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Font size Thus, if the traced curve be the involute of a circle concentric with the
given circle, the rolled curve is one of two similar logarithmic spirals.

If the curve traced be the spiral of Archimedes, the rolled curve may be
either the hyperbolic spiral or the straight line.

In the next case, one curve rolls on another and traces a circle.

Since the curve traced is a circle, the distance between the poles of the
fixed curve and the rolled curve is always the same; therefore, if we fix the
rolled curve and roll the fixed curve, the curve traced will still be a circle,
and, if we fix the poles of both the curves, we may roU them on each other
without friction.

Let a be the radius of the traced circle, then the sum or difference of
the radii of the other curves is equal to a, and the angles which they make
with the radius at the point of contact are equal,

.♦. n-=±(a±r,)andn^^ = r,^\

dO, _ ±(a±r^ dS,
drt~ r, dvi'

If we know the equation between ^j and r,, we may find ^— in terms of r„

substitute ± (a ± r,) for r„ multiply by ^ \ and integrate.

Thus, if the equation between 6^ and r^ be

r, = a sec \$,,

TEU: THEORY OF ROLLING CURVES. 17

which is the polar equation of a straight line touching the traced circle whose
equation is r = ay then

dd _ a

dr, ~ r, -Jr.'-a'
a

{r,±a)Jr,'±2r,a
dO^ r^±a a

dr, r, (r,±a) Jrf±2r^

a

_ 2a _ 2a

Now, since the rolling curve is a straight line, and the tracing point is

not in its direction, we may apply to this example the observations which

2a
Let, therefore, the curve ^ = ^ — 7 be denoted by A, its involute by B, and

the circle traced by C, then B is the tractory of C; therefore the involute

2a
of the curve ^ = ^ — r is the tractory of the circle, the equation of which is

^ = cos"' /— — I. The curve whose equation is ^'=s — ; seems to be among

spirals what the catenary is among curves whose equations are between rec-
tangular co-ordinates ; for, if we represent the vertical direction by the radius
vector, the tangent of the angle which the curve makes with this line is
proportional to the length of the curve reckoned from the origin ; the point
at the distance a from a straight line rolled on this curve generates a circle,
and when rolled on the catenary produces a straight line ; the involute of this
curve m the tractory of the circle, and that of the catenary is the tractory
of the straight line, and the tractory of the circle rolled on that of the straight
line traces the straight line ; if this curve is rolled on the catenary, it produces
the straight line touching the catenary at its vertex ; the method of drawing

18 THE THEORY OF ROLLING CURVES.

tangents is the same as in the catenary, namely, by describing a circle
radius is a on the production of the radius vector, and drawing a tangent to the
circle from the given point.

In the next case the rolled curve is the same as the fixed curve. It is
evident that the traced curve wiU be similar to the locus of the intersection
of the tangent with the perpendicular from the pole ; the magnitude, however,
of the traced curve will be double that of the other curve; therefore, if we
call n = <^o^o the equation to the fixed curve, r, = <f>,6, that of the traced curve,
we have

also, £^ = f.

SimUarly, r, = 2p, = 2r,f = A^ Ar, (^J, 0,^6,-2 cos- ^ .

Similarly, r„ = 2p„., = 2r„_, ^ &c. = 2^ (^^J ,
and ^^f.

^„ = ^„-7lC0S-f-\

'o

V
0n = 6. — ncos~^ -^ .

Let e, become 6^'; 0„ 6,' and ^ , ^. Let ^„^-^„ = a,

^„^ = ^;-ncos- ^,
» «.

a = ^„^- e„ = ^.^-^o-ncos-^ ^' +n cos-^ ^

-1 Pn -1 Pn O- , ^0 ~ ^0

\ cos ^ ^^-^ — COS * -^— = - 4 .

THE THEORY OF ROLLING CURVES. 19

Now, cos"^ — is the complement of the angle at which the curve cuts the

' n

radius vector, and cos"' — —cos"' -^ is the variation of this angle when 6^ varies
by an angle equal to a. Let this variation = (^ ; then if 6^ — 6 J = fi,

^ n n
Now, if n increases, <f> will diminish ; and if n becomes infinite,

<^ = ^ + ^ = when a and )8 are finite.

Therefore, when n is infinite, <}> vanishes ; therefore the curve cuts the radius
vector at a constant angle ; therefore the curve is the logarithmic spiral.

Therefore, if any curve be rolled on itself, and the operation repeated an
infinite number of times, the resulting curve is the logarithmic spiral

Hence we may find, analytically, the curve which, being rolled on itself,
traces itself.

For the curve which has this property, if rolled on itself, and the operation
repeated an infinite number of times, will still trace itself.

But, by this proposition, the resulting curve is the logarithmic spiral ;
therefore the curve required is the logarithmic spiral. As an example of a curve
rolling on itself, we will take the curve whose equation is

n=2"a(cos|)".
-1=2". (sing (oosf-;
2"a'(cos^")'"

.'. r^ = 2p,= 2

r, = 2

^2-a'(cosg%2-a^(sing (cosg"^'^

2"a cos — / n\ „+i

^^cos-j+(sm-j

20 THE THEORY OF ROLLING CURVES.

Now ^1-^0= -cos-^^"= -cos-' cos -" = -^,

" n+1
substituting this value of 6^ in the expression for r^,

r. = 2-'a^cos - J ,

similarly, if the operation be repeated ni times, the resulting curve is

*afcos— ^^y
\ n + mj

When n=l, the curve is

r = 2a cos 9,

the equation to a circle, the pole being in the circumference.

When n = 2, it is the equation to the cardioid

r = 4a (cos -J .

In order to obtain the cardioid from the circle, we roll the circle upon
itself, and thus obtain it by one operation ; but there is an operation which,
bei6g performed on a circle, and again on the resulting curve, will produce a
cardioid, and the intermediate curve between the circle and cardioid is

r = 2

> / 20\i

As the operation of rolling a curve on itself is represented by changing n
into (n + 1) in the equation, so this operation may be represented by changing n
into (w + i).

Similarly there may be many other fractional operations performed upon
the curves comprehended under the equation

r = 2"a(cos-j.

We may also find the curve, which, being rolled on itself, will produce a
given curve, by making 7i= — 1.

THE THEORY OF ROLLING CURVES. 21

We may likewise prove by the same method as before, that the result of
performing this inverse operation an infinite number of times is the logarithmic
spiral.

As an example of the inverse method, let the traced line be straight, let
its equation be

r<, = 2a sec d^,
then P^^p,^2a^2a_

therefore suppressing the suflSx,

= ar,

* • \d0j a '

dr
r

7i-''

■■&-')

- 2a
^~l-cos^'

the polar equation of the parabola whose parameter is 4rt.

The last case which we shall here consider affords the means of constructing
two wheels whose centres are fixed, and which shall roll on each other, so that
the angle described by the first shall be a given function of the angle described
by the second.

Let 0^ = (f}0i, then r^ + r^ = a, and -j^ = — ;

d0^ a-r^'

Let us take as an example, the pair of wheels which will represent the
angular motion of a comet in a parabola.

THE THEORY OF ROLLING CURVES.

Here 6^ = tan -^ ,

. ^_

2 cos' -^

a 2 + cos ^1 '

therefore the first wheel is an ellipse, whose major axis is equal to | of the
distance between the centres of the wheels, and in which the distance between
the foci is half the major axis.

Now since ^i = 2 tan"' B^ and r^ = a - r„

'• 1+ 1

a ^2(2-^)'

'-'-±;'

a
which is the equation to the wheel which revolves with constant angular velocity.

Before proceeding to give a list of examples of rolling curves, we shall
state a theorem which is almost self-evident after what has been shewn pre-
viously.

Let there be three curves. A, B, and C. Let the curve A, when rolled
on itself, produce the curve B, and when rolled on a straight line let it
produce the curve C, then, if the dimensions of C be doubled, and B be
rolled on it, it will trace a straight line.

A Collection of Examples of Rolling Curves.

First. Examples of a curve rolling on a straight line.

Ex. 1. When the rolling curve is a circle whose tracing-point is in the
circumference, the curve traced is a cycloid, and when the point is not in the
circumference, the cycloid becomes a trochoid.

Ex. 2. When the rolling curve is the involute of the circle whose radius
is 2a, the traced curve is a parabola whose parameter is 4a.

THE THEORY OF ROLLING CURVES. 23

Ex. 3. When the rolled curve is the parabola whose parameter is 4a, the
traced curv^e is a catenary whose parameter is a, and whose vertex is distant
a from the straight line.

Ex. 4. "When the rolled curve is a logarithmic spiral, the pole traces a
straight line which cuts the fixed line at the same angle as the spiral cuts

Ex. 5. When the rolled curve is the hyperbolic spiral, the traced curve
is the tractory of the straight line.

Ex. 6. When the rolled curve is the polar catenary

r 2a

the traced curve is a circle whose radius is a, and which touches the straight
line.

Ex. 7. When the equation of the rolled curve is

the traced curve is the hyperbola whose equation is

y' = d' + a^.

Second. In the examples of a straight Hne I'olling on a curve, we shall
use the letters A^ B, and C to denote the three curves treated of in page 22.

Ex. 1. When the curve ^ is a circle whose radius is a, then the cui-ve B
is the involute of that circle, and the curve C is the spiral of Archimedes, r = ad.

Ex. 2. When the curve ^ is a catenary whose equation is

the curve B is the tractory of the straight line, whose equation is

X I

y = a log , + JcL' — -f^,

a + V a' - ar"

and C is a straight line at a distance a from the vertex of the catenary.

24 THE THEORY OF ROLLING CURVES.

Ex. 3. When tKe curve A is the polar catenaxy
the curve B is the tractory of the circle

and the curve (7 is a circle of which the radius is - .

Third. Examples of one curve rolling on another, and tracing a straight
line.

Ex. 1. The curve whose equation is

= Ar-"* + &c. + Kr-' + Lr'^ + Jf log r + iVr + &c. + Zt^,
when rolled on the curve whose equation is

n — 1 71+ L

traces the axis of y.

Ex. 2. The circle whose equation is r = a cos ^ rolled on the circle whose
radius is a traces a diameter of the circle.

Ex. 3. The curve whose equation is

^=J'i-

1 — versm - ,
a

rolled on the circle whose radius is a, traces the tangent to the circle.

Ex. 4. If the fixed curve be a parabola whose parameter is 4a, and if we
roll on it the spiral of Archimedes r = ad, the pole will trace the axis of the
parabola.

Ex. 5. If we roll an equal parabola on it, the focus will trace the directrix
of the first parabola.

Ex. 6. If we roll on it the curve ^ = t^ t^® P^^® "^^ ^^^^ ^^® tangent
at the vertex of the parabola.

THE THEORY OF ROLLING CURVES. 25

Ex. 7. If we roll the curve whose equation is

r = a cos (t^)
on the ellipse whose equation is

the pole will trace the axis h.

Ex. 8. K we roll the curve whose equation ia

on the hyperbola whose equation is

the pole will trace the axis h.

Ex, 9. If we roll the lituus, whose equation is

on the hyperbola whose equation is

the pole will trace the asymptote.

Ex. 10. The cardioid whose equation is

r = a(H- cos ^),
rolled on the cycloid whose equation is

12 = a versin"' - + J2ax - ic*,
^ a

traces the base of the cycloid.

Ex. 11. The curve whose equation is

= versm-'- + 2^/ 1,

rolled on the cycloid, traces the tangent at the vertex.

26 THE THEORY OF ROLLING CURVES.

Ex. 12. The straight line whose equation is

r = a sec B,
rolled on a catenary whose parameter is a, traces a line whose distance from
the vertex is a.

Ex. 13. The part of the polar catenary whose equation is

rolled on the catenary, traces the tangent at the vertex.

Ex. 14. The other part of the polar catenary whose equation is

rolled on the catenary, traces a line whose distance from the vertex is equal to 2a.

Ex. 15. The tractory of the circle whose diameter is a, rolled on the
tractory of the straight line whose constant tangent is a, produces the straight
line.

Ex. 16. The hyperbolic spiral whose equation is

a

'■=5'

rolled on the logarithmic curve whose equation is

1 ^
2/ = alog-,

traces the axis of y or the asymptote.

Ex. 17. The involute of the circle whose radius is a, rolled on an orthogonal
trajectory of the catenary whose equation is

traces the axis of y.

Ex. 18. The curve whose equation is

THE THEORY OF ROLLING CURVES. 27

rolled on the witch, whose equation is

traces the asymptote.

Ex. 19. The curve whose equation is

r — a tan Q,
rolled on the curve whose equation is

traces the axis of y.

Ex. 20. The curve whose equation is

2r

e=

rolled on the curve whose equation is

y = / , or r = a tan \$,

traces the axis of y.

Ex. 21. The curve whose equation is

r = a (sec d — tan 0),
rolled on the curve whose equation is

2/ = alogg+l),
traces the axis of y.

Fourth. Examples of pairs of rolling curves which have their poles at a fixed
distance = a.

Ce straight line whose equation is ^=sec"'-
..„ , .

r

2a

The polar catenary whose equation is 0= ±fj I ±

Ex. 2. Two equal ellipses or hyperbolas centered at the foci.
Ex. 3. Two equal logarithmic spirals.

(Circle whose equation is r = 2a cos 6.

Curve whose equation is ^-/J^ — l + versin"^-.

Ex. 4.

28 THE THEORY OF ROLLING CURVES.

fCaxdioid whose equation is r=2a(l+co8^).

Ex. 5.

Ex. 6.

Ex. 7.

[Curve whose equation is ^ = sin"*- + log ,— — — .

(Conchoid, r = a ( secg- 1).

Icurve, ^ = >A-?

Spiral of Archimedes, r = a0.

T T

Curve, ^ = - + log

+ sec"^ -
a

a ° a

f Hyperbolic spiral, r=-Q

Ex. 8. -!

ICurve,

a

e'+l

1

Cpse whose equation is ^"^^2+ ~Q'

Ex. 10.

(Involute of circle, ^~Ja^^^ ®®^"^ a '

'curve, e^J^±2l±log(-±l+J^.±2'^.

Fifth. Examples of curves rolling on themselves.
Ex. 1. When the curve which rolls on itself is a circle, equation

r = a cos 6,
the traced curve is a cardioid, equation r = a(l+cos^).
Ex. 2. When it is the curve whose equation is

r = 2"a (cos-j ,
the equation of the traced curve is

Ex. 3. When it is the involute of the circle, the traced curve is the spiral
of Archimedes.

THE THEORY OF ROLLING CURVES. 29

Ex. 4. When it is a parabola, the focus traces the directrix, and the vertex
traces the cissoid.

Ex. 5. When it is the hyperbolic spiral, the traced curve is the tractory of
the circle.

Ex. 6. When it is the polar catenary, the equation of the traced curve is

J

2a , . ., r

1 — versin - .

r a

Ex. 7. When it is the curve whose equation is
the equation of the traced curve is r = a (e' — €~").

This paper commenced with an outline of the nature and history of the problem of rolling
curves, and it was shewn that the subject had been discussed previously, by several geometers,
amongst whom were De la Hire and Nicolfe in the Memoir es de I'Academie, Euler, Professor
Willis, in his Principles of Mechanism, and the Rev. H. Holditch in the Cambridge Philosophical
Transactions.

None of these authors, however, except the two last, had made any application of their
methods ; and the principal object of the present communication was to find how far the general
equations could be simplified in particular cases, and to apply the results to practice.

Several problems were then worked out, of which some were applicable to the generation
of curves, and some to wheelwork ; while others were interesting as shewing the relations which
exist between different curves ; and, finally, a collection of examples was added, as an illus-
tration of the fertihty of the methods employed.

[From the Transactions of the Royal Society of Edinburgh, Vol. XX. Part i,]

III. — On the Equilibrium of Elastic Solids.

There are few parts of mechanics in which theory has differed more from
experiment than in the theory of elastic sohds.

Mathematicians, setting out from very plausible assumptions with respect to
the constitution of bodies, and the laws of molecular action, came to conclusions
which were shewn to be erroneous by the observations of experimental philoso-
phers. The experiments of (Ersted proved to be at variance with the mathe-
matical theories of Navier, Poisson, and Lame and Clapeyron, and apparently
deprived this practically important branch of mechanics of all assistance from
mathematics.

The assumption on which these theories were founded may be stated thus : —

Solid bodies are composed of distinct ^molecules, which are kept at a certain
distance from each other by the opposing principles of attraction and heat. When
the distance between two molecules is changed, they act on each other with a force
whose direction is in the line joining the centres of the molecules, and whose
magnitude is equal to the change of distance multiplied into a function of the
distance which vanishes when that distance becomes sensible.

The equations of elasticity deduced from this assumption contain only one
coefficient, which varies with the nature of the substance.

The insufficiency of one coefficient may be proved from the existence of
bodies of different degrees of solidity.

No effort is required to retain a liquid in any form, if its volume remain
unchanged; but when the form of a solid is changed, a force is called into
action which tends to restore its former figure ; and this constitutes the differ-

THE EQUILIBRITJM OF ELASTIC SOLIDS. 31

ence between elastic solids and fluids. Both tend to recover their vohirne, but
fluids do not tend to recover their shape.

Now, since there are in nature bodies which are in every intermediate state
from perfect soHdity to perfect liquidity, these two elastic powers cannot exist
in every body in the same proportion, and therefore all theories which assign to
them an invariable ratio must be erroneous.

I have therefore substituted for the assumption of Navier the following
axioms as the results of experiments.

If three pressures in three rectangular axes be applied at a point in an
elastic solid, —

1. TTie sum of the three pressures is proportional to the sum of the com-
pressions ichich they produce.

2. The difference between two of the pressures is propo7'tional to the differ-
ence of the compressions which they produce.

The equations deduced from these axioms contain two coefficients, and differ
from those of Navier only in not assuming any invariable ratio between the
cubical and linear elasticity. They are the same as those obtained by Professor
Stokes from his equations of fluid motion, and they agree with all the laws of
elasticity which have been deduced from experiments.

In this paper pressures are expressed by the number of units of weight to
the unit of surface ; if in English measure, in pounds to the square inch, or
in atmospheres of 15 pounds to the square inch.

Compression is the proportional change of any dimension of the solid caused
by pressure, and is expressed by the quotient of the change of dimension divided
by the dimension compressed'".

Pressure will be understood to include tension, and compression dilatation ;
pressure and compression being reckoned positive.

Elasticity is the force which opposes pressure, and the equations of elasticity
are those which express the relation of pressure to compression f.

Of those who have treated of elastic solids, some have confined themselves
to the investigation of the laws of the bending and twisting of rods, without

* The laws of pressure and compression may be found in the Memoir of Lam6 and Clapeyrou. St^t-
note A.

t See note B.

32 THE EQUIUBRIUM OF ELASTIC SOLIDS.

considering the relation of the coefficients which occur in these two cases;
while others have treated of the general problem of a solid body exposed to
any forces.

The investigations of Leibnitz, Bernoulli, Euler, Varignon, Young, La Hire,
and Lagrange, are confined to the equilibrium of bent rods; but those of
Navier, Poisson, Lam^ and Clapeyron, Cauchy, Stokes, and Wertheim, are
principally directed to the formation and application of the general equations.

The investigations of Navier are contained in the seventh volume of the
Memoirs of the Institute, page 373; and in the AnnoUes de Chimie et de
Physique, 2^ Sdrie, xv. 264, and xxxviii. 435 ; L'AppUcati(m de la Micanique,
Tom. I.

Those of Poisson in Mem. de I'lnstitut, vm. 429 ; Annales de Chimie, 2"
S^rie, XXXVI, 334 ; xxxvii. 337 ; xxxvtil 338 ; xlu. Journal de VEcole
Polytechnique, cahier xx., with an abstract in Annales de Chimie for 1829.

The memoir of MM. Lam^ and Clapeyron is contained in Crelle's Mathe-
matical Journal, Vol. vii. ; and some observations on elasticity are to be found
in Lamp's Cours de Physique,

M. Cauchy's investigations are contained in his Exercices d! Analyse, Vol. in.
p. 180, published in 1828.

Instead of supposing each pressure proportional to the linear compression
which it produces, he supposes it to consist of two parts, one of which is pro-
portional to the linear compression in the direction of the pressure, while the
other is proportional to the diminution of volume. As this hypothesis admits
two coefficients, it differs from that of this paper only in the values of the
coefficients selected. They are denoted by K and h, and K^fi — ^m, k = m.

The theory of Professor Stokes is contained in Vol. vin. Part 3, of the
Cambridge Philosophical Transactions, and was read April 14, 1845.

He states his general principles thus : — " The capability which solids possess
of being put into a state of isochronous vibration, shews that the pressures
called into action by small displacements depend on homogeneous functions of
those displacements of one dimension. I shall suppose, moreover, according to
the general principle of the superposition of small quantities, that the pressures
due to different displacements are superimposed, and, consequently, that the
pressures are linear functions of the displacements."

THE EQUILIBRIUM OF ELASTIC SOLIDS. 33

Having assumed the proportionality of pressure to compression, he proceeds
to define his coefficients.— "Let -^8 be the pressures corresponding to a uniform
linear dilatation 8 when the solid is in equilibrium, and suppose that it becomes
mA8, in consequence of the heat developed when the solid is in a state of rapid
vibration. Suppose, also, that a displacement of shifting parallel to the plane
xy, for which 8x = kx, Sy= - hj, and hz = 0, calls into action a pressure - Bk
on a plane perpendicular to the axis of x, and a pressure Bk on a plane
perpendicular to the axis of y; the pressure on these planes being equal and
of contrary signs; that on a plane perpendicular to z being zero, and the tan-
gential forces on those planes being zero." The coefficients A and B, thus

defined, when expressed as in this paper, are ^ = 3/x,, B = -.

Professor Stokes does not enter into the solution of his equations, but gives
their results in some particular cases.

1. A body exposed to a uniform pressure on its whole surface.

2. A rod extended in the direction of its length.

3. A cylinder twisted by a statical couple.

He then points out the method of finding A and B from the last two cases.

While explaining why the equations of motion of the luminiferous ether are
the same as those of incompressible elastic solids, he has mentioned the property
of jylasticity or the tendency which a constrained body has to relieve itself
from a state of constraint, by its molecules assuming new positions of equi-
librium. This property is opposed to Hnear elasticity ; and these two properties
exist in all bodies, but in variable ratio.

M. Wertheim, in Annales de Chimie, 3« Sdrie, xxiii., has given the results
of some experiments on caoutchouc, from which he finds that K=k, or fi = ^m;
and concludes that k = K in all substances. In his equations, fi is therefore

The accounts of experimental researches on the values of the coefficients
are so numerous that I can mention only a few.

Canton, Perkins, (Ersted. Aime, CoUadon and Sturm, and Regnault, have
determined the cubical compressibilities of substances; Coulomb, Duleau, and
Giulio, have calculated the linear elasticity from the torsion of wires; and a
great many observations have been made on the elongation and bending of beams.

VOL. I. ^

34 THE EQUILIBRIUM OF ELASTIC SOLIDS.

I have found no account of any experiments on the relation between the

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