PREFACE

THIS book relates, not to the general theory of curves,but to the definite problem of ascertaining the form of a

curve given by its equat ion in Cartes ian coordinates, in

such cases as are likely to arise in the actual appl icat ions

of Analyt ical Geometry. The methods employed are exclu

sively algebraic, no knowledge of the D ifferential Calculus

on the part of the reader being as sumed .

I have endeavored to make the treatment of the sub

j ect thus restricted complete in all essential points, without

exceeding such limits as its importance would seem to

j ust ify. This it has seemed to me poss ible to do by intro

ducing at an early stage the device of the Analytical Tri

angle, and us ing it in connect ion with all the methods of

approx imat ion .

In constructing the triangle, which is essentially New

ton ’s parallelogram, I have adopted Cramer’s method of

represent ing the possible terms by points, with a dist in

guishing mark to indicate the actual presence of the term

in the equat ion . These point s were regarded by Cramer

as marking the centres of the squares in which , in New

PRE FACE

ton ’s parallelogram , the values of the terms were to be

inscribed ; but I have followed the usual pract ice, first

suggested, I believe, by Frost, of regarding them merelyas points referred to the s ides of the triangle as coordinate

axes . I t has, however, been thought best to return to

Newton ’s arrangement, in which these analyt ical axes are

in the usual pos it ion of coordinate axes, instead of placing

the third s ide of the triangle, like De Gua and Cramer, in

a horiz ontal pos it ion .

The third s ide of the Analyt ical Triangle bears the

same relat ion to the geometrical concept ion of the line at

infinity that the other sides bear to the coordinate axes .

I have aimed to bring out this connect ion in such a way

that the student who desires to take up the general theoryof curves may gain a clear view of this conception, and be

prepared to pass readily from the Cartesian system of

coordinates,in which one of the fundamental lines is the

line at infinity, to the generaliz ed system, in which all

three fundamental lines are taken at pleasure .

Lists of examples for pract ice will be found at the end

of each section . These examples have been selected from

various sources,and classified in accordance with the sub

j ects of the several sections .W. w. J .

U . S. NAVAL ACADEMY,

November, 1884.

CONTENTS

E quations solved for one variableD iametersL imiting tangents .

Asymptotes to an hyperbolaParabolasCurvi linear diametersEmployment of the ratio of the coordinatesPoints at infinity .

Asymptotes general methodSymmetry ofcurves

EXAMPLE S I

The analytical triangleIntersections with the axes

T he line at infinityAsymptotes parallel to one of the axes

Parabol ic branchesParallel asymptotesT angents at the originT angents at the points of intersection W 1th an

Intersection ofa curve w ith a tangentIntersection ofa cubic w i th its asymptotes

EXAMPL E S I I

CONTE NTS

Approximate forms ofcurvesApprox imate forms at infinityRadius ofcurvature at the originMethod ofdetermining the equations ofapprox imate curvesThe analytical polygonConstruction of the approx imating curvesS ides of the polygon representing more than one approx imate formImaginary approx imate forms

EXAMPLE S I I I

IV

Second approx imation when the s ide of the polygon%gives only

first approx imationSelection of the terms which determine the next approximationSuccessive approx imationAsymptotic parabolasContinuation of the process ofapprox imation

EXAMPLE S IV

Cases ofequal rootsCusps

TacnodesCusps at infinityRamphoid cusps

CircuitsAux iliary lociTangents at the intersect1ons ofaux il iary loc iLoci representing squared factorsPoints in which several aux iliary loci intersect

EXAMPLE S V

C U R V E T R A C I N G

E quations 50171841for One Var iaéle

1. THE equation of a curve i s in these pages supposed tobe given in Cartesian coordinates and the curve is said to belmcm’

, when the general form of it s several parts or branchesi s determined, and the pos it ion of those which are unlimited inextent i s indicated . In the diagrams the coordinate axes will,for convenience, be assumed rectangular ; but the methods areequally applicable to oblique axes . When it is poss ible to solvethe equat ion for one of the coordinates, so as to express itsvalue in terms of the other, the result ing form of the equat ionaffords the most obvious method of tracing the curve . Thismay be done for either variable when the equat ion is of thesecond degree in both variables, in which case the curve is acomb. For example, let the given equat ion be

2x2 — 2xy + y

2

Solving for y , we have

y = x

CUR VE TE ACI A/G [Art. 1

Thus, for any given value of x , we have two values of y ; andif we put

2: x

the equation of the curve becomes

r = r'rr V( 2x

D iameters

2 . E quat ion (3) represents a straight line, and equation (4)shows that the two ordinates for the curve may be found byadding to and subtract ing from the ordinate of the straightline the same quant ity. In other words, the chord j oining thetwo point s of the curve which have the same abscissa, that is ,any chord parallel to the axis of y , is bisected by the straightline . This line is therefore called a diameter of the curve .

The diameter represented by equat ion (3) is constructed inFig. I .

3 . The radical V(2x — x2

) , which is half the length of thechord

,varies with x

,and vanishes for the two values

x = o

the corresponding point s on the diameter are therefore alsopoints of the curve . Writ ing the radical in the form

it i s obvious that all values of x between 0 and 2 give real

values to the radical,s ince they render both of the factors

D I AME TE E S

under the radical S ign pos it ive, and that all values°

exterior tothese limits give imaginary values to the radical . Hence allthe real points of the curve lie between the straight lines x O

and x 2 , as represented in Fig . I . Theselimiting lines are evident ly tangent s to thecurve at the point s where they cut the diameter. Putt ing the radical in the form

Y

( I

we see that the maximum value of the rad ”9"

ical is unity, and corresponds to the value x 1 , or the middlepoint of the diameter. The lines pass ing through the corresponding point s of the curve parallel to the diameter are

obviously tangent s . The curve is an ellipse .

4 . As a second example let us take the curve

yz — xy + y

Solving for y ,

y = % (x I ) rt 236 -

3)

Putt ing x2

2x 3 0 , we find that the radical vanisheswhen x z : 3, and when x z 1 the equation may thereforebe written in the form

J )J2=(x 1 ) i %v[ (x 3) (x I ) ]

The straight line

y % (x I )

is therefor e a diameter, and the lines

as : x = 1

CURVE TRACIN G

are tangents ; but the radical i s, in thi s case, imaginary forvalues of x between the limit ing values, since such values makeone of the factors posit ive and the other negat ive

,and real for

values exterior to the limits . Hence the curve consists of twobranches touching the lines x z 3 and x z : I at their intersect ions with the diameter represented by equation as inFig. 2 .

Asymptotes

5 . Putting the rad ical in the form

V[ (x I ) 2

we see that its value increases indefinitely as x increases innumerical value, but that iti s always less than x I .

Thus the points of thecurve lie between the lines

y = % (x I ) i % (x + (4 )

that is to say, the lines

= x and y- 1 .

The curve approaches indefi

nitely to these lines, since the difference between x I andthe radical decreases without l imit as x increases numericallyThe lines are therefore asymp totes . The asymptotes and thetwo branches of the curve are const ructed in Fig. The curve

is an hyperbola.

AS YMP TOTE S

6 . It is evident from the above method of finding theasymptotes that the position of these lines will not be affectedby a change in the value of the absolute term . For example,if the absolute term in equation ( I ) be changed from 1 to 2 , theonly change will be that the radical will become V[(x I )

2

and the asymptotes will st ill be represented by equat ionMoreover

,the value of the absolute term may be such as to

make the equat ion identical with the equat ion which repre

sents the two asymptotes . In the present instance, the latterequation is (y — x ) (y -I—I ) O, Or y

2 —xy + y — x :-_ O, which

is equation ( I ) with the absolute term changed to z ero . If, onthe other hand, the absolute term is changed to I , the radicalbecomes s 2x 5) which cannot be made to vanish as inArt. 4, but is always numerically greater than x I . Thusthe curve will no longer cut the diameter, as in Fig. 2 , but willl ie on the other s ide of the asymptotes . It will, in fact, be theconj ugate of the hyperbola drawn in Fig. 2 .

7. If we solve equat ion ( I ) of Art. 4 for x , we have

x = y +

This form of the equation Shows that for each value of y therei s a single value of x , but this value i s infinite when y I .

The line y I i s therefore an asymptote . The value ofI

the fract iony I

i s posit ive for all values of y algebraically

greater than I , and negat ive for all values less than I ;

hence the part of the curve above the line y I lies on theright of the line y z x

,and that below the line y I on the

left of the line y z x . As the numerical value of y increasesindefinitely, that of the fract ion decreases indefinitely, whichShows that y x i s also an asymptote .

CURVE TRA (II /VG [Art 8

Pambolas

8 . It follows from what i s shown in Art . 6 that an equationof the second degree will represent an hyperbola when theterms of the second degree can be resolved into real factors .In this case, they const itute the difference of two squares . On

the other hand, the equation represent s an ellipse when theseterms const itute the sum of two squares, as in equation ( I ) ,Art. I . The intermediate case is that in which the terms forma perfect square . For example, let the equation be

x2 + 2xy + y

2 2x 6y + I = o.

Solving for y ,

hencey + x (fl

is the equation of the diameter bi secting chords parallel to theax is of y . The radical is real for all values of x algebraically

less than 2, and imaginary for all values

greater than 2 ; hence the curve lies on theleft of the line as 2

, which it touches atthe point (2 , I ) where it cuts the diameter.If we solve for x , we find, in like manner,

x + y— I = i 2 V:yy

hencex + y

— I = o

is the equation of the diameter bisecting chords parallel to theaxis of x , and the curve touches the line y O at the point

( I , O) . The value of the radical in either form of the equat ionincreases without limit

,but there is no asymptote . The curve

is a parabola.

§ I] CUR VIZINE AR D IAME TE E S

Carailinear D iameters

9 . We pass now to an example in which the curve is acubic

,although the equat ion is only of the second degree with

respect to either variable . Given the equat ion

2x 2

y x 2 + y2 2x

solving the equation as a quadrat i c for y , we have

._ x z:t V(x 4 + x 2 _

or, resolving the quantity under the rad ical S ign into factors,

x 2:t I ) (x 2 x

Putting y O in equat ion ( I ) , we find that the curve intersects the axis of a: at the origin , and at the point (2 , O) ; andputt ing x 0 , we find that it meets the axis of y at no pointexcept the origin . E quat ion (2 ) gives in general two values of

y for each value of x , and the corresponding points are equallydistant from that point of the parabola

x 2 ( 3)

which has the same abscissa . In other words, chords parallelto the axis of y are bisected by this parabola, which is therefore a diameter. This diametral parabola is constructed inFig. 4

10 . The quantity which occurs under the radical S ign inequation namely

x (x x

CURVE C’

L/VG [Art . I o

changes S ign only twice as x passes through all poss ible values,since it vanishes only when x O and when x z I . I t is

,in

fact, negative for all values of xbetween these limits, and pos it ivefor all other values of x . Thereis therefore no part of the curvebetween the lines 4: 0 andx I ; and these limit ing linestouch the curve at the pointswhere they intersect the diametral curve represented by equation

that is, at the origin and atthe point ( I , - I ) . See Fig . 4 .

11. If we solve equat ion ( I ) for x ,we have

z y— r

One value of x becomes infinite when 2} I O ; the othervalue takes an indeterminate form in equat ion but, referringto equation ( I ) , it is found to be re Hence the line

is an asymptote, cutt ing the curve at the point as inthe diagram. The express ion under the radical Sign evidentlyvanishes for y z : I and, putt ing the radical in the form

-

r) ( 1 + y 2rd ] ,

it is seen to be imaginary for all values of y greater than unity,and real for all other values . Hence the curve lies entirelybelow the line

y = 1 3

§ I ] E MPL OYME N T OF THE RA 77 0 OF coORB /NA TE S

and,subst itut ing this value in equation we find that this

l ine touches the curve at the point I ,

Employment of Me R atio of tne Coordinates

12 . If we denote the ratio Of the coordinates of any pointby m, thus

the value of m may be regarded as a new coordinate of thepoint

,which

,in conj unction with one of the other coordinates,

will serve to determine the posit ion of the point . The point swhich have a common value of m are s ituated upon the straightline

y mx

which passes through the origin ; and, when a point is determined by the values of m and x , it is virtually determined asthe intersect ion of a l ine of this character and a line parallelto the ax is of y .

If, now, we eliminate y from the equation of a curve bymeans of the equat ion y mx , we Shall have the equat ion ofthe curve in the form of a relat ion between m and x . Whenthis form of the equat ion can be solved for x , we can, by givingvalues to m, determine any desired number of points on thecurve

,and can trace its form with as much facility as when y is

expressed in terms of x .

13 . As an illustration, let the equation of the curve be

y3

CURVE TRACING' [Art 13

Putting mx for y , this becomes

(m3 I ) x3 ( 2m I ) x 2 o. ( 2 )

The factor x 2 in this equation indicates that the curve passestwice through the origin for a straight line generally cuts thecurve in three points, and equation (2 ) Shows that for all l inesof the form me two of these point s have the abscissax 1: O. R ej ecting this factor, and solving for r , we have

in which x is expressed in terms of the variable m and denotesthe abscissa of the third point in which the line y mx meet sthe curve . The ordinate Of this point, which we shall denoteby P , is

y = mx

and,assuming the value of m, we may determine the pos it ion

of P by means of any two Of the three coordinates x, y and m.

14 . Beginning with m 0 , equation (3) gives x I , and Pis at A,

the point ( I , O) , the line y mx coinciding with theaxis of x ; as m increases, the line rotatesabout the origin, coinciding with the axisof y when m is infinite . When m I ,

the values of x and y are both infinite ;thus

,as m varies from O to I , x increases

from I to infinity, P describing the infinitebranch AB in Fig. 5. Passing the valuem 1 , x changes S ign ; and as m passes

from I to 00,P describes the infinite branch C0 , arriving at the

origin when m z oo,s ince equation (4) then gives y z 0 .

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CURVE TRA (fl /VG [Art 16

Asymptotes

16 . A real point at infinity usually corresponds to an-

ymptote whose direct ion is determined by the value of m.

To illustrate the method of finding it s posit ion, let us determinethe asymptote to the curve draw n in Fig. 5. The equation ofthe curve, equation Art. 13, may be written in the form

y— x

Now,when the point P recedes indefinitely on the branch AB ,

x and y become infinite, but y 75 may nevertheless have afinite value . To find this value, put mx for

y in equat ion ( I ) , and then make m z I andx r: 00 ; or, what comes to the same thing,

put y x , and x z 00 . The process maybe expressed thus,

5

y— x

2

J’Z -i- xy

-i- xy= x = oo

in which the suffixed equat ion is to be understood to meanthat the rat io of y to x is one of equality when x is infinite .

The result shows that , when this is the case, the quantity y x

approaches the finite limit I . If, now, we draw the straightl ine

y x

in which this quantity has constantly the value I , it is evidentthat the point P moving along the curve approaches indefinitely to this line, which is therefore an asymptote, as repre

sented in Fig . 6 .

S YMME TR I CAL CUP VE S

17 . It i s obvious that , had the equat ion contained terms ofa degree lower than the second, they would have vanished inthe process of finding the asymptote . For example, the curve

y3

whose equat ion differs from that of the curve drawn in Fig. 6

only by containing the term x, will have the same asymptote

for we have

y— x — I .

Thus, if the equation of the curve is of the nth degree,the

asymptotes generally depend only upon the terms of the nth

and (n I ) th degrees .

Symmetry of Curnes

18 . In some cases, the form of the equation indicates thesymmetry of the curve in certain respects . For example, ifthe equat ion contains powers of y with even exponent s only,the curve is symmetrical to the axis of x ; for, in this case, if thepoint (a, 5) sat i sfies the equat ion , the point (a , o) symmet

rically s ituated with respect to this axis also sat isfies the equat ion . In like manner, if the equat ion contains powers of a

:

with even exponent s only, the curve is symmetrical to the axi sOfy . Again, if the sum of the exponent s of x and y in eachterm is an even number, or if it is in each term an odd num

ber, the curve is symmetrical with respect to the origin as acentre for, in this case, if the point (a, 5) sat isfies the equat ion ,the point a, 6) will also sat i sfy it . If a: and y are interchangeable, the curve is symmetrical to the line bisect ing theangle between the axes ; since, if the point (a , o) sat isfies the

ew eVE TA’Acave [Art 18

equat ion,the point (6, a) will also sat isfy it . These considera

tions will be useful in the case‘

ofsome of the following examples.

E xamp les I

Trace the curves whose equations are given below

1 . x2 + xy

—y2 + 3x o .

2 sxz

4xy + 4y2 Sr

—4 0

3 . x2 — 2xy + y + 2 O.

4 . x2 2xy + y

2 2x 2y + 1

5. 6x 2 + 4xy + y2 -

3x— 2y

— 2

6 . x2 8xy + l 6y

z —6x 1 z y + 9 O.

7. xyz — xZ —

y2 = 0 ,

8 . x 3 x -y O.

9 . az

y a zy a3 0 .

I O. y3 x 2 (x a ) .

1 1 . x 3 + x 2 —y2 = o .

1 2 . az

y 2axy a z x a2

y O.

1 3 . x 4 = y-’v (4a

zx 2 ) .

1 4 4y2

(x a) “ ( 5x a ) ’ o

1 5. x2

yz

a2

(x2+ y

2 ) .

1 6 . (x 2 a ) xy d (x a ) (x 3a) .

1 7. x 3

1 8 . x 4 3axy2 2 ay

3 o.

1 9 . y3 x 3 — y + 4x o .

2 0 . x 3 + y3 - x2 —

y2 = o.

2 1 . x 3 + y3

2 2 . x 3 y3 + ( 2y x ) 2 o .

2 3 (x + y ) (x 2 + y’

) 2 6W24 . y

4y3x x 3 2n o.

2 7. y3 x 2 (x a ) .

8 6x ( 1 - x )y = I 3x .

THE ANAL YTI CAL TRIANGLE

The Analytical Tr iang le

19 . LE T two intersect ing straight lines be drawn ; and leteach term in the complete equation of the nth degree be represented by a point whose coordinates with respect to these linesare the exponents of x and y respect ively in that term we Shallthus have a triangular arrangement of point s, there being n 1

points on each s ide of the triangle . Thus, if n z : 3, Bwe have the arrangement given in Fig. 7, in whichthere are four points on each s ide of the triangle .

The diagram thus formed is called the analytical

triang le ; the lines of reference are the analy ti Fig . 7cal axes of a: and y respect ively. The points upon 0A, the analytical axi s of x , represent the terms of the equation which donot contain y ; the points on OB represent the terms whichdo not contain x ; and the points on the third s ide, AB , of theanalytical triangle represent the terms Ofthe nth degree .

20 . The equation of a given curve is said to be p laced upon

tke analy tical triang le when the point s which represent theterms actually occurring in the equation are markedin some convenient manner. Thus the equation

2x’

y

of the curve traced in Fig. 4, page 8, is placed onthe analyt ical triangle in Fig. 8. In thus placing

the equation,no attention is paid to the coefficients, the obj ect

CURVE TRAcave [Art 2 0

being simply to indicate the presence of certain terms in theequat ion, and the absence of others . When the values of theterms are required, it is of course necessary to refer to the equation of the curve.

Intersections z oitfi t/ie Axes

21. If we put y O in the equation of a curve, and supposex to have a finite value, all the terms except those representedby points on the analyt i cal axi s of x w ill vanish . Hence theresult of equat ing these terms to z ero is an equation determining all the finite points in which the curve meets the axis of x .

This equat ion will generally have n roots real or imaginary ;thus, if n 3, its form will be

A + B x + Cx 2

+ Dx3

a cubic equation . If there be no absolute term in the givenequation, that is, if in equat ion (a) A 0 , one of the roots willbe z ero

,and the curve will pass through the origin . On the

other hand,if the term 0 x 3 be absent , the equation reduces to

one of the second degree, and determines but two finite intersect ions with the axis of x ; but, putt ing equation (a) in theform

x 3 x 2 x

I

which is a cub ic equation determining three values of we see

I

that,when D 0 , one of the values of Is z ero, and therefore

a:

one of the values Ofx is infinite . Hence, in this case, one of theintersect ions of the curve with the axis of x is said to be at

infinity . The geometrical meaning of this is that a straight

§ I I] I N TE RSE CTI ONS WI TH THE AXE S

line usually cuts the curve in three points but, when the line isbrought into coincidence with the axis Ofx , one of these pointsOf intersection recedes indefinitely and disappears . An inspec

t ion of Fig . 8 shows that the curve whose equat ion is thereplaced upon the analyt ical triangle cuts the axis Ofcc at infinity,at the origin, and at one other point .

22 . In like manner, an inspect ion of Fig. 8 Shows that thecurve cuts the axis of y at infinity, and tw ice at the origin it istherefore said to meet the axis of y in two coincident points atthe origin . The geometrical meaning Ofthis is that two of thethree point s in which a straight l ine cut s the curve come intocoincidence at the origin when the cutting line is broughtinto coincidence with the axi s of y . I t is evident that this willhappen whenever the curve has two branches passing throughthe origin, and also whenever the axis of y is a tangent to thecurve ; but, S ince in the present instance the curve cut s theaxis of a: but once at the origin, we infer that the curve has asingle branch passing through the origin and touching theaxi s of y .

23 . It is evident that there must always be at least one ofthe marked point s upon each analyt ical axis for otherwise theequation could be divided throughout by y or by x , and its locuswould not be a proper curve of the nth degree, but the combinat ion consist ing of the straight l ine y z 0 , or x O, and acurve of the (n I )th degree . If there is but one marked pointon an analyt ical axi s, it divides the side of the analyt ical triangleinto parts which indicate respect ively the number of t imes thecurve cuts the corresponding geometrical axis at the originand at infinity, and the curve cut s the axis in no other point .But, if there are two or more marked points upon the analyt icalaxis, the number of spaces betw een the most distant of thesepoints indicates the number of finite points of intersection with

CURVE TRACI NG’ [Art 2 3

the axi s dist inct from the origin . When there are two or moreof these roots, a pair of them may be imaginary, or they may bereal and equal . In the latter case, the curve cuts the geometrical ax is in two coincident points, which generally indicatestangency to the axis , as in the case Of the two z ero roots for ycons idered in Art. 2 2 .

Tile L ine at Infinity

24 . I t was Shown in Art. 1 5 that, if we put equal to z erothe group of terms of the highest degree in the equat ion of acurve, the result ing equat ion gives the values of m,

the rat io of

y to cc, for the points at infinity. These terms are those whoserepresentat ive point s are situated upon the third side, AB , ofthe analyt ical triangle ; and it is customary to speak of thisequat ion as determining tke intersections of Me curve w itk tke

line at infinity. E ach Of the real intersect ions generally determines

,as shown in Art. 16, the direct ion of an asymptote .

25 . The equation in quest ion is of the general form

Lac” Man—{y Ry

” = o,

*

which,being of the nth degree, determines n values of

“

it; thus,the line at infinity, l ike an ordinary straight line, i s said to cutthe curve in n real or imaginary points . If the term Ln”

,rep

resented by the vertex A of the analyt i cal triangle, is want ing

I f the equation of the curve is as usual rendered homogeneous by the introduction of the letter a , wh ich may be regarded as denoting the unit of length, thisequation can be derived from the equation of the curve by putting a 0 ; just as

the equation for the intersections w ith the axis of x is derived by putting y 0

Hence, as y o is the equation of the ax is of x, so the impossible equation a o

is regarded as the equation ofthe l ine at infinity.

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CUR VE TRACING

fore the equat ion of the asymptote . Compare A rt. I I . I t isevident that, in this process, all the terms vanish except thosecontaining the highest power of x which occurs in the equat ion,that i s to say, the terms represented by the points adj acent tothe vertex A of the analyt ical triangle ; hence the result isarrived at by s imply putting the sum of these terms, namely,z x

z

y x2

, equal to z ero, and rej ecting the common factor x 2

Paraéolic B rant /cos

27. The attempt to find an asymptote parallel to the axis of

y in the case of the curve considered in the preceding art iclesresults in an imposs ible equat ion . Thus, dividing equation ( I )by y ,

we have

+ I + 2

in which it i s imposs ible to make y infinite while x remainsfinite ; we infer, therefore, that the curve has parabolic branc/z esin the direct ion of the axi s of y ; that is, infinite brancheswhich, l ike the parabola, have no asymptote . This is owing tothe absence from the equation of the curve of the term repre

sented by the point on the third side A3 of the analyt ical , triangle adj acent to the vertex B . See Fig . 8, page 1 5. Now, inconsequence of the absence of this term, the equat ion for theintersect ion with the line at infinity has a double root x 0 ,

8 0 that the line at infinity is said to meet tke curve in two coinci

dentpoints where it crosses the axis of y . It is evident that thiswould also occur if the curve had two asymptotes parallel tothe axis of y ; but, s ince Fig . 8 shows that the curve meets theaxis of y but once at infinity, the diagram indicates the parabolicbranch .

28 . In general,the line at infinity is said to meet the curve

PARAB OLI C B RAN CHE S

in two coincident points whenever the equat ion for its intersect ions has a pair of equal root s, and this usually indicates a parabolic branch . For, let the equat ion of the curve be written inthe form

P , P. ( I )

where P,,denotes the sum of the terms of the nth degree,

the sum of the terms of the (n I )th degree, and SO on : then,if m

,is a double root of the equat ion P , O, equation ( I ) may

be written in the form

(y mix )Z

Qu - 2‘ l" Pn— I Pn—z ( 2 )

in which Q 18 an expression Of the (n 2 )th degree . Put

t ing y m,x in the expressions etc . , the equation

takes the form

( J , mlx )2xn—2

qn_ 2 xn— r

pn_ l xn—2

Pn_ 2

in which etc . , are numerical quant it ies . D ividing bya

rm- 2

, and making x infinite, it is plain that this gives an infinitevalue to y m,

x , except w/ien pn__I 0 . Hence in general wehave a parabolic branch , and Me line at infinity is reg arded as a

tang ent to the curve. This may be further explained as meaningthat, whereas the asymptote to an infinite branch is the tangentwhose point of contact i s at infinity, a parabolic branch is onefor which the tangent whose point of contact is at infinity is altogether at an infinite distance .

Parallel Asymptotes

29 . The except ional case ment ioned in the preceding art icleoccurs when the subst itut ion y m,

x reduces the expressionto z ero ; that is, when contains y m

,x as a factor.

CURVE TRACING [Art 29

When this is the case, the value of y m,x , when x and y are

infinite, may be determined by a quadrat ic equation. Forexample, let the equation of the curve be

x3 2x2

y xyz

an2

axy 2a2x a

’

y o,

which may be written in the form

x (x y )2

ax (x y ) 2a2x a

z

y o .

Dividing by x , and making y as when x is infinite, we havethe equation

(x — y )’

+ d (x -

y )

which g ives

a dx —y + —

2

and x —y + -

2

- ( I

the equations of two parallel asymptotes .The case here considered of course includes that in which

there are no terms of the (n I ) th degree ; that is, when0 . It is to be noticed that the asymptotes , being de

termined by a quadratic equation, may be imaginary, in which

case there will be no corresponding infinite branches .

30 . An inspection of Fig. 9, in which equation ( 1) is placedupon the analytical triangle, Shows that the curvecuts the axis of y at the origin and in two coincident points at infinity ; and, s ince it cut s the lineat infinity but once in the direct ion of the ax1s

A of y ,we infer that the ax is of y i s a tangent at

Fi9° 9 infinity, that is to say, an asymptote . The

method given in Art . 26 also shows that the asymptote is in

this case the axis itself.

PARALLE L AS YMP TOTE S

The intersect ions with the ax i s of x are determined bythe equation

The points indicated are constructed in 10 , together withthe asymptotes determined in the preceding art icles . I t is evident that a lineparallel to the parallel asymptotes cut sthe curve in two points at infinity, andhence each of these asymptotes meetsthe curve in t/z ree point s at infinity,and in no other point . Thus the curvecons ists of three disconnected branches

,

as represented in the diagram .

Tang ents

32 . The form of the curve i s more precisely ascertained bydetermining it s inclinat ion at known points . D ividing theequat ion of the curve, equat ion Art. 29, by 2 ,

we have

x z — z xy + y2+ ax — ay

x

Now suppose the point (x , y ) moving upon the curve to passthrough the origin ; the direct ion of it s motion at the instantwhen it reaches the origin depends upon the value of the rat io

2 at this instant , which may be called the direction ratio of theac

curve at this point . This value is determined by putting x O

and y O in equation The result is

- 2a2

a 2

CURVE TRACJN C [Art 32

in owhich all the terms of equat ion (2 ) have vanished exceptthose derived from the terms of the first degree in equat ionHence the direct ion of the curve at the origin is the same asthat of the straight line whose equat ion is formed by equat

ing to z ero the terms represented by the points in Fig. 9

adj acent to the analyt ical origin . Thus,

2a2x a zy o

,

y 2x )

is the equation of the tangent at the origin .

33 . The tangent s at the other points where the curve crossesthe axis of x may be found by a method similar to that em

ployed in the preceding art icle . Grouping together the termscontaining the same power of y , equat ion Art . 29, may bewritten in the form

x (x a ) (x 2 a) -

y (az ax — z u

z

) ocy2

0 .

Now,when the point (x , y ) , moving upon a branch of the curve,

passes through the point (a, its direction rat io will be the

J’

x a

the curve were referred to the point (a, O) as origin . From

equat ion we derive

value of the quant ity which would be . denoted by i if

whence , putting x z a and y r : O in the second member, we

have

TAN GE N TS

which determines the direction of the curve at the point (a,Hence the straight line

3 (x a ) ,

which passes through this point in the same d irect ion, is therequired tangent . In l ike manner, the equation of the tangentat 2a, 0 ) is found to be

5y 6 (x 2a ) .

34 . It was shown in Art. 17 that the posit ion of an asymp~

tote generally depends only upon the terms of the nth and ofthe (n I )th degrees that is to say, when the equat ion of thecurve is placed on the analyt ical triangle, the asymptote is determined by the terms represented by the marked points uponthe side AB of the analyt ical triangle and upon the adj acentparallel l ine . The process exemplified in the preceding articleShows that, in like manner, the posit ion of the tangent at apoint of intersect ion with the ax is of cc generally depends onlyupon the terms not containing y and those containing the firstpower ofy , which terms are represented by the marked pointson the analyt i cal axi s of a: and on the adj acent parallel line. If

the latter terms be wanting, the value of y will be infi23

'

( l a, o

nite, and the tangent will be the line a: z : a, parallel t o the axisof y . For example, the curve

x 3

cuts the axi s of x at the point — a, O) , and the equation doesnot contain the first power of y ; accordingly, we find

— a , 0

thus the line ac a is the tangent to the curve.

CURVE TRAC/NG [Art 35

35 . If the equat ion determining the intersect ions with theaxis of cc has a double root, the curve is said to meet the axis ofx in two coincident points . The numerator of the value of thedirect ion rat io, found as in Art. 33, will in this case containthe factor cc a , suppos ing (a, O) to be the intersection inquest ion hence it s value will generally be z ero, and the axis ofx will be the tangent . Thus, in the case of the curve

x 3 ocy2 2aoc2 a2x a

z

y O,

the equation for the intersect ions with the ax is of ac is

ofwhich x z : a is a double root, and we have

Hence y z 0 is the equat ion of the tangent, and the two coin

eident intersect ions indicate tangency to the axis ofx .

36 . If,however

,when x a is a double root , the terms

containing the first power Ofy be wanting, or if they contain

cc a as a factor, the value Ofx ita] can be determined by

a quadratic equation, as in the following example . Let theequation of the curve be

x3 -y3 “ (4x2 w 2T ) a=(4x 6y ) o . ( I )

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CUR VE TRACIN G [Art 37

When the roots are equal , we have a cusp, or point at whichtwo branches meet with a common tangent . When the rootsare imaginary, there are no real tangent s ; hence we have anisolated point or acnode. I t will be not iced that the case ofparallel asymptotes, explained in Art . 29, i s analogous to thenode ; accordingly, a curve with parallel asymptotes is said tohave a node at infinity. I f the asymptotes , or tangents at thenode at infinity, are found to be coincident, we have a cusp atinfinity ; and if they are found to be imaginary, the curve is saidto have an acnode at infinity.

I ntersection of a Curoe wit/t a Tang ent

38 . Since a tangent to a curve meets the curve in two coineident point s at the point of contact , it can meet the curve inonly n 2 other point s

,the curve being Of the nth degree.

Hence,if we eliminate one of the variables between the equa

t ions of the curve and a tangent, the result will be an equat ionof the (n 2 )th degree . This circumstance facilitates the determination Oi particular points which may be useful in tracingthe curve . In part icular

,if the curve be a cubic, the equation

will be of the first degree , and the s ingle point of intersect iondetermined will have rat ional coordinates . For example, theequat ion of the tangent at the origin to the curve constructedin Fig . 10 , page 23, was found in Art . 32 to be

y 2x .

Subst ituting in the equat ion Ofthe curve, equation Art . 29,

x 3 2n cry2 an? axy 2 a2x a zy 0

,

we have

I I] I N TE RSE CTI ON OF CURVE WI TH AS YMP TOTE

R ej ecting the factor ccz , which corresponds to the two coinci

dent points at the origin,we have cc a ; hence the point

(a, 2a) is a point of the curve.

I ntersection of a Curoe witfi an Asymptote

39 . Since an asymptote is a tangent whose point of contactis at infinity, the remarks made in the preceding art icle applyalso to the intersect ion Ofa curve with its asymptotes . In particular, a cubic cuts each of its asymptotes in a single point .When the cubic has three real asymptotes, it is convenient todetermine the three points of intersection at once by combining with the equation of the curve the equat ion whichrepresent s the three asymptotes . This equation will be acubic agreeing with the equat ion of the curve in the termsOf the third and of the second degree

,for it follows from

Art. 17 that curves having the same asymptotes must agreein the terms of the nth and (n I )th degrees . If

,therefore

,

we subtract this equat ion from that of the given curve, weshall have an equat ion of the first degree which must besat isfied by each of the three intersect ions ; in other words,the equat ion of a straight line cutt ing the three asymptotesin the required points .

40 . For example, let the equation of the curve be

x2

y + xy2 + 2x

2 —3xy

The equations of the asymptotes are

2 = 0, y 2

CURVE TRACI N G [Art 40

Multiplying these equations, we have, for the equat ion of thethree asymptotes in combinat ion,

-

3xy- 2y

2

Hence the equation Of the straight l ine pass ing through thepoints of intersect ionsI S

5x y 6 0 .

Fig . 11

a node at the origin, which

x — 2y = o,

2x + y = o ;

hence the form of the curve is that indicated in the figure .

E xamp les I I

I . oc(oc2+ y

2 ) d (y z o.

2 . y3

xz

y xy 2x 2

3 . x z

y—yex + oc2 4y

2O.

Combining this equa~

t ion with the separateequations of the asymptotes,we find the pointsof intersect ion to be

(2 , “g" 2 ) andThe asymptotes

and the line Of intersection are constructedin Fig. I I . The curve

tangents are

E XAMPLE S

4 . x2 2 + 3xy

3 + x 2 — xy— 2y

2 = o .

5. xz

y yzx x

2

4y2 0 .

6. a 5 2a3oc2

sa3xy 2a 3y

2

y5 0 .

7.

—y2 + x — y o .

8 . (x 2 a”

) (y2 R ) a

zbZ.

9 . x2 (y — x )

0 . (x2 _

y2 ) 2 a

2 (x2 + y

2 ) .

1 1 . x (y ac) ‘z a zy o .

1 2 . x 3 xyz

3axy ay2 o .

1 3 xy’

4 (x + y ) 8

14 . x 3 4.1cy2

324 1 2xy I 2y

2 8x 2y 4

1 5. (y 2x ) (y° x 2 ) aw x )?

402 06 y )

(y?

x"

2x ) a (4y2

9x2 ) Q

(r x ) 2 (y ex ) 4 (y x )?

1 60 2x ) 0

1 8 . (asg —

y2 ) (x a ) a zy o .

1 9 . y (y x ) 2 (y 2x ) 30 0 x ) x’ z a’x’ 0 .

2 0 . coz

y2

3xyz

4x2

z y2 0 .

CUR VE TRACIN G [Art 4 1

Approx imate Forms of Curr/es

4 1. WHE N the equat ion of a curve is placed upon the an

alytical triangle, the absence of the term represented by theanalyt ical origin indicates that the curve passes through theorigin of coordinates and we have seen in Art. 32 that the re

sult of equat ing to z ero the terms of the first degree is theequation of the tangent to the curve at the or ig in, that is tosay, the straight line which approx imates most closely to thecurve at that point . Accordingly, if only one of the terms ofthe first degree occurs in the equat ion of the curve, the tangentis one of the coordinate axes ; and it is now to be shown thatwe can, in this case, with equal facility determine a closer approximation to the form of the curve at the orig in .

42 . For example,let the given equat ion be

x2

3) an ( I )

which placed upon the analyt ical triangle in Fig. 1 2 . Thetangent at the origin is the line x z : 0 , the axis

of y . The ratio ifbeing infinite at the origin, thex

ifmay have a finite value when cc and yx

vanish s imultaneously. D ividing equat ion ( I ) by

§ I I I] APPR OXI MA TE FORMS A T THE ORI GI N

and, making x and y each equal z ero in this equation, we have

hence the equation

in which this rat io has constantly the same value as thatass igned by equat ion represents a curve approxi

mating to the form of the given curve at the origin .

This approximat ing curve is the parabola y2

ax ,

which touches the axi s of y and lies on it s righthence the form of the curve at the origin is as represented in Fig . 13.

Approx imate Forms at Infi nity

43 . The absence of the term represented by the vertex Aof the analytical triangle indicates , as explained in Art. 2 5, thatthe curve passes through the intersect ion of the axis of x withthe line at infinity ; and it is Shown in Art. 26 that, when thisis the case, the result of putting equal to z ero the terms containing the highest power of x which occurs in the equat ion isthe equat ion of the asymptote parallel to the axis of x . It willbe convenient to refer to this point at infin i ty as the point Athus, as in the case of the origin, the equat ion of the tangentat the point A cons ists of the terms represented by the pointsof the analyt ical triangle adj acent to the vertex A . Accordingly, when, as in Fig. 1 2

,the term represented by the point

on the analytical axi s of x adj acent to the vertex A i s absent,the tangent at A is the line y O,

the axi s of x , as explainedin Art. 30 ; but, in this case, as in the analogous case at theorigin, we can with equal facility determine a closer approxima

[Art 43

tion to the form of the curve at the infinitely distant point A .

Since, for a point P moving on the infinite branch passingthrough A, y becomes z ero as x becomes infinite, the value ofxy may remain finite . D ividing the equation of the curve,equation Art . 42 , throughout by x , we have

xy + a

Mak ing x 00 and y 0 in this equat ion, we find

Hence the hyperbola

xy— a

2 = o

approximates closely to the remote port ions of the infinitebranch pass ing through A . This hyperbola lies in the first

and third quadrant swe infer, therefore,that the infinite branchhaving the axi s of x

as an asymptote liesFig . 14 above the ax is on the

right and below it on the left, as in Fig. 14

44 . Fig. 1 2 shows that the curve passes also through theintersect ion of the line at infinity with the axi s of y , which weShall call the point B ; and the absence of the term representedby the point adj acent to B on the s ide AB of the analyt icaltriangle indicates

,as explained in Art. 27, that the line at in

finity i s the tangent at B ; in other words, that the infinitebranch pass ing through B is parabolic . In this case also wecan determine an approximation to the form of the infinite

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CURVE TRACING [Art 46

46 . The general form of the curve being indicated v by the

pos it ions of the approximating curves, it is of course unnecessary to construct the latter,unless an accurate drawing isdesired . It i s, however, useful to notice that, when the

X approximating curve at theorig in 1S a common parabola

,

the radius of curvature at theorigin is one half the coeffi

cient of the coordinate whoseFig-15 first power occurs, when the

coeffi cient of the square is unity. For example, equat ionArt. 42 , is

The equation of the circle whose radius is 5a, and whichtouches the axis of y at the origin, i s

y2 + x

2 == O.

The method of Art . 42 gives for this circle the same approximating parabola as for the given curve . The circle is thereforeas good an approximat ion as the parabola is to the given curve .

General Metéod of Determining tne E quations ofApprox imate Curnes

47. The process of finding the equat ions of approximating

curves, i llustrated in the preceding art icles, consist s in the

§ : I I I] E QUA TI ONS OF APPROXIMATE CURVE S

rej ection of certain terms of the given equat ion, which vanishat the origin or at one of the infinitely distant po ints A or B .

These three points , which are called tfie fundamental points,may be regarded as corresponding to the three vert ices of theanalytical triangle . We shall now show that, two of themarked points, when the equation of a curve is placed uponthe analyt ical triangle, being j oined by a straight line, iftnere are no marked points on Me same side of tke line w it/c

one of tne vertices of tbe analy tical triang le, tbe terms repre

sented by the points situated upon t/i is line w ill constitute the

equation of an approx imate curve at t/ce corresponding fundamental point.

48 . We shall prove this proposit ion first when the vertex in

question is the analyt ical origin . Sincethere is at least one marked point uponeach analyt ical axis, the line must in thiscase cut both analytical axes, l ike theline CD in Fig. 16 . Let p, q be the analytical coordinates of one of the two

Fig-16

points, and p r, q s those of the other ; so that the termsrepresented by these point s are of the form

Lx l’y4I f f

yq8.

R epresent the analytical coordinates of any other point uponthe l ine by p a , q B, in which a and B may be fractional .Let us suppose that the equation contains a term representedby this point ; then the terms represented by the three pointswill be of the form

E xcyq Mxfi‘

fy9

+ s Nxc-

aq ,

in which N 0 if the supposed term does not actually occur

CUR VE TRACING [Art 48

in the equation . Now let the equat ion of the curve be dividedthrough by xfy q ; these terms then become

L + M£ + N Wor?

From Fig. 16 we have, by S imilar triangles,

so that, putt ing ‘

y for the value of either member of this equation, we may write [3 7s, 0. yr, and the terms (2 ) may bewritten

L + Mfl + N

If, then, we suppose the ratio to remain finite when the point

P , moving upon the curve, passes through the origin, the'

B“2,

W Ill also rema in fin ite .x 0 .

49 . Now cons ider any marked point not s ituated upon theline : such a point i s by hypothesis on the upper S ide of theline ; it therefore represents a term containing a higher powerof y than it would if, while containing the same power of x ,

itwere represented by a point upon the line. Hence, after theequation is divided by xf’yQ, the term will be of the form

where t is pos it ive ; and, when x and y are each put equal toz ero, this term will vanish . The result ing equat ion will there

I I I] E QUATI ONS OF APPROX/MA TE CUR VE s

fore contain only the terms which are represented by pointson the line ; hence these terms const itute the equation of anapproximate form at the origin .

50 . In the next place, the line having the same pos it ion as

in Fig. 16, let us suppose all the marked points not upon theline to be upon the lower side of the line,as in Fig . 17. Then, us ing the same notation as before, when the equation isdivided through by xey

q, the terms repre

sented by point s on the line will take thesame form as in expression Art. 48 .

E very other term contains a lower powerof y than it would if, while containing the same power of x , itwere represented by a point on the line ; it will therefore takethe form

Now, if we suppose the rat io Z to remain finite when x and yx?

both become infinite, this term will vanish . The result ingequat ion will, as before, consist only of the terms representedby points on the line ; but it will in this case represent a format infinity.

5 1. Since the side AB of the analytical triangle must contain at least one marked point

,it must meet the line CD

between it s intersect ions, C and D ,with the analyt ical axes .

If r z s,AB will coincide with CD ,

and the process is in factthe same as that given in Art . 15 to determine the value of therat io of y to x when x and y are infinite . But suppose that sis less than r, as represented in Fig. I 7 ; in this case, when x

and y are both infinite and if is finite,Z is readily seen to bex”

x

CURVE TRA CI NG [Art 51

infinite, so that the infinite point on the branch in quest ion isthe fundamental point B . Accordingly we find that in thiscase (see Fig. I 7) B is the vertex of the analyt ical trianglewhich is on that side of the line CD on which there are nomarked points, in conformity with the enunciat ion of the proposition in Art . 47. Since x and y are both infinite, the resultindicates a parabolic branch at the fundamental point B ; and,in like manner, if s be greater than r, we shall have a parabolicbranch at the fundamental point A .

52. In the third place, let the line cut one of the analyt icalaxes produced ; for example, let it cut the analyt ical axi s of ybelow the origin, as in Fig. 18. Since the analyt ical axi s of

y must contain at least one marked point,all the marked point s not on the line are byhypothes is above it thus A is the vertex tobe considered, and the curve passes through

Fig . 18 the fundamental point A . Let the analyti

cal coordinates oi the point s be p, q and p r, q s ; theterms represented being of the form

Ln q c rye

s,

Denoting the analytical coordinates of any other point of the

line by p a , q B, the term which would be represented bythat point is of the form

NxP-t-ayQ

-t-B.

D ividing the equation through by xeyQ, the terms in quest ion

becomeL Mx’y

‘ Nx a

yfi,

which, S ince Q 9, may be written in the form

s r

L Mx ’

ys N (xr

ys) 7 .

I I I] E QUA TI ONS OF APPROXIMA TE CUR VE S

Now, when the point (x , y ) recedes from the origin indefinitelyon the branch pass ing through the fundamental point A, ifwe suppose that, as x becomes infinite, y becomes z ero but xty s

remains finite, x"‘

y‘3 will also remain finite .

53 . Now cons ider the term represented by any markedpoint not on the line . Such a point is by hypothes is abovethe line ; hence the term contains a higher power of y thanit would if, while containing the same power of x , it were represented by a point on the line . Thus, after the equation isdivided through by xf

’

yq, the term will take the form

N

and,when y becomes z ero, it will vanish . The result ing equa

t ion will therefore, as before, contain only the terms representedby point s on the line ; and it will in this case be an approximateform at the fundamental point A .

54 . Finally, suppo se the line to be parallel to one of theanalyt ical axes, for example to the axis of y , so that the curvepasses through the fundamental point A . Then, us ing the samenotat ion as in the preceding art icles, w e shall have r O, andexpress ion Art. 52 , will become

In this case we must regard y as remaining finite when x be

comes infinite ; and, s ince all the marked points not on the lineare on the left of it, the terms represented by these points contain powers of x lower than ccl’ ; hence, when x is made infiniteafter dividing by xi

’, these terms will vanish as before . The

result ing equat ion will determine one or more finite values of

y when x is infinite ; that is, one or more asymptotes parallel tothe axis of x , as in Art . 26 .

CURVE TRACING [Art 5’

5

Tfie Analytical Polyg on

55 . If, when the equation of a curve has been placed uponthe analyt ical triangle, the marked points are j oined by straightlines in such a manner as to form a convex polygon exterior towhich there is no marked point, the result i s called the analy tical potyg on. This polygon will have at least one vertex lyingin each s ide of the analytical triangle, and it may have a s idelying in either s ide of the triangle . In the case of such a sideof the polygon, the result of equating the corresponding termsto z ero determines s imply the intersections of the curve withone of the axes or with the line at infinity. In all other cases,the equation corresponding to a side of the polygon determinesan approximate form at one of the fundamental points, in ac

cordance with the theorem proved in the preceding art icles .In the example given in Art. 42 , the polygon reduces to a triangle having no side coinciding with a s ide of the analyt icaltriangle the curve accordingly passes through each fundamental point

,and each s ide of the polygon gives an approximate

form at one fundamental point .

Construction of tke Approx imating Curves

56 . If a S ide of the analyt ical polygon when produced cut sboth axes, as in Figs . 16 and 17, and contains no marked pointexcept its extremit ies , the equat ion of the corresponding ap

proximating curve will be of the form

Lacey? Mace r

ye8

(see Art. or,rej ecting a common factor,

Lx" Ill )” 0 .

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CUR VE TRACI rVG [Art 58

58 . Again, let the equat ion corresponding to a Side of thepolygon be

dz

}, 0,

Since x has the same sign as x 3, x and y must have the same

S ign, and the curve lies in the first andthird quadrants . The axis of x is the tangent at the origin ; hence the form is thatindicated in Fig. 2 0 ,

e the curve being inthis case a cubical parabola . If this be anapproximate form at the origin, the givencurve has a point of inflection at that point ;but, if it be a form at infinity, the given

curve has parabolic branches in the first and third quadrantstending to parallel ism to the axis of y , and it i s said to touchthe line at infinity at the fundamental point B .

Fig . 20

59 . If the S ide of the polygon when produced cuts one ofthe axes produced , as in Fig. 18, and contains no marked pointsexcept its extremit ies

,the equation of the approximating curve

is of the form

s ’y? i + r

yq +s

L Mx’y’ 0 .

Suppos ing, as before, that r and s are prime to one another,equat ion ( 1) represents one of the family of curves known astbc byperbolas, having both axes as asymptotes . If r z 1 ands z 1

,we have the common hyperbola, as in Fig. 15, page 36 .

Since r and s cannot both be even, the curve will always be

poss ible ;‘it will cons ist of two branches, and it is easy to

I I I] CONS TRUCTI ON OF APPR OXIMA TE CUR VE S

determine the quadrants in which these branches l ie . Forexample, let the equation be

xyz

2a3

2 623.

Since y2 is posit ive whatever he the s ign of x must be nega

t ive hence the branches lie in the second third quadrants,as in Fig . 2 1 . If this be an approxi

mate form for the fundamental pointA, it indicates that the given curve hasbranches approaching as an asymptotethe left end of the axi s of x on bothS ides but, if it be an approximateform for the fundamental point B , itindicates that the curve has branchesapproaching both ends of the axis of

y on the left side .

Sides of the Polyg on Representing more than One

Approx imate Form

60 . When a s ide of the polygon contains any other pointof the analyt ical triangle except its extremit ies , the corresponding equat ion representstwo or more approximate forms at the samefundamental point . For example, let theequat ion of the given curve be

zys

sxy’

x 5

Fig . 22which is placed upon the analyt ical triangle in Fig. 2 2 . Theside AC of the analyt ical polygon, which in this case reduces

S IDE S OF THE POL YGON [Art 60

to a triangle, passes through the point whose analytical co

ordinates are (3, The corresponding equation is

W x5

y2

which may be regarded as a quadrat ic equation to determinethe rat io of y to x

”. Solving, we have

y i vigr d cz

,

the equat ions of two approximate forms at theorigin . The form at the origin indicated by

this s ide is,therefore, that shown in Fig. 2 3.

Fig. 23

61. The s ide B C in Fig. 2 2 gives the approximate form

which indi cates another branch pass ing through the origin,touching the axis of y , and lying in the first and third quadrants. The s ide AB gives

51/2 .y x o

for the only real point at infinity, and absence of termsrepresented by points on the adjacentparallel line (see Art. 17) shows thatthe asymptote passes through the origin . The curve does not cut the axesor the asymptote except at the origin,and it is symmetrical in oppos ite quadrants ; hence its form is that repro

sented in Fig. 24.

Fis 24

§ I I I] RE PRE SE N TI N G MORE THAN ONE FORM

62 . In general, whenever a side of the analyt ical polygonis divided into segments by point s having integral coordinates,whether they be marked points or not, the number of thesesegments indicates the degree of the corresponding equat ion ,and hence the number of approximate forms represented bythe s ide .

63 . The equat ion corresponding to a side cons ist ing of twoor more segment s may have a pair of imaginaryroots . For example, let the equation of thecurve be

x2

y4 2 a

2xy

3 a 5y a6

for which the analyt ical polygon is drawn in0

Fig. 2 5. The S ide OC of the analyt ical polygongives

,for the approximate forms at the fundamental point A,

W 4 a6

o, ( 2 )

which Imposs ible, because it gives imaginary values to theproduct xy 2

. There are, therefore, no infinitebranches in the direct ion of the ax i s of x .

The side CD gives, for the fundamentalpoint B , the approximate form

2 4 2: ( 3)

indicat ing branches in the second and fourthquadrant s having the axis of y for anasymptote . The side DE gives, for the

same fundamental point,

(4 )

indicat ing branches in the first and fourthquadrants, which it i s readily seen are much closer to the axis

CURVE TRACING [Art 63

of y than those corresponding to equation The side E Ogives y z : a for the intersection with the axi s of y . The curvehas therefore the form represented in Fig. 26 . The position ofthe branches is more precisely determined by solving the equat ion forx and determining the l imit ing values of y these arefound to be

E xamples I I I

1 . y4 z any

3 ax3 o .

x3y—3xy

3

-

y3x x3 2x

2

y o .

4 . y4 i 6x 4 x

2

4xy o .

5. x4

6 . x6

az

y4 a

2x 3y a3xy

2o .

7. x4 xz

y y3 o .

8 . x 4 xz

y2 6ax¢y a

z

y2

o .

9 . x 5 bx 4 d 3yz

o.

1 0 . y’x x3 an

2ay

2o .

1 1 . y4 2 axy

2x4 0 .

M o x 5 y5

sax3y o .

1 5 x 5 5n + y5 o .

1 6 x4 y“ any

20 .

1 7. x 5 2 a2x¢

y a3y2

o.

1 8 . y5 d x4 a

zxy

20 .

1 9 “ 72

062

y2

a?

) c6.

2 0 Saxyz 8ax 3 0 .

2 1 . (x2

y ) ( y 2x )2

44 312,

2 2 . y2

(y2

x2

) 2 a2x2

.

2 3 903 xyz

dyz

0 .

2 4 ° ”2

052

J“

) 442962 6d?

2

25 fl aw xz

) (y 2x )

SE COIVD APPR OXIMATI ONS

Second Approx imations

64 . W E have seen that , when a side of the analyt ical polygon coincides w ith a side of the analyt ical triangle, the corresponding equat ion gives only the point s of intersect ion withone of the fundamental lines, that i s with one of the coordinateaxes , or with the line at infinity ; what i s really determined inthe latter case being the direct ion of an asymptote. Themethod of obtaining the posit ion of the tangent l ine or firstapproxima tion to the curve at one of these point s of intersect ionis explained in A rts . 16 and 33. But, when the intersect ion in

quest ion is one of the fundamental point s, we have a s ide of thepolygon which gives at once the first approximation or tangentat the fundamental point .We have also seen that , if the side of the polygon is par

allel to one s ide of the analytical triangle, so that it corresponds to the same number of intersect ions with two of thefundamental l ines, it gives the first approximat ion only ; butotherwise the fundamental line with which the side indicatesthe greatest number of intersect ions is itself the tangent, andthe side gives a second approx imation, determining, in fact ,upon which side of the tangent the curve lies in the ne ighborhood of the fundamental point . We have now to Show howthis second approx imat ion can be determined in those casesin which the side of the polygon gives only the first approxi

mat ion

CURVE TRA CI N G [Art 65

65 . Let us take for illustrat ion the curve whose equat ion

x2

y3 + xy

3 — x2

+ x -

y = o,

for which the analyt ical polygon is drawn in Fig. 27. TheS ide CD gives, for the tangent at the origin, the

F equation

x —y = o .

D E Now the value of the quant ity x y , when (x ,y )Fig . 27is a po int of the curve, determines the posit ion of

the curve with respect to the tangent ; s ince , obviously, if thisquantity i s positive, _

the curve is on the right of the tangent,while if it i s negat ive

,the curve is on the left of the tangent .

In the process of finding the equation of the tangent, as in

Art. 32 , we show that at the origin the rat io ?has the valuex — y x — yunity ; therefore O at that point , and which is

x x2

the rat io this last quant ity bears to x , may have a finite valuewhen x z y r: o. D ividing equation ( I ) by x

2

, we have

in which, making y z x 0 , we have

Fig . 2 8

and,since x 2 is essentially pos it ive, we infer that x y i s posi

t ive on both sides of the origin . Thus the form of the curveat the or igin i s that ind icated in Fig. 2 8.

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CUR VE TRACI N G [Art 67

67 . The side E F in Fig . 2 7 gives, for the tangents at thefundamental point A, that is to say, the asymptotes parallel tothe axis of x ,

of which y I is the only real root . Arranging the termsaccording to powers of x ,

equat ion ( I ) may be written

c os I ) xoi 1 ) 0 .

Now,in finding the asymptote y I r: O, as in Art . 26, we

Show that y I,which measures the distance of the curve

from the asymptote , vanishes when x becomes infinite ; hencex (y I ) may have a finite value when x increases withoutlimit . From equat ion (4) we have

xU I ) ( y’

+ y + I ) + y3 I

in which,putting y z 1 and x 00 we have

M y I ) 2

which therefore represent s a second approximation to the formof the curve when x is verygreat . In equat ion y 1

i s posit ive when x is negat ive,and negat ive when cc i s posit ive ; hence we infer that theposit ion of the curve with re

spect to its asymptote is that represented in Fig. 29 .

Fig . 29

68 . In like manner,the side PG of the polygon gives the

IV ] SE COND APPROXIMATI ONS A T I NFI N I TY

asymptote cc 1 . Arranging the terms of the equationaccording to powers of y , we have

y3x (x + 1 ) —y

D ividing by the highest power of y which occurs except in thefirst term, the equat ion takes the form

new I )

from which we see that, if we make y infinite and x

y2

(x 1) will have a finite value thus

— I = °

is the second approximation . Since y2 is essentially pos it ive,

this equat ion shows that cc I is negative both for pos it iveand for negat ive values of y ; in other words, the branches towhich x 1 is the asymptote lie on its left at both ends ofthe asymptote .

69 . To complete the construction of the curve,we have

the side CC of the polygon giving, for another approximateform at B ,

2 1 1,

which shows that the curve has branches asymptot ic to the axisof y in the first and fourth quadrants . The side DE gives theintersect ion ( 1 , O) with the axi s of x , and the tangent at thispoint i s found to be the line whose equation is

x + y = 1 .

The result s thus far determined are constructed in Fig. 30 .

CURVE TRAC/ MO [Art 70

70 . The curve is of the fifth degree ; but, since there arethree asymptotes parallel to the ax is of x , although two ofthem are imaginary, a line parallel to the ax i s of x meets thecurve three t imes at the fundamental point A, which is accord

ingly said to be a triple point ofthe curve . Hence the asymptote y I

, which is a tangentat A, can meet the curve inonly one other point . A ccordingly, putt ing y I in theequat ion of the curve

,equation

Art. 67, we obtain

an equation of the first degree,Fig . 30 giving the point (3, I ) as the

intersection of the curve with the asymptote.

71. In like manner,a line parallel to the axis of y meets

the curve twice at the fundamental point B , and can cut thecurve in only three other points . But the axis of y it self cutsthe curve in only one other point , and meets the branch towhich it is the tangent at B in three coincident points at thepoint of contact

,like the tangent at an ordinary point of in

flexion. The asymptote cc I has the same character ; andaccordingly, putt ing x 1 in the equat ion of the curve, wefind

giving the s ingle point of intersect ion

72 . There can be no doubt as to the method of j oiningthe branches now determined, except in the case of those in the

IV ] THE B RAN CHE S D E TE RM /N E E

first quadrant . To remove this remaining uncertainty, let usconsider the intersection of the curve with the tangent linewhose equat ion is

x + y = 1 .

The result of eliminating cc from the equation of the curve bymeans of this equation is

an equat ion which is easily shown to have no negative roots .But, writing it in the form

( y I ) 3 = yc

it obv iously has no posit ive root less than two . Its only realroot i s in fact about Hence the branches must be j oinedas in Fig . 30 .

Selection of the Terms which Determine the nex t

Approx imation

73 . It will be not iced that the addit ional terms employed,when we proceed to the next approximat ion above that givenby the side of the polygon, as in Arts . 65 and 67, are thoserepresented by point s si tuated upon the next adj acent lineparallel to the S ide of the polygon, exact ly as in the case offinding the first approx imat ion when the side of the polygongives only points of the curve . Compare Art . 34 . If thisparallel l ine contains no marked points, or if the terms repre

sented are divisible by the factor corresponding to the side ofthe polygon , we must employ the terms on the nearest parallelline for which they are not so divi s ible . This i s illustrated,in the case of both classes of S ides, in the following example.

CURVE TRACIN G [Art 74

74 . Let the equat ion of the curve be

M "963 x

” x2

y"

( I )

for which the analyt ical polygon is drawn in Fig . 31 . The s ideCD gives the tangents at the origin

x —y = o x + y = o.

The terms represented by the next parallel l ineare

903 — y3 (x x} + y2

) ( 2 )

Now in the case of the tangent x y 0 , we have, after putting x y inthe coefficients of x y in equation

2 (ac — y ) + 3x (x —y ) + 903 = 0

This equation Shows that, when x y z : o, x y has a finiteratio to x 3 ; for x (x y ) bears a vanishing rat io to x y .

Thus the term aris ing from express ion (2 ) vanishes, and wehave at the origin

2 = 0 ;

hence2 ( y x ) ( 3)

is the approximate form at the origin of the branch whose tangent is x y . E quat ion (3) shows that the curve lies abovethe tangent in the first and below it in the third quadrant .The second approx imat ion to the branch whose

tangent is x y O i s found, by putt ing y x

in equat ion ( I ) , to be

which shows that the curve lies below this tan ”9° 32

gent on both sides of the origin . Hence the complete format the origin is that indicated in Fig. 32 .

§ IV ] SE COND APPR OXIMATI ONS

75 . Consider next the side DE of the polygon, which byitself gives only the point of the curve 1 , Writing equation ( 1) according to powers of y ,

x’

(x I ) I ) -

y3 0 (4)

There are no marked points on the parallel line adj acent to theside DE of the polygon , which shows that x I i s the tangent, as in Art. 34. Moreover, the express ion y

2

(x2

I ) , corresponding to the next parallel l ine, contains the factor x I ,

which vanishes at the point in quest ion . Hence it is necessaryto include the term represented by the point C thus, putt ingx I in the coefficients of y and x 1 in equation

x + i —y3 = o .

In this equation, x I bears a finite ratio to y 3 at the point

for y2

(x I ) bears a vanishing ratio to1

1 . Hence the approximate form at th i s pomtY

fi = x + I ;

and, S ince y and x 1 have the same Sign, the Fig, 33

form of the branch is that indicated in Fig. 33.

76 . To complete the construction of the curve, we have thes ide FG, giving

x — yc

which indicates parabolic branches in the first and secondquadrants in the direction of the axis of y . The s ide EF gives

y:

" x ,

hence the curve has parabolic branches in the direct ion of theaxis of x in the second and third quadrants .

CURVE TRACI N G [Art 76

The s ide CC gives the intersection (O, I ) . Proceedingas in Art. 75, the tangent at this point is

and the second approximation is

y + 1

which Shows that the curve lies above the tangent, and thatthe radius of curvature at this point is 3. See Art. 46 .

77 . The inflexional tangent y z x in Fig. 32 meets thebranch to which it is tangent in three coincident point s, andcrosses the other branch at the node : it therefore meets thecurve infour coincident point s at the origin ; and, since the curveis a quartic, it cannot meet the curve again . The inflexionaltangent x 1 (Fig. 33) meets the curve in three coin

eident points at the point ofcontact, and meet s it also at thefundamental point B ; hence thecurve cannot again cross thistangent . It i s therefore necessary to j oin the branches determined as indicated in Fig. 34 .

The tangent y z x meet sthe curve in three coincidentpoints at the origin ; and, putting

y z x in the equat ion of thecurve

,the fourth point of inter

section is found to be 2,thus determining a point on the

infinite branch in the second quadrant . The l ine x 2

will be found to touch the curve at this point, and to meet itagain at 2 , I ) .

Fig . 34

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CURVE TRA CI NG [Art 78

may have a finite value at the point ( I , To ascertain thisvalue, divide equation (2 ) byy , thus

2x x I

y2

x2

J’ 0 °

B efore putt ing y z : 0 in this equation, we must subst itutefor x its approx imate value I y , derived from the first approximat ion, in the numerators of the fract ions, and retain all theterms which have a finite value when y z 0 . Thus we have

I )

2 (x - I ) — 2y+

2 (x

y r

x ..

and,s ince 1 at the point ( I , this becomes

x

Hence we infer that x 1 y i s pos it ive in the neighborhoodof the point ( I , O) , and the form of the curve istherefore that represented in Fig. 36 .

x E quat ion (4) may be used to determine theradius of curvature, as in Art. 66.

Fig' 3679 . We may, in a similar manner, determine a

second approximation when the side of the polygon gives onlyan intersect ion with the line at infinity and the first approximat ion gives the asymptote. For example, the s ide EF in Fig.

35 gives

SUCCE SS I VE APPROXIMATI ONS

determining a point at infinity. The equation of the curve is

x s x ) ( I 3 + yx2

) W 2x (5)

and the asymptote is found by divid ing by x 3 and then making

y x z : 00, while y x is assumed to have a finite value.

The result isy x 2 0

, ( 6)

the equat ion of the asymptote . Now, since equation (6) showsthat the quantity y x 2 vanishes when x is infinite, thequantity

My — x

may have a finite value when the point (x , y ) recedes indefinitely upon the branch to which equat ion (6) represents theasymptote . To ascertain this value, divide equat ion (5) by x

2

,

and put y z x 2 , as given by the first approximation, beforeputting x infinite . Thus we have

3 2 3 2

— x )x + 6x +

and,rej ecting terms which vanish when x is infinite, this

becomesx ( y — x — x )

or, s ince y x 2 when x i s infinite,

x ( y x 2 ) ( 7)

Hence we infer that y x . 2 i s pos itive when x i s pos it iveand great, and negative when x is negative and great ; thecurve is therefore above the asymptote for distant points inthe first quadrant

,and below it for distant points in the third

quadrant . See Fig . 37, page 64, in which the curve is traced .

80 . It should be not iced that, in the processes illustrated inArts . 78 and 79, while the terms employed in the first approx

CUR VE TRACING

imat ion are those represented by points on the s ide of thepolygon and on the adj acent parallel l ine, the addit ional termsintroduced in the succeeding approximation are represented bypoints on the next parallel l ine .

Asymptotic Parabolas

81. A process of successive approximation, similar to thatemployed in the preceding art icles, enables us to determinemore accurately the pos it ion of a parabolic branch at one ofthe fundamental points A or B . Such a branch is indicated bythe side PG of the analyt i cal polygon in Fig. 35. Let theequation of the curve be written in the form

_y’

(x2 —

y ) — x 3y— x

2

y 2x2

2x

the terms included in the first expression being those corresponding to the side of the polygon . These terms aloneproduce terms with finite values when the equation is dividedby y 3 the result being

x2

y co

from which we infer that the parabola

= I

x2

y ( 1 0 )

is an approximate form at the fundamental point B . Now,

s ince it follows from equation (9) that 00, the product

may have a finite value at B . This will, in fact, be found to bethe case when the equat ion contains terms represented by

points of the analyt ical triangle situated upon the parallel lineadj acent to the S ide of the polygon in quest ion . In Fig. 35

IV] AS YMP TOT1C PARAB OLAS’

there is one marked point so situated, namely the point E ;

accordingly, dividing equat ion (8) by yzx , we have

x y x x

2 fx J’ J ’

2

in which the only addit ional term that does not vanish a t thefundamental point B i s that represented by the point E . Theresult is that at B

x2

I O ;

hence we infer that the parabola

x2

y x o ( 1 2 )

i s a closer approximat ion to the curve than that representedby equat ion

xz —

y— x

x

damental point B ; hence the product of this quant ity by x ,

namely

82 . E quat ion ( 1 1 ) shows that 0 at the fun

2x _

y— x ,

may have a finite value at B . To find this value,divide equa

tion (8) byy , thus

2x —

y

In order to retain all the terms of this equation which have3

finite values at B , it i s necessary in the infinite term to subJ ’

stitute for x 2 its value y x as given by the second approximat ion, equat ion The result i s that at B

x (x x )

CUR VE TRACI N G [Art 82

xz —

y— x 1 =z o ;

hence the parabola-

x2 —a-

y- x — 2 = o ( 14)

is a st ill closer approximation than that represented by equation This curve is called the asymptoticparabola .

83 . Putting the equation of this parabola in the form

(x ZE,

we see that its vertex is at 23) and its parameter is unity.

ill

llllllllIllll

ll

The parabola is constructed in Fig. 37, together with thebranches determined in Arts . 78 and 79, and those correspond

AS YMPTOTI C PARAB OLAS

ing to the remaining S ides of the analytical polygon in Fig. 35.

In j oining the branches, it is obvious that the oblique asymptote must be crossed in the fourth quadrant, and also in thesecond quadrant ; and, s ince this asymptote cannot be crossedagain, the remaining branches must be j oined as in the figure .

84 . The process given in Arts . 8 1 and 82 may be continuedin order to determine the value of the quantity

x (xz —

y— x

which may be finite at the fundamental point B , S ince by equation ( 14) the quant ity in the parenthes is vanishes at B . Mult iplying equat ion ( 13) by x , we have the equation of the curve inthe form

x (x2

y ) o .

The last two terms vanish at B ; and, employing the value ofx 2

given by equation we have at B

x 2 ) (x2

x )x x

2

y )y

x (x’ -

y ) (x2

x )(x my

;” C 2 )

2 (x2

3x

and, rej ecting vanishing terms,

x (xz —

y— x — Z) = 4 » ( I S)

By equation the quant ity y x2

x 2 is the differ

ence between the ordinate of the curve and the corresponding

CURVE TRACIN G [Art 84

ordinate of the asymptotic parabola. Hence we infer fromequation ( 15) that the curve is below the parabola in the firstquadrant, and above it in the second quadrant, as representedin Fig. 37 ; the distance measured in the direct ion of the axis

of y being, when x is great, about4

x

E xamples IV

1 . x2

y3 + xy

4 -y3 - x

2

+ x o2 . x 4 y

4 2 ax 3y az

y2

a4 0 .

3 . x 4 z ay3

3a2

y2

2a2x2

a4 o.

4- x3 + x

2

5. x 4 2ay3 2 a

2x2

3a2

y2

0

6 . (T 4az

) (x’+ r

’

) 444 o .

7. x (x2+ y

2

) ( y 2x ) 4ax3 z ay ‘

3 z : o.

8 . xyz

yx2

ay2

azx o .

9 . ( y r )z

( y 2x ) M y 1 1x ) .

1 0 . y4 z ay

3 —1 2 ayzx 4a

2xy a

zx2

0 .

1 1 . x2 2

d (x 3 + y3 ) .

I 2 x(x y )2

4ax (x r) 44 9 0 ~

1 3 . x6

x 4yz

2 a2xy

3 a4x2

, 3a5y 0 .

14 . (x y ) (x2

4y ) 2 ax2

I oaxy 2 0 ay2

24a2x .

1 5. y“ 2 axy

2

3a2

y2

z a3x o .

1 6. xz

yz

2xy3 2ax

2

y 2 axy2+ a

2

(x y )2

0 .

1 7. x 4 x 3y xz

y2

2axy2

ay3 a

zx2

az

y2

o.

1 8. x 3y xz

y2

x3 xz

y xyz

y3 x

20 .

I“ T his result, as wel l as that found in Art. 79, equation g ives a good approx

imation only when the value of x is considerable . For instance, when x 5,

the ordinate of the parabola is 18, and that of the rectil inear asymptote is 7 ; thevalues ofy for the curve are about 1 5.5 and differing from the former byand respectively, whereas the differences g iven by equations ( 15) and (7) wouldfor th is value of x be .8 and .4 respectively.

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CUR VE TRAC/NC [Art 86

this line, that i s to say, a parabolic branch . It is to be not icedthat in these cases the process of approximation determines atonce the S ide of the tangent on which the curve lies, and inthe case of the line at infinity it determines in which of twooppos ite quadrants the paraboli c branches l ie. For example,take the curve whose equat ion is

x 4 2x2

y2

y4 2ax

2y ay

3 o,

for which Fig. 38 gives the analytical polygon . The s ide ABgives for the intersection with the line atinfinity

x4 y4 O

(x + y )2

(x r)”

Fig . 3 8an equat ion having two pa irs of equal roots .

Now an infinite branch for which 1 must lie in the first

or in the third quadrant, and from equation ( 1) we have

(x y )”

in which, putt ing y x, we have, when x is infinite,

(x y )z

The corresponding branches are parabolic because x y isinfinite when x is infinite ; and, since this quantity is real onlywhen x i s positive, the branches are in the first quadrant.

V ] PARAB OLI C B RAN CHE S

87 . The construction of the curve is readily completed ; forthe S ide AD of the polygongives the form at the origin

the s ide CD gives the tangents

y -x,

and the s ide B C gives the point

(0 , a) . Moreover, the equation containing powers of x witheven exponent s only, the curve is symmetrical to thehence it s form is that indicated in Fig . 39 .

Fig. 39

88 . Supposing, as in Art. 86, that the equat ion giving theintersect ion with one of the fundamental lines has equal roots,the singular case, corresponding to a node, is indicated by theoccurrence of a quadrat ic equat ion for the first approximat ion,as in Art. 36 , the roots of this quadratic being real or imaginary according as the node is a crunode or an acnode . In thecase of the line at infinity, the singular case is that of parallelasymptotes and these may be real forming a crunode at infinity, as in Art. 29, or imaginary forming an acnode at infinity,there being in the latter case, of course, no correspondinginfinite branches .

89 . Let us now suppose that equal roots occur in the quadratic equation determining the first approximation or tangentsat a node . This will usually const itute a case in which thecurve is real for values of x on one side, and imaginary for

CURVE TRACI N C [Art 89

values of x on the other side, of the part icular value of x inquest ion . For example, let the equat ion of the curve be

x2

(x2

y ) z ax3 2ax2

y axy2

a2

(x -y )

2

( I )

for which Fig. 40 gives the analyt ical polygon . The s ide CDgives , for the tangents at the node at the origin,the equat ion

(x r)”

Fig . 40which has equal roots, showing that both tan

gent s coincide with the line

y = x .

Proceeding to the next approximation, by dividing equat ion ( I )by x 3 and making x y z : 0 , we have at the origin

a’

(xo

,

hence we infer that near the origin the quantity y x is realwhen x is pos it ive, and imaginary when x is negative . More

over, in the former case, th is quantity, which is the d ifferencebetween corresponding ordinates of the curve and

the tangent,has two values, one pos it ive and the

other negat ive . Hence the curve lies on the rightof the origin and on both s ides of the tangent,the two branches forming a cusp at the origin, asin Fig. 4 1 .

90 . The s ide CA of the polygon in Fig. 40 gives, for theintersection w ith the axis of x , the equation

x“

z ax a2

which has a pair of equal roots . Arranging equation ( 1)according to powers of y , it i s

x2

(x a )2

2 ax (x a )y (x2

ax a2

)y2

o,

and the occurrence of the factor x a in the coefficient of yrenders this a quadrat ic equat ion for the rat io of y to x a atthe point (a, O) , as in Art . 36 ; hence there is a node at thispoint .

Putt ing x a , the quadratic i s

which has equal roots hence the point is a cusp at whichtangent is the line

x — a + y = a (9

9 1. The tangent at a cusp, l ike a tangent at an ordinarynode , meet s the curve in three coincident point s at the pointof contact thus , the curve being a quart ic, the tangent y z : x

can meet the curve in only one other point , which is found tobe (3a , 3a ) , and the tangent (3) passes through the same point .The curve has no infinite branches ;

for the Side AE of t he polygon gives

x2

y2

0,

indicat ing imaginary intersect ions withthe line at infinity ; and the side DEgives, for the parallel asymptotes ortangent s at the fundamental point B ,

2 2x “x a

Fig. 42

which has imaginary roots, so that B is an acnode .

The shape of the curve is therefore that Shown in Fig . 42 .

CURVE TRACING [Art 92

Tacnodes

92. When equal roots occur in the equation determiningthe tangents at a node, we have seen in A rt. 89 that we ordinarily have a cusp ; but here also a singular case, analogous tothat explained in Art. 85, may arise that is to say, the quant itywhich measures the distance of the curve from the tangentmay be real on both sides of the node, or imaginary on bothS ides of the node . For example, let the equation of the curvebe

xz

y’ w y (x

2 -T ) x (x -

y )’

( I )

for which Fig . 43 gives the analyt ical polygon . The side CD ,

indicat ing a node at the origin, gives an equationwith equal roots, and the line

x ( 2 )

is the only tangent at the origin, as in Art . 89 .

But, s ince in this case x y occurs as a factor inthe terms of the third degree also, we no longer obtain a finiterat io between (x y )

2 and x 3 in fact , putt ing in equat ion ( 1)y x ,

as determined by the first approximat ion, we have

x“ 40 902

00 —y ) 42

00 y )’

a quadrat i c equation for the rat io of x y to x"

. Solving, weobtain real roots, namely

“ (x — y ) 2 It V3) x"; ( 3)

hence the quantity cx y i s real on both sides ofthe origin

,having two values, both of which are

negat ive . Thus the form of the curve at theorig in is that represented in Fig. 44 . The two branches havinga common tangent but different curvatures are said to form a

Fig. 44

TA CNOD E S

tacnode. We have already had in Art . 60 an example of a tac~

node in which the common tangent is one of the coordinateaxes .

Cusps at Infinity

93 . The line DE in Fig. 43 corresponds to a node at thefundamental point A the equation is

y2 my a

2

which has equal root s hence the curve has two tangents at A,

or asymptotes , coincident with the line

y + a = o.

Arranging the terms according to powers of cc, the equation ofthe curve is

x2

(y a )2

2a2xy 2 ay

3 az

y2

0 .

The next approximation is found, as in Art . 67, by dividing byx and then making x 00

and y a as determinedby the first approx imation.

The result is

x (y a )2

2 4 3 0 ;

whence we infer that for theinfinite branches x must benegat ive, and that y a w illhave two values, one posit iveand the other negative .

Hence the curve has two infinite branches approaching theleft end of the asymptote, one on each side of it, as in Fig . 45,

forming a cusp at the fundamental point A,

Fig . 45

CUR VE TRAC/N C [Art 94

94 . To complete the construct ion of the curve,we have the

side . E P giving parabolic branches in the first and second

quadrants, and the side FC giving the point (0 , 3a) . The tangent y x meets the curve in four coincident points at theorigin, and therefore cannot meet it again . The asymptote,being a tangent at a cusp, meets the curve in three coincidentpoint s at infinity, and the fourth point of intersection is foundto be at 3a, a) .

R amphoid Cusps

95 . When a quadrat ic equat ion presents itself for the second approx imat ion at a node where the tangents are coineident , if the root s are real and dist inct , as in Art . 92 , the twobranches having the common tangent have different curvatures and these branches are real upon both sides of the nodeforming a tacnode . If the root s were imaginary, there would,of course, be no branches, and the node would be an isolatedpoint of the curve .

96 . Suppose, now,that this equation has equal root s ; the

two approximate forms determining the curvature come intocoincidence, and it will usually be found that the curve is realon one side and imaginary on the other s ide of the node, as inthe case of the ordinary cusp . For example, let the equat ionof the curve be

x 4 2x2

y 2xy2

y2 ( 1 )

which'

upon the analyt ical triangle in Fig. 46 . Thes ide AC,

which represent s a second approxi

mat ion at the orig in, the axis of x being the

tangent , gives the quadratic

Fig . 46x4 2n 0

,

which has equal roots, andx2

y 0

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CUR VE TRACI N G [Art 98

end . A branch of the latter character may be regarded as anarc of the curve terminated in each direct ion by One of thepoints in which the curve meets the line at infinity. But wemay regard the infinite branch as continuous at one of thesepoints with the other infinite branch which terminates at thatpoint, thus making the curve to cons i st ent irely of reentrantbranches or circuits, which may or may not be cut by the lineat infinity. For example, the conic cons ists always of a s inglecircuit, which in the case of the hyperbola is cut into two arcsby the line at infinity.

99 . A circuit i s odd or even accord ing as it is cut in an oddor an even number of points by a straight line . Obviously anodd circuit must cut every straight line in at least one point,and in part icular it must cut the line at infinity in at least onepoint . A curve of odd degree must contain at least one i

odd

circuit . Thus a cubic contains always an odd circuit whichcuts the line at infinity at least once, and it may in addit ioncontain an even circuit , which last may be a closed curve oroval not cutt ing the line at infinity.

100 . A circuit containing a crunode cons ists of two partsor loops

,the two extremit ies of a loop being at a common point

but not having a common tangent . For example, in Fig. 48

both extremit ies of the infinite branch in the first quadrant areat the fundamental point B but it does not form a complete circuit , because the tangent s at the extremit ies are the asymptoteand the line at infinity respect ively. The branch in the fourth

quadrant i s the other loop complet ing the circuit . An acnodeis a point at which an oval vanishes . Thus the nodal varietyof a curve is intermediate between variet ies which differ in thenumber of circuit s . As an illustrat ion, take the curve whoseequat ion is

(x2

I2

442

2’

V ] CI RCUI TS

The curve has no infinite branches, and is symmetrical to bothaxes . For its intersect ions with the ax is of x , we have

x2

x = i cz

) ;

and for its intersect ions with the axi s of y ,

(y’

az

)z

y rt ( MIt 62

The curve always cuts the axis of x at the distances c2

)from the origin ; and if c a , it cut s this axis also at thedistances c

z

) , and does not cut the axis of y . Inthis case the curve consists of two ovals . If c a, the curvecuts the axi s of x only at thedistances c

2

) and it cutsthe axi s of y at the distances

at

) from the origin . Inthis case it consist s of a singleoval . In the intermediate case,when c a , there is a crunode atthe origin . The three variet iesof the curve, which is known as Fig . 49the Cassinian, are drawn in Fig. 49. The nodal case is thelemniscata.

10 1. It i s evident that a circuit which does not cut eitherof the fundamental lines , namely the coordinate axes and the

line at infinity, w ill not be detected if w e employ only themethod of the analyt ical polygon . The port ion of the plane inwhich such a circuit may possibly lie may generally be greatlyl imited by the considerat ion of the intersect ions of the curvewith a straight line moving in such a manner as to sweep over

CURVE TRACI N G [Art 10 1

the ent ire plane . For example, in Fig . 48 the curve i s a quart ic and the origin is a node ; hence a line passing through theorigin meets the curve tw ice at the origin and in two otherpoints . If the line revolves in the posit ive direction fromcoincidence with the axis of x through it sweeps over theent ire plane, and we not ice that the two points are real unt i lthe line arrives at a posit ion in which it i s tangent to thebranch in the fourth quadrant . Hence, although there canobviously be no branch in the fourth quadrant other than thatdrawn, we might st ill suspect the ex istence of an oval in thesecond quadrant .

But, if we put y 2 mx in the equat ion of the curve,and solve

for x ,as in Art. 13, we find

x m( i m) : l: m i On

whence we see that the two values of x are real until mand imaginary for all values of m between 2 and 0there is no other circuit . The limit ing value, mx z 2 ; therefore (2 , 4) i s the point of contact of apass ing through the origin .

Aux iliary Loci

102 . When the equat ion of a curve is in such a form thateach of its members is readily resolved into simple factors, theloci of the result s of equat ing these factors separately to z erocan frequently be used in construct ing the curve in the mannerillustrated below .

For example,let the equation of the curve be

x(yz

d x ) (r2

(x a ) ( I )

The locus of y r: O is the axis of x , and that of y2

acc O isthe parabola constructed in Fig . 50 . I t is to be noticed thatthe value of the express ion in the first member for the point

A UXI LI ARY LOCI

(x , y ) vanishes and changes S ign whenever the point crosseseither of these lines, being posit ive intwo of the four regions into which thelines divide the plane, and negat ive in theother two .

The locus of x a 0 ,corresponding

to a factor in the second member,is for

dist inct ion constructed by a dotted line inthe figure . The factor y

2a2 i s always

posit ive , and there i s no correspondinglocus ; thus the expression in the secondnumber is posit ive for all points on the right of the dottedline, and negat ive for all point s on it s left .

Fig . 50

103 . Since each member of equat ion ( 1) reduces to z ero at apoint where the dotted line intersects a full line

,it i s evident

that the curve passes through every such point ; it is alsoev ident that the curve cannot meet either of the lines at anyother point . Moreover, if we mark each of the eight regionsinto which the lines divide the plane by the S ign oraccord ing as the expressions in the first and second membersof the equat ion have like or unlike signs in that region, it isevident that no part of the curve can lie in a region markedThus at each of the points of intersection a branch of the curvepasses from one of the vertically Opposite regions marked tothe other .

104 . The points of intersect ion are (a, O) , (a , a) and (a , a) .

The inclinat ion of the curve at either of these point s is readilydetermined in a manner equivalent to that which we employwhen a curve passes through the origin . Thus, for the point

(a, making x z : a and y r : O in the coefficients of the factors which vanish at this point in equat ion we have

az

y a2

(x

CURVE TRACI NG [Art 10 4

hence

y z a — x

is the equation of the first approximation or tangent at thispoint .

105 . Again, at the point (a, a) we have, in like manner,

z a (x a) ,

which the equation of a parabola approximat ing tothe given curve . If we des ire only todetermine the inclinat ion at (a, a), wewrite equation (3) in such a form that

y a or x a i s a factor of each term,

thus— a) i

and, again putt ing x a and y a, wehave

Fig. 51

ZU a ) 306

for the equat ion of the tangent . The process is equivalent totransferring the origin to the point (a, a) and retaining in theresult only the terms of the first degree .

In like manner we find the tangent at (a , a) to be

106 . When, as in the present case, only two branches entera region, it i s evident that these branches must be cont inuous,as represented in Fig. 5 1 . The construct ion of this branch ,which forms a complete circuit , i s readily completed after determ in ing the asymptote, which is found to be the line x : y .

The fundamental point A is an acnode ; and, cons idering straight,

AUXI L I AR Y LOCI

lines pass ing through this point, that is to say, l ines parallelto the axis of x, it is evident that the curve contains no othercircuit.

Loci repr esenting Squared Factors

107 . In the preceding example each of the auxi liary locirepresents a single factor, and separates regions oppos itelymarked . If, now, one of the loci represent s a squared factor,the expression containing this factor vanishes but does notchange Sign, when the point (x , y ) crosses this locus ; hencethe adj acent regions separated by it will be similarly marked .

Such a locus may be regarded as formed by two auxil iary loci ofthe same system coming into coincidence . Hence, at the pointwhere it crosses a s ingle locus of the other system the curvemeets the single locus in two points which have come into coincidence that is, it touches the S ingle locus in fact, it l ies onboth S ides of the double locus, and on that side of the s inglelocus where the adj acent regions are marked

108 . To illustrate, let us take the curve whose equation is

4y’

(x d )’

(w x ) x (xi

4dx 5492. ( I )

The auxil iary loci for the first member are the axis of x , the

l ine x a and the line x 2a, which are drawn as full linesin Fig. 52 . The loci for the second member are the axis ofyand the rectangular hyperbola

x2 —

y2

4d x saz

my r a.

( 2 )

which are drawn as dot ted lines . The origin i s a point at whicha single factor meets a double factor, and accordingly the curve

CURVE TRAC/N C [Art 10 8

touches the axis of y , lying on its right s ide where the regionsare marked In fact, if we put x O and y o in the co

effi cients of y2 and x in equation we have

8a3yz

z 5a4x

,

the process being equivalent to that of finding the approximateform at the origin .

Again, the s ingle locus x 2a cuts the hyperbola, which isa double locus, in the point s (2a, 11: a) hence the curve touchesthe line x 2a at these points .

equation ( 1) the result is

8a2 (x a)’

(x2 —

y2

4ax 5a2

)2

,

— a) :1: (x2 -

y2 —

4ax

Proceeding as in Art. 10 5, we have

1: . d (x a ) (x a!)2

z a (x a) (J’z

109 . At a point where two doubleloci, one belonging to each system,

meet, the four regions are all similarly marked ; and, supposing themmarked we have two branchescrossing each of the loci and forminga crunode . Thus, in the present example, the double locus x a cutsthe hyperbola, equation in thepoint s (a, j : a\/2 ) hence these pointsare nodes . To find the tangents atthe upper node, we put x z a and

y a\/2 in the coeffi cients of thefactors which vanish at this point in

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