Surendramohan Ganguli. # Lectures on the theory of plane curves ; delivered to post-graduate students in the University of Calcutta online

. **(page 1 of 13)**

Online Library → Surendramohan Ganguli → Lectures on the theory of plane curves ; delivered to post-graduate students in the University of Calcutta → online text (page 1 of 13)

Font size

THEOllY OV PLANE CURVES

PAllT II

CUBIC AND UUARTIC ClIttVES

LECTURES

ON THE

THEORY OF PLANE CURVES

Deltvehed to Post-Graduate Students

IX TiiK University of Calcutta

BY

SURENDRAMOHAN rxANGFLI, M.Sc.

LKCTURER IN PURR MATHEMATICS,

UNIVFRSITV OV (A I. (ITT A

PART II

PUBLISHED BY THE

UNIVERSITY OF CALCUTTA

1919

CALCUTTA.

Sole Agents

R. CAMBRAY & CO-

CALCUTTA.

PRINTKD BY ATDLCH AxNDKA HHATTACHARYYA

AT THB CAT.rUTTA TTNIVKRSFTY PRESB, SRNATK HOUSE, CALCUTT/

PREFACE

The subject of Analytical Geometry covers so extensive

a field that it is by no means easy to decide what to omit

and what to insert ; a w^ord is therefore necessary to

explain the plan adopted in this book, which, I trust, will

prove a useful introduction to the his/her branches of the

subject and will facilitate the study of a variety of al<yebraic

curves.

In the preparation of these lectures, I have endeavoured

to present the subject in clear and concise terms to the

student commencinpj a systematic study of the properties

of alojebraic curves, especially of Cubics and Uuartics. In

the portion of the book devoted to the discussion of

cubic curves, I have not confined myself exclusively to the

application of analytical methods, but have availed myself

of the methods of Geometry whenever simplicity could be

gained thereby. One prominent feature of the present

work is that properties of cubic curves have been exhaus-

tively discussed with special reference to points of inflexion

and harmonic polars ; in this connection, canonical forms

have at times been found of ojreat use, A separate

chapter has been allotted to the discussion of some special

cubic curves of historic importance, and their most general

properties ; but no systematic analysis has been attempted,

lest the young student should feel embarrassed ; only

^â€¢52988

VI PREFACE

cpeneral characteristics of these curves have been outlined

which will supply sufficient material for independent think-

ing in more advanced staoes.

The subject of quartie curves is too extensive to be

adequately considered in a small work like this. I have

therefore confined the discussion chiefly to the most

prominent characteristics of these curves. One chapter

has been devoted to the consideration of bicircular quartics

with special reference to their mode of generation. In

fact, this chapter^ together wnth a note in Appendix I, is

mainly based on the well-known Memoir on bicircular

quartics by Dr. Casey, published in the Transactions of the

Royal Irish Academy 1869. Circular cubics have been

studied with much advantage, regarded as degenerate

bicircular quartics. In the last cha])ter are considered

some well known quartie curves, most of which are bicir-

cular or are cartesians. A similar consideration, as in the

case of cubic curves, has led me to restrict my discourse

only to the general properties of these curves. I have

intentionally avoided the discussion of Roulettes, Cycloids,

etc., reserving the topics for a future occasion. The reader

who desires to study the subject from a higher stand-

point can conveniently consult the following works â€”

Clebsch â€” Le9ons sur la Geometric ; Chasles â€” Histoirede la

Geometric. In Appendix II, a note on Trinodal Quartics

has been inserted. This was communicated to me by Rai

A. C. Bose, Bahadur, M.A,, Controller of Examinations,

University of Calcutta. *

PREFACE Vll

In studying singular points on eubics and quartics, I

have retained the common phrase " non-singular " to

designate a curve which has no double point or midtiple

point, although it has been pointed out by Prof. Basset that

this is a misnomer ; for, he says, Pliicker had shown that

all algebraie curves except conies possess singularities and

accordingly he introduced the term " anautotomic " in

preference to the pharse *' non-singular " commonly in use.

In concluding this preface, I desire to say that, in

addition to the works of authors cited in the preface to the

tirst part, I have consulted, with much advantage, some

notes on Cubies and Quarties furnished by my colleague

Dr. H. D. Bagchi, M.A., Ph.D., and that I am indebted for

some valuable hints to Rai A. C. Bose, Bahadur, M.A.

Once more I must acknowledge myself in the highest

degree indebted to Sir Asutosh Mookerjee, Kt., President

of the Council of Post-Graduate Teaching in Arts, for his

extreme kindness in encouraging me to revise these

lecture-notes for the press, and to the authorities of the

University of Calcutta for publishing them. Finally, I

must thank the Staff of the Calcutta University Press, but

for whose untiring energy and ready co-operation, the

second part of the book could not have seen the light of

day before December next.

University of Calcutta^

S. M. GANGULI.

August, 1919,

CONTENTS

CHAPTER X.

Curves of the third order â€” Cubic Curves.

Page.

Classification of Cubic Curves ... ... 139

Equations of Cubic Curves

HO

Tangential of a point ...

I4a

Satellite ...

14.3

Properties of Cubic Curves

146

Generation of Cubic Curves

150

Sextactic point

156

CHAPTER XI.

Harmonic Properties of Cubic Curves.

Poles of a right line

158

Harmonic polar of a point of inflexion

161

Properties of points of inflexion

163

Inflexional triangles ... ... ...

lib

Configuration of harmonic polars

B

167

X CONTENTS

CHAPTER XII.

Canonical Porms.

Canonical equations of non-singular eubics

Co-ordinates of the nine points of inflexion

Equations of harmonic polars

Hesse's Theorem

The canonical form of a nodal cubic . . .

The canonical form of a cuspidal cubic

Salmon's Theorem

Harmonic and Equianharmonie Cubics

Conjugate Poles

Net of polar conies

The relation of the Hessian with the Cubic

The Cayleyan

The Pole Conic of a given line

The Pole Conic of the line at infinity

169

171

174

175

176

178

181

185

187

187

189

192

199

202

CHAPTER XIII.

Unicursal Cubics.

Unicursal cubics defined. . .

Unipartite cubics

Bipartite cubics ...

Parametric representation of nodal cubics

Parametric representation of cuspidal cubics

Points of inflexion on a unicursal cubic

204

206

207

208

210

213

CONTENTS XI

CHAPTER XIV.

Special Ctjbics.

Circular cubic ... ... ... 216

The equation of a circular cubic ... ... 216

The Logocyclic Cubic ... ... .. 218

The Trisectrix of Maclaurin ... ... 220

The Folium of Descartes... ... ... 2-23

The Cissoid ... ... ... ... 226

Newton's method of generating the Cissoid . . 228

The Cubical Parabola ... ... ... 229

The Semicubical Parabola ... ... 229

Foci of circular cubics ... ... ... 230

CHAPTER XY.

Invariants and Covariants of Cubic Curves.

Number of independent Invariants ... .. 232

The Invariant S ... \.. ... 233

The Invariant T ... ... ... 2S5

The Discriminant T Â» + 64S ^ ... ... 236

Covariants of cubics ... ... ... 238

The Sextie Covariant ... ... ... 240

Contravariants of cubics ... ... 242

CHAPTER XVI.

Curves of the fourth order â€” Quartic Curves,

Classification of Quartic Curves ... ... 244

Generation of Quartic Curves ... ... 248

Complex singularities ... ... ... 251

Harmonic properties of biflecnodes ... . 2.') 7

Bitangents ... ... ... ... 260

Salmon's Theorem on bitangential conies . . . 264

3|ll CONTENTS

CHAPTER XVII.

Tkinodal Quartics.

Equation of a Trinodal Quartic

Generation by Quadric Inversion

Properties of the nodes and nodal tangents

Properties of the points of inflexion, etc.

Tricuspidal Quartics

267

268

27a

274

CHAPTER XVIII.

BiciRCULAR Quartics.

Equation of a Bicircular Quartic ... ... 276

Generation of Bicircular Quartics ... ... 278

Focal conies ... ... ... 282

Centres of inversion ... ... ... 285

Foci of bicircular quartics ... ... 286

Cyclic Points ... ... ... 290

Double foci of bicircular quartics ... ... 291

Inverse of a bicircular quartic and its foci ... 292

Bitan gents of bicircular quartics ... ... 299

Bicircular quartics having a third finite node ... 301

CHAPTER XIX.

Circular Cubics as degenerate Bicircular

Quartics.

The Focal conic â€” a parabola

The centres of inversion ...

The inverse of a circular cubic

The foci of the focal parabolas

Intersections of a circle with a circular cubic

The real Asymptote of a circular cubic

806

307

309

810

311

314

CONTENTS

Xlll

CHAPTER XX.

Special Qtjartic Curves.

The Oval of Cassini

The equation of the Cassinian

Properties of Cassini's ovals

The Cartesian Ovals

The equation of the Cartesian

Generation of a Cartesian

Foci of Cartesians

Points of inflexion on Cai-tesians

The Lemniscate of Bernoulli

The Lima^on

The Cardioid

The Conchoid of Nicomedes

Trisection of an angle ...

316

316

317

318

319

321

321

323

324

325

326

327

828

Appendix I

Appendix II

Index

329

335

347

CHAPTER X.

Curves of the third order â€” Cubiu Curves.

139y The g-eneral equation of a curve of the third

order involves ten arbitrary constants. We may divide

out the whole expression by any one of the constants,

and hence the number of disposable constants in the

equation of a cubic curve is nine and a cubic can be made

to pass through any nine arbitrary points ; or nine arbi-

trary points; will determine a curve of the third order

uniquely.

The general equation of a cubic curve in Cartesian

co-ordinates can be written as â€”

n = o. ... ... ... ... (1)

Or, in any system of homogeneous co-ordinates, it may

be taken as a "" -{-Uc^y + 'dcAy^ J(.dy^ -{â– a\^^z + WAyz-\-b'y^z

+ 1 '~J+my.^+?i:^=o. ... ... ... (-2)

Or, symholicaWy, 2iQZ^+niz^ + u^z + Ms=o ... (3)

where n^ is a constant and u^, ^'^, n^, are homogeneous

expressions of the first, second and third orders resnectively

in X and y.

140. '^ We have seen, Â§ 40, that a curve of the third

order can have at most one double point, and no other

multiple point. Hence according to their deficiencies,

cubic curves may be divided into the following three

fundamental species : â€”

(1) Non-singular or anantotomic cnhics, which have

no double points.

(rZ) Nodal cubics â€” in which the double point is a node,

with two distinct tangents (real).

(3) Cuspidal cubics â€” in which the double point is a

G ISp.

140

By using the formulae of Â§ 97, we may calculate the

Pliicker's numbers for the three cases as follow : â€”

n=. 8= k= 7n= t= i= j) =

Case I

... 3

6

9

1

Case II

... y

1

4

3

Case III

... 3

1

3

1

141.* The trilinear equation of a cubic circumscribing

the triangle of reference is

x^u-{-y^v + z^w-\-lcxyz^o ... (1)

where u, v, w are linear functions of y, z', z, x ; and x, y

respectively and represent therefore' the tangents at the

vertices A, B, C respectively

If the vertex A be a double point on the cubic, the

equation should contain no x^ and x^ ^ and consequently

it lakes the form an^-\-u^=o, where m^ and m^ are

homogeneous functions of the second and third orders

respectively in y and :, If further the curve passes

through B and C, the equation cannot contain y^ aiad z^.

Therefore the equation of a cubic circumscribing the

fundamental triangle and having a double point at A is

AU^+yz{my + nz)=0 ... ... (2)

in which u^z=o is the equation of the tangents at A.

Hence the point A will be a node, a cusp, or a conjugate

point on the curve, according as '>' ^ represents two real and

distinct, coincident, or imaginary right lines.

142.*^ If the vertex A is a point of inflexion on the

curve, the tangent at A meets the cubic in three conse-

cutive points. Now the equation of a cubic passing

through A can be written as

141

If A is a point of inflexion, the tangent ?^i=o meets

the curve in three consecutive points at A. Therefore,

if ^i is made equal to zero in the equation, i,e., if y be

eliminated between ti^=o and the equation (3), the

resulting equation should have z^ as a factor, which re-

quires that the coefficient of ,v should vanish, i.e., u^ should

contain w^ as a factor. Thus the equation of a cubic

having a point of inflexion at A is

143.*^ We have proved in Â§ 21 that if a cubic curve

passes through eight points of intersection of two cubics,

it must pass through the ninth also. Asa particular case,

we may prove the following theorem : If two right lines

A and B meet a cubic in the points a, b, c and a' , h\ c'

respectively, then the lines aa', hh\ cc' respectively meet

the cubic again in three other colUnear points a", ^", c".

Let n â€” o and v â€” o\>q the equations of the two given

lines A and B and let /^'=o, v' =o, w' = o be the equations

of the lines aa'{k'), hh'{'^'), cc'{C) respectively. If w = o

be the equation of the line a"6"{C), then it must pass

through the point c". (Fig. 18).

The lines A', B', C constitute a cubic which intersects

the given cubic in nine points, and the lines A, B, C make up

a cubic passing through eight of these nine points. It

must therefore pass through the ninth j^oint c'^ also. But

this last point cannot lie on A or B, which already meet

the cubic each in three points. Therefore it must lie

onC.

Now, the equation of a cubic passing through the

intersection of two cubics \]=o and Y = o is of the form

JJâ€”kY^o. Therefore the equation of the given cubic'

can be written as uv'w'â€”kuvw=o, since it passes,

throusch the intersections of the cubics uvw and u'v'tv' ,

142

Cor : From what has been said above, it follows that

the equation of all eubics can be expressed in the form

tivw â– \- hu' v' lo' â– =â– . ... ... (5)

where Â«, f, lo u, v' , w', are linear functions of the variables,

and therefore represent right lines. The equation represents

a cubic passing through the nine intersections of (u, v, w)

and (//, ?/, iv').

144/ If u = v, i.e., if the lines A and B coincide, the

equation of the cubic takes the form

n'v'w' -\-hi^iv = o. ... ... (6)

The linos u\ v\ iv' become tangents to the cubic at the three

points where the line A meets it. Also, the three other

points {a'\ b'\ c") in which these tangents meet the cubic

again lie on the line lo^o. Hence we obtain the theorem: â€”

If a right line intersects a ciilic in three points, the

tangents at these points meet the cubic again i)i three other

collinear points.

Definition : (I) The point a'\ in which the tangent

at any point a meets the cubic again, is called the tan-

gential of the point a. This point is also called the ^'satellite

point" of the tangent, (Cayley, A memoir on curves of the

third orderâ€” Coll. Papers Vol. II, No. 146, p. 409).

{%) The line C on w-hich lie the tangentials of three

collinear points lying on a right line A is called the

satellite of A. We may thus state the above theorem as

follows : â€”

The tangentials of three collinear points are collinear.

145. ^ As an application of the above properties of a

cubic, we may prove the following theorem"^ : Having

* A. Cayley â€” Memoire sar les courbes du troisieme ordre â€” Journal

de Mathematiques Pares et Appliques (Liouville) tome IX (1884), or

Coll. Papers. Vol I, No. 26. p. 184,

143

given a curve of the third order which passes through the

six points of intersection of four right lines, the tangents

to the curve at opposite points interr^ecfc on the curve in

three collinear points. [Two points are said to be opposite

when one is the intersection of two lines and the other of

the remaining two].

Let 5 and 6 denote two points on the curve such that

the tangents at these points intersect at a point 5' or 6'

on the curve. Let 'I denote any other point on tlie curve

and the hues (5, i) and (6^ 2) intersect the curve at the

points denoted by 4 and 3 respectively. Then (5, 3) and

(6, 4) intersect on the curve at the point 1. The tangents

at 1 and %, also the tangents at 3 and 4, intersect in two

points on the curve which are collinear with the point 5'.

Fcr the tangentials of 5, 3, 1 lie on (5', 3', 1'), and those of

6, 2, 3 are on the line (6', 2', 3'), ie., the line (5', V , 2>'),

Hence the theorem. (Fig. 19).

146. The three points in ivhich a cubic intersects its

asymptotes lie on a right line.

We have defined that the asymptote of a curve is a

tangent whose point of contact is at infinity. But a

cubic curve has three asymptotes. Therefore, if we suppose

that in Â§ 143 the line u^o is at infinity, i.e., ?^ = I, the

equation of the curve becomes n'v'w' -\-k\'^ic=.Oy where I = o

is the line at infinity, and u\ ?/, w' are the tangents whore

points of contact lie on 1 = ^, i.e., u ^=-0, v' = o, to =q are the

asymptotes. The form of the equation shows that the points

in which these asymptotes meet the cubic again lie on the

line IV = 0.

The straight line to which passes through the points of

intersection of a cubic with its asymptotes is called the

satellite of the line at infinity.

144

147.' The product of the perpendiculars drawn from any

point on the, curve on to the asymptotes is in a constant ratio

to the perpendicular drawn from the same point on to the

satellite of the line at infinity.

This is only a geometrical interpretation of the equation

of the previous article. The equation of a cubic whose

asymptotes are ?/, v' , v/ and lo is the satellite of the line

at infinit}^ is n' v' w' ^=^kV-io â€” k'iD, where k' is a constant.

Now, u\ v' , tv' and iv are proportional to the lengths of the

perpendiculars drawn from any point of the curve on to

those lines and hence the theorem.

148."^ If two of the points of intersection of a line with

a cubic he points of infleion^ the third must also he a point

of inflexion.

L ;t the points a and h be points of inflexion on the

curve. If the line ah meets the curve at c, then c is also a

point of inflexion.

Now, the tangent at a point of inflexion has a three-

point ic contact with the curve. Consequently, the

tangential of a point of inflexion coincides with the

point itself. Thus the tangentials of a and h respectively

coincide with them. Therefore the satellite of the line

ah coincides with itself, and consequently the tangent at the

third point c, in which the line ah cuts the cubic, has a

contact of the second order, i.e., the point ^ is a point of

inflexion.

If we put u' =-v' = w' in the equation (5) of Â§ 143, it

becomes uvw-\- ku'^ =-o, which shows that the lines n, ?', w

have each a contact of the second order with the curve at the

points where u' intersects it. Hence we obtain the

theorem : â€” If a cuhic has three real points of infle.rion, they

lie on a right line.

145

149.'' We have seen that the tangents to a cubic at

three eollinear points meet the cubic again in three other

collinear points, or, what is the samething, that if tangents

be drawn to a cubic from three collinear points a, d, c,

on the curve, then the line joining the point of contact

of ani/ one of the tangents from a to the point of contact

of aiii/ one of the tangents from h, passes through tho

point of contact of ant/ one of the tangents from c. Now,

from any point on a non-singular cubic four tangents

can be drawn to it. Therefore the sixteen lines which join

the four points of contact of tangents drawn from V/ ' to

those of the tangents from h, must pass through the four

points of contact of the tangents drawn from c

Thus the twelve points of contact of these tangents lie on

sixteen lines, three on each, and through each point there

pass four of these sixteen lines.

Prom this it follows that, for a given line x^, there is

but one satellite to it ; but to a given line A there corres-

pond sixteen different lines, of which the given line is the

satellite. Herce we obtain the theorem : â€” A given line

has onl?/ one satellite y hut there are si teen different lines

of tvhich it is itself the satellite.

150. ' The four points of contact of tangents drawn

from any point A on a cnhic are the vertices of a quadri-

lateral, the three diagonal points of tvhich are the points of

contact of the tangents drawn from the tangential point erf A.

Consider a line which intersects the cubic in the three

])oints A, B, C. Let a^^a^, a^, a^,; b^, b^, ^3>^4 ; ^u^*2>

Cg, c^ be the points of contact of tangents drawn from A,

B, C respectively. Then these twelve points lie on sixteen

different lines. Let the points A and B coincide. Then

the points a^, a^, a^, a^ coincide with the points b^, b,^,

b^, b^ and one of the points c's (say c^) coincides with A.

146

Thus we see that the line joining c^, one of the points of

contact of tangents from a, to a ^ , one of the points of

contact of tangents from A, must pass through one of the

ot/ier points of contact of tangents from A, say a 2.

Similarly the line c^a^^ passes through a^. Thus the

sixteen lines reduce to six sides of the quadrangle a^a,^a^a^,

counted twice, and the four tangents at these points.

Hence the intersection of a pair of opposite sides is one of

the points Ci, ^'g, Cg, and the tangents at the vertices (T^i,

^2J ^'^sj ^4 nieet the curve at the same point A (c^), i.e.,

Ci, Cg, c^ are the diagonal points of the quadrilateral.

\bV: From this we easily deduce the truth of the theo-

rem: â€” If two tangents be draiviifrom any point A on a cahic,

the tangent at the third point in tohich the chord 0/ contact

meets the cubic cuts the tangent at A at a point on the curve.

We may analytically prove the theorem as follows : â€”

If we take the two tangents and their chord of contact as

the sides of the triangle reference^ the equation of the cubic

must be of the form yz{l'+my-\-7iz)-\-x'^{By-\-Cz) = o,

where By4-C~ = o is the tangent at A and lx-^wy-^nz = o

is the tangent at the point where the chord of contact x = o

meets the curve. These two lines intersect on the curve.

162.^ The chords of contact of tangents dratvnfrom

any point of a cubic are harmonic conjugates of the tangent

to the curve at their intersection and the liiic joining the

intersection ivith the point.

Let the line C be the satelHte of any line A, the

tangents at the points on A being L^ M, N. Then the

equation of the curve can be written as LMN â€” A2C = (?,

if A = y and C = <? represent those lines. Let B be any

other line of which C is the satellite, so that B passes

through the point of contact of N and those of two

other tangents L' and M' respectively which meet C in the

147

points where L and M respectively meet it. Then the

equation of the curve naay again be written as L'M'N

-B^C-o. Thus we obtain the identity N(LM-L'M')

= (A^~B2)C. Tne right-hand side represents three lines

A4: B and C, therefore the left-hand side must also represent

three right lines. Now the line N must be one of A + B,

and C must be one factor of LM â€” L'M', which is the

line (LL', MM'), and the other factor is the line (LM',

L'M) which is A + B. Therefore when C is a tangent,

so that L, M, L' and M' meet C at the same point on

the cubic, one of A + B becomes the line joining the

point of contact of C with that of N. But A, B,

A + B form a harmonic pencil. Hence the theorem.

(Fig. 20.)

153.*^ Any line drawn through any point A on a cubic

is cut harmonically in the two points P and Q tvhere it

meets the cubic again, and the tioo points L and M tvhere

it meets a pair of chords joining the points of contact of

tangents drawn from A.

Let rtTj, flg, Â«3, r/^ be the four points of contact of

tangents drawn from any point A on the curve. (Fig. 21.)

Then the lines joining a^, a^ and ^g, a^ intersect at a

point Ci on the curve. Let a line through A intersect

the cubic in P and Q and the tangent at Cj at D, and

the chords of contact at L and M respectively. By the

previous theorem, Ci(LAMD) is a harmonic pencil i.e.,

LM is a harmonic mean between LA and LD.

â€¢2_ + ^=_l_ (1)

LA LD LM ^ ^

By Maclaurin's another theorem (Â§ 53), since any

line through L intersects the curve at P, A, Q and the

tangents at three points (eollinear with L) in the three

148

points A, A and D, we have

LP LA LQ LA LA LD

'â€¢''â€¢ LP LQ LA LD LM *

.". PQLM are harmonic.

154.^ The theorem of Â§ 19 can be applied to the case

of the cubic when ?// = 8, and it then takes the form : â€”

Every curve of the nth degree ivhich pa^es thrnngh 8w â€” 1

fixed pohits on a cnhic passes through one other fi^ed point

on the cnrve.

This can be proved very easily with the help of the

theory of residuation. Let the i^roup of 3;^â€” 1 points

be denoted by P. Describe two curves of the n\kv degree

through these points P. Let them intersect the cubic in

the points Q and Q' respectively. Then [P + Q]=o and

PAllT II

CUBIC AND UUARTIC ClIttVES

LECTURES

ON THE

THEORY OF PLANE CURVES

Deltvehed to Post-Graduate Students

IX TiiK University of Calcutta

BY

SURENDRAMOHAN rxANGFLI, M.Sc.

LKCTURER IN PURR MATHEMATICS,

UNIVFRSITV OV (A I. (ITT A

PART II

PUBLISHED BY THE

UNIVERSITY OF CALCUTTA

1919

CALCUTTA.

Sole Agents

R. CAMBRAY & CO-

CALCUTTA.

PRINTKD BY ATDLCH AxNDKA HHATTACHARYYA

AT THB CAT.rUTTA TTNIVKRSFTY PRESB, SRNATK HOUSE, CALCUTT/

PREFACE

The subject of Analytical Geometry covers so extensive

a field that it is by no means easy to decide what to omit

and what to insert ; a w^ord is therefore necessary to

explain the plan adopted in this book, which, I trust, will

prove a useful introduction to the his/her branches of the

subject and will facilitate the study of a variety of al<yebraic

curves.

In the preparation of these lectures, I have endeavoured

to present the subject in clear and concise terms to the

student commencinpj a systematic study of the properties

of alojebraic curves, especially of Cubics and Uuartics. In

the portion of the book devoted to the discussion of

cubic curves, I have not confined myself exclusively to the

application of analytical methods, but have availed myself

of the methods of Geometry whenever simplicity could be

gained thereby. One prominent feature of the present

work is that properties of cubic curves have been exhaus-

tively discussed with special reference to points of inflexion

and harmonic polars ; in this connection, canonical forms

have at times been found of ojreat use, A separate

chapter has been allotted to the discussion of some special

cubic curves of historic importance, and their most general

properties ; but no systematic analysis has been attempted,

lest the young student should feel embarrassed ; only

^â€¢52988

VI PREFACE

cpeneral characteristics of these curves have been outlined

which will supply sufficient material for independent think-

ing in more advanced staoes.

The subject of quartie curves is too extensive to be

adequately considered in a small work like this. I have

therefore confined the discussion chiefly to the most

prominent characteristics of these curves. One chapter

has been devoted to the consideration of bicircular quartics

with special reference to their mode of generation. In

fact, this chapter^ together wnth a note in Appendix I, is

mainly based on the well-known Memoir on bicircular

quartics by Dr. Casey, published in the Transactions of the

Royal Irish Academy 1869. Circular cubics have been

studied with much advantage, regarded as degenerate

bicircular quartics. In the last cha])ter are considered

some well known quartie curves, most of which are bicir-

cular or are cartesians. A similar consideration, as in the

case of cubic curves, has led me to restrict my discourse

only to the general properties of these curves. I have

intentionally avoided the discussion of Roulettes, Cycloids,

etc., reserving the topics for a future occasion. The reader

who desires to study the subject from a higher stand-

point can conveniently consult the following works â€”

Clebsch â€” Le9ons sur la Geometric ; Chasles â€” Histoirede la

Geometric. In Appendix II, a note on Trinodal Quartics

has been inserted. This was communicated to me by Rai

A. C. Bose, Bahadur, M.A,, Controller of Examinations,

University of Calcutta. *

PREFACE Vll

In studying singular points on eubics and quartics, I

have retained the common phrase " non-singular " to

designate a curve which has no double point or midtiple

point, although it has been pointed out by Prof. Basset that

this is a misnomer ; for, he says, Pliicker had shown that

all algebraie curves except conies possess singularities and

accordingly he introduced the term " anautotomic " in

preference to the pharse *' non-singular " commonly in use.

In concluding this preface, I desire to say that, in

addition to the works of authors cited in the preface to the

tirst part, I have consulted, with much advantage, some

notes on Cubies and Quarties furnished by my colleague

Dr. H. D. Bagchi, M.A., Ph.D., and that I am indebted for

some valuable hints to Rai A. C. Bose, Bahadur, M.A.

Once more I must acknowledge myself in the highest

degree indebted to Sir Asutosh Mookerjee, Kt., President

of the Council of Post-Graduate Teaching in Arts, for his

extreme kindness in encouraging me to revise these

lecture-notes for the press, and to the authorities of the

University of Calcutta for publishing them. Finally, I

must thank the Staff of the Calcutta University Press, but

for whose untiring energy and ready co-operation, the

second part of the book could not have seen the light of

day before December next.

University of Calcutta^

S. M. GANGULI.

August, 1919,

CONTENTS

CHAPTER X.

Curves of the third order â€” Cubic Curves.

Page.

Classification of Cubic Curves ... ... 139

Equations of Cubic Curves

HO

Tangential of a point ...

I4a

Satellite ...

14.3

Properties of Cubic Curves

146

Generation of Cubic Curves

150

Sextactic point

156

CHAPTER XI.

Harmonic Properties of Cubic Curves.

Poles of a right line

158

Harmonic polar of a point of inflexion

161

Properties of points of inflexion

163

Inflexional triangles ... ... ...

lib

Configuration of harmonic polars

B

167

X CONTENTS

CHAPTER XII.

Canonical Porms.

Canonical equations of non-singular eubics

Co-ordinates of the nine points of inflexion

Equations of harmonic polars

Hesse's Theorem

The canonical form of a nodal cubic . . .

The canonical form of a cuspidal cubic

Salmon's Theorem

Harmonic and Equianharmonie Cubics

Conjugate Poles

Net of polar conies

The relation of the Hessian with the Cubic

The Cayleyan

The Pole Conic of a given line

The Pole Conic of the line at infinity

169

171

174

175

176

178

181

185

187

187

189

192

199

202

CHAPTER XIII.

Unicursal Cubics.

Unicursal cubics defined. . .

Unipartite cubics

Bipartite cubics ...

Parametric representation of nodal cubics

Parametric representation of cuspidal cubics

Points of inflexion on a unicursal cubic

204

206

207

208

210

213

CONTENTS XI

CHAPTER XIV.

Special Ctjbics.

Circular cubic ... ... ... 216

The equation of a circular cubic ... ... 216

The Logocyclic Cubic ... ... .. 218

The Trisectrix of Maclaurin ... ... 220

The Folium of Descartes... ... ... 2-23

The Cissoid ... ... ... ... 226

Newton's method of generating the Cissoid . . 228

The Cubical Parabola ... ... ... 229

The Semicubical Parabola ... ... 229

Foci of circular cubics ... ... ... 230

CHAPTER XY.

Invariants and Covariants of Cubic Curves.

Number of independent Invariants ... .. 232

The Invariant S ... \.. ... 233

The Invariant T ... ... ... 2S5

The Discriminant T Â» + 64S ^ ... ... 236

Covariants of cubics ... ... ... 238

The Sextie Covariant ... ... ... 240

Contravariants of cubics ... ... 242

CHAPTER XVI.

Curves of the fourth order â€” Quartic Curves,

Classification of Quartic Curves ... ... 244

Generation of Quartic Curves ... ... 248

Complex singularities ... ... ... 251

Harmonic properties of biflecnodes ... . 2.') 7

Bitangents ... ... ... ... 260

Salmon's Theorem on bitangential conies . . . 264

3|ll CONTENTS

CHAPTER XVII.

Tkinodal Quartics.

Equation of a Trinodal Quartic

Generation by Quadric Inversion

Properties of the nodes and nodal tangents

Properties of the points of inflexion, etc.

Tricuspidal Quartics

267

268

27a

274

CHAPTER XVIII.

BiciRCULAR Quartics.

Equation of a Bicircular Quartic ... ... 276

Generation of Bicircular Quartics ... ... 278

Focal conies ... ... ... 282

Centres of inversion ... ... ... 285

Foci of bicircular quartics ... ... 286

Cyclic Points ... ... ... 290

Double foci of bicircular quartics ... ... 291

Inverse of a bicircular quartic and its foci ... 292

Bitan gents of bicircular quartics ... ... 299

Bicircular quartics having a third finite node ... 301

CHAPTER XIX.

Circular Cubics as degenerate Bicircular

Quartics.

The Focal conic â€” a parabola

The centres of inversion ...

The inverse of a circular cubic

The foci of the focal parabolas

Intersections of a circle with a circular cubic

The real Asymptote of a circular cubic

806

307

309

810

311

314

CONTENTS

Xlll

CHAPTER XX.

Special Qtjartic Curves.

The Oval of Cassini

The equation of the Cassinian

Properties of Cassini's ovals

The Cartesian Ovals

The equation of the Cartesian

Generation of a Cartesian

Foci of Cartesians

Points of inflexion on Cai-tesians

The Lemniscate of Bernoulli

The Lima^on

The Cardioid

The Conchoid of Nicomedes

Trisection of an angle ...

316

316

317

318

319

321

321

323

324

325

326

327

828

Appendix I

Appendix II

Index

329

335

347

CHAPTER X.

Curves of the third order â€” Cubiu Curves.

139y The g-eneral equation of a curve of the third

order involves ten arbitrary constants. We may divide

out the whole expression by any one of the constants,

and hence the number of disposable constants in the

equation of a cubic curve is nine and a cubic can be made

to pass through any nine arbitrary points ; or nine arbi-

trary points; will determine a curve of the third order

uniquely.

The general equation of a cubic curve in Cartesian

co-ordinates can be written as â€”

n = o. ... ... ... ... (1)

Or, in any system of homogeneous co-ordinates, it may

be taken as a "" -{-Uc^y + 'dcAy^ J(.dy^ -{â– a\^^z + WAyz-\-b'y^z

+ 1 '~J+my.^+?i:^=o. ... ... ... (-2)

Or, symholicaWy, 2iQZ^+niz^ + u^z + Ms=o ... (3)

where n^ is a constant and u^, ^'^, n^, are homogeneous

expressions of the first, second and third orders resnectively

in X and y.

140. '^ We have seen, Â§ 40, that a curve of the third

order can have at most one double point, and no other

multiple point. Hence according to their deficiencies,

cubic curves may be divided into the following three

fundamental species : â€”

(1) Non-singular or anantotomic cnhics, which have

no double points.

(rZ) Nodal cubics â€” in which the double point is a node,

with two distinct tangents (real).

(3) Cuspidal cubics â€” in which the double point is a

G ISp.

140

By using the formulae of Â§ 97, we may calculate the

Pliicker's numbers for the three cases as follow : â€”

n=. 8= k= 7n= t= i= j) =

Case I

... 3

6

9

1

Case II

... y

1

4

3

Case III

... 3

1

3

1

141.* The trilinear equation of a cubic circumscribing

the triangle of reference is

x^u-{-y^v + z^w-\-lcxyz^o ... (1)

where u, v, w are linear functions of y, z', z, x ; and x, y

respectively and represent therefore' the tangents at the

vertices A, B, C respectively

If the vertex A be a double point on the cubic, the

equation should contain no x^ and x^ ^ and consequently

it lakes the form an^-\-u^=o, where m^ and m^ are

homogeneous functions of the second and third orders

respectively in y and :, If further the curve passes

through B and C, the equation cannot contain y^ aiad z^.

Therefore the equation of a cubic circumscribing the

fundamental triangle and having a double point at A is

AU^+yz{my + nz)=0 ... ... (2)

in which u^z=o is the equation of the tangents at A.

Hence the point A will be a node, a cusp, or a conjugate

point on the curve, according as '>' ^ represents two real and

distinct, coincident, or imaginary right lines.

142.*^ If the vertex A is a point of inflexion on the

curve, the tangent at A meets the cubic in three conse-

cutive points. Now the equation of a cubic passing

through A can be written as

141

If A is a point of inflexion, the tangent ?^i=o meets

the curve in three consecutive points at A. Therefore,

if ^i is made equal to zero in the equation, i,e., if y be

eliminated between ti^=o and the equation (3), the

resulting equation should have z^ as a factor, which re-

quires that the coefficient of ,v should vanish, i.e., u^ should

contain w^ as a factor. Thus the equation of a cubic

having a point of inflexion at A is

143.*^ We have proved in Â§ 21 that if a cubic curve

passes through eight points of intersection of two cubics,

it must pass through the ninth also. Asa particular case,

we may prove the following theorem : If two right lines

A and B meet a cubic in the points a, b, c and a' , h\ c'

respectively, then the lines aa', hh\ cc' respectively meet

the cubic again in three other colUnear points a", ^", c".

Let n â€” o and v â€” o\>q the equations of the two given

lines A and B and let /^'=o, v' =o, w' = o be the equations

of the lines aa'{k'), hh'{'^'), cc'{C) respectively. If w = o

be the equation of the line a"6"{C), then it must pass

through the point c". (Fig. 18).

The lines A', B', C constitute a cubic which intersects

the given cubic in nine points, and the lines A, B, C make up

a cubic passing through eight of these nine points. It

must therefore pass through the ninth j^oint c'^ also. But

this last point cannot lie on A or B, which already meet

the cubic each in three points. Therefore it must lie

onC.

Now, the equation of a cubic passing through the

intersection of two cubics \]=o and Y = o is of the form

JJâ€”kY^o. Therefore the equation of the given cubic'

can be written as uv'w'â€”kuvw=o, since it passes,

throusch the intersections of the cubics uvw and u'v'tv' ,

142

Cor : From what has been said above, it follows that

the equation of all eubics can be expressed in the form

tivw â– \- hu' v' lo' â– =â– . ... ... (5)

where Â«, f, lo u, v' , w', are linear functions of the variables,

and therefore represent right lines. The equation represents

a cubic passing through the nine intersections of (u, v, w)

and (//, ?/, iv').

144/ If u = v, i.e., if the lines A and B coincide, the

equation of the cubic takes the form

n'v'w' -\-hi^iv = o. ... ... (6)

The linos u\ v\ iv' become tangents to the cubic at the three

points where the line A meets it. Also, the three other

points {a'\ b'\ c") in which these tangents meet the cubic

again lie on the line lo^o. Hence we obtain the theorem: â€”

If a right line intersects a ciilic in three points, the

tangents at these points meet the cubic again i)i three other

collinear points.

Definition : (I) The point a'\ in which the tangent

at any point a meets the cubic again, is called the tan-

gential of the point a. This point is also called the ^'satellite

point" of the tangent, (Cayley, A memoir on curves of the

third orderâ€” Coll. Papers Vol. II, No. 146, p. 409).

{%) The line C on w-hich lie the tangentials of three

collinear points lying on a right line A is called the

satellite of A. We may thus state the above theorem as

follows : â€”

The tangentials of three collinear points are collinear.

145. ^ As an application of the above properties of a

cubic, we may prove the following theorem"^ : Having

* A. Cayley â€” Memoire sar les courbes du troisieme ordre â€” Journal

de Mathematiques Pares et Appliques (Liouville) tome IX (1884), or

Coll. Papers. Vol I, No. 26. p. 184,

143

given a curve of the third order which passes through the

six points of intersection of four right lines, the tangents

to the curve at opposite points interr^ecfc on the curve in

three collinear points. [Two points are said to be opposite

when one is the intersection of two lines and the other of

the remaining two].

Let 5 and 6 denote two points on the curve such that

the tangents at these points intersect at a point 5' or 6'

on the curve. Let 'I denote any other point on tlie curve

and the hues (5, i) and (6^ 2) intersect the curve at the

points denoted by 4 and 3 respectively. Then (5, 3) and

(6, 4) intersect on the curve at the point 1. The tangents

at 1 and %, also the tangents at 3 and 4, intersect in two

points on the curve which are collinear with the point 5'.

Fcr the tangentials of 5, 3, 1 lie on (5', 3', 1'), and those of

6, 2, 3 are on the line (6', 2', 3'), ie., the line (5', V , 2>'),

Hence the theorem. (Fig. 19).

146. The three points in ivhich a cubic intersects its

asymptotes lie on a right line.

We have defined that the asymptote of a curve is a

tangent whose point of contact is at infinity. But a

cubic curve has three asymptotes. Therefore, if we suppose

that in Â§ 143 the line u^o is at infinity, i.e., ?^ = I, the

equation of the curve becomes n'v'w' -\-k\'^ic=.Oy where I = o

is the line at infinity, and u\ ?/, w' are the tangents whore

points of contact lie on 1 = ^, i.e., u ^=-0, v' = o, to =q are the

asymptotes. The form of the equation shows that the points

in which these asymptotes meet the cubic again lie on the

line IV = 0.

The straight line to which passes through the points of

intersection of a cubic with its asymptotes is called the

satellite of the line at infinity.

144

147.' The product of the perpendiculars drawn from any

point on the, curve on to the asymptotes is in a constant ratio

to the perpendicular drawn from the same point on to the

satellite of the line at infinity.

This is only a geometrical interpretation of the equation

of the previous article. The equation of a cubic whose

asymptotes are ?/, v' , v/ and lo is the satellite of the line

at infinit}^ is n' v' w' ^=^kV-io â€” k'iD, where k' is a constant.

Now, u\ v' , tv' and iv are proportional to the lengths of the

perpendiculars drawn from any point of the curve on to

those lines and hence the theorem.

148."^ If two of the points of intersection of a line with

a cubic he points of infleion^ the third must also he a point

of inflexion.

L ;t the points a and h be points of inflexion on the

curve. If the line ah meets the curve at c, then c is also a

point of inflexion.

Now, the tangent at a point of inflexion has a three-

point ic contact with the curve. Consequently, the

tangential of a point of inflexion coincides with the

point itself. Thus the tangentials of a and h respectively

coincide with them. Therefore the satellite of the line

ah coincides with itself, and consequently the tangent at the

third point c, in which the line ah cuts the cubic, has a

contact of the second order, i.e., the point ^ is a point of

inflexion.

If we put u' =-v' = w' in the equation (5) of Â§ 143, it

becomes uvw-\- ku'^ =-o, which shows that the lines n, ?', w

have each a contact of the second order with the curve at the

points where u' intersects it. Hence we obtain the

theorem : â€” If a cuhic has three real points of infle.rion, they

lie on a right line.

145

149.'' We have seen that the tangents to a cubic at

three eollinear points meet the cubic again in three other

collinear points, or, what is the samething, that if tangents

be drawn to a cubic from three collinear points a, d, c,

on the curve, then the line joining the point of contact

of ani/ one of the tangents from a to the point of contact

of aiii/ one of the tangents from h, passes through tho

point of contact of ant/ one of the tangents from c. Now,

from any point on a non-singular cubic four tangents

can be drawn to it. Therefore the sixteen lines which join

the four points of contact of tangents drawn from V/ ' to

those of the tangents from h, must pass through the four

points of contact of the tangents drawn from c

Thus the twelve points of contact of these tangents lie on

sixteen lines, three on each, and through each point there

pass four of these sixteen lines.

Prom this it follows that, for a given line x^, there is

but one satellite to it ; but to a given line A there corres-

pond sixteen different lines, of which the given line is the

satellite. Herce we obtain the theorem : â€” A given line

has onl?/ one satellite y hut there are si teen different lines

of tvhich it is itself the satellite.

150. ' The four points of contact of tangents drawn

from any point A on a cnhic are the vertices of a quadri-

lateral, the three diagonal points of tvhich are the points of

contact of the tangents drawn from the tangential point erf A.

Consider a line which intersects the cubic in the three

])oints A, B, C. Let a^^a^, a^, a^,; b^, b^, ^3>^4 ; ^u^*2>

Cg, c^ be the points of contact of tangents drawn from A,

B, C respectively. Then these twelve points lie on sixteen

different lines. Let the points A and B coincide. Then

the points a^, a^, a^, a^ coincide with the points b^, b,^,

b^, b^ and one of the points c's (say c^) coincides with A.

146

Thus we see that the line joining c^, one of the points of

contact of tangents from a, to a ^ , one of the points of

contact of tangents from A, must pass through one of the

ot/ier points of contact of tangents from A, say a 2.

Similarly the line c^a^^ passes through a^. Thus the

sixteen lines reduce to six sides of the quadrangle a^a,^a^a^,

counted twice, and the four tangents at these points.

Hence the intersection of a pair of opposite sides is one of

the points Ci, ^'g, Cg, and the tangents at the vertices (T^i,

^2J ^'^sj ^4 nieet the curve at the same point A (c^), i.e.,

Ci, Cg, c^ are the diagonal points of the quadrilateral.

\bV: From this we easily deduce the truth of the theo-

rem: â€” If two tangents be draiviifrom any point A on a cahic,

the tangent at the third point in tohich the chord 0/ contact

meets the cubic cuts the tangent at A at a point on the curve.

We may analytically prove the theorem as follows : â€”

If we take the two tangents and their chord of contact as

the sides of the triangle reference^ the equation of the cubic

must be of the form yz{l'+my-\-7iz)-\-x'^{By-\-Cz) = o,

where By4-C~ = o is the tangent at A and lx-^wy-^nz = o

is the tangent at the point where the chord of contact x = o

meets the curve. These two lines intersect on the curve.

162.^ The chords of contact of tangents dratvnfrom

any point of a cubic are harmonic conjugates of the tangent

to the curve at their intersection and the liiic joining the

intersection ivith the point.

Let the line C be the satelHte of any line A, the

tangents at the points on A being L^ M, N. Then the

equation of the curve can be written as LMN â€” A2C = (?,

if A = y and C = <? represent those lines. Let B be any

other line of which C is the satellite, so that B passes

through the point of contact of N and those of two

other tangents L' and M' respectively which meet C in the

147

points where L and M respectively meet it. Then the

equation of the curve naay again be written as L'M'N

-B^C-o. Thus we obtain the identity N(LM-L'M')

= (A^~B2)C. Tne right-hand side represents three lines

A4: B and C, therefore the left-hand side must also represent

three right lines. Now the line N must be one of A + B,

and C must be one factor of LM â€” L'M', which is the

line (LL', MM'), and the other factor is the line (LM',

L'M) which is A + B. Therefore when C is a tangent,

so that L, M, L' and M' meet C at the same point on

the cubic, one of A + B becomes the line joining the

point of contact of C with that of N. But A, B,

A + B form a harmonic pencil. Hence the theorem.

(Fig. 20.)

153.*^ Any line drawn through any point A on a cubic

is cut harmonically in the two points P and Q tvhere it

meets the cubic again, and the tioo points L and M tvhere

it meets a pair of chords joining the points of contact of

tangents drawn from A.

Let rtTj, flg, Â«3, r/^ be the four points of contact of

tangents drawn from any point A on the curve. (Fig. 21.)

Then the lines joining a^, a^ and ^g, a^ intersect at a

point Ci on the curve. Let a line through A intersect

the cubic in P and Q and the tangent at Cj at D, and

the chords of contact at L and M respectively. By the

previous theorem, Ci(LAMD) is a harmonic pencil i.e.,

LM is a harmonic mean between LA and LD.

â€¢2_ + ^=_l_ (1)

LA LD LM ^ ^

By Maclaurin's another theorem (Â§ 53), since any

line through L intersects the curve at P, A, Q and the

tangents at three points (eollinear with L) in the three

148

points A, A and D, we have

LP LA LQ LA LA LD

'â€¢''â€¢ LP LQ LA LD LM *

.". PQLM are harmonic.

154.^ The theorem of Â§ 19 can be applied to the case

of the cubic when ?// = 8, and it then takes the form : â€”

Every curve of the nth degree ivhich pa^es thrnngh 8w â€” 1

fixed pohits on a cnhic passes through one other fi^ed point

on the cnrve.

This can be proved very easily with the help of the

theory of residuation. Let the i^roup of 3;^â€” 1 points

be denoted by P. Describe two curves of the n\kv degree

through these points P. Let them intersect the cubic in

the points Q and Q' respectively. Then [P + Q]=o and