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I. F. Quinby Horatio Nelson Robinson.

A new treatise on the elements of the differential and integral calculus

. (page 14 of 22)

d , rfr

r' -â€”r COS. ((? â€” ^0 = ^ â€” â€¢
dd ^ ^ do

Adopting a notation like that in the case of the tangent, (4)

becomes

do

U' = U COS. {d' â€” (?) â€” t^2 gin^ (^0f __ ^^ ^5^^

CbU

158. Let P be the polar point to which is referred the
curve BMSj and through P draw
vector PM; then MT being the
tangent, and MN the normal, to
the curve at the point il/^ the lines
MT, PT, MN, and PiV, are, re-
spectively, the polar tangent, sub-
tangent, normal, and sub-normal. "^T

To find the formulas for the lengths of these lines, put angle

PMT=i ^f and resume the equation tan.(3 z= r - - (Art. 156),

dr

do 1 r .

making - = â€” : whence tan. B = â€”. from which we find
^ dr r' r' '

r' T

cos. B = â€” sin./3=: â€” â€” â€”

Then the triangle PMT gives

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270 DIFFERENTIAL CALCULUS,

COS. PMT cos.fj r' ^ \ ^ \dr/'

FT=T, = Pif tan. FMT= r^- = Â±r'' *?-,

r' dr

PM PM

â– =J^.

COS. PMN sin. PMT ^ "^ ^ "^

PiVr:^ iV; zn PJf tan. PilfiV^z= r cot. ^ = r - = r' = ziz ^^.

^ r do

The polar sub-tangent is considered positive when it is on
the right, and negative when on the left, of the line PM; the
eye being supposed at P, and looking fromP toward M, The

dv

sign of the sub-tangent will then be the same as that of -^;

that is, positive when r is an increasing function of ^, and neg-
ative when r is a decreasing function of 0,

159. An asymptote to a curve referred to polar co-ordi-
nates is a tangent line, the polar sub-tangent to which remains
finite when the radius vector of the point of tangency becomes
infinite. Hence, to find the asymptotes to a polar curve, we

must seek the values of 6, which make r infinite while r^ -r-

' dr

remains finite. If m be a value of which satisfies these con-

ditions, the asymptote may be constructed by drawing through

the pole a line, making, with tlie initial line, the angle m, and

another line at right angles to this through the same point ;

laying ofi" on the latter, to the right or to the left according as

r^ T- is positive or negative, the distance represented by r^ -5-,

and through the extremity of this distance drawing a line par-
allel to the first line. The line last drawn will be the asymp-
tote.

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POLAR CO-ORDINATES,

271

Example. In the hyperbolic spiral, sa called because of tlio
similarity of its equation r = -, or rO =z a, to that of the hy-
perbola referred to its asymptotes, we have

,dd

^2

^^:7:: = -^X - = -a.

rfr ~~ a' ' ' ' dr 0'^ a

Hence the sub-tangent is constant, and equal to â€”a: but
0=^0 gives r =s: 00 ; whence the line parallel to the polar axis at
the distance from it equal to â€” a is an asymptote to the curve.

This curve, beginning at
an infinite distance, contin-
ually approaches the pole,
making an indefinite num-
ber of turns around without
ever reaching it.

IGO. When the curve in the vicinity of a tangent line at
any point, and the pole, lie on the same side of the tangent,
the curve at that point is concave to the pole ; but, if the
curve and the pole lie on opposite sides of the tangent, the
curve in the vicinity of the point of tangency is convex to
the pole.

Let Pp = j9 be a perpendicular
from the pole on .the tangent to the
curve at the point {0, r) : then it
is plain, that, if the curve at this
point is concave to the pole, p will
increase or decrease as r increases
or decreases ; that is, p is an in-
creasing function of r ; and, on
the contrary, if the curve is convex to the pole, ^ is a decreas-
ing function of r. Hence, when the curve is concave to tha

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272

DIFFERENTIAL CALCULUS.

dp . , dp

pole, ~ must be positive ; and, when convex, -/- must be neg-
dr dr

ative (Art. 51).

Cor. At a point of inflexion, the curve with reference to the
pole must change from concave to convex, or the reverse.

dp
Hence, for a point of inflexion, â€” = or oo .

We have (Art. 158)

sin. PMT= sin. ^ : *"

p = rsin.^ =

Make u = -: then 3^ = ^'-r:

r do r* dd

â€¢â€¢â€¢ 7 = ^' + U)'^^^ - i;^"rf. = (" + WVd^'
dedu du ^ V^ doy

dp ^dp du ]_dp_p^/ .d^u\

hence di^'^dud^^'''^du'~r^V'^W'/'

d^u
Therefore, at a point of inflexion, ^ + -^^ will generally change

its sign.

y" 161* Differential co-eflScient of

y s the arc of a plane curve.

Let F{x, y) = or y =/{x)

be the equation of the curve

BPS referred to the rectangular

axes OXf Oy ; and take, in this

curve, any point, P, of which the

co-ordinates are x, y. Denote the length of the curve estimated

yo

y

K

M'

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DIFFERENTIAL CO-EFFICIENTS OF ARCS. 273

from a fixed point to the point Phj a ; then, if x be increased
by MM' = AXf 5 is increased by the arc PP' = as, and it is

required to find the limit of the ratio â€” , or the differential

co-eflScient of the arc s, regarded as a function of x.

The tangent line to the curve at the point P meets the or-
dinate P'M', produced, if necessary, at Q; and makes, with the
axis of X, an angle of which the tangent, sine, and cosine are
respectively

Now, if, within the interval Ax, the curve is continually con-
cave or continually convex to the chord PP', it is evident/
that arc PP' > chord PP', and arc PP' <PQ+ QP'.

PN

But chord PP' = Vax'' + A.y^ PQ = 7r^r=^Vl+y'\

COS. QPN ' ^ '

QN=zPNta,n.QPN=y'Ax; .-. QP' =z7/ax^aj/:

hence, substituting in the preceding inequalities, we have

A8 > Vax^ + Ay2, A8<:^AxVl + y'^ + J/' AX â€” Ay.

Therefore

A5 . . , A?/2 AS ^ , , . Ay

AX ^ ~ AX^ AX^^ ^^y ^^ AX

At the limit, the second member of each of these inequalities
reduces to Vl + y'^ = ll -[- (-^ j : hence we have

,. AS ds

lim. â€” = â€”

AX ax

=vr+^ = jn-(|)"

for the difiFerential co-eflScient of the arc regarded as a func-
tion of the abscissa. This must be understood as expressing

ds
only the absolute value of -j- ; for the arc may be an increas-

CLX
85

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274 DIFFERENTIAL CALCULUS.

ing or a decreasing' function of the abscissa, according as it is
estimated in the direction of x positire or x negative : hence
the general value should be written

the sign -|- to be taken if 8 increases with aj, and the sign â€”
in the opposite case.
Cor. 1. Since

PP' ' VAaj^ + Ay*

Vl + y'' + y'-

4y

= lim ^= 1,

M^'

and the arc PP' is always included between the chord PP' and

the broken line PQ + QP\ it follows that lim. â€”. â€” = 1 ;

^ ' \^Lx'' + ^y^ '

and hence, when the arc is inftnitely small, it and its chord

become equal.

Cor, 2. Squaring both members of the equation

and multiplying through by ( â€” j , we find

Now, if a; and y are both functions of a third variable Â«, then

dx
dx __dx da ^ ^ dx _^ dz

dz

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DIFFERENTIAL CO-EFFICIENTS OF ARCS. 275

dy dy ds . dy dz

dz ds dz' * * ds ds '

di

doc dti
These values of -y-, -â€”, substituted in the preceding equa-
(XS cts

tion, give

S-=M(S)+(S)T-

162. Denote by a and ^ the angles that the tangent line at
the point P (figure of last article) makes with the axes of x
and y positive ; then

COS. a = lim. â€”/ ^ = . = â€” (Arts. 41, 161),

^/^x' + ^y' ^^l+y'" ds ^

A?/ dy dx dy

COS. jg = hm. ,- / ^ = -g - = -^ (Art. 42);
VAx'' + Ay^ dx ds ds ^ ^'

or, more generally, writing the sign zb before \^Ax'^-\-Ay'^^

. dx ^ dy

cos. a = Â± -~- , COS. /5 z= dt - ,
ds ds

in which the upper sign or the lower sign is to be used ac-
cording as the angle is that made with the axis by the tangent
produced from the point P in the direction in which the arc
increases from that point, or the opposite This refers to the
algebraic signs of cos. a, cos. ^: their essential signs are deter-
mined by combining their algebraic signs with the essential

cdx dy
''^""''^ds^ds'

163. Differential co-efl5cient of an arc referred to polar
co-ordinates.

For the transformation of rectangular into polar co-ordinates,

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27G

DIFFERENTIAL CALCULUS.

we have re = r cos. 0, y =^r sin. 0. We also have (Arts. 42,
161)

dO^di'dd'^ do S "^ Vdi/ "" N\rf^/ "^ \d^/ *

_ rfoj c?r . dy . dr

But -:;- = COS. ^ -, r sm. 0, - , = sin. ^ -;- + r cos. d;

do do 'do dd^ ' -

therefore -7- = ^ p ^ + ( "r P ai^d, in like
do \ \dol '

manner,

c?r

^ dOdr'~\ ^ \drj'

Cor. When /3 is the angle included between the radius vec-
tor of a curve at the point (r, 0) and the tangent line at that

do
point, we have (Art. 156) tan. |3 = r â€” ; and hence

and

sin. j3 =

COS. ^ =

do
dr

i+<

do
__ _ dr _ do
'~d~K^'~~d8 '~^da*

dr)

dr

1 ^
da'

164. DifiFerential co-efficient of the area of a plane curve.

The area enclosed by the arc
y ^y^ HP of a plane curve, a given

"* ordinate ABj the ordinate of any

point P of the curve, and the axis
Oa:,is obviously a function of the
abscissa of P, since the area va-
ries with the position of this point
on the curve.
Let ARPM = u, and give to a?, the abscissa of P, the increment

2Â£

M'

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DIFFERENTIAL CO-EFFICIENTS OF AREAS. 277

MM^ = Ax; then MPP^M* = aw is the corresponding incre-

* ... Au

ment of w, and we are to find the limit of the ratio â€” .

Aa?

Through P and P' draw parallels to Ox, and limited by tho
ordinates PM, P'M^; and suppose lx to be so small, that, be-
tween these ordinates, the ordinate of the curve constantly
increases or constantly decreases. The rectilinear areas
3IPP^M^j MN^P^M^j have yAa:, {y + Ay) Aa;, for their respective .
measures, and the curvilinear area MPP^M^ is constantly in-
cluded between these two ; that is,

Au^'yAx, Au<^{y-\-Ay)Ax:

AU

whence y < â€” < y + Ay ; or, by passing to the limit,

AX

du

â€” =:y, du=zydx.

If the co-ordinate axes are oblique, making with each other
the angle Â«, the above demonstration still applies, observing
that then the area Au lies between the areas of two parallelo-
grams, the sides of which are parallel to the axes ; and, since
the area of a parallelogram is measured by the product of its
adjacent sides multiplied by the sine of the included angle, we

du
should have â€”-^:Ly sin. Â«.
ax

165. When the curve is re-
ferred to polar co-ordinates, the
area considered is a sector em-
braced by a given radius vec-
tor PBj the radius vector Pif of
any point {0, r), and the arc BM.
Denote this area, which is a func-
tion of dy by u ; let 6 be increased
by Ad J by which the point Jf moves to 8^ and u is increased by

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278

DIFFERENTIAL CALCULUS.

the sector Pil//S'=Aw; and suppose A^ so small, that tlic radius
vector of the arc from Mto S is constantly increasing or con-
stantly decreasing. With P as a centre, and the radii vectores
P3I= r, PS=: r^j as radii, describe the arcs MI, 8K, limited
by these radii vectores. We have

sector PJ/7 <; a?^ < sector PSK;

or, since sector Pil/7i= o^^^^' ^"^ sector P/SÂ£'=^r'^ A (?,
But, at the limit, r' becomes equal to r: hence

The equations x = r cos. 0, 9/ = ?'sin. 0, give - = tan. ^: whence,
by differentiating with respect to Oj

X

dO

dx

do

x*

r^cos.^^

cos.'^(? cos.'-^^

- {xdy â€” ydx) = - r'^dd;

an expression in terms of rectangular co-ordinates for the dif

ferential of a polar area that is of frequent use.

IGG. Differential co-eflScient of the volume of a solid of

revolution.

If V represent the volmme
generated by the revolution of
the plane area ARPM about the
axis Ox^ and x be increased by
MM^ =zAx, the corresponding in-
crement AFof Fwill be thÂ« vol-
ume generated in the revolution
by the area MPP'M'. Now, if

A^ be so small, that y increases constantly from P to P',. A V will

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DIFFERENTIAL CO-EFFICIENTS OF VOLUMES. 279

bo included between the volumes of the cylinders generated

by the rectangles

MPNM, MN'P'M'.

HencCy denoting MP by y^ and M^P* by ^i, we have
Tty'^^x < aV< 7ty]Ax, or mf < â€” - < ^yf;

that is, â€” is comprised between the two quantities ny^^ ny^,

the second of which converges to equality with the first as

Lx diminishes. Hence, at the limit,

dV

â€”-z=.ny'^, dV=.7ty'^dx.

ox

1G7. Differential co-efficient of the surface of a solid of
revolution.

Let 8 represent the arc JRP (figure of last article), and 8
the surface generated by the revolution of this arc about the
axis Ox. For the increment M3P = ax of cc, 8 will be increased
by the arc PP^ = as ; and 8, by aS= the surface generated
by AÂ«. When ax is sufficiently small, the surface A/S'will be
comprised between the surface of the conical frustum gener-
ated by the chord PP% and the surface generated by the broken
line PQP^. The surfaces generated by the chord and by the
broken line are measured by

7t(2y + Ay)VAx^ + Ay\

^{^y+fx^^y^^^^+'^

â€” 27tyAy â€” n{Ayf:
hence, establishing the inequalities, and dividing through by
AXf we have

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280 DIFFERENTIAL CALCULUS.

At the limit, the second member of each of these inequalities
becomes equal to %iy ^ 1 + ( j~ ) â€¢ hence

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SECTION IV.

DIFFERENT ORDERS OP CONTACT OF PLANE CURVES. OSCULA-

TORY CURVES. â€” OSCULATORY CIRCLE. â€” RADIUS OP CURVA-
TURE. â€” EVOLUTES, INV0L1JTE8, AND ENVELOPES.

168. Suppose y = F{x)j y =i/{x), to be the equations of
the two curves RPN, Ii'PN\
which have a common point P;
and let us compare the ordi-
nates M'N, M'N', of these
curves corresponding to the
same abscissa OM^ = a; + A,
differing but little from the
abscissa OM = a; of the point P.
We have

M'N=z F{x + A), M'N' =f{x + A) : .
NN'=F(x + h)â€”f{x + Ji).

Developing each term in the vahie of NN' by the formula of
Art. 61, observing that F{x) =/{x) by hypothesis, we find

A" /d-F d-f\ A^*+^ f d-'^^F ^Â»+V \

AÂ»+"^

- (F^^'r''\x + Oh) â€” /C'* +2)(a; + ^^A)) ,

' 1.2.. .n + 2'
the last term of which may be written, â€”

86

281

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2S2 DIFFERENTIAL CALCULUS.

1.1^.. .71 + 1

i? being a quantity that vanishes with h: hence

AÂ» + i f d-'+^F d" + y "

If, in addition to F{x) :=/(x), wo Iiavo â€” = -â€” , the curves

have a common tangent, PT, at the point i^, and are said to
have a contact of the ^rst order : and if, at the same time,

-7â€”2 = ^' 2 7 ^^^^ contact is of the second order ; and, generally,

the contact is of the 7i^^^ order if n denotes the highest order
of the diiTerential co-efficients of the ordinates of the 'two
curves that become equal when in them the co-ordinates of the
common point are substituted.

169m When two curves have a contact of the 7i"* order, no
third curve can pass between them in the vicinity of their
common point, unless it have, with each of the two curves, a
contact of an order at least equal to the 7i"\ For y =^ F(x),
y ^zf(^x)j being the equations of two curves, BPN, R^PN^
(figure of the last article), whijch have at the point P a contact
of the 71*'* order, let y := q{x) bo the equation of a third curve,
B'^PIT"^ passing through P, and having with the first curve a
contact of tlic m*^* order, m being less than n. Then, 'by the
preceding article, we should have

^^ ^ , .dF dcp d"'F_d'^cp

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CONTACT, OSCULATIOXy Jc. 283

a and iZi being quantities which vanish with h: hence

i^^^(m + 2)...(n + l) ^ d^^F rf^+>

Since, as A converges towards the limit 0, B and Bi converge
towards the same limit, and reach it at the same time h does,

and since n > m, it follows that the ratio \^^^ can be made

as small as we please by giving to h a sufficiently small value ;
that is, when h is a very small quantity, NN' will be less than
AW''^, and the curve y = q,{x) cannot, in the vicinity of the
common point P, pass between the curves 7/ =F{x)j y =/(x).

It is evident that this reasoning holds when h is negative as
well as when it is positive.

Cor. When Ti is sufficiently small, the sign of the expression
for NN^ (Eq. h) will be the same as that of /i" + ^, and will
therefore change with that of h if n be even, but remain in-
variable if n be odd. Hence, if two curves have a contact of
an even order^ they will cross each other at the point of con-
tact, but not otherwise.

J 70. Osculatory curves. If the form of the function
F{x), and the constants which enter it, ai^e given, the equation
y = F{x) represents a curve fully determined in respect to
species, magnitude, and position ; but if the form of the func-
tion only is known, the constants which enter it being arbitrary,
the species of curve is all that the equation determines. Thus
the equation y = 6 it s/r^ â€” ^a; â€” of, when r, a, 6, are fixed in

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284 DIFFERENTIAL CALCULUS.

value, represents a circle that is completely known ; but if r, a,
and h are undetermined, the equation may represent every
possible circle lying in the plane of the co-ordinate axes.
It is then the equation of the species '* circle."

When a curve of a given species has a higher order of con-
tact than any other curve of that species with a given curve,
the former is said to be an dsculcUrix to the latter.

Suppose /(xi, yi, a, 6, c. . . ) = (1), involving n-\-l arbi-
trary constants, to be the equation of the species of curve that
is to be made an osculatrix to the curve of which y=zF(x) (2)
is the equation. By means of the n + 1 constants in (1), we
can satisfy the n + 1 equations

or, in other words, these equations will determine the values
of a, 5, c. . ., which, substituted in (1), will make it the equa-
tion of a curve having a contact of the n^^ order with the
curve represented by (2) ; and it will be an osculatrix, since,
in general, a higher order of contact cannot be imposed. "We
conclude from the above, that the number denoting the order
of contact of an osculatory curve is one less than the number
of constants entering the equation of the curve.

Example. The form of the equation of a straight line is
%fz=iax-\'h; and since this equation contains two constants,
a and 6, we may so determine them as to cause the line to
have a contact of the first order with a given curve at a given
point. Suppose y = F{x) to be the equation of the curve,
and that x=:m,y=:n, are the co-ordinates of the point ; then
the equations to be satisfied are

am-\-b = F(m)j a=iF^{m),
which determine a and 6.

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OSCULATORY CIRCLE, 285

17 !â€¢ Osculatory circle, or circle of curvature.

Assume the co-ordinate axes to be rectangular, and let
y = F{x) (1) be the equation of the given curve; then, since
{xi â€” ay + (yi â€” by =p^ (2) is the general equation of the
circle, and contains three constants, the osculatory circle will
have, with the given curve, a contact of the second order.
From (2) we get, by two successive diflferentiations,

a?i-o + (y.-6)^ = o

(3);

and, because the circle is to be the circle of curvature, we
must have

" ^" dx .dxi' dx^ dx\ ^ ''
These values of y^, -,-', Vt> substituted in Eqs. 3, give

._â€ž + (,_i)| = 0, 1 + (!)â– + to -6)g = (5):
therefore

y-b = - }f'') ,x-a = i \HAÂ± (6).

dx^ dx^

By substituting these values of y â€” 6, a? â€” a, for y^ â€” J, Xi â€” a,
respectively, in Eq. 2, we find

_J-(^'i

dx^
Eqs. 6 will determine the position of the centre; and Eq.
7, the length of the radius of the osculatory circle to the
given curve at any point. When the curve at the point of

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286

DIFFERENTIAL CALCULUS.

osculation is concave to the axis of x, as is the case if y is

positive and . ^ negative, then, to make p positive, wo must

take the minus sign written before tlie second member of (7).
The first of Eqs. 5 indicates that the centre of the circle is
in the normal to the curve at the point of osculation ; and from
the secoud of these equations we conclude that y â€” b and

â€”, must have opposite signs, and hence that the centre of

the circle is always on the concave side of the curve, since
y â€” 6 is the difference between the ordinate of the point of
contact and the ordinate of the centre of the osculatory circle.

In general, the contact of an osculatory circle is of the
second order, that is, of an even order ; and consequently it
crosses the curve at the point of contact, except at particular
points where the contact is of an order higher than the second.

The osculatory circle is often called circZc of curvature; and

172. As an application of the formulas of the preceding
article, let it be required to find the radius of curvature of
a conic section at any point of the curve. If the curve be
referred to one of its axes, and to the tangent through its ver-
tex, as co-ordinate axes, its equation
will bo

y^ rz: 2px -{- qx^^

which, by two differentiations, gives

dy p -f- qx

dx

y

d'y

y dx^

In the last of these, substituting for

â€” its value taken from the first, we
ctx

have, â€”

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Vdx^-^

y'

whence

d'y

p\

and for p

we

have

P

y'(i

P'

The numerator of this value of p is the cube of the normal
NN^; for from the triangle MNN^ we have

and P=p-^-

Therefore the radius of curvature at any point of a conic sec-
tion is equal to the cube of the normal at that point divided
by the square of the semi-parameter.

The value of p expressed in terms of the constants of the
equation, and the abscissa of the point 2^, is

((q + q')^' + 2p(l + <z)x +p'^i
P= - p

175. The equation of the tangent line to a curve at the
point (Xj y) being

dy , .

the expression for the length of the perpendicular p let fall
from the origin of co-ordinates on the tangent is

' dx

'^J^

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288 DIFFERENTIAL CALCULUS,

whence, by diflferentiation and redaction,

dp _ dx' j ' ^ W ^ d x d x' \ dx V

h(i)T

['^^dxjdx-' 1/ dy'

m

= r0^4-;

l^/ - .^,* P\ "*dx,

;^ (Art. 171).

And, if r be the distance from the origin to the point of tan-
gency,

r^ = x^ + y^-,.'.r% = . + y2:

and, substituting this value of a; 4-y;r^ in the expression for

'^, we have

dp 1 dr dr

dx p ax dp

174. If X and y are both functions of a third variable, Â«,

then

dij d'^ydx d'^xdy

f=f, fy <^<^'J~f<^' (I); (Art. 129);
dx dx dx^ /dx\^ \ /^ \ /y

ds \^ /

and these vahies of -,^, ^-^, put in the formula for p (Art.
171), give

^'-' d^dx^iPx dy ^ ^'
ds^ ds ds^ ds
Supposing 8 to be the arc of the curve estimated from a fixed

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point, we find,/rom the formula ;7~=^14"(;t') ^^ -^^' ^^^f

^ 1 1 _ (i^y dx __ d^ dy ,

^'^d^dx d^dy' ^"d? ds 'ds^ da ^ ^*
ds^ ds ds'^ ds

From (3), by diiferentiating, we get

dxd'^x dyd^y _

ds di^^ds ds'^ ^ ^'

Squaring (4) and (5), and adding, we find

p2 - \dsy "^ \ds^) '

d^ X d v
Eliminating -7-7, -r^t in turn, between (4) and (5), observing

that

we also find

'l)V(f)=..

fdÂ£

V

d'^y d^X

p dx "" dy
ds ds

175. To find the expression for the radius of curvature in
terms of the polar co-ordinates of the curve, we substitute

in the value of p, Art. 171, the values ^^ ;r-j ^ %} given in
Art. 131, thus getting

87

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290 DIFFERENTIAL CALCULUS.

and, when r = , we have
u

dd^^u^dd' dd^'^y} \dd) "~ w^ rf^'
and these values, substitnted in the above value of p, give

P

"â– (Â»+S)

176 â€¢ The chord of curvature at any point of a curve is the
portion of a secant line through that point that is included be-
tween the point and the arc of the circle of curvature at the
same point.

The chord of curvature that, produced if necessary, passes
through the pole, is obtained by multiplying 2p by the cosine
of the angle included between the radius vector and the
normal to the curve at the point ; but if r is the radius vec-
tor, and p the perpendicular let fall from the pole on the

tangent to the curve, ^ is the cosine of the angle included be-
tween the radius vector and the normal. But the value of ^
is readily found to be - ^ . - ; hence the chord of cur-

J"

,+C'y

vature through the pole is equal to

2"? = 2;, -^ = yj (Arts. 173, 175).

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17 7 â€¢ Denoting by a the angle which the tangent to a curve
at any point makes with the axis of abscissae, we have

tan. a = â€” ^ a = tan .-' -_^:
ax dx

therefore

da dx'^ dx dx'^

da

1 +

(!)â– "" l'+S)T'

 Using the text of ebook A new treatise on the elements of the differential and integral calculus by I. F. Quinby Horatio Nelson Robinson active link like:read the ebook A new treatise on the elements of the differential and integral calculus is obligatory