I. F. Quinby Horatio Nelson Robinson.

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

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fj — X -^^ z= a* — xa^Ia = oo for x = oo, but = for as == —
dx

and, for a; = — oo , / = a'la = — =: 0.

ax a'^



00



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ASYMPTOTES OF PLANE CURVES, 251

Hence the axis of a? is an asymptote to the curve, and ap-
proaches the curve without limit on the side of x negative*
In this reasoning, we have supposed a> 1. If a < 1, the axis
of X is still an asymptote ; but, in this case, the curve ap-
proaches the axis on the side of a? positive.

14:8. An asymptote to a curve may be defined as the line
which the curve continually approaches, but which it can
never meet. An investigation, based on this definition, may
be given that difiers somewhat from the preceding.

Let y z:^ ax-\- ^ be the equation of a straight line, and
y •=. ax -^ ^ -^ V the equation of a curve, v being a function of
X and yj which vanishes when x and y are made infinite, or,
at least, when one of these variables is made infinite ; then
the straight line is an asymptote to the curve. For the formic
la for the perpendicular distance from the point (a:, y) to the

y — (XX — Q V

straight line is — / - — - = .- — ^ when the point is a
va^ + l va^ + 1

point of the curve. Hence when v vanishes, as it does, by
hypothesis, for one or both of the values x = oo ^ y =z oo , the
straight line is an asymptote to the curve.

From the equation y=:az -{- B -\-Vj we have y =za-\- ^-J"— -

x ^ -

whence a is the limit of - when x and ?/ are increased without

x

y

limit. In general, for these values of x and y, - takes the

X

dy

dx dtt
form ~ : but its true value is -^ = -,-. So, also, B is the limit

du
of y — ax, and a is the limit of -^ ] therefore, in general, j3 is

ccx

dy

the limit of y ^x.

dx . _

When the value of a and j3 thus determined are substituted

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252 DIFFERENTIAL CALCULUS,

in the erjuation y =3 oo; -j- 13, it becomes the equation of an
asymptote to the curve.

140. When two curves are so related that the diflTerence
of the ordinate 8 answering to the same abscissa converges
towards zero as the abscissa is increased without limit, or the
difference of the abscissa© answering to the same ordinate
converges towards zero as the ordinate is increased without
limit, either curve is said to be an asymptote to the other.

Suppose we have a curve, the equation of which may be
made to take the form

then the curve represented by

y = aaj'» + aia;»-i+...+a,^ia? + a. (2)
will be an asymptote to the first curve.
So also is that represented by

y = ax» + aix"-^ H (- a^^^x + «„ + - (3),

X

and



It is obvious, also, that of the curves represented by Eqs.
1, 2, 3 ... , any one is an asymptote to all the others.

Example. Fiad the asymptotes, rectilinear and curvilinear,
of the curve represented by

ar' — ocy'^ + ay^ = 0, or y = =t l__? ,

\x — a

The value of y may be put under the form y=:dca;fl — _) j
and, expanding this by the Binomial Theorem, we have

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SINGULAR POINTS. 253

which expresses the true relation between x and y for points
of the curve far removed from the origin ; for then - is less

X

than 1, and the series !-{- — -{- o— 2 + * • • converges to a fixed
(luite limit. Whence we conclude that the curve has two recti-
linear asymptotes represented by the equation y=::h(x-\ - \

and an unlimited number of curves, having for their equa-
tions

which are asymptotes to it and to each other.

ISO. Singular points of curves are those points which
offer some peculiarities inherent in the nature of the curve,
and independent of the position of the co-ordinate axes.

Firstj Conjugate or isolated points are those the
co-ordinates of which satisfy the equation of the curve, but
which have no contiguous points in the curve.

Ex. 1. ic^ + y^ = can be satisfied only for a? = 0, y = 0,
and represents therefore but a single point ; i.e., the origin of
co-ordinates.

Ex. 2. y^ = x\x^^a^). This is satisfied by cc = 0, y = 0,
and therefore the origin belongs to the curve : but there are
no points consecutive to it ; for values of x between the limits
x=i -{-a, a; = — a, make y imaginary. Hence the origin is
an isolated point.

Ex. 3. ay^ — x^+ bx^ = 0.

Second, Points d^arrSt are those at which the curves
suddenly stop.

Ex. 1. y = xlx. Here x = 0, y = 0, satisfy the^ equatioui ;

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254



DIFFERENTIAL CALCULUS.




but negative values of x make y imaginary. The origin is
therefore a point d'arret,

1
Ex. 2. 2/ = c~x. If ic be indefi-
nitely great, and positive or nega-
tive, y approaches the limit 1 ; but
if X be indefinitely small, and posi-
tive, y approaches the limit 0;
while, for negative and very small values of x, y approaches
-{- CO , The curve will be composed of two branches, as rep-
resented in the figure, and will have for the common asymptote
to these the parallel to the axis of x at the distance 1.

Third, Points saillant are those at which two branches
of a curve unite and stop, but do not have a common tangent
at that point.

Example.



y =



X



From this we find



1 + e'



dy __
dx



l + e^^



+



1^




If X be positive, and be dimin-



ished without limit, both y and



dy
dx



ultimately become zero ; but if x
be negative, and be numerically
diminished without limit, we have

ultimatel V y = ^, 7^ = 1 • Hence
" ^ ^ dx

the origin is a point of the curve at which two branches unite

having difi'erent tangents ; one branch having the axis of x for

its tangent, and the other a line inclined to the axis of a: at an

angle of 45^.



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SINGULAR, POINTS. 255

Fourth, JPoints de rebroussement, or cusps, are

points at which two branches of a curve meet a common tan-
gent, and stop at that point. The cusp is of the first species
if the two branches lie on opposite -sides of the tangent, and
of the second species if the branches lie on the same side of
the tangent.

Fifth, Multiple points are points at which two or more
branches of a curve meet, but do not all stop, or at which at
least three branches meet -and stop.

Ex. 1. y^ = x^{l — x^) represents a curve of two branches
which cross at the origin, at which the equations of the tan-
gents are y = — x, y =i x,

Ex. 2. The equation y^ =:a;*(l — ic^) is that of a curve
composed of two branches which meet at the origin, and have
the axis of x for a common tangent. The origin is a multiple
point.

Sixth, A point of inflexion is one at which the curve
and its tangent at that point cross each other.

151. We will now establish the analytical conditions by
which the existence and nature of singular points in a curve,
if it have any, may be generally recognized ; omitting, for the
present, the case in which the first differential co-efficient of
the ordinate of the curve becomes infinite.

If a curve has either a conjugate point, a point d'arret, a
point saillant, or a cusp of the first or second species, we may
pass through this point an indefinite number of straight lines,
such that, in the vicinity of this point, there is not on one side
of any one of these lines for the last three kinds of points just
named, or on either side for that first named, any point belong-
ing to the curve under consideration.



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256



DIFFERENTIAL CALCULUS,




This is illustrated
in the adjoining fig-
ure, in whichil/,Pjg.l,
is a conjugate point ;
Mj Fig. 2, is a point
d'arrgt; ilf, Fig. 3, a
point saillant ; and M,
Figs, 4 and 5, are
cusps of the first and
second species.
Now, if, for any one of these cases, two points, P, Q, be
taken on one of these lines, one on each side of the point Jf,
and however near to it, these points may be united by a curve
which has no point in common with the given curve AB,
Consequently, if u =/(^Xj y) = is the equation of ABj and u
is continuous, as is supposed, it cannot change sign, except at
zero : but no values of Xj y, belonging to PQ, can reduce u to
zero } for, if so, then that point would be common to AB and
PQ, Hence the values of ic, y, belonging to points of PQ,
make the sign of u constant ; while the values of a?, y, belong-
ing to the point M, reduce u to zero.

Since, then, tlie value of u at the point M is zero, and has
the same sign at P, on one side of this point, that it has at Q
on the other, these points being very near Jf, u must be a
maximum or minimum at M according as the sign of u at P
and Q is negative or positive. In either case, we must have



du . du dy
dy dx



dx



Again : denoting the tangent of the angle that the arbitrary
straight line PMQ makes with the axis of x by a, the equation
of this line, which the co-ordinates of M must satisfy, will be



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SINGULAR POINTS. 257

j/ = ax-^-b: whence -J^ =: a; and, substituting this above,

we have

da du

dx dy

But this last equation must hold for an indefinite number of
values for a, since the line JPMQ is arbitrary ; and therefore
we must have

dx ' dy

The co-ordinates of the four kinds of singular points under
consideration should then satisfy, at the same time, the three
equations

f. du ^ du ^

Two of these equations will determine values of x and y to
substitute in the third. If a set of these values x=.x^jy =:y^j
verifies the three equations, the corresponding point may be
a singular point, but not necessarily so.

To ascertain the nature of the point thus determined, let us

seek the value of -, - , which the equation -^- 4- .- -.— z=
dx dx dy dx

gives under the form - . The second difierential equation,

because of the conditions — =0, -^- = 0, reduces to

dx dy

dy' \dx) ^ dxdy dx ^ dx' ^ ^ ^'

Suppose, also, that, by the solution of u =/{Xf y) = 0, we
have found y ==. F{x) for the equation of the branch of the

83



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258 DIFFERENTIAL CALCULUS,

curve on which the point about which we are inquiring is sit-
uated. The solution of Eq. (a) with respect to -,- gives



dy
dx~


d'-u /d-'uV' d'ud^u

dxdy ~^ ■>/ \dxdy) dx' dy^

d'u



Hence, dy'^

I. From the definition of a conjugate point and these equa-
tions, we conclude that the point x=z Xq, y =zy^^j will be con-
jugate : first, if the two ordii)ates

are both imaginary ; second, if the curve at this point has no
tangent, which requires that

[d^yj ~ dx^ d^'^ '
unless we have

dx'^ ' dxdy ' dy"^

II. The point x = Xq, y == yQ, will be a point d'arret: frsf,
when only one of the ordinates y := F{xq -{- li), y = F(xQ—h)j
is imaginary ; second if the curve at this point has but one
tangent, which will be the case when the co-ordinates of the

point satisfy the equation -^— ^ = 0.

III. The point x = XQ,y zizy^, will be a point saillant : Jirst^
if to each of the abscissa) x^=iXQ-\-h, x^ix^ — A, there is but
one corresponding ordinate, differing but little from y^, or
if there are two, and but two ordinates, differing but little
from ?/o, corresponding to one of these abscissas, and none to
the other abscissa; second, if the curve at the point XQ,y^,
has two tangents, which requires that we have

\dxdy/ dx^ dy'^



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SINGULAR POINTS. 259

IV. The point x^^y^j will be a cusp, when, the first condi-
tion for a point saillant being fulfilled, the two tangents at
that point coincide ; which cannot be the case unless

\dxdy/ dx'^ dy^^ "" '

152. . To investigate the conditions for multiple points, let

the equation F{x, ^) = in rational form represent the curve ;

then

dF dFdy^^ (Art. 84).
ax ay ax

Since at least two branches of a curve pass through a mul-
tiple point, two or more tangents may be drawn at that point :

hence -_-, for such a point, must have more than one value.
ax

dF dF
But, since F{x,y) is supposed rational, , - ~, will each ad-

ax ay

mit of but one value for the values oi x^^y^, which determine
the point. Therefore -y- cannot have more than one value,

unless -r- = 0, — - = : and these are the conditions for the
dx dy

existence of a multiple point. The equation from which to

dx

d-F d^-F dy d^-F/dy\^_

dx' "^ dxdy dx "^ dy' \dx) ~ ^ ''

dy
which will give two real values for —- , if, for the values of x^

and yo.,

/d'F^ d^d'F

\dxdy) dx^ dy^ '
and in this case the multiple point is called a double point.



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260 DIFFERENTIAL CALCULUS.

^^ dx' "" ' dxdy " ' dy' "" '

then Eq. (6) becomes indeterminate, and we must pass to the
differential equation of the third order, which, after intro-
ducing the above conditions, i.e. - = 0, — - = . . . , is

d^F .^d'Fdy d^F (dy\^ d^F/dy\^_^

dx^ "^ dx:^dy dx "^ dxdy' \dx) "^ dy^ \dx) "~ ^ ^*

This cubic equation will give three values for -_: , which, if

all real, show that three tangents can be drawn to the curve
at the point {x^, y^) : the point is then called 2k triple point
If Eq. (rf) becomes indeterminate, we proceed to the differen-
tial equation of the fourth order, and thus get an equation of

the fourth degree for finding -,^ ; and, in general, if n branches

of a curve unite in a multiple point, the co-ordinates of such
point must verify the following equations :

dF_ 1^-0 £!^-o -^-0 ^-n
dx " ' dy " ' dx' " ^' dxdy "" ' dy' ^ ^'

d^-'F _ d - 'F _ d^''F _

dx''-'" ' dx'^-'dy^ •••' rfy^" '
and the v!^ differential equation of the curve would in general

determine n real values for -^.

dx

153. If a curve has a point of inflexion, the co-ordinates

of that point must verify the equation .-\ =. 0.

Suppose the equation of the curve has been put under the
form y = F{x) ; then the difference Ay of the ordinates corre-
sponding to the abscissas x and aj -(- A is (Art. 61)

Aj^ = hF'{x) + i^ F"{x) + . . . + j-^^ F^'^z + Oh).



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SINGULAR POINTS. 261

The difference of the ordinates corresponding to the same
abscissae of the tangent Hne at the point (x^ y) is a?^i = hF\x) :
hence, denoting Ay— Ay^ by 8, we have

« = o ^"(^) + M ^'"(•^) + • • • + o*-:^ ^""^"^ + "'*)•

When h is very small, the first term in the expression for 8
exceeds the sum of all the others ; and consequently the sign
of d for points in the vicinity of the point (a;, y) will be con-
stantly positive, or constantly negative, according as F^^{x) is
positive or negative: hence, if F"{x) does not vanish, the
curve cannot cross the tangent at the point {x, y), and there
can b0 no point of inflexion. If F^^{x) vanishes, then the first

term in the value of d is v-..-i7 F'^\x), if F'"{x) does not vanish

at the same time ; and the sign of this term will change from
positive to negative, or the reverse, as li changes from positive
to negative. This can only be the case when the curve crosses
the tangent at the point (x, y) ; and this point is therefore a
point of inflexion. If F'^'{x) = 0, then, by the same course of
reasoning, we prove that the co-ordinates of a point of inflex-
ion must verify the equation F'^'^{x) = 0, &c. Thus, to find
the co-ordinates of a point of inflexion, we seek the roots com-
mon to the equations

y = F{x), F"{x) = 0, ov/{x, y) = 0, g = 0.

A system a? = o;^, y = y^, of these roots, will be the co-ordi-
nates of such a point, if the first of the derivatives that does
not vanish for them is of an odd order.

154:. Throughout this investigation of the conditions for
singular points, we have supposed -F(x), and its derivatives
for values of a? and y in the vicinity of those corresponding to



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262 DIFFERENTIAL CALCULUS,

the point {x^^ y^)y to be continuous. But, if —- 1=00, ^ye ma}'

readily determine the nature of the point (x^, y^). Under
this hypothesis, the two quantities F{x^ -\- h)j F{Xq — h), may
both be real J or one may be real, and the other imaginary.

First, If both are real, and both greater or both less than
F{Xq)j the point (x^, y^) will be a cusp of the first species: if
one is greater and the other less than F(Xq), the point will be
a point of inflexion.

Second, If one of these quantities, say F{x^ — A), is real,
and the other imaginary, then, i{ F{Xf^ — h) has but one value,
the point will be a point d'arret : if F{Xq — h) has two values,
both of which are greater or both less than F{7J^), the point
will be a cusp of the second species ; but, if one of these values
is greater and the other less than F{x^), the point will be
simply a limit of the curve.

Third, If each, or but one, of the quantities
F{x, + h), F{x,-h),
has more than two values, the point {x^, y^) will be, in gen-
eral, both a multiple point and a point of inflexion.

In conclusion, to obtain the co-ordinates of singular points
of curves, we seek the values of x and y that will reduce the


differential co-efEcients to zero, to infinity, or to -• The na-

ture of the point is ascertained by inquiring how many
branches of the curve pass through the point, and determin-
ing the position of the tangent line or tangent lines corre-
sponding to the point.

155. The terms " concave ^^ and " convex " are employed
to express the sense or direction in which, starting from a
given point, the curve bends with reference to a given line



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CONCAVITY AND CONVEXITY.



263



from the tangent at that point. If it bends from the tangent
towards the line, it is said to be concave, or to have its con-
cavity turned towards the line ; but, if the sense in which it
bends from the tangent is from the line, it is said to be convex,
or to have its convexity turned towards the line.

To find the conditions of the concavity or convexity of a
curve towards a given line, take that line for the axis of x,
and let P, of which the co-ordinates are x and ^, be the point
at which the curve is to be examined with reference to these
properties. Draw the
tangent at P; then,
from our definition, if
at P the curve be con-
vex to the axis of x,
the ordinates of the
curve for the abscissas
X '\-hj X — hj must be
greater than the corresponding ordinates of the tangent at P;
h having any value between some small but finite limit and
zero. But, if the curve be concave towards the axis of x, the
reverse must be the case.

If the equation of the curve is y = F{x)j the ordinate cor-
responding to the abscissa a; + 7i is

y + ^y=^F{x) + liF\x)+~^F-{x) + -'




+



1.2...n



F^'^^x -\- dJi).



The equation of the tangent to the curve at the point {x, y)
hyi — y = F^{af) (a?i — a;), or yi = F{x)'\-XiF^{x)^xF^{x).
Observing that x, y, are the co-ordinates of the point of tan-



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264 DIFFERENTIAL CALCULUS.

gency, tho ordinate of the tangent corresponding to the aK
scissa x-^his

y, + Ayi = F{x) + xF'{x) + hF\x) — xF'{x)
= F{x) + hF'{x)i

hence, if d denote the diflFerence y+^y — (2^i + ^yi)j we
have

S = 1-2 i^"(^) + • • • + ~77^ -F'<"'(^ + Oh}-

The sign of this difference, when h is very small, is the same

as that of — F^^{x), which has the sign of F"{x) whether
I* JL

h be positive or negative : therefore, if F"{x) be positive, the
curve is convex to the axis of x; and it is concave if F"{x) be
negative.

We have supposed the point of the curve at which its con-
vexity or concavity was examined to be above the axis of a;,
or to have a positive ordinate. Had the point been below the
axis, F^'{x) positive would have indicated concavity, and
F^^{x) negative would have indicated convexity. To include
both cases in one enunciation, we say, " When a curve at any

point is convex to the axis of x, y -r-^ is positive at that

0/X

point ; w^hen it is concave to the axis of a?, y -r— ^ is negative."

(XX

Cor. Comparing this article with Art. 153, we conclude, that,
when a curve has a point of inflexion, it will be Convex to a
given line on one side of the point of inflexion, and concave
on the other.

EXAMPLES.

Find the asymptotes to the curves represented by the fol*
lowing equations : —



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EXAMPLES, 266

1. y' = x^(2a — x). Ans. y =: — x -{- -^.

2. y^=L{x — of (a? — c). Ans. y = x — J (2a + c).

3. x^y^ = 0^(0?^ — ^^). Ans. y = db a.

4. {y — 2x) (y^ — x^) — a(y — aj)^4- ^^^(^ + y) = «'•

Ans. y = a;, yzz:-jc + y, y = 2ir + |.

Find and describe the singular points in the curves of which
the following are the equations : —

X*

5. y = g — 2. There is a point of inflexion at the origin,

and also at the point having a; = db a y^3 for its abscissa.

6. y{a^ — V) = x{x^ ay — ocb*. There are two points of

inflexion corresponding to the abscissae x=zaj x=: — .

5

7. y^ z=(x-' ay (x — c). There is a cusp of the first spe-
cies at the point of which a; = a is the abscissa.

8. X* — ax^y — axy^ + ^^V^ = 0. There is a conjugate
point at the origin.

9. ay^ — ic' -f- hx^ = 0. There is a conjugate point aW the

origin, and a point of inflexion at the point having os=^— for

d

its abscissa.

84



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SECTION III.



POLAR CO-ORDINATES. — DIFFERENTIAL CO-EFFICIENTS OF THE ARCS
AND AREAS OP PLANE CURVES. — OF SOLIDS AND SURFACES OF
REVOLUTION.

136» Let the pole coincide with the origin of a system of
rectangular co-ordinate axes : denote the radius vector by r,
and the angle, called vectorial angle, that it makes with the
axis of X taken as the initial line, or polar axis, by ; then
the formulas by which an equation expressed in terms of rec-
tangular co-ordinates may be transformed into one expressed
in terms of polar co-ordinates are x=ir cos. 0, y :=.r sin. 0.

To express in polar co-ordinates the tangent of the angle
that a tangent line to a curve makes with tlje axis of x, we

dy
have, calling this angle r, tan. r = — ; and hence (Eqs. a,

ax



Art. 132)



tan.r =



. dr

sm. ^ -r- -4- r cos.
do '



dr

COS. 6 , r sm.

do



and from this we may readily find the expression for the tan-
gent of the angle that the tan-
gent line at any point makes
with the radius vector of that
point.

Let M be the point, P the
pole, MT the tangent line, and
Px the axis of a;, from which d
is estimated ; then




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POLAR CO-ORDINATES. 2G7

PMT = MTx — MPT:
hence, by the formula for the tangent of the difference of

two arcs,

dr

sin. 0~- -\- rcos.O

^, tan. d

dr
cos.^- r sm. ^

tan. PMT = -. -, z= r— .

, dr \ dr

tan. 6{ sm. 6-r- + r cos, ]
. . \ dO^ J

^ dr .

cos. d~ T Sin.

do

This may also be found directly as follows : Talvo on the cnrve

a second point, Q, the co-ordinates of which are r + Ar, ^ + aO,

and draw JfiV perpendicular to PQ; then MN^i r sin. aO, and

QN^= r -\- Ar — r cos. a<?; hence

r sin. aO



Un.NQM =



r -|- Ar — r cos. a^

Now let the point Q move towards 31. The limiting position of

the secant QM is the tangent MT, and the limit of the angle

NQMh the angle PMT. Call this angle § ; then

. ,. rsin.A^ ?'sin.A^

tan. p = lim. = lim.



r + Ar — r cos. a^ r» • o ^-^ .
' 2r sm. 2 [- Ar

r sin. A^



:lim. _ ^'



2rsin.^ —

2^ . A^

A^ A(?



. ,A^ , Ad

A sm.^ — - sin.—- ,^

The limit of = 1, lim. = lim. -— sm. -r- = 0»

A^ ' A^ A^ 2 *

and lim. — is denoted by — : therefore tan. 5 = r -p*
Ad -^ dd . "^ dr



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268



DIFFERENTIAL CALCULUS.



157 • To find the polar equations of the tangent and nor-
mal lines to a curve, we may assume the equations of these
lines referred to rectangular axes (Arts. 141, 142), and change
them into their equivalents in polar co-ordinates ; or we may
proceed thus : —

Let r and be the co-ordi-
nates of the point M; and r\ 0\
those of a second point, L, in
the tangent line : then from the
triangle PLM, making
PML = x,

we liave
r _ %m,PLM _ sin. ((^ — ^^ + t)
r* ~~ sin. FlUL "" sin. x

= sin. {0 — 0') cot. r + cos. {d - 0%

:?; = l^^sin.(^-(90 + cos.((9-^0 (1),
r' r do ^




or



1 \ dr

observing that cot.r = ^ = - — (Art. 156). Eq. 1 may

be written,

7-2 izzr'-f-r sin. ((9-^0 (2).
do ^ ^ ^ ^

Making: w = , t^' = - : then - —= - -: and hence, by

^ r' r'' r^dO dO ' ^

dividing both members of (1) by r, and substituting these val-
ues, we find

u' = u cos. {0 - 0') — ^ sin. {0 — d') (3).

To find the polar equation of the normal at any point of a
curve, denote by r and the co-ordinates of M; and by r', ^',
those of any point, Mj in the normal : then



Digitized by VjOOQIC



POLAR CO-ORDINATES, 269

PM __ sm. PRM _ \ 2 ) ^

FB "" sin. JPMB '. /Vt

sm. ( - — T

therefore -7 =: sin. (^^ — 0) tan. r -[- cos. (d' — ^)

= sin. («' - <?) -^ + COS. (<?' - 0) (4),

which may be written


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