dx 1 . n da
®^°^® ~l^^ =■-? 71 — ^T-7 ' therefore p = ^-.
Ht)Y
178» The co-ordinates of the points of the curve at which
the radius of curvature is a maximum or a minimum must be
found from the equation of the curve and the equation -^ =0;
the latter leading to
.K0)'I-01'+(D'|=««-
DiflFerentiating thp second of Eqs. 3, Art. 171, we find
^ dvi d'^Vt , ,, d^Vi
d*yi_ dxt dx'^ ^ \dx^J dx
by Eqs. 4 of the same article.
Comparing Eqs. 1 and 2, it is seen that — ^ z= — ^ : which
dx^ dx^
proves, that, at the points of maximum or minimum cur-
Digitized by VjOOQIC
292
jyiFFERKNTlAL CALCULUS,
vature, the osculatory circle has, with the given curve, a coii-
tact of the third order.
179 • If a perpendicular be let
fall from the origin of co-ordinates
on the tangents drawn to the differ
ent points of any curve, as SMS', the
locus of the intersections of the per-
pendiculars with the tangents will
be a new curve, the properties of
which will depend on those of the
given curve.
Denote the co-ordinates of the new curve by x^, t/i] then
will the length of the perpendicular j?i, from the origin to the
tangent drawn to this curve at the point corresponding to the
point {x, y) of the given curve, have for its expression
.=j-
-(I:)'
The equation of the tangent to the given curve is
fi and y being the general co-ordinates. Since the point (rci, y^)
is on this tangent,
The equation of Op \q ft = ^v; and, because Op isperpendio-
dx
X
ular to Mp, ^ = — ^: whence
Xi dy
or
(y 1 - y)y\ = — ^i(^i — ^),
yy\-\r^\ = ^\ -Vy\^
Digitized by VjOOQIC
E VOLUTES OF PLANE CURVES. 293
Differentiating this last with respect to x, we find
Substituting for -^ its value, , transposing and redu-
cing, Ave have
and, by means of tbis,^i becomes
2
», = ;^ r^r-t = j
^ Vx'^ + y' ^'
r being the distance from the origin to the point (a:, y) of the
given curve, and^ the perpendicular Op let fall from the ori-
gin on the tangent to the curve at the same point.
180. If /{Xy y) = be the equation of a curve, it has been
shown (Art. 171), that, calling ft, Vj the co-ordinates of the cen-
tre of curvature corresponding to the point (x, y) of the given
curve, we have
a:_^ + (y_.)g=0 (1),
By means of these equations, the equation of the curve, and its
first and second diflferential equations, Ave may eliminate x, y,
dtj d v
-^ ; -~ , and find a direct relation between ft and v. This will
be the equation of a new curve, called, with reference to the
given curve, tht evcHute; the given curve being the involute.
It is evident that fi and v may be considered as functions of
Digitized by VjOOQIC
294
DIFFERENTIAL CALCULUS,
x; and, if Eq. 1 be differentiated tinder this supposition, we
liavo
1 +
(^â– +(.-
dx
d>
dx dx '
and, through (2), this reduces to
dii dy^ dy^^. .^
dx dx dx ' * *
dji dx
dy
whence, by the substitution of tlio value of -,- derived from
* dx
the last of these equations, Eq. 1 becomes
dv . .
These relations show that the tangent to the evolute is a nor-
mal to the corresponding point of the involute, and the con-
verse.
A consequence of this property is, that the evolute of a
curve is the locus of the
intersections of the con-
secutive normals to this
curve. For take the two
normals MK^ M^K\vf\i\c\
by what precedes, are tan-
gent to the evolute at the
points K, K\ When the
point M^ is made to ap-
proach the point M, the line M^K* approaches the line MK,
and the points K^ and N tend to unite in the point K: hence
the point K may be regarded as the intersection of the normal
MK vf\i\i the normal indefinitely near or consecutive to it.
Another important consequence is, that the length of the
arc of the evolute between two centres of curvature is the
Digitized by VjOOQIC
EVOLUTES AXD INVOLUTES. 295
differenco of the corresponding radii of curvature. To prove
this, differentiate the equation
p2 z= (a; — fif + (2^ — 1')^
treating y, fi, v, and /> as functions of x : we thus have
which, by Eq. 1, reduces to
From Eq. 1 and the equation ;^- + ;t- -,? = 0, we get
^^ * ( fd^\- /dry
x — fi y —
d. d. _j [r.)+\T.) )>=±^^^ (6)
{x-i^f + iy-vf
when 8 denotes the length of the arc of the evolute estimated
from any point F ( Cor. 2, Art. 161) : whence
(^-'^^^ ^y^^Tx ^ids ,,
{x — iiy {ij-'^y p dx ^ '
And, by the combination of Eqs. a and c, we find -^ = dc -^ :
d(8:Tip)
wherefore, since — ^ = 0, it follows that 5 qp p is equal to
some constant which we will denote by I; that is,
»zpp=:?, 8^zpp^ = l; .*. 8 — 8^ =p —p\
or arc i^Z^' — arc FK = MK — M'K' = arc KK'.
Suppose a flexible but inextensible string, of a length equal
to 3I^K^ plus the arc K'KF, fastened by one of its ends to Fy
to envelope the curve FKK'^ and then pass in the direction of
the tangent to the curve at K' from K' to W. If this string
be unwound from this curve, its free end will describe the
Digitized by VjOOQIC
296 DIFFERENTIAL CALCULUS.
curve MM* 8. It is from this property that the terms "cvolnte"
and " involute " are derived. It is also seen that there may be
an unlimited number of involutes answering to the same evo-
lute FKK*, and that, to describe them, it is only necessary to
lengthen or shorten arbitrarily the part of the string that ex-
tends in the tangent to the evolute. Since the tangents to
the evolute are normals to all these involutes, it follows that
the latter curves have the same normals and the same centres
of curvature, and that the parts of the common normals in-
cluded between any two will be equal: hence one involute
enables us to find all the others.
181* Radius of curvature and evolute of the ellipse.
The equation of an ellipse, referred to its centre and axes, is
whence -f- = ^, -j-^. = j-^»
ax a^y^ dx^ a*y^
These values, substituted in the formula for the radius of
curvature. Art. 171, give
b'x^i
- X " ^ / _{h*x^ + aY)i
a'b'
a^y^
To find the equation of the evolute of the ellipse, resume
the equations
dv d^v
of Art. 180. Putting in Eq. 2, for -^^, -^, their values, it be-
comes
a*y» + Vx^y ^ a'^b\y ^ y) = :
Digitized by VjOOQIC
E VOLUTE OF THE ELLIPSE.
207
whence v-.- («V + ^^^')y _ «'y' + &'(«* &'- a V)
a'b
y
a'b
- b* y
Making a* — 6' = c', we find
Substituting in (1) the value of y — v just found, we get, after
reduction, and the elimination of y by means of the equation
y + ~ic' •••»' = ir (3).
of the ellipse,
(4).
Eq. 4 might have been derived from (3) by changing the sign
of the latter, and in it writing x for y, and a for b. This is a
consequence of the symmetry of the equation of the ellipse,
and the relation between a, 6, and c.
Put
X
^\* y
\
-=m, -T = n; then-z=(— ,7 = 1-
Writing the equation of the ellipse under the form
/V\
•A- I -I — 1
\o/
X 'H
this becomes, when the above values of -, ~, are substituted.
©•
+
for the equation of the evo-
lute. The form of this equa-
tion shows that the curve is
symmetrical with respect to
the axes of the ellipse. For
y = we have
1= zh m= zb-
a
The curve has, therefore, two
points, H, H^, in the transverse
88
Digitized by VjOOQIC
298 DIFFERENTIAL CALCULUS.
axis, situated between the foci, and equidistant from the cen-
tre. Making f* = 0, we find y = =b n = =b - for the distances
from the centre to the points E, E', at which the curve meets
the conjugate axis.
By two diflFerentiations, we find
m \m/ n \n/ dfi ~~ '
m'\mj n-\n) \diij ^ n\n) dii*~
whence
d,^ 31/A-i
n\n/
Since the numerator of this expression is positive, the sign of
d'v . . . d^v
-7—5 will be the same as that of the denominator ; that is, -,—5
dfi^ ' ' dfi^
and V will have the same sign. The evolute at all its points
will therefore be convex towards the axis of x (Art. 155).
Moreover, we have
^ — — \^/ ^^ __ /^\* ^ .
\w/ n
Since this differential co-efficient becomes «ero for v = 0, and
infinite for f* = 0, we conclude that the axes of the ellipse are
tangents to the evolute at the points Hj jEP, and E, E* ; and
that, in consequence of the symmetry of the curve with
respect to the axes, these points are cusps.
182. Radius of curvature, and evolute of the parabola.
When referred to the principal vertex as the origin of co-
Digitized by VjOOQIC
CURVATURE AND E VOLUTE OF PARABOLA. 299
ordinates, the equation of the parabola is y^ = 2px, from which
we find
dy jp cf^y 2^^*
dx y' dx'^ y^
and, by means of these, the general value of p, Art. 171, be-
comes, without respect to sign.
y"^^)^ _{y'+p'f
To get the equation of the e volute, we must substitute the
values of -^j ^-f , in
dx dx^
^-f^ + (y-^)^| = o.
i+ftY+(»-')S=o,
which thus become
dx''
^P
.{y-v)^^=.0.
The elimination of x and y between these equations and the
equation of the parabola leads to the equation of the evolute.
From the second, we find
and, putting this value of v in the first, wo have
^ — f^+i^H = ^'i .*. /*— ^ = 30?.
P
Therefore we have
Digitized by VjOOQIC
300
DIFFERENTIAL CALCULUS.
and
j'^'
'- ^y (^ -!>)', '*- 2^ (^-1')'
for the required equation.
If the origin of co-ordinates be transferred to a point at
t!ie distance p in the direction of positive abscissae, the new
being parallel to the primitive axes, the equation of the evo-
lute takes the form
21p^' S21p^
We readily recognize that this curve is symmetrical with re-
spect to the axis of abscissae, and that it extends without limit
in the direction of x positive.
By differentiation, we find
Therefore, at the origin of co-ordinates, the axis of a? is tangent
to the curve, and this point is a cusp ; and, since the sign of
-i-^ is the same as that of y, the curve is at all points convex
toAvards the axis of x.
183. The expression for
the radius of curvature and
the equation of the evohite
of the hyperbola may be de-
^ duced from those for the el-
lipse by changing 6^ into
— J^ Thus we have, for the
radius of curvature.
Digitized by VjOOQIC
CURVATURE AND EVOLUTE OF CYCLOIjp, 301
and, for the equation of the evolute,
after making c^ = a* + 6^, — = m, -r- = w.
The fonn of this equation shows that the evolute of the hy-
perbola is composed of two branches of unlimited extent, and
symmetrical with respect to both axes of the hyperbola. It
has two cusps situated on the transverse axis beyond the foci,
and is convex at all points towards the transverse axis.
184:, Radius of curvature and evolute of the cycloid.
dy
By squaring the value of — , which, for this curve, is
CLCC
dy \2tr v
-^ = I ^ (Art, 146), and diiferentiating, we find
dx \ y
dx dx^ y^ dx' * * dx'^ y^'
dtt c?^ 1/
Substituting these values of ^ , t— , in the general expres-
sion for p, we have
?
Xow, Pm = IN, and,
from th e ri gh t-an gled
triangle PNG, we
have
PN=\/GNxNI;
that is,P^=V2ry.
Hence the radius
= -^: r. p=^2^/2ry.
/
n
a
Nr77
f
I
c
(/
)^
^j^
^ ^
J
6«p
■^»
N
B ^ — â–
I
Digitized by VjOOQIC
302 DIFFERENTIAL CALCULUS.
of curvature at any point of the cycloid is twice the normal
at that point; and, if PN bo produced until NQ z=i PN, the
point Q will be the centre of curvature.
183. The property just demonstrated leads, by very sim-
ple deductions, to the determination of the evolute of the
cycloid.
Produce the vertical diameter GN of the generating circle
(figure last article), making NL = GN, and on NL, as a diame-
ter, describe a circle. Through i, the lower extremity of this
diameter, draw LE parallel to Ox, meeting the axis O^D, pro-
duced in E, The arcs PN, NQ, belonging to equal chords,
are equal; .•. arc iV'Q z= ON: but OZ) = arc NQL ; .'. arc
LQ=^ ND =i LE, Thus it is seen, that if two equal cir-
cles lying in the same plane be tangent to each other, and the
one be rolled on the common tangent while the other is rolled
on a parallel to it at the distance of the diameter of the circle,
the points of the two circumferences which are common at the
time of starting will, during the motion, generate two equal
cycloids ; that generated by the point in the circumference of
the second circle being the evolute of that generated by the
point in the first.
This relation between the two cycloids, generated as just
described, may also be inferred from the property of the sup-
plementary chords of the generating circle, which are drawn
through the extremities of the vertical diameter of this circle
in any of its positions, and the corresponding point of the cy-
cloid (Art. 14G). For, since PG is tangent to the cycloid
00' B at the point P, NQ, or PJ*7 produced, is tangent to the
cycloid OQE at the point Q. Hence this last curve is the
locus of the intersections of the consecutive normals to the cy-
cloid 00' B, and is therefore its evolute.
Digitized by VjOOQIC
E VOLUTE OF THE CYCLOID. 303
186 • The application of the formulas of Art. 180 leads to
the same result.
From the equation of the cycldid, we have
dy l2r— y d'^y __ r
dx 'N y ' dx^ y^*
dy d^y
Substituting these values of — - , -^-^ , in the equations
ax dx ,
dy
we find from the second
2r r
— -(y— ^)-2 = 0, or2ry — r(y — f)zi:0;
" if
'y=-f (1);
and from tho first
which, if we replace y by the value just found for it, and trans-
pose, becomes
a, = ,. + 2.J-— ^ (2).
The equation of the cycloid
ojzrrcos.-i !Lll^._ V2ry — y2 ^g)^
by the substitution of these values of x and y, becomes
^4-21' J -L- = rcos.-i — ^- V-2ri' — y2 (4)^
which is the equation of the evolute. But from Eqs. 1 and 2,
it is seen, as it also is from (4), that there are no points of the
Digitized by VjOOQIC
304
DIFFERENTIAL CALCULUS.
curve for which p is positive. Making y negative, transposing
and reducing, we have, finally,
fi =rcos.-^ 1- V2ri' — y* (5).
Now, let the reference of the
curve be changed from Ox, Oy,
to Ex', Ey' ; positive abscissaa
being estimated from E to-
wards x\ and positive ordi-
nates from E towards y', x^
and yi denoting the new co-or-
dinates of the evolute. Since
OD =. nr, DE=i 2r, we have
11= OD- DIz=i Ttr - aJi, v = IF— FQ=2r- y^.
Eq. 5, by the substitution of these values of fi and v, becomes
nr - x^ = r cos "^ ^ ~ (2r - yO _^ s/2r{2r — y^) - {2r — y{)\
which reduces to
nr — Xy=. rcos.~^ ^ \'^2ryi — yj,
or
^zrirt - cos.-^ — J — V2ryi — yj.
: introducing this in the
But cos.-i ^'^LZ:^ = TT - cos.-i "^ — ^':
r r
equation above, it becomes, finally,
Xi = r cos.~^ ^- — V2ryi — yj.
This equation difi'ers in no respect from (3), except in having
a?!, yi, instead of x and y ; which shows that the evolute of a
cycloid is an equal cycloid, situated, with reference to the axes
Ex*, Ey', as the involute is with respect to the axes Ox, Oy,
187, It has been proved (Art. 180) that the length of an
Digitized by
Google
ENVELOPES OF PLANE CURVES. 305
arc of the evolute to any curve is the diflFerence of the radii
of curvature corresponding to the extreme points of the arc.
In the cycloid at the point (last figure), p = 2\^2ry=:0-
hence PQ=.2PN i^ the length of the arc OQ; and, with re-
spect to the given cycloid, arc PO' ^=z 2P0.
To express the arc PO' in terms of the ordinate of the point
P, we have
arc PO' = 2PG = 2V2r X GC.
But GC^lr-^y: .*. arc PO' = 2 v/i/^'-^^^ry^
Making y = 0, in this value of PO^, we have arc 0' 0'^=:. 4r :
hence the entire arc of the cycloid is four times the diameter of
the generating cirde.
188, Envelopes. If one or more of the constants enter-
ing the equation of a curve be changed in value, we shall have
a new curve, differing in position and dimensions from the
given curve, but agreeing with it in kind : that is, if the given
curve be an ellipse, the new curve will be an ellipse ; if a pa-
rabola, the new curve will be a parabola. The constants which
thus change in value are called the variable param^ers of the
curve represented by the equation.
The locus of the intersections, if any, of the consecutive
curves of the same species, — that is, of curves whose equa*
tions are derived from a given equation by causing one or
more of its constants to vary by continuous degrees, — is
called an envelope.
Suppose F{x^ t/y a) =1 (1) to be the equation of a curve
involving, among others, the constant a; and let a bo taken as
the variable parameter. Changing a into a + A, the equation
becomes P(a;, y, a + A) = (2), which represents another
curve belonging to the family of that represented by (1).
89
Digitized by VjOOQIC
30G DIFFERENTIAL CALCULUS.
By Art. 5G, Lq. 2 may bo put under the form
F{x,y,a)+1iF^{x,y,a + (ih) = (3).
Observing that F' signifies the derivative, with respect to a,
of the function symbolized by F, Eqs. 1 and 3, when sunulta-
neous, are equivalent to
F{x, y, a) = 0, F' {x, y,a + Ch) =0 (4);
and the values of x and y, determined by the combination of
these equations, will be the co-ordinates of the intersection
of the curves of which (1) and (3) are the equations.
If /i. be diminished without limit, Eqs. 4 become
F{x,y,a) = 0, F^{x,y,a) = (5);
and t!ie point determined by these equations is the limit of
the intersections of the curves of which (1) and (2) are the
equations. Tlie equation which results from the elimination
of a between Eqn. 5 will evidently be the envelope of the
family of curves represented by the equation F{Xf y, a) = 0,
and of which the individual curves are formed by assigning
diiferent values to a.
The envelope touches each curve of the series at the point
common to the curve and the envelope. This is proved by
showing that the envelope and the curve, at the common
point, have the same tangent.
Since (1) becomes the equation of the envelope when in it
the value of a, deduced from the second of Eqs. 5, is substi-
tuted, let (1) be differentiated under this supposition, treating
X as the independent variable, and a as a function of x and y,
and we have for finding the value of V- for the envelope,
dF dF dy dFi ^ , ^ _^^ ? _ o (6).
dx"^ dy dx^ da\dx^ dy dx)
Digitized by VjOOQIC
ENVELOPES OF PLANE CURVES. 307
But, at the point of intersection of the envelope with the given
curve
^^ = F'{x,y,a) = Q',
hence (6) reduces to
which is the same as that obtained by the differentiation of
(1) : whence, at the common point, the tangent line to the en-
velope is also a tangent line to the given curve.
Ex. 1. Find the envelope of the family of straight lines
derived from the equation y z= ox -| — , by causing a to vary.
Differentiating with respect to a, x and y being constant,
we have
X -o — 0: .-.a^i -;
a* \x-
y =i::^2 VmXj y'^ = 4mx :
hence the envelope is a parabola.
Ex. 2. Find the envelope of the straight lines represented
by the equation y = ax + (6^a^ + c^)i, when a is made to
vary.
Differentiating with respect to a, we find
^ . h^a c X
0=x-\ 7-: ,*. a=
Substituting this value of a in the given equation, we have,
after reduction,
^ V y' - 1
62 "t- c2 — ^'
which is the equation of an ellipse referred to its centre and
axes.
In each of the examples just given, it has been required to
determine the curve from the general equation of the tangent
Digitized by VjOOQIC
308 DIFFERENTIAL CALCULUS.
line. This process, being the inverse of that for finding the
equation of the tangent line, is sometimes called " the iuverso
method of tangents."
If a point be taken on the axis of a; at a distance from the
origin equal to m, and a line be drawn through this point,
making, with the axis of x, an angle having for its tan-
a
gent, the equation of this line is y = — {x — m), and it inter-
a
sects the axis of y at the distance — from the origin. The
equation of the perpendicular to this line, at its point of inter-
section with the axis of y, is y = ax-{- ~ . Hence the geo-
metrical interpretation of Ex. 1 is, " From a point in the axis
of x, at the distance m from the origin, draw lines intersect-
ing the axis of y, and to these, at their points of intersection
with the axis of y, draw perpendiculars ; required the enve-
lope of these perpendiculars:" and that of Ex. 2, "To find the
envelope of a series of straight lines, so drawn that the product
of the two ordinates of any one of these lines corresponding
to the abscissae, -j- h, — t, shall be equal to c^."
Ex. 3. Find the envelope of all the parabolas given by the
equation y := ax — -~ — x^j by causing a to vary.
Differentiating with respect to a, we have j
= x ; ,*. a='^ :
^ X
whence, by substituting this value of a in the given equation,
we find, for the envelope,
^^ = 2p f I - y\ or x^ +- 2p2^ -^ = 0,
which is the equation of a parabola.
Digitized by VjOOQIC
ENVELOPES OF PLANE CURVES. 309
Ex. 4. Find the envelope of the normals drawn to the dif-
ferent points of a given curve.
Let the equation of the curve be y =:/{x) ; then the equa-
tion of the normal is
x,-x + iy,-y)f^ = (1),
in which ajj, j/i, are the running co-ordinates of the normal.
From the equation y =/(x), y and -^— can be expressed in
terms of x, and thus x becomes the variable parameter in
Eq. 1. Hence the equation of the required envelope may be
found by eliminatmg x between (1), and
-•+(^.-^)g-(i)=» (^>.
which we get by differentiating (1) with respect to x.
Comparing (1) and (2) with the formulas, Art. 171, it is
seen that a?i, yi,are the co-ordinates of the centre of curvature
of the point (x, y) of the given curve ; that is, the envelope
of the normals of a curve is the evolute of the curve.
189. When the equation, representing the family of curves
whose envelope is sought, involves several, say n variable pa-
rameters, and these parameters are connected by n — 1 inde-
pendent equations, instead of effecting the elimination of n — 1
parameters, and then differentiating with respect to that which
remains, we may proceed as follows : Let the equation of the
curve be
i^(a;,y,a,6, c...) = (1),
and let the n — 1 equations of condition for the parameter be
/i(a,b,c...) = 0^
f,{a,h,c...) =0
(2).
/n-i(a,^c...)i=0^
Digitized by VjOOQIC
310
DIFFERENTIAL CALCULUS.
By reason of Eqs. 2, n — 1 of the parameters may be regarded
as functions of tlio remaining; one taken as independent. Let
this bo a, and differentiate Eqs. 1 and 2 with respect to it,
thus getting
dF dFdb dFdc _
da^'db da^'did^^' ^''
da
+
da "^
db
df,
db
db
da
db
da
d/i dc
dc da
dA dc
dc da
+ •••=0
+
=
d/„-i , <y„-i db ^ d/„.
db da dc
+ •••=0
y (4).
da ' db da ' dc da
Now, it is plain that if, in Eqs. 1 and 3, all the variable para-
meters and their functions be expressed in terms of a, and a
be then eliminated between these two equations, the resulting
equation will be that of the envelope. To effect this elimina-
tion, we have 2n equations ; viz., the n given equations, and
their n differential equations : but there are only 2;i — 1 quan-
tities to eliminate; viz., the n quantities a,h,c,,., and the
n — 1 quantities -^} -r *"'- hence the elimination is possible.
Multiply the first of Eqs. 4 by the indeterminate Aj, the sec-
ond by >l2, and so on, and add the results and Eq. 3 together:
we thus get
da+^^a+ Va+ + ""' da
r(dF,^dAdA df„_Adb^
+ [-db+^'db+^-'db+ ^ "-'"db-Jda
,/dF dA ,, dA rf/»_,\ dc
^=0 (5).
Digitized by VjOOQIC
ENVELOPES OF PLANE CURVES, 311
By means of the n — 1 indeterminate multipliers ^i, ^2 • • •? ^n-n
we may satisfy n — 1 conditions. Let these be that the co-ef-
ficients of ., , ,- , in Eq. 5, shall reduce to zero. These,
da da
together with that expressed by Eq. 5 itself, lead to
â– (6).
We have now the 2n — 1 quantities a^b^ c. ,,, l^, ^2- • •; ^n-u
to eliminate between the 2n equations (1), (2), and (6); and
the result, being an equation between x and y only, will be the
equation of the required envelope.
190m When the general equation of the family of curves
contains only two variable parameters, and they are connected
by one equation, the process admits of the simplification, and
the result takes a form the same as those in Art. 128.
Ex. 1. Find the envelope to the different positions of a