B H ESQ b67 1 xrbe tlniversitp ot Cbicaao FOUNDED BY JOHN D. ROCKEFELLER LINEAR POLARS OF THE /^-HEDRON IN /2-SPACE A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (department of mathematics) BY HARRIS FRANKLIN MacNEISH L'r>ilv'C.f-.S2TY THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS Bgents THE BAKER & TAYLOR COMPANY NEW YORK THE CAMBRIDGE UNIVERSITY PRESS LONDON AND EDINBURGH ^be *Clniversiti? of Cbicago FOUNDED BY JOHN D. ROCKEFELLER LINEAR POLARS OF THE y^-HEDRON IN w-SPACE A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (department of mathematics) BY HARRIS FRANKLIN MacNEISH {^,/^^,.\vORi''ii THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS Copyright igi2 By The University of Chicago Published March 1912 1. « » ♦ « « , ' .' : 'J . ' • ♦ Composed and Printed By The University of Chicago Press Chicago, Illinois. U.S.A. INTRODUCTION The following general definition of the harmonic mean of a set of segments is given by C. MacLaurin in A Treatise of Algebra, Appendix Concerning the General Properties of Geometrical Lines, § 27: ''A segment PQ is the harmonic mean of a set of segments PPi, i=i, 2, . . . ., n; n if — = "V^ -^. Q is also called the harmonic center of P as to the ^ »=i ' set of points \Pi\y This generalization and its application to polar theory were known to Roger Cotes, who gives in Harmonia Mensurarum (1722) the following general theorem called Cotes' s Theorem: *'If a transversal intersecting a curve Cn of the w*** order in n points Pi, revolves about a fixed point P, the harmonic center (2 of P as to the set of points Pi describes a straight line." MacLaurin gives a proof of this theorem {op. cit., § 28). Poucelet, "Memoire sur les centres des Moyennes Harmoniques," Journal fur Mathematik, Vol. Ill, 1828, gives a treatment of the har- monic mean based upon MacLaurin's definition. E. de Jonquieres, "Memoire sur la theorie des poles et polaires," Liouville's Journal, ser. 2, Vol. II (1857), p. 249, applies the theory of the harmonic mean to the polar theory of curves of the third and fourth order. L. Cremona, " Introduzione ad una Teoria Geometrica delle Curve Plane," Memorie della Accademia delle Scienze delV Istituto di Bologna, ser. I, Vol. XII (1861), pp. 305-436, gives a resume of the preceding theory and an extensive treatment of the properties of curves and sur- faces of the »*^ order from a purely synthetic standpoint, including a treatment of polar theory based upon the idea of harmonic mean. The geometric definition of Linear Polar (see §3, Definition IV4, i) used in the following treatment occurs in the Collected Memoirs of E. Caporali, pp. 258-66. Caporali also considers the quadrangle- quadrilateral configuration which is generahzed in Part IV of this paper. The generalized configuration is considered by F. Morley in a paper on "Projective Co-ordinates," Transactions of the American Mathematical Society, IV (1905), 288. For a bibliography of the subject I refer to the Encyklopddie der mathematischen Wissenschaften, III, AB, 4a, §§ 24, 25, 26, and III, C, 4, §5- 111 251385 CONTENTS PAGE I. Synthetic Treatment i This section consists of a recursion sequence of geometric constructions for the linear polar of a point as to a linear ^-ad of points, as to a ;fe-line in a plane, and in general as to a ^-hedron in «-space— the pro- cess not being based upon the MacLaurin generalized definition of harmonic mean. II. Analytic Treatment 8 In this section I show analytically that the linear polars obtained synthetically in section I harmonize with the analytic polar theory for the M-ary ife-ic which is the product of k linear factors, and the linear polar of a linear point set is proved to satisfy the MacLaurin generalized definition of harmonic mean. III. Algebraic Loci ^3 In this section I give the application to the construction of the linear polars of algebraic curves, surfaces, and spreads. IV. Certain Configurations with Polarity Properties . . 14 In IVa the quandrangle-quadrilateral configuration in the plane is considered from the standpoint of linear polar theory. In IWb the quadrangle-quadrilateral configuration is generalized and a self-dual configuration in «-space is obtained consisting of an {n+2)- point and an («-|- 2)-hedron. The dual figures have interesting polarity and incidence relations, and each face of the («-f 2)-hedron contains the same configuration in space of (« — i) dimensions. In IVc the configuration is generalized to form an associated ^-point and )fe-hedron in «-space. In IWd the corresponding associated pair of ^-points on a line is considered. V. The Reciprocity of Certain Associated Linear Sets of Points ^9 Associated linear 3-points are proved to be reciprocal. Associated linear 4-points are not in general reciprocal and certain conditions on the invariants of the binary quartic representing the 4-point are developed under which reciprocity exists. These invariantive con- ditions lead to interesting geometric interpretations. VI. Concomitant Theory of the Associated 4-Point and 4-L1NE in the Plane 24 From the ternary point quartic representing the 4-point, the contra- variant representing the 4-line is obtained and the reciprocity of the figure is proved analytically. I. SYNTHETIC TREATMENT' § I. The treatment is based on the following assumptions for general projective geometry from Veblen and Young, "A Set of Assumptions for Projective Geometry," American Journal, XXX, 376, § 9. The point is an undefined element, and the line is regarded as an undefined class of points. Ai. // A and B are distinct points, there is at least one line containing both A and B. A2. If A and B are distinct points, there is not more than one line containing both A and B. A3. // A, B, C are points not belonging to the same line, and if a line 1 con- tains a point D of a line joining B and C and a point E, distinct from D, of a line joining C and A , then the line 1 contains a point F of a line joining A and B . Eo. There are at least three points on every line. Ej. There exists at least one line. H. For any three collinear points A, B, C there exists a unique harmonic conjugate^ point D (distinct from A, B, C) of point A as to the pair of points B, C. Definition of an ^-space: 5', i=2, 3, . . . . If F° and F'~' represent a point and an (i— i)-space, respectively, {F° not on F'~^), an i-space 5' is the set of all points \S°\ collinear with F° and the points of F'~^. A o- space is a point. A I -space is a line. Ei+i (i=i, 2, 3 . . . ., n—i). It is not true that every point lies on every i-space. § 2. In this treatment we consider a set of definitions and theorems concerning r 5-spaces in (5+i)-space which shall be numbered Ir,s, Hr, i, etc. The principal definition is the recursion definition IVr, ^ of the polar 5-space of a point as to an r-hedron in (5-f-i)-space and the theorems Irs, IIr,5, etc., lead up to this definition. ' The substance of §§ i, 2, 3, 4, 5 was developed in connection with Dr. Veblen's projective geometry course (Princeton, 1908-9). ^ The harmonic conjugate point is defined by the usual complete quandrangle construction. I 2 rilSlEAR POLARS OF THE yfe-HEDRON IN W-SPACE Definition I2, i : The polar line of a point as to a pair of lines is the harmonic conjugate line of the point as to the pair of lines. Theorem 13,1: The three polar lines of a point as to the pairs of lines of a triangle form a triangle perspective to the given triangle. Let P be a point and pi, p2, p^ a triangle with vertices P23, -P31, ^12. Let 9i, q2, qi be the polar lines of P as to p2 p^, pi pi, pi p2 respectively. qi and qz intersect on PP12, since the harmonic conjugate point of P as to the points Pi={PPi2, p^) and P12 is unique. Similarly ^2, q^ meet on PP23 and q^, qi on PP31. The triangle ^i, q2, q^ is called the cogredient triangle of P as to triangle Pi, p2, Pi- Theorem 113,1: The Desargues Theorem. The intersection points of the pairs of homologous sides of two perspective triangles are coUinear. Definition IV3, i : The polar line of a point as to a triangle is the line of perspective of the given triangle and the cogredient triangle. Definition IV3, ©t The linear polar point of a point as to a linear point triad. Given points P, Pi, P2, Pj on the line /». Through Pi,P2,P3 pass three non-concurring coplanar lines pi, pi, p^ distinct from p. The polar line 9 of P as to the triangle pi, P2, p^ intersects p in the point Q, called the linear polar point' of P as to the point triad Pi, P2, P^. Theorem III3, i: If two triangles are perspective, the two polar lines of a point on their line of perspective meet on their line of perspec- tive. Let the corresponding sides of the perspective triangles pi, p2, p^ and P'ti p2, Pi meet in the points Pi, P2, P3 of their line of perspective p. qi the polar line of P (any point on p) as to p2 p^ meets q'l the polar line of P as to p2 p^ in Qi on p, since the harmonic conjugate point of P as to P2, P^ is unique. Similarly ^2, ^2 meet in Q2 and q^, q'^ in Q3 on p. Quadrangles (^1^2), {piqi), (^i^z), (M^)^ and {p[p2), (piqi), {qiq^, {p'zq'z) have five pairs of corresponding sides meeting on p; therefore the sixth pair of sides, i.e., q, the polar of P as to pi, p2, pi and q' the polar of P as to p[, p2, Pi meet on p.'^ Points Qi, Q2, Qi are a fixed point triad associated with P, Pi, P2, P3 called the cogredient point triad of P as to Pi, P2, Pj. Theorem IV3, q: The Hnear polar point of a point as to a linear point triad is unique. ' By Theorem IV3, the linear polar point is independent of the auxiliary triangle and of the plane of the triangle. ^ Veblen and Young, op. ciL, Theorem 7. SYNTHETIC TREATMENT 3 From Theorem IIl3,i, the cogredient point triad Q^, Q^, Q^ are fixed points and the Hnear polar point Q is determined uniquely as the sixth point of the quadrangular set (Pi, P2, P; Q2, Qi, Q). The sixth point of a quadrangular set of which five points are given is indej^endent of the plane of the quadrangle, therefore, in finding the linear polar point the auxiliary triangle may be taken in any plane whatever passing through the given Hne. § 3. In order to generaUze inductively in the plane the theorems and definitions given in § 2 for the 3-line and Hnear 3-ad, the following definitions and theorems are assumed for the (yfe— i)-line and the linear point (k— i)-ad and are proved for the /fe-line and the linear point ife-ad for k^4.. Theorem lk,i: The k polar lines of a point as to the k (/^— i)-line figures of a ^-line form a ^-line perspective to the given y^-line. For point P and ^-line \pi\, (i=i, 2, . . . ., k) let qi be the polar line of P as to the (^-i)-line figure \ph\, {h=i, 2, . . . ., k; h^i). The ^-line \qi\ is called the cogredient k-\\ne io \pi\,{i= 1,2, . , . ., k) as to point P. Let Ri,st be the points of intersection of Hnes pi and PPsi, (i=i, 2, . . . ., k; i^s, t) where Pst=^(pspi). Then 9, the polar line of \pi\, (i=i, 2, . . . ., k; i^s) as to P and qt the polar line of \pi\, (i=i, 2, . . . ., k; id^t) as to P intersect in Qsi which is on PPst, because the linear polar point of P as to the {k— i)-ad Psh Ri.st, {i= 1,2, ... ., k; i^s, t) is unique (Theorem IV^-i.o), and the two ^-lines \ Pi \ and \ qi \ are perspective from P. Theorem II^, i: If two ^-lines are perspective from a point, the points of intersection of corresponding sides are collinear. Given two ^-lines \pi\ and \qi\, {i=i, 2, . . . ., k). Triangles pj, pj+i, pj+2 and qj, qj+i, qj+2 are perspective from P, so corresponding sides meet in points Aj, Aj+i, Aj+2 on a Hne aj, (J=i, Successive Hnes aj and aj+i have in common two points Aj+j, Aj+2, 0'=i> 2, . . . ., k—2), so that all the lines aj coincide and the intersec- tion points of corresponding sides of the two given yfe-lines are collinear on a Hne called the line of perspective. Definition TVk, 1 : The polar line of a point as to a k-line is the line of perspective of the ^-line and its cogredient ^-line as to the given point .^ ' Cremona, op. cit., p. 364. 4 LINEAR POLARS OF THE ^-HEDRON IN W-SPACE Theorem Illjfe, i : If two ^-line figures are perspective from a point, the two polar lines of a point on their line of perspective meet on their line of perspective. Let p be the line of perspective of the ^-lines \pi\, \pi\. For a point F on p let \ qi \ and \ q'l \ be the cogredient ^-lines and q and q' the polar lines of P as to \pi\ and \pi\ respectively {i=i, 2, . . . ., k). qi and q'l {i=i, 2, . . . ., k) meet on p, for they are the polar lines of P as to the (^— i)-lines \pj\ and \Pj\, (j= i, 2, . . . ., k; j^i) (Theorem III;fe_i,i), therefore the cogredient ^-lines \qi\ and \q'i\ have p as line of perspective. Then qi and q'i meet in Qi on p and \Qi\ is called the cogredient point k-ad of P as to \Pi\, {i=i, 2, . . . ., k). The quadrangles P„, (Mr), Qrs, (psqs) and P'rs, (prq'r), Q'rs, iPaq's) have five pairs of sides meeting on line p, therefore the sixth pair of sides q and q' meet on p, (r, s=i, 2, . . . ., k; r^s). Definition IV^^.q: The linear polar point of a point as to a linear point k-ad^ Given points P, Pi, P2, . . . ., P^fe on line p. Through P,- draw coplanar lines pi distinct from p, no three concurring. The lines Pi determine a ^-line and the polar line 9 of P as to the ^-line \ pi \ inter- sects p in the point Q which is the polar point of P as to the linear ^-ad Theorem IV^^.q: The polar point of a point as to a linear point ^-ad is unique. Given P, Pi, P2, . . . ., Pk on line p. The cogredient point set \Qi\ of P as to \Pi\, {i=i, 2, . . . ., k) is determined by Theorem Illk.i, and the polar point ^ of P as to \Pi\ is determined uniquely as the sixth point of any one of the quadrangular sets (P^ Ps P; Qs Qt Q) § 4. /w space of 3 dimensions. Definition I2, 2 : The polar plane of a point as to a pair of planes is the harmonic conjugate plane of the point as to the pair of planes. Definition 1^,2: A k-hedron in space is a set of ^-planes no 4 of which have a common point. The following definitions and theorems are assumed for the (^— i)- hedron and given in full for the ^-hedron, ^^4. ff ft '^^ ^ I ' Poucelet, op. cit., p. 231, defines Q by the equation ^= ^ ^ Hp"- This defi- nition is the usual basis of treatments of linear polar theory. SYNTHETIC TREATMENT 5 Theorem 1;^, 2: The k polar planes of a point as to the k (k—i)- hedrons of a ^-hedron form a ^-hedron perspective to the given ife-hedron. For point P° and ^-plane |P?f, (i=i, 2, . . . ., k) let Q} be the polar plane of P° as to the (^— i)-hedron \P^a\, {h=i, 2, . . . ., k; h^j). I0h 0*=i> 2, . . . ., k) is called the cogredient k-hedron to the ^-hedron |P/|, (i=i, 2, . . . ., k) as to P°. Let R],st be the line of intersection of planes P? and P° P',, (*'= i, 2, . . . ., /fe; i^s,t) where PL=(P'P'). Then ^? is the polar plane of the (k— i)-hedron \P]\, (j=i, 2, . . . ., k> j^s) as to P° and is the polar plane of the (jfe— i)-hedron \Fj\, (j=i, 2, . . . ., k; j^t) as to P°. Q's and Q? intersect in line Ql^ which is on plane P° P]t because the polar line of P° as to the (^-i)-line P\i, R],st, (j=i> 2, . . . ., k; j^s, t) is uniquely defined, and the two ^-hedrons \Pl\ and \Q'i\ are perspective from P°. Theorem II^, 2 : If two ^-hedrons are perspective from a point the lines of intersection of corresponding planes are coplanar. For k=2, the theorem is evident. For ^^3. Any plane (not through a vertex of either yfe-hedron) through the point of perspective intersects the k intersection lines of pairs of homologous faces in collinear points by Theorem II;fe, i, therefore the k intersection lines of pairs of corresponding faces are coplanar and the plane is called the plane of perspective. Definition lYk, 2 : The polar plane of a point as to a k-hedron in space is the plane of perspective of the ^-hedron and its cogredient ^-hedron as to the given point. § 5. /w space of n dimensions. In order to prove inductively the theorems of §4 in w-space we assume in {n— i) -space Theorems lk-i,n-2 and llk-i,n-2, leading to the Definition IV^_i, n-2'- The polar {n— 2) -space of a point as to a (/fe— i)- hedron in («—i) -space is the (w— 2) -space of perspective of the {k—i)- hedron and its cogredient (^— i)-hedron. Definition Iz.n-i: The polar {n— i)-space of a point as to a pair of (n—i)-spaces is the harmonic conjugate («— i)-space of the point as to the pair of (n— i)-spaces and is determined as follows: Any line through the given point and not through the (w— 2) -space of intersection of the two given (w— i)-spaces intersects each (w— i)-space in a point. The 6 LINEAR POLARS OF THE ^-HEDRON IN W-SPACE harmonic conjugate point of the given point as to this pair of points and the (n — 2)-space of intersection of the two given (n— i)-spaces determine the harmonic conjugate (w— i)-space of the given point as to the pair of (n— i)-spaces. This determination can be proved to be unique. Definition Ik.n-i- An n-space k-hedron is a set of k (n— i)-spaces, no w+i of which have a common point. Definition Ilk,n-i' Twok-hedrons are perspective from a point if the («— 2)-space edges, in corresponding pairs, lie in (n— i)-spaces which pass through the point of perspectivity. Theorem lk,n-i' In w-space the k polar (w— i)-spaces of a given point as to the k (/fe— i)-hedrons of a ^-hedron form a ^-hedron perspec- tive to the given ^-hedron. For point P° and yfe-hedron \P'r'\, (i=i, 2, . . . ., k) let Q'J-' be the polar (w— i)-space of P° as to the (^— i)-hedron \P2~'\, {h = I, 2, . . . ., k; h^j). \Qj^^\f (y=ij 2, . . . ., k) is called the cogredient ^-hedron of Jfe-hedron \P'r'\, (i=i, 2, . . . ., k) as to P°. Let i?"T/ be the (w— 2)-space of intersection of (w— i)-spaces P"~' and P^P'^r, {i=i, 2, . . . ., k; i^s, t) where P':r=(P'r'Pr'). Then QT' is the polar («-i)-space of (/fe-i)-hedron \Py~'\, {j=i, 2, . . . ., k; y=t=5) as to P". And Q"~' is the polar (»— i)-space of (^— i)-hedron \P)~'\, (j=i, 2, . . . ., k; j^t) as to P°. Q"~^ and Q"~' intersect in (w— 2)-space Qlr'' which is on {n—i)~ space P° P"t~^ since the polar (w— 2)-space of P° as to (w— i)-space (*— i)-hedron P'^r% Rj.Tf, 0*=i, 2, . . . ., k; j^s, t) is uniquely defined, and the two ^-hedrons {P""^] and \Q'!~''\ are perspective from P°. Definition lHk,n-i' A complete k (w— i)-space is a ^ (w— i)- space with no {n-\-i) (w— i)-spaces through the same point such that each (w— i)-space cuts every other in an (w— 2)-space. Lemma: A complete k (n— i)-space is an w-space ^-hedron, i.e., has all of its elements in an w-space. Every (w— i)-space of a complete k (w— i)-space intersects every other (w— i)-space in an (w— 2)-space, therefore the «-space determined by one pair of (w— i)-spaces contains all the remaining (w— i)-spaces, since it contains two distinct (w— 2)-space of every one that remains. SYNTHETIC TREATMENT Theorem llk,n-i'- The Desargues Theorem for n-space. If two ife-hedrons are perspective from a point, corresponding (w— i)-space faces meet in (w— 2) -spaces of the same (w— i)-space. If \A"-'\ and \B'r'\, {i=i, 2, . . . ., k) are perspective k- hedrons, any pair of corresponding («— 2)-space edges A",'^^=A"~\ A"-' and E'C^'^BT', B"~' lie in the same (w-i)-space C"~' and therefore intersect in an (w— 3)-space C"T^ Then C"~^=A"''\ B"~', and C"~^=A"-% B"~' contain C"rJ and in general any pair of (w— 2)- spaces C"~% CT^ which are intersections of corresponding pairs of («— i)-space faces of the given ^-hedrons have a common (w— 3)-space, therefore the whole intersection figure is a complete k («— 2)-space and hence must lie in an (w-i)-space (by the Lemma) which is called the (w— i)-space of perspective. Definition IV;fe,„-i: The polar {n—i)-space of a point as tea k-hedron in n-space is the (w—i) -space of perspective of the ^-hedron and its cogredient ^-hedron as to the given point. Thereoms III;fe,„-i and lYk,n-2 are unnecessary for n>2, as the uniqueness of the linear polar is evident from the construction except in the case of linear polars of linear point sets. All the theorems and constructions of this section may be dualized. II. ANALYTIC TREATMENT § 6. It is possible to extend the set of assumptions given in § i to form a sufficient basis for a system of homogeneous co-ordinates and to proceed analytically (Veblen and Young, op. cit., § 2, p. 352). For an w-ary linear form we use the Clebsch notation: n and we indicate the factored «-ary ^-ic where^ n ig^g The polar operator* is written 8 X OX In z = i and the polar operator repeated r times is indicated The {k—xy^ polar or linear polar of the point x'={xi, Xj, . . . ., x„)4= (o, o, . . . ., o) with respect to /^"^ where /";'^^o may be written in the form § 7. The ■polar line of a point as to a 2-line. Given point P:{x'„ x'^, x'^) and lines pi:a^^^x=o; p^:ai%=o. The line PP,2=p is /l'3) /7<3) "•I, X "■2, X t*i, a; "■2. a! ' The superscript («) is omitted when no ambiguity arises. " The subscript n is omitted when no ambiguity arises. 8 ANALYTIC TREATMENT The line a:3 = o intersects p, p^, p2 in Po, Pi, P2. The harmonic conjugate of the point given by Ka'^^^+Ka^^.x=^ as to the points al:l,=o and at^=o is given by the equation Then the harmonic conjugate Q of Po as to Pi, P2 is then the line q=QPi2 is "■1, a; I "'2, a; "■I, a; ""2, a; ""I, X I "•»■ a; t*i, X ""2. a; the linear polar of P as to lines pi and /^j. § 8. The polar line of a point as to a k-lineJ- Given point P {x[, X2, x'^) and lines \pi\ : fli,x = o, (j=i, 2, . . . .,^^). For purposes of an inductive development we assume that the polar line of a point P {x[, x^, x'^) as to a (^-i)-line \li\ : hi,x = o, (i=i, 2, . . . ., k—i) is given by the equation then the cogredient yfe-line of ^-line \pi\ as to P will be |g4 : 'V ^^ = 0, (i=i, 2, . . . ., ^). Any line through the point of intersection of p{ and qi is given by the equation 7 ^^^+aiai,x = o, ii= I,2j • • • aj /£/ where a,- are arbitrary constants. These equations are all identical for ai= so that all the points 'Cf. Cayley, "Sur quelques th6oremes de la geometric de position," Collected Works, I, 360. lO LINEAR POLARS OF THE ^-HEDRON IN W-SPACE {pi qi), (i=i, 2, . . . ., k) are collinear on the line 1 = 1 which is the equation of the polar line of P as to the ^-line \pi\. § 9. The linear polar point of a point as to a linear point ^-ad. Given point P : (x'l, x'z, o) and points Pi : a,fa;=o, (j=i, 2, . . . ., li) on line p : 0:3 = 0. Pass the lines pi : 0^5^ = through the points Pi. The polar line of P as to the ^-line {pi\ is the Hne « = i and q intersects p in the point •^ "■!. a;' t = i which is the equation of the polar point of P as to the ^-ad \Pi\. § 10. The MacLaurin generalized definition of harmonic mean. Let OQ=y = — represent the distance from some fixed point taken as origin on a given line to any point Q of the line af^ For the point Pi : af^x=o y= — ^ . ai, I x[ For the point P {%[, x'z) 0P=—, . For a general point Q {xi, x^ 0Q=— . Oi/2 "•J. aj QPi=OQ-OPi= PPi=OP-OPi= If Q is the polar point of P as to \Pi\, (i^i, 2, . . . ., k), which reduces to ^pPi ° ANALYTIC TREATMENT II Whence sr^ PPj-PQ 2^ pPi or ^\PQ PPi) °' or n _ y^ I PQ~ 2^JPi' so that PQ is the harmonic mean of the segments PPi according to the MacLaurin generalized definition. §11. The polar plane of the k-hedron in space as to a given point. Given point P° {x'l, xi, x^, x'^) and ^-plane \P^\ : ai''x=o, (i=i, 2, ,k). For purposes of an inductive development we assume that the polar plane of a point P° (yi, y^, yj, y4) as to a (^— i)-plane \R^i\ : 6,-f'^=o, (i=i, 2, . . . ., k—i) is given by the equation: k-i X ^ A (4) M (4) = y Then the cogredient ^-plane of the ^-plane |P!j as to point P° is ,(4) m ■ X #- Any plane through the line (Pf' Qf') is given by the equation: ^ al- ^-^+a,ai^i=o -V', X' where a; are arbitrary constants. These planes are all identical for fl,-=-^r, so that the polar plane of P° as to the ^-plane |P?^ is given by the equation: § 12. The polar {n— 1)- space of a k-hedron in n-space as to a given point. Given point P° {x[, xi, . . . ., x',+0 and /fe-hedron \P'r'\ : ai"+'^ = o, {i=i, 2, . . . ., k). 12 LINEAR POLARS OF THE ^-HEDRON IN W-SPACE For purposes of an inductive development we assume that the polar line of a point P° (ji, y^, . . . ., Jn+i) as to a (^— i)-hedron in«-space \Rl~^\ '. bi"^'^=o is given by the equation 1=1 Then the cogredient ^-hedron of {P" '\ as to P° is Any (w— i)-space through the (w— 2)-space (P" ' Q" ') is given by the equation where a,- are arbitrary constants and these (n— i)-spaces are all identical for ai=-^r+i, so that the polar («—i) -space of P° as to the ^-hedron \Pi~'\ is given by the equation («+i) '•^ =o a^:^'' III. ALGEBRAIC LOCI § 13. From Section II we can prove "Cotes's Theorem."' Theorem I: "Any transversal line through a point intersects its polar line as to a curve of the w*^ order in the polar point of the linear point w-ad determined by the curve on the transversal"; and the generalization to «-space: Theorem II: Any transversal line through a point intersects its polar (w— i)-space as to an w-space spread of the k^^ order in the polar point of the linear point ^-ad determined by the spread on the transversal. From Theorem I we obtain the following method for constructing the polar line^ of a point P as to a curve of the w*^ order C„. Through P pass any two transversals pi, P2 intersecting C„ in points Pi, I, Pi, 2, («"=!, 2, . . . ., n). Connect the points P,-, j and P,-, 2 by the lines pi forming w-line \pi\. (This can be done in n" ways by changing the notation for the points.) Then the polar Kne 9 of P as to the w-line \pi\ is the polar line of P as to the curve C„, because q has two points, one on each transversal common with the polar line of Cn, by Theorem I. Likewise from Theorem II we obtain the general method of construct- ing the polar («— i)-space of a point P° as to a spread Qk of the k^^ order in w-space. Through P° pass any n transversal lines P} (J=i, 2, . . . ., n) not in the same (n— i)-space, intersecting Qk in points P°y, {i= 1, 2, . . . ., k). Let the points Plj, (j=i, 2, . . . .,n) determine (w— i)-space P7~', whence for (i=i, 2, , . . ., ^) we get the w-space ^-hedron {P""'}. (This can be done in k" ways by changing the notation for the points P°,j.) Then the polar (w-i)-space Q"-' of P° as to the ^-hedron \P7~^\ is the polar (w—i) -space of P° as to the spread Qk, since Q"~' has n points, one on each transversal common with the polar (w— i)- space of P° as to Qk, by Theorem II. ' MacLaurin, op. cit., § 28. " For the cubic see Salmon, Higher Plane Curves, 3d ed., p. 143; Durege, Ctirven Dritten Ordnung, pp. 167, 168. 13 IV. CERTAIN CONFIGURATIONS WITH POLARITY PROPERTIES c) THE ASSOCIATED 4-POINT AND 4-LINE IN THE PLANE § 14. Let pi be the polar line of the point Pi of a given 4-point figure \Pi\ in a plane taken with respect to the triangle formed by the other three points {i=i, 2, 3, 4). We then have associated with the 4-point \Pi\ the 4-line \Pi\. The two figures form a complete quadrangle and complete quadrilateral with a common diagonal triangle. In homogeneous co-ordinates with the common diagonal triangle as triangle of reference, if one of the four points Pi is taken as unity point, the corresponding points and lines of the two figures have the same co-ordinates: P^ — Xi-\-X2-\-X3=0 P^ (-1, I, l) p^ Xi — rC^-f 3^3 = P2 (l, -I, l) P3 Xi ~r" X2 ^3 ^ A (l, I, -l) Pa Xi-^X2-{-X3 = P^ (l, I, l) From the duality of these equations it is evident that the configura- tion is self -reciprocal. In supernumerary co-ordinates with 2x, = o pi : Xi=o Pi : (»£=-3, Xj=i), (j=i, 2, 3, 4; j^i) for j=i, 2, 3, 4. The group of collineations under which the configuration is invariant is the permutation group G^i and the 24 transformations are given by the following equations in supernumerary co-ordinates Xi^Xf^, (t^i, 2, 3j 4/ (r„ r^, rj, r^ distinct=i, 2, 3, 4) b) THE ASSOCIATED (w+2)-P0INT AND (w-|-2)-rLAT IN W-SPACE' § 15. The n-space configuration. An /-space is incident with an w-space if, for /m the /-space contains the w-space. An w-space 'The contents of Sections IVa and IVb are in substance given in MacNeish, A Self Dual Configuration in n-Space, Master's Thesis, University of Chicago, 1904 (written in connection with Dr. Moore's projective geometry course, 1902), deposited in Library of the Department of Mathematics of the University of Chicago. 14 CERTAIN CONFIGURATIONS WITH POLARITY PROPERTIES 1$ configuration is a system of n sets of ^-spaces {k = o, i, . . . ., n—i); Go points, fli lines, and in general an ^-spaces such that every g-space is incident with the same number agh of A-spaces {g, h = o, i, 2, . . . ., For ^-spaces we use the notation : (^ = 0, I, . . , ., n — i) A. i^ j^ . . . . i^. V'j^^ I J 2, . . . 'J dk lor J = 1, 2, . . . ., K , ijd^ij. lor j^f) The numbers c;, agh are written as a square matrix called the configura- tion specification^ as follows: {ogh), {g,h = o,i,....,n-i; agg = ag) The elements of the main diagonal agg = ag specify the number of g-spaces, and any element agh specifies the number of g-spaces incident with each A-space. It can be proved that between the numbers of a configuration speci- fication, the following relations hold: dij o.jj=^aji an , {i,j = 0,1, . . . .,n — i) The dual configuration to a given configuration in «-space is defined by interchanging the words g-space and (w—g — I ) -space (g = o, I, . . . ., n — i) in the definition of the given configuration. An w-space configuration dual to itself is called a self n-space dual configuration. GENERAL DEFINITION OF CIRCUMSCRIPTION IN fl-SPACE In w-space one configuration (ai/) circumscribes another (6,7) index n—k, (n—i ^k^i) if the a^ r-spaces ^^^,^ ,> of the first for r = k, k-\-i, . . . ., n—i are in one-to-one correspondence with the 6; r-spaces Bj J y. , of the second for r = r— k in such a way that corresponding r-spaces and r-spaces are incident. § 16. The associated (n-\-2)-point and {n-\-2)-flat in n-space. Given n-\-2 points A°, (i=i, 2, . . . ., n-\-2) in w-space (no k-\-2 of them in a ;fe-space for k = i, 2, . . . ., n—i). Let AT'' be the polar (w — i)-space of A°, taken with respect to the («+i)-hedron \A°\, (/=i, 2, . . . ., n-\-2\ j^i) whose vertices are the remaining given points (see §5, Definition I\lk,n-i)- We then have associated with the («+2)-point an (w-{-2)-flat. ' Cf. E. H. Moore, "Tactical Memoranda I," American Journal of Mathematics, XVIII (1896), 264. 1 6 LINEAR POLARS OF THE ^-HEDRON IN W-SPACE In the (w+2)-point figure \A°\ any ^-space is denoted A^t^,^ ,• and it contains every element of lower dimensions whose subscripts are all of the set ii, iz, . . . ., ik-\-f In the (»+2)-flat figure ] A""^ \ any ^-space is denoted ^t^ ,^ ,-^^_^ and it lies in every element of higher dimensions whose subscripts are all of the set ii, ij, . . . ., in-k- n+i In supernumerary co-ordinates in w-space, where y^x, = o t=i Ir' :Xi=o A° : {xi=-in+i),Xj=i), 0' = i) 2, . . . ., n-\-2; j=^i) for i=i, 2, • . . ., w+2. From the duality of the co-ordinates (i.e., point co-ordinates and »+2 («— i)-space co-ordinates) since ^^Xi = o, it follows that the configura- »=i tion is self -reciprocal. The group' of (»+i)-ary coUineations under which the configuration is invariant is simply isomorphic to the symmetric group on w-|-2 letters and the equations of the coUineations are of the form: r : %\=Xr. {i=i, 2, . . . ., n-{-2) where r=(fi, rj, . . . ., r„+2) is a permutation of (i, 2, . . . ., n-\-2). §17. Theorem: The («+2)-flat is inscribed index n—i in the (»+2)-point. A iX '„ is represented by Xi^^^= Xi^^^ ^?«+ii«„+2 is represented by Xi ,=0 *«+i Xi ,=0 Therefore A7j\...i„ of the («-|-2)-gon contains A"^_^^i^_^^ of the (w-|-2)-flat. And in general A7-^, ,„_^^^ of the (;z+2)-gon is represented by See E. H. Moore, "Concerning Klein's Group of (»4-i)! w-ary CoUineations," American Journal of Mathematics, XXII (1900), 336. CERTAIN CONFIGURATIONS WITH POLARITY PROPERTIES 1 7 and A"~^7^ , , , .■^ of the («+2)-fiat is determined by the k-\-i equations: ^'«+3 =° SO that ^j;7* :„_^+xOf the (»+2)-point contains ^r„_V. ■«-^+3 '«+. of the (w4-2)-flat, and the (»+2)-point circumscribes the («+2)-flat index n—i. § i8. The associated («+2)-point and (»+2)-flat in w-space form a configuration whose specification is the matrix: ifl&h), {g,h = o, I, . . . ., w-i) where and and n-\-2\ /n-\-2 g+i/ \n-g agg=\ . . _) + %^=(7!t')'"'>^ where ( ] denotes uCv, the number of combinations of v things taken from u things. u>v. Theorem: In the polar (w— i)-space A"~'' of the point Aj, as to the (w+i)-point \A°\, (i=i, 2, . . . ., w+2; i^j) in w-space, the section of the («+2)-point \A°\, (i=i, 2, . . . ., n+2) is the («+i)- point (w+i)-flat configuration in (w— i)-space. In a supernumerary co-ordinate system A°i is represented by (a;,= — (w+i), Xj=i ior j=i, 2, . . . ., n+2; j^i) and _ A"~^ is represented hy Xi = o For simplicity consider the section of the configuration in («— i)- space A'l''^ : Xi = o and in order to have a supernumerary system in this (w— I ) -space we will omit the variable x^ and call Xi = yi-iy (i = 2, 3, n+i . . . ., w-}- 2) whence ^ yi = o. 1 8 LINEAR POLARS OF THE ^-HEDRON IN W-SPACE Line A\k intersects Ai~^ in point Bl-j (k = 2, 7,, . . . ., n-\-2) with co-ordinates ()';fe_i= —«, 7^=1 /or 7 = I, 2, . . . ., w+i; j^k — i) A'k~^ k = 2, 3, . . . ., w+2 intersects ^""^ in B^Zi given by the pair of equations :Vi=o, X;fe_i = o or simply by x;fe_i = o (^ = 2, 3, , . . ., w+2). Then (w-{-i)-point |-B^_i^ and (w+i)-flat {B'^Zi} have precisely the co-ordinates of the associated (»+i)-point (»-|-i)-flat configuration in (« — i)-space (see § 17). The same can be proved of the sections in the (« — i)-spaces ^"~', (r = 2, 3, . . . .,n-{-2). c) THE ASSOCIATED r-POINT AND r-FLAT IN W-SPACE § 19. (i) For r = «+2: Given (w-j-2)-point |P^(, (^'=i, 2, . . . ., n-\-2) in w-space. Call P"~^ the polar (w — i)-space of point P°, as to (n-|-i)-point \P°\, (i' = i, 2, . . . ., n-\-2; i'^i). An («+i)-point in w-space is also an (w+i)-flat. Then \P'i~^\ is the (w+2)-flat associated with the (w-f 2)- point \P°\. The properties of this configuration are discussed in §§16,17,18. (2) For r = n-\-2,'- Given («4-3)-point \P°\, ii=i, 2, . . . ., w-f 3) in w-space. With (w+2)-point \P°\, (i' = i, 2, . . . , w-f3; i'^i) is associated (n-\-2)- flat P'l-.'\ by § 18 (i). Call PT' the polar (w-i)-flat of the point P°, as to the (w+2)-flat \P'll'\, (i' = i, 2, . . . .,^+3; i'^i). Then \Pi~^\ is the («-|-3)-flat associated with the («+3)-point \P°\ (i=i, 2, . . . ., w+3). (3) In general: Given r-point \Pi\, {i=i, 2, . . . ., r) in w-space r^n-\-2. With (r— i)-point \P°\, (i'=i, 2, . . . ., r; i'^i) is associated an (r—i)- flat \P'l'7'\ obtained by successive application of the method of § 19 (2) above. Call P"~^ the polar (w— i)-space of point P°, as to (r— i)-flat jPH'^L (^'=1, 2, . . . ., r; i'^i). Then \P'r'\ is the r-flat asso- ciated with r-point \P°\. d) ASSOCIATED POINT SETS ON A LINE Given r-point \P°\, (i = i, 2, . . . ., r) on a line P\ To any sub- set of r—i of these points \Pj\, (j=i, 2, . . . ., r; j^k) there is a cogredient set (see § 3, Theorem lVk,o) of r—i points \P°j\, (7=1,2, . . . ., r; j^k) as to the point P°. Call the polar point of Pi as to the (f— i)-point \P°/\, Q°k. The r-point \Q°\ is the associated r-point to \P°i\, {i=i, 2, . . . .,r). V. THE RECIPROCITY OF CERTAIN ASSOCIATED LINEAR SETS OF POINTS § 20. Let the linear equation ax = aiXi-\-a2X2 = o represent the point (fla,— a,) on some fundamental line. We use the co-ordinates {ui, U2) to represent a point in a manner analogous to the method of writing point and line co-ordinates in the plane. Then aM = ^1^1+ ^2^2 = represents the point (d, a^ and the equations aiXi-\-a2X2 = o and O2M1— ai«2 = o represent the same point. We will consider certain sets of points given by their co-ordinates and write their equations in Ui, U2', while certain sets of points associated with them will be given by equations in Xi, X2. Throughout Section V, the notation for the concomitants of Binary Forms will be that of Clebsch, Theorie der bindren algebraischen Formen. §21. Associated linear 2,-points. For a linear point triple represented by a binary cubic fu = o, we designate as the associated point triple, the triple consisting of the har- monic conjugate points of each point as to the remaining pair. The associated point triple is represented by the cubic covariant of /„, i.e., Qu = o (cf. Clebsch, op. ciL, pp. 115, 134), or by the contra variant Qx = o obtained by changing Ut to —X2, W2 to Xj in Qu. Now Qu{Qx) = —Rl fu where R is the Discriminant of /"„ (cf. Clebsch, op. cit., p. 123); therefore the two point triples are reciprocal. The two point triples form 3 pairs of points belonging to a quadratic involution and the double points are represented by the Hessian Hu of /«. § 22. Associated linear 4-points. Let P' be the linear polar point of Piy'i, y2) as to the point triple Ap, Bp, Cp associated with the triple A, B, C represented by a binary cubic fu=o. Ap, Bp, Cp (cf. § 21) are represented by Qx=o. Then P' is given by the equation: (l) dxl ' = ir points A : a„ = o B : Ui=o C : «2=o D : Ui-{-U2 = 19 20 LINEAR POLARS OF THE ^-HEDRON IN W-SPACE then fu = auUiU2(ui-hu2)=aiulu2+(ai-}-a2)ulul-\-a2UiUl = o represents the four points A, B, C, D. By formula (i) we can obtain the equation for point A' , the polar point of A as to the triple 5a, C^, Dp, associated with B, C, D. Similarly points B', C, D' can be obtained. A', B', C, D' form the 4-point asso- ciated with 4-point A, B, C, D. A' : Xi(2al — 2aia2—al)—X2(al-\-2aia2—2al)=o B' : Xi(at-\-a2) (flj — 2^2) (201 — O2) — 01^2(^1— 4aia2+fl2)=o C : a2Xi(al—4ata2-{-aV)—X2{ai-]-a2) (fli — 202) (2C1— a2)=o D' : a2Xi{al-\-2aia2 — 2al) — aiX2i2al — 2aia2—al)=o Then if /^ is the product of these four linear expressions /^ = o represents the four points A', B', C , D' . From /„ we obtain : Hx== — -^j[2,alx\ — Aa2{ai-\-a2) xlx2-\-2{2a\-\- aia2-\- 20^x1001— /^ai{ai-\-a^XiXl +Za\x'^ where Hx is the Hessian of fx- The two invariants I and / of fx are : I=\{al—a^a2+a''^ /= — 7V(«i+«2) (ai — 2a2) (201 — 02) Then/^ is expressible as a function oifx, Hx, I, J: fx = 8 . 6^\24J{i2J'-P)Hx+P{I'+42J')M (2) § 23. The self-reciprocal 4-point. If either / = o or i2j^—P = o; /^ = o represents the same 4 points as fu=o and the 4-point is self -reciprocal. For J = 0, the 4 points are harmonic and each point goes into itself, so that the 4-point is identically self -reciprocal. For i2j'—P = o, the 4 points are operated on by the substitutions (AB) (CD); (AC) {DB); (AD) (BC). Therefore the two cases in which the 4-point is self-reciprocal con- stitute the substitutions of the subgroup G4 of the symmetric group G41 on 4 letters. It can be proved that 12/^— P = o is the necessary and sufficient condition that A, A'; B,B'; C,C'; D,D' are pairs of a quadratic involu- tion. 12/^— /3 = 36/H(Clebsch,o^.a/.,p. 141, note); therefore 12/^— /3 = o is the condition that the 4 points represented by the Hessian of fu are RECIPROCITY OF CERTAIN ASSOCIATED LINEAR SETS OF POINTS 21 harmonic; therefore the necessary and sufficient condition that a 4-point be self-reciprocal is that either fu = o ot Hu = o shall represent harmonic points. /= —J^{k-\-i) {k — 2) (2^ — 1) where k is the cross ratio of the four roots oi fu = o. If k is rationally expressible in the coefficients of /„, then / is rationally factorable into factors linear in the coefficients of /„. i2j^— 7^=7h= — tVC^+i) (^~2) (2A— i) where h is the cross ratio of the four roots of Hx = o. If h is rationally expressible in terms of the coefficients of Hx, /h = o will be rationally factorable into three factors linear in the coefficients of Hx and therefore quadratic in the coefficients of/«. § 24. Cubic covariant theory connected with the self-reciprocal 4-point. We will consider what function of the concomitants of the cubic representing three given distinct points, determines a set of points any one of which taken with the original set of three points constitutes a self-reciprocal 4-point. Suppose gu = o is the cubic representing three given distinct points. Qu = o represents the three 4**^ harmonic points to the triple represented by gu = o; this corresponds to / = o for the quartic /m = o (cf. §23). Therefore there are precisely three points which may be taken with a given point triple to form an identically self-reciprocal 4-point. In § 23, — = ^ is the cross ratio of the four points A, B,C, D, there- di fore: 12j^-P={2k^-2k-l) {k^-\-2k-2) {k^-/^k-\-i) Let three given points be P : Wi = o; Q : U2 = o) R : bu = biUi-\-h2U2 = 0, then: gu = ibiUlu2-\-2)^2UiUl For any 4*^ point X : (xi, X2) the cross ratio of the 4 points P, Q, R, X • T 00x02 IS k = — r • X2O1 Then (2)fe=»— 2^ — i) {k'-\-2k — 2) (yfe^— 4/^+1) reduces to 2blx{— 6biblx\x2 — isb\bix\xl-\- ^oblblxlxl — i'-fb\blx\xi — ()b\b2XiXl-\- 2b\xl, which in terms of the concomitants of gu is equal to — lyi?^^-]- 14^^ — 5 A^. Then the 6 points represented by -I'jRgl+i^Ql-S^l^o (3) have the property that any one of them taken with the three points represented by gu = o form a non-identically self-reciprocal four point. 22 LINEAR POLARS OF THE ife-HEDRON IN W-SPACE If 4 points SO obtained are represented by /« = o then if the cross ratio of the 4 points represented by Hu = o is rationally expressible in terms of the coefficients of Hu, the sextic equation (3) will be factorable into rational quadratic factors (cf. § 23). §25. The linear 4-point and its associated 4-point are reciprocal for 1 = 0. Let/„ = o represent A, B,C, D. Then f'^ = 2^J{i2p-D)H:,-\-P{I^-\-^2p)f:, = o represents A' , B' , C',D' {ci. §22). For 1 = 0, fx = o is equivalent to Hx — o. . P Since I of Hx is — (cf. Clebsch, op. cit., p. 141), if / = o, / of Hx is zero. Therefore A", B" , C" , D" will be represented by the Hessian of Hx, 'i.e.,f'J=-fx—7Hx = o (cf. Clebsch, op. cit., p. 139). But since I = o f'J = o reduces to/x = o and the sets A, B, C, D and A', B' , C, D' are reciprocal. If gu = ois a cubic representing three distinct points, ^u — o, its Hessian represents the two points either of which taken with the original three given points form a quartic for which I = o, i.e., form a reciprocal four-point. § 26. yl 4-point and its associated 4-point are not in general reciprocal. The 4-point A : Oa = o; B : Ui = o; C : M2 = o; D : Ui-\-U2 = o is repre- sented by : fu = Wi W2 (wi -{- W2) (fliWi -\- aiU^ = o and the associated 4-point A', B' , C , D' is represented by: /^ = P(/3-}-42/%+24/(l2p-/3)Zf, = Let yfe=P(/3-f427^) /= 24/(12/^-/3) Then the 4-point A", B" , C" , D" associated with A', B' , C , D' is repre- sented by: f:!=k%-\-i'm=o where yfe'=r^(r3+42/'o and r = 24/'(i2/'^-7'3) H'u=(hl-\--Afu-\-(k'-^-AH^, (cf. Clebsch, op. cit., p. 139). RECIPROCITY OF CERTAIN ASSOCIATED LINEAR SETS OF POINTS 23 Therefore ■^2^\l2j'^-I'^)(^-kl-\--l^ \fu+\ 24/V(7'3+427'»)(l2p-P) If /,'/ = o reduces to /„ = o, the coefl&cient of Hu must vanish identi- cally, since I and J are independent. We shall therefore consider the relation: 24/V(/'3+42r^)(i2/^-/3) + 247'(i2r^-r3)(y^»-^;*)=o or k'i-[-i'{k'-y)=o o I'=Ik'-j-2Jkl+—l^ (cf. Clebsch, op. cit., p. 141, note). Then /' = /a[/9-i-i32/6y^- 1980/3/4+38016/'*] k'-lp = /[/9 _ I 2/6/^+4068/3/4 - 13824/6] 6 j> ^j]ii^llkH+—kl^+{—-—}\U (cf. Clebsch, op. ciL, p. 141, note). 2 2 \3 30/ Then in /' the term of highest degree in / is —iiP^J. The term of (/'3+42/'^) of highest degree in / is /33. The term of (12/'^— /'3) of highest degree in / is — /33. Then the term of k'l+l'{k^--l^) of highest degree in / is io/s»/, therefore the coefficient of Hu in/'/ does not vanish identically and/'/ = o is not equivalent to/„ = o, i.e., the 4-points A, B,C, D and A', B', C, D' are not in general reciprocal. VI. CONCOMITANT THEORY OF THE ASSOCIATED 4-POINT AND 4-LINE IN THE PLANE Let A : au= aiUi-\-a2U2-\-a3Ui= o B : Ui = o C : «2 = o D : «3 = o be four distinct coplanar points, then fu = aiUlu2U3-j-a2UiU'2Ui-\-a^UiU2ul = o represents the 4-point A, B,C, D. By taking the polar line of each point as to the triangle formed by the remaining three points we obtain the associated 4-line a', b', c', d'. a' : Xia2C3+ 0:2(1301+ a;3aia2 = o b' : — 3X1^2^3+ X2(iiCii-\- x^aia2 = o c' : rCiaaOs— 3.1^203(351+ Xiaia2 = o d' : 0:10203+ X203ai— 30;3OiO2 = o Then /i= —3 / ^x\aia\-\-^ / ^a:^o:20iO|o^+i4 / ^x\xlalalai— 20 2. o^i0;2o:30io2o3 = o For the general ternary point quartic/M = o in symbolic notation, let: I tt ^^H ^2i • • • • Invariant A = {abcY Contra variant Ix={abxY = x\ = x'^^= .... Co variant Su = (o-/3uy = si = t^,= .... and Contra variant Wx=(stxy. Then SiW^- iSi A'Ix= 2^ - 3^ - f^ For the ternary line quartic/^ = o representing 4-line a', b', c', d' let the corresponding concomitants be denoted A, lu, Wu, Sx- ^=32 . 2'^ . alalal Iu = 2T ' $aiaia*i j ^^ a\u\-\-2 ^^ o^02W?M2+3 ^^ alalulul — 8 ^ ^ Oi0203^iZ<2^3 [ 24 • o • ••• ♦ •• THEORY OF ASSOCIATED 4-POINT AND 4-LINE IN THE PLANE 25 Sx = 2'^ • S(^i^^^^3 ] 9 ^. aiaivj — 12 ^^ aiak3:»^:J:;2+ 2 14 ^^ aiaafli^i^J^j — 196 ^^alalalxlxiX^ [ I^«=233 . 3c?°afaf j 181 7 ^aje^j+36.2 ^aiaJ2fi?fi+543 ^aiaa^iM^ — 680 ^ alaiaiUlUiUi [ Then /;/ = 8iW«-i8i^^/« = 24i . 2,^afafaf ^ a.ulu^u^ = 2^' . 3^ . aVafaffu This verifies analytically the fact that the quadrangle quadrilateral configuration is reciprocal. Due two week^ after date. .A.. ; ]y5b ILU 30m-T.'l^ BERKELEY UBRAR C0bl355b4f r m-