Digitized by tine Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/trilinearcoordinOOwhit TRILINEAR COORDINATES. erambritise: PKINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. V V TRILINEAR COORDINATES AND OTHER METHODS OP MODERN ANALYTICAL GEOMETRY OF TWO DIMENSIONS: AN ELEMENTARY TREATISE, BY THE REV. WILLIAM ALLEN WHIT WORTH, M.A. PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, LIVERPOOL, AND LATE SCHOLAR OF ST JOHN'S COLLEGE, CAMBRIDGE. MATH, PEPT* CAMBRIDGE: ' '^ '"*^- DEIGHTON, BELL, AND CO. LONDON : BELL AND DALDY. 1866. b5^ ?9 PREFACE. Modern Analytical Geometry excels the method of Des Cartes in the precision with which it deals with the Infinite and the Imaginary. So soon, therefore, as the student has become fa- miliar with the meaning of equations and the significance of their combinations, as exemplified in the simplest Cartesian treatment of Conic Sections, it seems advisable that he should at once take up the modern methods rather than apply a less suitable treatment to researches for which these methods are especially adapted. By this plan he will best obtain fixed and definite notions of what is signified by the words infinite and imaginary, and much light will be thereby thrown upon his knowledge of Algebra, while at the same time, his facility in that most important sub- ject will be greatly increased by the wonderful variety of expe- dient in the combination of algebraical equations which the methods of modern analytical geometry present, or suggest. With this view I have endeavoured, in the following pages, to make my subject intelligible to those whose knowledge of the processes of analysis may be very limited; and I have de- voted especial care to the preparation of the chapters on Infinite and Imaginary space, so as to render them suitable for those whose ideas of geometry have as yet been confined to the region of the Ileal and the Finite. W. J VI PREFACE. I have souglit to exhibit methods rather than results, — to furnish the student with the means of establishing properties for himself rather than to present him with a repertory of iso- lated propositions ready proved. Thus I have not hesitated in some cases to give a variety of investigations of the same theorem, when it seemed well so to compare different methods, and on the other hand interesting propositions have sometimes been placed among the exercises rather than inserted in the text, when they have not been required in illustration of any par- ticular process or method of proof In compiling the prolegomenon, I have derived considerable assistance from a valuable paper which Professor Tait contri- buted five years ago to the Messenger of Mathematics. My thanks are due to Professor Tait for his kindness in placing that paper at my disposal for the purposes of the present work, as well as to other friends for their trouble in revising proofs and collecting examples illustrative of my subject from Uni- versity and College Examination Papers. Liverpool, 15 September, 1866. CONTENTS. PAGE Peolegombnon. Of Determinants ix Chapter I. Of Perpendicular Coordinates referred to two axes ... 1 Exercises on Chapter I 7 Chapter II. Trilinear Coordinates. The Point 9 Exercises on Chapter II 20 Chapter III. Trilinear Coordinates. The Straight Line ............ 21 Exercises on Chapter III 34 Chapter IV. The Intersection of Straight Lines. Parallelism. Infinity 36 Exercises on Chapter IV 54 Chapter V. The Straight Line. The Equation in terms of the Perpendiculars 58 Exercises on Chapter V 69 Chapter VI. The Equations of the Straight Line in terms of the direction sines 71 Exercises on Chapter VI 92 Chapter VIL Modifications of the System of Trilinear Coordinates. Areal and TriangTilar Coordinates.. 92 Table of Formulae 96 Exercises on Chapter VII 101 Chapter VIII. Abridged Notation of the Straight Line 104 Exercises on Chapter VIII 115 Chapter IX. Imaginary Points and Straight Lines 117 Exercises on Chapter IX 130 Chapter X. Anharmonic and Harmonic Section 132 Exercises on Chapter X 145 Chapter XL Transformation of Coordinates 147 Exercises on Chapter XI 153 Chapter XII. Sections of Cones 154 Exercises on Chapter XII 163 Chapter XIII. Abridged Notation of the Second Degree...^ 165 Exercises on Chapter XIII 171 52 via CONTENTS. PAGE Chapter XIY. Conies referred to a Self-conjugate Triangle 173 Exercises on Chapter XIV 180 Chapter XV. Conies referred to an Inscribed Triangle 192 Exercises on Chapter XV 204 Chapter XVI. Conies referred to a Circumscribed Triangle 206 Exercises on Chapter XVI 219 Introduction to Chapter XVII. Notation, &c 221 Chapter XVII. The General Equation of the Second Degree 22S Exercises on Chapter XVII 253 Chapter XVIII. The General Equation of the Second Degree continued 255 Exercises on Chapter XVIII 285 Chapter XIX. Circles 287 Exercises on Chapter XIX 304 Chapter XX. Quadrilinear Coordinates 307 Exercises on Chapter XX 321 Chapter XXI. Certain Conies related to a Quadrilateral 325 Exercises on Chapter XXI 331 Chapter XXII. Tangential Coordinates. The Straight Line and Point 332 Exercises on Chapter XXII 343 Chapter XXIII. Tangential Coordinates. Conic Sections 345 Exercises on Chapter XXIII 364 Chapter XXIV. Polar Reciprocals 368 Exercises on Chapter XXIV 387 Chapter XXV. Conies determmed by Assigned Conditions 390 Exercises on Chapter XXV 400 Chapter XXVI. Equations of the Third Degree 401 Exercises on Chapter XXVI 421 Introduction to Chapter XXVII. General properties of Homo- geneous Functions 426 Chapter XXVII. The General Equation of the «* Degree 431 Exercises on Chapter XXVII 451 Miscellaneous Exercises 455 Notes on the Exercises. Results and occasional Hints 484 Index 499 ERRATA. ge 5, line 4 from bottom, for OY, OX read CY, CX. , 15, „ 9 „ for hHin read k'Hmn. 16, ,, 4 ,, for APB read APC. , 24, „ 12 „ for as, read in. , 48, ,, 3 ,, for 2Zw cos C read 2foi cos (7. , 58, „ 8 „ for Ihcrp cos B read 2carp cos B. , 73, „ 7 „ for a read a'. , 78, ,, 3 ,, f or s'm A sin B sin C read 2 sin A sin B sin , 95, „ 6 „ /o}" difference read reference. ' , 134, Art. 119, QOP is (three times) misprinted ior AOQ. QOQ is (twice) misprinted for AOP, , 202, line 7, for — 9'eacZ ^ , and read equations (1), (2), (3) as follows : 73«2 &7z 7273 « ^ /3 ^ 7 aaai a^§3 yia^ So the first determinant ought to be ■(1). ■(2). .(3). 73^2, /3372J 7273 aaai, ai^g, 7103 = 0. Page 281, Une 10, fm- '^ read 2AH PROLEGOMENON. OF DETERMINANTS. § 1. Introduction. 1. If we have m equations involving a lesser number n of un- known quantities, we may determine the unknown quantities from n of the equations, and, substituting these values in the remaining m — n equations, obtain m — n relations amongst the coefficients of the m equations. In other words, if we eliminate n quantities from m equations, there will remain m — n equations. 2. If the equations are all simple equations, the solution can always be effected and the m — n equations practically obtained. The notation of Determinants supplies the means of conveniently expressing the results of such elimination, and the study of their properties facilitates the operation of reducing the results to their sim- plest forms. 3. It must be observed, however, tha.t if the equations be homo- geneous in the unknown quantities, or, in the case of simple equa- tions, if every term of each equation involve one of the unknown quantities, the equations do not then involve the actual values of the unknown quantities at all, but only the ratios which they bear one to another. Thus the equations 3a; -t- ^y — 5z = 0, 5a3 + 5^/ - 7s = 0, ' ■ X OF DETERMINANTS. are satisfied if x, y, z are proportional to 3, 4, 5, but they do not involve any statement as to the actual values of x, y, z. In this case the number of independent magnitudes, concerning which anything is predicated in the equations, is one less than the number of u.nknown quantities involved in the equations. Thus each of the equations just instanced, involving the three unknown quanti- ties X, y, z, speaks not of the actual magnitudes of those quantities, but of their ratios one to another, which are only two independent mag- nitudes, as is immediately seen by writing the equations in the form z z 5-4- 6^ = 7, z z where the equations are exhibited as connecting the two independent X ^ ratios - and — . z z § 2. Of Determinants of the second order, 4. Def. The symbol a^ , b^ is used to express the algebraical quantity a^h^ — b^a^, and is called a determinant of the second order. The separate quantities a^, &j, a^, b^ are called the elements of the determinant, and may themselves be algebraically either simple or complex quantities. Any horizontal line of elements in a determinant is called a row, and a vertical line is called a cohimn. Thus the determinant above written has two rows a,, b^ and a,, 6g, and two columns a^, a^ and 6^, b^. 5. It follows from the definition that K K = CI'fi,j — «,&! Hence a detea'minant of the second order is not altered by chang- ing roios into columns and columns into rows. OF DETERMINANTS. 6. It follows similarly from the definition that XI «2. K »i5 ^ = {%h - afi,) = - {a A - «2^,) = «2» K Hence in a determinant of the second order the interchange of the two rows changes the sign of the determinomt. So the interchange of the two colwmns changes the sign of the deter- minmfit. 1. If ax +hy = 0, OAid a'x + h'y = 0, he two consistent equations, then will a, h a, y = 0. Multiplying the first equation by h' and the second by 5, and subtracting, we get (ah' — a'h) a? = 0, therefore ah' — a'h — 0, or a, h =0. a, h' \ 8. If ax -hby + cz = 0, arid a'x + h'y + c'z = 0, then will \. E. D. X y z b, c c, a a, h h', c c, a a, V For if we multiply the first equation by c' and the second by c, and subtract, we get {ca — ca') X + (he — h'c) 3/ = 0, (ca' - c'a) X = (be' — h'c) y, X y or or be — b'c ca — c'a ' Xll OF DETERMINANTS. which may be ■written y h, G b', c' and therefore, by symmetry, c, a c', a X y h, c G, a b', c' c', a' a, b a', b' Q. E. D. 9. If and then will h, G h', C a, b a, b' ax +by = c, a'x + b'y = c', , and y = — c, a c\ a a, b a, b' This follows from the last proposition by writing — 1 for z. § 3. Of Determino/nts of the third order. 10. Dep. The symbol a, b, c a', b', G a", b", c" is used to denote the expression a b', c -b a', G + c a, b' b", c" a", c" a", b" and is called a determinant of the third order. 11. // a'od ax +by + cz = 0, a'x + b'y + c'z — 0, a"x + b"y + g'z = 0, OF DETERMINANTS. xm then will a, h, c a, h\ c 0. a , , c For the second and third equations give, by Art. 8, X y z K c' G, a' «; h K g" c", a" a". h" Substituting these values in the first equation, we get 6', c h", c" c, a' I + c I a, h' It ti \ ir -III c , a I \ a , h', c -h a, c + c a', h' h", c" a", c" a", h" -0, = 0, a, h, G a', h', G a'\ h" c" = 0. Q. E. D. 12. In the foregoing proposition we eliminated the two ratios X '. y : z from the three given equations, and found the result in the form of a determinant. We might have proceeded otherwise as follows : Multiplying the three equations by X, jx, v (at present undeter- mined multipliers) and adding, we get {a\ + a' II + a"v) x + (6X + V [i + h"v) y + {gX + g'ij. + c"v) z = 0, which must be true for all values of X, fx, v. Now by Art. 8 we know that if \ /X V b', h" h", h b, b c\ c" c". G c, d XIV OF DETERMINANTS. then the coefficients of y and z in the last equation will vanish, and the equation will reduce to {ok + a'/x + d'v) 33 = 0, so that we must have ak + «'/x. + d'v — 0, or substituting the values of A. : /x : v a 6', h" + a b", b + a" b, b' c, c" c", c C, G or a, a y a = 0, h, h\ h" c, c, c" which is ther( }fore the resn It of the elir uina tion. = 0, But this result must be equivalent to the result obtained by the other method. Hence the two equations a, h, c = and a, h', c a" y, c" a, a, a" h, h\ b" c, c\ g" must be identical, and therefore their first members must either be identical or difier only by a constant multiplier. But the coefficient of the term ab'c' in each is seen to be + 1. Hence the two deter- minants are identical, or a, a, a" b, b\ b" c, c', c" a, b, c = a', v, g' a", h", c" 13. Cor. A determinant of the third order is not affected by changing the rows into columns and the columns into rows. Care must, however, be taken that the first column becomes the first row, the second column the second row, and so on, and vice versd. OF DETERMINANTS. XV 14. Since the result of the elimination is tlie same in whatever order the equations be taken, it follows that the resulting equation a, b, c =0 a, b', c a", b" , c" is not altered in whatever order the rows of the determinant be written. Hence the determinants a, h, c J a, h\ c a", b", c" a'. b", c" a, h', c a, b, G a", b", c" . a\ b\ c" , &c. a, b, G a, b', G can only differ by some numerical multipliers, and since the coefficient of every term in the expansion of each of them is either + 1 or — 1, they can therefore only differ by the algebraical sign of the whole. a, ^, c ) a", h", c" a\ v, c Since a, b, c a, b', c' a", b" , g" = a b', c -b a, G + G a, b' h", g" a", g" a", b" and a, h, G = a c, b' -b o', a' + G K a' o!', h", g" o", b" c", a" h", a" a', b', g' it follows from Art. 6 that a, b, c II -III II a , b , c a, b', g' are of opposite algebraical sign. Hence the sign of a determinant of the third order is changed by interchanging its last two rows. It will be seen on examination that the effect is the same if we change any other two adjacent rows, or two adjacent columns. a, h, G and a, h\ C a", h", c" XVI OF DETERMINANTS. That is, the sign of the determinant is changed when any two adjacent rows are interchanged, or when any two adjacent columns are interchanged. 15. But any derangement whatever of the rows or columns may be made by a series of transpositions of adjacent rows or columns. Such a derangement will or will not affect the sign of the deter- minant according as it requires an odd or an even number of transpo- sitions of adjacent rows or columns to effect it, thus a, b, c = - «', b ', c' = a ', I ', c a'', b", c" a', b', G a, b, c a", b", c" a', b', G a", b", c" a", b", c" a, b, G a, b, G ■ a", c", b" a', c', b' a, c, b ^^ c", a", b" c', a', b' c, a, b c", b", a" c', b', a' G, b, a = &c. an Simil y othe arly w r dera e may Qgeme ascer nt of 1 bain tt bhe co^ le sign umns of the determinant formed by or rows. 16. If a row or a column of a determinant be multiplied through- out by any number, the value of the determinant is multiplied by the same number. For the determinant may be deranged till the row or column in question becomes the first row. Now + /AC j a', h' I a", b" fjca, jxb, fic = ixa b', c' + ixb c', a' a', b', c' b", g" c", a" a", b", c" = IX a, b, c a', U, c a", b", g" which proves the proposition. 17. If two rows of the determinant be identical, or if two columns be identical, the determinant vanishes. OF DETERMINANTS. XVll For tlie rows and columns may be deranged until the last two rows are identical, and the determinant takes the form ziz a, a' h, h', h', c c' c' g' =f=b a', d ±c a\ b' c' a', g' a, y = ±a = 0, by Art. 4. Therefore, &c. Q. E. D, 18. Cor. If one row be a multiple of another row or one column of another column, the determinant vanishes. For the mul- tiplier may be divided out by Art. 16. 19. To shew that a + x, b + y, c + , a, b' , g' a, b", c" a, b, c a, b', G a", b", g" + \ X, ?/, z a, b', G a", b", c" Expanding the first determinant, it takes the form {a + x) h', G - {b + y) a, G ! + (.+.) a ', b' b", g" a , c \ \ a , b a b', G b", g" -b a', G 1 "'" ^ a, b' a", b" + x b', G -y «'; c + z a\ b' b", c" a", c" a", b" a, b, c + X, y, z a, b', G a, 6', c' a", b " c" c i', b", g" Therefore, &c. Q. E. d. XV 111 OF DETERMINANTS. Similarly, a + a, b, c a + a, h', c a + a , , c a, h, c a, b', c a", b", c" a, b, G a, b', c a", b', c' And so if each element of an^ column or row be divided into two parts, the original determinant is equal to the sum of the two deter- minants formed by substituting for each divided element first one of its parts and then the other. But note that this operation cannot always be performed at once on more than one column or row. Conversely, if a series of determinants are identical except as regards one column or one row in each, their sum is equal to the new determinant formed by retaining in their places the rows or columns that are identical, and adding together the corresponding elements of the row or column which differs. 20. If any row of a determinant be increased by multi'ples of any other rows, or if any column be increased by multi'ples of any other columns, the value of the determinant is not altered. For, by Art. 19, a, b, c a', b', c a", b", c" a, b, c a', b', c' a + mb + nc, b, c a' + mb' + nc, b', c a" + mb" + nc", b", c" + mb, b, c mb', b', c mb", b", c" , by Art. 18, nc. h, c nc. y, c' nc", h", c" a", b", c" which proves the proposition. This theorem is of the greatest use in reducing determinants. OF DETEKMINANTS. XIX 21. // and then will ax +hy + cz + du = 0, ax +h'y + c'z + d'u — 0, a'x + h"y + c"z + d"u = 0, X h, G, d h', c\ d' h", 0", d" a, G, d a', c, d' a", g", d" a, b, d a', K d' a", h", d" a, h, G a', v, c' a", h", g" For if we multiply the first equation by I c, c" j d', d" , the second by G, C d, d" the third by c, c d, d' , and add, we get a, a', a" x + c, c, g" d, d\ d" h, h', h" y + G, g', g" d, d', d" G, C, C G, c', g" d, d', d" z + d, d\ d" G, c, c" d, d' , d" or, in virtue of Art. 17, a, a , a G, G, G d, d', d" a, c, d a', c , d' a", g", d" x + h, h', h" c, c , g" d, d\ d" 2/=0, x-\- h, c, d h' , c, d' , G , d -y 2/ = 0, h, c, d h\ c', d' b", o", d" a, c, d a, c\ d' a", c", d" u = 0, and similarly the other equations may be established. XX OF DETERMINANTS. 22. // and then ivill ax +hy + cz =d, a'x + h'y + cz = d', a"x + h"y + c"z = d", h, c, d b', c, d' X^ h", c", d" a, b, c a, b', d a", b", c" y = c, a, d a, d! c", a", d" a, h, c a, b', c a!', b", c" a, h, d a, b', d' a", b", d" a, b, c a, b', c a", b", c" This follows from tlie last proposition by writing — 1 for u. [It will be observed tliat these values of x, y, z obtained by solving the three simultaneous equations might have been written down by the method of cross multiplication in Algebra.] § 4. Of Determinants of the fourth order. 23. Def. The symbol a, b, c, d a', b', c', d' a", b", c", d" a'", b'", c"\ d'" is used to denote the expression + c b', o', d' -b a', c', d' b", c", d" a", c", d' U", c"\ d!" a'", c'", d a', b', d' -d a", b", d" a'", b'", d'" a', b', c a", b", c" a'", b'", c'" and is called a determinant of the fonrth order. OF DETERMINANTS. XXI 24. If then will ax +hy +CZ + du = 0, a'x + h'y + cz + d'u = 0, d'x + y'y + cz + d'u = 0, d"x + 6'"3/ + d"z + cr'w = 0, a, 6, c, c? a', 6', c , c?' a , , c , d lf> Til) /// 7/A a ^ ^ c ^ d = 0. For the second, third, and fourth equations give X -y z -u b', c', d' b", c", d" h'", o'", d'" a, c, d' a". c", d" a", c", d'" a', h', d' a", h", d" a"',h"', d" d, h' , c a", h", c" /// 7 /// /ff a ,o,c Substituting these values in the first equation, we get b', c', d' -b b", c". d" b'", c"\ d'" a, c, d' a , c , d f// frf -If, a , c , d + c a', b', d' // 7 If 7// a , , d a , , d -d a, b, c " Iff ff a , , c a, b, c, d = 0. Q. E. D. a, b', c, d' a", b", c", d" "f T,'" J" J'" a , , G , a Precisely as in the case of the determinant of the third order (Art. 13), we may shew that the value of a determinant of the fourth order is not affected by changing the rows into columns and the columns into rows. So the results obtained in Arts. 14—20, wUl be seen to depend upon general principles, and to hold for determinants of the fouith oi'der. W. c xxu OF DETERMINANTS. 25. // cmd then will ax +b9/ +CZ +du + ev = 0, ax + h'y + cz + d'u + e'v = 0, ax + h"y + cz + d"u -v e"v — 0, a:"x + h"'y+c"z+d"'u+e"'v=0, -y ^ I 6, c', d", e" ri a, c , d", e" \ \ a, h\ d", e" \ \ a, h', c", e" ~ I a, h', c", d'" I ' where ] a, h\ c", d'" \ denotes the determinant a, 6, c, d a, V, c, d' " T." " J" a , , c , a /// 7/// /// T/// a , , c , d For if we multiply the four equations respectively by c', C, c' } c, c, c, J c, c, c » c, c, c d", d\ d" d", d" d" d', d', d' d', d', d' e"\ e", e'" «'", e'", e" e'", e'", e" e", e", e" and add, we obtain a, a, a , a Cy C y C y C dy d\ d'\ d*' f It ttt X + whence X h, c, d, e V, c, d\ e h", c", d", e" V", c", d"\ e" h, V, h", b" d, d', d", d"' Gy 6 J C y 6 ~y y = o. a, c, d, e a', c, d', e «", c", d", e" a", c", d"\ e" and similarly the other equations may be established. OF DETEEMINANTS. XXIU § 5. Of Determinants of the n^^ order. 2%. The student will have observed that the reasoning of the last two sections is perfectly general. He will have recognised the law by which a determinant of any order is defined with reference to those of the next lower order, and he will have perceived that the proofs of properties of determinants of the 3rd order given at length in § 3 will apply mutatis mutandis to establish corresponding properties for determinants of any order whatever, if they can be assumed to hold for the next lower order. It follows, therefore, by the principle of mathematical induction, that all those properties may be attributed to any determinant whatever. - 27. Def. If we strike out one of the columns and one of the rows of a determinant of the w'^ order, we shall obtain a new deter- minant of the [n— If^ order, which is called the minor of the original determinant with respect to that element which was common to the column and row. Thus the minor of the determinant a, h, c, d a', b', c, d' a", b", c", d" /// 7 /// /// 7/// a , , c , a *fec. with respect to the element c" is the determinant a, 5, d ... a', h' , d' ... a'", h"\ d'" .. 28. Def. The element which is common to the p^^ column and q^ row of a determinant is said to occupy a positive or negative place according as p + q h even or odd. ' c2 XXIV OF DETERMINANTS. 29. TJie coefficient of any element of a dete7'minant is the minor with respect to that element affected with the sign + or — according as the element occupies a positive or negative place. Let the determinant be ■ ' ' a, h, a\ h', .... .. : : &c. and let the element x occupy ttie q^ place in tlie p^^ column. By making q—1 transpositions of adjacent rows and p—1 trans- positions of adjacent columns, the determinant may be written (-1)"+'-^'! X ! : «j b, ■■■ : (i'l b', ■■■ I ; : : ^^c. whence we see that the coefficient of x is a, h, ... a', b', ... '■ [ &c. the sign being positive or negative according as ^ -f g is even or odd, which proves the proposition. § 6. Of Determinatits of unequal columns and rows. 30. Su.ppose we have to eliminate n unknown quantities from a greater number m of simple equations. As we shewed in Art. 1, we shall obtain m — n equations of relation amongst the coefficients. One method of writing down these m — n equations would be to take the first n of the given equations, and associate with them in order each of the remaining m — n equations. From each of the m — n groups of w + 1 equations thus form- ed, we might eliminate the n unknown quantities. Expressing OF DETEEMINANTS. XXV the result by means of a determinant of the {n+iy^ order. We should thus have m — n determinants of the (n+lf^ order each equated to zero, constituting the m — n equations sought. But it is plain we might have taken any m — n combinations whatever oi n + 1 equations that could have been formed out of the m equations, provided we took care to introduce all the original equations. Thus there will always be a variety of forms in which the result of such elimination may be expressed. 31. Suppose, for instance, that we have to eliminate the two ratios x \y \ z from the five equations Oj^x + h^y + c^z = 0, »2^ + ^4!j + c^' = *^' a^ + h^j + c^ =0, ttrpo + h^y + c^z = 0. As the result of the elimination we shall have the three equations «,, ^2, «3 = 0, K h, K «!' C2, «3 ^1, ^2^ h^ Cl, C2» (^^7 = 0, ^o K h^ 0, or for any one of these equations we may substitute any other equa- tion obtained by elimination from a different set of three equations, such as a, , a, , fti Kl K ^4 c.. c. 0, or a^, a^, a^ = 0. But it is convenient to express the result of this elimination briefly by the form a^, ^2, «3, Oli, ttg ^, *2, &3» ^4> h ^i> ^2} ^3' ^iy ^5 = 0, XXVI OF DETERMINANTS. where the determinant of five cohjbmns and three rows indicates that we may select any three of the five columns to form a square determinant and equate it to zero, and tlie trifle vertical lines indicate that three such equations 'may he independently for'med. 32. The example given iu the last paragraph will suffice to sug- gest to the reader the interpretation of any unequal determinant. In most general terms the definition will stand as follows : The compound symbol ttj, a^, ^3 ... a, fO^, «2, ^3 h = 0, where the number of the quantities a, h, c.lc iB n (less than m) and the number of vertical lines bounding the determinant is r, is to he understood as expressing a system of r independent equatio7is ohtained by equating to zero r several determinants each formed hy taking n of the rows of the given unequal determinant. It will be observed, that if the system, expresses the result of the elimination of n quantities out of m independent equations, we must have r = m — n. The notation is, however, found convenient in cases when the original equations, altliough m in number, are only equiva- lent to some lesser number m' of independent equations. In such a case we shall have r = m'—n. 33. As an examjjle of the case last considered, suppose we have to eliminate the two ratios x : y : z from the four equations {a ~ a ) X + (h - b' ) y + (c - c ) z = 0, (a — a" )x + (J) —b")y + (c — c" ) z=0, [a" — a") x+ (h" - b'") y + (c" — c") z = 0, {a!" — a )x-\-(b"'—b )y + {c"' — c )z — 0. These ai'e equivalent to only tliree independent equations, since any one of thorn may be obtained from the others by simple addition. OP DETERMINANTS. XXVll There will therefore be only one resulting equation, which, may be obtained by elimiaating from any three of the given equations. If, however, it be desired to have a result recognising symmetrically all the four equations, we may write it a — a, a— a, a —a , a —a h-h', b'-h", h"-h"', h"'-h I I n It III III c — c, c —c , c — c , c — c = 0. § 7. Examvples. Example A. To evaluate the determinant a^, a, 1 ^^ A 1 Subtracting the third row from each of the others, the determi- nant becomes a'-y', a-y, («-7)(^-y) a — y , a — y = (a-y)(^-y)(a-P).. a + y, 1 ^ + y, 1 Example A'. By a similar method we may shew that a^ a, 1 =(a-y)(/3-y)(a-^)(a + ^ + y). Example B. To shew thai x + a, x + b, x + c =0. y + a, y + b, y + c z + a, z + b, z + c xxvm OF DETERMINANTS. If a = 5 or b = c or c = a, two columns become identical and the determinant vanishes. Therefore if the determinant be not identi- cally zero, a — b, b — c, c — a are factors. Similarly, if a? = 2/ or y = z or z = x, two rows become identical, and so y — z, z — x, x — y are factors. But the determinant is only of the 3rd order and cannot have more than three factors. Hence it must vanish identically. Q. E. D. Example C. To evaluate the determinant JL/u JU ~~ Oj %JU ~~' (J _ z — b, z — a, z Subtracting the first column from each of the others, we get X, —c, —b y — c, c, c — a z — b, b~ a, b whence we see that the coefficient of x is c, c — a , or ab + ac— a^. b — a, b By symmetry, the coefficients of y and z must be respectively bc-k-ab — ¥ and ca + bc— = ^.. m. ^3' ^3 ^3. mj if,= '^a' h > ^^> ^3, m. ^' ^l K, m^ 1 »*2' h ^3=1 ^1. »*, I K, ^2 then we have l^L^ + rn^Jf^ + »^^i\^^ - ?,Z^ + m^if, + n^N^ = ^3X3 + m3if3 + n^N h^ ^h^ '^2 h^ »^3' '^3 XXX So also OF DETERMINANTS. \L^ + m^M^ + n^N_^ = And so l^L^ + m^M^ + n^N^ = 0, and four other like relations. And similarly, m^Z, + mjj^ + mjj^ = 0, and five other like relations. Example F. To shew that = 0. M., N. -h h m,, n^ K^ K h, ^2> ^2 h, ^3 J % This may either be proved by multiplying out, or as follows. Suppose the three equations l^x + m^y + n^z = 0. (1), l^x + m^y+n^z=^ (2), l.^x + m^y^n^z = ^ (3), coexist, (3) and (1) by the elimination of x gives us i)f,2/+^.^=0 (4). Similarly (1) and (2) gives us M,y+N^z=^ (5). The condition that (4) and (5) coexist must be identical with the condition of coexistence of (1), (2), (3), that is, the equation must be identical with the equation 0. OF DETEEMINANTS. XXXI And therefore tliese two determinants can only differ by a con- stant multiplier, which is determined by compaidng the coefficient of Wg in each, and found to be l^, so that ^%, K =h h, m^, n^ ^s, ^s h, "*2 5 '\ h^ 77*3, '\ E. D. Example G-, To shew that A, ^^,, N. = { A, ^.> ^. ^3, K, ■^3 We have A. -3^x. ^1 4. ^3, ^3 ^a» »^a' »*3 1, M^, J^^ ;,byEx. E, 0, J^L, N.^ h^ »*3: »^3 -K{ ^2» »*3' ^^2 ^3' «*3' ^^3 ^, by the last example. xxxn C )F DI ]TEE MINANTS. therefore ' K ^.^ N. = { l^, m,, n^ A. M,, K h' ^2^ % A, ^3, ^3 h^ «*3» ^3 Q. E. D. Example H. To eliminate xfrom the two equations ax^ + bx+ c = aV + h'x + c'=0 MultiiDlying each equation by x throughout, we get ax^ + hx^ + cx = aV + h'x^ + cx = Q •(1), .(2). •(3), •(4), and eliminating x^, x^, x from the four equations (1), (2), (3), (4), we have = 0, 0, a, h, c 0, a\ h', c' a, by c, a', h\ c', a, c 2 _ a, h h, c a', c' a', V K d Ex. J. If uxx' + vyi/ + wzz' + u' {yz' + y'z) + v' {zx' + z'x) + w' {xy' + x'y) he zero for all values of x, y, z, then will uvw — uu'^ — vv'^ — ww'^ + 2u'v'w' = 0. Eor since the given expression vanishes for all values of x, y, z, the coefficients of x, y, z must severally vanish. Therefore ux! + w'y' + v'^ = 0, w'xl + vy' + wV = 0, v'x' -f u'y' + wz' - 0, XXXlll ). E. D. OF DETEEMINANTS. and eliminating a;' : y" : z', we have u, lo', v' = 0, w', V, v! v', u', w or uvw — uu'^ — vv'^ — ww'^ + 2u'v'w' = 0. Example K. To expand the expression ax + ly, c'x + ny, h'x + m'y c"x + fi'y, hx + my, a'x + l'y h"x + 'm"y, a"x + l"y, ex + ny according to powers of x and y. Putting 2/ = 0, "we obtain tiie term involving x^, viz. a, c', b' c", h, a' h", a'\ c So putting £0=0, we obtain the term I, n', r>i' n", m, V m", l'\ n Suppose the y, y, y in the three columns distinguished by suflSxes, so that the determinant becomes ax +ly^, c'x+n'y^, h'x + m'y^ =0. c"x-irn"y^, hx -\-my^, a'x-vVy.^ h"x+ m"y^, a"x + l"y^, ex + ny^ Putting 2/2=0 and y^ ~ 0, we obtain for the term involving x'^y^, I, c\ h' n", h, a' m", a", c x-y^. XSxiv OF DETEEMINANTS. Similarly, we find the terms a, n', h' c", m, a' K I", c x^y and a, c , m c", b, I' h'\ a", n ^Vs Hence the whole coefficient of x^y is I, c', h' n", b, a m", a", c a, n', V c", m, a' b", I", c a, G , 7)1 c", 6, I' b", a", n Similarly the coefficient of xy^ is I, n', b' n", m, a' m", I", c a, n', m' + I, c', ml c", m, V n'\ b, V b", r, n m", a", n Hence we arrive at the result, that ax + ly, c'x + n'y, b'x + m'y g"x + n"y, bx + my, a'x + I'y b"x + 7)11' y^ al'x + l"y, ex + 7iy a, c', b' c", b a' + { + { I, c', b' n", b a' ml', a", c a, n', ml c", m, I' b", I", n the expansion required. b , a , c a, n', V d', m, a' b", I", G + I I, c', m' n", b, I' m", a", n I, n' ml n", m, I' m", I", n a, c', 1)1 c", b, V b", a" n I, n', b' n", m, a' m", I", c }«V ]xy^ OP DETERMINANTS. XXXV Example L, As a particular case of the last theorem consider the determinant U + KO?, W' + Kob, v' + KttC w' + KttJ, TJ + k6^, u' + lii>G v' + Kac, u' + kLc, w + KC^ Expanding in powers of k, we obtain u, w', v' w\ V, v! v', u, w + K { a^, w', v' + u, ah, v' w', h^, u' V, bo, w the coefficients of k and k vanishing since each determinant therein contains at least two identical columns. But further, the coefficient of k may be written 0, a, b, c a, u, w', v' b, u/, V, u' c, v', u', w a\ w', v' + ab, V, u ac, u', w u, w', ac vo\ Vy be % u', c' Hence the whole determinant becomes 1, 0, 0, a, u, w', v' 0, w', V, u' c, v\ u', w 0, a, b c a, u, w', v' 6, w' , V, v! c, v', u', w — , a, b, G K a, u, w' V b, w', V, u' c, v' . u', w XXXVl OF DETERMINANTS. Example M. To eliminate h and hfrom the equations tt = ha^ + A;w^ 2w' = h {a' -b'-c') + 2kvw, v=h}f + kv\ 2^ = A {If - c' - a') + 2^^w, w = h(? + hn\ 2m/ = A (c' - a' - 5') + 2^wv. where u=u + v' + V)', v = v + w' + u', W = 'W + U' + v'. ' The resulting equations will be all the independent equations included in the system a^ 6^ c^ «^-6^-c^ 6^-c^-a^ c'-a'-b' =0. w^, i?, ^o^ 2vw, 2wu, 2uv u, V, w, 2u', 2v\ 2^' But the first column increased by half the sum of the fifth and sixth gives us (it + V + w) w Vj or, dividing by v, throughout, u-^-v-Vw 1 •which, being perfectly symmetrica], might have been equally obtained by combining the second, fourth and sixth, or the third, fourth and fifth columns. This shews that the original equations were not independent, but equivalent to only four equations and the result consists therefore of the two equations a\ h\ c\ u^, v^, y?, u + v + w u, V, w, 1 = 0. [Exercises on Determinants will be found on pages 455, 456.] TRILINEAE COORDINATES AND OTHER METHODS OP MODERN ANALYTICAL GEOMETRY. CHAPTER I. OF PERPENDICULAE COORDINATES REFERRED TO TWO AXES. Before proceeding to explain the method of trilinear coordi- nates, "we will introduce a system which may be regarded as a connecting link between the usual Cartesian system, and the trilinear system : — it is the method of perpendicular coordinates referred to two oblique axes or lines of reference. 1", In the ordinary system of oblique coordinates, the position of any point P with reference to a pair of axes GX, CY, is de- termined by the lengths of two lines FM, PN measured parallel to eitlier axis to meet the other : that is, if x, y be the oblique coordinates of P referred to GX, GT, then x = PM, and y = PN. It is obvious that the position of the point would be equally well determined, if it were agreed to take as coordinates the perpendicular distances of the points from each axis, instead of measuring the distance from each axis parallel to the other. Thus, if we let fall the perpendiculars PM' and PN' on GB and GA, we observe that PM' and PN' might be used, as lawfully w. 1 XXXVl OF DETERMINANTS. Example M. To eliminate h and hfrom the equations u = ha^ + ku^, 2u' = h {a^ - 6* - c^ + 2Jcvw, v=hb^ + hv% 2v' = A (6^ - c' - o?) + IhJou, w = hc^ +hw^, 2k/ = h {c^ - a^ - b^) + 2kuv . where u = u + v' + io', v = v + w' + u', tv = w + u' + v'. The resulting equations will be all the independent equations included in the system a\ h\ c\ a'-¥-G\ V'-c'-a^ c'-a'-h' =0. u^, v", vfy 2vw, 2wu, 2uv u, V, w, 2u', 2v', 2w' But the first column increased by half the sum of the fifth and sixth gives us (m + V + w;) M u or, dividing by u throughout, u + v + w 1 ■which, being perfectly symmetrical, might have been equally obtained by combining the second, fourth and sixth, or the third, fourth and fifth columns. This shews that the original equations were not independent, but equivalent to only four equations and the result consists therefore of the two equations a\ h\ c\ u , v^, w~, u + v + w U, V, w, 1 = 0. [Exercises on Determinants will be found on pages 455, 456.1 TRILINEAR COORDINATES AND OTHER METHODS OF MODERN ANALYTICAL GEOMETRY. CHAPTER I. OF PERPENDICULAE COORDINATES REFERRED TO TWO AXES. Before proceeding to explain the method of trilinear coordi- nates, "we will introduce a system which may be regarded as a connecting link between the usual Cartesian system, and the trilinear system : — it is the method of perpendicular coordinates referred to two oblique axes or lines of reference. 1", In the ordinary system of oblique coordinates, the position of any point P with reference to a pair of axes CX, GY, is de- termined by the lengths of two lines PM, PN measured parallel to eitlier axis to meet the other : that is, if x, y be the oblique coordinates of P referred to OX, CF, then x = PM, and y = PN. It is obvious that the position of the point would be equally well determined, if it were agreed to take as coordinates the perpendicular distances of the points from each axis, instead of measuring the distance from each axis parallel to the other. Thus, if we let fall the perpendiculars PM' and PN' on GB and GA, we observe that PM' and PN' might be used, as lawfully w. 1 2 OF PERPENDICULAR COORDINATES as PM and PN, to determine tlie position of P, provided it be specified beforehand wliich system of measm^ement is intended. We shall speak of PJ/and PN Si,s the oblique coordinates, and P3I' and PN' as the perpendicular coordinates of the point P referred to the same axes CX and CY. We shall use a and /9 to denote perpendicular coordinates, reserving x and 3/ to denote, as usual, the oblique coordinates. Thus for the point P x^PM, y = PN: a = PM', ^ = PN'. When the axes are rectangular these two systems of mea- surement will indeed coincide, but in the case we have intro- duced of oblique axes they will be distinct. Fiff. 1. The same convention as to the algebraical signs of the coor- dinates will hold equally in the perpendicular as in the oblique coordinates. Thus we shall consider as positive the distances from CX of all points lying on the same side with Y, and con- sequently the distances of all points on the opposite side will be negative. So also the positive side of CY will be that on which X lies, and the negative side the side remote from X. EEFEREED TO TWO AXES. 3 2. If the angle XGYhe denoted by G, then we have PM' . ^ .FN' . ^ -p^ = sm (J and -^ = sm U. Hence if a, /3 be the perpendicular coordinates of any point whose oblique coordinates are x, y, we shall have a = aj sin G and ^ = ysmG; or x= a cosec G and ?/ = /3 cosec G. Consequently, if we have any relation holding good between the oblique coordinates of all points on a locus, we can, by the substitution of x = a cosec G, y = ^ cosec G, obtain a relation holding good between the perpendicular coor- dinates of the same locus. In other words, we may transform by this substitution the oblique equation of any locus into an equation in perpendicular coordinates rej)resenting the same locus. For example, the equation in oblique coordinates to the straight line AB cutting off intercepts GA — h and CB= a from the axes is known to be X ti .- + •^ = 1. Q a Hence the equation to the same line in perpendicular coordinates will be a cosec /3 cosec (7 _ a or aa + 5/3 = ah sin G; and in a similar way any other equation might be transformed. 3. Or, instead of taking an equation, we might by the same substitution transform any function lohatever of the oblique coor- dinates into an equivalent function of the perpendicular coordi- 1—2 4 OF PEEPENDICULAR COORDINATES nates. For example, writing tlie equation to tlie same straight line AB in the form ah — ax — "by = 0, we can at once write down the expression for the perpendicular distance of anj point {x , y) from it, viz. oh — ax — hit . ™ + , ^ sm G. sjcb + If — 2ah cos G Hence if a', yS' he the perpendicular coordinates of the same point, we shall have as the expression for the distance ah sin G— aoi'-hB' + — • ~ Va^+&''-2a5cos(7' or if c denote the distance AB, and A the area of the triangle ABG so that c^ = a^ + 5^ — 2a5 cos (7, and 2A = ah sin G, then the expression (for the perpendicular distance from AB of the point whose coordinates are a and ^') becomes , 2A-aa'-5/3' ~ c ' This is given here merely as an example of transformation of coordinates from the one system to the other, but the result is one which will be seen hereafter to have an important bearing on trilinear coordinates. 4. The student will do well to examine at this stage of the subject the interpretation of some of the simpler equations con- necting a and yS. (1) Consider the equation a = 0. It is evidently satisfied by all points on the line CFand by no other : it is therefore the equation to this axis. REFERRED TO TWO AXES. 5 (2) Consider the equation a = £?, where J is a constant. It is equally obvious that this is satis- fied at any point on a line parallel to GB at a distance d from it on the side towards X. Similarly the equation a = ~d is satisfied at any point on a parallel line at an equal distance on the other side oi CY. (3) Consider the equation a = /3. The equation speaks to us of points whose perpendicular distance from GX is equal iG the perpendicular distance from GY, and of the same algebraical sign. The locus of such points is the interior bisector of the angle XGY, which is therefore the locus of the equation. (4) Consider the equation This speaks of points whose perpendicular distances from the axes are equal, but of opposite sign. These points will be seen to lie on the exterior bisector of the angle XGY, which will therefore be the locus of the equation. (5) Consider the equation a = ml3. Suppose P to be a point on this line, join PG and draw PM' , PN' perpendiculars on Y, OX, then (by the equation) PM' = m. PN', , , . PM' PN' and therefore -tttt = *^ • -jrry > sin PGY=m.&mPGX, 6 OF PERPENDICULAR COORDINATES or (in "words), Plies on a straight line dividing the angle XOT into two parts PGY, PGX such that their sines are in the ratio m : 1. This straight line will therefore be the loons of the equation. 5. To find the area of a triangle the perpendicular coordi- nates of lohose angular points are given with respect to a pair of oblique axes. Tig. 2. C Q' F R' AX Let PQB be the triangle and (a^, /3J, {a^, /3^, {a^, /Sg) the perpendicular coordinates of the angular points P, Q, R referred to the oblique axes GX, GY. Parallel to GY draw PP', QQ', BR to meet GX in P', Then A PQR = trapezium PQ QF + trapezium PRR 'P' — trapezium QRR'Q. But the trapezium PQQ'P' stands on a base F Q' equal to («! — ocj cosec G and it has a mean altitude equal to \ {/3^ + /3J. EICFEERED TO TWO AXES. Hence its area = - cosec C {^^ + /3J (oCj — aj . Similarly, area PRR'F = ~ cosec G {^^ + /3J (a^ — aj, and area QUE Q = -^ cosec G (/S^ + ^^ {a^ - aj ; therefore the area of the triangle PQR = Jcosec(7J(^,+ ^J(a,-a,) + (^3+/3J(a3~aJ-(/3,+/93)(a3-aJ I cosec G |a, (/3, - /33) + a, (^3 - /3J + a^ (/3, - /3,)| , 2 or with determinant notation = - cosec G 2 «!, «2' «3 A, A„ /Sa 1, 1, 1 Exercises on Chapter I. (1) If {a, /3) he the perpendicular coordinates of tlie point whose oblique coordinates (referred to the same axes) are {x, y), and {a, /8') the perpendicular coordinates of the point whose oblique coordinates are {x, y'), shew that (a + /ca', /3 + /c/3') will be the perpendicular coordinates of the point whose oblique coordinates are (x + kx, y + «;?/'). (2) Shew that the points whose perpendicular coordinates are (a, ^), (a', /3'), (a -a, yS-/3') and (a + a', /3 + /S') lie all on one straight line, provided a/S' = a'j5. 8 EXEECISES ON CHAPTER I. (3) Find tlie equation in perpendicular coordinates to the straight line drawn from the origin at right angles to the straight line CX. (4) Write down the equation to the other diagonal of the parallelogram two of whose sides are the axes, and one of whose diagonals has the equation aa+h^ = ah sin C. What is the area of this parallelogram ? (5) The distance between two points in terms of their oblique coordinates {x, y) and (cc', 3/') is given by the formula p^ = {x-xf-\- {y-yY-\-2 {x-x) {y-y) cos G; hence write down the corresponding formula for perpendicular coordinates. (6) Find the area of the triangle one of whose angles is at the origin, and the other two at the points whose perpendicular coordinates are (a, y8) and (a', /3'). (7) Find the area of a triangle whose base is of length d in the axis GX, and whose vertex is at the point whose perpen- dicular coordinates are (a, /3). (8) Find the perpendicular coordinates of the point bisect- ing the straight line joining the points whose coordinates are (a, /S) and (a', /S'). (9) Find the equation in perpendicular coordinates to the straight line drawn from the origin, at right angles to the straight line whose equation is aa + h^ = ah sin G. CHAPTEE II. TEILINEAR COORDINATES. — THE POINT. 6. We will renew our construction before proceeding to the next step in the development of the system of trilinear coordinates. Let BG, GA, AB he any three fixed straight lines forming a triangle, and let P denote some point in the plane of the triangle. We have seen that if the perpendicular distances of P from any two fixed straight lines (CB and GA for instance) be given the point P is determinate, and that these distances may be regarded as the coordinates of P. 10 TEILINEAR COORDINATES. Now let a, (3, y denote perpendicular distances of F from the three fixed straight lines BG, CA, AB. Then, as we have seen, any two of these (e.g. a, /3) may he regarded as the coordinates of P with respect to the corresponding axes {BG, GA), and the remaining perpendicular (7) may be expressed in terms of these two (a and /3), and known constants, by the method of the ex- ample, Art. 3 (or more simply, as we shall shew presently). But there are advantages, as the sequel will shew, in regard- ing all the three perpendiculars as coordinates of P, and thus expressing P's position at once with respect to the three fixed axes, or lines of reference (as it is more usual to call them). We shall regard as positive the distances from any line of reference to points on the same side with the intersection of the other two lines of reference, and consequently the distances of points on the other side will be negative ; thus A lies on the positive side of BG, B GA, G AB. Thus all points within the triangle formed by the lines of refer- ence (which we shall briefly call the triangle of reference) have aZ? their coordinates positive, and points without the triangle of reference have either one or two coordinates negative. It will be observed that it is impossible to find a point on the negative side of all the lines of reference, that is, no point has all its three coordinates negative. 7. We have observed that when two of the trilinear coor- dinates of a point are given the third may be calculated. We proceed now to investigate a simple equation, which we shall find connects these three coordinates. Let P be any point whose coordinates are a, /3, 7 and join FA, FB, FG, and draw FF, FE, FF perpendiculars on BG, GA, AB respectively. THE POINT. 11 First, suppose P toitliin the triangle ABC as in fig. 3 ; then the triangles FBG-\- PGA + P^P = whole triangle ABG. But BG.PD or aa is the double of the area of the triangle PBG: so J/3 and 07 are the doubles of the triangles PGA and PAB) .-. aa+5/3 + C7=2A, where A denotes the area of the triangle of reference. Secondly, suppose P without the triangle. Let it be on the negative side of PC and on the positive sides of GA, AB, as in figure 4. Fig. 4. In this case the triangles PGA + PAB -PBG= whole triangle ABG. But in this case a is negative, so that the length PP = — a ; and therefore — a-x represents the double of the area of the tri- angle PBG. Hence, as before, aa + h/3 + cy = 2A. 12 TEILINEAE COOEDINATES. Again, suppose P on the negative side of both BG and AB, as in fig. 5. Fig. 5. In this case the triangles PGA - PAB -PBG = the triangle ABG. But in this case both a and 7 are negative, and therefore — aa and — cy represent the double areas of the triangles PBG and PAB. Hence, as before, aa + J/S + C7 = 2A. Hence we see that if a, /3, 7 be the coordinates of any point whatever, they are connected by the relation aa. + hl3 + 07 = 2 A. 8. We might have deduced the result just obtained at once from the result in Art. 3, taking care to determine the ambigu- ous sign so that the perpendicular from the origin G should be positive in accordance with the convention of Art. 6. THE POINT. 13 Thus we should have written at once 2A - aa - hB ry = , or aa + h^ + cy = 2A, but the proof given in Art. 7 is more simple in its character. 9. Let p be the radius of the circle circumscribing the triangle ABC. Then by trigonometry, a h c s,inA &\nB sin C ^P) hence the equation obtained in Art. 7 may be written a sin ^ + /3 sin 5 + 7 sin 0= —= S suppose. P 10. The equation of Art. 7, or the equivalent form just obtained plays a very important part in trilinear coordinates. It enables us to make every equation involving a, /3, 7 homoge- neous, for, since 2i±||±^=l(Art.7), we are at liberty to multiply any term we please in an equation by the fraction :-—r , thus raising by unity the order of the term. By repeating this operation we can raise every term of an equation up to the same order as the term of highest order, and thus render our equation homogeneous. For example, if we have the equation a' + 3a7 + 5/3 = 1, 14 TRILINEAR COORDINATES. we can raise every term to the third order: thus we get the homogeneous equation a^ + 3a7^^i^||±^ + 5^ ^ +^^f + ^^) aa. + h^ + cj 2A which we might proceed to simplify. 11. If the ratios of the coordinates of any point be given, the point is determinate, and the actual values of the coordi- nates can be found by means of the relation aa + h^ + cy= 2 A. We may proceed thus : Let the coordinates be proportional to X : /jl : v, then a _/3_ 7 A, fA, V ^ and therefore each of these ratios _aa + h^ + cy 2A Hence aX + hfjb -\- cv a\ + h/jb+ cv' 2XA O'^ ^ 7 5 aA, + 6/i + cv /3= 2/^^ 7 = 2yA oX + hiM + cv 12. We may however observe that in practice we very rarely require the absolute values of the coordhiates. For advantage is almost universally taken of the principle detailed in Art. 10, by means of which our equations in trilinear coordi- THE POINT. 15 nates are homogeneous. And it is scarcely necessary to point out that a homogeneous equation in a, /3, 7 will not involve in any way the actual values of the quantities, but will only in- volve their ratios. For example, may be written ^7/ V7/ 7 a B . where only the ratios - and — are involved. 7 7 Again, if we have to substitute the coordinates of a point in a homogeneous equation it is not necessary to know more than the ratios of the coordinates. For suppose the equation were a/e7-3a'/3+5yQV + 7' = 0, and suppose the coordinates of the point were known to be pro- portional to I : m : n. The actual values of the coordinates may be supposed to be kl, hm, kn, but it is not necessary generally to know the value of the multiplier k; for if we substitute kl, km, kn in the given equation, we get knJ^ 2,kH'm + 5k%i\ + kSt" = 0, or, dividing by F throughout, Imn — ^ISn + bm^n + r^ = 0, so k disappears from the final result, and therefore a knowledge of its value was unnecessary. We shall conclude this chapter with some examples in which we shall determine the coordinates of several points re- lated to the triangle of reference, leading to results which are continually required in the solution of problems. 16 TEILINEAE COORDINATES. 13. To find the coordinates of the angular points of the triangle of reference. For the point A it is evident that /9 = and 7=0. Also acL = 2A, hence we can write down the coordinates oiA, 2A a ' 0, 0. So, of B, 0, 2A 0, and of G, 0, 0, 2A e N.B. The angular points of the triangle of reference are conveniently spoken of as " the points of reference." 14. To find the coordinates of the middle point q/'BC. Fiff. 6. Let P be the middle point, and suppose a, /3, 7 the coordi- nates of P. Since P lies on 5(7 we have a = 0. Also IjB = twice the triangle APB = the triangle ABC, since APB, APC on equal bases and of the same altitude are equal. THE POINT Therefore ^=i and so A Hence the coordinates are 0, A A b ' c' 17 15. To find the coordinates of the foot of the perpendicular from A upon BC. JjQtAA' be the perpendicular and let a, /3, 7 be the coordinates of A'. Draw A'H, A'K perpendicular to GA, AB respectively ; then a=:0, ^ = A'H, r^ = A'K. But —rr-r = COS AA'H= COS ; A A ' .-. A'H=A'AcobC; /D 2A „ or /S = — cos (7 : a so 7 = — cos B. a Hence the coordinates are ^ 2A ^ 2A 0, — cos 6, — cos B. a a 16. To find the coordinates of the centre of the inscribed circle of the triangle of reference. This point is equally distant from the three lines of re- ference ; •'• a = ^ = 7; a _^ _y _aa + h^+ cy _ 2A 111 a + b + c a + b + c' 18 TKILINEAE COOKDINATES. Hence each of the coordinates is — , where s denotes, as in s Trigonometry, half the sum of the sides of the triangle. 17. To find the coordinates of the centre of the circle cir- cumscrihing the triangle of reference. Fig. 7. Let he the centre, a, /3, 7 its coordinates ; join OB, OCand draw OP perpendicular on BC; then (Euclid, iii. 2), BG \b bisected in P. Hence the triangles OPB, OPG are equal in all respects. Now the angle BOG 2ii centre = twice angle BA G at circum- ference. Hence ^BOP=^BOG = A; OP BP = cot^, or OP =BP cot A, So i.e. a = - cot A. ^ = - cot 5, 7 = - cot C, which give the required coordinates. THE POINT. 19 CoE, If p be the radius of the circumscribed circle, these coordinates may be expressed thus : a=pcos^, ^ = pcosB, 7=pcosC7. 18. To find the coordinates of tJie jpoint which divides in a given ratio the straight line joining two points whose coordinates are given. Fiff. 8. Let P„ P^ be the given points and (a^, /9„ 7J, (a^, 13^, 7^ their coordinates, m : n the given ratio. Suppose P the point required, and let (a, /3, 7) be its required coordinates ; then P^P : PP^ =:m : n. Draw PD, P^D^, ^J^^ perpendiculars on BG, and through Pand Pj draw PH and P^Z" parallel to BO, and meeting P^D^ and PD respectively in H and K. Then by similar triangles PK : P,Zr= PP, : P,P = m : n, i. e. a — «! : Kg — cc = m : ?2, whence wa — na^ = ma^ — ma ; or [m + n) a = ma^ -{-na^-, or a = m + n 2—2 20 TRILINEAR COORDINATES. m +n Similarly we may shew that , my, + ny, and 7 = -^-^ , m -f n which give the required coordinates. Cor. The coordinates of the middle point between {a^ , yS, , 7,) and {a„ ^^, 7,) are «i + ^2 ^, + ^s 7i+72 2 ' 2 ' 2 ■ Exercises on Chapter II. (10) Find the coordinates of the points of trisection of the sides of the triangle of reference. (11) If A' be the middle point in the side BC of the triangle of reference ABG, find the coordinates of a point Pin AA' , such that^P=2^'P. .-K^f'-L^ , c^x^i^^ - <^L^^3) (12) If A A be the perpendicular from the point of reference A upon the opposite side BC, find the coordinates of a point P, so dividing this line that AP : PA = cos J. : cos B . cos G. (13) Find the coordinates of the centres of the circles escribed to the triangle of reference. (14) Render the following system of equations in trilinear coordinates homogeneous : {Ja^ + mW + nc') 0? - AMaa + 4A'^ = {la^ + ml/ + no') fi' - AAmb/3 + iA'm = {la^ + mV + nc^) rf - 4 Anc7 + 4A'w. CHAPTER III. TRILINEAR COOEDINATES. THE STRAIGHT LINE. 19. To find the area of a triangle, the trilinear coordinates of whose angular points are given. Let (a^, /3^, 7J, {a^, ^^, 72J 73 "a, A? 73 a perfectly symmetrical expression for the area of the triangle. 20. To find the condition that three points whose trilinear coordinates are given should lie on one straight line. Let (a,, /3,, 7J, (a^, /3„ 7,), {a^, /3^, %) be the three points. That they should lie on a straight line is the same thing as that the area of the triangle formed by them should be zero. Hence, by the last article, the condition is = 0. «1, A, 7i "a' /32, 72 «3, /^a, 73 21. It follows from Art. 20 that the equation a, /3, 71=0 ax, A, 7i «a, ^2^ 72 speaks to us of a variable point (a, ^, 7) which lies on one straight line with the points (otj, ySj^, 7J, {a^, /Sg, 7^. It is manifest that the equation will be satisfied if (a, /3, 7) denote any point whatever on this straight line : and that it cannot be satisfied if (a, /3, 7) lie elsewhere. THE STRAIGHT LINE. Hence the equation a, /S, 7 =0 «!» A. 7l is the equation of the straight line joining the points If L, M, N be equal to, or 'proportional to the minors 23 /3x, 7, A' 72 7i, a, of the above determinant the equation becomes La + M^ + Ny = 0. CoE. Every straight line is represented by a homogeneous equation of the first order in a, /3, 7. We proceed to shew- that the converse of this proposition is also true. 22. Every liomogeneous equation of the first order represents a straight line. Let loL + mjB +ny = be any homogeneous equation of the first order in a, ^, 7. It shall represent a straight line. By giving 7 any value (7j suppose) in the system of equa- tions la + m/3 -I- W7 = 0, aa + 5/3 + 07 = 2A, we shall get corresponding values (a^, (3^ suppose) for a and /?. Thus we can find coordinates (a^ , /S, , 7J representing a point upon the locus of the given equation. Similarly by giving 7 another value (73 suppose) we can find the coordinates (a^, ^^, «. «. A /^a, 7. 72 > «2 0^2 , A where (a,, A; 7i)j (<^2> A) 'i^y denote points upon the line, or we may say that they are proportional to ap, hq, cr, where p, q, r are the perpendicular distances of the line from the points of reference. THE STKAIGHT LINE. 27 25. It should be noticed that the equation , /3, 7 2, A' 72 = will not be altered if we substitute for a^, /3j, and 7^, or for ofg, /Sg, and 7^5 ^^7 quantities proportional to them. For this is only equivalent to multiplying the equation throughout by a fixed ratio. Hence it is not necessary, in order to form the equation to the straight line joining two points, to know the actual co- ordinates of the points, but it will suffice if the ratios of the coordinates are given. Thus if two points be determined by the equations - = ^ = 1 and ^> = — = -,, K fJb V the equation to the straight line joining them will be a, /3, 7 X, //., V -\ ' ' ' A, , fl , V 26. To find the condition that three straight lines lohose equations are given should pass through one point. Let la + W2/3 + n If m n\ V l\ m' are equal, therefore each of them is equal to aa + h^ + cy a, h, I, m, n I , m , n or 2A a, h, c I, m, n l\ on'y n Hence a = 2A m, n m, 71 a, h, C I, m, n V , m\ n' with similar expressions for ^ and 7. 35. To find the condition that two straight lines whose equations are given may he jyarallel. Let ?a + w/3 + ny = 0, Ho. + rri^ + w'7 = 0, be the two given equations. If the two lines are parallel their point of intersection lies at an infinite distance from the triangle of reference. Hence the common denominator in the expressions for the coordinates of the point of intersection, obtained in Art. 34, must be zero. That is a, 5, c ?, m, n l\ m, n' = 0. 38 THE INTERSECTION OP STEAIGHT LINES. 36. To interpret the equation aa + b/3 + cy = 0. We shall shew first that the locus of this equation includes no point other than at infinity; and, secondly, that it includes every point at infinity. In order to find the coordinates of points on the locus of any given equation, we have to determine values for a, /3, j, which will satisfy both the given equation and the perpetual relation In the present case the two equations which have to be com- bined are inconsistent for all finite values of the variables. For, if a, yS, 7 are finite, we get, by subtraction, = 2A, which is contrary to our original hypothesis. But looking at the equations a little more generally, and remembering that a, /S, 7 may have infinite values, it appears that the result of the subtraction ought strictly to be written 0.a + 0./3 + 0.7 = 2A, an equation which requires that one or more of tlie variables a, y8, 7 should be infinite. And from considering either of the original equations, we observe that tivo at least of these variables must be infinite, since if only one were infinite, we should have aa + h/3 + cy = CO . But it may be asked, how can the equation «a + Z>/3 + 07 = be satisfied by points anywhere situate, since we know by the geometrical construction, Art. 7, that if a, y8, 7 are the coordi- nates of any point whatever, aa + h/3 + cy will rejoresent the double of the area of the original triangle? True. But when we take any point in the plane of the tri- angle of reference to represent (a, /3, 7), we necessarily take it at some finite distance or other from the triangle. We can make PARALLELISM. INFINITY. 39 tins distance as great as we please, but we can never actually make it infinite. So when we say that the equation aa + 5/3 + C7 = represents a locus lying altogether at infinity, we are not contra- dicting, but rather asserting this fact. For to say that every point upon the locus lies at infinity is in fact saying that no point can he found or drawn which shall satisfy the equation. But the statement further implies the following: that al- though no finite point can be found to satisfy the equation la. + wi/3 + ^7 = when Z, m, n are proportional to a, 5, c, yet when the ratios of /, m, n difier from those of a, 5, c by the least possible difierence, then such points can be found ; and by making the difierence as small as we please, the locus will recede as far as we please from the points of reference. This is exactly the meaning which is attached to the term "infinity" in Algebra, where (for instance) the statement -+- + - + &c. to an infinite number of terms = 1 does not mean that any number of terms which we can actually take will amount to unity, but that by taking as many terms as we please, we can make the sum as near unity as we please. But, secondly, any point lying at an infinite distance from the triangle of reference may be regarded as lying upon this locus. For, let X be any point at an infinite distance, and let P. be any finite point, then we can conceive a straight line joining PX, and by Art. 21, Cor. it will have an equation of the form ?a + ?n;S + W7 = ( 1 ) . Now let Q be another finite point not in the straight line PX, and let the equation to QX be ?'a + w'/3 + w'7 = (2). 40 THE INTERSECTION OF STRAIGHT LINES. Then since PX and QX intersect at infinity tliej are pa- rallel, and therefore their equations must satisfy the condition investigated in Art. 35, i. e. a, h, c =0. I, m, n V ' I I , m , n But this equation expresses the condition that the three equations a2+h^ +cy =0, la + w^/3 +n 72 72 > «2 «2, A respectively, then will g, - «, _ ^1 - A _ 7.-7. _ 1 For 6, c &, c = M,N c, a K L a, b Z, M 2A therefore and similarly 2A 0, -c, h «!» A, % «2. Aj 7. 0, -c, a., A, 2A a^, A» 2A = 2A(a,-aJj a, — a„ ^, c M,N A -A .. 7,-7, c, a a, ^>, N, L L, M PAEALLELISM. INFINITY. 45 42. By comparing these relations with the result of Article 40, it is seen that the equation to the straight line through the point (a^, ^1, 7j parallel to the straight line joining the points (as>/52,7j, and (a,, ^3, 73) IS a, ^, 7 Ca-^a, ^,-^3, 7^-73 43. To find the distance hetween two joints whose triUnear coordinates are given. Fig. 13. Let P, Q be the two points whose given coordinates are («!? /^i5 7i)j (02 J A J 72)? a^^ Ist P ^6 ^^6 distance hetween them. On PQ as diameter describe a circle, and in it draw QA' , QB' parallel to GB, GA. Join PA\ PB\ A'B' and through A' draw a diameter A'X. Join XB', then A'B" =FA" + PB" - 2PA'. PB' cos A 'PB' = PA" + PB"+2PA'.PB'co&G (1). 46 THE INTERSECTION OF STRAIGHT LINES. But the angles A'XB\ A'QB' in the same segment are equal ; .-. aA'XB'=aC, and .-. A'B' = A'Xsin (7= PQ sin G = psin C; also PA' = of^ — Kg) and PB' = /3^ - /S^. Substituting these values oi A'B', PA', PB' in (1), we get p^ sin^ G={a,- a J + (/3, - /3,)^ + 2 (a^ - «,) (^, - /3,) cos a Similarly we have p^ sin^ ^ = (A - ^.)-^ + (7, - 7.)' + 2 (/3, - A) (7, " 7.) cos A, p" sin' B={y^-ry) + {a^- a^f + 2 (7^ - 7,) (a, - a,) cos B. Thus we have three expressions for the required distance, each of them symmetrical with respect to two of the coordinates of the given points. By combining these expressions in various ways, among themselves or with the identity a (a, - ffa) + J (/3i - ^2) + c (7, - 72) = 0, we might obtain a variety of expressions for the distance, sym- metrical with respect to all the three coordinates of each point. Several such expressions will be found in Chapter VI. 44. To find the distance hetween the two points whose coor- dinates are (a^, ^^, 7^), {p.^, ^^, 7^) in a form symmetrical with resjtect to the determinants A. 7. 7l, «! "i. A ^2, r. J 7.> «. ' «2» ^2 Let L, M, N denote these determinants, then retaining the notation of the last article, the required distance is given by p" sin'^ = [13, - ^,Y + (7, - 7,)' + 2 (^. - A) (7, - 7.) cos A. PAEALLELISM. INFINITY. 47 But by Art. 41, c, a N, L a, b L, M 2A therefore 4Ay sin'^ = c, a 2 + a, h 2 + 2 c, a a, h N, L L, M N, L L, M cos^ = a' {U + M^ + JV' - 2MN cos A - 2NL cos B - 2LM cos C], or remembering tliat 2A sin A = Sa, (Art. 9) p'' = ^AL' + M' + N'-2MNcosA-2NLcobB-2LMcobC], p = ^^[L'+ 11'+ N'-2MNco& A -2NL cos B-2LMcosC}. 45. To find the perpendicular distance of the point whose coordinates are (a, /8, 7) from the straight line joining the two points whose coordinates are (a^, yS^, 7 J and (a^, /3.^, 7J. Let^ be the perpendicular distance required, and p the dis- tance between the last two points, then pp = twice area of the triangle formed by joining the three points ; .'.p = -. (this double area), and therefore in virtue of Arts. 19 and 44, a, ^, 7 a 5 /3,, 7i P - s/[I^'+ M'+ N-'- 2MNcos A - 2NL cosB-2LM cos C\ ^ Ta + Mi3 + Ny ^/{L'+ M^+ N"- 2MNco& A - 2NL cos B - 2LM cos C] ' an expression for the perpendicular required. 48 THE INTERSECTION OF STEAIGHT LINES. 46. To find an expression for the perpendicular distance of the point (a, yS', f^') from the straight line whose equation is la + m^ + ^7 = 0. Let (a^, y8j, 7j, (c^, /S^, yj be two points on the given line, and let L, M, N denote the determinants A, 7. 7i, «i «„ A ^,, 72 J 72^ «2 5 a2, /5, Then by the last article the required perpendicular is given by La' + M^' + Ny P~ ^[L' + M' + N'- 2iliiVcos A - 2NL cos B - 2 LM cos C } * But the equation to the straight line joining (c^, ^,, 7,), (a^j /^a' 72) may be written La + if/3 + Nry = 0, which must therefore be identical with the given equation la + m^ + ^7 = 0. Hence L_M^N I m n ' in virtue of which the expression for the perpendicular becomes la + m^ + n = V{^^ + in^ + n^ — "^"nm cos A — 2nl cos B — 2hn cos G\ Other methods of arriving at this result will be found in Chapters V. and vi. Note. The expression l^ ^-rr^ ■\-r^ — 2mn cos A — 2nl cos B- 2ln cos C is of such frequent occurrence that it will be convenient to denote it briefly by the symbol [I, on, nY. PARALLELISM. INFINITY. 49 47. To find the inclination to the lines of reference of the straight line whose equation is la + m^ + ny = (1). Let be tlie inclination of the given line to tlie line of reference BC. And let a = A;/3 (2), be the equation to tlie parallel straight line through C. Then is the inclination of this line to BG, and therefore by Art. 4, (5), sin 0=Jc sin {0-$) = h (sin (7 cos ^ — cos G sin 6), {l+h cos G) BinO = h sin G cos 9, h sin G tan^ = 1 + Zj cos (7 But since (1) and (2) are parallel, we have (Art. 35) 1, -k, I, m, n a, h, c or (mc — hn) = {na — lc)k; therefore substituting in (3), .(3). tan^ = {mc — hn) sin G {na — Ic) + {mc — bn) cos G sin G mc — hn c ' m cos G + 71 cos B—l m sin G — n sin B m cos + w cos B—l' Similarly if ^ and -v/r are the inclinations of the same line to GA and AB, we shall have tan^ = sin ^ — Z sin G n cos A-^l cos G ■ ■m I sm B — m sm A tan -vlr = -z ^ — ■ -; . ^ I cos B + m cos A—n w. 50 THE INTERSECTION OF STRAIGHT LINES. 48. To find the tangent of the angle between the two straight lines represented hg the equations lot. + m^ + W7 = 0, la + m^' + ni = 0. Let 6, 6' be their inclinations to the line of reference BC. Then if D denote the required angle between the straight lines, "vve have T^ / /I /1/N tan 6 — tan 6' tan Z) = + tan (9 -6') = + — ^- — 3? / ^ - 1 + tan ^ tan ^ (m sin C-n sin £) (m'cos C+n cos B—r)—(m' sin C—n' sin B) (m cos G+ n cos B—l) (m cosC+n cosB —I) (m'cos C+n cos B~l') - (rn sin G-n sin B) (m'sin C—n' sin B) l{m' sinC —n' sinB)+m{n' sinA—V sinC)+nil' sinB—m' mi.A) ll'+ mm'+ nn'— {mn+ m'n) cos A — {nl'+ n'l) cos B — {lm'+ I'm) cos (7 ' or (as we may write it), I, m, n r, m, n' sin^, sin 5, sin C I ll'-¥ mm'+ nn — (mn+ m'n) cos J. — {nl'+ n'l) cosB— (lm+ I'm) cos (7 ' 49. Cor. 1. The straight lines whose equations are la + m^ + ny = 0, ' and I'ci. + m'/S + n'y = 0, are at right angles to one another provided W + mm' + nn — {mn + m'n) cos A — [nT + n'l) cos B — ijm' + I'm) cos (7=0. Cor. 2. If the equation U(x + v^^ + wy" + 2m'/S7 + Ivy a + 2ty'ayQ = represent two straight lines, they will he at right angles ^provided M + V + w - 2u' cos ^ — 2y' cos B - 2w' cos (7=0. PAEALLELISM. INFINITY. 51 50. Obs, We shall in the course of the work give several other methods of finding the expression for the angle between two lines whose equations are given. The method in the fore- going article is generally thought to be the most convenient; but the student is recommended not to pass over, simply because they lead only to results already obtained, those other methods which we shall give, but to read them as very suggestive exam- ples of the application of trilinear coordinates. The methods given in Chapters V. and Yi. in particular are offered as very good illustrations of the use which may be made of those forms of equations which it is the special object of those two chapters to develop. 51. To determine the sines of the angles of a triangle the trilinear coordinates of whose angular points are given. Let P, Q, R be the angular points of the triangle and their coordinates. Then therefore PQ .PR Bin P= 2 area PQR ; 2 area P^i? Bin P = PQ.PR =^S tta' /^g' 73 I {L„M,,N,]{L,,M„N,} (Arts. 19, 44) ; where L„ ~ h^ /^a. 73 , M,= 73 » ^3 , N,^ «3' A A, 7. 7,, «! a.,, A A, 7i , M,= 7n «! , iV3 = «!, A /5a, 7. 72. «2 «a, A 4—2 52 THE INTEESECTION OF STRAIGHT LINES. But Hence a3» ^8J 73 «! /3x 7, «i-a2> ^i-^2J 7i-72 ai-a^j ^i-^3» 7i-73 2A a 2 A, ^„ 7x 0, ^i-A. 7i-7a 0, A -^3, 7i-73 A -^2' 7i-72 A-^8> 7i-73 1 1 c, a 2Aa ^3, A > c, a ^., A * a, 5 4, ^3 a, h A, M. (Art. 41) 2A a, 5, c 1 - s 4. ^3, ^3 sin^, sin 5, sin C N. sinP= sin A, sin B, sin (7 4» ^2. ^. 14, IT,, i\rj|i;^, jf^, i^T}' and similar expressions may Ibe written down for sin Q, and sin R. 52. To find the sine of the angle between two straight lines whose equations are given. Let the given equations be h + ni^ + ny = 0, la + in ^ + w'7 = 0. PARALLELISM. INFINITY. 53 Let («! , /3i , 7i) denote the point of intersection of these two lines, and let [a^, /3^^ obtained in the last article might have been deduced from the expression for tan D obtained in Art. 48 ; but the process of squaring and adding the numera- tor and denominator of that expression and resolving the result into its factors would have been tedious, so that it is perhaps more convenient to investigate the sine and tangent indepen- dently. 54 EXERCISES ON CHAPTER IV. From a comparison of the results of Arts. 48 and 52 we can immediately write down the expression for the cosine of the angle, viz. cos-D U'+mm-\- nn'— {mn'+m'n) cos J.— {nl'+n'l) cob B—{h7i+Tm) cos G ~ [l, m, n] [I', m, n] Exercises on Chapter IV. (24) Find the coordinates of the point of intersection of the two straight lines whose equations are a = 7 cos B, ^ = 7 cos A ; and find the equation of the straight line joining this point with the point of reference G. (25) If the sides QB, BP, PQ of a triangle PQB be repre- sented respectively hy the equations 7)1/3 + ny — 2?a = 0, ny + loL — 2m/3 = 0, loL + »w/3 — 2n7 = 0; find the equations to all the straight lines joining the points P, Q^ B with the points of reference. (26) Shew that the straight lines (a + f?) a + (& + (7)y8 + C7 = 0, and [a + d)oi. + {b ~ d) jB + cy = Q, are at right angles to each other. EXERCISES ON CHAPTER IV. 55 (27) Shew that the straight lines a sin 5 + yS sin (5 - G) + 7sin C cos G= 0, acos5 + /Scos(5- 0) +7sin'(7 =0, are parallel, and that each is parallel to the straight line a sin {A— (7) + /3 sin yl + 7 sin (7 cos (7 = 0. (28) Shew that the equations a cosec A-\- ^ cosec B = 0, a cos A+ ^ cos B — y cos C = 0, represent parallel straight lines. (29) Find the condition that the straight line la + m^ + W7 = may be parallel to the side BG oi the triangle of reference. (30) Find the condition that the straight line loL + m^ + ^7 = may be parallel to the bisector of the angle A of the triangle of reference. ',- (31) Shew that the straight lines whose equations are a + 7 cos B =0, yS + 7 cos ^ = 0, are parallel. (32) Find the angle between the straight lines whose equa- tions are a — 7 cos B = 0, /8 — 7 cos ^ = 0. (33) The perpendiculars from the middle points of the sides of the triangle of reference are given by the equations /8 sin 5 - 7 sin + a sin {B - G) = 0, y sin G — asm A + ^ sm {G — A) = 0, a sin A— ^ sin B+j sin {A — B) = 0. 56 EXERCISES ON CHAPTER IV. (34) Straight lines are drawn from the angular points of the triangle of reference so as to pass through the point given by la = m/3 = ny, and so as to meet the opposite sides in the points A', B', C: find the equations to the sides of the triangle A'B' C. (35) Find the equations to the sides of the triangle whose angular points are given Tby (a = 0, and/3+ ly =0), (j8 = 0, and 7 + ma = 0), (7 = 0, and a + n/3 = 0), respectively. (36) If be the centre of the circle circumscribing the triangle of reference, amd. if AO, BO, GO he produced to meet the opposite sides in A'B'C, shew that three of the four straight lines represented by the equations a sec J. + iS sec jB + 7 sec (7 = are the sides of the triangle A'B'C; and construct the fourth straight line. (37) Draw the four straight lines represented by the equations a cos ^ ± /3 cos 5 + 7 cos (7 = 0. (38) Draw the four straight lines represented by the equations a ± yS ± 7 = 0. (39) Interpret the equations a sin -4 + /S sin 5 + 7 sin (7 = 0. (40) Of the four straight lines whose equations are la ± w/3 ± ny = EXERCISES ON CHAPTER IV. 57 two intersect in P, and the other two in P; two intersect in Q, and the other two in Q'; two intersect in B, and the other two in H'; find the coordinates of the middle points of FF, QQ', BB! ; and shew that they lie on one straight line. And find the equation to this straight line. (41) On the three sides of a triangle ^5 C triangles FBG, QCA, BAB are described so that the angles QAC, BAB are equal, the angles BBA, FBG are equal, and the angles FOB, QCA are equal; prove that the straight lines, AF, BQ, CB pass through one point. (42) Shew that the point determined hy aa _ h^ _ cq _^_ 2 A I m n {I, m, n] ' 64 and THE STRAIGHT LINE. a£^ _'bq_cr__ 2A V ~ m'~ n ^ ~ [11, m', n] ' But if D be the angle "between tlie lines, we have by the last article therefore sin 2) = ^i^^=2A 2A J), q, r t I I 1, 1, 1 {Z, ?«, w}{Z', m', w'} 2A ahc {?, w, w}{Z', 7?i', w'} I in n a' b' ~c 7/ » / i m n a' T' "c 1, 1, 1 I, m, n I , m, n a, h, c h m, n l\ m\ n sin^, sin B, sin G [I, m, n] [I', m, n] the same expression which we otherwise obtained in Art. 52. 62. To find the altitude of the triangle whose base is given by the equation apa. + bq^ + cr r v\ ?'. r' ^ apa + hq^ 4- cry /'> ?", r" ^- 2A 1, 1, 1 /' 4^ > r P\ / 1 q, r p", II II q , r 1, 1, 1 1, ] I, 1 p^ S'' r p', q, r /'' ^"' r" •1 II II p , q , r 64. To find the area of the same triangle. We have only to express half the rectangle contained hy the base and the altitude. Therefore by the last article area = A p, q, r I 1 1 p, 2, r II II II p , q , r 2 p, q, r p, q, r 1, 1, 1 1 1 ] 3, q, r II II It ^ , q , r [, 1, 1 ih q, »• 1, 1, 1 65. Cor. The expression just obtained is homogeneous with respect to p, q, r, and of zero dimensions, hence it will not be altered if we substitute for p, q^ r any quantities pro- portional to them. Now suppose that the equation to the base, instead of being given in the form apa + hq^ + cr > c, a a, h or (which is the same thing), where X, /a, v satisfy the relations aK^-l)fi + cv = 0, and Z\ + m^ ■\-nv=0. IN TERMS OF THE DIRECTION SINES. 73 67. It follows that if a', ^', 7 be the coordinates of any point, the system of equations a — a' _ ^ — ^' _ 7 - 7' \ fJb V will represent a straight line provided a\ + h^ + cv = 0, and if the equation to the same straight line in the ordinary form be la. + m/3 + W7 = 0, the ratios I : m : n will be determined by the equations \l + fim + vn = 0, a I + /S'm + y'n = ; that iSj the equation in the ordinary form will be a* ^, 7 a, ^, 7 = 0. 68. We proceed to obtain the equations to a straight line in the form a — a' _ y8 — /3' _ 7 — 7' without reference- to the ordinary form. Let OP be the straight line whose equations are to be found, and let a', y8', 7' be the coordinates of the fixed point 0, and a, /3, 7 those of any point P upon the straight line : and let p be the distance between these two points. It will be observed that p like a, y8, 7 is a variable quan- tity dependent upon the position of P. Through draw Oa, Oh, Oc parallel to the lines of refer- ence BG, CA, AB respectively, and so that the angles hOo, 74 THE EQUATIONS OP THE STRAIGHT LINE cOa, aOh may be the supplements of the angles A, B, G respectively. Fig. 17. Let 6, <}>, ■ylr denote the angles POa, POh, P Oc all measured In the same direction from the initial line OP. then Draw the perpendiculars PA', PB', PC, PA' a -a' sin^ = OP Also, due regard heing had to the algebraical signs, and sin^ = sin ^fr = p 7-7 IN TEEMS OP THE DIRECTION SINES. 75 Hence, if X, jx, v be proportional to the sines of the angles 6, ^, yjr, we have o: - a' _ ^-^ _ y-y ' \ fJb V relations among the coordinates (a, /S, 7^ of any point on the given line, and therefore representing the given line. And further, if X, fx, v be not onlj proportional but actually equal to the sines of the angles 6, <^, i/r, we may write a. — a _ ^ — _ 7-7' _ where p is the distance between the points (a, ^, 7) and («', ^', 7')- 69. Def. The sines of the angles which any straight line makes with the three straight lines of reference may conveni- ently be termed the direction sines of the straight line. 70., To find the relations among the direction sines of any straight line. Let \ = sin ^, /* = sin , v= sin i/r be the direction sines of any straight line, then (Art. 68), j>-e = ir-C, and '\lr — ^ = 7r — A. Hence we have sin ^ = — sin ( + ^), and sin t/t = sin {A — (f>). Consequently we may write \ = -sm{G+(f)),'] fi — sin <^, } V = sin [A — , ^' be the angles which the straight lines make with the line of reference GA, Then referring to Art. 70, we have sin <}> = fl, and sin {A—) = v, or sin A cos (f) — cos -4 sin ^ = v, sin A cos (j) — /M cos A = v. Therefore and sin A cos j> = v-\- fx cos A, , v + LL cos A cos = V- — ^ . ^ sm^ Similarly, we have sin 0' = fjb, ., v' -i- it' cos ^ cos d) = V- — 5 . Bin A Now if i) denote the required angle between the given straight lines therefore smZ> = sm (<^~(^ ) = + -^ 4— -| sin -4 = + fiv — fl^ sin J. .(1). Or we may write it sinZ> = 1 /i, V 1 V, X 1 \, fl sin J. ~" sin B V , A. ~ sin C \, fl So also n /JL ^'\ {v + fi cos A) {v + fl ^os A) cos D — cos (d> — 6) = ^^ Af-i — + ^^ ^ ' smM fifi + vv + ijiv + f^v) cos A fifi sin'J. w. •(2), 6 82 THE EQUATIONS OF THE STRAIGHT LINE and "by symmetry _ vv' + W + jvX' + v"^) cos JB _ W' + /jL/ji' + {X/jb + \'/ji) COS C ~ sin^C 76. To find the angle between the straight lines a -a' = t = ^ -i \ V a -dt /3 -/8' _ry -7 where X, /*, y a?icZ X', //,', y' are not equal hut only 'proportional to the direction sines. The expressions of the last article were obtained on the sup- position that X, [X, V and X', /*', v were actual direction sines. But if we could reduce them to a form in which they would be of zero dimensions in X, /*, v and also in X', yu,', v\ they would still be true when these quantities are only proportional to the direction sines. Now we have (Art. 71) _ V/i''* + v^ + 'iiJbv cos A _ slv^ + X^ + ^v\ cos B sin A sm B VX^ + A<-^ + 2X//. cos ~G and similar expressions connecting X', yu.', v\ in virtue of which the results of the last article may be written sin D = (/>.'-/.V)5in^ ^ ^^^ V (^'^ + Z^^ + 2yU,y cos A) (/a" + I^" + 2yaV COS ^) ■ V (|U,' + i^' + 2/iv COS A) (/Li" + z;" + 2/aV' COS ^) IN TEEMS OF THE DIRECTION SINES. 83 whence tanZ) {[jbv — fjlv) sin A piJi + vv + [fiv + im'v) cos a = &c. and these expressions for the sine, cosine and tangent of D are of zero dimensions in X, //,, v, and also in X,', //,', i/', and are there- fore still true in the case before us when X, /a, v and X', /u.', v' are only proportional to the direction sines. 77. To deduce exjpressions for the angle between two lines whose equations are given in the form ■ la + m^ + W7 = 0, I'a + m'/3 + w'7 = 0. (Compare Arts. 48, 52, 61). If (a', /3', 7') be the point of intersection of these two lines, the lines may be expressed (Art. ^&) by the equations 7-7' a — a /3-^' m, n n, I h, c c, a I, m a, h and a — a' ^- -^' 7-7' m', n 1 n, I' ~ /', m' h, c c, a a, h which are of the form of the given equations of the last article. We must consider what the functions fJbV — [Jb'v, fji/jb + vv + {{Jbv + fJ^v) cos A, and ij?+v^ + 2iJbvco^A, become when we substitute X = 6—2 m, n /* = n, I v = ?, m b, c c, a a, b and similar expressions for X', /^', v . 84 THE EQUATIONS OP THE STEAIGHT LINE I. We get fxv — fJb'v = /^5 V = n, I I, m /*'j V C; a ? a, h n, I' I', m c, a J a, h — a T I I L , 7n , n a, b, c See Prolegomenon, Example F. II. fjbfi + w' + {jjbv' + fiv) cos A = {na — Ic) {n'a — re) + (7c — mh) {I'c — m'h) + {{na — Ic) {I'c — m'b) + {h — mh) {n'a — Tc)] cos A = c^\ll' + mm + nr^ — {mn + m'n) cos A — {nV 4- n'l) cos jB— (Zw'+ Im) cos CK III. ti^-\-v^+ 2fiv cos A is the same expression as tlie last with X,', /*', v written for X, fjL, V respectively. Therefore it reduces to c^\P ->rm^+n — 2mn cos A — 2nl cos B— 2lm cos C], or with the notation of a former chapter, a^{l, m, w}^ Hence we can write down the following expressions for the trigonometrical ratios of the angle between the two straight lines whose equations are la + m/S + ny =0, I'a + m'^ + n = I, m, n I', m, n sin -4, sin B, sin G {I, m, n] {^', m', n} cosi> _ ZZ'+ mm'+ nn— {mn'+ m'n) cos -4— {nl'+ ri I) cos B— {lm'-\- I'm) cos G {I, m, n}{l', m, n] ' and tan D I, m, n I', m', n' sin -4, sin 5, sin G U'+mm'+nn — {mn'+m'n) cosA— [nl'+n'l) cosB— (Im'+l'm) cos G* 78. To find the direction sines of the straight line whose equation is la + m/3 + W7 = 0. Let \, /A, V be the direction sines required. The angle which this straight line makes with the straight line Vol + m!^ +ny^O was shewn, Art. 77, to be given by ' sin D= + I, m, n, h m' , n' sin^, sin 5, sin G \l, m, n] jT, m', n'} THE EQUATIONS OF THE STRAIGHT LINE Hence writing m =0, n = 0, and dividing numerator and denominator of the fraction by X, we get \ = ifn, n sin B, sin G [I, m, n] So writing Z' = and n = 0, we get 11 = sin 0, sin^ [l, m, n] I, m sin -4, sin 5 and similarly v 1^^ ,^^ ^j » which give the direction sines required. 79. Cor. 1. With the same notation the expression for sin D may be written sm ^ _ l'\ + nifju + n'v which therefore gives us an expression for the angle between the two lines a — a' _ /8 — /8' _ 7—7' _ and A, At V ^' Xa. + m'/3 + w'7 = 0. Cor. 2. The two straight lines expresed in Cor. 1 are at right angles provided IX + m fjb + nv = + [I , m , n], 80. To find the perpendicular distance of the point [a, 0, 7') from the straight line whose equation is la + m/S + n>y = 0. IN TERMS OF THE DIRECTION SINES. 87 Let ^=^^ = lr:^=p (1) be the equation to the perpendicular from (a, ^', 7') on the given line. Then the length of the intercept, or the value of p at the point of intersection of the two lines, is obtained by combining their equations ; thus (1) gives us a = a' + X/3, ^ = ^' + /Xp, J = ry' + vp, and substituting in the given equation, we get la + tn^' + W7' + (l\ + m/^ + nv) p = 0, , la! + mB' + rvy whence p = ^s: . ' IX + m[M 4- nv But since the lines are at right angles, we have by the last Cor. IX + QiiiJb + 'nv= ± [l, m, n], ., p , la + m/3' + nry' thereiore P = + — n — t — , •^ ~ [t, m, n\ as before in Arts. 46, 59. Cor. The distance of the point (a', /3', 7') from the straight line apa + hq^ + cry = is given by p = ± ^ («pa + h^ 4- cry), since it was shewn in Art. 55 that [a^, hq, or} = 2A. 81. To find the equations to the perpendicular from (a, /3', 7') on the straight line whose equation is la + m^ + 7iy = 0. Let X, fi, V be the direction sines of the required line, and X', (j! , V those of the given line, then from the expression for cos D in Art. 75, 88 THE EQUATIONS OF THE STRAIGHT LINE since the cosine of a right angle is zero, = /iyLt' + vv + [iJbv + fJLv) cos A, or 0=/jb{/jb +v' cos A) +v [v -\- fM cos A). But therefore substituting H' V n, I I, m c, a a, h = fi I, m cos A — n a, bcosA — c + v\ I, m^n cos A \ a, h — c cos A But 5 — c cos J. = a cos C, arid c — & cos -4 = a cos B, there- fore dividing hj a, we get = fM(n — I cos B— m cos A) +v (J cos C+ncos A — m), or ^ V m — n cos ^ — ^ cos G n — l cos 5 — in cos . and therefore I —m cos — w cos B Hence the equations to the perpendicular will be Z — m cos G — n cos 5 m — n cos ^ — ? cos (7 ^ 7-7' ^ w — ? cos B-^m cos J. ° 82. The equations of the last article I— m cos C — w cos B m—n cos -<4 — ^ cos G — I cos B — m cos -d .(1), IN TERMS OF THE DIRECTION SINES. 89 express the conditions that X, fj,, v m&j be the direction sines of a line at right angles to the line la + m^ + ny = 0. But since they imply the relation a\ + h/j, -\- cv = 0, they express only one further condition. To find this one condition in a symmetrical form, we have from the first of the equations in (1), I {fi + \ cos C) — m{X + /J, cos (7) + w (//, cos /3 — X cos A) — 0, whence dividing by G, and remembering that a\ + h/M + cv = 0, we get m , - {fji, cos B—v cos G) + -J- {v cos C — X. cos A) + -{\cosA — fi cosB) = 0, Or = 0, or XcosA, ficosB, V cos G 1, 1, 1 Z, m, n \ sin 2A, fi sin 2B, w sin 2 sin -4, sin -5, sin G = 0; a result which the student acquainted with the difierential cal- culus could have written down at sight from the consideration that l\ + mfi + nv had to be made a minimum subject to the relations \^ sin 2A + fM^ sin 2B+v^sm2G=2 sin A sin B sin G, (equation 3 of Art. 71) 90 EXEECISES ON CHAPTEE VI. and XsinA + /jbsinB + vsmC=0, whence we must have ISX + mSyti + nBv = 0, \ sin 2A.h\ + fji sin 2B . B/ju + v sm2G .Bv=0, sin -<4 . SA, + sin B .S/j, + sin G .Sv = 0, and eliminating the differentials, the result is obtained. Exercises on Chapter VI. (57) The straight line whose direction sines are \, [i, v meets the line at infinity in the point given by the equations ^ > (58) Find the coordinates of the point at which the sides of the triangle of reference subtend equal angles. If through this point three straight lines be drawn each parallel to a side and terminated by the other two sides, the rect- angles contained by their segments are equal. (59) From the point (a', ^', 7') the straight line is drawn whose direction sines are \, fi, v: find the length intercepted upon this line, between the straight lines whose equations are loi + m/3 + ny = 0, and I' a. + w'/S + ny = 0. (60) Shew that if from any fixed point there be drawn three straight lines OP, OP', OP", whose lengths are p, p, p , and whose direction sines are (A-, /i, v), (X', pi, v), (V, //,", v') respectively, then the area of the triangle PP'P" will be 2A 1, /*P, vp a 1» /^V, 1 1 vp ■• II II II 1 V p EXEECISES ON CHAPTER VI. 91 (61) From tlie middle points of the sides of the triangle of reference perpendiculars are drawn proportional in length to the sides ; and their extremities are joined to the opposite angular points of the triangle. Shew that the three joining lines will meet in a point whose coordinates (a, ^Q, y) are con- nected by the equation sin {B - C) sin {G-A) sin (A - B) ^ + ^-^ + ^ = 0. (62) From the point 0, (a, /3', 7') a straight line is drawn in any direction to meet the straight lines la. + ml3 + ny = 0, I' a + m'yS + vj = 0, {I + l')a + (m + m) /3 + {n + n') 7 = 0, in points P, Q, B. Shew that the ratio OP.QB : OQ.PB is equal to la + m/B' + ny ~ I'a +m'/3'+n'y" whatever be the direction of the transversal. CHAPTER Vli. MODIFICATIONS OF THE SYSTEM OF TEILINEAE COOEDINATES. AEEAL AND TEIANGULAE COOEDINATES. 83. The great principle which distinguishes the modern methods of analytical geometry from the old Cartesian methods is, as we have seen, the adoption of three coordinates instead of two to represent the position of a point, and the recognition of the power thus gained of rendering all our equations homoge- neous. This homogeneity of equations will be always attainable whatever quantities x, y, z we may use as coordinates of a point, provided the third, s, be connected with the other two by a linear equation, Ax ■\- By ■\- Gz = D, Ax + By+ Cz ^ ^ =1; for (exactly as in the case of Art. 10, page 13) any term in an equation which is of a lower order than another may be raised by multiplying it by Ax + By+ Gz D ' (since this is equal to unity) and we may repeat the operation AEEAL AND TRIANGULAR COORDINATES. 93 till every term is raised to the order of tlie highest term, and the equation is thus homogeneous, 84. We have hitherto used the perpendicular distances of the point P from the lines of reference as the coordinates of P, and we have established the relation (1), aa + 5/3+C7=2A , connecting the coordinates of any point. The position of the point would be equally determinate if we used any constant multiples of these perpendicular distances as coordinates. For instance, we might call the coordinates of P, a, /3', 7, where a' = Xa, ^' = ^/3, 7 = vy, and the relation (1) connecting the coordinates of any point would then become ,(2). The particular case in which \=a, /J' = h, v = c will present the advantage of a very simple relation among the coordinates, for the equation (2) reduces in this case to a' + ^' + 7'=2A (3). Fig. 19. 94 MODIFICATIONS OF TKILINEAR COORDINATES. And the quantities a , ^' , 7', which in this case will be the coordinates of the point P, are capable of a simple geometrical interpretation. For if PD be the perpendicular from P on JBG (fig. 19), we have oi:=aa = BG.PD = 2APBC, so /3' = 2A PGA, and j' = 2APAB. The coordinates a, /3', y of the point P are therefore the double areas of the triangles having P as vertex, and the sides of the triangle of reference as bases. 85. If a", /3", 7" denote the halves of a', /3', 7', the equa- tion (3) of Art. 84 gives us a" + ^" + y" = A (4), as the relation connecting a", j3", 7" if they be taken as the coordinates of P. These coordinates represent the areas of the triangles BPG, PGA, PAB, and used often to be called indiffer- ently the ureal or triangular coordinates of P with respect to the triangle ABC. These terms areal and triangular have however more recently been applied to tlie system of coordinates described in the next article, and authors are not uniform in their use of the expressions. It seems convenient to describe these coordinates a", ^", 7" which represent the actual areas of the triangles PBG, PGA, PAB as areal coordinates, observing that as they represent areas they are of two dimensions in linear magnitude. We can thus reserve the term triangular for the system now about to be described, although it would certainly be preferable to invent a name for them which should indicate the fact (which will immediately appear) that they are of zero dimensions in linear magnitude, expressing not lines nor areas but simply ratios. VIZ. AREAL AND TRIANGULAR COORDINATES. 95 86. The relation among the trilinear coordinates a, ^, 7, aa + 5yS + C7 = 2 A may be written 2A"^2A"^2A If therefore x, y, z denote the ratios aa 5/3 cy 2A' 2A' 2A' they will be subject to the very simple relation X + 1/ + z = 1 .., , (5). But since x, ?/, z bear constant ratios to, a, ^, 7 they may be used as the coordinates of P (Art. 84) : and on account of the simplicity of the relation (5) just obtained, very great advan- tages attend their use. It will be observed that these coordinates {x, y, z) represent the ratios of the triangles PBG, PGA, PAB severally to the triangle of reference ABC. They are (not very appropriately) often spoken of as the areal or triangular coordinates of P, but as we said in the last article, we shall call them triangular coor- dinates, reserving the term areal for the system described in that article. In speaking of the areas of the triangles PBG, PGA, PAB, the same convention with respect to algebraical sign will have to be adopted as in the case of the perpendicular distances of P from the lines of difference. Thus (as in Art. 6, page 10) the triangle PBG will be considered positive when it lies on the same side of the base BG SiS does the triangle of reference, and so for the other triangles. 87. It is important to observe that if the triangle of refer- ence be the same, the triangular coordinates {x, y, z) and the 96 TABLE OF FORMULA. trilinear coordinates (a, /S, 7) of any point P, are connected "by the relations 03 _ 2/ _ s l^ aa~5^~c7~2A' so that we can at once transform any equation or expression from the one system to the other. To exemplify this, and for convenience of reference, we append a table of the principal results which we have already obtained in trilinear coordinates, together with the correspond- ing results for triangular coordinates. TABLE OF FORMULA AND OTHER RESULTS. In trilinear coordinates. \ In triangular eoordinates. kJ (i) The coordinates of any point are connected by the rela- tion (Art. 7), aa + 2'/8 + 07 = 2A. x-\-y-\-z = l. (ii) The coordinates of the middle point of BG are (Art, 14), ^ ^ I oil (iii) The coordinates of the foot of the perpendicular from A upon BG are (Art. 15), „ 2 A ^ 2 A „ I ^ 5 cos c cos 5 0, — cos G, — cos B. 0, , . a a a a TABLE OF FORMUL:^. 97 In trilinear coordinates. | In triangular coordinates. (iv) The centre of the inscribed circle is given (Art. 16) by a = /3=:ry = 2A 1 a + b + c ^ _y _z _ a h c a + b + c \y (v) The middle point between the two points («r ^v 7i) and {a^, ^„ 7,), is \ {x^, y^, z^ and (cc„ y^, s,), is ^aj_-Mf, A + /3 , Tt + 7A ^ /^Hi^. .^1 + 3/2 ^. + ^ 2 ' 2 ' 2 (vi) The area of a triangle whose angular points are given (Art. 19), is A- «i. /5i» 7i 1 «2' A. 72 ^ i "s' /^3» 73 ) ^=A ^15 3/1' ^l •^2 ' 3^2 ' ^2 ^3' 3^3» ^3 (vii) The equation to a straight line joining two points whose coordinates are given (Art. 21), is a, /3, 7 «!» /^i, 7l «2' A' 72 = 0. a?, y, ^ x„ y., ^x ^■,, 3/2. ^2 = 0. (viii) The equation to the straight line whose distances from the points of reference are^, q^ r (Art. 23), is apo. + bq^(3 + cr' II II II p , q , r p> q, r p, q, r /': II r" y» q', r' It II II p , q , r y. 9.'^ r 1, 1, 1 1, 1, 1 1, 1, 1 (xviii) The condition that the two straight lines h + W2/3 + W7 = 0, I Ix + my + nz =0, I'a + m'/3 + n'y = 0, I I'x + my + nz = 0, should be at right angles is (Art. 49) ll'+ mm'+ nn' I Ud^ + mrnlf + nn'c^ — [mn + m'n) cos A < — [mn + m'n) he cos A — [nl! + n'T) cos B \ — {nT + riT) ca cos B — {Im + l!m) cos C = 0. ^ — (???/ + tm) ah cos (7=0. (xix) Any straight line drawn through the given point \ / («', ^', 7'). may be represented by the equations (Art. 68) a-a' /3-/3' 7-7' /* a; — a; _y — y _z — z EXEECISES ON CHAPTER VII. 101 In trilinear coordinates. \ In triangular coordinates, where X, fx, v are proportional to the coordinates of the point where the straight line meets infinity and are subject to the relation a\ + 5//, + cy = 0. 5 A- + /A + z/ = 0. (xx) Each member of the equations in (xix) will be equal to the distance {p) from the given point to the point («» A 7)» I (a?, y, z), provided X, /it, v satisfy a further condition which is expressed by any one of the equations (Arts. 70, 71) fi^-\-v^-\- 2fjbv cos A = sin^^, a^fjiv + 5V\ + c^Xfi + 1 = 0, y^ + X^ + 2v\ cos B = sin^jS, X^Jc cos A + fju^ca cos B X^ + fi^ + 2\fji cos C= sin^O. + i^ab cos C= 1. Exercises on Chapter VII. (63) Shew that the straight lines {m + n)x+{n + l)y + {l + m) z = 0, and Ix + my + ws = 0, are parallel, and find their inclination to the straight line (w — w) a; + (*^ — ?) 3/ + (? — wi) s = 0. (64) If a series of dificrent values be given to p, the equa- tion X (cos^'a +^ sin'^a) + y (cos'^/S +^ sin'^yS) + % (cos^ +i> sinV) = will represent a series of parallel straight lines. 102 EXEECISES ON CHAPTEK VII. (55) Shew that the straight line bisecting at right angles the side BG oi the triangle of reference is given Iby the equa- tion (x — y + z) cot B = {x ■\- y — z) Qot C. Hence shew that the three straight lines, bisecting at right angles the three sides of the triangle of reference, meet in one point, (66) Draw the four straight lines represented by the equa- tions in triangular coordinates, xc,otA±ycoiB±zQoiG = 0, (67) Draw the four straight lines represented by the equa- tions X cosec A±y cosec B ±z cosec C = 0, (68) Interpret the equations X ± y ± z = Q. (69) Find the area of the triangle whose sides are given by the equations Ix + my + nz = 0, mx + ly — nz = 0, 2f = 0. (70) Find the area of the triangle whose angular points are given respectively by x = and y = Iz, y = and z = mx, z = and x = ny. If mn = 1, the result is independent of I. Interpret this cir- cumstance geometrically. EXERCISES ON CHAPTER VII. 103 (71) Of the four straight lines whose equations are Ix + my ±nz = 0, two intersect in P and the other two in P', two intersect in Q and the other two in Q', two intersect in R and the other two in B,'. Find the triangular coordinates of the middle points of PP', Q Q\ RB', and shew that thej lie on one straight line. Find the equation to this straight line, and compare the result with that of Ex. (40), page 56. (72) Prove that the straight line represented in trilinear coordinates by the equation is parallel to the straight line represented in triangular coordi- nates by the equation XQ.oiA+yQ,oiB-\-zQ,oiG=0. (73) ABC is a triangle : through the angular points A, B, G, straight lines he, ca, ah are drawn forming a second triangle ahc : and through a, h, c straight lines are drawn paral- lel to P(7, GA, AB respectively, so as to form a third triangle, and so on. Shew that the areas of the triangles thus formed are in geometrical progression. CHAPTEK VIII. ABRIDGED NOTATION OP THE STRAIGHT LINE. 88. In Chapter iii. Art. 27, we used u = and v = to represent the equations to two straight lines^ regarding u and v as abridgements or symbols of the expressions la. + m^ + w^ and Z'a + m'/3 + n'7. But it will have been observed that the reasoning of that article would have been equally valid if m = 0, -y = had re- presented the equations to two straight lines expressed in tri- angular coordinates, or in the ordinary Cartesian system. We may therefore write the result of that article in the following more general form. If u = 0, V = he the equations to two straight lines express- ed in any system hi which a point is determined hy coordinates, then will the equation u + kv = represent a straight line pass- ing through the intersection of the former lines. And by giving a suitable value to k this equation can he made to represent any straight line whatever passing through that point of intersection. 89. The following principle will often be assumed. If u = 0, v = 0, to = he the equations to three straight lines forming a triangle expressed in any system in which a point ABEIDGED NOTATION OF THE STRAIGHT LINE. 105 is determined hy coordinates, then any straight line whatever will he represented hy an equation of the form \u ■\- ybV -\- vw = 0. Fig. 20. F G For let QB,, RP, PQ be the straight lines whose equations are w = 0, v = 0, w = Q respectively, and let FG be any straight line whatever whose equation it is required to find. Since QR, RP, PQ are not parallel {hypoth.), one of them can be found which being produced will meet FG, Let it be QR. Let QR produced meet FG in F. Join FP. ' Then, by Art. 88, since FP passes through the intersection of V = 0, w = 0, it will have an equation of the form V + KW = 0, or, as we may write it, /A'y + yit! = 0. But since FG passes through the intersection of the straight line whose equations are m = and ixv ■\-vw= 0, it will by the same article (88) have an equation of the form Xw + /Ltv + rw = 0, which was to be proved. 106 ABRIDGED NOTATION OF THE STRAIGHT LINE. We have supposed that the three original straight lines form a triangle. All that is necessary however for the validity of the proof is that they should not he all parallel, and that they should not meet in a 'point. If they were all parallel, one could not necessarily be found to intersect P Q, and if they met in a point, FF and FQ would coincide, and the equation to FQ could not then be determined as passing through the intersection of FF and FQ. But the theorem is perfectly true if two of the original straight lines be parallel and intersect the third. 90. If w + w + t c. Hence in stating our result we must say that there will be a point of intersec- tion only when h> c. In the language of analysis which we have just introduced we shall be able to state the result more generally: we shall be able to speak of the point of intersection as always existing, but we shall observe that it is real or imaginary according as 5 > c or h 9, ^' h',f f\9' equations which determine Z : w : w, and shew that only one solution is possible. Therefore every imaginary point lies on one and only one real straight line. Q. E. D. CoE. 1. The real straight line 'passing through the imaginary point whose coordinates are f + f^-^, g + gW~l, h-Vh'4^ is represented hy the equation a, A 7 / 9^ h /'= 9\ h' = 0. 122 IMAGINAEY POINTS AND STRAIGHT LINES. Cor. 2. The imaginary point whose coordinates are lies on the straight line Joining the point {/, g, h) with the point at infinity [f : g : h'). Cor. 3. The same real line passes through the two imagi- nary points f+fW^, g + gW^, h + hW^, and f-f'^-h ^-^'V-1, h-hW-1. Cor. 4. If two curves intersect in a series of imaginary 2Joints, they will lie two and two upon real straight lines. This follows immediately from Cor. 3, when we remember that imaginary roots can only enter into an equation by pairs, the two members of every pair differing only in the sign of the imaginary part. 105. We have said (Cor. to Art. 101) that the coordinates of an imaginary point at infinity will be /+/'V=ri, ^+yv=T, A + A'V^nr, where /, g, h, /', g\ Ji satisfy the relations af-\- hg -\-ch = 0, af + 5/+ ch' = 0. This statement requires a little consideration. Let us return for a moment to real points, and suppose \ fjb, V are numbers satisfying the relation aX + &ft + cy = (1), then we are accustomed to say that the equations ?=^ = ^ (2) IMAGINAEY POINTS AND STRAIGHT LINES. 123 represent a point lying at infinity, for if we suppose a', /3', 7' to be the coordinates of the point determined by (2), then, since in virtue of (2), a', /3', 7' are proportional to X, yu,, v, it follows from (1) that aa' + 5/3' + C7' = 0, which shews that a', /3', 7' satisfy the equation to the straight line at infinity. But if we consider what are the actual values of these coor- dinates a', /S', 7', we perceive (Art. 36, page 38) that two at least and generally all three of them are infinite. But no difiiculty practically ensues from this, because we never want the actual coordinates of such a point, it being sufficient to know that the finite quantities \, /j,, v are proportional to them : and it is very convenient to speak of the point at infinity whose coordinates are thus proportional to \ fi, v, as the point (A-, /j,, v), since the quantities X, ^, v, or any quantities proportional to them, satisfy all practical conditions of the coordinates of the point. For, as we have already seen, so long as we have to do with homoge- neous equations we never require the actual values of the coor- dinates of any point, but only the ratios of those actual values, except in theorems connected with the distance of the point from other points ; consequently we shall not expect ever to require the actual coordinates of a point at infinity, since its distances from all finite points are infinite and cannot therefore generally be introduced into problems. (It should be noticed that if only two of the actual coordi- nates of a point be infinite and the third be finite, then two only of the quantities A,, yu,, v will have a finite magnitude, and the third will be zero). But, to return to the imaginary points, it follows from what we have said, that whether be the actual coordinates of a point at infinity or only propor- tional to them, we must still have 124 IMAGINARY POINTS AND STRAIGHT LINES. . af + hg + ch = 0, af+lg' + ch'^O. And we shall find it very convenient to speak of such a point as the point if + f'^'^, 9+9 ^~h h + hW^l), whether the quantities f+fj~\, g+g'J^l, h + h'J^ Tbe the coordinates, or be only proportional to the coordinates of the point. And indeed, since it is more convenient to deal with finite than with infinite quantities, we shall always suppose that the expressions /+/V=^, 9+9 J^, li + yJ^ do represent quantities only jproportional to the actual coordi- nates. In other words, if we speak of the point (m, v, w) as a point at infinity, we mean the point at infinity determined by the equations - = ^ = ^ U V w' 106. Two imaginary straight lines are said to be parallel when they intersect in a point on the straight line at infinity. Hence the condition investigated in Art. 35 may be applied to imaginary straight lines. 107. Any equation of the form la" + majS + njS'' = (i), represents two straight lines intersecting in the point G of the triangle of reference. For if //,, , /j,^ be the roots of the quad- ratic Ifj,'^ + m/x + n = 0, IMAGINARY POINTS AND STEAIGHT LINES. 125 the equation (i) can be written (a-/.,/3)(a-/./) = 0, which shews that it represents the two straight lines whose separate equations are a - /A^^ = 0, a-fi^^ = 0. The two lines will be real if /a^ and fi^ are real, and they will be imaginary if /u,^ and fi^ are imaginary. Or, assuming the condition investigated in Algebra, the straight lines will be imaginary or real according as 4.ln is or is not greater than m^. For example, consider the equation Since 1 > cos'"* C, it follows that this will represent two ima- ginary/ straight lines. The following proposition is interesting. 108. The two straight lines represented by the equation /8'' + 7' + 2/37 cos -4 - (1), are parallel, each to each, to the two straight lines represented hy rf+o?-^2^aQO^B=0 (2), and also parallel, each to each, to the two straight lines repre- sented hy a' + /3' + 2a/3cosa=0 (3). For the straight lines (1) meet the line at infinity in the two points determined by /3' + 7'+2/37COS^ = 0] aa + 5/8 + C7 = o) 126 IMAGINARY POINTS AND STRAIGHT LINES. the latter of wliicli gives us by the transposition, or 2;S7 = N,5^/3^+cV-aV he Substituting this value of /Sy in the first equation, we get he (/3' + 7=^) + {V^' + cV - a\') cos ^ = 0, or aV cos A + ^"^h cos B + rfc cos (7=0. Hence the two straight lines (1) meet the line at infinity in the two points given by the symmetrical equations a^a cos A + ^"^h cos B + ^fc cos (7 = aa + 5/S + C7 = By symmetry the straight lines (2) will meet the line at infinity in the two points given by the same two equations. Hence the straight lines (1) and the straight lines (2) pass through the same two points at infinity, and therefore are parallel. And similarly, each pair is parallel to the two straight lines (3). q.e.d. 109. Def. We observe that the six straight lines repre- sented by the equations (1), (2), (3), pass three and three through two imaginary points on the straight line at infinity. These two imaginary points will be found hereafter to have some very important and curious properties. We have indeed to refer to them so often that it is convenient to have a special name by which to distinguish them : and on account of properties which we shall shew hereafter to belong to them it is deemed appro- priate to term them the two '^ eireular" points at infinity. By this name we shall continually refer to them. IMAGINAEY POINTS AND STEAIGHT LINES. 127 110. To find the ratios of the coordinates of the circular points at infinity. The circular points are given by the equations yS' + 7"+2;S7Cos^ = (1), and aa + 5/3 + C7 = (2). The equation (1) determines the ratio of /3 to 7, thus yS" + 2/37 cos ^ + 7' cos'^ = - 7^^ sin'^, ^ + 7 cos J. = + 7 y^— 1 sin ^, ^= - 7 {cos A ± J- 1 sin J.}, or in virtue of the equation (2), ^ = _Z. =: ? cosJ. ±7-lsin-4 -1 cos^ + ^^sin^' the upper signs going together and the lower together. Hence the ratios are known. 111. From considerations of symmetry we at once per- ceive that the result of the last article, giving the ratios of the coordinates of the two circular points as infinity may he ex- pressed in any one of the following forms, « ^ ' ^ ^ 7 — 1 cos (7 + n/^ sin C cos 5 + J— 1 sin B ' a /S 7 cos G ± J— 1 sin (7 — 1 cos J. + J— 1 sin A cos ___^^ ' ^ 7 B + J- 1 sin B cos ^ + V- 1 sin ^ - 1 By simple addition we can express these ratios in a form symmetrical with respect to the three axes : but such form is more complicated, and one of the forms already written down will generally be more useful. 128 IMAGINARY POINTS AND STEAIGHT LINES. By multiplication we obtain a ^' cos {B-C)±J-1 sin {B- C) cos {G-A) ± J-lsin {C-A) ^ 7^__ cos {A-B)±J-1 sin {A-B)' a result symmetrical as far as it goes, but when we come to ex- tract the cube roots by Demoivre's Theorem we lose the symme- try, as it is found upon examination that we cannot take similar cube roots and write «___ ^ B-C , /— - . B-C~ G-A ,— . G-A cos — - — + V— 1 sm — - — cos — - — + J— 1 sm — - — A-B /^ cos . /— r • A-B' + V - 1 sin — - — 3 - 3 but we must take dissimilar roots as a B-G ,—-. B-G cos — h V— 1 sm — - — ^ cos r + V— 1 sm o o A.IT + A-B , /—r . ^17 + A-B' cos — — + V— 1 sm a much less convenient form than those already obtained. 112. Throughout the present chapter we have spoken of trilinear coordinates, and proved the properties of imaginary points and straight lines by the aid of them. But all that we have said applies mutatis mutandis to tri- angular coordinates : and in order to adapt our arguments to IMAGINARY POINTS AND STRAIGHT LINES. 129 this system it is in most cases only necessary to read unity for a, 5, and c severally, as well as for 2 A. Thus our results will take the following form. In Triangular Coordinates. (i) If f+g + 1i = \, and f'+g'+h' = 0, then f + f'^^, 9 + g''^^, h + hW^, are said to be the coordinates of an imaginary point lying within a finite distance of the lines of reference. (ii) If f+g + h=0, and f' + 9' + ^^' — ^> then f+f^^, g+9''^^7 h + hW^, are said to be the coordinates of an imaginary point lying at an infinite distance from the triangle of reference. (iii) The results of Arts. 101—104, 106, 107 will remain unchanged. (iv) The two circular points at infinity are represented by the equations oc _ y s — a Z> cos (7 + V— 1 Z> sin - c cos B + V— 1 c sin 5 or or X _ y z a cos (7 + V— 1 a sin "~ ^ c cos -4 + V— 1 c sin ^4 X y _ ^ a cos W. B ± V— 1 a sin 5 h cos ^ + V— 1 6 sin ^ —^ 130 EXERCISES ON CHAPTER IX* Exercises on Chapter IX. (83) If the imaginary straight lines u + v V^ = and u + vW^ = have a real point of intersection, then the four real straight lines u = 0, v = 0, u'=0, v=0 are concurrent. (84) The straight line joining the real point (a', ^', 7') with the imaginary point (/+/V=T), ff + gW^, h + hW~l is represented by the equation + a, /3, 7 V- 1 = 0, = 0, a, A 7 a, /3, 7 / 9, ^ and will be real, provided a, /5', 7' f, 9, 1^ f, 9'i 1^ (85) If the value of the multiplier h vary, the locus of the imaginary point {f+hfj~\, g + hg'J^, Ti+Mj~[) is a real straight line. (86) Shew that the equation in trilinear coordinates a' + ^'+7' + 2/37C03(i?-C)+27acos(a-^) + 2ayScos(^-i?) = 0, EXERCISES ON CHAPTER IX. 131 represents two imaginary straight lines parallel to each other and to the real straight line a cos J. + yS cos -B + 7 cos (7= 0. (87) Given u = 0, v=0, tv = the equations to three real straight lines in any system of coordinates, and 6, (f>, ■y^ three angles together equal to three right angles, shew that the equa- tion u^ + v^ + w'^ + 2vw sin 6 + 2wu sin (p + 2uv sin -v/r = represents two imaginary straight lines, and find their separate equations. (88) Shew that the equation in triangular coordinates (x^'+f+zYix + iz + zy+AxT/z {x^'+f+z^) {x + y+z) + 8xy2''=0, represents six imaginary straight lines parallel two and two to the three lines of reference. 9—2 CHAPTER X. ANHARMONIC AND HARMONIC SECTION. 113. Definitions. (1) Let a straight line AB he divided at P into two parts in the ratio m : 1, and be divided at Q in the ratio n : 1, then the ratio m : n is called the anharmonic ratio of the section of AB in P and Q. (2) Let an angle AOB he divided by OP into two parts whose sines are in the ratio m : 1, and be divided hy OQ into two parts whose sines are in the ratio w : 1 ; then the ratio m : w is called the anharmonic ratio of the section of the angle AOB hy OP and OQ. 114. It will be observed that if hotJi the sections be external or hoih internal m and n will be of the same sign, and therefore the anharmonic ratio of the section will be positive. If one section be external and the other internal, m and n will be of opposite signs and the anharmonic ratio of the section will be negative. 115. For the sake of brevity the anharmonic ratio of the section of AB in P and Q is often spoken of as the anharmonic ratio of the range of points APBQ, and it is expressed by the symbol [APBQ]. ANHARMONIC AND HARMONIC SECTION. 133 So the anliarmonic ratio of the section of the angle A OB by OP and OQ is spoken of as the anharmonic ratio of the pencil of straight lines OA, OP, OB, OQ, and it is expressed by the symbol {O.AFBQ]. 116. When we speak of the range of points APBQ it must not be inferred that the points necessarily occur in the order in which we read them: it must be understood that AB (terminated by the points mentioned first and third) is the line which we suppose divided, and P and Q (mentioned second and fourth) are the points of section. The sections may either of them be inter- nal or external, but we read the letters in the order in which they would come if the first section were internal and the second external. It is found most convenient to adopt this system because in a particular case of most frequent occurrence (the case of harmonic section described below) one section is always internal and the other external. 117. In expressing the ratios of lines it must be understood that AB and BA denote lengths equal in magnitude but opposite in sign. Th ^P_ AP_PA_ PA ihus, ^p- pjg-p^- _gp- 118. The anharmonic ratio of the range APBQ -UPBO]-^^'^-^^^^ and the anharmonic ratio of the pencil OA, OP, OB, OQ -\^'^^^Q\- ^^^BOP- ^ix^BOQ sin AOP.smBOQ '^sinAOQ.sinBOP' 134 ANHAEMONIC AND HAEMONIC SECTION. 119. If the 'pencil OA, OF, OB, OQ cut any transverse straight line in the range APBQ, the anharmontc ratio of the pencil is the same as that of the range. Fig. 25. ^mQOP _AP ^mAOQ AQ -^^^ sinP^O PO &inPAO~ QO' therefore &mQOP ^AP QO sin QOQ~AQ'PO' ^. „ , sin BOP BP QO ^^^^^^^■^^' ^i^BOQ=BQ'PO' therefore hj division, sin QOP. sin BOQ _ AP.BQ sin QOQ . sinBOP~ AQ . BP' or [O.APBQ} = [APBQ], q. e. d. 120. To sheio that we have only to write the values of the several anharmonic ratios as in Art. 118. ANHAEMONIC AND HAEMONIC SECTION. 135 Thus [APBQ]=^^=[BQAP]', and [AQBP] = ^^= [BPA Q] ; ■which prove the proposition. 121. Similarly we may ])rove that [APBQ] = [PAQB] = [QBPA] = [BQAP]. 122. If the angle G of the triangle of reference he divided hy two straight lines GP, GQ whose eqiiations are respectively ^ = ma and /3 = na, then the anharmonic ratio of the pencil GA, GP, CB, GQ is m : n. For hj Art. 4, sin^CP , sin AG Q = m and -. — ^ ^, ,-, = n. sin BCP sin BGQ therefore by division * {APBQ]=m : n. q.e.d. . 123. It follows that the anharmonic ratio of the section of the same angle by the two straight linos Za' + 2maj3 + 72/3' = 0, («i ± Vm' - Inf IS In which is real when m' > In, i. e. when the straight lines them- selves are real. If 771^ < In the straight lines are imaginary, and unless w = 0, the anharmonic ratio is also imaginary. But if »n = and I and n have the same sign the two straight lines become imaginary, but the anharmonic ratio of their sec- tion of the angle G is real and equal to negative unity. 136 ANHAEMONIC AND HARMONIC SECTION. Cor. The two straight lines, whether real or imaginary, which are represented by the equation divide the angle of the triangle of reference so that the an- harmonic ratio of the section is negative unity. 124. If the angle between tlie straight lines w= and v = he divided hy the straight lines w + /cv = and u + k'v = 0, the anharmonic ratio of the section is k : k. Let OA, OB be the two straight lines represented by w = and v = 0, and OP, OQ the two straight lines represented by u + Kv = and u-\- k'v = 0. Through any point S whose coordinates are («', yS', 7') draw a transversal SAPBQ cutting OA, OB in A and B, and OP, 0(9 in P and Q. And let u'y v' be what u, v become when a', y8', 7' are written for a, yS, 7. Also let \, fi, V be the direction-cosines of the straight line SAB, and let m, n be what w, -y become when \, fi, v are written for a, /3, 7. Then we have (as in Art. 80) SA=--, 8P==-'^-^^, SB = ~-, SQ^-^'^"'"' therefore m + Kn u' u' + Kv'\ (V U + KV . J pT>ri\ _ AP .BQ _ \??^ m + KnJ \n m + /c'n ^ ^^~ AQ.BP~ /v^_u' + k'v'\ Jv u' + KV \m m + Kn/ \n m+KU K {nil — wiv') {mv — nu) K [nu — mv) {mv' — nu') ' or {APBQ] = -,. Q.E.D. ANHARMONIC AND HARMONIC SECTION. 137 125. To find the anharmonic ratio of the 'pencil formed hy the four straight lines, tl + KV = 0, U + \V — 0, U + /jiV = 0, u + vv=0. Put = u + Kv and i|r = w + fxv, then = —^ ^- and V = 2- — L. , K — fl K — jJ, So the four given equations become <^ = 0, K-^ — IJ, + \{(l} — '\}r)=0, ^ = 0, /Ci/r - /^^ + 1/ (^ - i/r) = 0, ^ or ■< i> = o, , K — \ , I K—V, or Hence by the last article the anharmonic ratio is K—\ fC— V \ — fM ' V — fJ,* (k — \) {fX — v) {k — v) {/m — X)' 126. Def. If the anharmonic ratio of any pencil or range be equal to — 1 the pencil or range is called harmonic. An example of this is obtained if the angle C of the triangle A GB be bisected internally and externally by straight lines (XP, CQ meeting the base AB in P and Q. 138 ANHAEMONIC AND HARMONIC SECTION. For by Euclid vi. 3, AF:FB = AC:BC, AQ:BQ = AC:BC: whence AP^AQ PB BQ' AP.BQ °'' PB.AQ~ ' and therefore 7?p' A ~ ~^'' i.e. the range APBQ is harmonic, and therefore Tby Art. 119 the pencil GA, CP, CB, CQ is harmonic. 127. By reference to Art. 114, it will be seen that if a line or angle be divided harmonically one of the sections must be internal and the other external. Thus if APBQ be a harmonic range one of the points P, Q will lie between A and B and the other beyond them : the four points will in fact occur either in the order in which they are read in the numerator of the fraction AP.BQ AQ.BP' (which expresses the ratio) or else in the order in which they are read in the denominator. 128. From Arts. 120, 121, it appears that if APBQ be a harmonic range then {APBQ} = [PBQA] = [BQAP] = { QAPB} = { QBPA] = [A QBP] = {PA QB] = {BPA Q] : in other words, we may read the four letters in any order in which neither A and B arc contiguous, nor P and Q. ANHAEMONIC AND HARMONIC SECTION. 139 129. It follows immediately from Art. 122 that the straight lines whose equations are yS — ma. = 0, and /3 + ma. = 0, divide the angle C harmonically. Or more generally, from Art. 124, the four straight lines M = 0, U — KV —0, V = 0, U + KV — O, form a harmonic pencil. 130. To find the equation to a straight line lohich shall form with the three straight lines u+ Kv = 0, u-\-\v = 0, ic + /J,V = 0, a harmonic pencil. Let u + vv = 0, he the equation required. Then hy Art. 125, {K-\){fl-v) ^ 2 {k — v) (jjb — X) ' or • {k- \) {fjb — v) + {K~ v) {iji — \) = 0, or (« — 2X + yu.) V + Xyu- — 2fiK + k\ = 0, \a — 2ixK + kX or v= ~ ■ . a; — 2a, + /A Hence the straight line required will be represented by the equation w (/c — 2X + ^) —V (X/.L — 2fiK + k\) = 0. 131. To establish relations among the different anharmonic ratios obtained hy taking a range of four points, or a pencil of four straight lines in various orders. 140 ANHAEMONIC AND HAEMONIC SECTION. Four letters K, L, M, N can be written in 24 diiFerent orders. We have seen however, in Art. 121, that there are four different orders in which any range of points can be taken with- out affecting their anharmonic ratio. Hence we cannot obtain more than six different anharmonic ratios by taking the points in different orders. We observe also from Art. 120, that the reciprocal of any anharmonic ratio can be obtained by taking the points in a different order. Hence we cannot expect more than three different anharmonic ratios and their reciprocals. We may shew this more formally as follows : Let be the equations to four straight lines OK, OL, OM, ON inter- secting in the point which is given by u=v = 0. And let I— k\ + fMV, m = Kjj, -{■ v\ 71 = KV+ Xfl. Then by Art. 125, {KMIN} = ^^^-=4r^-"-^' or {KLMN} = -'^ (1), 80 {KMNL] = -^^ (2), and {KNLM} = -'^-^ (3), and the ratios {RMfL}, [KLNM], [KMLN] are (Art. 120) the reciprocals of these three, therefore {ZTVifi} = - ^^^ (4), ANHAEMONIC AND HARMONIC SECTION. 141 (£tOTf) = -^ ; (5), (™^^i=-F^^ («)• It will be seen (by Art. 120 or independently) that if the letters K, L, M, N be taken in any other order besides these six, they will still give one of these sanxe six ratios. Hence we observe, (i) that hy taking a range of four points in different orders we can only get six different anharmonic ratios. (ii) that of these six ratios, three are the reciprocals of the other three. (iii) that the ratio compounded of the first three is negative unity, and so is the ratio compounded of the other three. 132. Cor. If {KLMN] he any harmonic ratio, any an- harmonic ratio obtained hy taking the points K, L, M, N in different order will he equal either to +2 or to +-. For since {KL3IN] is harmonic, therefore by equation (1) of the last article, on — '^ , or in — n = l — m. And since m — n and l — m are equal, each is equal to half their sum ; that is, m — n l — m n — l Hence the equations (2), (3), (5), (6) of the last article give us [KMNL] = [KMLN] = 2, and [KNLM] = [KLNM] = \ , and the equation (4) shews that [KNML] is harmonic. 142 ANHARMONIC AND HARMONIC SECTION. Conversely. If the anharmonic ratio of a pencil or range be ler 2 or - , we may obtain a harmonic p taking the lines or points in a different order. either 2 or - , we may obtain a harmonic pencil or range by 133. We proceed to establish some important harmonic pro- perties of a quadrilateral. Let the straight line A'B' C meet the three straight lines JBC, GA, AB in the points A\ B' , C respectively, so as to form a quadrilateral. Let the diagonals AA\ BB', CC be drawn and produced so as to form a triangle ahc. Fig, 27. Let M = 0, ^^ = 0, w = 0, be the equations to BC, GA, AB respectively, where u, v, w in- clude such constant multipliers that the equation to A'B'G' may be (Art. 89) u + V + w = 0. Then, as we shewed in Art. 90, the equation v + iv = represents the line AA', and v — w — Aa: ANHARMONIC AND HARMONIC SECTION. 143 therefore loj Art. 129, AA' and Aa divide harmonicallj the angle contained by the straight lines v = 0...{AC), 10 = 0... (AB). Hence the pencil {A.BA'Ga} is harmonic, and therefore (Art. 119) the range, in which this pencil is cut by BB' will be harmonic, that is, {BcB'a] = -l. And similarly, [CaCh] and [AhA'c] are harmonic ranges. Again, since the range [BcB'a] is harmonic ; therefore the pencil {A . BcB'a] is harmonic, and therefore the range in which this pencil is cut by the straight line AB'C is harmonic, i.e. if Aa meet A'B'C in X, then {C'^'5'X} is harmonic. These properties may be extended and multiplied almost without limit. 134. The following geometrical constructions are sometimes useful. I. Given three joints in a straight line to find a fourth point completing the harmonic range. Let -4, P, 5 be the given points, and through AB describe any circle, ARBO. Bisect the arc ABB in R, and join RP and produce it to meet the circumference again in ; and through draw OQ Sit right angles to RO meeting AB in Q. Then Q shall be the point required. Join AO, BO, then by Eucl. ill. 27, the straight line OR bisects the interior angle A OB ; therefore ^ at right angles to it bisects the exterior angle ; therefore by Art. 126, {APBQ} is harmonic, and Q is the point required. 144 ANHARMONIC AND HARMONIC SECTION. Fig. 28, II. Given three concurrent straight lines to find a fourth line completing the harmonic pencil. Let OA, OP, OB be the three given straight lines, and let them be cut by any transversal in the points A, P, B. Find a point Q completing this harmonic range and join OQ, then (Art. 119) the straight line OQ forms a harmonic pencil with OA, OP, OB, and is therefore the line required. 135. The constructions of the last article can be made by the ruler alone, without the introduction of the circle, by apply- ing the properties proved in Art. 134. Thus : Fig. 29. EXERCISES ON CHAPTER X. 145 In OA take anj point P', and let OP, BP' intersect in A'; then let AA\ OB intersect in c, and let cF, AB intersect in Q. Join OQ. Then, applying to the quadrilateral APA'F the proparties proved in Art. 133, [APBQ] is a harmonic range, and therefore {O.APBQ} is a harmonic pencil. Hence Q is the point, and Q the line, required. Exercises on Chapter X. (89) If two straight lines OZ'and OK' intersect a system of parallel straight lines KK' , LL\ MM\ N2^' in K, L, M, N and K', L', M, N' respectively, then will {KLMN} = {K'L'IfN'}. (90) A point is taken within a triangle ABC, and OA, OB, OC are drawn; and through A, B, C straight lines B'C, C'A', AB' are so drawn that each of the angles of the original triangle is cut harmonically. Shew that the points of intersec- tion of BG and B'C, CA and C'A', AB and A'B' are col- linear. (91) If through the vertex of a triangle two straight lines be drawn, one bisecting the base and the other parallel to it, they will divide the vertical angle harmonically. (92) Any two straight lines at right angles to one another form a harmonic pencil with the straight lines joining their point of intersection with the circular points at infinity. w. 10 146 EXERCISES ON CHAPTER X. (93) If «, \, jx be in aritlimetical progression the straight line v = will form a harmonic pencil with the three straight lines U+ KV = Oj M + \v = 0, w + /iV = 0. (94) If K, \, [i he in harmonical progression the straight line M = will form a harmonic pencil with the three straight lines (95) The four straight lines represented by the eq^uations w = 0, V = 0, Iv^ + 2muv + nv^ = 0, will form a harmonic pencil if 8w^ = din. (96) The angle between the two straight lines 2u^ + 2muv + nv' = is divided harmonically by the two straight lines Tu^ + '2m' uv + w V = 0, provided In, mm, nT are in arithmetical progression. (97) The anharmonic ratio of the pencil which the two straight lines Iv^ + ^muv + nv^ = 0, form with the two straight lines Tu^ + 2m' uv + nV = 0, is equal to In - 2mm' + nl' ±2'^ (m' - In) {m"' - I'n) In - 2mm' + wf + 2 V(m' - In) {m^ - I'n) ' CHAPTER XI. TRANSFOEMATION OF COORDINATES. 136. Suppose we have the equation to any locus referred to a triangle ABG, and suppose we wish to find the equation to the same locus referred to a new triangle A'B' C. The method of transformation will depend upon how the new triangle is given, and two cases immediately present them- selves ; first, the case in which the new triangle is given by the coordinates of its angular points being assigned, and secondly, the case in which the equations of its sides are given. We proceed to discuss these two cases separately. 137. Case I. When the coordinates of the new points of reference are given. Let the coordinates of A\ B\ C referred to the original tri- angle be (a„ ^1, 7j, (a^, ^^, 7J, (0(3, ^3, 73) ; and let a\ h', c' denote the sides of the new triangle A'B'C, and A' its area. Also let a, /3', 7' denote the new coordinates of any point whose old coordinates were a, /S, 7. Then a^, a^, a^ are the distances of 5 (7 from the three points A'^B, G' respectively; therefore the equation to the straight line BG referred to the new triangle A'B'G' is (by Art. 23) a,aV + a,5'/3' + G3cV = (i). 10—2 148 TRANSFOEMATION OF COORDINATES. rig. 30. A Q C And therefore (Art. 58) tlie perpendicular distance of from tlie line BG\b 2A 7(aXa' + a//3' + a3cV). Bat this perpendicular distance is the coordinate a; there- fore and so and 2A' .(ii) equations which express the old coordinates a, /3, 7 of any point explicitly in terms of the new coordinates a', yS', 7' of the same point. If therefore an equation is given connecting the old coordi- nates a, /3, 7 of any point on some locus, by writing the three expressions given by (ii) instead of a, yS, 7, we at once obtain a new equation connecting the new coordinates a', /3', 7' of any point on the same locus : that is, we obtain the equation to the locus referred to the new triangle A'B' C, r---. TEANSFOEMATION OP COORDINATES. 149 Thus if the equation to any locus referred to the triangle ABGhe /(a,^,7)=0, the equation to the same locus referred to the triangle A'B' C is fC- 2A' ' 2A' 2A' But if, as is nearly always the case, the given equations be homogeneous, then 2A' will divide out, and therefore If any locus referred to the triangle ABC he represented hy the homogeneous equation /(«,A7)=0, the same locus referred to the triangle A! B' C will have the equation /(a/a' + aj)'^ + a.c'i, ^^dd + ^p' ^ + ^^dr^', 7/a' + 7//3' + 73cV) = 0. 138. Cor. It will be observed that the equation just ob- tained is necessarily of the same degree as the original equation. Hence the degree of an equation is not altered hy transformation of coordinates. This is a very important result. 139. This case of transformation of coordinates becomes very mucb simpler when triangular coordinates are used. For if {x, y, z) be the triangular coordinates of any point referred to the old triangle of reference, and {x , y , z) the tri- angular coordinates referred to the new triangle, the angular points of the latter being (^15 Vx') ^l)j (^2? Vii ^2/5 V^ZI Vzl ^3/? 150 TEANSFOEMATION OF COOEDINATES. then the equations (ii) of the last article become X = cc^x' + x^' + x^z' j z = z,x + z^y + z^z'\ And the equation f{x, y,^z) = therefore transforms into /(a?/ + oc,y' + x^z', y^x' + y.^y + y/, z^x + z^' + z^z) = 0. 140. Case II. When the equations of the new lines of re- ference are given. If the sides of the new triangle be represented by the equa- tions in terms of the perpendiculars from A, B, C; viz. by the equations p^aa + q^h/3 + r,cy = 0, p./i(x + q,h^ + r^cy = 0, ^^aa + qj)^ + r^ey = 0, then the coordinates of A, B, C referred to the new triangle are respectively : new coordinates of J. , (^j , ^^ , ^g) , of -S. fe'^^J ^s)* of C, {r„ r„ 7-3). Hence, if (as before) any point have the old coordinates (a, yS, 7) and the new coordinates {a, /3', y'), then a represents the perpendicular from (a, ^', y) on the line joining [q^, q.^, q^ and (r^, r^, r^. And ad is the double area of the triangle whose angular points in the new coordinates are (a'j ^'; 7') J fc' 2'2' S'a), (^i> »*2, O- Hence we have aa = 2/S" TRANSFORMATION OF COORDINATES. 151 a = 2aS' , ^ = 26aS' 2C/S" a, P, 7 ^n ^2» ^3 Therefore, ^^Ae 7 ?25 ^3 = 0. 141. Tlie cases whicli occur in practice are generally very simple. The following is an example. . Let an equation connecting the coordinates a, /S, y involve as terms or factors the three expressions ?a + wi/3 + M7, Xo. + m'/3 + w 7, X'lx + rn' ^ + ^"7, and suppose we have to transform it to the new triangle of reference whose sides are represented by the equations Za + w^ + W7 = 0, Z'a + m';Q + w'7 = 0, Ta + m"/3 + w' 7 = 0. 152 TRANSFORMATION OF COORDINATES. Then if a', ^', y denote the new coordinates of the point whose old coordinates are a, /3, 7, we have (Art. 46) r _ la + m^ + ny ^, _ I'a + m'/3 + w 7 [I, m, n\ ' [I', m', n] Zit , II a I II .__ a + m p + n y [I", m", n"} Hence, in effecting onr transformation, wherever the expres- sions la + 7W/3 + ny, l'a + m'^ + n'y, I" a + m"/3 + n"y occur, we have only to substitute for them K(X, Kp , K y , where k, k', k' are constants and represent the expressions {?, m, w}, {r, m', w'}, {Z", m\ n"}. It follows that if the original equation be made up entirely/ of the expressions la.+ m^ + nr/, I' a + m'^ + n'y, Vol + m!'^ + ri'y, the transformation will generally simplify it very much. It should also be borne in mind, if aa+h^ -\- cy occur as a factor or a term in the original equation, that since it is known to represent a constant quantity it cannot be trans- formed into an expression which would denote a variable quan- tity. It can therefore take no other form than /c{a'a'+h'^ + c'y'), where a, h', c' are the sides of the new triangle of reference, and K is a constant expressing the ratio of the areas of the old and new triangle. exercises on chapter xi. 153 142. Example. The equation (loL + m^ + nj) (aa. + 5/5 + cyY = K {I'a + m'/3 + n'rf) [V'a + m"/3 + n"r^)\ may be transformed into the much simpler form Tby taking the straight lines la. + m^ + niy = 0, ta + m'^ + n'y = 0, ra+m"/3+n"y=0, as lines of reference. We shall very often have recourse to such a transformation as this in order to simplify the equations of curves. Exercises on Chapter XI. (98) Transform the equation 0.^ + 0' + ^+ 2j3y cos (f) sin t/t + 27a cos ^|r sin + 2a/S cos ^ sin ^ = to tlie new triangle of reference formed by the straight lines /3 cos <^ + 7 sin i/r = 0, 7 cos ■^/r + a sin ^ = 0, a cos ^ + /3 sin ^ = 0. (99) Transform the equation wa" + v^" + wr/+ 2ufiy + 2vr/a + 2w'a^ = to the new triangle of reference formed by the straight lines ua. + w'0 + vy — 0, w'/3 + v'y = 0, (w'^— li'w') /3 — {ww' — u'v) 7 = 0. CHAPTER XII. SECTIONS OF CONES. 143. Having explained the principles of Trilinear Coordi- nates and exhibited the application of the method to the inves- tigation of the properties of straight lines, we now pass on to apply the same method to curved lines, commencing with the conic sections. We shall endeavour to make our investigation of these curves as independent as possible of the knowledge which we have acquired of their properties loj pm-ely geometrical and other methods. At the same time the student must not expect to find in these brief chapters anything like a complete treatise on the properties of the curves, as it is our object rather to set before him such properties as can be advantageously treated of by trilinear coordinates, and to leave for treatment by other methods those properties to which other methods are specially applicable. Success in the solution of a problem generally depends in a great measure on the selection of the most appro- priate method of approaching it ; many properties of conic sec- tions (for instance) being demonstrable by a few steps of pure geometry which would involve the most laborious operations with trilinear coordinates, while other properties are almost self-evi- dent under the method of trilinear coordinates, which it would perhaps be actually impossible to prove by the old geometry. We shall strive to set before the student such a series of propo- sitions as shall best illustrate the use of trilinear coordinates, and SECTIONS OF CONES. 155 at tlie same time put him into possession of sucli properties and results as are most often called for in the solution of problems. 144. Any 'plane section of a right circular cone wlien referred to suitable lines of reference may he expressed hy an equation of the form la' + m^' + ny^ = Oy where I, m, n are not all of the same sign. rig. 31. Let "be the vertex, OZ the axis of any right circular cone, and XYZ any plane cutting the cone in a curve, one of whose points is P. Through draw two straight lines OX, F at right angles to OZ and to one another, meeting the plane of the section in X, r, and let 6, ^, o/r be the angles which OX, OY, OZ make with the perpendicular upon XYZ. 156 SECTIONS OF CONES. Take XYZ as triangle of reference, and let be the trillnear coordinates of any point P on the curve. Let PL, PM, PN be the perpendicular distances of P from the planes OYZ, OZX, OXY. Then we have PL . a PM . , PN . , ,.. -p^=sm0, p^=sin<^, -^ = sin'»^ (i). But if CO be the semivertical angle of the cone PN= ONAsLna, and since ON' = PL" + P3P, (Euclid, i. 47) PA^ = {PL^ + PM') tan^ « ; and therefore, in virtue of (i), y' sin'^A/r = (a'^ sin''^ + ^' sin"^) tan'^w, or a' sin'9 tan^w + ^' 8m'(j> tan'^o - r/^ sin^^/r = 0, which may be written la' + m/S' + nr/ = 0, it being observed that n is of the opposite algebraical sign to I and m ; but the absolute ratios of I, m, n are any whatever, since angles can be found with their sines in any assigned ratio. 145. Any section of a right circular cone, to whatever lities it he referred, will he represented hy a homogeneous equation of the second degree in trilinear coordinates. For we have shewn in the last article that when suitable lines of reference are chosen, the conic section will be represented by an equation of the form lo? + m^'' + nrf=Q, a homogeneous equation of the second degree. SECTIONS OF CONES. 157 Now if we transform our coordinates to any other lines of reference (Art. 138, page 149), the degree of the equation will not be altered. Hence, whatever be the lines of reference, the conic will be represented hj an equation of the second degree. Q. E. D. 146. Conversely. Every equation of the second degree in trilinear coordinates represents some conic section. For any equation of the second degree (being rendered homo- geneous, Art. 10, page 13) may be written in the form wJ^ + v^' + wf + 2m'/37 + 2v^a + 2w;'a/3 = 0. Multiplying by u and re-arranging, we get {uoL + w'^ + v' (a3> A> %)^ («45 ^4J 74), («5» /^s' 75) be the coordinates of the five given points. And suppose ua? + v^ + w^'^ + 2m'/37 + 2v^ot. + 2M?'a/3 = to be the equation to the conic. Then, since the five points lie upon it, we have ua^^ + v^^ + w;7i' + 2i*';S.7, + 2v'-/,a, + 2wa,^, = 0, wa/ + vyS/ + W7/ + 2ii'yS,7, + 2v\a, + 2w'aJ3^ = 0, uo.^ + v^^' + wr^^ + 2u'^,% + 2v'73«3 + 2w;'«3^3 = 0, wa/ + v^: + W7/ + 2m'/3,7, + 2y'7,a, + 2wol^, = 0, wa,^ + v/3/ + wy,^ + 2ul3,y, + 2^'7,a, + 2wa,^, = 0. SECTIONS OP CONES. 159 Hence, eliminating u '. v : w : u : v '. w , yjQ get ^\ ^\ J% ^7, 7«, «)S =0 (^^^ A'j 7i', A7i» 7iai, "A tta', ^a'j 72'j A72> 72aa, Og/^a "a'j ^3'> 73'> A735 73^3 5 «3^3 a/» ^/» 7^ ^474> 74«4' "A as'j ^5^ 7^ /3576> 76a6r cCfi/^* which will be the equation required, 149. To find the condition that six points whose coordinates are given should lie upon one conic. Let the given coordinates he [a^, /3^, y^, (a^, /S^, 7J, Kj Aj 73)> («4» ^45 74)5 ("s' ^5» 75)^ («e5 /^s, 76)- The points will lie upon one conic if the coordinates of one of them satisfy the equation to the conic through the other five. Hence "by the last article the condition is a^ A'> 7^ /3i7i, 7i«i5 tti^i = 0. a2^ A'j 72^ ^.72 > 72^2 » ^2^2 °^z, ^st y^, A73, 73^3 > «A a?, ^4^ 74"^ ^474 » 74a4' ^A "s'^ ^B^ %^ ^575J 76<'5> "As «/. /^e', 76^, /3676. 76«6> o^A 150. Every straight line meets every conic section in two real or imaginary points, distinct or coincident. Let uo^ + v^+wy^+ 2u^'^ + 2vyaL-\-2w'oi^==(} (1) Ibe the equation to any conic section, and la + m^ + ny = (2) the equation to any straight line. 160 SECTIONS OF CONES. The coordinates of their points of intersection must satisfy both the equations as well as the relation «a + Jy8 + C7 = 2A (3). Hence, to find the coordinates we may proceed theoretically thus: From (2) and (3), which are simple equations, we may express yS and 7 as functions of a of the first degree. We may then substitute these values in (1), which thus becomes a quadra- tic equation in a. Being a quadratic it will give. two values for a, real or imaginary, equal or unequal, and the simple equations (2) and (3) will then give a value for /3 and a value for 7 corre- sponding to each value of a. Thus there will be determined two and only two points of intersection, which may however be real or imaginary, coincident or distinct 151. If in the argument of the last article the straight line be at infinity the reasoning still applies. Hence every conic cuts the straight line at infinity in two real or imaginary points, either coincident or distinct. If these two points be real and coincident the conic section is called a Parabola. If they be real and distinct it is called a Hyperbola. If they be imaginary it is called an Ellipse. 152. Def. Tangents which do not lie altogether at infinity but have their contact at infinity are called Asymptotes. It follows that an ellipse has two imaginary asymptotes and a hyperbola two real ones. In the parabola, since the straight line at infinity meets it in two coincident points, that line is a tangent, and there can be no other tangent touching at infinity. Hence the parabola has no asymptote. 153. Students who have not commenced Analytical Geo- metry of Three Dimensions may omit the remainder of this chapter. SECTIONS OF CONES. 161 But those who have made any progress in that subject will observe that the first article in this chapter is but a particular case of the following more general theorem. If f{x,y,z)=Q (1) • he the equation to any surface in rectangular coordinates of three dimensions, and if a? cos ^ + 3/ cos^ + z cos'\^=p (2) he the equation to any plane, then the equation to the section of the surface (1) hy the plane (2) will he f {a sin 6, /3 sin 0, , 7 sim/^) = is of the %*^ order : that is, Every plane section of a surface of the %*'^ order is a curve of the n^'^ order. The following particular case is important. Every plane section of a surface of the second order is a conic section. 156. The identical relation among the coordinates a, yS, 7 may Ibe obtained directly from the equation to the plane, thus X cos 6 +y cos + 2; cos l/r =p, but a? = a sin ^, 3/ = /3 sin 0, 5; = 7 sin i/r ; therefore substituting a cos ^ sin ^ + yS cos ^ sin + 7 cos -v/r sin -v/r =p, or a sin 20 + fi sin 2^ + 7 sin 2i/r = 2p, Cor. !Z%e equation to the straight line at infinity in this plane will he a sin 26 + fi sin 2^ + 7 sin 2'\lr = 0. 157. To fnd the section of the ellipsoid whose equation is x^ i^ z^ a c hy the plane whose equation is X cos 6 +y cos 4> + ^ cos •>/r —p. The equation to the section in trilinear coordinates is (Art. 153) a' sin^^ /3^ s'm^cf) r/sinSfr_ a' "^ H' ■*" c^ ~-^' SECTIONS OF CONES. 163 which has to be rendered homogeneous by means of the relation a cos 6 sin + y cos sin ^ + s cos -«//• sin ^ =p. Hence the equation becomes a^ SAV^O ^ sin^0 rf sin^^/r a c (a sin 2^ + /3 sin 2S' — KUV = 0, being of tlie second order, represents some conic section. To find where the straight line u= meets it, we have to substitute u = in the equation S — kuv = : whence we must have >^ = ; i.e. the points of intersection of the straight line ^t = with the conic to be investigated lie upon the given conic 8; i.e. M = is a common chord of the two conies. So iJ = is a common chord of the two conies. Hence 8 — kuv = represents a conic intersecting the conic >S'= in the four points which lie upOn the lines u = 0, v = 0. 166. To interpret the equation S-KU''=0. We may regard this as a particular case of the last equation, u = and v = coinciding, and so hy reasoning similar to that in Art. 161 we conclude that it represents a conic touching the conic ;Sf ^ in two points, u=0 being the equation to the chord of contact. 167. It will be observed that all the equations which we liave considered in the present chapter have been found to re- present conies passing through some four fixed points. Each equation has involved an undetermined constant k, which may receive different values distinguishing the different conies which can be drawn through the same four points. 170 ABETDGED NOTATION And by giving k a suitable value, we can make the equation represent any conic whatever passing through the four points. For since five points determine a conic section, any conic through the four points will be determined by one point' more, and the condition that the equation may be satisfied at this point is an equation from which k may be determined. Hence by giving k a suitable value, the equation uv — kwx = will represent any conic circumscribing the quadrilateral whose opposite sides are u, v and w, x. So uv — kw^ = will represent any conic having m = 0, v = as tangents, and io = as chord of contact, and so in the other cases. 168. If 8=0 and 8'=0 represent the equations to two conic sections, then will the equation S + kS' = represent a conic section passing through all the points of intersection of the first two. And hy giving a suitable value to k, this equation can he made to represent any conic whatever passing through those points of intersection. For the coordinates of any point of intersection of /S = and >S' = satisfy both the equations: i.e. they make 8 and 8' seve- rally zero: therefore they make 8 + k8' zero : i.e. they satisfy the equation 8 + k8' = 0. Hence the locus of this equation passes through every point of intersection of the given conies. And since 8 and 8' are of the second degree, >S'4- k8' = is of the second degree, and therefore 8+k8' = represents a conic section. Further, by giving a suitable value to k this equation will represent any conic passing through the four points of intersec- tion of the first two. For since five points determine a conic section, any conic passing through the four points of intersection will be determined if one other point upon it be determined. It is only necessary therefore to shew that by giving k a suitable value, the equation 8+ /c/S" = can be made to pass through any one assigned point ; which follows immediately as OF THE SECOND DEGREE. I7l in the last article. For if s and s be what S and S' become when the coordinates of the assigned point are substituted for the current coordinates, the equation s + ks' = will be the con- dition that the conic should pass through the assigned point. s s' will be fulfilled. Hence if we give k the value — ? , the required condition ExEECiSES ON Chapter XIII. (105) Prove that the equation (aa + b^ + cjY = (la + m/S + ny) [I'a + m ^ + w'7) represents a hyperbola, and find its centre. (106) Interpret the equations, (i) ka? = ^{aa + h^+ci). (ii) {la + m/3 + ^7)^ = (acos^ + /Scos 5 + 7 cos C) (asin^ + /3sin5 + 7sin G). (iii) W(^ + a^/3^ = 2aa {aa + 07) . (107) Shew that whatever be the value of k, the locus of the equation will pass through four fixed points, and find their coordinates. (108) If s = be the equation to the straight line at infi- nity, and M = 0, V =Q, w = represent three other real straight lines, the equation uv + sw = will generally represent a hyperbola ; but if u and v be parallel, it will represent a parabola. 172 EXERCISES ON CHAPTER XIII. (109) With the notation of the last exercise the equation represents a parabola, and the equation u^ + v^ + sw = 0, where u and v are not parallel, represents an ellipse. (1 10) If two conies have double contact with a third, the chords of contact are concurrent with a pair of common chords of the first two conies, and form with them a harmonic pencil. (111) If three conic sections have one chord common to all, their three other common chords are concurrent. (112) The straight line u — 2kv + k^w = is a tangent to the conic uw = v^. (113) The straight line M+v + w = Oisa common tangent to the three conies u^ = Avw, v^ = 4:wu, v? = 4wy. (114) Three conic sections are drawn through a point P, and each touches two sides of the triangle ABG at the extremi- ties of the third side, shew that each of the tangents at P makes a harmonic pencil with the straight lines joining P to the an- gular points of the triangle. CHAPTER XIV. CONICS EEFEEKEI) TO A SELF-CQNJUGATE TKIANGLE. 169. We shewed in Chapter xii. that by selecting suitable lines of reference any conic section might be represented by an equation of the form M-^m^''+nrf = (1). It is quite evident that if ?, m, n are all positive, since the squares a^, /S^, 7^ are necessarily positive and cannot be all zero, the expression Id^ + m^^ + 717^ must be positive and therefore greater than zero. Hence the coordinates of no real point can satisfy the equation, and the locus must be entirely imaginary. If I, m, n are all negative, we may change the signs through- out, and thus arrive at the case just considered. Hence if the equation have a real locus, two of the coeffi- cients must be of one sign and the remaining one of the opposite sign: and therefore (by changing the signs throughout, if neces- sary) we can suppose two of the coefficients positive and the third negative. We will suppose that I and m are positive and n negative, and we may write l=L\ m = M\ n = -N\ so that the equation becomes ' ra' + M'^'-Ny = (2). 174 CONICS REFERRED TO A ■ This may be written U^ + (ilf/3 + Ni) [M^ - Nry) = 0, whence we at once conclude, Art. 161, Chap. xiii. that the lines ilf/3 + Ny = 0, are tangents, and the line their chord of contact. That is, the side BC oi the triangle of reference is the chord of contact of tangents from the opposite angular point A. Similarly, by writing the equation (2) in the form M'^' + {La. + iV7) [La - Ny) = 0, we conclude that the lines LoL + Ny = 0, La-Ny=0, are tangents, and the line their chord of contact. That is, the side CA of the triangle of reference is the chord of contact of tangents from the opposite angular point B. But further, the equation (2) might be written {La + M/3\/^^l) {Lx - MjB^^ - Ny = 0, from which form we see that the two imaginary straight lines La + iI//3 V^ = 0, ia - if/3 V =1=0, (which both pass through the real point C) are tangents, and the; real straight line 7 = their chord of contact. SELF-CONJUGATE TEIANGLE. 175 That is, the side AB of the triangle of reference is the chord of contact of imaginary tangents from the opposite angular point G. Hence the conic is so related to the triangle of reference that each side is the chord of contact of the (real or imaginary) tan- gents from the opposite vertex. This is represented in figm-e 35. Fia-. 35. 170. Definitions. The chord of contact of real or imagi- nary tangents from a fixed point to a conic is called the polar of -the point with respect to the conic. Also "a point is said to be the pole of that line which is its polar. We may therefore express the result of the last article as follows : The equation Id' + m/3^ + nrf = represents a conic such that with respect to it each vertex of the triangle of reference is the pole of the opposite side. This is often briefly expressed by saying that the triangle is self -conjugate with respect to the conic. 176 CONICS REFERRED TO A 171. From a given point a straight line is draion in a given direction to meet the conic U + m/8' + W7' = 0, it is required to find the lengths intercepted hy the curve upon this straight line. Let (a', /3', 7') be the given point, and X., /x, v the sines of the given direction (see Chap. VI.), then the equations to the straight line are a — a'_/3 — y8'_7— 7'_ whence a. = Cl! +\p, ^ = ^ +IXp, ry=ry' + pp. If we substitute these values of a, /3, 7 which are true for any point on the straight line in the equation to the conic, the resulting equation, viz. I (a' + \pY + m (/3' + fipY + n{y'+ vpf = 0, will give the values of p at the points of intersection, that is, the lengths of the intercepts required, measured from the given point {a, /3', 7')- The equation is a quadratic, and may he written {lX% miM^+ nv^) p'+ 2 (Z\a'+?nya/3'+ nv^) p + [la^+ml3'^+n\ '>^ r \ ^A -7 H 1 — 182. Cor. 1. If the conic be a parabola, we have (Art. 177) a« V & ^ T + - + - = ^' L m n and the equations to determine the foci give one point at infinity, and reduce for the other to «a - A _ Z>/5 - A _ C7 - A I m n ' each of which fractions must be equal to -A 1 + 771+ n' Hence aa = A , , ^, , h/3 = ^J-^ — , C7 = A or l + m + n' l+m + n' l+m + n' aa 5/3 _ cy m + n n + l l + m' COE. 2. In the case of the parabola, smce -7 H 1 — = 0, it follows that the coordinates of the finite focus of any parabola SELF-CONJUGATE TRIANGLE. 187 with respect to which the triangle of reference is self-conjugate, will satisfy the equation aa - A h^-i^ 07 - A or T-E-. + —. rn + h^ + cy — aci. c) + w (a' + Xp) {13' + /xp) = 0, which may be written {Ifiv + mv\ + nX/ji) p^ + {l^'j + mj'oi' + nx'^') + [\ {my + nl3') + /x {no! + ly) + v {1/3' + ma')] p = 0, a quadratic giving two values for p, expressing the length of the two intercepts. Cor. If the point (a, /3', 7') be on the conic, so that ^^y + my a + na'/3' = 0, one of the intercepts is zero, and the other is given by {l/xv + mvk + nXix" p+\ {my + n^') + p, {no! + ly) + v (7/3' + mo!) = 0. w. 13 194 CONICS EEFEREED TO AN INSCRIBED TRIANGLE. 189. To find the equation to the tangent at any 'point on the conic. Let \ fi, V be the direction sines of the tangent at (a', yS', 7') and let (a, /3, 7) be any point on the tangent, then (Chap. VI.) we have ^r-=-"7 ^^P ^^^' and the length of the chord in this direction by the last corollary- is given by {Ifiv + mvk + wXft) /3+X, {my'+n^') +fi {na'+ Z7') + v (?/3'+ ma') = 0. But since the direction is that of the tangent, the length of the chord must be zero : therefore X (my + w/3') + fi {no! + ly') + v (Z/3' + mo:) = 0, or in virtue of (1), (a - a') [my' + n^') + (/3 - /3') {na' + ly') + {y- y) (ZyS' + ma') = 0, or a {my + w/9') + yS {na' + ly) + 7 {l^' 4- ma) = 2 {l^'y + my a! + wa'/3') . But since («', /3', 7') lies on the conic, we have l^'y' + my'a' + na'^' = (2), and the equation becomes a {my + n/S') + y8 (71a' + ly) + y {l^' + ma) = 0. This equation expresses a relation among the coordinates of any point (a, /9, 7) on the tangent at (a', ^' , y) and is therefore the equation to that tangent. Cor. The equation may be written a fm n\ 6 fn l\ ^ y /l m\ But (2) gives us I m n -, + ^, + - = 0, a 13 y CONICS REFERRED TO AN INSCRIBED TRIANGLE. 195 therefore the equation will take the form a a p p 7 7 or _+_+_ = 0, a form of which we shall presently give an independent investi- gation (Art. 198). 1 90. To find the equation to the chord of contact of tangents drawn from a given point to the conic, or, to find the polar of a given point with respect to the conic. Let (a, IB', y') be the given point, then we may shew, pre- cisely as in Art. 173, that the required equation is of the same form as that of the tangent at a point on the curve. That is, the equation a {my + n/3') + /3 {no! + ly) + y (Z/3' + ma) = 0, which, when (a, /3', 7') is a point on the curve, represents the tangent thereat, will, when (a, /3', 7') is a point not on the curve, represent the polar of that point. 191. To find the condition that any straight line whose equa- tion is given should he a tangent to the conic. Let fa+g;3 + hy = (1) he the given equation to the straight line and suppose (a', ^', y) its point of contact with the curve. The tangent at (a', ^', y) is given by a {my + n^') + jS {na + ly) + 7 (?/S' + ma ) = 0, which must be identical with (1). Therefore my + n^' _ no! + ly _ 10 + ma — 7— =— 7-=— X~' a' y' or I {If— mg — nh) m {mg — nh — lf) n {nh — If— mg) ' 13—2 196 CONICS REFERRED TO AN INSCRIBED TRIANGLE. But (a, /S', 7) must also lie on the locus of (1), whence Hence eliminating a , /3', 7' we get lf{lf— vug — nh) + mg {mg — nh — If) + nh {nh — If— mg) = 0, or rf^ + my + nW-2mng7i-2nlhf-2hnfg = (2), which is therefore the condition required. Note. The equation of condition just obtained may be written in the form \/^+^/mg + \/nk = (3), as will be seen by clearing this latter equation of radicals, when it will be found to take the form (2). 192. Cor. The equation in trilinear coordinates Ifiy + onycf. + na^ = 0, will represent B: parabola provided yal + yhm + Ncn — 0. And the equation in triangular coordinates lyz + mzx + nxy = 0, will represent a parabola provided The remarks made in Art. 176 will apply here, 193. To find the locus of the middle points of a series of parallel chords in the conic whose equation is l^y + mya. + 7ia/3 = 0. Let \, IX, V be the direction sines of the parallel chords, and let (o(, /3, 7) be the middle point of any one of them. CONICS KEFERRED TO AN INSCRIBED TRIANGLE. 197 Then (Art. 188) tlie lengths of the intercepts measured from (a, /8, j) to the curve in the direction (A,, fi, v) are given by the equation (Jfjiv + 7nv\ + nkiJb) p^ + (l^y + my a. + wa/3) + {\ (w7 + n/3) + /J, (?^a + ly) +v (//3 + ma)]p = 0. But since (a, yS, 7) is the middle point of the chord, the two values of p, representing the intercepts, must be equal in magni- tude and opposite in sign. Therefore the coefficient of p must vanish in the quadratic, and therefore \ {my + n/3) + [x (wa +ly) +v (1/3 + ma) = 0, or a {mv + 72//,) + /3 {n\ + lv) + 'y {I/jl + m\) = 0, a relation among the coordinates a, /S, 7, of the middle point of any one of the chords, and therefore the equation to the locus of the middle points. 194. To find the centre of the conic. Let (a', /3', 7') be the centre. Then the lengths of the intercepts measured from the centre in the direction (\, /*, v) are given by the quadratic (Ifiv + mv\ + n\[x) p^ + {l^'y + my (a + wa'/3') + {A, [my + w/3') + jjl (wa' + ly) + v {1/3' + ma')] p = (1). But since all chords are bisected in the centre, the two roots of this quadratic must be equal in magnitude and opposite in sign ; therefore the coefficient of p must vanish, and therefore 'K{my+n/3') + fM {na + ly') + v {l^' + ma') = (2), for all values oi\ : /j, i v, subject to the relation a\ + b(i + cv=0. Hence, we must have my + n^' _ na' + ly _ ?/3' + ma 198 CONICS EEFERRED TO AN INSCEIBED TEIANGLE. «' ^' _ 7 or I {la — mh — nc) m [mh — nc— la) n [nc —la — inb) ^ 2A ~ ^V + m^¥ + 7^(? — 2mnbc — 2nlca — 2lmab ' "whicli determine (a', /3', 7') the coordinates of the centre. N.B. If the coordinates are triangular instead of trilinear, we find that the centre of the conic whose equation is l^z + mzx + nxy = 0, is given "by the equations til £C _ y _ S 1(1 — m—n) m{m — 7i — l) n{n—l — m) 1 P + m^+ n^— 2mn — 2nl — 2lm ' Cor. The centre of a parabola is at infinity. 195. To find the conditions that the equation Z/S7 + m'yci. + wa/3 = 0, should represent a circle. Let (a, /3', 7') he the centre, then the length of the semi- diameter in any direction is given by equation (1) of the last article, which in virtue of (2) reduces to p^ [Ifiv + mv\ + nXfj,) + l^'ry' + my a! + noi/3' = 0. Hence all the diameters will be equal, provided Ifiv + mvX + nXfi be constant for all directions. But wc know (Art. 71, page 78), that IJiv sin A + vX sin B + Xfi sin G CONICS KEFEREED TO AN INSCRIBED TRIANGLE. 199 is constant for all directions. Hence the diametars will be all equal, provided I m _ n sin A sin B sin (7 ' wliicli therefore express the conditions that the conic should be a circle. Hence the equation ^7 sin ^ + 7a sin 5 + a/3 sin (7 = 0, or, al3 2 ipcmn + mnl + aUm). (138) The tangents to the conic l^fy + nvyoL + na^ = 0, parallel to the line of reference BC are represented by ? (/S + 7) + {Jm± JnYa.-= 0, the coordinates being triangular. (139) The chord of contact of the tangents (whether the coordinates be trilinear or triangular) is l{^ — y) + {m — n) a = 0. CHAPTER XVI. CONICS REFEREED TO A CIRCUMSCRIBED TRIANGLE. 202. The equation ZV + m^^'' + wV - 2W2W/37 - Inl^a - 2lma^ = (1) may be written {la, + m^ - njY - Alma^ = 0, and therefore (Art. 161) represents a conic section to which a = and ^ = are tangents, and la + m^ — W7 = 0, the chord of contact. Similarly, W2/3 + ny — la = 0, is the chord of contact of tangents /8 = and 7 = 0, and ny+ la — m^ = 0, the chord of contact of tangents 7=0 and a = 0. Hence the equation ZV + nt"^ + n^i^ - 2mnl3y - 2nlya - 2lmoi/3 = (1 ) represents a conic section, to which the lines of reference are tangents, and — la + ?n/3 + ny = 0, la — m^ + W7 = 0, la + m^ — ny=0, the chords joining the point of contact. CONICS REFEEEED TO A CIECUMSCRIBED TRIANGLE. 207 203. It should be observed that if we write — I for I, the equation (1) takes the form ZV + w'/3' + -n^rf - 2mn^x + "^^h^ + 2?wayS = (2), and the chords of contact now become loL + m/3 + tvy = 0, loL + on^ — nj = 0, loL — m/3 + ny = 0. So also if the equation to the conic be written ZV + m^/3^ + wV + 2mn0y - 2nlyoL + 2hna/3 = (3) , the chords of contact will be given by la + m^ —ny = 0, la + myS +ny = 0, — la + m^ ->fny=Q; and if the equation to the conic be written l\^ + rri'^'' + wV + 2'mn/37 + 2nlyx - 2lma^ = (4), the chords of contact will be given bj loL -~ m/3 + ny -^ 0, — la + 9W/3 + ny = 0, la + m/3 + ny — 0. Hence the four equations (1), (2), (3), (4) represent conies inscribed in the triangle of reference, and so related that all the twelve points of contact lie three and three on the four straight lines given by ± ?a + m/3 + 727—0. This reasoning applies equally whether the coordinates be regarded as trilinear or triangular. 208 CONICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 204. The last two articles shew that every equation of the form ?'a^ + w^'/S' + wV ± 2mnl3y ± 2nly(x ± 2lma^ = 0, where we take either one only or all of the doubtful signs as negative, represents a conic inscribed in the triangle of reference. It will be observed, that if the doubtful signs he otherwise deter- mined, the first member will become a perfect square and the equation will reduce to one of the forms {la + m^ + nyy=0, {-la + mj3+nyY = 0, {h-ml3 + nyY=0, [la + m/3- nyf = 0. In each of these cases, the locus of the equation consists of two coincident straight lines, the limiting form of a conic section when the plane of section becomes tangential to the cone along a generating line. Such a locus will moreover meet any straight line in two coincident points, and will therefore, like an inscribed conic, meet each side of the triangle of reference in two coincident points. It cannot however be said to touch those sides in anj geometrical sense. 205. Conversely, every conic section referred to a circum- scribed triangle will he represented hy an equation of the form V-c^ + w2'/3' + wV ± 2mw/37 + 2nly^ ± 2lmo.^ = 0, where, the douhtful signs must he either all negative, or one negative and tivo positive. For any conic section may be represented (Art. 145) by tlie equation uo? + v^"^ +wrf-\- 2ul3y + 2v'y:t. + 2w'o.^ = 0. But if the triangle of reference be circumscribed, a = re- presents a tangent, and therefore we must find two identical CONICS EEFEREED TO A CIRCUMSCRIBED TRIANGLE. 209 solutions when we combine a = with the equation to the conic. Therefore the quadratic vjS" + 2i//37 + w;7' = 0, must have two equal roots. And therefore u^ = vio, or u' = + yvw. Similarly, since /3 = and 7 = are tangents, we have V = ± Wwu, and w = ± Vmv. Hence the equation takes the, form uoc' + v^^ + wy^ ± 2 JuvjSj ± 2 Jwuyu ± 2 Juva^ = 0, or, writing F, m^, rt for w, v^w^ V(k + m^'/S' + wV + 2?nw/37 + 2nZ7a ± 2lmo.^ = 0. We thus see that every conic inscribed in the triangle of reference has an equation of this form : and the doubtful signs must be either all negative or only one negative, since we found in the last Article that if they were otherwise determined, the equation would represent two coincident straight lines. 206. It will be observed that if two of the doubtful signs be positive and one negative, we can immediately make all three negative without altering the rest of the equation, hy changing the sign of one of the quantities ?, m, n. We may therefore always assume the equation to a conic referred to a circum- scribed conic to be of the form . ZV' + w'/3^ + wY - 25ww^7 - 2nlr^0i - 2lma/3 = 0, where I, m, n may be positive or negative quantities. w. 14 210 CONICS EEFEEEED TO A CIRCUMSCRIBED TRIANGLE. It should be noticed that the equation ±jTa± JmS± Jn^=0, when cleared of radicals, takes the form of the equation just written down. So the equations (2), (3), (4) of Art. 203 are the rationalised forms of the equations + V-Za± Vm^ ± VJ27 = 0, + V/a 4 V- mj3 ± Vwy = 0, + VZa ± ^m^ ±'J-ny = 0. Thus we may always write the equation to a conic inscribed in the triangle of reference in the form Via + V»2/3 + Vw7 = 0, the coefficients I, m, n being either positive or negative, and double signs being understood before the radicals. 207. From a given point a straight line is drawn in a given direction to meet the conic l\^ + m'/3' + wV - 2wn/87 - 2n^ • B ,- . G V - a cos — + V/S sin— + V7 sm — = 0, r . A ,—-. B /- . C Va sm— + V - /3 cos — + V7 sm — = 0, r . A ,77 . ^ / G Vasm— + V/asin--+v -7COS — = 0. A ^ 2 215. To find the 'pole of any given straight line with re- spect to the conic JlcL + Jm^ + Jwy = 0. Let fa+g^ + hy = (1) be the equation to the given straight line, and suppose (a', /3', 7') the pole. The polar of this point is given bj la {la — w/3' — ny) + m^ (m/3' — ny — la!) + ny {ny — la — m/3') = (2), which must be identical with (1). Therefore / ^ ff _ , ^ , I {la — in^' — ny) m {m^' — ny — Id) n [n^ — la' ~ m^') ' ' a' ' a P 7 or = — = — ng + mh lh+ nf mf ■\- Ig ' which determine the coordinates of the pole required. 216. If we assume the geometrical property that every conic has a pair of foci situated at equal distances on opposite sides of the centre, and such that the rectangle contained by the perpendiculars from them upon any tangent is constant, we can 216 CONICS EEFEEEED TO A CIRCUMSCRIBED TRIANGLE. readily write down equations to give the trilinear coordinates of the foci of the conic Jla. + Jm^ + Vwy = 0. For let (a, /3, 7) be the coordinates of a focus, and let (a', /3', 7') he those of the centre. Then, since the centre bisects the line joining the foci, the sum of the two values of a is the double of a' (Art. 18. Cor.). But since the line of reference BG is a tangent, the rectangle represented by the product of the two values of a is equal to a constant, ¥ suppose. Therefore the two values of a are the roots of the quadratic But similarly yS and 7 are given by the quadratics ^2_2/3')S + ;^^ = 0, and 7' - 277 + ^' = 0. Hence a« - 2a'a = /3' - 2/3'y8 = 7^ - 2^7, or substituting the values of a', yS', 7 (Art. 213), (5cZ + cam + abn) 0? — 2 A {bn + cm) a = {bcl + cam + dbn) /3^ - 2 A (cZ + an) /3 = {l)cl + cam + dbn) 7^ — 2A {am + hi) 7, (I m n\ „ 2Aa /m n or - + T + -^*-- — r + - \a cj a \o c (I ^rn ^ n\ 2A/3M ^ l\ \a h c) b \c a) (I m n\ ^ 2A7 fl . m \a b cJ c \a equations to determine the two values of a, /3, 7 for the two foci. CONICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 217 Obs. In triangular coordinates, the foci of the conic nIx + Wmy + ynz = 0, are given by 1 / a m-^n \ _ 1 / 2 n + l \ o" V l + m + n^J~¥v~ l + m + n V c^ V l + m + n J' 217. To find the condition that it should he possible to find a conic touching the three straight lines of reference and three other given straight lines. Let /a+^,/3 + A,7 = a (1), /a+^,/3 + A,7 = (2), /3«+^a/3 + % = (3), be the three given straight lines. Any conic inscribed in the triangle of reference will be repre- sented by the equation If the straight line (1) be a tangent we must have (Art. 210) I m n ^ ,,. /i ffi K Similarly if (2) and (3) be tangents, I m n ^ ... I'-J.^K^'- <'^' I . in n ^ ,„. :^ + - + -r = (6). /a 9b h. Hence, eliminating I : m : n from the equations (4), (5), (6), we have 218 CONICS EEFEEEED TO A CIKCUMSCEIBED TEI ANGLE. ILL fV 9.' K L L L /;' 9: K L L L /s' 9z' K whicli will be tlie condition required, = 0, 218. BeianCHOn's Theoeem. If a hexagon he described about a conic section, the three diagonals formed by joining o^j^o- site angular points will be concurrent. Let PQ'RFQR' be the hexagon. Take three alternate sides, QR' , RF, PQ' produced, for lines of reference, and let the equations of the other sides be {Q'R),f^+g,^i-Ky = o (1), {R'P),fa+g,/3 + h,y = (2), {PQ),f<^+g,^ + hy = (3). Then P is given by 7 = 0, ^a + g^ = 0, and P' is given by /3 = 0, fa + h^y = 0, therefore the diagonal PP is represented by the equation so Q Q will have the equation and RR' will have the equation Kf,^ + Kg^ + KK'y = o. Hence the condition of concurrence of the three diagonals is gJi^ 9z9^^ 9 A = 0, EXEfiCISES ON CHAPTER XVL 219 or 1 1 1 /.' f: /a 1 1 1 9: 3^ ^3 1 1 1 K K K =0, ■which is the condition that the three straight lines (I), (2), (3) should touch the same conic with the lines of reference; which proves the proposition and its converse. Exercises on Chapter XVI. (140) The conies Zy37 + «i7a + wa/3 = 0, and Z'/S7 + w'Ya + na.^ — 0, intersect in the points of reference and in the point given by w, n = /3 w, I = 7 I, m m', 1 n ^', I' L m! (141) The straight line \oi + /jl^ + vy = will be cut har- monically by the conies l^y + mjci + wa/3 = 0, and T^y + m'ya + n'o.^ = 0, provided ll'X^ + mm! fj? + nnv^ — (m^' + m'w) /az' — (w^ + n'l) vX — (Im' + I'm) Xfi = 0. (142) The imaginary triangle whose sides are is self-conjugate with respect to the conic 1+1+1=0. U V w 220 EXERCISES ON CHAPTER XVI. (143) A triangle ABC is inscribed in a conic, and from each angular point straight lines are drawn parallel to the oppo- site sides to meet the conic again in P, Q, B; prove that QB, BP, PQ are parallel to the tangents sX A, B, G. (144) The triangle whose sides are will be self-conjugate with respect to the conic provided I in n - - + 3 + - = 0» a /3 7 ' 0, 0, 0, I, m, ti, fi^ fh fz^ 9A^ QK 9h 5'l^ 9l^ 9z^ ^1/1 ' KU Kfz V, K^ Ky /i^i' f^9,> fz9s = 0. (145) Interpret the equation in trilinear coordinates ^/aa + ^/^ + 7-C7 = 0, and find the coordinates of the foci of its locus. (146) If a parabola touch the sides of a triangle its focus will lie on the circle which circumscribes the triangle. INTRODUCTION TO CHAPTER XVII. NOTATION, ETC. 219. Students who have not read the Differential Calculus are recommended to pay particular attention to the notation which we now introduce. Those who have read the Differential Calculus will accept it without explanation. Let fix) denote any function whatever of x. Then the dfix) symbol ■• . (which must be regarded as a single expression ctoo not capable of resolution into numerator and denominator) is used to denote the expression derived from f{x) by substituting for every power of x (suppose x^), the next lower power multii plied by the original index (i.e. nx"~^), and omitting altogether the terms which do not involve x. Thus X will be replaced by x^ ox 1, x^ by 2x, x* by 4«', and so on. dfix) For example, if/ {x) denote x^ + Zax^ + Za^x + a^, then - _} ^ will denote Sx'^ + 6ax + 3(^^ So also, if / (a, ^, df df „ df _ which is the condition that Q should lie on the polar of P. .*. &c. Q.E.D. 235. If a straight line p pass through the pole of a .straight line q, the straight line c[ will pass through the pole of the straight line p. For if the equations to the two straight lines be la + m^ + ^7 = (/>), Z'a + m'yS + /i'7 = (§), then, by Art. 233, the equation = w. w, V, I w V, u, m V, u, w, n I' rn n, expresses equally the condition that the pole of ^ should lie on 2, and that the pole of q^ should lie on p, which proves the proposition. _ . 236. The two preceding articles express the same proposi- tion in different forms. The following corollaries follow from either article. COE. 1. If a point lie on a fixed straight line, its polar will pass through a fixed point (viz. the pole of the fixed straight line). Or, if a series of points be collinear, their polars are con- current. OF THE SECOND DEGREE. 237 Cor. 2. If a straight line pass through a fixed point, its pole will lie upon a fixed straight line (viz. the polar of the fixed point). Or, if a series of lines be concurrent, their poles are col- linear. 237. To find the equation to the two tangents drawn from a given external point to the conic whose equation is /(a,/3,7)=0. Let (a', /3', 7') be the given point P, and let (a^, ^^, 7^) be any point Q on either tangent. Then PQ being a tangent, passes through its own pole; or P, Q and the pole of PQ are coUinear. Therefore PQ is concurrent with the polars of P and Q, (Art. 236, Cor. 1.) But the equation to PQ is, (Art. 21) a, /3, 7 a, /3, 7 = 0, and the equations to the polars of Q and P are, (Art. 231) oc -^ + /3 -^ + 7 -^ = d(x, "^ d^o (ho ' df , r,df ^ df ^ and therefore by the condition of concurrence, (Art. 26) /3o, 7o ^, 7 K do.^' df_ do!' 7o» ao 7', a' df M. a', /3' d% d£ d'y' = 0, 238 THE GENERAL EQUATION a relation among the coordinates of any point {a^, ^^, 7o) ^^ either tangent. Hence, suppressing the subscripts, we have the equation to the two tangents /8, 7 7, a a, ^ /3', 7' J 7', a' 5 a', /3' da.' do!' 61 d^' d^ c?7 df_ d df and i^ - ^ (^'''' + *'*'^ + ^V) + ^' (^^ + wz^ + W7). Hence (Art. 88) the equations represent three straight lines passing through the point of inter- section of the straight lines la + m^ + ^7 = 0, and I' a + m^ +n'y = ; that is, the three straight lines ^/ .^ , jQ=wa + v^ + uy = Q, ■j-=v'a + u'^ + t^j7 = 0, are concurrent; and therefore (Art. 26), u, w , V w', V, u' v' , u\ w = 0, which will be the required condition. 246 THE GENERAL EQUATION 246. To find the equation to the common chords of two conies whose equations are given. Let i^(a, /3, 7) s JJ^ + V^ + W + ^ Z//37 + 2 F'7a + 2 TF'«^ = 0, and / (a, /?, 7) = wa* + v^^ + w^ + 2w'y87 + 2v'7a + 2M;'ayS = 0, be the two conies. Any pair of common chords constitute a locus of the second order passing through points of intersection of the two conies, and must therefore be represented by an equation of the form i^(«, A 7) + /c/(a, A 7) = (1), where k must be so determined that this equation may satisfy the condition of representing two straight lines. That is, K must be determined by the equation (Art. 245) = 0, (2), JJ +KU, W' + KW, V+KV W' + KW', V +KV, U' + KV! V +a:u', U' +KI1, W+KW a cubic equation giving three values of k for the three pairs of common chords (Art. 163). Obs. The equation (2) may be written U, W V W, V, U' V\ U, W + /c{w (JW- U") + v{WU- V") + w{UV- W") + 2u'{rw'-uu') + 2v'{wu'-vv') + 2io'{zrv'-ww')] + K^ { U{vw - u") + V{wu - v'^) + W{uv - w'^) + 2 C/^ {vw - uu') + 2 F' {w'u - vv) + 2W' {u'v - ww')] + k' Uy W , V w', V, u' V, U, W = 0. OP THE SECOND DEGREE. 247 247. Cor. 1. In the particular case when the first conic consists of the two coincident straight lines (Za + m^+ nyY = 0, the equation for k reduces to u, io\ r I - w', % u\ m v, u', '10, n I, m, n, W , V, u v', u, w K = 0. Hence the tangents whose chord of contact is la, + m^ + W7 = 0, have the equation u, w, v', I /{a, ^, y) + w', V, u', m v', u', w, n I, m, n, u, w, v' w', V, u' v', u, w iloL + ?w/3 + nyy= 0. Cor. 2. Since the asymptotes are the two tangents whose chord of contact is at infinity they will be represented by the equation w, w\ V', a w, V, u', h 1 u w, c a, h, c, /(«,A7) + w, V, u v', u, w if the coordinates are triUnear, and by the equation w, w', V, 1 /(a, /S, 7) + w , V, u', 1 v , u', w, 1 1, 1, 1, if the coordinates are triangular. u, w , V W, V, u' v', u', w (a + /3 + 7r = 0, ^48 THE GENERAL EQUATION GOR. 3. If the conic be a parabola the asjMnptotes lie alto- gether at infinity : therefore in the equations of the last corollary we must have = 0, u, w', V, a = 0, or w, V, u', b v', tl, w, c a, h, c, u, io\ v\ 1 to', V, u, 1 v', u, w, 1 h I, h according as the coordinates are trilinear or triangular. These will therefore be the respective forms of the condition that the equation should represent a parabola. But we shall arrive at this result more directly in the next article. Cor. 4. The asymptotes will be at right angles to one another (Art. 49, Cor. 2) provided u + v + w — 2u' cos A — 2v' cos B — 2w' cos = 0,; when the coordinates are trilinear. This is therefore the condition that the general equation of the second degree should represent a rectangular hyperbola. When the coordinates are triangular this condition becomes uc^ 4- vW + w& — 2u'hc cos A — ^v'ca cos B — 2w'db cos G = 0, or a^{u + u —v' — w') +b^(v + v — w —u) + c^ (w + 10 — u — v) = 0. 248. To find the condition that the general equation of the second degree u^ + VyS' + wf + 2x1 ^-i + 2vY-^ + 2io'a^ = should represent a hyperbola, parabola, or 6lli2yse. I. Suppose the coordinates are trilinear. We shall find the coordinates of the points where the locus meets infinity by solving the given equation simultaneously with the equation aa + ^y8 + -^7 = 0. OF THE SECOND DEGREE. 249 Eliminating 7 we get c^ (wa' + v;8^+ livcn^) - 2c {aa + 5/3) {v'a + u (3) + w{aa-\- h^Y = 0, or (toa^ + lid' - 2v'ac) a' + {vc^ + wV - 2ubc) /S' + [w'c^ + wah — v'hc — u'ac) 2a^ = 0, a quadratic whose roots are unequal, equal, or imaginary, accord- ing as > {wah + w'c^ — v'hc — uacf = {wa^ + u6^ — Iv'ac) {vd^ + wW — 2u'hc), that is, according as > = 0. < u, ■w\ a w, V, u', h 1 n', w, c a, h c, Hence the given equation represents a hyperbola, parabola, or ellipse, according as the determinant just written is positive, zero, or negative. II. Suppose the coordinates are triangular. The reason- ing will be the same as in the other case, with unity substi- tuted for a, h, c severally. Hence the equation will represent a hyperbola, parabola, or ellipse, according as u, w', v', 1 w', V, u', 1 -v', u', w, 1 1, 1, 1, is positive, zero, or negative. 249. We may arrange the proof of the last article in a different form as follows : Let a-a' _/3~0 _y-y _ = P be a straight line drawn from any point (a', ^', y) so as to meet 250 THE GENERAL EQUATION the conic at an infinite distance. Then since one of the inter- cepts on this line is infinite, we must have / {\ H', v) = 0. Therefore the conic will have one, two, or no directions in which one of the radii from a finite point is infinite, according as the equation f{\, f^,v) = gives real and equal, real and unequal, or unreal solutions. Eliminating v by means of the relation a\ + bfi-\- cv = 0, [trilinear we get \^ {wd^ + uc^ - 'iv'ca) + //," {vc^ + wTf - 2ubc) + 2\/jb {wah + w'c^ - u'ac -v'hc) = (1). Hence the conic is a hyperbola, parabola, or ellipse, (Art. 228) according as {wah + w'c^ — u'ac — v'hc) = (wa^ + u(? — 2v'ca) (vc^ + w¥ — 2u'hc), that is, according as > = 0, < u, 10 , V , a w', V, w', h V, u, w, c a, h, c, as in the previous article. Obs. If the coordinates are triangular the equation (1) takes the form V (wj + w - 2v') +fJt?(v + io- 2u) + 2\/A [w + w' — v! — v) = 0, OF THE SECOND DEGREE. 251 and the final condition becomes > = 0. < ^«, w , V , 1 lo', V, u'f 1 v', u, w, 1 1, 1, 1, Cor. Bj reference to Art. 242, we see that if the conic is a parabola the centre is at infinity, and the diameters are therefore parallel. Hence the following proposition arises. 250. To find the, direction of the diameters of the parabola wa^ + v/3^ + wrf + 2ul3r^ + 2v'7a + 2w'a/3 = 0. Let (\, fjb, v) be the required direction. The equations con- necting X, fjb, V may be expressed in a variety of forms derivable from one another in virtue of the relation among the coeificients expressing the condition that the conic is a parabola. • But one of the most useful forms may be obtained as follows : One of the diameters is represented by the equation in tri- linear coordinates 1 d£_\ df_ a da h d^^ ua. + w'B + v'v w'cL + v/8 4- ^'7 or = 1 '- , a Now X, fjb, V are proportional to the coordinates of the point where the diameter meets the straight line at infinity. Hence we have vX + w' jj, + v'v w'\ + v/A + u'v a ^ h ' and a\ + l[jb-\- cv — 0, whence eliminating v, we get \ [uhc + u'a^ — v'db — w'ac) = [m {vca + v'b^ — iv'bc — u'ah), and by symmetry = v {wab + w'o^ — u'ca — vhc), which determine the ratios \ : fM : v required. 252 THE GENERAL EQUATION OF THE SECOND DEGEEE. Obs. In triangular coordinates the result will become \ {u + U — v' — w') = jji {v + v' — w' — u') = v [lo + w' — u — v). 251. Cor. 1. The equations to the diameter of the parabola, through the point (a', 0, 9, ^h (where f, g, h are the coefficients of a, /3, 7 in the identical rela- tion/a + g^ + ^7=1, connecting the coordinates of any point) is called the bordered Discriminant of the function u(^ + v^^ + wrf + 2m'/S7 + 2v'7a + 2wj'a^, and will be denoted by the letter K. In triangidar coordinates /, g, h are each unity, and we have K= ot u, w', V, 1 tv', V, u, 1 v\ u, w, 1 1, 1, 1, In trilinear coordinates /, g, h become and we get K= 4A'^ a he 2A' 2A' 2A' u, w , V , a w, V, u, h V , u , w, c a, h, c, or -K=-^^F{a, h, c). SECOND DEGREE CONTINUED. 257 256. The minors of the bordered discriminant with respect to the terms /, g, h will be denoted by Aj B, C, so that we have in trilinear coordinates C = w', V, u' v', u', w a, h, c v'i u', V) u, w , v' a, h, c 2A ' ^- 2A and in triangular coordinates, u, w', v' w.. V, u' a, h, G 2A A = w, V, u > i^ = V V, r u, w u h 1, 1 1= w C = W , V h 1 u, w', v w\ V, v! 1> h 1 257. It should be observed that In trilinear coordinates K has — 2 linear dimensions and A, B, G have each *^ 1, while H is of zero dimensions. : Bat in triangular coordinates all these functions are of zero dimensions, giving a great advantage to the triangular system. It will be seen that the expressions in triangular coordinates throughout this chapter will be mostly derivable from the ex- pressions in trilinear coordinates by writing 1 for 2A or ^ for A. So, conversely, the expressions in trilinear coordinates may often be derived from the corresponding expressions in triangular coor- dinates by multiplying each term by such a power of 2A as will produce homogeneity. 258, The student can easily verify the following results, which it is convenient to collect here for future reference* we have and L For the conic whose equation is K— — {mn + «? 4- Im), aJ^mn + h^nl + cHm \K=- ^A--* W, \triangular [trilinear 17 2m THE GENERAL EQUATION OF THE II, For the conic whose equation is l^y + myoL + n^.^ = 0, we have and < H=- Imn, 4 1/r. K = -{t^ + in' + n" -2mn -2nl- 2hn) , K= [triangular aH^+ Vir^ + cV — ibcmn — 2canl — 2ahhn \trilinear 16A^ IIL For the conic whose equation is we have {K= — 4Zmn (l + m + n), \K = — -r^ {Ibc + mca + nah) , IV. For the conic whose equation is and [triangular \trilinear we have and iK^¥+2h, a^k^^2lch \K = 4A'' [triangular [trilinear 259. To express the equation of the straight line at infinity in terms of the derived functions d£ df dl da.' dl3' dy lohen /(a, /S, 7) = u-j^ + v^"" + wrf + 2k'/37 + 2vyo. + 2to'a/3. ^'' ^1 + ^1 + 47^' be the equation required. SECOND DEGREE CONTINUED. 25^ The first member may be written - , {la + mw + nv) 2a + Qw ■{■ mv + nu) 2/3 + ijv + mu + nw) 2% whicli must be identical with if the coordinates be trilinear, where ^ is some constant. We have, therefore, III + mw + nv = ak, Iw + mv + nu = hk, Iv + mu -f nw =■ ch, , I m n 2Ak whence A^B^C-^'H^ and therefore the equation to the straight line at infinity- becomes which holds equally whether the coordinates be trilinear or tri- angular, 260. Cor. 1. The identical relation connecting the trili- near coordinates of any finite point becomes So also in triangular coordinates, dx dp «7 261. Cor. 2. The result of the last Cor. may be written (by Art. 221, equation ii), "l4 + ^i+-f-^'. 17-2 260 THE GENERAL EQUATION OF THE yihich. being a relation among the coordinates of any point whatever must be identical with or a + /3 + 7 = 1. Hence in trilinear coordinates, d£_aH §£^'b^ dA~ I^' dB A ' and in triangular coordinates, dC~ A = dA'dB'dC [trilinear [triangular .262. Given to express L, M, -AT explicitly in terms of I, m, n. Substituting their values for ^ df ^ da' d^' dy' and equating coefficients of a, y8, 7, we obtain 1=2 {uL +v)'M+vN)\ m-2(w'L+vM +u'N)> w = 2 {v'L + uM + wN)) and solving for L, M, N, we obtain L = ^{Ul+W'm + rn) M^—iWl+Vm + Wn) N = ~(V'l+U'm+Wn) the required results. (1)= .(2), SECOND DEGREE CONTINUED. 261 263. By reference to Art. 242 it will be seen that the coordinates of the centre of the conic whose equation is /(a, A 7) =0, are _A B_ _G K' K' K whether the coordinates be trilinear or triangular. Hence in trilinear coordinates, (Art. 261) 1 dl^\ 8^^\ dl^_H^ a doi~ h d^~ c drf~ iCA ' and in triangular coordinates do. d^ dl~ K ' 264. By Art. 245 the condition that the equation /(a, /3, 7) —0 should represent two straight lines is 265. By Art. 247, Cor. 2, the asymptotes of the conic represented by the general equation of the second degree /(a, /3, 7) = are given by /(«,A7)+f=0. 266. By Art. 247, Cor. 3, the condition that the equation /(a, )8, 7) = should represent a parabola is By Art. 248 the equation will represent an ellipse or a hyperbola according as K is positive or negative. 267. We found in Art. 247, Cor. 4, the condition that the equation / (a, /3, 7) =0 should represent a rectangular hyperbola* We shall write this condition 262 THE GENERAL EQUATION OF THE SO that E represents tlie function w + V + w — 2?/ cos A — Iv cos B — ^lo cos C if the coordinates are trilinear, or a^ {u + It — v' — w')-\- h^ (tJ + v' — tv — u) +c^{w-\-w' — u' — v) if thej are triangular. 268. If (a, /S, 7) he the centre of the conic whose equation is f{a., /S, 7) = 0, then will For by Art. 261, we have identicallj a€+b€ + c€=2k del. d^ d ry I I a' > 0-, P, 7 a, /3, 7 d^^ v, u\ w A, B, G a, P, 7 dy = 0. Cor. 2. The general equation to a diameter may be written -0, a» ^, 7 A, B, C X, /*, V or A, B, C X, yli, V dj_ d(J. + W\ V, U' A, B, C V, U', W A, B, C d'y = 0. 270. The polar of any point on a diameter is parallel to the tangent at the extremity of the diameter. For let A,, IX, v be proportional to the coordinates of a point on a diameter, then its polar is given bj (1) ^ df ^ df ^ df ^ da. '^ dp dy ), and the equation to the diameter can be written a, A 7 = a \, /i, V A, B, C Now let (a, /3', 7') be the coordinates of the extremity of the diameter, then we have a', ^', 7' 1 = 0. X, fl, V A, B, 264^ THE GENERAL EQUATION OP TfiE whicli expresses that the straight line (1) is parallel to the straight line which is the tangent at (a', , y). Thus the proposition is established. 271. To find the condition that the diameter = a, ^, 7 A, B, G X, /A, V should he conjugate to the diameter a, ^, 7 A, B, G -V / 1 r = 0. The first equation (1) may be written •0) ,(2). A, B, r G \ /A, V W',V, U A, B, G \ /t, V, = ....(3), and the second bisects chords parallel to the straight line F', U', W df A, B, G dy \, fX, V ^l-^-'l+^'l-- .(*)• If the diameters be conjugate these equations (3) and (4) must represent parallel straight lines ; hence we must have u, w, v W\ V, U' y\ u', W A, B, G > A, B, G, , A, B, G \, IX, V X, /i, V X, fx, V A, B, G V, 1 p = 0. SECOND DEGREE CONTINUED. Or, if I, m, n denote the determinants 265 B, G C, A A, B IX, V ' V, \ J \, fi le determinants B, C C,A I A, B 1 1 /J., V } v\X' > V, A*' the condition takes the form VU + Vmrd ^- Wnri + TT (ww' + n^n) + r {nl + n'l) + Wilm' + I'm) = 0. Cor. 1. From the symmetry of this result we infer that if one diameter be conjugate to a second, the second is also conju- gate to the first. Cor. 2. The equations la + m^ + n7 = 0, ?a + m'i8 + W7=0, will represent a pair of conjugate diameters of the conic /(ot,/3,7)=0, Al + Bm+ Cn = Oj Ar + Bm'+Cn' = 0, ' UU' + Vmm' + Wnn + XT (m»' + m'w) + V {nl' + n'T) + W {Im! + Im) = 0. 272. To find the equation to the diameter parallel to the tangent at a given point. Let (a, ^, 7') be the given point, (a, /S. 7) the centre, and («» /5? 7) ^"7 point on the diameter whose equation is required. ^ provided 266 THE GENERAL EQUATION OF THE The tangent at (a, /?', 7') is given bj df n. df df ^ and the two points {■^, /9, 7), (a, /3, 7) are equidistant from this tangent, therefore df ^ df df - df y:df - df <2TT = .- -^ (Arts. 261 and 263), which can be rendered homogeneous as in Art. 10. 273. To estahlish equations determining the foci of the conic f{y.,^,7) = oy [Def. The foci are a pair of points, equidistant on opposite sides of the centre, such that the rectangle contained by the per- pendiculars i'rom them on any tangent is constant.] Let {y.^, /S,, 7,), and {a,, ^^, 7J be the foci, and let (a, /3', 7) be any point on the conic. The tangent at this point is df ^ df df ^ and the rectangle contained by the perpendiculars upon it from (^^1, ^1, %): («2, /52> 7.) is, (Art. 46) "^f . o^f . df\( df ^ ^ df df df d'/d" df} But by definition this is equal to a constant area (Jc^, sup- pose), hence we obtain SECOND DEGREE CONTINUED. '267 + («A + "A + 2''' cos C) ^, f^ = 0. Now tliis is a relation between the coordinates of any point whatever on the conic, and must therefore (accents suppressed) be tlie equation to the conic, and identical with the given equa- tion which maj be written (Art. 222) <£)--Q'--'^S c//3 dtW 274:. Definitions. A point on a conic at which the tangent is at right angles to the diameter is called a vertex, and the diameter through a vertex is called an axis. 275. To Jlnd equations to determine the vertices of the (ionic whose equation is /(«,A7)-0. Let (a', /8', 7') be the vertex ; then the tangent is at right angles to the diameter a, A 7 a, ^', 7' A, B, C = 0. 270 THE GENERAL EQUATION OF THE Hence if the coordinates be trilinear (Art. 49, Cor. 1). d£ d-x A, B, G 1, —cos C, —cos B + d^' J, B, G ^cos G, 1, — cos A + df_ dy oU /3', 7 A, B, G — coaB, —cos A, 1 = or if they be triangular, df a, ^, 7 J, 5, G a, — &cos(7, — ccos-4 (1), +b''^f «', ^\ 7 ^"dfl ^, ^, — a cos C, 5, — c cos J. dy /3', 7 A, B, G — acoaB, —hcoaA, c = 0. .(1). ' This equation, together with the relation /(a',/3',7') = 0, "will be sufficient to determine the ratios of the coordinates. And since each equation is of the second order there will hefow solutions indicating ^itr vertices. 276. To find the equation to the axes of the conic. The equation (1) of the last article is a relation among the coordinates a', yQ', 7' of any vertex of the conic. SECOND DEGREE CONTINUED. '271 Therefore if we suppress the accents it will represent a locus of the second order passing through tlie four vertices. But it is satisfied also at the point ■ ABC that is, at the centre. Hence it will represent a locus of the second order passing through the four vertices and the centre. But through these five points there can be only one locus of the second order (Art. 147), and the two axes constitute such a locus. Hence the ecjuation will represent the axes. 277^ The equation to the axes may be directly obtained in another form which is sometimes useful, by the following method, which depends upon the property that a conic is sym- metrical with respect to an axis, and tiierefore the two tangents from any point on an axis are equal in length. Suppose the coordinates trilinear. Let X, fj,, V, X', /jb, v be the direction sines of the two tan- gents drawn from a point (a', /S', 7') to the conic. The length of the first tangent is given by the equation /(a' + Xp, /3' + yCA/), 7'+z//j) = 0, and the roots of this equation must be equal, therefore -^ Similarly, (^' I + ''' S + ^' ©'=*/(«'' ^'' ^V(v, /^', .'). But if (a , /S', 7') be any point on either axis of the conic the two tangents are equal, and therefore /(X, ^, v) =/(V, ii, I/').....,... .......... ..(1), consequently ^ df . df ^ dfV f,df ,df' , df\' ,,,, 272 THE GENERAL EQUATION OP THE Also by the identical relations which exist among the direc- tion sines of any straight line (Art. 70), {a\ + hfi + cvY={a\' + hfx.' + cv'y (3), y + »/' + 2fMv cos A = fi'^ + v'^ + 2/iV cos ^ (4), j/' + X'+2z/\cos5 = i;" + X"+2z/'\'cos5 (5), \'+/A'' + 2Vcos(7=\" + /A"+2Xycos (7 (6). Eliminating the six quantities X2 > '2 2 '2 2 '2 ' > ^ K > -v "v ' ' — A., /A— /*, V —V\ [IV — flV , V\— VX, X/t — \/Lt from these six equations, and suppressing the accents'*^ on the coordinates, we obtain da]' a\ u, 0, h 1, W' U7/' d^dr^' dy d' = 4A^ir^ 280 THE GENERAL EQUATION OF THE ^ quadratic equation, giving two values for p^, expressing the squares on the two semi-axes. ''- Hence, if ^ and 23 denote these two semi-axes, we have and 'i ab c EH ^+^~ 16A* ^^ <^+a£> /^ _^. . 286. CoE. Considering the quadratic in p^, we observe that the two values of p^ will be of opposite sign, and therefore one value of + p real and the other imaginary ii K> 0, — the condi- tion that the conic should be a hyperhola. The two values of p^ will be of the same sign, and there* fore hoth values of + p real or both imaginary if K<0, — the con- dition that the conic should be an ellipse. And further, the ellipse will be real when E and IT are of opposite sign, and imaginary when they are of the same sign. Again, both the values of p^, and therefore both values of + p will be infinite if jK'= 0, — the condition that the conic should be a parabola. Similarly, if ^=0, the values of p^ are equal and of oppo- site sign, and the conic is a rectangular hyperbola, as we saw in Art. 247, Cor. 4. So, if 11= 0, the two values of p" are zero, and the conic degenerates into two straight lines, as we proved in Art. 245. 287. To find the length of the latus rectum in a 'parabola represented hy the equation ud' + ?;/3' + wrf + 2m'/37 + 2v'r^0L + 2w'a^ = 0. and : SECOND DEGEEE CONTINUED. 281 In the general case, we have by the preceding articles, — 7— * TT^ ? TT~ \r % ^S, ^3. 73 "i^ ^1. 7i O3J ^3' % «1 » A. 7l a^, ^2' 72 (166) Each of a series of parallel chords in a conic is divided so that the rectangle under the segments is constant. Shew that the points of section lie upon a similar, similarly situated and concentric conic. (167) Each of a series of parallel chords in a conic is divided so that the algebraical sum of the reciprocals of the seg- ments is constant. Shew that the points of section lie upon a similar and similarly situated conic which cuts the original conic at the extremities of the diameter bisecting the parallel chords. (168) A straight line is drawn from the focus of a conic to meet the tangent at a constant angle ; find the locus of the 286 EXERCISES ON CHAPTER XVIII. point of intersection, and sliew that, in the case of the parabola, the locus will always touch it, but in the case of the other two curves it will only touch them (in one or two points) under cer- tain conditions. (169) If A', B', C, K' be the values of the functions A, B, C, K (Arts. 255, 256) for the equation /(ot, /3, 7) + 2 (aa + 5/3 + C7) (?a -\-m^ + n'^ + ^^^"+^"^=^ ^'^' and it will therefore pass through the point given by and wcr' + M'o- = (4), whatever be the value oih. Hence the polars of the poiat {a, ^', 7') with respect to all the conies are concurrent. 304. For convenience of reference we will now recapitulate the results arrived at in the foregoing chapters respecting some particular cases of circles (see Chapters xiv. xv. xvi.). I. The circle with respect to which the triangle of reference is self-conjugate is represented (Art. 179) by the equation in trili- near coordinates a^ sin 2^ + /S' sin 25 + 7' sin 2 (7 = 0, which is obviously a particular case of the equation of Art. 293. The centre of this circle is given by a cos A = 13 cos J3 = y cos C, 19—2 292 CIRCLES. and is the point of intersection of perpendiculars from the angu- lar points of the triangle of reference on the opposite sides. In triangular coordinates the same circle is represented by the equation a' cot ^ + /3' cot 5 + 7' cot 0=0, and its centre is given hy a cot -4 = ^ cot 5 = 7 cot C. II. The circle passing through the three points of reference is given (Art. 195) by the equation in trilinear coordinates ay87 + &7a + ca/S = 0, and its centre is given by the equations « ^ /3 ^ 7 cos -4 COS^ COS C The same circle is represented in triangular coordinates by the equation and its centre is given by « ^ /3 ^ 7 sin 2^ sin 25 sin 2(7' III. The circle inscribed in the triangle of reference is repre- sented (Art. 214) by the equation in trilinear coordinates /- A ,-. B r C ^ Va cos — + V/3 cos — + V7 cos — = 0, ^ A A and its centre is given by a = /3 = 7. In triangular coordinates the same circle is represented by the equation y a cot ~ + y ^ cot -- + y 7 cot - = 0, and the centre is given by a /3 7 CIRCLES. 293 IV. The circles escribed to the triangle of reference (i. e. touching one side of the triangle and the other sides produced) are represented (Art. 214) Iby the equations , A ,-^ . B ,- . G ^ V- a cos — + V^ sin — + Vy sm — = 0, A Jj Ji f- . A ,—^ B r- ■ ^ s/a sm — + n/-^ cos - + V7 sin ^ = 0, ,- . A ,^ . B , — . C ^ V a sin — + V/3 sm — + V— 7 sm — = 0, 2k JL 2i and their centres are given respectively by -a= ^= 7, a= /3=-7. The same circles are represented in triangular coordinates by the equations y^- a cot - + y ^ tan - + y^7 tan ^ = 0» y a tan - + y - /3 cot - + y 7 tan -= 0, f^ a tan - + y^/3 tan - + /\/- 7 cot - = 0, and their centres are given respectively by a a h c a /3 7 a h~ c a /3_ 7 b G f294: CIRCLES. 305. It should "be observed that (^ sin 2^ + yS' sin 2^ + 7^ sin 2 (7 + 2 (yS7 sin J[ + 7a sin 5 + a/3 sin (7) = 2 (a sin J. + )S sin J5 + 7 sin G) (a cos J. + ^ cos j5 + 7 cos C). Hence the straight line represented in trilinear coordinates "by the equation a cos ^ + /3 cos 5 + 7 cos (7=0, is the radical axis of the circle circumscribing the triangle of reference, and the circle with respect to which that triangle is self-conjugate. 306. To find the equation to the circle which passes through the middle points of the sides of the triangle of reference. Let the equation be (Art. 293) 0? sin 2 A + ^ sin 2B+ 7" sin 2 (7 = {la. + m^ + W7) (a sin J. + /3 sin J5 + 7 sin C). Since the circle passes through the middle point of BC it must be satisfied loj a = 0, /3 sin ^= 7 sin C, hence we get cot 5+ cot C = -^-^ + sin B sin (7 ' or sin ^ = m sin C+n sin B, Similarly, since the circle bisects OA and AB, sin B=n sin A + I sin C, sin C=lsinB + m sin A, these equations give I = cos A, m = cosB, n=cosG. Hence the required equation is a^ sin 2 A + j3'' sin 2B + rf sin 2 G = (a cos^ + /8 cos^+7 cos (7)(a sinA+ ^ sin j5 + 7 sin (7)....(1), CIECLES. 295 or, in virtue of the identity of Art. 305, a^ sin 2^ + yS' sin 25 + 7' sin 2 C = 2 (/87 sin ^ + 7a sin 5 + a/3 sin (7) (2), or again, by comparison of (1) and (2),' 2 (/S7 sin ^ + 7a sin 5 + a^ sin G) = (a cos ^ + /3 cos i? + 7 cos (7)(asin^ + /3sin5 + 7sin (7)...(3). All these three forms of the equation are useful. The second shews that the circle passes through the points of intersection of the circumscribed circle, and the circle with respect to which the triangle of reference is self-conjugate: the first and third shew that its radical axis with respect to either of these circles is the straight line a cos A+^ cos B+ry cos C=0. 307. To find where the circle which bisects the sides of the triangle of reference meets those sides again. The equation to the circle is a** sin 2^ + /3' sin 25 + 7^ sin 2 a — 2^87 sin A — 27a sin B — 2cl^ sin = 0. ■ Putting a = 0, we get /3' sin 2B + rf sin 2 C- 2/37 sin A = 0, or /8^ sin BcosB + 7^ sin (7 cos (7 — /Sj (sin BcobC+ sin (7 cos B) = 0, or (/3 sin 5 — 7 sin G) (/S cos 5 — 7 cos B) = 0, shewing that the circle meets BG in the two points deter- mined by a = 0, /8 sin 5 = 7 sin G, and a = 0, /3 cos B = y cos G, that is (Arts. 14, 15), in the middle point of BC, and in the foot of the perpendicular from A. And similarly for the other sides. -296 CIECLES. Hence the circle wMch hisects the sides of the triangle of reference passes also through the feet of the perpendiculars from the oppo- site angles. 308. But this circle has other significant properties with respect to the triangle. Thus, let A, B', C (fig. 37) be the middle points of the sides of the triangle ABC, and let AA", BB", GC" be the perpendi- culars from the opposite angles, and their point of intersection. Bisect OA, OB, OCin a, b, c respectively. Fig. 37. Then, since A", B' , C" are the feet of the perpendiculars from the angular points on the sides of the triangle OBC, there- fore by the last article the circle through A!' , B" , G" will bisect the sides of the triangle OBG, and tlierefore will pass through h and c. Similarly it will pass through a. Hence the same circle which passes through A' , B' , C' , A", B' , G" passes also through a, h, c. In consequence of all these nine points lying on CIECLES. 297 its circumference, this circle is called the nine-points' circle of the triangle ABC. 309. It is easily seen that the property last proved is only a particular case of the following more general theorem. Any straight line drawn from 0, the point of intersection of the perpendiculars of the triangle ABO to meet the circumference of the circumscribed circle is bisected by the nine-points^ circle. 310. Any circle may be represented (Art. 296) by the equation «/37 + br^a + cayS + {la + mjS + nj) (aa + 5/S + cj) = 0...(1). Hence if ua!" + v/3^ + wy^ + 2u^y + 2vyOL + 2w'a^ =0 (2) , be the equation to a circle, this equation must be identical with k [a^y + hyy. + ca/9) + (^ + ^ + ^) («« + 6/3 + cy) = 0...(3). 311. From the result stated in the last article we may readily deduce the conditions (already established by a different method in Art. 244), that the general equation of the second degree should represent a circle. For, comparing the equations (2) and (3), and equating coefficients, we get 2u=ka + -^-\-- — , 6 c ^ I, J J wa uc 2v =hh + — + — , c a t, , J ub , va a ' or ; — Tcabc = v& + wl? — 2uhc = loa^ + uc^ — 2v'ca = uV + va^ — 2uah, the same conditions as we found in the article referred to. 298 CIRCLES. 312. The equation in trilinear coordinates to tlie circle inscribed in the triangle of reference may be written a' cos^ ^ + /S'' cos* f + 7' cos* ^ - 2/37 cos^ f cos^ ? - &c. = 0. Hence (Art. 310) it must take the form h {a^y + hya + ca/3) where (by Art. 311), — ahck = c^ cos* -^ + h^ cos* — + 2hc cos^ — cos^ — - Ji 2i a u = ( 2-^.7 2 ^ c cos 77 + 6 cos — s denoting as in trigonometry ^ (« + 2* + c) ; therefore the equa- tion to the circle becomes a/37 + 57a + cayS = ~ jaa (s - af + hj3{s- by + cy (s - c) 4 (aa + b/3 + cy). By a similar method we may shew that the equations to the escribed circle opposite to A may be written a^y + lya. + ca/3 ahc fa .A 3 . .B y • aG\ , = irW [a <=°^ 2- + f ^'" 2 + ^'-^ 2 j ("« + ^^ + <^). or a/37 + ^7<^ + ca/3 = ^ |«as' + 5/3 (s - c)'' + C7 (s - 5)4 (aa + 5/3 + 07) , CHICLES. 299 and by symmetry the other two escribed circles may be repre- sented by the equations = — |aa (s - cf + &/3/ + cj (s - afX (aa + h^ + cy), and a^y + hja + col^ = -^\aa{s-hy+ hj3 (s - af + cys^'l [m + 5/3 + cy). 313. CoE. The radical axes of the circumscribed circle and the several inscribed (or escribed) circles are represented in trilinear coordinates by the equations acL{s-af-\-h^(s-hf+cy{s-cf = 0, ms'' + 5yS (s - cY +cy{s- hf = 0, aa. (s - cf + h^g" + C7 (s - af = 0, aa (s - by + h/3{s- of + 075' = 0, and therefore in triangular coordinates by the equations a («-«)'+ /8(s-&)^ + 7(s-c)^ = 0, as'' + /3(s-c)''+7(s-5)'=0, a(s-c)' + y8s' + 7(s-a)' = 0, a(s-5)^ + /3(s-a)'+7/ = 0. 314. The equation to the nine-points' circle may be writ- ten (Art. 306) a/37 + 57a + ca/3 1 = - (a cos ^ + /3 cos 5 + 7 cos G) (aa + 5/3 + 07), and the equation to the inscribed circle is a^y + 57a + ca/3 = ^ |aa (s - af +h/3{s- If + C7 (s - c)4 (aa + 5/3 + 07), BOO CIRCLES. therefore their radical axis is represented by ajcos^-- A^|+^|cos5--i^|4-7|cos(7--^ = 0, or £(c-a)(a-5)+£(a-5)(&-c) + ^(5-c)(c-a) = 0, aa hi3 cy or 7 1 --1 r=^ W- — c c — a a — ^ ' Similarly tlie radical axes of tlie nine-points'' circle with each of the several escribed circles will be seen to have the equations _^ + J^__^ = (2) ace hjS cy _ ^__M_ + 3L=o (4). b + c c + a a — b 315. It will be observed that the equation (1) of the last article satisfies the condition (Art. 210) of tangency to the in- scribed circle. - Hen^e the radical axis of the nine-points' circle and the inscribed circle is a tangent to the latter, and therefore the circles touch one another, and the radical axis is the tangent at the point of contact. So the equations (2), (3), (4) of the last article will be seen to represent tangents to the escribed circles ; hence the nine- points' circle touches these circles also. That is, the nine-points' circle of any triangle touches all the four circles which touch the sides of the trioMgle, or the sides pro- duced. CIRCLES. 301 316. It should be observed that the trilinear equation satisfies all the conditions of representing a circle. Its radius is infinite, and the equations to give the centre reduce to - = ^ = ^ 0' shewing that the centre is indeterminate. It represents therefore the circle spoken of in Art. 38, being the limiting form of a circle drawn from any centre at a distance which is indefinitely increased. Every point at infinity lies upon the locus, which indeed is geometrically identical with the straight line at infinity as ex- plained in Art. 38, but analytically it must rather be regarded as equivalent to two straight lines, both lying altogether at infinity. We shall have much more to say about this circle in the chapters on tangential coordinates and polar reciprocals. . 317. The equation (aa4-5/9 + C7)(Za + myS+%7)=0 (1), also satisfies the conditions of representing a circle. To explain this, consider any straight line Za + m/3 + W7 = (2), and suppose a circle is described from any centre so as to touch this straight line. If the distance of the centre from the straight line be in- creased, the circumference in contact with the straight line becomes less curved and tends to coincide with the straight line ;• and by increasing the radius indefinitely the circumference will coincide as nearly as we please with the straight line. Hence ultimately the part of the circle remaining in finite space coin- cides with the straight line, while there is still another part, at 302 CIRCLES. an infinite distance, wliicli since its curvature is indefinitely small will ultimately coincide with the straight line at infinity. Hence the equation (1) which we know represents the straight line (2) and the straight line at infinity, represents the • ultimate form of an infinite circle : — which accounts for its satis- fying the circular criteria. 318. Before we leave the subject of circles, we ought to observe as a particular case of Art. 292 that the equation to the straight lines joining any real point to the circular points at infinity satisfies all the conditions of representing a circle. To take the very simplest case in ordinary Cartesian coordi- nates, the equation a?+y^ = Q may be said to represent two imaginary straight lines through the origin, or to represent an indefinitely small circle at the origin. So in trilinear coordinates the equation /S'+7' + 2/37Cos^ = 0, represents the two imaginary straight lines joining the point of reference A to the circular points at infinity, but it is also the equation to the indefinitely small circle round A. Perhaps this circumstance is best represented by the follow- ing statement : When the radius of a circle is indefinitely decreased the real ^art of the circle degenerates into a point, and the imaginary hranches into two straight lines joining that point to the two cir- cular points at infinity. 319. To find the equation to an indefinitely small circle at the point (a', 13', y). The equations to the straight lines joining this point to the circular points are by Arts. 21 and 110, — 1, cos C±J —1 sin C, cos B + J— 1 sin B a, A 7 a', /3', y = 0, CIECLES. 303 or if A, B, C denote respectively the determinants /3, 7 > 7, a J a, /? ^',7 1 1 7, a a', ^' -^+5cos(7+acos^±7-l(-5sina- asin5) = 0, and therefore tlie two straight lines are given hy {A -Bcos G- Ocos By + (5 sin C-Csin Bf^Q, or [A, B, CY=0. But by the last article these two imaginary straight lines coincide with the indefinitely small circle at {a, /3', 7'), therefore that circle is given by the equation = 0. 320. CoE. 1. A circle such that {a, /3', 7') is the pole of the straight line l{a + a') + m{/3 + /3') + w (7 + 7) = 0, is represented by the equation ^, 7 > 7, a 5 a, ^ /3',J J, a ^, ^' /3, 7 > 7, a 5 a, yS /3', 7 7, a a', /3' + ^ (Za + W2)8 + ny) {aa + 5/3 + 07) = 0. 321. CoE, 2. A particular case of considerable importance is that of the indefinitely small circle at the intersection of the perpendiculars of the triangle of reference. This point is given by the equations a cos J. = /3 cos 5 = 7 cos G, and the equation to the circle reduces to {aacos^, h^cosB, 07 cos (7}^ =0, or in triangular coordinates {a cos A, 13 cos B, 7 cos Cf = 0. 304 EXERCISES ON CHAPTER XIX. 322. Cor. 3. The equation to any circle may be written {aa cos A, h^ cos B, cS = l (2) may be written {A- I))a+{B-I))^+{G-D)y ==!.... (3). But a, /3, 7 are multiples of the trilinear coordinates of the point ; hence the equation (3) expresses implicitly an identical relation amongst the trilinear coordinates of the point, and there- fore must be equivalent to the other forms in which the identical relation can be written (Art. 84), therefore the three terms {A-J))a, {B-JD)^, {G-D)j, must represent respectively the ratios of the triangles OBG, OGA, GAB to the whole triangle ABG. So by eliminating a we should prove that {B-A)^, {G-A)y, {D-A)S, represent the ratios of the triangles GAB', GAG', GB'G' to the whole triangle AB' G'. Similarly, {A-B)a, iG-B)y, {B-B)B, represent the ratios of the triangles Ovl'-S, OBG', GG' A' to the whole triangle O'BG'. And (A-C)a, (B-C)^, {D-G)S, represent the ratios of the triangles OA' G, OB' G, OA'B' to the whole triangle AB' G. The student will observe that these results hold while a, ^, 7, B are the quadrilinear coordinates of any point whatever. 312 QUADEILINEAE COORDINATES. 330. It will be easily seen tliat the coordinates of tlie six angular points of reference are as follows : of ^ ...... a = -S = 2-3;^ » /3 = 7 = 0; of 5 ^=^-^ = B^' 7 = a = 0; of (7 7 = -S=^^, a = /3 = 0; of^' /3=-7=;g:^, a=S = 0; of5' ^ = -« = -oi:i:' /5 = S = 0; of 0' a = -/3=j^, 7 = S = 0. Cor. The coordinates of the middle points of the diagonal AA will be (by Art. 18), 1 p 1 -1 ;v -1 2(^-Z>)' ""^ 2{B-G)' ' 2{B-C)' 2{A-D)' so the middle point of BB is given by -1 p 1 1 ^ -1 .2((7-^)' ^ 2{B-1))' ' 2{G-A)' 2{B-D)' and the middle point of CO is given by 1 p -1 1 ;v -1 ^ = 17T-A eTn ' P^^Tm D\' 7 = ^7777 tT N » o=- 2{A-B)' "^ 2{A-B)' ' 2{C-I))' 2{G-D)' 331. We proceed to interpret some of the simplest equa- tions connecting quadrilinear coordinates. In virtue of equation (1) of Art. 326, the equations /S + 7 = and a + S = (1) must be identically equivalent, and therefore represent the same straight line. But from their form, the first represents a straight QUADRILINEAR COORDINATES. 313 line through A, and the second represents a straight line through A. Hence either equation must represent the diagonal AA. Similarly either of the equations 7 + a = and /3+S = (2) represents the diagonal BB' , and either of the equations a + /3 = and 7-!-S = (3) represents the diagonal GC , Again, the locus of the equation yS — 7 = must pass through the intersection of BB\ CC as well as through the point A. Hence /8 — 7 = represents Aa. So 7-a = Bh, a-/S = Cc, a-S = A'a, yS-S = B'h, and 7-S = C'c. 332. The equations in the last article immediately lead us to some of the most important harmonic properties of a quadri- lateral. Thus from the form of the equations we observe (Art. 129) that the lines AA' and Aa divide the angle at A harmonically. So also the pencils {B.AB'Cb], [G.BG'Ac] as well as the pencils [A' . GAB a], [B . ABG'h], [ G'. BGA'c} are harmonic. 333. From the form of the equations in Art. 331, we observe that the straight lines Aa, B\ Cc ^14 QUADEILINEAR COORDINATES. are concurrent. So A'a, m, C'c, A'a, B'h, Cc, Aa, Bb, Cc, are concurrent systems. Let a (Fig. 39) be the point of intersection of tlie first set, /9, 7, S those of the other sets respectively, Fig. 39. 51 then the points a, ^, 7, S are given respectively by the systems of equations ^=7=S, 7 = S = a, a =^ = 7. 334. To find the condition that three points whose guadrilinear coordinates are given should he coUinear. Let (a', ^', 7', B'), (a", /3", 7", S"), (a'", /3"', 7", 8'"), be the three points. QUADRILINEAE COORDINATES. 315 Suppose they lie upon a straight line whose equation free from d is h +WI/3 +ny =0. Then la! +m^' +^7 =0, hi' +m^" ^ni' =0, la:"+m^"'+ni"=0, therefore a', /3', 7' a'", r. 7'" ,(1). So if we had written the equation to the straight line free from 7 we should have found the condition in the form = 0, a, /3', S' a", /Q"> S" «'", /3"', S'" which is obviously identical with (1), since a' +y8' +7' +S' =0, a" +/9" + 7" + 8"=0, a'" + y8"'+7'" + S'"=0. Thus it will be convenient to write the condition in the form a', /S', 7', S' =0, a , p , 7,6 til r\"i I" s;'" the unequal determinant denoting that we may take any three of the four columns to form a determinant equal to zero. 335. The middle points of the three diagonals of a quadri- lateral are collinear. Taking the sides of the quadrilateral as lines of reference and using the coordinates of the middle points obtained in 316 QUADRILINEAR COORDINATES. Art. 330, the condition that the points should he coUinear will be (bj the last article) 1 1 -1 A-B' B-C B-G 1 1 -1 A-C B-D' A-G 1 -1 1 A-B' A-B' G-D = 0, or B-G, A-D, D-A D-B, C-A, B-D G-D, B-G, A-B which is seen to be an identity by the addition of its rows. .*. &C. Q.E.D. 336. It may be shewn, as in trilinear coordinates, that the polar of the point (a, /3', 7', h') with respect to a conic whose equation is /(a, /3, 7, S)=0, , is represented by the equation Obs. If (a', /3', 7', h') lie upon the conic this equation will represent the tangent thereat. 337. Keferring to Art. 159 we observe that the general equation in quadrilinear coordinates to a conic passing through the four points B, B, G, G', is So the general equation to a conic through G, G' , A, A', is 7a + k/38 = 0, and to a conic through A, A', B, B' QUADRILINEAR COORDINATES. 317 338. As an example we may prove the well known theorem : If a system of conies pass through four fixed points, the jpolars with respect to them of any fixed- point are concurrent. Take the four fixed points as the angular points B, C, B' , C of the quadrilateral of reference. Then the equation to any conic in the system may be written ^'y + lca.h = 0. Let (a', /3', 7', S') be the fixed point, then the polar is given by the equation and therefore it passes always through the point determined by ^ + ^=0, and -, + g.= 0, a fixed point since a', /3', 7', B' are constant. Hence if a system of conies pass through four fixed points the polars with respect to them of any fixed point are con- current. Cor. Let two of the four points be the circular points at infinity, then the theorem reduces to the following, which we proved otherwise in Art. 303. If a system of circles have a common radical axis the polars with respect to them of any fixed point are concurrent. 339. To find the equations to the tangents at B, C, B', C to the conic whose equation is fiy + JcaB = 0. The coordinates oi B are (Art. 329) 318 QUADRILINEAE COORDINATES. Hence, applying Art. 336 the equation to the tangent at B will he 7 — ^a = 0. So the tangent at B' will be given by Similarly, the tangents at G, C will he given hy j3-JcoL = 0, y-hB = 0. 340. CoE. From the form of the equations to the tangents w:e observe, that in ani/ conic passing through B, C, B', G', the tans'ents at B and G] , . ° „ „> meet on Aa; B ... C'l ,, B' ... OJ ' G ... O'J ^^' A great number of well known properties follow from these results. "We will enunciate two of them. I. If a quadrilateral {BGB'G') he inscribed in a conic, and another quadrilateral {^oljS) he described touching the conic in the angular points of the former one, the four interior diagonals of the two quadrilaterals meet in one point (a) and the two exterior diagonals coincide {AA'). II. If a quadrilateral he inscribed in a conic, the points of intersection of opposite sides and the points of intersection of the tangents at opposite angles are collinear. (Camb. Math. Tripos, 1847.) 341. The following proposition exemplifies the use oi per- pendicular quadrilinear coordinates (Art. 325). To shew that the circles circumscribing the triangles AB'G', A'BG', A'B'G, ABC pass all through one point. QUADRILINEAR COORDINATES. 819 In perpendicular coordinates, all the distances being positive towards the interior of the quadrilateral BOB' G' (fig. 39), the equations to the four circles will be (Art. 195) sin G' sin B' sin A {AB'O, "-^ + — = 0, ^ ^' a 7 S ' ^ 'a p tAT,ns ^VQ.A sin 5 sin (7 - {ABG), + — 0- + = 0. And these equations will be satisfied by the same values of a, /9, 7, h, provided = 0. ,(1). 0, sin G', sin B', — sin A sin G', 0, — sin ^', sin B sin 5', sin J', 0, sin G sin A, sin B, sin (7, But if «, 5, c, c? denote the sides BG, GB', G'B, B'G' respectively of the closed figure BGB'G', we obtain by project- ing the sides upon lines at right angles to each of them in order — hsinG+csmB + d sin A' = 0, — asinG+eslnA + d sin B'=0, — a sin ^ + & sin ^ + £? sin (7' = 0, a sin ^' - 5 sin ^ + c sin G' = 0, whence eliminating a : h : c : d, 0, sin G, sin B, sin A' sin G, 0, sin A , sin B' sin B, — sin vl, 0, sin G' — sin A', sin ^, sin (7', =;=0. .(2). 320 QUADRILINEAR COORDINATES. The equations (1) and (2) are tlie same, each of the determi- nants being equal to (sin A sin A' — sin B sin B' + sin C sin C')". Hence the condition (1) is satisfied, and therefore the four circles meet in a point. 342. To find the anharmonic ratio of the range in which the lines of reference are cut by a given line. Let loL + m^ + ?i7 + rS = be the given line, and let it meet a = 0, /3 = 0, 7 = 0, S = in L, M, N, B. Then the equations to BL, BM, BN, BR will be a = 0, (Z — r) a + (»i — »') 7 = 0, 7 = 0, (Z — w) a + (w — 7n) 7 = 0. Hence [LMNR] = ^. f)y { . {L — r){L — m) 343. To find the anharmonic ratio of the pencil formed hy joining any given 'point to the four points of reference B, C, B, C. Let (a', /S', 7, B') be the given point 0. Then the four straight lines are represented loy (OB), V^ = 0, (OB), |-| = 0, iOC), ^-1 = 0, (00-), 1-^ = 0. EXERCISES ON CHAPTER XX. 321 Now let u=:—, — J and v = -^, — kt . 7 ex P <> Then the equations to the four straight lines become M = 0, v = 0, 7% — S'i? = 0, /S'w — av = 0. Hence [O.BCB'0'] = ^. 344. Cor. If lie upon the conic ^87 + KOih — 0, so that then we have or, ^^e anharmonic ratio of the pencil formed hy joining any point on a conic to four fixed points on the same is constant. And conversely, if the anharmonic ratio of the pencil formed hy joining a variable point to four fixed points is constant, the locus of the variable point is a conic passing through the four fixed points. Exercises on Chapter XX. (187) The four coordinates of a point cannot he all positive in the ordinary system of quadrilinear coordinates. But in per- pendicular coordinates the four coordinates of a point may be of the same sign. (188) The general equation lo a conic inscribed in the qua- drilateral of reference may be written (/x. - vf (^7 + o.h) +{v- \f (7a + /3a) + (X - fJ^Y (a^ + 7S) = 0. w. 21 322 EXEECISES ON CHAPTER XX. (189) The equation XjSy + VaS + fjuya -\- fi'/SB + va/S + v'jB = will represent a parabola, provided 0, 1, 1, 1, 1, =0. 1, 0, \, fjb, V, A 1, \, 0, v\ fu,', B 1, ^, v, 0, V, G 1, V, /i', V, 0, D 0, A, B, C, D, (190) The equation /37 + /caS = will represent a parabola, provided K^{B-GY + K[{A-Bf+{G-Df+{A-CY+{B-Df] + {A-i)y = o. (191) Through any four points on a parabola another para- bola can be drawn unless the four points lie on two parallel straight lines. (192) The general equation of a conic circumscribing the triangle formed by the three diagonals of the quadrilateral of reference may be written X (/37 - aS) + /i (7a -^B) + v (a/3 - 78) = 0. (193) The equation to the parabola inscribed in the quadri- lateral of reference is {B- Gy {A -BY (/37+ aS) + {G-Ay {B-By (7a + /5S) + {A- By {G- By (a^ + ryS) = 0. (194) The conic passing through the four points of reference B, G, B, G' and through the fifth point (a', y8', 7', B'), is repre- sented by the equation I3'y' a'B' ~ ^' and its tangent at the point (a', /3', 7', B') is represented by a B ry B ^ — — -4 — = a p 7 6 EXERCISES ON CHAPTER XX. 323 (195) If two conies circumscribing GAG' A', and ABAB', intersect in a point 0, the tangents at divide the angle "be- tween OA, OA' harmonically. (196) If the point of intersection of a pair of common chords of two conies be joined to the points of contact of a common tangent, the pencil thus formed is harmonic. (197) If fom' common tangents be drawn to a pair of conies which intersect in real points, and if the four points of contact with one of the conies be joined in all possible ways bj straight lines, the three points of intersection of these straight lines coincide with the points of intersection of the six common chords of the two conies. 21—2 CHAPTER XXI. CERTAIN CONICS RELATED TO A QUADRILATERAL. 345. We shall use the term tetragram to describe the figure contained by four straight lines indefinitely produced, and not regarded in any particular order. We shall use the word quadrilateral when we speak of the four-sided figure contained by four straight lines taken in a par- ticular order. Thus a tetragram has three diagonals, but a q^uadrilateral has two proper diagonals and an exterior diagonal. Thus any four straight lines a, ^, 7, B forming a tetragram, form three quadrilaterals 'according to the order in which we take the sides, viz. : (1) 7c ** would have no efiect. This is a direct consequence of the circumstance that the coordinates p, q, r of any line are identically connected by a relation of the second order, a^{p — q) [p — r) + h^{q — r) {q—p) + c^{r —p) (r — q)= 4A^, (Art. 74) and not like the coordinates of a point by a simple equation. 362. To find the equation in trilinear coordinates to the straight line whose tangential coordinates are p., q, r. Since p, q, r are the perpendicular distances of the straight line from the points of reference, therefore by Chap, v., the equation to the straight line is apo. + hq^ + cr 9.2> *'2)> ^^^ suppose (a, ^, 7) the trilinear coordinates of their point of intersection. Then since (a, yS, 7) is a point on a straight line whose per- pendicular distances from the points of reference are p^, q^, r^, therefore (Chap, v.) ap^OL 4- Iq^^ + cri7 = 0. Similarly, ap^a + hq^ + cr,r^ = 0. Therefore we have an &/3 cy ^2» ^2 ^1: P, Fi^ 9.x A» 2'2 equations which determine the ratios of the coordinates required. COE. In virtue of Art. 360 it follows that the tangential equation to the same point is ' = 0, K^ 2'2» '^2 a result which we presently establish (Art. 366) without reference to the trilinear system. 364. To find the coordinates of the straight line joining two points whose equations are given. Let Ip + mq + wr = 0, and I'p + m'q + n'r — 0, be the equations to the two points. 336 TANGENTIAL COORDINATES. Then tHe coordinates of the straight line joining them must satisfy both equations (Art. 359), and therefore their ratios are given by v 9. r m, n n, I L, m m\ n w', V T, m To find the absolute values of these coordinates we shall have to substitute their ratios in the relation which we found in Chap. vi. (Arts. 73, 74) connecting the perpendiculars upon any straight line. That relation, as we there shewed, can be written in any of the various forms, c^p^ + W(f + cV^ — Ihcgr cos A — 2carp cos B — 2ahpq cos G = 4A^ a' {p -q){p-r) + ¥{q - r) [q -p>) + c" {r-p) {r - q) = 4A^ {q - rf cot A+{r - pf cot B + {p - qf cot (7= 2A, or with the notation of Art. 46, it may be written [ap, hq, cr}^ = 4A^, the form in which we shall generally quote it. 365. It appears from the foregoing article that by solving together the equations of any two points we may determine the coordinates of the straight line joining them. Hence any two equations of the first degree taken simul- taneously will determine a straight line, viz. the straight line joining the two points which the equations represent separately. Therefore two equations may be spoken of as the equations of a straight line. For example, the straight line {p , q, r') may be said to be given by the equations or p_ _£ and q r p' q q r 21- -1- r p q r THE STRAIGHT LINE AND POINT. 337 ■ 366. To find the equation of the point of intersection of two straight lines whose coordinates are given. In other words, to find the relation among the perpendiculars p, q, r from the points of reference upon any straight line pass- ing through the point of intersection of the given straight lines. Let {pi, q^, r^), {p^, q^^ '^^ he the given straight lines, and suppose ■ - - lp-\- 'mq + nr = (l), the equation of their point of intersection 0. Then this equation expresses a relation satisfied by the coor- dinates of any straight line passing through 0. But {p^, q^, rj passes through 0, therefore Jp^-^mqj^-\-nr^ = (2). Similarly, [p^, q^^^^ passes through 0, and therefore lp^->r mq^+ nr^ = (3). Hence eliminating I: m : n from the equations (1), (2), (3), we get 2' '2 = 0, a relation among the coordinates p, q, r, and therefore the equa- tion of the point 0. ■ r ' Cor. 1. The equation just obtained will not be affected if Pi, q^, r^ or p^, q^, r^ be multiplied by a,ny constant ratio. Hence if the coordinates of one straight line be only proportional to three given quantities p^^, q^, r^, and those of another straight line proportional to p^, q^, r^, the equation of their point of in- tersection is -still p, q, r = 0. w. 22 338 TANGEN'TIAL COOEfilNATES. 367. To find the equation of the point where a given straight line meets the line at infinity. Let {p^, 2'iJ ^i) ^® ^^ given straiglit line, tlien {p^ + h, q^ + h, r, + h) will be a parallel straight line (fig. 41), and these will therefore intersect in the point required. Q: P' B! Hence by the last article the equation required is or Pi, 2'u ^, Pi + h, q^+h, r^-^h = 0, = 0, p, <1, r Ft, 9.x, ^ K h, h or p, q, r Px, ^x, '^x 1, 1, 1 = 0. THE STRAIGHT LINE AND POINT. 339 368. Cor. The last equation is satisfied ii p=q = r, for then the first and third rows of the determinant become identical. Hence all points at infinity lie upon the straight line given Hence p = q = r are the equations of the straight line at infinity » 369. To find the coordinates of a straight line passing through a given point and parallel to a given straight line. Let lp-\-mq + nr = (1) be the equation to the given point, and let (p^, q^, r^ be the given straight line. The coordinates of any straight line parallel to {p^, q^, r^ may be written (fig. 41) p^ +h, q^ + h, r^ + h» If this straight line pass through the point (1) we must have ipi + mq^ -f- nr^ + (? + w + w) 7* = 0, therefore ^^_ ?P. + "'?■ + '"•■ ■ 1 + m + n therefore . p, + A = ^^iiAZ^^iAll^l , ^' l+m-\-"- ' n ^ l + m + n * which are therefore the coordinates required. 370. To find the distance of the point whose equation is Ip +mq + nr = (l) from the straight line whose coordinates are {p', q', r). 22—2 340"^ .■ TANGENTIAL COORDINATES. Let Ti be the distance, then the line paralleLto {-p, g^, t) through the given point will have the coordinates {p ±71, q ±li, r'±h). These must satisfy the equation (1), therefore Ip' + mq +nr' ± {l+m + n)h = 0, • , Ip' + mq + nr - • or A = ± ^^j-^ — ^-; . 371., To find the equation to a point wMch divides in a given ratio the straight line joining two given points. . Let 1 : ^ be the given ratio, and Ip + mq + nr = 0, and Ip + m'q -V nr — 0, the equations to the given points. Suppose p, q, r the coordinates of any straight line through . the required point. Then the perpendicular distances of the given points from this line are in the given ratio. Therefore by^tbe last article, Ip + mq + nr -^ _ I'p + inJq + n'r _ ^ . 7 L+m+n t +m +n the two expressions for the distance having opposite signs, since the two points are on opposite sides of the straight line. Therefore ylp+ vnq -^ nr l']) + m'q 4- nr _ ; l-\-m +n I + m +n a relation among the coordinates of any straight line through the required point, and therefore the equation to the required point. Cor. The middle point between the points . Ip + mq -{■ nr = 0, and Ip + m'q + n'r = 0, . is given by the equation Ip + mq + nr Ip + m'q + n'r l+'m + n V + rn-\-n'^ . " • THE STEAIGHT LINE AND POINT. 341 372.. The principles of abridged notation explained in Chapter viii. for trilinear coordinates are equally applicable to tangential coordinates. . ■ ^ As we there iised m = 0> v = 0, w = to represent equations to straight lines expressed in their most general form, so now we shall use the same, expressions to .denote the most general forms of the equation to the point in tangential coord.inates. 373. If u—0, V = he equations to two -points in tangentia coordinates, then u + icv = 0, {where k is an arbitrary constant) will represent a point lying on the straight line joining the two points. For if ^, q, r be the coordinates of this point thej satisfy the equations m = and v =Q ; that is, their substitution makes u and ■v severally vanish, therefore it must make m + «?; vanish; that is, p, q, r satisfy the equation , ; ; U+ KV = 0, ;, . .: and therefore this equation represents a point on the line [p, q, r). 'q.E.B. 374. If the line joining the points u = and v — ^he divided hy the points u + kv = 0, and u -{■ k'v = 0, the anharmonic ratio of the section is k : k. Let A, B \i& the two points represented by w == and v = 0, and P, Q the two points represented hj u + kv = Q and u-\-kv = 0. Let {p, q, r) be the coordinates of any straight line what- ever, and let u, v' be what u, v become when p, q, r' are written for p, q, r, and let m, n be what w, v become when unity is written for each of "these letters^, q, r. Then the perpendicular distances of the points A, B, P, Q from the straight line [p, q, r) are respectively II v' U + KV U + KV in ' n ' m + kh ' m + Kn ' 342 TANGENTIAL COORDINATES. and therefore the distances AP, AQ, BP, BQ are proportional (bj similar triangles) to the differences U U + KV U U + KV V U + KV V U + KV m m + Kn ^ m m + ku ' n m + ku' n m + k'u ' therefore ^U U + KV AP.BQ \m m + Kn \ fV M +/Ci?\ \)'\n m-\-K'n) [APBQ] ^^ ^ ^p- ^ u' + k'v' \ fv u-vi^ \ni m-^- Kn) \n m + ktiJ K [nu — mv) {mv — nu') K {nu — mv) [mv — nu) ' K or {APBQ]=-,. Q.E.D. K 375. To find the anTiarmonio ratio of the range of the four points whose equations are U + KV = Q, w + Xv = 0, u-\- iiv = 0, u-\-vv=0. The proof of Article 125 (p. 137) applies verbatim. Thus we find that the anharmonic ratio required is {k — \)(/jl — v) (k — v){fi — X)' 376. It follows, as in Art. 123, that the line joining the points u=0, V = 0, is divided by the two points lu^ + 2muv + nv^ = ; so that the anharmonic ratio is (m ±Jni^ — lnY In EXERCISES ON CHAPTER XXII. 343 Exercises on Chapter XXII. (201) The coordinates of the line of reference BG are 2A 0, 0, — . a (202) The coordinates of the perpendicular from A on BG are 0, ±5 cos (7, + c cos B. (203) The coordinates of the straight line through A paral- lel to BG are 2A 2A v. . . a a (204) The straight line joining A to the middle point of BG is given hy p = 0, 2' + r = 0. (205) The equation ^ + r = represents the middle point of the side BG oi the triangle of reference. (206) The equation q tan B + r tan (7 = represents the foot of the perpendicular from the point of reference A upon BG, (207) The equation 'mq-\-nr = Q represents a point P in the line BG such that BF:FG=n:m. (208) The equation mq — nr = represents a point P in the line BG produced, such that FB : FG = n : m. (209) The equation q — r = represents the point of inter- section (at infinity) of straight lines parallel to BG.^ (210) The equation p + q+r = represents the point of intersection of the straight lines which join the angular points of the triangle of reference to the middle points of the opposite sides. 344 JEXERCISES ON CHAPTER XXII. (211) The equation p tan -4 + 2' tan B-\-r tan (7=0 represents the point of intersectidn of the perpendiculars from the angular points on the opposite sides of the triangle of reference. - \ ■ -l (212) The equations ° ' • - +psin-4 + 2'sin5±rsin C=0 represent the centres of the inscribed and escribed circles of the triangle of reference.= . - (213) The equation ^ sin 2^ + g^-sin 25 + r sin 2 = represents the centre of the circle passing through the points of reference. . . , ... (214) The equation \ ... (2' + r)sin2^+ (r + p) sin2B+ (^ + g') sin 2(7= represents the centre of the nine-points' circle of the triangle of reference. . . _ (215) Apply tangential coordinates to shew that the middle points of the three diagonals of a complete quadrilateral are CoUinear. (216) The straight line joining the points Vp-\- mq + Tir = 0, I'p + m'q + n'r = is divided harmonically in the points . - " Ip ■\- mq + nr I'p + m'q + n'r ■ l->tm-\-n ~ I + m +n' . ' CHAPTER XXIir. ^ rrA-NiiE:NTiAi; cooEt)iNA^ES. conic ~sE'ci:ioNs. 377. Definition.: The equation to a curve In tangential coordinates is a relation among the coordinates of any straight line which touches the curve. . . . ■ The equation to a curve is therefore satisfied by the coor- dinates of any tangent to the curve ; and any straight line whose coordinates satisfy the equation is a tangent to the curve. 378. We have already seen that the identical relati'on Con- necting the coordinates of any straight line may be written in any of the forms ,; , /, , . :j (^ - ry cot A + {r -j,f cot B+{p- qf cot (7= 2A, «^ (p - 2) {p -'>') + y" {q - r) {1 -p) + ^ (** -p) (** - 2) = 4^^ or (op, Iq^, cr}^ = 4A^ It should be noticed, that if • then -^ = 2a {ap — 5^' cos G— crcos B)^ . -J- = 2b {hq — crcos A — ap -cos C), -T~ — 2c {cr — ap cos B — hq cos, A) , , dO dO dO ^ ■ - ' and — -+_^+ =0. . dp ctq dr 346 TANGENTIAL COOEDINATES. 379. To find the equation to the circle whose centre is at the point Ip -\- mq + nr = 0, and whose radius is p. Let p, q, r be the coordinates of any tangent to the circle. Then since p is the distance of the tangent from the centre, we have (Art. 370) _ lp-\- mq + nr ^ ^~- l+m+n ' and rendering this homogeneous hy the relation [ap, hq, crY = 4kA\ we get te.„.r=f(to±i7 a relation among the coordinates of any tangent and therefore the equation to the circle. 380. The general equation to a circle is therefore {ap, hq, erf ={\p-{- fiq + vrY, and its radius is 2A X+fJU + v' and the equation to its centre is \p -\- fiq + vr = 0. For comparing the equation just written down with the form which we investigated, we have 2A Ip+mq +nr \p + fiq + vr= — -£-7- — ^- J -^ ^ p l+m + n therefore Xp _ I /*P _ wi ^P _ ** 2K~ l + m + n' 2K~l + m + n' 2K~ l + m + n' and hy addition, (\+fjL + v)p , 2A ^^ ^-r — ^^-^=1, or p = - . 2A ' ^ \+/j,+ v CONIC SECTIONS. 347 381. A particular case of the equation to a circle occurs when X = fjb = v = 0, or when p = go . In this case the equation takes the form [ap, hq, crY = 0, or a' {j>-q) {p-r) + h^{q-r){q -p) +d'(r- p) {r -q) = 0. . . (1), which is evidently satisfied when p = q=^r, shewing that any- straight line lying altogether at infinity is a tangent. But since the coordinates p, q, r of any Jinite straight line satisfy the relation a^p -q)[p-r) + V {q -r){q-p)+ c\r -p) {r-q) = 4A', which is inconsistent with (1), we see that no finite straight line is a tangent to the circle. The circle is in fact that described in Article 38, and would be represented in trilinear coordinates by the equation (aoL + hl3 + cyY = 0. The centre is given by Op+ Oq+ Or = 0, and is indeter- minate : the radius p is infinite. We shall speak of this circle briefly as the great circle. 382. Some writers speak of the equation «' [p-g) {p-r)^- If {q-r) {q-p) + c'{i' -p) {r-q)==0, as representing only the two circular points at infinity: and some correct results are deduced from giving it this interpretation. The discrepancy is precisely analogous to that which attaches to the interpretation of the trilinear equation /3' + 7' + 2/37 cos J. = 0, or to the Cartesian equation It has already been pointed out (Art. 318) that either of these equations represents two imaginary straight lines intersect- '348. TANGENTIAL COORDINATES. ing in a real point, Ibut is also the limiting form of the equa- tion to an evanescent circle at that real point. We explained that a complete description of the locus of such an equation of the second order must recognise the fact that when the real part of a conic section degenerates into a point, the imaginary .branches become two straight lines through the point; and th^ equation to any two imaginary straight lines, intersecting in a real point-^so soon as it is regarded as representing a locus of the second order at all-^-must be regarded as representing the 'ultiniate conic evanescent at the real point and having the two straight lines as imaginary branches. In the present case we have toi deal with the ultimate conic at. the opposite limit. Instead of the diameters, becoming inde- finitely small they have become indefinitely great : but as before the asymptotes are imaginary, and in' the limit the imaginary branches of the curve coincide with them. And just as in the former case, the equation to the conic could in a partial view be regarded as only representing the imaginary asymptotes, so in this case the tangential equation to the conic may be regarded 'as representing only the two circular points at infinity, which are at the same time the points of -contact of the asymptotes and their polars with respect to the curve. We must again refer to the chapter on reciprocal polars, where this point is more fully discussed. 383. To find the equation to the conic section whose foci are at the points Ip + mq -\- nr = Q, Tp + m'q + n'r = 0, and whose conjugate or minor axis is 2p. Let {p, q, r) be any tangent to the conic ; 'then since p^ is equal to the rectangle under ,tlie focal perpendiculars on any tan- gent, we have 2 Ip + mq + nr I'p + m'q -f- n'r ^ l + m +n ' I' + m' + n' ' CONIC SECTIONS. 319; and, rendering this homogeneous Tby the relation ' " {ap, bq, crY = 4A^, ' we get ' - fan hn rr]^ - ^^' ^^'P + ^^ + ^^^) (^> + ^^'g + ^'^) {ap, bq, c^] -.^, (^ + m + n)(Z'T^7T^0~' a relation among the coordinates of any tangent, and therefore the equation to the conic. " ' ' 384. The general equation to a conic may therefore he written up^ + v^ + wr^ + 2u'qr + 2v'rp + Iw'jpq = 0, and the foci are given by the equation ujp^ + vq + vr) {X'p + fi'q + v'r) = 0. Hence the four tangents pass through one or other of the points represented by this equation, i. e. through one or other of the foci of the conic. Therefore, &c. Q. B. D, '391. CoE. 1, We may adopt the following definition of the foci of a conic. The four common tangents to any conic and the great circle at infinity intersect in six ^points which are called the foci of the conic. Two of these six foci are the circular points, as we saw in Art. 385. Hence every real or imaginary tangent to the great circle passes through one or other of the circular points, 392. CoE, 2. The common tangents to two confocal conies pass two and two through the foci and touch the great circle at infinity, ■ 352; TANaENTIAL COORDINATES. ' 393. Tor find the equation to the centre of the conic iBhose tangential equation is f{p, q, r)=0. ■-' Let (t?*^, q', r) "be the coordinates of any diametei*, and sup- pose (p + h, q + h, r +h) a parallel tangent. Then, since (p + h, q' + h, r + A) is a tangent these coordinates must satisfy the equation to the curve, therefore - -^ f{p'^fi, q+h, r+}i)=0, an eq^uation to determine /?. . , We may write it , . /(^■.g>')+A(|+| + f) + iy(l,I,l)=a; .- and since the two values of h must be equal and of opposite sign we have . " dp dq dr But {p, q, r) is any diameter: therefore every diameter passes through the point whose equation is ^ + ^/+^=o - dp dq ' dr ^ - ■ therefore this is the equation to the centre which was required. 394. CoE. I. If the equation to the conic be Written [ap, hq, crY_ + h {Ip + mq-h nr) {I'p + m'q + n'r) = 0, the equation to the centre becomes . . ' Ip + mq + nr Tp + rn'q + n'r _ >. ' l + m + n I' + m' + n' ~ ' a result which we might have inferred a priori from the pro- perty that the centre bisects the line joining the foci. CONIC SECTIONS. 353 395. CoE. 2. If we write the equation to the conic in the general form up^ + v^ + wr^ + 2uqr + 2v'rp + 2w'pq = 0, the equation to the centre takes the form up + vq-Y wr = 0, where u = u-]-v' + w, v = v + w+u', m=io + u' + v. 396. CoE. 3. If f(p, q,r)=0 represent a circle this equa- tion must (Art. 380) be identical with [ap, bq, crY — Jc (up + vq + iJorY = 0. Hence we must have (see Prolegomenon,) u, V, w, 1 u^, v^, w^, u + v + w ,2 7.2 ^2 A = 0, which, therefore, express the conditions that the general equa- tion of the second degree should represent a circle. 397. To find the coordinates of the diameter parallel to a given straight line. Let {p', q, r) Tbe the given straight line, and suppose {p'-\-h, q+h, r' + h), " : ' ■ the parallel diameter. Then these coordinates must satisfy the equation to the centre, therefore u{p' + h) +v {q +h) + w {r +h)=0, 7, _ up + vq + wr ^ - il + v + w ' hence the required coordinates are ^ lp'~- q)-\-w{'p —r) w{q—r) + u{q—p') u{r —p') + v(r -q) u + v + w w. u + v + w u + v + w 23 354 TANGENTIAL COORDINATES. 398. To find the condition that the equation f iPy 2'5 ^) = '"'P^ + '^3^ + ^^^ + '^uqr + 2vrp + 2wpq = 0, should represent a parabola. Tlie necessary and sufficient condition is that the line at infinity should be a tangent. Therefore p = q = r must satisfy the equation. Therefore /(I, 1, 1) = 0, or u + v + w + 2u' + 2v +2w-=0, or u + v + w = 0, the required condition. 399. COE. If the equation to the conic he written {ap, hq, erf + {\p + ^tg + vr) (\'p + fi'q + v'r) = 0, the condition becomes i^ + ix^-v) (V + /a' + v) = 0, shewing that a conic is a parabola if either focus lie at infinity. 400. To interpret the equation i'f-^^f+^f=» « with respect to the conic f(p, q, r) =0. I. Suppose that the straight line {p, q, r) is a tangent to the conic. Then f{Prq',r) = 0, or, as we may write it, , df , , df , df . which shews that the equation (1) represents some point on the tangent (^Z, q, r'). CONIC SECTIONS. 355 Now let {p", q', r") be tlie other tangent from this point. Then since it passes through the point (1), we have „ df ndf „ df which shews that {p\ q^, r') passes through the point given by ^f+^f-'-#=o (^)- But since {p", g", r") is a tangent, we have f{p",f,T")=0, P dp"^^ di'^"^ dr" ^' which shews that {p", q' , r") also passes through the point (2). Hence the point (2) is the point of intersection of the tan- gents {p, q, r') and (/', q", r") ; that is, it coincides with the point (1), therefore the equations (1) and (2) are identical. Therefore, ^ ^ ^ dp^ _ dd __ dr^ df dq dr" of which a solution (and since they are simple equations, the only solution) is evidently p _4 _'!!_ or the tangents {p\ q, r), {p", q", r") coincide. Hence the given equation represents the point of contact of the tangent 23—2 356 TANGENTIAL COORDINATES. But II. suppose {p\ q, r) be not a tangent, then let iP\^ S'lJ ^il') iPz^ 9.2^ ^2) ^6 *^® tangents at the points where {p't 4^ **') Di^ets the conic. Then by the Case I. their points of contact are given by the equations ^ dp^ ^ dq^ dr^ ' ^ dp,^ ^dq^ ^ dr^~ And since these points both lie upon {p\ q, r), we have , df , df ' ^f _ f^ ^ dp^ ^ dq^ dr^ and or and , df , J/ , J/" ^ ^ -^ dp^ ^ dq^ dr^ ' df df df ^ ^^ dp ^^ dq * dr df df df ^ which. shew that (^j, q^^ r^, (p^, q^, r^ pass through the point given by df df df ^ dp ^ dq dr ' df_ df ^_^^o ^ dp " dq dr that is, the equation < represents the point of intersection of tangents at the extremities of the chord (p', q\ r). Therefore always — The pole of the straight line [p, q , r) is represented hy the equation df df df ^ dp ^ dq dr ~ ' CONIC SECTIONS. 357 401. CoE. 1. If the equation to the conic be written uj^ + v^f _i (df df df\ _ ^ dp ^ dq dr \dp dq dr) The conjugate diameter joins these two points : hence its coordinates will be obtained by solving together these two equa- tions. Hence the coordinates are given bj and or by ,df , df ,df ■^ dp ^ dq dr df df .df ^ \ dp dq dr •(1), dp dq _ dr r. — p p' — q ,(2). The equations (2), with the identical relation (Art. 364), determine the coordinates p, q, r required. Cor. The first of the equations (1) shews that the con- jugate diameter passes through the pole (at infinity) of the original diameter. Hence we might express the definition of conjugate diameters thus : Two diameters of a come are said to he conjugate when each passes through the pole of the other. 405. If the equation to the conic be written in the form up^ + v^ + wr^ + 2uqr + 2v'rp + 2w'pq = 0, the equations to determine the diameter conjugate to a diameter {p, q, r) become up + v'r + w'q _vq + w'p + ur _ wr +uq + v'p q —r r —pi ~~ P ~ 9.' ^ CONIC SECTIONS. 35a or P q —r , w , V r — p, V, u p — g', u, w q —r , V , u I r r I r —p f u , w jp — q, w, v' r q -r, u, w' 1 r 1 r -p, w, V p'-q, v\ 1 u Cor. If the equation / (p, g, r) = represent a parabola, (^5 q^ r) will be a diameter, provided r _r —p _p V w 406. To find the asymptotes of the conic f{p, q, r) = 0. Let [p, 2'', /) be an asymptote. Then, since {p\ g'', r) is a tangent wliose point of contact is at infinity, these coordinates must satisfy the equation to the conic, and the coordinates of infinity must satisfy the equation, to the pole of this tangent. Hence and dp dq dr The first of these is a quadratic, and the second is a simple equation ; the coordinates of the two asymptotes will therefore be obtained by solving them together. Cor. It appears therefore that the coordinates of the asymp- totes of a conic are obtained by solving together the equation to the conic and the equation to its centre. Hence (Art. 386) the asymptotes are the tangents to the curve from its centre, 407. To shew that the equation Iqr + mrp + npq = represents a conic inscribed in the triangle ofi reference. 360 TANGENTIAL COORDINATES. The equation is satisfied i? q—0, r = are satisfied. But these equations represent the side BG. Hence jBC is a tangent to the conic. So the other sides are tangents. Therefore &c. Q. e. d. 408. To shew that the triangle of reference is self-conjugate with resjpect to the conic Ip^ + mg^ + nr"^ = 0. By Art. 400, the equation to the pole of the line {p, q', r) is Ipp' + mq(f + nrr = 0. Hence, putting q =0, r — 0, the pole of the side BG of the triangle of reference is given hy p = 0, that is, it is the point A. Hence each side of the triangle of reference is the polar of tlie opposite angular point. Therefore &c. Q. e. d. 409. To find the general equation to a conic circumscribing the triangle of reference. Let up^ + v^ + wr^ + lu'qr + 2vrp + 2w'pq = be the equation of a conic passing through the points of re- ference. The tangents from ^ = are given hy vq^ + wr^ + ^uqr = 0, and these must be coincident ; therefore u=±nvw^ so v' = + VtOM, and w' = + yuv. - CONIC SECTIONS. 361 Hence writing f, m^, n^ for u, v, w, the equation becomes rp' + m^q^ + n^r^ ± 2mnqr ± 2nlrp ± 2l'mpq = 0, and, as in Art. 205, the doubtful signs must be taken either all negative or only one negative, or else the equation would degenerate into two simple equations. Hence the general equation of a conic circumscribing the triangle of reference may be written wlp + Nmq + ^nr = 0. 410. If 8= he the equation to a conic, and u = 0, v = the equations to two points, it is required to interpret the equation JS + KUV = 0, where k is an arhitrary constant. Let {p, q, r) be one of the tangents from the point w = to the conic /S" = 0. ' Then [p, q, r) satisfy both the equations >S'=0 and m=0, and therefore {p, q, r) satisfy the equation S-\- KUV = 0. Hence either tangent from w = to >S^=0 is a tangent to the conic 8 + KUV = 0. Similarly, either tangent from v = to S=-0 is a tangent to the same conic. Hence the equation represents a conic section, so related to the given . conic that two of the common tangents intersect in (w = 0), and the other two in ^(v 5=^0). 362 TANGENTIAL COOKDINATES. 411. To interpret the equation uv + icwx = 0, where w = 0, v = 0, w = 0, x = are the equations to points. Suppose {p, q, t) the straight line joining the points w = 0, w = 0, then these coordinates satisfy both the equations w = 0, t« = Oj and therefore satisfy the equation uv + KWX = 0. But this equation "being of the second order represents a conic section. Hence it represents a conic section touching the straight line joining the points - w = Oj w = 0. Similarly, the conic touches the line joining t^ = 0, flj = 0, and the line joining v = 0, w = 0, and the line joining v = Q, x=0. Hence it represents a conic inscribed in the quadrilateral whose angular points are u = Oy w = 0, v = 0, x=0 in order. 41 2. To interpret the equation uv + KW"^ = 0. As in the last case, this is a conic touching the lines joining the points (w = 0, w = Q>) and {v = 0, w= 0). CONIC SECTIONS. 363 = 0, '=0 J ' Moreover, the tangents from m = to the curve, are given by M = w'' = and therefore are coincident. Hence (m = 0) lies on the curve. Similarly, {v = 0) lies on the curve. Hence the equation represents a conic section passing through the points u=0, v =0, and whose tangents at those points intersect in w; = 0. 413. To interpret the equation Ivw + mwu + nuv = 0, where w = 0, v = 0, w =0 are the equations of three points. Being of the second order the equation represents some conic. The equation is satisfied when v = and w = 0. Hence the straight line joining v= and t« = is a tangent to the conic. Similarly, the straight line joining wr= and w = 0, and the straight line joining u =0, v = 0, are tangents. Hence the equation represents a conic inscribed in the tri- angle whose angular points are m = 0, v = 0, w = 0. 414. By comparison with Art. 408, it will be seen that the equation Iv^ + mv^ + nw^ = 0> represents a conic, with respect to which the triangle formed by joining the points w = 0, V = 0, w=0, is self-conjugate. So it may be shewn as in Art. 409, that the equation tjlu 4- Jmv + sfnw = 0, represents a conic circumscribing the same triangle. 364 EXERCISES ON CHAPTER XXIII. 415. It will be necessary for the student to distinguisli between a curve of the n^^ order and a curve of the n^^ class. The following definitions are usually given. Def. 1. A curve is said to be of the n^^ order when any straight line meets it in n real or imaginary points. Dep. 2. A curve is said to be of the w**" class when from any point there can be drawn to it n real or imaginary tan- gents. A curve of the n^^ order will therefore be represented by an equation of the w*'* degree in trilinear or triangular coordinates, and a curve of the ti*^ class will be represented by an equation of the w*** degree in tangential coordinates. We have shewn that every conic section is both of the second order (Art. 145) and of the second class (Art. 230). Exercises on Chapter XXIII. (217) The equation / tan ^ + / tan J5 + r' tan (7= 0, represents the circle with respect to which the triangle of refer- ence is self-conjugate. (218) The circle circumscribing the triangle of reference has the equation Vp sin ^ + Vg- sin 5 + Vr sin (7 = 0. (219) The circles escribed to the triangle of reference are given by the equations - s^r + (s- c) rj? + («- 5)^2- =0, [s — c)qr — srp + {s — a)pq = 0, (s — h)qr+{s — a) rp — spq = 0. EXERCISES ON CHAPTER XXIII. B65 (220) The circle inscribed in the triangle of reference is given by {s — a)qr+{s— h) rp+{s— c) pq = 0. (221) The equation to the nine-points' circle of the triangle of reference is {ap, hq, crY= [op cos {B—C)+ hq cos {G — A) + cr cos {A —B)]^, or a '\fq + r + h'^r +p + c '^p + q = 0. , (222) The general equation to a conic bisecting the sides of the triangle of reference is {m — nfif + (re — If , a, 5, c, d be the tangents to a conic at the points P, A, B, G, D respectively, then will [p.ahcd] = [P.ABCD]. 430. With respect to a circle, any circle reciprocates into a conic, having a focus at the centre of reciprocation. Take a triangle self-conjugate with respect to the circle of reciprocation as triangle of reference. Then in triangular coordinates the circle of reciprocation has the equation (Art. 179), a" cot ^ + /3' cot ^ + 7' cot (7=0 (1), 382 POLAR RECIPROCALS. and any other circle may be represented by the equation (Art. 322), {acos-4, l3cosB, y cos Cf = {Iol + m^ + nj) {a+ ^ +j)... {2). Now by Art. 424 the reciprocal of the circle (2) with respect to the circle (1) is represented in tangential coordinates by the equation {^ sin -4 , q sin B, r sin CY = (Ip tan A + mqt9inA+ nr tan A) {p tan ^ + $- tan 5 + r tan C), or {ap, iq, erf 2 i-^ (IptSiYiA+rnqtsinB+nrtsai (7)(ptan^+2'tan5+rtan C). sm But (Art. 383) this represents a conic whose foci are given by Ip tan A + mq tan B + nr tan 0=0, and p tan A + q tan B+r tan C=0 ; the latter of which is the equation to the centre of recipro- cation. Hence the reciprocal of a circle with respect to a circle is a conic, having a focus at the centre of reciprocation. Cor. Conversely, any conic reciprocated with respect to a circle having a focus as centre, corresponds to a cii-cle. 431. We can tabulate our results as follows: Keciprocal Loci with respect to a Circle. (xl). _ I _ (xl). A hyperbola having the < A circle having the centre centre of reciprocation as fo" s of reciprocation without it. cus. I See (xxxiii). POLAR RECIPROCALS. 383 (xli). An ellipse having the centre of reciprocation as focus. (xlii). _ A parabola having the cen- tre of reciprocation as focus. (xliii). The directrix of the conic. (xliv). The great circle at infinity. The circular points at in- finity. (xlvi). The foci of a conic. See Art. 391. (xlvii). The focus of reciprocation. (xH). A circle having the centre of reciprocation within it. (See xxxiv). (xKi). A circle passing through the centre of reciprocation. See (xxxv). (xliii). The centre of the circle. (xliv). The evanescent circle at the centre of reciprocation. (xlv). _ The straight lines joining the centre of reciprocation to the circular points, (xlvi). The chords joining the four points in which the correspond- ing conic is cut by radii from the centre of reciprocation to the circular points, (xlvii). The straight line at infinity. 432. Observing that the polar of a point P with respect to a circle whose centre is 0, is the common chord of that circle, and the circle on OP as diameter, and is therefore at right angles to OP, it follows that the angle which two points subtend at the centre is equal to the angle between their polars. Hence, when we reciprocate with respect to a circle, the angle hetween two straight lines is equal to the angle which the corresponding points subtend at the centre of reciprocation. 433. Moreover the distances of a point and its polar from the centre of the circle contain a rectangle equal to the square on the radius. 384 POLAR RECIPROCALS. Hence, when we reciprocate with respect to a circle, the distances of different points from the centre of reciprocation are inversely proportional to the distances of the corresponding lines. 434. By the aid of this property it is easj to calculate the magnitude of the conic corresponding to anj circle with respect to another circle. For let h be the radius of the circle of reciprocation, r the radius of the circle to be reciprocated, and h the distance be- tween their centres. And suppose a and h the semi-axes of the conic, and e its excentricity. By symmetry, the line joining the centres of the circles must be the axis of the conic, and the perpendiculars on the tangents at the vertices lie along this line. We have, therefore, P h? ail + e) = 7 , and a {\ — e)= ^ , Ur P h whence, a = ~^ — p , 6 = -j-r^ — r^ , e = - . r — h v(* h) T 435. It thus appc^ars tha,t the excentricity of the reciprocal conic is independent of the radius of the circle of reciprocation. The magnitude of this circle tlierefore only affects the magni- tude, not the form of the resulting figure. Thus it happens in many cases that the magnitude of the circle of reciprocation does not affect a proposition, and it is therefore often con- venient to speak briefly of reciprocation with respect to a point 0, when we mean reciprocation with respect to a circle drawn at an undefined distance from the centre 0. We will now give some examples of the manner in which the method of polar reciprocals is applied in the solution of problems. 436. Four fixed tangents are drawn to a conic: to prove that the anharmonic ratio of the points in lohich they are cut hy any variable tangent is constant. POLAR RECIPROCALS. 385 Let a, h, c, d denote four fixed tangents to a conic, and let p and q be any other tangents. Reciprocate the figure with respect to d, focus: then the tangents a, 5, c, d correspond to four fixed points A^ B, (7, Z> on a circle, and p, q to any other points P, Q on the same circle. Now, by Eucl. iii. 21, the chords joining A^ B, G, D sub- tend the same angles at P as at Q. Hence, [P. ABCD] = [Q . ABGD] ; therefore by Art. 428, [p . abed] = [q . abed]- Q; E. D; 437. Four fixed points are taken on a conic: to prove that tlie anharmonic ratio of the pencil joining them to any variable point on the same conic is constant. Let A, B, C, D denote four fixed points on a conic, and let P, Q be any other points. Reciprocate the figure with respect to any point; then the points A, Pj G, P, Pj Q correspond to tangents a, b, c, d, p, g to another conic, and therefore by the last proposition {p . abed] = [q . abcd]i , Hence by Art. 428, {P. ABCD] = {Q. ABGD]. q. e. d. 438. An ellipse is inscribed in a quadrilateral: to prove, that any two opposite sides subtend supplementary angles at either focus. Reciprocate the whole figure with respect to a circle having the focus as centre. Then, by Art. 430, the conic corresponds to a circle, and the circutnscribed quadrilateral to an inscribed quadrilateral. By Micl iii. 22, any two opposite angles of this quadrilateral are equal to two right angles. Hence, by Art. 432, any two opposite sides of the corresponding quadrilateral subtend at the centre of reciprocation angles which are together equal to two right angles. Hence the proposition is proved, w. 25 386 POLAR RECIPROCALS, 439. The following "corresponding theorems" will suffice to shew how the principal properties of conic sections may be deduced from the simplest properties of the circle by the method of polar reciprocals : Two tangents to a circle are equally inclined to their chord of contact. Two tangents to a circle are equally inclined to the diameter through their poiut of inter- section. Parallel tangents to a circle touch it at the extremities of a diameter. A chord which subtends a right angle at a fixed point on a circle passes through the centre. In any circle the sum of the perpendiculars from a fixed point on a pair of parallel tangents is constant. If chords of a circle be drawn through a fixed point, the rect- angle contained by the segments is constant. Two tangents to a conic mea- sured from their point of inter- section subtend equal angles at a focus. The segments of any chord of a conic, measured from the direc- trix subtend equal angles at a focus. Tangents at the extremities of a focal choi'd intersect in the directrix. Tangents to a parabola at right angles to one another intersect on the directrix. In any conic the sum of the reciprocals of the segments of any focal chord is constant. The rectangle contained by the perj)endiculars from the focus of a conic on a pair of parallel tangents is constant. 440. The following corresponding theorems illustrate the natm'e of the great circle at infinity: All real points on an evanes- cent conic coincide. All real tangents to an evan- escent conic meet in a point. All real tangents to the great circle coincide with the straight hne at infinity. All real points on the great circle lie on the straight line at infinity. POLAR RECIPROCALS. 387 All imaginary points on an evanescent conic lie on one of two imaginary straight lines. All imaginary tangents to the great circle pass thi-ough one of two imaginary points (viz. the circular points). 441. The following will be also seen to be reciprocal theorems : Pascal's Theorem. If a hex- agon be inscribed in a conic the points of intersection of opposite sides are coUinear. If a quadrilateral circumscribe a conic, the intersections of its opposite sides and of its diago- nals will be the vertices of a self- conjugate triangle. If two triangles be polar reci- procals with respect to any conic, the intersections of the correspond- ing sides lie on a straight line. Brianchon's Theorem. If a hexagon circumscribe a conic the straight lines joining opposite ver- tices are concurrent. If a quadrilateral be inscribed in a conic, the intersections of its opposite sides and of its diagonals will be the vertices of a self-con- jugate triangle. If two triangles be polar reci- procals with respect to any conic, the straight lines which join their corresponding vertices meet in a point. Exercises on Chapter XXIV. (236) If a conic touch the sides BO, CA, AB of a triangle in the points A!, B, C, then at either focus BG, C'A, AB sub- tend equal angles : so also do CA, AB', BO', and so do AB, BO,GA\ (237) If two tangents to a parabola meet the directrix in Z, Z', and if 8 be the focus, the angle Z8Z' or its supplement is double of the angle between the tangents. (238) In the plane of the figure 38 (page 310) anj point is taken, and through A, A', a straight lines AB, A'P', op are drawn so as to make the pencils {A.BPOO}, {A'.BP'O'O}, {a,h2)c0} harmonic. Shew that these three straight, lines are concurrent. ' 25—2 388 EXERCISES ON CHAPTER XXIV. (239) If two conies have a common focus and directrix, the tangent and focal radius at any point on the exterior conic divide harmonically the tangents from that point to the interior conic. (240) With a given point as focus four conies can be drawn so as to pass through three given points, and another conic can he described having the same focus and touching the first four conies. (241) S is the common focus of two conies, and S^, S^ are the poles with respect to either of the directrices of the other. Shew that S, S^, 8^ are collinear. (242) Four conies are described each touching the three sides of one of the four triangles ABC, BCD, GAB, ABB, and all having a common focus S : shew that they all have a common tangent. (243) The reciprocal of a parabola with regard to a point on the directrix is an equilateral hyperbola. (244) The intersection of perpendiculars of a triangle cir- cumscribing a parabola is a point on the directrix. (245) The intersection of perpendiculars of a triangle in- scribed in an equilateral hyperbola lies on the curve. (246) The tangents from any point to two confocal conies are equally inclined to each other. (247) The locus of the pole of a fi:xed line with regard to a series of confocal conies is a straight line. (248) On a fixed tangent to a conic are taken a fixed point A and two moveable points F, Q, such that AP, A Q subtend equal angles at a fixed point 0. From P, Q are drawn two other tangents to the conic, prove that the locus of their point of intersection is a straight line. (249) Chords are drawn to a conic, subtending a right angle at a fixed point; prove that they all touch a conic, of which tliat point is a focus. EXERCISES ON CHAPTER XXIV. 389 (250) Prove that two ellipses which have a common focus cannot intersect in more than two points. (251) OA, OB are common tangents to two conies which have a common focus S, and A SB is a focal chord. Shew that if the second tangents from A and B to one conic meet in G, and those to the other conic meet in D, then G, i>, >S are colli- near, (252) If two conies circumscribe a quadrilateral and have double contact with another conic, the tangents at the extremi- ties of the chords of contact intersect in two points which divide harmonically one of the diagonals of the quadrilateral. (253) Three conic sections have a common tangent, and each touches two sides of the triangle ABC at the extremities of the third side ; shew that if the sides of this triangle meet the common tangent in A', B^ C, each of the points of contact of that tangent will form with A\ B^ C a harmonic range. (254) A triangle ABC is inscribed in a conic, and the tan- gents at the angular points A, B, G are produced to meet the opposite sides in P, Q, B. From these points other tangents are drawn to touch the conic in A\ B! , G'. Shew that if the tangents at A, B, G form a triangle «5c, and the tangents at A', B , G' form a triangle a'b'c\ then A, a, a! are coUinear, so are B, b, h', and so are G, c, c. CHAPTER XXV. CONICS DETERMINED BY ASSIGNED CONDITIONS. 442. We shewed in Art. 147 that a conic can generally be found to satisfy five simple conditions, each condition giving rise to an equation connecting the coefficients in the general equation to a conic. It will, however, be observed that if any of these equations are of the second or a higher order, we shall have two or more solutions indicating two or more conies satis- fying the given conditions. Again, in Art. 201 we gave an example of a double con- dition, when we shewed that if the centre of a conic be assigned this is equivalent to two simple conditions being given: and it will presently be seen that conditions may occur equivalent to three or four or five simple conditions. In order therefore that we may in all cases be able to judge of the sufficiency of any assigned conditions to determine a conic, it will be desirable (1**) To determine what conditions shall be regarded as simple conditions, classifying them according to the nature of the relations to which they give rise, among the coefficients of the general equation. (2") To consider how many conies can be drawn to fulfil five simple conditions when the classes of those conditions are assigned, and CONICS DETERMINED BY ASSIGNED CONDITIONS. 391 (3°) To analyse more complicated conditions, and to deter- mine to how many simple conditions they are equivalent, assign- ing the class of those simple conditions. 443. We shall only find it necessary to make two classes of simple conditions, which we shall distinguish as jjoint-condi- tions and line-conditions. We shall find that all other conditions of common occurrence may be regarded either as particular cases of these two, or as made up of repetitions of them. 444. Def. We shall call two points conjugate with re- spect to a conic when each lies on the polar of the other, and we shall call two straight lines conjugate when each passes through the pole of the other. 445. Let /(a? ^, i) = ur^' + ^'yS' + W + 2?*'/37 + 2w'7a + 2w;'a/3 = be the equation in trilinear coordinates to a conic section, and let (Kj, /3j, 7J, (ttg, ySg, 72) be any two points in the same plane. The equation wa,o(2 + ^J^A + t«7i72 + w' (A72 + /327i) + ^' (Ti^a + IP-^ + w?'(aA + a2^x) = 0, may be written in either of the forms df ,^ df d.f ^ df „ df df ^ "^i. + ^'^ + '>'«^ = »' and expresses the condition (Art. 232) that each of the points (a^, /3i, 7i), (0^2' ^^2' 72) ^^^^ ^^ ^^^® polar of the other with respect to the conic, or that the two points are conjugate with respect to the conic. It will be observed that when the two points ia^,^^,jj, (a^, ^^, 72) are given, the condition that they should be conjugate furnishes us with a simple equation, connecting the six co- efficients in the general equation to a conic. Five such con- 392 CONICS DETERMINED BY ASSIGNED CONDITIONS. ditions will therefore suffice to determine the five ratios of the coefficients in the equation, and therefore to determine the conic. 446. Let l^a + m^l3 + n^y = 0, and Ijx. + WgjS + n^y = 0, be two straight lines. Then the equation u, w, v, Z, =0 w, V, u, m^ v, u , w, n^ ?„ Wg, w,, will express the condition (Art. 233) that each passes through the pole of the other, or that the two straight lines are conjugate with respect to the conic. It will be seen that this equation is a quadratic in w, v, w, u, v, w. Hence when two straight lines are given, the con- dition that they should be conjugate furnishes us with a quad- ratic equation connecting the six coefficients in the general equation to the conic. Therefore if a condition such as this be substituted for one of the conditions in the case last con- sidered, there will be an ambiguity in the determination of the coefficients of the trilinear equation unless it happen that the quadratic have equal roots. 447. Let f{p,q,r)= up^ + vq^ + wr^ ■+ %u'qr + 2t;'rp + 2w'pq = 0, be the equation to a conic section in tangential coordinates, and let (p,, q^, rj, (^2, $'2, r^ be any two straight lines in the same plane. The equation n\P, + n^ ^2 + '^^'J, + u {q;i-^ + q,r^) + v (r^ p^ + 7\p^ + w {p^q^+p^q;)=0, may be written in either of the forms CONICS DETEKMINED BY ASSIGNED CONDITIONS. 393 or §L ^ ^ df df df _ P'^^,^^' dq^ + n dr. 0, and expresses the condition (Art. 232) that each of the straight lines (Pj, 2'i3 ^i)j (^2 5 2'2' ^2) Passes through the pole of the other with respect to the conic, or that the two straight lines are con- jugate with respect to the conic. It will be observed that when the two straight lines (^^ , q^ , rj, {p^, q^, r^ are given, the condition that they should be con- jugate furnishes us with a simple equation connecting the six coefficients in the general equation to a conic. Five such con- ditions will therefore suffice to determine the five ratios of the six coefficients in the equation, and therefore to determine the conic. 448. Let l^p +m^q + njr = 0, and ^2 P + ^2^ + n^r = 0, be two points. Then the equation u, w, v, \ =0 will express the condition (Art. 233) that each lies on the polar of the other, or that the two points are conjugate with respect to the conic. It will be seen that this equation is a quadratic in u, v, w, u, v, w'. Hence when two points are given, the condition that they should be conjugate furnishes us with a quadratic equation connecting the six coefficients in the general equation to a conic. Therefore if a condition such as this be substituted for one of the conditions in the case last considered, there will be an ambiguity in the determination of the coefficients of the tangential equation unless it happen that the quadratic have equal roots. 394 CONICS DETEEMINED BY ASSIGNED CONDITIONS. 449. Def. The condition that a conic be such that with respect to it two given points are conjugate, is called a point- condition. The condition that a conic be such that with respect to it two given straight lines are conjugate, is called a line-con- dition. 450. To fulfil five point-conditions there can he drawn one and only one conic. For, using trilinear coordinates, each of the five conditions will furnish us with a simple equation (Art. 445) connecting the coefficients of the general equation to a conic. These five equa- tions will determine the five ratios of the coefficients without ambiguity, and therefore will determine one and only one conic fulfilling the given conditions. 451. To fulfil four point-conditions and- one line-condition there cannot he drawn more than two conies. For, using trilinear coordinates, each of the four point- conditions will furnish us with a simple equation (Art. 445) connecting the coefficients of the general equation. And the line-condition will furnish us with a fifth equation, a quadratic (Art. 446), connecting the same coefficients. These five equa- tions will determine the five ratios of the coefficients, but since one is a quadratic there will in general be two solutions, indi- cating two conies fulfilling the given conditions. 452. To fulfil three point-conditions and two line-conditions there cannot he drawn more than four conies. For, using trilinear coordinates, each of the three point-con- ditions will furnish us with a simple equation (Art. 445) con- necting the coefficients of the general equation and the two line- conditions will furnish us with two more equations, quadratics (Art. 446), connecting the same coefficients. These five equa- tions will determine the five ratios of the coefficients, but since two are quadratics there will in general be four solutions, indi- cating four conica fulfilling the given conditions. CONICS DETERMINED BY ASSIGNED CONDITIONS. 395 453. To fulfil three line-conditions and two point-conditions there cannot he drawn more than four conies. For, using tangential coordinates, each of the three line- conditions will furnish us with a simple equation (Art. 447) connecting the coefficients of the general equation, and the two point-conditions will furnish ^us with two more equations, quadratics (Art. 448), connecting the same coefficients. These five equations will determine the five ratios of the coefficients, but since two are quadratics there will in general be four solutions, indicating four conies fulfilling the given conditions. 454. To fulfil four line-conditions and one point-condition there cannot he drawn more than two conies. For, using tangential coordinates, each of the four line-con- ditions will furnish us with a simple equation (Art. 447) con- necting the coefficients of the general equation. And the point- condition will furnish us with a fifth equation, a quadratic (Art. 448), connecting the same coefficients. These five equations will determine the five ratios of the coefficients, but since one is a quadratic there will in general be two solutions, indicating two conies fulfilling the given conditions. 455. To fulfil five line-conditions there can he drawn one and only one conic. For, using tangential coordinates, each of the five con- ditions will furnish us with a simple equation (Art. 447) con- necting the coefficients of the general equation to a conic. These five equations will determine the five ratios of the coeffi- cients without ambiguity, and therefore will determine one and only one conic fulfilling the given conditions. 456. It remains that we should analyse the conditions most usually assigned, and determine to how many point- or line- conditions they may severally be equivalent. We shall then be able to apply the five preceding articles to determine how many conies (at most) can be drawn in cases where such conditions are given. 396 CONICS DETERMINED BY ASSIGNED CONDITIONS. I. Given a point on a conic. Since a point on a conic lies on its own polar it is conjugate to itself. This therefore is equivalent to one point-condition. II. Given a tangent to a conic. Since a tangent to a conic passes through its own pole it is conjugate to itself. This therefore is equivalent to one line-con- dition. ' III. Given a diameter. Any diameter passes through the centre, which is the pole of the straight line at infinity. Hence a diameter and the straight line at infinity are conjugate lines. This therefore is equivalent to one line-condition. IV. Let a given point he the pole of a given straight line with respect to a conic. Let P be the given point and QR the given straight line. Then the polar of P passes through Q, which is one point-con- dition ; and the polar of P passes through R, which is another. Hence the data are equivalent to two point-conditions. Or we may reason thus : the pole of QR lies on PQ, which is one line-condition, and the pole of QR lies on PR, which is another. Hence the data are equivalent to two line-conditions. Therefore the pole of a given straight line being given may be regarded as equivalent to two point^conditions or two line-con- ditions. V. Given a point on a conic and the tangent thereat. This is a particular instance of the last case, the given pole lying on the given polar. It is therefore equivalent to two point-conditions or two line-conditions. VI. Given an asymptote. This is an instance of the last case, the given point being at CONICS DETERMINED BY ASSIGNED CONDITIONS. 397 infinity. It is therefore equivalent to two 'point-conditions or two line-conditions. VII. Given the direction of an asymptote. In this case one of the points in which the conic meets the straight line at infinity is given. It is therefore an instance of (i) and is ec[uivalent to one point-condition. VIII. Oiven that the conic is a parabola, Or that the line at infinity is a tangent. This is an in-- stance of (ii) and is therefore equivalent to one line-condition. IX. Given that the conic is a circle, Or that it passes through the two circular points. By (i) this is equivalent to two point-conditions, X. Given the centre. The centre is the pole of the straight line at infinity; hence this case is an instance of (iv) and is therefore equivalent to two point-conditions or two line-conditions, XI. Given a self-conjugate triangle. A triangle is self-conjugate if each pair of angular points are conjugate. Hence this case is equivalent to three point-con- ditions. Or agaiuy a triangle is self-conjugate if each pair of sides are conjugate lines. Hence it is equivalent to three line-con-: ditions. Therefore a self-conjugate triangle being given, constitutes three point-conditions or three line-conditions. XII. Given in position {not in magnitude) a pair of con- jugate diameters. A pair of conjugate diameters form with the straight line at infinity a self-conjugate triangle. Hence this is an instance of (xi) and is equivalent to three point-conditions or three line- conditions. 398 CONICS DETEEMINED BY ASSIGNED CONDITIONS. XIII. Given the directions of a pair of conjugate diameters. The points where any lines in these directions meet the line at infinity are conjugate points. Hence this is equivalent to one point-condition. XIV. Given in position an axis. The axis is a diameter, and this being given is equivalent to one line-condition. But the direction of the conjugate dia- meter is known to be at right angles to this, which gives by (xiii) a point-condition. Therefore that an axis be given in position is equivalent to one point-condition and one line-condition. XV. Given in position the two axes. This is no more than a case of (xiv) and is equivalent to three point-conditions or three line-conditions. XVI. Given a focus. The two tangents from the given point to the great circle at infinity are tangents to the conic. Hence two tangents are given, and therefore by (ii) the data are equivalent to two line-conditions. XVII. Given a similar and similarly situated conic. Since similar and similarly situated conies are those which meet the straight line at infinity in the same points, this is equi- valent to two points being given. Hence by (i) it may be treated as two point-conditions. 4:57. When a conic has to be drawn subject to conditions having reference to another conic, we may often estimate the value of the conditions by considering the particular case in which the latter conic reduces to two straight lines. Thus : XVIII. Given a conic having double contact with the re- quired one. Consider the case when the given conic reduces to two straight lines. Then we have two tangents given, furnishing CONICS DETERMINED BY ASSIGNED CONDITIONS. 399 two line-conditions. Hence we may infer that generally a conic having double contact with the required one being given is equivalent to two line-conditions. The following are examples of the application of our results : 458. Only one parabola can he inscribed in a given qua- drilateral. That the required conic is a parabola is one line-condition (viii) ; that it touch the sides of the quadrilateral gives four more. Hence we have five line-conditions, and therefore (Art. 455) the conic is absolutely determined. 459. Not more than two 'parabolas can be described about a given quadrilateral. That the required conic is a parabola is a line-condition (viii); that it circumscribe the quadrilateral gives four point- conditions (i). Hence (Art. 451) not more than two solutions are possible. 460. Two conies can generally be described with given foci and passing through a given point. For the foci give four line-conditions (xvi) ; and the point gives a point-condition (i). Hence (Art. 454) there will gene- rally be two solutions. 461. Only one conic can be described with given foci so as to touch a given straight line. For the foci give four line-conditions (xvi), and the tangent gives a fifth (ii). Hence (Art. 455) there is only one solution. 462. Only one conic can be described with a given centre, with respect to which a given triangle shall be self-conjugate. For the self-conjugate triangle may be regarded as giving three point-conditions (xi), and the given centre as giving two more (x). Hence (Art. 450) there will be only one solution. 400 EXERCISES ON CHAPTER XXV. Exercises on Chapter XXV. (255) Given two tangents and their chord of contact, shew that only one conic can he described so as to touch a given straight line. (256) Given two tangents and their chord of contact, shew that only one conic can be described so as to pass through a given point. (257) Two confocal conies cannot have a common tangent. (258) Three confocal conies cannot have a common point. (259) Two concentric conies cannot circumscribe the same triangle. (260) Two concentric conies cannot be inscribed in the same triangle. (261) Only two conies can be described about a triangle having an axis in a given straight line. (262) Only two conies can be inscribed in a triangle and have an axis in a given straight line. (263) Four circles can generally be described through a given point so as to have double contact with a given conic. (264) Four circles can generally be described so as to touch a given straight line and have double contact with a given conic. (265) Only one conic can be described having double con- tact with a given conic, and such that a given triangle is self- conjugate with respect to it. (266) One conic can generally be inscribed in a given quad- rilateral so as to have its centre on a given straight line. CHAPTER XXVI. EQUATIONS OF THE _THIKD DEGREE. 463. Definitions. The curve determined by an equation of the third degree in trilinear coordinates, or in any other system in which a point is represented by coordinates, is called a cubic curve, or a cubic locus. The curve determined by an equation of the third degree in tangential coordinates where a' straight line is represented by coordinates, is. called a cubic envelope. 464. Every straight line meets a cubic locus in three points^ real or imaginary, coincident or distinct: and from every point there can be drawn to a cubic envelope three tangents real or ima- ginary, coincident or distinct. For. to find the points of intersection (or the tangents) we have to solve together the equation to the given straight line (or the given point), which is of the first degree, and the equation to the curve, which is of the third degree. Hence we shall have three solutions real or imaginary, equal or unequal. All the solutions however cannot be imaginary, since ima- ginary roots enter into an equation by pairs. One at least must be real, and the other two either both real or both imaginary. Hence every straight line meets any cubic locus in one or three real points, and from every point there can be drawn either one or three real tangents to any cubic envelope. CoE. By Art. 415, a cubic locus is a curve of the third order, and a cubic envelope is a curve of the third class. w. 26 402 equations op the third degeee. 465. Definitions. I. A point in wliicli two brandies of a curve intersect, or at which there are two distinct tangents, is called a double point. II. A point in which more than two branches intersect, or at which there are more than two distinct tangents, is called a multiple jpoint. A multiple point is said to be of the w* order when n branches intersect in it, or when n tangents can be drawn at it. III. When a closed branch of a curve becomes indefinitely small so as to constitute an isolated point satisfying the condi- tions of a point on the curve, it is called a conjugate point. From the consideration that a conjugate point is an indefinitely small oval, it follows that any straight line through it must be regarded as the ultimate position of a chord of the oval. Any such straight line will therefore satisfy the condition of meeting the curve in two coincident points. IV. A cusp is a point on a curve at which two branches meet a common tangent and stop at that point. Any straight line through a cusp must be regarded as cutting both branches at the cusp, and therefore satisfies the condition of meeting the curve in two coincident points. V. If the two branches having the common tangent be con- tinued through the point, then the point is called a point of oscu- lation. VI. A point at which a curve crosses its tangent, is called a point of inflexion. If P be a point of inflexion and Q be another point on the curve very near to P, the straight line QP being produced through P, will meet the curve again in another point Q\ very near to P. If this straight line turn about the fixed point P until it ultimately coincide with the tangent, since it must ulti- mately be a tangent to the branch on each side of the point P, it follows that as Q approaches P so also will Q\ and that they will EQUATIONS OF THE THIRD DEGREE. 403 both simultaneously arrive at coincidence with P. Hence the tangent at a point of inflexion may be regarded as meeting the curve in three coincident points. All the foregoing are often classed together as singular jpoints. VII. A double tangent to a curve is a tangent which touches the curve in two distinct points. 466. It will be observed from the definitions in the last article, that a double point, cusp, and conjugate point are marked by the same property, that any straight line through such a point meets the curve in two coincident points, and that a tangent thereat meets the curve in three coincident points. But they are distinguished by the property that the two tangents at a double point are distinct, at a cusp — -coincident, and at a conjugate point — imaginary. Again, a cusp and a point of inflexion are both characterised by the property that the tangent at such a point meets the curve in three coincident points, but they are distinguished by the fact that a straight line other than the tangent meets the curve in only one point at a point of inflexion, but in two points at a cusp. 467. A cubic curve cannot have more than one double pointy cusp or conjugate points For, if possible, let it have two such points P and Q^ and join them by a straight line. Then this straight line cuts the curve in two coincident points at P, and in two coincident points at Q (Art. 465), i. e. in four points altogether, Which is impossible (Art. 464). 468. A cubic curve cannot have a double tangent. For such a tangent, touching at P and at Q^ would meet the curve in two coincident points at P and in two coincident points at Q, i. e. in four points altogether, Which is impossible (Art. 464). 28—2 ^4 " EQUATIONS OF THE THIED DEGREE. 469. A cubic curve cannot have a point of osculation. For the tangent at a point of osculation, toucliing both branches of the curve, would meet it altogether in four points. Which is impossible (Art. 464). 470. The general homogeneous equation of the third degree in three coordinates consists of ten terms, viz. the three terms whose arguments are . the six, o^y.) x^z ; ifz, y^x ; s^aj, z^y ; and the one, xyz. If the coefficient of any one of these terms be arbitrarily assigned, those of the remaining nine will be undetermined constants. Hence the general equation of the third degree involves nine undetermined constants, and can therefore generally be made to satisfy nine independent conditions. Hence a curve represented by an equation of the third degree can generally be drawn through nine given points, or otherwise made to satisfy nine given conditions. 471. If the nine conditions be given, the equation to the cm've can generally be determined. If any less number (r suppose) of conditions be given, a series of curves can gene- rally be drawn to satisfy them, and their general equation will involve the complementary number (9 — r) of undetermined con- stants. For example, we shall shew in the next article that the general equation in trilinear coordinates to a curve of the third order, circumscribing the triangle of reference -45(7, and whose tangents at -4, 5, C are represented by the equations w = 0, V = 0, w; = 0, respectively, has for its equation «/37 + lu^ + mv/3" + nw («> ^, i) _ -^ (a, /3, 7) <^ (a, /3', 7') ^ («', /3', 7') (1) EXERCISES ON CHAPTER XX VL 421 must represent it. For since the functions (f) (a, /3, 7) and yjr (a, /3, 7) are of the third degree, this equation is of the third degree and therefore represents some cubic. But it is satisfied at the point (a, ^, 7), and also at all points of intersection of the two cubics (a, /3, 7) = and ■\jr (a, /3, 7) = 0, and therefore at the eight given points. But the two cubics- '' denotes the expression obtained by expanding (^ d d dy as if each of the expressions.^-^, M'-joi v -j- were an alge- braical term, and then replacing every such term as which occurs, by ■\n-p-g,.P„g ^J GENERAL PROPERTIES OF HOMOGENEOUS FUNCTIONS. 429 So also will denote 510. If f[a, /3, 7) &e « homogeneous function of a, y5, 7 o/" ^Ae ( p + qf^ degree, then I f d d dy ^, ^ . 1 / ^W^^§=o w- But since (a", j8", 7") lies on the locus of the equation (2), we have a. —a. _ f3 —p _ 7 —7 X fjt, V ' in virtue of which, the equation (4) becomes (,"_„.) ^, + (^"_^.)^^ + (y'_y)^=o (5). But from (3), by the property of homogeneous functions (Art. 508), we have „n df , o'r df „ df _ hence the equation (5) becomes «'|;+^'f + -';$=» w- THE GENERAL EQUATION OF THE n^^ DEGREE. 433 But (a", /S", y") is the point of contact of any tangent from (a? /S'j 7) to the curve; hence all the points of contact of tan- gents from (a', /6', 7') to the locus of the equation /(a, iS, 7) = lie upon the locus of the equation .(7). But again, since (a', ^', y) is any point upon the tangent at (a", /3", 7"), therefore the equation (6) may be also read as stating that the equation to the tangent at any 'point (a", /3", 7") on the locus of the equation /(a,/3,7) = is represented hy the equation df ^ df df ^ .(8). 514. If \, fX; V he the direction sines of the tangent at any point (a, yQ', 7') on the curve f {a, /3, 7), then will 1^ df df d^' dy' sin 5, sin G df df dy ' dal sin C, sin A df df dd' d^ sin J^, sin 5 1 [df df \dd' d^" c ly'] For by equation (4) of the last article, we have ^df ^ df ^ df ^ and by the identical relation (Chap, vi.), X sin ^ + yu, sin 5 + i^ sin (7 = 0. W. 28 434 THE GENEEAL EQUATION OP THE n^ DEGEEE. Therefore H' df df d^" dy' sin B, sin C df df dy" da' sin C, sin A df da' df d/B' sin^, sin^ .(1). But . /ji^ + v^+ 2iJbv cos A = sin^^, whence each of the equal fractions in (1) df df dfy da' J/3' dri) the coordinates being trilinear. Therefore, &c. Q. e. d. 615. The locus of the equation (7) of Art. 513, containing all the points of contact of tangents from {a, /3', y) to the curve represented bj the equation may be conveniently called the first polar curve of the point (a', /3', 7') with respect to the original curve. It will be observed that this polar curve is of an order lower by unity than the original curve, and when the original curve is a conic section, the first polar curve of any point, becomes the straight line which we have been accustomed in previous chapters to speak of as the polar of that point with respect to the conic. 516. If we take the equation to the first polar of the point (a, /5', 7') with respect to the given curve, and form the equa- tion to the first polar of the same point with respect to this new curve, the locus of this equation is called the second polar of the point (a', yS', 7') with respect to the given curve. Similarly THE GENERAL EQUATION OF THE n^^ DEGREE. 435 the polar with respect to the second polar is called the third polar, and so on. The equation to the first polar of the point (a, /3', 7') being «'f+'^| + v|=o (1,, the equation to the second polar will be or, as we may write it, (4 + '3'| + T'3/(«,A7)=0 (2). So the equation to the third polar may be written (4a + '3-| + v|)V(«,A7) = (3), and the equation to the r^ polar, 517. It has been observed that the first polar is of an order one less than that of the original curve. So each polar is of an order one less than that of the preceding polar, and therefore, the original curve being of the rfi^ order, its (n — 1)^^ polar will be a straight line represented by the equation 518. As our condition of tangency we have simply express- ed that a line should meet a curve in two coincident points. It is obvious that this condition will be satisfied by any line through a cusp, multiple point, or conjugate point. Hence every cusp, multiple point, or conjugate point in any curve will lie upon the polar curve of any point whatever. 28—2 436 THE GENERAL EQUATION OF THE W*'' DEGEEE. Therefore the cusps, multiple points, and conjugate points of the curve whose equation is /(a,/3,7)=0 lie upon the curve where the ratios a : yS' : 7' may have any values whatever. Hence the coordinates of all these singular points must satisfy simultaneously the equations da. "' d^ ' c?7 519. If a point lie u])on a fixed straight line, its first polar with respect to a curve of the n^"^ order will pass through [n — 1)^ fixed points. Let the point (a', /S', 7') lie upon the straight line la + w/3 + W7 = 0. Its first polar curve with respect to the curve / (a, ^S, 7) = 0, is represented by Therefore, since la! + m^' + n (a, /3,2, the determinant 3?- COj^ , OC^ a^ J * * * "^1 ^'a fl7„ — a^ , OC^ — flj ; * ■ * ^3 ^n = 0. a, 13, 7 a, P, 7 a , /3 , 7 456 MISCELLANEOUS EXERCISES. (318) Prove that 0, 1, 1, 1 1, 0, z\ f 1, z\ 0, ic" 1, y, x\ = [x + y + z) [x - y - z) {y - z - x) {z - X - y). (319) Shew that if H = u, w\ 1 V w, V, u 1 1 u , IV and K= tlien will w, w , V i a w, V, u, h v , u , w, c a, h, c, Ea^ + Ku, Hah + Kw, Eca + Kv = 0. Hab + Kw', HV' + Kv, Hbc + Ku Hca + Kv, Hhc + Ku\ He" + Kw (320) Prove that the determinant of the (« + 1)*"^ order 0, 1, 1, 1, ... 1, 0, a + b, a + c, ... 1, b + a, 0, b + c, ... I, c + a, c + b, 0, ':':': • &c. = -(-2)-«Jo.,.(i + ^+i + ...). 2(a + b + cy (321) Prove that b + c a a ' b+c' b + c b c + a b c + a' b ' c + a c c a + b a + b ' a+b ib + c){c + a) {a + b)' MISCELLANEOUS EXERCISES. 457 (322) If (a^, ^^, 7j, (a^, ^^, ryj, (a3, /^g, 73) be the trilinear coordinates of three points, the ratios of the sides of the triangle of reference are given by the equations 1, 1, 1 1, 1, 1 1, 1, 1 A: ^.. /3s 7i> %, % «!, a^, Ctg 7i» 725 73 «!, a„ a^ /3l. ^2. -^3 (323) If (a, A 7), (a, z^', 7')^ («", /3", 7") be the coordinates of three points, and (/ca, Ka, Ka"), {k'/S, k'^', /c'/3"), {kj, k"j, k'j") the coordinates of three other points, shew that a h c - - + - + — = a + o + c. Also prove that if the first three points are collinear, so also are the other three. (324) If in a homogeneous equation in trilinear coordinates the sum of the coefficients on each side of the equation be the same, then the equation will be satisfied by the coordinates of the centre of the circle inscribed in the triangle of reference. (325) ABC is a triangle, right-angled at G: draw AE, BF perpendicular and equal to AC, BC respectively; join AF, BE, and draw CD perpendicular to AB. Then the three lines AF, BE, CD will be concurrent. (326) The straight line whose equation is Za + m^ -\-nviiB Xahc&mA ahc sin G 2{c^-\ab) ' 2Qf-\ah)' \ab-& ' (384) Prove that if w = 0, v = 0, t« = be the equations of the sides of a triangle, the equation of a conic section circum- scribed about the triangle will be I m n ^ - + - + - = 0, U V w w. 30 466 MISCELLANEOUS EXERCISES. and that the equations of the tangents of the conic section at the three vertices of the triangle will he m. n ^ 11 I ^ I m ^ - + - = 0, _ + - = 0, - + - = 0. V W W U U V (385) The equation to the self-conjugate rectangular hyper- bola passing through (/, g^ h) is (386) If ABC he a triangle such that the angular points are the poles of the opposite sides with respect to a conic, and ahc he another triangle possessing the same properties with respect to the same conic, then that one conic will circumscribe the two triangles. (387) If- + -7s + - = 0, -+-^ + — = Obe two comes, find ^ ^ a p 7 a ytf 7 the equations of the several lines joining the centre of the circle inscribed in the triangle of reference with the four points of intersection of the two conies. (388) If A', B', C be the middle points of the sides of a triangle ABC, and a parabola drawn through A', B', C meet the sides again in A", B'\ C"— then will the lines AA", BB", CC" be parallel to each other. (389) Conies circumscribing a triangle have a common tangent at the vertex; through this point a straight line is drawn: shew that the tangents at the various points where it cuts the curves all intersect on the base. (390) OA, OB are tangents to a conic section at the points A, B; and C is any point on the curve, li AC, BC be joined and OPQ be drawn to intersect AC, BC (or these lines produced) in P and Q, prove that BP, A Q intersect on the curve. (391) AP, BP, CP are drawn to meet a conic circumscribing ABC in DEF. The tangents at DEF meet BC, AC, AB'm A'B'C. Prove that A'B'C lie on a straight line. (392) A conic is described about a triangle so that the normals at the angular points bisect the angles. Shew that the MISCELLANEOUS EXERCISES. - 467 distances of the centre from the sides are inversely proportional to the radii of the escribed circles. (393) Find the equation to the conic section circumscribing the triangle of reference and bisecting the exterior angles of the triangle. (394) The diameter of a conic circulnsCribing ABC which bisects the chords parallel to AP, BP, CP where P is a given point, meet the tangents to the conic at A, B,0 in DBF, prove that DBF lie on the polar of P. (395) The tangents to a conic at ABC meet the opposite sides of the triangle produced in PQB. The other tangents from Q and B being drawn meet AB and CA respectively in 2, r; prove that Pqr lie on a straight line. (396) A conic section is inscribed in the triangle ABC and touches the sides opposite to A, B, G in A', B' , C respectively, any point P is taken in B' G' and GP, BP meet AB, AC in c,b respectively; prove that be is a tangent to the inscribed conic. (397) If perpendiculars be drawn from the angular points of a triangle on the opposite sides, an ellipse can be drawn touching the sides at the feet of the perpendiculars ; construct it. (3.98) If a conic touch a triangle at the feet of the perpen- diculars from the angular points, the distance of the centre from the feet is proportional to the length of the sides. (399) Two conies touch each other in two points A, B, If be any point in the straight line AB and if GPP Q Q be any chord cutting the two conies in P, Q and P, Q' respectively, prove that 1 J_ 1_ 1 OP^ OQ' OP^ OQ" (400) The four common tangents to two conies intersect two and two on the sides of their common conjugate triad. 30—2 468 MISCELLANEOUS EXERCISES. (401) Shew that the general equation to a circle in trilinear coordinates is S= (aa + h/3 + cy) (?a + ml3 + ny) - {a^y + Jrya + c(x/3) = 0, and that the square of the tangents drawn to it from a point whose trilinear coordinates are a', /3', y is —^ S': where a, h, c are the sides and A the area of the triangle of reference. (402) The self conjugate, the nine-points', and the circum- scribing circle of a triangle have a common radical axis, which is the polar of the centre of gravity with respect to the self-con- jugate circle. (403) The radical axes of the circles (areal) {u, V, w, u, V, w) [o-^yf = 0, {P,^, ^, P, i, r) (a/37)' = 0, will be represented by ua + ?;/3 + wy _ pa + q^ + ry u-\-v + w — u' —V —w p + q + r —p — q —r ' (404) Three circles described on the chords of a complete quadrilateral as diameter have a common radical axis. (405) Shew that the equation to any circle that passes through the points B^ G of the triangle of reference, may be expressed in the form ^y sin J. + 7a sin jB+ a/3 sin G + ^a(a sin -4 4- ;8 sin 5 + 7 sin (7) = ; and determine the value of the constant h in order that the circle may touch the side AB. (406) Shew that the equation of the fourth tangent common to the circle inscribed in the triangle of reference, and to the escribed circle that touches BG externally is A ,^ , . B-C ^ acos — + (^-7) sm— - — = 0. . MISCELLANEOUS EXERCISES. 469 (407) The two points at which the escribed circles of a triangle subtend equal angles, lie on the straight line whose equation in trilinear coordinates referred to the triangle is acos A {b - c) +^ cos -S (c — a) 4- 7 cos G {a — h) = 0. (408) If T be the intersection of perpendiculars from A, B, G, on the opposite sides of the triangle ABG and L the middle point of BO, and if TL be produced to meet the circle circumscribing ABG in A' ; shew that AA! is a diameter of the circle. (409) Prove that four fixed points on a conic subtend at any other point on the curve a pencil of constant anharmonic ratio, which is harmonic if the line joining two of the points wliich are conjugate passes through the pole of the line joining the other two. (410) The anharmonic ratio of the pencil formed by joining a point on one of two conies to their four points of intersection is equal to the anharmonic range formed on a tangent to the other by their four common tangents. (411) Pp, Qq, Br, Ss are four chords of a conic passing through the same point, shew that a conic can be drawn touching SB, BQ, PQ, sr, rq, qp. (412) Having given five tangents to a conic, shew how to determine their points of contact. (413) The equation of the line passing through the feet of the perpendiculars from a point (a^, ^1, %) of the circle a^y + hay + cafi = on the sides of the fundamental triangle, may be put in the form c/8, + hy, ay,+ccc, ^^ ^ ha^ + a^, ^^ ^^cosG-y^cosB 7,cosA-a,cosO ^ a^cosB- /3^cos G 470 MISCELLANEOUS EXERCISES. (414) AP, BP, CP are drawn to meet a conic circumscrib- ing tlie triangle ABG in D, E, F; EF, FO, DE meet BG, CA, AB in A^, B^, G^ respectively. Shew that these three points are in a straight line, which is the polar of P with regard to the conic. (415) One conic touches OA, OB in A and B, and a second conic touches OB, OG'va. B and G: prove that the other common tangents to the two conies intersect on A G. (416) Two conies touch each other, and through the point of contact any chord is drawn : if the tangents to the conies at the other extremities of the chord meet on the common tangent, the common chord of the conies will pass through their inter- section. (417) Two rectangular hyperbolas intersect in four points, shew that each point is in the intersection of perpendiculars from the angles on the sides of the triangles formed by joining the other three. (418) If three conies be drawn each touching two sides of a triangle and having the third for their chord of contact, shew that the three chords of intersection pass through a point. (419) If three parabolas are drawn having two of the sides of a triangle for tangents and the third for their chord of con- tact, shew that their other three points of intersection form a triangle similar to the original one and of one-ninth its area. (420) If a triangle is self-conjugate with respect to each of a series of parabolas, the lines joining the middle points of its sides will be tangents: all the directrices will pass through the centre of the circumscribing circle : and the focal chords, which are the polars of 0, will envelope an ellipse inscribed in the given triangle which has the nine-points' circle for its auxili- ary circle. MISCELLANEOUS EXERCISES. 471 (421) Shew that there are two points P, Q in the polar of with respect to a conic, such that PO is perpendicular to the polar of P, and QO to the polar of Q and that then PO ^ is a right angle. (422) Through a point P within the triangle ABG a line is drawn parallel to each side. Prove that the sum of the rect- angles contained bj the segments into which each of these lines is divided by the point P is equal to E^— OP^, B being the radius of the circumscribed circle, its centre. (423) The diameter of the circumscribing circle of the tri- angle ABG sin 2^ sin 25 sin 2(7' where a', jS', 7' are the perpendiculars on any tangent from A, B, a (424) Similar circular arcs are described on the sides of a triangle ABG, their convexities being towards the interior of the triangle; shew that the locus of the radical centres of these three circles is the rectangular hyperbola sin {B- G) mi{G-A) ^m{A-B) _ a, /S, 7 being the trilinear coordinates of a point with respect to the sides of the triangle. (425) The pole of a tangent to a fixed circle with re- spect to another fixed circle will have a conic section for its locus. (426) A conic circumscribes a triangle ABG, the tangents at the angular points meeting the opposite sides on a straight line DEF. The lines joining any point Pto A, B, and G meet the conic again in A', B', G': shew that the triangle A'B'G' envelopes a fixed conic inscribed in ABG, and having double contact with the given conic at the points where they are met 472 MISCELLANEOUS EXEECISES. by DEF. Also the tangents at A', B', C to the original conic meet B'C, G'A', AB' in points lying on DEF. (427) If straight lines be drawn from the angular points of a triangle ABC, through a point P, to meet the opposite sides in a, /3, 7, shew that if P moves on a conic, the intersection of PA and ^ and the coordinates of the middle points of QQ, RR' can be written down by symmetry. These three points lie on the straight line la niB ny a o c (42) The equations to the straight lines joining the point of reference A to the two given points at infinity can be written down, and the condition that they should be at right angles can be reduced to the given form. (43) The line through A will have the equation h^ {q —_p) + cy {r —p) = 0. (44) aa. (2p — q — r) + h/3 (2q —T — 'p)-\-cy (2r —p — q) — ^- (45) (aa — 5/3) cos -4 = 7 (5 + a cos C) . (46) 2Aa+d{aoi + h/3+cy)=0. (47) 4A. ,^g^ 4APmV ^ A {nl + Im — mn) [Im + fn-n — nl) {inn + nl — hn) ' ^ 4 ' (53) The straight line is tlie perpendicular from C on BA. EESULTS AND OCCASIONAL HINTS. 487 (55) [For r = 0, read r = d.] Two straight lines will satisfy the conditions, and their equations are d {aa + h^ +c=^G-A, then will '^-^ = A-B, and the equation may be written {a cos^+ V C0& B + w cos Cf+ (u sin ^ + v sin B + w sin Cy= 0. (88) The equation may be written {x' + f + z''+ 2yz) (of + f + z' + 2zx) [x^ + y'' + z" + 2xy) = 0, each factor of which, when equated to zero, represents a pair of imaginary straight lines parallel to a line of reference. (90) If la = m^ = ny be the point 0, the points of inter- section lie on the straight line la + 7W/3 + ny = 0. (91) See Art. 129. (92) Use one of the equations of Art. 108. (93), (94) Apply Art. 130. (95), (96), (97) Apply Art. 125. (100) One system of lines satisfying the required conditions has the equations a = 0, ^ = 1 ?^ + i + ^ = o. X. jjb V RESULTS AND OCCASIONAL HINTS. 489 (101) Form the determinant as in Art. 149. The sum of three rows will be identically equal to the sum of the other three. Hence the determinant vanishes and the condition is fulfilled. (102) See Art. 97. (103) Z{x-yy-4.z[x-\-y)+z^=0. (104) Apply Art. 149. The conic is the circle of Art. 307. (105), The straight lines la + m^ + %7 = 0, V a + m! ^ + ny= — 1. (110) See Arts. 161 and 129. (114) If (a', /S', 7') be the coordinates of P, the three tan- gents have the equations /Q'^^y a' ' 7'"^a' ^" a! '^ ^' 7" (116) With the notation of Art. 91, the three conies have the equations u^ = v^+ w\ v^ = w^ + u\ w^=u'+ v\ See (1 19). (117) A conic with respect to which two particular triangles are self-conjugate. (120) Two imaginary straight lines dividing the right angle harmonically. (123) The resulting equation should be (ka + mfBy + {Ik"" + m) nj^ = 0. (124) (^ + 1' + ^) {la' + m/3^ + ^7^) - {fa +g^ + hjY = 0. (126) a'yz + Wzx + c'xy = A (a;' cot A-\-y''coiB+ z' cot C) . 490 RESULTS AND OCCASIONAL HINTS. (130) Either condition is V/A, + Vm/i, + Vwv = 0. (132) If - +?+-=0 be the conic, -^-\ 1-- = is the straight line. i m n ° (134) Take the triangle as triangle of reference, and let (x, y, z) he the fixed point. The centre will lie on the conic — + •^+-=0, X y z (147) The polar of the point («', ^ ^ S'. (252) Reciprocate (110). (253) Reciprocate with respect to any point, (254) Extend Exercise (132), applying Art. 95. Then reciprocate. 496 EESULTS AND OCCASIONAL HINTS. (266) That the centre lies on a given straight line is equi- valent to saying that the given straight line and the straight line at infinity are conjugate. Hence this is a line condition. (269) The double point (w = 0, ^ = 0). (270) The double tangent {u = 0, v- 0). (271) At the point (w = 0, v = 0). (272) The six points [I : m : n)] (in : n : t), (n : I : m), {n : m : l), {I : n : m), (m : I : n). (275) If ;S'= be the equation to one cubic, and u = the straight line joining the points of contact, the equation to the cubic will be /S' + u\ = 0, where v = represents a straight line on which the other points of intersection lie. (276) Form the equations as in (275). (277) See Art. 481. (279) Apply (275). (280) The equation to the curve is a (P/3^ + Q^y + W) = /37 (^/S + My). (282) The equation to the cubic is {h + m^ + ny) (ma + 7i/3 + ly) {na + l^ + my) = Imn (Xa + m^ + 1^7) (^ + " + ^) (« + /3 + 7)- (283) The equation to the cubic is (Ta-lS- y) {m'/3 - 7 - a) (7^^ -a-/3) = {a+/3+y) {la + m^ + nyf. (286) The six points {a :h : c), {h : c : a), (c : a : h), {c : b : a), (a : c : h), {h : a \ c) on a conic, and the three points in which the straight line ic + ?/ + ^ = cuts the lines of reference. RESULTS AND OCCASIONAL HINTS. 497 (288) A particular case of the next exercise. (289) If a = be the asymptote, the cubic will have the equation a ./(a, /3, 7) = {la + ?w/3 + W7) (a+ /S + 7) in triangular coordinates, where ,/(«, A7)=0 is the equation to a conic. The equation to the required locus will be which represents a hyperbola. In the particular case when l=:m = n, the locus reduces to the straight line (290) If a? = 0, 3/ = 0, s = be the tangents, and m = the line of contact, the equation must take the form of Art. 483. (292) Using the equation of Art. 472, the chords are represented by n2/3 + m37 = 0, 47 + w^a = 0, w«^a + Z^/S = 0, and the tangents by m,/3 + w,7 = 0, ^27 + \a. = 0, \a. + m^^ = 0. The condition that either system should be concurrent is ZgWigWj + Zgm^Wg = 0. (304) Apply the last result. (306) The coefficients X, /a, v cannot be all of one sign- The real or imaginary tangents at the points of reference have the equations ^ + l- = 0, ^ + ^=0, T + - = 0- W. 32 498 - RESULTS AND OCCASIONAL HINTS. (307), (308) The equation may be written 1 1 , 1 ^ I m n m n I n I m a~/3 "7 ^ 7 a 7 a ^ (311) Take tlie equation of (309) and apply llie condition that each equation of (310) may represent a pair of coincident straight lines. INDEX. [The numerals refer to pages.] Abridged Notation'. The straight line, 104 The straight line in terms of the equations to three other straight lines, 105 Condition of concurrence of straight lines, 113 Conic Sections, 165 — 171 Abridged notation for the circle, 288 Conic sections in tangential coordi- nates, 361—363 Curves of the third degree and curves of the third order, 410—413 Angle Between given line and line of refer- ence, 49 Between two straight lines in trilinear coordiuates : the tangent, 50, 85 the sine, 52, 64, 85 the cosine, 54, 85 In terms of perpendicular distances of the lines from three points of reference, 62 In terms of the direction sines of the straight lines, 81, 82 Anhaemonic Eatio. Definitions, 132 Anharmonic ratio of straight lines whose equations are given, 135-137 Different ratios obtained by taking range of points in different orders, 139 Anharmonic property of a conic, 321 Anharmonic ratio of range of points whose tangential equations are given, 341 Anharmonic ratio of a range of points is the same as that of the pencil formed by their polars with respect to any conic, 381 Anharmonic property of tangents to a conic, 384 Area Of a triangle in terms of two perpen- dicular coordiuates of each angular point, 7 Of a triangle in terms of trilinear co- ordinates of each angular point, 21 Of a triangle when equations to its sides are given, 68 Area of an ellipse whose equation is given, 281 Asymptote. Definition, 160 Equation to the asymptotes of given conic, 247 Tangential coordinates of asymptotes, 359 Its polar reciprocal, 375 500 INDEX. Given asymptote of a conic equivalent to two simple conditions, 397 Asymptotes of cubic curves, 416 Parabolic asymptote, 418 General equation to asymptote, 445 Axis of a Conic. Definition, 269 Equation to the axes, 270, 272 Lengths of the axes, 279 Axis of a conic given in position equivalent to one point- and one line-condition, 398 Bbianchon's Theorem Enunciated and proved, 218 The reciprocal of Pascal's Theorem, 387 Centre op a Conic. Its coordinates, 241 Its coordinates in terms of the dis- cruninants, 261 Equation to centre in tangential coor- dinates, 352 Its polar reciprocal, 375 Polar reciprocal of concentric conies, 375 Circle. Equation to a circle referred to a self- conjugate triangle, 181 Equation to a circle referred to an inscribed triangle, 198 Equation to a circle referred to a cir- cumscribed triangle, 214 Conditions that the general equation of the second degree should repre- sent a circle, 243 Equation to circle -whose centre and radius are given, 287 Every circle passes through the cir- cular points at infinity, 289 The intersection of circles (see Radi- cal Axis), 290 The nine-points' circle, 296 Evanescent circles, 302 (iuneral equation in tangential coor- dinates, 346 Condition that equation should repre- sent a circle in tangential coordi- nates, 353 Circles reciprocate into conies having a focus at the centre of reciproca- tion, 381 Circular points at infinity. Definition, 126 Their coordinates, 127, 129 Every circle passes through them, 289 Every conic v^hich passes through them is a circle, 289 Tangential equation, 347 Class of a Curve. Definition, 364 Class of a curve the same as the order of its reciprocal, 372 -Curves of the third class, 401 — 413 Coaxial Triangles. Definition, 109 Arc co-polar, 110 CoLLiNEAR Points. Definition, 109 Condition in trilinear coordinates, 22 Condition in quadrilinear coordinates, .314 Common Chords. Definition of pair of common chords, 168 Two conies have three pairs of com- mon chord.s, 168 Equation to common chords of two conies whose equations are given, 246 Common chord of circles (see Radi- cal Axis), 290 Common Tangents. To conic and great circle pass through the foci, 351 Tangential equations, 361 Concurrent Straight Lines. Definition, 109 INDEX. 501 Condition in trilinear coordinates, 27 The perpendiculars from vertices on sides of any triangle are concur- rent, 33 " Conditions. A conic will satisfy five simple condi- tions, 158 Point-conditions and line-conditions defined, 394 One conic fulfilling five line-condi- tions or five point-conditions, 391 Two conies fulfilling four line and one point-condition, or four point and one line-condition, 394 Four conies fulfilling three line and two point-conditions, or three pomt and one line-condition, 394 Analysis of compound conditions, 896—398 Cubic curve will satisfy nine condi- tions, 404 Curve of nth. order will satisfy — ^-^ conditions, 449 CONFOCAL CONICS Are inscribed in the same unaginary quadrilateral, 351 Conic Section. Every conic section is a curve of the second order, and conversely, 156 Equation to conic through five given points, 168 Equation to conic referred to a self- conjugate triangle, 173 Equation to conic referred to an in- sciibed triangle, 192 Equation to conic referred to a cir- cumscribed triangle, 206 Tangential equation to conic with given foci, 348 General tangential equation, 349 Tangential equation to conic referred to a self -conjugate triangle, 360 Tangential equation to conic referred to a circumscribed triangle, 359 Tangential equation to conic referred to an inscribed triangle, 361 Polar reciprocal of conic section, 375 Conjugate Conics. Definition, 273 Are similar and similarly situated, 274 Two conjugate conics cannot be bolh real unless they be hyperbolas, 275 Conjugate Point on a Cueve. Definition, 402 Equation to the imaginary tangents at a conjugate point, 437 Cubic can have only one conjugate point, 403 Conjugate Points, and Conjugate Lines. Definition, 391 Condition that two given points should be conjugate (trilinear coordinates) 391 Condition that two given lines should be conjugate (trilinear coordinates) 392 Condition that two given points should be conjugate (tangential coordi- nates) 392 Condition that two given lines should be conjugate (tangential coordi- nates) 393 Coordinates. A system of perpendicular coordinates referred to two axes, 1 Trilinear coordinates, 10 Their ratios often sufiicient, 27 Areal and triangular coordinates, 94, Quadrilinear coordinates, 307 Tangential coordinates, 332 Co-POLAB Triangles. Definition, 109 Are co-axial, 111 502 INDEX. Cubic Loci and Cubic Envelopes Defined, 401 General equation, 405 Cubic loci through eight fixed points pass through a ninth, 420 Cusp. Definition, 402 Cubic can have only one cusp, 403 Cusped curve of the third order is of the third class, 407 General equation to a tangent at a cusp, 440 At a cusp aU first polars touch the curve, 441 CUEVATUEE. General expression for radius of cur- vature at any point on a curve, 446 Coordinates of centre of curvature, 448 Diameter of a Conic. Definition, 240 Its equation, 240, 262 Conjugate diameters defined, 240 Direction of diameter of given para- bola, 251 Equation to diameter of parabola, 252 Condition that equations should re- present a pair of conjugate dia- meters, 265 Tangential coordinates of conjugate diameters, 358 Polar reciprocal of pair of conjugate diameters, 375 Given diameter equivalent to one line-concUtion, 396 Conjugate diameters given in position equivalent to three simple condi- tions, 31J7 DiEECTiON Sines. The e({uations to a straight line, 73 Relations connecting the direction sines, 75 Symmetrical forms of these relations, 77 Direction sines of a straight line in terms of the coefficients of its ordi- dinary equation, 86 Proportional to the coordinates of the point where the straight line meets the straight line at infinity, 229 DiSCEIMINANT. Discriminant {H) of function of second degree, 255 Definition of bordered discriminant {K), 256 Meaning of the conditions H=Q, ^=0,261 Distance between Points. Expression in trilinear coordinates, 45 Distance op Point feom Steaight Line. In trilinear coordinates, 48, 61, 87 In tangential coordinates, 339 Double Point. Defioition, 402 Reciprocates into double tangent, 374 Cubic can have only one double point, 403 General criteria for double points, 436 Tangents at a double point, 437 Double Tangent. Definition, 403 Reciprocates into double point, 374 Duality. Principle of duality, 368 Example of double interpretation, 369 Ellipse. Definition, 160 Condition that general equation of second degree should represt;nt an ellipse, 249 Polar reciprocal of ellipse, 376 INDEX. 503 Equation. Equation of first degree in trilinear coordinates, 23 Equation of second degree in trilinear coordinates, ]57 Discussion of its general form in tri- linear coordinates, 226 Equation of third degree in trilinear coordinates, 401 Equation of nth. degree in trilinear coordinates, 431 Equation of first degree in tangential coordinates, 333 Equation of second degree in tan- gential coordinates, 345, 349 Equi-anharmonic Range Defined, 329 Focus OF A CONIO. Definition, 183, 216, 2Q6 Coordinates of the foci of a conic, 266 Real and imaginary foci, 350 Given focus of a conic equivalent to two line-conditions, 398 Great Circle at Infinity. Explained, 42 Its tangential equation, 347 Common tangents to great circle and any conic intersect in the foci of the conic, 351 Great circle reciprocates into evane- scent conic at the centre of recipro- cation, 380 Harmonic Ratio. Definition, 137 Equations to straight lines forming harmonic pencils, 139 Harmonic properties of a quadrilate- ral, 142 Fourth harmonic to three given points or lines, 144, 143 Homogeneous. How to render a trilinear equation homogeneous, 15 Properties of homogeneous functions of the second degree, 223 General properties of homogeneous functions, 426—430. Hyperbola. Definition, 160 Condition that general equation of second degree should represent a hyperbola, 249 Polar reciprocal of hyperbola, 376 Identical Equation. Connecting the trilinear coordinates of any point, 11 Connecting the triangular coordinates of any point, Connecting the distances of a straight line from three given points, 58, 80 Connecting the direction sines of a straight line, 77 Connecting the quadrilinear coordi- nates of any point, 308 Imaginary Points and Lines. Definitions, 117, 119, 129 Every such line passes through one real point, 120 Every such point lies on one real straight line, 121 Imaginary point at infinity, 122 Imaginary branches of an evanescent conic, 284 Imaginary tangents, 351 Infinity. The straight line at infinity, its equa- tion, 38 Parallel to any other straight line, 42 Its equation in quadrilinear coordi- nates, 310 Equations in tangential coordinates, " 339 Infinity reciprocates into the centre of reciprocation, 374 Infinite branches of cubic curves, 415 Inflexion. Definition, 402 S04 INDEX. lieciprocal of point of inflexion, 174 Real points of inflexion on a cubic arc collinear, 409 Point of inflexion at infinity, 420 General criteria for points of inflexion, 442 Inteeskotion. Form of equation to straight line through the point of intersection of given straight lines, 28 Coordinates of point of intersection of given straight lines, 36 Intersection of conies (four points), 167 Equation to conic through the points of intersection of given conies, 170 Tangential equation to point of inter- section of given straight lines, 337 Multiple Points Defined, 402 General criteria for a multiple point, 443 Equation to tangents at a multiple point, 444 Nine-Points' Cikcle. Its properties, 296 Equation to nine-points' circle of the triangle of reference, 294 It touches the inscribed and escribed circles, 300 Normal. Equations to normal to a conic, 228 OiiDEE OF A Curve. Definition, 364 Order of a curve the same as the class of its reciprocal, 372 Curves of the third order, 401 — 421 Osculation. Definition, 402 Point of osculation reciprocates into a point of osculation, 374 Parabola. Definition, 160 Condition that general equation of second degree should represent a parabola, 249 Diameter of parabola, 251 Condition in tangential coordinates, 354 Polar reciprocal of parabola, 376 Parabolic asymptote, 418 Parallel Straight Lines. Condition in trilinear coordinates, 37 Equation to straight line parallel to given line, 43 Parallelism of imaginary straight lines, 124 Polar reciprocal of parallel straight lines, 375 PasCxVl's Theorem Enunciated and proved, 201 The reciprocal of Brianchon's Theo- rem, 387 Perpendicular Straight Lines. Condition in trilinear coordinates, 50 Equations to straight line perpendi- cular to given line, 87 Point. Represented by trilinear coordinates, 9 Coordinates of point dividing given straight line in given ratio, 19 Point represented by equation in tangential coordinates, 333 Tangential equation to a point at in- finity, 338 Tangential equation to point dividing given straight line in a given ratio, 340 Polar. Polar curve of any point with re- spect to a given curve defined, 434 Any polar curve passes through all the singular points, 436 INDEX. 505 Polar Eecipkocals Explained, 372 Table of reciprocal loci with respect to a conic, 373—376, 380 Equations to reciprocal conies, 376 Table of reciprocal loci with respect to a circle, 382—383 Reciprocation of angular magnitude, 383 Reciprocation of distances, 384 Conies reciprocated with respect to a focus, 381 Pole and Polab. Definition, 233 Equation to polar of given point with respect to a given conic, 233 Coordinates of pole of given straight line, 234 Reciprocal properties of poles and polars, 235, 236 Tangential equation to pole of given straight line, 355 Polar of a finite point with respect to great circle is at infinity, 357 Quadrilateral. For 'complete quadrilateral' see te- iragram. Harmonic properties of a quadrila- teral, 313 Conies circumscribing a quadrilateral, 316, 325 Conies inscribed in a quadrilateral, 325, 326 Radical Axis op Two Circles. Definitions, 290 Three radical axes of three circles con- current, 290 Self-Conjugate Triangle. Definition, 175 Equation to a conic referred to a self- conjugate triangle, 173 Triangle self -conjugate with respect to each of a series of conies having four common points, 189 W. A given self-conjugate triangle equiva- lent to three simple conditions, 397 Similar and Similarly Situated CONICS. Definition, 272 Equation to conic similar and simi- larly situated to a given conic, 273 Their bordered discriminants are equal, 283 Their discriminants are in the dupli- cate ratio of their linear dimen- sions, 283 Singular Points. Defined, 403 Cubic having a singular point, 406 Tangents at singular point, 407, 437 Singular point at infinity, 419 Straight Lines. Straight fine parallel to a line of re- ference, 4 Straight line bisecting angle between lines of reference, 5 Straight Une dividing angle into two parts whose sines are in a given ratio, 5 Perpendicular of the triangle of refer- ence, 31 Straight line joining two points, 23 Equation in terms of perpendicular distances from points of reference, 25,60 Equations in terms of direction sines, 73 Condition that general equation of the second degree may represent two straight lines, 245 Two straight lines may be regarded as a limiting case of a conic section, 284 Tangent. Equation to tangent to a conic, 227 Its direction, 230 Two tangents can be drawn from any point to a conic, 231 33 506 INDEX. Equation to pair of tangents from any point, 237, 238 Tangents at a singular point, 407, 437 General equation to a tangent to any curve, 433 Teteageam. Hai-monic properties of a tetragram, 313 The middle points of the diagonals are coUinear, 315 Properties of the fourteen-points' co- nic, 329 Teansfokmation op Coordinates From an oblique Cartesian system to a perpendicular system, 3 From trilinear to triangular coordi- nates, and vice versa, 96 From one trilinear or triangular sys- tern to another, 146 — 152 From trilinear to tangential coordinates for points and straight lines, 334 Vertex. Definition, 269 Equations to determine coordinates of vertices of a conic, 269 CAMBRIDGE : PRINTED AT THE UNIVERSITY PRESS. Date Due JftK £7'5l m^i is'^^ f) BOSTON COLLEGE ^20 3 9031 01550312 1 ij ■ ^J.0 Boston College Library Chestnut Hill 67, Mass. Books may be kept for two weeks unless a shorter time is specified. Two cents a day is charged for each 2-week book kept overtime; 25 cents a day for each over- night book. If you cannot find what you want, inquire at the delivery desk for assisstance.