Horace Lamb.

An elementary course of infinitesimal calculus online

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10. If ds, aV be corresponding elements of a curve and its
inverse, ds : ds' - r 2 : k 2 = A? : r* f

where r, r' are the radii.

11. The pedal of a parabola with respect to its vertex is the
cissoid (Art. 133 (16)).

12. If two tangents to a curve make a constant angle with
one another, the locus of their intersection (P) touches the circle
through P and the two points of contact.

13. Prove that the area of a pedal curve is given by the
formula \ fp*d\l/.

14. Prove that the arc of a pedal curve is expressed by


15. The area of the pedal of an ellipse, the centre being pole,
is \ir (a 3 + ft 2 ),

where a, b are the semi-axes.

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16. The pedal of the hyperbola

a* 6*
with respect to the centre consists of two loops, each of area

J(a a + 6 f ) + J(a»-.6 a )tan- 1 J.

17. If p , p x be perpendiculars on the tangent to a curve
from the origin of (rectangular) coordinates, and from the point
(xy f y x ) respectively, prove that

Pi =/> - ^ cos f- yi »n T*f
where ^ is the inclination of the perpendiculars to the axis of x.

18. If A v , A x be the areas of the pedals of a closed oval
curve with respect to the origin and with respect to the point
(iCj, y x ), both these points being within the curve, prove that

AOOBfty-fyJ p sin ^r + tt (a^ 1 + yi 1 ).

19. Prove that the locus of a point such that the pedal of a
given closed oval curve with respect to it as pole has a given
constant area is a circle; and that the circles corresponding to
different values of the constant are concentric.

Also that, if be the common centre, the area of the pedal
with respect to any other point P exceeds the area of the pedal
with respect to by the area of the circle whose radius is OP.

20. The negative pedal of the parabola

with respect to the vertex is the curve

27ay a = (*-4a) 8 .

21. In what case is


22. Prove that the curve for which

p = a sin iff cos ifr
is the astroid

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23. State what property follows by differentiation with
respect to the arc («) from the equation

«* + !* = £!
and verify the result geometrically.

24. Prove the following construction for the normal at any
point P of a Cassini's oval : In PS, PS' take points Q, Q', re-
spectively, such that PQ = PS', and PQ' = PS ; the line joining P
to the middle point of QQ 1 is the required normal.

25. A system of parallel rays is to be reflected so as to pass
through a fixed point ; prove that the reflecting curve must be a

26. A system of parallel rays is to be refracted so that their
directions pass through a fixed point ; prove that the refracting
curve must be a conic, and that the eccentricity of the conic will
be equal to the ratio of the refractive indices.

27. Prove that the equation of a Cartesian oval, referred to
either focus as pole, is of the form

r 2 - 2 (a + b cos 6) r + c 2 = 0.

28. Prove that a Cartesian oval is necessarily closed, if we
except the case where the curve is a branch of a hyperbola.

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160. Measure of Curvature.

As regards the applications of the Calculus to the theory
of plane curves we have so far been concerned chiefly with
the direction of the tangent at various points. We have not
considered specially the manner in which this direction varies
from point to point.

The subject of curvature, to which we now proceed, can be
treated from several independent stand-points, and although
all the methods lead to identically the same formulae, it is
important for the student to observe that they are in their
foundations logically distinct.

In the first of these methods*, we begin by defining the
' total ' or • integral ' curvature of an arc of a curve as the
angle (Syfr) through which the tangent turns as the point of
contact travels from one end of the arc to the other.

The ' mean curvature ' of the arc is defined as the ratio of
the total curvature to the length (&) of the arc ; it is there-
fore equal to

The ' curvature at a point ' P of a curve is defined as the
mean curvature of an infinitely small arc terminated by that

* Other methods are explained in Arts. 158, 154.

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150] CURVATURE. 395

point. In conformity with the previous notation it is
denoted by

£ o>

In a circle of radius R we have & = Rlfy, and therefore

ds ~~R'

i.e. the curvature of a circle is measured by the reciprocal of
its radius. Hence, if p be the radius of the circle which has
the same curvature as the given curve at the point P, we

>-£ »•

A circle of this radius, having the same tangent at P,
and its concavity turned the same way, as in the given curve,
is called the 'circle of curvature/ its radius is called the
' radius of curvature/ and its centre the ' centre of curvature/

The length intercepted by this circle on a straight line
drawn through P in any specified direction is called the
'chord of curvature' in that direction. If be the angle
which the direction makes with the normal, the length (q) of
the chord is given by

2 = 2/>cos0 (3).

If £, 17 be the rectangular coordinates of the centre of
curvature, we have by orthogonal projections

f = # — psin^r, *7 = y + pcos^r (4^

provided the zero of yfr be when the tangent is parallel to the
axis of x.

The centre of curvature is the intersection of two con-
secutive normals to the given curve. For if PC, P'C be the
normals at two consecutive points, including an angle 8^,
and if Bs be the arc PP\ then drawing the chord PP' we
have (see Fig. 114)

OP sin OPT
PP'" sinty '

08 sinoy oy

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When P' is taken infinitely near to P, the limiting value
of each factor on the right hand, except the last, is unity.
Hence, ultimately, CP = ds/d^ = p.

Fig. 114.

In modern geometry a curve is regarded as generated in a
two-fold way, first as the locus of a point, and secondly as the
envelope of a straight line (see Art. 158). Considering any
continuous succession of these associated elements, the straight
line is at any instant rotating about the point, and the point is
travelling along the straight line; and the curvature df/ds
expresses the relation between these two motions.

If at any point the curvature is zero, the rotation of the
tangent is momentarily arrested, and we have what is called a
'stationary tangent/ The simplest instance of this is at a point
of inflexion (Art 68), where the direction of the rotation of the
tangent is reversed after the stoppage.

If at any point the radius of curvature (ds/cty) vanishes, the
motion of the point along the line is momentarily arrested, and
we have a ' stationary point.' The simplest instance of this is at
a ' cusp ' such as we have met with in Figs. 75, 79, 84, 88, etc.
The direction of motion of the point is in such cases reversed
after the stoppage. In the examples of Art. 133 a cusp was
regarded as due to the evanescence of a loop: this shews in
another way why the radius of curvature should vanish there.

The consideration of curvature is of importance in numerous
dynamical and physical problems. For example, in Dynamics, if
the force acting on a moving particle be resolved into two

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150-151] CURVATURE. 397

components, along the tangent and normal to the path, re-
spectively, the former component affects the velocity, and the
latter the direction of motion. If from a fixed origin we draw a
vector OV to represent the velocity at any instant, the polar
coordinates of V may be taken to be v, \ft, where v = d8Jdt.
Hence the radial and transverse velocities of V will (see Art. 110
(8)) be

a"* "3 < 5 >>

respectively. These are the rates of change of the velocity
estimated in the direction of the tangent and normal to the path
of the particle. Since

V dt =V di*-p (6) '

the latter component is equal to the product of the curvature
into the square of the velocity.

151. Intrinsic Equation of a Curve.

The formula

P = d+ (1 >

is of course most immediately applicable when the relation
between 8 and yfr for the curve in question is given in the

•-/<*) (2)-

This is called the 'intrinsic' equation of the curve, for the
reason that its form does not depend materially on space-
elements extraneous to the curve. The only arbitrary
elements are the origin of 8 and the origin of ^r, and a change
in either of these origins merely adds a constant to the
corresponding variable.

Ex. 1. In the catenary we have

« = atan ty (3),

whence p = asec 3 ^ = y sec^r (4),

the notation being as in Art. 134. On reference to the figure
there given it appears that the radius of curvature is equal to the
normal PG .

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Ex. 2. In the cycloid (Art. 136) we have

8 = 4aBiu.\j/ (5),

and therefore p = 4a cos^r (6).

Hence in Fig. 84, p. 347, we have p = 2PI, or the radius of
curvature is double the normal.

A(a + b)b . a . mx

«= „ sm ^To^ ( 7 )>

Ex. 3. Again, in the epicycloid we have (Art. 137 (11))
i + b)b . a

'— Sill jr

a a + 26

and therefore

4(« + 6)6 a , 4(a + 6)6 1JL , ft ,

P = — - — sr- cos — oi^= — or- C08 W (8).

r a + 26 a + 26 T a + 2b ST ^ v '

Hence, on reference to Fig. 86, p. 351, it appears that

2(a + 6)

p ~ a+26 ^ W '

where PI is the length of the normal between the tracing point
and the fixed circle.

If the intrinsic equation be not known, we may employ
one or other of the formulae of Art. 152; or we may, in
particular cases, have recourse to special artifices.

Ex. 4. In the parabola y* = iax we have, by Art. 53 (9),
y = 2acot^r (10),

whence sin \b = -^ = — r-r-: -7- »

T as sin* \ff as

p = -^ <">•

the negative sign representing the fact that \ff diminishes as 8

Ex. 5. If the ellipse

x = acos<f>, y = 6sin </> (12)

be supposed derived by homogeneous strain (or by orthogonal
projection) from the circle

x = acoB<f>, y = asin </> (13),

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151] CURVATURE. 399

we have -77 = ft (14),

where ft is the semi-conjugate diameter. For the element of arc
is altered from a8</> to $8, and the parallel radius from a to ft.
Also since \fPty and £a 8 8# represent corresponding elements of

area, we have fifty = - x a'fy,

<%> F /1K\

^=S < 15) '

Hence p== _ = __ s= _ (16).

If p be the perpendicular from the centre on the tangent-line,
we have p/3 = oi, so that our result may also be written

P=p' or '"p" ( 17 >-

Since p 8 = a 2 cos 8 ^ + If sin 8 ^r = a 8 (1 - e 8 sin 2 \ft) f
the last form is equivalent to

1 -e 2

' =a (i-«w*)i (18 >-

This formula leads to an important result in Geodesy. The
figure of the Earth being taken to be an ellipsoid of revolution,
the expression for the radius of curvature in terms of the latitude
i//, is, if we neglect e 4 ,

^ = a(l-6 2 + |« 8 sin 8 ^) = a(l-|€-f€COs2^)...(19),

where c = (a — b)ja = Je 8 ; that is, c denotes the ' ellipticity ' of the
meridian. Integrating (19) we find, for the length of an arc of
the meridian, from the equator to latitude if/ y

« = a(l-j€)^-ja€sin2^ (20).

Ex. 6. In the equiangular spiral (Art. 140), we have

^r = + a (21),

whence d\j//ds = dOjds = (sin a)/r,

or P = -^— (22).

r sina v /

Hence the radius of curvature subtends a right angle at the

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152. Formula for the Radius of Curvature.

The expression dyfr/ds for the curvature is easily translated
into a variety of other forms.

1°. In rectangular Cartesian coordinates, we have

and therefore

»*+-& <»>

8ecr ds ds\dx) da? ds COSY <&»'
whence -= - /J.aAi (2)-


This form shews, again, that the curvature vanishes at a point
of inflexion.

Ex. 1. In the catenary

y = acoshx/a (3)

we have

dx a' dx* a a* \dxj a 9

whence p = a cosh 2 xja = y^ja (4).

Since y ^ a sec ^r, this agrees with Art. 151, Ex. 1.

When dyjdx is a small quantity the formula (2) gives,

\~% <«.

the proportional error being of the second order.

The form (5) is an obvious transcript of dtyjds, since when iff
is small we may write dyjdx (= tan \j/) for ^r, and djdx for d/ds.


The approximate formula (5) has many important practical
applications, e.g. to the theory of flexure of bars (see Art. 130).
If the axis of a be parallel to the length, and if y be the lateral

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152] CURVATURE. 401

deflection at any point, the * bending moment,' or 'flexural couple,

"*% < 6 >-

2°. It was proved in Art. 147 that the projection (t) of
the radius on the tangent is given by

<-$ ; <'>

If OU, OU' be the perpendiculars from the origin on two
consecutive normals PC, P'C, and if OU' meet PC in iV, we
have, ultimately,

0U'-0U=U'N=CNh+, or ht = CNhf.

The limiting value of CU or CN is therefore dt/dyfr, whence

P-ap-oi + ou-p + £- P + *jL (8).



Fig. 115.

3°. With the notation of Arts. 110, 147 we have

t , dr

- = cos6= j-

r ^ as


d*^_ds dr dp _ ds dr
ctyr "" dr dp d^r "" drdp'


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this gives P = r fy < 10 )'

a form which is very convenient of application when the
tangential-polar equation (Art. 143) is given.

Ex. 2. In the parabola

r=?l* (11)

we have p = r |=^ = ^ (12).

Ex. 3. In the central conies we have (Art 143, Ex. 4)

*$=»***+ < 18 >..

and therefore P = ± — (14).

Ct Art. 151, Ex. 5.

153. Newton's Method.

In another method of treating curvature, employed by
Newton*, a circle is described touching the given curve at P,
and passing through a neighbouring point Q on it, and we

investigate the limiting value of the radius of this circle
when Q is taken infinitely near to P.

• Principia, lib. i., prop, vi., oor. 8.

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1 52-1 53] CURVATURE. 403

We can easily shew that in the limit the circle becomes
identical with the ' circle of curvature ' at P, as defined in
Art. 150. For if G be the centre, then, since CP = CQ, there
will be some point (P') on the curve, between P and Q, such
that its distance from is a maximum or minimum, and
therefore* such that CP' is normal to the curve. In the
limit P' approaches P indefinitely, and C t being the inter-
section of consecutive normals, will coincide with the ' centre
of curvature ' (Art. 150).

Newton's method leads to a very simple formula for the
radius of curvature. Let Q'QT be drawn perpendicular to
the tangent at P, meeting the circle again in Q', and the
tangent in T. Since

TP* = TQ.TQ\

we have 2p = lim TQ' = lim ^ (1).

If Q'QT be drawn at a definite inclination to the normal
at P, instead of parallel to this normal, the limiting value of
the same fraction gives the chord of curvature in the corre-
sponding direction. It occasionally happens that the chord
of curvature in some particular direction can be found with
special facility ; the radius of curvature can then be inferred
by the formula (3) of Art. 150.

Ex. 1. In the parabola, let QR be a chord drawn parallel
to the tangent at P, to meet the diameter through P in V; see
Fig. 117. We have, then, from the geometry of the curve,

QV* = iSP.PV,
where S is the focus. Hence, for the chord of curvature (q)
parallel to the axis,

* = lim££=4«> (2).

If $ be the angle which the normal at P makes with the axis,
we have cos = SZ/SP, where SZ is the perpendicular from the
focus on the tangent at P. Hence

'-^••••-•Sr'SS ■■•<«•

since SZ* = SA . SP, A being the vertex.
* See Art. 55, Ex. 2.


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[CH. X

Pig. 117.

Ex. 2. In the ellipse (or hyperbola), if QR y drawn parallel
to the tangent at either extremity of the diameter PCP\ meet
this diameter in V, we have

QV*:PV. rP' = CD*:CI»,

Big. 118.

where CD is the semi-diameter conjugate to CP. Hence, for the
chord of curvature (q) through the centre,

r QV* .. CD* __, Q CD*


CP %



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153] CURVATURE. 405

If GZ be the perpendicular from the centre on the tangent at
P, and $ the angle which CP makes with the normal, we have
cos $ = CZjCP, and therefore

p = J?sec0 = -£jr (5),

in agreement with Art. 151 (17).

Again, if ff be the inclination of either focal distance to the
normal at P, it is known that cos & = GZjGA, where A is an ex-
tremity of the major axis. The chord of curvature (q r ) through
either focus is therefore given by

S^ = 2 P 008^ = 2^ (6).

Ex. 3. To find the radius of curvature (p ) at the vertex of
the cycloid

» = a(0 + sin0), y = a(l -costf) (7).

We have

whence p = lim^o «- = 4a (8).

Newton's method, combined with the result of Art. 66, 2°,
leads to a general formula for the chord of curvature parallel
to the axis of y, and thence to the Cartesian expression for
the radius of curvature. Denoting the chord in question by
q, we have, in Fig. 43, p. 155,

^ = lim^ = lim^.cos-f = i^(a)cos«^...(9),

where yfr is the inclination of the tangent at P to the axis of
x. Since

q = 2pco8^fr } tan^r = £'(a),
it follows that

J-fw-^-p-jfl^ w

This is identical, except as to notation, with the formula
(2) of Art. 152.

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164. Osculating Circle.

A slightly different way of treating the matter is based
on the notion of the ' osculating circle/ If Q and R be two
neighbouring points of the curve, one on each side of P, we
consider the limiting value of the radius of the circle PQR 9
when Q and JR are taken infinitely close to P.

We can shew that if the curvature of the given curve be
continuous at P, this circle coincides in the limit with the
* circle of curvature/ For if C be the centre of the circle
PQR, there will be a point P', between P and Q, such that
CP' is normal to the given curve, and a point P", between P
and R, such that CP is normal to the curve. Let P'C and
P"C meet the normal at P in the points C and (7",
respectively. Under the condition stated, C and C" will
ultimately coincide with the centre of curvature at P, and,
since CO < C'C", C will & fortiori ultimately coincide with
the same point.

Fig. 119.

Since, before the limit, the circle PQR crosses the given
curve three times in the neighbourhood of P, it appears that
the osculating circle will in general cross the curve at the
point of contact. See Fig. 123, p. 422.

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154] CURVATURE. 407

Ex. If in Fig. 1 17, p. 404, the circle PQR meet PV in W, we

have QV. VR = PV. VW f and therefore VW=4SP.

Hence the chord of curvature parallel to the axis of the parabola

A similar argument may be used to find the chord of
curvature through the centre, in the case of the ellipse (Fig. 118,
p. 404).

If in Fig. 42, p. 153, QV meet the circle through P, Q, P'
again in W, we have


and therefore, for the chord of curvature of the curve
y = (f>(x) t parallel to the axis of y,

- = limJfp = Um^pCos 8 ^ = |^ / (a)(M)B»^,
as in Art. 153 (9).


1. Prove that the circle is the only curve whose curvature is

2. Prove that the intrinsic equation of an equiangular spiral
is of the form

3. The intrinsic equation of the tractrix is

8 = a log sin if/.
Prove that in the tractrix the curvature varies as the normal.

4. By differentiation of the formulas

dx . dy . .

d*x (dy d*y jdx

... 1 d*x (dy

provethat _ = __/_ =

d# di'


p*-\d*J + Uv*


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5. If a curve be defined by the equations

* = *(& y =./(«),
prove that " l^jT^L*

p (x'*+ y y>

where the accents denote differentiations with respect to t.

6. Apply the preceding formula to the cases of the ellipse

x = acoB<f>, y = b sin ^,
and the hyperbola

x = a cosh u 9 y = b sinh u,

7. Prove that the curve whose intrinsic equation is

8 = &sin^r
is a cycloid. (Use the method of Art. 134 (3).)

8. Given that in the * catenary of equal strength '

p = kaec\l/,

where ^ is the inclination to the horizontal, prove that if the
origin be at the lowest point

x = fyt y = k\ogaeci//,
the axes of x and y being horizontal and vertical.

9. Given that the intrinsic equation of a curve is

8-k sin* ^r,
deduce the Cartesian equation


10. If the coordinates x, y of a point on a curve be given
functions of t> prove that

d % x d*s f l/<fo\* . f
d*y dh . . l/*\«

d? = de m + %(*)"»*'

and give the kinematical interpretation of these results.
Hence shew that


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11. Shew how to express the coordinates re, y of a point on a
curve whose Cartesian equation is given in terms of the inclina-
tion (\j/) of the tangent, and prove that


12. Prove that, in the astroid

x = a cos* By y = a sin* 6,

and thence shew that

p = 3a sin 6 cos 9.

13. Prove from the Cartesian formula of Art. 152 (2) that
in the rectangular hyperbola xy = &

14. Also that, in the ellipse

p = — ^6 *

15. Also that, in the hyperbola

* £-\

^ ab •

16. Also that, in the parabola y' = Aax,

2(a + x)i

17. Also that, in the semi-cubical parabola ay* = z>,

(4a + 9rc)l;e*
'' 6a '

18. Also that, in the cubical parabola a*y = x*,


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19. Also that, in the astroid

p = -3(aa?y)*.

20. Shew by differentiating the expression


for the square of the distance of a variable point (x, y) of a curve
from a fixed point (£, rj) that when this distance is stationary
the point (&, y) must be at the foot of a normal from (£, 17) to
the curve.

Also that the distance is then a minimum or maximum
according as the point (£, rj) is nearer to or further from the curve
than the centre of curvature.

21. The curvature at any point of an ellipse is

rr' '

when r, r are the focal distances, and <f> is the angle between

22. In the rectangular hyperbola r* cos 20 = a*,

p = t*la\

23. In the lemniscate r 1 = a 3 cos 20,

24. In the curve r* = a m cos m$ ,

25. Apply the formula p = rdr/dp to find the radius of
curvature at any point of an epicycloid. (See Ex. 25, p. 376.)

Examine the case of the involute of a circle.

26. If the equation of a curve be given in the form r =f(p),
the chord of curvature through the pole is

9 dr
2 *V

cardioid is 1 J times the radius vector.

Prove that the chord of curvature through the pole of a


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27. Prove that the chord of curvature, through the pole, at
any point of the curve *■"* = a™ cos m$ is 2r/(wi + 1).

28. Prove that the curvature of the pedal of a curve r =/{p)
with respect to the origin is

2 p
where r, p, p refer to the original curve.

29. Prove that the curvature at any point of the pedal of an
ellipse of semi-axes a, b with respect to the centre is equal to

3 a" + P

where r is the radius vector of the corresponding point of the

30. Prove the formula

1(1 1 /dry c?V| f MVV*
p-\r r\ds)- d*r I \ds)f '
and apply it to deduce the conclusions of Ex. 20.

31. Prove that in polar coordinates the condition for a
stationary tangent is

dhi A

^ + W = °'
where w=l/r.

32. From the formula

^fl + ^fl + cot- 1 ^

T r rod

deduce the formula for curvature in polar coordinates :

- ©♦•)*{• ♦ear-

where u = 1/r.

33. With the same notation, prove that the chord of curvature
through the origin is



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34. The radius of curvature of the curve

ay* = (x-a)(x-Pf
at the point (a, 0) is (a - 0)'/2a.

35. Prove by Newton's method that the radius of curvature
at the vertex of the catenary

y = a cosh xja
is equal to a.

36. The radius of curvature of the curve

y* = a*(a + x)jx
at the point ( - a, 0) is \a.

37. The radius of curvature of the ( witch '

y* = a 2 (a-x)lx
at its vertex is \a.

38. The radii of curvature of the trochoid

x = a$ + k sin 0, y = a — k cos
at the points where it is nearest to and furthest from the base are


30. Prove that in the meridian-curve (t* = a* cos 0) of the
1 solid of greatest attraction' (see Ex. 18, p. 375) the radii of

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