LIBRARY OF THE University of California. Class Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/anharmoniccoordiOOhimerich ANHARMONIC COORDINATES BY THE SAME AUTHOR. THE OUTLINES OF QUATERNIONS. Crown 8vo, 10s. GUNPOWDER AND AMMUNITION : their Origin and Progress. 8vo, 9s. net. HISTORY OF THE ROYAL REGIMENT OF ARTILLERY, 1815-1853. 8vo, 6s. net. LONGMANS, GREEN, AND CO., LONDON, NEW YORK, BOMBAY, AND CALCUTTA. ANHARMONIC COORDINATES BY LiEUT.-CoLONEL HENRY W. L. HIME (late) royal artillery ^' ^^ OF THE UNIVERSITY Of LONGMANS, GREEN, AND CO 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY, AND CALCUTTA 1910 AU rights reserved c> mmi PREFACE Although fifty years have passed since the invention of Anharmonic Coordinates, no book, I believe, has hitherto been written on the subject. The explanation of them given by their inventor. Sir W. R. Hamilton, in his Elements of Quaternions, is short; the space devoted to them by Professor P. G. Tait and Mr. C. J. Joly in their works on Quaternions is still shorter; and they are not referred to at all in ordinary books on Coordinate Geometry. Whatever value be assigned to them, we ought not to allow a method devised by a great British mathe- matician to be altogether forgotten. These considerations may justify the publication of the present attempt to fill in the details of Hamilton's outline. The book lays no claim to originality, and confines itself to the application of the method to well-known geometrical theorems. Mistakes will no doubt be detected, but I trust they will be few and unimportant. 19^^ June, 1910. 208548 Of THE UNIVERSITY OP SECT. PAGE CONTENTS CHAPTER I PLANE GEOMETRIC NETS r. General principles of a net 1 2°. The first construction 3 3°. The second construction 3 4". The third construction. Invariable characteristics of the vectors of rational points 3 5°. Vectors of irrational points 4 . CHAPTER II THE POINT 1°. Definition of the anharmonic coordinates of a point - - 6 2°. The coordinates of various rational points - - - - 6 3°. The coordinates of various irrational points ... 7 CHAPTER HI THE STRAIGHT LINE 1°. Coordinates of a fixed point cutting the join of two given points in a given ratio 8 2°. Coordinates of a variable point always collinear with two given points 10 3°. Anharmonic equation of a straight line - - - - 11 4°. Equations of various lines of the net 11 5°. Equation of Aoo , the line at infinity in the plane - - 12 6°. Coordinates of the cross of two straight lines - - - 13 7°. Condition for the concurrence of three straight lines - - 13 ▼iii CONTENTS 8SOT. PAGE 8°. Coordinates of any line passing through the cross of two given lines 14 9". Relation between the coordinates of two parallel lines - 14 10°. The angle between two given lines and condition of perpendicularity 15 IV. Coordinates of a line which passes through a given point and is perpendicular to a given line - - - - 17 12*. Connexion between anharmonic and trilinear coordinates - 18 CHAPTER IV DISTANCES, AREAS AND ANGULAR FUNCTIONS 1". The distance between two given points - - - - 20 2*. The distances of the corners of the given triangle from a straight line 22 3°. The distance of the origin from a straight line - - - 22 4°. The distance between two parallel lines - - - - 23 5". The distance of a point from a straight line - - - 24 6". The area of a triangle 24 7°. The sine of an angle 25 CHAPTER V THE GENERAL EQUATION OF THE SECOND DEGREE 1°. The general equation in anharmonic coordinates represents a conic section 26 2". The first derivatives of the general equation - - - 26 3°. The discriminant, A 26 4°. The points of section of a conic by a straight line - - 27 5°. The equation of a tangent 28 6°. Condition that a straight line shall touch a conic - - 28 7°. Equation of the polar of a point 29 8°. Coordinates of the pole of a line 29 9*. Some properties of poles and polars 30 10°. Ratio of the segments in which a finite straight line is cut by a conic 31 11°. A line which passes through a given point, and cuts a conic, is divided harmonically by the conic, the point and its polar 32 CONTENTS ix SECT. PAOI 12 . The centre of a conic, (A, By C) 32 13°. The bordered discriminant, D 33 14°. Species of conic determined hy B 33 15°. Equation of a diameter 34 16°. Equation of a diameter conjugate to a given diameter - 34 17°. The polar of any point on a diameter is parallel to the tangent at its extremity 35 18°. Equation of a pair of tangents 35 19°. Equation of the asymptotes 37 20°. Condition for rectangular hyperbola 38 21°. The meaning of the absence of any term from the general equation 39 22°. Equation of a conic in terms of two tangents and the chord of contact 40 23°. Parallel tangents 40 CHAPTER VI SPECIAL CONICS 1°. The inconic - 42 2°. The inconic touches the sides of the triangle in A', B\ C - 42 3°. The inconic is an ellipse or circle when is within the triangle 42 4°. When is outside the triangle, the species of the conic depends on the ratios l:m:n 42 5°. The circumconic 43 6°. The terms ' inconic ' and ' circumconic ' - - - - 44 7°. The polar conic 45 8°. Conies which have a common self-conjugate triangle - - 46 CHAPTER VII TANGENTIAL EQUATIONS 1°. The principle of duality 48 2°. The tangential symbol of a line 48 3°. The tangential equation of a point - - - - - 49 4°. The tangential equation of the cyclic points - - - 49 5°. Coordinates of the join of two points 49 X CONTENTS 8BCT. PACK 6°. Equation of the cross of two lines 50 7°. Parallel lines 50 8°. Equation of a circle 50 9°. Transformation of local and tangential equations (Hamilton) 51 10°. Discriminant of a tangential equation of the second degree 53 11°. Only two tangents can be drawn from a point to a curve of the second class 54 12°. Equation of the point of contact of a tangent - - - 54 13°. Equation of the polar of a line 54 14°. Equation of the centre of a conic 55 15°. Coordinates of the polar of a point 55 16°. Equation of the points in which a conic is cut by a straight line 56 17°. Coordinates of the tangents drawn from a point to a conic 57 18°. Criterion to determine whether a point lies on a conic or not 58 19°. Coordinates of the tangent at a given point - - - 58 20°. Coordinates of the asymptotes 58 CHAPTER VIII CROSS RATIO 1°. Definition of (ABCD). General expression for a pair of harmonic conjugates 60 2°. The permutations of {ABCD) 61 3°. (a) Cross ratio of a row in terms of the coordinates of its points 62 (6) Cross ratio of a pencil in terms of the coordinates of its rays 63 (c) Cross ratio of any four lines cut by a transversal - - 63 4°. (a) Cross ratio of a pencil in terms of its vertex and the points in which it is cut by any transversal - - 64 (6) Cross ratio of a pencil in terms of its vertex and any four points on its rays 64 5°. (a) If any two homographic pencils, with different vertices, have a corresponding ray in common, the crosses of the remaining corresponding rays are collinear - - 65 (6) If two homographic rows have a corresponding point in common, the joins of the remaining corresponding points are concurrent 66 6°. (a) Homographic divisions of a line ----- 66 (6) Homographic divisions of two lines - - - - 67 CONTENTS xi SECT. ,^0, T. {LANB)={LA'NB') 68 8". Equation of the directive axis 68 9°. Coordinates of the directive centre 70 10°. Finite points corresponding to points at infinity - - 72 11°. Determination of double points 73 12°. Given 4 points, P^, P^, P^, F^, no three of which are col- linear ; to find the locus of a fifth point P, such that P -PiPg A A =a constant 74 13°. Given 4 straight lines, Ai, Ag, Ag, A4, no three of which are concurrent ; to find the envelope of a fifth line A, such that the an harmonic ratio of its crosses with the four given lines is constant 74 14°. Every triangle which circumscribes the inconic is inscribed in the circumconic 75 15°. The points in which a circle is cut by conjugate chords form a harmonic group 76 1G\ MA.MA'^MB.MB ..,=k^ 77 17°. Given ^, -4' and jB, i?'; to find J/" geometrically - - 77 18°. The various species of involution 78 19°. Numerical calculation of the centre, if - - - - 80 20°. Deductions from the general equation, 16° - - - 80 21°. Connexion between the coordinates of a system of points in involution 82 22°. The cross ratio of 4 collinear points is the cross ratio of the pencil formed by their polars . ... 83 CHAPTER IX TRANSFORMATION OF COORDINATES 1°. General Remarks 85 2°. Relation between the ratios in which the rays of two points cut the sides of a triangle 85 3°. Change of origin to a rational point 86 4°. Change of origin to an irrational point . - - - 86 5°. Change of both origin and triangle (Hamilton) - - - 87 6°. Geometric illustration of 5° 88 7°. Simplification of Hamilton's equations - - - - 90 8°. The modulus of transformation 92 xii CONTENTS CHAPTER X THE CIRCLE 1°. Condition th&t yz+zx+xi/ = shall represent a circle - 93 2°. The coordinates of the cyclic points, / and ./ - - - 94 3". Metric equation of a circle 94 4°. All circles pass through both the cyclic points - - - 95 5°. The circle is the only conic which passes through both cyclic points 95 6*. General equations of the circle 95 T. Condition that (^{xyz) shall represent a circle - - - 96 8°. Circle through three given points 97 9^ The incircle 98 10°. The nine-point, or IX circle 99 11°. The polar circle - - - 99 12°. The in- and IX circles touch 100 13°. Length of tangent from a given point to a circle - - 102 CHAPTER XI THE FOCI OF A CONIC 1°. Property of circle which has double contact with a conic - 103 2°. A focus may be regarded as an evanescent circle - - 104 3°. The foci may be regarded as the intersections of imaginary tangents to the conic from / and J, and they are 4 in number 104 4°. In general, two foci are real and two imaginary, the latter two lying on a real line which bisects at right angles the join of the real foci 104 5°. Sir William Hamilton's method of finding the coordinates of the foci ; ^ : m : 71=1 ; a=6 = c . . . . 107 6°. Transformation of coordinates from a scalene to an equi- lateral triangle 110 T. Property of the tangential equation, T-\-kuv==0, when it resolves into equations of points 110 8°. Determination of the values of k which resolve the tan- gential equation, T+kuv=0, into equations of points - 111 9°. Examples 112 CONTENTS xiii CHAPTER XII MISCELLANEOUS THEOREMS SECT. PAO« , 1°. The harmonic properties of a plane net - - - - 118 2". A theorem by Roger Cotes 119 3°. Coordinates of the isogonal and isotomic conjugates of a given point 120 4°. The isogonal conjugate of every point on the circumcircle is at infinity 121 5'. Pascal's theorem 122 6°. Brianchon's theorem 122 7*. A homogeneous equation of the second degree in terms of its derived functions 123 Index 126 CONVENTIONAL SIGNS 1. A = any straight line. 2. A„ =the line at infinity in the plane (saves 27 letters). 3. AB^ etc., is occasionally used to distinguish the vector AB from the Euclidean line AB. 4. AB'GD^\hQ cross of the line AB and CD. 5. ^2 + ^2 + yj2 = 2?2. (^ + ^ + 7^)2 = 22^. 6. Si = s — a, S2 = ^~^> ^3 = *~<^- Area of triangle = ^^55^5^ s^-^s^ = a^ etc., etc. 7. ^ i^xyz) = (ji {x, y, z) = w^2 ^ ^2/- + 1^^^ + 2u'yz + 2?;'0^ 4- 'ifw'xy. 8. i^Cp^rr) = C^p2 + pr^2 + TFr2 + 2 C^'^r + 2 F'/p + 2 PT'p^. 9. A is the discriminant of ^{xyz). 10. Z) is the bordered discriminant of <^{pcyz). 11. -4, J5, (7 are the coordinates of the centre of cfi(a^z) = 0. In the places in which they occur they cannot be confounded with the corners of the given triangle, ABC. 12. Z^ is a certain function of the coordinates of a straight line. 13. 12^ is the tangential equation of the cyclic points. 14. The nine-points circle is occasionally referred to shortly as the IX circle. 15. II, 5° means Chapter II, section 5. II, (5) means Chapter II, equation 5. (5) alone means equation (5) of the chapter in which the reference occurs. ^^ Of THE UNIVERSITY CHAPTER I PLANE GEOMETRIC NETS 1". In framing his method of Anharmonic Coordinates, Sir William Hamilton made use of a plane geometric net constructed somewhat on the plan of Prof. Mobius.* Regarding every point of the net as the term of a vector drawn from the origin, he deduced a general vector expres- sion which, by a suitable choice of certain coefficients, would represent the vector of any one of these points, which he called " the rational points " of the net. He then proceeded to show how this general expression could be very simply modified so as to represent any point in the plane, not included in the net. These points he called " the irrational points " of the net. Let any four points. A, B, G and (fig. 1), no three of which are collinear, be taken in the plane, and let the six lines, OA, OB, OC, CA, AB, BG be drawn. Then if the vectors OA, OB, OC be called a, /3, y, three scalars I, m, n can always be found such that • la-\'m^-\-ny = 0', .../. (1) and if a, jS, y produced meet the sides of the triangle ABG in A\ B\ a, BG_± CA^_m ^_n, ,^. Conversely, if three coinitial vectors a, ^, y, when pro- **'It was by combining some parts of (Mobius' Barycentric) Calculus with Quaternions that I happened to form the conception." MS. C, 1860, 64, Trinity College, Dublin, p. 51, kindly lent to the British Museum for my use by Dr. Abbot, Librarian, T.C.D, A large part of this MS., which consists of letters from Sir W. R. Hamilton to Dr. (Sir) Andrew Hart, is devoted to the anharmonic treatment of cubic curves. t Outlines of Qiiaternions, by the present writer, p. 14. H.C. A PLANE GEOMETRIC NETS duced, cut the sides of the triangle formed by their terms in points A\ B, C such that Ba_l^ CA^__m AE _n, CA^m' A'B~n' RC~l' then ^a + 7>i/5 + 7iy = (3) 0, A, B, C are the cardinal points of the net, and ABC is the given triangle. ^ B' C Pig. 1. If lies without the triangle, two of the ratios of (2) are negative. In this case we may take one of the three scalars as negative and the other two as positive. The values of I, m, n are subject to certain limitations. First, all three of them must have an actual value. For suppose that one of them, say n, is zero. Then, la+m^ = 0, and since a and /? are not parallel vectors, ^ = 0, m = 0, and the net shrinks to the point 0. Secondly, we must have 2+mH-ti =1=0. For let CHAPTER I Then (&g. 2) = ^a+m^-(^+m)y=^(aZ-0O) + m(a5-0a) = lGA+mGB, and ^ = ^^ CA m Therefore CB is parallel to CA, or B lies somewhere upon the inde- finite line CA, and the net shrinks to the line CA. Consequently, I, m, n must be actual scalars such that l+m-\-n=\=0. 2°. The first construction is to draw the intersections OA ' BC, OB ' CA, OC'AB. To find the vector of the point OA'BC,ovA\ and CA'=^OA'^OG=OA'^y) A'S^fi-OA') CA' m A'B n and Similarly, Hence {m-\-n)OA' = mfi-\-ny 7n-\-n n + L 64-m , .(4) 3°. The second construction is to draw the intersections BC ' BV\ CA ' CA\ AB • A'B', OA • B'C, OB • CA', OC • A'B. By pursuing the plan indicated in 2°, we get m^n OC": l — TYl ^ ~ 2l+m-\-n ' ~ ^+2m+7i ' OC la-{-7nl3-{-2ny l-{-7ii-{-2n .(5) 4°. A third construction would give 84 new points, and the process might be carried on indefinitely — Hamilton investigated some thousands of points ; but however far it PLANE GEOMETRIC NETS be continued the vectors of the rational points of the net are all of the form : •(6) __ xla + ym^ -f zny _ l^xla ^~~ xl+ym+zn ~ Xxl' where x, y, z are whole numbers (or proportional to whole numbers) and the denominator is the algebraic sum of the coefficients. 5°. Let R (fig. 3) be a rational point, the lines through it from the corners of the triangle cutting the opposite sides in -B^, i^g, -Kg. Thus, (6), Qj^^ ^ xla-\-ym^-^zny ^ xl-\-ym-{-zn ' and 0=^xl(a — p)-\-ym{/3—p)-hzn{y — p) = xlRA + ymRB + znRG, Therefore, (2), BRo «^. CR^_ym^ AR^ zn R^A ym' Rfi zn' Rfi xV .(7) Fig. 3. Now suppose R to be an irrational point whose position in respect to the given triangle is given by the ratios : BR^^p. CR^^q. AR^^r R^A q' R^B~r' Rfi~~ 'p Then, (3), =pRA + qRB + tRG =^(a-^) + g(/3-p)+r(y-p), and Oi2=p=^^±^^ '^ p + q+r (8) CHAPTER I 5 Comparing this expression with the standard form (6), x=pl~^; y — q7n~^\ z = m~^.j Substituting these values of x, y, z in (6), we get for the vector of the irrational point i2, ^~ (2?^i)i+(gm-i)m+('r7i-i)7i ^ ^ The vector of any point in the plane may be thus reduced to the standard form. CHAPTER II THE POINT 1°. The anharmonic function of any four collinear points, A, B, G, D, is defined to be (ABCD)^^^ W Let OJ; = '^^"y^+^^y , (fig. 3). Then AB is cut in C in the ratio I : m, and by R^ in the ratio xl : ym ; GA and BG being divided in corresponding ratios. Hence G' AOBR = {AG'BR.)=^^ ^=^-'^ ^ "^^ I ym y A'BOGR==(BAVR,)=- ^ = ^ TYh zn z B ' GOAR = {GEAR.) = i ^ = £. ^ ^^ n xl X •(2) The product of these three anharmonic functions is unity, and any two of them suffice to determine the position of R when the triangle ABG and the origin are given. Hence the name Anharmonic Coordinates. Definition. The three coefficients x, y, z, or any scalars proportional to them, are the anharmonic coordinates of the point R. The point R is denoted by the symbol R = {xyz). 2°. The 13 rational points shown in fig. 1 are symbolised as follows. The vector of the origin (from itself to itself) is zero. Now the standard expression, I, (6), becomes zero when x — y = z, since la + m^ -\-ny — ^. Consequently, = ( 1 , 1 , 1 ), or any three equal numbers. For the point A, p — a\ and to reduce the standard ex- pression to this value we have merely to equate x to unity (or any multiple of 1), 2/ to and 2; to 0. Consequently, (3) CHAPTEK II 7 A = (1, 0, 0). Similarly, B = (010) and (7= (001)— omitting the commas. For A\ we have, I, (4), ^^ ^Z^+^V Consequently 771' "^ ih ^' = (011). Similarly, ^' = (101), C' = (110). For A'\ I, (5), ^ = ^/^-^y . and ^'' = (OlT)-the minus sign being put above the line to save space. Similarly, ^' = (101), (7" = (110). For A'", I. (5), .=^^15^; a,d ^^^^=(211). Similarly, ^'^ = (121), C"" = (112). And so on. To recapitulate : = (111) ^=(100) 5 = (010) (7= (001) ^' = (011) 5' = (101) (7' = (110) ^" = (011) ^" = (101) ^" = (110) ^"' = (211) 5'" = (121) C"" = (112)j 3*". Irrational points are symbolised in a similar way. For instance, let 31^ be the middle point of BG. Then, ^'(^^)' -mr _ |g+y _ (m-^)m/3+(^-^)^y Hence M^ = (om - % - ^) = (oTi^n). Similarly for the middle point of GA,M^^ = (l-''on-'') = (nol) ; M^=-(l-^m-^o) = (mlo\ Again, lines through the incentre, /, cut BG in the ratio a : b, etc., etc. Therefore, I, (9), x = al-'^; y = bm-'^; z = cn-\ and I=(al-'^, hm-'^, cn-'^). The following are the coordinates of some irrational points : Mean Point, M . (l~'^m-'^n~'^). Incentre, / . . (a^"\ 6m -\ cn-'^). 6-excentre, /& . . (al-\ —hm~'^, cn-'^). Symmedian Point, S (aH-\ ¥m-\ c^n''^). Brocard Points-^ ^ . 0707 / 79 9 1 92 i\ Orthocentre, P \ (V Han ^ , m - Han 5, ti " Han C). Circumcentre, Q . (V^sin 2A, m-^sin 25, n-'^ sin 2G). Midcentre (IX circle) { ^ " ^ (tan A + X tans), m-i(tan 5+2 tans), {n-\ta>n (7+S tans)}.^ (4) CHAPTEE III THE STRAIGHT LINE Let OA=p,^^ and OB=p,=U (fig. 4) be two given constant vectors, and let a third constant vector, OR = p = -yr-j-y cut BA so that BR :RA=f: g. What are the coordinates of the point R in terms of A and B ? By an elementary principle of vectors,* (f-^9)p=fpi+9P2 p= {fx-^^-\-gx^x^)la 4- {fy^x^ + gy^x^)m^-^ ... {fx^x^-\-gx^x^)l^{fy^x^-\rgy^x^)m-^ ... ' Tj . xla-i-ym^-^zny X>U t p — J— ; . '^ xL + y7n-{-zx Therefore x —fx^x^ + gx^x^, \ y=fy^x4.-\-gy^J.x^, \ (1) z^fz^x4.-\-gz^x^, J the sought coordinates. Ex. 1. The coordinates of A', which cuts BG in the ratio m : n. a;i = 0, 2/1 = 1, % = ; ^Ix^ = m. a52=^» 2/2 = 0' 2^2 = 1 ; 2^ = n. * Outlines of Qitatemions, p. 12. (r = 0, 2/ = T/iTi, z = m'?i. ^' = (0,7^72,7X1^) = (011). CHAPTER III Ex. 2. The coordinates of M^, the middle point of GA. iCj = 0, 2/1 = 0, 01 = 1 ; ^lx^ = n. M=^{nol)=^{l-\o,n-^). Ex. 3. The coordinates of R, the term of /a = f y. R = (n, n, 2l+2m+Sn), 051 = 1,2/1 = 1,01 = 1; 2^iCi = 2Z. a^2 = ^'2/2=^> 2^2=1; 2ia;2 = '^- /=l;sr = 2. The following is a method of determining the co- ordinates of a multiple or submultiple of a given vector, ^¥fl ' ^ being a proper or improper fraction, or a whole number. ■p , xla + 2/^/3 -h ^tly _ t fla + grm/3 + /my ^^^ 2SZ "^ Wl * Dividing across by z and eliminating y by means of the equation la-^rrip-\-ny=0, we get an equation of the form whence (M-P)a = (Q-N)^. Therefore, since a and /? are not parallel, if-P = 0; Q-i\r=o, X y two equations to determine the value of - and -. ^ z z It will be found ultimately that x-.y.z =(^j-\)yi+m ■■ (^j-i)vi+g^l : {~-l)2fl+h-2l. ...(2) Ex. 1. Let l:m :n = S:l :2. To find the coordinates of Ja. Here/=l,^ = 0,/i = 0; 2/^ = 3; 21 = 6; |-1 = 2. Therefore aj = 2x3 + 6; y = 2xS; = 2x3, and x:y:z = 2:l:l. ^ ^, , 2^a4-m^4-'^2'y 6a+j8 + 2y Consequently, ia = o;_^.,^^ = o • 2l + irb-\-n Verification. 2la + ml3 + ny _ {la-\-m/3 + ny) + la _Sa _1 ^ 2l+m+n ~ 9 9 3^* 10 THE STKAIGHT LINE Ex. 2. The coordinates of — a. Let l:m:n=^\. Here - — 1=— 2 and a; : 2/ :2; = 1 : — 2 :--2. Verificaticnt. „+^^^^=?(2±|±V> = 0. — o o Ex. 3. The coordinates of the unit- vector of a, Ua or -, a being the tensor of a. Here - — 1 = a — 1 and x:y :z = al-\'ra-\'n:{a—\)l :(a — 1)Z. or if ^ :m:'r2, = l, ^7^^ (^ + ^)« + (^-l)i^+(«-l)y ^ 2a-g-y ^a 3a 3a a* Similarly, f,(_„) = (jL:i2)£±(«+M±(^±l)y. The coordinates of a point can only be obtained from the expression of its vector when this expression is in the standard form, I, (6). 2°. Instead of being a fixed point, let jR be a variable point with the indefinite straight line AB for its locus. In this case / and g may be any two scalars whatever, and the coordinates of any and every point upon AB are of the form _ / _i_ ^ y = ty^-\-vy^, - (3) where t and v are arbitrary scalars. Conversely, any point in the plane whose coordinates are of this form is collinear with A=(Xiy^z^) and B = {x^y^z^. By hypothesis, ^ (tx-^ + vx^la + {ty^ + vy 2)^^/3 + (^^1 + '^^2)^7 ^ (tx^ + vx^jl-^ (ty^ -\- ^2/2)^+ (^% + ^^2)'^ __ t^^la + v^xjLa "" tJ^X-^l + vI^x^l (t^x^l + v^Xcp)p — t1.x^a — vLxj^a — 0. CHAPTER III 11 But and P2 llxjia (by n Therefore {tlx^l + vl.xj.) p - tp^ljcj, - vp^xj. = 0. Now the sum of the coefficients of these three coinitial vectors is zero. Therefore R, A and B are collinear * 3°. If t and v be eliminated from the three equations of (3), we get which may be written px-\-qy-\-rz=^0, or X x^ Xa 0. (4) (5) (6) y ^ Vi ^1 2 2/2 ^2 Equations (4), (5) and (6) are the equations of a straight line, since they express the condition that the variable point (xyz) shall be always collinear with the two fixed points J. =(iCj2/i%) ^^^ ^ = (^22/2^2)- "^^^ coefficients of (5) are the anharmonic coefficients of the line, and the line is denoted by the symbol 4°. The equations and symbols of the lines of the net (fig. 1) are as follows : BC passes through B = {010) and O=(001), II, (3). Consequently, (6), 0; X y z 1 1 or, x = Oy the equation of BC. lines are similarly obtained. The equations of the other Lines. Equations. Symbols. BC x = (100) CA 2/ = (010) AB z = (001) OA y-z = (Oil) * Outlines of Quaternions, p. 12. 12 THE STRAIGHT LINE Lines. Equations. Symbols, OB z-x = (101) OC x-y=^0 (ITO) AA" 2/4-^=0 (Oil) BR' z+x=-0 (101) CC' a;+2/ = (110) A'^RV' x+y+z = (111) RC y-\-z-x^O (in) C'A' z-hx-y^O (111) A'R x + y-z = (111) R'V'' 2/+^-3a; = (311) 0"'^'" 04-a;~32/ = O (131) A'^'B'"' x-hy-Sz = (113) A« ia;^-m2/ + '?^2; = (Imn) 5°. If we have three vectors OA=a, 0B = /3, 0(7= y, as in fis:. 2, and if i . rx I and 771 being constant; then the point G lies on the line AB, which it cuts in the ratio -.HI and m are variables, ^ xa-\-yfi ^ x + y expresses that the locus of C is the indefinite line AB.* In a similar way, when x, y, z are constants and the denominator of I, (6) happens to be zero, the expression is the vector of a point R which is infinitely distant ; and when X, y, z vary, it implies that the locus of R is the line at infinity, A^. Hence the linear equation Ix + Tny -i-nz^O (7) is the equation of Aoo, being a constant relation between the coordinates of every infinitely distant point. To illustrate this geometrically: let the point P = (xyz) recede to infinity (fig. 5). At the limit, AP2 and Pfi become parallel, and BP^ _BA _ BP^'-AP^ _ BP^ P^G^AP' AP^ ~-PA *See Outlines of Quaternions, p. 13. CHAPTER III 18 Therefore, I, (7), and ym~ ym ' lx-\-my-^nz = ^. Fig. 5. 6°. The coordinates of the cross of two given straight lines {p^q{r^) and {p^qc^r^). The sought coordinates {tuv) must satisfy both the given equations. Therefore Consequently, t _ u ^ V Therefore the coordinates of the cross are the cofactors of X, y, z in the matrix X y z Pi ^1 n P2 92 ^2 Ex. The cross of (pqr) and A.. .(8) X y z p ? r I m n The cofactors of x, y and z and the coordinates of the cross are {nq — mr, Ir—nr, mp — lq). T. The coordinates of the cross of two given lines, (i^i^'i'^i) ^^^ (p2?2'^2)' i^ust satisfy the equation of any third line {p^q^r^ which passes through it. Therefore ^3(^1^2 - ^2^1) + ^3(^1^2 - '^2Pl) + ^3(:Pi5'2 -P29l) = ; or, the condition that the three lines shall be concurrent is, Pi
  • i+'?^i52)a;+(^^i+'yg'2)2/+(^^i+W2)2; (10) Therefore the coordinates of any straight line passing through the cross of two given straight lines, (PiqiTi) and (P2^2^2)> i^ust be reducible to the form, {tpi-\-vp2, tq^+vq^, tr-^ + vr^}. And the converse. Ex. If we take ^ = 1 and v = — 2, we find that one of the lines passing through the cross of ^"-B'' = (lll) and ^-B=(001) (fig. 1) is x-^-y-zr^O, which is A'F. 9°. If ■^i = (Piqi^i) and A = (pqr) are parallel, they concur in Aao ={1^^.71). Therefore the coordinates of A^ must be reducible in the form, {tp + vl, tq-{-vm, tr-\-vn},hy 8° (11) Conversely, any two lines whose coordinates are of the form (pqr) and {tp-\-vl, tq-^vm, tr+vn} are parallel. If the line {tp + vl, tq-\-vm, tr-{-vn] passes through a known point, (fgh), we have {tp + vl)f-\- {tq -f vm)g + (tr +vn)h = 0, CHAPTER III 16 and consequently, t _ --(fl+gm-^hn) v" fp-\-gq+hr ' by means of which relation we can calculate the coordinates of the parallel to (pqr) through (fgh). Ex. 1. The equation of a line through B, parallel to CA (fig.i). Since the equation of CA is y — 0, any line parallel to it must be of the form, lx + {t+m)y+nz = (a) In the present case this equation must be satisfied by the coordinates of B = (010). Therefore t+m — O, and (a) becomes lx+nz=^ 0. Verification. This parallel, CA and A„ are concurrent. Therefore I TYi n 1 =znl-nl = 0. Ion Ex. 2. The equation of a parallel through C to OA, lx-\'{t+m)y + (-t+n)z = 0. This equation must be satisfied by the coordinates of C=(001). Therefore ^t-{-n = and t = n. Consequently, lx+(m-{-n)y = 0, the required equation. Cor. A„ is parallel to every straight line in the plane. 10°. The angle contained by two given straight lines {^g. 6). (a) Let the two lines be (par) and (pW), which pass through the corner B of the given triangle. These lines cut CA in P = (rop), and F = (ropy Therefore pa^-jT^ PV-~W" 16 THE STRAIGHT LINE Let the angles which BP and BF respectively make with AB be B and 0'. Then AP csin^ AF csinO' PC a sin (B'-ey FCamniB-'e')' B Fig. 6. Consequently, . ^ anp sin B . ^, anp' sin B tan 6= ^-^ J- ; tan = , ^ ^ p-, . aTip cos B — clr anp cos i) — cLr If ^ be the angle between the given lines, (jt — O'^O', and . . J. //^ /i/x tan 0— tan 0' Substituting in this equation the values of tan 6 and tan 0' given above, we get , . nacl sin Birp' — r'p) .-.. tanrf.= ±y2-2— 7- 2-2 ^ 7 p/ / , . v (12) 7- t^c^rr -\-n^ayp —nlca COS B{rp +rp) ^ ^ If the two lines are at right angles, tan ^ = oo and Pc^rr' + n^a'^p' — nlca cos B {rp' -f r'p) = 0, ......( 1 3) the relation between the coordinates of two straight lines which intersect at right angles in B. (b) Let (Piq^i^i) and {P'^fl-f'^ intersect in any point in the plane. The equations of parallels to them through B are {Iq^ — mp^x-^ (nq^ — mr^)z = 0, (Iq^ - mp2)x + (nq^ - mr2)z = 0. Substituting the coefficients of x and z in this equation for r and p, r' and p' , in (12), I 'ffi n iTnnca sin B tan0 = P2 92 — hnn { the cos A {q^r^ + qc{r^ + mca cos B(r^p2 + ''"2^1) + 7ia6 cos C^(jPi^2+i^29'i)} .(14) CHAPTER III 17 If the two lines are 2/ = and px-\-qy-\-rz = 0, , J . mc sin ^ (7109 — Cr) _., tan0=± 7^ T[~i — T- (16) ^ mna 008 Cp — nlhq-^lmc cos Ar ^ ^ (c) If the two lines are rectangular, tan ^ = oo and — lmn{lbc cos A (q^r^ + q^r^) + mca cos B{r^p^ + r^p^) ■^nahco8C{p^q^-\-p^q^)}, (16) which may be written mna^{(mp — lq){np' — Ir') + {mp' — lq')(np — Ir)} + nlh'^{{nq - mr){lq' — mp') + (ti^'' — mr'){lq - mj?)} + l7nc^{(lr — np){mr' — nq') + (Zr' — np'){mr — nq)} = ; or if the given triangle be equilateral and its mean point the origin, 2pp' + 2qq' + 2rr' = qr' + g V + rp' + '»"'p -\-pq' ^pq- It appears from these expressions that A«, is perpendicular to every straight line in the plane. Ex, Let lines be drawn from the corners A and G of the given triangle to some point {x'yz'), with the condition that these lines shall be at right angles. What is the relation between the coordinates of ^X and GX under this con- dition ? The equations of the lines are z'y — y'z = 0, and y'x—x'y — O^ and by (16) — nH^y^z'x' — iTYinilhc cos Ax'y — mca cos By^ + nab cos Gy'z') = 0. Suppose X to be a variable point, and omitting the dashes, we have m^ca cos By^ — mnah cos Gyz — nlhhx — Imhc cos Axy = 0, the equation of a circle with the line GA for a diameter. 11°. If a line {p'q'r), perpendicular to a given line {pqr), passes through a given point (fgh), we have the equation, 2-fjf.^n-\.h = 0. Thecondition (16) gives another equation r r to determine the ratios ^, K Solving these equations we get r r 18 THE STRAIGHT LINE 'p' = l^(nq — mr)(7ng + nh)hc cos A — lm^g{lr — np)ca cos B — nHh(mp — lq)ab cos C, q'=— l^mf{nq — mr) be cos A -{'m^(lr-np)(nh-{-lf)caco9B} (17) — 7nn^h{7np — lq)ah cos C, r' = — nl^f(nq — mr)hc cos ^ — m^ng(lr — np)ca cos J5 +'M2(7?ip — lq)(lf+7ng)ab cos (7,, the coordinates of a line which passes through the point (/^A) and is at right angles to the line (pqr). This equation holds good whether the point (fgh) lies on or off the line (pqr). Owing to the complexity of these expressions, which are often wanted, it is frequently simpler to let fall a perpen- dicular on (pqr) from one of the corners of the given triangle and find the equation of a parallel to it through 12°. The connexion between Anharmonic and Trilinear Coordinates (fig. 7). Let ABG be the given triangle and the given origin. / Let P be any point in the plane \Jixyz) its anharmonic coordinates; PP^ = a, PP^ = ^, PP^ = y, its trilinear co- ordinates ; 00^ = S, 00^ = 6, OO3 = f, the trilinear coordinates of the origin 0. Then l:m:n = OBG:OGA:OAB = aS:b€:cn .^^. and S:e:^=bcl:cam:abn. f Again. Iximy: nz = PBG:PGA :PAB = aa:b/3:cy. ...(19) Hence a : /3 : y = bclx : camy : abnz, \ x:y:z = ,^a:^SP:S€y. / .(20) CHAPTER III 19 Ex. 1. The trilinear coordinates of the circumcentre are a = Ja cot ^, /S = J6 cot -B, y = Jc cot G. By (20), we get Q = (Jm?iSfa'/3') Now ^2= _d2^ -Pg^i'; «'= -«'. etc.; S0y = bcco8 A, etc.* Therefore d^2Hx^XHx^ = m%2^y ^ ^2^252^2 _^ /2^2^2^2 — 2l7nn(lqrhc cos ^ + Tnrpca cos 5 + npqab cos 0) „ = mna^(7np — lq)(np — Zr) + nlb'^(nq ~ mr)(lq — mp) + lmc^{lr — np)(7nr — nq).. Let the right-hand member, which occurs frequently be Z^ and , -h7 ^^ = 2^, (3) If ^ : 771 : '}^ = 1 and a — h = c, Z^=p^ + q^-{-r^ — qr-'rp—pq = (p-{-coq'i-co^r)(p+w^q'\-oor), where w and co^ are cube roots of unity. Ex. 1. The distance from J5 to C (fig. 8). Here p =1, ^ = 0, r = 0; 2Za;i = m, 'Zlx2 = n; Z=inna; and consequently, (iwi^i = mna, and, c? = a. ^aj. 2. The distance from to ^. p = 0,q = l,r=-l; 11x^ = 1.1,21x^ = 1: Z^ = ^2(72,252 ^ ^2^2 ^ 2m'yi6c cos ^ ) ; Ti^b^ + TTi^c^ + 2'??i'?i?)c cos A (2) and 0^2. XH * 5/3'7' is the scalar of the quaternion ^'y' ; Outlines of Quaternions, p. 47. 22 LENGTHS, AKEAS AND ANGULAE FUNCTIONS If / (incentre) be taken as origin, limm^aibiCf and J. __ 26c cos J J. ~ a + h-hc * If Q (circumcentre) be taken as origin, l:m:n = 8m2A:sm2B : sin 2(7, and ^ .^_ 6^(cos^jB+cos^ (7+ 2 cos ^ cos ^ cos G) _ ¥ _ pg ^ 4sinMsin2J5 "4sin25"" ' 2°. The perpendicular distance of the comers of the given triangle from a given line A = (pqr) = (fig. 9). Produce AB, AG to meet A in B = {qfo) and C' = {rop). Let the perpendiculars from A, B, G he d^, d^, d^, and let the function of the coordinates of BG' be Z. By (3) we get the following lengths : Fio, 9. AR^f^^^; BR=r=I^' ^C'=^; Iq — mp Iq^mp np — lr GG'= "^^^ ' B'G'= ^^ np—lr' {Iq — Tnp) (np — Ir)' d^ . B'G' = 2 area ABV =-AB' .AG\ sin A, and ^ mnphc sin A ,.. ^1= —^ — w t>4. da BB J do GG , J _ '^^qcGb sin B -J Imrah sin G 3*. The distance from the origin, 0, to any given line, (pqr)=-0. CHAPTER IV 23 Since la-\-7n^-{-ny = 0, is the complex mean point of the system of points A, B, C, weighted with the given scalars, I, 7n, n. Consequently, the perpendicular distance from to any line is the complex mean of the distances oi A,B, C from it ; that is, T _^ld^-\'7nd2-{-nd^_l7nnhc sin A^p .^. "*" Tl '" ZTl ^^^ Ex. The distance from to CA. The equation of CA being y = Oj 22? = ! and Z=nlh. Therefore mc sin A d n - „ . , . , , . . , 6c sin ^ If the mcentre be origm, d= , i , =y' Ti? i.1- • i 1. . . , c sin ^ If the mean point be origin, a= — ^ — • r„, ,. . /. ^ . ..A/TV/ . Slmnhc sin A ^. ,, The distance from to A"B' is j^j^ . If the triangle be equilateral and its mean point the origin, this expression becomes ^ = ^- But in this case Z=0; therefore cZ = oo . 4°. The perpendicular distance between two parallel lines. Let e^ and e^ be the distances of the two lines from 0. Then whatever be the position of 0, Let Z^ be the function of A^ and Z^ the function of Ag. ^, , ^ ^ Imnhc sin A I^p^ Then, (6), e^ = ^^Z Now, since A^ is parallel to Aj, its coordinates are of the form (tp^-{-l, tq^+'rn, tr^+n). Consequently, Xp.^^tlp^-^^l, and it will be found that ^2 = ^-^1- — 1 Imnhc sin A //»v Therefore d^^e^'-e^^—^ ^ W 24 LENGTHS, AREAS AND ANGULAR FUNCTIONS Ex. Let the parallels be GA and lx^my-{-nz = 0, a line which passes through (cmTTi), the midpoint of BG. Then t = — 2m and Z^ = nib. Therefore , _ 1 Imnbc sin A _c sin J. ""2771 -71^6 " 2 5°. The distance from any point to a given straight line. Find the value of the factor t for a parallel to the given line through the given point and apply (6). Let the given point be (fgh) and the given line (pqr). Then t = ^fP Therefore ^^^. ^mn^n^ ^^^ Ex, The distance of the symmedian point from BG. q2 52 q2. q} Here /= p 5r = - /i = -; p = l, g = 0, r = 0; 2/jp = j; 2/^ = 2a2; Z^^inna. Therefore ■J _ CI?' bnnhc sin J. _ a6c sin il lHa'^ mnna ~ 2a'^ 6°. The area of a triangle in terms of the coordinates of its corners. Let the corners of the triangle EFG be E={x^y^z^), F= {x^y^z^), G = (x^y^z^), and let the function of FG be Z^. The equation of FG is (2/2^3 - 2/32^2)^ + (^2^3 - ^3^2)2/ + («^22/3 - «^32/2)^ = ^• Therefore ^1 = 2/22^3-2/3^2; 5'i = 2;2a;3-03a;2; n =^22/3 -^32/2- The length of i^^G^ is , ^ . For a parallel to i^G^ through £', ^^ -^K ^ -llx^ . spio^i i«^i2/2%r and the perpendicular from E on i^(? is , _ Imnbc sin A \ x^y^z^ \ ^"" Z^ElXj^ * A rtrtry ^ 7 n/^ Imnbc SlTi A .„. Area^^(? = Jd.i'e = 22;^^^S;2Z^ «=, y, ., . ...(8) X, 2/i 2^1 ^2 2/2 ^2 a?3 2/3 ^3 CHAPTER IV 2& As verification, (8) becomes — ^ — when EFQ is the given triangle. V. The sine of an angle — the angle E of the triangle EFG, 6°. Let the functions of the coordinates of EQ and EF be ^2 and Z^. Then the length of the perpendicular from F on EO is _ Imnbc sin A \ x^y^z^ \ The length of ^^ is ^1^^. Therefore p bnnhc sin A2lx^\x^y^s\ . ''^^=SF= zj, (^> CHAPTER V THE GENERAL EQUATION OF THE SECOND DEGREE 1°. The general equation of the second degree, ux^ + vy^ + wz^ + 2u'yz + Iv'zx + 2w'xy = 0, (1) represents in general a conic section, because it is cut in two, and only two, points by every straight line in the plane. 2°. Differentiating successively with respect to x, y, z, (2) ^-^=nx-\'wy'\-vz=^4>^y ldd> 1 d / , / , :^'^ = vx-{-uy+wz = z. Obviously, X(t>^-\-y(l>y+Z(l>;, = (l>{xyz) (3) Multiplying the 3 equations of (2) respectively by x\ y\ z\ = {ux-\' w'y + v'z)x' + {w'x + ^2/ + 'W''^)^' •\-{vx-\- vJy + wz) z' „ = {nx! + w'y' 4- v'z')x + {yo'x! + vy' + v!z')y + {v'x' + 'oJy' + wz')z » =x^+y,, (4) 3°. Suppose that (j>{xyz) = {]px-\-qy+rz){'p'x + q'y-\-r'z)=^(), the product of two straight lines. Then z=p ip'^ + q'y + ^'^) +?' iv^ -^qy-^ '^^^ y= q(p'x-^q'y-\-rz)+ q'ipx + qy-^rz), 02 = r{p'x + q'y + r'z) + r'(px + qy + rz). CHAPTER V 27 Hence the three equations, (p^ = 0, ^y = 0, {fgh)t^+2{f^+H,)+{x'y'zy Now since {x'y'z') is on the curve /F ^^^ ^^ ct>{x'y'z') — 0, and, consequently, one root of the quadratic (corresponding to X') is zero. The other root is -2(/0^+5r0j^ + A0^) 28 GENERAL EQUATION OF THE SECOND DEGREE ^^fx' + y(t>y + zr - ^(^Yz) = 0, or since (f>{x'y'z') — {) when ^Va; + 2/Vy + ^V2 = ^ (^) This is the equation of the tangent to the conic at {x'y'z'), 6". The condition that a straight line shall touch a given conic. Let {pqr) be the line and {fgh) its point of contact. The equation of the tangent at this point is, (7), fl30/+2/^^+2^0A = O. But 'px-^-qy-^-rz^O. Therefore ^^=^ = ^=:(say)-Aj. Therefore u/+ w'g + v'h -^pk — 0, w'f+vg-{'U'h-\-qk = Oy v'f'\' u'g '\-wh+ rk — 0. Also 'pf+qg'\''^^=^* since {fgh) lies upon the line {pqr). CHAPTER V Therefore = p q r u w' v' p w V u q v' v! w r + 2V'rp-{-2W'pq ...(9) ==F(pqT) is the condition for the tangency of the given line (pqr). T. Let F={fgh) be a fixed point, and let a straight line passing through it cut a conic in ^i = (x-^yiZj) and -^2 = (^22/2^2)- ^^^ tangents at these points are, (8), ^iy + 2^20z = ; and for their cross, Vl^l-y^^l 2^1^2-2^2^! a^l2/2~^22/l But since i^, Zj, Zg are collinear. From (a) and (b), fx+g4>y+Hz = ^> (l(>) the equation of the polar of {fgh) in respect to the conic fX-\-(Pgy + (f>j,z = 0, (a> » (/y^Ois ^f'X + g>y + (j)h'Z = (6) If (fgh) lies on (6), = ^g + ^n'^^ = 0// + gg' + Kh\ which is the condition that (fg'h') should lie on (a). (fgh) and (f'g'h') are conjugate points. (h) If a straight line (pqr) passes through the pole of the line (p'q'r'), then (p'q'r) passes through the pole of (pqr). Let (fgh) be the pole of (p'qV). Then the polar of /^ + (pgy + (phZ = p'x + qy + r'z = 0. pqr uf-\-w'g-\-v'h-\-pk = 0, wf-\- vg + u% + q'k — 0, vf-\- u'g ■\-wk-\- r'k = 0. Also pf+qg+rh =0, because the line (pqr) passes through (fgh), the pole of (p'qV). (fgh) is and Consequently, Therefore Therefore u w' V p w' V u' ^' v' llf w r' V = is the condition that (pqr) should pass through the pole of (p'cfr'). = 0, CHAPTER V 31 But this matrix is identically equal to w' V u' q v' vJ w r 'p' q' t' which for similar reasons is the condition that {p'q'r') should pass through the pole of {'pqr). (pqr) and {pq^r) are conjugate lines. (c) The cross of two straight lines A^, Ag is the pole of the join of their poles, A. Since the pole of A^ lies on A, the pole of A lies on A^, (b). Similarly, the pole of A lies on Ag. Therefore the only point which A^ and Ag have in common, their cross, is the pole of A. {d) If a number of points are collinear, their polars are concurrent. Let the points P^...Pn lie on A. Then {a) since A passes through P^, the polar of P^ passes through the pole of A. Similarly, the polars of P^..^ Pn pass through the pole of A, which is the common cross of the polars of these points. (e) Conversely, if a number of lines are concurrent, their poles are collinear. Let Aj . . . A^ concur in P. Then, by (c), p, the polar of P, is the join of the poles of Aj and Ag, Ai and Ag, Ag and Ag, etc. Or p is the locus of the poles of the given lines. 10°. To find the ratios of the segments into which a given finite straight line, FP, is cut by a conic. Let F=(fgh), P = (pqr), and let the sought ratio be ^: L Then the vector of the point of section is _ tOF-\- OP _ tfla + tgm/3 + thny pla + qm/3 + my ^~ <+l ~(t + l)(fl+gm-^hn^(t + l){pl-i-qm-\-my _ (tflpl+p2fl)laHt9^pl + qWl)'^I^Hth2fl-{-r'Efl)ny (t-hl)lfllpl Consequently, the coordinates of the point of section are {(tflpl-^plfl), (tg2pl-\-q2fl), {th1pl+ryi)Y Now this point lies upon the conic. Substituting its coordinates in the general equation of the second degree^ we get I,^l,+ qf„+ri>,)t + 2yi<}>(p, q,r) = (12) 32 GENERAL EQUATION OF THE SECOND DEGREE The roots of this quadratic are the values of t for the two points X,,Xo, in which the line FP is cut by the conic, . FX, , FX. ,,e. ^ and ^. 11°. Let the roots of (12) be real and their sum zero. Then the coefficient of the second term vanishes and which shows that the equation of the polar of the point F, is satisfied by the coordinates of the point P. Therefore P lies on the polar of F when the sum of the roots of (12) is zero, and .^Ifi l and ^i^=(J'XPZ,)= -1 (13) ^^^=(^X,PZ,)=-1. FP is thus the harmonic mean between FX^ and FX^y P being a point upon the polar of F. Therefore a line which passes through a given point and cuts a given conic, is divided harmonically by the point, its polar and the conic, whether the point lies without or within the conic. 12^ Let ^g. 11 represent a central conic. Let F^...Fn be points on Aoo and let Ai...A„ be their polars. Since the given points are collinear, their polars concur in K, the pole of A«„ 9° {d). By (13) all chords drawn in the An directions of the infinitely distant points F^...Fn are bisected respec- tively by Ai...A„. Consequently, all chords passing through their common cross are bisected in K, which is the centre of the curve. CHAPTER V 33 Since the centre of the conic, K={xyz\ is the pole of A«, by (11) its coordinates are = ^. (14) X -y z w' v' I u v' I u w' I V u' m w' vf m w' V m u' w n v' w n v' n' n 13°. The three matrices of (14) are the cofactors of I, m and n in the matrix I m n u w' v' w' V vf v' iv' w o I m n = A which is the discriminant bordered by I, m, n. In future this bordered discriminant will be called D, and its minors (14) will be called A, B, C. The coordinates of the centre of a conic may consequently be written {x,y,z) = {A,B,G) (15) If B be expanded, we get the determinant D = {vw — vf^)l^ + {wu — v'^)7n} + {uv — w^)n^ + 2 {v'w' — uu') mn + 2 (w'u' — vv) nl + 2 {u'v' — ivw') Im „ = m^+Vm^+Wn^-h2U'mn-{-2V'nl + 2W'lm (16) The determinants oi A, B, C are A=m-{-W'm-j-r7i; B=W'li-Vm+Un; C^V'l^-U'm+Wn (17) Evidently lA+mB^nG=B (18) On expanding and arranging the function, it will be found that 0, the conic is cut in H.C. c B, (20) 34 GENERAL EQUATION OF THE SECOND DEGREE two imaginary points and is an ellipse or circle. If D = 0, the conic is touched in two real and coincident points by A« and is a parabola. Since the vector of the centre is Ala + Bm/3^Gny lA-^mB+nC ' and since, for the parabola, = i) = U + mJ5+^0, the centre of this curve is at infinity. 15°. Chords which pass through the centre are diameters, the loci of the midpoints of parallel chords. If (x'y'z') be any point upon a diameter, its equation is {yV-z'B)x-\-(z'A-xV)y + (xB-y'A)z = 0. ...(21) Conjugate diameters are such that either is parallel to the tangents at the extremities of the other, and therefore passes through its pole. Only central conies possess such diameters, all diameters of the parabola being parallel because the centre is at infinity. 16°. The equation of a diameter conjugate to a given diameter, px-\-qy+rz = 0. The sought diameter passes through the centre and the pole of the given diameter. Its coordinates are therefore given by the matrix Up+W'q+Vr, W'p-^-Vq+Ur, Vp-[-U'q-\-Wr Ul -f W'm + Tn, W'l + Vm + U'n, V'l + Urn + Wn On expanding and simplifying the determinants, it will be found that the coordinates of the sought diameter are X = u{nq — mr) + v'{7np — Iq) + w\lr — np), ^ y = v{lr — np)-{-w'(nq — mr)-{-u(mp — 7iq)\ .-.(22) z = w{mp — Iq) 4- u'{lr — np)-\- v\nq — Ir). j Ex. Let the conic be the inscribed conic, x^-hy^+z^-2yz'-2zx-2xy = 0, with l:m:n = 2:S:2. This conic touches AB in C" = (110), and its centre K is (545). The diameter CK is consequently (551), and its conjugate is, (22), A = 10iK! + 152/-322; = 0. CHAPTER V 35 This equation is satisfied by the coordinates of the pole of CKy (320), and A is parallel to AB, a tangent at the extremity of G'K. For (bearing in mind that the equation of A«, is 2ic + 32/ + 22; = 0) 10 15 -22 1 2 3 2 0. Any two diameters, {jpqr) and (p'q'r), will be conjugate if Upp'-{- Vqq'+ Wrr'+ U'(qr'-\-q'r)'\- VXrp'+r'p) + W\pq'+p'q) = 0..,.(2S) 17°. The polar of any point (fgh) upon a diameter is parallel to the tangents at its extremities. In this theorem we shall denote Aoo by the equation A^+B^y+C^^=0, in its quality of polar to the centre of the conic. Let one extremity of the diameter be {x'y'z'). The tangent at this point is x'x^-gy+Hz\ and A«, is J 9, h){p, q, r)--(pf+q,-hr,f=p (24) Since the roots are equal, the points of section of FP (fig. 12) are either both internal (as shown) or both external. In either case the two values of t, ^^ and j-p, can only become equal when the points X^ and X^ coalesce, which happens when FP revolves in the plane round F until its 36 GENERAL EQUATION OF THE SECOND DEGREE direction coincides with FQ or FR, the tangents at Q and R. When this occurs the two ratios are equal, whatever be the position of P on the line FX^. Since the position of Fig. 12. P is immaterial, we may eliminate its coordinates from (24) 1 p ji i- x — f y — Q z — h by means oi the equations p = — -^, q = ^ ^ , r = —j—. Equation (24) then becomes But 0{(a:-/), {y-g\ {z-h)} = (p(x, y, z)-2{f(l>^-\-g(l,y^hcl>z) + (l>{f, g, h). Therefore (fgh), q= Wf{f-g)- Uh{g-h)+ Uy{h-f)+ vyg+ gh-2hf) - WXh-f)±(h-f)J-A{fgh). Ex. Let the conic be yz-\-zx + xy = 0, and the point (11 1 ). Then f-g=-2, g-h = 0, h-f=2; hf+fg-2gh=--4>', fg-\-gh-2hf= -2; gh-{-hf-2fg=^0. U=V=W=-1; U'= V'= Tf ' = 1. A = 2; 9!>(Ill)=-2; x/-A0(Ill) = ^^ = 2. Consequently, the coordinates of one tangent are (-8, 0,-8) = (101), and of the other, ( - 8, - 8, 0) = (110). Therefore xy = is the equation of a pair of tangents from G to an inconic as it ought to be, since the equations of GA and GB are 2/ = and x = 0. 19°. Suppose F to be the centre of the conic. Then (25) becomes ^(^^ ^ q^^^^^ y^ z)^{Acj>^+Bcl>y+Gcf>^f=^0. Now (f,{A, B, G) = DA, and on expansion and rearrange- ment it will be found that A(xyz). To find the co- ordinates of the tangent parallel to A. CHAPTER V 41 Every parallel to px+qy-\-rz = must be of the form {tp + l)x + {tq + m)y-\-(tr + n)z = 0. The condition that a parallel line should be itself a tangent is obtained by substituting tp + l for p, tq-\-'m for g, and tT-\-n for r in the matrix of 6°, and the result is F{fqr)f + ^Ap-\-Bq-^Cr)t + D=^Q (29) Now F(pqr)=0, this being the condition that A should be a tangent to the curve. One of the roots of (29) is consequently intinite. But when t = oo, the distance be- tween the parallels is zero, IV, (6) ; or every straight line is parallel to itself. If D = 0, as in the case of the parabola, the other root of (29) is zero, and the distance between the two tangents is infinite. In other words, if an arbitrary tangent be drawn to a parabola, the only other tangent parallel to it is A^o. When D has an actual value, as in the case of central conies, the second root of (29) is t- ^ - 2(Ap + Bq + Cry and the coordinates of the tangent parallel to A are {2l{Ap-\-Bq + Cr)-Dp, 2m{Ap+Bq + Cr)-Dq, 2n{Ap+Bq + Cr)-Dr} (30) CHAPTER VI SPECIAL CONICS V. The locus of the term of the variable vector _ tHa + u^m^ -\- v^ny ^ tH + u^ra + v^n with the condition t-\-u-\-v — 0. Comparing this expression with the standard form, t^=x, v? = y, v^ = z. Eliminating t, u, v from these three equations and <-f u+t; = 0, we get x^ + y^-{-z^-2yz-2zx-2xy = (1) 2°. (I>x = x-y-z, 0, m>0, n0, 7i + Z>0; CHAPTER VI 43 on which suppositions llm may be = 0. For n (a negative number) must be either greater, equal to, or less than ~ ^ --Im l + m If 7i>'T^7 — , '?n7iH-tii>> — ^7)1, and 7nn-\-nl-\-lm'^0. „ = „ , 7nn+nl-{-lm = 0. „ < „ ,mn+nl<^ — lm,a,ndmn-^nl+lm r-{-p-q, p + q—r). Putting p + q + r = 2v, we get F-^={2v—p, 2v — q, 2v — r], P^={2v'-2p, 2v-2q, 2v-2r). Therefore pp ^ _V^^ + ^^P + ^^y inereiore ur^- 2vn-np ' 2- 2vn-21lp OP and -yp^ = a scalar. Therefore P^, 0, P^ are collinear. Hence KK' and P^Pa intersect in 0. 6°. The term * circumconic ' is used to denote the family of conies, of whatever species, represented by the equation yz-{-zx-\-xy = 0. CHAPTER VI 45 Similarly, the term ' inconic ' denotes here the family of conies represented by the equation x^-\-y^ + z^-2yz-2zx-2xy=^0, (a) all of which touch the given triangle in A\ B\ C An indefinite number of conies of all species, inscribed and escribed, touch the sides of the triangle in other points, but such conies are not represented by (a). For instance, the ellipse ajH 92/2 +4^2-1 22/^ ~40aj-6aj^ = O (6) touches the sides, BG in D = (023), GA in ^=(201), AB in i^=(310). By taking the point (623), in which the lines AD, BE, GF concur, for origin, (6) may be transformed into (a). In this case, IX, (3), x' \y' :z'=f-^x:g-^y :h-'^z==x:Sy: 2z; and x = x\ y = iy\ z — \z. Substituting the values for x, y, z in (6), we get x'^ + y'^ + z"^ - 22/ V - 2zx' - 2x'y' = 0. But at the same time the conic (a), which does not touch the sides in the new A\ B\ G' (namely, D, E, F), becomes 36x'^ 4- 42/^2 + 9;2'2 _ 1 2y'z' - SQz'x' - 2^x'y' = 0. In a word, any conic represented by (a) is here called ' the inconic,' while any conic such as (6) is called 'an inconic/ 7°. It was pointed out in V, 21°, that when vf, v', w' do not appear in the general equation, each side of the given triangle is harmonically cut by the two other sides and the conic. This may be illustrated by the curve x'-y''-z^=:0 (6) This equation represents a conic because A = 1. and the conic is a hyperbola, parabola or ellipse according as D is negative, null or positive. The ellipse is shown in tig. 14. It is easy to show that the curve cuts AB in G' and G'\ and since ^^,Ja+rn^ QcJ'^-, G' and G" are harmonic conjugates of A and B, VIII, (5). 46 SPECIAL CONICS Similarly, the curve cuts GA in R and B", which are harmonic conjugates of C and A, and it cuts BG harmoni- cally in two imaginary points (0, V^l) and (0, -V^l). The tangents to the conic at these two imaginary points, 2/^^ + 2^ = 0; -ys/'^-{-z = 0, intersect in (2 sf^, 0, 0) = (100) = A. Fig. 14. Since the tangents from A to the curve are imaginary, A lies within the curve; and since BG cuts the curve in imaginary points, it lies wholly without the curve. The lines BE, BB'\ GG\ GG" are tangents to the conic; hence B is the pole of GA, G the pole of AB. Therefore, V, 9°, (c), A is the pole of BG. The triangle is consequently self -conjugate, or autopolar, in respect to (5). The coordinates of the centre are ( — ^, m, n). If l — 7n — n = (), B"G' will be a diameter of the ellipse (fig. 14), and if OA be produced to meet B'V" in X, OX will be trisected in A. When a parabola, the curve touches the lines M^M^, M.^M^, M^M^ drawn through the middle points of the sides of ABG, 8°. In general, if PQRS be any quadrilateral whose internal diagonals meet in Y and whose opposite sides meet PS and QR in X, and PQ and SR in Z; then the triangle XYZ is self -conjugate to any conic whatever which passes through P, Q, R and S. As an illustration, let BA'OG' be the quadrilateral (fig. 1). Then X is A, Fis R'' (121) and Z is G. The equations of CHAPTER VI 47 OA' and OC being respectively y — z — ^ and a? — 2/ = ^» we have for the equation of any conic passing through 5, A\ and 0', z(y'-z) + kx{x — y) = — kx^ — z^-^yz—kxy. The polar of A with respect to this conic is 2x — y = 0, or £"'0 ; the polar of R'' is y = 0, or (7 J. ; and the polar of Gi8y-2z = 0,OTAR'\ CHAPTEK VII TANGENTIAL EQUATIONS 1°. By the principle of duality the equation, px-\-qy+rz = 0, admits of a double interpretation. When the set p, q, r are constant and the set x, y, z are variable, as in the preceding chapters, the equation means that a variable point {xyz) lies somewhere on the fixed straight line {pqr). When the set p, q, r are variable and the set xyz are constant, the equation means that a variable line {pqr) passes in some direction through the fixed point {xyz). The hypothesis of a variable point and a locus are discarded here and replaced by the hypothesis of a variable line and an envelope. 2°. A straight line K=px-\-qy + rz = (fig. 15) cuts the sides of the given triangle in D = {qpo)j E = {orq), F = (rop). Then {BEGA") = ^, {CFAB")=^-. The product of these three anharmonic func- tions is unity, and any two of them suflace to determine the position of A with respect to the given triangle. The tangential co- ordinates of A are , . (pgr), Fig. 15. CHAPTER VII 49 which are the coordinates of its local equation. A line is fully represented by this symbol and it has no equation. The coordinates of BG are (100); of A"B'\ (111); of A«„ (JiTnn). 3°. As the local equation of a straight line is a relation between the coordinates of a variable point which in every position lies on the line, so the tangential equation of a point is a relation between the coordinates of a variable line which in every position passes through the point. Thus the tangential equation xp-\-yq + zr = 0, (1) where x, y, z are constant and _p, g, r are variable, is the equation of a fixed point whose anharmonic coordinates are (xyz). To obtain the tangential equation of any particular point, we have merely to substitute its anharmonic co- ordinates for X, y, z in (1). Thus, for A, p = A", q-r = 0, p-\-q-\-r = 0. B, q = B", r—p = MeabuVoint, mnp-{-nlq + lmr=0. C, r = C, p — q = Incentre, mnap+nlhq+lmcr^O. 4°. The tangential equations of the cyclic points are, A-X. , (d), j^ mnap -\- nl (ce*^ ^a)q — Imcre^^ = 0. J, mnap + nl (ce ~^^-a)q — Imcre ~ *^ = 0. Multiplying these two equations together, we get for the two points — 2lmn(lhc cos Aqr-\- mca cos Brp + nah cos Cpq) = 0, „ = mna^(mp — lq)(np — Ir) -f- nlh\nq — mr){lq — mp) -f Imc^ (Ir—np) (mr — nq) = 0., These equations are identical in form with Z^, IV, (2) ; but in the latter p, q, r are constant, while in the present case they are variable. When the triangle is equilateral and its mean point the origin, (2) becomes 0=p^-^q^-\-r^ — qr — rp —pq = (^ + wg + uyh-)(p + oo^q+tar). (3) 5°. Let {p'q'r) be a line which passes through the two ^iven points x'p+y'q-\-z'r = and x"p-^y"q-\-z"r = 0. H.C. D K2) 80 TANGENTIAL EQUATIONS Since the coordinates of the line must satisfy the equations of both points a:y+2/Y+^V = 0, Therefore -i-jr- — tti— , „ r yzi' — yz; ysd' — z"x' x'y" — x"y' ' or the coordinates of the join of two points are {y'zf'^y"z\ z'x"'-'z"x\ xy-xY) (4) 6°. Let the equation of the cross of two given lines, (p'gV) and {f(i'r"), be x'jp-\-y'q-{-z'r=0. Since this equation must be satisfied by the coordinates of both lines ajy+2/Y+^V = 0, x'f-\'y'q"+z'r"=^{). Therefore ^ y ,»_" qr—qr rp—rp pq—pq or the coordinates of the cross of two straight lines are (qy'^q"r\ ry^rY, p^f-pY) (5) 7°. Let x'p-^y'q-\'Z'r = be the equation of the point in which two parallel lines, (p'qV) and {p"q"r"), concur with Aoo- Then, since this equation must be satisfied by the coordinates of the three ^^^®®' x'p' + y'q' + zV = 0, xy -f- y'x=p = ux-{-wy + vz, J^zf= (pz = r = vx-{- vfy + wz, 0=px-\-qy-{-rz, where p, q, r are used instead of Hamilton's ly m, n, which are otherwise required. Hence P u w' v' p =Up^-^Vq^+Wr^-{-2U'qr+2V'rp -\-2W'pq = F(pqr), (b) To transform F(pqr) into {xyz). (c) The tangential equation of the incircle is s^lqr + s^mrp + s^npq = 0. To find its local equation we have the equations Fp = x = s^nq-{-s^mr, Fg = y = 8^np-\-8jlr, Fr=z= s^mp + sj^q, = xp+yq + zr. = Hence X 8^n — ^s^s^TYinyz — 28^s{alzx — 28^8J,mxy, y z 8^n s^m X „ »i^ y 82m 8^1 Z the local equation of the incircle. The utility of the method depends mainly, as shown above, on the equation, px-\-qy-{-rz = 0, "which may at pleasure be considered as expressing, either that the variable point (xyz) is situated somewhere on the given right line (pqr\ or else that the variable line (pqr) passes in some direction through the given point (xyz) " (Hamilton). 10°. As a local equation of the second degree may be the product of the equations of two straight lines, so a tangential equation of the second degree may be the product of the equations of two points. The criterion in the latter case is strictly analogous to that in the former. The equation Up^+Vq^-\-Wr^+2U'qr+2V'rp-h2W'pq = will be the product of two equations of the first degree if discriminant U W r W V U' T U' w =0. •(7) 54 TANGENTIAL EQUATIONS 11°. Since the coordinates of a tangent {p'q'r') drawn from a given point to the curve, F(pqr)=:0, must satisfy the equations of both the point and the curve, we can determine the ratios of the coordinates of the tangent from these two equations. If we solve for ^„ we obtain a quadratic equation. Therefore the ratios p' :q' : r have two and only- two sets of values, or, only two tangents can be drawn from the given point to the curve. 12°. Let ^i = (2?'?V) be a tangent to F{pqr)^^. Then, p'Fp-\-q'F^-\rr'F^ — ^, and t^ evidently passes through some point, P =pFp + qFq. + rF^ = 0. Let the second tangent from P to the curve be t, = {p"q'y'). Since <2 passes through P, p"Fp^q''F^^r''Fr = ^p'Fp,-\-q'F^.. + r'Fr'. and -\'rFr"^0. But since t^ is a tangent, p"Fp.'\-q"F^>-\-r"F^,=zO, and ^g also passes through Q. Since then t^ and t^^ both pass through P and Q, these two points must be identical and Fp- _ Fqt _ Fr Fp-F^-p;.' Consequently £. = 1. = £ , p q r that is, the two tangents are identical, and pFp,-\-qFg, + rFr--0 (8) is the equation of the point of contact of the tangent (p'qy). 13°. Let (PiqiTj) and (|?2^2^2) ^® tangents to F{pqr) at the points in which it is"^ cut by any line (p'qV). The points of contact of the two tangents are, (8), pFp,+qFg^-^-rFr,=^0 and pFp^+qFg^-hrFr,=^0; and since both these points lie on the line (p'qV), PiFp,-^q^F^-{-r^Fr = and P2Fp>'hq2Fg'+r^r'=-0, CHAPTER VII 56 Therefore both tangents, {Piqir-^) and (2>25'2^2)> P^^ through the point pFj,-^qFg>+rFr. = 0, (9) which is consequently the pole of {p'q'r). It is immaterial whether (p'qV) cuts the conic in real or imaginary points. For example, the line (Oil) lies altogether outside the income, qr-\-rp+pq = 0. Here F^=q'^r' = 2, i?',, = r'+p' = l, F^=:p'+q' = l', and the pole of (Oil) is 2^+^+r = 0, a point which lies inside the conic since the tangents from it to the curve are imaginary. 14'. It follows from (9) that the pole of (Lmn), or A,, is 0==pF,-hqFrn-{-rF, ,,=(Ul+ W'm'\- V'n)p-\-{W'l+ Vm+ TJ'n)q -\-{y'l-\-U'm-\-Wn)r ,,=Ap+Bq-\'Cr, the tangential equation of the centre of the conic. 15°. Let {p'q'r') be the polar of x'p-\-y'q+2fr^(^. For the pole of (p'q'r') we have the two equations pFp^-^qFg^-^rFr^O, x'p + y'q+z'r=0. (10) Therefore, or Fp^ _ ^ __ ^ _ x' " y'" z Up' + W'q' + 7 V + x'k = 0, Fy+Fg'+CrV+^'A; = 0, Tp' + U'q' + Wr' + zk = 0. k\ Therefore V w T x' V v y' U' w z' -q r' U V x' W U' y' y F z U w V y' U' z' or, treating the constants x\ y\ z' as variables, the tangential coordinates of the polar of x'p^-y'q-^z'r-^^ are (^^, ^2/', M (11) 66 TANGENTIAL EQUATIONS just as the local coordinates of the pole of pX'{-qy-\-rz = are, V, 8^ ^^^^ ^^^ ^^y 16°. In the preceding sections the following corre- spondences have been established : Local. The symbol of a point. „ equation of a line. » „ „ tangent. The polar of a point. „ pole of a line. Tangential. The symbol of a line. „ equation of a point. „ „ „ the point of contact of a tangent. The pole of a line. „ polar of a point. We may therefore, when convenient, transform expres- sions in one system into corresponding expressions in the other directly, without calculation. Take for example the local equation of a pair of tangents drawn to a conic from apoint^=(/^^),V,(25), Let A be the chord of contact of the tangents. Then (p(xyz) becomes F(pqr), the tangential equation of the conic. f{fgh) becomes F(fgh); the local function of the point (jgh) becoming the tangential function of the line (fgh). f^x+g(t>y+^^zy the local expression for the polar of F, becomes the tangential expression for the pole of A. Finally, the equation F(fgh)F(pqr)^(fF^=gF,+hFry=^0 (12) is the equation of the two points in which a conic is cut by any straight line (fgh); for since the tangential equation of the point of contact of one tangent corresponds to the local equation of the tangent, the tangential equation of the points of contact of a pair of tangents, i.e. the points in which the conic is cut by the chord of contact, must corre- spond to the local equation of a pair of tangents. For the discriminant of (12), see XII, 7°. Ex. 1. For (Imn), (12) becomes (IFp -h mFq + nFrf - F(lmn)F{pqr) = 0. The equation of the incircle is, s^lqr -h s^mrp -f- s^npq = 0, and lFp+mFg'\-nFr==mnap-{-nlbq-{-lmcr; F{lmn) = 2lm8. CHAPTER VII 67 Therefore — 2lmn{lbc cos Aqr-{- mca cos Brp + nab cos Cpq) = Q^ i.e. A« cuts the incircle in the cyclic points. Ex. 2. Local. The equation of the pair of tangents drawn to the inconic, x^+^^+z^- %2 - 2zx - 2^ = 0, from the point (22T), i.e. the pole of (115), is „ = 2^2 + 2i/'^ - 4:z'^ - 2i/z - 2zx - bxy „=(a;-2y-2^)(2^-y + 24 two tangents which touch the conic in (411) and (141). Tangential. The equation of the pair of points in which the inconic, 'iqr + 2rp + 2pq = 0, is cut by the line (115), i.e. the polar of (221), is = I8(2qr+2rp + 2pq) +(-4p-4q + 2rf = 4p2 + 45^2 + ^2 + 5gr + brp+Vlpq = {4p + q-\rr){p + 'iq + r\ two points which locally are (411) and (141). 17°. Since by definition the coordinates of a tangent must satisfy the equation of a conic; to obtain the co- ordinates of the two tangents which can be drawn from a point to a conic, we have merely to determine the ratios p:q:r from the equations of the point and the curve. Let a5'p4-2/'g+2;V=0 be the point and F{pqr) = the conic. Then solving for ^, we get + ( Uz'^ + Wx'^ - 2 Vz'x') = and ultimately, writing X U .(13) 5 = v y w V z' y w w .(14) the ratios of the coordinates are ^= - F;2V+ TJ'x'y'-- Vy^+ Wy/±y's/^: g= - Uyz'- U'x'^+ Tx'y'-\- W'z'x^x' s/ -S, r=^Uy'^'{-Vx'^'-2W'xy\ ,.(15) 68 TANGENTIAL EQUATIONS Ex. The two tangents from ^ — r = (J.") to Here a;'=0, y' = \, ^'=-1; Therefore ^ = 4or0; ^ = 1; r=l, and the coordinates of the two tangents are (411) and (Oil). 18**. The value of 6 in the last section determines whether a given point lies on a given conic or not. If the point lies on the curve, the two tangents become one and the same, and the roots of equation (13), which may be written must be equal. Therefore If then (5=0, the point lies on the curve. Obviously, A" does not lie on the circumconic in the preceding section, for <5 = — 4. Does the point p = ? It will be found that ^ = 1 — 1 = 0, and jp — lies on the curve. 19°. The coordinates of the tangent at a given point on the conic are given by (13), a^^ -1-26^+0 = 0. For in this case the roots are equal and r a Ex. The tangent at the point i5-f-5' + 4r = to gr+rp-f^2 = 0. Here a;' = l, y' — l, s' = 4; 0'=F=Tf=0; W=^V'=W'^\; a=-2, 6= -4. r r and_ the coordinates of the tangent at the given point are (221). 20°. If the point from which the two tangents are drawn is the centre, (13) gives the coordinates of the asymptotes. In this case, if 8 be positive the asymptotes will be imaginary and the conic will be an ellipse or circle. If CHAPTEE VII 69 ^ = 0, (13) will have equal roots and the conic will be a parabola; and if S be negative the conic will be a hyperbola. Ex. The centre of the hyperbola 7p2 ^ 7^2 + ^2qr +2rp-\-S2pq = 0, with the condition, l:m:n==2:l:2, is p-{-2q + r = 0, and it will be found that the coordinates of its asymptotes ^^^ (113) and (311). To calculate the equations of the asymptotes from the local equation of this curve, lQx^-Sy^+16z^-\'12yz-S2zx-\'12xy = 0; hy V, (25), S6(16x^'-Sy^+l6z^-{'l2yz-S2zx+l2xy) -{{iex+6y-16z)+2(6x-Sy + 6z)-\-{-l6x+6y + iez)}^ = 0, which gives 0== -Sx^+y^" Sz^ - 2yz + lO^o; - 2xy „=(a:+2/-32)(-3a;+2/ + 3). CHAPTER VIII CROSS RATIO 1°. Let OX = X and ON=v he two vectors (fig. 16). If a third vector OX = p cut LN so that LX:XN=y:x, xk-\-yv then P = x-\'y (1) If Z' be another point on LN such that LX''.X'N=y':x\ x'+y' .(2) y The ratio ^ will of course be positive or negative X according as the definite line LN is cut internally or ex- ternally (as in fig. 16) by the point X\ Since LX=y^. XN=^-^. x-\-y x+y LT^y^Z^). TN^'^^; x'-hy x+y' LX LT LX.NT _yx' XN''X'N~XN.X'L''xy'' CHAPTER VIII 61 If we define the anharmonic function of any four collinear points A, B, G, D to be, II, V, i^B(^^)--BG'-UD^BC7DA^ (3) where the cyclical order of the letters — Ay B, G, D and B, G, jD, a — is preserved above and below, we have for '""' <™--=ii^.=0 <« If {LXNX')= - 1, then — ^ =^, and (1) and (2) become xX+yv , xK-yv ,,, P = ^+y' P'^^^y-' w the general expression for a pair of harmonic conjugates to X and y. AB.GD _ BA.DG _GD.AB_ nG.BA , ^' BG.DA~AD,GB'~DA,BG~GB,AD' that is, {ABGD) = {BABG) = {GDAB) = (DGBA)= k .p^^., J5a.i)^_J5a.j[)^__l__l . . ^^^^^^-GD.AB-AB,GD-(ABGD)-k ^""^ AG.BD {AB±BG){BG±GD) _ BG{AB-\-BG-^GD) AB.GD " ~ GB.DA ^GB.DA BG.AD , AB.GD _^ , ... " "GB.DA'^GB.BA'^"^ ^ ^ The reciprocal of a function, (a), is obtained by con- tinuing the cyclic progression one stage: (BGDA) is the reciprocal of (ABGD). By reversing the order of the two central letters, (6), we obtain a function which is unity minus the original function : (AGBD)= l-iABGD). 1-^ ,AnJ,r^^,DnnA^ A G.BD BG. DA -Y-=^^^^^^^^^^^^^GBTDAGD7AB 62 CEOSS KATIO Therefore {ACDB) = —~ and j^=^(CDBA). If (A BCD) is harmonic, (DOB A) is harmonic; and all the cyclic permutations of both are harmonic : - 1 = (ABGD) = (BCD A) = (GDAB) = (DABC) = (DOB A) = (GBA D) = (BADG) = (^2)C5). The foregoing results are collected for convenience. When 1. iABCD)=(BADC)==(CDAB)^{DCBA)=L 2. (ADCB)=(BCDA)^(CBAD)=(DABC)=^^. 3. (ACBD)=(BDAC)=(CADB)=(DBCA) = l-k. 4. (^Z)5(7)=(ig(7.1Z))=((75i)^)=(Z)^(7J?)=^. 5. {ACDB)={BDCA)=(CABD)={DBAC)=^. 6. (ABDC)=(BACD)==(CDBA)=(DCAB)=j^. = -1 = 2 = 2 = i 3°. (a) li A, B, Gy D are any four collinear points, and if ^ =(iCj2/i^i) ^^d ^=(^32/3^3) ; then, III, (3), the coordinates of J5 and D must be of the form {tx^+ux^, ty^-^-uy^, tz^+uz^) and {t\+u%, <'2/i+^'2/3» ^'^i+'W-'^?,), or, for shortness, {t, u) and (t\ vf). Let ^aj J + m2/i + ti^j = 2^1 = a-^ , S^iCg = 0-3 . Then AB=OB--OA _(tx^+ux^)la-\-(tyi+uy^)m/3+etc. xJ,a + y{inP'{-etc , _u{(x^(r^'-Xj(T^)la'i-(y^ar^-y^(r^)mp+{z^(ri'-z^(r^)ny} o-iita-i + ua-^) uO Similarly, te 0-3(^0-1 + 14^3) o-^ittr^ + ^c^s) ^i(^<^i + "^s) CHAPTER VIII 63 Therefore m^-iABOI))^^, (6) Ex. Let the row be (Til), (100), (211), (322). Calcu- lating from the coordinates of the first and third points the values of t and u for the second and of f and u' for the fourth, we get ^ = T-' ^ = 3' ^=3' ^=3' and (ABCD)==^, (b) The cross ratio of pencils is strictly analogous. If two rays, VA and VC, be (PiqiT^) and (p^q^r^), the co- ordinates of the second and fourth, VB and VD, must be of the form (t, u) and (f, u'), III, 8^ Let (pqr) be any transversal. Its intersections with the rays are : for VA^{qr^'-q^r, rp^-r^p, Pqi^Piq) H<^Aci)> „ VC-lqr^-q^r, rp^-r^p, pqz-p£}-={afi^c^), „ FB— {toj-f-'M^g, tb^+uh^, tCj^+uc^}, Hence for the four points of intersection, K, L, M, N, F.^5OT=(irZilfi^=^=[£ii4!^ (7) (c) If the four lines cut by the transversal are not concurrent, equation (7) still holds true. Let the lines (Piqiri) ...(Piq^Ti) be cut by (pqr). Then the coordinates of the cross of (pqr) and (2>i2i^i) are (\qr^\, |rpi|, \pqi\), with corresponding results for the remainder. Calculating the values of t and u for the second point, and of f and u' for the fourth, from the co- ordinates of the first and third points, we get t^lqr^Wrp^l-lqr^Wrp^l^rlpq^r^l u = \qr^\\rp^\-\qr^\\rp^\==r\pq^r^\, f=-\qr^\\rp,\^\qr^\\rp^\=r\pq^r^\, u'=\qr,\\rp^\-\qr^\\rp^\^r\pq,ril a„d ^JPM2\\PMa\ 64 CROSS RATIO Ex, The four lines, no three of which are concurrent, are cut by y = ; to find the cross ratio of the intersections. Here (pgr) = (010), (Mi^i) = (001), {m,^.} = {\^1\ (M3^3) = (321), (M4n) = (100); and |p^ir2| = l; lF9'2^3l = 8; |pg3r4| = l; |pg4nl=-l- Consequently, the cross ratio is ~1 8 ' 4°. (a) The cross ratio of a pencil in terms of the vertex, ^=(^o2/o^o)» ^^^ ^^® points in which the rays are cut by a transversal, p^ = (a,^y^,^)...p^ = («,^2/^^^). P P P P •^ -t 1-t 2-' 3-^ 4~~\''^1-' 2-^ S^i/^p p PP _ sinP,FP,.smP3FP; _ - sin P, FP3 . sin P, FP, - ''°''^'*"* (6) The cross ratio of a pencil in terms of its vertex V and any four points upon its rays, P^, P^, P^,P^, Fig. 17. Let the pencil be V-P^PJP^P^ (fig. 17); V={x^y^z^)\ A = (a^i^/i^i) • . . -P4 = (^42/4^4)- The transversal PJP^ cuts FPg and FP^ in F^ = {t, u) and P'4 = (f, u'). CHAPTER VIII Then 4°, (a), V-P^F^^F,^ ut' tvf' Now, since F, P2, P\ and also F, P^, P\ are collinear, = '-t\x^y^z^\-\-u\xQy^z^\ OCq 01/2 ^X't ~t~ UOCn ^=12/0 2/2 ^2/1 + ^2/3 Xq x^ fx^-^u'x^ I and 0= 2/0 2/4 i'Vi^'^'y^ M'|a5o2/4%|-'M''ko2/32'4|. Therefore ^= F.P,P,P3P, = p^i^^ 1 1 ^0^3^^ (9) ^02/2^3 1 Fo2/4%l (Hamilton.) It may be observed that if the points {x^y^z^ and {x^^^, or {x^y^z^ and (a;42/42^4) coincide, the anharmonic function vanishes. If {x^y^^) and {x^y^z^\ or (0542/4^4) and (a?i2/i2^i) coincide, the function becomes infinite ; and if {x^y^z^ and {^zHz^z)^ o^ {^^J'f^ a^^ (^42/4^4) coincide, it becomes unity. 5°. (a) If two homographic pencils, Y'A'Bai) (tig. 18) (a), have different vertices and a cor- responding ray in common, the crosses of the remaining rays are collinear. Let the common ray be yBY'\ let the first and third rays meet in A and G ; and let the two remaining rays meet the line AG va. D and D', their cross E not lying on the line AG, Then, by hypothesis, AB,GJ) and Fig. 18 (o). BG.DA and Therefore UABA D'G~ DG' H.C. A'E. G 'U BG'.UA' A'F. G'D B'G'.DA' 9A. HG' %'. DV=DG; D'G-DG^D'D^O. HOMOGRAPHIC DIVISION Therefore D' is D, a point on the line AC. (b) If two homographic rows have a corresponding point in common, the joins oi the remaining points are concur- rent (fig. 18) (b). Let B be the common point, and let A A' and G'C meet in F, through which D'D does not pass. By hypothesis, Fxa. 18(6). (ABGD) = {A' ECU), and since AG and A'G' are transversals of V'ABGD, by 4° (a), (ABGD)={A'BV'E). rp, f EA' D'A' C'A' G'A' Therefore ^,= 5^; ^^ = ^.5 EG'=D'G') EG'-UG' = EU^O. Therefore iy = E, and JLJ.', G'G and D'i) are concurrent. 6°. (a) When 05' = hx and 2/' = Aj'2/, h and ^' being constants^ equation (4) of 1° becomes {LXNX') = ^, (10) When, therefore, - varies under the conditions of X equations (1) and (2) of 1°, _x\-\-yv ,_hx\-\-k'yv ^~ x-\-y ' ^ ~ hx-\-k'y the points X and X' form two homographic divisions on the indefinite line XiV", L and N being the double points of the system. For let the successive positions of X and X' be A and A\ B and B, etc.; the successive values of ^ being ^ for A^ ' ' x ^ x^ * ^ for 5, etc. Then ^,=(LANA') = (LBNR)=(LGNG') = etc (11) CHAPTER VIII ^ Now AB=OB 0,1= ^2^+^2^ ^i^-^V i^^ l^^iVzK^-^) ^2 + y2 ^l + 2/l (^l + 2/l)(i»2 + 2/2)* Writing out the values of the four segments, (a;i+2/i)(«2 + 2/2)' («3+2/3)(^4+2/4)' ^ + ^2)^+2/3)' (aJ4+2/4)(^i + 2/i)' (A;a;i + k'y^){hx^ + ^g) ' /^/<^' 1 0:32/4 1 (^ - X) ^/^. ^ (A^2 + %2)(^3 + ^3) . 2)-^!^^ ^^^^'1a;^yil(»^-^) Therefore {ABGD)J^^^^^^^^\^{A'EG'U) (12) ih) Let the given equations be _xk'\-yv ,_kx\' + k'yv ^~ a; + 2/ ' ^ "~ kx+h'y where X — OL\ v=ON' (fig. 19); the variable points Z and X' now moving on different lines LN and L'N\ Fig. 19. As - varies, X will assume on LN successive positions X A, By etc., such that ^^^^^^-\x,y,\\x,y,y and at the same time X' will assume on UN' successive positions A\ B\ etc., such that 68 HOMOGRAPHIC DIVISION 7°. From the ratios given in 6° (a), «l + 2/l ' «3 + 2/3 Combining these values with those of -45 and BG, A'B' and EC, given in the same section, {LABG) = :^yilM^ = (LA' EG'). y^\^iy2\ I (13) Similarly, {NABC) = ^^ImA = {NA'EG'). Since (LAGB)==(LA'G'B') and {NAGB) = (NA'G'E), LA.GB_ LA '.G'E AG.BL^A'G'.EL' NA.GB_NA'.G'E AG,BN~A'G'.B'N' LA.BN LA'. EN LA.NB LA'.NE Dividing, NA.BL~NA'.EL' AN.BL~ A'N.EL' (LANB) = (LA'NB') (14) 8°. If two homographic rows have no common point, and if all the points which do not correspond are joined — A and Fig. 20. B', A' and B, and so on — the joining lines intersect on a straight line, the directive axis, A (fig. 20). Let the points A, B, C on A be CHAPTEE VIII 69 and let the corresponding points on Ag be Let |2//il = ai 1 2/^3 1 = ^i 1 2/3^'i I = ^i It/s^'sh^i- l^yil = «^3 la^i^/'sh^s \Xzy\\ = Gz Ns/sh^s- It will be found in the usual way that the equation of the line through L = AR'A'B and M=AC'A'G is {tw\ — uvc^x + {tw\ — uvc^y + (^^63 — uvc^z = 0. . . .(15) This is the directive axis, A. If any fourth arbitrary point D be taken on A^ and joined to any one of the three points on Ag, say C, cutting A in T; the point corre- sponding to D on A2 is found by drawing a line from G (corresponding to C) through T. The point D' in which it cuts A2 corresponds to D. For let D be {tx^-^u'x^, ^Vi+'^Vs* ^'^i+'W-'s^g), and let any point whatever on Ag be P — {v'x\'\-iv'x'^y v'y\-\''^'y'zy v'z\-\'W'z'^. It will be found that the intersections of A'B and AF with A are -{ iw I a^^ \'-twY\ 62C3 l+uv I CgOtg I, tW I ^361 l — tW-p-l ^3<^l \ + '^v\ Cs^i I, r w' e = U'M;|a263|-iw^|62Cs|+'?^'v|c2a3|: tw 1 ^361 1 — ut; — 1 63C1 1 -f ^v I Cja^ |, ilPzWz\ (^^ '^') 5 and /i = k/3l 9i=^\qz^\\ /2 = klip's I S'2=k3/ll fz = \Viq'z\ 9z = \Pb9\\ L is the directive centre. CHAPTER VIII 71 If VA, VB, VG are given rays of the F-pencil and the -corresponding rays of the F'-pencil are V'A\ V'B\ V'G\ and if an arbitrary fourth ray of the first, FD, be drawn ; the corresponding ray of the second is found by joining the point VC'-VD to the directive centre. The point in which this join cuts VG is the cross of VG with the sought ray TD\ Ex. Let there be 2 pencils of 3 rays each in which F^=(211)^ jF'^' = (lTl) F5 = (Tll)[ correspond to | V'E = {121) Fa=(199)J i F'(7'==(131) From these data we find that /i = 4. /2 = 3, /3 = 5; and the directive centre is (7, 7, 24). Now let a fourth arbitrary ray, VD==(011), be drawn to the first pencil. VD cuts V'C in (411); the join of (411) and the directive centre cuts VG in (279, 75, —44); and the fourth ray of the second pencil, V'D\ is (15, —47, 15). The two pencils will be homographic ; for — 1 2 i'^-jf' ^'""17' '^="~^' w=16 , uf w v' and I— ? = >• I'd V w Given three corresponding pairs of points or rays, if we select a fourth point or ray in one system we are enabled to draw the corresponding point or ray of the other system by means of the directive axis or centre. But we can calculate the coordinates of the fourth corresponding point or ray, without the assistance of either, by the equation For since (ABGD)=^(A'B'GD), t-, = — > and — = - — -„ ^ ' ^ ' tu vw w twu which gives the sought point or ray. Let the two rows be (111), (100), (211), (322) and (Oil), (Oil), (031). 72 HOMOGRAPHIC DIVISION — 1 115 Then for the first row, t = -^, f^^^y ^ =o' '^' = oJ ^^^ — 2 —1 for the second, v = —^, w = -k-. Consequently -y' _ uvf _ — 2 tt;' ~ twu' ~ 5 and the fourth point of the second row is (0, 15, 3) = (051). The directive axis, however, enables us to find easily the point on one axis which corresponds to infinity on the other. 10°. The point on Ag (fig. 20), corresponding to the point at infinity on A^, is obtained in the same way as any other corresponding point. Let / and J (not used in this con- nexion as symbols of the circular points) be the points at infinity on A^ and Ag respectively. Draw a line from A' to / (that is, parallel to A^), cutting A in (say) X, and the line AX will cut A., in /', the point corresponding to / on Aj. Ex. Let the two axes be the sides AB, AG oi the given triangle, and let the points C\ B, G" on AB correspond to B\ C E' on AG (fig. 22). Fio. 22. Since BE and GG' cross in and EG" and G'E' in A\ the directive axis is 0A\ uIjB cuts Aoo in (m, — Z, o) = J; -40 cuts A„ in { — n, o, l) = J; CHAPTER VIII 73 Therefore EI = (I, m, - 1) and Bl^ OA' = {l-^m, I, I) = M, Similarly, G'J= (I, - 1, n) and CV- OA' = {l-'n, I, I) = N, C'M={l,-l,m) and G'M-AC=(-m, o,l)==r, RN={1, 71, - 1) and B'N-AB = {n, -1,0) = J\ Since {G'BJ'I) = {RGJn 2^ = S = S (1^> To verify this : C'B = ^^^~") • BJ' = ^^^"^ ^ • l-\-m ' m — 71 ' /;fD/^ ^(«~y) . D.^._ 7i(^ + m)(a-Y) 71 + ^ ' (71 + 0(^-^)* Therefore S| = ^^™^ = ^. 11°. Given two homographic rows on an axis, to de- termine the double points, L and N. By (13), {LBGD) = {LECU\ and RG\ GD .LB.LIX^BG . CD'. LR. LD. Let AL = x, and assuming that L lies to the left of A (fig. 16), LB = AB-x, LU^AU-x, LR=AB'-x, LD = AD-x. Hence B'G\GDiAB-x)(AD'-x) = BG.G'D'(AR-x){AD-x). (20) This quadratic will give two values for AL. One will be the value of AL, the other the value of A^; for not only is (LBGD) = (LRG'D% but {NBGD)={NRG'U\ {ny Ex. Let AB = 1, BG=2, GD = 4>, DD' = 10, DV' = 4>, C'R = 8, RA' = 16. Then 8x4(l-a;)(l7-a;) = 2x4(29-aj)(7-a;), and a;2_ 12a; -45 = 0. Therefore a; =—3 or 15; L being 3 units to the left, iV 15 units to the right of A. 74 HOMOGRAPHIC DIVISION Verification. (LANB) = \i^^-„ (X^'i\rF) = f tV=iV, and {LANB):={LA'NB'). 12°. Given four fixed points, no three of which are collinear, P^, P^, P3, P^ (fig. 23); to find the locus of a fifth point P = {xyz\ subject to the condition P • P^F^F^F^ — —J- > a constant. Let Pi = (flJi2/i^i) . . . P4 = (^^42/42^4)- -»^^» -r^^^l <«> / 1 «'2/i2=2 1 1 ^Vz^i I + ^ I ^2/2^3 1 1 ^V^^x I = 0> which is evidently an equation of the second degree. The locus of P therefore is a conic, and it passes through Pi, P2, P3, P^; for if we substitute for the variables in this equation the coordinates of any one of the four points, say P3, the second and third matrices vanish and the equa- tion becomes identically, = 0. This theorem shows that a conic must pass through any %N^ arbitrary points, no three of which are collinear. A sixth point, P\ will only lie upon it if equation (a) remains true when the coordinates of P' are substituted in it for those of P. From a geometric point of view, P' will lie on the curve if the intersections PP^-P^F, P,P,-P^„ PP,-P,F are collinear, as shown in fig. 23. It follows that the cross ratio of any four points on a conic is constant. 13°. Given four fixed lines, Aj, Ag, A3, A4, no three of which are concurrent; to find the envelope of a fifth line A = {pqr), such that the cross ratios of its intersections with the four fixed lines is constant, -^. CHAPTER VIII 75 Let Ai = (^igiri)...A4 = (^4^4r,). Then, by 3° (c), the cross ratio of the intersections for some fixed position of A is Now let p, q,r vary under the condition of this equation, and we have a tangential equation of the second order, which consequently represents a conic. The four fixed lines touch the curve ; for if the coordinates of any one of them, say {p^fl^'r'^, are substituted for those of the variable, the equation becomes identically zero. Ex. Let the four given lines be (2T2), (001), (010), (100), and let ^=L Substituting these values for the constants above, the equation becomes g'r+rp4-j5g = 0, the tangential equation of the inconic. It follows that the cross ratio of the intersections of a variable tangent to a conic with four fixed tangents is constant. 14°. Every triangle which circumscribes the inconic x^ + y'^ + z^'-2yz''2zx-2xy = is inscribed in the circumconic yz-^zx-\-xy = (fig. 24). From any point D on the circumconic draw two tangents to the inconic cutting BC in d, CA in c and AB m. b. From E, the point in which Dh cuts the circumconic, draw a tangent cutting BC in e and meeting Dd in F. Then F lies on the circumconic. It will be observed that the four fixed tangents, AB, AG, FD, FE, cut the tangent BC in d and e, and the tangent DE in b and c. Therefore, 13°, (BCde) = (bcDE) = const. F'BCDE= A 'BCDE = con8t Therefore, F lies on the circumconic, 12^ 76 HOMOGRAPHIC DIVISION This proposition only holds good for conies which are represented by the equation x^ ...— 2xy = 0, i.e. inscribed or escribed conies which touch the sides in A\ B', (J. For example, the ellipse x^+9y^-^4>z^-12yz-4!zx-exy = touches the sides of the given triangle, and on this curve lie the points (24, 2, 3), (683) and (316). If tangents be drawn at these points, it will be found that they cross in the points (12, 4, 3), (323) and (613) ; and the coordinates of the two latter points do not satisfy the equation of the circumeonic. Consequently the triangle of which these three points are the corners is not inscribed in the circum- eonic. 15°. The points in which a circle is cut by conjugate chords form a harmonic group. Let L and iV be any two points upon a circle (fig. 25), and let the tangents at these points meet in M. Then LN and any secant MA A' are conjugate chords, V, 9° (6). Let LN and MA cross in E. Let P be any point on the circle and join it to L, A, N, A'. Since M is the pole of LN, {MAEA')^-\, V, ir, and consequently L-MANA'^^-'-X. lMLA = lLFA, lNLA'^ lNFA\ lALN= lAPN, lA'LM= lATL. Therefore L'MANA' = P'LANA'= -1, and {LANA')=-1. The points in which a circle is cut by any diameter form a harmonic group with / and J. For / and / lie on every circle and I J and any diameter are conjugate chords, either passing through the pole of the other. If a number of secants be drawn through M cutting the circle in B, B ; (7, C" ; D, jy, etc. ; we have {LAN A') = {LBNE) = {LONG) = etc. = - 1, and MA.MA' = MB.MR^MC.MCr==etc.=^LM^. Pairs of points thus related form a system in involution. Fio. 25. But CHAPTER VIII 77 16°. It was shown in 1" that the harmonic conjugates of X and V are . ^~ x + y ' ^ ~ x--y Let M be the centre of LN (fig. 16). Then ^ 2 ' aj + 2/ 2 2/+a5 2 ' j^j^,^ x\-yv X-hv^ y + x v-\ x^y 2 2/""^ 2 Therefore jgZ . FZ'= ('^)' = ^^^^. If ^> 1, Z and X will lie to the right of if ; if < 1, X both points will lie to the left of M. Let the successive positions of X and X\ as ^ varies, be ^, ^' ; 5, B\ etc., and ^ MA.MA' = MB:MB' = etc.==MN^ (21) Thus the variable points form divisions in involution on the indefinite line LK The points L and iV are the foci of the involution, and M (the conjugate of the point at infinity) is the centre. 17°. Given two pairs of points, A, A' and B, B' on a straight line ; to find a point M such that MA,MA'=MB.MB' Fxo. 26. Draw a circle through AA\ as in fig. 26, and draw another circle through BB' and any point P on the first 78 INVOLUTION circle. The cross of PQ (the radical axis of the two circles) and the axis is the sought point M\ for MP.MQ = MA.MA' = MB.ME. To find the conjugate of any fifth point C; draw a circle through P, Q and G and it will cut the axis in G\ the conjugate of G. 18". The position of the radical axis, PQ (fig. 26), with respect to the axis, or the position of M on the axis, depends upon the relative position of the points A, A' ', B, B\ etc. M may lie {a) outside the circles ; or (6) it may coincide with Q on the axis; or (c) it may lie within the circles. (a) When M lies without the circles we have the hyper- bolic involution of 16° and fig. 26, where k^ is positive. Since the points and their conjugates lie on the same side of M, and since A' moves towards M as A moves from M, one pair of conjugates must ultimately meet in a point F such that, MP . MQ=OF\ OF is therefore equal in length to a tangent from M to any of the circles of fig. 26 ; all such tangents being equal because M is a point on their common radical axis PQ. To find F, we draw a circle through P and Q, touching the axis. Two such circles can in general be drawn, one of which touches the axis in P, the other in F' (fig. 26). Obviously, MF=F'M. In a hyperbolic involution we have, 16°, (F'AFA') = {FBFR) = etc. = - 1, in addition to the general equation of involution, (21). (6) If Q happens to lie on the axis, M must co- incide with it, as also must F\ P and A, B, G, and in this case (fig. 27) MP.MQ==MA.MA' ^MB.ME^B^O. This is parabolic involu- tion. pjo 27. (^) When M lies within the circles the foci are imaginary; algebraically, because ¥ is negative, the points CHAPTER VIII 79 and their conjugates lying on opposite sides of M (fig. 28) ; geometrically, because the foci are the points of contact in which circles through P and Q touch the axis, and in the present case these circles are imaginary. Fig. 28. Pio. 29. This is elliptic involution and MP .MQ = MA .MA' = MB .MB^ = etc.= -kK (d) When the radical axis is bisected at right angles by the axis we have circular involution (fig. 29), in which the segments AA\ BB\ etc., subtend right angles at P and Q. As in (c), MP.MQ = MA.MA'=^MB.MB'=^QtQ.=^^k\ The peculiar property of this species is, that each ray of the pencil P'ABA'E is at right angles to its conjugate, whatever the number of points. It also enables us to introduce the imaginary focal rays; for, 15°, p. lAJA' = p. IBJR = etc. = - 1. It will be observed that in elliptic involution the segments AA\ BB\ etc., overlap, that in parabolic invo- lution they have one point in common, and that in hyper- bolic involution they lie wholly within or without one another. An involution may have, (a), two real and distinct foci, or, (b), two real and coincident foci, or, (c) and (d), two imaginary foci. Every involution has a centre, and in all cases the product of the distances of any two conjugate points from the centre is constant. This distance is MF^=MA.MA' = k\ „ = „ =0, for hyperbolic involution.! „ parabolic „ r(22) „ elliptic „ J 80 INVOLUTION 19°. To calculate the position of the centre of involution M, given the distances between four collinear points (a) and (6) (fig. 30). (h) ^J^ K 1 — 5' Fig. 30. Since the segment BB' lies wholly within AA\ (a) is a case of hyperbolic evolution; while (6) is elliptic, the segments overlapping one another. Let AM= X. Then, since MA.MA' = MB. MB\ ■^x{AA' -x)=^{AB'-x){AF --x) , AB.AB' ,«Qx In (a) let AB=^1, AB' = Q, AA' = ^ andaj=-3. In (6) let ^5= -1, AF = \2, AA' = 15 and x = Z. MA.MA' = MB.MB'=±S6. If y be the distance from the centre to either focus in (a), we have, 16°, y^ = MA,MA' = S6 and 2/= ±6. 20°. We have now to draw certain deductions from the general equation MA . MA'=MB.MB' = MG, i/0' = etc. = constant. Since MA_MB^ MA-MB _ MB'-'MA' , AB _BM MB ~ MA' ' MB ~ MA' A'B'~MA" BG with corresponding expressions for r,,^,, etc. B G Again, MA MB MA -MB' MB -MA' , AB' MR and MB'~MA" MB' ~ MA' BA' MA" BC CB' BC with corresponding expressions for 77^7 > ®^ CHAPTEE VIII 81 Writing out the two series for clearness and convenience: (1) (2) (3) (4) AB A'E BG B'C CD G'U DA BM\ MA" CM MB" DM MC" AM .(a) D^A'~MD"I (5) (6) (V) (8) AF BA'' BC CF' CD' DC DA' AD MB' \ MA" MC^ MB" MD' MC" MA' ' MD"/ .(6) The product of the first and third expressions of (a) divided by the product of the second and fourth is AB.CD.B'C'.UA' MB.MB'.MD.M U BC.DA. A'B'. CD' " MC . MC. MA . MA' ~ Therefore (ABCD==(A'B'CD') (24) Multiplying together the left-hand expressions of (6), and also the right-hand expressions, AB'. BC. CD'. DA' MB'. MC. MU. MA' 1 .(25) A'B . B'G . CD . D'A ~ MA'. MB'. MC. MD' and AB'. BC. CD'. DA' = A'B.B'C. CD . D'A. By (1) and (3) of (a), AB.CD'MB.MC CD.A'B'~MD.MA'' By (6) and (8) of (6), BC. D'A -MC. -MD' _ MB . MC EC. DA'" MA'. MB' "MD.MA'' Therefore w^=wutSz'' ^" (^50T')=(^'^aD)....(26) By (1) and (4) of (a), AB A'B'.MB.MD' DA By (6) and (7) of (h), CD CU.MF BC B.C. D'A'. MA. MA FC.MD' F 88 INVOLUTION AB . CTD A'E. CD'. MB . ME Therefore BG\ DA ~ B'C . D'A\ MA . MA" and {ABG'D^{A'B'Gn) (27) 21*. The connexion between the coordinates of a system of points in involution. Let {ABCD)={A'EG'U) and {AFGD)={A'BG'D'). AB,GD A'E.G'U BG.DA~RG\D'A'' AE.GD _ A'B,G'U EG.DA" BG\D'A'' By division, AB.EGA'E.BG' A'B.GE AE.G'B BG . AB' ~ EG'. A'B ' BG.EA'~ EG'. BA' and {A'BGE) = (AB'G'B). Therefore (A'GBB') = (AG'B'B), A'G.BE AG'. BE and Therefore GB.B'A'" G'B'.BA' Let A = (x^ViZ^) and A' = (x^y^^^) ; „ ^ = (^3,^3); G' = (t^, u^). Then if wejjcalculate the values of the various vectors AB ... G'A, we get, (28), ^3t^4 1^1^2 1 which may be more conveniently written p^=l, (29) as the condition that the six points shall be in involution. Ex. The transversal, cc — 42/ + 22; = 0, cuts the sides and internal diagonals of the quadrilateral AG'A'G (fig. 1) as follows : G'A' in (213), GA in (201), AG' in (410), CC in (211), AA' in (223), A'G in (012), CHAPTER VIII 83 and connecting these coordinates with the letters of (28), A, B, (7, C\ R, A\ (213) (201) (410) (211) (223) (012) Calculating the values of t and u for 5, G, R, G\ from the coordinates of A and A\ we get, for jB, :g-^y:h-^z. (3) Conversely, x\y\z—fx'\gy'\}iz' (4) 4°. The commonest case is that in which the new origin is irrational. Let (y ^\^,—,—\ its rays cutting the sides as f:g:K Then (^(7'BP3)=||=-'; (B^'CP,)=^=2/'; d \y' \z' —f-Hx •.g~'^my :h-'^nz, (5) x:y :z = l-'^fx' :m-'^gy' :n~%z' (6) Ex. 1. To transform the equation of the circumcircle mna^yz+nlbhx+lmc^xy = (a) from origin to origin 8, the symmedian point, ( -r, — , — ). ^""^ \r m' n) '' r^Pr^«^^^ ^^^ ^^ (^>' x:y :z==-tX : — y : -z. CHAPTER IX 87 Consequently, = mna?yz + nlWzx + Imc^xy = mnaP' y'z' + nlW — r- z'x' q2})2 ^lmc^^x'y\ In fact we have merely to substitute a^, b\ c^ for I, m, n in (a). j&a;. 2. Let the converse problem be considered: to transform to origin the equation yz+zx+xy = 0, which represents the circumcircle when S is origin. As S is irrational in respect to 0, the symbol of when S is origin is f-g, Ti> -z)- These coordinates now represent \j, — , — j above, and we have, (6), Ix' my' nz' Therefore 0=^yz-{-zx+xy = ^^yz -¥^^zx ^-^^^y , „ = „ = mna^y'sf + nWz'x' + Imc^x'y'. 5°. In the foregoing sections the origin only was changed, the triangle remaining the same. We have now to con- sider the case in which both the origin and the triangle are changed. Let any point 0' be chosen for the new origin and any three points A^.B^^C^ for the corners of the new triangle ; and let their old coordinates (for the triangle ABG and origin 0) be 0'^{x^y^z^\ A^ = (x^y^z^\ B^^ix^y^z^), G^ = {x^y^z^). Let P be any point whose old coordinates are {xyz) : it is required to find its new coordinates (x'y'z') with respect to the new triangle A^Bfi^ and the new origin 0'. TRANSFORMATION OF COORDINATES By II, l^ -, = {C^'Afi'B^P\ and by VIII, (9), the value of this pencil is {G,.Afi'BJ>)=^ H Vz ^z ^3 VS ^Z X y z ^0 Vo ^0 ^1 Vi ^1 <^2 2/2 2^2 ^2 y2 ^2 ^3 2/3 % ^0 Vo ^0 X y z ^S 2/3 % a^i 2/1 ^1 ^z Vs H ^Z 2/3 H X y z a^o 2/o «o ^0 Vo ^0 X y z a^3 2/3 ^8 a'2 2/2 2^2 ^2 2/2 ^2 ^1 2/i ^1 «i 2/i 2^1 ^S 2/3 % or, V |a52/3^il|a'o2/22^3l Similarly, y'^I^MiJiMifd. ^^^ l^2/i^2ll^o2/2^3 2^' |a52/22^2ll^02/32^ir ^'^ |a'2/22^3ll«'o2/l2^2 From these three ratios we have .(7) 05=^2/2^3 I Fo2/3^i I Fo2/l^2 2/=F2/32^i z' = \xy^z^\ ^oyi^2\\^oy^s aJo2/2^3 11^02/32^1 (Hamilton.) ;}(8) 6°. Let the point (xyz) be C = (112) (fig. 1); let the points chosen for the comers of the new triangle be (121), (Oil), (1X0) ; and let the new origin be (131). Here aj = l Xq = 1 fl?i = l x^ = x^ = l 2/ = l 2/0= -3 2/1 = 2 2/2 = 1 2/3= -1 z = 2 00=1 -1 0o=-l Substituting these values in (8) we get, aj' = — 4, 2/' = 8, — 2 s;' = -o- ; and putting these numbers in continued proportion, x':y':zf = 6: -12:1, or the new coordinates of P are (6, —12, 1). We may verify this result without any reference to the equations of (8). The points taken for the new triangle are D^RC'GR'' =(121), ^" = (011) and (7" = (lT0) (fig. 33), an extension of fig. 1. For the new origin, an UNIVERSITY OF CHAPTER IX m auxiliary point H=BG'AD = (021) was determined, and the point BB'- G"H= (131) was chosen for 0\ FxG. 33. The old is chosen so that l\m'.n = l : 5 : 2. so TEANSFORMATION OF COORDINATES It will be found that O'D, 0'A'\ OV" cut the sides of I)A"G" in the ratios A''E l-\'2m-n 3 ED~ m- -n "1' C"F m- n 1 ~4' DG GO"' 4 FA"' --3(^ -m) "3* V:m'', ',n' = 3:1 :4. Hence Lines drawn from G"' through D, A" and C meet the opposite sides of DA'V' in r^=^(lS4>), F2 = (152), P'3 = 5' = (101). Mence p,^^ "my" -2(m-7i)- -2' C'P'i my 4(m-^) -3. DF\ n'z' 2 or ZV:my:7iV = 9:-6:2. But i':m':%' = 3:l:4. Therefore, aj' : 2/' : 0' = 6 : — 12 : 1, as before, or ^ = (6, -12,1). 7". We may somewhat simplify Hamilton's equations. Let the nine minors of \x{ij^^ |, in the usual terminology, and let | x^y^z^ \ \ x^y^z^ \^\, I «'o2/2% 1 1^02/3^1 1 = ^3- Then the equations of (8) become x' = h^{L^x-\'M^y^-N^z\\ y'^k^{L^^M^y^N^)\ (9) z' = \{L^-JtM^y-\-N^z).] X- 11 o CHAPTER IX nerefore, ; y= ^^ ^> 1 ; 2^= M, N, 1 ^^ ^^ 1 M, N, 1 ^3 ^3 1 91 4 ^3 1^ Evidently, But Consequently, x=\M^,\^+\MsN,\l+\M,N,\^. /Cj /Cg A/j ^3^ll = «'2|a'l2/2^3l; |-^l-^2l = «'3ki2/2^3l- AJj /Cg A^g 2;. Similarly, 2/ = r' ^'+f-' 2/'+f-'^', /Cj /Cg /Cg .(10) /Cj /tg /Cg For the triangles of fig. 28, A:, = 3; 0^1 = 1; y^ = 2; ^, = 1; Zi = T; ifi = T; N^ = l. k^=S; 0^2=0; 2/2=!; 2^2 = !; 4=1; ^2=1; ^2=3. A;3=l; a;3=l; 2/3=!; 2^3=0; Z3 = T; if3=l; i\r3 = l. Equations (9) and (10) consequently become x' = S{X'\-y-\-z); x—'-^x^-^-z' — x' — Zz'. \ y=-3(l + 2/ + 30); 2/=~K-"i2/'-^' = 2a;'+y'+30'.Hll) From these equations we get the new coordinates of the corners of the old triangle ^ = (331), J? = (331), C=(391), = (9, 15,1). From the value of aj', it is clear that the equation of the side G"A" of the new triangle, x' — 0, is in the old coordinates 92 TEANSFORMATION OF COORDINATES the axis of perspective of the old triangle, according to construction. The axis of perspective of the new triangle in the old coordinates is , k /^ A circumconic of the old triangle, yz+zx-^xy = 0, in the new coordinates is x^-\-y^-\-9z^-{-Syz-{-Szx-^Sxy==0. S°. The matrix formed from the coefficients of the transformed values of x, y, z,(ll), 1 3 2 1 3 1 1 is the modulus of transformation, and the invariants of two conies are calculated in the same way as in other systems of coordinates. Let the equations of two conies be -8y^-\-2yz+4>zx+2xy = 0, 2yz-^2zx-^2xy = 0. The invariants of these equations are A = 36, 9 = 42, e'=16, A' = 2. The transformed equations are Sex^-\-10y^-\-dOz^+4iSyz+96zx+4iOxy = 0, 2x^'\'2y^-hl8z^-}-6yz-{'6zx-\'6xy=0; their invariants are Ai = 1296, 91 = 1512, 9'i = 576, A'i = 72; , Ai 9i e\ A\ o« ^^^ A=9=9^=A^ = ^^ = 1 3 2 1 3 1 1 CHAPTER X THE CIRCLE 1°. The condition that the circumconic, yz-\-zx-^xy = Oy shall be a circle (fig. 34). Br T, "~ — — » •——.«• m^ ^.m '-y^ "Tv^" ■--/ '<'-'^ c R y •X 1 / V \ f ^>.J K 1 ' \^ K M -^ A^ ^ Fig. 34. The tangents to the curve at Ay B and C meet in Ti = (lll), T2 = (lll), T^ = {lll\ and K the centre of the curve is (m+'?i — ^, ^i+^ — tti, ^-f'^ — '^). Therefore the equations of KT-^, KT^, KT^ are (m — 71)0; + 7711/ --nz — 0, — ?ic + (ti — i^) 2/ + 712; = 0, ^aj — my + (i — m)0 = 0. These lines cut the triangle respectively in {onm), (not), {mid), the middle points of the sides, and when the conic is a circle they are perpendicular, KT^ to BG, KT^ to GA, KT^ to AB. Applying the condition of perpendicularity, III, 10°, (c), we get i(62-c2)-ma2+^a2 = 0, ^62+m(c3~a2)-7i62 = o, whence l:m:n = a^:h^:c^ (1) 94 THE CIECLE Therefore the equation yz+zx-^-xy^O represents a circle when the symmedian point is origin. Changing the origin from the symmedian point to 0, we get, IX, 4°, mna^yz-\-nlb^yZ'\'lmchx = (2) as the equation of the circumcircle. 2°. The coordinates of the points in which A« cuts (2) have three forms : (1) x:y :z = mna : 712(06^*^— a) : ^lmce'^*^.\ (2) „:„:„=- mnae^'^ : nib : Imiae^^^^- h)X . . .(3> (3) „:„:„ = mn{he^^^ - c) : -nlhe^^"^ : Imc. J These are the Cyclic Points at infinity, / and J. If an angle of the given triangle, say A, happens to be 90°, the lines AI and A J will be found to be the harmonic conjugates of AB and AG. Ifa=6 = c = l and l:m:n:lf the 1^* form becomes (., -i.^. -i.^} <« three of the cube roots of unity, which will be as usual written (loooo^) and (Iw^oo). It will be observed that 3°. A metric equation of the circle may be obtained as follows. Let d be the constant distance of a variable point (xyz) from a fixed point F—(fgh). Then the given triangle being equilateral and its mean point the origin, the distance between the points is d^l^f.H^x^^p^+q^+r^-qr-rp-pq = FI.FJ, because ^ = % — 2/^, q^fz—hx, r^gx—fy. Consequently, FLFJ-'d^I.^.I^^x^O (5) is the equation of a circle with (fgh) for centre and d for radius. CHAPTER X 96 The tangential equation, VII, (6), 72_2^ (Imnhc sin Ay corresponds to the local equation (5), .^_ FI,FJ If c? = 0, the local equation becomes FI,FJ=0, the pro- duct of the equations of two imaginary lines ; the tangential equation becomes Ifp = 0, the equation of the centre, which is the cross of these two imaginary lines. If d = oo , the tangential equation becomes 0^ = 0, the product of the equations of two imaginary points ; the local equation becomes 2ic = 0, the equation of the (analytically) real line Aoo , the join of these two imaginary points. 4°. Equation (5) represents a circle, and its form shows it is in terms of two tangents, FI=0, and FJ=0, which touch the curve at I and / respectively, and the chord of contact aj-|-2/+2^ = A„ = 0, V, 22°. Since the pair of tan- gents are drawn from the centre and touch the curve at infinity, they are the (imaginary) asymptotes of the circle. The value of d and the position of F being arbitrary, the general conclusion is that all circles pass through the two cyclic points / and J. 5°. The circle is the only conic which passes through both / and ./. Every parabola meets Aoo iu two real and co- incident points : every hyperbola is cut by it in two real and distinct points. No ellipse can pass through both. The coordinates of the points of intersection of two conies are derived from two quadratic equations, and consequently have four, and only four, sets of values; or, two conies intersect in four points only. Suppose a certain ellipse to pass through I and J. Let any three points P, Q, R be taken on the curve and let a circle be drawn through them. Then the two conies intersect in P, Q, R and also in I and J, that is, in five points ; which is impossible. There- fore no ellipse can pass through both / and J.* 6°. It follows from the foregoing that if A represent any straight line, S any circle, and if k be an arbitrary constant, AA^-^kS=0 (6) * Whitworth, Modem Analytic Geometry, p. 289. 96 THE CIRCLE represents some other circle, &. First, being of the second degree, (6) represents some conic. Secondly, S' passes through the two points in which 8 is cut by Aoo and the two points in which it is cut by Q, But >Sf is cut by A^o in / and J. Therefore 8' is a circle, the only conic that passes through these two points. Since 8 and 8' are cut by A in the same two points, A is the radical axis of 8 and 8\ T. The condition that ^(a?, 2/1 2;) = shall represent a circle. Let A=^a;+gi/+r2; = 0, and let 8 represent the circum- circle in (6). Then, (^ 4- gy + rz){lx + my + nz) + kimnahfz-\-nlWzx-\' lmnB' Jtsm 0/ Therefore, IX, (6), _ Ix' ^ my' ^ nz' ^'^ '^~ J-ioxiA' ^tan B ' ^tan C Consequently, -ajH2/H^2 = cot-4^V2 + cot5mY2+cota7^V2 = 0, (17) which is real when A is obtuse. rtM, A. £ i 1. • 1 • /tan A tan B tan G\ , , Ine centre oi the circle is — j — , , ), the orthocentre, which lies outside the triangle because A is obtuse and tan A negative. Equation (17) may be written, 6ccosu!lZV+etc. = 0, and can be thrown into the form Aoo (he cos Alx-\-ca cos Bmy + ah cos Gnz) — /Sf = 0, where B represents the circumcircle. Now the equation of the IX circle, (16), may be written, Aoo (he cos Alx-\'Ca cos Bmy + ah cos Gnz) — 28=0. Therefore the polar and the IX circles have the same radical axis in respect to the circumcircle; or the three circles intersect each other in the same two points. 12^ Since a = S^ + 8^y 6 = S2 + Si, C = Si-^S2y BJldC08A=C08^A^8iD!^iA, the equation (a) of the IX circle may be written, Aoo {(s^i — s^s^lx + (ssg — s^s^my + (ssg — s^s^nz} — 28=0. The equation of the incircle in the same form is Aoo {s-^Hx 4- S2^my + s^^nz} — 8=0. Now, if two circles are given, a=AooA+A;^=0 and G' = A^A' + k'8^0, then (7= Aoo (a - 1 A') + 1 C\ CHAPTER X 101 k . where A — p A' = is the radical axis of G and C, Applying this result to these equations of the in- and IX circles, we 5^^^„+^ = (18) as their radical axis. Let the Gergonne point, (SgSg, s^s^, s^s^), be taken for origin. Then this equation of the radical axis becomes ^ ' -0; (a) and at the same time the equation of the incircle become^ Now, for this equation, U= V=W=0] U'=T= W' = 2. Therefore the condition that (a) shall be a tangent to the incircle is, V, 6°, Si(b — c)+S2(c — a) + 8^(a — b) = 0. But this equation is identically zero. Therefore the radical axis (18) is a tangent to the incircle, and consequently to the IX circle. The point of contact is {8,%h-cy, si{c-af, s,\a-hf} (19) to the Gergonne point as origin, and to origin 0, isjy-cf 8^{c-af 8^{a-hf \ I — I ' ^r~' n J ^^"> These results may be reached more easily as follows. The tangential coordinates of the radical axis of the IX and incircles are, (18), / I m n \ \6 — c' c — a' a — hJ and the tangential equation of the incircle is Is-^qr + ms^rp + ns^'pq = 0. For the Gergonne point as origin, these expressions become / ^2^3 ^3^1 ^1^2 \ \h — c^ c — oJ a — hi and gr+'>l>+M = ^- 102 THE CIRCLE Substituting the coordinates of the radical axis in the equation of the circle, we get Si(6 — c)+S2(c — (x) + S3(a — 6) = 0, which is identically zero. Therefore the radical axis is a tangent to the incircle and consequently to the IX circle. 13°. If X' — {x'y'z) be any point without a circle with centre F— (Jgh) and radius r, the length of a tangent from X' is, (5), f'^FX'^^rK If we regard FX' as the radius FI FT of another circle with F for centre, FX'^ = (P = • ^' ^^ . Therefore <2 = g^_^2_ (21) the length of a tangent from any external point to a circle, with the conditions, a = h = c, and l:7n:n = l. CHAPTER XI THE FOCI OF A CONIC 1°. Let A = be the equation of a fixed line and S=0 the local equation of a conic. Then the equation, /Sf— A^ = 0, represents a conic S' which has double contact with S in the two points in which it is cut by the chord of contact A, whether A cuts 8 in real or imaginary points. Consequently S' and S have two common tangents, which are real in the first case and imaginary in the second. Let >S^ be a circle with a fixed point F=(fgh) for centre and an arbitrary radius r^ ; and let A be (pqr), the function of its coordinates being Z^ as usual, IV, (2). Let a = b = c and l:7n:n = l. Then the equation, >Sf-A2 = 0, may be written, X, (5), Now by X, (21), the first term in brackets is the length squared of the tangent drawn from any point, X = (xyz) on S' to the circle S, say r^ ; qv2/ ^® ^ constant, say e^ ; and the second term in brackets is, lY, (7), the perpendicular squared from X to (pqr), say a^. Hence, r and cr being variables, for every point on S\ T2 = eV (2) In words, if a circle S have double contact with a conic S\ the tangent drawn to the circle from any point X on the conic is in a constant ratio to the perpendicular from the point to the chord of contact. 104 THE FOCI OF A CONIC 2°. Let A cut S in imaginary points and let r^ approach zero. At the limit (1) becomes ^"SyS^a; ^ ^Z^^^x ^^ The first term now represents the distance squared from X to ^, say p^, and the equation may be written = p2_g2^2^ (4) the known equation of a conic section, p being the distance of the variable point from a fixed point (Jgh) and o- its perpendicular distance from a fixed line (pqr). According as 6 = 1, the curve is an ellipse, parabola or hyperbola. The focus (fgh), then, may be considered as an infinitely small circle which touches the conic in the two imaginary points in which it is cut by the directrix. 3°. Substituting for e^ its value oy2/' equation (3) becomes 0==FI.FJ-^^x (6) Now the form of this expression shows that it is the equation of a conic in terms of two (imaginary) tangents, FI and FJ, and their (real) chord of contact. Consequently a focus, still regarded as an evanescent circle, is the cross of two imaginary tangents to the conic, the one from /, the other from J. But four such tangents may be imagined as drawn, two from / and two from J, intersecting in four points which form a quadrangle. Therefore a conic has four foci. In the case of the parabola two of these imaginary tangents, one from I and one from J, coincide with Aoo , which is itself a tangent to the curve. Since FI and FJ remain the same in (5) whatever (pqr) may be, it follows that all conies which have a common focus have two common imaginary tangents ; and if they have two (real) foci in common, they have four common imaginary tangents. 4°. We have now to enquire into the nature and position of the four foci. In V, 18°, (6), were given the separate coordinates of the two tangents from a point (fgh) to a conic. If we suppose that {fgh) becomes successively I={lcio(a^) and J=(l(o^w), the CHAPTER XI 105 first set will con tain the quantity s/ — ^^(Iwa)^) and the second v — A0(lo)2ft)), which it is necessary to expand. Let u+2u' = a, v-\-2v' = b, w-{-2w' = c. Then ^/■i 2\ . z. 2 . 2a — b — c .(6 — c),^3 0(10)0)2) = a4-t>f«>+ CO) = ^ ^^ 9~' 0(lo)^O)) = a + OO) + CO)2 = ^ f-^^ n ' Therefore __. / V(a^ + 6^+c2-6c-ca-a6)-(2a-6^ = s/P-WQ' Similarly, V0(1^'«) = V^+VQ- Consequently, >v/-A0(lo)O)2) _ ^Q^ + ^ VPA, V-A0(lo)2o))= -^QA4-iV?A. The coordinates of the two tangents from /, when reduced, are, p = 6(- D"+ r + F0H=2VaPA + i^S{4^V-4>W+2r-2W'±2s/QA}, -^i^S{2U-2W-2U'-^6W-4^W'i^(JSPA + ^QA)}, r = 6(- U+ V- U'+ F0±(V3PA-3VQA) -^iJ^{-2U+2V^-2U'-\-^T-QW'±{JWK-jQA)}' The two tangents from / are / = 6(-£7'+F'+F')±2V3PA -V3{4F-4r+2r-2F'=F2VQA}, g' = 6(F-Cr-£^'+r)+(>v/3PA-3VQA) __ -V3{2[7-2Tf-2f^'+6r-4Tr'±(V3PA + ^QA)}, ^' = 6(- U+ F- ^'+ Tr')+(N/3PA-3VQA) -V3{-2i[7+2F+2(7' + 4F'-6F'+(x/3PA->v/QA)}. 106 THE FOCI OF A CONIC The coordinates of the four tangents may be written : From I 1^1 = (^ + ^'^' f'^^^^ ^ + ^^)' ^ ' \t^ = {d'+e'i, f+g\ h'-^k'i). (6) (8) From /, [t, = {d'-e'i, f^g\ h'^k'i), \t^ = (d — ei, f—gi, h — ki). It is evident from these expressions, that F=t^-t^ = (gh-fk, dk-eh, ef-dg\ \ .^. F' =zt^.t^ = {g'h' —fk\ d'k' — eh\ ej' — d'g'.] (fK-fh+gk'-g'k+i{gh'+g'h-fk'--fk), Y=t^'t^ = \M -h'd+ke' -k'e + i{dk' -^d'k-ehf -e'h\ \df - d'f-\- eg' - e'g -{-iief + e/~ dg' - d'g) ; [fh' -fh+gh' - g'k - i(gh'-{-g'h -fk' -fk), F = ^2- h = \hd'- h'd + ke' - k'e - i{dk + dik + eh! - e'h), [df-d'f-^eg'-e'g-i{ef+ey-dg'-d'g). ) It appears from the equations, (7), that F and F' are real points, and from (8) that Y and Y' are imaginary. But the line YY' is real, as is evident from the form of the coordinates of F and Y\ (p + qi, r+si, t-\-vi), (p — qi, r—si, t — vi). These are general conclusions, and F, F' may be any two points in the plane. Let, then, F={x{y^z^, F' — {x^^z^\ the equation of FF' being as usual, px-{-qy-{-rz = 0, (a) where p-={y^z^-y2^i\ q^{Zv^2-^2^i)> ^ = (^1^2 - ^Jg^/i). Forming the equations of FI, F'J^ etc., we get FI={z^u>-y^w^, x^aP'-z^, y-^-x^w), FJ={z^(£>^ — y^(ay x-^w — z^, y^^x^(£p-)y FJ=^{z^w^-y^(a, x^w-z^, y^-^z^^)* F'I={z^w-y^w^, x^ui^-z^, y^-x^w). From these equations, F= FLFJ^ {( _ 2p + g + r) ^i{x^(r^ + a'2^i)>v/3, (^ - 2^ + r) + ^(2/10-2 + 2/2^1) V3, {p-\-q-'2.r + i{z^(r^-\-z^(T^)J^). CHAPTEE XI 107 where cr^ = x^-\-y^-[-z^ and o-^^x^ + y^ + z^. Let '-2p + q + r = a, p — 2q + r = h, pi-q — 2r = c, «1<^2 + ^2<^1 = ^1' 2/1^2 + 2/2^1 = ^2' %0-2+ 02^1 = ^/3; and the equation of YY' will be the real line (bM^-cM2)x-\-(cM^-aM^)y-^(aJ\^^-.hM^)z = 0. ...(b) Now III, 1°, the middle point of the line FF' is (i/i, ifg, -^3), and Therefore the real line YY' bisects FF\ Again, since the given triangle is equilateral and its mean point the origin, the condition of perpendicularity, III, 10°, is 2pp'-\-2qq'-^2rr'-(qr'+q'r-\-7p'-\-rp'}-pq'-^p'q) = 0. Applying this test to (a) and (6), the equations of FF' and YY\ we get = 2p(bM^ - cM^) + 2q{cM^ ~ aM^) + 2r{aM^ - bM^) - q {aM^ - bM^ - r{cM^ - aM^) - r(bM^ - cM^) —p{aM^ — bM^ —p{cM^ — xM^ — q{bM^ — cM^ „ = (6if3 - cM^){2p - g - r) 4- {cM^ - aM^){ -p-\-2q- r) -{-(aM^-bM^)(-p-q-\-2r) „ = - a(bM^ - cM^) -b(cMi - aM^) - ciaM^ - bM^) = 0, identically. The two imaginary foci, then, are situated on a real line which bisects at right angles the line joining the two real foci. 5°. The coordinates of the foci given in 4** are ill-adapted for calculation, and we have now to consider other methods which will give these coordinates in a less complicated form The following method of finding the coordinates of the foci, in the case when the given triangle is equilateral and its mean point the origin, is given by Sir William Hamilton. 108 THE FOCI OF A CONIC Writing for (p(xyz), I for x, m for — /uL — C(i) — v (9) By means of these three equations we can determine the four points in which the two pair of tangents from / and / intersect. Let^, g, r be any three constants such that p + g+r = 0. ^^^"^ p(a0~X) + g(60-^)4-r(c0-.) = O (10) represents a conic passing through the four foci. Let p — b — c, q = c — a, r = {a — h), and this equation becomes (b-c){P+2mn) + (c-a)(m^-^2nl) +(a-6)(7i2+2^m) = 0, (11) where a, h, c are known and real constants (the given conic being real by hypothesis), and I, tr, n represent real and homogeneous functions of 0(a?, y, z). This equation breaks up into two real straight lines. For let h^ = a^ -{-b^ + c^ — be — ca — ab, which is real since the conic is real, and (11) is equivalent to = {{b-c)l+(a+b)m + (c-a)n + h(m-n)} X {(b-'c)l-\-(a-b)m-^(c-a)n + h{n-m)}, ...(12) CHAPTER XI 109 the product of two real and distinct straight lines on which the foci are situated. These two lines are consequently the axes of the conic ; their cross is the centre ; their inter- sections with (10) are the four foci ; and their intersections with x=-y-^z; m = 0j,= -a; + 82/-0; n = ,= ~2x-'y, X = 4a;2 - 1 52/2 + 4^2 + 62/0 + 4>zx - SOxy^ luL = x^+6ey^+z^-12yz+10zx-12xy,\ V = 4£C2 — 14^/2 _J. 42;2 _ ^()yz + ^ZX + Qxy.) Substituting the above values of a, h, c in (12), we get = (l-n)(l-2m-\-n) = y{x-'z). The axes of the conic are therefore y = and a; — = 0. The cross of these lines (101) is the centre. The foci are the intersections of the axes with (10). Let p = 2, q= —1, r= —1, and we ultimately get for this equation x^-lly^+z^ + Uyz-10zx-22xy = (13) The intersections oi y = with this conic will be found ^^ (1 + -^/f . 0, 1 - VI) and (1 - VI, 0, 1 + Ji). These are the real foci. The imaginary foci are the intersections of ic — = with (13), namely (11,6^/^-4,11) and (11, -(6 V^ + 4), 11). The intersections of 1/ = with the conic give two of the vertices, (100) and (001). The intersections ofa; — 2; = with the conic give the other two, (111) and (212). The conic is an ellipse since it has four real vertices. Since G and A are two opposite vertices and CA = 1 by hypothesis, the length of one semiaxis is J. The distance of the centre (101) from either of the other vertices is ^^ — -7=, the length of the minor semiaxis. From the lengths of the semiaxes, the eccentricity is found to be ^^f. 110 THE FOCI OF A CONIC 6°. When the given triangle is scalene and l\iyi\ n=\=l, we may transform the coordinates by selecting an equilateral triangle for the new triangle and taking its mean point for the new origin. If, as generally happens, the figure under consideration contains no equilateral triangle, we may construct on the base of the given triangle an equilateral triangle ADC, the points B and D lying on the same side of GA. The reader will find little difficulty in proving that the coordinates of D are given by x:y \z = mna(smG—cosG^S) : nlh^S : ^mc(sin^ — cos A ^3), (14) and that the coordinates of M, the mean point of ADG, are x:y :z = mna(sm G^S — cos G) : nib : ^mc (sin A ^3 — cos A). (15) We then proceed to Hamilton's equations, 5°. This transformation of coordinates, owing to the form of the coordinates of D and M, is tedious, and the following method is in general preferable. 7°. If T—0 be the tangential equation of a conic and u — and v = the equations of any two points, then T+kuv = (16) is the equation of a conic T' so related to T that two of their common tangents pass through u and the other two through V. For if (pqr) be either of the tangents from u to T, its coordinates must satisfy the equations of both u and T, and consequently satisfy (16), the equation of T\ Therefore (pqr) is a tangent to T\ YII, 9°. In like manner the coordinates of the other tangent from u and those of both the tangents from v to T satisfy (16), and all three of them are consequently tangents to T\ Therefore T and T' are both inscribed in the quadrilateral formed by the intersections of their common tangents from u and v. Now the equation of T' possesses this property, that when, for certain values of the constant k, it breaks up into the equations of pairs of points, these points are the opposite corners of the quadrilateral in which T and T' are inscribed. This may be illustrated simply. The lines BA and BG CHAPTER XI 111 of the given triangle (fig. 36) are tangents to the inconic, qr-^rp+pq = 0, and B''A is also a tangent. The second tangent f rom 5'' cuts jB(7 in Z) = 2g + r = and BA in E = p + 2q = 0. For the conic which has four tan- gents passing through B and B'' in common with the inconic, fiq. se. we have = qr-\-Tp-\-pq-\-kq(r—p) = (l-\-k)qr-\-rp-\-(l—k)pq. To obtain the values of k for which this equation breaks up into pairs of points, we must equate its discriminant to zero and solve for k. , 1-k 1 , l-\-k, 2(1— ^2) = 0, the roots of which are ± 1. For k = l, the given equation becomes = 2qr-\-rp = r{p-{-2q) = CxE. ForA:=— 1, = rp-^2pq=p(2q-\-r) = AxD. 8°. To obtain the equations of the foci, I and J are taken for u and v, and we deduce the values of k from the dis- criminant of the equation equated to zero. This discriminant is 1/+ kmhi^a^ , W - Mmn^ab cos C, V - klmHca cos B W - Umn^ah cos C, F+ hiH%^ , U'- kl^mnhc cos ^ = 0. ( 1 7) V — klrri^nca cos B , U' — kPmnbc cos A , W+ kl^m^c^ (a) {h) {c) (d) (e) if) This matrix can be resolved into eight matrices of the third order, formed from the columns of the above which have been lettered for ease of reference. The matrix (ace) = A^, A being the discriminant of the local equation of the conic, (p(xyz) = 0. The matrix (bdf), which involves the third power of k, vanishes. 112 THE FOCI OF A CONIC The matrices {adf), (bcf) and (bde) involve k\ and their determinants are (adf) = Pm^n^Ah^c^ sm^Ak\ (bcf) = Pm^Ti^Bbh^ sinMyb2, {bde) = Pm^n^C¥c^ sin^Ak^ Consequently the coefficient of k^ is l^m^n^b^c^8inU{lA -\-7nB+nG) = l^m^nWbh^8m^A. (18) The matrices (bee), {ode) and {aef) involve k. {bee) = Ak{u7n^nV — v'lm^nca cos B — w'lmn^ab cos (7), {ade) = Ak{vnH^b^ — w'lmn^ab cos G—u'l^mnbc cos J.), (ac/) = Ak{wl^m^c^ — u'Pmnbc cos -4 — v'lm^nca cos 5). Therefore the coefficient of A: is A {urn?n^a^ + vnH^b^ + wl^m^c^ — 2lmn{ulbe cos J. + v'mca cos jB + w'nab cos C)} (19) = A [mtia^ {umn '-l{ — u'l-\- v'tyi -\- w'tyi)] + 71^6^ {'y-y^i — 7n{u'l — -v'm + tt;^)} + Imc^ {wlm — 7i(it7 + v'm — ty'm)}] = Ae', (20) 0' being the ordinary invariant symbol. Consequently the complete determinant of the original matrix is l^m^nWb^c^8m^Ak^-\-A&k-\-A^ = 0. (21) In this equation A is the discriminant of {xyz) = Q, the local form of the tangential equation T=F{pqr) = 0. D is the bordered discriminant of zx — 2xy = 0, (a) a = b = c; l:m:n=l have been already calculated from this, its local equation, example of 6°. CHAPTER XI 113 They will now be calculated from its tangential form, Since Q^ = rn?n^a^p^ ... - 2lmnHh cos Cpq T-hkQ^ = (k-'l)p^-^{k-4>)q^-]-(k-'l)r^-{k-4>)qr -(k-S4>)rp-(k-4>)pq = 0. (h) From (a), A =-36, D = S6, sin^ = ^. 9' = um^nV ... — 2w'lmn^ah cos G Therefore the equation for k is, (21), = 36xp2-12x36A;+36 = A;2-16A;+48; and A; =4 or 12. Substituting 4 for k in (6), 0=p24.r2+10r29={(l + VS)i> + (l-V*M The two real foci are therefore (l + Vf)2>+(l->v/l> = and (l-Vt)i>+(H-x/f)^ = 0, or locally, (1 + Jh 0, 1 - Vf ) and (1 - Vf , 0, 1 + Ji). Putting 12 for k in (6), = ll_p2+8^2^11r2-8gr-22rp-8^g, r . 6x/^^-4 , \r 6V^=^+4 ^ \ " = F+"^1 — ^+^||i^ 11 — ^+7' or locally, {11, 6x/^-4, 11}{11, -6^/^-4, 11}, the focoids. ^oj. 2. The foci of the conic the triangle being scalene and l:m:n = a^ :¥ :c^, or the origin being the symmedian point S. For this curve, A=-4, and D = Ha%^-\-h^c^-hcV) = 4^i:a^b^. H.C. H 114 THE FOCI OF A CONIC Since B is positive the conic is an ellipse. Its tangential equation is ^^ 4^^ _^ 4^^ ^ ^^ ^ ^ 0' = aWc\la%^ + 4a2&c sin B sin G) = a26V(2ct262 4.46V sin^^). Let 2^262 = 0-, and 6Vsin2^=4(triangle)2 = 4^l Then the equation for k, (21), is ' 4a*6V^VA;2 - a^b^c'-{(T + 16^^)^ + 4 = 0, ^""^ ^^^a%\H^ """^ ^2cv- Q2 = ^26*6*2^2 _|. ^^452^4^2 ^ C6*6*c V2 - 2a*63c^ cos ^ qr — 2a^6*c^ cos 5rp — 2aWd^ cos (Tpg ; and taking the second value of /c, T-\- ^Q^ ig 6Vp2 + c Vg2 + a26V2 + {h\^ + a*) gr + {cV + 6*) rp + (a262 + c*)^g = 0, that is, (62p + c\ + aV) (c^^ + oJ^q + ^V) = 0. The tangential equations of the foci therefore are h^'p-\-c\-\-a^r = and c^p + a'^g + 6V = 0, and their local coordinates are (to origin 8) (62, c\ a2) and (c2, a2, h''). To origin these coordinates are /a262 62^2 g2^2\ /c2^2 ^2^2 52^2v V-T' ^' ^) ^^^ V-T' -^' -^J' which are the Brocard points. The given conic is the Brocard ellipse; an inconic which touches the sides of the triangle in the points in which they are cut by lines drawn from the corners through the symmedian point. Ex. 3. x^-\-y'^-{-z^-'2yz-2zx-2xy-=0, ^:'m:'^ = 3: -2:6; a = 6 = c = l. This equation represents a conic because A = — 4 is actual and a parabola because D = 0. It is a 6-escribed conic, and its tangential form is T = 4gr' + 4-)^ + 4pg = 0. Q2 _ 77^2^2^2p2 _ 2lmn^ah cos C „ =lUp^-\-S24>q^-\-S6r^-hl0Sqr+72rp-\-2l6pq = 0. CHAPTER XI 115 Therefore r+M2^A;(144pH324g2+36r'2)+(4 + 108A;)gr+(4-72/o)rp + (4 + 216%g (a) Since D = 0, one root of (21) is infinite, and for the other 9' = umH^ ..." w'lmn^ = 7x36, and k=-^. Therefore (a) becomes = (p-\-q-^r){4>p-^9q+r). One focus therefore is ^+^+r = 0, the origin. The local symbol for the other is (491), and its vector is _ 4>la-^9ml3+ny _ 2a-'Sl3-\-y ^~ U-\-9m-\-n ~ 2-3 + 1 * Since the denominator is zero, the second focus is at an infinite distance, although real. The axis of the parabola, which passes through the two foci, is (853) and its vertex is (16, 1, 25). The directrix, the polar of the focus (111), is the axis of perspective of the given triangle, x-]-y-\-z = 0. Since the given triangle is a triangle of tangents to the curve, its orthocentre (231) lies on the directrix, and the circumcircle passes through the focus. The axis cuts the directrix in P = (2, 13, 11). Con- sequently the distance from the focus, P=0 = (111) to P ought to be twice the distance from to the vertex, V= (16, 1, 25) or 20V= OP. To ascertain the coordinates of 20 F we may employ the method III, l^ (2), x:y :z = (|-l)2/^+/2^ : g-l)2/i+^Si : (j-l)2fl+hlL Here E/^ = 196. - = 2 and ^-l=-i; 2^ = 3-2 + 6 = 7 ;/=16,^ = l,;i = 25; 116 THE FOCI OF A CONIC Therefore x:y :z = 2: -IS.U and ^•^p. ^ 16Za+m^+257iy _ 2^a~13m^ + llr^y ^^ zuv-z i6i+m+25^ ~ 2^-13m+ll'ri ~^^' Ex. 4. x^ + 9z^ - 202/0 - lO^^a; + 4aJ2/ = 0, with l:m:n = l:2:S and a = 5, 6 = 4, c = 3. This equation represents a conic because A = 64 is actual and a hyperbola because D=— 3x64 is negative. Its tangential form is T=- 100^2 _ 16^2 _ 4^2 _ 40^^ ^ Q4,pq = o, Q2 = 25 X 36jp2+9 X 16g2+4 x 9r2- 24 x 9rp-m x I6pq = 0, -(54/c + 10)rp-16(9^-l)2?g = 0. (a) 0' = 32 X 36. Therefore the equation for k is = fc^-Afc__l^, and k = l or :=^. For k = ^, equation (a) becomes = r2), or r = 0'and p = 0. Therefore the real foci are the corners G and A of the given triangle. Fork = ^, ,, = {(3~4x/^)2? + 2^V^+r} {(3 + 4>v/^i?-2g>v/3I+r}. The hyperbola cuts d in if=(901) and M' = (101), the vertices of the curve. Since GA = 4 and i : m : 'W- = 1 : 2 : 3, the length of the transverse axis, MM\ is 2. It will be found that the eccentricity is 2. The equation for the asymptotes, A {Ix + my + nzy — D (xyz) = 0, gives = x^-\-y^-^9z^ — l 2yz — 600? + 4a;2/, „ = {a;+(2 + V3)y-30}{a;+(2-V3)2/-3«}. CHAPTER XI 117 Ex. 5. The conic, x^ — y^ — z^ = 0, with the conditions For this conic A = 1, 2) = — 7, and consequently the curve is a hyperbola. 9 = 24, and A; is ^r^r or -=^ ; the first beinej the value for the real foci. ^^ ^^ Q2 = 16 x8|)2+16g2+16r2+ 16 x4r^+ 16x4^)^ = 0; and T=p2_g2_^2^0. Therefore !r+W2 = 9^2^4^^^4^^ = 0=p(92) + 4g'+4r); and the equations of the foci are ^9 = 0; 9pH-4g'H-4r = 0; or locally (100); (944). The centre is (122) and the asymptotes are {(2 + 2^7), -(4 + V7),3} and {(2-2^7), -(4-^7X3}, or locally — Saj^ + g^/^ + ^z^ — Syz — 4>zx — 4!xy = 0. The given triangle being self -conjugate to the conic, the side BG is the polar of the focus A, and is consequently a directrix. CHAPTER XII MISCELLANEOUS THEOREMS 1°. The harmonic properties of a plane net (^g. 1). ^^m^ 0F=^^; Oa = ^±^, (1) m — -ji ' 7^ — 6 ' 6 — m Therefore -4' and A'' are the harmonic conjugates of B and (7, B „ J5" „ „ C „ -4, 0' „ C „ „ J. „ 5; or {ACBC") = (BA'GA") = (GRAB') = - 1 (2) Again, let OA/ = a, 0F==l3\0a = y. Then -^-^ 2^a + m/3 + '^^y ^ (/^g + m,8) + ('yiy + ^«) ^ ~ 2^+m+ti (^+m)+(w + (l-\-m)+(n-hl) ' m/3 — ny (la-\-m^) — (ny±la) ^ ^ 7/1— '}^ (^4-'m/) — ('yi + O (l-{-m)-(n-i-l) Therefore ^''' and -4'' are the harmonic conjugates of B' and G\ Similarly, B'" and B'' are the harmonic conjugates of G' and J.', 0'" „ (7" „ „ A' „ 5'; or {A'G"'EG") = (^^'"C"^") = {G'B^A'F') = - L . . .(3) CHAPTER XII 119 Since {B'-AG'BC'') = (B''AA"VA), AO is cut harmonically in A'" and A'; while BV is cut harmonically in A"' and A'' (8), and CB is cut harmonically in A' and A'' (2). Therefore each of the three diagonals of the complete quadrilateral AC OB' is cut harmonically by its two other diagonals. Let B'B be produced to meet ^"C" in D, Then A"G"B" is the harmonic triangle, and F'B'^D the diagonal triangle, of the quadrilateral A'EG'B. 2°. A theorem by Roger Cotes. If a straight line revolve in the plane round a fixed point 0, cutting the sides of a given triangle in R^, R^, R^, and if a point R be taken on this transversal such that 0R~ OR^"^ ORi^ OR^' then the locus of i^ is a straight line. Let be the origin, let the triangle be the given triangle ABC, and let the transversal be px-\-qy-\-rz = 0. Since the line passes through the origin p+q+r = (1) It will be found that R^ = {orq), Rc^ = (rop), R^=(qpo). Then by the aid of (1) and la-\-m0-^ny = O, we get 1 rm — qn ^P ^rla+pny _ rmp — qny ^~ pn — rl pn — rl 3~ ql—pm ~~ ql—pm Let rm^ — qny = 6. Then 3 rm—qn pn—rl ql—pm OR" e "^""e"""*" e (q — r)l-\-(r-p)m+{p — q)n - e ' 3 (rml3 — qny) and OR (q-r)l-\-(r-p)m-\-{p-q)n {q-r)la + (r-p)mp + (p-q)ny (q-r)l + {r-p)m-h{p-q)n 120 MISCELLANEOUS THEOREMS Comparing this expression with the standard form, we '^^^^ q — r — x, r—p — y, p — q = z, and since (5' — '^) + (^ ^1^) + (p — ^) = 0, x-\-y-\-z = 0, the equation of the axis of perspective, or polar, of the given triangle. 3°. Let Q and R (fig. 37) be the isogonal and isotomic conjugates of the rational point P = (fgh). Then the ratios of the various segments of the sides of the triangle are Fig. 37. P^A mg BQ^ _ mna^gh QgZ" nlh%f BR^ _^mngh R^A^ nlhf Consequently, mg nh 9Ei P,B CQ^^ nlbVif Q,B Imcjy GR^^nl hf R^B ~ Imfg AP. nh "~lf P^G AQ,^ Imcjg Qfi mna^gh AR^^ Imfg RjJ TYingK Q, the isogonal conjugate of P, is (— 7*1-, — ^, ^t^)» | \ isotomic (gh hi fy\ \l^' m2' nV' If P be an irrational point, (7, — , - ), ^ \6 m n/ r._(^gh mf c^\^ '*^~\ I ' m ' n J' j._{gh hf fy\ (1) .(2) CHAPTER XII 121 Ex. 1. The isogonal conjugate of the symmedian point, /a2 62 c2\ . (a^h^c^ h^c^a^ cVh^\ , , , . \l m nJ \ I m n / ^ ■ the mean point. Ex. 2. The Gergonne point of the triangle is -^, -^, -^ j, the point in which concur the lines drawn from the points of contact of the three escribed circles to the opposite corners of the triangle. Ex. 3. Any two lines whose equations are of the form px + qy-^rz = 0, and p-'^x-\-q-'^y-\-r~'^z = 0, cut the sides of the triangle isotomically. Ex. 4. The Brocard points, II, (4), are isogonal con- jugates, as also are the orthocentre and circumcentre. 4°. The isogonal conjugate of every point upon the circumcircle is at infinity. Let the point be P — (pqr). Since P is on the circumcircle, mna^qr-\-nlh^rp+l7nc^pq = and p= \_^^^^2g^ ' The point P may therefore be written * / —mna^qr \ \l(nb^r+mc^qy ^' V' the isogonal conjugate of which is, (1), ^ f — (n¥r+mc^q) nb^r mc^q \ The vector of Q consequently is — (nh^r + mc^q) a + nh^r/3 + mc^qy OQ — {nb^r 4- mc^q)-{- nWr + mc^q which is infinitely long because its denominator is zero. Therefore the isogonal conjugate of every point on the circumcircle is at infinity. * See * Conventional Signs ' at the beginning of the book. 122 MISCELLANEOUS THEOREMS 5°. Pascal's Theorem. The crosses of the oppo- site sides of a hexagon inscribed in a conic are collinear (fig. 38). 6°. Brianchon's theorem. The joins of the opposite corners of a hexagon circum- scribed to a conic are con- current (fig. 38). Let ABC be the given triangle, and let D=(x^y^z^), » ^=(a'32/3^3)- The equation of the conic IS yz-\'ZX-\-xy = 0. The condition that the points D, E, F shall lie on the conic is 1 1 ^1 2/i 1 1 ^2 2/2 1 1 ^Z 2/3 Pig. 38. Let ABC be the given triangle, and let D^x^'p + y-^q-^-z^r^O, „ E=x^]p + y^q-)rZc^r==0, „ F=x^p-hy^q-i-z^r=0. The equation of the conic is p^+q^+T^- 2qr - 2rp - 2pq=0. The six points, A ... F are the points of contact of six tangents, the sides of the circumhexagon. The coordin- ates of these six tangents are a = (011), 6 = (101), c = (110); ^ = (2/i+%. Zi+x^, a;i+2/i)> e = (2/2+^2> 2;2+«'2> ^2+y2)> /=(2/3 + 2;3' ^3 + «^3> <^3 + ys)' The condition that the lines d, e, f shall touch the conic is 1 = 0. CHAPTER XII 123 If the three crosses of the opposite sides be calculated, it will be ultimately found that the condition that they shall be collinear is If the three joins of the opposite corners be calculated, it will be ultimately found that the condition that they shall be concurrent is x^ 2/i 1 1 x^ 2/2 1 1 2/3 = 0, which is the condition that the hexagon shall be in- scribed in the conic. which is the condition that the hexagon shall be circum- scribed to the conic. 7°. To express a homogeneous equation of the second degree, F(Jgh)= Uf+... + 2W'fg^Q, in terms of its derived functions, i^, Fg, F^. F(fgh)=fF,+gF,+hF„ F,= Uf+W'g+V'h F,= W'f+Vg+U'h F,= V'f+U'g+Wh.) Eliminating /, g and h from these four equations, we get F,, Fg, F„ F(fgh) 0= U, W\ r, Fy =(VW'U'^)F^... W\ V, U\ Fg J^2{WV'-W W')FyFg - A^F(fgh) ; r, U\ W, F, A^F(fgh) = A(uF/... + 2wT,Fg); AF(fgh) = „). The first of these two equations is met with in calculating the discriminant of the equation F(fgh)F(pqT)--(pF,-{-qFg+rF,f = 0, in order to verify the conclusion drawn in VII, 16°, that this is the equation of two points, not of a conic. 124 MISCELLANEOUS THEOREMS Putting F(fgh) = k, Ff=a, Fg = hy Fj, = c, we have A:( C72)2 . + 2 Tr>g) - (ay. . . + 2a6??g) = 0, the discriminant of which is kU-a\ kW'-ab, kV'-ca A= kW'-ab, kV-b\ kU'-hc kr-ca, kU'-bc, kW-c^ Four of the matrices of the third order into which this matrix resolves are zero. The determinants of the remaining four give A = ^^ A2 - ak^A {ua + w'h + v'c) - h¥A {yd a + v6 + u'c) — cl^bk. (y'a + v^h + wc) „ =BA{kA'-i^hc)} = AF\fgh) x{AF(fgh)^(F,,F„F,)} = 0. In conclusion may be quoted the opinion of M. Laissant about the Quaternion method, which seems to be applicable to Anharmonic Coordinates : " la methode d'Hamilton n'est pas d'une application universelle, non plus qu'aucune autre ; mais elle me semble presenter dans des cas nombreux de reels avantages. ... Ce serait un tort, a mon sens, de se priver de ressources nouvelles, sous pretexte que ces ressources ne sont pas d'un usage constant." * * Applications Mecaniques du CcUcul des Quaternions, Paris, 1877. INDEX (The references are to pages) {A, B, G), coordinates of centre of conic, 33. Angle, Sine of, 25. Tangent of, 16. Area of triangle, 24. Asymptotes, Local equation of, 37. Rectangular, 38. Tangential coordinates of, 58. Axis, Directive, 68. of Perspective, A\ B', G\ fig. 1, 12. Radical, 96. Brianchon's Theorem, 122. Brocard's circle, 97. ellipse, 113, Ex. 2. points, 7. Centre, Directive, 70. of conic, Coordinates of, 33. of conic. Equation of, 55. of a system of points in involution, 77, 80. Circle, Every, passes through both cyclic points, 95. General equation of, 95. Metric equation of, 50. through three points, 97. with given diameter, 17, Ex. Circles — Brocard's, 97. Circumcircle, 86, 93. In, 53, 98. Nine-points (IX), 99. Polar, 99. Circular involution, 79. Conic, Circumscribed, 43. General equation of, local, 26. General equation of, tangential, 52. Inscribed, 42, 75. Conic, Polar, 45. Ratio of segments of a finite straight line cut by a, 31. Species of, determined by the bor- dered discriminant, Z>, 33. Conies, Confocal, 104. Conjugate points and lines, 30. Coordinates, Anharmonic and Tri- linear. Relation between, 18. Anharmonic, of a line defined, 11. Anharmonic, of a point defined, 6. Tangential, of a line defined, 48. of the term of the multiple or sub- multiple of a given vector, 9. Cotes, Roger, Theorem by, 119. Cross of two straight lines, Coor- dinates of, 13. Cross of two straight lines, Equation of, 50. Cross ratio of four points on a conic, 74* Cyclic points, / and /, Coordinates of, 94. Cyclic points, / and /, Equation of, 49. Derivatives, First, of general equation of the second degree, 26. Diameters, Conjugate, of conic, 34. Directrix, 84, 117. Discriminant, A, of general equation of the second degree, 27, 53. Discriminant, Bordered, D, of general equation, 33. Division, nomographic, 66. Duality, Principle of, 48. Ellipse, Brocard's, 113, Ex. 2. Elliptic involution, 78. 126 INDEX Envelope of variable tangent cut by four fixed tangents, 74. Equation, Homogeneous, of the second degree, in terms of its derived functions, 123. Equations, Transformation of Anhar- monic and Trilinear, 18. Equations, Transformation of Local and Tangential, 51. Foci of a system of points in involu- tion, 77. Geometric meaning of the vanishing of a coefficient in the general equation of the second degree, 39. Hamilton, Sir W. R., 1, 3, 43, 51, 53, 65, 88, 107. Hyperbola, Rectangular, Condition for, 38. Hyperbolic involution, 78. Infinity, Cyclic points at, 49. Line at, Coordinates of, 49. Line at. Equation of, 12. Line at, Section of conic by, 33. Point conjugate to a point at, 72. Invariants calculated as in other methods, 92. Involution, Relation between the co- ordinates of a system in, 82. Involution, Various species of, 77. Isogonal conjugates, 120. Isotomic conjugates, 120. Join of two points. Coordinates of, 49. Laissant, M., 124. Line, straight, cut by conic, Ratio of segments of, 31. Lines, Coordinates of certain straight, 11. Lines, Coordinates of cross of two straight, 13. Lines, Equation of cross of two straight, 50. Lines, Parallel, 14. Four, cut by a transversal, Cross ratio of, 63. ?, wi, n. Conditions restricting the values of, 2. Locus of vertex of pencil with con- stant cross ratio, 74. Modulus of transformation of coor- dinates, 92. Nets, Construction of geometric, 3. Harmonic properties of, 118. Origin, Change of, 86. 02, 49. Parabola, 34. Some properties of, 41, 115. Parabolic involution, 78. Parallel lines, Relation between the coordinates of two, 14. Parallel lines, Distance between two, 23. Parallel tangents, 41. Pascal's theorem, 122. Pencil, Cross ratio of a flat, 63. Pencils, Property of two, with one common ray, 65. Perpendicular lines, Relation between the coordinates of, 17. Point, Anharmonic coordinates of a, 6. conjugate to a point at infinity, 72. Equation of a, 49. Points, Coordinates of certain, 7. Coordinates of the join of two, 49. Distance between two, 20. Rational and irrational, 1. Polar, Coordinates of, 55. Equation of, 29. circle, 99. conic, 45. Pole, Coordinates of, 55. Equation of, 29. and polar, Some properties of, 30. Quadrangle, Conic circumscribed to a, 40, 47. Quadrilateral, Conic inscribed in a, 110. Quadrilateral cut by a tranversal in involution, 82. Radical axis, 96. Row, Cross ratio of a, 61. Rows, Property of two, with a com- mon point, 66. Tangent, Equation of, to conic, 28. Length of, from a point to a circle, 102. INDEX 127 Tangents, Equation of a pair of, 36. Four fixed, cut by a variable tan- gent, 74. Parallel, 41. Tangential equation. General, of the second degree, 52. Transformation of anharmonic and trilinear equations, 18. Transformation of local and tangential equations, 51. Transformation, Modulus of, of co- ordinates, 92. Triangle, Area of a, 24. circumscribed to the inconic is inscribed in the circumconic, 75. Given, Change of, 87. Self -conjugate, 46. Vector, Coordinates of term of mul- tiple or submultiple of a, 9. Vector, Standard form of, 4. ^2 21. GLASGOW : PRINTED AT THE UNIVBRSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. UNIVERSITV OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. Th„ book . DUE on the last date stamped below. ^ANi«fwaP^^'|g^^ 13Uec56BC REC'D LD NOV 29 1956 e:d Fee 12 '69 -6 p tOAN DEPT. ttCe t989o ^Jbo UD Q£C 2 ' i9-eP« '^^C'o , ^Oig^ FEB 2 4 1984 i LD21-l00m-9.'47(A5702sl6)476 / 20854G