LIBRARY THE UNIVERSITY OF CALIFORNIA SANTA BARBARA IN MEMORY OF PHILIP FRANCIS SIFF ■ f MODERN ANALYTICAL GEOMETRY AN ITsTTRODUCTORY ACCOUNT OF CERTAIN MODERN IDEAS AND METHODS IN PLANE ANALYTICAL GEOMETRY BY CHARLOTTE ANGAS SCOTT, D.Sc. // OIRTON COLLEOE, CAMBRIDGE; PROFESSOR OF 'MATHEMATICS IN BRYN MAWR COLLEGE, PENNSYLVANIA MAC MILL AN AND CO. AND NEW YORK 189 V ^^^ A// riiiJitu reserved PREFACE. In the following pages I have assumed on the part of the reader as much acquaintance with the processes of Cartesian Geometry and the Differential Calculus as can be obtained from any elementary text-books in these subjects ; and starting from this, I have endeavoured to give a systematic account of certain ideas and methods, a familiarity with which is tacitly assumed in higher mathematics, while no adequate means of acquiring this familiarity is provided in existing English works. Among these ideas, one of the most important is that of Correspondence, and on this, in a few of its many manifestations, I have dwelt at some length. My desire has been to refrain from encroaching on what properly belongs to the theory of Higher Plane Curves — a theory so extensive, and, as is now acknowledged, so much less simple than it appeared some few years ago, that an introductory study of its fundamental conceptions may well be undertaken as a preliminary. vi PREFACE. To a certain small extent the iield here marked out coincides with that already occupied by the later chapters of Salmon's Conic Sections. Eecognizing that every English- speaking student of mathematics must of necessity acquire an intimate knowledge of Dr. Salmon's incomparable treatises, I have gladly refrained from any discussion of this part, and have simply referred the reader to those chapters, adding occasionally the few words of explanation that seemed necessitated by the different order of treatment here adopted. It has not been my ambition to add another to the many excellent collections of problems already existing, but I trust the examples scattered through the pages will be found sufficient for purposes of illustration. As these, (many of which contain results of independent importance,) are placed in general immediately after the account of the theorems on which they depend, their position sutticiently indicates the process of solution, and I have therefore in- cluded a number of them in the index. Regarding this work as strictly introductory, I have pre- ferred not to give too many references. Those that do appear have been given, some because they are perhaps not just in the line of reading that is usually followed, some because of special felicity of statement, a few for their historical interest. Thus the frequency of reference to any one author is not to be interpreted as an attempt to indicate the extent of my indebtedness. Had the re- ferences been so adjusted, it would have been alike my duty and my pleasure to write on every page the name of Professor Cayley. PREFACE. vii My hearty thanks are due to various friends ; to ]\Iiss I. Maddison and Miss H, S. Pearson, for help in seeing' the book through the press ; to Mr. F. Morley, for vahiable suggestions while the work was in progress ; and to Mr. J. Harkness, for his great kindness in reading the whole, not only in proof, but also in manuscript. C. A. SCOTT. Bryn Mawr, Pennsylvania. May, 1894. CONTENTS. CHAPTER I. POINT AND LINE COORDINATES. Introductory — General Idea of Coordinates — Homogeneous Point Co- ordinates — Hoinogeneous Line Coordinates — Relation of the Two Systems — Distance from a Point to a Line — Pole and Polar with regard to a Triangle — Examples, ----- pp. 1-24 CHAPTER IL INFINITY. TRANSFORMATION OF COORDINATES. Parallel Lines — The Special Line at Infinity — Relation of Cartesians and Homogeneovis Coordinates — Change of the Triangle of Reference — Examples, --------- pp. 25-34 CHAPTER HI. FIGURES DETERMINED BY FOUR ELEMENTS. Collinear Points and Concurrent Lines — Tlie Six Cross-ratios of Four Elements — The Complete Quadrilateral and Quadrangle — Pairs of Points, Harmonically related — Imaginary Elements — Examples, pp. 35-50 X CONTENTS. CHAPTER IV. THE PRINCIPLE OF DUALITY. Correspondence hitherto noted — Curves in the two Theories — Dual Inter- pretation of Algebraic Work — Examples, - - - pp. 51-56 CHAPTER Y. DESCPvIPTIVE PROPERTIES OF CURVES. General Principles — Equations of the Second Degree, satisfying Three assigned Conditions — Equation of the Derived Secondary Element — Formation of the Reciprocal Equation — Poles and Polars — Ex- amples — Conies with Four assigned Elements — Examples — On the Number of Conditions determining a Conic — Examples — Condition that Six Elements may belong to a Conic — Examples — Joachimsthal's Method — Examples — Curves with Singular Points and Lines — Ex- amples, ... - pp. 57-100 CHAPTER VI. METRIC PROPERTIES OF CURVES ; THE LINE INFINITY. Introductory — Points at Infinity. Asymptotes — Diameters and Centre of a Conic— Exanii)les, ------ pp. 101-109 CHAPTER VII. METRIC PROPERTIES OF CURVES; THE CIRCULAR POINTS. Two Special Imaginary Points at Infinity — Condition of Perpendicularity — Relation of a Conic to the Special Points — The Circle — The Rectangular Hypeibola— Foci — Examples, - - pp. 110-128 CONTENTS. xi CHAPTER VIIT. UNICURSAL CURVES. TRACING OF CURVES. Unicursal Curves — Examples — The Deficiency of a Curve — Curve-tracing in Homogeneous Point Coordinates — Examjjles, - pp. 129-143 CHAPTER IX. CROSS-RATIO, HOMOCRAPHY, AND INVOLUTION. Projection — Alteration of Magnitudes by Projection — The Group of Six Cross-ratios, Algebraically considered — The Group of Six Cross- ratios, Geometrically considered — Homographic Ranges and Pencils — Homographic Systems with the same Base — Homographic Systems with different Bases — Involution — Double Elements of an Involu- tion — Involutions determined Algebraically — Common Elements of two Involutions — Involution determined by a Quadrangle — Ex- amples — Desargues' Theorem — General Idea of Involution — Involu- tion Properties of Conies — Examples — Systems of Conies — Determin- ation of a System of Conies by Pairs of Co)iiugates — Examples — Homographic Correspondence on Curves, - - - pp. 144-188 CHAPTER X. PROJECTION AND LINEAR TRANSFORMATION. Effect of Projection — Possibilities of Projection — Comparison of Different Projections — Alteration in Appearance caused by Projection — Analytical Aspect of Projection — General Linear Transformation — Comparison of Projection and Linear Transformation — Canonical Forms, - - pp. 189-209 CHAPTER XI. THEORY OF CORRESPONDENCE. Special Cases of (1, 1) Correspondence — Collineaticm — Examples— General Theory of Correspondence — General (1, 1) Quadric Correspondence— Quadric Inversion — Effect of Inversion on Singularities— Effect of Inversion on a Curve as a Whole — Beciprocation — The. Dualistic Transformation — Birational Ti'ansformation of a Curve, pp. 210-242 xii CONTENTS. CHAPTER XII. THE ABSOLUTE. Resume of the Argument — Degenerate Conies — The Absolute — Relation of a Curve to the Absolute — Correspondence of Asymptotes and Foci — Correspondence of Linear and Angular Magnitude — The Generalized Normal and Evolute — General Considerations, pp. 243-259 CHAPTER XIII. INVARIANTS AND COVARIANTS. Groups of Transformations — Linear Transformations — Binary Quantics — Ternary Quantics, - - pp. 260-278 CHAPTER I. POINT AND LINE COORDINATES. Introd'iictovy. 1. In analytical geometry the subject-matter is geometry while the language is algebraic. For progress and pleasure it is of primary importance that the language be properlj^ adjusted to the subject ; elasticity must be preserved and unnecessary restrictions cast aside. We begin therefore by examining our conceptions in analytical geometry, recognizing natural limitations, but rejecting artificial limitations except in so far as these can be shown to serve some good purpose. We begin, that is, by generalizing our conceptions and their expression as much as possible. General Idea of Coordinates. 2. The whole of analytical geometry as hitherto studied depends on the possibility of representing the position of a point in a plane by two coordinates, with the dependent possibility of representing the position of a point in ordinary space by three coordinates. These coordinates in the case of plane geometry were regarded initially as the distances from the point to two selected lines, these distances being- measured in assigned directions, viz., parallel to the selected lines. But other systems were occasionally used ; for ex- ample, polar coordinates, where the distance from a fixed pole to the point, and the direction of the line joining the lixed pole to the point, were the two determining quantities ; dipolar coordinates, where the two coordinates of the point were its distances from two fixed poles ; the system of co- ordinates arranged by means of confocal conies ; etc. The fundamental idea of coordinates derived from plane geometry is therefore that they are any two quantities that serve to determine the position of a point in a plane. Here there are implied certain limitations, which may be accidental ; for 2 POINT AND LINE COORDINATES. there are geometrical elements other than a point, whose position we may wisli to specify ; and we have no assurance that the number of coordinates required is necessarily two. We generalize therefore by dropping these limitations ; and we say : — Coordinates are quantities that determine the 'position of a, geometrical element. The nature of these quantities will depend on (]) the space assumed, (2) the problem considered, (3) the element selected. 3. We here recognize that the primary element may possibly not be a point. The point certainly presents itself naturally to our minds as the element, par excellence, probably because all our drawing is done with a point. But the straight line is essentially as simple ; and it is possible to imagine that we might have learnt to do all our drawing with a straight-edge instead of a point. We should then regard a point as a secondary element, uniquely determined by two straight lines ; and this secondary element, the point, would suggest to us an infinity of straight lines passing- through it, just as with our present ideas the secondary element, the straight line, unicjuely determined by two points, suggests to us an infinity of points lying on it. We shall constantly have occasion to notice in detail the correspondence between the two geometrical theories ; the two that is in which, the field being restricted to the plane, the primary elements are respectively the point and straight line, the secondary elements the straight line and point. The element, then, need not be a point : it may l)e some otlier geometrical entity. 4. In the next place we consider liow many coordinates are necessary. Our primary element is regarded as having position in space; we consider it therefore as able to change its posi- tion in certain ways, reducible to a certain nundjer of independent ways ; or we may say, the element has a certain number of degrees of freedom. Suppose, for definiteness, we take for element a point, and for space a line, straight or curved. There is only one possible way in which the point can move, viz., along the line ; it may of course move forward or backward, but these differ simply as positive and negative, that is they differ only in sense. The point has therefore only one degree of freedom ; its position on the line is determined by one coordinate, e.g. its distance from some fixed point POINT AND LINE COORDINATES. 3 on the line ; its freedom consists in the possibility of vary- ing this one coordinate ; if the value of this one coordinate were given, the one degree of freedom would be destroyed. Now imagine the point to be at P on any surface, for example a plane, and free to move to any other position Q on the plane. For this it is not essential that the point be able to move directly from P to Q\ the step can be accomplished by steps in two selected directions ; for example, by PM, MQ parallel to Ox and Oy (Fig. 1); thus two degrees of freedom allow for all conceivable motions of tlie point on the plane. These two degrees of freedom can Ije algebraically expressed as the variations of two coordi- nates, X and y ; and if the value of either of the'se coordinates were given, one degree of freedom would be destroyed ; if both coordinates were given, that is, if the position of the point were given, both degrees of freedom would be destroyed. Similarly a point in ordinary space has three degrees of freedom, and to destroy the three, that is, to fix the posi- tion of the point, three independent coordinates must be given. 5. This same idea may be differently expressed. A point (an element) with one degree of freedom can assume a singly infinite number of positions ; its one coordinate is susceptible of numerical values ranging from — jo through to + GO . A point (an element) with two degrees of freedom can assume a doubly infinite nundoer of positions (oo^) ; each of its two coordinates is susceptible of numerical values ranging from — go through to -f- oc ; and in general, an element with a degrees of freedom, that is, with a independent coordinates, can assume x" positions. 6. These fundamental conceptions arc expressed in various ways ; we speak of space of one, two, three, . . . ., a 4 POINT AND LINE COORDINATES. dimensions ; of a one, two, three, or a- way spread. But it must be kept in inind that the number of dimensions of any assumed space depends on the selected element. Thus, for example, tlie point being the element, a plane is space of two dimensions ; a circle with a fixed centre being the element, the plane is space of one dimension. In ordinary space, planes through a line are singly infinite in number ; that is, the line is to be regarded as one-dimensional, when the plane is the element ; and simi- larly the line is one-dimensional when the point is the element. Ordinary space itself is three-dimensional, when the point is element ; it is likewise three-dimensional when the plane is element. Thus we detect a correspondence between figures determined by points and figures deter- mined by planes. The points in a plane form a two-fold infinity ; the planes through a point form a two-fold infinity. Thus the point being element, the plane is a two-dimensional space contained in the three-dimensional space ; and the plane being element, the point is a two- dimensional space contained in the three-dimensional space. Now two points in space determine a line ; two planes likewise determine a line ; thus the line takes the same part in the two theories. Ordinary space is seen to be four-dimensional in lines; this appears from the fact that every line can be obtained b}'' joining every point in one plane to every point in another plane. Thus in Solid Geometry we have assemblages of points, planes, and lines ; and we have corresponding assemblages of planes, points, and lines. Ex. Considering the five regular solids as determined (1) by their vertices, (2) by their faces, show that one corres2)onds to itself, and that the others are in corresponding pairs. But now confining ourselves to Plane Geometry, we can no longer regard the plane as element ; we have only points and lines. The plane is two-dimensional as regards its points ; it is also two-dimensional as regards its lines ; for the position of a straight line depends on two independent quantities, for example, the intercepts made on the axes; or again, the number of straight lines in a plane is doubly infinite, for we obtain all straight lines by joining every point on one line to every point on another line. Thus tlie plane is of the same nature whether the point or line be regarded as element. Further, as regards aggregates contained in the plane ; tlie point being the element, the straight line is one- POINT AND LINE COORDINATES. 5 dimensional ; and the straight hne being the element, the point is one-dimensional. This last appears from various considerations ; a line through a fixed point has one degree of freedom, for it can rotate ; one coordinate, for example, the angle made with a fixed direction, serves to determine the position of the line ; the number of straight lines through a point is singly infinite. We have thus one more step in tracing the correspond- ence already referred to (§3) between two special theories in plane geometry, — the two in which the point and the straight line are respectively taken as primary element. Note. For a discussion of the different aggregates of elements iu any assumed space, see Reye, Die dt eonietrie der Lage, Cli. I., or Section II. of Professor Henrici's Article on Geometry in the Encyclopcedia Britannica. 7. We have seen that the number of independent co- ordinates required depends on the space and on the element. But in many cases it is convenient to use more than this necessary number of coordinates, connected by identical relations. Thus for instance we sometimes use Cartesians and polar coordinates in one piece of work, that is, four coordinates with two identical relations x = T cos 0, y = r sin 0. Homogeneous Point Coordinates. 8. We shall now confine the work, for the present, to geometry in a plane, using for element sometimes the point and sometimes the line. In either case the position of the element is determined by two independent coordinates ; but we are at liberty to use more than two coordinates con- nected by relations. In Cartesian geometry, we obtain the two coordinates of a point by means of two fixed lines ; we here begin by assum- ing an undetermined number of fixed lines, a, b, c, etc. Let the distances from a point P to these lines be denoted by a, (3, y, etc. ; using the ordinary convention of signs, each line has a positive side and a negative side, which may be initially arbitrarily assigned. Any one of these distances, e.g. a, determines P as lying on a line parallel to the line a ; the ratio of any two, a : ^, determines P as lying on a line through the intersection ah. The position of P is there- fore determined uniquely by two distances, a and /3, or by two ratios a : /3, a : y, by which a third /? : y is implied. If then we select three lines, a> h, c, not concurrent, the position 6 POINT AND LINE COORDINATES. of P is determined by any two of the ratios a: (3 : y. It is essential that the three lines a, h, c (Fig. 2) be not concurrent, for otherwise the two lines P{ab), F{ac), whose intersection has to give P, would be identical (Fig. 3). Let the lengths of the sides of the triangle be denoted by a, b, c, and its area by A ; and for detiniteness let the positive and negative sides of thr thivc lines be assigned so iliat a point inside the tri- aiigle is on the positive side of every one of these lines. We have then , i o , o v For denoting the vertices of the triaiigle l)y A, B, C, we POINT AND LINE COOEDINATES. 7 have for a point inside the triangle a relation among the areas, viz. 2.PBC+2.PCA + 2.PAB = 2.ABG, that is, BC.PD + CA. PE+AB . PF= 2 A, that is, aa + ^/3 + cy = 2 A ; and for a point outside the triangle, e.g. P', 2 . P'BC- 2 . P'CA + 2.P'AB=2. ABC, that is, BG.P'D'-CA.rE' + AB.P'F' = 2A, Fk;. 4. that is, aa + h(3 + cy = 2A, as before, since /3 is now (Fig. 4) represented by —P'E'. Thus we have in general aa + 6/3 + cy = 2A, and consequently when the ratios u: (3 :y are given, the values of a, /3, y are known. Now one advantage of using the ratios is that all our ecjuations can be made homogeneous in a, (3, y. For the two independent coordinates of P being a : y, /S : y, ^ny f;ict about the position of P is expressed by an ctpiation Fi- , ) = ^\ ^jf degree n ; multiplying throughout by y", this becomes homo- geneous. Or again, if a, /3 be taken as coordinates, then by means of the relation «a + 6/^ + cy = 2A, which can be written in the form ^ra+ -rr/3 4-^* y = l, any non-homogeneous ex- 2A 2A 2 A pression can be made homogeneous: for tlie terms of lower 8 POINT AND LINE COORDINATES. degree can be multiplied by any desired power of the unit multiplier 9. From our definitions it follows that for any point on the line a we have a = 0; that is, the line a is denoted by a — 0] thus the equations of the three fundamental lines are a = 0, ;8 = 0, y = (). Now consider any fourth line S = 0. By means of the triangle ahc we proved that a, /3, y must satisfy a relation which may be written in the form a a + h'ji + c'y = 1 . Similarly by means of the triangle ahcl we find that a, ^, S must satisfy a relation a"a + b"(3 + d"S=l. Subtracting, we see that a, (3, y, 8 must satisfy a relation (a - a")a + (b' - b")j3 + c'y - cV'S = 0, 1 , a —a" , b' — b" rt , «' whence S= -^„— a+ -"J^^^ + ^^'7^ that is, 8 —fa + g^ + h y. JVote. If the line 8 = pass through the intersection of the lines a = 0, /3 = 0, there is not any triangle abd ; but there is a triangle acd. Thus any fourth distance, 8, is a linear function of a, /3, y, and the equation of any fourth line, 8 = 0, when expressed in terms of a, /3, y, becomes fa+g^ + hy = 0; that is, in this system of coordinates any straight line ii> represented by a linear equation. 10. Tlicre is no occasion to limit ourselves unduly in tliis choice of coordinates ; the position of P will be eipially determined if instead of a : /3 : y we know la : m^ : ny, wlicre I, m, n are any multipliers, initially arbitrarily chosen. We write therefore x, y, 2 = la, rn^, ny; by this change the fundamental identical relation aa + b^+cy = '2A becomes a b c - . I m/ n 2A^ 2Anv' 2A7?. whicli may be written %->' + b^y + c^z = l-, POINT AND LINE COOIIDINATES. 9 and the equation of any line fa+g/S + hy^O becomes f'x + g'y + Jtz = 0, that is, it remains a homogeneous hnear equation. Recalhng to mind the significance of" S, we see that the distance from any point x, y, z to a line fx+gy + hz = is a multiple of fx + gy + hz; that is, the distance frum any point x, y, z to a line % = () is a niidti'ple of the corre- sponding value of the expression u. 11. It may be convenient occasionally to retain more than three coordinates, with homogeneous identical relations ; that is, any number of coordinates S may be kept, with relations S = la-\-mli-\-ny. Thus, for example, if the line x+y + z = be an important one in a problem, it may be convenient to write —u for x-\-y + z, so that we have four coordinates X, y, z, u, connected by the identity x-\-y + z-\-u = (). Suppose we have two lines u = (), v = (), where u, v, can when we please be written as linear functions of x, y, z. Then u-{-kv=:0 is a line through the intersection of it = 0, v = {). For the values of x, y, z, that make the linear ex- pressions u, V, vanish, make u + kv vanish; that is, the point that lies on each of the lines u = 0, v = 0, is a point on u + kv = 0. JVote. Letters such as u, v, etc., will be used not only as abbreviations for linear (or other) expressions, but also for referring to the diagram. Thus the line u means the line marked with a letter it in the diagram ; and when the equation of this line is written in abridged form, it will be written u = 0. The point 2iv is the intersection of the lines n = 0, v = ; e.g. in the fundamental triangle A, B, C are respectively i/z, z.r, xy. Homogeneous Line Coordinates. 12. Now regarding the line as element, let us assign its position in the corresponding way. The position of the point was referred to fixed fundamental lines : Ave refer the position of the line to fixed fundamental points. Let the distances to the line from these fixed points A, B, C, etc. be denoted by ^>, q, r, etc., so that the statement p = () means that the line passes through A, and so on. Regarding the line, as usual, as having a positive and a negative side, we see that any two distances, q, r, nmst bi; considered as being of the same sign wlien /^, C, are on tlie same side of the line. The absolute values of two distances, q, r, determine tlie line as one of the four common tangents to two circles whose centres are B, G, and whose ratlii are q, r: a com- 10 POINT AND LINE COORDINATES. parison of the signs of q, r shows that the line is one of two; in Fig. 5 q, r have the same sign for 1, 2, and opposite signs for o, 4. Thus in general tlie values of two distances q, r, determine the line as one of two. Fifi. 5. But if we deal witli ratios of distances, the position of tlie line is determined by means of the points in which it meets BC, CA, AB (the three fundamental points A, B, C, having been chosen non-collinear), just as when we are dealing with ratios of distances from a point P to three non-concurrent fundamental lines the position of P is deter- mined by means of the lines joining it to the points he, ca, ah. For if the line meet BG in A', we have (Fig. 5) by similar triangles, rj-vr nv ^ ' BX : C A =q :r, and the position of A' is uniquely determined by the ratio q:r. Hence two such ratios, q:r, r-.p (which imply the third, p : q) determine two points A", Y on tlie line, and thus they determine the line itself. That is, the lino is determined by the ratios j) : q : r. 13. Let Tis now compare the determination of the position of the line by means of ^), q, r, with the results obtained in the system of homogeneous point coordinates already dis- cussed. In order to keep the connection between the two theories as clear as possible, and the transition from one to the other as simple as possible, we take the points A, B, C, determined by tlie triangle of reference in the system of ])oint coordinates, for fundamental points in the system of line coordinates now under investigation. Let the line re- ferred to the system of point coordinates have the ecpuition the line is therefore determined by /, (j, Ji : it is e(|ually de- termined by p, q, r: conse(|uently these two sets of (piantities must be expressible in terms of one another. POINT AND LINE COORDINATES. 11 Let A' (Fig. 6) be a^, /3j, y^ then «, = (), and therefore from the assigned equation of tlie line A'FZ, ,7/3j + /iyi = 0. From the figure, X being on the negative side of C'A and on the positive side of AB, q:r = BX:GX= area BXA : area CXA = cy^ : - hfi^, therefore hq : cr = y^^ : —^^=g:h ; and similarly (q^ : hq —f : g ; hence f:g:h = ap : hq : cr. Thus the coefficients in the ecjuation of the line in the system of point coordinates are linearly expressiljle in terms of the coordinates of the line in the system of line coordinates. 14. Tliis same I'esult may be expressed in a form sliglitly different and more significant. Let a, /3, y be any point on the line, then fa + gl3 + hy = 0. But regarding the line as determined by ]), q, r, we write this condition by what has just been proved in the form apa-\-bq^ + cry = 0: that is, the point a, /3, y and the line p, q, '^' are united in position if the condition ap)u-\-hq^-\-cry = () be satisfied. 15. We have already noticed that we are at liberty to take for coordinates of the point any nuiltiples we please of a, ^, y, viz., X, y, z = la, m/3, ny. Similarly for coordinates of the line we are at liberty to take any multiples we please of p, q, r, viz., £, >;, ^—Xj^, /xq, vr. Making tliese changes, the 12 POINT AND LINE COORDINATES. relation just found becomes: — The point x, y,z and the line ^, ;/, ^ are united in position if the condition be satisfied. Since we constantly use both systems of co- ordinates in one investigation, there is a decided practical convenience in adjusting the two systems of multipliers I, m, n and X, JUL, V so that this condition may reduce to the simplest possible form : we therefore choose these quantities so that ct _ 5 _ c . that is, we take ^X : w/x : nv = a :h:c; and the relation becomes : — TJte point x, y, z and the line ^, j;, ^ are united in position if xi+yr^ + z^=0. Relation of the Two Systems. IG. The equations l\ : nifx : nv = a : h : c leave us free to choose either set of ratios l:ni:n or \: fx :v, but they then determine the other set. We may therefore choose our system of point coordinates, but then the corresponding system of line coordinates is at once deduced. As particular systems of point coordinates we have I. Tvilinears. Here x, y, z are simply proportional to a, /3, y, i.e. x : y : z = a : ^ : y, therefore l = ni = n. In this system the fundamental identical relation is ax + hy + cz = constant. The associated system of line coordinates is determined l)y ^X : vijUL -.nv — a'-h : c, that is, by \: jx: v = a:h : c, and thus i '• '1 '• ^= ('1> '• '^9' • ^"''• II. Areals, also called triangular coordinates. In this system the coordinates of a point P arc the ratios of the triangles PBC, PGA, FAB to the whole triangle ABC. Now 2 area PBG= au, therefore _ aa _ hjB _ c'y ^"'2A' ^~2A' ^~2A' ,, , . , a h c that IS, l,m,n = ,^^-^. ^A' 2 A" POINT AND LINE COORDINATKS. 13 The fundamental identical relation is now x-{-y-\-z=l. The associated system of line coordinates is simply for the equations giving X, ja., v reduce to X = /x = V. 17. The condition a;^ + 2/>; + c^'=0, expressing the union of the point x, y, z and the line ^, >/, ^, is of primary importance. Writing /, (j, Jc for ^, ;/, ^ it may be stated as follows : — If the equation of a line be fx+yu+Jiz — O, the coordinates of the line are f g, h. It thus enables us to pass from the expression of a line in point coordinates to its expression in line coordinates. But again it may be read differently : — The line whose coordinates are ^, ?/, ^ j^^'^''^*^^ througli the point x, y, z if that is, any line with coordinates ^, ?/, ^ passes through the point /, (J, h if f^ + gi] + h^=0. Thus in order that a variable line may pass through a fixed point the coordinates of the line must satisfy an equation of the first degree. This equa- tion is called the equation of the point ; it is the equation that must be satisfied by the coordinates of all primary elements that are united with the assigned secondary element ; it is therefore exactly analogous to the equation of a line in point coordinates, for this is simply the equation that must be satisfied by tlie coordinates of all points that lie on the assigned line. //' then f g, h he the 'point coordinates of a point, its equation in line coordinates is f^+gi] + Ji^=0. The distance from a point I, ni, n to a line ^x-{-]]y + ^z = was shown in § 10 to be a multiple of ^l -\- ipn + ^7i ; hence the distance from a point l^+7nrj-{-n^={) to a line ^, ;/, ^ is a multiple of l^+m}] + n^; that is, tJte distance to any line ^, >/, ^ from a point ct = is a midtiple of the corresponding value of the expression oX. 18. Just as in the case of point coordinates, we are at liberty to use more than three fundamental points ; let any additional one be = 0, then 6 can at any stage of the work be written as a linear function of ^, ri, ^. Suppose we have any two points wliose equations are CT = 0, /3 = 0, where V5, p are linear functions of £, j/, ^. Then To + kp — being linear is the equation of some point ; now coordinates ^, r], ^ that make the linear expressions w, p vanish u POINT AND LINE COORDINATES. simultaneously, make ^-\-l'p vanish; thus the line that passes throuo-h each of the points ~ = (), p — 0, passes through the point C7 + /.'p = (): I.e. ~ + kp = is a point on the line nr^. (See §11.) Note. Letters such as 6, p, etc., will be used not only as abbrevia- tions for linear (or other) expressions in ^, r], ^, bnt also for referring to the diagram. Thus the point p means the point marked with a letter p in the diagram ; and when the equation of this point is written in abridged form, it will be written p = ; the line TTTp is the join of the points try = 0, p = 0; e.g. in the fundamental triangle, J, B, C are the points ^, ■>], (, their equations being ^=0, 'q — 0, C=0; a, h, c are the lines >;{', {'^, ^^]. Ordinary Roman capitals will however frequently be used simply to denote point-positions in the diagrams. 19. Here we have an important step in the correspond- ence already noted between the two o-eometrics in a plane, the two, that is, in which the primary element is taken to be (1) the point, (2) the straight line. These two theories now run as follows : — We may regard the point as eleiuent ; two points determine a line ; an indefinite nund)er of points lie on the line. The position of a point in the ])lane is detei^mined by two inde- ))endent coordinates ; but it is convenient to make use of three coordinates. By the equation of a line we mean the relation satisfied by the coordinates of all jjoints on the line ; this is of the first degree f.r+gij + hz = Q. The e(piation of the line deter- mined by the jwints (,'j, //,, ?,), (./:,, 7/,, z.;) is We may regard the line as element ; two lines determine a point ; an indefinite number of lines 2>ass through the point. The position of a line in the plane is determined by two inde- pendent coordinates ; but it is convenient to make use of three coordinates. By the equation of a point we mean the relation satisfied by the coordinates of all lines through the point ; this is of the first degree The equation of the point de- termined by the lines (^i, ir]^, ^j), (^'o, ^/,, Q 'is X y ■>/■> "1 = 0. fe2 Tliis is obtaineil Ijy considering wliich has to be satisfied by two sets The condition that three points 1, 2, 3 be coliinear is the vanishing of the determinant ■ 1 Vx the linear equation f.v+gif + Iiz = Q, of quantities {x^, y^, z{), {oc.^, y.^, z.^). The condition that three lines 1, 2, 3 be concurrent is the vanish- ing of the determinant 1 ^1 V, Ci ■ ^2 Vz C2 I ^3 V3 C3 I POINT AND LINE COORDINATES. 15 For tlie three linear equations y-''i + ,9'^i + /'%= 0, f.r2 + g>/.,+ /tz.i=0, fx:i+ gyn + hz., = 0, must be satisfied ; eliminating/, g, h, the result fullows. The coordinates of the join of the two points are gh' - (/'h, hf - ///, f- + g,jj + h.f = 0, be concurrent is the vanishing of the determinant /i 'Ji K I- f-2 'J-2 ^>2 fa 'Js K ' The condition that three points be collinear is the vanishing of the determinant /i 'Ji h f-2 Oi ^^2 fo 0; h. This is obtained by eliminating the variables from the given cquati(jns. The coordinates of any point on The coordinates of any line the join of {a\, ?/,, z^), (.r^, y^, z-^) are through the intersection of expressible in the form {^^, r;^, ^^), (^^, 7/.,, ^.,) are ex- h\ + mx.^, ly^+mij.,, b-^ + m.:.-,. pressible in the form l$i + m$,, h,^ + mg,. 11^ + ml,. For (.r, ?/, ,::), (.Cj, _?/j, z^\ {.c.,, 1/.^, z./) are collinear if, and only if .v y z =0. 'Vo y., z.. Hence determining I, m, so that the determinant shows that X — li\ - m:r., =0, that is, unless y^z,, — y.^^ =0, x = Li\ + mx.j^. This possible interference, yi^-y-fx = ^., would however be noticed earlier, for it woidd prevent the determination of /, in as directed. But if Vx-fV-i^x — ^-: tjertainly x^z.^^- x.>Zy^yQ^ for otherwise the ratios .»'j :^i : rj M'ould be the same as x.,\y.^:z.^, i.e. the ])oints 1, 2 would be the same. Thus in this case we determine ?, m from the equations Ix^ + mx.^ = .r, Iz^ + mz., = z. The underlying principle manifested in this correspondence is known as the Principle of Duality. Tlie nieanino- of tlie name, the importance of the principle, and the utility" of the correspondence, will appear more plainly in the following chapters. 16 POINT AND LINE COORDTNATES. 20. We have foiiml that the actual distances from a point to the sides of the triangle of reference satisfy a permanent relation, viz., aa + b^ + cy=2A; and that hence the actual values of the coordinates in any homogeneous point system satisfy a permanent relation of the form Similarly the actual distances to a line from the vertices of the triangle of reference satisfy a permanent relation, viz., a'-^y' + h''(f + c-r- — 'Ihq . cr . cos A — 2ci- . ap . cos B — 2(12) ■ ^1 • ^^^ ^— 4^"- Let the line make with BO, CA, AB, angles 6, (p, \/r (Fig. 7), so that ^|r-(p = A, e-i^ = B, e- = 7r-C. Considering the quadrilaterals BCFE etc., we have area ABC= BCFE - CFDA + A BED, therefore 2 A = (r/ + v)FE- {r + 'p)FD + (p + q)ED = -p.FE+q.FD-v.ED. Now FE = a cos (tt — 6), Fl) — U c( )s 0, ED = c cos \jr, therefore dp c* )s + hq cos f/> — cr cos \f/ = 2A ( 1 ). Also, from the identity p(q-r) + q{r-j,)+r{p-q)^0, by means of the relations q — r — ds'mO, r — p = hiim^, p — q=—cs{n\ly, we obtain dp sin Q-\-hq sin — crsin \/y = (2). POINT AND LINE COORDINATES. 17 From (1) and (2), by squaring and adding, we iind a-p" + Ircf' + ch'" — 'Ihq . cr (cos cos i//- + sin sin i/r) — 2 cr . wp (cos Q cos \j/- + sin 6 sin i/r) + 'lap . hq (cos cos + sin sin 0) = 4 A'-. The expressions within brackets being- cos (\//-—0), cos(0 — \/r), and cos(0 — 0), that is, cos A , cos i^, and — cos C, this relation reduces to a^2^^ -{- b-q' -\- c~'r^ — 2hq . cr . cos A — 2cr . aj) . cos B — 2a^> . hq . cos C' = 4 A'-. The fundamental identical relation is therefore of a different form in the two systems considered ; that is, the correspondence Idtherto noted between the two theories does not apj)ear to hold. This will be investigated later (Ch. ^^II. and XII.) : for the present we shall not use the Principle of Duality in work that depends on the fundamental identical relation, that is, in work that recjuires us to use actual values of the coordinates ; we shall not use it in investigating metric relations. When the point coordinates are trilinears, we have agreed to use as the associated line system ^, >/, ^—ap, hq, cr. With these coordinates the relation just found becomes i~ + ^f + r^ - ^4 cos A -2.1^ cos B - 2^>j cos C = 4 Al Similarly for any other system of coordinates we have a modified form of each of the two fundamental relations, wliich can be deduced at once from the natural forms aa + hj3 + cy = 2A, a-p" + //-(/- + c'-'i-- — 'Ihq.cr cos A — 'lev . ap . cos B — 2(ip . hq . cos (7= 4A^. JJlstdiice from o, Point to a Line. 21. General homogeneous coordinates do not lend them- selves readily to the direct investigation of purely metric properties. Fornuilse on which these investigations depend {e.g. for the distance from one point to another, for the angle made by one line with another, etc.) can of course be obtained, and may be found, e.g. in Ferrers' Triliuear Goordvnatea. But the principle here adopted of using only suitable methods for any assigned problem, and applying given methods only to suitable problems, forbiils their introduction. One particvdar synnnetrical metric expression must however be found, viz., tliat for the dis- tance from a point to a line. Tliis is of importance in S.G. IJ 18 POINT AND LINE COORDINATES. tlie discussion of the metric relations involved in the Prin- ciple of Duality (Ch. II., VII., and XII.), and moreover it relates to the conception that has been made the founda- tion of our double system of coordinates. Let OM be the distance from 0, the given point a, (3, y, to the given line p, q, r, this given line making with CB the angle 6. Draw from A, 0, lines parallel and perpen- dicular to BC, meeting in E, and join OB (Fig. 8). Then we have (indicating signs of magnitudes by signs prefixed, and not l)y the order of the letters), OM=q- OB sin (6 + DBO), that is 0M= q - OB cos {0 - BOD) : also OM =r-OC cos (6- COD), and Now therefore hence OM=p-OA sin (^Oil/- j). A0M-\-A0D = '7r-\-e, AOM-'^^ = '^^+0-AOD: 03I=p-0AcoH{e-A0D). From the tliree expressions liere found for OM we obtain p - OM = -OE. cos e+AE. sin 0, q - OM = OD . cos e + Bl). sin 0, V - 0M= OD . cos + CD . sin 6 ; whence;, multiplying by da, bjS, cy, and adding, ((dp -{- hft = iJ'' 9'- "^^ ^ ^ t ni^ n the expression for the distance from the point x, y, z to the line ^, rj, ^ becomes . 2A(xi +yrj + zC) . ( / * + - 2/ + ^){^'i~ + ■^'^ V + '^^^^^ — 2mnr]^ cos A — 2nl^^ cos B - 2lm^,j cos C)^ The particular forms assumed when (1) trilinears, (2) areals, are used should be noticed. 20 POINT AND LINE COORDINATES. Pole and Polar with regard to a Triangle. 22. It is usually convenient to determine the nmltipliers I, m, n so that a particular line of the dia^-rani may have a particular equation, or so that a particular point may have particular coordinates. Suppose e.g. we wish the centroid of the triangle to be 1 9 \ 1, 1, 1. The actual distances being a, /3, y, we have a = Q ' o a 2A 2A 2A etc.: we must therefore take Ix"^^- =mx'^ =nx^ , in oft 36 So order that this point may have x = y = z. These equations give I -.in : n = <( -.b : c, hence x : y : : = aa : h/B : cy, and we must use areals. It should be noticed that the condition a given 'point has fipecijied coordinates determines only the ratios l:m:n: it gives us a result of the form x:y:z — Aa:B/3:Cy, and not x = Aa, etc. Practically, we hardly ever require actual co- ordinates ; in fact, if we use the fundamental identical rela- tions properly, we never require them. A point /, (/, h may be taken as having coordinates /./, l-g, kh, where h is any nuiltiplier we please : this follows from the fact already noticed that all e(|uations used can be made homogeneous. 23. Whatever system of coordinates we employ, the point 1, 1, 1 and the line 1, 1, 1, — or, more generally, the point /, g, It and the line -,., -, j , — have a simple geometrical connection. Let P be the point /, g,h ; let AP, BP, OP meet BC, GA, AB in A', B',C' •. and let B'C meet BC in A", and similarly let B", C" be determined : then A", B", C" are collinear, and the line they determine is the line -:, -, j (Fig. 9). M'hc lines AP, BP, CP have equations 2/_^=o --■'■=0 ''-y=o- hence Ji' is the intersection of ?/ = an, (JE, OF are equal ; they therefore represent the coordinates of 0. The coordinates of any other point P are rejjresented by PL, PM, PN drawn parallel to OD, OE, OF. Similarly if O be the point /, g, h ; determine points Z>, E, F so that 01), OE, OF represent /', g, h on any scale ; the coordinates of any other point P are rei)resented by PL, PM, PN drawn jjarallel to OD, OE, OF (Fig. 10). So also a ^loint with any assigned coordinates can be accurately inserted in the diagram ; and any desired line can be drawn, by means of the points in which it meets the sides. A better method, however, follows from principles to be explained in Chaijter III. Examples. 1. Determine the distances to the sides of the triano-le of reference from the following j^oints : — (i.) the centroid ; (ii.) the orthocentre ; (iii.) the centre of the circumscribed circle ; (iv.) the centre of the inscribed circle ; (v.), (vi.), (vii.) the centres of the three escribed circles. Give the coordinates of these seven points {a) in trilinears, {h) in areals. 2. Determine the seven different systems of ctionlinates in which these points shall be 1, 1, 1 ; and determine in every case the associated system of line coordinates. Verify that the condition of collinearity is satisfied by the points (i.), (ii.), (iii.). 04 POINT AND LINE COORDINATES. 3. Determine the distances from A, B, to (i.) the internal bisectors of the angles A, B, C ; (ii.) the external bisectors of these angles ; (iii.) the lines through A,B,C bisecting BC, CA, AB ; (iv.) the lines through A, B, G perpendicular to BC,CA,AB. Give the coordinates of these four sets of lines in the two line systems that are associated with (a) trilinears, (b) areals. 4. Having obtained the coordinates of these sets of lines, write down their equations. Verify that the condition of concurrence is satistied by the lines in sets (i.), (iii.), (iv.). 5. Find the point coordinates of the intersection of the lines whose line coordinates are /^, ni-^, -Ji^, and Z.,, m^, ')i.2- State the result with reference to the equations of the lines and their intersection. 6. SI low (rt) by point coordinates, (h) by line coordinates, that if the Joins of vertices of two triangles be concurrent, then the intersections of sides are collinear. Note. Trian^^fles thus situated are said to he in perspective ; the jwint and line are the centre and axis of perspective ; or again, the relation of the triangles is spoken of as homology ; the point and line are the centre and axis of homology. 7. Find tile equations of lines through A, B, C making witli AB, BO, CA (inside the triangle) angles w. F'ind the value of to if these be concurrent ; and the coordinates of their connnon point. Show that the same value of oj ensures the concurrence of lines through A, B, making with .16', BA, CB angles w. Xotc. These two points are the Brocard points of the triangle ; w is the Brocard angle. CHAPTER 11. INFINITY. TRANSFORMATION OF COORDINATES. Parallel Lines. 25. Taking any two lines whose equations in point co- ordinates are Ix + my + nz = 0, I'x + niy + n'z = 0, and solving for the coordinates of their point of intersection we find X :y : z = mn — m'n : nV — n'l : Im — I'm. To obtain the actual values of the coordinates, we make use of the fundamental identical relation, which we suppose to be written in the form This gives mn' — mn _nl'—'iil hii' — l'm x=- jj , y= ~j) , -= D ' where D stands for the determinant I m n , . r m n Thus we have ordinarily a set of finite values for x, y, z, giving for the intersection of the two lines a point at a finite distance from every side of the triangle of reference. But if D = 0, those fractions whose numerators do not also vanish become infinite. Now the vanisliing of two of the numerators entails the vanishing of the third ; for n ■ '^'^ '>^ 1 r 'I n • n I mn— 7/111 = <>-ives -,= „ and nl-nL = i) M-ives -,= >>; I m ^'^ ^^ ^ n I hence r,= n that is, Im —l'')n = (). But thesi; give simply I : '111 : n = r : m : n' , which would make the two lines identical. Hence not more than one of the numerators can vanislu Thus of the coordinates x, y, rj, certainly two and possibly ■2C) INFINITY. TRANSFORMATION OF COORDINATES. all three are infinite, and the point x, y, z is at infinity. We see then that two lines may be so situated that instead of a finite intersection they have their intersection at infinity. This atj^rees with the properties of parallel lines that we have already employed : and by means of this we formulate the definition as follows : — Lines that meet in a ijoint at infinity are said to he ])arallel. The condition of parallelism is therefore I in n =0, i m n I «.Q Oq Cq I where af^x + h^y + c^z—l is the fundamental identical relation. The Special Line at Infinity. 'Id. The form of this condition recalls the condition that three lines be concurrent. It is in fact the condition for the concurrence of Lc + my + nz = , l'x + m'y-\- n'z — O, %x-hb^y-\-CoZ = 0: that is, of the two given lines, and a line which is the same whatever pair {I, m, n), (V, m/, n) be chosen. We naturally consider therefore what is the significance of this special line a^x + h^y-\-c^z = (). One peculiarity in the etpiation is at once evident ; it is at variance with the fundamental identical relation. Noav this Fundamental relation was obtained by a process certainly valid for finite values of a, ^, y. but not so obviously ad- missible if any of the three a, (i, y happen to be infinite. We shall find that this ex})lains the apparent paradox ; the line we are to consider, viz. aQX-\-b^^y-\-CQZ = 0, is a line lying entirely at infinity, and accounting entirely for infinity in point coordinates. T. Every point on this line is at ivfinity. For to determine the actual coordinates ot" a point on the line, wc lia\e to determine x, y, z to satisfy the two ecpiations (6oa; + 6o?/ + CoS = (1), a^x + b^,y+c^z=] (2), I.e. to satisfy the etjuation obtained by subtraction, 0.x + 0.y + 0.z=\ ( ;3 ). Now (8) re(iuires one at least of the quantities x, y, z to be iiitinite; and then, by equation (1), certainly one other must INFINITY. TRANSFORMATION OF COORDINATES. 27 be infinite ; thus two of the coordinate.s of any point on the line are infinite, i.e. every point on the line is at infinity. II. But further, this line is the complete i^oint representa- tive of infinity ; that is, every point at infinity lies on it. For let X be a point at infinity, and let P, Q be points at a finite distance, chosen so that FQ does not pass thi'ouo-li A'. Let the equations of PA"", QX be Ix + my + ')?5; = 0, I'x + rn'y -\-n'z = 0. These lines, meeting at X, are by definition parallel, and consequently = 0: I m n I' m n % K Co that is, the three lines lx + my + nz — 0, l'x + m'y-{-n'z = 0, aQ« + 6^^ + Cq-s = arc concurrent, and consequently a^x + b,^y + c^s = passes through X. It appears as though this special line might be regarded as parallel to every line u, for it meets u at infinity, and there- fore satisfies our definition of parallelism. This would also make it perpendicular to every line u, being parallel to a line perpendicular to u. But for reasons that will appear later, the idea of direction must not be associated with this line ; any attempt at assigning direction to it leads to absolute indetermination. 27. Now the fundamental identical relation in point co- ordinates, being obtained from aa -1-6/3 -hey = 2 A, may have for the constant on the right any finite (juantity, though we generally arrange the equation so that the value of this constant may be unity. The relation is therefore essentially of the form a(,x + b^y + c^z^\=0: hence we have the conclusion : — In general the coordinates of a point in a pLtne are conditioned by a linear inequality a^x -\- b^y -{- c^^z^i^O, bid there are exceptional points in the plane which make a^x -\- b^y -{- CqZ = ; these exceptional points are the totality of points on a certain straight line tvhich lies entirely at infinity, and has for its equation aQX-\-b^)y-\-C(,z = 0. Note. It i.s well to notice that a similar statement may ])e made with reference to any linear function lo = Ix + m u + nz. Taking aiiv 28 INFINITY. TRANSFORMATION OF COORDINATKS. point. /' at ramloin, this does not lie on the line u ; that is, in general the coordinates of a point in tlie plane make u^\=0 ; l)ut there are points for which « = (), viz., the totality of points lying on the line >'. The bearing of this remark will appear in (!ha})ter X. 28. Using trilinears, the line iutinity is ax + hy-{-cz = 0. Now the expression For the distance from a point x, y, z to a line ^, >/, ^ lias been shown to be 2A(^-^+?/>? + gO _^ {ax -\-by + c^Xf- + ;/' + ^- - 2;/^' cos A -2^^ cos B-2^tj cos C)* When we compare distances from different points to a fixed line, f, >y, ^ are constant quantities, and therefore 2A ( f + '/' + r' - -4 cos A -2^^ cos B - 2^,; cos 6')^ is simply a constant multiplier /;. Thus the distance from ■ ^^ • , , J^ 1 T 7 • j Ix + '1111/ -\- HZ a variable point x, y, z to a lixed line I, on, n is Ic , , -— - - > i- '^' ax+by + cz where /.; depends on I, m, n. If then we take for I, m, n, A multiplied by a, h, c, the numerator in this fraction becomes \(ax + hy + cz), and consequently ax + by + cz divides out. Tluis the expression for the distance from any point x, y, z to the special line ax-\-by -{- cz = is kX, where kX does not depend on x, y, z ; consequently all ordinary points in the plane must be regarded as being at an absolutely constant distance from the special line ax + by + cz = 0. It is obvious that this constant has not a finite value ; and in fact h in- volves in its denominator the square root of i" + n^ + r - 2';^ cos A - 2^^ cos B - 2^,y cos C, where ^, ?;, ^ are respectively a, b, c. Now a- + //- + c- — 26c cos A — 2ca cos B—2 ab cos G = a' + b-' + r' - (/;- + c^ - (r)-(<^^ + a' - b'') - (a^ + ¥ - c^) = 0, ami tlms the absolute constant /'A has an infinite value. 29. We have here shown that the line coordinates of the s])ecial line ax + by + cz = contradict the fundamental identical relation in line coordinates, tor they make a certain expression vanish, while the identical relation asserts that this expression has a constant value different from zero. By what has been pointed out already regarding the applicability of the l^i'inciple of Duality, we know that we must not expect the al)ove investigation to apply exactly to line coordinates ; but we have here a hint that there INFINITY. TRANSFORMATION OF COORDINATES. 29 is an analogous paradox and presumably an explanation. To this we shall return when considering expressions of the second degree. (See Ch. VII.) Rdation of Cd't-te.sia'ns rmd liomofjeneoux Coordinates. oO. With the help of the line infinity, Cartesians may be exhibited as a special case of homogeneous coordinates, and simple formula) of transition can be found. Taking the Cartesian axes, and adjoining to tliem this special line, we have three non-concurrent lines, which ma}' therefore be taken as lines of reference. Let the Cartt\sian coordinates be A^, Y, and take the line A" = 0, i.e. the axis of Y, for the line a = (); take Y=() for /3 = 0, and the lino infinity for y=:(). Then we have, from Fig. 11, A'' sin (jd = a, Y sin to = /5 : and y = a constant (,^ 28). Let the homogeneous coordinates be as usual X, y, z = la, m^, ny : then x — l'X, y = 'm'Y, s = constant. Now although y is infinite, yet it has been sliown to be an absolute constant ; therefore by properly choosing n , ny (i.e. z) can be made to assume any finite value we please, e.g. unity; our formula} of transition fi'oni homogeneous coordinates to Cartesians are therefore a; — any nmltiple of X, . y = any (other) multiple of ]", z — any convenient constant : that is, if we choose, x = X, y=Y, z=l; and we pass from Cartesians to homogeneous coordinates by introducing in the various terms such powers of z as will make the i'(jua- tion homogeneous. 80 INFIMTV. Tr{ANSF()R>rATTOX OF COORDINATES. The line iiitinity is here s = (), i.e. 0..r + 0. ?/ + c = : wliich ill Cartesians is o.A'+o. y+('=i). that is, 6' = 0. Tims tlie line infinity presents itself in Cartesians under the paradoxical I'oini finite constant = 0. 81. The effect of this transformation on the implied line coonlinates is noteworthy. The points A, B are at infinity; let a line cut BC, CA, in L, M (Fio-. 12). Xow MA and LB are distances from M, L to the line infinity, an-^ = 0, x cos /3 + y sin /3 — p., = *^ etc., and let us use the letters a, /3, etc., as abbreviations for X cos o + ,v sin a - ])^, etc., then a = represents the line a, and the value of a at any point P o'ives the distance from P to the line = Lu + L'i-\-L\u, etc., where 1) is the ddcnn'mant of transformation, or more often the ■modiiiu.s of transfurin- ation ; the non-evanescence of D is secured by the condition that u, V, lu form a triangle. Since only the ratios of x:y:z are required, the factor D may be dropped, and the resulting fornndfe of transform- ation are X— ill -\-L'v +L"u', y = Mil -f M'v -f M"iv, 'z = Nh+N'v + N"w. These formid.e being linear, the degree of an equation in point coordinates is not affected by any change in the funda- mental triangle. Similarly the triangle of I'eference can be changed in line coordinates. Let the vertices of the new triangle be = (l, = 0, x//- = 0, where INFINITY. TEANSFOEMATION OF COOEDINATES. 33 Solving these equations, ^, rj, ^ are found as linear functions of the new coordinates 6, (p, x/y. 34. But frequently the two sets of coordinates are in use together, so that forniulffi are required for the change of the fundamental triangle in the double system ; the two sets of coefficients (I, rn, n, I', etc.), (A, /x, v, \', etc.) are no longer independent. Consider any line whose equation referred to the original triangle is ^x + i]y-\-^z = {), so that ^, >/, ^ are the original coordinates of the line. The transformed equation is to be 6it + ] + N^, y = Mil + M'v + M"w, = L'i + M'lj + N'^, z = Nn + N'v + W'lv ; ^Ir = L"^+ M"i^ + N"^ : which may also be written u — lx + 'iny-\- nz, etc. , ^^ie + l'cp+l"x}y, etc. Two substitutions related as above are said to be m verse ; one expresses a system of old variables in terms of new variables, the other expresses a related system of new variables in terms of old variables ; the coefficients are the same, and if read in the one case by columns, and in the other case by rows, they are in the same order. Examples. 1. Using areals, lind the equation of the line bisecting two sides of the triangle of reference, and sliow tliat it is parallel to the third side. 34 INFINITY. TRANSFORMATION OF COORDINATES. 2. Find ((0 in triliiuvirs, (b) in areals, the equation of the line tlirout,^! a vertex parallel to the opposite side. 3. Representing- the line infinity in any system by 8 = 0, prove that the four lines u, v, u + ls, v + ls form a parallelo- gram, and find its diag-onals. Interpret this if 6' = () be a hne not at nifinity. 4. Taking any four lines, no three of which are concurrent, as x = {), y = 0, z = 0, u = 0, where x-\-y + z + u = 0, determine what lines are represented by x±y = (), x±z = 0, etc. Construct the general diagram ; and a special one for the case when x, y, z, are areal coordinates. CHAPTER III. FIGUEES DETERMINED BY FOUR ELEMENTS. Gollinear Points and Concurrent Lines. 35. Among- figures determined by assigned points or lines those in wliich the points or lines are collinear or concurrent are naturally considered first. Points lying on a line are spoken of as a range, lines pass- ing through a point are spoken of as a pencil. In considering a range (quantitatively, we are not concerned with actual distances from point to point, but, for a reason whose full force appears in a later chapter (Ch. IX.), with functions of the ratios of these distances. Any such function depends on four points, taken in a determinate order. Let the four points be A, B, P,Q: consider the segment AB as divided, first by P, and consequently in the ratio AP.PB: secondly by Q, and consequently in the ratio A Q : QB. The ratio of these two ratios, that is, AP AQ PB ■ ~QB' is called the Cross-ratio (Anharmonic Ratio, Doppelverhiilt- niss) of the points; it is denoted by {AB, PQ), a symbol in which both the grouping and the order of the points are indicated; or by {APBQ}, in which, unless by a clearly specified convention, the grouping and order are not indicated. 36. To obtain the corresponding function for a pencil of four lines, let the rays a, h, 'p, q, meeting in 0, be met by a transversal in points A, B, P, Q; then the cross-ratio of these points can be expressed in terms of the angles determined by the lines. For if h be the distance from to the transversal, Fig. 13 shows that h.AP = 2xiiniii()AP = OA . OP . sin ((,p, h.PB = OP .OB.smpb, 30 FIGUEES DETERMINED BY FOUR ELEMENTS. ,, ,. AP OA sinap PB OB sm 2)0 o- ., 1 AQ OA sinaq Sninhuiy 7T^ = 7Tr) ' ^^ f' '^ QB OB sm qb T ,, j; AP AQ sinap sin aq and theretore ^^^ : 7^^ = -. — ^ : — f- PB QB sm jw sm qb Fio. 13. The cross-ratio determined b}^ the pencil on the transversal is therefore the same for all positions of the transversal ; this cross-ratio, sin ap , sin aq sin pb ' sin qb' is called the cross-ratio of the pencil. It is denoted b}' {ab, pq), or by {apbq} ; or, when explicit reference to the vertex is desired, by {0 . APBQ}. Hence, given a range, the cross-ratio of the pencil formed by taking as vertex any point in the plane is known, for it is equal to the cross-ratio of the range ; and given a pencil, the cross-ratio of the range determined by any transversal is known, for it is equal to the cross-ratio of the pencil. The Six Cross-ratios of Four Elements. 37. By taking the four given elements in different orders a number of different cross-ratios are obtained. There are twenty-four different orders, but the cross-ratios reduce to six different ones. For, by definition, any one is the ratio of a product of segments AP . QB to a complementary product AQ . PB. Now the four points determine six segments, and therefore three products of the admissible type. Writing these in the order AB.PQ, AP.QB, AQ.BP, it appears that any cross-ratio is tlie ratio of two of these, FIGUEES DETERMINED BY FOUR ELEMENTS. 37 taken with a negative sign. Hence calling these I, m , n, the six cross-ratios are m n I _ n i m n I in m n I But these six are not independent ; they are in reciprocal pairs, and the product of the three in a row is —1. Moreover l+m + n^AB.PQ+AP.QB + {AP+PQXBq + QP) = AB.PQ + AP.QB-(AP + PQ)(QB+PQ) = AB.PQ-PQ(AP + PQ + QB) = AB.PQ-PQ.AB = 0. Til 1 Hence = 1, etc.; that is, if h be any cross-ratio, 1 — k is also a cross-ratio. The six members of a group of cross-ratios are, therefore, ''' l-yfc' ^ F 1 1-F-^ yfc' ' l-Jc Ex. Show that the permissible changes in the order of the elements determining any sjjecial cross-ratio are expressed by the formula : — Any two elements may be interchanged, provided that the other two be interchanged also. 38. An important special case is that of harmonic division, where AB la divided internally and externally in the same ratio; hence AP.PB and AQ:QB ai*e equal in value but opposite in sign. This gives {AB, PQ)= —I. The scheme just given for the cross-ratios now becomes -14 2 — 1 2 1 — 1, .^j 2' Hence (PQ, AB)== —1, which shows that if AB be har- monically divided by PQ, then PQ is harmonically divided by AB. The harmonic relation therefore involves the two pairs of points synnnetrically. 31). When one of the four points (e.q. Q) is at infinity, any cross-ratio reduces to a simple ratio. I'or -^jyu'^Y, becomes AP ^ pn : — 1, since AQ, QB are equal in magnitude (being infinite) 38 FIGUEES DETEKMINEJ) BY FOUR ELEMENTS. but opposite ill diroctioii. Tluis (AB, Px ) = AP : BP. If now tlie 2)ciirs AB, PQ be harmonic, tliis gives us (AB,Px)=-l: that is, AP:PB=-\-l, showing tliat AB is bisected at P. The rehition thus ex- hilnted between bisection and harmonic division lias im- portant consequences later, for it brings a certain class of metric properties within the legitimate application of homo- geneous coordinates. 40. It has been shown that any line through the inter- section of u = {), u = has an equation of the form w — kv = 0; and that the value of n at any point P is some multiple of the perpendicular from P to the line u — 0, that is, u = la, and similarly u = on^. Writing p for u — kv, Fig. 14 shows that since at P, u — kv = 0. Similarly taking another line q= n — k'v — O, sin uq sin qu sin up sin uq . by the two pairs Now the equa- Hence ' :" "^^' • "\-^^ ^ /. . ]/ sin pv sm qv ' that is, the cross-ratio of the pencil formet u = 0, v = 0: u — }iv = {), u — k'v = (): is k:k'. tions of any four concurrent lines can be thrown into this Umu: hence the cross-ratio is known. Since l:m does not appear in the result, it is immaterial what ^•alues the multi- pliers may have in the expressions u = la, v = m^. The pairs of lines will be harmonic if k:k'= —1, that is, if their equations can be reduced to u = 0,v = 0: u±kv = 0. Similarly if 9 = 0, = be any two points, H, K: it has been shown that 0-k(l, = is a point on the line Oc/>: call the ])oint P. Taking any line through P, draw the perpendicu- lars from H, K ; these are proportional to d, 0, that is, e = l. HM, (p = m. KK. (Fig. 1 5.) FIGURES DETERMINED BY FOUR ELEMENTS. 39 Hence the equation of P, viz., gives I. HM - km . KN = 0. But HM:KN = HP:KP, therefore HP : KP = km : /, , HP m-, whence plT^ ~ 7^- Fid. IJ. Similarly taking another point R, given by p — Q — k'(ji — 0, HR_ m,, RK~ r- „ HP HR , ,, Hence PK^RK^^''^' and the cross-ratio of the range formed by the two pairs = 0,0 = 0: - /t-0 = 0, - //0 = ; is k: k'. Now the e(iuations of any four collinear points can be thrown into this form, and thus their cross-ratio is known. Hence if u, v be linear functions, whether in point or line coordinates, the pairs u = 0, t? = ; ii-\-kv = 0, u- + k'v = ; give a configuration whose cross-ratio is k : k'. Now all four expressions are here of the same type, u + Av, X having the values 0, x, k, k'. The result may therefore be stated in the symbolical form {()-j^,kk') = k:k'; or, by means of the equalities among the twenty-four cross- ratios (see § 37), (kk', Ox) ) = /.; : k' (compare § 39). 41. The pairs of elements discussed in the last section were supposed given by their equations. Passing to the expression by means of coordinates, let V =f.f^ 4- g.{ij + h,z, (or f\^ + g,r] + h^^), 40 FIGURES DETERMINED BY FOUR ELEMENTS. then iL, V are respectively f\, g^, \ ; /o, ^2- ^'-2 '■ u + kv is f^-\-kf.,, K+^Al (A+^-y2> ^i+%2' h,+k'K) ; determine a conliguration of cross-ratio k : k'. And here, just as in § 40, the result may be stated sym- bolically : — The cross-ratio of the configuration (kk', Ox ) is k -.k'. One particular aspect of the one-dimensional geometry here considered deserves special mention. Instead of considering points in a plane, limited to a line of that j)lane, and assigned by three homogeneous coordinates, we may confine ourselves to the one line on which the points lie. Let A, B he two fixed points on this line : the position of a variable point F depends on the ratio of -4P to FB, i.e. if AP be called x, and FB y, on x:y ; and we have a system of two homogeneous coordinates for the geometry of points on a line. Now let the segment F-^P.j be divided in the ratio X : 1 by the point P ; then ^^^^ = A, and '^^^i = A, «2-*' 2/2-2/ therefoi'e x:y — x^-\- Xx.-, : y-^ -\- \y.^. Thus the conclusions of §§ 40, 41 apply to this case. Ex. Sliow that the cross-ratio of the coufigii ration {L-k\ W) is , ,, : J, — ^,, whether we are dealing with ec^uations or coordinates. The Complete Qitadr Hate rat and Quadrangle. 42. The correspondence between point and line configura- tions is gradually exhibiting itself as depending on a double interpretation of sets of quantities, and hence of eciuations. Coordinates may be interpreted as referring to points or to lines ; equations of tlie first degree then repre- sent lines or points ; the cross-ratio of four elements whose e( [nations are of a particular form can be determined without any knowledge as to whether the elements are lines or points. Thus by this dual interpretation a piece of algebraic work can be made to prove two theorems at f)nce. This may be illustrated by certain theorems that will now be given relating to configurations detei-miiu'd by four non-concurrent lines or non-collinear points. FIGURES DETERMINED BY FOUR ELEMENTS. 41 43. Four lines deteniiine a quadrilateral, sometimes calle<] a four-side ; the intersection ot* any two sides is a vertex, there are therefore six vertices ; as each vertex accounts for two sides, the six vertices fall into three pairs, the joins of these pairs are the three diagonals, forming the diagonal triangle. Let the sides be (1), (2), (3), (4); call the vertices (12), (13), (14), D, E, F, and the complementary vertices (34), (24), (23), G, H, K ; then the three diagonals a, b, c are DG, EH, FK. The entire configuration is spoken of as a complete quadrilateral ; it is interesting on account of its harmonic lyroperties. These are expressed by the theorem : — Any diagonal is harmonically divided by the other tuv. Note. The diaf^oual liere referred to as having a deterniiiiate length is the segment determined by two vertices. To prove this, take abc as triangle of reference, and choose coordinates so that the equation of the side (1) may be x + y + z = 0. Then (2), (3), (4) have equations of the form lx + y + z = 0, x-\- my -\-z = 0, x-\-y-\-nz = i) : but as (3), (4) are to meet on .^' = 0, we must have m)i = l, and similarly nl=l, Zm = l; consequently lmn= ±1. If we choose the upper sign, we find 1 = 1, '?h = 1, ';( = !, values which make (2), (3), (4) the same as (1); the proper solution is therefore Imn = — 1 , whence ^=—1, m= —1, n= —1. The sides are therefore (1) x + y + z = 0, (2) -x + y + z = (), (3) x-y + z = 0, (4) x + y-z = 0. Note. We have liere proved that by a proper choice of point coordi- nates the equations of four lines no three of which are concurrent can be thrown into the form ±x±y±z = Q ; and therefore also tliat by a proper choice of line cooixlinates, the coordinates of the four lines ciin be made to be ±1, ±1, ±1. The e(|uation of AD is at once found (Fig. 16) ; this is the line joining A to the intersection of (1), (2): it is there- fore obtained by the elimination of x from the equations x + y + z = {), -x + y + z = Q, that is, AD is y-\-z = 0\ and similarly AG is y — z = 0. These two lines are harmonic with ivgard to y = 0, c = 0; that is, DG is divided harmonically by BC. 42 FIGURES DETERMINED I'.Y FOUR ELEMENTS. 44. A complete quadrangle (four-point) is determined by four points (1), (2), (o), (4); these joined in pairs (12), (13), (14): (84), (24), (23) give six sides d, e, f; r/, h, k; these in pairs determine three diagonal points A, B, C, forming the diagonal triangle. In this diagram the line- })air (hj is harmonic with regard to be (Fig. 17). This is Fii;. IT. proved by exactly the work already used, the ./;, y, z now being interpreted as line coordinates ; and incidentally it is FIGUEES DETERMINED BY FOUE ELEMENTS. 43 proved that the c(|uatioii,s of" four points can be ina/±^=0, and that the coordinates of four points can be made to be ± 1 , ± 1 , ± 1 . To make this dual interpretation clear, the algebraic work is now written in an abbreviated form, with the two inter- pretations at the two sides. Symbols I., II., III. are used in the two cases for the vertices and the sides of the triangle of reference. Comparison of Figs. IG and 17 shows that the fact expressed by the two theorems is the same. (12) (34); (13) (24); (U) (23) determine the triangle of reference. DEF Let equation of (1) be .v+ y+ 3 = 0, DKH then (2) is lr+ y+ ,j = 0, GKE (3) is .v+mij+ i = 0, GHF (4) is x+ i/ + 7iz = 0. (3) and (4) are (3) and (4) are to give x = ; .". mn = l. to meet on the Similarly nl = l, Im — 1 ; line a. .'. {lmiif = 1 . The solution lmn= + 1 is inadmissible, .*. hnn— -I, and therefore l = m = n= - 1. Hence (1 ) is x + )j + 2 = 0, (2) is -x+'y+z = 0, (3) is x-y + z = 0, (4) is x+y-z = 0. Hence I. (12) is y + ,: = 0, L(34)is 'y-z^O, .'. (12), (34) are harmonic with regard to II., III. dvf dkh yke The join of (3), (4) is to pass through A. AD AG •. />, 6r are har- monic with re- gard to Z», C. ad ag .'. d, (J aie har- monic with re- gard to ^, c. On account of the harmonic properties here proNX'd, the diagonal triangle is called the harmonic tvianyle of the quadrilateral (jr quadrangle. 45. The theorem just proved affords a construction by the ruler only for the fourth point Q in a harmonic range, when A , B, and F are given : — Draw through A any two lines, cutting any line through B in (1), (2). Join F to either of these points, (1), and let this join meet the second line through A in (o) ; let 7^(3) meet the first line through A in (4); then (24) passes through Q. Other constructions may be derixed ; the essential thing in all is to constinict a quadrilateral with two vertices at A, B, and one diagonal through F: the other diagonal determines Q. 44 FIGURES DETERMINED BY FOUR ELEMENTS. 46. By means of the conception of cross-ratio we can insert accurately any line we please. To determine Li: + mj/ + nz = 0, it suffices to find the points in which this line meets two sides, e.g. .v=0, y = 0. It meets ,r = where mij + nz = Q ; thus we simply require to know how to draw any line 2 = Ay. Our choice of coordinates has virtually determined one line z = X^f ; for if the point be 1, 1, 1, then xU) is z=y. Let therefore AD (Fig. 18) be a known line s = Xit>/ ; it is required to insert z = X.y. DraAV frnni B a i)arallel to AL\ meeting the known line in D'. Then consider a pencil with vertex .1, and three known rays through B, C, D ; the transversal cuts these in B, oc , D'. Mark off on IH)' a length BF, determine.l bv BF : BD' = k : K- Then (Z/x , 1"1)') = BP' ■ BD' = k : Ao- If then AF meet BC in 1\ {BC, FD) = {Bo3, Pn) = X: Ao, whence it follows that AF is z = Xij. Any ])oint can be inserted by means of the lines joining it to any two vertices of the triangle. Pdirs ('/ Faints, llarinonicallij related. 47. The hcirmoiiic relation is essentially a relation between two pairs of elements, — for definiteness, between two pairs of points. It may also be regarded as a relation between two segments, each determined by one pair of points. J'oints on a line, represented by their distanctNs from a ti.xed origin on the line, can be represented by an equation, n()n-homogene(ms in one variable x; or again by an e(|na- tion, homogeneous in two variables x, y. A pair of points is thus given by a quadratic equation. Note. In saying that a group of n points is given by an equation of degree n, we imply that no distinction is made in tlie treatment of the points. If they are to be treated separately, they must be given by separate linear equations. FIGURES DETERMINED BY FOUR ELEMENTS. 45 Consider two pairs on a line, given by the two quad- ratics ax" + 26.r + c = 0, ax' + 2h'x + c' = 0. These will not be harmonic unless a certain conditi(jn be satisfied : this, expressed o-eometrically, is PB ' QB Let the points be at distances from equal to x^, x.,; x{, x'^. This condition becomes ^A/•\ tAJ\ <^2 2 "• l/^'2 t^'-i lA^2 t^/]^ that is, '2x^x.2 — {x^ + ^aX^^i + ^'O + 2iCi'a?.{ = 0, whence ac' + a'c — 266' = 0. If then the first pair AB be given, and we assume any point P as one of the second pair, the remaining point Q is determined linearly ; we have a series of pairs P, Q ; P', Q' ; etc.; all harmonic with regard to A, B. Ex. 1. Let .2,"- — 36 = give the first pair; and let .r — 2 = give P, .T — h = give (^. The harmonic condition just found shows that /i = 18. Ex. 2. Let the bisection of FQ be ,r = 4, the points ^1, B being given as in Ex. 1. Then the quadratic for PQ is a'.v^ + 2h':r + c' = 0, witli the cnmlitious ~=-4, c'-36rf' = 0, a hence the quadratic is ,r- — 8.r + 36 = 0, an equation witli imaginary roots. Thus a i)air of imaginary vahies is found to be liarmonic with regard to a given pair of real values. Imaginary Elements. 48. This confronts us with the question of imaginarj^ elements. Are they to be admitted ? If so, on what terms ? Imaginaries present themselves naturally in the solution of algebraic ecjuations, and are then recognized for the sake of continuity. If now we refuse to admit them into algebraic geometry, we shall have to examine tlie work at every step, to see whether it has a legitimate application in geometry ; our symbolical language will no longer ha\-e an exact relation to the subject matter. But there are even more cogent reasons why we should recognize imaginaries in algebraic geometry. One of the fundamental principles of the subject is that it takes two equations to represent a point ; one equation can represent 46 FKUTRES DETERMINED V,Y FOUR ELEMENTS. only a locus (i.e. in point coordinates). But if we confine ourselves to real points, we have to say that such an e<[uation as x'-\-y'^ = represents the orioin only, that is, one equation represents a point. The alternative is to admit imaginary elements, and say that this equation represents the two imaginary lines x-]-iy — 0, x — iy = 0, tliese having for their intersection the real point, the origin (Salmon's Conic Sections, §§ 78 and 82). We choose this alternative, and recognize imaginary elements, that is, elements whose coordinates have imaginar}" values, or whose equations have imaginary coefficients. Xnte. Tlie introihictioii of imaginary elements into Pure Geometry depends on diffei'ent considerations, and requii'es independent justifica- tion. 4!). Using trilinears for definiteness, let f+ if, g + ig, h+ilt be values of a, /5, y that satisfy the fundamental identical relation aa + bl3 + cy = 2A. The real and imaginary parts give (if + hg+ch =2A (1), <>f + J>g' + ch'^0 (2). Of these, (2) shows that if two of the three quantities /') f/, J^' vanish, the third does also; that is, when two' per- pendiculars are real, all three are real. Hence one may be real, and two imaginary: or all three may be imaginary. But the coordinates can always be written as if one were real, foi- we are at liberty to nndtiply the three by any ([uantity we please. K.r. a = 3, 6 = 4, c = .") ; whence 2d = 12. Tlie equations to be satisfied are Sf +4f/ +5/i =12, 3/' + 4,9' + 5// = 0. One set of values satisfying these is 4 - 4^■, 5 + 8?:, - 4 - 4?' ; multiplying by 1+?', the.se become S, -3 + 1 3/, - 8/ ; which may therefore be taken for the coordinates of tlie imaginai'v point. 50. 1. ThfoiigJi every imaginary point there passes one real line. Note. It is jtlain that tlicrc cannot l)e two, foi- the intersection of two real lines is a real point. FIGURES DETEEMINED BY FOUR ELEMENTS. 47 Let the point be f+if, fj + if/, h+ik': the line will pass throiio'h this it* If +ing -\-nh =0, lf + ong' + nh' = 0: that is, if I -.vi : n — gJt —g'h: Itf — h' f : fg -fg. Thus the line is determined uniquely. Ex. On every imaginaiy line there is one real point. II. A real line contains an indefinite nuinher of imaghiav]/ points, arranged in conjugate pairs. For f+if, g-\-ig', h + ih' lies on the real line ^a + m/5 + 7iy = it" /;/'+ mg + nh + i(lf' + mg' + nJi) = 0. Hence /', g, h, f, g', h' must be determined to satisfy If +mg +nh =0, c(f +bg +ch =2A: If' + mg' + nJi =0, «/ + %' +c/i' =0. But these conditions being satisfied,* not only will f+if, g + ig', h + ili lie on the line, but also /'—'?'/', g — ig', h — ili: thus the points are in conjugate pairs. That the number is indefinite appears from the fact that we have six quantities wherewith to satisfy four equations. III. Imaginaries that -present themselves through real algebraic equations are in conjugate pairs ; for f+ if being- a root of a real equation, f—if is also a root. A pair of imaginaries always means a conjugate pair, not simpl}^ any two ; the line joining a pair of imaginary points (or the intersection of a pair of imaginary lines) is necessarilj' real, its equation being aigh' - g'h) + li{hf - h'f) + y{fg' -fg) = 0. IV. Conjugate imaginary lines pass through conjugate imaginary points; for if f+if, 9 + '>[/, h+ih' satisfy u + iv = 0, then f—if, g — '^g', h — ih' must satisfy u — iv = {). 51. Just as four real points determine a quadrangle, four imaginary points may be taken as determining elements : in fact, the investigations in ^.^ 43, 44 apply whether the elements be real or imaginary. But it is of some inq")()rtance to know how the real and imaginary parts in the resulting conHgura- tion are arranged. Let the quadrangle be determined b}' two paii's of imagin- 48 FIGURES DETERMINED BY FOUR ELEMENTS. ary points, AA', BB' : the lines AA', BB' are real, and no other real line can go through these imaginary points. Con- sequently the four cross-joirrs AB, etc., are imaginary; but they are in conjugate pairs, viz., AB, A'B' form one pair, and give therefore a real intersection: similarly for AB', A'B. Thus tlie complete quadrangle determined by two pairs of imaginary points has its sides, one pair real, two pairs imaginary; and the diagonal triangle is real. In Fig. 19 real lines are represented by solid lines, conjugate imaginaries by the same kind of broken line ; the diagonal points are P, Q, R Similarly for the complete quadrilateral determined b}' two pairs of imaginary lines. Fig. 1 9 represents this also ; the given pairs are QB, QB' and BB, BB' : tlie real vertices are Q, R; the pairs of imaginar}^ vertices are A A', BB'. The three diagonal lines QB, AA', BB' are real. E.v. Discuss the configuration wlion tlic determining elements are one pair real, and one pair iniaginaiy. 52. It was proved in R^ 43, 44 tliat tlie coordinates of four elements can ])e made to be 1, ±1, ±1 by a suitable choice of coordinates, the diagonal ti'iangle being taken for triano-le of reference. If the elements be imaginary, this involves the use of imaginary multipliers I, in, n in the relations x = la, etc. This, while causing no difficulty in general, is inadmissible if we wisli to discriminate between real and imaginary iji the final results. We have seen that if the four elements be two imaginary pairs, the diagonal triangle is real, and can thei'efore l)e taken as triangle of reference. Let the given elements be points, for detiniteness, viz., the pairs AA', BB' in Fig. 19; then PQR is to be taken as triangle of reference. The lines A A', BB' are real, and they liave been shown to be harmonic with ivgard to Ph', PQ : if FIGURES DETERMINED BY FOUR ELEMENTS. 49 then coordinates y, z be chosen so that A A' is ?/ + 5; = 0, BB' is y — z = 0. Hence the coordinates of A hemgf,g,h, we have g-j-h = \ thus A is /, g, —g, where /, g are imaginary. Hence dividing throughout by g, A is \ + i/uL, 1, —1 : and A', being conjugate to A, is X — i/m., 1, —1. Now QA, QA' are conjugate imaginary lines, harmonic with regard to QR, QP ; their equations must therefore be of the forms .c ± /.•/s = ; comparing these with the actual forms x + i\ + iiuL)z = 0, x + (X-i/uL)z = 0, it appears that A = 0. Hence the points A, A' are ±i/u,l, — 1 ; and the numerical multiplier involved in the x being still undetermined, can be used so as to give to jul the value unity: the points A, A' are now ±i, 1, —1. The point B is the intersection of AR (i.e. x — iy — 0) and BB' (i.e. y — z = ()): hence i? is +i, 1, 1; and B' is — /, 1, 1. Thus without obliterating the distinction between real and imaginary, the coordinates of two pairs of imaginary elements can be made to be ± /, 1, ±1 : or, if preferred, i, ±1, ±1. Examples. 1. If {AB, PQ)=-l, then jp + ^^Zg' 2. Any two lines are harmonic with regard to the bisectors of their angles. 8. If two equal ranges have one point in common, the joins of the other pairs of points are concurrent. 4. If two equal pencils have one ray in connnon, the inter- sections of the other pairs of rays are collinear. 5. Let the sides BG, GA, AB of a triangle be divided in the ratios h:l, k:l, 1:1; then the points of division are collinear if JiJd= —1 ; and the lines joining the points of division to the vertices are concurrent if hld= +1- 6. The bisections of the diagonals of a complete quadri- lateral are collinear. 7. The six lines joining the vertices of the diagonal triangle to the vertices of the complete quadrilateral (determined liy (1), (2), (3), (4) ) are concurrent in threes. Calling the four points of concurrence a, /?, y, S, show that a, ^, y, S are the poles of (1), (2), (3), (4) with regard to the diagonal triangle. 8. Show by constructing the diagram that the diagonal triangle of the complete quadrilateral (1)(2)(3)(4) is the diagonal triangle of the complete (piadrangle ajiyS. 50 FIGURES DETERMINED BY FOTTR ELEMENTS. 9. State, for the complete ([uadraiigle, tlie theorem corre- spondino- to that of Ex. 7 : and show by means of the diao-ram in Ex. 8 that it has ah-eady been virtnally proved. 10. Sliow that the four points (lines) .7 - h, h -f, f- g: !/-h. ~ {h +/), /'+ g : g + h, h -f, -(f+g): - {g + //), h +/, /- g : are collinear (concurrent) : and that the cross-ratios of the confio-ni'ation are the six expressions •^^' etc ^^.,_^^„etc. 11. Find the equations of the lines joining /, g, h to the four points 1, ±1, ±1 ; and determine the cross-ratios of the pencil. 12. If the four points be I, ±?7^, ±7?,, find the lines joining them to /, g, It ; and the cross-ratios of the pencil. 1 3. Show that the pencil determined at /, g, li hy \, ± 1 , ± 1 , is equal to the pencil determined at p, q, r by d, ±h, ±c, and to the range determined on the line jj, q, v by the lines a, ±h, ±c, if the following relations hold ; a-\-b-\-c = {), af- + hg- + chr = 0, hciy'- + caq- + ahr~ = ; and that the six cross-ratios are b c a c a h ~? ~a' ~b ' ~6' ~c' ~a CHAPTER IV. THE PRINCIPLE OF DUALITY. Gorresjjondence Jiitherto noted. 53. In the preceding- sections attention has continually been drawn to the correspondence between the geometrical theories in whicli the element is (i.) the point: (ii.) the straight line. This correspondence is a manifestation of the Principle of Duality ; this appears more clearly if we speak of the primary element and the secondary element. In the one system these are to be interpreted as point and line ; in the other as line and point. The point The primary element Tlie line has three cooidinates :v, _?/, z. Two points Two primary elements Two lines determine a line a secondary element a ])oint. United with this line secondary element point we have an indefinite number of points, primary elements, lines, but the coordinates of all these satisfy one equation of the first degree, called the .• r . 1 I- ( etr nation of the secondai'\l ^- r ^, ■ ^ equation of the Ime....- ' , , - - equation of the ixiuit. ^ ( element. J • ■ The line The secondary element The point determined by :i\, ?/,, 4j ; .v.,, ^2, z^, has for its equation X y z =0. X\ Vx 2i The condition that three points three primary elements three lines lie on be united with pass througli a line one secondary element a point is (.fi3/.i3) = 0- 52 THE PRINCIPLE OF DUALITY. Any point Any piiniary clement Any line , ." , ,. 1 , • "x ii T 1 i fpassinof tliroutjli lyuii;- on the hne ...l)el(ingnig to the secondai'y element...- ^|^^ ,),,hit determined by .rj, //,, ^i ; j:,, y.,, Zo ; has coordinates .ri + A.r.2, .t/i + \>J->, z-^ + Xz-^; and if we take another such ])oint such in'imarv element sneh line the cross-i'atio of the range configuration pencil is A : A'. A point A primary element .A line lying on united with passing through a fixed line a fixed secondary element a fixed 2)oint has one degree of freedom ; it can move it can move it can rotate along the line in the secondary element about the point. Curves in the iivo Theories. 54. We now consider liow curves present themselves in these two theories. Tlie o-eneral idea of a curve is that it is a succession of points arranged according to some hiw, so that after every point there is one next point: ordinarily the point after A, B does not lie on the straight line AB, that is, ordinarily not more than two consecutive points lie on a straight line. This law is algebraically expressed by an equation F{x, y, z) = 0: and if F be of degree ii, any straight line in the plane meets the curve in n points, and the curve is said to be of order n. The line joining any point A to the consecutive point B is a tangent to the curve ; consecutive tangents AB, BG intersect in a point B on the curve. Stating this in general terms, there is a succession of primary elements arranged according to some law, so that after every primary element there is one next element : ordinarily not more than two consecutive primary elements belong to one secondary element. This law is algebraically expressed by an c(iuation in the coordinates of the primary element; this e( [nation being of degree oi, a secondary element has in connnon with the system li, primar}' elements. The derived secondary element determined by two consecutive primary elements is closely related to the system ; two consecutive derived secondary elements deter- mine a primary element of the system. Translating this into terms of lines and points, there is a succession of lines arranged according to some law, so that after every line there is one next line ; ordinarily not more than two consecutive lines pass through a point. This law is algebraically expressed by an ('(|uation THE PRINCIPLE OF DUALITY. 53 ^d' '?' D — ^'i ^^^^^ i^" ^ ^6 o^ degree n, through any point in the plane there pass n lines of" the system. The point determined by two consecutive lines of* the system is closely related to the system ; two consecutive points determine a line of the system. It appears therefore that the lines of the system are regarded as tangents to some curve, their envelope, the points of this curve being determined as the intersections of consecutive tangents. The number of tangents from a point being n, the envelope is of class n. Thus the locus of order n, and the envelope of class n, are corresponding conceptions. For instance, if n be unity, the locus of order n is a straight line, and the envelope of class oi is a point : the conception of the straight line in the point theory (i.e. of the straight line qua locus of points) corre- sponds to the conception of the point in the line theory (-}.e. of the point qua envelope of lines). 55. Since the system of primary elements affords a derived system of secondary elements, from which again the primary elements may be derived, it follows that a curve may be considered under either aspect. It has points {ineunts, Cayley) from which the tangents may be derived, viz., by joining every point to the next ; it has lines {tangents) from which the points may be derived, viz., by marking the intersection of every line with the next. The point system or the line system may be regarded as the primary system ; the other is then the derived secondary system ; thus the curve may be regarded as a locus or as an envelope. Every curve has therefore two diiferent equations, one in point coordinates, F{x, y, z) = 0, and one in line coordinates, ^{^, ?/, 0~^^ ^^ these be respectively of degrees m and n, the curve is of order 'in and class n ; according to the nature of the proposed investigation, one equation or the other will be the more suitable. " The equation of a curve in point-coordinates, or as it may be termed the point-equation of the curve, is the rela- tion which exists between the point-coordinates of any ineunt of the curve. The e(|uation of a curve in line-coordinates, or line-e(|uation of the curve, is the relation which exists between tlu' line- coordinates of any tangent of the curve." (Cayley, Collected Papers, vol. ii., No. 158 ; 1859.) For many purposes it is more convenient to consider the curve as on the one hand traced by a moving point, on the other enveloped by a moving line : the tracing point must then be regarded as moving along the tangent, the enveloping 54 THE PR1N(;IPLE OF DUALITY. line as rotating about the point of contact. Tliis dual con- ception of the nature of a curve was first formulated by Pliicker {Theorie der Algebraischen Gurven, p. 200 : 1889): — " If a point move continuously along a straight line, "while the straight line rotates continuously about the point, one and the same curve is enveloped by the line and di'scribed by the point." 56. Plainly the case n = 1 is an exception to the statement just made regarding the dual aspect of a curve ; a point, which is an envelope of class 1, cannot be regarded as fur- nishing a system of points, that is, it cannot be regarded as a locus : it has not a point ecpiation : a line, Avhich is a locus of order 1 , cannot be regarded as furnishing a system of lines ; that is, it cannot be regarded as an envelope : it has not a line equation. 57. Now an algebraic expression may be a product of fac- tors : that is, the equation may split up into equations of lower degree : the curve considered under either aspect may be degenerate. A degenerate locus is composed of loci of lower order; thus a degenerate locus of the second order can only be two loci of the first order, and therefore a pair of straight lines. A degenerate envelope is composed of en- velopes of lower class : thus a degenerate envelope of the second class can only be two envelopes of the first class, and therefore a pair of points. Dual Interpretation of Ahjehraic Work. 58. The analytical discussion of any geometrical theorem consists of four parts : — I. The statement of the geometrical data : there are certain elements, whose positions are controlled by given conditions : II. The algebraic expression of these conditions, by wliich certain ecpiations are o])tained : III. Algebraic com])inations and transfc^rmations applied to these equations : IV. Geometrical interpretation of the results of these alge- braic operations. In tile purely algebraic parts of the discussion, II. and 111., no attention is paid to the significance of the symbols; these may l)e point coordinates, or they niay be line coordin- at(3S. But the whole pi'oof of the theorem is in III. Hence in proving any theorem regarding })oints and lines we do at the same time prove another theorem regarding lines and THE PRINCIPLE OF DUALITY. 55 points ; there is no question of deducing the one theorem i'roni the other ; the two are proved simultaneously. It is in this dual iiiterpretatiun of the algebraic tvork that the Principle of Duality is algebraically exhibited. " This reciprocity " (between point and line in a plane) " can be formulated in the following manner ; starting from the point, we obtain the straight line by joining two points ; starting from the straight line, we obtain the point by the intersection of two lines. The simplest geometrical figure for the point is the straight line which it describes in mov- ing along it ; for the straight line it is the point which it envelopes in rotating about it.... The aggregate of these relations is expressed by the Principle of Duality ; this principle asserts that certain theorems holding for point configurations can be transferred to line configurations ; this reciprocity applies to the join of two points and the intersection of two lines, and extends to all constructions resulting from operations of this nature. It does not hold when other auxiliary means have to be employed, e.g. when a metric determination enters into the problem. But the principle is not thereby limited, for we shall find that all such metric relations can be expressed in terms of the others." (Clebsch, Vorlesungen ilber Geometrie, t. i., p. 28 ; ed. Lindemann, 1876.) Examples. Point or line coordinates are to be used (or both together) according to the nature of the jproblem ; and all residts Quust be homogeneous. 1. Find the envelope of a line moving so that the })roduct of its distances from two fixed points is constant. Verify from a priori considerations that this envelope is of the second class. 2. Find the locus of a point moving so that tlie product of its distances from two fixed lines is constant. 3. Find the locus of a point moving so that the sum of its distances from n fixed lines is constant. 4. Find the envelope of a line moving so that the sum of its distances from n fixed points is constant. 5. Find the envelope of a line whose distance from a ti.xed point is constant. (i. Lines BP, GP are drawn to meet the sides of the triangle of reference in B', C" ; if P move along a line thn)Ugh A, show that B'C passes through a fixed point on BC. 56 THE PRINCIPLE OP DUALITY. 7. In Ex. G, if P move along a tixed line lx-\-my-\-'nz = 0, tin«l the line e(|nation of the envelope of B'C. S. If B'C pass through a fixed point fi-\-]-\-hi'^0, find the point ecjuation of the locus of P. 9. A point moves along a fixed line : show that the en- velope of its polar with regard to a triangle is of the second class.. 10. A line passes through a fixed point : show that the locus of its pole with regard to a triangle is of the second order. CHAPTER V. DESCRIPTIVE PROPERTIES OF CURVES. General Prmciples. 59. When homogeneous coordinates are employed, the investigation may proceed without any reference to the fundamental identical relations, the properties discussed being purely descriptive ; on the other hand the properties being metric, the fundamental relations have to be taken into account. The general discussion of curves falls there- fore naturally into two principal divisions ; the second being subdivided according as the fundamental identical relation involved is that in point coordinates or that in line coordi- nates. 60. Eliminating z between an equation of degree n, F{x, y, z) = 0, and a linear equation fx+gy + hz = 0, the result is a homogeneous equation of degree n in x, y. It represents therefore 7i straight lines through the vertex G; from the way in which the equation was obtained, these lines pass through the intersections of the curve F=0 and the line. Hence the number of these intersections is 'ii, and the curve F=0 is of order n. Similarly the elimination of ^ between the line ecjuation of a curve = is of class 7^. JVote. The locus /Mias also a line equation whose degree, at this stage unknown, will be considered later ; and the envelope <1» has a point equation, to be similarly determined (§ 68). 61. A principle that is of wide application is the follow- 58 DESCiMlTIVE rROPEliTIES OF CURVES. iiiu-: — If /'=0, f' = be any two equation.s ot" the same (len-ree, then /+^:/' = i« satisfied by all the common elements oi" /=0, ,/" = (): tliat is, if /=0, /' = 0, be the point e(|uations of two curves, then /+/"/' = passes through all their common points: and if = 0, 0' = O, be the line equa- tions of two curves, then 0-|-/i;0' = O touches all their counnon tangents. The curves /'-|-i/' = 0, through the connnon points of /"=0, /' = 0, form a 'pencil: the curves f/) + /i'0' = O, touching the connnon tangents of = 0, 0' = O, may conveniently be spoken of as a range. The curves forming either of these configurations, depending on one parameter h, form a singly infinite system ; or, using the phraseology of § 4, the curve element of the conhgura- tion has one degree of freedom : the one parameter may be regarded as the one independent coordinate of the curve. If now /, /', split up, this principle takes the foi'm : — The curve ici^-' + r?/ = (in point coordinates) passes through all points common to uu and vv ; i.e. through all intersec- tions of the pairs u, v ; u, v' ■. u, v ; iv, v' ; and in line coordinates, uii -\-vv' — ^ touches all the common tangents of the pairs u, v ; ii, v ; \i , v : u', v. As a special case, let ii, V, IV, ^y' be linear functions : then uvs -j- wiv't = passes through (among others) the points uw, uw'. If now w approach iv indefinitely, the equation becomes uvs-\-uH = ; the points uiu, utv' (W . W in Fig. 20) approach one another indefinitely, and u becomes a taiigent, whose point of contact is liw. Similarly the curve touches the line v at v'W ; the two lines u, v are tangents, iuid w is the chord of contact. The interpretation of the equation iivs-\-uiH = in line coordinates is that it represents a curve passing through the ])oints u, V, the tangents at these points intersecting at the point w (Fig. 20). The argument just used does not re- (|uire U; v, tu to be linear: the curve nn-\rtv^t = 0, in point coordinates, touclies tlie curve u = at every point common to the curves u = 0, %u = 0: and in line coordinates, this cuts the curve tb = on every tangent common to a = 0, 10 = 0. Again, no assumption having been made as t(^ the nature of the coordinates used, they may be homogeneous or Cartesian. Kid. 20. DESCRIPTIVE PROPERTIES OF CURVES. )9 Ex. 1. x'^-4v/ = 36, that is, (.r - 'iy){x + 2y) = d'. This touches ,i-2_y = 0, .i; + 2y/ = 0, on the line 6' = 0, i.e. at infinity. It is a hyperbola with x-2ij = Q, .*' + 2// = 0, for asymptotes. Ex. 2. y/'- = 4.c; that is, x.c=i/-. This touches u-' = 0, c = 0, on tiie line .y = 0. Ex 3. x>/ = (x- + 7/- - u^y\ This touches .'=0 wliere it meets .i'^+y2-«'- = 0, i.e. at the two puints ^• = 0, i/=u ; .*' = 0, v/= -« ; and again it touches // = at ,'; = '^, // = U ; x= —a, i/ = 0. These examples relate to Cartesian coordinates ; examples in homo- geneous coordinates, point and line, occur in the following pages. Eqiudions of the Second Degree, satisfy iny Three assigned, (JondAtions. G2. We .sliull now consider e(|uations of the second deo-ree more especially, tlioiio-h not exclusively, since occasionally equations of hig-her degree attbrd better illustrations of the methods. The general equation of the second degree in point co- ordinates is ax- + by- -f cz- + '2fyz + 2gzx + 2hxy = 0. This locus will pass through the vertex A of the triangle of reference if the coordinates of yl, 1, 0, 0, satisfy the equation, that is, if (t = 0. Similarly it will pass through B, (J if b — i), c = 0. Hence the locus of the second order through the vertices of the triangle of reference is fyz+gzx + hxy^O; and the envelope of the second class touching tlie sides of the triangle of reference is The tangent at A can be determined by a direct method. Any line through A is y = kz. The lines joining Ji to the intersections of this line and the locus fyz + gzx-^hxy = are obtained by eliminating y : they are therefore fkz--\-{g + hk)xz = 0: that is, z — 0, which represents BA, and fkz + {g + hk)x = (), '""■''■ which represents 5J.' (Fig. 21). For y = kz to be a tangent, the points A, A' must coincide, i.e. BA' must be the same 60 DESCRIPTIVE PEOPERTIES OF CURVES. as BA : hence (j-\-lil' = 0. The tano:ent at A is therefore Similarly the equation of the point of contact of BC with the envelope is /"/+^^=0. Ex. 1. If three tangents be drawn to a locus of the second order, they meet the chords of contact in collinear points. Ex. 2. If three points be taken on an envelope of the second class, the lines joining respectively one of these to the intersection of tangents at the other two are concurrent. In the argument of this section x, y, z are simply any three lines ; thus if f, Y, TT be a triangle other than the triangle of reference, having for sides the lines it = 0, f = 0, iy = 0, any locus of the second order through JJ , V, W, is fviu + g wu + h. u V = 0. And even if u, v, w be functions not linear, fvw + giuu + liu v = () represents a locus through the intersections of the pairs v, iv ; u\ 'a; u, v; e.g. if n, v, iv be of the second degree, this is a locus of the fourth order. Ex. Find the general equation of a locus of the third Lirder through A, B, C. Here two of the three values given by .t^^O are to be .'/ = 0, z = ; therefore the equation is of the form xic + )/zf = ; where the whole expression being, by requirement, of degree 3, n is of the second degree and v of the first. Similarly v/ = is to give x = or i = ; therefore u is of the form sy + tz, where s and t are linear. The three assigned conditions have now been satisfied, and the cubic locus has for its equation yzv + zxt + .vys = 0, where r, t, and s are any linear functions. This result can also be obtained by taking the general cubic eijuation, «.r3 + . . . + 3a'.>;2i; + . . . + Sa".vz^ +... + Hdxyz = 0, and expressing that this is .satisfied by the sets of values 1, 0, U ; U, 1, ; 0, 0, 1 ; it a])i)ears at once that « = 0, 6 = 0, c = 0. 08. In the last section tlie configuration of the second degree was considered as having three assigned primary ele- ments. It will now be considered as having three assigned secondary elements. The conditions that the locus of tlie second older touch the sides of the triangle of reference are most simply determined from the consideration: — The line x = touches the locus if DESCRIPTIVE PROPERTIES OF CURVES. 61 the result of making x' = in the equation be a perfect H(|uare ; for this result gives the equation of the lines joining A to the two intersections. Hence the locus touches BG if be a perfect square, that is, if /- = 6c; and it touches the other sides of the triangle of reference if ^i + c)vi : these beino- (1) point co- ordinates, (2) line coordinates. IF the ('(juation be Ix + 7717/ + iiz = (1), I, m, n nmst satisfy h\ + my^ + i}z,=0 (2), Kx^ + (^.^'i ) + ^" C'/i + ^IJx^ + ''K n + -^^i) = <>> from wliich, l)y snbtraction, Ux^ + m8y^ + n8z^ = (3). From (2) and (8), l:m: n = yySz^ — z^Sy^ : z-^8x^ — x^Sz-^^ : x^Sy^ — y^Sx^. If now F=i) be the homogeneous equation of the curve, and F^ — , - , — be written for the values of F, ^:—, etc., ^ dx^ dy^ dz^ dx at the point ,r^, y■^, z^, we have, by Euler's theorem, dF , c>F ^ dF. „, ,. and since x.^ + Sx^, Vi + ^Vv 2^1 + ^^i i^ "n the curve, dF 'dF dF therefore dF -dF dF . „ . . . ,, ' , dF dF dF H encc I : m •.n = ..—-: - — : ;r— , dx^ oi/j dz^ and equation (1) becomes dF^ dF ^ dF ^. dx^ dy^ dz^ which is therefore the c'(|uatioii of the tann'cMit to F=() at the \>o\ut x„ y^, Zy Similarly the ('(|uati()n of the point of contact with = () of the line ^",, i]^, ^'j, is 05. Thus when the point c(iuation of a curve, F=(), is known, the ecpiation of the tangent at any point is known; the coordinates of the tangent are thei'el'ore known ; and hence, theoretically, the relation to which these coordinates DESCRIPTIVE PROPERTIES OF CURVES. 63 are subject can be deduced ; that is, the hue equation of the curve can be found. This line equation is often called the tangential equation of the curve. As an exanq^le of the process here outlined, consider tlie special locus of the second order ax' + hlf + ('Z- = (K The tangent at x^, y^, z^ is ax^ + hy^ij + ('z^z = 0, hence the coordinates of the tangent are i=(^ix^, i] = hy^, ^~cz^. We have to find the relation satisfied b}' ^, »;, ^. Writing these equations in the form _^ _n _t ^^~iC ^i~6' ^i""c' and making use of the relation expressing that x^, y^, z^ is on the given locus, viz., ax^-{-hi/'^-\-cz^ — 0, the equation in ^, J], ^ is found to be ^- + 1 + ^ =0. a b c In applying this process to the general curve F=^, it is necessary to eliminate x^, nj^, z^ from three equations of degree , . dF . dF ' dF . , ^, ,. „ ,, -7? — 1, VIZ., --=^, — =,^, ^;~^S' ^^^^ ^"^^ equation i'\ = (). ox^ oy^ oz-^ Now since dF ^ dF , dF „ ^3*^ dy^ '■dz-^ ^ the equation F^ = may be replaced by Even with this simplification the difficult}^ of the elimination in the general case is practicalh^ prohibitive. But if n = 2, the equations are all linear, and the elimination can be at once performed. In this case F = ax^ + by'^ + cz- + 2fyz + 2/^ + cz^ + -Ifi/z + 2r/:.c + '2h:ry = is Ibund to be + ^, = 0, the point equation is Aa-^ + Bif + C'^- + 2 Fy^ + 2 Gzx + 2H.ry = 0. 06. This result may be obtained in a slightly dift'crent form. Instead of solving equations (1) for x-^, y^, z^ and substituting in (2), we may simply eliminate x^, y^, z^ from equations (1) and (2). Till' result is a h (J ^ ^ 0, h h f , fJ f c ^'• i ^ '; ^^ the saniL' »'(]uation as before, Imt now expressed as a deter- minant. This (Mjuation, in either of its forms, can be interpreted as a condition. Regard ^, ;;, ^ as a known line, I, m, n: the con- dition that the Hne lx + my-\- az — O touch the locus a.r2 + hy'^ + cz^ -f 2/}/; + 2(/zx -\- 2hxy = IS (I Ji (J I h h f m '- - ac)(c- — hd). In the present example he - ad, &2 _ ac, (? - U = ^{^ + f - C% $rC, e-'lC' •> consequently the line ^.r + '>pj + C^ = is a tangent if Plainly ^=0 does not make the line a tangent ; the factor {'', introduced in forming the equation, is irrelevant. The line equation of X^ + ?/ + z^' = is therefore that is, e + v' +c'- '^^iX' - K'e - ^ev' = 0. This order-cubic is therefore a class-sextic ; and similarly the class-cubic f + r + r = o, is the order-sextic x6 -f 7/6 + -6 _ 2f^ - 2,^%3 _ 2xY = 0. If then the order m be 3, the class n may be as much as 6 or as little as 3. Thus the class does not dei)end solely on the order, and the order does not depend solely on the class. G9. If F,X^, y, z) = be the point equation of a curve, the above process gives the line equation ^n(i, »;, D = 0- Now corresponding to the locus Fm(x, y, z) = {) there is by the principle of duality an envelope F^H, '/, ^) = ; and the point equation of this, found by the same algebraic work as before, is «J>,i(*') ?/> z) = ^). Instead of discussing the two equations Fj^x, y, 0) = O, $,,(^, ,,, = for the one curve, it is often convenient to discuss the two distinct curves F„J.x, y, z) = (), ^n{'x, y, z) = {), which are called veciprocal curves. I'hus instead of attend- ing simultaneously to the points and lines of the one curve, we consider the points of the two curves, knowing that all the line properties of F,„(x, y, z) = are exactly represented DESCRIPTIVE PROPERTIES OF CURVES. 67 by the point properties of ^n(x, y, ^) = 0. These two reciprocal curves will now be spoken of as F and $. Carrying the general comparison of Chapter IV. a step further, these two curves, F, ^, are corresponding curves. To a double point on F, with tangents distinct or coincident, corresponds on $ a double line, with points of contact dis- tinct or coincident. For in each case the primary element presents itself twice in the same position, and we dis- criminate by means of the derived secondary elements. Now coincident tangents at a double point may be due to a cusp, which involves the turning back of the tracing point along the tangent ; or to a tacnode, where there is contact of two distinct branches. In the latter case, the two branches have the same point and the same tangent ; in the reciprocal there are therefore two branches with the same tangent and the same point of contact ; that is, there is a tacnode on the reciprocal. At a cusp, the tracing point turns back along the tangent : that is, the primary element reverses the sense of its motion in the derived secondary element, and therefore in the reciprocal tlie enveloping line changes the direction of its rotation about the point of contact (Fig. 22) ; it is an inflexional tangent ; the point of contact, viz., the inflexion, corresponds to the cuspidal tangent. Thus the cusp and the inflexional tangent are properly stationary elements, while the node and the double tangent are simply double elements. Ex. The line equation of F=x^ + kyh = is f +p/'Y--=0 (Ex. 2, 5^68). Hence the reciprocal curve is The curve F passes through B, C ; to see how it lies at C, let this be made the origin of Cartesian coordinates. Wi'iting a.)' + bi/ + c for c (§ 32) the equation becomes x^ + k7/-{(u: + h>/ + c) = 0, exhibiting a cusp at C (i.e. at 0, 0, 1), the cuspidal tangent being y = (i.e. the line 0, 1, 0). On the reciprocal $ we are therefore to consider the line 0, 0, 1, and 68 DESCRIPTIVE PROPERTIES OF CURVES. the point 0, 1, ; that is, the line z=0 and the point B. Making B the origin, the equation of the reciprocal 4' in Cartesian coordinates .r, z is x^+pz{l.r + 7nz + ny- = 0, a form which shows that there is an inflexion at B, the line z = being the inflexional tangent. Thus to the cusp and cuspidal tangent on F there correspond on the inflexional tangent and the inflexion. Poles and Polars. 70. From any point x', y', z , n tangents can be drawn to a curve of class n. If the line equation of the curve be known, the coordinates of these n tangents can be determined (§ 60). If the point equation of the curve be given, the question is how to determine the equations of these n tangents. Let a\, 2/i, z^ be the point of contact of any one ; then the equation dF dF dF of the tangent is a'—;- + 2/^ — h z~ = 0. The condition that this is to pass through x', y', z shows that x^, y^, z^ nuist satisfy the equation of degree m — 1, 9«j "^ ?>y^ 'dz-^ and as .r^ y-^, z^ is b}^ hypothesis a point on the given curve, it is an intersection of the two curves F=0 (1), ,'dF ^ ,dF ^ ,'dF ^^ ^^+2/^-+5^^r = ^^ (-)• ox oy ?)z ^ ^ Tliis deri^•ed curve (2) is called the tirst polar of x, y', z with regard to the given curve F. But if there be a point at which ^— = 0, ^^ =^-~=l\)^ at ox ^y ?)Z tliis point equation (2) is satisfied, and (1) is also satisfied, for 3i'\ ^F ^ -dF dx ^dy dz and therefore F—0. But the ecpiation for the tangent is now illusor}' ; and therefore in determining the points of contact of tangents from x,y',z' by means of tlie intersections of (1) and (2), these exceptional points must be excluded. The result must therefore be stated : — The intersections of a curve and the Jirst polar of a point ivith regard to the curve give (1) the ^points of contact of all tangents from the point; (2) all p)Oints, if such there he, at which ' ^' » "^ vanish simultaneously. DESCRIPTIVE PROPERTIES OF CURVES. 69 JVote. Two equations of degrees m, p, homogeneous in three variables, have mp sets of roots ; this is jjroved in works on algebi'a (see Sahuon's Higher A Igehra, § 73) ; hence two curves of onlers m, p have mp common points ; and since the class of a curve of order m cannot exceed the num- ber of intersections of two curves of orders ?», »?-l, n'l(- in{m — \) ; and reciprocally, m 1c n{n— 1). The expression for the number of intersections of two curves will how- ever not be assumed, it is simply noted here for purposes of illustration ; when the number of intersections of two curves is required in an ai'gu- ment, an independent proof is given. 71. For the special case ot* tlie conic, the first polar — or more simply, the polar — is a straight line. This meets the conic in two points ; before it can be asserted that these are points of contact of tangents from x', y', z', the possibility of the simultaneous vanishing- of tt-, t:—, ^r- must be examined. ^ ?)x ?)y dz The question is, can the three equations ax + Jty+gz = 0, hx-\-hy-\-fz^Q, gx + fy + cz = Q, be simultaneously satisfied? Not rniless A = 0, where A is the determinant a It g h h f g f c This is the possibility that presented itself in ^ Go, and the result of that section agrees with the result now obtained, viz., the curve of the second order is also of the second class, unless A = 0. The significance of this condition appears from the fact that the polar of any point with regard to a line-pair passes through the intersection of the lines, this general con- currence of the polars being moreover a sufiicient condition for the degeneration of the locus of the second order. For the polar of x , y',z' \v\i\\ regard to yz — is yz'-\-zy' = 0, which passes through the point yz ; and if all polars pass througli a point, take this as yz; then the values 1, 0, for x, y, z make the linear expressions — , — , — vanish ; consequently no one of these contains x, and so the expression F does not contain X, but is homogeneous of the second degree in y, z, and is there- fore the product of linear factors. Hence the condition A = (), which primarily expresses the concurrence of all polars, is the condition that the linear expression ax^ + hy' + cz' + ^'yz + ^yzx -f -Ihxy split up into factors : it is the condition that the locus of the 70 DESCRIPl^IVE PROPERTIES OF CURVES. second order degenerate to a line-pair, or that tlie envelope of the second class degenerate to a point-pair. Since a line, or a pair of lines, cannot be regarded as an envelope, and reciprocally the point, or the pair of points, cannot be regarded as a locus, it is right and to be expected that the ordinary process for finding the reciprocal equation should, in this case, prove illusory. 72. Dealing with a proper conic, there are two tangents from any point. The construction for the polar of a given point (pole) is therefore : — Draw the two tangents, mark their points of contact, the join of these is the polar. The reciprocal construction starts from a line and leads to a point : — Mark the two intersections of the line with the conic, draw the tangents at these, their intersection gives the point. Comparing these, the line and point in the second construction are seen to be polar and pole : the two diagrams are the same ; each is its own reciprocal, for pole and polar are reciprocal. Since the tangents or intersections may be conjugate imaginaries, these constructions are not generally available : otliers will be given later. (See § 78). 73. The polar of j\, y^, s^ is x{ax^ + hy^ + gz^) + y{]tx^ + hy^ +/; j ) + c{>jj:^ +fy^ + cz^) = ; this is synunetrical in the two sets x^, y^, z-^ ; x, y, z. Hence if the iiolar of P ^>a6vs' throiujh Q, the iiolar of Q jnisses throwjh P ; and similarly if the pole of p lie on q, the pole of q lies on p. Pairs of elements related in this way are said to be conjugate. (See ,^ 78.) Hence a point has an indefinite number of conjugates, all lying on the polar; and a line has an indefinite number of conjugates, all passing through the pole. The pole and polar are also spoken of as conjugate : taking any figure composed of points and lines, tlie polars and poles of these with refer- ence to any fundamental conic form the conjugate figure ; tlius a triangle gives a conjugate triangle ; a complete (|uadrilateral gives as conjugate a complete quadrangle. If the fundamental conic be the imaginary conic i«^+2/' + ^" = 0, the point /', ' and F' are identically the same, consequently the equation of the pair of tangents is FF'-p'' = 0. Similarly if = () be the line equation of a conic, and trr = the equation of the pole of the line p (^', >/, ^'), any conic having double contact with $, ^ being the pole of the chord of contact, is and the equation of the intersections of ^', >/, ^' and «I> is #$'-^2 = 0. Examples. 1. Show that the lines l^, ii\, n^ ; l^, m^, n., are conjugate with regard to ax^ -\- by- -\- cz' + '2fyz -f 2gzx + 2hxy = 0, if (6c -p)liL 4- {ca-(f)m{m.^ + {ah - Jt')n^v^ + (gh-af){m{)} ., + mwn j) + (¥- klX'^hh + "2^1) + ify ~ ohKkm,, + If 11^) = 6. 72 DESCRIPTIVE PROPERTIES OF CURVES. 2. Find the coordinates of the point of intersection of a line-pair <;iven by a point equation of the second degree. Find also the equation of this point. 3. State the results reciprocal to those found in Ex. 2. 4. Apply the condition A = to show that ^2 _^ ,^2 ^ ^2 _ 2,^^ cos A -2^^ cos B - 2^,; cos 6' = represents a pair of points ; and show that the equation of the line joining these points is ax + hy-\-cz = 0. 5. Let u, V, v be three lines ; denote the points vv , uv, uv by p, ~, (^'. Show that the line equation of ti^ = Jew' is of the form p' = XuT~'. 6. Find the reciprocal to x"'y' = s'"+". 7. If the conic reduce to a line-pair, show that any point in the plane is conjugate to the intersection of the lines. 8. With respect to a line-pair every point has a polar, but not every line has a pole. 9. Discuss the results for a point-pair analogous to those given in Ex. 7 and Ex. 8 for a line-pair. Conies with Four ass^lgned Elements. 75. The equation of a conic through three points, or touch- ing three lines, left two constants to be determined, viz. (§ 62), f:g:h. The equation of a conic through four points, or touching four lines, contains therefore one undetermined quantity. Let the four points be the intersections of lines a = 0, ^ = 0, y = (), (5 = 0, these lines being taken in the order given, so that the four points are P{a/3), Q(/3y), HiyS), S(8a). . One conic through FQRS is ay = 0; another is /3^ = 0. The conic ay = kf3S passes through all the intersections of ay, /3o, that is, through P, Q, R, S ; and since this equation contains one undetermined quantity, it is the most general equation of a conic tlu'ough the four given points. Similarly, if a, ^, y, S be linear in ^, r], ^, so that a = (), etc. represent points, the line equation of the conic touching a/3, /3y, y^, Sa, is ay = k^S. 76. The coordinates of the four elements were here supposed given by means of etpiations ; but tlie elements themselves being given, and the choice of coordinates left unrestricted, the equation of the conic can be found in a useful form. DESCRIPTIVE PROPERTIES OF CURVES. 73 The four given elements detennine one of the configura- tions considered in §§ 43, 44 ; taking the diagonal triangle for triangle of reference, and choosing coordinates properly, the four elements have coordinates 1, ±1, ±1. The general equation of the second degree is satisfied by 1, 1, 1 if « + /j + c+2/+2ry + 2/i = (1), -1, 1, 1 if a + h + c+-lf--l(j--lk = i) (2), 1, -1, 1 if ii + h + c-y+'2g-^h = (3), 1, 1, -1 if (6 + ?-> + c-2/-2r/ + 2A=0 (4). Now (1) and (2) give r/ + /i = 0, (3) and (4) give + c = () ; and the line equation of the conic touching the four lines 1, ±1, ±1 is with the condition a-\-h-\-c — Q. Note. If the elements be taken as p, ±q, +r, the equations obtained are of the same form, with the condition ap^ + bq'-^ + cr- = 0. Now the point equation of a^'^-\-br]'^-\-c^'^ = is bcx'^ + cay^ + abz^ = : and the lines whose coordinates are 1, ±1, ±1 have ecjuations x±y±z = 0; consequently the point equation of the conic touching the four lines x±y ±z = is bcx^ + cciy'^ + abz^ = 0, with the condition a + b + c = 0; or it may be written Ax^ + By'--{-Gz^ = 0, with the condition . + ,,+ ^ = 0. ABC Ex. Two conies being drawn through four ])oints, ])rove that their eight tangents at these points all touch one conic. Choosing coordinates as above, the conies are 2M:- + qf + rz^ = (1), p'x- + q'f + r'z^^O (2), with the conditions p + q + r = 0, j)' + q' + r' = (.3). The tangent to (1) at 1, 1, 1 is p,t: + q7/ + rz = ; and similar equations are found for the other tangents. Hence the eight lines to be considered have coordinates p, ±q, ±r; p>\ ±q', ±r' ; 74 DESCRIPTIVE PROPERTIES OF CURVES. and it is to be sliowii that these satisfy a quadratic relation f/f- + Inf + (f- + 2/7?^+ 2(iC^ + 2hiy = 0. The first set tjf four will satisfy this if /=0, g = 0, h = (4), ap- + bq~ + cr^--=0 (5); the second set will satisfy this if in addition up'- + bq"^ + cr'"- = (6). Hence «, 6, c must be determined from the linear equations (5), (6) ; these give (/, i, c = qV^ - q'h\ r-p- - r"-p\ p'q"^ - p'-(f ; but the conditions (3) give (//•' - qr = rp' - r'p =pq' -p'q \ hence f^ 6, c = qr' + qr, rp' + r'p, pq' +p'q, and the line equation of the conic touched by all eight lines is iqr' + q'r)^^ + {rp' + r'p)->f + {pq'+p'q)C^ = 0. E.r. 1. A conic can be draw^n to touch the six lines that join the vertices of a triangle to the points in which the opposite sides are cut by any conic. Kr. 2. A conic can be drawn through the six points in which the sides of a triangle are met liy the tangents drawn from the opposite vertices to any conic. 77. The equation of* a conic through four points 1, ±1, ±1, being ax^ + by^ + cz- = (), with the condition a + b + c = 0, there is one undetermined constant in the equation, and this para- meter is involved hnearly ; the conies form that particular singly infinite system defined as a pencil. For one special conic of the system, obtained by taking a = 0, is y'' — Z" = {), and another is x^ — z'^ = 0; and the general equation of the system can be written a{x^ - z^) + 6(2/2 - z') + (a + 6 + c)z' = 0, which, by the condition ct-j-6 + c = (), is reduced to a{x'-z-') + h{y'-z') = 0, that is, to (t + A'' = 0, the typical form for a pencil of curves. Siiiiilarl}' the conies touching four given lines are ivpresented ' by f/) + A\//- = 0, and therefore form a range. 7(S. The e({uation written in the form {y/ax + x/ — byX>Jax — J — l>y)^{'J — cz)-, shows that >Jax±>J — hy = {) are tangents, = being the cliord of contact. Hence eacli vertex of the triangle of reference is tlie pole of the opposite side ; that is, a, h, c, DESCEIPTIVE PROPERTIES OF CURVES. 75 A, B, G are conjugate to A, B, C, a, b, c: the triangle is its own conjugate, and is said to be self -conjugate with regard to the conic. For the conic to be real, the coefficients a, b, c must be not all of the same sign : let c be negative ; write l'^, m^, — u- for a, b, c, so that the equation can be written in the forms l^x^ + m^y^ = n^z^, n^z^ — m?y" = Px-, showing that the tangents from C are imaginary, while those from A, B are real. Thus for a real self-conjugate triangle, the vertices are one inside, two outside the conic ; and the sides cut the conic, one in imaginary points, two in real points. From the properties of the triangle ABC, which was chosen by means of the four points P, Q, R, S, a construc- tion for the polar of a point A is deduced. Draw through A any two chords, PQ, MS ; join the four points so deter- mined in the two remaining possible ways, so determining points B, G; BC is the polar of A (Fig. 23). This con- struction applies whether A is inside or outside the conic. Fig. 23. It should be noticed in Fig. 23 that taking different positions of the chords APQ, A SB, different positions can be obtained for B, C, but all lie on the line there deter- mined as BG. The construction here given shows that the join of conjugate points is cut harmonically by the conic. For let A, A', be conjugates; take -^1^1' for one chord through A, e.g. APQ] then since A' is conjugate to A, the polar BG passes through A'; hence by the harmonic properties of the figure. A, A' are harmonic with respect to P, Q, and therefore with respect to the ccmic. It is on account of this property that the pairs of elements are said to be 76 DESCRIPTIVE PROPERTIES OF CURVES. conjugate with respect to the conic ; they are harmonic covjugates. Similarly two conjugate lines, AB, AG are harmonic conjugates with regard to the tangents from their intersection A. The two points, two lines, or point and line might therefore with advantage be called harmonic instead of conjugate. To construct a self -conjugate triangle, that is, a harmonic triangle, any point A is chosen at random ; B is then chosen as any point on the polar of A, and the triangle is thereby completely determined. Hence all the triangles tliat are self-conjugate with regard to a given conic form a three-fold infinity. Xote. Ill general, the conic liaving the triangle uvw as a self-conjugate triangle is bf'- + mv'^ + nw' = ; hence the conies with regard to which a given triangle is self-conjugate form a two-fold infinity. 79. Tliat any two conies intersect in four points, real or imaginary, appears from the fact that the elimination of z between the two equations leads to an equation of degree 4 in X : y. Similarly any two have four conniion tangents. From the four common points, or from the four common tangents, a triangle can be constructed self-conjugate with regard to each of the conies. E.K. Show that the two constructions lead to the same triangle. The four common points being all real, or all imaginary, the self-conjugate triangle is real (§ 51). Taking it as triangle of reference, the two conies are ax^ + b7/+cz^~ = 0, a'x^ + h'f-\-c'z^ = 0. Th(.' liiu; e(juations of these conies are />c^-+ catf+ a6^'- = 0, h'c'}-^ + ca,f + a'b'^-^ = : consefpiently the coordinates of the common tangents, deter- mined by solving these equations, are given by ^2 . ,ji . ^2 _ aa\hc — h'c) : hh'(c(i — c'a) : cc'{ah' — a'h). If all the expressions on the right have the same sign, the four values of ^, >/, ^ are real ; if the expressions have not all the same sign, the sets ^, >;, ^ are imaginary. Hence all four intersections being of the same nature (all real or all imaginary), the four common tangents are all real or all imaginary : and reciprocally, all four common tangents being' DESCRIPTIVE PROPERTIES OF CURVES. 77 of the same nature, the four coinnion points are all real or all imaginary. The only cases omitted here are 07ie, for the nature of the points, and one, for the nature of the lines. These must therefore go together : that is, the intersections being two real and two imaginary, the common tangents are two real and two imaginary. This case is of no special interest : Ijut the cases where the four fundamental elements are all real or all imaginary require more detailed investigation. SO. Let there be four points, all real ; their coordinates may therefore be taken as 1, ±1, ±1. Any conic of the pencil is with the condition a-\-h + c — 0. Hence one of the three coefficients has the opposite sign to the other two ; the real conies of the pencil fall into thri'i- sets, for which the signs of the coefficients are (i.) a — , h and v-\-, (ii.) & — , G and « + , (iii.) c — , a and h-\- ; these three sets are geometrically distinguished by being met in imaginary points by the lines x, y, z, respectively. The coordinates of the connnon tangents to two conies of the pencil are given l)y ^'^ : }f : ^'- = auj'ihc — b'c) : l>b'(ca' — c'a) : cc'{ah' — ah). I. Let the two conies belong to the same set, for example the third, so that c, c' are negative. There is no loss of generality in choosing numerical values for c, c' ; take therefore c = — 1 , whence a +h = 1 ; f' = — 1 , whence a' -\-h' =:1. Since a, h, a', h' and the product co are all posit^^■e, the signs of the expressions on the right are the same as those of — h-\-b\ —a'-\-((, ab' — a'b, that is, of a—l-\-l—a', —d-' + a, «(1 — u') — rr(l — «), and therefore of ((. — a', a—a, (t — (t'. These are all of the same sign, and consequently ^, >/, ^, are real ; that is, members of the same set have real connnon tangents. II. Let the two conies belong to ditterent sets, for exainpU>, the third and second ; a, b are now positive, and c is negative : a, c are positive, b' is negative. 78 DESCRIPTIVE PROPERTIES OF CURVES. Take c = — 1 , whence « + 6 =--1 : c—-\-l, whence a-\-h'= —I. The signs now depend on tliose oi' h + h', —( — a —a), — («/>' — ab) , that is, of — (a + (('), + {) or in Fig. 25 (li) is a nest. Thus if two pairs of imaginary elements be given for the determination of a system of conies, the real conies fall into two nests, which visibly connect or intersect according as the given imaginary elements are points or lines. Examples. 1. Show that there is one point conjugate to any given point with respect to each of two conies ; but that if the given point have one of three particular positions, the conjugate becomes indeterminate, being any point on a certain straight line. 2. Show that points conjugate with respect to each of two conies are conjugate with respect to all conies of the pencil. *1 Find the locus of the pole of a fixed line, and the envelope of the polar of a fixed point, with respect to (i.) a pencil, (ii.) a range, of conies. 4. The points P, Q are conjugate with respect to a pencil of conies ; show that if P describe a straight line, Q describes a conic ; and that this conic is the locus of the pole of the given line with respect to the pencil. How is it situated with regard to tlie pencil ? (Poncelet.) 5. Find the envelope of the tangents at the points where a fixed transversal meets a pencil of conies. 6. Find the locus of the points of contact of tangents from a fixed point to a range of conies. 7. The polars of a point with regard to four conies of a pencil form a pencil whose cross-ratio is independent of the position of the point. Note. This cross-ratio is called the cross-ratio of the conies ; it may be detennined as the cross-i-atio of the four tangents at any one of the common points. (Chasles.) 8. Sliow that three conies of a pencil are line-pairs, and that of these three certainly one is real. Discuss the reciprocal idea. 9. Can a pencil of conies ever be a range ? *See Mr. Hulburt's account of the paper by Hilbert (J/«^A. Annalen, t. xxxviii. ; 1891) in the BuUeiin of the yew York Mathematicd Society, vol. i. p. 197. DESCEIPTIVE PROPERTIES OF CURVES. 81 10. Takiiit;- any three conies, tlie polars of a point P are not ordinarily concurrent : find the condition to be satisfied by P in order that these polars may be concurrent : and apply this to prove that pairs of points harmonic witli respect to three conies lie on a curve of order three. 11. Show that 25<^ii^ts harmonic with respect to the three conies 11 = 0, v = 0, '?f = 0, are harmonic with respect to every conic of the net lu-\-riiiv-\-mv==0. 12. If the line equations of the two conies U=0, V=i) be $ = 0, ^ = 0, the line equation of the pencil U-\-kV=0 is $ + /v2 + Jc^^ = ; and the point equation of the range $ + X^ = is U'+\S+\^V=i), where S, 1,, are expressions of the second degree. Show that the line equation of the pencil represents a range if 'Z = A^ + B^. Hence find the conditions that the conies ax^ + bi/ + cz"" + 2fijs + 2gzx + 2hxy = 0, a'x^ + h'}f 4- f's- + If II : + Ig'zx + "Hixy = 0, may have double contact. On the Nwniber of Gonditions determining a Conic. 83. The general equation of the second degree contains six terms, and therefore it involves five disposable constants, viz., the ratios of the six coefficients ; hence a conic, whether regarded as a locus or as an envelope, is determined by five conditions. Thus the three conditions of passing through A, B, C, or touching a, h, c, left ttuo constants to be deter- mined ; the four conditions of passing through four points, or touching four lines, left one. Similarly the general equation of degree m contains h('in + \)(7ii-\-2) terms, the number of disposable constants is therefore h(m-\-l){m + 2)-l, i.e. |m(m-|-3); the general curve of order m, or of class m, is determined by |-m(m-i-3) independent conditions. Considering now the conic, determined by five conditions, let these be simply that certain point or line elements are given. I. Given Jive points, or five lines. The coefficients have to satisfy five equations such as ax^ + hy'i + csf + 2///^^, + -Igz^r^ + 2/i.r,7/, = : that is, they are given by five linear e(|uations, and are therefore determined uniquely. 82 DESCRIPTIVE PROPERTIES OF CURVES. Note. It should be obsei'ved that h linear equations, homogeneous in /•+ 1 quantities, cannot be inconsistent, for a reason that a single example will make clear. Eijuations such as 3.r + 2y+l=0, 6.r + 4_y-3 = 0, are called inconsistent in some books on elementary algebra, on account of the result obtained by the ordinary method of solution. But writing them in the homogeneous form, 3.r + 2?/ + i = 0, 6.V + 4 y - Zz = 0, the solution is seen to be ^ = 0, .r :;y = 2 : — 3 ; hence tlie equations are not inconsistent. Compare the discussion of parallel lines in ,§ 25. Thus tlie locus of" the second order is determined uniquely by five points. Now the general locus of the second order is also the general envelope of the second class, this is therefore determined uniquel}' by five points : and by the principle of duality five lines determine the conic uniquely whether it be regarded as an envelope or as a locus. Hence one conic can be drawn to pass through five points, or to touch five lines. II. Given four points and one line, or four lines and one i^oint. Take the diagonal triangle of the four elements as triangle of reference, and choose coordinates so that the four elements are 1, ±1, ±1. Then the equation is with the condition a + /; + c = (1), and the condition of contact with the given line px + qy + rz = 0. This last requires hep- + ca(f + ahv- = (2) : hence a, b, c must be determined from equations (1) and (2) ; these give two sets of values for a :b :c: consequently two conies can be drawn to pass tli rough four points and touch one line, or to touch four lines and pass tlu'ougli one point. Thus the five conditions detoniiiiu' the conic, tliough they determine it as one of two. R/\ To what condition nui.st the line be subject in order that the two conies may coincide ? III. Given three p>oints and two lines, or three lines and tivo points. Let the triangle of ivference be tlie one determined by the three given elements, then the conic is /.'/ - + fISX + //,r ?/ = 0. DESCRIPTIVE PEOPERTIES OF CURVES. 83 From the conditions of contact with tlie Hnes px-\-qij + rz = 0, !>'■'' + -2ffjpq =0 (1), ■yy 2 ^ ,j2^j'i + /,2^/2 _ 2gkqV - 2hfry - -Ifgp'q' = (2). Equations (1) and (2) give four sets of vahics /':_(y :/<. Fonv conies can therefore be drawn to pass throu^'li three p< 1 SO). 85, PascaVs Theorem. The intersections of 02)posite sides of a hexagon in a conic are collinear. Brianchon's Tlieorem. The joins of opposite vertices of a hexagon about a conic are concurrent. In the two theorems the six given elements, wliicli by hypo- thesis belong to a conic, may be condjined in any order to foi'm the hexagon : there are therefore 60 difierent hexagons, to 84 DESCRIPTIVE PROPERTIES OF CURVES. every one of" wliich the theorem applies: consequently six points on a conic give GO Pascal lines : and six tangents to a conic give 60 Brianchon points. The theorems can be proved by a direct use of coordinates, point coordinates for Pascal's theorem, line coordinates for Brianchon's theorem. Take for triangle of reference that determined by the alter- nate elements 1, 2, 3 (Fig. 26); let the opposite elements ] ', 2', 8' have coordinates x^, y-^, z^ ; x^, y.j, z^ The equations of 23', 2'3 are 2/3. X X.-, '3 •^S therefore the derived element X X. I. is 1, the derived element II. is the derived element III. is 1(2, ?/2 ?/3 The determinant T) formed with these coordinates is 2/i 2/2 1 2/2 ^3 = X{IJ 2'^3 2/i 1_ V-i Now by hypothesis the three elements 1', 2', 3' belong to a conic 1, 2, 3 : consequently a relation lyz-\- mzx + 7ixy = 0, that is, I . 7)i n ,, -+ +-=0, x y z IS satisfied by every set x^, y^, z^ ; from the tliree equations so obtained /, m, n can be eliminated, and the result is D = 0. Hence the three derived elements I, II., III. are imited with a .single secondaiy element: they are relatetl as stated by the theorems. (SO. By means of Pascal's theoi'eiii any iiniiibev of points on a c(jnic 1 2 2' 3 3' can be linearly constructed. For 23', 2'3 detei-mine I.: take any line tln-ough 1. meeting 13', 12' in IL, III. : then 3 II. and 2 III. intersect' in I '. if the tangent at 3 is to be c(jnsti'ucted, V must be made the same as DESCRIPTIVE PROPERTIES OF CURVES. 85 3, and tlien 81' II. (i.e. 8 II.) is the tangent ; the construction is therefore; 23', 2':3 determine I. ; (12', 1'2 i.e.) 12', 23 determine III. ; hence the Pascal line I. III. is known, and this meets 13' in II. : thus 3 II. is constructed. 87. A better proof of these theorems depends on the ecjua- tion of a conic with four given elements (§ 75) ; this proof (from Salmon's Conic Sections, % 267) is here given for Pascal's theorem : a few simple verbal changes will adapt it to Brianchon's theorem. Let the six points taken in order (Fig. 26) be 1 3' 2 T3 2', and let the sides be denoted by u=^i}, etc., as shown in the diagram. Any conic through 1 3' 2 1' has for its equation V'W = kpw, where 'p = is the line 1 1'. The conic F througli the six points is one of this system ; let the multipliers involved in u, V, w be adjusted so that this particular conic is F—civ—2>ii — i). Similarly any conic through 1 2' 3 1' is v\v' = k'pu' ; and the multipliers in a', v\ lu being still undetermined may be chosen so that the particular conic F oi this system has for its equation F = v\v' — 'pvb' =^i). Hence the expression F can be written in each of the forms vw—pu, v'w'—pu', 86 IlESC^RTPTIVE PrvOPERTIES OF CURVES. these are therefore identically the same ; that is, viu —pu = v'w' —pu', from which viu — v'w'=p{ii — u'), showiiii;- that the expression ?;!(;— yW splits np into factors, p and a — u. Hence the conic viv — v'lv' = is made uj) of the line -p = (i.e. 1 1'), and a line ii — u,' = {), which is some line through I. But this ccmic passes through all the intersections of vw = and vw' = i): that is, through the four points 1, 1', II., III. The line /> = accounts for 1, 1'; the line u — 16' = must therefore accoimt for II., III. : hence I., II., III. are collinear. 88. Chasles' Theorem is given in his Traits des Sections Goniqaes (1865), pp. 2, 3. He states it in a form involving synnnetrically five tangents and five points : — Four points on a conic determine at any fifth point of the conic a pencil whose cross-ratio is equal to that of the range determined on any fifth tangent by the tangents at the four points ;* and then deduces two fundamental properties of conies :— Four points on a conic determine at any fifth point a pencil of constant cross-ratio ; and four tangents determine on any fifth tangent a range of constant cross-ratio ; that is, four elements determine tuith any fifth element a confi.guration of constant cross-ratio. The materials for a direct proof are contained in Examples lO-l:^ of Chapter III. The four points being 1, ±1, ±1 the conic is ax' + hy' + cz^^O (1), with the condition a.-\-h-\-c = () (2). Hence one cross-ratio of the pencil deteiMuiiicd by a fifth point X, y, z is 2 — 2- Multiplying (2) by x', and sid)tract- ing from (1), b{y--x-)-\-c{z^-x^) = 0: ^, ,. • x^ — y'^ c tnereiore o o=— y x^ — z^ b which is the same for all points x, y, z on tlic conic. * " iSi par (luatrc points d'luie conique on nicnc k-s tangeiitos et quatre aiitres droitcs al)outis,saiit a uii cincpiienie point quelcoiique de la courbe : le raj)p<)rt anhainionique de ces quatre droites sera 6gal a eelui des (puitre points de rencontre des quatre tangentes et d'une cinquiemc tangente quelconque." Tins is perhaps as good an oppoi'tunity as any foi' acknowledging my general indebtedness to this fascinating work of M. Chasles. DESCRIPTIVE PROPERTIES OF CURVES. 87 The tangents at the four points are ax ±by ± cz — 0, i.e. a, ±h, ±c. Let any fifth tangent be ■px-\-qy-\-rz = (), then hcp'^ + caq^ + ahr^- = {) (3). The cross-ratio determined on -p, q, r by a, ± b, ± c is p^ r2 a2 c2 Multiplying (3) by a, (2) by hcp'^, and subtracting, therefore ((2 52 c p2 .^.2 ^' which is the same for all tangents p>, q, r ; and is the same as that found from the points. A simple proof depending on a different special form of the equation of a conic is to be found in Salmon's Conic iSections, §§ 274, 275. 89. Chasles' two fundamental properties of conies are direct interpretations of the equation ay = k/BS, a, /3, y, S being regarded (i.) as point coordinates, (ii.) as line coordinates. (i.) For detiniteness, let these be actual perpendiculars. Let the four points be A, B, C, D, and let any fifth point be P. Use {AB) to denote the angle subtended at P by ^, B. Then 2 x area PAB = uxAB, and also = FA.FB. Hm(AB) ; therefore a.AB = FA.PB. sm(AB), y.CD=FG.PI).sm{CD), ^.BG=FB.PC .sm(BG), S.DA=FD.FA.sin(DA). ^ ay.AB.CD _jAB.CD """^^ (3S . BG . DA ~ "bG . DT and is therefore constant ; , sin(^iAsin(aP) . , , hence -^-/w^x-- )i^4\ is constant; sin{BG)tim{DA) that is, the pencil {F . ABGB} is constant. 88 DESCRITTTVE PROPERTIES OF CURVES. (ii.) Now considering; tan^vnts a, b, c, d, with a iit'tli tano-ent 2>, let (ah) denote the len^-th intercepted on 2> by a, h. The equations of the intersections ah, he, cd, da are ^=0, >/ = (), ^=0, = {), where ^, i], ^, 6 are actual perpendiculars; the conic is ^^=J>'r]0. A diagram shows that 2 area^^a^ = ^{ah), and also = (a6)"sin jm sin ph -i- sin ah ; therefore ^ sin ah = (ab) sin pa sin ^6, ^ sin cd — (cd) sin p>c sin jx:?, ;;sin 6c ={bc) sinj96 sin^^c, sincZa = (f?ft)sin pd mnjM. Now sin ((6, sin cd, etc., are constant, hence tinally ). . / , , is constant, "^ {hc){da) that is, the range {^^^.a^cc^} is constant. Note. The cross-ratio of the pencil deteninnuil in a conic by four |ioint8 on the conic is spoken of as the cross-ratio of tht' points ; hut an implied refeienoe to the conic must be understood, the cross-ratio being ditfeient for ditierent conies through the four points. Examples. 1. 'leaking two triangles in perspective, })rove that the joins of non-corresponding vertices (six lines in all) touch a conic, and that the intersections of non-corivspunding sides lie on a conic. 2. Show that two triangles tliat are conjugate with respect to a conic are in perspectiNc. 3. Two triangles are self-conjugate with respect to a conic. Are they in perspective ? 4. The six sides of two inscribed triangles touch a conic, and tlie six vertices of two circumscribed triangles lie on a conic. r). Two triangles are self-conjugatt' with respect to a conic. Sliow that their six vertices lie on a conic, and that their six sides touch a conic. 6. Two triangles are to be di-awn such that a conic can be described having each as a self-conjugate triangle. Choosing one triangle arbitrarily, how many of the deter- mining elements of the other are at our disposal '! 7. Considering the four points determined on a conic by two chords, prove that these ])()ints will he harmonic if the chords be conjugate. DESCRIPTIVE PROPERTIES OF CURVES. 89 8. Show that the problem " to draw throu^'h four points a conic such that the pencil determined in it by the four points shall be harmonic " has three real solutions if the four points be all real, two real solutions if they be all imaginary, one I'eal solution if the points be two real and two imaginary. 9. Show that the three harmonic conies determined by four real points belong to the three diti'erent sets of the pencil. JoachimsthaVs Method. 90. The question of the connnon elements of two curves can be discussed with special facility when the coordinates of an element of the one curve are expressed in terms of a single parameter, while the other curve is represented by an equation. The general question as to the possibility of expressing the coordinates of a point on a curve para- metrically is considered briefly in Chapter VIII. : but there is one case that can now be discussed with advantage, and used to exhibit the general method employed, viz., that in which the equation of one of the two curves is linear. For definiteness, point coordinates only will be explicitly referred to ; the curves considered are therefore a curve of order n, and a straight line. The coordinates of a point on the line can always be expressed in terms of a single parameter ; the points in which the line meets the curve are then determined by the values of the parameter given by an equation of degree n ; and any particular relation of the line to the curve, being necessarily a particular relation of the points of intersection, is expressible by a relation in the roots of this equation. 91. Let the line Vje assigned by tlic two points on it, x', y', z \ x" , y", z' \ and let x, y, z divide the Vnw joining x , y', z to x" , y" , z" in the ratio i)i:l (or 1 :/). Then x:y '.2 = Ix' + ')nx" : ly' + my" : Iz + » t^:". Consider the intersections of this line and a eur\e F—0. I. Let F = ax^ + by' + cz- + 2fyz + 2gzx + '2hxy. By substituting for x, y, z the expressions just given we limit ourselves to such points on F—^ as are also on tlie line joining x', y' , z to x\ y", z" ; tliat is, to the intersections of the line and the conic. The resulting eijuation, arranged in terms of I, m, is l%ax' + hy"^ +...) + 2lmiax'x' +...) + n}r{ax"'- +...) = 0. 90 DESCRIPTTVE PROPERTIES OF (TTEVES. Writing x, y, z for x", ij", z", so that the points determining the line are x', y\ z' ; x, y, z, this equation becomes wliere p is written for x{ax' + ]ty' + (jz') + y{hx' + by +fz') + z{'-HAF+im''-H-^A"F+ ... + ^l"A>'F^O. 2! 'Ill Dut by synmietry in x, y, z : x, y\ z\ this niiglit have been obtained in tlie form lnF'-\-l»-hnA'F'-\- h"--m^A'-'F'-\-... + ^,m"A'"F' = 0, 1. n\ where A' is wi'itten for '"'^''^y-^'^^z'' It should lie noticed that identically A"F==n\F' ; A"-^F=n^li\' /'" : etc. If now /" = (), one value of r)i:l = 0, and the point x, y',z' is on the curve. If in addition A'F' = 0, a second value of in -.1 = 0; the line joining ,/■', y', z to «, y, z meets the curve in DESCRIPTIVE PROPERTIES OF CURVES. 91 two consecutive points at x\ y', z , and is therefore a tangent. Hence the equation of the tangent at x , y', z is 92. If X, y', z' be not on the curve, consider the tangents drawn from x , y , z to the curve. If the point of contact of any one of these be x^, y^, z-^, that tangent is 3i^ , dF , dF ^ that is, A^F^ = 0. This is to pass through the known point x', y' , z ; hence *i' ^1' % must satisfy ,dF^ ,'dF^ ,dF „ oXy dy-^ dz^ that is, x^, 2/i, % must lie on tlie curve ^ p ,3^ , .dF , ,dF ,^ AF=x— + y':^-~-i-z'— = 0, dx "^ dy dz the first polar of x\ y', z with resj^ect to F (^ 70). The first polar of X , y\ z with respect to this curve, viz., A(Ai^) = 0, is called the second polar of x\ y', z with respect to F : it is plainly of order n — 2. Ex. Find, with respect to the curve x^ -ojz'^ -y-z = 0, the successive polars of (i.) ^, 0, 1 ; (ii.) |, 0, 1 ; (iii.) 0, 0, 1. 93. In the case of the conic a special arrangement of the investigation is more interesting. There are, on the line con- sidered, two pairs of points, viz., the determining points Q{o', y, z), Q'{x' , y' , z), and the points determined by tlie equation l-F' + '2lmp + m-F=i); call these R, R'. Two pairs of points are susceptible of a particular relation of position : tliey may be harmonic. By the definition of harmonic division the two values of l:ia are numerically equal but opposite in sign: hence ^ +-- =0: that is, the sum of tlie roots in the (piadratie equation vanishes. Hence the couditi(m that Q, (/ be harmonic con- jugates with respect to tlie conic is p = 0, that is, x{ax + hy' + yz) + y{hx' + by' +fz) + z{yx' +fy' -f- cz) = 0, 92 DESCRIPTIVK PROPERTIES OF CURVES. and the locus of all points conjugate to (/ with respect to the conic is the straight line p = x{ax + hy' -\-f/z') + //(A.''' + by' +fz') + z{gx' +fy' + cz') = 0. This is called the polar of x, y',z' ; and the line being here denoted by j), the point x, y' , z (Q') will now be called P. All properties of poles and polars with respect to conies follow from this fundamental harmonic property : — I. If the polar of F pass through Q, then F, Q are harmonic with respect to the .conic, and therefore the polar of Q passes through F. IT. If R, R be real, they separate and are separated by F, Q. Hence as R, R' approach one another indefinitely, the line PRQK becoming a tangent, one of the two points P, Q — Q, suppose — becomes coincident with RR' ; hence the polar of P passes through the points of contact of tangents from P. III. If P be on the curve, so coinciding with R, the con- dition of harmonic division requires that K also coincide with R ; and then Q is simply any point on the line FRR' ; the locus of Q is therefore this line FRR', which is the tangent at P ; that is to say, the polar of a poi^it on the conic is the tangent at that point. IV. The construction for the polar, given in § 78, is at once seen to be correct, owing to the harmonic properties of a com- plete (juadrangle. Hence the theory of self -conjugate triangles (liariii()iii(* triangles) also follows. 94. If X, y', z be not on the curv(> F, let it be required to make the line a tangent. This requires that two roots of the equation in ^ : m be made coincident. Hiis applied to the special case of the conic gives the condition which is therefore the ecpiation of the two tangents that can be drawn from x , y', z' to the conic F= 0. A'.'-. Dc'teriuiiK! tliu tliit'c tangents that can Ix' drawn from I, 1,1 to the cubic .'••' + /;y-i = U. Sole. It was shown in j^ 6S tliat this particnhir order-cubic is a class- cubic. 95. Tills method can also be applied to the determination of the iiiti'rsections of two curves, by finding the lines joining any arbitrary point x , y' , z to the connnon points of the given curves (t = 0, '?' = 0. Let these be of ordei's p, q; let a line through x', y', z meet them in 7"'^, F.,, ...,Fp\ Q.^, Q.„ ..., Q,^. For this line to pass through a conniion point A, one of the DESCRIPTIVE PROPERTIES OP CURVES. 93 points P must coincide with one of the points Q. But these two sets of points are determined by the two equations l'Pii+lp-hni::^'ii-{- . . . = 0, I'iv' +Vi~'^m/l'v'+ ...={) : hence the line considered will pass throug-h an intersection A if these two equations have a common root. The condition is obtained by the elimination of l-.m: it is therefore ex- pressed in terms of u', A'u',... : v', AV.... Hence it involves x', y', z, a known point, and x, y, z, a variable point. It is therefore an equation in x, y, z, and by the mode of its formation it represents the lines joining x, y', z to all the intersections of tt, v. Applying the process to two conies u = 0, v = 0, let the two polars of x , y', z be p, q : the two equations are I'hi + 'Ibivp + m^u = 0, l~r' + 2lmq + inhf = 0, and the equation obtained by eliminating I : m is {uv' — 11 vf = 4(m-(/ — }'p){v'p — 11 q ) : this equation of degree 4 represents the four lines that join x', y', z' to the intersections of the conies «,, v. 96. This method for dealing symmetrically with the inter- sections of straight lines and curves is due to Joachimsthal. It should ])e noticed that for its employment it may possibly not be essential that the coordinates of the various points be the same multiples of the actual j^erpendiculars, thouo-h they must of course belong to the same system. For let the system of coordinates be given by ^, y, s = Aa, iJ.fi, vy. Suppose that x', y' , z are ecpial to \a' , ij.(i\ vy' multi])lied by /', and that x" , y", z" are equal to Xa", //;5", vy" , nuiltiplied by /", etc. Then if «, /5, y divide a, /3', y \ a, /3", y" : in the ratio m : /, ^a'-f-TOa" a = — r~, > titc. t-t-m Hence, x, y, z being equal to \a, fx^, vy nuiltiplied by /", 1 X X "^ f^ ./^ etc ,1 , • /' lf"x'-\-r>if'x' , tnat IS, x = J^„ . ^^-r-; — — , etc. , whence x:y:z = If'x' + mf'x" : Ify' + mfy" : If'z' -f wfz!' ; 94 DESCRIPTIVE PROPERTIES OF CURVES. and writing' I', 'ni for If" , mf, these become ,,,• ■.y:z = I'x + m'x" : I'y' + mfy" : I'z + m'z". Tluis tlie effect of the different miiltiplier,s / is to alter all values of I : m in the same ratio. If then we propose to deal with these ratios directly, obtaining- results that depend on their actual values (for example, if we wish a line to be bisected), these different nniltipliers / are not admissible : the coordinates of the various points nnist l)e the same multiples of the actual perpendiculars. But if the desired results depend only on a comparison of the values of I : m, the actual values them- selves not entering- into the final expressions, the different multipliers / have no effect: in this case it is not essential th;it the coordinates of the various points be the same multiples of the actual perpendiculars. Examples. 1. Show that two lines can be drawn through an}" point so as to be harmonically divided by two conies ; and that if the point be an intersection of the conies, the two lines are the tangents to the two conies at this point. '2. Hence show that the enveloj^e of a line harmonically divided by two conies is a conic : and the common self- conjugate triangle being taken as triangle of reference, find the line equation of this envelope. Find also the point e(iuation. (Von Staudt. Salmon.) 3. Find the equation of this envelope if the equations of the conies be in the general form. 4. Show that any common tangent to the conies u±Xv = is cut harmonically by the conies u = i), v = 0, for all values of X ; and apph' this to find the equation of the envelope of a line harinonically divided by two conies. (F. Morlcy.) 0. Show that the locus of a point harmonically subtended* * In the system of geometry in which the point is tiie ])rimary element, the line the secondary element, the point is an entity, witliout ]iarts, incajiable of division ; the line is an aggregate of ))oints, and therefore capable of division ; .segments on a line are determinefl by their bounding pciint.s. If however the line be the primary element, it is the entity incajjable of division ; the point is regarded as an aggregate of lines, and therefore capable of division ; the divisions being determined by the bounding lines. Hence linear magnitude and anguhir magnitude relate respectively to divisions of a line and of a ])f)int. Thus if two pairs of points on a line be haiinoiiic, the line is harmonically divided ; and if two ]>airs of lines through a point be harmonic, tiie ])oint may be said to be harmonically divided. In Ex. 2 the line is spoken of as divided by the conies ; hence withdut any explicit statement it is known tliaf the conies DESCRIPTIVE PROPERTIES OF CURVES. 95 by two conies is a conic, passing through the eight points of contact of the connnon tangents to the two conies. Determine the point equation of this conic, the two given conies being referred to their common self-conjugate triangle. Find also the line equation. (Von Staudt. Salmon.) Curves loith t^lnrjular Points and Lines. 97. The process given for finding the tangent to any curve at a point (§ 64) involves the tacit assumption that only one line can be drawn to meet the curve in two indefinitely near points. And as a matter of fact, the process leads to an equation SXj dy^ 30J which is determinate unless --, -~, — vanish tosfether. 3«i 3^/1 dZj ^ This same possibility presents itself as a source of disturbance in the determination of the reciprocal to a curve. The mean- ing of this will now be briefly considered. 98. Suppose that the assumption as to there being only one line through a point, meeting the curve in two consecutive points there, is not justified. Let there be two such lines, u = 0, v — i). Then since u meets the curve in two points on V, the equation must be of the form Also -^ = is to give u- = 0, hence u$ = u^X + uvQ, and the equation of the curve F is u^X + uvS-i-v-^^O.. (1). But this being the equation, every line through uv, ^•iz., u + \v = ii, meets the curve in two points at uv. Hence if more than one line through a point meet the curve in two points there, then every line through the point has this property ; the point is a double point. In the e(iuation it-|-Av = 0, X is still undetermined; it is therefore possible to make the line meet the curve in three points at uv ; the are regai'ded as point systems, i.e. as loci. In Ex. 5 the jjoint might be s|joken of as " hamionicall}' divided by two conies"; the j)oint being divided, the impHed element is the line ; the conies are regardeil as line systems, and each has two line elements in common with the point. Thus there are two pairs of lines through the point, and these pairs are to be harmonic. But as these lines are tangents to the conic, the same idea may be conveyed by the jjhraseology adopted in the text. no DESCRTPTTYE PROPERTIES OF CURVES. ('((uation for A, exactly as in Cartesians, is a qnadratie, and there are tlieivfore two tang-ents at the double point. For example, if tiie equation of the curve be of the form 11 = o-ives /•^ = 0, and therefore u = is one tano-ent at the double point w-?' ; and if the equation be of the form u=:0 o-ives v^ = 0, and 7; = p-ives u-^ = 0: hence the lines u = 0, v = are the two tangents at the double point ur. The general principles of § Gl can be applied to the case of double points. The equation can be regarded as the final form of uuX + uvQ + vv'^ = 0, when u, v' coincide, and also v, v'. Hence two of the points connnon to F, u coincide, or else are consecutive on v ; and two of the points common to F, r coincide, or else are con- secutive on u. These conditions ai-e harmonized by the double point at wi In this argument it is not assumed that u, v are linear : they may be expressions of any degree, and what has just been proved is that ever}'- intersection of the curves ?t = (), v = is a double point on the curve Er. Till' curve (.'/- - •^')- + (f - •^)(-^' - y)0/ - 1 ) + (.r - y)2(.r2 +f) = Iia.s a double point at every intersection oi i/^ — ,v = 0, X — 7/ — ; that is, at 0, Oaiulut 1, 1. Similai'ly there is a triple point at -?(/• if tlie eipiation be reducible to u3<|> + iih'^ 4- ^^^'-X + ^-'^f ) = 0, for every line u-\-\v = now meets the curve in three points at uv ; the tangents are defined as lines that meet the curve in four points, and are determined by a cubic equation in X ; there are thei'cfore three tangents at the tri])'^' point. 90. Since the double point makiis itself felt in the course of the work l)y I'cndering tlie ])rocess for finding the tangent migatory, tliis must be by cansing the ('(|uation -dF ^ -dF , dF ^ orKj oy^ dz^ DESCRIPTIVE PROPERTIES OF CURVES. 97 to become illusory. The conflitions for a (loul)le point at X, y, z are therefore 1^=0. 1^=0, 1^=0. ox oy oz Note. These are essentially the same as the ordinary Cartesian con- ditions. For let the Cartesian equation be /(.r, v/) = ; the conditions for a double point are ox oy Let the equation made homogeneous as directed in § 30 be F{Xy y, z) = 0. Then the condition /=0 gives ^=0 ; ~^ = 0, made homogeneous, gives ^- =0 ; ox O.r ^ = 0, made homogeneous, gives _ =0. oy ' oy Also, by Euler's theorem of homogeneous functions, OX ' oy oz 7) P 7)1^ hence the condition F=0, with the help of _=0, ^ =0, is reduced to ox oy |?=0. oz The elimination of x:y:z from these three homogeneous equations leaves one condition to be satisfied by the co- efficients. Hence the general curve of any assigned order has not any double points (nodes or cusps) ; the presence of even one double point requires the equation to be specialized ; a certain function of the coefficients nuist vanish. Reciprocally, the general curve of any assigned class lias not any double lines (double tangents or inflexional tangents); for the existence of these tlie equation must be specialized by the vanishing of a certain function of the coefficients. Note. In this connection it may be remarked that if the reciprocal to the general curve of order m be of order n (the assigned general curve being of class m), it is not the general curve of order n. For example, the reciprocal to the general cubic is a sextic ; now the general equation of a cubic involves 9 disposable constants ; and the coefficients of the recip- rocal sextic are expressed in terms of these 9 quantities. But in the general equation of a sextic there are 28 terms, and consequently 27 arbitrary constants. Thus the sextic is specialized ; the 27 independent constants that belong to the general sextic are expressed in terms of 9 quantities, and by eliminating these 9 quantities in the various possible ways from the 27 equations, it is seen that certain fiinctions of the coefficients of the sextic vanish. The discussion of the tangents from a point (§§ 70, 92) shows that their points of contact are determined as inter- 98 DESCRIPTIVE PROPEETIES OF CURVES. sections of tlie curve and the first polar of the point, with the proviso that ponits at which - -, -^, and ,^ vanish are not ^ ox oy cz to be counted : the effect of these points is therefore to diminisli the number of tangents that can be drawn from a point. These singuhrr points liave now been found to be double points ; hence the presence of double points (nodes or cusps) on a curve of given order causes a diminution in the class ; and the presence of double lines (double tangents or inflexional tangents) on a curve of given class causes a diminution in the order. For example, in § 68 two order-cubics were found to be, the one of class 6, the other of class 3 ; and it was shown in § (i9 that one of these has a cusp. The investigation of the law of diminution belongs however to the theory of Higher Plane Curves. 100. The effect of an inflexion on the point equation, and reciprocally of a cuspidal tangent on the line equation, must be noticed. An inflexional tangent meets the curve in three points, instead of two ; hence by the process of ;^§ 61, 98, if %b = be an inflexional tangent whose point of contact is on -^ = 0, the equation of the curve F must be U0 + v^xlr — 0. 101. As an example of the direct processes that can be employed consider a quartic. This has a double point at A if the equation be of the form ?A/. + 2/s^/r + s2x = 0, where 0, yjy, ^ are general expressions of the second degree ; hence there is a double point at A if the terms be absent fi'om the general equation of a quartic. Similarly there is a double point at B if the terms y^, xy^, y^z be absent ; and at C if the terms 2^, xz^, yz^ be absent. Hence the most general equation of a quartic with double points at A, B, G is a7/%2 + hz~x^ + cxhf + ifxhjz + ^gxy^-z + ^hxyz"- = 0. To determine the tangents at ^, X is chosen so that y = \z meets the curve in three points at A. The equation for intersections is aXV + 2(r/XH/'XV:3H(^X-+2/A + %V = 0, DESCRIPTIVE PROPERTIES OF CURVES. 99 showing that for all values of X, two intersections are given by 3 — 0. To make z^ a factor, X must satisfy cX' + 2fX + b^0. Hence the tangents at A are determined by this equation ; the equation of the tangents themselves is obtained by writing \ = - ; hence the tangents at A, B, and C are az^ 4- 2/ + /»y-, where A is ah-eady known to be the same as a; hence taking B^—2h, all the conditions are satisfied. The conclusion is therefore : — The quartic aijh^ + hz'-x^ + cxhf + 2fxhjz + 2gxy'^z + 2hxyz^ = has nodes at A, B, C: and the six nodal tangents touch the conic Examples. 1. Writing the general equation of a cubic in the form ax^ + h]f + cd^ + 3a x-y + 36' yH + 3c' z^x + 3a"a;?/2 + 36"?/0- + '^c"zx^ + Gc^cct/^ = 0, show that the conditions for a double point at C are c = 0, c' = 0, ?>" = 0; and that GA, GB will be the tangents at this double point if 6' = 0, c" = 0. 2. Show that the conditions for a cusp at G are c = 0, c' = 0, 6" = 0, d^ = h'c": and that if x — y = be the tangent at the cusp, c" = b', d= —b'. 3. Find the equation of a cubic, touching the lines tt = 0, v = 0,w = on the line s = 0. Hence show that, P, Q, R being collinear points on a cubic, the points P', Q', R' in Avhich the tangents at P, Q, R meet the cubic again are collinear. 4. Find the equation of a cubic having y = 0, = 0, as in- flexional tangents at points lying on the line x-\-y-\-z = 0. Show that this cubic has a third inflexion, also on the line a; + 2/ + s = 0. 5. Find the equation of a cubic with a cusp at viv, tangent to w = 0, and an inflexion at lov, tangent to u = 0. 6. Show that a quartic can be found having cusps at A, B, G; and tliat tlie cuspidal tangents arc concurrent. CHAPTER VI. METRIC PROPERTIES OF CURVES; THE LINE INFINITY. Introductory. 102. The fundamental identical relation in point coordi- nates leads US to consider a certain straight line lying entirely at infinity. In considering therefore properties of curves that depend on the actual values of the coordinates, that is, on the identical relation in point coordinates, we naturally consider the relation of the curve to this special line. This line being specialized only in position, not in nature, all general investigations on the relation of a curve to a line are applicable. Points at Infinity. Asymptotes. lOo. In the first place let the curve considered be a conic. The only specialization in the relation of a conic to a line is that expressed by contact ; the two points of intersection coincide. Hence a classification of conies pre- sents itself, according as the line infinity is or is not a tangent. One division of conies into species has already been found, viz., into proper and degenerate ; that how- ever is simply a cross-division ; two straight lines, not parallel, give distinct points at infinity ; parallel straight lines give coincident points. But as regards the points at infinity, the eye introduces another distinction, that between real and imaginary ; this division is in a sense accidental, but is recognized in the classification of conies. These fall therefore into three species : — I. (i.) The points (it infinity being coincident, the conic is a parabola. ' (ii.) The points at infinity being real and distinct, the Tj conic is a hypei'bola. ■"* (iii.) The points at infinity being imaginary, the conic is an ellipse. 102 METRIC PROPERTIES OF CURVES; Hence a line-pair comes under the heading parabola, hyperbohi, ellipse, according as the lines are parallel, real and distinct, or imaginary. Ex. Using trilineai's, the conies represented by fyz+gzx + hxi/ = 0, are distinguished by means of ax + hi/ + cz = 0. The intersections are given by hhy- + {hg + ch - af)yz + cgz- = hence the conic is a hyperbola, a parabola, or an ellipse, according as d-f~ + h-g- + c7r - ibcgh — 2culif- 2ubfg is + , 0, or - . Ex. Using areals, the nature of the conic depends on f-+g- + /r - 2gh - 2hf- 2fg. 104. Hence to be told that a conic is a parabola is to be given one simple condition ; for one tangent is given. Consequently (§ 83) four more tangents determine the conic uniquely, four points determine it as one of two ; thus one parabola can be drawn to touch four lines ; two para- bolas can be drawn to pass through four points. Ex. 1. The line intinity being cit^x + b,))/ + CaZ — O, find the condition that the conic ax- + by- + cz- + Ifyz + "^gzx -\- ihxy = be a pai'abola. Deduce the ordinary condition in Cartesians. Ex. 2. Using actual perpendiculars for line coordinates, find the con- dition that the general envelope of the second class be a parabola. lOo. Any curve of order n cuts the line infinity in n points ; some may be real, some imaginary ; and there may be coincidences. The position of any point on the line intinity is indicated by its direction, all lines in this direction passing through the point. If the curve touch the line intinity -at P, no one of these parallel lines (lines through P) has any special relation to the curve ; but if the curve cut the line intinity at P, the tangent is a line through P, distinct from the line infinity, and is therefore one of the system of parallel lines, that is, it is an ordinary line. Such a line is called an asymptote to the curve ; an asymptote is therefore defined as a tangent ivhose point of contact ?'« at infinity, the tangent itself not lying entirely at infinity. Thus the definition excludes the line infinity from the asymptotes. A parabola has no asymptotes, there are no points where the curve simply cuts the line infinity ; a hyperbola and an ellipse have each two asymptotes, respec- tively real and imaginary. A curve of order n may have THE LINE INFINITY. 103 n asymptotes ; but tliis nniaber is diminished by contacts with the line infinity. If k points be used in accounting for these contacts, the number of asymptotes is n — h. lOG. Let 8 = be the hne infinity, and let x, y' , z be its pole with regard to a conic F=0; by § T-i the equation of the asymptotes of the conic is If now s be a tangent, the pole is on s, and also on F, hence F' = {), and the equation reduces to 8- = U, the line infinity counted twice, but not to be counted as an asymptote. £x. Using areals, find the a.symjjtotes of 2?/i + 22.x' + 2.i'?/ = (1). The line infinity is .v + i/ + z = (2). The polar of a point x', y\ z is xiij' + 2') +y(2' + .*;') + zf^ +y') = 0, and this is the same as (2) if y' + z' = z' + x' — x-' + 9/' = l, whence x' =y' = ^ = \. Hence the asymptotes are i^yz + ^zx + 2.^'y)(2 . ^ . i + 2 . i . I + 2 . i • i) - (^'- + J/ + -f = 0, that is, (,(• + y + z^ -2>((yz-\-zx-\- xy) = 0, that is, X'^+y- + z--yz — zx-xy = 0. But in any particular example it is generally more con- venient to adopt the other process of § 74, and simply express that u,j..-2_o is a pair of straight lines. Ax. k{x+y + zy + 2(yz + zx + xy)=0 is a line-ijair if abc + 2fg/i — af' — b(/'- — c/r = 0, that is, if P + 2(/t + 1 y - U{k + 1 )' = 0, which gives simply '3k + 2 = 0. Hence the asymptotes ;u-e, as before, .?■- +y- + z' — yz — zx — xy = 0, which may be written (x + o)y + to-z)(x + dry + wz) = 0, where oj is an imaginary cube root of unity. If then the conic be not a parabola, it has a })air of asymptotes, real or imaginary, u, ii ; and the equation of these is ' cr , ? .> ^ uu = i' + Ics- = when k is properly determined. Hence F ~ llu — ks^. 1 04 METRIC PROPERTIES OF CURVES ; When this i.s expressed in Cartesians, the term ka^ is a constant; hence, as is known, the equation of a conic, F=0, differs from the equation of its asymptotes, au' = 0, only by a constant. 107. Similarly for curves of higher order. Let F=0 be a cubic, havino- its three points at infinity distinct ; there are therefore three asymptotes, ^i = 0, ^-2 = 0, ^3 = 0. By the principle of § Gl a cubic toucliing these three lines on the line s — has an e(|uation of the form where I is a linear function. In passing to Cartesians, s be- comes a constant, and the equation is F= tyt.2t^-\-2i linear function = 0. Hence the ec^uation of the cubic F—0 differs from the equation of its asymptotes t^.jt-^^^ only in the terms of degree > 1. Moreover, f^, t.^, t.^ meet the cubic again where ^ = 0; that is, the (finite) points in which the asymptotes of a cubic cut the curve lie on a straight line ^ = 0. (Compare Ex. 3 at the end of Ch. V.) A quartic with all four points at infinity distinct has for its equation F=t^tj^^t^ + s^u.;^ = 0, showing that the intersections of the curve and its asymptotes lie on a conic tt., = ; and that the Cartesian e(|uation of the ([uartic differs from that of its asynq^totes only in the terms of degree :^ 2. And in general i^„ = f/./.5...f„ + .s'-'tt,i.2 = is the equation of a curve of order n with n distinct points at infinity ; hence the remaining intersections of the curve and its asymptotes lie on a curve of order n — l, Uh-2 = 0; and the Cartesian equations of the curve and its asymptotes are alike as to the terms of the two highest degrees. 108. From the general principles here used (§^ (J I, 98, 107) the ordinary rules for the determination of asynq:)totes in Cartesian coordinates ^()ll()^\■ at once. Let the triangle of reference be that made by the Cartesian axes, a; = 0, y = 0, and the line infinity, s = 0. Let the equation be arranged according to powers of z: denote tlie terms of degree n in X, y by u,i, etc. Then the ecpiation of tlie curve F=() is Un + z lln - 1 + z-U;, . 2 4- . . . + s" tto = 0. Since Un is a homogeneous expression in x, y, it splits up into n factors, l-^, I.,, ...,1^; now the lines joining xy to the intersections of F={) and z = {) are given by ^l,^ = {)■, that is, by ^, = 0, l^^O,...,l„ = 0. THE LINE INFINITY. lOo I. Let l^ be a non-repeated factor of m„, and let the point l-^^z be L^. There is therefore through L^ an asymptote, wliose equation is 1-^ + /ulz = 0, where jul is to be determined so that the line shall be a tangent with its point of contact on z — 0. The equation ^=0 must therefore be of the form {l, + fxz)V-\-z^W=0. (a) If now l-^ be a factor in u,i-i, the equation is at once expressible in this form; for u,i, and itjt-i contain /^ and the remaining terms contain z^ ; hence the equation of the curve is l,V+zW^O, and consequently the line 1^ = is itself the asymptote. Hence the rule :— A non-repeated factor of u„, if also a factor of %,i_i, gives an asymptote when equated to zero. (6) If however l-^ be not a factor in Un-\, jj. must be deter- mined by substitution, as in the ordinary Cartesian process. II. But if any factor in u,,. be repeated, I suppose, so that Un is l^v,^_ 2, the equation is F=l^Vn-2 + ZUn-l-i-Z^'Un--i-^----\-Z''Uo^O. {a) If I be not a factor in Un~i, this equation is of the form lW+zW=0, showing that z = is the tangent at the point Iz, that is, the line infinity is itself the tangent. Hence a repeated factor of iin, which is not a factor of Un-i, indicates contact with the line infinity in the direction determined by equating the factor to zero. (b) If I be a factor in itu-i, the equation is of tlie form lW+zlU+z'W = 0, showing that there is a double point at Iz. There are therefore two lines through Iz to be determined, 1-\-iul^z = 0, l + luL^z — O; and these are to be determined so as to have three points in common with the curve. Thus the rules for determining asymptotes in Cartesians are : — (i.) Any non-i'epeated factor I of iv^ indicates an asymptote 1-\-/ul = 0; this will pass through the orif/in C, (i.e. iU = 0) if I he a factor in Un-\- (ii.) Any repeated factor I of Un, if not a factor of «,i_i, indicates contact luith infinity in the direction 1 = 0. (iii.) Any repeated factor I of u,i, if a factor of u„-i 106 MRTPvIC PROPERTIES OF CURVES; indicates a double 'point at injinity in the direction 1 = 0; and there are two parallel asyinp)totes to be determined. Similarly a three-fold factor in 26,^ is accounted for in dif- ferent ways aecordino- as it occurs or not in u„_i, w.,1-2; and in general tlie explanation is more easily seen when the equation is thrown into the homogeneous form. Diameters and Centre of a Conic. 109. From a line and a conic a particular point is derived, viz., the pole of the line with respect to the conic. We have therefore to consider with respect to a conic a special point, the pole of the line infinity ; and in connection with this, the polars of all points at infinity, these polars being- concurrent in the special point. Let P be any point at infinity, and consider chords KK' passing throiigh P. If any such chord meet the polar of P in P', we know that KK' is harmonically divided in PP'. Hence P being at infinity, KK is bisected at P' ; that is, the locus of the bisections of parallel chords is a straight line, the polar of the point at infinity through which the chords pass. Any such line is called a diameter; and as diameters are the polars of points at infinity, that is of collinear points, all diameters are concurrent in R, the pole of the line infinity. Here two cases must be distinguished according as the line infinity (1) is, (2) is not, a tangent to the conic. (1) R is on the line infinity; all diameters of a parabola are parallel. (2) B is at a finite distance; all diameters of an ellipse or hyperbola meet in a point R. The detining property of diameters shows that every chord through R is bisected there ; the point R is called the centre of the conic. Thus the classification of conies based on their jiosition with regard to the line infinity is exactly indicated by the terms central (ellipse and hyperbola) and non-c(Mitral (para- bola). 'Vo detei'mine the centre ol' a conic it is necessary simply to find the pole of the line iiilinity. If lliis line be ao^ + />„v/ + (V- = 0, comparison with the eipiation of the \)()\;iv of x', y' , z' with respect to the conic ax~ + by- + cz'^ + 'Ifyz + '20 C(, hence the centre is uniquely determined. For the transition to Cartesians, the Hne infinity is 5 = 0, that is, .x + i) .y + z = 0: hence a^ = 0, ^o ~ '^ • ^^^^^^ ^he equations found become ax + hy-\-g = 0, hx + hy+f=0, the ordinary Cartesian equations for the centre of a conic. 110. The conjugate relation of points (or lines) leads to the theory of conjugate diameters. In the general theorem (§ 73) "if the polar of P pass through Q, the polai- of Q passes through P," let P, Q be at infinity ; their polars j), <1 become diameters intersecting in R ; since p passes through Q, it is the line RQ, and q is the line RP ; hence RP bisects chords parallel to RQ, and RQ bisects chords parallel to RP. Thus the diameters are arranged in pairs of covj agate cUametera (Fig. 27). Let p (I.e. RQ) meet the conic in Z, Z\ then the tangents at Z, Z' pass through P, that is, they are parallel to the chords bisected. Thus all the ordinary properties of conjugate diameters follow from the theory of poles and polars. 111. RPQ is a self -conjugate triangle: hence taking it as triangle of reference, the conic is \'>j?--\-qy'^-\-vz^ = K), where s = is the line PQ, that is, the line infinity. The transition to Cartesians is therefore accomplished by writing 5; ~ 1 : the equation becomes 108 METRIC PROPERTIES OF CURVES ; hence the equation of a central conic referred to any pair of conjugate diameters as Cartesian axes is p.icr -\- qy- = constant. Now the two central conies, ellipse and hyperbola, are discriminated by means of the points at inlinity ; these are given by that is, by ^jj;^ + 52/^ = 0. They are therefore imaginary, that is, the conic is an ellipse, if p and q have the same sign ; and the conic is a hyperbola if p and q have opposite signs. Hence by taking any pair of conjugate diameters of an ellipse as axes the equation is reduced to a form which shows that all diameters meet the ellipse in real points ; and by taking any pair of conjugate diameters of a hyperbola as axes, the equation becomes 9 9 showing that conjugate diameters meet the curve, one in real points, one in imaginary points. Tliis reduction of the ecjuation, depending on the point R, is inapplicable to the parabola. But in this case taking for x = any tangent, and for ^ = the line joining the point of contact of this tangent to the point of contact of the line infinity {i.e. of 2 = 0), the equation becomes y- = kxz : hence the Cartesian equation of the parabola, with any diameter and the tangent at its vertex as axes, is y'^=2}x. Examples. 1. An imaginary conic can have real points in number 4, 2, or 0. 2. Write down the equation of (i.) an imaginary ellipse ; (ii.) an imaginary parabola; (iii.) an imaginary hyperbola. 3. Determine the locus of the centre of a conic through four points {i.e. the centre-locus of a pencil of conies). THE LINE INFINITY, 109 4. Determine the centre-locus of a range of conies. 5. Show that to be given the centre of a conic amounts to two conditions. 6. From the general condition for conjugate lines deduce the ordinary Cartesian condition for conjugate diameters, the centre being origin. - 7. Show that in the case of a pencil of conies through four real points, two of the three sets must necessarily be composed of hyperbolas. 8. Show how (a) the three line-pairs, (b) the two parabolas, present themselves in the pencil whose fundamental points are (1) imaginary, (2) real ; noticing especially the case of the points forming a parallelogram. CHAPTER VII. METRIC PROPERTIES OF CURVES; THE CIRCULAR POINTS. Tiuo Special Imaginary Points at Infinity. 112. P)efore cntei'ing on the discussion of properties of curves that depend on the fundamental identical relation in line coordinates, the investigation of the significance of this relation referred to in i:^ 29 must be given. In dealing with point coordinates it was found that although in general x, y, z are subject to a condition of inequality which in trilinears is ax + hy + cz =|= 0, yet there are points not conditioned by this, viz., all points lying on a certain straight line. Similarly ^, i], ^ arc subject to a condition of inequality which may be written e + 'r + ^' - 2>/^ COS A - m cos B - 2^,; cos C =1= ; but certainly one line exists not conditioned by this, viz., acc-\-hy-\-cz = 0, for which ^, >/, ^ — a, b, c. Note. (Tsiiig ;vi-e;ils tliis line is ./■ + ?/ + s = 0, wlieuce ^, >y, C= b b b values which similarly do not satisfy the corresponding condition of ine(iuality, a-^-' + h-iy- + c-(~ - '2hc cos A.i](- 2ca cos Ji.(^ - '2ab cos C.^y =|= 0. Since then at any rate one line exists not subject to the ordinary condition, the course that naturally suggests itself is to consider all the lines that escape this condition ; that is, the lines whose coordinates make f + rf + 1" - 2'/^ cos A - 2^^ cos B - 2^>i cos C = 0. This equation being of the second degree, all the lines now considered form a particular .system of the second class. But the expression on the left splits up into linear factors ; the system therefore degenei^ates into two of the first class, ajid the env(!l()pe is a pair of ]i()ints. Consequently THE CIRCULAR POINTS. HI the exceptional lines under consideration pass through one or other of two lixed points, w = 0, (»/ = (), where w, w' are the linear factors of the expression on the left. These factors being imaginary, the points are conjugate imaginary points ; the line joining them is real, and its coordinates, beino- given by f — cos G .}] — cos B . ^—0, — cos C . ^+ >] — cos A.^= 0, (see § 71, and Ex. 4, after S 74) are ^:t] : ^= sin A :sini? : sin C=a: h:c\ the line is therefore ax-\-hy-{-cz = i), the special line at in- finity ; the two imaginary points w, w are at infinity. Since only one real line can pass through an imaginary point, all lines through w, w, other than the line infinity, are imaginary: through any real point p two of these lines pass, viz., po}, and pw : and these are conjugate imaginary lines. Thus corresponding to the statement of § 27 relating to point coordinates, the conclusion here arrived at may be stated as follows : — In general the coordinates of a line are subject to a quadratic inequality, f + r +^-- ^4 cos A-1^^ cos B - 2c, J cos G H= 0, but there are lines at variance tvith this condition ; these are the totiditij of lines 'passing through one or other of two fixed imaginary points at infinity ; through every 2^oi')'it in the 'plane there pass tivo of these exceptioncd lines. Note. A remark analogous to that in the Note to § 27 may here be made ; but the result is better stated in a more general form witli reference to any quadratic expression in line coordinates, not necessarily a product of linear factors. If this expression be ^, taking an}' line p at random, this does not touch the conic ; that is, in general the coordinates of a line in the plane make <^=|=0 ; Iwt there ai'e lines for which ^ = 0, viz., the totality of lines touching the conic <^. The importance of this extension will appear in Chapter XII. llo. The coordinates of the two exceptional lines through any point p are obtained by combining the equations /) = 0, that is, A^+^,y + 1/^=0, and ^- + ir + i' - -']i cos A -2^^ cos B - 2^,j cos G = 0. Thus the two lines through G are obtained from the Inst equation combined with ^=0: that is, from ^=0, and ^•'-2^;/cos6'+,;-^ = () (1). Solving these, we have the coordinates of the lines ; if however their equations be required, either one is ^if + >/// = (), 112 METRIC PEOPERTIES OF CURVES ; where ^, >/, are determined from (1); the equation of the two is therefore found by writing;' i-n = y--^' m (1): hence it is x- + 2xy cos C + ^-^ = (2). As a special case, let C be a right angle ; equation (2) becomes x--}-y'' = {). Hence in rectangular Cartesians, the lines joining the origin to w, w are «- + 7/'- = 0. If Ave attempt to treat these exceptional lines as ordinary lines, and regard them as having direction, we are led to a paradoxical result. The angle made by a line y = hx with y — mx beino- 6, the ordinary formula gives tan 6= -, — , — If now y = kx be one of the lines x^ + y- — 0, so that /■ = ?', this becomes „ i — tn i(i — 7)i) tan 6 = r^——. — = A = i. 1 + im i — m Thus m does not appear in the expression for tan 0. If we interpret this result without considering the mean- ing of the symbols, it is tan = constant ; that is, the excep- tional element makes the same angle with
?' + r - ^H cos A - m cos B - 2^f] cos C ' Now the linear expressions co, w are by delinition the factors of i" + r + r - 2>/^ cos A - 2^^ cos 5 - ^n cos C ; hence the numerator and denominator of the fraction in- volved in the expression just written are identically the same. The product of the distances is therefore /•/.:', which does not involve the coordinates of the variable line, and is therefore an absolute constant. It is infinite, for a:, y, z referring to w, make ax-\-hy-\-cz = 'd\ hence k is infinite, as also h'. Condition of Perpendicularity. 115. The fact that there are two exceptional lines through any point shows that a pair of ordinary lines may have a special relation. For the ordinary lines and the excep- tional lines form two pairs ; and these may be harmonic. Thus, for example, using rectangular Cartesians, the lines ax^-\-2hxy + by" = are harmonic with regaixl to the isotropic lines through their intersection, x^ + y^ = 0, if (§ 47) « + 6 = (): but this is the Cartesian condition of perpendicularity. We have used the conception of perpendicidarity at intervals in the preceding pages, chiefly in illustrative examples, but have not hitherto formulated this conception, nor given any systematic discussion of properties dependent on it. It now however presents itself naturally as expressing the special relation that a ])air of lines may hold with regard to the exceptional elements, and wo arc lead to the defini- tion :- — Lines harmonic ivitk respect to the isotropic lines through their intersection are said to he perpendicular. Considering the pencil formed by these two pairs of lines as cut by the line infinity, this may be otherwise stated : — Lines that divide uxa' harmonically are pcrpon- 114 METRIC PROPERTIES OF CURVES ; (licular. Or again, remembering that w, w' present them- selves as a degenerate conic, and recalling the harmonic properties of conjugate points and lines (§ 78) : — Lines conjugate with regard to ^2 _^ ^2 _f. ^2 _ 2,;^ cos A -2^^ cos B - 2|,; cos G=0 are said to be perpendicular. Hence the lines l^x + 'in{i/ + n-j^z = 0, l^x + m^y + n^z = 0, where the point coordinates are trilinears, that is, the lines l^, 7i\, Wj ; l^, Wo, 71.2 ' ^^6 perpendicular if IJ.^ + ')n■^m.-, + n^n.-, — (m^n.^ + wio'?? i) cos A — (n-J..2 + nd^) cos B — {({in.^ + ?2''"i) cos C'= 0. Ex. Find the condition that two lines whose equations are given in areals be perpendiculai'. Relation of a Conic to the Special Points, 116. Just as conies were differentiated by their relation to the line infinity, they may be further differentiated by their relation to to, co'. These being conjugate imaginary points, a real conic through one passes through the other also. The principal distinction is therefore that the conic (1) passes through «, w \ (2) does not pass through co, w' ; but since there are two jiairs of points at infinity to be considered (viz., m, w, and the intersections of the line infinity and the conic) taking into account the special rela- tion that may be held by these two pairs, we have to consider that the conic may divide wco' harmonically. The Circle. 117. Since the points oo, w' are imaginary, a conic througli ftj, ft)' is an ellipse, by the broad classKication of § lOo. It is a special ellipse, and we are now considering sub-divisions. Any conic througli w, w' has a line equation of the form wlicre p = is the pole of the line cckd' (that is, the line in- finity), and is therefore the centre of the conic. Substitut- ing for p, COO)' thei]' (expressions in terms of ^, >;, ^ this becomes {fi+9r] + hO^ = k{e-^rj^+^-'-2,j^cosA-2^icoHB-2i>}C08G). Here /, g, h are the point cooi'dinates of the centre p ; the THE CIECULAR POINTS. 115 equation can be made homogeneous in /, g, h by means of the identical relation a/+6ry + c/i = 2A; it is then ^^\fi+()ri+Hf ^7. {af+ bg + ch)%i^+ ri'+^^-^nt cos A - ^ cos B - 2^^, cos G) Writing ->•- for k, this expresses that the distance from the point /, g, li to the line ^, ?;, ^ is constant an; sin (7 - ^ sin Bf, these equations are seen to be - i+ >Kcos G-\-i sin C) + ^"^(cos B - i sin B) = 0, _ ^_l_ ,^(cos G-i sin C) + ^\cos B + / sin 7?) = ; that is, -i+rje'"^ +^e--^^ = 0, -i+^ie-'^+^e^' =0; hence the coordinates of the circvTkxr points are (_1, e'C e -'■«), (-1, e-^; e'^^ which ma}^ also be written in either of the forms 120. From the fact tliat all circles pass through the same two points at infinity, it follows that the ecjuation of a circle in point coordinates is of a special form. Let s = be the line infinity, and let S = be any one circle ; any conic through the intersections of S and s has an equation of the form S'^S + sv = 0, where v is linear ; hence the equation of any circle is S + sv = 0, where 8 = is any particular circle. Thus all that is neces- sary is to find the e(]uation of some one circle ; for example, the circumscribing circle. Any conic through A, B, C is fyz + gzx + hxy = ; this is a circle if it pass through the points w, w', whose coordinates were found in the last section. Writing the equation in the form X y z these conditions give -f+ge-'^+he'^ =0, -f+ge'^ +he-'^ = 0; therefore g(eJ^-e-'^) = /t(e'^ -e' '"), that is, g s\nC = /i sin B: whence f:g :/t = sin A : sin Ji :sin G=5 Ho); this line BC is in any case real (§ 51). If era-' be real, the points B, C are imaginary; but if crcr' be iniaginary, the pencil is as represented in Fig. 24 (6), but with the conies all circles ; the points B, G are real, and the circles are in two nests about B, G. Now the line-pair Bco, Bco' is a conic of the pencil, and is a circle ; but it is a degenerate circle : being degenerate in point coordinates, it must be regarded as an intinitely small circle (compare §§48, 117); it is the innermost circle of the nest about B ; similarly the nest about C is limited by the degenerate circle Gu), Geo'. These two points B, G, which are the limits of the system, are called the limiting 'points. (Poncelet. See Salmon's Conic Sections, §§ 109-112.) Since they are vertices of the self- conjugate triangle, the polar of B with regard to any circle of the system is GA, and the polar of G is AB; since A is at infinity on the radical axis v, this may be stated : — The polar of either limiting point with respect to any circle of the system is a line through the other limiting point, parallel to the radical axis. The Rectangular Hyperbola. 125. Let the points in which the conic meets infinity be harmonic with respect to w, cd' ; the conic, if real, is a hyperbola. For the lines joining G to w, w' are, by § 113, x^ + 2xy cos G + y^ = ; let the lines joining G to the points at infinity on the conic be ax^ -f- 'ifixy + hy^ = ; these pairs are harmonic if a-f-6-2/icos6' = 0, that is, if m cos'^O = {a -h hf, which gives 4(/t- - al)) = 4/^- sin-6' + {a - 6)-, hence a, b, h being real, k~ — ab is positive, and the points at infinity on the conic are real. The conic is therefore a hyperbola ; its asymptotes divide coco' harmonically, and are therefore at right angles, in consequence of w liicii the cur\e is called a rectangular hyperbola. To be told that a conic is a rectangular hyperbola is to be given one condition; for one pair of conjugate points. o), to)', is given. 120 METEIC PROPERTIES OF CURVES ; 126. In general, the condition for a rectangular hyperbola is found by expressing that the lines joining any point to the points at infinity on the conic are at right angles. For example, the coordinates being trilinears, is a rectangular hyperbola if For the line infinity is ax-\-hy-\-cz = 0; the elimination of z gives the lines joining C to the points in which the conic cuts the line infinity. Expressing that these lines «2(c2/'+ d%) + 'Ixyahh + y\c^g + hHt) = are harmonic with respect to a;2 + 2^;7/cos6'+2/- = 0, the condition is found to be C-/+ a^h + C7/ + Jj-'h - 2abh cos C = 0, that is, /+(y4-/i = (). Hence any conic through the intersections of two rect- angular liyperbolas is a rectangular hyperbola. For taking the connnon self -conjugate triangle as triangle of reference, the given hyperbolas are S'=fx'^+fjY+Iiz^ = 0, with the conditions f+!j + h = 0,f + g' + h' = 0. Hence for the general conic of the pencil, the sum of the coefficients, which is M./+// + /0 + /\'(./"+//-f/0, =0. In this pencil of conies three line-pairs are included; and since all the conies of the pencil are rectangular hyperbolas, each pair must be composed of lines at right angles. Hence calling the four points F, Q, R, S, it is seen tliat FQ is perpendicular to RS, PR is perpendicular to QS, RS is perpendicular to QR, (Fig. 30): considering tlie triangle PQli, tliis sliows that lines througli the vertices pei'pendicular to tlie opposite sides are concurrent in S ; S is the orthocentre of the triangle RQR ; and similarly any other of the four points is the orthocentre of the triangle THE CIKCULAE POINTS. 121 foniied by the remainino- three. Here we have proved two theorems : — (i.) The Knes through the vertices of a triangle perpendic- ular to the opposite sides are concurrent ; (ii.) If a rectangular hyperbola circumscribe a triangle, it passes through the orthocentre. 127. The centre locus for any pencil is known to be a conic : if the pencil be the one just considered, this conic is of special interest. The line infinity is ax-{-hy + cz = 0. The pole of this with respect to 1 fx ay hz IS given by ' — = *^ = — • ^ "^ a b c Now/4-(y + /i = 0, hence the locus of the pole is "+''- + ^ = 0, X y z that is, ayz + bzx + cxy = 0, which is the circle circumscribing the triangle of reference. Fig. 30 shows that the three points A, B,G are the " centres" for the three line-pairs PQ, RS; PR, QS; PS, QR. Let the circle cut PQ again in A'. Since some particular conic of the a/ c" \ A/ /-^ / S^ ^ N$'\ c c Fio. 30. pencil has A' for its centre, A' must bisect PQ : similarly the bisections of the remaining five segments, B\ C on PR, QR ; A", B", G" on 8R, 8Q, SP, lie on the circle. Stating these relations with reference to the triangle PQR, we have the two theorems : — (i.) The bisections of the sides of a triangU', tlic feet of the perpendicidars from the vertices, and the bisections of the lines joining the vertices to the orthocentre (nine points in 122 METRIC PEOPERTIES OF CURVES; all) lie on a circle, called the Nine Points Circle (n.P.C.) of the triangle ; (ii.) If a rectangular hyperbola circumscribe a triangle, the locus of its centre is the n.p.c. Xote. All these theorems relate properly not to a triangle P(^li, Ijut to a jiartieular configuration of four jioints J'<^RiS', wliicli may be called an orthoceiitric (juadrangle ; this is determined by two pairs of perpendicu- lar lines. The theorems may be stated with reference to any one of the four triangles determined by the four points. ] 28. The equation of the n.p.c. of the triangle of reference is at once found from the fact that it passes through the bisections of the sides. These in trilinears are ((), y, -j etc., hence the circle being {Ix + my + nz){ax + hy + cz) — 2(ayz + hzx + cxy) = 0, I, m, n must satisfy m n\ a therefore the equations for I, m, ot are mca + nab — d^, Ihc -^nab — h", Ihc + mca =c^; these give I, m, n = cosA, cos 5, cos (7, and the N.p.c. is (x cos A+y cos B-^z cos G)(ax + by + cz)-2{ayz + bzx + cxy) — 0, that is, (x cos A+y cos B-i-z cos C}{x sin ^1 + y sin 7^ + - «in 0) — 2(yz^iuiA-\-zx^mB + xyH'u\C) = 0. Foci. 121). Kaving found that there ai-e certain exceptional lines, a (juestion naturally presents itself with regard to any line system, viz.. What exceptional lines are included in the system ? A curve being given, this regarded as an envelope gives a singly infinite system of lines : we have to determine how many of these are isotropic, and how they are situated. The class of the curve being 'ii, there are n tangents from co and n from to' ; these are all imaginary, but they are in conju- gate paii's, aiifl have therefore //. real intej-seetions, n(n — \) imaginaiy intersections. Though these exceptional line ele- ments behjnging to the system cannot be directly represented in the diagram, being imaginary, yet they can be exactly THE CIRCULAR POINTS. 123 indicated by means of their real intersections. Similar!}^ the exceptional point elements of a curve cannot be marked on the diagram, for being at infinity they are beyond the limits : but they can be indicated by means of the asymptotes. The points thus used to mark the exceptional line elements be- longing to a system are called foci ; in general a focus of a curve is the intersection of mi oo-tangent and an w' -tangent ; that is, of two isotropic tangents (Pllicker). It is not neces- sary that the tangents be conjugate, the name focus is properly applied to the n(7i — l) imaginary intersections as well as to the n real intersections ; but practicalh^ the former are very rarely taken into account (§§ 1*31, 270). 130. Let the curve be a conic ; the four isotropic tangents determine a complete (quadrilateral. This has one pair of real vertices, p, p, and two pairs of conjugate imaginary vertices, a, (t', to, w'. The lines pp', acr', coco' are real, and form a self- conjugate triangle ; the intersection of pp', aa, is conse- quently the pole of am', that is, is the centre of the conic, and pp, crar' are conjugate diameters. The harmonic pro- perties of the complete quadrilateral show that pp , era-' divide (aoci harmonically, and are therefore at right angles ; hence the conic has a pair of conjugate diameters at right angles ; these are called the axes. Note. If the conic be a j^arabola, the concepticjii of conjugate dia- meters is not applicable ; if it be a circle, every pair of conjugate diameters is harmonic with respect to ojoj', and therefore at right angles ; hence any paii' of conjugate diameters may be regarded as axes. These two conies are excluded from the present discussion. The real foci, p, p , are on one axis, and tlie liai-monic pro- perties of the quadrilateral show that pp is bisected at : the imaginary foci 5 18-i). 135. Ex. If the centre of the inscribed conic describe a fixed line, the foci describe a cubic circumscribing the triangle of reference. Using trilinears and the associated line system, the line equation of an inscribed conic is M+ffC^+'tiv=o (1). The pole with respect to this of the line ax + b7/ + cz = 0, that is, of a, b, c, is {gc + hh)^ + {ha +fc)-<} + (fb+ga)C = 0, that is, gc + hb, ha+fc, fh+ga. Writing the equation of the straight line described by the centre in the form lax- + mbi/ + ncz = 0, the sul)stitution of the coordinates just found gives the condition Ia{gc + hb) + mb{ha +fc) + nc{f}) +ga) = 0, th at is, -^{m + n) + %i + l)+ -{I + m) = 0, a ('. which may l)e written fl' +gm' + kn' = (2). If a jjuir of foci h& p, />', the line equation of the conic is (oto' = kpp', which, if p = A^ + /i,7/ + i'^. p' = X^ + iJLi]-\-v'(, is ^2 + T' + C--2'/Ceos.l-2C^cosiS-2£7/cosC-(A^ + /x»/ + vO(A'^' + /^S/ + i''0=0. Comparing this with (1), wliicli is the known lino equation for the conic, we find AA' = 1, jifj! = 1, I'v' = 1 ; that is, if one focus of the inscribed conic l)e A, /v., r, tlie other is 111. A /x V u V and - H 1-2 cos A = f. etc. ; THE CIRCULAR POINTS. 127 therefore, taking .-r, _■?/, z for either focxis, / . . / _3/^ + '^y^ cos J. + 2^ _ s2 ^ 2s:t7 cos B + A'- . x^ + 2^17/ cos C +^2 yz zx xy whence f:g: h = xiy"- + 2yz cos A + ^2) ; y[zi + 2s.-' cos B + .r^) : 3(.r2 + 2xy cos C+.y2). It has been found that /, g, h satisfy a linear relation (2) ; hence substituting these values for /, g, h^ the locus of any focus of the conic (that is, the locus of the foci) is found to be Vx{y''- + ^yz cos ^i + 3-) + ni!y{f- + ^zx cos B + .r-) + iilzix^ + 2.r_y cos C+y'^) = 0, a cubic circuniscribins' the triangle of reference. Examples. 1. Find (a) the point equation, (6) the line equation, of the circle with respect to which the triangle of reference is self -conjugate. 2. Hence show that the point equation of every circle can be thrown into the form x^ sin 2 1 + 2/2 sin 25 + z^ sin 2C' + {x sin A-\-y sin B-\-z sin G){lx + 'my + nz) = 0. 3. Show that the equation of the N.p.C. can be written x^sm2A+y'^H\\\'2.B +z^ sin2C'— 2{yz imiA+zx sin B+xynmC) — 0. 4. Find the line equation and the point equation of the inscribed circle. 5. Find the line equation of the circumscribing circle. 6. Show that the n.p.c, the circumscribing circle, and the circle with respect to which the triangle of reference is self- conjugate, have a common radical axis. 7. The three radical axes of three circles taken in pairs are concurrent. 8. The polars of a fixed point with regard to a coaxal system are concurrent. How are the poles of a fixed line arranged ? 9. State with regard to a confocal system the results corresponding to those in Ex. 8. 10. Every conic that passes through all the foci of a conic is a rectangular hyperbola. 11. Show how to determine the foci of a conic in trilinears. 12. What sort of (a) pencil, (6) range, is determined by two conies, these being both (1) circles, (2) parabolas, (3) rectangular hyperbolas i 128 METKIC PROPERTIES OF CURVES. 13. How many (1) circles, ('I) parabolas, (8) rectangular hyperbolas, must be contained in («) a pencil, (b) a range ? What is the effect, in every case, of there being more than this necessary number ? 14. If the circle circumscribing the triangle formed by three tangents to a conic pass through a focus, the conic is a parabola. 15. Find the equation of the line infinity if tlu^ (juadrangle 1, ±1, ±1, be orthocentric. Also the e(|uation if/, ±[j, ±]i, be orthocentric. 10. Find the envelope of the asymptotes of a rectangular hyperbola through three fixed points. CHAPTER YIII. UNICURSAL CURVES. TRACING OF CURVES. 136. The parametric expression of the coordinates of a point on a curve depends on the fact that considering the points of the curve as elements, the curve itself as the space, we are dealing with one-dimensional space, this being- selected out of the general two-dimensional space, the plane ; that is, the statement that the point lies on the curve de- stroys one degree of freedom. Thus for example in polar coordinates, r being given, the point is restricted to lie on a certain circle whose centre is the origin ; it is limited to this one-dimensional space ; and its position in this space is determined by one more coordinate, for example, the vec- torial angle 6. Thus the two general independent coordinates of a point are involved in the two statements " the point lies on a certain curve," " the point has a particular position on this curve " ; hence the homogeneous coordinates a\ y, z and these implied coordinates must be expressible in terms of one another. Must he expressible ; that is, theoretically : the two sets of quantities depend on one another, but it does not follow that this dependence can be actually expressed with any convenience or simplicity. 137. Now suppose that x, y, z are expressed in terms of the implied coordinates 0, ^i, where = constant limits the point to lie on the particular curve, and /x determines its position on that curve ; e.g. taking the above example, a; = 9' cos 0, y=^r^\nO, r = a limits the point to lie on a particular circle. If then the point be limited to a circle of radius a, whose centre is the origin, the coordinates of the point are x = a cos 6, y = a sin 6 ; they are expressed in terms of 6, the one coordinate required S.G. I 130 UNICUESAL CURVES. TRACING OF CURVES. in the one-dimensional space to which the point is limited ; the constraint by which the point is kept to the curve is exhibited by the tVn'm of the expressions of x, 7 in terms of e. In the more general case, the relations being x-.y.z^ F^{(l), fi) : /'o(<^, p.) : F.^^^, p.), so long as f/>, ju are susceptible of any arbitrary values, the point has two degrees of freedom, it can range all over the plane. But if a constant value, a, be given to (p, one degree of freedom is destroyed, the point is confined to the curve ^ = a, its position on that curve is determined by the one coordinate jm. ; thus the two coordinates of the point are involved in the form of the expressions for the coordinates, and the value of the parameter jo. ; x:y:s=f^(fi):f,{fx):fl{fx). The elimination of fx from these equations leaves the position of the point on the curve undetermined, but does not interfere with the fact that the point is on the curve; that is, the elimination of /x gives the equation of the curve. Thus, for example, the elimination of 6 from the equations x = aGOH 6, y = a sin 6, gives x^-\-y'^ = a^, the equa- tion of the circle to which the point is confined by the form of the expressions for x, y in terms of 6. Unicursal Curves. 138. We now consider exclusively the case when /^ /!„ /^ are rational integral algebraic expressions in ,u. Let the degree be not greater than n ; then X : y :z = aQitx''-\-a^iuL''-^+. . . +a„ : hy+ ... + h,,: ^oM''+ • • • +c„, or writing these in the homogeneous form, X = tto^" + (i^Xiul" -^+...+ rt„X", We thus discriminate a special family of curves, charac- terized Ijy the property that the coordinates of any point can be expressed rationally in terms of a single parameter fjL ; any such curve is said to be unicursal. Similarly a curve may be unicursal qua envelope. (Compare § 140). In considering the intersections of two curves, one of which is unicursal, the method explained in § 90 is appli- cal)](! : X, y, z for a point on the one curve are expressed in terms of fx ; for a point on the other curve they are connected by an equation F ,„{x, y> z) = 0. Combining these UNICURSAL CURVES. TRACING OF CURVES. 181 two, the intersections are determined by means of the vakies of jj. given by an equation of degree ran. The order of the unicursal curve is at once determined by means of an arbitrary line fx+gy + hz = (). The equation giving the values of /x at the common points is (fao+gh + ^^Co)m" + . . . + {fctn+U^'n + hc^i) = 0, which is of degree n. Hence an arbitrary line meets the curve in precisely n points, and the locus is of order n. Similarly if the coordinates of a line, and therefore also the equation of the line, involve an indeterminate /j., this entering in the 7?*'^ degree, the envelope is of class n, and is, by definition, unicursal. The converse does not hold ; it is not true in general that the coordinates of a point on a curve of order n are algebraic expressions of degree n in a single variable. We shall prove (§§ 141, 142) that for n = \ or 2 the curve is unicursal ; a single example will show that the general n-\c is not unicursal. 139. One property of unicursal curves is at once apparent: all the real points of any such curve are arranged in a single series. For the two independent coordinates x:y:z being rational integral algebraic functions of /x, we obtain all points by giving yu all values. Now a^, a^, etc. being real, aQ/x" + ...+«„ etc. cannot be imaginary for a real fi ; consequently the real ^'s from through x to give a series of points, ending where it began, that is, a single circuit. This may pass through infinity, but that does not interfere with the continuous description of the circuit by a point. Thus for example an ellipse or a hyperbola equally consists of a single circuit. As to imaginary /x's, 0,^^"+... +^(„ may possibly assume a real value for yu, = a + /3i, but then it assumes the same value for /x = a — /3?'; now there can only be a finite number of such pairs of values iu = a±^i that will make x:y:z real, and these will exist only under certain conditions; that is, there may be a finite number of real intersections of imaginary branches ; these are simply isolated points (acnodes, conjugate points) that cannot be included in the description of the curve by a real tracing- point. Hence the unicursal curve consists of a single cir- cuit; it is unipartite. Note. If any of the coefficients «q, a^, etc. be imaginary, ival values of X : y : z can be given only by ,'^peeial imaginary values of ^i ; tlie curve consists entirely of a finite numbei' of isolated points. 132 UNICURSAL CURVES. TRACING OF CURVES. Now the cubic —x^-\-x — y- = 0, which can easily be drawn by points obtained by taking a sufficient number of arbitrary values for x, is seen to consist of two parts, one included between .r — 1=0 and x = 0, the other between x -{-!=() and iutinity ; it is bipartite, and hence it is not unicursal. Thus the statement of § 138 that the general -Ji-ic is not unicursal is proved. Jjut further, a unipartite curve is not necessarily unicursal. The truth of this statement is made evident by a comparison of the cubic x^+x — y^ = (which, drawn by points, is at once seen to be unipartite) with the cubic — x^ + a; — ?/'^ = 0, just discussed and shown not to be unicursal. Writing in the homogeneous form, the two cubics to be compared are x^ + xz^ — y^z — 0, —x^ + xz'^ — y~z = 0. If x^-\-xz"—y^z = were unicursal, there would be expres- sions for X, y, z of the form x = aQjUL^ + c\\iu.^ + a^X V + ^^3^^ =/i > y = by + b,\p.^ + 6,X V + ^3^' =/2. z = cy + c^Am- + ^^2^ V + CgX^ =/3 ; and the elimination of X, /x from these would give the result x^x + z- — y^-z = 0. This is a purely algebraic statement as to the expressibility of certain connected quantities in a certain form, and has no concern with the reality or otherwise of the quantities involved. Hence writing x', s/iy', iz for x, y, z, we see that the elimination of X, /j. from would give x'^ + x'{iz')' — (s/iy')-(iz') = 0, that is - x^ + .r'c'- - y'^z' = 0, and thus the coordinates of a point on the curve —x'^-\-x — y'' = would be rational integral algebraic expressions involving a single parameter fx:\; this curve would therefore be uni- cursal. But it has Vjeen shown to be bipartite, and therefore not nnicui'sal : lieiice the cui-nc x-- + .r-y-' = 0, tiiougii unipartite, is nut unicursal. The term unipartite has reference simply to the appearance of the curve, and relates to the distinction between real and imaginary: the term unicursal relates to the algebraic law of being of the cui've, and d(ws not refer to the distinction between real and imaginary ; there is this much connection UNICUESAL CURVES. TRACING OF CURVES. 133 between the two, that if the unicur.sal curve have any real part other than isolated points (p. 131), it is composed of a single circuit ; the unicursal curve is unipartite. 140. If a curve be unicursal qua locus, it is also unicursal qua envelope. For the coordinates of a point arc given by and an adjacent point is f-^{fx + Ojj) : /'o(/x + Sfx) : f./fA. + o/a). Hence the line joining the two is X y that is, z h 0, = 0, X y z h U U f'l .n n and the coordinates of the tangent are therefore that is, they are rational integral algebraic expressions in JUL, and the curve, qua envelope, is unicursal. £x. Show that the degree to which [x occurs in the cooixlinate.s of the tangent cannot exceed 2(« — 1), and may fall short of this number; and that hence the class of a unicursal curve of order m is not greater than 2(m - 1 ), and the order of a unicursal curve of class n is not greater than 2(11-1). 141. Special examples of these principles are familiar. The -point f\ + kf.,, (Ji + hj.,, li^ + hli., lies on the line X y z [ = 0; /i r/i K j f-1 ih h I the \\\\Q f\-\-hf.^, .7i + ^''^i- I'l + f^'f'-j, I'asses through the point ^ n ^ =0; /i Ui K I f-2 0-1 K I the line ii-\-hv = 0, that is passes through the point ?t = 0, v = 0. Moreover, the coordinates of any point on the line joining l;34 UNICURSAL CURVES. TRACING OF CURVES. *i' Uv ^1 ^o ''■>' .V-" ^■' ^^^ ^^ expressed in the form x^ + lcx.^, etc., and similarfy for lines. Thus if 9^ = 1, the locus or envelope is necessarily unicursal. 142. Taking the case u = 2, the direct theorem is proved (for the general value of n) in § 1 38 ; viz., if the coordinates of a point (line) involve an indeterminate in the second degree, the locus of the point (envelope of the line) is a conic. This may be presented in a slightly different form, using equations instead of coordinates ; if the equation of a line involve an indeterminate in the second degree, the envelope of the line is a conic. The direct special proof is the follow- ing (Salmon, Conic Sections, p. 248). Consider the line luhi + 2yuu' -\-v = 0, where u, v, tv are linear functions of x, y, z : and find the point equation of the envelope, that is, the locus of the inter- section of consecutive lines. The consecutive line is (/x + ^/>i)% -f 2(/A + (5^)ty -h V = 0, whence (2/x^^ -f 8ijc)ii + 'ISixw = 0, and therefore ij.u -\-w = (): that is, the line fx-u, + 2/x'?y -|- v = has contact with its envelope on ij.'ii + w = Q. Eliminating ^u, the required envelope is found to be the conic iiv = If . Conversely, every conic is unicursal. For uv — W" is simply the equation referred to two tangents and their chord of con- tact. Hence the equation of any conic can be thrown into this form, and that in a doul)ly infinite number of ways. Writing jj. for , the ccjuation gives ij.v = iv. Hence x, y, z are determined by u — fxw = 0, -zy — /A y = : that is, by /,/■ -^I'y -\-l'z — fxi^ax +n'y -\-n"z) = 0, nx + n'y + n"z — ij.{mx -\- ni'y + m"s) = ; whence expressions for x, y, z are obtained of the form (I' - /j.n){n" — fxm") — {n — ixni){l" — ixn"), etc., whence x:y:z = a^ix- + a^p. + (t.^ : b^^iu'' + 6j/x + h, : c^^a- -)- c^fi -f c.,. The ordinary expressions for a point on an ellipse or hyper- bola by means of the eccentric angle can be exhibited in an algebraic form. For the ellipse a; = ttcosc/), i/ = 6sinr/i. Write 0=20, tan0 = m; UNICUESAL CURVES. TRACING OF CURVES. 135 ,, „^ l—tan^O 1— m^ then x = a cos W = a^--L — ??^ = c^i , ~9' 1 + tsin^d 1 -h m^ , . ^„ J 2 tan 6 -, 2m ■^ l+tan^0 1+m^ Examples. 1. Express the coordinates of a point on the hyperbola x = a sec (j), y = h tan 0, algebraically. 2. Express «, y, z in the equation as algebraic functions of a single variable. Deduce the eccentric angle expressions for the ellipse and hyperbola. 3. Find the line equation of the envelope of y = mx-\ Find also the point equation, and the coordinates of the point of contact. 4. Folding a leaf of a book so that the corner moves along an opposite edge, the crease will envelope a parabola. 5. Apply the process of § 188 to show that a conic meets a curve of order m in 2m points. Hence show that the general cubic is of class (J. 6. A line of constant length slides with its extremities on two fixed lines at right angles. Express the coordinates of the line rationally in terms of a single parameter, and hence show that the curve enveloped is of class 4 and order G. Find the line equation of this envelope. The Deficiency of a Curve. 143. It has been shown that a curve of assigned order (or class) may or may not be unicursal. A few wortls may now be added as to the conditions for this, though no proofs can be given for the general case, the (juestion belonging properly to the theory of Higher Plane Curves, where it is proved that the necessary and suthcient con- dition for a curve of assigned order to be unicursal is that it have its maximum number of double points. Thus, for example, a cubic can have one dp (double point), but not more; for if it had two, the line joining them would meet the cubic in four points. It is shown in § 101 that a quartic can have three dps ; if a quartic have four dps, A, B, G, D, take these four points and any other point P 13U UNICUESAL CURVES. TRACING OF CURVES. on the curve as determining points for a conic : this meets the quartic in exactly eight points. But it meets the quartic in two points at ^1, B, C, 1), and in one point at P, therefore in nine points, which is impossible. Consequently the assumption as to the possibility of four dps on a quartic is incorrect : a quartic cannot have more than three dps. We shall show by examples how to express algebraically in terms of a single variable the coordinates of a point on a cubic with one dp or on a quartic with three dps. Note that some of the dps may come together ; thus x'^ + x^y — y^ = ^ is a quartic having at the origin a triple point caused by the crossing of three branches, tangent to ^ = 0, x-\-y = 0, x — y — 0. Geometrically this is ecjuivalent to three crossings of branches, and therefore to three dps : and it uses up all the dps that the quartic can have, for a line joining the triple point to a double point would meet the quartic in five points. The number by which the actual number of dps (nodes or cusps, separate or in composition) possessed by a given curve falls short of the possible maximum for curves of that order is called the deficiency of the curve; and the theorem above stated without proof is that the necessary and sufficient condition for a curve to be unicursal is that the deficiency be zero. It is proved in § 140 that a curve if unicursal qua locus is unicursal qua envelope ; combining these two, the conclusion is : — // the 'point deficiency he zero, the line deficiency is also zero ; that is, if a curve have all the double points possible for a curve of that order, it has also all the double lines possible for a curve of that class. A^ote. Tliis does not inemi that the presence of the double jjoints causes the double lines to ajjpear, for the eifect is exactly the reverse ; a curve of any specified order with a double point has fewer double lines than a curve of that order without a double ]>oint. But the presence of the maximum number of double ]>oints on a curve of any specified order reduces the class to such an extent that the possible number of douljle lines is thereby diminished and made the same as the actual number. Similarly if the class be given the occurrence of the maximum nundjci- of double lines reduces the order to such an extent that the possible number of double ])oints is made the same as the actual uumbei'. Thus we have found liiat the non-singular cubic is of class (5 (Ex. 5, >$ 142) ; now a curve of order 6 can have a certain number of double ])oint.s, and a curve of class can have a certain number of double lines. But if the cubic be the one considered in i^ 68, Ex. 2, and again in i;; 09, that is, one with a cusp and an inflexion, the recijjrocal is of class 3. The cubic has jxnnt deficiency zero, for it has one dp, the cusp ; and its liiu; deficiency is also zero, for being only of class 3, it can have only one double line, and that ])resents itself as the inflexional taui^ent. UNICUKSAL CUKVES. TEACING OF CURVES. 137 This theorem is a particular case of the more general tlieorem that for any curve the point deficiency and the line deficiency are the same ; and this again is but a special case of a much more general theorem (§ 288). These theorems are here referred to, — though the proofs, depending on the theory of Higher Plane Curves, cannot be given, — for the sake of drawing attention to the import- ance of the conception of the deficiency, a characteristic number ^9 which applies equally * to the point system and the line system derived from the curve of order m and class n. Curves may be classified by order, class, and deficiency (m, n ; |9) ; and the special family of curves considered in § 138 and the following sections is the family p = 0.t For cubics p = or 1 ; for quartics, 2> = 0, 1, 2, or 3. 144. As an example, consider the cubic x^-\-y'^ — oxy = 0, which in the homogeneous form is x^ + y^ — dxyz = 0. This has a dp at xy ; take a line through xy, x = \y: of the three intersections of this and the cubic we know two, for two come at the dp. Hence the third must be given by a linear equation. The equation for intersections is {\^+l),f-2Xy^2^0, that is, y'-^ = 0, referring to the two intersections at xy, and (X3+l)7y-3Xs = 0. Hence for the variable intersection x:y = \-A, and 2/ :s = oA : A.^+1, therefore x\y\z — oA" : 3A : X'^ + 1 ; and returning to Cartesians, _J^ _ 3X ^~\^^v y x^+i" Ex. 1. Find the coordinates of the tangent in terms (if A; hence show that tliis cubic is of class 4, and find the reciprocal equation. From the fact that tlie line deficiency is tlie same as the point deficiency, show that this cubic must have three inflexional tangents. Ex. 2. Express tlie coordinates of a point on the qnartic by the same process ; determine the coordinates of the tangent, and hence find the class of this qnartic. * See Clifford, Synthetic Proof of Miquel's Thedreni ; Mathematical Papers, pp. 39-41. f Curves of deficiency are of "genre jo = 0," " Geschleclit jp = 0." 138 UNICUESAL CUEVES. TEACING OF CUEVES. As another example, consider the tricuspiclal qnartic, which by a proper choice of coordinates can be written 2/ V + sV- + jfly- - 2xhjz - 2xyh - 2xyz^ = 0, or more conveniently 1 1,1 2 2 2 ,, 3,2 yi ^l y2, ZX Xy Comparing this with the conic X-2 + F ^ + Z' - 2 YZ - 2ZX - 2Z F= 0, it is seen that the expression of A'^, Y, Z in terms of a single parameter /n gives the desired expression of x, y, z, by means of the relations 2 1 1 •^j 2/' 2^ — "Y' Y' Z This conic is (4X + F- Zf = X Y : and writing 2/xX = A^ + F— Z, we have ^(X+ F-^) = 2F, therefore X : Y : Z=l : fji^ -.{1- fx.y. Hence x:y:z—l:^: -.^ ^., pr (l-/x) that is, x:y:z = ju% 1 — /x)"^ : ( 1 ~ m)"^ : jur. E.'\ 3. Show that the tricuspidal quartic is of class 3, and tiiid its line equation. 145. The special process here applied to the qiiartic is not universally applicable. The general process is analogous to that adopted for the cubic, where a pencil of lines through the double point was used. In the case of the (piartic, a pencil of conies is required ; the three dps give three base points, and for the fourth any point whatever on the quartic may be taken; then of the eight intersections of the (juartic and a conic of the pencil, seven are known ; consequently there must be a linear equation giving the remaining intersection in terms of the parameter of the pencil. For the tricuspidal quartic, we take the fourth point adjacent to A, that is, we take the conies of the pencil to touch the tangent to the (juartic at A. This tangent is ?/ — 5 = 0; hence the equation of any conic is {y - z){ax +hy + c) + y~ = 0, with the conditions that x = must give yz = 0, since the conic is to go through B, G ; hence 6+1 = 0, c = (), and the conic is {y-z)(ax-y) + y~ = 0, that is, x{y — z) = \yz (1). UNICURSAL CURVES. TRACING OF CURVES. 139 The intersections of this and the quartic 1 1 1 2 :i 2 ^-h\+ .-- =0, x^ 2/ ^ y^ ^^ ^y which can be written X" y'z^ x\y zj are fviven by "^ ^—cr^- • '^—^~ = 0, x^y^z^ X yz that is, by a+\'')yz = 2x{y + z) (2); (1) and (2) give (1 + Xfyz = ^xy, ( 1 - Xfyz = ^xz, 1 1 1 therefore x:y:z = V\\-\Y{\^Xf (l-X^)2:4(l+X)■^:4(l-A)^ which can be reduced to the form first found by writing 1 — 2^ for X. Gurve-traciyuj in Homoyeneov.s Point Coordindtet^. 146. When we wish to trace a curve whose equation is given in Cartesian coordinates, we determine the points in which it meets the axes, and the shape at the origin it' tliis be a point on the curve, not because these points are specially im- portant, but because their determination involves less algebraic work than for any other points. Here one advantage of homogeneous coordinates shows itself; we have three lines instead of two, and we have three points, any one of which may lie on the curve ; hence the form of the equation gives more information than in Cartesians. On the other hand there is the disadvantage that homogeneous coordinates are not well adapted to the measurement of actual lengths, though this can be overcome by the process of § 46. Ex: 1. .r* — .r?/% — ?/%'-^ = 0. Since x — O gives y/%- = 0, the curve passes through /J and C. Every term of the equation is of the second degree in .v, z coni])int'il, therefore there is a dp at xs, i.e. at B ; and simihirly there is a dp at ('. Since // = gives .r* = 0, all four points on ^IC are at C ; and siniilarlv all four points on AB are at B. To determine the shape at C, we have to deal with points in the immediate vicinity of C, hence J\ >/ are infinitesimal ; z is finite, subject to changes which are infinitesimal and therefore negligible in com])arison with z, hence z may be regarded as a constant. Since no attempt is made to determine actual lengths in finding the shape of a curve at a point, we may take any value we please for this constant ; we there- 140 UNICURSAL CURVES. TRACING OF CURVES. fore write 2 = 1 ; and the terms that give the shape at Care, exactly as in Cartesians, those of lowest order in The ordinary i)rocess shows that these are x^-y"^, and that therefore the curve approximates to .r*-?/-^ = ; hence there are two branches .>;-±?/ = 0, foi-ming a tacnode (j^ 69). Similarly to find the shai)e at B, put y = l, and select terms from x* - xz - z^ = i). The two tangents at the origin are 2 = 0, x-\-z = 0; and the shape of the branch that touches s = is given by x^-z = 0. The shape of the other branch may be found by continuiug the expansion z=—x ; but we can see at once that there must be inflexional contact, for .r + 2=0 meets the curve in four points at B ; one is accounted for by cutting the other branch ; three are to be accounted for by contact with its own l)ranch. Moreover, z{x+z)=x^, hence 2 and x + z must have the same sign in the immediate neighbour- hood of B. Considering the curve as a whole, yh{x + z) = x\ hence the curve can be only in certain divisions of the plane, viz., those in which z and x + z have the same sign. Collecting these results in a diagram, shading parts of the plane in which the curve cannot lie (Fig. 33), and noting that the curve cannot cross any of the lines ^ = 0, »/ = 0, 2 = 0, .r + 2 = 0, except at /?, C, it is at once evident that the parts of the curve are joined as indicated by the dotted lines, the two ends P joining through inliiiity, and similarly for the two ends (^. If it be requircil to determine more exactly how this junction is etl'ected, the asymptotes must be f()un i, the four points in which the line y=-vz meets the curve are imaginarv. Fin. 34. The curve is therefore as represented in Fig. 34, where as in Fig. 33 the shaded portions are those that are obviously excluded by the form of the equation. For an accurate diagram, the lines s=— 4?/, j= ±.7'v8 must be inserted by the process of § 46. Ex. 3. The curve ;r(.r2 — 45-) = ?/(?/- -f-) is easily drawn when tlie lines .r^ — 42^ = 0, _y2 — 2^ = 0, are inserted. Drawing any line through A for _y — 2 = 0, ?/-f-2 = is known, for y' — z" = are harmonic with respect to yz = ; and drawing any line through B for x-2z = 0, ,r + 2x = is known. The point .r = 20, y = z {i.e. the point 2, 1, 1) has here been determined, and therefore all j^oints and lines can be marked in by § 46. The lines already drawn divide the plane into parts, the curve being excluded from the alternate divisions, since .v{x-2z){x + 2z) and y{y-z){y + z) must have the same sign. Writing the equation in the form z\^x-y) = x^-y^, that is, z\Ax ~y) = {x — y){x — ioy){x — M-y), we see that the curve meets 3 = in cue real point, and that the tangent 142 T^NTCURSAL CURVES. TRACING OF CURVES. there i.s x-y = ; and since xy is on the curve, writing „~ = 1 we find that the tangent at .vy is 4.r-y = 0, and that this is an inflexional tangent, for it meets the curve in three points at xy. For tlie sake of clearness, the line Ax-y = Q is not indicated in Fig. 35, liut all the other lines used are there shown. 1 17. If an equation be given in Cartesian coordinates, it can be made homogeneous by using the line infinity for the third side of the triangle of reference. Thiis, for example, ,f3=_y'- can be discussed under the form ;i form whicli e.\iiil)its tlie inflexion at induity on the line y — O, having .i=0, that is, the line infinity, for its tangent. But the distinct Cartesian equation written in the homo ^yz- and is therefore the same, with ?/, z interchanged. The cusp is now at iidinity, and the line infinity is the tangent there. And finally the third (squation written in tlic homogeneous form being xy^=z^, is still essentially the .same, though now both cusp and inflexion are at infinity, the tangents being re.spectively y = and .r = 0. Similarly Ex. 1, in § 146, gives information regarding ( 1 ) .;•' - .r/ - y"- = ; (2) .?/» - xy - ;/2 = ; (3) 1 - xy^ - x-y"- = 0. UNICURSAL CURVES. TRACING OF CURVES. 148 Examples. 1. Draw the following curves, and in every case give the three Cartesian equations derived from the homogeneous form. Draw these curv^es referred to Cartesian axes. (1) x^ + xhf-y-'z-=^Q. (2) x^-xhj -yh-^0. (3) x^ — x'^z —y^^ =^- 2. Express in terms of a single parameter the coordinates of a point on the following curves. (1) y- — x^ — xy'^. (2) x^-xy-f- = 0. (3) l~xy^-xhf = 0. (4) y' = x^4.xHf. (5) f = x^{x-r). 3. By means of the expressions for x, y in terms of a single parameter, draw the curves in Ex. 2 by points. 4. Determine the coordinates of the tangent to the curves in Ex. 2 ; hence find the class of these curves. 5. Find the equation of the reciprocal to (5) in Ex. 2 : draw the curve either from its equation or from the expression of the coordinates in terms of a single parameter. CHAPTER IX. CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. Projection. 148. In the preceding chapters on properties of curves an algebraic classification has presented itself, depending on whether the algebraic work involved has or has not reference to the actual values of the coordinates, the properties being in the two cases metric and descriptive. The geometrical significance of this classification is now to be considered ; it is found by means of the theory of Projection. 149. Given any plane figure, take any point V not in the plane ; draw lines from V to all points of the given figure, and cut the conical surface so obtained by a second plane. To a point F in the first figure corresponds in the second the point P' determined by the cutting plane and the line VP ; to a line AB in the first figure corresponds the line A'B' determined by the cutting plane and the plane VAB ; collinear points A, B, G give collinear points A', B', C ; concurrent lines a, h, c give concurrent lines a, h', c ; a curve cut by a straight line in m points gives a curve cut by a straight line in m points : and so on. Thus the second figure has a certain general resemblance to the first figure. l.")(). The conical .surface obtained by joining the points of the given figure to V is called by the Germans the " Schein " of the figure (Reye, Geometrie der Lage) ; the second figure, the section of the " Schein " by the second plane, is what we call the " Projection " of the first figure ; in German this is often called the " Schnitt " ; the point V is the centre or vertex of projection. Thus a figure has any number of projections, since any point can be taken as vertex, and from this the figure can be projected on to any plane. It is at once evident that these projections may differ considerably in appearance from one another and CEOSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 145 from the original ; though the persistence of properties of collinearity and concurrence shows that a certain general resemblance will be maintained. Geometrical properties are therefore divided into projective and non-projective according as they remain unaltered or not by projection. We proceed to investigate the nature of the properties that fall under these headings. Alteration of Magnitudes by Projection. 151. It is at once apparent that lengths of lines and magnitudes of angles (Figs. Si), 37) can be altered by pro- jection ; and yet there must be some connection, for the one tigure does depend on the otlier. The first thing to notice is that any segment XA can be projected so as to have any desired length. From X draw any line not in the plane (1), measure on this the desired length A'' J.'; take any point V on the line ^yl', and project from V on to any plane (2) containing XA'. The projections of A'', A being X, A', the segment XA becomes XA\ which is of the desired length. Note. The planes (1), (2) are not really needed here ; the whole con- struction is in the one plane that ct)ntains VXAA' ; in general, when tlie points to be projected are all in one straight line AB, the plane VAB contains the whole figure. Evidently the degree of choice here allowed enables us to project XA, AB, two contiguous segments on a line, so as to have desired lengths. For if XA', A'B' be taken of the desired lengths, the point V is the intersection of A A', BE. This shows that not only are the lengths of lines altered by projection, but also the ratio of these lengths is altered, for it can be made to assume any desired value ; and we have to determine what it is that controls the alteration in this ratio, this being evidently not the same for all pairs of lines. Instead of measuring XAB, XA'B' from tlie. point of intersection of the lines AB, A'B', the more general question will be considered, as to the relation of AB : BC to A'B' : B'C (Fig. :56 («)). The whole diagram is in one plane, as represented in Fig. 36 (6). Draw AM, ON, A'M', C"N' perpendicidar to VB; then AB:BG=AM: NO, A'B' : B'C = A'M' : N'C : therefore AB A/B'_AM A'M' _ AM NO _VA VC_VA VA' BC '■ B'C ~ NG • X'C~A'3r ' N'C~ VA' '' VC~ VC " VC ' 146 CROSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. Thus the relation of the two ratios can be expressed in terms of the distances from V to A, C, A', C. If now a fourth point D be taken, also on the line AC, and VD be produced to meet A'C in D' , AD A'n'_VA VA/ ■ ])'C'~ VG ■ VG" A'B'_AD A'jy DG ■ D'C" therefore that is, DG AB BG AB BG B'G' AD DG A/B' A/D' 'bv'-d'G'- Thus although the relative position of three points on a line can be altered by projection to any desired extent, the position of any fourth point D is reoulated by the fact that .,., : ..,, is unalterable by projection. This is a combination of lengths AB, etc. and therefore depends on metric (pian- tities : but it is a projective mietric co^nbination. 152. Similarity the angles determined by concurrent lines a, b, c, can be altered to a certain extent, viz., the two angles ab, be can be made to assume any desired magnitudes in the projection : but then tlie position of any fourth line d is determined. For let (I, b, c, d meet in : their projections meet in 0', the prr)jection of 0. Take any transversal ABGD in the CROSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. 147 first plane: its projection gives a transversal A'B'C'D' in the second plane (Fig. o7). We know that AB AD_A^' A'D' BC' DG~ B'C'- DV Fic. 37. Let h be the perpendicular from to ABCD. Then as in § 3G OA.OB.H\nah = AB.li, etc. sin a?> iAnad_AB AD ■ M • DC whence Therefore sin he sin dc sin ah sin ad is unalterable by projection : it is a sin he ' sin dx py'ojective raetric coinhmation. 153. It appears now that all descriptive propertief^ and some metric properties are projective. Projective metric properties are those that depend on the combinations just considered. These combniations -g-^, : -^^, ^^^ : ^j^^^ are what were defined in §§ 85, 3G as the cross- ratios of the range and of the pencil : the importance of the conception of cross-ratio consists in what has just been proved, viz., that cross-ratio is unalterable by projection.^ Theorems stated with reference to cross-ratio are metric theorems stated in projective form. 148 CROSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. Compare the remark in § 40, ' Since I : m does not appear in the result, it is immaterial what values the multipliers may have'; that is, cross- ratio does not depend on the nature of the coordinates. Hence the difference in the properties of curves, tir.st noticed in the algebraic work, is the difference between projective and non-projective : and in determining to which division to assign a property that does not appear to be purely descriptive, it is necessary to consider it from the point of view of cross- ratio. Note. Thouyh the conception of cross-ratio is liei'e defined by means of meti'ic quantities, which are combined in such a way that the metric quality is eliminated, the conception itself is descriptive, not metric, and in |)ure geometrical reasoning it ought to be defined accordingly. See Von Staudt, Beitrilge zur Geometrie dcr Lage, pp. 131 etc. ; 1856-1860. The Group of Six Cross-ratios, Algebraically considered. 154. Since the four elements considered may be taken in any order, they afford a number of cross-ratios. It was shown in ^ 87 that the 24 different orders give G different cross-ratios, viz., , — ,, etc., where I -\-rn -i-v =0 : that 7? t is, the different cross-ratios are /■ -A_ 1 1 k: ' 1-/;' where the product of members of a column is -f-l, and the product of members of a row is —1. One set of values is inadmissible, except in special cases, viz., /i;=l, 0, 00, for these values indicate the coincidence of some of the points considered. If for example AB AD_ AB_Ai) BC'- DC~ ' BC" DC" and therefore J) coincides with B. Again, at any rate one of the set of six, if real, is positive and less than luiity. For if k be negative, 1—/.; is positive; and if 1—/.-, being positive, be found greater than unity, its reciprocal which is also positive is less than unity. Let therefore /.' be one of the group that is positiv^c and less than unity. Tlio six values are now + <1, +>1, -, + >1, +<1, -. Tfence for a real value of /;, there cannot be equalities between members of the same row. CROSS-EATIO, HOMOGRAPHY, AND INVOLUTION. 149 But there can be equalities between members of the two rows. It" then /i'= 1 —h, h=h, and the scheme of values is i 2 - ] accounting for all possible equalities when the cross-ratios are real. The fact that the value — 1 is included in the scheme shows that the division is harmonic. For AB AD__ BC '■ I)C~ shows that AB : BC= -AD : DC. If now we wish k's Irom the same row to be equal, the cross-ratios must be imaginary. Let k = - — J, therefore /,;-— /.;-f 1 = 0, that is, /.: is either J- — ru imaginary cube root of —1; hence k=—w, where w is an imaginary cube root of -|-1. The scheme now becomes — CO, —0), —w, 2 2 2 — (jO , — CO , —CO . The arrangement of points that gives this scheme of cross- ratios is called equianharmonic. 155. If the four points be given by a quartic equation, it is not possible to distinguish among the six cross-ratios, for there is no way of specifying any order among the points. Hence any equation found for one of the cross-ratios will give all the others, that is, the cross-ratios will be given by a sextic equation. This sextic is most easily constructed when the conditions that the points be (i.) harmonic, (ii.) eciuianliar- monic, are known. (i.) It was shown in § 47 that the pairs of points given by ax" -{- 2hx + c = (), ax- + Ih'x -h o' = 0, are harmonic if ac'-\-(t'c — 2biy = 0. Hence the points given by a^fc* -f 4, — w"-)'^. Hence some value of the one quantity involved nuist throw (3) into the form {(0 + l)(0-2)( 0-^)P = O, that is, into {(0+ l)(0-2)(20- 1 )}- = () ; and some other value must give the form {(0 + «X0 + co-)}=^=(); hence (3) must be reducible to A{(0 + l)(0-2)(20-l)}-^ + /x{(0 + aO(0 + .r)}^-O....(4). Qomparing coefficients, we find 4X + ;x = l, -3A + 6^ = r7, hence X and /x are linearly determinable, and the form (4) 152 CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. may be adopted instead of (3). We have now to determine X, ij. so that the group of cross-ratios given by (4) shall be the group belonging to the quartic a^x^ + ^a^x^ + 6a.2«^ + ^a^x + a^ = 0. Since ix = gives the harmonic group, the v^anishing of ijl must imply ^'3 = 0, and nothing else; hence ijL = inig.J. Again, \ = gives the equianharmonic group, hence the vanishing of A nuist imply //2~^' ^^^'^ nothing else; therefore \ = lg.^. Now (/g is of degree 3, g.2^ of degree 2 in the coefficients ; equation (4) must be homogeneous in the coefficients cIq, a-^, etc., therefore 3j = 2/; hence i = Sh, j — '2h, and writing for the numerical nniltiplier — -. the single letter J/, equation (4) becomes where (/.^ = a^a^ — '^a^a^ + '^a.f', To determine h and M, take a special quartic, having roots 0, 1, l-l', 00. One cross-ratio is ^ — z — : — ^ — , that is, /.;. 0—1 0— X The quartic is x{x — \){x — 1 + /.) = 0, that is, . x^ + .^;3 - ( 2 - /.•>■- + ( 1 - k)x = 0. Writing this as . «■' + 12^j3 _ 1 2(2 - /.•><;- + 1 2(1 - h)x = 0, we have ctQ = 0, ' = l'\ hence, identically, 123/<(/,-^_/,+ l):i/'{(/,+ j)(/,_2)(2/.--l)}--^ = il/.■t-''{(/.;+l)(/,•-2)(2/■-l);-M(/• + fo)(/.■ + a)-)}^ that is, 123/'(/,2_/,.+ l)3/.|(/,+ l)(/,_2)(0/,_l)|-2 ^J/.42A((/,+ l)(/,-2)(2A;-l)}2V.'--/.' + lf ; therefore A = 1 , \-I^ = M x 42, whence ^1/= 4 x 27, CEOSS-EATIO, HOMOGEAPHY, AND INVOLUTION. 153 and consequently the cross-ratio sextic (5) is ^/{(0+l)(0-2X20-l)}- = 4x27. that is, i//{(0 + l)(96-2)(0-i)p = 27^3-M(0 + co)(0 + c«-O}^. The Group of Six Cross-ratios, Geometrically considered. 157. The twenty-four orders for the four letters A, B, G, D were obtained in § 37 by associating the points in pairs AB, CD; AG, BD; AD, BG, thus obtaining three groups of eight ; the eight different orders in a group were found to give two different cross-ratios, these being reciprocal, and thus six values were obtained. The same result might be arrived at by considering (1) the arrangements in which A stands first, six in number ; (2), (3), (4), similar groups in which B, G, D occupy the first place. The six members of the first group, viz., (1) ABGD, (2) ADBG, (3) AGDB, (ly ADGB, (2)' AGBD, (S)' ABI)G, give the six different values, corresponding to the scheme in § 154. Thus, in general, if we keep A in the first place, no chano^e in the order of the remaining letters is admissible, agreeing with the rule : — Any tiuo letters may he interchcmyed if at the same time the other two be interchanged. It may however happen that in special cases other interchanges are admissible. I. Suppose that (1) is unaltered by the interchange of G, D. This interchano-e rearrano-es the six orders as follows : — (o)', (2)', (ly, (o), (2), (1). Hence we must have (3)' = (1); and therefore also (1)' = (3), and hence (2)' = (2); thus the group is now the harmonic group, and the interchangeable letters determine one of the pair of segments with regard to which the harmonic relation holds. In the case supposed, G, D being interchangeable, the segments AB, GD are harmonic. II. Suppose that (f) is unaltered by a cyclic interchange of B, G, D, that is, [ABGD] = [ADBG] (i.). If then we interchange the letters occupying any two positions in the left-hand member of this, and make the same interchange on the right, the results will be the same ; 154 CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. therefore {ADCB} = {ACBD] (ii)., {ACDB} = {ABCD} (iii.), and {ABDG} = {ADCB} (iv.). From (i.) ami (lii.), {ABCB} = {ADBG} = {AC1)B}, from (ii.) ami (iv.), {ADCB} = {ACBD} = {ABDC} ; that is, (1)=(2)=(3), (l)' = (2y = (3)'. The group is now the equianharraonic group already con- sidered ; and as we have considered all possible interchanges of letters, there are no more special cases to consider. Thus the distinguishing characteristic of the harmonic arrangement is that a single interchange is permissible ; while for the equianharmonic arrangement a cyclic interchange of three letters is jjermissible. Sole. The symbol {A BCD] means properly the group of six cross- ratios ; now this group is the same in whatever order the points may be taken, and consequently {ABCD} and {ACBI)\ contain the same six values. But we do not write {ABCI)} = {ACBD}, unless the interchange of B, C is admissible, that is, unless the segments AB, BC Are harmonic. Homographic Ranges and Pencils. 158. The projection of a range has been shown to be an 'equal' range, the ranges being estimated quantitatively by cross-ratios. This second range can now be moved in space, so that the two are no longer in projective position ; but they remain equal. Instead of .speaking of the ranges as eqiud, which suggests metric determinations, they may be spoken of in all positions as projective, a term which has no metric reference ; a special term is then required for the ranges in projective position, and for this perspective is used. Thus })rojective I'anges in space of three dimensions are in per- spective when the joins of corresponding points are con- curi'ont ; and projective pencils are in perspective when the intersections of corresponding rays are coUinear. (From ^5 152, with the help of the fact that any line and its pro- jection meet on the line of intersection of tlie planes.) Considering the limiting case, when the two planes are coincident, and the point V lies on them, this gives us ranges and pencils in perspective in a plane ; Init the double idea here involved is simply what can be derived by the principle of duality from the idea of ranges in perspective in a plane. CROSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. 155 Thus there is no necessity to appeal to three-dimensional geometry for the idea of perspective, though a gain in clear- ness is thereby obtained. 159. The ranges must now be considered as comprising an indefinite number of points ; and from the fact that the two ranges are projective it follows that the cross-ratio of any four points in the one is equal to the cross-ratio of their correspondents in the other ; this is expressed by the notation {ABODE ...} =^ {A'B'C'D'E'...] ; another notation is frequently used, especially when the purely descriptive view of the subject is adopted, the ranges being regarded as projective, viz., ABODE . . . -A'B'O'D'E. . . . (Reye, Geometrie der Lagc.) Projective ranges are in perspective . if three joins of corre- sponding points be concurrent. For if AA', BB', 00' meet in V, the ranges are necessarily in one plane, the plane VAB \ and if DD' do not pass through V, let VD meet A'B' in D". Then, by projection from V, {ABOD\ = {A'B'0'D"]- and by the given relation of the ranges, [ABOD\ = [A'B'O'U } ; therefore [A'B'O'D") - [A'B'O'D)' , whence A 'D" : D"B' = A'D': D'B', that is, the segment A'B' is divided in the same ratio by the two points D' , D" ; hence D" is the same as D'. Thus the line joining any two correspondents D, D' passes through V. A particular case of this is the theorem : — If one pair of correspondents in two projective ranges co- incide, the joins of corresponding points are concurrent; and similarly, if one pair of corresponding rays in two projective pencils coincide, the intersections of corresponding rays are collinear. 160. Projective ranges or pencils are also called homo- graphic ; that is, tivo ranges or 'pencils are homogniphlc when the cross-ratio of any four elements of the one is equal to the corresponding cross-ratio formed from the other. And this definition applies also to the case of two different configurations; a range and a pencil can l)e lioino- graphic. 156 CROSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. But homography can be considered from an entirely different point of view. The one-dmiensional Jijj'ures we are concerned luith are homograijhic wJien there is a one- one correspondence hetiueen their elements. When we say that there is a (1, 1) correspondence between the elements of two configurations, we mean that there is a construction, geometrical or algebraic, by means of which an element of one configuration can be derived from an element of the other configuration ; and that this construction is of such a nature that to one element in the one configuration there corresponds one element in the other configuration. Thus, for example, there is a (1, 1) correspondence between two ranges in perspective, and the construction is by means of lines through V. But if we take a straight line and a conic in a plane, and a point V not on either locus, though the points of the two can be connected — made to correspond — by means of lines through V, the correspond- ence is not (1, 1); one point on the line now gives two points on the conic, while one point on the conic leads to one point on the line; the correspondence thus instituted between the straight line and the conic is therefore a (1, 2) correspondence. JVote. This does not assert that there cannot be a (1, 1) correspond- ence between the line and the conic ; such a correspondence can in fact Ije instituted, for example, by taking the point T on the conic. 1()I. 'iliis second definition of homography is now to be proved eijuivalent to the one first adopted. For simplicity, the proof is worded so as to apply to the case of two ranges ; but as it depends only on the determination of the elements of the configuration by a single coordinate, it can be at once applied to the other cases. Let the points of one range be given by their distances, X, from a fixed origin on the line ; and let y, 0' belong- to the second range. Then since there is a correspondence, X is a function of y, and y is a function of x. But x must be a direct function of y, for an inverse function (sin " hj, sfy, etc.) is not one-valued ; and // must be a direct function of x. The only possible way of satisfying these two conditions is to have a; = one linear function of y (li\ided by another; that is. x^^^'^'""', ry + s ,. , . , —sx-]rq. irom which y= •'; •^ rx—p CROSS-EATIO, HOMOGEAPHY, AND INVOLUTION. 157 hence the relation between x, y is of the form linear in x, y separately. Let x-^, x^, x^, x^ be four points of the first range, y^, y.„ y.^, y^ their correspondents. To show that ■^i~^ 2 . ^1 ~ -^4 . _ 2/1 ~ 2/2 . 2/1 ~ 2/4 ^•2~^'i ^i~^3 y2~yz 2/4~"2/,3 express (x.^^ — x.y)(x^ — x.^) in terms of y. We have Pl/i + q vv-i + q therefore ■^ r^i + .s' ry^ + s ^ ( VVi + ? )( ' 7/2 + ■->•)- ( / ni., + q )( ry^ + .9) (r2/i + s)(r2/2 + s) ^{ VS-qr){y^-y.^ {ry^ + 8){ry^-\-sy ix,-x^(x-x^ - (p'-^'f(y ^-y,)(y.-y^ (ry^ + s){ry., + s){;ry^ + s){7-y^ + *.•) ' and similarly (r - T vr -t\- (p^'-9'''y(y-2-y-:d(yi-y4) . ix, x,)^x, "^^^-^yy^ + sXry. + .Xry. + sXry. + sy dividing one by the other, (-^1 - x.2 ) {x, - X 3) ^ (2/1 - y 2 )(2/4 - ys ) . (a;^ - x^)(x^ - X,) (2/, - 2/3)(y 1 - 2/4) ' that is, (1, 1) correspondence of one-dimensional fujures invplies equality of corresponding cross-ratios. 162. The correspondents to any three points can be chosen arbitrarily ; but the correspondent to any other point is thereby determined. P^or from the first definition, {ABGD} = {A'B'CrB'}: hence if A', B', C be chosen arbitrarily to coi-respond to A, B, G, the correspondent to 1) is known. From the second definition, the two parameters x, y are connected by a relation axy + ^x -\-yy-\-S — 0; here there are three quantities to be determined, o : /3 : y : (5, and three sets of values for x, y determine tliese : hence any fourth y is known, in terms of x. 158 CROSS-EATIO, HOMOGRAPH Y, AND INVOLUTION. In the linear relation between x, y the origins for the two systems are any points 0, 0' ; these can be changed to any other points without altering the form of the rela- tion, though the values of a : /3 : y : (5 will be altered. Homographic Systems ivitli frte same Base. 108. Since the relative position of projective ranges can bo altered to any extent, the two lines on which they are marked can be supposed to coincide : and since either origin can l)e changed to any desired point, the same origin can be adopted for the two systems ; let the para- meters of the two systems be now denoted by x, x' ; these are connected by a relation axx +^x-\- yx +(5 = 0. At any point of the line there is a point of the first range, P, and also a point of the second range, Q' ; this can be expressed by saying that every point of the line is counted twice, once for each range. Let the point of the second range that corresponds to P be P' ; and similarly let Q in the first range correspond to Q' in the second. Let ()P = X. ci' , Bx + 8 ^ yx'+S bnice X = — , — , and x= — '--, — ^, ax-\ry ax -\-p therefore OP' = - ^±^, and 0Q= - ^^+4 : and thus P' and Q do not come together ; that is, the correspondent to any point of the line is ditt'erent accord- ing as the point is regarded as belonging to the first range or to the second. 164. A (piestion that naturally occurs is: — Can a point correspond to itself ? This recjuires x' = x; the ecpiation connecting x, x' becomes aa;2-f(/3-t-y)a' + (5 = 0, showing that there are two such points, real or imaginary ; these are the double points of the system. Wlicther these are real or imaginary, the point midway between them is real ; if this be taken as origin, ft + y = 0, and the homograpliic relation between the two systems is expressed by axx + ft{x — x') + = 0. CROSS-EATIO, HOMOGRAPH Y, AND INVOLUTION. 159 Calling the double points F^, Fc,, we have, since each corresponds to itself, {ABC. . F^F,} = {A'B'i". . . F,F.^. Similarly homographic pencils with a common vertex form a system with two double lines. Homogra'phic Sydems with different Bases. 1G5. From the second definition of homography it is at once evident that if u, v be elements of the same nature, as also u, v', the two systems u + \v, u' -\-\v' are -homo- graphic. (Compare ,^s; 40, 41, where it is shown that the cross-ratio of the configuration (u, v; u + kv, u-\-h'v) is /■://.) Hence the locus of the intersection of corresponding rays of two homographic pencils is obtained by eliminating X from the equations u + \v = 0, a' -\-\v' = 0; it is therefore the conic uv' — u'v=0 ; that is, the locus of the intersection of corresponding- rays of two homographic pencils is a conic through the vertices of the pencils. And by the same algebraic work, the envelope of the line joining corresponding points of two homographic ranges is a conic, touching the lines on which the ranges are marked. (Compare with Chasles' two fundamental properties, § 88.) 166. The purely geometrical proof i.s interesting-, for it is by niean.s of projective pencils and ranges in non-pi'ojective jiosition that conies are introdviced into desciiptive geometry. The intersections of corresponding i-ays of the pencils form a singly infinite series of points in a determinate order, that is, a curve ; to determine the order of this curve, consider any transversal ; this cuts the two ])encils in projective ranges, ABCD ..., A'B'C'J)' ... . Now a point in which the transversal meets the curve is an intersection of corresponding rays, and is therefore a point which coinciiles with its correspondent. Hence the transversal meets the curve in the double points of the system ABC ... A'B'C ... \ that is, in two points. The locus is therefore of the second order. (Reye, Oeometrie der Lage.) Involution. 1G7. The correspondence considered in .^i^ 168, 1G4- is between the points on a line taken in one way and the points on that lino taken in another way ; it is not strictly a correspondence between points of the line. For a point A, regarded as P, gives a certain correspondent P' ; regarded as Q', this same point A gives a ditibrent corrc- 1(50 CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. spoiulent Q. A (1, 1) correspondence between points of the line would associate them in pairs AA\ so that A corresponds to A', and xl' to A ; luhen the elements of a one-dimensional sj^ace are thus associated in pairs of correspondents, they are said, to form an Involution ; and any finite number of these pairs of elements are spoken of as being in involution. Counting all the points of the line twice, as is done when two homographic ranges are marked on one line, the arrangement just defined as an involution appears to be a special case of homography ; it is a (1, I) correspond- ence between the two linear aggregates of points, the ordinary equation expressing the homographic relation, viz., axx-\-l3x-\-YX+S = (1), being specialized so that the correspondent to a point A is the same whether A is regarded as P or as Q'. Hence (§ 163) ^X + S ^^^^^ yX + S aX + y aX + /3 must be the same for all values of X. This recpiires /3 = y, and (1) becomes axx' + ^{x-\-x) + S = ( 2 ) . But although this reduction of involution to a special case of homography by the device of counting all points of the line (all elements of the one-dimensional sjmce) twice is often convenient, it yet entirely disguises the real difference between the two conceptions. Homography is a (1, 1) cor- respondence hetiveen the elements of tivo differe/id spaces; involution is a (1, 1) correspondevce hetiveen pairs of ele- ments of one space. 108. Applying the conclusions already obtained for homo- graphic systems on a line to the case of involution, or working them out afresh from equation (2) of the last section, it is seen that the cross-ratio of any four points on the lino is ef[ual to that of their correspondents. Stattid with reference to points A, B, and their correspondents A', B', this tells us nothing about tlie points. For {ABA'B'} = {A'B'AB} idriitically, whether the points are regarded as belonging to an involution or not. But if three points A, B, C and tlieir correspondents enter into the relation it becomes {ABCA'} = {A'B'C'A}, and thus the correspondent to C is determined when two pairs of correspondents A, A': B, B' arc known. Hence CROSS-EATIO, HOMOGRAPHY, AND INVOLUTKJN. 161 two pairs of points, or two segments, determine an involu- tion ; three pairs of points are not in involution unless they satisfy a certain condition. 169. If in two homographic ranges on a line one point, other than a double point, can he found that has the same correspondent whether regarded as belonging to the first range or the second, then the ranges are in involution. For if the point be A, and its correspondent //, where \==iul, x=\ is to give x' = jx, and x = fx is to give x =\. Hence from the general homographic equation, X, jj. must satisfy the two equations aX/x + /3A + y/u + = 0, aX/jL + /5/'t. + yX + = : from which, by subtraction, (^-y)(X-/x) = 0, that is, /^ = y> and therefore the equation is the ecpiation expressing in- volution. Any two homographic ranges can be placed so as to be in involution : and this can be done in two ways. For let 0^ in the first correspond to infinity in the second, and let Og in the second correspond to infinity in the first. Bring the bases of the two ranges together, placing 0^ and ^2 together, at 0. Then the point 0, whether considered as belonging to the first range or the second, has the same correspondent, viz., tlie point at infinity ; the ranges ai'e therefore in invohition ; and as the required placing of the bases can be accomplished in two ways, the second part of the statement follows. JJouble Elements of an Involidum . 170. If an element coincide with its correspondent, tliere is a double element of the involution. The condition x' = x reduces the equation axx + ft{x + x') + = to ax-+'l(ix + o = (), showing that there are two tlouble eleuients, real or imagin- ary ; that is, the involution of lines through a point contains two double lines, the involution of points on a line contains two double points. Dealing with this last case, the point 162 CROSS-EATIO, HOMOCITJAPHY, AND INVOLUTION. midway between the two double points is real ; taking it as origin, /3 = 0, and the involution is expressed by axx^ + <5 = ^), that is, by xx = h, and the double points are given by 01? = k. This special point, the centre, corresponds to the point at infinity on the line: for x = gives x'=(X). Two cases arise according as /; is (i.) positive, (ii.) negative. (i.) If k be positive, the double points are real ; x, x' have the same sign, and therefore corresponding points are on the same side of the centre. (ii.) If k be negative, the double points are imaginary : X, x' have different signs, and therefore corresponding points are on opposite sides of the centre. jVote. Imagine a point P to describe tlie line, tlien its con-espondent P' also describes the line. Let P start from 0, the centre, P' therefore starts from infinity ; let P move in the ])ositive direction from through infinity to 0. In case (i.) since P and P' are on the same side of 0, P' travels to meet P ; the two coincide scjmewhere between and infinity, at i^j ; i^j is a double point. Similarly P having jmssed through infinity, and simul- taneously /*' through 0, P, P' are now to the left of 0, and are travelling towards one another ; they meet at Fo, the second double point. In case (ii.) since P and P' are to be on opposite sides of 0, they jnirsue each other along the line, but do not coincide in real points ; they remain in the two distinct segments bounded by and infinity ; as /-• changes from the fii'st of these to the second, P' changes from the second to the first. Similarly in an involution of lines with real double lines, corresjjond- ents revolve in opposite directions ; in an involution with imaginary double lines, correspondents revolve in the same direction, and do not overtake each other. The dou])le points (or lines) are sometimes called foci (or focal lines). When these are real, the involution is hyper- bolic ; when imaginary, the involution is elliptic. An elliptic involution is overlapping; a hyperbolic involution is non- overlapping. In Fig. 88, the involution determined on the lower transvci-sal by tlie overlapping segments A A', BB' is elliptic: the involution determined on the upper transversal by the non-overlapping segments A A', BB' is hyperbolic, and in this there are segments such as CC, which is entirely contained by BB'; GC and BB' are also non-overlapping. 171. Ill the involution AA', BB', let the centre be 0; this corresponds to infinity, therefore {AA'n()]=\A'Ali'^], CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 163 that is, BA' AJ) BO A'A B'A A'c B'zc therefore AO_AR BO~BA" whence is determined. But a construction that is practically more convenient than this equation can be obtained by means of circles. Take any point G off the line (Fi^-. 38) and describe circles AA'G, BB'O ; Pi(i. 38. these have one real intersection G, and they have therefore another real intersection G'] let GG' meet the base in 0. By the properties of circles, OA.OA'=OG. OG' - OB . OB', therefore is the centre of the involution. If now be not between A, A', it is outside the circles GAA', GBB' (Fig. 38, upper part) ; draw a tangent OT, then OT' = OB.OB' = OA .OA', therefore marking off on the line OF, = OT, and OF., = OT, F^, F.^ are the double points of the involution. If be between A, A', that is, inside the circle, no real 164 CROSS-EATIO, HOMOGRAPH Y, AND INVOLUTION. tangents can be drawn, and the double points are imaginary. In this case, describing a. circle on A A' as diameter, let the double ordinate through be S^S., (Fig. 39). The two points ^\, So are of service in constructions.* From the circle we have OS, . OS, = OA . 0A\ and OS, = - OS,^, therefore OF,^ = OA . OA' = - OS,^. Xote. The constructions liere given, while practically convenient, are open to an important theoretical objection. The idea of an involution with its double elements is ])urely descriptive ; and though the centre cannot be determined by purely descrijjtive constructions, yet it dejjends only on the line infinity ; but these constructions dej^end on the circular points. A purely descrijjtive construction is given in § 195. 172. Any number of pairs of correspondents can be inserted when one pair and the centre are known. For if any line through meet any circle through A A' in //', points P, P' on the base determined so that OP=OJ, OP'=OJ\ give OP . OP' = OJ . OJ' = OA . 0A\ and are therefore correspondents in the imolution. This construction is available for an involution of either kind ; if howev^er the involution be elliptic, a special construction can be used which is interesting as illustrating the relation of involution and homography. In Fig. 39, P0.0P' = 0S,^ therefore PS,P' is a right angle. Hence if a rio-ht anole revolve about its vertex, the two legs describe an involution on any transvei'sal ; pairs of corre- spondents can be inserted by means of perpendicular lines through S,. Now suppose any constant angle PSP' to revolve about its fixed vertex S (Fig. 40) : the two legs describe homographic ranges on any transversal, Init these are not in involution. For wlien the first leg passes through P', the position of the second leg is determined by taking i.P'sq'=-PSP', * If, liowever, we adopt the ordinary two-dimensional repi-esentation of imaginary values, ami represent a + fti by the point a, [i, the double points are re()resented by ; + ("/' + //'A) = 0. CEOSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 167 Eliminating X from these, a quadratic in x is obtained whose two roots give the two double elements, viz., ax + «' _ ax + a" that is, {ah' — a'h)x" + («^" — a"h)x + {ah" — a"h') = 0. The value of A that reduces the quadratic to a linear equation gives inlinity and its correspondent, that is, the centre. Hence for the centre, _ 1 a"h — ah" 2 ah' — a'h' 175. The two quadratics it = 0, f = (), determine an involu- tion u+Af = whether their roots be real or imaginary; and this involution certainly contains real pairs. Some real values of A will give imaginary pairs, but this can happen only if the involution be hyperbolic. For let the centre be origin, then therefore the quadratics are reducible to the form a;^ + 2a'a;-l-/u = 0, and u + \v = is then 1 + A th at is, OCT' + 2|;a; + k = 0. Hence if k be negative, so that the involution is elliptic, a real j9, which implies a real A, gives real points. Tliis can be stated : — Pairs of imaginaries can occur only in a hyperbolic in^•olu- tion ; the imaginaries that occur in an elliptic in^'olution are not pairs, in the special sense of § 50. Hence the involution determined by the (juadratics it = 0, v = i) will be hyperbolic imless both quadratics have real roots ; in this case it may be hyperbolic or it may be elliptic. Common Elements of tivo Invohiiions. 176. Two involutions on a conuuon base have necessarily one pair of common elements, determined as the pair har- monic to each of the two pairs of (lou])lc elements. Reference to § 175 shows that these common elements will certainly be 1()S CK0S8-EATI0, HOMOGEArilY, AND INVOLUTION. real unless both the given involutions are hyperbolic, in which case they may be real or they may be imaginary. Considering this case first, let the two pairs of double elements be F-^, F., ; $p $.,. These are real, by hypothesis ; hence the required points are determined as the double points of the involution F-^F.,, ^^^o ; they are imaginary or real according as the segments (F), ($) do or do not overlap. Now let the first involution be elliptic, i^^ F., are imaginary, and the construction just given is inapplicable. Let the centres of the two involutions be 0, Q, and let a pair of conjugates in each be D, D' ; A, A'. Let K, K' be the connnon elements, therefore OK . OK' = OD . OU, and UK . QK' = QA . QA'. By h3^pothesis, is between I), D' ; therefore a circle on DD' as diameter contains : draw the double ordinate S^OS.^- Join QaS'j, and take on this a point S[ such that QS^.QS[ = nA.nA'. The circle ^^lS.2Sl cuts OQ in the required points K, K' ; these are in every case real, since >S^, 8.^ are on opposite sides of the line. This construction applies whether the second involution is hyperbolic or elliptic. If it be elliptic, drawing the double ordinate S^QSo, the circle ^^S.-i^^.^ is the one required in the constructio)!. Note. Tliis roiistniction, like tliose in S 171, is open to tlie theoretical objection that it uses circles in a j)uiely descriptive problem. It is however a simple ])ractical construction. A purely descriptive con- struction is i^nven in § 195. Involution (letennined by a Quadrangle. ill. The sinqilest purely descriptive construction for cor- respondents in an involution is afibrded by tlie tlieorem : — Tlic, tJiree />a/r.s of sides of a complete quadravglc are cut in involution by any transversal. Let the transversal cut the sides in XX', YY', ZZ' (Fig. tl). Tlu'ii {^1 . liODZ] = {C. BGDZ), and as e(iual pcm-ils determine e([ual I'anges on any line, the ranges deteiniined on the ti-ansvi-rsal are e(|ual : therefore {YZX'Z} = {XZYZ], that is, { A" YZZ} = { X YZZ } , therefore the cross-ratio of foui- points is e(jual to tliat of their conjugates, wliieh shows that the three segments A'A", YY, ZZ are in involution. CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 169 Taking for the transversal the special position EF, the points A'^, X' come together at E, and the points Y, Y' at F; these are therefore the double points of the involution on EF \ they are harmonic with respect to the points in which EF is met by BD, A C, agreeing with what we already know as to the harmonic properties of the figure. Hence to construct Z', the conjugate to Z in the involution determined by XX', YY', it is necessary to construct a complete quadrangle whose sides shall pass through tlie points. For this the lines tlirougli Z, Y, Y' may be taken arljitrai-ily, but then the points B, I) are known, and these nuist be joined to A'', A"' (in either order). Tlius the points A, C are found, and the line AC passes through Z'. The construction can be differently arranged, and in any special problem the lines of the figure should be utilized as far as possible : the essential thing is to get tlie three pairs of sides of a complete quadrangle associating tlie pcjints in paii's as assigned. 178. This tlieorem shows clearly that tlie conception of involution is purely descriptive ; it depends simply on con- structions with collinear points and concurrent lines, and requires no metric determination. The conception of har- monic division is also purely descriptive, as was pointed out in § 45 ; and harmonic division might be defined accordinglv : If four collinear points be such that a quadrilateral can be described with vertices at two of them, and two diagonals passing through the other two, the points are said to be harmonic. It is then shown (e.g. by means of triangles in perspective) 170 CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. that this determines uniquely the fourth point, associated with a specified one of the three ; and that the segments determined by the two pairs necessarily overlap. (Reye, Geometrie der Lage.) 179. That equianharmonic division is also purely descriptive appears from a construction now to be given. Let 1, 2, 3 be any three collinear points ; determine 1', 2', 3' so that 11' may be harmonic with respect to 23, etc. : then (a) ir are harmonic witli respect to 2'3', etc.; (h) 11', 22', 33' are in involution; (c) either triad of points and either double point of this involution form an equianharmonic system. The analytical proof of this construction illustrates a process of combination that can be used with cross-ratios. By hypothesis, (11', 23) is harmonic, as is also (22', 13), therefore {1231'} - {2132'} (i.). Continue the range {1231'} with 2' and then with 3', determining the corresponding points on the right. Let {1231'2'} = {2132'A'} ; for the determination of X there is any equation made from corresponding cross-ratios on the two sides ; therefore { 1 232' } = { 2 1 3 A" } , that is, (22', 13) = (1 A, 23), hence {IX, 23) is harmonic, and X is 1'; therefore {1231'2'} = {21321'} (ii.). iVote. If (i.) be written {1231'} = {2312'}, which is permissible by liarnionic proi^erties, the point A' in {1231'2'} = {2312'J:} is deterniined In- { 1232'} = {231A'}, and is thei'efore 3'. Hence { 1231'2'J ={2312'3'1 ; and similarly a number of other relations can be found. Continuing (ii.) with 3' on the left, {1231'2'3'} = {2132'1'F}; therefore { 1 233'} = { 21 3 F} , that is, (12, 33') = (21, 3 F), but (12, 33') = (21, 33'), being harmonic, and therefore (21, 33') = (21, 3F), sliowing that F is 3'; hence {1231'2'3'; = {2132'1'3'} (iii.). CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 17] (a) From (iii.), {31'2':r} = {32'r;3'}, therefore (33', 1'2') is harmonic: and similarly (11', 2'3'), (22', 1'3') are harmonic. Hence starting with 1', 2', 3' the construction leads to 1, 2, 3 ; that is, the two triads of points are symmetrically involved. (b) Since (11', 23) and (11', 2'3') are harmonic, {1231'} = {1'2'3'1}; that is, the cross-ratio of four of the points is etpal to tluit of their conjugates ; hence the three pairs of points are in involution. (c) Let x^, X.2 be the double points of this involution, tliey are therefore harmonic with respect to every pair 11', 22', 33'. Now 12, 1'2' are also harmonic with respect to 33', therefore x-^^x.j, 12, V2' are in involution. Hence {121'.«J = {212'«,} = {2\21} (iv.), by permissible interchanges. Therefore {121\x,} = {2'x.^ly} , where y is determined by {n\x,J^{2'21y}, that is, by ( 1 1', x\x.^ = (2'2, ly). The left hand is harmonic, hence y must be 3, showing that {121'a;i«J-{2'a;.2213} (v.). From (v.), {UVx^x.^} - {2'xl21Sz}, where {Ul'S} = {2'x^2z} , that is, (ir, 23) = (2'2, a;2^). The left hand is harmonic, hence z is x^, and therefore {l2Vx^x.;i} ^ {2\2V3x^} (vi.); and similarly it can be shown that {212'a;ia;23} = {raJ2l23«J (vii.). From (vi.), {«il23} = {12'a-2a-J (viii.) ; from (vii.), {.7-,123} - {2x^rx^}, that is, {2x,rS} ^ {r2x.^'^} (ix.). Now since a;^, x.^ are the double points of the involution 11', 22', {12'a;,a;J = {l'2a;,«J; hence (viii.) and (ix.) give {a;il23} = {2a;J3} = K231} (by permissible interchanges), which proves that {,i'il23} is equianharmonic, where x-^^ is either double point. 172 CROSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. And from the involution 11', 22', *33', whose double points are x^, x,^, [x^l22,} = {x^V-2"^'\. Having found that {ir^l23} is equianharnionic, equation (viii.) shows that {I'^'x^x-^} is also equianharinonic ; that is, the two double points with two non-corresponding points from the two triads are equianharnionic. Hence an equianharnionic range depends on the construc- tions for harmonic ranges and the double points of an involution ; it is therefore purely descriptive. Examples. 1. Discuss the contents of this section with reference to the range determined on the line inhnity by the three sides of an equilateral triangle and the lines through the vertices per- pendicular to the sides. 2. Taking three concurrent lines at equal angles, show that either isotropic line through their intersection completes the equianharnionic pencil. 8. Show that two lines at an angle of 30° and the two isotropic lines through their intersection form an equian- harnionic pencil. 4. If the double lines of a pencil in involution be at right angles, they are the bisectors of the angles determined by any pair of conjugates. 5. Show that every pencil in involution contains one pair of orthogonal rays. Desai 'gucH Theoi 'eii ^. ISO. Tlie theorem of 1^177 on tlic inxolution properties of a (juaih-angle is a particular case of Desai-gues' Theorem, which was originally stated with rcrcreiice to a conic and two pairs of sides of an inscribed (quadrangle, but in its general form ivlates to a pencil (^f c(mics. Tliis theorem is: — Any transversal is cut in iiivolatlon />// a pencil of conies; and reciprocally. Any jjoint is suhtendecl (p. f)4, footnote) in involution by a range of conies. Using point coordinates, let the base conies Ik^ a — 0, v = 0, where the transversal is s = 0, then any conic of the pencil is u + Xr = (), and the pairs of ]M)iiits in Avliicli tin's is in(>t by tlie transversal are given by ax- -t- '2hxi/ + by' + \{(i'x~ + '-^h'xy + b'y-} = 0, and arc tlicrcfore bv !^ 174 in involution. CEOSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 173 If now the conic be chosen to touch the line, tlie two intersections coincide ; the point of contact is a double point of the involution ; hence as seen in § 83 two conies can l;e drawn to pass through four points and touch one line ; and two conies can be drawn to touch four lines and pass through one point. Since the pencil contains three line-pairs, the involution properties of a complete quadrangle are included under Desargues' theorem ; and using two of the line-pairs and one proper conic, the original form of the theorem is obtained. Let the transversal meet the conic in XX', and the line- pairs AB, CD ; AC, BD ; in PF, QQ'. The fact that tlie three pairs PP', QQ', XX' are in involution makes the position of X' depend on X, PP', QQ' ; that is, on A,B,C, D, X. Thus Desargues' theorem agrees with Pascal's theorem and Chasles' theorem in expressing the dependence of any sixth point of a conic on the determining five points ; the three theorems express the same truth under ditt'erent aspects, and any one can be deduced from any other. Ex. 1. Deduce the proces.s for fiiuliiiL; the centre of an invokitiun (§ 171) from Desargues' theorem. £x. 2. Examine the construction for the common elements of two invohitions (S 176) with the help of Desargues' theorem. Ex. 3. Two conies are determined by five points each, of which three are common. Determine the fourth common jjoint. Genei'al Idea of Involution. 181. Though a formal proof of Desargues' theorem has just been given, yet this is not really necessary; for the definition of involution (§ 1G7) makes the theorem intuitive. Any conic of the pencil determines a pair of points on the line ; and since four points on the conic are known, one point of the pair determines the conic, and therefore determines the other point; hence the elements of the one-dimensional space are associated in pairs, which is the one charactei-istic of an involution. The idea here involved has been generalized in various directions. Instead of groups of two points tiiich, we may imagine the points of the line grouped by -v? 's ; in the case hitherto considered, two pairs being given by it = 0, r = (), any pair is given by u-\-\v = ; in this generalization, any two groups being given by ib = 0, v — 0, the involution consists of all groups n -j- \v = 0. Since only one parameter, A, is involved, a single point determines the group. Such a system as this, a singly infinite system of groups ol' n eloineuts 174 CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. linearly determined by two groups of the system is called an involution of degree n. For example, a pencil of cubics, u.^ + \v.^ = 0, determines on any transversal an involution of degree 3. The involution of pairs of points is therefore of the second degree. Again, the expressions u, v may be of degree n in any number of variables ; that is, instead of representing a group of n points, u = may represent a curve or a surface of order n ; or it may be homogeneous in more than four variables. Thus the pencil of conies 162 + Xv2 = may be re- garded as a system of conies in involution. Just as in the simj^lest involution there are double elements, so in an involution of degree n there will be certain groups in which some of the elements fall together ; and just as in an involution of conies there are three line-pairs, that is, three conies that have double points, so in an involution of curves of higher order there will be certain curves that have double points. Note. These extensions of the theory of invohition can be fonnd in Fiedler, Die Darstellende Geometrie, t. iii., pp. 218, 256, 261, etc., and in Clebscli, Vorlesungeii iiher Geometrie, t. i., pp. 203, 207-210; where other references will be found. Another extension depends on an increase in the number of base- expressions u, V, etc. (Cayley, On the Theory of Invokition, Trans. Camh. Phil. Soc, 1866 ; No. 348 in Collected Papers, vol. v.), the general element in the invohition is then X^l + fJ^,v + vy'+etc. 182. There is no very convenient distinctive notation in use for involution ; occasionally (u, v) may be advantageously^ used for the involution ii-\-\v. If the involution be deter- mined by pairs of points, we may use {AA' , BB\ CC); and the double points being F^, F.^, this can be indicated by a natural extension of the symbolism, (AA', ... , F^~, F.^^) ; or we may denote the whole involution by (F-^^, F.^). It may occasionally be found convenient to write F={AA', JiB')'-, as a symbolic expression of the fact that F is a 'double element. Involution Froperties of Conies. 183. The intimate association of the theory of involution (of the 2nd degree) with the conic, implied in its definition, leads naturally to a series of theorems expressing this association. For instance, pairs of conjugate points on a line are in involution ; this comes from the definition : thci known synnnetric relation of conjugates proves the (1, 1) correspondence on which involution depends; any point on a conic has itself for one of its conjugates, hence the double CROSS-EATIO, HOMOGRAPH Y, AND INVOLUTION. 175 points of the involution are the points in which the trans- versal meets the conic. Or the theorem can be proved from the fact that conjugate points are harmonic with respect to the two points in which the line joining- them cuts the conic. Similarly pairs of conjugate lines througli a fixed point form a pencil in involution, and the double lines are the tangents from the point to the conic. As a particular case of this, conjugate diameters form a pencil in involution, of which the asymptotes are the double lines ; these are real or imaginary, that is, the invo- lution is hyperbolic or elliptic, according as the conic is a hyperbola or an ellipse. The involution is circular if the conic be a circle, in which case every pair of conjugate diameters is harmonic with respect to the circular points, that is, every pair of conjugate diameters is at right angles (see § 172). Examples. 1. From the fact that two involutions wnth a common base have one pair of connnon elements, prove that every central conic has one pair of conjugate diameters at right angles, and that these are necessarily real. 2. Given two pairs of conjugate diameters (in position, not in magnitude), construct the axes and the asjnnptotes, dis- tinguishing the two cases that occur. 3. Apply Desargues' theorem to show that every conic through the intersections of two rectangular hyperbolas is a rectangular hyperbola. 4. If a pencil contain a circle, the axes of the two parabolas it contains are at right angles ; and the axes of all conies of the pencil are in two fixed directions. (Steiner.) 5. If a circle intersect a conic, the pairs of chords of intei'- section make equal angles with an axis. 184. To obtain the focal properties of conies, consider the two pairs of tangents from w, co , and use Desargues' theorem in the reciprocal form. Two conies with tlie given tangents can be drawn through any point P ; their tangents at P are the double lines of the involution formed by the tangents from P to the conies of the range. One conic (degenerate) of the range is the point-pair co, w' ; hence the tangents to the two conies through 7^ are harmonic with regard to the isotropic lines through P, tliat is, they are at right angles. Hence confocal conies are orthogonal (compare ,§ 134). Again, another degenerate conic of the range is the point- 176 CROSS-EATIO, HOMOGRAPHY, AND INVOLUTION. pair F, F' (the real foci) ; the two perpendicular tangents at P are therefore harmonic with respect to PF, PF', that is, they are the bisectors of the angle FPF' ; hence tlie tangent at any point is equally inclined to the focal distances. Taking a point T not on the conic considered, let TP, TP' be tangents ; the lines joining T to ww', FF\ PP' are in involution : therefore (i.) the double lines of this involution are at right angles, and consequently (ii.) they bisect the angle FTP' and also the angle PTP', therefore lFTP = -FTP\ hence the two tangents that can be drawn from any point to a conic are equally inclined to the focal distances of the j)oint. 1S.5. If any problem depend on the determination of a single point or a single line, we expect to obtain the solution by a linear construction. Not so if the point to be determined be one of a pair; for this we must expect to need a conic. A pair of points, real or imaginary, on a known line, can always serve as the double points of an involution determined by two real pairs, and this is frequently the most convenient way of dealing with a pair that may possibl}^ be imaginary; or again the two points may occur as the double points of two houiographic ranges. A construction of this nature, that is, a construction for a pair of elements, is a construction of the second degree. Ex. Find the points in wliicli a L^nven line meets tlie conit' dctci-niined by the five points A , D, C, /), E. Let one of the required ])oints be l\ then bv Cliasles' theorem the pencils {A.CDEP], {B.CDEP] are equal. Let AC\ AD, AE meet the given line in 1, 2,3, and let liC\ BJ), HE meet it in 1', •!', 3' ; /' is either doulile pnint of the homo- graphic ranges {1 i ?>...], {V -1' 3' ...:. Ex. 1. Apply De-sargues' theorem to tiiis prnljicm. Ex. 2. Two conies are determined by live ])oints, (if which two are common. Find the line joining the other two common jwints by a linear construction, and the i)oints themselves by a quadratic construction. 18G. A nund)er of relations among segments in involution can be found ; these all follow from the two ])roperties, (i.) the cross-ratio of any four points is equal to that of their correspondents ; (ii.) the double points ai-e harmonic with respect to every pair of conjugates. .Similarly relations can be found for the segments determined l)y homographic ranges. CEOSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 177 For tliese reference should be made to Chasles, (ieomHrie Superieure, Chapter's IX. to XIII. ; or to Russell, Pure Geometry^ Chapters X. and XVII. In Ch. XXV. of the latter work there will be found a number of interesting constructions of the second detrree. Examples. 1. Construct tlie polar of a point P with regard to tlie conic determined hy A, B, C, I), E. 2. Given a pole and polar, and three points on a conic, construct the conic. 3. Given two poles and polars, and one point on the conic, con.struct the conic. 4. Given two pairs of conjugates, AA', BB', show that the two points determined by the intersections of the cross-joins are conjugates. (Hesse.) 5. Show that the conies passing through three points and having one assigned pair of conjugates form a pencil ; and find a linear construction foi- the fourth common point. 6. Given four points on a conic, and one pair of conjugates, determine the points in which the conic cuts the join of the conjugates ; hence construct the conic. 7. Consider a triangle, and a line-pair ; on any side of the triangle there are now two point-pairs, viz., the two vertices, and the intersections with the line-pair ; let the doul)le points of the involution on BC be A^, A. 2'- etc. Show that the six points A.^, A. 2, B^, B.,, C\, C, lie by threes on four straight lines. 8. Apply Uesargues' theorem to a conic and a pair of tangents, showing that one double point of the involution determined on any transversal is on the chord of contact of the tangents. 9. Three points and two tangents are given for the deter- mination of a conic ; show that the chord of contact is one of four lines ; hence construct the four possible conies. Apply this construction to the case when two of the given points are the circular points, discriminating carefully as to real and imaginary in the construction. 10. Each vertex of a pentagon is the pole of thr ojjposite side with respect to a conic; show that this statement in- volves exactly the right nund)er of conditions for detcniiining the conic ; and find the construction for the conic. Note. TJiC folloiving four examples relate in tlie peneil determined by iivo conla^ all of icJiose inter seel ion fi are S.G. M 178 CROSS-EATIO, HOMOGRAPHY, AND INVOLUTION. imaginary. All deiiencl on ajyplications of JJesaiy^ien' theorem. 11. Draw the conic of the pencil tliat passes tlironii,'!! a given point. 12. Draw tlie conies of the pencil that touch a given line : (a) if the line touch one conic, and cut the otlier in (i.) real, (ii.) imaginary points ; (h) if the line cut one conic in real, one in iniaginar}' points ; (c) if the line cut both conies in imaginary points. 13. If the two given conies belong to one "nest," determine one conic of the other nest. 14. Hence construct the common self-conjugate triangle for two conies, all of whose connnon j)oints and common tangents are imaginary. 15. Take a fixed conic u, a fixed line p, and a fixed point 0, these having no special position with regard to one another. Let any variable line through cut the fixed line in 0', and the conic in (I, U' ; let X, X' be the double elements of the involution {OW, UU') ; the locus of X, X' is a conic, 0, having 0, p as pole and polar. 16. Let V be any other conic having 0, p as pole and polar ; the intersections of v, (J) lie on two lines through 0. 17. Hence show that through any point two lines can be drawn to be cut harmonically by two conies u, v. 18. Find tlu! envelope ot* a line cut hai-monically by two conies. 19. Find tlic locus of a point sTd)tcndt'(| liurmonicall}' b}' two conies. 20. Find the locus of the intersection of perpendicular tangents to a conic. 21. Any line through a vertex of the self -conjugate triangle of a pencil of conies is cut in involution by the pencil. 22. If tlie lianiioiiic conic dctciMiiinccl by tlie segiiieiits PP', QQ' divide 711/1'' liarmouically, then tlic relation of the three segments is synnnetrical. (Clifloi'd.) 23. If P, P' be conjugate with regard to the pi'ueil of conies determined by an ortlioccntric (piadi-angle, then with P, P' as foci a conic can be inscribed in the harmonic triangle of the <|U;i(b'angIe. CEOSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 179 24. Show that throiio-h any point three lines can be drawn to be cut in invokition by three conies u, v, v\ which do not belong to a pencil. Determine the lines when the three conies have a conniion self -conjugate triangle. 25. Show that any line cut in involution by three conies u, V, w is cut in involution by every conic of the net \u + ixv + vw = 0. 26. Hence show that the envelope of a line cut in involution by a net of conies is a curve of the third class. Systems of Conies. 187. A number of the examples and theorems discussed have related to special systems of conies ; for example, among systems that are singly infinite we have considered a pencil and a range ; but there are other singly infinite systems that may be considered, systems of conies determined by three points and one line, by two points and two lines, by one point and three lines. The characteristic of the pencil is that the point equation of any member is linearly expressible in terms of any two members ; the fundamental idea in the range is reciprocal to this ; the line equation of any member is linearl}" expressible in terms of any two members. If then we know that the conies considered in any problem form a singly infinite system, we cannot therefore infer that they form a pencil, or a range. But if the conditions imposed be of such a nature that u, v l)eing members of the system, 'ii-\-\v is also a member, then all members of the s^^stem are given by u + Xv. For if there be any one, cf>, not included in this, then by the given conditions all conies \lr + J<-(J) are included in the system, where xfr is any one of the system u-\-Xv\ hence the system is not a one-fold infinity, for it includes u + Xv + k^, that is, it depends on two independent parameters. Hence we see that all conies of the system are given b}^ u-\-\v; if therefore u, v be expressed in point coordinates, the system considered is a pencil ; and if ii, v be expressed in line coordinates, the system is a range. For instance, given four pairs of conjugate points, that is, four conditions, the conies are singly infinite in number. Let n, v be any two satisfying these conditions : Desargues' tlieorem shows that all conies u-\-\v belong to the system; hence all conies of the system are included in u-{-\v, an(, the pairs of points in which the join of the conjugates meets them are expressed by three quad- ratics, u = (), v = 0, ty = 0, obtained by writing z = in = 0, \fj- = 0, x~^- '^ii^ce these are by hypothesis in involution, w = an -\- hv. Taking any other conic of the net, \cl)-\- fj-xfr+vx^^, and writing z — 0, the pair of points is given by \vb + iJ.v + vio — (i, that is, by lu-\-iinv = 0. Hence this pair is in the involution (vi, v), and therefore the given conjugates are conjugate with regard to the conic \(}) + lxylr + vx = ^\ that is, with regard to any conic of the net. 182 CROSS-EATIO, HOMOGRAPHY, AND INVOLUTION. Hence the locus of the points that are conjugate with regard to a net is a certain order-cubic. Again, in Ex. 26, after § 186, it is stated that the envelope of a line cut in involution by a net of conies, that is, of a line joining a pair of conjugates, is a class-cubic. Thus associated witli the net there is an order-cubic and also a class-cubic : and the whole theory of nets of conies is most satisfactorily studied in connection with curves of the third order or class; not that these higher curves are required for the proofs, but because the results obtained with regard to the net can by their means be more clearly stated and realized. The theory of cubics as depending on nets of conies is systematically developed geometrically by Schroter, in liis Tkeorie der Ehenen Kiirven dritter Ordnung ; it is algebraicalh' treated by Clebsch, Vorlesu7igen, t. i., pp. 519-527. A detailed account of singly infinite systems of conies, and some discussion of doubly infinite systems, is to be found in Steiner's Synthetische (Jeometrie, t. ii. (Sciiroter) ; and the paper "On Some Geo- metrical Constructions" by H. J. 8. Smith (Proc. Lond. Math. Soc, vol. ii., pp. 85-100 ; 1868) is devoted to a discussion of " systems of order 1, 2, 3, 4" {i.e. pencil, net, etc.). But this paper presupposes some knowledge of the theory of Invariants, as does also the treatment adopted by Clelisch. 190. It has been hitherto assumed that the conies u, v, w have not any connnon points ; they may however have in common, one, two, or three fixed points, but not four, for that would reduce the net to a pencil, since plainly any conic \u + /ulv -f j/iv = goes through all points connnon to a, v, and iv. The various theorems stated still hold, except with regard to the fixed points. All conies of the net througli any other point form a pencil ; but one, two, or three of the base points (jf the pencil are now fixed : two points, not the fixed points, deter- mine a conic uniquely, and so on. Determination of a System of Conies hy Pairs of Conjugates. 191. We now consider the determination of a conic by means of five conditions, of whicli a certain lunnber are supphed })y pairs ol" conjugates. Tliis is in close connec- tion with the theory of pencils and nets, and involves the determination of a pencil by lour ])airs of conjugates, of a net by thr(!(! pairs of conjugates : and it leads to de Jonquieres' solution of Chasles' problem: — To determine a conic that shall cut Jive given segments harmonically.* * Terquem et Gerono, Annales dc Matlmnatiqmx^ t. xiv., p. 435 ; 1855. CBOSS-RATIO, HOMOGRAPH Y, AND INVOLUTION. 188 The proofs of some of the constructions are omitted, as they can easily be supplied from the principles already discussed. I. Given four points and one pair of conjugates. The pencil determined by the four points cuts tlie join of the conjugates in an involution ; the given conjugates themselves determine on this line a second involution, of which they are the double elements ; the common elements of these two involutions are points on the required conic, and with the given four points the determination is complete. II. Given three points and tivo 'pairs of conjugates. Let the given points be P, Q, R, and the conjugates AA', BB'. Describe conies through P, Q, R to touch A A' at A and A' respectively ; let S be the fourth intersection of these conies. Any conic of the pencil FQRS has A A' for conjugates, and one conic of the pencil has BB' for conjugates. Hence the required conic is known. It is better to determine S by a linear construction. Let QR meet AA' in P^, and let F^ be conjugate to F^ with respect to AA' ; similarly determine Q.^, R.2- Then QR, FF.^ is one conic of the pencil : therefore FF.^ passes through >S*, as do also QQ.2, RR.,. It is here shown that three points and one pair of con- jugates determine a pencil. III. Given tiuo points and three pairs of conjugates. We first show that two points and tw^o pairs of conjugates determine a pencil. Let the points be F, Q, and let the conjugates AA', BB', lie on lines a, h. Let a, h meet in 2) ; determine S^, S^ on a, h, conjugate to S with regard to A A', BB'. Then the conic PQS^^/S'^ satisfies the given conditions. Let FQ cut a, h in L, if; determine L', M' the conjugates to L, M with regard to AA' , BE: then the line-pair FQ, L'M' satisfies the given conditions. Let L'M' meet the conic FQllS^S., in X, Y, then any conic through F, Q, X, V satisfies the given conditions, and all conies are included in this pencil. The one conic of this pencil that has the third pair of conjugates GC is the conic determined by the five conditions. Ex. ]. Show how to construct a conic through the intersections of two given conies, and having one pair of conjugates, in the case when (1) two, (2) four of the intersei-tions of the given conies are imaginary, allowing for the possibility of the line joining the given conjugates meeting one or both of the given conies in imaginary points. Ex. 2. Describe a circle, when three pairs of conjugates are given. IV. Given one point, and four pairs of conjugates. That tlie four pairs of conjugates determine a pencil was shown in § 187 ; but no construction was given for the 184 CROSS-HATIO, HOMOGRAPHY, AND INVOLUTION. pencil. Let the pairs of conjugates be AA', BB', CC, DD'. Consider the lines AB, A'B\ intersecting in X. Any line- pair that is hannonic witli respect to tliese has AA', BB' for conjugates: one such line-pair can be constructed that has GC for conjugates. Similarly one line-pair can be con- structed, with its vertex at A" (the intersection of AB' , A'B), tliat has AA' , BB', GC for conjugates : any conic of the pencil determined by these two line-pairs has AA' , BB' , GG', for conjugates : and one conic of the pencil lias also DD' for conjugates. Thus one conic is found having the assigned four pairs of conjugates. Now grouping the pairs ditterently, another conic is found: and the pencil determined by these two conies has the assigned four pairs of conjugates : it is the required pencil. One conic of this pencil passes through any given point : hence a conic through one point and having four pairs of conjugates is determined: and one conic of this pencil has a given fifth pair of conjugates : hence V. Given jive pairs of conjugates, the conic is determined. The construction just given can be applied to case II T. The three pairs of conjugates can be grouped in three ways : hence three pencils of conies are found having these three pairs of conjugates. Now one point and three pairs of conjugates determine a pencil ; hence (8 188) three pairs of conjugates determine a net ; hence the three pencils just found, being three pencils of a net, are not independent, but this does not interfere with the construction. We have to find the conic of this net that passes through the two given points, P, Q. The point P determines one conic of the first pencil and one of the second : these two determine the pencil through P: and the recpn'rcd conic is the one of this pencil that passes through Q. Hence as concerns the determination of a eonie, a pencil, or a net, a pair of conjugates is a single linear condition, equivalent in its effect to a given point. N^ote. This is at once apparent algebraically ; tlic ((nidition tliat two points be conjugate with respect to the general conic imposes a linear condition on the coefficients a, b, etc. Ex AM I' I. ES. 1. A])ply the linear construction for the loui'tli bnsc-point of the pencil detei'iiiined by three points and one pair of conjugate points to the case of rectangular hy])erbolas througli three ti.xed jioints. 2. Sliow that the locus of tlic ]n>\r of a h'.xed line with CROSS-RATIO, HOMOGRAPHY, AND INVOLUTION. 185 respect to a pencil of conies is a conic through five kncnvn points, of which three are fixed, and two lie on the line. 3. Apply tlie result reciprocal to tliat obtained in Ex. 2 to show that four tangents and one pair of conjugate points determine a conic as one of two. Construct these conies. 4. Construct the conies determined by (i.) one tangent and four pairs of conjugate points ; (ii.) three points, one tangent, and one pair of con- jugate points ; (iii.) two points, one tangent, and two pairs of con- jugate points ; (iv.) one point, one tangent, and three })airs of con- jugate points. Homographic Gorrefiponclence on Curves. 192. The general definition of homography (>; IGO) simply requires the comparison of two one-dimensional spaces ; there is to be a (1, 1) correspondence between the elements. Each element being indicated rationally by means of a single para- meter IX, Hi', the correspondence is expressed by a bilinear relation a/X/U.' -|- /3/x -|- y/i' -f- ^ = ( 1 ), and the cross-ratio being estimated by means of the values of m, it follows that the cross-ratio of any four elements is ecjual to that of their correspondents. This conception is directly applicable to any two unicm-sal one-dimensional spaces. Thus for example the cubic X^ -\- y^ — oxyz = was shown in § 144 to be unicursal ; the coordinates of any point are X : y -.2 = 2111' : o/ul : jX^+l. Again, the conic ^^ = i]^ is unicursal, the coordinates of any tangent are r- . . ? ' . 1 . '2 ^. )/ . ^=/X . i .^t . A (1, 1) correspondence can be instituted between these two one-dimensional aggregates of elements by means of a relation of the form (1). It should be noticed that the bilinear rela- tion between the two parameters does not imply a bilinear relation between the two sets of coordinates, though there may happen to be such a relation. For instance, in the example just given, X , P y >i 186 CROSS-EATIO, IIOMOCiRAPHY, AND INVOLUTION. therefore (1) becomes '<.'f +/?.'-•;; + y//^+^,V'/ = 0; but the relation fsatishcd by y : z and >; : ^ is of a different form. 193. Tlie two liomograpliic systems may be in the same one-dimensional space ; for example, we may have homo- <4'raphic ranges on a conic, just as we have homograpliic ranges on a line. And instituting a (1, 1) correspondence between the points of a conic, we have an involution on the conic. It was sliown in ^142 that the coordinates of a point on a conic can be expressed in terms of a single parameter in a doubly infinite number of ways. But whatever system of expression may be adopted, the cross-ratio of the parameters of four points is the same, and is the cross-ratio of the pencil determined in the conic. For let the expressions be x:y:z = a/ur + " V + ^^" • ^Z^" + ^V + ^" '■ ''M'^ + (^'V + ^" '■ knowing that the pencil determined at any fifth point of the conic is the same whatever point be taken, we can take the pencil subtended at jul = 0. The ray of this that passes through a point fx is X y z =0 ; u fx.'^ -{- d' jUL -\- a" hix~-{-h' ij.-\-l>" c/x"- + c'^ -f c" a" b" c" subtractnig the last row from the second, and then dividing the second row by /x, this becomes X y z (i/ii-\-a' hfji + h' r'/x + c' (t" h" c" 0. Hence any ray of the pencil is it + /uLO, wliere a = x{h'c" — b"c') + ij{c ct" — (;"«') -t- i{a'b" — (t"b'), v=zx{bc" —b"c) + >j{ca" —c"a) +z{(ib" —a"b), and the pencil is therefore iiu^fi.,, iij...fx^), that is, l)y §41, a pencil of cross-ratio /^i /^:{ . Mi /^ 1 /x.j — fX., //j — IH.2 In dealing with systems of points on a conic, we are there- fore at liberty to a., meet in P (Fig. 42); project on to a plane parallel to PV: let Pv P'i. ii^eet the line of intersection in X, Y. By elementary solid geometry, planes VPX, VPY are cut ])y tlie plane parallel to VP in parallel lines [>[, pL Moreoxcr tlu' pro- jection of P is the intersection of VP with the })lane parallel to VP. Hence the concurrent lines p^, />., are projected into parallel lines, their intersection being projected to infinity. Now let there be a line through P parallel to XY: this is projected by a plane parallel to the plane of ]irojeetion. 190 PROJECTION AND LINEAR TRANSFORMATION. and ON'eiy point on it is projected to infinity : but no other point in the phme PXY is projected to infinity, for every other point lias a finite projection. Hence we see that in a phine, infinit}' is a straiglit line. In a diagram the line in each plane that represents infinity in the other plane is of importance. Fig. 42. From the reciprocal nature of the two figures in Fig. 42, it is seen that parallel lines project into a pencil of lines. In (ji'der to project a pencil of lines into parallel lines, the cutting plane must be parallel to VP : and to project a second pencil of lines with vertex Q into parallel lines, the cutting plane nmst be parallel to VPQ. This leaves us free to choose V arbitrarily, but then the direction of the cutting plane is determined. Hence any line can be pro- jected to infinity from an arbitrarily chosen centre. For example, a quadrilateral can be projected into a parallelogram : in Fig. 41 let EF be projected to infinity, ABCl) l)ecomes a parallelogram, and consequently BD, AC l)isect each other in G. Let HI) meet KF in //: then the relation can l)e stated in a projective form, for H being at infinity, {BD, (HI) is harmonic. Thus the harmonic pro- perties of a complete quadrilateral are deduced from the properties of a pai^allelogram. Possihiliiica of Projcctiov . 190. We know that the magnitudes of lines and of angles are altered by projection; lait we have just seen that we can PROJECTION AND LINEAR TRANSFORMATION. ]()] to a certain extent determine beforehand Avliat tliis alteration shall be. We consider therefore to what extent we can in general control this alteration. It was shown in § 151 that the ratio of two segments on a line can be altered to any desired extent, but that other seo-- nients on the line are then determined. Now let there be segments not all on one line; let ABC be on one Vme, PQR on another; we can project so that AB:BC and PQ:QR may assume any desired values A and //. To do this, take J) on ABG and S on PQR, so that (A G, BD) =-X, (PR, QS) =-fx, and project DS to infinity. Ex. Project so tliat two iiou-collinear segments may be bi-sected at their intersection. 200. The extent to which we can control the alterations in linear and angular magnitudes is assigned in the most generally convenient form by the theorem : — Any straight line can he projected to ivjinity, and at the same time any tivo angles into given angles. For in any projection let tlie plane through V parallel to the cutting plane meet the original plane in r'. Let the lines AB, A(J containing an angle .1 cut r, r' in Z/'. VV 192 PROJECTION AND LINEAR TRANSFORMATION. (Fig. 43). Then VY', VZ' are parallel to A'Y, A'Z, and there- fore the angle A' is equal to the angle Y'VZ' ; also v' is pi'ojected to infinity. Hence to project so that a particular line v' may go to infinity, while two angles BAC, QPR assume given magnitudes 6, cj) : — Let the sides AB, AG cut v' in Z', Y\ and let PQ, PR cut v' in R', Q' ; take any plane through v', in this on Y'Z' describe a segment of a circle containing an angle 6, and on Q'R' describe a segment containing an angle

tiMiisfdrniation is a(coni])lished l)y a real j)rojection. Ex. 2. A (luadrilateral can be j)rojccte(l into a square by a real projection. One diagonal must be ))rojected to infinity, and the angles formed by (1) the remaining diagonals and (2) one \y,i\Y of sides must be jn-o- jected into right angles. E.i:. 3. Any conic with any pnint can be prDJected intn a circle and its centre. Take any two pairs of conjugate lines throiigli the point. Project so that these may be at right angles, wliilc thr polar of the point goes to infbiity. Tlie conic remains a conic; tlic point becomes the pole of infinity, that is, the centre ; the pairs of conjugate lines become con- jugate diametei's ; but a conic with two jiairs of conjugate diameters at right angles is a circle ; hence Ihc projection is ai-com])Iished, but it is not necessarily real. PROJECTION AND LINEAR TRANSFORMATION. 193 201. This last example gives the important result that any two 'points can he 'projected into the circular points. For taking the line joining the points, and any conic through the points, the result is attained by projecting so that the conic becomes a circle while the line goes to infinity. Ex. 1. Show that if the two points be real, tlie pi'ojection is imaginary, and vice versa. Ex. 2. Show that three angles with different vertices can be projected into right an tiles. Gomparison of Different Projections. 202. In the proof of the theorem in i^ 200 there is a certain margin of choice in the construction : — (i.) take anij plane through v ; (ii.) project on to any plane parallel to F-?/. Two ditferent determinations under the heading (ii.) give similar figures, the ratio of their linear dimensions being as the distances of the planes from V. It remains to examine the effect of a different determination under the heading (i.). Let the centres be V^ and V.^ ; the planes on to which we project are parallel to V{v' and V^v' , and may be taken to meet the original plane in tlie same line v. Since the figures in the planes V{v', V^v' are exactly the same, one can be brought to coincidence with the other by a rotation about v as an axis. We have to show that a similar thing is true about the two projections. Constructing a diagram like Fig. 43 for each projection, and using suffixes for the points in the projections, we have ZA^:V^Z'^ZA -.AZ', ZA,:V.,Z' = ZA:AZ': now V.,Z'=V^Z\ therefore ZA.^ = ZAy Also lA^ZY=-1\Z'Y', lA,ZY==^V,Z'Y'- now lV^Z'Y' = i.V^Z'Y', therefore lA^ZY=^A^ZY\ and similarly for other lines and angles in the two pro- jections. Hence the second projection is simply the first with its plane turned through a certain angle. Thus the apparent choice relates simply to size and to posi- S.G. N 194 PROJECTION AND LINEAR TRANSFORMATION. tion in space. The projection is absolutely determinate as to shape * when the magnitudes of two angles, and the line that is projected to infinity, are given. The plane of projection may make any angle with the original plane without any alteration in the resulting figure. 203. Finally we may suppose the plane on to which we project to revolve so as to come into coincidence with the original plane ; then the Vv' plane also coincides with this. The whole figure is now in one plane, with its general pro- perties unaltered ; corresponding lines intersect on v, the axis, and corresponding points connect through V, the centre. Thus we are led to the theorem : — When two figures in a iilane are related so that the joins of corresponding 'points are concurrent, then the intersections of corresponding lines are colUnear. The point and line are Poncelet's centre and axis of homology; thus the theory of homology, or perspective, is rendered self-evident by means of perspective in space. Fio. 45. As an example of this plane projection, consider the trans- formation of a (piadrilateral into a s(niar('. *Tlie segments of circles used for the deteiiniiiation of 1' may have their intersections, when these are real, on opposite sides of the line, in which ease only one of the intersections satisfies the conditions of the prolilem unless the .segments be semicircles ; or on the same side of the line. In this case there are two points 1" that might be taken as centre of projection ; and the projection is one of two. This cannot occur when we project so that two angles may become right angles, which is the usual projection. PEOJECTION AND LINEAR TRANSFOEMATION. 195 Let the opposite sides of the quadrilateral ABCD (Fig. 45) meet in E, F, and let the diagonals, which intersect in G, meet EF in N, K. The line EF must be projected to infinity, while the anodes FBE, HGK become rioht auQ-les. (Jn EF, HK, describe semicircles, let these intersect in F; V is the centre of perspective. For axis of perspective any line parallel to EF is to be taken ; take for example the line through B parallel to EF. Let AD cut this in A' ; then A'U goes through X, and is parallel to VE ; hence A'U is con- structed. Similarly the other lines are constructed, and the resulting figure is a square. In Fig. 45, the projections of the lines AD, BA are marked, their intersection is the point A' , which lies on VA, as it should; and completing the figure by means of the projections of BC, CD, the points C, D' will be found to lie on VC, VD. 204. Two distinct projections of a figure, with a common line for the three planes, are projections of one another ; for the common line is the axis of perspectiv^e. Let the two centres be F^, V.^, and let the derived centre be V ; these three centres are collinear. For if the two projections of AB be A^B-^, A.,B.,, then AB, A-^B^, A.,B., meet on the axis of perspective ; therefore the triangles AA^A.,, BB^B., are in per- spective ; and therefore the points {A^A,, B,B.^, {A,A, B,B), {AA„ BB^) are collinear ; that is, V, V^, V.^ are collinear. Alteration mi Appearance caused by Frojection. 205. Each of the two planes is divided into three com- partments by the line of intersection, the line infinity, and the line that is the projection of infinity on the other plane ; and these three compartments of one plane project into the three compartments of the other taken in a different order. In Fig. 46, x J, JX, Xcc in one plane project into /'oo , X X, XI'. The appearance of the figure is most altered by projection when it cuts either the line infinity or the line that corre- sponds to infinity in the projection. To insert the projection of any point 7^ accurately, when one pair of lines FX, FX is drawn, let BM parallel to v cut FX in M : let MV cut F'X in M'; let M'R' parallel to v cut RV in K ■ R is the projection of R. In Fig. 46 two points R, S, off the line FQ, are marked in this way ; these have been chosen in about the same positions relative to F, Q respectively : but being in different compartments, the projections present a diftl'Vent 196 PROJECTION AND LINEAR TRANSFORMATION. appearance. By inserting a few points in the ncighbonrhood of /, on the two sides of the line, the eti'ect of projection (M when the line projected to inhiiity cuts the figun^ will be made apparent. Tlie (!Hect should be noticed for the various PROJECTION AND LINEAR TRANSFORMATION. cases that arise. The line through J may (i.) cut the curve ; let PQ be the tangent : (ii.) touch the curve : let the point of contact be on PQ. (i.) The point may be (ft) an ordinary point, Fig. 47, (i) ft, ft ; (b) an inflexion, (i) h, h ; (c) a crunode ; {d) a cusp, (i) d, d. 197 (i) (ii) (ii.) The line may be (ft) an ordinary tangent, Fig. 47, (ii) ft, ct ; (6) an inflexional tangent, (ii) h, h : (c) a tangent at a crunode : {d) a tangent at a cusp, (ii) d, d. (i.)(c) is not represented in the diagram, as it is simply (i.) («) taken twice ; and (ii.) (c) is a combination of (i.) {a) and (ii.) {a). 20G. Since any conic can be projected into a circle while any desired line goes to infinity, in proving any descriptive theorem for a conic it is suflicient to prove it for a circle. Thus, for example, Pascal's theorem need only be proved for a circle, and the inscribed hexagon can be taken with two pairs of opposite sides parallel ; the truth of the theorem is at once evident, for the third pair of sides is necessai-ily parallel. Again, the constancy of the cross-ratio of a pencil in. a conic is deduced from the constancy of the angles in a segment of a circle. The use of projection in proving properties of conies is fully treated in Salmon's Conic Sections, Chapter XVII., where, in ^ 366, there is the formal proof by elementary geometry that any conic can be projected into a circle. It should be noticed that a system of conies with two common points can be projected into a system of circles, and a pencil of conies into coaxal circles : and that a range of conies can be projected into a system of confocals. Ex. 1. Draw tlie cui-ve {i/z - 2x^){i/z - h.v') =.rv/%. Apply § 205 to determine the a])pearance according as one i>r othei' 198 PEOJECTION AND LINEAR TRANSFOEMATION. of the tliree lines .r, y, z is projected to infinity ; and verify by drawing the carves whose Cartesian equations are obtained In' writing 1 for .e, ?/, z respectively. Ex. 2. Do the same thing with the curves in § 147, Ex. 1. Analytical Aspect of Projection. 207. We have now to consider how projection presents itself as a method in analytical geometry. Let the sides of the fundamental triangle be a, b, c. Sup- pose we are working with actual perpendiculars, then the present line infinity is rta + 6/S + Cy = 0. We project SO that pa + g^+ry = may go to intinity, and so that the triangle may have sides a, h', c. Any straight line projects into a straight line, that is, Aa + B/3 + Gy = becomes A 'a' + i//3' + G'y = ; hence any linear function of a, /^, y becomes a linear function of a, ^', y ; we have therefore a linear transformation. It is a special linear transformation, for the lines a=(), fS=-0, y=0, are to become u' = 0, /3' = 0, y' = ; hence the transformation is of the form a =Xa , ft = julIS , y ~ vy . By this the line is to become 2Ja + (//3 + ry = therefore pK : qfx : tv = a' : b' : c, that is, hence the traiisf( jrm . a b' c A : // : 1/ = - : - : - ; ^ p q r ation is a a! , // c p ^ q^ " r Now a, b', c\ ij:q:r may initially be chosen arbitrarily; consequently , -, can Ik; made to assume anv values A '' J) q '"' l,m,n; and the formulae of transformation arc a = la', ft = on ft', y = n y. Hence the use of ])rojection in space shows that if wo prove a projective theorem using actual perpendiculars, the same PEOJECTION AND LINEAR TEANSFORMATION. 199 work proves the theorem when tlie coordinates are any multiples we please of the perpendiculars. We are therefore independent of the nature of the coordinates in dealing with descriptive and projective metric properties ; that is, we may take any equation we please for the line infinity. 208. In the case of plane projection, it is better to refer both figures to the same fundamental triangle. Let the axis be the line x goes through the intersection of this and the line x, hence X = px + A(/b + (jij + ]iz)^ y'=qy+^l{f'''+oy+^'-)^ (i-); z' = Tz + v{fx^-ijy + hz),] the multipliers implied in x, y', z may be chosen so that \, fx, V, =1, and then the formulae of transformation become x'^fx^gy ^Uz,\ y'=f^ -^gy-^hz, ^ (h.). z'=fx +gy +h'z] The line joining the point A' {i.e. y' — 0, z = 0) to the point A is {9-!/)y = {^''-f>'')z^ and similar equations hold for BB', GC" ; hence the centre of projection is given by the equations (/-/>== (y-i/')2/ = (/^-/0^ (iii-)- The interpretation of these formulae is that if F{x, y, z) = i) be any curve, then the equation of the curve obtained by a certain projection is F{x', y', z) = 0, where x, y', z are ex- pressed linearly in terms of x, y, z by means of equations (ii.). Hence projection is accomplished by means of linear trans- formation. 209. As an example, let it be required to project the curve x^-{-'lxy--2x^-'Mj^-\-x^0 (i.), so that x-\-y — \ = may go to infinity, the curve remaining unaltered in the immediate neighbourhood of (J. To do this, we (1) change to homogeneous coordinates, and then (2) change the triangle of reference in part so that tlie assigned line may be the third side; and finally (o) project this third side to infinity. By (1) equation (i.) becomes x^-\r1xy^-'2.x^z-'6yH-\-xz^ = (ii.). 200 PROJECTION AND LINEAR TRANSFORMATION. The line x-\-y — z = is to be taken as the line s' ; hence we have the equations X = \x', y = fiy', x-\-y — z = vz. As the equation is to be unaltered for small values of x and y, X=l, ^ = 1. Noticing how the line z, that is, x-{-y — z — (), lies, it is seen that in order that a point inside the new triangle may have its three coordin- ates positive, V must be negative ; for x-^y — z is negative at a point inside this triangle. Write v= — p, and the formulae of transformation become Fig. 48. , , , , , x = x, y = y, z = x+y +2^z. Making these substitutions and dropping the accents, equa- tion (ii.) becomes x^ + 2xy-'-2x%x + y+2y'^)-Sy\x + y+ir,) + x{x + y+p-y = 0, which reduces to -iif + (2xy-Sy^)pz + x{pzf = (iii.). This is the result of step (2) in the process ; (3) is accomplished by w^riting for z any convenient constant (see § 30) ; we write therefore j:;s = l, and the equation of the projection becomes Sy'^ - 2xy + 'M/-x = {) (iv.). This can be verified by drawing the curves (i.) and (iv.), and applying § 205, by means of which it will be seen that the change in shape is what it should be for the given pro- jection. The second curve has a node at infinity, the two tangents being the line infinity and a line parallel to the axis of x. In general, unless the line that is to go to infinity passes through 0, it is best to keep x, y unchanged. Let the line to be projected to infinity be ax-\-hy = cz, then the forinuhe of transformation are ,'• = X, y = y, - ax - by -}-cz= 2)z', from which ez = ax' -f- hy'-\-2^^' ! here the sign of 2> nuist be carefully doteruiinod, as in the ex- ample given above ; the numerical value is iruliff'erent, for at the end any convenient constant is written for z, and there- Fore for pz'. Thus 2? can be taken numerically unity, but it may be +1 or — 1. PROJECTION AND LINEAR TRANSFORMATION. 201 Ex. Find formuhe for projecting (i.) 3.i+3y + l=0, (ii.) .f-y + 2 = 0, (iii.) ^■- 1 =0, to infinity ; and apply these as follows : — (i. ) to x{x - yf + Ix^ - Qxi/ + 'ix + 3_y + 1 = ; (ii.) to x{x -yf + Ix- - Ixy 4- 3.?; + 3j/ + 1=0; (iii.) to x{x — yy^ + x^—y^+y = 0. Verify the results by drawing the curves, and applying § 205. General Linear Transformathn. 210. Projection has here presented itself as a specialized linear transformation ; before considering to what extent the transformation is specialized, we must consider the effect of linear transformation in general. In this we write x' — l^x + ni^y + 'i^iS,"! y' = l.,x + m.,y + n.,s, - (i.), where the coefficients are independent. But these are the formulse for changing the fundamental triangle from xyz to x'y'z' , where a;' = is the line lYtl^m{^J^n^z = K^. For example, let it be required to transform so that the new triangle of reference may be formed by the tangents 2X A^B, G to the conic F=^fyz-\-yzx + kxy=^(}. ii formation can be written These tangents are -+ r = 0, etc., hence the fornmlse of trans- ^ (I k ^ + ^-2Aa;', -.+ ,--2^7/', +-/ = 2,/5 X whence ,. = — X.i'' + /x//' + vz, etc. Suppose for clearness that /, g, It are all positive, st) that tlie jiosition of the new triangle with regard to the old is that repi'esented in Fig. 49 (a). A point inside the original triangle is also inside the new triangle, and therefore A, /a, v are all positive. Making these substitutions, and arranging, the eipuition of the conic becomes F' = X'x'-' + fjC'y'-^ + v'z^ - lixvyz - 2vXz'x' - 2\iuxy' = 0, the ordinary form for the ecjuation of a conic inscribed in the triangle x'y'z . 202 PROJECTION AND LINEAR TRANSFORMATION. These two equations F=0, F' — have been found as two different equations of one curve, with two different triangles of reference. But (h'opj^ing the accents from x', y', s', the second equation has a meaning when referred to the original triangle ; it represents a conic inscribed in this triangle. Thus the two equations ^=0, F' = 0, may be connected in two different ways, as represented in Fig. 49 ((0 and (6). 211. Adopting the second way of regarding the two e(|ua- tions, we proceed to consider more carefully the corre- spondence of the two figures, these not being restricted to the special example of the last section. In the one figure F, we have a point x, y, z ; from this, by equations (i.) of !^ 210, we derive a point x , y', z' : hence point corresponds to point. Moreover, the equations of trans- formation being linear, straight line corresponds to straight line. Hence all properties of collinearity, concurrence, order, class, etc., that is, all descriptive properties, arc the same for the two figures. Further, cross-ratio is unaltered ; for if linear expressions u, v become i'/, v , expressions n-^-hv, u-\-lv become u' + kv', u'-\-lv; and the cross-ratio of the configura- tion, being k : I, is unaltered. Hence when from any figure another is derived by linear transt'ovmution, the two agree as to all projective properties. Comparison of Projection and Linear Transformation. 212. The number of constants involved in the general linear transformation being \), the number of disposable constants is 8, for we are concerned only with ratios. Jn th(j foniudae PROJECTION AND LINEAR TRANSFORMATION. 203 for projective transformation, § 208, (ii.), the number of co- efficients is 0, and there are therefore 5 disposable constants; projection presents itself as a special case of linear trans- formation. But the difference is simply one of position. Let a tigure $ be derived from F by linear transformation. Suppose that F by projection becomes i^^, move F-^ aljout in the plane in the most general way possible, that is, turn F^ through an arbitrary angle Q, and bring any assigned point of F^ to an arbitrary point whose (Cartesian) coordinates are a, b. Let F-^ in this new position be called F'. Hence F has become (1) projection, involving 5 independent constants, (2) displacement, involving 3 independent constants ; that is to say, F has become F' by a linear transformation involving 8 independent constants ; and these can be deter- mined so that the transformation may be the one by which is derived from F. Hence , derived from i^ by a general linear transformation, differs only in position from F^, a projection of F: and the general linear transformation, as a process for discovering geometrical theorems, is not one whit more general than projection. For drawing a figure on a different sheet of paper, or on a difi'erent part of the same sheet, tells us nothing new about it. For example, take the case of a quadrilateral and a square. Draw a square (i/3y8 of any size anywhere in the jjlane of the quadrilateral ; by rotation and translation this can be brought into perspective with the quadrilateral. Construct for T as before ; turn the square al)out so that 0y may he parallel to VE (Fig-. 45) ; move the square parallel to itself so that f3 shall remain on VB ; slide it along VB until y is on VC ; then the square is in perspective with the quadrilateral. Ex. Show that any two quadrilaterals can be placed in perspective. 213. Hence we see that the methods of linear transforma- tion and projection can be connected in one of two ways. From an equation ^=0 let a second equation F' = be derived by a linear transformation. Draw separately the two curves C, C, represented by these two equations. Then (1) C, C can be placed in space, or in a plane, so that one is the projection of the other : (2) placing the two diagrams in the same plane, with the lines X, y, z in the same position, a geometrical construction in the plane can be found by which the points of C" are obtained from the points of C. 204 PROJECTION AND LINEAR TRANSFORMATION. But also the two equations F—0, F' = can be regarded as representing the same curve, with a ditterent triangle of reference. 214. Stating the plane construction for projection in the simplest possible way, it is : — Let be the centre, so that all lines through correspond to themselves ; let any line through meet the line that is to go to infinity in N ; let NP meet the axis in J, through J draw a parallel to ON meeting OP in P' : then P' is the projec- tion of P (Fig. 50). The line taken for ON may be any line through ; the same point P' will in any case be found. It is often convenient to take the axis to pass through 0. The assumption that all lines through can be made self -correspondents is justified by the possibilities of projec- tion. For let any three lines through cut the line that projects to infinity in X, Y, Z\ we can then, by the general theorem of § 200, project so that the angles XOY, YOZ retain their values, while the assigned line goes to infinity. Let OW be any fourth line: the pencil {O.XYZW} "is unalterable by projection; and consequently the angle ZOW is unaltered. 2L5. This construction enables us to determine the formulae of transformation moi'e simply than by the process of j^ 209. Let tlie line that is to be projected to infinity, and the line that is taken as axis, be ax + J)y = I , ax -f l)y = c. Take the axis of x for ON; let P be x, y and use X, Y for current coordinates. The equations of the lines of the dia- gram, in the order used in the construction of § 214, are; 1 NP It Y x— '^ a JP'. Y{ax + })y - 1 ) = ij{c - 1 ) X y ' OP, PROJECTION AND LINEAR TRANSFORMATION. 205 the last two give the coordinates of P' : writing x', y' for these, the relations are , x(c — l) , vic—X) ax -\-hy — V "^ ax + hy — 1 ' and X = y ax'-\-hy' — c-\-V "^ ax' -\-hy' — c-\-\' if the axis be taken to pass through 0, f = 0, and the relations become _ a?' _ y' ax -\-hy' -{-V "^ ax -{-hy' -^-X Thus in the example of § 209, x' y' 00 = —, ; , V = —r ' x'+y' + V '^~x'+y' + l Again, to project Zx + Zy+\ =0 to infinity, we apply tlie transformation x' if y= -3^'-3y + l' ^ -3a,-'-3/+l' and to project x = \ to infinity, x' _ y' '^~x' + V ^^~x' + \ 216. It is important to notice that a Cartesian transform- ation is limited by the condition that the line infinity is to be unaltered. Changing to homogeneous coordinates as directed in § 30, this is expressed by z' = \z. Hence we have only six constants at our disposal, two involved in the determination of each new axis, one for the multiplier implied in each coordinate ; though in the strict Cartesian use of coordinates, these implied multipliers are not available. If the transform- ation be further limited by the condition that it is to be orthogonal, — that is, that the axes, being initially at light angles, are to remain so, — one degree of choice is destroj'od, and we have three constants at our disposal. The condition that the transformation is an orthogonal Cartesian transform- ation may be expressed in the form — the circular points are to be unchanged. For they are the intersections oi lines parallel to x^ + y'^ = 0, and it is known that this equation is unchanged if the rectangular axes be turned through any angle. Canonical Forms. 217. One object of projection is to reduce the figure to its simplest form, a conic to a circle, a (juadri lateral to a square, harmonic section to bisection, etc. Similarly one object of linear transformation is to I'educe an ecpiation to 206 PEO.TECTION AND LINEAR TRANSFORMATION. the most manageable form : tliat is, to clioose the best triaiiii'le for fundamental triana'le, and tlic Ijest system of coordinates. To tell a priori whether one ecjuation can be reduced to anotlier, we must compare the number of disposable con- stants in the two. For example, using Cartesians, the general equation of the second degree contains live constants : this same numljei- is contained in each of the forms (^x-ay + {y-l3f = {px + qy + ry (1), {0^'-a)H (.7-/9)^4+ {G>--«? + (y-/3?}' = ^ (2): but the forms (ax + byf = cx + dy + e (8), (ax + hij + c)(a.r + h'y + (■') = (4), contain only four disposable constants ; and (x-ay + (y-^f = r^ (5), ix-ay' + {y-l3)- = ax + by + c (6), contain only three. Hence the general equation of the second degree can be written in the form (1), which gives a proof of the focus and directrix properties ; and it can be written in the form (2), which shows that the sum or difference of the focal distances is constant : but it cannot be written in the other forms. 21(S. In homogeneous coordinates the question presents itself in a slightly disguised form ; the disposable constants are in- volved in the implied change of the tiiangle of reference. When we say that F(x, y, z) — can be written in the form #(.'c, y, s) = we mean that a linear transformation X — l{x -f m^y' -\- n^z, z = l^x -{■ ')n.^]j' -f n.^z, can be found by means of which F{x, y, s) = 1)ecomes ^{x', y', z') = 0, that is, dropping the accents, '\>{x, y, z) = {). In this eight C(jnstants are involved, and the (piestion is whether these can be determined so as to change F into $. For example, x^ = yz is a perfectly general form for a conic. For this is {l^x -I- m{y -{- v^zf ^{l.,x + vwy + n.,z){L^.c -\- vi.^y + n.^z), that is, A (x -\-\y + fizf = (x + A'// + ij.'z )( ,r + A"// + ,/'^), and it therefore contains seven disposable constants where- PROJECTION AND LINEAE TRANSFORMATION. 207 with to satisfy the five equations obtained by coniparin<;' it with any given equation of the second degree. The reduction can therefore be accomplislied in a doubly infinite nuujber of ways, as is apparent geometrically, all that is necessar}' being to take two tangents for the lines y, z, and their chord of contact for the line x. Again, the constants implied in ^..2 + 2/2 +,.2^0 are involved in such a way as to give eight disposable constants. Hence the reduction can be acc, -1. The ordinary fdrniula for the tangent gives - imoy.v + oj-// + i — 0, that is, --2rnorx + >/ + o):=i), which meets the cubic where (// + (o:)-' = 0, that is, at an inflexion. Tims there are nine inflexions, lying by threes on the three lines .r, y, z ; but only three are real, viz., those lying on tlie line .v+y + z = 0. The formulai of transfoi'mation can be written .'■+ y+ r = 8X.r', \ .'•+ w// + w--=8^.?/', \ (i.): x-\-ury-\- wz = '^vz\ J where A, ju, v have any values Ave please, sa}' unit3\ The equations give « = .'"'+ y'+ :', \ qj = x' + (,<'y'+ (oz'. ^, (ii.). z = x'-{- o)y'-\-ort', j Now x^'+y^+z^Sxyz becomes 27x'y'z', by multiplying together equations (i.): and by multiplying together (Mpiations (ii.), xyz becomes x'^-{-y''^-\-z'^ — Hx'y'z'. The given equation can be written .'-■■'■ + '^■' + -'2^- Sxyz + ({im -\-'A):ryz = 0, PROJECTION AND LINEAR TRANSFORMATION. 209 and tlie transfonued e(:|uation is therefore ^jyy'z + 8( 2vu + 1 )( ^'y' + ]/'' + z'^ - ^xy'z) = 0, that is x^ + if^ + z"-" + G^^^x'y'z' = 0. 1 + 2m "^ Hence, just as before, the nine inflexions lie b}' tlii'ees on the lines :>■', y', z ; that is, on the lines ^•+ y-^ 2^ = 0, .'■ + ("-?/+ (o~ = 0. Ea\ Find what tlie ec[uatioii , r-^ + y ^ + 2^ + Qmxyz = becomes, when the triangle of reference is that formed by the lines — 'imx+y + z = 0, .v-2my + z = 0, x+y -2mz = 0. CHAPTER XL THEOKY OF COEEESPONDENCE. Special Cases of (1, 1) Goi'respondence. 221. The formulEe for projection do not depend on any- special figure that is to be projected: corresponding to any . point P in one plane there is a point P' in the other plane, obtained by means of a line VP through a fixed point V; and in the limiting case when the two planes coincide there is a construction, depending on a fixed point and a fixed line, by means of which P' is derived from P. Now a correspondence of this nature, in which one point of one system corresponds to one point of another, has been con- sidered already (§§ 160 etc.) under the heading homography. A (1, 1) correspondence between the elements of two one- dimensional spaces is homographic ; and in the theory of projection and linear transformation we have the extension of the idea of homography to two-dimensional spaces ; and similarly a homographic correspondence can be introduced between the elements of two a-dimensional spaces. More- over, the elements compared need not be of the same nature ; the two-fold infinity of points in a plane may be associated with the two-fold infinity of points, or of lines, in a plane ; or with the conies of a net, or in sfeneral with the elements or any two-dimensional space. 222. We shall now consider certain cases of (1, 1) corre- spcjiidence between two planes, and shall in general suppose the planes superimposed. Projection, tliat is, a specialized linear transformation in point or line coordinates, institutes a correspondence of point to point, and straight line to straight line; this is a special case of the general (1, 1) linear correspondence, the Collincntion of Mobius {Der barycentrische (Jalctd, 1827 : WerJ,-e, t. i., p. 266). A second case that we shall consider is tliat where point corresponds to point, but a strai ; then P' describes a conic whose focus is 0, the directrix being the line through N parallel to the axis. The proof of this by elementary geometry is ])erfectly simple; the proof by modern geometiy is here given. The circle cuts the present line infinity at the circular points oj, oj', and Ow, Oca' are tangents, coco' being the chord of contact. Lines through transform into themselves, and the line om transforms into the line through JV parallel to the axis ; hence thougli the circular jjoints aie changed, the isotropic lines through are unchanged. By the linear transformation in question, the circle 1)ecomes a conic ; is the intersection of isotropic tangents, and is therefore a focus ; and the specified line through ^V, being the polar of the focus 0, is the directrix. We have here seen that the points of a plane can be associated in pairs. When the elements of any two-dimen- sional space are associated in pairs, they may be said to be in involution ; and, exactly as in one-dimensional geometry, involution can be regarded as a special case of homography by the device of counting every element of the space twice, though this disguises the fundamental idea in involution. 224. The harmonic transformation, just given, is the only linear transformation by which the points of a plane can be arranged in involution. For without making the assump- tion that two figures are in perspective, let A A', BE', etc., be pairs of correspondents. The transformation being linear, the line AA' corresponds to itself as a whole, and similarly for BB': hence the correspondent to 0, the intersection of these lines, must lie on A A' and on BB'\ is therefore its own correspondent. It is consecpiently a double point of the involution on AA', and of the involution on BB': and by the harmonic properties of double points, the other self-coi're- spoiidents on AA' and BB' are consti'ucted by means of a diagonal of the (|u;idi-aiigle AA'BB'. Let them be A', ]'': on THEORY OF CORRESPONDENCE. 213 this diagonal there is an invohition, with X, Y as double points : and the line corresponds to itself as a whole. Hence the point in which XY meets AB corresponds to the point in which XY meets A'E. But these points coincide at C (Fig. 52). Thus in the involution on XF one point, other than the double points, coincides with its correspondent, and therefore every point on the line is its own correspondent, and consequently every line through is its own corre- spondent. To construct the correspondent to any point P, let OP meet XY in iv : 0, R are the double points of the Fig. 'yl. involution on OP, in wliich PP' are correspondents ; hence {OR, PP') is harmonic, and the points of the plane arc con- nected by the harmonic transformation, as are also the lines of the plane. 225. From § 223 we see that if it be known that two figures are in perspective, then the fact that one pair of points are conjugates proves that the figures are in involution ; and § 224 shows that if it be not known that the figures are in perspective, but only that there is between them a (1, 1) linear correspondence, then the fact that two pairs of points are conjugates proves that the hgures are in involution. In one-dimensional geometry, two homographic i-anges can always be placed so as to be in involution : but there is no corresponding theorem in the present case. Homograpliic figures can be placed in perspective, but we lia\e seen tliat figures in perspective are not necessarily in involution. 214 THEORY OF CORRESPONDENCE. 226. In homographic systems in the same one-dimensional space there are two elements that coincide with their cor- respondents. We now consider what elements are self- correspondent in the general linear transformation. Since both point and line coordinates are subject to linear transformation (§ 84), we nmst expect a certain number of points and the same number of lines to be self-correspondent. Now ii A, B correspond to A', B', the line AB as a whole cor- responds to A'B' as a whole: thus the self-corresponding lines are obtained by joining the self-corresponding points in pairs. Let A, B be two self-corresponding points : on ^4 B we have two homographic ranges with A, B as double points ; hence on the line AB there are ordinarily no more self-correspond- ing points, that is, not more than two points on a line or two lines through a point can be self-correspondent. Hence if there be any self-corresponding points and lines, there nmst ordinarily be three of each, these being the vertices and sides of a triangle J. i?C'; two of the points, situated on a real line, may be imaginary, and then the sides opposite to them are imaginary with a real intersection. But now suppose that a third point on the line AB corre- sponds to itself ; then every point on A B corresponds to itself. If there be no self-corresponding point oft" this line, the triangle ABC of the general case has its three sides co- incident. If there be a self -corresponding point C off the line, then every line through C corresponds to itself ; the sides a, b and the vertices A, B oi the triangle ABC are in- determinate, but the side c and the vertex C are determinate ; this last case is simply projection. The fact tliat tliere are self-corresponding elements appears at once from the equations for linear transformation. These give x:y:z = l^x -\- iii^ij -f- n^z : l^x' + m.jf + n.^z : l.^x + m.^y' -\- n.^z\ and we are to liave x:y:z = x':y':z'\ therefore lyX+7 )i^y + n.^z _ l^x + mc,y + n<^z _ l^x + m^y + "J^s^; _ .^ X y z ~ ' that is, (^'1 - A).'- -f m^y -I- n^z = 0, Lr)o-\-{m.., — \)y-\- 7i.,z = 0, l^x + riuy -f {ii., -\)z= 0, from wbiclt ])y eliminating x, y, z a cubic equation is obtained foi- A. The tJn-ee roots, of wliich one is necessarily real, give THEORY OF CORRESPONDENCE. 215 three sets of values for x, y, z, and therefore three self-corre- sponding points A, B, G. Taking the triangle ABC for tri- angle of reference, x = corresponds to a;' = 0, etc., hence the formulas of transformation reduce to a^^fa', y=9y', z^hz'. But if the cubic in X have two, or three, equal roots, a certain degree of indeterminateness is introduced. To illus- trate this point, consider the transformation x — x' + az', y = y'-\- hz', z = z. The cubic in X has its three roots equal ; the three points ABG are indeterminate, all lying on = 0. Again, consider the projective transformation x = kx! -{■ y'+ z', 2/= x' + ky+ z', z= x'+ y' + hz. The cubic in X has now two equal roots, ^ — 1 , the third root being k + 2. The non-repeated root gives a definite point G (1, 1, 1); the repeated root gives simply x-\-y + z = (}, the line c. Examples. 1. Show that the eliect of rotation on a hgure can be represented by a linear transformation, in which the three fixed points are the circular points and the centre of rotation. 2. Discuss the case of translation. 3. Show that in any rotation the line infinity has its points displaced along itself; and that in any translation the line infinity is entirely unaltered. 4. In the case of harmonic transformation, if the axis be at infinity, the centre is a centre of symmetry; and if the centre be at infinity, the axis is an axis of symmetry. General Theory of Gorrespondence. 227. The general theory of correspondence may be presented in a slightly different way. Let there be a system of elements, forming a p-iold inhnity ; any one of these may be indicated by the ratios of p-\-l parameters; calling these X, /x, »/,..., cr, the element is \, ix, v, cr. Taking any other p-io\d infinity, the element X, n, v, ..., o- of one system can be regarded as corresponding to the element X, ^t, i^, ..., cr of the other. For example, let tS-^, S^, S^ be conies ; consider the net ^16 THEOHY OF CORRESPONDENCE. in this any conic can be re<;ar(le(l as the element X,iul,v\ it can be considered as corresponding to a point \, ju, v, or to a, line or more generally to a curve This view of correspondence is somewhat disguised in the ordinary presentation of projection ; it appears more plainly in the interchange of point and line coordinates. In this the marks X, ju, v are attached on the one hand to a point, on the other hand to a line, as coordinates or as coefficients. The algebraic statement of a theorem relating to one set of elements with characteristics X, fx, v, ..., a- may be exactly the same as for a totally different set of elements with the same characteristics (compare § 53) : we have then a means of generalizing. From a geometrical theorem relating, for example, to lines \x + ljiy^-vz = 0, we pass to a theorem relating to conies X>S'j + IJ.S.2 + vS^ — 0. This transition is not by means of deduction, but it depends on an implied reference to a second interpretation of the algebraic work by which we know that the theorem can be proved, even though the proof actually given may be geo- metrical. 228. But we can regard this principle of correspondence in a more exclusively geometrical way, without even this implied reference to analysis. A figure in any space is generated by a moving element coinciding successively with certain of the ultimate atoms of the space, according to some law of selection. Now let there be another space, with ultimate atoms of a different kind (or of the same kind), the ordei' of the manifold- ness being howe\ei' the same ; and imagine a moving element to coincide successively with cei-tain of these, the law of selection being the same as before. The behaviour of the second generating element and the laws oi the second con- figuration can to a certain extent be deduced from the behaviour of the first generating element and the laws of the first configuration : tlieorems proved foi' the first configuration afford tlieoi'eiiis relating t(^ the second configuration. The practical ditficulty that presents itself here is in the ex})ression of the law of selection. This iinist be expressed in terms of the elements of the space. Suppose for example that the correspondence considered is between the points of a THEORY OF COKRESPONDENCE. 217 plane and the lines of a plane ; and that it is of such a nature that to a line in the first plane corresponds a point in the second ; this may be called a linear aggi'egate of elements. Now let a point describe a conic in the first plane ; this can be expressed in terms of the elements of the plane. The moving element coincides successively with a singly infinite immber of the fixed elements, these being chosen so that there are two belonging to every linear aggregate. The law of selection, so stated, is referable to the second system, and gives an envelope of the second class. (Compare § 54.) 229. We may if we choose concentrate our attention entirely on the laws and operations manifested in these ditierent geometrical figures ; these figures with their related theories are then unimportant transi- tory incarnations of an underlying unchangeable princijjle ; the differ- ences perceived by the eye are neglected and tlie figures are regarded as the same, inasmuch as the}"^ express the same sequence and connection of elements. From this point of view a curve and all its projections, or a curve and its reciprocal, are essentially identical ; a system of lines in a plane may be identical with a system of conies in a net. This view pervades a great deal of recent work ; it is formulated by Professor Klein in his address Vergleichende Betrachhingen Uber neuere geometrische Forschungen, 1872*: — " Streifen wir jetzt das mathema- tisch unwesentliche sinnliche Bild ab ... . Aber die projectivische Geometrie erwuchs erst, als man sich gewohnte, die ursprlingliche Figur mit alien aus ihr projectivisch ableitbaren als wesentlich identisch zu erachten.... Wenn wir im Texte die raumliche An- schauung als etwas Beilaufiges bezeichnen, so ist dies mit Bezug auf den rein mathematischen Inhalt der zu formulirenden Betrachtungen gemeint. Die Anschauung hat ftir ihn nur den Werth der Veran- schauliclmng." We are however here concerned with the manifestations of the underlying piinciples and operations ; these last exist for us only as the cause of the correspondence that we consider. General (1,1) Quadric Correspondence. 230. Important examples of this deduction of one theorem from another occur in the use of a particulai' (1, 1) corre- spondence, geometrically represented by Quadric Inversion. This presents itself as the next simplest case to projection ; it is a correspondence of point to point, and of straight line to conic, that is, it is a (1, 1) (juadric correspondence. Since the number of straight lines in a plane is doubly infinite, the conies considered must form a two-fold iniinity; the equation of a line being * For a reprint of tliis address, see \). 63, t. xliii. uf tlie Jlat/wimi- tische Annalen ; for a translation, see }>. 215, vol. ii. of the Bulletin of the New York Mathematical Society. 218 THEORY OF CORRESPONDENCE. where Z^ Z^, X.^ are three particular lines corresponding to the conies >S'j, *S'^, S.^, the corresponding- conic is But by the given conditions, as regards points the corre- spondence is (1, 1) ; now the intersection of two lines corre- sponds to the intersection of the corresponding conies ; hence of the four intersections of any two conies, all but one must be automatically excluded : three must be fixed, and one variable. Hence all the conies considered nuist pass through three fixed points. These points ordinarily form a triangle, but there are special cases that may be considered, arising from coincidences. At j)resent we consider only the general case. Taking this triangle as triangle of reference in the conic- plane, the conies ^S^^, So, S^ have equations hence to the line \L^ + /ulL.^-\-i/L.^ = there corresponds the conic that is, the conic ( Vi + f^f-2 + j/s)?/^ + i^'Ji + 1^0-2 + ^iJ^-'' + (^^^1 + M'''2 + v^f-i)'^y = 0- Now change the triangle of reference in the line-plane by means of the formulae h=h^^+9-2y'+K^'^ the correspondence is then between the straight line ( Vi + m/2 + j^/sK + 0^9i + ^92 + vy^y' + (A/' 1 + /x/i, + vh.,)z = 0, and the conic (X/i + iJif.y + v^Qijr. + (\9i + Kh + v, Bi, D.^. Let I be any line in ttj ; a, b, I determine a hyperboloid of one sheet, and this is cut by -rr^ in a conic ; thus the lines in ttj are projected into conies in 7r2. Now whatever line I we take we can draw from J^ a line to meet b and I ; hence the conic projection of every line goes through A.,-, and similarly it goes through B.^^. Also the line A^B^ meets a, b, and I ; if therefore AyB^ meet tt.j^ in C^, C.^ is a point on the conic projection of I. Hence all the conies considered jjass through three fixed points A.,, B.,, G-i- Special cases^ arise owing to special relations among the given fixed elements ; for example, if the two given lines a, b meet in a point I', this skew projection reduces to the ordinary, or conical projection. The transfoi'mation of figures thus obtained is the Steiner transformation ; it was fully explained by Steiner in his Systematische Entv:icktiuny tier Abliiingigkeit geometrischer Gestalten von cinmider, Berlin, 1832. Ues. Werke, i. i., pp. 409, 421. Quadric Inversion. 232. We now consider a plane construction by means of which this quadric correspondence can be produced ; this was given by Dr. Hirst, in the Proceedings of the Royal Society, 1865. The correspondence of point to point is effected by means of a fixed origin 0, and a fixed fundamental conic, the base : points that are collinear with the origin, and con- jugate with respect to the base, are said to be inverse : if for the fundamental conic and the origin we take a circle and its centre, the points are the ordinary inverse points with regard to a circle, hence the process is simply circular inversion generalized, that is, it is quadric inversion. As regards points, the correspondence is (1,1); moreover it is reversible, and it associates the points of the plane in pairs PP' ] that is, to a point P of the plane there corresponds definitely a point P'. There are however exceptional ele- ments ; for let the tangents from to the base be 01, OJ ; by the definition, P' , inverse to P, is the intersection of OP and the polar of P ; this is indeterminate (1) if P be at ; then P' is any point on /,/ ; (2) if P be at / ; then P' is any point on 01 ; (3) if P be at J\ then P' is any point on OJ. Hence if P be at any vertex or on any side of the triangle 220 THEORY OF COERESPONDENCE. OIJ, tlie ordinary laws of the correspondence are not appli- cable. For the present these special positions of F are not taken into account. To consecutive points correspond consecutive points ; hence an arc of a curve gives an arc of a curve, and arcs intersecting or touching give arcs intersecting or touching. Let F^, F.y, P3, P4 be points on a line q, meeting the base \\\ A, B; let" the pole of q be Q, and let the polars of F^ etc. be p^ etc., so that F[ is the intersection of OF^ and i\. Then [F,F,F,P,] = {p^p.,p,lh)- now the four lines Pv 2^i^ Ps^ 'Pi ^o^'^^^ ^ pencil whose vertex is Q, and F[, etc. are points on these lines : hence the relation just given can be written {O.F,P,F,F,} = {Q.F{F!,FiF',}, that is, {0 . F[F'F',F',] = {Q . F{F',F',F',]. This shows that the points P', inverse to collinear points P, lie on a conic through 0, Q] and as any point on the base is its own inverse, the points A, B belong to this conic. Let the line q cut 0/ in X ; by the general construction the inverse to A'' is at /, hence the conic passes through /, and through J. The inverse to any line q is therefore a conic through the three principal points. Moreover we have here shown that the cross-ratio of the four points on the line is equal to the cross-ratio of the pencil determined in the inverse conic by the inverse points. Hence points in involution on a straight line give rise to points in involution on the inverse conic : the transformation is in fact homographic. Note. is till' origin, neressurilv iral : /, •/ are the fiuulameiital points, tlu'v may he real or imaginary ; 0, /, ./ are the principal points. The three sides of this triangle are the principal lines, and of these (>/, U-) may be distinguished as the fundamental lines (Hirst). 283. Hence the correspondence is that of straight line to conic, and the conies satisfy the condition of § 230. Take 0/, OJ, IJ for the sides x, y, z of the triangle of reference, and let the coordinates be chosen so that the fundamental conic is z^_^^ = 0. Let F' be a;', y\ z : tlic polar of F' is _.,._y'_,y,,.'4-2::' = 0, and the ('(inatioii df UF' is THEORY OF CORRESPONDENCE. 221 hence the coordinates of F are given by x:y:z = M :y : ^, = x'z' : y'z' : xy'. Applying to x', y\ z a linear transformation X :y : z =y^:x^: z^, the formulae become 111 x:y:z = — : — :— : «i 2/1 -1 hence the geometrical transformation now under investigation gives the general (1,1) quadric correspondence of § 230. 234. In the analytical theory special cases arise owing to special positions of tlie three principal points ; (i.) the three may be distinct: (ii.) two may come together : (iii.) the three may come together. In the geometrical theory, we naturally discriminate accord- ing as the fundamental conic is proper or degenerate, and according as it does not or does pass through 0. Hence four cases present themselves : (1) the base may be a proper conic not passing through the origin : (2) the base may be a degenerate conic, not passing through the origin : (3) the base may be a proper conic passing tlu'ough the origin : (4) the base may be a degenerate conic, composed of two straight lines, of which (a) one passes through the origin : tliis is simply harmonic transformation (^ 223) ; for the base being JO, JV, the inverse to P is the intersection of OP with the polar of P; let OP meet JV at V, then (OF, PP') is harmonic; (6) both pass through the origin ; the ti'ansforina- tion becomes indeterminate, for if P be at 0, P' is any point in the plane : and if P be not at O, P' is at 0. The first three cases correspond in order to the tlnve cases in the analytical theoiy. Foi' in (1), the tln-ee points (), /, J are distinct : in (2), the points /, J coincide, as may be seen 222 THEORY OF CORRESPONDENCE. by considering a hyperbola on the point of degenerating into two straiglit lines; and in (8), the points /, 0, J are con- secutive points on the fundamental conic, a fact which is made evident by a diagram showing tangents 01, OJ drawn to a conic from a point just oft" the conic. The equations for (1) are given in § 233 ; to deal with case (2), in which 7, J coincide, take for the sides x, z, y, the line OJ, the polar of 0, and any arbitrary line through 0. The fundamental conic is a pair of lines throvigh xz, harmonic with respect to X, z[ hence by a proper choice of coordinates its equation is made to be x^-z^ = 0. The polar of P'ix, y', z) is x'x — z'z = 0, and the equation of OP' is y'x — x'y = 0, hence the point P is given by x:y -.z^x' :y' \ — , = xz :y'z' : a?'- ; from which x -.y : z =xz:yz\ a?. In case (3), taking for x, y, z the tangent at 0, any cliord through 0, and the tangent at the other point where this chord meets the conic, the equation of the conic is of the form ky^-~2xz = Q, where /.' is at our disposal ; if P' be x, y', z, the polar of P' is zx-ky'y + xz = 0, and the lint' OP' is y'x — x'y = 0. Hence the point P is given by x:y:z = x^-\ x'y' : ky'^ — x'z', fi-om which x' : y' : z = X" : xy : k\f- — xz. 235. We have seen that in general the inverse to a straight line is a conic through 0, I, J. But if the straight line })ass through 0, then the pole Q is on 7./, and the conic OIJQAB has three points on the line PI ; it is therefore degenerate, composed of /./ and the given line q. IJ presents itself here simply because the iuNei-sc to is indeterminate, being any point on /./ : and similai'ly whenever a curve to be invei'ted passes throngli 0, the line /•/ presents itself as part THEORY OF CORRESPONDENCE. 223 of the inverse. Also a line through /, meeting the base again in H, has for its inverse the degenerate conic com- posed of 01 and JH\ for the point Q is now on 01, hence the points 0, I, Q on the conic are accounted for by the line 01, and the remaining points J, H give the other line of the degenerate conic; and similarly if a curve pass througli / (or J) the line 01 (or OJ) presents itself as part of the inverse, this being necessitated by the fact that the point inverse to / is indeterminate, being any point on 01. These factors thus occurring in the inverse are not counted as part of the proper inverse ; they are rejected, and only the residual inverse is counted. Thus for example if a conic through OIJ cut the fundamental conic again in AB, the formulas of transformation give for the inverse the four lines 01, OJ, IJ, AB ; the proper inverse is simply the line AB. Tlie conic is fyz+yzj: + h.cii = 0, and the inverse is fx'y'h' +gx''^y'z' + h.v'y'z"' = 0, that is, x'y'z'{fy' +gx' + hz') = 0. 236. The fact that lines through /, ./ intersecting on the base are inverse affords the most generally convenient way of determining graphically the inverse to a point P when /, / are distinct. Let IP meet the base in H, then JH meets OP in P'. By this means we can divide the plane into pairs of inverse compartments 11', 22', etc., and when a point P passes from 1 to 2, the inverse point P' passes from 1' to 2'. To see clearly the effect of inversion on a curve, it is advisable to draw the conic that corresponds to the line infinity. The point Q of § 232 is now C, the centre of the fundamental conic; the points A, B are the points at in- finity on the fundamental conic. The point at intiiiit}' on the line IJ being il, the polar of il passes through 0, and the line 00, is parallel to //, hence the inverse to il is con- secutive to on the line Oil ; that is, the tangent at to the required conic is parallel to I J. Hence OC is the diameter conjugate to //, and the tangent at C is parallel to I J : these conditions are more than sufficient to determine the conic, but it may also be noticed that drawing from / a line parallel to 01 to meet the base in A", IK is the tan- gent at /, and similarly the tangent at J can be constructed. In Fig. 53 the conic inverse to infinity is represented by a broken line. Note. One advantage of this division into compartments is that in the next few diagrams tliere is no occasion to rejjieseiit tlie fundamental conic. It is advisable to consider the arrangement of the compartments not only for the ellipse, shown in Fig. 5:3, but also for the hyperbola, and 224 THEORY OF CORRESPONDENCE. then, making /, J^ approach one another indefinitely, for the line-pair. It appears further on that for one important use to which the method of quadric inversion is put, tlie line-pair is the hest fundamental conic. Effect of Inversion on Singulni'ifics. 237. The law of construction sliows that an ordinaiy point inverts into a single point : but tliis may be an inflexion. For at the ordinary })oint three consecutive points are not coUinear; but if these tlu'ei' points an=o oz ■^ A ^^ y=o z\ Fig. or. Since a single inversion may replace the singularity by a singularity whose penultimate form is not known, it may be necessary to resolve this second singularity by inversion. For instance, in the example just used, we find yi=x-^-. This has a singularity at X{y-^^, with x^ = for tangent; hence the formulfB of transformation for this case are ^iV'h 2/i = 2/2' and the second inverse is that is, 2/2 — *2^- Thus by two inversions the arc with the singularity is reduced to an ordinary arc. The steps are represented in the right-hand part of Fig. 57. jVote. The formuhe of transformation are applied without any explicit reference to the fundamental (degenerate) conic ; in the exnmple just used, the two inversions are accomplished by means of two diiierent conies. When a number of inversions have to be performeil, it is con- venient to represent them as in the right-hand part of Fig. oT, not using the triangle at all. In passing from one set of axes to another, care shoukl be taken in noticing the positive sides of the new axes ; and it nuist be noticed that the formuhe depend on the tangent ; u = being the tangent, and i' = any other line through the jioint considered, the formula' are V = v', u = v!v'. 243. The transformation here explained is the ordinary (juadric trans- formation employed in investigations in the theory of algebraic functions ; it is used in the form just given, for purely algebraic purposes, in papers by Brill and Nother in the Muthematischc Annalen, tt. vii., ix., xxiii., etc. It was first used by Newton in his Eiimneratio Linearum Tertii Ordinis, 1704 ; the curve obtained is called a hyperbolism of the original curve, and it is the whole curve that is considered, not any special point. The same transformation is vised by Cramer in his Analyse des Lignes Courhcs, 1750, for the analysis of singularities ; but the geometrical connection is somewhat obscure, the inverse being referred to the same axes as the 230 THEORY OF CORRESPONDENCE. original. The sjjecial geometrical form here given to the transformation by means of Dr. Hirst's method of qnadric inversion is to Ije fonnd in the American Journal of Mathematics, vol. xiv., p. 301, and vol. xv., p. 221. 244. As regards any specialty of position with respect to 01, the results are very similar. Contact with 01 at X, on the line JH, gives a cusp at /, IH being the tangent, etc. Moreover, it is seen from Fig. 53 that contact with 01 at / (compartments 1', 7', for example) gives contact with 0/ at /; and contact with 01 at (2', 13', for example) gives contact with IJ at /. But if the principal points be not distinct, the results are slightly ditlerent: the form they assume will be made plain by the construction of the diagram corresponding to Fig. 53. For instance, if the fundamental conic be a line-pair, a branch cutting 01 gives a branch touching IJ at 1 : hence a branch cutting 01 in two points gives a tacnode at / ; and a branch touching 01 gives a cusp of the second species at /. Effect of Inversion on a Curve as a Whole. 245. We have here used inversion as a method for analysing singularities : but as it institutes a correspondence between two sets of elements in a plane, it is applicable to the in- vestigation of the properties of a curve as a whole. As an example, consider the theorem :-^If a conic be in- scribed in a triangle, the lines joining the points of contact to the opposite vertices are concurrent. Let tlie triangle be OIJ, and the point of concurrence M. Invert with respect to the conic that touches 01, OJ at /, J, and passes through M. There is now a cusp at 0, with OM as tangent, and likewise at /, J; since the original does not pass through 0, I, J the inverse does not cut //, 01, OJ, except at the points 0, I, J already considered. Hence the inverse is a (piartic witli tliree cusps, and the cuspidal tangents are con- current. Arranged in this way, this does not prove that in any quartic with three cusps the cuspidal tangents are con- current ; to prove this, let 0, I, J be the cusps, whicli are certainly not collinear: invert with respect to any conic toucliing 01, OJ at /, J. Let the cuspidal tangent at meet IJ at 0' ; let the tangents at /, /meet the conic of inversion 'At A, B, and let J A, IB meet 01, OJ at /', J'. Then owing to the cusps at 0, /, J the inverse has contact with IJ, 01, OJ at 0', I', J'; now the inverse lias no othoi- points on the sides of OIJ, it is thei-efore a conic inscribed in OIJ, tuid consequently 00\ JI\ /./' are concui'i'ent ; tlieii' inverses, THEOKY OF CORRESPONDENCE. 23l that is, 00', I A, JB, are therefore concurrent: hence in a tricuspidal quartic the cuspidal tangents are concurrent. Similarly considering a conic cutting the sides of the tri- angle in three pairs of real points, the existence of a trinodal quartic is proved, and it is . shown that the three pairs of nodal tangents touch a conic. And considering a conic in the various possible positions with regard to the triangle, meeting the three sides in all possible ways, the existence of different varieties of quartics with three double points is made evident. 246. These theorems depend on the correspondence of point to point, and of straight line through a principal point to straight line through a principal point; but to a straight line in general corresponds a conic through OIJ. Consider the general conic and the quartic derived from it by inversion ; the tangents to the conic invert into conies touching the quartic and passing through 0, I, J; the intersection of any two tangents inverts into the intersection of the tangent conies. Hence there follows the theorem : — Through any point two conies can be drawn to touch a quartic with three double points and pass through the double points. Ex. 1. Obtain the general equati(jn of a trinodal quartic (i^ lOJ) by inversion. Ex. 2. Show that thi-ee pairs of tangents to a quartic can be drawn from the three nodes ; and that these six lines touch a conic. 247. Since the conic and quartic are inverse, the order of a curve may be doubled or halved by inversion ; and these are the extreme cases. The application of the formulae of transformation gives for the inverse an equation whose degree is twice that of the primitive equation ; but as all factors that are simply powers of x, y, z are to be rejected (.^ 235), the degree of the inverse equation may be lowered. But as inversion applied to this derived e(|uation is to restore the primitive, the degree cannot be lower than one half that of the primitive. Let the primitive curve of order tyi have at /, J, multiple points of orders i, j, k (where any of the numbers i, j, k may be zero) ; and let it cut 01, OJ, IJ in i' , j', k' points, so that i' + i-{-k = m, / -\-j -\-k = 1)1, k! + i +j = m ; that is, i' = m — i — k, f = m —j — k, k' — tn - i —j. The inverse has then at /, ./, multiple points of ordei-s i\ f, k', and cuts 01, OJ, IJ in v, ;/, /• points : hence m' ^i' + k' + i =/ + // +7 = i' +j + k : 232 THEORY OF CORRESPONDENCE. tliat is, m=2m—i—j — k, i' = m — i — li, /= m-j-k, h' = m — i —j : from which also m = 2m — i' —f — k', etc. Thus for example, a conic through /, J inverts into a conic through I, J ; a, conic through one principal point, 0, inverts into a cubic with a double point at and ordinary points at /, J. The conic inverts therefore into a cubic with one double point, or a quartic with three double points, each of which is a curve of deficiency zero ; and on inverting any curve, it will be found that the deficiency is unaltered by the transformation (compare §§ 143, 288). 248. Since lines through either fundamental point invert into lines through the other fundamental point, it follows that in circular inversion, where the circular points are the funda- mental points, isotropic lines invert into isotropic lines ; every one inverting into a line through the other circular point, the two of a pair are interchanged, but as they enter only by pairs this does not afi'ect the final result. Hence a focus F inverts into a focus F' . If however 0, the centre of the circle of inversion, be itself a focus, 01, OJ being tangents there are cusps at /, / on the inverse, and for this curve is not necessarily a focus. Ex. Di.scuss tlie inverse of a conic witli res>ard to (1) a focus, (2) the centre, determining all particnlars as to order ; number, situation, and nature of double points ; number and situation of foci. Recip7'ocafion. 249. We have now to consider the association of the doubly infinite system of points in a plane with the doubly infinite system of lines in a plane. We shall find that all the results obtainable by this were arrived at in the earlier chapters by means of the principle of duality. In v5 73 a hint was given as to a way in which a geo- metrical connection can be instituted between the points and lines of a plane. Let the polar of P with respect to any proper conic F be denoted by p ; thus to the points P, Q,..., corresjjond the lines p, q,...'. and since PQ is the polar of pq, to tlui lines PQ, PR, ..., correspond the points 'pq, pr,... . Moreover, collinear points P, Q, R, ..., give con- current lines p, q, r, ..., and the two configurations are homo- graphic : their con-espondence is precisely that afibrded by THEUEY OF CUERESPONDENCE. 238 the principle of duality. The two figures thus obtained by means of poles and polars, being reciprocal in their rela- tion, are called Reciprocal Polars ; and the process by which one is derived from the other is called Reciprocation ; the conic used as a foundation for the process is the auxiliary conic. If this auxiliary conic be the imaginary conic x' + f~ + z^^O, the polar of /, (j, h is hence to the point /, g, h corresponds the line /, g, h ; to the locus of the point /, g, h corresponds the envelope of the line /, g, h ; the two curves thus connected by reciprocation with respect to the special auxiliary conic are the curves defined as reciprocal in i^ 60. 250. The correspondence hitherto studied by means of the principle of duality has affected only non-metric properties. But in reciprocating with respect to a specified conic, all the properties of one figure are derived from those of the other, and consequently it is possible to pass from the metric properties of a figure to those of its reciprocal ; but the only case in which this can be done with facility is when the auxiliary conic is a circle. This however does not limit the generality of the method. The auxiliary conic being a circle, the centre is the origin of reciprocation. The polar of any point P is constructed by taking on OP a point M, given by Oil/. OP = (radius)-, and drawing through M a line perpendicular to OP ; and similarly the pole of a line is constructed. Hence lines belonging to the primitive that pass through give points at infinity on the reciprocal curve ; for instance, the tangents from to any curve of the primitive figure and tlieir })()ints of contact with this curve reciprocate into points at infinity on the reciprocal and the tangents at these points ; that is, the points of contact of tangents from to any curve re- ciprocate into the asymptotes of the reciprocal curve. 251. The theory of reciprocation with respect to a circle is fully treated in Salmon's Conic Sediovi<, Chapter XV., and there is therefore no occasion to go into details here. But one special example may be given, for tlie sake ol" showing the connection between § 124 and Ji 134. A pencil of conies reciprocates into a range ; let the origin of reciprocation be taken at a vertex of the self -conjugate triangle, then as the connnon points P, Q, R, S are in two 234 THEORY OF CORRESPONDENCE. pairs on lines through 0, the common tangents which deter- mine the reciprocal range are parallel in pairs. If now the pencil be a pencil of circles, that is, a system of coaxal circles, two of the intersections are the circular points ; the origin of reciprocation is one of the limiting points of the system. The four intersections P, Q, w, no connect in pairs through 0, viz., Poo and Qw pass through ; the circle passes through «, and CO therefore reciprocates into the tangent at w, that is, into Oo), while P reciprocates into another line through to ; similarly oo, Q reciprocate into lines through w' ; let the lines reciprocal to P, Q intersect in F. The reciprocal to the system of coaxal circles when taken with respect to either limiting point is therefore a system of confocal conies having that limiting point as one focus ; the other focus is the point F. Ex. 1. Show that the auxiliary circle can be chosen so that the two limiting points may be the foci of the reciprocal system. Ex. 2. Discuss the case of a coaxal system with imaginary limiting points. 252. In connection with reciprocal curves it was seen that a cusp witli its tangent and an inflexional tangent with its point of contact are reciprocal. This fact is brought out by the upper part of Fig. 58. Four arcs are drawn, meeting at A, and all having the same tangent h: the reciprocal arcs have as their tangent a, the polar of A, and the point of contact is B, the pole of b. An arc and its reciprocal are marked with the same number; thus 1,3, which make an inflexion at A, make a cusp at B. The lower part of Fig. 58 shows the reciprocation of a node with a loop. Corresponding to the node iY, with the two tangents 'p, 5', there is a double tangent n with the two points of contact P, Q. From a real tangent OT can be drawn to the loop, this occurs between p ^iiid q as we travel round the loop; hence the reciprocal passes tln-ough infinity, P and (^ being separated by the point at infinity ; tlie asym- ptote, /, is the reciprocal to T. In § 239 it was sliown tliat a cusp may be regarded as the final form of a node with a loop, when the two tangents close up and tlie loop disappears. Now when p, q approach coincidence and NT \anishes the points P, Q approach coincidence and also the Hues 'it, t. Hence we have the two arcs that make an inflexional branch, togetlier witli all points on the line that is the inflexional tangent ; tliese being reciprocal to the two arcs .tliat make tlie cuspidal branch, together witli all lines through the cusp. Now considering the evanescent loop as an envelope, we see that it does give rise to the two arcs and this assemblage THEOKY OF CORRESPONDENCE. 235 of lines, and just as this last is not counted as part of the cuspidal brancli (envelope), the assemblage of points on the inflexional tangent is not counted as part of the in- flexional branch (locus). Fic. 5«. The Dualistic Transformation. 253. We have seen tluit projection presents itself as a special case of linear transformation, specialized however only by position ; and that (piadric inversion, at first sight a special quadric transformation, diflers from the general re- versible quadric transformation only by a linear transforma- tion. We liave now to show that reciprocation, ])rescntin<'- itself as a special case of the linear dualistic transformation, diflers from this only by a linear transformation. 236 THEORY OF CORRESPONDENCE. The theory of* linear dualistic traiisfonuatioii is sometimes called the theory of skew reciprocation. This name has reference to the fact, now to be proved, that the essentials of the theory are to be found in reciprocation with regard to a fundamental conic ; the other name has reference to the underlying essential, the principle of duality, and regards the relation to the auxiliary conic as purely accidental. Note. This correspondence has also been called Correlation, the figures being correlative (Chasles), but the name is not in general use. 254. In the general dualistic transformation the coordinates of a line are general functions of the coordinates of the corre- sponding point ; the transformation is linear, when these ex- pressions arc linear. Hence the formulae of transformation are and to a point P{x, y, z) in the first system corresponds a line ■p{^, ij, t,) in the second system ; the two systems may be represented in different planes, or the two planes may be superimposed so that we have the two systems in one plane, with it may be different triangles of reference. Now subject the first system to a linear (point) trans- formation, x = (lyic + a.^y + a-^z, y'=h^x-\-Ky-\-b.^z, z'= c^x-^ c.,y+ e.^z, by which from a point F(x, y, z) there is derived a point P'{x', //', z'). There is now a correspondence between this derived system (P') and the second system (p), expressed by the transformation i = x', >] = y, ^=z'; these two systems are therefore reciprocal with respect to the auxiliary conic which shows that the general linear dualistic transform- ation differs from the interchange of point and line coordin- ates only l)y a collineation. /Vote. The term linear transformation is [)roperly used wlienever the formulie of transformation are linear, whether they express point and line coordinates in tei'nis df ))oint and line coordinates or in terms of line and point coordinates ; that is, linear transformations include col- lineations and linear dualistic transformations. THEORY OF CORRESPONDENCE. 237 255. Hence so far as the geometrical properties of a figure are concerned, nothing more can be learnt by means of the most general linear dualistic transformation than is recog- nized intuitively by means of the principle of duality. But considering the correspondence of point and line hereby' insti- tuted in its relation to the general theory of (1, 1) cori-e- spondence, one or two points require investigation. Let the two systems be represented in one plane, with the same triangle of reference. To a point x, y, z of the first system corresponds a line ^, ;/, ^ (jf the second system, where ^= a-^^x -f do]/ + a.^z, T] = h^x + h.yy + h^z, ^=c^x+c^y+c^z: equations which may also be written where a^, a^, etc. are the minors of a^, «.,, etc. in the deter- minant {a^^c^, and this determinant is rejected us a factoi' in the expressions for x, y, z. Hence to a point of the second system, corresponds the line of the first system l{a^ ■\- (i.{ij + a.^) + m(b^x + h.,y + h.^z) + n{Cyi' + c^y -f- c._^z) = 0, that is, to the point I, m, n considered as belonging to tlio second system there corresponds the line (a-J, + h^in 4- c{ii)x -f (a.,Z -|- h.{i)i -\- c.{ii)y + {((./ -\- h.^m -(- ti^7? ): = 0. Thus the point x, y, z has two different lines for corre- spondent when considered as belonging to the two different systems, and the coordinates of these lines are respectively a^x + a^y + a^z, \x + h.^y + b^z, c-^x+ c^y + c^z; a^x-{-h^y+ c^z, a^x + Ky + c^z, a.^x + h.^y + c^z. These two lines coincide if a-^x -f fu^y + a^z _ h^x + b^y -\- b^z _ c\x 4- c.^y + c.^z _^ a^x + b^y + c-^z ~ a.^ -f b.^j + c.^^z ~ a.p' + b.^y -\- c.^z where X satisfies the equation, obtained by eliiuinnting ./•, //, :, • b., c.X — b.^ This is of the form X=* -f ^ A' - ^ X - 1 = 0, a^\ — <(^ b,\ a.,\ — 6 J b,\ a.J^-<\ b,X 238 THEORY OF CORRESPONDENCE. hence one solution is X = 1 , wliich gives x : y : s = 63 — c, : c^ — a.^ : a., — b^, and there are two others, real or imaginary. There are tlierefore three points and their corresponding lines which are associated with one another regardless of the system to which they belong. Thus the general linear dualistic transformation does not definitely associate the points of the plane with the lines of the plane, though this can be done by a special dualistic transformation. The Iwo lines corresponding to a point coincide for every point if the equations ayX+ a^y + a^z _ h^x + h^y-\-h^z _ c-i X+c^y + c^ z a^«; + b{ij + G-i^z a^fic + h^j + c^ a^x + h.^y + c^z hold for all values oi x, y, z : hence we have the conditions showing that the points of the plane are associated with the lines of the plane only if the equations of transformation assume the form ^=ax-\-hy-{-gz, ri = hx+hy+fz, ^=gx+fy+cz, expressing simpl}^ that the point and line are pole and polar with respect to a conic ax- + by- + cz- + 2fyz + 2gzx + 2hxy = 0. 256. Corresponding elements being different in nature, the question as to elements that coincide with their correspond- ents (§ 226) is replaced by the question as to elements that are united with their correspondents. The line ^, >/, {' is united with the point x, y, z if hence the locus of points that lie on theii' corresponding lines is the conic ("r'' + "o// + '';i^)'' + ( byV + J>^y + /a,:)// -|- (r^x + c.^j + e^z)z = ; that is, (t^x' + b.,ij' + c.j'.- + ( /aj + c.^y. + (('1 + «;J :a' + («2 + ^)a'i/ = ^h which is called the pole conic. Also the envelope of the lines is a different conic, the polar conic. Kor the coordinates of a point are given in terms of the coordinates of the corre- sponding line by the e([uations . ^^ = ] + 7^i, etc.. THEOEY OF CORRESPONDENCE. 239 and therefore the envelope of lines that pass through their correspondents is «if+/32r+y3r+(y2+/33)'/f+(«.+yi){^X/8i + «2)f'/ = <>> that is, (62C3- 6362)^-+ {c^a^- c^a.^)rf + {i\K- aJ)^X" + j]^(%\- a^\+ c^a.^ c^i^) which is called the polar conic. We have here been dealing- with points of the first system and lines of the second ; the equations expressing the corre- spondence between points of the second system and lines of the first being (§ 255) ^=aiX + b^y + G-^z, etc., the pole conic and the polar conic are the same as before. To compare these two conies, both must be expressed in line coordinates or in point coordinates. Writing the equation of the pole conic in line coordinates (§ 65) the coefficients A, F are proportional to that is, {h.,^ — c^f — ^(^^Cg — h.f.^), and 2ai(63 + c.^ — (c^ + ('■;^){a., + h^), that is, (cj — a^{a.2 — b^) — -(ajj^ — a-J)^ + c^a.^ — c.,a^). Hence writing the polar conic in the form ^ = 4(6,c, - \c.^^-^ +... + ... = (), the line equation of the pole conic is $ = ((63 - c.^i+ (c, - a,)>j + {a, - h,)^y - ^ = 0. This form shows that the two conies are tlifferent, unless the transformation considered is reciprocal ; and that they have double contact, the intersection of the common tan- gents being the point 63 — Cg, c^ — a^, «2~^i' ^^^^ chord of contact itself being the line P-^ — y-i, yi~«:i' m — ["iy Let this line, p, meet the conies in Q, R, and let the intersection of q, r (the tangents at Q, R), be P. Let A'' be an^^ point on the pole conic, so that the correspondent to A" passes through X ; by the definition of the polar conic, it is touched by the line corresponding to X ; hence the two correspondents to X are the two tangents from A'' to the polar conic. If therefore X be at Q or at R, the two correspondents coinciile : the points Q, R are two of tlie points determined in ^ 255, the point F being the third. Hence of the three points con- 240 THEORY OF CORRESPONDENCE. sidered two lie on their correspondint;" lines, and the third does not. If now the triangle PQR be the triangle of* reference, ^=0 corresponds to .X' = 0, biit ;y = and ^ = correspond respectively to = and ^ = 0. Hence the fornnil;\} of transformation are Note. For a fuller discussion of ski'W reciprocation see Salmon's Higher Plane Curves, §§ 332-342 ; there i.s a tyi)o_oraphical mistake in § 335, where the line ecjuation of the pole conic is given instead of the line equation of the })olar conic. Birational Transformation of a Curve. 257. The transformations hitherto considered are birational transformations of the whole plane ; tliey are Cremona trans- formations. But there are transformations that are birational only as regards a curve of the plane. For instance, to the locus of a point P there corresponds by reciprocation the envelope of a line p ; let F' be the point of contact of ^j with its envelope, then P' is determined by F, and P b}^ P'. Hence there is a (1, 1) correspondence between the points of the two curves, and also between the lines of the two curves. For example, the reciprocal to cc^ + i/ + z' = (1), is (§ 68) «« + f + s° - 2fy' - l^j? - -Ixhf = (2). Let «;^, 3/j, z-^ be a point on (1) ; the tangent at this point is x^x + y;-u-\-z^~z = 0, and to this line corresponds the point x.,:y^_:z^^==x^':y;':zy- (3), a point on (2). The tangent to (2) at this point is {x.i'-x^y.i- x^-zi)x + {yj' - yiz.f-yix.f)y^ {z.^ - 0./.r./ - z^y^)z = 0, which can be written x^i^xi - u.^x + y.ri^yi - u.^y + z^C^z^ - u.^z = 0, where u., stands for x./' + y.^'^-^-z./'. Now ,«p y-^, z^ corresponds to this line ; hence «i •• Vi ■■ -^1 = «2^(2«.2^ - y^-z) ■ yii-'ji' - ^^h) ■ H^i'^^-i - '^2) • • • (4)- Thus fl^2' 2/2' H ^^^ given rationally in terms of x^, y^, z^ bj" equations (3), and cc^ ^/j, z^ in terms of x.^, 2/2. ^2 ^Y equa- tions (4). THEORY OF CORRESPONDENCE. 241 These two sets of eciuations are equivalent, in virtue of equation (1) or (2) ; for substituting in (4) the values of j:,, y.^, z.^ given by (3), the result is •t'l : y 1 : si = .riX2.ri« - ,/;/' - y i« - i/) :...:... =.V(..-i'' + 2yiV-.ri«) :...:.. . that IS, an identity. An algebraic transforimition that is hiraiional as regards the 'points of tioo curves but not as regards the ivhole plane is called a Rleinann transforriiation. That is, a Rieinaiin transformation of a curve F{x, y, z)=0 is expressed ])y tlie equations ^■v- ^=./i(-^''> 2/'. '0 -.W^ y'> •^') ■U.-'''^ y'^ •^')' wliere f^, /„ /^ are lioniogeneous polynomials of the same degree /.• that have no common factor, and are such that (hy means of the equation F=0) x, y, z' can be obtained in the form X : y : z'==i + cz = 0, then the equation of the circle in the homogeneous form is x^ + ^f- 2{gx +fy){ax + hy-\- cz) + (ff +f - r'){ax ^-hy + czf = 0. Hence the point equation of a circle of infinite radius, obtained by writing in the above r = oo , is {ax-\-hy + czf = 0, that is, the line infinity taken twice. We are therefore led to consider the question of degenerate conies. Note. The degenerate conies that present tlieniselves most readily to our remembrance are evanescent conies, such as .t-+y'- = ; but we have just seen that a conic can degenerate by becoming infinite ; the question to be considered is therefore the general one of degenerate conies in point or line coordinates. Degenerate Conies. 268. A degenerate locus of the second order is two loci of the first order, and is therefore a line-pair ; and similarly a degenerate envelope of the second class is two envelopes of the first class, that is, a point-pair. Now a line has not a line equation ; hence we cainiot expect the line-pair to have a line equation ; and yet from one point of view the equation u = ax' -f by- -f cz' + %fyz -f 2r/-.f + -Ihxy = 0, even when j)^i+ 2{fg - cli)irj = 0. a It (j =0, we have also h h f j 9 f c \ But since a It ), {<■)■, (d) form a consecutive series, representing the conversion of an ellipse (a) into a liyperbola (d), by the vanishing of one axis, the quantity 6" changing from ])ositive to negative through zero. Fid. 00. The Absolute. 266. Thus it appears that the special points co, to', which are fully represented in line coordinates by a degenerate ecpiation of the second degree, cannot be exactly represented in point coordinates ; they give simply the line infinity taken twice. The special line that we are led to consider in the use of point coordinates presents itself therefore as a part of the special configuration that we have to consider when using line coordinates ; and the circular points are of more fundamental importance than the line infinity. We have seen that the degree of choice allowed in projection enables us to choose m, oo' arbitrarily, for we can project so that any two points become the circular points (§ 201), but then the figure is determined. Determined, that is to say, as to shape ; not as to size, for projection from a plane on to any parallel plane alters the size, but not the shape, and does not ati'ect w, co' ; not as to position, for w, w are affected neither by tran.slation nor rotation ; but absolutely deter- mined as to the metric relations that the parts bear to one THE ABSOLUTE. 249 another. Thus the circular points are the ahsohite elements in the plane, all other eli'iiients may be considered as de- pendent on these. Now this absolute configuration presenting itseli' as a degenerate conic, the natural generalization is to replace it by a proper conic. This proper conic is called the Absolute : the notion of the Absolute was introduced by Professor Cayley in his Sixth Memoir upon Quantics, 185f) (Collected Papers, vol. ii., No. 158). If, therefore, we investigate the purely descriptive relations of a system to a general conic, the Absolute, and then make this conic degenerate in the particular way just considered, we shall obtain the relations of the system to the special configuration composed of the circular points and the line infinity taken twice ; and interpreting such of these relations as do not prove illusory, we shall obtain the metric properties of the system. Relation of a Carve to the Absolute. 267. A point at which a- curve cuts the Absolute gives a point at infinity on the curve : but the double occurrence of the line infinity in the degenerate Absolute has to be taken into account. For instance, a straight line meets the Absolute in two points ; and it is of interest to see what trace is left of this when the Absolute becomes ordinary infinity. We are dealing with questions involving metric (juantities, that is, linear and angular magnitude ; and we are examining how far our conceptions of these are in accordance with con- clusions drawn from the principle of duality. We nnist consider therefore in what way linear and angular magnitude correspond. Imagine a line p to revolve about one extremity 0, and let it be indefinite in extent in one direction from 0. Let it meet a fixed line in P, and in its initial position a, that is, OA (Fig. 61), let it be perpendicular to this fixed line. As p revolves about 0, so describing angular magnitude, P moves along the line, so describing linear magnituile. When p has described one right angle, P has described the line from A to infinity, on the upper side. As p continues its revolution, describing the second quadrant, P reappears to the left of A ; but the line p being terminated at 0, we reach the point P' by travelling along the line p, through infinity to P' on the lower side of the line, and accordingly P' describes the line from infinity to A', on the lower side. 250 THE ABSOLUTE. Similarly as p describes the third quadrant, the lower side of the line from A' to infinity is described ; and as p describes the fourth quadrant, so completing the revolution, P describes the upper side of the line from infinity to A, ho completing the description of the double line. f / Yp' o -f >l \,,^^ • a A 7^ \P ^/P' A' Fig. 61. Hence in order to exhibit a conqjlete correspondence between linear and angular magnitude, the line must be regarded as double, the upper edge being continuous with the lower edge through infinity. 268. A tangent connnon to the curve and the Absolute gives rise to an isotropic tangent. If the curve be of class n, there are 2n connnon tangents, falling into n pairs of conjugates. The intersections that do not come at w, w' give the foci, of which n are real, n{n — \) imaginary. But if the line infinity be a tangent to the curve, the curve has contact with the Absolute, which is a singularity of position to be considered separately. 2(j!). Contact with the Absolute is represented by contact with the line infinity (Fig. (j2, A): or, since every line through either circulai- j^oint is a tangent at that point, by passage thi'ougli one of the points w, co'. Jiut as oo, w arc conjugate imaginary points, they cannot present themselves separately ; there is consequently double contact with the Absolute. For THE ABSOLUTE. 251 the case of a conic this is the only way in whicli double contact can occur ; for contact at B, B', where BB' makes with (Oft)' a vanishing angle, gives a conic that degenerates into the line infinity taken twice ; and contact at C, C, where C'C" makes with coo) a finite angle, gives rise to a double point when C, C come together. Thus a parabola is a conic having contact with the Absolute ; a circle is a conic having double contact with the Absolute. ^ote. Although the letters oj, o/ appear in Fig. 62, it is important to notice that the points w, oj' have no definite existence until the Absolute is regarded as degenerate. Govres'pondence of Asymptotes and Foci. 270. It now appears that the asymptotes of a locus of order n correspond to the foci of an envelope of class n (compare § 129). For a curve of order w cuts the Absolute in 2;/. points ; considering the Absolute as a very flat conic, thougli not actually degenerate, it is seen that these 2n points are in n pairs PF', QQ' (Fig. 62). A line determined by a pair, FF', is ultimately a tangent to the given curve /, and therefore an asymptote; a line determined by points that are not a pair, FQ', or FQ, is ultimately the line infinity. Let the equation of the Absolute be the curve f—uv = 0, where v is any expression of degree n — 2, is a curve of order n passing through all the intersections of / and u. If therefore v be chosen so that this may split up into n linear factors, it represents one of the sets of n lines deter- mined by the '2n points ; let v be chosen so that these n lines may be FF', QQ', etc., and let their equations be i, = 0, t.,^0, ..., ^„ = 0; then t^t.2 ... tn=f—'av. If now the Absolute become ordinary infinity, its point equation, it — 0, becomes s" — 0, where s = is the line intiuity : and the equation /= becomes f=t^t., ... f„ + s-y„_o = 0, which is the ordinary expression for the curve in terms of the asymptotes. Reciprocally, let the line equation of an envelope of class n be (p — 0, and let the line equation of the Absolute be = 0. Then the curve , ^ , ^ (p — 6Y = 0, 252 THE ABSOLUTE. where xf/- is any expression of degree 91 — 2, is a curve of class 11 touching all the connnon tangents of (p and 6. Choosing \fr so that this may split up into n linear factors, it represents one of the sets of n points determined by the 2n lines. Now tangents to the Absolute are ultimately lines through CO, w', unless they coincide with the line intinity ; but this alternative is excluded, for we are not supposing the curve to have contact with the Absolute. Hence the 2n tangents fall into conjugate pairs, giving n real inter- sections : and we may suppose xj/- to be chosen so that the n points given by tlie linear factors of f() — 0\fy are these n real points. Let these factors be p^, p.,, ..., p^: and let the Absolute become ordinary infinity, so that becomes coco' : then we have as the eipiation of the curve in terms of the real foci. Correspondence of Linear and Angular Magnitude. 271. Since the metric properties of a system are descriptive properties of tlie extended system obtained by combining the Absolute with the given system, it must be possible to give a descriptive definition of linear and angular magnitude ; and if there be an exact quantitative correspondence between the two conceptions, we may expect to discover it at this point. 272. It was shown in § 115 that lines, perpendicular accord- ing to the ordinary conception, are harmonic witli I'espect to the isotropic lines througli their intersection : and accord- ingly this was adopted as the definition of perpendicularity. In generalizing this, the isotropic lines are replaced by tangents to the Absolute ; now lines through a point har- monic with respect to tht; tangents from that point to a conic, are conjugate with respect to the conic. Hence lines conjuf/ate with, respect to the Absolute are said to he per- pendicular. To obtain a general definition of tlie angle between two lines OP, OQ that shall be consistent with the ordinary Cartesian conception, consider wliat is in\<)l\ed in tliis idea. Take tlie line ()(^ an2A,where Z)=|=0, hence of the two equations (1), (2), each implies tlie othei'. INVARIANTS AND CO VARIANTS. 261 283. In this abstract consideration of the subject, the con- ception of a group of operations (in this case, transformations) presents itself.* A system of operations is a group when the operation resulting from the combination of any number of the contained operations is itself a member of the system. If a body be moved from any position A to another position B, and then from B to G, the total effect is simply that it is moved from A to C ; the combination of two movements makes a movement, and nothing else ; all movements form a group. Dividing movements into translations and rotations (and thus not considering a translation as rotation about a point at infinity), the translations form a group by them- selves, for a translation combined with a translation gives a translation ; but the rotations do not form a group, since two rotations may result in a translation. The group of translations extended by the rotations, these not foi'ming a group, forms the whole group of movements : and the whole group of movements contains certain smaller groups, as for example the group of translations, and the group of rotations about any arbitrarily chosen point. 284. Confining ourselves now to the plane, the group of translations and rotations leaves unaltered the circular points, and therefore also the line infinity as a whole, but not the separate points on it. The sub-group of translations leaves unaltered every point on the line infinity; the sub-group of rotations about a fixed point leaves unaltered the circular points and the fixed point. Thus the sub-groups contained in a given group are difierentiated by leaving unaltered some configuration that is not left unaltered by all mend)(^rs of the given group. Two collineations produce a collineation : hence collineations form a group. This includes the group of movements, which is characterized by leaving the circular points unaltereil : and as before, other sub-groups may be selected by the property of leaving unaltered an arbitrary configuration, for it is plain that if transformations A and B leave certain elements unchanged, then the transformation resulting from the two leaves these elements unchanged. The combination of two dualistic transfoi-mations is not a dualistic transformation ; it is a collineation. Hence the dualistic transformations do not form a group by themselves : but as the combination of a dualistic transformation and a collineation is a dualistic transformation, we may extend the *See §§ 1, 2 of Professor Klein'.s Vcnjleiclu-ndr netrachtioigen, already refeiTed to in § 229. 262 INVARIANTS AND COVARIANTS. group of collineations by adding to it all the linear dualistic transformations ; we thus obtain the group of linear trans- formations. 285. Since different points and lines are left unchanged by different linear transformations, none are unchanged by the complete group. If we wish a curve to be unchanged as a whole (considering at present only the points, not the lines), let the original equation be F =(«!, a.-^, ...lx,y, zY = 0, then if the fornmla3 of transformation be x = l^x' + '^n^y' -\-n^z , etc. the transformed equation is F' = («j, rf.,, . . . Kj.r' 4- m{y' + n^z', , f = 0, that is, F' = {a[, a!,, . . . ^x, y', z'f = 0, where a(, a'^, etc. involve the coefficients of transformation. Since the form of the equation is to be unaltered by the transformation, these coefficients l-^^, ii\, 72^ Z.,, etc. must satisfy the equations a[: a^ — ai: a.^^ ...etc (1). We have therefore 8 quantities wherewith to satisfy equa- tions (1) : thus we have more than sufficient if the desired stationary curve be a straight line or a conic, the number of equations in these two cases being 2 and 5 respectively. The number of equations for a stationary cubic is 9 ; but as we have no assurance that these are independent (for every cubic), we cannot say without further examination that they cannot be satisfied : this question we leave. Returning to the conic, since there are 8 quantities, and only 5 equations, the number of solutions is certainly triply infinite : and if the equations be not independent, it will be greater. But the equations are independent ; for if they be not independent in the general case, they cannot be inde- pendent in a special case. Now using rectangular Cartesian coordinates, the circular points are unaltered by any trans- formation : but here are three constants involved, viz., the coordinates of the new origin, and the angle through which the axes are turned ; hence in this special case a three-fold infinity of transformations leaves a special conic unaltered, and the equations are seen to be independent. Thus in general the transformations that leave a given conic un- altered foi-m a three-fold infinity. Ex. 1. Show that the transformations tliat leave unaltered a conic and a straight line are sin,i(ly infinite in number ; and that a ti'iangle is unaltered by a two-fold iiifiiiit\' of transformations. INVARIANTS AND COVARIANTS. 203 Ex. 2. By means of a certain linear point tran.sfonmation the points on a conic become other points on the conic ; show that by the associated line transformation (§ 34) tangents to the conic become other tangents to that same conic. Ej: 3. By a certain linear dnalistic transformation the points on a conic become tangents to that conic ; show that the tangents to the conic become points on the conic. 286. The considerations here presented show that the ordin- ary orthogonal transformations (§ 216) form the inckided three-fold group that leaves the Absolute unchanged. Now in considering the metric properties of a system of curves directly, we confine ourselves to rectangular Cartesians ; the properties are unchanged by this group. But we have seen that instead of confining ourselves to the direct investiga- tion of metric properties, we may use projective methods, and investigate the relation of the system to the Absolute. Dealing with projective properties, the group is now the more extensive group of linear transformations. Thus we may extend the group, if at the same time we adjoin the Absolute. We now pass on to consider the group of linear transfor- mations, these including collineations and dualistic transfor- mations ; and as regards any system of curves, we have to investigate (i.) the properties of the system by itself ; (ii.) the properties of the system extended by the adjunc- tion of the Absolute. L Ineai' Tran^form.ations. 287. The question of linear transformation may be con- sidered purely algebraically, without any reference to possible geometrical interpretations. When a system of equations in any number of variables is subjected to a linear transforma- tion, certain related expressions are unaltered ; a complete knowledge of all the unalterable expressions of the system is tantamount to the knowledge of everything essential in the system. The subject is treated under this aspect in works on Modern Higher Algebra ; the Theory of Binary Forms, the Theory of Ternary Forms, the Theory of In- variants, are a few of the special titles given to such works. It is here considered on account of its geometrical signifi- cance, and only in such detail as is necessary to show its connection with the foregoing chapters. Hence, though the general algebraic theory applies to homogeneous expi-essions in any nundjer of variables, the only parts to be taken into account are those relating to binary and ternary (piantics, 264 INVARIANTS AND COVAEIANTS. these finding their geometrical interpretation in the one and two-dimensional geometries that have been considered. 288. When a curve is subjected to any transformation, we consider naturally which of its properties are altered, and which remain. If the transformation be linear, order, class, number and nature of singular points and lines, all remain unaltered : if the transformation be not linear, these will be altered ; for instance, if the curve be inverted with regard to a conic, order and class are altered, multiple points are gained and lost, inflexional and double tangents are gained and lost (§§ 237, 247). Thus these numbers associated with the original curve are altered. But it will be found in every case that the number expressing the deficiency is unaltered by quadric inversion ; a conic, which is a curve of deficiency 0, inverts into a conic, a cubic with one double point, or a quartic with three double points (§ 247) ; that is, into a curve of deficiency 0. It is found that the dejiciency is unaltered by any birational transformation ; that is, any two curves that have a (1, 1) correspondence have the same deficiency (§ 143) : but for the proof of this theorem we must refer to works on Higher Plane Curves and the Theory of Functions. If from any system of expressions there be derived an ex- pression, numerical or literal, endowed with the property of remaining invariable when the system is subjected to the transformations of a group, this expression may be called an invariant of the system for the group : the problem is there- fore (Klein, loc. cit.) to develop for the system the theory of invariants relating to the group of transformations. Thus the numbers expressing order, class, etc., being in- variable under linear transformations might be called in- variants : but the name in this case is not so used : it has a specialized meaning and is used only as referring to a special class of derived algebraic expressions. A71 invariant of a system is a function of the coeffi'cients ivhose vanishing expresses a projective property of the system. 289. The way in which invariants present theinsel\'es may be shown by the example already referred to in § 282. Let the expression F= ax" -f hy- -f <-z^ -f 2///c -|- "iyzx -f- 2hxy be subjected to a linear ti'aiisfonnation, X = J^x -\- m dj -(- n^z , y = /.,x' + m.,y' + n.^z', z = l^x' + rn^y'-\-n./, INVARIANTS AND COVARIANTS. 265 by which F becomes F'. Form the expression A from the coefficients of F, and similarly A' from the coefficients of F' ; then will A' contain A as a factor. For a' = al^^ + bl,~ + cJ^^ + 2/1/^ + 2glJ^ + 2)dJ,, h' — am^" + , c' = an-^^ + ; /' = am-^n-^ + b7ri./)}.^ + C'm.^7i.^-\-f(')n.2i}^ + ni.^nq,) (/ = an-^l^ + , Ji = al-^m^ + ; hence writing \ = al^ + hi., + (jl.^, K = hl^+bl.>+fl.,, the expressions for the transformed coefficients become a=l-^\ + l.X2 + k\i, c — 71^1/^ + 11 ^i.^ + '''^•s^-s ; (/ = '/? 1X1 + n.X, + n.^-^ = l^u^ + Lt'o + f'^i^s, It = l^fx^ + kiuL.2 + /a/X3 = -"i-i Ai + m.,\ + mgX.j. The expression A', written as a determinant, is therefore, by means of the two forms for/' etc., I li\ + l.X> + ls\ hf^i + k^'-i + k^^.i h^'i + fi^'i + ^s^'s mjAj + 7?i2X.2 + '''';jX;{ m^jUL^ + iii^/u^ + m.^iiJi^i m^i'j + iu.^r.^ + m.^i/g I 7i-i\^ + n.X2-\-n.^\ ii^^^ + n^iuL.^ + v.^^.,. '^^1/1 + 110^2+ 7*31/3 which is nt.. 11: \ Ml X., jULo \i Ms The second determinant in this product is itself the product of l. I. l.^ and a h 9 h h f U f a -2. t\ n^ n^ hence writing D for the determinant (l^m^n.^), 290. This determinant 1), formed with the coefficients of transformation, is the modulus of transformation (§ 34). It 266 INVAraANTS AND COVARIANTS. is important to notice that in no circumstances can D vanish ; for the vanisliing- of D implies the concurrence of the lines X, y, z. Moreover, the non-evanescence of D is the only limitation in the choice of values for l^, m^, ... ; hence D can be made to have any assigned value other than zero ; and any other function of l^, rti^, ..., can be made to vanish by properly choosing Zj, m^, ...; i) is therefore the only ex- pression in Zp m^ ..., that does not vanish for some trans- formation. 291. Let the equations of a system be /S'j = 0, So = 0, etc., and let the various sets of coefficients involved be a^, \, ..., ctg, ^2' •••• ^^^ ^^^y ^ become S' by linear transformation, that is, let the substitution of l^x' + m-^y' + ... for x, Ux -\- m.^y' + ... for y, etc., change {a, h, ...\x, y, ,..)" into {a, h', ...][a;', y', ...)"; hence (a, h', . . .\x, y'y . . .)" EE (a, h, . . .)}jyC^m^y'^ ..., Ux'+m.{y'^-. ..,.. .)" ; comparing coefficients, a={a, h, ...)X?i, m-^, ..., U, m.,, ...)", etc., that is, the new coefficients in an equation of degree n are linear in the old coefficients and of degree n in the coefficients of transformation. Since we are concerned only with the ratios of the coefficients in any one of the expressions S, every expression that we have to deal with is homogeneous in every set of coefficients separately. Let (r be a function of the coefficients of the system, whose vanishing expresses tliat the system has some projective property, and let G' be the same function of the new coefficients ; let the degree in the several sets of coefficients be j>9p yj.2, .... Then G' can be expressed in terms of the old coefficients and the coefficients of transformation. Thus expressed, G' is G rtiultiplied by a power of the modidus of transformation. For by hypothesis the vanishing of G entails the vanishing of G' : hence G' = MG (1). Consider any one set of coefficients, a, b, ... : the expressions for (t, 6', ... in terms of a, 6, ...slunv that tlie degree in which a, b, ... ai'c foutid in G' is tlie same as the degree in which they are found in (j: hence M does not contain a, b, ..., and therefore M is a function simply of the coefficients of transformation, homogeneous and of (legi"e(! 'Zn■^^'p^. Now the vanishing of G' is to iniply the vanishing of G, and nothing INVARIANTS AND COVARIANTS. 267 else; hence M cannot vanish for any possible transformation, and consequently, by § 290, M can only be a numerical multiple of a power of D. The number of variables beinj^ k, the determinant I) is of order k, that is, i) is of degree k in the coefficients of transformation ; hence M is a numerical multiple of Z) ^ ; that is, G' = KD k G (2). To determine the numerical multiplier K, consider the identical transformation x = x', y = y',... for which D = l : this shows that K=l ; and consequently G' = D k G (8). This equation is the algebraic expression of invariance : tlie algebraic definition of an invariant is the following : — Any function of the coejjicients that is imclianged by linear transformation, save as to a power of the modulus of transforTnation, is called an Invariant. 292. If we have two invariants /, J, belonging to the same system, let r=D>i, j'=D'j, and let f=ih, (J=jh, where h is the greatest connnon measure oifg. Then I'J = JjiJhlJ^ J'i ^ JJiJhJi^ I'^ P . . . I' hence ^i — ~ri> ^i^<^ writing G for ,!> this shows that G'=G; that is, the function G is absolutely unalterable by linear transformation ; G is an absolute invariant. Plainly an absolute invariant is of zero dimensions, for otherwise it would be affected by the transformation ■x — Ix', y = ly', .... From the tw^o invariants /, J, another can be derived. For if H=I' + J^, then H' = rJ + /'' = D'M^ + J)^"J^ = D^'^H, and therefore H is an invariant. But being expressil)le rationally in terms of / and J, H is not counted as a distinct invariant. 293. Given a system of curves, there may be a certain locus projectively related to the system. For instance, we have seen that the locus of a point harmonically subtended 2G8 INVARIANTS AND COVARIANTS. by two conies is a conic ; and as the harmonic property is projective, the relation of this conic to tlie given conies is nnaltered by projection. Let the given conies be S^ = 0, ^2 = 0, and let this derived conic be F=0; let S-^^, S.^ by linear transformation become Si, S!>, and let V be derived from Si, So exactly as V from S-,^, S^, then F' is what V be- comes by linear transformation ; for V, the locus derived from Si, S!,, is the projection of V, the locus derived from S-^, S.^. A locus, thus 'projectively connected tuitk a system of loci, and reciprocally, an ^envelope projectively connected ■with a system of envelopes, is ct covariant of the system. Hence a covariant involves the variables as well as the coefficients. Let the order of the covariant (that is, the degree in the variables) be q, and let the degree in the several sets of coefficients be /9^, yx_,, . . . , which may be sym- bolically expressed by writing then V = (((0^'K«2) ''-••• (^«, y, ■■■ )''■ But by the linear transformation in question, V is to become V, save as to a factor whose one characteristic is that it cannot vanish. Applying to V the linear transformation, it becomes y^^i^a^yi{a.y>^...{l^x' + ni^y'+ ... , i,x' -\- ul^' + ..., ...)'/, and we are to have V'^MV, (1), when V and F^ are expressed in terms of the same (piantities. Expressing in terms of a^, h^, ... , x, y' , ... , a comparison of dei'-rees on the two sides of this equation shows that M contains only the coefficients of transformation ; let the de- gree of M in these coefficients be A : the degree on the right hand side is therefore h-\-q, and on the left it is "^n^p^: hence h = ^n^l\-q. Since M is a homogeneous function of the coefficients of transformation which cannot vanish for any possible trans- formation, it is a numo-ical nuiltiple of a power of D\ and as the degree of i) in ^j, ra^, ... is h, h M=KD', h and therefore V' = KD'V, (2). Tlie suffix in \\ sini])ly indicates that .i\y,... occurring in V art! expressed in terms of ./,)/',...: it may tlierefore be discarded. The value of K, found as in § 291, is unity; and INVARIANTS AND COVAEIANTS. 269 the fact that V is a covariant of de^Tees p^,jj.^,... and of order q, is expressed by the equation V' = D'^^^"'''-'h' Ci). Though the idea of a covariant is here introduced by means of curves, that is, with reference to ternary quantics, the aroument does not require that the number of variables be specified ; we have arrived at the general algebraic idea oi a covariant, which is formulated in the definition : — Any function of the coejfficients and variables that is un- changed by linear transformation, save as to a poiuer of the niodulus of transformation, is called a Covariant. From the meaning of invariants and covariants it is at once evident that an invariant or covariant of a covariant is an invariant or covariant of the original system. Binary Quantics. 294. One geometrical interpretation of the binary quantio being b}" points on a line, the n-ic represents n points. If two of these points coincide, they coincide in any projection ; the condition for the coincidence of two points, being sinqily the condition for equal roots in the non-homogeneous ecjuation fix) = 0, is obtained by the elimination of x from the two equations f(x) = 0, |J; = 0, or, transferring to the homogeneous form (compare 5^ 99), I)}' the elimination of x, y from the two homogeneous equations ¥='■ ¥='■ ox oy The expression thus found, whose vanishing expresses the pro- jective property of coincidence, is called the discriminant of tlie quantic : and we see that the discriminant is an in\ariant. If n = '2 this is the only invariant ; the only pi'ojective relation possible for two points is that of coincidence. Writ- ing the quantic in the form (a, />, c\x, y)'-, or in the non- homogeneous form, {a, h, cjx, 1)", that is, ax^-\-2hx-{-c, the discriminant is ac — Jr: and it is at once seen that a'c — V" = {l^m.y — l.,m^)-(ac — h'^), so agreeing with equation (8) of § 291. Tliere is not any covariant, for there are no definite points specially associ- ated with a pair of points. 270 INVARIANTS AND COVARIANTS. 295. If w = 3, there is again only the one invariant, viz., A = akl^ + iac^ - Gahcd + ^bhl - S^^c^ ; and in accordance Math the formula, A' = D^A. There are in this case two covariants : for taking the three points given b}^ the cubic as the 1, 2, 3 of § 179, and completing the harmonic range in the three possible ways, we obtain a second triad of points 1', 2', 3' ; hence there is a co variant of order 3. Also § 179 proves the existence of a pair of points projectively related to the cubic, either of the pair serving to complete the equi- anharmonic range ; hence there is a covariant of order 2. Let x' be one of the triad, that is, let the points given by the quartic (x — x')(ax^ + Sbx^ + Sex + d) = be harmonic. The condition for this was found in ^ 155, (i.); applying that condition, g^ = a^fi^a^ + ^a^a^a.^ — a^a^ — a-{-a^ — a.^ = 0, to the present case it is found that x' must satisfy the cubic equation (ahl-Sahc + 2h^, abd + h-c-2ac\ 2h~d-acd-hc\ 2bcd-ad^-2c^lx,lf = 0. Similarly we obtain the covariant of order 2 b}' applying the condition g.2 = a^a^ — 4aja3 + Sa,- = (§ 155, (ii.)) to the quartic (x - x){ax^ + ohx^" + Sex + d) = 0, obtaining the result that x' must satisfy the equation {ac — lr, ad — be, bd — c^\x, 1)^ = 0, where the arrow-head on the bracket indicates that the equa- tion is to be written without binomial coefficients. Since an invariant or covariant of a covariant is an in- variant or covariant of the original quantic, we consider what we obtain from the cubicovariant and (|uadricovariant. As regards the lattei- it is at once evident that the only invai'iant, the discriminant, is simply the discriminant of the original cubic ; and that this must be so is evident a priori ; for coincidence of the points aj^, x,^ (§ 179) can only be due to a coincidence among 1, 2, 3. But also nothing INVARIANTS AND COVAEIANTS. 271 new is derived from the cubicovariant : a coincidence amonj^ 1', 2', 3' implies a coincidence amonti- 1, 2, 8, slunvn alf^e- braically by the fact tliat the discriminant of the cubico- variant is a power of the discriminant of the cubic : the symmetric relation between the two triads of points pi'oved in § 179 shows that the cubicovariant of tlie cubicovariant is not different from the original cubic, and it is seen that it is the original cubic multiplied by the square of the dis- criminant ; and the fact that the points a\, a;., are the double points of the involution (11', 22', 88') shows that the same two points will be found if we start from the cubicovariant. 296. As regards the quartic, (a, h, c, d, e^x, ly, we know three invariants (§§ 294, 155) : — (i.) the discriminant A : (ii.) the expression g^, —ae — 4'hd + ^C", whose vanishing expresses that the roots are equianharmonic ; (iii.) the expression g^, =ace-]-2bcd — ad^ — hh — c^, whose vanishing expresses that the roots are harmonic : but these three are not independent. For the group of cross-ratios is determined by the equation (.§ 156) ^2'{(0 + l)(0-2)(0-*)P-27//3^{(^ + ''^)('/' + ^')}'; now a coincidence among the four points makes two of the cross-ratios equal to 0, and two equal to go ; and as tlie sextic equation is (g-^ - 27^3- )(0'^ + 1 ) - 3((/./ - 27g,'W + 0) + ^4 (4>' + ') + ^>'/>' = «. the necessary and sufficient condition for a coincitlcnce of points is g,'-27g,' = 0. Hence g^^ — ^^g^" is a numerical multiple of a power of A This expression is of degree 6 ; and A, obtained by the elimination of x from ax* -f -ihx^ -f Gcx^ + -idx -f e = 0, ax^-i-'Sbx^ + Scx -\-d = 0, that is, from ax^ + 2bx^ + Sex +d^0, hx^ + Scx^ + Sdx+e=0, is also of degree 6. Now as the term a^e^ occurs in gS' — 27g.f with coefficient unity, there is no numerical f.-ictoi- to be rejected from this expression ; we find therefon- A=^./-27/y,^ and A does not count as a distinct invariant. 272 INVARIANTS AND COVAEIANTS. Using any two of the tlireo invariants A, g.^, (j.^-, which are all of degree 6, the quotient is an absolute invariant ; but of course one only of the six here indicated must be counted. Writing, for example, J for the quotient of //./ by A, we have and tlie cross-i'atio sextic can be written in an}^ one of the forms J : J- 1 : 1 = 4(f ^ - + 1 )' : (20=^ - 30- - 30 + 2)^ : 270-(0 - 1 f. Note. Any one of the six cross-ratios determined by tlie four points is an invariant ; but not being expressible rationally in terms of the coefficients, it is an irrational invariant ; and the invariants g.^, g^, A are symmetric functions of these six irrational invariants. Let the roots of the qnartic be 1, 2, 3, 4, and write / = (1 -1^X3-4), m={l-3){4-2), « =(1 - 4)(-2 -3), so that / + m +n = ; then the six irrational invariants are the fractions _m _n _n _l _1 _m V V m on ii n The expressions for g,,, g.^, A are q., = — (r- + nr + v")= - — (mn +iil + /m), ;u 24 li! 256 See Cayley, A Fifth Memoir upon Quantics, 1858 ; (Collected Papers, vol. ii., No. 156); and Klein, Theorie der EUiptischen Modulfunctionen, t. i., pp. 3-15. In Professor Cayley's Memoir, the invariants g-it g^i are denoted by /, ./. The quartic certainly has covariants ; for calling the points 1, 2, 3,4, we can associate these in ( liferent ways, so obtaining three involutions (12,34), (13,42), (14,23); the three pairs of double points are symmetrically derived from 1, 2, 3, 4, hence the sextic e({uation by which they are determined has for its coefficients rational homogeneous ex- pressions in a, h, c, d, e ; they are projectively connected with 1, 2, 3, 4, hence the sextic is a covariant; and consider- ing the mode of formation it is plain that the six points are ordinarily distinct, hence this is not a power of any lower covariant. The (piartic has one more covai-iant : but no special purpose would be served by attempting a conq)lete enumeration in tliis way: the exanq^les gi\'en suffice to show how invariants and covariants present themselves in con- nection with a single binary (juantic. In detecting invariants INVARIANTS AND COVARIANTS. '27'-^ or covariants by this process, two things iinist be attended to ; in order tliat the derived function may be rationally expressible in terms of the coefficients of tlie <;iven quantic, all the points nuist be involved symmetrically ; and in order that the function may be endowed with the propert}' of invariance, all the relations used must be projective. 297. As an example of invariants and covariants of a system, consider two quadratics, ax^ + 2hx + c = 0, ax^ + 2b' x + c' = 0. Here we have two pairs of points to consider ; hence there is an invariant, ac'-\-a'c — 2hb', whose vanishing expresses that the two pairs are harmonic. And as the two pairs determine an involution, whose double elements are projectively related to the system, there is a co variant, (ah' — ab)x^ + (ac' — ac)x -\-{bc' — b'c) , which equated to zero gives the double elements of the involution (§ 174). Similarly with regard to a system of three quadratics there is an invariant, a b c a' b' d a b" c" whose vanishing expresses that the three pairs of points are in involution (§ 174). Ex. Sliow by geometrical considerations that if an n-ic have a tjuadri- covariant, it has an «-ic covariant, wliich is not a product of lower covariants. Ternary Quantics. 298. In dealing with ternary quantics, whose interpretation is by curves in a plane, we have to consider invarianU — ex- I pressions whose vanishing indicates some permanent propei'ty of one curve, or relation of a system of curves; covar'mnts — expressions which equated to zero give loci, having some permanent relation to the loci of the system (this system being supposed given in point coordinates, for deliniteness) : and also contravariants — expressions involving line coordin- ates, which equated to zero give envelopes having some permanent relation to tlie given system. For exanq)le, the envelope of a line cut harmonically by two conies aS'^, >S'.,, is a conic «!>; this conic is a curve having a projective relation to the given pair of conies ; if then we express its equation in terms of .r, y, z, we have a covariant of the system of two s.a s 274 INVARIANTS AND COVARIANTS. conies. But its equation presents itself most naturally in line coordinates ^, rj, ^. In this form it is not an invariant of the given system, for it is not expressed in terms of the coefficients only ; it is not a covariant, for it does not involve the variables x, y, z\ nevertheless it is endowed with the property of invariance, and it is called a contravariant (French, Forme adjointe; German, Z uyeJtorige Fo)-ni). Re- ciprocally, if a system be given in line coordinates, invariant expressions involving ^, r], ^ are covariants, and invariant expressions involving x, y, z are contra variants. The general term invariant, explicitly referring to the permanence of relation, is often used as including covariants and contravariants. In this general sense, an invariant of an invariant is an invariant ; in the special sense, an invariant of a covariant or contravariant is an invariant; a covariant of a covariant, or a contravariant of a contravariant, is a covariant ; and a covariant of a contravariant, or a contra- variant of a covariant, is a contravariant. 299. The idea of two distinct sets of coordinates, to which we are led in analytical geometry by the principle of duality, is deliberately accepted and generalized in the algebra of linear transformations ; that is, as stated in § 285, the group of collineations is extended by the addition of linear dualistic transformations. Reference to § 84 shows that the relation of the two sets of coordinates exhibits itself in linear trans- formation in the fact that they are transformed by inverse substitutions ; that is, the formulae of transformation for the one set being X = l^x' -\- m,^y' -\- ..., y = Ux-\-m.,y'+..., etc , those for the otlier set are }]' = vi^^-\- on. ,)]+..., etc , and the two sets are connected by the identical relation «f +2/';+ «^+ • • . = a;'f + Z/V + 5^T + • • • • This is adopted as the defining property in algebra : sets of (piantities that are transformed by the same substitution are called cog^redicnt ; sets that are transformed by inverse substitutions are contragredAent. Hence sets of point co- ordinates are cogredient : sets of line coordinates are co- gredient ; but point and line coordinates are contragredient. INVARIANTS AND COVAETANTS. 275 Thus modern algebra does not stop with the projective two-dimensional geometry we are now considering, it admits any number of variables ; " its group consists of the totality of linear and dualistic transformations of the variables em- ployed to represent individual configurations in the mani- foldness ; it is the generalization of projective geometry " (Klein). 300. Any contravariant expression can, however, be re- garded in a different way. Taking the example alread}' used, it was required to find the envelope of a line cut harmonically l)y two conies. But this may be stated in the form : — What condition must l:m: n satisfy in order that the line L = Ix -\- ony -\- nz = may be cut harmonically by the conies S^ = 0, 8.-, = ? This is a question as to the permanent relation of the line L and the conies /S'^, S2', the answer is that a certain invariant of the system >Sj, S.^, L must vanish. Thus when we regard ^, rj, t, as known, $ is an invariant of the system S^, S.,, L ; when we regard ^, >/, ^ no longer as known quantities, but as line coordinates, $ is a contra- variant of the system >S\, 8.-,^ ; and when we express the equation of the conic $ in point coordinates, so obtaining F, V is a covariant of the system S^, 8.^. Hence there is no absolute necessity for considering contra variants in the theory of ternary quantics ; a contravariant is simply the reciprocal form of a covariant : the reason for admitting the two con- ceptions is the same as the reason for using both point and line coordinates. Any curve ^ = has a line equation 2 = 0, that is, in dealing with ternary quantics, to every single expression there is certainl}^ one contravariant. As regards binary quantics, contravariants have no special significance. The connecting equation for the two sets of coordinates being in thi.s case x^-\-yri — 0, we have ^:i] = y- —x, and nothing is to be gained by the use of ^, >;. 301. Finall}^ both sets of coordinates may enter into an invariant expression; it is then called a mixed concomitant (French, Forme mixie ; German, Zivischenform). For exaniple, the equation of the pair of tangents to a conic ;S at the points where it is met by a line L involves the coefficients in 8, the coordinates x, y, z, and the coefficients I, on, n in L ; if I, m, n be regarded as known, the result is a covariant of the system *Si, L ; but if I, m, n be regarded as coordinates of the line, and written ^, >/, ^, the result is a mixed concomitant. 302. When considering the properties of two or tln-ee 276 INVARIANTS AND COVAEIANTS. conies, it was found that some relate to all conies of the pencil or net. For example, if a line be cut in involution by three conies S^, >Sf.,, S.^, it is cut in invohition by every conic of the net. Rc^vardino- the line L as known, the fact that it is cut in involution by the three conies, beino- projective, is expressed by the vanishing of a certain expression G; G is an invariant of the system S^, S.2, S^, L ; and as regards the conies >Sfj, 8.-,, S^, it is a comhinant ; and when the coefficients in L are regarded as line coordinates, so that G is a contra- variant of *S'p So, ^3, it is still a comhinant. " An invariant of a system of quantics of the same degree is a comhinant if it be unaltered when for any of the quantics is substituted a linear function of the quantics " (Salmon). Combinants present themselves in the theory of binary quantics. Consider two quadratics, u = ax" + 26^ -j- c, v = ax- -\- 2b'x + c'. These have a covariant (§ 297), fz= ((ih' — ab)x' + (ac — ac)x + (be' — b'c), which equated to zero gives the double points of the in- volution determined by the two pairs %i = 0, v = 0. But every pair ih-\-\v = is in this involution; hence the covari- ant / is a comhinant. Similarly the three pairs u = 0, ^ = 0, w = are in involution if a certain invariant vanish (§ 297); and in this case all pairs u + \v + ij.id = belong to the in- volution ; the invariant is a comhinant. 303. Considering a single conic, there is one invariant, A, whose vanishing expresses that the conic is degenerate ; and there is a contravariant, the reciprocal equation of the conic. A proper conic by itself has no projective property ; and there is no locus and no envelope projectively connected with it. To obtain the metric properties of a conic we have to consider the projective relations of two conies, and then make one degenerate into a pair of points. Hence the im- portance of the theory of the invariants of a system of two conies. Supposing the conies to be given in point coordinates, a number of the special relations of position can be expressed in terms of the four common points A, B, G, D. Now these common points arc most simply assigned by means of straight lines joining tlKsm in pairs; we require therefoi'e the conniion eh(ji"ds of the conies, that is, the line-pairs included in the pencil /.•>S'-f-^"=:0. The three values of /.; giving the three line-pairs are the I'oots of the cubic ^/,^ + ()//^ + ()7, + ^' = (1), INVARIANTS AND CO VARIANTS. 277 where >S', /S" being written in the ordinaiy form, A, A' are the discriminants, and e =Aa + Bb' + Cc' + 2Ff + 2Gg'-\- 2Hh', 9' = A'a + B'b + Cc + 2F'f+ 2 G'g + 2H'h, A,B,... A',B',... being written for the minors of (i.,b,... a,b',... in the determinants A, A'. N^ote. The notation here used is that of (_'ha]i. XVIII. in Salmon's Conic Sections ; this work will now be referred to as C. If kS-\-8' be degenerate, it is degenerate after linear trans- formation ; hence the values of h given by equation (1) are unalterable by transformation ; the coefficients in the equation are invariants ; the vanishing of any one of these coefficients expresses some projective property of the system of two conies. The signihcance of A = 0, A' = is known; for the meaning of the vanishing of G, 0' respectively see C. % 375. The conies have contact if two of their connnon points coincide; since the coincidence of A, B causes the line-pair AC, BD to coincide with BC, AD, the condition of contact is found by expressing that equation (1) has equal roots {G. § 372) ; and since every conic of the pencil passes through A, B, any two conies of the pencil have contact. The in- variant whose vanishing expresses contact, that is, the tact- invariant, is therefore a combinant. 304. Since the special importance of the system of two conies consists in its affording a means of investigating the metric properties of one conic, the second conic of the system is to be degenerate, in line coordinates. If 8' be degenerate, A' vanishes, and equation (1) reduces to the quadratic A/^He/- + H' = (2). Supposing the equations to be in point coordinates, 8' = is a line-pair, AG, BD. The other line-pairs are AB, GD and AD, BG. If A, B coincide, the pair AD, BG is tlie same as 8', and the intersection of the two lines 8' is the point A (or B), that is, a point on >S'. Now for the pair AD, BG to be the same as 8', the corresponding value of h nuist be zero, hence O' = 0. Thus if >S" be a line-pair, and B' = 0, the intersection of the lines 8' is a point on 8 (C. § 373). If however = 0, the lines 8' are conjugate with respect to 8 (G. § 373). Reciprocally, if the equations be in line coordinates, S' being degenerate is a point-pair; 0' = O is the condition that the conic 8 touch the line joining the points 8'; and = is the condition that the points 8' be conjugate with respect 278 INVARIANTS AND COVARIANTS. to the conic S. Tlius if S' = represent the circular points, the condition that the conic be a parabola is 9' = 0, and the condition for a rectangular hyperbola is G = 0, these con- ditions being formed from the line equation of the conic. Bx. Deduce the ordinary Cartesian conditions for a parabola and a rectangular hyperbola. 305. A full discussion of the invariants of a system of two conies is contained in Chapter XVIII. of Salmon's Conic Sections. Reference may also be made with advantage to Chapter XV. of Casey's Analytical Geometry (edition of 1893), where, on p. 517, there will be found a complete list of the concomitants of two conies ; and to Clebsch, Vorles- ungen ilber Geometric, t. i., pp. 265-304. These books being well known there is no occasion to repeat the discussion here ; the object of this chapter is simply to show that the language of the algebra of linear transformations does ex- actly express the projective geometry to which the preceding chapters are devoted. INDEX. ( The numbers refer to the pages ; ivhen a number is jyrhUed in italics, the reference is to an examj)le on the j/age indicated. ) Absolute, the, defined, 249. common tangents of curve and, 250. contact of curve with, 250. double contact of curve with, 250. intersection of curve witli, 249. use in generalizing metric concep- tions, 252, 257, 263, 276. Angle, generalized idea of, 253. Angular magnitude, comparison of, with linear, 94, 249, 252, 255. Anharmonic ratio, see Cross-ratio. Areals, 12. Asymptotes, 102. compared with foci, 231. equation of, for conies, 103. equation of, for any curve, 104. number of, 103. of a conic are the double lines of the involution of conjugate diameters, 175. rules for determining, 105. Axes, of conic, 123, 175. Axis, radical, 118, 1S7. Ball, Sir R., 259. Binary quantics, 269. Birational transformation, 240. Brianchon's Theorem, 83. Brill, 229. Brocard angle and points, ;^4- Canonical form, 205. Cartesian coordinates, compared with homogeneous, 29. equation of circle in, 1 17. equation of isotropic lines in, 112, equation of line infinity in, 30. how made homogeneous, 29, 31. line coordinates, 30. Casey, 278. Cayley, 53, 124, 174, 242, 249, 259. Centre, see Diameter. Change of triangle of reference, in point coordinates, 32, 201, 208, 200. in line coordinates, 32. in point and line coordinates to- gether, 33. Chasles, SO, 80, 177, 182, 236, 242. Chasles' Theorem, S3, 86. Circle, 115. double contact with the Absolute, 251. equation of, in line coordinates, 115. ecpiation of, in point coordinates, 116, 117, /-37. indefinitely small, 115. indefinitely great, 1 15, 245. nine-points, 121, 1^7. radical axis of two, 118, 1^7. system of, coaxal, 118. Circular points, a necessary conception, 110. absolute elements in the plane, 249. 280 INDEX. Circular points, classification of conies by means of, 114. coordinates of, 115. generalization of, 249. more important than the line in- finity, 248. product of distances from any ordinary line is constant, 112. relation of conic to, 114, 249, 276. unchanged by orthogonal trans- formation, 205. unchanged by rotation or ti'ansla- tion, 210. See aho Absolute, Foci, Isotropic. Class of curve, 53, 57, 65. Clebsch, 55, 174, 182, 278. Cliff"ord, 137, 17S, 259. Coaxal circles, 118, 233. Cogredient, 274. CoUineation, 210, 211. See also Trans- formation, linear. Combinant, 276. Comparison of equation and coordinates of a line, 13. equation and coordinates of a point, 13. linear and angular magnituile, D4, 249, 252, 255. point and line coordinates, 10. Complex variable, 164, 259. Concomitant, 275. Condition, conditions, for degenerate conic, 69. for double contact, SI. for double point, 97. for line-pair, 69. for parabola, 278. for point-pair, 69. for rectangular hyperbola, 278. number of, determining a circle, 1 15. number of, determining a conic, 81. number of, determining a paral)ola, 102. number of, determining an »(-ic, 81. simple or multiple, S3, S.l. that six elements may belong to a conic, 83. Condition, conditions, that system be equianharmonic, 150. that system be harmonic, 44, 149. that three lines be concurrent, 14. that three points be coUinear, 14. that two lines be conjugate with respect to a conic, 71. that two points be conjugate with respect to a conic, 70, 75. See also Construction of conies. Confocal conies, form a range, 126. orthogonal, 175. reciprocal to coaxal circles, 233. Conic, conies, asymptotes of, 103, 175. axes of, 123, 175. centre of, 1U6. class and order the same, ()5. common self- conjugate triangle of two, 76, 17S. condition for degenerate, 69. condition that a line touch, 64. conditions for double contact, 81. confocal, are orthogonal, 175. conjugate chords of, determine harmonic points in, S8. conjugate diameters of, 107, 175. cross-ratio of extremities of two conjugate chords harmonic, 8S. cross-ratio of four, SO. cross-ratio of four points in, 88. degenerate, 69, 245. diameters of, 106. eccentric angle, 134. envelope of line cut harmonically by two, OJf, 178, 273. envelope of line cut in involution by three, 179, 182. cvpiation of circumscribed, in point coordinates, 59. equation of circumscribed, in line coordinates, 61. ecjuation of inscribed, in line co- ordinates, 59. e(|uation of inscribed, in point co- ordinates, 61. equation of reciprocal to, 64. INDEX. 281 Conic, conies, equation of, referred to self-con- jugate triangle, 74. equation of, through four points or touching four lines, 7'2. flat, 247. foci of, 123-126. harmonic, S9, 176'. homothetic, 250. imaginary, lOS. involution properties of, 174. locus of intersection of correspond- ing rays of two homographic pencils, 159. locus of point harmonically sub- tended by two, 94, 17S, 267. locus of points harmonic with re- spect to net, is a cubic, SI, 181. metric properties of, 106, 114-122, 125, 276. net, SI, 180. pencil, 74, 76-79. range, 74, 76-79. reciprocal equation, 64. reduction of equation, 107. similar, 256. system of, through four points or touching four lines, 76. system of, when four conditions are given, 179. tangents fi'oni a point to, 71. unicursal, 134. See also Condition, Conjugate, Con- struction, Pencil, Polar, Range, Self -conjugate triangle. Conjugate, imaginaries, 47. lines, with respect to a conic, 77, SS, 174. points, as determining elements for a conic, 182-184. points, with respect to a conic, 70, 75, 174. points, with respect to a conic, Hesse's Theorem, 177. points, with respect to a pencil of conies, SO. See also Diameters, Pole, polar, Self-conjugate triangle. Connection of point and line co- ordinates, 70. Construction, for equianharnionic elements, 170, 17 J. for harmonic elements, 43. for involution, 163, 167, 169, 187. of accurate diagram, 23, 44. when linear, 176. when of second degree, 176. Construction of conic determined by five conditions, — five points or five lines, 81, I7S. four points and one line, or four lines and one point, 82, 17S. three points and two lines, or three lines and two points, 82, 177. tliree points, a pole and polar, 177. one point, two poles and polars, 177. four points, one pair of conjugates, 183, 177. three points, two pairs of con- jugates, 183. two points, three pairs of con- jugates, 183. one point, four pairs of conjugates, 183. five pairs of conjugates, 182. four tangents, one pair of con- jugates, ISo. one tangent, four pairs of con- jugates, IS/i. three points, one tangent, one pair of conjugates, 1S.5. two points, one tangent, two pairs of conjugates, ISo. one point, one tangent, three pairs of conjugates, ISo. a self-conjugate pentagon, 177. Contragredient, 274. Contravariant, 273, 275. Coordinates, defined, 2. general idea of, 1. homogeneous line, 9, 11. homogeneous point, 5, 8. line, in Cartesians, 30. number of, 2, 5. 282 INDEX. Coordinates, relation of Cartesians and homo- geneous, 29, 31. Coordinates and equations, of four points and four lines, 41, 43. of four imaginary elements, 48. Correlation, correlative, 236. Correspondence, defined, 156. equivalent to homograpiiy, 156. general idea of, 210, 215. geometrical idea of, 216. identity of corresponding figures, 217. of asymptotes and foci, 251. of cusp and inflexional tangent, 67, 234. of linear and angular magnitude, 252. of node and double tangent, 67, 234. of point and line coordinates, 40. of point and line figures, 4, 14, 40, 43, 232. of point and line theories, limita- tions of, 244. of points on a curve (Chasles), 242. of projective figures, 202. one-one, 156. one-one linear, 210. one-one quadric, 211. one-one (^uadric, skew projection for, 219. See also Duality, principle of. In- version, Projection, Keciproca- tion. Covariant, algebraic definition of, 269. general idea of, 268. of binary cubic, 270. of two binary quadrics, 273. of ternary quantics, 273. Cramer, 229. Cremona transformation, 240. Cross-ratio, delincd, for a pencil, 35. detined, for a range, 36. elements, four, determine six cross- ratios, 36. Cross-ratio, elements, how interchangeable, 153. equalities among the six cross- ratios, 149. given by a sextic equation, 151, 272. idea is descriptive, not metric, 148. not altered by projection, 147. notation for, 35, 154. of configurations, (kk', Oco ), (k/c', W), 38, 39, 40. of four conies, SO. of four points in a conic, 88. of special pencil or range, 50. when equianharmonic, 149, 150. when harmonic, 37, 149. Cubic, asymptotes of, 104. equation of, under assigned condi- tions, 60, JIM. inflexions of, 208. linear transformation applied to, 208. reciprocal to, 65, 66, 97, 135, 136. special unicursal, 137. theory of, depends on nets of conies, 182. Curve, deficiency of, 135. degenerate, 54. e(£uation of, defined, 53. formation of reciprocal ccjuation, 65. general idea of, 52, 54. has two different equations, 53, 57. how affected by inversion, 231. order and class, 53, 57. order and class in general diH'erent, 65. pencil and range, 58. reciprocal defined, 66, 71. reciprocal, how found, 65. reciprocal is specialized, 97. tangential equation, 63. tracing of, in homogeneous co- ordinates, 139. unicursal, 130. See also Conic, Cubic, Quartic, Unicursal. Cusp, nee Singular points and lines. INDEX. 283 Deficiency, an invariant, 137, 264. defined, 136. not affected by inversion, 232. not affected by birational transfor- mation, 264. the same for a curve and its recip- rocal, 136. zero for nnicursal curves, 135. Degrees of freedom, 2, 129. Desargues' Theorem, 83, 172. applications of, 17S. ajjplied to focal properties, 175. constructions depending on, 175. Descriptive, distinction between descriptive and metric properties, 57, 147. distinction between descriptive and metric properties obliterated, 257. Determinant of transformation, 32, 265. Diameters and centre, 106. conjugate diameters derived from poles and polars, 107. conjugate diameters in involution, 175. conjugate diameters, one pair at right angles, 123. Dimensions, 4. Discriminant, of binary quantics, 269, 270, 271. of ternary quadric (conic), 69, 276. of ternary quantic in genei-al, 97. Distance, from a point to a line, 9, 17- generalized, between two points, 254. Double point, double tangent, .see Singular points and lines. Dualistic transformation, 211. formulae for linear, 236. not a gi'oup, 261. reciprocation, a sjiecial case of, 235. three elements united witii thoii- correspondents, 238. Duality, principle of, 15, 17, 43, 51, 257. statement of, 55. Duality, principle of. See al-io Dualistic transformation, Reciprocation. Element, nature of, 2, 216. primary, 5, 14, 51, 52, 53, 257. secondary, 51, 52, .53, 257. Ellipse, 101. Envelope (compare Locus), degenerate, 54. line not an envelope, 54. of a line cut harmonically by two conies, 94, 178, 273. of a line cut in involution by three conies (net), 17D, 182. Equation of, a line, 8. a point, 13. line through intersection of two lines, 9. point of contact in line coordinates, 61. point on join of two points, 13. reciprocal to a given curve, 65. tangent in point coordinates, 61. See also Conic, Cubic, Quartic. Equianharnionic, 149. condition that four points be, 150. constructions for, 170, /?,.'. elements, how interchangeable, 153. idea is descriptive, 170. Evolute, generalized, 256. Ferrers, 17. Flat conic, 247. Focus, 122, 124. confocal conies, 126, 175. directrix is polar of, 123. effect of quadric inversion on, 2,32. focal properties of conies, 125. foci conqjared with asymptotes, 251. Four points or lines, coordinates of, 41, 43. ec^uations of, 41, 43. treatment of, when imaginary, 48. Fundamental identical relation, in line coordinates, 16. in point coordinates, 6. 284 INDEX. Fundamental identical relation, significance of, in line coordinates, 111. significance of, in point coordinates, 27. See al-io Absolute, Circular points, Infinity. Group, "2(31. contains sub-groups which leave something unaltered, 261. may be extended, 263, 274. of collineations, 261. of linear transformations, 261, 263. of movements, 261. of operations, 261. of orthogonal transformations, 263. of transformations, 261. of translations, 261. Harkness, 164. Harmonic, 37. bisection depends on harmonic divi- sion, 38. condition that four points be, ^9, 149. condition that two pairs of points be, 44. conic, 89, 178. construction for, 43. division of line, 04. division of point, 04. elements, how interchangeable, 153. idea is descriptive, 169. properties of complete quadrilateral and (piadrangle, 41, 42. properties of poles and polars, 91. relation of harmonic division and bisection, 38. transformation, 212, Jlo. triangle, 76. Henrici, 5. Hesse, 177. Hexagon, .see Briauciions Theorem, Pascal's Theorem. Hilbert, 80. Hirst, 219, 230. Homograpliy, 1")."). compared witli involution, 160. Homograpliy, equivalent to ( 1 , 1 ) correspondence, 156. homogi-aphic correspondence on curves, 185. homographic pencils generate a conic, 159. homographic ranges generate a conic, 159. homographic systems with the same base, 158. homographic systems, double ele- ments, 15S. Homology, J4, 194. Homothetic conies, 256. Hulburt, SO. Hyperbola, 101. rectangular, 119, 120, 121, 278. Hyperbolism, 229. Imaginary elements, 45, 47. coordinates and equations of four, 48. pencil or range of conies determined by four, 76. quadrangle and quadrilateral deter- mined by four, 47. self- conjugate triangle determined by four, 178. Infinity, a point in spherical geometry, 257. a special line in projective geometry, 26. at the same distance from all ordi- nai'y points, 28. direction not to be associated with, 27. equation of, in Cartesians, 30. relation of conic to, 101. relation of curve to, 102. j^ee a/so Absolute, Asymptotes, Diameters and centre. Inflexion, .see Singular jioints and lines. Intersection of, line and conic, 89. line and curve, 57, 90. two conies, 76, 93. two curves, 69, 92. INDEX. 285 Intersection of, two lines, 25. Invariant, 264. absolute, 267. algebraic definition of, 267. discriminant, 269, 271. condition for parabola, 278. condition for rectangular hyperbola, 278. irrational, 272. of binary quartic, 271. of two binary quadrics, 273. of two conies, 276. tact-invariant, 277. Inverse substitutions, 33, 274. Inversion, circular, 219. Inversion, quadric, 211, 217. analysis of singularities by, 225. applied to a curve as a whole, 230. construction for inverse points, 223. effect of, on deficiency, 232. effect of, on double points and double lines, 224. effect of, on focus, 232. formuhe for, 221, 222. plane construction for, 219. Involution, 160. centre, 162. circular, 165. common elements of two involu- tions, 167, 115, 188. compared with homograpliy, 160. constructions for, 163, 167, 169, 187. determined algebraically, 165. determined by two pairs, 160. double elements, 161, 162, 188. elliptic, 162. extension of idea of, 173. harmonic property of double ele- ments, 165. hyperbolic, 162. idea is descriptive, 169. notation for, 174. on a conic, 187. pairs of imaginaries, 167. pencil contains one pair of ortho- gonal rays, Hi. properties of quadrangle, 168, 173. Involution, jjroperties of conies, 174. See. al-io Desargues' Theorem. Involution-position, 212. Isotropic lines, 112. all pass through two fi.xed points, 111. have no direction, 112, 255. two through every point. 111. Joachimsthars method, 89. nature of coordinates not generally important, 93. Joncp'.iercs (de), 182. Klein, 217, 259, 261, 264, 272, 275. Laguerre, 259. Limiting points, 1 1 9, 234. Line-pair, condition for, 69. reciprocal to, 245. Linear magnitude, comparison of, with angular, 94, 249, 252, 255. Linear transformation, see Transform- ation. Locus (compare Envelope), degenerate, 54. of intersection of perpendicular tan- gents to a conic, 17S. of pairs of points harmonic with respect to three conies, SI, 181. of a point subtended harmoiiii'aliy by two conies, 9^, 17S, 2()7. of the pole of a fi.xed line with respect to a pencil of conies, SU, IS4. point not a locus, 54. Metric, distinction between doserii)tive and metric properties, 57, 147. distinction between descriptive and metric properties obliterated. 257. origin of metric relations, 244. projective metric combinations, 146, 147. properties of a conic, 249. 27(). 286 INDEX. Metric, properties of a curve, 249. See also Circuliir Points, Infinity. Mixed concomitant, 275. Modulus of transformation, .S2, 265. Mubius, 210. Morley, 94, 1G4. Nest of curves or conies, 80. Net, 180. of conies, SI, 179, ISO, 182. tangential net, ISO. Newton, 220. Node, .see Singular points and lines. NOther, 229. Normal, generalized, 256. One-one, see Correspondence. Order of Curve, 53, 57, 65. Orthocentre, of a triangle, 121. Orthocentric (juadrangle, 122, 17S. Parabola, 101, 278. has contact with the Absolute, 251. Parallel lines, 26, 254. Parameter, expression of coordinates in terms of, 89, 129. jS'ee also Unicursal. Pascal's Theorem, 83, 197. constructions by means of, 84. Pencil (compare Pvange), 35, 58, 1 79. connected with range by reciproca- tion, 23.3. harmonic conies of a pencil, three in number, S9. homographic pencils, 155. homographic pencils generate a conic, 159. line equation of, SI. of conies, 74, 76-80, 179. of conies determined by three points and one pair of conjugates, 177, 183. of conies determined by two points and two pairs of conjugates, 18.3. of conies detcsrmined l)y four pairs of conjugates, 179, 183. Pencil (compare Range), of lines, when equal, projective, perspective, 154. See also Desargues' Theorem, Locus, Envelope. Pentagon, self-conjugate with respect to a conic, 177. Perpendicular lines, 113, 114,252. Perspective, J4, 154, 194. Pliicker, 54, 123. Point-pair, condition for, 69. reciprocal to, 245. Pole, polar, 68. polar conic, pole conic, in dualistic transformation, 238. pole and polar with respect to a line-pair, 73. pole and polar with respect to a triangle, 20. reciprocal polars, 233. theory of poles and polars with respect to conies, 70, 75, 91. Poncelet, 80, 119, 194. i 'rejection, 144, 189. alters metric properties, 145. analytical view of, in space, 198. analytical view of, in a plane, 199. any conic can be projected into a circle, 192. any two points can be pi'ojected into the circular points, 193. concurrent lines can be made parallel, 189. correspondence of projective figures, 202. diagram for, 195. does not alter cross-ratio, 147. formula' of transformation, 204. general theorem on, 191. harmonic transformation, 212. linear transformation specialized, 201, 203. parallel lines become concurrent, 189. plane construction for, 204. plane projection, 194. skew projection for (|uadric corre- spondence, 219. INDEX. 287 Quadric invei'sion, .see Inversion. Quadrangle, complete, 42. determined by imaginary elements, 47. harmonic properties of, 42. involution properties of, 168, 173. orthocentric, 122, 17S. Quadrilateral, complete, 41. bisections of diagonals, ^,9. determined by imaginary elements, 47. harmonic properties of, 41. Quaitic, cannot have more than three double points, 13(j. tricuspidal, lUO, 138, 230. trinodal, 98, 231. Radical axis, 118, 137. Range (compare Pencil), 35, 58, 179. connected witli pencil by reciproca- tion, 23.3. homograpliic ranges, 155. homographic ranges generate a conic, 159. of conies, 74, 70-80, 179. of points, when equal, projective, perspective, 154. point equation of, SI. See also Desargues' Theorem, Locus, Envelope. Rational transformation, .see Bi- rational. Reciprocal, curves, 67. equation, formation of, 65. polars, 233. to conic, 64. to cubic, 65, 66, 97, 135, 1.36. to line-pair and point-pair, 247. relation of confocal conies and co- axal circles, 233. Reciprocation, 211, 232. a special dualistic transformation, 235. diagram for, 234. skew, 236. See also Dualistic transformation. Rectangular hyperbola, 119, 120, 121, 278. Reye, 5, 144, 155, 159, 170. Riemann transformation, 241. Russell, 177, 188. Salmon, 46, 85, 87, 94, 9.j, 119, 124, 1.34, 187, 188, 197, 2.33, 240, 276, 277, 278. Schrciter, 182. Self-conjugate triangle, 75. of pencil of conies, 178. of two conies with real common points, or with real common tangents, 70. of two conies with imaginary common points and imaginary common tangents, 17S. system of two, . linear, 201, 2.S6, 20.3. linear, special cases of, 204, 205, :U5. quadric, 229. Kiemann, 241. Steiner, 219. See alxo Dualistic transformation, Inversion, Projection. Trilinears, 12. Unicursal, 1.30. curve is unipartite, 181. curve of zero deficiency is, 1 35. distinction between unipartite and, 132. every conic is, 134. general curve is not, 132. reciprocal is, when curve is, 133. unipartite curve not necessaril}', 132. Union, of point and line, 12. Unipartite, 131, 132. United points, 242. Web, 180. END. OLASaOW: PBl.VTEP AT THE UNIVEKSri'V PRESS BY ROBURT MACLEHOSE. A 3 1205 01966 8175 UC SOUTHERN REGIONAL LIBRARY FACILITY A A 001 430 504 9