Surendramohan Ganguli.

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

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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

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



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 —
Clebsch — Le9ons sur la Geometric ; Chasles — 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. *


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^

August, 1919,



Curves of the third order — Cubic Curves.

Classification of Cubic Curves ... ... 139

Equations of Cubic Curves


Tangential of a point ...


Satellite ...


Properties of Cubic Curves


Generation of Cubic Curves


Sextactic point



Harmonic Properties of Cubic Curves.

Poles of a right line


Harmonic polar of a point of inflexion


Properties of points of inflexion


Inflexional triangles ... ... ...


Configuration of harmonic polars





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
















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



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

Invariants and Covariants of Cubic Curves.

Number of independent Invariants ... .. 232

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

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

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

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

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

Contravariants of cubics ... ... 242


Curves of the fourth order — 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


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




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


Circular Cubics as degenerate Bicircular

The Focal conic — 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




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 ...


Appendix I
Appendix II



Curves of the third order — 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

The general equation of a cubic curve in Cartesian
co-ordinates can be written as —

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

Or, in any system of homogeneous co-ordinates, it may
be taken as a "" -{-Uc^y + 'dcAy^ J(.dy^ -{■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, § 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 : —

(1) Non-singular or anantotomic cnhics, which have
no double points.

(rZ) Nodal cubics — in which the double point is a node,

with two distinct tangents (real).
(3) Cuspidal cubics — in which the double point is a

G ISp.


By using the formulae of § 97, we may calculate the
Pliicker's numbers for the three cases as follow : —

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

Case I

... 3




Case II

... y




Case III

... 3




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


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 § 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 — o and v — 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

Now, the equation of a cubic passing through the
intersection of two cubics \]=o and Y = o is of the form
JJ—kY^o. Therefore the equation of the given cubic'
can be written as uv'w'—kuvw=o, since it passes,
throusch the intersections of the cubics uvw and u'v'tv' ,


Cor : From what has been said above, it follows that
the equation of all eubics can be expressed in the form

tivw ■\- hu' v' lo' ■=■ . ... ... (5)

where «, 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: —

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 order— 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 : —

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 — Memoire sar les courbes du troisieme ordre — Journal
de Mathematiques Pares et Appliques (Liouville) tome IX (1884), or
Coll. Papers. Vol I, No. 26. p. 184,


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 § 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.


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 — 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

If we put u' =-v' = w' in the equation (5) of § 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 : — If a cuhic has three real points of infle.rion, they
lie on a right line.


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 : — 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.


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: — 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 : —

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 — 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


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 — 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, «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.

•2_ + ^=_l_ (1)

LA LD LM ^ ^

By Maclaurin's another theorem (§ 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


points A, A and D, we have


'•''• LP LQ LA LD LM *

.". PQLM are harmonic.

154.^ The theorem of § 19 can be applied to the case
of the cubic when ?// = 8, and it then takes the form : —
Every curve of the nth degree ivhich pa^es thrnngh 8w — 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;^— 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

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