LONGMANS' MODERN MA THEMA TICAL SERIES General Editors P. ABBOTT, B.A., C. S. JACKSON, M.A. F. S. MACAULAY, M.A., D.Sc. PROJECTIVE GEOMETRY LONGMANS' MODERNMA THEMA TICAL SERIES. General Editors: P. ABBOTT, B.A., C. S. JACKSON, M.A., and F. S. MACAULAY, M.A., D.Sc. A SCHOOL COURSE IN GEOMETRY (INCLUDING THE ELEMENTS OF TRIGONOMETRY AND MENSURATION AND AN INTRODUCTION TO THE METHODS OF COORDINATE GEOMETRY). By W. J. DOBBS, M.A., sometime Foundation Scholar of St. John's College, Cambridge. With 36i Diagrams. Cr. 8vo. Without Answers, 3S. 6d. With Answers, 4s. Answers Separately, 6d. THE TEACHING OF ALGEBRA (INCLUDING TRIGONOMETRY). By T. PERCY NUNN, M.A., D.Sc., Professor of Education in the University of London. Crown 8vo. 7s. 6d. EXERCISES IN ALGEBRA (INCLUDINGTRIGONOMETRY). By T. PERCY NUNN, M.A., D.Sc. Crown 8vo. Part I., 3s. 6d.; With Answers, 4s. Part II., 6s.; With Answers. 6s. 6d. These are companion books dealing with the teaching of Algebra and Elementary Trigonometry on modern lines. Thefirst book is written for the teacher and deals with the general teaching of the subject, and also in detail with the work in the companion book of Exercises which isfor the use of thelupil. THE TEACHING OF ARITHMETIC. By P. ABBOTT, B.A. *** A Handbookfor the teacher, to accompany the Exercises. EXERCISES IN ARITHMETIC AND MENSURA. TION. By P. ABBOTT, B.A., Head of the Mathematical Department, The Polytechnic, Regent Street, London, W. With Diagrams. Crown 8vo. Without Answers, 3s. 6d.; With Answers, 4s. THE GROUNDWORK OF ARITHMETIC. By MARGARET PUNNETT, B.A., Vice-Principal, London Day Training College, Southampton Row, London, W.C. Crown 8vo. 3s. 6d. EXERCISES IN ARITHMETIC. By MARGARET PUNNETT, B.A. Book I., Book II., Book III. THE TEACHING OF GEOMETRY. By G. E. ST. L. CARSON, B.A., B.Sc., Head Mathematical Master, Tonbridge School. SLIDE-RULE NOTES. By COLONEL H. C. DUNLOP, late Professor of Gunnery, Ordnance College, Woolwich, and C. S. JACKSON, M.A. Crown 8vo. 2S. 6d. net. PROJECTIVE GEOMETRY. By G. B. MATHEWS, M.A., F.R.S. 5s. EXAMPLES IN DIFFERENTIAL AND INTEGRAL CALCULUS. By C. S. JACKSON, M.A., and S. de J. LENFESTY, M.A. NON-EUCLIDIAN GEOMETRY. By H. S. CARSLAW, M.A., D.Sc., Professor of Mathematics in the University of Sydney. LONGMANS, GREEN, & CO., LONDON, NEW YORK, BOMBAY, CALCUTTA, AND MADRAS. lonoman' Mlobern jflatbemattcal Series PROJECTIVE GEOMETRY BY G. B. MATHEWS, M.A, F.R.S. LECTURER IN PURE MATHEMATICS IN THE UNIVERSITY COLLEGE OF NORTH WALES, BANGOR We are no other than a moving row Of Magic Shadow-shapes that come and go Round with the Sun-illumined Lantern held In Midnight by the Master of the Show. (E. FITZGERALD, after OMAR KHAYYAM.) WITH DIAGRAMS LONGMANS, GREEN AND CO. 39 PATERNOSTER ROW, LONDON FOURTH AVENUE & 30TH STREET, NEW YORK BOMBAY, CALCUTTA, AND MADRAS 1914 Made in Great Britain PREFACE THE two main objects of this work have been to develop the principles of projective geometry without making use of the theory of distance, and to give a satisfactory discussion of complex elements of space, up to three dimensions. No attempt has been made to start with the smallest sufficient number of indefinable terms and primitive propositions; there are ten primitive propositions, involving quite a large number of indefinables (besides point, plane, line, the indefinable entities), and these propositions have been chosen because they seem to be obvious to the ordinary intuition, and fall into five correlative pairs. Thus the principle of duality appears at the outset, and is emphasized more and more as we proceed. The proof of the fundamental theorem (after that of Liroth and Zeuthen) involves the notion of a converging sequence of points, as well as that of order: so it is by no means the most rigorous that could be given, but I believe it is as near to a rigorous proof as an ordinary student is likely to appreciate. The beginner is recommended to assume the theorem of Art. 51, and treat the rest of Chap. VII. as something to ruminate upon. In dealing with complex elements I have constantly consulted v. Staudt and Liiroth (Math. Ann. viii.); I hope it will be found that my treatment is in some respects vi PREFACE more elementary than that of either of them. Of course, the facts are wholly due to v. Staudt: it is only a question of how they should be presented. It is very rash to claim any theorems in projective geometry as absolutely new; but I may venture to say that the theory of projective metrics in the last chapter but one is at any rate original, and seems to me to be one which can be logically defended on the base of a strictly limited set of projective axioms. It also affords a very interesting illustration of the principle of duality. I am indebted to various friends for helpful criticisms and suggestions; to Dr. F. S. Macaulay, who very carefully read the earlier part both in MS. and in print, and saved me from making some serious mistakes; to Prof. F. S. Carey, who read some of the proofs, and made valuable suggestions about the scope of the book; to my colleague, Mr. W. E. H. Berwick, who has read nearly all the proofs and much of the MS. with great care, and has contributed a number of examples. With regard to the examples, those which Mr. Berwick and I have not made up have been taken from various sources which seemed to me by this time common property, and I have seldom attempted to name the original authors. Messrs. Veblen and Young and their publishers have very kindly allowed me to take examples from their Projective Geometry; these examples have been marked V.Y. In the same way, such references as G. 17 or B. 27 mean ~ 17 of v. Staudt's Geometrie der Lage, or ~ 27 of the same author's Beitrdge respectively. In preparing the diagrams for reproduction I have received most valuable help from my friend and former pupil, Mr. W. A. Jones, B.A. (Wales), who ungrudgingly spent much time and trouble in constructing diagrams from PREFACE vii my sketches and instructions. Every one who has tried knows that it is not an easy thing to draw such figures as these, in the space available, without either making them cramped or leaving them incomplete. Finally, I should like to say that my first acquaintance with the true theory of projective geometry began when I attended Professor Henrici's lectures at University College in the session I878-9. Whatever merits this book may possess are very greatly due to the interest in the subject which I then acquired; and I take this opportunity, as an old pupil, of expressing my gratitude to the teacher who first made me realise that mathematics is an inductive science, and not a set of rules and formulae. G. B. M. CONTENTS PAGE BIBLIOGRAPHY - - - - - - - - xiii CHAPTER I. ELEMENTARY FORMS. PRINCIPLE OF DUALITY - I CHAPTER II. ELEMENTS AT INFINITY 9 CHAPTER III. ORDER AND SENSE 14 CHAPTER IV. PROJECTION AND SECTION. ELEMENTARY PERSPECTIVITIES i8 CHAPTER V. PERSPECTIVE TRIANGLES. DESARGUES' THEOREM - 25 CHAPTER VI. HARMONIC ELEMENTARY FIGURES - - 30 CHAPTER VII. THE FUNDAMENTAL THEOREM OF PROJECTIVITY - - - 39 CHAPTER VIII. ELEMENTARY CASES OF PROJECTIVITY 49 x CONTENTS CHAPTER IX. PAGE PLANE HOMOLOGY. ELATION - 62 CHAPTER X. PROJECTIVITY OF PLANE FIELDS AND BUNDLES - - 68 CHAPTER XI. GROUP THEORY. 76 CHAPTER XII. INVOLUTION - - 82 CHAPTER XIII. PROJECTIVE GENERATION OF CONICS - 90 CHAPTER XIV. PASCAL'S THEOREM - 97 CHAPTER XV. POLES AND POLARS 105 CHAPTER XVI. RULED QUADRICS - I I15 CHAPTER XVII. EXTENDED THEORY OF PERSPECTIVE - 20 CHAPTER XVIII. PROJECTIVE RELATIONS OF CONICS TO EACH OTHER - - 131 CHAPTER XIX. PLANAR INVOLUTIONS. POLAR SYSTEMS 141 CONTENTS CHAPTER XX. ELLIPTIC INVOLUTIONS. COMPLEX ELEMENTS CHAPTER XXI. GENERALISED THEORY OF CONICS - CHAPTER XXII. THE THEORY OF CASTS CHAPTER XXIII. CROSS-RATIOS CHAPTER XXIV. CENTRAL PROPERTIES OF CONICS - - CHAPTER XXV. ORTHOGONAL PROPERTIES OF CONICS - - CHAPTER XXVI. PROJECTIVITIES IN SPACE - - - CHAPTER XXVII. QUADRIC SURFACES CHAPTER XXVIII. NULL-SYSTEMS - - - CHAPTER XXIX. SKEW INVOLUTIONS CHAPTER XXX. LINE GEOMETRY - - - xi PAGE I50 - 63 -69 206 - 213 - 218 225 242 253 263 - 275 xii CONTENTS CHAPTER XXXI. PAGE SPECIAL METHODS - - - 290 CHAPTER XXXII. PROJECTIVE PROBLEMS - - 312 APPENDIX. COMPLEX ELEMENTS AT INFINITY - 324 EXERCISES -- - 326 INDEX -- 347 BIBLIOGRAPHY M. Chasles, ' Aper9u historique sur l'origine et le d6veloppement des m6thodes en g6om6trie' (Paris, 1837. 2nd ed. 1875)., ' Rapport sur les progres de la geometrie' (Paris, I870). -, 'Trait6 de g6ometrie sup6rieure' (Paris, 1852. 2nd ed. i880). L. Cremona, 'Elements of Projective Geometry' (trs. C. Leudesdorf. Oxford, 1893. Original ist ed. Rome, 1872). W. Fiedler, 'Die darstellende Geometrie in organischer Verbindung mit der Geometrie der Lage' (Leipzig, I883-88). J. V. Poncelet, ' Traite des propri6t6s projectives des figures' (Paris, I822. 2nd ed. 1865-6). Th. Reye, ' Die Geometrie der Lage ' (Hannover, 1866-7. 3rd ed. Leipzig, 1886-92; 4th ed. 1898- ). H. Schrdter, 'Die Theorie der Kegelschnitte, gestiitzt auf projectivische Eigenschaften' (Part II. of Jacob Steiner's 'Vorlesungen iAber Synthetische Geometrie.' Leipzig, i866. 3rd ed. I898). --—,' Theorie der Oberflichen zweiter Ordnung und der Raumkurven dritter Ordnung als Erzeugnisse projectivischer Gebilde' (Leipzig, i880). K. G. Ch. v. Staudt, 'Geometrie der Lage ' (Niirnberg, I847).,' Beitrage zur Geometrie der Lage' (Niirnberg, 1856-7). J. Steiner, 'Systematische Entwickelung der Abhangigkeit geometrischer Gestalten von einander' (Berlin, 1832= Werke, herausgegeben von C. Weierstrass, i. pp. 229 ff.). xiv BIBLIOGRAPHY O. Veblen and J. W. Young, ' Projective Geometry,' vol. i. (Boston, etc., I9I0). A. N. Whitehead, 'The Axioms of Projective Geometry,' and ' The Axioms of Descriptive Geometry ' (Cambridge, I907). For a mixed analytical and geometrical treatment of the subject (so far as real elements are concerned), G. Salmon's treatises (' Conics,' ' Solid Geometry,' ' Higher Plane Curves,' and 'Modem Higher Algebra') are still unsurpassed; Grace and Young's 'Algebra of Invariants' gives many important theorems in an elegant analytical form. C. M. Jessop's 'Treatise on the Line Complex' is the only English work on line geometry; before consulting it, the student is recommended to read Plicker's ' System der Geometrie des Raumes ' (Diisseldorf, I846); this is still the classical work on this theory. Students who are interested in the history of projective geometry will find help in the prolegomena to C. Taylor's 'Introduction to the Ancient and Modern Geometry of Conics'; in the works of Apollonius, edited by Heiberg, and in the paraphrase of them by T. L. Heath. All the works of the last-named author are valuable, because he has really succeeded in appreciating the attitude of Greek mathematicians towards their science. Abundant references to papers and text-books will be found in the ' Encyclopadie der mathematischen Wissenschaften' (III. A B 5 A. Schoenflies: 'Projectivische Geometrie'); by consulting this article, the student will be able to follow up the original sources as far as he may care to do. Among modern writers on pure geometry, Castelnuovo and Enriques deserve speciallattention. CHAPTER I. ELEMENTARY FORMS. PRINCIPLE OF DUALITY. 1. Elements. The elements taken for granted in projective geometry are the plane, the point, and the straight line, or, as we shall usually call it, the line. Most people have acquired a fair conception of the meaning of these terms from things that they constantly see, such as bricks, walls, ceilings, mirrors, and so on. Perhaps the best object to suggest the right ideas is a natural crystal: its corners, edges, and faces are what, in ordinary language, we should call points, straight lines, and planes respectively. More accurately, an edge of a crystal suggests a segment of a line, and a face of it suggests a plane polygon. A complete line is obtained by supposing a segment extended as far as possible without destroying its straightness; and in the same way the complete plane may be thought of as the utmost enlargement, with preservation of flatness, of any finite plane lamina. From this intuitive point of view the line and plane are of infinite length and area, and have no definite boundaries. 2. Elementary Propositions. Without making any attempt to define these primitive terms plane, point, and line, we proceed to state certain propositions about them, and their relations to each other. These theorems are not all independent, but form a convenient starting-point for the subject, and may be regarded as equally obvious to the ordinary intuition. M.P.G. A 2 PROJECTIVE GEOMETRY We shall generally denote planes by small Greek letters a, p, y, etc., points by capital letters A, B, C, etc., and lines by small letters a, b, o, etc. The propositions are intentionally stated in general terms; apparent exceptions will be considered in the next chapter (i) A plane contains an unlimited number of points and lines. (ii) Through a point can be drawn an unlimited number of planes and lines. (iii) A line contains an unlimited number of points. (iv) Through a line can be drawn an unlimited number of planes. (v) Any two planes a, /, intersect in a line which may be denoted by ao., and called the meet of the planes. Every point on this line, and no other point, is contained in both a and /3. (vi) There is one and only one line which contains two given points A, B. This is denoted by AB, and called the join of A and B. Every plane through this line, and no other plane, contains both A and B. (vii) Given a plane P and a line b, then either b is contained in p or else it meets it in one and only one point, b/3. (viii) Given appoint B and a line b, then either B lies on b, or else there is one and only one plane Bb, which contains them both. (ix) Two lines a, b, in general, do not meet: if they do, there is a single point ab common to both, and a single plane ab containing them both. (x) Two lines, in general, are not in the same plane: if they are, they meet in a point. Of course, every word in these statements, except those italicised, is supposed to have its meaning known by the reader. Thus, "in," "through," "meet," "contain," are intended to suggest the ideas generally associated with them as the result of intuition. It is impossible to give a geometrical definition of what is meant by a point being " in " a plane: all we can do is to appeal to intuition by ^ /'I 'I, ELEMENTARY FORMS 3 means of diagrams, models, and so on, until the proper conception has been formed. 3. Conjunction. There is, however, a way of bringing propositions (i) to (x) more or less within the scope of a single idea. Taking any two elements of different names (plane, point; plane, line; point, line), we may distinguish them as being disjoint or conjoint. For instance, a point and a plane are conjoint if the plane contains the point, or, which is the same thing, if the point is in the plane. Similarly, a line and a point on it are conjoint; and a plane and a line in it are conjoint. Propositions (i) to (x) can be all stated so as to involve this idea of conjunction; for instance (vii) is equivalent to saying that " a plane and a line are conjoint, or else there is a single point which is conjoint with both." It will be observed that conjunction and disjunction are complementary terms: two dissimilar elements are either conjoint or disjoint, and the latter is the more general relation. 4. Duality. An examination of the ten elementary propositions will show that there is a kind of correspondence between (i) and (ii), (iii) and (iv),...., (ix) and (x), of the following character. Whenever in one theorem of the two the term " plane " occurs, the term " point " occurs in the other, and vice versa; while " line " in the one corresponds to " line " in the other. If we introduce the term " conjunction" the correspondence becomes still more evident: thus, taking (vii) and (viii), we have, as equivalent statements of these theorems: Given a plane [3 and a line b, there is a single point b1i conjoint with both, or else P and b are conjoint. Given a point B and a line b, there is a single plane Bb conjoint with both, or else B and b are conjoint. This fact we express by saying that the ten elementary propositions fall into five correlative pairs. Since there is this correspondence between our elementary propositions, it follows logically that there must be a similar correspondence between all deductions we can make from them. mrs is (0) iA<s 6^g ^(^Sl to f s An/ r 1> cY/b/^A i5H bna.l-< 0 PROJECTIVE GEOMETRY the basis of the principle of duality, one of the most beautiful features of projective geometryC$>From the statement and proof of any projective proposition we can deduce by a proper change of terms the statement and proof of the correlative proposition. It may happen that a proposition is its own correlative, but this is exceptional; so that practically the principle of duality halves our labour, because all we have to do is to translate, so to speak, the enunciation of a proved proposition into that of its correlative, and then infer the latter at once. The force of this argument will become more evident as we proceed. 5. Definitions. Collinear, etc. Points conjoint with the same line are said to be collinear: planes conjoint with the same line are coaxial: elements conjoint with the same plane are coplanar: elements conjoint with the same point are concurrent. Two lines which are not concurrent are said to be skew. 6. Two planes a, P are always coaxial. If three planes a, fp, y are coaxial, the meets fy, ya, aft coincide; if these three lines do not coincide, they must be all different, because if, for instance, fly, ya coincide, this means that a, f meet y in the same line I, and therefore I is conjoint with both a and i/, and is consequently identical with at3. When a, f3, y are not coaxial, a/f contains all the points conjoint with a and /, and hence apu.y is the only point conjoint with a, /, and y. This same point may be denoted by /?y.a or y/3.a or ya./3, etc.; so we arrive at the conclusion that Three planes a, /3, y which are not coaxial are conjoint with a single point, which may be denoted by apy (or fay, etc., all permutations of a, /3, y being allowable). The correlative proposition is that Three points A, B, C which are not collinear are conjoint with a single plane, which may be denoted by ABC (or BAC, etc.). In ordinary language, these propositions amount to saying that three planes which enclose a trihedral angle ELEMENTARY FORMS 5 meet at a singleypoint; and three points which are the vertices of a triangle lie in a single plane. 7. Postulates of Construction. We assume the possibility of " constructing" any element for which sufficient pro, jective data are provided; for instance, given any three points which are not collinear, we assume the possibility of constructing the plane which contains them all. Practically, we are confined, for the most part, to plane constructions: such diagrams as are given to illustrate constructions in three dimensions are merely pictorial makeshifts, and the student must practise the art of mentally visualising the actual constructions in space. 8. Correlative Vocabulary. The principle of duality is so important that we give here a brief vocabulary of correlative terms, arranged in parallel columns. It will be observed that there is no distinction between the columns; if any term (t) is entered in the left-hand column with the correlative term (t') in the right-hand column, this implies an entry t' on the left opposite t on the right, unless t, t' happen to coincide. But for the sake of brevity, when we have made a (t, t') entry, the corresponding (t', t) entry has been omitted. Plane a. Line a. Line a/3. Point a/3y. Point ab. Plane ab, containing two coplanar lines a, b. Coaxial planes. Coplanar lines (points). Three planes, the faces of a trihedral angle. All the points in a plane a. All the points on a line. Point A. Line a'. Line AB. Plane ABC. Plane Ab'. Point a'b', the intersection of two concurrent lines a', b'. Collinear points. Concurrent lines (planes). Three points, the vertices of a triangle. All the planes through a point A. All the planes through a line. 6 PROJECTIVE GEOMETRY By way of exercise, the student may cover up either column and practise the translation of the uncovered one into its correlative. Ultimately this process of dualisation becomes almost mechanical, except in the case of very complicated theorems. Besides projective geometry in three dimensions, there is a projective geometry of the plane. In this the system of correlative terms is different, so a specimen of it is given below. Point A. Line a. Line AB. Point ab. Collinear points. Concurrent lines. Three points, A, B, C, the Three lines, a, b, c, the vertices of a triangle. sides of a triangle. The locus of a point. The envelope of a line. Order of a locus. Class of an envelope. Whenever, in what follows, the text is arranged in parallel columns, the statements made in them are correlative, for two or three dimensions as the case may be: the reader will easily see which correspondence is intended. 9. Row of Points. A line g contains an unlimited number of points (2. iii). If we give names A, B, C,... to these points, and consider them as individuals, we have a set of points which is called a row of points, or simply a row. The row is complete if it contains all the points of the line: we obtain a partial row if we select some of the points on the line. For instance, the divisions on a graduated scale define a partial row on its edge. As a rule, we shall understand by row a complete row. Any two points A, B determine a row, of which they are elements. The line on which a row is situated is called its base, and must be carefully distinguished from the row itself. Axial Pencil of Planes. Through a line g can be drawn an unlimited number of planes (2. iv). Calling these a, /, y... we have a figure consisting of all the planes through g. This is called an axial pencil, and may be denoted by (a/{y...) or more briefly by (at), or even by (g). The line ELEMENTARY FORMS 7 g is called the axis of the pencil. Any one line, or any two planes, or any two points, determine an axial pencil. An axial pencil fills up the whole of space, and may be considered as the result of rotating a plane about any line in it. A way of partially realising it is to look at the fly of a clock which is striking the hour. The fly is a metal rectangle which, when the clock strikes, spins quickly about an axis, and -looks like a sort of shadowy cylinder, owing to the persistence of vision. The row and the axial pencil are correlative figures in space. Flat Pencil. Let (ABC...) be a row on g, and S any external point (that is, one which is not on g): then the lines SA, SB, SC... all lie in the plane Sg, and all pass through S. These lines, which fill up the whole of the plane sg, form what is called a flat pencil. The general definition of a flat pencil is that it is a set of lines of which every three are concurrent and also coplanar: thus it is associated with a definite point, and also with a definite plane. The point is called its centre, and the plane its base. Conversely, a point and a plane which are conjoint determine a flat pencil. The spokes of a wheel or a chimney-sweep's brush give some idea of a flat pencil. Correlatively, a flat pencil may be obtained by cutting an axial pencil (a/py...) by an external plane o. The result is a flat pencil o(a./3y..) of which Ca is the base, and ra/3 the centre: its rays are (ra, a-/, cry, etc. The correlative figure, in space, corresponding to a flat pencil, is also a flat pencil: this follows from the definition given above. The fact is that just as the line has a threefold aspect, first as an independent element (a), secondly, as a join (AB), and, thirdly, as a meet (a/3), so a flat pencil has a three-fold aspect, first as an independent figure, secondly as a form S(ABC...), derived from a point and a row, and thirdly as a form r-(a//y...) derived from a plane and an axial pencil. In the plane a flat pencil corresponds to a row. Here the /.rz. Aser..is-A CII b' &.'? 1 8S PROJECTIVE GEOMETRY axial pencil does not exist, and the flat pencil is simply a \ set of concurrent lines. 10. Elementary Forms. The axial pencil, the row, and heflat pencil are said to be elementary forms of the first rank, or nemens elementary forms. Besides these there are two elementary forms of the second rank, called the plane field and the bundle respectively. Plane Field. Taking any plane a, the assemblage of all points, lines, rows, and flat pencils contained in it, is called a plane field, and may be denoted by (a). From a geometrical point of view this is more complex than any elementary form of the first rank: for instance, every point in a is the centre of a flat pencil belonging to (a), and every line in a is the base of a row belonging to (a). Bundle. Taking any point A, the planes and lines which pass through A are the elements of what is called a bundle. Denoting the bundle by (A), this contains every flat pencil which has its centre at A, and every axial pencil whose axis goes through A. The plane field and the bundle are correlative forms. The plane a is called the base of the field (a), and the point A is called the centre of the bundle (A). It should be noticed that a plane field contains no axial pencils, and that a bundle contains no rows. The assemblage of all elements and all elementary forms of the first and second ranks is called the elementaryform of the third rank. Assuming that our classification is now complete, we have in all six elementary forms, three of the first rank, two of the second, and one of the third. CHAPTER II. ELEMENTS AT INFINITY. 11. Consider a flat pencil (abc...), and a line I in its plane, which does not go through 8, the centre of the pencil (Fig. I). According to our elementary propositions each point R upon I determines a ray SR of the pencil, and con\\^s ft FI. A. \ c versely each ray r of the pencil meets I in a definite point/ rl. Now this is contrary to the usual theory of parallels, according to which there is just one ray, p, of the pencil, which is parallel to I, and does not meet it at all. For practical purposes, it is desirable to keep to the ordinary theory of parallels, but, on the other hand, it would be very tedious to state, in terms of parallels, all the exceptional cases of our general theorems; for instance, we should have to say " Two coplanar lines either meet in a point, or are parallel," and so on. 10 PROJECTIVE GEOMETRY We avoid this difficulty by a certain convention of language, which is justified by its usefulness and consistency. In the figure that we are considering we say that the parallel lines p, I " meet at infinity ": and since there is only one parallel to I through S we say that there is one, and only one, point at infinity on 1. And although, in the ordinary sense, this point does not exist, we can introduce a symbol P, or pi, of the same type as those used for actual points on I, and use SP as an alternative symbol for p. The main reason why this does not lead to any confusion is that if two lines a, b are each parallel to a third line c, they are parallel to each other. From our present point of view, this is strictly analogous to the fact that if two lines a, c meet in P, and b, c also meet in P, then a, b meet in P and in no other point. In fact, we come to regard the first theorem as a particular case of the second. 12. Parallel Flat Pencils. If, in the plane of the figure, we draw all the lines, such as p, which are parallel to I, we obtain a flat pencil of parallels: this we regard as a limiting case of an ordinary flat pencil whose centre passes away to infinity. There are as many pencils of parallels in a plane as there are directions of lines in it. 13. Line at Infinity in a Plane. Consider, now, a plane f3 and a point S outside of it. Every plane but one that can be drawn through S actually meets P in a line: the exceptional plane (2, say) is parallel to /f. Here, again, we adopt a convention, and say that z intersects /f in a line at infinity. The fact that only one plane through S is parallel to /3 is expressed by saying that P contains one, and only one, line at infinity. This is consistent with what has gone before, because if we use the symbol u for the line at infinity in /, and I is any other line in /, the one point at infinity on I may be regarded as lu, its intersection with u. 14. Exceptional Axial Pencils and Bundles. The Plane at Infinity. All the planes in space which are parallel to a given plane P pass through the same line at infinity and form a pencil of parallel planes. This we regard as the ELEMENTS AT INFINITY 11 limiting form of an ordinary axial pencil, whose axis passes away to infinity. A parallel bundle consists of all the lines and planes which are parallel to a given line. This is what becomes of an ordinary bundle when its centre moves away to infinity. Finally, we regard all the " infinitely distant " elements of space as those of a special plane field whose base is the plane at infinity. Every other plane meets the plane at infinity in a line at infinity and every ordinary line meets the plane at infinity in a single point at infinity. 15. Illustrations. To see how this convention works in practice, it is instructive to take one or two particular propositions, and find out how they are modified when some of the given elements are at infinity. For instance, consider the theorem: " Three points A, B, C which are not collinear, determine a plane which contains them all ": and suppose first that c is at infinity while A, B are not. To fix C we draw a line c in such a direction that C is the point at infinity on it: this line must not be parallel to AB, otherwise A, B, C would be collinear. The theorem now becomes, in ordinary language: " Two points A, B, and a line c not parallel to AB, determine a plane which passes through A and B and is parallel to c." By supposing B as well as C to go to infinity, we obtain the theorem: " Given a point A and two lines b, c which are not parallel, there is a unique plane through A which is parallel to both b and c." Again, take the following problem, of which the solution is often required: " Through a given point A draw a line to meet two given skew lines b, c both external to A." The required line is perfectly definite, being the meet of the planes Ab, Ac. If A is at infinity, the problem is: " Draw a line parallel to a given line a so as to meet 12 PROJECTIVE GEOMETRY two given skew lines b, c, neither of which is parallel to a." The required line is the meet of planes through b, c respectively, each parallel to a. The cases (i) when b is at infinity, and (2) when A, b are both at infinity, are left as exercises for the reader. 16. The Line as a closed Figure. Let us return to Fig. I where we have the flat pencil (abc...) cut by I in the row (ABC...). Starting with any ray SR (other than p) we may suppose it to rotate about S in the positive sense: as it turns through two right angles it describes the whole of the pencil. What happens to R, the join of the moving ray with I? At first it moves along I in the sense RAB..., taking up the positions R, A, B, etc.; and as the ray comes back to its initial position, the intersection reappears to the left of R, travelling in the same sense as before. Now, on account of the one-one correspondence between rays and points, we naturally try to think of R describing the whole row in one and the same sense, as r describes the whole pencil in one and the same sense. We can do this, if we make proper allowance for the peculiar properties of the point at infinity on I. Suppose we start with the moveable ray in the position p, when it is parallel to I. We speak of the intersection pi, and use a symbol P for it, but we cannot think of P as being either to the right or to the left of any ordinary point, such as R, upon 1. But however little we turn the ray round in the positive sense, we get a definite intersection to the left of R, and after this everything is normal until the moving ray comes into coincidence with p, after turning through two right angles. So in a certain sense of the words we may regard a line as a closed figure, when we include among its points the one point at infinity, as we have defined it. But we must be careful only to apply this idea so far as it is really valid, remembering that it depends upon our convention about elements at infinity in general. To assist the ideas, suppose a circle drawn through S ELEMENTS AT INFINITY 13 (Fig. 2), and let the rays SA, SB, SC, etc. cut the circle in A', B', C', etc. For each point R upon I there is one definite corresponding point R' on the circle and we may call A'B'C'... the circular image of the row ABC.... Observe that p meets the circle in P', the image of the point at infinity on I, and that if the tangent to the circle at S cuts I in T, we must put S=T'. The advantage of the circular image is that it is actually closed, and completely visible, while the ~__ __ P,>,"'"' S =T' / FIG. 2. order of any points A, B, C, etc., on I is similar to that of their images on the circle. 17. For the present, the reader will do well to regard the whole system of " elements at infinity" as a set of conventional terms, introduced for the purpose of securing a valuable economy of thought and language. The idea that there is a special plane field " at infinity" in the sense of being inaccessible or the like, is not theoretically necessary: it is a convenience, because we can thus adopt the ordinary theory of parallels, and make applications of projective geometry to practical things, such as drawing and engineering and mechanics. But projective geometry is not necessarily restricted in this way; it may, in fact, be constructed for non-Euclidean spaces, namely, those for which Euclid's theory of parallels is false. CHAPTER III. ORDER AND SENSE. 18. Complete Angles. Order of Rays in a Pencil. Two lines a, b intersecting in S are said to be the sides of two complete plane angles. Each of these complete angles is made up of what we usually call two vertically opposite angles. We shall say that the two complete angles are complementary, and denoting either of them by zab, we shall call the other L b. a. A variable straight line may be supposed to rotate about S so as to describe the flat pencil of which S is the centre and a, b are two rays. The rotation may be of two kinds, which we distinguish as being of the positive and negative sense respectively. Let the variable ray r start from the position a and rotate in the positive sense until it first reaches the position b. We shall say that r has described the positive angle ab. If it continues its rotation until it first resumes the position a, it describes the positive angle b.a. Let c be any ray of the pencil which r passes over as it describes the positive angle ab. We shall say that a, c, b occur in the positive order acb. Similarly, if d is any ray which r passes over as it describes the positive angle b.a, we shall say that b, d, a occur in the positive order bda. If the angle ab is described in the negative sense, so that r starts in the position b, and rotates in the negative sense ORDER AND SENSE 15 until it first reaches the position a, we have an order boa the opposite of acb. If acb is a positive order, then b is outside the positive angle ac, and therefore within the positive angle c.a. Therefore oba is a positive order. Similarly a is within b.c, and therefore bac is a positive order. If, on the other hand, acb is a negative order, it can be shown in the same way that cba, bac are negative orders. The general result, which may be verified by drawing figures for all possible cases, is that abc, boa, cab are orders of the same sense, and that acb, cba, bac are orders of the opposite sense. Which orders are positive and which negative depends upon the definition of positive rotation, and the relative positions of a, b, c. Exactly similar results hold good for an axial pencil: we have only to replace a, b, c by a, /3, y and consider dihedral angles instead of plane angles. 19. Order of Points on a Row. The base of a row may be considered as being divided by any two points A, B upon it into two complementary segments AB, B.A, and we may suppose the first of these to be finite. There are two opposite senses of motion along the line, either of which we may call positive, the other then being negative. Suppose the senses defined so that motion from A to B along AB is positive: then so is that from B to A along B.A. If a point C is on AB, we have positive orders ACB, CBA, BAC (the two latter involving passage through infinity) and negative orders BCA, CAB, ABC. On the other hand, if C is on B.A, we have positive orders BOA, CAB, ABC and negative orders ACB, CBA, BAC. 16 PROJECTIVE GEOMETRY Similar results hold if motion from A to B along AB is supposed to be negative: so that we have results agreeing with those obtained for pencils. 20. Let (ABC...) be any row in a plane and 8, T any two points in the plane on opposite sides of the base of the row. By joining 8, T to the points of the row we obtain two flat pencils (SA, SB, SC...) and (TA, TB, TC...). In whichever sense the row is described, the two pencils will be described in opposite senses. Thus all the flat pencils in the plane, except those which have centres on the base of the row, fall into two classes, those whose sense agrees with that of the row, and those whose sense does not. Points A, B, C, D, E... of a row are said to occur in that order, or to form an order ABODE..., when it is possible for a point to travel along the base of the row, continually in the same sense, and starting from A, so as to describe successive segments AB, BC, CD, DE, etc., no two of which overlap. One of these segments, but not more than one, may be infinite. The opposite order is...EDCBA. Similarly, rays a, b, c, d, e... of a flat pencil form an order abode..., when it is possible for a line to rotate round the centre of the pencil, always in the same sense, so as to describe successive angles ab, be, cd, de, etc., no two of which overlap. The opposite order is...edcba. 21. Separated Pairs of Elements. Let the points A, B be taken on a line I so as to define the complementary segments AB, B.A. If C is any point on AB and D any point on B.A, then C, D are said to be separated by A, B. In fact, we cannot go along the line from C to D without passing through either A or B: this is not the case if C, D are both on AB, or both on B.A. If C, D are separated by A, B, then A, B are separated by C, D. For, if not, one of the segments AB, B.A would contain neither C nor D, and therefore the complementary segment (B.A or AB) would contain both C and D, contrary to ORDER AND SENSE 17 hypothesis. The proposition is intuitively obvious if we take the circular image of the row (Art. I6). If A, B separate C, D the four points occur in the order ACBD, and conversely. Similar theorems are true for pencils, and may be proved in the same way. M.P.G, CHAPTER IV. PROJECTION AND SECTION. ELEMENTARY PERSPECTIVITIES. 22. Projection and Section. Let (ABC...) be any row, and S any external point. By joining SA, SB, SC, etc., we obtain a flat pencil called the projection of (ABC...) from 8, and denoted by S(ABC...). Let (abc...) be a flat pencil, and S any point not in its plane: then the planes Sa, Sb, So, etc., form an axial pencil, which is called the projection of (abc...) from 8, and denoted by S(abc...). Let (ABC...) be a row, and g any line skew to its base: then the planes gA, gB, go, etc., form an axial pencil called the projection of (ABC...) from g, and denoted by g(ABC..). Let (a/y...) be any axial pencil, and o- any external plane. The lines c-a, o-/, -ry, etc., form a flat pencil, called the section of (a/iy...) by ra and denoted by (a(a7y...). Let (abc...) be a flat pencil, and a- any plane not passing through its centre. The points a-a, a-b, a-c, etc., form a row which is called the section of (abc...) by -( and denoted by ra(abc...). Let (aP/y...) be an axial pencil, and g any line skew to its base. The points ga, g/, gy, etc., form a row called the section of (a/gy...) by g, and denoted by g(a/3y...). The last three of these propositions are correlative to the first three. PROJECTION AND SECTION 19 23. Elementary Perspectivities. Consider a row (ABC...) and its projection S(ABC...) from a point S. We can set up a one-to-one correspondence between the points of the row and the rays of the pencil by saying that to any point A of the row corresponds that ray, SA, of the pencil which passes through it, and conversely, to any ray a of the pencil corresponds that point, A, of the row which lies upon it. In other words, two elements of the row and the pencil correspond when they are conjoint (Art. 3). This correspondence is called a perspectivity; and, when it is established, the row (ABC...) and the pencil S(ABC...) are said to be put in perspective. We shall often say that a row is perspective to a pencil when it is a section of the pencil, it being understood that this tacitly implies the correspondence of conjoint elements. Perspectivity, connecting a row and a pencil, is only a particular case of one-to-one correspondence. For instance, we might set up a correspondence such that any point A of the row corresponds to that ray of the pencil which is perpendicular to SA: but this is not a perspectivity. An axial pencil and a row which is a section of it are put into perspective by making each point of the row correspond to that one plane of the pencil which passes through it. An axial pencil and a flat pencil which is a section of it are put into perspective by making each ray of the flat pencil correspond to that one plane of the axial pencil which passes through it. Thus there are three elementary perspectivities, each connecting two dissimilar forms by associating conjoint elements. 24. Compound Perspectivities. It is convenient to extend the definition of perspectivity so as to include cases where the associated forms are similar. This is done by the following definitions. Two rows are said to be perspective when both are perspective to the same flat pencil. If (abc...) is the pencil and (ABC...), (A'B'C'...) are the rows, corresponding points 20 PROJECTIVE GEOMETRY A, A' are those which lie on the same ray, a, of the pencil. The centre of the pencil must be external to the rows. Two axial pencils are said to be perspective when each is perspective to the same flat pencil. If (abc...) is the flat pencil and (af/y...), (a'/3'y'...) the axial pencils, corresponding planes a, a' are those which pass through the same ray, a, of the flat pencil. The plane of the flat pencil cuts the axes of the axial pencils at one and the same point. Two flat pencils are perspective when they are either perspective to the same axial pencil, or perspective to the same row. Let (abc...), (a'b'c'...) be both perspective with (a/py...): then corresponding rays a, a' are those which lie in the same plane a of the axial pencil. Similarly, if the flat pencils are both perspective to (ABC...) corresponding rays a, a' are those which pass through the same point A of the row. There are, in fact, three distinct cases to consider. The perspective flat pencils (abc...), (a'b'c'...) may be coplanar without being concentric: in this case they are perspective to the same row, without being perspective to the same axial pencil. Secondly, they may be concentric without being coplanar: in this case they are sections of the same axial pencil by planes which meet its axis in the same point, and the flat pencils are not perspective to the same row. Thirdly, the perspective flat pencils may be neither coplanar nor concentric: in this case they are perspective to the same row (ABC...) and the same axial pencil (a/y...). For suppose S, T the centres of the flat pencils: then assuming them both perspective to (ABC...) they are also perspective to (STA, STB, STC...), which is an axial pencil whose axis is ST. Similarly, if cr, r are the planes of the flat pencils, and they are both perspective to (ay...), then they are also perspective to the row (o-Ta, o-r, -ry,...), the base of which is or. Fig. 3 is intended to suggest an axial pencil (a/py...) and a flat pencil (abc...) perspective to it: (a/py...) is the projection of (abc...) from B. It will be noticed how much more difficult this is to suggest than the correlative PROJECTION AND SECTION 21 figure. Very often, when we are dealing with correlative theorems, we shall only give a figure for one of them, leaving the reader to construct, or mentally visualise, the other. 25. Plane Field and Bundle. A bundle (8) is cut by an external plane c- in a plane field (cr). If with each ray a of the bundle we associate the point A in which it is cut by FIG. 3, o, the bundle and plane field are put into perspective. To every axial pencil in (S) corresponds a perspective flat pencil in (cr); to every flat pencil in (S) corresponds a perspective row in ((r); and vice versa. It may be noted that a bundle does not contain any rows, just as a plane field contains no axial pencils. Two plane fields ((r), (a') are in perspective when they are both perspective to the same bundle (S). Corresponding points A, A' are those which lie on the same ray of (S). Every point on cT-' corresponds to itself. To every elementary form in (c) corresponds a perspective elementary form of the same kind in (o'). Two bundles (8), (S') are in perspective when they are 22 PROJECTIVE GEOMETRY both perspective to the same plane field (o). Corresponding rays a, a' are those which meet at the same point of (r). Every plane through SS' corresponds to itself. To every elementary form in (S) corresponds a perspective form of the same kind in (S'). 26. Vanishing Points and Lines. When two rows (ABC...) and (A'B'C'...) are in perspective, AA', BB', CC', etc., the joins of corresponding points, meet in a single point S. (See Fig. 4.) Suppose that I, I' are the bases of XP A /R B C /1A FIG. 4. the rows. If we draw through S a line parallel to I', this will, in general, meet I in an ordinary point P, which we take to correspond to the point at infinity (P') on I'. Similarly, a line through S parallel to I generally meets I' in a point Q' corresponding to the point at infinity (Q) on I. The points P, Q' may be called the vanishing points of the rows on I, 1' respectively. In the exceptional case I, I' are parallel: P, Q' are now coincident, each being the point at infinity where I, I' meet. When two plane fields are in perspective, and their bases are not parallel, the line at infinity in either corresponds to an ordinary line in the other: this is called the vanishing line in its plane, and is the meet of that plane with the plane through 8, the centre of perspective, drawn parallel to the PROJECTION AND SECTION 23 other plane. The vanishing lines are both parallel to the line which is common to the two fields. 27. Data Fixing Perspectivities. A perspective between two coplanar rows is established by taking any two points A, B on the first row, and any two points A', B' to correspond to them on the second row. Namely, if AA', BB' meet in S we may put each row into perspective with the flat pencil S(AB...). It is understood that none of the four chosen points is the intersection of the bases of the rows. Given two coplanar flat pencils, we can put them into perspective so that two arbitrary rays a, b of the one may correspond to two arbitrary rays a', b' of the other; namely, by putting them both into perspective with the row (aa', bb'...). None of the four selected rays, however, can be the join of the centres of the pencils. Given two axial pencils, whose axes intersect, we can put them into perspective so that any two planes a, /3 of the one may correspond to any two planes a', /' of the other: namely, by putting each into perspective with the flat pencil (aa', /3'...). None of the four selected planes, however, can be the plane containing the axes of the given pencils. 28. Data for two Plane Fields. Given two plane fields (a), (-') there are various ways of fixing a perspectivity between them by choosing elements to correspond. Perhaps one of the simplest is this: on cro' take any point P, and through it draw any two lines I, I' in o-, a-' respectively: now take any two points A, B on 1, and any two points A', B' on I', and let AA', BB' meet in S. Then by putting (cr), (a') both into perspective with (S), we have a perspectivity determined by the three corresponding pairs (I, I'), (A, A'), (B, B'). Another way is the following. In ar, a' draw any two lines u, v' each parallel to a —'. Then we can put the fields into perspective in such a way that u, v' are the vanishing lines in their planes. In fact, the centre of perspective may be anywhere on the meet of the planes through u, v' 24 PROJECTIVE GEOMETRY parallel to o-', o- respectively. If, on v', we assign a point P' to correspond to a definite point at infinity in o-, the centre of perspective is determined. The reader should have no difficulty in verifying these statements, which are of considerable importance in pictorial perspective. CHAPTER V. PERSPECTIVE TRIANGLES. DESARGUES' THEOREM. 29. We shall now prove a theorem which is of great importance for all that follows, and is a typical example of a projective proposition. FIG. 5. Let there be two triangles ABC, A'B'C', in different planes, and such that the lines AA', BB', CC' meet in a point S: 26 PROJECTIVE GEOMETRY then the pairs of corresponding sides (BC, B'C'), (Ct. C'A'), (AB, A'B') are respectively coplanar, and their joins P, Q, R are collinear. (Fig. 5.) To prove this we observe that since BB', CC' intersect they determine a plane SBC: now BC, B'C' are lines in this plane, and therefore meet in a point P. Similarly, CA, C'A' meet in a point Q, and AB, A'B' in a point R. Now P is in the plane ABC and also in the plane A'B'C'; therefore it lies on I, their line of intersection. Similarly, Q, R lie on I, and hence P, Q, R are collinear. Conversely, if the pairs of corresponding sides intersect, the intersections must be on 1, and are therefore collinear Moreover, the lines AA', BB', CC' must be concurrent: for since BC, B'C' are coplanar, BB', CC' will meet, and, similarly, CC', AA' and AA', BB' will meet. Finally, since AA', BB', CC' are not coplanar, and each two of them meet, they must concur in one and the same point S. The triangles ABC, A'B'C' are said to be in perspective: S is the centre, and I the axis of the perspective. The triangles are, in fact, perspective sections of the trihedral angle SABC contained in the bundle (8). As an example of dualisation, the correlative theorem may be stated, though it is hardly ever used. Let there be two trihedral angles a/3y, a'/3'y' with different angular points, and such that the lines aa', 3/3', yy' enclose a plane triangle; then the pairs of lines (ly, P'y'), (ya, y'a'), (a3, a'3') are respectively coplanar, and their planes all pass through the same line, namely, the join of the vertices of the given trihedral angles. The two trihedral angles are, in fact, projections of one and the same triangle. The converse theorem is also true. 30. Case of Coplanar Triangles. Consider, now, two coplanar triangles ABC, A'B'C', such that AA', BB', CC' are concurrent in S: it will still be true that the pairs of corresponding sides meet in three collinear points P, Q, R. (Fig. 6.) To prove this, we draw any line through S that is not in the plane of the figure, and from any two points U, V on PERSPECTIVE TRIANGLES 27 this line project the vertices of the triangles ABC, A'B'C' respectively. Since UV, AA' meet (in S) they are coplanar: hence UA, VA' meet in a point A". Similarly, UB, VB' meet in a point B", and UC, VC' in a point C". The triangles 'i//... X S * _ FIG. 6. ABC, A"B"C" are in perspective with U, so, by the earlier proposition, their pairs of corresponding sides intersect in three points on the meet (I) of the planes ABC, A"B"C". But the same thing (with a change of U to V) is true of the triangles A'B'C', A"B"C", and the points of concurrence P, Q, R are the same as before, namely, the points where I meets the sides of A"B"C". Therefore BC, B'C' meet at P; and similarly for the other pairs of sides. The converse theorem is most easily proved by the indirect method. Suppose that P, Q, R, the meets of (BC, 28 PROJECTIVE GEOMETRY B'C'), etc., are collinear, and let AA', BB' meet in S. Then, if SC did not pass through C' it would cut B'C' in some other point D'. It would follow by the theorem just proved that A'D', AC meet PR in the same point; but this is impossible unless D' coincide with C'. It should be noticed that, in the plane, the theorem and its converse are correlative. The coplanar triangles ABC, A'B'C' are usually said to be in perspective: S being the centre, and I the axis of the perspective. This is an extension of the term ' perspective,' but need not lead to confusion; another current term is homologous. 31. The theorem about coplanar triangles in perspective is generally called Desargues' theorem. In order to prove it we have used a figure constructed in three dimensions: this is, in fact, the only way to obtain a purely projective demonstration. Various consistent systems of plane geometry have been devised 1 in which Desargues' theorem is false, though the usual projective axioms about points and lines in a plane are valid. Other proofs have been given, using a plane figure, but they all apply principles such as those of proportion, which are not of a projective kind. 32. It is a very good exercise to see what becomes of the last two figures when some of their elements go to infinity: we thus obtain a number of special theorems. For instance, two similar and similarly situated coplanar triangles are in perspective, the axis of perspective being at infinity; if, in addition, the triangles are congruent (equal in all respects), the centre of perspective is also at infinity. We have used Fig. 6 as a sort of picture of a threedimensional configuration: but it is actually a plane diagram and, as such, has remarkable properties. Starting with any two perspective triangles ABC, A'B'C' in a plane, we deduce another triangle A"B"C" perspective to each of 1 A very simple one, due to R. F. Moulton, is explained in Whitehead's Axioms of Projective Geometry, p. II. Another one is given by D. Hilbert in his Grundlagen der Geometrie. PERSPECTIVE TRIANGLES 29 them: the axis of perspective being the same for each pair, and the centres of perspective (S, U, V) being collinear. There are other perspectives in the figure: for instance, the triangles BB'B", CC'C" are in perspective, P being the centre, and UV the axis. The student should draw the correlative plane figure, and then, by applying the general principle of duality, obtain the corresponding two figures and theorems for the bundle. CHAPTER VI. HARMONIC ELEMENTARY FIGURES. 33. Complete Quadrilateral, etc. Four lines in a plane, no three of which are concurrent, form what is called a complete quadrilateral. The lines are its sides, and their six intersections are its vertices. One, two, or even three, of its vertices may be at infinity: when two are at infinity, we have a complete parallelogram. The correlative figure in the plane is the complete quadrangle consisting of four points, no three of which are collinear, and the six lines joining them in pairs. The four points are the vertices, and the six lines are the sides of the quadrangle. Correlatively, in the bundle, we have the completefour-face (or tetrahedral angle) with six edges, and the complete fouredge with six faces joining its edges two by two. In the bundle, the four-face is the correlative of the quadrangle in the plane. 34. Harmonic Conjugates. In Fig. 7 p, q, r, s are the sides of a complete quadrilateral of which K, L, M, N, A, B are the vertices: the lines AB, KM, LN are called its diagonals, and 0, C, D its diagonal points. Two vertices which lie on the same diagonal are said to be opposite. Through A draw any two lines p', r', and let any line through C meet them in K', M'. Join BM', BK' by lines q', s' meeting p', r' in L', N' respectively. We shall prove that L'N' goes through D. HARMONIC ELEMENTARY FIGURES 31 By construction, the triangles KLM, K'L'M' are such that the meets of corresponding sides are collinear: hence the lines KK', LL', MM' (not shown in the figure) are concurrent. Similarly, KK', MM', NN' are concurrent. But the points of concurrency are the same, namely, the intersection of KK', FIG. 7. MM': hence KK', LL', NN' are concurrent, the triangles KLN, K'L'N' are in perspective, and the intersection of LN, L'N' is therefore collinear with A, B. Hence L'N' goes through D, as asserted. It is not essential that the quadrilateral p'q'r's' should be coplanar with pqrs: it may be drawn in any plane which passes through AB. The result may be stated as follows: Given three collinear points A, C, B ze can construct, in an unlimited number of ways, a complete quadrilateral pqrs, of which two sides p, r go through A, the other two sides q, s 32 PROJECTIVE GEOMETRY through B, and one diagonal (ps, qr) through C. However this is done, the diagonal (pq, rs) goes through a fixed point D on the line AB. This point D is said to be the harmonic conjugate of C with respect to A, B (or B, A, since A, B play symmetrical parts in the construction). C is also, by definition, the harmonic conjugate of D with respect to A, B. Moreover, C and D are separated by A and B: because if C lies between A and B on the finite segment AB, the notation may be so arranged that M lies within the triangle KAB, and then L, N are on the finite segments KA, KB, so that LN must meet the side AB produced. Consequently, also (Art. 2z), A, B are separated by C, D. 35. Motion of Conjugate Pairs. Suppose that, in the figure, we keep the lines AK, AN, KB fixed, and let C move in the direction CB. For any position of C, M is determined as the intersection of AN, CK, and L as the intersection of BM, AK. As C moves towards B, the point M moves towards N, and L towards K. Since D is the intersection of AB with NL, and N is a fixed point, it follows intuitively that D moves towards B in a sense contrary to that of the motion of C: and that when C is at B, D is also at B. Similarly, if C moves towards A, then D moves away from B in the contrary sense, ultimately reaching A from the left when C reaches it from the right. Thus we see, in an intuitive way, that as C describes the whole finite segment AB in either sense, its conjugate D describes the whole of the complementary segment B.A in the opposite sense. On the other hand, let us keep A, C fixed, and vary B, D subject to the condition that C, D separate A, B harmonically. In the figure we may take ACK as a fixed triangle and L a fixed point: then, given B, we find M as the intersection of BL, CK, next N as that of AM, BK and finally D as that of LN, AB. Now, if B moves in the sense ACB, M moves in the sense CMK, and the lines AM, KN rotate in the same sense: therefore LN rotates in that same sense, and finally D moves in the sense ACB, agreeing with that of B's motion. HARMONIC ELEMENTARY FIGURES 33 These results will subsequently prove of very great importance: it will be seen that the proofs given here are mainly intuitive. 36. Bisection as a Harmonic Property. Considerations of symmetry at once suggest that when C bisects AB its conjugate D is at infinity. This is true, but we cannot prove it projectively. However, if we assume that the medians of a triangle are concurrent, and that the line bisecting two sides of a triangle is parallel to the other side, the proposition follows from Fig. 8, where C, L, N are the midD FIG. 8. points of AB, AK, KB respectively. It is interesting to compare the special figure with the general one: to make this easier, the same letters have been put at corresponding points. 37. Symmetry of the Harmonic Relation. If C, D separate A, B harmonically, then A, B separate C, D harmonically. Starting with a quadrilateral KLMN related as before to the points A, B, C, D, let KM, LN meet in O (Fig. 9); let AO meet LM, KN in G, E, and let BO meet KL, MN in F, H. The quadrilateral OGMH has two sides meeting at A, the other two meeting at B, and one diagonal OM going through C: hence (Art. 34) the diagonal GH goes through D. Similarly, FE goes through D, and FG, EH go through C. It follows from the quadrilateral EFGH that A, B are harmonic conjugates with respect to C, D, so that the M.P.G. C 34 PROJECTIVE GEOMETRY relation between the point-pairs (A, B) and (C, D) is symmetrical. We shall say that, in the figure, ABCD is a harmonic range. In the symbol used we write first the signs (A, B) of two conjugate points, and then the signs (C, D) of the other points. Either pair may be taken first, and the order of writing the letters of each pair is indifferent: so there are eight ways of saying that four points form a harmonic C D FIG. 9. range in a definite way. For instance, to say that DCAB is harmonic is the same thing as to say that ABCD is harmonic. On the other hand, if ABCD is harmonic neither ACBD, nor ADBC will be harmonic, because the separation of (A, B) by (C, D) is inconsistent with the separation of (A, C) by (B, D) or of (A, D) by (B, C). If ABCD is harmonic, its elements occur in the orders ACBD, DBCA, and their cyclical permutations (CBDA, etc.): not in the order ABCD. Usage in this matter is not uniform, so the student who consults different books and memoirs should pay careful attention to the notation used in this connexion. 38. Harmonic Pencils. We will now consider the correlative, in the plane, of Fig. 7, and the corresponding new theorem. HARMONIC ELEMENTARY FIGURES 35 Let a, b, c (Fig. io) be any three concurrent rays meeting in L. On a take any two points P, R, and on c take any point O. Let PO, RO meet b in 8, Q respectively, and let PQ, RS meet in T. Then, however the construction is varied, the ray LT, or d, is always the same, and is called the harmonic conjugate of c with respect to a, b. We also P R\ FIG. 10. say that abed is a harmonic pencil. It is unnecessary to write out the proof of the theorem, because this may be obtained, if we wish, by dualising that of the former theorem (Art. 34). The rays a, b, c being given, we may complete the figure, after assuming three arbitrary non-collinear points P, 0, Q upon a, c, b respectively. Now, if c meets PQ in U, it follows from the quadrilateral RLSO that PQTU is a harmonic range: and since P, Q are any two points upon a, b, except L, it follows that The section of a harmonic pencil by any external line is a harmonic range. In a similar way we can infer from Fig. 7 that The projection of a harmonic range from any external point is a harmonic pencil. 36 PROJECTIVE GEOMETRY 39. Let ABCD, AB'C'D' (Fig. 11) be harmonic ranges on bases I, I', having the same element A in common: then the lines BB', CC', DD' will be concurrent. For if S is the intersection of BB', CC' the pencil S(ABCD) is harmonic, and its section by I' is therefore a harmonic range. Three of the points of this section are A, B', C': therefore the fourth must be D', and DD' goes through 8. 1' A C\ B D FIG. II. Conversely, if ABCD is harmonic, and BB', CC', DD' are concurrent, then AB'C'D' is harmonic. For in this case S(ABCD) is harmonic, and AB'C'D' is a section of it. 40. Harmonic Axial Pencil. Let ABCD be a harmonic range, and I any line that does not meet its base. Then the planes IA, IB, IC, ID are said to form a harmonic pencil of planes I(ABCD), or a harmonic axial pencil. The section of a harmonic axial pencil by any external line is a harmonic range. Let the pencil be I(ABCD) and the section A'B'C'D'. If any one of these last points, such as A', coincide with its corresponding point A, the ranges ABCD, AB'C'D' are coplanar; and, if we cut the axial pencil by the plane of the two sections, the theorem follows by Art. 39. If not, let the line AD' meet the planes IB, IC in B", C" respectively. Then by the first case we infer that AB"C"D' is harmonic, and thence that A'B'C'D' is harmonic. HARMONIC ELEMENTARY FIGURES 37 Thus, given any three coaxial planes a, /, y, there is a definite fourth plane 8 coaxial with them, such that a/3y8 is harmonic. The pair (a, /3) is separated by (y, 8); and every section of the pencil by an external plane is a harmonic flat pencil. We may also obtain a harmonic pencil FIG. 12. of planes by projecting a harmonic flat pencil from any point that is not in its plane. We may sum up many of the results of this chapter by saying that the projections and sections of elementary harmonic figures are also harmonic figures. 41. As a practical illustration we may take the following problem. Given in a plane a point P and two lines I, I', the intersection of which is not accessible, it is required to draw through P a line concurrent with I, I'. The solution is as follows: Through P (Fig. 12) draw any two lines APC, BPD meeting I, I' in A, C and B, D respectively: join AD, BC and let them meet in S. Now through S draw any line SFE meeting I in 38 PROJECTIVE GEOMETRY E, and I' in F: let BF, CE meet in Q, and join PQ: then PQ is the required line. To prove this, we note that P, S are two of the diagonal points of the complete quadrangle ABCD, and are therefore (cf. Fig. IO) harmonically separated by AB, DC. Similarly, Q, S are harmonically separated by CF, BE. Hence, if O is the (inaccessible) intersection of I, I', the points P, Q both lie on the harmonic conjugate of OS with respect to I, I': therefore PQ goes through O as required. CHAPTER VII. THE FUNDAMENTAL THEOREM OF PROJECTIVITY. 42. Chain of Perspectives in a Plane. Let us suppose the following construction carried out in a single plane. Start with a line I and a row (ABC...) upon it; and project it from an external point S1, so as to produce the pencil 18(ABC...) perspective to (ABC...). Now cut the pencil by a line 1,, so as to obtain a row (A^BC...) perspective to it. Again, project (ABOC...) from S2, so as to obtain a pencil S2(AIBC,...) perspective to it, and cut this pencil by a line 12, so as to obtain a row (A2B2C...), and so on. We thus have a sequence of alternate rows and pencils which we may denote by (1), (SI), (11), (S2), (12).. each form being perspective to the one next following. By this chain of perspectives each element in any one of the forms is definitely associated with a single corresponding element in any other form: in (Ir), Ar, Br, Cr... correspond to A, B, C... in (I) respectively, and vice versa: similarly, in (Ir), (St), Ar and StAt-i correspond, and so on. The points and lines Sm, Im need not be all different. Suppose that In coincides with 1. Then we have, on I, two associated rows (ABC...), (AnBnCn...) put into a oneone correspondence with each other. It will be convenient to write (A'B'C'...) instead of (AnBnCn...). The two rows (ABC...), (A'B'C'...) are such that to any 40 PROJECTIVE GEOMETRY harmonic range in the one corresponds a harmonic range in the other. This follows from Chapter VI. (cf. Art. 40, end). 43. Self-corresponding Elements. It may happen that the cobasal rows (ABC..), (A'B'C'...) coincide: but this is not generally the case. However, a limited number of points may coincide with their corresponding points; and the question arises, how many of these self-corresponding elements can exist, when the rows are not identical? The answer to this is that there are two at most, and this is one way of stating the fundamental theorem of projective geometry. Staudt expressed it in the following form (G. Io6): If two cobasal rows (ABC...), (A'B'C'...) are such that there is a one-one correspondence between them, and to every harmonic range in one corresponds a harmonic range in the other, then, if there are three self-corresponding elements, every element is self-corresponding, and the rows are identical. 44. Before proceeding to the rigorous proof of the theorem, it will be well to give Staudt's demonstration, and show precisely where it is defective. He says (substantially): "Suppose A, B and every point of the segment AB are self-corresponding points: then every point harmonically separated from a point of this segment by A and B, and therefore every point of B.A (cf. Art. 35) corresponds to itself. Supposing, however, that the two rows have no continuous succession of self-corresponding elements, but more than two of them, then, if A, B are two consecutive self-corresponding elements, one of the segments AB, B.A will contain no such element, and the other at least one, say P. But this leads to a contradiction, because, if Q is the harmonic conjugate of P with regard to A, B, one of the points P, Q is in the segment AB, and the other in B.A (Art. 35), and each of them is a self-corresponding point." There are two undefined terms here which, as was subsequently discovered, deprive Staudt's argument of strict THEOREM OF PROJECTIVITY 41 validity. One is "consecutive," and the other "continuous." Staudt assumes as a logical disjunction that either there is a segment AB every point of which is selfcorresponding, or that if A is any self-corresponding point there is a "consecutive" self-corresponding point, if A is not the only one. This amounts to saying that there is a segment AB, terminated by self-corresponding points, within which there is no self-corresponding point. But this assumption is not legitimate. The term "continuous" begs an even larger number of questions. Dedekind and Cantor succeeded in giving a definition of arithmetical continuity by means of which there is a definite sense in which we can speak of a variable x increasing continuously from a to b, passing through all intermediate values. But when they tried to translate this into the continuous " motion" of a variable point X from a point A to a point B along the segment AB, passing through every intermediate point of AB, and travelling in the same sense, they found that they could not do it without the assumption of an axiom called the Cantor-Dedekind axiom, which amounts to saying that if 0, I are any two points on a line, and we associate them with the numbers o, I, then it is possible to associate with every number x a definite point X on the line, and conversely. In other words, the " continuous motion " of a point along a segment AB is assumed to be the exact image of the " continuous variation" of a number x over an interval (a, b), and conversely. It should be particularly noticed that arithmetical "variation" is the definite term, and geometrical " motion," or change of position, the indefinite term, assumed to correspond to the former. In the arithmetical theory, every interval must be terminated by two definite numbers: in virtue of the axiom, every segment must be terminated by two definite points. Without the axiom, or some equivalent form of it, it seems impossible to arrive at this geometrical conclusion, in a sense that is perfectly free from ambiguity. 42 PROJECTIVE GEOMETRY 45. Harmonic Derivative. Take any three points A, B, C on a line, and find points A', B', C' such that BCAA', CABB', ABCC' are all harmonic ranges. We thus obtain six different points, one order of which is AC'BA'CB' (or C'BA'CB'A, etc.), and the other order B'CA'BC'A (cf. Art. 20). Taking every triad out of the six points and repeating the construction, we obtain a number of new points A", B", C", D", etc. For instance, the harmonic conjugates of B, A', C, B' with respect to the consecutive points A, C' give us four new points within the segment AC'. By continuing this process indefinitely, we obtain on the line (I) a set of points which we may call the harmonic derivative, or simply the derivative of ABC and denote by [ABC]. At any finite stage of the process we have a finite set of points, and each point of this set will have a " consecutive " point on each side of it. But in the limit, this ceases to be the case, because we can prove that Between any two given points of the set there exist an indefinite number of other points of the set. Suppose that, at any stage of the process, P, Q are consecutive points, and that PQ is the segment that contains, so far, no point of the set. Then Q.P contains at least one point, R say, of the set already obtained, and if R' is the harmonic conjugate of R with respect to P, Q, then R' belongs to the set, and is within PQ. In other words, if at any stage P, Q are consecutive points for one of the segments PQ, Q.P, then there is a subsequent stage of the construction for which P, Q are not consecutive either for the segment PQ or for the segment Q.P. Hence no two points of the set are ultimately consecutive, and the proposition follows from this. For consider any segment PQ terminated by points of the set: then there is at least one point P1 of the system within PQ, and hence at least one point P2 within that segment PP1 which is a part of PQ (i.e. for which PP2P1 is the same order as PP1Q). So we can get a sequence in the order QP1P2P3... Pn... P along THEOREM OF PROJECTIVITY 43 the segment QP, and by taking the harmonic conjugates with respect to P, Q we get a sequence on P.Q. This property, that on either segment joining any two points of the system there exists at least one, and therefore an indefinite number of points of the system, is expressed by saying that the set [ABC] is compact. 46. The reader must be careful not to assume that the set [ABC] contains all-the points on I. In fact, this is not the case, and it can be proved that if A, B, C are associated with the numbers o, I, Co (or any three distinct rational numbers a, b, c) then every point of [ABC] can be associated with a rational number. Every point, then, on I, associated with an irrational number, does not belong to [ABC]. Now, in the arithmetical theory, every irrational number can be expressed as the limit of a converging sequence of rational numbers (P1, P2, p3,... Pn..), so correspondingly every point of I either belongs to [ABC] or else is the limiting point of a sequence of points (P1, P2, P3.. P...) which do belong to [ABC]. But since we do not wish to appeal to arithmetical ideas if we can help it, we adopt a course explained in the following Article. 47. There is no segment of I within which points of [ABC] do not lie. The proof of this (which does not follow from Art. 45) was independently discovered by Liiroth and Zeuthen (cf. Math. Ann., vii. (1874), 531, or Whitehead, I.c., p. 30). Some comments on the proof are given in Art. 48. The annexed diagram (Fig. 13, next page) is merely schematic, and is only given to help the reader to realise certain statements about order of elements. Suppose, if possible, that there is a segment of I, within which no points of the derivative lie. Let this segment be enlarged as much as possible, without destroying the property in question. We assume that the final result is a segment FG terminated by two definite points F, G on I. (This amounts to assuming the Dedekind-Cantor axiom in one of its least obviously arithmetical forms.) 44 PROJECTIVE GEOMETRY These points cannot both belong to the derivative, because otherwise, as we have just seen, FG could not be free from derivative points. But since FG cannot be produced in either direction without passing over derivative points, either F or G, or both, must be a limiting point of the derivative set: that is to say, if G, for instance, is not a derivative point, every segment GP belonging to G.F, and with one extremity at G, must contain at least one derivative point. Similarly, for F: we shall suppose, first, that F, G are both limiting points. _A,..., F.,0 L 9 _ e QJ FIG. 13. On G.F take any derivative point A, and let H, J be those points of I which make AHFG and AGFJ harmonic: or, in a convenient notation, let AH harm. FG,...............................(I) AG harm. FJ................................(2) Then (Arts. 35, 37) AFHG, AFGJ are orders and they agree in sense. Hence also AFHGJ is an order of the same sense. In the figure this sense is from left to right, and we take it as the positive one. Since F is a limiting point, we can find a point B of the derivative within the segment AFH, and we may suppose B as near to F as we like. Take the point K such that AH harm. BK...............................(3) Comparing (3) with (i) we see that as we take B nearer and nearer to F, the point K takes up positions nearer and nearer to G (cf. Art. 35). Moreover, ABHK is an order, so we have an order ABFHGKJ. Take any derivative point C on the segment GKJ. We may suppose the point B in such a position that the orders GKC, KCJ are both positive. For by taking a sequence (B,, B, B... ) of derivative points converging to F, the relation (3) gives a sequence of points (K1, K2, K3,...) converging to G, so that we must ultimately reach a point Km THEOREM OF PROJECTIVITY 45 within the finite segment GO; and if we now put Km = K, Bm = B, we have a positive order ABFHGKCJ. Now take the points L, D such that AL harm. BJ.............................. (4) AD harm. BC.........................(5) Comparing (4) with (3), we infer, by Art. 35, that the order AHL agrees with the order BKJ, and is therefore positive; and by comparing (4) with (2) that ALG agrees with JBF, and is therefore positive. Since AHL, ALG are positive orders, AHLG is a positive order; and since FHG, HLG are positive orders, FHLG is a positive order. From (3), (4) and (5) it follows that the order HDL agrees with KCJ, and is therefore positive; and finally, the positive orders HDL, FHLG imply a positive order FHDLG, whence it follows that D is within the segment FHG. But since A, B, C are derivative points, it follows from (5) that D is a derivative point; so that the hypothesis that the segment FHG is free from derivative points is contradicted. The proposition amounts to showing that every point of the line is either a derivative point, or else a limiting point of the derivative set. We have supposed both F and G limiting points: if G is a limiting point, but F is not, then F must be a derivative point; we then take B at F, and argue as before. We may suppose either F or G at infinity: this does not affect the argument, as the reader will see by taking the circular image of the row (Art. i6). 48. It will be seen that the foregoing proof depends upon three main premises. First, there is the assumption that F, G actually exist as determinate points: secondly, the notion of order, and the propositions of Art. 35 about the relative motions of two points of a harmonic range, when the other two are fixed-here we appeal to geometrical intuition: thirdly, the use made of the clustering of the derivative points near F, so as to obtain the derivative point B in such a position as, so to speak, to force 46 PROJECTIVE GEOMETRY D, the conjugate of A with respect to B, C, into the segment FHG. The spirit of the proof, without its rigour, may be illustrated by the following observations. Let us make the same assumption as before about the existence of F and G, and take the case when they are both limiting points, and neither of them at infinity. Suppose also that it is the finite segment FG which is void of derivative points. Let the derivative set be constructed from A, B, C, any three points of 1; then F, G are both distinct from these points, and the finite segments AF, AG are of the same sense. Let H be the harmonic conjugate of A with respect to F,: this is on the finite segment FG and the segments FH, HG are fixed and finite. Now, since F, G are limiting points, we can take, on the infinite segment G.F, derivative points F', G' as near as we please to F, G respectively. Let H' be the harmonic conjugate of A with respect to F', G'; then if H' is within FHG, the proposition is proved: if not, it must lie on one of the finite segments F'F, GG', and at a distance from H greater than the least of the fixed distances FH, HG. Now if we actually draw a figure in which F'F, GG' are both very small compared with FG, we find that we can easily bring H' within FG: so from this intuitive point of view the theorem seems obvious enough. But the reason why this argument is not sound is that in projective geometry we have nothing to do with the comparative sizes of segments, at any rate when, as here, the comparison is made by ratios. And the notion of distance, as such, is entirely foreign to the subject, in its elementary stages. 49. From three rays a, b, c of a flat pencil we can form a harmonic derivative set [abc], and from three planes a, /f, y of an axial pencil we can form the set [sPyi]. As in the case of a row, each complete pencil consists of the derivative set, and its limiting set, and of no other elements. It should be noticed that, if ABC, A'B'C' are different triads, the derivatives [ABC] [A'B'C'] are, in general, different. THEOREM OF PROJECTIVITY 47 50. Let us return, now, to the problem stated at the beginning of the chapter, where two rows (ABC...), (A'B'C'...) on the same line are connected by a chain of projections and sections. Suppose A', B', C' coincide with A, B, C respectively: then since every harmonic range in (A'B'C'...) corresponds to a definite harmonic range in (ABC...), and conversely, it follows that the set [A'B'C'] must coincide, element by element, with the set [ABC]. Now when two sets of points are identical their limiting points are identical too. Hence the complete rows (ABC...), (A'B'C'...), must be identical, in the sense that each point of either is self-corresponding. Hence, if the rows are really different, there cannot be more than two self-corresponding points. 51. Practical Application. Another Aspect of the Fundamental Theorem. Any two elementary forms that can be connected by a chain of projections and sections are said to be projective to each other, or, more briefly, projective. Suppose the figures to be of the first rank, and let us denote them by (abc...), (a'b''...), where the correspondence a, a', etc., has been established by a perspective chain C1. This chain makes any fourth element b of the first form correspond to a definite element b' of the second form. The fundamental theorem leads to the conclusion that every perspective chain, connecting the forms in such a way as to make a' correspond to a, b' to b' and r' to c, associates any element b with an element b' which is always the same. In other words, however the perspective chains are altered, subject to the condition stated, the correspondence they establish is definite and unique. To see this, suppose, if possible, that another chain C2 brought about a correspondence (abb...), (a'b''T"...) with b" different from b'. The chain C1, and the chain C2 taken in the reverse order, may be considered as a single chain connecting (a'b''b'...) with (a'b'c'"...): but this is impossible, unless b', i" coincide. If the elements a', 48 PROJECTIVE GEOMETRY etc., are points, this follows from Art. 50: if they are rays or planes, we take a line cutting them, and thus obtain two rows (A'B'C'D'...), (A'B'C'D"...) from which we infer that D', D" coincide, and thence that b', b" coincide. 52. Triad-theorem. We may sum up these results in the following statement: Given two elementary forms of the first rank, which are required to be connected by a chain of projections and sections: then we cannot assume more than three arbitrary pairs of corresponding elements. We shall presently see that it is always possible to solve the problem when three pairs of corresponding elements, or, as we may say, two corresponding triads, are assigned. The number of possible constructions is unlimited, but the resulting correspondence between the given forms is always the same. The fact that this is so is of vital importance in everything that follows. CHAPTER VIII. ELEMENTARY CASES OF PROJECTIVITY. 53. Projectivity in General. We have already explained (Art. 51) what is meant by saying that two elementary forms of the first rank are projective. We now extend the definition so as to apply to elementary forms of the second rank, or to figures, in the usual sense, such as polygons, curves, etc. Suppose that we have a sequence of forms F1, F2... Fn, such that each is in perspective with the one before it. In general, the forms F1, Fn are not in perspective; but by means of the (n - 1) perspectivities we can set up a one-one correspondence between their elements. Let el be any element of F1 (point, line or plane, as the case may be); then the first perspectivity associates this with a definite element e2 of F2: the second perspectivity associates e2 with an element e3 of F3, and so on: finally we arrive at a definite element en of Fn. Taking the chain of perspectives in the opposite order, we come back from en to e1. The correspondence (e1, en), (fi, fn), etc., thus set up between the elements of F1 and those of Fn is a one-one reversible correspondence: we call it a projectivity, and say that F1 is projective to Fn. In symbols, we write F1 7 Fn (read " F1 is projective to Fn"). We also write (elfgl...) w (enfngn..) M.P.G. D 50 PROJECTIVE GEOMETRY in the sense that there is a perspective chain connecting (elflgl...) with (enfngn....) in such a way that (el, en), (fi, fn), (gl, gn) are pairs of corresponding elements. 54. The following propositions follow from the definition: (i) Figures in perspective are also projective, but not (in general) conversely. (ii) If F 7 G, then G ~ F. (iii) If F A G, and G 7 H, then F 7 H. By hypothesis, F can be connected with G by a perspective chain, say C,, and also G with H by a chain C2. Then C, followed by C2 is a chain connecting F projectively with H. (iv) If F, G are any two elementary forms of the same rank, and F 7C G, then to any elementary harmonic figure in F -FI. Q. FIG. 14. corresponds an elementary harmonic figure in G (cf. Art. 40o, end). 55. Given two triads of points ABC, A'B'C' on two lines I, I' which do not meet: we can set up a projectivity between the rozes (ABC...), (A'B'C'...) such that (A, A'), (B, B'), (C, C') are pairs of corresponding points. (Fig. I4). To prove this, we join AA', BB', CC', and through any point P of AA' draw the line I" meeting BB', CC' in Q, R ELEMENTARY CASES OF PROJECTIVITY 51 respectively. Then the axial pencil I"(ABC...) is perspective to (ABC...), and its section by I' is a row (A'B'C'...) projective to (ABC...) and satisfying the required conditions. The projectivity thus set up is unique: that is to say, if the projective construction we have set up makes (ABCD...) '7 (A'B'C'D'...), D being any element of (ABC...), and any other projective construction makes (ABCD...) 'A (A'B'C'DD"...), then D" must coincide with D'. This follows from the fundamental theorem (Art. 43) and the argument of Art. 51. It is easy to see that we have an unlimited number of constructions at our disposal: for instance, P may be any point on AA': this illustrates the importance of the fundamental theorem and its consequences. To recapitulate the argument for the present case, the projective correspondence of the triads ABC, A'B'C'.implies the projective correspondence of the derivative sets [ABC], [A'B'C'], and hence of the complete rows (ABC...), (A'B'C'...), in a perfectly definite sense. Thus we may speak of the projectivity (ABC...) '7 (A'B'C'...), meaning that definite projectivity in which (A, A'),(B, B'), (C, C') are pairs of corresponding elements. Similarly, for any two projective elementary forms of the first rank (ee2e3...), (fifSf3...), any projective construction leading to (ele2e3e4...) w (f1f2f3f,...) leads to a definite correspondence [ele2e3] 7 [flf2f3] and hence to a definite unique correspondence (ele2e3e4...) 7 (flf2f3f4...) where f4 is uniquely determined by (el, e2, e3, fl, f, f3, e4). 56. Given any two elementary forms of the first rank (ele2e3..), (fif2f3...): we can set up a projectivity (ele2e3.. A (fif2f3...), and this is unique, although the number of possible constructions is unlimited. If either of the forms is not a row, take a perspective section of it by a line. We now have two rows (EEE3...), 52 PROJECTIVE GEOMETRY (FF2F3.. ), and if their bases are skew, we make them projective to each other by the process of Art. 55. We now have (ee123...) ' (E1EE3...).' (F1F2F3...) (ffA...) whence (ele2e3...) A (ff2f3...). The uniqueness of the correspondence is proved as before. If the bases of (E1E2E3...), (F1F2F3...) are coplanar or coincident, take any line skew to them, and a triad G1G2G3 upon it: now by Art. 55 set up the projectivities (EE2E3...) l D ~ ~ m j/b FIG. 15. 7 (GiG2QG..) and (GG2G...) 7 (F.F2F3...). We now have (e.12e3...) 'A (EIE2E3...).A (G1G2G3...) 7A (FiF2F3...) (flf2f... ). whence also the required projectivity (ele2e3...) 7^ (ff2f3..) has been set up. As before, we can prove that this projectivity is unique. 57. Plane Constructions. Practically, all our constructions have to be in one plane; so we give alternative solutions of the triad-to-triad problem, when the six elements are coplanar. ELEMENTARY CASES OF PROJECTIVITY 53 (i) Let ABC, A'B'C' be two triads on different coplanar lines I, I': it is required to establish, by a construction in their plane, the projectivity (ABC...) 7 (A'B'C'...) between the rows on I, I'. On AA' (Fig. 15) take any two points S, T: let SB, TB' meet in B", SC, TC' in C", and let B"C" cut ST in A". Then by means of the pencil (S) we can put (ABC...) into perspective with (A"B"C"...) and by means of the pencil (T) C QQ' Q-P' ' F B' C0'. FIG. i6. we can put (A"B"C"...) into perspective with (A'B'C'...). The result is a projectivity between (ABC...) and (A'B'C'...). To find the point on I' corresponding to D on I, find the point D" where SD cuts I", the base of (A"B"C"...): then D' is the point whereTD" cuts I'. The rows (ABC...), (A'B'C'...) may happen to be in perspective: if so, I" goes through II', and the point II' corresponds to itself (cf. Art. 24): also AA', BB', CC' are concurrent. Fig. i6 shows what happens if we take T at A and S at A'. The line I" is now the join of (AB', A'B) and (AC', A'C). If it meets I in P and I' in Q', the point P corresponds to II', considered as a point on I', while Q' corresponds to II' considered as a point on I. Now the projectivity being unique, P, Q' are fixed points, I" is a fixed line, and all such points as 54 PROJECTIVE GEOMETRY (AB', A'B), (AC', A'C), (BC', B'C), etc., are collinear and lie upon I". The fixed line I" may be called the axis of the projectivity: if AA', BB', CC' concur, I" goes through II', and is the harmonic conjugate with respect to I, I' of the line joining Il' to (AA', BB'). FIG. 17. Incidentally, we see that if ABC, A'B'C' are any two triads on coplanar lines, the three points (BC', B'C), (CA', C'A), (AB', A'B) are collinear. This is a special case of Pascal's theorem for conics. (ii) Let abc, a'b'c' be tzo triads of rays concurring in the different points L, L'. it is required to establish, by construction, the projectivity (abc...) (a'b'c'...). (Fig. I7.) The given figure being supposed in one plane, draw any lines s, t through aa'. Let b", c" be the joins of (sb, tb') and (so, to') respectively. Then the rows s(abc...), t(a'b'c'...) can be put into perspective by means of the pencil (L"), where L" is the point b"c". Thus (abc...) becomes perspective to (a"b"c"...), and the latter to ELEMENTARY CASES OF PROJECTIVITY 55 (a'b'c'...); so the result is a construction for the projectivity (abc...) X- (a'b'c'...). To find the ray d' of (L') corresponding to d in (L), join ds to L" by the line d": then the line joining L' to the point td" is the required ray d'. If we take a for t and a' for s, the result is a construction correlative to that of Fig. I6. The lines (ab', a'b), (ac', a'c), etc., concur in a fixed point L", which we may call the centre of the projectivity. LL" in (L) corresponds to L'L in (L'), and L'L" in (L') corresponds to LL' in (L). The drawing of this figure is left to the reader as an exercise. (iii) To establish the projectivity (ABC...) W (A'B'C'...) by construction, when the triads ABC, A'B'C' are on the same line 1. Project A'B'C' on to any other line I', so as to obtain a perspective triad A"B"C". Then, by (i) set up the projectivity (AB"C"...) w (ABC...). (iv) To establish the projectivity (abc..) W (a'b'o'...), when the triads abc, a'b'c' concur in the same point L. Take any other point L', and draw three lines a", b", o" through it. By (ii) find constructions for the projectivities (abo...) (a"b"c"...) and (a"b"c"...) -(a'b''...). Compounding them, we obtain a construction for the projectivity (abc...) W (a'b'c'...). If we like, we can choose (a"b"c"...) so as to be perspective with (a'b'c'...): the construction is now correlative to that given in (iii). These are the four most frequent cases that occur. As an exercise, the reader may find a construction for the projectivity (abc...) w (ABC...) when ABC is a triad on one of the rays of the flat pencil (abc...). Every such problem in the bundle may be solved by taking a section of the bundle by a plane. 58. Illustrations. We will now give a few illustrations of the foregoing theorems and constructions. (i) Let a triangle A'B'C' be inscribed in a triangle ABC: then there is an unlimited number of triangles PQR simultaneously circumscribed to ABC and inscribed in A'B'C'. (Fig. I8.) 56 PROJECTIVE GEOMETRY Draw any line through A meeting A'C', A'B' in Q, R respectively, and let BR meet B'C' in P. As Q moves, so does R, and they are corresponding points of perspective rows. Similarly, R, P, as corresponding points, describe C*4/X8 FIG. 18. two perspective rows. Therefore P, Q are corresponding points of two projective rows on C'B', C'A'. But when Q is at C', it follows from the construction that P is also at C', so that C' corresponds to itself. Hence the rows described by P, Q are perspective, and PQ goes through a fixed point. When Q is at A', R is also at A', and PQ coincides with BC: when R is at B', P is also at B', and Q is the meet (A'C', AB'): therefore PQ coincides with AC. Thus the fixed point through which PQ passes is the intersection of BC, AC, that is, C; and this proves the theorem. (ii) Let there be four concurrent lines a, b, c, d meeting in S, and let a quadrangle ABCD be taken with its vertices A, B, C, D on a, b, c, d respectively, and its sides AB, BC, CD passing respectively through fixed points P, Q, R. Then the side AD will always pass through a fixed point x. In the figure Al, A2, A3 are any three positions of A, from which the corresponding positions of B, C, D are deduced. ELEMENTARY CASES OF PROJECTIVITY 57 The row (A1A2A3...) is perspective to (B1B2B3...), the latter to (C1C2C3...) and this to (D1D2D3..). Hence (A1A2A3..) W7 (DD.D..). But in this projectivity S corresponds to itself, because it does so in each of the above-mentioned perspectivities. lb FIG. 19. Therefore (A1A2A3...), (D1D2D3D...) are in perspective, and the lines AlDl, A2D2, A3D3, etc., all meet in the same point X. When a, b, o, d, P, Q, R are given, X is determined by constructing any two quadrangles satisfying the given conditions: namely, it is the intersection of the sides A.D., A2D2. By the same argument it follows that the other sides AC, BD describe flat pencils about two other fixed points Y, Z (not shown in the figure). 58 PROJECTIVE GEOMETRY The theorem may be extended to the case of an n-angle whose vertices lie on n concurrent lines, while the (n -- i) sides A, A, AA,.. An_ An pass each through a given point. The remaining I (n - i) (n - 2) sides each pass through a fixed point, deducible by linear construction from the given fixed lines and points. The given concurrent lines need not be coplanar. From the case where they are we deduce the correlative plane theorem. (iii) Let there be four collinear points A, B, C, D on a line s, and let a quadrilateral abed be taken, whose sides a, b, c, d pass through A, B, C, D respectively, and whose vertices ab, be, cd lie on three fixed lines p, q, r respectively. Then the three remaining vertices ad, ac, bd will lie respectively on three fixed lines x, y, z. This may be extended to the case of an n-lateral. In the enunciation of (ii), and the corresponding figure, we have four different lines a, b, c, d. But the proof, so far as AD is concerned, will not be altered if we suppose that d coincides with b. However, in this case BD does not describe a pencil, and there is no corresponding fixed point Z. Similar remarks apply to the extension to an n-angle. 59. Projective Properties of a Tetrad. If A, B, C, D are any four points of a row, then ABCD W BADC W CDAB W DCBA. (Taking ABCD as made up of the pairs AB, CD, the symbol CDAB is derived from it by interchange of pairs. If in these two symbols we change the order of elements in both pairs, we obtain the remaining symbols. Or again, if we suppose the notation such that ABCD is an order, the other symbols are orders, CDAB agreeing with ABCD, and the others being opposite to it.) To prove the theorem, draw a line through D, and project A, B, C upon it from a point M, so as to give the points E, F, G. Let MC, AF meet in N. Then, as sections of the pencil (F), ABCD W NMCG; and, as sections of the pencil (A), NMCG W FEDG. Therefore ABCD w FEDG. But, as ELEMENTARY CASES OF PROJECTIVITY 59 sections of the pencil (M), FEDG W BADC; therefore, finally, ABCD W BADC. In the same way we can prove that CDAB W DCBA. By producing CF to meet MA in P, we see, by similar reasoning, that ABCD W EFGD 7 EPMA W DCBA (projecting successively from M, C, F). Hence the theorem A 7 IG.0 A /B O D FIG. 20. follows. To avoid risk of confusion, the point P has been omitted in the figure. 60. If ABCD W CBAD, then A, C are harmonic conjugates with respect to B, D. (Fig. 2I, next page.) Take M, E, F, G as before: let AG meet MB in S, and let ES meet AD in C,. Then by the pencils (M), (S) we have successively ABCD w EFGD, and EFGD w C1BAD: therefore ABCD w7 CBAD. Thus the hypothesis leads to C1BAD A7 CBAD, from which we infer, by the fundamental theorem, that C, coincides with C. The proposition now follows from the quadrilateral MESG. The connection between ABCD and CBAD may be expressed by saying that, if the notation is arranged so that ABCD is an order, then CBAD is the opposite order starting with the point C, which is separated from A by the other two points. 60 PROJECTIVE GEOMETRY 61. From the relation MNAB 7' MNA'B', supposed to exist, we can deduce MNAA'WA MNBB'. (Fig. 22.) FIG. 21. Draw any line through M, and take on it two points S, S'. Let SA, S'A' meet in A", and join NA" by I", cutting It Is. FIG. 22. MS in M". From the hypothesis we have S(MNAB) W S'(MNA'B'), and therefore the sections of these pencils by I" are projective. But these sections have three self ELEMENTARY CASES OF PROJECTIVITY 61 corresponding points M", A", N and are therefore identical: hence SB, S'B' must meet in a point B" on I". Now from the pencil (A") we have MNAA'7^ MM"SS', and from the pencil (B") we have MM"SS'7$ MNBB': hence MNAA'7\ MNBB', as asserted. It should be noted that if, as in the figure, A, A' are both in MN, and B, B' both in NM, the sense MAA' is opposite to that of MBB'. 62. From the theorems proved in Arts. 59-61 we can at once infer corresponding theorems for each of the other elementary forms of the first rank. Thus, if a, b, c, d are any four rays of a pencil, abed 7 badc, and so on. CHAPTER IX. PLANE HOMOLOGY. ELATION. 63. Take two planes /3, y, a line p skew to py, and points U, V on p, each external to / and y. By projecting the plane field (/) on to the plane y from U and V, we obtain, FIG. 23. in y, two cobasal plane fields (y), (y)', which are said to be homologous (or perspective). Let S be the point p7: B" any point on P, and B, B' its PLANE HOMOLOGY. ELATION 63 projections on y from U, V respectively. Since B, B' are both in the plane pB", the line BB' must meet p, and therefore goes through the fixed point S. The point 8, and every point on /y, corresponds to itself. Beside these there are, in the homology, no other selfcorresponding points. If A, B are any two points in (y), and A', B' the corresponding points in (7)', the line AB corresponds to the line A'B'. Suppose AB cuts f/y in C: then C corresponds to itself, and must lie on A'B'. Hence Two distinct corresponding lines meet at a point on /3y. Every line through S corresponds to itself. This follows from the construction. On every such line there are two, and only two, self-corresponding points: namely, S and the point where the line cuts Py. 64. A plane homology is a projectivity. In fact, the fields (y), (y)' are connected by the perspective chain -(7), (U), (/3), (v), (7)' consisting of alternate fields and bundles. Hence to every row or pencil in (y) corresponds a projective row or pencil in (7)'. There is one self-corresponding pencil, namely (S), and one self-corresponding row (Py): "self-corresponding pencil" here meaning a pencil, every ray of which corresponds to itself. The point S is called the centre, and the line fy the axis, of the homology. 65. In any given plane 7 take a line I, a point S outside it, and any two points A, A' both external to I, but collinear with: then there is a definite homology in y for which 8 is the centre, I the axis, and A, A' a pair of corresponding points. Draw any line p through 8, external to y, and on it take any two points U, V. Let UA, VA' meet in A", and let 3 be the plane A"1. Then the chain (y), (U), (/i), (V), (y)' establishes an homology satisfying the conditions stated. To show that the homology is unique, we prove, by a plane construction, that if B is any point external to I and different from 8, then its corresponding point B' is 64 PROJECTIVE GEOMETRY definite. Suppose, first, that B is not on SAA': and let AB cut I in T. Then (Art. 63) B' must lie on A'T, and also on SB: so that B' is determined as the intersection of SB, A'T. Suppose now we take another point C in (y); we can determine C' either from A, A' or from B, B': but by Desargues' theorem for coplanar triangles it follows that both constructions lead to the same point C'. Hence the plane construction pairs off corresponding points without ambiguity. Having thus fixed the correspondence for all points external to SAA', let Q be any point of (y) on this line. Take any other pair, such as B, B' outside of it, and let QB meet I in R. Then Q' is determined as the intersection of RB' with SAA', and it can be shown as before that this is always the same point, however B, B' are chosen. In fact, we have on SAA', if this meets I in w, a projectivity (SAW...) 7 (SA'W...), because S, W are self-corresponding points; and this determines the point Q' corresponding to Q, independently of any special construction. 66. Vanishing Lines. Through S draw any line SP distinct from SAA': let AQ, parallel to SP, cut I in Q, and let A'Q AV' R7A u R\ / P\ FIG. 24. cut SP in V'. Then, since A'Q, AQ are corresponding lines, V' in (y)' corresponds to the point at infinity on SP in (y). PLANE HOMOLOGY. ELATION 65 The point at infinity on I corresponds to itself: hence the line v' drawn through v' parallel to I corresponds to the line at infinity in (y). Similarly, if A'R be drawn parallel to SP, cutting I in R, and RA cuts SP in U, the line u drawn through U parallel to I corresponds to the line at infinity in (y)'. The lines u, v' are called the vanishing lines in (y), (y)' respectively. To any pencil of (y) with its centre on u corresponds in (y)' a parallel pencil, and conversely. Similarly, to a pencil of parallel rays in (7) corresponds a pencil with its centre on v.' It may happen that u coincides with v'. In this case the points U, V' in Fig. 24 coincide, and from the quadrilateral ARA'Q we see that S, P are divided harmonically by U and the point at infinity on SP; in other words, SP is bisected at U. 67. Harmonic Homology. Suppose that we have a homology with two pairs A, A' and B, B' forming the vertices A FIG. 25. of a quadrangle. Let us consider A' as an element of (y), and give it, as such, the name C. To find the corresponding M.P.G. E 66 PROJECTIVE GEOMETRY point C', let CB, that is, A'B, cut I in Q; then B'Q cuts SC, that is, SA', in the point C'. In general, this is different from A; but it may, as in the figure, coincide with it. Let SA cut I in R: then we see that A, A' separate S, R harmonically. We also see that if this is true for any two corresponding points A, A' it is true for any other two B, B'. Thus (A, A'), (B, B'), etc., are pairs in the homology, independently of the fields in which A, B, etc., are taken to lie. This very important special case may be called, for the present, a harmonic homology. When A' is at infinity, A bisects SR: so we have the case, referred to in last Article, when u, v' coincide. 68. When the homology (S, I; A, A') connects two plane systems (y), (y)', we may imagine (y)' rotated about I so as to assume a new position in space, leaving (y) unmoved., When this has been done, the fields are in perspective to the same bundle; because if (A, A'), (B, B'), (C, C') are any three pairs of corresponding points, BC and B'C' still meet on I, and so on; therefore AA', BB', CC' are concurrent, and since this is true for any three such lines, the proposition follows. Hence Each of two homologous fields may be regarded as the projection of the other from a different plane. Conversely, if two fields (/), (y) are perspective to the same bundle, they become homologous when their bases are brought into coincidence by rotation about/ py. Suppose we draw a circle in (y): then the homologous figure in (y)' must be the section of a circular cone. Generally, to a curve or envelope in (y) will correspond a curve or envelope in (y)' of the same order and class. Any given plane homology (S, I; A, A') may be derived by projection in an unlimited number of ways. Namely, take two points U, V on any line through S external to 8: then UA, VA' will meet in a point A", and if the plane IA" be called P, each of the given homologous fields can be put into perspective with (/3) by means of the bundles (U), (V). This follows from Art. 63. PLANE HOMOLOGY. ELATION 67 69. Elation. In the construction of Art. 63, suppose that p, instead of being skew to /y, meets it in a point S. We can still project (/) from two points U, V of p so as to obtain two related fields (y), (7)'. This is a limiting form of a homology, when the centre is on the axis, and has been called an elation. As before, any two corresponding points in y are collinear with S, any two corresponding lines are concurrent with I: every point on I, and every line through S is self-corresponding, and so on. The main distinctive property of an elation is that if we take any line through S distinct from I, S is the only self-corresponding point upon it, or, if we prefer to say so, represents two coincident self-corresponding points. Similarly, if we take a pencil with its centre at any point P on I distinct from 8, there is only one selfcorresponding ray, namely I. Usually we shall employ the term homology in the wider sense, as meaning either an elation or an homology in the narrower sense. CHAPTER X. PROJECTIVITY OF PLANE FIELDS AND BUNDLES. 70. Projective Fields. As explained in Art. 53, two plane fields (o), ((r'), whether cobasal or not, are projective, when they can be connected by a perspective chain (or), (P1,) (rv-), (P)..., (Pn), (a'), consisting of alternate fields and bundles. In addition to the theorems of Art. 54 we have the following: If A', B' in (o-') correspond to A, B in (cr), then the line A'B' corresponds to AB. Similarly, if a', b' correspond to a, b, the point a'b' corresponds to the point ab. To every row or pencil in (u-) corresponds a row or pencil, projective to it, in (ro'). These results are obvious from the definitions and theorems which have been already proved. Similar propositions hold for projective bundles: but, by the principle of duality, it is enough to consider projective plane fields. 71. We have seen that a projectivity between two elementary forms of the first rank is established when any three pairs of corresponding elements are assigned.. A corresponding theorem for two plane fields is that: If, in the field (o-) we take any four points A, B, C, D, which are the vertices of a quadrangle, and in (o-') any four points A', B', C', D', which are the vertices of a quadrangle, then there is a definite projectivity ((r) w (a-') in which (A, A'), (B, B'), (C, C'), (D, D') are four pairs of corresponding points. We may call this the projectivity (ABCD...) W (A'B'C'D'...). PLANE FIELDS AND BUNDLES 69 We shall prove this by first showing that a projectivity satisfying these conditions exists, and then showing that it is unique, although it can be established by an unlimited number of constructions. 72. The first part of the proof will consist of a chain of propositions, leading up from special cases to the general one. (i) Let ABCP, ABCP' be two quadrangles in the same plane, such that P, P' are collinear with A; there is a projectivity (ABCP...) A (ABCP'...). Namely, the plane homology, of which A is the centre, BC the axis, and P, P' a pair of corresponding points (Art. 65). (ii) Let ABCP, ABCP' be two quadrangles as before, but P, P' not collinear with A, B, or C: there is a projectivity (ABCP...) A (ABCP'...). Let AP, BP' meet in P": then since A, P, P" are collinear, and also B, P", P', we can find, by case (i), two homologies (ABCP...) W (ABCP"...), and (ABCP"...) 7 (ABCP'...). By compounding them (that is, applying them in succession), we obtain a projectivity (ABCP...) w\ (ABCP'...). (iii) Given any six coplanar points A, B, P, P', Q, Q', such that ABPQ, ABP'Q' are quadrangles, there is a projectivity (ABPQ...) W (ABP'Q'...). Let C be any point on PP'. Then taking P, P' as corresponding points, there is a homology, centre C, axis AB, which makes (ABCPQ...) 7 (ABCP'Q"...), where Q" is on CQ. If Q" coincides with Q', the theorem is proved: if not, ABP'Q', ABP'Q" are quadrangles, and by case (ii) there is a homology, making (ABP'Q"...) 7 (ABP'Q'...). The proposition now follows by compounding the homologies (C, of course, comes into a new position C', generally speaking). (iv) Consider, now, two quadrangles ABCD, A'B'C'D' in different planes Or, a'. For the sake of generality, suppose that none of the pairs (A, A'), etc., coincide. (There must be at least two non-coincident pairs in every case.) Through A draw any plane r different from oa, and project A'B'C'D' on it from any point of AA', so as to produce a new quad 70 PROJECTIVE GEOMETRY rangle AB1O1D1 in the plane r. Next, from any point of BB1 project ABlOlD1 on to the plane ac. We now have in the plane a two quadrangles ABCD and ABC2D2 having two vertices A, B in common. By case (iii) there is a projectivity (ABCD...) A7 (ABC2D2...) in the plane a, and since, by construction, there is a projectivity (ABC2D2...) 7 (A'B'C'D'...), it follows (Art. 54) that there is a projectivity (ABCD...) 7- (A'B'C'D'...). Every other case may now be reduced to (iv). For instance, take two quadrangles, in a general position, in the same plane a-. By projecting one of them from an arbitrary point P on to an arbitrary plane r, the problem may be reduced to a case of (iv). Thus the first part of the fundamental proposition has been proved. 73. We have now to show that the perspectivity (ABCD...) w (A'B'C'D'...) is unique: that is to say, it is a determinate correspondence independent of the construction by which it has been obtained. Since the three rays AB, AC, AD of the pencil (A) correspond definitely and projectively to the three rays A'B', A'C', A'D' of the pencil (A'), it follows by Chapter VIII. that we have a definite projectivity (A) w (A'). Similarly, whatever the construction may be, the projectivities (B) 7 (B'), (C) 7w (C'), (D) 7 (D') are perfectly definite. Now if P is any point of (c), it can always be defined as the intersection of two rays of some two of the pencils (A), (B), (C), (D): hence P' must be the intersection of the corresponding rays of the corresponding pencils in (a'), and is therefore uniquely defined. Similarly, to every line p in (o) corresponds a perfectly definite line p' in (ao'). The same argument shows that to any element, or elementary form, in (a-') corresponds a definite element, or projective elementary form, in (o). This merely amounts to reversing any perspective chain connecting (a-) with (cr'). 74. Plane Constructions. The most important case, practically, is when the two assigned corresponding quad PLANE FIELDS AND BUNDLES 71 rangles ABCD, A'B'C'D' are coplanar. We can then, by a plane construction (Art. 57 (ii)), establish the relations A(BCD...) 7 A'(B'C'D'...) and B(ACD...) 7 B'(A'C'D'...), which can co-exist, because each makes AB, A'B' correspondinglines. Anypoint P that is not in AB may be considered as the intersection of two rays AP, BP: hence P' is determined as the intersection of the corresponding rays A'P', B'P'. For any point on AB we may use the projectivities C(DAB...) -\ C'(D'A'B'...) and D(CAB...) 7w D'(C'A'B'...). To find the line p' corresponding to a given line p, we can take any two points H, K on p, and find their correspondents H', K': then H'K' is the required line p'. 75. Alternative Forms of the Projective Data. There are various other ways of stating necessary and sufficient conditions for a definite projectivity (cr) w (-'): they may all be reduced to that of the assignment of two corresponding quadrangles. (i) Take any quadrilateral abed in (or) and any quadrilateral a'b'c'd' in (o-'). Then the correspondence (a, a'), etc., establishes a definite projectivity between (c), (cr'). The points ab, be, cd, da are the vertices of a quadrangle, and they must correspond to a'b', b'c', c'd', d'a' respectively. Take the projectivity established by these quadrangles: then it satisfies the given conditions, and is the only one that does so. (ii) Take any two points A, B in (cr), and two points A', B' in (a-') to correspond to them. Now make the pencils (A), (A') projective in any way consistent with AB, A'B' being corresponding rays: and then make (B), (B') projective in any way consistent with BA, B'A' being corresponding rays. Then a definite projectivity between (o), (a-') has been established. In fact, we may take two arbitrary lines p, q, drawn through A, and different from AB, to correspond to two arbitrary lines p', q', different from A'B', drawn through A'. Similarly, we may assign two rays r, s through B to 72 PROJECTIVE GEOMETRY correspond to two rays r', s' through B'. We now have a projectivity for which pqrs, p'q'r's' are corresponding quadrilaterals. (iii) Take any two lines a, b in (a-), and two lines a', b' in (a-') to correspond to them. Now make the rows (a), (a') projective in any way consistent with ab, a'b' being corresponding points: and then make (b), (b') projective in any way consistent with ba, b'a' being corresponding points. Then a definite projectivity between (C), (oa') has been established. This is the planar correlative of (ii), so it is needless to write out the proof. 76. Collineation. When (or) 7 (c'), any element A (or a) in (a-) corresponds to a similar element A' (or a') in (a-'), and, moreover, to any join (or intersection) AB (or ab) in (o) corresponds the join A'B' (or intersection a'b') in ((-'). The combination of these two properties is expressed by saying that (a) is collinear to (a-'). The same thing holds for two projective bundles, which are therefore also said to be collinear. Every projectivity which associates similar elements (point to point, etc.) is a collineation, but the converse must not be assumed. 77. Collinear Plane Nets. Take a projectivity (ABCD...) 7 (A'B'C'D'...) deduced from two assigned quadrangles ABCD, A'B'C'D'. Since the relation is a collineation, we have three pairs of corresponding points given by (AB, CD), (AC, BD), (AD, BC) 7 (A'B', C'D'), (A'C', B'D') (A'D', B'C'); these are the diagonal points of the given quadrangles, and can be found by linear construction. Let them be called (E, E'), (F, F') (G, G') respectively. We have now in (ar) a configuration of six lines and seven points. If we join EF, FG, GH, each of these lines cuts two sides of ABCD in two new points: so we have now nine (6 + 3) lines and thirteen (7 +6) points (cf. Fig. 9, starting from the quadrangle KLMN: the seven points are KLMNOAB, and the PLANE FIELDS AND BUNDLES 73 thirteen points are KLMNOABEFGHCD). Each of these twenty-two elements has a definite correspondent in (o-'), obtained by the corresponding linear construction. The process may be continued indefinitely, because however far we go, there will always be points not yet joined: we thus obtain what we may call two projective collinear nets [ABCD], [A'B'C'D']. At any stage we shall always have at least one quadrangle (such as OGMH in Fig. 9), with its diagonal points not all determined: so we obtain at least one new point by completing the quadrangle in question. It is not difficult to show that [ABCD] contains the four harmonic derivative pencils A[BCD], B[CDA], etc. (cf. Chapter VII.): hence the set of points in [ABCD] is compact, and every point in (o-) is either a point of this set, or a limiting point of it. In particular, there is no area in r which does not contain an unlimited number of points, and none that is not crossed by an unlimited number of lines, belonging to [ABCD]. 78. Correlative Plane Fields. Suppose that in (a-) we take any quadrangle ABCD, and in (ra') any quadrilateral abed. We can, in one way only, make the pencil A(BCD...) projective to the row a(bcd...): similarly, we can make B(ACD...) projective to b(acd...), and these two projectivities can co-exist, because each of them makes AB correspond to ab. We now have a correspondence between (oa), (r-') such that to every point in (Cr) corresponds a line in (ao'), and to every line in (a-) a point in (a-'). Conversely, to every point or line in (a-') corresponds one and only one line or point in (a-): because otherwise we could find two projective fields in ac, not identical, but having A, B, C, D as self-corresponding points. The fields (C), (a-') thus related are said to be correlative (or reciprocal). The essence of a correlation between two fields is that it is a one-to-one correspondence, such that to each element of either field corresponds a correlative element in the other, and the join (meet) of any two similar elements in either field corresponds to the meet (join) of the 74 PROJECTIVE GEOMETRY corresponding elements of the other field. From these properties alone it follows that, if two dissimilar elements (P, p) of (C) are conjoint, that is, if P is on p, the corresponding elements (p', P') are also conjoint. Further, to every harmonic figure in (-) corresponds a harmonic figure in (a'): to every row or pencil in ((r) a pencil or row in (a'), and so on. If we set up a homology in (r, there is a correlative homology in ': the centre and axis of the latter correspond to the axis and centre of the former; and if one homology is an elation, so is the other. It is not difficult to prove that any projectivity in C- can be obtained by compounding a chain of homologies in -r. The correlative chain in oa' gives, by composition, what we may call the corresponding projectivity in a'. More generally, if we have any plane construction which establishes a projectivity in o, we can, by dualisation, find an analogous construction setting up the corresponding projectivity in r'. 79. Correlation between two plane fields essentially depends upon the principle of duality. The only direct projective relation between two plane fields is that of being both perspective sections of the same bundle: this is a collinearity, and not a correlation, and the same is true if we connect two plane fields by a perspective chain of any extent. Nevertheless, it is convenient to include correlations as well as projectivities proper, in the class of projective relations (or correspondences). 80. Correlative Bundles. Everything said about correlative fields applies to correlative bundles, if we change "point " into "plane." For instance, if (S), (S') are two bundles, we can set up a correlation between them by taking any four-face a/3y8 in (S) to correspond to any fouredge abed in (s'). The sections of two correlative bundles by two planes (or one plane) are correlative fields: conversely, the projections of two correlative fields from two points (or one point) are two correlative bundles. It may PLANE FIELDS AND BUNDLES 75 be noticed that a flat pencil in one of two correlative bundles corresponds to an axial pencil in the other. 81. There are various other simple ways of setting up a correlation between forms of the second rank, but they are all, in the end, equivalent to making four similar elements a, b, c, d in the one correspond to four correlative elements a, /3, y, 8 in the other. Neither tetrad of elements must belong to the same form of the first rank: otherwise they are arbitrary, and we may write (abcd...) 7 (a/3y...) to indicate the definite correlation set up in this particular way. CHAPTER XI. GROUP THEORY. 82. Definition of Group. Let A, B, C, etc., denote a set of operations which can be applied to a given object or set of objects 12, and let A(12) be the result of applying A to Q2. Suppose that if A, B are any two of the operations, A can be applied to B(S2) and also B to A(Q2). We shall put, in order to avoid brackets, A{B(f2)}- AB2, B{A(2) } = BA2, and each of the symbols AB, BA may be regarded as denoting a single compound operation applied to Q2. The operation AB is not necessarily the same as BA: for instance, Q may be a solid body, and A, B finite rotations, in which case AB is generally different from BA. Next let us suppose that BA, AB are both operations of the original set; say BA=K, AB=L; and, finally, let us suppose that for every operation A there is one definite unique operation A-1 of the set, such that A-1A(Q) -AA-1(Q) =2. We shall say that A-1 is the inverse of A: clearly A is the inverse of A-~. Also AB is the product, or resultant, of A, B in the sense that the operation B is followed by the operation A. We shall write AA-1=I, regarding I here as a symbol of "the identical operation," which leaves 2 unaltered. From the nature of the case, A(BC) =(AB)C, etc., GROUP THEORY 77 so the associative law of multiplication is valid and we may write ABC for either symbol; moreover, if we write AA=A2, A-1A-1=A-2, etc., it is easy to show that the index law AmAn =Am+n is true for all positive and negative integral values of m and n. All such symbolical equations, however, only have a " real" interpretation when we have a "real" object 2 to which all the operations can be applied. If we are indifferent to the existence of Q2, we can still work out an abstract group-theory, which then becomes a branch of pure mathematics. If the symbols A, B, C, etc., obey the above laws, they are said to form a group. In the applications we intend to make, 12 is a geometric form, and A, B, etc., are projective relations. 83. Deductions: (i). If AB=AC, or BA =CA, then B=C. For if AB =AC, then A-1(AB) =A-1(AC), that is, (A-A)B = (A-'A)C, whence B =C: and similarly, if BA =CA, it follows that BAA- =CAA-1, and hence that B=C. (ii). The inverse of ABC is C-'B-A-1. In fact ABC.C-B-1A-1 = AB.CC-1.B-1A-1 =AB.B-'A- = A.BB-1.A-1 =AA-1 =I, and since ABC has only one inverse, the proposition follows. Similarly, the inverse of A1A2... A1n An is An An-1....Aa1. (iii). The sequence A, A2, A3,.... An,... is either unlimited, or else there is a least positive integer p such that AP =I. Proof. If the operations A, A2, etc., are not all different, there must be two least positive integers a, p such that Aa =Aa+P; but this leads to A-aA =A-Aa.Aa+p that is, I =AP, 78 PROJECTIVE GEOMETRY and p is the least index for which this is true. The operation A is then said to be of period p; the operations I, A, A2,... AP-1 are all different, because if we had Am = An with m, n both less than p and m>n, we should have Am-n = with (m -n) less than p. 84. Subgroups. Tetrahedral Group. It may happen that a part of a group may itself be a group: in this case it is said to be a subgroup of the original group. For instance, if A is of period p, the operations I, A, A2,.... AP —1 form a group, AP-r being the inverse of Ar. In projective geometry the most important group is that which can be represented by the permutations of four symbols A, B, C, D. This is a finite group of order 24 (since there are 24 distinct permutations), and the analysis of it will help the reader to understand those parts of group-theory that we shall require. The easiest way is to start with a regular tetrahedron ABCD, and consider those rotations about axes through its centre which permute the vertices. One such rotation is that of I80o about the line bisecting AB, CD: this changes ABCD into BADC and is of period 2. Similarly, if we draw a line through A perpendicular to BCD, a rotation of I20~ about it changes ABCD into ACDB. Altogether, since the latter rotation is of period 3, we have 3.I+4.2=II rotations, which, with the identical operation, make up a group of order 12. Tabulating them according to the way they were found, we may write U = ABCD S1i2=BADC, S2 2=CDAB, S32=DCBA S2 = 2.2 = S32 = I T1,2 =ACDB, T12Q =ADBC T2Q = DBAC, T22_2 =CBDA T3J = BDCA, Ts2f = DACB T4S =CABD, T42Q = BCAD T13 =T23 =T33 -T3 = I. GROUP THEORY 79 These form what is called the tetrahedral group; by interchanging A, B in each of them we get the remaining 12 permutations of ABCD. The operations (I, SI, S2, S3) form a subgroup; in fact, S2(S1&2) =Sl(S22) =S,32, etc., so we have the multiplication table I 1 S 2 S3 I I S S2 S3 S, Sz I S3 S2 S2 S2 S3 I SI l. _. S3 83 I S2 I where the entry for SmSn is in the row to which Sm is prefixed and in the column headed by Sn. We have already encountered this group in Art. 59. By interchanging A, B in each of these four permutations we obtain four new ones, and the eight thus found form a subgroup of the 24 permutations of ABCD. This group has presented itself in Art. 37. 85. Projective Groups. A projective relation sets up a one-to-one correspondence between two geometrical forms, and the elements they contain. Let F, F' be the forms; then F' can be connected with F by a set of projections, sections, correlations, and so on. This construction may be varied without changing the relation of F' to F: on the other hand, if F, F' are given, there will be, if they are of the same rank, an unlimited number of different projective relations between them. We shall write F' =OF to express that F' is connected with F by a definite projective relation, denoted by A, which may also be regarded as an operation transforming F into F'. The advantage of this notation is that by using different symbols, A, p, etc., we are able to denote different relations 80 PROJECTIVE GEOMETRY connecting the same two figures; it also emphasizes the relation in itself, apart from any special construction by which it is established. When F' =F, we shall also write F =r-1F', and regard zw- as an operation inverse to m. This is legitimate, because if any actual construction leads from F to F' we can reverse it, so as to go from F' to F. Since zr(z-wF')= WF = F', we are entitled to put ~-1=I; and similarly -a- 1' = I. Suppose now that F and ZF are cobasal forms; then in the abstract they are identical, in the sense that each contains the same set of elements. But each element has a different name according' as we regard it as belonging to F or OF. For example, if we have a projectivity (ABC..). (A'B'C'...) on the same line, so that z5A = A', etc., then A' as an element of (ABC...) will have a name such as P, and the corresponding point in (A'B'C'...) will be called P'. This kind of projectivity may be called a permutation. For any given base the set of all possible permutations forms a group. This follows from the definitions of Art. 82. Suppose, for instance, we have two permutations, mz, Iz2 converting the row (ABC...) into (A'B'C'...) and (A"B"C"...) respectively: then 1aZ2(ABC... ) =a1(A"B"C"...), and this is a permutation, a' say, of (ABC...) because we are dealing throughout with the same set of points. More generally, let there be any three forms F,, F2, F3 of the same rank, such that F1 F2, and F2AF3; then F 7 F and we can write F2= aFl, F3= zF2, F3= aF1. But we also have, symbolically, F3 = z2(1F) = z2WzlF1: hence we may write w3 = Wr2, to express that Z3 is the resultant of zI followed by zv2. From what we have already proved in previous chapters, it follows that the resultant of two projectivities is always a projectivity, even after including homologies and correlations. It should be observed GROUP THEORY 81 that the resultant of two correlations is a collineation: thus if (p', P') =mz1(P, p) and (P", p") =r2(p, P'), then (P", p") = r2zM(P, p), and this is a collineation. For the present we shall be mainly concerned with (projective) permutations: the first and most obvious problem is the discussion of permutations which satisfy the relation z2 = I. These are called involutions, and will be the subject of the next chapter. M.P.G. CHAPTER XII. INVOLUTION. 86. Let ABC, A'B'C' be triads on the same line I, with A' distinct from A. Then there is a definite projectivity (ABC...) A (A'B'C'...) connecting two rows upon I. Let this be denoted by A, so that (A'B'C'...) = z(ABC..). Then M2(ABC...) = (A'B'C'...) = (A"B"C"...), where A" is, in general, different from A. If, however, A" coincides with A, we can prove that r2 = I, that is to say, the rows (A"B"C"...), (ABC...) are identical. To prove this, let P be any other point on (ABC...) and put zP = P', zP' = P": then AA'PP' r (AA'PP') AA'A"P'P" A'AP'P", since A" =A. But by Art. 59 A'AP'P" -AA'P"P': hence AA'PP' AA'P"P', and therefore, by the fundamental theorem, P" coincides with P, and 2 = I, as asserted. When,2 =I, the relation mP =P' leads to zP' =P; and conversely, these last two relations both holding for an arbitrary point P lead to za2P=P, and hence to 2 =I. INVOLUTION 83 For a special position of P, we may have Mp = P; P is then a double point of the involution. 87. Involutionary Row. The permutation zA, satisfying M2I, is said to produce an involution of points on I. The essential fact is that if A' in the second row corresponds to A in the first, then A' (-= Q, say) in the first corresponds to A(=Q') in the second. Thus, apart from selfcorresponding points, all other points of I are grouped into conjugate pairs such as A, A'. Now although, from the original point of view, we have on I two different rows in a certain special projective relation, it is for many purposes more convenient and important to think of the result as one row, the elements of which are paired off in a definite way. We can always, if necessary, go back to the other point of view. From the later standpoint we shall speak of an involutionary row, or an involution of points, or simply an involution (on I). 88. An involution on a line is determined by any two conjugate pairs (A, A'), (B, B'). Namely, there is a unique projectivity (AA'B..) =M(A'AB'...) which makes zrA =A' and zA' = A, whence also 2A = A. Now A is not a self-corresponding point of a; hence, by Art. 86, we have rZ2P =P, if P is any point on the line. Hence 2 = I and the relation is an involution. We cannot have more than one such involution, because the correspondence of the triads AA'B, A'AB' fixes the projectivity. 89. Let K, L, M, N be the vertices of a complete quadrangle; and let any line I cut KL, MN in A, A', KM, LN in B, B' and KN, LM in C, C'. Then (A, A'), (B, B'), (C, C') are three pairs of an involution on I. Let KM, LN meet in 0. Then by projection from K on to LB' we have ABCB' TLONB', 84 PROJECTIVE GEOMETRY and now by projection from M upon I LONB' -\C'BA'B'. Hence ABCB' - C'BA'B'' A'B'C'B by Art. 59. Writing (ABCB') =Z(A'B'C'B), we have Z5B = B', and MB' = B; that is, z2B = B, whence also (Art. 86),2A-A, Az =2CC, as may be proved independently from ~A Z 4' E//A FIG. 26. the other diagonal points of KLMN. This proves the proposition. 90. An involution on I being given by two pairs (A, A'), (B, B'); it is required to find the conjugate of any other point C. Draw any two lines through A, A' and any line through C cutting them in K, N; let B'N, BK meet AK, A'N in L, M: then the intersection of LM with I is the required point C'. This follows from last Article. 91. If an involution on I contains a double point M, there is another double point N, and all such ranges as MNAA', MNBB' are harmonic. Take N so that MNAA' is harmonic, and let N' be the conjugate of N in the involution. Then MNAA' AMN'A'A, INVOLUTION 85 since M corresponds to itself. But since MNAA' is harmonic, MNAA' - MNA'A: hence MNA'A-AMN'A'A, and therefore N' = N, and N is a double point. An involution cannot have more than two double points; when they exist, they are often called the foci of the involution. 92. Involutionary Pencils. By projecting an involutionary row from an external point or line, we obtain an involutionary flat pencil or axial pencil respectively. These have properties exactly analogous to those of the row; and we may, if we like, prove them independently. The projections and sections of involutionary forms of the first rank are themselves involutionary. If two involutions are projective, and one of them has double elements, so has the other. 93. By dualising Art. 89 we find that if abcd is a complete quadrilateral and L any external point in its plane, the six lines joining L to the vertices of the quadrilateral form three pairs of rays in an involutionary pencil, corresponding rays of a pair being those which pass through opposite vertices of the quadrilateral, such as ad, be. It is a good exercise to write out the correlative proof. By dualising from plane to bundle the student can obtain two other correlative theorems. 94. Notation. An involution of points is determined by two pairs, one double point and a pair, or two double points. Thus we may speak of the involution (AA'.BB'), or(M2.AA'), or (M2.N2). To express that (A, A'), (B, B'), (C, C') are conjugate pairs, in the same involution, we shall write ABC-A'B'C'. Any accent on the right may be removed to the corresponding letter on the left; thus AB'C-A'BC' means the same thing as ABC-oA'B'C'. If C, or B and C, be selfcorresponding, we write CABt-CA'B', BCA BCA', the last formula being a way of expressing that BCAA' is 86 PROJECTIVE GEOMETRY a harmonic range. Similar notation may be used for pencils. 95. Consider an involution in which (A, A'), (B, B'), (C, C') are three pairs. The first two pairs determine the involution, and also two corresponding segments ABA', A'B'A. If, then, C describes the segment ABA' in that sense, C' simultaneously describes A'B'A. Take the circular A0 N M FIG. 27. image of the line (Fig. 27) and suppose, in the first place, that B, B' are not separated by A, A'. Then B, B' are on the same one of the complementary segments AA', A'.A, and therefore ABA', A'B'A represent the same segment described in opposite senses. Therefore on this segment there must be a position of C which coincides with C'; that is, there is a focus N upon it. Similarly, there is a focus M on the segment complementary to ABA'. In a similar way we can prove that if B, B' are separated by A, A' there cannot be a focus. For in this case the segments ABA', A'B'A are of the same sense and complementary, so that C can never coincide with C'. An involution without double elements is said to be elliptic: one with double elements is called hyperbolic. What we have proved amounts to the statement that any two pairs of an elliptic involution separate each other, and that no two pairs of a hyperbolic involution separate each other. INVOLUTION 87 One focus of an involutionary row may be at infinity: in this case the other focus bisects each finite segment AA'. 96. Involutionary Fields and Bundles. A harmonic homology (8, I: A, A') may be regarded as an involution. In fact (cf. Arts. 67-8) we have on every line such as SAA' an involution of which S is one focus and the meet (AA', I) the other. Similarly, every point P on I is the centre of an involutionary pencil, of which I, SP are double rays. Apart, then, from (S) and (I), all the points and lines in the plane are grouped into conjugate pairs, and the homologic transformation (r, say) satisfies the equation Z2 =I. By projecting an involutionary field from an external point we obtain an involutionary bundle. 97. The following proposition is of great theoretical interest: A projectivity connecting two cobasal rows can always be expressed as the resultant of two involution;. Let -z be the projectivity, A, A' any two distinct corresponding points, so that MA =A'. Let MA'= A"; then if A" =A, A is an involution. If not, let il be the involution (A'2, AA"): then lW1(AA') = i(A'A") = A'A. Therefore 1lz is an involution; call it 12. Then ll = 12, and hence, since 12 = I, M = 1i2 = 112. With the same notation, we have Dh(A'A") = M(A'A) = A"A'; therefore z3a is an involution, say t3, and Zt1l = t3, whence T = rz12 =- 31. Thus, unless the given projectivity is an involution, we can always write = 11t2 = 1311 where il is an involution with a focus at any point A' which is not a self-corresponding point of t. 88 PROJECTIVE GEOMETRY 98. Commutative Involutions. Two involutions hi, 12 are said to be commutative, if ~2t1 = 112. If 11, 12 are commutative involutions, then 1t2 is an involution, and conversely. Suppose t112 = t21; then (1lt2)2 = 1112-112 = 1112-1211= _1112.11 = 12 = I, so that l112 is an involution. Conversely, if 1112 =3, with Li2 = 12232=I, we have t11213 =, and hence 121 = 1211l 11213 - 12.112.1213 = 12.2t 3 1223 3 13 1112' 99. If A, A' are any two distinct conjugate points of an involution i, then the involution i' of which A, A' are foci is commutative with i. In fact 1'(AA') = t(AA') = A'A, hence 1t' is an involution, and the theorem follows from Art. 98. 100. If BCAA', CABB', ABCC' are all harmonic ranges, then ABC AA'B'C'. We have ABCC'W CABB' ACBB', whence ABC'A-ACB'. Now AA'BC is harmonic by hypothesis, so that A' is the other focus of this involution, and therefore, since B', C' are conjugate points, B'C'A'A is harmonic. Similarly C'A'B'B and A'B'C'C are harmonic. Therefore C'A'B'BA CABB', from which we infer ABC AA'B'C'. The reader may prove in a similar way that ABB'7AA'C'C, BAA'^B'C'C, and CAA'WC'B'B. The theorems of this article have an interesting analogue in algebra. Let the homogeneous coordinates of A, B, C be determined by a cubic equation f(x, y) =o: then those of A', B', C' are the roots of the cubicovariant of f. Also the roots of the Hessian off are the coordinates of the foci of the first of the involutions defined above. 101. Let there be two projective rows (ABC...), (A'B'C'...) on two coplanar bases I, I', and let i" be the axis of the pro INVOLUTION 89 jectivity (cf. Art. 57) Then if S is any point on I", we have S(ABC...)S(A'B'C'...). This follows, in fact, immediately from the nature of the axis of projectivity. The pencils S(ABC...), S(A'B'C'...) are projective, and if SA cuts I' in P', SA' cuts I in P. Putting, then, A'S = z(AS), we have a2(AS) = r(A'S) = ((SP) ==SP' = AS; hence z2=I, and the proposition is proved. The reader should deduce the correlative plane theorem. CHAPTER XIII. PROJECTIVE GENERATION OF CONICS. 102. Suppose we have a projectivity (a'b'o'...)= r(abo...) connecting two coplanar pencils which are neither concentric nor perspective; then the points aa', bb', etc., in which corresponding rays intersect will occupy a certain curve, the properties of which we may proceed to examine. No line can meet the curve in more than two points. Let S, S' be the centres of the pencils. Then any line I disjoint to S and S' cuts the pencils in rows (PQR...), (P'Q'R'...) which the given projectivity makes projective to each other. Now any point of the curve which is on I is a self-corresponding point in the projectivity connecting the rows; and there cannot be more than two such points. The points s, 8' lie on the curve. In fact, to S'S considered as a ray of (S') corresponds a different ray of the pencil (): these two rays meet at 8, which therefore lies on the curve. Similarly S' lies on the curve. There are lines which meet the curve in two points. Namely, if (a, a'), (b, b') are any two pairs of rays, the join (aa', bb') meets the curve in the two points aa', bb', and no others. Any line, such as p, through 8, in general meets the curve in one other point pp'. If, however, it happens that p' is S'S, p will not meet the curve at any point distinct from S. This special ray of (S) is called the tangent at S. Similarly there is a tangent at S'. PROJECTIVE GENERATION OF CONICS 91 On account of these different properties, the locus of aa' is called a curve of the second order. 103. Envelope of the Second Class. By dualising Art. Io2, we arrive at the following results: If there is a projectivity (A'B'C'...) =zr(ABC...) connecting two coplanar rows which are neither cobasal nor perspective, the lines AA', BB', CC', etc., form an envelope of the second class. No point can lie on more than two lines of the envelope. If s, s' are the bases of the rows, then to s's considered as a point of (s') corresponds a point P on s which is called the point of contact of s. No other line of the envelope goes through P. Similarly, there is a point of contact, Q', upon s', and we have SS' =P' =Q. We shall prove in a short time that there is a tangent at every point of a locus of the second order, and that these tangents form an envelope of the second class. The converse is also true: so we are ultimately dealing with the same figure from correlative points of view. It is convenient therefore to use the term conic to denote, in a strict sense, a locus of the second order, with its associated envelope of tangents: and in a looser sense, as either the locus or the envelope separately. The reader will find that this rather vague usage does not lead to confusion and saves words. 104. Construction of the Locus. Suppose the projectivity (S') =z(S) is defined by the assignment of two triads of rays abc, a'b'c'. We have already given a construction (Art. 57, ii) for finding d', the ray of (8'), which corresponds to any given ray d of (S): but it will be convenient to reproduce it here, and develop its consequences. In the figure * 8, 8' are the centres of the pencils, and u, u' two lines through aa' cutting the pencils (S), (S') in the rows (ABC...), (A'B'C'...) respectively. As perspective sections of (S), (8'), we have (ABC...)W (A'B'C'...), and since A = A', these rows are perspective. * Figures 28, 29 are taken from Reye's Geometrie der Lage. 92 PROJECTIVE GEOMETRY Let CC', BB' meet in S2: then 82 is the centre of the perspective. Now let any other ray d of (S) cut u in D: then if S2D cut u' in D', the line S'D' (=d') is the corresponding ray of (8'), and dd' (=P) is a point on the locus. By taking a set of points such as D upon u we deduce a corresponding set of points P on the locus. Practically we choose so many FIG. 28. points D in appropriate positions as to enable us to draw a ' fair curve ' through the corresponding points P. Let 838, S2S' cut u', u in L', M respectively: then it follows from the construction that L', M are points on the locus. Incidentally we have solved the following problems: Given any ray through S or S', to find its second intersection with the curve: Given any line through aa', to find its other intersection with the curve. To construct the tangent at S, let S'S cut u' in T', and PROJECTIVE GENERATION OF CONICS 93 82T' cut u in T: then ST is the tangent at S. Similarly for the tangent at 8'. (This construction has not been carried out in the figure.) 105. Suppose the conic has been completely described, as above, from (S), (S') and the auxiliary lines u, u'. Clearly all such points as A, L', M, P are on the same footing; but it might be supposed that S, S' have special properties as centres of the generating pencils. We propose to show that this is not the case, and that all the points of the conic are coordinate. In the figure, let us keep S, M, P, L', S' fixed: then S2 is also fixed, being the meet of SL', S'M. Now suppose A to move along the curve: then D describes a row along the fixed line SP, and D' describes a row along the fixed line S'P. Also DD' goes through the fixed point S2; so we have a projectivity D'= D connecting the rows. Therefore MD, L'D' describe projective pencils, and these pencils generate the curve. Thus any two points on the conic, distinct from S and S', may be taken as centres of generating pencils, and S, S' have no peculiar properties, so far as the conic is concerned. 106. Construction of the Envelope. Given two triads ABC, A'B'C' on the lines s, s', there is a single projectivity (A'B'C'...) ==z(ABC...) connecting the rows of which s, s' are the bases. When this is not a perspectivity the lines AA', BB', etc., envelope a conic, and we can construct the envelope, line by line, by dualising the process of Art. 104. Thus (Fig. 29) we take on AA' any two points U, U': then the pencils U(ABC...), U'(A'B'C'...) are perspective, because UA coincides with U'A', and determine an axis 82, the join of (UB, U'B') and (UC, U'C'). Let D be any other point of (ABC...); then the join of U' to the meet of UD with s8 meets s' in the point D' corresponding to D. Hence DD' is the line of the envelope, distinct from s, which passes through D. By varying the position of D we can construct as many lines of the envelope as we desire. 94 PROJECTIVE GEOMETRY The lines s, s' are elements of the envelope. Also, if ss2 be called L, and s's2 be called M', the lines U'L, UM' belong to the envelope. All lines of the envelope are coordinate, and any two of them may be taken as the bases of two projective rows which generate the conic. lb=l FIG. 29. The student is strongly recommended to draw the figures for himself, varying the position of the data until he is quite familiar with the constructions. He should also supply the omitted proofs of the later propositions of the present Article. 107. We are now able to draw the following important conclusions: x x t '" ", x,, x 'xX xI ly SI~~~~ C I,.// PROJECTIVE GENERATION OF CONICS 95 By joining any two points of a conic to all the other points of it, we obtain two projective pencils, in which corresponding rays are those which intersect on the conic. By finding the intersections of any two tangents to a conic with all its other tangents, we obtain two projective rows, in which corresponding points are those whose join touches the conic. If A, B, C, D, P are five If a, b, c, d, p are five points on a conic, such that tangents to a conic, such P(ABCD) is a harmonic pen- that p(abcd) is a harmonic cil, then if Q is any other range, then if q is any other point on the conic, Q(ABCD) tangent to the conic, q(abod) is also a harmonic pencil. is also a harmonic range. The set A, B, C, D is said The set a, b, c, d is said to be harmonic, and any one to be harmonic, and any point is determined by the one tangent is determined other three. by the other three. At each point of a locus On each line of an enveof the second order there is lope of the second class a tangent. there is a point of contact. Five points, of which no Five lines, of which no three are collinear, deter- three are concurrent, determine a single conic which mine a single conic which goes through them all. touches them all. Suppose, for instance, we have five points 8, 8', A, B, C. The definite projectivity S(ABC...) S'(A'B'C'...) generates one conic through the five points: and there cannot be any other such conic, since otherwise we should have two distinct projectivities with the pairs (SA, S'A), (SB, S'B), (SC, S'C) in common. Similarly for the correlative theorem. 108. Cones. By projection of Figs. 28, 29 from an external point, or independently, we can prove the following theorems: Two axial pencils in the If (abc...), (a'b'c'...) are same bundle, which are pro- two flat pencils in the same 96 PROJECTIVE GEOMETRY jective, but not perspective bundle, which are projecor coaxial, generate a cone tive, but neither perspec(aa', 3', yy'...) whose tive nor coplanar, the planes rays are the intersections (aa', bb'...) form a conical of corresponding planes. envelope of the second class. This cone is of the second order. It will ultimately appear that this envelope of the second class consists of the tangent planes of a cone of the second order, and conversely; so we use the term " cone" (or "quadric cone") in a loose sense for the conical regulus, or its set of tangent planes, or the assemblage of both, according to the context. The reader will easily extend to cones the theorems proved for conics. Thus a quadric cone is determined by five concurrent lines, no three of which are coplanar, and so on. The section of a cone by any plane disjoint to its vertex is a conic: in fact this is why the term "conic" (short for " conic section ") is used for a curve of the second order. By supposing the vertex of a cone to go to infinity, we obtain a cylinder; so we can state a number of projective theorems for quadric cylinders analogous to those which hold for cones and conics. The epithet "quadric" used here, although borrowed from analysis, is conveniently applied to a locus or envelope (whether plane or solid) to express that it is of the second order or second class respectively. CHAPTER XIV. PASCAL'S THEOREM. 109. Let I, 2,... 6 denote six coplanar points, no three of which are collinear. By 'the simple hexagon 123456' we shall understand the figure obtained by drawing the lines 12, 23, 34, 45, 56, 6i. These lines are called the sides of the hexagon; they may be grouped into three pairs (12, 45), (23, 56), (34, 6i) each consisting of two opposite sides. Cyclical permutations of 123456, 65432I, such as 345612, 432165, represent the same hexagon; any other permutation represents a different hexagon. Hence from the six points we can construct ~(6 1 6), that is, 60 different hexagons. If the six points lie on one and the same conic, their positions are not independent; because any five of them determine a conic, and the condition of being on this is a restriction upon the remaining point. Now Blaise Pascal (I623-I662) discovered, at the age of sixteen, the genuine projective way of stating the necessary and sufficient condition that six points may lie on the same conic; namely, If six points on a conic are named (in any order) I, 2, 3, 4, 5, 6, then the three points (12, 45), (23, 56), (34, 6i), in which pairs of opposite sides of the hexagon 123456 intersect, are collinear. Conversely, if any hexagon I23456 satisfies this condition, the six vertices I, 2,... 6 lie on the same conic (Pascal's Theorem). M.P.G. G 98 PROJECTIVE GEOMETRY The proof follows immediately from Fig. 30. Here we have the hexagon SPS'MAL': the three intersections of opposite sides are (SP, MA)=D, (PS', AL') = D', and (S'M, L'S)=82, and we proved in Art. Io4 that D, D', 82 are collinear. Conversely, suppose the hexagon SPS'MAL' does in this way give rise to three collinear points D, D', 82. Then the pencil (S2) determines on AM, AL' two projective FIG. 30. rows each perspective to it, and the projections of these from 8, S' respectively generate a conic. But this conic passes through 8, 8', A, M, L', P-the last point because D, D', S2 are collinear. Pascal did not discover this proof: in fact, he proved the theorem (metrically) for a circle, and then inferred it for a conic by the limited theory of conical projection then known, which was mainly derived from Greek geometers, such as Apollonius. Nevertheless, Pascal's theorem PASCAL'S THEOREM 99 is of an epoch-making character, and is the foundation of all the modern projective theory of the conic sections. By drawing a circle to represent the conic, the student can actually verify the theorem with very little labour. He should specially see how by drawing different hexagons for the same six points on the circle, he obtains different Pascal lines (I2.45, 23.56). There are 60 Pascal lines for a given set of six points, and they form a set with very interesting properties; many of these are stated in Salmon's Conic Sections. 110. Brianchon's Theorem. The planar correlative of Pascal's theorem is due to Brianchon, who obtained it by the method of reciprocal polars. It is as follows: Six lines I, 2,... 6 are tangents to the same conic, if the three lines (12, 45), (23, 56), (34, 6i) are concurrent: and conversely. The proof of this follows from Fig. 29 just as that of Pascal's theorem did from Fig. 28. Brianchon's theorem is often rather vaguely stated in the form " If a hexagon circumscribes a conic, its three diagonals meet in a point," and there are equally vague statements of Pascal's theorem. 111. Corollaries. One reason for the great fertility of Pascal's theorem is that we can obtain a number of important corollaries by making one pair, or more, of the vertices of the hexagon move up to coincidence. If in Fig. 30 we make P move up to coincidence with 8, the limiting position of SP is the tangent at S; so we see that by taking limiting cases of Pascal's theorem we obtain tangent properties of a conic. Thus, (i) If A, B, C, D, E are five points on a conic, the intersection of the tangent at A with CD is collinear with the intersections (BC, EA) and (DE, BA). This follows by putting A =I =2, B, C, D, E=3, 4, 5, 6 in the statement of Pascal's theorem. It may also be proved independently from Fig. 30. Incidentally, we see that if a conic is defined by five points A, B, C, D, E, we can draw the tangent at any one of them, such as A. Namely, 100 PROJECTIVE GEOMETRY let BC, EA meet at P, and DE, BA at Q: then if PQ meets CD at T, AT is the tangent required. Correlatively, (ii) If a, b, c, d, e are five tangents to a conic, the point of contact of a joined to cd gives a line concurrent with the lines (bo, ea) and (de, ba). If a conic (envelope) is given by means of a, b, c, d, e, we can find a's point of contact T. Namely, find the lines (be, ea) = p, (de, ba) =q, then T is the point where the line (pq, cd) meets a. (iii) If A, B, C, D are four points on a conic, the tangents at A, C intersect in a point on the line joining (AB, CD) to (AD, BC). This follows from Pascal's theorem by putting A = =2 B=3, C=4=5, D=6. (iv) If a, b, c, d, are four tangents to a conic, the points of contact of a, c lie on a line concurrent with (ab, cd) and (ad, be). This is the correlative of the last. Finally, let us put A = =2, B=3=4, C =5 =6; then Pascal's theorem becomes (v) If A, B, C are three points on a conic, and a, b, c are the tangents thereat, the points (BC, a), (CA, b), (AB, c) are collinear. Correlatively, (vi) If a, b, c are three tangents to a conic, and A, B, C their points of contact, the lines (be, A), (ca, B), (ab, C) are concurrent. Of these six corollaries, (iii) and (iv) are by far the most important, because they lead, as we shall see, to the theory of poles and polars, and the proof that an envelope of the second class consists of the tangents of a, locus of the second order, which for convenience we have been assuming. The reader should note that the converses of the six corollaries are also valid. 112. It will be noticed that as corollaries of the fact that a conic is (in general) determined by five points, we PASCAL'S THEOREM 101 infer that a conic is determined by four points and the tangent at one of them, or again by three points and the tangents at two of them. In each case we can find other points on the curve by the Pascal construction, exhibited in Fig. 30. For practical purposes it is convenient to give the following rule: Calling the five points I, 2, 3, 4, 5, find 8,, the intersection of I2, 45: draw any line through I cutting 34 in D', and let -,i (K FIG. 31. S2D' cut 23 in D. Then the lines ID', 5D meet on the curve (in Fig. 30, 12345 =S'MAL'S). The case when the data are three points K, L, M and the tangents k, I at K, L (Fig. 31) is so important, that we shall give one of the Pascal constructions for it in full. We put K = =2, L=4=5, M =3, and the construction derived from the rule is the following: Let k, I intersect at R, and let any line through K cut LM in Q. Join RQ, meeting KM in O; then LO meets KQ in a 102 PROJECTIVE GEOMETRY point N on the curve. In the figure, K, L are opposite vertices of the simple quadrangle KMLN, so we have a diagram illustrating Art. inI (iii). Finally if, in the projective pencils (K), (L), which generate the conic, we put KM, KN =a, b and LM, LN =a', b', then 0 =ab', Q =a'b, and R is the centre of the projectivity (Art. 57 (ii)). The student should work out the correlative case for the envelope: the data being three tangents k, I, m and K, L the points of contact of k,I. 113. Equivalence of Locus and Envelope. We are now able to prove that the tangents of a locus of the second FIG. 32. order form an envelope of the second class, and conversely, the points of contact of an envelope of the second class form a locus of the second order. Let K, L, M, N (Fig. 32) be four points on a quadric locus, and let the tangents there form the complete quadrilateral klmn of which the vertices are marked ABCDEF in the figure. Complete the quadrangle KLMN, and let X, Y, Z be its diagonal points. Consider the simple quadrangle KLMN, of which K, M are opposite vertices: then by Art. inI (iii) the points C, X, Z, F are collinear. Similarly, from the simple quad PASCAL'S THEOREM 103 rangles KMLN, KMNL we infer that A, Y, D, Z are collinear, and that B, Y, E, X are collinear. Thus: If we construct, from four points K, L, M, N on a locus of the second order, a complete quadrangle KLMN, and from the corresponding tangents k, I, m, n a complete quadrilateral kimn, then X, Y, Z being the diagonal points of the quadrangle, each of the lines YZ, ZX, XY contains two vertices of the quadrilateral. Conversely, if these conditions are satisfied by a quadrangle KLMN and a quadrilateral kimn, there will be one and only one locus of the second order which passes through K, L, M, N and has k, I, m, nfor the tangents there. With regard to the converse, it is enough to observe that there is one and only one locus of the second order going through K, L, M and having k, I as tangents at K, L. But the relations of the quadrangle and quadrilateral show, with the help of Arts. III, II2, that the locus thus determined also passes through N, and has m, n as tangents at M, N respectively. Correlatively: If we construct from four lines k, I, m, n of an envelope of the second class, a complete quadrilateral kimn, and from the corresponding points of contact K, L, M, N a complete quadrangle KLMN, then x, y, z being the diagonals of the quadrilateral, the points yz, zx, xy each lie on two sides of the quadrangle. Conversely, if these conditions are satisfied, there will be an envelope of the second class touching k, I, m, n at K, L, M, N. Now a consideration of the figure shows that the conditions satisfied by kimn, KLMN in these correlative theorems are the same. Hence, whenever KLMN are four points on a quadric locus, and kimn their tangents, kimn are four tangents and KLMN the corresponding points of contact of an envelope of the second class. But the locus and envelope are each, in such a case, uniquely determined without any reference to N or n. Namely, we may, for instance, define the locus as that which passes through KLM, and has k, I as the tangents at K, L; and the envelope as that which touches k, I, m and has K, L as the points of 104 PROJECTIVE GEOMETRY contact of k, I. Therefore every point (N) on the locus must be the point of contact of a line (n) of the envelope, and conversely. 114. Alternative Proof. Chasles' Theorem. Since the foregoing proof may be thought rather hard to follow, we give another, which has the advantage of also proving a theorem which appears to be due to Chasles, though he did not give a strict demonstration of it. Starting with the locus, supposed to be given, let us suppose L, M, N, and consequently I, m, n to be fixed, while K, k vary. In the figure B=nk and A =lk, so these points also vary, but they describe projective rows on n, I, because E, D are fixed and BE, DA intersect on the fixed line NL. Consequently as K describes the locus generated by N(NML...)7 L(NML...), k envelopes an envelope of the second class generated by two projective rows on n, I. The pencil (M) is projective to the pencil (E) by means of the row (Y), on LN, perspective to both: hence the row described by B, that is (EY, n), is projective to the pencil described by MK. Here we deduce Chasles' theorem: Let M be any point, and n any tangent of a conic. Then if K is a variable point on the conic, k its tangent, and we make the line MK correspond to the point nk, the pencil described by MK is projective to the row described by nk. 115. As an illustration of the extension of the theory to cones and cylinders, we merely state the following theorems, leaving the reader to supply the proofs. The tangent-planes to a quadric cone form a quadric conical envelope, and conversely. Any two tangent-planes of a quadric cone are cut by the remaining tangent-planes in two projective flat pencils. A quadric cone is determined by five rays, or by five tangent-planes, or by three tangent-planes and the lines of contact of two of them, etc., etc. CHAPTER XV. POLES AND POLARS. 116. Let K be any given conic: P any point in its plane that is not on the curve. Through P (Fig. 33) draw any FIG. 33. two lines cutting K in K, L and M, N respectively; and let 0, Q be the intersections (KM, LN), (KN, LM). Finally, let the line OQ be called p. 106 PROJECTIVE GEOMETRY If the tangents at K, L meet in R, then by Art. iii (iii) it follows that R is on p. Similarly for S, the meet of the tangents at M, N. If p meets PK in P' the range KLPP' is harmonic. This follows at once from the quadrangle QNOM. Hence the line p can be constructed from any one line, such as PKL, which cuts the conic in distinct points. Namely, we draw the tangents at K, L meeting in R, and find P', the harmonic conjugate of P with respect to K, L: then RP' is the line p. It follows from what has been proved that we obtain the same line by this last construction, whatever secant we draw through P. Thus, in the figure, the quadrangle QKOL shows that p cuts MN in a point P" such that MNPP" is harmonic. Thus, if through P we draw any number of secants, the following points all lie on one and the same line p: (I) The meets (KM, LN) and (KN, ML) of pairs of opposite sides of a simple quadrangle KLMN inscribed in K, whose opposite sides KL, MN meet in P. (2) The point on every secant which is harmonically separated from P by the curve. (3) The meet of any two tangents, whose points of contact are collinear with P. The line p which enjoys these properties is said to be the polar of P with respect to K. 117. Correlatively, let us start with any line p which does not touch K. On p there are an unlimited number of points at which two tangents intersect: let R, S be two of them, K, L the points of contact of the tangents through R, and M, N those of the tangents through S. Let KL, MN meet in P: and denote the lines RK, RL, SM, SN by k, I, m, n respectively. Then applying Art. iii (iv) we see that the lines (km, In) and (kn, Im), not shown in this figure, but cf. Fig. 34, pass through P, and that if p cuts KL in P' the pencil R(KLPP'), or klpp', is harmonic. Thus, if from points on p, we draw any number of tangent POLES AND POLARS 107 pairs, the following lines all pass through one and the same point P: (I) The joins (km, In) and (kn, Im) of pairs of opposite vertices of any simple quadrilateral klmn circumscribed to K, whose opposite vertices kl, mn lie on p. (2) The line concurrent with any tangent-pair such as k, I which is the harmonic conjugate of p with respect to k and I. (Cf. Fig. 34, putting YZ = p, x =P.) E// / FIG. 34. (3) The chord of contact of any two tangents which meet on p. The point P which enjoys these properties is called the pole of p with respect to K. 118. For the present, in speaking of pole and polar, we shall omit the words " with respect to K," which must be understood in every case. (I) If P is the pole of p, then p is the polar of P, and conversely. This follows at once from the definitions, and the figure. (2) If P moves up to the curve, so as ultimately to reach a point Q upon it, then the limiting position of the polar p is the tangent q which touches K at Q. This follows, for instance, from Art. II6 (3). When P =Q, a point on the conic at which q is the tangent, let 108 PROJECTIVE GEOMETRY r, s be any two other tangents, then r, q and s, q are pairs of tangents whose points of contact are collinear with Q, that is with P, and the join (rq, sq) is q, that is, the tangent at Q. We may, if we like, define the polar of a point on K as the tangent there, and the pole of a tangent as its point of contact. On the whole, this is preferable, as it avoids discussion of continuity. (3) If p, the polar of P, cuts K in two points T, T', then PT, PT' are tangents to K. Suppose, if possible, that PT cut the conic in another point U: take P' such that TUPP' is harmonic, then P' is on p, and also distinct from T, which is impossible. Hence we have a linear construction for drawing tangents to K from P, when this is possible. Namely, draw two secants PKL, PNM as in Fig. 33; find the intersections (KM, LN), (KN, LM), and their join p. Then if p cuts K in two real points T, T', the lines PT, PT' are the tangents required. The student should carry out this construction when K is a circle and P an external point. The figure gives everything except the actual joining of PT, PT'. The correlative problem is: Given an envelope of the second class, find the points of contact which lie on a given line. 119. Conjugacy. (I) If P lies on q, the polar of Q, then Q lies on p, the polar of P, and conversely. As in Fig. 33, draw any secant PKL through P: evidently this may be chosen so that QK, QL are not tangents to K, and we may suppose that QK, QL meet the conic again in N, M respectively. Let KM, LN meet in 0. Then by Art. 116 (I) O is on the polar of Q and since, by hypothesis, P is on q, PO must coincide with q. Hence M, N must be collinear with P, by the converse of Art II6 (I), and therefore p is the line OQ which passes through Q. The points P, Q, each of which lies on the polar of the other, are said to be conjugate with regard to K. Two points are conjugate when they are diagonal points of a quadrangle whose vertices are on K, and conversely. Thus, in the figure, POLES AND POLARS 109 the three pairs (P, Q), (Q, 0), (0, P) are conjugate, and each side of the triangle OPQ is the polar of the opposite vertex. Such a triangle is called a polar (or self-polar) triangle with respect to ic. To construct a polar triangle, we may start with any point P not on K, and then take Q, any point on its polar p which is not on the curve. Then 0 is determined as the intersection pq of the polars of P, Q. Correlatively, if we start with any line p, not a tangent of K, and draw any line q through its pole P, then if o is the polar of pq, opq is a polar triangle. (2) Incidentally we have proved that if p, q are the polars of any two points P, Q then pq is the pole of PQ. For since pq lies simultaneously on p and q, its polar must simultaneously pass through P and Q; that is, it must be PQ. In particular, if P, Q are conjugate, (pq, P, Q) are the vertices of a polar triangle, and conversely. Two lines p, q are said to be conjugate, if each passes through the pole of the other. 120. Points Inside and Outside a Conic. If P is any point not on K, the pencil (P) will contain either two tangents to K or none at all. In the first case we shall say that P is outside the conic: in the second, that it is inside. Suppose we take P inside: then every line through P meets K in two distinct real points, because clearly there are some such lines, and on account of the continuity of I/ we cannot have an abrupt change from two intersections to none at all. Let KPL be any chord through P, and let k, I the tangents at K, L meet at Q. Then Q is on p, the polar of P, and is outside the conic. Now as KPL turns round, and describes the whole pencil (P), kl describes the whole line p: consequently every point of p is outside K, and p does not cut the conic, because k, I never coincide. Thus if P is inside K, its polar does not cut the curve. Hence if PQR is any polar triangle with P inside the conic, Q and R are both outside. Conversely, if a line p does not meet K, its pole P must be inside the curve. On the other hand, if p meets K in 110 PROJECTIVE GEOMETRY two distinct points, its pole P is by definition outside the curve. As a special case, if p touches the curve, its pole is its point of contact P, and is on the curve. Thus, in general, if P is a point, and p its polar, either p cuts the curve and P is an outside point, or p does not cut the curve at all, and P is an internal point: and conversely. 121. Projectivity of Polar Rows and Pencils. On any line p in the plane of K we have a correspondence of conjugate points A, A'. We proceed to show that this is a projectivity. Let P (Fig. 33) be the pole of p, and through P draw any secant PKL. Let M be a variable point on the conic: then KM, LM describe projective pencils. But the rays KM, LM cut p in the conjugate points 0, Q: thus as M varies, 0, Q describe projective rows. Hence it follows that as PO describes the pencil (P) its pole Q describes a projective row on p, and conversely. In the pencil (P) we have a correspondence of conjugate rays a, a': this is also a projectivity, as we see by projecting from P the rows described by A, A': or we may prove the theorem independently. As a matter of fact, the projectivities A' = 1A and a' = 2a are both involutions: for the construction shows that ZA' =A and z2a' =a. It may be worth noticing that the projective pencils (K), (L) give the projective rows on p twice over. This is on account of the involutionary nature of the correspondence. 122. Method of Reciprocal Polars. By the theory of poles and polars, a given conic K sets up a one-to-one correspondence (P, p) between dissimilar elements of its plane. To PQ corresponds pq, and if P, q are conjoint, so are p, Q. Thus we have a correlation, or more particularly a polar correlation; and we can thus justify by a roundabout procedure the principle of duality in a plane. Suppose a conic A is generated by two projective pencils (S), (S'): then if 8, s' are the polars of S, 8' with respect to K, we have on them two rows projective to (8), (S') respec POLES AND POLARS 111 tively and these generate a conic X' correlative to X. To any tangent at a point P on X corresponds the point of contact of the corresponding tangent to A', and so on: in particular four harmonic points on X correspond to four harmonic tangents to A'. More generally, if L is any locus in the plane of K, the polars of its points with respect to K have an envelope E, the " reciprocal" of L, and to every characteristic of L corresponds a correlative characteristic FIG. 35. of E. For instance, the class of E is the same as the order of L: if L has an ordinary double point or cusp, E has a corresponding double tangent or inflexional tangent: the " deficiencies" of L, E are the same, and so on. But it is much better to deduce these theorems from the general principle of duality. 123. Two conjugate lines p, p' which cut K do so in four harmonic points. (Fig. 35.) Let p, p' cut K in A, C and B, D respectively: let AC, BD meet in 0, and let P be the pole of p. Then P is on BD, and AP is the tangent at A. The points 0, P separate B, D harmonically, and hence (AC, AP, AB, AD) is a harmonic pencil. But this is the limiting form of a pencil M(CABD), 112 PROJECTIVE GEOMETRY projecting C, A, B, D from a point M on the conic, as M moves up to A along the curve; and since any two such pencils are projective (Art. Io7), it follows in the present case that they are all harmonic. Conversely, if ACBD is a harmonic set, the lines AC, BD are conjugate. Correlatively, if P, P' are conjugate points both outside FIG. 36. K, and (a, o), (b, d) are the tangents from them, acbd is a harmonic set, and conversely. If ACBD is a harmonic set of points on K the corresponding set of tangents acbd thereat is harmonic. 124. Staudt's Theorem. A very important theorem, due to Staudt, is the following: Let ABC be any triangle inscribed in K, and S the pole of BC: then any line through S meets AB, AC in conjugate points, and conversely. (Fig. 36.) POLES AND POLARS 113 Draw any line SPP' through S, cutting AB, AC in P, P' respectively; and let CP cut K in D. We have a Pascal hexagon by putting A =, B =2==3, D = 4, C=5= 6. The collinear Pascal points are (12, 45) =P, (23, 56) =S, (34, 6I) =(BD, AC): hence (BD, AC)=P'. Consequently we have a simple inscribed quadrilateral ABDC with (AB, DC)= P, (AC, BD) =P'; therefore P, P' are conjugate by Art. II9. Otherwise thus: the sections of (S) by AB, AC are projective, and in these sections corresponding pairs are (A, A), (B, BS. AC), (C, CS. AB) in which each pair is a conjugate one. Hence the correspondence (P, P') can only be that of conjugate points on AB, AC. (Arts. 12I, 52.) Conversely, if P, P' are any two conjugate points on a line through 8, then PB, P'C (and also PC, P'B) will meet on the conic. This follows by the indirect method of proof. 125. Degenerate Conics. The reader should notice here that although the rows described by P, P' on AB and AC are projective, the envelope of PP' is not a proper conic, because the rows are perspective, as well as being projective. Now, as a limiting case of a projectivity leading to an envelope, we have both S and A, because any line through either may be regarded as joining two corresponding points. Similarly, if (S), (S') are two perspective pencils, we may regard them as generating not only the line s, on which any two different rays intersect, but also the line SS', which as a ray of (8) corresponds to 8'S, that is, the same line, regarded as a ray of (8'). Thus we have a line-pair occurring as a degenerate case of a quadric locus, and a point-pair (correlatively) as a degenerate case of a quadric envelope. The student should carefully bear these correlative cases in mind; for although it is easy enough to show, in a figure, the transition of a hyperbola into two straight lines, it is not so easy to show the transition of a conic (envelope) into two points. In the light of the foregoing observations we are able to say (in a language easily understood) that The complete locus generated by two projective pencils is either a proper conic, or else two straight lines: and the M.P.G. H 114 PROJECTIVE GEOMETRY complete envelope generated by two projective rows is either a proper conic, or else two points. In the degenerate locus or envelope the order or class respectively remains 2, but the class in the first case, and order in the second reduce to zero, being replaced, in the characteristic-theory, by the occurrence of one double point and one double tangent respectively. A more accurate way of describing the degenerate locus and envelope is to say that they consist of a pair of rows and a pair of flat pencils respectively. CHAPTER XVI. RULED QUADRICS. 126. Let (ABC...), (A'B'C'...) be two projective rows, of which the bases u, u' are askew. Then of the lines AA', BB', etc., joining corresponding points, no two can meet, and the whole set of them will occupy a curved FIG. 37. surface (p). We shall say that the set of lines AA', BB', etc., forms a regulus, of which each of them is a ray, and u, u' are directrices. The axial pencils u'(ABC...), u(A'B'C'...) are projective, and AA' is the meet of corresponding planes u'A, uA': hence any regulus generated by projective rows can 116 PROJECTIVE GEOMETRY also be generated by projective axial pencils. Conversely, let (a/y...), (a'(/'y'...) be two projective axial pencils whose axes u, u' are askew; then the rows u'(a/y...), u(a'/'y'...) are projective, and generate a regulus identical with {aa', /33', Y7'..}. 127. Conjugate Reguli. Take any point A" on AA', and through it draw the line u" which meets BB' and CC'. Let B", C" be the intersections; then, as previously proved (Art. 55), the line A"B"C" meets every ray of the regulus, and the rows (ABC...), (A'B'C'...) are perspective sections of the axial pencil (u"). Since every triad such as AA'A" is collinear, it follows that the rows (A'B'C'...), (A"B"C"...) are perspective sections of the axial pencil (u), and hence that they are projective to each other. We shall say that u" is a directrix of the regulus. By making A" describe the whole of the line AA', we obtain a set of directrices, which occupy the same surface (p) as the regulus. Every directrix cuts all the rays of the regulus, and the latter meets any two directrices in two projective rows; corresponding points being those on the same ray. The system of directrices is, itself, a regulus. For consider any three of the directrices, such as u, u', u". Every ray, r, of the regulus meets u, u', u", and conversely: so that the regulus may be defined as the locus of a line r which moves under the condition of meeting three fixed lines u, u', u". But now if we take any three rays r, r', r" of the regulus, the set of directrices is the system of lines which meet them all, and is therefore a regulus. Thus on the surface p we have two conjugate reguli: any ray of either is a directrix of the other, and any two rays of the one meet the lines of the other in two projective rows. A ray of either regulus is called a generator of p. Through each point P on p pass two conjugate generators p, p' belonging to the conjugate reguli. The accompanying figure is intended to suggest the two sets of generators. 128. The section of p by any plane a is a conic or a linepair. RULED QUADRICS 117 Suppose, first, that a contains no generator of p. Take two generators p, q of the same system (i.e. on the same regulus): then p may be generated by projective axial pencils (p), (q). Let pa=P, qa=Q: then the section of the figure by a gives two projective, and not perspective, FIG. 38. flat pencils (P), (Q) which generate a conic pa, common to p and a. Secondly, suppose that a contains a generator p. Take q, any other generator of the same system, and consider p as generated by projective pencils (p), (q). The plane a now belongs to (p), and there is a definite plane a' corresponding to it in (q). Let aa' = p', then p' is on p, and the line-pair (p, p') is the complete intersection of a with p. No line, except a generator, can meet p in more than two points. This follows from the above, and may be proved independently. Thus if u is any line, and p is generated by two projective pencils of planes, q meets these pencils in two projective rows, which cannot, in general, have more 118 PROJECTIVE GEOMETRY than two points of coincidence. Hence also, if a line meets p in at least three points, it must be a generator. 129. Tangent Cones. Take either regulus of p, and let it meet any two lines of the conjugate regulus in the projective rows (ABC...), (A'B'C'...). If we project all the rays of the first regulus from any external point P, we obtain a system of planes which is the same as (PAA', PBB', PCC',...). But since the rows are projective, this set of planes (by Art. Io8) forms a conical envelope. Each plane such as PAA' cuts p in the line AA' and another generator: if these generators are called p, p', then the line joining P to pp' meets p in two coincident points, and is a line of contact for the envelope of PAA'. The cone we obtain in this way is called the tangent cone from P to p. Every plane which meets p in a line-pair is said to be the tangent plane at the point where the lines meet: consequently, there is one tangent plane at every point of p, and we have just proved that those tangent planes which pass through a given point P envelope a quadric cone. We obtain the same quadric cone whichever of the conjugate reguli we project from P. Through any line that is not a generator we cannot draw more than two tangent planes to p. Let u be the line, and (ABC...), (A'B'C'...) any two projective rows from which p can be generated. Projecting these rows from u we have two projective coaxial pencils u(ABC...), u(A'B'C'...): and there cannot be more than two planes of coincidence. On the other hand, if u is a generator, every plane i that contains it is a tangent plane. Moreover, if B is ft's point of contact, the relation P/= =c(B) is a projectivity, because if v is any other generator of the same system as u, the row v(/i'/3"...) is perspective to (f/W'"...) and also projective (Art. 127) to the row (BB'B"...). 130. The points of contact of a tangent cone lie on a conic: and the tangent planes at points on a plane section envelope a cone. RULED QUADRICS 119 Let P be the vertex of the cone, a, /3, y any three of its tangent planes, a, b, o their lines of contact with the cone, ABC their points of contact with p. The plane ABC cuts p in a conic K, and the cone in a conic X: but these conics must coincide because they have in common three points A, B, C and the tangents there. Consider, for instance, the line where a cuts ABC: this is clearly the tangent to X at A. Now since this line goes through A and also lies in the plane (a) which touches p there, it meets p, and therefore K, in two coincident points at A; that is, it is the tangent to K at A. The second of the above propositions may be proved by a correlative argument. 131. A surface p, such as we have been discussing, is called a ruled quadric (or ruled conicoid). Being, as proved above, of the second order and second class, it is the analogue, in three dimensions, of a conic in two. If a hyperbola is rotated about its conjugate principal axis, it generates a ruled quadric of revolution. Other objects which give an idea of the shape of the surface are a dice-box, and the collapsible frameworks used to put round flower-pots. The latter are particularly instructive, because they form the skeleton, so to speak, of two conjugate reguli, hinged to one another so as to admit of motion with one degree of freedom. If a ruled quadric touches the plane at infinity, it is called a hyperbolic paraboloid; if not, it is called a hyperboloid of one sheet. To avoid misunderstanding, it should be stated that whereas, in the plane, we can generate any conic whatever by projective rows or pencils, we cannot generate every kind of quadric surface by the construction of Art. 126. For instance, a sphere is a quadric surface which cannot be generated in this way. CHAPTER XVII. EXTENDED THEORY OF PERSPECTIVE. 132. We have now become acquainted with five new geometrical forms: the conic locus and envelope, the conical locus (of lines) and envelope (of planes), and finally the quadric regulus. Each of them is a set of primitive elements, and each may be regarded as being on a certain base; namely the first two on a conic, the next two on a cone, and the last one on a hyperboloid or on a hyperbolic paraboloid. Although the bases of these forms are not elements, still, considered as sets of elements, the figures have so many properties analogous to those of rows and pencils, that it is convenient to annex them to our previous list of elementary forms. When we wish to be precise, we may refer to them and the three elementary forms of the first rank as one-fold elementary forms; meaning by this that their elements may be regarded as a set which may be ordered in a way analogous to that of any way in which the points of a row or pencil can be ordered. For instance (Fig. 39), consider a conic K, and any fixed point S upon it. Let A be a variable point on K; then SA is a variable line a through S and for every ray a which passes through 8 there is a definite point A where a meets the conic again, unless a is the tangent at 8, in which case we may regard A as coinciding with S in a special limiting way. As a describes the whole pencil once in a definite EXTENDED THEORY OF PERSPECTIVE 121 sense, we may regard the corresponding point A as describing the whole conic in a definite sense. For the present it will be convenient to denote by {e1e2e3...}, with braces instead of brackets, a one-fold set whose base is not a primitive element. In the case we are considering, we have a pencil (abc...) S a b FIG. 39. and a form {ABC...} which we can bring into a one-one relation by making two conjoint elements, such as (a, A), correspond. As another example, take a regulus {abc...} and its section {ABC...} by an arbitrary plane. Taking any point S on this section we have a pencil S{ABC...} which is an elementary form of the first rank, and now there is a one-one correspondence between S{ABC... } and {abc...}: namely SA corresponds to a when a is conjoint with A. We are thus led to extend the definition of perspectivity so as to include the following cases: the reader will remember that we have expressly omitted any general definition of perspectivity. (i). A pencil (abc...) is (ii). A row (ABC...) is perperspective to a conic spective to a conic {abc...} {ABC....} when 8, the when s, the base of the row, centre of the pencil, is on touches the conic, and each 122 PROJECTIVE GEOMETRY the conic, and each ray (b) of the pencil corresponds to the point (B) where it meets the conic again. (iii). A flat pencil (abc...) is perspective to a cone {a/y...} when ac, the plane of the pencil, touches the cone, and each ray (b) of the pencil corresponds to the point (B) of the row corresponds to the line (b) which is the other tangent from B to the conic. (Fig. 40.) (iv). An axial pencil (apy...) is perspective to a cone {abc...} when s, the axis of the pencil, is a generator of the cone, and each plane (/3) of the pencil FIG. 40. other tangent plane (P) of the cone, which passes through b. (v). A regulus {abo...} is perspective to the (proper) section a{abc... } by an arbitrary plane a, when any ray (b) of the regulus is made to correspond to (ba) its intersection with a. (vii). A regulus {abc...} is perspective to a row (ABC...), when u, the base of the row, is a directrix of the regulus and any ray corresponds to the line (b) in which it meets the cone again. (vi). A regulus {abo... } is perspective to the (proper) tangent cone A{ab... } from an arbitrary point A when any ray (b) of the regulus is made to correspond to (Ab), the tangent plane through A which contains b. (viii). A regulus {abc...} is perspective to an axial pencil (af/3y...) when u, the axis of the pencil, is a directrix of the regulus and EXTENDED THEORY OF PERSPECTIVE 123 b of the regulus is made any ray b of the regulus to correspond to the point ub is made to correspond to the where it meets u. plane ub. Since we have now 8(=3 +5) one-fold forms to consider, the reader will expect 36 (82 +8=28 +8) cases of perspectivity, in the wider sense. They do, as a matter of fact, exist: we have stated 14 of them, namely the 6( =32 + 3) for the original elementary forms (cf. Chap. IV.), and the 8 which have just been enunciated. The reader ought not to have any great difficulty in discussing as many of the remaining cases as he likes, on the model of those which have been given. 133. Any two one-fold forms are said to be projective (in an extended sense), when there is a one-one correspondence between their elements, which can be established by a chain of perspectives in the wider sense. We have already (Art. Io7) defined a harmonic set of points on a conic; namely, they are such that, if we join them to any other point on the conic, we obtain a harmonic pencil. The tangents at four harmonic points form what is called a harmonic set of tangents: by Chasles' theorem (Art. II4), they cut any other tangent in a harmonic range. By projection from a point we derive corresponding definitions of a set of four harmonic rays, or four harmonic tangent planes of a cone or cylinder. Four rays of a regulus are said to be harmonic, when they are met by any, and therefore by every, directrix in a harmonic range. With these definitions we are able to prove that in every projectivity connecting two one-fold forms, every harmonic set in one form corresponds to a harmonic set in the other. Any one of the 8 forms that is not a row can be put into perspective with a row, flat pencil or axial pencil. Thus, for instance, a cone may be put into perspective with an axial pencil whose axis is any ray, u say, of the cone. If a is any other ray of the cone, ua is the corresponding plane of the pencil. Again a regulus may be put 124 PROJECTIVE GEOMETRY into perspective with a row on any one of its directrices: for if u is the directrix, and a any ray of the regulus, we can take ua as the point corresponding to a. Any flat or axial pencil can be put into perspective with a row. Hence, given any one-fold form which is not a row, we can set up a projectivity between it and a row by means of a perspectivity, or else by a chain of two perspectivities. Now let P1, P2 be any two of the 8 forms: fi, gl, hi three assigned elements of P1, and f2, g2, h2 three assigned elements of P2. Find a row, R1, projective to P1, and let F1, GI, H1, be the points of Ri which correspond to f,, gl, hl. Similarly find a row R2 projective to F2 and let F2, G2, H2 be the points of R2 which correspond to f2, g2, h2. We can set up a unique projectivity between Ri and R2 which makes (F1, F2), (GI, G2), (H1, H2) three pairs of corresponding points. Compounding this with the previous projectivities, we obtain a projectivity connecting P1, P2 in such a way that (fi, f2), (gl, g2), (hl, h2) are three pairs of corresponding elements. Of course if P1 is a row we may take R1 = P, and similarly for P2. By the usual argument we can prove that, given the three pairs (f1, f2), etc., the projectivity set up is unique: hence Any two one-fold forms may be made projective to each other in such a way that any three assigned elements of the one correspond respectively to three assigned elements of the other. There is only one such projectivity, though it may be established by an unlimited number of perspective chains. 134. Projectivities on a Conic. The arguments of Art. I33 still apply when the forms P1, P2 are cobasal. We shall now consider an important special case. Given two triads of points ABC, A'B'C' on a conic K: it is required to establish, by construction, the projectivity {A'B'C'...} ={ABC...}. (Fig. 4I.) Put the pencils (A), (A') into perspective with the forms {A'B'C'...}, {ABC...} respectively; we thus get two perspective pencils, because in the relation A{A'B'C'...}7 A'{ABC...} EXTENDED THEORY OF PERSPECTIVE 125 AA' corresponds to itself. Let AB', A'B meet in X, and AC', A'C in Y: then if XY =u, this line u is the axis of the perspectivity. We can now find the point D', corresponding to an assigned point D, by joining A to the intersection of u and A'D: this line meets K in D'. We might have begun with the perspectivity B(A'B'C'..) 7B'(ABC...), but we should obtain the same axis u as FIG. 41. before: for Pascal's theorem applied to the hexagon AB'CA'BC' shows that the points (AB', A'B), (BC', B'C) and (CA', C'A) are collinear. If, then, {A'B'C'D'...} =z{ABCD...} is a projective permutation of points on a conic, all points such as (AB', A'B), (BD', B'D), obtained from two pairs of corresponding points, are collinear, and conversely. If the axis u meets K in H, K these points are self-corresponding. This is an obvious corollary from the construction. Moreover K being actually given, as we suppose here throughout, we can find H, K if they exist, from any three given pairs of corresponding points (A, A'), (B, B'), (C, C'): namely, we find u as above by a linear construction, and then find its intersection with K, if they exist. The permutation is said to be hyperbolic if H, K exist and are distinct: parabolic, if u touches K, so that H, K coincide: elliptic, if u does not meet the conic at all. 126 PROJECTIVE GEOMETRY A permutation is determined (i) by the axis and one pair of distinct corresponding points, or again (2) by two pairs and one self-corresponding point. 135. Correlatively, given two triads abc, a'b'c' of lines all touching the same conic, we can establish the permutation {a'b'c'...} =z{abc...} among the whole set of tangents by finding the point U, in which, by Brianchon's theorem, the lines (ab', a'b), (ac', a'c), (be', b'c) concur. Then d'=z(d) if (ad', a'd) goes through U. This point is the centre of the projectivity, and we can write down, if we like, theorems correlative to all those of Art. I34. In particular, the tangents through U, if they exist, are the self-corresponding elements. Suppose, now, that A, A' are the points of contact of a, a', and so on. Then we have two projectivities, say {A'B'C'...} =zi{ABC...} and {a'b'c'...} =T2{abc...}. Let u be the axis of the first, and U the centre of the second: then U is the pole of u with respect to the conic. For suppose AB', A'B' meet at X: then ab', a'b both lie on x, the polar of X, and (ab', a'b) is the polar of (AB', A'B). Similarly (ac', a'c) is the polar of (AC', A'C), and finally the meet of (ab', a'b) and (ac', a'c), that is U, must be the pole of u. For many purposes we may regard a,, M2 as the same projectivity, permuting " conjoint elements" of the conic, each consisting of a tangent and its point of contact. 136. The Involutionary Case. Suppose that, in our figure, we regard B' as a point of (ABC...}, and, as such, call it P. To find P', we join A'P, that is, A'B', and find Q, its intersection with u; then P' is the intersection of AQ with the conic, and is in general distinct from B. But it will coincide with B if A'B', AB meet on u: and in this case AA', BB' both pass through U, the pole of u. Now (A, A'), (B, B') may be supposed to be any two corresponding pairs: hence putting B' =z (B), we see that z (B')=B implies that all the lines AA', BB', etc., joining corresponding points, concur in U, the pole of u, and that zrA' =A, etc.; so that we have an involution on K. It is easily seen, by drawing EXTENDED THEORY OF PERSPECTIVE 127 a figure, that this case cannot occur when u touches the conic; but it can in every other case. Thus If through any point U that is not on a conic we draw secants meeting it in point-pairs (A, A'), (B, B'), etc., these are conjugate pairs of an involution of points on the conic, and conversely. If tangents can be drawn from U to the conic, their points of contact M, N are the foci of the involution, and all such sets as MNAA' are harmonic. Correlatively, if from points on a line u, not tangent to a given conic, we draw tangent-pairs (a, a'), (b, b'), etc., to the conic, these are conjugate pairs of an involution among all the tangents of the curve. Further, if A, A' are the points of contact of a, a', the line AA' goes through a fixed point U, the pole of u: so we have an associated involution of points on the conic. Finally, we may regard (A, A'), (a, a'), etc., as conjugate pairs of points and lines in a harmonic homology, or plane involution, of which U is the centre, and u the axis. This homology transforms the conic into itself. Conversely, any plane homology which transforms a conic into itself must have for its centre and axis a point and its polar with respect to the curve. 137. Projectivities on other Forms. Projectivities on cones, cylinders, and ruled quadrics, may be discussed independently, but are most easily treated by taking plane sections. As an example, which leads to some important results, take the following: Given two conjugate reguli {abc...}, {a'b'c'...} on the same quadric, it is required to establish, by construction, the projectivity {a'b'o'...} =m{abc.. }. Draw any plane f3 cutting the quadric in a proper conic K, and let ABC, A'B'C' be the points on K where 3 meets abc, a'b'c' respectively. In the plane / we can find a construction for the projectivity {A'B'C'...} =f{ABC...} on K. In this let P, P' be corresponding points, so that P'= P; let p be the element of {ab,..} which goes 128 PROJECTIVE GEOMETRY through P, and p' the element of {a'b'c'...} which goes through P'; then p'=zsp. For the reguli are perspective to {ABC...}, {A'B'C'...} respectively, and the proper projectivity between these two forms has been established. Similarly, we may find constructions for projective permutations of the rays of a single regulus, and so on. FIG. 42. 138. We add two interesting deductions: (i) If the projectivity {a'b'c'...} =z7{abc... } connects the conjugate reguli {ab...}, {a'b'c'...} then the points aa', bb', cc', etc., all lie on the same conic. The three points aa', bb', cc' determine a plane: the section of the reguli by this plane gives a projectivity {ABCD'...} =~{ABCD...} on a conic, with three selfcorresponding points A, B, C=aa', bb', cc'. Hence D'=D, and the theorem follows. (ii) If {A'B'C'...} =z{ABC...} is a projectivity of points on a conic, then, in general, the lines AA', BB', etc., envelope a conic. (Fig. 42.) To prove this, draw through A, A' any two lines a, p EXTENDED THEORY OF PERSPECTIVE 129 respectively, cutting the plane of the conic, and not intersecting one another. Then a{A'B'C'...}, p{ABC...} are two projective pencils which generate a regulus p. Let p' be the conjugate regulus: a, b, c the directrices of p which pass through A, B, C, and a', b', c' the directrices of p' which pass through A', B', C'. Then by (i) the projectivity {a'b'c'...} =z{abc...} determines a conic on K, the ruled quadric which is the common base of p, p'. The tangent planes to K at points on this section envelope a cone (Art. I30): now AA', BB', etc., are the sections of these tangent planes by the plane of the given conic, and therefore envelope a conic, unless this plane contains the vertex of the tangent cone to K: in this special case, the given projectivity is an involution. The degenerate case of a conic envelope is, in general, two points (more correctly, two flat pencils): in this particular example the points (pencils) coincide. If the projectivity is hyperbolic, there are two double points H, K, and the envelope of AA' touches the given conic at H and K. The proof of the theorem given here is a good illustration of the value of three-dimensional constructions, even when we are dealing with two-dimensional problems. Other, and planar, proofs can be given, both geometrical and analytical, but none is so natural and simple as the one here adopted. Memoirs by Segre and others make it perfectly certain that if we could gain an intuition of a fourdimensional space analogous to the spaces with which we are familiar, or fancy ourselves familiar (so that, for instance, our " solid " space would be a " plane " section of the higher space), many three-dimensional problems, for which we can only find laborious and artificial solutions, would admit of comparatively simple and natural " intuitive " solutions, especially in the case of projective geometry. As it is, if we have found a really natural solution by algebra, we can generally put it into the proper geometrical language for the four-dimensional solution: unfortunately, this M.P.G. I 130 PROJECTIVE GEOMETRY does not help us to realise the proper geometrical figures. The student is recommended to dualise theorems (i) and (ii). 139. Incidentally, we have used the following theorem, the truth of which is obvious: Two skew lines a, b and a conic {ABC... } which meets each of them in a single point (A, B), determine a regulus perspective to the conic, of which a, b are directrices: Namely, that generated by the projective pencils a{ABCDE...} and b{ABCDE...}. If we take a, b concurrent, but not in the plane of the curve, the regulus degenerates into a cone (or cylinder). Correlatively A conical envelope {a[y...} and two skew lines a, b which lie respectively in a, /3, but do not pass through the vertex of the cone, determine a regulus, perspective to the envelope, of which a, b are directrices: Namely, that generated by the projective rows a{afly38... } and b{a/lyS..... If a, b are concurrent, but not at the vertex of the cone, the regulus degenerates into a conic envelope in the plane ab. This is a good example of the principle of duality: note that the cone in the one case of degeneration corresponds to the plane conic in the other. CHAPTER XVIII. PROJECTIVE RELATIONS OF CONICS TO EACH OTHER. 140. In this chapter we shall, at first, follow Staudt (G. 264 sqq.) rather closely, in order to show the power and elegance of his methods, and to induce the reader to study his work. Given two conics K, K', it is required to construct the projectivity {A'B'C'...} =zr{ABC...} from the three given pairs of corresponding points (A, A'), etc., where ABC is any triad on K, and A'B'C' any triad on K'. Let P be the pole of AB, with respect to K, and P' the pole of A'B' with respect to K'. There is a definite collineation (P'A'B'C') -=Z(PABC) since PABC, P'A'B'C' are quadrangles. This makes K' =t(K), because K is determined by A, B, C and the tangents PA, PB, so that 5(K) is determined by A', B', C' and the tangents P'A', P'B': 'that is, (r(K) coincides with K'. The collineation being definite, to any point Q on K will correspond a definite point Q' on K' It will be observed that the argument does not imply that K, K' are in the same plane. If they are, we can get a plane construction for zr, and hence, for instance, construct K', if K is actually drawn, and A', B', C', P' are given. Correlatively, we can establish the projectivity {a'b'c'.. } =?r{abc...} between any two conic envelopes by means of two quadrilaterals pabo, p'a'b'c', where p, p' are the polars of ab, a'b'. 141. Let P, Q' be any two points on two projective conics 132 PROJECTIVE GEOMETRY {ABC... }, {A'B'C'...: then the pencils P{ABC...}, Q'A'B'C'... } are projective. Namely, let P' be the point corresponding to P: then P'{A'B'C'...} -P{ABC...}: but P'{A'B'C'...} 7Q'{A'B'C'...}: hence the theorem. Similarly for the correlative theorem. It should be noticed that if any two conics K, K' are projective, then a fortiori the plane fields in which they lie are (definitely) projective. A projectivity between two conics K, K' may also be fixed in the following way: Take any two points P, P' on K, K' respectively: then the tangents at P, P' must correspond, and now if PA, PB are any two chords of K through P, and P'A', P'B' any chords through P', the projectivity P'{P'A'B'...} =z. P{PAB...) fixes a projectivity on the conics, it being understood that PP means the tangent to K at P and P'P' the tangent to K' at P'. Here, of course, the conics are both supposed to be drawn. We omit the correlative theorem, as we shall generally do in what follows. 142. (i) If S is a point common to two coplanar conics K, K', the pencil (S), being perspective to both, fixes a projectivity between them: namely, that in which corresponding points A, A' are collinear with S. (Fig. 43.) Correlatively, if two conics K, K' have a common tangent s, there is a projectivity between them in which two corresponding tangents (a, a') intersect upon s. This holds even when the conics are not coplanar. (ii) Consider, now, the case when the conics touch at 8, and let s be the common tangent there. (Fig. 44.) Draw any two secants SAA', SBB' and let P, P' be the poles of AB, A'B' with respect to K, Kt respectively. Then 8, P, P' are collinear, because SP, SP' are each the harmonic conjugate of s with respect to SA, SB. Hence the triangles PAB, P'A'B' are perspective, and the points (PA, P'A'), (PB, P'B'), (AB, A'B') are collinear. Now we may suppose K determined PROJECTIVE RELATIONS OF CONICS 133 by the points S, A, B and the tangents s, PA; and similarly K' by the points 8, A', B' and the tangents s, P'A': these data determine the line {(PA, P'A'), (AB, A'B')}, on which therefore the tangents (PB, P'B') at any two corresponding A S A 0' FIG. 43. points intersect. In fact, the two conics thus present themselves as corresponding figures in a plane homology, of which S is the centre, and the line (PA.P'A', AB.A'B') is the axis (u). If the conics have another common point T, FIG. 44. the axis must pass through it and hence meet them again in another common point U, or else touch them both at T. Thus: If two coplanar conics touch at S, then by putting them both into perspective with (S), they become two corresponding 134 PROJECTIVE GEOMETRY figures in a plane homology, of which S is the centre and whose axis is the locus of points where corresponding chords (AB, A'B') and hence corresponding tangents (a, a') intersect. If the conics have any other common points, the axis goes through them. By dualising this, putting K, K' into perspective with s, we obtain another homology, of which s is the axis, and whose centre V is the point of concurrence of all such lines as QQ', joining the points of contact of tangents from a point on s. FIG. 45. The conics have either no common tangent but s, or else they have two more, t, u (which may coincide); in this case V=tu. (Fig. 45.) By rotating one of the conics about s, and making use of Art. 68, we infer: (iii) Two conics in different planes a, / which touch a/ in the same point are sections of one and the same quadric cone. 143. Let the conics K, K', in different planes, cut the meet of the planes in the same two points X, Y; then there are two quadric cones, each of which passes through both K and K', if every point of XY that is within K is also within K'. PROJECTIVE RELATIONS OF CONICS 135 Put XY = u, and let P, P' be the poles of u with respect to K, K' respectively. Draw a line through P cutting / in A, B and u in E: then the line P'E will cut K' in two points A', B'. Take on u the point F which is conjugate to E with respect to K, and therefore also to K': also let G, H be any other pair of conjugate points on u. Then GHAB, GHA'B' are two quadrangles which determine a projectivity between the plane fields containing K, K'. Now AA,' BB' meet in a point U: so this projectivity is a perspective (U, u) with centre U and axis u. Since (PEAB), (P'EA'B') are harmonic, PP' goes through U, and P, P' are corresponding points. Now let K'< be the conic into which K is projected from U on to the plane of K': the conics K', K" both touch FA, FB in A, B and also (by Staudt's theorem, Art. I24), pass through the intersection of GA, HB. Therefore K', K" coincide. Similarly, if AB', A'B meet in V, we can prove that V is the vertex of a cone on which K, K' both lie. The proof still holds good, word for word, when u, the meet of the planes of K, K', does not meet either conic at all, but they determine upon it the same involution of conjugate points. The points P, P' are now within the conics, instead of being outside, as in the former case. The conditions satisfied by K, K' when the involution on u is hyperbolic, are briefly expressed by saying that K, K' enclose the same segment on u: the enclosed segment may be either the finite segment XY, or its complement Y.X. The fact is that if a cone is cut in two conics by different planes a, 3, the conics determine on a/3 the same involution of conjugate points, and if a/3 cuts the cone, the conics enclose the same segment on a/3. What we have done is to prove the converse, with the addition that there are two cones containing the conics. Two conics may have two points in common, without enclosing a common segment on their join: thus K may enclose XY, while K' encloses Y.X. By drawing figures, the reader may convince himself that in this case K, K' 136 PROJECTIVE GEOMETRY cannot lie on one and the same (real) cone. We shall return to this at a later stage. By rotating the planes a, P about u so that K, K' become coplanar, we infer: Two coplanar conics K, K' which either enclose the same segment on a line u, or else determine on it the same elliptic involution of conjugate points, are in two ways homologous figures in a plane perspective of which u is the axis. If P, P' are the poles of u with respect to K, K', then U, V, the centres of the perspectives, are on PP', and PP'UV is a harmonic range. Two coplanar conics may intersect in as many as four different points: so we may have as many as twelve different ways of thus making two given conics corresponding figures in a plane homology. The reader should dualise the theorems of the present article in as many ways as he can. 144. The following (G. 277, where n + should be 2n + i) is an interesting extension of Pascal's theorem: If a simple (4n +2)-gon is inscribed in a conic, and of the (2n + i) intersections of pairs of opposite sides, 2n lie on the same line u, then the remaining intersection is also on u. Consider, for instance, a decagon AB'CD'EA'BC'DE': then the five intersections are (AB', A'B), (BC', B'C), (CD', C'D), (DE', D'E), (EA', E'A). Let us suppose the first four lie on u. This shows that if we take the definite projectivity {A'B'C'...} =z{ABC...} on the conic, u is its axis, and (D, D'), (E, E') are pairs of corresponding points. But now, since (A, A') and (E, E') are pairs of corresponding points, (EA', E'A). must lie on u. The argument is quite general. The correlative theorem is left as an exercise. 145. Through four given points A, B, C, D, the vertices of a quadrangle, we can draw three conics on which ABCD, ACDB, ADBC respectively are harmonic sets of points. Through A draw a line AT such that A(TBCD) is harmonic: PROJECTIVE RELATIONS OF CONICS 137 then on the conic which touches AT at A and passes through B, C, D, the set ABCD is harmonic. Similarly for the other cases. The correlative theorems are left to the reader. 146. Let K be a conic and H, P two points which, jointly and severally, are in a general position with respect to K: then there is a determinate conic X passing through P such FIG. 46. that every line through H has the same pole with respect to K and X. (Fig. 46.) Let h be the polar of H with respect to K: then we are to have H, h pole and polar with respect to A, and moreover K, A are to have the same involution of conjugate lines through H, and the same involution of conjugate points on h. Let Q be the pole of HP with respect to K: then since P is to be on A, QP must touch A at P. Suppose HP cuts h in Q', and that P' is taken so that HQ'PP' is harmonic; then P' is also on X, and QP' is the tangent there. Let 138 PROJECTIVE GEOMETRY (R, R') be any pair of the involution which K determines on h, and let S, S' be the remaining vertices of the complete quadrilateral PR'P'R: then as (R, R') vary, S and S' describe one and the same conic, touching QP, QP' at P, P' respectively, and for which, by Staudt's theorem, (R, R') in each of its positions is a pair of conjugate points. If h meets K in two points X, Y, the conic X touches K at X and Y: so we have, in fact, found the solution of the problem; construct the conic (X) which passes through a given point P and touches a given conic (K) at two given points X, Y upon it. Of course, there are simpler solutions of this problem, at any rate if we suppose the tangents to K at X, Y already drawn; but for many reasons the above is the preferable proof. The reader should investigate the correlative theorem. If any line cuts h in M, and K, X in A, A' and B, B' respectively, then M2.AA'.BB' is an involution. For, suppose M' the conjugate point of M upon h: then M'H is the polar of M with respect to each conic, and if it cuts AA' in N, the ranges MNAA', MNBB' are both harmonic: hence the proposition (Art. 9I). As a special case, suppose A'=A: then the range MABB' is harmonic; thus, if any tangent to K be drawn, its point of contact (A) and its intersection with h are harmonic conjugates with respect to its intersections with X. Hence also K, X have no common tangents other than those whose points of contact lie on h, and therefore coincide: that is, K, A either have real double contact, or else have no intersections, and no common tangents. 147. From the theorem correlative to that of Art. 146 Staudt deduces a very pretty proof of the proposition (Art. 138) that if z5(ABC...) =(A'B'C'...) is a projectivity on a conic K, which is not an involution, then AA' envelops a conic. Let h be the axis of the projectivity: then there is a definite conic X which touches AA', such that every point on h has the same polar with respect to K, A; so all we have to do is to show that BB' touches A. PROJECTIVE RELATIONS OF CONICS 139 Let AB', A'B meet h in L: then the polar of L with respect to K, and therefore to A, is the join of the points (AA', BB') and (AB, A'B'). Call these points M, N: then they are conjugate with respect to X as well as to i/, and the conjugate lines ML, MN are harmonically separated by AA', BB'. Since, then, AA' is a tangent to A, BB' must also be a tangent to A. A comparison of this proof with that of Art. I38 is very instructive: especially as this is probably the best of all planar proofs. 148. (i) If ABCD, A'B'C'D' are complete quadrangles having the same set of diagonal points, their eight vertices lie on the same conic. (Fig. 47.) G E B FIG. 47. Let (AD, BC)=E, (AC, BD)=F, (AB, CD)=G. There is a definite conic K passing through A, B, C, D, A', and of this EFG is a polar triangle. Now by the quadrangle A'B'C'D', the pencil E(A'B'FG) is harmonic, and therefore its section by A'G is harmonic: but this is (A'B'G'G), where G' is the meet of A'GB' and EF. Now G being the pole of EF, this proves that B' is on K. Similarly for C', D'. If A' is on any side of ABCD, K degenerates into a line-pair. (ii) If two conics are inscribed in the same quadrilateral abod, the eight points of contact lie on a conic. 140 PROJECTIVE GEOMETRY Let ABCD, A'B'C'D' be the quadrangles on the two conics whose vertices are the points of contact, AA' being a, and so on. Then, as proved in Art. 113, the diagonal triangle of abed has vertices which are the diagonal points of ABCD, and also of A'B'C'D'. The theorem now follows from (i): of course the derived conic may be a line-pair. The correlative theorems are proved in a similar way. 149. If two triangles are both self-conjugate with respect to a given conic, their six vertices lie on a conic, and their six sides touch a conic. (Fig. 48.) BE E B E FIG. 48. Let ABC, DEF be the triangles; (BC, DE) =B', (BC, DF) =C', (AB, EF) =E', (AC, EF) =F'. Since A, F are the poles of BC, DE, the point B' is the pole of AF: similarly, C', E', F' are the poles of AE, DC, DB. Hence (Art. I2I). BCB'C'A A(CBFE) A F'E'FE T'E'F'EF (Art. 59). Thus the two sides BC, EF are cut projectively by the other four; and this proves the second part of the theorem. The first part follows from (cf. Art. I2I) A(CBFE) 'XBCB'C' D(BCB'C') D (BCEF) A D(CBFE). CHAPTER XIX. PLANAR INVOLUTIONS. POLAR SYSTEMS. 150. A planar involution, that is to say, a projectivity in a plane which satisfies the symbolical equation r2 =I, is either a collineation or a correlation. We have had an example of the former in the harmonic homology, of the latter in the pole-to-polar correspondence established by a conic. We shall now discuss involutions of this kind from a more general point of view: the reader will remember that for every theorem stated there is a correlative theorem for the bundle; as a rule this will be omitted. 151. Every collinear involution is a harmonic homology. Let A, A' be any distinct pair of corresponding points: then the line AA' corresponds to itself, because z(AA') =A'A. Now let B, B' be any conjugate pair not on AA'; then AA'BB' is a quadrangle, and we have a definite projectivity AA'BB' = (A'AB'B). But this is established by the harmonic homology of which (AA', BB') is the centre, and the join of (AB, A'B') to (AB', A'B) is the axis. On the axis, each point corresponds to itself: similarly every ray through the centre corresponds to itself. There are no other self-corresponding points or lines. Incidentally we see that A collinear involution in a plane is determined by two pairs of conjugate points which form the vertices of a quadrangle; 142 PROJECTIVE GEOMETRY or again by any two pairs of conjugate lines which form the sides of a quadrilateral. A given quadrangle KLMN leads to three involutions; for if we put K=A, we may take either L, M or N for A'. The three involutions are related to each other somewhat in the same way as the three conics of Art. I45. 152. Polar Systems. When the involution is a correlation, it is said to be a polarity, and to establish a polar \_y __ R \ FIG. 49. system, or field. If p, P correspond, p is called the polar of P, and P the pole of p. Through P, any point disjoint to its polar p, draw any line q meeting p in R; and let r, the polar of R, meet p in Q. Then Q is the polar of q. In fact, putting p =(P), r = =(R), we have since z2p =P, a2R =R, by hypothesis, q = PR = (r2p, 2R) = M(P, Z-R) = M(pr) = (Q). Thus in the triangle PQR each vertex is the pole of its opposite side. Such a triangle is called a self-conjugate, or polar triangle of the system. (In Fig. 49 the polarity is supposed established by a conic; but the proof of the theorem does not depend upon this.) PLANAR INVOLUTIONS. POLAR SYSTEMS 143 Since q is any line through P, we have an unlimited number of polar triangles PQR with a common vertex at P. As q varies, the points (Q, R) vary, and in doing so define an involution on p, of which (Q, R), in any position, form a conjugate pair. Any two points such as Q, R, each of which lies on the polar of the other, are said to be conjugate. Similarly two lines (such as q, r) each of which goes through the pole of the other, are said to be conjugate. Conversely: If we have a correlation such that the vertices of a triangle ABC correspond to the opposite sides, then this must be a polarity (G. 234). By hypothesis z(A), m(B), z(C) = BC, CA, AB: it follows that r(AB) =(zA, ZB) = (BC, CA) = C, and similarly M(BC) =A, z(CA) =B. Now take any point P on BC: its polar p must pass through A. Let it meet BC in Q, and let q be the polar of Q, which must also pass through A. Then because the correlation is projective (BCPQ) - (bopq) Aa(bopq) A(CBQP'), where P' is the point where q meets BC. But (BCPQ) ^(CBQP), by Art. 59: hence (CBQP) (CBQP'), and P, P' coincide, so that wz(p) =zr(AQ) =P, whence also v"2(P)=P. Similarly for every point X on CA or AB, we have Z2(X) =X: and for every line x through A, B, or C we have r2(x) =x. Any other line I will cut two sides of ABC in points X, Y, and now z(2(1) = 2(XY) = (r2X, Z2Y) = XY = I. Finally, any point P may be regarded as the intersection of two lines I, m, and hence z52(p) =zi2(Im) =(zs21, z2m) = Im P. 153. The foregoing argument is based on the assumption that in a polar system we can find at least one point P disjoint to its polar p, and on p at least one point Q disjoint to its polar q. To justify this, suppose, if possible, that four lines, a, b, c, d forming a quadrilateral pass through their poles A, B, C, D: then it is easy to prove that abed, ABCD 144 PROJECTIVE GEOMETRY have a common diagonal triangle, whence there is a conic touching abed at ABCD which fixes by its pole-to-polar relation the correlation abed =zr(ABCD). Any line which does not touch this conic is disjoint to its pole. Moreover, when we have a point P disjoint to its polar p, we cannot have more than two points on p conjoint to their polars. Suppose, for instance, there are two points A, B on p conjoint to their polars, which are therefore PA, PB. FIG. 50. On PA take any point G, and let AH, its polar, cut PB in H. Then since H =gb, GB must be the polar of H; so if GB, AH meet in Q, and AB, GH in C, it follows that q = (gb, ah) =HG, and therefore the pole of PQ is (AB, HG), that is, C. But (by the quadrangle HQGP) C, PQ are harmonically separated by PA, PB, and are therefore disjoint (G. 235). It is easy to arrange the figure so that PQ is any line through P distinct from PA and PB. 154. Data fixing Polarities. A polarity is determined by a correlation abcp =zM(ABCP), where abop is a quadrilateral and ABCP a quadrangle, so related that be, ca, ab =A, B, C respectively. In other words, we may assume any triangle ABC as a PLANAR INVOLUTIONS. POLAR SYSTEMS 145 polar triangle, and then take any point P to be the polar of any line p, provided that PABC, pabc are a quadrangle and quadrilateral respectively. This follows from Arts. 152, 78. Suppose now that p cuts BC, CA, AB, the sides of the finite triangle ABC in X, Y, Z respectively, and let us put (AP, BC) =X', (BP, CA) =Y', (CP, AB) =Z'. Then the three couples of pairs (BC, XX'), (CA, YY'), (AB, ZZ') determine the involutions of conjugate points on BC, CA, AB respectively. It is easy to draw a figure in which these involutions are all elliptic: for instance, take P within ABC and let p cut all itssides externally. In this case the polarity cannot be effected by FIG. 51. a conic, and no point can be conjoint with its polar. Suppose, for instance, we take Q so that AQ cuts BC internally, while BQ, CQ cut AC, AB externally: then q must cut BC externally and AC, AB internally, and no such line can be drawn through Q. Similarly for any other position of Q. However we take P, p, we find that either all the involutions are elliptic, or else one is elliptic and the others hyperbolic. Hence, If ABC is a triangle self-conjugate to a conic, two of its sides cut the conic and the other does not. 155. Every simple plane pentagon ABCDE determines a polarity in which each vertex of the pentagon is the pole of the opposite side. (Fig. 5I.) Let (AB, CD)=F: then the polarity in question is that M.P.G. K 146 PROJECTIVE GEOMETRY for which ADF is a polar triangle and E is the pole of BC. For this makes zS(B) = 5(AF, BC) =DE, and similarly zr(C) =AE. 156. Let (aa', bb'...) be an involution of rays, centre H; and let any line h cut it in the perspective point-involution (AA', BB'...). Take any point P external to h: let PH cut h in Q, and let Q' be conjugate to Q in the point-involution. Finally draw any line p, cutting h in Q', and not passing through H. We shall have now a definite polarity, in which p =z(P) and (aa', bb'...) =zM(A'A, B'B...). Namely, take on h any two conjugate points A, A' distinct from Q, Q'. There is a definite polarity for which HAA' is a polar triangle, and p is the polar of P. But this makes the rays (HQ, HQ') correspond to (Q', Q) as well as (HA, HA') to (A', A): consequently it involves the whole correspondence (aa', bb'...) =z(A'A, B'B...). 157. Relative Polar Triangles. In a given polar field take a triangle ABC, of which no two elements are conjugate. Let a, b, c be the polars of A, B, C, and let us put be, ca, ab =A', B', C'; then the triangles ABC, A'B'C' are said to be relative polar triangles. This relation is a reciprocal one, because, since A' = be, the polar of A' must be BC, and so on. Moreover no two elements of A'B'C' can be conjugate. Any two corresponding triangles ABC, A'B'C' in a plane homology, which have no common elements, determine a polarity in which they are relative polar triangles (G. 241). Let H be the point where AA', BB', CC' meet, and h the axis of the perspective (Fig. 52, next page). Let h meet BC, CA, AB in P, Q, R respectively, and also HA, HB, HC in P', Q', R' respectively. Then from the complete quadrangle HABC we see that (PP', QQ', RR') is an involution (Art. 89); and now if we take HPP' as a polar triangle, and B'C' as the polar of A, we have a definite polarity w, in which, as appeared in the previous article, r(HQ) =Q', and so on. Hence M(B') =z(B'C', HQ') = (B'C', zHQ') =AQ, that is, AC and similarly z(C') =AB, zr(A') = B, PLANAR INVOLUTIONS. POLAR SYSTEMS 147 If, in a given polar field, ABC is a triangle of whose six elements no two are conjugate, it is perspective to its relative polar triangle A'B'C'. \h FIG. 52. This is the converse to the theorem last proved, and is most easily established by the indirect method. The centre of the perspective is the pole of the axis. 158. If, in a given polar field, ABC is a triangle such that AB, AC are conjugate lines, then for the relative polar triangle A'B'C' the points B', C' will be conjugate, and lie on AB, AC respectively. The point H of last Article now coincides with A, and the line h with B'C'. Although the triangles are not homologous, it is still true that AA', BB', CC' are concurrent, and that the intersections (BC, B'C'), (CA, C'A'), (AB, A'B') are collinear. 159. Desargues' Theorem. Let K, L, M, N be four 148 PROJECTIVE GEOMETRY points on a given conic: and let any line u cut the conic in P, P'. Then (P, P') is a conjugate pair in the involution on u determined by any two of the three pairs of opposite sides of the complete quadrangle KLM N. (Fig. 53.) FIG. 53 -Let u(KL, MN, KN, LM) =A, A', B, B' as in the figure. By the projective properties of the conic K(LNPP') AM(LNPP'), hence the sections of these pencils by u are projective: thus (ABPP') w (B'A'PP') W (A'B'P'P), and this proves that ABP —A'B'P'. We have previously proved that if KM, LN cut u in C, C', (C, C') is a conjugate pair in the involution (AA'.BB'). Correlatively: If k, I, m, n are four tangents to a given conic, and from any point U are drawn two tangents p, p' to the conic: then (p, p') is a conjugate pair of rays in the involution of lines through U determined by any two of the three pairs of opposite vertices of the complete quadrilateral klmn. The reader is recommended to consider the special cases of Desargues' theorem which arise (I) when two vertices of the quadrangle move up to coincidence; (2) when K = N, and L = M; (3) when u touches the conic, and PLANAR INVOLUTIONS. POLAR SYSTEMS 149 so on. The correlative theorem may be examined in the same way. Assuming Desargues' theorem to be true, and also its correlative, almost, if not quite, the whole projective theory of conics could be deduced. 160. Pencils and Ranges of Conics. In the figure any point P on u determines the conic PKLMN, and hence the point P'. Thus, if we make P traverse the whole of u, we obtain an unlimited number of conics through K, L, M, N each occurring twice. More generally, taking X, any point in the plane, we have a definite conic XKLMN: and by giving X all possible positions we obtain all possible conics through K, L, M, N: each conic occurring an unlimited number of times. This set of conics, having four base-points in common, is called a pencil of conics. Correlatively, if kimn is a given quadrilateral, the whole set of conics which touch its sides is called a range of conics, of which k, I, m, n are the base-lines. In the pencil of conics there are three and only three which degenerate into line-pairs, namely, (KL, MN), (KM, NL), (KN, LM). Similarly, in the range of conics there are three and only three which degenerate into point-pairs, namely, (kl, mn), (km, nl), (kn, Im). 161. Given a quadrangle KLMN and a line u disjoint to its vertices: then either two conics can be drawn through K, L, M, N which touch u, or none at all. Namely, the pencil of conics {KLMN}, as we have seen, determines an involution on u. If this is hyperbolic, and P, Q are its foci, then the conics PKLMN, QKLMN are those in question: if it is elliptic, no such conics can be found. Correlatively, either two or no conics can be drawn touching four given lines and passing through a given point external to them all. CHAPTER XX. ELLIPTIC INVOLUTIONS. COMPLEX ELEMENTS. 162. In considering the properties of a single involution, we may suppose it to exist among the elements of any one of the eight elementary forms. For many reasons, it is convenient, in the first place, to consider it as an involution of points on a conic; and, for practical purposes, we take the conic to be a circle. The reader may suppose this done in what follows, but it will be seen that nothing said involves this specialisation. Given a conic (circle) K, any two separated pairs (A, A') (B, B') of points upon it determine an involution. If AA', BB' meet in 8, then every line through S cuts K in two conjugate points P, P' and conversely. Two conjugate points never coincide. Any point S within K determines upon it an elliptic involution, such that any two corresponding points are collinear with S. This is merely repeating, for the sake of clearness, what has been already proved. In what follows we are speaking of an elliptic involution, unless the contrary is expressly stated. 163. Sense of an Elliptic Involution. The points A, A' being separated by B, B', a point P which travels round K in the same sense will pass over the four points either in the order ABA'B'( = BA'B'A, etc.) or in the opposite order A'BAB'. We shall regard ABA'B', A'BAB' as symbols COMPLEX ELEMENTS 151 representing two conjugate involutions which are geometrically equivalent, but associated with two opposite senses ABA', A'BA. In this connexion we shall not use such a symbol as AA'BB', in which the symbols for two conjugate points are adjacent. It is to be observed that ABA'B' = BA'B'A =A'B'AB = B'ABA' and A'BAB'= BAB'A' =AB'A'B = B'A'BA, but the first set is not equivalent to the second. 164. Harmonic Representation. Given any pair (A, A') there is a definite pair (B, B') such that the set (AA'BB') is harmonic. Namely the chord BSB', which is the conjugate of ASA' with respect to K, determines the pair (B, B') in question. Having found B, B' in this way we shall say that ABA'B', AB'A'B are the harmonic representations, or harmonic images, of the conjugate involutions, starting with A. All the foregoing results and symbols may be extended to an involution on any one-fold elementary form. 165. Introduction of Complex Elements. Geometrically, if we take a conic K and a line u, then u may cut K in two points, or touch it, or not meet it at all. Now in analytical geometry we have as the image of K a quadratic equation k(x, y, z) =o homogeneous in x, y, z and with real coefficients: the image of u is a homogeneous linear equation X(x, y, z) =o. Whenever u does not cut or touch K the simultaneous equations 4 =o, X =o have two complex solutions, conjugate to each other. Now clearly it would be a great advantage if we could deduce from the geometrical figure in this case something real and geometrical which we could call, in a geometrical sense, " two conjugate complex points." Whatever these entities are, they are not points in the ordinary sense, because u does not cut K. Now the essential connexion of u, K whether u cuts K or not, is that either u touches K, or else K< determines on u an involution of conjugate points. When this involution is hyperbolic, its foci are the points where u cuts K: the question is, can we interpret the elliptic involution on u, i52 PROJECTIVE GEOMETRY when u does not cut or touch the conic, so as to provide a geometrical equivalent for the corresponding algebraic solution, and further, can we make an extended definition of the term " point," so as to maintain the validity of the projective axioms? Staudt soon saw that the answer must be contained in the involution on u: the trouble was to see how the single involution could yield two geometric entities. By his discrimination of the senses of an elliptic involution he was able to overcome the difficulty, and thus open a new chapter in the book of pure geometry. We shall proceed deductively as, in fact, Staudt does in his treatise; but the foregoing remarks may help to show what was, in all probability, the base of his inductive procedure. 166. Complex Points and Lines. DEF. (i). An elliptic involution (AA'.BB'...) of points on a line, associated with the sense ABA', is said to represent, or to be, a complex point. The same involution, associated with the sense A'BA, is said to be the complex point conjugate to the preceding one. DEF. (ii). An elliptic involution (aa'.bb'...) of coplanar concurrent lines, associated with the sense aba', is said to be a complex line. The same involution, associated with the sense a'ba, is said to be the conjugate line. In order to justify these definitions, we must contrive not to contradict propositions (i)-(x) of Art. 2. This is done by a set of subsidiary definitions and theorems. First of all, we have definitions relating to conjointness, for elements in the same plane: DEF. (iii). A complex point lies on one and only one real line, namely, the base of the involution which defines the point. DEF. (iv). A complex line passes through one and only one real point, namely the centre of the involutionary pencil which defines the line. DEF. (V). A complex point defined by (AA'.BB'...) and a complex line defined by (aa'.bb'...) are conjoint when the involutionary row defining the first can be put into perspective COMPLEX ELEMENTS 153 with the involutionary pencil defining the second. In other words, we can, by a proper choice of notation, make the row (AA'BB'...) a perspective section of the pencil (aa'bb'...). 167. From these definitions we derive the following theorems: PROP. (i). A real point A and a complex point B are conjoint with one and only one line. Namely, if B is defined by the involution (PP'.QQ'...) and the sense PQP', a fact which we may denote by writing B =PQP'Q', and A is not on the base of B, the line in question is that defined by AP.AQ. AP'.AQ'; while if A is on the base of B, the line in question is that base. It should be noticed that, in the general case, the sense of PQP'Q' determines the sense of A(PQP'Q'); and that the line conjoint to A and the conjugate of B is A(P'QPQ'), the conjugate of the other line. It is easily seen that no other line, real or complex, can satisfy the definition of being conjoint with both A and B. PROP. (ii). There is one and only one point which is conjoint with a real line a and a complex line b. Namely, if b = pqp'q', and a does not pass through the centre of b, the point in question is ap.aq.ap'.aq'. If a goes through the centre of b, that centre is the point in question. PROP. (iii). Two complex points determine one and only one line conjoint to both. Case (a). If the complex points have the same real base, that is the line in question. Namely, by Def. (iii) this is conjoint to the given points; and we cannot have, on the same real line, two different involutions perspective to the same involutionary pencil, so the line in question is unique. Case (P). If the complex points P, Q have different bases u, v, let uv =A. Starting from A, let ABA1B1 be the harmonic image of P, and ACA2C2 the harmonic image of Q. Then the lines A1A2, BC, B1C2 meet in a point 8, and the complex line S(ABA1Bl) is conjoint with both P and Q. 154 PROJECTIVE GEOMETRY With the help of the fundamental theorem we can prove that there is no other such line. PROP. (iv). Two complex lines determine one and only one point conjoint to both. The construction and proof are correlative to those for Prop. (iii). 168. Fig. 54 shows how the construction for Prop. (iii) may be actually carried out. In the plane of the figure iIG. 54. we draw any circle, and take a fixed point X upon it. Suppose now a complex point on u is given by RQRQl: we join XQ, XR, XQ1, XR1 cutting the circle in q, r, q', r' respectively: then o, the meet of qq', rr', is the centre of the involution (qq'.rr'...) on the circle. As above, let uv =A, and let XA cut the circle in a; then if ao meets the circle in a', and we find b, b' such that (aa'bb') is harmonic (Art. I64), the pencil X(aba'b') will cut u in ABA1B1, a harmonic image of the complex point on u. Similarly COMPLEX ELEMENTS 155 for the harmonic image ACA2C2 of the point on v (the construction is omitted in the figure): and now the join of BC, A1A2 gives S, the centre of the required line. It should be noticed that S is also the centre of the line conjoint with A1BAB1, A2CAC2, and that this line is conjugate to the other one. Moreover, if BC2 (or BC) cut A1A2 in S', the point 8' is the centre of two conjugate lines, of which one is conjoint with ABA1B1, AC2A2C(=A2CAC2), and the other with A1BABl, A2C2AC(=ACA2C2). Thus, taking two complex points M, N and their conjugates M0, No on the bases u, v, we have two conjugate complex lines conjoint with M, N and Mo, No respectively; and two other conjugate complex lines conjoint with M, No and Mo, N respectively. The centres (S, S') of the two pairs of conjugate lines are separated harmonically by u and v. 169. We are now able to say that, in the plane, Any two points A, B, whether real or complex, determine a single line conjoint to both which we may denote by AB or BA, and call their join. Any two lines a, b, whether real or complex, determine a single point conjoint to both, which we may denote by ab or ba, and call their meet. We have also shown how the join or meet can be actually constructed, at least when neither of the given elements is at infinity. The reader will find it a good exercise to see how the constructions have to be modified, in a practical sense, when one of the elements is at infinity, or both of them are. Theoretically, there is no difficulty, so we have obtained a complete generalisation of Props. (vi), (ix), (x) of Art. 2, so far as they relate to a plane field. We have also enlarged props. (i), (ii), (iii) of the same Article: for instance, a real line contains, besides an unlimited number of real points, an additional unlimited number of complex points, namely, a conjugate pair for every distinct elliptic involution that can be constructed on the line: or, which comes to the same thing, two conjugate complex points for every harmonic set (AA1BB1), 156 PROJECTIVE GEOMETRY starting from a fixed point A on the line. Every such set is determined by assigning the two real points Al, B on the line; so, in a certain sense, if v is a symbol for the number of real points on the line, we shall have v2/2 pairs of conjugate lines, each occurring twice, because the pair A1, B1 defines the same harmonic set as A1, B. Finally, then, we have v real points and v2 complex points, or v2+v points in all. This is analogous to the relation of complex numbers x +yi to real numbers x. The reader should remember, however, that the cardinal number of all the points on a line is unaffected by our generalisation, being, as before, that of the arithmetical continuum. Similarly, and in the. same sense, a real point lies on v2 + v lines, and a plane field contains v4 + 1,3 + V2 + v points, and the same number of lines; and here, again, the cardinal number of elements of either kind is that of the continuum. 170. Complex Planes. Proceeding by analogy, we define a complex plane as follows: DEF. (i). Given an axial pencil (aa'.P'./'...) in involution, with no real self-corresponding elements, this is taken to represent two conjugate complex planes, associated with the senses a/3a', a'/3a respectively, and denoted by the symbols a/3a'P', a'/3a3' respectively. As in the case of a complex point or line, we can suppose, if we like, that ap/a'/', a'3a/p' are harmonic images of the planes. Each real element a determines the harmonic image a/a'3' or a'/3a/' of the corresponding plane. DEF. (ii). A complex plane contains one and only one real line, namely, the axis of the pencil which defines it. DEF. (iii). A complex plane contains all the real points on its real line, and no other real points. DEF. (iv). Let =a/P(a'3' be a complex plane, and o- a real plane. If cr goes through the axis of $, we shall say that this axis is the meet of o- and f. If not, Ca cuts (ap/a'P') in a set (aba'b') of concurrent lines, which define a complex line. This line we shall call the meet of a- and $. Thus PROP. (i). A real plane o- and a complex plane $ meet in COMPLEX ELEMENTS 157 a definite line ro- (or -cr). This is the axis of $, if cr goes through it: if not, it is a complex line in o- with its centre on the axis of e. DEF. (V). Let u be a real line distinct from the axis of the complex plane $. If u meets. the axis of $, we call this the meet of u and $. If not, let apa'/3' be any representation of: then ABA'B' = u(ca'3') is a representation of a definite complex point on u, and this we shall call the meet of u and $. Hence PROP. (ii). A real line u and a complex plane $ have a definite meet u$, which is real if u cuts the axis of b, but is in all other cases complex. (Defs. ii, v.) It is supposed here that u is distinct from the axis of $. DEF. (vi). If two complex planes have a common axis, that axis is said to be their meet. PROP. (iii). Two complex planes, whose axes intersect, define a complex line contained in both. Let the axes be u, v and put uv=a. Then we can find a definite harmonic image apa33' of the first plane $, and a definite harmonic image aya"-y" of the second plane p. The pencils (aa'PP3'), (aa/'y") are harmonic and have a common element a: hence they are perspective, and if v be the real plane a'a'"./y, this will intersect 6, p in the same complex line aba'b' (Prop. i), which has its centre at uv, the point where u, v intersect. DEF. (viii). Let $ be a real plane and u a complex line in a real plane p, with its centre disjoint to $. Let aba'b' be any representation of u, and $(aba'b') =ABA'B': then the point ABA'B' on up is defined to be the meet of $, u and denoted by $u. This complex point is definite: hence PROP. (iv). A real plane $ and a complex line u in a different real plane p meet in a definite point $u. This point is generally complex, but is real when $p is a ray of the pencil which defines u; u is then the centre of that pencil. PROP. (v). A complex line u in a real plane p, and a real 158 PROJECTIVE GEOMETRY point P external to p, determine a plane Pu, containing them both, PROP. (vi). Let $ be a complex plane, and u a complex line in a real plane p; there is a definite point $u common to t, u, and this lies on the line $p. Let v be the axis of i, and first let p cut v: then 4u is the join of the complex lines u, $p which are both in p. Secondly, let p contain v: then $u is the meet of u, v, and this is on $p, because in this case p = v. It is supposed, of course, that u does not lie on $. PROP. (vii). Let P be a complex point; u a complex line, centre R, in the real plane p. There is a definite plane Pu containing both P and u, whenever they are not conjoint. Let p be the base of P, and first suppose that R is not on p. Then u, PR are two different complex lines with the same centre, and therefore, by the correlative of Prop. (iv), Art. 167, determine a plane conjoint to both, and therefore to u and P. The plane Pu may coincide with p. In the more general case, let the real planes containing u, PR intersect in a. Take harmonic images aba'b', aca"c" of u, PR respectively: then the planes be, a'a", b'c" are concurrent and determine the axis of the plane Pu. PROP. (viii). A real line u and a complex point P external to it, determine a plane Pu. This is real, if the base of P meets u: otherwise it is complex, and has u for its axis. (Cf. Prop. ii). 171. The Bicomplex Line. Although we have now discussed a variety of cases of joins and meets, there is still one problem to solve before we can say that any two points determine a line, and any two planes determine a line. What are we to understand by the join of two complex points whose bases are skew, or the meet of two complex planes whose axes are skew? This cannot be either a real or complex line of the kind hitherto considered; so the introduction of a new entity is indispensable. This new element was invented by Staudt, who called it " the imaginary straight line of the second kind ": for brevity COMPLEX ELEMENTS 159 we shall call it " the bicomplex line," although such a term as "skew complex line" might perhaps be more appropriate. We will, in the first place, pursue an inductive line of thought. Suppose that on two lines, skew to each other, we have two elliptic involutions. Let (AA', BB') be any two pairs of the first, such that B, B' separate A, A' harmonically, and (A1A,', BB,') any such two pairs of the second. There is now a definite projectivity (ABA'B'..) = zT(A1B1iA'B'...) connecting the rows, and if we set it up by the construction of Art. 55, we obtain the given involutions as the intersections of their bases u, v with an axial pencil w(ABA'B'...) where w is any line meeting AA1, BB1, A'A1' (and therefore also B'Bi'). In this projectivity, the lines AA1, BB1, etc., form a quadric regulus p, and all the axes w form the conjugate regulus p' on the same ruled quadric as p. The generators of p are, by construction, paired in involution, and determine an involution on each axis such as w. Now suppose we take ABA'B' as the harmonic image of a complex point on u, and A1B1AB'B1' as the harmonic image of a complex point on v. Let the four lines AA1, BB1, etc., cut w in P, Q, P', Q' respectively; then PQP'Q' is the harmonic image of a definite complex point on w. We now agree to say that this point on w is collinear with the points on u, v. Thus on each generator of p' there is a definite complex point collinear with the given ones. They all lie on the same hyperboloid, and form what Staudt calls a "chain " connecting the given points. Denoting the given points by X, Y, the chain thus determined does not exhaust all the points collinear with X and Y. Namely, we can take any harmonic set (AA'BB') in the involution on u, and any harmonic set (A1Al'B1B,') in the involution on v: each such choice determines a chain. Using v as above (Art. I69), we have v2 choices, since A determines 160 PROJECTIVE GEOMETRY A', B, B', and thus apparently v2 chains of v points in all. But this is a delusion, because we obtain all the distinct hyperboloids containing collinear chains by keeping (AA'BB') fixed, and varying the other set: so we are left with v2 collinear points contained in v chains. This agrees with algebra, because if ($, q, C) ($) ', ', ) are any two analytical points, the points collinear with them are (1l + m$', lr + mrI', li+ m') where 1, m are any two complex quantities such that + m = I: and this gives v2 points. 172. Suppose we take K, any one of the hyperboloids considered, and cut it by a real plane a. The involution of lines on K determines an involution of points on the conic Ka. Let S be its centre, and s the polar of S with regard to Ka. Then on s we have an involution of points conjugate to the conic, and with a proper choice of sense this can be interpreted as the intersection of a with " the line XY." Similarly, if A is a real point, the projection of p from A is an involutionary conical envelope, and we can deduce from it an interpretation of the join of A to the line XY and show that it is a complex plane. But in all this, it is necessary to show that we reach the same result, whichever of the hyperboloids K we choose, and to prove this would lead us beyond the scope of the present book. We shall content ourselves with stating Staudt's ultimate definition in its most convenient form: Let a, b, a', b' be four generators of the same ruled quadric, on which there is an elliptic involution (aa'.bb'...). Then aba'b' may be taken as the symbol of a bicomplex line in a sense analogous to that in which aba'b' is taken as the symbol of a complex line when a, b, a', b' are coplanar and concurrent. To justify this definition, it is necessary to show that there is a criterion for the equivalence of two symbols aba'b' and pqp'q', and also that the projective axioms are not violated. Into this we shall not enter: as a matter of fact, Staudt completely solved the problem, so far as points, planes, lines, and quadric surfaces are concerned. The main reason for introducing this rather abstract COMPLEX ELEMENTS 161 theory here is that although the results of Arts. I62-169 enable us to establish the law of duality, and carry out all projective constructions in one plane, we cannot prove any projective proposition without the use of Desargues' theorem for every kind of triangle in the plane. Possibly the theorem for real elements may lead to a proof of it for complex elements without any further appeal to threedimensional constructions: if so, we can completely generalise the projective geometry of the plane, without introducing the notion of a bicomplex line. But there are a considerable number of cases to consider, because a triangle ABC may have one, two, or three complex vertices, and in particular two complex vertices may be conjugate. We shall simply assume, without proof, the validity of Desargues' theorem for coplanar triangles, real or complex: this is actually the case, and it follows that all plane propositions previously proved are still true, and can be proved by the same arguments. 173. Complex Points as Foci. Let X, X0 be the conjugate complex points defined by the involution (AA'.BB'...): then any two conjugate real elements (such as A, A') of the involution are harmonically separated by x, X0 (B. I45). Any one of the real elements being A, let ABA'B' (Fig. 55, next page), be the harmonic image of X, starting from A. Through A draw a real line v and on it take points C, A", C" such that (AA"CC") is harmonic; then ACA"C" is a harmonic image of a point Y on v, and AC"A"C a harmonic image of its conjugate Y0. The lines BC, A'A", B'C" concur in a real point 8, and BC", A'A", B'C in a real point T. Now consider the complete quadrangle STYYo: two of its sides SY, TYo meet in X, two sides SYO, TY meet in X0, the side YYO goes through A, and the side ST goes through A': hence X, X0 separate A, A' harmonically. In a hyperbolic involution the elements which separate every pair harmonically are the double points: so we are led to consider the conjugate complex points defined by an elliptic involution as being the double points of the involuM.P.G. L 162 PROJECTIVE GEOMETRY tion. The reader may verify that they actually are self-corresponding elements by considering the following problem: given on u an involution of real points (AA'.BB'...) FIG. 55. and PQP'Q', the harmonic image of any complex point X on u: construct a harmonic image of the point X' which is conjugate to X in the given involution. Since it is still true that projections and sections of harmonic sets are harmonic sets, we can state an analogous theorem for every involution on an elementary figure. For instance, an elliptic involution among the lines of a quadric regulus defines two complex (or rather bicomplex) lines, which may be regarded as the double lines of that involution, and so on. CHAPTER XXI. GENERALISED THEORY OF CONICS. 174. Complex Elements of Real Conics. Let us take, in a real plane, a real conic K and a real line u which does not touch it. Then K determines on u an involution of conjugate points, and if it is hyperbolic its double points M, N are the intersections of u with K. When the involution is elliptic, we can define its associated pair of complex points as the intersections of u with K. Thus, if K is a real conic, and u a real line, u either touches K or meets it in two definite points which are either real and distinct or else conjugate complex points. In each case they are the self-corresponding elements of the involution which K determines on u, and separate any real pair of coniugate points on u harmonically. Correlatively, if K is a real conic, and U a real point, U is either on K, or else lies on two definite lines which are tangents to K, and are either both real or complex and conjugate. In each case they are the self-corresponding rays of the involutionary pencil, of conjugate pairs through S, determined by the conic. If U is the pole of u with respect to K, and M, N are the intersections of u and K, then UM, UN are the tangents to K through U. If M is complex, N = M0. 175. Let S be a real point on a real conic K, and aba'b' any complex line u through S. The point S is on the conic and also on the line: we shall show that there is a definite 164 PROJECTIVE GEOMETRY complex point conjoint with u and K, which we may consider to be the remaining intersection of u and K. Let a, b, a', b' meet K again in A, B, A', B': let AA', BB' meet in T, and let t be the polar ofT. By Staudt's theorem (Art. I24) SA, SA' meet t in two points P, P' which are conjugate with respect to K. Similarly SB, SB' meet t in two points Q, Q' which are conjugate with respect to K. Therefore the point PQP'Q' is on K (Art. I74), and it is also on u (Art. I66, def. v.). By an indirect argument we can show that no other complex element of K can lie on u. Correlatively, if ABA'B' is a complex point U, whose base s touches K, we can draw through U one other tangent to K. This is complex, and can be found by a construction correlative to the above. 176. Common Pair of Two Involutions. Let (AA'.BB'...), (PP'.QQ'...) be two distinct involutions of real points on a conic: then if AA', BB' meet in S, and PP', QQ' meet in T, the line ST meets the conic in two points X, X' which are a conjugate pair in each involution. These points are real, if at least one of the involutions is elliptic: if both are hyperbolic, X, X' may be conjugate complex points. A similar theorem holds for two involutions of points on the same line, two involutions of lines through the same point, and two involutions of tangents to the same conic. All these, in fact, may be reduced to the above, by using an auxiliary conic. 177. Tangents to a Conic from a Complex Point. Let K (Fig. 56) be a real conic, and let s contain the complex point ABA'B', not upon K. The involution (AA'.BB'...) and that which K determines on s are therefore distinct, but they have in common (Art. I76) a real pair of conjugate points. We may suppose these to be A, A' so that if S is the pole of s, SAA' is a self-polar triangle. Let b be the polar of B: then the pencil (B) is projective to the row (b) when we make any ray through B correspond to its pole on b. Now B' is not on b: hence the projective pencils (B) and B'(b) generate a conic X which goes through GENERALISED THEORY OF CONICS 165 B, B'. This conic also touches SB, SB', because when the ray through B is s, the corresponding ray through B' is B'S, and when the ray through B is BS, the corresponding X FIG. 56. ray through B' is s (cf. Art. Io2). One of the lines SA, SA' (say SA) will cut X in two real points F, G: we shall now prove that the complex lines F(ABA'B'), and G(ABA'B') are tangents to K. In fact, from the way that X was con 166 PROJECTIVE GEOMETRY structed it follows that FB, FB' are conjugate with respect to K, and since F is on SA, of which A' is the pole, it follows that FA, FA' are conjugate with respect to K. Hence F(ABA'B') is a tangent to K, and similarly G(ABA'B') is a tangent. Since only one of the lines SA, SA' cuts X in real points, there are only two tangents through the given point. We have supposed that s does not touch K: if it does, the problem is solved by Art. I75. By a correlative construction we can find the points of intersection of K with a given complex line. It should be noticed that since s is the polar of S with regard to X, the points F, G in the figure are harmonically separated by 8, A. 178. Generalised Definition of a Conic. A real conic defines a polarity in its plane, and the conic presents itself, from this point of view, either as the locus of points conjoint with their polars, or as the envelope of lines conjoint with their poles. Staudt expresses this by saying that the conic is the Ordnungscurve of the polar system. We shall translate "Ordnungscurve" by "nucleus," understanding by this either the conic locus, or the conic envelope, or the set of line-elements Pp consisting of a point P and its polar p, when they are conjoint. The context will show which particular sense is intended. Now, as we saw in Chap. XIX., a polarity in a plane can be defined and discussed without any reference to conics, and all the fundamental theorems hold good even when we include the complex elements of the field. There will still be a set of lines conjoint with their poles, and these determine a figure which we may call the nucleus of the polar field. We define this in every case to be a conic: but this nucleus may not have any real elements at all, even when the polarity is " real"; that is to say, when to every real line corresponds a real point. This is analogous to the fact that a quadratic equation with real coefficients may represent an " imaginary " curve or envelope. The GENERALISED THEORY OF CONICS 167 theory is made still more complicated by the fact that a polarity need not be " real": for instance, we may take one defined by a polar triangle of which one vertex is real and the others two conjugate complex points, together with a real point P assigned as the pole of a real line p. The final result (though we shall not develop it in detail) is that Staudt's theory covers the whole range of analytical geometry, when both coefficients and coordinates may be complex, and no equation is of higher degree than two in the variables. 179. Data Fixing Real Conics. There are some important propositions (B. I79-18I) which are special extensions of theorems already proved for real elements in Chap. XIII. (i) Let S(ABA'B') be a given complex line u, ABA'B' a given complex point Q upon it, and let P be a real point distinct from 8, and not on the real line QQO. Then there is a determinate real conic which goes through P and touches u, u, at Q, Qo respectively. We may suppose that ABA'B' is a harmonic image of Q, and that SA goes through P (Art. I64). There is a definite polarity (Art. 154) for which SBB' is a polar triangle, and P is the pole of PA'. The nucleus of this polarity is the conic in question. Clearly this is a conic, and the only one, which satisfies the conditions of the question: all we have to do is to show that it is real. Let P' be the harmonic conjugate of P with respect to S, A: then P' is on the conic, and now by Staudt's theorem (Art. 124) the real intersections (PB, P'B'), (PB', P'B), etc., lie on the conic, so the real part of it is in fact generated by the pencils P(X), P'(X'), where (X, X') is a variable pair of the involution (AA'.BB'...). (ii). A real conic is determined by three real points, and two complex conjugate points, when no three of the five points are collinear. Let A, B, C be the real points, Q, Q0 the complex points on the real line QQO. Let AB cut QQO in X, and let X' correspond to X in the involution defining Q. Then X, X' must be conjugate with respect to the required conic. Now 168 PROJECTIVE GEOMETRY take Y, the harmonic conjugate of X with regard to A, B: then Y must be on the polar of X, and therefore X'Y must be the polar of X. Let XC cut X'Y in Z, and find D, the harmonic conjugate of C with respect to X, Z: then D must be a point on the conic. Similarly, by producing AC to meet QQO, and carrying out the same construction as before, we find another real point E on the conic, which is therefore determined by the five real elements A, B, C, D, E. (iii). One and only one real conic can be drawn to touch three given real lines and two given conjugate complex lines: no three of the five being concurrent. This is the correlative of (ii), and may be proved in a corresponding way. (iv). One and only one real conic can be drawn so as to pass through one given real point, and two pairs of conjugate complex points; no three of the five being collinear. Let A be the real point, P, Po and Q, Q0 the pairs of complex points; and let F, G, H be the real meets (PPO, QQO), (PQ, PoQ0), (PQ0, PoQ). Then FGH must be a polar triangle of the required conic: if, then, AF cuts GH in F', and we take B, the harmonic conjugate of A with respect to FF', this point B must be on the conic. Similarly, we can find on AG a real point C which must be on the conic. Now by (ii) above, there is a definite conic ABCPPo, and we see from the construction that this also goes through Q, Q0. (v). One and only one real conic can be drawn to touch one given real line, and two pairs of conjugate complex lines, no three of thefive being concurrent. This is the correlative of the last. CHAPTER XXII. THE THEORY OF CASTS. 180. Let A, B, C, D be four elements of the same elementary form K, considered in a definite order of writing, and also with reference to K. These elements are said to form a cast on (or in) K. Altogether, by permuting letters, we have 24 casts ABCD, BACD, etc., all on K, and defined by the same four elements. For the present we consider casts of real elements of real forms, such as four rays of a flat pencil or a regulus, four points on a conic, and so on. Through four points A, B, C, D, the vertices of a plane quadrangle, we can draw an unlimited number of conics K, X, etc. On each of these the four points determine a set of casts, and when necessary we can distinguish them as (ABCD)K, (ABCD)X, etc.; but this notation will rarely be required. 181. Projective Casts. Suppose two elementary forms K, KI are connected by a projectivity A, so that K' =c(K); then if ABCD is any cast on <, there is a corresponding cast A'B'C'D' on K,', and we may write A'B'C'D' =ZT(ABCD), and say that A'B'C'D' is projective to ABCD. A cast on any elementary form can always be made projective to a cast of four points on a line, or four points on a conic. Now for four collinear points A, B, C, D it has been proved that ABCD BADC A CDAB' DCBA; 170 PROJECTIVE GEOMETRY hence the same theorem is true for any cast. Moreover, the group of 24 casts ABCD, etc., contains either 8 harmonic casts or none at all. We shall say that ABCD is a harmonic cast, when A, C separate B, D harmonically. By supposing two elements of a cast to coincide, we obtain three kinds of improper casts, of the types (i) ABCA or BAAC (ii) ABCB or BABC (iii) ABCC or CCAB. Two casts are said to be equal, when they are projective, whether they are on the same elementary form or not. In particular, all harmonic casts are equal. 182. Let A, B, C be any three fixed points on a conic, and D a variable point; then we have an unlimited number of casts ABCD which are all different, and any other cast A'B'C'D' of points on the conic must be equal to one of them. In fact, when A', B', C' are assigned, there is on the conic a definite projectivity (A'B'C'...) =M(ABC...), and if, in this, D' corresponds to D, we have A'B'C'D'= ABCD. We may call A, B, C the base points of the series of casts ABCD. 183. Addition of Casts. Let ABCD1, ABCD2 (Fig. 57) be two casts on the same conic. Produce D1D2 to meet the tangent at C in T, and join AT cutting the conic in S. Then the cast ABCS is defined to be the sum of the casts ABCD1, ABCD2, and we shall write ABCS = ABCD1 + ABCD2. The following facts should be noted as following from this definition. (I) The projective determination of S is the condition that (CC.D1D2.AS) is an involution. (2) The point B is not used in the construction. (3) The construction is symmetrical so far as D,, D2 are THE THEORY OF CASTS 171 concerned: hence the commutative law of addition is satisfied, namely, ABCD1 + ABCD2 ABCD2 + ABCD1. AB / /2 A FIG. 57. Suppose one of the given casts is improper. First let D2 coincide with A: then S coincides with D1, and we have ABCA+ABCDI =ABCD...............(i) Secondly, let D2 coincide with C: then T and S both coincide with C, and hence ABCC + ABCD = ABCC...............(ii) These relations show that the improper casts ABCA, ABCC have properties analogous to those of o and oo in arithmetic: we shall presently see that ABCB has properties analogous to those of I. 184. The Associative Law of Addition. Suppose that, in Fig. 58, ABCD1, ABCD2, ABCD3 are three proper casts, which, for brevity, we may call cl, 02, C3. Find the point K such that ABCK = 2+C3, and the point S such that ABCS = C1 +ABCK: thus ABCS = (C2 + ( +). There are two auxiliary points H, L on the tangent at C, where it is met by D2DO, D1K respectively. Now we wish to prove that C1 + (02 + 3) = (o+ +) + 3; and this amounts to showing that if D1D2 meets CH in M, 172 PROJECTIVE GEOMETRY the lines AM, D3L meet on the conic. Let AM, D3L meet in N, and consider the hexagon AKD1D2D3N: then the meets of pairs of opposite sides are (AK, DD) = H, (KD1, D3N) = L, and (D1D2, AN) =M, and these are collinear, so that by Pascal's theorem N lies on the conic AKD1D2D3. C L MIM H FIc;. 58. The reader may prove that the associative law is still valid when one (or more) of the given casts is improper. Given two proper casts c, c, there is always a proper cast c2 such that 1 + 02 =c. This cast will be denoted by (c - C1). To construct it (cf. Fig. 57) let c=ABCS, c=ABCD1: then if AS cuts the tangent at C in T, and TD1 cuts the conic again in D2, we have ABCD2-= -C0. As a limiting case, suppose c = ABCA: then T is the intersection of the tangents at C, A, and D1, D2 separate A, C harmonically. In this case, if we put ABCD2= - ABCD1, it will be found that all the usual algebraic laws are satisfied: for instance, -(-) =c, Cl-(-C2) =C + C2, and so on. To fill up the details will afford the reader a good set of exercises. 185. Multiplication of Casts. Let ABCD1, ABCD2 (Fig. 59) be two proper casts on a conic: let D1D2 meet AC in O, and OB cut the conic in P. Then the cast ABCP is, by definition, the product of the casts ABCD1, ABCD2, and we write ABCP = c0 x 0c, where c%, c2 denote the given casts. THE THEORY OF CASTS 173 The following points should be noted: (I) The projective definition of P is that (AC.D1D2.BP) is an involution. (2) The point B is used in the construction. (3) The construction is symmetrical so far as D1, D2 are concerned: hence the commutative law of multiplication is satisfied, namely, co x c2 = c2 x cl. 0 P D2 FIG. 59. Suppose one of the casts is improper, and first suppose D2 coincides with A. Then 0, P both coincide with A, and hence ABCDi x ABCA =ABCA................. (i) Next let D2 coincide with B. Then O is the meet of BD1, AC, and P coincides with D1: thus ABCD1 x ABCB =ABCDI................ (ii) Finally let D2 coincide with C. Then 0, P both coincide with C: thus ABCD1 x ABCC =ABCC................ (iii) The only exception to (iii) is when D1 coincides with A as well as D2 with C: the line D1D2 now coincides with AC, and the construction fails. This is analogous to the arithmetical theorem that the product o x oo is indeterminate. Proposition (ii) shows that, as far as multiplication is concerned, the cast ABCB behaves like I in arithmetic. 174 PROJECTIVE GEOMETRY 186. The Associative Law of Multiplication. Let cl, c2, C3 be three proper casts on a conic (Fig. 60) represented by ABCD1, ABCD2, ABCD3. Find K so that ABCK =C xC2, and then P such that ABCP = 3 x ABCK=(c x 2) x 0. This gives two auxiliary points 0, L on AC, and we wish to show that (c x C2) x c3 = Ce x (02 x c3). This amounts to proving that if M is the point at which D1L meets the FIG. 60. conic again, BM, D2D3 meet on AC. Consider the hexagon BMD1D2D3K inscribed in the conic: L=(MD1, DK), and O = (D1D2, KB) are two intersections of opposite sides: hence by Pascal's theorem (BM, D2D3), the remaining meet of opposite sides, must lie on OL, that is AC. This proves the theorem. Given two proper casts c, cl there is a definite proper cast c2 such that c xc= c. This follows from Fig. 59, if we suppose ABCP =c and ABCD1 =cl; we find 0, the meet of BP, AC and then join OD, cutting the conic in D2: then c2= ABCD2. We shall write, when c0 x c2 = c, the equivalent c2 = co c 2 or c2 =- c/c. THE THEORY OF CASTS 175 As limiting cases let P coincide with A, B or C. In the first case, O and D2 coincide with A: thus ABCA ABCD1 = ABCA.................(i) In the second case, 0 is the meet of AC and the tangent at B, and D1D2 goes through this point: in this case we shall say that c,, c2 are reciprocal, and write 02=c.-~. Clearly (0-)-i = c, and if c- =, we must either have c= ABCB, or else c=ABCB', where B' is the harmonic conjugate of B with respect to AC, that is to say (Art. 184) c= - ABCB. More generally, if ABCP =c, and OT1, OT2 are tangents to the conic from the meet (BP, AC), then ABCT, = - ABCT2, and each of them is a cast satisfying the equation X2=o, and beside them there are no others. We only obtain "real" solutions, however, if BP, AC intersect outside the conic: so the full discussion must be deferred. Thirdly, let P coincide with C: then D2 coincides with C, and we have ABCC ABCD = ABCC,.............. (ii) whenever D1 is distinct from C. This again shows that ABCC is analogous to oo. We have next to consider the cases when ABCP is proper, but ABCD1 is improper. The reader will easily verify the following statements: ABCP + ABCA = ABCC0 ABCP ABCB = ABCP.............. (iii) ABCP + ABCC =ABCAJ 187. The Distributive Law of Multiplication. In order to make our calculus agree with ordinary algebra, it remains to prove that if c, Ci, c2 are any three casts, c(o0 + c2) =cc + cc2. In Fig. 6i, let ABCD =c, ABCDi=C0, ABCD2 =. Construct ABCS =o +c2 by the auxiliary point L, and now construct ABCP =c(co + -2) by the auxiliary point (SD, AC), that is, O in the figure. Next construct ABCP1 = cc by the auxiliary point H, and ABCP2 =cc2 by the auxiliary point K. We have to prove that ABCP + ABCP2=ABCP, or in other 176 PROJECTIVE GEOMETRY words (Art. 183) that P1P2 and AP meet CL in the same point. Consider the perspective pencils B(HKO..), D(HKO..): they establish on the conic a definite projectivity (PiP2PAC...) =-(D1D2SAC..), of which A, C are self-corresponding points. Hence we have Z(DID2) =P1P2, zr(AS) =AP, and r(CC) =CC =CL, the FIG. 6i. tangent at C. But D1D2, AS, CL are concurrent by construction: therefore P1P2, AP, CL must be concurrent, and this proves the theorem. (It should be noticed that the projectivity on the conic establishes a definite collineation in the whole plane (cf. Art. 71) because of the relation (P1P2PA) — (D1D2SA) between two quadrangles.) The reader may amuse himself by showing that the distributive law is valid, for all practical purposes, when one or more of the casts c, c,, c2 becomes improper. THE THEORY OF CASTS 177 188. Numerical Values of Casts. Indices. We have seen that the casts ABCA, ABCB, ABCC behave in our calculus like the symbols o, I, oo in arithmetic: we shall say that o, I, oo are the numerical values of the casts, and the indices of the base points A, B, C respectively. We shall also write ABCA =o, ABCB=I, ABCC =oo. / FIG. 62. Now (Fig. 62) by applying the rule of addition, we can find on the conic points 2, 3, 4, etc., such that ABC2 =ABCB + ABCB, ABC3 = ABC2 + ABCB, ABC4 =ABC3 + ABCB, and so on, continually adding ABCB. To these points we assign the indices 2, 3, 4, etc., and the values 2, 3, 4, etc., to the corresponding casts. The points form an unlimited ordered sequence of which C is the limiting point. Let the tangents at A, C meet in R: then the lines RB, R2, R3, etc., meet the conic again in points B', 2', 3', etc., to which we assign the indices -I, -2, -3, etc., because, from the construction, ABCB' +ABCB =ABCA =0, M.P.G. M 178 PROJECTIVE GEOMETRY and so for the rest (cf. Art. 184). The points B', 2', 3', etc., form an unlimited ordered sequence of which C is the limiting point. Let the tangent at B meet AC in 8. Then the lines S2, 83, S4, etc., meet the conic again in a series of points 2", 3", 4", etc. (not shown in the figure), to which we assign the indices 2, 3, i, etc., because, for instance (cf. Art. I86). ABC2 x ABC2" =ABCB =I. These points form an unlimited ordered sequence of which A is the limiting point. The sense of the order is opposite to that of the points 2, 3, 4, etc. Similarly, if R2", R3", R4", etc., meet the conic again in 2"', 3/", 4"', etc., the indices of these points must be -, -, -, etc., and they form an unlimited ordered sequence converging to A.,The order is of the same sense as that of 2, 3, 4, etc. By geometry, the points 2', 2"' are collinear with S: hence we have I (- - ) = - 2, as we ought to do, and so, in general, I + ( - pl) = - p, when the symbols stand for casts. Now let p, q be any two positive integers: then we can find a point X on the conic such that ABCX =ABCP. ABCQ, where P, Q are the points whose indices are p, q. To this point we assign the index p/q. If RX meets the conic again in X', we assign to X' the index - (p/q). Thus every positive or negative rational number corresponds to a point on the conic which has that number for its index; and we can prove without difficulty that equivalent numbers such as 3/4, 6/8 correspond to the same point, while numbers that are not equivalent correspond to different points. 189. Let X, p/ be two positive rational numbers, and let ABCL, ABCM be the casts which have X, P for their values: then L, M are both on the arc ABC of the conic. Construct the points D, D' (Fig. 63) such that ABCD =ABCM - ABCL, ABCD'=ABCL - ABCM. THE THEORY OF CASTS 179 Clearly ABCD +ABCD'=o, so DD' goes through the pole of AC, and D, D' are separated by A, C. Suppose that /> X: then - A, the value of ABCD, is positive, and D must be on the arc ABC. Now we see from the figure that this can only be when A, M separate L, C; or, in other words, when ALMC is an order. Conversely, whenever ABCL, ABCM are positive casts, and ALMC is an order, the value of ABCM exceeds that of ABCL numerically. It follows from this that when the points whose indices are positive and rational are plotted on the conic, they have C B FIG. 63. an order of position corresponding to the order of magnitude of their indices; and this order of position is of the same sense as the order ABC. Now let (rl, r2, r3,...) be any rational sequence converging to an irrational number p. Then if R1, R2, R3, etc., be the points on the conic which have rl, r2, r3, etc., for their indices, the sequence (R1, R2, R3,..) must have a limiting point R, to which we assign the index p. Thus for every rational or irrational number we have a corresponding point on the conic of which that number is the index. We assume that all the real points on the conic are now exhausted: the result is that if a variable point D, starting from A, describes the conic in the sense ABC, the value of the cast ABCD varies continuously, going first through all 180 PROJECTIVE GEOMETRY positive values in increasing order, and then through all negative values in (algebraically) increasing order. Observe that in this statement we regard oo, like o, as signless, so there is no breach of continuity as D passes through C, any more than there is when it passes through A (cf. the remarks in Art. I6). 190. Permutation of Base-points. In the foregoing theorems and constructions we have made use of three \/c FIG. 64. fixed base-points A, B, C with the indices o, I, oc respectively. Now without changing the set of base-points, we may interchange their parts in the constructions: in other words, we may permute the indices o, I, oo. The question is: how does this affect the index of any point D on the conic, supposed in the first instance to be distinct from A, B, C? There are six cases to consider. (i). Suppose we interchange the indices o, oo: that is, let us take C, B, A instead of the original A, B, C. The index of D will now be the value of the cast CBAD, and we have to connect this with the value of ABCD. There is, on the conic, a definite projectivity (CBA...) = (ABC..) THE THEORY OF CASTS 181 and now, if D = YD', we shall have CBAD =z(ABCD') =ABCD'. Now the projectivity is evidently an involution the centre of which is Q (Fig. 64), the meet of AC and the tangent at B: hence Q, D, D' are collinear, and CBAD = ABCD' = I ABCD. (Art. i86). Hence CBAD xABCD =I..............(i) (ii). Let us interchange o, i: that is, let us take B, A, C instead of A, B, C. Take the projectivity (BAC...) =a(ABC...), then BACD =ABCD', where zrD'= D. Now the projectivity is the involution whose centre is the point where AB meets the tangent at C: hence DD' goes through this point, and (Art. 184) ABCD' + ABCD =ABCA + ABCB=I. Thus BACD +ABCD =................. (ii) (iii). Let us interchange i, oo: that is, take A, C, B instead of A, B, C. The projectivity (ACB...) =(ABC...) is an involution with its centre at the meet of BC and the tangent at A. This results from the projectivity in (i) if we interchange A, B. Since A, B, C, D may be any four points on the conic, we deduce from equation (i) CABD x BACD =I, that is, by equation (ii), CABD(I -ABCD) =............... (iii) Proceeding in this way, we arrive at the following formulae, putting ABCD =X: ABCD =X BACD =- X CBAD = XBCAD=(X-I)/X() ACBD =D/(X- I) CABD =I/(I -- X) 182 PROJECTIVE GEOMETRY In these six equations, each symbol on the left is to be interpreted as the index of D when the three points preceding it in the symbol have the indices o, i, oo in the order in which they are written. It does not seem worth while to introduce a new symbol for the value of a cast distinct from that used for the cast itself: because in all additions, multiplications, etc., of casts we have to use the same set of base-points in some one fixed order, and now any symbolical equation connecting casts gives the same equation connecting their values. We have already seen that ABCD =BADC = CDAB = DCBA, hence of the 24 cast-symbols derived from ABCD there are in general four which have the same value A, and there are five other sets of four which have the values i-X X-I A I X' A AX-' I-A respectively, in accordance with formulae (iv). As will be found by equating pairs of values in (iv), the only exceptions are the following: A= o a A= I; values o, I, oo each counted twice: X = o A= in = -i; values I, -, 2 each counted twice: A= +,/3 X=; values I + i/3, each counted thrice. 2 } Of these three cases, the first involves improper casts, the third complex casts, which we have not yet considered: the second is the harmonic case, the most important of all. We have defined a harmonic cast ABCD to be one in which B, D separate A, C harmonically: in this case its value is THE THEORY OF CASTS 183 - I: if A, B separate C, D harmonically the value of ABCD is I, and if A, D separate B, C harmonically its value is 2. All this follows from (iv), and the previous proof that ABCD = - I in the first case. (Cf. Fig. 62.) 191. Linear Group of Six Casts. There is another way of looking at the matter, which is very instructive. In FIG. 65. Fig. 65 let P, Q, R be the points where the tangents at A, B, C meet BC, CA, AB respectively. Let D1 be any other point on the conic, and let D2, D3,... D6 be determined by making the following triads of points collinear: (Q, D, D6), (R, D1, D2), (Q, D2, D5), (R, D5, D4), (Q, D4, D3): that is to say, let QD1 cut the conic in D6, and so on. Then the following triads are also collinear: (R, D3, D6), (P, D2, D3), (P, D4, DI), (P, D5, D6). 184 PROJECTIVE GEOMETRY Moreover, if X is the value of any one of the casts ABCDi, the values of the other five will be I -X, X-1, (X -I)/X, I/(I - X), X/(X - I). For instance, we may consistently put, in the figure, ABCD 1 =X, ABCD2 = I - X, ABCD3 = (X - I)/X, ABCD4 = X/(X - I), ABCD5 = I/( - X), ABCD6 = /X. R FIG. 66. On account of these relations the six casts are said to form a group; we have, in fact, a geometrical image of a well-known group of linear substitutions of a variable X. Two of the special cases considered in Art. I90 follow at once from the figure. Suppose the points D1, D2 coincide: this may happen in two ways. First they may coincide at C; in this case D3, D4 coincide at B, and D5, D6 at A, so the six values of the casts reduce to oo, I, o, each reckoned twice. Secondly, D1, D2 may coincide at C', the point of contact of the other tangent from R to the conic: and now if B', A' are the points of contact of the other tangents from Q, P THE THEORY OF CASTS 185 it is easy to prove that D3, D4 coincide at B', while D5, D6 coincide at A'. Moreover ABCC'=~, ABCB'=- I, ABCA' =2. The three involutions whose centres are P, Q, R respectively have for their cast-equations XX -X -x =0 XX - I=, X + X -I =O. Fig. 66 shows what becomes of Fig. 65 when the conic is a circle, and ABC is an equilateral triangle. For some purposes this is a convenient standard figure: it will be observed that the points P, Q, R are all at infinity. 192. Complex Casts. Let X be a complex point on a real conic; then taking, as before, a set of real base-points A, B, C on the conic, we may regard ABCX as the symbol of a cast defining the position of X on the conic, and call it a complex cast. If X, Y are any two points on the conic, whether real or complex, we can still apply the processes of Arts. 183-7 to the casts ABCX, ABCY, so as to find their sum, product, etc. This follows from Art. I75, I77, and the results contained in Chapter XX. Generally, if x, y, z... represent given proper casts, and (xx, y, z,..), (x, y, z,...) are any two polynomials with rational integral coefficients, we can find a definite cast f satisfying the symbolical equation equation f=(x, y, z,...)/+(x, y, z...), unless it happens that +(x, y, z,...) = +(x, y, z,...) =ABCA or ABCC, in each of which cases f is indeterminate. The actual construction for f may be varied in an unlimited number of ways; but since the laws of algebraic combination are satisfied, we always arrive at the same final result. 193. Supplementary Theorem. If two casts ABCX, ABCY satisfy the relation ABCX.ABCY =ABCA, and neither of them is ABCC, then one of them must be ABCA. This corresponds to the arithmetical theorem that, if the numbers x, y are both finite, and xy = o, then either x = o or y = o. 186 PROJECTIVE GEOMETRY The truth of the theorem follows from Art. 185. Thus, let XY meet AC in Z: then the hypothesis implies that BZ meets the conic again in A; but (whether we suppose Z to be real or to be complex) this cannot be unless Z coincides with A. But now we have A, X, Y as the symbols of three collinear points which are all on the conic: this is impossible, unless two of the symbols stand for the same point (cf. Arts. 175, I77). Therefore, if X, Y are distinct, one of them must coincide with A; and if we suppose X =Y, then the tangent at X must go through A, and therefore X must coincide with A, for the only tangent that goes through A is the real tangent at A, and this does not meet the conic elsewhere. The proposition has now been proved. It may be shown in a similar way that if the product of two casts is ABCC, one at least of the factors must be ABCC. This theorem, however, is of comparatively slight importance. 194. Let X, X0 be two conjugate complex points on the conic; then the sum and product of the casts ABCX, ABCXo are each real. Conversely, if the sum and product of ABCX, ABCXo are each real, the points X, Xo are either real, or else conjugate complex points. If X and X0 are conjugate, the line XXo is real; call it h. Let h meet AC in 0, and the tangent at C in T; then if AT, BO cut the conic in S, P respectively, the real casts ABCS, ABCP are, by definition, the sum and product of the casts ABCX, ABCXo. Conversely, let the sum and product be the given real casts ABCS, ABCP. Let AS meet the tangent at C in T, and let BP meet AC in O; then the points X, Xo must lie on the real line OT, and are therefore its intersections with the conic. These are either real (and possibly coincident), or else complex and conjugate. 195. Numerical Values of Complex Casts. We have just shown how to find two casts which have an assigned real sum, and an assigned real product. The solution is always possible when the sum and product are each finite; that THE THEORY OF CASTS 187 is, distinct from ABCC. Now this problem is analogous to the arithmetical one of finding x and y from the equations x +y=s, xy=p, where s, p are assigned real numbers. The solution is either a pair of real numbers, or else a pair of conjugate complex numbers; moreover, we are at the same time finding the solution of the quadratic equation z2 -sz +p =o. Let s, p be the indices of the points S, P on the conic, defining the assigned sum and product of two casts which have to be found. If the solution leads to two real casts ABCX, ABCY, these will have real values x, y, the indices of the real points X, Y. There is no difficulty in proving that these numbers x, y are the roots of the numerical equation Z2 - Z +p = O, so the analogy between the geometrical problem and the arithmetical one is complete. If we are to maintain the analogy in the case when the geometrical solution leads to two complex casts ABCX, ABCX0, we must assign to these casts values which are the roots of z2 -sz+p =o; the question is, which value is to be assigned to which cast? When we have only to deal with a single pair of conjugate casts, it does not matter how we decide; but when we try to extend the arithmetical analogy to all complex casts, we have to discover a rule which makes the assignment of complex indices consistent. Without attempting to justify it for the present, we shall state it in what seems to be its most convenient form as follows: Let PQP'Q' be a representation of a complex point of the conic, lying on the real line h. Let H be the pole of h: this is within the conic, and there is a definite sense H(PQP') in the pencil (H) which either agrees with the sense H(ABC) or is opposite to it. In the former, case we say that the index of the complex point has a positive imaginary part; in the other case that it has a negative imaginary part. 188 PROJECTIVE GEOMETRY 196. The Complex Unit Casts. Suppose that in Fig. 62 the lines AC, BB' meet in J. If any line through J meets the conic in x, x' and x, x' are the indices of these points. we have (Art. I85) XX'+I =o. The double points of this involution are the points whose indices satisfy the equation X2 + I = 0, and since they are the points of contact of the tangents from J to the conic, they are the intersections of the conic with j, the polar of J, that is to say, the line RS in the figure. We therefore assign to these points the indices + i, - i, discriminating the signs according to the rule just indicated. 197. In the arithmetical theory we can build up everything from the two units I, i: and, however we do this, we have an analogous way of building up the theory of casts. As an example: having fixed the point whose index is i, we can, by Art. 185, construct all the points whose indices are of the form qi, where q is real; and now by Art. 183 we can construct all points whose indices are of the form p + qi, where p and q are real. Thus we do actually derive from (I, i) in this way a definite field of points with known indices + qi, which exhaust all the complex arithmetical field in which p, q are finite. With the help of the Cantor-Dedekind axiom, we can justify ourselves in asserting that we have now exhausted all the (finite) points on the conic. But there are several outstanding difficulties. In the first place, the arithmetical identity (p + qi) + (p' + q'i) = (p + p') + (q + q')i, ought to be satisfied by any two casts. As a matter of fact, this is true, if we adopt the rule of sign given in Art. I95. Another difficulty, which is more serious, is that, by the four fundamental operations of addition, subtraction, multiplication, and division, we can build up a complete set of THE THEORY OF CASTS 189 indices (p +qi), with p, q rational, in different ways. It is essential to prove that all these different ways lead to the same result; that is to say, that when we have once fixed the points whose indices are o, i, oo, i, the indices of all other points on the conic are determined. To do this in a rigorous manner would far exceed the compass of this book. We must content ourselves with the following remarks, which are intended to show the reader the main outlines of the argument. The properties of the casts whose values are o, I, oo, i, so far as we have developed them, are consistent with those of the corresponding numbers. Moreover, the laws of combination, together with the definition of I and i, and the supplementary theorem (Art. 193) can be shown to include all the laws which we actually use in proving any arithmetical equivalence, so that any objection made to an arithmetical theorem (in the complex field) is equally valid in the field of casts on a conic, and conversely. For the elaborate analysis of the arithmetical theory we refer to Russell and Whitehead's Principia Mathematica (Cambridge Press); if the reader will study this carefully, he will see how the theory of casts can be put upon an equally logical basis, possibly after introducing certain axioms which we have not explicitly stated. As an example of the sort of equivalence intended, take the arithmetical equivalence and let us prove the corresponding cast-equivalence, supposing fractional casts defined by the multiplication rule (Art. I88). Put x=-~+,y=; then, by applying the proved laws of combination, 3X=3. 3. +3 =I + I =2 (by definition); also 3Y=2; therefore 3x = 3x, and hence 3 (x - y) = o, (Art. 184) 190 PROJECTIVE GEOMETRY and by the supplementary theorem, since 3 is different from o, we infer that x -y =o, and thence that x = y, as easily follows by a geometrical argument. Any other special case may be proved in a similar way; and this is all that we can say about arithmetical equivalences. 198. Correspondences on a Conic. Let x, x' be the indices of two points P, P' on the conic; let +(x, x') be a polynomial in (that is, a rational integral function of) x and x', which we may suppose to be of degree m in x and degree n in x'. Then, if x, x' vary, subject to the condition (x, x') =o, the points P, P' will vary also. For any given position of P there will be, in general, n positions of P', and for any given position of P' there will be m positions of P. We say that the equation (x, x') = o fixes an m-to-n (algebraical) correspondence of points on the conic. As an example, take two ellipses, one wholly inside the other; then, if any tangent to the inner one cuts the outer one in P and P', there is a 2-to-2 correspondence between P and P', as we see by geometry; and although we cannot prove it now, this is an algebraic correspondence of the kind that,we are considering. As we shall deal exclusively with algebraic correspondences, the qualifying term ' algebraic' will generally be omitted. 199. If a, b, c, d are any four constants, such that (ad - be) is not zero, the equation axx' + bx + cx' + d = o.................... (i) fixes a projectivity on the conic, and conversely. We shall prove this important theorem by a chain of auxiliary theorems, which are special cases of it. (i) If h is a finite constant, the relation x' =x+h fixes a projectivity. THE THEORY OF CASTS 191 Let H be the point whose index is h, and P, P' the points whose indices are x, x'. Then (Fig. 57) the lines AP', HP meet the tangent at C in the same point T; the pencils A(P'), H(P) are therefore perspective, and their sections by the conic are projective. This projectivity is parabolic, because C is the only self-corresponding point. (2) If m is a finite constant, the relation x' =mx fixes a projectivity. Using P, P' as before, let M be the point whose index is m: then (Fig. 59) the lines MP, BP' must meet on AC, and we have a projectivity for which AC is the axis, and A, C are the self-corresponding points. (3) If m is a finite constant, the relation x' =m/x fixes a projectivity. In this case, with the same notation as in (2), the line PP' passes through the fixed point O (Fig. 59) where BM and AC intersect; so we have an involution of which the centre is 0. The double points are the points of contact of the tangents from 0, and have for their indices \/m and - ^/m. (4) The relation (I) may be replaced by b, bx+d Xl -. —, ax + c' and now, if a is not zero, we have b bc -ad b x =. - -.1 —f, a a(ax+ c) a bc - ad (bc - ad)/a2 (bc - ad)/a2 say, where xl = = =-, say, where XI =a(ax + c) x + c/a x say, where x2 = x + c/a. Thus we have a chain of projectivities connecting (x', x1), (x1, X2), (x2, x) respectively; the given relation follows from them by composition, and therefore also represents a projectivity. When a =o, c must be different from o, and the reduction is even simpler than the above. The reason why 192 PROJECTIVE GEOMETRY (ad -bc) must not be zero is that, if it were, the relation (I) would reduce to (ax' + b) (x + c/a) = o, and would not give any connexion between x and x'. Subject to this condition, a, b, c, d may be any finite constants, real or complex. To prove the converse, let us suppose any projectivity fixed by making three points P1, P2, P3 correspond to three points Q1, Q2, Q3 respectively. Let x1, X2, x3 be the indices of the first three points, and yi, Y2, y3 those of the other three. Construct a determinant of which the first row is xy, x, y, I, and the other rows XnYn, XQ, yn, I, with n =I, 2, 3. By equating this determinant to zero, and expanding it as a linear function of the elements of the first row, we have a relation of the form (I), with y, for convenience, instead of x'. Purely algebraical transformations show that the condition ad bc is satisfied; so the converse of the proposition has been proved. As a corollary, we see that if PlP2P3P4 and Q1Q2Q3Q4 are casts projective to each other, I nyn, Xn, yn, I I =, (n= I, 2, 3, 4) and conversely: the symbol on the left being put for a determinant of which a typical row is given, and x,, yn being the indices of P,, Qn respectively. We shall subsequently be able to express this condition in a different but equivalent form (Art. 213). 200. Group of Linear Substitutions. For certain purposes it is convenient to modify our notation in the following way. Taking any four constants a, /3, y, 8, for which aS8 /y, we put ____ we put x'=-+P= a(, ' (x) =S(x), yx+8 \y, 8/' / and say that x' is derived from x by means of the linear substitution S. THE THEORY OF CASTS 193 Let us take another similar relation a/.+P' x"= X],+ _ T (x'), then, by expressing x' here in terms of x, we find X = '+ = U(x), y"x+8" where a" =a'a+3'y, f3" =a'p + 3'8, y"= y'a + 8'y, 8" = y' + 8'8. Now, symbolically, x"=T(x') =T(S(x)) =TS(x), so we write U =TS, or, in a more explicit notation, (a', 3'\ a, P/3\ a'a + /3'y, Ta'/ + 3+/\ a", 3"\) y'8, 8' y, 'a + 8'y, /y + J, 8'+8 " The symbols used here to represent substitutions are, in the abstract, what are called matrices: so we may regard the equivalence (I) either as a rule for compounding linear substitutions, or as a rule for multiplying matrices. The rule is analogous to that used for expressing the product of two determinants: but there is a restriction which the student must always most carefully remember. Each element of the product-matrix (e.g. a'a+ /3'y) is derived (or compounded) from a row of the matrix-factor first written and a column of the other factor. Thus, a'a + /'y is derived from the row (a', /3') and the column containing (a, y). It is easily verified that a"t - 3"y" = (a'6' - /'') (a - py), identically, and is therefore different from zero. Thus, if S, T are linear substitutions, the resultant TS (and in like manner ST) is a linear substitution; and we can now easily prove that the whole set of substitutions form a group, if we include I.X+, o O.X+I 0, I as the unit, or identical substitution. For if, with the M.P.G. N 194 PROJECTIVE GEOMETRY above notation, we suppose a, /3, y, 8 given, and put a", 3", " =I,o,,, we find (as- ly) a'=8, (aS -/y)y'= - 7, (as-/3y)/3'= -/3, (a3-/3y) 8' =a: these values give (a', A'\ a, 3\= (i, 0) =, and it can also be verified that aY, )\y, 8' = I; so every operation S has an inverse S-1 satisfying SS1 =S -S = I. Thus all the conditions for the existence of a group are fulfilled. Since the substitutions form a group, and each of them corresponds to a projectivity on the conic in such a way that if 8, T are two substitutions and ac, r the corresponding projectivities, the substitution ST corresponds to the projectivity or, there is what is called in group-theory an isomorphism between the group of substitutions and the group of projectivities. But it must be remembered that if m is any constant, neither o nor co, the substitutions y),, \my, m8 which are formally different, and for some purposes may be regarded as distinct, all correspond to the same projectivity. If we choose all the substitutions for which a8 -fy= i, these form a sub-group of the general linear group: calling the latter L, this particular sub-group may be called L1, and now any definite projectivity corresponds to two and only two elements of L1, which will be of the forms (a A) (_- -/) There is now a 2-to-I correspondence between the substitutions and the projectivities: this is expressed by saying THE THEORY OF CASTS 195 that II, the group of projectivities, is merihedrically isomorphic to L1. 201. Involutionary Case. The relation (I) of Art. I99 represents an involution if b =c. We can easily show that the converse is true: for if we write down the same relation for x', x" and suppose that x"=x identically, we have simultaneously axx' + bx + cx' + d = o ax'x +bx' +cx +d= o for any two corresponding points; and this can only be if b=c. The student should prove as an exercise that the resultant of two involutions is not, in general, an involution. He should also find the algebraic condition that this may be the case, and interpret the result geometrically. 202. Sub-groups of II. The group of projective permutations on a conic is of infinite order, and the theory of its sub-groups is too complicated for us to discuss. However, there are one or two sub-groups so important for our subject that a few words must be said about them. The reader will easily verify the following theorems whenever the proofs are omitted. (I) The projectivities defined by bx + cx' + d = o (bc o) form a sub-group of II. These can be expressed in the alternative forms x' =px +q, x=p' + q', (p o, p' o), and they all have C as a self-corresponding point. (2) The projectivities defined by axx' + bx + cx' = o (be o) form a sub-group of II. These can be expressed in the alternative forms I p+ q I ' p ), and they all have A as a self-corresponding point. and they all have A as a self-corresponding point. 196 PROJECTIVE GEOMETRY (3) The projectivities defined by P +q, (po, qEoo) x' -a x-a form a sub-group of II, and they all have as a self-corresponding point the point whose index is a. It is supposed here that a is fixed, and p, q variable, subject to the conditions that p is not zero and q is not infinite. All these sub-groups are coordinate, each being fixed by the assignment of a common self-corresponding point: but owing to the peculiar properties of the numbers o, oo they have to be expressed in different analytical forms (4) The self-corresponding points of the relation (3) are given by I p ~= l-q; x-a x-a that is to say, x=a, or else I-p x=a+-. q Thus the correspondence is parabolic if p=I, and then reduces to the form I I =-+q (q>~o). X -a x-a By taking a fixed, and q variable, we have a sub-group of II consisting of all those parabolic permutations which have a common self-corresponding point, of which the index is a. The coordinate groups of projectivities when a is o or oo assume the forms I I x=-+q, x =x+q respectively. 203. Homogeneous Coordinates. We now introduce a notation which a reader unacquainted with it will probably think unnecessary and highly artificial; he will, however, realise its convenience as we proceed. We shall say that (x1, X2) is a homogeneous representation of the real or complex number x, whenever the ratio xl/x2 is equal to x, and THE THEORY OF CASTS 197 x1, x2 are both finite real or complex numbers. If (xI, x2) is any one such representation of x, all the rest are included in the set (pxl, px2), where p is any number different from o and oo; and two symbols (x,, x2), (Y1, Y2) represent the same number, or are equivalent, whenever xy2 - x2yI = o, and conversely. Every symbol (o, x2) is taken to represent o: every symbol (x,, o) to represent oo. The only exceptional symbol is (o, o), which is not employed: symbols such as (x1, o~), (o0, x2), (o, oo) are excluded by definition. Keeping our usual notation, let x be the value of any cast ABCP on a conic, or in other words the index of P; then if (xi, x2) is any homogeneous representation of x, we shall say that x,, x2 are homogeneous coordinates of P. We shall often, to save words, say " let (xl, x2) be the homogeneous coordinates of P," instead of saying " let (xl, x2) be any one homogeneous representation of the index of P." It will be found that this does not lead to any confusion: we may also write P = (xl, x2) or ABCP = (xl, x2) whenever (xl, x2) is a homogeneous representation of the index of P. In particular, we may write A=(o, I), B=(I, I), C=(I, o);..............(I) also if ABCP = (xl, x2), ABCQ = (y1, Y2) then ABCP + ABCQ = (x2 Y1, YY2)........(2) ABCP. ABCQ = (x1x2, Y1Y2) The formulae (2) are subject to no exceptions, and include all the cases considered in Arts. 183, 185; this already shows one advantage of the new notation. The general equation of a projectivity is, in its homogeneous form, either axly1 + bx1y2 + cx.y + dx2y2 =................(3) or Y _= ax1 + /3X2 Yor y=, + x2.......................... (4) a 2 1 i a s. and this includes all special cases. 198 PROJECTIVE GEOMETRY Any projectivity of which (a,, a2) is a self-corresponding point may be written in the form (cf. Art. 202 (4)) l(xla2 - x2a) (yla2 -y2a1) + my2 (xla2 - x2al) + nx2 (Ya2 -2a) =..0. (5) where 1, m, n are any three finite constants. The projectivity (5) is parabolic if m + n = o, and neither m nor n is zero; so the general equation of a parabolic projectivity with its self-corresponding point at (a,, a2) is I (x1a2 - x2a1) (Yla2 -y2a1) + m {Y2 (Xx1a2 - x2a) - x2 (ya2 -y2a)} = o, or, which is the same thing, I (xla2 - x2al) (yja2 -y2a1) + nma2(xly2 - x2Y1) = o, or, again, in a more symmetrical shape, 1 (X1Y2 - x2Y1) + m(Xla2 - x2al) (yxa2 -y2al) = o,....(6) with lm o. If we put al = I, a2 =o, we obtain l(xy2 - x2y) + x2y2 = o as the general equation of a parabolic projectivity with its self-corresponding point at C; and this agrees with Art. I99 (I). Similarly for other special cases of the formulae (3)-(6) in this Article. 204. Generalised Theory of the Lineo-linear Equation. Let {abc..}, {a'b'c'...} be any two elementary forms in the wider sense; to fix the ideas, suppose them to be flat pencils in the same plane. Let a, b, c be taken as baseelements of the one form, and a', b', c' as base-elements of the other. Take elements d, d' in the forms such that a'b'c'd' 'abd: then the index of d' is equal to that of d, and conversely, if the indices of d, d' are equal, then the casts abed, a'b'c'd' are equal, and a'b'c'd' abcd. Thus a projectivity between two elementary forms is established by assigning the indices o, I, co to any three elements of the one, and the same indices to any three elements of the other. The projectivity is now expressed by the relation x' =x, connecting the values of corresponding casts. THE THEORY OF CASTS 199 - But, more generally, if the values x, x' are connected by the relation axx' +bx + cx' + d =o, (ad - bco)....(I) this fixes a projectivity between the pencils, because it may be regarded as the resultant of ax'x" + bx" + cx' + d = o, fixing a permutation of the elements of {a'b'c'...} among themselves (Art. 199), and the relation x=x" connecting the two forms projectively. 205. Triangular Coordinates. Let A, B, C, G be the vertices of a quadrangle; and for the sake of clearness let us suppose that G is within the triangle ABC. In the pencil (A) take AC, AG, AB as base-rays with indices o, I, oo respectively; and in the pencil (B) take BC, BG, BA as base-rays with indices o, I, co respectively. If P is any point, in the plane ABC, that is disjoint to AB, the rays AP, BP will have definite finite indices as elements of (A), (B) respectively: these are also the values of the casts A(CGBP), B(CGAP). Conversely, if the values of these casts are assigned, and not both infinite, the positions of AP, BP and their meet P are determined. Now let finite quantities x, y, z be assigned in any way consistent with writing B(CGAP) =x/z, A(CGBP) =y/z: then we say that (x, y, z) are homogeneous coordinates of P referred to the triangle ABC and the unit-point G. If m is any finite quantity (neither o nor oo), the point (mx, my, mz) is the same as the point (x, y, z). We may write P = (x, y, z) to express that P is the point for which x, y, z are coordinates; in this notation C=(o, o, I), G = (i, I, I), and it only remains to find coordinates for points on AB. For every such point we have z =o, and conversely; so we may regard this as the equation of AB. Let CG meet AB in L; let Q be any other point on AB, and let (x, y) be its homogeneous coordinates derived from the base-points 200 PROJECTIVE GEOMETRY B, L, A, as explained in Art. 203. Then we take (x, y, o) as the coordinates of Q. We are now able to write A=(I, o, o), B=(o, I, o), L=(i, I, o). With the exception of (o, o, o), every triad (x, y, z) now corresponds to a definite point: conversely to every point corresponds a set of equivalent triads (mx, my, mz). For instance, we may put G = (m, m, m) as well as G = (I, I, I); but in practice we suppress any such explicit common factor. 206. Equations of Loci. Suppose we make AP, BP corresponding rays in a projectivity connecting the pencils (A), (B); then by Art. 204 we conclude that the coordinates of P satisfy a relation a...+b.-+c.-+d=o (bc-ad'o) Z 2z or axy + bxz + cyz + dz2 = o,......................... (I) and conversely. Now the locus of P is either a proper conic, or else a line-pair of which AB is one member. For every point on AB we have z =o; so the degenerate case can only occur when (I) is satisfied identically on putting z = o. The condition for this is a=o; the equation now becomes z (bx + cy + dz) =o, and since z =o is one line of the locus, the equation of the other must be bx + cy + dz = o. (bc o)...............(2) If the restriction be ' o is removed, the equation (2) will still represent a line, if any one of the constants b, c, d (supposed all different from oo) is different from zero. Suppose, for example, that b =o: the equation reduces to cy +dz =o, expressing that y/z is a determinate constant. But this means (Art. 203) that P is on a definite ray of the pencil (A); so the equation may be taken to represent that ray, considered as a locus. Similarly, if c =o. Taking b, c, d in (2) as undetermined constants, the equation is sufficiently general to represent any line whatever, THE THEORY OF CASTS 201 because we can make it go through any two assigned points (xl, Y1, zl), (x2, Y2, z2). Namely, the line in question is X y z =0, xl Yi Z1 x2 Y2 Z2 or (yl1Z) x + (Zlx2)y + (xy2)Z =o.................. (3) where (ylz2) stands for (yz -y2zl), and so on. In a similar way we can remove the restriction be - ad - o from equation (I), and prove that it is now sufficiently general to represent any (proper or improper) conic locus passing through A and B. This, however, is of minor importance: the really important results, with a change of notation, may be recapitulated as follows: Having fixed a system of homogeneous coordinates by means of a triangle of reference ABC and a unit-point G, every linear equation,x + cy + z = o, in which no one of the constants $, r, g is oo, and some one at least is different from o, represents a definite straight line; and conversely. In particular, x=o, y = o, z = o are the equations of BC, CA, AB respectively. We are now in a position to verify all our previous results and discover new ones by the ordinary processes of analytical geometry; so we shall in future use analytical processes whenever they seem to be preferable. 207. Line Coordinates. By dualising the arguments and processes of Arts. 203-6 we can fix a system of homogeneous coordinates derived from a quadrilateral abcg, in which we regard abc as a triangle of reference, and g as the unit line. To every triad ($, ar, i) will now correspond a definite line p, and conversely to every line p will correspond a set of equivalent triads (m$, my, mI); so that ($, r, C) may be called homogeneous coordinates of p. It will be a very good exercise for the student to work out this analogy in detail: but we shall obtain equivalent results by a different method which has various interesting features of its own. 202 PROJECTIVE GEOMETRY Taking any fixed system (ABC: G) of homogeneous point coordinates, we have seen that the linear equation x + y + Z=o,.................................() represents a line, and conversely. The equation m$x + m-qy + mz- = o, where m is neither o nor oo, represents the same line. We shall say that all the equivalent triads (mn, mqr, m() are homogeneous co-ordinates of the line whose point-equation is (I). This fixes a one-to-one correspondence between lines and sets of equivalent triads. In particular, BC=(I, o, o), CA= (o, I, o), AB= (o,o, I). A FIG. 67. Let AG, BG, CG (Fig. 67) meet BC, CA, AB in H, K, L respectively: then we can prove analytically that the equations of KL, LH, HK are respectively y +z -x=o, x-, z +x-y +o, xy-z=o. Now let KL, LH, HK meet BC, CA, AB in P, Q, R respectively: then we can prove analytically that P, Q, R are collinear, and lie on the line +y + z =o; that is, the unit line (I, I, I). If, now, we put BC, CA, AB, PQR =a, b, c, g, THE THEORY OF CASTS 203 the reader will see, without much difficulty, that if we derive a system of point-coordinates (x, y, z) from (ABC: G), and a system of line-coordinates ($, vr, () from (abc: g) by the dualistic process, then the condition for the conjunction of (X, y, z), (d, 1, 0) is,x + i y + Sz= o, so in this way we arrive at the definition of line-coordinates given above. We shall say that when our systems of point and line coordinates are thus related, we have taken a normal system of (point and line) coordinates. Conversely, whenever the conjunction of the point (x, y, z) and the line ($, r, () is expressed by the relation x +vqy+ Z = o, the coordinate system must be normal. The point G is sometimes called the pole of g with respect to the triangle ABC. The ranges BCHP, CAKQ, ABLR are harmonic, by construction: so if G is the centroid of ABC, the line g is at infinity (Art. 36): in this case we are said to have taken a system of areal coordinates, because the coordinates of any point P are then proportional to the areas PBC, PCA, PAB, each taken with its proper sign. 208. Law of Duality. In the condition of conjunction (for a normal system of coordinates) x + y + z= o................................(I) let us consider ($, q, () fixed and x, y, z variable. Then we can regard (i) as a point locus, namely, the equation of the row of points situated on the given line ($,, C). We may also, as is usually done in analytical geometry, regard (i) as the equation, in point-coordinates, of the base of the row in question: this is not so accurate, but is rather more convenient in practice. If, on the other hand, we regard (x, y, z) as fixed and (n, a, ) as variable, equation (i) may be regarded as representing the set of all lines conjoint with (x, y, z): that is to say, the flat pencil which has its centre at (x, y, z). 204 PROJECTIVE GEOMETRY Here, again, it is rather more convenient to regard (I) as the line-equation x+ y + z= o of the point (x, y, z), although this is really a less accurate way of interpreting the equation. The real advantage of this point of view arises when we consider homogeneous equations of the form, (,, )= o, where (b(i, a, ) is a polynomial in 5, rj, C. This represents, when the degree of ~ is greater than I, an algebraic envelope, just as 4 (x, y, z) = o represents an algebraic locus: and it is, on the whole, more convenient to think of the linear cases as being degenerate. Whenever (x + 4qy + Cz = o is the condition of conjunction, we may regard ( (x, y, z) = o and 4 ( C, l, ) = o as representing normally correlated figures. From this point of view, the law of duality is merely a change of notation, in which (x, y, z) are interchanged with (f, r, C). 209. Tetrahedral Coordinates. There is no difficulty in extending these results to three dimensions; so we merely state the principal theorems without supplying the proofs. (I) A tetrahedron ABCD, and a point G within it, define a system of homogeneous coordinates (x, y, z, t), in which the coordinates of G are (I, I, I, i), and those of A, B, C, D are (I, o, o, o), (o, I, o, o), (o, o, I, o), (o, o, o, I) respectively. (2) The point-equation of a plane is of the form 5x + fy + Cz + rt = 0, and we may call (5, r, T, T) the homogeneous coordinates of the plane of which the above is the point-equation. (3) Taking x, y, z, t as fixed, and C, r, ~ r as variable, the equation x5+y'+Z+&=0, equation fx + yrl + zi+ tr = o, may be regarded as the equation of the point (x, y, z, I) in planar coordinates. (4) The ratios x/t, y/t, z/t may, if we wish, be taken as non-homogeneous coordinates of any point (x, y, z, t) for THE THEORY OF CASTS 205 which t is not zero: that is to say, which is not in the plane ABC. These coordinates, if P is the point belonging to them, are the indices of the planes PBC, PCA, PAB in the axial pencils (BC), (CA), (AB) referred to appropriate baseplanes. (5) When G is at the centroid of the tetrahedron of reference, the unit-plane (I, i, i, i) is the plane at infinity. The coordinates are now said to be volume-coordinates. CHAPTER XXIII. CROSS-RATIOS. 210. In order to show the real object of the following discussion, we shall assume, for the moment, the ordinary theory of proportion and measurement according to which the position of a point on a line is fixed by the numerical measure, taken with the proper sign, of its distance from a fixed point O on the line, taken as the origin of abscissae. If a, b are the abscissae of any two real points A, B, we shall write, as is usually done in works on analytical geometry, OA=a, OB=b, AB=OB-OA=b -a, the last formula giving the sign of AB as well as its magnitude. Consistently with this notation, we have the identical relations BC +CA +AB =b BC.AD +CA.BD +AB.CD =o;........... the latter, for instance, being merely a quasi-geometrical way of expressing the algebraic identity (c-b) (d-a)+(a-c) (d-b)+(b-a) (d-c)=o......(2) Taking any triad A, B, C with abscissae a, b, c, we have AC c-a CB b-c' and this is defined, in every case, as the ratio in which C divides AB. It is independent of the origin, and is positive or negative according as C is within or without the finite segment AB. If C passes away to infinity, its limiting value is -I. CROSS-RATIOS 207 Now take a fourth point D with abscissa d: this divides AB in the ratio AD/DB, of which the measure is (d - a)/(b - d). We now write P AC AD c-a d-a R (ABCD) -CB ' DB b-c ' b- d AC.DB (c - a)(b-d) CB.AD -(b - c) (d - a) and call this a cross-ratio. By permutation of symbols, we can deduce from A, B, C, D a set of 24 cross-ratios, but these are not independent, nor are they all different in value. The importance of the cross-ratio is this. Take any two tetrads ABCD and A'B'C'D' on the line: then, in general, they are not projective to each other; but if they are, we shall have 1(ABCD) =1(A'B'C'D'), and conversely. Thus we have a metrical way of expressing the projective condition that ABCD7AA'B'C'D'. Our aim being to deduce, as far as possible, metrical properties from projective ones, we proceed to give the purely arithmetical theory of cross-ratios, afterwards bringing it into connection with the theory of casts. 211. Let a, b, c, d be any (real or complex) numerical quantities; we define the cross-ratio R (abcd) by the following equivalence: c-a d-a (c -a) (b -d) R(abcd)b - c a -d - (b - c) (d -a)............() It follows from the definition that R3 (abcd) = I (badc) = R (cdab) = -R (dcba).......... (2) also that, if we put I3(abcd) =X, B, (abdc) = R (bacd) = R (cdba) = 1 (dcab) =.... (3) Taking the identity (b -c) (a -d) + (c -a) (b -d) + (a -b) (c -d) =o, (c -a) (b -d) (a -b) (c -d) we deduce I+ + =o; (b - c) (a - d) (b -c)(a - d) that is, I - R (abed) - R (acbd) = o; so we have U (acbd) = I -, (abcd) = I - X.................... (4) 208 PROJECTIVE GEOMETRY Proceeding in this way, we deduce the following set of formulae: R (abcd) = X, R (bead) = (A - i)/X, R (bacd) = I/, R (cbad)= /(X - )......... (5) R (acbd) = I - X, (cabd) = I/(I - X). These may be both compared and contrasted with those of Art. I90 (iv.). As important particular cases, we have L(oI oo d) = (d - i)ld, K (i oo od) = I/(I - d), } (I oo d) = d/(d- I), (oo Iod) = I - d,.....(6) 1(o oo Id)= /d, ( ooid) = d. J If we express a, b, c, d in the homogeneous forms (al, a2), etc. (Art. 203), we have (alc2) (ad()- (adc2) (dlb2),(abcd)(CAb2) (dlb2) (clb2) (ad2). (7) writing (alc2) for (alc2 - a2c1), and so on. 212. Let a', b', c', d' be derived from a, b, c, d by the same linear substitution; that is to say, let aa+3 ab+13 'yib+c' etc., a', b, c, -ya +8' yb + etc., with a8 - fly X o: then R(a'b'c'd') = R(abcd). This follows at once by substituting, in the expression on the left, the values of a', b', c', d' in terms of a, b, c, d. For instance, /, aC+,- aa +/ (a8- P3)(c-a) c ya + c ya c+- (yc++ ) (ya + S) ' and so on: putting these into the expression for,(a'b'c'd'), everything cancels out except the differences (c-a), etc., and the result is that stated in the theorem. 213. If a, b, c and a', b', c' are two distinct triads of numerical quantities, and x, x' are two variable numbers, the relation R(abcx) = (a'b'c'x')............................ (I) is a lineo-linear equation axx' + x + yx' + 8 = o, CROSS-RATIOS 209 for which a - 3y o, and (a, a'), (b, b'), (c, c') are three corresponding pairs of values of (x, x'). The first part of this statement follows from (i) by multiplying up, and transposing: the second from the fact that the substitutions (x, x') =(a, a'), (b, b'), (c, c') make both cross-ratios in (I) equal to oo, o, I respectively (cf. Art. 211 (I) ). Hence equation (r) of the present Article must be algebraically equivalent to the condition XX, X, X', I =0. aa',, a, I bb' b, b', I CC', C, C', I already found in Art. I99. 214. Cross-ratio of a Cast. Now let p, q, r, s be the indices of four points P, Q, R, S on a conic, referred to any set of base-points; we put R1(PQRS) =iT(pqrs), and define this to be the cross-ratio of the cast PQRS. Its value does not depend on the set of base-points chosen, because, if we take a new set of base-points, the new indices (p', q', r', s') of P, Q, R, S are derived from the old ones by the same linear substitution, and hence, by Art. 212, R(p'q'r's') =R(pqrs). In particular, if p, q, r = o, i, o, so that P, Q, R are taken as base-points, R(PQRS) = 3(oI oo s) = (s - I)/,................... (I) where s is the index of S referred to the base (PQR). By a little reflection the reader will see that by a modification of the theory of casts, or of the definition of a cross-ratio, it is possible to make 1R(PQRS) = s, instead of (s - i)/s; we have preferred not to make this modification for two reasons. The first is that to alter the theory of casts would make Staudt's own exposition more difficult to follow for a reader acquainted with the present work; while, on the other hand, by altering the definition of a cross-ratio, we should be departing from what is, on the whole, the most M.P.G. 0 210 PROJECTIVE GEOMETRY familiar and natural notation. The second reason is that if s is the value of any given cross-ratio, (s - I)/s is the value of an associated cross-ratio, namely, one of the set derived from s by the processes of Art. 211. The results of next Article will lend additional weight to this second consideration. 215. The Cross-ratio Sextic. We have seen that if A, B, C, D are any four points on a conic (or any four elements of a one-fold elementary form) their indices a, b, c, d give rise, in general, to six different cross-ratios, all expressible as rational (linear) functions of any one of them, according to Art. 211 (2), (5). Hence any symmetrical function of the six cross-ratios is a projective invariant for the tetrad ABCD; that is to say, its value is unaltered for any permutation of A, B, C and D, and is also the same for any of these permutations as for any permutations of A', B', C', D', provided that A'B'C'D' AABCD. The six coordinate values, if A is any one of them, are X, I/A, i-A, (X-I)/X, X/(X-I), I/(I-A); their simplest symmetrical combination is their sum, but this has the constant value 3, and is not the kind of function we require. On the other hand, the sum of their squares is an actual function of A, and we find that I + (I -)2 + (X I+)2 X2 (), h2 X(2 (X - I)2 (I - )2 where (I - )2X2.(X) = (26 - 6X5 + 9X4 - 8X3 + 9X2 - 6X +2) =2 (X2 - X + I)8 _ 3X2(I - X)2, so if (after H. Weber) we write h (A) _ 28(' ( ) ) - I '.................... we have ~(X) = 2-7i(X) - 3, and j(X), as well as +p(X), is a projective invariant. The function j(X), or any numerical multiple of it, is the funda CROSS-RATIOS 211 mental invariant in this connexion: the reader may verify, as an algebraic identity, that j:j- I728:I =28(1 - X+ X2)3:26(2 - X)2(I - 2X)2(I + A)2:X2(I - X)2,............................ (2) so that the equations j =o, j= I728, and j=oo give respectively the special values of X tabulated in Art. I90. Any prescribed value of j gives a sextic (X2- +)3 j A2(X- )2 - 256 *............ to find the corresponding six values of X; and if we can find any one of them, the five others can be found by the formulae of Art. I9o (iv.) or Art. 2II (5): and it does not matter which set of formulae we use, although the results have different interpretations. For this reason the discrepancy noted above between Staudt's theory of casts and the usual definition of cross-ratios is of very slight importance. The value of a cast is that of one of six cross-ratios derived from the cast, and since these are coordinate, the choice of a particular one is more or less arbitrary. It should be observed, however, that we define ABCD, metrically, to be a harmonic range when the ratios AC/CB and AD/DB are equal and opposite: that is, when 1 (ABCD) = - I. In this case the value of the cast ABCD is ~ (cf. Art. I90, end, and Art. 214 (I)). 216. The reader should particularly observe that we have only given a metrical definition of 1(PQRS) when P, Q, R, S are four collinear real points. On the other hand, if p, q, r, s are any four real or complex numbers, we have given a definition of 1 (pqrs), and if P, Q, R, S are any four real or complex elements of an elementary form, we have given a projective definition of iR(PQRS), because we have been able, in the last case, to associate P, Q, R, S with four real or complex indices p, q, r, s. 212 PROJECTIVE GEOMETRY 217. Equation of a Conic in the form xy - 2 = o. We have already seen that if a conic passes through the vertices A, B of the triangle of reference ABC, its equation is of the form (Art. 206) axy + bxz + cyz + dz2 =o. Supposing any conic given, take two points A, B upon it, and let the tangents at A, B meet in C. Let G be any third point on the conic, and take this as the unit point: then the reader can easily prove that in the homogeneous system of coordinates derived from (ABC:G) the equation of the conic reduces to xy-z2 =o..................................(I) For many reasons, this is the simplest and most convenient form of equation for any one conic. Taking (t, u) as a variable parameter in homogeneous form, we may put x:y: = t2: u2: tu,..........................(2) whence we have, in a non-homogeneous form, x:y: z=t2: I: t. We can speak of a point (t, u) or a point (t) on the conic, and we have the following fundamental results: (i) The chord joining (t,, u1) to (t2, u2) is l2X + t t2Y - (tlu2 + t2u1)z = o.................(3) (2) The tangent at (t, u) is 2x + t2y - 2t =...........................(4) (3) The tangential equation of the conic, that is, its equation in line-coordinates, is 4 - 2 =................................(5) Other useful formulae in this connexion will be found in the examples. CHAPTER XXIV. CENTRAL PROPERTIES OF CONICS. 218. If we define a conic as a plane section of a complete right circular cone, a little experimenting convinces us that so far as general appearance goes, there are three classes of conics to consider. If the cutting plane meets all the rays of the cone on the same side of the vertex, the section is a closed oval figure, more or less differing from a circle, as we alter the position of the plane. If, however, the cutting plane does not meet all the rays of the cone on the same side of the vertex, the section, so far as we can trace it, consists of what we naturally call two distinct " branches," each of which we can extend as far as we like. One of the most important inventions of the Greek geometers was to consider, explicitly or implicitly, the sections of a complete cone, so as to make the two branches of a hyperbola parts of one and the same curve. As a boundary case, the plane of the section may be parallel to a generator of the cone; in this case the intersection of the plane and that generator cannot be considered as being on either side of the vertex (cf. Art. I6). All other points on the section are on the same side of the vertex, and we have a single branch, extending to infinity in either direction, which we call a parabola. When we define a conic by means of projective pencils or rows, it presents itself as a closed figure in the same sense as a row or a pencil is a closed figure. The above-mentioned 214 PROJECTIVE GEOMETRY three cases may now be discriminated by the following definition: A real conic is said to be an ellipse, a hyperbola, or a parabola, according as it meets the line at infinity in its plane in two conjugate complex points, or in two distinct real points, or in two coincident real points. For any given real conic, one and only one of these cases must occur: so we have a definite distribution of real conics into three species. The critical reader may suggest that we ought to say " the real line at infinity "; as a matter of fact, this is not necessary, because, as we shall show in an excursus, the real line at infinity in a real plane may be considered to be conjoint with all points " at infinity" in that plane. A parabola may also be defined as a conic which touches the line at infinity in its plane. 219. Definition of Centre. The pole of the line at infinity (in its plane: this will be omitted in this context) with respect to any conic is said to be the CENTRE of the conic. Thus, the centre of a parabola is a definite point at infinity: the centre of a real ellipse or hyperbola is a definite real point that is not at infinity. Real ellipses and real hyperbolas are conveniently, if rather improperly, distinguished as central conics. In what immediately follows we shall assume the metrical theorem of Art. 36 (if the reader prefers he can take this as a projective definition of bisection). We specialise the theorems of Arts. II6, 117 by supposing that P or p, as the case may be, is an element at infinity. The result may be stated as follows: (I) Let KL, MN be two parallel chords of a conic: then the meets (KM, LN), (KN, LM) lie on a line which passes through C, the centre of the conic. This line is the polar of the point at infinity (KL, MN). (2) The mid-points of any set of parallel chords KK', LL', etc., lie on a line which is the polar of that point at infinity CENTRAL PROPERTIES OF CONICS 215 which is the centre of the parallel pencil (KK', LL',...). This line goes through the centre C. (3) If KK', LL' are any two parallel chords of the conic, the meet of the tangents at K, K' and also the meet of the tangents at L, L' lie on the polar of the meet of KK', LL'; and since this meet is at infinity, the polar goes through the centre C. Propositions (I), (2), (3) of Art. 117 may be similarly specialised. 220. Diameters. Conjugate Diameters. Asymptotes. Any chord of a conic which goes through C, the centre of the conic, is called a diameter. All the diameters of a parabola are parallel: for other conics the diameters form a proper pencil (C). We shall consider this more general case first. Two diameters d, d' are said to be conjugate, if, either, and therefore each, goes through the pole of the other: it follows by Arts. II6, II7, 36, that if two diameters are conjugate, the mid-points of chords which are parallel to either diameter lie on the other. In particular, the tangents parallel to d' are those at the extremities of the conjugate diameter d, and conversely. In the pencil (C) we have an involution in which conjugate diameters are corresponding rays (Art. 121). The double rays of this involution are the tangents from C to the conic: these are called the asymptotes of the conic, and are real, or complex and conjugate, according as the conic (here supposed to be real) is a hyperbola or an ellipse. In every case they are the double lines of the involution (dd'.ee') defined by any two pairs of conjugate diameters; and the asymptotes therefore separate any pair of conjugate diameters harmonically (Art. 9I). If we take as a triangle of reference (x = o, y = o, z = o), that for which x = o, y = o are the asymptotes, and z = o is the line at infinity, and if we also take the unit-point on the conic, the equation of the conic assumes the form (Art. 217) xy - z2 o.................................(I) 216 PROJECTIVE GEOMETRY When the asymptotes are real, this corresponds to the ordinary Cartesian equation xy = I for a hyperbola referred to its asympotes: but it is equally available for an ellipse. In the latter case, however, it is usually more convenient to write x + iy, x - iy for x, y, so that the equation becomes 2 +y.2 =2.............................. (2) From our present point of view, this is merely a change of notation; it may also be regarded as a change of the triangle of reference from (x =o,y=o,z=o) to (x+iy=o, x-iy=o, z=o). For a parabola the asymptotes coincide with the line at infinity. It is still true that the mid-points of a set of parallel chords lie on a diameter, but we no longer have a system of conjugate diameters. Instead of this, we have for each diameter a conjugate direction, namely, that of the chords which the diameter bisects. If we represent the line at infinity by z = o, and any other tangent to the parabola by x = o, then, taking as usual the unit-point to be on the curve, the equation of the parabola reduces to zx -y2 = o................................(3) where y =o is the diameter through the point of contact of x =o. This corresponds to the Cartesian equation x =y2 of a parabola referred to a tangent (x = o), and the diameter (y = o) through its point of contact. 221. Every determination of a conic from a sufficient set of projective data may be expressed in a special form when some of the data involve elements at infinity. For instance, it is a definite problem to construct a conic which passes through three given accessible points and has its asymptotes parallel to two given lines (either real or complex). This is only a special case of drawing a conic through five given points, two of which, in the particular case, are definite points at infinity. The Exercises at the end of the book give other illustrations of this remark. 222. Similar and Similarly Situated Conics. In general, two conics in the same plane cut the line at infinity in two CENTRAL PROPERTIES OF CONICS 217 distinct pairs of points (P, Q), (P', Q'). When these pairs coincide, so that we may put P' = P, and Q' =Q, we shall say that the conics are similar and similarly situated, or, more briefly, that they are homothetic. If, in addition, we have P=Q, the conics are both parabolas. Thus two parabolas are homothetic when any diameter of one is parallel to any (and therefore every) diameter of the other. Two central conics are homothetic when the asymptotes of one are parallel to the asymptotes of the other. The equations of any two homothetic central conics can be simultaneously reduced to the forms xy - (lx+my+nz)z=o. (I) xy - ('x +m'y + n')z=o, f where (1, m, n), (1', ', n') are any two sets of absolute constants, either of which, if we like, may be taken to be (I, I, I): this amounts to fixing the unit-line after fixing the triangle of reference: the system of coordinates is then determinate, because we can construct the unit-point from the unit-line (Art. 207). In connection with equations (i), the equations x=o, y =o may represent any two lines through P, Q respectively (each different from z =o): if these lines meet the conics again in A, A' and B, B' respectively, then Ix r+my + nz =o, and 'x + m'y + n'z = o represent the lines AB and A'B' respectively. The equations (I), as might be expected, are each equivalent to an equation such as (I) of Art. 206; in fact, we are merely dealing here with a particular case of finding the general equation of a conic through two given points. CHAPTER XXV. ORTHOGONAL PROPERTIES OF CONICS. 223. The central properties of conics, discussed in the previous chapter, are not strictly projective, at least if we use the term " bisection" in its ordinary metrical sense. There is another set of properties of a still more metrical character, which, for convenience, we may call orthogonal properties, because they depend on the theory of angles, and in particular on the properties of right angles. The object of the present chapter is to show, in a comparatively simple way, the connection of these properties of conics with the strictly projective ones. We shall assume as much as is necessary of the ordinary theory of angles, as it is found, for instance, in the Elements of Euclid, and the introductory chapters of works on trigonometry. 224. Congruentially Projective Pencils. Let S, A, B, C, D by any five points on a conic k, and let SA, SB, SC, SD be joined. We may suppose the conic rotated in its own plane about S so as to come into a new position SA'B'C'D': we assume that the new figure is equal in all respects (congruent) to the original one. In particular, LA'SB' = ASB, BSC =.B'SC', and so on. Now on the conic k we have a cast (ABCD)k and on the conic k' a cast (A'B'C'D')k'. We assume, as sufficiently obvious, that the values of these casts must be equal. Moreover (Art. I40), there is a definite projectivity between ORTHOGONAL PROPERTIES OF CONICS 219 the conics in which (A, A'), (B, B'), (C, C') are pairs of corresponding points, and our assumption that (A'B'C'D')k' = (ABCD)k leads to the conclusion that D' corresponds to D. Hence, (ABCD)k W (A'B'C'D')k, and consequently S(ABCD) AS(A'B'C'D'). That is to say, Two concentric fiat pencils S(ABCD...), S(A'B'C'D'...) are projective, if every angle such as ASD in the one is equal to, and of the same sense as, the corresponding angle A'SD' in the other. C D A B' FIG. 68. Consider, now, two flat pencils (S), (T) in perspective with the line at infinity. Any two corresponding rays are parallel, and hence any two corresponding angles are equal, and may be taken to be of the same sense. Combining these results, we see that, whether they are concentric or not, two flat pencils are projective when every angle in one is equal to, and of the same sense as, the corresponding angle in the other. By a suitable rotation, followed, if necessary, by a translation, we can make every ray of one pencil coincide with the corresponding ray of the other. Two such pencils may be called congruentially projective; for brevity, we may say that they are congruent, the projective relation being understood. 220 PROJECTIVE GEOMETRY 225. The Circle as a Conic. Assuming now the ordinary metrical properties of a circle, we are able to show that it is a conic according to our projective definition. Thus, let 8, T be any two fixed points on a circle, and let them be joined to other points A, B, C,... on the curve: then, in the ordinary sense of the words, any two angles such as ASB, ATB are either equal and of the same sense, or else (like ASD, ATD in the figure) supplementary and of opposite EDA B FIG. 69. sense. But by producing the chords through S and T the discrepancy disappears: for instance, in the figure, we have the angles ASD, A'TD equal and of the same sense. The fact is that if we suppose A to travel round the circle always in the same sense, the complete lines SA, TA turn round 8, T respectively at the same rate and in the same sense, and describe two congruent projective pencils. 226. Orthogonal Involution of Rays. Let O be the centre of a circle, S any fixed point on the circumference, AOA' a variable diameter. Then, because the circle is a conic, and O a fixed point, SA, SA' are conjugate rays of an involutionary pencil (S), by Art. I36. Now, by the metrical ORTHOGONAL PROPERTIES OF CONICS 221 properties of the circle, SA is perpendicular to SA': hence we deduce the very important theorem: A pencil of rays is put into involution by making each ray correspond to that one ray which is perpendicular to it. Such an involution is called an orthogonal involution (of rays). The reader should particularly notice here that from axioms of congruency and bisection alone we can obtain a definition of a right angle, and hence show that if in a pencil we make each ray correspond to that one which is perpendicular to it, we have a one-one correspondence W5 which satisfies the relation M2 = I; but there is nothing in this to show that the relation is a projective one. 227. The Focoids. An orthogonal involution of rays is obviously elliptical, because no real line can be at right angles to itself. Hence it defines a pair of complex lines through its centre, and since each of these (cf. Art. 173) has to be considered as a double ray, we arrive at the conclusion that through any real point of a plane can be drawn two conjugate complex lines each of which is perpendicular to itself. We shall call a line perpendicular to itself an isotropic line. All isotropic lines are complex. Every isotropic line passes through a fixed point at infinity. This is a most important theorem, the proof of which is now very easy. Consider two orthogonal involutions of rays of the same sense, with different real centres. The only way to put them into perspective is to make each ray of one correspond to the parallel ray of the other (otherwise they would generate a circle), and now, if a ray a of the one describes its pencil in either sense, the parallel ray a' of the other describes its pencil in the same sense, and the lines which are defined by the pencils meet the line at infinity in the same point (Art. I68, iv.). They cannot meet in any other point. According as we give to an orthogonal involution one or the other sense, it represents a complex line going through one or the other of two fixed complex points on the line at infinity. These points we shall call the focoids, and denote by J, Jo. 222 PROJECTIVE GEOMETRY Every circle goes through the focoids, and conversely. The first part of this proposition follows at once from Art. 226; because if we assign either sense to the orthogonal involution (S) the line which it determines meets the circle again in the corresponding point where the polar of 0 (the centre of the circle) meets the curve: that is, in one of the points where the line at infinity cuts the circle (cf. Art. I75). In virtue of this proposition, the points J, Jo are frequently called the circular points at infinity. The converse of the theorem follows from the fact that if a conic passes through J, J0 any one of its diameters must subtend a right angle at any point of its circumference. A conic which passes through one of the focoids, but not the other, is necessarily unreal: it might be called a hemicycle, and has some rather curious properties. The converse accounts for the facts that a real circle is determined by three real points, that two real circles cannot meet in more than two real points, and so on. 228. Principal Diameters. Let C be the centre of an ellipse or hyperbola; then the pairs of conjugate diameters form an involution of rays, and by Art. 176 this has, in general, one pair in common with the orthogonal involution whose centre is C. Thus: An ellipse or hyperbola has one, and only one, pair of conjugate diameters which cut at right angles. These are called its principal diameters. The only exception occurs when the ellipse becomes a circle, in which case every diameter is perpendicular to its conjugate. The case of the parabola is a little different. Here all the diameters are parallel and pass through a fixed point P at infinity. Each diameter d determines a set of parallel chords which it bisects, and these meet the line at infinity in the same point Q. As the line d alters its position, so does Q, and there will be one position of d for which P, Q are harmonic conjugates with respect to the focoids; that is to say: ORTHOGONAL PROPERTIES OF CONICS 223 A parabola has one and only one diameter which is perpendicular to the chords which it bisects. This is called the principal diameter. 229. Foci and Directrices. If we take any point A disjoint to a given conic, we have an involution of pairs of conjugate lines which meet at A (Art. I2I); and, as in last Article, we see that in general we have only one pair perpendicular to each other. But it may happen that every ray of the pencil (A) is perpendicular to its conjugate ray; this means (cf. Art. I74) that the tangents from A to the conic pass through the focoids. When this is the case, A is said to be a focus of the conic. In general, a conic has four foci; and when the conic is real, two of the foci are real and two are complex and conjugate. To prove this, let t, u be the tangents from J to the conic, and t', u' the tangents from JO. Then, in general, these four tangents form a quadrilateral, and its vertices tt', uu', tu', t'u are the foci in question. For a real conic, the tangents through J are respectively conjugate to the tangents through J0, and we may put t', u'=to, uo; we now have two real foci tto, uu0 and two conjugate complex foci tuo, tou. Each principal axis contains two foci. This follows at once from the fact that the principal axes are the two diagonals (tt', uu'), (tu', t'u) of the quadrilateral tt'uu'. Namely (cf. Fig. 32, putting J, J0 for F, C, and so on), these diagonals meet at the pole of JJ0, that is, the centre; they are conjugate with respect to the conic, and separate J, JO harmonically. The foci of a circle all coincide at its centre, which is accordingly a sort of quadruple focus. When the conic is a parabola, two sides of tt'uu' (say t, t') coincide with JJ0, and there is only one definite focus: this lies on the principal diameter. A hemicycle (Art. 227) has only two foci, each of which is a sort of double one. The polar of a focus is called a directrix. From Fig. 32 the reader will see that each directrix is perpendicular to the diameter which contains the corresponding focus. 224 PROJECTIVE GEOMETRY From our present point of view, the following is the main theorem on which the focal properties of conics depend: If the tangent at P to a conic meets any directrix s in Z, and if S is the corresponding focus, then the angle PSZ is a right angle. In fact, since Z lies on s as well as the tangent at P, the line SP is the polar of Z; therefore SP, SZ are conjugate lines, and hence at right angles, because S is a focus. Other focal properties of conics are given in the Examples. 230. Confocal Conics. With the same notation as before, consider the range of conics (Art. I6o) inscribed in the (proper) quadrilateral tut'u'. These all have the same foci, and are said to form a system of confocal conics. From Arts. I60-I6I we can draw the following conclusions: The tangent-pairs from any point P to a set of confocal conics form an involution, of which the double rays are the tangents at P to the two confocals that pass through P. Moreover, we can show that If two of a set of confocal conics meet, they cut orthogonally at each intersection. Let two of the confocal conics meet at P, and let p, p' be the tangents to them there. Then (Art. 9I) p, p' are harmonic conjugates with respect to every tangent-pair from P. Now (cf. Art. I60) the range of confocals contains three degenerate point-pairs, one of these being J, J0. The tangent-pair from P is now PJ, PJd, and since these are divided harmonically by p, p', the latter must be at right angles to each other. Since every point in the plane lies on two and only two conics of the system, we can pick out such a large number of the conics as to cut up the plane into small curvilinear rectangles, which ultimately differ very little from plane rectangles. It can be shown analytically that we can choose the conics so as to make these small rectangles approximately squares. CHAPTER XXVI. PROJECTIVITIES IN SPACE. 231. By projectivities in space we shall understand projective relations (including correlations) which affect all the elements of the elementary form of the third rank (Art. Io). Thus, in a certain sense, any two corresponding forms are cobasal, and we are dealing with a group of projective permutations (Art. 85). All the strictly projective correspondences will be collineations; the others will be correlations. Let A, B, C, D, G be any five points such that no four of them are coplanar, and let A', B', C', D', G' be another set of such points. Then there is a definite collineation (A'B'C'D'G'...) =zr(ABCDG...), in which (A, A'), (B, B'),... (G, G') are five pairs of corresponding points. Namely, we can make the bundles (A), (A') projective to each other by putting A(BCDG...) AA'(B'C'D'G'...), and at the same time make the bundles (B), (B') projective by putting B(ACDG..)7 B'(A'C'D'G'...); these projectivities are definite, and consistent with each other, because they both make A'B' correspond to AB (cf. Art. 73). If, now, P is any point disjoint to AB, the corresponding point P' is determined as the meet of the rays A'P', B'P' corresponding to AP, BP. Consequently, the correspondence between the bundles (C), (C') and (D), (D') is established, and if Q is any point on AB, the corresponding point Q' is the meet of C'Q', D'Q' the lines corresponding to CQ, DQ. M.P.G. P 226 PROJECTIVE GEOMETRY It ought, perhaps, to be noticed that if two rays AP, BP of the bundles (A), (B) are coplanar, the corresponding rays must be coplanar. The reason for this is that the correspondence set up between the bundles (A), (A') and between (B), (B') establishes, ipso facto, a correspondence between the axial pencils (AB), (A'B'), so that to every plane through AB corresponds a definite plane through A'B'. In fact, we have, in consequence of the construction, AB(CDG...) WA'B' (C'D'G'...). Another useful way of looking at the matter is this. The five points ABCDG determine a system of coordinates for which ABCD is the tetrahedron of reference and G the unit point. Similarly, we may take A'B'C'D' as a tetrahedron of reference, and G' as the unit point. If (x, y, z, t) are the coordinates of any point P in the first system, there is a definite point P' which has the same coordinates in the second system, and there is a one-one reversible correspondence between the points P and the points P'. From Art. 204 the reader will be able to see that this is precisely the projective correspondence established above. By joining AB, BC, CD, DG, GA we obtain what we may call a simple skew pentagon. We may refer to the above theorem as the pentad-theorem, and express it by saying that two simple skew pentagons determine a collinear projectivity. As many as four pairs of vertices of the pentagons may coincide. It can be shown algebraically that, in general, from two given corresponding pentagons we can find two corresponding pentagons which have four vertices in common. Let these be A, B, C, D, and the other points G, G'. Then, if we refer G' to the coordinate system (ABCD: G), its coordinates will be (a, b, c, d) where a, b, c, d are not all equal; and the result of the analysis is that if (x, y, z, t) are the coordinates of any point P, those of the corresponding point P' are (ax, by, cz, dt). This is, then, what we may call the canonical way of expressing, by algebra, the most general kind of collineation. It will be noticed that the PROJECTIVITIES IN SPACE 227 collineation involves three independent numerical constants, besides the twelve constants required to fix the four self-corresponding points. 232. Consequences of the Pentad-Theorem. By dualising the pentad-theorem, we infer that if a/3fyS are five planes, no four of which are concurrent, and a'/'y'6'xc' five other planes of the same character, there is a definite collineation (aIpYK...) X (a'P'y''K'...). Again, if ABCDG is a general pentad of points, and a/pyK a general pentad of planes, there is a definite correlation in which (A, a), (B, P),... (G, K) are corresponding pairs. Here, again, we shall write (ABCDG...) 7 (a3ySK....), in agreement with what we have previously done (cf. Arts. 79-8I). An important application of the pentad-theorem is contained in a theorem which Staudt enunciates as follows (B. 8): To set up a projective relation in space so that a given triad of lines a, b, c, each pair of which is skew, may correspond to another such triad a', b', c', we may choose any three directrices p, q, r of the regulus (abc...) and suppose them to correspond respectively to p', q', r', any three directrices of the regulus (a'b'c'...). The relation is now determined, except that we may make it either a collineation or a correlation. Suppose, for instance, we make the pentad of points ap, aq, bp, bq, or correspond respectively to the points a'p', a'q', b'p', b'q', c'r'; this fixes a collineation for which a, b, p, q obviously correspond to a', b', p', q'. Moreover, since c goes through or and meets p and q, its correspondent must go through c'r' and meet p' and q': that is, it must be c'. Similarly, r and r' must correspond. If, on the other hand, we make the point ap correspond to the plane a'p', and so on, we have a correlation in which (a, a'), (b, b'), etc., are pairs of corresponding lines, as before. The pentad-theorem enables us to prove that after choosing the additional six lines, the two projectivities above established are the only possible ones. 228 PROJECTIVE GEOMETRY 233. Homology. We have seen the importance of homology and involution in fields of two dimensions. We shall now discuss analogous relations in three dimensions. Given a point 8, a plane o- disjoint to it, and two points A, A' collinear with S, but each disjoint to o-; there is a definite projectivity in which A' corresponds to A, and every element of the plane field (o-) as well as every element of the bundle (S) corresponds to itself. We call this the homology (, o-; A, A'). Consider, for instance, any plane /i passing through SAA'; this cuts o in a line pr, which we may call b. In the plane,/ we can set up a definite homology (8, b; A, A'), and by doing this for every plane of the axial pencil (AA') we have a definite one-one correspondence of points, which is a projective one in every plane through AA'. Now, let (B, B'), (C, C') be any two pairs of corresponding points not all in the same plane with A, A'. Let the lines SAA', SBB', SCC' meet cr in P, Q, R respectively; then, from the construction, SPAA' '7 SQBB', and SPAA' A7 SRCC', whence SQBB'w7 SRCC', and from this it is easy to prove that the correspondence in the plane SBC is a homology. Thus in every plane through S there is a projective correspondence; and similarly every point on Cr is the centre of a bundle in which there is a projective correspondence of elements. For this reason we may regard the whole relation as being projective; but it must be remembered that here, as in most of what follows, the so-called projectivity cannot be established by a chain of projections and sections. If we treat the matter algebraically, however, we find a complete analogy, so that we are tempted to infer that if we could attain to an intuition of the right sort of four-dimensional space, we could actually realise a set of projections and sections which would produce such relations as the one here considered. (Since the theorem that in a plane homology (8, a; A, A') we have SPAA' A SQBB', where P, Q are the points where SA, SB meet a, does not seem to have been proved, we insert it here. The lines AB, A'B' meet a in the same point T; hence the rows SPAA', SQBB' are perspective PROJECTIVITIES IN SPACE 229 sections of the same pencil T(SPAA'), and this proves the theorem.) Since all the casts of which SPAA' is the type are projective to each other, they all have the same value; hence, also, the cross-ratio U(SPAA') is invariable. This gives another way of defining a homology. If SPAA' is a harmonic range (that is, if SAPA' is a harmonic cast) the relation is an involution (cf. Arts. 67, 96). After choosing the point S and the plane o-, the homology is determined, except for the numerical constant,(SPAA'). Thus a homology in three (or two) dimensions involves one intrinsic constant, which may be called its parameter. When the parameter is - I, the homology is an involution. It should be observed that either or both of the points A, A' may be complex, and the same thing applies to 8, cr. If, for instance, S, ra are real, and we take A, A' to be such that R(SPAA') = o, where w is a primitive root of o" = I, we have a correspondence?z which satisfies the equation an = I, and no similar equation of lower degree. The involutionary case corresponds to n = 2; what we may call the equianharmonic case corresponds to n =6, and the reader will find it a good exercise to work this out in detail. 234. Converse Theorems. (i) If in a planar projective collineation we have a row of self-corresponding points, then we have a pencil of self-corresponding rays, and conversely. Moreover, the projectivity is in this case a homology. Suppose the row of self-corresponding points is on the line a. There cannot be any other row of self-corresponding points, otherwise we could find a self-corresponding quadrangle, and the projectivity would reduce to identity (cf. Art. 71). Hence there are at least two pairs of distinct corresponding points (A, A'), (B, B'). If AB meets a in P, then A'B' must meet a in P, because P corresponds to itself. Now take on a two points H, K, such that HKAB, HKA'B' are quadrangles; then (Art. 7I) there is a definite projectivity (HKAB...) (HKA'B'...), and this is established by the 230 PROJECTIVE GEOMETRY homology (S, a; A, A'), where 8 is the meet of AA' and BB'. We now have a self-corresponding pencil (8). The converse can be proved by dualisation. (ii) If in a projective collineation in space we have a plane field (o) of self-corresponding elements, then we have a bundle of self-corresponding elements, and conversely. Moreover, the projectivity is in this case a homology. There must be at least two distinct pairs (A, A'), (B, B') disjoint to o-: otherwise we could find two self-corresponding general pentads, and the relation would reduce to identity (Art. 231). Now AB, A'B' are corresponding lines and must therefore meet cr in the same point P. Hence AA', BB' are coplanar, and meet in a point S. If we take in - three points H, K, L such that HKLAB, HKLA'B' are general pentads, there is a definite projectivity (HKLAB...) 7 (HKLA'B'...), and this can only be the homology (S, a-: A, A'). The converse theorem may be proved by dualisation. 235. Classification of Projective Collineations. Let (x, y, z, t), (x', y', z', t') be the coordinates of two points P, P' referred to the same coordinate system. Take sixteen arbitrary constants ai, and let p be an undetermined multiplier; then, if we put px' = allx + a1y + a, z + al4t py' =a21x + a22Y + a23Z + a24t................... (I) pz' = a31x + a32y + a3z + a34t pt' = a4x + a42y + a4 + a44t we have a point-to-point correspondence, which is one-toone and reversible, provided that the determinant l ail is different from zero; we shall suppose this to be the case. Equations (I), under this condition, represent a collineation; and the classification of collineations corresponds to that of this type of equation-system. The different cases are arrived at by a discussion of the self-corresponding points. If, in (I), we put x', y', z', t' =x, y, z, t and then eliminate the coordinates, we arrive at the so-called characteristic equation PROJECTIVITIES IN SPACE 231 a11 -p, a12, a,3, a14 =o..........(2) a21, a22 - P, a23, 24 31, a32, a33 -, a34 a41, a42, a43, a44 - In the most general case, this has four different roots; there are four distinct double points A, B, C, D which are the vertices of a tetrahedron, and we can prove algebraically that if we take ABCD as a tetrahedron of reference, the equations of correspondence reduce to the canonical form x':y': z': t'=ax: by: cz: dt, as previously stated. But when the characteristic equation has multiple roots, various other possibilities arise. For instance, in the case of a homology (S,; P, P') we can take as a tetrahedron of reference SBCD, where BCD is any triangle in ac; and we can show that the equations of correspondence are now of the form x':y: ' =xy:: x:y: z: t, where m is a constant different from i. The characteristic equation is now (p - I)3 (p - m) = o, with one simple, and one triple root. For a detailed discussion of plane collineations the reader is referred to Newson's paper " A New Theory of Collineations," etc. (Amer. Journ. of Math., vol. xxiv., p. o09). The number of types of collineation is considerable; even in plane collineations there are five distinct types. The equations for a planar collineation are, of course, obtained from (I) by omitting the last equation and the terms in t in the other equations. There is no particular difficulty in proving that, in the plane, every projective collineation can be represented by a linear algebraical correspondence, and conversely. To do the same thing in a similar way for three dimensions we should have to consider projections and sections in a field of four dimensions. However, if we start with any given projective collineation, we can always find for it an algebraic representation of the type (I); moreover, 232 PROJECTIVE GEOMETRY the algebraic correspondence preserves the values of casts and cross-ratios, so we are justified in considering all the algebraic correspondences represented by (I) as projective. They evidently form a group (of infinite order): two of them compound into a resultant according to the matrix rule (Art. 200), thus, if we have two correspondences, x':y': z': t'= allx + a12y az+ alaz + a4t:. and x" y": z": t" = blx' + b12y' + bz' + b4t':.. they compound into X": ff: Z: t = llx + C12Y + C3Z + C14t: ~ ~, where ql = blall + b1a2a2 + b3aa + b14a41, and so on; that is, putting (a), (b), (c) for the matrices of the three substitutions, we have (c) = (b) (a). An interesting transformation in space is that in which we make every real element correspond to itself, and every complex element correspond to its conjugate. This satisfies the geometrical definition of a collineation, but is not in any sense of the term a projective transformation; for instance, a cast with a complex value transforms into one with the conjugate complex value. By compounding this operation with the group of projective collineations, we obtain a new group of correspondences. 236. Correlations. Polarities. We can establish a correlation in space by taking five points A, B, C, D, G, no four of which are coplanar, and assuming as their corresponding elements any five planes a, f/, y, 8, K, no four of which are concurrent. By making the four rays AB, AC, AD, AG in the bundle (A) correspond to the lines a/?, ay, aS, aK in the plane field (a) we establish a projectivity between them (Arts. 80, 8i). Similarly, we can set up a projectivity between (B) and () ): this is consistent with the other one, since in each case AB corresponds to af3. The whole correlation is now determined. PROJECTIVITIES IN SPACE 233 We obtain the analytical equations for a correlation from those given in Art. 235 for a collineation by changing one set of point-coordinates into plane-coordinates. Thus we may take either of the sets P/ = alx1 + ai2x2 + a3x3 + ai4x,.................(I) OXi= a 1ill + aii22 + ai3$3 + ai444....................(2) (i =I, 2, 3, 4), where for convenience we have denoted the coordinates by suffixed letters. In order that (i) and (2) may represent the same correlation, certain conditions have to be satisfied. Let us suppose that our coordinate system is normal, so that ixi=o is the condition for a point and plane being conjoint. Then, since conjoint elements must correspond to conjoint elements, we must have 1Ei'xi' =-Iixi identically, where f/ is some constant. We find from this that aij is proportional to a certain first minor of the determinant l aij, namely, that which is the coefficient of aji. If, on the other hand, we solve equation (I) for x, y, z, i, we obtain a set of equations MXi = Pill' + Pi22' + Ai303' + Pi44.',................ (3) where fiji is proportional to the coefficient of ai in laij. If in (3), we accent the xi, and write $i for ri', we clearly have a correlation, the inverse of (i) or (2). If the original correlation is an involution, it is identical with its inverse; we conclude from this that (after adjusting the multipliers a/, ro) we must have /3i=aij, and hence that aj=aij; that is to say, the coefficient-matrix (a) must be symmetrical. We thus have the important theorem that In order that a correlation may be an involution, it is necessary and sufficient that, in the linear equations which define it, the coefficient-matrix of the system be symmetrical. An involutionary correlation is called a polarity. The conditions aj =aij reduce the possible number of distinct coefficients from sixteen to ten, and in order to avoid 234 PROJECTIVE GEOMETRY suffixes we may (after Salmon) write the typical equations of a polarity in the form p$' =ax + hy +gz + t, pr'l = hx + by +fz +mt, I M pC'=gx+fy+cz+nt, (4) pr' = x + my + nz + t, The nucleus of a polarity is the locus of points conjoint with their polar planes (cf. Art. I78). Assuming (as we shall always do, unless the contrary is stated) that the coordinate system is normal, we deduce from (4) that the nucleus of that polarity is the surface of the second order ax2 + 2hy + by2+.. + dt2= o.................. (5) This is the most general form of a homogeneous quadratic equation in four variables; so every surface of the second order may be obtained as the nucleus of a polarity. Conversely, every given surface of the second order is the nucleus of a definite and unique polarity, as we see from (4) and (5). It should be observed, however, that even if the polarity has real coefficients, its nucleus may contain no real elements at all; for instance, if we put $': r': (': r' = x:y z: t, the nucleus x2 +y2 + 2 +t2 = o consists entirely of complex elements. 237. Degenerate Cases. There is one important restriction to the foregoing statement. In order that the relation (4) may be reversible, the determinant a hg h bf m gf c n I mI n d which we shall denote by A, must be different from zero. This quantity A is called the discriminant of the corresponding nucleus. Now we may have a surface of the second order for which A =o; in fact, this is the case when the surface is a cone or a plane-pair. In this case we can form the system (4) and obtain a definite point-to-plane corre PROJECTIVITIES IN SPACE 235 spondence-except for the vertex of the cone, or a point on the meet of the pair of planes-but this is not reversible, and we have a relation, which is a sort of degenerate polarity. If we confine the term polarity to reversible involutions, we must say that to every polarity corresponds a definite nucleus, which is a surface of the second order, with a non-vanishing discriminant; while conversely every such surface determines a polar field of which it is the nucleus. 238. Let ABCD be a tetrahedron; P a point disjoint to all its faces, and 7r a plane disjoint to all its vertices. Then there is a definite polarity for which each vertex of the tetrahedron is the pole of the opposite face, and P is the pole of r. This may be proved geometrically in a way analogous to that of Art. 152. In fact, if we denote the planes BCD, CDA, DAB, ABC by a, /, y, 6, there is a definite correlation (a//87r...) W (ABCDP..), and a little consideration shows that this must be an involution. However, for the sake of variety, we give an algebraical proof. Take ABCD as a tetrahedron of reference: then the four correspondences (A, a),... (D, 6) show that the equations (I) of Art. 236 must reduce to ': ': f': ' = ax: by: cz: dt.......................(i) Now, let (1, m, n, r) be the coordinates of P, and (X, /, v, p) those of 7: then the correspondence (P, 7r) determines the ratios a: b: c: d, and we have, in fact, lI ': mr': n ': rr'= Xx: py: v: pvt,....................(2) expressing the required polarity. From this proof we can at once derive several important corollaries. (i) Every correlation in which there is a tetrahedron of which each vertex corresponds to the opposite face is a polarity. (ii) In connection with each such tetrahedron we have oo3 polarities; because each is defined by four homogeneous constants (a, b, c, d). 236 PROJECTIVE GEOMETRY (iii) Taking such a tetrahedron as a tetrahedron of reference, the system of polarities leads to a set of nuclear surfaces, represented by ax2 +by +cz2 +dt2 o........................(3) where a, b, c, d are arbitrary constants. 239. Polar Tetrahedra. Such a tetrahedron as that considered in last Article is called a self-conjugate, self-polar, or simply polar tetrahedron. We proceed to show that every polar field in space contains an unlimited number of polar tetrahedra. We shall first prove some preliminary theorems, -mostly by algebra, as being the method most easy to follow. (i) If Q lies on rt, the polar of P, then P lies on K, the polar of Q. Let (x, y, z, t) be the coordinates of P, and (x', y', z', t') the coordinates of Q. The coefficients of the polarity being (a, b, c,....), as in Art. 236 (4), the condition that Q lies on r is (ax +hy +gz+lt)x' +( )y' +( )z' + )t' =o;......... (I) but this may also be written in the form (ax' +hy'+gz' +lt')x + ( )y +( )z + ( )t = o, which shows that P lies on K. The reader should notice how the symmetry of the coefficient-matrix comes in here; but for it, the argument would fail. Two points, each of which lies on the polar of the other, are said to be conjugate. Conjugacy is a reciprocal relation. (ii) If P, Q are any two points, and rr, K their polar planes, then the polar of any point on PQ passes through 7rK, and conversely. This follows immediately from the last. (iii) If a variable point X describes the row (PQ), its polar $ describes the axial pencil (7rK), and the relation $= 5(X) is a projective one. This follows from (ii) and the projective nature of the correspondence. We can now extend the notion of conjugacy by saying that two planes are conjugate when each goes through the PROJECTIVITIES IN SPACE 237 pole of the other, and two lines are conjugate when the polars of any two points on the one meet in the other. A point or plane has an unlimited number of conjugate elements, but a line, in general, has only one conjugate. Take a point P disjoint to the nucleus, and let wr be its polar plane. Take any point Q in 7, and let g be the line conjugate to PQ. This is in 7r, being, in fact, 7rK, where K is the polar of Q; and if we take Q disjoint to the section of the nucleus by ar, g will be disjoint to Q. Take on g any point R: then there will be on g a definite point S, conjugate to R, namely, gp, where p is the polar of R. It is now easy to show that PQRS is a polar tetrahedron. Since the choice of P involves the fixing of three absolute constants, we may say that P may be chosen in oo 3 different ways. After fixing P, the point Q may, in the same sense, be chosen in 00 2 different ways: finally, after choosing P, Q there are oo ways of choosing R, and this fixes S. So in a convenient notation we may say that in a (proper) polar system there are oo06 polar tetrahedra. The process given leads to all of these the same finite number of times; this does not affect the conclusion if we regard moo 6 as equivalent to oo 6, when m is a finite integer. Propositions about relative polar tetrahedra may be proved in a manner analogous to that employed for plane polarities (Art. I57). The discussion of them is left to the reader as an exercise. 240. Theory of Poles and Polars with respect to a Surface of the Second Order. Let u be any line, in a polar field, which is skew to its conjugate line u'. Then, if P is any point on u, its polar 7r cuts u in a conjugate point P', and as in Art. 121, we can show that the relation P' = (P) is an involution. This involution has two real or complex foci which divide every pair such as (P, P') harmonically (Arts. 9I, I73). These foci can only be the points where u meets the nucleus; for it follows from Art. 239 (i) that a self-conjugate point is a point on the nucleus. Hence 238 PROJECTIVE GEOMETRY If P is any point in a polar field disjoint to the nucleus, and u is any line through P, then, in general, u meets the nucleus in two real or complex points, which divide harmonically every pair of conjugate points on. Hence also the polar plane of P may be defined as the locus of points harmonically separated from P by the nucleus. On the basis of this theorem we can construct a theory of poles and polars precisely analogous to that previously given for plane conics (Chapter XV.); some of the main propositions will be given presently. 241. Tangential Equation of the Nucleus. Keeping the same notation as before, let a polarity be defined by the equations p~' = ax + hy +gz + It, etc.; then, since the polarity is an involution, it may also be expressed in the form o-x' = Ad + Hy + GC + Lr (Ty' = H + B- + FC+ + MT o-z' = G +F + CC +N........ o-t' = L M + M + N + Dr J where A is the coefficient of a in A, and so for the rest (cf. Art. 236). The coefficients A, B, etc., are usually called the reciprocal coefficients corresponding to a, b, etc. By the theory of determinants, we infer that the determinant of the reciprocal coefficients is equal to A3. If the plane ($, -I, f, r) is conjoint with its pole, we infer that (A, B, C..., C, T)2=o,~................... (2) an equation corresponding to (5), Art. 236, with a change from point to plane coordinates, and of coefficients into the reciprocal coefficients. Now each equation expresses the same fact, namely, that a point and its polar are conjoint. Hence (2) above is the tangential equation of the envelope of planes conjoint with their poles. We define the polar plane of a point on the nucleus to be the tangent-plane at that point; therefore PROJECTIVITIES IN SPACE 239 The envelope to a surface locus of the second order is a surface-envelope of the second class, and conversely. This is the analogue of Art. 113 for conics, and on the strength of it we are able to dualise every theorem about a surface of the second order. 242. Plane Sections of a Quadric. Let N be the nucleus of a given polarity, and o- any plane which is not a tangent to N; the section of the nucleus by a- is a conic. Let S be the pole of a-; then (Art. 239) we can take a triangle TUV in a such that STUV is a polar tetrahedron. Let g, any line through 8, meet a- in G, and let the polar plane of G meet a- in g'. There is no difficulty in proving that the correlation (G, g') in a- is a polarity, of which TUV is a polar triangle: the nucleus of this is a conic, which can only be the section of N by -. Correlatively, the tangent-planes to N from S envelope a quadric cone, and the section of this by cr consists of the tangent-lines to the conic aN. These results may be proved very easily by algebra, if we suppose, as we may do, that the plane cr coincides with the plane t=o. For by putting t=o in the equation of N, we obtain (a, b, c, f, g, h x, y, z)2 =o, which, in the plane t = o, represents a conic. Consider, next, the case when a- is a tangent-plane to N; and let S be its point of contact. Take P, any other point in a; then, in general, its polar plane wr will meet cr in a line rr- distinct from SP and conjugate to it. If Q is any point on ro-, its polar plane will go through SP. Hence in C we have an involution of conjugate lines; the double lines of this involution are the intersection of a- with N. Suppose, for instance, we take a point R on either of these lines; then since RS is conjugate to itself, the polar plane of R must go through RS (Art. 239), and therefore contains R. Thus R is conjoint with its polar plane, and therefore on the nucleus. It is easily proved that no other points of cr are on the nucleus; hence 240 PROJECTIVE GEOMETRY The section of a quadric surface by a tangent-plane consists of a pair of lines meeting at the point of contact. These lines may be either real or complex, according to circumstances. Suppose that N is a real surface, and that one of its tangent-planes cuts it in a real line-pair (p, p'). Then every real plane through p cuts N in another real line q'; hence the axial pencil (p) cuts N in a regulus of real lines q'. Similarly, the axial pencil (p') cuts N in a regulus of real lines q, of which p is one. The reality of p involves the reality of p': hence we infer that if a real conicoid has one real generator, it has two chains of real generators. Clearly, if N meets the plane at infinity in a complex conic, all its generators are complex. We can hardly draw any further conclusions from a purely projective point of view. As a matter of fact, a hyperboloid may or may not have real generators; a paraboloid has real or complex generators according as it is hyperbolic or elliptic. In every case, however, we can say that a real quadric surface (with A 'o) contains two conjugate reguli, either real or complex. Analysis enables us to generalise this theorem for any conicoid whether real or complex. The tangent-cone to N from a point P upon it degenerates into the two axial pencils, whose axes are the generators through P; or, from another point of view, into the tangentplane at P counted twice over. Analytically, we obtain one or the other result according as we start with a planecoordinate or a point-coordinate equation to represent N. The reader should note this, because similar cases of different degenerations constantly occur in analytical geometry, and they are not usually emphasised in the text-books. Similarly, the section of N by a tangent-plane may be regarded either as two rows of points, or its point of contact, taken twice over. 243. General Equation of a Conic. We have repeatedly assumed that the general equation of a conic is the general homogeneous quadratic equation in three homogeneous variables. It is easy to justify this now: for, if, in the PROJECTIVITIES IN SPACE 241 analytical formulae of the present chapter, we omit one of the coordinates (say t, and in like manner r) we have an algebraical image of all the planar geometrical theory. We can clearly generalise the analytical theory so as to apply to a system of n variables (xl x2,... x); we thus have an image of what we may call " projective geometry in n dimensions." For different values of n we have various interesting special (so-called " degenerate ") cases. Is it too much to hope that we may gradually attain to a really geometrical intuition of these higher systems? M,P G, Q CHAPTER XXVII. QUADRIC SURFACES. 244. For convenience we shall use the term quadric surface, or simply quadric, to denote the geometrical figure represented, in either point or plane coordinates, by a homogeneous quadratic equation in four variables. A proper quadric defines, and is the nucleus of, a proper polarity in space; for distinction, we may call a proper quadric a conicoid. Thus cones of the second order, cylinders of the second order, and plane-pairs, are quadrics, but not conicoids. We have just seen that every conicoid contains two sets of rectilineal generators; each element of either set meeting all the elements of the other. Hence the results we have obtained for quadrics with real generators are equally true for all conicoids, when we include complex elements of space. 245. Projective Generation of a Quadric by two Bundles. The projective generation of a conicoid by two axial pencils is always possible, but is not easy to realise when the axes are complex. For this, and other reasons, we give here another construction, which is of great importance in the development of the theory. Let (A), (B) be two bundles (with different centres) connected by a correlation, so that if r is any ray of (A) there is a corresponding plane p in (B). Then the locus of the point rp is a surface of the second order, QUADRIC SURFACES 243 Take a tetrahedron of reference ABCD, of which A, B are the vertices (I, o, o, o), (o, I, o, o). Any plane through B will have coordinates of the form ($, o, C, r); let this meet the corresponding ray of (A) in the point P, with coordinates (x, y, z, t). The direction of the ray is fixed by the ratios y: z: t, and the correlation is represented algebraically by a set of equations::: -r=ay+bz+ct: a'y+b'z+c't: a"y +b"z +c"t. The condition that P may be conjoint with (4, o, (, r) is x + Cz + if =o; hence, in virtue of the correlation, the locus of P is (ay + bz + ct)x + (a'y + b'z + c't)z + (a"y + b"z + c"t)t =o; or S = b'z2 +c"t2 +a'yz + bzx +axy + cxt +a"yt + (b" +c')zt =o. (I) The equation 4 = o is the most general form of a quadratic in which the terms in x2, y2 are missing; hence it is the most general equation of a quadric passing through A, B. Moreover, if we start with # =o as given, it determines the correlation between (A), (B) except for the coefficients b", c', of which only the sum is given; in other words, if m is a variable parameter, the oo correlations 4:: r=ay +bz +ct: a'y +b'z+ (c' +m)t: a"y + (b" -m)z +c"t all generate the same quadric / =o. We may also use a purely geometrical method analogous to that employed for conics in Chapter XIII. Thus, starting with the correlation between (A) and (B) we may investigate the properties of the locus of P. The section of it by every plane through AB is obviously a conic (or exceptionally a line-pair); but it is a little difficult to show that the points A, B are not exceptional points on the locus, and that every plane section of the surface is a curve of the second order. For these reasons we have preferred to give an algebraic proof; the geometrical theory, from this point of view, will be found in Reye's Geometric der Lage, Part II., Lecture 5. 244 PROJECTIVE GEOMETRY 246. General Equation of a Plane. Since the equation s=o is the most general quadric through A, B, we will enquire under what circumstances part of the locus may be a plane through AB. Since the unit point is arbitrary, there is no loss of generality in supposing that this plane is z + t = o. If this is part of the locus, then (z + t) must be a factor of <, and the conditions for this are a=o, b=c, a'=a", b'+c"=b"+c', in which case we have ( = (z + t) (b'z + c"t + bx a'y) identically. Thus the remaining part of the locus is represented by bx + a'y + b'z + c"t = o, and from the nature of the case this must represent a plane. Thus we have a projective verification, indeed a proof, of the fact that the general linear equation in point coordinates represents a plane (or more strictly a plane field of points). We have so often assumed that a linear equation always represents a plane, that the reader may suspect a vicious circle in the argument; but this really does not exist. From the theory of casts we can prove that any linear equation not involving all the coordinates represents a plane, which is either a face of the tetrahedron of reference, or contains one of its edges, or is conjoint with one of its vertices; and correlatively for plane coordinates. The correlation used in Art. 245 only involves two triads of coordinates ($,, r), (y, z, t), so the condition of conjunction, $x + Cz + rt = o, may be legitimately applied, without assuming the theorem about the general plane. Finally, we can show, geometrically, that there is a limiting case of the correlation, in which AB corresponds to a plane through BA, one plane through AB is part of the locus, and the remaining part of the locus is a plane. One thing, however, should be noted. When c breaks up into two factors, as above, the equations connecting (C, T, r) with (y, z, t) assume the form (with a slight change of notation) QUADRIC SURFACES 245 $:: = bz +bt: a'y +(k+l)z + (k + )t: a'y + (k -m)z + (k - )t; and this is not a reversible correspondence, because o, b, b a', k+l, k+m =o, a', k-m, k-I identically. We may regard it, however, as a limiting case of a correspondence that is reversible. The student is advised to work out the proofs of the theorems correlative to those of the present and the preceding Article. 247. Central Properties. The polar plane of a point P with respect to a quadric is the locus of points harmonically separated from P by the surface (Art. 240); and we have a correlative definition of the pole of a given plane. If we take a conicoid which does not touch the plane at infinity, this plane will have a definite pole 0, not at infinity, which we call the centre of the conic. The following theorems now follow without any difficulty (cf. Arts. 2I9, 220). (i) The centre bisects every chord that passes through it. These chords are called diameters. (ii) If any set of parallel chords are drawn, their midpoints all lie on a diametral plane, which is, in fact, the polar of the point at infinity where the chords meet. The diametral plane is said to be conjugate to the diameter which is parallel to the bisected chords. (iii) By Art. 239 we can find a triangle PQR at infinity such that OPQR is a polar tetrahedron. The intercepts AOA', BOB', COC' made on OP, OQ, OR by the surface are called a set of conjugate diameters. Each is conjugate to the plane of the other two. There are,o 3 sets of conjugate diameters. (iv) Referred to any polar tetrahedron, the equation of a conicoid must be of the form ax2 + by2 + c2 dt2 = o, 246 PROJECTIVE GEOMETRY and by supposing t=o to be the plane at infinity, we have an equation analogous to the ordinary standard equation found by Cartesian methods. We may call this a canonical equation of the conicoid (whatever polar tetrahedron we choose). On the other hand, the conicoid may touch the plane at infinity, and so be a paraboloid. It will still be true that the locus of mid-points of a set of parallel finite chords will be a (conjugate) plane, but the planes we obtain by varying the directions of the chords all pass through the same point at infinity, namely, the point where the plane at infinity touches the surface. So in this case there is no centre, properly so called, at least from a metrical point of view. The centre of a cone coincides with its vertex. We have here, also, 0o3 sets of conjugate axes through the vertex, and canonical equations of the form ax2 + by2 + c2 = o. More generally, any homogeneous equation in three of the coordinates is that of a cone, with its vertex at one apex of the tetrahedron of reference. For instance, take (a,b,c,f,g, hx,y, z)2=o; then, if (x, y, z, t) is a point on it, so is (x, y, z, mt), where m is any constant; that is to say, the surface contains the line joining (x, y, z, t) to (o,, o, I), or D, say. Hence the surface is a cone, vertex D. The central properties of cylinders and plane-pairs are left for the reader to investigate. 248. Orthogonal Properties. For the sake of comparison with Cartesian methods, we give a brief outline of the orthogonal properties of quadrics, without giving detailed proofs of the propositions. (i) By assuming the metrical properties of a sphere we can prove that it is a quadric surface according to our projective definition. If we carry out the general theory far enough, it is sufficient to combine the facts that every QUADRIC SURFACES 247 plane section of a sphere is a circle, and that a circle is a conic. (ii) By the metrical theory, every diameter of a sphere is perpendicular to its conjugate diametral plane. Hence we infer the main proposition of this theory: Two concentric bundles are connected by an involutionary correlation, when each ray of either is made to correspond to that plane of the other which is perpendicular to it. We may, if we like, regard this as a polarity among the elements of one and the same bundle; and from this point of view we may call it an orthogonal (polar) bundle. (iii) Any two distinct orthogonal bundles can be put into perspective with the same polar field at infinity. All we have to do is to make parallel rays correspond. This polar field is independent of the particular pair of bundles we choose in order to define it. (iv) The nucleus of this polar field at infinity is a conic, which we shall call the focoidal circle, and denote by 'P. It is usually called the imaginary circle at infinity. (v) Every plane distinct from the plane at infinity cuts the focoidal circle in the focoids of the plane. (vi) Two lines are perpendicular when their points at infinity are conjugate with respect to the focoidal circle; and similarly for two planes. A line is perpendicular to a plane when the line at infinity in the plane is the polar, with respect to 'r, of the point at infinity in the line. 249. Principal Axes of a Central Conicoid. A conicoid that is not a paraboloid meets the plane at infinity in a proper conic X. In general, X meets the focoidal circle in four distinct points P, Q, R, S (this is most easily proved by algebra). Complete the quadrangle PQRS, and let X, Y, Z be its diagonal points. Then (cf. Fig. 32 and Art. II9) XYZ is a polar triangle both for X and for 'P; therefore, if 0 is the centre of the conicoid, the lines OX, OY, OZ form a set of conjugate axes, each pair of which are perpendicular. Since X and P are the nuclei of two distinct polar fields 248 PROJECTIVE GEOMETRY they cannot have more than one common polar triangle (Art. I54). We infer that A central conicoid has, in general, one and only one set of conjugate diameters, each pair of which cut at right angles. These are called the principal diameters, and their conjugate planes the principal diametral planes. Each principal plane is a plane of symmetry for the surface. It can be proved algebraically that if the coefficients in the equation of the quadric (referred to a real tetrahedron) are all real, then the principal axes are all real. In particular, a real central conicoid always has three real principal axes. 250. Circular Sections. With the same notation as in last Article, consider the (parallel) axial pencil (PQ). Since P, Q are the focoids of every plane which contains them, we see that the sections of the conicoid by the planes of the pencil consist of a set of circles in parallel planes. Each side of the complete quadrangle PQRS gives rise to such a set; therefore A central conicoid has, in general, six sets of parallel circular sections. For a real conicoid, the four points P, Q, R, S fall into two conjugate pairs, say P, Po and Q, Q0: there are consequently only two real sets of circular sections corresponding to the real pencils (PPo) and (QQ0). The reader will easily see from the figure that the real planes OPP0, OQQo are harmonically separated by two of the three principal planes; this corresponds to the fact that the real circular sections through the centre are equally inclined to two of the principal planes. 251. Special Cases. Surfaces of Revolution. Coincidences among the points P, Q, R, S lead to various special cases. Of these we shall only discuss the one which is important for real conicoids. We now have two conjugate pairs P, P0 and Q, Q0, so the only possible case of coincidence is P =Q, whence also P0=Q0. That is to say, the section of the conicoid by the plane at infinity has double contact with QUIADRIC SURFACES 249 'he focoidal circle, and the points of contact are conjugate womplex points. Let Z be the pole of PPO with respect to the touching conics; then O being the centre of the surface, OZ is a principal axis, being perpendicular to the conjugate plane OPPo. But the section of the surface by this plane is a circle, because P, Po are its focoids; and now any two perpendicular diameters of this circle form with OZ a set of principal axes. The conicoid is in this case called a surface of revolution, because it can be generated by revolving a conic about the axis OZ. The reader is left to work out for himself the analogous orthogonal properties of paraboloids. What they are is stated in current treatises on analytical geometry. 252. Foci. Let p be any tangent to the focoidal circle. Then if p' is the line conjugate to it with respect to any given conicoid /, the planes joining p to the meets of p' and P will be the two tangent-planes T, T' which can be drawn to /3 through p. As p's point of contact travels along the focoidal circle, the planes r, T' will envelope one and the same algebraical surface circumscribed to the conicoid. By analysis it can be proved that this surface is a developable; it will consequently have a cuspidal edge. In the present case this cuspidal edge consists of three conics, one in each principal plane; these are called the focal conics of the conicoid, and every point on them is called a focus. Another way of looking at it is this. For simplicity, suppose that /3 is a real surface. Take any point P and let 7r be the tangent cone from P to /; then P is said to be a focus of P, if r has double contact with the focoidal circle. This imposes two conditions upon the coordinates of P, and therefore we may expect a curve locus of foci. It is not very difficult to see that this second way of looking at the matter is equivalent to the first. We conclude that the real foci of a real conicoid are vertices of cones of revolution which are tangent-cones to the surface. 250 PROJECTIVE GEOMETRY 'With the notation of Art. 249 we can proceed a little further. Taking F, ~ as a point and the tangent-cone from it to p, the intersection of C with the plane at infinity will be a conic pa, which has double contact with X, the conic in which f is cut by the plane at infinity. If F is a focus, /t must also have double contact with 'P. If we take XYZ as a triangle of reference, the equations of I, X respectively may be taken to be x2 +yy +2 =0,............................ ax2 + by2 + - z2 = o,............................(2) with a, b, c all unequal, since we suppose P, Q, R, S to be distinct. Now if p/=o is a conic having double contact with (I) and (2), we must have identically MI = p (x2 +y2 z2) - (lx + my + nz)2 = p'(a x2 + cz2) - ('x + m'y + n'z)2. The reader will find that if we suppose lmn ` o, this leads to an inconsistent set of equations. Suppose I=o; then we must have mn=m'n', p=p'a, p-m2=p'b- '2, p - n2 = 'c-n'2 four equations in six homogeneous variables which have a singly infinite number of solutions. Now to say that = o means that the polar plane of F goes through X: hence F is in the principal plane OYZ: and we conclude that the foci consist of three curves, one in each of the three principal planes. We shall not go into further detail, because a proper discussion essentially involves the theory of higher curves and surfaces. 253. Confocal Surfaces. Assuming the theorem that a conicoid and the focoidal circle determine a developable circumscribed to the conicoid, all conicoids inscribed in this developable will have the same foci, and form what is called a confocal system. From the geometry we can conclude that any two surfaces of the system cut everywhere at right angles: the analytical theory leads to the conclusion that through every real point pass three conicoids of a real confocal system, and that by appropriately choosing a QUADRIC SURFACES 251 (partial) set of confocal quadrics we can cut up the whole of space into curvilinear rectangular solids, which are ultimately cubes (cf. Art. 230). 254. Alternative Method. Reye obtains the focal conics of a polar system by other considerations; and although his method cannot be generalised for surfaces of higher order, it is so elegant that we give a short account of it here. Let P be any point and 7 its polar plane. Then through P we can draw one and only one line p perpendicular to 7r; for the plane 7r, in general, contains two focoids J, JO, and now if P' is the pole of JJ0 with respect to the focoidal circle, PP' is the line p in question. This construction holds even in the limiting case when J and JO coincide. Thus for every plane 7r in space we have a corresponding line p which we may call its normal conjugate (in the given polar field). Now consider the section of all these pairs (?r, p) by one of the three principal planes; say, the one called OXY in Art. 249. We have line-point pairs (q, Q), which give a line to point correspondence in the plane OXY, and we can prove that this is a plane polarity. Consider, for instance, a particular pair (q, Q). To the axial pencil of planes (q) will correspond a row of points, which are their poles, on the conjugate line q'; and by joining them to Q, we have a flat pencil (Q) projective to (q). Since q is in a principal plane, it is perpendicular to q', and hence the pencil (Q) is in the plane Qq' drawn through Q perpendicular to q. Consider, next, the normal conjugates of the planes through q. From the general construction it follows that they form a flat pencil; in the present case this is (Q). In fact, besides the normal conjugates p and wr with which we started, the perpendicular from Q to the line q and the plane OXY belong to the set of normal conjugates. Hence also (cf. Art. 88) the normal conjugate of every plane through q is the perpendicular to it from Q. Consequently, in the plane OXY the correspondence (Q, q) is one-one and reversible (unlike the correspondence (7r, p) in space). 252 PROJECTIVE GEOMETRY There is no difficulty now in proving that this correspondence is a plane polarity: we define the nucleus of this polarity to be a focal conic of the original polar field. In this way we arrive independently at each of the focal conics, and all the usual focal properties of conicoids can be deduced by pure geometry. CHAPTER XXVIII. NULL-SYSTEMS. 255. There is an important type of correlation in which every point is conjoint with its associate plane. The reader will easily verify that the most general equations defining a relation of this kind are of the form p4 =. ay - bz +ct pr = - ax. + dz + et pC= bx -dy. +ft......... p = - cx - ey -fz. The determinant of the coefficient-matrix has the value (af +be + cd)2, so, in order to make the correspondence reversible, we assume that af +be + cd is not zero. Subject to this condition, equations (I) are said to establish a nullsystem (a, b, c, d, e, f). There are oo 5 null-systems in space. Considered as a whole, a null-system is correlative to itself; so, by solving equations (I) for x, y, z, t, we must arrive at a similar set of equations with crx, cry, crz, rot on the left-hand. In fact, these equations are oX =. f - eC +dr r-y =-. + + br a-z= e$ -crc. + a ut = - d - br -a. so we may call (f, e, d, c, b, a) the coefficients reciprocal to (a, b, c, d, e,f). 254 PROJECTIVE GEOMETRY 256. The null-system is involutionary, and is, in fact, a kind of polarity; this follows from the foregoing equations. Hence we may use the terms pole and polar for any point and plane considered in relation to their associate elements. A null-system is determined by a simple skew pentagon ABODE, no four vertices of which are coplanar (G. 325). In fact (Art. 232), there is a definite correlation in which the planes EAB, ABC, BCD, CDE, DEA correspond to A, B, C, D, E respectively. This can only be a null-system. Consider, for instance, any point P disjoint to AB, and let wr be its polar. Then to the line PA will correspond the meet of 7r and the plane EAB; and to PB will correspond the meet of wr and ABC. Hence, also, the pole of the plane PAB is the meet of the three planes wr, EAB, ABC, that is, it is the point where 7r cuts AB, and is therefore in PAB. Thus every plane of the axial pencil (AB) has a pole conjoint with it on the line AB. Suppose, now, that P is disjoint to the plane ABC, and let Q, R be the points on AB, BC which are the poles of PAB, PBC. Then the line QR is the conjugate of PB, and hence the polar plane of P must be PQR, which is conjoint with P. Since P, if disjoint with all the sides of the pentagon, must be disjoint with at least one of the planes ABC, BCD, etc., it follows that every point is conjoint with its polar. 257. Conjugate and Self-conjugate Lines. Let p be any line; A, B any two points upon it, and a, /3 their polar planes. Putting a3 = p', the line p' is, in general, skew to p; and if we suppose a point P to describe a row on p, its polar 7r will describe a projective axial pencil through p', and conversely; moreover, this relation is a reciprocal one. Two such lines p, p' are said to be conjugate with respect to the null-system. Let P be any point, and R its polar plane. Then we have in g a flat pencil (P), every ray of which is conjugate to itself. Thus, let Q be any other point in 7r, and < its polar plane; then K goes through Q by definition, and it also goes through P, because Q is in 7r. Therefore rK = PQ, and NULL-SYSTEMS 255 the line PQ is self-conjugate. It is left to the reader to show that there are no other self-conjugate lines in 7r. We have now proved that In a null-system every plane and every point is associated with a fiat pencil of self-conjugate lines conjoint with it. Staudt calls the self-conjugate lines the directrices of the null-system. There are oo3 of them from the analytical point of view; because each of the oo3 planes of space contains co 1 of them, while, on the other hand, each one of them lies in o 1 planes. The whole system of self-conjugate lines forms what is called a linear complex, and we shall give some of its properties subsequently. 258. Data fixing a Null-System. (i) Let p, p', d be three lines which are mutually skew; then there is a definite null-system in which p, p' are conjugate, and d is a directrix. Through p' draw any two planes p'AE and p'CD, cutting p in A, C, and d in E, D respectively; and let B be an arbitrary point on p'; then, by Art. 256, the skew pentagon ABODE determines a null-system, and this satisfies the conditions laid down. For, by construction, the conjugate of p, that is AC, is the meet of the planes BAE, BCD; and this is p'. Also, each side of ABODE, and in particular DE, is a directrix. To find the polar of any point P from p, p', d alone, we proceed as follows. Let the plane Pd meet p, p' in Q and Q'; then the meet of QQ' and d is the pole of Pd. Call this point R; then PR is a directrix; so also is the line PS drawn through P to meet p and p'. Hence the plane PRS is the polar plane of P. In a similar way we can find the pole of any given plane; and since we make no use here of the point B, it follows that the data fix the null-system uniquely. (ii) Five lines a, b, c, d, e, in a general position, determine a null-system of which they are directrices. (Pliicker.) To prove this we require the following lemma: (a) If a, b, c, d are four lines in a general position, there are two lines which meet them all. 256 PROJECTIVE GEOMETRY The system of lines which meet a, b, and c form a regulus p occupying a conicoid ro (Art. 127). The line d meets C in two real or complex points P, Q; and now, if p, q are the directrices of p which pass through P, Q respectively, these are the lines in question. If a, b, c, d are all real, then p, q are either real (possibly coincident), or complex conjugates. But the argument is quite general: the construction only fails if some three of the lines are concurrent or coplanar, or if the four lines are elements of the same regulus. Returning now to the main theorem: let p, p' be the lines which meet a, b, c, d; then by (i) there is a definite null-system for which p, p' are conjugate, and e is a directrix, and this satisfies the conditions of the question, because each of the lines a, b, c, d meets two conjugate lines, and is therefore a directrix. We suppose, of course, that e does not meet either p or p'; this is involved in the term " general position " in the enunciation. When p, p' are complex we can replace them by a pair of real conjugates q, q' as follows. Supposing the given lines are all real, the real reguli (abe...) (ode...) consist entirely of directrices. Now we can always find a real line which cuts two lines a', b' of the first regulus and also two lines c', d' of the second regulus. Let this be q; then there will be another real line q' which also meets a', b', c', d', and we may take q, q' instead of p, p'. In the light of our previous results about complex elements, it is only for constructional purposes that the replacement of p, p' by q, q' is of any interest. Exceptions to the main theorem occur when any three of the given lines are concurrent or coplanar, or when any four of them are on the same regulus. These are, in fact, included in the assumption that e meets one of the lines p, p' which meet a, b, c, d. Incidentally, we see that if e meets one of the lines which cut a, b, c, d, then a must meet one of the lines which meet b, c, d, e, and so on. NULL-SYSTEMS 257 (iii) If p, p', q, q' are four lines on the same regulus, there is a definite null-system for which (p, p') and (q, q') are two pairs of conjugate lines. Let d, e be two arbitrary lines meeting p, p' and q, q' respectively; and let a, b, c be three arbitrary directrices of the regulus pqp'q'. We can prove (if necessary, by algebra) that the five lines a, b, c, d, e may be assumed to be in a general position; they then determine a nullsystem, of which they are directrices, by theorem (ii) above. In this system (p, p') are conjugate lines, because each meets the four directrices a, b, c, d; similarly (q, q') are conjugate. But the two pairs of conjugates determine the pole of an arbitrary plane r, because, if 7 cuts the lines in P, P', Q, Q', its pole must be the meet of PP', QQ'; and similarly for the polar of an arbitrary point. (iv) An involution (aa', bb'...) on a regulus determines a null-system, in which the pole of an arbitrary plane is the centre of the involution on the conic in which the plane cuts the regulus. This follows from (iii). The pole of any tangent plane through a is the point where it meets a'. Every ray of the conjugate regulus is a directrix of the null-system. 259. Metrical Properties. In any given null-system the plane at infinity has a pole O (also at infinity), which we may call its centre. All the ordinary rays of the bundle (O) form a parallel bundle, each ray of which we shall call a diameter of the system. Every diameter has a definite conjugate line at infinity. If a point describes a row on any diameter, its polar describes a parallel axial pencil, whose axis is the line conjugate to the diameter; and conversely. Now consider the axial pencil of planes which cut all the diameters at right angles. This will have a conjugate diameter c, perpendicular to all its elements; we call c the central axis of the system. It is a definite line, disjoint to the plane at infinity, if, as we suppose throughout, O is disjoint to the focoidal circle A; this is always the case for a real null-system. M.P.G. R 258 PROJECTIVE GEOMETRY Every plane perpendicular to c contains a flat pencil of directrices, with its centre on c. (i) Given two skew lines c, d, there is a definite null-system in which d is a directrix, and c the central axis. The pencil of planes perpendicular to c meet in a definite line c' at infinity. By Art. 258 (i) there is a definite nullsystem in which c, c' are conjugate lines, and d is a directrix. This is the null-system in question. (ii) A null-system is unaltered by a translation parallel to the central axis. Draw any plane parallel to the central axis c. Then, since it passes through the centre, its pole must be at infinity, and the pencil of directrices it contains is a parallel one. Translation parallel to c leaves every plane parallel to c in the same position as before, and produces a (parabolic) permutation among the directrices in such a plane. The whole result is a permutation of the directrices, and the correlation in space is unaltered. (iii) A null-system is unaltered by rotation about the central axis. Draw any plane y, meeting the central axis at right angles in C; then C is the pole of y in the null-system, and the plane field (y) is correlative to the bundle (C). In the bundle take a right circular cone, of which c, the central axis, is the axis of revolution. To this cone will correspond (Art. 139, end) a conic in y, of which C is the centre, because y, being the polar plane of c with respect to the cone, C must be the pole, with respect to the conic, of the line c' which, in the null-system, is conjugate to c; and this line c' is the line at infinity in y. Consider, now, any two lines in y which cut at right angles in C; they are self-conjugate in the null-system, and conjugate to each other with respect to the cone. It follows that they must be conjugate with respect to the conic; and this conic must therefore be a circle, centre C. All the planes of the bundle (C), except y and the planes through c, can be grouped into envelopes of right circular NULL-SYSTEMS 259 cones, axis c; and we have in y a corresponding set of concentric circles. This arrangement is unaffected by rotation about c, and this proves the theorem; because the set of cones and circles is deduced from, and implies, the nullsystem. Strictly speaking, the rotation produces a permutation of elements. (iv) Let r be the distance of any point P from the central axis c, and let ~ be the angle which the polar plane of P makes with c. Then, for all positions of P, the quantity r tan 4 is invariable. (Mobius.) C FIG. 70. To prove this we draw any line u (Fig. 70) cutting c at right angles in C, and set off equal segments CP, CP' on u, c respectively. By varying the length of the segments we obtain on u, c two congruent and perspective rows. Let T, wr' be the polar planes of P, P' respectively. Then, as P describes the row on u, 7 describes an axial pencil (u) projective to it; and as P' describes the row on c, 7r' describes a (parallel) axial pencil (c'), where c' is the line at infinity conjugate to c. The projectivity between P and P' establishes a projectivity between the axial pencils. Now the axes of the pencils intersect at infinity; and the pencils 260 PROJECTIVE GEOMETRY have one plane (y) in common, namely uc', the polar plane of C. Hence (Arts. 27, 24) the axial pencils are in perspective and generate a flat pencil of parallel rays 7rr'. One of these rays is the line at infinity in the plane uc, as we see by taking P and P' at infinity; hence the plane of the rays 7r/' is parallel to the plane uc, and therefore at a fixed distance m from it. Suppose, now, that we draw from P' a perpendicular to 7rrr'; this is also perpendicular to c, and to the plane POP', so its length must be m. But we see from the figure that its length is also CP' tan; and since CP'=CP=r, this proves the theorem. The quantity m, which is a linear magnitude, may be called the parameter of the null-system. Thus a null-system is completely determined by its central axis and its parameter. If in the plane wr we draw through P a line d perpendicular to PC, we have a directrix whose shortest distance from c is CP, and whose angle with it is ~. We conclude that if d is any directrix, we shall have rtan =m, where r is now the shortest distance between c, d, and 4 is the angle between them; that is to say, the angle between any two concurrent lines drawn parallel to c, d respectively. 260. The properties of a null-system have important applications in statics, and Mobius was, in fact, led to the invention of the null-system by statical considerations. A general system of forces and couples can be reduced to a wrench (G, R; c), consisting of a single force R in a definite line c, and a couple G whose vector image is parallel to c. Now let u be any line skew to c, and let r, 4 be the shortest distance and the angle between u and c. By resolving the vectors representing R and G in directions parallel and perpendicular to u, we see that the total moment of the wrench about u is Rr sin G - G cos k, and if this vanishes we have tan G/R. r tan - = O/R. In other words, the lines for which the resultant moment vanishes are the directrices of a null-system whose central NULL-SYSTEMS 261 axis is that of the wrench, and whose parameter is G/R. 261. In Arts. 258, 259 we have substantially followed the exposition of Reye (Part II., Lecture I8); we add a few propositions taken from Staudt (B. 93-99). (i) Let (pqr...) be a regulus, S and o- a point and plane conjoint with each other, but both disjoint to the regulus. There is a definite null-system for which S is the pole of o-, and every line of the regulus is a directrix. The plane C cuts the regulus in a conic K, and on this we have a definite involution (AA'.BB'...), of which the centre is S. Perspective to this we have a definite involution (aa'.bb'...) on the conjugate regulus, and the theorem now follows by Art. 258 (iv). (ii) Every set of three mutually skew directrices of a given null-system determine a regulus, every ray of which is a directrix of the system; and the null-system determines an involution on the conjugate regulus. Let the given directrices be a, b, c; and take two others d, e, so that the set of five are in a general position. Let p, p' be the two lines that meet a, b, c, d, and q, q' the two lines that meet a, b, c, e. Then we see, as in Art. 258 (iii), that (pp'.qq'...) is an involution of lines on the conjugate regulus, for which every pair is conjugate in the nullsystem; while each ray of the regulus abc is conjugate to itself. (iii) Let m, n be two skew lines; A, B, C any three points on m, and a, P/, y any three planes through m. There is a definite null-system in which m, n are directrices, and A, B, C are the poles of a, /3, y respectively. Let p, q, r be the lines joining A, B, C to the points na, n.B, ny respectively; and let a, a' be any two directrices of the regulus pqr, such that mnaa' is a harmonic set. Then we have on the conjugate regulus an involution (m2.n2.aa'...) 262 PROJECTIVE GEOMETRY which determines a null-system, and this is the one in question. For it is clear that m, n, p, q, r are all directrices, and hence the points mp, mq, mr must be the poles of the planes mp, mq, mr (Art. 257); that is, A, B, C must be the poles of a, P, y. It is easy to see that only one nullsystem satisfies the required conditions. CHAPTER XXIX. SKEW INVOLUTIONS. 262. We have seen that, for the most general type of collineation in space, we have four self-corresponding points, the vertices of a tetrahedron, while, on the other hand, in a harmonic homology we have a self-corresponding bundle, and a self-corresponding plane field. It has also been pointed out that in the first of these cases the characteristic equation has four different roots, while in the second it has a triple root (Art. 235). We shall now discuss a new kind of collinear involution, for which, as we shall presently see, the characteristic equation reduces to the form (p2 m-2)2 =o. This is called a skew involution. 263. Let m, n be any two skew lines; then we can establish a point-to-point correspondence in space in the following manner. Let P be any point disjoint to m and n; draw through P the line which meets both m and n, and take upon it the point P', which is the harmonic conjugate of P with respect to the points of intersection. We define P' as the point corresponding to P; clearly P is the point corresponding to P', and it is not difficult to show that we have now an involutionary collineation, if we make every point on m and n correspond to itself. To save space, we give an analytical proof. Take a tetrahedron of reference for which m, n are a pair of opposite edges (z=t=o; x=y=o); then the above construction leads to the equations x': y': z: t' =x:y: -z: -t,.....................(I) 264 PROJECTIVE GEOMETRY which is clearly an involution, as asserted. The characteristic equation is (p2 - )2 =o, and if ($, A, I, T), ($', V', ' T') are two corresponding planes disjoint to m and n we have ':r':f: T=:: X1 -: -7,..................(2) whence we conclude that if a plane 7r cuts m, n in M, N, the corresponding plane 7r' goes through MN, and m, n separate rr, 7r' harmonically; that is to say, the planes mMN, nMN are harmonic conjugates with respect to 7r, 7r'. Every plane through m or n corresponds to itself. Let p be any line skew to m and n. The lines m, n, p determine a regulus, and if, on this, the ray p' is the harmonic conjugate of p with respect to m, n, the lines p, p' are conjugate. This follows because every directrix of the regulus meets mnpp' in a harmonic set of points. Let p meet m in M, but be skew to n. Let the plane pm, which is conjugate to itself, meet n in N. Then the rays m, MN are self-conjugate, and if we take p', the harmonic conjugate of p with respect to them, this is the conjugate of p. Every line that meets both m and n is a self-conjugate line; and if M, N are the meets, the line contains an involution of conjugate points, of which M, N are the foci. Conversely, every self-conjugate line distinct from m, n must meet them both; and there are no self-conjugate points or planes except those conjoint with either m or n. Every self-conjugate plane contains a flat pencil of self-conjugate lines. 264. Derivation from a Conicoid. Suppose we have an involution (aa'.bb'...) among the rays of any given regulus; we can set up a point-to-point correspondence in the following way. Let P be any point disjoint to the regulus; then its polar plane wr will cut the surface in a conic, on which there is an involution perspective to (aa'.bb'...). Let P' be the centre of this involution: then we may take P' as the point corresponding to P. If the involution (aa'.bb'...) is hyperbolic, we shall have two real double SKEW INVOLUTIONS 265 elements m, n, and we can prove geometrically that the correspondence is the same as that derived from m, n by the process of last Article. Staudt's theory eventually leads to the conclusion that when (aa'.bb'...) is elliptic, the construction of the present Article leads to a real skew involution in space, which may be taken to represent two conjugate bicomplex lines. The construction of the present Article may be put into an algebraical form as follows. Take the equation of the conicoid in the form xt-yz=o, then we have upon it a regulus, obtained by varying X in the equations = y, z =t. Let (x', y', z', t') be the point P: then its polar plane is t'x - z'y -y'z + x't = o. This cuts the generator (X) in the point whose coordinates are given by xi: i: 1:= X(Xy' - x'): Xy' - ': X(Xt' - '): ' - z'. An involution on the regulus is defined by an equation of the form aX - b(X +/) +c=o, and now the point where the polar plane of P cuts the generator (p) is given by x2: Y2: Z2: t2 = M1(y' ) - X: M (ty' - z): t('' - z'. In virtue of the relation between X and /A, we find by algebra that the line joining (xlyxzlt) to (x2Y2z2t2) passes through the point P', whose coordinates are given by x": y": z" t" =cy' -bx': by' -ax': ct' -bz': bt' -az'. From these equations it follows that the locus of selfcorresponding points is the set of points common to the three surfaces ax2 - 2bxy + cy2 = o, az2- 2bzt + ct2 = o, xt -yz =; in other words, they are the lines (x = ay, z = at), (x = fy, z = 3t), 266 PROJECTIVE GEOMETRY where a, /3 are the roots of the quadratic am2 - 2bm+c =o. This result holds good, whether the roots of the quadratic are real or complex. 265. If we take a, b, c any three constants such that ac b2, the equations x':y': ': t=cy-bx: by-ax: ct-bz: bt-az......(I) define, as we have just seen, a skew involution associated with two real or complex generators of the surface xt -yz = o. The characteristic equation is (p2 b2 +ac)2=o; and by analytical transformations into which we shall not enter here it can be proved that whenever a collinear involution has a characteristic equation of the form (p2 -A)2 =o, with A different from zero, there is at least one tetrahedron of reference for which the involution is expressed by equations of the form (I). Consequently, every such involution is associated with a definite pair of real or conjugate bicomplex lines, if we suppose that a, b, c are real constants. Equations (I) involve the plane-to-plane transformationequations ': rA': (; T'= - b~ -a,: c~ + bq: -b -ar: c +br,...(2) which are of the same type as (i); as, of course, might be expected. 266. Reguli contained in an Involution. Suppose we take any two skew lines, defined by the equations u=o, v = o and u' = o, v' =o respectively; the separate equations u = o, etc., denoting planes. By arguments similar to those of Art. 206 we can prove that the most general quadric surface containing the lines will satisfy an equation auu' + Puv' + yuZ'v + 8vv' = o.....................(I) where, for a proper conicoid, as -/ly o. The given lines establish a skew involution, such that for every surface such as (i), one set of its generators (namely those that meet the given lines) are self-conjugate, while those of the other set form an involution of conjugate SKEW INVOLUTIONS 267 pairs. We shall say that all these surfaces, and their reguli, are contained in, or belong to, the involution. Thus a given involution contains oo 3 conicoids, and their reguli. If all the planes u, v, u', v' are real, we obtain a set of real conicoids from (I) by taking a, /3, y, 8 to be real. Suppose, on the other hand, that we take u = o, v = o to be complex, and u' =o, v'=o to be the planes conjugate to them. Then we obtain a set of real conicoids from (I), by taking a, 8 to be real, and /i, y conjugate complex quantities. Thus, with a change of notation, we have the real system XUUo +!UVo + poV + VVVO =o................... (2) It is convenient to transform this, when the involution is given by x':y': z': t'=cy -bx: by -ax: ct -bz: bt -az. We find without any difficulty that the system of real conicoids is a(ax2 - 2bxy + cy2) + P/(az2 - 2bzt + ct2) + y(axz - byz - bxt + cyt) + 8(xt -yz) = o,..... (3) where a, f3, y, 8 are any four real constants. Conversely, any three real constants a, b, c simultaneously define the system (3) and the skew involution in which it is contained. It should be remarked that this holds good, whether the lines on which self-conjugate points lie are real, or bicomplex and conjugate. 267. Sense of a Skew Involution. Take the system of surfaces defined by (3) of last Article, and cut it by an arbitrary plane. In this plane we now have a system of conics, on each of which there is an involution of points. If any one of these involutions is hyperbolic, so are all the rest, and the conics all pass through the same two real points. Suppose, however, that all the involutions are elliptic; then there are two senses in which any one of them can be taken. Now it is possible to assign geometrically what we may call a concordant sense to all the involutions; and the following seems to be the simplest way of 268 PROJECTIVE GEOMETRY proceeding, though it involves some ideas which are not, strictly speaking, projective. Let (AA'.BB') be the involution of points on any one of the conics, and S the centre of the involution, so that S is inside the conic. If a point move along the conic in the sense ABA', the line joining it to S will describe the pencil (S) in a definite sense of rotation. We assume that we can intuitively distinguish two kinds of rotation of lines in the plane such that a flat pencil can be described continuously by a ray rotating in either the positive or the negative sense. If, now, as above, a point describes the conic in the sense ABA', we say that this is the positive or negative sense, according as the corresponding sense of the pencil S(ABA'...) is positive or negative. We may now give a concordant sense to the involutions on the conics by taking them all in the positive sense, or all in the negative sense. As soon as we have done so, we have a corresponding sense of description for all the involutions of reguli on the o 3 conicoids contained in the skew involution we started with; reversing the sense of the pointinvolutions reverses that of the line involutions. We express this result by saying that to an elliptic skew involution we can attribute one or other of two opposite senses. This is how Staudt discriminates between two conjugate bicomplex lines; any bicomplex line is represented by (we may even say is) an elliptic skew involution associated with a definite sense. 268. Involution fixed by two complex Points or Planes. Let (AA'.BB'...), (CC'.DD'...) be involutions of points on two skew lines m, n; then there is a system of oo 1 collineations for each of which (A, A'), (B, B'), (C, C'), (D, D'), etc., are corresponding points. Among these collineations there are two, and only two, which are involutions. In any collineation which satisfies the conditions stated, the lines m, n are self-corresponding, and n(AA'. BB'...), m(CC'.DD'...) are two involutions of corresponding planes. By taking m, n as opposite edges of a tetrahedron of reference, we may suppose the given involutions defined by SKEW INVOLUTIONS 269 z=o, t=o, axx'-b(xy'+x'y) cyy' =o (.) x=o, y=o, a'zz' -b'(zt' +'t) +c'tt' =oJ Putting b2-ac=A, b'2-a'c'=A', the quantities A, A' are each different from zero, and, in general, unequal. If we take any two constants X, /u and write x': y': z': t'= X(cy - bx): X(by -ax): (c't - b'z): (b't-a'z),........(2) we have a collineation of the kind specified, and it can be easily proved that all such collineations are obtained from (2) by varying X and /. By repeating the transformation we arrive at a point (x", y", z", ft") =(x', y', z', t') =2(x, y, z, t), whose coordinates are given by x": y": z": t"= X2A: X2y: 2A'z: A't,.........(3) so we have an involution if, and only if, X2A = k2.A'; this gives two values for X//, and each of them leads to an involution, as asserted. Suppose that m, n are real lines, and (a, b, c) (a', b', c') real constants: then the two involutions corresponding to X2A=tA2A' are real or imaginary according as AA' is positive or negative; that is to say, they are real if the involutions on m, n are both hyperbolic, or both elliptic, and imaginary in the other two cases. The most interesting case for us is when the involutions on m, n are both elliptic. They then define two pairs of conjugate complex points (M, Mo) and (N, No) on m, n respectively. The two associated skew involutions in space define two pairs of conjugate bicomplex lines (p, po) and (q, qO), and these are, in fact, the joins (MN, MONo) and (MNO, MON). The correspondence of symbols p =MN, etc., must be established by a geometrical concordance of sense (cf. Art. 267). This we effect by actually giving a sense to one of the skew involutions in space derived from m, n; this will then, of 270 PROJECTIVE GEOMETRY itself, fix the sense of each of the point-involutions on m and n. We are now able to say that Two given complex points M, N whose bases are skew determine one and only one (bicomplex) line conjoint to both. This we may call their join, and denote by MN or NM. By a correlative argument we can prove that two given complex planes p/, v, whose axes are skew, determine uniquely a line conjoint to both. This we may call their meet, and denote by [uv or v/k. It is, of course, bicomplex. These two theorems help to complete the analogy of the complex theory with the real one, so far as joins and meets of simple elements are concerned. 269. Tetrad-symbols for a Bicomplex Line. Take a skew involution in space with a definite sense attached to it. Then (Art. 266), we have a three-fold system of contained reguli, each of which forms an involution with a definite sense. Let (aa'.bb'...) be any one of these reguli, and let aba' be the associated sense. We take aba'b' as a symbol for the bicomplex line represented by the involution; and we are entitled to do this, because from any one such tetrad of rays as a, b, a', b' we can construct the whole involutionary regulus (aa'.bb'...) on which it lies, and this determines a skew involution through the whole of space. The latter defines two conjugate lines, in which (aa'.bb'...) is associated with the senses aba', ab'a' respectively; so, without ambiguity, we may represent the lines by aba'b' and ab'a'b (or a'bab'). Any two pairs of conjugate rays on any one of the involutionary reguli may be used to form a tetrad-symbol for each of the bicomplex lines. Conversely, if aba'b', pqp'q' are symbols for the same line, the involutionary reguli (aa'.bb'...), (pp'.qq'...) must be contained in the same skew involution, and in that involution the senses aba', pqp' must be concordant. In such a symbol as aba'b', we may, if we like, suppose that a, a' separate b, b' harmonically; we then have a harmonic representation of a bicomplex line, starting with a (cf. Art. I64). We may SKEW INVOLUTIONS 271 start with any real line a whatever, that is not selfconjugate; because the surface represented by (3) in Art. 266 can be determined so as to contain any given real line. 270. Chains. Any such regulus as (aa'.bb'...) is met by any ray of the conjugate regulus in a point-involution (AA'.BB'...). The complex point ABA'B' is defined to be conjoint with the bicomplex line aba'b'. On any one such conjugate regulus we have oo 1 points like ABA'B', all lying on aba'b', and forming what Staudt calls a chain of points. Let P, Q, R, S be four such points, and let p, q, r, s be their real bases. Then, if, as above, our bicomplex line is represented by aba'b', the four point-tetrads p(aba'b'), q(aba'b'), r(aba'b'), s(aba'b') represent the four collinear points P, Q, R, S. Now these points lie respectively in the real planes ap, aq, ar, as (Art. 166); hence PQRS is the section of the real pencil a(pqrs) by the bicomplex line. The real pencil a(pqrs) is cut by the real line b in the real range of points (bp.bq.br.bs); hence, as sections of two lines by the same axial pencil, we have (PQRS) (bp.bq.br.bs). Hence, by definition (Art. I8I), the corresponding casts are equal: and by choosing corresponding sets of base-elements, we can make the values of the casts equal. In particular, we may write (cf. p. 182, first paragraph) PQRS = bp.bq.br.bs. The value of the cast on the right hand is a real quantity: hence so is that of PQRS, and we infer that If P, Q, R, S are four points on a bicomplex line which belong to a chain, the value of the cast PQRS is a real quantity. Any three points P, Q, R on a given bicomplex line determine a chain to which they belong; we may denote this accordingly by ch(PQR). Let p, q, r be the real bases of P, Q, R; each pair of these is necessarily skew, so that we have a determinate real regulus (pqr...), and on each ray of this we have a point collinear with P, Q, R. We also see that if S is any point collinear with P, Q, R, such that the cast PQRS has a real 272 PROJECTIVE GEOMETRY value, the real base of S must be an element of the regulus (pqr.. ); that is to say, S is an element of ch(PQR). There is a corresponding theory of chains for any complete one-fold elementary form. Thus, let any triad of base-elements be taken, and let x1, x2, x3 be the indices of any assigned triad of elements. If we put k (X1X2X3x4) = m, where m is an arbitrary real quantity, we have a set of indices (x4) which has a one-to-one correspondence with the real arithmetic continuum. The corresponding elements are said to form the chain ch(x.x2X3). For instance, consider the complete axial pencil, which consists of all the planes conjoint with a given line u. When u is real, we have one real chain, namely, what in the real theory we denote by" the axial pencil (u) ". If u is simply complex, we have no real chain; but we can obtain the simplest possible chains in the following manner. Let o- be the plane base, and S the centre of the given line u. Through S draw any two lines p, q disjoint to ac, and take the planes cr, pu, qu as base-elements. Then we have a chain consisting of those planes ru, whose real axes (r) are rays of the flat pencil determined by p and q. If u is bicomplex, take any one of the associated involutionary reguli (aa'.bb'...); then, if p is any ray of the conjugate regulus (pqr...), p(aa'.bb'...) is the complex plane pu, and by giving to p all its possible positions we obtain a chain in the axial pencil (u). It may be denoted by ch(up.uq.ur) or chu(pqr), where p, q, r are any three real rays of the regulus (pqr...). Similarly, the points on a real conic form what we may call " the simplest chain " of elements on the corresponding complete conic (which is necessarily determined by its real part). 271. Illustrations. We conclude this chapter by giving complete proofs of one or two elementary propositions which are either axiomatic, or have been proved, in the real theory. SKEW INVOLUTIONS 273 (i) A point A and a line u define a unique plane which is conjoint to both, and may be denoted by Au or uA; unless A is conjoint with u. There are six cases to consider, because A may be either real or complex, and u may be real or complex or bicomplex. (I) If A, u are real, the proposition is axiomatic (Art. 2, viii). (2) If A is real, and u is simply complex, A may be either conjoint or disjoint with the plane of u (that is the real plane in which the pencil defining u is situated). In the first case, the plane Au is the real plane containing u: in the second case, the proposition follows from Art. 170 (v); the plane is complex, and represented by A(aba'b'), where (aba'b') is any representation of u. (3) Let A be real, and u bicomplex. Since no real point can be conjoint to u, A will have a real conjugate A' in the involution defining u. In this involution AA' is a selfcorresponding line, and the axial pencil (AA') is paired in involution with a definite sense: it therefore defines a complex plane Au conjoint with A and u. (4) Let A be complex, and u real. If a, the real base of A, is coplanar with u, the plane Au coincides with the plane au. If not, let PQP'Q' be any representation of A; then, by projection from u, we obtain a complex plane of which one representation is u(PQP'Q'), and this is conjoint with A, u as required (Art. 170, viii). (5) Let A be complex, and u (simply) complex. Let S, r be the real point and plane conjoint with u; and let a be the real base of A. If a is in o-, but disjoint to S, the plane Au is o-. If a is conjoint with S, but cuts o-, let (pqp'q') be any representation of u; then Au is the complex plane of which one representation is a(pqp'q'). Finally, let a cut C- in a point P distinct from S, and put SP ==p. Let PQP'Q', pqp'q' be harmonic representations of A, u, starting from P, p respectively; and let T be the common point of the planes Qq, P'p', Q'q'; then Au is the complex plane of which ST(PQP'Q') is one representation. M.P.G, S 274 PROJECTIVE GEOMETRY (6) Let A be complex, and u bicomplex. Through A draw any real plane o-, and find P, the complex point in which C cuts u. This can be done by means of any one of the involutionary reguli associated with u; this will be cut by - in an involutionary conic, and the axis of this planar involution contains P, which is one of the intersections of the axis with the conic (cf. Art. I74). Now in the plane -, find S, the real point of the complex line PA (Art. 168). Then the plane Su can be found by case (3), and this is conjoint with A, because SP contains A by construction (B. I30). The reader will find it a good exercise to prove the correlative theorem. (ii) Given two skew lines u, v and a point A disjoint to both; one and only one line can be drawn through A which meets both u and v. By the foregoing theorem we have two definite planes Au, Av, and their intersection is the line in question. If we go into detail, there is a large number of cases to consider; for instance, if A is real, and u, v are conjugate bicomplex lines, the meet of Au, Av is the real line AA', where A' is the conjugate of A in the involution from which u and v are derived. (iii) A real conicoid is, in general, determined by the condition of containing two real skew lines u, v, and three points A, B, Bo, of which the first is real, and the others are complex and conjugate. Let the real plane ABBo cut u, V in U, V respectively; then (Art. I79) the points AUVBBo determine a real conic K. By Art. I39, this conic and the lines u, v determine a real regulus, and this proves the theorem. The proposition fails if A, B, Bo are collinear, or if the plane ABBo contains either U or v. CHAPTER XXX. LINE GEOMETRY. 272. In discussing the null-system and the skew involution we have come across sets of lines more general than those which form bundles, or envelopes, or reguli; thus we have the set of directrices of a null-system, the set of selfconjugate lines of a skew involution, and so on. The study of sets of lines is best pursued by a method substantially due to Pliicker; in this theory the line is regarded (ultimately) as the primary element of space, the plane and point being derivative ideas. We shall, however, consider a line as a join or a meet, and from this obtain what we shall call the coordinates of the line, when we regard it as a single element. 273. Coordinates of a Line. In the present chapter we shall use suffixes for the coordinates of a point or a plane; thus, the point (x) will mean the point whose coordinates are (xl, x2, X3, X4) and the plane (u), the one whose coordinates are (um, u2 3, u4). We suppose that the tetrahedron of reference is ABCD, and use the symbols a, /3, y, 8 for the planes BCD, CDA, DAB, ABC. Finally, we suppose the coordinate system to be normal, so that the point (x) and the plane (u) are conjoint if tx = x + I 2i 2 +22 +u3X3 + i4X4 = 0, where ux is a convenient symbol for uixi. Let X, Y be any two points distinct from A, B, C, D, and not collinear with any one of the latter. Then we have 276 PROJECTIVE GEOMETRY four definite planes (not necessarily all distinct), each containing X, Y and one vertex of ABCD. Let (xl, x2, x3, x4) be the coordinates of X, and (Y1 Y, y, y y4) those of Y: then the planes in question have the coordinates (o, XIY2 -X2Y1, x1y3 - x3y, x1Y4 -X4Y1), (x2Y1 -x1y2, o, x2Y -x3y2, x2y4 -x4y2), (x3Y1-x1Y3, x3Y2 -2Y3, o, X3Y4-X4y3), (x41 - x1Y4, x4y2 - x2Y4, x4y - xy4, o). W e now put - yi = (iy) = ij.........................(I) thus pii=o, and pji +pij=o, when i, j are unequal. The six quantities P12, P23, P31, P14, P24, P34 satisfy the identical relation ~~~P2P +PP2+relationPa234=~.............1+......... (2) and the other six quantities only differ in signs from these. In this notation the four planes AXY, etc., are (0~ P12, -P31, P14) (-P12, 0, P23, P24)) ( 2 -P23o, p ) a,...................... (3 (P31, -P23, 0, P34) (-P14 - P24, -p34, ) Conversely, if we take any six quantities satisfying the identity (2), the ratios p23: 31:.. P 34 fix the planes (3), and we can prove that these planes determine a single line through which they all pass. We accordingly say that (P23, 31, P12, P14, P24, p34) are the homogeneous radial coordinates of the line (p). Six homogeneous coordinates give rise to oo 5 elements: but since the identity (2) has to be satisfied, our field of lines contains oo 4 elements. In this sense we may say that all the lines in space make up an elementary form of the fourth rank. Starting from a correlative point of view, we obtain a different set of coordinates for a line. Let (u), (v) be any two planes in an arbitrary position; then the four points, LINE GEOMETRY 277 where their line of intersection meets the planes a, /, y, 8 respectively, have coordinates (-w712, o 723 7724).............. (4) (731 W- 23, 0, w n734) ( -14' - 24, - 34-, ) J where rij = UiVj - jvi. As before, we have the identity W237r14 + 7731724 + 7712734 = 0, and if this is satisfied the points (4) determine a line on which they lie. We shall say that (7r3,... 7734) are the homogeneous axial coordinates of the line. Combine the first two planes of (3), expressed in pointcoordinates, with the equation 4 = o; then for the common point we have p122 -p3X3 =, -Pl12Xl +p23X3 =o, whence x1: x2: x3 =P23: p31: 12. Comparing this with (4), we see that 7714: 7T24: r34=P23: 31: p12, and by proceeding in the same way, we eventually find that 23: 31: r12: 714: 7r24: 734 =P14: P24: 34: P23: P31: 12 (5) So a change from radial to axial coordinates, or vice versa, merely amounts to a change of order. 274. Condition of Conjunction. The condition that two lines (p), (q) may be conjoint is that P23q14 +Pl4q23 + 31q24 +4 +P4q31 + 12q34 + P34q12 ==......(I) Take any two points (x), (y) on (p), and any two points (z), (t) on (q): these four points must be coplanar, and consequently (xY2z3t4) =o, where the symbol on the left stands for a determinant of which the first row is (x1, x x, x4), and so on. If we expand the determinant as a linear function of minors 278 PROJECTIVE GEOMETRY taken from the first two rows, the condition assumes the form stated above. We have supposed (p), (q) given by radial coordinates; but the result is similar if they are given by axial coordinates. If one is given by radial, and the other by axial coordinates, we have a slightly different formula, obtained by combining (i) with the relations (5) of last Article. If two given lines are conjoint, we can find their common point by taking any two of the planes in the set (3) of Art. 273 and combining them with one of the corresponding set connected with the other line (q). Similarly, we can find the plane containing the given lines. Practically, however, we hardly ever have to make these determinations. 275. Collinear Transformations. Suppose we take a collinearity defined by the equations px'1 = alxz + a2x2 + a3X3 + a4X4 px'2 = blx +b2x + bs +b4x 4...... px'3=clX1 +c2x2 + c33 +-c4x4... px'4 = d1x, + d2x2 + d3X3 + d4xJ then from these we can deduce the relations between the coordinates of corresponding lines. For instance, we have P2(xlY2)' = (alb2) (xlY) + (alb3) (x1y3) + *= (aibj) (xiyj) ( P2 (XY3) = (aicj) (Xiyj), and so on. Thus we have a linear correspondence between the coordinates of two associated lines; not, of course, a general one. We may, if we like, regard the coefficients (aibj), etc., as the coordinates of six lines determined by the collineation: namely, those of the six edges of the new tetrahedron of reference referred to the original one. When the collineation can be reduced to the form x':y': z': t' =ax: by: cz: dt the six radial coordinates of a line are multiplied by be, ca, ab, ad, bd, cd respectively. LINE GEOMETRY 279 276. Complexes and Congruences. When the coordinates of a line satisfy, in addition to the fundamental identity, an integral, homogeneous, algebraic equation F(p23, p3, p.. 34) =0, it is said to belong to the complex represented by F = o. All such lines form a three-fold set containing 0o3 elements. The lines common to two complexes which have no complex in common form what is called a congruence. The lines of a congruence form a set of oo 2 elements. Three complexes, in general, determine a set of co 1 lines common to them all; these lines occupy a ruled surface. When the equation F =o, defining a complex, is of degree n, we shall say that the complex is of degree n. In a certain sense, as we shall presently see, the complex is of class n and order n; so the single term degree may be taken to imply both order and class. In a complex of degree n there are, in general, 2n lines which meet three given lines. By Art. 274 any such line has to satisfy three linear equations, besides the equation F = o of degree n, and the fundamental identity P23P14 + P31P24 + P12P34 =, which is of degree 2. By the theory of algebraical equations we infer that, in general, there are 2n lines satisfying the conditions stated. Suppose, however, that the three given lines are concurrent in (a, b, c, d), but not coplanar; then every line that meets them all must go through (a, b, c, d), and therefore must have radial coordinates of the form (bz - cy, cx - az,... ct - dz), and if this belongs to the complex, we must have F(bz-cy,..) =o.........................() supposing that F=o is an equation in radial coordinates. 280 PROJECTIVE GEOMETRY But this equation is a cone of the nth order, with its vertex at (a, b, c, d); hence The lines of a complex of the nth degree, which pass through a given point, form, in general, the rays of a cone of order n. Correlatively, the lines of a complex of the nth degree, which lie in a given plane,form, in general, the tangents of an envelope of the nth class. It should be noticed that if F(p23, P31,... 34) =o is the equation of any complex in radial coordinates, then, by Art. 273 (5), the equation F(r14, 7124,.... 12) = o represents the same complex in axial coordinates; so the function F is, so to speak, its own correlative, and this fact is the expression of the law of duality for linesystems. Finally, let two of the given lines, say p, q, be conjoint, and the third, r, be disjoint to them. Let p, q meet in S and lie in the plane r; and let r meet Co in T. In the plane or we have an envelope of class n, of which n lines go through T; and through the point S we have a cone of order n, of which n rays meet r; so we have 2n lines, as in the general case, but these separate into two sets of n each, correlative to each other. 277. The Linear Complex. The simplest kind of complex is the linear complex of degree I. Its equation is of the form ap = a23p23 + a3131 +. + a34P4 =o, where the coefficients are arbitrary. Thus the space of lines contains oo5 linear complexes, and a linear complex is determined by five lines in a general position. The rays of a linear complex which lie in a given plane form a flat pencil; so do those which pass through a given point. Every general linear complex determines a nullsystem in which the pole of any plane is the centre of the flat pencil it has in common with the complex, and the polar of any point is the plane of those rays of the complex which pass through the point. LINE GEOMETRY 281 Suppose (aij) are the coefficients, or, as we may call them, the coordinates of the complex, when the pij are radial, the equations of the corresponding null-system are pul =. a12x2 - a3ax3 + a14x4 p2 = -a12x1. +a23X3+a24x4. (I pu3= a31x1 -a23x2. + a344 pU4 = - a14x -a24x - a34x3, By taking any two points (x), (y), and their corresponding planes (u), (v), we obtain the following equations connecting the radial coordinates of any line and the axial coordinates of its corresponding line. We put a23a14 + a31a24 + a12a34 = A................... (2) and write a. for Eaijpij. Then 7rr2 = al2ap - AP34 rr74 = al4aP - Ap23 r,723 =a23ap - Ap4 7rr24 =a24a -Ap3 a *.......(3) T7r31 =a31a - Ap24 rr34 = a34ap - Ap12 From these equations we see that to have a proper nullsystem A must not be zero; and that the self-corresponding lines of the null-system are precisely those which satisfy ap = o. Since, when A is not zero, the equations (i) represent the most general null-system, we see that in every case a null-system determines a linear complex, and vice versa, except when the coefficients of a complex make A =o. 278. Special Linear Complexes. The quantity denoted by A is called the invariant of the complex (a). By means of equations (2), Art. 275, we can show that if two linear complexes (a), (a') are connected by a collineation, and A, A' are their invariants, we have identically A' = (alb2c3d4)A =AA, where A is the determinant of the collineation; thus A satisfies the analytical definition of invariance. When A = o, we shall say that (a) is a special linear complex. The coefficients of a special complex determine a line of which they are the axial coordinates; supposing, then, that a. = o 282 PROJECTIVE GEOMETRY the equation of the complex, is in radial coordinates, this relation expresses that the variable line (P12,... 34) meets the fixed line whose axial coordinates are (a2,... a34). In other words: The elements of a special linear complex consist of all the lines which meet a fixed line h. Conversely, every line fixes a special linear complex. Any plane disjoint to h contains a flat pencil of rays belonging to the complex; and similarly, every point disjoint to h is conjoint with a flat pencil of rays of the complex. But every plane through h contains a field of oo2 rays of the complex; and every point on h is the centre of a bundle every ray of which belongs to the complex. It should be noticed that each of the equations ij =o represents a special linear complex, consisting of all the lines which meet one edge of the tetrahedron of reference. 279. Pencils of Complexes. Let (X, /j) be a homogeneous parameter, and a = o, bp =o two given complexes; then the system ap +/bp =o............................... (I) is said to form a pencil of complexes. Let A, B be the invariants of ap, bp, and let us find the condition that the complex Xap + lbp =o may be a special one. The condition is found, by calculation, to be AX2 +M u +B o2 =,..........................(2) where M = a2b34 +a23b14 + a3b24 + 14b23 +a24b31 + a34b12.....(3) Now A, B are invariants, and clearly the ratio X//I, obtained from (2), must be independent of the tetrahedron of reference; hence M is also an invariant. We shall call it the mutual invariant of the complexes (a) and (b). Any linear substitution changes A, B into AA, AB (Art. 278): hence the same substitution must change M into AM. In general, then, a pencil of complexes contains two special complexes; these determine two lines, which may be called the directrices of the pencil. LINE GEOMETRY 283 Two special cases require consideration. First, suppose that the equation (2) has equal roots: this is the case if M 2 -4A B =o...............................(4) Let the double root be X/1/ =h/k, and take cp = hap + kbp instead of ap (or bp, as the case may be; but there is no loss of generality in supposing that k is finite, and we then proceed as follows). Then the pencil of complexes may be put into the form Xbp +~Cp = o, and if we now form the equation (2) it assumes the form X2B + XMMb, = 0, since C = o. But this must have two equal roots, and therefore Mbc, the mutual invariant of the complexes (b), (c) must vanish. In other words: When a pencil of linear complexes contains only one special complex, this has to be reckoned twice, and if we denote it by c= o, and bp=o is any other complex oJ the pencil, the invariant Mb, must vanish. Conversely, if cp =o is a special complex of the pencil, and Mb, vanishes for every non-special complex (b), then cp =o is the only special complex in the pencil, and has to be reckoned twice. Moreover, since Mb, = o, the line defined by the special complex c = o belongs to every complex of the whole pencil. Suppose, now, that equation (2) reduces to an identity: that is to say, suppose A =M =B =o. The complexes (a), (b) are now both special, and their mutual invariant vanishes; that is to say, the lines defined by the complexes (a), (b) intersect. The pencil of complexes Xap +/Jb=o consists entirely of special complexes, and for any particular value of A//, the corresponding complex may be defined as follows. The two complexes (a), (b) define two lines u, v and their intersection uv, which we may call (xI, x2, x3, x4). If, now, (yi, Y2, y3, y4) is any other point on u, and (zX, Z2, Z3, z4) any other point on v, the radial coordinates of the line joining (Xy1 +Az,,...) to (xl, x2, x3, x4) will be of 284 PROJECTIVE GEOMETRY the form X(x1y2) + t(xz2), etc., and hence be those of the line defined by the special complex Xap +,bp =o. Hence If ap =o, bp =o are two special complexes such that the lines associated with them intersect, then every complex of the pencil Xap +~tbp =o is a special one, and the lines associated with the complexes of the pencil form a flat pencil of rays. 280. Apolarity. Two linear complexes (a), (b) are said to be apolar (to each other) when their mutual invariant vanishes. We have found the geometrical meaning of this when one at least of the complexes is special; we now proceed to find it when both of them are general. Let A, M, B be the invariants of the complexes. Each of the complexes defines a null-system, and by Art. 277 (3) if (r12,... 7r34), (K12,.. '(34) are the axial coordinates of the lines corresponding to the same line p, we have equations of the form rrij =aija - Apln, fKij =b bjbp - B............ (I) Let us put (r12,... r34) =$(p12,.. P34), and (K12,... K34) =~(P12,.. P34), and let us now find f(r12,....r 4), that is to say, (P12,. P34)-. First of all, we note that, by (I), r (b3412- +... * b12 34) =Map - Abp, and hence, if (q2,... q34) is the required line, we have, from the equations correlative to (I), pq12 =b34(Map -Abp) - B (a34a- A12)......() =Mb4ap - Ab4bp - Ba34ap + ABp12 and so on. If (rl,... 34) =b{(p12,... P34), we find in the same way that pr12 = Ma34bp - A b4b - Ba34ap + ABpl2, and so on. Hence, if M =o, the lines (q), (r) are identical, and conversely: that is to say, LINE GEOMETRY 285 If the mutual invariant of two general linear complexes is zero, then the null-correlations of lines which they establish are commutative, and conversely. Given five linear complexes, it is, in general, possible to find one and only one linear complex which is apolar to them all. 281. Klein's Canonical Coordinates. Let us put xl -P12 +iP34, X3=P23 +iP14, X5 =31 +i24 X2 =iP12 +p34, X 4=iP23 +P14, X6 =iP31 +P24; then the fundamental identity becomes Xl +X2 2 +2 +X 2 + 2X52 +X62 = 0,.............. (I) and any set (xi) satisfying this relation may be taken as the coordinates of a line. The six complexes xi=o have the remarkable property that every two of them are apolar. There is an infinite group of linear substitutions transforming EXi2 into a similar sum of squares Zyi2; so we have an unlimited number of coordinate systems of this type. We shall call them canonical systems. In canonical coordinates the condition that two lines (x), (y) may meet is Exiyi =o; the invariants of two linear complexes a. = o and bx =o are Eai2, lbi2, and 2Eaibi. 282. Linear Congruences. A linear congruence is the set of lines which simultaneously satisfy two linear equations ap=o and bp=o. If (X, p), (X', ju') are any two different homogeneous constants (i.e. such that XJ/' - X'/ is not zero), we may replace the given equations by Xap + fbp = o, X'ap +Jl'bp==o. Suppose now that the pencil of complexes containing ap, b2, comprises two different special complexes; then the congruence consists of all the lines common to the two special complexes. That is to say (Art. 279), the congruence consists of all the lines which meet the directrices of the pencil of complexes. Therefore A general linear congruence consists of all the lines which meet two fixed lines, skew to each other. These lines are called the directrices of the congruence. 286 PROJECTIVE GEOMETRY A plane disjoint to both directrices contains just one line of the congruence; a plane through one directrix contains a flat pencil of rays of the congruence. Similarly, through a point disjoint to the directrices can be drawn one ray of the congruence, while a point on a directrix is the centre of a flat pencil contained in the congruence. As a limiting case, we have what we may call a reducible congruence, whose directrices intersect. It breaks up into a plane field of lines, and a bundle; the plane of the field, and the centre of the bundle, being the plane and point which are conjoint to both the directrices. This corresponds to the case when all the complexes of the pencil Xap + ubp = o are special. Finally, we have a special linear congruence, consisting of all the lines, belonging to a general linear complex, which meet a fixed ray c belonging to it. If bp = o is the complex, and c =o the special complex associated with c, the pencil Abp + pp = o contains no special complex except c,; and the line c may be regarded as a double directrix of the congruence, obtained by moving into coincidence two rays of the complex bp. Every plane through c contains a flat pencil of rays of the congruence, and every point of c is the centre of such a pencil. A good way of realising a special congruence is to take a conicoid, and consider all the lines which touch it at points on a given generator; these form a special linear congruence. To prove this, let xt - yz = o be the conicoid, and x=o, y=o the given generator. Any point on this generator may be taken to be (o, o, z1, t,) and any point in its tangent-plane z1y - tax = o to be (zl, t,, h, k). Calculating the radial coordinates of the corresponding tangent line, we find them to be, in the order (p23, P31, P12, P14, P24, 34), ( - ztl, z2, o, - Ztl, - tl2, zlk - htl), from which we deduce the linear relations 2 =o, p23-P14 =o.......................... () LINE GEOMETRY 287 which, as the reader will easily verify, define a special linear congruence. 283. Congruence associated with a Skew Involution. Since any pair of skew lines fixes a linear congruence, and also a skew involution, there is a one-to-one correspondence between the involutions and the congruences. We proceed to express this in an analytical form. By Arts. 264, 265 we may suppose that a given involution is represented by the equations x': y' ': t'=cy -bx: by -ax: ct - bz: bt -az, hence the line joining any point to its corresponding point has for its radial coordinates P12 = ax2 - 2bxy + cy2, P14 - 2bxt + azx + cyt\ P23=2byz-azx -cyt, p24=a(yz -xt),....(I) P31 = - C(yz - xt), p34 = aZ2 - 2bzt + ct2 whence we deduce the relations ca(p23 + 14) = - 2abp31 = 2bcp24.................. (2) which define a linear congruence. If we take its equations in the form aP23 + apl4 - 2bp24 = 0 () ap31 + c24 = 0 the invariants and joint invariants of these complexes are A =a2, B=ac, M= -2ab; so the equation to find the special complexes of the pencil (aP23 + apl4 - 2bp24) + 4(ap31 + cp24) = reduces to a(aX2 - 2bXk/L + cr2) = 0.. (4) a(aI 2 -2b A. +c/c 2) =......................... (4) Conversely, let a,=o and bp=o be any given general linear complexes with invariants A, B, M; then we can infer that there is a tetrahedron of reference for which the equations of the given complexes are of the form A (p23 + p14) + Mp24 =0..o (5) Ap31 +Bp24o 3. 5 288 PROJECTIVE GEOMETRY and for which the equations of the skew involution corresponding to the congruence a = bp = o are XI ': y': z': t' =2By +Mx: -My -2Ax: 2Bt+Mz: -Mt -2Az.....(6) 284. Line-equation of a Quadric Surface. Let the equation of a quadric surface be F(xl, x2, x3, X4) = allX2 +... + a4442 + 2a12X1X2 +. + 2a34X3X4 = 0, and let (x), (y) be any two points disjoint to it. The point (Xx1 + yi,... Xx4 + fy4) will be on the surface, if X2Fx +- 2XxPxy + I 2Fy = o, where F =F(x1, 2, x3, x4), Fy=F(y1, Y2, Y3, Y4), and Pxy = axlly, +... + a12(x y2 + x2y. ) +... This quadratic, then, determines the points where the line (xy) meets the surface. The points coincide, that is, the line touches the surface, if FxFy - Pxy2 = 0. Clearly, this can only depend on the combinations (xiyj) or pij which are the radial coordinates of the line. By picking out the coefficients of x12y22 and x1x.y22, and using the principle of symmetry, we find the condition in the form 2 (aiiajj - aij2) pj2 + 2Z (ajaki - ailajk) Pikpjl =0. This may be called the line-equation of the quadric; it is a very special form of quadratic complex. 285. There is a point in the general theory of congruences to which attention should be called. Just as the intersection of two surfaces may break up into two distinct algebraic curves, so the common elements of two complexes may break up into two or more distinct congruences, and it may be impossible to define one of these congruences as the complete intersection of two complexes; just as a twisted cubic curve cannot be obtained as the complete intersection of two surfaces. We can, however, when a LINE GEOMETRY 289 complete congruence forms only part of the intersection of two complexes = o, X= o, find an additional set of complexes X = o, X2 = o,... X, = o, finite in number, such that the congruence considered is contained in each of the (m + 2) complexes, and comprises all the elements which they have in common. Thus we have, so to speak, isolated the congruence as a single algebraico-geometrical form. M.P.G. T CHAPTER XXXI. SPECIAL METHODS. 286. The present chapter has been written with some hesitation, because a student may regard it as a set of " tips " enabling him to prove projective theorems by a sort of sleight-of-hand, without being able to give, or even appreciate, a general demonstration. On the other hand, when he has really mastered the general theory, these special methods will be useful as well as amusing, because they not only reduce general theorems to special metrical ones, but they enable us to infer the general theorem for which any given metrical theorem is a special case. Practically, all these special methods depend on the theorem (Art. 71) that two plane fields can be made projective to each other by making the vertices of any quadrangle in the one correspond to the vertices of any quadrangle in the other. By suitably choosing one of the quadrangles, and especially by taking two of its vertices to be the focoids in its plane, the projective properties of a figure in the one field can be interpreted as metrical properties of the corresponding figure in the other, and conversely. Before going into detail we will give an example. If two triangles ABC, STU are circumscribed to the same conic, their six vertices lie on a conic. The general proof is as follows. Let the pencil S(BCTU) meet BC in the points BCB'C', and let the pencil A(BCTU) meet TU in the points T'U'TU. The range BCB'C' is that in SPECIAL METHODS 291 which the tangent BC is cut by four other tangents BA, AC, ST, SU; and the range T'U'TU is that in which the tangent TU is cut by the same four tangents in the same order. Hence (Art. 103) BCB'C' T'U'TU, and therefore, by projection from S, A, S(BCTU) A(BCTU), which proves that ABCSTU lie on a conic (Art. 107). To obtain a special theorem, we suppose that T, U are focoids. The conic is now a parabola, because the tangent TU is at infinity; the conic ABCSTU is a circle, and S is the focus of the parabola. Hence If ABC is a triangle circumscribed to a parabola of which S is the focus, the points A, B, C, S are concyclic. This, of course, is a well-known metrical theorem; the important point to notice here is that not only can we deduce the metrical theorem from the general one, but we are entitled to infer the latter from the former. The reason is that neither the focoids, nor a parabola, have per se any special properties of a projective kind; and if we take them together, we have a figure projectively equivalent tL - y conic and any two points on any one of its tangents (each distinct from the point of contact). 287. Laguerre's Definition of an Angle. Taking ordinary rectangular coordinates, consider the equations y-xi=o, y-xtanO=o y+xi=o, y-xtan=o.................... Of these the two y ~ xi = o are the isotropic lines through the origin. By definition (Art. 2II) 3(tan6,.tan, (, -i=(i - tan 0) (tan i + i) (tan, - i) ( - i - tan 0) = e2(0-)i.............................(2) on reduction, making use of the relation cos x + i sin x = eix. Hence 0 - = -log l1(tan 0, tan q, i, -i),.............. (3) and we thus have the analytical result that 292 PROJECTIVE GEOMETRY If s, So are the isotropic rays of a fiat pencil, and a, b are any two other rays, we can define the angle between a and b by the equation z ab = z log l1(absso)........................ (4) There is still a certain ambiguity about this definition for two reasons. In the first place, if we interchange s, so, the value of L ab changes sign; this is connected with the senses of rotation in the plane, and there will be no ambiguity if we take s, so in a definite order. But, in the second place, since a logarithm is only determinate up to a multiple of 2ri, the value of Lab is only determinate up to a multiple of rr. This corresponds to the determination of an angle by means of its tangent, which is the only way available in analytical geometry, so long as we avoid irrationalities. However, there will be just one value of the logarithm for which, when a, b are real, o < ab < r, and this value of Lab we shall call its principal positive value. We can now easily prove that if Lab and Lbc have principal positive values, L ab + b =L ac +,.................... (5) where L ac is a principal positive value, and E=o or i according as Lab+ z bo is, or is not, less than 7r. This agrees with the ordinary theory of the addition of angles. For complex angles the principal positive values are those for which the real part P satisfies the conditions o </3 <7r; these values are perfectly definite. 288. Analytical Definition of Distance. We might expect to find a correlative definition of distance in terms of a cross-ratio; but we cannot find this by an analogous process if we start with rectangular point-coordinates. The reason is that although they lead to two special points (the focoids), they lead to only one special line (the line at infinity), so that there are no two special points, as a rule, SPECIAL METHODS 293 collinear with two given points, in the same way as there are, in general, two special rays concurrent with two given rays. However, by a rather artificial process, we can more or less establish an analogy as follows. Let + y2 _r2 =...............................(I) be any circle, and (xl, y) (x2, y,2) any points. Putting X: y: I = XX1 + PX2: AY1 + 1.Y2: A + M and expressing that (x, y) is on the circle, we obtain the equation X2 (X12 + y12 - r2) + 2X1j (XIx2 + Y1Y2 - r2) + p2 (x22 + y22 - r2) = 0, which, for brevity, we shall write S1A2 + 2P12APi + S212 = 0....................... (2) Let (X1, PlI), (A2, 12) be the roots of this equation; then if A, B are the given points, and P, Q are the points where AB cuts the circle, we have ')11.2 -- P12 + --- JP2 - _1 R=R(ABPQ) =A122= 1P2_ P -S-S2 - API1- - P12 T 12 -.(lk) Now suppose that r is very large compared with x12 + y12, x22+ y22, and xxx2 + YY2. Then we can expand the righthand side of (3) in descending powers of r; the result is ~I i+ rl-(X1 - X2)2 + (Y1 - Y2) +Br-2 +... I T r-1/(xl - x2)2 + (Yi - y2)2 + B'r-2 +... = I + 2J(x1 - X2)2 + (Y1 - Y2)2/r + C/r2 +.... Hence in the limit when r becomes indefinitely large Lt log 1 (ABPQ) = + /(X1 - x2)2 + (Y - Y2)2.......(4) It is enough to suppose that r goes to its infinite limit through real values; the result is the same as if we suppose r 1, the absolute value of r, to become infinite in any way. The ambiguity of sign in (4) arises from the fact that we cannot, in the rational field of coordinates and coefficients, distinguish P from Q, and by definition (ABPQ).(ABQP)= I. 294 PROJECTIVE GEOMETRY It is curious, however, to see how the ambiguity persists; because the points P, Q are ultimately coincident, from a projective point of view. The reason seems to be that we have made P, Q coincide in a special way. It should be noticed that in (4) the logarithm is a definite function and not the general one. In fact, if we put I (ABPQ) = +z, the value of z is ultimately very small, and the logarithm intended is that expressed by the onevalued function z - z2 + z3 _... otherwise the expression on the left of (4) could not have a finite limit. Equation (4), then, shows how the analytical expression for the distance between two points may be transformed to an expression which is the limiting value of the product of the logarithm of a cross-ratio by a parameter (~r) which it contains. Another point to be noticed is that since, in this connection, the circle has to be considered in the form ( + i) (x - iy)(xiy) r2 =0, the ultimate circle presents itself as the line at infinity counted twice; and we may still consider the focoids as being special points upon it. An analogous case is presented by the real radical axis of two circles considered as the limiting case of a circle of the coaxial pencil defined by the two given circles. Its intersections with the circles will be, in this connection, special points upon it. In the case we are dealing with, the speciality of the focoids consists in the fact that if two points are collinear with one of the focoids, their distance vanishes, and conversely. The most vivid way of expressing this result is to say that the sections of an orthogonal involution of rays by all the real lines in its plane, give, when a sense is ascribed to the involution, a set of collinear points which are such that, although they are distinct, the distance between any pair is zero (cf. Art. 227). Of course, with a real system of SPECIAL METHODS 295 rectangular coordinates, the distance between two real points cannot vanish unless they coincide; but this does not affect the theorem. It may be noted that the distance of any complex point from its conjugate complex point is a pure imaginary, but not conversely. From the identity, in which A, B, C, P, Q are collinear points, J3(BCPQ) R(CAPQ) 1R(ABPQ)= I, we conclude, in general, that log 13(BCPQ) + log 1R(CAPQ) + log IR(ABPQ) = 2k7ri, where k is an integer. Let r be the radius of any circle through P, Q, and let the logarithms be so chosen that rlog](BCPQ),tc, etc., have finite limits: then k must be zero, and if we multiply by ~r, and proceed to the limit (r->oo), we have BC + CA + AB=o connecting the distances of three collinear points, taken with appropriate signs. The reader should work out the analogous theory in three dimensions: here we take a set of concentric spheres instead of concentric circles. We find in the same way that Lt x2 + y +2 ) may be considered as the plane at infinity reckoned twice over, with the focoidal circle as a special curve upon it. A line that goes through a focoid (that is, an isotropic line) may also be called a line of zero segments. The whole set of such lines consists of the lines that meet the focoidal circle; they form a degenerate case of the special quadratic complex of lines which touch an ellipsoid of revolution, when the ellipsoid flattens down into a circular disk (reckoned twice). 289. Symmetrical Theory. Let S, E be two proper conicoids, in a general position to each other, the first being considered as a point-locus, and the second as a planeenvelope. 296 PROJECTIVE GEOMETRY If A, B are any two points disjoint to 8, the line AB will meet S in two points P, Q. Let m be an arbitrary constant; we define the projective distance AB by the equation AB = m log I(ABPQ)....................... (I) This function of A, B, S has a period 2m7ri, and by choosing m appropriately we can make it equal to any assigned period o; in fact, m = o/2ri. Similarly, let a, / be any two planes disjoint to 2: we define the projective angle a/3 by the equation L al = n log 1(ap/rK),..................... (2) where n is any constant, and ay, K are the two planes of I which pass through the line a/3. This function has a period 2n7ri, which we may suppose equal to an assigned quantity o'. We shall say that (S, E; o, o') define a projective metrical system of reference. We now at once infer the theorem If A, B, C are three collinear points BC + CA + ABo (mod ),................... the notation meaning that BC + CA + AB is a multiple of w. Similarly If a, /, y are three coaxial planes Lz p3 + ya + L aa/3 (mod o')................. (2) Suppose, now, that a, b are two intersecting lines. The plane ab will intersect S in a conic; let t, u be the tangents to it from the point ab. We can form a definition of what we may call the angle Sab by putting L ab =m' log l(abtu),.......................(3) where m' is any constant. Taking the same two lines a, b, from the point ab we have a tangent-cone to Z: let t', u', be the lines in which this cone is cut by the plane ab. Then, if we put L/ ab =n'log R(abt'u'),.........................(4) where n' is any constant, we have a definition of an angle:ab which we may call the angle correlative to z sab in SPECIAL METHODS 297 the system (S, I; w, 'o) for the multipliers mi', n'. These functions zLSab, z 2ab will have periods 27rim', 2rrin', and in order that they may have the same period it is necessary and sufficient that m' =n'. This does not, however, make zLSab=z lab for all positions of the coplanar pair a, b; so we do not gain any essential simplification. We have two relations similar to (I) and (2) above for three concurrent and coplanar rays a, b, c. Let us now suppose that ==o consists of the tangent planes to S =o, so that in a certain sense we have only one conicoid of reference. The effect of this is that, on putting mn' n', the quantities z Eab, z Sab have the same value for any pair of coplanar lines, and we may denote each of them by z ab. Except for its sign, this is a determinate quantity, up to multiples of 2mn'ri. We still have numerous discrepancies with the ordinary theory, since we still have a symmetric system. There are three independent periods, say 01, w2, (12, for distances, dihedral angles, and plane angles respectively; and, if S is real, there will be complex distances and angles defined by real pairs of elements. For instance, let (o be real; then the multiplier m is a pure imaginary, and the distance AB is only real if the corresponding cross-ratio (ABPQ) is negative; that is to say, if A and B are separated by the surface S. The points P, Q are both at an infinite distance from A, because RI(APPQ) = oo and R(AQPQ) = o. Thus every point on the surface 8, and no other point, is at infinity in the metrical sense of being at an infinite distance from all other points of space. The reader should particularly notice how different this is from the elementary theory, in which all the points at infinity lie in a plane. The only way of reconciling the two points of view is to suppose that S =o represents the plane at infinity, reckoned twice over. But here we are met at the outset by two difficulties, one analytical, and the other, in a way, geometrical. If we put S = (alX1 + 2f2 + a 33 + a4x4)2 298 PROJECTIVE GEOMETRY where a =o is the plane at infinity, the corresponding equation E =o reduces to the identity o =o, so there is no definition of angles. Moreover, the cross-ratio 1R(ABPQ) defining the distance AB becomes (ABPP), which is I, so every distance, so long as m is finite, reduces to zero, and the theory of distances also breaks down. The difficulty is removed in the following way. To fix the ideas let S = o be a conicoid cutting the plane at infinity in a proper (real or complex) conic K. If t = o is the equation of the plane at infinity, the equation S + t2 = o,...............................(5) where (X, /) is a homogeneous parameter, represents a set of conicoids touching S =o at every point of K. The equation of the surface AS+ t2=o in plane-coordinates is found to be X2(X + ) =o,.............................(6) where; =o is the equation of S in plane-coordinates, and if (al, a2, a3, a4) are the coordinates of t, and (al, a1,... a34) the coefficients of S, ( = (a122 - 1al12) (a2u1 - au2) 2 +...; in other words (Art. 284) I =o is the condition that the line (au) may touch S. That is to say, 4 = o is the equation of the conic K in plane-coordinates. Let us divide (6) by X2, and proceed, after this, to the limit X=o. The result is the definite conic 4 =o, which will serve as well as a proper quadric envelope to define angles between lines (as limits of L Zab) and planes. But in order to obtain a limiting definition of a distance from (5) we must first find the general expression when X, / are finite, and expand it in ascending powers of X//. The result is a convergent series of the form log R(ABPQ) 2dl(XA//)i + d2X/! +... and leads to Lt. (/AX) ilog R(ABPQ) =d................... (7) \ so SPECIAL METHODS 299 where dl2 is a definite function of the coordinates of A, B, in general finite in value. This, or any finite multiple of it, we may take to be the square of the distance AB by definition. We shall still have the relation BC + CA + AB = o for three collinear points, if we adjust the signs of the distances properly. The analysis, for rectangular coordinates, has been given in Art. 288. Correlatively, we have a special case when the pointlocus S=o is a cone, so that: =o degenerates into its vertex taken twice over. From this we can deduce a special projective metrical system in which dihedral angles are definite, except for their sign, whereas distances and linear angles are periodic. There are no other limiting cases of interest to be considered. It should be remarked that the arguments of the present article (unlike those of Arts. 287, 288) are projective, because although we use coordinates, we may suppose them projectively defined by means of casts. Thus there is no vicious circle in defining angles and distances by cross-ratios, as there would be if we began, in the usual way, by defining a cross-ratio of four collinear points by means of ratios of segments. Cayley was the first to deduce an analytical theory of metrics by a method analogous to that employed in Art. 288; it was afterwards pointed out by Klein that unless the coordinates used had projective definitions, the results could not be regarded as forming a projective theory of metrics, in the proper sense of the term. The theory of casts rebuts the objection. 290. A few words may be added as to the geometrical aspect of the limiting case when S=o degenerates into a double plane. Consider, in a plane, the pencil XE + t2 =o............................... (I) where E is a real ellipse, and t a real line, meeting E in the real points H, K. The pencil consists of conics having double contact with E at H, K, and it is clear that when X//, is positive, the corresponding conic is an ellipse. 300 PROJECTIVE GEOMETRY Suppose, for instance, we take X, [A both positive, and at the same time choose the sign of E in such a way that (x, y) is outside the ellipse when E(x, y) >o. Then to satisfy (I) we must have E (x, y) <o for all real points on the locus, which is consequently an ellipse wholly within E except at the points of contact. If we now make A//u go to the limit o through positive values, intuition suggests that the result is the finite segment H K considered as an exceptional ellipse with vanishing transverse axis. Similarly, by making X/A/ approach the limit o through negative values, we seem to arrive at the complementary segment K.H, reckoned twice over, as the limit of the hyperbolas of the system. But now we have the whole line HK reckoned twice, that is, a locus of the second order, appearing as a combination of two conics, which is a locus of the fourth order. Moreover, if we make X//i go to the limit zero in any way, the ultimate form of (I) is t2 = o, a locus of the second order. The reason of the paradox is this. Consider any real flat pencil, say the real lines perpendicular to HK; then, so long as A//h is different from zero, these lines will meet the conic (I) in a set of points which are partly real, and partly complex. In the limit X//=o these points become all real, and consist of all the real points of the complete line HK, each reckoned twice; the points on K.H arising from the coincidence of two conjugate complex points by the vanishing of the imaginary parts of their coordinates. Thus: If on a given line we mark two points H, K, the whole line, reckoned twice, may be regarded as the limiting element of any pencil of conics which have double contact at H, K. As being a limiting form, the line contains two special points H, K. We are now able to see that in the metrical theory when S =o is a double plane with a special conic K marked upon it, the whole plane, reckoned twice, may be regarded as the limiting element of a pencil of conicoids which touch at every point of K. We obtain the same result whether we " flatten " ellipsoids or hyperboloids of the system, because as soon as we completely flatten an ellipsoid, its comple SPECIAL METHODS 301 mentary flat hyperboloid at once starts into existence as a locus of real points on the ellipsoid, and vice versa. The reader is probably familiar with a corresponding phenomenon in the case of a system of confocal conicoids: here the three principal planes, each reckoned twice, and with a focal conic as a special curve upon it, form the three limiting surfaces of the system. Similar remarks apply to the correlative special system when S is a cone. E = o is now a point reckoned twice, with a special cone of rays passing through it. 291. Perpendicularity. Returning to the general case, we shall say that two points A, B are perpendicular when they are harmonic conjugates with respect to S; and that two planes a, / are perpendicular when they are harmonic conjugates with respect to E. In the first case AB = ol, and in the second case L ap/=- 2. In the case when S degenerates into a cone, the definition of perpendicular points remain valid, but that of perpendicular planes breaks down; vice versa, when Z =o degenerates into a conic, the definition of perpendicular planes is still valid, but that of perpendicular points breaks down. Taking the case when Z is a conic (or more strictly a flat conicoid) and putting n'=n, the reader will prove without difficulty that if any two planes a, / are cut by a plane y perpendicular to both, then L a/ = L >ab, where a, b stand for the lines ay, /py. Properly speaking, the angle sab does not exist; so we may, if we like, put / ab for / Eab. When I =o is the focoidal circle, this is a well-known metrical theorem; there is, of course, a correlative theorem in the general case. 292. Parallels. In the special case when Z is a flat conicoid, it follows by definition that two concurrent lines which make a zero angle with each other meet on the plane of E, and conversely. Hence, if the metrical condition of meeting at a zero angle is to be equivalent to the 302 PROJECTIVE GEOMETRY quasi-projective condition of meeting at infinity, we must take the plane of Z as being the plane at infinity. This, of course, follows if we start with the ordinary rectangular Cartesian system of coordinates; but it should be noted that in the projective theory of metrics, if we take any plane to be the plane at infinity, and any conic in it to be the focoidal circle, then they determine a special projective metrical system which, by a proper choice of coordinates, can be made an exact algebraic image of the ordinary rectangular Cartesian system. 293. We are now in a position to generalise almost any metrical theorem, or translate a general theorem into various special metrical theorems. In the examples which follow we shall, for brevity, say that a figure F can be projected into a figure F' when there is at least one collineation (or correlation) which transforms F into F'. For plane collineations the term is strictly accurate, because (Art. 71) any such collineation may be effected by a perspective chain; and it is convenient to use the same term for the other cases. (i) If A, B, C, D are any four fixed points on a conic, and S a variable point on it, the value of the cast S(ABCD) is constant. In fact, if S, T are any two positions of S, the conic STABC is generated by putting S(ABC...) 7T(ABC...), and since D is on the conic, by hypothesis, we have S(ABCD) 7T(ABCD). By projecting C, D into focoids, we deduce: If A, B are fixed points on a circle, and S a variable point on it, the lines SA, SB include a constant angle. If, in the original figure, CD is conjugate to AB, the set (ABCD) is harmonic, and AB goes through the pole of CD: hence, as a special case, since AB becomes a diameter of the circle, The angle in a semicircle is a right angle. (ii) Any two conics in the same plane, and in a general position, may be simultaneously projected into two circles. We see, by algebra, that the two conics intersect in four different points A, B, C, D. By projecting any two of these SPECIAL METHODS 303 into focoids, the conics become circles. Suppose that C, D become focoids, and A, B become any two points H, K. The casts S(ABCD), T(ABCD). where S, T are arbitrary points on the given conics, are projectively invariable, hence in the transformed figure H K subtends angles at the circumference of the circles which are determined by the given conics. Now suppose that the smallest positive values of these angles are complementary (in the usual sense): then the circles cut at right angles, and the corresponding condition for the conics is that, if t, u are the tangents at A, the pencil (t, u, AC, AD) must be harmonic. We shall return to this case presently. (iii) As another example, let us generalise the theorem that tangents to a parabola from any point on the directrix are at right angles. The result is that If the lines SJ, SJo, JJ0 touch a conic at A, B, C respectively, the tangents to the conic from any point on AB are separated harmonically by J, Jo. The special theorem follows by supposing that J, Jo are the focoids. To prove the general theorem, let P be any point on AB; and let x, x', the tangents from it, meet JJO in X, X'. By construction, the whole set of tangents is paired in an involution (Art. I36) of which AB is the axis: hence the section of it (XX'.YY'...) by JJ0 is an involution (Art. 132 (ii)). The double points of this last involution are J, Jo; hence (Art. 9I) the range JJ0XX' is harmonic. (iv) The orthocentre of any triangle circumscribed to a parabola lies on the directrix (Steiner). A very elegant proof of this is due to J. C. Moore (Salmon's Conics, Art. 268). Let a, b, c be the given tangents; a', b', c' the tangents perpendicular to them; f the line at infinity, which is also a tangent. Applying Brianchon's theorem to the hexagon abco'fa', we infer that the three joins (ab, c'f), (bo, fa'), (aa', cc') are concurrent; now the first two of these are perpendiculars of the triangle abc, and the third is the directrix. This proves the theorem; 304 PROJECTIVE GEOMETRY not only so, but it provides a proof that the three perpendiculars of any triangle are concurrent. For if ABC is the triangle, there is a range of parabolas inscribed in it; taking any one of them, and proceeding as above, we prove that if h1, h2, h3 are the perpendiculars, and d the directrix of the parabola, the three triads h2h3d, h3h1d, hlh2d are concurrent, whence also hlh2h3 is a concurrent triad. As a corollary we infer that the directrices of a range of parabolas form a flat pencil. 294. Canonical Equations of Two Conics. Let S, S' be two conics in a general position, so that they cut in four distinct (real or complex) points A, B, C, D forming the vertices of a quadrangle. Let xyz be the diagonal triangle of ABCD; this (Art. II9) is a common self-conjugate triangle for the conics, and the only one. Taking it as a triangle of reference the equations of the conics must reduce to S=ax2 +by2 +cz2 =o1 (I) S' =a'x2 + b'y2 + c'z2 = o.................... with corresponding tangential equations Z=bcu2 +cav2 +abw2 =0o (2 i = b'c'u2 + c'a'v2 + a'b'w2 = o. ( By suitably choosing the unit point we may reduce the equation of either conic to the form x2+y2+z2=o; but this produces a certain loss of symmetry, and may lead the beginner into error, when he deals with invariants. The discriminants of AS + jS' and AX + pE' are respectively A (XS + MkS') = (Xa + lia') (Xb + /b') (Xc + Ic') A (X + EZ') = (Abc + pb'c') (Aca + Mc'a') (Xab + a'b'),.(3 hence the condition that AS + /cS' = o may represent a linepair is X3abc + X2plZbca' + XA/2b'c'a + /3a'b'c' = o...........(4) Clearly we shall have a corresponding equation for a given pair of conics, whatever the triangle of reference may be: it is usually written AX3 + OX21A + O'AA2 + A2 + A = 0,..................(5) SPECIAL METHODS 305 and A, 0, 0', A' are called the invariants of the system. The reason is that if 8, S' are connected with two other conics T, T' by the same collineation, the equation A(XT + IT') =o, agrees with A(XS+ /-S') =o, except for a numerical factor 86, where 8 is the determinant of the linear substitution defining the collineation. Suppose that we replace equations (i) by pS= o, S' = o; the geometrical system is the same, but the invariants become p3A, p2o-O, pcr20, 2 r3A', so we arrive at the conclusion that there is a definite triangle of reference for which the invariants of two given conics may be assumed to be of the form A, 0, 0', A'=p3abc, p2o-Ebca', pO-2b'c'a, o-3a'b'c' where (p, a) is a homogeneous parameter, neither zero nor infinite. If, now, we take a homogeneous polynomial F(A, 0, 0', A') such that F(p3A, p20o, pa-20, -,3A) _ pmicnF(A,, 0, ',') identically, we obtain what we may call a joint geometrical invariant of S, S', because the equation F=o persists, not only if we submit 8, 8' to the same collineation, but also if we replace S =o, S' =o by the equations pS=o, rS' =o. For instance, AA' - ' is such an invariant, with =3. Under these circumstances the equation F = o will represent some projective property of the system of two conics. The condition satisfied by F may be called the homogeneity condition: it involves the condition that F is homogeneous in the ordinary sense. 295. Rational and Irrational Invariants. Let P be any point on: then I3P(ABCD) is invariable, as P moves along the conic S. But this is not a rational invariant; that is to M.P.G. U 306 PROJECTIVE GEOMETRY say, it cannot be expressed as a rational function of the coefficients of S and S'. The reason is that the coefficients of A, B, C, D, starting with a general form of the equations S=o, S'=o are determined by a biquadratic equation which is, in general, irreducible in the field of the coefficients, together with a linear equation which is rational when we adjoin any one root of the biquadratic, i.e. when we regard that one root as rational. Hence we may consider 13P(ABCD) as an irrational invariant of the system. By permuting A, B, C, D in all possible ways we obtain six different values of 1RP(ABCD), in general, and these will be the roots of a sextic (Art. 215), 256 (X2 - X + )3 = jX2 (X _ -)2 where j is a rational geometrical invariant. We can find j in terms of A, e, 0', A' by taking the equations of S, S' in the canonical forms s=ax2 +by2 +cz2 =o1 (I) s' =a'2 + b'y2 + c2 =................ and calculating RP(ABCD) from the limiting value BA(ABCD). Let us take X, 1/, v with determinate signs so that X2: 2 2 = bc' - b'c: ca' - c'a: ab' - a'b...........(2) then the coordinates of A, B, C, D may be taken to be (X /A, v), (- X,, v), (XA, -), - ), (X,, - v), respectively. The equations of the tangent at A and the lines AB, AC, AD are respectively aXx + bIty + cvz = o, vy- z=o, X-vx= o, /-x-Xy=o. Hence we obtain the equations of AC, AD in the forms v(aXx + bly + cvz) - b/ (vy - hiz) = 0, p (aXx + bly + cvz) + cv (vy - 1fz) = o, whence we deduce the cross-ratio (AB) b cv b (ca' -c'a) IRA (ABCD)= v - a -- a v A c(ab' -a'b) SPECIAL METHODS 307 Calling this a-, and calculating the expression (0.2 - a + I)3/c-2 ( - 1)2, we find (T2 - + I)3 (b22a'2 _-abc bc'a')3 () 2 (o. - I)2 - a2b2c2(bc.)2(ca.)2(ab')2 '.. and since the denominator can only vanish when A=o, or when A(XS + AS') =o has equal roots, we conclude that (o-2 - o-+ 1)3 G-2(a- 1)2 (02 - 3Ao,)3 A2(020'2 + I8AA'00 - 4AO'3 - 4A'e3 - 27A2A'2)' As a verification, we find that r + i =o is a double solution if 203 - 9Ae/' + 27A2A' =.....................(5) This is the condition, then, that the intersections of S with S' may (in a proper order) form a harmonic cast upon S. If the denominator of (4) vanishes, and A is not zero, the conics touch; if the numerator vanishes, we have the equianharmonic case. If, in these formulae, we change A, 0, 0', A' into A', o', 0, A respectively, we have corresponding results for IRQ(ABCD), where Q is now any point on S'. We obtain a group of correlative formulae by starting with the tangential equations E=o, Y'=o given in Art. 294. It is unnecessary to go through the work in detail, if we observe that A(XA + /'f) = A2X3 + 'X2/ + A'XA + A 2 + A2/;....... (6) all we have to do is to change A, 0, 0', A' into A2, AO', A'0, A'2 respectively. If, then, r is any one of the six cross-ratios of the points where the common tangents of S, S' are cut by any other tangent to 8, we find from (4) above that (T2 - T+ I)3 72(7 - 1)2 (0'2 - 3A'0)3 A'2(02012 + IAA'00e -- 4AO'3 - 4A'03 - 27A2A'2)' 7 308 PROJECTIVE GEOMETRY whence also the harmonic case occurs if 2e'3 -9A\' ' + 27AA'2=o..................(8) Comparing results, we arrive at the remarkable theorem that the four common points and the four common tangents of two proper conics in a general position are so related that the values of the six casts determined by the four points on either conic are the roots of the same sextic as that which gives the values of the six casts determined by the four tangents to the other conic. In particular, the two sextics (4) and (7) coincide if A/2(O'2 - 3A'O)3 - A/'2(2 - 3AO')3 =o.............. (9) This is, then, the invariant condition that the four common points and the four common tangents to two conics may determine on each conic the same sextet of associated cast-values. Of course, if A, B, C, D are the common points, the value of the cast ABCD on S will be different to its value on S'; otherwise the conics would be identical. But we may have, for instance, ABCD on S equal to ABDC on S'; the invariant condition (9) will then be satisfied. Looking at the matter from a rather different point of view, let ABCD be a given quadrangle; then we have in connection with it a pencil of conics {ABCD}. Let SABCD be any conic of the pencils; we may regard it as the locus of a point P for which the cast P(ABCD) =S(ABCD) =o-, where o- is a constant, determined by S. Conversely, every constant o- determines a conic of the pencil in this way, and we may speak of " the conic (o-)." Now let (o-, r2,2... *6) be the roots of any equation of the form 256(o-2 - a+ I)3 - ja2(r - I)2 = o, (cf. Art. 215) where j a prescribed constant; then we have, in general, an associated sextet of conics of the pencil, each pair of which satisfy the invariant relation (9). When j = o we have a dyad of conics reckoned thrice; when j=1728 we have a triad of conics reckoned twice. SPECIAL METHODS 309 296. The relation 00'-AA'o o. We return now to problem (ii) of Art. 293. Let the conics be taken, as before, in the forms S=ax2 by2+cz2 =o S = ax2 + by2 + cz2 = o S' =a'x2+ b'y2 + c'z2 =o, and let A, B, C, D be their intersections (X, /, v), (-A, A, v), (X, - A, v), (X, ju, - v), as in Art. 295. Let t, t' be the tangents at A to S and S', so that their equations are akx + buy + cvz =o, a'Xx + b'uy + c'vz =o. The equation of AB is vy - z = o, which is equivalent to a'(aXx + biy + cvz) - a(a'Xx + b'y + c'vz) = o, and similarly the equation of AC is b'(aXx + b/uy + cvz) - b(a'Xx + b'uy + c'vz) = o. Hence we have R(t.t'.AB.AC) =ab'/a'b, and similarly R(t.t'.AC.AD) =bc'/b'c, 1(t.t'.AD.AB) =ca'/c'a. If any one of these has the value -I, (bc' + b'c) (ca' + c'a) (ab' + a'b) = o, which, by Art. 294, reduces to 00' - AA/'= o.......................... (I) Suppose, in particular, that R(t.t'.AC.AD) = -I; then by projecting C, D to the focoids, we make the conics circles, and these circles cut orthogonally. Conversely, whenever (I) is satisfied, we can project the conics into two orthogonal circles. It should be remarked, however, that we cannot take any pair out of A, B, C, D to project into the focoids: thus, in the case when R(t.t'.AC.AD)= -I, we may take (C, D) or (A, B) for the pair, but if we take (A, C) or (B, D) for the pair, the conics in each case become circles, but they are not orthogonal. 310 PROJECTIVE GEOMETRY If we take Cartesian equations of any two real circles in the form x2 + y2 _r2=o (x_-d)2 + y2 2=o, the condition 00' - AA' = o reduces to (d2 _ r2 - S2) (d2 - 2r2 _ 2s2) = o. If the first factor vanishes, the circles cut orthogonally. If the second factor vanishes, the circles meet in two conjugate complex points A, AO besides the focoids J, JO; and the meaning of the invariant relation is that if we draw the (complex) tangents to the circles at A, they are harmonic conjugates with respect to AAo and one of the lines AJ, AJo. It is rather interesting to verify this algebraically. 297. Generality of the Theory. In earlier chapters we have had numerous cases of problems which, from an elementary point of view, may or may not admit of solution. For instance, take the problem of drawing a conic through four given points to touch a given line. When the four points are the vertices of a real quadrangle, and the line is real and disjoint to them all, there may be (Art. I6I) two, one, or no real conics satisfying the conditions of the problem, if we assume that the points of contact with the line are to be real. But if we remove the restrictions of reality we may say that if A, B, C, D are the vertices of a quadrangle, and u is any line disjoint to them all, there are, in general, two definite conics which pass through the four points and touch the line; the only exception being that the two conics may coincide. The fact is that the four points determine in every case a pencil of conics through them, and this pencil determines on u a definite involution of points, which has two double points E, E' which may exceptionally coincide; and now the real or complex conics ABCDE, ABODE' satisfy the conditions of the problem and always exist, but may possibly be coincident. To suit particular cases we may, if we like, translate the conditions into other terms by means of the " real " theory of involution. This may be useful when we wish to realise the visible facts connected with the different solutions; just as, when SPECIAL METHODS 311 certain elements are at infinity, it may be convenient to introduce the term " parallel." But the great value of the complex theory is that it includes in one enunciation a large number of cases which would otherwise have to be expressed in different, and sometimes very cumbrous, terminology. Another example is the theorem of Art. I43. We are now able to say that if K, K' are any two conics, in different planes, which cut the meet of the planes in the same points X, Y, there are always two cones which pass through both K and K'. In this and other similar examples, the simplest formal proof is generally an analytical one; the theory of casts fully justifies us in adopting this method, if we choose. But the student should remember that although analysis is an excellent tool for verifying a known theorem, it is not the instrument of mathematical research. All mathematical discovery proceeds by induction, and the fact that it is convenient to put its results into a deductive form should not be misinterpreted. And even in this particular theorem, the student will find it much more instructive to see that the proof given in Art. 143 still holds, word for word, whether K, K', X, Y are real or not, than to give a neat algebraic proof, which can be condensed into half a dozen lines of writing. CHAPTER XXXII. PROJECTIVE PROBLEMS. 298. In the preceding chapters various problems have been stated, and their solutions have been given; but since each, as a rule, has been considered in a particular context, we have not had any suggestion of the natural classification of the problems of projective geometry. To give an outline of this is the object of this final chapter. The most definite, and perhaps the only definite, way of discussing the matter is to make use of analysis. By introducing coordinates, the conditions of every projective problem can be expressed by means of a system of equations; the unknowns in these equations being either coordinates of elements which have to be determined, or coefficients of equations of loci or envelopes which have to satisfy certain conditions; or, possibly, a combination of all these kinds of unknowns. For instance, to determine the conic which passes through five given points, we assume an equation ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = o to represent the conic; and by expressing that this conic goes through five given points (xi, yi, zi) we obtain five linear equations in a, b, c, f, g, h, from which, in general, the equation of the conic is uniquely determined. 299. Whenever a problem is solved in the analytical way, the algebraic solutions must always be examined, to see PROJECTIVE PROBLEMS 313 whether they really correspond to the conditions laid down. For instance, in the problem just considered, suppose we take the equation of the conic in line-coordinates; then the conditions of passing through the five points give five quadratic equations if the coefficients, and we have, in general, 32 solutions instead of one. The reason of this apparent paradox is as follows. Let the five given points be A, B. C, D, E; and for simplicity suppose them in a general position, so that there is one point-conic passing through them all. Each of the flat pencils (A), (B), etc., is representable by a linear equation in line-coordinates (cf. Arts. 125, 208); and every pair of pencils selected from them may be regarded as a degenerate envelope satisfying the conditions of the problem. There are 15 such pencil-pairs; so we have, in fant, i6 solutions, of which 15 are degenerate. The analytical solution gives each of them twice over; and the number 32 is thus accounted for. We have here an instance of some of the advantages of analysis. The algebra compels us to realise that the problems " to find a quadric locus passing through five given points" and " to find a quadric envelope for which five given points are points of contact" are really different in character, although the unique solution of the first corresponds to the only proper solution of the second. It is always worth while to try to account for what appear to be extraneous or irrelevant solutions of a system of equations derived from a geometrical problem. 300. Apart from certain abstruse arithmetical considerations, the theory of a system of rational integral equations has now reached a high degree of perfection; and we have a corresponding gain in applications of analysis to geometry. The most important result may be stated in the following form: Let al, a2, etc., be the given constants of a system of rational integral equations; let x be any one of the unknowns. If the system admits of a finite number of distinct solutions, we can, by a process of elimination and a 314 PROJECTIVE GEOMETRY subsequent process of resolution into factors, arrive at a set of rational equations fi(x; a, a2,...) = o, f2(x; a,, a,...) =o,... f(x; a,. -2..) = 0, each of which is irreducible in the field (ai); that is to say, fh cannot be resolved into a product of polynomials each rational in x and the ai. Any further step in the solution necessarily involves the introduction of new constants derived from auxiliary equations. For instance, if ax2 + bx + c =o is irreducible in the field (a, b, c), we can only solve it by introducing (or as Galois says, adjoining) the quantity t defined by t2 =b2 - 4ac. If we do this, the roots are given by 2ax + b= + t. 301. Every geometrical problem that can be made-equivalent to a system of rational equations may be said to belong to the same class as that of the analytical system. Since the classification of such analytical systems may be said to be complete, we thus have a complete classification of geometrical problems from an analytical point of view. But we can, in various ways, select classes of problems from a geometrical or a constructional point of view. For instance, the problems considered in Euclid's Elements are those which can be solved, constructionally, by means of an ungraduated straight-edge, and a compass which, as De Morgan used to say, shuts up as soon as it is removed from the paper-so that the transference of a distance by a divider is not allowed. With these limitations, the determination of points by construction can only be (i) the determination of one point as the meet of two lines, (ii) the determination of two points as the meets of a line with a circle, or of one circle with another circle. The analytical correlative is a system of two linear equations, or of a linear and a quadratic equation, or of two quadratic equations which have two known solutions in common. Conversely, if we take any such algebraic system we can always inter PROJECTIVE PROBLEMS 315 pret it as the equations of two lines, or of a line and a circle or of two circles. 302. Suppose, now, that an equation f(x)=o can be solved in the following way. Let (c) be the rational field of the coefficients of f: that is to say, the one which consists of all rational (not merely integral) functions of the coefficients. Let there be a set of auxiliary equations,ixi2 + qixi + ri = o (some of which may be linear), such that pi, qi, ri are rational in the field {(C), (pl, ql, rj)... (pi-,l qi-, Yi-1), X1, X2.... xi- }, and, in particular (p1, ql, rl) are rational in the field (c); and suppose that f(x) =o has a root which is rational in the field {(c), x,, x2...xi}: then there is a corresponding geometrical solution requiring only Euclid's constructions. Clearly Euclid's constructions cannot but correspond to such a system of equations: hence, if the analytical solution of a geometrical problem depends upon an irreducible equation of a degree higher than the second, which cannot be reduced by adjoining a root of an equation of lower degree than the third, it is impossible to solve the problem by Euclidean methods. This conclusion gives a final answer to a famous problem: that of trisecting an angle by Euclidean methods. In general, this is impossible: for it leads to an irreducible cubic equation, and, as first (we believe) conclusively proved by 0. Holder (Math. Ann. xxxviii.), no irreducible cubic can be solved by means of a set of auxiliary quadratic equations, such as the above. On the other hand, the trisection of a right angle is possible, and, as Gauss was the first to show, the division of a right angle into 17 equal parts is possible by Euclidean constructions, although that of the general angle is not. 303. A little further discussion of the trisection problem leads to some other interesting conclusions. Let AB be any arc of a given circle; take a point C on it, such that the arc BC is one-third of the arc BA, and two other points D, E on 316 PROJECTIVE GEOMETRY the circle, such that CDE is an equilateral triangle. The four points A, D,, may, in various ways, be determined as the intersections of the circle with a conic. One of these conics is a hyperbola of eccentricity 2, with one focus at B, and the perpendicular bisector of AB for the corresponding directrix. The interesting thing here is that the auxiliary curve is of the lowest possible degree, and its intersections with the circle give three points (C, D, E) which correspond to the cubic equation to which we are led by analysis, and one point (A) which is, in a certain sense, irrelevant. Algebraically, if we eliminate y from the equations x2+y2=I, (X —I)2 +y2=4(xsin~' -y cos10)2, we obtain 4x4 - 4x3 cos 0 - 3x2 + 2x cos 0 + cos20 =; that is to say, (x - cos 0) (4x3 - 3x - cos 0) =o, where the roots of the second factor are really what we want to find. A cubic may, of course, be solved by finding the intersections of a line with a cubic curve; but, as we see by this example, it may also be solved by means of two conics, and from a geometrical point of view this is simpler, though it leads to a point extraneous to the problem. It should be noticed also how the construction given above, although, so to speak, irreducible, leads to the solution of a reducible quartic equation. So although the analytical solution of a problem may be said to be perfectly definite, it does not always provide what we may consider the simplest geometrical solution, when we determine points, for instance, as intersections of loci. Simplicity, of course, is a relative term; here it is, practically, the easiness of constructing auxiliary loci or envelopes. 304. Linear Problems. In the case of projective geometry the simplest type of problem is that which can be solved with the aid of a straight-edge alone. A typical example is that of constructing the conic which passes through five given points (Art. 104): we can find as many PROJECTIVE PROBLEMS 317 points on the curve as we like, each separately, with the help of a straight-edge alone; or, as we say, by linear construction. What is a linear problem from one point of view may not be so from another. Take, for instance, the problem of drawing tangents to a given conic from a given point. If the conic is real and completely drawn, and the point is real and outside of it, we can, by a linear construction (Art. II8), find the polar of the points, and thus intuitively find the intersections of this polar with the conic: joining these to the point, we have the tangents required. But suppose the conic is not drawn, but only fixed by certain data; for instance, let it be required to draw the tangents from P to the conic which passes through five given points A, B, C, D, E; the six points here named being the only actual data. We can still, by a linear construction, find the polar of P; namely, by the Pascal construction we can find the points A', B' where PA, PB meet the conic again, and the polar of P is the join of the points (AB, A'B') and (AB', A'B). Moreover, we can do this whether the six given points are all real, or not; but we cannot go any further by linear construction alone, unless it accidentally happens that the polar of P goes through one of the six given points. Problems of this kind, then, are not strictly linear, although in some cases they may appear to be so. If we use the term " a given conic " to mean a conic fixed by five assigned points, or by five assigned tangents, the following are linear problems in the strict sense of the term, and so are their correlatives: (i) To find the tangent at a given point on a given conic. (ii) To find where an arbitrary line through a given point on the conic meets it again. (iii) To find the polar of any assigned point with respect to the conic. (iv) To find the remaining intersection of two given conics.ABCDE, ABCD'E' circumscribed to the same triangle ABC. 318 PROJECTIVE GEOMETRY The reader will find it a good exercise to make a list of all the problems discussed in previous chapters which are either strictly linear, or quasi-linear, like the problem of drawing tangents to a conic from a point. 305. We can extend the idea of linearity to problems in three dimensions by supposing that, besides a straight-edge, we have an (ideal) instrument, which we will call a " flatpiece"; namely, a moveable solid with a plane face, just as a straight-edge is a moveable solid with a straight edge. The flat-piece enables us to "construct" any plane for which projective data are assigned; and we shall say that a problem in three dimensions is linear, when it can be solved by means of the straight-edge and flat-piece alone. Here again we have a distinction between strictly linear and quasi-linear problems: for instance, to construct the tangent-cone from a given point to a given quadric surface, and so on. No problem or theorem is strictly linear unless the figure belonging to it can be constructed by a linear process; that is, by the two projective mathematical instruments aforesaid. The main use of the flat-piece is that it enables us to construct the meet of two planes, each given, but not actually drawn. 306. Problems of the Second Class. It may be said with considerable confidence that every problem which is linear in the geometrical sense defined above is linear from an algebraical point of view: that is to say, it can be solved by a system of linear equations. There is another class of problems which we shall call problems of the second class; namely, those which depend on the construction of selfcorresponding elements in cobasal projective elementary forms of the first rank, and in particular those which depend on the construction of the foci (or double elements) of involutions. It will be convenient to include cases where more than one of such constructions have to be made, and where the rth construction can only be carried out after the (r- i)th has been completed. We shall suppose also PROJECTIVE PROBLEMS 319 that the total number of constructions required for any one final result is finite; some of the constructions may be linear, but not all of them. The nature of such problems will become clearer if we take a typical example. Take the following: In a given conic, completely drawn, it is required to inscribe a triangle whose sides pass respectively through three assigned points P, Q, R, each disjoint to the conic (Fig. 7I). X C..13 E Z. FIG. 71. Take any point A on the conic, and draw the chords ARD, DPE, EQA'. If A' coincides with A, the problem has been solved; but suppose this is not the case. Take another point B: repeat the construction, so as to arrive at the point B', which we suppose distinct from B; and, in like manner, let C lead to the point C' distinct from it. We now have two triads ABC, A'B'C' on the conic connected by a projectivity: namely, the resultant of the involutions whose centres are R, P, Q respectively. By joining (AB', A'B) to (AC', A'C) we obtain the axis of the projectivity (Art. I34) 320 PROJECTIVE GEOMETRY let this line meet the conic in X1 and X2; then X1, X2, as shown in the figure, are vertices of two triangles X~YZl, X2Y2z2 which satisfy the conditions of the problem. It may happen, of course, that X1 and X2 coincide; in this case, we call the solution a double one, because the general solution leads to two triangles. The reason why we regard this problem as of the second class is because, when we translate it into algebra, the solution depends upon a quadratic equation. The projectivity (A'B'C'...) 7 (ABC...) leads to an index-equation axx' + bx + cx' + d = o, and its double points have for their indices the roots of ax2 + (b + c) d =o, and as this is, in general, irreducible, we cannot expect a linear solution. It should be noticed that three unsuccessful trials give us more information than one successful one; because the former suggest the possibility of two solutions, which the latter does not. The complex theory enables us to assert that there are always two solutions, possibly coincident. 307. Reduction of Problems of the Second Class to Quasilinear ones. Let us suppose that, in a plane, we have one conic completely drawn. For instance, let us draw the real part of an arbitrary circle; this determines all its complex elements. By using this circle as an auxiliary figure, every problem of the second class can be solved by linear construction; provided that we assume that the intersections of the auxiliary circle with an arbitrary line are intuitively evident. This follows because every cobasal correspondence in the plane can be projected into a correspondence on the auxiliary circle. The axis of this correspondence, like X1X2 in Art. 306, can be found by linear construction; its intersections with the circle correspond to the double points of the original correspondence, so the latter are determined. Of course, when the given correspondence is on a completely drawn conic, we need not use any other conic. PROJECTIVE PROBLEMS 321 308. On the whole, it seems best to regard the determination of the meets of a given line with a given conic as essentially a problem of the second class. Suppose, for instance, we take a real line u, and a real conic K determined by five real points A, B, C, D, E. The six sides of the complete quadrangle ABCD meet u in three pairs of the involution in which u is cut by the pencil of conics {ABCD}. Similarly, from the complete quadrangle BCDE we can find on u three pairs of points in the involution determined by the pencil of conics {BCDE}. We now project the two involutions on to an auxiliary circle, find their common pair, and project the latter on to u. The points thus found lie on a conic which goes through ABCD and also through BCDE; hence they are the intersections of u with the conic ABODE. The algebraic problem, which corresponds to that of finding the common pair of two cobasal correspondences, is that of solving the simultaneous equations axlx2 + bxly2 + cx2y1 + dyly2 = o, a'xx2 + b'xly2 + c'x2y1 + d'Y1Y2 =o whence (ax, + cyl)(b'xl + d'yl) - (bx, + dyl) (a'xl + c'yl) = o, a quadratic equation in x1/yl. Conversely, every such system of equations may be interpreted geometrically as representing two cobasal correspondences whose common pairs of elements have to be found. 309. Poristic Problems. The conditions of a problem may be such that either a solution is impossible, or else there are an unlimited number of solutions. In this case the problem is said to be poristic. A typical example is the following: Given two conics S, S' it is required to inscribe a triangle in the first which is a self-conjugate triangle for the second. Suppose, if possible, that there is one such triangle ABC. Take any point P on 8, and let its polar with respect to S' meet S in Q and R. Let the polar of Q with respect to S' meet QR in R'; then PQR' is a polar triangle for S'. Now, M.P.G. X 322 PROJECTIVE GEOMETRY by Art. 149, the six points ABCPQR' lie on a conic; and this conic is the one determined by ABCPQ, that is to say, it is S. Therefore R' coincides with R, and every point on S is the vertex of a definite triangle satisfying the prescribed conditions. Taking any such triangle as a triangle of reference, the equations of the conics will be of the forms S = 2fyz + 2gzx + 2hxy = o, S' =ax2 + by2 + z2 -o, whence the discriminant of KS + XS' is 2fghK3 - (af2 + bg2 + ch2) K2X + abc3, where 0', the coefficient of KX2, is zero. Hence O' =o is the condition that triangles can be inscribed in S which are selfconjugate with respect to S'. Similarly 0 = o is the condition that triangles can be inscribed in S' which are self-conjugate with regard to S. 310. Problems of the Fourth Class. Some important projective problems which admit of four solutions can be solved by the ordinary methods, because the quartic equation to which analysis leads us has some special properties. As an example we take the following (G. 313): It is required to inscribe a conic in a given triangle ABC, so as to pass through two given points D, E disjoint to the sides of the triangle. Let p, p' be the double rays of the involution A(BC.DE), and q, q' those of the involution B(AC.DE). If we put pq, P'q', pq', p'q =M, N, S, T respectively, the lines MN, ST go through C, and CM, CS are the double rays of the involution C(AB.DE); this is most easily proved by projecting D, E to the focoids, so that M, N, 8, T become the centres of the inscribed and escribed circles of the triangle. Returning to the general figure, draw the conic which touches MD, ME at D, E and also touches AB; this must also touch BC, CA, and gives one solution of the problem. In fact, AM, AN are conjugate with respect to this conic, because M is the pole of DE, and AM, AN cut DE harmonically; and PROJECTIVE PROBLEMS 323 since A(BCMN) is harmonic, AB a tangent, and AM, AN conjugate, it follows that AC is also a tangent. Similarly, BC is a tangent. Each of the points M, N, 8, T leads in this way to a conic satisfying the assigned conditions; and these conics are distinct, unless DE goes through a vertex of ABC, in which case there are only two proper solutions. The reader will see from the above that this problem is equivalent to that of drawing a circle to touch three given straight lines which enclose a triangle. Algebraically, if we take ABC as a triangle of reference, and use tangential coordinates, we have two quadratic equations to be satisfied by the coefficients of the equation of the conic. The result presents itself in the form: (d2e3 ~ d3e2)2vz + (d3el ~ dle3)2wu + (dje2 ~ d2e,)2uv = o where (d12, d2, d32), (e12, 2, ee32) are the point-coordinates of D, E respectively, and the ambiguities are subject to a condition which makes the number of solutions four, and not eight, as they would seem to be. If DE goes through A, there are two improper solutions, each consisting of a pair of flat pencils, of which (A) is one. It is instructive to solve this problem by point-coordinates. The coefficients of the point-equation of the conic have to satisfy three quadratic and two linear conditions, so there are analytically eight solutions. In the general case these consist of the four proper solutions, and of the line DE reckoned twice; this last being, algebraically, a solution to be counted four times. APPENDIX. COMPLEX ELEMENTS AT INFINITY. So long as we confine ourselves to the purely geometrical theory, there is no difficulty about elements at infinity. All those which are complex may be regarded as being points and lines in the real plane at infinity, and there is no need to consider complex planes at infinity, unless we like to give that name to complex planes whose real axes are at infinity. But in the theory of casts and their values there is a point which deserves some further discussion. Suppose that we start with A, B, C, three real base points on a circle, and as in Arts. 188-197 construct the casts which have the values p +qi. We may say that the quantity p +qi tends to infinity when its norm, p2 q2, tends to infinity; it can do this in an indefinite number of ways, owing to the independence of p and q. In spite of this, we may regard all the limits as equivalent, and corresponding to the one real point C on the circle. For if we construct (Art. I95) the real line containing the points whose indices are p ~ qi, we find that whenever p2 + q2 is very large, the line makes a very small angle with the tangent at C, and cuts AC very near to C; so that in the limit it always becomes the tangent at C. This result is analogous to the fact that in the theory of analytical functions of a complex variable there is only one infinite value to be considered. APPENDIX 325 As another illustration, let the circle X2 + y2 = be cut by the lines x =p and y = q; the meets are (p, + i Jp2 _ I) and (~ i /F2, q). So long as p, q are finite, these are distinct pairs of points; but if p, q tend to infinity, we may regard each point-pair as ultimately coinciding with the focoids through which the circle passes. EXERCISES. 1. Let C be the point at infinity on a given line, and A, B any other two points on the line. Taking A, B, C as basepoints with indices o, I, oo respectively, show, with the smallest possible number of metrical assumptions, that the points whose indices are (o, + i,:~2,... n,... oo ) form what is usually called a scale of equal parts on the given line. (Art. 36.) 2. Two complete quadrangles ABCD, A'B'C'D' are such that five of the intersections such as (AB, A'B') are collinear. Prove that the sixth intersection is a fixed point collinear with the other five. (Art. 90.) 3. P, Q, R are three points on a line which meets the sides of a triangle ABC in L, M, N, such that PL, QM, RN are all bisected at the same point 0. Prove that AP, BQ, CR are concurrent. 4. Show that (i) the theorem of Menelaus concerning the ratios of the segments in which the sides of a triangle are divided by any transversal, (ii) the theorem that the angles of a triangle are together equal to two right angles, have the property that each can be deduced from the other by projection and dualisation. (Art. 287.) 5. Parallel plane sections of a conicoid are similar conics. 6. The intercept made by the asymptotes on any tangent to a hyperbola is bisected at the point of contact. (Art. II9.) 7. From a point P perpendiculars PM, PN are drawn to two given lines. If P describes a row of points, MN envelopes a parabola touching the given lines. What is the general theorem of which this is a particular case? 8. Any two concurrent lines, and the bisectors of the angles between them, form a harmonic pencil. (Draw a circle through EXERCISES 327 the point of concurrence, and prove that this meets the four lines in a harmonic set.) Deduce, metrically, the theorem that if ABCD is a harmonic range, AC: CB=AD: BD. 9. Let (AA'.BB'...) be an elliptic involution on a line; then the circles of which AA', BB' are diameters will meet in two real points S, S'. Let SS' meet AA' in M; then AM.MA' = BM.MB', and all such products have the same value. Prove also that if (AA'.BB') is a hyperbolic involution, there is a point M such that MA.MA'=MB.MB', and that all such products are equal. (In each case M is usually called the centre of the involution: it is conjugate to the point at infinity.) 10. Two angles of constant magnitude rotate about their vertices in such a way that the meet of two of their sides describes a line: prove that the meet of the other sides describes a conic (Newton's "organic," i.e. mechanical, method of drawing conics). Can every real conic be drawn in this way? Generalise the above by the projective theory of metrics. What is the correlative theorem, both in its general form, and more specially? (Arts. 225, 287.) 11. The sides of a triangle rotate each about a fixed point in such a way that two of its vertices slide along two fixed lines. Prove that, in general, the locus of the third vertex is a conic. (This is MacLaurin's method of constructing conics; every conic, real or complex, can be constructed in this way.) 12. Construct a hyperbola from the following data: (I) two asymptotes and one point, (2) one asymptote and three points, (3) the centre, one asymptote, and two points, (4) a circumscribed triangle and the directions of the asymptotes. 13. Construct an ellipse passing through a given point and having a given involution of conjugate diameters. (Art. 179.) 14. Construct a parabola, given (I) four tangents, (2) four points, (3) three points and the direction of the axis, (4) the axis and two points. 15. Given four points A, B, C, D, construct that conic of the pencil {ABCD} on which the cast ABCD has a prescribed value. 328 EXERCISES 16. Two lines rotate, each about a fixed point, with equal and opposite velocities; prove that their meet describes a rectangular hyperbola, for which the fixed points are ends of a diameter. 17. Let OA, OB be fixed radii of a circle, and let OP, OQ be variable radii within the angle AOB, such that the angle AOP is half of the angle QOB. Prove that the meet of OP, BQ describes a rectangular hyperbola. Hence deduce a construction for trisecting a given angle. 18. Construct a conic of which three points and one focus are given (four solutions). 19. Given eight points A, B, C, D, E, C', D', E'. Find a construction for the two remaining intersections of the conics ABODE, ABC'D'E'. 20. ABC is a triangle inscribed in a conic and the tangents at A, B, C meet BC, CA, AB respectively at A', B', C'. Prove that A', B', C' are collinear (a deduction from Pascal's theorem). The line A'B'C' may be called the satellite of the triangle ABC with regard to the conic. 21. The tangent and normal at any point on a central conic bisect the angles between the lines joining the point to the foci. What is the corresponding theorem for a parabola? 22. Let any chord AB of a conic meet a directrix in P; let S be the corresponding focus, and T the pole of the chord. Prove that SP, ST are at right angles. Next prove that, if lines are drawn through A, B parallel to ST meeting the directrix in A', B' respectively, then SA: AA':: SB: BB'; and finally deduce the ordinary relation between the focal radius and the perpendicular to the directrix (Reye). 23. A is a point inside a circle; PAQ any chord through it; and PCR the diameter through P. Prove that QR envelopes a conic. (R and Q are connected by a chain of two involutions.) 24. A circular piece of paper is folded over and creased in such a way that the curved edge of the folded part passes through a fixed point on the paper. Prove that the crease envelopes a conic of which the fixed point is a focus. Where is the other real focus? 25. Any line meets a hyperbola in P, P' and its asymptotes in Q, Q'. Prove that PP', QQ' are bisected at the same point. EXERCISES 329 Deduce a metrical construction for a hyperbola of which the asymptotes and one point are given. 26. Let A, B, C, D be four points, and U a line disjoint to them all. Draw the two conics through the points which touch the line, and let X, Y be the points of contact. Prove that X, Y are conjugate with respect to every conic of the pencil {ABCD}. (Arts. 159, 9I.) 27. Deduce from the last exercise that the polars of a point with respect to a pencil of conics form a flat pencil. Dualise this theorem, and hence prove (i) that the centres of a range of conics are collinear, and (ii) that the poles of any line with respect to a system of confocal conics are collinear (Art. 230). 28. Let P be a point which is disjoint to a given central conic, and distinct from its centre C. Draw any diameter d and through P draw the line p which is perpendicular to the conjugate diameter. Then (i) the locus of the point pd is a rectangular hyperbola, passing through C, P, and having asymptotes parallel to the principal axes of the given conic; (ii) the intersections of this hyperbola with the given conic are the feet of the normals which can be drawn from P to the given conic. (This hyperbola was discovered by Apollonius in connection with the problem of drawing normals, and is hence known as Apollonius's hyperbola.) 29. On a given conic any point 0 is taken, and through it are drawn any two perpendicular chords OP, OQ. Prove that PQ passes through a fixed point on the normal at O (called the Fregier point, from its discoverer). 30. If AB is a given chord of a conic, and PQ a variable chord, such that the cast ABPQ has a constant value, the envelope of PQ is a conic having double contact with the given one. (The relation RJ(abxy) =constant is lineo-linear in x, y.) 31. If a conic S be inscribed in a polar triangle of a conic 8', polar triangles of S can be inscribed in 8'. (Show that the same invariant relation must be satisfied.) 32. Two involutions of points on a conic are commutative if their centres are conjugate with respect to the conic. 33. Let ABCD, A'B'C'D' be two tetrads of points on two concentric spheres, centre S; no three points of either tetrad being on the same great circle. In the bundle (8) we can set 330 EXERCISES up a definite projectivity S(A'B'C'D') = S(ABCD), and this determines a definite one-one correspondence of diameters of the spheres. What sort of correspondence have we between points on the spheres? 34. Two spheres are to be connected by a projective real collineation: how many arbitrary pairs of corresponding real points can we choose? 35. If we put I-Xy.I I+xy X+y a=, =, 13= — x-y x ' z-y we have a parametric representation of points on the sphere a2 + /2 + y2 = i, for which x = constant gives one set of generators, and y = constant the other (Darboux). 36. Show that, with the notation of last example, 4dx dy da2 + dI2 + dy2 = (x y)dy ( x-y)2 and hence that a linear substitution (x, ') (m, n') (x, y) represents a projectivity on the sphere which leaves lengths of arcs unaltered. What are the conditions that this may be a real rotation of the sphere about a real diameter? 37. Let A, B, C, D be four points on a circle, such that ABCD is an order: prove that the value of the cast ABCD is -AD.BC/AB.CD. Show the connection of this with Ptolemy's theorem (AB.CD + BC.AD =AC.BD). 38. Two spheres are projectively related in such a way that the points A, B, C, D on one correspond to the points A', B', C', D' on the other: prove that AB.CD: AC.BD: AD.BC=A'B'.C'D': A'C'.B'D': A'D'.B'C' (B. 6oi. Let CD meet the tangent planes at A, B in P, Q: then PC/PD = AC2/AD2, and PDQC = AC2. BD2/AD2. BC2. Similarly for the corresponding cast P'D'Q'C'.) 39. Construct a conicoid circumscribed to a given tetrahedron, and having two given concurrent lines as generators (B, 37). EXERCISES 331 40. Given a tetrahedron ABCD and two lines U, V, such that the casts U(ABCD), V(ABCD) are equal, there is a definite polarity for which ABCD is a self-polar tetrahedron, and U, V are conjugate lines (B. 39). 41. Assuming that the diagonal points of a complete real plane quadrangle are never collinear, prove the same theorem for any plane quadrangle (B. 142). 42. Given eight points Pi and eight lines pi in a plane; find, geometrically, the collineation for which the eight points corresponding to the given ones lie respectively on the given lines. (See Burmeister, Math. Ann. xiv. The problem is due to Clebsch.) 43. Two plane systems (r), (cr') are in perspective with a point S. Keeping one system fixed, let the base of the other be rotated about ao'. Then in each position the systems are in perspective: prove that the locus of the centre of perspective is a circle. 44. Two plane systems (a), (a-'), in different non-parallel planes, are put in perspective from a centre S. Through S draw the plane rr, which is perpendicular to the line (r-', and in this plane draw the lines h, k through S, which are parallel to the bisectors of the dihedral angle ccr'. Let h, k meet ain F, G, and a' in F', G' respectively; and let F, G be called the foci in a, and F', G' the foci in a'. Prove that (i) to an angle at either focus of C corresponds an equal angle at the corresponding focus of ar'; (ii) the coaxial system of circles in o- of which F, G are the limiting points, corresponds to the coaxial system of circles in a' of which F', G' are the limiting points; (iii) the conics in a, of which F, G are foci, correspond to conics in cr' of which F', G' are foci; (iv) if P, P' are any two corresponding points, FP. G'P' + F'P'. GP = o. (H. J. S. Smith, Proc. L.M.S. ii.) 45. Show that in a plane homology the centre S counts as a double focus, and that there are, in general, two other noncoincident foci, which coincide in the involutionary case. In the general case, find the foci by construction. (Draw through S a line perpendicular to the axis; in the correspondence of points on this line there will be one pair equidistant from the axis.) 332 EXERCISES 46. In a plane homology there are, in general, two corresponding lines, such that to any segment of either corresponds an equal segment of the other. (Namely, those two corresponding lines parallel to the axis which are equidistant from the centre.) 47. Let ABC be a triangle; h a line disjoint to all its vertices, and H a point disjoint to all its sides. Prove that, if H is disjoint to h, the correlation H(ABC)7h(BC.CA.AB) connecting the pencil (H) with the row (h) is a polarity (G. 22I). 48. Let a line U meet the sides BC, CA, AB of a triangle in three distinct points A', B', C'; and let V be any line which meets U in 8, but cuts the plane ABC. Prove that the casts SA'B'C' and v(UABC) are equal (B. 34). 49. A line U meets the faces BCD, CDA, DAB, ABC of a tetrahedron in A', B', C', D' respectively; prove that the casts A'B'C'D' and U(ABCD) are equal (B. 35). 50. Show that all the special properties of a system of confocal conics can be derived from those of a set of coaxial circles and conversely. 51. Let -=o be the tangential equation of a conic referred to a triangle whose sides are two rectangular axes and the line at infinity; prove that, if t is a variable parameter, c + t( + 2) = o represents the system of conics confocal with the given one. Generalise this theorem for three dimensions. 52. When a conicoid degenerates into a cone, its focal conics become three line-pairs, of which only one is real, when the cone is real. 53. From a fixed point are let fall perpendiculars on conjugate rays of a pencil in involution; prove that the join of the feet of these perpendiculars passes through a fixed point. 54. Two conjugate points A, A' of a row in involution being joined to a fixed point 0, lines through A, A' perpendicular to OA, OA' meet in a point which lies on a fixed line. 55. Through a given point are drawn chords PP', QQ' of a given conic, so as both to touch a confocal conic; prove that the meets (PQ, P'Q') and (PQ', P'Q) are both fixed. Dualise this theorem so as to give a property of a system of coaxial circles. EXERCISES 333 56. If U be an umbilic of a conicoid, i.e. the point of contact of a tangent plane parallel to a set of circular sections, then every chord of the surface which meets the normal at U passes through a fixed point K on it. K is the Fr6gier point of every normal section through U. 57. If AP, BQ, XY are three pairs of points in involution, and if A'P, B'Q, XY are also three pairs of points in involution, so also are AB', A'B, XY. (Use Pascal's theorem.) 58. A circle cuts a hyperbola in P, Q, R, S; PQ meets the asymptotes in X and Y, and RS meets them in Z and W. Show that X, Y, Z, W lie on a circle concentric with PQRS. 59. A circle cuts each diagonal of a parallelogram harmonically. Show that the parallelogram is a rectangle. 60. 0 is a fixed point on a conic and A, B, C three variable points on it. Show that the circle through the feet of the perpendiculars from O on the sides of the triangle ABC passes through a fixed point. [Solution. Let F, Fo be the focoids and OF, OFo meet the conic in G, Go. Then the sides of ABC, OGGo touch a conic 2, and the circle through the feet of the perpendiculars from O on the sides of ABC is the auxiliary circle of 2, since O is a focus of 2. This circle passes through the foot of the perpendicular from O on GGo-a fixed point.] 61. The polar of a point O with respect to a conic meets the curve in P, Q, and PR is any chord of the curve through P. Through O a line is drawn parallel to PR meeting the curve in U, V. Show that QR bisects UV. 62. If PQR is a self-conjugate triangle of a conic, of which ZQX and XRY are chords, then YPZ is also a chord. If ABC is a triangle inscribed within a conic, then an infinite number of self-conjugate triangles PQR can be described in such a way that P lies on BC, Q on CA, and R on AB. Also AP, BQ, CR meet in a point. What is the locus of this point? (Math. Trip. I906). 63. A variable circle passes through two fixed points A and B and meets two fixed lines through A in P and Q. Show that PQ envelopes a parabola. Generalise by projection and dualisation. 334 EXERCISES 64. Two points P, P' are conjugate with respect to a given conic, and their join passes through a fixed point S. Show that there is, in general, a one-one correspondence between P, P'; and that, if ax2+by2+cz-=o is the given conic, and (h, k, 1) the fixed point, the correspondence is, in general, given by either of the relations x': y': z' = x(bky + clz) - bhy2 - chz2: y(clz + ahx) - ckz2 - akx2: z(ahx + bky) - alx2 - bly2, and that which is obtained by changing x', y', z' into x, y, z, and vice versa. (This is a case of a reversible quadratic transformation, and includes the theory of inversion in plane geometry.) 65. Two points P, P' are conjugate with respect to a given conicoid ax + by2 + cz2 + dt2 =o, and their join passes through a fixed point (p, q, r, s). Express the coordinates of P' in terms of those of P and vice versa. 66. Six fixed points A, B, C, A', B', C' are taken on a conic so that B'ABCABA'B'C', and P, P' are any other points on the conic, such that PABCT P'A'B'C'; prove that AA', BB', CC', PP' are concurrent. (Arts. 86, 136.) 67. A conic meets the sides BC, CA, AB of a triangle in P, P', Q, Q', R, R' respectively. Prove that AP, AP', BQ, BQ', CR, CR' touch a conic. 68. The points D, E are disjoint to each of the sides of a triangle ABC. By joining them to the vertices we obtain six lines which meet the sides of the triangle again in six points; prove that these six points lie on a conic. 69. Any three pairs of points which divide the three diagonals of a quadrilateral harmonically lie on one conic. 70. The pairs of tangents to a parabola from points in the same straight line are parallel to conjugate pairs of rays of an involutionary pencil. 71. The locus of the middle point of a chord of a conic which passes through a fixed point is another conic. 72. Circumscribe to a given conic a polygon having each of its summits upon a given straight line. In particular, circumscribe a triangle having one of its summits onga'real given line, and the other summits on two given conjugate complex lines. EXERCISES 335 73. On the same line there are two projectivities (A'B'C'...) = Z(ABC...) and (A'P'Q'...) = i,(APQ...), which have the given pair (A, A') in common. Find, by construction, the other pair which they have in common. 74. Given two projectivities (A'B'C'.. )=Z'(ABC...), and (P'Q'R'...) = -(PQR...) on the same line, find, by construction, the solution of X' = ZX = bX. 75. If POP', QOQ', ROR', SOS' are any four concurrent chords of a conic, the conics OPQRS, OP'Q'R'S' have a common tangent at O. 76. If a conic touches the sides SF and SF' of a given triangle SFF', and also two other given lines, the second tangent to it from F and F' meet on a fixed straight line. (The pencils of tangents (F), (F') are projective; and also perspective.) What is the special form of this theorem when F, F' are the focoids? 77. If a point P on a conic be joined to two fixed points F, F' in its plane, all the chords divided harmonically by PF, PF' are concurrent; and the locus of the point of concourse, as P varies, is a conic touching the first at two points on FF'. (Let PF, PF' meet the conic in Q, Q': then the point of concourse is the pole of QQ'. As P varies, QQ' envelopes a conic, and its pole describes another conic.) Specialise this theorem by supposing that F, F' are focoids; and deduce the theorem that the locus of all the Fregier points derived from a given conic (Ex. 29) is a homothetic conic. 78. If a triangle is self-polar to a parabola, the three lines joining the middle points of its sides touch the parabola, and conversely. 79. Given two conics, find a conic with respect to which they are polar reciprocals. (Begin by finding the common polar triangle of the given conics; this must be a polar triangle of the required conic. An algebraic method is most appropriate here.) 80. The locus of poles of a given line with respect to a pencil of conics {ABCD} is, in general, a conic. (This is most easily proved by analysis.) 336 EXERCISES 81. If we put, with a parameter t, x: y: z = at2 + bt + c: a't + b't + c': a"t + b"t + c", the locus of (x, y, z) is, in general, a conic. What are the exceptions? 82. If we put, with parameters u, v, x y: y: t=fi(u, v):f2(u, v):f8(u, v): f(u, v), where fi (iu, v) is a lineo-linear function of the form aiuv + bu + cyv + di, then the locus of (x, y, z, t) is, in general, a quadric surface, and by putting u=constant, or v=constant, we obtain the two sets of generators. Prove that every quadric surface can be obtained in this way; and find such a representation for 2+y2+z2 -t2=o. 83. Let ABC be a triangle, and let the perpendiculars from A, B to BC, CA meet in L. Then the line-pairs (AL, BC) and (BL, CA) are special conics of the pencil {ABCL}, and determine on the line at infinity an involution of which the focoids are double points. Deduce from this (i) that the three perpendiculars of a triangle are concurrent; (ii) that every rectangular hyperbola circumscribed to a triangle goes through its orthocentre; (iii) that the pencil {ABCL} consists entirely of rectangular hyperbolas. Generalise by projection and dualisation. 84. If one focus of an involution of points is at infinity, the other focus is equidistant from any two corresponding points. 85. The circles whose diameters are the three diagonals of a complete quadrilateral are coaxial, and cut orthogonally the circle through the meets of the diagonals. 86. The nine-point circle of every triangle self-polar to a parabola passes through the focus. 87. If ABCDE be a pentagon circumscribing a parabola, the parallels from B to CD and from A to DE meet on CE. (A case of Brianchon's theorem.) 88. Tangents PT, PT' are drawn to a conic, of which 8, S' are the foci; prove that the bisectors of the angles TPT', SPS' coincide. Obtain other theorems from this by projection and dualisation. EXERCISES 337 89. If two conics are such that a single triangle can be inscribed in one, and circumscribed to the other, an unlimited number of such triangles can be drawn; and the invariant condition for this porism is 62= 4A/'. (Art. 286.) 90. A common tangent touches two conics in F, F' respectively. Prove that every conic through the four meets of the conics divide FF' harmonically (Art. I59). Obtain a special theorem from this by supposing one of the conics to be a circle which osculates the other conic. 91. Two lines a, b are cut by a third line C in the points A, A'; and P, P' are any other points on C. Prove that the chord of contact of any conic through P, P' touching a, b passes through one focus of the involution (AA'.PP'). Hence show that there are four conics passing through three given points and touching two given lines. (Apply Art. I59 to the conic and the line-pair ab considered as a degenerate conic). 92. A line b and a plane 3 meet at right angles in a fixed point S. Prove that if b describes a quadric cone, /3 will envelope a quadric cone (Art. 248). Generalise this proposition by supposing that b describes a cone of the nth order, with assigned singularities. 93. Show that there is a law of duality for constructions on the surface of a sphere, in which points correspond to great circles, described in a definite sense. For example, let the north pole of the earth (regarded as a sphere) correspond to the equator described from east to west, and the south pole to the equator described from west to east. How does this fact affect the analytical theory of spherical trigonometry? Investigate a corresponding theory for constructions on any conicoid; taking, instead of great circles, sections of the conicoid by planes which pass through a fixed point. 94. Show that any conic in space may be projected into the focoidal circle (Art. I40). 95. Any proper surface of the second order (including cones and cylinders) is projectively equivalent to a sphere. 96. If part of the intersection of two conicoids is a conic, the rest of it must be a plane curve of the second order. Dualise this theorem. M.P.G. Y 338 EXERCISES 97. Generalise the theorems relating to the radical plane of two spheres, and the radical centre of three spheres. 98. What is the projective problem corresponding to that of describing a sphere that touches each face of a given tetrahedron (cf. Art. 3Io)? 99. The envelope of a transversal cut harmonically by two homothetic parabolas is another parabola. 100. If a conic passes through two given points, and touches a given conic at a given point, its chord of intersection with the given conic passes through a fixed point. 101. The envelope of the axes of a conic which touches four fixed tangents to a circle is a parabola. 102. If S, T are fixed points, and HK a segment of given length supposed to slide along a fixed line disjoint to S and T, prove that the meet of SH, TK describes a hyperbola of which the fixed line is an asymptote. Deduce that if 8, T are any two fixed points on a hyperbola, and P a variable point on it, the distance between the meets of SP, TP with an asymptote is invariable. 103. The problem to inscribe in a given conic a 2n-gon, whose n pairs of opposite sides shall pass in any assigned order through n given points, disjoint to the conic, is either impossible or poristic (Townsend). 104. If two triangles circumscribed (or inscribed) to a conic are in perspective, every ray through their centre of perspective meets their sides in three pairs of points in involution. (Let S be the centre of the perspective, s its polar with respect to the conic. There is a definite homology for which S is the centre and S the axis, which transforms the conic into itself; the proposition follows from this.) Dualise this, and deduce Steiner's theorem that the orthocentre of a triangle circumscribed to a parabola lies on the directrix. 105. There is a threefold infinite system of conicoids for which two given skew lines are generators of the same system (Art. 266). Investigate the system of their remaining intersections. EXERCISES 339 106. Prove that every projective collineation in a plane may be reduced to one of the following types: (i) x': y': z'= ax: by: cz, with a, b, c all unequal; (ii) x': y': z' = ax: by: cy + bz, with a, b unequal and c not zero; (iii) x': y': z'= ax: bx + ay: dx + cy + az, with a, b, c different from zero; (iv) x': y': z'=ax: ay: bz, with a, b different from zero; (v) x': ' = ax: ay: by + az, with a different from zero. Discuss the self-corresponding elements and characteristic equation for each case. Which case corresponds (i) to a homology, (ii) a harmonic homology, (iii) an elation? (V. Y.) 107. Special pencils (and ranges) of conics are obtained by making some of the common points (tangents) coincide. Prove that all types of pencils can be reduced to one of the following forms: (i) x2 _y2+X(y2 2) = o, (ii) y2 - z2+kzx = o, (iii) y2 + 2zX + Xyz = o, (iv) x2+ Xyz =o, (v) y2+ 2zx + Xz2=o, where X is a variable parameter. Draw the corresponding figures for any two conics of the pencil, and point out the nature of the coincidences involved. (V. Y.) 108. Representing any point T on the conic y- - zx = o by the equations x y: z=t2: t: r, prove the following theorems: (i) If T1, T2, T3, T4 correspond to t,, t2, t4, t4, then RI(TT2T 3T4) =R(3(tlt2t3t); (ii) The pole of the chord T1T2 is {tt2, (tl + t), I } (iii) If ad ' be, the relation att'+ bt + ct'+ d = o 340 EXERCISES corresponds to a projectivity on the conic, and the envelope of TT' is {ax + (b + c)y +dz}2+ 4(ad - bc)(y2 - z) =o. Interpret this result when b=c, so that the projectivity is an involution. 109. A, B, C, P, Q, R, X are points on a conic, and the values of the casts ABCP, ABCQ, ABCR, ABCX respectively are p, q, r, x. Prove that the value of PQRX is (q - r) (x - ) 110. Given a conic and three ea e real base points A, B, C on it, construct the points X such that x, the value of the cast ABCX, satisfies the equations (i) 2+I =o, (iv) x2-X-i=o, (ii) 3- = o, (v) 5-I = o, (iii) x4 + I = o, (vi) X4 2+ 2 = o. 111. Representing points on the conic y2-ZX=o by the homogeneous parameter ': $2 in the form x: y: z=$2 $12: 22, a homogeneous binary form in $1, $2 f($1, $2)- (alas... anl ) can be taken to represent the n points on the curve whose parameters satisfy f($1, 2) =o. Show that there is a one-one correspondence between all lines in the plane and all quadratic forms (two forms being taken to be identical when ao: o' = aa: a' =etc.). 112. If the line corresponding to the form f ao0l$ + 2a+1$2$ + a222 passes through a fixed point P, show that the joint invariant aoa'a - 2ala'1 + a2a'0 of f and a fixed quadratic form f ao0'$ + 2a1$2 + a2'$22 vanishes, and that the line corresponding to this fixed quadratic form is the polar of P. Hence show that two lines are conjugate with regard to the conic when the joint invariant of the corresponding forms is zero. EXERCISES 341 113. If three points P, Q, R on the conic y - ZX = o are represented by the binary cubic form f(t, $2) = ao13 + 3a1$12$ + 3a2$1$22 + a33a3, the line corresponding to the Hessian of f($1, $2), H($1, 2) - (ao0a - a2) 12 + (a0a3 - aja2) $1,2 + (alas - a,2) 22, is the satellite of PQR (Ex. 20). The points represented by the cubicovariant of f($1a2), T(1i, 2) = (a02a3 - 3a0ala2 + 2a13)$13 + 3(aoala3 - 2aoa22 + al2a2)l12$2 + 3( - aoa2a3 + 2a12a3 - aa,2)122 + ( - aoa32 + 3ala2a3 - 2a3)23, are P', Q', R', the points on the conic, such that (PP', QR), (QQ', RP), (RR',PQ) are harmonic pairs. A cubic and its cubicovariant having the same Hessian, it follows that the satellites of the triangles PQR, P'Q'R' are the same. 114. If a binary quartic form f(~$, $2) represents the four vertices of a quadrangle inscribed in a conic, show that the diagonals of the quadrangle meet the conic in six points represented by the sextic covariant of f($1, $2). 115. Show that four points A, B, C, D on a conic represented by the quartic form f($1, 2) = aoS04 + 4a113$2 + 6aa21'22 + 4a,$122 + a,2 form a harmonic system (j = 1728), if s = a0aa - aoa,a - a 0a3 - a23 + 2ala2a3 = o, and an equi-anharmonic system (j =o), if T - ao0a - 4ala3 + 3a22 = o. 116. If the determinant la/jl is not zero, show that the parametric equations X = a03 + ai202+ai30+ai4 (i= I, 2, 3, 4) define a space curve which meets every plane in three points -from which property it derives its name, a " twisted cubic." By changing the tetrahedron of reference reduce the parametric equations to the canonical form Y1=-3, Y2=02, y:=0, Y4=I. 342 EXERCISES Show that the equation of the osculating plane at 6 is Yi - 3Y20+ 3Y32 - y403 = o, so that three osculating planes pass through a given point. Show also that the line coordinates of the chord (0, )) are, in the order (P12, P31, p14, P23, P24, P34), (02-2, -4 (6+) +, 02 + 6+ 2, 020, 0 +, ), and those of the tangent line at 0 are (69, -203, 302, 02, 20, I). Deduce that one chord of the curve of the curve passes through a given point, that four tangent lines intersect a given line, and that all the tangent lines belong to a linear complex. 117. Writing the parametric equations of a twisted cubic in the homogeneous form Xl: X2:X s: =: - 12 2 l: $l22 2 23, show that there is a one-one correspondence between all planes in space and all cubic forms f ($, $2) - a013 + 3a,11$2 + 3a2$$22 + a3,23, and that if a plane passes through a fixed point O the lineolinear invariant aobs - 3alb2 + 3a2bl - a3b0 of the corresponding cubic form and a fixed cubic form ($1, $2)= bo013 + 3b1122 + 3b2$1$22 + b323 is zero. Show also that the points represented by the form 2($1, $2) are the points of contact of the osculating planes through 0, and that these points are coplanar with 0. Again, the points given by the Hessian of ($2, $2) are on the single chord of the curve passing through 0. What points are given by the cubicovariant of 4($i, $2)? 118. If P and Q are corresponding points of a space collineation, with four distinct self-corresponding points, and PQ passes through a fixed point, show that the locus of P is a twisted cubic. 119. Regarding two points in a plane as the finite intersections of the circles X1(X2 +y2) + X2X + y + 4 = o, y(x2 + y2) + Y2 +Y3Y +Y4 = 0, EXERCISES 343 show that there is a one-one correspondence between geometry of lines in space and geometry of point-pairs in a plane. Taking the homogeneous coordinates of a point-pair to be P12, P31, P14 P23, P24, P34, (pij=Xiyj-Xjyi), deduce the following results: (i) Two point-pairs are concyclic if Pl2q34 + P3lq24 + Pl4q23 + P23ql4 + p24q31 + P34q12=o. (ii) If ABCD, ABEF, CDEF are three concylic quadrangles, then either ABCDEF are all concylic or AB, CD, EF are pairs of inverse points with regard to the same circle. (iii) Given four point-pairs in a plane, there are, in general, two more point-pairs concyclic with each of them. 120. If two linear complexes are apolar, the nul-planes of a point with regard to them separate harmonically the planes through the point and the directrices of the congruence of lines common to the two complexes. 121. Reciprocally the nul-points of a plane with regard to two apolar complexes, are harmonically separated by the points of intersection of the plane with the directrices. Examine the special case when the directrices are coincident. 122. There is a quintuply infinite number of tetrahedra with reference to which the equations of two general linear complexes reduce to the form A(p23+ P14) +Mp24=o, Ap31 + Bp24 = o. 123. Reduce the equations of two linear complexes, whose invariants are A, M, B, to the forms Ap31 + P24=0, M- /A M + /A 2 + 2A P2 where A= M2 - 4AB. 124. Prove that every special linear congruence consists of all the tangent lines to a conicoid at points on a generator (Art. 282). 344 EXERCISES 125. Two flat pencils (abc...), (ab'c'...) are projective, and have in common a self-corresponding ray a, but are not coplanar. Prove that the set of lines which meet two corresponding rays of the pencils form a linear complex (Sylvester). 126. Show from Laguerre's definition of an angle (Art. 287) that two angles, in the ordinary sense, are equal or supplementary, if each side of the one is parallel to a side of the other. 127. A circle and a rectangular hyperbola being given by 2 +y2 = y2 X2 Z2 =2 z=0 Y=o0 referred to rectangular axes, prove both analytically and geometrically that the circle and hyperbola are sections of a quadric cone which has its vertex at a focoid in the plane x = o. 128. Show that in Fig. 65 (p. 183) the points P, Q, R are collinear, and that the line through them meets the circle in points whose indices are the roots of x2 -x + I =o. 129. Taking the symbols to stand for casts, verify by construction the following identities: (a+b) (a-b)=a2-b2, (a+b)2+(a-b)2=2(a2+b62), and make up other similar examples. 130. Let j be any complex cast, and W any other cast; prove, without using the theory of indices, that there are two determinate real casts U, V, such that =U + vj (B. 278). 131. If (MN.AC.BB) is an involution, then (B. 280) (MANB)2 = MANC. 132. Let M, N, A, B, A', B' be six points on a conic, and let AA', BB' meet in S; then (B. 284), S(MANB) = (MANB)(MA'NB'). 133. If M, N, A, C, E are five points on a conic, such that MNACEWMNCEA, then M, N are determinate when A, C, E are given, and C, E are determinate when M, N, A are given; also (MANC)3=(MCNA)3 =I. Hence solve the cast equation X3 = C where C is any given cast (B. 286-29I). EXERCISES 345 134. Let A, B, C be three fixed points on a conic, and P, Q variable points on it, such that ABCQ = (ABCP) 2; construct the envelope of PQ, and prove algebraically that it is a curve of the third class and fourth order. Show also that every such curve may be obtained in this way; and dualise the theorem. 135. Show that the envelope of the line t(I +t2)x + ( +t2)y -t(3 -t2)a=o is the three-cusped hypocycloid (X2 +y2)2 + 8ax(3y2 - x2) + i8a2(x2 +y2) - 27a4 = o; that the tangential equation is a -(2 - 3,2) - ($2 + 2) = o; and that by projecting the curve so that two of the cusps become focoids, we arrive at a cardioid. 136. The pedal of x2-y2=c2 with respect to its centre is the lemniscate (x2+y2)2=c2(x2-y2), which has nodes at the origin and the focoids. Verify this; generalise by projection, and show how it leads to a projective theory of all trinodal quartics and their correlatives. 137. Show that, with the notation of Art. 289, the only way of making LSab =L2ab for every coplanar pair of lines a, b is to suppose that S =o, Z=o represent the same surface, when S = o is a proper conicoid. Also, that if = o degenerates into a proper conic, we must have S = o degenerating into the plane of that conic, and correlatively. 138. With the analysis of Art. 288, suppose we define the distance AB by the equation AB= Lt -R{(ABPQ)}; — )..oo2 would this entitle us to infer that BC+CA+AB=o for three collinear points? In this case, also, what is the locus of points at an infinite distance from a given point A? 139. Two conics can be projected into the circles of Euclid's Elements, Bk. I., Prop. I, if 8A2A'2(020'2 + I8AA'We' - 4AO'3 - 4A'03 - 27A2A'2) = 3A2(0'2 - 3A'O)3 = 3A'2(02 - 3AW')3, and conversely. 346 EXERCISES 140. Show that any two coplanar conics, which do not touch, may be projected in three ways, and only three, so that in the new figure the intersections of the conics are the vertices of a square. Can two conics be projected into conics inscribed in a square? 141. Let S, H be the real foci of a real conic, and let any two circles through S, H cut the transverse axis in P, P' and Q, Q' respectively; prove that PQP'Q' is a representation of one of the complex foci. Construct the corresponding directrix. 142. Given the coordinates (pi) and (q,) of two concurrent lines; find, in a symmetrical form, the coordinates of their common point. INDEX. (The numbers refer to the pages.) Angle, projectively defined, 291, 296 Apolarity of two linear complexes, 284 Asymptotes, 215 Axial pencil, 6 Base, 6 Bicomplex line, 158, 270 Brianchon's theorem, 99 Bundle, 8 Canonical equations of conicoid, 245; cone, 246; two conics, 304 Casts, I69; addition of, 170; multiplication of, 172; values of, 177; group of six, 183; complex, I85 Centre of conic, 214 Chain of elements, I59, 271 Characteristic equation, 230 Chasles' theorem, 104 Circular sections of quadric, 248 Collinear, 4; plane nets, 72 Collineation, 72; classification of, 230 Complex of lines, 279; linear, 280; pencil of, 282 Cone, 95; containing two conics, I34, 3II Confocal conics, 224; conicoids, 250 Congruence, 279; linear, 285; associated with skew involution, 287 Congruent pencils, 218 Conic, 9I; through given point and touching given conic at two given points, I37; tangents to, from complex point, I64; as nucleus of polarity, I66; standard forms of equation, 212, 304; classification, 214 Conjugate points and lines with respect to a conic, o08 Conjunction, 3 Construction of projective rows and pencils from two triads, 50; of projective fields from two quadrangles, 70 Coordinates, I96; of a line in space, 275 Correlation, analytical form of, 233 Correlative vocabulary, 5; plane fields, 73; bundles, 74 Cross-ratio, 207; of a cast, 209; sextic, 210 Degenerate conics, II3 Desargues' theorem for triangles, 25; for conic and quadrangle, I47 348 INDEX Diameter of conic, 215; principal, 222; of central conicoid, 247 Directrix of conic, 223; of regulus, 15; of null-system, 255 Distance, projectively defined, 292, 296 Duality, law of, 3 Elation, 67 Elementary forms, I-8, 120 Elements, i Flat pencil, 7 Foci of conic, 223; of conicoid, 249 Focoids, 22i; focoidal circle, 247 Group, tetrahedral, 78; projective, 79; of linear substitutions, 192 Harmonic range, 30; pencil, 34; derivative, 42; set on a conic, 95, iI; on a regulus, I23; representation of involution, 15I Homology, centre and axis of, 63; harmonic, 65; connecting two conics, I32; in three dimensions, 228 Homothetic conics, 217 Infinity, lines and plane at, Io Invariant, cross-ratio (j), 2Io; of linear complex, 281; of two linear complexes, 282; invariants of a system of two conics, 304 Involution, 82; foci of, 84, 6I; notation, 85; commutative, 88; on a conic, I26, 150; in a plane, I4; orthogonal, 220, 247 Line as closed figure, I2; meeting four given lines, 255; six coordinates of, 276, 285 Line-equation of quadric, 288 Mobius's theorem for null-system, 259 Normal system of coordinates, 203 Nucleus, 166 Null-system, 253; data fixing, 255; parameter of, 260 Pascal's theorem, 97; corollaries, 99; rule for construction derived from, IoI; extension of, I36 Pencil of conics, 149 Pentad-theorem, 225 Perspectivities, elementary, I9; data fixing, 23; extensions of, 121 Plane field, 8 Polar, io6; in null-system, 254 Polar triangle, og9; two have vertices on same conic, 140 relative, 146 Polarity, I42; data fixing, I44, 235; analytical form of, 233; nucleus is a quadric surface, 237 Pole, I07; in null-system, 254 Projection and section, 22 Projectivity, 49; expressible as product of two involutions, 87; of polar rows and pencils, I o; on a conic; 124; between any two elementary forms, I27; analytical expression of, 190 Quadrangle, complete, 30; two with vertices on same conic, I39 Quadric, 96 Quadrilateral, complete, 30 Range of conics, I49 Reciprocal polars, I o INDEX 349 Regulus, 115; conjugate, II6 Row, 6 Sense of elliptic involution, 50; of skew involution, 267 Separated pairs of elements, 16 Skew involution, 263; equations of, 266; sense of, 267 Staudt's theorem, II2 Tangential equation of quadric, 238 Tetrad, projective properties of, 58 Triad-theorem, 48 Trisection of angle, 315 Vanishing points and lines, 22, 64 GLASGOW: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. 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