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Trilinear Coordinates and Other Methods of Modern Analytical CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall ..CORNELL UNIVERSITY LIBRARY 3 1924 059 323 034 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059323034 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. 1991. TEILINEAR COOEDIMTES. PRINTED BY C. J. CLAY, M.A. AT THE CNIVEBSITY PEES& : TRILINEAR COORDINATES AND OTHER METHODS OF MODERN ANALYTICAL GEOMETRY OF TWO DIMENSIONS: AN ELEMENTARY TREATISE, REV. WILLIAM ALLEN WHITWORTH. PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, LIVERPOOL, AND LATE SCHOLAR OF ST JOHN'S COLLEGE, CAMBRIDGE. CAMBRIDGE DEIGHTON, BELL, AND CO. LONDON BELL AND DALPY. 1866. 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 modem 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 modem 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 Keal and the Finite. w. ^ — VI PREFACE. I have sought 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. LiVBEPOOL, 15 September, 1866. CONTENTS. PAOS Prolegomenon. Of Determinants ix Chapter I. Of Perpendicular Coordinates referred to two axes ... 1 Exercises on Chapter 1 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 Pei-pendiculars 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 VII. Modifications of the System of Trilinear Coordinates. Areal and Triangular 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 12 VUl CONTENTS. FAOB Chapter XIV. 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 Introductio:* to Chapter XVII. Notation, &c 221 Chapter XVII. The General Equation of the Second Degree 226 Exercises on Chapter XVII 253 Chapter XVIII. The General Equation of the Second Degree continued 255 Exercises on Chapter XVIII 285 Ch.\pter XIX. Circles 287 Exercises on Chapter XIX 30t 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 determuied 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 n"" Degree 431 Exercises on Chapter XXVII 451 Miscellaneous Exercises 455 Notes on the Exercises. Results and occasional Hints 484 Index 499 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 vn — 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, that 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; + 4y - 5« = 0, 5a; -I- 5j/ - 78 = 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 unknown quantities involved in the equations. Thus each of the equations just instanced, involving the three unknown quanti- ties X, y, z, sjjeaks 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 3^ + 4^ = 5, z z 5^+5^ = 7, z z where the equations are exhibited as connecting the two independent X It ratios - and — §2. Of D^erminants of the second order. 4. Def. The symbol is used to express the algebraical quantity afi^ — h^a^, and is called a determinant of the second order. The separate quantities a,, 6j, a^, 6, are called the elements of the determinant, and may themselves be algebraically either simple or complex quantities. Any hoiizontal line of elements in a determinant is called a row, and a vertical line is called a column. Thus the determinant above written has two rows ct,, 6^ and fflj, 6j, and two columns a,, a^ and 6,, 6,. 5. It follows from the definition that = aA-aA = Hence a determinant of the second order is not altered by chang- ing rows into columns and columns into rows. OP DETERMINANTS. XI 6. It follows similarly from the definition that - = (a,6, - afi^) = - (».6, - a,6,) = Hence in a determinant of the second order the interchange of the tvoo rows changes the sign of the determinant. So the intercluxmge of the two colwmns changes the sign of the deter- minant. 7. IJ ax +by = 0, and a'x + b'y = 0, be two consistent eqtcations, then will a, b = 0. a', b' Multiplying the first equation by b' and the second by 6, and subtracting, we get {ab'-a'b)x=0, therefore ab' — a'b = 0, or a, b = 0. a, b' Q. E. D. 8. // ase +by + ez =0, and a'x + b'y + c'z = 0, then will X Xll OF DETERMINANTS, which may be written X OF DETERMINANTS.
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