a globe ABD, of radius
K, and attracting accoi-ding to the law of nature ; let ^ be the
accelerating force of gravity at the surface of the globe. Then
the attraction exerted at a imit of distance, if the whole
mass of the globe were collected at a point, would be ^r R^
Fig. 147.
Illustrating the motion of a body descending from rest towards a globo
attracting according to the law of natorc.
Let a particle falling fi-om rest at E i-each the point P i
time t ; and let AE = H, and CP = x. Then the ecjuation
of motion is
GRAPHICAL USE OF CYCLOIDAL CURVES. 219
giving
SO that, since the particle staiiis from rest at a distance
(R + H) fix)m C, we have
For convenience write D for (E. + H) ; then we have
(f:)'=..=..E.(i-i)
2(7R2 /D_a;
D V
-:-')• <■)
Thus
2g dt X
D dx \/~t>x — x^
Integrating, we have
R ^y^^ . t = vDa;-aj2 ---^cos-^ (^-^^) + 0.
Dtt
But when < = 0, oj = D ; so that C = o" >
hence we have
R ^§t = ^ir. -i^ + 5 COB-, ('-""g-?), (2)
(where D is equal to the radius of the glohe added to the
height from which the particle is let fall).
Equation (1) gives the velocity acquired in falling (from
rest) from a height H to a distance x from the centre, and
(2) gives the time of falling to that distance. The geo-
metrical illustration to which I have referred, relates to
•the deduction of (2) from (1). We see from (1) that at the
jioint P
220 GEOMETRY OF CYCLOIDS,
2(7R^ /!> -X'
•■=^^H'^')-
Bisect CE in F, and describe the semicircle CDE ; then if
DE is a tangent to the circle DAB, and if DM is drawn
])erpendicular to CE,
^^ — CE ~" D ^
80 that
But if close by G, either on the tangent GH or on the arc
GE, we take G' and draw GT' perpendicular to CE, and
G n peq)endicular to GP, we have
GG^ 4- Gn GF H- FP __CP^
PP " GP ■■ v^CP . P£
CP
PE*
Hence, fi-om (a),
V PP'
=v;
V2g .CM GG + G ?i '
so that
r the vel. 1 . r velocity acquired in falling thi-ough 1
\ at P J ' \ space CM, under const, accel. force g j
, , f elem. space "1 . f sum of elementaiy 1
. . > pp/ I . I gp^^^ QQ g^^j G 71 / •
Therefoi*e the falling particle traverses the space PP' in the
same time that a particle travelling with the velocity acquired
in falling through space CM under constant accelerating
force g, would traverse the space (GG' + G n). It follows
that the time in falling from E to P is the same as would be
occupied by a particle in traversing (arc EG + GP) with the
velocity acquired in falling through the space CM under a
constant accelerating force g. In other words.
GRAPHICAL USE OF CYCLOID AL CURVES. 221
PG + arc GE
< =
V2g , CM
or
R \f^ . t = >/PE . PC + OF arc GE
D /2aj — Dn
a= \/(D -a;)a; + YCOS-» ( — JJ — j'
as before.
The relation here considered affords a very convenient
construction for determining the time of descent in any given
case. For, if PG be produced to Q so that GQ = arc GE,
Q lies on a semi-cycloid KQC, having CE as diameter ; and
the relative time of flight from E to any point in AE is at
once indicated by drawing through the point an ordinate
parallel to CK. The actual time of flight in any given case
can also be readily indicated. For let T be the time in
which LC would be desciibed with the velocity acquired in
falling through a distance equal to LC under accelerating
force g, and on LM describe the semicircle L m M ; then
clearly C m (= '/CL . CM) will be the space described in
time T with the velocity acquired in falling through the
space CM under accelerating force g ; and we have only to
divide C m into parts corresponding to the known time-
interval T, and to measure ofl* distances equal to these parts
on PQ to find the time of traversing PQ with this uniform
velocity, i.e., the time in which the particle falls jfrom E to P.
The division in the figure illustrates such measm'ements in
the case of the sun, the value of T being taken as 18f minutes.
Moreover it is not necessary to construct a cycloid for
each case. One carefully constructed cycloid will serve for
all cases, the radius C A being made the geometrical variable.
As an instance of this method of construction, I will take
Professor Young's I'emarkable observation of a solar out-
323 OEO^fETRY OF CTCLOIVS.
hoist, preiniBing that I only give the couBtruction as ai
tnitiou, mid that a proper ciilculation follows.
GRAPHICAL USE OF Cl'CLOlDAL CURVES. -'23
Oil September 7, 1871, Professoi' Young saw wisps of
liydrogeucarriedin ten minuteB from a height of 100,000 milea
to a height exceeding 200,000 miles from the sun's suifiw*.
L;4- f"II account of his ob»erviitions is given iu the second and
I
thii-deditionaof my treatise on the sun. Figa. 148, 149, 150,
and IBl, with the times noted, indicate the progress of the
changes, I assumed in wlmt follows that thei-ewas no foi'e-
hhoi-teniug. The height, 100,000 miles (upper part of cloud
in fig. 148), was determined by estimation ; but the ultimate
height reached bj- the hydrogen wisps (that is, the eieviition
224 GEOMETRY OF CYCLOIDS.
at which they vanished as by a gradual dissolution) results
from the mean of three carefully executed and closely ac-
cordant measures. This mean was 7 49", corresponding to
a height of 2 10,000 miles (highest filaments in ^g, 1 49). We
may safely take 100,000 miles as the vertical range actually
traversed, and 200,000 miles as the extreme limit attained.
We need not inquire whether the hydrogen wisps were
themselves projected fix)m the photosphere, — most probably
they were not, — but if not, yet beyond question there was
propelled from the sun some matter which by its own motion
caused the hydrogen to traverse the above-mentioned range
in the time named, or caused the hydrogen already at those
heights to glow with intense lustre. We shall be under-
rating the velocity of expulsion, in regarding this matter
as something solid propelled through a non-resisting me-
dium, and attaining an extreme range of 200,000 miles.
What follows will show whether this supposition is ad-
missible.
Now g for the sun, with a mile as the unit of length and
a second for the unit of time, is 0*169, and "R for the sun is
425,000. Thus the velocity acquired in traversing R under
imiform force g,
= V2g . R
s= V338 X 425
= 379, very nearly.
(This is also the velocity acquired imder the sun's actual
attraction by a body moving from an infinite distance to the
sun's surface.)
And a distance 425,000 would be traversed with this
velocity in 18°» 40» (= T).
Let KQE, fig. 152, be our semi-cycloid (available for
GRAPHICAL USE OF CYCLOID AL CURVES. 226
many suooessive constructions if these be only pencilled), and
ODE half the generating circle.
Then the following is our construction : — Divide EC into
6;^ equal portions, and let EP, PA be two of these parts, so
that £A represents 200,000 miles and CA 425,000 miles
(the sun's radius). Describe the semicircle ADL about the
centre C ^d draw DM perpendicular to EC ; describe the half
circle M m L. Then m C represents T where the ordinate PQ
represents the time of fisdling from E to P.
Fio. 152.
IHafitrating the oonstrnotion for determining time of descent of a particle from
reet towards a globe attracting according to the law of nature.
T = 18^ 50™, and PQ (carefully measured) is found to
correspond to about twenty-six minutes.
Thus a body propelled upwards from A to E would
traverse the distance PE in twenty-six minutes. But the
hydrogen wisps watched by Professor Yoimg traversed the
distance represented by PE in ten minutes. Hence either
E was not the true limit of their upward motion^ or they
Q
236
GEOMETRY OF CYCLOIDS.
were retarded by the resistance of the solar atmosphere.
Of course if their actual flight was to any extent fore-
shortened, we should only the more obviously be forced to
adopt one or other of these conclusions.
But now let us suppose that the former is the correct
solution ; and let us inquire what change in the estimated
Hmit of the uprush will give ten minutes as the time of
moving (without resistance) from a height of 100,000 to a
height of 200,000 miles. Here we shall find the advanta^
Fig. 153.
niiistrating the oonstraction for determining time of desuent between given lerels
when a body descends from rest at a given height towards a globe attracting aooord'
ing to the law of nature.
of the constructive method ] for to test the matter by calcu-
lation would be a long process, whereas each construction
can be completed in a few minutes.
Let us try 375,000 miles as the vertical range. This
gives CE = 800,000 miles, and our construction assumes the
appearance shown in ^. 153. We have AC =425,000 miles;
GRAPHICAL USE OF CYCLOID AX CURVES. ^Stl
AP=PF = 100,000 miles ; and Q Z or (PQ-FQ') to repre-
sent the time of flight from P to P'.
The semicircles ADL, M m L, give us m C to represent
T or 18™ 50» ; and QL carefully measured is found to corre-
spond to rather less than ten minutes. It is, however, near
enough for our purpose.
It appears, then, that if we set aside the probability, or
rather the certainty, that the sun's atmosphere exerts a
retarding influence, we must infer that the matter projected
from the sun reached a height of 375,000 miles, or there-
abouts. This implies an initial velocity of about 265 miles
per second.*
.. But it will be well to make an exact calculation, — not
that any very great nicety of calculation is really required,
but in order to illustrate the method to be employed in such
cases, as well as to confirm the accuracy (^ the above con-
structions.
In equation (2) put >v/2 ^r E = 379; R = 425,000;
D = 625,000 ; and x == 525,000 ; values corresponding to
Professor Young's observations. It thus becomes —
V
III (379) t = a/( 100,000) (525,000)
. Q10KAA 1 /1050-625\
+ 312,500 co8-i(^ 625— J'
♦ The value is of course deduced directly from (1), p. 219 ; but it
is worthy of notice that it can be deduced at once from fig. 153, by
drawing A a parallel to KG, and w/ parallel to <i E ; then C /repre-
sents the required velocity, CL representing 379 miles per second.
A similar construction will give the velocity at P, F, &c. Applied
to fig. 1 47, it gives 0/ to represent the velocity at A, C /' to represent
the velocity at P ; m/and mf being parallel to /* E and GE re-
spectively. Applied to the case dealt with in fig. 152, we get C/t
represent the velocity at A, where E is the limit of flight : C/i
found to be rather more than | of CL ; so that the velocity at A is
rather more than 210 miles per second.
Q 2
« \
228 OEOMETRY OF CYCLOIDS.
or
379 VT7 . t = 250,000 V5T + 1,562,500 cos-i f ~Y
1562-7 1 = 1,145,100 + 1,285,800 = 2,430,900,
t = 1,556« = 25« 66-.
This then is the time which would have been occupied in
the flight of matter from a height of 100,000 to a height of
200,000 miles, if the latter height had been the limit of
vertical propulsion in a non-resisting medium.
In order to deduce the time of flight t between the same
levels, for the case where the total vertical range is 375,000
miles, we have, putting ti for the time of faU to 200,000
miles above the sun's surface, and t^ for the time of fall to
100,000 miles, the equation,
'125
V
800 (379) «,= a/(175,000) (625,000)
+ (400,000) cos -•(125^^^
1^(379) <j = V (275,000) (525,000)
V
+ (400,000) COS-' (i^S^j^
giving (since <2 — ^j = *')
^ /m (379) «' = 25,000 {VlFxTT- Vrx25}
V oOO
+ 400,000 I cos-' (^)- cos-' ^Aj J
276-25 1' = 49,250 + 111,816 = 161,066,
t' = 583- = 9°» 43-.
This is very near to Professor Young's ten minutes. I had
foimd that an extreme height of 400,000 miles gave 9™ 24*
for the time of flight between vertical altitudes 100,000
GRAPHICAL USE OF CYCLOID AL CURVES. 229
miles and 200,000 miles. It will be found that a height
of 360,000 miles gives 9" 58", which is sufficiently near to
Professor Young's time.
Now to attain a height of 360,000 miles a projectile from
the Sim's surfeuse must have an initial velocity
^-—— 7360,000 „^^ /72"
= ^'2^^. V 785:000 = 3^^ V 157
= 257 miles per second.
The eruptive velocity, then, at the sun's surface, cannot
possibly have been less than this. When we consider, how-
ever, that the observed uprushing matter was va^rous,
and not very greatly compressed (for otherwise the spectrum
of the hydrogen would have been continuous and the
spectroscope would have given no indications of the phe-
nomenon), we cannot but believe that the resisting action of
the solar atmosphere must have enormously reduced the
velocity of uprush before a height of 100,000 miles was
attained, as well as during the observed motion to the
height of 200,000 miles. It would be safer indeed to assimie
that the initial velocity was a considerable multiple of the
above-mentioned velocity, than only in excess of it in some
moderate proportion. Those who are acquainted with the
action of our own atmosphere on the flight of cannon-balls
(whereby the range becomes a mere fraction of that due to
the velocity of propulsion), will be ready to admit that hy-
drogen rushing through 100,000 miles even of a rare atmo-
$^here, with a velocity so great as to leave a residue sufficient
to carry the hydrogen 100,000 miles in the next ten minutes,
must have been propelled from the sun's surface with a
velocity many times exceeding 257 miles per second, the
result calculated for an unresisted projectile. Nor need we
wonder that the spectroscope suppHes no evidence of such
230 GEOMETRY OF CYCLOIDS.
yelocities, since if motions so rapid exist, others of all
degrees of rapidity down to such comparatively moderate
velocities as twenty or thirty miles per second also exist,
and the spectral lines of the hydrogen so moving would
be too greatly widened to be discerned.
Now the point to be specially noticed is, that supposing
matter more condensed than the upflung hydrogen to be
propelled from the sun during these eruptions, such matter
would retain a much larger proportion of the velocity origi-
nally imparted. Setting the velocity of outrush, in the case
we have been considering, at only twice the amount deduced
on the hypothesis of no resistance (and it is incredible that
tiie proportion can be so small), we have a velocity of pro-
jection of more than 500 miles per second ; and if the more
condensed erupted matter retained but that portion of its
velocity correspondi^ to three-fourths of this initial velocity
(which may fairly be admitted when we are supposing the
hydrogen to retain the portion corresponding to so much as
half of the initial velocity), then such more condensed
erupted matter would pass away from the sun's rule never to
return.
The question may suggest itself, however, whether the
eruption witnessed by Professor Young might not have been
a wholly exceptional phenomenon, and so the inference
respecting the possible extrusion of matter from, the sun's
globe be admissible only as relating to occasions few and
far between. On this point I would remark, in the first
place, that an eruption very much less noteworthy would
fairly authorise the inference that matter had been ejected
from the sun. I can scarcely conceive that the eruptions
witnessed quite frequently by Respighi, Secchi, and Younjg;
— such eruptions as suffice to carry hydrogen 80,000 or
100,000 miles from the sim's surface — can be accounted for
GRAPHICAL USE OF CYCLOID AL CURVES. 231
without admitting a velocity of outrush exceeding consider-
ably the 379 miles per second necessary for the actual rejeo
tion of matter from the sun. But apart from this it should
be remembered that we only see those prominences which
happen to lie round the rim of the sun's visible disk, and
that thus many mighty eruptions must escape our notice
even though we could keep a continual watch upon the
whole circle of the sierra and prominences (which unfortu-
nately is very £ai* from being the case).
It is worthy of notice that the great outrush witnessed
by Professor Young was not accompanied by any marked
signs of magnetic disturbance. Five hours later, however, a
magnetic storm began suddenly, which lasted for more than
a day ; and on the evening of September 7, there was a dis-
play of aurora borealis. Whether the occurrence of these
signs of magnetic disturbance was associated with the
appearance (on the visible half of the sun) of the great spot
which was approaching or crossing the eastern limb at the
time of Young's observation, cannot at present be deter-
mined.
I would remark, however, that so &r as is yet known
the disturbance of terrestrial magnetism by solar influences
would appear to depend on the condition of the photosphere,
and therefore to be only associated with the occurrence of
great eruptions in so far as these aflect the condition of the
photosphere. In this case an eruption occurring close by the
limb could not be expected to exercise any great influence on
the earth's magnetism ; and if the scene of the eruption were
beyond the limb, however slightly, we could not expect any
magnetic disturbance at all, though the observed phenomena
of eruption might be extremely magnificent.
In this connection I venture to quote from a letter
582 GEOMETRY OF CYCLOIDS.
addressed to me by Sir J. Hersohel in March 1871 (a few
weeks only before his lamented decease). The lettei* bears
throughout on the subject of this paper, and tiierdore I
quote more than relates to the assodaticm between terreetrial
magnetism and disturbances of the solar photoi^)here.
After referring to Mr. Brothers' photqgra{^ oi the eorcma
(remarking that ' the corona is certainly aos^ro-atmosphmc
and ti/^o-lunar '), Sir John Herschel proceeds thus i —
' I can very well conceive great outbursts <^ vaporous
matter from below the photosphere, and can admit at least
the possibility of such vapour being tossed up to very great
heights ; but I am hardly yet exalted to such a point as to
conceive a positive ejection of erupted particles with a
velocity of two or three hundred miles per second. But
now the great question of all arises : what is the photo-
8p/iere? what are those intensely radiant things — scenes,
flakes, or whatever else they be — ^which really do give out
all (or at least -^ths of) the total light and heat of the
Sim ) and if the prominences, &c,, be eruptive, why does not
the eruptive force scatter upwards and outwards this lu-
minous matter) . . . Through the kindness of the Kew
observers I have had heliographs of the two great outburst-
ing spots which I think I mentioned to you as having been
n(Hi-existent on the 9th, and in full development on the 10th,
both [being] large and conspicuous, and including an area of
disturbance at least 2' (54,000 miles) across. They were both
nearly absorbed, or in rapid process of absorption, ou the
11th. In my own mind I h^A set it down as pretty certain
that the outbreak must have taken place verj/ suddenly at
somewhere about the intervening midnight. Well, now !
The magnet's declination ciu-ves at Kew have been sent me,
and, lo ! while they had been going on as smoothly as
GRAPHICAL USE OF CYCZOIDAZ CURVES, 2S8
possible on the 6th, 7th, 8th, and 9th, and up to 11^ p.h. on
the latter day (9th), suddenly a great downward jerk in the
curve, forming a gap as far as 3^ a.m. on the 10th. Then
comparative tranquillity till 11 A.M., and then (corresponding
to the re-absorption of the spots) a furious and convulsive
state of disturbance extending over the 11th and the greater
part of the 12th. I wonder whether anything was shot out
of those holes on that occasion I and, if so, what is going on
in the inside of the sun t '
234 GEOMETRY OF CYCLOIHS.
EXAMPLES.
All the examples which have no name appended to them are
original, except four or five familiar ones (as 125, 126, &c.), the
authors of which are not known.
1. A chord of a cycloid parallel to the base is equal in
length to the perimeter of the uppermost of the two seg-
ments into which the chord divides the generating circle.
2. ATB' is the generating circle through P on the cy-
cloidal arc APD ; A'B' diametral ; and equal arcs P q and
P q' are taken on A'PB'. Show that straight lines drawn
from q and q', parallel to the base, to meet APD, are equal.
3. AQB is a semicircle on diameter AB ; and from Q, QL is
drawn perp. to AB, and produced to P, so that QP =s= arc
AQ. Show that the locus of P is a cycloid having a cusp at
A, and AB as secondary axis.
4. If B'P (fig. 4, p. 8), the normal at P, be produced to
meet AA' produced, in G, then PB' . PG = A'P>.
5. If the tangent A'P (fig. 4, p. 8), produced, meet the
tangent at D in T, show that AT : A B :: arc PB' : PB.
6. Show that the rectangle under PG {^, 4, p. 8) and
the diameter of curvature at P = (arc AP)*.
7. Show that the chord in which the tangent at P (fig. 4,
p. 8) intersects the circle on B'G as diameter, is equal to the
arc AP.
8. VQ'p is the tracing diameter of P on the cydoidal arc
r\
EXAMPLES, 235
D'APD. If p P', parallel to the base, meet the arc D'A
in P , show that the tangents and normals at P and P' form
a rectangle.
9. An equilateral triangle AQC is described on AG
(fig. 4, p. 8) as side ; show that QP, parallel to the base of the
cycloid, bisects the arc APD in P.
10. If through C, CP be drawn parallel to the base, to
meet the cycloid in P, show that (arc AD) ' = 2 (arc AP)*.
11. K there are two cycloids APD and AP''D placed as
in fig. 3, p. 6, and the straight line drawn from any point P
in one to a point Q in the other, P and Q lying on different
sides of C c, is equal to the diameter of the generating circle,
show that the circle on PQ as diameter touches BD and AE.
12. When the angle BAQ {0%, 4, p. 8) is equal to two
thirds of a right angle, then in the limit when P' moves up
to P,
PF= 2 MN, and ^ F = 2 n 7 = 2 ^w.
13. When the angle BAQ = one-third of a light angle,
then in the limit
PF = gF = 2wg=r|Zn.
14. In fig. 3, p. 6, if arc AQB intersect arc AP' D in
E, show that
area AQRF' = area BED.
15. APD, AP'D are two equal semi-cycloids placed as in
fig' 8, p. 17 ; show that every generating cii'cle A'PB' divides
the area APDP' into three parts, which are equal each to each
to the three parts into which the area of the circle A'PB' is
divided by the arcs APD, AP'D.
16. In the sameca^e, if two generating circles P'RATB'
and p'r apb cut APD in R, P and r, p, respectively, and
AP'D in F, p', show that
area P'R r y= difference of areas RAT, rap.
236 GEOMETRY OF CYCLOIDS.
17. In ^g. 6, p. 10, Area BD ss area AQC'T.
18. In fig. 11, p. 22,
Area AQBy — area Ey D = ^ generating circle.
19. If in fig. 5, p. 10, BJ is drawn perp. to BD, and a
quadrant AIC about T as centre, show that
area BJD = aiw AQC'I.
20. If CQP parallel to base BD cut the central genera-
-ting circle in Q and meet the cycloid in P, show that the area
AQP is equal to the triangle ABQ.
21. A semi-cycloid having BA as axis, B as vertex, cuts
the semi-cycloid APD (A vertex, AB axis, and D cusp) in P,
and AQB is the central generating circle, Q lying on the
same side of AB as P ; show that the area AQBP is equal to
the square inscribed in the circle AQB.
22. The normal at any point of a cycloidal arc divides
the area of a generating circle through the point, and the area
of the cycloid, in the same ratio.
23. In Example 20, show that
(arc AP)2 = i (arc APD)^.
24. If a cycloidal arc DAD' is divided into any two parts
in P, and PB' is the normal at P (B' on the base), show that
arc DP . arc PD'= 4 (PB') 2.
25. D is the cusp of a cycloid APD, C the centre of the
tracing circle PKB' through P. If DC cut the tracing
circle PKB' in K, and DP = 2 arc PK, show that DP
touches the tracing circle at P.
26. If APD is a semi-cycloid, having axis AB and base
BD ; AP'D the quadrant of an ellipse having semi-axes AB,
BD ; and AP"D the arc of a parabola, having AB as axis,
show that
area APDP I area AP'DB : area AF'DB: : 9 : 3 tt : 8.
EXAMPLES. 237
27. With the same construction, the radii of curvature of
the three curves at A are in the ratio 16 : 2 ?r* : ir^.
28. On the generating circle AQB the arc AQ = ^ dr-
cumference is taken, and through Q a straight line parallel
to the base is drawn, cutting the cycloid in the point P ;
show that the radius of curvature at P is equal to the
axis AB.
29. The axis AB of a cycloid APD is divided into four
equal parts in the points D, C, and E, through which straight
lines, are drawn parallel to the base, meeting the cycloid in the
points Pi, P2, and P3; if the radii of curvature at A, Pj, Pj,
and P3, are respectively equal to pj, p2> p3> ^^d p4, show that
pi • 92 • r3 • r4 • • * • '^ • ^ • A.
30. 01 (fig. 14, p. 27) is produced to a point J, such that
IJ = 2 OK, and on OJ as base a cycloid is described ; show
that radius of curvature at vertex of this cycloid = LG'.
31. If a cycloid roll on the tangent at the vertex, the
locus of the centre of curvature at the point of contact is a
semicircle of radius 4 K.
32. K a cycloidal arc be regarded as made up of a great
number of very small straight rods jointed at their extremities,
and each such rod has its normal (terminated on the base of
the cycloid) rigidly attached to it, show that if the arc be
drawn into a straight line, the extremities of the normals
will lie in a semi-ellipse, whose major axis = 8 K, and minor
axis = 4 K.
33. PB' and FB" are the normals at two points P, F,
close together on a cycloidal arc, and PQ parallel to the base
BD' meets the central generating circle in Q ; show that if
PP' is of given length, B'B" varies inversely as the chord
BQ.
34. From different points of a cycloidal arc, whose axis is
238 GEOMETRY OF CYCLOIDS.
vertical, particles are let fall down the normals through those
}X)ints ; show that they will reach the base simultaneously in
time 2 A / — .
V g
If they still continue to fall along the normals pro-
duced, they will reach the evolute simultaneously in time
w—
35. If the distance of P on semi-cycloidal arc APD (^.
10, p. 21) from base BD = | AB, show tjiat
3 moment of PD about AE := 14 moment of AC about AE.
36. In same case,*if PM parallel to BD meet AB in M,
show that
moment of PD about AE = f (AB)i [(AB)t -(AM)I ].
37. Show that the moment of arc AP (fig. 10, p. 21)
about AB
= 2 (NQ + arc AQ) AQ-| AB^ (AB* -BMf ).
38. If equal rolling circles on the same fixed circle
trace out an epicycloid and hypocycloid having coincident
cusps, the points of contact of the rolling circles with the
fixed circles coinciding throughout the motion, show that
the tangents through the simultaneous positions of the tracing
point intersect on the simultaneous common tangent to the
three circles.
39. A tangent at a point P on an epicycloidal arc APD is
parallel to AB the axis, and a circular arc PQ about O as
centre intersects the central generating circle in Q ; show
that
Arc AQ : arc BQ : : F : 2 R.
40. Two tangents P'T, PT to the same epicycloidal are
DT'APD intersect in T at right angles, and through P' and
EXAMPLES, 239
F drcular arcs P'Q' and PQ are drawn around Q as centre
to meet the central generating circle in Q and Q, neither arc
cutting this circle ; show that
arc Q'AQ : a semicircle : : F : F + 2 E.
41. If the rolling circle by which an epicycloid is traced
out travel uniformly round the fixed circle, the angular ve-
locity of the point of contact about centre of fixed circle being
tDy show that the directions of the normal of the tangent also
F 4- 2E
diange imiformly with angular velocity — o"^p — w.
42. On the same assumption, the direction of the tracing
^. c^ ^^^, ^ ^U, ..loe,., £±« ..
43. If the rolling circle by which a hypocycloid is traced
out travel uniformly round the fixed circle, the angular
velocity of the point of contact about centre of fixed circle
being a>, show that the direction of the normal and of the
tangent also change uniformly with angidar velocity
F-2E
2R ^-
44. On the same assimiption the direction of the tracing
F — R.
radius changes uniformly with angular velocity — p — w.
45. A is the vertex of a hypocycloidal arc APDP', D the
cusp, P' a point on the next arc ; and the tangent at P' is
parallel to the axis AB. If a circular arc P'Q around as
centre intersect the remoter half of the central generating
circle in Q, show that
Arc ABQ : arc BQ :: F : 2 R.
46. Two tangents P'T, PT to the same hypocycloidal arc
DT'APD, the base D'D less than a quadrant, intersect in. T
at right angles ; and through P' and P circular arcs P'Q' and
240 GEOMETRY OF CYCLOIDS.
PQ are drawn around as centre to meet (without cutting)
the central genei*ating circle in Q' and Q ] show that
Arc Q'AQ : a semicircle : : F : F-2 R
47. AQ, QB are quadrants of the central generating
circle of an epicycloid or a hypocydoid, and the circular arc
Q£ about as centre meets APD in P ; show that
Area APQ : triangleABQ : : CO : BO.
48. In last example, show that (arc AP)* = ^ (arc APD)^.
49. At any point B' in the base of an epicycloid DAD'
a tangent PB'P is drawn to the fixed circle, meeting the
epicycloid in P and P ; show that
PB' < arc DB', and P'B' < arc D'B'.
50. With the same construction, show that PBT' has its
greatest value when B' is at B, the foot of the axis AB.
51. At P, a point on the epicycloid DAD', a tangent
PKD' is drawn cutting the fixed circle in K and K', and the
normal PB'6' cutting the fixed circle in B' and h' (B' on the
base DBD') ; show that
PK . PK' : (PB')2 : : F + R : B : : Q^h'f : pk . pk'.
52. "With the same construction if OM be drawn perp.
to PKF, show that
OM : PB' : P6' :: F + 2 E : 2 E : 2 (F + e).
53. If tangent at P to epicycloid DAD' touches the
fixed cirde, and TB'b' the normal at P meets the fixed circle
in B' and b' (B' on the base DBD'), show that
PB' (F -f 2 E) = 2 E2 ; and P 6' (F + 2 E) = 2 E (F + E) .
54. If tangent at P to epicycloid DAD' toudies the fixed
circle and cuts the rolling circle in A', then
(A'P)2 : (2 E)2 :: (F + E) (F + 3E) : (F + 2E)»
EXAMPLES, 241
' • 55. In figs. 21 and 22 (pp. 44, 45) the points P, B', 6, lie
in a straight line.
56. In figs. 21 and 22, the tangent to DP at P cuts
hOc' produced in a point a such that ha ^2hc',
57. At D the cusp of an epicycloid DAD (^g, 19, fix)n-
tispiece) a tangent D < to the fixed circle DBD' meets D'AD
in t, and from t another tangent^/ K is drawn meeting the
fixed circle in K ; show that D f is always less than the arc
DBK if the radius of the rolling circle is finite.
58. ACB is the axis of an epicycloid DAD' ; D, D' its
cusps ; CQ, O q radii, of central generating circle and fixed
circle respectively, perp. to ABO and on same side of it.
If C 9^ cut Q q parallel to CO in K, and a sti'aight line d K d'
through K parallel to ^ is the generating base of a prolate
cycloid having AQB as centi-al generating circle, show that
the ai*ea between the epicycloid DAD' and its base DD' is
equal to the area between the prolate cycloid cZ A cZ' and its
base d d\
59. ACB is the axis of a hypocycloid DAD ; D, D' its
cusps ; CQ, O q radii of central generating circle and fixed
circle perp. to BAO and on the same side of it. If C ^ cut
Q q parallel to CO in K, and a straight line d'Kd' through
K parallel to ^ is the generating basis of a cui-tate cycloid
having AQB as central generating circle, show that the
ai-ea between the hypocycloid DAD' and its base DD' is
equal to the area between the curtate cycloid d Ad' and its
base d d'.
60. The area between the cardioid and its base is equal
to five times the area of the fixed circle.
61. The area between the cardioid and a circle concenti-ic
with the fixed circle, touching the cardioid at the vertex, is
equal to three times the area of the fixed circle.
R
242 GEOMETRY OF CYCLOIDS.
62. The area of a circle touching the cardioid at the
vertex and concentric with the base, is divided into three
equal parts by the arc of the cardioid and the axis produced
to meet the circle.
63. AreaAoP{fig. 39, p. 74) = 3R(6it + arc B6).
64. If = Z. BO ft {^. 39, p. 74)
Area PSA = E« (3 + 4 sin 6 + i sin 2 0).
65. The area between one arc of the triciis[>id epicycloid
and the base is equal to 3§ times the area of the generating
circle.
66. A complete focal chord is di'awn to a cardioid.
Show that the lesser of the two segments into which the
focus divides the chord, is equal to the portion intercepted
between the fixed circle and the tracing circle through the
extremity of the longer s^ment.
67. A circle is described on the axial focal chord as
diameter, show that the segments of a complete focal chord
intercepted between the curve and this circle are equal.
(Purkiaa,)
68. Lines perp. to focal radii vectores through their ex-
tremities have a circle for envelope. (Purkiss,)
69. From S, the focus of cardioid, a perp. SQ to a com-
plete focal chord PSP', is drawn, meetuig the fixed circle in
Q ; show that SQ is a mean proportional between SP and
SF.
70. If SP be any focal radius vector of a cardioid whose
vertex is A, and the. bisector of the angle PSA meet the
circle on SA in Q, SQ will be a mean propoiiional between
SP and SA. (Purkiss.)
71. PSP' is a complete focal chord of a cardioid ; SQAQ'
a circle on SA as diameter ; SQ, SQ^ bisectors of the angles
EXAMPLES. 243
PSA, P'SA respectively ; and S q perp. to PSP' meats circle
SQA in q ; show that
SQ :S^:: SB :SQ'.
72. The pedal of a cardioid with respect to the focus is
also the locus of the vertex of a parabola which is coufocal
with the cardioid and touches the cu'cle on SA as diameter.
{P^ia'ki88,)
The demonstration of this will be more easily effected by taking
for the cardioid the locus of it, fig. 39 (see p. 75). From n draw
uy dk parallel to bf, then S y, perp. to n y, gives y a point on the
pedal of this cardioid with respect to IS. It can readily be shown
that a parabola having S as focus and y as vertex touches the
circle B 6 S in A.
73. From a fixed point A any arc AQ is taken and bi*
sected in Q^ If P is a point on the chord QQ' such that
QP = 2 QT, show that the locos of P is a cai*dioid.
74. If rays divei^ from a point on the circumference of
a circle and be reflected at the circumference, the caustic will
be a cardioid. (Coddington's 'Optics/ or Parkinson's * Optics/
Art. 72, which see.)
If S ft, fig. 39, p. 74, represent path of a ray, to circle B ft S, re-
flected ray ft y is in the line Pfty, normal to the caustic APS, and
therefore the envelope of the reflected rays is the" evolute of the
cardioid APS, or is a cardioid having its /ertex at S, SO diametral
and linear dimensions one third those of APS. This, however, is
not a direct proof. The preceding proposition will be found to
supply a direct proof. For if from A two rays proceed to neighbour-
ing points Q, q^ and thence respectively after reflection to neigh-
bouring points Q' and q', arc Q,' q' = 2 arc Q q ; and the point of in-
tersection of QQ' and q q' therefore lies on QQ' (equal to AQ), at a
point ultimately equal to one-third of the distance QQ' from Q.
75. A series of parallel rays are incident on a reflecting
semicircular mirror and in the plane of the semicircle ; show
that the caustic curre is one half (from vertex to vertex) of
n 2
244 GEOMETRY OF CYCLOIDS.
a bicuspid epicycloid or nephroid, (Coddington's ' Optics/ or
Parkinson's * Optics/ Art. 71, which see.)
76. A series of rays are incident on the concave side
of a reflecting cycloidal mirror to whose axis they are
parallel and in whose plane they lie ; show that the caustic
curve* consists of two equal cycloids each having one half of
the base of the cycloidal mirror for base, and the axis of this
larger cycloid as the tangent at their cusp of contact.
77. The linear dimensions of the e volute of the bicuspid
epicycloid (or nephroid) are -^ those of the curve itself.
78. The area between one aix; of the nephroid and the
base is equal to four times the generating circle.
79. The evolute of a nephroid is drawn, the evolute of
this evolute, the evolute of this second evolute, and so on
continually : show that the sum of all the areas between
all the evolute nephroids, and their respective base-circles,
are together equal to one-third of the area between the
original nephroid and its base-circle.
80. If in the epicycloid m R = n F, show that the linear
dimensions of the evolute are to those of the epicycloid as
m ', Til -\- 2n,
81. K i?i R = w F, area between an arc of epicycloid and
.. , 3m-^2n _ (3 m -f- 27i) w^ « ,
its base = ^ . gen. © = -^ -^ — ^ — . fixed 0.
82. If PB'o Q is the diameter of curvature at the point
P of an epicycloid, o the centre of ciu-vature, B' a point of the
base, then
Area of epicycloid ; area of gen. © : : QB' : B'o.
83. If the arc of an epicycloid, from cusp to cusp = a,i
and m R. = n F, show that a -f arc of evolute from cusp to
cusp -I- arc of e volute's evolute from cusp to cusp, and so On
ad infinitum^
{m -4- 2n) a
2"n
EXAMPLES. 245
84. If the area between an epicycloid and its base = A,
and m R = n F, show that A + area between an arc of the
evolute and its baise + ai*ea between an arc of the evolute's
evolate and its base/ and so on oc^ infinitum^
_ f m -4- 2 nfA?
85. If in the hypocyloid m R = n F, show that the linear
dimensions of the evolute are to those of the hypocjdoid as
f» : »i— 2n.
F
Interpret this result when R =i: ^.
86. If m R =r « F, area between an arc of hypocycloid
and its base = gen. = ^ ~ — L — . fixed 0.
m m*
87. If PB'o Q is the diameter of curvature at the point
P of a hypocycloid, o the centre of curvature, B' a point on
the base,
QB : B'o::3CF-2R : F.
88. If the arc of a hypocycloid from cusp to cusp=a, and
m R ^ w F, show that a -f arc of hypocycloid of which the
given hypocycloid is the evolute + ai*c of hypocycloid of
which this hypocycloid is the evolute, and so on oc? infinitum,
m
= --- a.
2n
89. If the area between a hypocycloid and its base = A,
and m R = 7i F, show that A -f the area between one arc of
the hypocycloid of which the given hypocycloid is the evolute,
and its base + the area between one arc of the hypocycloid
of which this hypocycloid is the evolute and its base, and
60 on oc^ infinitum,
4 n{m^n)
S>46 GEOMETRY OF CYCLOIDS.
90. lyAD is an arc of a tricui^d epicjcloidy from easp
to cusp, ACB the axis, AQB the central generating circle, G
its centre, OBOA diametral ; show that an angle may. be tri-
sected by the following consbnictioa : — Let ACQ be the
angle to be trisected. Join QB, QO; about O as centre
describe arc QP meeting D^AD in P (on AD) : join PO ;
make the angle OPB equal to the angle OQB, and towards
the same side, PB' meeting the base D'BD in B' ; and join
B'O. Then the angle BOB' is equal to one-third ci the
angle ACQ.
91. D'AI> is an arc of a tricuspid hypocycloid from
cusp to cusp ; ACB the axis ; AQB the central g^ioii^ting
circle, its centre, OAOB diametral. Show that an angle
may be trisected by the following construction. Let ACQ
be the angle to be trisected. Join QB, QO ; about O as
centre describe arc QP meeting D AD in P (on AD) ; join
PO and make the angle OPB' equal to the angle OQB, and
towards the same side, PB' meeting the base D BD in B';
and join BO. Then the angle BOB' is equal to one-third of
the angle ACQ.
92. D'AD is an arc of an epicycloid from cusp to cusp ;
ACB the axis ; AQB the central generating circle, C its
centre ; OBCA diametiul. A radius CQ is drawn to AQB ;
and BQ, OQ are joined. About O as centre the arc QP is
di'awn meeting D'AD in P (on AD) ; VO is joined, and the
angle OPB is made equal to the angle OQB and towards the
same side, PB' meeting the base D'BD in B'. If OB' is
joined, show that
angle BOB' = ? . angle ACQ,
so that, by means of a suitable epicycloid, an angle may bQ
divided in any required ratio.
93. D'AD is an arc of a hypocycloid from cusp to
EXAMPLES, 247
eusp; ACB the axis; AQP the central generating circle,
O its centre ; OACB diametraL From C a radius CQ is
drawn to AQB ; and BQ, OQ are joined. About O as centre
the arc QP is drawn meeting J)' AD in P (on AD) ; PO is
joined; and the angle OPB' is made equal to the angle
OQB, and towards the same side, PB' meeting the base
D'BD in B'. If OB' is joined, show that
angle BOB = — . angle ACQ,
so that by means of a suitable hypocjcloid an angle may be
divided in any required ratio.
94. If PC p is the tracing diameter at P on an epicycloid
or hypocycloid APD (vertex at A), o the centime of curvature
at P, show that op produced meets the tangent at P in a
point T such that TP is equal to the arc AP.
95. If an epicycloid roll upon the tangent at the vertex,
show that the locus of the centre of curvature at the point
of contact is a semi-ellipse having semi-axes
4R(Z±m a^d i|-Y#±£V
• 96. If a hypocycloid roll upon the tangent at the vertex,
show that the locus of the centre of curvatiu'e at the point of
contact is a semi-ellipse having semi-axes
iAVIL:L5^ and ^^(^"^) .
F vf-2r; F
97. An arc DAD of the bicuspid epicycloid, or nephroid,
has its axis AB coincident in position with A \ the axis of a
cycloid whose vertex is at A ; but AB = f A 5. If the
nephroid and the cycloid roll on T'AT, the common tangent
at A, in such sort that they simultaneously touch the same
{K)int on T'T, show that the centre of curvature of the
nephiYiid at the point of contact will trace out the same
curve as the foot of normal to the cycloid at the point of
248 GEOMETRY OF CYCLOIDS,
ff
* ' *
contact (the foot of normal being understood to mean tbd
intersection of the normal with l^e base).
98. If a quadricuspid h jpocycloid (radius of fixed circld
F) is orthogonally projected oh a plane through two opposite
cusps, in such sort that the distance 2 F between the other.
two cusps is prcgected into distance 2/, show that the pro-
jected curve is the evolute d^ -an ellipse having axes equal to
99. Show that the ai'C of the projected curve in 98, from
cusp to cusp,
_ F» + F/+/^
F +/ "
100. AOAVBCB^ are the major and minor axes of an
ellipse, G its centre; and a Ba' B' is a similar elUpse having
BOB as major axis ; if the elKpse ABA'B^ is c»i)hogonally
projected into a circle, show that the evolute of a. B a'^B' will
be projected into a quadricuspid hypocycloid, and determine
its dimensions.
101. With the same construction, show (independently)
that the portion of the projection of any noimal of a B a B ,
intercepted between the projections of A A' and BB', is of
constant length. (This will be found to follow readily from
Propos. X. and XIV. of Drew's * Conies,' chapter ii.)
Note. — This propositiorif demonstrated geometrically, combined
with what is shown at pp, 72, 73, affords a geometrical deinon-
stration of the natii/re of the evolute to the ellipse. See Jiext
problem,
102. Let AC A', BCB be the major and minor axes of an
ellipse, hOh' the orthogonal projection of BCB on a plane
through ACA', so situated that h b' : BB' : : BB : AA .
From B draw BL perp. to AB to meet A'C in L; and about
EXAMPLES. 2W
C in the p''ane A 6 A , describe a circle with radius LA' cutting
CA, CA , C 6, and C 5', in K, K', A;, and k , respectively. Draw
a four-pointed hypocycloid, having cusps at K, k\ K', and k.
Then a plane perpendicular to the plane A h A!h', through
any tangent to the hypocycloid K k'YJk, will intersect the
plane ABA'B' in a normal to the ellipse ABA'B', and a
right hypocycloidal cylinder on K k'YJk as base, will inter-
sect ABA'B' in the evolute of this ellipse.
103. Two straight lines intersect at right angles in a
plane perpendicular to the sun's rays, one of the lines being
horizontal. If the extremities of a finite straight line slide
along the fixed straight lines, and the shadow of all three
lines be projected on a horizontal plane, show that the
envelope of the projection of the sliding line is the evolute of
an ellipse. Determine the position and dimensions of this
ellipse.
If the sun's altitude is o, and the length of the sliding line /,
then taking for axis of x the shadow of the horizontal fixed line,
the equation to the envelope is a?* + y2 sin^osZ*; and the
equation to the involute ellipse is a?* cos* o + y* sin* o cos* o= I*.
104. At P a point on the hypocycloid DP AD' the tan-
gent KPK' is drawn, meeting the fixed circle in K and K',
and the normal 6'PB' meeting the fixed circle in h' and B'
(B' on the base DBD') ; show that
KP . PK' : (PB')2 : : F-R : R : : {Vh'f : KP . PK'.
105. With the same construction, OM is drawn perp. to
KPK' ; show that
OM : PB' : P h' : : F-2 R : 2 R : 2 (F-R).
106. If the tangent to the cardioid at P touches the fixed
oii-cle, and cuts the rolling circle in A', and the normal at
P cuts the fixed circle in B' and 6', then
260 GEOMETRY OF CYCLOIDS.
PB' = f B ; P 6' = i^;aiid A'P**^ K.
107. In the trochoid, if R 6', the normal at jp, meets the
generating base in B', and the tangent at p meets the tangent
at vertex in T, a'h' being diametral to tracing circle ; show
that triangle TB'p' is similar to triangle a'h'p,
108. With same construction
Z TBV = z 6> B' = z Tpa'.
109. In ^, 48, triangle C 6 ^" = — sector h C q".
r
110. In ^g. 48, p. 96, show that
loopy rc^r =2^ ^arca5NL^" + 2 ^ rect. Nn.
111. Show that the result obtained in the last example
agrees with that obtained in Prop. IX., Section III.
112. If in Q' q", fig. 48, produced, a point X is taken such
that (CX)2 = rect. a B a C, and a circular arc XY (less than
semicircle) with C as centre and CX as radius cuts a h pro-
duced in Y, show that
loop p"rd^2 segment X Y — rect. N n.
113. In fig. 48, p" X is drawn parallel to q"b to meet the
base bd m^; show that
area ydrp" : seg. q"\k 6 * * a B : a C.
114. From B (fig. 45, frontispiece) a straight Hne ^qq[
is drawn cutting the central tracing circle in q and ^', and
straight lines qp and c^p' pai'allel to the base meet the arc
a dvap and p'\ show that the tangent at j9 is parallel to the
tangent at p'*
115. P and P' are two points on an epitrochoid or hypo-
trochoid, C and C the corresponding positions of the centre
of generating circle, O the fixed centre, OA, OB the apo-
•
central and pericentral distances. If OP . OP' = OA . OBl,
EXAMPLES, ?ol
show that the tangents at P and P' make equal angles with
OC and OC respectively.
116. A cycloid on base BD {^, 45, frontispiece) has its
cusps at B and D ; show that it touches the prolate cycloid
a j9 (/ at a point of inflexion.
117. A series of pi-olate cycloids have the same line of
centres, their axes in the same straight line, and their bases
equal. Show that their envelope is a pair of arcs of a cycloid
having its base equal to half the base of each prolate cycloid
of the system, and the line of their axes as a secondary axis.
118. If the normals at p and q, two points on a prolate
cycloid apqd, are parallel, and meet the generating base in
h' and h" respectively, then p and // being the radii of cur-
vatiu^e at p and q respectively,
9'9':\{ph'Y\{qh")\
119. If p is the radius of curvature at the point where a
curtate cycloid cuts the generating base, and /i is a mean
proportional between the radii of curvature at the vertex
and at d on the base, show that p^ =s ur,
120. Show that that involute of the central gene-
rating circle of a cycloid which has its cusp at the vertex
passes through the cusps of the cycloid.
121. That involute of any generating circle of a cycloid,
which has its cusp at the tracing point, passes through the
cusps of the cycloid.
122. The sum of the two nearest arcs of the involute of
the circle, cut off by any tangent to the circle, is least when
the tangent touches the circle at the farther extremity of
the diameter through the cusp of the involute.
123. If the rolling straight line by which the involute of
a circle of radius/ is traced out has rolled over an arc a from
/
the cusp, show that the arc traced out = h a*.
252 GEOMETRY OF CYCLOIDS.
124. If the rolling straight line by which a spiral of
Archimedes is tiuced out, has rolled over an arc a from first
position, when the extremity of perp. carried with it was
at the centre of the fixed circle (radius/), show that
arc traced out= o j " "^^ + u* + log (a + \/l + a*) > .
125. All involutes of circles are similar.
126. All spirals of Archimedes are similar.
127. If a straight line carrying a perp. of length d roll on
a circle of radius/ and another straight line carrying a perp.
of length D (on same side with reference to centre of fixed
circle) roll on a circle of radius F, show that the curves
traced out by the extremities of these perps* will be similar
if P:;?::F:/.
128. In the spiral of Ai'chimedes the subtangent is equal
to that arc of a circle whose radius is the radius vector,
which is subtended by the spiral angle. (Frost's * Newton ').
The subtangent is the portion of a perp. to radius vectori
through pole, intercepted between pole and tangent at extremity of
radius vector. What is required to be shown in this example is
that if p'p (fig. 72, p. 130), produced, meet B'O produced in Z, OZ
is equal to the arc corresponding to DQB' in a circle of radius 0^.
1 29. Establish the following construction for determining
the centre of curvature at point p (fig. 72, p. 130) of a spii*al
of Archimedes. Draw radius OB' to fixed circle, perp. to
O p ; join p B' ; and draw OL perp. to p B'. Then if B'L is
divided in o so that
Bo : oL::B> : B'L,
o is the centre of curvature at p,
130. From this construction (established geometrically)
show that, taking the usual polar equation to the spiral of
Archimedes, viz., r=adf
_ a{l 4- H>)^
EXAMPLES. 263.
131. A straight line turns uniformly in a plane round a
fixed point, while the foot of a perpendicular of length I
moves uniformly along the revolving line; show that the
other end of this perpendicular will tiuce out one of the
spirals described at pp. 128, 129.
132. If the angular velocity in preceding problem is w,
the linear velocity of the foot of perpendicular t?, and I •=• — ,
the perpendicular lying on the side towards which the revolv-
ing line is advancing, show that the other extremity of the
perpendicular will describe the involute of the circle.
133. KDT, fig. 42, p. 82, rolls on the circle DQB of radius
a, and a point initially on DO and distant h from D is carried
with DT to tiuce out a spiral in the manner described at
pp. 128, 129, show that the polar equation to the spiral, OQ
being taken as initial line, and the rolling taking place in the
usual positive direction, is
e = ^ - + tan-> — -- — •
a ' " V r^ — (a — 6)2
134. Show that the constiniction given in Example 129
foi determining the centre of curvature at a point on the
spiral of Archimedes is applicable to all the spirals of Ex-
amples 131 and 133.
135. In the case of one of these spirals, patting the arc
over which the rolling line has passed from its initial
position := (f'f show that
136. The locus of the foot of perpendicular fix)m a point
on a cycloid upon the diametral of the generating circle
through the point is the companion to the cycloid.
254 GEOMETRY OF CYCLOIDS.
137. From D, the cusp of an inverted cycloid, and P, a
point near D, two particles roll down the smooth arc to the
vertex ; show that in the limit the path of either relatively
to the other is a semicircle.
138. A particle is projected with given velocity from the
vertex of a cycloid whose axis is vertical, and vertex upper-
most; find where it will leave the curve, and the latus
rectum of its future parabolic path. — (Tait and Steele's
* Dynamics.')
139. A particle £silling from rest at a point in an in*
vei-ted cycloid has its velocity suddenly annihilated when it
has passed over half its vertical height above the lowest
point; then proceeds, again losing its velocity when half-
way down from its last position of no velocity, and so on
continually. Show that it will be at jr^^ of its original
height above the vertex after n times the time it would have
taken to fall to the vertex undisturbed. — (Tait and Steele's
* Dynamics.')
140. If a curve of any form is rolling upon another
curve in the same plane, and jp is a point on the curve
traced by any given point carried with the rolling curve and
in the same plane with it, h the point of contact of the fixed
and rolling curves, show that the following relation exists
between Pi, pj* the radii of curvature of the fixed and rolling
cuives at 6, and p^ the radius of curvature of the traced
curve at p (putting pb = n and the angle between pb and
the normal of fixed curve at 6 = ft),
[n (pi + P2) - pi p2 cos e}p3 = n^{pi + pj).
141. A tube of uniform cross section, small compared with
its length, is bent into the form of a cycloid, its open ends
EXAMPLES. 265
lying at the cusps, and this cycloid is placed with its axis
vertical and its vertex downwai^s. Equal quantities of
fluids of specific ^avity 9^ and a^ *^ poured in at the two
cusps, the quantity of each being such as would fill a length
a of the tube (a being the length of the cycloid's axis, so that
4a is the length of the tube). If the fluids do not mix and
the distance of the upper levels of the fluids from the vertex
(measured along the cycloidal arc) be ajj, X2 resi)ectively,
show that f
4a3,(<ri -f (Tj) = a{(Ti + 3 9^^
and Ax2{(f\ + ©"a) = a(3 tr^ +• ^a).
142. If in problem 141 an equal quantity of a third fluid
of specific gravity tr^ is poured in upon the free surface of
the second fluid (sp. gr. (Tj), and a?,, x^, are the respective
distances of the free surfaces of the first and third fluids
from the vertex (measui^ed along the cycloidal arc), show
that • '
4a;i(/ri + ffj + ffj) = a(«ri 4- 3 Cj + 6 ca),
and 4 X2{*r\ + ©"g + ©"a) = «(5 o*! + 3 (Tj + o'a)*
Under what condition will either the first or third fluid run
over]
143. If n fluids are poured in, as in Ex. 141, the specific
gravities of Ist, 2nd, 3rd, &c., to the nth, being o-,, o-j, 0-3,
<fec., to <T„, respectively, the arcs occupied by the respective
fluids being Zj, Zg, ^3> • • • Ki ^^^ ^o fluid overflowing; and if
X is the distance of the free surface of the first fluid fix)m the
vei-tex (measured along the cycloidal arc), show that
+ <^3(V + 2^1^3 + 2^3)+ .. .
+ (T„(^„2 + 2^,^,+2ya . . . + 2l,M,
360 GEOMETRY OF CTCLOlhS.
Y-y.. I4t. V\.:. lo! El. 144. I'T.^.
3 2044 018 806 8
This book should bQ returoed to
the Library on or before the last date
stamped below.
A fine of five aents a day is inourred
by retaining it beyond the apeoifled
Please return promptly.
