known functions of the independent variables x and y, and of
the dependent variable «, and 9 {v) an arbitrary function of v.
DijBTerentiating w =: g}(t;) first with respect to x and «, and then
dz dz
with respect to y and z, and, for brevity, making -r-^ip^-^ z=iq^
we shall have
du ^ du , , , /dv . dv\
du , du , , , /dv dv\
Dividing these equations member by member, we have
du y du dv dv
dx dz dx ' -^ dz
du du dv dv'
Ty'^^'dz Ty^'^Tz
Clearing of fractions, and making
j^_^dudv ^du dv ^__dudv du dv ^^dudv dudv
'^ dy dz dz dy' ^'^dz dx^dx ^* '^dxdy'^dydx'
we find that the partial differential co-eflScients of the first
order are connected by the equation
Pp+Qq-R: .
and this equation is in no wise dependent upon the form of the
function characterized by cp ; in other words, this function has
been eliminated. ^
139 • Suppose c and Cj to be two known functions of a?, y, «,
Digitized by VjOOQIC
ELIMINA TION OF CONSTANTS AND FUNCTIONS. 231
expressed bj c=z/{x, y, z), Ci=/i{x, y, 2); and that, in the
equation
F(x,7/,z,(r.{c),cp,{ci)) = (1),
qp, g)j, denote arbitrary functions. Let us see if it be possible
to pass from (1) to a differential equation which shall not con-
tain g)(c), (fiici), or their derivatives.
The equations
dW d^F d'^F
i#=». I^,='>' ^='' <^)'
that we get by differentiating (I), will contain the unknown
functions ^'^ {c), gp/(ci), ^'^(c), qr/'(ci), which, with ^(c), gPi(Ci),
make six quantities to be eliminated between Eq. 1 and the
five equations of groups (2) and (3), which are generally in-
sufficient. Passing to the equations
d?F_. d'F _^ d'F _^ cPF_. ,,.
wo introduce two additional arbitrary functions q>"^{c)j (pi^\Ci),
and only these two. We shall now have ten equations, viz.
Eq. 1, and those of groups (2), (3), (4), and but eight arbitrary
functions to eliminate : hence the elimination can be effected,
and Avo may have two resulting differential equations of the
third order.
We have said, that, in the case supposed above, it is gener-
ally impossible to effect the desired eHminations without pass-
ing to Eqs. 4. It may happen, however, that the forms of the
functions/(a;, y, «),/i(a;, y, z), are such that Eqs. 1, 2, 3, will
prove sufficient.
Digitized by VjOOQIC
232 DIFFERENTIAL CALCULUS.
Suppose 2 = qp (x + ay) + qPi (a; — ay) ; then
dz
_ = 9' (a; + a^ ) + (jp /(a; — ay),
dz
-^ z=acp'{x+a7/)^a(p'{X'-ay),
d^z
dx^ ~ ^''^"^ + "^^ + ^^''(^ â– "^^^'
d^z
— = a2(3p^'(x + ay) + a^(pi''{x - ay).
From the last two of these equations, Ave find
dhi_ ^dh
dy'^ dx'^'
140* Suppose that we have two functions,
in which c is an implicit function of x, y, z, and g)(c), g'i(c) . . .,
are arbitrary functions of c. It is proposed by successive dif-
ferentiations to eh'minate c and the arbitrary functions. To
accomplish this, z and c must be considered as functions of the
independent variables x, y ; then, having differentiated the
given equations a number of times successively with respect
to X, and also with respect to y, we must eliminate the quan-
tities
dc dc d^c d^c d^c ,^v
^' di' dy 3^2' ^^y' dp'^' ^ ^'
T(c), g)'(c), <(c)..., cp,{c), qp/(c), (]Pi''{c)... (2),
between the given and the difierential equations.
Let m denote the number of arbitrary functions
end n any positive integer; then, if we stop with partial
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ELIMINA TION OF CONSTANTS AND FUNCTIONS. 233
derivatives of c and of g)(c), qpi(c), of the n*^ order, the num-
ber of terms in series (1) will be expressed by
1 + 2+3 + 4 + .. .n + l = i!i+iy!i±2).
The number of the arbitrary functions
g)(c), qp'(c)..., (3p(">(c), <jPi(c), <]p/(c)..., (jPi^^^^) . . .,
will be equal to m(n4"l)' Again: since each of the given
equations will give rise to two derived equations of the first
order, three of the second, four of the third, and so on, the
number of given and derived equations together will be equal
to (n + 1) (n -f- 2). Hence to be able, in the general case, to
eliminate c and its arbitrary functions, and their derivatives
up to the v}^ order, we must have
(n + l)(n + 2)> (^ + ^>^^"+^^ + (n + l)m,or|+l>m.
This condition will be satisfied if n = 2m — 1, which will give
2m(2m+ 1) equations between which to eliminate Am'^-\-m
quantities. The number of equations exceeds the number of
quantities to be eliminated by m ; hence there will be, in gen-
eral, m resulting differential equations.
When the proposed equations contain but one arbitrary func-
tion, g)(c), of c, they become
F(x, y, z, c, g)(c)) = 0, F,(x, y, z, c, g)(c)) =0,
each of which gives two partial derived equations of the first
order ; and we shall thus have, including the given equations,
six equations between the quantities
dz dz dc dc , . ,, ,
^' ^' '''^=Tx' ^ = dy' '' di' Ty' '^(')' "^^'^^
the elimination of the last five of which will lead to a single
80
(
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234 DIFFERENTIAL CALCULUS,
partial differential equation of the first order between the
variables x, y, z, of which x and y are independent.
If there are but two arbitrary functions g)(c), cpi{c) of c, wo
should find that the given equations
i^(x«,?/,2,c,g)(c),g^i(c)) =0, F,(x,y, z, c, g)(c), g5i(c)) = 0,
with their partial derived equations of the first order, making
in all twelve equations, would involve twelve quantities to be
eliminated; viz.,
dc dc d-c d^c d^c
^' diJ' dy' dp' d^dy' d^'-'
^{c)y ^'(c), cp''{c), qri(c), (p[{c), g)l'(c):
hence the elimination cannot bo effected, except in special
cases. Passing to the partial derived equations of the third
order, wo should then have in all twenty equations, with
eighteen quantities to be eliminated; viz., the twelve above
given, and
d^ d^c d^c d^c ,, . ,,f.
dx^' db^dy' dM^' dif ^ ^^^' ^' ^^^'
additional : and we may therefore have for our results two par-
tial differential equations of the third order between ic, y, z;
the latter being the dependent variable.
In certain cases, it is unnecessary to make as many differen-
tiations as have been indicated to enable us to effect the de-
sired eliminations. Suppose, for example, "that the given
equations contain but three arbitrary functions, 9(c), gti(c),
92(0) : in this case, m == 3, 2m — 1 = 5; and, to effect the
eliminations, it would be generally necessary to form the de-
rived equations of the fifth order, and we should have for our
results three partial differential equations of the fifth order
between cc, y, z. But if the arbitrary functions are bo related
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ELIMINA TION OF CONSTANTS AND FUNCTIONS, 235
that g'i(c) =;: qp'(c), 9)2(0) = 9^^(c), the proposed equations be-
come
F\ X, y, z, c, g)(c), qr'(c), ^'^(c) | = 0,
â– ^1 1 ^, Vj ^j C; ^(c), <3p'(c), t''(c) } = ;
and these, with their derived equations of the first and second
orders, make twelve equations, involving the eleven quan-
tities
dc dc d'c d'^c d'^c
^' dx' ly' dF^ d^' 5p'
(3P(c), g^^(c), (p-(c), <(c), (3p - (c);
and the elimination will lead to a single partial differential
equation of the second order.
If the value c bo found, as it may be, theoretically at least,
from one, say the second, of the equations
F\x,yy z, c, g)(c), cp^{c) \=0, F^\x, y, 2, c, g)(c), g)i(c) | — 0,
and this value be substituted in the first, we should have for
our result an equation of the form
F\x, y, z, rp{x, y, z), ^p^{x, y,z)\ = 0,
which is evidently equivalent to the two proposed equations.
By Art. 139, wo shall generally be unable to eliminate the two
arbitrary functions of^, t/^j, with this equivalent equation and
its derived equations of the first and second orders ; but it
would be necessary to pass to the third derived equations to
effect the elimination.
EXAMPLES.
1. Eliminate the constant a from the equation
Vl - a?2 + \/n=^2 =a{x- y).
V 1 — aj2 dx
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236 DIFFERENTIAL CALCULUS,
2. Eliminate c from the equation
OJ^ + y^ =:CX.
Ans. y^ — 2xy -^^ — a;* = 0.
3. Eliminate the functions e"^ and cos. a; from
y — e' cos. a; = 0.
4. From y = asin.a; +icos.a; eliminate the functions sin.o;,
COS. X,
70
5. Ify = ce''""'', prove that
6. If y = Je^cos. {nx -\- c), show that
e* -1- g— *
7. From the equation y = ^ _^ eliminate the exponen-
tial functions.
8. From is; = g) (e* sin. ^) eliminate the arbitrary function
characterized by cp,
. , dz ^2 A
Ans. sni. y -. cos. y -y- z= 0.
^ ay ^ ax
X 'Z/^ z^
9. From — 2+;li + ~2 — 1 = eliminate the
a2 ^ 6"^ ^ c
a, J, c.
constants
IstAns. 0:2 — + a:f^)-2~=0.
d^z /dz\^ dz
2d Ans. V2 -T— , + V(^) — «-r=0.
^ dy'^^\dy) dy
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, ELIMINA TION OF CONSTANTS AND FUNCTIONS. 237
10. From u = x/(-j '\- (p{ooy) eliminate the functions
f(l\<pi-y)'
(J^U db XL
11. Eliminate the functions from
w =/(a; + y) + a:y 9 (a: — y).
d^u d^u d^u d^u 2 /d^u d'^u\
Ans. f - ( )=0.
dx^ dx^dy dxdy^ dy^ x-{-y\dx^ dy'^/
12. From
eliminate the arbitrary functions /, <f, i//.
/ ,d'a rf'2\ , / dz dz\/- dz dz\ ^
d^X dp"!!
13. From the equations -^-y = (f(x^ y), -i-y = V(^; y)j elinoi-
nate the variable z ; i.e., change the independent variable from
z to X.
Ans.2,(.,y) = ^ dry ^-
14. Eliminate the arbitrary functions from
. d'^z , o d'^z , ^d'^z , dz , dz « ^
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DIFFERENTIAL CALCULUS.
'Fj^:rt sEOonsriD-
GEOMETRICAL APPLICATIONS.
SECTION I.
TANGENTS, NORMALS, SUB-TANGENTS, AND SUB-NORMALS TO PLANE
CURVES.
Idl* The tangent line to a ciirv- e at a given point is
the limiting position of a secant line passing through that point,
or it is what the- secant lino becomes Avhen another of its
points of intersection with the curve unites with the given
point. It is now proposed to find the form of the equation of
tangent lines to plane curves.
Let y =^/{x) be the equa-
tion of the curve RPQ, and
take on this curve any point,
as P, of which the co-ordi-
nates, referred to the rectan-
^ gular co-ordinate axes Ox, Oy,
are a; and y. This point will
be briefly designated as point
{Xj y). Give to x, taken as the
independent variable, the in-
crement ^x, y will receive a corresponding increment Ay, and
238
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TANGENTS AND NORMALS. 239
X + Au;, y 4" ^Vj are the coordinates of a second point, Q,
on the curve; then, if Xi, y,, denote the general or running
co-ordinates of a straight line passing through P and Q, the
equation of this line will be
or
AX
Now, conceive the point Q gradually to approach the point P,
— ^ will, at the same time, gradually approach its limit ^- = y',
and finally become equal to this limit when Q unites with P;
but then the secant line becomes the tangent line. Hence the
equation of the tangent line is
in which -p is the tangent of the angle tljat the tangent line
makes with the axis of abscissae. Calhng this angle r, we
have
tan. r = -, - = v > cot. r = -y- == — r >
dx ^ ' dy_ y''
dx
-4-1 a. 1
COS. T = ifc ■, r: = dr — - , = ,
— -4-
y y' , dx
Bin. r = rb — -?-^-_ =: it
>+(^'
i4J?, The normal line to a curve at any point is the
straight line passing through the point at right angles to the
tangent line at that point.
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240 DIFFERENTIAL CALCULUS,
Since the normal and tangent lines at a given point are per-
pendicular to each other, denoting the angle that the former
makes with the axis of x by y, wo have
tan = - -^_ - 1 =:-i - ^.
tan. T """ y' "~ ^y "~ dy'
dx
and, if x^jyi, represent the general co-ordinates of the nor-
mal line, its equation is
yi-y='- yA^i- ^)y or y 1 - y = - ~^^{x^ - x).
Cor, When the equation of the curve is in the form
F{x, y) = 0, or the ordinate y is an implicit function of the
dF
abscissa, we have (Art. 84)-^= — -^ hence the equation
'dy
of the tangent line becomes
and that of the normal
143. To find the equation of the tangent line passing
through a given point out of the curve represented by the
equation F{x, y) = 0, we should make Xi^y^ in the equation
of the tangent, equal to the co-ordinates of the given point.
Then, since the point of tangency is common to curve and
tangent, the co-ordinates of this point must satisfy both the
equation of the curve and the equation of the tangent : hence
these two equations will determine x and y, the co-ordinates
of the point or points of tangency. In the same way, we may
find the equation of a normal line passing through a point not
in the curve.
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TANGENTS AND NORMALS, 241
Now, if we liave two curves, of which the equations are
F{x, y) = 0, F{x,y) — c = 0, respectively, the equations of
the tangent and of the normal to the first curve will be iden-
tically the same as those of the corresponding lines to the
second (Art. 142, cor.). Hence, if for given values of Xi^ y^
and any assumed value of c, the values of a? and y be deduced
from the equations
such values will be the co-ordinates of the points of tangency
of the tangent line drawn through the point (xuyi). In like
manner, the combination of the equations
Fix,y)-c = 0, (x,-x)^-{y,-y)^=0,
will determine the points of intersection with the curves of the
normal lines drawn from the point {xi,yi).
Since the equation
.dF , , ^dF' ^
(^^"^^^ +(2/^-2^)^ = *^
is independent of c, it will represent a line which is the geo-
metrical locus of the points of tangency of the tangent lines
drawn from the point {Xi,yi), with all the curves which, by
ascribing different values to c, can be represented by the
equation F{x, y) — c = 0. So also
â– .dF . .dF ^
IB the equation of the geometrical locus of the intersections
of the normal lines drawn through the point (a?i,yi) with the
same curves. Hence, if these geometrical loci be constructed
from their equations, their intersection with the curve answer-
ing to an assigned value of c will be the points common to the
curve and tangents, or normals, as the case may be.
81
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242
DIFFERENTIAL CALCULUS.
144. Formulas for the distances called the tangent^ the
sub'tangent, the normal, and the sub-normaL
Def. 1. The tangent referred to either axis of co-ordi-
nates is that portion of the tangent line to a curve which is
included between the point of tangency and the axis.
Def 2. The sub^tangent is that portion of the axis
which is included between the intersection of the tangent line
with the axis and the foot of that ordinate to the ixis, which
is drawn from the point of tangency.
Def. 3. The normal is the part of the normal line in-
cluded between the point of tangency and the intersection of
the normal with the axis.
Def 4. The aub-normal is the part of the axis in-
cluded between the normal and the foot of the ordinate of the
j)oint of tangency. The relation of sub-normal to normal is
the same as that of sub-tangent to tangent.
In the figure, let P be the
point of tangency ; then, with
reference to the axis of x, PM
being the ordinate of ,P, Ft
is the tangent, Mt the sub-tan-
gent, PJVthe normal, and MN
the sub-normal. With refer^
ence to the axis of y, the
lines of the same name are
PT, IP'T, PN', and M'^N', ro^
8pectively.
tan. Ptx = ~y-:
ax
m = MP^ = MP^,
dy dy'
dx
or
Mt = sub-tangent :
dx
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TANGENTS, NORMALS, §-d 243
. Again :
^J^= tan. MPN^ tan. Ptx = ^^ :
MP ax
MN^ sub-normal z=iy j-,
Pt =2 tangent = y (;)-) + l,
and PN'=PM\m'=y^ + r(^^^=y^ W+^\
PN^nomwl = y J(^) + 1-
Grouping these formulas, we have
J/dx\^ dx
( -7- j 4" 1- Sub-tangent = y —-,
Normal =i y ( jt" ) + 1- Sub-normal =^ y ~.
145* A curve may be given analytically by two equations
of the form y = qp (t), x =:\fj(t)j which, by the elimination of t
between them, may be reduced to one of the form y =z/{x).
Without effecting this elimination, the equation of the tangent
line will be
(yi-y)|'-(^i-^)f = 0;
and that of the normal,
(y.-y)§ + (a=i-a;)§=o.
When the co-ordinate axes are oblique, making with each
other an angle «, the limit of the ratio - - or - ^- no longer
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244 DIFFERENTIAL CALCULUS.
expresses tan. r, but — . — .- ,-. In this case, the investi-
sin. (oj — t) '
gation and the form of the equation of the tangent line remain
Tinchanged ; but the equation of the normal line becomes
1 + ^- COS. w
y'-y=-dfv — ^^^-^')'
EXAMPLES.
1. The equation {x■^ — x)x + (y, — y)y = of the tangent
line to the circle can be put under the form
which, if X and y arc variable, and x^ and y^ constant, is the
X 7y
equation of a circle, the centre of which, having -^^ ^\ for its
co-ordinates, is the middle point of the line drawn from the
point (x,, ^i) to the centre of the given circle. The radius of
this circle is equal to - s/ x" -}- y^ . Now, for assigned values
of iTj, 2/i, the points of contact with the given circumference
of the tangent lines drawn from the point {x.^^ y{) must be in
the circumferences of both of the circles ; and, since (1) is in-
dependent of r, the circumference of the circle of which it is
the equation is the geometrical locus of all the points of con-
tact with the given circumference of the tangent lines drawn
from the point {x^, yi) to the different circles that we get by
causing r to vary in the equation x^ -{- y'^ = r^
2. The general equation of lines of the second order (conic
Bections) is
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TANGENTS, NORMALS, (fc 245
:-. ^ = By + 2Cx+E, ^ = Bx + 2Ji,-{.D;
and the equation of the tangent is
which the given equation reduces to
x,{By + 2Cx + E) + y,{Bx + 2Ai/ + D)+Di/ + Ex + 2F=0.
3. The logarithmic curve is that which has'y = pZx
for its equation. For it we have -^ - = -y-; and the equations
CbX XlOi
of its tangent and normal lines are
a^K^i — y) — (^1 - a;) = 0, xla{x^ — x) + (yi - y) = 0.
The sub-tangent on the axis of y is expressed by x-^-=z — -,
and is therefore constant, and equal to the modulus of the
system of logarithms.
4. The logarithmic spiral is a curve having
I = tan. ^ ^^'^ + y\ or tan.-^^ = V^M=^' - IB,
X J-li X
for its equation ; whence
dy , d}f
ax ^ __ ^ ^ dx dy ^x + y ^
x'^-^y^ x'^-\-y'^ dx x — y'
and the equations of the tangent and of the normal are
(xj - x) (x + y) + (y 1 - y) (y - a;) = 0,
(Xx — x){y — x)-' {yi - 'y){x+y)= 0.
When Xx, yu are considered constant, and x, y, are made to
vary, these last equations represent two circles, the circumfer-
ences of which cut the spiral in the points of contact of the
^ngents to the spiral which are drawn from tlie point (x^, y^),
5. Denoting the tangent by T, sub-tangent by T,, normal
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246 DIFFERENTIAL CALCULUS.
by N^ and sub-normal by N,, determino these lines for the fol-
lowing curves : —
First, The circle : x'^ + 7f =z r\
dj X
Second, The ellipse or hyperbola : %±:^-, = ±: 1.
-=|(i*-)('-S)h - -i;!^.=-(^')-
Third J The parabola: y^ = 2px,
T= 2x^(x +|y, T. = 2x, N=pi (p + 2x)\ N, =p.
The sub-norraal in the parabola is constant, and equal to the
semi-parameter ; the sub-tangent is double the abscissa of the
point of tangency.
Fourth, The logarithmic curve : x= j-li/.
2
Ixla
In this curve, the sub-tangent on the axis of a; ia constant, and
equal to the modulus of the system of logarithms.
146. The Cycloid is a curve which is generated by a
point in the circumference of a circle, while the circle is
rolled on a line tangent to its circumference, and kept con-
stantly in the same plane.
Suppose the circle of \yhich C is the centre, and which is
tangent to the line Ox at the point 0, to roll on this line from
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THE CYCLOID, 247
O towards x, Whilo the point of contact is passing from
to Nj the radius CO, which, at the origin of the motion, was
6!a it B
perpendicular to Ox, will turn about the centre of the circle
through the angle NC'P ; and the generating point will move
from O to P, describing the arc OP of the cycloid. To find
the equation of this curve, take Ox, Oy, for the co-ordinate
axes. Let r = CO = radius of the generating circle ; co i= NC^P
the variable angle ; and x = OR, y — PR^ the co-ordinates of
the point P : then
x=, OR = ON-'RN=i arc PN—PQ izzru — r sin. o = r (w — sin. w),
y = PR = ON— C'Qzur— r cos, o=ir (1 — cos. o).
From y =zr(l — cos. co), we have
COS. CO = , sin. 0) = dr - V 2r2^ — yS <» = cos. "* ^;
and these values of w, sin. w, substituted in the equation
a; = r (w — sin. w), give
X
= r ^cos.-» ^- ^j =F V2ry — y\
which is the equation of the cycloid. The minus sign before
the radical must be used for points in the arc OPO^ which is
described while the points in the semi-circumference OLK
are brought successively in contact with the line Ox; and the
plus sign must be used for points in the arc O^B, The point
(y is called the vertex of the cycloid, or rather the vertex of
the branch OOB ; since, by continuing the motion of the gen-
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248 DIFFERENTIAL CALCULUS.
erating circle on the indefinite line Ox, we should have an
unlimited number of curves in all respects equal to 0(yB,
From the equation of the cycloid, we get
dx
y . dy_ I 2r —
2r — y' * * dx \ y
y
dy Ni
Hence the equation of the tangent line at any point is
l2r — y,
yi-y=^^—^{^i-^)'^
and of the normal,
If, in this last equation, we make y^ = 0, we find
Xi — xzzz \/y(2r ^ y) =z r sin. o = UN.
dti dx
Substituting the values of -^-, — , in the general formulas,
cLx (ty
for tangent, sub-tangent, normal, and sub-normal (Art. 144), we
have for the cycloid
N= V2^, N, = \/y{2r-y) ;
which last agrees with what was found above; and from which
we conclude, that, if supplementary chords be drawn through
the extremities of the vertical diameter of the generating cir-
cle in any of its positions and the corresponding point of the
cycloid, the lower of these chords will be the normal, and the
upper the tangent, to the cycloid at that point.
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Google
SECTION II.
ASYMPTOTES OF PLANE CURVES. — SINGULAR POINTS. — CONCAVITY
AND CONVEXITY.
14:7* When a plane curve is such, that, as the point of
tangency of a tangent line is moved to a greater and greater
distance from the origin, the tangent h'ne continually ap-
proaches coincidence with a certain fixed line, but cannot be
made actually to coincide with it until at least one of the
co-ordinates of the point of tangency is made infinite, such
fixed line is said to be an asymptote to the curve. Hence
we may define the asymptote of a curve to be the limiting
position of a tangent line when the point of tangency moves
to an infinite distance from the origin of co-ordinates.
To establish rules for finding the asymptotes of curves, re-
sume the general equation of a tangent line
yi-y = ^(^1 - *) (^^*- 1^1)^
and find from it the expressions for the distances from the
origin at which the tangent intersects the co-ordinate axes.
These are,
doc
Xi = a; — 2/ y- = distance on axis of a; (1 ),
yi=^y — x~~- =: distance on axis of ^ (2).
Now, there may be two cases in which asymptotes will ex»
doc diTj
ist : 1st, Both a: — y -p, and y — x -^, may remain finite for the
cty (X'OC
82 249
Digitized by VjOOQIC
250 DIFFERENTIAL CALCULUS,
values X =zao J y = oo . 2(1, One of these expressions may re-
main finite while the other becomes infinite. If the expression
for the distance on the axis of x is finite while that for the
distance on the axis of y is infinite, the asymptote is parallel
to the axis of y ; and it is parallel to the axis of x when the
distance on the axis of y is finite, and that on the axis of x is
infinite.
Ex. 1. The equation of the parabola is
v'-2nx' • ^2^- VjP dx_^2x^
and, for these values, expressions (1) and (2) for a;=:Qo, y = oo,
are both infinite. The parabola, therefore, has no asymptote.
Ex, 2. The equation of the hyperbola is
^2^2 _ j2^2 -_. — a^6^, or y = it - >/x^ — a^,
dv y hx J dx x^ — a^ a}
_ = =t — — , and X — y '~=:x — = — ,
dx ay/x'-^a dy x x'
d\i
which reduces to for ic = oo : y —x -^ will also become zero
dx
when a; = oo , and - - becomes i -. Hence the hyperbola has
dx a
two asymptotes passing through its centre, and making equal
angles with the transverse axis on opposite sides.
Ex. 3. The exponential curve :
dx ,1 1
X — y-^ =z X — a
dy a^la la
.dy
= X — ~-= ztico for a; = d=
00