ganguli, theory of plane curves, chapter 13

8/14/2019 Ganguli, Theory of Plane Curves, Chapter 13

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CHAPTER XIII

HIGHER SINGULARITIES ON CURVIS.257. The theory of singularities on plane curves was first

studied by Plucker in his great work the Theorie derAlgebraischen Om'ven (1839), in which he considered some ofthe higher singularities. But the importance of the analysisof higher singularities has been recognised from the timeof Cramer.* The subject has been studied by Cayley,tHalphen.j H. J. Smith, Brill and Noether.j] Finally,Scott,' in her well-known paper, gave a number of highlyinteresting geometrical methods of dissolving higher singularpoints on curves by means of a number of simpleillustrations. Other workers on the subject are Bertini,Zeuthen, Segre, Cremona, etc.

In Chapter VII we have discussed the six equations ofPlucker with regard to the nodes, cusps, etc., which aretermed "ordinary singularities" of curves. But there areother kinds of singular points of a complex nature whichare called "higher singularities," as distinguished fromthe ordinary singularities .

Cramer-Introduction a I'analyse des IigneaCourbes, Geneva(1705).tCayley-On the Higher Singularities of a Plane Curve-Quarterly

Journal of Mathematics, Vol. VII (1866), pp. 212-22.t Halphen-Comptes Rendus, Vol. 78 (1874), p, 1105, and Vol. 80

(1875), p. 638. H. J. Smith-On the Higher Singularities of Plane Ourvea-e-Proc.

of the London }lath. Soc.,Vol. VI (1873-74), p. 153.I I Brill & Noether-A number of papers published in the Mathe.

matischen Annalen, Bd. IX (1876), XVI (1880), XXII (1884).'Scott-On the Higher Bingularities of Plane Curves-Amerioan

Journal of Mathematics, Vol. XIV (1892), pp. 3013=5,


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322 THEORY OF PLANE CURVESIn this Chapter we shall discuss the nature of different

species of singularit.ies and their effects on Plucker'snumbers.

258. SPECIES OF Cusps:

We have seen in 48 that the cuspidal tangent touchesat the cusp both the branches which, however, may lie onthe same side or opposite sides of the tangent. Cusps areaccordingly divided into two species, as shown in the accom-panying figures.

Case I: The two branches AP, AQ of the curve touchthe tangent AT at A and lie on opposite

~

sides of it. A is then called a cusp of. the first species,' or, a keratoid cusp (i.e.,

., CllSp like a horn).Case II: The two branches touch the tangent AT on the

same side. This is called a cusp--'j . .Rr of the second species, or a Ramphoidcusp (i.e., cusp like a beak).

Eg). Consider the nature of the origin on the curve (y-III')=g)"Any positive values of IIIgive real values of y. Writing the equation

in the form y = III'III-, it is seen that the values of y will bepositive for small values of g),whether the upper or lower sign is taken,since the second term is less than the preceding when IIIis small.

The g).axis is a tangent and the two branches lie on the upper side.The origin is therefore a mmph01:d cusp.

The analytical triangle gives the approximate form near the originas that of the curves (y-x')' =0, wbich represent two coincidentparabolas for the two branches. For a second approximation anotherterm is taken into account and the branches are given by-

The axis of IIItouches both the branches and, in fact, has fonr-pointiecontact with the curve. It is regarded as equivalent to four tangentstpat can be drawn from the origin to the curve.


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HIGHER SINGULA.RITIES ON CURVES 828259. DOUBLE CUSPS ~It may also happen that the two branches of the curve,

instead of extending towards one extremity of the tangent;extend towards both extremities, as shown in the adjoiningdiagrams. In these cases, there is a double cusp, which is

formed by the two branches of a curve touching at the point.Prof. Cayley calls it a tacnode. This point is indeed adistinct singularity, different in nature from ordinarysingularities; because the tangent at such a point has infaot four points along the curve, q ,namely, two points on each branch. ~wThese may arise when a. curve F f~Tconsists of bwo curves c p and", of lower orders, touching eachother at a number of points A, B, etc.

It may further happen that a double-cusp is of the firstspecies towards one extremity and of the second species onthe opposite extremity of the tangent. When the cusp is ofdifferent species towards opposite extremities of the tangent,Cramer calls the point a point of oscul-inflexion,

In this case, the point is a point of inflexion on onebranch of the curve. It is evident

~

. .' then that all these properly belong tothe class of higher singularitieswhich we proceed to consider pre-sently.


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324 THEORY OP PLANE CURVESE. Oonsider tbe curve y+2:v3y+.'=O

The curve is Dot symmetrical about either axis. It passes throughthe origin, but does not cut the axis again.

x

Evidently, there is a cusp at the orrgm with y=O as the cnspidaltangent; and in fact, there is a double-cusp at the origin, which isof the first species on the negative side of the y-axis and of the secondspecies on the positive side. The point is an oscul-Inflexion, and itsshape is shown in the diagram.

260. CLASSIFICATION OF TRIPLE POINTS:Triple points are classified into two main divisions

according as the three tangents are-(a) all real, (b ) one real(0 (2)

~

and two imaginary. The class of three real tangents may


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HIGHER SINGULARITIES ON CUllVES 325again be subdivided into three species, according as thetangents are (1) all three distinct, (2) two coincident,(3) all three coincident. Thus there are in all [our species oftriple points. But a triple point may be regarded as arisingfrom the union of three double points. The accompanyingfigures show the penultimate forms of these points and howthese double points are about to unite to form the triplepoint. in the different cases. It is formed by the union ofthree crnnodes in the first case, two crunodes and a cusp inthe second, and a crunode and two cusps in the third case.In this last case, however, the point does not visibly differfrom an ordinary point on the curve.In the case (b), the triple point is formed by the unionof an acnode with an ordinary point of a curve, i.e., a realbranch of the curve passes through an acnode, and thesingular point does not appear to differ from any ordinarypoint on the curve.

261. EQUIVALEST SmGULARITIES :Prof. Cayley has shown that any higher singularity

whatever may be regarded as equivalent, in a perfectlydefinite manner, to a certain number of the simple singulari-ties-the node, the ordinary cusp, the double tangent, andthe inflexion. It must be noted, however, as Halphen hasshown, that this equivalence is possible under certainconditions and within certain limits. vVe thereforerequire to determine how for any given singularity thevalues of these numbers are to be ascertained, so as toproduce the same deficiency and the same effect on theclass of the curve as the singularity in question. Whenthis is done, we shall have to see how Plucker's equationsare to be applicable to any singularities whatever of a planecurv:e. There are, in general, two principal methods ofstudying the subject-(l) by successive quadric transfor-mations,-(2) by expansions. We shall in this Chapterexplain the essentials of bath these processes, one after


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826 THBORY OF PLANE CURVESthe other; but it is convenient at the outset to exhibithe effeot of such singularities by means of a simplellustration.E(/}. Consider the curve " , < + u: -4x'y + y' =0The curve belongs to the class represented in 190 and possesses

a tacuode at the origin. The class of the curve can be determined bycounting the number of tangents passing through any point, the pointat infinity on the axis of lIJ, for example. If the tangent at the originis regarded as a tangent to the two branches, and each proper bitangentis counted as two, the class is found to be 8. But the 01as8of a curveof order 4 is, in general, 12.

This diminution of 4 tangents is due to the presence of a highersingularity at the origin, and the effect is the same as due to thepresence of two nodes, which unite to give rise to the tacnode, assuming,of course, there is no other singular point.

Two proper bitangents coincide with the tangent at the origin, and

...............

.,. . . . . ~ . . . . . .I .

'e . .... !,there are six other bitangents. Hence the origin is not an ordinarysingularity, nor the ",axis an ordinary singular tangent. In fact, theorigin is a singular point both in point and line singularities.

That the class is reduced by 4 can also be seen from the fact that thefirst polar of any point meets both the branches in two points coin-ciding with the origin.

262. ANALYSIS OF HIGHER SINGULARITIES :The principles of the theory of analysis of higher

singularities of a plane curve are contained in the followingtheorem;

Every irreducible algebraic cun:e can be transformed by abiraiional transformation into one having no singularities erceptdouble points with distinct tangents.


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HIGHER SINGULARITIES ON CURVES 327Several proofs < of this fundamental theorem have been

given by different mat.hematicians since the time ofKronecker who was the first to state the theorem, althoughin a slightly different form. G. A. Bliss of Chicago in apaper comments on the different proofs, and holds that theproofs given by Hensel-Landsberg and Walker are to beregarded as the best, although both of them are complicatedand lengthy, Bliss in another paper,t however, gives aproof which he claims to be simpler than any of its pre-decessors and is an extension of t he method of Kronecker.

Prior to all this, Nether t gave the following theorem:Each irreducible algebraic plane curve can be transformed

into another which possesses only ordinary singularities,i.e., multiple points with distinct tangents, by a aeries ofquadric transformations of the plane, i.e., by Cremona.transformations.

That the number of these transformations is finite hasbeen shown by Hamburger, while Bertini I I gave a geometri-cal proof.

Krollooker-Crelle, Bd. 91 (1881), p. 301. Henael-Landaberg-e-Theorie der algebraisohen Functionen, 2, p. 402. Also Hensel-Encyolopadie der mathematischen Wissenschaften II. c. 5, 25.Halphen-Comptes Rendus, Vol. 80 (1875), p. 638. and also J. deMath. Bd. 2 (3) (1876), p. 87. Bertini-Rivista di Matematica, Vol. I(1891), p.22, or Math. Ann, Bd. 44 (1894), p. 158. Walker-On theresolution of bigher singularities of algebraic curves into ordinarynodes-Dissertation, Chicago (1906). G.A. Bliss-'l'he reduction ofsingo.larities of plane curves by birational transformation-Bull. of theAm. Math. Soc., Vol. 29 (1923), pp.161.183.t G. A. Bliss-" Birational transformetiou simplifying singularities

of algebraic curves "-Traus. of the A m . Math. Soc., Vol. 24 (1922).t Nother-Gottiugen Nachrichten (1871), p. 267, also Math. Ann.Bd. 9 (1876), p. 166, and Bd. 23 (1884), p.311.

Hamburger-Zeitschrift fllr Mathematik und Phyeik-Bd. 16(1871), p. ~l.I I Hertini-Reale Istituto Lombardo di Scienze e Lettere-Rendiconti


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328 THEORY OF PLANE CURVES26:3. SUCCESSIVE TRANSFORMATIONS:In order to resolve a k-ple point A on the curve 1=0

we take A and any two other points Band 0 as the verticesof the fundamental triangle of a quadric transformation.Then the transform l' has at the fundamental points A', B/,O f of the transformed plane ordinary multiple points, whilethe singularities of f, other than A, are transformedinto multiple points of l' of the same order and form asthe original ( 214) . If some of the k-tangents at A coincidein the lines tu ti' ts ," " then there is on the line B'O' anequal number of points A'uA'., ... (other than B' or 0')which are multiple points onf' of orders (say) k" k" ks,""such that ki+k,+ks+ ... $,k. These points, however,can all of them unite to form a single k-ple point A'.Again, we apply on f' another quadric transformation withone fundamental point at A', and so on. By a finite numberof such operations, we may obtain from A multiple pointsof lower orders; and finally a curve c p will be obtained onwhich the points corresponding to A are all ordinary points.The ordinary multiple points of orders ku k.,ks, .. .indefinite-ly adjacent to the k-ple point A on the tangents tu t" ts,'"are said to form the" neighbourhood " of the first order on f.In a similar manner, the neighbourhood of the second orderis formed by the points of orders kill kl2l k'll kll, ...whichare indefinitely adjacent to A'i, A't, A's'" onf, and so on.

The numbers k, k , k , . . . obtained by the coincidence ofmultiple points depend on the succeeding but not on thepreceding transformations.

264. Oramer used the form y=v.n * in particular cases,and showed that certain singularities with coincidenttangents occur as final forms of singularities with distinct

* Newton calls the curve obtained by the transformation Y=f1JY ahyperbo!ism of the original curve-Enumeratio Iinearum tertii Ordinis(1706). Cramer uses the same transformation-Analyse des LignesOourbss (1750).


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HIGHER sINGULARITIES ON CURVES 329tangents involving a number of ultimately vanishing loops.Thus the cusp appears as a node with a vanishing loop.Noether"" also uses the same form in developing the analyti-cal theory without any reference to geometry, whichconsequently fails to show clearly the existence of thevarious elements of the compound singularities. This defectis, however, removed by using the geometrical method ofQuadric Inversion.El. Consider the singularity at the origin on the curve

(1)Cramer (p. 636) uses the transformation (y= UII) in the form

This gives a new curve (2)Referriug both the curves to the same axes, corresponding points canbe easily constructed. The curve (2) meets the axis of y where II,=0,and consequently from equation (1) we obtain v: =0, which gives thefour correspondents of the points where II,=0 meets (2), i.e., an arc of(2) cut off by w , =0 corresponds to a loop of (1) closed at the origin.If, however, the four values of y, obtained from (2) are equal, thecurve (2) has a "serpentement" of the appearance of ordinary contact,and (1) has the appearance of a simple cusp at the origin, whichcontains three vanishing loops and is really a quadruple point.

265. PRACTICAL ApPLICATIONS:We shall now show by means of a few illustratiTe

examples how, by the use of successive quadric transforma-tions, a compound singularity may be resolved into ordinarysingular points with distinct tangents." For furtherinformation and details, the student is referred to theoriginal papers 011 the subject quoted before.

E. 1. Discuss the nature of the singularity at the point 0(11, y)on the curve y2Z=W

Apply the transforma.tion O I l : y : Z=II'Z' : y'z' : x" ( 218) Noether-Math. Ann. Bd. 9 (1876), pp. 166182.

42


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330 THEORY OF PLANE VURV~SThe transformed curve becomeslal"y"z"=al"z", and consequeutly

the proper inverse is y" =QI"z', and the singular point C is nowtransformed to O'(y', Ii').

The nature of the point C', by writing ill'= 1 in the equation of tbetransform is obtained from the equation y " =:1

Thus there is an inflexion at C' on the transformed curve, with theline z'(AB) as the tangent. Therefore the singularity at C on theoriginal curve is a triple point (2J 3).

Ea>. 2. Consider the curve y'=al'.Applying Nother's transformation a>= al" Y = al,y" the transformed

curve isThe transformation QI,=Ql2>y. =a>.y. gives for the second transform

y'.=:u" which is a parabola and is unicursal, Consequently tbeoriginal curve is also unicursa.l. In fact, the first transform (QI,'=y,')has a cusp at (:u" v.) with y, =0 for the tangent. Hence the singula-rityat (:u, y) is a complex singularity with two inflexions and a cusp atthe origin, and there is an inflexion at infinity.

Ea>.3. The curve y'=a>' has a triple point at the origin having anapparent appearance of a point of inflexion. This can be shown byapplying two transformations successively. Thns a>=a>" y=0,y,j!ives y,'=al,', which again by the transformation y, =Y2' "', =QI.Y.is reduced to x.'=y. which ia a parabola with the tangent y.=O(see the figure, EQI. 1 219).

Ea>.4. Take the ourve ill' ~y7.This curve has a compound singularity at the point C(QI, V). The

fir!t transform (QI=""y" Y~Y,) is "','Y,'=y, r, i.e., .,'=y,.'


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HIGHER SINGULAltITIES ON CURVES 331Here y, =0 IS the tangent at the triple point (:II" y,). Hence

we put :11,-"'., y,""y.aI., and the second transform is y."=z., i.e.,the point (z y,) is a point of inflexion, and consequently (III" y,)is a triple point with two evanescent loops, and (z,:r) is therefore aquadruple point with a neighbouring triple point.

266. LINEAR AND SUPERLINEAR BRANCHES:1, as explained above, we apply successive quadric

transformations so that ultimately a k-ple point A onI isresolved into ordinary points P, Q, ... on a curve 4 > , then to thepoints in the neighbourhood of one of these points P,Q, ...on4 > correspond on I the points of a certain domain about A,which are then said to form a "branch"t< with A as origin.

By means of a birational transformation of I, any branchis transformed into another. The principles are expressedin the following theorem:

The co-ordinates (r, y) of the points of a branch can beexpressed in series of positive integral powers of aparameter t, which is again a rational function of thoseco-ordinates. All points of the branch will be obtained byusing Gauss-plane (Argand's Diagram), if t is allowed tomove within the circle of convergence of the power series.

The principal properties of the branches are obtained byconsidering the intersections of a branch with an algebraiccurve passing through its origin A. For simplicity, we takethe origin of co-ordinates at A, which therefore correspondsto the value 0 of the parameter t.

Let F(.c, y )=O be an algebraic curve through A. 1inthis equation we now put for x and y two power series withthe argument t, then the exponent (>0) of the least powerof t denotes the number of intersections of the branch withthe curve F.

In general, the number a of intersections of a straightline through A. with any branch is called the order of the

Halphen calls it a Oycle.


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332 1'HEORY OF PLANE CURVESbranch. A branch is said to be linear or eicperlinear accord-ing as a=1 or >1. If, however, any line l through Ameets the branch in a+a'(a'>O) points, it is called thesingular tangent to the branch at A.

Taking this line for the axis of z, the branch can berepresented in the form :

y=a'ta.+a.' + (I)The numberhere a, a', ... are constants different from O .

a' is called the class of the branch.The tangent at a point of the branch is reciprocal to the

point, and the numbers a and a' correspond reciprocally.a' is the number of tangents coinciding with l, which passthrough any point of l (other than A), a+ a' is thenumber of tangents coinciding with l which pass through A.*

Halphen t states these facts in the following theorem:If a variable line is indefinitely near the origin of a

cycle, among the points of intersection with the curve,there are points indefinitely near this origin belonging tothe cycle. The number of such points is tae order of thecycle, when the line makes a finite angle with the tangent.On the other hand, if the line does not differ from thetangent, this number is equal to the sum of the order andthe class.

From (I) y can be expressed in a series of positive1 1- -integral powers of za . , where in each term xa . is to be

replaced by its a values, i.e., in the form: (w =)

* Ca.yley gave this theorem in Qua.rterly Journal Vol. 7 (1866),p. 212, but the proof was supplied by Halphen-Comp. Rendust. 78 (1874), p. 1105, and by Stolz- Math. Ann. Bd. 8 (1875), p. 415.Segre gave a geometrical proof-Introduzione, etc., '10'43.t Halphen-E'tude sur lea pointe singuliers, 7.


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HIGHER SINGULARITIES ON CURVES 333The branch is said to be quadric, etc., according as

a=2, 3,.... The expression for y has precisely a values1

obtained by putting for xa . each of its a values. Thebranches corresponding to these a values are termed byCayley" Partial Branches." *

If the branch is real, the co-efficients in (1) are real,and real values of t correspond to real points on the branch. -Hence it follows that there are the four following types ofbranches, according to the nature of the point A on thebranch.t

(1) a odd, a' odd, A is an ordinary point,(2) a odd, a' even, A is an inflexion,(3) a even, a' odd, A is a cusp of the first species,(4) a even, a' even, A is a cusp of the second species.Therefore, from general considerations of nodes and

cusps, we obtain the following:A node is of the k-th. species, if it has two simple points

in its neighbourhood of the k-th. order, and consequentlyit can be analysed into simple points by k quadric trans-formations. A cusp, on the other hand, is of the k-th species,if it has one simple point in its neighbourhood of the k-tiiorder, and consequently it can be transformed into anordinary point by k quadric transformations. Each nodeis the origin of two branches of the first order, and a cuspthat of one branch of the second order.

Halphen has studied the properties of cycles and theirexpansions in E'tude Bur les points singuliers, Part I,pp. 540-557.

One method of finding these expansions was given by Newton-Analysis per eqnationes numero terminorum infinitas (1669). AlsoPuiseux-Lionville, t. 15 (1) (1850), p. 365, and t. 16 (1851)-, p. 228.t Stolz-Math. Ann. Bd. 8 (1875), p. 433.


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334 THEORY OF PLANE CURVES267. ApPLICATION OF THE METHOD OF EXPANSIONS :If, instead of the a-axis, the line y=m,r is a tangent.,

the expansion for a branch of order a;S;,k is obtained in the"form:

where tbe origin is a k-ple point on the curve, and m=f=O .Then, f3's are all positive integers in ascending order

of magnitude and a is the least common multiple of all thedenominators (f3 i >a ) and w is anyone of the a roots ofx a=1. Negative exponents can, however, occur in thisexpansion, if the axis of y meets the curve at infinity, butwe exclude that case from our discussion.

The entire portion of the curve near the origin obtainedby putting for w every a-th. root of unity in turn is then calleda superlinear branch of order a, having Y =m:J; for thetangent. .I'he different superlinear branches of a curve at ak-ple point give in all ~a=k expansions, but the individualbranches can have different or any number of commontangents at the point, i.e., in the expansions of the branches,the co-efficients A" A., ...may, some or all of them, beidentical, or any two or more expansions can be identicalup to a certain finite number of terms. In this case wemust take the expansion until the two branches separate.

It follows then that if a point is an ordinary point onthe curve, only one linear branch (the curve itself) passesthrough it. If it be an ordinary k-ple point with distincttangents, there pass k linear branches through the point.If, however, two or more tangents coincide, we havesuperlinear branches.

The following illustration will clearly explain how expan-sions are useful in studying the nature of higher singu-larities:


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HIGHER SINGULARITIES ON CURVES 335Ez. Consider a Ramphoid cusp, i.e., a CU8pof the second species

at the origin on a curve.There is a superlinear branch of order ,,= 2, whose expansion may

be written 8S : y=mOl +A,,,,tOl> + A.",oOl> + A.","Ol> + (I)where y=fnal is the tangent, ,,=2 and "'= 1.

Hence the expansion (1) becomes

If the origm is a cusp of the first species, the expansion of thebranch (of order ,,=2) is-

3 y=mmA,,,,> + A.aI' A,II' (2)1-which evidently differs from (1) only by the term ..

268. PRACTICAL METHOD:The above expansions and other connected formulre are

proved in works on the Theory of Functions to whichbranch they properly belong. Without going further intothe details of the theory, we shall now exhibit how theseexpansions are obtained in actual practice for determiningthe intersections of curves at higher singular points.

E " , . 1. Find an expansion for the branch near the origin of thecurve y-",-aly+2y'=0.

Assume y=aal + bz- +CilDS + diIDt + ...Substituting this expansion for y in the given equation, we obtain

+2(a" + b , , ' + c",' + ...)'=0.i.e., aal + (b-l),,- + (c-a + 2a3)aI' + (d-b + 2ob). + ... =0Equating the co-efflcients of "', Ol', aI', ai', to zero, we find

a=O, b=l, c=a-2a3=0, a=b-2a'b=1, and 110 on.

. The required expansion for the branch i8-


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336 THEORY OF PLANE CURVES

Ez.2. Expand (y+z')'=4y'(z+y') near the origin.Newton's diagram gives for the first approximation y + QI' =0.The next approximation is given by y + Z2 = 2y-vQI' + yO

Assuming y=-Ql'+az'+bx+ ... ,we obtain-

(aQl3 +bx' + ...)'=4{ -z' +ax3 +b2l' + ...P{z + (-ilJ' +a.3 + ...)!}whence a' =4, 2ab = -Sa, b' + 2ac=4+ 4a'.i.e., a= 2, b= -4, c= 9, etc.

E2I. 3. Find expansions for the branches of the following curvesnear the origin:

(iii) (Y_Ql3)' =zy'.

269. EXPANSION OF A FUNCTION:Let the implicit function F(.!:, y) be of order n, and

suppose that the origin is a k-ple point and that neitheraxis is a tangent to the curve.

Let y=CP,(X), y=cp.(r),be the expansions corresponding to the k linear branchesat the k-ple point. These are, in fact, the expansions fork of the 11 values of y obtained by putting );=0 in theequation F (x, y) =0. Besides the singular point, the curveintersects the y-axis in n-k other finite and distinct points,for each of which there is a similar expansion, having aconstant added to it. This constant evidently representsthe distance of the point from the origin.Thus, we obtain (n-kl expansions of the form-

which is an ordinary power series, representing the implicitfunction F(x, y).


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HIGHER SINGULARITIES ON CURVES 337These series are all convergent within a certain region,

and indeed are absolutely convergent when each term isreplaced by its absolute value." The product of two suchseries is also absolutely convergent,t and consequently, wemay represent the equation of the curve as the product ofsuch series:

270. DISCRIMINANTAL INDEX:We have seen in 82 that the first polar of any point

passes through the points of contact of tangents drawn fromthat point to the curve and the class of a curve 'is deter-mined by the number of intersections of the curve withits first polar, which, however, is reduced by the existenceof nodes and cusps ( 121). Thus, for determining theeffect of higher singularities on the class of a curve, werequire to find the number of intersections of the curve withits first polar at such a singular point.

Let F=O be the equation of an n-ic of class ii1., andconsider its intersections with the first polar of any point,(0, 1, 0) for example. If, therefore, we eliminate y (say)between F=0 and the equation a F/ a y =0 of the firstpolar, we obtain the resultant in the form of an equation0(x)=0, where 0(x) is, in general, of order n(n-l) in e,Since the class of the curve is rn, the equation 0::1:)=0will have m simple roots x' corresponding to the points ofcontact of the m tangents which can be drawn from thepoint to the curve. Dividing the equation by these mfactors II(.v-x'), the remaining' factor 0(x)/II(;v-;c') isof degree n(n-l)-m, and when equated to zero givesn(n-l)-m roots, which correspond to the intersectionsat the singular points. Thus the singular points count

Townsend-Functions of a complex variable. 47.t Cauchy-Analyse Aigebrique, Chap. V[.

43


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338 l'Hl!:ORl Or PLANE CURVESas n(n-1)-m intersections of the curve with its firstpolar, and when there are only nodes and cusps on thecurve and their numbers are 8 and K respectively, theycount as 28+3K intersections ( 121).

The number J=n(n-I)-m=28+3K is called the total" Discriminantal Index" of the singularity.

The roots of the equation (v) =0 give the so-calledbranch points of the function. '1'he root with the leastmodulus gives the limit of convergence of the above series,which, therefore, converges absolutely up to the nearestbranch-point, But the discriminant of F is the productII(y; - Y r) ' of the squared differences of the roots ofF =0 regarded as an equation in y. '1'herefore, withinthis convergence limit, we can consider the discriminantl,!:) equal to the product of the squared differences ofthe series c p and if taken two at a time.

271. From what has been stated above, the reductionIII the class of a curve due to the existence of highersingularities can now be easily determined.

The discriminant (z)EII(cp-if)' can be divided intotwo parts-one part, called the "variable" factor corres-ponding to the 7l( proper tangents, and the other the"fixed" factor of order n(n-1)-m corresponding to thesingular point. This fixed factor therefore is the productII c p ; - c p r) 2 of the tk( k-1) differences corresponding tothe k partial branches through the point taken two a.t atime. The number of roots of the equation II( c p ; - C P r )2 =0is therefore equal to twice the number of intersections ofall the partial branches with one another, each root beingcounted twice, i.e., equal to twice the number of inter-sections of the curve with itself at the singular point.

But twice the number of intersections is equal tothe discriminantal index J=28+3K. For the discriminant(.r) is the result of eliminating y between F=O and


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HIGHl!JR SINGULARITIES ON CURVES 339

oFj8y=O. Therefore the number J is equal to twice thenumber of intersections of the curve with its first polar at :the singular point.

Combining the results obtained, we may state thefollowing theorem*' :

The number J= 28+ 3K , which represents the reduction: in theclass oj a curve due to the existence of hiqher singularities isequal to twice the number of intersections of the curve withitself at the singular point.

Ex. Consider the curve Ql' + y' -4m'y + y' =0.The origin is a triple point consisting of a cusp and two nodes and

there is a third node. . 6= 3, IC= 1.There are two expansions :-

and y=~'+ ...Hence J=wice the number of intersections = 2 t3= 9=2~+ 3IC.

272. INTERSECTIONS OF Two CL'RVES AT A SINGULAI~ POI~T :From what has been said above with regard to the

intersections of a curve with its first polar at a singularpoint, we can easily determine them for the general case ofany two curves, i.e., how many of their intersections coincideat the singular point.

Let F .(x, y)=O and Fn' (x, y)=O be the equations of anytwo curves of orders nand n', having at the origin multiplepoints of orders k and k ' respectively.

Assuming as before, the curve F n =0 admits of n-kexpansions of the type . t { I and k expansions of the type cp,so that

Fn =ll(y-cp) X ll(y-t{l)=0 (1)* Halphen-Bulletin de la Soc. Math de France, Vol. I, p. 133.

Al.o Zeuthen-H Sur lea singularities des courbes planes," Math. Ann.Ed. 10(1876), f' 213.


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340 THEORY 01') X II (y-tf; ') =0 (2)

If we eliminate y between (L) and (2) we obtain theresultant (x), of order nn' in x, which may be written inthe form of the difference-product

(:I) ::II(cJ>-cJ>') X II(cJ>-tf;') X II(tf;-cJ>') X IT(tf;-tf;') =0The zero roots of this equation will give the intersections

at the origin.The factors IT(cJ>-cJ>') of the kk' differences all contain!'

in a certain power ,\ as a factor ; and if we proceed with allthe differences, we obtain

II( cJ> - cJ>'}L.=o ~ =constant.Accordingly we obtain the number of intersections oftwo

curves at a singular point by adding together the differentvalues of,\ corresponding to the kk' partial branches eachdefined by the equation

y-y'L.=o -2- =constant,xi.e., it is equal to the sum of the numbers of intersectionsof one curve with the branches of the other.*

We may state the above facts in the following theorem:-If y = cJ>i and y=f;i be the expomsion of y in terms of .r

for branches of two curves passing through the singular pointat the origin, the number of intersection of the two curveswhich coincide at the ongin is equal to the index of the productof the lowest powers of x in all possible erpression of theform cJ>i- tf;i.

Halpheu-e-Bull. de 19 0 Soc. Math. de France t. 4 (1875), p. 59.Journal de Math.-Bd. 2(3) (1876), pp. 257, 371.

Bee also Brill-Sitzungsberichte der math-phys. Klaase, etc., znKiinohen, Vol. 18 (1888), p. 81.


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HIGHER SINGULARITIES ON CURVES 341

EID. l. Consider a curve with a cusp at the origin, and anotherpassing through the cusp and touching the first curve there.

Here the expansions for the two curves are respectively:

3y=alD+blD+... = < / > ,!!.y=-alD'+bx- ... = v ( 1)

y=ax' + f31D'+ ... (2)

Then,

Hence 1 / 1 , meets each of < / > , and < / > . in ! points, and the totalintersections=! x 2=3.

E, 2. Consider the intersections of a curve with its Hessian ata double point.

Let F=(y-mlD)(y-plD) H.+ ... =0 be the equation of a givencurve, having a node at the origin.

It has at that point the two expansions :-

y,=mx+m,ID!+ ... =,

The Hessian has at this point a double point with the same tanienta.Therefore, for the Hessian we have-

whence it is easily seen that the differences < / > , - 1 / 1 , and < P . -+ , eaohgives one intersection, while the differences < / > , - 1 / 1 , and < P . -1/1, giveseach two intersections, since the branches touch.

Thus, we get altogether 1+ 1+ 2 + 2= 6 intersections, as wasotherwise found in 103.

At a cusp, however, the curve has the expansions-~y, =mlD' + ... = / > , ~y.=-mIlJ2+ ... = < P .

The expansions for the Hessian are-y,'=m,,+ ... : : : :+, 8y.'=-mx+ ... ;:I/1.

y.'=alD+bx~+ ... : = . 1 / 1 3 (for the ordinary branch)


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342 TH~ORY OF PLANE CURVESThe third branch 1 / 1 . intersects the two partial branches < / > , and < / > ,in two points (each partial branch once), while the four partialbranches < /> " < / > 1 / 1 " 1 / 1 . meet each other in ~ points. Tl:ns we obtainthe total number of intersections =t x 4 + 2=8.

Ex. 3. Consider a curve with a simple cnsp at the origin andanother with a cusp of the second species, having the same tangent ..The expansion for the curve is y=x' +.Hwhich is a snperlinear branch of order 3, consisting. of the partialbranohea-

y=x' + ... =, '). I11=",:8+ ... = < / > . ~ where ",'=1 .. . (1) Iy=",0;U8 + ... = < / > . )

For the ramphoid cusp, we have'l. ')i.e., y=",'+"'"+." = 1 / 1 , IJ (2)y=;U2_X2+ ... = 1 / 1 .

Each of the branches 1 / 1 , and 1 / 1 , meets the partial branches < /> " < /> ., < / > ;in t points,since Q=I, c . . , Wi.Hence the total intersections =ix 3 + t x 3=8.

E:. 4. Consider two cnrves having respectively k and k' linearbranches through the origin, with all distinct tangents.

In this case the product n(tp,-1/I;) contains kk' factors of thetype aa:+bx'+cx+ ...

Hence the curves meet in kk' points at the origin. If, howeverthe cnrves have the same tangents, the product n(tp,-1/I;) containsk(k-l) factors of the type am+b",' +ca:3 .. and k factors of the type

Hence the curves meet k(k-1) + 2k=k(k + 1) times at the origin.It follows, in particular, that the t.angent to the super linear branch

of 267 meets it in fJ 1pointe.


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HIGHER SINGULARITIES ON CURVES 343273. EXPANSIONS IN LINE CO ORDINATES :Just as a curve regarded as a locus of points has point

singularities, regarded as an em-elope of lines, it hasline singularities. In order to examine this closely, werequire to obtain an expansion in line co-ordinates of thebranch of a curve near the singular point.

We shall suppose that the axis of .r, and not the :I-axis,is a tangent at the origin, which is supposed to be a singularpoint. Therefore no proper fraction will appear in theexpansion of the branch in point co-ordinates, which may bewritten in the form:

f3, 8.y=a1x a +a.x a + ... ({3 i> a) (1)

where {3, denotes the number of points which the tangenthas common with the branch at the origin. We ml1y writethe equation of the tangent at any point of the curve in theform ~G+y+~=O , which expresses the fact that the point(.c, y, 1) lies on the line (~, 1, ~). The co-ordinates of thetangent are given by-

~=-Oy0 . 1 ) (2)If the axis of z, i.e., the line y=O is the tangent, we have

~=O, 1)=1, ~=O.We shall now find an expansion of ~ in terms of ~ in the

form-f3.' f3.'

~=AJ~+A.i7 + (3)The least common multiple a' of the denominators in the

exponents of this series is called the order of the branchregarded as an envelope of tangents, i.e., its class, anddenotes the number of tangents drawn to the curve from apoint on the singular tangent and coinciding with it (exclu-ding the tangent to the other branch, if any). {3l' is the


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344 THEORY OF PLANE CURVESnumber of tangents passing through the singulal' pointand coinciding with the singular tangent.

From (1) we obtain

(4)

and { 3 {3. e,'=(a, _, .z-;+ ...)-(a,x Cl + ... )a(3 -a ~=-a,. _,_-. X Cl +...a (5)

Eliminating x between (4) and (5), we may obtain therequired reJation between' and~. Now, from the series (4)

1 1- (3-Clwe can deduce a series expressing .t Cl in powers of ~ 1 ,

and put this in (5). Thus WP. obtain-

(6)This series must be the same as the series (3).

Since we have taken only the first term from theseries (4), we cannot. at once say that a'={3,-a, or, thatto a given value of ~, {3,-a values of 'correspond. For,it may happen that in (6) all terms having the leastcommon denominator are absent. But in every case it iscertain that

{3.. . l . . '= ~ , a '< {3 , -a , and therefore, {3.'


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HIGHER SINGULARITIES ON CURVES 345It follows then that {3.'={3u a={3,'-a', a'={31-a.Combining all these results we may state the following

theorem s:-The number {31of the coincident intersections of the tangent

of a superlinear branch is equal to the number {31' of thetangents at the singular point which coincide with thesinqula tangent. If a and a' are the order and class respec-tively of the branch, then always

a+a'={31 ={3,'.If there are more than one branch passing through the

singular point, having the same tangent, the result can beobtained by addition of the results just established for asingle branch.

Ex.!. Consider the cusp on the curve 1 J = " ' \ - 'We have ch 4 le = - _ =--XB andBQJ 3

Eliminating x, we obtain (=~~4a'={3,-a=4-3=1

i.e., there is only a single tangent, and the cusp of the first species isonly a point. singularity .

E " , . 2. Consider the cusp of the second species.The expansion for the branch is of the form :..y=xQJ+",8 ... ( 266)

Now, taking the branch 5y=x' +X2 +x3 + ... (1)we obtain B y ~~=-_ =-2x-!x+ ...B '" (2)

e s(=(2QJ'+%x2+ ... )_(Ql2+X2+W'+ ... )=QlO+%W2+... (3)

Zeuthen-" Note sur les singularities des courbes planes"- Math.Ann. Bd. 10 (1876), pp .210.220.

44


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846 THEORY OF PLANE CURVES

Now,we shall have to eliminate ill between (2) and (3), and in doingthis we have to express ill in terms of ~ by the process known-as.eve?sion o f series.

where a" b" ell'" are to be determined from the series (2).Substituting in (2) we obtain-

=-2al'~-(4albl +ta18)~2+ ...Equating the co-efficientson both sides, we get- .

andwhence i"1'= - - - ="/2 and

(4)

Substituting this value in (3), we obtain -

e ( i 5 3 i ) t- + 4__ ._+_.-_ ~ +...4 ,2"/2 16 2 4-"/2s

~. . ~2=- -~---+..4 4"/2= ( , { ) " - i ( . l ) t +2 2'"

where, it is seen, only negative values of ~ gives real values of (.Hence,putting - ~for f, we get the reciprocal expansion in the form


27/40

HIGHER SINGULARITIES ON CURVES 347274. POLAR RECIPROCAL OF A SUI'ERLINEAR BRANCH: * 'If, in the expansion (3) of & 273, we put IIand y for

~ and ~ respectively, we obtain the equation of the polarreciprocal of the branch with respect to the conic x' +2y=O( 114). Thus the polar reciprocal of the branch is-

i3, ' {3 2y=.A1x T+A.xd + ...

which is again a superlinear branch of order a ', and is metby the tangent in f3.' points coinciding with the origin.Hence, we may state the theorem;

If the polar reciprocal of a superlinear branch of order a atany point 0 whose tangent meets it in f 3 1 points coincidingwith 0 is a superlinear branch of order a' at any other point0' whose tangent meets it in f 3 1 ' points coinciding with 0',then a+a' =f3.' =f31'

Again, by the properties of reciprocal singularities, thenumber of tangents drawn to a superlinear branch at 0,coinciding with the singular tangent at 0, is equal to thenumber of intersections of the reciprocal branch with itstangent coinciding with the singularity 0'. The number oftangents from a point on the tangent at coinciding withthis tangent is equal to the number of intersections of a linethrough 0' with the reciprocal branch coinciding with 0'.

Hence, from the above relations, i.e., a+a'=f31 =f3.', weat once deduce the following theorem:

If a superlinear branch of order a at 0meets its tangentin f31 points coinciding with 0, then f31 tangents from 0 to thebranch coincide with the tangent at 0, and f31- a tangents tothe branch. from any point on the tangent at 0 coincide withthe tangent at O .

Since the origin is a k-ple point, the tangent y=O meetsthe ourve in 1.-+1 points at the origin. But again there

O J , Hilton-Plane Alg. Curves, Chap. VI, 5.See also-A. B. Basset-Quart. J. of Math. Vol. 45 (19H.) pp. 5265.


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348 THEOkY OF PLANE CURVESis a enperlinear branch of order a, and consequently the liney=O meets the other branches in k-a points at the origin.Hence it meets the superlinear branch in a + 1 points.

:. (31 =a+l and a'=l, ,8'=a+lor, in other words,

The polaT reciprocai of a superlinear branch. of order a is,in general, a Linear branch. having (a+1-pointic contact withits tangent.

Ee, Consider the cusp of the first species at the origin with 1 1 =0as the tangent.

The expansions are 3 5Y= aill' + bill"Cill' + ...The expression in line OIl-ordinates, obtained by the method of 273

beoomes-(=-~e-~ ~'+ ...27a ' Sla'

whence the polar reciprocal w.r.t. ill' + 2y=O of the branch is--y=_,! 213_ 16b ill'+27a" SIa'

which shows that it is a linear branch with an inflexion at the origin.ThUR we have a proof of the fact that to a ousp corresponds aninflexional tangent an the reciprocal curve.

275. CUSPIDAL INDEX:Existence of higher singular points as well reduces thedeficiency of a curve. To find the effect, we have to

determine an equivalent number 8+K , which has the sameeffect on the deficiency as the singularities. We shall nowfind the value of K :

Consider a curve S with a. superlinear branch of order aa.t any point O. Let S' be another curve of the same orderand class with 8 nodes and K cusps, and having a one-to-onerelation with S. If now A and A' are two fixed points inthe same plane, and P, P' two corresponding points on 8 and8' respectively, thQ order and class of the locus C of theintersection of AP and A'P' can be determined by themethod of 152. The order of Cis 2n, with an n-ple point


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HIGHER SINGULARITIES ON CURVES 349at each of A and A'. To determine the class, we count thenumber of tangents which can be drawn from A or A'.There are 2n tangents at A or A', and m tangents whichare tangents to S or S'. Now, consider the line AO. Sincethe point 0 on S corresponds to 0. consecutive points onS', AO has o.-pointic contact with C at any point H. Wehave now to consider two cases: (1) the 0. consecutivepoints lie on a simple branch of S', so that all cusps of S'correspond to only ordinary points on S. In this case Chas a singular point at H, where AH meets it in 0. points,but A'H in only one point. Hence, by the formula of 274,the number of tangents coinciding with AH is 0.-1, andconsequently the class of C is 2n+m+0.-1. On the otherhand, to each line through A' and a cusp of S' correspondsa tangent to C through A', so that the class of C is

2n+m+K,Hence,(2) The 0. consecutive points on S' may coincide with

a cusp on S'. Then the singularity H on C is met by AHin 0. points, but by A'H in two points only, and conse-quently, 0.-2 tangents coincide with AH. Hence theclass of Cis 2n+m+0.-2. Again, since a line through acusp is not a tangent, counting the number of tangentsfrom A', the class of C becomes 2n+m+K-l,

whenceThe number K IS called the "cuspidal index"'" of the

singularity at 0, and we have-The cuspidal inde c K of a singularity is one less than the

order 0..tSimilarly, in the line system t =0.'-1. Smith-Proc. Land. Math Soc., Vol. VI (1873.76); Nother calls it

Verzweigung "-Math. Ann. Bd. 9 (1876), p.166.t For a geometrical demonstration see Bertini-Lomb. Ist, Rend.,


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350 THEORY OF PLANE CURVES276. EXTENSION OP PLUCKER'S FORMULAE:The first polar of any point is an adjoint curve ( 227)

which has at each k-plepoint on the curve a (k-I)-plepoint.But each k-pZepoint A counts as .'.Sk(k-I) intersections ofa curve with any adjoint, where the sum extends over all ofNother's component points. Therefore, for a first polar(adjoint) this number is to be increased by .'.S(a-I), where.'.Sextends over the orders a of the superlinear branchespassing through A, and .'.S(a-I) is called the "CuspidalIndex," as explained above, of the singular point.

Hence, the class of an n-ic is given by ( 146)m=n(n-I)-.'.Sk(k-I)-l( a-I) (1)and reciprocally,

n=m(m-l)-lk'(k'-I)-l(a'-I) (2)whe~e k ' and a' represent respectively the order of themultiple tangent and the class of the superlinear branch.

Since a curve and its reciprocal have the same deficiency,we may writen(n-3)-.'.Sk(k-I) =m(m-3)-lk'(k'-I) (3)2(p-I)=n(n-3)-lk(k-I)=l(a-I)+m-2n

=m(m-3) -lk'(k'-I) =l(a'-I) +n-2m.whence l(a-a')=3(n-m)

l(2a+a'-3)=3(n+2p-2)l(a+2a'-3) =3( m+ 2p- 2).

where, in the last equation, the sum extends to all thebranches 01' which aa'> l.

All theRe investigations properly belong to the theoryof functions, and without going further into the details,we shall conclude the topic by mentioning the importantfact that Pliicker's Formulae are applicable to curves withhigher singularities, if we regard the singular point orsingular tangent as equivalent to a number of nodes and


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HIGHER SINGULARITJES ON CURVES 351

cusps or bitangents and inflexional tangents respectively.The four equivalent numbers 1 5 " { " 1 5 / , {1',-which Zeuthencalls "Principal Equivalence," and Smith terms 1 51and (I"Nodal and cuspidal index "-are connected by threeindependent equations, to which the expression for deficiencyis added, so that the four can be determined.

Thus, for a superlinear branch C a , a'), which diminishesthe class by J, the discriminantal index, and the order byJI, these numbers are determined by the equations

The four numbers are connected by the relation:1 51-SI'=t({1 -(1')({1 +(/-1)

Brill'" has shown that each higher singularity can beshown as limiting cases of ordinary singularities by meansof a series of deformation processes, which can, however,be effected by means of quadratic transformations as Scott thas shown. Prof. Basset in a number of papers, t hasJetermined the point and line constituents of certainsingularities, and specially considered the resolution ofmultiple points with tacnodal branches, etc.

For a real curve of order n and class n', with Wireal inflexions, f' real bitangents with imaginary pointsof contact, 1.1 real cusps and d" real double points withimaginary tangents (acnode i, F. Klein established therelation-

n+Wi+ 2t" =n'+ 1.1+ 2d"For a curve with higher singular points, see Juel-Math.Ann. Bd. 61(1905) p. 77.

Brill-Math. Ann. Bd. 16 (1880), p. 348.t Scott-Am. J. of Math. Vol. 14 (1892), p. 301, & Vol. 15 (1893),

p.22l.t A. B. Basset-Quart. J. Math., Vol. 37 (1906), p. 313, and Vol. 43

(1912), p. 15l. F. Klein-Math. Ann. Bd. 10 (1876), p. 199.


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:152 'l'HI!lORY OF PLANB CURVES277. CURVES OF CLOS8ST CONTACT: OSCULATING CURVES:The process of expansions described in this Chapter

affords a very convenient method of studying intersectionsof curves at multiple points, whether of point-singularitissor line-singularities, and in particular, of studying contactof curves at any given point. At the outset, however, wemay state the following theorem :

When two curves have linear branches through the origin,and the expansions oj y in terms of J' for the branches areidentical as [a r as the term. with z ", the CUtTeS hare (1' + 1)poi'1,ticcontact at the origin.

For in this case the difference y, -yo = c p , - " ' , ( 272)contains :1:'+' as a factor, which shows that the curvesintersect (1'+1) times at the origin, i.e., they have Cr+1)pointic contact at the origin.

The same process applies to the general case of anypoint on a curve, the point being taken as the origin ofco-ordinates.

We may use this theorem for finding curves havingclosest contact with a given curve possible for a curve ofthat order.

Let the gIven curve be an n-ic, and we require to findan m-ic having at the origin the closest possible contactwith the given curve.

Now, since the m-ic is determined by tm(m+3) points,only tm( m+ 3) points can be assumed on the n-ic and them-ic is completely determined. Hence, the m-ic of closestpossible contact can have at the most tm(m+8)-pointiccontact with the given curve. If, however, the m-ic issubjected to satisfy other conditions, the order of contactwill be reduced.

Thus, a circle can have only three-pointic contact, aparabola can have four-pointio contact, and a general comecan have five-pointic contact, and so on.


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HIGHElt 8INGULAltITIlI8 ON CUltVES 353DEFINITION: Curves having closest possible contact with

a given curve at a given point on the latter is called theosculaiinq curve at that point. Thus, the circle of curvatureat any point is the osculating circle, etc.

Ez. 1. Find the osculating conic at the origin of the curve

Let y=rnz + Ax' + 2H,,,y + By' be the reqnired conic, andassume Y=f7.


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-854 THEORY OF PLANE CURVES

" The branch is Y= #D' j3a+... Writing the equation of theosculating conic in the form y=Ax'+2H#Dy+By', and proceeding asin Efil. I, the expansion for the conic becomes-

y=Ax' + 2HAx + (2H'A+A 'B)x' + ...Henoe, comparing the terms with those of the branch of the CUrTe

we have A= Ij3a, H=B=0i.e., the osculating conic is y=fII'j3a or 3ay=x'.Similarly, for the other branch, the osculating conic is 3afll=y'.

Be, 3. Find the conic of closest contact at the origin of tbe curve""+!3y'+Y'=O

E, 4. The equation of an n-ic with a tangent of n-pointiccontact and a superlinear branch ma.y be put in the form

278. CONICS WITH FOUR-POINTJC CONTACT:The locus of centres of all conics having a four-point

contact with a curve at a given point is a straight linethrough the point.

Taking the given point as origin, the equation ofthecurve may be written as-

y=al+ln:' +cx' +dx +...The equation of a conic through the origin having the

same tangent may be written 8.S-y=aJ:+Ax' +2H.ey+By'

The expansion for tbis is found to be-y=aJ:+ (A+ 2aH + Ba' );c'

+2(H+aB)(A+2aR+Ba');cS + .._Since the conic has four-pointic contact, the co-efficients

of ;v , x', XS must be identical in the equations of the curveand the conic... A+2aH+Ba'=b and 2(H+aB)(A+2aH+Ba')=c,

whence, H + aB=cJ2b, A+aH=b-ac/2b.But the centre of the conic is given by-

2Ax+2Hy+a=O, 2H.c+2By-l=O


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1UGHER SINGULARITIES ON CURVES 355whence x(A+aH)+y(H+aB)=O, i.e., (2b'-ac).I'+cy=O,which is the locus of the centres, and is called the "axi~of aberrancy."

279. TRANSON'S THEORY OF ABERRANCY:"From what has been said in Chap. IX with regard to

.the approximate forms of a curve in the neighbourhood of a.point, it is clear that the form is, in general, defined by thecircle of closest contact, i.e., the circle of curvature. Butthe form may be further defined by means of the osculatingconic.

Let the normal at a given point 0 on the curve meetat P the infinitesimal chord AB drawn parallel to thetangent at O. Then the arcs OA, OB, as also the lines PA,PB, regarded as small magnitudes of the first order, differby magnitudes of the second order and may, therefore,be regarded as equal, i,e., if N is the middle point of AB,then NP is a small quantity of the second order.Similarly, OP is also of the second order. The angleNOP=tan-1 ~~ is consequently a finite angle, i.e., the lineON is inclined to the normal OP at a finite angle. In thecase of a circle, ON and OP coincide, and therefore thedeviation from the circular form is measured by this angle,which is called aberrancy,t and the line ON is called the" axis of aberrancy."

Weare then led to the following definition:The measure of deviation of a curve at a given point of

it from the circular form is a finite angle, and is called" aberrancy," and the line which forms this angle with thenormal is the axis of aberrancy .

Transon-Reoherohes snr la oourbure des !ignes et des surfa.oes,Liouville, t. VI (1841), pp. 191207t Transon usee the term "deviation," but the term If aberranoy II

is generally preferred.


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356 THEORY OF PLAN~ CURVES

When this is a conic, in particular, the axis of aberrancyIS the diameter through the point and the aberrancy is t.heinclination of this diameter to the normal.

If at a given point of a curve a conic is drawn havingfour-pain tic contact with it, it is clearly seen that the curveand the conic have the same axis of aberrancy, and conse-quently, the centres of all such conics lie on the axis ofaberrancy, as has been directly shown in the preceding article.

The point where the axis of aberrancy at a given pointof a curve meets the axis of aberrancy at a consecutivepoint is called the" centre of aberrancy;" consequently, thecentre of aberrancy of a curve at a given point is the centreC of the conic having five-pointic contact with the curveat the given point 0, i.e., the centre of the osculating conic."The length OC is called the radius of aberrancy.

280. ANGLE OF ABERRANCY:We shall conclude the discussion by referring to only

one other important fact in this connection, namely, that theangle of aberrancy 8 at any point of a curve is given bythe formula t

tan 8=p_(l+p")r3q"where p, q, I' are the first, second and third differentialco-efficients of y in regard to .r.

* The investigations of these properties properly belong to thedifferential geometry, and there is no anfflcient scope for them in thepresent work. But in view of the interesting nature of such investiga-tions, it has been considered desirable to refer to some most importantpoints. For further details, the student is required to consult 'I'rauson'spaper-Liouville Journal VI. (1841), pp. 191.207, and the several papersof A. Mukhopadhyay, Journal of the Asiatic Society of Beng-al,Vo\. 57, Part II (1888), and Vol. 59, Part II (1890).t For a second proof of the formula, see the paper by A. Mukho

padhyay, Differential Equation of a "parabola ", J. A. S. of Bengal,Vol. 67, Part II, p. 319.


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lnGHER SINGULARITt]t;S ON CURVES 357Taking the origin at the given point on the curve, the

equation of a conic passing through the origin may bewritten as-

(1)If y be expanded in powers of .c , by Maclaurin's Theorem,

we have-

where P q, r ,.. are respectively the values at the origin of

Since the conic has a contact of the third order, thevalues of p, q, r are the same for the conic and for the curve.

Substituting in the equation o the conic, we have

:1:'=mx+a3lI+2hx(px+q2 !+ ... )Xl+b(px+q2 !+ ... )

Equating the co-efficients,we obtain-m=p, a+2hp+bp" =l, hq+bpq=i!

whence, h+bp=;q and (a+hp)+p(h+bp)= ~

s.e, a+hp=!L-'l!'! and a+1i,p ~ ( 3q' _ p ) (2)2 6q h+bp rNow, the centre othe conic (1) is given by-

2ax+2hy+m=O2hx+2by-l=O

The line joining the origin to the centre i8-x(a+mh) +y(h+bm)=O (3)


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358 THEORY OF PLANE CURVESIf 8 be the angle which it makes with the normalmy+.r=O, i.e., if 8 be the aberrancy, we have

tan 0a s-mh. 1h + b m - ' ; ; ; ;1+ a+mhm(h+bm)

(l+p4) ,.=p- 3q>281 . ABERRANCY CURVE:DEFINITION: The locus of the centres of osculating

conics at points of a given curve is called the aberrancycurve.

Taking the tangent and normal at any point of a givencurve as axes of co-ordinates, the co-ordinates of the centreof aberrancy may be expressed as-

X =R sino, Y=R.coB O .where R is the radius of aberrancy, and 0 is the angle ofaberrancy.

But, from the relation tan Il =p - (1+3P g)r ,we obtainq "3pqO-r(l+pO)sin Il = _ _ 1

.vl+p {r" +(rp-3q")'}"3q"cOBIl = __ 1.vl+p {r' +(rp-3q')'}T

X =3q{3pq'-r(1+p')}.vl+p (3qs-5rll)

where s is the fourth differential co-efficient of y with'regard to : 1 1 .


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HIGHER SINGULARI1'IES ON CURVES 359If, however, instead of taking the tangent and normal

as axes of co-ordinates, we take the axes such that theaxis of a; makes an angle ()with the tangent, we have

tan () =_OY=_pOJ:sin 0 =~ cos 0 1_.vl+p" .vl+p~

and the new co-ordinates (a, f3) of the centre of aberrancyare given by the two expressions-

a=X cos ()+Y sin ()=3 -3q1'q8-51"f3=-X sin 0+ Y cos()= -3q(p1'-3ql)

3'1s-51"Therefore, when the origin is taken anywhere, the

co-ordinates of the centre of aberrancy at any given point( : 1 : , y) of the curve are given in the most general form bythe formulae *

a=,t- ~3q$-51"f3=y- 3q(pr-3q')3q8-5r"

A. Mnkhopadhyay calculated these formulae-J. A. S. B., Vol. 57(1888), Part II p. 324, and deduced from them a number of veryinteresting results, which led to the remarkable geometrical interpreta-tion of Monge's differential equation to all conics, namely,

9q' t - 45qrs + 4Or" =0which he denoted hy T=O. It will be interesting to recall in thisconnection the remarks of Dr. Boole with regard to the geometricalinterpretation of Monge's equation (Differential Equations, p.20)-" But here our powers of geometrical interpretation fail, and resultssuch as this can scarcely be otherwise useful than as a registry ofintegrable forms." Since then two attempts had been made-one byLt.-Col Cunningham, R.E., and the other by Prof. Sylvester-to supplya true geometric interpretation to the Mongian, but Mukhopadhyaypointed out the futility of both these interpretations, and gave


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860 THEORY OJ' PLANE CURVES

Be, 1. Find the aberrancy curve for the cubicy=alD3 + 3bll> ' + 3e '" + d

(The aberrancy ourve is Y= All>+ 3Ba:' + 30iIJ+D.where A=-ku, B=-kb,

a'd-b3D=-kd+(I+k) --. ,a-k=125

64s.e., the aberrancy curve is then a cubic of the same class.

At the common points of the curves, we have (a " , + b)' =0, whichshows that the two curves have only one oommon point of' intersectionwhich is a point of inflexion on both.-A. Mukhopadhyay, Journal ofthe Asiatic Bociety of Bengal, Vol. 59, Part II (1890), pp. 61-63.]

El. 2. If p and pi are the radii of curvature of It conic and itsevolute, show that the aberrancy in the conic is given by tan =ip'fp(cf. Mukhopadhyay, J. A. S. B., Vol. 57, Part II (1888), pp. 317-319).

E ll> . 3. The envelope of the axes of all conics having a fonr-pointiecontact with a cnrve at a given point is a parabola having the axis ofaberranoy for directrix.

E . 4. Find the coordinates of the oentre and the radius ofaberrancy for the oubio y-.' at the point (1, 1).

[The centre of aberranoy is the point ({, -8), and the radius is 81i'~,i.6., the length joining the centre of aberrancy to the point (1, 1).J

El l>. 5. Prove that the aberrancy curve for the curve y'=z' is32y" =511' (Math. Tripos, 1891).

EIlI. 6. Show that the centre of aberrancy of the curve y=",- atthe point (II, y) is- ( 2 n+l- - : 1 > ,211.-1

the following true geometrical interpretation of the Morigian-"The radius of CUTVllture of the Aberrancy Ourve vanishes at everypoint of et'er1l cO'llic."- J. A. S. B., Vol. 58 (1889), Part I.

ganguli, theory of plane curves, chapter 13

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