A treatise on the higher plane curves: intended as a sequel to A treatise on conic sections

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HIGHER PLANE CURVHS.

A TREATISE

ON THE

HIGHER PLANE CURVES:

INTENDED AS A SEQUEL

TO

A TREATISE ON CONIC SECTIONS.

BY

GEORGE SALMON, D.D., D.C.L., LL.D., F.BS.,

REGIUS PROFESSOR OF DIVINITY IN THE UNIVERSITY OF DUBLIN.

THIRD EDITION.

Dublin :
HODGES, FOSTER, AND FIGGIS, GRAFTON STREET,

BOOKSELLERS TO THE UNIVERSITY.

MDCOCLXXIX,

fapagua?

cae
stages.

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ae _ CAMBRIDGE: |
PRINTED BY W, METCALFE AND SON, TRINITY STREET, ee:

PREFACE TO THE SECOND EDITION.

Tae first edition of this treatise has been for
several years out of print, and I had for sometime
given up the idea of reprinting it. The work, having
been written at a time when the Modern Higher
Algebra was still in its infancy, required extensive
alterations in order to bring it up to the present
state of the science; and, as I had failed to bring out
a new edition before my appointment to the office
which I now hold, I judged it impossible to do so,
now that other engagements left me no leisure to
make acquaintance with recent mathematical dis-
coveries, or even to keep up my memory of what
I had previously known. When, however, years
passed and mine still remained the only work in
English professing to give a systematic account of
the modern theory of curves, I began to consider
whether republication might not be possible, if I

could obtain the assistance of some younger mathe-

b
Mee OS71

vi PREFACE.

matician competent to contribute additional sections
representing the later progress of the science.
Consulting Professor Cayley on this subject I was
much and agreeably surprised by his offering himself
to give me the help I required. It is needless to
say how gladly I embraced a proposal calculated
to add so much to the value of my book; and the
only scruple I have felt in profiting by it is lest
the time and labour which Professor Cayley has
devoted to the work of another may, for a time
at least, have deprived the mathematical world
of a better work on the same subject by himself.
My original plan for the division of the labour was
that Professor Cayley should contribute certain new
sections or chapters, of which he should take the
entire responsibility, while I should content myself
with revising the older part of the book; and
accordingly the first chapter is entirely Professor
Cayley’s. But I found it would be impossible in
this method to give the book the unity it ought to
possess; and actually our work has been combined
in a manner that makes it not easy to separate
our respective shares. Professor Cayley has carefully
gone over the whole, and there is scarcely a page
that has not in some way been influenced by his
suggestions; on the other hand, I have completely

re-written many of his contributions either for the

A PREFACE. Vili

purpose of making them fit in better with the rest
of the book, or if I thought I could make some
simplification in his process or some addition to
his results. I have in fact dealt in the same
manner with*some of the manuscript materials which
he was so good as to place at my disposal, as I
have done with published memoirs of his, the results
of which I have incorporated in the work. On
looking through the pages the parts which I re-
cognize as taken from Professor Cayley, with but
‘slight or with no alteration, are Chap. I.; the account
of the forms of triple points, Art. 40; Art. 47,
the view taken in which I have not myself in
other places fully accepted; Ex. 6, p. 48; and Arts.
56—58, 87—89, 138, 189, 151, 198, 243, 270,
282—291, 407, 408. Besides these I have worked
into Chap. III. a manuscript of his on envelopes,
including the theory of evolutes and quasi-evolutes
and .of parallel curves; from another manuscript
of his I obtained my knowledge of Sylvester’s
theory of residuation; and I have used one on
the classification of quartics and one on the bi-
tangents of quartics. The additions made to the
chapter in the former edition on the transformation
of curves are almost entirely derived from a manv-
script of Professor Cayley’s, from which Arts. 370
to the end are taken nearly without alteration ;

Vili ‘aS PREFACE.

Arts. 401—406 are founded on a manuscript of his
on Steiner’s theory of polar curves. |

The first edition of this work contained a chapter
on the application of the Integral Calculus to the
theory of curves; this I have now omitted principally
on account of the extension which this subject has
since received. Such a chapter now, in order
to have any pretensions to completeness, ought to
contain an account of the applications which the
lamented Clebsch, in continuation of Riemann’s
researches, made of elliptic and Abelian integrals
to the theory of curves. But it seems impossible
that those subjects could be. done justice to, except
in a work having the Integral Calculus as its main
object; and as such works ordinarily contain chapters
on the theory of curves, I have thought that this
branch of the theory might safely be omitted from

the present treatise.

The causes which delayed the publication of
the Second Edition have also retarded the issue
of this Third, and have prevented me from doing
all that might be desired in the way of including
recent investigations. My friend Mr. Cathcart, to
whose help in correcting the press on this as on

former occasions I am greatly indebted, had called

PREFACE. 1X

my attention while the printing was in progress
to various points which needed fuller treatment.
These I had hoped to deal with in an Appendix
at the end, but all I have found time to do has.
reduced itself to the addition of a few references.
Professor Cayley, it will be observed, has kindly
given me one or two new contributions.

TRINITY COLLEGE, DUBLIN,
July, 1879.

COM CN So.

CHAPTER I.
COORDINATES,

_ Descriptive and metrical theorems ; ‘ . : .
General definition of trilinear coordinates < ¥ : F
Relation between point (1, 1, 1) and line r+ y¥+2=0 ‘ : °
Particular case of trilinear coordinates an : ‘ :

circular coordinates ‘ ‘“ ‘ ' i ‘
Circular points at infinity . : ° ‘ ° ‘
LINE COORDINATES . ‘ F ‘ ‘ » .
Their relation to trilinear coordinates “
Particular cases of line-coordinates ; ‘ Fi P
Geometrical duality . P P ‘
CHAPTER II.

GENERAL PROPERTIBS OF ALGEBRAIC CURVES.

SECTION I. NUMBER OF TERMS IN THE EQUATION ‘ y

All forms which are general must have as many independent constants as the
general equation . . . es 3

Number of terms in the general eauntic ‘ : . . :

Number of points which determine a n-ic ° :

A single curve determined by these conditions : : . .

In what cases this number of points fails to determine a n-ic

If one less than this number of ee be given, the curve passes through whee
fixed points . . ° :

If of intersections of two n-ic’s, a lie on a sicko the emnidivles lie on a (” — p)-ic

Extension of Pascal’s theorem . ; : : ;

13

13
15
15
16
17

18
18
19

Xil CONTENTS.

Steiner’s and Kirkman’s theorems on the hexagon .

Theorems concerning the intersections of two curves

Section II, MULTIPLE POINTS AND TANGENTS | . :
Equatiou of tangent at origin . ‘ . ‘ .
The origin, a double point . . : ‘

Three kinds of double points . ee ‘ ‘

Their relation illustrated . ‘ ° ‘ ‘
Triple points, their species ° ‘ :

Number of nodes equivalent to a multiple votnt ; :

Multiple point equivalent to how many conditions .

Limit to number of nodes on a proper curve

Deficiency of acurve . : ; ° .
Fundamental property of caleeneat curves

Consideration of the case where the axis is a multiple tangpesit
Stationary tangents and inflexions . :

Two consecutive tangents coincide at a stationary bangeat

Correspondence of reciprocal singularities ‘ °

Curve crosses the tangent at an inflexion . . .

Measure of inclination of curve to the axis ° ‘ .
Points of undulation . - ° ° °
Relation between points where tangents meet the curve aqua :
Equation of asymptotes, how formed ° . : °
Examples ° . . ° .
Section III. TRACING OF CURVES ° :

Newton’s process for determining form of curve at a singular eae
Keratoid and ramphoid cusps, e : ; ;

SecTION IV. PoLES AND POLARS .
Joachimsthal’s method of determining points nae a tine meets a curve
Polar curves . ° .

of origin .
Every right line has ie —1)? les . : ; °
Multiple points and cusps, how related to polar curves ° °
_SEcTion V, GENERAL THEORY OF MULTIPLE POINTS AND TANGENTS
All polars of point on curve touch at that point ‘
Points of contact of tangents from a point how determined
Degree of reciprocal of acurve . . :
Effect or singularities on degree of reciprocal . . .
Discriminant of a curve

If the first polar of A has a doutsle point B, polar conic of B bite a aisle point -

Hessian and Steinerian defined . . :
Conditions for a cusp ° . °
Number of points of inflexion . .

how affected by multiple points .
Equation of system of tangents from a point
Application to the case of a cubic
Number of tangents from a multiple point
Section VI. RECIPROCAL CURVES .
What singularities to be counted ordinary :
Pliicker’s equations . °
Number of conditions, the same for curve and its sactpeodal
Deficiency, the same for both ;

Cayley’s modification of Pliicker’s equations s

CONTENTS.
CHAPTER III.

ENVELOPES,

Two forms in which problem of envelopes presents itself ’
Envelope of curve whose equation contains a single parameter

Envelope of acos"0+dsin"0=c . ; > .
Equation of parallel toaconic . . . .
Envelope of right line containing a single parameter algebraically : its aa
istics 4 4 . . . ° .
Envelope of curve with related parameters : ‘ :
Method of indeterminate multipliers . . .
Envelope of curve whose equation contains independent re RS a a
Explanation of difficulty in theory of envelopes : ,
RECIPROCAL CURVES . : ° .
Method of finding equation of oa! ‘ ‘ ‘
Reciprocal of a cubic :
Symbolical form of equation of ieipesaal, ; ,
Reciprocal of a quartic . ° ; .

Equation of system of tangents from a saiat
Equation of reciprocal in polar coordinates

TAOCT-INVARIANT of two curves P

Its order in the coefficients of each curve .

EVOLUTES ‘ ‘

Defined as envelope of aoenaie ° . , ‘ :

Coordinates of centre of curvature .
General expression for radius of curvature
Length of arc of evolute :

Radius of curvature in polar coordinates . 4 F .
Evolutes of curves given by tangential equation » .
Quasi-normals and quasi-evolutes ° ° >
Quasi-evolute of conic ° ‘

General form of equation of canelnenall ;

Quasi-evolute when the absolute is a conic ‘ . :
Normal of a point at infinity . . , , .
Characteristics of evolute . °

Deficiency of evolute . ° : .
Condition that four consecutive points on a curve should be seaibules
CAUSTICS e . . i ’ :
Caustic by reflection hi a circle ° ‘ j .
Quetelet’s method ‘é : ‘ . . .
Pedal of a curve ;

Caustic by refraction of right line si circle ; : :
Evolute of Cartesian ‘ A ; :
PARALLEL CURVES AND NEGATIVE PEDALS ,

Cayley’s formulee for characteristics of — curves :
Problem of negative pedals : ‘ ‘
Roberts’s method :

Method of inversion . ; : ‘
Characteristics of inverse curve ind of iiial i ; ‘
Caustic by reflexion of parabola ‘ d F :
Negative pedal of central and focal ellipse - ‘

Xili

XIV CONTENTS.

CHAPTER IY.
METRICAL PROPERTIES,

Newton's theorem of constant ratio of rectangles

Carnot’s theorem of transversals . ° ° . :

Three inflexions of cubics lie on a right line , ; °

Two kinds of bitangents of a ; . .

DIAMETERS : ° .

Newton’s i craligation of seldon of Machen ‘. ° .

Theorem concerning intercept made between curves and asymptotes . °

Curvilinear diameters . : > > , .

POLES AND POLARS . , ee ie : ;

Cotes’s theorem of harmonic means of radii ‘ ‘ pk ;

Polar curves ° ° ° ; ‘ .

Polars of points at na ;

Pole of line atinfinity ©. :

Mac Laurin’s extension of Newton’s euacieen : , :

Pole of line at infinity with respect to curve of n*} class .

Its metrical property : centre of mean distances of contacts of serntiel tangents

Foct ° . ° : .

General definition of foci ; ; . . . ,

Number of foci possessed by a curve . ; , : .

Antipoints . A : : ‘ ;

Coordinates of foci awe found .

Locus of a point the tangents from which ‘inks with a fxd line 1260 hohe
sum is constant ; é

Every focus of a curve is a focus of its evolute .

Theorems concerning focal perpendiculars on tangents

concerning angles between focal radii and tangent
concerning focal distances of point on curve

Locus of double focus of circular curve determined by N — 3 pointes

Locus of focus of curve of n*® class determined by WV — 1 tangents

Miquel’s theorem as to foci of parabolas touching 4 of 5 given lines

CHAPTER V.
CUBICS.

General division of cubics > fs ‘

SECTION I. INTERSECTION OF A CUBIC WITH OTHER CURVES

Tangential of a point and satellite of a line defined : °
Asymptotes meet curve in three collinear finite points . , ,
Three points of inflexion lie on a right line °

Four points of contact of tangents from any point on curve, how velated
Mac Laurin’s theory of correspondence of points on acubic , °
Coresidual of four points on a cubic P

To draw a conic having four-point contact and elsewhere fouching a cubic
Conic of 5-point contact, how constructed
Sextactic points on cubic, how found
Sylvester’s theory of residuation

129
130
130
131
131
132
133
134
135
135
135
136

CONTENTS.

Two coresidual points must coincide ‘ . i A ACC RE
Two systems coresidual to the same are coresidual to each other
Analogues in theory of cubics to anharmonic theorems of conics ‘ :
Locus of common vertex of two triangles whose bases are given and vertical

angles equal, or having a given difference ; ; GENE
SEcTION II. POLES AND POLARS. ‘ : ‘ ; ‘
Construction for polar of a point with respect to a triangle ; :

Construction by the ruler for polar of a point with respect to a cubic . oi?

Anharmonic ratio. constant of pencil of four tangents from any point on a cubic

Two classes of non-singular cubics . : $ :
Sixteen foci of a circular cubic lie on four circles ‘ :
Chords through a point on cubic cut harmonically by patie ex conic
Harmonic polar of point of inflexion . . ° : ‘
All cubics through nine points of inflexion have thése for inflexions
Correspondence of two points on Hessian ; ; ; ‘
Steinerian of a cubic identical with its Hessian 3 : 3
Cayleyan, different definitions of ‘ é ‘ ‘ ,
Polar line with respect to cubic of point on Hessian touches Hessian A
Common tangents of cubic and Hessian ‘ . :
Stationary tangents touch the Hessian . . ‘ ‘
Tangents to Hessian at corresponding points meet on Hessian
Three cubics have common Hessian ° ‘ :
Rule for finding point of contact of any Sehiiont to Cayleyan — ie
Points of contact of stationary tangents with Cayleyan F °
Coordinates of tangential of point on cubic, how found : : .
Polar conic of line with respect to cubic P ; ‘
How related to triangle formed by tangents where line meets duties ‘ .
Double points, how situated with regard to polar conics of lines . é
Polar conic of line infinity 3 : : ‘ , ,
Another method of obtaining tangential equation of cubic
Polar conic of a line when reduces to a point “ . . .
Points, whose polar with respect to two cubics are the same
Critic centres of system of cubics ‘ , . ° ;
Locus of nodes of nodal cubics through seven points . . :
Pliicker’s classification of cubics ; . . ‘ ;
SECTION III. CLASSIFICATION OF CUBICS . : : :
Every cubic may be projected into one of five divergent parabolas ‘ e
and into one of five central cubics : :
Classification of cubic cones ‘ ° R : ;
No real tangents can be drawn from oval ; ‘ . ee
Unipartite and bipartite cubics " ‘ ‘ ‘ .
Species of cubics mak : R ‘
Newton’s method of roddednad the general equation ‘ : F
Pliicker’s groups . d ‘ : :

' Section 1V. UNICURSAL CUBICS . ‘ ‘ 5 ;
Inscription of polygons in unicursal cubics J ‘ , .
Cissoid, its properties ‘ ‘ ; :
Acnodal cubic has real fiedbain Siapiaaiat’ imaginary
Construction for acnode given three inflexional tangents ‘ °

General expression for coordinates in terms of parameter
SECTION V. INVARIANTS AND COVARIANTS OF CUBICS
Canonical form of cubic “ ; ;

XVi CONTENTS.

Notation for general equation . ‘ ‘
General equation of Hessian and of adleyen : ‘
Invariant S, and its symbolical form ‘ ‘ ‘
Invariant 7 “ . ° ‘~
General equation of ectansel ‘ ° . .
Calculation of invariants by the differential equation . °

Discriminant expressed in terms of fundamental invariants
Hessian of AU + mH ‘

Conditions that general equation duaia ienanice three right =e

Reduction of general equation to canonical form i .
Expression of discriminant in terms of fundamental invariants
Of anharmonic ratio of four tangents from any point onecurve .

Covariant cubics expressed in form AU + »H ¥ °
Sextic covariants . : ° ° ;
The skew covariant . : ‘ :
Equation of nine inflexional biugents , . .
Equation of Cayleyan in point coordinates pr eh
Identical equation in theory of cubics . ‘ .
Conic through five consecutive points on cubic ‘ .
Equation expressed in four line coordinates . ° ‘
Conditions that cubic should represent conic and line
Discriminant of cubic expressed as determinant ;
Hessian of PU and UV ‘ ‘ ‘ .

CHAPTER VI.

QUARTICS.

Genera of quartics : : , ; .
Special forms of quartics . ; : , :
Illustration of the different forms re 4 .
Distinction of realandimaginary . ; ‘ .
Flecnodes and biflecnodes : :
Quartic may have four real points of piuléiéon ‘ ‘
Quartics may be quadripartite . . : ‘
Zeuthen’s classification of quartics . ‘ ‘ ‘
Number of real bitangents ; > ‘

Inflexions of quartics, how many real

Classification of quartics in pean of their infinite aie:

THE BITANGENTS ° . .
Discussion of equation U W = = ye -

There are 315 conics passing through eight Sata of at ike
Scheme of these conics : . . ;
Hease’s algorithm for the bitangents

Geiser’s method of connecting bitangents with solid NOLEN

Cayley’s rule of bifid substitution . : ‘
Bitangents whose contacts lie on a cubic . :
Aronhold’s discussion of the bilangents ‘ , :
From 7 bitangents the rest can be found by linear constructions
Aronhold’s algebraic investigation . P

BINODAL AND BICIRCULAR QUARTICS
Tangents from nodes of a binodal are béithogtaphtic
Foci of bicircular quartic lie on four circles

213
214
215
217

Mae | §

218
219
220
220
221
222
223
224
227
228
230
231
232
234
234
237
238
240
241
242

CONTENTS. XVI

Casey’s generation of bicircular quartics ; ‘ : é "O43
Two classes of bicircular quartics. ‘ : ° : . 246
Relations.connecting focal distances of point on bicircular ° ‘ 246
Confocal bicirculars cut at right angles’. : . . . 247
Hart’s investigation « . . . ‘ ‘ 248
Cartesians ‘i ‘ Ne re . . 250
The limagon and the asic « ‘ . ° ° 252
Focal properties obtained by inversion . . : ‘ « 252
Inscription of polygons i in binodal quartics . P ‘ ‘ 253
UNICURSAL QUARTICS ‘ ° ‘ ‘ . . 254
Correspondence between conics and trinodal quartics . . é 254
Tangents at or from nodes touch the same conic . ’ . - 256
Tacnodal and oscnodal quartics ‘ : ° ° ‘ 258
Triple points Pe : ; . . . . 269
Expression of coordinates by a parameter ‘ é . * 260 ©
INVARIANTS AND COVARIANTS OF QUARTICS ; ‘i ; « 268
General quartic cannot be reduced to sum of five fourth powers . p 265
Covariant quartics . ‘ ° . . ‘ . 269
Examination of special case “ . ° . . 269
Covariant conics y ‘ : ‘ : : ». 273
CHAPTER VII.
TRANSCENDENTAL CURVES,
The cocload . “. : . . : . 275
Geometric investigation of its properties .. . ‘ ° « 276
Epicycloids and epitrochoids. . : . . : 278
Their evolutes are similar curves ‘ ‘ , : . 281
Examples of special cases . ° ‘ ; ° ‘ 282
Limagon generated as epicycloid : : : . . 282
Steiner’s envelope ; ‘ ‘ i ‘ 283
Reciprocal of epicycloid . : . : ‘ - 283
Radius of curvature of roulettes . ° F : 284
Trigonometric curves : eae . : : . 285
Logarithmic curves ° : : ‘ ‘ ; 286
Catenary : : : : . ‘ . 284
Tractrix and syntractrix . ‘ . , . , 288
Curves of pursuit . i . . ° ° - 290
Involute of circle e ° : ‘ . ° 290
Spirals , ‘ ‘ ; ae oe - 291
CHAPTER VIII.
TRANSFORMATION OF CURVES.
LINEAR TRANSFORMATION : : ‘ ; . 295
Anharmonic ratio unaltered by linear transformation e R - 296
Three points unaltered by linear transformation . . < 297

Projective transformation : ‘ ‘ = ig ; . 298

XViil CONTENTS.

Homographic transformation may be reduced to projection . ‘
INTERCHANGE OF LINE AND POINT COORDINATES . ‘ : :
Method of skew reciprocals . . :
Skew reciprocals reducible to dba ettadnobadl ° ‘ ; .
QUADRIO TRANSFORMATIONS ‘ " . . °
Inversion, a case of quadric transformation ° . . °
Applications of method of inversion . . ‘ ° ‘
RATIONAL TRANSFORMATION . : ‘ ° .
Roberts’s transformation . P ‘ . . :
Cremona’s rational transformation " ; : ‘ :
If three curves have common point their Jacobian passes ites it 3

Deficiency unaltered by Cremona transformation
Every Cremona transformation may be reduced to a succession of quadrie

transformations . . ete . . :
TRANSFORMATION OF AGIVEN CURVE . . : : :
Rational transformation between two curves . . :
Deficiency unaltered by rational transformation . . :
Transformation, so that the order of the transformed curve ony be as low as
possible . ° ° . :
Expression of coordinates by means of elliptic functions whieh B= .
and by means of hyper-elliptic functions when D = 2 ; "
Theorem of constant deficiency derived from theory of elimination . :
CORRESPONDENCE OF POINTSONA OURVE : : ,
Collinear correspondence of points ‘ ° ° ° :
Correspondence on a unicursal curve . . ° .
Number of united points ° . . . ° °
Correspondence on curves in genera]. . . . :
Inscription of polygons in conics s . ; ie :
in cubics . . . . . ‘ :
CHAPTER IX.

GENERAL THEORY OF CURVES.

Cayley’s method of solving the general problem of bitan ant F ;
_ Order of bitangential curve : ; ;
Hesse’s reduction of bitangential sciatton ° : ‘ ;
Bitangential of aquartic . : ; °
Second method of solving problem of double ancects ; . ‘
Formation of equation of tangential curve ° . ° ,
Application to quartic . . : . . .
POLES AND POLARS . ° .
Jacobian, properties of ‘ ° ° ‘ ° ;
Steiner’s theorems on systemsofcurves > ,
Tact-invariants ° ;
Discriminant of discriminant of Xe + pv aha of Au + pv + vw , ;
Condition for point of undulation . . .
For coincidence of double aud stationary tangent ° ° .
Steinerian of acurve , . . ° . °
Its characteristics ° . . . .

The Cayleyan or Bteinen Hedatiin
Its characteristics ;:

321

822
824
824
826

329
830
330
831
331
3381
332
333
334
837
337

CONTENTS. Xix

PAGE

Generalization of the theory : : ; é rf 365
O3CULATING CONICS . ; . ‘ ° e . 868
Aberrancy of curvature . . F , ° ° 368
Investigation of conic of 5-point contact . . . . . 870
Determination of number of sextactic points . p . ° 372
SYSTEMS OF CURVES . . . , . . . 872
Chasles’ method . ° . ‘ . ‘ ee” 373
Characteristics of systems of conics . : . . . 874
Number of conics which touch five given curves A ; ‘ 875
Zeuthen’s method . ° . s ° ‘ Pe i i
Degenerate curves : é ° ° ° . 877
Cayley’s table of results ° ° : : ‘ - 3880
Number of conics satisfying five conditions of contact with other curves . 382
Professor Cayley’s note on degenerate forms of curves . ° . 383
NOTES,

Professor Cayley on the bitangents of a quartic ; ; ‘ 387

Sa
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a.

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-¢

HIGHER PLANE CURVES.

CHAPTER L*

COORDINATES.

POINT-€OORDINATES.

1. We have in the plane a special line, the line infinity ;
and on this line two special (imaginary) points, the circular
points at infinity. A geometrical theorem has either no re-
lation to the special line and points, and it is then descriptive ;
or it has a relation to them, and it is then meérical.

2. The coordinates used for determining the position of
a point in the plane are Cartesian (rectangular or oblique}
or else trilinear; the latter, however, including as a particular
ease the former. Speaking generally we may say that the
Cartesian (rectangular) coordinates are best adapted for the
discussion of metrical properties; trilinear coordinates for that
of descriptive properties; but for metrical properties there is
often great convenience in using the notation of trilinear
coordinates, the equation of a curve being presented as a
homogeneous equation in (a, y, z), where a, y are ordinary
rectangular coordinates, and z is = 1.

It is proper to consider in some detail the theory of the
foregoing kinds of coordinates, .

3. As defined Conics, Art. 62, the trilinear coordinates of a
point are its perpendicular distances (p, q, 7) from three given
lines: it is assumed that the lines form a triangle (viz. that

* This chapter is by Professor Cayley.

2 POIN ~COORDINATES. ;
nw lkat bAag oS nee att per? Show m4 pte

no two of them are parallel), and then if (a, d, c) are the sides
of this triangle, and A its area, and if, moreover, the co-
ordinates (p, 9,7) are taken to be positive for a point within
the triangle, the coordinates p, g, 7 satisfy the relation (Conics,

Art. 63)
ap+bq+er=2A.

By means of this relation, an equation, not originally homo-
geneous, can be made homogeneous; and it is always assumed
that this has been done, and, in fact, the equations made use of
are always homogeneous.

4, But a more general definition of trilinear coordinates
is advantageous; viz., without in anywise fixing the absolute
magnitudes of the coordinates (x, y, z), we may take them to
be proportional to given multiples (ap, Bq, yr) of the original
trilinear coordinates (p, q, 7).

Observing that the distance measured in a given direction
is a given multiple of the perpendicular distance of a point from
a line, the definition may be stated with equivalent generality
in several forms as follows: the trilinear coordinates (a, y, 2)
of a point in the plane are proportional to

given multiples of the perpendicular distances—
given multiples of the distances measured in given direc-
tions—
given multiples of the distances measured in one and the
same given direction—
the distances measured in given directions—
of the point from three given lines.

The three given lines, say the lines x=0, y=0, z=0, are —
said to be the axes of coordinates, or simply the axes; and the
triangle formed by them, the fundamental triangle, or simply the
triangle.

Observe that while the quantities (a, y, z) remain indeter-
minate as regards absolute magnitude, there can be no identical
relation connecting them; and the equations which we use,
being necessarily homogeneous, express relations between the
mutual ratios of the coordinates.

POINT-COORDINATES., 3

5. It is not in general desirable to do so, but we may,
if we please, fix the absolute magnitudes of the coordinates,
and say (x, y, 2) are equal to (ap, Sq, yr) respectively; the
coordinates are in this case connected by the relation

| a” B
which relation serves to determine the absolute magnitudes of
the coordinates (a, y, z) of any particular point when their
ratios are known.

It is scarcely necessary to remark that the distance of a
point from a line is considered to change its sign as the point
passes from one to the other side of the line. The selection
of the positive and negative sides might be made at pleasure
for each of the three lines, but it is in general convenient to
fix them in suchwise that for a point within the triangle
the ratios (x: y: 2), or (when these are determinate in absolute
magnitude) the coordinates (a, y, 2), shall be positive. |

6. Taking the lines x =0, y=0, 2=0 to be given lines, the
values of the ratios x: y: 2 depend upon those of the implicit
constants a, 8, y, and are thus not as yet completely defined;
but we can fix them so that for a given point the ratios (a: y : 2)
shall have given values. Thus, if for the given point whose
perpendicular distances are p,, ¢,, 7, the ratios are to have the
given values 2,:y,:2,, this completes the determination of
the coordinates, viz., we have

beats

We Shs eae
oo pe 4 Cie

Again, what is nearly the same thing, we can choose our co-
ordinates so that a given linear equation Ax + By+Cz=0
shall represent a given line. In fact, if the equation of the
given line in terms of the coordinates (p, g, r) is ap + bg +er = 0,
then we have thus the determination

: a b Cc
B:yie= Api agi ar.

r.

_It is not in general desirable to make any use of the equations
just written down; the convenient course is to consider the

4 POINT-COORDINATES.

coordinates to have been fixed in ‘suchwise that the point
(1: 1:1) shall be a given point of the figure, or that the line _
x2+y+z=0 shall be a given line of the figure.

7. It is to be observed that we may properly speak of the
point (a,/8, y), meaning thereby the point, the coordinates of
which have the mutual ratios 7: y: 2 equal toa: @8:y. And
when we speak of the coordinates of a point as being (a, 8, 1),

or of (x, y, z) as being equal to (a, 8, y), we mean the same

thing ; that is to say, we only assert the equality of ratios, for
the very reason that the absolute magnitudes are indeterminate.
Thus, in the last paragraph, instead of the point (1:1: 1),
we might have spoken of the point (1, 1, 1).

8. The point (J, 1,1) and line x+y+2=0 (or generally
the point (a, 8, y) and line ~ + 5 + —=0) stand in a well-

known geometrical relation to the fundamental triangle, viz.
if the point be O, the line
will be ZMN which joins
the intersections with the
sides of the fundamental
triangle ABC of the cor-
_ responding sides of the
triangle DEF formed by
the points where the lines N A F B
joining O to the vertices of the fundamental triangle meet the
opposite sides; or, conversely, if the line LIZN is given, we
geometrically construct the point O by joining the points L,
M, N where the line intersects the sides of the fundamental
triangle to the opposite vertices of that triangle; the joining
lines form a new triangle, and the lines joining its vertices to
the corresponding vertices of the fundamental triangle meet in
the point O. The line and point are in fact “ harmonics,” or,
as will be hereafter explained, they are “pole and polar” in
regard to the triangle considered as a cubic curve, or we may
say simply in regard to the triangle. ‘Thus, if either the point
or the line be given, the other is known, and it is the same

POINT-COORDINATES. 5

thing whether we assume the point 4, 1,1) to bea Bie point,
or the line x+y¥+2=0 to be a given bie:

Considering the line «c+ y+2=0 as a given line, we
have in all four given lines, and writing for convenience
x+y+z2+w=0 (that is, considering w as standing for —x—y—2),
the determination of the coordinates is such that «=0, oe 0,
2=(0, w=0 are given lines.

9, The coordinates may be such that the point (1, 1, 1) .
shall be the centre of gravity of the triangle; or, what is the
same thing, that the line 2+ y+2=0 shall be the line infinity.
Reverting to the equation ap+bg+cr=2A, this comes to
assuming «:y:z=ap:bq:cr; viz. if we join the point to
the three vertices, so dividing the fundamental triangle into
_ three triangles, then the coordinates x, y, 2 are proportional to
the three component triangles (or, what is the same thing, each ~
coordinate is proportional to the perpendicular distance from
a side, divided by the perpendicular distance of the opposite
vertex from the same side). And it may be noticed that if,
fixing the absolute magnitudes of the coordinates, we assume

ap bq or
2, y, 2= oe, 2A? 2A?

that is, take x, y, 2 to be equal to the component triangles, each
divided by the fundamental triangle; then the relation satisfied
by the coordinates will be ~+y+z2=1.

10. A particular case is when the fundamental triangle is
equilateral ; here if x, y, 2 be proportional to the perpendicular
distances from the sides, (1, 1, 1) is the centre of the figure,
and x+y+2z=0 is the line infinity; if, fixing the absolute
magnitudes, we take (a, y, z) to be equal to the perpendicular
distances, and moreover take as unity the perpendicular distance
of a vertex from the opposite side, then the coordinates of the
centre of the figure are (4, 4, 4), and the relation between the
coordinates is a+ y+2=1. =

In this case, where the fundamental triangle is equilateral
and «+y+2=0 the line infinity, the coordinates of the cir-
cular points at infinity are 7: y:z2=1:@:° and 1: *: a,

6 POINT-COORDINATES.

where » is an imaginary cube root of unity; in fact, taking
X, Y as Cartesian (rectangular) coordinates, the origin being
at the vertex (c=0, y=0) of the triangle, and the coordinate
X being along the side «=0, we have

X/3-Y 2-X/3-Y
2 : 2

L,Y; 2=Y, respectively.
But for the circular points at infinity X and Y are infinite and
X+tY=0 (where ¢=/(- 1), as usual) ; wherefore

—147173 —117¥3

e:yi:2z=1: 9 : 9 ’

—1-73
2
Meise: ys sel: ws or H82 wo 2 w.

—14+7/38
2 ’]

or taking w to be = , and therefore w* =

11. Let one of the axes, say that of z, be the line infinity :
the distance r has here the value 0, which must be regarded
as an infinite constant; yr is therefore a constant, which may
be made finite, and without loss of generality put =1; we
have therefore x: y¥: z=ap: 8q:1, where the coefficients a, 8 .
may be so determined that ap, 8q shall represent the dis-
tances from the line «=0 and from the line y=0, each
measured in the direction parallel to the other of these lines;
that is, if X, Y are the Cartesian coordinates of the point,
then a2: y:z2=Y:X:1; or, what is the same thing, fixing
the absolute magnitudes of the coordinates, z, y and z=1, will
be the Cartesian coordinates of the point referred to any two
axes of coordinates. :

12. In what just precedes we have used only the line
infinity, not the circular points at infinity; and the resulting
Cartesian coordinates are in general oblique, but they may
be rectangular; viz. taking the lines x=0, y=0 as any two
lines harmonically related to the circular points at infinity; or,
what is the same ‘thing, at right angles to each other, then the
coordinates will be rectangular. The harmonic relation re-
ferred to is that the two lines meet the line infinity in a pair
of points forming with the circular points at infinity a range

CIRCULAR POINTS AT INFINITY. 7

of four harmonic points; or, what is the same thing, the two
lines and the lines from their intersection to the circular points
at infinity form a harmonic pencil. (See Conics, Art. 356).

13. It is in some cases convenient to use the imaginary
coordinates §=a2+ ty, n=a—ty, and z=1: these may be
called circular coordinates.

CIRCULAR POINTS AT INFINITY.

14. Fora given system of trilinear coordinates, the coordi-
nates of the circular points at infinity may be obtained as.
follows. Suppose, first, that the coordinates w, y, z denote the
perpendicular distances from the sides of the fundamental
triangle; then taking an arbitrary origin O and system of
rectangular axes OX, OY, if p, g, 7 are the perpendicular dis-
tances of O from the sides of the triangle, and A, pw, v the
inclinations of these distances to the axis OX, the relations
between the two sets of coordinates (a, y, z) and (X, Y), are

e=X cosrX4+Y sind—p,

y= X coswt+Y sinu —4q,

z=X cosy + Y sinv —r.
Write for shortness cosA+7 sind, cosu+7 sing, cosy+7 siny
(or e, cu, cv) = L, M, N respectively; then taking X and Y
infinite, and X+7Y=0, we have for the two circular points

respectively
gory ase

eiy:e=L: M:N and eiyie=FTiggin:

Writing A, B, C for the angles of the fundamental triangle, we
have between A, B, Cand X, p, v a set of relations such as

A= wr+p-y,

B=-aTwW+v—-N2X,

C= Tt+rA-y,
and hence writing cosA +7 sin A, cosB+7 sin B, cosC +7 sinC
(or e4, ¢3, e'°) =a, B, y respectively, we find

M N L
detersns * B=- Acie 7 &, aBy=—1,

8 CIRCULAR POINTS AT INFINITY.

and the coordinates of the circular points at infinity are thus

1 | 1
Z:y:2=-—1: —: and z:y:z=-—1: 25
y y B ) y Y B ]
1 1
= ys =: = = ‘s oo 2
1 1
pty eae =P.
the three expressions for each set of coordinates being of course
identical in virtue of the relation a@y = - 1.

The same formule obviously apply to the case where the
coordinates x, y, 2, instead of being equal, are only proportional
to the perpendicular distances from the sides of the triangle ;
and they are thus the formule belonging to the system of
coordinates for which the equation to the line infinity is

«sin A+y sinB+zcosC=0.

15. It may be added, that the original system of relations
between a, y, 2 and X, Y, gives

(y +49) & + ie sin A + (2+7) (a+p) sinB+ (x+p)(y+q) sind —
=sinAd sinB sinC(X’?+Y”),

or, what is the same thing, we have
yz sinA+2e snB+ay snC=sind sinB sinC (X*+ Y")

+ linear function of X, Y, 1,
viz. the equation yz sinA + zx sinB+ ay sinC=0 is the equa-
tion of a circle, and this being so, it is obviously the equation
of the circle circumscribed about the fundamental triangle;
and the formula holds good in the case where 2, y, z are

proportional to the perpendicular distances; the circular points
at infinity are therefore the intersections of the circle

yz snA+ze sn B+ zy sinC=0,
by the line infinity
xsinA+ysinB+z2sinC=0, :

(compare Conics, Art. 359), and it is easy to verify that the fore-
going expressions of the coordinates of the circular points at

LINE-COORDINATES. 9

infinity in fact satisfy these two equations. It is to be re-
marked also, that the general equation of a circle is

(yz sin.A + ze sinB+ ay sinC)
+ (Px+Qy+ Rz) (x sin 4 +y sinB+z sinC)=0,

where P, Q, FR are arbitrary coefficients.

16. In the system of coordinates wherein a, y, 2 are pro-
portional to the perpendicular distances, each multiplied by the
corresponding side, or where the equation of the line infinity is
2+y+2z=0, we have only in place of the foregoing a, y, 2

x y ee
to write dt ae ant?

points are therefore given by

the coordinates of the circular

A es eS ary Le B
sinA’sinB’ sinC ee
1
= vy -—1: a

1
“= 3 * ask,
Mo Pe Gt pale
a sind‘ sinB* sinC ree a

1

1

= er
oO ?

and the general equation of a circle is

(yz sin’ A + zx sin’ B+ xy sin’C) + (Px + Qy + Rz)(a@+y + 2) =0.

LINE-COORDINATES.

17. The coordinates above considered are coordinates for
determining the position of a point; say they are point-
coordinates. We have also line-coordinates (tangential co-
ordinates, see Conics, Art. 70) for determining the position of
a line; viz. if with any given system of trilinear coordinates
(x, y, 2), the equation of the line is e+ny+€z=0, then
we have a corresponding system of line-coordinates, wherein

C

10 LINE-COORDINATES.

(£, 7, £) are said to be the coordinates (line-coordinates) of
the line in question. Observe that according to this definition
(E, n, ¢) are given as to their ratios only, their absolute magni-
tudes are indeterminate; herein resembling point-coordinates
according to their most general definition.

18. The coordinates (&, 7, €) belong to a line; a linear
equation a&+bn+cf=0 between these coordinates refers to
the whole series of lines, the coordinates of any one of which
satisfy this equation; but all these lines pass through a point,
viz. the point whose coordinates in the corresponding system
of point-coordinates (x, y, 2) are (a, b,c); the linear equation
af +bn+cC=0 in fact expresses that the equation in point-
coordinates a + ny + Cz =0 issatisfied on writing therein (a, 6, e)
for (x, y,2). The conclusion is, that in the line-coordinates
(£, m, €), the equation a+ 6yn+c§=0 represents a point, viz.
the point whose trilinear coordinates in the corresponding
system are (a, b,c). And, generally, any homogeneous equa-
tion in the line-coordinates (£, », €) represents the curve which
is the envelope of all the lines && + yy + ¢=0, which are such
that the coefficients (&, , €) satisfy the relation in question;
and this relation is said to be the line- or tangential equation
of this envelope; in other words, the line-equation of a curve
is the equation between (&, 7, ), which expresses that the line
Ea +ny + &=0 is a tangent to the curve.

19. In what precedes the line-coordinates (&, 9, €) are
defined by means of a corresponding system of trilinear co-
ordinates (x, y, 2), the signification of the ratios &: 1: ¢ being
thereby in effect completely determined. This is the most con-
venient course; but, not so much for any application thereof,
as in order to more fully establish the analogy between the
two kinds of coordinates, it is proper to give an independent
quantitative definition of line-coordinates. We may say that
the trilinear coordinates (&, 7, €) of a line are proportional
to given multiples of the distances measured in given directions
of the line from three given points. Suppose, to fix the ideas,
we take them proportional to the perpendicular distances of
the line from the three given points. If referring the figure

LINE-COORDINATES. 11

to Cartesian coordinates, the coordinates of the points are
(a, 8), (a’, 8’), (a”, 8"), and the equation of the line is

AX+BY+C=0,

then we have

E:n:€=Aa+BB+C: Aa’ + BE'+C: Aa” + BB" 4+C,
or, what is thesame thing, the equation of the line is

Ay Sp lets
&, a, By 1
n, a, B, 1
c a ge 1
the coefficients of £, , € are here given linear functions of
(X, Y, 1), and denoting these coefficients by (a, y, z) we shall
have (x, y, z) a system of trilinear coordinates, and the equation
will be &e+ ny + &=0; the definition thus agrees with the one
given above.

We may in like manner, as in Art. 6, determine the line-
coordinates (£, 7, €), so that the line (1: 1:1) shall be a givén

line of the figure, or that the point €+7+¢=0 shall be a
given point of the figure.

20. Some particular systems may be mentioned. Let a, B, y
denote respectively the distances Cc
in a given direction of the vari-
able line from the points A, B,
C, viz. (a= Aa, B= Bb, y=Co);
then the coordinates &, 7, € may
be taken proportional to these
distances, £:97:€=a:8:¥. @ S c
Imagine the point C to move off to infinity in the given
direction; y has an infinite value which must be regarded as

a constant ; and writing &: 7: : =a:B:1, we may, instead

of the original coordinates, &, 7, €, take as coordinates &, 7, 2 ;

that is, a, 8, 1. We have here a system of two coordinates
a, 8, which are respectively equal to the distances in a giver
direction of the line from two fixed points.

12 LINE-COORDINATES.

21. Again, in the annexed figure we have
oe) ee ta B

y Cp?y Cy’
or, what is the same thing,
oe ee
Ap’ Ba’ * Gp’ Cq
Imagine A, B to go off to infinity
in the given directions pC, gC re-
spectively; Ap, Bg have infinite g P
values which must be regarded as constants; and instead of
coordinates proportional to a, 8, y, we may take coordinates

. C

roportional to —~ Lids y; that is, we may take as co-
prop Ap’ Bq j ’ 'Y

ici et
ordinates —- 1; we have thus a system of two coordinates,

Cp ’ Cy ’
which are respectively the reciprocals of the distances in two
given directions of the line from a fixed point.

22. There is little occasion for any explicit use of line-
coordinates, but the theory is very important; it serves in
fact to show that in demonstrating by point-coordinates any
descriptive theorem whatever, we demonstrate the correlative
theorem deducible from it by the theory of reciprocal polars
(or that of geometrical duality), viz. we do not demonstrate
the first theorem and deduce from it the other, but we do
at one and the same time demonstrate the two theorems;
our (x, y, 2) instead of meaning point-coordinates may mean
line-coordinates, and the demonstration is in every step thereof
a demonstration of the correlative theorem, | |

23. And in like manner when any theorem is demonstrated
by line-coordinates, this is also a demonstration of the corre-
lative theorem; the only difference is that we here pass from the
somewhat less familiar theory of line-coordinates to the more
familiar one of point-coordinates; the transition is rendered
clearer if we consider the original line-coordinates (&, 9, ¢) as
being the point-coordinates of the point which is the pole of
the line in regard to the conic 2+ y°+ 2° =0.

y CHAPTER IL

ON THE GENERAL PROPERTIES OF CURVES OF THE n't DEGREE.

SECT. I.—ON THE NUMBER OF TERMS IN THE GENERAL EQUATION.

24. The first step towards obtaining a knowledge of the
general properties of curves of the n™ degree is the ascertaining
the number of terms in the general equation. We should thereby
be enabled, on being given any equation of the n™ degree,
by simply counting the number of independent constants in the
equation, to know whether or not the given form were one to
which all equations of the n™ degree could be reduced. For
example, the general equation of the second degree contains
five independent constants.. If, then, we were given any other
equation of the second degree, containing five constants, for
instance,

(w—a)' + (y—B)'= (ax + by + ¢)",

or i(w—a)'+ (y—B)'}¥ + (e—a)'+ (y—-BY}=«,

we could expand, and comparing the equation (as at Conics,
Art. 77) with the general equation of the second degree, should
obtain a sufficient number of equations to determine a, 8, &c.,
in terms of the coefficients of the general equation. We see,
then, that any equation of the second degree may, in general,
be reduced to either of the above forms, and we might thus
obtain a proof of the properties of the foci and of the directrix.
The equation

(ax + by +c)’ = (au + b'y 4c’) (a"a + b"y +c")
contains seven independent constants. ‘The problem, therefore,
to express these in terms of the coefficients in the general

equation is indeterminate, as is also geometrically evident,
since the equation may be thrown into this form by taking

axt+ by +e, aa + "y+e"

14 ON THE NUMBER OF TERMS

to represent any two tangents, and az+by+e their chord of
contact. The equations

(ax + by) =cu+ dy+e,

(ax + by +1) (vx + 'y +1) =90,

contain each but four independent constants, and must, therefore,
implicitly involve one other condition; or, in other words, the
general equation cannot be thrown into either of these forms,
unless one other condition be fulfilled. This is geometrically
‘evident, since the first equation denotes a parabola and the
second two right lines. The general equation of a circle,

(w—a)'+(y—-By =",
containing but three expressed constants, must implicity involve

- two conditions, or the general equation cannot be thrown into

this form unless two conditions be fulfilled. And so, again,
the equation

S—kS'=0,
(where S, S’ are given quadric functions of the coordinates)
containing but one expressed constant must imply four con-
ditions; as we otherwise know, sitce the conic expressed by
this equation passes through four fixed points.

25. Some caution must be used in the application of these

principles. Thus, the equation
(w—a)’?+(y—B8)*=ax+by+e
appears to contain five constants, and, therefore, to be a form to
which every equation of the second degree is reducible. But
if we expand, we shall see that the constants do not enter into
the highest terms of the equation, and that there are but three
equations available to determine a, 8, &c. The equation can,
therefore, not be thrown into this form unless two other con-
ditions be fulfilled. In like manner, the equation
aS,+08,+c¢S,+d8S,+ eS,+fS,=0,

where S,, &c., are six conics, is a form to which the equation
of any conic may be reduced; but suppose three of the equations
of these conics to be connected by the relation S,=4S,+/8,;
substituting this value, the equation would be found to contain
but four independent constants, and the general equation could

IN THE GENERAL EQUATION. 15

not be reduced to this form unless some one condition were

fulfilled.

26. Having thus endeavoured to give the reader an idea of
the nature of the advantage to be gained by a knowledge of
the number of terms in the general equation of the n™ degree,
we proceed to dn investigation of this problem. The general
equation of the x degree between two variables may be written,

A
+ Be +Cy
+ Da’ + Exy + Fy’

+ Pa* + Qa" y +...4+ Ray" + Sy" =0.
And the number of terms in this equation is plainly the sum
of the series 1+2+4+34...4(n+4+ 1), and is therefore equal to
4 (n+ 1) (n+ 2), as has been already proved (Conics, Art. 78).

We shall sometimes write the general equation in the

abbreviated form,

U+tU,+tUz+.--+ U, =),
where uw, denotes the absolute term, and u,, w,, u,, &c., denote
the terms of the first, second, n“, &c., degrees in # and y.

We shall also sometimes employ the equation in trilinear
coordinates, which only differs from that just written in having
a third variable 2 introduced, so as to make the equation homo-
geneous, ViZ.,

Uz +ue +u ze +...4-u,2+4u,=0.
The number of terms is evidently the same as in the preceding

case (Conics, Art. 289).

27. The number of conditions necessary to determine a
curve of the n degree is one less than the number of terms
in the general equation, or is equal to 4n(n43). For the
equation represents the same curve if it be multiplied or divided
by any constant; we may therefore divide by A, and the curve
is ‘completely determined if we can determine the 4n (n+3)

&e,

We
quantities eur ©.

16 ON THE NUMBER OF TERMS

Thus a curve of the n™ degree is in general determined when
we are given $n(n+3) points on it; for the coordinates of each
point through which the curve passes, substituted in the general
equation, give a linear relation between the coefficients. We
have, therefore, 4x(n+3) equations of the first degree to
determine the same number of unknown quantities, a problem
which admits in general of but one solution. We learn, then,
that a curve of the third degree can be described through nine
points, one of the fourth degree through fourteen points, and
in general through 4n(n+3) points can be described one, and
but one, curve of the n™ degree.

28. When we say that 4n(n+3) points determine a curve
of the x degree, we would not be understood to mean that
they always determine a proper curve of that degree. All
that we have proved is, that there exists an equation of the n™
degree satisfied for the given poimrts, but this equation may be
the product of two or more others of lower dimensions. Thus,
_ five points in general determine a conic, but if three of them
lie on a right line, the conic is the improper quadric curve
formed by this right line and the line joining the other two
points. And, in general, it is evident that, if of the }n(x+3).
points more than xp lie on a curve of the p™ degree (p being
less than x), a proper curve of the x degree cannot be described
through the points, for we should then have the absurdity of
two curves of the n“ and p™ degrees intersecting in more than
np points (Conics, Art. 238). The system of the n™ degree through
such a set of points is the curve of the p™ degree, together with
a curve of the (n—p)™ degree through the remaining points.

We may even fix a lower limit to the number of points
determining a proper curve of the x“ degree which can lie on
a curve of the p™ degree, and can show that this number
cannot be greater than np —4( p—1)(p—2). For if we suppose
that one more of the points (viz. np — 4 (p—1)(p —2)+1) lie
on a curve of the p™ degree, subtracting this number from
jn(n+3), it will be found that the number of remaining
points is } (n—p) (n—p +8), and that, therefore, a curve of the
(xn —p)™ degree can be described through them. This with the
curve of the p™ degree forms a system of the n™ degree through

IN THE GENERAL EQUATION. 17

the points; and it follows from the last Article that it is in
general impossible to describe through them any other.

29. There are cases, however, in which the solution of
Art. 27 fails: a very simple instance will show that this is so.
The number of points required for the determination of a cubic
curve is nine; but nine points do not in every case determine
a single cubic, for any two cubics intersect in nine points; and
through these nine points there pass the two cubics; as will
presently appear, there are in fact through the nine points an
infinity of cubics. The explanation is that although m linear
equations are in general sufficient to determine m unknown
quantities, the equations may be not all of them independent,
and they will in this case be insufficient for the determination
of the unknown quantities. The given points are then in-
sufficient to determine the curve, and through them can be
described an intinity of curves of the n™ degree. The geo-
metrical reason why such cases occur requires to be further
explained.

Let us, for simplicity, commence with the example of curves
of the third degree. Let U=0, V=0, be the equations of two
such curves, both passing through ae given points; then the
equation of any curve of the third degree passing through these
points must be of the form U-—kV=0. _ For this equation,
from its form, denotes a curve of the third degree passing
through the eight given points, and it contains an arbitrary
constant & which can be so determined that the curve shall pass

through any ninth point. We should, in fact, have k=,
where U’, V’ are the results of substituting the coordinates of the
ninth point in Uand V. This gives a determinate value for &
in every case but one, viz. when the ninth point lies on both U
and V; for since two curves of the m™ and n™ degrees intersect
in mn points, U and V intersect not only in the eight given
points, but also in one other. For the coordinates of this poiut

k takes the value 3 and indeed the form of the equation suffi-

ciently shows that every curve represented by the equation
U-kV=0 passes through ail the intersections of U and V.
D

18 ON THE NUMBER OF TERMS

Hence we have the important theorem, All curves of the third
degree which pass through eight fixed points pass also through
a ninth. And we perceive that nine points are not always”
sufficient to determine a curve of the third degree; for we can
describe a curve of the third degree through the intersections of
two such curves, and through any tenth point.

30. The same reasoning applies to curves of any degree. If
there be given a number of points one less than that which will
determine the curve {4n(z+3)—1}, then U-kV=0 (where U
and V are any two particular curves of the system) is the most
general equation of a curve of the n™ degree passing through
these points. For the equation contains one arbitrary constant,
to which we can assign such a value that the curve shall pass
through any remaining point, and be therefore completely de-
termined. But the form of the equation shows that the curve
must pass through all the n” points common to U and J, and
therefore not only through the 4n(n+3)-—1 given points, but
also through as many more as will make up the entire number
to n°. Hence, All curves of the n™ degree which pass through
gn(n+3)—1 fixed points pass also through 4(n—1)(n— 2)
other fixed points.

31. The following is a useful deduction from the preceding
theorem: If of the n® points of intersection of two curves of the
n™ degree, np lie on a eurve of the p” degree ( p being less than n),
the remaining n(n—p) will lie on a eurve of the (n— 5) ae
degree. For describe a curve of the (n—p)™ degree through
4(n—p)(n—p+t3) of these remaining points, and this, together
with the curve of the p degree, form a curve of the n™ degree
passing through 4 (n—- p) (n—p+3)+ np points; and since this
number {being equal to $n (n+ 3) -1+4(p—1)(p—2)} cannot
be less than 4n(n+3)—1, this curve will pass through all the
remaining points; but, obviously, the remaining points do not any
of them lie on the curve of the p™ degree, and therefore they
lie all of them on the curve of the (n — p)™ degree.

It is to be understood in these theorems concerning the
intersections of curves of the n™ degree, that the curves need not
be proper curves of that degree, for the demonstration in Art. 30

IN THE GENERAL EQUATION. 19

holds equally even though U or V be resolvable into factors.
As an illustration of the theorem of this Article, we add the
following: If a polygon of 2n sides be inscribed in a conic, the
n(n—2) points, where each odd side intersects the non-adjacent even
sides, will lie on a curve of the (n—2)" degree. For the product
of all the odd sides forms one system of the n™ degree, and the
product of all.the even sides another; these systems intersect
in 2’ points, viz. since each odd side has two adjacent and n—2
non-adjacent even sides, in the 2n vertices of the polygon, and
the n(n—2) points, which are the subject of the present theorem.
But since, by hypothesis, the 2n vertices lie on a conic, the
remaining n(x—2) points, by this Article, lie on a curve of
the (n — 2)” degree.

32. Pascal’s theorem is a particular case of the theorem just
given, but on account of the importance that the learner should
clearly understand the principle of the foregoing demonstrations,
we think it advisable to repeat in other words the proof already
given. :

Denote the sides of the hexagon by the first six letters of
the alphabet 4=0, &c.; then ACH-ABDF=0 is the equa-
tion of a system of curves of the third degree passing through
AB, BC, CD, DE, EF, FA, and also through AD, BE, CF.
If the first six points lie on a conic S, then the curve of the
system determined by the condition that it shall pass through
any seventh point of the conic S must give ACH-k'BDF= SL.
For it cannot be a proper curve of the third degree, since no
such curve can have more than six points common with 8.
The right line Z will therefore contain the three points AD,
BE, CF.

We may add, that it is this proof of Pascal’s theorem which
leads most readily to Steiner’s and Kirkman’s theorems (Contes,
p- 361). Thus, let

12.34.56 — 45.61.23 = SZ,

where 12 denotes the line joining the vertices 1, 2, &c.; and
where Z consequently denotes the line through the intersections
of the opposite sides, 12, 45; 34, 61; 56, 235 and let

12.34.56 - 36.25,14= SM;

20 ON THE NUMBER OF TERMS

then, obviously,
45.61.23 — 36.25.14=S(M- L);

or the Pascal line indicated by the latter equation passes
through the intersection of the other two.

It may, however, be remarked that the theorem of Art. 31,
in the case in question n = 3, is a particular case of the theorem
of Art. 30; viz., the system of the three odd sides is one of the
cubics, and the system of the three even sides the other of the
cubics U=0, V=0 of Art. 30. And we may deduce Pascal’s
theorem directly from that theorem; viz., considering the conic
through the six vertices, and the line joining two of the three
points of intersection of the opposite sides, the conic and line
form a cubic through eight of these nine points, and therefore
through the ninth point; that is, the line passes through the
remaining one of the three points of intersection of the opposite —
sides ; viz., these three points lie in a line. j

33. It has been proved that, although two curves of the
n” degree intersect in n® points, yet n” points, taken arbitrarily,
will not be the intersections of two such curves; but that
n° —4(n—1)(n—2) of them being given, the rest will be deter-
mined, A similar theorem holds with regard to the np points
of intersection of two curves of the n and p degrees. Thus,
though a curve of the third degree intersects one of the fourth
- in twelve points, yet through twelve points taken arbitrarily
on a curve of the third degree, it will, in_general, be impossible
to describe a proper curve of the fourth degree. For the
system of the fourth degree through these twelve and any
other two points will, in general, be no other than the curve
of the third degree and the line joining the two points. And,
generally, very curve of the n° degree which is drawn through
np —4(p—1)(p—2) points on a curve of the p® degree (p being
less than n) meets this curve in 4 (p—1) (p—2) other fixed points.
For we had occasion in Art. 31 to see that

np—4(p—1)(p—2)+2(n—p) (n- p+3)=4n(n+8)—1;
therefore, by Art. 30, every system of the n* degree described
through the given points, and 4(n—p) (n—p+83) others, passes
through $(n— 1) (n— 2) other fixed points. But one system of

a Zon, Qaven kN \,

IN THE GENERAL EQUATION. 21

the n‘® degree which can be described through the points is
the given curve of the p degree and one of the (n—p)™
through the additional assumed points. The 4 (n—1)(n—2)
new points must therefore lie, some on one, some ou the other
of these two curves. And it is evident that these points must
be so distributed between them as to make up the total number
of points, in the first case, to np, in the second to n(n—p).
Hence the truth of the theorem enunciated is manifest.

34, A further extension of this theorem has been given by

Prof. Cayley: ‘ Any curve of the r™ degree (r being greater than
m or n, but not greater than m+n—3), which passes through all
but 4 (m+n—r—1) (m+n—r—2) of the mn intersections of two
curves of the m™ and n™ degree, will pass also through the
_ remaining intersections.”
_ The reader will more easily understand the spirit of the
general proof we are about to give by applying it first to a
particular example. “Any curve of the fifth degree which
passes through fifteen of the intersections of two curves of the
fourth degree will also pass through the remaining intersection.”
For take two arbitrary points on each of the curves of the
fourth degree. These four, with the fifteen given points, make
nineteen points, through which, if several curves of the fifth
degree pass, they will (by Art. 30) pass through six other fixed
points. But each curve of the fourth degree, together with
the line joining the two arbitrary points on the other curve,
forms a system of the fifth degree through the nineteen points.
Hence all the intersections of the given curves of the fourth
degree lie on every curve of the fifth degree through the
points. Q.E.D.

So, in general, take 4 (r —m)(r— m+ 3) arbitrary points on
the curve of the n™ degree, and through them draw a curve of
the (r—m)™ degree; and take $(7—n) (r—n+3) points on
the curve of the m‘* degree, and through them draw a curve of
the (r—n)™ degree; take as many of the mn points of inter-
section as with the arbitrary points make up $r (7+3)—1; then,
since the curves of the (7 —m)™ and m‘* degree make one system
of the r degree through the points, and the curves of the
(r—n)™ and n make another, the intersection of these two

22 MULTIPLE POINTS AND TANGENTS OF CURVES.

systems will be common to every curve of the 7 degree through
the points. But

br (r-+8)—1-4 (r—m) (r— m8) —$ (rn) (r—n +8)
=mn-4(m+n—r—1) (m+n—r—29),

as the reader may verify without difficulty. Hence the truth
of the theorem appears. ‘To make the proof applicable 7 must
be at least equal to the greater of m or n; and also r—m
must be less than n, since otherwise it would not be possible
to describe, through the assumed points on the curve of the
n degree, a curve of the (r— m)™ degree, distinct, from or not
including as part of itself the curve of the n* degree; and, since
the theorem is nugatory for r=m+n—1 or m+n—2, the
condition is 7 not greater than m+n —3.*

SECT. II.—ON THE NATURE OF THE MULTIPLE POINTS AND

TANGENTS OF CURVES.

35. The simplest method of introducing to the reader the
subject of the singular points and lines connected with curves
seems to be, first, to illustrate by particular examples the nature
of these points and lines, and afterwards to lay down rules by
which their existence may be detected in general.

We shall employ the Cartesian equation given in Art. 26.

* Euler appears first to have noticed the paradox, that two curves of the n™ degree
may intersect in a greater number of points than are sufficient to determine such a
curve (see a Memoir in the Berlin Transactions for 1748, “On an apparent Contra-
diction in the Theory of Curves”). The same difficulty is pointed out by Cramer,
in his “Introduction 4 l’Analyse des Lignes courbes algébriques,” published in the
year 1750. It was only comparatively recently, however, that the important geo-
metrical theorems were observed, which are derived from this principle. In the year
1827 M. Gergonne gave the theorem of Art. 31 (Annales, vol. Xv1I., p. 220). The
general theorem of Art. 80 was given about the same time by M. Pliicker (Entwicke-
lungen, vol. I., p. 228; and Gergonne’s Annales, vol. X1X., pp. 97, 129). It was some
years afterwards that the cases were discussed of the relation which exists between
the points of intersection of curves and surfaces of different degrees (as in Art. 33),
These cases were discussed in two papers sent at the same time for publication in
Crelle’s Journal, one by M. Jacobi (vol. Xv., p. 285), the other by M. Pliicker
(vol. XVI., p.47). Besides the papers just mentioned, the reader may also consult
a Memoir by Prof. Cayley (Cambridge Math. Journal, vol. 111., p. 211), The historical
sketch given in the present note is taken from Pliicker’s Zheorie der Algebraischen
Curven, p. 18.

MULTIPLE POINTS AND TANGENTS OF CURVES. 23

If we transform this equation to polar coordinates, by sub-
stituting p cos@, p sin@ for « and y (or if the axes be not
rectangular, mp, np, as at Conics, Art. 136), we get an equation
of the n™ degree in p, whose roots are the distances from the
origin of the n points, where the curve is met by a line drawn
through the origin, making an angle @ with the axis of x.

4

36. If in the general equation the absolute term 4 =0,
then the origin is a point on the curve; for the equation is
evidently satisfied by the values ~=0, y=0, that is, by the
coordinates of the origin.

The same thing appears from the equation expressed in polar
coordinates,

| (B cos 0+ C sin@) p+ (D cos’@+ Ecos @ sind + F sin’ 6) p?4+ Ke.=03

for this equation being divisible by p, one of its roots must be
p=0, whatever be the value of 6, and therefore one of the
n points, in which every line drawn through the origin meets
the curve, will, in this case, coincide with the origin itself. :

The other (n—1) points will in general be distinct from the
origin; there is, however, one value of 0, for which a second
point will coincide with the origin, viz., if @ be such that

B cos6+ C sind =0.

The equation then becoming
(D cos’@ + E sin @ cos @ + F sin’6) p? + Ke. = 0,

is divisible by p*, and has, therefore, for two of its roots, p=0.
The line, therefore, answering to this value of 6, meets the
curve in two coincident points, or is the tangent at the origin.

Since we have.a simple equation to determine tan@, we see
that at a given point on a curve there can, in general, be drawn
but one tangent. Its equation is evidently

p(B cos0+ Csin@)=0, or Be+ Cy=0.

Hence tf the equation of a curve be u,+u,+&e.=0 (the origin
being a point on the curve), then u,=0 is the equation of the
tangent.

If B=0, the axis of x isa tangent; if C=0, the axis of y.

24 MULTIPLE POINTS AND TANGENTS OF CURVES.

37. Let us now, however, suppose that A, B, C are all =0;
the coefficients of p will then =0, whatever be the value of 0;
in this case, therefore, every right line drawn through the origin
meets the curve in two points which coincide with the origin.
The origin is then said to be a double point.

We may see now, exactly as in the last Article, that it is in
this case possible to draw through the origin lines which meet the
curve in three coincident points. Tor let @ be such as to render
the coefficient of p*=0, or D cos*0+ # sin@ cos@+ F'sin’@=0,
then the equation becomes divisible by p*, and three values of p
are =(. Since we have a quadratic to determine tan@, it
follows that there can be drawn through a double point two right
lines, each of which meets the curve in three coincident points;
their equation is

p’ (D cos*@ + E sin @ cos@ + F'sin’@)=0, or Da’ + Exy + Fy’ =0.

We learn hence that although every line through a double
point meets the curve in two coincident points, yet there are
two of these lines which have besides contact (viz., a conse-
cutive point common) with the curve at that point; so that it
is usual to say that at a double point on a curve there can be
drawn two tangents. If the equation of the curve (the origin
being a double point) be written w,+u,+&c.=0, then u, =0
is the equation of the pair of tangents at the origin.

38. It is necessary to distinguish three species of double
points, according as the lines represented by u,=0 are real,
imaginary, or coincident. ;

I. In the first case the tangents are both real; the double
point or node is such as that represented in the second figure
(Art. 39) 5 an inspection of the curve shows that there are at the
node two branches each with its own proper tangent; and the
foregoing quadratic equation in fact determines the directions of
these two tangents: such a point is termed a crunode.

A simple illustration of such double points occurs when the
given equation is the product of two equations of lower dimen-
sions, or U= PQ. The equation U=0 then represents the two
curves denoted by P=0 and Q=0. But if these two be con-
sidered as making up a complex curve of the n™ degree, this

MULTIPLE POINTS AND TANGENTS OF CURVES. 20

curve must be said to have pg double points (the points, namely,
where / intersects Q); and at each of these points there are
evidently two tangents (viz., the tangents to Pand Q).

II. The equation u,=0 may have both its roots imaginary.

In this case no real point is consecutive to the origin, which
is then called a conjugate point or acnode. Its coordinates satisfy
the equation of the curve, but it does not appear to lie on the
curve, and, in fact, the existence of such points can only be
made manifest Rearnettalls by showing that there are points,
no line through which can meet the curve in more than n—2
points.

III. The equation u, may be a perfect square; in this case
the tangents at the double point coincide, and the curve takes
the form represented in the fourth figure (Art. 39). Such
points are called cusps or spinodes. ‘They are also sometimes
called stationary points; for if we imagine the curve to be
generated by the motion of a point, at every such cusp the
motion in one direction is brought to a stop, and is exchanged
for a motion in the opposite direction.

The reader might suppose that we could illustrate these
points, as in the last paragraph, by supposing the curve U to
break up into two, P and Q, which touch; for ;
every point of contact will be a double setat the a
tangents at which coincide. But such a point
must be classed among singularities of a higher
order than those which we are now considering ; Naa
for the tangent has at it four points along
the complex curve, viz., two on each of the simple curves,
while at the cusps we are considering we have seen that the
tangent generally meets the curve in only three consecutive
points. In order that the tangent at a cusp should meet the
curve in four consecutive points, it is necessary not merely that
u, should be a perfect square, but further, that its square root
should be a factor ‘in u,; that is to say, that the equation should
be of the form

0, +vv,+u,+ &e. = 0.

Such points arise from the union of two double points, as

the reader will readily perceive from the example which we
E

26 MULTIPLE POINTS AND TANGENTS OF CURVES.

have already given, for when the curves P and Q touch, the
point of contact takes the place of two points of intersection.

It is proper to remark that the crunode and the acnode are
varieties of the node, and varieties of the same generality, the
difference being that of real and imaginary. The cusp has in
the investigation presented itself as a particular case of the
node, but it is really a distinct singularity; the force of this
remark will appear in the sequel.

389. As the learner may probably find some difficulty in
conceiving the relation of conjugate points to the curve, we
shall illustrate the subject by the following example. Let us

take the curve
y' = (w—a) (wb) (@- ),

where a is less, and c greater than b. This curve is evidently:
symmetrical on both sides of the axis of x, since every value of x
gives equal and opposite values to y. The curve meets the axis
of x at the three points x=a, x=b, x=c. When =~ is less than
a, y’ is negative, and therefore y imaginary; y’ becomes positive
for values of x between a and 8; negative again for values
between & and c; and, finally, positive for all values of a

exceeding c. The curve therefore consists of an oval lying
between A and B, and a branch

commencing at C, and extending |
indefinitely beyond it. a -

Let us now suppose b=c and AINE A Me

the equation will become

y' = (x —a) («— by,
where 4 is greater than a. The point B has now closed up to C3
as B approaches to C, the oval and infinite branch sharpen out

towards each other, and when ulti-

mately the two points are united

together the oval has joined the in- : B

finite branch, and the point B has
become a double point, with branches Fale
cutting at an angle.

But, on the other hand, let =a, then the equation
becomes

e
y' = (x — a)! (x 6),

MULTIPLE POINTS AND TANGENTS OF CURVES. 27

where a is less than be the oval has shrunk into the point A,

and the curve is of the annexed form.
This example sufficiently shows the

analogy between conjugate points and A ©

double points, the tangents at which are

real. If we suppose a=b=c, the equation becomes y’=(x—a)’,

the point A beedmes a cusp, as in III. of oe
last Article, and the tangent at the cusp Bagi et
meets the curve in three coincident points oat
A, B,C,

_ 40. Ifin the general equation A, B, C, D, L, F were all =0,
then the origin would be a triple point, every line through the
origin meeting the curve in three coincident points; and it is easy
to see, as before, that at a triple point there are three tangents,
which are the three lines represented by the equation u, =0.

We may also, as before, distinguish four species of triple
points, according as the three tangents are (a) all three real
and (1) all three distinct, (2) two coincident, (3) all three co-
incident, or (b) one real and two imaginary. A triple point
may be regarded as arising from the union of three double
points: viz. in the cases (a) these are (1) three crunodes, (2) two
crunodes and a cusp, (3) a crunode and two cusps; as illustrated
in the annexed figures, which exhibit the three double points
as they are about to unite
into atriple point. The @ (2)
case (3) scarcely differs
visibly from an ordinary
point on the curve, but
when the figure is drawn accurately there is a certain sharpness
of bend at the singular point. In the case (b), there is in like
manner a real branch which comes to pass through an acnode:
to the eye the singular point does not appear to differ from any
other point on the curve.

We may, in like manner, investizate the conditions that the
origin should be a aiultiple point of any higher degree (4).
The coefficients of all terms of a degree below & will vanish,
and the equation will be of the form

U, + u,,, + Ke. = 0.

(3)

28 MULTIPLE POINTS AND TANGENTS OF CURVES.

At the multiple point there can be drawn & tangents, represented
by the equation u,=0; and the nature of the multiple point
varies according as the roots of this equation are all real and
unequal, or two or more of them equal or imaginary.

A multiple point of the order & may be considered as
resulting from the union of $4(4—1) double points. This may
be illustrated by the case of & right lines, which must be
regarded as a system having 44(4—1) double points, namely,
- the mutual intersections of the lines. But if all the lines pass
through the same point, this is in the system a multiple point
of the order &, and takes the place of all the double points.
And the peindnle 4 is the same whether the lines which intersect
be straight or curved. A curve by the mutual crossing of
k; branches may have 4k(k—1) double points, but if all the
branches pass through the same point, these double points are
replaced by a multiple point of the order &, |

41. To be given that a particular point is a double point
of a curve is equivalent to three conditions. For if we take it
for the origin, three terms of the equation vanish (Art. 37),
and the constants at our disposal are three less than in the
general case. If we are further given the tangents at the
double point, this is equivalent to two conditions more; for in
addition to A =0, B=0, C=0, we are now also given the ratios
Pee D 2 f.

Being given a triple point is equivalent to six conditions 3
for, making it the origin, the six lowest terms of the equation
vanish; and so in general if it is given that a certain point is
a multiple point of the order 4, this is equivalent to $& (k +1)
conditions,

42, There is a limit to the number of double points which
a curve of the n” degree can possess, when it does not break
up into others of lower dimensions.

For example, a curve of the third degree cannot have two
double points; for if it had, the line joining them must be con-
sidered as meeting the curve in four points ;. but more than three
points of a curve of the third degree cannot lie on a right line,
unless the curve consist of this right line and a conic,

MULTIPLE POINTS AND TANGENTS OF CURVES. 29

Again, a curve of the fourth degree cannot have four double
points; for if it had, the conic determined by these and any
fifth point of the curve must be considered as meeting the curve
in nine* points; whereas no conic, distinct from the curve, can
meet it in more than 2 x 4 points. And, in general, a curve of
the n degree cannot have more than 4(n—1)(n—2) double
points; for if it had one more, through these 4 (mn — 1) (n—2)+1
and n—8 other points of the curve, we could describe a curve
of the degree n—2 (Art. 27), which must be considered as
meeting the given curve in 2 {4 (n—1) (n—2)+1}+n—83 points,
or in n(n—2)+1 points, which is impossible if the given curve
be a proper curve. Of course, the demonstration given only
shows that curves cannot have more than a certain number of
double points, and does not show (what in fact is the case)
that they can always have so many.

43. If the curve have multiple points of higher order, the
same criterion applies, each multiple point of order k being
counted as equivalent to $k(k—1) double points. But there
are limitations to the possibility of substituting for a certain
number of double points a multiple point of higher order.
Thus a curve of the fifth degree may have six double points,
and three of these may be replaced by a triple point; but
in this case the other three cannot be replaced by a second

* Tf a point of intersection of two curves be a double point on one of them, that
intersection must be reckoned as two, and the curves can only intersect in np — 2 other
points. If it be a double point on both, the intersection must be reckoned as four,
And in general if it be on the one curve a multiple point of the degree /, and on the
other of the degree 7, that intersection must be counted as //, Thus, for example, a
system of / right lines meets a system of / right lines in #/ points; but if all the lines
of the first system pass through a point on a line of the second system, that point
clearly counts as / intersections, and the lines intersect only in & (J ~~ 1) other points,
And if every line of both systems pass through the same point, that point counts as
kl intersections, and the lines meet nowhere else,

If two curves touch at their point of intersection, the point of contact will, of
course, count as two intersections, since they have two coincident points common,
If the point of intersection be a multiple point on one or both curves, and if one
of the tangents at the multiple point be common to both curves, we must add one
to the number of intersections to which it has been already shown that the multiple
point is equivalent; for, besides the points just proved to be common, they have a
consecutive point in common on one of the branches through the multiple point.

The reader will have no difficulty in seeing the effect of any combination of
tangents and multiple points.

30 MULTIPLE POINTS AND TANGENTS OF CURVES.

triple point, since the line joining the two would meet the
curve in more points than five. Or, generally, if a curve have
a multiple point of the order n —2, it can have no other higher
than a double point, and of these according to the criterion not
more than n — 2.

44. We call the deficiency of a curve the number D, by
which its number of double points is short of the maximum 3
this number playing a very important part in the theory of curves.
If D=0, that is, if a curve have tts maximum number of double
points, the coordinates of any point on the curve can be expressed
as rational algebraic functions of a variable parameter. For
the 4 (n—- 1) (n—2) double points, and n — 3 other assumed points
on the curve, making together 4 (n+1)(m—- 2)—1 points, or one
less than enough to determine a curve of degree n— 2, we can
describe through these points a system of such curves included
in the equation U=X~AV. Now if we eliminate either variable
between this equation and that of the given curve, we get
to determine the other coordinate for their points of intersection,
an equation of the n(n—2) degree in which 2 enters in the
n degree. But of this equation all the roots but one are
known; for the intersections of the curves consist of the double
points counted twice, of the n—3 assumed points, and only of
one other point, since

(n —1) (n—2)+(n—3)4+1=n(n—2).
Dividing out, then, the known factors of the equation, the only
unknown root remains determined as an algebraic function of
the ‘n degree in A.

It is true, conversely, that if the coordinates can be expressed
as rational functions of a parameter, the curve has the maximum
number of double points. Curves of this sort are called wnicursal
curves. When we are given @, y, 2 respectively proportional to
an” + &e., ar" 4+ &e., a’d" + &e., the actual elimination of A is
easily performed dialytically. Writing down the three equations

6x=ar"+&e., Oy=a'r" + &e., O2=a"r" 4+ &e.,
and multiplying each successively by A, d’,...A"", we shall have
8n equations, exactly enough to eliminate linearly all the
quantities 0, OA, &e., A, *, Ke. The equation of the curve,

MULTIPLE POINTS AND TANGENTS OF CURVES. 31

then, appears in the form of a determinant of the order 3n,
but only 2 rows will contain the variables; the curve therefore
will be of the x order, and its equation will involve the co-
efficients a, b, &c., in the 2n degree. All this will be more
clearly understood if we actually write down the result for the
case n=2. We have, then, the three equations

6x=anN+bN +c, Oy=aN+br4 Cc, Oz=a"N+b'X4C".
Multiplying each by 2X, and then eliminating linearly from the

six equations the quantities 0, OA, »*, A”, A, the result appears
as the determinant

, ! ?
Ys Bapbeyiey

2, a,b,c |=0.
_ This is the same as the final equation, Higher Algebra, Art. 193.

45. It appears from Art. 41, that any three points taken
arbitrarily may be double points on a curve of the fourth
degree; for the three are equivalent to but nine conditions.
But the tangents at all these double points cannot also be
assumed arbitrarily; for being given the three double points
and these three pairs of tangents is equivalent to fifteen con-
ditions, one more than enough to determine the curve. There
must then be some relation connecting these tangents; and in
fact, we shall prove afterwards that these six tangents all touch the
same conic section, so that, given five, the sixth is determined.

Twenty conditions determine a curve of the fifth degree.
We may then assume arbitrarily its six double points, and also
the pair of tangents at any one of them; but the curve is then
completely determined, and therefore also the pairs of tangents
at the other five.

‘T'wenty-seven conditions determine a curve of the sixth
degree. It would therefore, at first sight, appear that such
a curve might be described, having for double points nine points
assumed arbitrarily. But this is not so, for there is through
the nine points a determinate cubic curve U=0; and then

32 MULTIPLE POINTS AND TANGENTS OF CURVES.

a curve of the sixth order having the nine points for double
points, and in general the only such curve is U* =0, viz. the cubic
twice repeated.

And so in like manner for curves of higher degrees, when
they have their maximum, or even some number less than their
maximum, number of double points there must be relations
connecting them. Except in the case of curves of the fourth
degree, we are not aware that any attempt has been made to
express these relations geometrically, but there must remain an
extensive class of theorems of this nature still to be discovered.

46. What has been. said is sufficient to enable the reader to
form a conception of the nature of multiple points on curves.
We shall now proceed to show that a curve may in like manner
have multiple tangents; or, in other words, that there may be
lines which touch the curve in two or more points, or which
have with the curve a contactof the second or higher order.
What are commonly called the “singular points” of curves may .
be reduced to the two classes, either of multiple points, or of
points of contact of multiple tangents. As we introduced
multiple points to the reader by an examination of the particular
case where the origin was a multiple point, so it will be more
simple to commence our discussion of multiple tangents by
examining the condition that the axis (y=0) should be a
multiple tangent.

We find in general the points where this line meets the curve
by making y= 0 in the general equation, whence we get

A+ Bat Dx’ + Gu’ +...Px"=0,
an equation which can be reduced to the form
P («a —a) (x —b) (w-—c) (w- d) &. =0,
where a, 6, &c., are the values of z for the points where the
axis meets the curve.

The axis will be a tangent when two of these points coincide,
that is, when there is between the roots a single equality a=.
The equation here is

P (x— a)’ (w—c) &e. =0.
The axis then touches the curve at the pointy=0,2=a. If
A=0, B=0, the axis touches the curve at the origin. We

MULTIPLE POINTS AND TANGENTS OF CURVES. 3S

consider only the case a real, because the equation being real, an
equality a=} between two imaginary roots would imply another
equality c=d between two other imaginary roots. |

The axis is a double tangent if we have between the roots
two equalities c=a, d=; the equation is then

P (x — a)" (% — b)? (x —e) &e. = 0.

We have here the two cases

I. a and 0 each of them real, when the axis is a tangent |
at the two real points, z=a,x=06. It is evident that such a

tangent, meeting the curve in two pairs ¥ Wf
of coincident points, cannot occurinany ___ EX JS \
a c

curve of a degree lower than the fourth.

Il. a and d imaginary, viz., the equation is here
P(a* + pxt g)* (x —e) &e. =0,
and we have a double tangent with two imaginary points of
contact.

Again, we may have between the roots an equality a=b=c.
Here the equation is of the form, a being supposed real,

P (a — a)’ (x — d) &e. =0.

The axis then meets the eurve in three consecutive points.
In general, taking three consecutive points on a curve, the line
joining the first and second of these is a tangent, and the line
joining the second and third is the consecutive tangent. In
the present case, therefore, two consecutive tangents coincide.
Hence too, in such a case, the axis may be called a stationary
tangent ; for if we consider the curve as the envelope of a move-
_ able line, in this case two consecutive positions of the moveable
line coincide. The point of contact of a stationary tangent is
called a point of inflexion.

_ If 4=0, B=0, D=0, the origin is a Me 6

point of inflexion, and y= 0 the tangent at it,

since then the equation is of the form oN
Px’ («@—c) &e. = 0.

47. ‘The crunode and acnode (Art. 38) correspond precisely
to the double tangent with real contacts and the double tangent
FEF

34 MULTIPLE POINTS AND TANGENTS OF CURYES.

with imaginary contacts; the cusp or stationary point also
corresponds precisely with the stationary tangent. But there
is no correspondence in the analytical theories; for the cusp we
have an equality a=, which is a particular case of the unequal
values (a, 6), which belong to the crunode and to the acnode;
for the inflexion we have a double equality a=b=c, which is
a relation distinct in kind from the equalities a= 0, c= d, which
belong to the double tangent with real or imaginary contacts.
The double point was discussed with point-coordinates; to make
the analytical theories agree, the double tangent should have
been discussed with line-coordinates—the stationary tangent
would then have presented itself as a particular case of the
double tangent. But in what precedes the stationary tangent
presents itself as a distinct singularity from the double tangent :
so with lime-coordinates the cusp would have presented itself as a
distinct singularity from the double point; and in reference
hereto the remark was made, Art 38, that the cusp was really
a distinct singularity. The singularities then mutually corre-
spond as follows =:

To a double point or nede A double tangent (contacts,
(crunode or acnede), real or imaginary),

To a cusp, spinode, or sta- A stationary tangent, or tan-
tionary point, gent at inflexion;

and it is only in @ certain point of view that the cusp is a
particular case of the double point, and in a different point of
view (the reciprocal one) that the stationary tangent is a parti-
cular case of the double tangent.

Considering the curve as described by a point which moves
along a line at the same time that the line revolves round the
point: there is at the cusp a real peculiarity in the motion, the
point first becomes stationary, and then reverses the sense of
its motion; and so at the inflexion, the line first becomes
stationary and then reverses the sense of its motion. At a
double point there is no peculiarity in the motion, all that
happens is that the point in its course comes twice into the
same position; and so, for the double tangent, there is no
peculiarity in the motion; all that happens is, that the line in
its course comes twice into the same position. The cusp and

MULTIPLE POINTS AND TANGENTS OF CURVES. 85

stationary tangent are singularities in a more precise sense than
are the double point and the double tangent.

48. In ordinary cases the curve lies altogether at the same
side of the tangent, but at a point of inflexion the curve crosses
the tangent, and lies part on one side and part on the other.

This is a particular case of the following more general
theorem: Zwo curves which have common an even number of
consecutive points touch without cutting ; those which have common
an odd number of consecutive at cross one another at their
point of meeting.

Let the equations of the two curves be y=¢a, y= Wa; let
them intersect at the point a=a; then, by Taylor’s theorem,
the values of the ordinates of the two curves, for the point
x=a+h, are

dph. dd h
sigh eas 1.2

dph dy h’
hae a har 7 1.2

ip i
dx 1.2.3
dnp h’
dx 1.2.3

+ + &e.

+ —— + &e.

dbx

where ¢, ¥, .#, &e., are the values of da, Wa, ses &e.,

when «=a. Now, by hypothesis, ¢ =, since the curves inter-
sect at the point «=a; therefore .
dp _dp\h (dp adp\h (do ayp\ bh’

IGu= (= - Si (ga dx’ his (s3- iss ag t he
Now, by the principles of the differential calculus, when / is in-
definitely small, the sign of the sum of this series is the same as
the sign of its first term, but the sign of this term is changed
when the sign of hf is changed; therefore, if at the infinitely
near point (v=a+h), the ordinate of the curve ¢ be greater
than that of the curve y, it will be less at the point (2=a—A).
Hence if two curves have ene point common, in general, that
which is uppermost at one side of the point will be undermost
at the other.

But now suppose that oe ou the first term of the series
ap dp h’ :
will then be (S- 4 oe which does noé change sign

when / changes sign. The same curve, therefore, which is

36 MULTIPLE POINTS AND* TANGENTS OF CURVES.

uppermost on one side of the given point, will be uppermost also

on the other. But when -. — the curves are manifestly

fe: de ‘
closer to each other than in the previous case, since the difference
of the ordinates no longer involves the first power of h; which
is equivalent to what is expressed geometrically, by saying that
the curves have two consecutive points common. Or the same

s)="02

thing may be shown thus: a'y’, 2"y" being the coordinates to

au

U
rectangular axes of any two points on a curve, ‘ —a is plainly

the tangent of the angle which the chord joining them makes
with the axis of x; but if the points coincide, we learn that

the value o ey or the given potnt expresses the tangent of the
me given p ip J

angle which the line joining it to the consecutive point (i.e. the
tangent) makes with the axis of x; consequently, if two curves

Z for that point the same for both

curves, it follows that the consecutive point is also common.

have a point common, and

49, When the curves have three consecutive points common,

we shall have - = ae ; the first term of the series for y, — y,
dh ad’ ia
s 5S - ae ay. which does change its sign with , and

therefore, as before, the curves cross at the given point. And
so, in general, if the expansion of y,—y,, commence with an
even power of h, it will not change sign with A, and therefore
the curves touch without crossing; but if it commence with an
odd power of h, the sign will change with , and therefore the
curves cross at the given point.

The reader has already had an illustration of this, in the case
of the circle which osculates a conic at any point, ang which, in
general, having three points common with the curve, touches
and crosses the curye (Conics, Art. 239); but at the extremities
of the axes the osculating circle passes through four consecutive
points, and touches without crossing.

‘The same investigation applies when one of the curves
becomes. a right line. A tangent, therefore, at a point of in-

MULTIPLE POINTS AND TANGENTS OF CURVES. 37

flexion, or any line meeting the curve in an odd number of
consecutive points, is crossed by the curve; but a tangent which
meets the curve in an even number of consecutive points has
the neighbouring part of the curve all at the same side of it.

50. The axis y=0 will be a triple tangent when the equa-
tion which determines the points where it meets the curve is
of. the form 3
P (x- a)’ (x —b)’ (x—c)? (x— d) &e. = 0.

It is evident such a tangent cannot occur in a curve of any
degree lower than the sixth. We may, as in Art. 40, dis-
tinguish four species of triple tangents according as the points
of contact are real and distinct, one real and two imaginary,
one real and two coincident, or all three coincident. The last
will be the case when the equation is of the form
P(a—a)*(w—b)&e.=0;
and the axis meets the curve in four coincident points: the point
of contact of such a tangent is called a point of undulation. In
like manner there may be multiple tangents of still higher
orders, or again, points of undulation of higher orders, arising
when a line meets the curve in more than four coincident points.
Cramer calls those points at which the tangent meets the curve
in an odd number of consecutive points, points of visible inflewion,
to distinguish them from those points de serpentement, or points
of undulation, which do not, to the eye, differ from ordinary
points on the curve.

51. We have hitherto only illustrated the case where the
origin is a multiple point, or one of the axes a multiple tangent ;
it is evident, however, that the form of the equation might, in
like manner, show the existence of multiple points and tangents
situated anywhere.

I. For instance, if the equation be of the form
ap + By =0, :
where a, 8 are linear functions of the coordinates, and ¢$, >

are any functions of the coordinates, then a8 is one point on the
curve. ‘The equation of the tangent at this point is

ap’ + Py’ =0,

38 MULTIPLE POINTS AND TANGENTS OF CURVES.

where ¢', y’ are the values which ¢ and yr assume when we
introduce the conditions a=0, 8@=0. For if we seek the n—1
points, in which any line through «8, («=8) meets the curve,
we get an equation of the form

B {k (¢'+ MB + NB’ + &.) + (v' + MB + N'B’ + &e.)} =0;

and in order that a second root of this should be 8=0, we must

have kf’ ++’ =0; whence, substituting for & its value B? we
get for the equation ef the tangent

ap’ + By’ =0.

II. In general the curve represented by
aByd &e. =a,8 75, &e.
passes through the points

aa, a8, ay, &c., Ba, BB, By,, &e., ya, 8, vy, &e.

III. If the equation be of the form
ap + Bp =0,

we see (as at Conics, Art. 252), that a is the tangent at the point
a8, for two of the points in which this line meets the curve
coincide.
Or again, if the curve be
ttt,...t, + B’p=0,
t,, &c. are the tangents at the n points, where 8 meets the curve.
The form of the equation shows that ¢f the points of contact of

a tangents lie on a right line B, the remaining points where these
tangents meet the curve lie on the curve of the (n—2)™ degree ¢.

IV. If the equation be of the form
arp + aby + Bx =0,
and if we seck the points where any line («=£8) through «8 meets
the curve, we find that two of these always coincide with a8,
and therefore that this is a double point. It appears precisely as
in I., and in Art. 37, that the tangents at this double point are
ag + aByp'+ Bty'=0,
where ¢', wy’, x’ are the values which these functions take for
the coordinates of the point a=0, 8B =0.

MULTIPLE POINTS AND TANGENTS OF CURVES. 39

V. So again, if the equation be of the form
aedt+aBy+ab y+ B’o=0,
the point @@ is a triple point; the three tangents being given ty
the equation
ad’ +a°Ry’ + a6’y'+ Bw’ =0.
VI. If the equation be of the form
ap + By = 0,
a is a double tangent at the points a8, ay.

VII. If the equation be of the form
ap + BY =

a8 is a point of inflexion, and a the tangent at it.

52. We shall first illustrate the last Article by showing how
the equation enables us to discern the nature of the points of
the curve at an infinite distance. ‘The trilinear equation is
(Art. 26)

U,+U, 2+ U, + Ke. =0.

n-1
Writing herein z= 0, the directions of the n points at infinity
are found from the equation vu, =0, which, solved for y : a, is of
the form
(y — m,x) (y — me) (y — mx) (&e.) (y — mx) = 0.

A curve of the n degree has, in general, n asymptotes, namely,
the tangents at the points, where z, the line at infinity, meets
the curve. We can find their equations readily as follows, when
the equation w,=0 has been solved for y: #. It appears, from
III. of the last Article, that if the equation were reduced to
the form

ti +2°¢=0,
t,, &c. would be the be cent But the given equation
(y — m,x) (y — mx) &e. + zu,_, + 2°u,_, + Ke. =0

may always be reduced to the form

(y —m,x + 2,2) (y — mw + A,2) Ke. = 2h;
for the terms of the n“* degree in x and y are obviously the same
for both equations, and the » arbitraries, X,, &c., in the second,

can be so determined as to make the nm terms of the (n—1)™
degree the same for both equations.

40 MULTIPLE POINTS AND TANGENTS OF CURVES.

The reader will have no difficulty in understanding this method,
if he tries to apply it to a particular example; for instance,

(a+ y) (Qe4+ y) (Buty) + 17x’ + lay + 2y? + 12% 4 10y + 36 =0,
which it is desired to throw into the form
(e+ y+r,) (Qa+y4+X,) (8a+y+A,)+ Ax+ By +C=0.
To determine 2,, A,, X, we should then have the three equations
6A, + 5A, + 2A,=17, 5A, +40, + 3A, =11, AFA, +A, =23
and the equation may be reduced to the form
(a+ y+ 4) (Qe+y—3) (8a+ y¥+1)+43x421y+48=0.

Observe that the values ,, A,, >, are such that we have
identically

17x? + llxy + 27? r r r

1 2 3

(e@t+y) Quty) Quty) ety Bet+y Baty’

and so in general the values 2,, X,,... are determined by decom-
posing w,_,+, into its simple fractions.

53. If two roots of the equation u,=0 be equal (m,=™,),
the general equation takes the form (y—m,x)’ $+ 2p~=0; two
of the points where z meets the curve coincide, and the line at
infinity is therefore a tangent to the curve. But if the factor
y—m,x is also a factor in w,,, then the curve has a double
point at infinity; for the equation is of the form

(y—m,x) o6+2(y—m wx) pte'y=0.

Should three roots of the equation u,=0 be equal, the line
at infinity meets the curve in three coincident points, and there-
fore touches at a point of inflexion.

If in the general equation the coefficient of y" be = 0, the axis
of y passes through a point at infinity, and we have evidently
only an equation of the (n--1) degree to determine the re-
maining points where it meets the curve.

Should the coefficient of y"* also vanish, the axis of y will be
an asymptote. :

54. We shall in a future section show how the singular
points of a curve may, in general, be found. But the application
of the general methods being usually a work of some difficulty,

MULTIPLE POINTS AND TANGENTS OF CURVES. 41

the examples given in works on the differential calculus are, for
the most part, cases where the existence of the singular points
more readily appears from mere inspection of the equations; a
selection, including all the most difficult of these examples,
will therefore serve to illustrate the preceding Articles. (See
Gregory’s Examples, p. 170, &c.)

Ex. 1. z*— aaty + bys = 0.

Ex. 2, xt — 2ax?y + 2a7y? + ayS + yt = 0.

In both cases the origin is a triple point. The tangents of the first are given by

the equation ax’y = by*; and of the second by the equation 2a%y=y3, By Art. 43
neither curve can have any other multiple point.

Ex. 3. ay? — 2° + bz? = 0.

The origin is a double point, whose tangents are given by the equation ay?+bz2?=0,

If the sign be given positive, the origin is a conjugate point.

Ex. 4. (a? — a?)? = ay? (2y + 8a), or (x — a)? (2 + a)? = ay? (2y + 3a).

Here evidently (« —a, y) and (a+, y) are double points. To get the tangents
at the first, we are to make «=a, y=0 in the parts which multiply (x — a)?, y’,

and we get
4 (aw — a)? = 3y?,

In like manner for the tangents at the other double point,

4 (@ + a)? = 3y?.
The curve has a third double point, whose existence can be shown by throwing the
equation into the form
x? (a? — 2a”) =a (2y — a) (y + a).
Hence, (w, y + a) is a double point, and the tangents at it are

2x? = 3 (y + a).
Having found these three, we know, by Art. 42, that the curve can have no other
multiple point.

Ex. 5. (dy — cx)? = (4 — a)5.

The point (dy — cx, x — a) is a cusp of such a nature that the tangent at it meets
the curve in five consecutive points,

Ex. 6. x (« + 6) = aty?,

The origin is a double point, the tangent at which meets the curve in four
consecutive points. There is a triple point at infinity, to which the line at infinity is
the only tangent. The line «+0 touches the curve where it meets the axis of 2,
and also at a point of inflexion at infinity.

Ex. 7, ike 0,

This equation, cleared of radicals, becomes

(a? + y? + 27)3 = 27ar7y"2? ;
and in this form the existence of six cusps is manifest, for each of the points where
x meets y?+2? is a double point, and z the only tangent at it. Similarly for
(y, x? + 2”) and (z, x? + y*). But the cusps are all imaginary.
The curve has also four double points, viz. (t#+y=0, «+2=0).
This can be proved by putting yf «=u, 2x =v; and therefore
Y¥=Ute, z=vtisz.

G

42 TRACING OF CURVES.

Substituting these values in the given equation, it is of the form
wp + uv + vx.

The tangents at any of the double points will be found to be given by the equation

w+tuv+v?=0,
and therefore the double points in question are conjugate points; and, in fact, these
are the only real points of the curve.

Or again, the equation may be written
9a? {at — a (y? + 27) + yt — ye? + a — (20? — y? — 27/8 = 0,7

which is one of three like forms, viz. writing &, , ¢ = y* — 2’, 2? — a, w — y?, the
form is 9x? (yn? + nf + ¢?) —(n— 0% = 0; putting in evidence the double points y = 0,
¢=0; or, what is the same thing, £=0, n=0, ¢=0, that is, 2? = y? = 2.

SECT. HI.—TRACING OF CURVES.

55. It is proper to give some examples of the method of
tracing the figure of a curve from its equation. If we give any
value (a) to either of the variables x, the resulting numerical
equation can be solved (at least approximately) for y, and will
determine the points in which the line 2=a meets the curve,
By repeating this process for different values of x, as at Conics,
Art. 16, we can obtain a number of points on the curve; and,
by drawing a line freely through them, can obtain a good idea
of its figure. By taking notice what values of x render any
of the values of y imaginary, we can perceive the existence of
ovals, or can observe whether the curve is limited mm any
direction; and we have already shown (Art. 52) how to find
whether the curve has infinite branches, and how to determine
its asymptotes. It will be shewn in the next section how to find
its multiple points and points of inflexion. The value of =
at any point gives the direction of the tangent at that point
(Art. 48); and if we examine for what points “d <0, or =o
we shall have the points at which the course of the curve is
parallel or perpendicular to the axis of a.

In practice we must, of course, take advantage of any
simplifications which the equation of the curve suggests. Thus,
if we consider a series of lines parallel to one of the asymptotes
(or a series of lines passing through a point on the curve), the
equation which determines the other points in which each of
them meets the curve is of a degree one lower than the degree
of the curve. If the equation shows that the curve has a double

?

TRACING OF CURVES. 43

or other multiple point, it is advantageous to consider a series
of lines drawn through this point, since then the equation in
question will lose two or more dimensions.

There is scarcely any exercise more instructive for a student
than the tracing of curves, and more particularly those in which
the equation contains one or more parameters which assume a
succession of different values. In the case of a single parameter,
this may be conceived of as an ordinate z in the third dimension
of space, and the problem thus, in effect, is to find the form of
the several parallel sections of a surface.

It will suffice to add a few examples to those which will
incidentally occur in the course of these pages. We refer
the reader who may wish for further illustration, to Gregory’s
Examples, Chap. X1.; or, if still unsatisfied, to the source
whence all later writers on the subject have drawn largely.
Cramer’s Introduction to the Analysis of Curves.

Ex. 1. xt — axy + by3 = 0 (see Ex. 1, p. 41).

Here, the origin being a triple point, it is advan-
tageous to consider a series of lines drawn through it.
Substituting y= ma, we find a*= m (a — bm?), a func-
tion which, as m passes from 0 to + , increases from 0,
when m = 0, toa maximum value when a — 3mb?=0;
then decreases, and vanishes when a — bm? = 0, and has
an indefinitely increasing negative value as m increases
further. The curve is manifestly symmetrical in re-
gard to the axis of y. Hence the figure is that here
represented.

Hx. 2. (a? — a*)? = ay? (3a + 2y), (see Ex. 4, p. 41),

Hence x? = a? + J{ay? (83a + 2y)}. The curve is plainly symmetrical in regard to
the axis of y. It has on each side two branches, corresponding to the two signs
which may be given to the radical. The two branches intersect when y= 0, and ac-
cordingly we have seen that there are on the axis of x two double points at the distance
x=+a. As y increases positively, the radical increases indefinitely ; hence the value
of. x, corresponding to the one branch, increases
indefinitely ; that corresponding to the other de-
creases, until we? come to the value of y corre- +
sponding to the single positive root of the equation ig

2ay® + 3a*y? = at, (2y=a), beyond which this

branch can extend no higher. For negative values
of y, the radical increases to a maximum value

when y+a=0; the one pair of branches then
intersect in a double point on the axis of y, and
the other pair is at its furthest distance from that
axis. Evidently neither branch can proceed lower

44 TRACING OF CURVES.

than the value 8a+2y=0. Hence the shape of the curve is that represented in
the figure,

Ex. 3. Given base of a triangle 2¢ and rectangle under sides m?, the locus of vertex
is Cassini’s oval, whose equation is, the origin
being the middle point of base,

(x? + 9? — 7)? — 4c?x? = m4,

The accompanying diagram represents the
figure for different values of m. The dark
curve represents the figure for m = c, the curve
being then known as the lemniscate of Ber-
nouilli. When m is less than ¢, Cassini’s curve
consists of two conjugate ovals within the parts of this figure: when m is greater
than c, of one continuous oval outside it.

Ex, 4. On the radius vector from a fixed point O to a fixed line MN a portion
RP of given length is taken on either side of the right line. The locus of P is a
curve called the conchoid of Nicomedes, invented by that geometer for the solution
of the problem of finding two mean proportionals,

If OA=p, RP =m, the polar equation is (9 +m) cosw =p, and the rectangular
equation

my’ = (p — y)? (x +9"),

The line MN (p=y) touches at a singular point at infinity, and there meets the
curve in four consecutive points.
The point 0 is also a double point, the tangents at which are given by the equations

pat + (p? — m?) y= 0.

It will therefore be a node, conjugate point, or cusp, according as m is greater, less
than, or equal to py. The continuous line represents the case when m is greater than
p; the dotted line that when m is less than p.

Ex, 5. In like manner on the radius vector to a fixed circle from a fixed point on it
a portion of fixed length is taken on either side of the circle. The curve is called
Pascals limagon. The polar equation is p=pcosw+m; and the rectangular
(x? + y* — px)* = m?* (x + y”). The origin is evidently a double point and is a node
or conjugate point according as p is greater or less than m. When p =m, the origin
is a cusp, and the curve is of the form of a heart, and is called the cardioide, This
is represented by the dark curve in the figure, the inner and outer curves repre-

_ senting the forms with a node and with a conjugate point respectively.

TRACING OF CURVES. 45

go al OIE Nee de
te

Ex. 6. (x? — a?)? + (y? — b*)?= ct, where 4 is supposed less thana. When ¢=0,
the curve consists of the four conjugate points +a, +0. The figures represent the
cases, (1) ¢ less than b, (2) c=}, (8) ¢ intermediate between 6 and a, (4) c=a,
(5) c>a, < (at +04), (6) c= 4{(a* + 54). When ¢ has a greater value, the curve
is of similar form, but without the conjugate point at the origin. Whenc=a=6,

the double points of (2) and (4) present themselves simultaneously, and the curve in
fact breaks up into two ellipses as in (7).

(1)

det) ke

ae
Aw akS,
4G.)

46 TRACING OF CURVES.

oh 7 ee

ees Ge.
> =
&

6
(5) (6)

56. If a curve pass through the origin, then if this be an
ordinary point on the curve, y may be developed in the form
y=Ax+ Bx’ +...; when the origin is a singular point, the form is
y= Aa*+ Bx? + &e., where @ is positive and @ and all the indices
which follow are greater than a; it is for determining the nature
of the singular point, and the form of the curve in its neighbour-
hood, very convenient to find even the first term of this develop-
ment; in fact, in the neighbourhood of the origin the figure
resembles that of the curve y= Az*, which can easily be con-
structed. In order to effect such a development, we can employ
the process given by Newton,* which is most conveniently
used in the following form. Write in the equation y = Ax’, and
determine the positive quantity a by the condition that the
indices of two or more terms shall be equal, and less than the
index of any other of the terms. This can always be done
by trial, by equating the indices of each pair of terms, and
observing whether the resulting value of a is positive, and
the equal indices not greater than the indices of some other
term. Having thus found a, we determine A by equating to
zero the quantity multiplying the terms with equal index.

* See Methodus Fluxionum et Serierum infinitarum, Gc., under the heading
De reductione affectarum equationum (Opusc. ed. Castillon, vol. 1. p. 37). See also
a paper by Professor De Morgan, Quarterly Journal, vol. 1, p. 1, and Transactions
of the Cambridge Philosophical Society, vol. 1X., p. 608. Newton gives the rule
by means of a diagram of squares, in a form different from that given above,

TRACING OF CURVES. 47

We can then carry on the expansion by substituting y=Aw*+ Bx’,
where A and a have the values already found; and § and B are
determined, if need be, by a similar process; but it usually
happens that after the first term or terms the indices will
proceed in a regular order, and the coefficients will be each of
them linearly determined. Thus, for example, let the curve be
a + y° — 3axy =0; where the origin is a double point having the
two axes for tangents; then, writing y=d* the equation becomes
a + A’a™* —3aAa*™ = 0.

We are now to make two indices equal. Trying first 3 = 3a,
or a=1, we reject this value because it makes the equal indices
greater than the index «+1 of the other term. ‘Trying next
3=a+1, or a=2, we find that this value will make the equal
indices less than that of the third term. The equation will
become (1 — 3aA) a*°+ A’x®=0, and determining A so as to
make the coefficient of a* vanish, we see that the equation may

be expressed in the form y= 52 + &e, where the indices of

the remaining terms are greater than 2; and we learn that the
form of one branch of the curve at the origin resembles that
of the parabola 3ay=2*. And in the third place equating
the indices 3a, a+1, we find a=}. Here again, the equal
indices are the lowest and the coefficients of the two terms are
A*, —3aA, whence A =,/(3a), and the branch is y=,/(3a)a*+ &e.,
wherefore near the origin the form approaches to that of the
parabola 7*=3ax. It is not necessary for our present purpose,
but if we desire to continue the expansion we should substitute

, ees
aero xz’ + Bx*®. The lowest terms would then be

: -s a® 4+ on xt? — 3a Bue" = 0.
We can then make the indices of two terms equal, and lower
than the remaining one, by making 8=5, whence Baa.
We have shown, then, that if we trace in the y
neighbourhood of the origin the two parabolas Ne V o

3ay = 2°, y= 3ax, we have approximately the :
figure in that neighbourhood of the curve we wish \
to construct.

48 TRACING OF CURVES.

57. The same process will lead to a determination of the
infinite branches of the curve. We must then expand y in
descending powers of x, and the only dif-
ference in the process is that we now make AQ
the equal indices greater than that of any ‘
other term. Thus, in the example already a
given, equating the indices 3, 3a, we have
a=1, and their coefficient A*+1. Attending Nx
only to the real value for A (=—1) we sub- |
stitute y=—a#+Bex*, and find in like manner B=0, B=- a.
We thus get the expression y=—a2—a+&c., and we see that
the line ~+y+a=0 is an asymptote. The figure is as in the
diagram.

58. In the case of the simple cusp of which we have had an
example, see Art. 39, the two branches which meet at the cusp
lie on opposite sides of the common tangent, and have their
convexities opposed to each other; but there is a cusp (which
is a singularity of higher order) in which the branches lie on
the same side of the tangent. ‘Thus, in the curve m(ay—2’)*=2°,

it is plain that any positive values of # give real values for y;
5

and if we write the equation in the form ay =x" + = , then since

the last term is less than the preceding when « is small, we see
that, whether we use the upper

or lower sign, the value of y will

be positive for small values of x.

The axis of a, then, is a tangent

and both branches lie on the | ee ee
upper side of it. The figure is ~~
as here represented. These two kinds of cusps have been
called keratoid and ramphoid from a fancied resemblance to the
forms of a horn and a beak. We have seen (p. 27) that
ordinary multiple points of higher order may be regarded as
resulting from the union of a number of double points. Professor
Cayley has shewn (Quarterly Journal, vol. vit. p. 212) that
any higher singularity whatever may be considered as
equivalent to a certain number of the simple singularities, the
node, the ordinary cusp, the double tangent, and the in-

POLES AND POLARS. AY

flexion. Thus, a cusp of the kind described in this article is
equivalent to one node,
one cusp, one double
tangent, and one inflex-
ion, as will appear from
the annexed figure which
exhibits the node and
cusp on the point of uniting themselves into the higher sin-
gularity in question.

SECT. IV.—POLES AND POLARS.

59. The method that we shall presently use in investigating
the conditions that a curve should have multiple points or
tangents, and in ascertaining their position, is the same as that
already employed in the case of the origin.. We shall consider
a series of radius vectors drawn through :a given point; we
shall form the equation which determines the coordinates of
the n points where any such radius vector meets the curve, and
we shall examine the conditions that one or more of these
points may coincide with the given point itself. In order to
determine the coordinates of these nm points we shall use
Joachimsthal’s method explained Contcs, Art. 290. Since the
trilinear coordinates of any point on the line joining two points
x'y'z', x'y"2" are of the form Ax’ + pa", rAy'+ py", Az’ + 2",
the points where the joining line meets any curve are found
by substituting these values for a, y, 2, and then determining the
ratio X: w by the resulting equation. And it will be a necessary
preliminary to the following investigation to discuss carefully
the functions which present themselves in this substitution.

If then in U, which is a homogeneous function of the n™ order
in 2X, ¥, 2, we substitute Aw + pa’, Ay+ wy’, AZ+ pz’ for a, y, Zz,
it is evident by T'aylor’s theorem that the coefficient of X” will
be U, and that of A” will be
om aU A aU | 2! dU

da 7 dy dz
using the abbreviations U,, U,, U, or L, M, N (as the case may
be) for the differential coefficients. We shall use the symbol A

H

,or £U,4+7'U,4+2'U, or a’ L + y'M+2'N,

50 POLES AND POLARS.

to denote the operation pass yrree, and the coefficient
of A" may thus be written AU. In like manner the coeffi-
cient of X”*y? will be half
Bau, @U. 80, EU Sy Ag ee
dat | apg ** da * Y* oyde* #2 Tede "4 dady’
which may be written
j 2

(o£ ty +25) OT or A’U.
The second differential coefficients are often written with double
suffixes U, U,,, U,,, U,. U,, U,., but we find it more con-
venient to use the letters, a, b,c, f,g,h, and so to write A*°U
in the form we have used in expressing the general equation
of a conic

ax” + by’ + ca? + 2fy2 + 2gux + 2hxy.

In like manner the coefficient of ’”*y* in the expansion is
1 eee
1.2.3 List jes
It is evident however from the symmetry of the substitution
that this coefficient will be U’, and in general, that the co-
efficients of any two corresponding terms A“y’, A’u", only differ
by an interchange of accented and unaccented letters. We
see thus that A”’U only differs by a numerical factor from

xU',+yU',+2U’,, and generally that
np p
(woty 5+" 5) U, (eBtyptes) sae
only differ by a numerical factor, We may write the last
function A’U’, the accent on the U serving to mark the inter-

change of accented and unaccented letters.

A*U, and so on; the last coefficient being

60. The curve of the (n— 1) degree AU=0 is called the
first polar of the point 2’y’z’, with respect to U. In like
manner A’?U=0 is called the second polar, and so on, the
degrees of the successive polar curves regularly diminishing by
one, the (n—2)™ polar being a conic, and the (n—1)™ a right
line. And, from the remark just made, it is plain that the
equations of the polar line and conic are respectively

i es a a

POLES AND POLARS. 51

Since A’U is obtained by performing the operation A upon
AU, it is plain that the second polar of 2'y'z', with respect to U,
is the first polar of the same point with respect to AU; and
generally that the polar curve of any rank is also a polar of the
same point with respect to all polar curves of a rank lower than
its own; as is.evident from the equation A‘ (A'U) = A*'U.

For the origin, for which 2 and y' vanish, the operation
A reduces to differentiating with respect to z. If the ordinary
Cartesian equation be made homogeneous by the introduction
of the linear unit z (Conics, Art. 69), it may be written

ue +42 +u,2"*?+&c.=0,

and we find without difficulty, by differei:tiating with respect to 2,
that the equations of the polar line, conic, &c. of the origin are

nue+u,=0, $n(n—1)u,2°+(n—l)uztu,=0, &e.

61. The locus of all the points whose polar lines pass through
a given point is the first polar of that point.

The equation 2U'+yU;+2U,,=0 expresses a relation
between xyz the coordinates of any point on the polar line,
and a'y'z' those of the pole. And, as in Conécs, Art. 89, we
indicate that the former coordinates are known and the latter
variable, by accentuating the former and removing the accent
from the latter coordinates, when the equation becomes
x U,+y'U,+2'U,=0. There are (n- 1)’ points, whose polar
lines with respect to U will coincide with any given line, or,
more briefly, every right line has (n—1)* poles. For take any
two points on it, the poles of the right line must lie on the
first polar of each of these points; therefore they are ‘the
intersections of these curves. Also the first polars of all the
points of a right line have (n-—1)’ common points, viz. the (n— 1)"
poles of the right line.

In like manner, the locus of points whose polar conics
pass through a given point is the second polar of the point;
and so on.

If the polar line (or any other polar) of a point pass through
the point, that point will be on the curve. for if we substi-
tute wy'z’ for xyz in the equation of the polar, it becomes
identical with the equation of the curve, since the operation

52 THEORY OF MULTIPLE POINTS AND TANGENTS.

| d
one LA pa ; ]
ty hy gaan performed on a homogeneous function only

affects it with a numerical factor.

62. [fa curve have a multiple point of the order k, that point
will be a multiple point of the order k—1 on every first polar,
of the order k—2 on every second polar, and so on. For if the
origin be at the multiple point, the lowest terms in a and ¥
will be of the degree £3; in the first polar, which involves only
first differentials of U, the lowest terms in 2 and y will be of
the degree £- 1, and therefore the origin will be a multiple
point of that order; the equation of the second polar, involving
second differentials of U, will contain x and y at lowest in the
degree /— 2, and so on.

If two tangents at the multiple point in the curve coincide,
the coincident tangent will be a tangent to the first polar.
For the lowest term u, is of the form a’bcd..., where a, b,...
represent linear functions of the coordinates, and hence its
differentials will contain @ as a factor, and therefore the
lowest terms in the equation of the polar contain a as a factor.
And, in general, if 7 tangents to the multiple point on the:
curve coincide, 7—1 of them will be coincident tangents at
the multiple point on the first polar, 7—2 at the multiple point
on the second polar, and so on. For if u, have any factor
in the 7 degree, that factor will be one of the (7— 1) degree
in all the first differentials of u,; of the (c— 2) in all the
second differentials, &c.

SECT. V.—GENERAL THEORY OF MULTIPLE POINTS AND
TANGENTS.

63. We proceed now to apply the method indicated in
Art. 59 to the investigation of the multiple points and tangents
of curves. In order to find where the line joining the points
xy'z', xy'z' meets the curve, we substitute in the equation
Aza’ + pa" for x, &e., and we get in order to determine the ratio
X : #, an equation which we may refer to as A=0, and whigh
may be written

AU +r" tpAU' +4r" WAU + &e. = 0,

it being supposed that in AU’, &c., as previously written, v"y"2”"

THEORY OF MULTIPLE POINTS AND TANGENTS. 53

have been substituted for xyz. In order that one of the points
Aa’ + po", ry’ + py", Az’ + we" should coincide with a'y’z’, it is
obviously necessary that one of the roots of the equation A=0
should be »=0. But this clearly will not be the case unless
U'=0; and it is otherwise evident that the condition that
x'y'z' should be on the curve is, that its coordinates substituted
in the equation of the curve should satisfy it.

64. Two of the points in which the line meets the curve
will coincide with z'y'z’, if the above equation be divisible by
pb’; that is, if not only U'=0 but also AU’=0: now it is plain

FES RE < 88

that if the line joining 2'y'z' a point on the curve to 2”y"z" meet
the curve in two points which coincide with a'y'z’, then ay'z"
must lie on the tangent (or tangents if more than one) which can

f fi-}

be drawn to the curve at 2’y'z': but we have now proved that in
this case a''y"z" must satisfy the equation «U,'+ yU+2U=0.
Hence, in general, at a given point on the curve there is but
one tangent, whose equation is that just written. It appears
thus that the polar line of a point on the curve ts the tangent.

All the other polar curves of the point x'y'z' will touch the
curve at that point. For it was proved (Art. 60) that the polar
line with respect to the curve U will also be the polar line
with respect to each of the polar curves; and (Art. 61) the
coordinates «'y'z' satisfy the equation of each of the polar
curves; and therefore, by what has been just proved, the polar
line with respect to any of them will coincide with the tangent.

65. The points of contact of tangents drawn to a curve from
any point lie on the first polar of that point. This is a particular
case of what was proved in Art. 61, or it may be established
directly in the same way. ‘The equation of the tangent at the
point 2'y'z’ having been shewn to be eU,'+ yU,'+2U,' =0, then
by an interchange of accented and unaccented letters we in-
dicate that the coordinates of a point on the tangent are sup-
posed to be known, and those of the point of contact unknown;
and we see that the latter coordinates must satisfy the equation
a U,+y'U,+2'U,=0. The curve and its first polar clearly
intersect in n(n—1) points, and since at each of these inter-
sections U=0, AU=0 will be satisfied, we see that from a

54 THEORY OF MULTIPLE POINTS AND TANGENTS.

given point there can be drawn n(n—1) tangents to a curve of
the n degree. Or, again, (Conics, Art. 303) the degree of the
reciprocal of a curve of the n™ degree is in general n(n—1).

66. If, however, the curve have a double point, it was
proved (Art. 62) that the first polar of any given point must
pass through that double point. The double point, therefore
(see note, p. 29), counts for two among the intersections of the
curve with its first polar. But the line joining the point a’y"z”
to the double point is not a tangent in the ordinary sense of
the word, though it is indeed included among the solutions to
the problem we have been discussing (viz., to draw a line
through zy"z", so as to meet the curve in two coincident
points); for we have shewn that every line through the double
point must be considered as there meeting the curve in two
coincident points. Now the entire number of solutions to this
problem being always x” (n—1) (viz., the intersections of U and
AU), the number of tangents, properly so called, which can be
drawn to the curve is diminished by two for every double point
on the curve; or the degree of the reciprocal of a curve of the
a4 degree having 6 double points is n (n — 1) — 26.

67. Ifthe curve have a cusp, we have proved (Art. 62) that
the first polar not only passes through the cusp, but also has its
tangent the same with the tangent at the cusp. Hence (see
note, p. 29) this cusp counts as three among the intersections
of the curve with its first polar, and the remaining intersections
are consequently diminished by three for every cusp on the
curve. Hence the degree of the reciprocal of a curve having 6
ordinary double points and x cusps, ws

n(n—1)—26—3x.*

* According to Poncelet, Waring was the first who investigated the problem
of the number of tangents which can be drawn from a given point to a curve of the
mn degree. (Miscellanea Analytica, p.100). This number he fixed as at most n?.
Poncelet shewed (Gergonne’s Annales, vol. VIII. p. 213) that this limit was fixed
too high; that the points of contact lie on a curve of the (nm — 1)th degree, and that
their number cannot exceed n(n—1). Finally, Pliicker established the formula
in the text, and thereby fully explained (as we shall do further on) why it is that
only n tangents can be drawn to the reciprocal of a curve of the nth degree, though
that reciprocal is, in general, of the degree n (mn — 1).

THEORY OF MULTIPLE POINTS AND TANGENTS. 55

68. The same principles would shew the effect of any higher
multiple point on the degree of the reciprocal. A multiple point
of the order & would (Art. 62) be a multiple point of the order
k —1 on the first polar, avd therefore the number of remaining
intersections, and consequently the degree of the reciprocal,
would be diminished by k(/—1).

We have shewn (p. 28) that a multiple point of the order
k is equivalent to 44 (k4—1) double points, each of which would
diminish the degree of the reciprocal by two. And the result
we have now obtained may be stated: the effect of a multiple
point on the degree of the reciprocal is the same as that of the
equivalent number of double points. And so generally (see
Art. 58) for a multiple point equivalent to 6’ double points, «’
cusps, rt double tangents, and v’ inflexions, the effect on the
degree of the reciprocal is = 28’ + 3x’,

69. We have already seen that the line joining a‘y’z’ and
x'y"z' will meet the curve in two points which coincide with
x'y'z' if U'=0, and if xyz" be so taken as to satisfy the
equation 2”"U'+ y"U/+2"U/=0. But if it should happen
that the coordinates a'y’'z’ satisfy the three equations U,=0,
U,=0, U,=0, then the second condition 2” U)'+ y"U,'+2"U,/=0
is satisfied, no matter what 2”y"z” may be. The point a’y’z' is
then a double point, and every line drawn through it meets the
curve in two coincident points.

We see then that the curve expressed by the general equa-
tion in Cartesian or trilinear coordinates will not have any
double point unless the coefficients be connected by a certain
relation. For the three curves U.=0, U,=0, U,=0 will not in
general have any point common to all three, and therefore the
functions U,, U,, U, cannot all be made to vanish together. IEf
between these three equations we eliminate 2: ¥:2, we shall have
a relation between the coefficients, which will be the condition
that these three polars should intersect, or that the curve U
should have a double point. This condition is called the dis-
criminant of the equation of the curve. Thus (Conics, Art. 292)
we found the discriminant of a conic by eliminating a: ¥: z

between the three equations
ax+hy+gz=0, ha+ byt+fze=0, gut+fyt+cz=0,

56 THEORY OF MULTIPLE POINTS AND TANGENTS.

each of which must be satisfied by the coordinates of the double
point if the curve have one, and we found
abe + 2fgh — af” — bg’ — ch? =0.

In general the discriminant will be of the degree 3 (n— 1)’
in the coefficients of the given equation; for (see Higher
Algebra, Art. 76) since the three derived equations are each of
the degree n—1, their resultant contains the coefficients of each
in the degree (n—1)*, but the coefficients of the derived equa-
tions are each of the first degree in the coefficients of the original
equation. See also Higher Algebra, Art. 105.

70. We may apply these principles to examine the con-
ditions which must be satisfied when the first polar of any point
A, xy'z', has a double point. Differentiating the equation
x U,+y'U,+2'U,=0, and using for the second differentials the
. notation of Art. 59, we see that if there be a double point B,
its coordinates must satisfy the three equations

ax'+hy'+gz2'=0, hx' + by'+ fz =0, ga'+fy'+cz’=0.
These are three relations connecting 2'y'z’, the coordinates of
the point A with xyz, the coordinates of the double point B,
of which coordinates a, 6, &c. are functions each of the (n—2)®
degree. But on comparing these equations with those cited

in the last article, we see that if we write the polar conic of
the point B
ax’ + by” + c2z* + 2fyz + 2gax+ 2hay =0,

the three relations are exactly the conditions that must be
fulfilled when A or a'y’'z' is a double point on the polar conic.
Hence we infer, ¢f the first polar of any point A has a double
point B, then the polar conte of B has a double point A; and
vice versa. :

Between the three equations we can eliminate a’y’z’, and
obtain as a relation which must be satisfied by zyz,

abe + 2fgh — af* — bg’ — ch? = 0.
This equation then is the equation of the locus of points B, and
it appears from what has been said, that it may be described

either as the locus of points which are double points on first
polar curves, or as the locus of points whose polar conics break

THEORY OF MULTIPLE POINTS AND TANGENTS. 57

-up into two right lines. Since the second differentials a, b, &e.
are each of the order n— 2 in xyz, the equation just written is
of the order 3 (n—2). The curve which it represents has im-
portant relations to the given curve, of which it is a covariant
(Higher Algebra, p. 124). On account of its having been first
studied by Hesse, it is called the Hessian of U.

If between the three equations we eliminated ayz, the re-
sulting equation in a'y'z’ would give the locus of points A,
which may be described either as the locus of points whose
first polar has a double point, or of points which are double
points on polar conics. This locus we shall call after the_
geometer Steiner, the Steinerian of U. In order actually to
perform the elimination in any case, it would be necessary to
write out a, b, &c., explicitly ; but we can easily see that the
degree of the resulting equation is 3(n— 2)’, since it is the
resultant of three equations each of the degree n— 2, and each
containing x, y, 2 in the first degree.

71. Returning now to the equation A=0, we sce that it will
have three roots 7» =0, or that the line in question will meet
the curve in three points coincident. with a'y'z’, if the three
conditions are satisfied U'=0, AU'’=0, A’U'=0. Let us con-
sider first the case when 2'y’z’ is a double point ; then, as we have
seen, U' and AU' vanish independently of «’y"z", and the third

oF. WRF

condition expresses that 2'y'z” must be on the polar conic of
zy'z'. But clearly the point a”y"z” may be any point on either
of the two tangents at the double point, since each of these
meets the curve in three coincident points. Hence the polar
conic of 2'y'z’ must be identical with these two lines; or, in
other words, the equation of the pair of tangents at the double
point is A*U'=0, or

. ax? + Dy’ +2? + 2f'yz + 2q'2x + 2h'xy = 0.
The double point, being one whose polar conic has thus been
proved to break up into two right lines, is a point in the
Hessian; and we shew directly that it satisfies its equation.
For, by the theorem of homogeneous functions, the three
equations U'=0, U'=0, U{=0, which are satisfied for the
double point, may be written

a'a' + h'y' + 9'2' =0, h'a' + b'y' +f'2 =0, gz + f'y' +2 =0,

{

58 THEORY OF MULTIPLE POINTS AND TANGENTS.

whence eliminating 2'y'z’ we see that the equation of the
Hessian is satisfied for the double point.

72. The double point will be a cusp if the equation which
represents the two tangents be a perfect square; that is, if
be =f", ca=g’,ab=h*. These three are only equivalent to
one new condition, for if any one of these be satisfied, and the |
coordinates ’y'z’ of the double point have any finite magnitude,
the others must also be satisfied. For, solving for the ratios
a’: 2', y':2', successively from each pair of the equations at
the end of the last article, we have

a hf bg. Berd. Semen

2. a-h. feta. ghawe

y _gh-af _fg-—ch _ ca-g’

z ab—-h* hf—bg gh—af’
Hence if ab=h’, and neither of the ratios is infinite, both
numerator and denominator of every one of these fractions
must vanish.

73. The origin will be a triple point if all the second dif-
ferential coefficients a, b, &c., vanish; for then A’U’ vanishes
independently of a''y''z", and if the second differential coefficients
vanish, the theorem of homogeneous functions shews that the
first differential coefficients vanish likewise, and therefore AU"
also vanishes. Hence every line through a'y'z’ meets the curve
in three coincident points; and it is obvious that the three
tangents at that point are given by the equation A*U'=0.

There is no difficulty in extending the same considerations
to higher multiple points. The point 2’y’z’ is a multiple point
of the order &, if all the differential coefficients of the order
k—1 vanish for that point, and the tangents at the multiple
point are given by the equation A*U'=0.

74. Let us now examine in what case a line can be drawn
through a point a'y'z’ on the curve (but which is not a double
point) so as to meet the curve in three points coincident with
x'y'z': to fix the ideas we may in the first instance assume
that the curve has no multiple points. We have seen, Art. 71,

that every abe on such a line must fulfil the sunididons
AU'=0, 4°U'=

THEORY OF MULTIPLE POINTS AND TANGENTS. 59

The first condition expresses that the line must coincide with
the tangent at a'y’z’, as is geometrically evident; the second
condition expresses that every point on it satisfies the equation
of the polar conic. The polar conic A*U’ must therefore, in
this case, contain the line AU’ as a factor; and therefore the
point «'y'z’ must be one of the points whose polar conics break
up into factors; that is to say, it must be a point on the
Hessian (Art. 70). And, conversely, every point where the
Hessian meets Uis a point at which a line can be drawn to
meet the curve ,in three coincident points; in other words, is
a point of inflexion. For (Art. 64) the polar conic of every
point on U touches U at that point; and if the point be also
on the Hessian H, and the polar conic consequently break up
into factors, one of these factors must be the tangent at a'y'z’.
Any point on that tangent will then satisfy both the conditions
AU'=0, A*U'=0. It follows, then, that every one of the in-
tersections of the curves U, H will be a point of inflexion on
U, and since H is of the degree 3 (n—2), that a curve of the
n® degree has in general 3n (n—2) points of tnflexion.

75. If the curve, however, have multiple points, the number
of points of inflexion will be reduced. We have already shewn
(Art. 71) that every double point on the curve is a point on
the Hessian, but we shall now shew that it is a double point
on that curve, and more generally that every multiple point
on the curve of the order & is a multiple point of the order
8k—4 on the Hessian. ‘The casiest way to shew this is to
suppose that the multiple point has been taken for the origin,
and consequently that the equation contains no terms in x and
y below the degree &. Let us examine, then, the degree of the
lowest terms in # and y in the second differential coefficients ;
then evidently where there have been two differentiations with
respect to x or y, the order of the lowest terms will be k—- 2;
where there has been one differentiation with respect to x or y
and one with respect to z, the order will be 4—1, and where
both have been with ‘respect to z, the order will be &; that is
to say, the order of the lowest terms will be

k-—2, k-2, k, k-1, k-1, k-2
in @5 Oy ey -f- yg ) -& respectively,

60 THEORY OF MULTIPLE POINTS AND TANGENTS.

And combining these, we see that the order of the lowest

terms in w and y, in every term of
abe + 2fgh — af” — bg" — ch’,
will be 34-4.

But further, we say that every tangent at a multiple point
on U will be also a tangent at the multiple point on H. For
suppose the line x to be a tangent at the origin, and therefore
(Art. 40), that the lowest terms in x and y all contain 2 as a
factor, then evidently a will also be a factor in the lowest
terms of each of the second differential coefficients in which
there has been no differentiation with respect to x; that is to
say, it will be a factor in 4, c, and f. But, on inspection, it
appears that every term of :

abe + 2fyh — af* — bg’ — ch’

contains either 8, ¢, or f.

76. We are now in a position to calculate the amount of
reduction in the number of points of inflexion which occurs
when U has multiple points. If U has a double point, this
will also be a double point on H, and the two tangents will
be common to both curves; but (see note, p. 29) when two
curves have a common double point and the tangents at it also
common, this point counts for six in the number of their inter-
sections. ‘he number of intersections therefore of U and H
distinct from the double point will be reduced by 6, and we
infer that if a curve have 6 double points, the number of its
points of inflexion will be 3x (n —2) — 66.

Similarly, if U have a multiple point of order 4, we have
seen that it is a multiple of the order 3k—4 on H, and that
there are k tangents common to the two curves. ‘The multiple
point therefore counts among the intersections as

k (8k —4)+k=6 x $k (k—-1).
But we have seen (Art. 40) that the multiple point is equi-

valent to 44 (4—1) double points; hence our present result may
be stated, the multiple point has exactly the same effect in re-

* It is a useful exercise on the method of Art. 56 to show-that at a double point

Pee and the curve touch the tangents on opposite sides (Clebsch, Vorlesungen,
p- 325).

THEORY OF MULTIPLE POINTS AND TANGENTS. 61

ducing the number of points of inflexion as the equivalent number

of double points.

77. The case of a cusp on U requires special consideration.
Let it be taken for origin and let «=0 be the tangent at it, so
that the equation is of the form a’z"*+ u,z"*+ &c.=03; then it
will be seen that the orders of the lowest terms in the second
differential coefficients are 0, 1, 2,2, 1, 1 respectively; the terms
in fact being

a=22"", b= a mS, C= (n—2) (n—3) xe",
du, n—4 er n~3 re du, n-8
psa ee 59 =2 (n—2) xz ei

It will be found then that the order of the lowest terms in
abe + 2fgh — af® — bg’ — ch’

is three, and that only in the terms abe and dg” is the order so
low, but each of these terms contains 2 as a factor. The point
on # is thus a triple point arising from a cuspidal point with
a simple branch passing through it; and the two coincident
tangents (or cuspidal tangent) coincide with the cuspidal
tangent of U. Now when two curves have a common point
which is double on one and triple on the other, that point counts
for six intersections; and_if, moreover, two tangents at the
double point are also tangents at the triple point, the curves
have two more consecutive points common, and therefore this
point counts for eight intersections. Hence if a curve have 6
double points and « cusps, the number of its inflexions will
be = 3n(n — 2) — 66 — 8x. |

78. We shall hereafter shew how to use the equation A=0
to discuss the conditions for double tangents; but the investi-
gation being a little difficult, we postpone it for the present.
We shall shew presently that the results already obtained,
combined with the theory of reciprocal curves, are sufficient
to determine indirectly the number of double tangents of a
curve of the n“ order.

The equation of the system of tangents which can be drawn
to the curve from any point a’y'z', may be derived from the
equation A=0 by the method used (Conics, Arts, 92, 294). Any

Jn

{ 2%

62 THEORY OF MULTIPLE POINTS AND TANGENTS.

point on one of these tangents is obviously such that the line
joining it to a’y'z' meets the curve in two consecutive points,
and in such a case the equation A =0 will have two equal roots.
We obtain then the equation of the system of tangents, by
equating to zero the discriminant of A considered as a binary
quantic in A, pw. |

Thus, for example, let U be of the third order. Then A is

MU'+ AA + AWA + pU=0,
where, for brevity, we have written A’ and A for AU’ and AU.
The discriminant of A equated to zero is
(27UU" + 4% —18AA'U') U=(A" -40U’') &’.

Now JU, A, A’ are respectively of the third, second, and first
degrees in xyz; the preceding equation then, being of the sixth
degree, shews that six tangents can be drawn from a'y’z' to U,
as we know already.

The form of the equation shews that it represents a locus
touching Uin the points where U meets A. ‘The other points
where U meets the locus lie on the curve A*—4AU'=0.
Hence, ¢f from any point six tangents be drawn to a curve of the
third order, their six points of contact lie on a conic A=0, and
the six remaining points, where these tangents meet the curve, lie
on another conic A” —4Q4U'=0, which two conics have evidently
double contact with each other in the points A=0, A'=0.

If a'y'z’ be on the curve U'=0, then A reduces itself to
NA'+AwA+y'U: equating the discriminant to zero, we have
A*=4A'U, an equation of the fourth degree in vyz. Hence
through a point on a curve of the third order can be drawn
in general only four tangents. The tangent at the point in
fact counts for two.

79. And so in like manner in general. The discriminant of
A or of w"U4+ pA + pA’ + Ke. is of the degree n (n — 1)
in xyz, and (Higher Algebra, Art. 111) is of the form kU+ (A)*¢,
where ¢ is the discriminant of A deprived of the first term.
Hence the locus touches U at its points of intersection with A,
as it plainly ought to do. |

Each of the »(m—1) tangents meets the curve again in
n—2 points, and the form of the discriminant shews that these
n (n—1) (n—2) points lie on acurve ¢ of the order (n —1) (” - 2).

RECIPROUAL CURVES. 63

Moreover, ¢ is itself of the form h’A + (A’)*. Hence the two
curves @ and yf touch each other at the points where the first
and second polars of «'y’z’ intersect.

Writing A, \°U'+ A" “*ywA' + &e. we see that the discrimi-
nant may also be written in the form £U'+(A')’ $3 hence if
x'y'z' is on the curve, and therefore U'=0, the discriminant
contains the double factor A”, or the system of tangents con-
sists of the tangent at a'y'z’ counted twice, and n? —n— 2 other
tangents represented by 6=0. In the same way ¢ is itself of
the form hA’+(A”)’y. If then w'y'z’ be a double point, and
therefore not only U' but A’=0, ¢, which was already of
the degree n’—n-— 2, contains the double factor (A”)*; that is
to say, among the n’—n—2 tangents are included the two tan-
gents at the double point, each counted twice, and therefore only
n° —n—6 other tangents represented by y~=0. And so, in like
manner, we can prove that the number of tangents which can
be drawn from a multiple point of the order & is n’—n—k (k +1).

The theory already given of the effect of multiple points
upon the number of tangents which can be drawn from any
point to a curve shews that the discriminant of A, which in
general represents the n (n — 1) tangents, will include as factors
the square of the line joining 2'y’z' to every double point of the
curve, the cube of the line joining it to every cusp, the sixth
power of the line joining it to every triple point, and so on.

SECT. VI,—RECIPROCAL CURVES.

80. We have seen (Conics, Art. 303) that the degree of the
reciprocal curve is always the same as the class of the given
curve, and vice versd. It is evident also, that to a double point,
on either curve will correspond a double tangent on the other ;
that to a stationary point on one curve corresponds a stationary
tangent on the other; and, in general, that to a multiple point
of the k order corresponds a multiple tangent of the same
order; that the & points of contact of the multiple tangent
correspond to the & tangents at the multiple point ; and that if
two or more of these last coincide, so will the corresponding
points of contact.

81. We have seen that the general equation in Cartesian

64 RECIPROCAL CURVES.

or trilinear coordinates represents a curve which has no double
or other multiple point, unless certain conditions be fulfilled.
But the general equation represents a curve which ordinarily
must have double and stationary tangents. For the abscisse
of the points, where the curve is met by any line y=az + J, are
found by substituting the value for y in the equation of the
curve; and since we have two arbitrary constants a and 6b at
our disposal, we can determine them so that the resulting equa-
tion shall fulfil any two conditions we please. With one
constant at our disposal, we could make the equation fulfil any
one condition; for instance, have a pair of equal roots. The
problem “ given a to determine 0, so that the resulting equation
should have a pair of equal roots,” is no other than the problem
to draw a tangent parallel to y=az. With the two constants
at our disposal, we can either cause the resulting equation to
have two distinct pairs of equal roots, or three roots all equal to
each other. ‘The first is the problem of double tangents, the
second that of stationary tangents and points of inflexion.
Thus the double and stationary tangents may be counted as the
ordinary singularities of a curve whose equation is expressed in
point coordinates; all higher multiple tangents and all multiple
points being extraordinary singularities which a curve will not
possess except for special values of the coefficients of its equa-
tion. But this is reversed if the equation be expressed in tan-
gential coordinates. ‘Then the curve represented by the general
equation ordinarily has double and stationary points and cusps,
but no singular tangents. Hence double and stationary points
on the one hand, and double and stationary tangents on the
other hand, are equally entitled to be ranked among the ordinary
singularities of curves; they are such, that if any curve possess
the one its reciprocal will possess the other.

82. We shall now denote

the degree of a curve by m™,
its class ee
the number of its double points eS
Sisnsevulvnrvea reves double tangents so tiie
whnaseaeese agus ws... Stationary points i 1G

ovécesivcdteseadecers: MeatlIonEry tangents: :4/ 4,

RECIPROCAL CURVES. 65

and the corresponding numbers for the reciprocal curve are
found by interchanging m and n, 6 and 7,4 and «x. We have
already obtained (Arts. 67, 77) the values of m and ¢ in terms
of m, 8, «; hence, from the reciprocal curve we have the
values of m and « in terms of nm, 7, ¢; and from these four
equations (equivalent, as will presently be seen, to three equa-
tions only) we ean obtain the value of rt in terms of m, 4, «,
and that of 8 in terms of n, 7, +. We have thus Pliicker’s six
equations, viz, these are

(1) n=m’—m—25- 38x.
(2) +t=38m*— 6m — 65 — 8x.
(3) 27 =m (m —2) (m? —9) —2 (m? — m — 6) (26 + 38x)
+48 (S—1)+128¢+9« («—1).
(4) m=n'—n—-2Q7r—38..
(5) «=38n"—6n—6r—8u.
(6) 26=n (n—2) (n*—9) —2 (n? —n—6) (27 + 32)
+47 (7-1) +1270 +91 (c- 1).
If from (1) and (2) we eliminate 6, or from (8) and (4) we

Lf

+f

eliminate 7, the result is in each case

(7) u-K=3 (n—™m),
shewing that the four equations are equivalent to three only.
This may also be written in the forms

38m —K=3n—1, and 8m+4=38n+k«.
By taking the difference of the equations (1) and (4), we obtain
mv —26—3K =n" — 27 - Be. .

Whence, replacing «— « by its value from (7), we obtain

(8) 2(r-—8) =(n—m) (n+ m- 9).
The last preceding equation, substituting therein for x and éj

or for m and « their values, gives the foregoing equations (3)
and (6). From (7) and (8) we obtain also

(9) $m (m+3)— 6- 2e=4n (n+ 3)-—7T-2e,
(10) 4(m—1)(m—2) —8—xK=} (n— 1) (n—-2)-7-1,
(11) m*-26-8«=n"—-27r-30=m+n.

The entire system of equations is, of course, equivalent to
K

66 RECIPROCAL CURVES.

three equations only, and by means of it given any three of
the six quantities m, , 5, x, 7, 4, we can determine the remain-
ing three; thus m, 5, « being given, m is given by (1), « by (2),
or more easily by (7), and + by (3), or more easily by (8).

Ex. Suppose we were given m= 6, 6=4, x =6; then, by (1), »=4; therefore
m—-n=2, n—m=-— 2,
Hence (5) e—x=6, or: =0;

n+m—9=1; thereforer—dO=—1; therefore 7 =3.

83. Since when a curve is given its reciprocal is determined,
it is evident that the same number of conditions must suffice
to determine each. Now to be given that a curve has 6 double
points is equivalent to 6 conditions. Thus, for example, a conic
is determined by five conditions; but if it have a double point,
that is, if it reduce to a system of two right lines, it is deter-
mined by four conditions; by two points for instance on each
of the right lines. So, again, to be given that a curve has a
cusp is equivalent to two conditions. Hence (and Art. 27)
a curve of the m‘* degree with 6 double points and « cusps is
determined by 4m (m+ 3)—6—2« conditions, and its reciprocal
by 4n(n4+8)—7—2¢ conditions. And the foregoing equation
(9) shews that these two numbers are in fact equal.

The foregoing equation (10) shews that the deficvency (Art. 44)
is the same for a curve and its reciprocal. In a subsequent
chapter it will be proved that this is true for all curves derived
one from the other in such a way that to any point of one
answers a single point or tangent of the other.

If (with Prof. Cayley) we write 3m+., =3n+x«, =a, then
everything may be expressed in terms of (m, n, @), viz. we have

K=a—3n,
t=a—3m,
25=m’ — m+ 8n — 3a,
2r=n'— n+ 8m-— 3a.

The meaning of equation (11) will appear in the following
chapter.

CHAPTER III.

ENVELOPES.

84, Ira curve depend in any manner upon a single variable
parameter, so that giving to the parameter a series of values,
we have a series of curves; these all touch a certain curve,
which is called the envelope of the system. ach curve is
intersected by the consecutive curve in a set of points depend-
ing on the parameter, and the locus of these points is the
envelope. See Conics, Arts. 283, &c., where the problem of
envelopes is considered in the case where the variable curve
is a right line.

Analytically, the equation of the curve may contain a single
variable parameter, or it may contain two or more variable
parameters connected by an equation or equations, so as to
represent a single variable parameter. ‘The two cases are
essentially equivalent, but it is often convenient to treat the
second in a different manner, by a method of indeterminate
multipliers, which we shall presently explain. The form of the
second case, which is of most frequent occurrence, is when
the equation of a curve contains the coordinates of a variable
point, limited however to a fixed curve; or, as we may say,
when the variable curve depends on a parametric point moving
on a given parametric curve. For example, it was shewn
(Conics, Art. 321) that the problem to find the reciprocal, with
respect to 2+ y’+ 2°, of a given curve, is the same as to find
the envelope of ax + By +z, where a, 8, y satisfy the equation
of the given curve. Here the equation of the variable line
contains the two variable parameters a: y, 8: y, these two
ratios being connected by the equation of the given curve.

85. Suppose, first, that the equation of the curve, say 7’=0,
contains a single variable parameter ¢. ‘The curves belonging

68 ENVELOPES.

to the consecutive values t, t-+dt, may be represented by the
equations 7=0, 7,=0. These equations, or the equivalent
equations 7=0, 7,—7'=0, determine therefore the coordinates
of the points of intersection of the two consecutive curves. We
have 7,=7+4,T.dt+&c., or T. — T=d,T.dt+ &c., where dt
being infinitesimal, the terms after the first are to be neglected.
The equations become therefore 7=0, d,7’'=0, which equations
determine a set of points depending on the parameter ¢; and
eliminating ¢ from these equations we get the equation of the
locus of all points of intersection of consecutive curves of the
system; that is to say, the equation of the envelope.

An important case is where the equation contains ¢ rationally ;
we may then, without loss of generality, take Z'to be an integral
as well as rational function of ¢, and the process described for
finding the equation of the envelope is equivalent to forming
the discriminant of Z considered as a function of ¢, and equating
it to zero. Thus, if a, b, c, &c. be any functions of the
coordinates, and if 7’ be

at” + nbt"* +4n (n-1) ct"? + &e.,
the equations of the envelope for the cases of most common

occurrence, viz. n=2, 3, and 4, are respectively (see Higher
Algebra, Arts. 193, 195, 207),

(2) ac—b°=0,
(3) ad’ + 4ac’ + 4b°d — babcd — 3b°c? = 0,
(4) (ae—4bd + 3c’) — 27 (ace + 2bed — ad’ — Be — c*)? =0,

and in using the last of these equations, when we desire to infer
its order in the coordinates from knowing the order in which
they enter into a, b, &c., it is useful to remember that when
the equation is developed, the terms containing c° and c‘bd
respectively cancel each other, so that the order of the envelope
may happen to be lower than that of either of the two members
of which the equation, as written above, consists.

If we substitute in 7’ the coordinates of any point, and solve
for ¢ the resulting equation a’t’+nb't’*+ &e.=0, there will
evidently be n solutions; that is to say, the system of curves
represented by 7’is such, that of them can be drawn to pass
through any fixed point; and, from what has been just said, it

ENVELOPES. 69

appears that if the fixed point be on the envelope two of these n
curves will coincide.

The case where 7’ depends on a parametric point may be
reduced to that just considered if the parametric curve be a line,
conic, or any other unicursal curve; for then (Art. 44) the
coordinates of the parametric point can be expressed as rational
functions of a parameter.

Ex. 1. To find the envelope of at” + dt? +c = 0, where, as well as in the other
examples a, b, &c. are supposed to be any functions of the coordinates. Combining
the given equation with its differential with respect to ¢, we have

nat” P + pb=0, (n—p) bt? +ne=9),
whence, eliminating t, we have
nraPcrP + pP (n — p)P) §n = 0,
where the sign + is to be used when z is odd and — when it is even.

Ex. 2. To find the envelope of a cos"@ + 4 sin"@ = c, where 0 is the parameter,

We have = d,T = — a cos™@ sin8 + b sin™!0 cos = 0;
Fe —L a
n-2 hn-2 n-2
whence tand =", cos 0 =, ——, sin = = see
bn-2 d{a** aS bn-?) ita? a aed

Substituting these values, and reducing, we find the equation of the envelope
2 2 2

In particular (as we saw, Conics, Art. 283), the envelope of a cos@+6sin@=ce is
a* + b?=¢?, Conversely, any tangent to the curve 2™ + y™ = c™ may be expressed by
2m—1) 2(m-1)
xcos ™ @+ysin ™ 6=<¢,
2 2
the coordinates of the point of contact being x = ¢ cos”0, y = ¢ sin™0,

This example might have been stated as an example of an envelope depending
on a parametric point lying on a unicursal curve. For if we write cos@=a,
sin@ = B, then a, B are the coordinates of a point lying on the circle a? + 6? = 1,
and the circle being a unicursal curve, these coordinates can be expressed rationally
in terms of a parameter. Thus if t be cos0+7sin0, we may write for @ or cos0,
; (« + +) , and for £ or sin 8, 7 (: ~ ;) , and the equation, for example, aa + 0B =c,
becomes : :

(a — bi) t? — 2ct + (a + bi) = 9,
whose envelope, as before, is
(a + bi) (a — bt) = c?, or a? + 2 = c?,
If we desired to avoid the introduction of imaginaries we might write tan 40=1t,
and (as at Conics, Art. 283) express cos 0, sin @ rationally in terms of ¢.

Ex. 3. Let the curve be
a cos20+ 6 sin20+¢cos0+dsind+e=0, "
Putting ¢ = cos@ + 7sin0, this becomes

1 =) co ge 1 . 1 a
a(# +5) bi (e-3) +e(¢+;)-ai(¢—,) +2¢=0,

70 ENVELOPES.

or (a — bt) t + (c — di) + 2et? + (¢ + di) t+ (a+ bi) = 0.
And applying to this the form already given for the discriminant of a quartic written
with binomial coefficients, we have

fa? +B? — 3 (c? +d?) + Je™} = 27 {3 (a2 +2) e+ ah (02 + a) e— Ja (0? — @) —Ied — he)’;
or, clearing of fractions,
{12 (a? + b?) — 3(c? + d?) + 4e7}8 = {72 (a7+ 67) e+ 9 (c?+ d?)e — 27a (ce? — d*) — 54bcd —8e5}? ;
and, again, it is useful to remark that the expanded result will contain neither of
the terms e°, (c? + d*) e,

Ex. 4. To find the envelope of the chords of curvature of the points of a conic,
The equation of the chord is (Conics, Art. 244, Ex. 1)

x ‘ z
— cosa — y sina = cos2a.
a b

- ae y? 3 2 2
The envelope is therefore (= + i 4) +27 (5 _ ) = 0.

Ex. 5. To find the equation of the curve parallel to a conic; that is to say, the
curve obtained by measuring from the conic on each normal a distance equal to 7.
This problem has been already solved (Conics, Art. 872, Ex. 2) by considering the
parallel curve as the locus of the centre of a circle of constant radius touching the
given conic. But it is easy to see that the parallel curve may also be considered as
the envelope of a circle of constant radius whose centre is on the given conic; that is
to say, we are to seek the envelope of (a — a)? + (y— )?—r?, where the parametric
point af lies on the conic; and the conic being a beanie curve, this may be reduced

to the case already discussed. Thus, let the conic be = = ee = 1, and write for a,

1
62
acos0@, for B, d sin@, when

a? + B? — 2ax — 2By + 2? + y?— 41?
becomes (a? — 6?) cos20 — 4ax cos @ — 4by sin @ + 2 (x? + y”) + a? + b? — 2r?,

a form included under the last example, by the help of which we should obtain a
result which, when expanded, is identical with that given, Conics, Art. 372,

86. A little further notice may fitly be given to the case
where 7’ is algebraic in ¢, and of the first degree in the
coordinates, so as to denote a right line; that is to say, to the
envelope of at"+nbt"*+&c. where a, 6, &c. are all linear in
the coordinates. In this case the envelope is clearly a curve
of the n™ class, being such that m tangents can be drawn
through any assumed point (Art. 85); and since the discriminant
of at’ + &e. is of the order 2(n—1) in the coefficients a, b, ke.
(Higher Algebra, Art. 105), which each contain the coordinates
in the first degree, the order of the envelope is 2 (n—1). Two
other characteristics of the envelope can easily be obtained.
It has ordinarily no points of inflexion. Ata point of inflexion
two consecutive tangents coincide; and therefore Z7' and d,T
represent the same right line; but in order that two linear

ENVELOPES. 71

equations should represent the same right line, two conditions
must be fulfilled, and it will generally not be possible to de-
termine the single parameter ¢ at our disposal, so as to satisfy
both conditions.

The number of cusps on the envelope is 3(n—2). As the
tangent at a point of inflexion on a curve contains three con-
secutive points, so,reciprocally a cusp is the point of intersection
of three consecutive tangents. At a cusp, therefore, on the
envelope the three equations will be satisfied, 7’=0, d,7’=0,
d,’T’=0, which may easily be reduced to

T., = at" + (n— 2) bt'* +4 (n—2) (n— 38) ct" * + &e. =0,

T,, = bt" + (n—2) ct" * +4 (n— 2) (n —3) dt”* + &e. =0,

T,, = ct” + (n— 2) dt"* +4 (n —2) (n— 8) et” *+ &. =0,
T,,, Z;., T,, being the three second differential coefficients if 7;
considered as a binary quantic, had been made homogeneous by
the introduction of a second variable. Now, if from these
equations we eliminate 2 and y, which enter in the first degree
into each, the resulting equation in ¢ will be of the degree
3(n—2). If in fact we write 7, xU+yV'+2W, where U, V, W
contain only ¢ and constants, we have obviously the determinant

U5) Via Ww
O45 Ves ake
O25 Ko Wy ae 0,

which gives the values of ¢ corresponding to the 3 (n — 2) cusps.

The problem of finding the number of double points on the
envelope is the same as that of finding the order of the system
of conditions that 7’ should have two distinct pairs of equal
roots (Higher Algebra, Art. 264), and the problem of finding the
number of double tangents is the same as that of finding the
order of the system of conditions that 7’ should represent the
same line for different values of ¢; or, in other words, the
number of ways in which it is possible to find a pair of values
t', t’, for which we shall have the equality of ratios

Gis pt = Fs VW
It is not necessary for us, however, to deal with these problems

directly, since we have already more than enough of conditions
to determine 6 and 7, by Pliicker’s equations, Art. 82. Sub-

72 ENVELOPES.

stituting in these equations 2(n—1) and m for the order and
class of the curve, and putting «=0, we find

k=3(n—2), 6=2(n—2)(n—3), t=4 (n—1) (n—2).

87. Let us now consider the case where the equation contains
k parameters connected by &—1 equations. To fix the ideas,
- suppose that we have the equation U=0 containing the three
parameters a, 8, y connected by the two equations V=0,
W=0. We may, if we please, regard 8, y, as functions of a,
determined by the two equations V=0, W=0. ‘The process,
in its original form, would then consist in the elimination of «
from the given equation, and

Ag dU dg , dU dy_
dB da’ dy da

Here ‘s = are functions of a determined by

Lae dB dV dy

dB da* dy da

ae WwW aW dW.dB dW dy

dB da’ dy da

and from these three oe we have y = 0, where

=0;

4 aU a dd

does ‘da’? dB? dy
AV. AV 0S,
da? dB? dy
iw aw aw |_,
da? AB gl st

and the final result is got by eliminating a, 8, y between
U=0, V=0, W=0, vy =0.
But v =0 is obviously the result of eliminating 2, « between
the equations
dU. AVG AW

a a aa?
Re a amp
dp’ dp ap-

aD ear aw

Tere eg ee

ENVELOPES. _ 73

so that the result may be got by eliminating «, 8, y, A, u
between the last three equations and those originally given.
This is the method of indeterminate multipliers referred to
(Art. 84).

88. An important case is where U is homogeneous in &+1
parameters connected by 4—1 other homogeneous equations.
This is really equivalent to the foregoing, since the £+1
parameters may be replaced by the ratios which any & of them
bear to the remaining one. But it is more symmetrical to retain
all the &+1 equations given by the method of indeterminate
multipliers, which equations in virtue of the theorem of homo-
geneous equations are connected by a relation making them really
equivalent to only & equations. ‘Thus, let U contain homo-
geneously a, 8, y the coordinates of a parametric point moving on
the parametric curve V=0; the method of indeterminate multi-
pliers gives us, in addition to the two original equations,

ao. OV: aU 2d ¥ dt aVe

+r =0 7B = dy ta

da. da ’ dB EJ 6s

But these three are really equivalent to two, since if we multiply
them by a, 8, y respectively, we get mU+2n V=0, which
is included in the equations U=0, V=0. We have then four
equations from which on account of the homogeneity we can
eliminate the four quantities a, 8, y, X, and so obtain the equa-
tion of the envelope.

Ex. To find the envelope of U=(Aa)™ + (BB)™ + (Cy)™ =0, where a, B, y are

connected by the relation V = (aa)" + (08)"+ (cy)" = 0.
The method of indeterminate multipliers gives us

mA™a™1 + \na"a™! = 0, MB™B™! + And"B™! = 0, MOMy™! + Ancty"1 = 0;
ay An
whence, writing for shortness oe pm", we have

n n n

a mn b \ mn Cc m-t
Aa=1 (5) ’ Bp=n(5) ’ Cy =u (5) ;
and substituting these values in U, we have the envelope required, viz.

mn mn mn

74 ENVELOPES.

89. Prof. Cayley has considered the case of a curve U=0
the equation of which contains two or more independent para-
meters. If, for instance, there are the two parameters a, A,
then from the equations

eliminating a, 8, we have the equation of an envelope. But
observe that we can from these same equations eliminate the
coordinates (2, y), and that the equations thus imply a relation
¢ (a, 8) =0 between the parameters. This gives in the double
system of curves U=0, a single system wherein the parameters
satisfy this relation. ‘Taking any curve of the double system
and the consecutive curve belonging to the values a+da,
8+d8 of the parameters, the two curves intersect in a set of
points depending in general on the value of the ratio d@ : da of
the increments. But if the curve belong to the single system,
then the set of points will be independent of the ratio in
question ; the coordinates of the points of intersection satisfy

the equations U=0, ue =0, Ee es and consequently the
1 da dg
dU
equation U7 ae 8 7, + =3 48 = 0, whatever be the value of the

ratio eae re we thus see that a curve of the single
series is intersected by every consecutive curve of the double
series in one and the same set of points, and that the locus of
these points is the envelope. In the case of a single parameter,
the envelope is the locus of a set of points on every curve of
the system, and it may be termed a “ general envelope ”; in the
case of the two parameters, the envelope is the locus of a set of
points not on every curve of the system, but only on the curves
of the single system wherein the parameters satisfy the equation
d (4, 8) =0, and it may be termed a “special envelope.” And
the like theory applies to the case of any number whatever of
parameters: there is always a resulting single system of curves.

89 (a). A difficulty in the theory of envelopes as given in
Art. 84 has been explained by Prof. Cayley. In that article we

ENVELOPES. 75

have considered an envelope as the locus of the intersections of a
variable curve with consecutive curves of the system. But each
curve has with the consecutive a number of common tangents
depending on its parameter, and the envelope of these lines is
also the envelope ; viz., each common tangent of the curve and
its consecutive curve is at a common point of the same two
curves atangent,of the envelope. Bat if the variable curve be of
the order m and the class , the number of common points is = m’,
and the number of common tangents = n”; and yet the common
points and common tangents have to correspond to each other
in pairs. The explanation depends on the singularities of the
variable curve. Suppose this has in general 6 double points,
« cusps T double tangents and «¢ inflexions; then, as is easily
seen, the curve meets the consecutive curve in 2 points contiguous
to each double point and in 3 points contiguous to each cusp
(viz. there are thus 26+ 3« intersections), and besides in
m*—26—3« points, and reciprocally the curve has with the
consecutive curve 2 common tangents contiguous to each double
tangent and 3 common tangents contiguous to each stationary
tangent (viz., there are thus 27 + 3. common tangents), and there
are besides n? — 27 — 34 common tangents: we have, see Art. 82,
m' —28-3K =n" —27-—38r=m+n; each of the m*—25—3«
points is (not a point of contact but) an ordinary intersection of
the two curves, but it has contiguous to it one of the
n’—27r—3+ common tangents of the two curves; and the
envelope is thus cotemporaneously the locus of the m* — 26 —3«
(=m-+n) points, and the envelope of the n*—27- 34 (=m-+n)
tangents.

It may be added that the complete envelope of the variable
curve consists of the proper envelope as just explained together
with (1) the locus of the double points twice, (2) the locus of
the cusps three times, (3) the envelope of the double tangents
twice, and (4) the envelope of the stationary tangents three
times.

In what precedes, the numbers m, n, 6, «, T, e apply to the
curve corresponding to the general value of the variable para-
meter; for particular values of the parameter, the variable curve
may acquire or lose point- or line- singularities, and the several
numbers be thus altered.

76 RECIPROCAL CURVES,

RECIPROCAL CURVES.

90. Let it be required to find the envelope of a lme
ax+ By +z, being given that a, 8, y are connected by a re-
lation ==0. In other words, let there be given Y=0 the
tangential equation of a curve, or its equation in line coor-
dinates, and let it be required to pass to the equation in point-
coordinates. Here then we have the two equations 2=0,
ax-+By+yz=0, and the method of Art. 88 shews that the
result is to be obtained by eliminating a, 8, y, \ from the two
given equations combined with

d= d= ad

Fi Paces 0, a3t= 0, 7
The solution of the reciprocal problem, given the point-equation
S=0, to pass to the tangential equation, depends on a precisely
similar elimination; namely, to eliminate x, y, 2, % between
S=0, ax+ By +yz2=0, and

dS dS dS

a system of equations which would also present itself naturally
from the consideration that if av+ By+yz be identical with
the tangent at the point zyz, then the well-known form of
the equation of the tangent (Art. 64) shews that a, 8, y must
be respectively proportional to — aS ln

dx’ dy’? dz’

. It has been mentioned (Art. 84, and Conics, Art. 321) that the
problem of passing from the Soiait equation of a curve to its
tangential equation is the same as that of finding its polar
reciprocal with regard to 2*+y’+27=0.

+Az=0.

Ex, To find the tangential equation of (ax)™ + (by)™ + (cz)™"=0, We have here

r r r
(az +X 20, bymrt4%F=0, Cmr+ > Yao,

whence immediately

r+ +@"=0

91. The method just indicated, however, is not always the
most convenient one for finding the equation of the reciprocal.
Let the equation of the curve be uv, +u,_,2+ 4,2 + &e. =0,

RECIPROCAL CURVES. T7

then we eliminate z by the equation aw + By+yz=0, and get
fu, — 7% (a + By)u, + (ax + By)", .— Ke. = 0,
which is now homogeneous in x and ¥; and the discriminant
of this considered as a binary quantic, equated to zero
gives the equation of the reciprocal curve, multiplied however
by the irrelevant factor , Ae!
Thus, for example, if it were required to find the reciprocal of

e+y+2+b6meyz=0,
eliminating z, it becomes
(ac + By) + 6mayy* (aw + By) — 9 (w +y") =

or (a — 4°, B+ 2may’, a8’ + 2mBy’, B* — y"La, y)* =0,*
the discriminant of which is divisible by y*, the quotient being
a’ + B° + 9° — (2 + 32m*) (B’y* + 9°a* + 2°86")

— 24m’ ary (a? + 8° +9°) -— (24m + 48m") a? By’ = 0.
In precisely the same way may be found the reciprocals of the

cubic or quartic given by the general equation, the results of
which are given at full length in subsequent chapters.

92. One chief advantage of the foregoing method of
obtaining the equation of the reciprocal is that it enables us
immediately to write down the equation of the reciprocal in
the symbolical form explained, Higher Algebra, chap. XIV.
If a ternary quantic be reduced to a binary by eliminating
z by the help of the equation ax+ y+ yz, we have imme-
diately the following rules for the differentials of the binary
quantic with respect to x and y,

A. 4 €¢ 4: 4° Ba

Siri Cer

* We use the notation (a, 0, ¢, .. La, y)* for the binary quantic written with
binomial coefficients ax” + nba* ly + 2” (n —1) ca”*y? + &e.; using the notation
(a, b, c, ....02, y)” when the quantic is written without binomial coefficients (see
Higher Alg@ra, Art, 104),

78 RECIPROCAL CURVES,

it becomes

1{,(2 22 2), 9(4 aa a
Bla, wed, de) Pe ee

2
@i@ord «ai

UGE, a, de
or, in other words, the symbol applied to the binary quantic
differs only by the factor y from the contravariant symbol («12)
applied to the ternary. Hence, if a line ax+ Py+yz cut a
curve so that the points of section satisfy any invariant relation
whose symbolical form is known, we can at once write down
in the same form the tangential equation of its envelope. For
instance, the symbolical form of the discriminant of a binary
cubic is known to be (12)* (34)? (13) (24); hence, if a line
ax+ By+yz cut a cubic curve in three points whose discrimi-
nant vanishes, that is to say, if it touch the curve, we must
have (a12)’ (a34)* (413) (a24)=0. In like manner the discrimi-
nant of a binary quartic is known to be of the form S*= 277”,
where S and 7’ are two invariants, whose symbolical form is
(12)*, and (12)”(23)’ (31)” respectively. It follows that the
equation of the reciprocal of a quartic is of the form $* = 277”,
where S is (a12)*, and Z’is (412)? (423) (a431)”, where S=0 de-
notes the curve of the fourth class which is the envelope of lines
cutting the quartic in four points for which the invariant S
vanishes, and 7’=0 denotes the curve of the sixth class which is
the envelope of lines cut harmonically by the curve, and for
which therefore the invariant 7’ vanishes.

93. We have already (Art. 78) given one method of forming
the equation of tangents drawn from any point 2'y'z’ to the
curve, but the problem is in effect solved when we are in
possession of the equation of the reciprocal, or, in other words,
of the condition that ax+ 8y+ yz should touch the curve. For
we have only to substitute in that condition for a, 8, y respec-
tively yz'—2y', zx’—axz', xy'—yx', when we shall have the
condition that the line joining the points xyz, a’y’z’ shall touch
the curve, a condition which obviously must be satisfied when
xyz is a point on any tangent through a’y’z’ (see Conics,
Art. 294).

RECIPROCAL CURVES. 79

Conversely, the equation of the system of tangents as found
by the process explained (Art. 63), is readily obtained in the
form, homogeneous function of (yz' — y'z, za! — 2'x, xy'—a'y)=0;
and then, substituting for these quantities a, 8, y, we have the
equation of the reciprocal curve.

94, We have: then immediately a theorem corresponding
to that of Art. 92, that when we are in possession of the tan-
gential equation of a curve, we can at once write down
symbolically the equation of the locus of a point, such that the
system of tangents from it to the curve shall satisfy any given
invariant relation. If we make z=0 in the equation of the system
of tangents, we have the equation of a system of lines parallel
to the tangents through the point wy, which will satisfy the
same invariant relation. But from the method just given for
forming the equation of the system of tangents we have

MRI a BA Cahn a aw

whence, as before

0. 8 6) a ae o. 6
bh -bb-+leb-a2
CUS Bye SEN. La & qa
ov tee ye
so that we have at once the rule, for every factor (12) in the
invariant symbol required to be satisfied by the system of

tangents to substitute (x12) and operate on the equation of the
reciprocal curve.

95. When the equation of a curve is given in polar co-
ordinates, that of its reciprocal with regard to a circle whose
centre is the pole may be found directly. If on any radius
vector OP there be taken a portion OP’ equal to the con-
secutive radius vector OQ, then obviously PP’ = dp, P’Q=pdo,
tanOPQ="O*, and psinOPQ is the perpendicular on the
tangent. ‘Thus let the curve be p” =a" cosmw; take the loga-
rithmic differential, and we have

d
a tanmeado ; = — cotma,
p Pp

80 THE TACT-INVARIANT OF TWO CURVES.

and if 6 be the acute angle made by the radius vector with the
tangent 9=90°—ma, and the perpendicular on the tangent
=psind=pcosmw. The angle between the perpendicular and
the radius vector =m, and between the perpendicular and the
line from which » is measured is (m+1). But the radius
vector of the reciprocal curve is the reciprocal of the perpen-
dicular on the tangent; hence it is easy to see that the equation
of the reciprocal curve is also of the form p"= a" cosmo, the
new m being equal to — oe . This family of curves in-
cludes several important species; for instance, the circle (m = 1),
the right line (”=—1), the common lemniscate (m=2), the
equilateral hyperbola (m=-—2), the cardioide (m= 4), the para-
bola (m=— 4) &e.

THE TACT-INVARIANT OF TWO CURVES.

96. It was remarked (Art. 90) that the problem of finding
the equation of the reciprocal curve is the same as that of find-
ing the condition that a right. line should touch the given curve,
both being solved by finding the envelope of ax+fy+ 2,
where a, 8, y are parameters satisfying the equation of the
curve. More generally, the problem of finding the condition
that two curves U, V should touch (which condition is called
their tact-invariant) is the same as that of finding the envelope
of either, the coordinates being regarded as variable parameters
satisfying also the equation of the other. For if the two curves
touch, the coordinates of the point of contact ay satisfy the
equation of both; and also since the tangents are the same, we
must have at that point the differential coefficients of JU,
respectively proportional to those of V. The condition of
contact is then found by eliminating a, 8B, y, A, between
U= 0, V= 0, and

a ee Bs ee

da 8 ae ap = dB il dy” dy
but these are the equations given (Art. 88) for solving the
problem of the envelope.

THE TACT-INVARIANT OF TWO CURVES. 81

97. Let the degrees of U and V be m, m' respectively, and
let it be required to determine the order in which the coeffi-
cients of either curve, say V, enter into the condition of contact.
Let the coefficients in V be a’, 5’, c’, &c., and let us take another
curve W of the same order whose coefficients are a’, 6", c’, Kc.
Then if in the condition of contact we substitute for each coeffi-
cient a’, a’ + ka", &c., we shall have the condition that V+ kW
should touch U, which will plainly contain & in the same degree
as the order in which the coefficients of V enter into the
condition of contact. This latter order, therefore, is the same
as the number of curves of the form V+W, which can be
drawn so as to touch U. But, as before, the point of contact
must satisfy the equations

Vi+kW,=AU, V,+kW, =U, V+ kW,=AU,,

whence eliminating , 2,

and the intersections of vy with U determine the points on U
which can be points of contact with curves of the form V+W.
Since the orders of U,, V,, W,, &c., are respectively m—1, m'— 1,
m'—1, the order of VV is m+2m'—3, and the number of inter-
sections is m(m-+2m'—3). ‘This then is the order in which the
coefficients of V enter into the tact-invariant, and in like
manner the coefficients of U enter in the order m' (2m + m'— 3).
By making m’=1 we have the result already obtained that the
condition that aw + Py + yz should touch a curve contains a, £, y,
in the degree m (m—1), and the coefficients of the curve in the
degree 2(m— 1). See also Conics, Art. 372.

If U have a double point, then since we have already seen
that U,, U,, U, pass through that point, and that if that point be
a cusp they have there the same tangent, the same things are
true for V5 and we see that the order of the condition of contact
in the coefficients of V must be diminished by two for every
double point, and by three for every cusp on U. The order is
therefore m (m+ 2m'—3)- 26-3 or n+ 2m (m'—1).

M

82 EVOLUTES.

98. These results might have been otherwise obtained thus:
Take any arbitrary line ax+by+cz, and equate to zero the
determinant
| ee ie ae «

U, U, U,

Vin Van Ve
This equation represents the locus of a point, such that its polars
with respect to U and V intersect on the assumed line. Now
at a point common to U and J, the polars are the two tangents
intersecting in the common point; there are, therefore, plainly
only two cases in which a point common to U and V can
lie also on vy; viz. either the assumed line passes through an
intersection of U, V, or at that point the two curves have a
common tangent. If then we eliminate between vy, U, V, the
resultant will contain as factors the condition that ax + by + cz
should pass through an intersection of U, V, and the condition
that U and V should touch. But since in the resultant of three
equations, the order in which the coefficients of each enter is
the product of the orders of the other two equations, and since
the orders of vy, U, V are respectively m+ m'—2,m, m’, the
order of a, b,c in the resultant is mm’, of the coefficients of U,
is mm’ + m' (m+ m' —2) =m’ (2m+ m' —2), and of the coefficients
of V,m(2m'+m—2). Similarly the orders of the resultant of
ax+by+cz, U, V, in the several coefficients are respectively
mm', m',m. Subtracting these numbers from the preceding,
we find, as before, that the orders of the condition of contact
are m' (2m +m'—3), and m(2m'+m-—8) in the coefficients of
U and V.

EVOLUTES.

99. We have hitherto only dealt with descriptive theorems,
and have postponed the consideration of any questions belonging
to the class described as metrical (Art. 1). The relation of
perpendicularity belongs to the latter class, since, as explained
(Conics, Art. 356), two perpendicular lines may be considered
as lines which cut harmonically the line joining the two imaginary
circular points at infinity. It is convenient not to exclude from
this chapter the discussion of some important cases of envelopes

EVOLUTES. 83

4

which involve the relation of perpendicularity, and the theorems
may be made descriptive if we substitute for the two circular
points at infinity any assumed points J, J, and wherever, in our
theorems lines at right angles occur, substitute lines cutting J, J
harmonically.

One of the most important and the earliest investigated class
of envelopes is that of the evolutes of curves. We have defined
the evolute of a curve (Conics, Art. 248) as the locus of the
centres of curvature of the curve; but the evolute may also be
defined as the envelope of all the normals of the curve. For the
circle of curvature is that which passes through three consecutive
points of the curve, and its centre is the intersection of perpen-
diculars at the middle points of the sides of the triangle formed
by the points. But the lines joining the first and second, and the
second and third points, are two consecutive tangents to the curve 5
and the perpendiculars to them just mentioned are two consecutive
normals; the centre of curvature is therefore the intersection
of two consecutive normals; and the locus of all the centres of
curvature must be the same as the envelope of all the normals.

2 2
Ex. 1. To find the evolute of at - wn 1.

2
The normal is (Conics, Art. 180) oe ey = = ¢,

or, writing z’=a cos¢, y’=d sin ¢,

an ee A

coop sing ”

an equation of the class considered Art. 85, Ex. 2, whose envelope is therefore

asx? + b3y3 c3,
Ex. 2, The normal to a parabola is (Conics, Art, 213),
P(y—y') +: 2y' (c—2’) =0,
or 2y/ + ( p? — 2pa) y’— p’y=0,
an equation of the class considered Art. 85, Ex. 1, whose envelope, y’ being the
parameter, is
2 (p— 2x) + 27py?=0.
Ex. 3. To find the evolute of the semicubical parabola py? =<’.
The equation of the normal is
32"? (y—y') + 2py' (ea!) =0,

Substitute for y’ in terms of 2’ from the equation of the curve, divide by a'?, and
(putting zt = t), the equation becomes

Bt! + 2pt? — Spt yt —Qpa = 0,
whose envelope is |
P(p—18x)* = (4px + 72 y? + p?)?,

84 EVOLUTES.

Ex. 4. To find the evolute of the cubical parabola p?y=2%,
The equation of the normal is

32’? (y—y!) +p? (w—2’)=0,
or 32’ — 3p? yx’? + pta'— ptx=0.
Now the envelope of
at’ + 10dt? + 5et ++ f=0
is (af? — 12d?e)? + 128 (2e?—3df) (ae? — adef—9d*)=0.
Therefore the envelope in the present case is
3p? (x? — 735y")? + 438 (2p? — Say) (dp* — Ep’wy — Z65y*) = 0.
Ex. 5. To find the evolute of the cissoid (x?+ y?) «= ay?.
This is a unicursal curve, and writing the equation in the form (a—z)y?=2',

it is at once seen that this is satisfied by the values = The

a a a
1+ 04> 6 (1+ 6%)"
equation of the tangent at the point in question is easily seen to be
20%y —380’24+a—2=0,
equation of the normal is therefore

208+ (14868) y= 2 OF)

or 264x + 36?y —267a+ by —-a=0.
Forming the discriminant of this it will be found to contain as a factor («+ 4a)?+y?,
the remaining factor giving the equation of the evolute proper, viz.
yt + 32a? + Deas =0.
Ex. 6. To find the evolute of «3 + ys =as, For any point of this curve we may

write (see Art. 85, Ex. 2) 2’=acos*¢, y’=asin'¢. The tangent at that point
will be

i + ae =a
cosp sing .
and the normal x cosm — y Sing = a cos2¢q,
or (x + y) (cos — sind) + (a — y) (cos + sin Pf) = 2a (cos? — sin? dp),
x+y c—y £

or 2°a,

sin(p +4) | cos(p+ 4m)
whose envelope is (Art. 85, Ex. 2)

(wt yy + (@—y)b = 208.

100. The following investigation leads to the expressions for
the coordinates of the centre of curvature, and for the radius
of curvature ordinarily given in books on the Differential
Calculus. In this and the next article we use Cartesian rect-
angular coordinates. If a, @ be the coordinates of any point
on the tangent, 2 and y those of its point of contact, the

equation of the tangent is B—y= a (a— 2x); where _ which

EVOLUTES. 85

we shall call for shortness p, is to be found from the equation of
the curve. For the tangent passes through the point xy, and
makes with the axis of 2 an angle whose tangent is p (Art. 38).
The normal then being a perpendicular to this at the point ay,
has for its equation

| (ga) + (Bm ye 0 a cscccanevcoun sts (1).
We have now to’ find the envelope of this line which contains
the parameters x and y, which is given in terms of x by the

equation of the curve. Differentiating then with respect to a,
2

and writing oe =q, we see that the point of contact of the line

with its envelope is found by combining the equation with its
differential

Solving for a—a and 8—y from these equations, we have

—p(l+p’) l+p
a4-Zz= B a =
Gh ese
and the radius of curvature is given by the equation

?

1+p°)
R=y{(a-a)'+ (@—y}=O4PL
The values which have been obtained for the intersection of
two consecutive normals might have been found for the same
point considered as the centre of curvature.
Take the equation of any circle

(w —a)'+(y—B) =f,

and differentiate it twice, when we have
d
(w- a) +(y—8) * =0,

dy\" ay _
1+ () + (y— 8) 73 = 0.

But if the circle osculate a curve at any point, then (Art. 48)

PC an. .
at that point “ ; “a , have the same values for both. We
may therefore in these equations write for the differential coeffi-
cients, the values p and g obtained from the equation of the
curve, when they become identical with equations (1) and (2)
already obtained from other considerations.

86 EVOLUTES.

101. Since in practice y is not given explicitly in terms of a,
but both are connected by an equation U=0, it is convenient
to substitute for these expressions in terms of p and q, expres-
sions in terms of the differential coefficients of U. Let us write
as before

dU dU d°U d*U ad°*U

oak, 5 aM, Sea, Se =, oe

dix dy dx , dy 9 dxdy ?
then, since the coefficients of 2 and y in the equation of the
tangent are L and M respectively, the equation of the normal is

M (a — ©) —L(B— y) =O. cerecrreceeceees (1),

whence differentiating

(44952) (a— a) — (a4n$ ) (8-9) - - M+ LB =o.

But from the equation of the curve, 1+ M = 0, whence sub-

d dx
stituting for “i we have

(Lb — Mh) (a — x) — (Lh — Ma) (8 - y) + L?'+ M*=0...(2).

Solving, then, between equations (1) and (2), we have

Aer ed Arial eid -~M(L+M)

= 7M? —-2hLM+bL? 7 oM®-2hLM+oL?
2 2%

whence R= a(t mt)

aM* —2h LM + bL?*

102. This expression can be made to assume a more symme-
trical form by introducing the linear unit z, so as to give
the equation the trilinear form. For, by the theorem of homo-
geneous functions, |

(n—1) L=an+hy+gz, (n—1)M=he+ by +fz,
(n—1) N=gx+fy+cez, :
whence (n—1) (bL—hM) =(ab—h’) «+ (bg —fh)z,

(n— 1) (aM — hL) = (ab — h’) y + (af —gh) z.
Multiplying the first equation by L, the second by M, and adding
(a — 1) (BL? —2hL M+ aM’) = (ab —h’) (2L + yM)

| +2 (bg fh) L + (af gh) M},

EVOLUTES. 87

or since, by the equation of the curve 7 +yM+2N=0,

=—2{(fh—bg) L+ (gh—af) M+ (ab—h’) N}.
Substitute for Z, 1, N their values given above, and we have
(n—1)?(bL’— 2hIM+ aM’) =—2*(abe —af*—bg’—ch’ + 2fgh)=—Hz2’,
and the expression for the radius of curvature becomes

(n—1)? (LZ? + M”)3
i 2H ,

R=

For any point whose coordinates satisfy the equation H=0,
the radius of curvature becomes infinite, and the centre of
curvature at an infinite distance. This will take place when
three consecutive points of the curve are on a right line, for
then the circle through them becomes a right line, and its
centre becomes at an infinite distance. We might then, from
this value of the radius of curvature, arrive, independently of
Art. 74, at the conclusion that the intersections of U and H are
points of inflexion. The above equation gives us as conditions
that two curves should osculate, that we should have in addition
to the condition for ordinary contact L = @L’', M= 0M’, also

ee ee
G1 WF

The double sign in the value of the radius of curvature is
analogous to that in the value of the perpendicular on a right
line ( Conics, Art. 34); and, of course, if we agree to use the sign +
when the radius of curvature, and therefore the concavity of the
curve, is turned in one direction, we must use the sign — when
it is turned in the opposite direction. Since every algebraic
function changes sign in passing through zero, we see that at a
point of inflexion the radius of curvature changes sign, and
that as we pass such a point the concavity of the curve changes
to convexity, and vice versd (see fig. Art. 45). At a double point
the radius of curvature assumes the form : , and its value must
be determined by the ordinary rules in such cases. In fact,
each branch of the curve has its own curvature at the point.
At a cusp it will be found that the radius of curvature vanishes.

88 EVOLUTES.

103. The length of any arc of the evolute is equal to the difference
of the radit of curvature at its extremities,

For, draw any three consecutive nor- zp
mals to the original curve: let C be the
point of intersection of the first and
second, C’ of the second and third; then
since, ultimately, CR =CS, O'S=C'T;
CC’, which is the increment of the are
of the evolute, is also the increment of
the radius of curvature.

Hence, if a flexible thread be supposed rolled round the
evolute, and wound off, any point of it will describe an ¢nvolute
of the curve CC’; that is, a curve of which CC" is the evolute.
It was from this point of view that Huyghens, the inventor of
evolutes, first considered them, and it was hence that the name
evolute was given.

<2

=
-
mn

ic’
i
i

104. We add here a formula which is sometimes useful
for finding the radius of curvature of a curve given by polar
coordinates. The polar equation p=/f(w), can be transformed
into one of the form p=f(p), where p is the perpendicular
from the pole on the tangent, and is given by the equations
(Art. 95),

, dw
p=p sing, sph ie ae ;

Let the distance from the pole to the centre of curvature be p,,
and the radius of curvature 2, then (Euclid 11. 13)

p, =p +’ 2kp.
If we pass to the consecutive point of the given curve, p, and &

remain constant, and differentiating, we have k= poe which

is the required expression for the radius of curvature.
When & has been thus expressed in terms of p, p, if we
eliminate p, » between the equations

p=f(p)) pi=p + B—2kp, p,’=p*—p',
the last of which is obviously true, we shall have the relation
which subsists between the p, and p, of the evolute; but it is
not always easy to pass hence to the relation between the p,
and the w, of the evolute.

- EVOLUTES. 89

As an example take the curve p"=a”™ cos mw, we find here
p=p cosmm, and hence p”’=a"p, for the relation between p

Pao ws a” for
(m+1)p? (m+1) p™

and p. And we then have R=

the radius of curvature.
The equations

vr pl =p’ + B’—2Rp,

a zoe p- —p°
give at once p,”, p, each as a function of p, and thus virtually
the equation of the evolute in the form p,=¢ (p,), but the
elimination cannot be actually performed.

It is however easy to find the equation of the reciprocal of
the evolute in regard to a circle described about the pole as its
centre. ‘Taking for convenience the radius of the circle to be
=a; then if p, is the radius vector for the reciprocal curve, and
w, the inclination to a line at right angles to that from which
w is measured, we have p, =p sinmo, and then |

a a

Fer

gue ay
De COS"™mw SIN Nw

Moreover (Art. 95), ,=(m+1)@j; wherefore the relation
between p,, @,, or equation of the reciprocal of the evolute is

Lice OE, a
is cos ga l ma tL =a.
It will readily appear that the locus of the extremity of the
polar subtungent (see Conics, Art. 192) of any curve is the
reciprocal of the evolute of the reciprocal curve. Thus this
locus is a right line for the focal conics, since the evolute of the

reciprocal then reduces to a point.

105. When we are given the tangential equation of a curve
u=0, we can obtain directly the line coordinates of the normal
and the tangential equation of the evolute. Tor if a'6’y' be the
line coordinates of any tangent, then « = ;+B a +y¥ = 0 is
the equation of the point of contact ; and if v=0 be — taaigental |

dv’
a 7 dy
N

equation of any pair of points JJ, then ad 7+ BS wei

90) EVOLUTES.

is the equation of the pole of the given tangent with respect to
LJ; or, in other words, of the harmonic conjugate in respect to
these points of the point where J is met by the given tangent.
When JJ are the circular points at infinity, the second equation
represents the point at infinity on the normal; the two together
determine the line coordinates of the normal; and if between
them and the equation of the curve we eliminate a'@’y’, we shall
have the equation of the evolute. In the system of tangential
coordinates which answers to ordinary rectangular coordinates,
the equation which represents the circular points LJ is a’ + 6’ =0,
(see Conics, Art. 385), and the second equation a a +8 oa +y S
is the well-known condition of perpendicularity aa’ + BR’ =0.

Ex. To find the equation of the evolute of a central conic given by its tangential
equation (see Conics, Art. 169, Ex. 1) a2a? + 6?82= 1. Here the two equations which
determine the coordinates of the normal are a?aa’ + 6786’ = 1, aa’ + BP’ = 0, whence

1

aa’ = — Bf’ = 3 Substituting for a’ and f’ in a?a’”? + 0B’ = 1, we get the tangential
; a PB
equation of the evolute-.+ —=c',
oie «

106. We give next some examples of the more general
problem in which that of evolutes is included, viz. (see Art, 99)
to find the envelope of the harmonic conjugate of the tangent to
a curve with respect to the lines joining its point of contact to
two fixed points J, J. This line may be called the quasi-normal
and its envelope the quasi-evolute.

Ex. 1. Let the curve bea conic. Take the line JJ as the base of the triangle of
reference, and let its vertex be the pole of this line with respect to the conic, then
the equation of the conic will be of the form (ax + y) (x + by) = 2’, and that of any
tangent will be

6? (ax + y) — 202 + (a + by) = 90.

The equation then of any line which together with this and the lines a, y, divides
z harmonically will be of the form

62 (ax — y) + (@ — by) = Mz.
We determine M from the consideration that the line is to pass through the point
of contact, for which we have @ (ax + y) = z, 62 =x + by. whence
a Ow) __ 2 (a6?—1).
~ Fao —1)?7~ O(@b—1)?

2 (b= ab)

and we find Fig 1h 8 ‘

EVOLUTES. 91

If we write then az —y = Y, « — by = X, 8z = (ab — 1) Z, the equation of the quasi-
normal becomes
altZ + 46°5Y + 40X —bZ=0,
and the envelope is a curve of the fourth class whose equation is
(abZ? + 4XY)8 + 272? (aX? — bY")? = 0,
which represents a curve of the sixth degree having the points XZ, YZ for cusps, 2

being their common tangent, and besides four other cusps at the intersections of
abZ? + 4XY, aX? — bY

Ex. 2. Let the conic pass through one of the points J, J; or, as we may say,
let it be semicircular. Then we have say d= 0, and zz is on the curve, « being the
tangent, The equation of the quasi-normal then becomes

a®Z +42Y +4X=0,

and the envelope is only of the third class, its equation being 64Y? + 27a?XZ?= 0,
which represents a cubic having YZ for a cusp and X¥ for a point of inflexion.

If the curve pass through both J and J; making a and 5 both = 0, we see that the
equation of the quasi-normal reduces to 6?7Y+ X, and that the line therefore passes
through a fixed point ; namely, the intersection of X, Y, the tangents at J, J.

Ex. 3. Let the conic touch the line JJ, The most convenient lines of reference
then to choose are this line together with the two other tangents through J, J, and the
equation of the conic is
x? + y? + 2? —Qyz — zx — Qxy = 0,
or @ (2e+ 2y— 2) = (wx — y)*.

The equation of the tangent then is
2x + 2y —2— 20 (x —y) + Pz = 0,
and we have for the point of contact
e—y = Oz, 2e+ 2y— 2 = Oz,
The equation of the quasi-normal then is
x—y—O(e@+y)=2z {0-30 (1 + &)},
or z— 6 (2n+2y+2)+2 (x-—y)=9,
and the envelope is also of the third class, viz. the cuspidal cubic whose equation is
27z (a — y)? = (2a + 2y +z),

Ex. 4. The three preceding examples might also have been investigated by
supposing the conic to have been given by its general equation. The tangent then
at any point aGy being

(aa +hB+gy) «+t (ha+bB+fy) 4+ (ga+fB + cy) 2=9,
the quasi-normal is
¥ {aa + hB + gy) x — (ha + 6B +fy) y} = (aa? — bp? + gay — fBy) ¢.
We have then to find the envelops of
aza® — ba? + (fy — ga) y? + (by — fz — ha) By + (hy + gz — az) ya,
where a, B, , are parameters, also satisfying the condition
aa? + bB? + ey? + 2fBy + 2gya t+ 2haB = 0.

And (Art. 96) the envelope is formed by the process given (Conics, Art. 872) for
finding the condition of contact of two conics. We must form then the invariants

92 EVOLUTES.

of this system of quadratic functions, and the discriminant of the first is 22S,
where S is

(ab — h?) (ax? — by?) + (bg? — af?) 22 + 2b (gh — af) yz — 2a (hf — bg) az.
We have
© = — (ab — h?) (ax* — 2hay + by”) + (Baf? + 38bg? — 4abe — 2fgh) 2?

+ (4bgh — 2abf — 2fh*) yz + (4abf — 2abg — 2gh*) xz.
©’ vanishes and the envelope is therefore 27 Az?S? = 63, which, as before, is of the
sixth degree having six cusps, two of which lie on z. But first let z touch the
conic, then ab—h?=0, and S and © take the form Lz, Mz where L and La are
linear and the envelope takes the form zl? = M3, and is a cuspidal cubic having z
as a stationary tangent. Secondly, let the conic pass, say through J or yz, then
a=0, S becomes 6 (hy + gz), and © takes the form (hy+gz) M. The equation
then becomes divisible by (hy + gz)’, and the envelope is of the form 2? (hy + gz) = IZ*.
It will be observed that hy + gz is the tangent to the conic at the point J, and that
it is an inflexional tangent of the envelope.

107. In general, as Professor Cayley has remarked, if
Lx+My+ Nz be the tangent at any point 2’y’z', and aBy,
a'B'y' the coordinates of J, J, the equation of the quasi-normal is

%, Y, & XH, Y;, %
(La! + Mp’ + Ny)| a’, y', «| + (La+ MB +My) |a!, y, 2'| =0.
a, B, é f a, B, ry

For the two determinants, which we shall call for the moment
A, 4’, severally represent the lines joining 2'y’z’ to J and J, and
since the tangent passes through their intersection we must

‘have an identity of the form Lx+ My+Nz=AA- Bd’.
Substitute successively in this identity a'@’y' and aSy for ayz,
and we determine A and Bas proportional to La’ + Mp’ + Ny’
and La+ MB + Ny, and therefore the equation of the harmonic
conjugate of the tangent with respect to A, A’ is of the form
written above.

108. Let us examine more particularly the case where one
of the points a@y is in the curve, and, for simplicity, we take
its coordinates 1, 0,0; that is to say, we suppose the point to
be yz; and we take the line z to be the tangent at it; and we
shall prove that the envelope contains z as a factor. We
may also without loss of generality take the second point
as 0,0, 1 oray. Making 8 andy, a’ and #’ =0 in the preceding

— equation, it becomes

N (yz! —2y') + L (xy — yx) = 0.

EVOLUTES. 93

Let us suppose now that 2’, y’, 2’ are expressed in terms of a
parameter ¢, the point 2, 8, y auswering to the value ¢=0, and
we must have ¢ as a factor in the expression for y’, and ¢ in
that for 2’, in order that the equation of the tangent may
reduce to z=0. In general, since the tangent is the line joining
the point «'y'z’ to the consecutive a’ + dx’, y' + dy’, 2'+ dz’, its
equation is F
x (y'dz' — z'dy’) + y (z'dx' — a'dz') + 2 (a'dy' — y'dz') =0.

L, M, N are the coefficients of x, y, z in this equation, and ¢ is
a factor in M, and ¢#’in LZ. If then the equation of the quasi-
normal be arranged according to the powers of ¢, it will be
found that there is no term independent of ¢, and that z is
a factor in the coefficients both of ¢ and of ¢. Now the
discriminant of a function A+ B¢+ Ct°+&e. is of the form
Ag+ By (Higher Algebra, Art. 107), and therefore a factor
which enters into both A and B will be a factor in the
discriminant. Also if in the discriminant we make B= 0, the
remainder will be of the form A(Af+ C*W): thus it appears
that the envelope will have z for an inflexional tangent (compare
Art. 99, Tux. 4).

109. It has been remarked (Conics, Art. 385) that the
relation of perpendicularity may be further extended by
substituting for the points J, J, a fixed conic, and by regarding
two lines as perpendicular if each pass through the pole of the
other with regard to that conic. In this extension then, what
answers to the normal, is the line joining any point on a curve
to the pole of its tangent with respect to the fixed conic; or, in
other words, the line joining the point to the corresponding
point on the reciprocal curve with regard to the fixed conic.
Thus the curve and its reciprocal have the same normals. For
example, taking the fixed conic as 2° + y* +2”, the coordinates of
the pole of any tangent to a curve are L, M, N, and the
equation of the line answering to the normal is

a (Mz' — Ny’) + y (Nx' - Lz’) + 2 (Ly' — Ma’) =0.
If the curve were a conic, this equation would be of the second

degree in w'y’z', and the envelope would be found as in Ex. 4,
Art. 106,

94 EVOLUTES.

110. The following remarks are a useful preliminary to the
investigation of the characteristics of the evolute of any curve.
The normal at any point of a curve at infinity coincides with the
line at infinity ttself. It has been already remarked (Art. 105)
that we may generalize the conception of a normal by substi-
tuting for the two circular points at infinity two finite points
I, J, and that then if the tangent at any point P meet L/ in J,
and if M' be the harmonic conjugate of J with respect to J, J,
the line PM" may be regarded as the normal. From this
construction it appears at once, that if the point P be on the line
IJ, then PM’ will coincide with that line. An exception occurs
where the puint P coincides with either J or J; then the points
M, M' coincide, and the normal coincides with the tangent (see
Conics, Art. 382, note). Thus, then, ¢f the curve pass through
either of the circular points at infinity, the normal at that point
will coincide with the tangent.

111. We proceed now to determine the class of the evolute
of a given curve; or in other words, the number of normals to
the curve (tangents to the evolute) which caz be drawn through
any point. By the law of continuity, the number of normals
is the same, whatever be the point through which they pass. It
is enough, therefore, to examine the case when the point is at
infinity. But the number of normals, distinct from the line at
infinity itself, which can be drawn parallel to a given line, is
equal to the number of tangents which can be drawn parallel to
a given line, that is, to the class of the curve. And we have
seen in the last article that the m normals, corresponding to
the m points of the curve at infinity, coincide with the line
at infinity, and therefore also pass through the assumed point.
Thus then the number of normals which can be drawn to the curve
from any point, is equal to the sum of the order and class of
the curve—or, what is the same thing, the sum of the orders of the
curve and its reciprocal. If the line at infinity be a tangent to
the curve, then the number of finite tangents which can be
drawn through a point at infinity, is plainly one less than in the
general case, and therefore the number of normals is also one
less. Thus four normals can be drawn from a given point to a
conic in general, but only three to a parabola.

EVOLUTES. 95

Again, if the curve pass through either circular point, we
saw (Art. 110) that the normal at that point does not coincide
with the line at infinity, and therefore, that for every passage
through a circular point, the number of normals is one less
than in general. Thus in the case of the circle which passes
through the two points J, J, the number of normals through
a point is reduced*by two, and is two instead of four. Thus
then if m and » be the degree and class of a curve which passes
f times through a circular point, and touches the line at infinity
g times, the class of the evolute is

n=m+n—f—g.

These results might equally have been obtained from the con-
sideration that if in the equation of the normal M (a—x)=L(8—-y)
we suppose a, 8 given and a, y variable, we shall have the
equation of a curve of the m degree, whose intersection with
the given curve determines the points the normals at which
pass through a, 8. If the curve have no multiple points, the
number of intersections will be evidently m* or m+n: and there
is no difficulty in showing, that in the general case of 6 double
points and « cusps, the order is m’ — 26 — 3x, that is m+n.

112. We next examine the degree of the evolute, and again
it suffices to examine the number of points in which the line at
infinity meets the evolute. Now if two consecutive normals
to the original curve be parallel, the corresponding tangents will
coincide; the points at infinity therefore on the evolute arise
in general from the points of inflexion on the given curve.
But to these must be added those arising from points at infinity
on the given curve, which points (Art. 111) also give rise to
points at infinity on the evolute. But we say, moreover, that
these will be cusps on the evolute having the line at infinity for
their tangent. Let M be any point on the line Z/, and ’ its
harmonic conjugate, then we saw that the line answering to the
normal at JM is the line //: but if the consecutive points of the
curve, antecedent and subsequent to M be Z and N, their
normals are LM’, NM’. Hence &M’ is a point through which
three consecutive tangents to the evolute pass, and is therefore
a cusp having /J for its tangent. Since then the tangent at a
cusp meets the curve in three consecutive points, the m points

96 EVOLUTES.

at infinity of the given curve, give rise to the same number
of cusps on the evolute which are met by the line at infinity in
3m points. If we add these to those already obtained, we find
the degree of the evolute =4+3m, or the number which we
have called a (Art. 83).

If the curve pass through either point Z, J, we have seen
that these give rise to no points at infinity on the evolute, and
therefore the degree will be less by three.

If the line LJ touch the curve, the normals for the two
consecutive points in which it meets the curve coincide with L/;
we have therefore two consecutive tangents to the evolute
coincident, or a point of inflexion on the evolute having LJ for
its tangent. As this takes the place of two cusps which we
have when JJ meets the curve in distinct points, the degree of
the evolute is reduced by three; and if we use f and g in the
same sense as in the last article, we have for the degree of the
evolute

m'=a—3(f+g).*
The values given show that the degree and class are the same
of the evolute of a curve and of its reciprocal as Art. 109 might
lead us to expect.

113. There will in general be no points of inflexion on the
evolute. For if there be such a point, two consecutive tangents
to the evolute (normals to the curve) coincide; but it is plain,
on considering the figure, that two consecutive normals cannot
coincide unless the corresponding tangents coincide with their
normals and with each other, which could only happen in the
exceptional case where the original curve had an inflexional
tangent passing through J or J.

If, however, the curve touch LJ, we have seen (Art. 112)
that there is a point of inflexion at infinity, and if the curve
pass through J or J (Art. 108), that the evolute has an
inflexional tangent passing through the same point. We have
thus conditions enough to determine all the characteristics of

the evolute, viz. :
m'=a-3(f+g),n=mt+n—(f+g),¢=(f+9);

* Some particular examples show that these formule must be modified when J or
J is a multiple point at which two or more tangents coincide. Thus if either be a
cusp, the diminution of degree is 4 not 6,

Ss

CO Pea Mags Asse,
ON ig eK ey hee es ae de lok. San tie Ay x

EVOLUTES., 97

whence by Pliicker’s formula «’ =3a—3 (m+n)—5 (f+),
a’ =38a—8(f+g); and we can in like manner write down the
number of double points of the evolute, and of its double
tangents; these double tangents are, it is clear, double normals
of the original curve. |

The “deficiency” (Art 44) of the evolute is the same as
that of the original curve, as may be verified by using the
expression for the deficiency 4 {a —2(m+n)}+1.*

114. The number of cusps on the evolute may also be
investigated directly. We shall have a cusp on the evolute,
when three of its consecutive tangents (normals to the curve)
meet in a point; or, in other words, when four consecutive
points of the curve lie on a circle. If this be ‘the case the
radius of curvature remains constant when we pass to a con-
secutive point. Differentiating then the expression given
(Art. 102) we have } r

(L'+ M) (a dnt dy) = 3H ((aL-+hM) de+ (LL +b1)dy),
and eliminating dx: dy by the equation Ldx+ Mdy=0, we
have

rat: 7) = 3H {(a—b) LM +h(M? — L)}.

Since # is of the order 3 (m—2), Z and M of the order m-— 1,
and a, 6, h of order m—2, this equation represents a curve of
the order 6m—10, whose intersections with the given curve are
the points where the osculating circle has contact of the third
order.t Ifthe curve have no multiple points, these m (6m — 10)
points together with m points at infinity give rise to m (6m —9)’
cusps on the evolute, a number in accordance with the

preceding formule.
We might, in like manner, investigate the characteristics of

(i? 4+ WP) (a1

* In general the deficiency of two curves is the same, if one is derived from the
other by such a process that to one point on either curve answers one point on the
other.

+ In a subsequent part of the work the question of conics having with the curve
contact of a higher order than the second is more fully considered, and a formula
given for the aberrancy of curvatwre or deviation of the curve from the circular
form,

O

98 CAUSTICS.

the evolute in the more general sense of the word indicated
Art. 109, and we should find that the formule we have already
obtained will apply, f being now the number of contacts of
the curve with the fixed conic, and there being no aingolarity
answering to g.

CAUSTICS.

115. As a further illustration of envelopes, we add some
mention of caustics, the investigation of which, though suggested
to mathematicians by the science of optics, belongs purely to
the theory of curves. ‘The subject has some historical interest,
caustics being among the earliest questions, involving the
problem of envelopes, actually discussed.*

If light be incident on a curve from any point, the reflected
ray is found by drawing a line, making with the normal the
same angle which is made with it by the incident ray; the
envelope of all these reflected rays is the caustic by reflection.

It is easy to form the general equation of the reflected ray.
Let the equations of the tangent and normal at the point of
incidence be 77=0, N=0; then the equation of the incident
ray is J’N— TN'=0, where 7’N' are the results of substituting
the coordinates of the radiant point in 7’ and WN; the
reflected ray then, which is the fourth harmonic to these three
lines, will have for its equation

T'N+ TN'=
and the envelope can then be found by the preceding rules.

Ex. To find the caustic by reflexion of a circle.

The reflected ray is, by the preceding (a8 being the coordinates of the radiant
point, and the tangent and normal being x cos@ + y sin@—r, and a sin 6 — y cos8),
(a cos0 + 8 sin 6 — r) (x sin 8 — y cos6) + (x cos@ + y sin @ — ”) (a sin @ —'B cos 6) = 0,
or (ay + Bx) cos26 + (By — ax) sin20+r («+ a) sind —r (y+ 8) cos6 =0,
whose envelope is (Ex. 3, Art. 85)
[4 (a? + B*) (a? + 9?) — 9? {((@ + a)? + (y+ B)*}]}* = 27 (Be — ay)? (2? + 9? — a? — f)?,

116. Instead of finding directly the envelope of the reflected

ray, M. Quetelet has given a method, which is more convenient
in practice, of reducing the problem to that of evolutes; since
the caustic would be sufficiently determined if we knew the curve
of which it was the evolute.

* The subject of caustics was introduced by Tschirnhausen, Acta Eruditorum
1682, referred to by Gregory, Examples, p. 224.

CAUSTICS. 99

“Tf with each point successively of the reflecting curve as
centre, and its distance from the radiant point as radius, we
describe a series of circles, the envelope of all these circles will
be a curve, the evolute of which will be the caustic required.”
‘The following (due to M. Dandelin) is a more convenient form
of stating the same theorem: Jf we let fall from the radiant
point O the perpendicular OP on the tangent, and produce tt
so that PR = OP, then the caustic is the evolute of the locus of ft.

For RT is evidently the di-

f
rection of the reflected ray, and KN
if we draw the consecutive ray,
then, since OT, TV; O71", T'V, P| __\T T
make equal angles with 77", =| oe
OT+ TV = OT' + T'V (Conics,
Art. 392); therefore VR = VR’, 0 ¥

and therefore V£ is normal to the locus of 2.

The locus of /, the foot of the perpendicular on the tangent,
we call the pedal of the given curve. The locus of £& is plainly
a similar curve, and its equation can always be written down
when the equation of the reciprocal of the given curve with

regard to 0 is known, by substituting ; for p in the polar

equation of that reciprocal. Thus the caustic by reflexion, of a
circle, is the evolute of the /imagon, (see Ex. 5, Art. 55), since its
equation (the radiant point being pole) as found by the rule
just given is of the form

p=p(itecosa).

117. If light be incident from any point on a curve, the
refracted ray is found by drawing a line, making with the normal
an angle whose sine is in a constant ratio to that of the angle
made with the normal by the incident ray, and the envelope of
all these rays is the caustic by refraction.

M. Quetelet has reduced in like manner these caustics to
evolutes by the following theorem, the truth of which it is easy
to see. “If with each point successively of the refracting curve
as centre, and a length in a constant ratio to its distance from
the radiant point as radius, we describe a series of circles, the
envelope of all these circles will be a curve whose evolute is the

100 | CAUSTICS.

caustic by refraction.” In fact, the method of infinitesimals
readily shows that, in consequence of the law of refraction, the
increments of the incident and refracted rays are connected by
the relation mdp+dp'=0, it follows then that if, on the
refracted ray produced, 7 be taken =mOT, T'R'=mOT",
then VR = VA’, and therefore the refracted ray is normal to
the locus of &.

We add geometrical investigations in relation to two
"interesting cases of caustics by refraction.

(1) Zo find the caustic by refraction of a right line. |
Let fall a perpendicular on the line, and produce it so that

AP=PB; and let a circle be described L

through A, B, and the point of incidence A
fi; let LR be the refracted ray; then

obviously the angle ALB is bisected, and aN

O
AL+LB; AB:: AL: AO \
: ::sn AOL: sn ALO; y
but AOL is the angle which the re- a
fracted ray makes with the perpendicular to the line, and
ALO=BLO=BAL is the angle which the incident ray makes
with the perpendicular; the ratio of AL +Z8B to AB is there-
fore given; the locus of Z is an ellipse, of which A and B are
the foci, to which ZA is normal, and of which, therefore, the
caustic is the evolute.

(2) Yo find the caustic by refraction of a circle.

Let a circle be described through A, the radiant point, and
2, the point of incidence, to touch Of; then
the point Bis given, since OA. OB= OL’.
The ratio RA: RB is by similar triangles ™,
equal to the givenratio04: O08. The ratio
RA: RM is equal to sn kbA: sn khBM;
but RBA = PRA, the angle which the in- /
cident ray makes with the normal to the |
curve, and RBM=PRM, the angle which
the refracted ray makes with the same %
normal; hence the ratio RA: LM is also
given. Now since

AM. RB+ MB. AR= RM. AB,

A

ee ee

oe oe ee

PARALLEL CURVES AND NEGATIVE. PEDALS. 101

if we denote the distances of M from A and B by, p, p’, these
distances are connected by the relation

RB rat
Rut Ru? = 4:

Now, a Cartesian is defined as the locus of a point whose
distances from two, given foci are connected by the relation
mp + np'= c; and it is proved precisely as at Conics, Art. 392, that
the normal to such a curve divides the angle between the focal
radii into parts whose sines are in the ratio m:n. Hence the
locus of WM is a Cartesian, of which A and B are foci, and
it is obvious that JZ is normal to the locus, and therefore the
caustic is the evolute of this curve.*

The ellipse in (1) and the Cartesian in (2) are curves cutting
at right angles the refracted rays;:the curve cutting at right
angles the reflected or refracted rays is termed the secondary
caustic.

PARALLEL CURVES AND NEGATIVE PEDALS.

117 (a). It remains briefly to notice one or two other classes
of envelopes. We have already mentioned the problem of
finding the curve parallel to a given one.. This may either
be treated as that of finding the envelope of a tangent parallel
to each tangent of the given curve, and at a fixed distance
from it, and so of finding the envelope of

Lae + My + Nz =kz /(L’ + MU”),

or else, as we have already seen, it may be regarded as
that of finding the envelope of the circle of given radius
(w—a)’+(y— 8)*=k’, whose centre a satisfies the equation of
the curve, or, what is the same thing, of finding the condition
that this circle should touch the given curve. The result will
evidently be a function of k*. In some exceptional cases to be
mentioned presently, the result can be resolved into factors, as
for instance, the parallel at a distance & to a circle of radius @
consists of a pair of circles of radii a+%. But, ordinarily, such a
resolution is not possible, and the two tangents at the distance

* This proof was communicated to me by Dr. Atkins,

102 PARALLEL GURVES AND NEGATIVE PEDALS.

+ from any tangent will touch the same parallel curve.
Hence, the number of tangents which can be drawn parallel
to any given line is double that which can be so drawn to the
original curve, or n'=2n. In like manner, to each inflexional
tangent on the original correspond two on the parallel curve,
or ?’=2s. ‘To find the order of the parallel it suffices to make
k =0 in its equation, which will not affect the terms of highest
dimensions in the equation; but what was proved for the conic
(Conics, Art. 372, Ex. 2) is true in general, that the result of
writing &=0 in the equation of the parallel is the original curve
counted twice, together with the two sets of n tangents drawn
from the points J, J to the curve. The order then is 2 (m+n).
There is no difficulty in seeing how these numbers are modified
if the original curve touch the line at infinity or pass through
the points J, J. We arrive in this way at Professor Cayley’s
formule

m' =2(m+n)—2( f+), n' =2n, v =2¢=— 6m + 2a,

Ke =2a—6 (f+ 9): fi =2 (n—g), 9 = 29.
The parallel curve and the original have the same normals and
the same evolute, but every normal to the parallel curve is so
generally in two places, answering to the values + h.

Ex. 1. To find the parallel to the ellipse or parabola. See Conics, Art, 372.

2
Ex. 2. To find the parallel to 23 + ye ~a*, The equation of any tangent is

(see Art. 99, Ex. 6)
xcosp+ysing =asin¢ cos¢.
Hence, that of a parallel at the distance & is
xcosp+ysing=k+asing cosq,
whose envelope is (see Art. 85, Ex. 3)
{3 (a? + y? — a®) — 4h?}8 + {27axy — 9k (x? + y?) — 180k + 8k)? = 0.
This is one of the cases where the parallels answering to the values + & are different
curves and not different branches of the same curve.

The curve whose equation has been just obtained is the envelope of a line on
which a constant intercept is made by two fixed lines. If the lines are at right
angles, taking them for axes it is seen immediately that the equation of a line. whose
length is a inclined at an angle ¢ to the axis of ris x sing + y cos@ = acos® sinlp,

ccm

whose envelope is x* + y* =a*. But consider fora moment a diameter and a parallel
chord of a circle, and it is evident that if a line whose length is a subtend a right
angle at any point, a parallel line at a distance Ja cos@ will make an intercept
a sin ¢ on a pair of lines including an angle @, and equally inclined to the rectangular
lines. Hence, obviously the envelope of a line whose length is a sing intercepted
between the oblique lines is a parallel (answering to the value k = 3a cos) to the

2 2 2
envelope for the rectangular lines, z° + y° =a*,

eg EE Oe ae a a ee ae

PARALLEL CURVES AND NEGATIVE PEDALS. 103

118. If ax+ Py++¥ be a tangent to a curve (the equation
being expressed in ordinary rectangular coordinates), then
evidently ax+By+y+k/(?+ 8’) is a tangent to the parallel
curve; and it follows at once, that if we have the tangential
equation of the given curve, we obtain that of the parallel by
writing in it for y,y+p where p is /(a’+ 8’). Hence the
tangential equation of the parallel to a curve whose tangential
equation is V=0 is

V+kp Tt+a nl kp” oo

The equation is cleared of radicals by transposing to one side
the terms containing the odd powers of p and squaring, when
we obtain an equation the order of which is double that of the
original tangential equation, in conformity with what was proved
in the last article.

+ &e. = 0.

Ex. 1. To find the tangential equation of the parallel to — ane Y= me Be ke

tangential equation of the ellipse is (see Conics, Art. 169, Ex, 1) a®a? + “86? = y’, whence
that of the parallel is

ara + U9 = (y + kp)’,
or {(a? — A?) a? + (0? — hi?) B? — 7}? = 4h? (a? + 2) vy

Ex. 2. To find the tangential equation of the parallel to the parabola y? = pa.
The corresponding tangential equation is pB? = 4ay; hence that of the parallel is

(pP? — dary)? = 16470" (a? + B*).

Ex. 3. To find the tangential equation of the parallel to a circle. The tangential
equation to the circle whose centre is the point a, b, and radius c, is (Conics, Art, 86)
(aa + bB + y)*? = c? (a? + B?); therefore that of the parallel is

(aa + bB + y + kp)? = ¢’p?,
which breaks up into factors, and gives
aat+bB+yt+kp=+e;
whence, clearing of radicals,
(aa + 6B + y)? = (¢ + k)? (a? + A),

representing a pair of concentric circles whose radii arec+4%, as is geometrically
evident,

119. In precisely the same. manner, as in the last example,
it is proved that if the tangential equation of a curve be of the
form wv’ (2° + 8”) =v’, the parallel will break up into two factors
of like form with the original, the parallels answering to the

104 PARALLEL CURVES AND NEGATIVE PEDALS.

values + being distinct curves, and not different branches of.
the same curve. For suppose that by the substitution of y + kp
for y, u becomes u+whkp +u'k'p* + &e., and similarly for v;
then up” = v" becomes

(wtwhkp +u'k'p® + &e.) p? =(v+ukp + v'k'p? + &e.)’,

which is at once resolvable into factors which can be rationalized
separately, giving the result

fu + u'kip? + &e. + (vk +0'"k'p? + &e.)}*p”
={v+v'k'p’ + &e. + (whkp’ + wu h'p* + &e.)}*.

Thus the equation given for the parallel of a conic is of the
form considered in this article, and it can be now easily verified
that the parallel to that parallel at the distance £' consists of
the two parallels to the conic at the distances 4 +k’, as manifestly
ought to be the case. ‘Take again the curve already mentioned,
ao + ye =a, whose tangential equation is (a?+*)7= a*a' 8,
which being of the form here considered, shows that the parallel
breaks up into factors. The tangential equation of the parallel
is in fact (a° + 8") y*={aaB+hk (a’+ *)’}.

If we take for uw and v respectively the most general functions
of the first and second degrees in a, 8, y, u’p*=v" denotes a
curve of the fourth class having two double tangents, and
which is therefore of the eighth order. But these functions may
be so taken that the double tangents shall become stationary
tangents, and that the curve may have another double or
stationary tangent, and in this way we can form the equation
of a curve of the third or fourth order whose parallels break
up into factors. Of this kind is the reciprocal of a Cartesian, as
will afterwards be shown.

120. If we had been using trilinear instead of rectangular
equations, it follows, from Conics, Art. 61, that the equation of a
parallel to ax+Py+ yz, at a constant distance from it, is of

the form
ax+ By+yz+m(esinA + ysinB+ zsin C) /(S)=0,
where S is
a + B? +9°- 2By cos.A — 2ya cos B— 2a/8 cos C,

PARALLEL CURVES AND NEGATIVE PEDALS. 105

and we see that if in the tangential equation of a curve we
write for a, 8, y,

atmsinA /(S),8+msinB (8), y+msin Cy/(S),

we shall have the tangential equation of a parallel curve. We
saw, Conics, Art. 382, that S= 0 is the tangential equation of the
points JJ; and jt is at once suggested, that if S=0 be the
tangential equation of any two points, and ax+by+cz=0
the line joining them, then considering the circular points at
infinity as replaced by the two points in question, the envelope
of ax+Py+yz2, and of ax+ Byt+yzt+(ax+ by +cz) /(S) are
quasi-parallel curves.

121. We called (Art. 116) the locus of the foot of the
perpendicular on the tangent from a given pole or centre, the
pedal of the given curve. Having found the pedal we may
find its pedal again, &c., and so have a series of second, third, &c.,
pedals of the given curve. Or we may continue the series
the other way, the curve of which the given curve is the pedal
being the first negative pedal, and so on. The problem of
finding the negative pedal is that of finding the envelope of a
line drawn perpendicular to the radius vector through its
extremity ; or, in other words, it is that of finding the envelope of

ON + By = a+ Bp,
where a, 8 satisfy the equation of the curve. We have just

- seen that the problem of finding the parallel curve is that of
finding the envelope of

2an + 2By + kh? —2?- y’=a' + B’,
subject to the same conditions; and accordingly Mr. Roberts
has remarked that the two geometrical problems are both

reducible to the same analytical problem, viz. that of finding
an envelope of the form
Aa + BB+ C=a' +p’,
and that if we had the equation of the parallel curve we could
deduce that of the negative pedal, by writing in it h*= 2+ 7’,
and then writing 42, 4y for x and y. Ordinarily, indeed, the
problem of finding the parallel curve is the more difficult of the
two; but this method gives immediately the negative pedal of
P

106 PARALLEL CURVES AND NEGATIVE PEDALS.

the right line or circle. For the parallel to a right line is a
pair of equidistant parallel lines, and the parallel to a circle
of radius a is two concentric circles of radii atk. In either of
these cases, then, the equation of the parallel curve can be
written down without calculation, and the negative pedal thence
derived by the process just indicated.

122. If for any curve there is taken on each radius vector
OP from an arbitrary origin or centre of inversion a portion
OP’ equal to the reciprocal of OP, the locus of P’ is said
to be the inverse of the given curve. From this definition it
is easily inferred that the pedal of a curve is the inverse of
its polar reciprocal, and that the first negative pedal is the
polar reciprocal of its inverse; the reciprocation being per-
formed in regard to a circle described about the origin or centre
of inversion as its centre.

There is no difficulty in deducing, by reasoning similar to
that used in other similar cases, the characteristics of the curve
inverse to a given one, and hence those of the pedal and of the
negative pedal respectively, and it is sufficient to give the
results. We use f and g in the same sense as before to denote
the number of times that the curve passes through a point J or
J, or that it touches the line Z/; f’ and g' denote the reciprocal
singularities, viz. the number of times the curve touches a line »
OI or OJ, or that it passes through the origin; p and g denote
the number of coincidences of tangents when the origin or when
a point J or J is a multiple point [for example, we should have
py =1, if the origin were a cusp], and p’, q' denote the reciprocal
singularities; then for the inverse curve we have

M=2m—f—g', N=n+2m—2(f+g')—-(f+9)+ (pt);
F=2m~f- 2%, G=p, F=q, @=m—f, P=g, Q=f.
Hence we must have for the pedal
M=2n- f'—g,N=m+2%n—-2(g+f)-9F + f)tP +4,
F=2n-29-f', G=p', f=, G=n—-f', P= 9, V=f,
and for the negative pedal
M=n+2m—2(f+g)-(f+g9)+pt+q N=2m-f-g,
F=q,G=m—f, F=2m-f—27', G=p,P=g9,U=f.

PARALLEL CURVES AND NEGATIVE PEDALS. 107

£x, 1. To find the negative pedal of the parabola, the pole being at the focus.*
Let the equation be y2=4 (mx+m?). We may then express any point on the
curve by x + m = A*m, y = 2m, and the equation az + By = a? + B? becomes
(A? — 1) w + 2QAy = (A? +1)? m.
The invariants of this quartic in \ are
S=3 (x +4m)*, T= (x + 4m)5 — 54m (a? + y?).

The discriminant therefore $*— 277? becomes divisible by 2? + y? and gives the
equation
(x + 4m)? = 27m (x? + y?).

This is equivalent to the polar equation ‘, cos }w = m®, which might have been other-
wise obtained, since it immediately follows, from Art. 95, that if the equation of any
curve can be expressed in the form p”™ = a™ cosmw, the equations of its pedal and

negative pedal are of the same form, the new m being i ge and oom respectively,

1

it may be remarked that the equation of the tangent to a parallel to this curve is
(A? — 1) & + 2Ay = (A? +1)? m+ (A? +1) K,

the envelope of which is of the fifth order, the curves answering to the values + *

being distinct. And so in general the parallels will be unicursal of curves, the
equation of whose tangent is

(A? — 1) @ + Wy= ¢ (A).

If we take @ (A) = m)3 we get a curve of the third class and fourth order touched
by the line at infinity and passing through the points J, J.

2
a?
Writing as usual for the coordinates of any point a cos@ and b sing, we have to
find the envelope of

2
Ex, 2, To find the negative pedal of = - a = 1, the pole being at the centre,

ax cos + by sing = a? cos*p + 0? sin’ = 4 (a? + 4?) + 3 (a? — 6%) cos2¢.

Hence, writing for the moment } (a? + 0?) = m, } (a? — 6?) =n, the envelope is (see
Art. 85, Ex. 3).
{3 (aa? + b?y?) — 4 (m? + 3n?)}3+- {9 (m—3n) a?a?+9(m+3n) b?y? — 8m (m*—9n?)}?= 0,
For Professor Cayley’s solution of the same problem, see Geometry of Three Dimen-
sions, (Art. 481).

Ex. 3. To find the negative pedal of the ellipse, the pole being at the focus.
The x measured from the focus is e+ a cos@ and the focal radius vector a + ¢ cos@.
We have therefore to find the envelope of

z{c+acos¢) + yb sing = (a+ ¢ cos ¢)?’,
or of ce? cos2 + a (4c — 2x) cos — 2by sin p + (2a? + c? — 2cxr) = 0
and the envelope is
{8b? (x? + y?) — (2b? + cx)?}5 + 90? (a? — cx + 2c?) (x? + y”) — (2b? + cx)3}? = 0,

which, when expanded, will plainly be divisible by x? + y? and will represent a curve
of the fourth degree, having the lines x? + y? as stationary tangents,

_ ™* Tt may easily be seen that this is the same problem as to find the caustic by
reflexion, the rays being perpendicular to the axis.

( 108 )

CHAPTER IV.

METRICAL PROPERTIES OF CURVES.

123. In this chapter we shall give some of the more
important of the metrical properties of curves. In the investi-
gation of such properties Cartesian rectangular coordinates are
most advantageously employed; then, as we saw in Art. 35, by
substituting p cos@ and p sin@ for x and y, we obtain the lengths
of the segments made by the curve on any line through the
origin; and so on any line whatever, since by transformation
of coordinates any point may be taken for origin.

The theorem given (Conics, Art. 148) may be generalized as
follows: If through any point O two chords be drawn, meeting
a curve of the n™ degree in the points f,h,...R,, S8,...8,, then
OE: OES oe will be constant, what-
ever be the position of the point O, “provided that the sidelined of
the lines OR, OS be constant.*

And the enh is the same as that already given in the case
of conic sections. From the polar equation of the curve, Art. 26,
we see that the product of all the values of the radius vector on a
line through the origin making an angle @ with the axis of « is

A
~ Pcos"6 + Q cos” 0 sind + &e.?
and the same product for any other line is
m A
~ Pos"? + Q cos” 6, sind, + &e. °
The ratio is therefore

P cos"@ + Q cos"’@ sin8 + &c.
Pos", + Q cos” ’@ sind + &e. *

the ratio of the products

* This theorem was first given by Newton, in his Enuwmeratio Linearum Tertia
Ordinis.

lee
te ee

ee ee ee ee

Te ee ee ee aye ee,

METRICAL PROPERTIES OF CURVES. 109

But we have seen (Conitcs, Art. 134) that, by a transformation to
any parallel axes, the coefficients of the highest powers of the
variables, and therefore this ratio, will be unaltered.

We may (as at Conics, Art. 148) express the same theorem

thus: Jf through two fixed points, O and o, any two parallel lines
be drawn, then the ratio of the products Oh,.OR,.OR,...&c.
2 or,.or,.0r,, &c. will be constant, whatever be the common direction
of these lines.
P cos" + &e.
A' is the absolute term when o is made the origin; and the ratio
of the products is A : A’, and independent of 6. We have seen
(Conics, Art. 134) that the new absolute term will be the result of
substituting the coordinates of 0 in the given equation. We see,
therefore, that the result of such a substitution is always propor-
tional to the product of the segments intercepted between o and
the curve on a line whose direction is given (Conics, Art. 262).

For the value of the second product is

, where

124. From the preceding theorem is deduced at once
Carnot’s theorem, of which we have given a particular case
(Conics, Art. 313). Let each of the sides of a polygon ABC, &e.,
meet a curve of the n™ degree in n real points. We shall
denote by (B)' the continued product of the n segments made on
the side BC between B and the curve; by '(B) the product of
the segments made on the side BA. Then

(A)' (BY (C)' (DY &e. ="(A) "(B) "(C) (WD) &e.
For through any point draw radii vectores parallel to the sides
of the polygon, and denote the continued product of the seg-

ments on each of these lines by (a), (0), (c), &c., then, disregarding
signs,

(B) = (BY: she (6),
(@) 2 (@)' 3: ): ),
(D) : (DY: (c) : (d),-

&e.,

and, compounding all these ratios, the truth of the theorem is
evident.

110 METRICAL PROPERTIES OF CURVES.

125. Some ambiguity will be avoided by attention to the
sign +. Considering the segments on the line AB, we have
(A)’ the product of n segments measured from A to B; and
‘(B) the product of m segments measured from B to A, and
therefore according to the rule of signs (Conzcs, Art. 7), each term
in the latter product is to be regarded as of an opposite sign
from each term in the former, so that if we give to (A)’
the sign +, we must give to ‘(B) the sign (—)"; that is to
say, + when z is even and — when it is odd. And if & be the
number of sides of the polygon, then since each side of the
equation of the last article consists of & factors such as (A)’, that
equation must be written

(A)’ (BY! (CY! &e. = (—)" (A) "(B) '(C) &e. 5
that is to say, the right-hand side will have the sign + when
either the degree of the curve or the number of sides of the
polygon is even; but when both are odd, the sign — is to be
used.*

Ex. 1. Let a right line meet the sides of a triangle AB, BC, CA, in the points
c,a,6. Then
Ac, Ba. Cb =— Ab. Be. Ca (Conics, Art. 42),
and the sign shows that, if it cut two sides internally, it must cut the third externally,

The equation
Ac,.Ba.Cb=+ Ab. Be,. Ca (Contes, Art. 48)

will be fulfilled if the three lines Aa, Bb, Cc, meetin a point; and the line AB is
cut harmonically in the points ¢ and ¢,.

Ex, 2. Let each side of the triangle touch a conic in the points a, 4,¢, Carnot’s
theorem gives us
Ac, Ba®, Co? = + Ab*. Be?, Ca? ;
and, therefore, Ac. Ba,Cb=+ Ab.Be.Ca.

The lower sign cannot be used, since no line can meet a conic in three points: we
learn then that if a conic be inscribed in a triangle, the lines joining each vertex to
the opposite point of contact meet in a point,

Ex. 3. Let a, 6, c be points of inflexion on a curve of the third degree, at which
BC, CA, AB, are tangents; then by Carnot’s theorem,
Ae’. Ba’, Cb3 = — Ab, Bc. Ca,
the only real root of which is
Ac. Ba. Ch = — Ab. Be. Ca.

Hence, if a curve of the third degree have three real points of inflexion, they must lie
on one right line. Hence, too, a curve of the third degree can have only three

* See Pliicker’s System der Analytischen Geometric, p, 44.

See

SE ak igh yen eee ree

METRICAL PROPERTIES OF CURVES. 111

real points of inflexion; for this argument would show that ali the real points of
inflexion must lie on a right line; and aright line can only meet the curve in three
points.

The same reasoning proves that if any curve of an odd degree n have three real
points, at each of which the tangent meets the curve in n points, these three points
must lie on one right line,

Ex, 4, Let a curve of the fourth degree have three double tangents; we have
Ac? Ac?. Ba®. Ba?. C8?. Cb? = Ab?. Ab,?. Be?. Be. Ca. Ca?,
whence Ac. Ac,. Ba, Ba,. Cb. Cb, =+ Ab. Ab,.Be.Be,.Ca. Ca, ;

but on account of the double sign we can only infer that “if a curve of the fourth
degree have three double tangents, the conic through five of the points of contact
will either pass through the sixth, or through the point which, with the sixth, divides
harmonically the side of the triangle on which the sixth lies.” There are thus two
distinct kinds of triads of double tangents, according as one or the other of these
geometrical relations holds good,

126. There are some particular cases for which Carnot’s
theorem requires to be modified. First, if one of the angles
(A) of the polygon were at infinity, that is to say, if two
adjacent sides be parallel, then (4)’ ultimately = '(4), and we
still have the equation

(BY (C) &e. ='(B)'(C) &e.

Secondly, if one of the angles (A) were on the curve; then
one of the m terms vanishes in each of the products (A)' and '(A) ;
AR snkh'A
AR snRRA? “~
may substitute for the ratio of these two vanishing sides the
ratio of the sines of the angles which the sides of the polygon at
A' make with the tangent at A, and the theorem becomes

(Ay (BY (0) &e. _ (A) (B)"(0) &e.

sin a sin a’

but now, since the ratio of any two lines

where (A)', '(A) have each but n— 1 factors, and where a, a’ are
the angles which the sides on which (A)’, '(A) are measured
make with the tangent at A. In this manner we can deduce
that, “if any polygon be inscribed in a conic the continued
product of the sines of the angles, which each side makes with
the tangent at its right-hand extremity, is equal to the similar
product of the sines of the angles made with the tangent at the
other extremity.”

112 DIAMETERS.

DIAMETERS.

127. Ifthere be points in a right line, a point on the line,
such that the algebraic sum of its distances from these points
shall vanish, is called the centre of mean distances of the given
points. Let the distance of the centre from any assumed point
on the line be y, let that of the other points be y,, ¥,, y,, &c.,
then the distances of the centre from the given points are y—y,,
y—y,, &c., and the condition given by the definition is

= (y—y,) =9, or ny— S(y,)=05

whence we learn that the distance of any assumed point from the
centre is equal to the sum of the distances of the assumed point
from the given points, divided by the number of these points;
or is equal to the mean distance of the assumed point from the
given points. ‘Thus, if there be only two given points, the
centre of mean distances is the middle point of the line joining
them, and the distance of any point on the line from the
middle point is half the sum of its distances from the two
given points.

The well-known properties of the diameters of conics have
been generalized by Newton into the following theorem, true for
all algebraic curves: Jf on each of a system of parallel chords
of a curve of the n™ degree there be taken the centre of mean
distances of the n points where the chord meets the curve, the locus
of this centre is a right line, which may be called the diameter
corresponding to the given system of parallel chords.

To prove this theorem, we adopt the same method of inves-
tigation as in the case of conic sections (Conics, Art. 141). ‘The
origin would be the centre of mean distances for a chord making
an angle @ with the axis of «, if, when we transform to polar
coordinates by substituting p cos 0, p sin @ (or in case of oblique
axes, mp, np), for « and y, @ be such as to cause the coefficient
of p"* to vanish. If we seek then the condition that any other
point z'y' should be the centre of mean distances for a parallel
chord, we must examine what relation should exist between
a’, y', in order that when we transform the axes to this point
the new coefficient of p””* should vanish for the same value
of 6. But when the given equation U=0 is transformed to

DIAMETERS. 113

parallel axes by substituting a+a’, y+y', for x and y, it
becomes

aa pau: Ee inl Per sy 8 es
U+a' + G+4 (ee +2a'y daay *? Tyr) t &e-= 05
only the three Re. terms can contain powers of the variables as
high as the (n— 1)", and since these involve a'y’ only in the first
degree, the required locus must be a right line. Its equation is,
in fact,

ie + ee —*+4u,,=9
da 7 dy )
where, in w,, w,_,, cos@ and sin@ (or, if the axes be oblique, m

and 7) have sein substituted for 2 and y.

128. Newton has also remarked, that if any chord cut the
curve and its asymptotes, the same point will be the centre of
mean distances for both, and that therefore the algebraic sum of
the intercepts between the curve and its asymptotes=0. This
is the extension of the well-known theorem (Conics, Art. 197).
The truth of it follows at once from the equation of a diameter
given m the last Article, and from what was proved (Art. 52)
that the terms u,, u,_,, are the same in the equation of the curve
and in that of its n asymptotes.

129. We may in like manner seek the locus of a point such
that the sum of the products in pairs of the intercepts, measured
in a given direction between it and the curve, shall vanish.
The origin would be such a point if the coefficient of p””
vanished for the given value of 0, and the locus is found, as in
Art. 127, by examining what relation must exist between 2’ and
zy’ in order that the coefficient of p"” in the transformed equa-
tion should vanish. But since the terms of the (n—2)” degree
in x and y involve no powers higher than the second of a’ and y/,
the locus will be a conic section, which we shall call the
diametral conic.

Its equation is a seen to be

du veda ~o! du, 2 d*u,,
+e-5— daz +Y ae +a (eS EAM aay! 9 a)=%

where, in u,_.,

Uns

fog a and sin@ have been substituted for
Q

114 DIAMETERS.

wand y. The distance of any point from either point on the
diametral conic being y, and from the curve y,, y,, &c., we have,
by the definition,
= (y—-4,) (y¥- 4.) = 0.

The number of terms in this sum is the same as the number
of combinations in pairs of n things, and is therefore = $n (n—1).
This, therefore, will be the coefficient of y? when we multiply out
each of these products and add them together. In the same
case the coefficient of y will consist of 4n(n—1) terms, each of
the form —(y,+y,), and since it must involve the nm quantities
Ii Yo, KC., symmetrically, it must be - (n—1)=(y). Hence

2 (y— 4%) (y—Yq) = 30 (n— 1) y'— (n—1) y2(y,) + 2 (Y,,) =O
This quadratic gives the distances of any point from the diame-
tral conic when we know its distances from the curve. $n (n—1)
times the product of these two distances = 3 (y,y,), or the product
of the distances from the diametral conic is equal to the mean
product in pairs of the distances from the curve, since there
are $n(n—1) such products. The sum of the distances from

: : 2 ; ;
the diametral conic = 2 =(y). The mean distance is then the

same for both curves, since there are two such distances in
the one case, and m in the other; and the two curves have
the same diameter.

130. There is no difficulty in seeing that a curve of the n™
degree may have other curvilinear diameters of any degree up
to the (n—1). Thus the locus of a point such that the sum
of the products in threes of its distances from the curve should
vanish, is found by putting the coefficient of p”®* in the trans-
formed equation=0; and since this coefficient involves no
higher than the third powers of the variables, the locus will
be of the third degree. We may see too, in like manner, that

= (y—y,) (Y¥—Ya) (Y¥— Ya) = 3" (m — 1) (n—2) y |
Pee 4 (n eee L) (n aes 2) y= (y,) ay (n oF 2) yz (YY) soe (Y,YYs)9

and we can readily infer hence that the curve and its cubical

diameter will have the same mean distance, mean product in

pairs, and mean product in threes of the distances; so in like
manner for diameters of higher dimensions. More light will

8

POLES AND POLARS. 115

be thrown on the subject of these curvilinear diameters by con-
siderations which we shall explain presently.

131. To the mention we have made of diameters we may
add some notice of centres. If all the terms of the degree n—1
were wanting in the equation, then the algebraic sum of all the
radii vectores through the origin would vanish, and the origin
might in one sense be called a centre.

The name centre, however, is ordinarily only applied to the
case where every value of the radius vector is accompanied by
an equal and opposite one. In this case, if the equation be
transformed to polar coordinates, it must be a function of p*
only. Ifthe curve then be of an even degree, its equation in
x and y, referred to the centre, can contain none of the odd
powers of the variables, and must be of the form

u,tu,+tu,+&e.=0.
If the curve be of an odd degree, its polar equation must be
reducible to a function of p” by dividing by p; and the x and y

equation can contain none of the even powers of the variables,
but must be of the form

U,+u,+u,+&c.=0.

This form shows that if a curve of an odd degree have a
centre, that centre must be a point of inflexion. It is also
evident that it is only in exceptional cases that a curve of any
degree above the second will have a centre; since it is not
generally possible, by transformation of coordinates, to remove
so many terms from the equation as to bring it to either of
the forms given above.

_ POLES AND POLARS.

132. We pass now to an important theorem, first given by
Cotes in his Harmonia Mensurarum: If on each radius vector,
through a fixed point O, there be taken a point R, such that

eae U oe + pan &e
On On * OR t Ott”
then the locus of R will be a right line.

116 POLES AND POLARS.

For, making O the origin, the equation which determines
OLf,, &c., is of the form

A 4B cos 04 Csind)
p p

+ (D cos’? + E cos @ sin 6+ F sin’@) = + &. = 0.

owe _n __ (Bcos6+ Csin 8)

Ok A ‘
or, returning to x and y coordinates,
: Bu + Cy+nA=0.
This is the equation found (Art. 60) for the polar line of the
origin, and the property just proved is the extension of the

well-known harmonic property of poles and polars of conic
sections (see Conics, Art. 146).

133. The preceding property may also be established with-
out taking the point O as the origin, by a method corresponding

to that used, Conics, Art. 92. We have seen (Art. 63) that
given two points O, 2'y'z', and R, xyz, then the equation
A= 0, or
MO +A *WAU' + 4A" WA?U' + &e. = 0,

determines the ratios RR,: OR,, &c., in which the line joining
these two points is cut by the curve. It follows then from
the theory of equations, that AU’=0 expresses the condition
that the sum of the roots of the equation A=0 should vanish :
that is to say, AU’ =0 is the locus of a point #, such that

RR Ree
OR OE

But writing for RR, OR,- OR, &c., this equation is at once

seen to be
7 1 1
G2 awe ont

2

134, It can be seen in like manner that the polar conic
A’ U' = 0 is the locus of a point, such that

RE, RSX. 5 1 1 Aw ae
Ga GR) Bi era (or A OR) (on OR) an
and similarly for polar curves of higher order. ‘The polar curve

peor

Se a Maen lar ee ee ae

POLES AND POLARS. 117

of the % order possesses the properties (if OR denote a radius
vector to the curve, and Or to the polar curve)

1 1 1 1

a” OR oR OF’
LP sig Re as ee
n(n—1)> OR,.OR, k(k—1)~ Or,.Or,’
1.9.3 Ph $9.3 1

n(n—1)(n—2) . OR,. OR,. OR, kik—1)(k—2) Of OF Oe.
135. If the point O be at infinity, then the distances OF,

OR,, &c., may be regarded as having to each other the ratio of
LR,

R,
OR, , OR, ,
&c., may be considered as equal. ‘The property then of the

equality, and the denominators in all the fractions

polar line = ——! pias = 0, reduces, when OQ is at infinity, to 2 (2R,) =

OR,
or the sum vanishes of the intercepts between the polar and the
curve on the parallel chords which meet at O. ‘Thus then the
polar line of a point at an infinite distance is the diameter of the
system of parallel chords which are directed to that infinitely
distant potnt.

OR, OR
reduces when O is infinitely distant to = (RR,.RR,) =0, or
=(OR-OR,) (OR- OR,) =0, the equation (Art. 129) shih
determines the diametral conic. And so in general, the curvilinear
diameter of any order ws identical with the polar curve of the
same order of the infinitely distant point on the system of parallel
chords to which the given diametral curve corresponds.

So again for the polar conic. The equation = Ge rae ‘= 0

136. Mac Laurin has given a theorem, which is the extension
of Newton’s theorem (Art. 128): “Jf through any point O a
line be drawn meeting the curve inn points, and at these points
tangents be drawn, and if any other line through O cut the curve
nh, f,, a and the system of n tangents in r,, 7, de., then

1

2 OR? oe

It is evident that two points determine the polar line; that,
therefore, if two lines through O meet two curves in the same

118 POLES AND POLARS.

points, #,, L,, &e., S, S,, Ke. the polar of O, with regard
to both curves, must be the same, since two points of it,
f and S, are the same for both. This will be equally true
if the two lines Of, OS coincide, that is to say: “If two
curves of the n‘ degree touch each other at n points in a right
line, then the polar of any point on that right line will be the
same for both curves; and therefore if any radius vector through
such a point meet both curves, we must have = ee > = #

OR Or

137. We know that the centre of a conic may be regarded
as the pole of the line at infinity with respect to the curve.
With respect to curves of higher order, however, every right
line has (n—1)’ poles (Art. 61), and there is therefore no
unique point for a curve of higher order answering to the centre
of a conic section. But it is different if we consider curves of
higher class. The preceding investigations are evidently appli-
cable also to tangential coordinates; and thus every right line
has a pole, a polar curve of the second, third, &c. class, and,
finally, a polar curve of the (n—1)™ class, touched by the n
tangents at the points where the right line meets the curve.
And if we thus by tangential coordinates seek the pole of the
line at infinity we find a unique point.

Let us examine what metrical property is possessed by the
pole of a line expressed in tangential coordinates, and, in par-
ticular, by the pole of the line at infinity. We take the system
of Art. 19, in which the coordinates of a line are proportional
to the perpendiculars let fall on it from three fixed points; and
then it may be seen, without difficulty, that 7: m denotes the
ratio of the sines of the angles, into which the angle between
two lines aBy, a’B'y' is divided by the line la + ma’, (8 + mf’,
ly+my'. The equation then which answers to A = 0 determines
the ratio of the sines of the parts into which the angle
between any two lines is divided by each of the tangents which
can be drawn through their intersection to a curve of the n
class. And, as in Art. 133, the pole & of any line possesses the
property = (a ZP6) = 0, where Pisa variable point on the

given line; &,, 2,, &c., the points of contact of tangents from

FOCI. 119

the point P, O any fixed point on the given line. Thus for a
eurve of the second class the relation is

sn RPR, , sin kPR,
sin PO sin k,PO

that is to say, “if from any point JP, on a fixed line OP, we
draw tangents PR, PR,, to a conic, and draw PF so that
{P. OR, RR,} shall be a harmonic pencil, then OF passes through
a fixed point.” This is the fundamental definition of pole and
polar with regard to aconic considered as a curve of the second
class.
We may write the relation
(Ps) =0 in the form = (eo) =0,

where J, is the foot of the perpendicular from £&, on the line
RP, and O, the foot of the perpendicular from the same point
on the line OP. Now let the line OP go off to infinity, then
all the denominators in this latter sum tend to equality, and we
have simply > (M_R,) =0; or the sum vanishes of the perpen-
diculars let fall from the points of contact of any system of
parallel tangents on a parallel line through &. In other words
then, the centre of mean distances of the points of contact of any
system of parallel tangents to a given curve is a fixed point, which
may be regarded asa centre of the curve. ‘Thus in a conic the
middle point of the line joining the points of contact of parallel
tangents is a fixed point; in a curve of the third class, the
centre of gravity of the triangle formed by them, &c. This,
theorem is due to M. Chasles ( Quetelet, v1. 8).

= 0,

FOCI,

138. It was shown (Conics, p. 228) that the foci of conics
possess the property that the lines joining them to the circular
points at infinity touch the curve. Hence we are led to the
following definition of foci in general: A point F' is said to
be a focus of a curve, if the lines /'/, FJ both touch the curve,
or, as we may say, when it is the intersection of an J-tangent
with a J-tangent.* A curve of the n“ class has in general n’

* This conception is Pliicker’s, Credle, vol. x. p, 84,

120 FOCI.

foci, namely the points of intersection of the m tangents with
the n J-tangents. But the curve being real, m and only x of
these foci are real; in fact the equation of one of the J-tangents
being 4+7B=0 (where A and Bare linear functions of the
coordinates), that of one of the J-tangents will be A —-7B=0, and
these intersect in the real point 4 =0, B= 0, and there is not on
either of these tangents any other real point. ‘Thus a conic
(n =2) has 4 foci, two of them real.

In what precedes it is assumed that the points J, J have no
special position with respect to the curve. Let us now suppose
that the line JJ is an ordinary, or singular, tangent at one or
more points A, B, &c., which for the present we suppose to be
distinct from the points J, J; say that JJ reckons g times
among the tangents from J or J to the curve; then the
J-tangents are made up of the line JJ counting g times, and
of n—g other tangents; and similarly for the J-tangents.
Then the only foci which do not lie at infinity evidently consist
of the intersections of the n—g J-tangents with the n—g
J-tangents, and there are (n —g)’ finite foci, of which, as before,
only n—g arereal. The total number of n’ foci is made up of
these (n—g)* foci, together with the point J counting g (n—g)
times (namely, as the intersection of each of the n—g J-tangents
with each of the g J-tangents which coincide with L/) ; similarly,
of the point J counting g (n—g) times, and lastly of the g’
intersections of the g J-tangents coincident with L/ with the
g J-tangents coincident with LJ. In this last case any J-tangent
IA must be regarded as intersecting the corresponding J-tangent
JA at the point of contact A, but its intersection with any
other J tangent JB will be indeterminate. ‘Thus, if the line at
infinity touch the curve in g real points, there will still be n
real foci, viz. x —g finite foci, and the g points of contact of LJ
with the curve.* For instance, the parabola (n=2, g=1) has
one finite focus, the other real focus being infinitely distant in
the direction of the axis.

Again, let the point J be on the curve; then assuming the
curve to be real, the point J is also on the curve, and if Z

* Prof. Cayley thinks that the preferable view is that the only foci are the (n — g)z
foci, and consequently that the only real foci are the (n — g) foci,

FOCI. 121

be a singular point, J will have the same kind of singularity.
Confining our attention for the moment to the case where both
_are ordinary points, the n —g J-tangents consist of the tangent
at J counted twice, together with n—g—2 other tangents; and
similarly for the J-tangents. Then the (n—g)’ foci are made
up as follows: the real intersection of the tangents at Z and J
counting as four; the n—g—2 imaginary intersections of the
tangent at £ with the n—g—2 J-tangents, each counting for
two; the n—g-— 2 imaginary intersections of the tangent at J
with the n—g-— 2 J-tangents, each counting for two; and lastly,
the (n —g — 2)’ intersections of the two sets of n—g—2 tangents.
Of these last, as before, n—g—2 and only n-g—2 are real,
and the intersection of the tangents at J and J takes the place of
two of the n—g real foci. Paying attention then only to real
foci, this point is commonly called a double focus; and we find
it convenient to use this language, though, as we have just seen,
if we considered imaginary as well as real foci, it ought to
be called a quadruple focus. Thus, in the case of the circle,
the only focus is the centre, which must be regarded as a
quadruple focus, if we consider that it takes the place of the four
foci which conics in general possess, but which may be spoken
of as a double focus if we only pay attention to the two real foci.

Similarly, if each of the points J, J is an f-tuple point on the
curve, it is seen in the same way that there are f” foci, which
each count for four and of which f are real; 2f(n—g—2f)
imaginary foci which each count as two, and (n—g—2f)’ single
foci of which n—g- 2f are real. Considering then both real
and imaginary foci, we should say that there are f’ quadruple,
2f(n—g-—2f) double, and (n—g-2f)? single foci; but con-
sidering real foci only, we may say that there are f double,
n—g— 2f single foci, and g foci at infinity.

If f and J be each of them an inflexion, or each a cusp, then
the tangent at J or J counts three times among the J or J-tan-
gents; ard there are from each point n—g—3 other tangents.
The (n—g)’ foci are then as before seen to be made up of one
which counts as nine, of (n—g-—3)+(n-—g-— 8) which each
count as three, and (n—g-—3)* single foci. Of these last
n—g-—38 are real, and the only other real focus is the intersec-
tion of the tangents at J and J, which is commonly called a

R

122 FOCI.

triple focus as counting for three among the real foci, though
if we took into account imaginary as well as real foci, it ought
to be regarded as a 9-tuple focus. There is no difficulty in
extending the theory to the cases where J and J are multiple |
points of higher order at which several tangents coincide, or
where they are points at which the tangent has contact with the
curve of a higher order than the second, or where they are
ordinary or singular points having LJ for their common tangent.

139. Given any two real foci A, A’ of a curve, the lines
Al, AJ; A'I, A’J, meet in two imaginary points B, B’ which
are also foci of the curve; and the relation between the two
pairs of points is, that the lines 4A’, Bb" bisect each other at
right angles in a point O, such that OA (= OA’) is equal to
iOB (=iOB'). The points A, A’ and B, Bb’ have been termed
anti-points.” The relation is one of frequent occurrence in
plane geometry; thus a conic has two pairs of foci, which
are anti-points of each other; any circle through A, A’ cuts
at right angles any circle through B, B', &. It is to be added,
that being given the n real foci, we form with these 4n (m—1)
pairs, each giving rise to a pair of anti-points, and thus obtain
the remaining n*— n foci.

140. The coordinates of the foci of a curve are obtained by
forming the equation of the tangents which can be drawn from
the point J to the curve. This will be of the form P+7Q=0,
the corresponding equation for the point J will be P—7Q=0,
and the intersection of the two systems of tangents are given by
the equations P=0, Q=0. Thus denoting the first differential
coefficients with respect to « and y by U,, U,; the second by
U5 U4, U,.) &e.; then, by Art. 78, the equation of the system
of tangents from 1, 2,0 is got by forming the discriminant of
MU+0"* (U4 7U,) 4+ $n"? (U,, + 2¢U,,— U,,) +&e.=0. Thus,
if the curve be a conic, the discriminant is

{U?—UY—2U(U,, —U,)} + 2¢(U,U, — 200),
and the foci are got by equating the real and imaginary parts
separately to zero. By combining these equations, we get the
equation of the two right lines, the axes, on which the foci
lie, viz.

U,, (U2 -0;3) —(U,, - U,,) 0,0, =0.

Emap ba i tat aliearpipen ek

FOCI. 123

The very same equations determine the foci of a cubic passing
through the points J, J; of a quartic having these points for
double points, &c.; for in any of these cases it is easy to see that
all the terms but those written above vanish of the equation
whose discriminant is to be found.

141. We canvalso determine the foci, as at Conics, Art. 258,
Ex., by expressing the condition that e—a'+7(y—y') should touch
the curve; or, in other words, by substituting in the tangential
equation, 1, 7, — (#’+ dy’) for a, 8, y. The real and imaginary
parts of the equation then separately equated to zero determine
the coordinates of the foci. It is not difficult to find a real
geometric interpretation of each of these equations. Let the
condition that «—a'+ p(y—y’) should touch the curve be written

ap” + bp" * + cp"* + &e. =0,
where a, 6, &c. are functions of wx’, y'; then by the theory of
equations Hee = &c. are the sum, sum of products in pairs,

&e. of the tangents of the angles, which the tangents to the
curve through wy’ make with the axis of a If now we write
p=t, and equate to zero the real and imaginary parts of the
equation, we get the two equations

a—c+e—&.=0, b-—d+f—&.=0;

the second of which, by the well-known formula for the tangent
of the sum of several angles, expresses that the sum of the
angles made with the axis of a by the tangents through ay’
is either zero or is some multiple of 7; and the first of
the equations expresses that the sum of the angles is some odd
multiple of 47. Hence the locus of a point such that the sum
of the angles made with a fixed line by the tangents through it
to a curve of the n™ class shall be given is a curve of the
n” degree, whose equation, the fixed line being taken for axis
of x, is easily seen to be

(a—c+e-&c.) tand=b-d+f- &e.

Whatever be the fixed line jor the angle, the locus will pass
through the foci of the curve. This may appear paradoxical,
since it follows hence, that the sum of the angles made with

124 FOCI.

any line by the tangents from a focus may be equal to any
given quantity. The reason of this is that the tangents of two
of these angles are +7, and the tangent of their difference assumes

the form : , and may be any assignable quantity. In fact, if

tand=7, ¢ may be regarded as an infinite angle, since it pos-
sesses the properties sn@d=cosP=co and tan(¢?+a)=tandg,
and the difference of two infinites is indeterminate.

We have seen (Art. 110) that a tangent through one of the
points J, J coincides with the normal; and hence every focus of
a curve is also a focus of its involutes and evolute.

142. An important property of the perpendiculars let fall
from the foci on any tangent is at once derived from the
equation expressed in that system of line-coordinates (Art. 19
and Conics, p. 364) in which the variables are the perpendi-
culars let fall from three fixed points on any line. Let a, 8, y, 5,
&c. be the n foci: let ww’ denote the points J, J; then, since
the lines aw, aw’, &e. are to be tangents to the curve, the
tangential equation must be of the form afyé kc. = we'd,
where ¢ is a function of the order n—2 in the line-coordinates.
For curves of the second class, this at once gives the property
that the product of the perpendiculars from the two foci on any
tangent 1s constant, since it was proved (Conics, p. 363) that
- for aw’ we may substitute a constant.

Similarly, replacing ww’ by a constant, the general equation
of curves of the third class is a8y =k6d, where a, 8, y denote the
three foci, and 8 a certain fourth point: viz., we may from
each focus draw to the curve (besides the two tangents through
I, J respectively) a single tangent; and the form of the
equation shows that the three tangents from the points a, 8, ¥
respectively mect in a point 6.* We learn, then, that the
product of the three focal perpendiculars on any tangent to
a curve of the third class is in a constant ratio to the per-
pendicular on the same tangent from the point 6. If the
curve pass through the points J, J, there is a double focus,

* The veciprocal theorem for curves of the third order cut by any two lines
is given post, Art. 148,

FOCI. 125

and the equation takes the form 2°8 =ké, the interpretation of
which is obvious. Ifa focus A is at infinity, we can see how
the formula is to be modified, by first using for the coordinate «
the perpendicular distance of A from any tangent divided by
AB; and then, when A goes to infinity in the direction AB,
it is easy to see that a will be cos@ where @ is the angle made
by AB with the direction of the perpendiculars on the tangent.
Thus the formula for a conic, 48 =’, becomes in the case of
the parabola where A passes to infinity, 8 cos@=h, showing
that the locus of the foot of the perpendicular from the focus 8
in a tangent is aright line. In like manner for a curve of the
third class the formula a8y=45 becomes Py cosO6=k8, which
may be written By=46', if we understand by 3’ the intercept
made by the variable tangent on a line drawn through D
parallel to AB.

For curves of the fourth class the equation is a@yi=k’h
where ¢ is the conic section which, as the equation shows, is
touched by the eight focal tangents which do not pass through
I,J. But if the foci of this conic be ¢, &, the equation may be
put into the form a8y5=4%ef+l', the geometrical interpreta-
tion of which is obvious. This equation includes the form
aB8yd5=l* or =@*w”, which represents a curve on which the
foci a, 8, y, 5 are double foci; the form a8 =o*w” in which
I, J are points of inflexion, &c.

And so in general the tangential equation of a curve of the
n class gives a relation of the first degree connecting the
product of the x focal perpendiculars, of n—2 other perpen-
diculars, of » —4 other perpendiculars, &c., and so on until we
come either to a single perpendicular or a constant term.

143. From relations connecting the focal perpendiculars on
the tangent can be deduced relations connecting the angles
between the focal radii and the tangent. I*or if AP be the
perpendicular a on the tangent at any point £& of the curve,
and if db be the angle between two consecutive tangents,
we have da=RPdd. Similarly d8=hP'dd, &e. So that
if we differentiate the relation connecting the perpendi-
culars, we may substitute for each da, HP the corresponding
intercept on the tangent between the foot of the focal per-

126 FOCI.

pendicular and the point of contact. Thus from afy=k'5
we deduce i ae a
By + 2B Rs > si a 0,
Be TS tae
AP BP CP" DP
or cot 8 + cotd’ + cot 0” — cot 0” =0,
where 0 is ARP, the angle of inclination of the tangent to the
focal radius vector AL, &c.

whence

144. The example of conics would lead us to expect to find
simple relations connecting the distances. of any point on the
curve from the foci. There does not appear to be any general
theory of such relations, but we can without difficulty find
particular curves for which they exist, for we have only to
write down any relation connecting the distances of a variable
point from fixed points, and find the locus for which it is
satisfied. Hach distance, if expressed in terms of the coor-
dinates, involves a square root; and if, as will commonly
happen, the equation when cleared of radicals is of the form
up =wv", the two imaginary lines denoted by p’=0 are tan.
gents to the curve, and the fixed point / is a focus. In
this way we might study the relations p+ mp'=d, for which
the locus is an ellipse or hyperbola when m=+1, a circle
when d=0, and in other cases a Cartesian: lo+mp'+ np" =0
for which the locus is in general a quartic having the points LJ
for double points, or, as we may say, a bicircular quartic; but
when /imin=0, the curve is a cubic passing through the
points JJ, or, as we may say, a circular cubic: pp'=d’, for which
the locus is a Cassinian (see Art. 55, Ex. 3); or, more generally,
ap’ + bpp'+cp”=d"*, which is in general a quartic, but is a
cubic if a+b+c=0, that is to say, if the left-hand side of the
equation is divisible by pip’, &c. We postpone the further
discussion of this subject until we come to treat of the curves
referred to.

From a relation connecting the focal distances we can infer
a relation connecting the angles which the focal radii make
with the tangent; for it is proved, as in Art. 95, that each
dp =cos@ds, where @ is the angle between the focal radius and

FOCI. 127

the tangent. Thus from p+ mp'=d we infer cos6+m cos6'=0,
&c. From the value given in the last article for da, &c. we
may infer Rda=pdp, &c., where # is the radius of curvature.
Thus, for example, if we are given that la+ m@+&c. is con-
stant, we can infer that /p’ + mp” + &c. is constant.

145. Denoting by N the number of conditions (Art. 27)
necessary to determine a curve of the n™ order, then if we
are given that such a curve is circular, that is to say, that it
passes through the points J, J; and if we are given N—3 other
points on the curve, the locus of the double focus (or inter-
section of the tangents at J, J) is a circle. For since but one
curve of the n™ order can be described to pass though N
points, if in addition to the above conditions we are given
a consecutive point at J, that is to say, if we are given TJ
the tangent at J, the curve will be completely determined,
and therefore FJ the tangent at J is determined. The point
f is then the intersection of corresponding lines of two homo-
graphic pencils (Conics, Art. 831), that is to say, two pencils
such that to any line of one answers one and only one line of
the other. The locus of / is therefore a conic passing through
the vertices of the pencils J, J, that is to say, it is a circle,
This conic breaks up into the line JJ and another line, when to
the line L/ of one pencil answers the line JZ of the other. This
will be the case in the present example when z=2, since lJ
cannot be a tangent to a conic passing through the points J, J,
unless the conic break up into two right lines, and the theorem
then is that for the circles which pass through two fixed points,
the locus of the centres is a line; but when z is greater than 2,
the locus will in general be a circle.

146. In like manner if we are given N—1 tangents to a
curve of the n™ class, the curve is completely determined if one
more tangent FZ be given. The reasoning of the last article
will apply, and the locus of the focus will be a circle, if the con-
ditions are such that when the curve is determined, only one
tangent can be drawn to it from the point J. ‘This will be the
case, if among the given conditions is, that the line L/ is a
tangent of the multiplicity n — 1, since then but one more tangent

128 FOCI.

can be drawn to the curve from any point on that line. We
have seen, Art. 41, that to be given that a point is a multiple point
of the order 4, is the same as if $£(k+1) points were given
Similarly to be given that JJ is an (n—1)-tuple tangent, is
equivalent to being given $n(n—1) tangents. Observing then
that N—4n(n—1)=2n, we infer that if we are given 2n—1
tangents of a curve of the nz" class, and also that the line at
infinity is an (n—1)-tuple tangent, the locus of the focus (in
this case there being but one focus) is a circle. Thus being
given three tangents to a parabola, the locus of the focus
is a circle. Again, the locus of the focus is a circle if we
are given five tangents to a curve of the third class, among
whose tangents the line at infinity counts for two. A particular
curve of this system is the complex made up of the point at
infinity on any of the five tangents, and the parabola touching
the other four; the focus of the parabola being the focus of the
complex. Hence we have Miquel’s theorem (Conics, Art. 268,
Note) that the foei of the five parabolas which touch any four
of five given lines lie on a circle.*

* This proof of Miquel’s theorem is Mr, Clifford’s, for whose other inferences from
the same principle, see Messenger of Mathematics, Vol, V., p. 137.

Lanes
> aa

( 199 )

CHAPTER V.

CURVES OF THE THIRD ORDER.

147. Ir has been proved (Art. 42) that a curve of the third
order, or, as we shall for shortness call it, a cubic, may have one
double point, but cannot have any other multiple point. Hence
is suggested the fundamental division of cubics into non-singular,
having no double point; nodal, having a double point at which
the tangents are distinct, and cuspidal, having a double point
at which the tangents coincide. Pliicker’s numbers (Art. 82)
for the three cases respectively are:

mi 8 @imr ie
50.0 Que Os
eae Ge a OF Ss
Soe ROS OE

It thus appears that the curves are of the sixth, fourth, and
third class respectively, or are such that six, four, or three
tangents respectively can be drawn to the curve from an
arbitrary point. If the point be on the curve, the tangent at
the point counts for two among these tangents (Art. 79), and
the number of tangents distinct from the tangent at the point
is four, two, or one. If the point be a point of inflexion, the
stationary tangent counts for three, and the number of other
tangents which can be drawn through the point of inflexion
is further reduced by one.

Nodal cubics may obviously be subdivided (Art. 38) into
crunodal and acnodal, according as the tangents at the double
point are real or imaginary. We shall hereafter see that there
is a parallel subdivision of non-singular ecubics. But for the
present we postpone the further discussion of the classification
of cubics, as the reader will be able to follow it with more
intelligence when he has first been put in possession of some of
the general properties of these curves. We likewise postpone

8

130 INTERSECTION OF A GIVEN CUBIC

the discussion of the general equation and the examination
of its invariants, and we commence by applying to the case
of cubics theorems we have already obtained for curves of any
degree, beginning with the theorems on the intersection of
curves established in the first Section of Chapter II.

SECT. I.—INTERSECTION OF A GIVEN CUBIC WITH OTHER CURVES.

148. It has been proved (Art. 29) that all cubics which pass
through eight fixed points on a given cubic also pass through
a ninth fixed point on the curve. This is a fundamental
theorem leading to the greater part of the properties of cubic
curves. In particular we infer that if two right lines whose
equations are Ad=0, B=0, meet a cubic in points a, a’, a’,
b, b', b" respectively, and if the lines ad, a’d', ab” (whose
equations we write D=0, H=0, #=0), meet the cubic in
points c, c’, c’, then the line cc'(C=0) joining two of those
points will pass through the third. For the lines D, £, F
make up a cubic passing through the nine points; the lines
A, B, C make up a cubic passing through eight of these points,
therefore it will pass through the ninth c”, and since this point
cannot lie on either of the lines A, B which already meet the
~ curve each in three points, it must lie on C. Since the given cubic
passes through the intersection of the cubics ABC =0, DEF=0,
its equation must be capable of being written in the form

DEF-kABC=0.

149. Let us suppose that the lines A, B coincide, then we
deduce as a particular case of the preceding theorem, that if a
right line, 4 =0, meet the curve in three points a, a’, a”, the
tangents at these points, D=0, H=0, /=0, meet the curve in
points c, c’, c’ respectively, which he on a right line C=0,
and the equation of the curve may in that case be written
DEF-—kA’C=0. The point c, in which the tangent at any
point a meets the curve again is called the tangential of the point
a; and the line C on which le the tangentials of the three
points a is called the satellite of the line A. We shall hereafter
show how when the equation of A is given, aw +Py+yz=0,
the equation of C can be formed. The line A will have a real
satellite, even though instead of meeting the curve in three real

WITH OTHER CURVES. 131

points it meets it in one real and two imaginary points. The
equations of the tangents at the imaginary points will be of the
form P+7Q=0; their product will be real; and the equation of
the curve can be written in the form D (P?+ Q’)=kA’C.

Two cases of the theorem of this article deserve to be
noticed. First, let the line A be at infinity, then the tangents
D, E, F at the points where it meets the curve are the three
asymptotes; each asymptote meets the curve in one finite point,
and we learn that these three points lie on a right line C,
the satellite of the line at infinity. In this case the equation of
the curve is reducible to the form DEF =kC, and we have the
theorem that the product of the perpendiculars from any point
of the curve on the three asymptotes is in a constant ratio
to the perpendicular from the same point on the line C.

Secondly, let the points a, a be points of inflexion; then
evidently the tangentials of these points coincide with the
points themselves; the satellite line C therefore coincides with
A, and consequently the third point @” in which it meets the
curve is also a point of inflexion (see Art. 125, Ex. 3). The equa-
tion of the curve is thus reducible to the form DH/F'= kA’, where
A=0 is the equation of the line through the three inflexions,
and D=0, H=0, #=0 are the equations of the tangents at
these three points respectively.

150. The theorem of Art. 149 may be otherwise stated,
starting with the line C instead of with A; viz. given three
collinear points c, c’, c’ of a cubic, the line joining a the point
of contact of any of the tangents from c, to a’ the point of
contact of any of the tangents from ec’ will pass through the
point of contact of one of the tangents from c’. Only one
tangent can be drawn aé a point of a curve, and therefore to
any position of A corresponds but one position of C; but in
the case of a non-singular cubic four tangents can be drawn
from any point on the curve, and therefore to any position of
C correspond sixteen positions of A. The twelve points of
contact lie on the sixteen lines A, viz. each line A contains
three points of contact, and through each point of contact there
pass four lines A.

Let us consider more particularly the case where C touches

132 INTERSECTION OF A GIVEN CUBIC

the curve, and let us suppose the points c, c’ to coincide.
Then we see that the line joining a,”, one of the points of
contact of tangents drawn from c’, to a, one of the points
of contact of tangents from c, must pass through one of the
other points of contact from ¢, say a, In like manner, the line
joining a,"a, passes through a, We have then the following
theorem: The four points a,a,a,a, which are the points of
contact of tangents from any point c of the curve are the vertices
of a quadrangle, the three centres of which are also points on
the curve, and are such that the tangents at these points and
the tangent at c all meet the curve in the same point.

151. Returning to the case where C does not touch the
curve, we have the tangents from c touching at the points
Gy Ay AG A, and the tangents from c’ touching at the points
a, a,,4,,4,. Attending only to two points, say a,, a, of the
first tetrad, it appears that separating the points of the second
tetrad into pairs in a definite manner, say these are a,', a, and
a, a, then combining the pair a,, a, jirst with the pair a,', a,,,
the lines a,a,', a,a, meet in a point on the curve, and also the

i417

lines a,a,', 4,a,, meet in a point on the curve; and secondly with

the pair a,'a,', the lines a,a,', a,a, meet in a point on the curve,
and also the lines a,a,', a,a, meet in a point on the curve: viz.
the four new points are the points of contact of the tangents
from c’ to the curve. Any two points such that the tangents
at these points respectively meet on the curve may be said to be
“ corresponding points ;”’ thus any two of the points a, a,, a@,, a,
are corresponding points; and so any two of the points a,', a,’ 5
a,, a, are corresponding points. But starting with the two
points a,, a,, the points a,’, a,’ (as also the points a,', a,’) may be
said to be corresponding points of the same kind with a,, a,: viz.
the property is that, given two pairs of the same kind, if w
form a quadrilateral by joining each point of the one pair with
each point of the other pair, the two new vertices of the quadri-
lateral are points on the curve (they are in fact corresponding
points of the same kind with the original two pairs). It is
obvious that there are three kinds of corresponding points,
viz. those of the kind aa, or a,a,, the kind a,a, or a,a,,
and the kind a,a, or a,a, And, moreover, starting with the

WITH OTHER CURVES. 133

pair a,a,, to obtain the whole system of corresponding points of
the same kind, we have only to take on the curve a variable
point K, and joining it with the two points a,, a, respectively,
these lines again meet the curve in a pair of corresponding
points of the kind a,a,. It may be mentioned that the envelope
of the line joining two corresponding points of a given kind is
a curve of the third class. The theory is, for the most part,
due to Maclaurin (see the “ De Linearum Geometricarum Pro-
prietatibus Generalibus Tractatus,” published with the 5th edition
of his Algebra), and it may appropriately be called Maclaurin’s

Theory of corresponding points on a cubic curve.

152. In further consideration of the case where C does not
touch the curve, let D,, L,, #, be tangents through the points
c, ¢, c’ respectively, and we have seen that the equation of the
curve may be written in the form DL, F\—A,*C=0. Let D,, L,
be another pair of tangents through c¢, c’, such that their
chord of contact passes through the point of contact of F,
and fhe equation of the curve may also be written in the
form DHF —AC=0. Hence we can deduce an identity
(DE, - DE.) F,=(A-A,’)C. The right-hand side of the
equation denotes three right lines, therefore the left-hand side
must denote the same three lines. One of the factors therefore
of D,L,—D,E, must be CG, which passes through the points
DD, LE, The other factor which joins the points DZ,
DE, must be A,+A,, F, being A,+A,. We see, then, that
the latter two lines and the two chords A,, A, form a harmonic
pencil, whose vertex is the point of contact of /. We shall
afterwards apply this theorem to the case where the points ¢, ¢
are the imaginary points at infinity J, J; the points D.Z,, DE,
are then foci, and /, is a tangent parallel to the single real
asymptote of the curve.

If the points ¢, c’ coincide, the line joining c to the point
of contact of F', F, itself, and the two chords A,, A, form a
harmonic pencil.

153. Hence can be deduced another theorem of Maclaurin’s.
Any line drawn through a point A on a cubic is cut harmonically
in the two points @, y, where it meets the cubic again, and the

134 INTERSECTION OF A GIVEN CUBIC

‘two points 8, 8’, where it meets a pair of chords joining the
points of contact of tangents from A. Let the line meet the
tangent C in the point e, then, since it meets A, and B, at A,
by Art. 136,

1 1 1 2 1

54° 38 * By BAT

1 1 L 1

3B *e 54 a 5A + Tin

But, by the last Article, 58’ is a harmonic mean between 64
and de, therefore also between 68 and dy. Q.E.D.

When the curve has a double point, only two tangents can
be drawn to the curve; but the theorem of.this Article will be
_ still true, if for the chord D’ we substitute the line joining the
double point to the point where the ehord D meets the curve
again.

or

154. We add one more application of the theorem, that
all cubics which pass through eight fixed points on a cubic
pass also through a ninth fixed point. Jf any conic be described
through four fixed points on a cubic, the chord joining the two
remaining intersections of the conic with the cubic will pass
through a fixed point on the cubic. Consider any conic through
the four points («) and meeting the curve in two other points
(8), and a second conic through the points (a) and two other
points (8’), then the conic through a, 8 and the right line
joining the two points 6’ make up a cubic system through the
eight points a, 8, 6’; the conic through a, 8’ and the right
line joining 8 make up a second system through the same
eight points; hence the ninth point of intersection with the
curve must be common to both systems; that is to say, the
lines joining the points 8, 8’ meet the curve in the same point,
Q.E.D. This point was in the first edition called the opposite
of the system of four given points; but now, in conformity
with the nomenclature of Prof. Sylvester’s remarkable theory
of residuation, which will be presently explained, is called the
coresidual of the system of four points. This point is easily
constructed by taking for the conic through the four points
a pair of lines. Let the line joining the points 1, 2 and
the line joining the points 3, 4 meet the cubic-in points 5 and 6

WITH OTHER CURVES. 135

respectively, then the line joining 5, 6 meets the curve in the
coresidual required. And since the grouping of the four points
is arbitrary, the construction can, it is clear, be performed in
three different ways.

Hence, for example, we infer that through four points on
a cubic four conics can be drawn to touch the curve elsewhere,
viz. the conics passing through the points of contact of the four
tangents which can be drawn from the coresidual.

155. Let us apply the rule just given to construct the point
coresidual to four consecutive points on the curve. The line
joining the points 1, 2 is then a tangent, and the point 5 in
which it meets the curve is the tangential of the point 1;
similarly, the line 34 meets the curve in a point 6, which is
consecutive to the point 5; it follows that the coresidual re-
quired is the point where the tangent at the tangential point 5
meets the curve again; that is to say, it is the tangential of
the tangential, or, as we shall say, the second tangential.

If then, for example, it be required to draw a conic passing
through the four consecutive points, or, as we may say, having
a four-point contact with the curve, and elsewhere touching
the curve, the point of contact is, as we have seen, a point
of contact of tangents from the second tangential to the curve.
One of these is the tangential‘of the point (1), and the corre-
sponding conic degenerates into two right lines; the remaining
three give solutions of the problem.

Again, if it be required to describe a conic passing through
five consecutive points of the curve (or having a five-point
contact with the curve), this is done by constructing the sixth
point in which the conic meets the cubic, viz. this is the point
where the line joining the point (1) to its second tangential
meets the curve again. In order that this point should coincide
with the point (1) it is necessary that the line last named should
touch the curve at (1); or, what is the same thing, it is
_ necessary that the first and second tangential should coincide.
Now a point which coincides with its tangential is a point of
inflexion; hence, on a non-singular cubic there are twenty-seven
points at each of which a conic can be drawn, having a six-
point contact with the curve ; viz. these are the points of contact

136 INTERSECTION OF A GIVEN CUBIC

of the three tangents which can be drawn from the nine ports of
ainflexion.

156. The theorem (Art. 29) as to the intersection of two
cubics was generalized in Art. 33. The theorem there given
_is applied to the case of the cubic by writing p=3, and it then
- becomes every curve of the n™ degree which passes through 3n—1
fixed points on a cubic passes through one other fixed point on
the cubic. It is to be observed, that for n=1, or n=2, one and
only one curve of the n™ degree can be described passing
through 3n—1 points on a cubic, and the theorem asserts
nothing; when nm is greater than 2, more than one such
curve can be described, and the curves all pass through one
other fixed point on the curve, as has been just stated. And,
as was explained in Art. 33, if it were attempted to describe a
curve of the n™ order through 3x” points taken arbitrarily on
a cubic, n being greater than 2, the curve so described would
in general not be a proper curve, but would be a complex
consisting of the cubic itself, and a curve of the order n- 3.

157. If of the 3 (m+ n) intersections of a curve of the (m +n)™
order with a cubic, 3m lie on a curve of the m™ order U., the
remaining 3n lie on a curve of the n™ order. For, as has been
just remarked, through 3n—1 of these 3n points, a curve of
the n™ order U, can always be described; and this, together
with U makes up a system of the order m+n which (Art. 156)
passes through the remaining point, and since this point cannot
_lie on U_, which already meets the cubic in 3m points, it must

lie on U..

158. We shall now explain the nomenclature introduced by
Prof, Sylvester, and in conformity with it re-state and extend
some of the preceding propositions. If two systems of points
a, 8, together make up the complete intersection with the cubic
of a curve of any order, one of these systems is said to be
the residual of the other. Since the total number of intersec-
tions of a cubic with any curve must be a multiple of three,
it is evident that if the number of points in the system a be
of the form 3p+1, that in the system 8 must be of the form
3q—1, and vice versa. We may call these positive and negative

WITH OTHER CURVES: 137

systems respectively, and say that the residual of a positive
system is a negative system, and vice versa. The simplest
positive system consists of a single point, answering to p=0;
the simplest negative system of a pair of points, answering to
q=1. In this case, evidently the one is the residual of the
other when the three points are on a right line. Since through
a given system of points a, an infinity of curves of different
orders may be described, it is evident that a given system of
poimts a has an infinity of residuals B, 6’, 8”, &e. Two
systems of points 8, 8’ are said to be coresidual if both are
residuals of the same system a. For example, in Art. 154
through four points a on a cubic we supposed conics to be
described meeting the curve again in pairs of points 8, 8’, Kc. ;
then any one of these pairs is a residual of a, and any two of
them are coresidual. Again, if the line joining the pair 8
meet the curve again in a point a’, this point, as well as the
four original points, is a residual of the group f, and this point
a’ is therefore, as we already called it, coresidual with the four
points a. It is obvious that two coresidual systems of points
must either be both positive or both negative.

The theorem of Art. 156 may be stated thus: two points
which are coresidual must coincide. In fact, we there saw that
if through 3p—- 1 points a we describe a curve U, meeting the
cubic in the residual point 8, and if through the same points
a we describe a second curve of the p” order meeting the
cubic again in a point §’, the coresidual points 8, 8’ arrived
at by the two processes, are one and the same point.

159. If two systems B, B' be coresidual, any system a’ which
is a residual of one will be a residual of the other. Say that
through any system « two curves U,, U, are described meeting
the cubic again in systems 8, §’, then these two systems are
by definition coresidual; and what is now asserted is that if
through 8’ be drawn any curve U, meeting the cubic again in
a system of points a’, then the points 8 and a’ also make up
the complete intersection of a curve with the cubic. For since
the systems a and 8 together make up the intersection of a
curve U,, with the cubic, and a’ and #’ make up its intersection
with a curve U,, the four together make up the intersection

T

138 INTERSECTION OF A GIVEN CUBIC.

with the cubic of a curve whose order is +r: but the systems
a and #' together make up the intersection with the curve U,
of the order g, therefore (Art. 157) the systems a’ and 8 together
make up the complete intersection of the cubic with a curve
whose order is p+r—q.

Hence also two systems which are coresidual to the same are
coresidual to each other. If 8 and §' are coresidual as having
a common residual a, and if 8’ and @” have a common residual
a’, then by what has been just proved a is a residual also of
B",and a’ of B: that is, if 8, B” are each of them coresidual
with 8’, then 8, 8” are coresidual with each other, for a, a’ are
each of them a common residual of (3, 6”.

160. We can now give for the theorem of Art. 154 a proof
which will at once suggest Prof. Sylvester’s generalization of
that theorem. The conic through four points « on a cubic
meets the curve in two points 8, which are a residual of the
system a. The line through the two points 8 meets the curve
in a point @ which is residual to 8, and therefore coresidual
toa. Ifthe same process were repeated with a different conic
we should arrive at a point a”, also coresidual to the system
a, and therefore to the point « ; and the two points a’, «” being
coresidual must coincide (Art. 158).

Now, in the first place, it is evident that the same proof
would hold good, if instead of four points we started with any
positive system of 3n+1 points P. A curve through them of
order p + 1 meets the cubic again in two other points, and the
line joining these meets the curve in a point coresidual to P,
and which is the same point whatever be the curve of order p+.
But, in the second place, instead of proceeding from the group
P to the coresidual point by two stages, we might employ any
even number of stages. ‘Thus through the 39+ 1 points P de-
scribe a curve U,,,, and the residual is the negative system NV
of 37-1 points. Through N describe a curve U,,,, and we get
a residual P’ of 3s+1 points. In like manner, from P’ we
can derive a residual of 3¢—1 points, and so on. And at
this or any subsequent stage where we have a negative
system of 3t—1 points, by describing through them a curve
U, we can obtain a residual of a single point. Prof. Sylvester’s

WITH OTHER CURVES. 139

theorem is, that this point is in all cases the same, no matter
what the process of residuation by which it is arrived at.
In fact, the system N is a residual of P; P’ is a residual
of .V, and is coresidual of P; N’ is a residual of P’, coresidual
therefore with NV, and therefore residual also to P, and so on,
Any positive system in the series is residual to every negative
system, and coresidual to every positive system. The point
therefore at which we ultimately arrive, is coresidual to the
original positive system, and must be identical with the point
coresidual of the same system obtained by any other process.
For example, if through four points we describe a cubic meeting
the curve in five other points; through these five another cubic
giving a residual of four other points, through these four a
quartic giving a residual of eight points; finally, through these
eight a cubic meeting the curve in one other point, this point is
the same as that obtained from the original four by the process
of Art. 154. And similarly, starting with any negative system
of 3¢g-1 points N, we may after any odd number of stages
arrive at a single point, which will be the residual of the original
system, and as such, independent of the particular process of
residuation.

161. The principles just established, enable us to find by
linear constructions, the point residual or coresidual to a given
negative or positive system. For example, if it were required
to find the point residual to eight given points, join them any
way in pairs, and the joining lines form a quartic system meet-
ing the curve in four new points residual to the given eight:
join these again in pairs, and we obtain a system of two points
coresidual to the given eight; the point where the line joining
these meets the curve is the residual point required. Or,
again, we may replace any four of the given points by their
coresidual point, constructed as in Art. 154, and the problem
is reduced to finding the residual of a system of five points;
and similarly, replacing any four of these by their coresidual,
reduce the problem to finding the residual of a system of two.
It is in any of these ways easily seen, that the residual of a
system of eight consecutive points at a given point of the cubic
is the third tangential of the given point.

140 INTERSECTION OF A GIVEN CUBIC.

In this method of finding by linear construction the ninth
point common to all cubics which pass through eight given
points, it is assumed that one cubic through the eight points
is given; and thus the question is not the same as that of
finding the ninth point when only the eight points are given.
Dr. Hart has shown, that in the latter question the ninth point
can also be found by linear construction, though by a more
difficult process.*

162. We conclude this section with a few remarks as to
systems of cubics having several pointscommon. If we are given
eight points on a cubic, or eight linear relations between the
coefficients in the general equation, we can eliminate all the
coefficients but one, so as to bring the equation to the form
U+kV=0. Similarly, if we are given seven points, or seven
linear relations, the general form of the equation can be reduced
to U+kV+lW=0, U, V, W being three cubics fulfilling the
seven given conditions, and the two constants /, Z still at our
disposal, enabling us to fulfil any two other conditions. And so
again if we are given six points, the general form of the equa-
tion is U+kV+1W+mS=0. We may take for U, V, &e.
systems of three lines passing each through two of the given
points. Thus, the six points being a, d, ¢, d, e, f, and ab=0
denoting the equation of the line joining a, 4, one form of the
equation of the required cubic is

ab.cd.ef + k.ac.be.df + l.ad.bf.ce + m.ae.bd.cf =0.

Since this equation contains three indeterminates, every other
cubic through the six points (for example, af.dc.de) must be
capable of being expressed in the above form, and the pre-
ceding equation would gain no generality if we were to add to
it a term n.af.bc.de, since this itself must be the sum of the
preceding four terms multiplied each by some factor.

In precisely the same manner as (Conics, Art. 259) we derived
the anharmonic property of the points of a conic from the equa-
tion ab.cd =k.ac.bd, we can derive from the equation just
written the following, which is the extension of the anharmonic
theorem to curves of the third degree: “If six given points on

* Cambridge and Dublin Mathematical Journal, vol. vi. p. 181.

WITH OTHER CURVES. 141

such a curve be joined to any seventh, and if any transversal
meet this pencil in points a, d, c, d, e, f, then the relation holds

ab.cd.ef +k.ac.be.df+l.ad.bf.ce+ m.ae.bd.cf=0,

where k, 7, m are constants, whose value is the same for each
particular curve through the six points.” The reader can easily
conceive the number of particular theorems which may be
derived from this (as in Conics, Art. 326), by examining the
cases where some of the points are at an infinite distance.

163. We saw (Art. 41) that to be given a double point was
equivalent to three conditions. If then we have a double point
and five other points, one more condition will determine the
curve, which may, therefore, be expressed by an equation of
the form S—S'=0, where S, S' are two particular curves of
the system. We may write it in the form

(oabed) oe — k (oabce) od = 0,

where {oabcd) denotes the conic through the double point o and
the four points abcd.

In like manner we may write the equation of the cubic
through the double point and four other points

oa.0b.cd + k.ob.oc.ad + l.oc.oa.bd =03

and, as in the last Article, the same relation holds between the
intercepts on any transversal by the line joining these points to
any point of the curve.

164, By the help of the same method (Conzes, Art. 259) of
expressing the anharmonic ratio of a pencil in terms of the perpen-
diculars let fall from its vertex on the sides of any quadrilateral
whose vertices lie each on a leg of the pencil, we can find the
locus of the common vertex of two pencils, whose anharmonic
ratio is the same, and whose legs pass through fixed points,
two of the fixed points being common to both pencils. For if
ab=0 denote the equation of the line joining the points ab, we
get an equation of the form

or ao.bp.cd = ab.co.dp.

142 POLES AND POLARS,

When 9, p are the two circular points at infinity, this gives us
( Conics, Art. 358) the locus of the common vertex of two triangles
whose bases are given and vertical angles are equal, and we
see that it is a curve of the third degree passing through those
circular points.

If the difference of the vertical angles were given, this would
be equivalent (Conics, Art. 358) to the ratio of two anharmonic
functions, and we should be led to an equation of the form

ao.bp _ > co.dp
ap.bo _ cp.do’

which represents a curve of the fourth degree, having the two
circular points for double points.

SECT. II.—POLES AND POLARS.

165. We next recapitulate and apply to the cubic the
theorems about poles and polars which we have already
obtained. Every point O (2’, 7’, 2’) has, with respect to a cubic,
a polar line and a polar conic, whose equations respectively are

OT AM: 50:d WD dU, dU, dU _
© ae 8 dy at ae 8 aga F
The equation of the polar conic may also be arranged according
to the powers of a, y, z, and will then be
a'x’ + b'y' + 2" + 2f'y2 + 2q'zx + 2h'xy =0,
where a’, b', &c. represent the second differential coefficients
written with the accented letters.

The polar conic is the locus of the poles of all right lines
which can be drawn through O, and thus every right line has,
with respect to a non-singular cubic, four poles, namely the
intersections of the polar conics of any two points on the line.
The polar conic passes through the points of contact of the six
tangents which can in general be drawn from O. In the case
of a nodal cubic, the polar conic passes through the double
point and meets the curve elsewhere only in four points; and
every line has but three poles; since the two polar conics (each
passing through the double point) intersect in only three other
points. In the case of a cuspidal cubic, the polar conic passes
through the cusp, touches the cuspidal tangent and meets the

POLES AND POLARS. 143

curve elsewhere only in three points; and every line has but
two poles. Ifthe cubic break up into a conic and a right line,
the polar conic of a point O passes through their intersections,
and every line has but two poles. The polar conic also passes
through the intersection of the conic with the polar of O with
respect to it; for it is easily seen that if we perform on LS,

yy

the operation A or 2 +y Eta z' — ee , the result is L'S+ LAS.

If the cubic reduce to three fo tive xyz=0, every polar
conic passes through the vertices of the triangle formed by
them, and every right line has but one pole. In this case the
equations of the polar line and polar conic are respectively
xy'2' + y2'x' +2a'y'=0, w'yet+y'zx+2zxy=0,
or Seo Bisk chdin Bivied:
ee Gee eis” Raleet

The equation just given affords at once a geometrical con-
struction for the polar line, M

since it appears from Conics,
Ep
A F B

Art. 60, that if the point O in
xy is (Contcs, Art. 127) = , + = = 0, and is therefore constructed

the figure be «'y’'z’, the line
IMN will be that whose
equation has been just
written. The tangent to
the polarconic at anyvertex N

by joining the vertex xy ; the point where the polar line meets
the opposite side z.

166. If any line through O meet the cubic in points A, B, C,
the point P in which meets the polar line is determined, since
1 1 1 ;
— 04+ OBt OG’ If a second line
through O meet the cubic in points A’, B’, C’, the point P’ in
which the polar meets this line is also determined, and therefore
the polar line itself, which must be the same for all cubics pass-
ing through the six points 4, B, C, A’, B’, C’. Thus then we
can by the ruler alone construct the polar line of O with respect
to the cubic; for we have only to draw two radii through O,

(Art. 132) we have

144 POLES AND POLARS.

and construct, by Art. 165, the polar of O with respect to the
triangle formed by 44’, BB’, CC".

The metrical relations, given Art. 134, shew also that when
the points A, B, C are given the two points in which the line
OA meets the polar conic are likewise given. We see then,
as before, that if we draw three radii through the origin meet-
ing the curve in A, B, C, A’, BY, C’, A”, B’, C", the polar
conic of O is the same with regard to all cubics passing through
these nine points. The points A, A’, A” may be taken as
the points in which any transversal meets the curve, and the
problem of constructing the polar conic of O with respect to
a cubic may be reduced to constructing it with regard to the
' system made up of the line 4A’A”, and the conic through the
six remaining points.

We consider now in more detail the cases (1) where O isa
point on the curve, (2) where it is a point on the Hessian.

167. If from two consecutive points 0, O' of the curve we
draw the two sets of tangents OA, OB, OC, OD; O'A, O'B,
O'C, O'D, any tangent OA intersects the consecutive tangent
O'A in its point of contact. Now the four points of contact
A, B, C, D lie on the polar conic of O, which also touches the
cubic at the point O (Art. 64); hence the six points OO'ABCD
lie on the same conic, and therefore the anharmonic ratio of
the pencil {O.ABCD} is the same as that of the pencil
{O'.ABCD}. Since then this ratio remains the same when we
pass from one point of the curve to the consecutive one, we leartf
that the anharmonic ratio is constant of the pencil formed by the
four tangents which can be drawn from any point of the curve.

We shall afterwards give an algebraical proof of this
theorem, by shewing that the anharmonic ratio of four lines
given by a homogeneous biquadratic in 2 and y, can be ex-
pressed in terms of the ratio of the invariants S* and 7” of the
biquadratic, and that when the four lines are tangents drawn
from a point on a cubic, this absolute invariant of the pencil can
be expressed in terms of an absolute invariant of the cubic, so
as to be the same, no matter where the point be taken. This
invariant is a numerical characteristic of the cubic unaltered by
projection or any other linear transformation. It was shown

“aN

POLES AND POLARS. 145

(Higher Algebra, Art. 213) that by the value of this invariant of
a biquadratic, we can discriminate those whose roots are two
real and two imaginary, from those whose roots are either
all real or all imaginary. Consequently, if from any point of a
cubic the four tangents which can be drawn to the curve are
two real and two imaginary, the same will be the case from
every point of the curve; and, in like manner, if the tangents
from any point are either all real or all imaginary, the tangents
from every point are either all real or all imaginary. On this
is founded a fundamental division of non-singular cubics into
two classes, those to which from each of their points can be drawn
two and only two real tangents, and those to which the tangents
may be either all real or all imaginary. This remark will
be further developed in the section on the classification of cubics,
and it will there be shewn that, in the second case the cubic
consists of two distinct portions, from every point on one of
which portions the tangents are all real, and on the other
portion are all imaginary.

168. It follows, from Art. 167, that, if O, Pbe any two poin’s
of the curve, through these points can be drawn a conic passing
through the four points where each of the tangents from the
first point meets the corresponding tangent from the second.
The anharmonic ratio of four points abcd is unaltered by writing
them in the order bade or cdab or dcba; hence, by taking the
legs of the second pencil successively in each of these four
orders, we see that the sixteen points of intersection of the
first set of tangents with the second, lie on four conics, each
passing through the points OP.

Let the cubic be circular, that is to say, let it pass through
the imaginary points J, J at infinity; then by taking these
for the points O, P we see that the’sixteen foci of a circular
cubic lie on four circles, four on each circle.*

169. When O ts a point on the curve, every chord through ‘tt
ts cut harmonically by the curve and by the polar conic of O.

* This theorem was first otherwise obtained by Dr. Hart, and thence was
suggested to me the theorem of Art, 167,

U

146 POLES AND POLARS.

We saw (Art. 78) that the intersections with the curve of the line
joining any two points are determined by the equation
MU' + pA’ +rAWA + pw U=0.

~ When a’y’z’ is on the curve, U'=0, and the preceding equation
becomes divisible by w, and if further, the points xyz, a'y'z’ are
connected by the relation A = 0, the remaining quadratic is of
the form 2° A’+ u* U=0, the roots of which being equal and
opposite, we see, as at Conics, Art. 91, that the line joining the
two points is cut harmonically by the curve. The same thing
may also be proved by taking the point O for the origin, and
finding the locus of harmonic means of all radii vectores through
O. We proceed exactly as in Art. 132, making first A =0,
and we find immediately

2 (Bu + Cy) + Da’ + Exy + Fy’ =0,
which is the equation of the polar conic of the origin. -

It is proved (as in Art. 136) that the tangent to the polar
conic at the point where any chord meets it passes through
the intersection of the tangents to the cubic at the points where
it is met by the same chord, and is the harmonic conjugate to
the line joining their intersection to the point O.

170. Let us now consider more particularly the case where
O is a point of inflexion. It was shewn (Art. 74) that the
polar conic of a point of inflexion breaks up into two right
lines, one of them being the tangent at the point. And the
same thing would appear from the equation of the polar conie
of the origin just given. For, in order that the origin should
be a point of inflexion and the axis of y the tangent at it, we.
must have (see Art. 46) d=0, B=0, D=0, when the equation
of the polar conic (Art. 169) reduces to |

2Cy + Hay + Fy’ =0.

The factor y is evidently irrelevant to the problem of the locus
of harmonic means; we learn therefore that if radi vectores be

drawn through a point of inflexion, the locus of harmonic means
will be a right line.* And, conversely, if the locus of harmonic

a
f

* This theorem is Maclaurin’s; De Linearum Geometricarum Proprietatibus E
Generalibus, Sec. 111. Prop, 9, E

ne ie a ee

~~
f

POLES AND POLARS. 147.

means be a right line, the point O is a point of inflexion. For,
Art. 74, the only other case in which the polar conic can break
up into two right lines is when O is a double point, and that
case does not apply to the present problem, since a line
through the double point must meet the curve only in one
other point.

We shall callthe line just found the harmonic polar of the
point O, to distinguish it from the ordinary polar line which
is the tangent at O.

171. The point O possesses, with regard to the harmonic
polar, properties precisely analogous to those of poles and polars:
in the conic sections. Thus if two lines be drawn through 0,
and their extremities be joined directly and transversely, the
joining lines must intersect on the harmonic polar. This is an
immediate consequence of the harmonic properties of a quad-
rilateral.

Hence again, as a particular case of the last, tangents at the
extremities of any radius vector through O must meet on the
harmonic polar.

The harmonic polar must pass through the points of contact
of tangents which can be drawn through 0, for, since OL’ RR"
is cut harmonically, if 2’ coincide with &”, it must coincide
with &. Hence through a point of inflexion but three tan-
gents can be drawn, and their points of contact lie on a
right line.

If the curve have a double point, it is proved, in precisely
the same way, that it must lie on the harmonic polar.

The first theorem of this Article may be otherwise stated
thus: if three points A’B’C’ lie on a right line, and the lines
joining O to them meet the curve again in A”B"C", these will
also lie on a right line, and the two lines will meet the harmonic
polar in the same point. Ifnow we suppose 4’,B’,C' to coincide,
we arrive again at the theorems that the line joining two points
of inflexion must pass through a third, and that the tangents at
any two meet on the harmonic polar of the remaining one.

| 172. If through any point of inflexion O there be drawn
three right lines meeting the curve in A,, A,; B, B,; C,, C,,

2? 2

148 POLES AND POLARS.

then every curve of the third degree through the seven points
OA,A,B.B,C,C, will have O for a point of inflection. For let
the three lines meet the harmonic polar in A, B, C, then these
points are also common to the loci of harmonic means of the
point O, with regard to all curves through the seven points.
This locus, then, which would in general be a conic, must,
since these three points of it are in a right line, be for all these
curves this same right line; and therefore (Art. 170) the point
O must be a point of inflexion.

173. We have seen (Art. 74) that the points of inflexion of a
curve of the third degree are the intersections of the curve JU
with the curve H, which is also a curve of the third degree.
Every curve of the third degree has therefore, in general, nine
points of inflection, only three of which, however, are real (see
Art. 125, Ex. 3). Since, also, we have proved that the line
joining two points of inflexion must pass through a third,
through each point of inflexion can be drawn four lines, which
will contain the other eight points. It follows then, as a par-
ticular case of the last Article, that any curve of the third degree,
described through the nine points of inflexion, will have these
points for points of inflexion.*

174. Of the lines which each contain three points of inflexion,
since four pass through each point of inflexion, there must be in
all 4 (4x 9) =12.T

If we attempt to form a scheme of these lines, it will be found
that it can only differ in notation from the following:

123, 456,789; 147, 258, 3693}
159, 267, 848; 168, 249, 357.
Hence it will follow that any cubic passing through any seven

* This theorem is due to Hesse, who showed that if U be a cubic, # its
Hessian, aU +bH=0 the equation of any cubic through their intersections, then
the equation of its Hessian is of the same form. The method of proof here
adopted is Dr. Hart’s.

+ It is easy to see that we may have nine real points lying by threes in ten
lines, but not in a greater number of lines: thus the nine points of inflexion cannot
be all real, which agrees with the remark, Art. 178.

t+ Clebsch has remarked that if we arrange the nine elements 1, 2,3 the systems

4, 5, 6

7, 8, 9
of lines are the three rows, the three columns, those forming positive, and those
forming negative, elements of the determinants, . :

?

POLES AND POLARS. 149

of the points of inflexion will have one of these for a point of
inflexion; for, take any seven (say the first seven), and it will
appear from the above table that they lie on three right lines
(147, 267, 357), intersecting in a common point on the curve,
and therefore, by Art. 172, that common point (7) is a
point of inflexion on them all.

From the manner in which these lines have been written, it
appears that they may be divided into four sets of three lines,
each set passing through all the nine points; or that, if we form
the equation U+ 7XH=0, there are four values of A, for which
the equation reduces itself to a system of three right lines.
For a direct proof of this, see the last section of this Chapter.

175. Let us now consider the case (2) where 2’y'z’ is on the
Hessian, and where its polar conic therefore breaks up into two
right lines. It was proved in general (Art. 70) that if the first
polar of any point A has a double point B, the polar conic of B
has a double point A. But in the case of cubics, the first polar
is the polar conic, and this theorem becomes, [f the polar conic
of A breaks up into two lines interseeting in B, the polar conic of
B breaks up into two right lines intersecting in A. In fact, if the
polar conic of a'y'z’ breaks up into two right lines, the coor-
dinates of their intersection xyz satisfy the three equations
got by differentiating the equation of the polar conic. But
(Art. 165) this last equation may be written in either of the
equivalent forms |

Ua! + Uy! + Uz =0,

or ax’ + by’? +2? + 2f'yz + 2g'zx + 2h'xy =0,
and the differentials may therefore be written in either of the
equivalent forms

ax’ +hy'+ gz'=0, ha’ + by'+fze'=90, gu’ +fy'+cz'=0,

adethy+g2=0, ha+by+fz=0, gat+tfyt+cz=0,
whence we see that these equations are symmetrical between
xyz and 2'y'z’, and therefore that the relation between those
points is reciprocal. Both A and B are evidently points on the
Hessian, on which they are said to be corresponding points,
and it will presently be shewn that they are so also in the
sense explained, Art. 151, that is, the tangents to the Hessian

150 POLES AND POLARS.

at the points A, B respectively meet in a point of the Hessian.*
In the case of the cubic, therefore, the curve called the Steinerian
(Art. 70) is identical with the Hessian.

176. The equation of the polar conic of any point what-
ever &nf being EU,+7U,+¢U,=0, the whole system of polar
conics form a system of conics such as that discussed, Conées,
Art. 388, viz. the equation of which involves linearly two in-
determinates. The equation of the polar of the point A with
regard to any conic of the system is

E (aa! + hy’ + ga) +m (ha' + by’ + fa’) + O(ga' + fy’ + c2') =0,
which is satisfied by the coordinates of B, whence we see that
the polar of either point A, B passes through the other, and
that therefore the Hessian of the cubic is the Jacobian (Conics,
Art. 388) of the system of polar conics. Since A and B are
conjugate with regard to any conic of the system, the line
joining them is cut harmonically by every one of these conics,
and the points in which the conics meet that line form a system
in involution of which A and JB are the foci. The two points
in which any of these conics meets the line AB can only coin-
cide at either of the points A, B; and, consequently, if any of
the conics break up into two right lines intersecting on AB,
the point of intersection must be either A or B, unless AB
be itself one of the lines. Now since the Hessian of a cubic
is itself a cubic, AB meets it in three points; that is to say,
in a third point C besides the points A, B. Every point on
the Hessian is, as we have seen, the intersection of the two
lines into which some polar conic of the system breaks up, and
it follows from what has been just proved, that of the two
lines which intersect in C one must be AL. Thus, then, from
the system of points whose locus is the Hessian we may derive
a system of lines, viz. by taking the pairs of lines which are
the polar conics of each point on the Hessian. ach line of
the system meets the Hessian in three points; two of them

* It will subsequently be shown that there are three cubic curves having each
of them the same Hessian: the correspondence of the points A, B on the Hessian is
of one or another of the three kinds of correspondence according as the cubic curve
is one or another of the three cubics.

ae jail ste

ee» Ee ae

PARTY eek Oe PT RRR Te See ke Le ee ee ©, wee. eS See a oe eee ee, eee

ee ee SS eee

POLES AND POLARS. 151

A, B are corresponding points on the Hessian, and the third, C,
which we may call the complementary point, is the point in
which the line meets the conjugate line.

177. The curve which is the envelope of the system of
lines just mentioned has been studied by Prof. Cayley, and
has on-that accoynt been called by Cremona the Cayleyan of
the cubic.* It is of the third class, as we see by examining
how many of these lines can pass through an arbitrary point P.
Any point 1M whose polar conic passes through P must lie
on the polar line of P (Art. 61), and in order that the polar
conic should break up into lines, JJ must be on the Hessian.
There are then evidently three points MW, whose polar conic
reduces to a pair of lines, one of which passes through P. There
is not any double or stationary tangent, and the curve is there-
fore of the sixth order.

Every line of the system joins corresponding points on the
Hessian (Art. 176); therefore the Cayleyan may at pleasure
be considered as the envelope of the lines into which the polar
conics of the points of the Hessian break up, or as the envelope
of the lines joining corresponding points on the Hessian. In
the case, however, of curves of higher degree, the envelope of
the lines joining the corresponding points A, B (Art. 70) is
distinct from the envelcpe of the lines into which polar conics
may break up. |

The Cayleyan may also be regarded (Art. 176) as the
envelope of lines which are cut in involution by the system of
polar conics. It was shewn, Conics. (Art. 3882), how the equation
of the envelope regarded from this point of view may be written
down, and that the curve is of the third class,

178. Let us now examine what are the four poles with
respect to the cubic of the tangent to the Hessian at any point A.
The four poles in question are the intersections of the polar conic
of A with the polar conic of the consecutive point A’ on the
Hessian. ‘The polar conic of A is the pair of lines BL, BN (see

fig. p. 153), and the polar conic of A’ is a pair of lines consecutive

* It was denoted by Prof. Cayley himself by the letter P, and called by him
the Pippian.

152 POLES AND POLARS.

to these. Now BL meets the line consecutive to BN in the
point B; BN meets the line consecutive to BL in the same
point; and BL, BN meet the lines respectively consecutive
to them in their points of contact with their envelope. The
four poles in question are thus the point B counted twice, and
the points of contact with the Cayleyan of the lines BZ, BN.
Thus, in particular, the polar line with respect to the cubic of
any point on the Hessian is the tangent to the Hessian at the
corresponding point. It may be directly inferred from what
has been said, that the Cayleyan is, as stated above, of the
sixth order. For the equation of the locus of the poles with
respect to the cubic of the tangents to the Hessian, is found
by expressing the condition that «U,+yU,+2U, should touch
the Hessian. This condition involves the quantities U,, U,, U,
in the sixth degree, and the locus is therefore of the twelfth
order. But, from what has been proved, the Hessian must
enter doubly as a factor into this equation; the remaining
factor therefore, which is the Cayleyan, is of the sixth order.

179. The locus of points whose polar lines with regard to
one curve U touch another curve V, evidently meets U at its
points of contact with the common tangents to U and V3 for
the polar of any point on U is the tangent to U at the point,
and if it is also a point on the locus, the polar by hypothesis
touches V. We have just seen that when JU is a cubic and
V its Hessian, the locus consists of the Cayleyan together with
the Hessian itself counted twice. The cubic and the Hessian
being each of the sixth class have thirty-six common tangents,
And we now see that these common tangents consist of the
tangents to U at the 18 points where it is met by the Cayleyan,
and of the tangents to Uat the points where it is met by the
Hessian; (that is to say, of the nine stationary tangents) these
last tangents each counting for two; and in fact it was remarked
(Art. 46, p. 33), that each stationary tangent to a curve

may be regarded asa double tangent, as joining both the first —

to the second, and the second to the third of three consecutive
points.*

* Reasons were given (Art. 47) for treating the cusp and the node, the stationary
and double tangent, as distinct singularities; but in counting the intersections of

a ot .
So a ee 5 Woh

POLES AND POLARS. 153

The polar conic of a point of inflexion A consists (Art. 170)
of the inflexional tangent itself, together with the harmonic polar
of A; and the point B corresponding to A is therefore the point
in which the inflexional tangent meets the harmonic polar.
And the tangent to the Hessian at B is the polar of A with
respect to the cubic; that is to say, is the inflexional tangent
itself. Hence, then, the nine points where the stationary tan-
gents touch the Hessian are the points where each stationary
tangent meets the corresponding harmonic polar.

It may be inferred from what has been just proved, and it
will afterwards be shewn independently (see note p. 150), that
the problem to find a cubic, of which a given cubic shall be the
Hessian, admits of three solutions. For the points of inflexion
being common to both curves (Art. 173), we are given nine points
(equivalent to eight conditions) through which the required cubic
is to pass, and if we were given the tangent at any of these
points .A, the cubic would be completely determined. But what
has been just proved shews that this tangent may be any one of the
three tangents (Art. 171) which can be drawn from A to the curve.

180. The tangents to the Hessian at corresponding points
A, B, meet on the
Hessian. Let the 4
polar conic of A |
be BL, BN, and
of Bhe AR, AN;
then L, M, N, &
are the four poles
of the line AB,
and the polar conic
of every point of
AB passes through
these four points.
If, therefore, this
polar conic breaks
up into two right lines, these lines must be LR, MN; and

two curves, a cusp or node on one of them alike counts for two; and a stationary
or double tangent to one of them alike counts for two among their common
tangents,

x

154 POLES AND POLARS.

we see that D is a point on the Hessian, and that it cor-
responds to the point Cin which AB meets the Hessian again.
But the tangent at B to the Hessian is.the polar of A with
respect to the cubic, which must also be its polar (Art. 60) with
respect to the polar conic. of A (BL, BN); therefore, by the
harmonic properties of a quadrilateral, this tangent is the line
BLD; and in like manner the tangent at A is the line AD.

If we are given the Hessian and a point on it A, the
problem to find the corresponding point B admits of three
solutions (see Art. 151). For if we draw the tangent at A
meeting the curve again in D, B may be the point of contact of
any of the three other tangents besides 4D, which can be drawn
from D to the curve. These three solutions answer to the

three different cubics, of which the given curve may be the
Hessian.

181. The points of contact with the Cayleyan of the four lines
BL, BN, AR, AN lie on a right line. The poles of AD with
respect to the cubie are the intersections of the polar conics
of A and D; the former is the pair of lines BL, BN; the latter
consists of the line 4B and a conjugate line passing through C.
The four poles are therefore the point B counted twice, and the
two points where Ca meets BL, BN. But AD being a tangent
to the Hessian, it appears, from Art. 178, that the latter two
poles are the points of contact of the lines BZ, BN, with their
envelopes. In like manner the points of contact of AR, AN
with their envelope lie on the same right line. This right line
is itself a tangent to the Cayleyan, therefore the six points
where it meets the Cayleyan are completely accounted for. In
other words, any tangent to the Cayleyan is one of a pair of
lines into which some polar conic breaks up; the other line
of the pair joins two corresponding points on the Hessian;
the four lines which make up the polar conics of these two
points pass respectively through the four points where the
given tangent meets the Cayleyan again.

Again, to find the point of contact of any given tangent.
with the Cayleyan, the rule we have arrived at is to take what
we have called the complementary point on the given tangent,
and join it to the corresponding point on the Hessian; the line

POLES AND POLARS. 156

conjugate to this meets the given tangent in the point required.
But we may hence deduce a simpler rule: for since the two
lines last mentioned make up a polar conic, and since every
polar conic divides harmonically the line joining two corre-
sponding points, the rule is to take the three points in which
the given tangent meets the Hessian, consisting of two corre-
sponding points arid one complementary, and to take the har-
monic conjugate of the complementary point with respect to
the two corresponding points.

182. Let us apply the preceding rules to the case where
A is a point of inflexion, and B, the corresponding point, is the
point in which the inflexional tangent meets the harmonic polar.
The polar conic of B is then a pair of lines through A, and the
polar conic of A is the inflexional tangent together with the
harmonic polar. In order to find the points in which these
four lines touch the Cayleyan, we take the point in which the
line AB meets the Hessian again; but this is the point B, since
AB touches the Hessian; and the line through B conjugate to
AB, on which the four points of contact lie, is the harmonic
polar. ‘T’hus, then, the point of contact of the inflexional tangent
with the Cayleyan is the point where it meets the harmonic
polar; or (Art. 179) the Cayleyan and the Hessian touch each
other, having the nine inflexional tangents for their common
tangents. ‘The Cayleyan, as a non-singular curve of the third
class, has nine cusps, and the construction just given shews
that the harmonic polars are the nine cuspidal tangents.

183. It has been shown that the tangent to the Hessian at
any point A meets the Hessian again in the point D, where it
meets the polar of A with respect to the cubic. It follows that
the tangent to a cubic at any point A meets the cubic again
in the point where it meets the polar of A with respect to a
cubic having the given cubic for its Hessian. Now such a cubic
passes through the inflexions of the given cubic, and therefore
its equation will be of the form aU+ 6H=0, and the equation
of the polar of any point with respect to it will be of the form

Diy AF or) a ( OH; 2 dit. 20 allt

(ear ty Gy t* a) © a TY ag Tan)

156 POLES AND POLARS.

It follows, then, that the point where any tangent meets the
cubic again is found by combining the equations
dU" ae ae OF 2 peti 5 « BAER SRS So

Rae TY Gy t tag eo Oe he Gee oe

In other words, the tangential of a point 2’y'z’ on the cubic is
the intersection of the tangent to the cubic at that point with
the polar of the same point with regard to the Hessian; and
hence may immediately be derived expressions for the coor-
dinates 2, y, 2 of the tangential in terms of a’, 9’, 2’, viz. they
are proportional to U,H,—U,H,, U,H,-—UH,, U,H,— UA,
functions of the fourth degree in 2’, y’, z’.

= 0.

184. The polar lines of the points on a given line axt+Byty2
envelope a conic, which we call the polar conic of the given line.
The equation of the polar of any point 2'y'z’ may be written

ax” + by” + ca" + 2fy'2' + 2gz'a' + 2ha'y' =0,
and the problem of finding the envelope of this, subject to the
condition az’ + By'+yz'=0, is the same (Art. 96) as that of

finding the condition that a line should touch a conic. The
equation of the envelope required is therefore

Ad’ + BB’ +Cy' + 2 By +2Gya+2HaB =0,

where A, B, &c. have the same meaning as in the Conics,
viz. be—f*, ca—g’, &c. They are therefore functions of the
second degree in the coordinates x, y, z. It is obvious that the ©
polar conic of a line might have also been defined as the locus
of points whose polar conics touch the given line.

If the method of Art. 88 had been applied to find this
envelope, the solution would be found to depend on the
equations

ax’ +hy' + gz'=da, ha' + by'+fze'=rB, ga’ +fy' + cz'=ry.

But these are the equations by which (Conics, Art. 293) we
should determine the pole of the given line with regard to
x'U,+y'U,+2'U,. Hence, as might also be seen from geo-
metrical considerations, the polar conic of a line is also the locus
of the poles of the line with respect to the polar conics of all
the points of the line. pe

POLES AND POLARS, 157

185. Since the polar line of any point on a line is the same
as if taken with regard to the three tangents at the points
where that line meets the curve, the polar conic of a line is
the same as if taken with regard to those three tangents. Let
their equation be xyz=0. ‘Then to find the polar conic of a
line is (Art. 165) to find the envelope of ay’z' + yz'x' + za'y'=0,
subject to the condition ax’ +Py'+yz2,=0; and this is (see
Conics, Art. 127) |

vi (aa) + V (By) + V(2) =

It follows that if the given line meet the cubic in the points
P, Q, &, the tangents at
esc points forming the
triangle ABC, then the
polar conic of the line
touches the sides of this
triangle in the points D,
i, Ff, which are the har-
monics of the points P,
Q, & in respect to the
point-pairs BO, CA, AB & |
respectively. It is alia a priort that the polar conic is
touched by the tangents to the cubic at P, Q, R, these being
particular positions of the line whose pis eo is ‘ought.

186. It follows from the definition that the tangents which
can be drawn from any point to the polar conic of a right line
are the polars of the two points where the polar conic of the
point meets the right line. Hence the polar conic of a point
meets a right line in real or imaginary points’ according as the
point is outside or inside the polar conic of the line; a point
being said to be outside a conic when from it real tangents can
be drawn to the conic. It has been already remarked, that if
a point lie on the polar conic of a line, its polar conic toiiche
the line:

In particular, since the hes conic of a double point is the
pair of tangents at that double point, the polar conic of every
line with regard to a crunodal cubic has the node outside the
conic, and with regard to an acnodal cubic has the conjugate

158 POLES AND POLARS.

point within it. If the cubic be cuspidal, the polar conic of
every line passes through the cusp.

187. It follows from the foregoing definitions, and from
Art. 135, that if the given line be at infinity, its polar conic
may be defined either as the envelope of the diameters of the
cubic, or as the locus of the centres of the diametral conics
of the cubic, or as the locus of points whose polar conic is a
parabola. Its equation is found by making a and B=0 in
the formula of Art. 184, and is C=0, or ab—h’=0; that 18
to say,

au @U (@#Uun.

ak * dt ~ (dady)
And it appears, from Art. 185, that this is the equation of the
ellipse touching at their middle points the three sides of the
triangle formed by the asymptotes.

188. If the given line touch the cubic, then since the polar
of the point of contact is the line itself, that line coincides
with one of the positions of the enveloped line of Art. 184,
and therefore touches the polar conic; and in no other case
can a line be touched by its polar conic with regard to a non-
singular cubic. Accordingly this principle has been used to
form the tangential equation of a cubic. Since 4, B, &c. are
functions in the coordinates of the second degree, the equa-
tion of the polar conic, Ag” pha =0, may be written in
the form

A's? + By? + C'2? + 2F"y2+2G'zx+ 2H'axy =0,
where A’, &c. are functions of the second degree in a, 8, y, and
then the condition that this should touch the given line is
(B'C'- fF”) a’ +&c.=0, which is of the sixth degree in a,
8, y, and is the required condition that the given line should
touch the cubic.

If the given line touch the Cayleyan, then since it, togiltedl
with another line makes up the polar conic of a certain point,
the polar line of every point on the line passes through that
point, and the envelope of Art. 184 accordingly reduces to a
point. :

POLES AND POLARS. 159

189. We next consider two cubics U, V, and investigate the
problem to find a point whose polar with respect to each shall
be the same; or, what is the same thing, whose polar with
regard to any cubic U+AV=0 shall be the same. In order
that «U,+yU,+2U, and xV,+yV,+2V, may represent the
same line, we must have

or U,V,— U,V,=0, U,V,— U,V,=0, UV,— UV,=0.
From the first form in which the equations were written, it is
plain that the three equations are equivalent to two; and that
the curves of the fourth degree represented by the equations
written in the second form have common points. But all their
points of intersection are not common, for any values which make
the numerator and denominator of any of the three fractions to
vanish, satisfy two of the resulting equations but not the third.
Subtracting then from the sixteen points common to the quartics
represented by the first two equations the four points common to
U,, V,, there remain twelve points common to all three quartics,*
and these are the points required.

190. Since the discriminant of a cubic is of the twelfth degree
in the coefficients (Art. 69), there are in general twelve values
of A, for which the discriminant of U+2AV will vanish; for
if in the general expression for the discriminant we substitute
for each coefficient a, a+ Aa’, we have evidently an equation of
the twelfth degree to determine A (see Conics, Art. 250). The
coordinates of the double point on any of these cubics satisfy
the three equations (Art. 69)

U,+2V,=0, U,+dV,=0, U,+AV,=0.

And the system of equations obtained by eliminating \ between
each pair of these equations is the same as that considered

* So generally if U,, U,, U; be functions of the mth degree in the coordinates, and
V, V2, V3 functions of the nth degree, the system of equations
pa ee
Bye Wer’,
represents three curves of the order m+n, having m?+mn +n? common points
(see Higher Algebra, Art, 257).

160 POLES AND POLARS.

in the last article. Hence, through the intersections of two cubics
U, V there can be drawn twelve nodal cubics, and the polar of
any of the twelve double points will be the same with regard to
all cubics of the system U+XV. ‘These points have been called
the critic centres of the system of cubics. ,

191. If we are given three cubics U, V, W, then the >
coordinates of the double point of any cubic of the system,
~U+pV+vW=0), satisfy the equations

AU, + pV +vW,=0, XU,+ wV,+ v W, =0, 0U,+ wVi+vW,=0;
therefore eliminating A, “, v we see that the locus of the double
points is the Jacobian
U, (V,W,- V,W,) + U, (V,W,- V,W,) + 0, (V,W,—V,W)=0.
If the three cubics have a common point, this is a double point
on the Jacobian; for if the lowest terms in x and y be in
U, V, W respectively ax + by, a’x+b'y, a'x+b"y, the terms in
the Jacobian below the second degree in w and y are easily
seen to be

a,b,ax+by

a,b,@at+by

"?

" " ”
a,b, ax+by

which vanishes identically. ‘Thus, then, the locus of double
points on all nodal cubics passing through seven fixed points
is a sextic having these seven points for double points, since
U, V, W may be taken for any three cubics through the seven
given points. So likewise the double points on the nodal cubics,
which can be drawn through eight points, are determined as the
intersections of the two sextic loci, which we get by leaving out
first one and then another of the eight given points. And since
these sextics have six double points common, the number of
their other intersections is 36 — 24 or 12, which agrees with the
result of the last article.

192. Of some of the twelve critic centres, the position can
in some cases be at once perceived. ‘Thus, in the system
Axyz +uvw =0, where u, v, w represent right lines, it is obvious
that xyz is one cubic of the system, having for double points
xy, yz, 2; im like manner wv, vw, wu are double points; there

POLES AND POLARS. 161

are therefore but six other critic centres. We shall more par-
ticularly study the system dXzyz+u’v=0, and will presently
show that this system has but three critic centres, exclusive of
the points xy, yz, zx, uv. Pliicker’s classification of cubics was
derived from the study of this equation for the case where uw
is the line at infinity, and consequently wv its satellite, and
x, y, 2 the three asymptotes. We may then for any position
of the lines a, y, z, v, study the forms which the curve assumes
as we give different values to the parameter X3 and it will be
readily understood, that each nodal curve in the series corre-
sponds to a change from one form of the curve to another.
‘Thus we have seen (Art. 39) that an acnodal cubic is the limiting
form of a cubic including an oval as part of the curve; and
again, if for one value of the constant, a cubic has two real
branches intersecting in a node, the example of conics makes
it easily understood, that for a small increase in the value of the
constant, the cubic will have separated portions in two of the
vertically opposite angles formed by the intersecting branches,
while for a small decrease in the constant it will have portions
in the other pair of vertically opposite angles. Hence the
importance of the critic centres in this mode of studying the
form of the cubic.

193. Since the polar of any point with regard to wv passes
through the point uv, any point which has the same polar with
regard to xyz must lie on the polar conic of wv with regard
to ayz, and it is therefore evident a prior?, that this is a locus on
which the critic centres lie. In order completely to determine
them, let us suppose that we have u=a+y+2, v=ax+by + cz;
and we get our result in a more convenient form, if before
differentiating Axyz+u*v we first divide all by u*. We then
have, differentiating successively with respect to x, ¥, 2,

Ayz (a - y — 2) rex (y —2 — a) Aay (2-a—y) _

(etyte ° (tytey) °. e@ryt+eP

ax ba Cz
whence = J a I
L-y-2@ y-2-“H 42-x-y

and the form of the equations shows that the problem has been

reduced to that of finding the critic centres of a system of two
Y

162 CLASSIFICATION OF CUBICS.

conics, and that the three points required are the vertices of
the common self-conjugate triangle of the conics

ax’ + by’ + cz’=0, and w+ y4*+ 2" — 2yz—2zx —2ay=0,
where it will be observed that the latter conic is the polar
conic of u with respect to xyz; that is to say, when w is at
infinity, it is the conic touching at their middle points the
sides of the triangle formed by the asymptotes. Two critic
centres will coincide in the point of contact when az’ + by’ + cz?=0
touches this conic ; hence, if » be regarded as variable, the locus
of double critic centres is the polar conic of w with respect to ayz.

The condition of contact of these two conics is easily seen, by
the ordinary rule, to be

(6¢+ca+ab)=270°0'c’, ora@?4+b4+4+¢%=0,
which is the tangential equation of the envelope of the satellite

of w when two critic centres coincide. This answers (Ex. Art. 90)
to the equation in point coordinates a! + y?+ 2! =0.*

194. Any point on Aryz+u7%v may be determined as the
intersection of z= @v with O6Azy+u*7=0. When w is at infinity,
the latter equation denotes a system of hyperbolas having z, y
for their asymptotes, and by the property of the hyperbola, the
chords intercepted by these hyperbolas on any line z= 6v have
a common middle point; namely, the point of contact of this
line with one of the hyperbolas of the system. Evidently, if z= Ov
either touch the cubic or pass through a double point on it, it
must touch the hyperbola, the critic centre being in the latter
case the point of contact. Hence, if any of the critic centres
be joined to the finite points where the asymptotes meet the
curve, the critic centres are the middle points of the chords
intercepted by the cubic on the joining lines.

SECT. III.—CLASSIFICATION OF CUBICS.

195. We shall shew in the first place that the equation of
every cubic may be brought to the form

zy’ =ax’ + 38bx"2 + 3cx2z" + dz’.

* For a fuller discussion of this theory, see papers by Prof. Cayley, “On a case
of the involution of cubic curves,” and “On the classification of cubic curves,’
Transactions of Cambridge Philosophical Society, vol, X1., 1864.

Moke

ee nT es ee

CLASSIFICATION OF CUBICS. 163

Every real cubic has at least one real point of inflexion, for
imaginaries enter by pairs, and the total number of points of
inflexion is odd, viz. either nine, three, or one (Art. 147). ‘If
we take for the line z the tangent at the point of inflexion, and
for x any other line through that point, the equation of the
curve (Art. 51, vit.) will be of the form z/= az’, where ¢ is
a function of the sécond degree, say
y? + Qlye + 2myn+ pa? + Wque+re*,

But now if we transform the lines of reference so as to take
y +lz+ mz for the new y, the terms in ¢ containing y only in
the first degree are made to disappear, and the equation takes
the form first written in this article. The geometric meaning
of the transformation we have made is that we take for 2 as
above stated the tangent at a real point of inflexion za, and
for y, the harmonic polar (Art. 170) of that point: for if we
examine where any line through the point of inflexion meets the
curve represented by the above equation, we find, on making
the substitution z= Aw, that we obtain for y values of the form
+x, shewing that the points where the line meets the curve
are harmonically conjugate with respect to the point where it
meets the line y, and to the point of inflexion.

196. In classifying curves those distinctions may be
regarded as fundamental which are unaffected by projection;
or, in other words, which separate not only curves, but cones,
of the same order. Among curves of the second order there
is no such distinction, for there is but one species of cone.
In order to ascertain whether such distinctions exist among
cubics, it suffices to take the form to which, as shown in the
last article, the equation of every cubic may be reduced, and to
examine whether any and what varieties, unaffected by projec-
tion, exist among the curves capable of being represented by
it. And since we are now only concerned with varieties
unaffected by projection, we may suppose the line 2 to be at
infinity, and discuss the form

y = ax’ + 3bx* + 3cx + d,

as one capable of representing a projection of any given cubic.
It will be observed that when a point of inflexion is at infinity,

164 CLASSIFICATION OF CUBICS.

a system of lines through it becomes a system of parallel ordi-
nates, and the harmonic polar becomes a diameter bisecting
them; and, in fact, for every value of x, the above equation
gives equal and opposite values of y.

The preceding equation has already been partially discussed
(Art. 39), and from what was there said, it appears that the
curves represented by it may be divided into the five following
principal classes :

The right-hand side of the equation may be resolvable into
three unequal factors, and (L.) these factors are all real. The
curve then consists (Art. 39) of an oval and an _ infinite
branch. Or (II.) the factors are one real and two ima-
ginary. The oval then disappears and the infinite branch
alone remains.

The right-hand side of the equation may be resolvable
into two equal and one unequal factors, being of the form
(a—«a)?(a—). Then we have the cases (III.), « less than 8
when the curve is acnodal (Art. 39), the oval being reduced to
a conjugate point; or (1V.), a greater than 8, when the curve is
crunodal, the oval and the infinite branch being each sharpened
out so as to form a continuous self-intersecting curve; (V.) the
factors of the right-hand side may be all equal, and the curve
is cuspidal (Art. 39).

Newton has given the name “divergent parabolas ” to the
curves considered in this article; and his theorem, which we
have just established, is that every cubic may ke projected
into one of the five divergent parabolas.

_ 197. Instead of, as in the last article, supposing the
stationary tangent to be projected to infinity, we may suppose
the harmonic polar to be so projected. The point of inflexion
will then become a centre, and every chord through it will be
bisected. Interchanging z and y in the equation of Art. 195,
and then putting z=1, the equation for this case becomes
y =ax + 3ba°y + 3cxy’* + dy’,

which is the equation of a central curve (Art. 131). As in
Art. 196, there are five kinds of central curves according to
the nature of the factors of the right-hand side of the equation,
and in this way is established Chasles’s supplement of Newton’s

CLASSIFICATION OF CUBICS. 165

theorem, viz. that every cubic may be projected into one of the
five central cubics.

198. Corresponding to these five kinds of cubic, there are
five essentially distinct species of cubic cones. A cone of any
order may comprise two forms of sheet, viz. (1) a twin-
pair sheet, or sheef: which meets a concentric sphere in a pair
‘of closed curves, such that each point of the one curve is
opposite to a point of the other curve (a cone of the second
order affords an example of such a sheet); and (2) a single
sheet, viz. one which meets a concentric sphere in a closed
curve, such that each point of the curve is opposite to another
point of the curve (the plane affords an example of such a
cone). Now corresponding to the parabola I. of Art. 196, we
have a cone consisting of a twin-pair sheet and a single sheet,
and corresponding to II., we have a cone consisting of a single
sheet only. It is evident that the crunodal, acnodal, and cus-
pidal singularities are reproduced in the corresponding cones.

The classification of cubic cones just made might, if we pleased,
be carried further. Not only is there but one species of cone of
the second order, but, with some limitations, any two curves of
that order may be regarded as sections of one and the same
cone. This is not so as regards cubics; for it has been proved
(Art. 167) that every cubic curve has a certain numerical cha-
racteristic, expressing the anharmonic ratio of the four tangents
which can be drawn from any point on the curve, and represented
by the ratio of the invariants S°: 7 of the biquadratic, which
determines those tangents. ‘This characteristic being unaltered
by projection, two curves, for which it is different, cannot be
cut from the same cone; and the parameter in question may
be regarded as a characteristic, not only of a cubic curve, but
also of every cone from which it can be cut. The five
‘kinds of cone we have enumerated might, therefore, be further
subdivided at pleasure, according to the values of this parameter,
Such subdivisions have in fact been made, but it is not thought
necessary to notice them here. In the last section of this
chapter, however, the cases S=0, Z'=0 will be discussed; and

it is now pointed out that these represent families not ail of
curves but of cones.

166 CLASSIFICATION OF CUBICS.

199. Let us now examine, more minutely than in Art. 39,
the figure of the cubic represented by the equation considered
in Art. 196, and it will be convenient to take the origin at
the middle point of the diameter of the oval, so that the
equation may be written

ay’ = (a'— m') (% — 2),
where z is greater than m. Differentiating, we find that the
values of « which correspond to maximum values of y, or to

points where the tangent is parallel to the axis of 2, are given
by the equation

. 3x” — 2na—m" =0; whence a=4 {nt /(n* +-3m’*)}.

If we give the negative value to the radical, we get the value
of x corresponding to the highest point of the oval, and since
this is negative, we see that the highest point on the oval
is on the side remote from the infinite branch, and that the
oval is therefore not, like the ellipse, symmetrical with regard
to two axes. This oval is symmetrical with regard to the axis
of x, and not with regard to the axis of y, but rises more
steeply on the one side and slopes more gradually on the other.
The greater 7 is for any given value of m, that is to say, the
greater in proportion the distance between the oval and the

infinite part the more nearly does the oval approach to the

elliptic form; while on the other hand, the difference is greatest
when the oval closes up to the infinite part, that is to say,
when the curve is crunodal. In this case the highest point
of the loop corresponds to the point of trisection of its axis.
If we give the positive value to the radical, the corre-
sponding value of x is intermediate between m and n, and the
corresponding value of y is imaginary. The form of the
equation shews that the point of contact with the curve of
the line at infinity is on the line 2=0, unlike the common
parabola y’= px, which is touched by the line at infinity on
_y=0. The infinite branches of the cubic, therefore, tend to
become parallel to the axis of y and not to the axis of a;
and there must be a finite point of inflexion on each side of
the diameter where the curve changes from being concave
to being convex towards the axis of w Hence the name
* divergent parabola.”

taht sl
{es tg gps ee a eR eta aia

CLASSIFICATION OF CUBICS. 167

The form of the curve is then represented by the oval and
the right-hand infinite branch on the
figure. If, however, we have in the
equation + m’* instead of — m”*, then there
will be no real oval, and the infinite
branch will be either of the left-hand
or right-hand form’ that is to say, there
will or will not be points for which y is
a maximum, and at which the tangent is parallel to the axis,
according as 3m” is less or greater than n*; and there is of
course the intermediate case 3m*=n", where there is on each
side of the axis of x a point of inflexion, the tangent at which
is parallel to this axis.

The figures of the crunodal, acnodal, and cuspidal forms do
not seem to require further discussion than was given in Art. 39.

200. Returning to the case where the curve has an oval,
it is plain that in general every right line must meet any
closed figure in an even number of real points, and therefore
that every line which meets the oval part of the cubic once,
must meet it once again and not oftener; since when a line
crosses to the inside of the oval, it must cross it again to come
out, and cannot meet the oval in four points. Every line,
therefore, must meet the infinite part of the curve once. It
follows that no tangent to the curve can meet the oval again,
and therefore that none of the points of inflexion can lie on
the oval. It is easy to see, on inspection of the figure, that from
any point outside the oval two tangents can be drawn to it.

- Thus, then, the oval is a continuous series of points, from
none of which can any real tangent, distinct from the tangent
at the point, be drawn to the curve. The cubic then, which
includes an oval, is of the class (Art. 167), the four tangents
from every point of which are either all real or all imaginary.
The tangents from every point on the oval are all imaginary,
and from every point on the infinite branch are all real; viz.
two can be drawn to the oval and two to the infinite branch
itself. In fact, the tangent at any point on the infinite branch
must meet that branch again, since the third point in which
it meets the curve cannot be on the oval.

168 CLASSIFICATION OF CUBICS.

201. What has been just said, may be used to illustrate
the essential property of unicursal curves (Art. 44). The co-
ordinates of any point on such a curve can be expressed
rationally as functions of a parameter, so that by giving to
this parameter values continuously increasing from negative
to positive infinity, we obtain all the points of the curve in
a continuous series, the coordinates being always real. In
the present example, on the contrary, it is geometrically
evident that if we commence with any point on the oval and
proceed on continuously, we return to the point whence
we set out, without. passing through any point on the in-
finite branch; and it is algebraically impossible to express
the coordinates of any point in terms of a parameter without
including a radical in the expression. For instance, we might

take z=1, e=0, y=/(a0?+300°+ 38c9+d). We shall then:

call the curve we have been considering a bipartite curve, as
consisting of two distinct continuous series of points.

A curve of the second kind considered, Art. 196, has no
oval, and is wndpartite, all the real points of the curve being
included in one continuous series; but the curve is not on
that account unicursal, for the coordinates of any point cannot
be rationally expressed in terms of a parameter, and a unipartite
curve is not necessarily unicursal, just as an equation having
only one real root is not necessarily a simple equation. A cru-
nodal cubic, on the other hand, is unicursal and unipartite; all
the points of the curve succeed each other in a definite order
forming a single series. The curve may, however, be regarded
as comprising a loop and an infinite branch consisting of two
parts separated by the loop. The argument used, Art. 200,
shews that no point of inflexion can lie on the loop, neither can
any tangent meet the loop. ‘The loop, therefore, includes a series
of points from none of which can any real tangent be drawn to
the curve, while from every other point on the curve, two real
tangents to it can be drawn, one of them to the loop, the other
to the infinite branch. So also an acnodal cubic and a cuspidal
cubic are each of them unicursal and unipartite.

202. Having thus divided cubics into five genera, we proceed
to subdivide these genera into species, according to the nature

mai”

CLASSIFICATION OF CUBICS. 169

of their infinite branches. And, obviously, we must have
at least four species under each genus, according as the line
infinity meets the curve, (a) in three real and distinct points,
(0) in one real and two imaginary points, (c) in one real and
two coincident points, (d@) in three coincident points. But in
the case of crunodal, acnodal, and cuspidal cubics, we must
distinguish under {c) whether the line infinity be properly a
tangent, or whether it pass through a double point; and in
the case of crunodal and cuspidal cubics we must distinguish
under (d) whether the line infinity be a tangent at a point of
inflexion or at the node or cusp. Further, in the case of
a bipartite or a crunodal cubic it is important to distinguish:
under (a) and (c) whether the three points in which infinity
meets the curve all belong to the infinite branch or whether
two of them belong: to the oval or loop and only the re-
maining one to the infinite branch. The differences thence
resulting in the figures of the curves are so great that the two
cases may properly be classed as distinct species. ‘These are the
only differences which are made in what follows, grounds of
distinction of species. The only other differences which would
seem to have equal claims to be put on the same level are that
the points of the curve at infinity may either all be ordinary
points, or else one or three of them may be points of inflexion.
But as the changes thus made in the figure of the curves are
slighter, and as it is desirable not to have more species than can
be easily remembered, I have preferred to class curves differing
only in the respect last mentioned, not as distinct species, but as
different varieties of the same species. It is obviously a good
deal arbitrary how many varieties of cubics may be counted,
and much depends on the point of view from which these
curves are discussed.

203. The figures for the case where the line infinity is a
stationary tangent have already been discussed, and the figure
for any other case may be regarded as a projection of one of
the figures for this case. Let us commence with bipartite cubics,
and consider first the projection of the oval. And it will be
readily understood that if the line projected to infinity do not
meet the oval, the projection of the oval will remain a closed

Z

170 CLASSIFICATION OF CUBICS.

curve, while if the line touch the oval, or if it meet it in two
real points, the projection will have the same kind of rough
resemblance to a parabola or a hyperbola respectively that the
oval itself has to an ellipse; that is to say, while the figures
have not the symmetry of the conic sections, the projection is in
the former case, like the parabola, a single curve whose branches
proceed to infinity in a common direction without approaching
to contact with any finite asymptote, and in the latter case
consists of a pair of curves having two common asymptotes, and
lying in two of the vertically opposite angles formed by them.
Such a pair we shall briefly refer to as a hyperbolic pair,
It will be observed that an ordinary asymptote to a curve has a
positive and negative branch at opposite sides of it. The
theory of projection teaches us to regard the extremities of a

line at positive and negative infinity as projections of the same

point, and similarly to regard the branches of a curve which
touch an asymptote at positive and negative infinity as con-
tinuous with each other. Thus, then, as when the oval is a closed
curve, its points form a continuous series, such that commencing
with any point we can proceed continuously round the curve till
we return to the point whence we set out; so this is equally true
of all projections of the oval, and the twin hyperbolic branches are
to be regarded as forming one continuous-curve, the part where
one branch touches an asymptote at its positive extremity being
regarded as continuous with the part where the other branch
‘ touches the same asymptote at its negative extremity.

204. Let us next consider the projection of the infinite part
of the curve (Art. 196) which must be met by every line either
in one or three real points. First, let the
line projected to infinity meet it only in one,
and then the branches of the projected curve,
instead of spreading out indefinitely, will os
approach to contact with a finite asymp- 62:
tote, as in the left-hand curve on the figure.
The curve, which will hereafter be briefly
referred to as the serpentine, must obviously
have three points of inflexion; for it is
convex towards the asymptote at positive infinity (since every

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CLASSIFICATION OF CUBICS. 171

curve is convex towards its tangent on both sides of the point of
contact) ; it must change this convexity into concavity in order
to cut the asymptote once again: having cut it, it must bend
again, else it would continually recede from the asymptote ;
and it must bend once more in order to become convex towards
the asymptote at negative infinity. The points in the curve
represented in the*figure form a continuous series, since it ap-
pears, from what was said in the last article,-that the branches
of the curve in contact with the asymptote at its opposite
extremities are to be regarded as continuous with each other.

In the above it was assumed that the point at infinity on
the serpentine is an ordinary point on the curve. If, however,
it be a point of inflexion, the difference is that instead of the
positive and negative infinite branches lying as usual on opposite
sides of the asymptote, they lie on the same side, as in the right-
hand curve on the figure. It is obvious that the curve has
then but two finite points of inflexion. We refer to this
as the conchoidal form.

205. Next, let the line projected to infinity meet the infinite
branch in three ordinary points; It may be seen that it will
always divide the curve into three parts, one of which has no
points of inflexion, another
one, and the other two.
The projection will consist
of three infinite branches ;
one, which we shall call a
simple hyperbola, having
no point of inflexion, and

not intersecting its asymp-
totes; the second, which a
we shall eall an ¢nflected hy.
perbola, crossing one asymp-
tote, and consequently hav-
ing one point of inflexion; and the last, which we shall call
a doubly inflected hyperbola crossing both asymptotes, and
having therefore two inflexions.* No two of these parts form

* Newton calls the first of these an inscribed, the third a circumscribed, and the
second an ambigenous hyperbola,

We: CLASSIFICATION OF CUBICS.

a hyperbolic pair, but the three together form a continuous
series. ‘Thus, in the figure, if we commence by descending
the vertical branch of the doubly inflected hyperbola, the path,
after passing through negative infinity on the vertical asymptote,
is continued from positive infinity on the same asymptote along
the singly inflected branch, until having passed to infinity on
the other asymptote it returns along the simple hyperbola, and
so back to the doubly inflected hyperbola.

If one of the points at infinity be a point of inflexion, either
the singly inflected hyperbola becomes simple or the doubly
inflected becomes singly inflected. If all three inflexions be at
infinity, the curve consists of three simple hyperbolas.

Cubics having three hyperbolic branches are called by
Newton redundant hyperbolas, as having one more than the
conic sections; those having but one infinite branch, as in
the last article, are called by him defective hyperbolas; and
those touched by the line at infinity, and having besides one
finite asymptote, are called parabolic hyperbolas.

206. We now enumerate the following species of bipartite
eubics. (1) The line projected
to infinity meets the oval twice
and the other part of the curve
once. Ifthe last point of meet-
ing be (a) an ordinary point,
the curve consists of a serpen-
tine and a hyperbolic pair, as in +
the figure. If it be (5) an in-
flexion, the only difference is,
that the serpentine is exchanged for the conchoidal form.

(2) The line infinity meets the curve in three real points,
none of which belong to the oval. If the points be (a) all
ordinary points, the figure is that of Art. 205. If one of the
points be an inflexion, the curve consists either (6) of an oval
with two simple and one doubly inflected hyperbolas, or else
(c) of an oval with one simple and two singly inflected hyper-
bolas. (d) If the three inflexions be at infinity, the curve
consists of an oval with three simple hyperbolas, In all these
cases the oval lies within the triangle formed by the asymptotes,

CLASSIFICATION OF CUBICS. ig

and the curves may be further distinguished according as the
hyperbolas lie in the angles which contain the asymptotic
triangle, or, as in the figure, in the vertically opposite angles.

(3) Infinity meets the curve in two imaginary points; and
we have an oval (a) with a serpentine, or (b) with
a conchoidal branch (see Art. 204).

(4) Infinity totiches the oval, which then as-
sumes the parabolic form, and is accompanied (a) “™,
with a serpentine, (4) with a conchoidal branch, We

(5) Infinity touches the other part of the curve.
The oval then remains a closed figure, while the
other part of the curve spreads into a parabolic
form. If (a) the remaining point at infinity be ordinary, one
branch crosses the asymptote and has two
inflexions, while the other branch has only
one. If (0) it be a point of inflexion, the ae
branches are both at the same side of the

asymptote, and each has only one in- o
flexion. zat)

(6) Infinity meets the curve in three eee:
coincident points. This is the case with |

‘which we set out (Art. 199),

207. We come next to the division of non-singular unipartite
cubics, and it is evident that we have now nothing corresponding
to the species 1 and 4 of the last article. We have, therefore,
only four species of such unipartite cubics, viz. redundant,
defective, and parabolic hyperbolas, and the divergent parabola ;
according as the points of the curve at infinity are all real and
distinct, two imaginary, two coincident, or all three coincident.
The same varieties of each may be counted as in the last article,
and the figures of the last article will serve by omission of
the oval; but for further illustration we give a figure for a
case where the satellite cuts the sides of the asymptotic triangle,
and where two critic centres (Art. 192) lie within that
triangle. We have, then, a portion of the doubly inflected
hyperbola in a purse-shaped form within that triangle;
and it is easy to conceive that by a change in the value of

174 CLASSIFICATION OF CUBICS. |

the constant the mouth of the purse closes, and we have a
double point at one of the critic centres, while, by a further
change, -we have a separate oval, at last shrinking into a
conjugate point at the other critic centre.

In like manner
we have the same y
four species of ac- 7
nodal cubics, to-
gether with a
fifth, for which the
acnode is at in-
finity. The figures
for bipartite cubics |
suffice to illustrate a
this class if we
suppose the oval
to shrink into a
conjugate point.
The figures for the case where the acnode is at infinity do not
strikingly differ from those where infinity meets the curve in
one real and two imaginary points.

208. Of crunodal cubics we have the following species:
(1) Infinity cuts the loop in two real points. We have, then,
two simple and one inflected hyperbola as in the left-hand
figure. It will be observed by tracing the curve in its
passages through infinity that the curve is unicursal. There

are two varieties according, as the remaining point is ordinary

CLASSIFICATION OF CUBICS. 1%

or an inflexion. In the latter case, all the hyperbolas are
simple.

(2) There are three real points at infinity, none of which
are on the loop, There are an inscribed, ambigenous, and
circumscribing hyperbola, the last forming a loop within
the asymptotic triangle. There are two varieties, according
as there is, or is ngt, an inflexion at infinity.

(3) Infinity meets the curve in two imaginary
points. ‘There are, as before, two varieties.

(4) Infinity touches the loop, and (5) infinity f
touches the spreading part of the curve. The ~~
figures explain themselves, and in the former case
there are two varieties, the curve lying all on
the same side of the asymptote when there is \
an inflexion at infinity.

a we

There is a double point at infinity, and consequently two
parallel asymptotes; and the remaining point at infinity is
(6) on the spreading part, (7) on the loop. In the former
case, the point of inflexion is outside the parallel asymptotes,
in the latter, between them. If the inflexion were also at
infinity,/ the two branches in the former case would lie on
the same side of the asymptote.

i

176 CLASSIFICATION OF CUBICS.

(8) Infinity touches at an inflexion, and we have the diver-
gent parabola of Art. 199.

(9) Infinity is a tangent at a double Fi
point, and we have a curve called the
trident, whose figure is here given.

209. Of cuspidal cubics there are
evidently no species answering to 1, 4,
7 of the last article. The species, then,
are (1) Three real points at infinity; two varieties. (2) One
real and two imaginary points at infinity; two varieties. (3)
Infinity an ordinary tangent; two varieties. (4) The cusp at
infinity ; two varieties. (5) Infinity, a stationary tangent. (6)
Infinity, a cuspidal tangent. The figures for the cases 1, 2, 3
can easily be conceived with the help of the figures of the last
article, by supposing the loop removed which is dotted in those
figures, and the double point replaced by a cusp. The figure for
case 4 is obtained from the left-hand figure (Art. 208) for
the case of two parallel asymptotes, by imagining those asymp-
totes united and the branch between them suppressed. We
have then a single asymptote with two infinite branches on
opposite sides, but at the same end of it.
The figure for case 5, the semi-cubical para- Ny
bola, my’=2",is given, Art. 39. Finally,
the figure for case 6, the cubical parabola, 5

my =x", is here represented.

210. Though we have here counted as many as thirty
species of cubics, it is not difficult to remember the classification,
if it is borne in mind that nothing has been done, but combine
the five-fold division of Art. 196 with the division of Art. 202,
depending on the nature of the points at infinity. It remains
to say something as to previous classifications of cubics. The
first was made by Newton, Enumeratio Linearum tertit ordinis,
whose classification is substantially the same as that here given,
except that what we have counted as varieties are made by
him distinct species; and that whereas in the case of a hyper-
bolic branch, touched by two asymptotes, we do not regard in
which of the vertically opposite angles formed by them the

CLASSIFICATION OF CUBICS. 177

branch lies, Newton discriminates the cases where it lies in the
angle crossed by the third asymptote, or in the opposite angle.
The cases where three real asymptotes meet in a point are
treated as distinct species. By attending to these distinctions
the number of species is made up to seventy-eight. Also,
whereas we have made the five-fold division primary, and that
depending on the’ infinite branches secondary, Newton’s course
of proceeding is the reverse.

Newton’s method of reducing the general equation is as
follows: one of the axes being taken parallel to the real
asymptote, the coefficient say of y’ vanishes, and the equation
of the curve is of the form

y (ax +b) +y (fa? + get+h)+ pe+qz' +re+s=0.
Now the locus of middle points of chords parallel to the asymp-
tote is obviously
2axy + 2by + fx’ + gxut+h=0;
and if we suppose the axes transformed to the asymptotes of
this hyperbola, the terms ), f, g evidently vanish, shewing that
the same transformation will bring the equation of the cubic to
the form
xy’ +hy=px'+qe+rxt+s,
or with Newton’s letters
cy" + ey = ax’ + bx? + ca4+d.
This is Newton’s most general form. If, however, in the
equation, as we have written it a and bd vanish, the locus is not
a hyperbola but a right line, and according as this is (1)
the line «=0, (2) an arbitrary line which may be taken
for y = 0, or (3) the line at infinity, the equation of the cubic is
similarly brought to the forms
xy =ax’ + bx’ + ca +d,
y = ax’ + bx’ + cx + d,
y = ax’ + ba? + cxut+d.

The only apparently different case is when in the equation, as
we have written it, a=0, and the locus a parabola; but in this
case there is another real asymptote, the locus of middle points
of chords parallel to which is a hyperbola, and the reduction

AA

178 CLASSIFICATION OF CUBICS.

proceeds as in the first case, only that the coefficient of x* vanishes
in the transformed equation. Newton’s results are obtained
from a discussion of these four forms. If y=¢(x) be the
equation of any curve, Newton calls the curve zy=¢(a) a
hyperbolism of that curve. Thus then he calls cubics which
have a double point at infinity, and whose equation can therefore
be brought to the form
xy’ + ey =cx+d,
hyperbolisms of the ellipse, hyperbola, or parabola, since the

equation just written is brought to that of a conic by writing
y for xy.

211. We have already noticed Pliicker’s discussion of cubic
curves, contained in his System der Analytischen Geometrie. In
this discussion the nature of the points at infinity is the primary
ground of classification. Commencing with the case of three
real asymptotes, when the equation is of the form ayz=ku’v,
the cases when the asymptotes meet in a point, or form a
triangle, are first distinguished; then all possible positions of
the satellite line v are examined; whether for instance it cross
the triangle, pass through a vertex, or meet all the sides
produced, whether two critic centres (Art. 192) coincide, and so
forth. All the curves capable of being represented by the
above equation for any given position of the lines 2, y, z, v, are
said to form a group, and by giving all possible values to 4,
the different species included under the same group are dis-
tinguished. This will be more readily understood from the
figure of Pliicker’s first group, which we reproduce on the next
page, and which answers to the case where the satellite line meets
the sides produced of the asymptotic triangle, and where we have
three real critic centres, one inside, two outside the triangle.
Vig. 1 represents a bipartite curve of the species in this volume
numbered I., 2. By a change in the value of & the oval shrinks
into a point, and we have (2) the acnodal curve III.,1. As
kis further changed, the curve becomes (3) unipartite II., 1; and
the branches recede further from their asymptotes. In (4) the
branches cross to the other asymptotes, and the curve becomes
erunodal, IV., 2. Fig. 5 is bipartite, I.,1. Fig. 6 is in our
enumeration of the same species as 5, 7 as 4, and 8 as 3, but the

ie
NE ote

UNICURSAL CUBICS. 179

position of the branches with regard to the asymptotic triangle

is different. Pliicker’s division into groups has been carefully
re-examined by Prof. Cayley, Zransactions of the Cambridge
Philosophical Society, 1864, who also gives a comparison of
Newton’s species with those of Pliicker, of which there are
two hundred and nineteen. It does not enter into the plan of
this treatise to give a more minute account of this classifica-
tion. It will suffice to mention, that in the case of the
parabolic curves an important part is played by the osculating
asymptotic parabola, or parabola which passes through five
consecutive points of the curve where it touches the line infinity.
The equation of the curve may be brought to the form

x (y° + 220+ 2") = 2 (ay + bz),

where obviously the parabola y’+2zx+ 2” meets the curve in
the point yz reckoned five times. The groups are then deter-
mined by the position of the osculating parabola with respect
to the linear asymptote «, and to the satellite line ay + bz.

SECT. IV.—UNICURSAL CUBICS.

212. We have seen (Conics, Art. 270) that computation is
facilitated when the coordinates of a point on a curve can_be

180 UNICURSAL CUBILS.

expressed in terms of a single parameter, and it has been
proved (Art. 44) that this is always possible in the case of
a unicursal curve. Of the application of this principle to cubics
we now give some examples. ‘The equation of a cuspidal
cubic can always be reduced to the form 2’z=y’, where xy is
the cusp, x the cuspidal tangent, and z the stationary tangent.
Any point on the curve may then be expressed as the inter-
section of Ox=y, @y=z;* or, in other words, the coordinates
of any point on the curve may be taken as 1, 0, 6°, where @ is
a variable parameter. ‘The line joining any two points on the
curve will then have for its equation, as may be easily verified,
60' (0+ O)x—(% 4+ 00+ 6") y+2=0.
Let 6 and 6’ coincide, and we have the equation of the tangent
202 —30y+2=0.
If we seek the points where any line az +by+cz=0 meets the
curve, substituting 1, 0, @ for x, y, 2, we have the equation
a+b6@+c#°=0, and as this equation in @ wants the second
term, the sum of its roots vanishes, and we learn that the para-
meters of three points on a right line are connected by the
relation 6+6'4+6"=0. Hence, in particular, the tangential
of the point 6 is—20, and the point of contact of the tangent
from @ is —40.

In like manner, if we make the substitution 1, 0, 6 for
x, Y, 2, in the equation of a curve of the p™ order, the term
6” * will be wanting in the equation, and the relation connecting
the parameters of the 3p points of intersection of the curve
with the cubic is that their sum vanishes. ‘Thus, then, the @
of the residual of a system of points is the negative sum, and
of the coresidual is the sum of the 6’s of the several points;
and generally the theorems concerning residuation, Art. 158, &c.,
are thus intuitively evident for cuspidal cubics. For instance,
denoting the parameters of the points by a, b, &c., the condition
that six points shall lie on a conic is

at+b+c+d+e+4+ f=),

* These equations considered as belonging to tangential coordinates give the

theorem “If J be the inflexion, C'the cusp, and 7 the intersection of tangents at
; IA? TB

these points, any tangent AB cuts the sides of the triangle JCT, so that AT27 k Bo’

aud when the line at infinity isa tangent 4= 1.” Compare Conics, Art. 327,

UNICURSAL CUBICS. 181

which at once gives the theorem (Art. 154), that given four
points on a cubic, the line joining the points e, 7, where any
conic through them meets the curve again, passes through the
fixed point (2+ 6+c+d); and that this point may be con-
structed by joining ad, cd, and joining the points where these
lines meet the curve again, since

— (a+b) —(c+d)+ (at+b+ce+d)=0.
So, again, various constructions for the ninth point where the

cubic through eight points meets the curve again are obtained
by inspection of the equation

(atb+e4+d)+(e+ftigt+h)+7=0.

213. The parameters of the points whose tangents pass
through a given point are found by substituting the coordi-
nates of that point in 26°~—36°y+2=0; and since in the
resulting cubic the coefficient of @ vanishes, the sum of the
reciprocals of the roots vanishes; or, three points whose tangents

: é Pea | 1 1
meet in a point are connected by the relation atat ger 0.
In like manner, since the condition that 26°~—3@y+2=0
should touch a curve of the p™ class is a relation of the p® |
order between the coefficients 26°, 36°, 1, and since such a
relation obviously does not contain the term @, it follows that
the 3p points where tangents touch a curve of the p™ class

; 1 Mee ‘
are connected by the relation = (5) =0. We give some illus-
trations of this application of the method to examples.

Ex. 1. To find the locus of the intersection of tangents whose chord of contact

passes through a fixed point on a cuspidal cubic.
This is to eliminate « and 8 between the three equations

2a3e — 8a’y+2= 0, 28x —8B’y+2=0, a+B+y=)9,
where y is known. We easily find y (2yx + 3y)? + 2az = 0, the equation of a conic.
Ex. 2. If a polygon of an even number of sides be inscribed in a cubic, and all
the sides but one pass through fixed points on the curve, the last side will also pass
through a fixed point on the curve.
Denote the parameters of the vertices by a), a, &c., and of the fixed points by

b,, b,, &c. We take the case of the quadrilateral for simplicity, but the proof
is general. We have then the equations

a,+6,+a,=90, a,+6,+a;=9,
ag+0;+a4=0, a +h+y=0 |

182 UNICURSAL CUBICS.

Adding, we have 4, + 5, = 6, + d,, shewing that the lines joining ,, 3,; 5,, 6, meet on
the curve, and that, when three of the points are known, the fourth is known also:
The theorem is true for all cubics, for the proof here given may easily be translated
into the language of the theory of residuation, shewing that the pairs of points 0,, 3, ;
5,, 6, are coresidual, a common residual being the system of vertices a), dz, @3, A.

It follows, as a particular case of this theorem, that if the sides of a polygon of an
odd number of sides pass through fixed points on the curve, the tangent at any
vertex passes through a fixed point on the curve; and hence, that the problem to
construct such a polygon whose sides pass through fixed points on a non-singular
cubic admits of four solutions.

Ex. 3. To find the quasi-evolute, the two fixed points being on the curve (see also
Ex. 5, Art. 99). The equation of the quasi-normal (Art. 107) is

(6? + BO — 267) {0a (0+ a) w — (0 + Oa + a?) y+ 2}

+ (a? + af — 26°) {08 (0+ B) x — (6? + 68 + B?) y+ 2} =0.
= Bh.
1-2
Art. 108, a biquadratic in A, in which the two extreme terms at each end respectively

differ only by a constant factor, and the discriminant, having as factors the equations
4th

If we transform this by writing 6@= , we get then. in conformity with

of the tangents at a and £, represents besides a curve only of the degree.

214, It remains to mention a few of the more remarkable
examples of cubics of the third class. We have already noticed
the semi-cubical parabola, which is the evolute of the parabola
of the second degree. In its equation, py*=2"*, the cusp is at
the origin, and the point of inflexion at infinity. In the cubical
parabola, on the other hand, p*°y=2*, the point of inflexion
is at the origin and the cusp at infinity. In the cubical para-
bola the origin is a centre, and all the diameters of the curve
coincide with the axis of y; for if we draw any line y= mx +n,
the sum of the values of x is=0.

To the cusped class also belongs the Cissoid of Diocles, a
curve imagined by that geometer for the solu- 3
tion of the problem of finding two mean pro- J 4
portionals. It may be defined as the locus of
a point JJ', where the radius vector to the Pe f
circle AM is cut by an ordinate, such that Reg |
AP'=BP, We must have Be

AM'= KM, and therefore p= AR- AM,
or p=2r secw — 2r cosw=2r tan sing;
or, in rectangular coordinates,

a(x" + y") =2ry’, or (2r.— 2).y" = 2,

~UNICURSAL CUBICS. 183

The origin is therefore a cusp, and 27 — x an asymptote meeting
the curve at an infinitely distant point of inflexion.

Newton has given the following elegant construction for the
description of this curve by continuous
motion: A right angle has the side G/’
of fixed length, the point / moves along r
the fixed line OJ, while the side GH
passes through the fixed point H; a # er f
pencil at the middle pointof GF will B Sf cP
describe the cissoid. ‘The proof we leave G
to the reader. (Lardner’s Algebraic Geometry, pp. 196, 472).

The cissoid is also the locus which we should find if we take
on each of the radii vectores from the vertex of a parabola a
portion equal to the reciprocal of its length. It is consequently
also the locus of the foot of a perpendicular let fall from the
vertex of a parabola on the tangent; or, in other words, if a
parabola roll on an equal one, the locus of the vertex of the
moving parabola will be the cissoid.

I

215. We can in like manner express in terms of a single
parameter the coordinates of any point on a crunodal or acnodal
cubic. The double point being the origin, the equation is of
the form

ax’ + 3ba*y + 3cxy’ + dy’ + 3fx? + gay + 3hy’ =0,

and if we put y = @x, we have immediately rational expressions
for x and y interms of @. ‘The discussion will, however, be
simpler if we suppose the equation transformed, as it always
may be, tothe form (a*+¥*)z=a". Here z is the tangent at
the one real point of inflexion which the curve must have:
x is the line joining the point of inflexion to the double point,
and x+y" are the tangents at the double point, the upper sign
belonging to the case of the acnodal, and the lower to that of
the crunodal cubic. ‘The coordinates then of any point on the
curve may be taken proportional to (1+ 6), 6(1+6°), 1. If
we substitute these values in the equation of an arbitrary
line Ax + py +vz=0, we get, in order to determine the para-
meters of the points where this line meets the cubic,

(A+) + wO+rO? + wh =0,

184 UNICURSAL CUBICS.

and these parameters are connected by the relation
6'6" a: 6’ Ee" a 0" @' ops + PE
If the line touch at a point of inflexion 6’= 0" = 6", and there-
fore 6 =+4. Hence, an acnodal cubic has three real points of
inflexion, and a crunodal cubic one real and two imaginary.
The equation of the line joining two points will be found
to be

(P+ 00' + O° +1) a— (04 6) y=+(14+) (14 0”) 2,
and therefore the equation of a tangent is
(30°41) x— 20y=4 (14 0’)%z,
whence we see that if four tangents meet in a point, the sum of
the corresponding parameters vanishes, and if two of the points
be given, we can at once form the quadratic which determines
the parameters of the other two. There is no difficulty in

applying this method to examples.
At Art. 122, Ex. 1 we have noticed the crunodal cubic, whose

polar equation is p* cos4@=m*, and whose rectangular equation
is 27 (a+ y")m=(4m-—<a)*; a curve having three points of
inflexion at infinity, oue real and the others being the two
circular points. The node is on the axis of z at the point a= —8m.

216. When a nodal cubic has three real points of inflexion, the
conjugate point ts the pole of the line joining these three points,
with regard to the triangle formed by the three tangents. Let
the equation of a cubic be

(e¢+y+z2)°=mryz;
then, if this has a double point, its coordinates must satisfy the
equations got by differentiation, viz.
3(a@+y +2)? = myz=mzx = mexy.
From these equations we get «= y =z, which (Art. 165) proves
the theorem enunciated, and we then have for the nodal cubic
m= 27, and the equation of the curve may be written in the form
a + y" +25=0.

In this case the coordinates of any point on the curve may be
taken proportional to 6°, (1—@)*,—1, and the equation of the
corresponding tangent is (1 — 0)’*a + @y+ @(1—@)*2=0.

sue
P Pre re!
ont Ade Ne GR fre ee

UNICURSAL CUBICS. 185

216(a). The subject of unicursal cubics may be otherwise
treated.* We may start with the most general expression for
the coordinates in terms of a parameter : p, Viz.

w=a +3) wt 3c Awd p,

y=a' > + 3b’ Nw+ 8c Ap? +d’ p’,

2 =a" + 80" + 8c"Aw? + dp,
and we can at once (as in Art. 44) write down in the form of a
determinant the equation of the resulting cubic. But agam,

there are in general three linear functions of x, y, %, whose
expressions in A, mu are perfect cubes. For if in the equation

La + My + Nz =(or+ By)’, ?
we substitute for 2, y, 2 their expressions in X: p, equate
coefficients of X°, A*w, &e. and linearly eliminate LZ, M, N from
the resulting equations, we get

8 ’ ”
a, a, a,a

GAR Foor
ay; Ge,
8°, d, d, d" \=0;

that is to say, we have a cubic for the determination of a: P,
which we may write

Aa’ + 3Ba'B + 3Cap’ + De’ =0,
A, 3B, 3C, D being the determinants of the system
fee Oy Oa
a are
Corresponding to the three values of a: 8, there are three .

values of Le+My+Nz; and if, writing down the three
equations

L'x + M'y + N'2=(ar+4 f'n)’, &e.,
we take the cube roots of both sides, and linearly eliminate 2: p,
we get the equation of the curve in the form of a linear relation

* For further developments of the method here explained see Igel, Math.
Annal., VI, 633; Haase, Math, Annal, 11, 526,

BB

186 UNICURSAL CUBICS.

between the cube roots of three linear functions. ‘This is
expressed in the simplest form by writing

X= (a! 046" w)*(a"B"— a"),
Y= (a+ 8" u)* (a"B’ -al B")',
Z=(a"0+8"n) (a B" — a" BY’
when we have the equation of the curve in the form (Art. 216)
Xi+4 Y4!+4Z4=0, which denotes a nodal cubic, X, Y, Z being

the three inflexional tangents, X¥+ Y+Z the line joining the
three inflexions, and X= Y=Z the node.

216 (4). We might arrive by another process at a cubic
identical with the Canonizant cubic of the last article. The
general condition that three points should be on a right line
being got by equating to zero the determinant formed with
the constituents 2’, y’, 2’, &c., if we substitute for a2’, av*+ &e.,
we get the condition that three points of the curve should be
on aright line. This is easily seen to be resolvable into partial
determinants, each of which is divisible by

(Vip” ae rp’) (A ” Dae Nps) haar ce rip’) ;
and the condition in question may be written
Ap'p'p + BN pp 4 a . je pe’)
of CO Ow =e p Nae Wa af 4 Vr”) + DBE ase 0,

where A, B, &c., have the same meaning as in the last article.
In other words, if the \: w of three points be determined by the
cubic

A'S + 3B’ 4+ 30D? + Dip? =0,
then the condition that these three points should be on a right
line is

(AD'— A'D) -—3(BC’- B’'C)=0.

The 2: w of a point of inflexion we get by writing V’=A"=A",
=p" =p" in the preceding equation, and we thus fall back
on the cubic

Ap’ + 3Brp’ + 30Nu+ Dr? =0.

We might arrive at the same cubic in a somewhat different
form. From the general determinant form of the equation of
the line joining two points, it follows that for a unicursal cubic,

UNICURSAL CUBICS. 187

in which we are given expressions for x, y, 2 in terms of a
parameter, the equation of the tangent at any point is

as a os
nat) Yn oN
Gy FF Mf = 0,
where the suffixes denote differentiation with regard to » or uw
of the expressions for x, y, or 2; and, in like manner, that the
condition that three consecutive points shall lie on a right
line is
wy Dy

uw? Dru 7 ry

“iu? Yuu "up | = 0.

Thus, then, for the case of the cubic which we are con-
sidering, the A : wu of the inflexions is given by the equation

arxtbp, brA+cp, crA+d pw
arX+0'p, UrA+c p, CA+d' pw
an +b"p, bX +“, oN +d" =0,

which may be seen (as Higher Algebra, Art. sce to be identical
with the cubic already mentioned.

216 (c). A node on the curve will. arise when the same
point answers to two different values of the ratio X:p. If
v3 w’, X’: w” be two values answering to the same point, then,
no matter what other point \”: yw” we take on the curve, the
condition of the last article (that it shall be on a right line with
the two coincident points of the node) must be fulfilled. ‘Thus,
equating separately to zero the parts in that condition multiplied

by '”, »’” respectively, we have
App " + B(vp" +2" p') 4 On’ Mt te aS 0,
Bu'w " re O(N pe" +: rw’) ni Dr ty, — 0,

and since, from the theory of equations, if the two values of
X:, corresponding to the node be given by a quadratic
equation, that equation must be

Vp" ie Ap (V'p” ~f. rp’) 4 wr nr’ pen 0 :

188 INVARIANTS AND COVARIANTS OF CUBICS.

eliminating p'u”’, &e., we get the quadratic, which determines
the values of the nodal parameter

dr’, —Vp, ue

Ay a

By. 55.27 158,
In other words (see Higher Algebra, Art. 195), the quadratic
which determines the two values of the nodal parameter is
the Hessian of the canonizant cubic.

If in the condition of the last article we write \": wp" =A: pw,
we get the relations connecting the parameter of any point with
that of its tangential, and it will be observed that the factors
multiplying 2’, w'” are the differentials of the cubic with
regard to A, mW.

- 216(d). In the preceding it has been assumed that the roots
of the canonizant cubic are unequal. ‘To consider in the
simplest form the case where- there are two equal roots let
x and y be two of the linear functions, which, expressed in terms
of the parameter, are perfect cubes; that is to say, let us
take v=D*, y=’, and if z=a"\? + 3b" 4 8c'Ap’? + dy’, the
canonizant cubic becomes af (b"8 —c"a)=0, which will have
two equal roots only, on the supposition that 6” or c’=0. In
this case we can, by linear transformation, bring the third
equation to the form z=", and the cubic will be z’=2’y; or,
in other words, it will have a cusp. Clebsch has shown
(Crelle, Lxty. 43), that in general the equation of the 3 (n—2)
degree, which determines the parameters of the points of in-
flexion, will have a pair of equal roots for every double point
which becomes a cusp.

If the canonizant have three equal roots, the curve breaks
up into a right line and a conic.

SECT. V.—INVARIANTS AND COVARIANTS OF CUBICS.

217. The equation of a non-singular cubic can always be
reduced to the canonical form

ety +2°+6mryz=0.

In this form x, y, 2 contain each implicitly three constants;

and these, together with the one expressed constant, make up

Be RR er ye ee ay RN Ep Tt yh

cpg
saat
frye

ten, the number of constants, which, according to the test of
Art. 24, a-form must contain if it be general enough to represent
any cubic. We shall presently shew how the equation of any
cubic can be reduced to the form just given. We may write it
(e+y —2mz) (wx + wy —2mz) (w*w + wy — 2mz) + (1+ 8m") 2’=0,
where is an imaginary cube root of unity. In this form it
is apparent that the line z joins three points of inflexion, and
the’same thing is proved in like manner for the lines @ and y.
Hence these three lines constitute one of the four systems of
three lines which we saw (Art. 174) can be drawn through
the nine points of inflexion; and we can foresee that the
problem to reduce the equation of any cubic to the canonical
form admits of four solutions.

The form here given is that which we shall generally use
in our investigation concerning cubics; but it is necessary
first to obtain the invariants for the equation in its general
form, which we write ;

%aa? + by’ + cz’ + 8a,0°y + 30,02 + 3b,y"a + 3b,y°2
+ 3¢,2°x + 3¢,2°y + 6mxyz =0.

218. We form now first the equation of the Hessian. The
second differential coefficients of the cubic, omitting the factor
6 common to all, are

a=anrt+a,y+a,2; f=me+b,y +625
b=baet by +b,23 g=ae+myt+czZ;3
oe oytce; h=aar+by+me.

* In Prof. Cayley’s Memoirs the coefficients of the terms yz, 2?x, xy, yz*, za, xy”,
are written respectively f, g, h, 7, 7, & In German Memoirs the variables are
usually denoted by xj, 2, x3, and the coefficients in question are written 93, 4331, @12)
@o33) M311) Ayo. The first notation has greatly the advantage in compactness; the
advantage of the second is that each coefficient shews on the face of it to which
term it belongs. In formule which we have much occasion to work with, the use
of suffixes is less convenient than a notation in which each coefficient is denoted by
a single character; but since the general equation of the cubic. is only used in the
articles immediately following, and there chiefly for purposes of reference, I have
thought the second advantage to be that which in this instance it was most important
to secure. The notation used in the text agrees with the German, replacing a1, yo,
a3 by a, 5, ¢, respectively. On the same principle the coefficients of 2%, y°, 2°, might
be written yy b,, Cz, and were so written in the first edition. I now omit {he
suffixes in the case of these three coefficients, not only for brevity but also to
diminish the risk of confounding any of them with one of the group of six coefficients,

INVARIANTS AND COVARIANTS OF CUBICS. | 189°

190 INVARIANTS AND COVARIANTS OF CUBICS.

Forming then H=abe+ 2fgh —af* —bg’ — ch’,
HZ is the cubic, the coefficients of which are respectively
a =ab,c,— am’ + 2ma,a, — b,a,' —¢,a,",
b = ba,c, —bm* + 2mb,b, — a,b,’ —¢,8,”,
2

ee mes 2 ie ge
ec =ca,b, — cm + 2mc,c, —a,c, —b,¢,’,

3a, = abc, — 2amb, + ab,c, — ba,’ + m’a, — b,c,a, + 2a,a,b, — ¢,0,",

3a, = ach, — 2ame, + ab,c, — ca,’ + m’a, — b,c,a, + 2a,a,¢, — b,0,"5

113 2°3°2 3°39
3b, = bac, — 2bma, + ba,c, — ab,’ + m*b, — c,a,b, + 2b,b,a, — ¢,0,",
3b, = bea, — 2bme, + ba,c, — cb? + mb, — ¢,a,b, + 2b,b,c, — a,b,’

3c, = cab, — 2cma, + ca,b, — ac,’ + m*c, — a,b,c

2
3 aC, + 2¢,¢,a, — b,c

74-2

3c, = cba, —2cmb, + ca,b, — bc,’ + mc, — a,b,c, + 2¢,¢,b, — A655

6m = abe — (ab,c, + be,a,+ ca,b,) + 2m* — 2m (b,c, + 6,4, + a,),)

+ 8 (a,b,c, + a,b,¢,).

As a particular case of the preceding, the Hessian of

ety t+2°+6mayz=0 is — m (w+ 4° +2°) + (14+2m’) cyz=00

219. We are also able to form the equation of the Cay-
leyan. ‘This contravariant expresses the condition that the line

ax + By +-yz shall be cut in involution by the system of conics
U,, Uy Uy, where

U,=ax'+by* +¢,2" + Qmyz + 2a,2u + 2a,xy,

U, =a, +b y' +0,2° + 2b,yz + 2mezx + 2d. cy,

U, =a," + by’ +.¢ 2’ + 2c, yz + 2¢,2x + 2may.
The method of forming this contravariant is given, Conies,
Art. 388a; and the result is there found in terms of the coeffi-

cients of the three conics. Applying the formule to the present
example, we find

P= Ao? + BB’ + Oy +34,0°B +3.4,0°y + 3B. 6a + 3B,B*y

+3C,7'a+ 30,78 + 6MaBy,
where

A =bem — be,c, — cb,b, — mb,c, + b,c," + ¢,b," ’

1°32
as 2 2
B= cam — ca, — aC,Cc, — MAC, + 4,0, + CAs

C=abm — ab,b, — ba,a, — mb,a, + 6,0," + 4,0,', ene

- ; beat
fia! i yy sega sa
is ties Se ae

See Shes

Pee Nae

INVARIANTS AND COVARIANTS OF CUBICS. 191
3A,=—bca,—cmb,+ be,?+2ca,b,+2m’c,—3mb,c, +¢,a,),+0,¢,0, —24,C,",
3A ,=—dca,—bme,+cb,"+2ba,c,+2m"b,—3me,b, +b,4,¢,+ b.c,b,—2a,),,

3B, =— cab,—cma,+ac,"+2ca,b,+2m’c, —3ma,c,+¢,d,),+4,6,¢,—25,¢,"

ins: 3 2°1°2 at EI
3B,=—cab,—ame,+ca,’+2ab,c,+2m'*a,—3me,a,+a,b,¢,+4,¢,4,—20,a,°,
3 C\=— abce,—bma,+ab,’+2ba,c,+2m'b,—3ma,b,+b,4,¢,+4,),b,—2¢,b,",
3 C= —abe,—amb,+ba,’+2ab,c,+2m'a,—3ma,b,+a,0,c,+4,a,),—2¢,0,'5

6M= abe — (ab,c, + bc,a, + ca,b,) — 4m* + 4m (b,c, + ¢,a, + 4,),)
— 3 (a,b,c, + a,0,¢,).

In particular, the Cayleyan of x’ + y°+ 2° + 6mayz is
m(a°+ 6° +y°*) + (1—4m’*) aBy =0.

220. If in the contravariant just found we substitute for
a, 8, y, symbols of differentiation with respect to x, y, 2 respec-
tively, and then operate on the given cubic JU, the result will

be an invariant (Higher Algebra, Art. 139).

This invariant, which we denote by 8, is of the fourth degree
in the coefficients, and is

S = abem — (bca,a, + cab,b, + abe,c,) — m (ab,c, + bc,a, + ca,b,)
+(ab,c,’+ac,b,'+ba,c,’+be,a,’+cb,a,'+ca,b,)—m'*+2m*(b,c,+¢,a,+4,),)
—3m(a,b,c,+a,b,¢,)—(0,"c7+¢,°a,?+a,"b,”)-+(c,a,4,b,+4,0,0,¢,+5,¢,¢,0,)-

22°38 Se Be tm ae

It amounts to the same thing to say that the equation of the
Cayleyan may be written

a ee d d d
3 baad > ited — bo
(« besa hh Oa. AV a he tas,

pe UR ar ON
+ Bas toes. +98 7, +081 7,) S=0.

We have explained, Higher Algebra, Art. 162, the symbolical
method by which Aronhold originally obtained this invariant
S; its symbolical notation being (123) (234) (341) (412), that of
its evectant, the Cayleyan, being (123) (#23) (a31) (412). For
the canonical form § is m—m*, and since S vanishes when
m=0; that is to say, when the equation is of the form
a+ y°+ 2°=0, it follows that S vanishes when the cubic function
equated to zero can be reduced to the sum of three cubes.

192 INVARIANTS AND COVARIANTS OF CUBICS.

221. When we have a quantic U= ax" + by" +z" + &e., and

a covariant V of the same degree ax” + by” +cz"+ &e., then if

we have any invariant of U, and if we form the corresponding ~

invariant of U+2XV, the coefficients of the several powers of

a
i

k|

.
&
=

» will obviously be invariants. We learn hence that, in the ©

case supposed, from any invariant of U we can form a new in- |

: : ; ‘ d d d
variant by performing on it the operation a Ags b tea &e.

Applying this principle to the cubic and its Hessian we can
from the invariant S derive a new invariant 7’ of the sixth order

in the coefficients; or, what amounts to the same thing, we can —

obtain 7’ by writing differential symbols for a, 8, y in the
Cayleyan, and then operating on the Hessian. We thus find ©

for JT the value

alc’ —6abc(ab,c, + be,a,+ca,b,) —20abem*+ 12abem (b,c,+ ¢,a,+ a,),)
+ 6abe(a,p,c, +a,b,c,)+-4 (a’be,+ a’cb,'+ b'ca,’+ bac,’ + Cab?+ cba’

273% 1 5956,
+ 36m’ (bea,a, + cab,b, + abc,c,)

— 24m (beb,a,' + bee,a,” + cac,b,’ + caa,b,” + aba,c,’ + abb,c,”)

— 3 (a’b,*c,"+ B'c7a,"+ €a,"b,") + 18 (bcb,c,a,4a,+ cac,a.b.b.--aba,b.c.c )

Pare ss 22°31 33-48

— 12 (bec, a,a,* + bcb,a,a,° + cac,b,b,? + caa,b,b,’ + aba,c,c,’ + abb,c,c,”)

2 372 17371 37178 271% 1°21
— 12m’* (ab,c, + bc,a, + ca,b,)

+ 12m’ (ab,c,? + ac,b,? + ba,c,” + be,a,2 + cb,a,? + ca,b,2)

— 60m (ab,6,c,c, + bc,c,a,a, + ca,a,b,5,)

+ 12m (aa,b,c,° + aa,¢,b, + 6b,c,a,' + bb,a,¢,’ + cc,a,b,” + cc,b,a,”)

2°3 3 8438

+ 6 (ab,c, + bc,a, + ca,b,) (a,b,c, + @,0,¢,)

23.1 3°43

+ 24 (ab b oy + acc b, + be.6,'Ay i baa, C ; + ca,a *b ? + cb,b,’a,)

tS A 12 2 ‘8.2 “2B

— 12 (aa,),c,° + aa,c,b,’ + bb,c,a,' + bb,a,¢," + cc,a,b,° + cc,b,a,°)

.

— 8m° + 24m* (b,c, + ¢,a, + a,b,) — 36m’* (a,b,c, + a,b,c.)

2°31 3°1°2

— 12m? (b,¢,¢,a, + C440, + a,,6,¢,) — 24m? (b,*c,* + c,"a,” + a,b,”)

> bt bat aed 2.28 Ss

e 36m (a,0,¢, Bs a,b,c, (d,¢, az C0, + ap.) 2% 8 (2,°c,° C, a," 7 a °b ny

8 ~s/

27 22% 27 2.2
— 27 (a,'b,’c," + a,b, c, ) — 6b,c,c,a,a,,
27.2
Ly ee 1 i? 628 2°°3°3 2° 2 Sao aad, Pack bog 3 Og Cys) s

— 12 (b,7c,7c,a,+ b,7¢,°a,b,+ ¢,7a,/a,),4+ ¢,°a,°b,c,+.a,b,'b,c, +a

Be a a pie oi

INVARIANTS AND COVARIANTS OF CUBICS. 193

For the canonical form this invariant reduces to 1 —20m*— 8m’.
Its symbolical form is (123) (124) (235) (316) (456), We can
; eee dT
derive from the invariant 7’ an evectant a ae + p° Fi + &. =0,
the coefficients of which it is needless to write at length. Tor
the canonical form, this contravariant, which we denote by Q, is
(1 — 10m*)'(a° + 6° + 4°) — (30m? + 24m") aby =0.

Every invariant of the cubic can be expressed as a rational
function of Sand 7. ‘This can be proved in the same way as
the corresponding theorem is proved (Higher Algebra, Art. 215)
for a binary quartic, there being much resemblance between
the theory of the binary quartic and that of the ternary cubic.

222. The method of finding the equation of the reciprocal of
a cubic has been explained (Arts. 91, 188). We give the result
for the general equation, only writing at length, however, those
terms the form of which is really distinct. The other coefficients
may be obtained from those we give by symmetrical inter-
change of letters.

a’ {b’c’ — 6beb,c, + 4be,° + 4cb,° — 3b,"c,"},

6a’B {—be’b, + 2beme, + beb,c, — 4meb,” + 3cc,b,b, — 2be,c,”

Pe ae |
+ 2mb,c,’ + b,’c,c, —- 2b,c,"},
3a‘ B? {2be’a, — 4mbec, + 3c°b,’ — 2bec,a, + 16m*cb, — 12meb,c,

+ 4be,"c,+ 4ca,b,” — 6ca,b,c, — 6cb,b,c, — 4c," — 8mb,c,c,

— b,%¢," — 2a,b,c," + 4a,c,° + 128,¢,c,"},

a 3:9 1712
bai By {be e Am’ + 5b,¢, Bre 2a,b, rag 2c,a,) 23 b (2me,c, a 4a,C," "Hs 30,c,’)
4 ; (2mb,b, + 4,5," Ls 3¢,b,") ee 8°b,c, Se 10m (b,°c, a2 c, 0.)
— 2a,c,b,* — 2a,b,c,” — 11b,5,c,c,},

3°2°8 2°32 1-312
20° 8° {— abc’ — 9c’a,b, + 3bec,a, + 3ach,c, — 2ac,’ — 2be,* — 16cm*
+ cm (18b,c,+ 18¢,a, —24a,5,) + 9c (a,b,c, + a,b,c,) +12m’c,c,

2°31 3°12
+ 6m (a,c,' + 0,¢,") + 6a,b,¢,c, — 18b,c,"c, — 18a,¢,¢,”

8-1-2 5 31% })

6a°B’y fabcc, + 6bema, — 4bea,c, — 2acb,’ + ab,c,” + 2mbe,* — 5be,c,a

1°23
+ 4em*b, — 10ema,b, + 2cb,a,b, — 6cb,’c, + 9ca,c,b, + 8n’c,

1°33 2 2.3
— 16m’*b,c, + 12ma,),c,

+ 10b,0,c,7 + 18a,,c,c, — 11a,),¢,"},

1°31 a ot 2 3°12

2 2
— 8ma,c,' — 2mb,c,c, — 4a,b,¢,

CC

194 INVARIANTS AND COVARIANTS OF CUBICS.

6a" B’ry* {— 4abem + (bea,a, + cab,b,tabe,c, —8m(ab,c, + be,a,+ca,,)

+ 5 (ab,c," + ac,b,” + bea, + ba,c,? + cba,’ + ca,b,”)

— 8m‘ + 4m” (b,c, + ca, + a,>,) + 18m (a,b,c, + a,b,c.)

+4 (b,7¢," + ¢,"a,’ + a,'b,”) — 19 (b,¢,c,a,+ ¢,a,a,0, + a,b,0,¢,)}.
The contravariant just formed is the second evectant of 7’; that
is to say, the equation of the reciprocal may be written

d d d d d d
Ue xe es Pak Se eet ee | Re ors
(« Steg Ma ae ee a
2 d 2 d 2 d ad 2 ad
+ Bag +a g-+ 8 7, +287 Z,) Tat

It has been mentioned, Art. 91, that the equation for the
canonical form is
o? eis (2 + 32m*) (B'y? + roe + a’ 3°)

— 24m*aBy (a + B° + 4°) — (24m + 48m") 0?B*y? = 0.

223. The invariants of a cubic may also be calculated by

means of the differential equations which invariants must satisfy

(Higher Algebra, Art. 143). For this purpose it is convenient
to arrange the equation according to one of the variables, and
to write it

re +3 (av+a,y) 2 + 3 (b2* + 2b. xy + b,y’)z
+ (c,x° + 8¢,2°y + 3c,cy" + ¢,y*) = 0.
If we desire then to form an invariant of any given order and

weight, the literal part may be written down without calculation.
For instance, we can foresee that S is of the form

r (cb) + (c’a’) + (eb"a) + (0°),
where by (c’b) we mean a function of the second degree in the
c, and of the first in the 6 coefficients; and we know also that
it must be an invariant of that order of b,x°+ &c., ¢2°+ Ke.,
considered as a binary quadratic and cubic. ‘The theory, there-
fore, of binary quantics enables us to foresee the form of this
term. Similarly for the others. And the invariant must
further satisfy the differential equation
at (24, 5 +a, 7) + (30, 5 +2, i +b ) =0.

2 de,

jie: as i AS at Shel aaah
ett ee mle on rer hs

INVARIANTS AND CONVARIANTS OF CUBICS. 195

In this way we find S to be
— 1 (c*b) + (c’a*) + (cb’a) — (0°)’,
where (c’d) = (c,c, — ¢,*) b, — (c,C, — ¢,0,) 8, + (¢,0, — C2) 9
(c?a*) = (c,c, — ©,°) a,” — (6,0, — Cy) 44% + (C10, — ©) A 9
(cb*a) = a,¢,0," — (c,a, + 3¢,a,) 6,5, + (a,c, + a,c,) (2b," + b,b,)
; — (a,c, + 8a,c,) 6,5, +.4,¢,,"

1°8°0)

(8°) =b,5, — 8,”
In like manner 7’ is
r” (ct) —6r (cba) +4 (c'a®) + 4r (0°) — 3 (c°b°a") — 12 (0°) (cb’a) +8(0*)",
where (c*) =0,%c,’ + 4,0," + 4¢,¢,” — 8¢,’c,” — 6¢,¢,0,C,,
(c*ba) = a,b, (¢,c, + 2¢, — 3¢,¢,¢,

+ (a,b, + 2a,0,) (2c,¢," — ¢,¢," — ¢,0,C,)

+ (a,b, + 2a,),) (2¢,¢," — ¢,¢," — ¢,¢,¢,)

+ a,b, (c,¢,' + 2c,” — 3¢,¢,¢,)

(c'a®) =a,’ (¢,0,' + 2¢,” — 3¢,¢,¢,)

2 ¢ 2 2
+ 3a,7a, (2¢,¢," — ¢,¢,' — 6,C,C,)
2 2 2
— 34,4, (2c,¢, es ag oe C,C,¢,)
3 ,2 ae
+4, (c,c, + 2¢,° —3c,¢,¢,),

(c°b°) —8 (0°) (c°b) = 0,7b,’ — 6c,c,b,5,” + 6c,¢,b, (26," — b,8,)

@:i° 1:3 0°22

+ ¢,C, (65,0,5, — 8b,°) + 9¢,°b,b," — 18¢,c,b,b,b

0-12 a: &.2°.0:-25:2

+ 6¢,¢,b, (20, — b,b,) + 9¢,"b,"b, —6¢,¢,0,5,"+¢, b,°

e649 2°8°170 8 °0?

(c'b¥a") = ¢,"b,"a," — 2¢,¢, (b,"a,4, + 26,5,a,’)

gt gene} Ls Fe SG

— 26,0, (b,b,a,2 + 2b,a,? — 100,,a,a, + 4b2a,")

0-2 02 1 be a 2

+ 2¢,c, (40,0,a,° + 4b,0,4,° — 60,"a,a, — 30,b,a,,)

0:,2>-4 I-20 Poet Bas | 0°2°0 1
+ ¢,” (8b,7a," + 90,"a,' — 12b,b,a,a, + 40,5,2,")

Ro en ee 6° 2°1
+ 2c,¢, (b,b,a,4, + 2b,*a,a, — 60,),a," — 6b,b,a,"

02°01 £2 ek 1°2°0 2,4, )

— 2c,c, (b,b,4,° + 2b,%a,* — 10b,b,a,a, + 4,74,"

02:0 i ee

+0, (8b,"a," + 9b,'a,’ — 12b,b,a,a, + 40,b,a,”)

bk OO" 4 o-2° 0
: = ne 20,0, (b,"a,%, + 2b,),a, ) + al a Pa
or, we may write,
(c°b°a”) = (cba)* + 4 (c°a*) (b*) —8 (c’b) (a*d),
where (cba) met C040, ae C, (2,0, + 2a,),) oe G, (4,0, a3 2a,6,) ss C,2,0,
(ab) = ba," — 2b,a,a, + 0,0’.

jad ak

196 INVARIANTS AND COVARIANTS OF CUBICS.

224. Ifthe curve have a double point, this point may be made
the origin; when we shall have 7, a,, a, all =0; S reduces to
— (6°)’ and 7'to 8 (d*)*; or, in the notation of Art. 217, S reduces to
— (a,b, —m*)* and T’to 8 (a,b, —m*)*.Wesee then that 7” + 648°

vanishes when the curve has a double point. This, therefore, is

the discriminant, as will afterwards be proved in other ways.
If the curve have a cusp (4”) vanishes, and therefore so do
both S and 7. For the canonical form, the discriminant
T” + 648° = (1 + 8m’)’.

225. In the articles next following we use the canonical
form. It has been proved, Art. 218, that the equation of the
Hessian of w+y°+2°+6mxyz=0 is of the same form with
a different value of m, and hence that the system of three lines
' ayz passes through the intersection of the curve and its Hessian,
as was otherwise shown, Art. 217. It appears also that the
equation of the Hessian of the Hessian is of the same form,
and hence that the points of inflexion of a cubic are in-
flexions also on its Hessian, as was otherwise proved, Art. 173.

Any equation of the form a (a +y*+2*) + Bxyz=0 can obvi-

ously be reduced to the form XU+pH=0. In fact we have

a+y+ 2+ 6mayz= U, —m (a+ y+ 2) + (14 2m’) ayz =H.

Solving, (1+ 8mm’) (a*+y°+ 2°) = (1+ 2m’) U—6mH,
(1 + 8m’) xyz =m’U+ H;
whence (1 + 8m*) AX =a (1+ 2m*) + Bm", (1+ 8m’) w=—6ma+ Bf.
Let us now form the equation of the Hessian of XU+ 64H;
that is to say, of
(A — 6m’) (x? + y° + 2°) +6 [Am + pw (1 +2m’*)} xyz =0,
and the result is :
— (= 6m) frm + p (14+ 2m')}* (a? + y" + 2"
+[{(A— 6m)’ +2 {frm + w (1+ 2m’*) Jaye =
and, by what has been just proved, this is of the for
rv’ U+ p Z=0, whence
(1 + 8m*) V’ = — (1+ 2m”) (A— 6um"*) {rm + pw (1 + 2m*)}?
+m’ [(X — 6um*)’ + 2 {Am + w (1 + 2m*)}*],
(14+ 8m’) p’ = 6m (A — 6wm*) {Am + pw (1 + 2m’*)}?
+ [(X — 6pm’) + 2 {Am 4+ pw (1+ 2m’)}*1.

ne ne ae
Lie
Rett

INVARIANTS AND COVARIANTS OF CUBICS. 197

Expanding, and remembering that we have
S=m—m, T=1—20m*’- 8m’,
these values may be written
NM =—28r7u— Trap? + 88"u*, w =r? + 12SAp? +2 Ty’.

The values of X' and uv’ being expressed in terms of the in-
variants, the expressions just given will hold good, no matter
how the equation be transformed, and therefore the Hessian
of 7U+6wH, where U and H have the general values of
Arts. 217, 218 is XU + pw’ H, d' and pw’ having the values just given.*

Thus when 2’: p’ is given, we have a cubic to determine
the ratio X: w; that is to say, there are, as has been already
stated, three cubics which have a given cubic as their Hessian.

Since, as a particular case of the foregoing, the second
Hessian

H (HU) =8 S'U+ 2TH,

it follows that 7’=0 expresses the condition that the second
Hessian shall be the original curve. If S=0; that is to say,
(Art. 220) if the equation is reducible to the sum of three
cubes, the Hessian coincides with its own Hessian, and there-
fore consists of three right lines, as the next article will show.

226. The Hessian meets a curve in the points of inflexion;
that is to say, in the places where three consecutive points of
the curve are on a right line. If, then, the curve be not a
proper curve, but a complex, including a right line as part of
it, every point on that line is a point on the Hessian; and
therefore when the curve consists of three right lines, these lines
constitute the Hessian. This may be verified by forming the
Hessian of xyz=0. Thus, then, the system of conditions that
the general equation shall represent three right lines is written
down by expressing that the coefficients in the equation of
the Hessian (Art. 218) are proportional to the corresponding
coefficients in the equation of the cubic, viz.

Poe <6 a. age be by. 08 ee

Fi ?
1

a system of forty-five equations, on the face of them equivalent

* This was proved by direct calculation in the first edition, and it was thus
that the values of S and T were there obtained.

198 INVARIANTS AND COVARIANTS OF CUBICS.

to nine, but which can be really equivalent only to three in-

dependent equations. For (Conics, Art. 78) only three con-—

ditions are necessary in order that an equation of the third

degree, containing nine independent constants, should represent —

a system of three lines involving only six constants. It may
be verified, by means of the values (Art. 218) of a, b, &c., that
the forty-five equations actually are equivalent to three, as has
been stated.

227. The Hessian of XU+ 6yuH being X'U + p'H, the former

‘ ; . 50 : , ; :
will represent three right lines if “ =o; which, introducing

the values (Art. 225) for 0, pw’, gives us the equation

M+ 249 py? + 8 Trp? — 485° u* = 0.
This being a biquadratic, we see that, as has been already more
than once stated, four systems of three right lines can be drawn
through the intersections of Uand H. This biquadratic, solved
by the ordinary methods (see Todhunter’s Theory of Equations,
Chap. XIII.), gives

* a(t) + V(t) + V4)

where ¢,, ¢,, ¢, are the roots of the equation
+ 128+ 48S°t— 7° =0, or (¢+48)*= 77+ 648".

Thus, then, the reduction of the equation of any non-singular
cubic to the canonical form can be effected. We first form
the equation of its Hessian (Art. 218), and calculate the values
of the invariants S and 7’ (Arts. 220, 221). The present article
then shows how we can form an equation \U+ 6uH=0, which
shall be resolvable into three linear factors. By solving a

cubic equation we can find these factors X, Y, Z And then
comparing the given equation with the form

aX? +bY°+ cZ°+6mXYZ=0,
we can determine a, 0, c, m, by equations of the first degree.

Ex. 1. Calculate the invariants of the cubic
ax (y? — 27) + by (2? — x?) +. cz (x? — 9?) = 0,

228. Of the four tangents which can be drawn from any
point of a cubic to the curve, two can coincide only when the

ae gine

SR ee eT Oe ee eS eS a eee

ea

INVARIANTS AND COVARIANTS OF CUBICS. — 199

curve has a double point, since a cubic has no double tangents.
The equation of the four tangents is (Art. 78) A? =4A’U, where
if V=x*+y'?+2°+ 6mzyz,

A =8 {a' (av? + 2myz)+y' (y? + 22x) +2' (2? + A2may)},

A’ = 8 {x («+ 2my'e') + y (y?+2me'x’) +2 (2?+2mz2'y’)}.
Making z= 0 in Ai=4A'U, we get the quartic, which determines
the four points in which the tangents meet the line 2, viz.

8 (vn? + yy" + 2mery "=A (ay?) {x (w"42my2!) + yly"+2me'a!)}
or (a + 8my'z') x* + 4 (y? — mz'x’) wy

— 6 (a'y' + 2m?2"*) ay? +4 (x — my'z'\ xy’ + (y? + 8m2'zr') y* = 0.
From what has been said it appears that the discriminant of
this quartic must contain as a factor the discriminant of the

cubic. Now remembering that v* +" +2" + 6mz'y'z'=0, we
find for the invariants s and ¢ of the quartic

= 12 (m* — m) 2 = — 1228,
t= — (1— 20m’ — 8m’) 2° =—2°T.
Hence the discriminant of the quartic, 272 —s*, is 272? (7"+ 648°) ;

and it is easy thence to see that the discriminant of the cubic is
7*1645",

229. The anharmonic function of the four points determined
by the quartic of the last article evidently is the same as the
anharmonic function of the pencil of four tangents. Now if the
roots be a, 8, y, 6, the anharmonic function of these roots is
any one of the mutual ratios of the quantities (4 — 8) (y— 9),
(a—+y)(8—8), (a—8)(8-—y). We can form by the method
of symmetric functions the equation which determines these
quantities; and if the coefficients of the quartic be a, 4), 6c,
4d, e, we find a®y’? —12asy+ 16 /(s°- 27¢?)=0. The mutual
ratios of the roots are not altered if we increase them all in the
same proportion, by substituting, say ay=2zs', when we see
that the anharmonic ratios are the mutual ratios of the roots of

27t" Y Se
ae tects: eee ae ; cores Peete
z e420, /(1 =) 0, or z 20+ /(1+ rm) 0

Thus, then, the anharmonic function depends solely on the ratio
ZT? : 8°, and is independent of the point whence the tangents

200 INVARIANTS AND COVARIANTS OF CUBICS.

are drawn (Art. 167). If Z’'=0, the equation just given reduces
to 2° —3z+ 2=0, of which two roots are equal; one, therefore, of
the ratios becomes unity, and the anharmonic becomes an
ordinary harmonic ratio. If S=0, the equation in y wants
its second term and becomes of the form y* = m*, whose roots are
of the form m, mw, mw’, where is an imaginary cube root of
unity; and the common ratio of the roots is w. This has been
called equi-anharmonic section.

230. By the help of the canonical form can be calculated, as
in Art. 225, the invariants S and 7’ of X\U+ 6u4, or of

(A — Gum”) (a + y? + 2°) + 6 {mr + pw (1+ 2m")} xyz,
and we find, without difficulty,
S (AU + 6¥H)=Sr+ Trip— 249° py? — 48ST Ap? — (1? +488" )u*,
T (XU + 6uH) = Tr — 96832 — GOSTAY? — 20 T?2p
+ 2405 Tr2u* — 48 (ST* + 96S") Ap — 8 (728° T+ 7) p.

And if, by the help of these, we form the discriminant & or
T” +-64S°, we find

RU + 6uH) = B (d+ 24 Sdéy? + 8 Trp? — 488%p')',

where the factor multiplying £& is the cube of the quartic function
of A, w, in Art 227; as might have been foreseen, since if the
cubic U have not a double point, the only cubics with double
points which can be drawn through the points of inflexion are
the four systems of right lines. The values just given for the
Sand T of \U+ 6H are covariants of this quartic function
of A, 3 differing only by the numerical factors 4 and 2 respec-
tively from the Hessian, and the covariant called J, (Higher
Algebra, Art. 209); and the coefficients of U and # in the value
of H(AU+6uH) differ only by numerical factors from the
differentials of the same quartic with respect to \ and yp.

All covariant cubics can be expressed in the form \U+ ywZH,
as is illustrated by the following examples:

Ex. 1. If a, d, c, &c. denote the second differential coefficients, and A, B, &c
denote bc —f*, &c., as Art. 184, and if a’, 0’, A’, B’, &c. denote the corresponding
quantities for the Ilessian then

Aa’ + Bb’ + Ce’ + 2Ef’ + 2Gq' + 2Hh’ = 0

INVARIANTS AND COVARIANTS OF CUBICS. 201

is a covariant cubic. We use the values
a=2, f=mx; A= yz— m2, F= myz — mz,
b=y,g=my; B= 2x — my, G = m2zx — my’,
e=2, h=mz; C= ay— m2?, H= mzy — mz’,
a’ =—6mx, f'= (1 +2m3)a ; A’ =36m4yz — (1+ 2m3)%x?, FY = (14+2m?)2y2z+-6m?(1+2m?)x?,

(0 =—6m’y, g’=(1+2m')y; B’ =36mitzx— (1 +2m?)*y?, G’ = (14+2m3)220+6m?(1+2m)y?,

e’=—6m%z, h’=(1+2m')z; C’=36mitay —(1+2m?)?22, A’ = (14+2m?)2ay+6m?(14+2m?)z?.
Hence the covariant in’ question is found to be —2SU. It might have been
foreseen that it could only differ by a numerical factor from SU, for it is a covariant
of the fifth degree in the coefficients; and, therefore, if it be of the form aU + 6H,
a must be of the fourth, and 4 of the second degree in the coefficients ; but there is
no invariant of the second degree, and S is the only one of the fourth,

Ex. 2, Calculate in like manner the covariant
A’a+ Bb+ We+2FF+2G’9+2Hh, Ans, —-TU+12SH.

231. The order in the variables of any covariant of a cubic
is a multiple of three ; and, generally, if the order of any ternary
quantic is a multiple of three, so is that of every covariant.
This appears at once from the symbolical method explained,
Migher Algebra, Chap. XIv., for every symbol (123) diminishes
by three the order of the function on which it operates, and
in the symbolical method the order of the function operated on
is a multiple of that of the given quantic.

It is easy to see that the equation of every cubic covariant
to a + y°+2° + 6mxyz = 0 is of the form a (a* + y°+2°)+Rayz=0,
which, as we have seen, is reducible to the form XU+ pwH=0.
In order, however, to express covariants of higher order, it is
necessary to have a third fundamental covariant. That which
we select may be defined as follows: consider the polar conic of
a point az’ +&c., and the polar conic of the same point with
regard to the Hessian a’a* + Ke. then there is eee Art. 378)
a conic covariant to these two, viz.

(BC+ BC-2FF’) 2 + &. =0;
and the condition that this conic passes through the original
point gives a covariant of the cubic. Since B, C, &c. contain
the variables each in the second degree, this covariant is of the
sixth degree in these variables; and since B, C are of the
second, and B’, C’ of the sixth degree in the coefficients, it is
of the eighth order in the coefficients. The actual value of this
covariant for the general equation has not been calculated, but
DD

202 INVARIANTS AND COVARIANTS OF CUBICS.

using the values for A, B, &c. given in the last article, we
find that for the canonical form the covariant is 40 where © is

3m* (1 + 2m’) (x? + y° + 2°)? — m (1 — 20m* — 8m*) (a* + y? + 2) xyz
— 3m” (1 — 20m® — 8°) a*y*z” — (1 + 8m’)? (y*2" + 2°a* + ay’),

or m’ (2+ m’) U?—m (14+ 2m’) UH
+ 8m? H* — (1 + 8m’)? (y°2* + 2°a* + a*y’),

There are two other covariants of the same order in the variables
and in the coefficients as ©, which had equal claims to be
selected as the fundamental covariant of the sixth order. The
first represents the locus of a point whose polar line with regard
to the Hessian touches the polar conic of the same point with
regard to the cubic, or

AL”? + BM” + CN” +2FM'N'+2GN'L' + 2HL'M,

where L’, M' N’, are the differential coefficients of the Hessian.
This covariant is expressed at once in terms of © by the help
of the formula (Conics, Art. 381, Ex. 1) OS'-Z. We are
here to write for ©, —2SU; for S', 6H; for /, 40; and thus the
covariant is found to be —4 (@ + 3SUH). In like manner there
is a covariant which represents the locus of a point, whose
polar with respect to the cubic touches the polar conic of the
same point with regard to the Hessian, or

A’? + BM’ + ON’ +22F'MN 4+ 2GNL 4+ 2H'IM =0.

Calculating this by the formula ©'S — /’( Contes, Art. 381), and
writing for 6’, -— 7U+12SH; for S, U; and for f, 40, the
covariant in question becomes

~ (TU* —128UH + 40).

232. Every covariant of 2° + y’ + 2° + 6mayz will plainly be
a symmetric function of x, y, z, and therefore capable of being
expressed in terms of a*°+y’+4+ 2°, ayz, y+ 2a +a°y’3 and
therefore in terms of U, H, ©, together with the invariants.
But a covariant is not necessarily a rational function of U,
H, ©. In fact, we can, as at Higher Algebra, Art. 223, form
a covariant of which the square, but not the covariant itself,

INVARIANTS AND COVARIANTS OF CUBICS. 203

is a rational function of these quantities. Let the coefficients
of the cubic ,

ph (1+ 8m’) (a? + y! +2") p”

+ (1 + 8m’)? (y°2? + 2°x* + ay") p —(1 + 8m’)x*y’2* =0,
be p, g, 7; then, by the theory of cubic equations, if J be
(1 + 8m°)* (y* — e) (2 — x’) (x — y’), we have

J* = pq? + 18pqr — 277" — 4q° — Arp’.
But p, g, 7 are each immediately expressible in terms of

U, H, ©, and substituting their values in the equation just
written, it becomes
J*=40°+ TU’
© (—- 48° U* + 28STU*H — 728° U*H* — 18 TUH® + 108 8H")
—169°WH-US TWH’ —47° UH + 54ST H*
— 432,S°UH*® —27TH*.
The identity just given may be written in the form
40 (0+ AU") (O+ pU")=J" + AO,
from which it appears that the system © (O+2U") (O + wU”) is
touched by H; that is to say, H either touches each of the curves
represented by the three factors, or passes through the inter-
sections of every two. But ©, Uand H have no point common
to all three, therefore © must be touched by H. The curve J
which passes through the points of contact consists of the
harmonic polars of the nine points of inflexion. We add an
example or two to illustrate the possibility of aes all
other covariants in terms of U, H, ©.

Ex. 1. To obtain the equation of the nine inflexional tangents. It was shewn
(Art. 217) that the inflexional tangents are U — (1+ 8m) a’, U— (1+ 8m’)y3,
U— (1+ 8m*) x. Multiplying together these three factors, we have
U3 — (1 + 8m) (x +.y8 + 2°) U2? + (1 + 83)? (323 + 233 + ay?) 0 — (1 + 8m?) 3x3y8z3 = 0,
Substituting for (1+ 87m?) (x + y% + 23), (1+ 8m)? (y23 + 283 + x3y3) and (1+ 8m’) ayz
their values previously given, we find, for the required equation of the nine tangents,

5SUW?H — H? — U8 = 0,
the form of the equation showing that H and @, which have been proved to touch
each other, have the nine tangents for their common tangents.

Ex. 2. To find the equation of the Cayleyan in point coordinates. We have to
form the reciprocal of the tangential equation of the Cayleyan, viz. (Art. 219)

m (a? +B? + 4) + (1 — 4m) aBy = 0,

204 INVARIANTS AND COVARIANTS OF CUBICS.

The reciprocal of this is formed by Art. 222, and the quantities 2° + y* + 2%, &e.
then expressed in terms of U, H, 0. The resulting equation of the Cayleyan is
480 — TH? —-16S°UH = 0.

233. In like manner every contravariant of the cubic can be
expressed in terms of three fundamental contravariants; and for
these three we may employ the three already mentioned, viz.
the evectants of S and 7’ (Arts. 219, 221), which we have called
Pand Q, in terms of which every contravariant cubic can be
expressed, and the reciprocal & (Art. 222). We can, as in
Art. 230, form the invariants of AP+ wQ, which for the canonical
form is
{mr+ (1—10m*) w} (a+ B°+ ry’) + [(1 —4m*) XN- 6m" (5 +4m?) apy,
and we find
S(AP+ wQ) = (1928° — T’) \*+ 7689" Te

+216 (387? — 648%) 2p? +216 (T° — 6478") Ap
— 1296 (5S8°T* + 648°) p', |
T(nP+ wQ) =(T?+5768°T) r° + 288 (58°72 — 1928") ru
+ 540 (38ST°— 3208*7T) r*p?+ 540 (Z'*— 448 9° T’) iy) |
— 19440 (78°T? — 649° 7) rout
~ 11664 (397 — 32,8*7? + 20488") Me
— 5832 (7° + 408° 7° + 25608°7') w°
R(AP+ wQ) = {SA + Tut T2087? w? + 1089!

+ 27 (TL? — 168") p*}’ BR,
and, as in Art. 230, the sacha and sextic functions of A, wu
which occur in the vshiien of S and 7 are the covariants of the
quartic function whose cube occurs in the value of £.

Again, H(AP+ “Q)

= (TD? + 1448°02u + 32497 du? + 108 (T? — 168°) w} P
— {494+ 37 y 4+ 1445°?n + 108STp*} Q,
the quantities multiplying P and @ respectively being the differ-
entials with respect to w and A of the same quartic function.

234. In like manner we can form the Pand @ of XU+6yH,
and we find

P(xU+ 64H) = Pr? + ig 129Pru?+ 4(SQ- TP) p’*,
Q(AU + 6h) = Qr’ + 608PAp — 30TPA'p* — 10TQX*p
- + 120(28°Q — STP) rp* + 24 {STQ — (1? + 248°) P} pw’

INVARIANTS AND COVARIANTS OF CUBICS. 205

Now if we denote by s and ¢ the S and T of 1U+ 6u4H,
as given Art. 230, these values differ only by the factors
3(7* +648") and (7? + a respectively guy

(48.9°P + 7) ~ + (3 TP— 48Q)

(48 S°P+ Fe ¢ (37P- 4gq) % mn

So again, forming the P and Q of AP+wQ, the results are
P(AP+ wQ) = (8S°U— TH) + 18d*p (STU + 88°)
+ 9rp*{( T° —328") U+ 128TH} —54y*{48°TU— (7? + 328°\H},
Q(AP+ pQ) =r {168° TU +4 (T” + 1928*) HZ}

+ 30A%p {[S( 7" - 648°) V+ 168° TH}

+ 15A%u? {7'(T? —3208°) U+4897°H}

— 270A7u? (16.8? T?U — T(T* — 648°) H}

— 1620Ap* {ST°U+ 48° (7? — 648°) H}

— 324y° {( 7" + 247°S* + 5125") U— 68ST (T? + 1288") 7},
and if we now write s and ¢ for the S and 7 of AP+yQ, as
given Art. 233, these values differ only by factors from

2 ds ds
(488°U+ 18TH) F + (TU-2408H) ©,

dt dt
oe (TU- 24.51) —— 5

To these formule may be added the reciprocal of XU + 6uH/,
which is
(At + 24.9 02n? + 8 Trp? — 489%") F—24y (n° 4+27y*) P?
— 24u" (A* — 4S") PQ — 8rp* Q",

and (48S9°U+ 18TH)

and of AP+ wQ, which is |

4 {SM + Tw t+ 728°r*p? + 108ST Ap? + 27 (7? — 168°) p4} ©
—{TA*+2168TA*p’ + 108 (7? — 648°) Au’ — 3888 7'S2u'} H?
— {16S°A*+ 3287 rA8u + 18 T?A*p? + 2169 (7? + 328°) uw} UH
+ {645°A'w + 1448? T 2 p?4+108ST"Ap?4 27 T( T+ 168") uw} 0".

235. We next mention a useful identical equation. If in a

206 INVARIANTS AND COVARIANTS OF CUBICS.

cubic U we substitute 2+’, y+rAy’, 2+Az2' for a, y, 2, let
the result be written

U+ nie BN P+2°U';

that is to say, let S and P denote the polar conic and polar line
of z'y'z' with respect to U; or, for the canonical form, let

S= (x +2myz )a' + (y’ +2mzex )y'+ (2 +2may ) 2’,
P=(x" + 2my'2') xv + (y? + 2me2'x’) y + (274+ Ama'y’) 2.
Similarly, let the result of a similar substitution in H be written
H+ 8r3+ 3011+ 1H’,

that is to say, let = and II denote the polar conic and polar line
of x'y'z’ with regard to the Hessian; then, by the help of the
canonical form, we can verify the following identical equation

3 (SI1—=P)= H'U- HU".
It follows hence, that when z'y’z' is on the curve, and therefore
VU’ =0, the equation U=0 may be written in the form

SiII—=SP=0.

From this form the following consequences immediately

follow :
(a) The lines P, IT intersect on the cubic; that is to say,

the tangential of the point «'y'z’, or the point where the tangent
P meets the cubic again, is the intersection of P with 1, the

polar of z'y'z' with respect to the Hessian (see Art. 183).

(0) The points of contact of tangents from z'y'z’ to the
cubic, which are known to be the intersections of S with J, are
also the intersections of S with 3, the polar conic of 2'y'z’ with
respect to the Hessian.

(c) The equation SIT—SP=0 is that which would be
obtained by eliminating an indeterminate 9 between S+ 62 =0,
P+6=0. The first denotes a conic through the intersections
of S, =; the second denotes the polar of a’'y'z’ with regard to
the same conic. Hence the given cubic may be generated as

the locus of the points of contact of tangents from a point a'y'z
to a system of conics passing through four fixed points.

(dq) If S+@= denote two right lines, P+6f1 obviously
passes through the intersection of these lines; this intersection

meee
cgi alia

INVARIANTS AND COVARIANTS OF CUBICS. 207

is therefore a point on the cubic, and P+ OI the tangent at it.
Hence the four points of contact of tangents to the cubic from
x'y'z’ form a quadrangle, the three centres of which are

on the cubic, and are the points cotangential with a'y'z
(see Art. 150).

(ec) If we consider the intersections of the curve and its
Hessian by any liné, for instance, z=0, the identity of this
article gives us

ab — ba = 3 (a,6, — b,a,),
that is to say, the invariant P of the two binary cubics vanishes.
Hence, again appears that the Hessian meets the curve in its
inflexions. For since P=0, the eliminant of the two binaries
is Q=0 (Higher Algebra, Art. 200); therefore at points of
intersection u + Av includes a perfect ,cube.

236. I have used this identical equation (Phil. Trrans., 1858,
p- 535) to form the equation of the conic through five con-
secutive points on the cubic. Since S touches the cubic, and
P is the common tangent, the general equation of a conic
touching U at a'y'z' is S-LP=0, where L=ax+ By+ yz is
an arbitrary right line. Now by means of the identity estab-
lished, the equation of the cubic may be written in the form

Il (S— LP) =P(z— LM).

Hence, the four points where S—ZP meets the cubic again are
its intersections with =- LII; and if the latter conic pass
through «'y'z’, the former will pass through three consecutive
points on the cubic. But on substituting a'y’s’ for ayz, we
have ='=Il'=H’, and the condition that =—ZII should pass
through 2'y'z’ is L'=1.

Next, in order that S— LP may pass through four consecutive
points, S— LII must have P for a tangent at the point a’y’2’.
Now the tangent to S— LI (being the polar of a’y’z’ with
respect to this function) is

20 — LT — LI’,
or (since L’=1, and Ml’ = H’) is I1— H'L, and since this is to be
1

proportional to P, we have L = @P+ ie

208 INVARIANTS AND COVARIANTS OF CUBICS.

The general equation, therefore, of a conic through four
consecutive points is

8— 6P*— 7 PM=0,
a $-9Ppa- 4 =
an we — oR =

passes through the two points where the former conic meets the
cubic again, the equation of the cubic being reducible to the
form
2 os — ————
n (S-@P*— 7, Pm) = P(2-ePn— 7A).

237. Since these two conics have P for a common tangent,
it will be possible, by adding the equations multiplied by suitable
constants, to obtain a result divisible by P, and the quotient
will represent the line joining the points where the conic meets
the cubic — It is necessary then to determine yp, so that
pS+3 ~F II’ may be divisible by P, which we do by equating
to nothing the discriminant of this quantity. Now this discrimi-

©’

nant when calculated will be found to be pe’ + 4p > . This
40’
rae
and since one of the factors is P, if we denote the other by J,
we have

quantity, therefore, will be divisible into factors if w=—

Le
By the help of this equation, the equation of the cubic given at
the end of the last article is transformed to
(1+ uP) (8 - oP - Pn) = Pp’ \u- f.n- (t+ nF).
The form of the equation shews that [I+ wP is the a at

the tangential of the given point on the cubic, and that If —F IT
passes through the second tangential of the given point (see
Art. 155).

INVARIANTS AND COVARIANTS OF CUBICS. 209

238. In order that the conic may pass through five con-
secutive points, the coordinates a’, y’, 2’ must satisfy the equation

_ The only difficulty is to determine the result of substituting the
coordinates 2’, 7’, z'in M. Now if we differentiate with regard
to x, y, or 2, the equation
and substitute a’, y’, 2’ for x, y, 2 in the result, observing that
wo dr ae atl
gt 2 aor) dal 2 gl + FP have M'=2y, and hence the

S$ $8

result of substituting w'y'z’ for xyz in

40’

is 4—OZ1'=0, and since w has been found to be =~ Fas

J

we have 0= mel and the problem is completely solved.

239. We next mention another general form to which the ©
equation of a cubic may be brought, viz.
| ax’ + by? +ca°+du’=0, where e+y+2+u=0.
The polar conic of any point 2’y’z'u’ onan

ax'a” + by'y" + cz'2" + du'u?

the polar conic of the point for which a’ =0, y/ Her is a pair of
lines passing through the point w=0, 2=0, &e.; and hence it
appears that the points ry, zw; xz, yu; xu, yz are pairs of cor-
responding points on the Hessian. ‘The form just written
contains implicitly eleven constants, and is one to which the

general equation of a cubic may be reduced in an infinity of
ways. ‘lhe values of the invariants for this form are S=— abcd,

T=b'c'd’ + c'd’a’ + d’a*b’ + a’b’c’ —2abed (ab+ac+ ad+ be+cd+ db).
The discriminant is formed from the three equations got by
differentiating with respect to , y, 2 respectively, viz.

aa’ =du', by'=du", cz’ =du’,
EE

210 INVARIANTS AND COVARIANTS OF CUBICS.

whence we have x, y, 2, w respectively proportional to the
reciprocals of ./(a), /(b), V(c), /(d). Substituting. these values
inx+y+2+u=0, we have the discriminant in the form

(bed) + »/ (eda) + / (dab) + /(abc) =
which cleared of radicals is, as before, R = T’ + 645° = 0.*

240. We conclude this chapter with a few remarks on the
case where the cubic breaks up into a conic and a right line,
If a curve have either two double points, or a cusp, not only
does its discriminant vanish, but also the functions obtained by
differentiating, with respect to any of the coefficients of the
original equation, the general expression for the discriminant
in terms of these coefficients. See Higher Algebra, Arts. 103, 113.
Now the expression for the discriminant of a cubic being of
the form 7*+64S*=0, its differentials are of the form

dT ys dT aT
20 +1928" , 22 +1928", &e.

If the curve have a cusp, we have S=0, 7'=0 (Art. 224), and
all these differentials vanish in conformity with the theory. If
the curve have two double points, that is to say, if the cubic
break up into a conic and right line, we have the equality
of ratios

dT d§_dT dS_daT dS ,
ie rt Oh ee ae ae

These equations if written at length would form a system of
equations, each of the eighth order in the coefficients, which are
the system of conditions that the™general equation of the
third degree should be resolvable into factors.

241. There is another form in which the foregoing conditions
may be written. In the first place we remark, that since a double
point on a curve is also a double point on its Hessian, the
coordinates of such a point satisfy the equations got by differen-
tiating with respect to a, y, 2, the equations both of the

* For the other covariants and contravariants when the equation is written in this
form, see Phil. Trans. 1860, p. 252 ; and for some remarks on the method of forming
invariants, &c., when the equation has been written with an additional variable con-
nected by a linear relation with the original variables, see Geometry of Three
Dimensions, Art, 538,

INVARIANTS AND COVARIANTS OF CUBICS. 211

curve and of the Hessian. In the case of the cubic, these six
differential equations are all of the second degree, and we
can linearly eliminate from them the six quantities 2’, y’, 2°,
yz, 22, xy, so as to obtain the discriminant in the form of
the determinant

died by, Ciy M, Ay a,
G, b, Coy by mM, 6,
Os, Dey Cy Cyy Cy ™
a,b, ¢, m, a, a,
a,, b, ¢,, b,, m, b,

as, b,, Cc, Coy Cy m =0.

We have seen also (Art. 226) that the conditions that the curve
should have three double points are expressed by taking any
of the first three rows, and the corresponding one of the second
three rows, and then equating to zero the determinant
formed with any two columns from these rows. So now in
like manner the conditions that the curve should have two
double points are expressed by taking any two of the first
three rows, and the two corresponding rows of the second
set, and equating to zero the determinant formed with any
four columns from these rows. In order to prove this it is
enough to observe that, as we shall show in the next article, if
U=PV, where V represents a conic, and P is ax+ By+ 2,
then the Hessian of Uis of the form XU+ uwP*. Consequently
we have

ie av _ a2 ae
Gn. giz dU dU
whence Pe a. a

shewing that the differentials of H and U, with respect to x
and y, are connected by a linear identical relation, and there-
fore that the determinant formed with the coefficients of four
corresponding terms in these equations vanishes.

242. The Hessian of PU, where P denotes the right line
ax + Sy+yz, and U is a function of any degree, may be found
in various ways. The second differential coefticients of PU are

212 INVARIANTS AND COVARIANTS OF CUBICS.

Pa+2aL, Ph+28M, Pe+2yN, Pf+BN+9M, Po+yL+aN,
Ph+aM+ BL, where L, M, N, as before, denote the first, and
a, b, &c. the second differential coefficients of U. Using these
values in forming the equation of the Hessian, and reducing by
means of the equations of homogeneous functions

(n—1) L=ax+ hy +92, &e.,
we get, for the Hessian of PU,

n

3 n
oar pone

where / denotes the quantity (dc —f”) a? + &c., Art. 184, which
geometrically represents the locus of points whose polar conics

touch the given line.

More generally the Hessian of UV is found by the same
process to be

(ntn'-1) 7.7,,, (n+n'-1) .,,
(n+n'—1) a evan , (n+n'—1) (n+ n'—2)
ms Peer Ey ee © + U’VO')/+ ace UVW,

where ©, ©’, as at Conics, Art. 370, denote (be—f’) a'+ &e.,
(b'c' —f”) a+ &c., and W denotes the covariant

(bc' + b'c — 2ff') LL' + &e.
The form just written shews that the intersections of U, V are

double points on the Hessian, the tangents at any such point
being the tangents to U and V respectively.*

* On the general theory of ternary cubic forms, see Aronhold’s Memoirs, Crelle,
vol. XXXIX., p. 140, 1850, and vol. Lv., p. 97, 1858; Professor Cayley’s “Third
and Seventh Memoirs on Quantics,” in the Philosophical Transactions, 1856 and
1861, and Clebsch and Gordan’s Memoirs in the Mathematische Annalen, vol. 1,
p. 56, 1869, and vol, vI., p. 436, 1873; also Gundelfinger, vol. Iv., p. 144, 1871.

3 oak rs
ii FATE

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CHAPTER VI.

CURVES OF THE FOURTH ORDER.

243. Ir will be remembered that we have classified curves
of the third order by combining a division founded on
characteristics unaltered by projection, with a division founded
on the nature of their infinite branches. The same principles
of classification are applicable to curves of the fourth order,
or, as we shall call them, quartics; but the number of
species is so great, and the labour of discussing their figures
so enormous, that it seems useless to undertake the task of
an enumeration. It will be sufficient here generally to direct
attention to the principal points that must be taken into
account in a complete enumeration. A quartic may be non-
singular having no multiple point; or it may have one,
two, or three double points, any or all of which may be
cusps. In this way we have ten genera, of which the
Pliickerian characteristics and the deficiency (Arts. 44, 82) are

Wn: Sele Meee, ae gaat seca 2

I, BOS OO: Te ae ae ee
II. f° Eo" FO 1S Te te. e
III. EOL Ie Tee
IV. SRO So Bs Bg
V. Bee ea
VI. en Bs Ouse ees Mar wg |
VIL. ye She daw ey ON Ay ©
Wit. 4.2 RCO SES
IX. Be E oe Bi Bee Ree
ae S00 O35 SoS ee

viz. in each of the last four cases the curve is unicursal.
Every quartic curve whatever may be considered as coming
- under one or other of these genera. But there are special forms,

214. CURVES OF THE FOURTH ORDER.

arising from the coincidence of nodes and cusps, which have to
be considered.

1’. Two nodes may coincide, giving rise to the singularity
called a tacnode; this is, in fact, an ordinary (two-pointic)
contact of two branches of the curve (see p. 28). It is to be
noticed that the common tangent counts twice as a double
tangent of the curve; thus, supposing that there is not (besides
the tacnode) any node or cusp, the curve belongs to the
genus IV., and its characteristics are as stated above; but 5=2
means the tacnode, and t=8 means that the double tangents
are the tangent at the tacnode counting twice, and 6 other
double tangents.

2°, A node and cusp may coincide, giving rise to the sin-
gularity on that account called node cusp, and called ramphoid-
cusp, Art. 58. It is to be noticed that the tangent counts once as
a double tangent, and once as a stationary tangent; thus, sup-
posing that there is not any other node or cusp, the curve
belongs to the genus V., and the characteristics are as above; but
o6=1, «=1 means the node-cusp; r=4 means the tangent.

at the cusp and 3 other double tangents; +=10 the tangent

at the cusp and 9 other stationary tangents.

3°. Three nodes may coincide as consecutive points of a
curve of finite curvature, giving rise, not to a triple point, but
to the singularity called an oscnode ; this is, in fact, an osculation
or three-pointic contact of two branches of the curve. The
tangent at the oscnode counts 3 times as a double tangent
of the curve; the genus is VIJ., and the characteristics are
as above, but 6=3 means here the oscnode; and r=4 means
the tangent at the oscnode counting 3 times, and 1 other
double tangent.

4°. Two nodes and a cusp, or a tacnode and a cusp, may
coincide as consecutive points of a curve of finite curvature
giving rise, not to a triple point, but to the singularity called
a tacnode-cusp; this is, in fact, an osculation or four-pointic
intersection of the two quasi-branches at a cusp. The genus is
VIII, and the characteristics are as above, 6=2, «=1 mean-
ing the cusp; t=2 the tangent at the cusp counting twice
as a double tangent; 1«=4 the tangent at the cusp, counting

CURVES OF THE FOURTH ORDER. 215

once as a stationary tangent, and three other stationary
tangents.

5°. Three nodes may coincide, as vertices of an infinitesimal
triangle, giving rise to a triple point (ordinary triple point with
three distinct tangents). The curve belongs to the genus VII.,
and the characteristics are as above, 6=3 meaning the triple
point. fe

6°. Two nodes and a cusp may coincide, giving rise to a
special triple point, at which an ordinary branch of the curve
passes through a cusp. The curve belongs to the genus VIIL,
and the characteristics are as above, 6=2, «=1 here meaning
the special triple point.

7. A node and two cusps may coincide, giving rise to a
special triple point not visibly different from an ordinary point
of the eurve. The curve belongs to the genus IX., and the
characteristics are as above, 6=1, «=2, here meaning the
special triple point.

244, In order to illustrate the distinction between the
different kinds of double points which we have enumerated,
let us suppose the origin to be a double point at which the
two tangents coincide with the line y=0, then the equa~
tion of the quartic will be of the form y*+u,+u,=0, where
U, = ax? + bx*y + cxy" + dy", u, = Fr" + Se y+ |

We proceed now as in Art. 56: In-order to determine the
form of the curve in the neighbourhood of the origin, we sub-
stitute y= max’, we determine 8, so that two or more of the
indices of x shall be equal and less than the index of any other
term; and we attend only to the terms of lowest dimension
inz. ‘Then letabenot=0. We find B=; the form of the
curve near the origin is the same as that of the curve y’+ ax*=0,
and the origin is an ordinary cusp.

(1) Let a=0. We then have 8=2, and m is determined
by the quadratic m*+bm+e=0. There are then two branches
whose forms near the origin are respectively the same as those
of the curves y=m,2", y=m,x", where m,, m, are the roots of
the above equation. The branches touch each other, and the
origin is a tacnode.

216 CURVES OF THE FOURTH OKDER.

(2) Let this quadratic have equal roots, the form of the —

equation then is
(y — ma*)* + cay’ + dy’ +fa'y + &e. =0,

and to the degree of approximation to which we have as yet
proceeded the two branches in the neighbourhood of the
origin coincide. In order to discriminate them we substitute
y= max" + nx", and determine n and y as before. We find then
y= and n*=—(cm*+fm). The form then of the curve near

the origin approaches to that of the curve y=ma* + nz", which
has been considered, Art. 58. The origin is then a ramphoid}
cusp or node-cusp.

(3) If, however, in addition to the preceding conditions we
have f=— cm, the equation of the curve is of the form

(y —mx*)’ + cay (y — max’) + dy’ + ga’y’ + &e. = 0,

and on substituting y= ma’+nz" we find y=3, and ” is de-
termined by the quadratic

n +omn+m'(dm+g)=0;

and if n,, », be the roots of the quadratic, the curve in the
neighbourhood of the origin consists of two osculating branches,
whose forms are represented by the equations y= mz" +n,2",
y=mx'+n,x*. Since the difference of these values of y com-
mences with an odd power of x, the branches cross as well as
touch at the origin. ‘The origin is now an oscnode.

(4) If, however, in addition to the former conditions we
have the roots of the last-mentioned quadratic equal, or
dm+g=tic’, the equation of the curve is of the form

(y — ma’ — cay — dy’)” = Any’ + By’,
and, as before, we find that its form near-the origin is given by

the equation y=ma*+cmz*+pa*. The origin is then a tac-
node-cusp. The node can have no higher singularity in a
proper quartic, for the next step would be to suppose A to
vanish, in which case the equation would break up into two
of the second degree. The case where the origin is a triple
point does not seem to require illustration.

aR

CURVES OF THE FOURTH ORDER. 217

245. We have thus far not attended to the distinction of
real and imaginary. Assuming that the quartic curve is real,
then imaginary nodes or cusps can present themselves only in
pairs, and we may distinguish the cases accordingly; thus we
may have one real node, two real or two imaginary nodes,
three real or one real and two imaginary nodes; and the like
for cusps. Again; any real node may be a crunode or an
acnode. The distinctions as to real and imaginary scarcely
present themselves in regard to the special singularities above
referred to (the condition that imaginaries must present them-
selves in pairs, implying for the most part that these singu-
larities are real); the only distinction seems to be in regard
to the ordinary triple point, which may be a point with three
real tangents, or with one real and two imaginary tangents,
viz. in the former case the point is the common intersection of
three real branches of the curve, in,the latter case it is the
common intersection of one real and two imaginary branches
of the curve; or, what is the same thing, we have a real
branch passing through an acnode. The point does not visibly
differ from an ordinary point of the curve, resembling in this
respect the special triple point 7° above referred to. The dif-
ference is, that in the case of an ordinary branch through an
acnode the tangents are one real and two imaginary; in the
case of the special triple point they are all real and coincident.

246. There are yet other specialties which may be taken
account of. A node may be in regard to one of the branches
through it a point of inflexion; that Is, the tangent to the
branch at the node may meet the branch in three consecutive
points (or the curve in four coincident points); or, again, the
node may be in regard to each of the branches through it a
point of inflexion. Such a node may be considered as the
union of an ordinary node with (in the first case) a point of
inflexion, and with (in the second case) two points of inflexion ;
and the node may be termed a flecnode or a biflecnode in the
two cases respectively. The point or points of inflexion thus
coinciding with the node must be reckoned among the inflexions
of the curve, and the number of the remaining inflexions
diminished accordingly. A biflecnode has properties analogous

¥F

218 CURVES OF THE FOURTH ORDER.

to those established (Art. 170, et seg.) for the inflexions of cubics.
In general, if we lock for the locus of harmonic means on
radii-vectores drawn through the origin, which is supposed
to be a double point on the quartic u,+u,+u,=0, we find
u,+2u,=0. When, therefore, u, is a factor in u,, the locus
becomes a right line, and the double point, having a harmonic
polar, has the properties established (Art. 170). The points
of contact of tangents from it lie on a right line, and the
curve may be projected so that this point shall become a
centre, or else so that all chords parallel to a given line
shall be bisected by a fixed diameter. In the latter case,
the form of the equation is in general

oY (w—a) (w@— 6) =+4 A(x —-c) (w— d) (w-e) (af).
There is no difficulty in discussing, as in Arts. 39, 199, the
different possible forms of curves included in this equation,
according to the reality, and to the relative magnitude of
a, b, &c.; and in deriving thence the different possible forms of
the projections of these curves. :

247. Once more, a quartic may have another kind of
singular point, of which account might be taken in the
classification, viz. a point of undulation, that is to say, one in
which the tangent meets the curve in four consecutive
points. The tangent at such a point replaces two stationary
tangents and one ordinary double tangent. A quartic may
have four real points of undulation, as we can see by writing
down the equation waz ‘ene where S is any conic touching
the four lines w, x, y, 2

248. We have not yet exhausted the list of characteristics
unaltered by projection which would have to be taken into
account in a complete classification of quartics. It will be
remembered that we divided non-singular cubics into unipartite
and bipartite according as all the real points of the curve are or
are not included in one continuous series; and it is natural to sup-
pose that similar distinctions exist in regard to quartic curves.

The possible forms of non-singular quartics have been studied
in detail by Zeuthen (Math. Annalen, vi. 411). He remarks
that the branches of a curve may be divided into those of odd

CURVES OF THE FOURTH ORDER. 219

order met by any line in an odd number of points, and those of
even order. ‘The latter are what we have called ovals (Art. 200),
using the word to include not only ovals in the ordinary sense
but also their projections. In this sense, for instance, all the
forms of conics would be described as ovals. Zeuthen shows
that a non-singular curve cannot have more than one branch of
odd order, and therefore that a curve of even degree cannot have
any. A quartic, therefore, can only have ovals. It is at once
apparent that if a quartic have two ovals, one wholly inside the
other, it can have no other real point. Tor if it had, the line .
joining this to a point inside the interior oval must cut the curve
in five points. For the same reason the interior oval cannot
have bitangents or inflexions. A quartic of this kind having
two ovals, one inside the other, is called an annular quartic.
This reasoning does not exclude the case of ovals exterior to
each other, but the quartic can at most have four such ovals; for
if it had any other real point the conic passing through this and
through points inside the four ovals respectively would meet the
curve in nine points. That a quartic may actually have four
such ovals appears as well from the curve (x? — a’)"+(y"— 0)’=c",
(c <b) considered p. 43, as from the following illustration which
Pliicker gives in order to show that the 28 tangents which a
non-singular quartic can have may be all real. Consider
the curve Q=+k, where

Qa (y'—2"\(a—1)(o- §)-2 fy’ +2(e—a)}

Now the equation © = 0 represents a
quartic having three double points as
shown in the dark curve in the annexed
figure ; and the equation 2 = denotes |
a curve not meeting { in any finite
point, which deviates less from the
form of the curve © the less we
suppose &, and which according to
the sign we give k is either altogether
within or altogether without the curve
©. When it is altogether without,
the curve is unipartite; when it is
altogether within, the curve in the first instance consists of four

220 CURVES OF THE FOURTH ORDER.

meniscus-shaped ovals, one in each of the compartments into
which the curve Q is divided. ach meniscus has one tangent
touching it doubly; and, besides, it will be seen that any two
ovals have four common tangents, and that there are six pairs of
ovals. It will readily be conceived that, as the value of the
constant is supposed to change, first one, then another of these
ovals becomes imaginary, so that non-singular quartics may be
either unipartite, bipartite, tripartite, or quadripartite. We can
in like manner conclude that a quartic having one double point
may be either unipartite, bipartite, or tripartite; and one having
two double points, either unipartite or bipartite.*

248 (a). Zeuthen takes as the basis of his classification of
quartics the real bitangents of the curve, which he divides into
two classes. When a quartic has a pair of ovals exterior to each
other, it is easy to see that (just as if they were two conics)
these ovals have four common tangents and cannot have more.
These common tangents are Zeuthen’s bitangents of the second
kind. Ifthe quartic have two ovals exterior to each other the ~
number of such bitangents is 4; if it have three such ovals the
number of such bitangents is 12; if it have four, the number
is 24. Zeuthen’s bitangents of the first kind may be either
(a) lines doubly touching a single branch of the curve; or
(b) bitangents, both of whose points of contact are imaginary.

Zeuthen has proved that every quartic has four real
bitangents of one or other of these two species, which four we
shall call the Zeuthen bitangents. The total number then of
real bitangents to a quartic is got by adding to these four
the 0, 4, 12, or 24 bitangents of the second kind ; and accordingly
is either 4, 8, 16 or 28. Zeuthen’s method of proof is to consider
the series of quartics, S+2S’, where S and S are any two
non-singular quartics. The number of real bitangents of a
curve of the series will only alter when A is such that the curve
has some singularity. Zeuthen shows that as » passes through
the value for which the curve has a double point, only real
bitangents of the second kind are lost or come into existence;
aad that for no ordinary singularity do bitangents of the first

* In general the maximum number of “ parts” of a curve is one more than the
* deficiency.”

CURVES OF THE FOURTH ORDER. 221

kind change into those of the second, or vice versd. But
consider a bitangent of the first kind touched by a branch in
two real points. As a parameter in the equation alters, these
points may approach each other and the intervening arc of the
curve become smaller. At last the points coincide and the curve
has'a point of undulation ; after that the bitangent has imaginary
points of contact..°Thus we see that at the value of A, for
which the curve has a point of undulation, Zeuthen bitangents
of the form (a) may change into the form (6), or vice versd.
The only change then that affects bitangents of the first kind
being an interchange of these two forms, the total number of such
bitangents is the same for the whole series of quartics included in
the form S+2S’, and therefore is the same for every quartic;
and Pliicker’s example shows that the number is four.

248 (b). When a branch has a tangent touching it in two
real points, it is obvious that the arc at each of these points
turns its convexity towards the tangent, and that there is an
intermediate part of the arc which turns its concavity towards
it, this concave part being separated by a point of inflexion at
each end from the convex parts. Kvery such bitangent then
implies two real points of inflexion; and it is not difficult to see
that the converse of this is also true. Since there can be at
most four such tangents, a guartic can have at most eight real
inflections. Zeuthen confines the name oval to a branch, having
no real bitangent or inflexions: one with a single real bitangent
he calls a unifolium; one with 2, 3, or 4 such bitangents, a
bifolium, trifolium or quadrifolium. ‘Chus the external curve in
Plicker’s figure is a quadrifolium; the four internal curves are
unifolia. The figure, p. 45 (3), represents two bifolia; p. 46 (5),
represents an annular quartic, quadrifolium with internal oval.

248 (c). Zeuthen further shows by the method of Art. 125,
Ex. 4, that the points of contact of any three of his bitangents
lie on a conic; and further, that it is the same conic which
passes through the contacts of all four bitangents. If then
w, ©, Y, 2, represent four lines, and Va conic, the equation of the
quartic must be of the form wayz=V*. Zeuthen’s analysis of
the possible forms of quartics is made by discussing the different

222 CURVES OF THE FOURTH ORDER.

positions which the intersections of the four lines with the conic
can have with respect to the quadrilateral found by them. Thus
when V meets all the lines in real points, he enumerates the
following cases: (1), annular quartic, quadrifolium and internal
oval; (2), quadrifolium and 2 ovals; (3), 4 unifolia; (4), trifolium,
unifolium and oval; (5), bifolium, 2 unifolia and oval; (6), 2
bifolia and oval; (7), 2 bifolia and 2 ovals; (8), bifolium and
2 unifolia; (9), trifolium, unifolium and 2 ovals. He enumerates
thus 36 cases in all, but the figures which he gives for the nine
cases just mentioned sufficiently illustrate the rest, a very slight
modification being enough to turn a unifolium into an oval, &e.
It will be observed that the classification just made rests solely
on projective properties and has no reference to the line infinity.
In Art. 249 we state the principles on which these classes may
be subdivided into species when the nature of the infinite branches
is taken account of.

248 (d). Zeuthen also applies his method of classification to
nodal quartics considered as limiting cases of non-singular quartics.
He enumerates and discusses the following cases: ‘a), conjugate
points considered as limiting cases of ovals; (b), nodes which
arise when in limiting cases of annular quartics the inner branch
comes to meet the outer;—in neither of these cases are the
Zeuthen bitangents affected; (c), nodes which arise when two
mutually external branches come to meet; (d), which arise when
a branch of even order breaks up into the intersection of two of
odd order; (e), the case of two imaginary double points. In the
cases where the Zeuthen bitangents are affected, the investigation
is carried on by considering the forms represented by the equa-
tion wxyz = V*, when V passes through the intersection of two
of the lines, or when two of the lines coincide with each other.

249. In order to see how quartics might be classified in
respect of their infinite branches, we observe that the line
infinity may meet a quartic, (2) in four real points, (4) in two
real and two imaginary, (c) in four imaginary points, (d) in
two coincident and two real points, (e) in two coincident and
two imaginary points, ( 7) twice in two coincident points, these
points being real, or (g) these points being imaginary, () in

THE BITANGENTS. 223

three coincident and one real point, (¢) in four coincident
points. Again, the cases (d), (e), (f), (g) would have to be
further distinguished according as the line infinity when meeting
the curve in two coincident points is simply a tangent or a line
passing through a double point, which double point may be
either crunode or acnode, cusp, or one of the special kinds above
mentioned. Similarly in the case (A), the line infinity may be
either an ordinary stationary tangent, or a tangent at a double
point or cusp, or it may pass through a triple point, and in
the case (¢) it may be either a tangent at a point of undulation,
a tangent at a double point of the special kind, or a tangent
ata triple point. Lastly, any of the points which count only as
single intersections of the line infinity with the curve may be
on the curve a point of inflexion or undulation, and where this
happens a difference in the figure will result which would have
to be taken into account in a complete classification of quartics.

250. We have already shown (Art. 70) low to form the
equation of the Hessian of a quartic, which is a curve of the
sixth degree, intersecting the quartic in the twenty-four points
of inflexion. We have also seen (Art. 92) that the equation of
the reciprocal of a quartic is of the form se = 7%, where 8
represents a curve of the fourth and 7’ of the sixth class,
and the form of the equation shows that both are touched by
the twenty-four stationary tangents. We have postponed to
another chapter the solution of the problem to form the equation
of a curve passing through the points of contact of double
tangents of a given curve. It will there be shown that,
in the case of the quartic, the equation of such a bitangential
curve may be written in the form @=3H®, where © is the
covariant AL” +&c., as in Art. 231; that is to say, L’ &e.
represent the first differential coefficients of the Hessian, and A
denotes bc—f", where a, 0, &c. are the second differential
coefficients of U. In like manner ® denotes Aa’ +&c., as in
Ex. 1, Art. 230.

THE BITANGENTS.

251. It is convenient to commence by studying a more
general theory in which that of the bitangent is included.
Let us then consider first the form UW=V", where U, V, W

224 THE BITANGENTS.

represent conics; a form containing implicitly sixteen constants,
and therefore one to which the equation of any quartic
may be reduced in a variety of ways, as we shall after-
wards more fully see. The form of the equation shows that
U and W each touch the quartic in four points, namely, the
points where they respectively meet V. Now we have already
discussed (see Conics, Art. 270, &c.) the equation UW = V*, when
U, V, W represent right lines, and the results hold good with
the proper alterations when they represent conics. It is merely
necessary to remember, that two conics represented by equations
of the form 1U+yV+vW=0, instead of intersecting in a
single point, intersect in four points; and that if we are given
one point on a conic whose equation is to be of this form,
' three other points are necessarily given; for if we have
AU’ + pV’ +vW’=0, the conic A\U+uV+4+vW=0 will, it is
clear, pass through the four points determined by the equations
LS NS ieee
oy Ww
Conics just cited, that the quartic UW= V” may be considered
as the envelope of the variable conic \°U+2\’\V+ W=0
where A is variable, and which touches the given quartic in the
four points determined by XU+ V=0, AV+ W=0. The two
sets of four points in which any two of the enveloping conics
touch the quartic lie on another conic, as appears by writing
the given equation in the form
(WU + 20V + W) (WU 4+ 2uV + W) = {frAp0+ (A+ pw) V+ WY’.
In like manner, the properties of poles and polars may be
extended to the curve under consideration. Through any point
(or, if we please, we may say through any set of four points)
may be drawn two conics of the system \°U+ 2XV + W, the two
sets of four points of contact lying on a conic UW’4+ WU’-2VV’,
which may be called the polar of the given point or set of
points, and the symmetry of the equation shows that the polar,
in this sense of the word, of any point on the latter conic
will pass through the given point. Conversely, any conic
aU+bV+cW meets the quartic in two sets of four points,
through each of which sets a quadruply tangent conic may be
drawn, the two intersecting in a set of points which constitute
in this sense the pole of aU+bV+cW.

It follows then from the discussion in the

THE BITANGENTS. 225

252. It is useful now to recall the properties established
(Conics, Art. 388, &c.) for a system of conics included in the
equation 2U+BV+yW=0. In the first place, if this equation
represents a pair of right lines, their intersection lies on a
fixed cubic, the Jacobian of U, V, W; a curve which may
also be defined as the locus of a point, whose polars with
respect to all conics of the system aU+@V+yW meet in
a point. If we consider two conics included in this system,
the equation of any conic through their intersections must
be of similar form; and hence, the intersection of each of
the three pairs of lines joining the four intersections of
the two conics must lie on the Jacobian. If the two conics
touch, two of these three intersections coincide with the
point of contact; and, therefore, if two conics of the system
aU+8V++W touch each other, the point of contact lies on
the Jacobian.

Secondly, the system aU+8V+yW may be regarded as
a system of polar conics of the variable point a@y with regard
to a certain fixed cubic, which has for its Hessian the Jacobian
of the system, and the equation of which can be formed when
those of the three conics are given.

Thirdly, if aU+S8V+yW represents a pair of right lines,
all such right lines touch a curve of the third class, the Cayleyan
of the cubic last mentioned.

253. Hence then, in particular, since any enveloping
conic U+2xXV+ W, and the conic through the four points
of contact are each included in the form aU+BV+yW,
if we draw the three pairs of lines connecting the points of
contact of any conic enveloping UW=V", the intersections
of each pair lie on a certain fixed cubic, viz. the Jacobian;
and the lines themselves are all touched by a fixed curve of
the third class, viz. the Cayleyan.

Again, if the two conics \U+ V, XV + W touch each other,
then the conic °U+2AV+ W, instead of touching the quartic
in four distinct points, has ordinary contact with it twice and
meets it once in four consecutive points. And from what we
have just seen, this point of contact of higher order lies on the
Jacobian. We infer then, that twelve conics of the system

GG

226 THE BITANGENTS.

vU+2XV+ W have this higher contact with the quartic,
namely, the twelve passing each through one of the intersections
of the Jacobian with the quartic.

254. Six conics of the system °U+2XV+ W reduce toa
pair of right lines; for the discriminant of this form being a
function of the third degree in its coefficients will be one of
the sixth degree in A, and therefore six values of \ can be found
for which it vanishes. When an enveloping conic reduces to
a pair of right lines, the four points of contact lie two on each
line, and each line is therefore a double tangent to the quartic.
It appears from Art. 249, that if ab, cd be any two of these
six pairs of bitangents, the equation of the quartic may be
transformed to abcd = V*, the eight points of contact lying on a
conic V. ‘Thus we see that the form U+2AV-+ W includes
six pairs of the bitangents of the quartic, these twelve bitangents
all touching a curve of the third class, viz. the Cayleyan of
the system, and the intersections of each pair lying on the
Jacobian. So again, if the points of contact of any of these
pairs of bitangents be joined directly or transversely, the joining
lines also touch the Cayleyan, and the intersection of each pair
lies on the Jacobian. This may be stated in a slightly
different form by considering the cubic S, of which U, V, W
are polar conics. ‘Then if the equation of a quartic is a function
of the second degree in U, V, W, since the vanishing of such a
function expresses the condition that the line cxU+yV+z2W=0
should touch a fixed conic, it is easy to see that the quartic
may be defined as the locus of a point whose polar with
respect to S touches a fixed conic, or, in other words, the locus
of the poles with respect to S of the tangents of that fixed
conic; or, it will come to the same thing if it be defined
as the envelope of the polar conics of the points of that conic.
The double tangents of the quartic correspond to the points
where the conic meets the Hessian of S.

255. Let us. now consider any two of the bitangents of a
quartic, which we take for the lines 2, y; then if we make
2=0, the equation of the quartic is to reduce to a perfect

THE BITANGENTS. 227

square, say (27+ ayz + by*)’, and if we make y=0, the equation
is to reduce to, say (2°+cxz+dzx’)*. Hence, evidently the
equation of the quartic must be of the form

xy U= (2° + ay2 + by’ + cxz + da’)’ 5
that is to say, of the form xy U=V’", which we have just discussed ;
an equation which may also be written

ay (VU + 2XV +4 xy) = (xy +rV)”.
There are, as we have seen, beside the value \ = 0, corresponding
to the pair of lines ay, five other values of A for which
MU+2AV+ay will represent a pair of lines; and thus in
five different ways the equation can be reduced’ to the form
wayz=V". Hence, through the four points of contact of any
two bitangents we can describe five conics, each of which passes
through the four points of contact of two other bitangents.

A non-singular quartic has 28 bitangents; and there are
therefore 4 (28.27), or 378 pairs of bitangents; each of these
pairs gives rise to five different conics, but each conic may arise
from any one of the six different pairs formed by the four
bitangents which correspond to that conic, hence there are in
all & (378) or 315 conics, each of which passes through the points
of contact of four bitangents of a quartic.*

256. We have seen that each pair of bitangents combines
with five other pairs to form a group of six pairs, the points of
contact of any two of which pairs lie on a conic. It follows
that the 378 pairs may be distributed into 63 such groups of six.
The twelve bitangents of each group touch the same curve of
the third class; and this is touched also by the lines joining
directly and transversely the points of contact of each pair.
The intersections of each pair of bitangents, and also those of
each pair of joining lines, lie on a cubic. Corresponding to each |
group there are twelve conics, each of which touches the quartic
twice with ordinary contact, and once so as to meet it in four

* Pliicker first noticed the possibility of bringing the equation of any quartic to
the form wayz= V*, but he hastily inferred that the six points of contact of any
three bitangents lie on a conic, and thence drew an erroneous conclusion as to the
total number of conics passing through eight points of contact of bitangents (see
the Theorie der Algebraischen Curven, p. 246).

228 THE BITANGENTS.

consecutive points, the twelve points of higher contact lying
on the cubic last mentioned. There being 63 groups, 756 such
conics may in all be drawn.

257. We shall show how to form a scheme of the 315
conics, and for that purpose we denote provisionally the first
26 bitangents by the letters of the alphabet, adding the symbols
g@ and W to denote the other two. We denote by abcd the
conic passing through the eight points of contact of the
bitangents a, 6, c, d. If now abcd, abef, be two of the 315
conics, the pairs ab, cd, ef belong to the same group, and from
what we have seen, cdef will be another of the conics. ‘This
may also be shown directly as follows. Let the equation of
the quartic be abcd = V*, or

ab (cd + 2NV + d’ab) = (V + Aad)’,

and we can determine X so that cd+2AV+Nab=ef. Solve
for V from this equation, and substitute in the equation of the
quartic, when it becomes

Nal’ + cd’ + ef” —2r*abed — 2M abef — 2cdef'= 0,
or 4cdef = (cd + ef — r’ab)’,

a form which proves the theorem stated. It appears thus, that
given three pairs of lines which are to be pairs of bitangents
of the same group of a quartic, the equation of the quartic will
be of the form /)/(ab)+m/(cd)+nV(ef)=0, so that if
two points were given in addition, a single quartic could be
found satisfying the prescribed conditions. Corresponding to
any group there are 15 conics, passing respectively through
the points of contact of each two of the six pairs of which
the group consists. ‘There would thus seem to be 63 x 15 = 945
conics; but then every conic abcd is counted three times over,
as belonging to the three groups ab, cd, &c., ac, bd, &e.,
ad, be, &c.; the total number is therefore 315 as before.

258. Consider any conic abcd, then the group abd, cd, &c.,
and the group ac, bd, &c., can have no other bitangent common,
the quartic being supposed to be non-singular. For example,

THE BITANGENTS. 229

if abef be a conic of the first group, aceg cannot be a conic of
the second. For (Art. 257) the equation of the conic through
the points of contact of a, b,c, d may be written in the form

dab + ¥ (ed — of )=0,

and if aceg be another conic, this must be identical with the form
1
pac + . (bd — eg) = 0.

From this identity we at once infer

(Ab — pc) («- = d) sé (—f- 9) ‘
It follows that e, being identical with one of the factors into
which the left-hand side breaks up, passes through the inter-
section either of 6 and or of a and d. But in either case the
point through which e is thus proved to pass will be a double
point on

4AMabcd = (Nab + cd - ef )’,

and therefore the quartic could not be non-singular.
In precisely the same way we see that if abef, acmn be two
conics, there is an identity

1

(Ab — pic) (a- 5, 4) = 9-5
and hence the diagonals of the quadrilateral efmn pass one
through ad, the other through dc; or, in other words, the inter-
sections of each pair of bitangents lie, according to a certain
rule, three by three on right lines. When once a scheme of
the 315 conics has been made, there is no difficulty in discri-
minating which diagonal passes through ad and which through
bc. For example, if it appears that aemu, afnv, aduv are conics
of the system, we infer in like manner that the diagonals of
the quadrilateral emfn pass through ad and uv; and thence we
infer that ad lies on the line joining en, fm. Thus then consider
any conic abed, this belongs to the three groups ab, cd, &c.,
ac, bd, &e., and ad, be, &ec., and it appears now that each of
the sixteen quadrilaterals formed by combining one of the four
other pairs belonging to the group ac, bd with a pair from

230 THE BITANGENTS.

the group ad, be, will have a diagonal passing through abd,
Now the pair ad belongs to five different conics, and therefore
there are eighty quadrilaterals having a diagonal passing
through ad. But it will be found, as we have intimated, that
these quadrilaterals may be distributed into pairs having a
common diagonal; hence, through each of the 378 points ab
can be drawn 40 lines, each passing through two others of
these points, and there are in all 5040 such lines.

259. We are now in a position to form a scheme of the
315 tangents, in which nothing but the notation shall be
arbitrary. Commence by writing down the group ad, cd, ef, gh,
aj, kl; then since the groups ac, bd; ad, be can have no
bitangent common with the preceding nor with each other,
these groups may be written, ac, bd, mn, op, qr, st; ad, be, wv,
we, yz, dvr. Proceed now to write down the group ae, df;
this must include no bitangent from the group ad; but in each
term one of the bitangents from the group ac will be combined
with one from the group ad. Now since it was free to us
to write down the pairs of each group in any order we pleased,
it is a mere matter of notation, and does not introduce any
geometrical condition, if we take this group to be ae, df, mu,
ow, gy, sp. In like manner, it is a mere matter of notation to
suppose that the bitangents have been so lettered, that ag and
mx, at and mz, ak and my shall respectively belong to the
same group. ‘This being assumed, it will be found that the
group af, be is necessarily nv, px, rz, ty, and we can thus
proceed, step by step, to write out the whole system. A table
of the 315 conics was accordingly given in the first edition,
but I do not occupy space with it now, because an algorithm
has been given by Hesse (Crelle, 1855, XLIx, 243), and more
minutely discussed by Professor Cayley (Credle, 1868, LXvIII,
176), which exhibits in an easily recognizable form the mutual
relations of the 28 tangents. Hesse’s method introduces
considerations from the geometry of three dimensions. He
equates to zero the discriminant of «U+8V+yW where
U, V, W denote quadric surfaces. ‘This discriminant being a
function of the fourth degree in a, 8, y, if these quantities
be regarded as variables, the equation denotes a plane quartic.

disth, a ne heneee

THE BITANGENTS. 231

But for any value of a, 8, y for which the discriminant vanishes,
aU+8V+W denotes a cone, so that to every point on the
plane quartic corresponds a point in space, namely, the vertex
of this cone; and Hesse’s method connects the double tangents
of the plane quartic with the lines connecting each pair of 8
points in space which are the intersections of three quadric
surfaces. We make no use here of any principles of solid
geometry, but merely borrow the notation which Hesse’s
method suggests.*

_ 260. Take then eight symbols 1, 2, 3, 4, 5,6, 7,8. Their
combination in pairs gives us 28 symbols 12, 13...78, which
we use to denote the 28 bitangents. ‘This notation, the symbols
being properly applied to the 28 bitangents, enables us correctly
to represent their geometrical relations, though it fails com-
pletely to exhibit the symmetry of the system. In fact, the
notation might suggest that the bitangent 12 was related in a
different manner to the bitangents 13, 14, &c., and to the
bitangents 34, 56, &c., whereas actually there is no geometric
difference between the relations of any pair of bitangents.
So again we suppose the symbols so applied, that 12, 34, 56, 78
shall denote bitangents whose 8 points of contact lie on a
conic. The same property will then belong to every tetrad
of bitangents represented by a like set of duads; that is,
by any four duads containing all the eight symbols. But
if we count, we shall find that we can only make 105 arrange-
ments of the 8 symbols into sets, such as 12, 34, 56, 78.
The remaining 210 conics correspond to four bitangents,
whose symbols are such as 12, 23, 34, 41; that is to say,
the duads are formed cyclically from any arrangement of
four of the eight symbols, and it will be found that we

* Another mode of connecting the theory of 28 bitangents with Solid Geometry
is used by Geiser, Mathematische Annalen 1. 129, as follows: From any point on a
cubic surface can be drawn a quartic cone touching the surface. This will be non-
singular, its bitangent planes being the tangent plane to the cubic at the vertex, and
the planes joining the vertex to the 27 lines on the surface. Zeuthen shows that his
classification of quartics with regard to the reality of their bitangents leads by a
different process to the results obtained by Schlafli in classifying cubic surfaces with
respect to the reality of their right lines.

232 . THE BITANGENTS.

can have 210 such tetrads. Thus then the group belonging
to the pair 12, 34, consists of 56, 78; 57, 68; 58, 67; 13, 24;
14, 23; and the group belonging to a pair such as 12, 13, is
24, 34; 25, 35; 26, 36; 27, 37; 28, 88. Thus the notation
shows completely how the bitangents are to be combined
in groups. It suggests, however, that the 105 conics of the
form 12, 34, 56, 78 differ in their properties from the 210
of the form 12, 23, 34, 41. This is not the case, the whole
315 tetrads forming an indissoluble system.

261. Professor Cayley remarks that Hesse’s researches
establish the following general rule: A bifid substitution
makes no alteration in the geometrical relations of the bitangents
denoted by any set of symbols. What is meant by a bifid
substitution is, that writing down such a symbol of substitution
as 1234°5678, we interchange everywhere the duads 12, 34; 13,
24; 14, 23; and again, 56, 78; 57, 68; 58, 67; but leave
unchanged such duads as 15, 36, where one of the first set
of symbols is combined with one of the second. ‘The number
of possible bifid substitutions is 35, or, if we add unity (viz.
no alteration of any duad) the number is 36.

For example, now if we apply the bifid substitution
1234°5678 to the pair 12, 34, we get the same pair in opposite
order; if we apply it to 12, 13, we get 34, 24, a pair of
the same type as 12, 13; if we apply it to 12, 15, we
get 34, 15, a pair of apparently a different type, but not
different in geometrical relations. Thus, then, if we apply
the same bifid substitution as before to the tetrad 15, 67,
28, 34, which is one of the set of 105 already referred to,
we get 15, 58, 82, 21, which is one of the set of 210,
and which, according to the rule, possesses the same geometrical
properties.

262. Professor Cayley has exhibited in the following table
the geometrical relations of the bitangents, taken singly in
twos, threes, or fours, and the number of terms belonging to
each type of arrangement of the symbols.

THE BITANGENTS. 233

Representattive

term. Wo. of terms. Geometrical character.

£12 28 28 | Bitangents.

4 ae a 378 | Pairs of bitangents.

LI | 12.23.34 Hag} 43 &o Triads of bitangents such that
II] | 12.34.56 840 6 points of contact are on conic.
A | 12.23.31 56 oe

Vi | 12.23.45 1680) 2016 Triads such that 6 points of con-
NY 12.18.14 aunt eae contact are not on conic.

III|_ | 12.34.56.78 | 105) | Tetradsof bitangentssuch that the
(] | 12.23.34.41 sat 8 points of contact are on conic.
‘WV | 12.84.56.67 | 2520

12.34.45.

7 19 3 oy . ie Sesah Tetrads such that 6 out of the
NV ; . ‘ > 5 .: os Salk 8 points of contact are on conic.
W\ | 12.13.14.45 | 3360

IA | 12.34.45.53 | 560 .

\V& | 12.13.14.15 | 280| go49 Tetrads such that no 6 points of
I\VV | 12.34.35.36 | 1680 contact are on conic.

VV | 12.13.45.46 | 2520 | 20475

In the above, for greater clearness, a geometrical symbol

has been attached to each term, viz. the symbols 1, 2, 3, 4,
5, 6, 7, 8 being regarded as points, when any two of these
are combined into a duad, this is indicated by a line being
drawn to join the two points; thus q is the symbol of the term
12.23.31. This is very convenient; we can for instance, by
mere inspection, see that the symbol of any partial set in the
set of 15120 terms, contains as part of itself one of the
symbols ||, ||, viz. that there are among the 8 bitangents
six such that their points of contact lie in a conic; whereas,
contrariwise in the symbols of the partial sets belonging to the
set of 5040, no one of these symbols contains as part of itself
either of the symbols |||, U.

HU

234 THE BITANGENTS.

To the foregoing may be joined the following two groups
of hexads of bitangents:
Representative term No. of terms
A /\ | 12.23.31.45.56.64 280
WY | | 12.34.35.36.37.38 63
Y V | 12.138.14.56.57.58 560

<> | 12.23.34.45.56.61 1680

[1 | 12.23.31.14.45.51 140
5040
VV | | 12 34 35 36 67.68 2520

These 1008 and 5040 hexads have been studied by Hesse
and Steiner as bitangents whose twelve points of contact lie
on a proper cubic, the former set having no six contacts
on a conic, but the twelve points of contact in the latter
case being divisible into two sets of six lying each on a
conic. It may be added, that the six tangents of each of
the 1008 hexads all touch the same conic, as will appear
from Aronhold’s investigations, which will be presently given.
The six tangents of each of the 5040 hexads may be dis-
tributed into three pairs, whose points of intersection lie on
a right line (see Art. 258).

263. We conclude this discussion of the bitangents with
an account of the method by which Aronhold has shewn
(see Berlin Manatsberichte, 1864, p. 499), that when seven
arbitrary lines are given, a quartic can be found having these
lines as bitangents, and of which the other bitangents can be
found by linear constructions. The method depends on pro-
perties of a system of curves of the third class having seven
common tangents, but it seems convenient to state them first
in the reciprocal form with which the reader is more familiar,
viz. as properties of a system of cubics passing through seven
given points. (1) Consider any one cubic of the system, then
if the eighth and ninth points in which it is intersected by any
other cubic of the system be joined, the joining line passes
through a fixed point on the assumed cubic, viz. the coresidual
of the seven given points (Art. 160). (2) Through any assumed
point 8 can be described one and but one cubic on which

THE BITANGENTS. 235

this point shall be the coresidual of the seven given points.
For all cubics of the system through the point 8 pass through
another fixed point 9, and, by definition, the coresidual is the
point where the line joining these points meets the curve again.
If, therefore, the coresidual is to coincide with the point 8,
the cubic must be that one which is determined by having the
line 89 as its tangent at the point 8. (3) Four cubics of the
system can be described to touch a given cubic of the system,
the points of contact being obviously the points of contact of
tangents drawn to the given cubic from the coresidual point
onit. (4) If the points 8,9 coincide, that is to say, if cubics
of the system touch, the envelope of the common tangent 89
is a curve of the fourth class. For consider how many such
lines can pass through any assumed point P. Suppose a cubic
described through P, and through the points 8, 9, then, by
definition, P is the coresidual point on that cubic, and by (2)
this cubic having P for the coresidual is a determinate known
cubic. We see then, from (3), that the envelope in question
is of the fourth class, the four tangents from any point P being
constructed by finding the cubic which has P for its coresidual,
and drawing the four tangents from Pto that cubic. (5) The
point P will be a point on the envelope curve, if two of the
tangents drawn from it coincide; but from the construction
just given, it appears that this can only happen when the
curve having P for its coresidual has a node; for in this case
two tangents coincide with the line joining P to the node.
Hence the envelope we are considering may also be defined as
the locus of the coresidual of the given system of points on all
the nodal cubics of the system. (6) If the cubic through the
seven points break up into a conic through five of them, and a
line joining the other two, it has two nodes, namely, the inter-
section of the line and conic. Any other cubic of the system
meets this complex cubic in two other points, one on the line,
one on the conic, and the coresidual is the point P where the
line joining these two meets the conic again. In this case,
then, P is a double point, the two tangents at it being the lines
joining it to the intersections of line and conic. Now seven
points can be divided in 21 different ways into a system of
two and of five. The curve we are considering has, therefore,

236 THE BITANGENTS.

21 double points, one on each of the 21 conics determined
by any five of the given points. (7) In addition, the seven
given points themselves are double points on the same curve.
For a cubic can be described through six of the given points
and having the remaining point for a double point, and it is
easy to see that the double point is the coresidual for that
cubic. The four tangents from it to the cubic reduce to two
pairs of coincident tangents, namely, the tangents to the cubic
at the double point. The envelope curve, therefore, has 28
double points, 7 of them being the seven given points, and
the pair of tangents at each of these seven points being the
same as those of the cubic of the system having that point
for a double point.

264. Reciprocally, then, if we have a system of curves of
the third class touching seven given lines, and consider any
one curve of the system, the eighth and ninth tangents common
to it with any other curve of the system, intersect on a fixed
tangent of the selected curve, which may be called the core-
sidual, for that curve, of the seven given tangents. (2) Cor-
responding to any arbitrary line, there is a curve of the system
having that line as the coresidual for it of the given tangents,
(3) Any fixed curve of the system is touched by four others,
the points of contact being the points where the coresidual
tangent again meets the curve, which, being a general curve
of the third class, is of the sixth degree. (4) The locus of
points where two curves of the system touch is a curve of
the fourth degree, the points where any line meets that locus
being the four points where it meets the curve for which it is
a coresidual tangent. (5) If the curve of the third class have
a bitangent, the coresidual for that curve touches the locus,
the point of contact being the intersection of the coresidual with
the bitangent. (6) If the curve consists of a conic touching
five of the given tangents together with a point, the intersec-
tion of the other two tangents; the coresidual for that system
will then be a bitangent to the locus. There will be 21 such
bitangents. (7) In addition, the seven given lines themselves
are bitangents, the points of contact being the same as those
in which any of them is touched by the curve of the third

THE BITANGENTS. 237

elass having that line for a bitangent and the six other given
lines as ordinary tangents.*

265. We can now, as has been stated, from the seven
given bitangents find the rest by linear constructions. We
have in fact to construct the coresidual tangents for the several
systems 12345, 67, &c., where 12345 denotes the conic touching the
first 5 lines, and 67 is the point of intersection of the other two.
Now the two systems 12345, 67 and 12346, 57 have obviously
seven common tangents, and the remaining common tangents
are the tangents to 12345 from the point 57, and to 12346 from
67. But Brianchon’s theorem enables us, when one point on a
tangent to a conic is given, to find by linear constructions
the remaining tangent. ‘These two tangents, then, having
been constructed, and their intersection found, the remaining
tangents drawn from it to each of the two conics in ques-
tion will be the two required coresiduals, and therefore two
of the bitangents. Or otherwise, if we consider the three
systems 12345, 67; 12346, 57; 12347, 56, and determine
in the manner just described the remaining eighth and ninth
tangent common to each pair of systems, the three intersec-
tions of these pairs of tangents will, when joined, give three
of the required bitangents. ‘The bitangent which is the core-
sidual for the system 12345, 67 may be called the bitangent
(67); and thus the twenty-one bitangents may be denoted by
combinations of the symbols 1, 2, 3, 4,5, 6,7. In addition we
have the seven given lines; and if introducing for symmetry
a new symbol 8, we denote these (18), (28), (38), (48), (58), (68),
(78), we are led by Aronhold’s method to an algorithm identical
with that of Hesse.

266. The intersection of the eighth and ninth tangents
common to any two curves of the system is a point through

* The point of contact of each of the seven given lines with the locus being thus
given, we have fourteen points on the quartic, which is thus completely determined,
and there is but one quartic satisfying the prescribed conditions. There may, however,
be several quartics having the seven given lines as bitangents; but the one deter-
mined by Aronhold’s method has them as unrelated bitangents, viz. such that no
- three of them belong to the same group.

238 THE BITANGENTS.

which passes the coresidual tangent for each of these curves.
Consider, then, the complex cubic systems 12, 34567 5; 34, 12567,
and one of the common tangents is the line joining the points
12, 34; that is to say, in the algorithm just referred to, the line
joining the intersections of the lines (18), (28); (38), (48); and
we now see that this line passes through the intersection of the
coresiduals of the two systems under consideration, that is to say,
through the point (12), (34). In this way we get the theorem
already proved (Art. 258), that the intersections of the lines
(18), (28); (88), (48); (12), (84), are in a right line; and
Art. 262 shows that by an ordinary or bifid substitution we
can find 5040 lines possessing the same property.

267. We conclude with Aronhold’s algebraic investigation
of the equation of the quartic generated according to his method.
Let us use tangential coordinates a, 8, y; and let u, v, w be
any linear functions of them, aa + 68 + cy, &c., then the equations

Bv—yu=0, yo- aw=0, au— Bv=0,

denote three conics having four tangents common, and of which
each touches one of the sides of the triangle of reference. And

a(Bv—yw)=0, B(yw—au)=0, y (au — Bv)=0,
denote three curves of the third class having seven common.
tangents, viz. the four common to the two conics, and the sides

of the triangle of reference. Any other cubic having the same
7 common tangents will be of the form
w/a (Bo — 0) + vB (yo — au) + wy (a — Bo) =0,

where uw’, v', w’ are arbitrary constants, which are supposed
to be of the form aa’ + 68’ + cy’, &c., where a’, 6’, 7 are the
coordinates of an arbitrary line. Writing the above equation
in the form

u, u, By

v, 8. ga |= 0,

w, w, a8
it is evidently satisfied by the coordinates a’6’y’, which therefore
are those of a tangent to this curve. And further, this tangent
is the coresidual for that curve ; for we shall find the other two
tangents through any point in that line, by substituting in the

THE BITANGENTS. 239

above Aa’+ wa” for a, &c. The equation then is divisible by yp,
and after division becomes

7 U ” Be 4 ’ ” Fe "oy? ’ ” UPL
u,u, PRY u,w, By +B U,U;

2 Ud ” ee A U ” CFR g noe 2 , ” if 208 a
WV, V, Ye |+AMI YL, VY, ya +yYa tw | Y, Vv, ya |=0,
’ ” Uj Ul U ” ‘gol wor ! ” "won

w,w,ap w,w',aB"+a'B W,W,a

and the symmetry pf the equation shows that the pairs of
tangents are the same which can be drawn from the intersection
of the lines a’8’y’, a’ By” to the curves

, 4,
u, wy By U, uy By
/, A
V, UV, ya | =0, v,v, ya |=0.
, 4?
w,w,ap w, w’, a8

Thus then the tangents a’§’y’, «By being respectively
the third tangents drawn to each curve from the intersection
of the eighth and ninth tangents common to both, are, by
definition, the coresidual tangents. The two curves will
touch provided that the quadratic equation in A, yp, has
equal roots; or if we write the coefficients of that quad-
ratic P, Q, R, provided we have @’=4PR. If we denote by

1s ee PRS 4 6 Ped

X, Y, Z the minor determinants vw’ —v'w’, wu’ —w'v,

uv’ —u''v’, we have

y om ty X 4. fa Y+ a’ BZ,
Q a (B’ry” 4 B’y) yu (y0” + ofa) yx (a B” 4 a’ B’) Z,
R ou Bie” sy + oy” of” 4 *. a” B” VA

,

Now for B’y” — B’9/, oa” — ya’, a RB” — a” — a8’ we may write
X,Y, 2, these being the point-coordinates of the point of inter-
section of the two lines a 8’9/, a’B’y". The equation Q@’=4PR
is then equivalent to

ve X? + y? VY? + 2°Z" —2yz2VYZ—22xZX —2ayXY=0,
or V(e2X) + V7 (y¥)+ 7 (2Z) =0.

It will be remembered that X stands for v’w” —v’w’, and if we
put for these their values

v souk ao of bp’ ." cy’, w sais aa + bp’ +} ey’,
a’ sine a a’ = b’B” + cy”, w’ oni ao” be 6” B” + oy;
we have X=(b'c” —b"c') w+ (ca” — ca’) y+ (wb” —a’S’) 2.

240 BINODAL AND BICIRCULAR QUARTICS.

Similarly Y= (0c — be’) x + (c’a— ca’) y+ (ab —ab") 2,
Z=(be —U'c)x+(ca —ca)y+ (ab —ab )z.
Thus X, Y, Z represent known lines. They are in fact the
sides of the triangle whose vertices are represented by w, v, w.
It will be observed that the coefficients in X, Y, Z are the
constituents of the determinant reciprocal to that formed by
the coefficients of u, v, w; so that if X, Y, Z had been originally
given, wu, v, w would be found by similar formule.

268. The same investigation would hold if the equations
of the three conics had been law=mBv=nyw. The values
of X, Y, Z would remain as before, but we should have

P=mnf'y X + nly'o¢ Y + lmd’ BZ, &e.,
and the equation would be

V (mnxX) + (nly Y) + / (lmzZ) =0.
This is the most general equation of a quartic having three
given pairs of lines x, X, &c., as pairs of bitangents of the same
group. If we were given a seventh bitangent, then /, m, n
would be completely determined by the equations supposed
to be satisfied by the coordinates of that bitangent, viz.,
la’u’ =m’ =ny'w’, whence mn, nl, ln are respectively pro-
portional to au’, Pv’, yw’. Thus, then, if we are required

Se ae ee Tee eT eee eT eT ct ee

to describe a quartic having seven given. lines as bitangents,

besides the one quartic determined (Art. 265) on the supposition
that no two of the tangents belong to the same group, we

can describe (7 x 15=) 105 others according to the method of

this article, by leaving out any one of the seven and dividing
the six remaining into three pairs, which can be done in fifteen
_ different ways.

BINODAL AND BICIRCULAR QUARTICS.

269. Except in connection with the bitangents, the theory
of non-singular quartics has been little studied, and what else
we have to state on this subject will be given in the concluding
section of this chapter, that on the Invariants and Covariants.
In order to complete the theory of the bitangents, we ought
to consider the modifications which that theory receives when

BINODAL AND BICIRCULAR QUARTICS. 241

the curve has one or more double points. ‘The case, however,
where the quartic has but one node has received no attention,
and will not be here discussed. Quartics with two nodes, in
the case where these are the circular points at infinity, have
been extensively studied under the name of bicircular quartics,*
and some of the principal results obtained will be here given.
All the projective properties obtained for bicircular quartics may
of course be stated and proved as properties of binodal quartics,
but we shall find it convenient to give several of them in their
original form, as the reader will have no difficulty in making
the proper generalization. Quartics having the two circular
points as cusps have also been much studied under the name of
Cartesians,t the properties of which may similarly be gene-
ralized and stated as properties of bicuspidal quartics. If a
quartic have one of the circular points as a cusp and the other
as a node, it cannot be real; consequently this case has been
little studied, and therefore we have little to state as to the
properties of quartics having one node and one cusp.

270. From each of the two nodes of a binodal quartic may
be drawn four tangents to the curve (Art. 79), and we shall
now prove that the anharmonic ratios of these two pencils are
equal. The general equation of a quartic having for nodes
the intersections of the line 2 with the lines a and y is

y+ Qayz (lee + my) + 2° (ax*+ by’+ cz*+ 2fyz + 2gza + 2hxy) = 0.
The pairs of tangents at the nodes are given by the equations
xv’ +2mxz+b2"=0, y’ + 2lyz + az" =0,

and we lose nothing in generality by supposing / and m to be
both =0, which is equivalent to assuming that for the lines a
and y have been taken the harmonic conjugate, with respect to
the pair of tangents at each node, of the line z which joins the
nodes. Arranging now the equation of the quartic

y” (x? + bz*) + Qy2? (fz + ha) + 2° (ax* + 2gzu + cz”) =0,

* See, in particular, Dr. Casey’s paper, Transactions of the Royal Irish Academy,
vol, XXIV. p. 457, 1869.

t See Chasles’ Apercu Historique, p. 350; Quetelet, Nowveaue Mémoires de
Bruxelles, tom, v.; Cayley, Liouville, vol. XV. p, 354,

bE

242 BINODAL AND BICIRCULAR QUARTICS.

we see immediately that the four tangents from the node zz are
given by the equation

(x” + bz”) (ax* + 2gzu + cz”) =z’ (fz + hz)’,
or ax*+ 2gx°2 + (c+ ab— h’) x’z"+ 2 (bg —hf) va + (be — f*) *=0.
The invariants of this quartic are
I= abe —af? — bg’ + foh + py (e+. ab —h’)’,
6J = (abe — af* — bg’ — 4 fgh) (c+ ab — h*) — 8h? (af* + bq’)
+ 3abfgh + 3%9° — 35 (ec +ab— hb’).
Now these values are symmetrical between a and 6, f and g, and
we see therefore that they are the same as the invariants of the

quartic which corresponds to the pencil of tangents from the
node yz, and that therefore the two pencils are homographic.

271. It follows at once, as in Art. 168, that a conic can be
drawn passing through the two nodes, and through the four
points where each of the tangents from one node meets the
corresponding tangent from the other; and further, since there
are four orders in which the legs of the second pencil can be
taken without altering the anharmonic ratio, that the sixteen
points of intersection of the first set of tangents with the second
lie on four conics, each passing through the two nodes. When
the quartic is bicircular, that is to say, when the two nodes
are the circular points at infinity, the theorem becomes that the
sixteen foct of a bicircular quartic lie on four circles, four on each
circle.* It is to be noted that any one of the conics through
the two nodes may degenerate into a right line together with
the line joining the nodes, so that four of the foci of a bicir-
cular quartic may lie on a right line.

272. We have already stated that the equation of any
quartic may, in an infinity of ways, be thrown into the form
aU*+bV"*+cW* + 2fVW+ 2qgWU+ 2hUV=0,
where U, V, W represent three conics. If the quartic is non-
singular, the three conics cannot have a common point, since it

* In point of fact, this theorem, which is due to Dr. Hart, was first obtained, and
the theorem of Art. 270 thence inferred. The proof given in Art. 270 is in substance
the same as Professor Cayley’s. See his Memoir on Polyzomal Curves, Edinburgh
Trans., 1869.

BINODAL AND BICIRCULAR QUARTICS. 243

is obvious that any point common to U, V, W must be a double
point on the quartic whose equation we have written. In the
cease of binodal quartics, U, V, W may be taken as three conics
passing each through the two nodes, and when these nodes are
the circular points at infinity, U, V, W are three circles. We
lose nothing in generality by confining our attention to the
equation UW= V?, to which, as in the theory of conics, the
preceding equation may in a variety of ways be reduced. It
may, for instance, be written
(aU+gW+hV) = (hk? — ab) V* +2 (gh—af) VW + (g’- ac) W*,
where the right-hand side of the equation breaks up into factors.
Bicircular, therefore, and binodal quartics may be discussed
by considering the form UW= V’, and by regarding the quartic
as the envelope of 7U+2AV+W=0, where U, V, W are in
the former case circles, and in the latter case conics passing
through the two nodes; and it is only necessary to examine
how this limitation modifies the results already obtained,
Arts. 251, &c.

273. When three conics have two points common, their
Jacobian breaks up into the line joining them, together with a
conic passing through the two points; and when the three
conics are circles, the Jacobian conic is the circle which cuts
them at right angles (Conics, Art. 388, Ex. 3). The Jacobian
being a determinant, the Jacobian of three conics whose equations
are of the form aU+ 8BV+yW=0 is the same as that of U, V,
W; and when U, V, W are circles, all circles included in this
form have a common orthogonal circle.

If U, V, W are circles, the coordinates of whose centres
are 2L,Y,2,, LY%., ,Y,2,, the coordinates of the centre of
U+2XV + W will be proportional to

n'a, 5 2r2, + sy rv 1 ox 2rY, Ys) A a 22, 3 3 @s)
and the locus of the centre, as % varies, is evidently a conic.

Hence the quartic UW= V* may be regarded as the envelope

of a circle whose centre moves on a fixed conic* /, and which

* Dr. Casey has shown that the foci of this fixed conic are the same as the double
foci of the quartic. In fact, if a tangent from a point J meets the conic F in two
consecutive points P, P’, the line /P will be a common normal to the two circles whose
centres are P, P’, and which pass through J, If then J be one of the circular points at

244 BINODAL AND BICIRCULAR QUARTICS.

cuts a fixed circle J orthogonally. And in the more general
case of the binodal quartic, where U, V, W are conics through
the fixed points, UW — V* is the envelope of the variable conic
U+2XV+ W, passing through the fixed points; all the
variable conics having a common Jacobian conic, and the pole,
with regard to any, of the line joining the fixed points moving
on a fixed conic £.

274. The nature of the quartic will be modified if any
special relations exist between the conic # and the Jacobian.
Thus, if / touch the Jacobian, the point of contact will be an
additional node on the quartic, and if / touches the Jacobian
twice, then each point of contact will be a node; that is, the
quartic will break up into two conics, each passing through the
fixed points. So if # pass through one of the fixed points, that
point instead of being a node of the quartic will be a cusp, and if
F' pass through both of the points both will be cusps, and we
have a bicuspidal quartic. Thus, in the case of bicircular quartics,
if the conic F' which is the locus of centres be a circle, the quartic,
having the points at infinity as cusps, will be a Cartesian.

If the conic £ touch the line joining the points, that line
becomes part of the quartic. Thus, in the case of bicircular
quartics, if the conic / be a parabola, the quartic will degenerate
into a circular cubic, together with the line at infinity.

If the centres of U, V, W lie on aright line, the Jacobian
reduces to the line joining the centres.

275. Let us now return to the equation UW=V*. We
have seen that there are in general six values of A, for which
WU+2XV+W breaks up into factors, and that the right lines
represented by the several factors are bitangents to the quartic
UW=V*. Now when JU, V, W all pass through fixed points,
wU+2XV+W, which denotes a curve passing through the
same points, must, if it denote right lines, denote two lines
passing one through each of the points, or else the line joining the
points together with another line. In the former case the two

infinity, it follows that the tangents from J to F are normals, and therefore tangents
to the quartic at J, The same argument holds, whatever be the curve /, or whatever
the law according to which the circles are described. Thus, the single foci of any
curve are double foci of any parallel curve,

BINODAL AND BICIRCULAR QUARTICS. 245

lines are not proper bitangents to the quartic UW=V", but
ordinary tangents passing through a node (any line passing
through a node being improperly a tangent) ; in the latter case
one of the two lines is a proper bitangent, the other is the line
joining the nodes. Of the six values of A, only two correspond
to the case of proper bitangents; for if Z be the chord common
to U, V, W, then V and W will be of the forms respectively
aU+IM, bU+LIN; and ’U+2\AV+ W will have Z for a
factor if X be one of the roots of X*>+2Aa+5=0. Thus, in the
case of bicircular quartics, when U, V, W all represent circles,
there are evidently two values of X for which the coefficient of
x+y" vanishes in 7U+2XV+ W=0, and for each of these
values the equation denotes a right line bitangent to the quartic
UW=V*. Or we may see the same thing geometrically
from the construction in Art. 273. Ifthe circle WU+2’V+W
becomes a right line, its centre passes to infinity, and must there-
fore be the point at infinity on one of the two asymptotes of the
conic /’; and the two bitangents are therefore the perpendiculars
let fall from the centre of the Jacobian on these asymptotes.

In each of the four other cases where the discriminant of
MU+2XV+ W=0 vanishes, the equation denotes a pair of
tangents to the quartic, passing each through one of the circular
points at infinity, and whose intersection therefore is a focus of
the quartic; or, what comes to the same thing, A’ U+2A V+ W is
an infinitely small circle whose centre is the focus, and which
has double contact with the quartic. If one of two orthogonal
circles reduce to a point, that point must lie on the other circle;
hence if YU+2AV+ W reduce to a point, that point must be
on the Jacobian circle of U, V, W. We have, therefore, obvi-
ously four foci, viz. the intersections of this Jacobian circle with
the conic /, which is the locus of centres of circles included in
the equation 1°U+2XV+ W=0, and which may, therefore, be
called a focal conic.

_ The four points in which the Jacobian circle meets the quartic
will be points in which circles of the system °U+2XV+ W
meet the quartic in four consecutive points (Art. 251).

There are four ways in which the equation of a given
_bicircular quartic can be reduced to the form UW=V’*; cor-
responding to each there are four foci, two bitangents and four

246 BINODAL AND BICIRCULAR QUARTICS.

cyclic points, or points on the eurve where four consecutive
points lie on a circle (see Art. 114); the quartic having in all
16 foci, 8 bitangents, and 16 cyclic points.

276. If one of the foci of the quartic be taken as origin,
the equation of the quartic must be of the form (27+ y")W=V?,
where V and W represent circles; and the quartic is the
envelope of 2*+y°+2\V+W=0. Besides the value \=0,
there are three other values of A, for which this variable circle
reduces to a point; and one of these values must be real. We
can then write the equation

(a? + y”) (a? +y°+2AV4+NW)= (2? +y?+rAV),

or, in other words, when we have a focus we can at once bring
the equation of the quartic to the form AB=V”", where A and
B are point-circles. Bicircular quartics may be divided into
two classes, according as the other two values of A, for which
A+2XV+d’B reduces to a point-circle, are real or imaginary,
or, in other words, according as the four real foci do or do not
lie on acircle. In the former case let C denote one of the two
point-circles, and, as in Art. 257, eliminate V between the
equations AB=V*, A+2AV+NB=C, and we see that
the equation of the quartic may be written in the form
L/(A) +m /(B)+n /(C) =0, that is to say, that the quartic is
the locus of a point whose distances from three fixed points
are connected by the relation /p + mp’ + np” = 0.

The condition that 1/(A)+m/(B)+n/(C) shall be touched

2 2 2
by AA +p46+4+ v7 is (Conics, Art. 130) : + oe + —= 03 and
when A, B, C are point-circles, and a, 0, ¢ the lengths of
the lines joining the points, it is easy to verify that the dis-
2 2 2

criminant of AA +pBb+vC vanishes if ~ + + =0. Tis
two equations just given determine A, pm, v, and therefore the
fourth focus.

We have seen (Conics, Art. 94) that if A, B, C, D be four point-
circles, we have identically bed. A+ cda.B+dab.C0+ abe. D=0,
where abc is the area of the triangle whose vertices are a, 0, c, &e.
Hence, i, mu, v are proportional to the areas of the triangles formed |
by the fourth focus and each pair of the other three foci. In the

BINODAL AND BICIRCULAR QUARTICS. 247

case where the three points a, 0, c are in a right line, it can
easily be proved that the squares of the distances from any point
of four points on a right line are connected by the equation
A B C D
ab.ac.ad ba.bc.bd ca.ch.ed* da.db. de
Hence we see that the reciprocals of 2, u, v are proportional
to ab.ac.ad, ba.be.bd, ca.ch.cd, and that we have the equation
Pab.ac.ad+m’ba. be.bd+n'’ca.ch.cd=0.
If we had Cab.ac+m’*ba.be + n’ca.cd=0,

the fourth focus would be at infinity, and the curve would be a
Cartesian.

= 0.

277. When we are given four concyclic foct of a bicircular
quartic, two such quartics can be described through any point, and
these cut each other at right angles. If we are given the fourth
focus, we are given the values of A, uw, v, for which AA+pB+vC
reduces to a point; and evidently two systems of values of

2 2 2

l,m, n can be found to satisfy the equations “Gs my + ~ = 0,

lp + mp’ + np” =0, where p, p’, p” or (A), /(B), V(C) denote
the distances from the three foci of a point on the curve sup-
posed to be given.
T'wo quartics
1 /(A)+m /(B) +n (2) =0, U f(A) +m! V(B)+n' /(C)=0
will be confocal if
a * (mn it Sa Mm” n°) + b? (n ‘d oe nT’) - Cc (lm? — 1”m*) =(0,

as appears immediately on eliminating A, w, v from the three
equations

A... Paap iv Vv a % tee
In order next to find the condition that the quartics should
cut at right angles, we first premise, and the reader can verify
without difficulty, that if A, B, C be point-circles, and a, 0, c have
the same meaning as before, the condition that AA + wB+yrC,
VA+p’B+yvC should cut each other at right angles is

a {py + pv) +0? (VN +d) + 0 (Ap +p) = 0.

248 BINODAL AND BICIRCULAR QUARTICS.

We observe further that, as at Conics, Art. 130, the quartic
L/(A)+m /(B)+n/(C) will be touched at any point for
which the values of ./(A), /(B), /(C) are p, p’, p”, by the circle
A + 7B + Fa C=0. The condition that this circle should cut

orthogonally the tangent circle to U’ /(A) +m’ /(B)+n' (C) is

7” mn’ + m'n +p nl +n/1 Le lm’ + Um ae
pp” p p pp’

But, solving between the two equations
Ip + mp’ “ np” ah 0, Up we mp’ es n'p” Nie 0,
we find p, p’, p” respectively proportional to mn’ — m’n, nl’ — nl,
ln’—Um. Substituting in the preceding equation, we find that
the condition that the quartics should be mutually orthogonal is
a’ (m*n® — mn’) +B (nl? — n0) + 2 (Pm — Tm) = 0,

the same as the condition already found that the quartics should
be confocal; and the theorem stated is therefore proved. It
does not appear to be necessary to the validity of this proof
that C should be real, and hence the theorem is true that con-
focal quartics cut at right angles, even though the four real
foci should not lie in a circle.

278. The theorem of Art. 277 was originally obtained from
geometrical considerations by Dr. Hart for the case of the
circular cubic. If we seek the locus of a point whose dis-
tances from three fixed points are connected by the relation
lp + mp’ + np” = 0, the coefficient of (a*+ y°)* will be found to be

(C+ m+n) (m+n—l) (n+l—m) (l+m—n).
Consequently, the locus, which is ordinarily a bicircular quartic,
reduces to a circular cubic if /imin=0, and the theorems
already here proved are true for circular cubics, which have also
sixteen foci lying in general in four circles. Dr. Hart’s proof,
which was given at length in the first edition, shews that if
O, P, Q be the centres of the quadrangle formed by the four foci
A, B, C, D, the cubic must pass through these points, the tan-
gents at any of these points O being one of the bisectors of
the angle made by the intersecting lines AC, BD, and being
parallel to the real asymptote of the cubic; and that the cubic

BINODAL AND BICIRCULAR QUARTICS. 249

also passes through & the centre of the focal circle, the tangent
at / being parallel to
the same asymptote.*
Since then O, P, Q, R Q
are points of contact
of tangents from the
same point of the
curve, the point where

OP meets QR (or the C
foot of the perpendi- S B

cular from O on QZ)

is also a point on the

curve (Art. 150), and

similarly the points *

0
\
Vv

where OQ meets PR,

and OR, PQ; and it

can be shewn that the —

tangents at each of

these points to the

two cubics which pass

through them cut at right angles. Thus the seven points common
to the two cubics having A, B, C,D for their foci, are determined
by simple constructions, and we may arrive by projection at
theorems, some of which have been already stated ; for instance
(see Art. 152), if corresponding tangents, taken in any order,
from two points J, J mutually intersect in points A, B, C, D,
the centres of the quadrangle formed by these points will be
also points en the cubic, having for a common tangential point
the point where JJ meets the curve again; and the point of
contact of the fourth tangent from this point will be the pole of
LJ with respect to the conic through the points A, B, C, D, J, J.

279. The method by which Dr. Hart proved these theorems
was by shewing that when the foci are given, the relations
established Art. 276, combined with the condition 7+n=m,
suffice to determine J, m, n, and that actually, denoting the

* Thus the centres of the four focal circles of a circular cubic are the points of
‘contact of tangents parallel to the real asymptote,

KK

250 BINODAL AND BICIRCULAR QUARTICS.

distances of O from the four foci by a, 5, c, d, the curve must
either have the property
(b+c)p+(a—8) p"=+(ate) p’, or (c—b) pt (a+b) p’=4 (ate) p’.
Each coefficient is given a double sign, because, when the equa-
tion Jp +mp’+np”=0 is cleared of radicals, it only contains
the squares of /, m,n. The two equations answer to two dif-
ferent cubics having the given points as foci; the different signs
answer to different branches of the same cubic. The upper
signs belong to a branch extending to infinity; for then the
equation is satisfied by the values p=p’=p”, which are true
for an infinitely distant point. The centre of the focal circle
obviously lies on this branch. The lower signs belong to an
oval, the equations then not being satisfied by p=p’= p.
The equations being satisfied by the values a, 6, ¢ for Pp; 2 rBs
we see that O is a point on the cubic.
In like manner we have the relations
(c—d)pt (a+d)p"=+ (at+c)p™ or (c+d) pt(a—d)p” = (ate) p”,
whence, combining the ce ia
p42 pee
ate b+a?
or the two cubics make up the locus of the intersection of two
similar conics whose foci are respectively A and C, B and D.
The similar conics which intersect at O have evidently as a
common tangent one of the bisectors of the angles at O;
these therefore are, as has been stated, the tangents to the
two cubics which constitute the locus, and which therefore cut
at right angles.

‘ih

dA

280. Bicuspidal quartics may be considered as a limiting case
of binodal quartics. In the case where the two cusps are the
circular points J, J at infinity, the curve is called a Cartesian.
Des Cartes studied this curve (thence known as the oval of
Des Cartes), as the locus of a point O, whose distances from
two fixed points A, B are connected by the relation /p + mp’ =c.
Chasles shewed, and it can be verified without difficulty, that
whenever this relation holds good, a third point C can be
found on the line AB, whose distance from O satisfies a
relation of the form /pinp”’=c’; in other words, that the

BINODAL AND BICIRCULAR QUARTICS. 251

oval possesses, besides the two foci considered by Des Cartes, a
third possessing the same property. We use the word Cartesian
here in a somewhat wider sense. We shall shew that when
a quartic has the two points J, J for cusps, it has three foci
lying on a right line. When these foci are real, the curve
is the same as that studied by Des Cartes; when two are
imaginary we still call the curve a Cartesian, though Des Cartes’
mode of generation is no longer applicable.

The equation of the Cartesian may generally be brought
to the form S*=°L, where S represents a circle and ZL a right
line, & being a constant (or, what is the same thing, /=0
being the right line at infinity), from which form it is evident
that the intersections of S and & are cusps, the cuspidal
tangents meeting in the centre of S, which is therefore the
triple focus of the Cartesian, while Z is evidently a bitangent
of the curve.* The curve is then obviously the envelope
of the variable circle WAL+2AS+h'=0, the centre of
which obviously moves along a right line perpendicular to
ZL; and equating the discriminant to zero, there are easily
seen to be three values of A, for which the circle reduces
to a point, and therefore three foci. rom the theory already
given, if A, B, CO be any three of the variable circles,
the equation of the envelope may be written in the form
1 /(A)+m /(B)+n/(C)=0; and therefore we have the property
lp + mp’ + np” =0, where p, p’, p” denote the distances from the
three foci; or, again, since &* is a circle of the system
answering to the value \=0, we have /p+mp’=nk.

A Cartesian may also be generated as the locus of the
vertex of a triangle, whose base angles move on two fixed
circles, while the two sides pass through the centres of the
circles, and the base passes through a fixed point on the line
joining them.

If any chord meet a Cartesian in four points, the sum of their
distances from any focus is constant; for the polar equation,
the focus being pole, is easily seen to be of the form

p' —2 (a+b cosw) p+c'=0,

* This equation has been studied by Prof. Cayley under the form
(a? + y? — a”)? + 164A (x — m) = 0.

252. BINODAL AND BICIRCULAR QUARTICS.

and if we eliminate w between this and the equation of an’
arbitrary line, we get for p a biquadratic of which — 4a is the
coefficient of the second term.

When, in the preceding c=0, the equation becomes
p=a+6 cosa, and in addition to the two cusps J, J, the curve
has the origin for a node. It is then called Pascal’s /émagon,
and may evidently be generated by taking a constant length on
the radii vectores to a circle from a point on it. If, further,
a= b,the curve becomes tricuspidal, and is called the cardiorde,
a curve generated by adding or subtracting a portion equal to
the diameter, on the radii vectores to a circle from a point on it.
The equation may be written in the form p? = m? cos 4a.

281. The focal properties we have been discussing may
be investigated by the method of inversion (Art. 122). It
is easy to shew, that to a focus of any curve corresponds
a focus of the inverse curve, and that the origin or centre
of inversion will be a focus if the points J, J at infinity
are cusps. Thus, for the Cartesian which has three col-
linear foci, the inverse with regard to any point is a bi-
circular quartic having three foci on a circle passing through
the origin, which is also a focus. In inverting, if O be the
origin, A, B any two points, a, > the inverse points, then for

the distance AB we are to substitute To any relation

ab
Oa.Ob*
then of the form AAP+ »LP=c will correspond one of the form
Napt p’bp=c Op, and thus by considering the bicircular quartic
as the inverse of a Cartesian we arrive at the fundamental property
of bicireular quartics; and, in like manner, from any relation of
the form AAP+ wbBP+vCP=0 may be deduced a relation
Nap+pbp+vcp=0. The inverse of a bicircular quartic from
any point on the curve is a circular cubic which, therefore,
possesses the same focal properties. A circular cubic or bi-
circular quartic is its own inverse with respect to any of the
points O, P, Q, & (p. 249). ‘The angle at which two curves cut
is not altered by inversion, and therefore the theorem as to
confocal curves cutting at right angles, if proved for cubics, is
proved also for quartics. The inverse of a conic is a bicircular
quartic having the origin for an additional node, and from

BINODAL AND BICIRCULAR QUARTICS. 253

the focal property of conics may be inferred that such quartics
have the property

where a and 3b are two foci and O the node. In like manner,
by inverting the focus and directrix property of conics, we
arrive at another method, given by Dr. Hart, for generating
this kind of quartic. If the radius vector from a fixed point
C to P meet a fixed circle passing through C in #, and if
A be another fixed point, the quartic is the locus of the point
P, for which PA = PE.

282. There exists for the binodal quartic® a theory of the
inscription of polygons, analogous to Poncelet’s theory in
regard to conics. Let A, B be the nodes: starting from a point
P of the curve, if we join this with A, the line AP meets the
curve in one other point, say Q; joining this with JB, the line
BQ meets the curve in one other point, say #; joining this
again with A, the line AR meets the curve in one other point,
say S; and so on. We have thus, in general, an unclosed
polygon PQRS..., of which the alternate sides PQ, RS, ...
pass through A, and the other alternate sides YL, ... pass
through B. For a binodal quartic taken at random, it is not
possible to find the point P, such that there shall be a closed
polygon of a given even number of sides; for instance, a
quadrilateral PQRSP, of which the sides PQ, LS pass through
A and the sides QR, SP pass through B. But the quartic
may be such that there exists a polygon of the kind in question
(as regards the quadrilateral this is obviously the case, since
considering a quadrilateral PQASP drawn at pleasure and
taking A for the intersection of PQ, &S, and B for that of
Qh, SP, we can describe a quartic passing through the points
P, Q, &, 8S, and having the points A, B for nodes), and when
this is so, that is, when there is one polygon, there are an
infinity of polygons; viz. any point P whatever of the curve may
be taken as the first summit, and the polygon, constructed as
above, will close of itself.

* Steiner, Geometrische Lehrsatze, Credle, vol. XXXII. p. 186 (1846).

254 UNICURSAL QUARTICS.

UNICURSAL QUARTICS.

283. Taking the nodes to be at the angular points of the
triangle of reference, the equation of the curve must be of
the form |

ay?” + ba*x* + cx*y” + 2fx’y2 + 2gy"aau + Zhz*ay =0,
which may be written
2 2
a(~) +2(-) +e(-) Lae bag 2 + 2h “s == 0,
x y z Ye 20 xy
Thus we see that the quartic may be generated from a conic by
writing, in the equation of the latter, for each coordinate its
reciprocal; a process which may be called “inversion,” using
the word in a wider sense than that in which we have already
employed it. It is easy to express this transformation by a
geometrical construction. Let the coordinates be proportional
to the perpendicular distances from the sides of the triangle of
reference, and let P, P’ be two points, whose coordinates are
connected by the reciprocal relations
esyieaeV Sree IY 5 Liy ite myer sm: ays

then we have seen, Conics, Art. 55, that the lines joining P, P’ to
the vertices of the triangle make equal angles with the sides; or
otherwise, Conics, p. 263, that if P be one focus of a conic touch-
ing x, y, Z, then P’ will be the other focus. In general, in this
method to any position of P corresponds a single definite posi-
tion of P’. If, however, we have x =0, or P’ anywhere on
the line BC, we have y and z both =0, and P coincides with A ;
and reciprocally to A corresponds any point on BC. It is to be
remarked, however, that when a =0, the corresponding values
of y and z, being respectively 2’a’, xy’, though evanescent, have
to each other the definite ratio 2’: y’; and therefore to any
point P’ on BC corresponds a definite element of direction
through A. We have, in fact, P indefinitely near to A, but in
a given definite direction, viz. such that (as in the general case)
AP, AP’ make equal angles with the sides. If now P describe
any locus, the other point P’ will describe a corresponding
locus; thus if the locus described by P be the right line
ax+by+cz=0, that described by P’ will be the conic
ay 2 + bax’ + ca’y’=0, and vice versd (compare Tonics, Art. 297,

UNICURSAL QUARTICS. 255

Ex. 13); if a=0, that is to say, if the line pass through A, the conie
reduces to 2” (bz’+cy’)=0, and leaving out the line « or BC,
we may say that to the line dy + cz corresponds the line bz’ + cy’;
and, as already mentioned, if the one locus be any conic, the
other will be a trinodal quartic.

284. The correspondence of the conic and quartic may be
examined in detail; the conic meets each side of the triangle,
say BC in two points; corresponding hereto we have through
A two elements of direction, viz. these are the tangents of the
quartic at its node A. MHence, according as the conic meets
BC in two imaginary points, touches it, or meets it in two
real points, the quartic has at A an acnode, cusp, or crunode,
and the like for the other sides. ‘Thus, if the conic be an
ellipse or, say, a circle, situate wholly within the triangle, the
quartic is a triacnodal curve composed of a trigonoid figure
within the triangle and of the three vertices as acnodes (fig. 1) 5
if the ellipse is inscribed in the triangle, the quartic is tricus-
pidal (fig. 2); if the ellipse cuts each side in two real points,
then the quartic is tricrunodal; viz. if on each side the inter-
sections are internal we have the fig. 3, whereas if the inter-
sections are external we have the fig. 4. It is to be observed,

Fig. (1). Fig. (2).

256 UNICURSAL QUARTICS.

that in the transition from the one form to the other the
ellipse must pass successively through the vertices of the tri-
angle; and that when the ellipse passes through a vertex
the corresponding quartic breaks up into a right line and a
cubic; the transition cannot be made (as at first sight it would
appear it might) through a quartic having a triple point.

The complete discussion of the different forms would be
interesting and not difficult, but it would occupy a good deal
of space; it would be necessary (in the present case of plane
curves) to consider the conics which in each figure correspond
to the line at infinity of the other figure. For the like theory,
as regards spherical figures, there are no such conics, and the
theory is considerably simplified.

285. The foregoing mode of generation of the trinodal
quartic leads at once to various properties of the curve. It
is well known that if a conic cuts the sides BC, CA, AB of
a triangle, and from each vertex we draw lines to the inter-
sections on the opposite sides, these six lines touch a conic;
and it is easy to shew further, that if instead of the two lines
through each vertex we consider the two inverse lines, these
meet the oppsite sides in six points lying on a conic; and
consequently that the six inverse lines also touch a conic.
In fact, if the lines (w=ay, xw=a’y), (y=Bz, y= 2),
(2=yx, z=y'a) meet the sides «=0, y=0, 2=0 respec-
tively in six points lying on a conic, it is easily seen that
ad’ 83’yy =1, a relation which remains unaltered when a, ~, y,
a’, 8, y are changed into their reciprocals. Now, if a conic
is transformed into a binodal quartic, then by what precedes
the tangents at a node A of the quartic are the inverses of
the lines from A to the intersections of BC with the conic;
hence, the tangents at the nodes A, B, C, touch one and the same
conic; a theorem which may also be derived directly from
the equation of the quartic.

286. Similarly, if from the points A, B, C we draw tangents
to a conic, then it may be shewn that the six inverse lines are
also tangents to a conic. But transforming the conic into a
trinodal quartic, the tangents from A to the conic are trans-

UNICURSAL QUARTICS. Zoe

formed into the tangents from the node A to the quartic (for a
curve of class n, the number of tangents from a node is =n —4,
and therefore for a trinodal quartic it is =2); and we have thus
the theorem, that the siz tangents from the three nodes to the
quartic touch one and the same conic.

287. To the bitangents of the quartic correspond conics
through A, B, C, having double contact with the conic; and
to the stationary tangents of the quartic correspond conics
through A, B, C, having stationary contact with the conic.
It can be shewn, that the numbers of such conics are 4 and 6
respectively, agreeing with r=4, .=6. But the result as to
the bitangents can immediately be obtained from the equation
of the curve, which may be written in the form

{y@ Va) + 2a /(b) + xy V/(c)}"
= 2xya |{v(be) —f} « + {W (ca)— 9} y + {W/(ab) — A}z],
where the factor multiplying 2vyz evidently denotes a bitangent,
and by changing the signs of the radicals, we have in all four
bitangents. Write for a moment fe+gy+hz=s, x /(bc)=l,
yV(ca)=m, z/(ab)=n, and if ©=0 denote the equation of
the four bitangents, we have
© = (s—l—m-—n) (s—l+m+n)(s+l—m+n)(s+l+m—n)
= (s?— 7? — m* — n”)’ —4 (mn? + n°? + Pm? + 2lmns)
= (s?— 2? — m? — nn’ — 4abeU.
In other words, the equation of the curve may be written
{( fet gy + hz)’ — beu* — cay’ — abz’}*- © =0,

shewing that the eight points of contact of the bitangents lie on
a@ conic.

_ Ifthe four bitangents be denoted by ¢, u, v, w, the equation
of the quartic may be written

f+ui+vt+wi=0,
or (04+ w+ v'+ w— 2tu — 2tv —2tw — 2vw — 2Qwu — 2uv)*? = 64tuvw.
In this form it is evident that ¢, wu, v, w are bitangents whose
points of contact lie on a conic, and it can be verified without
much difficulty, that (¢—u, v—w), (t-v, u—w), (¢—w, u—v)
are nodes.
LL

258 UNICURSAL QUARTICS.

288. We have just shewn how in one way the equation
of the quartic can be reduced to the form UW=V”*; and
generally if w,w, and v denote any two tangents to the conic
and their chord of contact, since the equation of the conic can
be written in the form uw=v’", that of the quartic is thence
immediately given in the form UW=V”", where U, V, W are
linear functions of yz, zx, xy.

In connecting the trinodal quartic as above with a conic,
we have also verified that the curve is unicursal. Since the
coordinates x’, ¥’, 2’ of a point on the conic can be expressed
as quadratic functions of a parameter @, the coordinates 72’,
za’, xy’ of the corresponding point on the quartic are imme-
diately given as biquadratic functions of the same parameter.

The preceding theory of trinodal quartics extends to the
case when any or all of the singular points are cusps. If all
are cusps the equation of the curve is reducible to the form
xt+y%+z2%=0, and the tangents at the cusps are g=y=z, which
meet ina point; as we may also see by reciprocation, the re-
ciprocal being a cubic whose equation may be written in the form
a+ yi+zi=0. When the curve has two cusps and a node,
the line joining the two points of inflexion, the line joining
the two cusps, and the bitangent all pass through the same
point. The cases of the higher singularities, described Art. 243,
require to be separately treated.

289. The equation of a quartic having a tacnode, as given
Art. 244, is

ye + ba'y2 + cxy’z + dy’2 + ea* + fa°y + ga°y’ + hay’ + iy* =0.
Let it also have a node, and since, in Art. 244, it was only
assumed that the point xy was the tacnode and the line y the
tangent at it, we may take the point zz as the other node,

In order that this point should be a node we must have d, h,
and 7=0, and the equation becomes

(yz)* + ba*.ye + cay .y2 + ex + fa?.ay+gax°y’=0.

We have written the equation so as to exhibit that it is a
quadratic function of ay, a*, yz. Hence, if in the general
equation of a conic we write xy, x’, yz for x, y, 2 respectively,

i Se

UNICURSAL QUARTICS. 259

we shall have the equation of a quartic with node and tacnode.
It will be seen that the relations

wiiy $e ey to see
imply reciprocally w:y:2=ay : 2": 72,
so that we have a like theory to that which exists for a quartic
with three distingt nodes. The constants may be determined
so that the node shall become a cusp, or the tacnode a node-
cusp, or that both these changes should take place, and the
theory thus extends to quartics having two distinct singular
points, one of them a node or cusp, the other a tacnode or
node-cusp.

290. ‘The equation of a quartic having an oscnode has been
given, Art. 244, as

(yz — max’) + cay (yz — max") + dy’z + gau°y? + hay’ + ty* =0.

It is obviously a quadratic function of yz—mz*, xy, y*. Now
the relations

esy se aay sys ye—mx
will be found to imply

RBiyi sary iy iy” t+m2z",
so that there is for the present case a theory analogous to that
established for trinodal quartics. ‘The constants may be parti-
cularized, so that the oscnode becomes a tacnode-cusp, and the
theory thus extends to the case of quartics having a tacnode
cusp. In all these foregoing cases we have expressed the
coordinates xz, y, 2 of any point on the quartic, as quadratic
functions of a’, y’, 2’, a variable point on a conic; and since
the latter coordinates can be expressed as quadratic functions
of a parameter 0, the former coordinates are expressed as
quartic functions of the same parameter.

291. In the remaining case of a quartic curve having a
triple point (general or of any special form), the mode of
treatment used in the last articles is not applicable, but we can
otherwise immediately express the coordinates as rational func-
tions of a parameter. ‘aking the point zy as the triple point,
the equation of the curve is of the form zu,=u,, where u,, wu,

260 UNICURSAL QUARTICS.

are homogeneous functions of the third and fourth degrees
respectively in a, y. If we now substitute y=@x, we get
zO,=20,, where ©,, ©, denote cubic and quartic functions of
6; and we have 2, y, z respectively proportional to ©,, 00,, ,.

The method here employed is exactly that suggested in
Art. 44. A variable line y = 6% drawn through the triple point
meets the curve in but one other point, the coordinates of which
are therefore rationally expressible in terms of @. And we should
be led to substantially the same results if we employed the
same method in the cases previously considered; for example,
if in the case of a trinodal quartic we determine each point
of the quartic as the intersection of the curve with a variable
conic passing through the three nodes, and through another
fixed point on the curve.

The special case of a quartic with a triple point z°y=z* may
be particularly noticed, as it can be treated by exactly the same
method as was used (Art. 212). The curve has, beside the
triple point, no singular point but a point of undulation, and
its reciprocal is a curve of like nature.

291 (a). Unicursal quartics may also be treated by the
method of Art. 216 (a). We may express the coordinates

czar +4dr pw +6crd7u? +4drpy* + ep",

y=ar* +40 rp + 60 pw? + 4d’ rp? + eps,

z2=a r+ 4b"r'p t+ 60'Ap? + 4d” rp? + eu’,
and can (Art. 44) write down the equation of the corresponding
quartic. ‘The equation determining the parameters of the points
of inflexion, and the relation between the parameters of three
points which lie in a right line, may be found as in the articles
referred to, or else as follows. Substituting the above written
values for the coordinates in lx +my+nz=0, we get a quartic

determining the parameters of the points in which that line
meets the curve.* ‘The theory of equations then enables us

* It is evident that by forming the discriminant of that quartic we get the
equation of the reciprocal, or tangential equation, in the form S*= T?,

UNICURSAL QUARTICS. 261

to write down

FA BF RIE

=f fh fe

— 4 (1b + mb! + nb”) = ry pl we pwr wl ye we ppl WO”,
6 (le+- me’ + nc”) =r ww ww rN” A pt yp VD”

3 ON we BOW,

ay (1 d +m d’ +n d”) is [Ps ng taly head ue hak i“ > ¥4 Pad Noss re Aw!”

la + ma’ + na”

le +-me +ne&? =ANA’N”,

From these equations, if we linearly eliminate 7, m,n, ”, w’”,
we get the relation connecting the parameters of three points on
a right line, viz.

, 4?
ee eres nee

Dh: = A BA

Gey Gey. Ga. CO, B
dd ad, —42", D, O
@, €4 €., ,D)=0,

where we have written
vps wp nw”, Fe ry’ pw” 4 x’ ran 4 i” be’,
C= pnd” if Wrn’'r at WN, D bh » he t42

If we make X: w=’: w’ =X": w”, we find that the para-
meters of the points of inflexion are determined by

a, @, a, pw,
Wap S ay Pee, Baa)
6c, 6c, 6c’, 3ur’, 3wr
oe Sas. ae ke, Oa
oe ak POE Cae |=0.

The first determinant expanded may be written

24 (ab’c’”) D® + 16 (ab’d’”) CD + 4 (ab’e”) (C* — BD)
+ 24 (ac’d”) BD + 6 (ac’e”) (BC -— AD) +96 (be'd”) AD
+4 (ad’e’”) (B*— AC) + 24 (dc'e’”) AC + 16 (bd'e”) AB

+24 (cd’e”) A? =0;

262 UNICURSAL QUARTICS.

and the second determinant expanded and divided by 24 gives,
for determining the inflexions, the sextic

(ab’c’) N° + 2 (ab’d”) Nw + {(ab’e’”) + 3 (ac’d”)} M*p?
re {2 (ac’e’”) es 4 (be'd’”)} Mu* + {(a d’ e”) “i 3 (bc , et Vu"
+2 (bd’e”) Ap? + (ed’e”) uw? = 0.

If in the preceding relation two of the parameters be made
equal, we get the relation connecting the parameter of any point
A with that of one of the points B where the tangent at A
meets the curve again, viz. writing for D, C, B, A respectively
Wr), 2ZApwr’ + Aw’, WA’ + 2m’, ww’, we have

N? [24 (ab’o”) Mt + 32 (ab’d”) Mu + {12 (ab’e”) + 24 (ac’d”)} Nu

+12 (ac’e’”) Ap? + 4 (ad’e”) p*]
+ 2n'p’ [8 (ab’d”) 0
+ {4 (ab’e”’) + 24 (ac’d”)} Mw + {12 (ac’e”) + 48 (be'd”\} Vu?
+ {4’ ad’é 4? |. 24 (bce a); AL + 8 (bd’e’”’) p'} ,
+ wh (able)
+ 12 (ac’e”) wt (12ad’e* + 24 (de'e’”)} Nu” + 32 (bd’e’”) r
+24 (cd’e’”) p*} =0,
from which equation we can determine the parameters, either of
the two points B answering to any point on the curve A, or of
the 4 points A answering to any point B. If we form the
condition that the equation in X’: w’ should have equal roots,
we get an octavic in X: mw, determining the: parameters of the
8 points of contact of the 4 bitangents of the quartic.

When it has been proved that it is possible to find four
linear functions ¢, wu, v, w of x, y, 2, which expressed in terms of
A, # are perfect squares, it is evident by extraction of roots and
linear elimination of X*, Aw, mw’, that the equation of the curve
ean be written in the form Aé? + But + Cv? + Dwt =

291 (4). Conditions to be satisfied by the parameters of a
node are obtained as in Art. 216 (c), from the consideration that
the relation connecting the parameters of three collinear points
must be satistied when two of these parameters correspond to the
same node, and the third to any point whatever on the curve,
Write pp” =a, Nw" +p’ =8, NA” =¥, then we have 4 =ya,

INVARIANTS AND COVARIANTS OF QUARTICS.

263

B=ra+ yu, C=rAB+ py, D=ry. Substituting these values in
the determinant of the last article, and equating separately to
zero the coefficients of X”, Au, w” we have the three conditions

6c,

€ 3;

edb, 4 Be

,
6c ,

— 4d, —4d', ~ 4d’,

i,
4

A
6c, 7,4

B
ef, y iO.

mee fa a oe ela var
— 4b, — 4b’, — 4b”, B, a — 4b, — 4b’, —4b", «
om, Ge, 60,4; 8 Go, GG) Gare aoe
Ad, —4d,—-4d% 4d, —4¢, — 4d, ¥, B
me eye, " ey aay eh lO
Gia Wy ees ot

Conditions which expanded are
24 (ab’c’’) y+ 16 (ab’d’) By + 4 (ab’e”) (8B? — wy) + 24 (ac’d”) ay
+6 (ac’e”) a8 + 4 (ad’e’”’) a’ =0,
4 (abe) y? + 6 (ac’e’”) By +4 (ad’e”) (B® — ay) +24 (dc'e”) ay
+16 (bd'e’’) a8 + 24 (cd’e”) a’ =0,
16 (ab'd") y’ + 4 (ab’e") By + 24 (ac'd") By + 6 (ac'e”) B?

+ 96 (be'd") ay + 4 (ad'e") a8 + 24 (bc'e"’) a8 + 16 (bd'e") a? = 0.
With these equations we combine the three obtained by mul-
tiplying the equation Wa—prAB+p'y=0 by a, 8, y respec-
tively, and linearly eliminating a’, 8’, y’, By, yx, a8 we get a
sextic for determining the parameters of the three nodes.

There is no difficulty in analysing, as in Art. 216(d), the
different cases where the sextic of the last article can have equal
roots, and so arriving at the different special cases of unicursal
quartics already enunciated.

INVARIANTS AND COVARIANTS OF QUARTICS.

292. When we have occasion to write the equation of a
quartic at length, we shall write it

aa* + by* + cz* + 6fy'2" + 6g2"x" + Cha*y’
+ 12la*yz + 12my*ex + 12nz*xy
+ 4a,a°y + 4a,0°2 + 4b,y°a + 4b,y°z + 40,2°@ + 40,2°y = 0.

264 INVARIANTS AND COVARIANTS OF QUARTICS.

The concomitant of lowest order in the coefficients is the con-
travariant (Art. 92) of the second order in the coefficients,
whose symbolical expression is (a412)*, and whose vanishing
expresses that the line ax+ y+ yz cuts the quartic in four
points, for which the invariant S vanishes. We shall call this
contravariant o; it is of the fourth order in the variables
a, 8, y, and its coefficients are

A =be+3f*—4b,c,, B=ca+3g°—4c,a,, C=ab + 3h’ —4a,b,,
F =af+gh+20 —2a,n—2a,m,

G =bg +hf +2m*— 2b, — 2b,n,

H =ch+fg + 2n’? —2c,m— 2c,l,

L =2fl —mn—gb,—he, +,¢,,

M =2gm —nl —he,- fa,+¢,a,,

N =2hn —Im —fa,—gb, + ab,
A, = 3mc,—3nf —cb, +b,c,, A,=8nb, — 3mf— be, + b,c,
B, =3na, —3lg —ac,+4a,¢,, B,=3le, - 38ng —ca,+ c,d,

C,=3lb, — 3mh—ba,+,a,, C,=3ma,- 3lh — ab,+a,p,-

293. The contravariant just mentioned is the evectant of
the simplest invariant A, which is of the third order in the
coefficients, and has for its symbolical expression (123)*; that
is to say, o is found by performing on A the operation |

a 2 +a5 +o 5 + By Gt &eu5
and conversely from the values already given for the coefficients
of o the value of A can be inferred. This is
A = abe +38 (af? + bg’ + ch’) —4 (abc, + be,a, + ca,b,)
+12 (f+ gm’ + hn’) + 6fgh —12lmn
— 12 (a,nft+ a,mf + b,ng + b,lg + cymh + ¢,lh)
+ 12 (lb,c, + me,a, + na,b,) + 4 (a,b,c, + a,b,¢,).
If we use the same notation as in Art. 223, the value of
A may be written

x (d’) + 4 (dea) + 3 (db*) — 12 (c’d),

or)
|

INVARIANTS AND COVARIANTS OF QUARTICS. 26:

where

(d*) =d.d,— 4d,d, + 3d’,
(dea) = a, {d,c, — 3d,c,+ 3d,c, — d,c,} + a, {d,c, — 3d,c, + 3d,c,—d,¢,},
(db*) = db,’ — 4d,b,b, + 4d,b,’ + 2d,b,b, — 4d,5,b, + db,"

cas 2.0.2 ‘3°01
(cd) soe b, (¢,¢, fg ¢,’) Siti b, (C,C, arr ,C,) aM b, (¢,c, Wie ¢,') ’

the invariants (d’),; (dca), &c., being all known in the theory
of the binary quantics.

294, The next simplest invariant B is of the sixth order
in the coefficients. It may be formed by taking the six
equations obtained by twice differentiating the given equation
with respect to x, y or z, and from these six equations elimi-
nating dialytically 2°, y’, 2’, yz, 2x, zy. We thus have B in
the form of a determinant | |

dy Ry og G a,
he ke

I) A Cy Coy Cy n
l, b

sy My, Cry Ny Ys l

ey) b, Ny Mm, l, h

We shall presently give the developed expression for B.
Meanwhile, we remark that Clebsch has used this invariant
to shew that the form

prgtr+s+=0,
where p, g, 7, s, ¢ are linear functions of the coordinates, is not
one to which the equation of every quartic can be reduced.
Since p, g, &c., each implicitly contain three constants, the
form just written involves fourteen independent constants, and
therefore, at first sight, seems capable of being used as a
canonical form sufficiently general to represent any quartic.
But on forming for the above equation the invariant B, it will
be found to vanish, and therefore this form will only represent
quartics for which B=0.* |

* This class of quartics has been studied by Liiroth, Mathematische Annalen,
vol, I, p. 87 (1870).

MM

266 INVARIANTS AND COVARIANTS OF QUARTICS.

295. In calculating the value of B, it is convenient to use ©
the following value for a symmetrical determinant of six rows
_ and columns, the constituents of which are denoted by a’, ad, ac,
&e., ba, b°, be, &e.
abcde’ f? — Sa’b'c'd® (ef)? + 23a°b’c’. de. ef . fd + Sa*b* (cd )? lef)?
— 23a°b".cd .de.ef . fo+ 23a’. be.cd.de. ef. fb — 23a’ (bc)* de.ef. fd
+23 (ab)’ cd.de.ef.fe — & (ab)? (cd) (ef )* — 2 3ab.hc.ed.de.ef. fa
+23ab.be.ca.de.ef.fd. |

The expanded value of B is as follows:

abe ( foh —fl — gm* — hn’ + 2lmn)

+ be {lt — Ugh +2 (gm— nl) al + 2 (hn — ml) al + (n® —fg) a,
4-(m* — fh) a,’ + 2 (fE— mn) a,a,}

+ ca {m* — mifh + 2 (fl — mn) bm + 2 (hn — ml) b,m + (n* — fg) b,”
+ (2 — gh) b, + 2 (gm — nl) b.b,}

+ ab {n* — nifg + 2 (fl— mn) en + 2 (gm — In) cn + (m? —fh) 0
+ (2 —gh) c,’ + 2 (hn — lm) c,c,}

— (af* + bg’ + ch®) (fgh —f$P — gm*® — hn? + Alin)

+ 3 (afm'n’ + bgn*l’ + chl’m’)

+ 2af” (b,gn + chm) + 2bg* (c,hl + a, fn) + 2ch" (a, fm + b,g!)

— 2af (b,n* + ¢,m*) — 2by (c,0° + a,n*) — 2ch (aym* + 6,0)

+ 2afl (b,n* + ¢,m*) + 2bgm (c,P + a,n*) + 2chn (a,m’ + bP)

_ —2afmn (bg + ¢,h) — 2dgln (c,h +a,f) — 2chlm (a, f+ b,g)

" — 2a (bmn* + c,m'n) — 2b (c,nl° + a,ln®) — 2c (a,lin® + bmi’)

+a (bign* + ¢,°hm*) + b (cZh? +.a,fn") + ¢(aZfim? + b,*g/*)

+ 2afl (mb,c, + nb,c,) + 2bgm (ne,a, + lc,a,) + 2chn (la,b, + ma,b,)

+ 2amn (mb,c, + nb,c,) + 2bnl (nc,a, + le,a,) + 2clm (la,b, + ma,b,)

— 2af (hnb,c, + gmb,c,) —2bg ( fle,a,+ hne,a,) — 2ch (gma,b, + fla,b,)

+ 2 ( fghtlmn)(ab,c,+ be,a,+¢a,b,) — 2afl’b,c,— 2hgm'c,a,—2chn'a,b,

~ 2 (af*lb,c, + bg’me,a, + ch’na,b,)

— 2ab,c, (b,gn + c,h) — 2be,a, (c,hl + a, fn) — 2ca,b, (a, fm + b,g1)

+ 2ab,c, (cym* + bn") + 2be,a, (a,n® + ¢,1") + 2ca,b, (b,0 + am’)

— 2al (inb,c,’ + ne,b,”) — 2bm (ne,a,” + la,c,”) — 2cn (a,b,” + b,a,")

INVARIANTS AND COVARIANTS OF QUARTICS. 267

+ a (hb,’c,* + gb,"c,") +b (fe,7a,” + he,?a,*) + ¢ (ga,’b,* + ha,*b,”)

1%)
+ afb,*c," + bgc,*a,” + cha,’b,’+ 2alb,c,b,c, + 2bmc,a,c,a, + 2cna,,a,),
— 2ab,c, (me,b, + nb,c,)— 2be,a, (na,c, + lc,a,) — 2ca,b, (lb,a, + ma,b,)
+ 2f%q°h* — fgh (f0 + gm* + hn’) + 10fghlmn — (fP + gm? + hn’)?
+ 2lmn (fl + gm? + hn*) — Pm?n?
+ 2 (b gn + chm) (gm® + hn® — 2 f0 — fgh — ln)
+2 (a, fn + c,hl) (hn* +f0 — 29m? — fgoh — inn)
+2 (a, fm + b,gl) (fl + gm? — 2hn? — foh — inn)
+ (gh—P)(b,9 — ofh)'+ (hf - m’) (oh —a,f)* + (fy - 0°)(a,f 8,9)
+ 2a,a,f° (2mn — fl) + 2b,b,9° (2nl — gm) + 2c,c,h (2lm — hn)
+ 2lb,c, (fgk + lan + fl — gm — hn’)
+ 2me,a, (fg + lmn + gm’ — hn? — fl’)
+ 2na,b, ( fgk + lmn + hn® —f0 — gm’)
— 2ghmnb,c, — 2hfnle,a, — 2fglma,b,
+ 2 (d,c,gm + b,c,hn) (gh + 20°) +2 (c,a,hn + ¢,a,,fl) (hf + 2m’)
+ 2 (a,b, fl + b,a, gm) (fg + 2n’)
— 2 (a,’c,f’m + b,'a,g’n + ¢,°b,h'1 + a2b, f?n +b,c,9'l + ¢,7a,h’m)
+ 2fmn (a,c, + a,'b,) + 2gln (b,7c, + b2a,) + 2hlmn (c.°b, + ¢,"a,)
— 2 (a,b,c, +a,b,¢,) (fl +gm*+hn?+lmn) _
— 2fa,a, (c,m” + b,n*) — 29b,), (c,0’ + a,n*) — 2he,c, (b,7 + a,m*)
+ 2 (fl— mn) (gb,c,a, + he,a,b,) + 2 (gm — nl) (he,a,b, +fa,b,¢,)
+ 2 (hn — Im) ( fa,b,c, + 9b,¢,4,)
— (Pb %c? + m°cZa,? + nab?)
+2 (b,c,a, —¢,a,),) (6,91 + chin + a, fn —c,hl — a, fm —b,gn)
+ 2 (0,c,a,0,f? + ¢,a,b,b,9° + a,b,c,c,h”)

— 29h (b,’a,c, + ¢,2a,b,) — 2hf (c,7b,a, + a,2b,c,) — 2fg (a,°c,b, + b,7c,a,)

22°71 | AD Lae 3°32 2°23 BONS Sas
+ (4/0 —2mn) c,a,a,b, + (4gm —2nl) a,b,b,c, + (4hn — 21m) be
+ 2 (a,b,c, + a,b,¢,) (1b,c, + mce,a, + na,b,) — (a,b,c, + 4,0,¢

2
3-1-2 2°34 0,0.) »

1%,

296. In the notation of Arts. 223, 293, the value of B is
r (d*) (b°) — x (d’c°b) + r (de*) — (d*) (ba*) + (d’c’a?) + 2 (d?cb*a)
— (6°) (d°b*) — 2 (de*ba) + (de*b*) — (c°d)’,
where (d°) =d,d,d, + 2d,d,d,—d,d — dd; —d,,

268 INVARIANTS AND COVARIANTS OF QUARTICS.

(d’c°b) = b, {c,” (d,d, — d,*) + 2¢,c, (d,d, — d,d,) + 2c,c, (d,d,'— d,”)

+ ¢," (d,d,— d,”) + 2¢,¢, (d,7, — d,d,) + ¢,” (d,d,— 4,7)
+ b, {c,” (dd, —d,’) + 2c,¢, (d,d, — d,d,) + 2c,c, (d,d, — d,”)

+c,” (d,d,— d,’) + 2¢,¢, (d,d, — d,d,) + ¢, (d,d,— d,’)}
— 2b, {c,¢, (d,d,- d,") + ¢,¢, (d,d, — d,d,) +. ¢,¢, (d,d, — d,”)

+c,’ (dd, —d,d,) + ¢,¢, (d,d, + d,d,— 2d,”)

+ ¢,¢, (d,d,— d,d,) + eu (ida dade) + Cfo (hey
(d’c’a’) is formed from (d’c’}) by writing a,”, a,’, 2,4, for ,, 5,, 4,
(de') = d, (c,c, — ¢,")? — 2d, (¢,c, — ¢,¢,) (¢,¢, — ¢,’)
+ d, {(¢,¢, — ¢,¢,)"+ 2 (6,6, — ¢,7)(¢,¢, — ¢,")} — 2d, (¢,¢, — ©") (¢,¢3- ©)
+ d,(c,c,—¢,")*,
(a*) = b,a,” — 26,a,a, + b,47

LisOn 2 Ot?
(d°cb*a) ae b, (a,c, +.4,¢,) + 4,4,¢,} P
+ {b,a,c, — 8, (a,c, + a,¢,) + 4,a,¢,} Q
+ {b,a,¢, — , (a,c, + aC, ) ss b,4,C,) fh,

where P=), (d,d,— d,’) — 6, (d,d,—,d,) +6, (dd, —d,"),
Q =, (d,d,—d,d,) — b, (d,?—d,d,) +.b, (dd, - d.d,),
R=b, (dd,—d,") —6b, (dd, - d,d,) + 6, (dd, — d,"),
(d°b") = (dd, — d,?) b2 + (dd, — d,’) 6° + dd, -— d,’) b,
+ 20.6, (d,d,— d,d,)'+ 2b,b, (d,d,— d,’) + 26,0, (d,d, — d.d,',
(de*ba) =a, {P(e,c, — ¢,”) + Q (c,¢, — 60.) + B (c,¢, - ¢,")}
+a, (P' (c0,— ¢,") + pe (¢,0, — G04) + Hi (0,0, — ¢,')53

where P= (c,d, ag diya let Pei

€) = ig hia oB +H od

Lt = b, (¢,d,- ace d,) +4, (ed

=b, (¢,d,—¢,d,) +8, fe “AN

pa b, (¢,d, - ee 608 et ia c,d,)

buted! cate + b, (ed, - ia
(dc*b") = d, {c,"b,b,? — 2¢,¢, ig + b,°) + io b,

c,” (b,0," + 3b,b,”) — 4¢,¢,b,0,? + 5,%c,"}

— 2d, (hh), — 0,0, (b,7b, + 26, x : + ¢,0,),°b, + 2c,"b,b,5, + ¢,¢,5,°b,

— 2c.¢, bb." nie D b2 +¢,¢,b,"}

sees

INVARIANTS AND COVARIANTS OF QUARTICS. 269

+ d, {c,'b,’ — 2c,c, (b,7b, + 2b,b,”) + 2¢,¢, (b,° + 6,5,4,)
— ¢,” (b,°b, + 2b,0,") + 2c,c, (b,° + 50,0,0,)
— 2¢,c, (b,b," + 2b,b,”) —c,* (b,b,7 + 20,0,”) +67}

— 2d, {c,°b,"b, - cc, (b,," + 25,0,") + ¢,¢,,b,7 + 2¢,7b,0,0, + ¢,¢,0,0,"

mace | ale | eee | te 8 Se moe 8
— 2¢,c,b,°b, — ¢,°b,b,° + ¢,c,),}
+ d, {c,"b,b,’ — 2c,¢, (6,0,2, + 5,") + 2c,¢,,"b, + ¢,° (b,°b, + 3b,5,")
se 4c,¢, 1”0

bb’ + B30,
(c’b) b, (¢,¢, Ri ¢,’) ad, (C50, es ¢,¢,) + b, (C,¢, me C,")«

297. We have seen (Art. 221) that if we had a covariant
quartic, we could, from the invariants already obtained, derive
a series of others. One such covariant can be at once obtained
by forming the equation of the locus of a point whose first
polar is a cubic for which the invariant S vanishes; in other
words, by equating to nothing the S of the polar cubic. The
symbolical expression for this covariant is (123) (234) (314) (124).
The covariant S of the guartic

aa* + by* + c2z*+ du* + evt=0
is of the form EN a Ser
ee 2 ae 8

Hence, as we have already seen, that the first form, though
apparently containing a sufficient number of constants, is a
special one to which the equation of a quartic cannot in general
be reduced; so is the second form also one to which the equa-
tion of a quartic cannot be brought unless a certain relation
between its invariants be satisfied.

There are other covariant quartics, but that just described is
of the lowest order in the coefficients. Any other covariant
quartic of the fourth order in the coefficients must be of the form
S+kAU, where & is a numerical constant and JA the first
invariant. ‘This may easily be verified with respect to the
covariant obtained by forming the contravariant of the contra-
variant of Art. 292.

298. The general values of the coefficients of S have not
- been calculated, nor have any of the higher invariants. I have
thought it worth while, however, to examine the special case

ax’ + hy’ + cz* 4 6fy?2” + 6gz "a + 6ha'y’ = (),

270 INVARIANTS AND COVARIANTS OF QUARTICS.

This form only implicitly contains eleven constants, and there-
fore is a very particular case of the general equation of the
quartic; but it lends itself easily to calculation, because the
covariant S is of the same form
aa* + by* + cz* + 6fy’2" + 6g2"a" + 6ha’y? = 0;

and, therefore (Art. 221), from any invariant can be derived

d
2a +b-= Be! &e., an
operation which we shall denote by the symbol ¢. Although
invariants which exist in general may vanish for the special case
here considered, yet invariants, which in this case are distinct,
will be distinct in general. By calculating the invariants for
the special case, we obtain all the terms of the general in-
variants which contain only the coefficients a, b, c, f, 9g, h

The values of the coefficients of S, for the form in question,
are

another by performing on it the operation a

a=x67°h", b=6hi/*, c=6f%q",
f =begh —f (bg" + ch’) — "gh,
g =cahf—g (ch’ + af") —fg"h,
h=abfg —h(af* + bg’) —fgh’.
It is convenient to remember, that for the same form the
values ot the coefficients of the contravariant o, Art. 292, are

A=be+3f*, B=ca+ 39°, C=ab+3f’,
=af+gh, G=bat+hf, H=ch+ fg.

299. We find it convenient to use the abbreviations
abe = L, af?’ + bg’+ ch® =P, beg*h* + cah®f? + abf’g?=Q, foh=R;
then the values of the invariants previously found are, for the
special case we are considering,

A=L4+3P+6Rh, B=LR+2h’?— PR; or B= AR-4PR-4R’.

The results of the operation ¢ on these several quantities are
O(Li= 60, d(P)=6LR-2PR-4Q4+18R’,
}(Q)=—-2PQ-—4hQ-6LAR + 12PR’ + 40PRK,
o()= Y—2PR— 3h’,

whence (A) =18B.

+ bt ‘4 » phe alias 5
somite as lecRe nee eee ENR Se Se

INVARIANTS AND COVARIANTS OF QUARTICS. 271

We can then obtain a new invariant of the ninth order in
the coefficients by performing on B the operation ¢ The
result is

$ (B)=C.=Q(L— P+ 14R)— LR(2P+ 9R) + R(2P*— 3PR-308’).

The invariant just found is not, however, the only independent
invariant of the ninth order in the coefficients. If we write the
general equation of a quartic u,+ w,z+ u,z" + u,z° + cz*=0, then
generally the highest power of ¢ which occurs in an invariant
of the ninth order will be the third, and c will be multiplied by
an invariant of the sixth order in the coefficients of the binary
quartic v,. This latter invariant must be of the form s°+ kt’;
and any assumed invariant of the ninth order can be resolved
into two parts, in one of which c’ will be multiplied by s°, and
in the other by ¢. ‘The former part can be expressed in the
form 14*+mAB+nC, where A, B, C, are the invariants
already calculated; for the expression of the latter a new in-
variant is necessary, and we proceed to give one of several ways
in which it may be obtained. It will first, however, be neces-
sary to mention some other covariants and contravariants.

300. The value of the Hessian for this case is
aghx’+ bhfy’+ ofgz°+ (abg + ahf— 3gh*)a*y’+ (ach + afg —3q°h) x*2"
+ (abf+ bgh—3fh*) y’x"+(bch+ bfg— 3f*h) y*2’+ (caf + chg—3fg")zx"
+ (beg + cfh — 3f"q) zy’ + (abe — 8af* — 8bg’ — 3ch’ + 18fgh) x*y?2*.
Again, it has been stated (Art. 92) that a quartic has also a
contravariant sextic, the symbol for which is (a12)* (a23)* (a31)*.
The value of this, for the case we are considering, is
(Lof —f°) a° + (cag — g*) B° + (abh — h?) yy?
+ (beg+ 6cfh—3f?9)a*B"+ (bch+ 6bfg—3f*h) aiy’-+(acft 6egh-39°f)B'a*
+ (ach + 6afg — 3g*h) By’ + (abf+ 6bgh — 3fh*) y*0?
+ (abg + 6afh — 3gh") y‘B’+ fabe—3(af*+ bg’+ ch?) + 48fgh} a® By’.
If, introducing differential symbols in either of these, we operate
on the other, the result is A*+576B8. If we operate on the
Hessian with the contravariant o, we get a covariant quadratic

of the fifth order in the coefficients; and if we operate on the
contravariant sextic with the quartic itself, we get a contra-

272 INVARIANTS AND COVARIANTS OF QUARTICS.

variant quadratic of the fourth order in the coefficients. The
values of these quadratics are respectively

(afx’ + bgy’ + chz’\(L+4+8P+30R)
+ (gha*+ hfy’+ fg2") (10L —6P— 12R) —4 (a®ftx®+ Big?y"+ ch'e!) +
(fo’ + 98° + hy’) (83L+ 5P+2R) — 8 (af*a? + bg? BR+ ch*y’)

+4 (begha’ + cahfP’ + abfgy’).
If we introduce differential symbols into either of these two

concomitants and operate on the other, the result is a new
invariant

C, = (80L — 32P+ 448R) 04+ 83P®—6P°L— 134P°R
+ 3PL* +128PLR — 60PR’ + 102L°R + 408 LR — 72K".
There appears to be for the quartic we are considering no
other independent invariant of the ninth order. If, for ex-
ample, we operate with the contravariant conic on the quartic
itself, the result is expressible in terms of the invariants
already found, being 3C,—- 80C,—180AB. We might perhaps
more simply have taken for the second independent invariant
3 (C, — 82 C,), or
C,=16QL+4 P*-2P°L—66P°R+ PL’ + 64PLR + 12PR’
+ 340°R + 232LR* + 2968",

301. We proceed next to form invariants of the twelfth
order in the coefficients. We can form the cubic invariant of
the quartic S by help of the formule
L' =216R*,
P'=6{Q-2PQR-42°Q4+2P°R’-2PLR’+4PRH+6LR+3K%4,
i= Q@-2LKhQ- PR -2PR+ PR +4LR — Fe,
whence L'+ 3P’ +6’ =6D,, where
D,=40'+ Q(-6PR—-2LR — 12K”)

+5P°R’- 6PLR +10PR + LR’ + 22D RR 4+ 44K*,
Again, by performing the operation ¢ on C., we get
. D,= 249’ + Q(4P* —4PL — 84PR — 202K — 248K”)
? —4P*R-14P*R* + 4PL'R + 144PLR + 444PR
— 18/7R*— 84 LR + 216K,

INVARIANTS AND COVARIANTS CF QUARTICS. 273

and, by combining these, we have D, —6D,=4D,, where
D,= Q(P*-— PL -12PR-—2LR-— 44h’) — P*R—-1UP*h

+ PI?R+45PLR + 96PR’ —- 62’ R’ — 54LR* —12K*.
In terms of these and of the other invariants already given

can be expressed the other invariants of the twelfth order, such
as $(C,), and the discriminant of the contravariant conic,

So, again, we can express in terms of the preceding the
invariants of the contravariant quartic; we have

D'=[?+3PL+9Q + 27f*,

f'=LR+ Q+PR+ Bh,

P'=3P*-5Q+6PR+ PL+6Lh + 9h",

Q'=3@'+ Q(3P?4+4PL4+24PR+L?-8LR + 6R’)
+12P°LR+18P°R’+ 4PL’R +10PLR’ + 36 PR’ -36L K+ 27K",
whence A’= A’+12B, B'=4D,+AC,+ A’B-12B".

302. It is to be noted, that though there is only one con-
travariant conic of the fourth order in the coefficients, there
are two covariant conics of the fifth, viz., in addition to that
already given, that obtained by operating with the contravariant
conic on the quartic itself, the result being
(3L+9P+ 10R) (afx* + bgy* + chz’)

+ (10L+4+2P+4R) (gh? + hfy’+ fgz) —12 (a f%x°+ Bg*y'+ ch'z"),
and if this be combined with that previously given, we can write
it in the simple form

4R (afx’ + bgy’ + chz*) + (L— P—2R) (ghx* + hfy’ +92’).

The discriminant of this last conic gives the simplest invariant
of the fifteenth order, viz., writing J — P—2h= WM,
E,=16MR'Q+4M?R’P+ MR’ + 6408";
or, at length,
E,=16 (L—P-2R) QR’ + RB {3P*- 5P*L£ + 10P*R
+ PD?—-4PLR+4PR + L’-6L’*R+76LK — 8B},
The other three invariants of the system of conics are, of course,

also invariants of the quartic of the same order, besides which
NN

274 INVARIANTS AND COVARIANTS OF QUARTICS.

we might also calculate 6D,, 6D,, &c. All these are expressible
in terms of £, and £,,* where

E,=16 (L- P-2R) @+(3P°-5P*L-6P*R

+ PL?—228PLR — 2172 Ph’+ L°+298L°R + 2636LR°—4296 B®) Q

4 R(-12P*4 44P*L —52P*L? + 20 PL’)
+ R? (348P* — 852 P°L + 308PL? + 324L’)
+ RB (1320P?— 416PL + 216L”) + 720PR* + 11376 R*— 864R.

There are also two independent invariants of the eighteenth
order, the first being the C, of the contravariant quartic, viz.

F = 128 Q°+ Q’(— 48P*+ 80PL + 368PR + 32L’— 528L KR - 160K’)
+Q(9P*-12P°L—108P°R—-2P*L’ + 324P°LR + 240P°R?
+4PL’ + 60PLI°R—- 288 PLR’ + 528 PR’? + L*— 201° R—400L7R*
— 2512 DR — 144R*) 4+ 18P°R—24P*DR+27P*R’-4P°LV’R
+180P° DR’ + 60P?R+8PPR+ 114 R + 716P* LR

+ 288P°R' + 2PL*R —44PL°R’ + 52PL°R® — 592PL Re

+ 288PR' — 21L*R’ — 60L°R’ —720L? R* — 2076 LR + 240R*.

F. = 1289 +@ (—8P* — 240PL —5312PR + 312L? + 9536LR

+ 11680R") + Q (-18P* + 54P°L4+1146P°R -54P°D’— 1978P°LR
+ 7548P*R? + 18PL’ + 262PL°R — 4432 PLR + 49272 PR

+ 5701°R + 1620L’°R? + 6648 LA + 77808 R*) + 24P°R
—T6P*LR —1224P*R*’ + 84P°L*R + 2622P°L RR — 13032P*R®
—36P’°D’R—- 946P°L’R’ + 8268P* LR — 30192 P?R* + 4PL‘R

— 822 PL°R® — 368PL*R*® — 73784 PLR — 5472 PR’ + 1140 R?

— 15240°R°— 147120 R* — 113904. R* + 25920R*.

It does not appear that, even in the special case we are
considering, the invariants of higher order that we have given
are linearly expressible in terms of those of lower order; nor
have I been able to find that, even in this case, the discriminant
is expressible in terms of lower invariants.

* The values of these and of the next two following invariants were calculated
for me by Mr, J. J. Walker.

ay ae

CHAPTER. VEL.

TRANSCENDENTAL CURVES.

303. We have hitherto exclusively discussed equations re-
ducible to a finite number of terms involving positive integer
powers of z and y; it remains to mention something of the pro-
perties of curves represented by transcendental equations. Since
these involve functions only expressible by an infinite series of
aleebraical terms, all transcendental curves may be considered
as curves of infinite degree; they may be cut by any right line
in an infinity of points, and must have an infinity of multiple
points and multiple tangents. There is, then, no room for a
general theory of the singularities of these curves, and it 1s
only necessary to mention the names and principal properties of
some of the most remarkable of them. We may notice, in
passing, a class of equations, called by Leibnitz cnterscendental,
or which involve the variables with exponents not commen-
surable with any rational number; for example, y=a*. Here,
as we successively substitute for ./2 the series of rational
fractions which approximately express the value of the radical,
we shall find a series of algebraic curves of constantly increasing
degree, more and more nearly resembling the figure of the
required curve, but not accurately expressing it as long as the
degree of the curve is finite. We pass on to the cyclord,
which holds the first place among transcendental curves, both
for historical interest and for the variety of its physical applica-
tions. This curve is generated by the motion of a point on
the circumference of a circle which rolls along a right line.
Let A be the point where the motion commences; then (see fig.
next page), in any position of the generating circle, if p be the
generating point, we must have the are pm = Am, and denoting

the angle poem by ¢, and cm, the radius of the circle, by a, we
shall have

y=a(1—cos®), a=a($—sing) ;

276 TRANSCENDENTAL CURVES.

whence, eliminating, we shall have the equation of the curve,

‘ +/(2ay — y’))
’

a

a—y=a cos

_—.

B

It is, however, generally more convenient to retain ¢, and to
consider the curve as represented by the two equations given
above. It is easily seen that the form of the curve is that
represented in the figure; and since the circle may roll on
indefinitely in either direction, that the curve consists of an
infinity of similar portions, and that there is a cusp at the point
of union of any two such portions.

Let MPN be the position of the generating circle correspond-
ing to the highest point of the cycloid, then, since Am = are pm,
AM= MPN, we have Mm=pP=arc PN; or the curve is gene-
rated by producing the ordinates of a circle until the produced
part be equal to the corresponding arc, measured from the extre-
mity of the diameter. Denoting the angle PCN by @, the curve
referred to the axes AM, MN is represented by the equations

y=a(1+cos@), x=a(@+sin8@).

£04. We can readily see how to draw a tangent to the curve,
for at any instant of the motion of the generating circle m (its
lowest point) is at rest, and the motion of every point of the
circle is for the moment the same as if it described a circle
alout m; hence the normal to the locus of » must pass through
m, and its tangent must always be parallel to NP. The same

thing appears analytically for oy = aa
gent therefore makes with the axis of # an angle the comple-
ment of CNP, which is $¢.

It is so easy to give geometrical proofs of some of the principal
properties of the cycloid that we add them here. The area of the

=cot$¢; the tan-

RS RS ee

TRANSCENDENTAL CURVES. 277
curve is three times the area of the generating circle. For the
element of the external area ( pp’rr’=pp'tt’= PP’ QQ’) is equal to
the element of the area of the circle; the whole external area
therefore, AHN/'B, is equal to the area of the circle; and therefore
the internal area ANB is three times the area of the circle.

The are Np of the cyclotd is double NP the chord of the circle.

For it is easy to see that the triangle PP’L is isosceles, and
therefore that if a perpendicular, MK, be let fall on the base,
PL, the increment of the are of the cycloid, is double PK, the
increment of the chord of the circle.

Hence, if s denote the arc of the cycloid, 5 the diameter of the
generating circle, z the abscissa N@ from the vertex, then the
equation of the curve is s* = 4bx, a form useful in Mechanics.

The radius of curvature is double the normal.

For the triangle formed by two consecutive normals has its
sides parallel to those of the triangle MPA’, but the base of the
first triangle is equal to PL, and, as we have just proved, is
double PX, the base of the second; hence the radius of cur-
vature is double /P.

The evolute of the cycloid ts an equal cycloid.

For if we suppose a circle touching the base at m, and passing
through & the centre of curvature, it is equal to the generating
circle, and the arc nf is equal to NP=nD; hence the locus of R
is the cycloid described by the circle mZn rolling on the base EF.*

N

’
F ’
X
A MAY, M L
*

\

’ \

‘ \

1 \
’ !
; !

’ H

‘ RY}

E ih D*< F

* The properties of the cycloid were much studied by the most eminent mathe-
maticians of Europe during the first half of the seventeenth century. Their attention
was first called to these problems by Mersenne; but Galileo claims to have inde-

278 _ TRANSCENDENTAL CURVES.

We might also seek the locus of any point in the plane of the
generating circle carried round with it; when the point is inside
the circle, the locus is called the prolate cycloid; when it is out-
side it is called the curtate cycloid; these loci are by some called
trochoids. There is no difficulty in calculating their equations or
in ascertaining their figures, but it does not seem worth while to
devote any space to them here. The method of drawing tangents
given for the cycloid applies equally to these curves. These
curves may (as the reader can easily see) be generated by a point
on the circumference of a circle, rolling so that the are pm shall
be en a constant ratio to the line Am.

305. When the properties of the cycloid had been investi-
gated, it was a natural extension to discuss the curve traced by
a point connected with a circle rolling on the circumference of
another. When the point is on the circumference of the rolling
circle, the curve generated is called an epicycloid or hypocycloid,
according as the circle rolls on the exterior or interior of the
fixed circle; if the generating point be not on the circumference,
the curve is called an epitrochoid or hypotrochoid.

Let us take for the axis of x that position of the common
diameter of the two circles which passes through the generating —
point; let CO be any other position of it, Q the generating
point ; let CN=a, ON=6, NCB=4, PON=¥, why d;
then since BN = NP, we ais

ab=by; OQM=180-($+W); 0
and the coordinates of Q are wie M we
y=(a+5) sing—d sin($+ 4), ot
x= (a+b) cos6—dcos(+ WW); /

or ifa+b=mb, 7 . a

y=mb sind —d sinmd,

x= mb cosh —d cosmd.

pendently imagined the description of this curve. Galileo, having failed in obtaining
the quadrature of the curve by geometrical methods, attempted to solve the problem
by weighing the area of the curve against that of the generating circle, and arrived
at the conclusion that the former area was nearly, but not exactly, three times the
latter. The problem of the quadrature was correctly solved by Roberval in 1634;
the method of drawing tangents was discovered by Des Cartes, the rectification by
Wren, the evolute by Huyghens; several other important properties by Pascal.

TRANSCENDENTAL CURVES. 279

Eliminating ¢ from these equations we obtain the equation
of the curve, which is not necessarily transcendental. In fact,
when the circumferences of the circles are commensurable, after
a certain number of revolutions, the generating point returns to
a former position, the curve is closed, and of finite algebraic
dimensions; but if they be not commensurable, the generating
point will not in any’ finite number of revolutions return to the
same position, and the curve will be transcendental.

To obtain the equations of the epicycloid we have only to
make d=+0, and we have

y=b(m sing + sinm®@),
x=b(m cosg +cosm®) ;
the lower sign answers to the case when the axis of @ passes

through the generating point when it is on the fixed circle; the
upper sign, when it is at its greatest distance from it.

306. The coordinates for the case of the hypotrochoid and
hypocycloid are found, as the reader can easily verify, by
changing the sign of 4 in the equations given above. ‘These will

be included in the equations which we shall use, by giving nega-
a—b

tive values to m, or by supposing m=—n, where n=——.

b

The equations given above, if we alter d into mb, and m

‘ 1
into — , become
m
son (- sath nab $)
hes re ‘mir

1 1
x = mb (— cos + cos — $) 5

and making ¢ = mvp, we see that these equations belong to the
same locus as the preceding. We can thus prove that the same
hypocycloid is generated whether we take 6=4(c+a). (Kuler
de duplici genesi Epicycloidum, Acta Petrop. 1784, referred
to by Peacock, Examples, p. 194). The hypocycloid, when
the radius of the moving circle is greater than that of the
fixed circle, may also be generated as an epicycloid, for then

a—b\. ge
m (= - — 1s positive.

280 TRANSCENDENTAL CURVES.

307. Tangents can easily be drawn to these curves, for by
the same reasoning as that used in Art. 304 the line NQ is
normal to the curve. We can thus see also that when a curve is
generated by a point on the circumference of one figure rolling
on another, there must be a cusp at every point where the
generating point meets the fixed curve. For by this construction,
at such a point the generating point approaches the fixed curve
in the direction of its normal, and recedes from it in the same
direction; hence it isa stationary point. An epicycloid then
consists of a number of similar portions, each united to the next
by a cusp; and the extreme radii, from the centre of the fixed

: : ea 2bar
circle to any such portion, are inclined at an angle =e
When the radii of the circles are commensurable and the curve
therefore algebraic, the number of cusps is finite, but when the
curve is transcendental, the number of cusps is infinite. Every
point of the base is in its turn a cusp, and therefore the base
may be said to be the locus of the cusps of the curve; but,
obviously, consecutive points of the base are not consecutive
points of the locus.

308. These curves have besides, as have epitrochoids in
general, a number of double points crunodal or acnodal, the
number being finite for algebraic curves and infinite for
transcendental, and all the nodal points being ranged in
circular loci. Consider the equations (Art. 305)

y=mb sing—dsinmd, x= mb cosd—d cosmd,

where ¢=0, corresponds to what we may regard as the initial
position of the generating point, viz. that where it is in a line
with the two centres, this line being taken as the axis of x, and
the initial distance of the origin from the generating point
being mb—d. But there are other positions of the moving
circle for which the generating point lies on the axis, the
values of ¢@ corresponding to these positions being found by
solving the equation mb sing =d sinmg. And setting aside the
root ¢=0, the other roots of this equation are obviously dis-
tributable into pairs equal with opposite signs, and for each pair
the value of x, mb cosp—d cosm®@, is the same. The corre-

TRANSCENDENTAL CURVES. 281

sponding points are therefore double points on the locus. The
value mb cosf—dcosmp@ may, by means of the condition
mb singd=d sinmd, be written in the form x sing=d sin (m—1) ¢.
Every time that the generating point returns to a similar
position with regard to the two centres we have a line on
which double points lie, the number of such lines being, as
has been stated,’ finite for algebraic curves and infinite for
transcendental.

309. The equations of the tangents to the epi- or hypo-
cycloids admit of being written in a very simple form. For
dy _ cosh+cosmp —— cos$(m+1)d sin$(m+1)¢
dz -—(sind+sinmd) sind (m+1)> cos$(m+1)o°
And, attending to the condition that the tangent must pass
through the point whose coordinates have been given in Art. 305,
the equation of the tangent becomes
«cost (m+i)o4+y sing (m+1)¢=(m+1)b cos4 (m— 1) ¢,
when the axis passes through the generating point at its greatest
distance from the centre of the fixed circle; and

x sind (m+1)¢—y cost (m+1)6=(m+1)6 sing (m—1) d,
when the axis of x passes through the generating point at its

least distance from the centre of the fixed circle.
The equation of the normal in the latter case is in the same

manner seen to be
xz cost (m+1)o+y sin}(m+ 1) ¢=(m—-1)b cost (m—1) ¢.
Comparing this with the first form of the equation of the
tangent, it follows that the evolute of an epicycloid is a similar
epicycloid, the radii of the circles being altered in the ratio
m—

, or else =

= and the generating point of the evolute being at its

greatest distance from the centre of the fixed circle when on the
same diameter on which the generating point of the original
curve is at its least distance.

The same remarks, of course, apply to the hypocycloid.

The equation of the tangent to an epitrochoid is in like manner

(b cosh— d cosmd) x+ (bd singd—d sin md) y

= {mb* + d* — (m+ 1) bd cos (m — 1) d}.
00

282 TRANSCENDENTAL CURVES.

310. We give examples of some of the simplest cases where
the equations of these curves are algebraic, and can be easily
formed. ‘These cases are (a) when the equation of the tangent
is included in the form

a cos204+6 sin28+ccos8+dsinO+e=0

the envelope of which is given, Ex. 3, p. 69; (b) when the equa-
tion of the tangent is included in the form

a cos 30 + b sin 30 + 3c cos 8+ 3d sin @=0,

an envelope, which when treated by the same method as that
just mentioned, is solved by forming the discriminant of a
cubic equation, the result being |

(a’+ b°)4+ 8 (ac?—bd’) — 24cd (ad—be) =3 (c*+ d’)’ +6 (a" +") (c?+d’) .

(c) when m is a fraction whose numerator and denominator
differ by one. If we square and add the equations

x=mb cosnp—d cos(n+1)¢, y=mb sinnd —d sin(n+1) ,
we have xv’ +y*=m'b’ + d* —2mbd cos ¢,

and by solving for cos ¢ from this equation, and substituting in
the value for x, the elimination is performed.

Ex. 1. To find the epitrochoid in general when d= mb. The equations are then
reducible to the form

x= 2dsink(m—1) psink (m+1) 9, y=2dsin}(m—1) pcos} (m+ 1) ¢,
whence obviously 4 (m+ 1) @ is the angle w made by the radius vector with the

a ge
axis of y; and the polar equation is p = 2d sin —— aa

Ex. 2. To find the equations of the epitrochoid and epicycloid when the radii
of the circles are equal, and therefore m = 2. Dealing, as in (c), with the equations
x=2bcosp—dcos2d, y= 2bsingd —dsin2¢,
we find (a? + y? — 26? — d?)? = 40? (b? + 2d? — 2da),
the equation of a Cartesian, having, as may be easily verified, y = 0, x= d, as a double
point ; the curve is therefore a limagon. Wesee from the theory already explained

that this point corresponds to the value cos @ = “ . When therefore d is greater than

b; that is to say, when the generating point is outside the moving circle, the node
corresponds to two real positions of the moving circle and is a crunode; but if the
generating point be inside the moving circle, the node corresponds to no real B gas
of that circle, and the curve is acnodal.
The case of the epicycloid is obtained by putting d = , when we have
(x? + y? — 367)? = 403 (36 — 22).
The double point now becomes a cusp, and the curve is a cardioide. It is plain from
‘what has been said that the evolute of a cardioide is a cardioide.

:
)

Ny

oe
te
#5
vi
+s

TRANSCENDENTAL CURVES. 283

Ex. 8. To find the equation of the epicycloid when the radius of the rolling circle
is half that of the fixed circle. The equation of the tangent is

x cos 20 + y sin 20 = 46 cos 8,
an equation included in the form p. 69, the envelope of which is
(x? + y? — 467)3 = 1080422,
Ex. 4. To find the hypotrochoid and hypocycloid when the radius of the rolling
circle is half that of the fixed circle. We have m =—1; the equations are
zx=fcosp+dcosd, y=bsing—dsing,
and the hypotrochoid is the ellipse

x2 i. a
G+aE* G-a~®
which reduces to the diameter y in the case of the hypocycloid where = d,
Ex. 5. To find the hypocycloid when the radius of the fixed circle is three times

that of the moving circle. Here m=-—2, and the equation of the tangent is of

the form
xcosm —y sing =) cos3q,

and the envelope is, by the form (b) given above,

(x? + y?)? + 8ba3 — 24bay? + 185? (x? + y?) = 2704,
the equation of a tricuspidal quartic, the tangents at the cusps meeting at the centre
of the fixed circle.

This curve has been studied by Steiner as the envelope of the line joining the
feet of the three perpendiculars on the sides of a triangle from any point on the
circumscribing circle. In fact, taking the centre of the circle as origin, and the
coordinates of the vertices cos2a, r sin 2a, &c., if the point from which the perpen-
diculars are let fall is r cos 2@, r sin 2@, the equation of the line joining the feet is

zsin(a+B+y—$)—ycos(at+Bt+y-— 9)
= 2 {sin(a+fp+y—3o)+sin(B+y—a—¢)+sin(y+a—B—)+sin(at+ B-y—-9)},
a form easily reducible to that considered in this example,

Ex. 6. To find the hypocycloid when the radius of the fixed circle is four times
that of the moving circle. We have here m=— 8; the equation of the tangent is

xz sing + y cos = 26 sin2¢, and that of the envelope at + y = (4b)8,

311. The equation of the reciprocal of an epicycloid is
readily obtained, for the tangent being
x cos$(m+1)¢+y sing (m+ 1) P6=(m+1)b cos} (m—1) d,
it is plain that the perpendicular on the tangent makes an
angle £(m+1)@ with the axis of x, and that its length is
(m+1)b cos} (m—1) $; the locus, therefore, of the foot of this
perpendicular is |

, m—1
p=(m+1)b cos("— w),

and the reciprocal curve is

m—1
p cos( w) =(m-+1)b.

284 TRANSCENDENTAL CURVES.

The radius of curvature is found by the formula R= oe

In the original curve we have

=a ty? =D" {m+ 1+ 2m cos (m — 1) d},

or p' =0? (m— 1)’+4mb’* cos’ $ (m—1) 4,
ae Rappers c 4in
eon (m+ 1p? °
Hence ye sic ¥
. — (m+ 1?

312. Another general expression for the radius of curvature
in roulettes (or curves generated by a point on a rolling curve)
may be found as follows: Let P, P’ be two consecutive points of
the curve, M the point of contact of the rolling with the fixed
curve, and # the centre of curvature; then PP’, the element of
the arc of the roulette, is= WP. PMP’; but, by considering the
curves as polygons of an infinite number of sides, we can see that
PMP’, the angle through which PY turns, is equal to the sum
(or difference) of the angles between two consecutive tangents to
the fixed and to the rolling curve. Hence, if do be the element
of the arc of the roulette, ds the common element of the ares of
the fixed and generating curves, p and p’ the radius of curvature
of each, we have

do = MP (= +"),
pp
but this element, do, is also equal to PA, the radius of curvature,
multiplied by the angle between two consecutive normals; and
if we call @ the angle O.MP, between the normals to the roulette
and to the fixed curve, then the angle between two consecutive
normals to the roulette is

cos dds

MR
MP+ME 1 (; *)
?

——— MP.MR ~ cos \p' p’

* The invention of epicycloids is attributed to the Danish astronomer, Roemer,
who, in the year 1674, was led to consider these curves in examining the best form
for the teeth of wheels. The rectification of these curves was given by Newton,
Principia, Book I., Prop. 49,

TRANSCENDENTAL CURVES. 285

MP? (- + ~)
and PR : a

uP (- + 7) _ “ey
(See Liouville, vol. x, p. 150.)

313. A large class of transcendental curves is obtained by
taking the ordinate some trigonometrical function of the abscissa.
There is no difficulty in deriving the shape of such curves from
their equation. For example, y=sinz has positive and constantly
increasing ordinates until e=47; the ordinates then decrease in
like manner until 2=7r, when the curve crosses the axis at an.
angle of 45°, and has a similar portion on the negative side of the
axis between w= and x=27. ‘The curve, therefore, consists
of an infinity of similar portions on alternate sides of the axis.

So again, y = tana represents a curve, of which the ordinates
increase regularly from «=0 to e=47, when y is infinite, and the
line x=47 an asymptote. For greater values of x, y alters from
negative infinity to 0 when a=7. ‘The curve then consists of
an infinity of infinite branches, having an infinity of asymptotes,
x=41n, «= 8m, &., and, as may be readily seen, points of
inflexion at 2=0, x=, x=2, &e.

In like manner the reader may discuss the figure of y =secz,
which also consists of a number of infinite branches, only that
each branch, instead of crossing the axis, as in the last case, lies
altogether at the same side of it. The branches lie alternately on
the positive and negative sides of the axis of a. ‘l’o the same
family belongs a curve called the companion to the cycloid. It is
generated by producing the ordinates of a circle, not as in the
case of the cycloid, until the produced part be equal to the are,
but until the entire be equal to the arc. If, then, the centre be
the origin, the curve is represented by the equations

y

e=acos0, y=al0, x=a cos = 5
a curve of the same family as the curve of sines.

314. Next, after curves depending on trigonometrical, we may
mention those depending on exponential functions, ‘The loga-
rithmic curve is characterized by the property that the abscissa is

286 TRANSCENDENTAL CURVES.

proportional to the logarithm of the ordinate, and its equation
therefore is

x=m logy, or.y=a’.

The curve then has the axis of x for an asymptote, since, if
x=-—0; y=0, it cuts the axis of y at a distance equal to the
unit of length, and w and y then increase together to positive
infinity. The subtangent of the logarithmic curve is constant;
ydax

dy

Some controversy has arisen as to the proper interpretation
of the equation of this curve y=e". Attention was at first only
paid to the branch of the curve on the positive side of the axis
of «x, arising from taking the single real positive value of e”, which
corresponds to every value of x Kuler, in his Analysis Infini-
torum, II. p. 290, contended for the necessity of attendine to the
multiplicity of values which the function admits of; and the
same subject has been more fully developed by M. Vincent
(Gergonne’s Annales, vol. xv. p. 1). Thus, if x be any fraction
with an even denominator, ¢° has a real negative as well as a
positive value, and therefore there must be a point corresponding
to this value of 2 on the negative side of the axis, but there is
no continuous branch on that side of the axis, since, when z is
a fraction with an odd denominator, e° can have only a real
positive value. The general expression, including all values of
the ordinate, is found by multiplying the numerical expression
for e”, by the imaginary roots of unity, whose general expression
is cos2mam +7 sin2max7, where m must be made to receive in
succession every integer value, and 7, as usual, denotes /(— 1),
This is equivalent to saying that the equation y=e* must be
considered as representing not only one real branch, but also an
infinity of imaginary branches included in the formula y=e"C"™™,
Any one of these imaginary branches contains a number of real
points where it meets the branch y=e"°°", and which must
be considered as conjugate points on the curve. There are an
infinity of such points, all lying either on the real branch of the
curve, or on the similar branch on the negative side of the axis
of z The latter branch is curious, since, though every point of
it may be considered as belonging to the logarithmic curve, no
two points of it are consecutive to each other, for two consecu-

for its value, being in general , becomes for this curve =m.

TRANSCENDENTAL CURVES. 287

tive points will belong to different branches. There is thus
formed what M. Vincent calls a “courbe pointillée.” In one
point, however, M. Vincent appears to me to have fallen into a
grave error. He says that the points of this branch are to be
carefully distinguished from conjugate points; for that at a con-
jugate point the differential coefficients have imaginary values,
but that at one of these points, on the negative side of the axis,
the differential coefficients, being all equal to e”, are all real, and
only differ in sign from those of the corresponding points on the
positive side of the axis. It is truly astonishing that M. Vincent
should have failed to observe that if the differential coefficients
were all real, it would follow from Taylor’s theorem that the
next consecutive point must be a real point on the curve, and so
that the negative branch would be an ordinary branch of the
curve. But, in fact, any one of these negative points must be
considered as belonging to a branch whose equation is of the
form y=e@O"™™, and the corresponding differential coefli-
cient will be y(1+2mm). Considering, then, an acnode in
general as the intersection of imaginary branches, in the same
manner as a crunode is the intersection of real branches, the
points here in question being points of intersection of imaginary
branches seem properly regarded as acnodal. We have already
seen that a transcendental curve may have an infinity of nodes
or acnodes, and, in the case of epitrochoids, that such points may
be ranged in a discontinuous manner on certain loci.*

315. The catenary is the form assumed by an inelastic chain
of uniform density when left at rest. Very simple mechanical
considerations lead to the property, which we shall take as the
mathematical definition of the curve, viz. that the arc, measured
from the lowest point, is proportional to the tangent of the
angle made with the horizontal tangent by the tangent at
the upper extremity. If, then, the axes be a vertical and a hori-

zontal line through the lowest point, we have sod - Now,

* The illustration here used is Dr. Hart’s. Some objections to M. Vincent’s views,
which are worth being considered, will be found in a paper by Mr. Gregory, Cambridge
Mathematical Journal, vol. 1. pp. 231, 264. Prof. Cayley considers that e* (which he
writes by preference exp.x) is a true one-valued function of 2, and that there is
nothing else than the real branch, the values being those of the function

x2 x3
1+ itpstpost &

288 TRANSCENDENTAL CURVES.

to rectangular axes the element of the are is the base of a
right-angled triangle, of which dx and dy are the sides, or
ds*=dx"* + dy’. By the equation of the curve we shall have,

therefore,
‘ , as’ cds
2 ee: Bede a
S+O =e aa, dx Verte)?

= log {24 ma

?

the constant being taken so that s and a shall vanish together.
Hence

fet ale eo Aas

So gia ga ulead p eg Pad

C c
But in like manner the equation of the curve gives
etc as’ sds

3 = 745 dy See a
s dy ‘/(s° + ¢°)
Hence y’=s°+ cc’, provided we suppose the axes so taken that
when s or x=0, y shall be =c. This value of y gives at once

the equation of the curve, viz. :

9 eee la _z
\ Rage’ (e+e).
A very convenient notation is
4(&+eé")=cosha, 4 (e°—e~)=sinhex

(read hyperbolic cosine and sine); we have then for the catenary

x ae
y= cosh — , s=csinh-.
c

316. We get from the equation of the curve
Oe ee

ax =} (e&-e *) , ms
Hence we are led to the follow-
ing construction. From the foot
of the ordinate / draw the tan-
gent MT to the circle described
with the centre C and radius c;
then MC=y, CT=c, MT=
V(y?—c*); tan MCT =tan MTL

v(y'—¢)

= “+; hence the tangent
c

TRANSCENDENTAL CURVES. 289

PS is parallel to MZ. The same values prove also that
PS=MT=the arc from P to the lowest point. The locus of
the point Sis therefore the involute of the. catenary, and SN
parallel to 7'C is its tangent, since PS must be normal to the
locus of S, being tangent to its evolute. The involute of the
catenary is therefore a curve such that the intercept SN, on
its tangent between the point of contact and a fixed right line,
is constant.* Such a curve is called the tractriz.

317. The equation of the tractrix can be obtained without
much difficulty. For the length between the foot of the ordinate
from § and the point N is /(c’—y’); it also is, by making y =0
in the equation of the tangent, — ee Hence the differential

equation of the curve is

which at once is made rational by putting 2’ =c’— y’, and gives

ip BIO) 0 ay:

Co — 2

We have then

Bice log | Mea} yey),

It will be readily seen that the curve consists of four similar por-
tions, as in the dotted curve on the figure; and the construction
of the last Article shows at once geometrically how to draw a
tangent to the curve.

Lhe syntractriz is the locus of a point Q on the tangent to
the tractrix, which divides into portions of given length the
constant line SN. Let the coordinates of the point on the
tractrix be ’y’, of those on the required locus zy; let the length
QN =d, then we shall have 7’d=yc; and

V(e—y")-V(@—-y)=2-2';

* The form of equilibrium of a flexible chain was first investigated by Galileo, who
pronounced the curve to be a parabola. His error was detected experimentally in 1669
by Joachim Jungius, a German geometer; but the true form of the catenary was only
obtained by James Bernoulli in 1691. Gregory (in his Examples, p. 234) refers to
what would seem to be an interesting memoir by Professor Wallace on this curve
(Edinburgh Transactions, vol, XIV. p. 625).

290 TRANSCENDENTAL CURVES.

and since, by the equation of the tractrix,

xe’ +(?—- y”)=c log are M,

that of the syntractrix will be
; d -y¥
e+ /(d’—y")\=c log | aa J : ;

The tractrix is a particular case of the general problem of
equi-tangential curves, where it is required to find a curve such
that the intercept on the tangent between the curve and a fixed
directrix shall be constant.

318. The problem of curves of pursuit was first presented
in the form-——To find the path described by a dog which runs
to overtake its master. It may be stated mathematically as
follows: The point A describes a known curve, and it is re-
quired to find the curve described by the point B, the motion
of which is always directed toward A. We suppose both
points to move with uniform velocities, and A to move along
a right line which we take for axis of y.* The intercept made

by the tangent on this axis of y is y — x = , and by hypothesis

the increment of this is to be proportional to the increment of
pee

the arc, or putting A =,

—- dp =h /(1+ p’) da,
log 2+ log {p+ (14+p’)}+log A=0,
2p = Ata" — Az’,
A ie ET dg

This curve will then be algebraic, except in the case when 4=1,
—n+1

when we have to substitute log x for — eeu s

319. The cnvolute of the circle is another transcendental curve
whose equation can be obtained without much difficulty. This

* See Bouguer, Mémoires de 1 Académie, 1732, Correspondance sur U école polytech-
mique, 11. 275. St. Laurent, Gergonne’s Annales, x111, 145.

TRANSCENDENTAL CURVES 291

is equivalent to the following problem: “If on the tangent at
any point P of a circle there be taken a portion PQ, such that
it shall be equal to the arc AP measured from any fixed point
A; to find the locus of Q.” Let the radius of the circle=a,
the centre being C and the radius vector

CQ=p; lett PCA=¢, QCA=0. Then \
PQ=(p’—a’); and it also =a¢ by hy- e

pothesis ; but lo

a]

=n Bd cos? —.
? p

Hence the polar equation of the locus is
2 2
v(p'—@) po @ + cos” . RN
a p
The involute of the circle is the locus of the intersection of tan-
gents drawn at the points where any ordinate to CA meets the
circle and the corresponding cycloid having its vertex at A.

320. We shall conclude this Chapter with some account of
spirals. In these curves referred to polar coordinates, the radius
vector is not a periodic function of the angle, but one which
gives an infinity of different values when we substitute w= 0,
wo=27r+6, o=4r+6, &. The same right line then meets
the curve in an infinity of points, and the curve is transcendental.
Let us first take the spiral of Archimedes, which is the path
described by a point receding uniformly from the origin, while
the radius vector on which it travels moves also uniformly round
the origin. The polar equation of the curve is then

p =a.
This spiral is the locus of the foot of the perpendicular on the
tangent to the involute discussed in the last Article. For, from
the nature of evolutes, the tangent to the locus of @ is per-
pendicular to PQ; and the length of the perpendicular on
that tangent from C will =PQ=ad, and ¢ is the angle this
perpendicular makes with a fixed line. Hence, too, the reci-
procal of the involute is the hyperbolic spiral pw =a, which we
shall discuss in the next Article. The spiral of Archimedes is
one of a family included in the general equation p=ao’, in all
which the tangent approaches more nearly to being perpendicular

292 TRANSCENDENTAL CURVES.

to the radius vector the further the point recedes from the origin.
For — = - ; therefore (Art. 95) the tangent of the angle made

by the radius vector with the tangent increases as w increases,
but does not actually become infinite until @ is infinite,

321. We have just mentioned the equation of the hyperbolic
spiral pw=a. ‘This spiral has an asymptote parallel to the
line from which » is measured; for the perpendicular from any

a sin@

point of the spiral on this line is p sinw = , which, when

w vanishes, and p becomes infinite, has the finite value a. Or,
again, we might calculate the length of the perpendicular from
the origin on the tangent. The tangent of the angle made by

the radius vector with the tangent is ae =—w; hence the
p
; ap : Hy
perpendicular 8 Ta +p) which, when p becomes infinite, is
=a. The form of the curve is
then as here given. ‘The polar A

subtangent of the hyperbolic spiral

is constant. The are AB of the

circle described with the radius

OA to any point of the curve is oY B
obviously constant.

Another spiral worth mentioning is the litwus po=a;
this also has an asymptote, viz., the line from which @ is
measured; for the distance of any point of it from this line,

a’ sin®

, decreases indefinitely as p increases, and

p sino =
consequently diminishes.

322. We shall mention in the last place the logarithmic
spiral, p=a®. In this curve p increases indefinitely with w; when
w is 0 it =1, and diminishes further for negative values of @,
but it does not vanish until w becomes negative infinity ; hence
the curve has an infinity of convolutions before reaching the
pole. One of the fundamental properties of this curve is, that

. 2 d
it cuts all the radii vectores at a constant angle, for abi becomes

TRANSCENDENTAL CURVES. 293

the modulus of the system of logarithms which has a for its
base; the angle, therefore, made by the radius vector with the
tangent always has this modulus for its tangent. From this
property we at once obtain the rectification of the curve ; for if
we consider the elementary triangle which has the element of
the are for its hypothenuse, and the increment of the radius
vector for one side; we see that the element of the are is equal
to the increment of the radius vector multiplied by the secant
of this constant angle, and hence that any are is equal to the
difference of the extreme radii vectores multiplied by the secant
of the same angle. ‘The entire length, measured from any point
P to the pole being p sec@, is constructed by erecting at the
pole O@ perpendicular to OP to meet the tangent at P;
PQ will then be the required length. The locus of Q will
evidently be an involute of the curve, but the angles of the
triangle OPQ being constant, OQ is proportional to OP,
and it makes with OP a right angle; the locus of @Q is
therefore also a logarithmic spiral, constructed by turning round
the radii vectores of the given curve through a right angle,
and altering them in a fixed ratio. Conversely, the evolute
of a logarithmic spiral is a logarithmic spiral. The locus
of the foot of the perpendicular on the tangent is likewise a
logarithmic spiral, for it also bears a fixed ratio to the radius
vector, and makes with it a constant angle. The caustics by
reflexion and refraction, the light being incident from the pole,
are likewise logarithmic spirals.*

* The logarithmic spiral was imagined by Des Cartes, and some of its properties
discovered by him. The properties of its reproducing itself in various ways, as stated
above, were discovered by James Bernoulli, and excited his warm admiration.

( 294 )

CHAPTER VIII.

TRANSFORMATION OF CURVES.

323. HAvING in former parts of this work explained par-
ticular methods by which the properties of one curve may be
derived from those of another, such as the methods of Projection,
of Reciprocal Polars, of Inversion, &c., we purpose in this
chapter to consider the general theory of such methods. In
such methods we have in general to consider the correspondence
of two points P, P’ which may be either in the same plane or in
different planes. In the latter case the two planes may be
regarded as existing in a common space, and the two points
P, P’ may be connected by geometrical relations in such space.
For example, in the method of Projection the line joining the
points P, P’ is subject to the condition of always passing through
a fixed point. O. Similarly, we should have another system of
transformation if the line PP’ were subject to the condition of
always meeting two fixed lines; and so forth. The development
of such theories belongs to solid geometry; here we consider the
two planes as existing zrrespectively of any common space. To
take the simplest example, suppose that we have a pair of axes
in one plane, and another pair of axes in the other plane; and
that the coordinates of P referred to the first pair of axes are to
be always respectively equal to the corresponding coordinates of
P” referred to the second pair of axes, we have evidently a system
in which to any point P in the first plane corresponds a point P”
in the second, and vice versa.

The two planes may be regarded as superimposed one on the
other, and so as forming a single plane. Supposing this done,
there will be theorems dependent on the superimposition of the
two planes; besides these there remain the theorems which
existed when the two planes were distinct, and the theory is not
really altered. Or, to express this otherwise, instead of two

LINEAR TRANSFORMATION. 295

figures in different planes, we have two figures in the same
plane, where by the word figure is meant any system of points,
lines, or curves; or, it may be, all the points of the plane. ‘The
kind of transformation chiefly studied has been the rational
transformation ; viz., where to a given position of P corresponds
in general a single position of P’, and to a single position of P’
a single position of P. The most simple instance of this is the
linear or homographic transformation, which we proceed to
consider in detail.

LINEAR TRANSFORMATION,

324, Let the coordinates of P referred to any system of
axes in the first plane be x, y, 23 and let those of P’ referred
to any system of axes in the other plane be a’, 7’, 2’; then
the correspondence of the two points is said to be linear if
the latter coordinates are proportional to linear functions of the
former

aeiy:2=axntbytce: dat Vy+cz2:a"at+b'y4+c"2,

by solving which equations we have evidently also linear
expressions for x, y, 2 in terms of a’, 7, 2’,

w:y:2= Ax + By +Ce: Ao’ + By +02: Ae 4+ BY 4 Cw.
It is easy to see that, properly assuming as well the funda-
mental triangles as the ratios of the implicit constants, these
equations may, without loss of generality, be written in the form
x: 4:2 =«:y:2. Thus then to any position of either point cor-
responds a single position of the other. If P describes any curve
¢ (x, y, 2) =0, by substituting in this equation the values of x, y, z
just written, we obtain the equation of the curve described by
P’. This latter equation is evidently of the same order as the
former ; therefore, to any curve in one plane corresponds a curve
of the same order in the other; in particular, to a right line
in one plane corresponds a right line on the other. It is
also obvious, that to a node or cusp on one curve will answer a
node or cusp on the other, so that two curves corresponding in
this method will have the same Pliickerian characteristics. Since
x’, y’, 2 expressed in terms of x, y, 2 contain each three con-
stants, there are nine constants employed in this method of

296 LINEAR TRANSFORMATION,

transformation; but since we are only concerned with the
mutual ratios of x’, y’, 2’, one constant may be divided out, and
the method of homographic transformation is to be regarded as
involving eight arbitrary constants.

325. To a pencil of four lines meeting in a point corresponds
a pencil whose anharmonic ratio is the same. For it was shewn
(Conzcs, Art. 59) that the anharmonic ratio of four lines a — 8,
a—/lB, a—mB, a—n§, is a function only of kh, 1, m, n, and
therefore is the same as the anharmonic ratio of a — kf’, &c.
Similarly to four points on a right line correspond four points
whose anharmonic function is the same. And it hence appears
how given any four points of the first figure and the correspond-
ing points A’, B’, C’, D’ of the second figure, we can construct
the point P’ which corresponds to any other point P of the first
figure. For the anharmonic ratio of the pencil A’ (B’, C’, D’, P’)
is equal to that of the pencil A (B, C, D, P), and we can hence
construct the line A’P’; similarly we can construct BYP’, C’P’,
YP", and the four lines will of course meet in a point which is
the point P’. The construction is applicable whether the two
planes are distinct or superimposed.

326. Let us now suppose the planes superimposed, and in-
vestigate another geometrical construction to express the relation
between corresponding lines and points. Let A,B, C be the ver-
tices of the triangle formed by the lines x, y, 2; and A’, B’, C’
those of the triangle formed by the corresponding lines 2’, y’, 2’ ;
then since all lines through A form a system homographic with the |
corresponding lines through A’, the locus of the intersection of
corresponding lines is a conic. Or, analytically, since the line
y + kz corresponds to y’ + kz’, eliminating 4, the locus of inter-
section is yz’=y’z. In like manner all lines through B and
through C meet the corresponding lines on the fixed conics
ze’ —az’, xy’— ya’. The construction thus assumes that in
addition to three pairs of corresponding points A, A’; Bb, B’;
C, C’, we are given three fixed conics each passing through a
pair of corresponding points; and the form of the equations

> ae 7 te 3 shows that these three conics have also three

&

LINEAR TRANSFORMATION. 297

common points. In order then to construct the point of the
second system corresponding to ay point P of the first, let
‘the line PA meet the curve yz — zy’ in the point J, then ‘AF
is the line corresponding to PA; similarly, let PB, PC meet
respectively the conics za’ — az’, ay’—yzx’ in points G, H; and
BG, C’H will be the respectively corresponding lines. The
three lines A’, B’G, C’H will have a common point P’, which
will be the required point corresponding to P. The line cor-
responding to any given one is constructed by constructing for
the points corresponding to any two points on it.

327. In the foregoing method the relation between two
points is in general not reciprocal; that is to say, if to P in the
first system corresponds P” in the second, it will not be true that
to P’ considered as a point in the first will correspond P in the
second. In fact, if we consider P as belonging to the second
system, we construct the corresponding point, as in the last
article, by joining P to A’, B’, C’: let the joining lines meet
the respective conics in 2”, G’, H’; then to PA’, PB’, PC’ will
correspond lines in the first system AZ”, BG’, CH’ meeting in
a point P” which will ordinarily not be identical with P’.

Consider, however, the three points LZ, IM, N which are
common to the three conics y’z—2’y, 2a@—a’z, ay—y'x, then
the construction shews that to the lines LA, LB, LC, answer
respectively the lines L.A’, LB’, ZC’. It follows that the two
systems have common the three points L, M, N; each of these

' points, considered as belonging to one system, having itself as

the corresponding point in the other system. In like manner
the lines joining these points are evidently the same for both
systems. And starting with the points Z, M, N as given, then
if we have a single pair of corresponding points we can at once,
in virtue of the theorem, Art. 325, construct the point in either
system corresponding to any point whatever of the other system.

If we express the equations in trilinear coordinates, assuming
these three lines ZW, MN, NZ as lines of reference, then since the
equations in the second system, answering to e=0, y=0, z=0 in
the first, are still to represent the same lines, they can only differ
from these by constant multipliers, and must be of the form
le=0, my=0,nz=0. Thus, then, by a suitable choice of lines

QQ

298 LINEAR TRANSFORMATION.

of reference, homographic correspondence. may always be
expressed in the form that to any point 2’, y’, 2’ in the first
system corresponds the point lx’, my’, nz’ in the second; and
homographic transformation is then effected by writing in the
equation of any curve lr, my, nz instead of x, y, 2 respectively.
We cannot here, as in Art. 324, write #39: 2’ =a:y: 2, for
the two figures would then be identical.

3828. The method of Projection is a case of this homo-
graphic transformation. In this method the line joining any
two corresponding points passes through a fixed point, viz.,
the vertex of the projecting cone; and any two corresponding
lines intersect on a certain fixed line, viz., the intersection of
the two planes of section. If one of the planes were turned
about this line so as to be brought to coincide with the other,
the figures would still have the property that the line joining
two corresponding points would pass through a fixed point;
for consider the triangles formed by three pairs of corresponding
lines ; and since the corresponding sides intersect in a right line,
the lines joining corresponding vertices meet in a point. It is
easy to form the most general equations of such a system. Let
ax + by +cz=0 be the equation of the line on which the cor-
responding lines intersect, then it is evident that the equations

be Re

of x’y’z’ (the lines corresponding to xyz) will be of the form

x =aua+by +cz =0,

y =ax +b y+cz =0,

2 =ax +by +¢z2=0,
a system involving three constants less than in the general case,
and therefore only five in all.

We shall call the point at which the lines joining corre-
sponding points meet, the pole of the system, and the line on
which corresponding lines intersect, the axis of the system. By
subtracting one from the other successively each pair of the
equations just written, it will be seen that the pole of the system
whose equations we have written is given by the equations

(a-—a@)x=(b-D')y=(c—Cc)z.
The simplest forms of the equations of projective trans-
formation are derived as follows: Any line passing through the

LINEAR TRANSFORMATION. . 299

‘pole is the same for the new figure; for any two points of
it have corresponding to them two points on the same line.
Hence if the pole be taken at the point wy, the two lines a
and y are unaltered by transformation; and any other line,
Az+ By+ Cz=0, has corresponding to it, Ax+ By+ Cf=0,
the two lines intersecting on the fixed axis, z—-€=0. Any
line Ax+ By =’ passing through the pole evidently remains
unchanged.

329. Conversely, if two homographic figures in the same
plane have the property that any corresponding lines intersect
on a fixed axis, one of the figures may be considered as a
projection of the other. For let the plane of one of the figures
be turned round this axis, and consider any three pairs of
corresponding points ABC, abc, the corresponding sides of these
triangles intersecting in L, WM, N. ‘Then, when the plane is
turned round, Aa, 6) must still intersect (since the lines AD,
ab intersect in NV, and are therefore in the same plane); and by
the theory of transversals Aa when produced is cut by 5d in the
same ratio as before the figures were turned round. But in
like manner Cc, and the line joining any other pair of cor-
responding points, meets Aa in the very same point.

330. The general homographic method of transformation,
containing three constants more than the projective method,
appears at first sight a more powerful instrument of research,
and we should expect to arrive, by its means, at extensions of
known theorems more general than those with which the method
of Projection had furnished us. It is obvious, however, that
if a figure were transferred bodily to some other position, we
should have a linear transformation, in which to every line of
the first figure would correspond a line of the second figure, but
yet which would give us no new geometrical information. Now
we owe to M. Magnus the remark, that the most general trans-
formation may be reduced to a projective transformation by
turning the figure round a given angle, and then moving it
for a given length along a given direction; these three latter
constants being just the number by which the transformation
appears to be more general than the projective.

300 LINEAR TRANSFORMATION.

To see this, we must first observe, that if a” figure be moved
in any direction without twisting, since all lines remain parallel
to their first position, the position of every point at infinity
remains unafiected by the operation.

Next, let the whole figure be made to turn round any fixed
point, and any system of parallel lines will still remain a system
of parallel lines, although no longer parallel to its former direc-
tion; hence, any point at infinity will still remain at infinity, and
therefore the line at infinity is the same for the figure in both its
positions, Moreover, since any circle will remain a circle, how-
ever it be moved, we see that the two circular points at infinity
will not be disturbed, no matter how the figure be moved.

If then it be required to move a figure so as to have a projec-
tive position with a given homographic figure, let the two circular
points be @, w’, the two corresponding points of the second figure
0, 0’, since no motion of the first figure can alter the position of
w and w’, the only possible position of the required pole of the
two figures is the point A, where the lines ow, o’w’ intersect. Let
then the first figure be moved so as to bring the point 2, which
corresponds to A, to coincide with it. Moreover, let the first
figure be turned about 7 so as to bring m, mw (any other pair of
corresponding points) into a line with 7; then we say that the
two figures will have a projective position, and the line joining
any other two corresponding points, 2, v, must also pass through J.
For the anharmonic ratio of {l.@w’puyv} = {l.00’mn} (Art. 325),

and since three lines of the system are the same for both, the

fourth must also be the same for both. M. Magnus’s theorem
has then been proved.

831. There is no difficulty in expressing analytically the
geometrical theory of the last article. Thus if it be required
to find the coordinates of the point 7 in the case of the general
transformation, we are, first, by the theory just laid down, to
find the line ow joining the point (# + dy, z) to

[jaw + by + cati(aae+bytez)}, act by +¢,2),
this will be

(b,—ta,) {(axe + by + cz) + t(a,e+b,y +¢,2)}
oa {a, + b+12(b, pas a)} (a,c + boy + ¢,2) =0,

Pe et om,

INTERCHANGE OF LINE AND POINT COORDINATES. 301

or (ab,— a,b) «+ (a,b, — a,b.) y + {(cb, — ¢,b) + (¢,4, — ¢,a,)} 2
+ ¢{(a,b, — b,a,) #+ (ab, — a,b) y + (¢,b, — b,c,) 2 + (ac,—ca,) 2} =0.

The line joining o’o’ will only differ from this in the sign of
the quantity multiplying 7 The point required is therefore the
intersection of the two lines found by putting the real and
imaginary parts of the equation separately = 0.

It is not necessary to dwell on particular species of linear
relation, such, for example, as similarity. We only mention
one kind of homographic relation, in which the area of any
space on the one figure is equal to that of the corresponding
space on the other figure. It is easy to see that such a transfor-
mation is possible. For let the triangle formed by xyz be equal
to that formed by x’y’2’, then, if we take any point O on the first
figure, it will be easy to determine a corresponding point o on the
second, such that Oxy=ox'y’ and Oxrz=oz2'2’; and therefore that
Oyz =oy'z’; and the triangle formed by any three points OPQ
will be equal to that formed by opg, the corresponding points
so determined. |

This species of homographic relation differs from orthogonal
projection just as the general collinear relation differs from
projection in general.

« . INTERCHANGE OF LINE AND POINT COORDINATES.

332. In the method of transformation just described and in
the others to be considered in this chapter, point corresponds to
point, and line to line; but there are transformations where a
point in the one figure corresponds to a curve in the other
figure. We have such a transformation in the method of
Reciprocal Polars, in which point corresponds to line and vice
versd. And the like is the case in the more general homo-
graphic transformation, or say in the theory of skew re-
ciprocals, which is as follows: Let there be any system of
point-coordinates xyz, and a system of line coordinates aPy,
in the same or in a different plane; then a point in the
first system corresponds to a line in the second, if the co-
ordinates x, y, 2 of the point are respectively proportional to
a, 8, y, the coordinates of the line. In the same case to
any line /e+my+nz in the first system corresponds the point
la+mB+ny in the second. Plainly, then, to four points in

302 INTERCHANGE OF LINE AND POINT COORDINATES.

a line will correspond a pencil of four lines having the same
anharmonic ratio; for the anharmonic ratio of y- lx, y—mz,
y — nx, y — px, is the same function of J, m, n, p, whether a and
y denote point- or line-coordinates. ‘The method now described
may be combined with any of the other transformations described
in this chapter; that is to say, in any of them, one of the
systems of coordinates may be supposed to be changed from
point- to line-coordinates; and in this way we can get all
possible transformations in which point answers to line and
line to point.

333. Let us now suppose the two systems to be in the same
plane, and let us endeavour to express the transformation
altogether in point-coordinates. To any point 2’y’z’ is to corre-
spond a line whose coordinates referred to a certain system of
line-coordinates a8y are 2’, y’, 2’. But this is equivalent to
saying that its equation is to be a X+y’/Y+2Z=0, where
X=0, Y=0, Z=0 denote the lines joining the points repre-
sented by a=0, B=0, y=0. And these being known lines,
the equation of the line answering to the point a’y’z’ must be
of the form
ac’ (a,x 2 by a9 C,2) + y (a,x + by + C,2) + z (a,x + by + C,2) =0.
This is an equation involving eight constants, and would
coincide with the equation of the polar of a point with regard
to a conic section, only if b,=a,, c,=a,, 6,=c,; the equation
in this case involving but five constants.

334. In the general case every point has a different line cor-
responding to it according as the point is considered as belonging
to the first or to the second system. Thus the equation just
written expresses the relation between any point 2‘y’z’ of the
first system and any point xyz on a corresponding line of the
second system. If now the latter point be fixed, and the former
variable, we have, for the equation of the line of the first
system corresponding to any point of the second,

(a, + by’ +¢,2)at+ (av + by’ + 0,2’) y + (ax + boy’ + ¢,2’) 2 =0.
In the case of reciprocals with regard to a conic, the same

line corresponds to a point, whether that point be considered as
belonging to the first or to the second system. 3

INTERCHANGE OF LINE AND POINT COORDINATES. 303

335. In order to give, in the general case, a geometric con-
struction for the line corresponding to any point, we shall first
seek for the locus of the points which lie on their corresponding
lines. ‘This is obviously
a,x" + (a, +b,) vy + by’ + (b, + ¢,) ye + (a,+¢,) v2 4 ¢,2°= U=0,
and is the same conic whether the point be considered as belong-
ing to the first or to the second system. We shall call this the
pole conic.

Next let us seek the envelope of lines which pass through
their corresponding points. The line Xa’ + wy’+ v2’ (where a’y’2’
is a point on the conic just written) touches (see Conics,
Art. 151)

(0, + ¢,' + 2b,c, — 4b,c,)

+ (44,0, + 4a,c, —2a,a, — 2a,c, —2b,a, — 2b,c,) wv
+ (a, + ¢,? + 2a,c, — 4a,c,) p” |

+ (4b,a, + 4b,c, — 2a,b, — 2a,c, — 2b,b, — 20,c,) vr
+ (a,’ +b," + 2a,b, — 4a,6,) v’

+ (4a,c, + 4b,c, — 2c,c, — 2a,b, — 2c,b, — 2c,a,) Aw = 0.
The envelope is therefore a conic, which we shall call the polar
conic, and which is also the same whether the lines in question
belong to the first or to the second system.

Using now the words pole and polar to express the kind of
correspondence we are here considering, we have at once the
polar of any point on the pole conic. For from that point draw
two tangents to the polar conic: one of these is the polar
when the given point is considered to belong to the first system ;
the other, when it is considered to belong to the second system.

Or, conversely, to find the pole of any tangent to the polar
conic. We have only to take the two points where this line
meets the pole conic; one of these points is its pole in the first,
and the other in the second system.

Let it be required now to find the polar of any point O.
Draw from it two tangents, OT’, OT., to the polar conic. Let
OT, meet the pole conic in the points A,A,, and let OZ, meet
it in the points BB, Then if A, be the point in the first
system which corresponds to O7,, and B, that which corresponds

304 INTERCHANGE OF LINE AND POINT COORDINATES.

to OT,, plainly A,B, is the line in the first system which
corresponds to O, considered as belonging to the second
system; that is, A,B, is one of the polars of 0. Similarly,
A,B, is the other polar of O.

Or, to find the pole of a given line meeting the pole conic in
the points 45, from these draw tangents AP, AP,, BQ, BQ,
to the polar conic; and if AP, BQ, be the lines in the first
system, which are the polars of A, , their intersection gives the
point in the first system, which is the pole of AB. And, in
like manner, the intersection of AP,, BQ, gives the point in
the second system, which is the pole of AB.

The reader will readily see how these constructions reduce
to the ordinary polar reciprocals if a,=0,, b,=c,, c,=a, The
pole and polar conic will then coincide; the polar of any point
on that conic is the tangent at that point, and the polar
of any other point is the same for both systems, and is the
line joining the points of contact of tangents from the point
to the conic.

336. It follows at once from these principles that in the
general case the pole conic and the polar conic have double
contact with each other. For, take any point of intersection,
its two polars coincide with the tangent at that point to the
polar conic; the two poles of this line must therefore coincide,
and therefore the two points where it meets the pole conic must
coincide, therefore the tangent to the polar conic at their inter-
section must touch the pole conic also. The same thing is
proved for their other point of intersection. Prof. Cayley has
proved the same thing analytically, by shewing that if U=0
be the equation of the pole conic, that of the polar conic (found
by putting for A, w, v their values, in the equation of the last
Article) may be thrown into the form

{x (2,0, Ras a,b, + aC, — a,C,) Ty (6,¢, ae b,c, 7 b,, si b.a;)
+ 2 (C1, — ,0, + c,5, at, ¢,,)}*
5

+4U. {@, (c,b, a b,C,) 7 a, (4,c, ge b,¢,) bi a, (,¢, i b,c,)} ae 0,

a form which shews at once that it has double contact with U.

aie
“’ 3 ae

”

INTERCHANGE OF LINE AND POINT COORDINATES. 305

337. There are, in the general case, three points whose polars
are the same with regard to both systems. For let the equations
of the polars in each system be

Au + py +v2=0, and Vx+ p’y+ v¥2=0,

then the system of equations

— ee i

is manifestly satisfied for three points; and the theory laid
down in the last Article shews at once what the three points are.
For the two points of contact of the pole and polar conics have
each the same polar in both systems, viz., the common tangents
at these points ; and the point at which these tangents intersect
has also the same polar in both systems, viz., the chord of
contact of the conics.

There are then three points which have the same polar in
both systems; and two of these points lie on their polars, but
the third does not.

338. It is desirable to shew that in the constructions which
we have given no ambiguity occurs, and that we need be at no
loss to know, of the two poles of a given line, which belongs to
the first, and which to the second system.

Since two conics having double contact may always be pro-
jected into two similar concentric conics, we use these in the
figure for greater simplicity.

Let A, B be the two poles of any
tangent to the polar conic, then of the
two poles of any other tangent J’, B’,
A’ will belong to the first system, since
if AL were moved round to coincide
with A’B’, A would coincide with 4’, and B with BY. The dis-
tinction between the points may be readily made by the help of
the following theorem: ‘ A’B and AB’ are parallel in the case
of two concentric conics; and by the method of projections, in
the general case, intersect on the chord of contact of the conics.”

Reciprocally, if we draw tangents to the polar conic from two
points on the pole conic, we must so number them, oa,, 0a,, pb,

RR

306 INTERCHANGE OF LINE AND POINT COORDINATES.

pb,, that the line joining the intersection of oa,, pb, to that of
oa,, pb, may pass through the pole of the chord of. contact of
the conics.

339. The number of constants in the case of skew recipro-
cals only exceeding by three the number of constants in the case
of reciprocals with regard to a conic, it is natural to inquire
whether the latter does not only differ from the former by
displacement of the figure. It is evident, at any rate, that the
skew reciprocal here considered is only a homographic trans-
formation of the reciprocal with regard to a conic, and that
therefore the use of skew reciprocals can lead to no geometric
theorem which we might not obtain by combining the use of
ordinary reciprocals with the method of projections.

It is very easy to see what must be the first step if it be
required to move the two figures into such a position that the
polar of every point may be the same, no matter to which system
that point be considered to belong. For, since the position of the
line at infinity is unaffected by any displacement of the figure,
we must begin by taking its pole in each system, and then
moving the systems so that these points shall be brought to
coincide. ‘The pole and polar conics will then become concentric
and similar, this point being their common centre.

340. Now we say, that if by turning the figures round their
common centre O, they can be given such a position that the
polar of any point A at infinity shall be the same line OB for
both systems; then if the polar of any other point C at infinity
be the line OD for the first system, it must be also so for the
second system. For the anharmonic ratio of the four points of
the first system ALCD is equal to the corresponding pencil
of the second system, viz. OB.OA.OD.OX; and since three
legs are the same in two pencils, OX must coincide with OC,
or the polar of the pomt ) must be the same whether it belong
to the first or second system; so also must then the polar of C.

Since now the circular points at infinity are unmoved by any
turning of the figure, we have only to take the two polars of
either ‘of these points, which im general will not pass through
the point, and turn either figure round, so as to bring these

INTERCHANGE OF LINE AND POINT COORDINATES. 307

polars to coincide; and then, from what has been just proved,
the polars of every other point will coincide.

341. We can readily obtain an expression for the ange
through which the figure is to be turned. ‘The two figures
being in a concentric position, and the origin being the centre,
it is readily seen that the most general equations of the two
polars of any point are

(a,2° + by’) x + (aa + by’) y + ¢,= 0,
and (a,x' +a,y') av + (b,x + b,y') y +c, = 0.
The two polars of the point at infinity, for which y’ = 72’, are
(a, +2b,) w+ (a, +2b,) y =0,
and (a,+ta,)e+(b,+2b,) y=0;
and the angle through which one of these lines must be turned

to coincide with the other is the difference of the angles whose
tangents are

a, + 0, a, + ta,
preys iets 7, Gane eee ee
a, +b, b, + tb,’

as . a,—b
but this is the real angle whose tangent is r. A i

x 2

342. Or the same result may more simply be obtained as
follows: If in general the line of the second system corre-
sponding to the point ’y’ in the first be

(a0 + by’) w+ (a,a' + dy’) y + ¢,=0;
then, when the second system is turned round an angle @, the
equation of this line will become

(a,x'+b,y') (a cosO—y sin@) + (a,x + b,y’) (a sin? + y cos@) +¢, = 0,
or {(a, cos0+a, sin@) x +(b, cos@+ 6, sin@) y'} x
+ {(a, cos 0 — a, sin @) x’ + (b, cos0 — 5, sin 8) y'} y + c, = 0.

But the locus of points of the first system whose polars pass
through «’y’, that is to say, the line corresponding to 2"y’,
considered as belonging to the transformed system, will be

{(a, cos? +a, sin@) x’ + (a, cos@ —a, sin@) y'} x
+ {(2, cos@ + b, sin @) x’ +(, cos? —d, sin@) y’} y+0,=0.

308 © QUADRIC TRANSFORMATION.

This line will always coincide with the other, if we have
b, cos@ + b, sin@ =a, cos@—a, sin 8;

| a, — b,

b,+a,

or, as before, tan é =

QUADRIC TRANSFORMATION.

343. Before proceeding to the general theory, it will be in-
structive to consider in detail one other special method, viz. when
the coordinates of the point P’ are functions of the second degree:
of the coordinates of P, or say in which 2:7’: 2=U:V: W.
Thus to the lines x=0, y=0, 2=0 will answer three conics
U=0, V=0, W=0; and, in general, to a curve of the n™
order will answer one of the 2n, whose equation is found’
by substituting U, V, W respectively for x, y, z in the given
equation. We have already used this method, Arts. 252, 272.
A simple example is when the relation between P’ and P is
expressed by the equations a: y’: 2 =a": y’: 2"; then to any
right line lde+my+nz will answer a conic lx? + my?+nz*
touching the sides of the triangle xyz, while to a right line in
the second figure answers also a conic in the first. To a
conic in the first figure (a, b, c, f, g, hax, y, z)* answers the
quartic

ax + by + ce + 2fytat + 2gztxt + 2haty? =0.

And, as the general equation of a conic may be written in the
form

Seg ie tae (ha BN ie a -}
Fog eB OAS, Sgr gG Ighl” NE Fgh) *
it follows that the equation of the corresponding quartic may be

written in the form az?+byt+cz++dwt=0. It is therefore
trinodal and has the lines x, y, z, w for bitangents.

344, The method of transformation just described, wherein
x :y':2=U:V: W is in general not rational. For, given
x, y, 2 we have 2’, y’, & rationally, but when z’, 7’, 2’ are given,
V
yf
represent conics having four common intersections, and therefore

U et
then to find x, y, z we have yy yd equations which

QUADRIC TRANSFORMATION. 309

to any position of the point a’y’z’ answer four positions of
the point xyz. If the conics U, V, W had a common point,
this point being independent of the position of the variable
point 2’y’2’ might be set aside; and to any position of the
one point would answer three of the other. Similarly, if U,
V, W had two common points; and finally, if they have three

4

: | AAS Vania States | ;
common points, the conics oo P, = ae have, besides the three

fixed points, only one other common point. The transformation
is therefore in this case rational, and to any position of either
point answers a single position of the other. It would be a
mere change of coordinates, if instead of the conics U, V, W
we took three conics of the form /1U+mV+n”W, making the
corresponding lines le +my-+mnz our new lines of reference.
There is therefore no loss of generality if we take for U, V, W
the three line-pairs got by joining each of the fixed points to
the two others. The most general rational quadric transfor-
mation is therefore that which we have already used, Art. 283,
where two corresponding points are connected by the reciprocal
relations
e:ysamya i axa say and a :y i 4 ye: 2x: wy.

345. It was stated, Art. 283, that to the point xy will cor-
respond any point on the line 2’ =0, &. If we transform
any curve, to each of the nm points where it meets the line 2’
will correspond the point xy, which will accordingly be a n-fold
point, or, more strictly, to each of the n points corresponds
the direction of a tangent at the n-fold pomt. There will be
a coincidence among these tangents should the original curve
touch the line 2’. To a curve therefore of the n™ degree, which
does not pass through any of the three fixed points 7/2’, 2a’, ay’,
will correspond a curve of the 2n™ degree having the three
points yz, zz, xy as n-fold points. Let us suppose, however,
that the curve passes through the point 4/2’, then the line «
must be part of the corresponding figure, and setting this aside
the order of the corresponding curve is reduced by unity. Also
since the line a passes once through each of the points za, xy,
the corresponding curve will only pass through each of these
points (n-1) times instead of m. And, in like manner, we

310 QUADRIC TRANSFORMATION,

see in general that to a curve of the n™ degree which passes.
through the three principal points, as we shall call them,.
J, g, and fh times respectively, will correspond a curve whose
order n’ is 2n—f—g-—h, and which passes through the three
principal points on the other figure /’, g’, and h’ times re-
spectively, where f’=n—g—h, gf =n-h-f, Kh =n—f—-g.

346. It is easy to verify that the numbers thus assigned
satisfy the reciprocal relation which exists between the corre-
sponding curves; that is to say, that we have also

n=2n' —f'— 9g — h'", f= n’ -g- h’, g= Pens Ag —f", h=n’ —f'-g.
We shall shew also that the two corresponding curves have the
same deficiency. Tor if a curve pass / times through a point,

this is equivalent to $f(f- 1) double points, (Art. 43). Hence
the deficiency of the first curve is

4 {(n—1) w—2)-f(F—-1)—g (9-1) -A(h- DN},
and using the values just obtained for n’, f, 9’, h’, it is easy
to verify that the number just written is equal to

4 {(n’ —1) (Ww —2)-f (fF -1)-¢9 (9 -I)-h'(h— 1)}.

347. A particular case of quadric transformation is the
method of inversion, or transformation by reciprocal radius
vectors, described Art. 122, and Conics, Art. 121 (c). In this
method we have a fixed point O; and corresponding points P,
FP’ lie on a line through O, at distances whose product is con-
stant; say OP.OP’=1. ‘Taking O as origin, it is easy to see
that the relations between the rectangular coordinates of P
and P’ are

ay f

es pe = 545; Z 5 OM
But these equations give
af + ty! = 1 af taf
x—ty’ Aer vy”

Hence, writing
X, Y, Z=a—ty, w+ty, 13 X’, Y’, M=a2' +, vc’ — wv, 1
we have AL Ade BAZ) Gases

QUADRIC TRANSFORMATION. 311

or the transformation is of the kind considered in this section.
The point O is called the centre of inversion; and the circle
whose radius is the square root of the given value of OP.OP’ is
called the circle of inversion, and if P describe any curve, the
curve described by LP” is called the inverse curve.

In particular, the inverse of a right line is a circle passing
through O; viz. if OA is the perpendicular on the line, and
A’ the point corresponding to A, the circle is that which has
OA’ for its diameter. ‘The point O corresponds to the point at
infinity on the line. Again, the inverse of any circle is a circle
(Conics, Art. 121(¢)), and in particular, the inverse of a circle C
which cuts at right angles the circle of inversion is this same
circle C; that is to say, the point P” corresponding to P lies on the
same circle, which is therefore its own inverse. We give this ex-
ample to illustrate a theory which will be more fully considered
In a separate section, where the general theory of transforma-
tion presents itself as a theory of correspondence of points on
a given curve. Here confining our attention to the circle C,
the points P, P’ on it correspond to each other; and in order
to find the point corresponding to a given one P, we have
only to join it to a fixed point O, and take the point where
OP meets the circle again.

348. To return to the general theory of inversion, it is
obvious that two pairs of corresponding points A, A’; B, B’,
lie on a circle which cuts orthogonally the circle of inversion ;
and by the property of a quadrilateral inscribed in a circle,
the line joining two points A, 6b makes the same angle with
the radius vector OA that the line joining the corresponding
points A’, B’ makes with the radius vector OB’. In the
limit, if AB be the tangent at any point A, the corresponding
tangent to the inverse curve makes the same angle with the
radius vector. It follows immediately that the angle which
two curves make with each other at any point is equal to the
angle which the inverse curves make with each other at the
corresponding point.

The inverse is immediately formed of curves included in
the equation p"=a" cosnw. Thus n=2, the lemniscate is the
inverse of the equilateral hyperbola; n=4, the cardioide is the

312 QUADRIC TRANSFORMATION.

inverse of a parabola having the origin for its focus, &e.
The inverse of a conic in general is a trinodal quartic, the
nodes being the origin and the circular points at infinity. If
the origin be the focus of the conic, the inverse is the limagon;
if the origin be on the curve, the inverse is a nodal circular
cubic, the origin being the node. Evidently in general to a
circle osculating one curve will correspond a circle osculating
the inverse curve; but if the circle passes through the origin
the inverse will be an inflexional tangent.

Ex. 1. The three points of inflexion of a nodal circular cubic lie on a right line.
Hence, through any point on a conic can be drawn three circles elsewhere osculating
the curve, and their points of contact lie on a circle passing through the given point.
The three points will be all real when the curve is an ellipse, but if it be a hyperbony
two will be imaginary.*

Ex. 2. In like manner, through any point on a circular cubic or bicircular quartic
can be described nine circles elsewhere osculating the curve, and of these circles three
will be real and their points of contact will lie on a circle passing through the given
point.

Ex. 3. “The feet of the perpendiculars on the sides of a triangle from any point
on the circumscribing circle lie in one right line.” ‘Inversely, if on three chords of a
circle, AB, AC, AD as diameters, circles be described, the points of intersection of
these circles with each other lie on a right line.

‘ Ex. 4. “The circle circumscribing a triangle whose sides touch a parabola passes
through the focus.” Inversely, if three circles be described through the cusp to ree
a cardioide, their points of intersection with each other lie on a right line.

Ex. 5. “Ifa right line meet a /imagon in four points, the sum of their distances
from the node is constant.” Inversely, if a circle through the focus meet a conic
in four points the sum of the reciprocals of their distances from the focus is constant. _

Ex. 6. To find the envelope of circles passing through a fixed point and whose
centres lie on a given curve. Take the fixed point for centre of inversion, and the
locus of the other extremity of the diameters passing through that point is evidently
a curve similar to the given one. It is easy then to see that the negative pedal
(Art. 121) of the inverse of this last curve is the inverse of the required envelope,
and, therefore (Art. 122), that the envelope is the inverse of the polar reciprocal
of that curve.t

349. It remains to mention the cases of rational quadric
transformation which cannot be reduced to the substitution
xi:y:z=y2:2x: ay. Of the three points common to the
conics U, V, W, two may coincide: let the line y be supposed

* This theorem is Steiner’s, see Conics, Art. 244, Ex.8. The proof here given is
Dr. Ingram’s.

+ This example is taken from Dr, Stubbs’s me on this method, Phil. Mag.
nis XXIII, 18,

er

QUADRIC TRANSFORMATION, 313

to be the common tangent to the conics at the point yz, and
let xz be the third point common to the three conics, then

the equation of each must be of the form ax’ + 2fyz+2hxy=0;

we may take a", yz, wy as the three conics, and the substitution
is that used Art. 289, a: 9: 2’ =axy: 2: yz, equations which
imply reciprocally «: y:z=a'y':x":7/2’. In this substitution,
as in the other, to the point az’ corresponds the line y; and
to any curve meeting this line in points will correspond a
curve having the point as a n-fold point. To the point «wy
corresponds the line x, but whatever be the point on this line,
the corresponding direction of tangency will be 7’=0. Toa
curve therefore meeting the line x in z points will correspond
a curve having the point a2’y’ as a n-fold point, at which all
the tangents coincide. ‘The theory, in short, is substantially
the same as before, only modified by the coincidence of two
of the principal points. Again, let all three points coincide, then
(Conics, Art. 239) the equations of the three conics must be of
the form by! + hay + 2f (y2 — me) = 0, and we are led to the
substitution used in Art. 290, viz. x ae a oo y: ye — mi’,
implying reciprocally a : yrt=ay iy”: ye +m2”.

350. Before discussing the general theory of rational trans-
formation, it is convenient to mention, in extension of what was
stated, Art. 347, that the general substitution of X", Y", Z"
for X, Y, Z assumes a simple form when the line Z is at
infinity, and X, Y pass through the two circular points. For,
transforming to polar coordinates, the equations of X and Y

become
p(cos@+7 sin@) =0;
and it is obvious that substituting for these functions their n>
powers is equivalent to substituting p" for p, and n@ for @.
This transformation is not rational, but it may conveniently
be applied to curves of the form p"=a" cosmo, which are
always thus transformed to curves of the same family. For
n=2 a circle becomes a Cassinian, and for n=4 a limacon.
Mr. Roberts has also noticed (Liouville, x11. 209) that the
angle at which two curves intersect is not altered by this
transformation. For the tangent of the angle which the tan-
gent to a curve makes with the radius vector is (Art. 95)
SS

314 GENERAL THEORY OF RATIONAL TRANSFORMATION. -

a and this is unaltered when we substitute ndw for dew

and we for 2 Thus the theorems given as examples of
inversion lead each to as many theorems as we choose to give
different values to x. Theorems also concerning the angles at
which curves cut are easily transformed by this method, as, for
instance, the theorems that a circle is the locus of intersection
of two right lines cutting at a fixed angle which each pass
through a fixed point; that a series of concentric circles are
cut orthogonally by lines through the common centre, &e,

THE GENERAL THEORY OF RATIONAL TRANSFORMATION.

351. We come now to the general theory of the rational
transformation, in which to any, system of values of ayz
corresponds a single system of values of a’y’2’; for example,
ve :y:2=U:V: W, where U, V, W are known functions of
x, ¥, 2, which we suppose to be of the n™ order; and, recipro-
eally, to any system of values for 2’y2 corresponds a single
system of values a: y:2=U’: V’: W’. When such mutual
expression is possible, U’, V’, W’ must be also of the n™ order
in wy'z. For to the 2 intersections of an arbitrary line
le+my+nz with any curve aU+bV+cW will correspond,
in the other system, the intersections of /U’+mV’+nW’ with
the line aa’ + by’ + cz’, which must also be in number n.

352. Let us now examine the conditions that such mutual
expression may be possible. In general, if we are given the
coordinates of a point in one system a : 7’: 2 =a:b:c, there
will correspond in the other system the intersections of the
curves U: V: W=a:b:c¢; and these will be »’ m number
if U, V, W are general curves of their order. If, how-
ever, U, V, W have p points common to all three, the curves
= = 2 = 4h will always pass through these points, and there
will be only n’—p variable points of intersection, which will be
the points in the other system corresponding to the given point.
Finally, if p=n*-1, there is but a single variable point of

GENERAL THEORY OF RATIONAL TRANSFORMATION. 315

intersection ; or, in other words, all but one of the intersections
of the curves*U: V: W=a: b: c being known, the coordinates
of the remaining intersection are uniquely determinate, and will
thus be rational functions of a, 6, c; that is to say, of a’, 7’, 2’,
and we have expressions of the form w: y: = U’: V’: W’.

353. Thus, then, one condition for rational transformation is,
that the curves U, V, W shall have n’?—1 common intersec-
tions; but there is a further condition. The system of curves
aU+bV+cW must be as general as the system of right lines
ax’ + by’ +z’ to which they correspond; that is to say, a curve
of the system must not be determinate unless two conditions are
given to determine the two expressed constants a: b:c. The
number of conditions, therefore, which U, V, W can be made to
satisfy must be at least two less than the number of conditions
necessary to determine a curve of the n™ order. For example,
if U, V, W be cubics, and if we subject them to the condition
of having eight distinct common points, they must also have
a ninth (Art.29); there would be no variable point of inter-
section, and the construction of Art. 352 would fail. But we
can still satisfy the conditions of the problem by supposing
the cubics U, V, W to have common one point, which is a node
on all, and four ordinary points. These are equivalent to but
seven conditions, since to be given a double point is only
equivalent to three conditions (Art. 41), and therefore two more
conditions are necessary to determine any curve aU4+bV+cW.
But the common points amount to eight intersections, since
a point which is a double point on two curves counts for four
intersections. And so, in general, we cannot take U, V, Was
curves of the » order, having n’—1 distinct common points,
because then (n being greater than two) they would have another
common point, and no variable point of intersection; but we
can satisfy the conditions of the problem by taking for U, V, W
curves having common a, ordinary points, a, double, a, triple,
&c., in such way that these are equivalent to n’—1 intersec-
tions, and that the number of conditions implied shall be less
by 2 than the number necessary to determine a curve of the n™
order. Remembering, then, that to be given a multiple point of
the order r is equivalent to $r(r+1) conditions, and that such

316 GENERAL THEORY OF RATIONAL TRANSFORMATION;

a point when common to two curves counts as 7” intersections,
we have the two equations

a,+4a,+9a,+... ra,=n'—1 00 tee

a, +3a,+ 6a,+... 47 (r+1)a,=4n(n4+ 3) —2.... (2).
Doubling the second equation and subtracting from it the first,
we get an equation which may conveniently be substituted
for (2)

@, + 2a, + 3a, +... 70, =3 (2 —1) ....00000 20(3).

We have then as many modes of transformation by curves of the.
n> order as there are solutions of these equations by positive
integer values of a,, a,, &c., provided always that the number,
of higher multiple points which the curves are supposed to.
possess is subject to the limitations assigned, Art. 43.*

354. The argument of Art. 353, strictly, only shews that in
equation (2) the left-hand side cannot be greater than the
value there written. But we can also shew that it cannot be
less, for add a term —¢ and subtracting equation (2) from (1)
we get

a,+3a,+...$7r (r—1)a,=4(n— 1) (n— 2) +4#.... (4).
Recollecting that a triple point is equivalent to three double
points, and an 7-fold multiple point to 4r(r— 1) double points, we
see that the left-hand side of the equation represents the number
of double points to which all the multiple points of any curve
aU+b6V+cW are equivalent. And since it was shewn (Art. 42)
that this number cannot exceed 4(n—1)(n—2), we must have.
t=0, then equation (4) asserts that the curves of the system
aU+bV+cW have each the maximum number of double
points, or, in other words, that they are unicursal. And it is
otherwise evident that this must be so, since these curves
answer to the right lines of the other system; and not only a
right line, but every unicursal curve will be transformed into a
unicursal curve; for if the coordinates of a point are rational’
functions of a parameter, the coordinates: of the corresponding

* This theory is due to Cremona, see his memoirs Sulle trasformazione geometriche
delle figure piane, Mem. di Bologna, t. 11. 1863, and t. V. 1865; see also Prof. Cayley’s
paper, Proceedings of the London Mathematical Society, vol. 111. 1870, pp, 127-180.

GENERAL THEORY OF RATIONAL TRANSFORMATION. 317

point being rational functions of these, must also be rational
functions of the same parameter.

355. We have seen that when n is greater than 2, the
equations (1) and (3) cannot be satisfied if the points common
to U, V, W are only simple intersections. We shall now shew,
in like manner, that if 2 is greater than 5, there must be
a multiple point of order higher than the second; and so on
generally. Let r be the highest index; multiply equation (3)
by 7, and subtract from it equation (1), and we have
(r—1)a,+2(r—2) a,+3(r —3) a,+...(r-1)a,,=(n —1)(87 —n—1).
Every term on the left-hand side is positive, therefore 7 cannot
be less than 4(n+1). We may take r equal to this number
in the case where 4 (+1) is an integer, that is to say, if n be
of the form 3p — 1, we may take r=p; but if so all the numbers
G,, @,-.., @,, must vanish, and the curves can have no common
points but the p-fold points; and we have pa,=3 (3p —2),
which cannot be satisfied by an integer value of a, if p exceed 3,
unless p=6. Except, then, when n=2, 5, 8, or 17, r must
be greater than 4(n+1); thus always for n greater than 5 there
must be a multiple point of higher than second order.

356. In the same manner is established a theorem from
which we shall presently draw an important inference, viz. that
if we take the three highest in order of the multiple points, the
sum of their orders must exceed n. Let the orders of the
three highest be 7, s, ¢, where s is supposed not greater than r
and ¢ not greater than s, then transferring the terms contributed
by the two former to the opposite sides of equations (1) and (3),
these equations become

a, +4a,+...04,=n7 -1—r'—s',

a, +20,+...6%,=3n—3-1r —S,
and, as before, we have a limit to the lowest admissible value
of ¢ from the consideration that if we multiply the second
equation by ¢ and subtract the first, the remainder is essentially
positive. Our business now is to shew that n—r—s is too low
a value for ¢, or that, in this case,

nW—l—r'—s’>t(8n—3-—r- 5).

318 GENERAL THEORY OF RATIONAL TRANSFORMATION,

Substituting 7+ s=n- #, this becomes
2rs—1+4+2nt—t?>t(2n—3 +42).
But since, by hypothesis, » and s are not less than ¢, the least

value the first quantity can have is found by putting r and s
both =¢, when the inequality becomes

t’ + 2nt—1> t+ 2nt—3t,
which is obviously true.

357. Cremona has tabulated as far as n=10 all the ad-
missible solutions of the system of equations we have been
considering. Some of his results will be given presently; but
enough has been said to shew that we can always take U, V, W
functions of the x order in xyz, such that the equations

ea Maw ew Pa ee tae | A
shall represent three curves having common certain fixed points,
equivalent to n*—1 intersections (which we call the principal

points), and one variable point, the coordinates of which ex-
pressed in terms of a’y’z’ give the converse system of equations
ei gies Fes Ww"

We have already seen that U’, V’, W’ are functions of the
n‘> order in x’y’z’, and it is plain that these also must represent
curves having common a number of fixed points satisfying the
conditions (1) and (2) already explained. It does not follow,
however, nor is it always true, that the same solution of the
system of equations is applicable in both cases; in other words,
the system of curves aU+bV+cW which answer to the right
lines of one system, and the system of curves aU’ +bV’+eW’
which answer to the right lines of the other system, have not
in general the same distribution of multiple points.

358. We have seen that, in the quadric transformation, to one
of the three principal points corresponds in the other figure
not a point but a line; and we shall now extend this theorem
by shewing that in general to any of the a, points corresponds
a unicursal curve of the r order. It is evident that the system
of equations

Gsye¥38aU2V:W

eyes

GENERAL THEORY OF RATIONAL TRANSFORMATION. 319

becomes illusory if we seek the point a‘y’z’ corresponding to
any point xyz common to the curves U, V, W. Now, first let
this be a point of simple intersection; and, by proceeding to a

| etl, a

consecutive point, we have a’y’z’ respectively proportional to

Ube +U,by +U,82, V,8n+V,by +V,8e, Wax + Wdy +W,8e,
where U,, &c., denote differential coefficients. We have thus
a different point ay’z’ corresponding to each element of direc-
tion at the assumed point ayz. But 7¢ three curves have a
common point their Jacobian passes through that point; as is
evident by writing the equations U=0, &c. in the form

Ue+UytUz=0, VerVyt+tVe=0, Wi2tWy+Wz=0,
and eliminating a2yz. We thus see that if we eliminate dz, dy
from the values just found for a’y’z’, dz will also disappear, and
all the points corresponding to xyz will lie on the right line

a (V,W,-V,W,) + (W,U,-W,U,) +2 (U,V,-0,7,) =0.

359. We proceed in like manner if the point common to
UVW be a multiple point. Let it, for example, be a double
point, then the values given, Art. 358, for a’y’z’ vanish; but
denoting the second differential coefficients as before by a, J, ¢,
&c., we have a‘y’z’ respectively proportional to
adz*+bdy?+cb242f8y82+29828a42hd.cby : a’ 8x74 &e. : a’ b2x"+ ke.
But the relation of the point xyz to UV W is such as to allow of
the simultaneous elimination from these equations of dz, dy, dz.
In fact, the above forms in dx, dy, Oz are only in appearance
ternary, but are really binary. For aa*+ by’+ cz’ + &c. equated
to zero denotes the pair of tangents to the curve U at the
double point, and is reducible to the form

a (a — mz)” + 2h (a — mz) (y— nz) +b (y— nz)’.
There are, therefore, but two quantities 6a — mdz, 8y—ndz to be
eliminated between the equations, and it will practically come

to the same thing if we write 5z=0, and eliminate da, dy.
And so for any multiple point we have a’, 7’, 2’ proportional to

(Goss 0 Ome DY) s. (Bo, ois 0 Ody OY) 3. (aS... FOR, Oy)"s
and dz, dy are eliminated in the manner explained, Art. 44,

and x’, 7’, 2 being rational functions of a parameter, are the
coordinates of a point on a unicursal curve of the r order.

320 GENERAL THEORY OF RATIONAL TRANSFORMATION.

360. The curves in one system which answer to the prin-
cipal points in the other may be called the principal curves,
and these curves together make up the Jacobian of the system
of curvesaU+bV+cW. For the Jacobian is the locus of the
new double point on such of the curves of that system as have
a double point in addition to the multiple principal points common
to all. But since each of these curves has already the maximum
number of double points, it can only acquire a new one by break-
ing up into inferior curves, and this will happen only when the
corresponding right line in the other system passes through one
of the principal points. In that case the curve aU+bV+cW
breaks up into the fixed 7'* curve corresponding to the principal
point, together with a residual curve variable with the line
through «,. Now, in general, if we have two unicursal curves, the
sum of whose orders 7 and 7’ is n, the aggregate multiplicity
arising from the singularities of the two curves and their in-
tersections is equivalent to 4 (r— 1) (r—2)4 4 (7-1) (r’-2) +77,
that is, to 4 (n—1)(n—2)+1 double points. Thus we see that
in the curve we are considering, the complex curve has besides
the principal points one new double point, which will be a point
of intersection of the fixed curve answering to a,, with the
residual variable curve; and the locus of such points is therefore
the fixed curve. ‘That the sum of the orders of all these prin-
cipal curves makes up the order of the Jacobian of the system
aU+bV+cW is expressed in equation (3), viz.

a, + 2a, + 3a,+...74,=3 (n— 1).

From the general theory of Jacobians, which will be more fully
entered into in the next chapter, it appears that the system
of principal curves passes through each of the points a, twice,
through each point a, five times, and through each point a,
3r—1times. There are other theorems which it is sufficient
to indicate as to the disposition of the principal curves with
respect to the principal points. For instance, take a right
line in one system which does not pass through a principal point
a,’, then the corresponding curve aU+bV+cW can have no
ordinary point in common with the principal curve a, and the
intersections of the two curves would be exclusively principal
points. In this way we can see that every principal right line

GENERAL THEORY OF RATIONAL TRANSFORMATION. 321

passes through two principal points, the sum of whose orders is
n, and every principal conic through five principal points, the
sum of whose orders is 2n.

361. We are now in a position to determine the charac-
teristics of the curve corresponding to a curve of the order &,
which we suppose not to pass through any of the principal
points. Evidently, if we write U, V, W for 2’, 7’, 2’ in a
function of the £™ order, we obtain one of the order nfs; and if
the curves U, V, W have a point @ in common, the line in the
other figure corresponding to a will meet the curve S in & points,
which will all correspond to a; this will, therefore, be a k-fold
point, and similarly, every one of the principal points a, will be
a rk-fold multiple point. If the original curve have no multiple
points, the transformed curve will have no multiple points other
than the principal points. ‘hus it appears that the transformed
curve will be of the order nk, the corresponding maximum
number of double points being 4(nk—1)(nk—2)3 and the
principal points will be multiple points, and the number of
double points to which they are equivalent will be

$a,k(k—1) + $a,2h (24 —1) +...44,rk (rk —1),
or $k? (a, + 4a, +...7°a,) —$h (a, + 2a,+...7ar),
or, in virtue of equations (1) and (3),

4 (n?- 1) k*-8(n-1)k.

Substituting, the deficiency of the transformed curve is
} (nke—1) (nk -2)— {} (n"— 1) k*— §(n—-1)}, =4 (0-1) (2), the
same as the deficiency of the original curve. Ifthe original curve
has multiple points other than the principle points, to these will
correspond in the transformed curves multiple points of the
same order, and the deficiencies of the two curves remain equal.

If the original curve pass through any of the principal points
a’, then for each time of passage the corresponding curve a
is part of the transformed curve, and the degree of the trans-
formed curve proper will be reduced accordingly. There will
be also a corresponding reduction in the number of passages
of the transformed curve through the principal points through
which a, passes. ‘The effect of this will still be to: preserve

the equality of the deficiencies of the two curves. Thus, for
TT

322 GENERAL THEORY OF RATIONAL TRANSFORMATION.

example, if the original curve passes through one of the points a,,
the transformed curve will include as part of itself a right line,
and the degree of the residual curve will be reduced from nk
to nk- 1, and there will be a consequent diminution of nk—2
in the maximum number of double points; so if the right
line pass through two points «,, @,, the number of passages
of the residual curve through these will be each reduced by 1,
and the number of equivalent double points will be reduced
by sk—1 and tk—1, or by nk—2, since s+t=n. It is unne-
cessary to enter into more detail, because we shall presently
arrive at the same results by another method.

562. Every Cremona-transformation may be reduced to a
succession of quadric transformations. Consider the most general
transformation in which to the right lines of one figure
answer in the other figure curves of the x order having
in common @, ordinary points, a, double points, &c. We have
seen (Art. 356) that there are three of those points, the sum of
whose orders exceeds n. ‘Take these as principal points
and effect a quadric transformation, the degree of the trans-
formed curve, being 2n—r—s-—#, is less than n. In like
manner, by a new quadric transformation, we can reduce the
degree of that curve; and so on until we have at length right
lines corresponding to the curves of the n™ order. Since it was
proved (Art. 346) that the deficiency is not altered by any
quadric transformation, the theorem of this article shews that
it is not altered by any Cremona-transformation. The following
particular example will illustrate the method, and will shew how
we can trace the disposition of the principal curves. Consider the
transformation in which right lines are transformed into quintics
having three ordinary points a,a,a,, three double points 0,5,4,,
and one triple point c. ‘l'ake cb,b, as principal points, and by a
quadric transformation the quintics become cubics, having 0,’ as
a double point, and a,a,a,c’ as ordinary points. Again, take
a,b,c’ as principal points, and apply a new quadric transfor-
mation when the cubics become conics passing through a,”a,"b,”,
and finally, a new transformation with these for principal points
brings them to right lines. In like manner we can see how
are transformed the right lines of the first system, or, more

GENERAL THEORY OF RATIONAL TRANSFORMATION. 323

generally, how are transformed curves of the £* order passing
a, times through the point @,, &c. After the first transformation
we have ;
kh =2k-c—b,-b,,
¢ =k-b,-b,, :
b/ =k-—c—}b,, b,=k—c—b,, b, =),,
G, $4, 4 =4,, a, =4,.
After the second transformation, in which a,’b,’c are the principal
points, we have
kA” =3k—2c—a,—b,-—b,-),,
e” =2k—c-a,—b,—b,-8,,
b,” =k—c—b,, b,’ =k-c—a,, 6,’ =k—c-b,,
a,’ =k—c-—}b,, a,"=a, a,’ =a,
Lastly, after the third transformation, the principal points being
a,a,"b,”, we have
K” = 5h — 30 — 2b, — 2b, — 2b, — a, — a, —
é” =2k—c—b,-b,-b,-a,,
a” =2k—c—b, —b,—b,—a,, af” =2k—-c—b,— b,—b,—a,, a,” =k—c—b,y
bf’=k—-c—-b,, b,/’=k—c-b,, bf” =38k—2c—b,—b,—b,- a,—4,— a.

And if we put k=1, and the other letters =0, we see that right
lines are transformed into quintics having common one triple,
three double, and three single points. Again, in order to trace
the correspondence of the principal points, we see that in the
first transformation to the point c corresponds the line 4,’, },';
to this in the second transformation corresponds a conic through
e"a,'b,"b,"b," ; and finally, to this a cubic having 6,”’ as a double
point, and the remaining six points as ordinary points. ‘The
following tables give the effects of the different kinds of
Cremona-transformation as far as n=6. ‘The values also in-
dicate the curves answering to the principal points. ‘Thus, in
Ex. 3, the value c= 3k—2c— (a) indicates that to c’ corre-
sponds a cubic having ¢ as a double point, and passing through
the points a.

Ex, 1. (i) #32, ea, = 8;

er i gee Bey 4 Po ae ee
ki = 2k —a,—a,—a3, W =kh—-a,— dy, a) =k—az;—a, a,’ =k-a,—a,

324 TRANSFORMATION OF A GIVEN CURVE.

ex. 2. (IL) «= 3, a, + 4,0, 1.
k! = 8k — 2b — a, — a, —a3—, b' = 2k —b — a, — a, — 3 — Ay, 4,’ =k —b — a4, ke.
Ex. 3, (IV. 1) n=4, a,=6, a,=0, a,=1. ?
k’ = 4k — 8c — & (a), ce’ = 8k — 2c — XZ (a), pia ccethy &e.
Vk. 4. (V.2) #24 ek 2 8.
k’ = 4k — 22 (b) — (a), ' = 2k — 2 (6) — a, — a, D,.'= ke, a,'=k —b,—b,, ao’ = ke.
Hx.:5. (V. 1) w= 5, a, = 8, a= 0, a, = 0,4, = 1.
kb’ = 5k —4d—& (a), d’=4k— 38d —2 (a), a’ =k-—d—a,.
Hx..6..(V.2) o= 5, ap 0, ee 8, =,
K’=5k—e—2 (b)—Z (a), C= 38k —2c—Z (b)—E (a), b)’=2k—c—a,—= (6), ay’=k-c—),.
a7. (¥.8) 2b, a, =), a = %, |
k’ = 5k — 2 (6), b,' = 2k — b, — bg — 0, — 5, — Dy, &e.
ma. 8, (VE, 1} a6, a, = 10, my oh
k’ = 6k — 5e ~ = (a), e’ = 5h —4e—X (a), a,’ =h—e— ay, ke.
peed. (71. 2).0=86, a, =1, 4,54, a = 32,
kK’ = 6h — 8E (c) — 2 (4) — a, oe,’ = 8k — 2c, — cg — 2 (b)—a,
by =2k-Z (c)-—b,-—b,—by, a’ =k—X(c).
Mx. 10. (VI. 8) a=6, «,=4, a, — 1, a, =8.
i’ = 6k —8E (ce) —26 -Z (a), d =4k — 22 (ce) —b—-Z (a),
b,’=2k-Z (c) —b—a,', b,,=&e., b,’/=ke., b= &e., a,’=k—c,—Cy, a,'= &e., ag= ke,
mex 3d. (Vi 4) aes, a, = 3, = 4, ag 0, oO, 1:
k’ = 6k — 4d — 2 (b) — & (a), ¢,! = 8k — 2d — Ed (b) — a, — ay, Cy’ = ke, ¢,/ = ke,
b’ = 2k -—d—X(b), a,’ =k-—d—b,, a,’ = &e., a,’ = &e., a, = be.

TRANSFORMATION OF A GIVEN CURVE.

363. The conditions assigned in the last section are neces-
sary for the general rational transformation between two planes,
so that to any point in either plane shall correspond a unique
point in the other. But they are not necessary to rational
transformation, if we consider only: the transformation of a
given curve S=0. Let us apply to the curve S a transforma-
tion 2’: y’: 2 =U: V: W, where U, V, W are functions of the
n> degree in x, y, 2, not necessarily satisfying Cremona’s
conditions; then, obviously, to any point in the first plane will
correspond a single point of the second, since 2’, y’, 2’ are given
as rational functions of z, y, z But according to the pre-
ceding theory, if U, V, W have common a, ordinary points,
a, double points, &c., then to any point in the second plane will
correspond n° — a, —4a,— &c. points in the first plane; and this
number, which we shall call 8, will ordinarily be different from

TRANSFORMATION OF A GIVEN CURVE. 325

unity. The locus of points in the second plane corresponding
to the points of the curve S will be a curve S’ corresponding
to S, and to any point P of the first curve will correspond a de-
finite point P’ of the second. Now, from what we have just said,
it appears that to P’ will correspond in the first figure, besides
the point P, @—1 other points; but these points will ordinarily
not lie on S, and the curve in the first figure corresponding
to S’ will consist of S together with a residuary curve, the
locus of the 9-1 points. And if we attend only to the points
on the curve S, we see that while to any point P of S cor-
responds a single point P’ on S’, so also to any point P’ on 8’
corresponds a single definite point P on S.

Thus then, though the equations a: 7:2 =U:V: W do
not by themselves suffice to give rational expressions for x, y, z
in terms of 2’, y’, 2’, it is otherwise when with these we combine
the equation S=0. If from all the equations we eliminate
xyz, we obtain an equation S’=0, which is the condition for
the co-existence of the system of equations. And when this
condition is satisfied, it was shewn (LHigher Algebra, Lesson X.)
that we can in general rationally determine the values for
x, y, 2, which will satisfy all the equations of the system. We
see, then, that when a given curve S is transformed by the
substitution of 2: y’: 2 = U:V: W, we can in general obtain
a rational converse expression a: y:2=U’':V’: W’.

Ex. Suppose that we are given 2’ :y':2’ syz+a?:yz+ay:yz+az, Here to
right lines in the second plane answer conics in the first, having common only two
points yx, zx; and therefore to a point in the second plane will generally answer two
points in the first plane. The general expressions for 2, y, z in terms of a’, y’, 2’
are easily found by observing that «—y,x2—z are respectively proportional to
x’ —y', x’ —2'; the geometrical meaning of which is, that the points ayz, a'y’2’,
considered as belonging to the same plane, are collinear with the point 1, 1, 1.

In other words, the equations are satisfied by writing c=a2’+A, y=y'+A,
z=2'+X, where Xd is determined by the quadratic
22+ (a’ +y'+2)rA+y'2/ =),

and plainly to any system of values for 2’y’z’ answer two systems of values for xyz,
But it is otherwise if we consider the transformation of a given curve. ‘Thus, take
a right line in the first plane ax + By + yz; then the relation between any point on
this line and the corresponding point in the second plane is given by the equations
a=a2' +X, &c., where (a+ 8+ y) X =— (av’ + By’ + yz’).

In like manner, if we have any conic S on the first plane, and if by the sub-
stitution «= 2’ +A, &c., S becomes 42+ PAX +8’, then the curve corresponding
to S is the quartic whose equation is obtained by eliminating between

P+ BA + S'=0, 2472+ (@’ + yo’ +2) A+y'2’=0;

326 TRANSFORMATION OF A GIVEN CURVE.

and the expression for x in terms of 2’ is obtained by taking for X the common root
of these equations given by the equation {2P — (a + y'+2')}X +28’ —y’2’=0.

364. The deficiency of a curve is unaltered, not only by
Cremona’s transformation, as already proved, but by any trans-
formation where to a point on either curve corresponds a
single point on the other.* This may be shewn as follows:

In the first place, it is to be observed that in the rational
transformation between two planes, where to a point A corre-
sponds a single point A’, if any curve pass twice through A the
corresponding curve must pass twice through A’, or to a double
point on one curve must correspond a double point on the
other. But if to A correspond more points than one, A’, B’, &e.,
then if the second curve pass through both A’ and B’, the
first curve will pass twice through A; that is to say, a double
point on one curve may correspond to a double point, but it
may also correspond to a pair of distinct points on the other.
In like manner, if the points A’, B’ coincide, we may have a
cusp on one curve corresponding either to a cusp or to a pair
of coincident points on the other.

Let us now consider two fixed corresponding points A, A’,
one on each of two corresponding curves S, S’, whose orders
we suppose to be m and m’, and which we suppose to be in
the same plane; let us consider also two variable corresponding
points M/, M’; and let us examine the degree of the locus of
the intersection of the lines AM, A’M’. Now take any fixed
position of the line AW, since it meets the first curve in m—1
points distinct from A, there are m—1 corresponding positions
of the line A’M’, and therefore AM meets the locus in m—1
points distinct from A. But if we consider the line AJ’, it
is easy to’see in like manner that it meets the locus in no
other points than the point A counted m’—1 times, and A’
counted m—1 times. Thus we see that the locus is of the

* This theorem was first derived by Riemann from the theory of Abelian
functions ; see Crelle, L1v. 1383. The proof here given is substantially the same as that
given by Zeuthen, Mathematische Annalen, 111. 150 ; but Iam informed by Dr. Fiedler
that it had been previously given by Bertini, Battaglini Giornale, v11. 105 (1869).
See also a direct proof in Clebsch and Gordan’s Theorie der Abelschen Functionen,
p. 54, for the case where the curves in one system answering to right lines in the
other have common no multiple points higher than the seconds

TRANSFORMATION OF A GIVEN CURVE. 326

degree m-+m’—2, the points A, A’ being multiple points of
the orders respectively m’— 1, m—1.
Let us next consider in what cases AM touches the locus.
This will be the case when two of the lines A’M’ corresponding
to AM coincide, without our having at the same time a coin-
cidence between two of the lines AM corresponding to A’M’;
for in the latter case the intersection of AM, A’M’ would be
a double point on the locus, and AJM would not be an ordinary
tangent. Now (1) if AM touch the curve S, AM will evidently
also touch the locus. (2) If AM pass through a double point
on S, then according as to that double point there corresponds
on S’ a double point or a pair of distinct points, we have
corresponding on the locus a double point or a pair of distinct
points, but in neither case is AJ an ordinary tangent. (3) If
AM pass through a cusp on S, then according as to that cusp cor-
responds a cusp on S’, or a pair of coincident points, AJ passes
through a cusp on the locus, or else is an ordinary tangent.
It appears from (1) and (3) that the number of ordinary
tangents from A, together with the number of cusps, is the
same for the locus and for the curve S. It is by expressing
this equality that we obtain the relation connecting the two
curves S, S’. It was shewn (Art. 79) that the number of
tangents which can be drawn to a curve of the m degree from
a multiple point of-the order r is m*—m-—r(r+1); or is
less than the class of the curve by 27. Hence, if N be the
class of the locus curve, the number of tangents which can
be drawn from A, which is a multiple point of order m’—1, 18
N-2(m’—1); and if we denote the number of cusps on the
locus curve by A, and the class of S by n, the equality we
desire to express is
N-2(m’ -1)+ K=n-24+k.

In like manner, considering the tangents from A’,
N-2(m—1)+ K=n'-24+k,

and we have therefore n—2m+K=n' —2m’'+,

or, writing for n its value m*® — m— 26 — 3k,

4 (m—1)(m—2)—8—K =} (m' —1)(m’-2)-O- xk. QE.D.*

* Zeuthen proves in like manner, that if, instead of the correspondence of the
curves being rational, a points on S correspond to any point on §’, and a’ points on

328 TRANSFORMATION OF A GIVEN CURVE.

365. It is proved, as in Art. 361, that if we transform a curve

S of the m* order by the transformation aw : y’: 2°=U:V: W,
where U, V, W are functions of the p™ order, then since the

points where an arbitrary line meets the transformed curve

correspond to the points where aU+8V+yW meets S, the
order of the transformed curve is mp — a, — 2a,, &c., where a,,
a,, &c. denote the number of single, double, &c. points common
to U, V, W, and which also lie on S. Let us now examine
rhow, by this transformation, we can reduce the order of the
transformed curve as low as possible. As in Art. 353, we
see that U, V, W may be made to satisfy two conditions less
than the number sufficient to determine a curve of the p™
order, that is to say, $p(p+3)—2; and we evidently apply
these conditions so as most to reduce the order of the transformed
curve, if we make U, V, W pass through as many as possible
of the double points of §. Let the deficiency of S be D, and
the number of its double points accordingly 4 (m’— 3m)- D+1;
and let us in the first place take p=m—1, in which case
we may make U, V, W pass through $(m*+m)—3 points.
We may, therefore, make the curves pass through all the
double points and through 2m+D—4 other points on S.
Writing, therefore, a,=2m+D-—4, a,=4(m" —- 3m)—-D+1,
p=m-—1, we find for the order of S$’, mp— a, -—2a,=D+ 2.
Let us next take p=m-—2, which of course implies that m
is greater than 2. Proceeding precisely as before, we see that
we may take a,=4(m’- 3m)—-D+1, a,=m+D-—4, and that
the order of the transformed curve will still be D+2. Once
more let us take p= m— 3, we may take a, =4(m*—3m)—D +1,
a,=D-—43%, provided always that D is greater than 2; and
we now find for the order of the transformed curve D+ 1.
The transformed curve has, as we have proved, the same
deficiency as the original, so that our result is, that a curve
of order m with deficiency D, or with 4 (m—3m)-D+1 double
points, may be transformed into a curve of order D+2 with de-
ficiency D, that is, with 4 (D’—D) double points; or, when D is

S’ to any point on S; and if ¢ and ¢’ denote the number of cases in which two of
these a or a’ points coincide, then

t —t’= 2a’ (D—1) — 2a (D’ — 1).

‘3

TRANSFORMATION OF A GIVEN CURVE. 329

greater than two, into a curve of order D+1 with 4 (D*- 3D)
double points.

Thus then, in particular, a curve may be transformed as
follows:

if D =0 into a conic,*

i a
5» 2 cubic,

»» quartic with one node,

») + & quartic,

»» 2 quintic with two nodes, &ce.,
», a sextic with five nodes,

7 with 9,

8 with 14 or 6 with 3.

27

Mot tt Bate
ee tee

y

366. The case of unicursal curves need not detain us.
Here D=0, and the transformed curve a conic; the coordinates
x’, y’, # are, as we know, expressible as quadric functions of
a parameter 0; therefore the coordinates x, y, z, which are
expressible as rational functions of x’, y’, 2’, can be expressed as
rational functions of @.

Let us then consider the case D=1. Here the transformed
curve is a cubic, and it is to be noted that, however the trans-
formation is effected, the resulting cubic will have always the
same absolute invariant; that is to say, the anharmonic ratio
of the four tangents from any point on the curve will be the
same (Art. 229). When D=1, the coordinates of any point
on the curve can be expressed as rational functions of a para-
meter 0, and of /(©) where © is a quartic function of @. It
is sufficient to shew this for the case of a cubic, since a, y, 2
can be expressed as rational functions of 2’, y’, 2’; and for
the case of the cubic, it appears at once by taking the cubic
to pass through the point wy, and then writing in the equation

* Although by the method just described the case D = 0 is only transformed into
a conic, yet by the Cremona transformation the conic can be further transformed

into a right line.
For some further developments see Jung and Armenante in Battaglini’s Giornale

vil, 235; and Brill and Noether, Math, Annal., vu. 298,
UU

330 TRANSFORMATION OF A GIVEN CURVE.

of the curve y=0@x, when the ratios w: y: 2 are immediately
obtained in the form in question. It is, moreover, clear that
the values of 6 for which © =0 are precisely those answering
to the four tangents from xy to the cubic.

We have thus seen that the coordinates of a point on the curve
for which D=1 can be expressed as rational functions of 0 and
/(@); and by a linear transformation of @ (that is to say, re-
placing @ by a properly determined function a6 +b+c0@+d) we
can bring /(©) to the form 4/(1— 6°) (1—4°6"). If we write
0=sinamw, this is cosamz Aamu, and we may say that the
coordinates of a curve, whose deficiency is 1, can be expressed
as elliptic functions of a parameter w.

867. There is a like theory where the deficiency is 2, and
where the curve is therefore reducible to a nodal quartic.
Taking the node of the quartic for the point ay and writing
y=Ox, we can immediately express the ratios e:y:z2 as
rational functions of @ and ./(©), where © is now a sextic
function of @; and this is equivalent to saying that the coor-
dinates are expressible as hyper-elliptic functions of the first
kind of a parameter wu. Jor higher values of D the coordinates
are irrational functions of a parameter, and it is only .in special
cases that they can be expressed by radicals. 7

368. Before quitting this part of the subject, another method
may be mentioned by which the same problem may be studied.
We may start with the equations connecting the coordinates
xyz, «yz; let these be A=0, B=0, C=0, each equation
being homogeneous both in ayz and 2’y’z’5 and being in
those variables of the orders a, b,c; a’, b’, ¢ respectively. If
between the three equations we eliminate a’y’z’, we obtain an
equation S=0 of the order ab’c’ + be'a’+ca’b’ in wyz, and if
we eliminate xyz, we obtain an equation S’=0 of the order
abe+Uca+cab in xyz’. The conditions S=0, S’=0 must
be satisfied in order that the equations d=0, B=0, C=0 may
co-exist; but for any system of values of xyz satisfying the
equation S=0, we can find a corresponding system of values
of a2’y'z satisfying equations d= 0, B=0, C=0, and therefore
also S’=0. The number of double points on the curve S may

al

CORRESPONDENCE OF POINTS OF A GIVEN CURVE. 331

be investigated by the methods explained in Higher Algebra,
Lesson XVIII., and the result I have obtained is

$0’c (b'c’ —1) a’ + dca’ (ca — 1) 0? + 4a'l’ (d’-1)
+ {(a’b’ — 1) (c’'a’ — 1) — 43 (a’ — 1) (a —2)} be
+ {(W’c’ — 1) (a - 1) — 3 (b’ —1) (WV -2)} ca
+ {(ca”— 1) (Ue — 1) -4 (¢ -1) (¢ —2)} ad,
and there is of course a similar expression with interchange of
accented and unaccented letters for the number of double points

on S’. In either case we find the deficiency to be 4(Q+2),
where

Q=a'l'd + Ba + Cad + abc 4+ beat cab
+ 2aa’ (be’ + b’c) + 2b0’ (ca’ + ca) + 2cc’ (ab’ + a’b)
— 3 (ab’c + bea’ + ca'l’ + abe + b’ca+ cab) s
so that again we have the theorem that the two curves have
the same deficiency.

CORRESPONDENCE OF POINTS ON A GIVEN CURVE.

369. What has been said may sufficiently illustrate the
theory of rational correspondence; in what follows we consider
the general correspondence of two points P, P’ on the same
curve, such that either determines the other. Suppose that to
a given position of P there correspond a’ positions of P’, and
to a given position of P’ya positions of P, the correspondence
is said to be an (a, a’) correspondence. When a=a'=1,
the correspondence is rational.

As a simple instance of correspondence on a given curve
of the m™ order, suppose the points P, P’ to be collinear with
a fixed point O (that is to say, that the line PP’ passes through
QO), then if P be given there are m—1 positions of P’, and
if P’ be given there are m—1 positions of P; or this is an
(m—1, m—1) correspondence. We have already noticed this
particular kind of correspondence in the case of the circle (see
Art. 347). This correspondence is evidently rational in the
case of the conic, or where m= 2.

If the point O is on the given curve, then to a given
position of either point there correspond m-— 2 positions of the

332 CORRESPONDENCE OF POINTS ON A GIVEN CURVE.

other point; or more generally, if O is an a-ple point of
the curve, then to a given position of either point there corre-
spond m—a—1 positions of the other point, viz. the corre-
spondence is a (m—a—1, m—a—1) correspondence. Observe
that we have in this way a (1, 1) correspondence of points
on a cubic (by taking O at pleasure on the curve), or on a
nodal quartic (by taking O at the node), but that we cannot
thus obtain a (1, 1) correspondence of points on a general
quartic.

370. In the foregoing instance the correspondence has been
a symmetrical one; viz. starting from either point the other
is obtained by the same construction, and of course a=a’.
But as an instance of a non-symmetric correspondence, suppose
that P’ is given as a tangential of P; here P being given, P”
is any one of the intersections of the tangent at P with the
curve (and thus to a given position of P there correspond m-— 2
positions of P’); but P’ being given, P is any one of the points
of contact of the tangents from P’ to the curve (and thus to
a given position of P’ there correspond n—2 positions of P, if n
be the class of the curve); and we have thus a (n—2, m— 2)
correspondence. It is hardly necessary to remark, that we
may have a=a without the correspondence being symmetrical.

371. In the case of a unicursal curve, to a given point on
the curve corresponds a single value of the parameter @; and
to a given value of @, a single point @n the curve (or extending
the notion of correspondence we might say that a point on the
curve and the parameter of such point have a (1, 1) corre-
spondence). It at once follows that if the point P has @ positions,
its parameter 9 must be given by an equation of the order a;
whence also, if as above, the points P, P’ have an (a, a’) corre-
spondence, the relation between their parameters @, &” must be
given by an equation of the form (0, 1)«(@’, 1)#’=0, viz. @ being
given the equation will be of the order @’ in 6’, but @& being
given it will be of the order a@ in @.

372. A point may correspond to itself, and it is then said
to be a united point; thus where the points P, P’ are collinear
with a fixed point O, it is clear that the point of contact of any

CORRESPONDENCE OF POINTS ON A GIVEN CURVE. 333

tangent from O to the curve is a united point; and if these are
the only united points, their number is =n.

The only other points which it might at first sik appear can
be united points are the nodes and cusps of the curve; in fact,
taking P at a node or a cusp the line OP meets the curve in the
point P, in the same point counting as one of the (m-— 1) inter-
sections, and in (m’— 2) other points; or, what is the same thing,
the line from O to the node or cusp meets the curve in the node
or cusp counting twice, and in (m-— 2) other points. But in the
case of the node, the two intersections at the node belong to
different branches of the curve, or we may say they are coinci-
dent, but non-consecutive points; in the case of the cusp they
are consecutive points: the distinction is well seen in the case of
a unicursal curve—here for a node we have two distinct values of
9, for each of which the coordinates have the same values; for
the cusp these two values of @ have become identical; or, what
is the same thing, the line from O to a cusp (although not a
proper tangent of the curve) is a tangent in a sense in which
the line from O to a node is noé a tangent to the curve. The
conclusion is, that a node is not a united point; in a special
sense a cusp is a united point; and we have, besides, the proper
united points, which are the points of contact from O to the
curve.

Reverting to the unicursal curve and to the equation
(0, 1)*(@', 1)" =0, at a united point we have 9=@’, and for
finding these points we have an equation (@, 1)¢+¢’=03; that is,
when the points P, P’ have an (a, a’) correspondence, the number
of united points is =a+a’,

Applying the theorem to the case where P, P’ are collinear
with the fixed point O, the correspondence is (m—1, m—1), or the
number of united points should be =2 (m—1). The number of
points of contact, or proper united points is =n, that of the
cusps or special united points is =«; or we ought to have

n+K=2(m-—1),

which is in fact the case for a unicursal curve with « cusps.

In the case where P” is a tangential of P, it has been seen
that the correspondence was (n—2, m— 2); and the number of
united points should be =m+n-—4. We have here as proper

334 CORRESPONDENCE OF POINTS ON A GIVEN CURVE.

united points the inflexions, and as special united points
the cusps; total number =c+«; and the theorem thus is
t+xK=m+n—A4, or what is the same thing 1=3 (m—2)—2«3
which is in fact the case for a unicursal curve with « cusps.

373. Consider the point P as given; the geometrical con-
struction for the determination of P’ comes in general to this,
that we have depending on P a certain curve © which, by its
intersections with the given curve, determines the points P”.
In some cases P’ is any one of the intersections in question;
but in others a certain number of them will in general coincide
with the given point P, and are to be excluded. Thus, in the
case where P, P’ are collinear with O, the curve © is the line
OP meeting the given curve in the point P counting once (to
be excluded) and in (m—1) other points. So when P’ is the
tangential of P, the curve © is the tangent at P meeting the
given curve in the point P, counting twice (to be excluded) and
in {m— 2) other points.

But further; the curve © may meet the given curve in
points forming two or more distinct classes, in such wise that
only the points of the one class are positions of the point
P’, Thus, in the last preceding instance, interchanging the
points P, P’, or now considering P’ as the point of contact of
a tangent from P to the curve, the curve © is the system of
n—2 tangents from P to the curve; each of these tangents
meets the curve in the point P counting once, in the point of
contact say P’ counting twice, and in m—3 other points say
P” (which are cotangentials of P, that is PP” touches the curve
at a point P’ distinct from P or P”). Or, what is the same
thing, the curve © of the order n—2 cuts the given curve in
the point P counting n—2 times, in 2-2 points P’ counting
each twice, and in (x — 2) (m—3) points P” counting each once.
The correspondence P, P’, as was seen, is (m—2, n—2); the

correspondence (P, P”) is clearly (n—2 m—3,n- 2 m-—3).

374, The theorem in regard to a unicursal curve suggests
the theorem that for a curve in general the number of united
points should be =a+a’+ multiple of the deficiency, or say
=ata+k.2D; but admitting that the curve © presents itself

es

CORRESPONDENCE OF POINTS ON A GIVEN CURVE. 335

in the problem, the last instance shews that there is a necessity
for considering the case where the curve © has with the given
curve distinct classes of intersection. The general theorem
is, that if for a given curve of deficiency D, the corresponding
points of Pare P’, P”, ..., and if P, P’ have an (a, a’) corre-
spondence, and the number of the united points is =a: P, P”
a (8, 8’) correspondence, and the number of their united points
is b : &c.5 and if the curve ©, which, by its intersections with
the given curve, determines the points P’, P”, ..., intersects the
given curve in the point P counting & times; in each of the
points /’ counting p times, each of the points P” counting
q times, and so on, then we have
p(a—a—a)+q(b—B-P’)+...=k.2D,

where of course in each of the different correspondences the
special united points (if any) must be taken into account.

Thus, in the instances above considered for a unicursal
curve; first, if P, P’ are collinear with O, we have

M+ = 2 (m—1) + 2D... .ccccccsccese (1).
Next, if P’ is a tangential of P,
bbK=M+EN—A+LAD 00... cccccceeceeee (2);

and in the case where P is a tangential of P’, and where
b, 8, B’ refer to the correspondence P, P” cotangentials,

b —2 (m—3) (n—2)+2(a—a-—a’)=(n—2) 2D,
where, by the example immediately preceding,

a—a—a =t+K—(m+n—4)=4D,
and therefore b—2(m-—3) (n—2)=(n—6) 2D.
The proper united points b are here the points of contact of
the double tangents, the number of which is 27; but we have
also as special united points the cusps each counted z— 3 times
(ct must be assumed that this ts so), and the result 1s
27 = 2 (m —3) (n— 2) + (n—- 6) 2D—(n—3) x .... (8).

The several equations (1), (2), (3) giving respectively the
class, the number of inflexions and the number of bitangents
of a curve of the order m with 6 nodes and « cusps agree with
the Pliickerian equations; they are most easily verified by
means of the expressions given, Art. 83, for the several quan-
tities in terms of m, n, and a= 3n+ K.

336. CORRESPONDENCE OF POINTS ON A GIVEN CIRCLE.

375. If on any.curve the points P, P’ have a (1, 1) cor-
respondence, the points (P’, P”) a (1, 1) correspondence...and
so on up to the points P™, P; then it is clear that the
points P, P have a (1,1) correspondence. And, conversely,
the points P, P“’ which have a (1, 1) correspondence may be
regarded as connected with each other through the series of
intermediate points P’, P’”...P°™. .

In the case of a unicursal curve, the (1, 1) correspondence
of the points P, P’ implies a like correspondence of the para-
meters 0, 0; viz. this is of the form (@, 1) (@, 1)=0, or what
is the same thing, a06’+00+c6’+d=0; that is, the para-
meters 6, & are homographically connected. The transfor-
mation depends upon three arbitrary parameters.

Taking the curve to be a conic, then if the points P, P’
have a (1, 1) correspondence, it is known that the line PP’
envelopes a conic having double contact with the given conic;
such enveloped conic, as satisfying the condition of double
contact, depends on three parameters. But if taking the points
A, B at pleasure, we take on the conic P, Q collinear with
A, and P’ collinear with B, Q, then the points P, P’ will have
a (1, 1) correspondence; this apparently depends upon four
parameters, and it follows that the points A, B can without
loss of generality be subjected to Spy oi
a single condition. Thus let the Au bac
correspondence P, P’ be given by ; Pe
means of the conic enveloped by
the line PP’; if on the chord of
contact we take at pleasure the point
A, draw PA to meet the conic in @
Q and QP’ to meet the chord in B, then (1, 1) correspondence
is also given by means of the points A, B; but here A may
be regarded as a determinate point on the chord of contact
(say its intersection with a fixed line), B is then found as
above, and we have the correspondence by means of these two
points, just as well as if A had been assumed at pleasure on
the chord of contact.

A case really included in the foregoing is when the corre-
spondence of P, P’ is such that the line PP’ passes through a
fixed point C’; viz. the enveloped conic regarded as a line-curve

CORRESPONDENCE OF POINTS ON A GIVEN CURVE. 337

is here the point C taken twice, regarded as a point-curve
it is the pair.of tangents from C to the given conic; that
is, the chord of contact is the polar of C, and the construc-
tion is the same as before, the points A, B, C forming, as
it is easy to see, a set of
conjugate points in regard
to the conic; the original
correspondence of P, P’ as
collinear with the given
point C, is here replaced by
a correspondence by means
of the two points A and B -
forming with C a system of
conjugate points.

The foregoing properties have reference to the problem of
the inscription in a conic of a polygon the sides of which either
pass through given points or touch conics having each of them
double contact with the given conic.

ff’

376. On a cubic curve (D=1) we have a (1, 1) corre-
spondence; this depends on a single parameter, but there are
two kinds of such correspondence, viz. (1) the points P, P’ are
collinear with a point A of the cubic. (2) The points P, P’
are such that P, Q are collinear with

a point A of the cubic and Q, P ¢ # z
collinear with a point B of the cubic;

this apparently depends on two para- , ‘3 -
meters, but really on a single one; "| . "i
for taking C a determinate point on

the cubic, join AC to meet the cubic

in O and BO to meet the cubic in

D; then the same corresponding point P’ will be obtained by
taking P, £ collinear with D, and &P’ collinear with C, that
is, by means of the single point D. It is, in fact, evident that
starting with P and constructing P’ as the intersection of the
lines QB, RC, then the cubic passing through A, B, C, D,
O, P, Q, R will also pass through P’, so that the points A, B
and the points D, C lead to the same point P’.

The theorem involved in the foregoing construction may be

xx

338 CORRESPONDENCE OF POINTS ON A GIVEN CURVE.

stated as follows: If on a cubic the points A, B, C, D are such
that the lines AC, BD meet in a point O of the cubic, then we
have inscribed in the cubic an infinity of quadrilaterals POP’ R,
the sides of which pass through A, Bb, C, D respectively; viz.
any point P whatever of the cubic may be taken as a vertex of
such quadrilateral.

377. More generally imagine inscribed in the cubic an
unclosed polygon PQ...X of 2n—1 sides, the sides of which
pass through fixed points on the cubic, then the points P, X will
have a (1, 1) correspondence of the first kind, that is, the closing
side XP will meet the cubic in a fixed point; that is, we have
inscribed in the cubic an infinity of 2n-gons, the sides of which
pass respectively through fixed points of the cubic. And of the
fixed points all but one are arbitrary, this one being determined
by constructing one such polygon.

378. This theory may be illustrated by the expression of
two points in a cubic by means of parameters, Art. 366. A (1, 1)
correspondence between two points on a cubic implies a rational
expression for the parameters sinamw’, cosamu’, Aamw’, in
terms of sinamw, cosamu, Aamu; and this again implies an
equation of one or other of the forms u+w’ = constant,
u—u =constant. Now when three points P, P’, A, are col-
linear, we have in general a relation w+u’+a=A where A is
a constant depending on the absolute invariant of the cubic.
A relation, then, of the form u+«’=constant, implies that P
and FP” are collinear with a fixed point A. If the relation
be of the form w—wu’=constant, say =b—a, we may write
utvt+a=A, v+b+w =A; and the geometrical meaning is,
that P, Q are collinear with a fixed point A and Q, P’ with
a fixed point B. We may evidently substitute for the points
A, B, two others D, C, provided we have 6—a=c—d, or
a+c=b+4d, that is to say, provided the lines AC, BD in-
tersect on the cubic. We have thus the results already
obtained.

879. For a binodal quartic (D=1) there isa like theory of
the (1, 1) correspondence; for a nodal quartic (D=2) there is a

CORRESPONDENCE UF POINTS ON A GIVEN CURVE. 339

(1, 1) correspondence not depending on any arbitrary parameter,
viz. the corresponding points P, P’ are collinear with the node.

There is an interesting theory of the (2, 2) correspondence
on a unicursal curve, and in particular on a conic. ‘The para-
meters which determine the position of the two points P, P’ are
here connected by an equation (0, 1)*(@,1)?=0. As regards
the conic we have Poncelet’s theorems as to the in-and-cir-
cumscribed polygons.

( 340 )

CHAPIER. 1X,

GENERAL THEORY OF CURVES.

380. In this Chapter we resume the general theory of curves
in continuation of Chap. II., and commence with the theory of
bitangents of a curve of the x" order postponed from Art. 78.
We shall explain two methods by which we can form the
equation of a curve whose intersections with a given curve shall
determine the points of contact of its bitangents.

The theory of the tangents of a curve was studied (Art. 64)
by means of the equation A =0, or

MU’ +A" UAU’ + dn" WAU’ + &e. = 0,

which determines the coordinates of the points in which the
line joining two given points meets the curve. We there saw
that if the point a’y’z’ be on the curve, and xyz anywhere on
the tangent, we must have U’=0, AU’=0, and if the tan-
gent meet in three consecutive points we must have besides
A’U’=0, if in four consecutive points we must have likewise
A’U’=0, and so on. If the tangent at a’y’z’ touch the curve
elsewhere, then making U’=0, AU’ =0, in the equation A=0,
the reduced equation of the (n—2)" degree must have equal
roots, and therefore, if the discriminant of that equation be Y,
the relation Y=0 must be satisfied by the coordinates a’y’e’,
xyz. In the case of points of inflexion where we have the two
conditions AU’ =0, A?U’ =0, the one being of the first degree
and the other of the second in xyz, and both satisfied for any
point on the tangent, it is evident, as was stated (Art. 74),
that AU’ =0 is the equation of the tangent, and that A*U’=0
must contain AU’ =0 as a factor. In like manner, in the case
of a bitangent, Y=0 must contain AU’=0 as a factor, and
by finding the condition that this shall be the case, we find the
condition that «’y’z’ shall be a point of contact of a bitangent.
The special method used, Art. 74, not being applicable to

GENERAL THEORY OF CURVES. 341

the general case, we employ the following method due to
Prof. Cayley, and it is convenient to begin with the follow-
ing lemma. :

381. Let the equations of two curves contain the variables
xyz in the degrees a, b respectively, and 2x‘y’z’ in the degrees
a’, b’; and let the ab points of intersection of the two curves
all coincide with 2’y’z’, it is required to find the order of the
further condition that must be fulfilled in order that they may
have other common points, which can only happen when there
is a factor common to U and V. When this is the case any
arbitrary line ax+fy+yz=0 must be sure to have a point
common to U and V; namely, the point or points where the
arbitrary line meets the curve represented by the common
factor. It follows that the result of elimination between U=0,
V=0, and the equation of the arbitrary line must, in this case,
vanish. This result contains aSy in the degree ab, a’y’z’ in
the degree ab’+a’b, and the coefficients of U, V in the de-
grees b, a respectively. But since the result of elimination
is obtained by multiplying together the results of substituting
in ax+ By +z the coordinates of each of the intersections of
U, V, and since by hypothesis these interesections all coincide
with xyz’, the resultant must be of the form II (ax’+ By’ +y2’)”.
The condition ax’ + By’ +2’ =0 merely indicates that the arbi-
trary line passes through a’y’z’, in which case it passes through
@ point common to U and V, whether they have a common
factor or not. Rejecting this factor, the remaining condition
11=0 is the sought condition that U and V may have a
common factor, and we see that it does not involve a@y, that
it is of the order ab’ + a’b—ab in xyz’, and of the orders 4, a
respectively in the coefficients of U and V.

382. When the method just described is applied to the inves-
tigation of the points of inflexion, that is, to the determination
of the condition that AU’, A*°U’ may have a common factor,
we have a=1, a =n—1, b6=2, b’ =n—2, and the formula just
obtained gives 3 (n — 2) for the order of [1 in 2’y’z’, which is the
order of the Hessian as already found. It appears also that IT
is of the second degree in the coefficients of AU’, and of the

342 GENERAL THEORY OF CURVES.

first in those of A’U’; and since each of these is of the first
degree in the coefficients. of the original equation, [I involves
these coefficients in the third degree, which also agrees with
previous results.

To proceed then to the case of the double tangents, since the
equation A =0 is reduced to the form $A*U’A"* +...4 Un"? =0,
a specimen term of its discriminant is (A*U’)"*U"*, whence we
see that Y is of the order (n+2)(n—3) in xyz, of the order
(n —2)(n—8) in a’y’z’, and of the order 2 (n—38) in the coeffi-
cients of the original equation. In the next place we can
show that all the intersections of Y and AU’ coincide with
x’y'2’; for the equation of the system of n*—n—2 tangents
through the point 2’y’z’ found by the method of Art. 78 is of
the form 4AU’+ Y(A*U’)’=0, and this system can evidently
be intersected by AU’ in no other point than 2’y’z’; therefore
making AU’=0 in the equation last written, we see that
AU’ can meet neither Y nor A’U’ in any other point than
ax’y’z’, We may then apply the method of Art. 381, writing
a=1, a =n—-1, b=(n+2)(n—3), b =(n—2) (n—3), whence
ab’ + a’b =(n? + 2n—4)(n—3). We have then for the order of
I] in xyz’, (n+3) (n—2) (n—3). It is of the order (n+ 2) (n—3)
in the coefficients of AU’, and of the first order in the coeffi-
cients of Y, and therefore of the order (n+ 4)(n—3) in the
coefficients of the original equation. The bitangential curve
II=0 meets the original curve U=0 in n(n +38) (n—2) (n— 8)
points, and since there are two of those points on each bitangent,
the number of bitangents is 4n(n—2)(n*—9) as found other-
wise, Art. 82.

383. The method of Art. 381 not only enables us to de-
termine the order of the required condition [[=0, but by the
actual performance of the operations indicated, to find the con-
dition itself. Thus 2’, y’, 2’ being, as before, the coordinates of
the point on the curve, in the case of points of inflexion we
have to eliminate between az + By +yz=0, AU’=0, A’U’=0,
and the last equations written at length are

Lae+ My + Nz=0,
ax + by’ + cze* + 2fyz + 2gzx + 2hay =0.

GENERAL THEORY OF CURVES. 343

It will be convenient, in order to avoid numerical multipliers, if
we suppose the original equation to have been written with
binomial coefficients, and the common multipliers to be removed -
after differentiation, so that Z, M, N denote the first differentials
of U’ divided by x; a, 5, &c., the second differentials of U’
divided by n(n—1); and the ordinary equations. of homo-
geneous functions will be La’+ My'+ Nz’=U’, ax’+ hy'+ gz'=L,
&e. | .

Now the condition that two lines shall intersect in a point
on a conic may be written in the form of a determinant

a, h, g, L,a@
h, b, f, UM, 8B

n tI & NY
L, M, N,

| a, B, %; =0,

for it may be verified, that this determinant expanded is the
same as the result of substituting in the equation of the conic, the
coordinates of the intersection of the two lines, viz. My— NB,
Na— Ly, LB —Ma. Now, in virtue of the equations of homo-
geneous functions, the above determinant may be reduced by
multiplying successively the first three limes and columns re-
spectively by 2’, y’, z’, and subtracting from the fourth. It
then becomes, if we denote ax’ + By’ + yz’ by R,

a,h,g, 9, ao
h, by f, 9, B

J) fi % 9%; ¥
6,-0.0, <7 U’, —k

a, B, 7, —#, 0 ’

a, h, g, 4 a, hy 9

or — U’ hb, fi P — fh} h, 5, f
Bo Sy & 9, fre
a, By ¥

After Clebsch we use the abbreviation ie for the determinant

multiplying U’, in which the matrix of the Hessian is bordered
vertically and horizontally by a, 8, y. In like manner the de-

344 GENERAL THEORY OF CURVES.

terminant with which we started, in which the same matrix is
twice bordered, by a, 8, y, and by the differential coefficients of

'U, would be written age and the equation we have
established is .

(9 )=-o'()-

U, |
When a’y’2’ make U’=0, the equation pe .) = 0 reducesto
?
H=0, as it ought.

384. In order to proceed by the same method to find the
equation of the bitangential curve, we have to find the result of
substituting My—N8, Na- Ly, LB—Ma for x, y, z respectively
in the discriminant of the equation A=0 (Art. 380), and our
course will be first to find the result of that substitution in the
several coefficients of that equation, viz. A*U’, A’U’, &c., or as
we shall more briefly write them A’, A®*, &c. The result of sub-
stitution in A’ has been calculated, (Art. 383), and Hesse has
shewn by the following process, that the result of substitution
in A* is of the form P,U’+ Q, (ax’ + By’ +2’), which when
x'y’z’ is on the curve reduces to Q,(aa’+By’+y2’)*, His
method shews that if this be true for two consecutive A*”, A‘,
it will be true for A’, and enables us to express P,,,, Q,,, in
terms of the corresponding previous coefficients. It will be
remembered, that by definition we have A**=A(A*), where

d d d Me ae
A denotes the operation x ait ¥ iy +2555 but in this it was
assumed that wyz, 2’y/z’ are independent quantities. In the
case now under consideration, where x is supposed to have
the value My —N~, and therefore to be implicitly a function
of xyz’, it must therefore be understood, that in the operation
A the differentiation only affects x’y’z’ as far as they appear
explicitly, and not as they are implicitly contained in ayz.
d
Si without
this restriction, then according to the gener Hy rule for deriving
differentials with regard to «‘y’z’ on the supposition that xyz
are variable from the differentials on the supposition that they

d
Let us denote by v the operation x pO hes +) +2

GENERAL THEORY OF CURVES. 345

are constant, we have in operating on any function S,

Y

ds dS ds
ME ADH op Veta vere ve

_ 885. The next step is to calculate the values of VL, VY, Ve
The result of operating with y on any function § is easily

Si S. 8,

3

LL, M, N

shat a, B,

xz or My — NB the sce is

| hy — 9B, by 18, f Fy - 0B
L, M, N

) | a) B, y ’

where the coefficient (n—1) arises from the condition we have

introduced, according to which the differentials of Z, &c. are

(x—1)a, &c. The determinant just written is then reduced by

the following process:

seen to be , and therefore when the function is

(n — 1)

1, J) ds C % I, Sy ©
0, hy-gB, by-JB, fy-c8 | _ |B, 4, 6, Sf
0, L, M, N oe) Oy Day Hh, 2ae
0, a, B, Y 0,4, 8, ¥
Y, Ci otey Aga: Ri — ax’, — ax’, — hx’, — gz’
oe B, Ry Ose y B, h, b, -
— (By' + yz), ax’, he’, ga’ apathy I; Jj ¢
0, a, By ¥ 0, ay B, Y
h, b, f ‘
=hi gf, ¢\+2 ( ).
a
a, B, ¥

If we denote (*) by =, and the halves of its several differ-
ie 8 :

entials with regard to a, 8, y, by %,, 2, %,, these last differ
only in sign from the determinants multiplying / in the values
of Vx, Vy, V2, and we have

eee ae eae

s-0(0) Bede)

be

346 GENERAL THEORY OF CURVES.

In particular let S=A*(V), where V is any function of the
order n’ in 2’y’z’, then since oS =k ve A**(V), we have

v (A'V) =a") —& (n-1) B ( 5 2%, at x5 ~) a** (V)

+k(n-1) (% ) (« att ate =) OY,

Since A*’V is a homogeneous function in 2’y’z’ of the degree
n’ —k+1, the last term reduces to

k(n— 1) (n’—k+1) C) At ().

386. It will be convenient to use the abbreviation y for

d

the operation &, sae ay? > kl and it will be observed

3 dz
also that
a, h, J; V,
te fe ¥. V
V ae ? 7/7) 2 or = B
HIM =| Oe (,)
a, By ¥

The result of operating with y on @ vanishes, as may easily be
seen by substituting in the last column of this determinant for
Vi, Vi, V,, 0 the values hy — 98, by —f8, fy — cB, By — 8, when
it at once resolves itself into two, each of which vanishes in con-
sequence of having two columns the same. The result then, of
operating, with yr on any function containing az, y, z, is the
same, whether or not these be regarded as constants. The
equation of the last article then, as applied to the quantities
A*, &c. which we desire to calculate, is

At = 9 (A) +4 (n-1) Ry (A™)- k(n- 1) (n—h +1) BA.

387. From the expression just found, we can shew that if
we have A** =P, ,U+ Q, 9, A*°=P,U0+ Q,R’, then A™ must
be of like form. For we have only to substitute these values
for A**, A* in the equation of the last article; and we must
observe that y(U) and y (#) both vanish, as at once appears
by substituting either Z, M, N, or a, B, y for S, S,, S, in

GENERAL THEORY OF CURVES. 347

Hence vy (A4‘)=Uy(P,)+F’v(Q,). We

S,, &,; 8
Ly M,N
» By ¥

ae by ny es nL, nM, nN, and a, B, ¥ respectively
for 8, 8, 8, in (° ); (UT) = nH1R, ¥ (R)=(%) , and therefore

ap At! = Ub (P,,) + Rr (Q,,) —nP,_,HR+2R3Q,

Collecting then the terms in the expression given for A‘
(Art. 386), we have A”’*= UP,,, + R’Q,,,, where

Py.=V (BP) —k(n- 1) (n—h+ 1) BP, , +2 (n= 1) RY(P,,);
Ques = 0 (Q,) —k (n= 1) (n-k-1) 3Q,,
+k (n—1) By (Q,,) - n(n—1) RP, HL

388. From these formule we are able to form a table of the
values of P,, Q,, &c. Thus to commence, it is obvious that
P.=0, Q,=0, and (Art. 383) PR=-—, Q,=-—H. Hence
P,=-A(2), Q,=-4 (H).

When the curve is a cubic A’ is no other than the cubic func-
tion itself, and the value just given for @, may be geometrically
interpreted as follows: If any line aw+ y+ yz meet a cubic,
and from each of the points of meeting four tangents be drawn
to the curve, the twelve points of contact lie on the quartic

a (H,N-H,M) +8 (HL — H.N)+y(H,M— HL) =0;

for this condition must, as we have seen, be fulfilled by any

point of the curve whose tangent intersects az + By+yz on

the curve. This result also immediately follows from Art. 183.
Proceeding now to Q,, we have (Art. 387)

Q,=—-V (4H)+4+3 (n—1) (n—4) 2H-3 (n—-1) Ry (H)4+3n(n-1)2H
=— V (AH) +6 (n—1) (n—2) SH—3 (n—1) Ry (HZ).

But in conformity with the result at the end of Art. 385, writing
k&=1, and denoting by n’ the degree of the Hessian, or 3 (n — 2),

v (AH) =A’°H—(n—1) Ry (HZ) + (n—- 1) nD.
Hence Q,=- A°H+(n-—1) nSH—-2(n—-1) Ry (A).

348 GENERAL THEORY OF CURVES.

389. We have now the materials for forming the equation
of the bitangential curve of a quartic. According to the
method explained ae 384) we are first to form the discrimi-

nant of A=0, or of 5 A+ oe AMM + 3G A*y’; and
then having seciien My — NB, &c. for x, &c. we must, by
the help of the equation of the curve, remove a, 8, y. By
making the substitution before forming the discriminant, the

équation cans
1 §
73 ae 1234 Ch =%

whose Aare differs only by a numerical factor from
Q,- 3Q,Q,, a function still containing a, 8, y in the second
degree, and therefore requiring further reduction. Jor this
purpose the following formula is useful.

QA +

390. If we border the matrix of the Hessian both hori-
zontally and vertically with three rows and columns, the
resulting determinant is clearly the product, with sign changed,
of the two determinants added horizontally and —
Thus in particular if V, W be functions of the orders n’, n’”’

we have =A(VA(W)=

a, h, g, a Vy Lb Pe Nee aia ee

h, 6, f, B, Ve, M Rye Oss Ja ec 0
Fs & hy ef ae ey Pee 0

a By, (ey Py Wy * Oy 9, — tt
W., W., W. Way Wo Wo Dy Oe re

or A(V) A(W)
” ; , W ” V 9 V a V
an'n’VW (2) wR )-n"WR( +B ( G40 (2,
and when 22/2’ satisfy the equation U=0, Me last term
vanishes. ‘Thus in particular
— V V
up are Tar bs 2 2
(AV anv? (%)\ ov VR( -) 4B (,),
or in the notation we have before used

2 2 12 2 / - H
Q; =(AH)' =n? HS — 2n' HR (H) +B fer

GENERAL THEORY OF CURVES. 349

the last term denoting the result of writing in 3, instead of
a, 8, y, the differential coefficients of H.
In precisely the same way we get a formula of reduction
for A?V by writing in the preceding determinant ’
da d
dx? dy ” de
and supposing the operation to be performed on V. In the
reduction, then, we have instead of n’V, and of n” W,
4 a + 7/ xa na ok a
* de’ 4 dy’ - dz’
and the formula becomes |
A°V=n! (n’ — 1)V (*) —2(n’-1)R co) + R* @ V,
a a 7 Sie aa

x

for V, VV, and for W., W, W,,

where. the last symbol denotes the result of substituting in =
symbols of differentiation instead of «, 8, y, and operating on V.
_ Introducing the value thus found for A’H into the value
given for Q, (Art. 388), we have :

Q,=- (n’ —n) 2H+4+2 (n’—n) Ry (HH) -L* ( fd,

zx

Thus, then, since Q, =— H we have in general

(n’—n) Q,-—2'9,0,= 2 {(0 —n) eS —nH o uh

x

and in the case of the quartic, for which n=4, n’ =6,

ar a0nm {) am) 3h

x

and accordingly the equation of the bitangential curve is

(i) (f=

x

that is to say, if = written at full length is
Ao? + BB’ + Oy’? +2F By +2Gya + 2HaB,
this equation is
aH” : ai” dH” 7d di fo dH dH dif

GH fH fH en os aH ea
=3H 7 ay Pat Cat ae OO Teds * E san SB)

a curve of the fourteenth order.

350 GENERAL THEORY OF CURVES.

391. The equation just obtained may be transformed by
the help of the expression given (Conics, Art. 381, Ex. 1), for
the condition that the polar line of a point, with regard to one
conic, may touch another. We there saw that if az*+&c.,
a’x* + &c. be the two conics, we have

(be —f*) (ax + h’'y + 9z)*+ &e.= {a’ (be —f")+&e.} {a’x*+ &e.}-F,

where F denotes a conic covariant to the two conics. And, in
like manner, that

(Bc —f”) (ax + hy + hots &e.= {a We I”) + &e.} {aa*+&e.}—F.

Now if a, b, c, &c. have the same meaning as before, and if
a’, &c. denote the second differential coefficients of the Hessian,
then, its degree being n’, (a’a + h’'y+ g’z) &c. are (n’— 1) times the
first differential coefficients, and (bc —/”) (a’a + h’y + g’z)* + &e.
is (n’ — 1)” times the covariant we have called © (Art. 231). We
may give the name ©’ to the corresponding covariant in which
the differential coefficients of the curve and of the Hessian
are interchanged, and whose vanishing expresses the condition
that the polar line of a point with respect to the curve should
touch the polar conic of the same point with regard to the
Hessian. In like manner, a’ (bc —f’)+ &e. is ® and a’a’* + &e.
is n’ (n’—1)H. We have then the identities

(n’-1)’@=n' (n’—-1) HO-F, O'= U0’-F,
(n’-1)’O—7n’ (n’ -—1) Hb=0'- UP’,
and in the particular case of the quartic where n’ = 6,
250 — 30H = 0 — Ue’”

Thus, then, the points of contact of bitangents are the inter-
sections with the curve, not only of © — 3H® as already obtained,
but also of 150-0’ or of O’—45H®; or, again, bitangential
curves might be expressed in terms of the covariant F.

392. Let us now proceed to the fifth order. We have
(Art. 387)

Q), =v (Q,) fe (n—1) (n—-5) = Q,+4 (n—1) iy (Q,) —4n (n—1) HP, ;

GENERAL THEORY OF CURVES. 351

and using the value of Q, last obtained, and employing the

abbreviations © for a and ® for i H, we have

Q,=—n' (n'—n)HA(2)—n'(n’—n)EA(H)+2(n’—n) RAW(H)—-L?A(#)
+ 4n(n—1) HA (3) +4 (n—1) (n— 5) SAH—4(n—1) Ry (AH)
= — 2 (n’? —13n +.18) HAS — 2 (n? — 3n +8) SA (H)
+4(n—3) RA (WH) —4 (n—1) Ry (A) — BA (®).

In particular when n = 5, we have
Q, =44HA (3) -362A (1) + 8RA (pH)—-16RYy(AH)—R'A(*).
In this case we have also

Q,=— 3621+ 8Ry (H)— Ro,

Q=-AH, Q,=-H.
In order to form the bitangential curve of a quintic, the quantity
to be calculated is

(27,9, — 5Q,9,)" = 5 (49, -9 ,9,) (5 9,’ — 129, Q,),

a quantity containing afy in the sixth order, and which it is
necessary, by the help of the equation of the curve, to shew to

be divisible by 2°. Now, in virtue of a formula already ob-
tained, we have

40°-90,0,=R?(40— H®).

It is also easy to shew that 27Q,0,—5Q,Q, and 5Q?— 120, Q,
are each divisible by &; but I have not been able to carry the
reduction further.

We shew elsewhere (Higher Algebra, Art. 295) how all these
calculations may be made by symbolical methods.

393. Another method* of solving the problem of double
tangents is suggested, by what was proved (Arts. 183, 235) that
the point where the tangent to a cubic meets it again is
determined by the intersection of the tangent with the line
cH, +yH,+2H,=90. It occurs to attempt to form in like
manner the equation of a curve of the order x — 2, which shall
pass through the (n—2) points where the tangent to a curve

* I gave this method in the Philosophical Magazine, Oct. 1858, and Quarterly
Journal of Mathematics, vol. ul. p. 317. See also Memoirs by Prof. Cayley,
Phil, Trans, (1859), p, 193, and (1861), p. 357,

352 GENERAL THEORY OF CURVES.

of the n™ order meets it again. If the equation of this tan-
gential curve were once formed, then, by forming the condi-
tion that the given tangent should touch this curve, we
should immediately have the equation of the bitangential.
Now, what has been proved already as to the order of the
bitangential will enable us to see what must be the order of
the tangential curve in a’y’z’ and in the coefficients. ‘The con-
dition that the line Lx+My+ Nz shall touch a curve of
the (n—2)" order is of the order (n—2) (n—8) in L, UV, N,
and of the order 2(n—3) in the coefficients of that curve.
Consequently, if the coefficients of the tangential curve con-
tain a’y’z’ in the order p, and the coefficients of the ori-
ginal in the order gq, the bitangential must be of the order
(n —1) (n — 2) (n- 3) 4+2p (n—3) in aw’yz’, and of the order
(n — 2) (n—3)+2q(n—3) in the coefficients of the original.
But actually the bitangential is of the order (n — 2)(n — 3)(n + 8)
in ay’z’, and of the order (n+4)(n—8) in the coefficients of
the original (Art. 382). It follows then that p =2 (n—2), g=3;
that is to say, that the tangential must be of the order 2 (n — 2)
in xyz’, and of the third order in the coefficients of the original.
Further, we know that if a’y’z’ be on the Hessian, the tan-
gential must pass through 2’y’z’, and therefore the substitution
of w‘y’z’ for ayz must reduce the tangential to H. ‘This con-
sideration and the known form of the tangential in the case
of the cubic suggests that the tangential in general is the
(n—2)” polar of a’y’z’ with regard to Hor A”"7H, for this is
a curve of the right order in wyz, in a’y’z', and in the coeffi-
cients, and it will pass through a’y’z’ when this point is on the
Hessian. Accordingly, in the next article we examine whether
the curve A**(Z) does pass through the points where the
tangent meets the curve again, and though the answer is found
to be in the negative, the process of examination leads to the
true form of the tangential.

394, Take then the origin on the curve, and the axis of
y as the tangent, and let the equation of the curve be

nby + $n (n —1) (c,x* + 2c.ay + c,y’)

1 ; :
+ 5” (n —1) (n — 2) (d,x*+ 3d,x°y+ 8d,xy’+ d,y’) + &e. - 0.

GENERAL THEORY OF CURVES. 1858

It is to be observed, and the remark will be useful in the
sequel, that the several polars of the origin, with regard to
the curve, are got by writing n—1, n—2, &c., for n in this
equation. Now, in order that a curve may pass through the
tangential points, its equation must be such that when we
make y=0 it will reduce to

y?

Ga c+ n(n —1) (n—2)d.a+ &e.=0.

1

2.3
‘Let us form then the equation of the Hessian, and since we
are about to form its polar curves with regard to the origin,
and then to make y =0, we need only concern ourselves with
those terms of the Hessian which do not contain y. The
second differeutial coefficients of the given curve are

a=c,+(n—2)d,c+4(n—2) (n—3)e2' + &e.,
b=c,+(n—2) d,x+ 4 (n—2) (n— 3) ea + &e.,
‘. ¢= | 4 (n— 2) (n—3) ca" + &e.,
f=b +(n-2)ea+4(n—2) (n- 3) daz’ + &e.,
= (n—-2) c+ 4 (n—2) (n—3) d,v* + &e.,
h=c,+(n- a exc’ + Ke.
The equation then of the Hessian is readily found to be
¢,b° + (n — 2) d,b’x + {4 (n— 2) (n— 3) eb’ + (n— 1) (n—2) P} x?
+ {k(n — 2) (n= 8) (n—4) f,B? + (n—1) (n- 2) Q
+(n—1) (n—2) (n—38) RB} w+ &. =0,
where for brevity we have written
2P=¢,0, —¢,¢, + 2be,d,—2bed,, 2Q=d,c, —2c¢,¢,d, + ¢,d,,
3h =c,0,d, — dc," + 2e,be, — 2c,be,,

0°2° 0

but the actual values of these quantities are not material to
our purpose. What is important is to notice that the equation
divides itself into groups of terms each having the same function
of n as a numerical coefficient, so that if we want to form
the equation of the Hessian of the first, second, &c., polar of
the given curve with regard to the origin we have only to
substitute n —1, n —2, &c., for m in the above equation.

Now the line polar, with regard to the origin of a curve
of the n™ degree u,t+ u,+&e.=0 being nu,+u,=0, the line

ZZ

354 GENERAL THEORY OF CURVES.

polar of the origin, with regard to the Hessian, which is a
curve of the order 3(n—2) is, from the preceding equation,
3c,+d,e=0, together with a term in y irrelevant to the present
question; and since this equation does not contain n, we see
that the polar of a point on a curve with respect to the Hessian
of either the curve itself or of its polar curves all meet the
tangent in the same point. In fact, the polar is in every
case the same line. When n=8, 3c,+d,a is the result of
making y= 0 in the equation of the curve; that is to say,
the polar with regard to the Hessian is the tangential, as we
have seen already.

The equation of the polar conic of the origin with regard
to a curve of the n™ order is 4n(n—1) u,+(n—1)u, +u,=03
and therefore the polar conic with regard to the Hessian is

3 (n — 2) (8n— 7) ¢,b° + (n—2) (8n-—7) d Dx
+ {4 (n— 2) (n—3) 6,0? + (n—- 1) (n—2) P} a? =0,

and it is evident, on inspection, that in the case of the quartic
this polar conic cannot be the tangential, because it contains
the group of terms P which do not similarly occur in the
equation of the curve. But we can readily form an equation
not containing these terms. Let A’H=0 denote the equation
we have just obtained, and let A’H, denote the polar conic
with respect to the Hessian of the first polar of the origin,
and as we have already seen, A’H, is derived from A*H by
writing n—1 form. Then it is easily verified that

(n— 3) A°H —(n—1) AH, =(n—- 3) 8 {6c,+4d,a+e2'}.

But when the given curve is of the fourth degree, the right-
hand side is what the equation of the given curve becomes when
we make y=0. It follows then that A’H-—38A°H, is the
required tangential of a quartic.

In precisely the same way the polar cubic of the origin,
with regard to the Hessian, is found to be

$ (3n — 6) (8n —7) (8n — 8) cb" + 4 (n — 2) (8n — 7) (82 — 8) dbx
+ 4 (n—2) (n— 3) (3n— 8) ¢,b°2? + (n— 1) (n—2) (3n— 8) Pre
+4(n-2)(n—8)(n—4) f,b'a°-+(n—1)(n—2)” Qa*+ (n—1)(n-2)(n-3) Ra’,

GENERAL THEORY OF CURVES. 355

and A*H,, A*H,, &e. are found by substituting (n — 1), (n — 2), &e.
for n. And we can verify that

(n — 3) (n—4) A°H— 2 (n—1) (n—4) A°A, + (n— 1) (n—2) A°H,
= 2 (n— 4) (100, + 10d,v + 5e,x* + fax’).
And when n=5 the right-hand side of the equation is what

the original equation becomes when we make in it y=0, and
therefore it follows, as before, that the tangential is

A’ H — 4A°H, + 6A°H, = 0.
When n=6 the tangential is in like manner
A‘H —5A‘H, + 10A‘H, = 0.
I was hence led, by induction, to the conclusion which
Professor Cayley has verified independently, that the tangential
is in general

A"? H — (n—1) A"°H, +4 (n—1) (n-2) A’? - &. = 0.

395. It is easy to establish what has been stated above,
that the polar lines of the origin are the same with regard to
its Hessian, and to the Hessian of any of the polar curves.
We hav oO = = ot + be, or employing the usual abbrevia-

tions A for be—f”, &e., we have

dH a(,@ fF @
dx Ty get baat Cae
Fi Z i?
ae 2H | U

with similar expressions for the differentials with regard to
y and z It is to be noted that these may be written in
the abbreviated form on (") . Now the differential
dx dx \d,
coefficients of the first polar 2’ U,+y'U,+2U, are got from
the corresponding coefficients of the original curve by per-
d

forming on them the operation 2’ = +y aie ae which
when we substitute 2’y’z’ for xyz is equivalent to multiplying
each by the factors n—1,n—2, &c. But the same numerical
factor being common to every term in the expression for 7,

356 GENERAL THEORY OF CURVES.

it-is plain that wH,+yH,+z2H, represents the same line
whether the polar be taken with regard to the Hessian of the
original, or to that of its first polar. And the same argument
applies to the other polar curves.

Let us proceed to the polar conic. If we differentiate the
expressions just given for H, &c., the differential will consist
of two groups of terms, viz. the differential on the supposition
that A, B, &e. are constant, together with the terms got by
differentiating these quantities. If we write, for shortness,

E,, €to denote the symbols of differentiation with regard to
X,Y, 2, we have

B= ARP + By’ +&e.} U + EF {a(nb’—1/C)+ b (S6'—6'€)*+ &e.} U,
it being understood that the accents in the last group of terms
may be dropped after the expansion, the term &£an’¢", for

he 8 lee okt 8 é
dady? dedz** The last equation may

instance, standing for a

be written in the abbreviated form
erp (§ » (EE
pu=—e (2) +8 (ep).

Thus then the equation of the polar conic of any point, with
regard to the Hessian, may be written V+ W=0, where V
denotes a group of terms in each of which a fourth differential
is multiplied by the product of two second differentials, and W
a group in each of which a second differential is multiplied by
the product of two third differentials. Now if we take the ~
Hessian of the first polar, then, as has been stated above, the

second, third, and fourth differentials become multiplied by
n—2,n—3, n—4 respectively, and the result is

A*H, = (n—2) (n-—4) V+ (n-—3)"W=0,
which when n =4 reduces to the latter group of terms. The
equation of the tangential of a quartic is then evidently of the

form V+kW=0, and may be transformed accordingly. Thus
it may be written in the form

d d dia Seen

d d OM! fered ye
43(a55 +9 7 +8 45) (4 ra + &e. | U'=0,

z da

POLES AND POLARS. 357

The equation of the bitangential curve is got by expressing

the condition that the tangent Zx+ My + Nz should touch the

conic just written; and it will evidently consist of three groups
of terms, since the eagalcos that a line should touch S+ 4S" is
of the fot =+kb+ #2’ =0. What answers here to 3 is the
covariant called ©’; and I have verified that the other two
groups of terms are also expressible in the form © + LH®.*

POLES AND POLARS.

396. It will be convenient to collect here some properties
of the Jacobian of a system of three curves, stated Higher
Algebra, Arts. 88 and 176, and elsewhere in this volume. The
Jacobian is the locus of points whose polar lines with regard

to three curves meet in a point, its equation being

Co) Us Us
W,) Wey Ws;

* I attempted in like manner to obtain the bitangential curve of a quintic
by writing down for the curve whose equation is given Art. 394, a covariant
of the right order, and such that the absolute term vanishes if the axis of x
touches the given curve a second time. For instance, if ~=40—-9H®, then

4 (3) + &. and (4 5 . i oa &e. are covariants of the right order. Although I

have not been successful, % may be useful for purposes of reference to give the
values I obtained for the covariants in this case. It will be seen that, without loss
of generality, we may suppose c, and c, to vanish, We have then
H = 0c + 8b? (doa + dy) + 3 (be) — 4bcd,) x? + 8 (2b%e, — Sbcd,) xy + 3 (b%e,— beds) y?
+ (b*fy — 16bce, + 18c?d,) x* + (8b2f, — 39bce, — 9b dod, + 9bd,? + 18c7d3) ay
+ (— 6bef, — 12bd,e, + 12be,d, + 18c7e, + 24ed,d, — 18cd,") «* + &e.,
© = 90? {(b4d,? + 60%c7d,) + (4b4d eg + 120%c?e, — 6b3edyd, — 57b7c%d2) x
+ (4b*d,e, + 1203c%e, — 28b%cdyd, + 31b%cd,? — 39b7c%d3) y
+ (204d fy + 4bte,? + 64%? f, + 6b3cdye, — 48b%cdye, — 105b7c%e, — 29307c?d,d,
+ 26907c%d,? + 36dc'd,) x? + &e.},
& = 66 [(03e,4+462cd,) +x (b3 f,-8b2ce,—38b07d,) +y {b°f\— 20ce,+27b? (d,?— dyd) —41 bcd}
+ x? (— 12b%cf, — 12b2d,e, + 12b%e,d, + 6be7e, — 162bcd,d,+ 168dcd,? — 6e8d3) + &e.].
Of the quantities A, B, &c. the only ones which contain terms independent of x and
y are A= 0?, F= bc; so that if any quantity w of the form 9+/H® written at
full length be 4 + Bor + Byy + Cyz? + &e., then the degree of w being 22, the

d?
absolute term in the covariant A (S) + &e. is 0?-B,? + 44bcAB,, and in A A ¢ + &c,

da?

is 20?C, + 42dcB,.

358 POLES AND POLARS.

We have seen, Art. 191, that the Jacobian is the locus of the
double points of curves of the system
Aut pu +vw =0.

If the three curves have a common point, ig point is on
the Jacobian. For, from the equations
rU+ YU,+ ZU,= MU, xv, + yv,+ 20, =m'v, ew, + yw,+ 20, =m",
(where m, m’, m”
tively), we have

are the degrees of the three curves respec-

Jae = mu (v,W, — ¥,0,) + mv (w,u, — W,u,) + mw (u,v, — U,,),
which we may write
Jx=mAu+m' Bo + m’ Cw,
whence evidently J vanishes for any values which make u, v, w
to vanish.
If the three curves be of the same aeoe, this common point

is a double point on the Jacobian. For differentiating with
respect to x, we have

Ta inn 4 iy et my Uo
La pad dla ne ca tea ay jigs y

but since Aw, + Bv,+ Cw,=J, we see that when m=m' =m’,

+ mAu, +m’ Bo, +m” Cw,;

Go itt vanish for any values which make wu, v, w and con-

dx

sequently J to vanish. So, again,
add GAs ORs @ 5 BO
—_—= meen ee Cicer a UR AP me
dy dy
which, since Aw, + Bv,+ Cw,=0, vanishes for any values that
make wu, v, w, J to vanish, when m=m’=m". In like manner
the other differential coefficient of J vanishes for the same point.
If only two of the curves be of the same degree, the
Jacobian touches the third curve at the common point. For
the equation written above, when we make m=m’, becomes

add dA dB dC
a tg Pe ali ae + mn wT + md + (m” —m) Cw,

+ mAu, + m’ Bo, +m’ Cw,,

J+a2

and for the common point, this reduces to «J, = (m” —m) Cw,;
and we have, in like manner,

xd, =(m"” —m) Cw,, xJ,=(m’ — m) Cw,;

POLES AND POLARS. 359

so that ad, + yJ,+2J,=0, xw, +yw,+2w, =0,

represent the same right line.

If in this case the common point be a double point on w,
it will also be a double point on J, having the same tangents
as those for the curve w.*

The values just, obtained for J,, J,, J, evidently vanish when
W,) Wy w, vanish. Differentiating again, and omitting the
terms which vanish as containing uw, v, w, J, J, or w,, Wy Wy
we have

ad dA dB =
eas m (u, Je tt i) + (m” —m) Cw,,.

But from the values previously found for A and B, we have

dA dB

U, dx a dx pails (VW, oe V,W,.) + YU, (20,,2, oe Ws)

and by eliminating xyz from the equations

-

LU, + YU, + 2uU,=0, xv, + yv,+2v,=0, rw, + yw,,+2w,,=0,

we have
U, (V,W,5 — V,W,.) + U, (00,0, — W,,4,) = — ,, (UU, — U0.) =— Cw.)
or xd, = (m” —2m) Cw,,,

and similarly the other second differential coefficients of J are
proportional to those of w; or the two curves have the same
tangents at their common double point.

397. It is proved, as in Art. 190, that there are

(m — 1)’ + (m—1) (m’ — 1) + (m’ — 1)?
points, whose polar lines, with respect to two curves w, v, are
the same, and through these points must pass the Jacobian of
u, v, and any third curve. It was shewn (Art. 97) that the
Jacobian intersects u in the points which can be points of
contact of w with curves of the system v+Aw. Hence, it
immediately follows that the locus of points, which can be
points of contact of curves of the system w+ Au’ with curves
of the system v+ mv’, where u and w’ are of the degree m, and
v and v’ of the degree m’ is a curve of the order 2m+ 2m’ — 3,

* Clebsch and Gordan, Abelsche Functionen, p. 62,

360 ‘POLES AND POLARS.

whose equation may be written in either of the equivalent
forms :*

Uy Uy. Us Uy) U., U,

vw U's, W's w', i ws, U's w’, = 0,
Vy Voy Us v's Va je
Yr Vy Us Vy Uy Us

UW} Vy Vey Vg | —U! Vy Vey Vy | = Oz
wy Uo U, Wy Wo ws

Again, it appears from the preceding that the points in
which curves of the systems w+ Aw’, v+pv, w+vw’, can all
three touch, are among the intersections of two curves of the
degrees respectively 2m+2m’—3, 2m+2m”—3. But among
these intersections are included the m” points w, w’; and the
3 (m—1)’ points common to the Jacobian of all curves of the
system u+Aw’. Deducting these numbers, we obtain for the
number of points in which the three curves can touch

A (mm! + m'm” + mm) — 6 (m +m’ +m”) 4+ 6.

398. We have seen (Art. 97) that the order of the condition
of contact of two curves u, v, or, as we shall call it, of their
tact-invariant, is in the coefficients of v, m (m+ 2m’— 3) — 26 - 3x
or n+2m(m’—1); and, in like manner, of the order n’+2m’(m-—1)
in the coefficients of uw. The tact-invariant, in the case of
two conics, was found (Conics, Art. 372) by forming the dis-
crimmant of w+2v, and then the discriminant of this con-
sidered as a function of ». By similar reasoning to that
used in the case of conics, it may be shewn that if the same
process be employed in the case of two curves of the m™
order, the tact-invariant is a factor in the result. In fact
if A be the tact-invariant, B=0 the condition that it may
be possible to determine » so that wt+Av may have two
double points, and C=O the condition that it may be possible
to determine X so that w+Av may have a cusp, then the
discriminant, with respect to A, of the discriminant of w+dz,

* Steiner has remarked that the number of curves of the system wu + Aw’, which
osculate curves of the system v + pv’ is 3 {(m +m’) (m +m’ — 6) + 2mm’ + 5}, Crelle,
vol. XLVII. p, 6. It will be remembered that we have seen, Art. 102, that the con-
dition for two curves osculating is, in addition to the conditions of ordinary contact,
that the ratio of H to L* shall be the same for both,

POLES AND POLARS. 361

is AB*C*. That Band C are factors appears by taking wu as
a curve which has either two double points or a cusp. In
this case, not only the discriminant of vanishes, but its
differentials, with respect to each of the coefficients of u (Higher
Algebra, pn 116); therefore, in the discriminant of u+ dz,
the term not containing » and the term containing its first
power both vanish, or A” is a factor in the discriminant ; therefore
its discriminant ganalilceed as a function of X vanishes.

Thus, if « and v be cubics, the discriminant of each contains
its coefficients in the twelfth degree, and these coefficients enter
in the one hundred and thirty-second degree into the dis-
criminant with respect to >. But the tact-invariant contains
the coefficients of each in the degree eighteen; and the invariants
which vanish when w+ Av can have a cusp, or a pair of double
points, contain the coefficients of each curve in the degrees
twenty-four and twenty-one respectively. Jor the degree in
the coefficients is the same as the number of curves of the form
u+hv+pw which have the singularities in question. In the
case of the cusp, this number is found by putting the inva-
riants S=0, 7’=0; giving thus an equation of the fourth and
one of the sixth degree to determine >, mw, and we have
twenty-four solutions. In the case of the two double points,
we may suppose uw, v, w to have seven points common, and
through these points we can have twenty-one systems of a
line and a conic. We have then 132 = 18 + 2 (21) + 3 (24).

399. In general the discriminant being of the degree
3(m—1)*, the discriminant with respect to X contains the co-
efficients of each curve in the degree 3 (m—1)’ (3m*— 6m + 2).
Now the tact-invariant contains the coefficients of each in the
degree 8m(m-—1), and from considerations afterwards to be
explained, it appears that the order of the condition that
w+ dv may have a pair of double points, (or, what is the same
thing, the number of curves of the system w+2Av+ ww, which
have two double points), is 3 (m— 1) (8m*—9m*—5m 4+ 22),
and the corresponding number for the case of the cusp is
12 (m—1)(m—2); and it may at once be verified that

3 (m— 1)" (8m" — 6m + 2)

=3m(m—1)-+3 (m—1) (3m°— 9m?— 5m + 22) + 36 (m2 — 1)(m — 2)-
; AAA

362 POLES AND POLARS.

In like manner, having formed the discriminant of Au + wo + vw,
where uw, v, w are curves of the same degree, we may form
the discriminant of this considered as a function of A, mw, V3
and this discriminant will contain as factors the resultant of
u, v, w, and the conditions that it may be possible that a curve
Au + wv+vw may have three nodes, or may have a node and
cusp, or may have a tacnode; the order of any of these
conditions in the coefficients of any of the curves being the
same as the number of curves of the form Au+ wo+ vw+t=0,
which have the singularity in question. When the curves
are all conics, the discriminant, considered as a function of
A, M, v, of the discriminant of Aw+puv+vw, is AB’, where A
is the resultant of wu, v, w, and B=0 is the condition that
Au+pv+vw=0 may be capable of representing two coin-
cident right lines, but I am not in possession of the general
theory.

400. In connection with this subject it may be observed
that, the tact-invariant of a curve and its Hessian being of the
order 3(m—2)'5m—9) in the coefficients of the former, and
of the order m(7m—15) in the coefficients of the latter, is of
the order 6 (6s*—17m+9) in the coefficients of the original.
When m= 3, this tact-invariant is the sixth power of the dis-
criminant; and assuming, therefore, that the sixth power of the
discriminant is always a factor, there remains a factor of the
order 6 (m— 3) (3m — 2), whose vanishing expresses the condition
that the curve has a point of undulation.

Again, take the condition that the curve, its Hessian and
bitangential have a common point; this condition being of
the orders respectively 3 (m—2)* (m?—9), m(m—2) (m*— 9),
3m (m—2) in the coefficients of these curves is of the order
3 (m — 2) (m —3) (8m + 8m — 6) in the coefficients of the original.
When m=4, this invariant seems only capable of being ac-
counted for as the twelfth power of the discriminant multiplied
by the square of the invariant last considered. And assuming
that the same factors are to be found in general, there remains
an invariant of the order 3 (m— 4) (8m* + 5m’ — 32m + 18),
which will vanish whenever the curve has an inflexional tangent
which elsewhere touches the curve.

ee ae vs

POLES AND POLARS. 363

401. As the Jacobian is the locus of points whose polar lines
with respect to three curves meet in a point, so we might
consider the locus of the points in which these polar lines
meet; or, what is the same thing, the locus of points whose
first polars with respect to the three curves have a common
point. We shall confine ourselves to the consideration of the
case when the three curves are the three first polars of a
given curve, in which case the Jacobian is the Hessian of that
curve, and the other locus now mentioned is its Steinerian (see
Art. 70), the theory now to be explained being the generalization
of that given for the cubic* (Art. 175, &c.).

To any point P, then, on the Steinerian corresponds a point
@ on the Hessian; the first polar of P has @ for a double
point, and the polar conic of @ consists of two right lines
intersecting in P. Consider two consecutive points P, P’ on
the Steinerian; then, as in Art. 178, the intersection of their
first polars will be the point @ counted twice, together with
the points of contact of the first polar with its envelope. Thus,
then, the polar, with regard to the curve, of any point Q on
the Hessian, is the tangent to the Steinerian at the corre-
sponding point P. In particular, if Q is a point of inflexion
on the curve, its polar will be the tangent at that point; thus
we see that the Steinerian is touched by the 3m(m—2) sta-
tionary tangents of the curve.

402. We have seen, Art. 70, that the orders of the Hessian
and Steinerian respectively are 3(m—2) and 3(m-— 2)’; the
Hessian ordinarily has no double point, and therefore its
Pliickerian characteristics are

#=3(m—2), 8=0, «=0, v=3(m—2) (3m—7),
T= 24 (m—1)(m— 2) (m— 3) (3m—8), 1=9 (m-— 2) (3m—8).
Since there is a (1, 1) correspondence between the Hessian
and Steinerian, the deficiencies of the two curves will be the

* The principal theorems of this section were given by Steiner in a paper read
before the Berlin Academy, 1848, and afterwards reprinted in Crelle, 1854, vol. XLVII.
The theory, as regards the cubic, was given by me in the former edition of this
work (1852) in ignorance of what Steiner had done, with which I only became
acquainted through Credle.

364 - POLES AND POLARS.

same. We have also the class of the Steinerian; for any tan-
gent thereof which passes through a fixed point J/, must have
its pole lying on the first polar of M, and since it must also
lie on the Hessian, it must be one of the 3(m—1) (m—2)
intersections of the two curves. The characteristics, therefore,
of the Steinerian are

b=3(m—2)’, v=3(m—1) (m2),
5 = 3 (m— 2) (m —3) (3m?—9m—5), «=12(m—2) (m—83),
T= 3 (m—2) (m—3)(3m*—3m—8), t=3(m—2) (4m- 9).

A point is a double point or cusp on the Steinerian, if it is a
point whose first polar has two double points or a cusp. The
numbers therefore 5 and « just obtained are the number of
first polars of points of the given curve which have the singu-
larities in question (see Art. 399).

403. If the first polars of any two points A, B touch at
a point @, having QP for their tangent, then two of the poles
of the line AB coincide with Q; and the first polar of any
point on AB (other than the intersection of AB with PQ)
will also touch QP at Q. ‘The first polar of the excepted
point or intersection of AB with PQ, will have Q for a double
point; Q will be a point on the Hessian, and P the corre-
sponding point on the Steinerian. ‘Thus the Steinerian is the
envelope of lines, two of whose poles coincide; and the Hessian
is the locus of such coincident poles. Steiner has investigated
the envelope of the line PQ, which joins two corresponding
points P, Q, or which is the common tangent of two first polars
which touch each other. This curve we shall call, as in the
case of cubics (Art. 177), the Cayleyan.* It has evidently
a (1, 1) correspondence with the Hessian, and with the Steinerian,
and has therefore the same deficiency.

In order to determine its class we use the principle estab-
lished, Art. 372, and Conics, Appendix, that if two points on a
line (or two lines through a point) havea (m, m’) correspon-
dence, there will be m+ m’ cases of coincidence of these points.

* Professor Cayley himself calls it the Steiner-Hessian,

POLES AND POLARS. 365

Consider, then, the lines joining any assumed point M to
two corresponding points P, Y. Then, since the Steinerian is
a curve of the order 3(m—2)’, if the line MP be fixed there
will be 3 (m—2)* positions of P and as many positions of MQ.
In like manner, to any position of MQ correspond 3 (m— 2)
positions of P. There are, therefore, 3 (m—2)*+3 (m—2) or
3 (m— 1) (m—2)’lines which can be drawn through J contain-
ing two corresponding points P, Q, and this is therefore the
class of the Cayleyan. It obviously touches the inflexional
tangents of the given curve. It has no inflexions, and its
characteristics therefore are

jp =3 (m—2) (5m—11), v=3(m— 1) (m—2),
5 = 3 (m — 2) (5m — 18) (5m? - 19m +16), «=18 (m—2) (2m—5),

T=2(m—2)"(m?—2m—1), +=0.

404, The definitions already given may be further extended,
by considering the double points not only on first polars, but on
any of the system of polar curves. The locus of a point, such
that its @-polar has a double point, is a curve of the order
30 (m— 6 —1)’, which is the 6-Steinerian; and the locus of the
double point is then a curve of the order 36° (m-— @-1), which
is the 0-Hessian. We know that if the @-polar of a point P
passes through a point Q, then the (m— 6) polar of @ passes
through P; and it is easy to see also that if the @-polar of a
point P has a double point Q, then the (m-—@-1) polar of
Q has a double point P. Hence the @-Steinerian is the same
curve as the (m—@—1) Hessian, and the @-Hessian the same
as the (m—@-—1) Steinerian. In like manner we might con-
sider the @-Cayleyan or envelope of the line joining corre-
sponding points on the 6-Steinerian and @-Hessian, the three
curves having the same deficiency. Except in the case of
@=1 these curves have not been much studied.

405. We have studied (Art. 184) the envelope of the polar
lines, with regard to a cubic, of the points on a right line,
which we have called the polar of that right line. So, in
general, if a point P moves along any directing curve S of the
order s, the envelope of its 6-polar, with regard to a given

366 POLES AND POLARS.

curve U of the order m, will be a curve which may
be called the @-polar of S, with regard to U. We saw
(Art. 96) that the envelope of a curve, whose equation con-
tains as parameters the coordinates of a point which moves
along a curve S, may be found by considering the parameters
as coordinates, and then expressing the condition that the
moving curve should touch S. Hence, the 6@-polar of 9 is
also the locus of points whose (m—8@) polars touch S. Using
then the expression (Art. 97) for the order of a tact-invariant,
we see that the @-polar of S is a curve of the order
s (s +2@—3) (m— @), this number to be diminished by 2 (m-— @)
for every double point, and by 3(m-—@) for every cusp
on S; or, if the class of S be s’, then the @-polar will be
of the order ,
(m — @) {s’ + 2s (@—-1)}.
It will be of the order 0(2s+ 6-8) in the coefficients of S.
Thus, in particular, if @6=1, the envelope of the first polars
of the points of a curve S is the same as the locus of the poles
of the tangents of S, its order being s’(m—1). If in this
case s=1, this order reduces to 0, as it ought, since the
envelope then reduces to the (m—1)* poles of the line S.
In general, it is obvious that each double tangent of S will,
by its (m—1)* poles, give rise to (m—1)’ double points on
the envelope, and that each stationary tangent of S will give
rise to (m-—1)* cusps on the envelope. We have, therefore,
for the class of the envelope
(m —1)"s—(m—1)s’-—2(m--1)?’r—3(m—1)* 0;
or, since s” —s’ —27 — 3u=s, the class of the 1-polar is
(m—1)(m—2)s’+(m—1)’s

If 0=m-—1, the envelope of the polar lines of the points
of a curve S, or locus of points whose first polars touch S,
is of the order s(s+2m-—5) or s’+2s(m—2). And since
the number of these polar lines which pass through an
arbitrary point I is the same as the number of intersections
with § of the first polar of M, the class of the envelope is
(m—1)s.

In general the number of double points on the baile of
S is (m—6)* times the number of (m-—1) polars of a point

POLES AND POLARS. 367

which touch the curve twice, and the number of cusps is
(m— @)* times the number of such polars which osculate the
given curve.

406. If the @-polar of a curve 8 be a curve R, then the
(m—@) polar of & must include, as part of itself, the curve S.
Thus, for example, if 9=m-—1, 2 is the envelope of the polar
line of a point P which moves on S; but since the pole of
this polar line may not only be the point P, but (m—1)’-1
other points besides, it follows that if we seek the locus of
the poles of the tangents of & (or, what is the same thing,
the envelope of the first polars of the points of #), we shall
get the curve S, together with another curve, which is the locus
of points copolar with the points of S; that is to say, having
the same polar lines. In this case, where 0=m-—1, we have
seen that the class of # is s(m—1); therefore, Art. 405, the
envelope of the first polars of the points of & is of the order
s(m—1)’; or, in addition to the curve S, there will be a
companion curve of the order sm(m—2). We have seen that
every point on the Hessian is a point at which coincide two
poles of a tangent to the Steinerian; consequently, the points
in which S meets the Hessian will be points on this companion
curve, which will, besides, meet S in 4s (m—2)(m—3) pairs of
copolar points.

If @=1, & is the locus of the poles of the tangents of S,
and since a given point has one polar, if we seck the envelope
of the polar lines of the points of R, we must fall back on the
curve S, and it would appear that there can be no companion
curve. It is to be noted, however, that the common tangents
of S, and of the Steinerian, form part of the envelope. In fact,
we have seen that to each of these common tangents there
correspond two coincident points on &, and therefore when
we employ the converse process, to these two points answer
_ two coincident lines, every point on either of which has a
right to be counted in the envelope. Further, the curve S-
must be reckoned in that envelope (m-— 1)’ times, because to
every tangent of S there answer (m-— 1)’ poles lying on &, and,
therefore, when we take conversely the polars of the points of
R, each tangent of S is counted (m—1)* times. Now we have

368 OSCULATING CONIUS.

seen that if the order and class of & ber and 7’, the order of
its (m —1) polar is 7’ + 2 (m—2)r, but
r =(m—1)(m—2) 8 +(m—1)?s, r=s' (m- 1);

hence, the order of the polar is 3 (m—1)(m—2)s’+(m—1)*s,
which agrees with what we have established, since, as the
Steinerian is of the class 3(m—1)(m—2z), the number of its
common tangents with S is 3 (m—1)(m—2)s’. There must
be a like general theory of the reciprocity when FR is the
6-polar of S, and S the (am—@) polar of &, but this has not
yet been investigated.

OSCULATING CONICS.

407. The form of a curve in the neighbourhood of a point
f thereof is defined by the circle of curvature, but it admits
of a further definition. In fact, drawing parallel to the tangent
at P an infinitesimal chord QR, then if the normal at P meets
this at N, the arcs PQ, PR, and the lines NQ, NR, regarded as
quantities of the first order, are equal to each other, but they
differ by quantities of the second order; in particular, NQ, NR
differ by a quantity of the second order; or, what is the same
thing, if Z be the maddle point of QR, then the distance NZ is
of the second order. But observe that PN is also of the
second order; hence the angle LPN, =tan’ZLN+PN is in
general a finite angle; that is, joining P with the middle point
of the chord QF (parallel to the tangent at P), we have a
line PL inclined at a finite angle to the normal. In the case
of the circle, PZ coincides with the normal; hence the angle in
question is a measure of the deviation from the circular form,
or we may call it the “aberrancy,” and the line PZ the axis
of aberrancy.*

In the case of a conic, the axis of aberrancy is the diameter
through P, and the aberrancy is the inclination of this diameter
-to the normal. And for a given curve, drawing any conic
having therewith a 4-pointic intersection at P, the curve and

* See Transon, “ Recherches sur la courbure des lignes et des surfaces,” Liouwv.,
t. VI. (1841); his term ‘déviation’ is in the text replaced by the more specific one
“ aberrancy.”

OSCULATING CONICS. 369

conic have the same axis of aberrancy; that is, the centres
of all the conics of 4-pointic intersection with the curve at P
lie on the axis of aberrancy at this point. Whence also the
axis of aberrancy at P and the axis of aberrancy at the con-
secutive point of the curve, intersect in a point, say the “ centre
of aberrancy,” which is the centre of the conic of 5-pointic
intersection with’ the curve at P; this conic is completely de-
termined by the conditions that its centre is this point, that
it touches the curve at P, and that it has there a curvature
equal to that of the curve.

It is easy to show that the aberrancy at the point P is given
by the formula

where p, g, 7 are the first, second, and third differential coeffi-
cients of y in regard to a.

408. Observe that the axis of aberrancy is a line having
reference to the line infinity, but independent of the circular
points at infinity; viz. if instead of these we had any two
points J, J, then the line in question is constructed by means
of the line JJ without any use of the points J, J themselves;
the chord Qf is taken so as to pass through the intersection
O of the tangent at P with the line JJ, and we have then
£ the harmonic of O in regard to the points Q, R

The theorem that the centres of the conics of 4-pointic
intersection lie in a line may be presented in a more general
form; the conics have, of course, a 4-pointic intersection with
each thes or, what is the same thing, they are conics having
all of aes four common tangents (viz. the tangent at P
taken four times); the general theorem is, that for the
system of conics touching four given lines, the poles of any
line in regard to the several conics of the system lie in a line ;
a theorem which is better known under the reciprocal form,
that for the conics passing through four given points, the polars
of any point in regard to the several conics pass all through
one and the same point.

In the case where the circular points at infinity are replaced

by a conic, there is not any analogous theory of aberrancy.
BBB

370 OSCULATING CONICS.

409. The investigation, Art. 236, of the equation of the
conic of 5-pointic contact at any point on a cubic may be ex-
tended to curves of any degree. Let S represent the polar
conic and 7’ the tangent at the point, then the equation of
any conic touching at the same point will be S—P7'=0,
where P is lx+my+nz; 1, m, n being still undetermined.
Then the equation of the lines joining to the point 2’y’z’, the
intersections of the conic and the curve is obtained by sub-
stituting in the equation of each curve a’ +a for x, &e., and
eliminating % between the two equations. The result of the sub-
stitution in the first equation is Z7’'+ 4X84 4r7A* 4+ sr? A* + Ke. 5
and the result of the substitution in the equation of the conic
is 2(n—1)7—-P’'T+2(S—PT); and if this last be written
67'+2V, the result of eliminating ’ between the two equations
becomes divisible by 7} the quotient being

V"*—40V""*S+40V"° TA’ — &. =0,

which represents the 2(n—1) lines joining the point a'y’z' to
the 2(n—1) other points common to the conic and curve. In
order that the conic should have a 3-pointic contact with the
curve, one of these lines must coincide with 7; or the equation
just written must be divisible by 7; and since every term,
except the two first, is so divisible, this condition is plainly
equivalent to 9=2, which, since 6=2(n—1)—P’, implies
P'=2(n—2).* Introducing this value of @, and performing
the division by 7, the equation reduces to

— PV"? +% V"*A® —4V""* Ta* + &. =0,
which represents the 2n — 3 lines joining the point a'y’z' to the
other points of intersection of the curve and conic.

The contact will be 4-pointic if this equation be again
divisible by 7 or if $A°—PS be divisible by Z. The con-
dition that this shall be the case is found, as in Art. 382, by
substituting in this quantity the coordinates of an arbitrary point
on 7, viz. My- NB, Na-— Ly, L8—Ma when it ought iden-
tically to vanish, and in this way we find immediately that P

aH.’ ai:

2 dH
must be of the form wD + ( « a a +2 i) where pu

* The problem of finding the circle of curvature at any point on a curve is
evidently that of describing a 3-pointic conic passing through two fixed points,

‘ OSCULATING CONICS. 371

is still indeterminate. Thus the chord of intersection with the
polar conic of every 4-pointic conic meets the tangent in the
fixed point, noticed Art. 394, where the tangent meets both
the polar cubic, and also the polar line of z'y'z', with regard
to the Hessian either of the curve itself or of any of the
polar curves. |

A
4

Let us denote by [1 the line A (« GH «adit o),

da +4 dy + * Gd
and allowing that we have the identical equation A°—IS=J7,
then, introducing the value for P, $11+ 7, the equation be-
comes divisible by 7, and gives for the equation of the 2n—4
lines, joining to 2’y'z'’ the other intersections of the curve and
conic
(37+ P?—pS) V"*—4V"*A* + &e. =0.

The condition for 5-pointic contact is, that this equation should
be divisible by 7, and we determine the value of » correspond-
ing to such contact, by substituting in the terms above written
My—-NB8, Na-Ly, LR - Ma for x, y, 2. From the identical
equation of Art. 235, we can infer what J is, and I have
found that, by the substitution just mentioned, J becomes

—3(n—1)(n—2) 34 Ae ee) i Eh H), where 3, f, and yH
have the same meaning as in say 386. The results of substitution

in 8, P, and in A‘ are Q and Q, respectively. Using

2
» 3H Ye
then the values of Arts. 390, 391, we have

pL” = % {3 (n— 1) (n- 2) TH —2 (n—1) Ry (A)}
—§ {9 (n—2" H3- 6 (n— 2) Tiny (1) + 2h

-$ 4-6 (n—2) (v3) BH +4 (n -3) BY (WL) - pO},

1
9H*

whence reducing, (4@ — 3H), and the 5-pointic conic

is determined.

410. Prof. Cayley has pursued the enquiry so as to ascertain
what condition must be fulfilled by the coordinates 2'y’z’ in order
that the contact may be 6-pointic (see Phil, Trans., 1865, p. 545).

372 SYSTEMS OF CURVES.

The investigation is too long to give here; his result is that
a'y'z' must satisfy the equation

(m — 2) (12% —27) HJ (U, H, ®) - 3 (m—1) HJ'(U, H, ®)

+40 (m—2)’J(U, H, ©)=0
where by J(U, H, ®) is meant the Jacobian of these three
functions, and by J’ is meant that, in taking the Jacobian,
® is to be differentiated on the supposition that the second
differential coefficients of H, which enter into the expression
for ®, are coristant. The equation here written represents a

curve of the order 12m-— 27 whose intersection with U deter-
mines m (12m — 27) sextactic points.

SYSTEMS OF CURVES.

411. The problem to find how many conics can have a
6-pointic contact with a given curve belongs to the class of
questions on which some remarks were made, Conics, Ap-
pendix on systems of conics satisfying four conditions. We
shall here somewhat develope the theory there indicated.
De Jonquitres, Liouville, t. VI. (1861), considered the properties
of a series of curves of the m‘ order satisfying 42 (m+ 3) —
conditions, that is to say, one less than the number sufficient
to determine the curve, the series being characterized by its
index N, where N is the number of curves of the series which
can. pass through an arbitrary point. Thus, if the equation
of the curve algebraically contains a parameter, N will be
the degree in which that parameter enters.* Chasles, in papers
in the Comptes Rendus, 1864—1867, on the number of conics
which satisfy four conditions, used, instead of De Jonquiéres’
single index, two characteristics, viz. ~ the number of curves
of the series which pass through an arbitrary point, and v the
number of them which touch an arbitrary line. This method

* Prof. Cayley has remarked that it is not true conversely that the equation of
a curve belonging to a series whose index is N, can be always expressed in this
form. For instance, the index will be plainly WN if the equation contain linearly
the coordinates of a parametric point limited to move on a plane curve of the order
NN, and unless the curve be unicursal, the equation cannot, without elevation of
order, be made an algebraic function of a single parameter. Or, more generally, the

equation may contain linearly the coordinates of a point limited to move on a curve
in space of & dimensions,

SYSTEMS OF CURVES. 373

is especially convenient as giving symmetrical results in the
case of conics which are curves of the same order and class.
A sketch of this method is given in Conics, J. c., and we
shall here repeat a few of the theorems, stating them for a
series of curves of any order.

412. The loets of the poles of a given line, with respect
to curves of the series, is a curve of the degree v. For this
is obviously the number of points in which the line itself can
meet the locus. ‘The envelope of the polars of a given point,
with respect to curves of the system, is, in like manner, a
curve of the class yp.

The locus of a point whose polar, with regard to a fixed
curve (whose order and class are m’, n’), coincides with its polar,
with respect to some curve of the system, is a curve of the order
v+m(m'—1). For, in order to determine how many points of
the locus lie on a given line, consider two points A, A’ on that
line, such that the polar of A, with regard to the fixed curve,
coincides with the polar of A’ with regard to some curve of
the system, and the problem is to know in how many cases
A and 4A’ can coincide. Now, first, if A be fixed, its polar,
with respect to the given curve, is also fixed, and the locus
of poles of this last line, with respect to curves of the system
being by the first theorem of the order v, we see that to any
position of A answer v positions of A’. Secondly, let A’ be
fixed, and since its polars, with respect to curves of the system,
envelope a curve of the class mw, and since the polars, with
respect to the given curve of the points of the given line,
envelope a curve of the class m’—1, Art. 405, there are yw (m'—1)
common tangents to the two envelopes, and therefore as many
positions of A answering to A’. The number then of coin-
cidences of the points A and A’ is v+m(m'—1), or this is the
degree of the locus in question. It is obvious that»this locus
meets the fixed curve in the points where it is touched by curves
of the system, and therefore that the number of these curves,
which touch the fixed curve, is m’ {v+ pu (m'— 1)}, or is m'v+n'p,

413. In general, the number of curves of the system which
satisfy any other condition will be of the form ywa+vf, and

374 SYSTEMS OF CURVES.

the numbers a, 8 may be taken as the characteristics of this
condition. If a curve be determined by a sufficient number of
conditions of any kind, and if these characteristics be given for
each condition, we can determine the number of curves satisfying
the prescribed conditions. "We exemplify this in the case of
conics. ‘The number of conics determined by five points, by
four points and a tangent, by three points and two tangents,
<— 1, 2, 4, 4, 2, 1,
and, consequently, the characteristics of the systems determined
by four points, three points and a tangent, &c. are
(1, 2), (2, 4), (4, 4), (4, 2), (2, 1).

The number then of conics satisfying the condition whose
characteristics are a, 8, and also passing through four points,
or through three points and touching a line, &c. are

a+28, 2a+48, 4a+48, 4a+28, 2a+8.

"

If we call these numbers pp”, v'", p’’, ao", 7” respectively,
we see that they are not independent, but we have

mt mt

y" = 2p", a= 20", p"=3 (v4 0").

The characteristics of the systems formed with the condition
a, 8 together with three points, or together with two points
and a line, &c. are plainly

(Mm, Vv"), (v'", pods (p""; a"), (o", 7)

And therefore the number of conics of these systems respec-
tively which satisfy a new condition a’, #’ is p”a'+v''B,
va +p''B', &. Or, writing at full length, if we have two
conditions whose characteristics are (a, 8), (a, 8’), and if we
denote by p”, v", p’, o” the number of conics which satisfy
these two conditions, and also pass through three points, or
pass through two points and touch a line, &c. we have

w= aa! +2 (Ba' +a6')+488', v"=2aa' +4 (Ba! + a6") +488",
p” =4aa' +4 (Ba'+ a8’) + 288", o” =4aa'+ 2(Ba'+a6')+ BP,
and it is to be noted that these numbers are connected by
the identical relation ,

p"— fu" +p" 0" =0.
In like manner the characteristics of the system of conics
satisfying the two conditions (a, 8), (a, 6’), and also passing

SYSTEMS OF CURVES. 375

through two points, or through a point and touching a line,
or touching two lines, are (u”, v’), (v", p”), (p"”, a”), and there-
fore the number of such conics which satisfy a third condition
a’, B" are p"a+v"8", &. Or, writing at full length, if we
denote by yw’, v’, p’ the number of conics which satisfy three
conditions (a, 8), (a’, 8’), (a”, 8”), and also pass through two
points, or through a point arid touch a line, &c. we have

bw =aa'a"” +23a0'B" + 4>a8'8" + 486'B",
vy’ = 2aa’a" + 43008" + 43a8'B" + 2868",
p =4aa'a"” + 43aa'B" + 23a6'B" + BR'B".
It is evident that the characteristics of the system formed by

adding to these three conditions a fourth, a”, B’’, are p’a’”+ v'B'",
va" + p'B'", or, at full length,

be —_ ad ae!” 4 2 aa' a" mr ay 4 3a0'B" "+ 45a8'B" B'"+288'B" ie
v= 2 aa’ oo!” mi 4>aa'a” mt af 43aa'B"R'"4+ 2>a8'B"R''+ BR'B"B"”.
And so finally, if we add a fifth condition, the number of conics

nt

satisfying all five is wa” + v8", or
a ono” 4 230010"! B"" + 43a0'a” id oA 43aa’B"R'"" Br"
a 2ZaB'B"B"B + BB'B"B'"B"".
Thus this formula gives the number of conics which touch five
given curves, by writing for a, 8, &c. the class and order
of each curve. And in like manner we could find the number
of curves of any order determined by the condition of touching
given curves if we knew the number in each case where the

conditions were only those of passing through points or touch-
ing lines.

414. In the preceding article, the conditions we considered
were each independent of the others, but we may have a con-
dition equivalent to two or more conditions, as for example,
the condition that a conic shall touch a given curve twice
or oftener, the condition that a curve shall osculate a curve
or have with it contact of higher order. A condition equi-
valent to two may be called two inseparable conditions. It
is found that the formule obtained in the last article for in-
dependent conditions are applicable with the necessary modi-
fications to inseparable conditions. Thus, if we have two

376 SYSTEMS OF CURVES.

inseparable conditions, the characteristics w", v", p”, o”, are
the number of conics determined when we combine with the
given two-fold condition three points, or two points and a
line, &c., and these numbers will be always connected by the
relation pw” — 3v"+3p"—c"=0. We proceed precisely as in
the last article to find the number of conics determined, when
with the two-fold condition are combined any three others.
In this way we obtain the following formule. If m”, n”, 7”, s”
are the characteristics of a second two-fold condition, then
the characteristics of the system of conics determined by the
pair of two-fold conditions are

ge A Sao | Ae t | or 07 ” ” now

mp" — 3 (win" + mv") + (r"w" + p"m") + Fn'v" — 3 (r'v" + n'p'),
o's" ca 3 (o"'7" +. fp) ~f- (v"s” “4. n'a’) we Tp"r" —4 (p''n" +r").
And if yw’, v', p’ be the characteristics of a three-fold condition,

the number of conics determined by the two-fold and three-
fold condition is

dy’ (20" a p') ats tp’ (2m” Gea vy") at sv’ {5 (w" on p') ni oe 6 (we mt o’)t.

415. Returning to the two characteristics w, v of a series
of curves of the m* order, satisfying one condition less than
the number sufficient to determine each curve, we may in-
vestigate as follows the relation between these two charac-
teristics. Consider the points A, A’, &c., in which a curve
of the series meets a given line; then, since w curves of the
series pass through A, each meeting the line in m—1 other
points, it is evident that to each point A corresponds p (m—1)
points A’, and in like manner to each point 4’, « (m-—1) points
A, And the number of united points of the correspond-
ence is therefore 24(m—1). This number will be v if the
united points can only arise when a curve of the series touches
the line AA’, but it may happen that a curve of the series
will be a complex containing a portion which counts twice,
and such a curve would give rise to united points which must
be deducted from 24(m—1) in order to give v the number
of proper tangencies. ‘Thus, in the case of conics which we
shall specially consider, let % be the number of conics of the
series which reduce to two coincident right lines, and we
have v=2u—2X.

SYSTEMS OF CURVES. ry:

416. A conic considered as a curve of the second order
may degenerate into a pair of lines, or line-pair; in this case
the tangential equation found by the ordinary rule becomes
a perfect square; or, geometrically, every line through the
common point of the line-pair is to be considered as doubly
a tangent to the curve. Similarly, a conic considered as a
curve of the setond class may degenerate into a pair of points,
or point-pair; and every point of the common line of the
point-pair may be considered as in a sense doubly belonging
to the curve. In the latter case, the point-pair may be con-
sidered as the limit of a conic whose tranverse axis is fixed,
and which flattens by the gradual diminution of its conjugate
axis, so as to tend to a terminated right line, the tangents of
the conic becoming more nearly lines through two fixed points,
viz. the terminating points of the line.

Thus then, if X be the number of point-pairs in the system,
and @ the number of line-pairs, we have

BM=2v—a, v=Q—-A, 3U=2A4+G, 3V=2T +A.

In Zeuthen’s researches, concerning systems of conics, the
numbers A, @ are substituted for Chasles’ characteristics p, v,
it being in most cases easier to ascertain the number of conics
of a given system which reduce to line-pairs or point-pairs,
than the number which pass through an arbitrary point or
touch an arbitrary line.

A. special case presents itself when the two points of a point-
pair coincide, the line of the pair continuing to exist as a definite
line; or, the two lines of a line-pair may coincide without
their common point ceasing to exist as a definite point. This
may be called a line-pair-point.

417. In a system of conics satisfying four conditions of
contact, it is comparatively easy to see what are the point-
pairs and line-pairs of the system; but in order to find the
values of \ and a, each of these pairs has to be counted, not
once, but a proper number of times, and it is in the deter-
mination of these multiplicities that the difficulty of the problem
consists. For this purpose Zeuthen uses the following con-
siderations: Take the elementary system of a conic determined

CCC

378 SYSTEMS OF CURVES.

by four points, then evidently the number of line-pairs is
three, and of point-pairs is 0, but since »=1, v=2, we have
X%=0, w=33 whence it is inferred that a pair of lines
joining, two by two, four given points counts once among the
number of line-pairs. But take a system ‘of conics determined
by three points and a tangent, here we may have three line-
pairs, viz. the line joining any two of the points, and the
line joining to the third point the intersection of the fixed
tangent with the line joining the first two points. There
are in this case no point-pairs. We have also p=2, v=4,
hence X=0, a =6; and it is inferred that a line-pair counts
for two if it consists of the line joining two given points,
together with the line joining to a third given point the in-
tersection of the first line with a given line.

Lastly, take the system of conics determined by two points
and two tangents, and there can be but a single line-pair, viz.
the pair joining the two points to the intersection of the two
tangents; but since in this case w=4, v=4, X=a=—4, it is
inferred that a line-pair counts for four if it joins to two
given points the intersection of two given lines. It is needless
to dwell on the reciprocal singularities.

The movement of a conic which touches a given curve may
be considered either a rotation round the point of contact or a
slipping along the tangent at that point; and hence it is in-
ferred in the case of a conic determined by touching four
given curves, that we are to count among the line-pairs, once,
(A’) a pair consisting of two lines, each being a common
tangent to the curves; that we count twice, (5’) a pair con-
sisting of a common tangent to two curves, and a tangent
drawn to a third curve from a point where this common tangent
meets the fourth curve, and that we count four times, (C’) a
pair consisting of tangents drawn to two curves from the in-
tersection of othertwo. Reciprocally, we count among the point-
pairs once (A) a line each of whose determinations is the inter-
section of two curves, twice (2) a tangent to a curve terminated
by another curve, and by the intersection of two other curves;
and four times (C) a double tangent to two curves terminated
on two other curves. In these cases for the intersection of
two curves, may be substituted the intersection of a curve with

SYSTEMS OF CURVES. 379

itself or a node, and for a common tangent to two curves
may be substituted a double tangent to a single curve.

418. Thus, for example, to find the number of line-pairs in
the system of conics which touch four given curves. We have
nn'n'’n” line-pairs consisting of one of the nn’ common tangents
to the first two, combined with one of the nn” common
tangents to the other two; and, since we can in three ways
form two pairs out of the four curves, the number JA’ is 3nn’n’n’”,
Again, there are nn’n”m’” pairs consisting of a common tangent
to the first two curves, and a tangent to the third from one
of the points where it meets the fourth; and, since we get
the same number if we take a common tangent to the second
and third, or to the first and third, we have B’=32nn’n’m’”’,

be OP 4 OOLL fs

Lastly, there are plainly Snn’m’m” pairs of tangents of the
kind C’. We have therefore

a = 8nn'n’n’” + 6Snn'n'’m” + 43nn'm’m”,

and, in ike manner,

N=43Enn'm"’m” + 6Snm'm’ im” + 38mm’ mn”,

and from these numbers are deduced the same values for mw,
and vy, as we have found already. |

419. We proceed in the same way if the conditions of the
problem are, that the conic shall touch the same curve more
than once, or shall have with it contact of higher order. Prof.
Cayley uses the following convenient notation. Let (1) denote
single contact, (1, 1) single contact with the same curve in
two places, (2) contact of the second order or 3-point contact,
and so on. Thus the system we have considered of conics
having single contact with four curves is denoted by (1), (1),
(1), (1). Let us now consider the system (1, 1), (1), (1), that
is to say, when the conics have double contact with a single
curve and touch two others. Then it is seen, precisely as
before, that A’ =tn'n”+nn’.nn”. We have also

Bat (n’m” + nm’) + nn’ (m—2) n” + nn” (m — 2) nr

SOs

+ nn'm” (n—1) + nnn’ (n—1) + n’'n'm (n—2),

Feed og

O" = 8n'n"” + mm! (n — 2) n” + mm” (n — 2) n' + mm" gn (n — 1).

380 SYSTEMS OF CURVES.

Lastly, we must count separately (D’) the «n’n” line-pairs,
consisting of a pair of tangents drawn from a cusp of the
first curve to the other two. Zeuthen shews that these last
count each for three, by writing in the formule in the first
instance an unknown multiplier 2, and determining w by an
examination of the elementary cases where the second and third
curves, reduce to points or lines. Collecting then the numbers
A’+2B’ +40’, and reducing, we find
@a=nn" (n> + 6mn —8n—4m4+ 7446 + 38x)

+2 (m'n” + mn’) (n? + 2mn—n -— 4m +7) + 2m'm’n (n—1),
and there is a corresponding expression for >. From these
we find expressions for pu, v, viz.

p= pm’ mM + pw” (m'n!” + mn’) + p'n'n”,
yaw mm” + Vv" (m'n!” + mn’) + ¥'n'n”,
where a 2m (m+n—3) +7,
bw” =v =2m(m+2n—5) +27,
p=" =2n (2m+n—5) +26,
v” =2n(m+n—3)+ 6.
And these numbers denote the number of conics determined by
the conditions of touching one curve twice, together with three
points, two points and a tangent, a point and two tangents, and
three tangents, respectively.

It is unnecessary to consider separately the case (1, 1), (1, 1),
see Art. 413, and the same principles are applicable to the cases
(3) (1), (4).

Referring for further details to Zeuthen’s memoir, which
may be most conveniently consulted, Nowvelles Annales, 1866,
and to Prof. Cayley’s memoirs, Phil. Trans., 1867, we give
the following table, in which Prof. Cayley has summed up the
simpler results expressed in terms of m, n, and a (see Art. 83).
(1,1, 1) po =2m* + 2m?’n + mn’ + gn* — 2m? — 8mn — $n’

— 20n — 29n + a(—3m— 3n+ 13),
v= 1m? + 2m’n + 2mn? + An? — m?— Amn —n?
— 48m — 48n+a(—3m—3n +20),
p = 4m? + m’n + 2mn? + 2n* — 4m? — 3mn — 2n*
— 29m — 29n +a (~ 3m —3n + 13),

SYSTEMS OF CURVES. 381

4 y
(1,1, 1,1) w= pgm*+ Zein + mn® + dinn® + ppm!
— 4m* — 3m’n — 2mn? — 1n°
— Tt ant — 21mn — 32,9n* + 191m + 492n
ee: 3 ae es 2 9 2
+a (— §m*— 8mn — $n? +48m +55n — 257) + Ba,
v= tm + 4m'nt mn’ + 2mn* + jn*- Lm — 2m'n
oe. _ pig PO See ey 2
3mn — n° — *2-9m* —21mn — 1,81n? + 493m

+134n + a (— $m*— 3mn —3n"+ 55m+43n— 352

7 gay
(2) pe’ =a, Vv’ = 2a, p” =2a, o”’ =a;
(2, 1) we =12m+12n+a(2m+ n—14),

v’ = 24m + 24n + a (2m + 2n — 24),
p =12m+12n+a( m+2n—14),
(2, 1, 1) fo = 24m* + 36mn + 12n? — 168m — 168n
: + a (m* + 2mn + 4n?—- 25m — 29n + 188) — 3a’,
v =12m* + 36mn + 24n? — 168m — 1687
+a (4m? + 2mn +n? — 29m —25n + 188) — 3a’,

(3, 2) b= 27m + 24n — 20a + $0’,
v = 24m + 27n — 20a + 407,
(3) fo =—4m—3n+3a, vr =—8m—8n+ 6a,
p =— 3m—4n+ 3a.
(3, 1) fe = — 8m*— 12mn — 8n’+ 56m+53n + a(6m + 3n—39),
vy =— 3m*— 12mn — 8n'+ 538m+56n + a(3m + 6n—39).
(4) #=—10m— 8n+ 6a, v=— 8m —10n+4+ 6a.

420. It still remains to give formule for the number of
conics satisfying five inseparable conditions, as for example (5)
the number of conics having contact of the fifth order with a
given curve. These numbers are found from an examination
of the case where a curve touched by the conics is a complex
of two other curves.. Thus the conics having contact of the
fifth order with a complex of two curves, are made up of the
conics having like contact with the separate curves, and there-

382 SYSTEMS OF CURVES.

fore the expression for (5) must be such a function of m, n,-a,
that

d(m+m, n+n',at+a)=h(m, n, a) + h(m’, rn’, &),
whence (5) is plainly of the form am+n+ca. From sym-
metry we must have a=), and knowing the number of
sextactic conics when m=3, we determine a and c, and find
(5) =— 15m=— 15n + 9a.

So, in like manner, the conics (4, 1) are made up of the
conics having this contact with each of the separate curves,
and of the conics having the contact 4 with one curve and the
contact 1 with the other. The number of these last conics
is found by the formule of the last article, so that we have
db(m+m,nt+n, at+a)—d(m, n, a)—h(m’, n’, a’) a known
function of m,n, a. By Ae process here tndieted Prof. Cayley
establishes the SCE

(4, 1) = — 8m*— 20mn — 8n'+ 104 (m+n) + 6a (m+n —11),
(3, 2) = 120(m+n)+a(—4m—4n—78) 4+ 32’,
(3, 2, 1) =— 3m*— 10m’n — 10mn* — 3n° +192m"

+116mn +192n*— 434m — 434n

+ a(3m"+ 6mnn + 3n*— &9m — ®9n + 291) —
(2, 2, 1) = 24m* + 54mn + 24n* — 468 (m+ n)

+ a(— 8m —8n + 327) + a” (4m + dn — 12),

(2,1, 1,1) = 62m"+30m'n+30mn'+6n'—17n (m+n)*+ 1320 (m+n)

+a(dm'+ m'n + mn'+ An? 15 m?—26mn— 15n? »

+ 238m + 388n — 960) + a” (— 3m — 3n + 28),
(1, 1, 1, 1, 1) = gb5 (m+n) + pymn (m* + x’) + fmin’ (m+n)

— xy (m* + n*) — 8mn (m* +n”) — 2m*n?

— 449 (m+ 2) — 22Pmn (m+n) + 128 (m+ nt

+ 583mn — 2129 (m+n)+ a(—4m*—8m'’n—3mn?

— tn'+ 22 m+ 23mn+ %P n’—232m—222n +486)

+a’ {2 (m+n) — 15}.

2
ga,

Zeuthen and Cayley have also investigated formule for the
cases where the conditions include contact with a curve at a
given point; and Cayley’s memoir contains investigations of

NOTE BY PROFESSOR CAYLEY ON ART. 416. 383

a formula of De Jonquiéres, giving the number of curves of
the order r having with a given curve of the order m, ¢ con-
tacts of the order a, 0, c, &c., and besides passing through p
points on the curve. But the subject is too extensive to be
here further treated of.

~

NOTE ‘BY PROFESSOR CAYLEY ON ART. 416.

Some remarks may be added as to the analytical theory
of the degenerate forms of curves. As regards conics, a line-
pair can be represented in point-coordinates by an equation
of the form xy=0; and reciprocally a point-pair can be re-
presented in line-coordinates by an equation &7=0, but we
have to consider how the point-pair can be represented in
point-coordinates: an equation 2*=0 is no adequate repre-
sentation of the point-pair, but merely represents (as a two-
fold or twice repeated line) the line joining the two points
of the point-pair, all traces of the points themselves being
lost in this representation: and it is to be noticed, that the
conic, or two-fold line a’=0, or say (aw+ By+-yz)*=0 is a
conic which, analytically, and (in an improper sense) geome-
trically, satisfies the condition of touching any line whatever ;
whereas the only proper tangents of a point-pair are the lines
which pass through one or other of the two points of the
point-pair. ,

The solution arises out of the notion of a point-pair, con-
sidered as the limit of a conic, or say as an indefinitely flat
conic; we have to consider conics certain of the coefficients
whereof are infinitesimals, and which when the infinitesimal
coefficients actually vanish reduce themselves to two-fold lines;
and it is, moreover, necessary to consider the evanescent co-
efficients as infinitesimals of different orders. Thus consider
the conics which pass through two given points, and touch two
given lines (four conditions) ; take y=0, 2=0 for the given
lines, x =0 for the line joining the given points, and (~7=0,
y—az=0), (2=0, y— Bz =0) for the given points; the equation
of a conic satisfying the required conditions and containing one
arbitrary parameter @, is

a” + 20ay + 26 /(a8) xz + & (y -— az) (y— Bz) =9;

384 NOTE BY PROFESSOR CAYLEY ON ART. 416.

or, what is the same thing,
fx + By +0 (a8) 2}* - 6 (a+ A) ye=05
and this equation, considering therein @ as an infinitesimal, say
of the first order, represents the flat conic or point-pair composed
of the two given points. Comparing with the general equation
(a, 6, o,f, 9, Aka, y, 2)’ =0,

we have

a=1, b=, c=@aB, f=-40' (a+), g=O (a8), h=8,
viz. a being taken to be finite, we have g and h infinitesimals
of the first order; 0, c, f infinitesimals of the second order; and
the four ratios /(d) : /(c): V(f):g: are so determined as to
satisfy the prescribed conditions.

Observe that the flat conic, considered as a conic passing
through the two given points and touching the two given
lines, is represented by a determinate equation, viz. consider-
ing the condition imposed upon @ (@=infinitesimal) as a de-
termination of 6, the equation is a completely determinate
one; but considering the flat conic merely as a conic passing
through the two given points, the equation would contain
two arbitrary parameters, determinable if the flat conic was
subjected to the condition of touching two given lines, or to
any other two conditions.

Generally we may consider the equation of a curve of
the order nx; such equation containing certain infinitesimal
coefficients, and when these vanish, reducing itself to a composite
equation P*Q*...=0; the equation in its original form represents
a curve which may be called the penultimate curve. Consider
the tangents from an arbitrary point to the penultimate curve ;
when this breaks up, the system of tangents reduces itself to
(1) the tangents from the fixed point to the several component
curves P=0, Q=0, &c. respectively ; (2) the lines through
the singular points of these same curves respectively; (3) the
lines through the points of intersection P=0, Q=0, &c. of each
two of the component curves; these points, each reckoned a
proper number of times, are called “ fixed summits;” (4) the
lines from the fixed point to certain determinate points
called “free summits” on the several component curves P=0,
Q=0, &c. respectively. We have thus a degenerate form

NOTE BY PROFESSOR CAYLEY ON ART. 416. 385

of the n-thic curve, which may be regarded as consisting of
the component curves, each its proper number of times, and
of the foregoing points called summits, and is consequently
only inadequately represented by the ultimate equation
P*Q%...=0; the number and distribution of the summits
is not arbitrary, but is regulated by laws arising from the
consideration of the penultimate curve, and there are of
course for any given value of m various forms of degenerate
curve, according to the different ultimate forms P*Q?...=0,
and to the number and distribution of the summits on the
different component curves. ‘The case of a quartic curve
having the ultimate form 2’y?=0 has been considered by
Cayley, Comptes Rendus, t. LXXIV., p. 708 (March, 1872),
who states his conclusion as follows: “there exists a quartic
curve the penultimate of z*y’=0, with nine free summits, three
of them on one of the lines (say the line y=0), and which are
three of the intersections of the quartic by this line (the fourth
intersection being indefinitely near to the point z=0, y=0),
six situate at pleasure on the other line 2=0; and three fixed
summits at the intersection of the two lines.”. Other forms
have been considered by Dr. Zeuthen, Comptes Rendus, t. LXXV.
pp. 703 and 950 (September and October, 1872), and some
other forms by Zeuthen; the whole question of the degenerate
forms of curves is one well deserving further investigation.

The question of the number of cubic curves satisfying given
elementary conditions (depending as it does on the consideration
of the degenerate forms of these curves) has been solved by
Maillard and Zeuthen; that of the number of quartic curves
has been solved by Dr. Zeuthen.

DDD

int

ake K

a rs
ys “As
Seah

NOTES.

Art, 58, p. 48. On the equivalence of higher singularitier of curves to ordinary
singularities, see Professor H. J. 8S, Smith, “On the higher singularities of plane
curves, Proceedings London Math. Soc, V1. 153; Zeuthen, Math, Ann., x, 212,

Art, 151, p. 132. In connection with this theory see Cremona (Nouvelles Annales,
1864, p. 23); also Schroter “on a mode of generating cubics”; Math, Ann, v. 50,
Durége “on a cubic considered as the locus of the foci of a system of conics,”
Math, Ann, v. 83; and Clebsch “on two methods of generating cubics,” Math, Ann.
v. 422. Grassmann (Credle, LI1. 254) has generated a cubic as the locus of a point
such that the lines joining it to three fixed points meet three fixed lines in points
which lie on a right line.

Art. 161, p. 139, Investigations of a nature kindred to those of Sylvester on
residuation were made about the same time by Brill and Noether, G@ttinger
Nachr., 1873, p. 116, An abstract is given by Fiedler in the notes to his translation
of this work,

p. 185. Add to the note “See also a dissertation by Rosenow Breslau, 1873,”
Art. 220, p. 191, The form in which S is written by Aronhold is as follows :
— S= (b,c, — m*)? + (cya — a5”) (cq — 55”) + (ab, — a7) (b3¢ — ¢,”)
+ (ga, — ma) (be — b,cz) + (agm — ab,) (b,c, + cb, — 2c,m)
+ (maz — G,C,) (b,c + cb — 2mbs),

p. 212. Add to the note, “In the paper last mentioned Gundelfinger writes down
the 34 forms which constitute the system of concomitants to a ternary cubic, in
conformity with Gordan’s theory, Math. Ann.1. 90. See also Gundelfinger’s paper
Math. Ann., Vit. 186, On the subject of cubic curves Clebsch ought also to be
consulted, Vorlesungen iiber Geometrie, p, 497.”

ON THE BITANGENTS OF A QUARTIO, BY PROFESSOR CAYLEY,

THE equations of the 28 bitangents of a quartic curve were obtained in a very
elegant form by Riemann in the paper “ Zur Theorie der Abelschen Functionen fiir
den Fall p = 3,” Werke, Leipzig, 1876, pp. 456—472; and see also Weber’s “Theorie
der Abelschen Functionen vom Geschlecht 3,” Berlin, 1876. Riemann connects the
several bitangents with the characteristics of the 28 odd functions, thus obtaining for
them an algorithm which it is worth while to explain, but they will be given also
with the algorithm employed p, 231 e¢ seg. of the present work, which is in fact the
more simple one, The characteristic of a triple @-function is a symbol of the form

apy,

a’ B’y’;
where each of the letters is = 0 or 1; there are thus in all 64 such symbols, but they
are considered as odd or even according as the sum aa’ + 6’ + yy’ is odd or even;

388

NOTES.

and the numbers of the odd and even characteristics are 28 and 36 respectively ; and,
as already mentioned, the 28 odd characteristics correspond to the 28 bitangents +

respectively.

We have «x, y, 2 trilinear coordinates, a, B, y, a’, 6’, y’ constants chosen at
pleasure, and then a’, 8”, y” determinate constants, such that the equations

Zt yt 2+ FE+ n+ 9 =9,

By #8.

SOP Rees gg
a'ntBytye+ 5444S <0
Bry 3 a B Y 9

” ? ” = n Pe
PERT ET DS Oa at inti

are equivalent to three independent equations; this being so, they determine &, n, ¢
each of them as a linear function of (2, y, z); and the equations of the bitangentg
of the curve J(x€) + J(yn) + J(2%) = 0 (see Weber, p. 100) are

18
28
38
23
13
12
48
14
58
15
68
16
78
17
24

34

111
lll

001
Oll

011
001

010
010

100
110

110
100

101
100

010
O11

100
101

011
010

110
010

001
101

010
110

101
001

100
111

110
101

#0,

cty+tz=0,
E+y+2=0,
ax + By + yz=0,
E

= + fy+yz=0,

a

a'x + ply + y'2=0,
e ’ yee
a + py t+ y’z=9,
ae + By + yz = 0,
i 3 ” $s ci,
a” + B’y + yz = 90,

ct+tnt+z=9,

ery+ = 0,

NOTES. 389

25 + ax + 3 +yz=0,

35 06 ax+ py + & =0,

vai 001 a’a + z + y'2=0,

- ou ae t py + = = 0,

27 | ig | a’e+ By + "2 =0,
ee ss

Mo a) Gee tic ye titers”

- ae

45 | OL co ity. sarees Ser =0,
110 a’ (1 em ‘) B’ (1 ry'a’) = y (l § ap’) =0,

The whole number of ways in which the equation of the curve can be expressed
in a form such as J(x£&) + J(yn) + (zg) = 0 is 1260; viz. the three pairs of bitangents
entering into the equation of the curve are of one of the types

12.34, 13.24, 14.23 By Nois 70
12.34, 13.24, 56.78 (1 I] » 680
13,23, 14.24, 15.25 & 9 | 560

1260

and it may be remarked that selecting at pleasure any two pairs out of a system
of three pairs the type is always O or |||, viz. (see p. 233) the four vena are
such that their points of contact are situate on a conic.

Art, 269, p. 241. In saying that the case of quartics with a single node had
received no attention I overlooked Brioschi’s paper, Math, Ann, Iv. 95, followed by
Cremona, p. 99, and Brill, Math. Ann. vi. 66 and Crelle, vol. 65.

Art. 276, p. 246. The method here employed had been indicated by Burnside,
Educational Times reprint Vit. 70.

Art, 287, p. 257. On this subject see a paper by Mr, Malet, Trans. Royal Irish
Academy, XXV1, 431 (1878).

INDEX.

Aberrancy of curvature, 97, 368,

Absolute invariant of a cubic, 144, 165,

Acnode, 25, 129.
of cubic constructed when stationary

tangents are given, 184.

Angle made by tangents with axis, 36.
with radius vector, 80.
sum of, given which tangents from a

point make with fixed line, 123.
between focal radii and tangent, 125.
Angle at which curves cut, unaltered by
certain transformations, 314.
Anharmonic, theorems of conics, their
analogues in cubics, 140.
ratio constant of pencil of tangents
from point on cubic, 144.
this ratio expressed in terms of fun-
damental invariants, 199.
ratio unaltered by linear transforma-
tion, 296.
ratios equal of tangents from two nodes
of quartic, 241.

Antipoints, 122,

Arc of evolute, length of, 88.

Archimedes, spiral of, 291.

Aronhold’s invariants of cubics, 191.
<r of bitangents of quartics,

Asymptotes, their equation how found, 40,
how cut by any transversal, 113.
of cubic, 170.

Atkins on caustics, 101.

Bernoulli, on lemniscate, catenary and
logarithmic spiral, 44, 289, 293.
Bertini on rational transformation, 326.
Bicircular quartics, 126, 142, 241,
Bifid substitution, 232,
Biflecnodes, 217.
Bipartite cubics, 168,
Bitangents, general theory of, 342, &c.
of quartics, 111, 220, 223.
eae Ts curve, of quartic, 223, 349,
Brill, on transformation of curves, 329.
on residuation, 387, 389.
Brioschi, on nodal quartics, 389.
Canonical form, of equation of cubic,
188, 196.
general equation of cubic how reduced
to, 198,

Cardioide, 44, 252, 282.
Carnot, theorem of transversals, 109.
Cartesians, 101, 104, 126, 241, 244, 250,
Cartesian coordinates, how related to
trilinear, 6.

Casey, on bicircular quartics, 241.
Cassini’s ovals, 44, 126,
Catenary, 287.
Caustics, 98, &c.

of parabola, 107.
Cayley on intersections of two curves,

on equivalence of higher singularities
to a union of simpler, 48.

modification of Pliicker’s equations,
66.

on envelope of equation containing
independent parameters, 74.

on quasi-evolutes, 92.

on acne of parallel curves,
102.

on problem of negative pedals, 107.

on foci, 120.

on involution, and classification of
cubics, 162, 179.

his notation for equation of cubic,
189

algorithm for bitangents of quartics,
230, 232.

on tangents from nodes of binodal
quartic, 241.

on cartesians, 251,

on logarithmic curve, 287,

on skew reciprocals, 304.

on transformation of curves, 316,

solution of problem of bitangents, ©

341, 351, 355.
on sextactie points, 371.
on systems of curves, 372, 379.
on degenerate forms of curves, 383.
note on bitangents of quartic, 387.
Cayleyan of cubic, different definitions of,
151

its equation, 190.
in point coordinates, 203,
of a system of conics, 225,
of a curve in general, 364,
Centres, 115.
Central cubics, 164.

Centre of mean distances, 112.
of contacts of parallel tangents, 119.

»

392

Characteristics of reciprocal, 65.
of evolute, 94.
of parallel, 102.
of inverse curve and pedals, 106.
of system of conics, 372.
Chasles on contact of parallel tangents, 119.
on projection of cubics into cen
cubics, 164,
on Cartesians, 241, 250.
on systems of curves, 372.
Circular points at infinity, 1, 83, 90, 119,124,
their coordinates, 7.
normal at, 94.
circular cubic, 126, 142, 248.
Circular coordinates, 7.
Cissoid, 84, 182.
Class of a curve how connected with its
order, 54.
Clebsch, on unicursal cubics, 188.
on canonical form of a quartic, 265.
on Jacobians, 359.
on generation of cubics, 387,
on symbolical notation, 343.
Clifford, on Miquel’s theorem, 128,
Conchoid of Nicomedes, 44.
Condition that curve should have a double
point, 55.
a cusp, 58.
a point of undulation, 362.
that two curves should touch, 80.
that four consecutive ‘points on curve
should lie in a circle, 97.
that cubic should be sum of three
cubes, 197.
should represent three lines, 197,
a conic and a line, 210.
that quartic should be sum of five
fourth powers, 265.
Contact of conics with cubics, 135, 207.
with curves in general, 368.
Contravariants of cubic, 190, 204,
of quartic, 264, 271, 273,
Coresiduals, 134.
Oe of two points on a cubic,
132.

on Hessian, 149.
general theory of, 255, 324, 331.
Cotes, theorem of harmonic means, 115.
Covariants of cubics, 189, 200.
of quartics, 264, 269, 273.
Cramer on intersections of two curves, 22.
_on points of visible inflexion, 37,
on tracing of curves, 43.
Cremona, on Cayleyans, 151.
on transformation of curves, 316,
on nodal quartics, 3.
Critic centres of system of cubics, 160,
’
of cubic and Hessian, 200.
Crunodes, 24, 129.
Curvature, centre and radius of, 84, 86.
of roulettes, 284
aberrancy of, 368.
Cusps, 25, 48, 58.
curvature at, 87.
Cuspidal cubics, 180,
Cycioid, 275...
Dandelin on caustics, 99.

INDEX.

Deficiency of a curve defined, 30.
same for curve and its reciprocal, 66.
or for any curve connected with it by
linear correspondence, 97.
unaltered by Cremona transforma-
tion, 321.
or any rational transformation,326,331.
Degenerate forms of curves, 377, 383.
De Jonquiéres on systems of curves,372,383.
De Morgan on Newton’s process for finding
figure of curve at multiple point, 46.
Des aero (see Cartesians), on the cycloid,
278

on the logarithmic spiral, 293,
Descriptive properties, 1, 82.
Diameters, 112.
Diocles, the cissoid, 182.
Discriminant of a curve defined, 55.
of a cubic expressed in terms of
fundamental invariants, 159, 196,
199, 210.
expressed as a determinant, 211.
of discriminant, 360.
Divergent parabolas, 164, 166, 173, 176.
Double points, their species, 24.
equivalent to how many conditions, 28,
limit to their number, 28.
Duality, geometrical, 12.
Durege, on cubic considered as locus of
foci, 387.

Envelopes, general theory of, 67.
of line whose equation is algebraic
function of parameter, 70.
of line whose intercept between two
lines is constant, 102, (see also 69,
84), 283.
of line joining feet of perpendiculars
from point on circle on sides of
inscribed triangle, 283,
of line joining corresponding points
on cubic, 133,
Equitangential curve, 290.
Epicycloids, 278.
Euler, on intersections of two curves, 22.
on epicycloids, 279.
on logarithmic curve, 286.
Evectants of invariants S and 7,191, 194.
Evolutes of conics, 41, 83.
of curves generally, 82.
tangential equation of, 89.
characteristics of, 94.
confocal with curve, 124,

Flecnodes, 217.
Foci, general theory of, 119.
locus of foci under certain couditions,
127.
of circular cubic lie on circles, 248.
of bicircular quartic, 242.

Galileo, on the cycloid, 277.
on the catenary, 289.
Geiser, on bitangents of quartics, 231.
Gergonne, on intersections of two curves,
22

Gordan, on number of concomitants to a
cubic, 387,

INDEX.

Grassman, on generation of cubics, 387,
Gregory, on tracing of curves, 43,

on logarithmic curve, 287.
Groups, of cubics, Pliicker’s, 178.
Guldenfinger, on concomitants of cubics,

Haase, on unicursal cubics, 185.
Harmonic mean of radii, 115.
pencil by chords of cubic, 133.
polar of point of inflexion of cubic,
146,203.
Hart, construction for ninth point common
to all cubics passing through eight,

140.

theorem that foci of a circular cubic
lie on circles, 145.

proof of Hesse’s theorem on inflexions
of cubics, 148.

on foci of bicircular quartic, 242.

theorem that confocals cut at right
angles, 248.

on logarithmic curve, 287.

Hesse, his theorem that inflexions of cubic
are also inflexions of Hessian, 148.
algorithm for bitangents of a quartic,

230, 234. ;
reduction of bitangential of quartic,
344

Hessian, defined, 57.
passes through points of inflexion,
5] e
of cubic, its equation, 190.
of quartic, 223,
of Hessian of cubic, 196.
of UV, 212.
Homographic, tangents from nodes of a
binodal quartic are, 241,
transformation, 295.
Huyghens, on evolutes, 88,
on the cycloid, 278.
Hyperbolas, cubical, 170, &c.
Hyperbolism of any curve, 178,
Hyperelliptic integrals, 330.

Identical equation for cubic, 205.
Tgel, on unicursal cubics, 185.
Independent parameters, envelope with, 74,
Infinity, pole of, 117,
normal at, 94.
satellite of, 151.
polar conic of, with respect to cubic,
158.
Inflexion, points of, 33.
tangent at it double, 34.
curve there crosses tangent, 35.
number of, 59,
three inflexions of cubics lie on a
right line, 110, 131.
inverse of this theorem, 312.
real for acnodal cubics, imaginary
for crunodal, 184.
of quartics, how many real, 221.
Inflexional tangents of cubic touch Hes-
sian, 152.
equation of system of, 203.
Ingram, on inversion, 312.
Interscendental curves, 275.

393
Intersections of curves, 16,

Inversion, 106.

characteristics of inverse curves, 106,
of parabola, 183.

my! to obtain focal properties,
in wider sense of word, 254.

a case of quadric transformation, 310.

applications of the method, 311,
Involute of circle, 290.

Jacobi, on intersection of two curves, 22,
Jacobian of three curves, 150.
of a system of conics, 225.
common point of three curves of same
degree is double point on, 160, 358.
properties of, 357.
Joachimsthal, his method of determining
point where line meets curve, 49.
Jungius, on catenary, 289.

Keratoid cusps, 48.
Kirkman, on Pascal’s hexagon, 19,

Leibnitz, on interscendental curves, 275.

Lemniscate, 44.

Limagon, 44, 99, 252, 282,

Line coordinates, 9

Linear transformation, 295.

Lituus, 292.

Locus, of common vertex of two triangles,
whose bases are given, and vertical
angles have given difference, 142.

of point whence tangents to a curve
have given invariant relation, 79.
whence tangents make with fixed
line angles whose sum is given,
123.
of nodes of all nodal cubics through
seven fixed points, 160.
Logarithmic curve, 286,
spiral, 292.
Liiroth, on special class of quartics, 265.

Mac Laurin’s, general theorem on curves,
117

theory of correspondence of points on
a cubic, 133,
on harmonic polars of inflexions of
cubic, 146.
Magnus, on reduction of homographic
transformation to projection, 299.
Maillard, on number of cubics satisfying
elementary conditions, 385.
Mersenne, on cycloid, 277.
Metrical theorems defined, 1, 108,
Miquel’s theorem, 128.
Multiple points, equivalent to how many
nodes, 28.
how related to polar curves, 52.
how affect points of inflexion, 60.
number of tangents from, 63.
Multiple tangents, 32, 52.

Newton’s process for finding figure of
curve at multiple point, 46.
theorem of ratio of rectangles, 108.
on diameters, 112, *

EEE

394

Newton, on intercept between curve and
asymptotes, 113.
theorem that a cubic may be projected
into one of the five parabolas, 164.
classification of cubics, 176.
description of cissoid by continuous
motion, 183,
Newton’s rectification of epicycloids, 284.
_ Nicomedes, conchoid of, 44
Node cusps, 214.
Normal, 89. .
of point at infinity, 94.
rae of terms in general equation,
1
of conditions which determine a
curve, 15.
of tangents to a curve from a given
point, 54,
of conics which touch five given
curves, 375.
satisfying any five conditions of con-
tact, 382,

Oscnodes, 216.
Osculating conics, 368, &c,
Oval, no real tangents can be drawn to
cubic from, 167.
@ quartic may have four, 219.

Parabola, cubical and semicubical, 83, 176.
divergent of the third degree, 164.
Parallel curve to a conic, equation of, 70.
tangential equation of, 103.
characteristics in general, 102.
Parallel tangents, have fixed point as
centre of mean distance of their
contacts, 119.
Parametric expression of point on unicursal
cubic, 185.
on cubic in general, 329, 338.
on unicursal quartic, 260,
on nodal quartic, 330.
Partitivity of cubics, 168,
of quartics, 219.
limit in general, 220.
Pascal, theorem of one derived from
‘theory of cubics, 1
limagon, 44, 99.
on cycloid, 278,
Pedal, of a curve, 99, a
negative, 105, 106
Perpendicularity, extension of relation,

Pippian of cubic, 151.
Pliicker, on intersection of curves, 22.
on degree of reciprocal, 54.
his equations connecting reciprocal
singularities, 65.
on theorem of transversals, 110,
on foci, 119.
classification of cubics, 161, 178.
on forms of quartics, 219.
on bitangents of quartics, 227.
Poles and polars,
general theory of, 49, 115, 357, &c.
in case of cubics, 142.
polar of point with regard to triangle,
“oe

INDEX.

Poles and polars,
of infinity with regard to a curve of
the n™ class, 119.
first polar contains points of conn
of tangents, 53.
polar conic of line with regard to
cubic, 156.
Polar coordinates, problems discussed in,
23, 79, 88, 108, 112, 116.
Polygons, problem of inscription of, in
conics, 253, 337
in cubics, 181, 338,
in quartics, 253.
Poncelet, on number of tangents to a
curve from any point, 54.
on vale Aad of polygons in curves,
253, 339
Projection, of cubics, 164, 169.
a homographic transformation, 298,
Pursuit, curves of, 290.

Quadrangle formed by contacts of tangents

from point on cubic, 132, 206.
Quasi evolutes and quasi normals, 90, 182.
Quetelet, on caustics, 99.

Ramphoid cusps, 48, 214.
Rational expression for coordinates of
point on unicursal curve, 30, 185, 260.
transformation, 308.
Reciprocal of a curve, its degree, 54,
characteristics of, 65.
method of finding equation of, 67, 76.
of a cubic, 76, 158, 193.
of a quartic, 78, 223.
in polar coordinates, 79.
skew reciprocals, 506.
Residuation, Sylvester’s theory of, (134,
for cuspidal cubics, 180.
Riemann, on constancy of ietrlenee, 326,
on bitangents to a quartic, 387.
Roberts, on problem of parallels and
negative pedals, 105,
on transformation of curves, 313.
Roberval, on the cycloid, 278,
Roemer, on epicycloids, 284.
Roulettes, 284,

Satellite of a line with respect to a cubic,

>
of line infinity, 131,
envelope of, 162,
used in classification, 161, 178,
Schroter, on generation of cubics, 387.
Sextactic points on cubics, 135,
on curves in general, 371.
Signs of coordinates, how determined, 3.
Singularities, higher equivalent to a union
of simpler, 49.
which to be counted ordinar, y, 64,
Sinusoid, 285.
Skew reciprocals, 306.
Smith, on singularities of Curves, 587.
Spinodesg, 25.
Spirals, 291.
Stationary points, 25,
tangents, 33.
of cubic touch Hessian, 153,

INDEX.

Stationary tengeneu ofsystem,203,
Steiner, on hexagon, 19
on inscription of polygons in quartics,
253.

on bitangents of quartics, 234.
on curve enveloping line joining feet
of three perpendiculars, 283.
on circles osculating conic and passing
through given point, 312,
on systems of curves, 360,
Steinerian defined, 57.
identical with Hessian in case of
cubic, 150.
its properties, 363,
Steiner-Hessian, 364,
Stubbs, on inversion, 312.
Sylvester’s theory of residuation, 134, —
se pe i, form of equation of reciprocal,

of locus of points, whence tangents
satisfy invariant relation, 79,
‘Systems of curves, 372,
Syntractrix, 289,

Tacnode, 214,
cusp, 214,
Tact-invariant of two curves, 80, 360,
Tangent, at origin, equation of, 23.
from any point, points of contact,
how determined, 53.
how specially related in case of cubic,
i. 182,

equation of system, 61, 78.

from a multiple point, 63.

locus of point if sum of angles made
with by a fixed line be constant, 123,

if tangents fulfil invariant relation, 79.

Tangential coordinates, 9.

particular cases of, 10.

equation of ev olutes, 89.

of a point with respect to a cubic,
130, 180, 206.

395

Tangential,
its coordinates, how found, 156,
points of a curve, how related, 38,
curve, mode of finding its equation,
352.

Tracing of curves, 40.
Tractrix, 289.
Transformation of curves, 294,
Transon, aberrancy of curvature, 368,
Tricuspidal quartics, 258,
Trident, 176.
Trinodal quartic, properties of, 254.

' tangents at or from nodes touch conic,

256.

Triple points, their species, 27.
Tschirnhausen on caustics, 98,
Twinpair sheet of cones, 165,

Undulation, point of, 37.
in case of quartics, 218.
general condition for, 362.
Unicursal curve, defined, 31, 69, 107.
cubics, 168, 179,
quartics, 254,
correspondence of points on, 332.
Unipartite cubics, 168,
United points of correspondence, 332.

Vincent, on logarithmic curve, 286.

Walker on invariants of quartics, 274.

Waring on number of tangents to a curve
from any point, 54,

Wallace on catenary, 288,

Weber, on Abelian functions, 387,

Wren on cycloid, 278.

Zeuthen, proof that deficiency is unaltered
by rational transformation, 326,
on bitangents to a quartic, 220,
on systems of curves, 377, 385.
on singularities of curves, 387.

THE END.

W. METCALFE AND S0N, PRINTERS, TRINITY STREET, CAMBRIDGE,

BY THE SAME AUTHOR.

A TREATISE ON CONIC SECTIONS.
Sixth Lédition.

London: LONGMANS, GREEN, AND Co.

A TREATISE ON THE GEOMETRY OF
THREE DIMENSIONS.

Third Edition.

Dublin: Hopces, Foster, AND FIGGIs.

LESSONS INTRODUCTORY TO THE
MODERN HIGHER ALGEBRA.

Third Edition.

Dublin: HonceEs, FosTrer, AND Fiaais.

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