twisted cubics associated with a space curve

Twisted Cubics Associated with a Space CurveAuthor(s): Louis GreenSource: American Journal of Mathematics, Vol. 62, No. 1 (1940), pp. 285-306Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371453 .

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TWISTED CUBICS ASSOCIATED WITH A SPACE CURVE.* t

By Louis GREEN.

1. Introduction. Various methods have been employed in investigating the projective differential properties of a curve immersed in ordinary space, each method having certain advantages. The procedure used here is to start with a pair of dual differential equations, to introduce certain transforma- tions of coordinates in order to obtain canonical power-series expansions for the curve considered, and to base the remainder of the paper on these expansions. The objectives of the paper are to characterize certain configurations associated with a curve, particularly the five-poinit twisted cubics, and to begin the problem of interpreting a duality formula in geometrical language.

2. Analytic basis. The differential equations of a twisted curve r, not belonging to a linear complex, may be written in the form

(1.1) Xiv + ax" + (a'- O)x' + cx = , (0 = con-st. + 0).

(1. 2)+ ae" + (a' + 0)e' + ce ?0 $ represents the osculating plane of r at the point x; differentiation is taken with respect to a properly chosenl parameter u; and a, c are scalar functions of u. The value of 0 can be chosen arbitrarily (+ 0); if 0 -1, these equations are the ones derived by Fubini and Cech; 1 if 0 4, then (1. 1) is the canonical form of Halphen.

When u is fixed at a suitable value uo, a poinit 0 (= x) on r is obtained, and a local tetrahedron of reference D1J{x, x', x", 8"'} is formed, with a unit point chosen so that any point whose coordinates in the original system are

x1x + X2X' + X3/Xt + 4X,X,

will have local co6rdinates proportionlal to x1, , X4. It follows readily that the local co6rdinates xi of a point P o0 117 "sufficiently near " 0 are

00

xi AInAU/n !n$ 1,** 4), n=0

where

Aio = 1, A20 A30 A40 0,

A,n+i A'ln CA4n-

A2,11+1 Ain + A'2n + (6- a')A4Z1, (n ? 0) A3,fn+i A2n + A'3n-aA4nn

A4,nl1= A3n + A"4n.

* Received February 13, 1939. t Presented to the Society, September 6, 1938. 1 Introduction a la Gfeometrie Projective Differentielle des Surfaces, 1931, p. 26.

285



286 LOUIS GREEN.

Halphen's local tetrahedron H1 which will be used throughout this paper can be obtained directly from D1. If local coordinates referred to H1 are denoted by yi, then the following relationis hold:

=YX/TXI + a12X2 + al3X3 + (14X4,

Y2/7- (Z222 + (23X3 + (24X4, ( 2. 1) = ~ 2xY37 (23X3 + )24X4, (r arbitrary)

=1/ ~ 33X3 + ~34$A,

Y4/7- 44X4,

a12 = /200, a13 (02 180a02)/60002,

a14 (03 - 1260a802 _ 10800a'03 + 1440004)/3600003,

922 a , 3 = Oq4/150, a24 = (02 - 420a02)q /60002,

=33 2 ', 34 == +q2/100, a44 =6 3

=100c 9a2 -30a", t (0/60q6)?, q6 #0 ?

If nioni-homogeneous co6rdinates x, y, z are definied in the customary way,

X y2/yl, y Y3/yl Z y44/yl

the equationis of F, relative to 0 as origini, are founld to be

00 00 (3. 1) 8~~= X2 + EpX ,, X3 + E nXn

7 6

The coefficienits p,,, q,n (n ? 7) are expressible in terms of q6 and the coefficienits in the differential equation (1. 1), and are understood, of course, to be evaluated at u = u0. The value of q6, as is the case with 0, can be chosenl arbitrarily (#/ 0), but since a numerical choice for q6 would prevent us from displaying the weights 2 of the coefficients, we merely specify that q6 be inidependent of the parameter u. The values of q7, q8, P7 are found from (2. 1) to be

q7 =/4200/4,

(4. 1) q8 (22 45'0"- 36Oa02)/040 005/0,

P7 (_ 2 - 600'O - 36OaO2) /1512000/50.

Let 7 (- ) be the plane osculating r at 0. Then a local plane tetrahedron- of reference D2{d, b', $", "'} with local coocrdinates proportiolnal to

el, *' 44 if plane cobrdinates ill the original system are

el4+ '

2e + 4 + Uffft

is formed from the differenitial equation (1. 2). We replace D2 by a new local plaiie tetrahedron If,, dual to H1. If local plane coc6rdinates referred to H2

are denoted by -q, then the relations between i% and -q are the same as those between x. and yi in (2. 1) except for the change in signi of 0.

2 The weights of p,, and q,, are n - 2 and t - 3 respectively.



TWISTED CUBICS ASSOCIATED WTITIT A SPACE CURVE. 287

Setting

we obtain the equationis of r in system H2:

00 00 (3. 2)

e 2 + E7-nen, e 3 + E Knen

7 6

where we choose

K6 = q6.

The values of K7, K8, w7 are obtained from q7, q8, p7 as given in equations (4. 1)

by changing the sign of 6. It theft follows that

(4. 2) K7=q7,

K8 =(7q6q8- l8q6p7- 72) /16, 77 = (24q6q8-63q6P p-28q72)/9q6

Returnin-g to tetrahedron H1 and equations (3. 1 ) we denote homogeneous

plane co6rdinates by t and non-homogeneous coo6rdinates by (, , (, where

t = (3/t4, ?7(2 , ( = 12/4

Then the plalie equations of r ini system HI are

j 2/3 + 2q6,(681 - 7q7/6 729 + (Sq8 - 21p7)47/2187 +

(5) - 3-6T/S -7P

( ) ( 27 + 5q6-6 729-2q7.7/729 + (Tqs,18p7)T8/6561 +

The transformation

= 27q6 34

9 82lq6 33 -189g6 2q7q4,

(6) =3 181q632 -2 378q62g7q3 + 441q6q72\,

= 2 189q62q7q2 + 441q6q3223 + ( 54q64 - 343q73) 4

carries (5) into (3. 2) and hence is the tralnsformation from H1 to H2. The

coordinates of the vertices of tetrahedroln H11 referred to system H1 are thus

found to be

(1 0,0,0), (7q7,q6,0,0),

(k) (49q72, 14q6q7, 3q62, 0), (343q73 - 54q64, 147q6q72, 63q62q7, 27q63).

The u-derivatives of the local point coordinates xi of system D1 are

obtained in the followiiig way. In the original co6rdinate system any poillt z

in space has co6rdinates

Z = X1x + X2X' + X3X" + X4X"'.

I-Tenice, from (1 1),

Z (X' - CX4 ) X + ( X'2 + X1 + OX4- a'X) X'

+ (X'3 + X2- aX4)x" + (X'4 + X- 3) x/,



288 LOUIS GREEN.

and placing z'i = 0 (i = , , 4), we get

x1 =CX4, x,2 X1 + (a" 0)X4, (8) rt3 rx-x2 + ax4, x'4 -x3.

The formulas for the derivatives of local co6rdinates yi of system H1 are found by differentiating equations (2. 1), in which we choose

Tr exp(f al2du),

and by replacing the x'i and the xi by their values in terms of yi as obtained from (8) and the inverse of (2. 1). The resulting equations are

=Y_ - 63P7Y2 - 36q62y3,

(9 rY'2 = - 3q6yl + 7q7y2 - 42P7Y3 - 18q62y4, (o 3q6/) 9

(y3 - 6q6y2 + 14q7y3 - 21p7y4,

Y'4 = - 9q6y3 + 21q7y4.

Now x =fAu + , so that at u u0,

y'f dy,/du = /dy,/dx.

Hence the x-derivatives of the local co6rdinates yi are gotten immediately. These have also been obtained by Miss Newton.3

3. Duality. The differential equations of r show that relative to r at 0 the dual of a point xi = fi (a, c, 0) in system D1 is the plane ei = fi (a, c, - 0)

in system D2. From the similarity of the canonical expansions (3. 1) and (3. 2) it follows that the dual of a point yi = fi (pa, qn,) referred to H1 is the plane =f (7r, K.) referred to 112. The co6rdinates ( of this plane referred to H1 are thein obtainable from transformatioil (6). The problem of finding the dual, relative to F at 0, of a given point is therefore solved.

But there appears to be a good deal more to the problem than this. For, the concept of duality considered here is quite different from the duality theory of projective geometry. Two coincident points, for example, may have distinct dual planes. Thus, the points

P(lq6q8 + mq6p7 + nq7l, q6q7, ? ) 0 m, n) ,', n)) P'(l'q6q8 -1- M'q6p7 -j- n'q7, q6q7~, 0, 0), (,m,n (1,i,n'

in system I1 both lie on the x-axis and for properly chosen values of 1', m', n' coincide. Yet their dual planes, which can be found by the method described above, are distinct. Furthermore, in order to obtain complete generality for

3 " Consecutive covariant configurations at a point of a space curve," Transactions of the American Mathematical Society, vol. 36 (1934), p. 61.



TWISTED CUBICS ASSOCIATED WITH A SPACE CURVE. 289

both the curve r and the position of the point 0 on r, we do not wish to specify the relations existing, at u = uo, among the coefficients in the equations (3. 1). We therefore regard P as one of a three-parameter family of points generating the x-axis as 1, m, n vary independently over all real numbers; i.e. we fail to consider the coincidence of P and P' unless 1, m, n 1', m', n' respectively.

Similarly, the point

(Iq7, q6, 0, 0)

referTed to tetrahedron H1 lies on the x-axis and coincides with P for proper choice of Ic. Yet their geometrical characterizations and their dual planes are completely unrelated, and the curves they generate as u varies (Ic, 1, m, n remaining independent of u), are entirely different.

The problem of characterizing geometrically the dual of a given point relative to r at 0 is completely untouched by the formal, analytic solution above. A special case of this problem, for example, is to determine how the point P, given above, is related geometrically to its dual plane under the assumption that 1, m, n are arbitrary numerical quantities. The simplest special case of the general problem is that of characterizing geometrically the dual of a point whose coordinates are expressible in terms of q6 and q7 alone. This problem is readily solved. For, we may write the coo5rdinates of such a point as

y f (q6, q7) ZZ

when referred to tetrahedron H1; hence the dual plane has coordinates

= f (K6, K7) =Z

when referred to H. By means of transformation (6) we find that this plane has the equation

27iq63z4yl + (81q63z3 - 189 q62q7Z4) Y2 + (8lq63z2 - 378q62q7Z3 + 44lq6q72Z4) Y3

+ [2 7 q63z -89 q2q7z2 + 44lq6q72z3 + (54q64-343q73)z4] = 0

when referred to H1. We have therefore proved the following result.

THEOREM 1. The dual, relative to r at 0, of a point whose coardinates are expressible in terms of q6 and q7 is the polar of the point with respect to a

quadric Q having the equation

(10) 54q63Yly4 + 162q63y2y3- 378q6 q7y32- 378q62q7y2y4 + 882q6q72y3y4

+ (54q64 - 343q73) y42 0.

Sannia has considered 4 a self-dual tetrahedron S whose vertices, referred to H1, are

4" Nuova trattazione della geometria proiettivo-differenziale delle curve sghembe," Annali di Matematica, IV, vol. 1 (1924), pp. 1-18; vol. 3 (1926), pp. 1-25.



290 LOUIS GREEN.

0, T(7q7, 2q6, 0, 0), N(49q72, 28q6q7, 12q62, 0), B (343q73 - 216q64, 294q6q72, 252q62q7, 216q63).

The quadric Q has three-point and three-plane contact with r at 0, and has

among its rulings the edges OT, ON, BT, BN of Sannia's tetrahedron. It is

contained in a one-parameter family of quadries having this property.

4. Fundamental tetrahedra. In system IH the osculatinig conic K2

of r at 0 is given by

(12. 1) 4y1Y3 3Y220 ? Y4,

and its dual, the osculating quiadric cone K'2, has the equation

(12. 2) Y32 Y2Y4 0.

Any tetrahedron { OPlPP>P3} with vertices defined in the following way will

be called a fundamenital tetrahedron of 1r at 0. One vertex is at 0, a second

at an arbitrary point P1 (3, t, 0, 0), t 4 0, on the x-axis; a third vertex

P2 (3, 2t; t2, 0) is the contact point of a tangent from P1 to the conic K2, and the fourth vertex P3 (1 - at3, t, t2, t3), for an arbitrary value of a, lies on the contact line of the cone K'2 with its tangent plane which passes through P,. The tetrahedra ]I, H2, S are fundamental tetrahedra determined by the following values of t, : oo,0; 3q6/7q7,2q6; 6q6/7q7,q6; moreover, the dlual

of anJy fundamental tetrahedron is another fundamental tetrahedron. Associatecl with the fundamental tetrahedra is a family of twisted clbies,

Ta, having five-point contact with r at 0, and expressed parametrically by

the equations (13) yl= 1 -t, y2 t y3 = t2, y4 t3.

All of these cubies belong to the same null system, lie on the cone K'<, andl

have Kf as osculating conic at 0.5

Each choice of the point P1, or of the plane 0P1P3, determines a slubset

of oo - fundamental tetrahedra; these can be placed in a one-to-one corre-

spondence with the cubies Ta by choosinig the vertex P3 as the intersection,

besides 0, of Ta and the plane 0P1P3. When this is done, the following relations exist:

The polars of the vertices of one of these tetrahedra with respect to the

common niull system of the cubies are the faces of the tetrahedra, the tangent to Ta at P3 is the edge P2P3, and the osculating plane to Ta at P3 is the face

P1P2P3.6 As P1 traces the x-axis, a remaining fixed, the edge P1P3 generates a cubic surface with the equation

5 Lane, Projective Differential Geometry of Curves and Surfaces, 1932, p. 29. 6 SU, " Note on the projective differential geometry of space cunves," Journal of

the Chinese Vllathematical Society, vol. 2 (1937), pp. 98-137.



TWVISTED CUBICS ASSOCIATED WITH A SPACE CURVE. 291

(14) YlY42 - 3Y2Y3Y4 + 2Y33 + ;y43 =0

The dual of an arbitrary point on the cubic Ta can be shown to be the

plalne which osculates the cubic Ta'

yS 1 atT3n 89 7n y3 7~2 y 3 y =l - r,,3 Y2T, 'Y3=r7., Y4=T 7-

for which

a, 2q -

at the poilnt determilled by

r 3q6t/ (7q7t-3q6).

THEOREM 2. The dual of the five-point cubic Ta is another five-potnt

cubic Ta' for- which a + at' 2q. The self-dual cubic is Tq,, and the only points of this cubic which lie in their- dual planes are the points 0 and B of Sannia's tetrahedron.

The self-dual cubic Tq6 harmonlically separates, with 0, the points of any

two dual five-point cubics. It is called the harmonic cubic by Fubinli andl

Cech,7 and the coinlcidence cubic by Kanitani.8 Theorem 2 shows that all

five-point cubics are also five-plane cubics, and since To is the six-point cubic,

then the six-plane cubic is Ta (a = 2q6).

5. The principal plane of a curve. Halphen's theorem 9 onl the princi-

pal plane at a poilnt of a curve has been extended by Bompiani 10 and dualizecl

by Sannia.11 We shall carry their results still further, basing all our cal-

culations on tetrahedron 1A. Let the tangent developables of r and of a five-point cubic Ta be cut by

an arbitrary plane 0P1P3 (# w) passing through the x-axis, and let the plane

curves of section be denoted by r' and T'a, the latter being a cusped cubic

Then the following conclusions hold:

THEOREM 3. 1. If a 7/ 4q6, the curves r' and T'a have exactly six-point

co;ttact at 0 for all planes OP,P3. If a = 4q6, these curves always' have just seven-point contact, with the sintgle exception that the plane givent by

(15. 1) 3q6y3 - 5q7y4 = 0

produces curves having eight-poitt contact.

7 Geometria Proiettiva Differenziale, vol. 1 (1926), p. 42. 8 Sur les reperes mobiles attachs a une courbe gauche," Memoirs of the Ryojun

College of Engineering, vol. 6 (1933), p. 106. 9 Sur les iinvariants diffhrentiels des courbes gauches," Journial de l'i9cole Poly-

techniique, vol. 28 (1880), p. 25. 10 " Sul contatto di due curve sghembe," Memorie della Reale Accademia delle

Scienze dell' Istituto di Bolognta, ser. 8, vol. 3 (1926), pp. 35-38. 11 Loc. cit.



292 LOUIS GREEN.

If a and the plane of section 0P1P3 are both arbitrary, the cusp of T'a

lies on the twisted cubic T,c, and is the vertex P3 of the fundamental tetrahedron determined by T,a and 0P1P3. The cusp-tangent is the edge P1P3 of this tetrahedron.

Bompiani's osculants 12 for the curve F', which has an inflexion at 0 for

arbitrary plane 0P1P3, are obtained immediately. His fourth-order neighbor-

hood of r' at 0 is the vertex P1 of the fundamental tetrahedra determined

by the plane 0P1P3, while his neighborhood of the fifth order is the edge OP3.

His neighborhood of the sixth order is the point P3 which lies on the five-

point twisted cubic Ta (a Q 4q6).13

We shall prove only the first part of our theorem. The tangent de-

velopable of r has the parametric equations

x U+ V,

(16) y u2 + p7u7 + + v(2u + 7p7u6 *), z u3 + q6 U6 + aq7z7 + q8,U8 +

+ v(3u2 + 6q6U5 + 7q7U6 +8q8u7+**U*)

It meets the plane 0P1P3, whose equation is

(17) ty-z=-,

in a curve F', represented bv (17) and

(18) y - 4x3/t -24X4/t2- 156X5/t3- (1072 + 128q6t3 ) X6/t4

- (7668 + 1728q6t3 + 320q7t4)x7/t5 + * ? . .

If the non-homogeneous equations of the cubic Ta are written in series

form, its tangent developable is seen to have the equations

x u + ? ,

(19) y 2812 au5 + 3a2U8 + + v(2u 5au4 + 24a2U7 +

z u3_ 2au6 + ?a29 + ** + v(3u2 -12aqu5 + 63at8? +

Its intersection with the plalne 0P1P3 is a curve T',, whose equations are

(17) and

(20) y - 4x3/t - 24x4/t2 - 156x5/t3 - (1072 + 32at3)x66/t4

- (7668 + 480at3)x7/t5 + ? - .

The desired results then follow from (18) and (20).

12 CC Per lo studio proiettivo-differenziale delle singolarita," Bollettino della Unione

Matematica Italiana, vol. 5 (1926), p. 118. 13 Su, loc. cit. See also his paper, " On certain twisted cubics projectively con-

nected with a space curve," Journal of the Chinese Mathematical Society, vol. 2

(1937), p. 59.




Dually we choose a point P1 (# 0) on the x-axis as the vertex of cones containing the curves r and Ta.

THEOREM 3. 2. If a 7 - 2q6, these cones have exactly six-plane contact along 7 for all points P1. If a = - 2q6, the cones always have just seven- plane contact, wilh the single exception that the point with coirdinates

(15. 2) (2q7,q6, O,0)

produces cones having eight-plane contact.

The next step would be to consider planes through 0 which do not contain the x-axis. Let II be such a plane, cutting the tangent developables of r and Ta in cusped curves F" anld Taj.

THEOREM 4. 1. The order of contact of F" and T." at 0 is greater for a -5q6/2 than for any other value of a, regardless of the position of the plane H. When a, 5q6/2, the order of contact is increased still further if II contains the line whose equations are

(21. 1) q6y2 -4q7y3 = 0 = Y4,

and is greatest for a uniquely determined position of II, namely

(22. 1) 3q62y2- 12q6q7y3 + (18q6p`7 - 7q6q8 + 20q72)y4 0.

We prefer to prove the dual theorem, and shall state here without proof several additional results of Theorem 4. 1. The osculating cusped cubics at 0 of r" and Taa" coincide if, and only if, a, = 5q6/2, independently of the plane II. For fixed but arbitrary a let Pa" be the osculating cusped cubic of Ta", and let II vary in the bundle of planes through 0. Then the inflexion point of TPa' generates the ruled surface (14) while the inflexion tangent of Ta" forms a congruence with the following properties. The focal sheets comprise an algebraic surface S of the sixth order whose asymptotic curves are twisted cubics; on each line of the congruence the harmonic conjugate of the inflexion point with respect to the two focal points lies in the plane 7r (y4 = 0); the developables of the congruence meet S in twisted cubics and meet 7r in a family of conics whose envelope is the conic K2. When, in particular, the inflexion tangent of TP," passes through a point P2 on K2, then the plane II is the face OP2P3 of the fundamental tetrahedron determined by Ta and P2, while the two focal points on this inflexion tangent coincide at the point P3 of this tetrahedron.

To obtaill the dual theorem, an arbitrary point P in 7r but not on the x-axis is chosen as the vertex of cones containing the curves r and T,,. For simplicity we cut these cones by the plane Y3 = 0, obtaining curves P and Ta. Then,



294 LOUIS GREEN.

THEOREM 4. 2. r and Ta have exactly six-point contact at 0 for all centers of projection P unless a, - q6/2. If ,= -q6/2, they have just seven-point contact unless P lies on the line whose equations are

(21. 2) 3q6y2 2q7y3 = 0 = Y4.

In this latter event the curves r and Ta have precisely eight-point contact at 0 except when P has coordinates

(22. 2) (3q6q8 - 4q72, - 2q6q7, - 3q62, 0),

when they have nine-point contact.

For, let r and Ta be projected upon the plane y3 = 0 from the point P

(1,m,n,0) (n#O).

It can be verified without difficulty that in the plane y3 = 0 the projections of r and of T have the respective equations

z = x3 + 3mx4/n + (9M2 - 2n)x5/n2 + (28M3 _ 13mn + q6n3)x6/n3

+ (90m4 - 64m2n + 5n2 + 6q6mn3 + q7n4)x7/n4 + (297m5 - 285m3n + 51mn2 + 27q6m2 n3

- 5q6n4 + 7q7mn 4 + q8n5)x8/n5 +

z = x3 + 3mx4/n + (9M2 - 2n) x5/n2 + (28M3 - 13mn - 2an 3) x?ln3

+ (90m4 - 64M2n + 5n2 - 15amn3)x7/n4 + (297M5 _ 285m3n + 51mn2 - 8Lxm2n3 + 12an4) x8/n5 +

The results follow immediately from these equations.

The Halphen-Bompiani theorem referred to above states:

THEOREM S3. 1. The locus of points projecting r and the six-point cubic To into cones having at least seven-plane contact is the principal plane of r at 0: (23.1) Y13= 0

If the center of projection lies on the line with equations

(24. 1) 2q6y2 + p7Y4 = 0 - y3,

these cones have at least eight-plane contact along the principal plane, while for a unique point W on this line nine-plane contact is obtained.

Theorem 3. 2 shows that all points on the x-axis, except 0 possibly, must

be excluded from the locus. Further examination indicates that the cones

projecting r and To from 0 have but five-plane contact along Xr, so that 0

must also be excluded. The dual of this theorem is




THEOREM S. 2. All planes 14 passing through the principal point of r at 0:

(23. 2) (7tC7, q6, 0, 0),

intersect the tangent developables of r and of the six-plane cubic Ta(a = 2q6) vt curves having at least seven-point contact. No other planes possess this property. If the plane of section contains the line whose equations are

(24. 2) 2q62yl - 14q6q7y2 + (21q6p7 - 8q6q8 + 42q72)y8 = 0 = Y4,

the curves have at least eight-point contact at the principal point, while for a unique plane Q throtugh this line nine-point contact is' obtained.

The tetrahedra H1 and H2 are characterized geometrically by the following dual theorems.

THEOREM 6. 1. The Halphen point H of r at 0 is the point of intersection, besides 0, of the six-point cubic To and the principal plane of r. Its co6rdinates are (25. 1) (0, 0, 0, );

the equation of the osculating plane ?' to To at this point is

(26. 1) y1 = 0.

THEOREM 6. 2. The Halphen plane ( of r at 0 is that osculating plane, besides 7r, of the six-plane cubic Ta (a-= 2q6) which contains the principal point (23. 2) of P. Its equation is

(25. 2) 27/q6y1 -189q62q7y2 + 441q6q72y3 + (54 q64 343q73)y4 = ;

the coordinates of the contact point H' of this cubic and this plane are

(26. 2) (343q73 - 54q64, 147q6q72, 63q62q7, 27q63).

Furthermore, the lines HH' and ??' are coplanar with the edge ON of

Sannia's tetrahedron,15 and together with the self-dual cubic Tq, serve to characterize this tetrahedron.

The principal plane of ir and of a five-point cubic Ta (` 7 0) is their common osculatinlg plane v-, while dually the principal point of r and of a five-plane cubic Ta (cc , 2q6) is their common point 0. Bompiani's extension of Halphen's theorem was obtained only when the principal plane of the two curves is distinct from the common osculating plane, while Theorems 3. 2 and 4. 2 cover the case where the two planes coincide. The complete results can thus be summarized as follows:

14 With the exception of the planes through the x-axis which must be excluded. 15 See footnote 7.



296 LOUIS GREEN.

The twisted cubic To determines an axis of Bompiani (24. 1) and a point 16 of Bompiani W in the principal plane (23. 1) of r; the cubic Ta (a = - q6/2) determines an axis of Bompiani (21. 2) and a point of

Bompiani (22. 2) in the osculating plane vr, and the cubic Ta (&= - 2q6)

determines a point of Bompiani (15. 2) on the tangent to r at 0. Dually, the cubic Ta (a = 2q6) determines a ray of Bompiani (24. 2) and a plane of Bompiani Q through the principal point (23. 2) of r; the cubic for which a = 5q6/2 determines a ray of Bompiani (21. 1) and a plane of Bompiani (22. 1) throutgh the point 0, and the cubic with a =- 4q6 determines a plane of Bompiami (15. 1) through the tangent to r at 0.

When a five-point cubic Ta is projected upon the plane y3 = 0 from a point P in the plane 7r, the projected curve is a cusped or a nodal cubic according as P is or is not on the conic K2. In either case an inflexion point is obtained at 0. Now, it follows from Bompiani's work 17 that a plane curve with an inflexion point sustains at this point a seven-point cusped cubic and oo1 eight-point cubics (two of which are nodal), but that ordinarily it possesses no eight-point cusped cubic and no nine-point osculating cubic. If the curve does happen to have a nine-point cubic, then every eight-point cubic is a nine-point cubic and there exists a unique ten-point cubic.

Theorem 4. 2 therefore states that the point where the line (21. 2) meets the conic K2, namely (27) (q72, 2q6q7, 3q62, 0),

projects F into a curve in the plane y3 = 0 which sustains an eight-point cusped cubic, and that the point (22. 2) projects r into a curve sustaining a ten-point cubic. The points (22. 2) and (27) are not the only points in the plane v-, however, with these properties. It can be shown, for example, that the locus of all points in v- projecting P into a curve in the plane y3 = 0

which sustains a ten-point cubic is the straight line joining the points (15. 2) and (22.2).

Theorems 3. 1, 4. 1, and 5. 2 are concerned with plane sections of the tangent developables of r and of Ta, and show that for properly chosen cubics Ta there are certain planes which yield curves of section having contact of higher orders than are obtained ordinarily. It is natural, then, to inquire into the nature of the curve of intersectionl of the tangent developables of F

and of Ta. The results are coiitained ill the following theorem:

16 We prefer this terminology to "principal point" since we wish to use the latter for the dual of the principal plane.

17 See footnote 12.




THEOREM 7. 1. The tangent developables of r and of Ts, intersect in the x-axis and in a residual curve C. If a = q6, then C consists of a single branch having four-point contact with r at 0. If a,= q6, then C has four branches three of which are linear, each of the three passing through 0 and having four-point contact with r. If a = 2q6, the fourth branch of C does not pass through 0 but through the princi,pal point (23. 2). Thtis branch is linear and has for tangent and for osculating plane at the pritncipal point the ray of Bompiani (24. 2) and the plane of Bompiani i. If a = 5q6/2, the fourth branch of C has a cusp at 0 wvith the ray of Bompiani (21. 1) as cusp-tangent and the plane of Bompiani (22. 1) as osculating plane. If a 4q6, the fourth branch of C has an inflexion point at 0 with the x-axis as tangent and the plane of Bompriani (15. 1) as os'culatiitg plane. For all other values of a the fourth branch of C is li,near, passes through 0, has two- point contact with r, and has 7r as osculating plane.

To prove this theorem we equate the space coordinates (16) of a point

on the tangent developable of r to the space co,6rdinates (19) of a point on the

tangent developable of Ta, after first replacing the parameters u, v in the

latter set of equations by pA, v. Elimination of v anid v from the three equations

obtained yields the equation 00

= + ? MkUk,

2

where, in particular, m2 [m23-2 ( a q6)] ?0

Three cases thus arise:

(a) m23 2(a -q6) #70;

(b) m 02 ? = q6;

(c) M.O 0, c~a q66

In case (b), m3 3q62/2q7,

while in (c),

m3=0, m4, a(2a + q6)/2(a-q6), m, a(4a -q6)q7/2(a -q6)2,

m6 [a(4a -q6) q72+ (a -q6){(q62 2aq6 8 P2)p7

+ (5 2 2aq6) q8}]/2 (a q6) 3.

We can now express v in terms of u.

(a) v rnM2n 2/2 +

(b) v -q7u2/3q6+

(c) three subeases must be considered:

5



298 LOUIS GREEN.

(C1) Oa 0, ac72q6: v= (a -q6)qu/3(2q6- a)

+ (q( 4x) q7 u2/9 (2q6 )'2 + 0,U3 +

where o, [3 (a 2q6) (5a-2q6) q, -- 9 (o.226) (4a -Y') p7

+ 7 (4a -q6) q72]/27 (2q6 - ) 3;

(c2) a= : v -u/6+ ; (C3) a = 2q6 V = q6/7q7 + (21q6p7 - 8q6q8 7q72)u/49q72 +

Substituting into (16) yields the equations

X u + m2u2/2 +

(a) y U2 + mu3 +**

Z U3+ 3m2U4/2 +

xU-u q7 U2 13q6 + ..

(b) y u2 2q7u3/3q6 + Z U3 q7 u4/q6 + * -

x = (5Sq6 - Ma ) u/3 ( 2qj - () + ( q6 - a ) q7 U2/9 ( 2q6 a ) 2 + UU3 + ***

(el) y=(4q6-2)u2/3 (2q6-) + 2 (q6 4a) q7 u3/9 (2q6- a) 2 + 2uU4 +

z-=q6t3/(2q6 a) + (q6- 4a) q7u4/3 (2q6- 2)2 + 3 o5 +

x 5u/6 + q7u2/36q6 +

(C2) y 2U2/3 + q7U3/18q6 +

Z = 3/2 + q7u4/12q6 + *

X q6/7q7 + (2lq6p'7 - 8q6q8 + 42q72)u/49q72 +

(C3) Y 2q6U/7q7 + (42q6p7 - 16q6q8 + 35q72)u2/49q72 +

z 3q6u12/7q7 + (63q6p7 - 24q6q8 + 28q72) U3/49q72 +

The theorem follows readily from these equations. The dual theorem is

THEOREM 7. 2. The platnes containing both a tangent to r and a tangent to Ta, form an axial pencil through the x-axis and a residual family C' of planes, whose edge of regression is a curve C". If a q6, then C"' consists of a single branch having four-plane contact with r at 0. If a # q6, then C" has four branches three of which are linear, each of the three passing through 0 and having four-plane contact with r. If a = 0, the fourth branch of CZ' passes thr-ouqh the point of Bompiani W of Theoremi 5. 1, and has for tangent and for osculatin'1,g plane at W the axis of Bompiani (24. 1) and the principal plane (23. 1). If a = - q6/2, the fourth branch of C" passes through the point of Bompiani (22. 2) where it has the axis of Bonpiaani (21. 2) as tangent and the plane 7r as singular- osculating plane. If a -2q6, the fourth branch of C" passes through the point of Bompiani (15. 2) where




it has the x-axis as tangent and the plane 7r as osculating plane. For all other values of a the fourth branch of C' is linear, passes through 0 and has two- plane contact with r.

There are certain interesting relations between these two dual theorems, but we shall not consider them here.

6. Osculating quadrics at a point of a curve. The self-dual quadric (10) was found to have both three-point and three-plane contact with r at 0. Although there are oo quadries with this property, there is no noll-singular quadric having both four-point and four-plane contact with r' at 0.

The general six-point quadric of r at 0 has the equation

(28) Mr1 (yly3 - Y2) + m2 (yly4 - y2Y3) + m3 (y32 - y2y4) + m4y42 0,

where the Mn are arbitrary. If this quadric contains the five-point cubic Ta, then

(XM1 = 0 == (Xm2 - M4X

while if the quadric is to have seven-point contact with r at 0, then

m2q6 + m4 0-.

Hence we have the following theorem characterizing the cubic T-q6.

THEOREM 8. 1. There exists a one-parameter family of quadrics having at least six-point contact with r at 0 and containing the five-point cubic Ta, (a 7/ 0). Neglecting the quadric cone K'2, seven-point contact is obtairted if, and only if, a -q,.

The dual theorem, which will be omitted, characterizes the cubic Ta, (a = 3q6). Another characterizatioin of T-q6 is due to Sannia.18 The QQ2

quadrics having seven-point contact vith 1r at 0 have in common just two points - 0 and the residual intersection

(q6, O, O, 1)

of the line OH ("5. 1) with the five-point cubic T-q6. Dually, the Co'2 quadrics having seven-plaane contact with r at 0 having in comnmon just two tangent planes -7r and the plane

27q683y 189 q62q7y2 + 44lq6q72y3 + (81q04-343q78)y4 = 0,

which is coaxial with 7w and ? (25. 2) and osculates the five-plane cubic Ta (a = 3q6).

18 Loc. cit.



300 LOUIS GREEN.

The fundamental tetrahedra are related to the seven-point quadrics according to the following theorem.

THEOREM 9. 1. Each point P2 (#/ 0) on the osculating conic K2 determines a unique seven-point quadric whose intersection wtth 7w is a conic tangent to K2 at 0 and at P2. The tangent plane to the quadric at P2 osculates the six-point cubic T0, the tangent plane to the quadric at 0 is the face OP,P3 common to all fundamental tetrahedra which have P2 for a vertex, and the polar qith respect to the quadric of the vertex P1 of these tetrahedra is the face OP2P3.

If, in particular, the point P2 is chosen at (0, 0, 1, 0), then the quadric reduces to the cone (29) y= x2

with vertex at the Halphen point. The dual theorem states:

THEOREM 9. 2. Each plane 0P2P3 ( w7r) tangent to the osculating quadric cone K',, determines a unique seven-plane quadric whose cone of tangents through, 0 touches K', along 7r and along 0P2P3. The contact point of the quadric with the plane 0P2P3 lies on the six-plane cubic Ta (a = 2q6), the contact point of the quadric with 7r is the vertex P1 common to all fundamental tetrahedra determined by the plane OP2P3, and the pole with respect to the quad'ric of the face OP1P3 of these tetrahedra is the vertex P2.

7. Consecutive configurations. A manifold M geometrically defined for each value of the parameter u of the curve r generates or envelopes another manifold M' as u varies. Several five-point twisted cubics can be characterized in this way.

As u varies, the five-point cubic T,a (oc = const.) generates a surface Sa and the osculating planes of the dual cubic Ta' (a + a' = 2q6) envelope the dual surface Sa'. The tangenit planes to S,a along Ta form a developable D"', while dually the osculating planes of Ta' are tangent to &S along a curve Da'.

Then,

THEOREM 10. 1. Dla is the tangent developable of a twisted cubic except in the following four cases. If oc= - 4q6, Dla is a cubic cone with vertex at the point whose coordinates are

(30. 1) (14q7, 5q6, 0,0).

[f =-2q6/3, Dla is a cubic cone with vertex at a point,

(31. 1) (49q72, 7Oq6q,, 75q62, 0),




lying on the osculating conic K2. If a = 0, Da is the quadric cone (29).19

If t = 6q6, Da is the osculating quadric cone K'2. That is, the characteristic curve at 0 determined by the set of all osculating cones K'2 is the cubic Tc, (a = 6q6) (and the x-axis) ; stated differently, the generators of K'2 form a congruence as the parameter u varies, and the surface S", (= 6q6) is one focal surface.20 The cubics Ta on Sa have an envelope other than r if, and only if, a = 6q6, the contact point on the envelope associated with 0 being at

(343q73 - 750q64, 245 q6q72, 175q62q7, 125q63).

The proof runs as follows. A point P o01 r near 0 determines a canonical tetrahedron HI (P). If local homogeneous co6rdinates referred to this tetrahedron are denoted by Yi, then the equations of the cubic Ta associated with P are (32) Y1=1 - r3 Y2 )= TY3==r2 Y4=r 3.

Now let P have non-homogeneous co6rdinates (h, k, 1) referred to tetrahedron H1 at 0. From (9) and the equation

Y Yyi + (dyi/dx)h+ + we have

py q6y1- (21P7y2 + 12q62y3)h + pY2 q6y2 - (q6y1 - 7q7y2/3 + 14P7Y3 + 6q6 2y4)h +

pY3 q6y3 -(2q6y2 -14a7y3/3 + 7p7y4)h + pY4 q6y4 -(3q6y3 - 7q7y4)h +

Solving for yi and using (32) we obtain

Ay1 q6 (1 - r3) + (21p7i + 12q6 22)h +

(33) AXY2 q67r+ (q6 - 7q77/3 + 14p772 + 6q62-3 aq6T3)h +

Ay3=q67-2 + (2q6T -14q7i2/3 + 7p773)h + -

AY4 =q673 + (3q6r2 -7q7-r3)h+ * -

as parametric equations of the surface Sa. When h 0, = t, the tangent plane to Sa has the form

q6(6q6 )t3y - (12q62 + 3zq6 - 7q7t)t2y2 + (6q62 + 9aq6 -14aq7t)ty

(5aq6 -7aq,t - 6acq62t3 + c22q6t3)y4 ?= 0

19 Newton, loc. cit. Also Tsuboko, " On the locus of the space cubics osculating a

space curve," Memoirs of the Ryojun College of Engineering, vol. 10 (1937), pp. 63-74. 20 WVilczynski. " General projective theory of space curves," Transactions of the

American Mathematical Society, vol. 6 (1905), p. 109. This result has also been

obtained independently by Kanitani and Newtoni, loc. cit. This cubic has been called the torsal cubic of r at 0 by Wilczynski.



302 LOUIS GREEN.

As t varies, this plane envelopes the developable surface Da whose edge of regression has the parametric equations

1/i = - 45aq63 (2q6 + 32) (4q6 + a) + 31 5 2q62(2q + a) q7t - 1470a3q6q72t2 + [6863q7,3+ 94q63(2q6+ 3a) (4q6+ a) (6q6- a)]t3,

y2 45aq63(2q, + 3a) (6q6 -)t + 2102 q62(6q6 -)q7t2 - 98 '2 q6(6q6- a) q72t3,

Y3 45aq63(4q6+ +a) (6q6 - a)t2 + 214q62(4q6+ + ) (6q6 -a)q7t3, Y4= 9q63(2q6 + 3a) (4q, + a) (6q6 -)t

These equations yield all but the last statement of the theorem. To complete

the proof we set

y=l I[ a3, y2 t, y3= t2, y4 =t3 in (33) and fincl

t r+ ( -7q7T/3q6 -7p72/q6)h+ .

t T + (1- 7q7T/3q6 7p7T2 q- 6q63 + aT?)h + t T + ( - 7q7-r/3q6- 7 -7T /q6 6q6T3 - 2aT3 + 7aq7T4/3q6

-l42p7T5/q6 6- q6r6 + a%6)h (1 + 2z3 ) +

The desired result then follows immediately. The dual theorem states:

THEOREMI 10. 2. The curve Da' is a tivsted cubic except in the following four cases. If a' G= q6, Da' is a curve of class three lyintg in the planle dual to the point (30. 1). If a' = 8q6/3, Da' is a curve of class three lytitg in the plane dual to the point (31. 1). If a' = 2q6, Da' is a conic lyintg in the Halphen plane (25. 2). If a' -4q6, Da' is the osculating conic K2 ; that is, the sur-face Sa is the locus of all osculating conics K2; 21 stated differ-enttly, the tangents to K2 form a contgruence (as the paLrameter u varies, and the surface Sa' (x' - 4q6) is one focal surface.

8. Projections of a space curve. When the space curve r and oiie of

its five-point twisted cubics Ta are projected from an arbitrary point P (h, k, 1), 1 # 0, upon the osculatiiig plalne v at 0, the projected curves r', T'i possess

certain interesting relations.

The equations of I' are readily foulnd to be

(34) y= - Icx3/l + 2hx4/l (3hk + l)X5/12

+ (7h2 + 21k q6kl)x6/11 + O=

while those of T'c, are

21 Tsuboko, " On the locus of the conlies osculating a space curve," Memoirs of the Ryojun College of Engineering, vol. 10 (1937), pp. 11-17.



TWITISTED CUBICS ASSOCIATED WITH A SPACE CURVE. 303

(35) y =X2- kx3/l + 2hx4/l- (3hk + ? + c12)x5/12

+ (7h2 + 2k+ 2kl)x612+ , z ==0.

These curves always have at least five-point colntact, and hence have at 0 a

common osculating conic K. Higher-order contact is obtained only when

a = 0, as seen from (34) and (35) or from the theorem that the principal

plane of IF ancl Ta (a # 0) is the osculating plane

The equations of K are

(36) y= %2-1.ry/l+ (2h1- 2)y- 2 2 z

This coniC has six-point contact with IF if, andl only if,

(37) /1p 12 + 2o3 -3hkl 0,

andl has six-poilnt contact with TWa if, and only if,

(38) 2=I1+ 13_O,

i. e. if, acd olily if, the center of projectioii P lies oli the cubic surface (14).

If P traces a line L through 0, the conic K remailns unchanged. More-

over, we can readily prove

THEOREM It. The co0nics K and K2 have double comttact if, and ontly if, the cenler of projection _P lies ont the quadric cone K'2. If this is the caise, the line OP and the chor-d of double contac/ of the comtics are edges of a common funtdamental tetrahedron.

We shall be concerned in this section with the projective normal and the

flex-rav of r' at 0,22 and in order that these be well-defined we must have a

nion-composite osculatilng inodal cubic of IF' at 0, which means that we must

assume that /i, #Z 0. This assumption furthermore prevents the center of

projection from lying on the six-point cubic To.

The osculating nodal cubic of IF' at 0 can be shown to have the equations

13,Llxy + l2ViY2 - 131LiX3 + 12(IkAJi - v_)x2y + l(Ik2/11 - thlpli + kv1)Xy2

+ ( I2 + k2Yv -2hlvl)y3 = 0-z where I,u is given by (37) alnd

v 8h1c21 h212 2l, 12 a _-_ q6kl3

Helice the projective normal of IF' at 0 is expressed by

(39) 1/11x + vly O = z

wh-ile the flex-ray is seen to be

22 The flex-ray of 1' at 0 is defined as the line of inflexions of the osculating nodal cubic of F' at 0.



304 LOUIS GREEN.

(40 ) 12 i2 + ljl (2vi + lj1p) x + (Jk2/J2 - 2hlj 2 + v2 ) y = 0 = z.

In the same way the projective normal at 0 of T'a is of the form

(41) l2x- + v2y 0=z,

where /L2 is given by (38), and

V2 = 8hk21 - h212 _ 21g12 - _ 2kIl3

while the flex-rav of T'a at 0 has the equations

(42) 12I22 + 1/2(101L2 I- 2V2)X + (k2 /2 2 2hl122 + V22)y 0 = z.

THEOREM 12. As the center of projection P traces a line L through 0, the projective normal of r' varies in the pencil at 0 unless L is a generator of the quartic cone (43) (y2_ X-z)2 + q6yZ3 = 0.

Neglecting the x-axis which must be excluded, this cone' meets the quadric cone K'2 in the z-axis. As P traces the z-axis, the projective norrmal of r' at 0 remains coincident with the y-axis.

As P traces a line L through 0, the projective normal of T',a varies unless L is a generator of K'2. As P traces a generator OP3 of K'2, the projective normal of T'a at 0 coincides wuith the edge OP2 of the fundamental tetrahedrct determined by OP3.

The proof offers nio difficulties. In (39) and in (41) we replace (h, 7c, 1) by homogenieous co6ordinates (h,, - - *, h4) and demand that the resulting equation be independent of h1. In both eases we obtain

h4x -2h3y = 0 z, ,so that

2lh3t + h4vi 0 (i 1, 2).

Replacing /iu and vi by their values we have the results immediately. The locus of all centers of projection P determining curves r' which

have at 0 a fixed projective normal, say

(44) x+ my=- z,

is a fourth-order surface S*r determined by the condition

(45 ) I/]1nM - vl O.

The only five-point twisted cubic Ta which lies on this surface is the one for which a = q6/2, this situation occurring only when the given pro jective normal of r' at 0 is the y-axis.




If the projective normal of T,a is chosen as the same line (44), then P generates another fourth-order surface S*a determined by the condition

(46) JL2M - V2 - 0

The surfaces S* r and S*,a coincide if, and only if, both a = q6/2 and (44) is the y-axis. If a =a q/2 but (44) is not the y-axis, these surfaces meet only in the plane 7r so that there is no center of projection yielding coincident projective normals at 0 for r' and T'a. If a 7/ q6/2 and (44) is the y-axis, the surfaces meet in the z-axis. In all other cases the surfaces intersect in a non-composite conic.

From (45) and (46) we obtain

(47) 1 (q6 - 2a) k/aM,

and upon substituting into (45) we find as the equations of the conic

(q6- 2a)y- acMz 0, q6 (q6 -2 a) 3,11Z- (q6 - 2 ) 4X2 _m2 (q6 -2c)2(3q6 + 2c)xz

+ cM[c2M3 (2q6 + c) + q6 (q6_ 2a) 3] Z2 0.

As m varies, a remaining fixed, these conics generate a surface whose equation is (48) q6yz2 + aX2Z2_ (3q6 + 2a)Xy2Z + (2q6 + a)y4 (a #).

-i- cq6yZ3 =_ 0

This surface is composite if a = - 2q6, and degenerates when e= 0 into the

plane y- 0, as is evident from (47). The flex-ray of r' at 0 is tangent to the osculating conic K of r' at the

point of intersection of flex-ray and projective normal. A similar statement holds for Tla. Hence, as P traces a line L through 0 the envelope of the flex-rays at 0 of the curves r' is the conic K unless L lies on the cone (43). The following theorem can be readily proved.

THEOREM 13. Let the center of projection P trace a five-point cubic Ta (aC 7/ 0). If a = 2q6/3, the flex-rays at 0 of the curves r' form a pencil through the point (0, 1, O, 0). If a = q6/3, or if a = q6, the flex-rays all pass through the point (0, O, 1, 0). In all other cases the flex-rays at 0 of r' envelope a non-degenerate conic. This conic has three-point contact with K2

at 0 if, and only if, a = 4q6/3.

If P traces a five-point cubic Tfl, the flex-ray at 0 of T'a (oc # /=) envelopes the conic K2 for all values of o, ,8.

Let the flex-ray at 0 of P' be a given line, say

(49) rx + sy + 1=- z.



306 LOUIS GREEN.

Then from (40) we find that the center of projection P lies on a space curve Cr which is the intersection of a quadric cone

(a0) 8h1- .5k2 + 2rkl- (r2- 4s)12 = 0

ancl a quadric surface (1+ 50q61k 0,

where

75X k + 25r17 + 42h2 + 13lr-hk + (r2- 4s)hl4 + (12r2 + 160s)1k2

(26r3 - lO14rs) kI.

Hence Cr has a node at 0 with the x-axis as one tangent. If the residual tangent at 0 lies on the coime K', then the flex-ray (49) is the edge P1P2

of the fundamental tetrahedra determined by the residual tangent, and con-

versely. Ordinarily C. is a quartic curve, but under the condition

(r2 -3s)2 -12q6r= 0

it consists of a generator of the cone (43) and a twisted cubic. If the flex-ray at 0 of T'"a is choseii as the same line (49), then the

center of projectioni P lies on a curve Ca which is the intersection of the cone (50) and a quadric surface

? + 75kl + 25a,l2 = 0.

The curves Cp and C,a coincide if, and onty if, both a c 2q6/3 and the flex- iay (49) passes through the point (0, 1, 0, 0).

If o = 2q6/3 but the flex-ray does not pass through (0, 1, 0, 0), there are 11o centers of projection P yielding coincident flex-rays at 0 for r' and T'V. If a # 2q6/3 ancl the flex-ray is the line y - y= 0, then the locus of P is

the z-axis. If a # 2q6/3 and the flex-ray passes through (0, 1, 0, 0) but not through (0, 0, 1, 0), then there are no points P. In all other cases there is a unique center of projection P which determines coincidcent flex-rays at 0 for P' and T'Ia, the locus of these points P being the surface (48).

Results of interest can also be obtainied by studying other elements associated with r' and TQa, such as the focal point on the projective normal 23

or on the flex-ray, the Halphen point, the condition for a coincidence point at 0, etc.

INDIANA TJNIVERSITY,

BLOOMINGTON, INDIANA.

23 Tsuboko, "Sur la courbure projective d'une courbe," Memoirs of the Ryojun

College of Engineering, Iiiouye Commemoration Volume (1934), pp. 59-74.



twisted cubics associated with a space curve

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