Ganguli, Theory Of Plane Curves, Chapter 11

CHAP'rl'JR

CURVES

UNICURSAL

231. It

PARA~[ETR[(;

has

unicursal

REPRESENTATION:

already

been

Chapter,

of all

curves,

expressed

however, the

that

will be discussed

co-ordinates

The real significance of freedom

a curve

certain

of whose

variable

that

(2)

The point

position

has a part.icular

Hence, the homogeneous of two independent

i.. I«. Is

Z

of freedom

=f

1

(A,

must be expressible

parameters

in the form

p.) : j

represent

2

rational,

(p.)

~=/l(A) :/.(A)

ibid,

parameter.

leaves the position

I.(A,

p.)

integral

IS

functions

given,

of

one

degree

lie

on the

to

are defined by :/.(A)

'I'he elimination

of A from

of the point

the curve

but still the point lies 'in the

the equation

" Brill, Math. Luroth,

p.) :

and the co-ordinates"

where A is a variable

obtain

(A,

a common factor.

z : y:

undetermined,

has two contained

on this locus.

is lost, and the point is restricted

these equations

of a point ill

a point

co-ordinates

When one of the parameters locus p.=const.

be

: locus;

order n without

can

in the plane, and is practically

The point lies on a certain

where

properties

points

of the two co-ordinates

(1)

x :y :

or

Tn the

parameter.

lies in the fact

in the two statements

in terms

rational

IS

is zero, and conversely.

in terms of a single

the Car-tesia.n system degrees

said

when its deficiency

present

Xl

011

curve,

and

we

of the cui-ve.f

Ann. Bd. 5(IS72),

Bd. 0, Pasch,

Rd. IS, anrl

p.4.01.

See

Humber-t,

also

the papers

by

Bnl!.

Soc. Math.

""

France t. 13( 11:>8;'),pp. 49 and SH.

t. Of. .J. ~;. Itowe·-Bull.

Am. Math. Soc., Vol. 2:1(I!H7), pp. 304-308.

288

THEORY

232. Or-ERseH

O}'

METHOD

If the n-ic is of zero double points.

PLANE

:

deficiency,

Through

these

tk(k+3)-Hn-l)(n-2) intersections,

which requires

other points

on the

Since each

if

through

other points

double

point

through

(k
the n-ic in

counts

as

tu.o

k=n-l we

or k=n-2.

construct.

the double

a, pencil points

will intersect

the n-ic in one point expressed

whose

co-ord inates can

if a pencil of (n-2)-ics

manner,

n-ie, each curve will intersect

the latter

can be similarly

Thus, by taking express

the

pencil

in terms of A.

the double points and n-3 ordinary

co-ordinates

(n-l)-ic8 2n-:::

of

and through

n-ic, then each curve of the

on the gi,en

In a similar

= kn

)(n-2)

be rationally

can

and curve

which intersects

+ t(n-l

that

Therefore,

through

Hn-l)~n-2)

has points

we must have

tk(k+8)

1t-AV=O

it

double

a curve of order k can be drawn, only kn points.

CURVES

be drawn

points

on

in a point

the

whose

expressed. of (n-l)-ics

a pencil co-ordinates

or (n-2)-ies

rationally

in

terms

once suggest

that

we of

a

parameter. The

above

considerations

at

always exists a k-ic which intersects number

of fixed points,

or has assigned

CaseI: ordinary

The n-ie has

no multiple


at those

intersections. points

other

than

nodes and cusps.

If k=n-l, connected relations,

singularities

count as nk-l

points, such that the points

there

n-ie in

the unicursal

the co-efficients

by

and the intersections 2

X

in the equation

of the k-ic are

Hn-l)(n-2)+2n-3=tk(k+3)-1

t(n-l)(n- 2)

count as-

+ (2n-3)=nk-l

points.

linear

VN~CURS~L

CURVES

Daee II.

The n-ic has ordinary multiple points and ~~e point at each r-ple p.?tn~on thll n-ic, and passes through (n-3) other fixed points. k-ic has an (r-l )-ple

If k=n-2,

since p=Hn-l)(n-2)-l,'Cr-l)=O,

the co-efficients in the equation of the k-ic are connected by l{-r(r-l)+

+ (n-3)

(n-3)=Hn-l)(n-2)

(§ 50)

=-}k(k+3)-1

linear relations, and the intersections count aslr(r-l)

+\n-3)=(n-l)(n-2) =nk-l

points.

In fact, the k-ic is an adjoint (n-2)-ic (n-3) ordinary points on the n-ic. 233.

THE

OF THE

ORDER

+( n-3)

UNIClJRSAf,

pas!li~g throu~ll

CURVE

:

Consider the curve;I):

s . Z=/1(A)

:/.(A)

... (1)

:/8(A)

where t.. I., Is are rational, integral, homogeneous functions of order n in A. To each point of the curve there corresponds a certain value of the parameter A, and c~pversely.· The functions 11' Is, I ~ cannot have any common factor. Any line lx + my points given by-

+ nz:;

III

°

intersects

the

+mi. +nf.~=0

curve

in

jt

.. , (2)

and consequently, its order is n. That the equations (1) represent a gen!lra1 purv,eof ~rfler n, having Hn-l)(n-2) double points, follows from the fact that the expressions (1) contai~ 3(~~: 1') ~rbitra.ry constants. By a linear transformation of the form".

1

A=PA'+q rA'+s

we may reduce the number to 3n-1. But

3n-l=-}n(n+3)-t(n-l)(n-2)

i.e., equal to the number required for determin,i~~ ~ ~!ln~~~~

37

290

THEORY OF PLANE CURVES

curve of order n with -Hn-I)(n-2) proves the proposition, 234,

double points, which

THE CLASS OF 'rHE UNICURSAL CURVE:

*

The equation of the curve in line co-ordinates is obtained by forming the discriminant of the equation (2), i.e., by eliminating .\ between the equations

lof, +m of, +n 0.\

0.\

If, (.\)+mf,(.\)

ob

0.\

=0

II

(3)

+oi, (~)=o)

The discriminant is, in general, of degree consequently, the curve is of class 2(n-I). If, however,

of

1

,

2(n-I), and

of. , of =i ' f ' f

0.\'0'\'8>.-"

S

"

(4) S

the two equations become identical, and the common routs of (4) reduce the class of the curve, Hence, if the equations (4) have K common roots, the class of the curve is m=2(n-I)-K. We shall see later on that the equations (4) give the parameters of cusps, and henoe, if there are K cusps, the class is m=2(n-I)-K. 235.

PARAllETRIC

REPRESENTATION

IN LINE

CO-ORDIN.A.TIB :

By eliminating l, m, n between the equations (3) and Z.e+my+nz=O, we obtain the equation of the tangent at any point in the form(5)

f. * AI1II.

OIebsch-Crelle,

Bd. 64 (1865), p, 43.

Bd. 2 (l870), p. 515.

See also

Haase,

Math.

UNICURSAL

291

CURVES

Hence, the co-ordinates of the tangent are given by-

i.e., the co-ordinates of the tangent are rational, integral, algebraic functions of a parameter. Hence we obtain the theorem: Ifa curve ts unicursal qua locus, it ill unicurllal q"a envelope. It is to be noted here that the parameters in the co-ordinates of the tangent cannot occur in degree higher than 2(n-I), but may be less, It. follows then that the class of the curve cannot be greater than 2(n-l). Jla:. 1. The

Consider the parabola

co-ordinates

form

a:=at',

shows that

of the

the

y=2at.

tangent at any point (t)

The equation of the the co-ordinates

tangent

parabola

2.

~= -1,

are

is unicursal

equation of the curve is easily Z.

y' ~ 40,•.

of any point on this curve can be expreased in the

qua envelope.

obtained in the

Express the co-ordinates

being ty-II-at' =0, C= -al', which

1/=t, form

The tangential an? = K.

of any point on the curve

(a: + 1/)311"=m'(11 +~)' rILtionally in terms of a parameter.

E.. 3.

Prove

that

the

order of a unicursal ourve of cla!1 m is no;

greater than 2( m.-l). Ea:. 4.

The locus of the poles of /tny normal to a given conic is, in

general, a unicnraal

quartic.

Discuss

the

case when

the

conic il

110

parabola. Ea:. 6.

If

f " f"

flare

polynomials

co-efficient of t"-l is wanting, tbe wbere any algebraio curve inteneot.

in t of degree n, in whioh the

Bum of the

parame;en

tbe ".le il eero.

of points

'T.

J

THEORY OF PLANE CURVES

~36. As already shown, at a double point the co-ordinates have the same values for two different values of the parameters ~ and ~', i.e., when ~=I=~',

f, (>..)/!, (>"')=f9 (A)/f~ (>"') =i, (>..)/fs (>..')

... (1)

The case of a node has been discussed in § 56, but in the case of a cusp, the equations of the tangents for the two branches mMting at the double point must be the same, i.e., the ce-ordinates of the two tangents must be proportional.

(fs 'OfI/OA- f. afs lo~) = k'(f s 'Of./8>'"- f.8fs 10>..') (f, afs lo>"-fsofl/a~) = k'( t,afs/a~'-1.8f, (f,of,/a)..-/,

18>"')

'Of.lOA)=k'(f .af,/a'>..'-/, 8/./'0>"')

where A, ~' are the parameters of the coincident points belonging to the two branches, whence we obtain the two foHowing determinant equations:

. 'Of, : a~' !

v, a~' o/s

ax'

'Of, I,(A) . =0, :I,(A'! a~ I

'Of,

'01-

t. (A)

=0

(2)

! i

of. aA

f. (A)

! f. (~') '0/.

afs ax

fs (A)

.I,(~')af. Is (A)

o~

f. (A)

aA

which show nothing but that ~, A' must be the double roots of the equation ~ ving the nodes. There are then two cases to be considered: (1) When ~ and ~' are different; there is no cusp but 'two nodes indefinitely near each other, such that their tangents coincide, and there is no reduction in the class. (2) When ~ and A' are equal and represent the pair of eqiia;l roots, which then give a OllSP, and the class of the cur-re is reduced by 0 lie.

~utilng ,\'::X.f£, approaching zero, £:'iiS

where

£

is a very snui.l1 quantity

from ; 2) we obtain~"ji. ~l

ax' aA fl a'l, al. I. a).' ax a 'Is afs f.

...

=0

(3)

ex

OA'

the double roots of whioh give the parameters of the OUlIpS. Thus the ousps are given by the equations

,,~ ax =

all

olfl

ax

I,

axa

f.

ax

0/.

aa/. ax·

I.

a/.

S·I.

ex

=0

I:>. =0

and

... (.•.)

at'

Second. method: The cusps may also be determined from the equations(5) For, at a node wehave equations (1) satisfied, and putting each ratio equal to k, we baveI,(X')-/,(X) _ (X'~X)fl (X) -

f .•(X')-/'JiX) (X'-X)I.(X)

_ I.(X')-I.(X) - ('A'-X)l.(X)

k-l

= (X'-X) But, by the mean value theorem, we havelJiXJ={l

().) =1' 1 (X+8[X' -X])

where () is a positive proper fraction.

2~4

THEORY

OF PLANE CURVES

Now, when the tangents at the node approach coinoidence A' approaches A, and we have in the limit' ~ 237.

/f,(A)

=

U:/f.(A)=-

~; b.(A).

INFLEXIONS:

If A, A', A" be the parameters of three on the curve, we have-

collinear

points

=0

lfl(A)+mf.(A)+nfa(A)

If, (A') +mf. (A') +nfs (A') =0 lfl (A") +mf.(A")

+nf.(>':') =0

where A, A', A" are all different. :. The oondition of collinearity is obtained by eliminating l, m, n from the above equations in the form-

1 (A-A')(A'-A")(>':'->")

f,(A)

f.(A)

fa (A)

fleA')

f.(>"') fa(>"')

=0

I,(A") f. (>':') t,(>"") If now A, A', A" approach equality, the above condition reduces to-

but

still

distinct,

.6. =0 2n(n-l)' Hence, the inflexions are given by the equation.6.

=0.

Thus at a cusp, we have both l:l. =0 and at a point of inflexion only .6. =0.

... (6)

al:l./O>..=o,

while

From equation (6) it follows that the number of inflexions is, in general, 3(n-2). But since a double root of (6) gives a cusp, the number of inflexions reduces to 3(n-2)-2K, if it has K double roots, i.e., if the curve has K cusps. In fa.ct,

t=3n(n-2)-3(n-l)(n-2)-2K =3(n-2)-2K

UNlCURSAT,

295

CURVES

This number gives an upper limit to the number of cusps which a curve can possess. For, since 3(n-2)-2K can never be negative, the number of cusps on a curve can Bflverexoeed t(n-2).* 238.

BITANGENTS

OF

UNICURSAL

CURVES

:

Bitangents and stationary tangents can be found axactly in the same manner, if the ourve is regarded as unicursal qua. envelope. As proved in § 235, the co-ordinates of a tangent axpresaad in terms of a parameter in the form : ~:

1] :

~=ep, (A)

can be

: ep.(A) : ep,(A)

Therefore, the parameters of the bitangents are obtained . ep,(A) ep,(A')

from

= ep.(A)

=ep3(A)

whellA-L.A'.

-r

4>.('11.')'

ep.(A')

The number of bitangents of a unicursaI curve is found to be 2(n-2)(n-3), as can be verified from the formulre of § 146. The parameters of inflexional tangents are given by-

at, aA

, fA.. =O
aA

'1'.

ex

'1','

and so on.

For a detailed discussion of the double tangents, etc., of a unicursal curve, the student is referred to the paper of Clebsch-Crelle, Bd. 64 (1865), pp. 53.54, and of HaaseMath. Ann. Bd. 2 (1870) p. 515. *

There

is

0.

further

limit to this.

The

1=3n(n-2)-63-8k

inflexions on an

n-ic o.re

.

.. For eo.ch n.ie, in genera.l, we must have 63+8K~3"(n-2). Henoe, if 5=0, the upper Iimij to the number of CUlpS i. tn(n-2). But,

t(n-l)(n-2)-jn(n-2)

~t
Hence, if n>4, all the double points on an n.ic can Dever be oueps See:rd. W. Haskell, Bull. Am. Math. Soc., Vol. 23 (1817), pp. 164-165.

TBEO~T OF P~AN¥ CU~VES Con.idor the ourve:

Ell:. I.

lI=t',

y=l +t",

.=t.

For double point~, we must have-

t"=:,~',

l+t"=lt~',

t'=t,

when

~+~'.

As in § 56, the nodes will be given by-

'I',=tt'-I=O

and

tf>.=tt'(t'"+tt'+t')-O

Whenoe eliminating t' we obtain parameter. for the nodes. But,

t' + t' + 1=0

which gives the

(t'. + t" + 1)= (to + t + I)(t' -t +1)=0 t"+t+l-0

giving t=1V,W',

the imaginary cube roots

of unity, and t"-t+l=O giving t=t( +l±i",a) i.e., the value. of the p&r&metersare imaginary. The :firstgives the aonode (1,-1, r) and the second gives the acnode I, 1). The cusps and inflexions are given by the determinant equation (4) § 286, whioh reduces to t'(t"-2)=0, whose roots are t=O,O, ± "'Z

( -1.

The two 0 value. of t give, in fact, a point of undulation at (0, 1,0). The cusps are given by the double roots, i.e., the common roots of this and t(t'-l)=O. Hence t=O is a common root and should give a cusp, but this is found to be a point of undulation. Therefore (0, 1,0) is not strictly speaking a cusp. If, however, we replace t by lit' in the original values of w, y and z, we find 11= l/t'., 1/=(1 + ~nlt'·, Z= lit'. Proceeding as before, the cusps are given by the double root. of t'1(1-2t'")=p. Henoe t'=O, i.e., t=oo giv~8 a cusp (I, 0, 0), Any line meeting the curve in points t and t' is(I-tt'). + lI(t"t'" +t"t' + t'"t) -z(tst" whence, by pntting t=t' is obtained in the form-

'J'!/,e f!pg~9F~~,8t1l1 .!?tp..e~llOi.p.t(t')

+2)_=0

"h#lI!-9!t

(I)

is-

(1-t'").+8t'~1I-2t'·(t'~ 8t

t tf/'1-t'~)=O,

the equation of the tangent at any point (t)

(l-t·).+3t'y-2t8(t·

We have

+ oe» +t' + tn'

Hl}z=O

'.'

(2)

bitangent when (4) and (5) represent the .ame line, i.e., if

w~ gel; It+t') (t" +(I-t't'")=O

and tt'=2

UNICURSAL are given by-

The b itangents

,I'

and

+ t'~ -t't" =0

( Thefirstgronpgives

tt'=2.

1=-1'=;,/2

The bitangent The other

297

CURVES

i.e.,

/'+2=t"+2=0.

x + 4y=0.

is

group gives

equal values of f and 1', and hence there is

no bitangent. Bitangents The

may be obtained otherwise

co-ordinates

of the

point

as follows: of (1) and (2) are

of intersection

given by---, y

2,,' t + t' =

where

If the

+ (4 + 2v' -6v)(·,,'

and

I"

tangents

is indeterminate.

tt'

-2u) + 4v'

= u,

coincide in a bitangent, Hence the

their point of intersection

denominators

must vanish, we get v-2=0

and

u=O,

in the above

also

n=v=O;

expressions but

".=v=O

gives t' =0, which gives an 1mdulation, as hail already been seen. Thus

,"=t+t'=O,

Ex. 2. (i)

v=tt'=2,

x=t+t-',

1/=1 + f'

x=l

Find the singular

(ii)

J' =

5.

Find the Pliickerian

y=f'(2-t)'

'1=1' +1' +t+ 1.

y=I',

.=1'-1

u=

z = cos 1>.

sin 31>,

characteristics

p=O,

0=1'=3,

of the curve

CU1've_

:

?J'z" = .".;

K=,=3.]

Show that the envelope of lines joining corresponding

on a unicursal

38

and we g-et

points and singular lines on the

cos 31>,

[n=m=5, Ell).

t' +2=0,

(ii) x=I(2-t),

+1+1',

(i) x=] +t',

E», 4.

whence

Find t.he singular points on the following curves:

(iii) E». 3.

and

a: + 4y + 0.

the bitangent

quartic and a conic is a olass-eextic,

pointe

eq

2!l8

THBORY

239.

SPECIAL

CLASS

O~'

PLAN};

OF RATIONAL

CURVBS

CURVES:

Among the rational curves there are those with an (n-l)-ple point, or with three ordinary points where the tangent has n-pointic cor-tact. To this class belong the curves called Triangular Symmetric Curves, represented by the equation a;u"+by",+cz" =0, where n is rational, positive or negative. For a detailed study, see Dal'boux-Oomp. lieud., Vol. 94 (1890), p. 930 and H08, Brusotti-Lomb. Ist. Rend., Vol. 37 (2) (1904), p. 888, Loria-Spezielle alg. und transzend. ebene Kurven, Vol. I (1910), p. 841 and Wieleitner-Spezielle ebene Kurven (1908). 240.

THE

CIRCUIT

OF A UNICURSAL

CURVE:

The co-ordinates of a point P on the curve being continuous functions of the variable parameter '\, the point P moves continuously as ,\ varies, provided, of course, P is finite. If the co-efficients in the equation of the curve are all real, real values of ,\ will give only real points P. Consequently the real values of ,\ from 0 to 0 through infinity give a series of real points, ending where it began, i.e., the points are arranged in a single circuit, which may pass through infinity, but the locus is still continuous, therefore, a unicursal curve consists of a single circuit, with its real points arranged in one continuous series, i.e., it is unipartite. If, however, ,\ is imaginary of the form a + i{3, it is also of the form a-i{3, i.e., the point P is a double point. Since a consecutive value of ,\ does not give a real point of the curve, there is no real point of the curve consecutive to P, i.e., P is an acnode. Hence it follows that there may be a finite number of real intersections of imaginary branches, which are isolated points (acnodes), but these cannot be included in the continuous description of the curve by a real tracing point. Thus the unicursal curve consists of a single circuit, i.e., it is unipartite.

UXICURSA

241.

UNII'AWI'TTr.:

CURVES

299

T. CURVES

NOT NECESSARILY

UNTCURSAL:

Although unicursal curves are unipartite, the converse theorem is not always true, i.e., all unipartite curves are not unicursal. For example, consider the curve ,1;3 +.c=y·, which consists of a single circuit, i.e., is unipartite : but it is not unicursal, If it is to be unicursal, the co-ordinates of its points must be expressible in terms of a single parameter, and the elimination of the parameter should give the equation of the curve. Since the co-efficients in the expressions may be either real or imaginary, by a proper substitution of the form x=.v', y= V'ii/, z=i;' the equation obtained is

X'3-X'Z'2 +y'2Z'=O

which is unicursal, but this is certainly bipartite, as can be easily verified, the branches lying between :1:' -1, x' and ;~'+1 and infinity. Hence the curve 'l:3+.c=y·, although unipartite, is not unicursal. * 242.

CURVES

WITH

UNIT

DEFICIENCY:

It has been shown in § 225 that a curve with unit deficiency can be transformed into a cubic having the same deficiency, and as will be shown later on, the anharmonic ratio of the four tangents drawn from any point on a cnbic is constant for all positions of the point. These facts enable us to express the co-ordinates of any point on a curve with unit deficiency as rational functions of a parameter e, and of V'®:where 0 is a quartic function of e. From what has been said above, it will be sufficient, if this can be established for a cubic curve . • It

is to be observed that the term umicursal. makes no distinotion

between

real

and

imaginary

points, and if a unicursal oarve has any

real parb, it consists of a single

circuit;

whereas

the

term

,."ip"'Tt'ite

refers to the appearance of the curve and takes cognizance of tbe real part only.

In fact, just as an equation

simple equation, a nnipartitc

having a real root is not neceBsarily a

curve is not necessarily unicursa l.

4

300

THEORY OF PLANE CURVES

For, in this case ,r, y, z can be expressed as rational functions of rc', y', z'. If the cubic is taken to pass through the point xy, we may write y=O,c in its equation, when x : y : z are obtained in the above form. The values of 0 given by @=O correspond to the four tangents which can be drawn from the point xy to the curve. Thus the co-ordinates of a point on the curve with unit deficiency can be expressed as rational functions of 0 and -V®. Now by a linear transformation of 0, -v@ can be brought to the form -V( l-O!) (l-k20'), and putting O=sn u, -v(l-O') (1_k002) becomes en u dnu and the co-ordinates are expressible in terms of the elliptic functions sn u, en u. and dnu of a parameter u. 243. CO-ORDINATES

IN TERMS

OF

ELLIPTIC

FUNCTIONS:

By asing a method similar to that used in § 54, we shall now directly establish the theorem: The co-ordinates of any point on a curt'e of unit deficiency can be erpressed rationally in terms of elliptic functions. Let /=0 be the equation of an n-ic with multiple of orders k 11 k., k s"" such that the deficiency

points

=} (n-I)(n-2)-~tk(k-I)=1

.. ~tk(k-I)=Hn-l)(n-2)-1=tn(n-3). Hence, a unique curve of order (n-3) the multiple points.

can be drawn through

Now take a system of (n-2)-ics adjoint '*' to the given n-ic, i.e., a system of (n-2)-ics having a (k-l)-ple point '" We may intersections satisfied

when

/1-2 other A. second

take the

fixed points must be eqnal N-ic

is an

adjoint

to nN - 2, which

(n-2).ic,

passing

is

through

fixed points, since lk(k-l)+(n-2)='Il(n-2)-2=nN-2. condition

~N(N+ 3)-Il'elations, since,

a pencil of N.ics subject to the condition that the

at the

1lk(k-l)

is that

the

co-efficients

must

be connected

which is again satisfied by the adjoint

+ (n-2) =·HlI-2)(n + 1) -I =tN(N -3)-1.

by

(n-2)-il",

301

UNICURSAL CURVES

at each k-ple point of j, and other fixed points on f.

passing through

(n-2)

But the number of arbitrary co-efficients in the equation of such an (n-2)-ic is Hn-2)(n+l), and the number of conditions assigned is~tk(k-l)+(n-2),

i.e., Hn-2)(n+l)-1

Hence the equation of the system will contain one arbitrary parameter and can therefore be written as (1) where u and v are any two particular system.

members of the

Now, the curves (1) intersectj, in general, in n(n-2) points, of which ~k(k-l)+(n-2), i.e., n(n-2)-2arefixed. There are then depend on .\.

points P and Q which

two variable

If we eliminate one of the variables y (say) between j=O and u+.\v=O, we obtain an equation of order n(n-2) in :v, n(n-2)-2 of whose roots are the abscissee of the fixed points and the remaining two are the abscissre of the two variable intersections P and Q. Removing the known factors, we have an equation of the second degree in al with the co-efficients rational in .\, whose roots give the abscissee of P and Q in the form1

:v=A±B02 where A and B are rational in .\, and 0 is a polynomial in .\. If we substitute either of these values in the equations of the 1H·C and (n-2)-ic, we obtain two equations in y whose common root gives the ordinate of one of the points P (say) 1

and is of the form A' + B'02, where A' and B' are both rational in .\. Since the values of .\ given by 0=0 correspond to the points of contact-" of the variable (n-2)-ics which touch it

These include cusps other than at the fixed points.

302

THEORY

OF PLAN~: CURVE:,;

/=0

at a point other than any of the fixed points, and there are four such curves, 0 must be a polynomial of order foul' in A. Thus the co-ordinates of any point on a curve of unit deficiency may be expressed rationally in terms of a parameter A and an expression of the form * 1

02

1

== {aA" +4bA' +6CA' +4dA+e}2

By a suitable transformation, both Aand 0 can be simultaneously expressed as rational functions of the elliptic functions of u- sn ~t, en u, dnze, whieh proves the proposition. 244.

SIMPLIFICATION

Weierstrass's

BY WEIERSTRASS'S

elliptic function

NOTATION:

'(5(~t) t is defined as-

and is connected with its derived functions by the relation

where g. and g, are invariants. The function '(5 u is dmtbly periodic. If wand w' are the periods, we obtain the same value of '(5~t, when u is replaced by u+mw+m'w', where m and m' are integers. Now, we have seen that the co-ordinates of a point can be 1

taken as A+B02, where A and B are rational in A,while o is of the form (aA4+4bA3+6cA'+4dA+e) 1-

In order to transform 02,

* Clebsch-Crelle,

t

we make the substitution-

Bd. 64 (1865) p. 217.

Goursat-Mathematical

Analysis, Vol. II, Part I, § 6H, p. 156.

UNlCuRSAL

CURVEs

where ~v and ~'v are defined by the relations-

a2~v= - Cae-b'),

a3~'v=a'd-3abe+2b'

g., Y3 are given by

and the invariants

a3Y3=aee+2bcd-ad'

and.

-eb'

1

a2l~(u)-~(u+v)],

to

while A and B are transformed and

_03•

1


Then

a'y, =ac-4bd+3c'

into rational

functions

of

~7t

~'ZL 1

Henes the co-ordinates

± Be"

A

are expressible

in i.erms of doubly periodic elliptic

where X and if! are rational Ex, 1.

in ~n and

rationally

in the form-

~'tt.

Consider the cubic y'=aw3

the cubic is transformed

But since

g23

-27g,

into

2

cubic and is satisfied by

w = -bfa

and

and

(2)

16g3=3abc-2b3-n'd.

+0, the equation

.0'

+ 4w'/a,

y" =4.0'" -g,x' -g3

16g.=12(b'-
where

(1)

+3bw' +3cw+d

y=4y'/a

Putting-

= ~ n,

=~'

y'

Ii,

(2) represents

a non- singular

where g 2' g" are the invariants,

y=~.'l),,,. a

whence

E». 2. ordinates

functions

Express

rationally

in terms

of any point on the curve

The curve

evidently

of elliptic functions

the co-

wz'=y(w-y)(k'x-y).

passes through

the point w=y=O,

which is a

point of inflexion with x=O as the inflexional tangent. The three lines y=O, w-y=O

and y-k'x=O

pass through

this point

and meet the curve elsewhere, where xz' =0, i.e., they are the tangents drawn

from

the

contact are situated satisfied, if we put

point of inflexion to the cnrve, and their on the pw=t",

line py=t,

z=O. pz=

This

equation

vl-t'

points

of

is identically

";i-=k't'-.

......•••

304

THEORY

01'

PLANE

CURVES

and

vl-k't'=dnu.

Taking the radical with a determinate

sign, we have-

Putbing

i=snll.,

vl-t'=cnu.,

Bn'l~+cn'u=l, p,v=sn3u,

Thus

We may consider

py=snn,

pz=cnu

intersection

:t1 ; Y ; z=sn3u

which gives E», 3.

the

k'sn'n+dn',,=l

can be expressed

of the curve with the line :t1=y,

; sn u ; cnu dnu.

Show that the co-ordinates y'=ag)'

dnn,

of any point on the quartic

+6cg)' +4dg)+e

in terms of Weierstrass's

fnnction ~1t in the formy=

~'v=~

where

va [~tt-(~U+'V)]

.

" E.'"

4.

Show tbat the invariants a'g,=ae+3c',

E",.

5.

where any

g" g 3 in

line meets

Express

rationally

3 are given by

a'g3=ace-c3-ad'.

Show that ijJ.e sum of the straight

E",.

a

arguments

non-singular

,of the cubic

three

points

is equal to the

period. E», 6.

in terms

of elliptic

functions

tbe co-

ordinates of any point on the curves:

(i) .3(",3 +y3)=",3y3

245.

THE

(ii)

y3 =(x-a)'(",-b)'

CONVERSE THEOREM:;;

If the co-ordinates of any point on a curve can be expressed rationally in terms of elliptic functions ~~~and ~ u, the curve is, in general, of unit deficiency. Let the expressions for co-ordinates be-

wherefl,ft,fa

are each of the form A+B~'(1t),

• Bee H. Hilton-Plane

A and B

Algebraic Curves, Ohap. X, § 8.

UNICURSAL

being polynomials in ~u. tion of the relation

305

CURVBS

Since, by successive differentia-

we may express the products of powers of ~u and ~/u occurring in A±B~'u linearly in terms of~u,

~/u,

~"u, ...

i.. f., fs

may each of them be taken as a linear function of these in the form a+b~(u) +C~'(1t)+d~"(U) + ...+p~~-·(u). Any straight line lot +my+nz=O

where

will meet this curve

If, (u) +mf. (u) +nfs (u) =0

The left-hand side of the equation has n poles, and therefore, n zeros. Consequently, the curve is of order n.* The equation of the tangent at any point of parameter u may be written as in § 55, and the co-ordinates of the tangent aref.f's -fs1'.,

fs!' ,-f,i'.,

f,!'. -f.!'

,.

Now, In each c£ these expressions, the terms of the type "-2

~

a-1

(u).~

(u).

cancel, and consequently, each, when reduced to linear form, can be written as-

The class of the curve cannot be greater than 2n. Hence, as in § 55, the reduction in the class of the curve cannot be less than n(n-l)-2n, or n(n-3), and consequently, the number of double points (excluding cusps) * Goursat-Math. Analysis, Vol. II, Part I, § 68.

39

306

THEORY OF PLANE CURVES

cannot be less than ~n(n-3) or t(n-l)(n-2)-I, deficiency cannot be greater than 1."

i,e., the

For a detailed discussion of the curves of unit deficiency, the reader is referred to the well-known paper of Clebsch"Ueber diejenigen Ourven, dereii co-ordinaten sich als elliptischen Functionen eines Parameters darstellen lassen "in Crelle's Journal, Bd 64. (1865), pp. 210-270. AlsoHarnack-Math. Ann. Bd. 9 (1876), p. 1, and PorterTrans. Am. Math. Soc., Vol. 2 (1901), p. 36,

• The deficiency is zero or unity. But since the functions I" I., I. will not usually be rational functions of a single parameter, we say that, in general, the deficiency is unity.