or < C a,
fig. 88 ; while the motion at a pericentre is always direct.
ScHOL. — K D=wE, there is a cusp at pericen. or apocen.
EPICVCLICS.
159
Prop. VII. — To determine the position of the points, if any,
where the motion of the radius 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^, or
D > n E, the motion, direct both at apoceatres and pericen-
tres, L«i direct throughout. For the motion to be retrograde
in part of the epicyclic, then, we require that D be < Ji E, or
CB < C a. Since the direction at ^ is pei'p. to Bjo, the mo-
FlG. 91.
Fig. 92.
tion will be directly towards or from centre if 'Bp is at right
angles to Op, for then 0^ will be the tangent at p. "We
have then the relations presented in fig. 91 for direct epi-
cyclic, and in fig. 92 for retrograde epicyclic.
O^ is the distance from 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 O jo is determined by describing a
semicircle on OB intersecting a'p h' in p. Now the angle
pQ'a' ^={n — 1) deferential angle (measured from apocen-
tral initial radius vector), say /_ p C'a' = (n — 1) 0, and we
might proceed by the epicyclic method of treatment to
160 GEOMETRY OF CYCLOIDS.
detennine ^ = i;;^- — ^^— ^ — ; — -^^ — IS known, and there-
^ KO D+ Ecos(n-l)0i
180'
fore, pOd = 0i + pOb' — is also known.
n- I
It can easily be shown that
sin (n- 1)^, = ^ (D'-E')(^'E^- D')
^ (-1 + n) DE
and ta.Tit)Oft=l ■ A'E'-
-D2
E*
EPICYCLICS. 161
Peop. VIII. — To determine the tangential, transverse, and
radial velocities, and the angular velocity around tJie
centre at any point of an epicydic curve.
Let Pi (figs. 86, 87, 88) be the position of the point on
the epicycle a j»]6. JoinOjOi and draw B A perp. to OjOj.
Then when C^i (^ E) represents the linear velocity in the
epicycle, Op^ represents the linear vel. at /7j in magnitude,
but is at right angles to the direction of motion at jb,.
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 p^,
carrying with it jOj B and p^ h,m the plane of the figure,
through a right angle, to coincidence with the direction of
joi's motion in the circle a pi. This includes all cases geo-
metrically, and the student will have no difficulty in efiect-
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 jo, is represented by p^ B in
magnitude and is in advancing direction shown by arrow
at jBi.
The transverse velocity of ^, is represented by p, A in
magnitude, and in direction by B h.
The radial velocity of j»i is represented by B/t in magni-
tude, and in direction by p^ h.
The angular velocity of ^j about O : uniform angular velo-
city of Pi about C :: ^ : ^^ :: p^h : Ojo,.
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 con-ect
for aU possible cases. Let then the angle p^Ga (fig. 86)
= (n—\) ^; that is, let the deferential angle=^; let the
linear velocity of the mean point (C) be V, wherefore 'the
linear velocity of the moving point in the epicycle -nY .-
This is what we have represented linearly by p^G in figs.
86, 87, and 88, so that siace jo, C = E, we have to afiect all
the above linear representations of velocity with the co-efii-
. .nY
cient — — :
Therefore, the tangential vel.
^"V fiY
= ^ •PiB= -g-^(P,C)»+(CB)2 + 2j9,C.OBcosjoGa.
& + — 2 + COS {n—\)(^
= gN/D2 +»i2E« + 2mDEco3(»4-l)f
The transverse vel. = .— . ^, B . cos B jOj O :
now,cosB;,.0=(M±(aO!t(BO)^ .-.^.B.cosB^.O
2jB, B.^1 O ' -r-i 11
£=+^' + 2— cos(ft-l)i^ + E2+D^+2DEcos(M-l)A-/'D-?V
and transverse vel. (direct)
^V D''+wE^+(n+ l)DEcos(n-l)0
D ■ y/~&~^-k' + 2 DE cos (n- 1)
Y
The radial vel. = — ^iB . sin Bp,0 :
EPICYCLICS. ]()3
8m;)iOB piB'
therefore,
V nj jOjO
and radial vel. (towards centre)
=(n-l)V Esin(m-1>____
VD!'+E''+2DEcos(m— 1)9
The angular velocity about O
_ transY. vel. _ V D=' + toE'' + (to+1) DEcos (to-1)(/ )
rad. vect. ~D" D* + E'' + 2DE cos(m-l) ^
The transverse vel. and the angular vel. about O vanish, if
D2 + n E2 + (w + 1) DE cos {n—\) ^ = 0,
the condition already obtained.
If V is the velocity in epicycle, v^Y _ , or Y^v —
D n &
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 p^ orp^ (figs. 86, &c.),
where a tangent from O meets the circle a pib; for then B h
or B h has its greatest 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 epicycUc
motion is directly towards or from the fixed centre, — that is,
at the points where a tangent from the fixed centre to the
epicycle meets this circle.
Cor. The angular vel. at apocentre > ^ or < angu-
lar vel. at pericentre, according as
oB > &B aB > aO
^0 C'B' . CI + (Cjo)!"
,._=-cos(«-l)^= ^C^B^^ 571^0^
Now by Cor. to Prop. XII., Sect, m., C'I= /"? V-i- D = ^
EPICYCLICS
166
to be regarded as negative for retrograde epicyclic. Hence
D D
Cos (n— 1) — ^r — dio.
w— 1
ScHOL. — The angular range round of the arc between
the points of flexure can be determined, as in case of arc of
retrogradation, see scholium to Prop. VII. "We have
ta^pOS' (figs. 95 aiid 96) = ^ = ^^ ^"7^^t \-T '
, aO D-|-Ecos(n— 1)^2
wherefore, if ^ ^/(7i2E«-D2)(D2-»i''E'')
^(^-1)^«= -^ n{\^l) BE
1^= —I ;
that is D' + n ' E^ = m (1 4- ii) DE
(the same condition, both for direct and retrograde epicyclic,
due account being taken of the sign of n) ;
or n {n^ B - B) E = (n" E - D) D
01- (nE - D)(7i2E - D) = 0,
which is satisfied, (i), if m = -, the condition (Schol. p. 158)
E
for a cusp (at pericentre in case of direct epicyclic, and at apo-
centre 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 retrogi'ade epicyclic. Com-
pare scholium to Prop. XII., Section III., from which the
relation between n*, D, and E, can be directly obtained.
Prop. XI. — To determiTie the radius of curvature, /j, at a
point on epicydic where deferential angle = E,, 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 wiU be found easy to deduce
any requii-ed 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 eflfected.
108 GEOMETRY OF CYCLOIDS.
Suppose we wish to determine when the tracing point
ceases to advance in the looped trochoid. We have, from
Prop. VII., in case of epicycKc,
cos (n — 1) 0, = — , ■ — ^i;-=,
^ ' ^' (1 + i) DE
and if m represents the ratio of linear velocities in epicycle and
deferent, ji = m — . Also n ip is the angle swept out in
E
epicycle, and when D becomes infinite is the same as (m — 1 )0,
so that the angle ^i (the angle a CL of fig. 48) is deter-
mined by the equation
D2 + m DE 1 , T> • • c •+
cos (4, ^ — - - — I — . -_^_ = — — when D is mnnite.
^' (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 ni 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 cii-ole 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 ofi" 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
HPICrCZICS. ]69
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 = d,
velocity of its extremity 1, and velocity of moving point m.
Then if oti = 1, the curve is the involute of the circle traced
out by the end of the revolving i-adius ; if to > 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 d=; 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 uniform angiilar 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 astronomers 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 : —
Prop. I. — Eegarding the planets as travelling uniformly in
circles about the sun as centre, and in the same plane, the
path of any planet P {fig. 97) with reference to any other
planet, p, regarded as at rest, is the same as the path of p
with reference to P regarded as at rest, the corresponding
radii veotores 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.
S jo 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, regarded as at rest, the planet P
has moved as if from P to Q ; while considered with refei--
ence to P, regarded as at rest, the planet p has moved as if
from ^ to g' : and since jo Q is equal and parallel to P y, the
path of the outer planet with reference to the inner, regarded
Eio. 97.
as at rest, is the same as the path of the inner planet with
reference to the outer regai'ded as at rest, — each path being,
however, turned round through 180° with regard to the
other.
Join p' q, P'P, p' p, and P'Q. Draw S s' parallel to
p'q, and SS' parallel to P'Q, and join s'q, s'P, S'^, and
S'Q. Also draw s m and S M parallel to SP, and complete
the parallelograms PMS'S, and p m s'S.
Then, by construction, the figures S'jj,^'Q, S'P, S g',
y P', and «' P', are parallelograms. "Wherefore jj S'= jo'S =
Sp; andz Sjr,S' = Zj»Sy; S'M = SP= SP'= S'Q and
Z MS'Q = Z. P9P' ; 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 S' in a circle about p as centre
EPICYCLICS. 171
(corresponding to the real motion of ^ to ^' around S
as centre), accompanied by the uniform motion of P (which,
if at rest, would have been carried to M), in a circle aroimd
the moving S as centre to Q, — that is, through the arc M Q
= P P'. Hence the motion of P with reference to p is that
of a direct epicycUc having D = S ^, E = S P, and
Aug. vel. of P round S
Ang. vel. of ^ 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 s around P as centre (corresponding to the
real motion of P to P' around S as centre), accompanied by
the uniform motion of jo (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^ p p . Hence the motion
of p with reference to P is that of a direct epicyclic having
D = SP, E = &p, and
Ang. vel. of p round S
n = — 2 £- •
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 = E, E = r, and
'=(?)'.
for the angular velocities of planets round the sun vary
inversely as the periods — that is, as the sesquipUcate power
of the mean distance.
Since j _ j^ > —,oin >
E'
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 scholium
to Prop. VII. of this section. The duration of the retrogra-
dation follows directly from the formula for determining
cos (rs — 1) 0j 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 formjila,
with the values above given for D, E, and n, becomes
or, putting P, p, for the respective periods of the planets,
cos i- (Ai ^ = ■
p ^^ R rl H- E,3 r R^ + H
Ri r\ a/Rt
E _ Ei ri + r - VR7- (R + r) ' ^""^
P-;,
sin 01= '/\ — cos (w— 1) <^, VI 4- cos {n — 1)^1
n/ ( R + r) ( R - 2 Ri ri + ?• )
R - Ri ri + r
_ ( Ri - ri ) a/R + r
E, _ Ri ri + r ■
Wherefore tan ^06' (see fig. 91, and schol. p. 160)
r ( Ri - r\ ) a/R + r
R (R - Ri ri + r) - Ri ri
r (Ri - ri) -/R + r
— R(R + r) -Ri ri (R + r) - B,5 VR + r
The arc of retrogradation, —
can be readily determined. Thus, the arc of retrogradation
EPICYCLICS. 173
= 2^^O6'-:5^-(360°-cos-i ,.=£^ \
^-P\ VRr-R-r/
= 2tan-'_ — ^
- -P— /i«no+>-os-' ^^ ^ (1)
This formula gives the arc of retrogradation. The angle
between pericentral and stationary radii vectores is half the
arc of retrogradation.
Thus the epicyclic path of a superior planet (period P)
with respect to an inferior planet (period p), or of latter
planet with respect to former, will have —
Apocentral distance = R + r ;
Pericentral distance = R — r j
Angle of descent = -n^ — • 180°.
" i^ — p
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 vmity.
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 epicycUc about the sun as fixed
centre.
AU the sateUites 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 epicycUc character. Our own
moon's path is but little inchned 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. II. — To determine under what conditions a satellite,
tra/oetling in a direct epicycle about the sun, wiM have its
motion {referred to the sum) looped, eusped, or direct
througlwut, or partly convex towards the sun, or just fail-
ing of hecoming convex at perihelion, or partly concave
towards the sun.
Let M be the sun's mass, m the primary's, E 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 -I- 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
wiU be found fully treated of.
BPICYCLICS. 175
given conditions. The most convenient way of doing this is
perhaps as follows : — Let V, « , be the respective velocities
of bodies moving in circles around the sun, and round the
primary, at the same distance, B, ; and let v be the velocity
of the satellite at distance r. Then we know that
V2 ,
R ■
,, 2
; g :: M:m,
or
Y w' :
; : -v/M : \/«i,
and
v' '. V ',
; : v^r : Vr.
.-.Y -.v :
; : a/m r : a/to r,
and
R-r '
:: a/Mt^ : Vm'B}
This is the ratio of the angular velocities of primary and
satellite in their respective orbits. It gives us
»i : 1 (:: P : ij) :: VmW> : a/Mt-s.
The path of the satellite will therefore be looped, cusped,
or direct throughout, according as
''^TES > R
M7-3 < r
mR^Mrjorjj^-.
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
toR8 > R
M r' <■ r'
The student will find no difficulty in obtaining formulse
for the range of the arc of retrogradation, if any, or of the
17G 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 = E, and E ^ r
p
and n^ — , as in the case of planetary motion, but that in
P
reducing the formula he must employ the relation
= v/
toR3
Mr3-
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 flexure, 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
limits of distance at which, in the case of these planets and
our earth, convexity towards the sun, or retrogradation,
wovdd occur.
M
In the case of our earth, — = 322,700 = (568)' about ;
and R = 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)
.^. ' A- , f ^v> ^v . 92,000,000
the moon s distance from the earth s centre were „,-,„ rmn
miles, or about 285 miles. That a moon shovild travel in a
path convex to the sun in perihelion, the distance should not
92,000,000
exceed ?»^ , 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
EPICrCLlCS. 177
R = 478,660,000 miles. Therefore a moon -would travel in
a cusped epicycle, or come exactly to rest in perihelion, if its
T . , 478,660,000
distance irom J upiter s centre were fTvJfi > *"'■ 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 ' ^i'
miles, or about 14,804,000 miles ^ which far exceeds the
distance even of the outermost moon.
M
In the case of Saturn — ^ 3,510 = (59)* about, and
m ^ '
R =. 877,570,000 miles. Hence a moon would travel in a
, . , .^ . ,. „ „ 877,570,000
cusped epicycle if its distance from Saturn were q"kTo
or about 250,700 miles. This is rather less than the distance
of his fourth satellite, Dione, 253,442 mUes ; and, owing to
the eccentricity of Saturn's orbit, it mu.st 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 Satm-n 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 to
escape convexity towards the sun, the satellite's distance
should be '~Kq^ miles, or about 14,874,000 miles.
Prop. III. — Regarding the planets as moving uniformly in
circles round the sun in the invariable plane, the projec-
tions of the patlis of t/te planets in space upon a fixed
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 Istst chapter of my
treatise on the sun.
Prop. IV. — To determine the tangential, transverse, atid
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 |)eiaod, 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
i-adial velocities in epicyclics, we have to put
D = Il;E = r;andn = (5)'=J;
but it will be convenient to retain n, remembering its value.
We may also conveniently write :^ = p, so that n = p-''
Iv
BPICYCLICS. 170
Moreover, with the units of distance and time in which E, r,
P, and p are expressed,
P~'
Also is the angle swept out around the sun by the planet
of reference since the last coig unction 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" a/I + p-' + 2 p* COS (w — 1) y .
The formula can obviously assume many forms, but per
haps this, which enables us at once to compare the tangential
velocity with V, the velocity of the planet of reference in
its orbit, is the most convenient.
The transverse velocity (direct)
a/R" +r> + 2Krcos(»i- l)p
_ -y - 1 + pi + (p-^ + p) cos {n — 1) ^
V f T^p" +^ p cos {n — Vff
The radial velocity (towards centre)
= (p-§-l)V. rsh>^{n-\)e
a/R2 + j-2 + "2 Rr cos {n - 1) f
(p-5 — p) sin (n — 1)0
= V
VI +p2 + 2pcos(»i- 1)^
* The conjunction must be such that the sun is between the two
planets. It is a convenient aid to the memoiy, 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.
h2
180 GEOMETRY OF CYCLOIDS.
The angular velocity of the planet about the planet of
reference
_ V i- i r" 4 - R'H^ ( p-i +1) Rrcos(w- 1)»
~R R^ -t-r2'+ 2r cos(n- 1)^
_ p^ 4 1 + (p~^ + p) cos (w — 1)
-" ■ 1+ pii + 2p cos (n - 1) ^
V
putting p = (1) =: angular velocity of the planet of reference
in its orbit.
Cor. 1. In conjunction (superior if moving planet is in-
ferior) ^ ^ ;
.". Angular velocity in superior conjunction
_ pi + 1 + p- i + p
- "" 1 + p2 + 2 p
(1 + p) X (1 + pi )
= '" (TT7? '
Cor. 2. Similarly since in opposition if the moAring planet
is superior, or in inferior conjunction if the moving planet is
inferior, (n—\) (p =■ 180°, angular velocity of a planet in op-
position or inferior conjunction
pi + 1 — p~i — P
= " l + P^-2p
(1 - p) - ^-l (1 - p) / 1 - p-i \
— W 1 — pi U)
\ p 1 — f A/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
Prop. V. — A planet transits the sun's disc at such a rate
that tlie sun's diameter S would he 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 i' = S, and the planet's
angular velocity about the earth when in inferior conjunction
-/p + p
Wherefore, the planet's retrograde gain on the sun (which
advances with angular velocity w)
-+ w,
= w / 1 + -v^P + p N _ S _ o i'
\ v'p + p / ^ ' '
or p+ A/p = p--— ^;
a quadratic giving
^P=--h± 2 = ^l± t'-t ~V'
ift+t VZt + t'\
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- Mercurial planet Vulcan. He failed for want
of such formulae as are here given.
182 GEOMETRY OF CYCLOIDS.
i-ate of transit was such that the sun's diameter would have
been traversed in rather more than four hours. Since 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-v'3)
= i (2 - 1-732) = 1 (0-268) = 0-134.
This is very near the estimated value of the imagined planet's
distance.
FORMS OP EPICYCLIC CURVES.
The relations discussed in the propositions of this section
enable us to determine the shape and general features of
epitrochoids or direct epicyclics and of hypotrochoids or re-
trogi-ade epicyclics, for varioiis 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-
cycUc 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.
I'm. ys.
PLATK 11.
Fig. 99.
Fig. 100.
Fio. 101.
Fig. 102.
Fig. 103.
Fig. 105.
Tig. lOe.
}'LATE III.
Fig. 107.
Fig. 108.
Fig. 109.
Fig. 110.
Fig. 111.
Fig. 112.
Fig. 113.
EPICYCLICS. 183
If n be infinite, whether positive or negative, we may
consider the defei-ential velocity zero, and that of the epicy-
clic finite, giving for the curve the direct epicycle itself if n
is positive, and the retrograde epicycle itself if n is negative.
When n is very great, we obtaia such a curve as is
shown in fig. 98, Plate II. (p. 184) if to is positive, and such
a curve as in fig. 99, if n is negative.
As n diminishes the angle of descent increases, the loops
separate and we obtain such forms as are shown in figs. 100
and 101, for n positive or negative respectively.
With the further reduction of n, the loops become
smaller, the point of intersection approachii^ the pericentre
when n is positive, the apocentre when n is n^ative, until
finally, when w^—, the loops disappear and we have peri-
central cusps as in figs. 102 and 104, or apocentral cusps as
in figs. 103 and 105, according as m is positive or negative.
In the former case the curve is the epicycloid, in the latter
the hypocycloid.
As n diminishes from --- towards unity the cusps disap-
pear and we have points of inflexion on either side of the
pericentres if »i is positive, or of the apocentres if n is nega-
tive, as shown respectively in figs. 106 and 107, Plate III.
As n further diminishes the points of inflexion draw
further apart for a while in case of direct epicyclic, and after-
wards approach until w^ = _, when they coincide again at
the pericentres, the curve being entirely concave towards the
centre for all smaller values of n. In the case of the retro-
grade epicyclic, the points of inflexion draw apai"t on either
side of the apocentres, and continue so to do till they meet
points of inflexion advancing from next apocentres on either
184
GEOMETRY OF CYCLOIDS.
side ; so that in this case, as in that of direct epicyclic, we
have when )i* = - two points of inflexion coinciding at
the pericentres. These two cases are illustrated in figs. 114
and 115. The former is a direct epicyclic; n-^b; and
D : E : : 25 : 1 ; (apocentral dist. : pericentral dist. : : D + E
: D— E : : 13 : 12. The latter is a retrograde epicydic;
M= — 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 » / _ the
'V E
angle of descent continually increases if n is positive and we
have curves of the form shown in fig. 108.
i'lG. 114. Fm. 115.
In diminishing from the value
, n passe; through
the value unity. When n = + 1 the curve is a circle hav-
ing the fixed point as centre, and having for radivis 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 i.s
correct on his assumption that the angle of descent is to be re-
EPICYCLIC^. 185
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 m=0 it becomes 180°, the curve being no'y 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', (O c being parallel to C P).
As n diminishes in absolute value from— \ / 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 n = — 1 the cui-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 n = — 1 for retrograde epicyclic. The semi-axes
are (D 4 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 n ^ 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 constantly advances 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 according 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° -i- {n -[■ 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 efiects of changes in the value of
• Prof. De Morgan strangely enough takes figa. 116 and 117 as
illustrating this case. But in both these figs. re=U; in fig. 117,
D = 5' E. In neither is E very nearly equal to D.
EPICYCLICS. 187
n, where D and E are supposed to remain unchanged through-
out. The number of apocentres and pericentres 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
value of — > teeping n constant. Since the angle of descent
is equal to 180° -i-(n— 1) ifnis positive, and to 180^ -t-
FiG.117.
E
(»i+ 1) if »i 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 n ^ {m + \), giving
all the direct epicyclios having m apocentres and m peri-
18K GEOMETRY OF CYCLOIDS.
centres, and secondly n= —{m—V) giving all the retrograde
epicyclics having m apocentres and m pericentres, for each
revolution of the deferent. (Of course, m is not necessarily
a whole number.)
I'IG. 118.
Suppose we take re= y , so that the angle of descent
(=180°-^f) is equal to fths of two right angles. Then if
K> -^ D we have such a curve as is shown in fig. 116. As
E diminishes untU E = -^^ D, the loops turn into cusps as
Fio. 119.
shown in fig. 117; as E diminishes still fiirther until E
= -^ D (that is n*=_ J, the cm-ve assumes the orthoidal
form shown in fig. 118. Again, take }i= — J. Then
EPICYCLICS. 189
when E is nearly equal to D the curve has such a form
as is shown in fig. 119, merging into the cuspidate form
as in fig. 120, when E = ^D; and into the orthoidal (or
straightened) form, as in fig. 121, when E = y"^ D (or
Fig. 120.
ji^ = —J. For further illustrations see p. 256.
E/
If we compare fig. 98 with fig. 122, we perceive that in
the former the loop between two successive whorls overlaps
Fig. 121.
two preceding loops, while in the latter each loop overlaps
but one preceding loop. A number of varieties arise in this
way. The determination of the condition under which any
given preceding loop may be just touched is not difficult ;
190 GEOMETRY OF CYCLOIDS.
Dut in no case does the condition lead to a formula giving n
directly in terms of D and E. The simplest of these cases is
dealt with in Prop. IX. of this section. (See fig. 160, p. 256.)
Figs. 123 and 124 illustrate eight- looped epicyclics direct
and retrograde. By noting the difierent proportions between
Fig. 122.
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
Fia. 123.
fig. 100, n = 11 ; in fig. 101, m = - 9 ; in fig. 123, n = 9,
and in fig. 124, w= - 7).
It is a useful exercise to take a series of epicyclics and
determine the value of I), 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 fig. 108 in
appearance; but in reality fig, 125 is a retrograde, -whereas
fig. 108 is a direct epicyclic. The character of the curve in
this respect is determined by tracing it directly from an}'
apocentre and noting that the next apocentre falls behind
Fia. 124.
the one from which we started. The values 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
Fig. 125.
by noting the angle between two neighboui-ing apocentres
(indicated best by the intersections 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 = one-tenth of 360°, so
192 GEOMETRY OF CYCLOIDS.
that the angle of descent is y^ths of 180° ; or n + 1 := y .
Thiis in absolute value n = ^, but n is negative.
In like manner we find that in fig. 126, n = — ^.
In each of the figs. 127, 128, and 129, n = 2, since
there is only one apocentre. In fig. 127, the trisectrix.
Fig. 126. . Fig. 127.
D = E ; in fig. 128, the cardioid, D = 2 E ; in fig. 129,
D = 3 E.
Figs. 130 and 131, Plate IV., illustrate some of the pleasing
combinations of curves which may be obtained by the use of
the geometiic chuck, the instrument with which alLthe curves
of the present part of this section have been drawn. In
Fig. 129.
fig. 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 m = 7, while for the inner »i = 15. In fig. 131
we have four direct epicycles, having (D + E) constant, but
ratio D : E difiereut in each. It will be found that there
Fio. 130.
PLATE IV. Fio. 131.
n:l::29:7. D:E-.:6:a.
Fm. 136. MARS.
i:l::13:8. D:K::10:7.
Fin. 137. JUNO.
n:l::2:l. D:E::3:2. n:l::13:3. D:li::li;3.
APPROXIMATE FORMS OF
Plate v.
Fig. 132.
Fio. 133.
Fig. 138. JUPITEE.
Fig. 139. SATURN.
ii:l::12:l. D:B::6:1.
Fio. 140. URANl'S.
n:l:;59;2. D:E::19:2.
Fig. 141. NEPTUiVE.
«:1::8S:1. D:B:;19:1.
n : 1::217: 1. D :E:.36 : 1.
THE PLA.VKTARY EPICYCLICS.
EPICYCLICS. 193
are 5^ apocentres in each circuit ; whence {n — 1) =
■^ . 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 (ji + 1) = 67|^, and
n= - 66i.
Figs. 132, 133, Plate V., are further examples for the
student.
The remaining eight figures of Plates IV. and Y., 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 return 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 remai-k that either of the ratios
15 ; 8 or 32 : 17 would have given a more satisfectory
approximation to the planet's epicychc 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 elUpticity 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 V. in symmetry of appearance. The loops are markedly
unequal, a relation corresponding of course to the observed
inequality of the area 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 biciroloid 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 radiai-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 versa, by making 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 = 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. K. & J. Beck, of Comhill), by which
all forms of epicyclics can be ocularly illustrated, A bright bead
EPICYCLICS. 195
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 » : 1 of
this section, but (n—X) : 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 /j-om 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 PA'P'
(fig. 81) as at rest, and therefore measure the motion of the tracing
radius ffP' from C'A', whereas in the mathematical way of viewing
the motions, C'a is regarded as the radius at rest, and the motion of
C'P is therefore measured from C'a. 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 iti mathematical treatises on epic3'clic 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 EIGHT TROCHOIDS.
Eight trochoids may be regarded 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
106 GEOMETRY OF CYCLOIDS.
plane of the circle, the point wUl 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 figure be projected on a plane at right
angles to its own, by parallel lines inclined half a right
angle to each plane (or perpendiculax 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 itself.
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 wUl be a point travelling 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 hehcoid 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 vifiw), ap])ears as a cycloid.
The helix is projected into other curves 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 shovm in fig. 144, plate VI., Abj^ f being a semi-
cycloid, and AbgT , A bg T', j« : Om :; Qto : arcQC
; ; sin QCC : circ. meas. of QCC. Hence the part A^ of
the companion to the cycloid is a curve of sines.
Produce Qn to meet AC'B in Q', draw M (^' p' paralle
Fig. 142.
to BD to meet the curve A^ D in jp' and AB in M', and draw
;)'m' perp. to COc. Then
Cm' = M>' = AC Q', and OC = AC
.-. O m' = arc C'Q' = arc C'Q = m ;
And p mJ = nQI = nQ =z p m.
Therefore the part 0^'D of the curve- hears precisely the
same relation to the line O c, which the part A.p O hears to
OC. Thus the entire curve is a curve of sines.
Area A^ OC = areaO/)' D c; wherefore, adding CODB,
area AODB = rect. CD = A rect. BE = circle AQB.
It is also obvious that the same curve D^' O^ A will be ob-
tained by taking E c' D as the generating semicircle, and
di-awing m' q' p=&TC q'T), mqp =&rc qq' T>; so that the
figure ED jo 0^ A is in all respects equal to the figure
'RA.pOp D.
JEPICrCLICS. 199
Since MQP = arc AQ + MQ ; and M ^ = arc AQ,
MQ=j9P;
so that an elementary rectangle QN = elementary rectangle
_p L of same breadth ; whence it follows that area A^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^ DP=^ circle AQB ;
.-. Area APDB = % circle AQB ;
and 2 area APDB = 3 circle AQB :
this is Eoberval's demonstration of the area of the cycloid.
Draw sr parallel and near to Qp, and ksh, C T, rl
perp. to OC ; then
IC := As; mC = AQ ; .*. ml= Qs; and
ml: nh:: Q« Inh:: CQ {=hk) : Qra (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 ApmG = rect. A n ; also
Area AOC = rect. CT ; and area ptnO ^ rect. n T.
Kepresenting angles by their circular measure : —
_0m\
r
«»»=rBjn-S — :=rBux : and rect. JiT=r' ( 1— cos i
r r V. »■ '
therefore, the proof that area pmO =■ rect. n T, may be re-
garded as a geometrical demonstration of the relation
^0
/:
sin X d X ^= \ — cos x
X
and similarly, since ^ m = r cos — -s = r cos , the proof
200 GEOMETRY OF CYCLOIDS.
tliat area A p m C = rect. A n may be regarded as a geome-
trical demonstration of the relation
J
cos xdx:= sin x.
X
It will easily be seen that for points on }}'!),
Area AOp' M' — rect. M' m' = rect. A n, or B n,
leading again to the relation
area AODB = rect. B c.
201
Section VI.
EQUATIONS TO CYCLOIBAL CURVES.
Although, properly speaking, the discussion of the equa-
tions to cycloidal curves belongs to the analytical treatment
of our 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 e, 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 of a;, C a for axis of y, and let the poiut start from a, in
direction aqb. When it has described an angle m about C,
the centre has advanced a distance e f along C c, and there-
fore, if X and y are the coordinates of the tracing point,
a!:=e^ + 6sinm^, y^e cos m. (1)
If we remove the origin to b, the centre of the base, taking
b d as axis of x and i a as axis of y, the equations are,
jc = e + e sin mf, y = e + e cos m . (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
X = e + e sin mf, y = e — e cos m f. (3)
202 GEOMETRY OF CYCLOIDS.
If we remove the origin to c', taking c' C as axis of x,
and c' d' as axis of y, the tracLng point starting from d in
the same direction as before, the equations are
a; = e — « sin ^, y = e cos m ^. (4)
If in this case we remove the origin to e', taking e'e as
axis of X and e' d' as axis of y, the equations are
a;^e^ — esin^, y = e + e cos m f. (5)
And lastly, if we remove the origin to d', taking d'd as
axis of X and d' e' as axis of y, we have the equations
x^ e. (6)
Eia. 143. (Join C>.)
If m = 1, these equations represent the right cycloid ; if
TO < 1 , they represent the prolate cycloid ; and if m > 1, they
represent the curtate cycloid.
For epioyclics, take O (fig. 143), the centre of fixed circle
as origin, OA through an apocentre A as axis of x, and a
perp. to OA through O as axis of y. Put 00, radius of defe-
rent = d ; GA, radius of epicycle = e (using italics as more
convenient in equations than capitals) ; Z. COG' = 0, and
angle a P = w 0. Then, if as and y are the co-ordinates of P
z=<2 cos ^-f e cos 71 ^, y=c2sin^-|-e sin n^. (7)
EQUATIONS TO CYCLOIDAZ 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'=d cos f — e cos n (p, y=d sin ^ — e sin « ^. (8)
For a retrograde epicyclic, angle aC"P'=n^, and the
equations (A being an apocentre ) are
x^d cos (p + e cos n (j>, y^d sin ^ — e sin n (j>. (9)
If B is a pericentre of retrograde epicyclic, the equations
are
x^d cos f — e cos n (j>, y^=d sin ^ + e sin n *^**
(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), n — -, the equations are those of
the epicycloid and hypocycloid respectively, when aai axis coin-
d
cides with the axis a; ; if, in equations (8) and (9), n^-, the
equations are those of the epicycloid and hypocycloid, respec-
tively, when a cusp falls on axis of x. it wUl be remembered
that if F is radius of fixed circle and R radius of rolling cii-cle,
d = E + F, and e = E. ; R being regarded as negative in case
of hypocycloid.
From (7) we get
a;2 + yS = r" = c?2 + e' -I- 2 rf e cos {n - 1) ^, (11)
. , rf cos (t -f- e cos w <4 ,,„,
andtane=, --j— ^— • --^; (12)
d sin.0 + e sin w ' ^ '
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 OC'P.]
For the epicycUc spirals, suppose OC, fig. 143 —/, and that
a tangent at C to circle CK, carrying with it the perp. BOA,
rolls over the arc CK, uniformly, tUl it is in contact at C ,
the angle C'OC being (f>. Then if AC = g, and x and y are
the rectangular coordinates of the point to which A tas
been carried, it is obvious (since CA in its new position is
parallel to OC ) that (taking projections on axes of x and y)
x=(/+ g) cos (b +f (pain cos f ; (13)
the equations to the epicydic spiral traced by A. The spiral
traced by B obviously has for its equations
x={f-g)cos cos tp. (14)
From (13) we get
x^ +y'i = r^={/+gy +fif; or
f. /-2— T"??:;— 2\ 4. a (f+ff) sin -f
f^= Vr»-(/"+^2). tan e=(/+^)cos^+/^sin^ (^^>
the polar equations to these spirals. See also Ex. 133, p. 253.
Ji g^Q, or the tracing point is on the tangent, equations
(13) become
X =■/ cos +fsva(j), y =/siii —f^cos(j> ; (16)
the equations to the involute of a circle. The polar equation
to this curve is (from 15),
/tan "^^'"-^ - ^^;^'=rp
^^0= -jM^^ __■ (17)
/tan Z1J-+ ^2-/2
It g ^ — /, equations (13) become
a;=/^sin^; y = — / (j) cos (j, ;
giving x^ + y^ =P ^* ; or r =/^ ;
7r
and taji 6 ^ — cot ^ ; or 6 ^ ^ — s >
EQUATIONS TO CYCLOIDAL CURVES. 205
whence r=/e+/^; (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 =/e. (19)
All the pairs of equations in rectangular coordinates can
readily, by eliminating (p, be reduced to a single equation
between x and y. Thus (1) becomes
e
— cos-
■m
■'(f)+ V«^-2/^ (20)
the general equation to the right trochoid.
From eqiiation (11)
a;' + 2/^ — \ \d(t>J ^\dJ f .
'> — y+ ^dx__^dy '
d df^ d(j>
where ? and >; ai-e 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 oc d ij
7— =e + mecosmffl: -7^ ^ — to e sin ni(ii :
d(j, ^' d(i> ■ '
•••(^)'+ (§!)'=«' (1 + 2TOCOSTO0 + TO^).
d 1/ d x
-J— 2=— w* ecosm^; 3—3 ^ — me sin to ^ ;
d^y dx d^x dy !!_«/ ^ , \
• — ^ r— s r^= _ e^m^icosTO + TO) ;
•' dq,^ d ^ y-r J,
, . (1 + 2 TO cos m',
and r]:=ek + e{l+km) cos m'^j' ;
from ■which we see that the evolute of the trochoid may be
regarded as traced by an epicycle of variable radius e ( 1 +km),
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 &om this axis is constantly equal to
e £ on the negative side of the axis of y.
If »» := 1 (or curve (1) becomes the cycloid), k^ — 2,
and equations (22) become
5 = e — e sin ; (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/rom the
concavity of the involute), and having vertices coincident
with the cusps of the involute cycloid.
From equations (7) we obtain
— =: —dshicb — ne sin ni>; -^^d cos d> + ne cos nOi:
d ^ ^' d ^ ^
/.{I^J+{'4f= ;
df'
.: ^^. i^^d^xdy _^2 + „3e2+ („. + n)dec^ln-\)^
d + ne cos n ^),
and ti = d sin ^ + e sin n i^ — A (cZ sin + w e sin 1 ^) j
or ^ := d(l— A)cos 4- e(l— OT^) cos »i^ 1 . ,„,»
and J/ = d(l— ^) sin^ + e(l— TO A;)sin W0 J
whence we see that the evolute may be regarded as traced
by an epicycle of variable radius e (1—nk) carried on a de-
ferent also of variable i-adius d (1 — ^).
It is easily seen (see p. 117, and figs. 63, 64), that
,_C'BY P^ \
CO' Vjo«-NB7"
When d = n e, so that the involute epicyclic is the epi-
cycloid or the hypocycloid (according as n is positive or ne-
2
gative), k reduces to , and the equations of the evolute
become
r n— 1 , ^ 71—1 J ^
4 = . d cos <6 — e cos n d>
n+l ^ n+1 ^
n = . d sm 0— r « sm n /2fl' (It X
V ij ■ 5^ = A/Da; -H^
Integrating, we have
R
But when < = 0, a; = D ; so that C = „-,
hence we have
R^g,= VD^^^ + 5eos- (^^-^), (2)
(where D is equal to the radius of the globe 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
point P
220 GEOMETRY OF CYCLOIDS.
g R' r D - x \
\j \ X y
Bisect CE in F, and describe the semicircle CDE ; then if
DE is a tangent to the circle DAB, and if DM is drawn
perpendicular to CE,
(CD)'' R2
^^- CE - D '
so 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 perpendicular to GP, we have
GG' + Gn GF + FP CP
PP - GP ~ a/6P . PE
-x/.
CP
PE'
Hence, from (u).
pp.
so that
fth
1 at P
V2g .cm. GG +Gn'
j the vel. 1 . f velocity acquired in falling through T
J ■ \ space CM, under const, accel. force g J
f elem. space 1 . f sum of elementaiy I
PP' / ■ \ spaces GG and G n
Therefore 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 CYCLOIDAL CURVES. 221
^ _ PG + arc GE .
Vlg.GK '
or
R \/^ . t = VYETPG + OF arc GE
D /'2a; — D\
= V'(D — a;)a;+ ycos-' I -^ 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 OK. 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 described with the velocity acquired in
falling through a distance equal to LC mider 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 cori'esponding to the known time-
interval T, and to measure off 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 from E to P.
Thfi division in the figure illustiutes such measm-ements in
the case of the sun, the value of T being taken as 1 8f minutes.
Moreover it is not necessary to construct a cycloid for
each case. One carefully constructed cycloid will serve for
all cases, the radius CA being made the geometrical variable.
As an instance of this method of construction, I will take
Professor Young's I'einarkable observation of a solar out-
i'22 GEOMETRY OF CYCLOIlJS.
buist, premising that I only give the construction as an ilhis-
tration, and that a proper calculation follows.
Fn. 148.
Wo"'.
GRAPHICAL USE OF CYCLOID AL CURVES. 223
On September 7, 1871, Professor Young saw wisps of
hydrogen carried in ten minutes from a height of 1 00,000 miles
to a height exceeding 200,000 miles from the sun's sm-face.
A full account of his observations is given in the second and
Fig. 150.
1".40"'
third editions of my treatise on the sun. Figs. 148, 149, 150,
and 151, with the times noted, indicate the progress of the
changes. I assumed in what follows that there was no fore-
FiG. 1.51.
r'.oo'".
shortening. The height, 100,000 miles (upper part of cloud
in fig. 148), was determined by estimation; but the ultimate
height reached by the hydi-ogen wisps (that is, the elevation
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 210,000 miles (highest filaments in fig. 149). 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 T^ere
themselves projected from 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 regai'ding this matter
as something solid propelled through a non-resisting me-
dium, and attaining an extreme range of 200,000 mUes.
What follows will show whether this supposition is ad-
missible.
Now g for the sun, with a mUe as the unit of length and
a second for the unit of time, is 0'169, and E, for the sun is
425,000. Thus the velocity acquired in traversing K under
imiform force g,
= y/'ig . R
= a/338 X 425
= 379, very nearly.
(This is also the velocity acquired under 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. 162, be our semi-cycloid (available for
GRAPHICAL USE OF CYCLOIDAL CURVES. 225
many successive constructions if these be only pencilled), and
CDE half the generating circle.
Then the foUowing is our construction : — Divide EC into
6^ equal portions, and let EP, PA be two of these parts, so
that EA represents 200,000 mUes and CA 425,000 miles
(the sun's radius). Describe the semicircle ADL about the
centre C and 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 falling from E to P.
Fio. 162.
^—
T^^y
e
Q^
^^""^
/^
y^
p
ab is/
>"
k ■
^'^
[y
5
/^
D£— —
M
f
/
^
1
l™K
<
c
10
5
*
L
Illustrating the construction for determining time of descent of a particle from
rest towards a globe attracting according to the law of nature.
T = IS*" 50™, and PQ (carefully measured) is found to
coiTespond 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 Young traversed the
distance represented by PE in ten minutes. Hence either
E was not the true limit of their upward motion, or they
Q
226
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
limit of the uprush will give ten miuutes as the time of
moving (without resistance) from a height of 100,000 to a
height of 200,000 mUes. Here we shall find the advantage
I"io. 153.
lUtistrating the oonstruction for determining time of descent between given levels
when a body descends from rest at a given height towards a globe attracting accord-
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 fig. 153. We have AC=425,000 miles;
GRAPHICAL USE OF CYCLOIDAL CURVES. 227
AP=PP' = 100,000 miles ; and Q i or (PQ-P'Q') 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 of the above con-
structions.
In equation (2) put V'2^R = 379; B, = 425,000;
D = 625,000 ; and x = 525,000 ; values corresponding to
Professor Young's observations. It thus becomes —
V
*|| (379) t = -/(100,000) (525,000)
D.JD
, QioKf^n I /1050-625\
-f 312,500 cos-' ^ 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
drg,wing Affi parallel to KG, and m/parallel to aE; then C /repre-
sents the required velocity, CL representing 379 miles per second.
A similar construction will give the velocity at P, P*, &c. Applied
to fig. 147, it gives 0/ to represent the velocity at A, C/' to represent
the velocity at P ; ra/and mf being parallel to a E and GE re-
spectively. Applied to the case dealt with in fig. 152, we get C/ 1
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.
a 2
228 GEOMETRY OF CYCLOIDS.
or
379 Vr! . t = 250,000 a/2T + 1,562,500 cos"' f~),
1562-7 t = 1,145,100 + 1,285,800 = 2,430,900,
t = 1,556« = 25" 56'.
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 (125_-80-j^
125 - SO
■f ytyjyjfVyjv) cos i
1^ (379) «2 = V (275,000) (525,000)
800
+ (400,000) cos-> (10^0),
giving (since t^ — ti ^ t')
^/|?|(379)«' = 25,000 {VII X 21 - VfxlB}
v 800
+ 400,000 I cos-' ^A^_ cos-' (J^^ I
276-25 «'= 49,250 + 111,816 = 161,066,
t' = 583' = 9" 43».
This is very near to Professor Young's ten minutes. I had
found that an extreme height of 400,000 miles gave 9™ 24»
for the time of flight between vertical altitudes 100,000
= ./2^E.V^i^ = 379v^j
GRAPSICAL USE OF CYCLOIDAL CUS.VES. 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 sun's surface must have an initial velocity
^2^
785,000""'" A/ 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 vaporous,
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 dming the observed motion to the
height of 200,000 miles. It would be safer indeed to assume
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-
sphere, 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 supplies no evidence of such
230 GEOMETRY OF CYCLOIDS.
velocities, 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
the 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 corresponding 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 woidd 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 Eespighi, Secchi, and Yoimg
— such eruptions as suffice to carry hydrogen 80,000 or
100,000 miles from the sun's surface — can be accounted for
GRAPHICAL VSE OF CYCLOIBAL CURVES. 231
without admitting a velocity of outrush exceeding consider-
ably the 379 miles per second necessary for the actual rejec-
tion of matter from the sun. But apaii; 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 fex 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 far 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 affect 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
232 GEOMETRY OF CYCLOIDS.
addressed to me by Sir J. Herschel in March 1871 (a few
weeks only before his lamented decease). The letter bears
throughout on the subject of this paper, and therefore I
quote more than relates to the association between terrestrial
magnetism and disturbances of the solar photosphere.
After referring to Mr. Brothers' photograph of the corona
(remarking that ' the corona is certainly ea!" D = ^ generating circle.
19. If in fig. 5, p. 10, EJ is drawn perp. to BD, and a
quadrant AIC about T as centre, show that
area EJD = ai-ea 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)" = 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') =.
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. K 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 : area AP'DB : area AF'DB: : 9 : 3 t : 8.
EXAMPLES. 237
27. With the same construction, the radii of curvature of
the three curves at A are in the ratio 16 : 2 w^ ; -n^.
28. On the generating circle AQB the arc AQ ^ ^ cir-
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 curvatirre 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, throiigh which straight
lines are drawn parallel to the base, meeting the cycloid in the
points Pj, P2, and P3; if the radii of curvature at A, Pj, P2,
and Pj, are respectively equal to p,, p^, ps, and p^, show that
Pi':p2':p3':p4'::'4:3:2: i.
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 B..
32. If 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 R, and minor
axis = 4 R.
33. PB' and FB" are the normals at two points P, P',
close together on a cycloidal arc, and PQ paiaUel 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 difierent points of a cycloidal arc, whose axis is
238 GEOMETRY OF CYCLOIDS.
vertical, partides are let fall down the normals through those
points ; show that they will reach the base simultaneously in
time 2a/ —
^ 9
If they still continue to fall along the normals pro-
duced, they will reach the evolute simultaneously in time
35. If the distance of P on semi-cycloidal arc APD (fig.
10, p. 21) from base BD = f AB, show that
3 moment of PD about AE =14 moment of AC about AE.
36. In same caae, if PM parallel to BD meet AB in M,
show that
moment of PD about AE = § (AB)i [(AB)} -(AM)! ].
37. Show that the moment of arc AP (fig. 10, p. 21)
about AB
= 2 (NQ+arc AQ) AQ-f AB* (AB* -BM! ).
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 K
40. Two tangents P'T, PT to the same epicycloidal arc
D'P'APD intersect in T at right angles, and through P' and
EXAMPLES. 239
P circular 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 R.
41. If the roUing 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
01, show that the directions of the normal of the tangent also
F + 2R
change uniformly with angular velocity — s-^k — w.
42. On the same assumption, the direction of the tracing
F + E
radius changes muformly with angular velocity — p — w.
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 in, show that the direction of the normal and of the
tangent also change uniformly with angular velocity
F-2R
44. On the same assumption the direction of the tracing
■p T>
radius changes uniformly with angular velocity — ^ — u.
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 O as
centre intersect the remoter half of the central generating
circle in Q, show that
Arc ABQ : arc BQ : : P : 2 R.
46. Two tangents P'T, PT to the same hypocycloidal arc
D'P'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 di-awn around O as centre to meet (without cutting)
the central generatLog 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
QB 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 PB'P' 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 V (B' on the
base DBD') ; show that
PK . PK' : (PB')" : : F + R : K : : (PVf : pk . pk'.
52. With the same construction if OM be drawn perp.
to PKP', show that
OM : PB' : P6' :: F + 2 R : 2 R : 2 (F + R).
53. If tangent at P to epicycloid DAD' touches the
fixed circle, and PB'6' the normal at P meets the fixed circle
in B' and 6' (B' on the base DBD'), show that '
PB' (F + 2 R) = 2 R2 ; and P 6' (F + 2 R) = 2 R (F + R).
54. If tangent at P to epicycloid DAD' touches the fixed
circle and cuts the rolling circle in A', then
(AT)": (2R)«:: (F + R) (F + 3R) : (F + 2R)"
EXAMPLES. 241
55. In figs. 21 and 22 (pp. 44, 45) the points P, B', 6, lie
in a str9,ight line.
56. In figs. 21 and 22, the tangent to DP at P cuts
60 c' produced in a point a such that 6 o = 2 6 c'.
57. At D the cusp of an epicycloid DAD (fig. 19, fron-
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 i 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 centnil generating circle and fixed
circle respectively, perp. to ABO and on same aide of it.
If C y cut Q q parallel to CO in K, and a straiglit line d'K.d'
through K parallel to O 5 is the generating base of a prolate
cycloid having AQB as central generating circle, show that
the ai-ea between the epicycloid DAD' and its base DD' is
equal to the area between the prolate cycloid dAd' and its
base d d'.
59. ACB is the axis of a hypocycloid DAD ; D, D' its
cusps ; CQ, q radii of central generating circle and fixed
circle perp. to BAO and on the same side of it. If C g cut
Q q parallel to CO in K, and a straight line d'K.d' through
K parallel to 5' is the generating basis of a cui-tate cycloid
having AQB as ceutitil generating circle, show that the
area between the hypocycloid DAD' and its base DD' is
equal to the area between the curtate cycloid dA.d' and itjf
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 concentric
with the fixed circle, touching the cardioid at the vertex, is
equal to three times the aiea of the fixed circle.
K
242 GEOMETRY OF CYCLOIBS.
62. The area of a circle touching the cardioid at the
Tertex and concentric with the base, is divided into three
equal parts hy the arc of the cardioid and the axia produced
to meet the circle.
63. Area A o P (fig. 39, p. 74) = 3 E (6;fe + arc B6).
64. If e = Z. BO 6 (fig. 39, p. 74)
Area PSA = R" (3 fl + 4 an 6 + l sin 3 fl).
65. The area between one arc of the tricus{>id epicycloid
and the base is equal to 3| times the area of the generating
circle.
66. A complete focal chord is dixiwn 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 segment.
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 cu-cle are equal.
(Purldss.)
68. Lines perp. to focal radii vectores through theii' ex-
tremities have a cii-cle for envelope. (Purhiss.)
69. Prom S, the focus of cardioid, a perp. SQ to a com-
plete focal chord PSP', is drawn, meeting the fixed circle in
Q ; show that SQ is a mean proportional between SP and
SP'.
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 wUl be a mean propoi-tional 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' mesta circle
SQA iu q ; show that
SQ :Sg:: 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 confocal
with the cardioid and touches the cu'cle on. SA as diameter.
i^Purkiss.)
The demonstration of this will be more easily effected by taking
for the cardioid the lociis of ti, fig. 39 (see p. 75). From n draw
ny a. parallel to bf, then S y, perp. to n y, gives y a point on the
pedal of this cardioid with respect to S. It can readily be shown
that a parabola having S as focus and y as vertex touches the
circle B i S in i.
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 Q'P, show that the locos of P is a cardioid.
74. If rays diverge 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,'
Ai't. 72, which see.)
If S b, fig. 39, p. 74, represent path of a ray, to circle B J S, re-
flected ray J y is in the line Vbg, 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 vertex 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' j' = 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 curve is one half (from vertex to vertex) of
E 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 evolute of the bicuspid
epicycloid (or nephroid) are \ those of the curve itself
78. The area between one arc 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 to R = n F, show that the linear
dimensions of the evolute are to those of the epicycloid as
m ; m + 2n.
81. IfmR^nF, area between an arc of epicycloid and
., , 3m + 2n _ (3 m + 2 w) m' „ , _
its base = ■ ■ . gen. = J^ !— = — '- — . hxed ©.
7)1 m'
82. If PB'o Q is the diameter of curvature at the point
P of an epicycloid, o the centre of curvature, B' a point of the
base, then
Area of epicycloid : area of gen. © : : QB' : Wo.
83. If the arc of an epicycloid, from cusp to cusp ^ a,
and m R ^ n F, show that a + arc of evolute from cusp to
cusp -I- arc of evolute's evolute from cusp to cusp, and so on
ad infinitum,
(m -\- 2n) a
2^
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 base + area between an ai-c of the evolute's
evohite and its base, and so on ad infinitum,
_ (m + inYA?
4 n {m, + n)
85. If in the hypocyloid m R ^ m F, show that the linear
dimensions of the evohite are to those of the hypocycloid as
m ; m — 2n.
F
Interpret this result when R = — ,
Z
86. If m R = n F, area between an arc of hypocycloid
J., , 3«i — 2n _ (3 m — 2 mW* ^ - j ^
and its base = gen. © = 1— 5— i nxed ©.
m nrn'
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=o, and
m R = m F, show that a + 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 ad infinitum,
m
= -?r- «•
2n
89. If the area between a hypocycloid and its base = A,
and TO R := »i F, show that A + 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
so on ad infinitum,
to'A
4 n{m—n)
L'46 GEOMETRY OF CYCLOIDS.
90. jyAJy is an arc of a tricuspid epicycloid, from cusp
to cusp, ACB the axis, AQB the central generating circle, C
its centre, OBCA diametral ; show that an angle may be tai-
sected by the following construction : — Let ACQ be the
angle to be trisected. Join QB, QO ; about as centre
describe are QP meeting D'AD iii 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 of the
angle ACQ.
91. D'AD- is an arc of a tricuspid hypocycloid from
cusp to cusp ; ACB the axis ; AQB the central generating
circle, C its centre, OACB diametral. Show that an angle
may be trisected by the following construction. Let ACQ
be the angle to be trisected. Join QB, QO ; altout 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 diameti-al. A radius CQ is drawn to AQB ;
and BQ, OQ are joined. About as centre the arc QP is
di-awn meeting D'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 epicycloid, an angle may be
divided in any required ratio.
93. D'AD is an arc of a hypocycloid from cusp to
EXAMPLES. 247
ciisp ; ACB the axis ; AQP the central generating circle,
C its centre ; OAOB 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 D'AD in P (on AD) ; PO is
joined ; and the angle OPB' is made equal to the angle
OQB, and towards the same side, FB' meeting the base
D'BD in B'. If OB' Ls joined, show that
angle BOB = ? . angle ACQ,
so that by means of a suitable hypocycloid 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 tiie centre of curvature
at P, show that op produced meets the tangent at P in a
point T such that TP is equal to the are 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
4B(Z±m and i|-Y#±^\
F F VF+2E,y
96. If a hypocycloid 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
iRVi:^Z^^ and ^^(^-^) .
F ^^F-2Ry F
' 97. An arc DAD of the bicuspid epicycloid, or nephroid,
has its axis AB coincident in position with A b, the axis of a
cycloid whose vertex is at A ; but AB = § A 6. If the
nephroid and the cycloid roll on T'AT, the common tangent
at A, in such sort that they simultaneously touch the same
point on T'T, show that the centre of curvature of the
nephroid 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.
contact (the foot of normal being understood to mean thd
intersection of the normal with the base).
98. If a quadricuspid hypocycloid (i-adius of fixed circle
F) is orthogonally projected on a plane through two opposite
cusps, in such sort that the distance 2 F between the other
two cusps is projected into distance if, show that the pi^o-
jected curve is the evolute of an ellipse having axes equal t»
99. Show that the arc of the projected curve in 98, from
cusp to cusp,
_ F'' + F/4-/'
F+/ ■
100. ACA', BOB' ai-e the major and minor axes of an
ellipse, C its centre ; and a B o' B' is a similar ellipse having
BOB as major axis ; if the ellipse ABA'B' is orthogonally
projected into a circle, show thut the evolute of a B o'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 noi-mal of a B o B ,
intercepted between the projections of AA' 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. — ThU propotition, d&moniitrated geometrically, eomlined
with what is shomn. at pp. 72, 73, affords a geometrical demon-
itration of t/ie natitre of the eeelute to the ellipse. See >iext
problem.
102. Let ACA', BOB be the major and minor axes of an
ellipse, hOb' tiie orthogonal projection of BCB on a plane
through ACA', so situated that 6 6' : BB' : : BB : AA .
From B draw BL perp. to AB to meet A'G in L ; and about
EXAMPLES 2t9
C in the p'ane A 6 A , describe a circle with radius LA' cutting
CA, CA , C h, and C 6', in K, K', k, and h , respectively. Draw
a four-pointed hypocycloid, having cusps at K, k', K', and k.
Then a plane perpendicular to the plane A b A'6', throvigh
any tangent to the hypocycloid K k'TS.'k, will intersect the
plane ABA'B' in a normal to the ellipse ABA'B', and a
right hypocycloidal cylinder on K A'K'A as base, will inter-
sect ABA'B' in the evolute of this ellipse.
103. Two straight lines intersect at right angles in a
plaaie perpendicular to the sun's rays, one of the lines being
hori2ontal. 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 a, and the length of the sliding line I,
then taking for axis of a: the shadow of the horizontal fixed line,
the equation to the envelope is x^ + y'' sin^o = P; and the
equation to the involute ellipse is x' cos* o + y" sin^ a cos' a=l'.
104. At P a point on the hypocycloid DPAD' the tan-
gent KPK' is drawn, meeting the fixed circle in K and K',
and the normal 6'PB' meeting the fixed circle in b' and B'
(B' on the base DBD') ; show that
KP . PK' : (PB')2 : : F-K, : R : : (Pb'y : KP . PK'.
'105. With the same construction, OM is drawn perp. to
KPK'; show that
OM : PB' : P 6' : : F-2 R : 2 R : 2 (F-R).
106. If the tangent to the cardioid at P touches the fixed
circle, and cuts the rolling circle in A', and the normal at
P cuts the fixed circle in B' and 6', then
250 GEOMETRY OF CYCLOIDS.
PB' = f R ; P 4' = ii?;aud A'P=i:^ R.
^ 3 ' 3
107. In the trochoid, if R h', the normal at p, 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'jo' is similar to triangle a'h'p.
108. With same construction
Z TBV = z 6> B' = Z T^ a'.
109. In fig. 48, triangle C 6 g" = — sector b C q".
r
110. In fig. 48, p. 96, show that
loop p' rdr= 2 "^—^ arc a 6 N L y " + 2^^ rect. N n.
111. Show that the result obtained in the last example
agrees with that obtained in Prop. IX., Section III.
112. If in Q' g", fig. 48, produced, a point X is taken such
that (CX)* = rect. o B a C, and a circular arc XY (less than
semicircle) with C as centre and CX as i-adius cuts a b 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 bdiay; show that
area ydrp" : seg. q"'L 6 ; : a B : oC.
114. Prom B (fig. 45, frontispiece) a straight line B g g'
is drawn cutting the central tracing circle in q and q' , and
straight lines qp and c^p' paiuUel to the base meet the arc
a dra. p and p'; show that the tangent at ^ is parallel to the
tangent at p'.
115. P and P' are two points on an epitrochoid or hypc-
trochoid, C and C the corresponding positions of the centre
of generating circle, the fixed centre, OA, OB the apo
central and pericentral distances. K OP . OP' = OA . OB,
EXAMPLES. 251
show that the tangents at P and P' make equal angles with
OC and OC respectively.
116. A cycloid on base BD (fig. 45, frontispiece) has its
cusps at B and D ; show that it touches the prolate cycloid
a^ rf at a point of inflexion.
117. A series of pi-olate cycloids have the same hne of
centres, their axes in the same straight line, and their bases
eq val. Show that their envelope is a pair of arcs of a cycloid
haWng 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 ^ and q, two points on a prolate
cycloid ap qd, are parallel, and meet the generating base in
h' and b" respectively, then p and f>' being the radii of cur-
vatiu-e at p and q respectively,
p:(.'::(ph'f:{qh")\
119. If p is the radius of curvatui-e at the poiat 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^ = fir.
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 = s a'.
252 GEOMETRY OF CYCLOIDS.
124. If the rolling straight line by which a spiral of
Archimedes is ti-aced 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 i " '^^ + "' + log (« + •v^l +"'')?■
125. All involutes of circles are similar.
126. All spirals of Archimedes are similar.
1 27. 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
ti-aced out by the extremities of these perps. will be similar
if V:p::Y:f.
128. In the spiral of Archimedes the subtangent is equal
to tliat 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 vector,
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 Op.
129. 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 ^ aO,
= «( 1 + fl') "^
'' 2 + 6»« •
EXAMPLES. 253
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 ti-ace out one of the
spii-als described at pp. 128, 129.
132. If the angular velocity in preceding problem is w,
the linear velocity of the foot of perpendicular v, and Z = — ,
the perpendicular lying on the fide towards which the revolv-
ing line is advancing, show that the other extremity of the
perpendicular will describe the involute of the circle.
1 33. If DT, fig. 42, p. 82, rolls on the circle DQB of radius
a, and a point initially on DO and distant b from D is cari'ied
with DT to trace 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
^r2 - {a-hf ^ , {a-h)
e = + tan-' '2 — 7 "t^'
a V r' — (a — by
134. Show that the construction 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, putting the arc
over which the rolling line has passed from its initial
position = If, show that
_ {a^f + V'Y
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