Constructive theory of the unicursal plane quartic by synthetic methods

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IN

MATHEMATICS
Vol. 1, No. 2, pp. 27-54, 31 text-figures September 24, 1912

CONSTRUCTIVE THEORY OF THE UNICURSAL
PLANE QUARTIC BY SYNTHETIC METHODS

BY
ANNIE DALE BIDDLE

INTRODUCTION

In the following discussion the unicursal quartic is regarded from two points
of view. Chapter I treats of the curve in its correspondence to a conic section
through a quadratic reciprocal transformation. This leads to an interesting
classification of unicursal quartics and affords a convenient and ready method
for determining the form of the curve. Incidentally, it brings to light a geomet-
rical application of the well known ‘‘Group of Four.’’ In Chapter II the curve
is defined as the locus of intersection of corresponding rays of two projective
pencils of the second order. This develops properties of the curve not readily
obtained in the other treatment. The discussion shows that the two definitions
are not independent, but that each is supplementary to the other.

CHAPTER I
THE UNICURSAL QUARTIC IN CORRESPONDENCE TO A CONIC
SECTION

1. A Quadratic Reciprocal Transformation—The following well known geo-
metrical construction of the general quadratic reciprocal transformation is of
fundamental importance for the study of the unicursal quartic. Consider in
the plane any two conic sections a and a’ and a point P. The polars p and p’
of P with respect to a and a’, respectively, will intersect in a point P’. The
polars of P’ must likewise intersect in P. To P corresponds P’, and vice versa.
We say that P and P’ are conjugate points of the transformation. By corre-
lating all such pairs of conjugate points an involutory transformation is
established between the points of the plane.

2. Theorem.—As a point P describes a straight line, its conjugate point P’
generates a conc section as the locus of the intersection of corresponding rays of
two projective pencils of the first order.

As P describes a straight line 1, the polars p and p’ of P with respect to a and
a’ describe projective pencils of rays Z and L’ of the first order.

3. Theorem.—As a point P describes a conic its conjugate point P’ generates
a unicursal quartic, as the locus of the intersection of corresponding rays of two
projective pencils of the second order.

28 Unwersity of California Publications in Mathematics | Vou. 1

As P describes a conic y, its polars p and p’ with respect to a and a’, respec-
tively, describe projective pencils of rays « and x’ of the second order. To de-
termine the degree of the locus of P’, the conjugate point of P, that is, the locus
of the point of intersection of corresponding rays of the two projective pencils
« and x’ of the second order, cut it with a straight line l. To l corresponds a
conic A. The points in which / intersects the locus of P’ correspond to the points
in which d intersects y, the locus of P. But A and y can intersect in at most four
points. The locus of P’ is, therefore, a quartic. In establishing a one-to-one
correspondence between the points of the quartic and a conic section, we have
shown that the quartic is unicursal.

4. Theorem.—Any curve of degree n is transformed into a curve of degree 2n
as the locus of the intersection of correspondings rays of two projective pencils
each of order n.

As a point P describes a curve of degree n, its polars p and p’ with respect to
a and a’, respectively, generate projective pencils of rays each of order n. To
determine the degree of the locus of the point of intersection of corresponding
rays, that is, of the point P’, conjugate to P, we proceed as before in § 3, cutting
the locus of P’ with a straight line /. To l corresponds a conic A. The inter-
sections of / and the locus of P’ correspond to the intersections of A and the locus
of P. But of these there can be at most 2n.

5. Theorem.—Any point on a side of the self-polar triangle of a and a’ corre-
sponds to the opposite vertex and vice versa. «

For if P lies on a side s, of the triangle S, S, S;, self-polar with respect to a
and a’, its polars p and p’ must meet in S,, the vertex opposite to s,.

6. Theorem.—A curve of degree n corresponds by the above transformation to
a curve of degree 2n having the vertices of the self-polar triangle as n- fold points.

For from §5 it follows that to each intersection of the curve of degree n
described by P with a side of the self-polar triangle corresponds the opposite
vertex as a point of the curve described by P’.

In special cases the curve of P may pass through a vertex of the self-polar
triangle. The curve of P’ then degenerates, the opposite side of the triangle
appearing as a part of the curve. In general, a curve of degree n which passes
in all k& times through the vertices of the self-polar triangle goes into a degen-
erate curve of degree 2n, which contains the sides of the triangle counted / times
and a curve of degree 2n —k. In particular, a line / passing through a vertex of
the triangle corresponds to a line l’ passing through the same vertex.

7. The Quartics Qa and Qa’.—To the conic a (or a’) itself corresponds a
quartic Qa (or Qa’) which passes through the four intersections of a and a’, and
also through the four points of contact on a’ (or a) of the tangents common to
aanda’. These curves can be traced at once for any given conics a and a’. They
are, therefore, of great assistance in the construction of any curve C’, for to the
intersections of C (the corresponding curve) and Qa (or Qa’) correspond the
intersections of O’ and a (or a’). These points of C’ on a (or a’) are the points
of contact on a (or a’) of one of the tangents drawn to a (or a’) from the inter-

sections of C and Qa (or Qa’).

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 29

8. Theorem.—Conjugate points P and P’ of the transformation project to any
vertex of the self-polar triangle in an involution of rays.

This follows from § 6, where it was shown that to a line J passing through a
vertex of the triangle corresponds a line l’ passing through the same vertex, and
vice versa. From this we see that it is possible to regard P’ as the point deter-
mined by the two rays which correspond to the rays in which a point P projects
to two arbitrary involution centers, S, and S8,. The line S, 8, will correspond to
some line s, at S, and to a line s, at S,, s, s, determining the point S,. S, S, S, is
then a singular triangle of which each vertex corresponds to the opposite side and
vice versa. The quadratic involutory transformation can be completely worked
out from this point of view.*

9. Theorem.—The double or focal rays of the involutions at the vertices of the
self-polar triangle are in each case the two common chords of a and a’ which inter-
sect at that vertex.

If this is true as P moves along c, a chord common to a and a’, P’ must also
describe c. The four intersections of a and a’ are the self-corresponding, or in-
variant, points of the transformation. c is then a line passing through a vertex
of the self-polar triangle and through two invariant points, J, and J,. It must
then correspond to a line passing through the same vertex and through J, and J,,
that is, to itself.

If a and a’ intersect in four real points, the double or focal rays are all real,
that is, the involutions at the three vertices of the self-polar triangle are all
hyperbolic; if they intersect in four imaginary points, at two of the three real
vertices the involution is elliptic, at the third it is hyperbolic; if they intersect
in two real and two imaginary points, the involution at the one real vertex is
hyperbole.

Hips

*See D. N. Lehmer, ‘‘On the Combination of Involutions,’’? Am. Math. Mo., vol. 18, no. 3
(March, 1911).

30 — University of California Publications in Mathematics [ Vo. 1

10. The Form of the Curve—If we regard the plane as divided into seven
regions by the self-polar or singular triangle, we can, knowing the character of
the involutions at the vertices of the triangle, determine in what region a point
P’ must lie that is conjugate to any given point P. Number the regions 1, 2, 3,
4, 5, 6, 7, and denote the vertices A, B, C, as in figure 1. If the involutions are
all hyperbolic, a ray projected from a point P in 1 corresponds at A to a ray
passing through the regions 1, 7, and 4; at B to a ray passing through the regions
2,4, and 1 or 5; at C toa ray passing through the regions 6, 4, and 1 or 38. Any
two of these three rays are sufficient to determine P’ as a point in 4 or 1. In
the same manner we ean locate P’ for any other point P in the plane, whether
the involutions are all hyperbolic, or two are elliptic. If the involution is hyper-
bolic at A, B, and C, we find that:

P in 1 or 4 projects to a ray
at A passing through 4, 7 and 1
at B passing through 4, 2 and 1 or5 -P’ is, therefore, in 4 or 1
at C passing through 4, 6 and 1 or 3

Pin2ors

A‘ — 2,6, 30r5
B—2,4,lor5 -P’ in2or5d
C— 2,7, and 5

Pw 3 or 6
A — 6, 2,5o0r3
B-— 6,7, and 3 -P in 6 or 3
C — 6,4, 1 or3

yee ad
A—7,1,4
Beate. oer a 4
C — 7, 3, 6

In the same manner we locate P’ for P in any region when the involution is
hyperbolic at C while elliptic at A and B, or hyperbolic at A while elliptic at B
and CO, or hyperbolic at B while elliptic at C and A. The complete results may
be tabulated as follows: |

Involution is hyperbolic at ABC C A B
lor 4 4orl 6or3 7 2or5
P lies in 2or7 P’ lies in 2 or 5 vi 6 or 3 4or1
3 or 6 6 or3 4orl 2 or5 { fe
vi e 2 or 5 4orl1 6 or 3

This investigation shows that the regions 1 and 4 are interchangeable; like-
wise, 2 and 5, 3 and 6. Indeed, we see that one can pass from 1 to 4 without
crossing a side of the triangle; similarly, from 2 to 5, and from 3 to 6. We may,
therefore, regard 1 and 4 as one region, R,; 2 and 5 as one region, R,; and 3

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 31

and 6 as one region, k,. 7 stands by itself as R,. The above table may then
be written:

Involution is hyperbolic at ABC C A B
R, R, R, R, ft,

P lies in R, P’ lies in dts yg Ri: R,
nee = R, R, R R,

R, | R, R, R, R,

The transformation of P indicated by the first column leaves R,, R., R,, R,
all unaltered; that is, it is effected by the substitution (1) (2) (3) (4). The
transformation of the second column is effected by the substitution (1, 3) (2, 4) ;
that of the third column by the substitution (1, 4) (2, 3); and that of the fourth
by the substitution (1, 2) (3,4). These four substitutions

(1) (2) (8) (4)

(1, 3) (2, 4)
(1) nee)
(1, 2) (3, 4)

constitute the well known ‘‘Group of Four.’’

Any given curve C intersects the sides of the triangle in a definite order, thus
establishing the order in which the corresponding curve C’ must pass through
the vertices. This enables us to determine for a given branch of the curve C
how the corresponding branch of the curve C’ must pass through the region in
which it hes. We ean, therefore, for a given curve C determine at once the form
of thé corresponding curve C’. Consider, for example, the quartie that will
correspond to the conic C in figure 2, where the involution is hyperbolic at every
vertex. Reading from right to left, C intersects the sides of the triangle in the
order abcabc. Denote the portions of C which he in 1, 2, 3, 4, 5, 6, by c,, c., c.,
C4, C;, and c,, respectively, and the corresponding portions of C’, c’,, c’,, c’,, ete.,

Fig. 2

32 University of California Publications in Mathematics [ Vou. 1

respectively. c, hes in 1 and between b and c. c’,, then, must lie in 4 or 1 and
between B and C. Moving continuously along C’ we may pass from B to C
directly so that c’, les entirely in 4, as in figure 1, or indirectly by way of infinity
so that c’, lies partly in 4 and partly in 1, as in figure 3. In the same way we
determine c’,, c’s, C’,, c’;, and c’,, and, excluding the possibility of infinite
branches of the kind shown in figure 3, we find that C’ must for the conie C in
figure 2 have the form given in figure 6.

oA

Fig. 3

11. Classification of Unicursal Trinodal Quartics—If the given curve C is a
conic or a curve of degree greater than the second, the position of the curve C
with reference to the triangle affords a basis for classification of the curves C’,
sinee the order in which the curve C intersects the sides of the triangle is the
order in which C’ must pass through the vertices. The unicursal quartic is of
particular interest here.

A eonie section according to its relative position may intersect the sides of
the triangle in five essentially different orders. These are:

joao boc
(20,0 COU
(2G 0-00 Cr
(4) aacbbe
(5)-6-Ca DGD

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 3

We may describe the position of the conic by noting the successive regions
through which it passes. Thus the position of the conic in figure 2 is given by
noting that it passes successively through the regions 1, 2, 3, 4, 5, 6.

The same order may be given by more than one position of the conic. Order
(1) is given by four positions, namely,

476

De oN
Ww
De WN

eye |

745

0%
order (2) by one position,

order (3) by three positions,

on)
On
po
oo

On
lop)
~]
bo

order (4) by four,

DWH Pp
beet OD DO OH

and order (5) by one position,

Any particular order represents more than one class of unicursal quarties,
since each of the four possible forms of the involution may give rise to a different
kind of curve for a given position of the conic. An analysis of all cases shows
that a change of the conic from one position to another representing the same
order is equivalent to a possible change in the character of the involution. It
follows that all the classes represented by one order may be obtained from any
one position of the conic which produces that order by changing the character
of the involution.

It is not difficult by the method indicated above to ascertain for the conie in
any given position and under any form of the involution the number of classes
represented by each order and the form of the curve (excluding the possibility
of infinite branches such as those shown in figure 3). The complete results may
be conveniently arranged in the following table:

34

University of California Publications in Mathematics

No. of

Order Classes | Position ant eee eerie
aabbece 2 476727 A,BandC Fig. 4
Cor A, or B Fig. 5
321272 C Fig. 4
A, ov Bor 4A, Bh and: C Fig 5
745434 A Fig. 4
B, or C, or A, Band C Fig. 5 .
567616 B Fig. 4
C,or A,or A, BandC Fig. 5
abcabc 2 654321 A, Band C Fig. 6
A; or B, or G Bigs?
ababce 3 745672 A, BandC Fig. 8
C Fig. 9
A, orb Fig. 10
476543 A Fig. 8
B Fig. 9
C,or A, Band C Fig. 10
654761 B Fig. 8
A Fig. 9
C, or A, Band C Fig. 10
aachbe | 323454 A, Band C Fig. 11
C Wig 12
A,orB Fig. 13
656121 A,BandC Big 11
C Fig. 12
A,orB Fig. 13
232767 A Bag ed
B Fig. 12
C617, band ©. Hipsi3
747212 B Pig it
A Fig. 12
C.or A, BandC Figs 13
acabcb 2 234761 A, Band C, or C Fig. 14

A,orB

Fig. 15

[ Vou. 1

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 35

B 5 B Ee ke)
Wal 969200 \) GACDOC
os % 2

Ke VY] a
/ A

Figs. 4-15

36 University of California Publications in Mathematics [ Von. 1

In all, twelve classes of unicursal trinodal quartices are obtained in this way.
Further distinction may be made, according as the intersections of the conie with
the sides of the triangle are real and distinct, real and coincident, or conjugate-
imaginary.

Special forms of the self-polar triangle give rise, of course, to special kinds
of curves.

CHAPTER II

THE UNICURSAL ‘QUARTIC. AS. THE LOCUS: OF TH aANTER-
SECTION OF CORRESPONDING RAYS OF TWO PROJECTIVE
PENCILS OF THE SECOND ORDER

1. The curve is defined as the locus of the points of intersection of corre-
sponding rays of two projective pencils of the second order.

2. Theorem.—The locus described is a wnicursal curve having, im general,
three nodes, and intersecting any line in the plane in at most four pornts.

See Chapter I, §§ 3 and 6.

3. Notations.—One pencil of the second order (and also the conie enveloped
by it) will be denoted by «; the other by x’. The rays of each pencil will be

denoted by a, B, y,.... and a’, B’, y’, . . . respectively. The tangents to «
and x’ from the points of the quartic and other than a, B, y, .... anda’, B’, y’,
p owillbesdenoted by 0, 0, 6). 4. -2oana: ad, bY ¢’, 2. 2 -respectively.-. The

quartie itself will be denoted by Q.

4. Theorem.—No point of Q les within x or x’.

This is obvious from the definition of the curve given in § 1.*

5. Theorem.—The quartic Q touches each conic x and x’ in at most four points.

Take a point S on « for the center of a pencil of the first order perspective
to x. This generates with x’ a cubic with a double point or node at S.j The
eubie can intersect x in at most four other points besides 8S. These are easily
seen to be points of the locus Q.

6. Problem.—Given x and x’ with three pairs of corresponding rays, to con-
struct Y. (See figure 16.)

Let the three pairs of rays be a, a’; B, B’; and y, y’. The intersection of a
and a’ will be a point A on QY. Likewise 8 and f’ determine B on @ and y and
y’,C on Q. Draw through A to « the other tangent a; also to x’ the other tangent
a’. Take a perspective to x and a’ perspective to x’. @ and a’ are two point-
rows in perspective position since they have a self-corresponding point, A. Their
center of perspectivity may be found by joining any two pairs of corresponding
points on a and a’, such as (a, 8) with (a’, 8’) and (a, y) with (a’, y’). Denote
this center of perspectivity by 3,4. Then to find the point XY of Q determined
by any ray é of x, we first join the point (é, @) with &,. This line will intersect
a’ in the point (, a’). The intersection of € and é’ is the desired point X.

* Unless otherwise stated, the sections referred to are those of Chapter II.

+ See Drasch, ‘‘Beitrag zur synthetischen Theorie der ebenen Curven dritter Ordnung mit
Doppelpunkt,’’ Wiener Berichte, vol. 85 (1882), p. 534. Also see D. N. Lehmer, ‘‘ Constructive
Theory of the Unicursal Cubie by Synthetic Methods,’’ Trans. Am. Math. Soc., vol. 3, p. 372.

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 37

Figs. 16 and 17

38 University of California Publications in Mathematics | Vou. 1

7. Theorem.—Every point P of the quartic Q has its corresponding point Sp.

The locus of all such points will be denoted by 3.

8. Problem.—Given « and x’ with one pair of corresponding rays and the
corresponding point of &, to construct Q.

Let the given pair of corresponding rays be a, a’. The intersection of a and
a’ will be the point A of Y. As in § 6, draw a and a’, taking them perspective
to x and x’, respectively. Using X,, which is given, as the center of perspectivity,
the ray of x’ corresponding to any ray of x, or vice versa, may at once be found,
as before. j

9. Theorem—The tangents from X, to x meet a’ in points of Q. Likewise,
the tangents from X, to x’ meet ain points of Q. (See figure 17.)

This is seen to be true by drawing the rays of «’ which correspond to the
tangents from , to «x; similarly, those of « which correspond to the tangents
from 3,.to x’.

10. Problem.—To find the fourth point of Q upon a or a’. (It is the point X
in figure 17.)

To do this we consider a as a ray € of «x. € meets a in the point of contact of
aandx. Join this point with S,. The line so obtained intersects a’ in the point
(é’, a’). The intersection of & with € (or a) is the desired point.

Instead of the point-rows a and a’ and their center of perspectivity, 4, we
may make use of any other such set, as b, b’ and Xs. In this case a, considered
as a ray € of x, appears as an ordinary ray and &’ is found in the usual manner
indicated in § 6.

The fourth point of Q upon a’ is found in precisely the same manner as the
fourth point of Q upon a.

11. We are now in a position to state the following:

Theorem.—The tangents from S, to x meet a’ in points of Q and a in points
of 3. Likewise, the tangents from Sx to x’ meet ain points of Y and a’ in points
of 3. The point of § ona (or a’) determined by one tangent from 4 to x (or x’)
corresponds to the point of Q on a’ (or a) in which a’ (or a) is met by the other
tangent from 3%, to x (or x’). (See figure 17.)

The first part of the theorem was proved in § 9.

Denote one tangent to x from 4 by 6 and the other one by «. Then it follows
from § 9 that (8, a’) is D and (e, a’) is EF. d’ and e’ eoincide with a’. In finding
Sp, draw the line (a, d—a’ d’). This isa itself. Therefore, 3p hes on a. Simi-
larly, we show that Sp lies on a.

Instead of S, start with Sp, drawing the tangents from Xp to x. One of these.
is a. The other must meet d’, which is a’, in a point of QY. Obviously, it cannot
meet d’ in A or D. Nor ean it meet d’ in the point XY found in § 10, since for X,
a’ is &’ and not x’. It must then meet d’ in FE and, therefore, the second tangent
from Sp tox ise. But « passes through 3,4. That is, 3, and Sp le one. In like
manner, starting with Xp, it may be shown that S, and 3%, lie on 6. Thus one
tangent § from 3, to « meets a’ in a point D of @ and a in the point x, while the
other tangent « from 3, to x meets a’ in the point # of Y and a in 3p.

In precisely the same manner the theorem is proved for the tangents from
>, to x’ instead of x.

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic

39

Figs. 18 and 19

40 University of California Publications in Mathematics [ Vou. 1

12. Problem.—Given «x and x’ with one pair of corresponding rays, and the
corresponding point of &, to find the point of Q corresponding to any particular
point of &.

Let the given pair of rays be @ 8’, and the corresponding point of 3%, Sp.
Since 8 and B’ are known, b and b’ are known. Let the particular point of &
be Xa. To find A, draw the tangents from 3, to « and x’, calling them 4, e, 0’,
and ’. Using Ss as a center of perspectivity of the point rows b and b’, con-
struct 8’, «’, 6, and vc, thus obtaining D, H, H, and J. D and E determine a’ and
H and I determine a. (a, a’) is A.

13. Theorem.—The locus of the points X, the centers of perspectivity corre-
sponding to the points of Q, is a conic section from which Q is generated by means
of a quadratic reciprocal transformation. (See figures 18 and 19.)

A quartic generated from a conic by means of the quadratic reciprocal trans-
formation discussed in Chapter I had its nodes at the vertices of the singular
triangle, the involution centers.

Consider the four tangents from a node of QY, two to « and two to x’. (See
figure 18.) They are two pairs of corresponding rays of the pencils « and x’.
Call them p, p’, and vy, v’. One pair determines the node as M, considered as lying
on one branch of the curve, the other determines it as N, considered as lying on
the other branch. Using p, p’, and y, v’, find 3, for A. This is done by marking
the intersection of the line (a, » — a’, »’) with the line (a, v—a’, v’). The points

Ay, SA5 (a, i) (a’, v’) } (a, v), (a’, p”)
are three pairs of opposite vertices of a complete quadrilateral and therefore
project to any point of the plane in an involution of rays. They project to the
node MN in three pairs of rays, two of which are p, v’, and p’, v. But these lines
are fixed and do not depend upon A or XS,q.

Now consider the involutions at any two of the three nodes of YQ. The lines
from any given point S, of S to the nodes correspond in involution to rays which
intersect in the point A of Q corresponding to S,4. As S, moves on &, A describes
the quartic Y. Thus & is exhibited as that curve from which the quarti¢ is gen-
erated by the quadratic reciprocal transformation and is, therefore, a conic
section. (Chapter I, §§ 3 and 8.)

14. Theorem.—The points of intersection of Q and & lie on the tangents com-
mon tox and x’. (See figure 20.)

Let a common tangent of « and x’ be a ray B in the pencil x. It corresponds
to a ray ~’ of x’, in general distinct from 8. b’ then coincides with 8. The
tangents from S, to « meet 8 in points of & and b’ in points of Y. But B and Db’
coincide.

15. Theorem.—« (or x’) and & are two conics such that a triangle circum-
scribed about « (or x’) is inscribed in &.

This follows from § 11, the tangents there denoted a, 6, and « forming such a
triangle, of which the vertices are Xp, 34, and Xp. (See figures 17 and 21, tri-
angles 34 Sp Sp and 3, Sr Sc.)

For the invariant relation connecting two conics related as in this theorem see
Salmon, Conic Sections, § 376.

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 4]

Figs. 20 and 21

42 University of California Publications in Mathematics [ Von. 1

16. Theorem.—When Xa, lees on x, a’ is tangent to Q and a is tangent to 3;
similarly, when Sa lies on x’, ars tangent to Q and a’ is tangent to &. (See figure
22.)

In this case the two tangents from X, to x, 6 and e, coincide. Consequently,
a’ meets Y in two coincident points, D and EH, and a meets & in two coincident
points S$, and Sp. Similarly, for S, on x’.

Differently stated, the latter part of this theorem tells us:

The tangents common to x (or x’) and & are obtained by drawing the tangents
to «x (or x’) from those points of = where the tangents to « (or x’) at the inter-
sections of « (or x’) and & again intersect &.

Thus « (or x’) and } have as many common tangents as intersections.

17. Theorem.—If « (or x’) and & do not intersect « (or x’) lies wholly within
>. (See figures 23 and 24.)

If « (or x’) and & do not intersect, either ¥ lies wholly within « (or x’) or
«x (or x’) lies wholly within §. Otherwise common tangents could be drawn. But
> cannot le wholly within « (or x’) since a triangle circumscribed about « (or x’)
is inscribed in &. Therefore, if « (or x’) and & do not intersect, « (or x’) lies
wholly within %.

18.—Theorem.—a and a’, a’ and a determine an involution of rays at A in
which A SX, corresponds to the tangent to Yat A. (See figures 24 and 25.)

Let D and FE be the points of Y on a determined by the tangents from X, to kx.
x’ and F and G the points of Q on a’ determined by the tangents from , to x.
Consider, then, the three pairs of points A and 3,, # and 3s, Ff and yp. They
are obviously not the three pairs of opposite vertices of a complete quadrilateral,
since A, S,, #H, and S, determine a quadrilateral of which D and Xp are the third
pair of opposite vertices; and A, Xa, / and X» determine a quadrilateral of which
G and Sq are the third pair of opposite vertices. The locus of points from which
the three given pairs of points are seen in involution is a general plane cubic C,
passing through all the six given points and through D, 3p, G and X¢.* From
§ 13 we know that the nodes of Q also satisfy the condition for points of C.

The cubic C is tangent to Y at A and to & at Sa.

In the quadratic reciprocal transformation a point A of @ goes into X, of &.
The tangent line to Q at A goes into a conic through the three involution centers,
the nodes of Q (see Chapter I, § 6), and tangent to = at 34. A point of the cubic
( passing through the three involution centers goes into the conjugate point
of C.+ A and X, are conjugate points of the cubic C. The tangent line to C at
A corresponds to a conic through the three involution centers and tangent to C
at 4. If the tangent lines to C and to Q at A coincide, the corresponding conics
must coincide; that is, if C is.tangent to Y at A it must be tangent to = at Xa.
The cubic C is tangent to & at Xa.

Construct the tangent to S at S, by means of Pascal’s theorem, using the
points 34, Sp, Sn, Sr, Sc, and numbering them 16, 2, 3, 4, 5, respectively. (See
figure 25.)

* See Schroeter, Ebene Curven Dritter Ordnung, Leipzig, 1888; $1, 6; § 2, 1-4.
+ See D. N. Lehmer, ‘‘On the Combination of Involutions,’’ Am. Math. Mo., vol. 18, no. 3
(March, 1911).

1912] Biddle: Constructwe Theory of the Unicursal Plane Quartic 43

Figs. 22 and 23

44 Unversity of California Publications in Mathematics [| Vou. 1

The tangent line to the cubic C at S, is that ray which corresponds to the ray

=a A in the involution determined at ¥, by two pairs of conjugate rays, as:
(Sa #) and (34 3e); (3a F) and (3, Sr).*

To obtain this tangent line to C at %,, construct a complete quadrilateral, two
of whose pairs of opposite vertices will project to XS, in (3, EZ) and (3, Xe) 3
(SX, F) and (3, Sr); and of the third pair, one vertex in (3, A). (See figure
25.) The other vertex must lie on the ray (3,4 3a), the tangent to C at 4. Such
a complete quadrilateral is determined by the four lines: a, a’, (Sp Se), and the
Pascal line found in constructing the tangent to § at S,. But the vertex which
projects to X,4 giving the tangent to C at 3%, is the intersection of the Pascal line
and the line (3, 4) in the construction of the tangent to % at Sa; that is to say,
the point which with 3,4 gave (6, 1), the tangent to % at X,. The tangent to
at Xs and the tangent to C at %, are, therefore, one and the same line; that is,
the cubic C is tangent to § at Sy. It is, therefore, tangent to Q at A.

Hence, to construct the tangent to Q at A, we need only to construct the
tangent to C at A. This is done, as in the case of the tangent to C at Sa, by con-
structing the ray which corresponds to the ray A XS, in the involution determined
at A by two pairs of conjugate rays, as:

(AE) =@. and (435) —«';

CASH) =04 OM Ae Sy) == a,
This may be done in the following manner. (See figures 24 and 25.) On the line
A Xx, select any point other than A. Through it draw two lines / and I’. Mark
the intersections of l with a and a’ and those of l’ with a and a’. The intersection
of the lines (la—l’a) and (la’—U’a’) must lie on the tangent to Q at A.

19. The Cubic C.—For every point of Q there is a cubie C such as that de-
~seribed in $18. It may be denoted by Cy, Cp, ete., the subscript indicating the
point of QY at which the cubic is tangent.

We observe that we have at once all the intersections of Cy, both with the
quartic Y and the conic &. The three nodes of QY account for six intersections of
@ and Cy, the tangency at A for two more, and the remaining four are at the
points denoted by D, FE, F, and Gin § 18. Cy, is tangent to = at S, and intersects
it in the four remaining points Sp, Sn, Sr, and Sq.

Incidentally, the existence of the cubies C gives a method of determining the
nodes of Q since through them all the cubies C must pass.

20. Theorem.—The class of Q is 6.

A tangent line to = from a node of Q, that is, from an involution center S,,
corresponds to a tangent to Q from S,. Since from any point in the plane only
two tangents may be drawn to a conic section, only two tangents may be drawn
from S, to & and, hence, only two from S, to Y. There are two tangents to Q
at S,, each of which counts as two tangents to Q from a point of the plane not
on Q. In all, then, at most four tangents may be drawn to Q from a node S,, or
at most six from an ordinary point of the plane.

Consistent with this is the number of tangents common to x (or «’) and Q.

* See Schroeter, op. cit.; § 2, 6.

1912]

Biddle:

Constructive Theory of the Unicursal Plane Quartic

45

Figs. 24 and 25

46 University of California Publications in Mathematics [ Von. 1

From § 16 we know that every intersection of «’ (or x) and & yields a tangent
common to «x (or x’) and Y. Since & and «’ (or x) can intersect in at most four
points, this gives rise to only four tangents common to «x (or x’) and Q. The
remaining eight lie at the four points where Q may touch x (or x’). (See § D.)

21. Theorem.—The six tangents drawn from the three nodes to the quartic are
tangent to one and the same conic. (See figure 26, conic I.)

The lines in question correspond by involution to the six tangents from the
three nodes to the X conic. Denote the nodes by S,, S8,, and S,; the tangents from
them to = by 1, 2; 3, 4; and 5, 6, respectively; and the lines corresponding to
1, 2, 3, 4, 5, 6, by 1’, 2’, 3’, 4’, 5’, 6’, respectively. Since the lines 1, 2, 3, 4, 5, 6
are tangent to a conic, the lines

(1, 2—4, 5), (2, 3—5, 6), (8, 4— 6, 1)
are concurrent. (Brianchon.) But these lines are

(S, — 4, 5), (S, — 5, 6), (S,; — 6, 1).

To them correspond

(S, — 4’, 5’), (S, — 5’, 6”), (S, — 6’, 1’),
respectively. Since the first three are concurrent the second three must be. But
the second three are

(1’, 2’ — 4’, 5’), (2’, 3’ — 5’, 6”), (3’, 4’ — 6’, 1’),

and therefore 1’, 2’, 3’, 4’, 5’, 6’ are tangent to one and the same econie section.
(Brianchon. )

22. Theorem.—The tangents to Q at a node correspond in involution to the
lines joining the node with those points of % which lie on the line joining the other
two nodes.

For from Chapter I, § 5, it follows that Sy and Sy, corresponding to the node
M N, lie on the side of the singular triangle opposite to M N, that is, on the line
joining the other two nodes. M XS» corresponds to M M, the tangent to Q at M,
and N Sy corresponds to N N, the tangent to Q at N.

23. Theorem.—The six tangent lines to Q at the three nodes are tangent to a
conic section. (See figure 26, conic II.) |

As before in § 21, denote the nodes of Y by S,, S,, and S,.. From Brianchon’s
theorem we know that if the six lines in question cirecumseribe a conic section
the three lines joining the opposite vertices of the hexagon they from all pass
through one and the same point. Number the two tangents to Y at S,, 1 and 2;
those at S,, 3 and 4; and those at 8,, 5 and 6. The lines

(1,2 —4, 5), (2, 3 — 5, 6), (8, 4 — 6,1)

join opposite vertices of the hexagon.

In the involutions at S,, S,, and S, there correspond to 1, 2, 3, 4, 5, 6, lines
which we shall eall 1’, 2’, 3’, 4’, 5’, and 6’, respectively.

If the lines 1, 2, 3, 4, 5, 6 are tangent to a cone section, the lines 1’, 2’, 3’, 4’,
5’, 6’ are also tangent to a conic section, and vice versa. For the point (4, 5) goes
into the point (4’, 5’) and the line

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 47

Figs. 26—28

48 Unversity of California Publications in Mathematics. [ Vou. 1

(S,—4,5) corresponds to (S,—4’,5’).
Similarly,
(S,—6,1) corresponds to (S,— 6’, 1’)
and
(S, — 2, 3) to (S, — 2’, 3’).
The intersection of any two of the three,
(S,— 4,5), (S,—6,1), (8S, —2, 3),
as (S, — 4,5) and (S, —6, 1),
corresponds to the intersection of the corresponding two, as
(S, — 4’, 5’) and (S, — 6’, 1’).
It follows that if the three lines
CS ai 0) Gh SO gk pS gi 2 tes
that is, the lines
CA A Oy Gah ea Oe A) ol a ee 284),
are concurrent, the three corresponding lines
(Sie) Ss 0, eo,
that is, the lines
(1,2—4, 5), (8,4—6, 1), (5, 6 — 2, 3)
are also concurrent, and vice versa. If the lines
(1’, 2’ — 4’, 5’), (3’, 4’ — 6’, 1’), (5’, 6’ — 2’, 3’)
are concurrent, the lines 1’, 2’, 3’, 4’, 5’, 6’ are tangent to a conic section. In
that case, the lines 1, 2, 3, 4, 5, 6 are also tangent to a conic section.

1’, 2’, 3’, 4’, 5’, 6 are lines joining the vertices of a triangle, each with the
points in which the opposite side intersects a given conic section. (See § 22.)
Any six such lines carcumbscribe a conic section. (See figure 27.)

Mark the six intersections of the three lines

(1’, 2’, — 4’, 5’), (2’, 3’ — 5’, 6’), (3’, 4’ — 6’, 1’)
with the given conic, numbering them in any order (1), (2), (3), (4), (5), (6);
say, so that (1) and (4) lie on the line (1’, 2’ — 4’, 5’), (2) and (5) on the line
(3’, 4’ — 6’, 1’), and (8) and (6) on the line (2’, 3’ — 5’, 6’). These six points.
form an inscribed hexagon whose opposite sides must, therefore, intersect in three
collinear points. (Paseal.) The three collinear points are:
ee a CO edaue) =) 0), fue) 4) 0)! CL)
Now make use of the following notation :*

(1) =A, (2)=B, {(2)(3) — (6) Q)}=G,

(4) =A’, (5) =B’, {(5)(6) — (8) (4)} =O,

(3) =A, (4)=B, {(2)(3) — (4) (9)} =C,

(6) =A’, (1) =B”, {(5)(6) —(1)(2)}=C”,

(5) =A, (6)=B, {(4)(5) — (6) (1)} =,

(2) =A’, (3) =B’, {(1)(2) — (3) (4)} =C’,

* Care must be taken to have corresponding vertices lie on the lines (1’, 2’ — 4’, 5’), (2’, 3’
— 5’, 6’), and (3’, 4’ — 6’, 1’).

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 49

A, B, C, and A’, B’, C’, are two triangles so situated that their corresponding
sides intersect in three collinear points. Therefore, the lines joining correspond-
ing vertices, that is, the lines

By Ae tl’, 2° 46"), Bu B == (3, 4 — 67, 1") and ©, 0’,
are concurrent. (Desargues.) For a similar reason the lines
AoA ee oe Ob ea tle a) nO Oe
are concurrent; likewise the lines
Aon waa ee a Oo Bb (eee we On) 5) and Oe © ox,
‘But the triangles C, C, C, and C’, C’, C’, also fulfill the condition for Desar-
gues’ theorem, and therefore the lines

ERE Gore Ge bia a: save le Opn 61
are concurrent. Accordingly, the lines
(1’, 2’ — 4’, 5’), (2’, 3’ — 9’, 6’), and (3’, 4’ — 6’, 1’)
are concurrent and the lines 1’, 2’, 3’, 4’, 5’, 6’ cireumscribe a conie¢ section. But
this is the condition that the lines 1, 2, 3, 4, 5, 6 should also circumscribe a conic
section.

24. Theorem.—If Q is a tricuspidal quartic the three cuspidal tangents meet
in one and the same point. (See figure 28.)

To a tricuspidal quartic corresponds a conic tangent to each of the three sides
of the triangle S, S, S8,. If the three points of contact on the sides s,, s,, s, be
denoted by L, M, and N, respectively, then to the lines S, L, S, M, and S, N corre-
spond the three cuspidal tangents. But S, LZ, S, M,and 8, N are three concurrent
lines (Brianchon), and therefore the three corresponding lines, the three cuspidal
tangents, are concurrent.

25.—Theorem.—If Q has three bitangents, their points of intersection may be
joined urth the three nodes, one with each node, in such a way that the three
jowning lines pass through one and the same point. (See figure 29.) If Q has four
bitangents their points of intersection in sets of three may be joined with the
nodes in such a way that the joining lines form a complete quadrangle of which
each pair of opposite sides intersects in a node. (See figure 30.)

Each bitangent of Q corresponds by the quadratic involutory transformation
to a conie passing through the three nodes, S,, S,, S,, and having double contact
with the } conic. Denote the four bitangents to Y by 1, 2, 3, 4; their points of
contact on @ by A, B; C, D; E, F; and G, H, respectively ; and their correspond-
ing conics through S,, S., S,, by I, u, ul, and tv, respectively. Denote the fourth
intersection of rand m1 by U; of 1 and 1m by V; of m1 andi by W; of 1 and Iv by
X ; of mand tv by Y; and of m1 and w by Z.

Consider first the case of three bitangents, 1, 2,3. Then the lines
ix SB, xc Xp,

and one side of the triangle S, S, S,, say S, S,, and S, U all pass through one and
the same point and form a harmonie pencil.* Similarly, for the lines

* See Salmon, Conic Sections, § 2638.

[ Vou. 1

University of California Publications in Mathematics

50

Figs. 29-31

1912} = Biddle: Constructive Theory of the Unicursal Plane Quartic af

Sc Sn, Sz Sr, 8S, S,, and 8S, V;
likewise, for the lines

ae ay, 24 oR, 0, 62, and 8, W.
From this it follows that the lines

S, U, 8S, V, and S, W

are concurrent. For on S, W the lines

2A Shy ho 2p. eees eG 8, U
determine four harmonic points. Likewise

Searle on, bs oa ee
determine four harmonic points on 8S, W. But on 8, W, Sc Sp in each case deter-
mines the same point; S, S, and S, S, both determine S,; 3,4 Sg and Sp, Sr deter-
mine the same point since the lines

Dp as a iy os BG ee
are concurrent. Therefore, S, U and S, V determine the same point on S, W;
that is, S, U, S, V, and 8S, W are concurrent.

But if these three lines are concurrent, the lines to which they correspond in
involution must be concurrent. That is, the lines
(8; 2192), (8, 2B Se
pass through one and the same point.
Consider now the case of four bitangents to Y, 1, 2, 3, 4. Proceeding as be-

fore, we find the following sets of harmonic lines:

For I, 1, and m1—

Tand Il II and Il ll and 1
Sos: Ser SS
Su: S,V S,W
sa 3B xc Xp <n ir
xc 2p Ln ir 2a XB
For I, 0, and 1v—
IT and II II and Iv Iv and I
SES Ses Sos:
8,U SY pe
2a 2B Xc Xp 2c 2H
Xc Xp xc 2H Xa 2B
For 1, m1, and Iv—
I and Ill Ill and Iv I and Iv
Sao S.S. Sie
Ss, W OA SEG
SE lr Sn ir Xe 2H
Xa 2B Xa 2H Sa 3p
For II, 11, and Iv—
II and Ill III and Iv Iv and II
Sas: SN: SS,
SV. 8,2 Sy
Xc Sp Xn lr xc Xp
Xn lr Xo 2H Xo 2H

52 Unwersity of California Publications in Mathematics [| Vou. 1

From this we obtain the following four sets of three concurrent lines:

8,U 8,U a7 SZ
8.V wx ae S,V
S,W S,Y S,W Spa

The configuration determined by these six lines is a complete quadrangle of
which each pair of opposite sides intersects in a node of Q. Each set of three
concurrent lines, one of which passes through each involution center, corresponds
in involution to another set of three concurrent lines, one of which passes through
each involution center. Therefore, the figure corresponding in involution to a
complete quadrangle of which each pair of opposite sides intersects in an invo-
lution center is another complete quadrangle of the same character; and the lines
S, U, 8S, V, 8S, W, 8S, Z, S, X, S,; Y correspond in involution to lines joining the
points of intersection of the bitangents of QY to the nodes.

26. Degenerate Cases.—In general, the pencils x and x’ have no self-corre-
sponding rays. That is to say, a tangent common to « and x’, considered as a ray
of «x, corresponds in general to some other ray of x’, and considered as a ray of
x’, to some other ray of x. But it may happen that of the three pairs of corre-
sponding rays of « and x’ given to construct Y (§$ 6), one, two, or all three may be
self-corresponding. Also, in general, the conics x and x’ are distinct. But the
two pencils of the second order may envelope the same base conic; and in that
case, too, there may or may not be self-corresponding rays. In all these special
cases both Y and & assume degenerate forms.

A. If « and x’ have one self-corresponding ray, a a’, Y consists of a cubic C and
the ray a a’; & consists of a line o and the ray a a’; points of Y on a a’ correspond
to points of on o and points of Q on C to points of S on aa’. (See figure 31.)

Since a and a’ coincide, the four tangents from S, to x and x’ meet a a’ in four
points of S, showing at once that all the points of a a’ are points of &, since more
than two of the points of a a’ are points of &. That is, } degenerates into two
straight lines, one of which is a a’.

If from points of & as 3p, Sn, =r, Bee on a a’ we draw tangents to x and x’,
such tangents must meet the lines d’, d, e’, e, f’, f, ete., in points of Y. But since
always a a’ itself will be a tangent from such points both to « and x’, it follows
that half of the points so obtained will lie on aa’. aa’ is then a part of Y. The
remaining part is, of course, a ecubie C.

The construction shows at once the correspondence between the different parts
of Q and &.

B. If «x and x’ have two self-corresponding rays, a a’ and B £’, @ consists of
the two lines a a’ and B pf’ and a conic A; & consists of the two lines a a’ and
B B’; points of Q on a a’ correspond to points of = on B f’ and points of @ on
B B’ to points of = on a a’, while the points of Q on the conic A all have their corre-
sponding points of & at the intersection of a a’ and B B’.

C. If « and x’ have three self-corresponding rays, a a’, B 8’, and y y’, Y con-
sists of the four tangents common to « and x’; & consists of the three vertices of
the triangle formed by a a’, B B’, and y y’. Points of Q on one of the three lines

1912] Biddle: Constructive Theory of the Unicursal Plane Quartic 53

aa’, 8 B’, or y y’ correspond to the opposite vertex of the triangle formed by
those lines. Points of % corresponding to points of QY on the fourth tangent
common to « and «’ are indeterminate.

To prove that the fourth tangent common to «x and x’ is a part of Q, find the
point P on a for which a is a ray z of x. p and p’ must coincide. Suppose they
do not. p, considered as a ray p of x, determines a point FR of Q, and p’, consid-
ered as a ray é’ of x’, determines a point X of Y. None of the points P, R, and
X can lie on any of the rays a a’, B B’, y y’, since every point of Q on a self-
corresponding ray is determined by that ray. Since Q is of the fourth degree,
P, R, and X must lie in one and the same straight line. This can happen only
if p and p’ coincide.

D. The two projective pencils of the second order, x’ and «’’, envelope the
same base conic x.

(1) If there are no self-corresponding rays, = degenerates into two straight
lines o, and o, tangent to x, and @Y into these same two lines and a conic sec-
tion A. The construction shows that points of Q on o, correspond to points of
>= on o,, and points of Y on o, to points of & on o,, while points of Q on X all
have their corresponding points of = at the intersection of o, and o,. The conic
A is tangent to « at the points of tangeney of o, and o,.

Here a coincides with a’, a’ with a, b with £’, b’ with B, ete. It follows that
the line (a, 8 —a’, 8’) coincides with (b’, a’ — b, a) ; (a, y —a’, y’) with (c’, a’—
c,a); and (b,y—b’, y’) with (c’, B’ —c, B). These are three lines joining the
opposite vertices of a hexagon circumscribed to a conic section and are therefore
concurrent. (Brianchon.) Therefore, S4, Sg, %c all coincide. The tangents from
Za, Xz, Sc, ete., to x determine points of ¥ on a, a’, B, 8’, y, y’, ete., and points of
Q ona, a’, b, b’, c, c’, ete. Because of the coincidences mentioned all these points
he on two lines, o, and o,, the tangents from Sazc, etc., tO x.

(2) If x’ and «’’ have one self-corresponding ray a a’, a a’ coincides with o,
or o,, and 3, is indeterminate. In other respects QY and & are as described in (1).

(3) If«’ and x’’ have two self-corresponding rays a a’ and 8 B’, a a’ and B fp’
are the lines o, and o,, and X, and XS, are indeterminate. In other respects Q
and & are as described in (1).

(4) If«’ and x’”’ have three self-corresponding rays, a a’, B B’, and y y’, every-
thing is indeterminate.

— 27. The Unicursal Curve of the Fourth Class.—It is of interest to apply the
principle of duality to the results of this chapter, considering then the unicursal
curve ’ defined as the envelope of lines joining corresponding points in two pro-
jective point rows of the second order, « and x’. It has, in general, three bitan-
gents and four nodes and is of the sixth order and fourth class. (Cf. $§ 1, 2, 20,
25.) Given three pairs of corresponding points in two projective point rows,
we can construct the curve (cf. §6) and establish properties entirely analogous
to all those of the curve of the fourth order. The more interesting and important
ones may be noted.

54 University of California Publications in Mathematics | Vou. 1

Every ray p of the rays enveloping @’ has its corresponding ray of perspec-
tivity &’p. The ensemble of rays 3’, envelope a curve 3’ of the second class.
(Cf. §§ 7 and 13.)

Any ray a joining corresponding points A, and A’, has a second point A, in
common with « and a second point A’, in common with x’. The lines joining the
points of intersection of the ray 3’ , and x with A’, are rays enveloping Q’; with
A,, are rays enveloping &’. Likewise, the lines joining the points of intersection
of the ray 3’ , and x’ with A, are rays enveloping Q’; with A’,, are rays envel-
oping 3’. The ray tangent to ¥’ which joins A, (or A’,) with one point of inter-
section of 3’ , and «x (or x’) corresponds to the ray tangent to Q’ which joins
A’, (or A,) with the other point of intersection of 3’, and x (or x’). (Cf. § 11.)

The common rays of the sets enveloping Q’ and 3’ pass through the points of
intersection of x and x’, two through each point. (Cf. § 14.)

The points A, and A’,, A’, and A, determine an involution of points on a in
which the point (a, 3’ «) corresponds to the point of tangency of a on Q’. (Cf.
$19.)

The six points of Q’ other than the points of tangeney which lie on the three
bitangents to Q’ all lie on one and the same conic section. (Cf. § 21.)

The six points of tangency on the three bitangents to Q’ all le on one and the
same conie section. (Cf. § 23.)

If the two points of tangency coincide on each bitangent the three points of
tangency all lie on one and the same straight line. (Cf. § 24.)

If Q’ has three nodes the lines joining them intersect the three bitangents, one
each bitagent, in such a way that the three points of intersection lie on one and
the same straight line. If Q’ has four nodes the lines joining them in sets of
three intersect the bitangents in such a way that the points of intersection form
a complete quadrilateral of which each pair of opposite vertices lies on a bitan-
gent. (Cf. § 25.)

Transmitted October 3, 1911.

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