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 ,ANE CUBICS WITH A GIVEN QUADRANGLE 
 OF INFLEXIONS 
 
 A DISSERTATION 
 
 Presented to the Faculty of Bryn Mawr College in Partial 
 
 Fulfillment of the Requirements for the Degree of 
 
 Doctor of Philosophy 
 
 JiY 
 
 BIRD MARGARET TURNER 
 
 1920 
 
 Reprinted from 
 
 American Journal of Mathematics 
 
 Vol. XLIV, No. 4, October, 1922 
 
<v 
 
PLANE CUBICS WITH A GIVEN QUADRANGLE 
 
 OF INFLEXIONS 
 
 A DISSERTATION 
 
 Presented to the Faculty of Bryn Mawr College in Partial 
 
 Fulfillment of the Requirements for the Degree of 
 
 Doctor of Philosophy 
 
 by 
 BIRD MARGARET TURNER 
 
 1920 
 
 Reprinted from 
 
 American Journal of Mathematics 
 
 Vol. XLIV, No. 4, October, 1922 
 
,e- 
 
 V 
 
 Reprinted from The American Journal of Mathematics, Vol. XLI,V, No. 4, October, 192;2 
 
 PLANE CUBICS WITH A GIVEN QUADRANGLE OF INFLEXIONS. 
 
 By B. M. Turner. 
 
 That every non-singular cubic has nine points of inflexion, lying in 
 related positions on the curve, is a classical fact in mathematics. Of these 
 points four may be chosen arbitrarily; and when such a quadrangle is 
 fixed, the finding of the positions of the remaining five presents a question 
 worthy of consideration. It appears that all the sets of five combine into 
 a group of fifteen points whose relative positions with respect to the given 
 four depend upon equianharmonic properties ; but that the equianharmonic 
 relations follow as a consequence of a combination of harmonic relations 
 and hence, in a number of cases, the points may be determined by linear 
 and quadratic constructions.* 
 
 It is also well known that four points of inflexion, no three collinear, 
 impose eight conditions on a cubic and determine it as one of a singly 
 infinite system; but, since only four of the conditions are linear while the 
 other four are of the third degree, the system is not a pencil. It will be 
 shown that the four points determine a system consisting of six pencils 
 and that every two of the six have a fifth point of inflexion in common, 
 that is, through every one of the fifteen points two of the pencils pass, and 
 have consequently an inflexion. 
 
 I. Determination and Construction of the Remaining Five Inflexions. 
 
 Four points of inflexion of a cubic may be chosen arbitrarily and the 
 conditions imposed by any one of the four are then independent of the 
 conditions imposed by the other three. For a real cubic, however, the 
 imaginary points of inflexion occur in conjugate pairs; hence for such a 
 cubic, if no more than four of the inflexions are involved in the selection, 
 the chosen quadrangle must consist of (1) two pairs of imaginary points, 
 or (2) two real points and one imaginary pair. As a system consisting 
 entirely of imaginary cubics is in itself of little interest, only these two 
 quadrangles supplemented for symmetry by a third with four real vertices 
 will be considered in this discussion. The results for the three cases can 
 be stated in identical terms, as shown in the theorems given in § 3 (pp. 
 277-8). 
 
 * That the whole set of nine points depends only on quadratic constructions was 
 virtually shown by Mobius (Gesammelte Werke, I, p. 437) in the determination of two 
 quadrangles in- and circumscribed to one another. The eight vertices with the addition 
 of the one common diagonal point form the desired set. 
 
 261 
 
 520137 
 
'262''. • / • • :] .Turner: Plane Cubics with Inflexions. 
 
 j'.l.VGiiyE&.'QllABR ANGLE — TWO PAIRS OF IMAGINARY POINTS. 
 
 Let the two pairs of imaginary points to be taken as points of inflexion 
 for a cubic be given as the intersections of two real lines with a conic (Fig. 1). 
 
 Fig. 1. 
 
 Then, as is known, the pencil of conies through the four points has a real 
 common self-polar triangle with A, the intersection of the two given lines, 
 as a vertex and BC, the polar line of A with respect to the given conic, as 
 a side. It is further known that the remaining vertices B, C may be 
 determined by a quadratic construction and hence, since every quadratic 
 construction can be performed by means of the one given conic, the self- 
 polar triangle may be constructed geometrically.* 
 
 As the relations to be noted are invariant under projection and any 
 four points forming a quadrangle may be projected into the vertices of any 
 other quadrangle, there is no loss of generality in choosing the points as 
 (i, ± 1, ± l).f Then the self-polar triangle is the triangle of reference, 
 y ± 2 = are the given lines, and the conic is one of the pencil 
 
 ax 2 + by 2 + cz 2 = 
 where — a + 6 + c = 0. 
 
 (i) Determination of the Five Points. 
 Since each component of the three pairs of sides of the quadrangle 
 y *-z 2 = 0, z 2 + x 2 = 0, x 2 + y 2 = 0, 
 
 passes through two of the given points, the six lines are inflexional axesj 
 
 * Two pairs of imaginary points can of course be given by two conies; but if so the 
 determination of the self-polar triangle, also of the real line pair, cannot be accomplished 
 by quadratic constructions. , 
 
 t Throughout this article the symbol » is used for V — 1 taken positively. In the 
 cases where an ambiguity enters, it is always preceded by the double sign. 
 
 X Inflexional axis: a line through three points of inflexion. 
 
Turner: Plane Cubics with Inflexions. 263 
 
 for every cubic inflected at the four points. For a real cubic it is known 
 that the real sides of the quadrangle determined by two pairs of its imaginary 
 points of inflexion are sides of the real inflexional triangle, the intersection 
 of the lines forming one pair of imaginary sides of the quadrangle is. a point 
 of inflexion, and the intersection of the lines forming the other pair of 
 imaginary sides is the point common to the three real harmonic polars. 
 Hence a real cubic with inflexions at the given four points has the lines 
 through A as sides of the real inflexional triangle, and a fifth inflexion lies 
 either at B or C. 
 
 If the fifth point of inflexion is at C, the third side of the real inflexional 
 triangle passes through this point and has an equation of the form 
 x + ay — 0, where a is a real number still to be determined. The point 
 B is common to the three real harmonic polars; and the "line of reals,"* 
 being the polar of B with respect to the triangle 
 
 y =b z = 0, x + ay = 0, 
 
 is x + Say = 0. Hence since the inflexional axes concurrent with the real 
 harmonic polars (z ± ix = 0) together with the line of reals form a second 
 inflexional triangle, the desired cubic is represented by 
 
 (z 2 + x 2 )(x + day) + X(*/ 2 - z 2 )(x + ay) = 0; 
 
 and the four remaining points of inflexion, being given by the intersections 
 of x + Say = with y 2 — z 2 = and of x + ay = with z 2 + ar 8 = 0, are 
 
 (3a, — 1, =b 1), (a, — 1, ± ia). 
 
 These nine points, namely the four given points, the point C, and the 
 four just found, are points of inflexion for a cubic if, and only if, they 
 satisfy the conditions of collineari'ty represented by the rows, columns, 
 three right- and three left-hand diagonals of the following scheme :f 
 
 (3a, - 1, - 1)> , ( i, 1, 1), (i, - 1, - 1), 
 
 .( *, 1, - 1), (3a, - 1, 1), (*, - 1, 1), 
 
 ( a, — 1, — ia), ( a, — 1, ia), (0, 0, 1). 
 
 This requires that 3a 2 = 1. Accordingly the four points are either 
 
 (V3, - 1, ± 1), (1, - V3, ± t) or (- V3, - 1, ± 1), (1, V3, ± i); 
 
 and the cubic is a member of one jof the two pencils: 
 
 (1) 
 
 (z 2 + z 2 ) (x + V3y) + \(y 2 - z 2 ) (x + ~. y \ - 0, 
 
 * The line through the three real points of inflexion. 
 
 t Hesse, Crelle's Journal (1849), Vol. 38, p. 257; also Clebsch, "Vorlesungen iiber 
 Geometrie," p. 506. 
 
264 Turner: Plane Cubics with Inflexions. 
 
 (2) (z 2 + .r 2 )(z - a/32/) + \{y 2 - z 2 ) (x - ±y\ - 0. 
 
 Similarly if a cubic have 5 as a fifth point of inflexion, the remaining 
 four inflexions are either 
 
 (V3, ± 1, - 1), (1, =fc i, - V3) or (- V§, ± 1, - 1), (1, ± i, V§); 
 and the corresponding pencils have equations 
 
 (3) (y 2 - z 2 )(z + <tix) + \(x*+y 2 )(z + l=x\= 0, 
 
 (4) (f - z*)(z - V3z) + X(a* + y 2 ) (z - ^x^ = 0. 
 
 For symmetry, A is considered as a fifth point of inflexion, and the set 
 of nine points is completed by either 
 
 (± 1, V3, i), (± 1, i, - V3) or (± 1, - V3, i), (± 1, i, V3); 
 but the pencils 
 
 (5) (*» + y 2 )(2/ + W3 2 ) + X(z 2 + x 2 ) (y + ±z\ = 0, 
 
 (6) (z 2 + 2/ 2 )(2/ - i V3 2 ) + X( 2 2 + *l{v - ~«) - 0, 
 
 are imaginary. 
 
 These results may be stated in the form of the theorem: 
 
 (1) If two pairs of imaginary points of inflexion of a plane cubic are 
 fixed, a real fifth point of inflexion is fixed as one of three, and the 
 complete group of nine is determined as one of six; 
 or in other words 
 
 (1') Two pairs of imaginary points of inflexion determine a system 
 of cubics consisting of six syzygetic pencils, four real and two imaginary; 
 and every two of the six have a fifth point of inflexion in common. 
 
 (ii) Relative Positions of the Inflexions. 
 
 The curves of the system of cubics have in all nineteen points of inflexion : 
 namely, the four common to all the curves and fifteen others of which every 
 one is common to two of the six pencils. When the four common points 
 are (i, ± 1, db 1), the fifteen are 
 
 ( 1, 0, 0), ( 0, 1, 0), (0, 0, 1), 
 ^ (V3, ±1,± 1), (db 1, V3, ± t), (d= 1, db i, V3). 
 
 These values show that the points determine certain quadranges. The 
 
 lines 
 
 y zk z = 0, zzt ix = 0, x d= iy = 
 
Tuener: Plane Cubics with Inflexions. 265 
 
 are the sides of the chosen quadrangle; while the sides of the determined 
 quadrangles are 
 
 y± 2 = 0, z±-pz=0, x±V§y = 0; 
 V3 
 
 y =fc i V3z = 0, z ± ix = 0, Z db -=» = 0: 
 
 V3 
 
 y zh — =2 = 0, 8 db V&c = 0, a; ± iy = ; 
 
 V3 
 
 Hence follows the theorem : 
 
 (2) The remaining points of inflexion of the cubics with two common 
 pairs of imaginary inflexions form a group of fifteen, consisting of the 
 three diagonal points of the common quadrangle and the vertices of 
 three other quadrangles with the same diagonal points, each of the three 
 new quadrangles having one pair of sides in common with the original 
 quadrangle. 
 The sides of the quadrangles and their common diagonal triangle are 
 further related. The six real lines through C considered as three pairs 
 
 x = 0, y = 0; x + V3y = 0, x =y = 0; x — V3w =0, x+ -= y = 
 
 V3 V3 
 
 form a pencil in elliptic involution with 
 
 x ± iy = 
 
 as double lines. The lines of any one of the three pairs are harmonic with 
 respect to the first lines of the other two pairs, and also with respect to the 
 last lines. Furthermore either triad of lines 
 
 x = 0, x zb V&/ =0 or y = 0, x±—y=0, 
 
 V3 
 
 together with either double line, forms an equianharmonic system.* Thus 
 the lines 
 
 x± J3y = 0, x±~y = 
 V3 
 
 satisfy a combination of harmonic and equianharmonic relations with 
 respect to x — 0, y = 0, x d= iy = 0. The equianharmonic properties, 
 however, are simply a consequence of harmonic properties; for if we write 
 
 x = 0, y = 0; x + Py = 0, x — ay = 0; x — (3y = 0, x + ay — 0, 
 
 * Consider x = 0, x * >/3y = 0, x + iy = 0. Let X = x + V%, Y = x — V3?/; 
 then x = 0, x + iy = are transformed respectively into X + Y = 0, X — «F = 0; 
 and the cross-ratio of the four lines X = 0, 7 = 0, Z + Y = 0, Z — uY = is — w 2 , 
 where w 3 = 1. Similarly for the other combinations. 
 
266 Turner : Plane Cubics ivith Inflexions. 
 
 and impose the condition that x = 0, x + (3y = be harmonic with respect 
 to x — fiy = 0, x + ay = 0, we find /3 = 3a ; and the condition that 
 x + 3a?/ = 0, x — ay = be harmonic with respect to x ± iy = shows 
 that 3a 2 = 1. 
 
 Similar statements hold for the lines through B; and also for the set 
 through A, except that in this case the involution is hyperbolic. Con- 
 sequently the sides of the three quadrangles, and hence the vertices, are 
 uniquely determined from the sides of the base quadrangle and its diagonal 
 triangle by means of harmonic properties. 
 
 The equianharmonic properties furnish an analytic means of deter- 
 mining the points, and also serve to show the relative positions of the three 
 quadrangles with respect to the four given points. The points equianhar- 
 monic to (1, 0, 0) with respect to (i, 1, 1), (i, — 1, 1) are (V3, 1, 1) and 
 (3, — 1, — 1); and those equianharmonic with respect to (i, — 1, 1), 
 (i, 1, — 1) are (V3, — 1, 1), (V3, 1, — 1). Thus the vertices of the real 
 quadrangle (\3, ±1, db 1) are the points on the lines y ± z — equi- 
 anharmonic to (1, 0, 0) with respect to (i, dz 1, db 1). Similarly (dz 1, 
 V3, dz i) are the points on z dz ix = and (1, dz i, V3) the points on 
 x dz iy = equianharmonic to (0, 1, 0) and (0, 0, 1) with respect to the four 
 given points. These results are expressed in the following theorem : 
 
 (3) The fifteen other possible points of inflexion of a cubic with 
 two fixed imaginary pairs are the three diagonal points of the fixed 
 quadrangle, and the two points on every one of the six sides of the 
 quadrangle equianharmonic to the diagonal point with respect to the 
 two fixed points on that side. 
 The vertices of the three determined quadrangles may also be obtained 
 analytically as the intersections of three conies. Three pairs of lines 
 
 y dz iz = 0, zd=£ = 0, xdb y = 
 
 are uniquely determined as being harmonic both with respect to the sides 
 of the given quadrangle (i, d= 1, d= 1) and with respect to the sides of the 
 diagonal triangle. Three conies having respectively these pairs of lines as 
 tangents, namely, 
 
 2x* - y % - z 2 = 0, x 2 - 2y 2 - z 2 = 0, x 2 - y 2 - 2z 2 = 
 
 pass, taken in the same order, through the pairs of quadrangles 
 
 (d=l, V3, ±i), (± 1, =fct, V3); 
 (dz 1, ±h V3), (V3, d= 1, d= 1); 
 (V3, dr 1, d=l), (dzl, V3, d=i). 
 
 Further any one of the three conies passes through the four intersections 
 
Turner: Plane Cubics with Inflexions. 267 
 
 of the pairs of tangents given for the other two. Thus any one of these 
 conies is uniquely determined by four points and two tangents, the 
 determining elements being fixed by means of harmonic properties with 
 respect to the given quadrangle. In turn the three conies uniquely deter- 
 mine the three quadrangles 
 
 (V3, ±1, ±1), (±1, V3, ±i), (±1, ±i, V3). 
 
 Accordingly the fifteen other possible points of inflexion of a cubic with 
 two fixed imaginary pairs are the three diagonal points of the fixed quad- 
 rangle, and the twelve intersections of three conies uniquely determined 
 analytically by the quadrangle. 
 
 (iii) Actual Construction of the Points. 
 
 It has been noted that when two pairs of imaginary points are given 
 as the intersections of two real lines with a conic, there follows a quadratic 
 construction for the triangle self-polar for the pencil of conies through the 
 four points.* It will now be shown that with the help of the triangle, the 
 fifteen other possible points of inflexion of a cubic inflected at the two pairs 
 of imaginary points may be determined by a series of constructions of which 
 one only is quadratic, the rest linear. 
 
 As A is the intersection of the two given lines, the given conic meets 
 BC certainly and either AB or CA in real points. Let it meet CA and call 
 the points D, D'. Denote the intersections of BC with the given lines 
 by E, E'. 
 
 The given points being («, =fc 1, ± 1), the object is to construct the lines 
 
 3+V3y = 0, «±iw = 0, z±V3z = 0, z±-Lx=0. 
 V3 a/3 
 
 The conic of the pencil through the four given points that meets y = 
 on the lines z ± V3x = is 3a; 2 + by 2 — z 2 = 0, with the condition that 
 — 3+6—1 = 0, that is, the conic 
 
 3z 2 + 4y 2 - z 2 = 0. 
 
 This conic meets x = where 4y — z = 0, hence the first step requires the 
 construction of the points of intersection of the lines 2y — z = 0, 2y + z = 
 with BC. Since 
 
 2y-z=0, z=0; y - z = 0, y = 0, 
 2y + z=0, z=0; ^+2=0, y=0 
 
 are two sets of harmonic lines, these points Pi, P 2 may be constructed 
 *.See page 262. 
 
268 Turner: Plane Cubics with Inflexions. 
 
 linearly. Draw PJ) intersecting the given conic in a second point D" 
 and the given lines in F, F'. As the conies of a pencil cut any line in 
 involution, P 3 the conjugate to Pi in the involution (D"D, FF') is another 
 point on the conic 
 
 Sx 2 + 4i/ 2 - z 2 = 0. 
 
 This conic is a member of a second pencil through two points Pi (APi 
 being a tangent at a given point), the point P 2 ,* and the point P 3 . The 
 pairs of lines 
 
 AP lt P2P3; P1P2, P1P3 
 
 are two other conies of the second pencil: hence this pencil cuts out on CA 
 the involution (AV, CD), where V is the intersection of P2P3 with CA. 
 The involution cut out on CA by the first pencil (the pencil through the 
 four given points) is defined by its two double points, and may be expressed 
 as (A 2 , C 2 ). It follows that Q, Q', the common points of the two involutions 
 
 (AV, CD), (A 2 , C 2 ), 
 
 found by means of the given conic (the one quadratic construction), are the 
 intersections of Sx 2 -{- 4y 2 — z 2 = with y = 0. Hence BQ, BQ' are the 
 desired lines 
 
 db V3z = 0. 
 
 Since 
 
 z - -l=.x = 0, z + V3z = 0; z = 0, z - V3x = 0, 
 V3 
 
 z + -^ F .x = 0, z-V3z=0; z=0, z + V&r = 
 V3 
 
 are two sets of harmonic lines, two other of the desired lines, 
 
 2±ia; = 0, 
 V3 
 
 may be constructed linearly: call them BK, BK'. The lines that join 
 the intersections of z db V3x = 0, 2 zfc —<=- x = with y ± 2 = to C are 
 
 x±^y= 0, x±^=y = 0. 
 V3 
 
 Hence the complete construction (Fig. 2) may be stated as follows: 
 
 Construct Pi, P2 the harmonic conjugates of B with respect to 
 E, C and E', C. Draw PJ) intersecting the given conic in a second 
 
 * The conic 3x 2 + 4y 2 — z 2 = also has AP t for a tangent, but the use of two points 
 Pi and two points P» would give an illusory construction. 
 
Turner: Plane Cubics with Inflexions. 
 
 269 
 
 point D" and the line-pair in F, F'. Construct P 3 the conjugate to 
 Pi in the involution {D"D, FF'). Draw P 2 P 3 intersecting CA in V. 
 Determine Q, Q' the common points * for the two involutions (AV 
 
 Fig. 2. 
 
 CD), (A 2 , C 2 ). Construct K, K' the harmonic conjugates of Q' with 
 respect to A, Q and of Q with respect to A, Q'. Draw BK, BK', BQ, 
 BQ' intersecting the given line-pair in a, a; (3, (3* '; y, y'; 5, 5'. Draw 
 a&', a/3, 75', y'd. 
 The seven real points of inflexion are 
 
 A, B, C, a, a', 13, /3 r . 
 
 That Q, Q' are real is shown by the analytical discussion. 
 
270 
 
 Turner: Plane Cvbics with Inflexions. 
 
 The eight imaginary points lie by pairs on the lines 
 
 77', 55', y 6', y'8, 
 
 where they are met by the two pairs of imaginary sides of the given 
 quadrangle. 
 
 (iv) Symmetrical Constructions. 
 When the two pairs of imaginary inflexions are given as the inter- 
 sections of two equal hyperbolas with the same pair of axes, the construction 
 (Fig. 3) of the fifteen points is unique and furnishes an illustration of the 
 three conies which intersect in the vertices of the determined quadrangles. 
 
 — *B 
 
 Fig. 3. 
 
 The hypothesis gives the axes and line at infinity, that is, the self-polar 
 triangle; and (keeping the lettering of the preceding construction) the point 
 A is the common center of the two hyperbolas. Draw the four other lines 
 joining the vertices of the hyperbolas; then AE, AE', the real line-pair 
 through the four given points,* bisect these lines. The lines APi, AP 2 
 
 * The equality of the hyperbolas accounts for the construction of these lines, in general 
 not possible by quadratic construction when the two pairs of imaginary points are given 
 by two conies. — See note, page 5. 
 
Turner: Plane Cubics with Inflexions. 271 
 
 are harmonic to AB with respect to AE, CA and AE ' , CA ; and the points 
 Q, Q' are determined as the vertices of the hyperbola through the four given 
 points having AP\, AP 2 as asymptotes.* Then the lines through Q, Q' 
 parallel to AE together with the lines joining their intersections with the 
 real line-pair are 77', 55', yb' , y f 5; and by means of the harmonic relations 
 between these lines and the axes AB, CA the lines aa, /3/3', a/3', a'/3 may 
 be constructed. 
 
 The above construction determines the vertices of the three quadrangles 
 as the intersections of lines. To show them as the intersections of conies, 
 let the two given hyperbolas be members of the pencil ax 2 + by 2 + cz 2 = 0, 
 where — a -{- b -{- c = 0, when (1) x = is the line at infinity and (2) 
 y = 0, 2=0 and y -\- z = 0, y — 2=0 are two pairs of perpendicular 
 lines. Then the three conies intersecting in the vertices of the determined 
 quadrangles are two equal, symmetrically placed ellipses, 
 
 2y 2 + 2 2 = x 2 , y 1 + 22 2 = x 2 , 
 and the circle 
 
 y 2 + 2 2 = 2.r 2 , 
 
 all three concentric with the hyperbolas. 
 
 Accordingly construct the two equal, symmetrically placed ellipses 
 through a, a , (3, /3'; and pass a circle through the four finite intersections 
 of the tangents to the ellipses at their vertices. Then the fifteen other 
 possible points of inflexion of a cubic inflected at the four imaginary inter- 
 sections of the two hyperbolas are the common center, the two points at 
 infinity on the axes, the four real intersections of the two ellipses, and the 
 eight imaginary intersections of the circle with the ellipses. 
 
 Another symmetrical construction is obtained by projecting one pair 
 of the given points into the circular points. Then, the pencil of conies 
 through the two pairs of imaginary points is a system of coaxial circles, the 
 real line-pair consists of the radical axis and the line at infinity, and two 
 vertices of the self-polar triangle are the limiting points of the system while 
 the third vertex is at infinity on the radical axis. A pair of circles, each 
 having one limiting point as a center and passing through the other limiting 
 point, intersect on the radical axis in two vertices of the quadrangle of real 
 points. A second pair of circles, having these vertices as centers and 
 passing through the limiting points, determine four other points (2) .on the 
 first pair. The lines joining these four points to the limiting points pass 
 through the remaining possible points of inflexion of cubics inflected at the 
 
 * Draw a line parallel to APi intersecting the real line-pair and the hyperbola with 
 AB as transverse axis. The center of the involution determined by the two pairs of points 
 of intersection is a point on the hyperbola having AP X and APi as asymptotes; and it is 
 known that a hyperbola can be constructed when the asymptotes and one point on the 
 curve are given. 
 
272 
 
 Turner: Plane Cubics with Inflexions. 
 
 four given imaginary points. This gives a unique construction when the 
 two pairs of imaginary points are taken as the intersections of two circles. 
 See (Fig. 4), where to complete the symmetry the two given circles are 
 drawn equal. 
 
 y 
 
 Fig. 4. 
 
 For the proof of the construction project (db i, 1, 1) into the circular 
 points and change from homogeneous to Cartesian coordinates. The 
 equation of the system of coaxial circles is then 
 
 x 2 + y> - 2X2/ + X = 0, 
 
 with 2y — 1 = as the radical axis and (0, 0), (0, 1) as the limiting points. 
 The two circles each having one limiting point as center and passing through 
 the other are 
 
 x 2 +y*= 1, z 2 + (y- 1) 2 = 1; 
 
 and these circles intersect on the radical axis in (± £ V3, £), or (zfc V3, 1, 
 — 1) in the homogeneous coordinates given by z = y — 1. The second 
 pair of circles having these two points as centers and passing through the 
 limiting points, namely, 
 
 (z-£V3) 2 + {y-\f= 1, (* + £V3) 2 + (y-h) 2 = h 
 
Turner: Plane Cubics with Inflexions. 273 
 
 determine on the first pair the points 
 
 (jV3,-i), (-*V3, -*), (|V3,|), (- *V3, I), 
 or 
 (1/3, -1,-3), (- V3, - 1, -3), (V3,3, 1), (-V3, 3, 1); 
 
 and the lines joining these four points to the limiting points are 
 
 a: db V3w = 0, x±-=y = 0, z ± V3s = 0, z±- ? =.x= 0. 
 V3 V3 
 
 § 2. Given Quadrangle — Two Real and a Pair of Imaginary Points. 
 
 Let four points, two real and one imaginary pair, to be taken as points 
 of inflexion for a cubic be determined geometrically as the intersections of 
 two real lines with a conic (Fig. 5). It is then known that the common 
 
 Fig. 5. 
 
 self-polar triangle for the pencil of conies through the four points has one 
 real vertex, the intersection of the two given lines, and one real side, the 
 polar of the real vertex with respect to the given conic; while the two 
 remaining vertices and sides of the triangle are imaginary. 
 
 The study of the cubics with two real and a pair of imaginary inflexions 
 fixed is correlated with the preceding study of the cubics with two fixed 
 pairs of imaginary inflexions, by choosing the four points as (V3, 1, ± 1), 
 (1, V3, ± * 
 
 (i) Determination of the Five Points. 
 
 The procedure followed in the case of the two pairs of imaginary points 
 shows that the cubics with the four given inflexions have six common 
 inflexional axes, namely, the three pairs of side of the given quadrangle, 
 
 x- V&/ = 0, x =y = 0, 
 
 V3 
 
 ico 2 x + coy'rb z = 0, — icox + u> 2 y ± z = 0, (co 3 = 1). 
 
 Also as before a fifth point of inflexion is one of three: the real point A 
 (0, 0, 1) or either of the pair. of imaginary points B (i, co, 0), C (—i, co 2 , 0). 
 * See page 263. 
 
274 Turner: Plane Cubics ivith Inflexions. 
 
 Consider first a cubic with the real point A as a fifth point of inflexion. 
 The cubic has two imaginary inflexional axes through this point; and the 
 equation is consequently of the form 
 
 (iuPx + cot/ + z) (ico 2 x + <ay — z)(x + ay) 
 
 + X(— icox -\- u?y + z)(— i(»x + oo 2 y — z)(x + fiy) = 0, 
 
 where a, /3 are a pair of complex numbers. Accordingly the remaining four 
 inflexions are at 
 
 (a, — 1, iua + co 2 ), (a, — 1, — icoa; — co 2 ), 
 (0, - 1, ioflS - co), (ft - 1, - irf(3 + co), 
 
 where, in order to satisfy the conditions of collinearity imposed on every 
 group of inflexions of a non-singular cubic, that is, the conditions repre- 
 sented by the scheme 
 
 ( V3, 1, 1), (0, - 1, ~ u/0 + co), (a, - 1, iua + co 2 ), 
 08, - 1, ico 2 /? -_ co), ( V3, 1, - 1), {a, - 1, - icoa - co 2 ), 
 (1, V3, 0, (1, V3, - i), (0, 0, 1) 
 
 either a = - i, |8 = i, or a = }(- i *- 4 V3), 0= |(t — 4 V3). 
 
 If a = — », j8 = i, the four points are (», ± 1, ± 1) in agreement with 
 the results in the preceding case. If a == y(— z — 4a/3), |8 = j(i — 4^3), 
 a second set of four points is obtained; but since the computations involved 
 are complicated it is advantageous to apply a linear transformation by 
 which the original four points become 
 
 (0, 1, - 1), (- 1, 0, 1), (1, - co, 0), (1, - co 2 , 0). 
 Then the fifth point is (1, — 1, 0); the remaining four are either 
 
 (0, 1, - co), (0, 1, - co 2 ), (- co, 0, 1), (- co 2 , 0, 1), 
 
 or 
 
 (- 1, co 2 - 1, 1), (- 1, co - 1, 1), (co 2 -1,-1, 1), (co - 1, - 1, 1): 
 
 and the corresponding pencils of cubics may be written as 
 
 3? + V 3 + z 3 + ^xyz = 0, 
 x 3 + y 3 + z 3 + 3z(^ + y 2 ) + Zz 2 (x +y) + \z(z + x)(y + z) = 0. 
 
 Similarly if either B or C is the fifth point of inflexion, there are two 
 distinct pencils of cubics; but, since the four pencils thus determined are 
 imaginary and consequently of interest in this discussion only to give sym- 
 metry to the results, their equations and the coordinates of their remaining 
 points of inflexion are omitted. 
 
Turner: Plane Cubics with Inflexions. 275 
 
 The cubics of the three pairs of pencils have in all only nineteen points 
 of inflexion, and these have the same relative positions as the nineteen for 
 the three pairs of pencils determined by two pairs of imaginary inflexions. 
 Hence, it follows that, with the proper interchanging of the words real and 
 imaginary, the theorems stated on pages 264, 265, and 266 hold for the cubics 
 with two real and an imaginary pair of fixed inflexions. The constructions, 
 however, because of the great number of imaginary elements involved 
 become almost entirely theoretical. , , 
 
 (ii) Related Quartic Curves. 
 The equations 
 
 -Pi = x 3 -f- y 3 + z 3 + \xyz = 0, 
 
 P 2 = x 3 + y 3 + z z + Sz(x 2 + y 2 ) + 3z\x + y) + \z(z + x) (x + y) = 
 
 represent two pencils of cubics having in common three real and a pair of 
 imaginary inflexions. Every value of X determines a definite curve in each 
 pencil, and the elimination of X between the two equations gives 
 
 z'[(x 3 + y 3 + z 3 ){x + y + z) - Sxy(x 2 + f) - Sxyz(x + y)] - 
 
 as the locus of the intersections of the two curves. Three of the fixed inter- 
 sections, namely, one real and the given pair of imaginary inflexions, are on 
 s=0; hence this line is a part of the locus only because of the two curves 
 of which it forms a part. The remaining two fixed inflexions and the four 
 variable intersections lie on the quartic 
 
 (x 3 + y 3 + z 3 )(x + y + z) — 3xy(x 2 + y 2 ) — 3xyz(x + y) = 0. 
 
 The remaining two fixed inflexions are (0, 1, — 1), (— 1, 0, 1), where the 
 quartic has nodes with tangents 
 
 =b ix + y + 2 = 0, x ± iy + z = 0; 
 
 and these lines are the inflexional tangents common to the two cubics 
 considered when X = ± Si. Thus the binodal quartic is the locus of the 
 four variable interesections of the two curves of Pi and P 2 having the same 
 parametric value. 
 
 The two sets of four points 
 
 (0, 1, - co), (0, 1, - co 2 ), (- co, 0, 1), ; (- co 2 , 0, 1); 
 (- 1, co 2 - 1, 1), (- 1, co - 1, 1), (co 2 -1,-1, 1), (co -1,-1, J), 
 
 completing the inflexional groups of Pi and P 2 are on the quartic ; and these 
 together with the two double points (0, 1, — 1), (— 1, 0, 1) are the complete 
 intersection of two quartics 
 
 (wx + y + z) (u 2 x + y + 2) (x + wy + z) (x + u 2 y + z) = 0, 
 xy(z+ x)(x+ y) = 0, 
 
276 Turner: Plane Cubics with Inflexions. 
 
 each composed of two pairs of lines, one pair through each of the double 
 points. Accordingly 
 
 (<ax + y + z)(u?x + y + z)(x + coy + z)(x + w 2 y + z) 
 
 + nxy(z + x)(x + y) = 
 
 is a pencil of binodal quartics through the eight points and two nodes. If 
 the second pencil of cubics is written 
 
 P 2 m x*+ tf+ z 3 + Sz(x*+ 2/ 2 ) + 3z 2 (z + y) + (X- n)z(z+ x)(y + z) = 0, 
 
 where n has any real value, a different quartic of the pencil corresponds to 
 each chosen value of n, that is, the manner of writing the equations of Pi 
 and P 2 can be so varied that every quartic of the pencil may be found as 
 the locus of the four variable intersections of the two cubics with the same 
 parametric value. The two reducible quartics corresponding to n infinite 
 or zero are obtained respectively when n = <x> or n = 3. If n = <x> , P 2 
 breaks up into three lines. If n = 3, the two cubics have the same tangents 
 at the two common imaginary inflexions (1, — co, 0), (1, — co 2 , 0) for every 
 value of X; and all their intersections lie at the five given points except 
 when the cubics have a common linear factor. In the study of non-singular 
 cubics these two cases are excluded, and hence the following result can be 
 stated : 
 
 There are two, and only two, pencils of cubics having in common three 
 real and a pair of imaginary inflexions; and the locus of the four variable 
 intersections of the two corresponding curves is a binodal quartic passing 
 through the remaining inflexions of the two pencils, the two nodes being 
 at the two real inflexions not collinear with the two common imaginary 
 inflexions. 
 
 In further consideration of the geometry on the pairs of cubics and the 
 resulting quartic curve it may be noted that three of the nine intersections 
 of the two cubics lie on a line, hence the remaining six, the six on the quartic, 
 lie on a conic. The equation 
 
 Pi - P 2 = 3z[> 2 + f + z(x + y) + \z{x + y + z) = 
 
 represents a pencil of cubics consisting of the line and a pencil of conies. 
 Every value of X determines a curve of Pi and P 2 and a conic through their 
 six intersections on the quartic. The pencil of conies passes through the 
 two nodes (0, 1, — 1), (— 1, 0, 1), and the two points (1, ± i, 0) where the 
 line z = is intersected by the nodal tangents 
 
 ± ix + y + z = 0, x db iy + z = 0. 
 
 Thus the four variable intersections of the two cubics are cut out on the 
 
Turner: Plane Cubics with Inflexions. 277 
 
 binodal quartic by a pencil of concis through the two nodes and the two 
 intersections of the nodal tangents collinear with the two common imaginary 
 inflexions. 
 
 When the points (1, ± i, 0) are projected into the circular points, the 
 special case arises where the four variable intersections lie on a circle through 
 two of the common real points of inflexion. It may also be noticed that, 
 since the four variable intersections of the two cubics and the two nodes 
 of the quartic lie on a conic, the four variable intersections subtend at the 
 two nodal points pencils of lines with the same cross-ratio. 
 
 § 3. General Conclusions. 
 
 For symmetry the cubics with a fixed quadrangle of real points of inflex- 
 ion are considered, although every such cubic is imaginary. Choose the 
 four points ( Af3, =fc 1, ± 1). Then the six fixed inflexional axes are 
 
 y±z=0, 2±is = 0, z±V3i/ = 0; 
 V3 
 
 and a fifth point of inflexion is any one of the three 
 
 (1, 0, 0), (0, 1, 0), (0, 0, 1). 
 
 It follows that a cubic of the system with an inflexion at (1, 0, 0) is a member 
 of one of the pencils 
 
 (z 2 - 3y*)(y + i V3z) + Ms 2 - i* 2 ) (y + ~«) = 0, 
 
 (z 2 - 3y*)(y - i&) + \(z 2 - J>) (y - ±z\ = 0; 
 
 and similar results hold with respect to (0, 1, 0) and (0, 0, 1). Furthermore 
 the six pencils have nineteen points of inflexion associated as in the two 
 preceding cases. Hence the theorems already stated (pp. 264, 265, 266) are 
 applicable to this case also, that is, for cubics with any fixed quadrangle of 
 inflexions* the following theorems hold: 
 
 f (1) If four points of inflexion of a plane cubic, no three collinear, are 
 fixed, a fifth point of inflexion is fixed as one of three, and the complete 
 group of nine is determined as one of six; or in other words, 
 
 (1 ) A quadrangle of inflexions determines a system of cubics consisting 
 of six syzygetic pencils, and every two of the six have a fifth point of inflexion 
 in common. 
 
 * Provided, as stated on page 261, that if imaginary the points enter by conjugate pairs. 
 t A. Wiman, Nyt Tiddskrift for Matematic (1894); also W. Burnside, Proc. London 
 Math. Soc. (1906-07). 
 
278 Turner: Plane Cubics with Inflexions. 
 
 (2) The remaining points of inflexion of the cubics with a common 
 quadrangle of inflexions form a group of fifteen, consisting of the three 
 diagonal points of the common quadrangle and the vertices of three other 
 quadrangles with the same diagonal points, each of the three new quad- 
 rangles having one pair of sides in common with the original quadrangle. 
 
 (3) The fifteen other possible points of inflexion of a cubic with a fixed 
 quadrangle of inflexions are the three diagonal points of the fixed quadrangle, 
 and the two points on every one of the six sides of the quadrangle equi- 
 anharmonic to the diagonal point with respect to the two fixed points on 
 that side. 
 
VITA 
 
 I, Bird Margaret Turner, was born in Moundsville, West Virginia, April 18, 1877. My father 
 was John Marion Turner and my mother Mary J. (Douglas) Turner. In 1893 I was graduated 
 from the Moundsville High School, in 1915 received the degree of Bachelor of Arts and in 1917 
 the degree of Master of Arts from the West Virginia University. 
 
 From 1900 to 1913 I was teacher of mathematics and in 1915-16 Principal of the Mounds- 
 ville High School; from 1913 to 1915 I was Student Assistant in Mathematics at the West Vir- 
 ginia University. During the year 1917-18 I was Assistant Director of the Phebe Anna Thorn 
 Model School and in 1918-19 part time Reader in Mathematics at Bryn Mawr College. 
 
 I entered Bryn Mawr College as Scholar in Mathematics in 1916, was granted the President 
 M. Carey Thomas European Fellowship for my first year's work, and was Resident Fellow in 
 1919-20. 
 
 At Bryn Mawr College I studied under the direction of Professors Charlotte A. Scott, Anna 
 J. Pell, Matilde Castro and Dr. Olive Clio Hazlett; at the West Virginia University under Pro- 
 fessors John Arndt Eiesland and Joseph Ellis Hodgson. It gives me great pleasure to have this 
 opportunity of expressing to all these professors my gratitude for their valuable instruction. In 
 particular my thanks are extended to Professor C. A. Scott for her constant encouragement 
 during my connection with Bryn Mawr College, and for her helpful criticism and unfailing inter- 
 est throughout the preparation of this dissertation. 
 
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