Sheldon & Company's Text~:Sooks, PROFESSOR OLNEY'S NEW MATHEMATICAL SERIES The success of Prof. Olney's series has been most wonderful. With all their admitted excellencies, both the Author and Pub- lishers have felt that it was possible to retain their many attractive features and yet adapt the books more perfectly to the special school-room wants. To accomplish this most desirable end, Professor Olney has been accumulating very valuable suggestions. He has also, for several years, had associated with him in the preparation of this new series, some of the best practical teachers in the country. The design is to present to the educational public the best and most teachable series of Mathematics ever published. The work is now so far advanced that the Publishers are able to make the above pleasing announcement, which they feel will be of great interest to all who are engaged in teaching. THE NEW SERIES EMBRACES: I. Olney' s First Lessons in Arithfuetic. Just Published. II. Olriey's Practical Arithinetic. This book has been published but a short time, but it has already had the most wonderful success. They are models of beauty and cheapness. For schools of a high grade, Professor Olney has prepared — III. The Science of Arithmetic. lY. The First Frincij^les of Algebra. An Introduction to the Author's Complete and University Algebras. V. Olney' s Complete Algebra, New Edition, in large type. This book is now entirely re-el ectrotyped in larger and more attractive type. Theexplanatory matter is greatly lessened. The attractive features of this book, which have made it the most popular Algebra ever published in this country, are all retained. .051 Sheldon & Company's Text-^Books^ OLNEY'S SERIES OF MATHEMATICS. Olney's First Lessons in Arithinetlc Illus- trated Olney's Practical Arithmetic Olney's Science of Arithmetic, (For High-Schools only.) Olney^s First Principles of Algebra Olney's Complete Algebra Olney's Book of Test Examples in Algebra.., Olney's University Algebra Olney's Elements Geom, 4& Trigonom. (Sch. Ed.) Olney's Elements of Geometry. Separate Olney's Elements of Trigonometry. Separate. . Olney's Elements of Geometry and Trigonom," etry. (Univ. Ed., with Tables of Logarithms.) Olney's Elements of Geometry and Trigonom- etry. (University Edition, without Tables.) Olney's General Geometry and Calculus The universal favor with which these books have been received by educators in all parts of the country, leads the publishers to think that they have supplied a felt want in our educational ap- pliances. There is one feature which characterizes this series, so unique, and yet so eminently practical, that we feel desirous of calling special attention to it. It is The facility with which the books can be used for classes of all grades, and in schools of the widest diversity of purpose. Each volume in the series is so constructed that it may be used with equal ease by the youngest and least disciplined who should fee pursuing its theme, and by those who in more mature years and with more ample preparation enter upon the study. Ij Library ciW.H.Metzleri Class Noi OLNEY'S MATHEMATICAL SERIES. A W. I' GENERAL GEOMETRY AND CALCULUS. INCIitTDING BOOK I. OF THE GENEEAIj GEOMETRY, TREATING OP LOCI IN A PLANE ; AND AN ELEMENTARY COURSE IN THE DIFFER- ENTIAL AND INTEGRAL CALCULUS. BT EDWAED OLNEY, FBOFESSOB OF MATHEMATICS IN THE Xmi^^^^F^^jpOMOeUXj NEW YOEK : SHELDON AND COMPANY. 1881. Entered according to Act of Congress in the year 1871, by SHELDON & COMPANY, In the Office of the Librarian of Congress at Washington. PROF. OLNEY'S MATHEMATICAL COURSE. INTRODUCTION TO ALGEBRA COMPLETE ALGEBRA KEY TO COMPLETE ALGEBRA --,---. UNIVERSITY ALGEBRA KEY TO UNIVERSITY ALGEBRA A VOLUME OF TEST EXAMPLES IN ALGEBRA ELEMENTS OF GEOMETRY AND TRIGONOMETRY ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University Edition - ELEMENTS OF GEOMETRY, separate - ELEMENTS OF TRIGONOMETRY, separate GENERAL GEOMETRY AND CALCULUS - - - - BELLOWS' TRIGONOMETRY PROF. OLNEY'S SERIES OF ARITHMETICS. PRIMARY ARITHMETIC ELEMENTS OP ARITHMETIC PRACTICAL ARITHMETIC SCIENCE OF ARITHMETIC - - - - - - - G. &. C. 150249 PREFACE. -•♦i- ^ This volume presents a course in the General Geometry and the Infinitesimal Calculus, which is thought to be as extended as is pr,acticable for the general student in the regular undergrad- uate course in our American colleges. If we can secure a suffi- ciently high grade of preparation, so that students in the Fresh- man year can complete a respectable course in Elementary Geometry, including Plane and Spherical Trigonometry, and in Algebra, it is thought that during the Sophomore year the 3on- tents of this volume can be readily mastered. Such is the purpose in this University ; and it is already well nigh reahzed. As to the propriety of including the study of both these sub- jects in the regular undergraduate course, there can be but one opinion among those competent to judge. No man can justly claim to have a good general education, who is ignorant of the elements of the processes by which all extended operations in the exact sciences are carried forward, and which are the foun- dation of all the arts based upon mathematical science. The man who is ignorant of the General Geometry and the Calculus, is not only a stranger in one of the subHmest realms of human thought, but knows nothing of the instruments in most familiar use by the engiueer, the astronomer, and the machinist in any of the higher walks of art. In short he is ignorant of the charac- teristic processes of the mathematician of his day. Nor is it impracticable for the majority of students to become intelligent in these subjects. They do not he beyond the reach of good common minds, nor require peculiar mental character- istics for their mastery. The difficulty hitherto has been in the methods of presentation, in the limited and totally inadequate amount of time assigned them, and more than all in the precon- ceived notion of their abstruseness. The mathematician will see in the plan of the first part of IV PREFACE, this volume, as well as in its title ( General Geometry), a recog- nition of the profound views of Comte upon the philosophy of the science. This science is a method of Geometrical reasoning. Its characteristic feature is that it represents form, as well as magnitude, by equations, and hence makes algebra its instru- ment. It is consequently indirect. Its ultimate object is breadth of comprehension, — the discussion of general problems. In accordance with this conception, the first purpose is to exhibit the method of translating geometrical forms into algebraic equations, i. 6. to show how loci are represented by equations. While the prominence is given to the Conic Sections which their importance in physical science demands, the student is not led to think that this is merely a scheme for treating these curves. He is taught to look upon it as a method of investiga- tion — as designed to embrace the discussion of all loci. For this purpose many Higher Plane Curves are treated. After the student has become famihar with the equation as the represent- ative of a locus, and has learned how to ]3roduce the equation of a locus from its definition, he has obtained the instrument. He is now to learn how to apply it for the purposes of Geometrical investigation. In carrying forward this part of his study the Calculus renders invaluable service. Moreover, this preparatory study of the General Geometry gives him exactly the needed means for illustrating the elementary processes of the Calculus. He has, therefore, come to a point where his further progress requires a knowledge of the Differential Calculus, and he has also the requisite preparation for its study. Hence, after having become famihar with the first three chapters of General Geom- etry, he reads the Differential Calculus. By this arrangement the Calculus is seen in its true relations, as an independent abstract science, grand and beautiful in itself, and rendering most efficient service in the more immediately practical science of Geometry, as it is afterwards seen to do in Physics. Having obtained the needed acquaintance with the Differential Calculus, the student returns to pursue his Geomet- rical studies, with equations of loci as his instruments, and the Calculus to aid in the manipulation of them. But the pecuhar features of the treatise are too numerous to be enumerated here, and can be seen in their true light only by a perusal of the work. PBEFACE. V In the treatment of the Calculus I have used the Infinitesi- mal method instead of the method of Limits, on account of its greater simplicity, as well as because it is the only conception which enables us to apply the Calculus to practical problems with any degree of facility. The general use of the method of limits in our text books has done not a little to prevent the com- mon study of this elegant and useful branch of mathematics. This method is not only exceedingly cumbrous, but it has the misfortune that its element, a differential coefficient, is a ratio. The abstract nature of a ratio, and the fact that it is a com- pound concept, pecuharly unfit it for elementary purposes. The beginner will never use it with satisfaction, for it does not give him simple, direct and clearly defined conceptions. But while I have adopted the infinitesimal theory, I have felt free to intro- duce the doctrine of Hmits, and to illustrate and apply it. The metaphysical objections to this method, if not rebutted by equal difficulties of a similar character encountered in the method of Hmits, are immensely overborne by its practical advantages ; for, let it be remembered that no writer adheres to the Newton- ian method throughout, but ghdes into the other in the Integral Calculus, and adopts it exclusively in most geometrical and physical applications. The sources from which the material has been drawn will be readily perceived by the mathematician, and need not be enum- erated here. That the treatise is sufficiently different from others of a similar purpose to justify its existence, the author feels more sure than that these differences will commend them- selves to his fellow laborers in the work of mathematical train- ing. One thing, however, is certain, nothing in matter, arrange- ment, or manner of treatment, has been introduced without careful reference to the capabilities and wants of such students as I have been accustomed to meet in the class room for more than twenty years ; and few things will be found in the volume but what have been put to the test of class room use many times over. A second volume, treating of Loci in Space, and affording a more extended course in the Calculus, will be published as soon as it can be prepared. The present is thought sufficient for all students except such as make mathematics a specialty ; and for the latter the other volume will be designed. VI PREFACE. In conclusion I must do myseK the pleasure to acknowledge mj indebtedness to mj accomplislied colleague and friend, Prof. J. C. Watson, Ph. D., for the original, direct, and simple method of demonstrating the rule for differentiating a logarithm, which is given on page 25, and which banishes from the Calculus the last necessity for resort to series to establish any of its funda- mental operations. I am also indebted to my friend and pupil, J. B. Webb, B. S., for many valuable suggestions, and much care- ful labor in reading both the manuscript and proof. To his quick and accurate eye, and his good taste and logical acumen, I am indebted for the ehmination of not a few defects which might otherwise have disfigured the work. That there is not much of the same sort of pruning yet needed, I have not the vanity to think. But, such as it is, I commend my work to the consideration of teacher and student, with the hope that it may contribute to aid the one in imparting, and the other in acquiring, a knowledge of the elements of two branches of science which, in their fuller developments, exhibit the profoundest and most sagacious workings of the human mind, and reach to the farthest verge of the hitherto explored realms of human thought. EDWAED OLNET. Ann Abbob, Mich., July, 1871. N. B. — A shorter course in tlie General Geometry, ivithout the Calculus, may he taken from this volume hy such as desire it. For this purjoose, the first three chapters are to he read, and then the course completed hy reading the XIV. and XV. Sections of Chap- ter IV. If time and purpose permit, Articles {194:, lOo) might he read loith profit hy such students. This ivill he found to comprise a course on Plane Co-ordinate Geometry somewhat more full than is found in our common text-hooks. CONTENTS. INTMOnUCTION. A BRIEF SURVEY OF THE OBJECTS OF PURE MATHEMATICS AND OP THE SEVERAL BRANCHES. PAGK PuBE Mathematics. — Definition (i) ; Brandies enumerated {2f 3) 1 Quantity. — Definition (4) 1 NuMBEK. — Definition {5) ; Discontinuous and continuous {6, 7f 8) 2, 3 Definition of the Several Branches of Mathematics. — Arithmetic {9 , ; Algebra (JO); Calculus {11); Geometry {12) \ Descriptive Geometry (15) 3, 5 General Geometry divided into Two Books {14:) 5 — -tt^^^ GENERAL GEOMETRY BOOK I. OF PLANE LOCI. CHAPTEE I. THE CABTESIAN METHOD OF CO-OBDINATES. SECTION L DEFINITIONS AND FUNDAMENTAL NOTIONS. Locus.— Definition {1) 6 General Geometry. —Definition {2) 6 Method of Co-ordinates.— What (5) ; Two Systems (4) ; Varieties of Eec- tUinear {5) ^' "^ Definitions.— Axes {6, 7) ; Origin {8) ; Co-ordinates {9, 10, 11) ; Illus- tration « • ' » " Notation.— Of Co-ordinates {12) ; The Four Angles {13) ; Signs of the Co-ordinates {14) ^' ^ OnvN-T-rr..-, —Constant and Variable US' ; Definition of each {16, 17 \ 9 El istration : Caution. Sch. 1 TlU CONTENTS. FAGS Indetebminate Analysis. — ^What, Sch. 2 9 To CoNSTEUCT AN EQUATION. — ^What (^18) 10 SECTION IL CONSTBTJCTING EQUATIONS, OE FINDING THEER LOCI. Deitnitions. — A Continuous Curve {19) ; Branch {20) ; Symmetry {21) ; Independent and Dependent Variables {25) 10-12 To LOCATE A Point {22) 10 To CONSTBUCT AN EQUATION {23) 11 Discussing an Equation. — Wliat ; Intersection ; Limits ; Symmetry {26) 12, 13 "R-yAATPT.-B^S ,^. 11-16 SECTION IIL THE POINT IN A PLANE. Deitnitions. — ^Equations of a Point {27) 17 Equations op a Point. — ^What {28) ; In different angles, In the axes, In the origin, ScKs. 1, 2 ; Points, how designa' ed, Sch. 3 17 Distance between two Points. — General Formulae {29) ; Special cases, Cor. and Sch , 18 Examples. . , , , , 17, 18 SECTION IV. THE EIGHT LINE IN A PLANE. Definit i on. — Equation of a Locus {30) 19 Equations of a Right Line. — Through Two Points {31) ; Through One Prnnt {32 i ; Common Form {33) ; Referred to Oblique Axes {34:) ; Meaning of ^,~^„ Cor.l 19-21 X — X Discussion oiy = ax -\- h, Sch's. 1 and 2 20 ^Methods of Constbuctestg y = ax -\- h 21 Locus of an Equation of the Ftest Degeee (55) 22 EXAfcLPLES. 21-23 SECTION K OF PLANE ANGLES, AND THE INTEESECTION OF LINES. Tangent op a Plane Angle. — Formulae for {36) 23 Equation of a Line making any given Angle with anothee Line. — Com- CONTENTS. IX PAGE mon form, When passing through a given point (57) ; When Parallel to a given Line, Comnion form, Passing through a given point {38) ; When Perpendicular to the given line, Common form. Passing through a given poiat {39) -. 24 Examples 25, 26 To Find the Intersection of Lines {40) 26 ExAMPiiES , 26-28 Distance feom a Point to a Line {41) ; Between Parallels, Cor 28 Examples 28, 29 SECTION VL OF THE CONIC SECTIONS. Boscovich's Definition {4=2) 29 To CONSTBTJCT A CoNIC SECTION {43) 29 Definitions. — Directrix, Focus, Focal Tangents, Transverse Axis, Conjugate Axis, Latus Kectum, Vertices, Focal Distances, Eccentricity {44) 29, 30 Axis of Hyperbola, Transverse, Conjugate, Conjugate Hyperbola, Equi- lateral Hyperbola {47) • 32, 33 Examples 30-34 Boscovich's Ratio = Eccentbicitt {48) 34 Fundamental Relations {45^ 46 , 49) 30, 31, 34, 35 To pass a Conic Section theough Thkee Points {50) 36 Examples 37 Equations of Conic Sections. — General Equation (5 J? ) ; Referred to their Axes, In terms of A and e {52), Common Forms {53y 54, 50, 57 f 59) ; Referred to Axis and Tangent at Vertex {55, 56, 57) \ Of Conjugate Hyperbola {58) 37-40 Comparison op Equations of Ellipse and Hypebbola {60) 41 Locus OF Equation of Second Degree {61) 41 Features op the Equation which chaeacteeize the dippebent Conic Sec- tions {62) ; Species dependent on A, B, C, {63) ; All varieties included in Aij^ + Or^ -{-I>y-\- Ex-j-F= {64) 42, 43 Examples 42, 43 Varieties.— Of Ellipse {65), Hyperbola {66), Parabola {67) ; Eccentricity of Circle {68) 43-45 Examples 46-49 Exercises in producing various forms of the equation of the Conic Sections directly from the definition 49-51 The Origin op the name Conic Section {69) 51 Five Points in the Curve determine a Conic Section {70) 52 Examples 52-54 Exercises in producing the equations of Conic Sections from various defini- tions. 54-57 X CONTENTS. SECTION VIL EQUATIONS OF HIGHER PLANE CURVES. PAGE Definitions. — Function {71)', Classes ofcT'^); Algebraic (75) ; Trigo- nometrical (7^) ; Circular (.75) ; Logarithmic (7^) ; Exponential {77) 57 Loci Ciassified.— Higher and Lower, Algebraic and Transcendental {78, 70) 58 Cisson). — Definition {80) ; Construction {81) ; Origin of name, Sch. 1 ; Mechanical method of Constructing, ISch. 2 ; Equation of {82 j ; Discus- sion of Equation, Sch. 1 ; Duplication of cube by means of, Sch. 2 . . . . 58-60 Conchoid. — Definition {83) ; Construction {84) ; Mechanical Construc- tion, Sch. ; Equation of {8S) ; Discussion of Equation, Sch. 1 ; Be- comes the equation of circle, Sch. 2 ; Trisection of an angle by means of, Sch. 3 60-62 Witch.— Definition {86) ; Construction {87) ; Equation of {88) ; Dis- cussion of Equation, Sch 62 Lemniscate, —Definition [89) ; Construction {90) ', Equation of {91) ; Discussion of Equation, Sdi. 1 ; How related to Equi-lateral Hyperbola, Sch. 2 63 Cycloid. — Definitions, of the Locus, Generatrix, Base, Axis, {92,93) ; To put the Generatrix in position {94:) : Equations of the Cycloid, 1st form {95), 2nd form (96) ; Discussion of Equation, Sch. to {95), and Cor. and Sch. 1 to (96) 64, 65 Equations of some i.oci written dikectly feom the definitions {98) .... 66 NuMBEB OF PLANE CUBVES INFINITE. — A fcw Suggested {99) 66 ■^♦» CHAPTEE II. THE METHOD OF POLAR CO-ORnU^ATES. SECTION L OF THE POINT IN A PLANE. How A Point is designated by Polar Co-ordinates {100) 67 Definitions. — Pole, Prime Radius, Eadius Vector, Variable Angle, Polar Co- ordinates ilOl) 67 Equations of a Point {102) ; Examples 67, 68 Distance between two Points {103) ; Examples 68 SECTION IL OF THE RIGHT LINE. Equations op the Right Line. — 1st form, 2nd form {104) ; Discussion of 1st form, Sch, 1 ; Diseussion of 2nd form, Sch. 2 ; Examples 68-70 CONTENTS. XI SUCTION III. OF THE CIRCLE. PAGE Equation "when the Pole is the Circumference, and the Polar Axis is a diame- ter {105) ; Discussion, Sch 70, 71 General Polab Equation {100) ; Discussion, Sch. ; Geometrical Illus- tration, ^.5 71-73 Examples 72, 73 SECTION IV, OF THE CONIC SECTIONS. Polar Equation op Conic Section {107) ; Of Parabola {108) ; Of El- lipse and Hyperbola {109) ; Discussion of Equation of Parabola, Sch. 1, Of Ellipse, Sch. 2, Of Hyperbola, Sch. 3 73-75 Examples 75, 76 SECTION V. OF HIGHER PLANE CURVES. Polar Equation oe Cissoro {110) ; Discussion, Sch 76 PoLAH Equation of Conchoid {111) ; Discussion, Sch 77 PoLAB Equation of Lemniscate {112) ; Discussion, Sch 77 OF PLANE SPIRALS. Definitions. — Of Spiral, Measuring Circle, Spire {113) 77 Spiral of Aechimedes. — Definition {lid) ; Construction {115) ; Equa- tion of {116) 78 Eecipsocal or Hyperbolic Spibal (J[j?7) ; Equation; Construction 78 The Lituus {118) ; Equation ; Construction 79 LoGAKiTHMic Spieal {119) ', Definition, Equation, Construction {119) 79 -♦-♦"^ CHAPTEE III. TMANSFOMMATION OF CO-ORiyiKATES. SECTION I PASSING FROM ONE SET OF RECTILINEAR AXES TO ANOTHER. Definitions. — Transformation ; Two aspects of the Problem ; Primitive Axes or System ; New Axes or System ; Illustba.tions {120) ; Practi- cal Advan'agec, Sch 80, 81 Xii CONTENTS. PAGE FoBMULaj FOB Passing from onb Eectelixear Set of Axes to Anotheb.— Ganeral Formulae {122)', From aay set to a Parallel set {123); From Eectangular to Oblique {124); From Rectangular to Rectangular {125); From Oblique to Rectangular {126); The foregoing where the origin is unchanged {127); From Oblique to Rectangular, when a and a' sig- nify the angles which the Oblique or Primitive axes make with the Rect- a igular, or New axis of x, Sch.. . 82-84 EXAMPIIES 84-90 SECTION IL PASSING FEOM RECTILINEAR TO POLAR CO-ORDINATES, AND VICE VERSA., Formula fob passing from Rectilineab to Polar {128) 90 Fobmuils; fob passing from Polab to Rectilineae {129) 91 ExAMPiiES « 91, 92 ^-♦-> — CHAPTER lY. TBOPERTIES OF PLANE LOCI INVESTIGATED BY 3IEANS OF THE EQUATIONS OF THOSE LOCI. SECTION I TANGENTS TO PLANE LOCI. (rt) BY RECTILmEAR CO-ORDINATES. Definitions. — Consecutive points {130); Tangent {131); Tangent has the same direction as the Curve, Cor. {132) 93 (111 Geometricai. Signification of -^ {133); A Tangent which makes any dx given angle with the axis of x, \Vhica is parallel, Which is perpendicular yl34:); Signification of ~ when the axes are oblique {135); Examples. 93-96 Equations of Tangenis. — Gensral Equation {136) ; Of the Ellipse, The Hyperbola, The ParaboU, The Circle?, and other Examples ; The Intercepts of the Axes by a tangent {137), Wi h the axis of x in Ellipse, Hyperbola, Parabola ; Other Examples ; To draw a tangent to an Ellipse {138), To an Hyperbola {139), To a Parabola {140) 96-101 SuBTANGExVTS. —Definition {141} ; General value of {142} ; Of an Ellipse, Hyperbola, Parabola, other Examples ; Use in drawing tangents 101, 102 Length of Tangent.— General formula {143) ; Of an Ellipse, Hyperbola, Parabola • 102, 103 Asymptotes (rectilinear).— Definition {144\ Illustrations; To examine a curve for A.sr/mptoies,— General Method (145 , By Inspection {148), By Develop^no: the function 149 ; An Asymptote the limiting position of a Tangent {146 ; Equation of (i^T* ; Examples 103-107 CONTENTS. XIU PAGS (6) TANGENTS TO POLAB CURVES. How Deteemi»ed {150) 107 SuBTANGENT.— Definition {151) ; General Value {152) ; Examples. .. 107-109 Asymptotes. — How Determined {153) ; Examples 109, 110 SECTION 11. NOEMALS TO PLANE LOCI. (a) BY RECTANGULAR CO-ORDINATES. Definition or Nobmal {154:) HO General Equation (155).— Signification of — — {156) ; Normal to El- lipse, Hyperbola, Parabola, and otber Examples 110, 111 Subnormal.— Definition {157) ; General Value {158) ; To Cycloid {159) ; To draw a Tangent to the Cycloid {160) ; To draw a Tangent making a given angle {161 ) ; Examples HI, 112 Length of Noemal {162) ; Examples 112 Pekpendiculaii upon a Tangent {163) ; From the focus of a Parabola {164) ; Examples • 112, 113 (&) NORMALS TO POLAR CURVES. SuBNOEMAii. — ^Definition {165} ; General Value {166) ; Examples ... 113, 114 Length of Normal to Polar Curve {167) H^ Length of Perpendicular from the Pole upon the Tangent op a Polar Curve {168) H^ SECTION IIL DIRECTION OF CURVATURE. (a) BY RECTANGULAR CO-ORDINATES Criteria for determining Direction of Curvature. Sign of p{ {169) ; Sign of ^' {170) ; Sign of yp{ {171) ; Ex- dxr ay- ax^ amples 114r-116 {h) BY POLAR CO-ORDINATES. Definition op Direction of Curvature of Polar Curves {172) 116 Ceitebia for determining {17 3 f 174) ; Examples 116, 117 SECTION IV. SINGULAR POINTS. Definition and Enumeration {175) 117 Maxima and Minima Ordinates.— Definition {176); To determine their XIV ^ CONTENTS. PAOS position and value {177) ', A. negative maximum or minimum {178) ; Examples 118, 119 Points of iNFiiExioN. — Definition {170), Illustration ; How determined by Kectangular Co-ordinates {180) ; By Polar Co-ordinates {181) ; Ex- amples 119-121 Multiple Points. — Definition, Species {182) ; How determined {183) ; Examples 121-123 Cusps. — Definition, Kinds {184) ; How determined {185) ; Examples 124, 125 Conjugate Points. — Definition {186) ; Two Criteria {187 > 188) ; How to examine a curve for Conjugate Points {180) ; Examples. , 125-127 Shooting Points. — Definition {100) ; Examples 127, 128 Stop Points. — Definition {101) ; Examples 128 SECTION K TRACING CURVES. Definition (102) 129 General Method {103, and Sch.) ; Examples. . , 129-132 To tbace a cubve of the Second Okdee. — By direct inspection of its equa- tion {104:) ; Examples ; By Transformation of Co-ordinates {105) ; Examples 132-135 To Tkace a Polab Cubve {106) ; Examples 135-137 SECTION VL RATE OF CURVATURE. Definitions. — Curvature {107), Illustration ; Osculatory Circle {108), Il- lustration ; Radius of Curvature, Centre of Curvature {100) ; Parameter {202) 137-140 Contact. — What, How closeness of Contact is characterized, Orders of Con- tact {200), Geometrical Illustration {201) ; Order of Contact dependent upon Parameters {203) ; Order of Contact of Eight Line {201', Of Circle, Of Parabola, ElHpse, Hyperbola {205, 206) ; Eestriction of these statements {207) ', Contact of a Eight Line at Point of Inflexion {215) ; Contact of Osculatory Circle at points of Maximum and Mini- mum Curvature {216), At the Vertices of the Conic Sections {217) ■ ■ 139-146 Eadius of Cuevatuee. —General Formula in terms of Eectangular Co-ordi- nates {208) ; Signification of the sign {200) ; Eadius of Curvature of the Conic Sections, At the vertices {210, 212), Varies how {211, 213) ; Centre of Curvature in the Normal {214) ; Examples 141-145 When Osculatory Curves intersect and when not (218) ; When the Os- culatory Circle Cuts a Conic Section (210) 146, 147 Eadius op Curvature of Polar Curves {220) ; Involving the Normal (^221) ; Examples 147, 148 CONTENTS, XV PAGX SECTION VIL EVOLUTES AND INVOLUTES. Definition (222), Illustration 148 To FIND THE EVOLUTE {223) ', Examples ; The Evolute of a Cycloid an Equal Cycloid {225) ; Same Geometrically, ^eh 149-151 NoBMAii TO Involute Tangent to Evolute {220) 151 Kadius of Cubvatuee vabies as Aug of Evolute {227) 151 A CURVE desckibed mechanically feom its Evolute {228) 152 a curve has but one evolute, but an evolute has an infinite numbeb of involutes {229) 152 SECTION VIIL ENVELOPES TO PLANE CURVES. Definition {230), Illustration 152, 153 To find the Envelope {231) ; Examples 153-159 Envelope tangent to the inteesecting seeies {232) 154 Caustics. — General Equation {233) ; When the incident rays are parallel to the axis of the reflector {23d), When perpendicular {235) ; Illustra- tion ; Examples 156-159 SECTION IX, RECTIFICATION OF PLANE CURVES. Definition {237) 159 By Eectangulae Co-oedinates. — General Formula {238) ', Examples ; Circumferences of Circles are to each other as the radii {240) ; Value of 7t [24:1 ) ; Arc of Cycloid equals twice the corresponding chord of the gen- eratrix i24:2) 159-162 By Polab Co-obdinates. — General Formula [24:3) ; Examples 163, 164 SECTION X. QUADRATURE OF PLANE SURFACES. Definition {244:) 164 By Kectangulab Co-oedinates.— General Formula {245) ; Examples ; Areas of Circles to each other as squares of radii [246) ; Area of Circle whose radius is 1 {247) ; Area of Circle = ^r X circumference {248) ; Area of Segment of Circle (249) ; Area of Ellipse compared with Circumscribed and Inscribed Circles (250) 164-168 B^ PoLAE Co-ordinates.— General Formula ^251) ; Examples 168, 169 XVl CONTENTS. SECTION XI QUADRATURE OF SURFACES OF REVOLUTION. Definition {252) ; Illustrations l(il) I^ENEEAL FoKMULA {233) ', Examples ; Surface of a Sphere =: 4 great Cir- • cl.-s, or Circumference X Diameter, Cor. 1 ; Area of Zone {254:) ; Sphere / and Circumscribed Cylinder, Sch 169-170 SECTION XII CUBATURE OF VOLUMES OF REVOLUTION. Geneeal Foemula {255) ; Examples ; Volume of a Sphere = the surface X 3 radius {256) ; Volumes of Spheres are to each other as the cubes of their radii {257) ', Volume of a Segment {258) ; Volume of Sphere and Circumscribed Cylinder {259) 171, iTa SECTION XIII EQUATIONS OF CURVES DEDUCED BY THE AID OF THE CALCULUS. Teacteix. — Dehuition {200) ; Equation (261) 172, 173 Locus WHOSE SUBNOEMAL IS CONSTANT [262) 173 Locus WHOSE NOEMAL IS CONSTANT {263) 174 Locus WHOSE SUBTANGENT IS CONSTANT {264) 174 Locus WHOSE SuBNOEMAL VAEIES AS THE SqUAEE OF ITS AbSCISSA {265) 174 Locus WHOSE Area is twice the peoduct of its Co-oedinates {266) 174 Locus WHOSE AeC VAEIES AS THE SQUAEE EOOT OF THE THIED POWEE Ol' I'rS abscissa {267) 174 SECTION XIV. OF TANGENTS AND NORMALS. [WITHOUT THE AID OF THE CALCULUS.] Tangents. — General Method of producing the equation of {268) ; Ex- amples, — Tangent t > Parabola, Ex. 1 ; Elhpse, Ex. 4 ; Hyperbola, Ex. 10 ; Tangent of the angle which a tangent to a Conic Section makes with the axis of X {260), Examples ; To find the point on a curve from which a tangent must be drawn to make a given angle with the axis of x, be paral- lel, be perpendicular {270) ; Examples 175-179 SuBTANGENTs.— Definition {271) ; To find the length of {272) ; Ex- amples, — Subtangent in Parabola, To draw a tangent by means of {273) ; Subtangent of Ellipse, To draw a tangent by means of {274, 275) ; Sub- CONTENTS. XVll PAGE tangent of Hyperbola {270), To draw a tangent by means of {277) ', Half either axis a mean proportional between its intercepts by a tangent and ordinate, Ex. 4 ; Analogy between the equations of the Conic Sections and the equations of their tangents {278) 179-181 Normals. — Definition {279) ; To produce the Equation of Normal {280) ; Tangent of angle which Normal makes with axis of x {281) ; Examples, Normal to Ellipse, Hyperbola, Parabola, Circle ; Expressions for tangent * of the angle which a Normal to a Conic Section makes with the axis of x {282) 181, 182 SuBNOEMAiiS. — Definition {283) ', Examples in the Conic Sections, Is con- stant in the Parabola and = p, To draw a tangent by means of the latter property. Ex's 1 and 2 183 The Peepediculae feom the focus or a Paeabola upon the tangent {284:) ; Cor. {285) ; To find the focus of a Parabola when the curve and its axis are given, Also to draw a tangent {280 f 287 ? 288) 183 SECTION XV, SPECIAL PROPERTIES OF THE CONIC SECTIONS. Eadh Vectoees. — Definition {289) ; Sum of in Ellipse and diflference in Hyperbola {290) ; Length of each {291) ; To construct an EUipse and Hyperbola on this principle {292) ; Eadii Vectores make equal angles with the tangent in Ellipse and Hyperbola {293) ; Corresponding property in Parabola {290) ; Angles included by the Eadii Vectores and Normal, in Ellipse and Hyperbola {294:) ; To draw a tangent upon these principles, 1st, from a point in the curve, 2nd, from a point without {295) ; Same problems in reference to the Parabola {298) 184-187 The eectangle of PEEPENDicuiiAES FEOM FOCI UPON Tangent {299) 187 The Semi-conjugate axis a mean peopoetional between focal distances {300) 187 Supplementaey Choeds and Conjugate Diameters. — Definition of Ordinate {301), Of Supplementary Chords {302), Of Conjugate Diameters {303) ; Fundamental property of Supplementary Chords {304, 305) ; When drawn on the Conjugate Axis {300) ; When drawn from a point in the Conjugate Hyperbola {307) ', This property in the Circle {308) ; Paral- lelism of Sup. Chords to the axes {309) ; The — sign m axi' = j {310) ; Discussion of the Angle included by Sup, Chords {311) ; Sup. Chords parallel to Tangent and Diameter {312 , 313)*; To draw Tangents by this property {314) ; Eelations between Conjugate Diameters and the Axes {318) ; Examples 188-194 Oedinates. — Eelation to each other, in Ellipse {319), in Hyberbola {322), in Parabola {333) ', Corresponding properties of oblique ordinates {325) ; Eelation of an ordinate to the corresponding segments of its diameter {320) ; Latus Eectum a third proportional to the axes {321 ) ; The rela- tion of ordinates in the circle {323) ; Eelation of ordinates to the conju- gate axis of Ellipse {324) ; Parallel chords bisected hj Diameter {320 , 334) ; To find the centre, axes and foci of a Conic Section when the curv- XVIU CONTENTS. PAGB ature is given {327 f 335) ; Ordiuates of different Ellipses on same axis {Ji28); Of Ellipse and Circle on same axis {329) ; The Trammel {330); Ordinates to different ellipses on same Conjugate Axis {331) ; Of Ellipse and Inscribed Circle {332) 195-199 EcjsNTEic Angle. —Definition (336) '■ Sine and cosine of this angle {337); Advantages, Sch. ; Equation of Tangent to Ellipse in terms of this angle {338) ; Eccentric angles of the vertices of the Conjugate Diameter {339} ; To draw a Conjugate Diameter on this principle {340) ; Kect- angle of Kadii Vectors = Square of Conjugate Diameter {341) ; Sum of the Squares of Conjugate Diameters constant {342) ; Examples 199-201 The Intercepts or a Secant between the Htpeebola and its Asymptotes {343) ; To construct an Hyperbola on this principle, 8ch 201, SOS Parajmetee to ai^ Diameter of a Conic Sectiok. — Definition (344) ; Distance from point in Parabola to focus {345) ; Parameter to any dia- meter of Parabola {340); Parameter to any diameter of Parabola a double ordinate through focus {347) ; Chord of Ellipse through focus {348) ; Sch. {347) not applicable to Ellipse {349) 202, 203 Chord of Curvature. — Definition {350) ; In the Parabola chord of cur- vature through focus a parameter to the diameter through potut of con- tact {351} ; Intercept on this diameter by the osculatory circle equals this chord of curvature {351) 203, 204 CONTENTS. XIX THE INFINITESIMAL CALCULUS. *♦* INTnonUCTION. PAGE Definitions. —Quantity {!) ; Number {2) ; Discontinuous and Continuous Number {3, 4:, 5), Illustrations ; An Infinite Quantity (6) ; An Infini- tesimal (7) ; Caution {8). 1, 2 Infinites and Infinitesimals eecipeocals of each other (9, 10) 3 Obders of Infinites and iNFiNiTESiMAiiS.— What {11} ; Belations to eacb other {12) 3 Axioms {IS, 14:, 15, 10, 17, 18) ; Illustrations ; Examples 4, 6 Constants and Variables. — What {19,20); Any expression containing a va- riable is a variable when taken as a whole {21) ; Distinction of Depend- ent and Independent Variables {22, 23, 24), Illustration ; Equicrescenfc Variable {2S) ; Contemporaneous Increments {20} ; Illustration 6, 7 Functions and their Forms. — Definition of Function {27), Illustration ; Exact limitation of the term, Sch. ; Functions classified as Algebraic and Transcendental, and the latter as Trigonometrical, Circular, Logarithmic and Exponential, with Definitions {28, 29, 30, 31, 32, 33) ; Functions Explicit or Implicit {34, 35, 36), Notation {37) ; Functions Increasing or Decreasing {38, 39, 40) 7-9 The Infinitesimal Calculus. — What {41) t Illustration; Two Branches {42) , 9,10 ^»» CHAPTER I. TSE DIFFERENTIAL CALCULUS, SECTION L DIFFEEENTIATION OF ALGEBRAIC FUNCTIONS. Definitions. — The Differential Calculus {43) ; A Differential {44) ; Con- secutive Values {45), Illustrations 11 Notation for a Differential {4G) 11 EULES FOR DIFFERENTIATING ALGEBRAIC FUNCTIONS. EuLE 1.— To Differentiate a Single Variable {47), Geometrical Illustration 12 Bule 2. — Constant Factors {48), Geometrical Illustration 12, 13 BuLE 3. — Constant Terms (49) ; Geometrical Illustration ; An infinite variety of functions may have the same differential {50) 13 BuLE 4. — The Sum of Several Variables (51^. Illustration ; Character of dr, dy, dz. etc , Sch 14 XX CONTENTS. PAGE; Rule 5. — The Product of Two Variables (52), Illustration ; Rate of Change 14 Rtile 6. — The Product of Several Vaiiables (33) 15 RT7ii£ 7. — Of a Fraction with variable numerator and denominator (54) , With constant numerator (55) ; With constant denominator, Sch 15, 16 Rule 8. — Of a Variable with exponent {36) : Square Root (57) ; Other special rules, Sch 16 ExEECiSES in differentiating 16-22 Tt.t.ustbative Examples showing the significance of differentiation 22-25 SECTION 11. DIFFERENTIATION OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS. Definition. — Modulus {S8) 25 To DiEEEBENTiATE A LoGABiTHM, Common, Napierian {SO) 25 To DiFFEBENTiATE EXPONENTIALS. — With Constant base {00), With reference to Napierian logarithms {01) ; When the base of the Exponential is the base of the system of logarithms {62) ; Of exponential with variable base (63) 26 Exercises 27, 28 DrFEEKENTIATING A VARIABLE WITH TmAGINABY EXPONENT {6S) 28 IujUSTBAtive Examples 28-30 SECTION IIL DIFFERENTIATION OF TRIGONOMETRICAL AND CIRCULAR FUNCTIONS. Of Teigonometeical Functions. — Of a sine {66) ; Of a cosine {67), Signifi- cance of the sign {68) ; Of a tangent {69) ; Of a cotangent (70) ; Of / a secant {71) ', Of a cosecant {72) ; Of a versed-sine {73) ; Of a co- ' versed-sine (74) 30-32 Exercises 32 , 33 Illustbative Examples 34, 35 Of Cieculab Functions. — In terms of sine (75), Relation io differentiating trigonometrical functions {76) ; In terms of cosine {77) ', In terms of tangent {70) ; In terms of cotangent {80) ; In terms of secant {81); In terms of cosecant {82) ; In terms of versed-sine {83) ; In terms of co- versed-sine {81:) 35, 36 Exercises ; Geometrical Illustration 37-39 SECTION IV, SUCCESSIVE DIFFERENTIATION AND DIFFERENTIAL COEFFICIENTS. Successive Differentiation. — Definiiions — Of successive differentials {87), IllustraMons ; Of Second, Third, etc., differentials {88} ; To produce cnccossivc diff^reiitial^ (89) 40, 41 CONTENTS. Xxi PAGR EXEECISES 41 , 42 DuTEEENTiAii CoEPFiciENTS. — A first, A second, A third {90) ; Illustration ; Differential coefficients generally variable, Sch 42-44 Exercises .......,,,,,,,,,,,,,,, 43, 44 SECTION V. FUNCTIONS OF SEVERAL VARIABLES, PARTIAL DIFFERENTIATION, AND DIFFERENTIATION OF IMPLICIT AND COMPOUND FUNCTIONS. Functions oi' Independent and of Dependent Vabiables {91) ; lUustra- j tions 44 Definitions.— Partial Differential (92) ; Total Differential (93) ; Illustra- tions ; Partial Differential Coefficient {94) ; Total Differential Coefficient {95) ; Equicrescence of variables, Sch 45 Total Deffseential equals the Sum or the Paetial Diffeeentials {97) ; Illustrations ; Exercises 45-48 Notation of Differential Coefficients {98) 48 Total Diffeeential Coefficient. — Of function of two variables, Formula, (99 ) ; Meaning of -i- in sucb cases, and distinction between I — and — , I ^ dx [dxj dx i Sch. ; Of three variables, Formula {100) ; "When u = f{y, z, w), and J y = q){x), z = cpiix), and w = qj^ {x) {101) ; Exercises 48-50 Implicit Functions. — To differentiate f{x, y) = {102) ; Why — p= 0, and —J or — not, Sch. ; Exercises 51, 52 dx dy Compound Functions. — Definition, and methods of expression {103) ; To differentiate u=f{y), when y = cp{x) {104:); Exercises; To differen- tiate u = cp{z), when z =f{x, y) {103) 52, 53 k SECTION VL SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO INDEPENDENT VARIABLES, AND OF IMPLICIT FUNCTIONS. Both Vaeiables may be Equiceescent {106) ; Illustration 53 Successive Paetial Diffeeentials. — ^Definition {107) '■> Notation {108) ; Partial Differential Coefficients, Sch. ; Examples ; Order of differentiation ■unimportant (109) ; Examples ; To form successive Partial Differentials of a function of two Independent Variables {110) ', Law of the formula, Sch 54-58 To FOEM Successive Diffeeential Coefficients of an Implicit Function of A Single Vaeiable {111) ; Examples 58-60 Deeived Equations. — What; Orders of; First and Second produced from u = =f{x, y) {112) 60, 61 XXU CONTEMS. SECTION VIL CHANGE OF INDEPENDENT VAEIABLE. PAGB Why necessaet {113) 61 FoBMS OS -^, -~, -— , -vrlien neither variable is equicrescent {114:) ; Ex- (XX CuC CfriC amples . . 62, 63 FoBMUIiaJ FOE CHANGING FROM X 10 p {116) 64 FoEMuxLiE FOE Inteoducing A NEW VAEiABiiE 6 as the equicrescent {117) 64 Examples 65, 66 To Express the Partial Duteeential Coefficients of u =f{x, y), in terms of r, and 6, when x = cp{r, 6), and 2/ = !entiax,3 136, 137 Teigonometrical DiFFEBENTiALS. —Of the forms dy = sm'»a; dx, dy = cos''x dx, dy = sin'" x cos" a; d.v {195) ; When this process is appHcable and the final forms, {196) ; Special method when the exponents are even (page 140) ; Of the form dy = ^^^^dx, Ex. 17 ; Examples 137-140 ^ cos»a; Of the forms dy = x" sin « dx and dy = x" cos x dx {197) ; Examples 141 p Of the form dy = sin"» x cos* x dx integrated in terms of multiple arcs {198) ; Examples 141, 142 CiECOLAB DiffeeentiaijS. — Of the forms dy =f{x) sin— ^ x dx, f{x) cos— i x dx, etc. {199) : Examples 142, 143 Of the Foems dy = e'^ sinj'xdx, dy = e«3=cos"xdx {200) ; Examples . . 143, 144 dy Op THE FoEM di/ = T — r-T r{201) 144, 145 ^ (a -f- fecosa;)" ^ SECTION K INTEGRATION BY INFINITE SERIES. Occasion foe this Method {202) 146 Examples illustrating the method 146, 147 SECTION VL SUCCESSIVE INTEGRATION. Geneeal Peoblem {203) ; Examples ; The constants strictly general, 8ch. 1 ; Condition of complete integration, 8ch. 2 147, 148 The nth Integbal Developed by Maclaurin's Formula {204) ; Example . . . 149 SECTION VIL DEFINITE INTEGRATION AND THE CONSTANTS OF INTEGRATION. Definitions. — An Indefinite Integral {205) ; A Corrected Integral (206) ; Integration between Limits {207) l A Definite Integral {208) ; Ex- amples 150, 151 Disposing op the Constant op Integbation. — Two methods {209) ; Ex- amples 151, 152 INTRODUCTION. 1 BRIEF SURVEY OF THE OBJECT OF PURE MATHEMATICS AND OF THE SEVERAL BRANCHES. 1, JPiire 3fathe7natics is a general term applied to several branches of science, which have for their object the inyestigation of the properties and relations of quantity — comprehending num- ber, and magnitude as the result of extension — and of form. 2, The Several branches of Pure Mathematics are Arith- metic, Algebra, Calculus, and Geometry. 3, Arithmetic, Algebra, and Calculus treat of number ; and Geometry treats of form and magnitude as the result of extension. 4, Quantity is the amount or extent of that which may be measured ; it comprehends number and magnitude. The term quantity is also conventionally applied to symbols used to represent quantity. Thus 25, m, xi, etc., are called quantities, although, strictly speaking, they are only representatives of quan- tities. ScH. 1. — It is not easy to give a philosophical account of the idea or ideas represented by the word Quantity as used in Mathematics ; and, doubtless, different persons use the word in somewhat different senses. It is obviously incorrect to say that "Quantity is anything which can be measured." Quantity may be affirmed of any such concept ; nevertheless, it is not the thing itself, but rather the amount or extent of it. Thus, a load of wood, or a piece of ground, can be measured ; but no one would think of the wood or the ground as being the quantity. The qicantiiy (of wood or ground) is rather, the amount or extent of it. The word is very convenient as a general term for mathematical concepts, when we wish to speak of them without indicating whether it is number or magnitude that is meant. Thus we say, *'i7i represents a certain quantity," and do not care to be more specific. As applied to number, perhaps the term conveys the idea of the whole, rather than of that whole as made up of parts. It is, therefore, scarcely proper to speak of multiplying by a quantity ; we should say, by a number. 2 INTRODUCTION. On the other hand, when we apply the term quantity to magnitude, it is -with the idea that magnitude may be measured, and thus expressed in number. The distinction between quantity and number is marked by the questions, " How much ?" and " How many ?" ScH. 2. — So, also, the word Magnitude, as used in 'mathematics, is not easily defined. Sometimes it has reference to quantity in the aggregate, or mass, and sometimes to the relation which one quantity bears to another. Thus, we speak of a line, a surface, or a solid, as a magnitude, simply mean- • ing thereby that these have extent, — are extended. A circle, a triangle, a cube, are magnitudes, — i. e., they have extension. Again, we speak of the magnitude of a circle, meaning its size, — area as compared with some other surface. The magnitude of a line is expressed by telling how many times it contains another Hne of known length. In like manner the magnitude of a surface or a volume is made known by comparing the surface with some unit of surface, and the volume with some unit of volume. In one aspect, therefore, number is an expression for the ratio of magnitudes. 5, WuiTlbev is quantity conceived as made up of parts, and answers to the question, "How many?' IiiiiUSTRATioN. — Thus, a distance is a quantity ; but if "we call that distance 5, we convert the notion into number, by indicating that the distance under consid- eration is made up of parts. Now, the distance may be just the same, whether we consider it as a whole, or think of it as 5, — L e.. as made up of 5 equal parts. Again, m may mean a value, as of a farm. We may or may not conceive it as a number (as of dollars). If we think of it simply in the aggregate, as the worth of a farm, m represents quantity ; but if we think of it as made up of parts (sis of dollars), it is a number. 6. Number is of two kinds. Discontinuous and Contin- uous. 7. Discontinuous JS^umber is number conceived as made up of finite parts ; or it is number which passes from one state of value to another by the successive additions or subtractions of finite units, — i. e., units of appreciable magnitude. 8, Continuous Number^ is number which is conceived as composed of infinitesimal parts ; or ib is number which passes from one state of value to another by passing through all intermediate values, or states. III. — The method of conceiving number with which the pupil has become familiar in arithmetic and algebra, characterizes discontinuous number. Thus the number 13 is conceived as produced from 5 by the successive additions of finite units, either integral or fractional. In either case we advance by succes- sive steps of finite length. If we say 5, G, 7, etc., till we reach 13, we pass by one kind of steps ; and, if we say 5.1, 5.2, 5.3, etc., till we reach 13, we pass OBJECT OF PURE MATHEMATICS. B Fig. 3. by another sort of steps (tenths), hut as really hy finite ones. If, however, we call the line A B, Mg. 1, x, and CD, x', and conceive AB to slide to the po- sition C D, increasing in length as it moTes so as to keep its extremities in the lines OM and ON, it will pass by infinitesimal elements of growth from ^ the value x, to the value rj'; or, it Vviil pass from one value to the other by passing through all inter- ^i^- !• mediate values, and thus becomes an illustration of continuous number. Again, if the line AB, Fig. 2, be considered as generated by a point moving from A to B, and vro AC B call the portion generated when the point has reached Fig. 2. C, X, and the whole hne x', x will pass to x' by re- ceiving infinitesimal increments, or by passing through aU states of value be- tween x and x\ A surface may be considered as generated by the motion of a hne, and thus afford another illustration of continuous number. Thus let the parallelogram AF be conceived as generated by the right lino A B moving parallel to itself from A B to E F. When A B has reached the position CD, call the surface traced, namely A BCD, X, and the entire surface ABEF, x'; then will x pass to x' by receiving infinitesimal increments^ or by passing through all intermediate values. Finally, as volumes may be conceived as generated by the motion of planes, all geometrical magnitudes afford illustrations of continuous number. "We usually conceive of time as discontinuous number, as when we think of it as made up of hours, days, weeks, etc. But it is easy to see that such is not the way in which time actually grows. A period of one day does not grow to be a period of one week by taking on a whole day at a time, or a whole hour, or even a whole second. It grows by imperceptible increments (additions). These inconceivably small parts of which continuous number is made up are called I:ifinitcsim,als. Motion and force afford other illustrations of continuous number. In fact, the conception which regards number as continuous, will be seen to be less artificial — more true to nature — than the conception of it as discontinuous. 9. A.rithmetic treats of Discontinuous Number, of its nature and properties, of the various methods of combining and resolving it, and of its appHcation to practical affairs. For an outUne of the topics of Arithmetic, see the Complete School Algebra, Akt. 9. 10. Algebra treats of the Equation, and is chiefly occupied in explaining its nature, and the methods of transforming and reducing it, and in exhibiting the manner of using it as an instru- ment for mfirhf^matioal investiiration. 4 INTRODUCTION. For a fall account of the province of Algebra, see the Complete School Algebra, Aet. 10. H, Calculus (The Infinitesimal Calculus) treats of Continu- ous Number, and is chiefly occupied in deducing the relations of the infinitesimal elements of such number from given relations be- tween finite values, and the converse process, and also in pointing out the nature of such infinitesimals and the methods of using them in mathematical investigation. 12* (jreOTyietvy treats of magnitude and form as the result of extension and position. ScH. 1. — The principal divisions of the science of Geometry are : 1. The AjQcient, Platonic, Special, Graphic, or Direct Geometry (the common Geometry of our schools), iucludiug Trigonoruetry, Conic Sec- tions, and all other geometrical inquiries conducted upon these methods. 2. The Analytical, Modern, Cartesian, General, or Indirect Geometry i^the theme of this volume), and 3. Descriptive Geometry. ScH. 2. — The first system of geometrical investigation probably took its rise as a science in the school of Plato (about 400 B. C), and was brought almost to its present state of perfection, as far as its methods are concerned, by the time of Euchd (about 300 B. C); hence it is called the Ancient or Platonic Geometry. As the argument is carried forward by a direct inspec- tion of the forms (figures) themselves, delineated before the eye, or held in the imagination, it is called the Direct or Graphic method. Inasmuch as it discusses particular instead of general problems it is properly charac- terized as Special. With this method of geometry the student is supposed to be acquainted before commencing the study of this volume. The fundamental notion of the Modem Geometry (a system of co- ordinates), was developed by Des Cartes in the earher part of the 17tli century, and hence the names Modern or Cartesian. The term Analytical has come to be applied in mathematics in the sense of Algebraical, all investigations carried forward chiefly by the aid of Algebra being called Arjalytical. This use of the term is quite unfortunate, inasmuch as the processes of Algebra are no more analytical, in the true sense of that term, than r.re those of the Special Geometry. Again, as a name for the General Geometry, even if used in the sense of algebraic, the term does not distinguish the system from any other api)lication of algebra to I geometry. The true character of the Modern Geometry is expressed by the terms Indirect, and General. This system of geometrical reasoning proposes the solution of general problems, and effects its purpose by first translat- ing geometrical forms into equations, then carrying forward the investi- gation by means of these equations, and finally returning to the geomet- rical forms by a re-translation. The indirectness of this method is appa- OBJECT OF PURE MATHEMATICS. 5 rent, and might iseem, in itself, a serious objection ; but it is found to be of great advantage, inasmuch as it makes the discussions much more comprehensive {general). To illustrate this general (comprehensive) char- acter of its discussions, we have only to notice some of its problems. Thus the Special Geometry discusses the problem of the tangent to a circle, and, on an independent basis, investigates the properties of a tangent to any other curve, making a special problem with respect to each sei3arate curve studied. On the other hand, the General Geometry proposes the problem in this way : To find a formula sufficiently general to embrace the 2'>roperties of tangents to all plane curves ; — in technical language. To find the equation of the tangent to any plane curve. Again, in the Special Ge- ometry, the area of a circle is obtained (approximately). But the General Geometry proposes to investigate the problem on a broader basis, and find a formula which shall be applicable in finding the area of any plane curve. IS, DescTiptive Geometry is that system of geometry which seeks the graphic solution of geometrical problems by means of projections upon auxiliary planes. This is the ordinary definition of the Descriptive Geometry, and it would be out of place to attempt any elucidation of it here. ScH. — ^From the definition of Geometry, as well as from the detailed study of its propositions, it will be seen to embrace two classes of prob- lems; viz., Problems relating to Position, and Problems relati7ig to Magni- tude. Problems of the latter class were solved by the aid of algebra before the time of Des Cartes ; but it was reserved for him to invent a method by which problems of both kinds could be so discussed. This system constitutes the foundation of the General Geometry. 14:, The inquiries in the General Geometry may be divided into two classes, viz. : 1. Concerning Plane Loci, 2. Concerning Loci in Space. In accordance with this division the present treatise is divided into Two Books.^ ScH. — This division is found especially convenient when the subject is treated by the aid of the Calculus, as it corresponds to the distinction between functions of a single variable, and functions of two variables. * The Second Book is reserved for another volume, which will also contain an advanced course in the Calculus. BOOK I. OF PLANE LOCI OHAPTEE 1. THE CARTJESIAK METHOD OF CO-OHDIHATES. snoTiojsr I. Definitions and Fundamental Notions. 1, The term Locus as used in geometry is nearly synonymous with geometrical figure, yet having a latitude in its use which the latter term does not possess. The locus of a point is the line (geometrical figure) generated by the motion of the point accord- ing to some given law. In the same manner, a surface is conceived as the locus of a line moving in some determinate manner. 2. TJie General QeoTnetry is a system of geometrical in- vestigation in which the loci under consideration are represented by equations, and the inquiries carried forward by means of these equations, the final object being the discussion of general problems. [Note. — ^WMle it is true that the only way to obtain a full comprehension of the nature of a science is by the detailed study of its parts, it is, nevertheless, important, at the outset, to com- prehend as clearly as possible the general aim of the science, in order that the tendency of the several steps in our progress may be perceived, and the symmetry and unity of the whole may appear . According to our definition it will be our first purpose to exhibit a scheme by whieh points, lines straight and curved, the magnitude of angles, surfaces, etc., which we have char- acterized as " geometrical forms " (loci), may be represented by equations. This will be done in Section 1st of this chapter. Section 2nd will then exhibit a method of constructing the geometrical figure represented by any given equation. Then will follow a series of sections showing how the equations of loci are derived from the definitions of the figures. This series of sections comprises •what may be termed the translation of geometrical forms into algebraic equations, and wiU answer such questions as : "What equations represent points? What straight lines? What circles? What ellipses ? etc., etc." Section 2nd, which shows how equations are translated into geometrical forms, might, perhaps, with strict logical propriety, follow instead of precede this series of sections ; but it is thought the present arrangement will promote clearness of conception. The first three chapters will be seen to be preparatory. It is not their purpose to develop geometrical truths, but DEFINITIONS AND FUNDxVMENTAL NOTIONS. 7 simply to prepare instruments (the equations of loci) to be subsequently used in conducting geo. metrical inquiries. In the fourth chapter it will be our purpose to show how geometrical truth can be developed by means of these equations.] 3, A device by means of which we are enabled to represent loci by equations is called a METHOD OF CO-ORDINATES. 4, There are two systems of co-ordinates in common use, viz.: 1. The system of BectiHnear Co-ordinates, 2. The system of Polar Co-ordinates. Sm There are two varieties of the rectihnear system of co-ordi- nates, the rectangular and the oblique. (In our study, the rectan- gular system will always be used unless otherwise specified. ) 0, In order to locate a point in a plane by the method of recti- linear co-ordinates, two lines intersecting each other are assumed as fixed in position. These hues are called A.xes of JtefevefiCCf or, simply. The A.xes. The system is called rectangular or obhque, according as these lines make a right or an oblique angle with each other. 7, One of these axes is called the Axis of Abscissas^ and the other is called the Axis of Ordinates, 8, The Ori(/ifl is the intersection of the axes. 9, TJie Co-ordinates of a point are its distances from the axes, the distance to either axis being measured on a line parallel to the other, or on that other axis. 10» The Abscissa of a point is the co-ordinate which is measured parallel to or on the axis of abscissas, and is the distance of the point from the axis of ordinates measured on a line parallel to the axis of abscissas. 11, The Ordinate of a point is the co-ordinate which is measured parallel to or on the axis of ordinates, and is the distance of the point from the axis of abscissas measured on a line parallel to the axis of ordinates. ScH. 1. — These lines, when spoken of separately, should be distinguished as abscissa and ordinate; but, when taken together, they are called co- ordinates. 8 THE CARTESIAN METHOD OF CO-ORDINATES. III. — Definitions 3 to 11 may be illuB- Y trated thus : Let the plane in which the loci r/ » are ^tuated be represented by the surface p' / , / of the paper, Fig. 4. In this plane assume / / / two fixed, indefinitely extended, straight D" / / D"' / I'nes, as XX' and YY, intersecting each / ^^j / D X other at A, and to which all points in the / / P'" plane are to be referred. These lines are p''" l'^" the Axes and A is the origin, i.e., the point /, at which the co-ordinates are conceived to originate, and from which they are reck- ^^' oned. One of these lines, as XX' (in ordinary use the horizontal one) is called the Axis of Abscissas, because abscissas are reckoned on it ; and the other, YY', is for a hke reason called the Axis of Ordinates. The system is called rectangular or obUque according as Y AX is a right or an oblique angle. It is evident that we can now define the position of any point in this plane by giving its distances from these two fixed lines, or axes. For convenience, we measure these distances on hues parallel to the axes. (In the case of rectangular axes, the co-ordinates will become the perpendicular distances of points from the axes.) Thus the location of the point P is determined by giving the lengths of PE, the abscissa of P, and of PD, the ordinate of the point. Usually, A D is called the abscissa, instead of P E. P D and A D taken together are called the co-ordinates of the point P. . ScH. 2. — The pupil will see that this device for locating points is not unlike the method of locating places on the earth's surface by means of latitude and longitude. 12, Abscissas are represented in the notation by the letter ar, and ordinates by y. 13. The four angles into which the plane is divided hy the axes are distinguished thus : The angle above the axis of abscissas and at the right of the axis of ordinates is called the First Angle; and the numbering proceeds from right to left. YAX is the First Angle, VAX' is the Second, X'AY' is the Third, and XAY' is the Fourth. 14:, In order to indicate in which of the four angles a point is located, the signs -f and — are used on the following principles : abscissas reckoned from the origin to the right are marked 4-, and those reckoned to the left are — • ; ordinates reckoned upward from the axis of abscissas are +, and those reckoned downwards are — . Accordingly, the abscissas of points in the 1st and 4th angles, as A D and AD'" arc +, while those in the 2nd and 3rd angles as, AD' and AD", arc — . Ordinates in the 1st and 2nd angles, as PD and P'D', are 4-^ and those in the 3rd and 4th, ar; P"D"aiid P"'D"' are — . DEFINITIONS AND FUNDAMENTAL NOTIONS. 9 IS* The quantities used in General Geometry are distinguished as Constant and Variable. 16, A. coiistaflt quantity is one which maintains the same value throughout the same discussion, and is represented in the no- tation by one of the leading letters of the alphabet. 17 » Variable quantities are such as may assume in the same discussion any value, within certain limits determined by the na- ture of the problem,* and are represented by the final letters of the alphabet. III. — In Fig. 5 let BECF be a circle whose radius is B, and XX' and YY' be the axes of reference. Eepre- Bent the abscissa of any point in this circumference by x and the corresponding ordinate by y ; bo that when x signifies AD, y shall represent PD ; when — xis AD', ?/ shall be P'D' ; when — a; is AD", — j/ shall be P"D", etc. Now, suppose it possible to represent the relation between cc, y and R by an equation so gen- eral as to be true for all points in this circumference, as P, P', P", etc. (It will subsequently appear that this equation is x- -f- ?/■ = R' •) lu such an equation B would be constant, for it remains the same for all positions of the point P ; and x and y would be variables, since they vary in value with every change of the position of P. In such a problem it is evident that x ox y could not exceed B, hence these variables could have all values- between the limits of -{- B and — B. Sen. 1. — Care should be taken not to confound the terms co7is^«wif and variable as here used, with known and unknown as used in aJgebra : esi^ecially as the notation would suggest an identity which does not exist. Botli the known and unknown quantities of Algebra are constants ; moreover the constants in General Geometry may be either knoimi or unknown; and the same, in a certain sense, may be said of the variables. ScH. 2. — In order that the variables may retain their peculiar charac- teristic, we cannot have as many equations arising from a particular prob- lem as there are variables ; thus, if there are two variables involved, we have but one equation. In algebra such problems are called indeterminate, since the equation does not determine definite values of the unknown quantities, but can be satisfied by an infinite variety of values . From this feature of the General Geometry it is sometimes called Indeterminate Analysis. The Calculus is also embraced under the same term, as its problems involve a like feature. * Our limits do not permit a discussion of the continuity of functions and the general geo- metrical interpretation of imaginary co-ordinates, and hence for simplicity we retain the con- ception of imaginaries as impossible quantities. 10 THE CARTESIAN METHOD OP CO-OBDINATES. IS. To construct an equation, or find its locus, is to draw the g&^ metrical figure represented by it. ■^♦»- SECTION IL Constructing Equations, or Finding their Loci. 19. A curve is continuous when its course is uninterrupted both in extent and in the character of its curvature. III. — A circle, an ellipse, and the curve in Mg. 6 are examples of curves continuous in extent and curvature. They may be traced throughout by the uninterrupted movement of a point. The curve Fig. 1, is discontinuous in extent ; and in Fig. S, we have an example of a curve discontinuous in curvature. Fig . 9 affords an example of discontinuity both in extent and curvature. 20, A. ^Branch is a continuous portion of a curve. In Figs. 7 and 8 the curves have two branches each. In Fig. 9 there are four branches. 21. A curve is symmetrical with respect to either axis, or to any line, when it has the same form on both sides of the hne, or when every point on one side of the line has a corresponding point on the other. The curves in Figs. 6, 7 and 9 are symmetrical with re- spect to the axis of abscissas, and the first two with respect to both axes. The curve in Fig. 8 is not S3^mmetrical with respect to any hne. Fig. 6. Fig. 7. Fig. 8 Fig. 9. 22 JProh. To locate a Point whose co-ordinates are given. Solution. — Lay off from the origin, on the axis of abscissas, a distance equal to the given abscissa, to the right if the abscissa is -}-, and to the left if it is — . Through the point thus found draw a line parallel to the axis of ordinates, and lay off on it a distance from the axis of abscissas equal to the given ordinate, above if the ordinate is -|-> and below if it is — . The point thus found wUl be the one required. CONSTRUCTING EQUATIONS. H Ex. 1. Locate the point x = 3, y= — 5. Solution. — ^Draw the axes XX' and YY'. Lay ^' off A B = 3 to the right, as a* is -\-, and draw BO parallel to YY'. Then take PB =5 helow the axis of abscissas as y is — , and P is the point required. Y' o ScH. — Points are usually designated by mentioning Fig. 10. simply their co-ordinates, as the point 3, — 5, for the point in the last example. The abscissa is mentioned first. Exs. 2 to 9. Locate —6, 2; —5, —7; —3, ; 0, — 3 ; 0, ; 5, —1; 2, ; 0, 4. Queries. — Where are points situated whose abscissas areO? Where are points situated whose ordinates are ? What are the co-ordinates of the origin ? In what line are 3, — 2 ; 3, 5 ; 3, ; and 3, 4 situated ? 23, JProb, To find the locus of an equation between two variables; i. e., to construct the equation. Solution. — Solve the equation with respect to one of the variables. Then, since the equation expresses the relation between the co-ordinates of all points in the locus, substitute for the other variable any values which give real values for the first, and locate the points thus determined. These will be points in the locus ; and, by determining a sufi&cient number, the locus can be sketched through them. Ex. 1. Construct the equation ^^ — - — = 1. o Solution. — Solving the equation for y, we have y = 2x -^ S. Now attributing arhitary values to x, we make the following table of corresponding values : When X =1, y =5, giving the point 1, 5 ; " x==2, y=7, " " " 2, 7 ; " a; = 3, 7/ = 9, " " " 3, 9 ; etc., etc., " " '* etc. Noticing that all positive values of x give real, positive, and single values to y, we discover that the locus has but one branch which extends to the right of the axis of ordinates, extends indefinitely, and lies above the axis of abscissas. Again, giving negative values to a', we have When X = — 1, y = 1, giving the point — 1, 1 ; " X = —2, y = —1, '' " " —2, — 1 ; «' ic = —3, y = —3, " " " —3, —3 ; and for all subsequent negative values of x, y has real, negative, and single values. Hence we learn that the locus has a single branch extending indefinitely in the third angle. If we make 2/ = 0, a; = — li ; whence we see that the locus cuts the axis of abscis- sas at — li, 0. If we make x = 0, y = S ; and hence the locus cuts the axis of ordinates at 0, 3. 12 THE CARTESIAN METHOD OF CO -OBDINATES. Finally, locating these points, as in Fig. 11, we find that the line M N includes all the points, and hence conclude that it is the required locus. ScH. — If any other values be attributed to x, either integral or fractional, positive or negative, and the corresponding values of 3/ deduced, the points thus determined will fall in the line M N. 0/3 X' A X Ex. 2. Find the locus of the equation Fig. 11. Result. A straight line cutting the axis of abscissas at 8, 0, and the axis of ordinates at 0, 2. ^ /)/ Ex. 3. Find the locus of the equation ^x — 1 = -—-. Result A right line passing through 0, 4, and 3, — 1. 24:. ScH. — ^If there is nothing in the nature of the equation to make an- other course preferable, it is customary to solve it for y, finding the value in terms of x, and constants. If, however, the equation is above the second degree with respect to either of the variables, it is expedient to solve it with reference to the variable which is least involved. Thus, in order to construct Zx — y- = 2j/3 — y — 5, we solve with reference to x, and then substitute arbitrary values for y, finding the corresponding values of x. 26. Def. — The Independent Variable is the one to which we assign arbitrary values, usually x. The other is called the Dependent Variable. This distinction is made simply for convenience, and is not founded in any difference in the nature of the variables : either variable may be treated as the independent variable. 2(>, ScH. — There are certain peculiarities of loci, which readily appear from the form of the equation. These should always be noted. Observing tiiein is called Discussing, or Interpreting the Equation. The following are some of these points : 1st. The Intersection of the locus loith the Axes. Where the locus cuts the axis of abscissas ?/ = 0; hence substituting this value (0) of y, in the equation, and finding the corresponding value or values of x, determines the intersec- tions with the axis of abscissas. In like manner, making x = 0, and finding the corresponding values of y, determines the intersections with the axis of ordinates. 2nd. The Limits between which the locus is comprised, and its continuity or discontinuity between these limits. These questions are to be determined with respect to each axis. The limits are discovered by determining the great- est and least values of the independent variable which give rea/ values to the CONSTRUCTING EQUATIONS. 13 dependent one. If all values of the independent variable between the Hmits observed in this way, give real values for the dependent variable, the locus is continuous in extent between these hmits. If, on the other hand, there are certain values oi the former which render the latter imaginary, the locus is discontinuous; and the limits of discontinuity are to be observed by find- in<^ the limits between which the values of the dependent variable are ima- ginary. 3rd. Whether the locus is symmetrical with respect to an axis, or with any line, or not The manner of determining this is as follows : If, for each real value of one variable, the other has two values, numerically equal but with contrary signs, there are points similarly situated on opposite sides of the axis from which the variable having two values is reckoned, and hence the locus is symmetrical with respect to that axis. Again, if there is any line so situated that the values of the intercepts of either of the co-ordinates be- tween It and the locus, on both sides of the line, are equal, the locus is ^symmetrical with respect to that Hne. [Note. — There are mauy other characteristic features of loci which appear more or less immedi- at^i^y from the forixi of the equation, and some of which will be noticed in a subsequent part ox the course. Those now mentioned are sufficient for our present purpose if the pupil becomes perfectly familiar with them. This famiharity can be attained only by 2 careful study of examples. In fact • it is hardly probable that the pupil can understand the full purport of the language of the last scholium until he has solved several examples. After studying a few which follow, he can retiim and read the scholiiim again, and be better able to see its meaning.] Ex. 4. Find the locus of the equation x^ + y'^ = 25. Solution. ?/ = ± ^^25 — a;2. For x = Q, y = 5 and — 5. Hence the locus cuts the axis of ordinates at (0, 5) and (0, — 5). For ?/ = 0, a; =: 5 and — 5. Heuce the locus cuts the axis of abscissas at (5, 0) and ( — 5, 0). Again, as every value of X between -f- 5 and — 5, gives two real values for y, numerically equal, but with opposite signs, the locus is symmetrical with respect to the axis of abscissas, and continuous between these limits. In hke manner, cc = it -n/25 ■ y~ shows that the locus is symmetrical with respect to the axis of ordinates, and continuous between y = 5, and ?/ =^ — 5. When x is numerically greater than 5 (either -f or — ), the values of 2/ bocome iwagrmarj/. Hence the locus is comprised between the limits ic = 5, and cc = — 5. From ic = dr ^25 — y', it appears, in like man- ner, that the limits in the direction of the axis of ordinates are y = 5, and — 5. Now assigning to x arbitrary values between -f- 5 and — 5, we find the following table of values, and points in the locus : When£e = l, y=±z v'24 = ±: 4.9 nearly ; and we have points (1, 4.9) and (1, — 4.9); «' £>;==2, y.-^=fc ^^21 ==h4.6 nearly; " " " " (2, 4.6) and (2,— 4.6); *' 05=^3, y=rfc^l6 = =fc4 " " " «' (3, 4,) and (3, --4); ** a = 4, 2/=.rt:v/9 =±3 " " ". « (4, 3,) and (4,— 3); For negative values of ;r the following points are found ( — 1, 4.9) and ( — 1, — 4.9); (—2, 4.6) and (—2, —4.6); (—3, 4) and (—3, —4); (—4, 3) and (—4, — 3). ii THE CARTESIAN METHOD OF CO-OKDINATES. Constructing the points thus determined they are found to be in the circumference of a circle whose radius is 5, and which is sym- metrical with the axes, as in Fig. 12. It is also to be observed that any values of x, fractional as well as integral, between the limits x = 5, and x = — 5, give values for y which locate points in the same circumference. Ex. 5. Construct and discuss the equa- tion 9y2 -j- 4:X^ = 36. Solution. — Solving the equation for y, y = db I'v/S' — x2. We now observe that for ic = 0, ?/ := ± 2 ; therefore the locus cuts the axis of ordinates at (0, 2) and (0, — 2). In like manner, making y = Q,x = zh^', and hence the locus cuts the axis of abscissas at (3, 0) and ( — 3, 0). Again, for each value of x which renders 9 — x-'^ 0, i. e. , for each value be- tween X = 3, and — 3, y is real and has two values, numerically equal, but with con- trary signs ; therefore the locus is symmetrical with reference to the axis of abscis- sas, and continuous between the hmits cc = 3, and x = — 3. Beyond these values of X, y becomes imaginary, and the locus is entirely comprised within x = 3 and X = — 3 along the axis of abscissas. In a similar manner from x = ±z ^ *>/ ^ — t/^, it appears that the locus is comprised between y = 2 and y = — 2, and is sym- metrical and continuous with respect to the axis of ordinates. Finding the values of y corresponding to a sufficient number of arbitrarily taken values of x, so as to enable me to sketch the curve, we have the following table of values : For a; = 0, ?/ = =fc 2, giving the points a, a' in J/ ig. 13 ; " " h, V " " " " c c' '^ " " d, d' " ** cc (c ee'" '* " g, g' " " h, h' " " Since the equation contains only the square of x, neg- ative values of x give the same values for y as positive values do, and the portion of the curve on the left of the axis of ordinates is sym- metrical with that on the right. Finally, locating the points, as made known in the table, and a similar set of points on the left of the axis of ordinates, we have an ellipse whose axes are 6 and 4, Fig. 13. Ex. 6. What is the locus of if- = 2x - For x = 0, 2/ = ± i i, givi X = ,5, y = ±: 1.97, X = 1, y = db 1.89, X =: 1.5, y = ± 1.73, ,1" = 2 y = ± 1.49, .1' = 2.5, y = : ±2 1.1 a- = 2.7£ \y== ±: .8, X = 2.9, y = : ±: .51, X = 3, y = = CONSTRUCTING EQUATIONS. 15 Ans. — It cuts the axis of abscissas at 3, 0, and lies wholly to the right of this point, extending indefinitely in two branches, one above the axis of abscissas and one below it ; and the two are sym- metrical with this axis. Fig. 14. The branch B M extends indefinitely in the 1st angle, and B M ' in the fourth. The locus is known as a Parabola. Iy' Fig. 14. Exs. 7 to 10. Construct the following equations : ^x -^ 2?/ = 4 ; 2x-\-3y=:0 ; Sx^ -\- 5y^ = 12 ; 2/« — ■ 6?/ -f ;y? = 16. Ex. 11. Find the locus of x^ — y^ = 10. Ans. — The locus is represented in Fig. 15. It is discontinuous "between x =^ n/10 and x = — v^lO ; but to the right and left of these points, it extends indefi- nitely. It is symmetrical with respect to both axes. The curve is known as an Hyperbola. ly/ Pig. 15. Exs. 12 to 23. Construct and discuss the following : y^~ 16 — j;2 . y-i =z 10^ — ^2 ; 2/2 = 1207 ; x^ — 6^ + 9 + 2/2 + 10y==0; 25{y + 4)^ + 16(^ — 5)^ = 400 ; y^ = 4 + 2 (07 — 3)2 ; 2/2 = 072 — 4 ^ y2 = Sx^ — 073 + 5 ; xy=16 ; y^ =x^ — 074 ; y'iz=x* — x^ ; y^ = x'* — x\ Ex. 24. Construct and discuss the equation x = log y. Results. — Assuming x as given in the following table of values (any convenient values of x maybe taken), the values of y can be found from a table of logarithms. For x= 0, y = 1. " X = .2, y = 1.58 nearly. " a;= .4, 2/ = 2.51 " » x = .e, t/ = 3.98 " " x = .8, y= 6.31 " " x= 1, 2/ = 10. etc. etc. etc. etc. Forx= — .1, 2/ =• 8 nearly. " a; = — .22, y—.6 " " « = — .4, y ^.4 " " x = —.7, 2/ = -2 " " x=— 1, y = .l '* " ic = — 2, 2/ — .01 ' * etc. etc. etc. etc. X' Fig. 16. Locating these values, we have the curve MN, Fig. 16, which is called the Zogarithmic Curve. It lies wholly above the axis of abscissas, as negative numbers have no logarithms. It extends on both sides of the axis of ordinates, and cuts it at (0, 1, ) a point through which all logarithmic curves pass, in whatever system the 16 THE CARTESIAN METHOD OF CO-ORDINATES. logarithms be taken, since log 1 = in all systems. The curve extends inde- finitely to the right and to the left ; but the portions are not symmetrical. Ex. 25. Construct x = log 2/, assuming 2 as the ha&e of the system of logarithms ; giving 2/ = 2"" . Tne values are, a:=0, y=\-^ x = l, y=2; x = 2, y = 4,-^x=3, y ==z 8; etc. Also, x = — 1, y= .5 ; x == — 2, y = .25 ; x = -3, y a 5 X = — 4, y = T¥> ^'^^• QuEKiES. — Locating this curve on the same axes with the preceding, what common point do they possess ? Does the right hand branch of this he to the right, or to the left of the former .'* Does the left hand branch approach the axis of abscissas more rapidly, or less rapidly in the latter than in the former ? What makes these differences ? How would it be with a base 100 ? Ex. 26. Construct and discuss y = sin x. SuGs. — The unit arc is a portion of the circumference equal to the radius. This arc is 57.3° nearly ; since, radius being unity, the semi-circumference is 3.1416, and 180^ Hence the following table. 3.1416 -""■" ^^"^^•'• For x= 0^ = 0, 2/= " £c = 103 = .17. 2/ = .17 " jc = 20o = .35. 2/ = .34 « x = 303 = .52, 2/ = .50 " £C = 403=.70, 2/ = .64 " £C = 50^ = .87, 2/ = .77 etc. etc. etc. etc. This curve is called the Sln- "usoid. Where does it cut the axis of abscissas? Is it hm- ited? What are the hmits of y ? What is the meaning of x = — 10^, x = — 200, etc. ? Exs. 27 to 33. Construct y = tan X ; y == cot x ; y = cos x y = versin x ; y = coversin^ y z= sec X ', y = cosec x. For x = 180° =3.14, " x = 1900 = 3.31, " a; = 2000 = 3.49, " a; = 2100 = 3.66, " ic = 2200 = 3.84, " a; = 230O = 4.01, etc. etc. etc. Y y = y = y = y = .17 .34 .50 2/ = — .64 y = -.ii etc. Y' Fig. 17. ScH. — These loci can be con- sfcracted with sufficient accura- cy without the numerical com- putations. Thus, taking the Ex. y = tana;, draw a circle ON, Fig. 18, with any convenient radius. Divide a quadrant, as M N, into equal parts, each so small that for practical purposes the chord and arc may be considered equal. 7 s 5 i 3 2 i. 1 Y A/1 1 MX' /' ■J / 1 / 12 3d5( / Y' >7 X Pig. 18. %. THE POINT IN A PLANE. 17 Estimating the tangents and arcs from M, and having dra^vnthe tangents as in the figure, lay off the arcs on the axis of abscissas. At the extremity of Al lay off an ordinate equal to tangent Mi, etc., etc. There are an infinite number of similar infinite branches to this curve. On the figure used for getting the tangents, when the arc passes 90° the tangents (and hence the brdinates) become negative. Strictly speaking, f negative values of x would be obtained by measuring the arcs on the circle from M downward, or from left to right ; so that, from x= to .r = —90'', the tangents (and hence the ordinates) are negative. From x = — 90° to .^ = —180°, the tangents (and hence the ordinates) are positive. Where do the branches cut the axis of abscissas ? At what values of x do the ordinates become infinite ? ^♦» . SUCTION IIL The Point in a Plane. 27* Def. — The Equations of a I^oint are the algebraic expressions which determine its position. 28. JProp, The Equations of a Point in a plane are x = a, and y = b, in which the signs of a and b are general. p1 B' Fr-pr D' Dem. — If, as in Fig. 19, we make AB = a, and through B draw DE parallel to YY', every point in DE will have its abscissa equal to a. In like manner make AC = 6, and draw FG parallel to XX', and every point in FG will have b for its ordinate. Hence the point P has a for its abscissa, and b for its ordinate ; and since two straight lines can meet in only one point, P is the onli/ point v/hich has these co-ordinates. Therefore x=a, and y = b, determine the position of a point, q. e. d. ScH. j^. — If we have x = — a, and t/ = b, P' is the point. 3/ = — b, P" is the point, etc. |Y E' Df pTT-Qr Y' Fig. 19. li x= — a, and ScH. 2. — If a; = a = 0, and y = h, the point is in the axis of ordinates. \i X = a, and y == 5 = 0, the point is in the axis of abscissas, x = 0, 2/ = characterizes the origin. ScH. 3. — A point is usually designated by simply naming its co-ordinates, '(h.Q abscissa being mentioned first. Thus the point (wi, ti) is the same as the point x = m, and y = n. Exs. 1 to 6. Locate the points x == — 3, y = 4 ; (5, — 7) ; (0,-5); (0,4); (0,0); (6,0). 18 THE CAETESIAN METHOD OP CO-ORDINATES. Exs. 7 to 10. How are the points (5, %) ; {% —6) ; (J, m) ; (— n, J) situated ? Answer to the first. — In a line parallel to the axis of ordinates and at a distance 5 from it. Any point in this line fulfills the conditions, since y = ^, i. e., is indeterminate. Exs. 11, 12. Construct the triangle whose yertices are ( — 3, 4) ; (5, — 1); and (2, — 6). Also the triangle whose vertices are (0, 3) ; (—5,0); and (0, 0). Exs. 13, 14. What figure is that the vertices of whose angles are (2, 3); (2, 8); (7, 8); and (7, 3)? What figure is that the vertices of whose angles are (2, 9); (—8, 9); (—8, 1); and (2, —1)? 29. JPvop, The Distance between two points in a plane is sJX^' — x"Y-\- {y' — y"y\ in v)hich {x', y',) and {x'\ y") arethe points. Dem. — Let the points (a;', y' ) and (x" y")be represented by P' and P", as in Fig. 20, and tlie distance between them, P'P", by D. Draw P"D parallel to AX. Then P"D = cc' — x", and P' D = y' — y". From the right angled triangle P' P" D, we have, — D = '-/{x' — x"Y -{- {y' — y' )-. Q. E. D. p7- B C Fig. 20. CoK. — ^ either of the points, as P'\ is at the origin, its co-ordinates are 0, 0, and D = Vx'^ + y'K QuEBiES. — When P" is in the axis of abscissas and at the right of the origin, what is the formula? The same with P" at the left of the origin, give D = ^/(x' + x")'2 -f-y'^. If P' is in the 1st angle and P" in the 3rd, what is the formula? If P' is in the axis of abscissas and P" in the axis of ordinates? If one is in the 2nd and the other in the 4th angle ? ScH. — Observe that the formulaD = ^[x — x"Y + [y' — y")'^ is strictly general, only noticing carefully the effect of the position of the points, upon the signs of their co-ordinates. Thus for a point P", in the 4th angle, we have x" , and — y" ; which, substituted in the formula, gives for P' in the 1st angle and P" in the 4th, D — - v {x — x"Y-\- {y' -f- y" Y' Examples. — Find the distances between the following points taken two and two: (3, 5); (2, 6); (—3, —2); (—1, 4); (—2, — 1); C-5,-7); (-3,0); (0,-4); (0,0); (-5,0). THE BIGHT LI^'E IN A PLANE. 19 SECTION IV, The Eight Line in a Plane, 30, Bep. — The Equation of a Locus is an equation which expresses the relation between the co-ordinates of every point in the locus. 31, JPvopm The Equation of a Eight Line passing through two given points is J — y' 77 (2: — x'), in which (x, y) is any point in the X — x line, and (x', y') and {x'\ j") are the given points. Dem. — Let M N be any right line referred to the rect- angular axes XX', YY'. Let P be any point in the line, and designate its co-ordinates, AD and PD, by a; andy. Let P' and P" be the given points whose co-ordinates are x', t/', and x", y", respectively. Now drawing P'E and P" F parallel to AX, x the triangles PEP' and P' FP" are similar, and m give PE : P'F : : P'E : P"F. But PE. = y—y\ P F =y' — y", P'E = x — x', and P" F = as' — x" ', hence, substituting these values, we ha-vte y—y '■ y—y X : X or y — y = y TT (a;" — x). Q. E. D. Cor. 1.— Since P'P'F = NGX, and y X X P'F P F V — y tan P'P"F, wehave^-7 = the tangent of the angle which the line X — X makes with the axis of abscissas. If x' X n y' — y" __ y' — y" . , „ ^ — GO, which being the tangent of •JO iv vl 90°, shows that the line is perpendicular to the axis of abscissas. This is as it should be, since if x' = x", the points P' and P" are equally distant from Y Y', and hence M N is perpendicular to XX'. If y = y", y'-y" 0, which being the tangent of 0°, X' X" X' X" shows that the line is parallel to (makes no angle with) the axis of abscissas. This is as it should be, since by obserying the figure, it appears, that when y' = y", M N is parallel to XX'. 32. Cor. %—The Equation of a Might Line passing through one given point. If a;' =x", and y' = y", we have 20 THE CABTESIAN METHOD OF CO-ORDINATES. y — 2/' = (^ — '^0» ^^^» by putting the indeterminate expression, ^, = a, 2/ — y' = a{x — x'). This is the equation of a straight line passing through a given point, since the conditions, x' = x", y' = y"j make P' and P" coincide. The a is indeterminate, as it should be, since, through one given point, an indefinite number of straight hues can be drawn. 33. Cor. 3. — 17ie Common Equation of a Right Line, If my — y = a{x — a:'), we make x' = 0, and designate the corres- ponding value of y' by h, so that the given point shall be the point in which the line cuts the axis of ordinates, we have, after reduction, y == ax -{- b, which is the common equation of the straight hne. In this equation a is the tangent of the angle which the line makes with the axis of abscissas, and b the distance from the origin to where the line intersects the axis of ordinates. ScH. 1. — Discussion of the Equation y = ax+b. If 5 be -f* the line cuts the axis of ordinates above the origin ; if — , below ; if 0, at the origin. In the latter case, we have y = ax, as the equation of a right line passing through the origin. If a be -f, the line makes an acute angle with the axis of abscissas, [i. e., it incHnes to the right, as the lines in Fig, 22), the tangent of an acute angle being 4-- If « be — , the angle is obtuse, [i. e. , the Hne inclines to the left, as in Fig. 23), since the tangent of an obtuse angle is — . If a = the Hne is parallel to the axis of abscissas, and if a = 00 , it is perpendicular, as will readily appear. X = ^H 2. 1 -If we solve b in which — is the a y =ax -\-h for x, we have tangent of the angle which the Hne makes with the axis of ordinates, since, in Fig. 21, the angle AHG = NHY =90° — NGX In this form, - is a the distance on the axis of abscissas from the origin to where the Hne cuts it (AG), since the base of a right-angled triangle is equal to the perpendicular divided by the tangent of the angle at the base. 34:, CoR. A.—TJie Equation of a Hight TAne referred to oblique axes. If the axes are oblique, we stiU have the same y' -, or a, signifies the ratio of the sines of forms, but in this case X' — X' the angles which the line makes with the axes, since the sines of the THE EIGHT LINE IN A PLANE. 21 angles of a plane triangle are to each other as the sides opposite. Thus in Fig. 24, P' F [ory'—y") : P"F (or ;r'-^") :: AH : AG : : sin AQH : sinAHG. Putting /? for the angle included by the ^^ axes, and a for the angle which the line makes with the axis of abscissas, we get sm a -, and, finally, y = sm (X. y' — y" __ x' — x" ~ sm(/i— a)' "'"^' —J, ^ — g-j^ ^^^ tion of a right line referred to oblique axes. Ex. 1. Construct the equation y = 2x -\- d. Fig. 24 -a) ■X-}- by as the equa- SoLUTioNs. — There are three methods of solution. 1st. Bp any two points. As it is known to be an equation of a right line from its form, if any two points be determined, as in the last section, the position of the Une will be known. For ex- ample, for a; = 3, y = 9, and for x = — 2, y = — 1 ; whence, locating these points and drawing a hne through them, we have the construction. 2nd. By the intersections with the axes. — -This is only a modification of the 1st method, merely making y = 0, whence x = — li, and making a; = 0, whence y = 3, constructing these intersections, and passing a line through them. (The pupil should execute the figures.) 3rd. By means of the tangent of the angle which the Une makes with the axis of ab- scissas. Since & = 3, we may lay off A C =3 above the origin, and thus determine C as a point in the line. Through C drawing C E parallel to AX and constructing the angle N C E so that its tangent shall be 2 (by taking CD any convenient length and erecting the perpendicular FD =2CD), the line NM is the one sought. — Or, having located C, take AG AC = iAC, whence tan AGC=— r--— -= 2, will AG give the construction. Or, again, drawing any Hne making with XX' an angle whose tangent is 2, and drawing a line parallel to it through C, the latter wiU be the hne sought. Y C / /" J D E "• ^ A X Fig. 25. ScH. — If the tangent were — , CD would be laid off to the left of O, or the perpendicular FD let faU below D. Ex. 2. Produce the equation of a line passing through ( — 3, 5), and (2, -1), Solution. — Here x' = — 3, x" =2, y' = 5 andt/" = — 1. Now, substituting _J/ y - (x — »'), and reducing to the form y =ax -j-h. these values in v — y' X — X we have y = — 1.2« -f- 1.4. The pupil should construct this equation, and then 22 THE CAETESIAN METHOD OF CO-ORDINATES. verify the result by locating the points ( — 3, 5), and (2, — 1), observing that, if the ■work is correct, they will fall in the line. Algebraically, we verify the result by sub- stituting in the equation y ■= — 1.2a; + I-^j successively for x and y, ( — 3, 5), and (2, — 1), each of which must satisfy the equation, as it expresses the relation between the co-ordinates of any point in the hne. Substituting, we get 5 =3.6 -f- 1.4, and — 1 = — 2.4 -j- 1.4, both of which are correct. Ex. 3. What angle does the line which passes through the points (3, 5), and ( — 7, 2) make with the axis of abscissas? Arts., 16° 42' nearly. Ex. 4. Produce the equation of a hne passing through the point (2, — 3), and making an angle with the axis of abscissas whose tan- gent is 4. Ans., y = 4:X — 11. Ex. 5. Produce the equation of a line passing through ( — 1, 0), and ( — 4, — 5), construct by the 3rd method, and verify the equation by locating the points. Ex. 6. Construct the triangle the equations of whose sides are y == |.J7 + 3, 1/ = — l-r 4- 4 and y == ^x — 1. Ex. 7. What is the equation of a line which cuts the axis of ordi- nates at 3 above, and the axis of abscissas at 5 to the left of the origin? (Notice that this is a case of a line passing through two points.) Ans., y = S.X ^ 3. Ex. 8. What hne ia y = 0.x ? What is x = 0.y^ How is y = 0.x + 4 situated ? How y = 0.x — 5^ Ex. 9. Find the angles which the following lines make with the axis of abscissas : viz., the hne passing through (3, 5), and ( — 1, — 4); through (5,-2), and (5, 3); through (—3, 2), and (7/2). How are these lines severally situated ? 3S, I^vop. Every Equation of the First Degree between two variables is an equation of a right line. Dem. — Every such equation may be put in the form Ay -^ Bx-\- C = 0, in which A and B are the collected coefficients of y and x, and C is the sum of the B absdlute terms. By transposition and division we have t/ = —x . Now B C putting — — = a and T == ^' there results the known form 2/ = aa; -{- 6. q. e. d. ScH. — If B and A have like signs, the line makes an obtuse angle with the axis of abscissas ; and if they have unlike signs, it makes an acute angle. li B =z the Hne is parallel to the axis of abscissas, and if J. = it is perpendicular. If A and C have like signs, the line cuts the axis of ordi- nates below, and, if unUke, above the origin. If (7 = the hne passes OF PLANE ANGLES, AND THE INTERSECTION OF LINES. 23 through the origin. In general, if an algebraic equation has no absolute term, the locus passes through the origin. (Why ?) Ex. 1. Keduce X y — 4t = 2x — lij X to the form Ay -f- 3 - — -^^ ^ JBx 4- C = 0, and describe the line according to the suggestions in the preceding scholium. Ans. — The equation is y — 13x — 21 = 0. A = 1, B := — 13, and = — 21. As A and B have unlike signs the line makes an acute angle with the axis of abscissas, the tangent of which is 13. It cuts the axis of ordinates above the orisfin at a distance of 21. Ex. 3. In like manner discuss 3 — 2 — y ^ + ?/ ^ + '^y — lOy Gx -j- y X — V , o 3^ + ?/ — F— ; — ^— + 2 = ?/ H — - — . Ex. 4. Construct the figure the equations of whose sides are 2y + 2x =3^ + 3 + 2/; ^^ — U = 2x — 6 — y ; dy+2x—6 2y X + 1 ; and x-{-y = — 3. "What is the figure inclosed ? -♦-♦-«»- 8JECTI0N V. Of Plane Angles, and, the Intersection of Lines. 3G» JPvop, The expression for the value of an angle included between two lines is tan V = ~, in which V is the angle included 1 4- aa' by the lines, and a and a' are the tangents of the angles which the lines make vjith the axis of abscissas. Dem. — Let MN and M'N', Fig. 26, be two lines whose equations are respectively y =: ax -^ h and y== a'x-\- h'. Now C BX being exterior to the triangle BCD, we have DCB = CBX — CDB, or by trigonometry tan CBX — tan CDB tanDCB — i ^ ^^n CBX X tan CDB But DCB —V, tanCBX=a', and tan CDB == a. .-. tan V: 1 + aa'' Q. E. D. X' D FiQ. 26. 24 THE CARTESIAN METHOD OF CO-ORDINATES. ScH. — ^In applying this formula to any particular example, we may obtain two results, numerically equal, but with opposite signs. Thus, if the two lines are 3/ = 2a7 + 4, and y = 3a; — 5, and we let a' =2, and a = 3, we 2—3 1 have tan V = - — , — ;:, = — --, But, if we let « = 2, and a' = 3, we have 1 + t) 7 XT- 3 — 2 1 tan V = -. -T = ij- This ambiguity is as it should be, since the two lines form, in general, two equal acute, and two equal obtuse angles with each other ; and as these angles are supplements of each other, they have tan- gents numerically equal but with opposite sigyis. 37 » J*VOb. To find the equation of a line which makes any required angle loith a given line. Solution. — Let y = ax -{-hloe the equation of the given line, y = a'x -j- h' he that of the required hne, and m the tangent of the required angle. As the relative directions of the hnes depend solely upon a, a', and m, the problem consists in finding the unkLio\\Ti a', in terms of the given tangents a and m. But m = — -; I -\- aa by the preceding proposition; whence a' = : andv== — ~ tc + feis the equation of the required Hne. ScH. — In this form b' is undetermined, as it should be, since there maybe an indefinite number of lines which will satisfy the condition, all having the same inclination to the axis of abscissas, but cutting the axis of ordinates at different points. CoK. 1. — If the required line is to pass through a given pioint (x', y'), ice , a 4- m , have J — y = -; (x — x'). •^ -^ 1 — am ^ ^ 38, CoE. 2. — If the required line is to be parallel to the given line, m = 0, and we have ai,'= a. The equations then become y = ax-\- b', and y — y' = a{x — x')^ both of which lines are parallel to y = ax -]- h. 30, CoR. 3. — If the lilies are to be perpendicular to each other, m = 00. a -\~ m in 1 .'. a' = - :=, ^ = , or 1 + act' = 0, which is called the 1 — am — am a equation of the condition of peiyendicularity. The equations of lines j)erpendioular io y = ax -{- b will therefore be y = ■ x + I'', and y — y' = (^ — ^'), the latter passing through {x', y'). * The principle upon which this reduction is effected is, that the finite terms a and 1 added to the infinites m and — am must be dropped. The axiom is, Suites added to infinites do not (apprecia- bly) affect the ratio of the infinites. The word appreciably is throwu in to aid the student's apprehenaion. It is not required, nor is it strictly correct. OF PLANE ANGLES, AND THE INTERSECTION OF LINES. 25 ScH. 2. — Two lines are parallel to each other when the two equations being reduced to the form y=a.c(7 + Z>, the coefficients of a; are the same in both; and they are perpendicular when these coefficients are reciprocals of each other with opposite signs. Ex. 1. Find the angle included between y = — ^ + 2, and y = dx 6. 3 + 1 Besult, Tan V= fTTQ '=~^- •*• ^^® ^^S^^ is ll6°34^ Ex. 2. What are the angles of the triangle the equations of whose sides are 2y — 5 = y — x; y -^ 4:X=8, and y=zix? Ans. I '^^® tangents of the angles are .6, —21, and 1.5. * 1 The angles are nearly 30°58', 92°44', and 56°19'. Ex. 3. Write the equations of three lines, each parallel to y = 2x 11, and construct the lines therefrom. Ex. 4. Write the equation of a line parallel to y — i,r = 5 and passing through (—6, 4). Keduce the equation of the parallel to the form y = ax -{- b, and then construct both lines from their equations. Verify the result by constructing the given point ; also by observino- that the ] coefficients of x in both equations are equal, and that the co-ordinates of the given point satisfy the equation of the parallel. Ex. 5. Write the equation of a line passing through ( — ^, 4), and parallel to ^x — ^y = 2. Verify as in Ex. 4. Besult The equation is ?/ = f^ -f 1^. Ex. G. Write the equations of three lines each perpendicular to ^y — 2x = 1, reduce them to the form y = ax -{- b, and verify the results by construction. Ex. 7. Write the equation of a line perpendicular to 2y 4= = x and passing through (1, — 3). Verify as before. Ex. 8. Write the equations of lines perpendicular to x^y .^J7 — 2, and severally passing through (—2,3); (0,-5); (0,0); and (—3, 0). Ex. 9. What is the angle included between y = 0.07, and y = 3x — 5 ? Between x = O.y, and y = 2x -{■ 1? Ex. 10. What is the equation of a line passing through ( 6, 0), and perpendicular to y = 0.x + 5 ? Ans., x = O.y 6. 26 THE CARTESIAN METHOD OF CO-OEDINAIES. Ex. 11. "What is the equation of a line passing through ( — 1, 3), and making an angle of 45° with y ^=.^x — 5 ? ^ns., y = — 3a:. Ex. 12. Produce the equation of a hne passing through ( — 4, — 5) and making an angle of 71° 34' with y == — 2a: + 7 ? Besult, y^=\x — 4f, calling tan 71°34', = 3. 4:0 • Pvohm To find the point or points of intersection of two lines. Solution. — For a common point tlie values of x and y are the same in both equations, and only for such a point Therefore, making the equations simultaneous restricts the values of x and y to the required point or points. Consequently, we have only to solve any two given equations for the values of x and y in order to find the point or points in which the loci intersect. ScH. 1. — The general formulcR for the value of the co-ordinates of the point of intersection of two straight Unes whose equations are y =^ ax -{-h^ h' — h ah' — ah and V = ax + &', are x = :, and y = ■, — . Upon these values ^ a — a ^ a — a ^ we may observe that for a = a', and b and b' unequal, the values of x and y become oo. This indicates that the hnes do not intersect, and hence that they are parallel. Therefore a = a' is the condition of parallelism of two straight lines. This may also be seen directly from the meaning of a and a. As these quantities are the tangents of the angles which the lines make with the axis of abscissas, it follows that when they are equal the lines make equal angles with this axis, and are, therefore, parallel. 2nd. Jlh = h', and a and a are unequal, we have x = 0, and y = h =:^h' . This is also evident from the meaning of h and h' . Both lines cut the axis of ordinates at the same point. 3rd. If a =z a and h = h' , x =%, and y = %, and the Hnes coincide. 4th. If a = 0, and 6 = 0, the first equation becomes ?/ = O.a; -f 0> h' or the equation of the axis of abscissas, and x= -, the pomt of inter- a section oiy = a'x -f h' with this axis, as it ought. Sch. 2, Art. 33. SCH. 3.— Any two equations between two variables being given, if the lines they represent are constructed, and the co-ordinates of the points of intersection measured, we have a graphic solution of the equations. Ex. 1. Where is the point of intersection of the lines ^x — \y=^\ and 2/ = — 2a: 4- 4 ? Ans., (2, 0). Ex. 2. What are the co-ordinates of the vertices of the triangle the equations of whose sides are y — 2a: + 3 = 0, — ^— ^ + 4: == ^y, and a: — iy = 2 ? Ex. 3. Where does a perpendicular from ( — 3, 8), to the line 3/ = \x — 5, intersect the latter? An^^ At (l-J, — 4|-). OF PLANE ANGLES AND THE INTEBSECTION OF LINES. 27 Ex. 4. "VVlaere does a perpendicular from the origin intersect 2ar — 3?/ = 4? Ex. 5. Given y = ^x — 3, y = — 4ar — 8, and y = — ^x + 10, as the equations of the sides of a triangle, required to find where a perpendicular from the angle included between the first two sides, intersects the third side. Besult, At the point 5 .5, 6. 34, nearly. Ex. 6. Eind the intersections of the loci y whose equations are 7{y — x) = 5 — 2x, and 2/2 + ^2 _|_ 9 =:::= Ig Qy^ q^j^^ COUStrUCt the figure. Results, At the points (.374 +, .981 +), P^J- and (—3.888 +, —2.063 +). The figure M is that given in the margin. Ex. 7. Eind the intersections of y^ = IQx, and x^ -j- t/2 = 144, and construct the loci, thus verifying the solution. Fig. 27. Ex. 8. Eind the intersections of 25?/2+ IBar^ = 1600, and IQy^ — Oo;' + 576 = 0. Results, At (9.12, 3.3), and (—9.12, —3.3), nearly. Ex. 9. Eind the intersections of x"^ — Sx -\- y"^ -\- Qy =^ 0, and y = \x -\- 1. Also of the first with 3?/ = 4ar. Also with 2/ ^= 3 — x. Results. — 1st. Imaginary results. No intersections. 2nd. A com- mon point at the origin. 3rd. Two points of intersection. ScH, 3. — The construction of loci represented by equations affords beau- tiful illustrations of principles in the theory of equations, concerning the number and character of the roots of an equation. Ex. 10. Eind the intersections of 25i/2 + 4a;2 = 100, by the following : 1st, y^ + a;' = 9 ; 2d, y^ + 2y + a;^ = 8 ; 3d, 2/2 + 41/+ ^2 = 5; 4th, 2/2 + 101/ +07'= y^^^ — 16 ; and 5th, i/^ + Vly + ^2 = — 27. Results.— Ui, At (2.44, 1.7); (—2.44, 1.7); (2.44, —1.7); (—2.44, —1.7) which affords an example of 4 real roots. 2nd, At (0.2) ; and (=fc 2.9, — f f ), which affords an example of what seems to be but three roots when there should he four. This is explained by the two values of xior y = 2, becoming -f and B X 28 THE CARTESIAN METHOD OF CO-ORDINATES. — 0, or practically, though not theoretically, one. 3rd, Gives two real and two imaginary points, illustrating that imaginary roots enter in pairs. 4th, Gives two equal real roots, both 0, and two imaginary, showing a point of contact. 5th, Four imaginary roots, showing no common point, the additional imaginary roots again entering in a pair. Ex. 11. Find the intersections of 1y^ — Lxy + '^x" — 3?/ — 2x — 8 = 0, by 4?/2+4^2 — 11 = 0. Also by y^-\-2y -^-x'^ — 6a? + 6 = 0. Also by 2/2+6?/+^2 — 4^+9=: 0. Results. — By the first in 4 points. By the second in 2 points. By the third not at all. The figure is seen in Fig. 29, in which a a a is the 1st locus, and 111, 2 2 2, and 3 3 3, the others, in order. 4:1, JPvob, To find the perpendicular distance from a given point to a given line. Method of So"littion. — First, find the equation of a line passing through the given point, and perpendicular to the given Hne {32 , 39). Second, Fig 29.. find the point in which this perpendicular inter- sects the given line {40). The problem then consists in finding the distance between two points {29. ) Cor. To find the distance between two parallels, wi'ite the equa- tion of a line perpendicular to the parallels {39), and find its intersec- tions with the parallels. The problem is then the same as {29). Ex. 1. Find the distances of the following points from each of the lines 2/ = 2j: — 3, and ^x — y = — 1, viz., 3, 2 ; — i, — 1; 0, — 6 ; 0, 0. Solution. — To find the distance from — 4, — 1, to ?/ = 2a; — 3, we have for the equation of a line passing through this point and perpendicular to this line y _j_ 1 = 4 (a; -}- 4), or 2/ = — l^ — 3. The intersection is at 0, — 3. The distance between —4, —1 and 0, —3 is D = >/l6 -}- 4, or ^20. Ex. 2. Find the sides, the angles, and the perpendicular distances from the angles to the opposite sides in the triangle the equations of whose sides are 36i/ — 4^ = 45, 3?/ + 3 = —x, and j/ = fa; — 3. f The sides are 12.79, 7.44, and 6.79. ' Besults.— ■] The angles are 24°46', 127°53', and 27°21'. (. The perpendiculars are 5.88, 3.12, and 5.36. OF THE CONIC SECTIONS. 29 Ex. 3. The yertices of a triangle are at 2, 8 ; — 6, 1 ; and 0, — 4 ; required the equations of the sides, of the hnes drawn from the vertices to the middle of the opposite sides, and of the lines drawn bisecting the angles and terminating in the opposite sides. ^ ^ ^ - SECTION VI Of the Oonic Sections. 42. Boscovich^s Definition of a Conic Section. — A Conic Section is a curve, the distance of any point in which from a given point, is to its distance from a given straight line, in a given ratio. If the distance to the point is equal to the distance to the line, the locus is a JParahola ; if less, an JEllipse ; if greater, an Hyperbola. If the distance to the line is infinite, the locus is a Circle ; but if the distance to the 'point is infinite, the locus is a Straight Line. 43. I*roh. To construct a Conic Section from Boscovich's de- finition. Solution. — Let F, Fig. 30, be the given point, A B the given line, and m : n the given ratio. Through F draw C K perpendicular, and G H parallel to AB. Take FG (= FH) : FC : : m : n, and draw CG and CH, i^roducing them indefinitel3\ Drav/ a series of parallels to G H , meeting the lines C M and C N . Now with the half of any one of these Hnes, as LT", for a radius, and the given point, F, as a centre, describe an arc cutting the parallel taken, as at P. Then is P a point in the curve. To prove that P is a point in the curve, join P and F, and draw PR parallel to C K. By similar triangles we then have Fig. 30. LT(= PF) : XC ( = PR) : : G F : FC (by construction) : : m : w. P R : : m : ?i. In like manner any required number of points in the curve may be determined, so that by connecting them the curve will be completely drawn. In this figure, as FG < FC the curve is an ellipse. Had FG been taken equal to FC, the curve would have been a Parabola. And if FG had been greater than FC, the curve would have been an Hyperbola. 44. Defs. — The fixed line, AB, is the Directrioc. The fixed point, F, is the Focus. CM and CN are the Focol Tan- gents. PF 30 THE CAETESIAN METHOD OF CO-OEDINATES. The portion of the perpendicular to the directrix through the focus, CK, intercepted by the curve is the Transverse or Jfajor A-Xis^ as I K. The centre of the transverse axis, O, is the centre of the curve. The perpendicular to the tranverse axis passing through the centre, and limited by the curve (in the ellipse), as DE, is the Conjugate, or 3£inor A.xis. The double ordinate passing through the focus, GH,is the Latus MectuTTi, I*rincipal I^arameter, or the parameter to the transverse axis. The extremities of the transverse axis, I and K, are the Vertices. The distances from the focus to the vertices are the Focal Distances. The JEccentricity, is the distance from the focus to the centre, divided by the semi-transverse axis. FO lO Ex. 1. Construct a parabola whose parameter is 12. CoNSTEUCTioN. —Let A B be the directrix. Draw C K perpendicular to it, take F at a distance 6 from the directrix, and through F draw G H parallel to A B. Take G F = H F = G, and draw C M, and C N through G and H . (The construction is then completed as in the problem above. ) To show that any point thus found, as P, is a point in a parabola whose parameter is 12, observe that LT (=PF): CT (= PR) ::GF :CF. ButGF = CF = 6, by construction. .-. PF = PR. Also GH, the parameter, *= 2G F = 12. QuEEiES. — Can the parabola ever return into itseK so as to inclose a space ? "Why ? Can j)ortions of this curve lie on both sides of the directrix ? Why ? r K Fig. 31. 4:S. CoE. 1. — If-^ =^the focal ordinate,!, e., one half the Latus Rectum, the vertex is at ^-pfrom the directrix, and the distance, if = ^i^, or ^ the parameter, GH. Ex. 2. Construct an ellipse in which the fixed ratio shall be ^, and the distance from the focus to the directrix 6. Also with the ratio |- and the distance from the focus to the directrix 5. QuEBiES, — With the same focus and directrix how does varying the ratio affect the form of the ciirve ? With the same ratio and directrix how does varying the position of the focus affect the form of the curve ? How does it appear from the first query that when the ratio is 0, the locus is a point ? Ex. 3. Construct an eUipse whose latus rectum shall be 6, and the fixed ratio f . OF THE CONIC SECTIONS. 31 Suo.— In Fig. 30, if Q F = 3 and the characteristic ratio of the curve is ^, what is OF? 46, CoK. 2 — From Fig. 30, by principles of construction it appears that the tangents at the vertices, viz., IS, and KM, are equal, respectively, to the focal distances I F, and F K- It also appears that the distance from the focus to the extremity of the conjugate axis in an ellipse, F D, equals the semi-transverse axis ; for F D = QO = -^'l I S + M K) = ^1 K. Ex. 4. Construct an ellipse whose transverse axis shall be 12, and conjugate 10 ; i. e., having given the axes, to construct the ellipse. Ex. 5. Construct an ellipse whose transverse axis shall be 10, and distance between the foci 8. Ex. 6. Having given the curve and the transverse axis, to find the foci and directrix of an ellipse. Solution. — Let ACBC, Fig 32, be the curve, and A B its transverse axis. Bisect the transverse axis with a perpendicular, and the portion of this perpendicular intercepted by the curve will be the conjugate axis. From either vertex of the conjugate axis as a centre, with a radius equal to the semi-transverse axis, describe arcs cutting the transverse axis ; these points will be the foci Fig. 32. {46). As there are two intersections, there are two foci. At each extremity of the transverse axis erect perpendiculars and make them severally equal to the adjacent focal distances, thus obtaining two points in the focal tangent (46). Draw the focal tangent, and where it intersects the transverse axis produced, erect a perpendicular to this axis, and this perpendicular wiU be the directrix. Queries.— How does it appear from the definition of the ellipse, that the curv^e can not lie on both sides of the directrix ? How does it appear that the curve cuts the axis beyond the focus ? Ex. 7. Letting A represent the semi-transverse axis, B the semi- conjugate, 2c the distance between the foci, and e the eccentricity^ show that c v/^2 i?2 ^—A~ I ' B^ and hence that 1 — ^^ ^= "7~ ' How does it appear from this that in the case of the ellipse e <^12 Ex. 8. Construct an ellipse whose transverse axis is 12, and eccen- tricity |-. SuG. First find the value of B, which is d^-, nearly. 32 THE CABTESIAN METHOD OF CO-ORDINATES. Ex. 9. Construct an hyperbola. Fig. 33. SoiiUTioN. — Let AB be the directrix, F the focus, and m : n the ratio, in which m >> n. Through F draw FK perpendicular to the directrix and GH parallel, producing both indefinitely. Take FG (= FH) : CF : : rn : n. Through C and G draw MM', and through H and C, NN', the focal tan- gents. (The process is exactly analogous throughout, to that pursued in construct- ing the ellipse, and hence need not be detailed. The student can supply it. It should be noticed, however, that the distance from the focus to any point in the curve, being greater than the distance from the same point to the directrix, there may be (are) points in the cui've on the opposite side of the directrix from the focus. These points are determined in the same manner as the others. Thus the point P'^" is found by taking T'N' as a radius, and from F as a centre drawing an arc cutting T'N' in P^'"^. In like manner other points in this branch are located.) The demonstration is as foUows : To prove that any point, as P , is in the curve, we have to prove that PF : PR : : wi : 7i ; {. e., the distance from any point in the curve to the focus, is to the distance of the same point from the directrix, in a constant ratio (m : n), which ratio is greater than 1, in the hyperbola. To prove that the construction gives this proportion, join P and F, and draw PR. parallel to TC. Now since PF = LT, and PR =: TC, and by reason of similar trian- gles, we have PF : PR : : LT : TC *• : GF '• FC : : w : n. In a similar manner any point on the other side of the directrix, found by the method described, as P^ii, is shown to be in the curve. Thus pviip ^^^ N'T' by construction, P"^"R' = T'C, and the triangles CFG and CT'N' are similar ; hence pv^F : P^"R' : : T'N' : T'C : : G F : FC : m : n. Q. E. D. 4:7. Def's. — Tlie Aocis of the Hyperbola is an infinite line drawn through the focus and perpendicular to the directrix, as TT' Fig. 33. The Transverse Aocis of the Hyperbola is that por- tion of the axis of the curve included between the vertices, as K I , Fig. 33. TJie Focal Distances are the distances from the focus to the yertices, as Fl, and F K, Fig. 33. TJie Conjugate Aocis of the Hyperbola is a perpendic- ular to the transverse axis at its centre, and is limited by an arc drawn from the vertex as a center, with a radius equal to the distance from the focus to the centre. Thus, in Fig. 33, D E represents the OF THE CONIC SECTIONS. 33 conjugate axis, the extremities D and E being determined by making the distances Dl and El each equal to OF- This definition is a convention adopted for the purpose of rendering more close the ana- logy between this curve and the ellipse. A Conjugate Hyperbola is an hyperbola having the conju- gate axis of a given hyperbola for its transverse axis, and the trans- verse axis of the given curve for its conjugate ; see Ex. 10, F%g. 34. Either of two hyperbolas thus related is conjugate to the other. They are sometimes distinguished as the X hyperbola and the Y hyper- bola, each taking the name of the co-ordinate axis upon which its transverse axis Hes. An Equilateral Hyperbola is one which has its conjugate -axis equal to its transverse. Ex. 10. To construct a pair of conjugate hyperbolas whose axes are 8 and 6. SuGS. — Draw two indefinite straight lines at right angles to each other, and take Ol == OK =4, andOD=OE=3. Having constructed the branches on the axis Kl, Fig- 34, as in Ex. 9, take O F' = OF (which = ID), and F' is the focus of the conjugate or Y hyperbola. Taking DS' = D F' and E L' = EF', and through S' and L' drawing a right line, it is one of the focal tangents. Having found the focal tangents the construction proceeds as before. Fig. 34. Ex 11. Construct an hyperbola whose transverse axis is G, and less focal distance 2. Eind also the conjugate axis, focus, and direc- trix of the conjugate hyperbola. Ex. 12. Letting e represent the eccentricity of an hyperbola, c the distance from the centre to the focus, A the semi-transverse axis, and ' B the semi-conjugate, show that 1^' Why is e e=z- = id!±J?!)I and hence that 1 — e» = — A A greater than 1 in the hyperbola? Ex. 13. "What is the eccentricity of an hyperbola whose axes aro i 34 THE CARTESIAN METHOD OP C0-0EDINATE8. 10 and 6 ? Wliat is the eccentricity of an hyperbola whose transverse axis is 12, and less focal distance 3? Ex. 14. The eccentricity being 1^ and the conjugate axis 4, what is the transverse axis ? What the focal distances ? What the charac- teristic ratio {4:2) ? Transverse Axis, 3.577 +. 4=S» ^TOp, Boscovich's ratio and the eccentricity are equal. Dem. — 1st. Let AB, Fig. 35, be the directrix of an ellipse, F the focus, CL the focal tangent, and O the centre. Draw GK parallel to CL- Then since LO = GO, and GF = IG = LK "\46), LO - LK - KO = GO - GF = FO. KO Therefore, — — - = the eccentricity {44). By defi- GO nition = = Boscovich's ratio. Now, by CG CG ' ^ Pig. 35 Birailar triangles, IG CG KO GO Q. E. D. 2nd. In the Hyperbola the demonstration is essentially the same. Thus, in Fig. 36, LO = EM - IG FE — FG 2 LO + LK = KO KO = GO. Hence and GO GF IG CG GO + FG == FO, = the eccentricity. By definition = the characteristic ratio (Bos- Now, hy similar triangles, Q. E. D. Fig. 36. CG covich's ratio) IG __ KO CG go' 3rd. In the Parabola we may call the eccen- tricity 1, from analogy ; or, better, we may conceive the parabola to be an ellipse with the centre, O, removed to an infinite distance from the vertex, G, Fvj. 35, whence the fraction -— — =1.* Q. e. d. QO 49, CoR. — The student should fix in memory the following relations, as they are fundamental, and of frequent use in the reduction offormulw. Letting A = the semi-transverse axis, B = the semi-conjugate axis, e = the eccentricity, and p == the semi-latus rectum, we have the fol- lowing * If the student has difficulty in understanding this statement, let him consider that, O being roraoved to infinity, the finite distance, GF, by which GO appears to be greater than FO, is of no appreciable vahie as compared vnth. the terms of the ratio, which are both infinite. OF THE CONIC SECTIONS. 35 POfDAMENTAL RELATIOIVS. From the definition {42) and {48), we see that, The distance from any point in the curve to the focus -^ e = the distance from the same point to the directrix. Also, The distance of any point in the curve from the directrix x e := the distance of the same point from the focus. Distances. Ellipse. Hyperbola. Parabola 1. ( Focus to extremity of Conj. -axis . . = ( Veetex " " = A 00 CO Ae 2. "Por.TTS to riKNT-RF. . Ae Ae cc 3. Focal Distances — A{1 =F e) A{e =F 1) hp 4. Vertices to Dieecteix. ... . — ^(1 =F e) e A(e ^ 1) e hp 5. Focus to Dieecteix — A{1 — e-^) e A{e'^~l) e p 6. r.-p.TxTTEE t*^ DtEK^'TKTX A e A e 00 7. 1— e^ _ A^ A-^ p 8. Semi-Latus Kectum, p = A A Dem. — For the ellipse see Fig. 30, for the hyperbola, Mg. 33, and for the parabola. Fig. 37. (1.) For Fllipse see {4:6). — For Hyperbola, by definition FO of eccentricity — - — = e. . • . F O = ^e = the distance from the vertex to the extremity of the conjugate axis, by the definition of the latter (47)- For Parabola, consider the curve as an ellipse with its centre removed to infinity. FO (2.) For Mlipse, — — =e by definition. FO =^e. Fig. 37. For Hyperbola and Parabola, see above. (3.) For Ellipse, \F = \0 — FO = A — Ae = A{1 — e). FK = OK -|-FO=^4-^e = ^(1 -|- e). For Hyperbola, IF= FO — \0 =^ Ae — A = ^(e — 1). F K = FO -f O K = ^e + -4 = ^(e -f 1). For Parabola see {4:5). (4.) For Ellipse, since I is a point in the curve IC= = 6 6 e) ^^_^KF_ A{l + e) ^ e e For Hyperbola, for same reason IC = Also, IF A{e—1) ; and KC KF ^(e+1) e e (5.) For Ellipse, FC : For Parabola, see {4S). A{l-e) IF+IC=^ — ^e + ^(1 - es) 36 THE CARTESIAN METHOD OF CO-OEDINATES. Ae A A(e^ — 1) For Par- For Hyperbola, FC= IF+ lO =Ae—A4- e e dbola, FC ^ G F = p, by definition. (6.) For Ellipse, D being a point in the curve wliose distance from the direc- trix is OC, we have OC = = — . e e For hyperbola, OC = Ol — CI A Ae = — . (The distance in the ellipse may be obtained in the same way.) For Parabola, same conception as in (1) above. (7.) For Ellipse and Hyperbola see Ex's. 7 and 12. For Parabola, 1 — e'^ = l — 1 = 0. ■ (8, ) For Ellipse, GF =p= FC X e = A{l — e'^) = -j-. For Hyperbola, G F A jB2 = p=FCXe = A{e^ — 1) = — . For Parabola, G F =p by definition. SO, ^Toh, To pass a conic section through three given points, so that it shall have a given focus ; and to determine its elements ; i. e., the axes, foci, directrix, eccentricity, etc., if an ellipse or hyperbola, or the latus rectum if a parabola. Solution. — Let M, N, and O, Fig. 38, be the given points, and F the given focus. Connect the points with the focus, and draw O N , and N M , and produce them towards the probable position of the directrix, as to L and K. Now, take a point R, on QL, such that OF : N F : : OR : NR,* and R is a point in the directrix. In like manner, take NF:MF::NS:MS, and S is another point in the directrix. Hence the direc- trix can be drawn. To prove that a line drawn through R, and S, as AB, is the directrix, we have to show that O^F _ N F _ MF OP ~ EiG. 38. OP, NQ, and MX, NQ M being perpendicular to AB. Now OP: NQ ::OR : NR. But by construc- O^F _ N^ p6 "~ NF ton OR:NR::OF:NF. In like manner NQ : M" .-. OP : OF :: NQ : NF, or N S : M S : : N F : M F. NQ MF NQ MT Q. E. D. To make the numerical computations requires much more labor than to effect the geometrical solution. We may proceed as foUows : Having the distances O N , NM, OF, N F, and MF given in numbers, compute the numerical values of N R and M S from the proportions used in the construction. The sides of the triangles O F N and N F M being known, their angles can be found by trigo- * This coustruction is effected tbiis: taking tlio proportion by division, (OF — N F), or OG : O F : : ( O R — N P. ), or O N : O R . From this proportion O R can be constructed, a.s the other terms are known. OF THE CONIC SECTIONS. 37 nometry; whence we get the angle R N S, as it equals 180° — (O N F -{- FN M). Then, in the triangle RNS, we shall have two sides and the included angle ; whence the angle N R S can be found, and from it N R Q becomes known. Now, in the right angled triangle N RQ, we know the hypothenuse and one acute angle, and can find N Q. Again, letting fall the perpendicula.r N E, forming the trian- gle FNE, we can compute EF, since FN is known and the angle FNE = FNR + RNQ— 90°. FC is therefore known, being equal to EF -f NQ. As NF NQ tricity, is known is now determined, the ratio e, or, what is the same thing, the eccen FH NF Taking a point, as H, upon FC, such that the HC NQ vertex is determined. In a similar manner the other vertex of an ellipse or hyper- bola can be found. Letting p be half the latus rectum, F U , it can be found from FN p NQ FC Ex. 1. Construct a conic section passing tlirough. the points O, M, and N, and having F for a focus, knowing that O F =6^, N F =3^, M F = 21, O N = v/18, N M = v/10. Let the geometrical con- struction be given, and also the numerical solution. The locus is a parabola whose latus rectum is 9. Ex. 2. Construct and compute as above, when OF =2.08, N F 1.08, M F = .46, ON = 1.12, and N M = .87. Ex. 3. Construct and compute as above, when O F = 10, N F "= G, M F = 3, ON = 6, and N M -= 4. SI, JPvob, To produce the general equation of a Conic Section referred to rectangular axes. Y M /" c \ \ V / 1 N \ \ A /S D K \^^ X /z' \B Fig. 39. Solution. — Let MN, Fig. 39, be an arc of any conic section, F the focus, C B the direc- trix, ZZ' the axis of the curve, and AX and AY the axes of reference. Let P bo v.ny point in the curve, and its ordinate P D : also draw the ordinate of the focus, FK. Draw from the origin AG perpendicular to CB. Draw DH parallel, and P L perpendicular to CB. X Join P and F, and draw PI parallel to AX. Let AG, the distance from the origin to the directrix, be represented by d ; the co-ordinates of the focTis, A K and F K, by m and n respectively ; the ratio mentioned irt the definition {4i2), P F PE axis of abscissas, ZSX = GAX nates, A D and PD, by a and y. , by e ; the angle which the axis of the curve makes with the LDP, by a ; and the general co-ordi- 38 THE CARTESIAN METHOD OF CO-ORDINATES. Now, PF2 =e2 . pE-^ But PF- = PI- + Fl'^ — (m — a;)2 -f {n — yY. Again, PE2 = (PL + AH — AG)-2 = (PD sin a -\- AD cos a — dy = {y sin a -f- ^ ^^^ ^ — ^^y'- Whence, substituting we have (Eq. A), (m — x)- -{• {n — yy^ = e'{y sin a -j- x cos a — d)^. S2. Cor. 1. — The equation of the ellipse and hyperbola re/erred to their axes is 2/2 + (1 — e^') x^ = A'-{\ — e=), in ichich A is the semi-transverse axis, and e the eccentricity, or the char- acteristic ratio {4:2). Dem. — Let the curve in Fig. 39 be conceived to change position so as to assume that in Fig. 40 or 41, the axis of the curve coinciding with the axis of abscissas, the origin at the centre, A, F the focus and CB the directrix. As the axis of the curve now coincides with the axis of abscissas, > 1. Therefore, 52 — 4J.(7= characterizes the Parabola ; ^ B^ — 4:A G characterizes the Hyperbola. J Ex. 1. Determine the species of the locus of the equation 2?/2 — Zxy + 5^2 _ 2i/ — 12 = 0. SuG.— As the equation is of the second degree the locus is a conic section. Again, in this case, ^ = 2, ^ = — 3, and 6'= 5. .-. B^ — 4A(7= 9 — 40 = — 31 and E depend upon m and n, and A and C do not, such values may be given to m and n, i. e., the origin may be so situated with respect to the focus, that D and E shall each be 0. Hence Ay- — Cx- -j- i^= 0, embraces all the varieties of the hyperbola. On this form we observe that if i^is positive, it gives by transposition Ay'^ — Ox^ If, = — ^ or the equation of the hyperbola referred to its axes, in which I -r, is the l-~F semi- transverse, and I is the semi-conjugate axis. If A and Care numeri- \ cally unequal these axes are unequal, and we have the common form of the hyper- bola. If A = C, the axes become equal and the locus is an Equilateral Hyperbola. Again, if i<^is negative the equation becomes Ay- — Cx- = F, which is the equation of the y hyperbola, since the real axis is on the axis of y ", and the imaginary one IX' on the axis of £C.* Finally, if 1^ = 0, we have vl?/' — Cx- = 0, or y=d:z l-x, which is the equation of two straight lines passing through the origin and making r^' re angles with the axis of x, whose tangents are respectively | -^ and — | — . There are, therefore, 3 varieties of loci embraced in the equation of the second degree between two variables, which fulfill the condition B- — 4cAC '^0, and are hence called varieties of the hyperbola ; viz., the Hyperbola with unequal axes, both on the axis of x, and on the axis of y, the Equilateral Hyperbola, and Two Bight Lines intersecting each other. 07* I^Toh, To determine the varieties of the parabola. '3oiiUTioN. — As the equation Ay- -f- G^' + ^If ~\~^^-\~ ^= ^j embraces all species and varieties of the conic sections, we have to determine only what loci it repre- sents when B^ — 4J.C=0. But as B = 0, this condition can be fulfilled only by * This form of expression is frequently used instead of " axis of ordinates," and " axis of abscis- sas." OF THE CONIC SECTIONS. 45 A = 0, or G=0. Now as the equation is symmetrical with respect to cc and y, it will be sufficient to examine the case in which C = 0, or the form Ay'- -{- Dij -{- JEx -\- F= 0. Kemembering that a has been made 0, and that e = l, we find D (which equals 2e'd sin a — 2)i) = — 2n. This can now be made by taking the axis of the curve for the axis of x, and the equation takes the form Ay^ -\- Ex -[- F=.Q. But in this case we cannot make E = (2e-cZ cos a — 2m = 2tZ — 2m) = 0, since that would require that d = m, which is absurd, since d is the distance from the origin to the directrix, and m is the distance from the origin to the focus. {Numerically d may equal m ; but E =: requires that they also have the same •sign). We therefore have to discuss the equation Ay"^ -f- -^^ "f" ^= ^' which includes all varieties of the parabola. As F depends upon m~ -\- n'^ — e-d^, and as n = 0, and e = 1, jPmay be made 0, by putting m = — d, which requires only that the E origin be at the vertex. The equation is thus reduced to y" = ±z —oc, the dr sign being given to E, as no restriction has been imposed upon it. This is the common E equation of the parabola, in which — = 2p. The -\- sign locates the curve at the right of the origin, and the — sign at the left, but both give the same variety. Again, if in Ay^ + Bxy + Cx^ _^ x>y + £r + F= 0, we make ^ = 0, JS = 0, and C = 0, the condition B- — 4 J. C = is fulfilled, and the locus is therefore sometimes called a variety of the parabola. This locus is evidently a right line, its equation, Dy -j- Ex -f- F=^ 0, being an equation of the first degree between two variables. * Finally, if an equation of the second degree between the two variables can be reduced to either of the forms ?/--[- '2xy-\-x- zb P{x -\- y)-\-S = 0, y" — 2xy -{- x- ± F' {y — x) 4- S' = 0, or t/2 ± 2xy-}-x-^ -\- S" =0, the condition B^ — 4.AG=0is still fulfilled, although the equation may bo reduced to the form y dzX = m ± Vp — q, which is the equation of two real or imaginary parallel right lines, "VVe have, therefore, 4 varieties of the Parabola ; viz., the Common Parahola, the Bight Line, Two ParUllel Bight Lines, and Two Parallel, Imaginary Bight Lines. OS* Cob. 1. — The eccentricity of the circle is 0, and the directrix is at infinity. Dem. — In obtaining the equation of the circle (OS), we made a = 0, and A= C. Hence 1 = 1 — e'~, or e = 0. Again, when the ellipse passes into the circle, the foci unite in the centre. Now, calling the distance from any point in the curve to the directrix s, and the radius of the circle B, we have — = (the distance from any point in the curve to the focus divided by its distance from the directrix equals the eccentricity) ; whence s = (X).+ * In reality this condition is not compatible with our fundamental hypothesis, which requires the equation to be of the second degree. Moreover, the conditions A = 0, B — 0, and C = 0, are inconsistent with the character of the coefficients A, B, and C, inasmuch as they require that 1 _ e^sin^a: = 0, 1 — e^cos^a = 0, and 2e-sinti:cosa: = 0, or three arbitrary conditions while there are but two arbitrary constants, e and OC, t If ^ = 0, the couditiou — = 0, is satisfied by any value of s. In this case the locus is a 46 THE CAETESIAN METHOD OF CO-OEDINATES. Ex. 1. Determine the species and situation of the locus i/« + Gy — 1207 + 33 = 0. Solution. — To determine the species of the locus, observe that the equation is of the second degree between two variables, and fulfills the condition B- 4:A0 = 0. Therefore, the locus is a parabola. To determine the situation of the locus, compare its equation with the general equation of the conic section : viz. , (1 — e"^ sin2 a)y' — 2e2 sin a cos axy-\-{l — e^ cos'^ a)x^ -f- (2e-cZ sin a — 2n)y + (2e''d cos a — 2m)x -f- m'^ -}- n- — e-d^ = 0. As in this example the coefficient of y'^ is 1, divide the general equation through by 1 — e- sin2 a before comparing the coefficients. The five equations from which a, e, m, n, and d are to be found are — (1) (2) (3) (5) — 2e2 sm a cos a ^ „ . ^ — :; ; = 0, or e2 sm a cos a =0. 1 — e2 sm2 a 1 — e2 cos2 a 1 — e2 sin2 a 2e^d sin a — 2n 1 — e2 sin2 a 2e^d cos a — 2m 1 — e^ sin2 a m2 -f ?i2 _ e2d2 = 0, or 1 — e2 cos2 a: = 0. = 6, or e-d sin — e2(^ 2e5d cos a — 2m = 0, or 1 — e^ cos2 a=^Q \ "- = 0, or e^d sin a — ji = ; = — 1, or 2e2d cos a — 2m = e^d^ m'^ n2. ,7i2 _|_ n^ _ e^d'^ The general equation is divided through by in--\-n^ — e^cZ^, since in this example the absolute term is 1. From (1) we have, directly, sin a = -, discarding the negative root for the rea- 1 1 /- d=-ve2 — 1. e son given in the preceding solution. . • . cos a = d= 1 1 Substituting this value of cos a in (3), we have e = v/ 2 = 1.4142-j-. Hence, cosa = i , 1-1 = ±^ \l-\=± l=±^v/2= ±.7071+, and 6t = 450 or 135°. Substituting the values of e and sin a in (4), we have n ^= \/~ld ; and OF THE CONIC SECTIONS. 49 by substitution in (2), we find m = zb 1. In this case, it will be seen that if we substitute the -|- value of cos a in (2), m be- comes imaginary, but the — value gives m real. Therefore, cos a = — -v 2, and a 1350. Finally, substituting in (5), we 1 — 3 — find d = — tV^^» ^^<1 tn/-^' or — .35, and 1.06; and, conse- 1 3 quently, n = — -, and -. (The locus is situated as in Fig. 49 ; but the construction is so close- ly analogous to the preceding, that the student will have no difficulty in effecting it.) 4. Determine the character and situation of the locus y^ — 2^^/ + 2^2 — 2j7 = 0. Results, e = .924, a = 58°17', d= 3.1 and — .37, m = 1.78 and .21, and n = 2.27 and — .27. S-ca. — In problems hke the preceding it is not admissible to divide the general equation through by a coefficient corresponding to one which is in the partic- ular case, inasmuch as this process would reduce each coefficient to infinity or inde- termination. Thus for Ex. 3, should we put the general equation in the form y2 2e2 sin a cos a ,1 — e^ cos^ a , 2e^d sin a — 2n -^y 4- :; -^-zr-^—^^ + -1 :;r - ., ^. y + etc. ; 1 — e^ sin'^ oc'^^ ' 1 — e^ sin^ a " ' 1 — e^ gin^ ^ each of the coefficients, when the application was made to the equation 2«2/ — jc -j- 1 = 0, would be infinite or indeterminate, since in this example 1 — e^ sin^ a = 0. EXEKCISES. [Note. — This list of exercises is designed to give the student an opportunity for making an effort to produce the equations himself. Nothing new is developed in them,- and the student need not necessarily tarry till he has mastered them all, though by doing so clearness and breadth of view will be promoted. Let every one understand, however, that ability to investigate— to reason for himself— is the proper object toward the attainment of which he should strive.] 1. To produce the equation of the eUipse referred to its own axes and in terms of its semi-axes, directly from the definition, with- out first obtaining the general equation of the conic section. r>o THE CARTESIAN METHOD OF CO-OllDINATES. Sug's. ad =x, PTD =y, AF = Ae, PF and— — =e. PF^ = e^XPE.^ PF2== 2/2 + {Ae + xy, and PE^ = { h ^) • ... 2/2 _|- (1 _ e2)a;2 = ^2(1 _ e^). por 1 — e^ substituting — , we have A^y'^ -f" ^2-j;2 =_ 2. In like manner produce the equation of the ellipse referred to its transyerse axis and a tangent at its left hand vertex, i! e., y^ = 3. Show that the equation of an ellipse referred to its conjugate A^ axis and a tangent at the upper vertex thereof is x'^ = — -z—i2By + y^). Sug's. AD=a^,PD: -y, ^p^=e,a.u6. C p F2 = e2 X P E2. Also P F^ = (x + Ae^ + (J5 + y)\ and PE^ = ("- + x)^ \ 6 / 4. Produce the common equation of the hyperbola, A^y^ — B^x^ = — A^B^, directly as above. c Y y/ A D E ^^ ~^"~--^ p • f^ ^ ^ G I F I 7 B "^ ^— Y^ Fig. 51. 5. In like manner as above produce the common equation of the parabola, y"^ = 2px, 6. Show that the equation of an ellipse i^A-{y — y^y + B'^{x — x-^)^ = A^B'^, when x^^ and y^ are the co-ordinates of the centre, and the axes of reference are parallel to the axes of the curve. •y Sug's. AL=iCx, OL=yi, AD=a5, PD = 2/' a^^ PF2 = e2 X PE2. Also h PF2=PR2-1- FR2, PR=y — y,, FR = x — xi + Ae, and PE = PH — H E = .r — Xi -] . Substituting, (?/ — y-^y + ic2 — 2x^x + 2Aex + Xj^ — 2AeXi -}- A^e^ = e^-x^ — 2e"x-iX -f 2Aex -\- e^-x^^ — ^ Transposing and collecting terms, c ^_ P E //- \ G B \ ^ y y^ K : L t 3 ■ X Fig. 52. OF THE CONIC SECTIONS. 51 (y — 2/i)' + (1 — e2)fl?2 — (1 — e2)2a;,a; + (1 — e-^)x,2 = A%1 — e'^), or (2/ — 2/i )^ + (1 — e2)(a; — Xiy = A\l — e^). Putting — for 1 — e% we have ^Hy — ^l)^ + JB%X — £Ci)2 = ^2J52. 7. Deduce from tlie general equation of the conic section (SI) the equation of a parabola whose parameter is 2p, referred to rectangular axes, the axis of the curve falling on the axis of abscissas, and the vertex of the curve being at ^i to the right of the origin. Also the equation when the vertex is at x^ to the left of the origin. Also the equation when the axis is parallel to the axis of x, and the vertex is at {x,, y^). The equations are y^ = 2p(^ — x^), y^ = '2p{x -\- x-^), and {y — ^/i)- = 2p{x — ^i). 69, Genebal Scholium. — It will appear hereafter that the conic sections are formed by the mutual intersection of a plane and a right cone with a circular base. It is from this fact that the name of these curves is derived ; and from this as a definition they we^e formerly studied. The different species arise from dif- ferent positions of the cutting plane. The plane which gives the parabola lies parallel to one of the elements of the cone, as O N P, and hence cuts but one nappe, and gives but one branch. To produce the ellipse the cutting plane lies between this position and perpendic- ular to the axis, as fi~rmn. To produce the hyperbola it lies between parallel to an element and parallel to the axis, as HIK and EGF, and hence cuts both nappes, giving two branches. Several of the varieties of the conic sections as heretofore considered, may be illustrated by means of this geometrical conception, and their mutual relations more clearly seen. Thus as the plane of the ellipse approaches perpendicularity to the axis, the eHipse approaches the form of a circle iijto which it passes when the plane becomes perpendicular to the axis. The circle is therefore a variety (or more properly a limit) of the ellipse. So also as the plane approaches the vertex, the ellipse diminishes, passing into its limit — a point — when the x^lane passes through the vertex. The hyper- bola becomes two intersecting straight lines when the cutting plane passes through the vertex afid is not parallel to an element. When it becomes parallel to an element and also passes through the vertex, it gives the limit both of the parabola and the hyperbola, which common limit is a right line. When the cone passes into a cylinder the parabola becomes two par- a.lel right lines, as also the hyperbola may, if it is conopived as produced by a cutting plane perpendicular to the base. If the cutting plane prod cing 52 THE CARTESIAN METHOD OF CO-OEDINATES. the hypeibola is conceived as oblique to the axis, the hyperbola passes into an elhpse when the cone passes into a cylinder. [Note. — Of course the above views are not give i as in any sense needed to ccmfirm the conclu- sions of the preceding discussions, but simply to give the student a Little further insight into the wonderful harmony which exists between algebraic /ormMte and geometrical loci.] or five arbitrary condi- 70, IPTOp, Through fi\:)e points in a plane one conic section may always he made to pass, and but one. Dem. — Dividing the general equation Ay"-{- Bxy-^Cx- -^Dy^Ejc-\- F= through by F, and distinguishing the new coefficients by accents, we have A'y'^ 4" -S'a;?/ -{-C'x^--\-D'y + E'x-\-l=0. Now let (x,, y,), {x.^, y.z), {x.^, y.^), {x^, y^\ and (a; 5, 2/o) be the five given points. Substituting, successively, in the last equation, the co-ordinates of these five points, for the general co-ordinates x and y, there result the five equations A'y,^-{-B'x,y, -^ Cx^- -{- D'y, -\-E'x,+l=X) A'y^^ + B'x,y.z 4- O'x,^ + Dy, -\-i:'x,+l=0 A'y,^ + B-x,y, + C'x,^ -f D'y, ^Kx,-^1=0 A'y,2 + B'x.y, + Cx,^ -f D'y, -f Ex, + 1=0 A'y,2 -f B'x.y, + Cx,^ + Dy, -{-Ex, +1=0 tions. This number of conditions is possible, since there are Jive arbitrary constants involved ; viz.. A', B', C, D, and E' . From these equations, asiC], 3/,, iCg, t/g* ^■ii 2/3J ^45 y^■> ^.TJ ^^d y,, are known quantities, the values of A' , B , C, B', and E can be determined. Having found the values of these coefficients, by substituting their values in the general equation A'y^ -\- B'xy -}- C'x" -)- D'y -\-Ex -|- 1 = 0, there results an equation of the second degree between two variables, or an equa- tion of a conic section. As this equation is satisfied by the co-ordinates of each of the five given points, the locus represented by them passes through these points. Finally, as the five equations are all of the first degree with respect to A', B' , C, D' , and E , but one set of values can be determined for these coefficients. There- fore, hut one conic section can be made to pass through the five given points, q. e. d. ScH. 1. — In this proposition the term Conic Section must be taken in its broadest sense, i. e. , as embracing all varieties of these loci, except the so- called imaginary loci. ScH. 2. — If the five points are so situated that the equation of the locus passing through them lacks some of the terms of a complete equation, it will not do to divide the general equation by the coefficient of such a term. If such an error has been made in the hypothesis in any solution, it will soon appear as the solution proceeds. This case is analogous to the one noticed in the suggestion under Ex. 4, page 49. Ex. 1. Produce the equation of a conic section passing through the five points (2, 3), (0, 4), (—1, 5), ( —2, —1), and (1, —2), and deter- mine its species. Solution. — The five equations which determine the coefficients A' , B', (7, D', and E , are OP THE CONIC SECTIONS. 53 (1) 9 A' -f 6B- 4- 46" -f 3i>' + 2^' -f- 1 =0 ; (2) 16^'4-4i)' 4-1=0 ; (3) 25A' — 55' 4- C -\-5lJ' — ^ + 1 = 0; (4) ^'4-25' -I- 4C^ — J>' —2^4-1=0; (5) 4.A' —2B' + C —2D' + ^4-1=0. „ /. ,, ,. . , ,, 169 ^, 220 ^, 89 _ 445 Solving these equations, we find ^ = — -^—, B = —:^, G = -— , D = — ^, 113 J?' = — -. Substituting these values in the general equation J.'t/2 -}- 5'ic?/ 4" C'a;^ 4- -O'y- 4" -^'^ 4~l=^j clearing of fractions and changing signs, we have 169?/- 4- 220xT/ — • 89a;2 — 445?/ — 113x — 924 == 0, which is the equation of a conic section passing through the five given points. This locus is an hyperbola, since B- — 4c.AC >0. Ex. 2. Produce the equation of a conic section passing through, the five points (1, 3), (4, —6), (0, 0), (9, —9), and (16, 12), and find its species. Suggestions. — As one of the given points is (0, 0), the locus passes through the origin ; and hence F=0. The form of the general equation used would, there- fore, be Ay^ -\- Bxy 4- Ox,''- -{- By -j- Ex = 0, which divided through by one of the coefficients, as A, gives the form y- -\-B'x^ -^C"x--\-B'y -]- U'x = 0. This equa- tion satisfied for the four points (1, 3), (4, — 6), (9, — 9), and (16, 12), in succession, gives rise to four equations from which the coefficients can be determined. The locus is a parabola whose equation is j^ = 9x. Ex. 3. Produce the equation of a conic section passing through (—4, —2), (2, 1), (—6, 8), (0, 0), and (2, —1), and determine its species. The equation is y = rp ^x. Ex. 4. Produce the equation of a conic section passing through (3, \/5), (—2, 0), (—4, — v/i2), (3, —Vl), and (2, 0), and deter- mine its species. The locus is an equilateral hyperbola. Ex. 5. Produce the equation of a conic section passing through i—h — i). (2, 1), (f, 2), (— f, —3)', and (f, — f) and determine its species. The locus is an ellipse whose equation is y~ — ■ 2xy 4- 3a72 -\- 2y — Ax — 3 = 0. Ex. 6. What is the equation of a circle whose radius is 5, referred to rectangular axes, and the origin at the centre ? When the origin is on the circumference and the axis of abscissas is a diameter? When the axes are tangent to the circumference ? Fquatio7is, y^-{-x-^ = 25, ?/2 = db 10j7 — x^, and y'^-^-x^ — lOy — • 10.77 -f 25 = 0. Ex. 7. What is the equation of an ellipse whose axes are 16 and 10, when referred to its own axes ? When referred to its transverse 54 THE CARTESIAN METHOD OF CO-ORDINATES. axis and a tangent at the left hand vertex ? The corresponding prob- lems in the case of the hyperbola. Equations, 64i/2 _j_ 25^2 = 1600. 64^/2 — 400^ + 25^72 = 0. 64?/2 — 25^72 = — 1600. 64i/2 + 400^ — 25^^ == 0. SuQS. — The results of tlae two preceding examples are readily written from the equations of the respective loci as given in (5S — 5T), and are designed to fami- liarize those most important forms. Ex. 8. Produce the equation of a parabola referred to rectangular axes, the vertex of the parabola being at ( — 3, — 2), the parameter, 6, and the axis of abscissas parallel to the axis of the curve. Equation, y^ + 4?/ — Qx — 14 = 0. Ex. 9. Produce the equation of an ellipse whose eccentricity is f, its major axis 18, the centre being at ( — 2, 3), and the axes of refer- ence being rectangular and parallel to the axes of the curve. Equation, 9v- + '6x-' — 54?/ + 20a7 — 304 = 0. Ex. 10. What are the following loci, and what their axes : viz., 9^2_|.4;j;2=,36? 7^^ + ll?/2 = 15 ? 100y2_ 25^2 =:_ 2,500? 11 x-^ — 252/2=:=— 116? EXERCISES IN PRODUCING THE EQUATIONS OF THE CONIC SECTIONS FROM OTHER DEFINITIONS. [Note. — These exercises may be omitted without destroying the integrity of the course. They are designed simply to lead the student to a more full comprehension of the jprocess of producing an equation of a locus from its definition, a subject of vital importance if one proposes to so master this method of geometrical investigation as to be independent in the use of it.] 1. To produce the common equation of the ellipse from the defini- tion : — The ellipse is a curve such that the sum of the distances from any point in the curve to two fixed points called the foci, is constant and equal to the major diameter. SuGS. AD =37, PD=2/, A& = A, AE.=B, A F = A F' = c. Then from the definition \/y^ -\-ic -\- xy- -\- Vy'^ -{-[c — xY = 2^. Whence A"y-^ + (^2 _ c2>2 = j^2 (^2 _ c;). But by definition, E.F = A , whence A^ — c^ = JB- ; and we have A^^ 2. In a manner similar to the above V produce the common equation of Fig. 54. the hyperbola, from the definition,— The hyperbola is a curve such that the difference of the didancesfrom any point in the curve to two fixed points is constant and equal to the transverse axis. OF THE CONIC! SECTIONS. 55 Sxjg's. — In this case it must be borne in mind that A^-\-B'^=c^ {.4:7), and hence that A^ — c^ = — B\ The equation is A^y^ — B^x^ = — A^B^, as before produced. 3. To produce the equation of the locus of a point moving so that the square of its distance from a fixed point is in a constant ratio to its distance from a fixed hne. Y P X* A D X Sug's. — Let the fixed line be taken as the axis of abscissas, and let a perpendicular to it through the fixed point, F, be taken as the axis of ordinates. Let P be any point in the locus. Then AD=.r, and PD=?/. As A F is constant, call it a, and let m repre- sent the ratio referred to in the definition in the example. The equation sought is {y — a)'^ -j" *'' Iv = my, or y- -f- x~ — (2a -f- 'm)y -f- a- = 0. This I''ig. 55. being an equation of the second degree, the locus is a conic section. Again, as -B2 — 4J.C<< 0, it is an ellipse. Finally, as the coefficients of y'^ and x^ are equal, it is an ellipse with equal axes, or a circle. To determine more fully the situation of this circle, notice that for y = 0, flj = =b \/— a'~, whence we see that, in general, the circle does not cut the axis of x. 2(7, -f- tn Making x = 0, ?/ iam -f- w^ Now, as every value of y in the ■A \^ 4 equation of this locus gives tv»^o values of x, numerically equal but with oppo- site signs, we see that the locus is symmetrical with the axis of y, and that the cen- tre of the circle lies in this axis. But the circle cuts this axis at ' >^ ,' t(a)i 4- m' - , 2rt 4- m -f- I 4 and at , : . Whence the diameter is the 4 2 \l 4: difi"erence between these values ; and letting r be the radius, we have 1 .- r =--^.1.6(7/1 -\- 'III , i. e., the radical part of the root. In the particular case in which a =0 ; i. e., when F is at A, the equation be- comes y^ -f- X' := my, which is the equation of a circle passing through the origin, and having its centre on the axis of y. 4. In the given right Hnes A P, A Q, intersecting at right angles, are taken variable points p, q, such that Ap I 7? P '.'. QiQ '. qA ; prove that the locus of the intersection of P^, Qj), is an ellipse which touches the right lines in P and Q. 56 THE CARTESIAN METHOD OF CO-ORDINATES. Sug's.— Let AP and AQ be taken as the axes of reference. Call A P = a, and AQ = 6. Then AD =x, and RD =2/- ^rom the similar trian- gles RPD, 5 PA, and RpD, QpA obtain the relation between x, y, a, and 5, This wiU be the equation sought. It is, when reduced, a\y—hY j^ l,i(^x — aY = «"^?>'^ — <^^^y- .ScH. — As this equation is of the second degree, and B^ — 4:AG^ — 3a2&2, the locus is an ellipse. As there is a term in xy, the axis of the curve is inclined to the axis of abscissas. For x = a, p y ^=h and ; hence the locus passes through [a, h), and P. I'or a; — 0, 3/ = 6 ; therefore the locus passes through Q. To effect the construction mechanically, take h.p '. pP : : Qg : gA, by- composition and alternation, giving AP : AQ : : Pi? : Ag. Now assuming p at any point in AX, we can find the corresponding value of Ag. After p passes P, Ag becomes —, and is laid off below A. So when Q_p' passes parallelism with AX, Ap becomes negative. 5. Required the locus of the middle point of a line moving with its extremities in two fixed lines at right angles with each other, while it passes through a fixed point. Sug's. —Take the fixed fines as axes of refer- ence. Let O ]3e the fixed point, and C B the line, the locus of whose centre, P, is to be de- termined. Calling AD «, and OD &, the equation of the locus is 'i.o^y — ay — 'bx=Q, which is the equation of an hyperbola, passing through the origin, since for .^ = 0, 2/ = ^; and also passing through O, since x = a, gives y = &. Let the piipil trace the curve. Fig. 57. 6. Required the locus of the point P, moving so that P D , bears a constant ratio to A D X D B ; A and B being fixed points. What is this ratio is 1 ? and the locus when The distance A B heing called 2a, and ^i»- ^^^ the ratio m, the equation is y2 = mx(2a — x), whence the locus is seen to he an ellipse. If m=l, it is a circle. EQUATIONS OF HIGHER PLANE CUEVES. 57 7. If P moves in Fig. 58, so that PD^ bears a constant ratio to A D, what is the locus ? -4-»-^ SUCTION VIZ Equations of Higher Plane Curves. 71, One variable is called a Function of another variable when it depends upon that other variable for its value. Thus the ordinate of a curve is a function of the abscissa. 72, Functions are classified as Algebraic and Transcen- defltalf and the latter are subdivided into TrigonOTnetvic. and Circular, Logarithmic, and Eocponential, 73, An Algebraic Function is one which involves only the elementary methods of combination, viz., addition, subtraction, mul- tiplication, division, involution and evolution. Thus in ?/ = ax^ — 3^^ and in all the equations hitherto discussed in this chapter, y is an algebraic function of x, except 24-33, Sec. II. 74=. A TrigonometJ'ical Function is one which involves sines, cosines, tangents, cotangents, etc., as 2/ = sin x, y= sin x tan X, etc. 75, A Circular Function is one in which the concept is an arc (in the trigonometrical the concept is a right line). These are written thus : 2/==sin~^^, read "?/ equals the arc whose sine is .a?"; 2/=tan~^j7, read "?/ equals the arc whose tangent is ^." Notice that in the expression 2/ = tan~'a:, it is the arc which we are to think of, while in the expression ^=tan?/ it is the tangent, which is a right line. Trigonometrical functions are right lines ; circular ^ functions are arcs. These functions are mutually convertible into \ each other ; thus, y = s,m.~'^x, is, equivalent to ^ r= sin ;?/, the only difference being that in the former we think of the arc, the sine being given to tell what arc, and in the latter, we think of its sine, the arc being given to tell what sine. The circular functions y = ^mr^x, y = cos~^^, y = seG~^x, etc., are often called the Inverse Trigonometrical Functions. 76, A LogaritJiTnic Function is one which involves loga- rithms ; as 2/ = log x, log^ i/ = 3 log ax, etc. 77, An Fxponential Function is one in which the varia- able occurs as an exponent ; as i/ = a"", z = x^, etc. 58 THE CAHTESIAN METHOD OF CO-ORDINATES. 78. Higher Plane Curves are loci whose equations are above the second degree, or which involve transcendental functions. As it has already been shown that loci of the equations of the 1st degree are right lines, and that loci of the 2nd degree are conic sec- tions, it follows that all other j^lane loci are higher plane carves. The former are called Lower Plane Loci. Of course the variety of higher plane loci is infinite. We can con- sider but a few, and these simply as specimens. 79. An Algebraic Curve is one whose equation contains only alge- braic functions. A Transcendental Curve is one whose equation con- tains transcendental functions ; when converted into algebraic forms their degree is infinite. THE CISSOID OF BIOCLES. 50. Def. If pairs of equal ordinates be drawn to the diameter of a circle, and through one extremity of this diameter and the point in the circumference through which one of the ordinates is let fall, a line be drawn, the locus of the intersection of this line and the equal ordinate, or that ordinate produced, is the Cissolcl of U lodes. 51. IPvoh. To construct the Cissoid. Solution. — Let AB be the diameter of a circle ; and ED, and ED' be equal ordinates. Through A and E' draw AE' inter- secting ED in p. Then is P a point in the locus. In hke manner draw AE and produce it till it meets E'D' produced in |. Then is | a point in the locus. In the same way other points are found both above and below AB. There are, therefore, two branches of the locus ACM and ACM', symmetrical with respect to the diameter AB- These branches evidently meet at A, pass through the extremities of the di- ameter CC, and have GG' as a common asymp- tote. ScH. 1. — The name Cissoid is from the Greek and signifies ivy-form. It was applied to the curve, probably, from its resemblance to the graceful outline presented by a growth of ivy upon a wall. The locus was invented by the Greek geometer whose name it bears, while he was seeking the solution of the celebrated problem of the Daplication of the Cube. Sen. 2. — Sir Isaac Newton gave tho following mechan- i'^a' m>rr)l of dMScrihinc^ this locus: Lot AI3 l)e the EQUATIONS OF HIGHER PLANE CURVES. 59 diameter of the circle from which the curve would be described by the definition ; at the centre O erect the perpendicular OL. and take AD = AO = OB. Now take a rectangular ruler FEC, whose leg CE = AB, and Avhile the extremity C moves in the line OL. let the leg FE sUde through the fixed point D, then will the middle point of CE» P> describe the cissoid. [The demonstration will afford a good exercise for the student. ] ScH. 3. — This curve is also the locus of the vertex of a common parabola rolling upon an equal parabola. LI Fig. 60. 82, JProh, — To produce the equation of the Cissoid of Diodes. Solution. — In Fig. 59 let AX and AY be the axes of reference, AB = 2a, the diameter of the circle referred to in the definition, and P any point in the curve. Then AD = x, and PD :=?/. Draw through P the ordinate ED, and also draw the equal ordinate E'D'. APE' is a straight Une by definition. We Squar- now have AD : P D : : A D ' ing and reducing, x- : y- : : 2a E'D', or ic :y : :2a — a*. : \/(2a X : X. yi X^ 'Aa — X is the equation sought. ScH. 1. — Since y = -, every real + value o/" x << 2a gives two real '2a — x' and numerically equal values of y, loiih contrary signs. Hence the locus is symmetrical with respect to the axis of x. For x = 2a, y = ±: go , whence the brandies are infinite, and GG' is an asymptote to both branches. For all values ()fx'^ 2a, and for :l negative, j is imaginary. Therefore the locus is comprised between the limits x = 0, x = 2a. ScH. 2. — By the Duplication of the Cube is meant finding the edge of a cube which shall have twice the volume of a cube whose edge is given. To efiTect this by means of this curve, lot AM bo any cissoid, AB the diam(!ter of the circle which pertains to it, and O the centre of that cii'cle. Take C0=20B, and draw CB. Let fall from the jDoint P, where C B cuts the curve, the per- pendicular P K. Then P K = 2 B K. Now a cube des- cribed on P K is twice one described on A K ; for since P K = y, A K = a;, and K B = 2a — x, we have ak' AK^ AK*. PK PK" = - — = 1: or iPK KB IPK' = 2AK . Finally, let a be the edge of any given cube ; fmd r/, so that a:ai :: AK : PK, whence a^ : c/i^ : AK^ : PK\ But A OK Fig. 01. PK =2AK. ai3 = 2a3. 60 THE CAETESIAN METHOD OF CO-OKDINATES. Bj taking CO =30B aud proceeding in a similar manner, we can tri- plicate the cube; or in the same way obtain the edge of a cube of any given number of times the volume of a given cube. (The pupil may show that |"k' =: 2Kb' ; also that Ak' = ^Tk^) THE CONCHOID OF MCOMEDES. 83. r>EF. — The Conchoid of Wicojuedes is the locus of a point in a line which revolves on and slides in a fixed pivot, so as to allow a constant portion of the line to project beyond a fixed right line. 84, JProh, To construct the Conchoid of Mcomedes. Solution. —Let O* be the fixed point, or pivot, X'X the fixed Hne, and A B the constant portion of the re- volving line. Draw a con- venient number of radiating lines through O, and on each lay ofl" above X'X the dis- tances CI, FP, E6, etc., equal to A B. Then will 1, 2, 3, 4, etc , be points in the locus ; and M B N will be the conchoid. ScH. — This locus is readily drawn by mechanical means. Let X'X and YY' be two bars fixed at right angles to each other. Let any one of the radiant lines, as OP, represent a ruler, grooved on the under side so as to slide on the head of a pin fixed in the bar YY', at O. Let there be a fixed pin on the under side of the ruler, as at F, which can slide in a groove on the upper side of the bar X'X. Now, placing the groove in the rnler on the head of the pin at O, and the pin in the ruler, in the groove in XX, any point in the ruler, as P, will describe the conchoid. 8S. JPvoh. To produce the equation of the Conchoid of Nicomedes. Solution. —Let P, Fig. 62, be any point in the locus referred to the axes XX', Y Y'; and let its co-ordinates A E and PE, be tc and y. Let A B = a, and AO = 6. Produce PE till it meets OD drawn parallel to AX. Now, by simi- lar triangles, PE :PD ::EF :OD; or y -.y-}-}) :: Va^ — 2/2 : x. Squarmg, 2/2 : (2/ -f- &)2 : : a-2 _ 2/2 : a;2. ... xHf- = {y -\- hy{a^ — y^). ScH. 1. — Since .-r = ± v/a^ — yy- j, for every positive value of y, numerically less than a, x has two nnmerically equal values with opposite signs ; which values increase as y di7ninishcs, and for ?/ == 0. .r =: dz ex. EQUATIONS OP HIGHER PLANE CURVES. 61 . • . This portion of the locus is symmetrical with respect to the axis of y, and has the axis of x for a common asymptote of its two branches. Again, as ail negative values of y, not numerically greater than a, give numerically equal values of x with opposite signs, there is a portion of the locus below the axis of .r, wliich is also symmetrical with respect to the axis of y. To discover the form of this portion, 1st consider a > h. Then f or y = — a, or — 5, a? = 0, but for values of y between these two limits, x has two nu-; merically equal values with opposite signs ; hence the locus between these two limits is an oval symmetrical with respect to the axis of y. For y nu- merically less than h, and negative, the values of x increase numerically till, at 3/ =: 0, they become =h oo ; hence between O and the axis of abscissas there arc two infinite branches, symmetrical with respect to the axis of y, and having the axis of x as a common asymptote. 2nd. When a^=b the oval disappears. These forms are described mechanically by taking the point on the moving ruler below the fixed line. ScH. 2. — When h = Q, the equation becomes .r-y^ ^^ y'^{a^ — y'^) ; or ic^ _|, y2 = a'^. This is the equation of the circle, as it evidently should be. ScH. 3. — This curve was invented by the geometer whose name it bears, for a purpose similar to that subserved by the cissoid. The problem of the Duplication of the cube and the Tri- section of an angle had been shown to be identical, as both depend upon the insertion of two means in a continued proportion between two extremes. Thus, letting a and b be the extremes, it is required to find x and y, so that a : X : y : b ; i. e., a : x : : x : y, and X : y : : y : b. This problem, viz., the insertion of two means between two extremes, is effected by the cissoid. In the cissoid, I^ig. 59, ED and AD' are the two means between AD and ID' : Fig. 63. Fig. 65. that is, AD : ED : AD' : ID'. The Triseciion of an angle by means of the conchoid is effected thus : Let COM, Fig. 66, be the angle to be trisected. From any point, D, in one leg let fall a perpendicular, DB, on the other. Take CB =2D0, and with O as the fixed point, X'X as the fixed line, and CO as the ruler with the constant portion CB projecting beyond X'X, construct the arc CR of the conchoid. Erect DP perpendicular to X'X, and draw PO. Then is POC one-third of COM. To prove this, bisect PH as at E, and draw 62 THE CARTESIAN METHOD OF CO-ORDINATES. DE. Draw also FE parallel to DH. Since PE = EH, PF= FD, and ED = PE = EH = DO- By reason of the isosceles triangles RED, and EDO, we have angle DEO = 2P =2POC. But DEO = EOD. .". 2EOC = EOD, or EOC = iCOM. [Note — This scholium is by no means necessary to tlie in- tegrity of the course. It is inserted merely as a matter of interest to the student, giving him a few hints upon a subject which has figured so prominently in the history of geometry. It vnll afford a good exercise for the student who has time and abiUty, to demonstrate fuUy the facts hinted at, and which are not demonstrated above. Thus, let him show why, in the cissoid, AD : ED : AD' : ID'; also how the insertion of two means enables us to obtain any multiple of the cube ; also how the conchoid effects the same pvirpose ; and that the two problems are in reality but one.] F ) c ^^^^ \ ^ ^ \ M\ F -\e x' C ^ B X Fig. 66. THE WITCH OF AGKESI. S(y. Def. — Hie Witch of Agnesi is the locus of the extrem- ity of an ordinate to a circle, produced until the produced ordinate is to the ordinate itself, as the diameter of a circle is to one of the segments into which the ordinate divides the diameter, — these seg- ments beinsf all taken on the same side. 87* I*TOh, To condruct the Witch of Agnesi. SoLTJTioK. — Let AB be the circle Draw a series of parallel ordinates 1 1 2 2, P'P, etc. To find a point in the locns, take P E : F E : : A B : A E, and P is siicli a point. In hke manner locate other points, a3 1, 2, etc. 88, I*TOh, To produce the equation of the Witch. Solution.— Letting the axes be as represented in Fig. 67, so that P being any point, A D = cc, P D = j/' ^^^ calling the diameter of the circle, A B = 2a, the equation is a>!?/ = 4a-(2a — y). [Let the student supply the demonstration.] ScH. — The Witch has but one portion, as represented in the figure ; it is symmetrical with respect to the axis YY', is comprised between y =0, and y = 2a, and has X'X for an asymptote. [Let the student give the proof.] EQUATIONS OP HIGHER PLANE CURVES. 63 THE LEMNISCATE OF BERNOUILLI. 89, Def. — The JLemniscate of JBernoiiilli is a curve such that the product of two Hues drawn from any point in it to two fixed points, called the foci, is equal to the square of half the distance b^ tween these foci. do. JProh. To construct the Lemniscale of BernouilU. Solution. — Let F andF' be the foci. From F' as a centre, with any convenient radius, as F' P, draw an arc, as P6, Find a third propor- tional to F'Pand F'A. Let this be PF. From F as a centre with this third proportional, draw an arc intersecting the former in P and 6. Then will P and 6 be points in the locns ; for, by construction F'P X PF = AF-. In like manner find other points. Fig. G9, shows a convenient method of find- ing these proportionals. GH = F'F, and XG = AF. Since TL : TG : : TG : Tl , XL : AF : : AF : T and T L and T I are the corresponding radii to be used in locating a point, as P. With one pair of distances four points can be found. 01, Pvoh. To produce the equation of the Lemniscate. Solution.— Assuming X'X' and YY' as axes of reference, letting P be any point whose co-ordinates AD and PD, are x and ^, and putting A F = A F' = c, we have the distance between the two points F' and P, or F'P = \/{x-\-cr-\-y-. In like manner FP = \/{c — xY -\- y-. Whence by definition \/i^~cjH^^ X V{G—xY-\-y^ = G% or (2/'-' -f x^f= 2c^(x'-^ — • y^). ScH. 1. — Let the student observe the symmetry and limits of the curve from its equation. Observe that in the construction F'B =:FC = TW- As FB X F'B =c2, and FB = AB — c, and F'B = AB + c, we find that AB =Cv/2. Putting AB =-a = c\/% 2c2 = a?, whence the equation of the curve in terms of its semi-axis is (3/2 J^x'^Y'= ^^{^^ — 3/-)- ScH. 2. — (To be read on review.) The equation of the Equilateral Hy- perbola whose semi-axis is a, and co-ordinates x', y' , is x''^ — y"^ ^a^. The equation of its tangent is xx — yy' = a^. The equation of a perpendicular x' from the centre upon the tangent isx = y. From these three equations, ehminating x' and y', that is finding the locus of the intersection of a per- 64 THE CARTESIAN METHOD OF CO-ORDINATES. pendicular from the centre upon tlie tangent, we find [x- -{- 3/2)2 = a-(x^ — y^). Therefore this lemniscate is the locus of the intersection of a perpendicular from the centre of an equilateral hyperbola upon its tangent, the axes of both loci being coincident. THE CYCLOID. 92. Def. — TJie Cycloid is the locus of a point in the circum- ference of a circle which rolls along a' fixed right line. ScH. — The cycloid can be constructed mechanically by rolling a wheel, as HPI, Fig. 70, along the edge of a fixed ruler, as AX. A point P in the circumference of the wheel describes the cycloid. Fig. 70. 03. Def's. — The circle H P I is called the Generating Circle, or, simply the G-eneratrix /AX is the J^ase, and is equal to the circumference of the generatrix ; and B F, erected perpendicular to the base at its centre, is the A-QCis, and is equal to the diameter of the generatrix. 04:. JProh, — Having given the cycloid, to put the generating circle in position. Solution. — There is given simply the curve A BX, Fig, 70. Draw the base AX, and bisect it by the perpendicular BF. BF is the axis. Bisect the axis by N K drawn parallel to the base. Now, to put the generating circle in the posi- tion it occupied when the generating point was at P. draw from P as a centre, with a radius equal to the radius of the generatrix (BO or OF), an arc cutting N K, as at C. C is the centre of the generatrix. Oo. I*roh. — To produce the equation of the cycloid referred to itx base and a perpendicular at the left hand vertex. Solution. — Let P, Fig. 70, be any point in the cycloid A BX, referred to A V, and AX as axes. Then AD =cc, and PD =2/. Call the radius of the generatrix r. Now A D =^ A I — D I . But by construction, A I = arc P I = versin — ' I L, Dl = PL or versin ~^t/, to a radius r. z= \/'2ry — y^. • . cc = versin—^ y — \/2ry — 2/) y2 ScH. — If y be negative, \/2ri/ — ?/-' becomes imaginary ; hence the curve lies on but one side of the base. For y =0, we have x r^r. versin -^0 = 0, EQUATIONS OF HIGHER PLANE CURVES. 65 2itr, 4iTtr, etc. , etc. Hence we see that there are an infinite number of arcs like ABX, belonging to the curve. This is also apparent from the defini- tion, as each revolution of the generatrix produces one arc, and there is no limit to the number of revolutions. For y = 2r, a; = versin ~^(2r) = ^r, S^rr, bTtr, etc. , etc. , as it should from the construction. OS, JProb. — To produce the equation of the cycloid referred to its axis as the axis of abscissas, and a tangent at the vertex of the axis ( B, ^ig. 71), as the axis of ordinates. Solution.— Let PM =?/, and BM =ic. Now PM=PL+ LM. But PL-=\/2r.^• — x^, and LM = IF = AF — Al=the semi-circumference of the generatrix — arc PI. Again, arc PI ==: versin — i (2r — x). .*. y = \/2rx — x"^ -\- 7tr — versia— 1 (2r — jc). 97. Cor. PR = arc BR, ■ing any point in the curve. P he- For PR = L M = A F — A I = arc HPI — arc PI =arc HP = arc BR. A I F Fig. 71. ScH. 1. — Considering the equation x = versin-'y — \/%-y ~~y^, we observe that there are an infinite number of values of x for every value of y. First of all, the term versin-^ is ambiguous as to its sign, since a nega- tive arc has the same versed-sine as the numerically equal positive arc. Moreover, whatever a versed-sine may be, there are not only the -j- and arcs less than 180°, and also the + and — arcs of 360° — the former, which corresponds to it, but these increased numerically by every multiple of lit. We are therefore to write the term versin~'y with the sign d=:, and under- stand that for every value of y it has an infinite number of numerical values, each succeeding value in the series, being numerically lit greater than the preceding. In the second place the term — s/'lry — y^, being a square V2ry — y2. - y2ry root is to be written — (d=:\/2?^y — y"^), or tion in this way we have x =^ ±i versin~'y siefnificance of these facts is as follows. Writing the equa- y'. The geometrical Let y have any value as PD, Fig, 70 ; then x has 1st, the positive value AD and gives the point P, and a numerically equal negative value, giving a point P' similarly situated on the left of AY, if we take versin"' y < 180° ; but if we take 360° — this arc, both -f and — as the value of versin— 'y, we get two other values of x, one -f and the other — . The former is where PM, Fig. 70, produced to the right meets the curve, and the other the corresponding point on the left of AY. Taking for values of versin— ly, the values now considered -\- l7t simply repeats the curves at a distance 27t both at the right and left. The equation in (.96) has a similar interpretation. 66 THE CARTESIAN METHOD OP CO-ORDINAIES. ScH. 2. — The Cycloid is a transcendental curve, and is next to the Conic Sections in importance among plane loci. OS, It frequently occurs that the equation of a locus can be writ- ten immediately from the definition. The Sinusoid is of this char- acter. The definition is, " The Sinusoid is the locus of a point whose abscissa is the arc, while its ordinate is the sine of the arc." Hence the equation is y = sin x. The other simple trigonometrical curves (page 16, Ex's 27 — 33) are of the same character, as is also the loga- rithmic curve, X = log y. 00, As has been remarked before, the number of plane curves is infinite. The foregoing have been given as specimens, from which it is hoped that the student will be able to learn how the equations of loci referred to rectangular axes are produced from the definitions of the loci. A great many kinds of curves are suggested by the study of the Properties of Curves. Some of these win be noticed hereafter. Again^ Mechan- ics and other branches of Physics, give rise to the study of curves, the production of whose equations requires a knowledge of the principles of Natural Philosophy. Thus, the Tractrix is the path described by a weight, W, Fig. 72, to which a cord, AW, is attached, and the extrem- ity, A, of the cord, made to pass over the path, AB, friction being supposed uniform and perfect. Again, the Catenary is the line which a perfectly flexible chain assumes when its ends are fastened at two points, as A and B, Fig. 73, nearer together Ak ^B than the length of the chain. Caustics are an interesting class of curves consequent upon the laws of reflected light. The next chapter gives several varieties of curves caUed Spirals. Fig. 73. CHAPTER II. THE METHOD OF POLAR CO-ORDINATES. SJECTIOJSr I. Of the Point in a Plane. 100. JPvop. — The Position of any Point in a plane can be designated j hy giving its Distance and Direction from a fixed point in the plane. In order to indicate direction, a fixed line has to be assumed. III.— Let A, Fig. 74, be the fixed point, and AX the fixed line. Let r represent the distance from the fixed point to the point to be desig- nated, as AP, AP', etc., and 6 the angle in- cluded between the fixed line and the line from the fixed point to the point to be designated, as ' " ' P'" PAX, P' AX, etc., etc. It is evident that by Pig. 74. giving all possible values to 0, and r, all points in the plane of the paper may be located. Thus, for P, we have =35o, r = 5 ; for P', = 1200, r = 10 • P'' 0=195° r= 8 ; and for P", 0=3450, r = 11. 101* Def's. — TheJPole is the assumed fixed point, as A. The JPrime Madius (called also the Initial Line, and the Polar Axis) is the assumed fixed hne, as AX. The Radius vector is the distance from the pole to the point to be designated, as A P, A P' etc. The Variable Angle {O) is the angle indicating the di- rection of the point from the pole. When 6 is reckoned around from right to left, it is called + ; when reckoned from left to right, . The radius is -f when estimated in the direction of the extremity of the arc measuring the variable angle ; and it is — when estimated in the opposite direction, r and are the Polar Co-ordinates. 102, JProp, — The Polar equations of a Point are, r = a, and = b ; since, by giving suitable values to r and d, all points in the plane can be located. Ex. 1. Locate r = 5, = \7t. Solution. — The radius being 1, it is the semi-circumference. Hence lit = 6O0. Now, lay off PAX = = 6O0, Fig. 74 ; and, taking AP = 5, P is the required point. 68 THE METHOD OF POLAB CO-ORDINATES. Exs. 2 to 6. Locate r = 3,0 = ^7r: r = 4., d = ^7r: r = 6, 0. r = —6, = 135° : r = 4t, 6 = —60°. ^TT Ex. 7. Show that r=lO,0 ^=135°. -45°, is the same point as r = — 10, 103, JPvob, — To find the Distance between two points given by their polar" co-ordinates. Solution. — Let the co-ordinates of P, Fig. 75, be (r, 6) ; and of P', (r', 6'). We are to find PP' in terms of r r' , 6, and 9'. Now, in the triangle PAP', AP = r, AP' = r' and the included angle PAP' = — 0'. Hence, representing PP' by D, we ^ have from principles of trigonometry D = \/r' -f- 1'"^ — '^rf' cos (6 — Q'), q. e. d. Ex. 1. Eind the distance between r = S, d = \7t, and r = 4, d = ^7t. Ex. 2. Find the distance between (8, f ;r), and (3, \^7t). Ex. 3. Find the distance between {\/\ 45°), and (1, 0°). Results in the last three examples, not in order, 7, 1, 5, Fig. 75. ■^♦» SUCTION IL Of the Eight Line. 10 4z* I^rob, — To produce the Polar Equation of the Right Line. Solution. — The form of this equation (like all others) depends upon the con- stants assumed. We will consider two forms. 1st. When the constants are the length of the perpendicular from the pole upon the line, and the angle which this perpendicular makes with the prime radius. Thus in Fig. 76, let M N be any line ; A, the pole ; AX, the prime radius ; the perpendicular from the pole upon the line, A D := p ; and the angle which the perpendicular makes with the prime radius, DAX = a. Let P be any point in the line M N , and its co-ordinates be (r, 0). Now, in the right angled triangle PAD, we have AD = A P cos PAD, or p = 7- cos (9 — a) ; V r = E. D. cos (9 — a) 2nd. When the constants are the intercept on the prime Fig. 76. OF THE RIGHT LINE. 69 radius, and the angle which the line makes with the prime radius. In Fig. 77, let M N be any line referred to the pole, A, and the prime radius, AX. Represent the intercept, AT, by c, and the angle NTX, by a. Let P be any point in the line, and its co-ordinates AP = r, and PAX = 0. ^\ The angle XP A = — a ; and, from the triangle PTA, we have AP : AT : :sinPTA:siu TPA, or r : c : : sin a : sin i^Q — a) ; sin a . • . r == —. t; c. Q. E. D. sm (6 — a) P ScH. 1. — Discussion of the equation r = . When 9 = 0, we have "^ ^ cos (9 — a) P r =. — . This is as it should be, for, when 9 = 0, r = AT, Fig. 76, cos ( OC) AD p which, from the trianffle ADT, is seen to be ~ , or ; ^,. The .cos DAT cos( — cc) — sign of (X. indicates that the radius vector falls upon the opposite side of the perpendicular from that assumed in producing the equation. . . . When r= a:, r = -J— = p, as it evidently should. . . . When 9 — ol = 90°, r = - = 00. In this case the radius vector becomes parallel to the line, and hence oo . . . . From 9 — a = 90° to 9 — a =. 270°, r is negative, as it should be ; since, in order to reach the line M N , it must be produced hackioard, i. e. , from the pole in a direction opposite to the extremity of the ■^ arc 9 measured from the prime radius around to the right. . . . From 9 — • K a = 270° to 9 — a = 360°, ?* is + ; and at 9 — a = 360°, r = p, as it should. P Also, ait 9 = 360°, r = ^ = AT. . . . When the line M N passes cos( — a) through the pole, r = , which is for all values of 9 except 9 =^ cos (9 — a) (90° + a), for which r = -, i. e., indeterminate. The values of r indi- cate that we have not to pass any distance from the pole to reach the line ; and the - value indicates that for all values of r, its extremity is in the line MN- . • . Finally, when a =:0, the line MN becomes perpendicular to the P prime radius, and its equation is r = cos 9 ScH. 2. — Discussion of the form r =: -; — — .c. For 9 = 0, r = — c ; sm (9 — a) which is evidently correct, as it is reckoned backward, and equals c, in length. . . . For all values of 9 <; a, r is negative, and hence is reckoned back- 70 THE METHOD OP POLAE CO-OBDINATES. ward. . . . For = a, y = c := oo, as it should, since it is then par- allel to the line. . . . For values between 6 >■ «:, and = 180° -f or, r is positive. ... At = 180° -\- a., r becomes infinite ; and, when passes ^ ^ . . . sin oc 180° -^ a, r IS again negative. . . . For = 180°, r = c = c, sin (180° — a) «. e., AT. . . . If the hne MN passes through the pole, c = 0, whence sin a ' c r = ~ — ; r = -: — ; = 0, for all values of except Q ^^a, in which sm (0 — a) sin (0 — a} ^ case r = -. These results are evidently correct, for in the former cases we have to go distance from the pole in the specified directions, in order to reach the Hne ; and, in the latter (when 6 = a) the radius vector falHng on the line MN, its extremity is equally in the line for all values of r. Ex. 1. What is the polar equation (first form) of a line the nearest point in vrhich is 6 from the pole, and the perpendicular to which makes an angle of 45° with the prime radius? Where does this line cut the prime radius ? For what values of ^ is r infinite ? What is the value of r for d= 75° ? For d = 15° ? Construct these values of r and verify them by drawing the line. Ex. 2. What is the polar equation (first form) of a right line per- pendicular to the prime radius, and which cuts it at 4 to the left of the pole ? What is the value of r when ^ = 60° ? Why is the sign of r negative in the latter case ? Between what values of 6 is r posi- tive ? What is the value of r when is 120° ? 4 The equation is r = . ^ cos d Ex. 3. Give the equation of a line parallel to the prime radius, and m 10 above it ; also at m below. The latter equation is, r == sin 6 SECTION III. Of the Circle. 105, I^roh, — To produce the Polar Equation of a Circle, the pole being in the circumference, and the polar axis being a diameter. or THE CIRCLE. 71 Solution. — Lot A, Fig. 78, be the pole ; B, the radius of the circle ; and P, any point in the circumference. Then AP=r, and PAB^Q. Now, from the right angled triangle APB, we have, by trigonometry, r = 2B cos 0. q. e. d. ScH. 1. — Discussion of ike Equation. If = 0, r = 2B. If 6 = ^7t, or f tt, r = 0. For Fig. 78. values of between j7t and f tt, r is negative, indicating that for these val- ues of the radius vector must be produced backward to meet the circum- ference. (The student should observe that the results obtained from the equation, accord with, the values as observed from the figure.) ScH. 2. — If the pole is at the centre, the equation is evidently r = R, for all values of 0. 106. IProh, — To produce the General Polar Equation of a Circle. Solution. —Let the constants be (r', 0'), the co-ordinates of the centre, C, Fig. 79 ; and E, the radius of the circle. Let P be any point in the circumference, and its co-ordinates, A P and the angle P AX , be r and 0. Now, since the dis- tance between the points C and P is constant (R), we have from {103)B—\/r^-\-r''^ — 2rr'cos(0 — 0'). Whence we have, r-2 _ 2r' r cos {6 — Q') = B^ — Fig. 79. 0- E. D. ScH. — Discussion of the Equation. — Solving the equation for r, we nave, 7- = r' cos (0 — 0') d= \/i?^— r'^sin^ (0—0'). This value of r is real only for such values of as render r"^ sin2(0 — 0') ^' cos (9 — 0'), r has different signs. In the former case the i^ole is without the circle, — in the latter within. These facts readily appear by solving the inequality. Thus, in the latter, R — r'2 sin2(9 — 9')> r'^ cos2(9 — 9'), or R > r'^ sin2(9 — 9') + r'^ cos2(9 — 9'). But the latter member reduces to r"^. .'. R^r', which puts the i^ole within the circle.* (For other cases, see examples below.) Ex. 1. What is the equation of a circle whose radius is 10, and the polar co-ordinates of whose centre are (15, ^tt) ? What values of in- dicate that the radius vector is tangent to the circle ? Between what limits of ^ is r real ? Between what imaginary ? What is the posi- tion, and what are the values of when the radius vector passes through the centre ? Construct the figure. The equation is r- — 30 sin Or = — 125 ; or r = 15 sin ± \/lOO — 225 cos* 0. The posilions of tangency are cos ^ = f , and cos = — |, when ?^ = 5v/5. Ex. 2. Give and discuss as above the polar equation of the circle whose centre is at (8, \7t), and whose radias is 10. Does this circle cut the polar axis ; and, if so, where ? How do you determine this jDoint from the equation ? Equation, r^ — 8\/2 (sin -{- cos 0)r = oQ. Condition of tangency, sin ^ + cos <9 = f v^ — 2. .'. The pole u within; as appeal's also from — ^ 1. Cuts the polar axis £j^ ?- = (4 ± \/34 ) v/ 2. Ex. 3. Show that the polar equation of a circle is r^ — 2r'r cos = JR2 — r'2 when the polar axis passes through the centre and the j)ole is without or within the circle. (Prove this directly from a figure without reference to the preceding forms. ) Ex. 4. Deduce from the general form in 100, the form in 10 S, and also the one in Ex. 3. R * This may also be observed from sin (Q — Q',) = + —, whicli is the condition of tangency. r' This is possible only when R <; r', i. e., when the pole is without the circle. When R =r' the two tangents become one. OF THE CONIC SECTIONS. 73 Ex. 5. Let the student show from the annexed figure that in the equation r =r'cos [d — e')±iV Ri—r'-^sm^ {6—0^ the rational part, r'cos {6 — 0'), is the chord, A D, of the circle described up- on A C (= r') as a diameter, and that the radical part, \^Ii^ — r'^ sin^ {0 — 0'), '"^^^^ ^^^^^ is P D = P' D, + for P D and — for P'D. B and B', are the points where the radical becomes 0. ■»♦■»■ SUCTION' IV. Of the Oonic Sections. 107 > I^voh, — To produce the Polar Equation of a Conic Section. Solution. — Let P be any point in tlie curve ; EC the e directrix ; A, the focus and pole ; and AX, the axis of the curve ana the polar axis. Let 2p be the latus rectum ; e, Boscovich's ratio, whence p=CAXe; AP = r, and PAX=0. Then AP = CD X e = (CA -f AD)e. But C A = — , and A D = r cos 6 ; whence, r = 6 ( f- r cos )e. P 1 — e cos Q. E. D. Fig. 80. 108. CoR l.—The JPolar JEquation of the I'arabola. Since in the parabola e = 1, its polar equation is P P 1 — cos 6 , or r == versin 0' 109. Cor. %—The JPolar Equation of the Mlipse and Hyperbola in terms of the semi-transverse axis and eccentricity. Since Boscovich's ratio (e) and the eccentricity are the same {48), and, numerically, p == A(l — e^), this equation is Ajl — e^) 1 — e cos 0' ScH. 1. — Discussion of the Polar Equaiicm of the Parabola. — ^For 9 = 0, 74 THE METHOD OF POLAR CO-OKDINATES. P ^ P ^ ^ 1 — COS e ^®^°^®^ ^' ^ flTT = '^> *'• ^•' *^® radius vector falling upon the axis does not meet the curve For any value of 6 > 0, however small, r is finite ; which shows that, if a line be drawn from the focus making any angle however small with the axis of the curve, it meets the curve at a finite distance. .... For © = 90°, r = p, as it should For B = 180°, r = P P m • -. ,. = -. . -. = ip- This is evidently correct, since r becomes A B, Fig. 80, when Q = 180°. But, in the parabola, AB = ip For 6 = 270°, r = p, as it should. (Let the student discuss in like manner the form P r == versin 6 ) ScH. 2. — DiscussioTi. of the equation r = u4(l — e2) :; for the Ellipse. — ^For 6 = 0, r = 1 — e cos 9 -^ A{l — e^) — — — = A(X -{- e) = A -\- Ae ) which makes AC = J. + Je, as it should {49) For the point P at the extremity of the conjugate axis, T-r- n^ „ AO Ae J^ig. 81, cos 6 = — — = — . Hence r = or r AP - Ae^ = ^(1 — e2). ^(1 — e2) Ae^ '■ r agreeably to [4=6] For 6 = 90 be. l + e r = ^(1 — e'^) = p For = = A — Ae, which is the value of A B, as it should ScH. 3. — Discussion of the JEqua- A{1 — e^ for the Hyper- tion r 1 — e cos bola. — Kemembering that in the hyperbola e > 1, we observe that ^(1 — e-) is essentially negative, and hence that the sign of r depends upon the sign of the denominator, 1—e cosQ ; r being + when e cos e>l, and — when ecose9O° and <;270°, r is negative, and the branch, AM', is traced while is passing from 90° to 180°, and the branch, A M, is traced a second time by the negative radius vector while passes from 180° to 270°. . . .While passes from 270° to 360°, r is positive and A M ' is traced a second time. Therefore the curve is traced twice by one revolution of the radius vector. OF HIGHEK PLAKE CUKVES. 77 111, ^Tob, To produce the Polar Equation of the Conchoid of Nicomedes. Solution. — In Fig. 62, draw F K perpendicular to OD. Let O be tlie pole, and O D parallel to AX, the polar axis. Let AO =6, and A B = a. Then P being any point in the curve, we have OP=r = OF-{-FP = OF-f-«. But O F = F K X cosec FO K = 6 cosec 0. Therefore, r = b cosec -\- a. q. e. d. ScH. — Discussion of the Equation. For = 0, ?• = co .... For = 90°, r = b-\-a For 6 = 180°, r = oo For > 180° and < 360°, cosec 9 is negative, and the lower branch is traced. The student should be careful to observe the several forms, as when a'^ b, a = b, a<^b. (See 85,) 112, JPvob, To produce the Polar Equation of the Lemniscate of Bernouilli. Solution. — Using the same notation as in Fig. 68, let A be the pole, and AX the polar axis. Let P be any point in the curve, and draw A P. Then A P = »*> and P AX = 6. The following is an outline of the solution : F' P' = c2 + r^ -f 2cr cos B. (1). FP"^ = 0*2 4- r2 — 2cr cos 0. (2). Multiplying (1) and (2) together, and remembering that F'PX FP = c2, we have, after a little reduction : r2 = 4.G^ cos2 — 2c2 = 2c2(2 cos2 — 1). But 2cos2 — 1 = cos 20. . • . r2 = 2c2 cos 20. q. e. d. ScH. — The pupil should discuss this equation as the preceding have been. [Note. — It is frequently more convenient to obtain the polar equation of a curve by transforming its rectilinear equation, according to a process to be explained in a subsequent chapter. We will close tb.is chapter with some account of Spirals, a class of curves of much historic interest ia con- sequence of the labor bestowed upon some of them by the old geometricians, and to which the method of polar co-ordinates is specially adapted.] i OF PLANE SPIRALS. 113. Def's. — A. I*lane Spiral is the locus of a point revolv- ing about a fixed point, and continually receding from it in such a manner that the radius vector is a function of the variable angle. Such a curve may cut a right line in an infinite number of points, which would render its rectiUnear equation of an infinite degree. Hence, these loci are transcendental. The Measuring Circle is the circle whose radius is the ra- dius vector at the end of one revolution of the generating point in the positive direction. A. Spire is the portion generated by any one revolution of the generating point. 78 THE METHOD OF POLAR CO-ORDINATES. 114. The Spiral of Archimedes is the locus of a point revolving around and receding from a fixed point so that the ratio of the radius vector to the angle through which it has moved from the polar axis, is constant. JProb, To construct the Spiral of Archimedes. Fig. 83, be the lis. Solution. — ^Let A? pole, and AX the prime radius. Through A draw any convenient num- ber of indefinite radial lines (say 8) making equal angles with each other. Since 6 and r are to vary alike, when = 0, r= 0, and the spiral begins at the pole. Take any distance, as Al, on Aa, twice this distance, as A 2, on Al*, three times the same distance, as A3 on Ac, etc., etc. Then will 1, 2, 3 17 be points in the spiral, q. e. d. III. — The dotted Hne ah cdefg is the measuring circle, and Al 2 3 the first spire. 8 9 10 11 15 16 is the second spire. ScH. — The several spires of this spiral are such that, the distance between any two consecutive ones measured on the radius vector is the same, and equal at all points to the radius of the measuring circle. 7 8 is 110. JProb. To produce the equation of the Spiral of Archimedes. Solution. — Letting a be the ratio of the radius vector to the variable angle, we have r = aO. Or, otherwise, calling the radius of the measuring circle, A 8, Fig. 83, 1, for this value of r, 6 = 27t. Let 3 be any point in the curve, whence A3 represents r, and SAX, or Sahc = Q. Now from the definition, r : 1 : : 6 : 27t. B . • . r = r— . Q. E. D. 27t 117. Cor. — Tlie Reciprocal or Hyperbolic Spii^al. This Spiral is naturally suggested by the Spiral of Archimedes, as in it the radius vector varies inversely as the variable angle. Hence the equation is lit CoNSTBUCTioN. — To construct theBecip- rocal Spiral, let A be the pole, and AX the polar axis. Draw any convenient number of radial lines through the pole, making equal angles with each other. Take Al any convenient length, A2 = ^Al, A3 = iAl, A4 = iAl, etc., etc. The points 123 4 8 — B are points Fig. 84. OF HIGHER PLANE CURVES. 79 in th6 curve. Since r can become only when 6 = go, this curve continues to ap- proach the pole as the radius vector revolves, but reaches it only upon an infinite number of revolutions. lis. The Lituus. — The equation of this spiral is r = — ^-. Let the stu- dent construct it and give the formal Fig. 85. definition. The form of the curve is given in Fig. 85. 119. TJie LogarithTnic Spiral. — In this spiral the radius vector increases in a geometrical ratio, while the variable angle in- creases in an arithmetical ratio. The equation is, therefore, r= a9. If a be the base of a system of logarithms, this equation becomes = log r. CoNSTEUCTioN. — To coustruct r=a^, let a = 2. Then for = 0, r = 1, which gives the point 0. For 0=1, i. e., the arc of *57.3o nearly, r=2^=2, which gives the point 1. For 6 = 2, I e., the arc of 114. 6° nearly, r = 2^ = 4:, which gives . the point 2. As increases r increases much more rapidly, so that with this small base (a =: 2), at the end of the first revolution, when 6 = 6.28 + , r= 26-28+= more than 64. Heuce, at one revolution the radius vector would be 64 times Ao. But, though r in- creases so very rapidly, it is easy to see that it does not become oo till 6 = oo . Again, letting the radius vector revolve in the nega- tive direction from AX, so that 6 is negative, we have for = — 1 = OAa, r = Aa = 2 — ^ = h For 6 = — 2, r = 2-2 = 1 Thus, it appears that as the radius vector revolves in this direction it generates a portion of the spiral which at first rapidly approaches the pole, but cannot reach it till 6 = cx) .... Were we to take a = 10, the base of the common system of logarithms, the change of r would be so rapid that we could represent but a small arc of tbo curve. OHAPTEE m. TRANSFORMATION OF CO-ORDINATES. SECTION L Methods of Passing from one Set of Eectilinear Axes to Another. 120. T>EF's.—Transfor7naHon of Co-ordinates is the process of changing the reference of a locus from one set of axes to another, or from one system of co-ordinates to another. The prob- lem presents itself under two different aspects which are nearly the converse of each other : 1st, Having given the equation of a locus re- ferred to one set of axes, or system of co-ordinates, to find the equa-- tion of the same locus when referred to another set or system. 2nd, Having given the equation of a locus referred to some known axes, to find the position of a new set, to which, when the locus is re- ferred, its equation will take some specified form. The axes, or sys- tem, to which reference is made in the given equation, may be called the Oldf or Primitive, Axes or System, and those to which the transformation is made, the Wew. Ill's. — The equations ic2 -j- 2/^ = 25, y'^ = .^(10 — x), and x(x — 4) -f- ViV — 6) = 12, may all be considered as equations of the same locus M O N , Fig. 87, but referred, respectively, to the three pairs of axes X,Xi', YiY,'; XiX,', Y2Y,' ;X.3X,', Y.3Y3'. Now, having given any one of these equations any oth- er of them niay be found if we know the position of the new axis with reference to the old. The process is transformation. Again, we are familiar with various methods of designating particular points Fig. 87. on the earth's surface, as by latitude and longitude, or by their distances and direc- tions from a given point. For example, we may give the position of Chicago by stating its latitude and longitude with reference to the meridian of Washington, or Y^ 0/- Y3 Y. x; A2 V A, A3 ) ^ X'3 Y3' y xa Y/ FROM ONE EECTILINEAR SYSTEM TO ANOTHER. 81 by giving its distances from the tropic of Cancer and the meridian of Greenwich, Eng. , or, in still another way, by giving its distance and direction from New York city. The process of converting one of these descriptions into any other of them, would furnish an analogy to the process of transformation of co-ordinates. The first two descriptions (considering the earth's surface a plane), would be equations of a point (Chicago) referred to rectangular co-ordinates ; the last would be an ex- ample of polar co-ordinates . "We will give one more illustration, as it is of the highest importance that the na- ture of the problem be understood from the outset. Let the student construct a pair of rectangular axes, XiXi', Y 1 Y J ', with the origin A i , and an- other pair, X2X2', Y2Y 2', also rect- angular, with the origin at A 2 [the point (0, — 1) when referred to the first axes], and the new axis of X, X2X2', making an angle with the primitive axis of — 45°, and the new axis of y making an angle of 450. Now, upon the first axes, con- struct ic2 — 6xy -\- y- — 6a; -f- 2?/ -j- 5 = 0, and upon the second axes con- struct 2/' — 2a;- = 2, when the two j y equations will be found to give the Fig. 88. same locus. The problem of transformation which affords this illustration may be stated thus ; To transform x- — 6xy -{- y- — 6x -{- 2?/ -|- 5 = 0, to a new system of rectangular axes having the new origin at (0, — 1), and the new axis of x mak- ing an angle with the primitive whose tangent is — 1. To illustrate the second form under which the problem of transformation pre- sents itself, the example of the last paragraph may be stated, — Having given the equation x'^ — Qxy + 2/^ — 6x -{- 2y -(- 5 =. 0, as the equation of a locus referred to rectangular axes, required to find the position of a new pair of axes to which, when the locus is referred, its equation will involve no terms in the first power, or in the rectangle of the variables. The result of the solution of this problem would be the determination of the position of new axes as given in the paragraph above. ScH. — As tliGfo7'7n (not the degree) of the equation of a locus, depends in a large measure upon the situation of the axes, or upon the system used, it will be readily seen that a set of axes in some particular position, or some particular system of co-ordinates, may be best suited to one class of pi'ob- lems, or of loci, and another set or system to another class. It is therefore de- sirable to be able to pass at will from any one set or system to any other. This transformation is effected by finding the values of the co-ordinates in the ^ven equation in terms of new co-ordinates (and certain constants) and substituting the latter for the former. The methods of doing this we will now explain. 82 TKANSFORMATION OF CO-OSDIXATES 122, JProh. — To produce the general formuloe for passing from one set of rectilinear co-ordinates to another. SoiiUTioN.— Let P, Fig. 89, be any point in a locus M N referred to the Frimitive Axes A i X , , A i Y ^ , the co-or- dinates being A i D = ^, and PD=:y. Let A2X2, A2Y2 be the New Axes, the CO ordinates of the point P, when referred to them being A2D'=a?2, and P D '=y2 . Let the angle included between the primitive axes, Yi A^Xi, be /3 ; the angle which the new axis of X makes with the primitive, X2 IXi, be a ; the angle which the new axis of y makes with the primitive axis of x, Y2l'Xi be a ; and the co-ordinates of the new origin, Aj, be AiG =m, AgG =: n. The problem now is, to find the values of the primitive co-ordinates x, y, in terms of the new co-ordinates x^, 2/2, and the constants m, n, a, a , and /?, so that the latter may be substituted for the former in the equation of a locus referred to the primitive axes, and the equation be thus transformed and made to represent the same locus in terms of the new co-ordinates ; i. e., referred to the new axes. We have a; =: AiD = AiG + AjE + D'F. But AiG = m ; and from the triangle AjD'E, AgE : ^-2 •• sinAjD'E : sinAgED', which becomes A2E : X2 : : sin{/3 — a) : sin /3, since AsD'E = D'ER — D'AgE = /3 — a, and sin D'EAg = sin D'ER = sin/5 (the sine of an angle equals the sine of Xz sin (/5 — a) Fig. 89. its supplement). From this proportion, A2E = manner, from the triangle PD'F, we have D'F = these equivalents in the value of x as given above ^2 sin (/5 — a) -\~ y^ sin (/3 sin p 2/2 sin (/5— a') Again, y = P D = A 2 G siny^ a') In the same Substituting D'E -f PF. (1). angles A2D'E, and PD'F, we have as before D'E = But A2G = n, and from the tri- x^ sin a sin/i ' and P F == ?/, sm a sin/i Hence y = n-\- X2 sin a -\~ y^ sin a sin/i ' (2). 123 • Cor. 1. — When the New Axes are parallel to the Primitive, the formulcB become X = m + X2, (1); and y = n + ja, (2); since in such a case a = 0, and a'== p ; v)hence sin /^ = 0. sin a' =^ sin /?, sin (/? — a)=^ sin /?, and sin (/? — a') = sin = 0. FllOM ONE BECTILINEAR SET TO ANOTHER. 83 124:, CoFw 2. — To pass from rectangular axes to oblique, we have X = m + Xa COS a -\- y^ cos a', (1) ; and y == n + Xa sin a -}- y^^ sin a', (2). These follow readily from the general formulse by observing that in this case P = 90°; whence sin (/? — a) = cos a, sin (/? — <3^') = cos a', and sin yj = 1. J.2S, CoE. 3. — To pass from one set of rectangular axes to another set also rectangular, but not parallel, we have X = ni + Xa cos a — j^ sin a, (1) ; and y = n + ^2 sin a + J2 cos a, (2). To deduce these f^om the general formulae {122 ), observe that fi = 90°, and ix' — az=z 90°, or a' = 90° + ^/ whence sin f3 = l, sin (/? — ^) = cos rectangular axes, to transform to new axes with the same origin, the FEOM ONE KECTILINEAR BET TO ANOTHEE. 85 tangents of tlie angles wliich the new axes of x and y make with the primitive axis of x, being 3 and — 3, respectively. Verify by a con- struction. Sug's. — The formulce are x = X2 cos a + 2/2 cos a', and 2/ = cca sin a + 2/2 ^i^ <^'. In this case tan a = S, v/lience sin a ^V^'ni^ ^^^ cos a =\/-^q. Also tan /j^(j. Introducing these values, the /onn- wZa? become x = \/-^u{x.2 — 2/2)5 ^'nd y = v^tuK'^i 4" 1^2)' The transformed equation is ^x^^ + ^^21/2 + ^l/z' = ^^ . Ex. 5. Given xy = 16 as the equation of a locus referred to rect- angular axes, to pass to new rectangular axes with the same origin, the new axis of x making an angle of 45° with the primitive axis of X. Equation, x^ — y^^ = 32. Ex. 6. By the same transformation as in Ex. 5, show that y* -\- x*-\- Gx^yi = 2, becomes ^2^ + 2/2'* = 1- Construct the locus, and both sets of axes, and observe the position of the locus with respect to the two sets of axes. Ex. 7. Given the equation y = ax -\-h (the common equation of the straight line), to pass to oblique axes with the same origin. Sug's. — The formulce for transforming are x = X2 cos a-\-p2 cos a', and y == x^ sin n: -f- 2/2 sin a'. Substituting and reducing, we have y-i = a cos a sm a •, ^2 + sm oc — a cos a: sm ol — a cos oc which is the equation of a right line referred to oblique axes making any angles (a, a') with the primitive axis of x. Now, if we desire simply the form, of the equation of a right hne referred to ob- Hque axes, we may consider the new axis of x as coinciding with the primitive, Tig. 90, and let the new axis of y make any angle, as y5, with this. Then a: = 0, and a' = ft ; whence the equation becomes a . 1) Vi = ttttj Trrrr-Tp ^2 + Fig. 90. sin fi — a cos (3 Again, letting the angle NT A] = aj, a = sin a^ sin (5 — a cos ft sin ai cos a-i whence sin/^ — a cos /J cos a^ sm OTi sm a^ sin/J Also, sma, cos 0:1 cos/? siu ft cos (Xi — cos ft sin cx.x h cos ax sin(/? — ai) sin/^ — a con ft sin (/? — aj : sin BCAi, or AiC : h :: sin (90o -j- a^) 6 sin (90° -f- <^ I ) hcosai = AiC, since AiC : Ai B : : sinCBAL sin (/i — a-i) sin(yS — a-i)' calling A 1 C ^ 5'. (See 34,) sin {ft — Substituting, we have 2/2 a-i) ; whence AjC sin a, sinift — OTi) X2 + h'. 86 TRAI^TSFOEMATION C? CO-ORDINATES Ex. 8; Given the equation y = ax-\-b,io find the position of a new set of axes parallel to the primitive, to which, when the locus is re- ferred, its equation shall have no absolute term. Sug's. — Substituting for y, y^ -\- n, and for x, x.^ -f- w, we have 2/2 = <^a*2 + ^'^ -{-6 — n. Now, if the new axes can be so situated that am, -|- & — n = 0, or ?i = a-m-Y'h^ the condition required will be fulfilled. But n and m are the co-ordinates cf the new origin, and the condition n = am -f- & (/i and m being co-ordinates) designates a point in the line y = ax -{-I). Hence the new origin is to be in the line y ^^ ax -\- h. (See 33 y Sch, 1.) Ex. 9. Transform A"y'^ + B-x'^ = A"B-^, to parallel axes with the new origin at ( — A, 0). Also to parallel axes with the origin at {A, 0). Also to parallel axes with the origin at ( — m, — n). (See SS») Ex. 10. Transform A-y^ + B-x^ = A'B^, to oblique axes with the B^ A^' terms of the diameters lying on the new axes. same origin, such that tan a tan a' = —^ and obtain the result in Sug's. — After making the substitutions and collecting terms, we have {A-sin^a -f- B-cos~a')y2 - -{- {A-sm"a -\- B'^cos"a)x2 ^ -{- 2{A-sma sina' -\~ B^cosacosa' ) ^2^/2 = A^B-, (1), which is the general equation of an ellipse referred to oblique axes, the origin being at the centre. If these axes are so situated that tan a sin a sin a' B' , . . , ^ , ^ -, ,-, . tan a = = -, A'^ sm a sm a -\- B- cos a cos a = 0, and the term cos a cos a A^ in Xzy-z disappears, and the equation is (^2 sia^a' -\- B^ cos^ a')y2^ 4- (J.2 sin2 a -\- B- cos'^ a.)x.2" = A-B^. This is the equation of the ellipse referred to the axes required, but it is not in the terms required^ it is the equation of the ellipse, as BPC, referred to the diameters A, Bg, AxDg, but is in terms of the semi-axes Aj B, AjG, and the angles a, a', which the new axes make with the primitive axis of x. Thus, P being any point in the curve, A 1 E rep- resents a-j, and RE 2/2. Nov/ in this equation, 4 when 2/2 =0, Xs becomes Aj B2. Hence calling Ai B2 Aj, we have ^.j- ^= - — -. -^ , A^Bin-^'a-^-B^cos^a A^B^ or A^sin^ a -\- B- cos^ a = — r^. In hke manner for x^ = 0, 2/2 becomes Ai D^. Ai^ A^B^ Hence calling Ai D, B,, we have B-,^ = , or A^ sin^ a' 4- ^ A^ sm^ a' + 252 cos2 a' ^ A^B^ B'cos-a' = -zr-T. Substituting these values of the coefficients of y^-, and iCg^, -"1 A 2 7-^2 A 2 7?2 we have —jj—y-z- -\ j-rajj^ = A'^B\ Finally, dividing by A'^B'^, and clearing of fractions, A^^y^^ + Bi^Xs^ = A^^B^^ q. e. t>. FROM ONE RECTILINEAR SET TO ANOTHER. 87 ScH. — ^Diameters so situated as to make tan a tan a' = — are Conju- gate Dia7aeters, as will appear hereafter. Hence the equation of the ellipse re- ferred to conjugate diameters, and in terms of those diameters, is of the same form as the equation of the curve referred to, and in terms of, its axes. I Ex. 11. Transform A^y"^ — B^x^ = — A^B^ to an oblique system with I B^ . tho same oriofin, such that tan a tan «' = -r-, and obtain the result in terms of the diameters lying on those axes. Sitg's. — The student is expected to recog- nize this as the equation of the hyperbola referred to its own axes. The transforma- tion is in all respects like the above, except that the diameter represented by B^ is im- aginary, i. e., does not meet the real branches of the curve, hence we call AiDa* ' Byy/^^n., or (A;^D2)2 = — B,^ The equation soiight is Ai-y^^ — ^I'^.-Cj- = — A,^B,^ Fig. 92. ScH. — In each of the two preceding examples, there were given, the equa- tion of the locus, and the position of the new axes, from which to find the form of the new equation. The converse of this problem is important ; i. e., Given the equation of the locus, and some specified form of its equa- tion, io find the position of the new axes. Thus, for example, — The origin remaining the same, what must be tho position of oblique axes, to which, when the eUipse is referred, its equation will take the same form as the common equation. To solve this problem, we first transform A'^y- + -S"a;2 = A'-B-^ to obhque axes, as in Ex. 10, and obtain the form (1) in the sugges- tions. It then remains to determine what values a aad a must have, i. e., how the new axes must be inclined, to make the equation take the primi- tive form. Now, the required form has no term in Xii/z, hence the coeffi- cient of .^2?/ '. must be 0, that is, J.2 sin a sin a + 5- cos a cos a = 0. From ; in (X sin a' B^ , ■, ,i , . i this, = tan a: tan a' = ; whence we learn that the new axes cos a cos a' A^ must be so situated that the rectangle of the tangents of the angles which they make with the primitive axis of x shall be — — , in other words they must be conjugate diameters. Putting tho resulting equation in the form 1, and maldng Xs = 0, A - sin2 a' 4- B^ cos^ a' A~ sin- a 4- B- cos" a A-B^ ^' + AB^ and 2/3=0, successively, we find that the squares of the new semi-diameters are J.2J52 A^B^ and -- This equation and these A^ sin2 a + B^ cos^ cc" A^ sin2 a-\- B'^ cos^ a values refer to any pair of conjugate diameters, as wiU appear hereafter. 88 TRANSFORMATION OF CO-ORDINATES 2pm =0. Ex. 12. Find the position of oblique axes to which when y^ = 2px is referred, the equa- tion will still have the same form. Sug's.— Passing to oblique axes in general, the equa- tion becomes {n + a^asin a:-}-2/2sin ay = 2p{m -{-Xscos a + yzcosa'); or, expanding and collecting terms srQ2a:'2/2 ^ + 2sin a sin a'a;22/2 + siusa a;, 2 -|- 2n sin a' j2/2 + 2n sin , which is the form required. Putting . ., ^^ sin-^a' sm^a 2/2^ = 2P2X2. 2p ScH. — The equation 3/2^ = - = 2p.2, we have -X2 leaves the problem indeterminate, sin2 a' inasmuch as a' is a function of 3/2 ; hence the new origin may be anyivhere on the curve. In reality the problem furnished four arbitrary constants and required but three conditions (the third and fourth being but one) ; hence we may impose another ; that is, we may put the origin where we please on the curve. Ex. 13. To transform A^y^ — B^x"^ = — A^B^ so that the hyperbola shall be referred to its asymptotes, i. e., to the produced diagonals of the rectangle drawn on the axes of the curve. FROM ONE RECTILINEAR SET TO ANOTHER. Yi 89 K Stjg's. — Let AjXi and A1.Y1 be the primitive axes, and the asymptotes A,X2, AtY2 be the new axes. As usual, let X1A1X2 = — oc, and XjAiYj = a'. Then, since CB = BD = jB, AiB=J., Ai C = A I D = ^y A' 4- B\ sin a = — B A . . B cos a = , and cos a' ^= A sm a = :, the — Fig 94. y/A^ 4- B-^ \/a^ + B^ sign being given to the value of sin a. since a is reckoned around the angular point Ai from left to right, and sin a is the sine of a negative arc less than 90°. Putting these values in the formulce for passing A from rectangular to obUque axes, we have x = (x.^ +2/2) B , and y == (2/2 — ^2)- V A-^ 4- B^ s/ A^ + B^ Now, substituting these values of x and y in the equation to be transformed, there results (2/2 cc2)2^2-B2 — (ccj -f y^y-A^B^ whence expanding and reducing Xg^/a = A' -f ii^J = — A^B-^ ; Since 4- — is constant, 4 - 4 we may represent it by c, and write the equation tCj^/a = c. In the case of the A^ equilateral hyperbola c = —, A being the semi-axis. Ex. 14. To find a system of oblique axes with the origin at the centre, to which when the hyperbola is referred, its eqiiation will take the form xy = c. Sug's. — The common equation, A-y^ — B^x'^ = — A^B^, becomes, when we pass to general oblique axes with the same origin ^22/2 4- ^^ sin2 a X2^ = — A'^B-^. — B- cos- a B^ cos''^ a' = 0, and A'^ sin^ a — ^ B Now, in order that A^sin^a yz'^ -\- 2 A- sin a sin a — B- cos2 a' — 2B^ cos a cos a' The conditions imposed are. A- sin^ a' B^ cos- a = ; whence tan a = , , and tan a' = — 7—. A A these values should indicate the positions of different lines, they must be taken with different signs. Thus the new axes are found to make angles with the primi- tive axis of x whose tangents are — — , and — , which relation characterizes asymp- totes. The equation then reduces to (2 J.2 sin a sin a' — 2B^ cos a cos a')x2y2 = — A'^B'^. But the conditions above give sin cc = — _. cos a :, sm oc B ^A-2 + B^ , and cosa' \/a^ + B^ v/^2 + 52 \/^2 _|. Bi These values substituted in the last equa- J.2 _L. £2 tion, give, after reduction, ^2^2 = 1 » which is the same form as found before. 90 TRANSFORMATION OF CO-ORDINATES Ex. 15. Letting a;^ — 6xy-i-y^ — 6a: + 2?/ + 5 = repiesent a locus referred to rectangular axes, required the equation when the refer- ence is to a new set of rectangular axes with the origin at (0, — 1), and the new axis of x makes an angle of — 45°, or 135°, with the primitive axis of x. Svg's.— The fonnulce for transformation become, in this case, x = s/Hxz + 2/2), and y = \/iiy.2 — Xz) — 1. The transformed equation is y^ — 2x2 = 2 (See Fig. 88, and the illustration accompanying it.) ^ »» SECTION IL Methods of Passing from Eectilinear to Polar Co-ordinates, and vice versa. 12s, IProh, — To produce the formulce for passing from a Rect- angular to a Polar system of co-ordinates. Solution. — Let P be any point in a locus M N referred to the rectangular axes A,Xi, AiYj, the co-ordinates of P being Ai D =x, and PD = y, when referred to these axes. Let the pole of the new, or polar system, be A 2, whose co-ordinates are m and n ; and let A2X2, or AgXj', be the polar axis making an angle a with AX, or what is the same thing, with A2 K parallel to AiXj. Let the polar co-ordinates of P be A, P = r, and PA2X; PA2X2' = 6. The angle Fig. 95. PA2 K will be -j- a: when the polar axis lies above A2 K, and — a, when the polar axis Hes below ; hence, in general, PA2K = zb a. Now x = Ai D = A,B + A2H. But, from the triangle PAgH, A2H = rcos PA2 K = r cos {Q ± a). . ' . x^=m -\-r cos (0 zt a), (1). In like manner y = n + r sin (0 ± a), (2), as y = PO = AzB -}- PV^, and P H = r sin (0 ± a) . If the pole is at the primitive origin, m = 0, n = 0, and X = r cos (0 zh a), (1 1 ) ; and y=zrsin{6±a), (2i). If the polar axis is parallel to the primitive axis of x, a = 0, and the formulce become cc = m -J- rcos 0, (Ig) ; 2/ = n -|- r sin 0, (^2) ', or, if the pole is at the prim- itive origin, x = r cos 0, (1 3) ; 2/ = rsine, (23). FEOM RECTILINEAR TO POLAR CO-ORDINATES, AND VICE VERSA. 91 12'9» J[*Vob, — To produce the formulae for passing from a Polar to a Rectangular system of co-ordinates. V— ' Solution.— From Fig. 95 we have PA2 = ^PH + A2H, or r = s/ {y — rt)2 •\- {X — m)2. From the same triangle we have also cos {B ±a) = ^ — fn, -, . ^ . V — ■^ , . , , and sm (6 =h a) = — — 1 , which are the \/{y — n)2 -\- {X— my^ \/{y — n)^ + (^c — m)2 formulce sought. When the polar axis is parallel to the primitive axis of x, the formulce are X — m r = \/(2/ — ?i)2 -\- (X — m)2, COS0 = — — . and sin = V{y — n)"^ + (aJ — w)2 HI Yi If at the same time the origin and pole coincide, ih.e formulas \\y — ny-\-{x — 7)1)2 are r = \/y^ -f~ ^^j ^^^ ^ = — ^==z > and sin 6 s/yi -^ cty^ s/y2 _|_ x2 Ex. 1. Transform 572 -f?/2= 5a j: to polar co-ordinates, the pole being at the origin, and the polar axis coincident with the axis of x. The equation is r = 5a cos 0. SuG. — The formulce are x = r cos 0, and y = r sin 0. Ex. 2. Transform {x^ + y-)'^ = a'^{x^ — y"^) to polar co-ordinates, the pole being at the origin, and the polar axis coincident with the axis of X. The equation is r~ = a^^cos^ — sin^ 6) = a' cos 2d. Ex. 3. Transform r^ = a^cos20 to {x^ + y^y = a^{x^ — y^). (See last example.) SuG. — First put the equation in the form r^ = a^{cos^ — sin^ 0). Ex. 4. Under the same conditions as above transform r^ cos 26 = a'-* to 072 — 2/3 = aK Also xy = a^, to r^ sin 20 = 2a^. Also ^2 _{_ ^/a = (2a — a;) 2, to r^ cos-|(? ■= a^. Also reverse these processes. Ex. 5. To deduce from A^y^ + B^x^ = A-^B^, the polar equation of the ellipse, in terms of the transverse axis and eccentricity, the pole being at the left hand focus and the polar axis falhng on the axis of the curve. Sug's. — The given equation being put in the form y^ = (1 — e^){A^ — x^), and the formulce for transformation in the form x = r cos — Ae, and y = r sin 0, an^ the substitutions made, we have r2sin2 = (1 — e2)(^9 — r2cos2 -\- 2J.ercos0 — AH^). Expanding this and reducing to a known form, „ 2Jecos 0(1 — e2) A^l — e^y , r^ _ : — r =. ; hence 1 — e2 cos2 1 — 6200820 92 TRANSrOllMATlON Or CO-Or.DINATES. _ AecosBjl — e2) rh V A'^jl — e^)-^(l — e^cos-e> -j- A^e^cos-Q^l — e'-J)^ 1 — e'^ cos-' 6 ^e cos 6(1 — e2) d= v/ ^-^(l — e^)^ _ ^e cos 9(1 — e^) ± ^(1 — e^) _ 1 — e^ cos'-' 1 — e2cos'-^6 ~ — ^^ — -. Now as neither e nor cos 9 can exceed 1, and as each is 1 — e- cos-^ 6 generally less than 1, r is positive only for e cos 6 + 1 5 hence we may reject the — sign m this factor and write r = ; = ... ° 1 — e- cos-2 6 1 — e cos 6 ^(1 e2) If the pole is taken at the right hand focus, x = rcosQ + Ae, and r = --^, — . ^ l-j-ecos0 (See 107 -109.) [Note. — There are expedients by which the algebraic reductions in this solution may be simph- fied ; but as our purpose is to exhibit simply the process of transformation, we do not think best to avoid the work by indirect means. Were the object merely to obtain the polar equation, the process of {107 — 109) would be much more simple and elegant.] P Ex. 6. Deduce the polar equation of the parabola, r = -, ^ 1 — cos from 2/2 = 2px. (See 108.) Ex. 7. Transform the equation of the cissoid, y^ = , to the 2a(l — cos2 96 PROPEETIES OF PLANE LOCI. 13S, Cor.— ^ the axes are oblique, -^ signifies the ratio of the sine of the angle which the tangent makes with the axis of x, to the sine of the angle which it makes with the axis of j, i. e., sin a sm (/j — a)' (See 34.) 136, I^f'Op.—Tne general equation of a tangent to a?iy plane curve is f in which (x', y') is the point of tangency, and x and j are the current co-ordinates of the tangent. Dem.— Let MN, Fig. 98, be any plane curve whose equation is y =f(x), and let P be the point of tangency whose co-ordinates are x', y'. Now, the equation of any line passing through {^x , y') is y — y' = a{x — x') (32). But, in order that this line should be tangent to M N at P, the tangent of the YL^T angle PTD, which is represented by a in " the formula, must be -^. Hence, substitu- ax Fig. 98. ting, we have y — y' =■ ^{x X'). l. E. D. Ex. 1. "What is tlie equation of a tangent to an ellipse referred to its axes? Solution. — The equation of the locus is A^y- -\- B^x^ = A^B'^ ; whence y^ = — B-x B'^x' dv —r—, which satisfied for the point (x\ y') is -—. Substituting this value of — in the general equation of a tangent (130), we have y — y' = jB%' A^y' X — .T Keducing, this becomes A^yy' -{- B^x' = A^y' ^ + B^x'-^. But as (x, y') is a point in the locus A^'^ -{- B^x''^ = A^B'^ ; hence, finally, A-t/t/' -|- B'^hcx' = A^B'^. dx _ ^ dy B^x QuEEiEs. — Li — = — , dx A^y , what are x and y co-ordinates of? In. y — y' = :{x — x'), what are x and y co-ordinates of? "What x' y' ? Of what degree with respect to the variables is A-yy' -f- B-xx = A-B-? Why should it be of this degi-ee (5«5) ? What are the variables? Notice that fo7^ the same tangent, x and y have all values, but x' and y' have fixed values : x', y' are general, i. e. , they rep- resent any point in the locus, but they do not represent all points at the frame x'), we had chanced to find time. If in our deductions from y — v' = -r-.i'X' ^ dx A^^ -f- B-x'^ could we have substituted for it A^B% as we did for A-y''^ -f- B-x'- ? Why not? TANGENTS- KECTILINEAR CO-ORDINATES. 97 Ex. 2. Produce the equation of a tangent to dy^ -{-x^ = 5, at a; = 1, and construct, first, the tangent from its equation, and, second, the curve from its equation. Solution. -In this locus -^ = ax < — — . This is the general value 32/ of the tangent of the angle ■which a tangent to this ellipse makes with the axis of x. For the particular point x = 1 (for which y = ±z 1.155, nearly), we have -p, = =F .29 approximately. Fig. 99. Substituting these values in the general equation of a tangent {136), we have y =F 1.155 = zp .2d[x — 1), or y = =p .2d.x ± l.M. There are, therefore, two tangents to this locus at a; = 1 ; one whose equation is ?/ = — .2dx -f- 1.44, and another whose equation is y = .29x — 1.44. RS, in the figure, represents the former; and R'S', the latter. Another solution of this example is obtained by observing that the locus Sy- -|- 35^ == 5 is an ellipse whose semi-axes are A = s/d, and B = v/f. But the equa- tion of a tangent to an ellipse is A^py' -\- B'xx = A^B^- ; whence, substituting, we have hyy' -j- ^xx = ^3^, or Zyy' -{- xx = 5, as the equation of any tangent to this eUipse. For the points (1, ± 1.155) this becomes yz=^ .29a; dr 1.44, as before. Ex. 3. Deduce the equation of a tangent to an hyperbola. Also of a circle. Also of a parabola. r The equation of a tangent to an hyperbola is 1 " " " " " a circle is yy' + xx' = R\ ^ " " " " " a parabola is y?/' = p(^+ ^')- Besults. Ex. 4. "What is the equation of a tangent to the parabola y^ = dx at .r = 4 ? Construct the tangent from its equation, and then con- struct the parabola as in Ux. 2. For (4, 6) the equation is y = f ^ + 3. Is there another tangent for a; == 4 ? Ex. 5. Produce the equation of a tangent to 3y^ — 2x^ = 10, at ^ = 4. Is there more than one tangent ? Construct the figure as above. Equation, y = dz .1121 x ± .8909. Ex. 6. What is the equation of a tangent to t/^ == 4 — .r', at a: = 3 ? "Why is the result imaginary ? 98 PKOPEKTLES Uv PLANE LOCI. Ex. 7. What is the equation of a tangent to y^ = , at a: = 2 ? Ans., y == 2x — 2, and y == — 2:r + 2. Ex. 8. Show that the equation of the tangent to the Napierian logarithmic curve {x = log y) is y = y\x — x' -f 1). Observe that the ordinate to this curve at any point, is the natural tangent of the angle which the tangent to the curve at that point makes with the axis of abscissas. Ex. 9. What is the equation of a tangent to an hyperbola referred y< to its asymptotes {xy = m) ? Ans.^ V = ,^ + 2?/'. y' Interpretation of the equatioji y = -,x -j- 2^/'. If this represents the tangent y' to an hyperbola referred to rectangular asymptotes, r, is the tangent of the angle which the tangent to the curve makes with the axis of x ; and 2y is the dis- tance from the origin at which the tangent cuts the axis of y, as in all equations of right hues referred to rectangular axes. But in this case the hyperbola is equi- lateral, since xy = m is the equation of an equilateral hyperbola when the asymp- totes are rectangular ; or, in other words, no hyperbola but an equilateral one has rectangular asymptotes To interpret v' 2/ = — —x 4- 2^/' for oblique axes, we CI/ observe that 2y' is AO, Fig. 100; and by making ?/ = we find that the intercept on the axis of x, AT", is 2x' . Now the coefi&cient of x, — - , is the ratio of the X' sine of the angle which the line (tangent) makes with the axis of x to the sine of the angle which it makes with the axis of y, by {34:). This fact accords Fig. 100. with the relations observable from the , . ^-^ AO sin A TO y' sin A TO figure. Thus, m the tnangle AOT, ^T^ sinAOT ' °' ^ = sin AOT* The minus sign is explained by observing that the line R S lies across the 1st angle when P is in this angle, and to pass to this position from that in the funda- mental figure. Fig. 24, the angle NGX of that figure becomes STX of this, an angle whose sine is +, and equal to sin ST A. But, by this change of the position of the line, the angle G H A of Fig. 2-i, first diminishes to and then re-appears generated from left to right and henc-^ is a negative angle. Therefore -f- sin ATO y' sin A O T is negative, and we have : — . ^_, = ;. ° — sm AOT X Ex. 10. Produce the equation of a tangent to the locus y^ = 2x -]r Zx^, at ^ = 2. Result, There are t^o tangents, viz. : i/ = ± |j7 ± -J^ TANGENTS — BY BECTILINEAll CO-OEDINATES. 99 Ex. 11. At "wliat angle does the line y n= i.^; -f- 1 cut the curve r/2 = 4a7? Ans., 10° 14', and 33° 4'. Sug's. — Find the point of intersection and the tangent to the curve at this point. Find the angle included between this tangent and y ■= hx -\- Ihj (36). Ex. 12. At what angle does y- = lOo; intersect x^ -]- y^ := 144 ? Ans., 71° 0' 58". Ex. 13. At what angle does 25?/2 -f 16^2 = 1600 intersect IGif — 9x^ = _ 576 ? Ans, 61° 58' 37". 13 7 • JPfop, — The general value of the intercepts^ of the axes by a tangent are .dx' 'dy' ^ — y\j.,,^ and Y=y' — x'£^ ; in which X is the intercept on the axis of x, and Y that on the axis of y, (x'j y') being the point of tangency. Dem.— The equation of a tangent being y — y' = -r-:(x — a;'), if we find where (JLvu this line cuts the axes by making 2/ = for the intercept on the axis of x, and finding the value of x ; and cc = and finding the value of y for the intercept on the axis of y, we have the results sought. Thus for y = 0, and a: = X, we have — y' = dv ' dx' -7^(X-=- a;'), or X = x' — v'-r-. For x = 0, and y = Y. we have Y — y' = dx dy -^,(0 — X'), or F= y' — a'^. Q. e. d. dic ^ ' -^ dx' ScH. — In solving an example we may either apply these formulae, ; or, first get the equation of the tangent and then make x and y successively = 0. This is but an application of [26., 1st). Ex. 1. From A^y"^ + B-x"- = A^B"^, show that a tangent to an ellipse A^ B-i cuts the axes at X -== — -, and F = — ^ ; i. e., If from any point in an ellipse a tangent and an ordinate be drawn to either axis, half that axis is a mean proportional between the distances of the intersections from the centre. * This is an abbreviated form of expression for "the distances from the origin to where the curve cuts the axis." 100 Hi-. PEOPERTIES OF PLANE LOCI. Sug's.— In the figure, AT = X, AD =x', PD=y , AO = Y, AB=A and AG = jB. Hence having obtained X = — , we have but to put it into the form X : ^ :: ^ : a;', to ob- serve the truth of the proposition. Also 52 r = -7 gives Y : B y JB :y' 138, ScH.- A^ -Since ^ = — . we see that X Fig. 101. the intercept of the axis of x does not depend upon the conjugate axis of the ellipse, so that, if on the same transverse axis, different ellipses be drawn, the intercejjts on this axis, hy tangents corresponding to the same abscissa are equal. That is, if x' and A remain the same, AT is the same. From this "we have a ready method of drawing a tangent to an ellipse geometrically. Thus, let it be required to draw a tangent to the ellipse Fig. 101, at the point P. Draw a circle (a variety of ellipse) upon the same transverse axis. Draw the ordinate PD and produce it to P'. Draw a tangent to the circle at P'. This fixes the intercept AT. Draw a line through P and T and it is the tangent sought. [Note — The student should make himself perfectly familiar with this, and all methods given for drawing tangents to loci geometrically.] Ex. 2. Show that in the hyperbola the intercepts on the axes made A" B^ ... by a tangent are X = — -, and Y= j, and that the proposition in X y Ex. 1, is true also of the hyperbola. 139, ScH. — This principle also affords a method of drawing a tangent to an hy- perbola geometrically. From the given point of tangency P, let fall the ordinate PD ; and upon the transverse axis HB, and the abscissa AD, draw semi-circum- ferences. From their intersection let fall LT a perpendicular upon the axis of X. Draw a line through P and T and it is tangent to the curve at P. Proof. Drawing AL and LD, we have Fig. 102. AD (or x) : AL (or J) : : AL (or A) : AT. intercept made by a tangent at P. Whence AT == — and is the x' Ex. 3. Prove that, if a tangent be drav^n to a parabola at any point, the intercept on the axis of x is equal to the abscissa of the point of tangency. 8UBTANGENTS — BY BECTILINEAR CO-OEDINATES. 101 14:0 • ScH. — The principle developed iu the solu- tion of this example affords the most simple method of drawing a tangent to the parabola, geometrically. Let it be required to draw a tangent at P, Fig. 103. Draw the ordinate PD, take AT = AD, and through P and T draw a line. This will be the tangent required. Ex. 4. To find where the tangent to y^x = 4(2 — x) (the witch of Agnesi, the radius of the fixed circle being 1), at ^= 2, cuts the axes. Besults, It cuts the axis of ^ at a; = 2, and the axis of y Sit y i. e., is parallel to the latter. [Note. — Observe from the last example that a tangent may cut the curve.] Fig 103. 00, 141, Def. — Tlie Subfangent is the portion of the axis of abscissas intercepted between the foot of the ordinate from the point of tangency, and the intersection of the tangent with this axis ; or it may be defined as the projection of the corresponding portion of the tangent upon the axis of abscissas. In each of the three preceding figures DT is the subtangent corresponding to P as the point of tangency. 14:2, I^TOp, — The general value of a subtangent is dx' Suht = v'-T", ^ dy' in which (x', y') is the point of tangency. Dem.— In auy of the three preceding figures we have from the triangle PTD, DX = PD X cot PXD. But DT" is subt, PD = y', and, as tan PTD is ^y' . ^-^^ . dx' ,, dx' -/-,, cot PXD is -;— ,. .-. suht = 2/ -r— . Q. E. D. dx dy dy' Ex. 1. What is the value of the subtangent of y^ = 3a;2 — 12, at a7 = 4? SuG s. — For a; = 4, V = ± 6. —■ = —• . * . 2/= ± 6, r— = -tx- =dtzi; and ^ dx y dy 12 dx' Suhi. =^y' -^ = 3. The pupil should construct the figure, if he is not sure that he fully comprehends the example without. Ex. 2. Find the value of the subtangent of the common parabola. Of the logarithmic curve. Results, 2x\ and m or 1. 102 PROPERTIES OF PLANE LOCI. Ex. 3. What is the value of the subtangent of 3/2 = 2^' at a: = 2 ? 5 S' Ex. 4. If upon the same transverse axis different ellipses be drawn, prove that the corresponding subtangents are equal. Sug's. — The general value is SuU. ■■ A^B^ — B^x'-^ A^ — £C'2 ■ B'X' , , , a result which B^x X does not depend upon B. This truth is q illustrated in Fig. 104, DT" being the common subtangent for all the ellipses, corresponding to the same value of a:, AD. This is essentially the same truth as was brought to light in Ex. 1, Art. 137. Fig. 104. Ex. 5. Find the subtangent to the hyperbola referred to oblique asymptotes. Result, Subt = x'. ScH. — In Fig. 100, P being the point of tangency {x, y'), DT = Subt. = X = AD. Now since PD is parallel to AO, PT = OP ; i. e., The inter- cepts of a tangent to a hyperbola between the point of tangency and the asymptotes are equal. This affords a method of drawing a tangent when the asymptotes are given. Thus let it be required to draw a tangent at P, Fig. 100. Draw PE parallel k) AX, and take AT = 2PE. Through P and T draw a Une and it is the tangent required. 14:3, JPvop, — The general expression for the length of a tangent, i. e., for the portion intercepted between the point of tangency and the axis of X, is Tan = y'Ax + — , in which (x', y') is the point of tangency. dy' Dem. — In any one of Figs. 101, 102, 103, we have from the right angled triangle PDT, PT = PD + DT , or PT dx' PD + DT PD=2/',and DT = 2/' dy'' rr. I dX^ , I , dx"- Now PT is Tan., dx"- — -. Q. E. D. Ex. 1. What is the length of the tangent to 1/2 = 2a: at ;r = 8 ? Ans., 4:VV7, ASYMPTOTES — BY RECTILINEAR CO-ORDINATES. Ex. 2. Show that In the eUipse, Tan = ^^'^^pZI^ . 103 In the hyperbola, Tan = y Ap^ -f py^ -\- Ay^ Ap'^ + py^ In the parabola, Tan = -vp'^ + 2/" ; p representing the semi- P parameter in each case. 14:4:, Def. — A.n ALsyaiptote is a line toward which a curve constantly approaches, but under such a law that they will never meet ; or^ what is the same thing, that they will meet only at an infinite distance from the origin. An asymptote is also conceived as a tangent to a curve at an infinite distance from the origin, which yet passes within a finite distance, i. €., cuts one or both axes making finite intercepts. Iiiii. — It is quite common for persons encounter- ing this idea for the first time, to repudiate it as an absurdity ; but the following illustrations will familiarize it. Let the law of the curve be such that, if ordinates Bb, Cc, Dd, Ee, Ff, Gg, etc., be drawn at equal distances from each other, each succeeding ordinate shall be a the preceding. It is evident that the curve will continually approach AX but under such a law that it can never absolutely reach it. {Practically such a curve will Boon become indistinguishable from the line, that is, will run into it.) AX is an asymptote to this curve. In a similar manner two curves may approach each other under such a law that the distance between them shall constantly diminish, and yet the curves never meet. Such curves are asymptotes to each other. Our present purpose embraces only rectiUnear asymptotes. B C D E F G Fig. 105. H X 14S, Pvoh, — To determine whether a plane curve has rectilinear asymptotes. Solution. —First determine whether the curve has infinite branches. If it has not an infinite branch it cannot have an asymptote, since an asymptote is a tangent Second, if there is an infinite branch, dx at an infinite distance from the origin determine the values of the intercepts of the axes by a tangent, X 7/-r-, and 104 PROPERTIES or PLANE LOCI. Y =y — x^, for x ox y=. cc. It wiU. be necessary to observe wlietlier both of the variables, or only one of them, vary continuously to infinity, and get the value or the intercepts in terms of that one which does vary continuously. If now one or both of the intercepts thus evaluated is finite, the branch has an asymptote. If both intercepts are infinite, the curve has no asymptote, since the tangent at co does not pass within a finite distance of the origin. Having ascertained that the branch which is being examined has an asymptote, it remains to determine its position. If the intercepts are both finite and not 0, their values fix the position of the asymptote. If one intercept is finite and the other infinite, the asymptote is parallel to that axis on which the intercept is infi- nite. Finally, if the intercepts are 0, i. e., if the asymptote passes through the dy origin, its direction is determined by evaluating — for a tangent at infinity. ax Ex. 1. Examine y^ = 6x^ + ^^ ^or asymptotes. Solution. — Since as x increases from positively, y increases continuously and ■without hmit, is positive and has but one real value, there is an infinite branch extending in the first angle. Now when x is —, we have y^ = 6x- — x"^, which gives positive values to y till x"^ = &x'^. After x^ ^ 6x'\ that is after ic ^ 6 and negative, y becomes negative and a branch is found extending in the 3rd angle to infinity. Either of these branches may have an asymptote, they may both have the same line for an asymptote, or they may have different asymptotes. To deter- mine what the facts are we find the intercepts made by a tangent. 2/2 4a;-^ _|_ a;3 — 2/3 4x'^ -^ x^ — Qx^ — x^ —2x2 X = x y-. 4x -\-x^ = + CO = — 2 ; 2.t2 4:X -\- x^ and Y = y — x 4dX-\-x^ A.x-\-x^ 4.T -f- X- They are, which for - , which for x = -\-ccz=2. y2 The branch in the first angle has an asymptote which cuts the axis of 2/ at 2 above the origin, and the axis of x at 2 on the left of the origin. The equation of this line is ?/ r=r ic -f- 2. Finally, as the intercepts have the same values for ic = — oc as for x =^ -\- cc, this line is an asymptote to both branches of this locur,. The curve is sketched in Fig. 106, in which M N is the asymptote. Fig. 106. Ex. 2. Show that y^ =^ a^ — x^ has an asymptote which is common to its two infinite branches, passes through the origin, and makes an angle with the axis of x of 135°. Ex. 3. Examine t/2 = 2^ + 3^72 for asymptotes. \ Ex. 4. Why has y^ ■= x'^ — x^ no asymptote ? Ex. 5. Examine y^ = ax'^ for asymptotes. Ex. 6. Examine the conic sections for asymptotes. ASYMPTOTES — BY KECTILINEAR CO-OEDINATES. 105 SoiiUTioN. — An ellipse or a circle can not have an asymptote as neither has an infinite branch. It remains, therefore, to examine the parabola and hyperbola. From 7/2 = 2px, we have -^ z= - ; hence X = ic — ?/— = — ^ = — ^ which is — 00 for 2/ = 00. Again Y =i y — x-~- = y — • ", which = 00 for y = oo. . ' . The parabola has no asymptote. To examine the hyperbola, we have from A-y^- — B'^x^' = for X = dy B^x dx A^y'- A"- ^'^'' ^r, = -Ai;, ' ^^^^^ ^ = ^ - 2/w^. = ^ - -«l7 = :;' ' ^^^i«^ dx A^y ±1 GO. Also T dy dy B^x'^ y-^^y B'-x B^ y which = for 2/ = rb cc, (In this curve both x and y vary continuously to infinity, hence the intercepts may be evaluated in terms of either.) .-. The hyperbola has two asymptotes, and they both pass through the centre. To determine the direction of the asymptotes we evaluate -^ B'^x A^y Bx fl; = 4- CO, Bx B ^ --=.-. For X A AVx^ Bx for a; = -+- A^ For B =r = -.. Whence we Fig. 107. AV'x-^ — A^ ^ As/x'^ learn that the asymptotes are the produced diagonals of the rectangle described upon the axes, as has been stated before. IdO, ScH. 1. — If, at successive points along an infinite branch of a curve, we draw tangents, these tangents wdll either approach some limiting position, or they will not. In the hyperbola. Fig, 107, it is evident that the successive tangents PX, P'X', FT' are approaching the limiting position SA. But in the par- abola, Fig. 108, it is equally evident, from the way in which the tangents are drawn, that there is no limiting position beyond which a tangent may not pass. Since AT = AD, AT' = AD', AT' r=: AD", the point of intersection wdth the axis recedes indefinitely as the point of tangency passes to the right. In a similar manner observing the method of drawing a tangent to an hyperbola, Fig. 102, it will appear that as P recedes, the inter- section L constantly (but more and more slowly) api^roaehes E but can never pass it ; and, consequently, that T can never pass A, however far P may recede. From these considerations, an asymp- tote is seen to be tJie limiting position toward lohich a tangent approaches as the point of tangency recedes to infinity. 14:7 • ScH. 2. — Having found the intercept on the axis of ordinates and the tangent of the angle which the asymptote makes with the axis of abscissas, we can readily write the equation of the asymptote by substitu- Fig. 108. 106 PROPERTIES OF PLANE LOCI. ting my = ax + b. Thus the equations of the asymptotes to the hyperbola B B are y — —x, and y = -x; or Ay = Bx, and Ay = — Bx. A a Ex. 7. Show that y = — x-\-- is the equation of the asymptote o to ?/3 = ax^ — x\ Ex. 8. Show that x = 2a, and y == ^ (^ + a) are asymptotes to y^-{x — 2a) = x^ — a\ Ex. 9. Show that the axis of abscissas {y = Ox) is an asymptote to y(a'> + ^2) = a^[a — x). Ex. 10. Show that ^ = 2a is the asymptote to the cissoid of Diodes. Ex. 11. Examine x = log y for asymptotes. Ex. 12. Examine y = tan x for asymptotes. - y- "^ _'vv Sug's. — As this curve is not continuous in the direction x = oo, vre evaluate the intercepts for ?/= oo, for which x = iTt, f^r, j7t, etc., and — ^Tt, — ^Tt, — |;f, etc., -^ = sec2£C = 1 -f- 2/2. .'. X = x — -— y — -. which for y= cc,=x = Itt, f ;r, f;r, dx 1 + y- etc, and —Iti:, — Itt, —^7t, etc. Y = y — x{l + y% which for y = oo, = oo. Hence there are an infinite number of asymptotes parallel to the axis of y. (See 23, M. 27, Sen., Fig. IS.) Ex. 13. Examine y==-. a3 - -\- c for asymptofe?',. {x — hy Sug's. — In examining this locus it is necessary to evaluate the intercepts both for ic = 00, {1 nd i/ = oo, as there are infinite branches running in both directions. The general form of the locus is given in the figure. The equations of the asymptotes are a- = h (the hne M N), and y = c (the line M'N'). 14:8, ScH. 3. — In very many cases there are more expeditious methods than the above for finding asymptotes. It is frequently the case that a simi)le inspec- tion of the equation of the curve will determine the fact. Thus, in the last example, ifc is evident that as x increases from to &, y increases, and when x = h, y becomes oo. .*. This branch of the curve approaches the line MN, parallel to the axis of y, and at a distance h from it, under the law required for an asymptote. So again when X passes .-r = ft, it is evident that y grows less, and the curve approaches the axis of x. But, as, when .r= ck, y = c, this branch extending to the \ TANGENTS TO I'OLAR OUEVES. 107 riglit can never come nearer the axis of x than y =■ c. In like manner when x=^ — 00, we see that y = c . • . M N ' is an asymptote. Ex. 14. Determine by inspection the asymptotes to xy = m. Ex. 15. Determine by inspection that x=^a, and y==b, are asymp- totes to xy — ay — hx =^ 0. bx . ay Stjg, — Observe that w == , and x = t. •^ X — a y — b 14:9, ScH. 4. — An elegant method of examining for asymptotes consists in expanding y =/{x), or x =/{y), into a series, by the binomial theorem, Maclaurin's formula, or some other method, when such development is practicable. This will be best illustrated by an example or two. Ex. 16. Determine the asymptotes of the locus x^ — xy^ + ay^ = 0, by developing y =f(x). a Now, if we take the first two terms of this development we have y = dz x zh -x, the equations of two straight lines. Comparing this value of ywith the entire value, which is the ordinate of the curve, we see that as x increases the terms fol- lowing — grow less and less, and consequently that the ordinate of the right line and the corresponding ordinate of the curve become more and more nearly equal ; that is, the curve is constantly approaching the lines ?/ = db ."^ ± jr. Now when a; = 00 this difference vanishes, as all the terms following — -, become 0. . • . The lines y = dzXzh ha are such that the given curve approaches them constantly but reaches them only at an infinite distance, and are therefore asymptotes. There is also an asymptote at cc = a, which can be discovered by inspection, as under the last scholium. Ex. 17. Show by developing y =^f{x), that y = =b ^ are asymp- totes to ?/- = X- — . Ex. 18. Sbow by developing y = /{x), that y = dz (x -{- a) are X'^ _L cix" asymptotes to y" = . Sua. — The value of y developed becomes y=± x{l -j \ \-, etc.). (5 ) TANGENTS TO POLAE CURVES. 150. It is found most convenient to determine tangents to polar curves by means of the subtangents. 151. Def. — Tlie Suhtangent to a ^olar Curve is the 108 PROPERTIES OF PLAKE LOCI. distance from the pole to the , tangent, measured on a perpendicular to the radius vector to the point of tangency. Thus in the figure let M N be a curve referred to P as its pole, S any point in the curve, and RT tangent at S- Then PT drawn through the pole, perpendicular to PS and hmited by the tan- gent, is the subtangent. 152. J^rop, — The general value of the subtangent to a polar curve is Subt. = r'^dd ^ Ir ' m which r is the radius vector and the variable angle. Fig. 110. Pem. In tlie last figure let R be a point on the curve consecutive with S (infi- nitely near it), so that the tangent RT is to be considered as coinciding with the curve between R and S. Draw PR, and with radius vector PS as a radius draw arc SQ, and also with radius Pb = 1, draw ah an arc of the measuring circle. Then RQ =dr, since RQ is an infinitesimal increment of the radius vector, contemporaneous with RS. So also RPS, or ab = dO. As SQ is infinitesimal it may be considered a right line, and it is perpendicular to PR. Again, as R approaches S, the triangle RQS approaches similarity with SPT ; and as it is the relation at the limit that we seek, we are to treat RQS as similar to SPT. Hence we have PT : PS : : QS : QR, or subt. : r : : QS : dr. But from the similar sectors QPS and aPb. we have QS = rdQ, and substitu- r~dQ ting, subt. :r::rdB :dr. . • . subt = -^. Q- e. d. Ex. 1. Find the value of the subtangent to the spiral of Archim- edes. Solution. — The equation is r = ^7- ; whence — = 27r. dr 27r = 7-7 X 27r = — . Subt = ^^- =r' X dr iLii. — The annexed figure furnishes an il- lustration. PT is subtangent for point S and = the square of the numerical value of 6 divided by the circumference of the circle whose radius is PB. The numerical value of in this instance 2|7r, since in passing from 9 = to S the radius vector makes 1§ revolutions. But one revolution = 2;r. Fig. 111. ASYMPTOTES TO POLAR CURVES. 109 Hence PT = 27t, or somewhat more than I4 times the circumference (247r)?_81 of the measuring circle. If B is the point of tangency, =t 27t, and subt PT' = 27t, or the circumference of the measuring circle. In this spiral the subtangent varies as the square of the measuring arc. Ex. 2. Prove that in the hyperbohc, or reciprocal spiral the sub- tangent is constant. "What is itsValue, and what the significance of its sign ? Construct the curve and tangents at a few points, and observe the subtangents. Ex. 3. Find the subtangent to the logarithmic spiral (r= a ), and show that the angle under which the curve meets the radius vector is constant. Sttg. — The tangent of the angle made by a tangent to the curve and the radius vector is equal to the subtangent divided by the radius vector. In this locus the tangent of the required angle is the modulus of the system of logarithms used in constructing the spiral. Ex. 4. Prove that -- -r3 a-* sin 20 lemniscate of Bernouilli. is the value of the subtangent to the 153, I^Tob, — To test polar curves for rectilinear asymptotes. Solution. — Any curve which continually revolves around the pole can have no rectiUnear asymptote ; for with respect to any fixed right line, such a curve will alternately approach and recede. But if for some finite value of 0, r becomes infinite, the curve ceases to revolve around the pole, and will have an asymptote if the tangent at r = co passes within a finite distance of the pole ; i. e., if the subtangent is finite, q. e. d. To construct the asymptote, we observe that the asymptote and radius vector are drawn from a point infinitely distant, to the extremities of a finite subtangent, and hence are to be considered parallel. We therefore determine the value of the subtangent for r = 00, and drawing the radius vector for that value of 6 which renders r = co, erect the subtangent, and through the extremity remote from the pole draw a line parallel to this radius vector. This line wiU be the asymptote. Ex. 1. Test the hyperbola for asymptotes by the polar method. Solution. — The polar equation, when the ^(1 — e2) pole is at F, is r = e cos Q — 1' Now for cos Q=-,r= 00 ; hence if there be an asymp- FR so drawn that From the equa- tote, it is parallel to cos R FX = - = — . d9 tion we have -7- = C4/- (e cos — Ae{l — e^) sm — ; whence Fig. 112. 110 PEOPERTEES OF PLANE LOCI. dQ ^2(1— e2)2 (e cos — 1)2 Ail — e^) , . , . - suht. = r-— = ^-7 X -rr—. ^— — t: = ^— 7r-> which, since for dr (e cos — 1)2 ^ Ae{l — e^) sm esm ' cos = -, sin = - s/e^ — 1, = — A\/e- — 1 = — B. There is therefore an asymp- e e tote. To construct it draw FR making RFX = cos-M-V FD perpendicular to R F and = B, and through D draw SX parallel to R F. ST is the asymp- tote. Moreover, since cos(— 0) = cos0. there is another asymptote similarly situated below the polar axis FX. Finally, as the angle which a diagonal upon the axes of an hyperbola makes with the axis of x is cos— * — . the asymp- VA^ -\- J?2 totes are parallel to these diagonals ; and since F D = ^, A F = \/A- -j- B^, and the asymptotes coincide with the diagonals. Ex. 2. Show by the polar method that the parabola has no asymp- tote. SuG. — In this case for = 0, r = oo ; but for this value of 9 suht = oo. Hence there is no asymptote. Ex. 3. Show that the hyperbolic spiral has an asymptote parallel to the polar axis and at a distance 27t from it. Ex. 4. Show that the polar axis is an asymptote to the lituus. -4-»">- SECTION 11. Normals to Plane Loci. (a) BY KECTANGULAE CO-OKDINATES. '154z» Def. — A Novinal to a plane curve is a perpendicular to a tangent at the point of tangency. 1^5. I^TOp, — The general equation of a normal to a plane curve is dx' y — y= — j-,{^ — ^')> in which (x', y') is the point in the curve to which the normal is drawn, and X and y are the general co-ordinates of the normal. Dem. — Letting [x , y') represent the point in the curve to which the normal is dy' , to be drawn, the equation of a tangent through this point is y — y' = -f-:i{^ — ^ )• Again the equation of any line passing through (a;', y') is y — t/' = a(x — x). Now, in order that the equation for this general line should become the equation NORMALS — BY RECTANGULAK CO-ORDINATES. Ill of a perpendicular to the tangent, a must = is the equation of a normal, q. e. d. dx,' dy' y — y dx dy {X— X) 156. Cor. — The general expression for the tangent of an angle vMch dx a normal makes with the axis of abscissas is — —- , (x, y) being the point .f ^y ~ in the curve to which the normal is drawn. Ex. 1. Produce the equations of the normals to the conic sections. Results, Ellipse, y — y' A^y' y ^l^(^ — ^0 ; Circle, 2/ = ^,^ ; A^y' y' Hyperbola, y~y'=—^/-,{x—x');'Parahola,y—y'=—^-(x—x'). ^ X p ScH. — Observe that these equations do not reduce to as simple and sym- metrical forms as do those of the tangents to the conic sections. The form of the equation of the normal to the circle shows that the normal of this locus always passes through the centre. It is, of course, the radius. Ex. 2. What is the equation of the normal to y^ = 2x^ x^ at x = l? Answer : At x=l, y=dr.l ; hence there are two points indicated. The equation of the normal at the former is y = 2x — 3, and at the latter y = — 2a; -f 3. Ex. 3. What is the equation of a normal to t/^ = 6.r • — 5 at 2/ = 5 ? What angle does this normal make with the axis of ^ ? Ex. 4. At what point in the ellipse whose axes are 12 and 8 must a normal be drawn to make an angle of 45° with the axis of ^? Ex. 5. At what point in the witch of Agnesi must a normal be drawn to be perpendicular to the axis of a; ? To be parallel ? To make an angle of 135° ? 157. 'Def. — The Subnormal is the. projection of the normal upon the axis of x ; or it is the distance from the foot of the ordinate let fall from the point in the curve to which the normal is drawn, to the intersection of the normal with the axis of x. 158. J^Toh. — To find the general value of the subnormal. Solution. — In Fig. 113 P E is the normal and D E the subnormal for the point P. Now in the triangle P D E, P D= y, and tan P E D = dx (numerically) tan PEX = by trigonometry. dy - °-=4 Fio. 113. 112 PEOPERTIES OF PLANE LOCI. TA D Fig. 114. Ex. 1. Show that the subnormal to the parabola is constant and equal to half the latus rectum. How can a tangent be drawn to the parabola, geometrically, upon this principle ? \ Ex. 2. Show that the subnormal to the cycloid is {2ry — y^) . 159. Con.— Since DC = PG =-v/CG X GS = \/y(2r — y)" = V 2ry — y2, tlie normal passes through the foot of the vertical diam- eter of the generating circle for the point to which the normal is drawn. Moreover^ since SPG is a right angle, the tangent passes through the other extremity of the vertical diameter. 160. ScH, 1. — This principle affords a ready method of constructing a tangent to the cycloid geometrically. Let P be the given point through which a tangent is to be drawn. Put the generating circle in position for this point {94), and draw the vertical diameter SC Through S and P draw a right line and it will be the required tangent. Also PC wiU be the normal to the curve at the point P. 161. ScH. 2. — To draw a tangent which shall make any given angle with the axis of x, draw the generating chcle on the axis HI, construct the angle LHI = the complement of the required angle, and through L, the point where this line intersects the circumference of the central generating circle, draw a parallel to the base of the cycloid. "Where this parallel cuts the curve P is the required point of tangency. Tlu'ough this point draw SPX parallel to HL, and it is the tangent required. 162, J^roh. — To find the length of the normal, i. e., the portion intercepted between the curve and the axis of x. Solution. — In Fuj. 113, from the right angled triangle, PDE we have PE = V */• PD-4-DE- = j2/> + 2,=-=2, 1+-. Q.E.D. 4 dif~ Ex. 1. Find the length of the normal in each of the conic sections. What is it in the circle ? K Ex. 2. In the cycloid the radius of whose generatrix is 2, what is the length of the normal at ?y = 1 ? Ans., 2. TGS, Cor. — The normal is hut a particular case of a perpendicular to a tangent. NORMALS TO POLAR CURVES. 113 Dem. — ^As y — y' = -r^{^ — ^') is the equation of a tangent at (x', y'), a perpen- dx dx' , dicular to this through the point {x", y") is y — y" = j-,ix — x"). Making cly the point through which this perpendicular to the tangent is to pass the point of tangency, the perpendicular becomes a normal, and its equation is 2/ — y' == dx dy -{X — x'), since in this case x"= x , and y" = y . For a perpendicular from dx' the origin on the tangent we have 2/" = 0, a;" = 0, and y = —x. Ex. 1. Show that the equation of a perpendicular from the focus V of the common parabola upon the tangent is ?/ = {^x — ^jy). P Ex. 2. Show that the perpendicular distance from the focus of an / hyperbola to the asymptote is B. 104:, CoR. — The perpendicular from the focus of a parabola upon a tangent meets the tangent in a tangent to the curve at the vertex (the axis oi y). Dem. — The equation of a tangent is yy' = p{x -\- x), and of the perpendicular v' from the focus upon this tangent 2/ = — —{x — Ip). "We have now but to find the intersection of these hues. Equating the values of y, we have — — (x — ip) = —{x -f- ic'), or — y'^{x — ip) = p-jc -j- p^^\ 01^ since y'^ = 2px', — 2pxx' -f- p^^' == p^o; -f-p2/);' ; whence {p-\-2x')x = 0. Now as x' can not be — ,p-^2x' can not become =0. Therefore to fulfill the condition {p -^2x')x = 0, x must =s 0. .• . The point of intersection is at ic = 0, or in the axis of y. Q. e. d. (b) NOEMALS TO POLAR CURVES. 16S, Def. — The Subnormal to a polar curve is the distance from the pole to the normal, measured on a line perpendicular to the radius vector to the point in the curve to which the normal is drawn. Thus E P is the subnormal of the curve M N corres- ponding to R, the pole being at P. 100, I^TOh, — To find the general value of the sub- normal to a polar curve. Solution. PT = r But tanPER-*^= PR ^d6 dr PR Tan PTR = -^ = -— = PT ifZ. r ~de dr rdff dr tan PER dr_ tan PTR Q. E. D. rdB dr' = -r-. .'. P E or subnormal = Fig. 115. 114 PKOPERTIES OF PLANE LOCI. Ex. 1. Show that is the value of the polar subnormal r of the lemniscate of Bernouilli. T Ex. 2. Show that the subnorraal to the logarithmic spiral is — , 7n being the modulus of the system ; and, consequently'', that in the Napierian logarithmic spiral the subnormal always equals the radius vector. (dr^ \i — + r^j . IGS, CoR. 2. — The length of a perpendicular from the pole upon a r2 tangent is p = /dr Vd^ Dem. — In FiQ. 115, let PS be the perpendicular from the pole upon the tangent, and consequently parallel to the normal R E. From the right angled triangle PST, PS = PT X cosSPT. But PT (the subtangent) = -j^ \ and cos SPX = cos RET = ftf^^^ = 1 = „ .c^^ , - sec RET ^ , X .„^_^xa /^ , r^dB^\k (l + tan-3RET)^ {} + -^) rHQ Whence p = — — : = — -. q. e. d. 0+'1?)' C^.+-)* ■#♦ » SUCTION' IIL Direction of Curvature. {a) BY KECTANGULAB CO-OEDINATES. d^Y ISO* JPvop.—At a point where -^ is positive, a curve is concave d^Y . , upward, and where j-^ is negative the curve is convex upward. Dem. — 1st. Let go be the angle which a tangent makes with the axis of x. When the curve is concave upward, as in B.g. 116, it is evident that as x increases (as from being the abscissa of P to being that of P' ), oa increases. In other words, if x takes the infinitesimal increment dx, the contemporaneous infinitesimal change in ca is DIRECTION OF CURVATURE. 115 -f- dao. Hence when the curve is con- cave upward, dx and doo have the same sign (a; and go are increasing functions of each other). In a similar manner it is evident that when the curve is convex upward, go decreases as x increases ; i. e.,ii x takes the increment dx, the cortemporaneous change in co is — dca. dy dx d ' tan go 2nd. P, dx^ doo d dx dx Bec2 GO— : Now, as sec" go is always (J/*C positive, -T^ is positive when x and go are increasing functions of each other (when dx and dGo have like signs), and negative when they are decreasing func- tions of each other. -f -r- mdi- dx^ cates that the curve is concave upward, and upward, q. e. d. Another Demonstration. — Let DD', and D'D", Fig. 118, represent consecutive equal infinitesimal increments of x, then P'E and P E' represent con- temporaneous infinitesimal increments of y. Eepre- sent them respectively by dyi and J?/2. The differ- ence between di/^ and dy2, is by definition d^y. Bat when a curve is concave upward it lies above its tangent. Hence dy2 >> dy-, and dyz — %i = -\- d-y. On the other hand, when the curve is con- vex upward, as in Fig. 119, it lies below its tangent and dy2 > «\/2. Ex. 2. Sliow r= a is always concave towards the pole. SUCTION IV, Singular Points, 17 S. "Def.— Singular Points of curves are points which possess some property not common to others. Of such points we shall notice : 1st, Points of maxima and minima ordinates ; 2nd, Points of inflexion ; 3rd, Multiple points ; 4th, Cusps ; 5th, Isolated or Conjugate points ; 6th, Stop points ; 7th, Shooting points. 118 PROPERTIES OF PLANE LOCI. MAlTTIffA AND MEVIMA ORDINATES. 17 6 • Def. — An ordinate is at a maximum when it is greater than the immediately preceding and the immediately succeeding values ; and at a minimum when it is less than the immediately preceding and immediately succeeding values. 17 7 • IPvoht — To find the position and values of maxima and min- ima ordinates. Solution. — As y=f{x), this problem is the ordinary one of maxima and minima of functions of a single variable, treated in the Calculus. Hence we find the values of x •which render — = or 00, as critical values, i. e., values to be examined, and at dx whicb the property exists, if it exist at all. To distingmsh between maxima and minima values we have the common test ; namely, -|- -y-^ characterizes a mini- mum, and ~ characterizes a maximum, subject to the conditions discussed m dx,' the Calculus. The value or values of y corresponding to the value or values of x found as above, will be the required maxima or minima ordinates. A Geometrical Solution. — If PD is a maximum, it is evident that at the left of P the tangent makes an acute angle with the axis of x, i. e. dv dv -- is 4- , and at the right -- is — . . • . dx, dx dy -- = is the point of change from dx -f- to — , or the point of maximum ordinate. In hke manner at the left of a point of minimum ordinate, as P', _FU__ A D D» >? dx is — , and at the right +. -- = locates also minimum ordinates.* dx Finally, since at a point of maximum ordinate the immediately preceding and suc- dv ceeding ordinates are less, the curve is concave downward, whence we have — -^ characterizing such a point. But, at a point of minimum ordinate, the immedi- ately preceding and succeeding values of y being greater, the curve is convex d^y downward and we have -|- — characterizing this point. 178, ScH. — If only the numerical values of the ordinates be considered, d^v . . . . d^7/ when P lies below the axis, will characterize a minimum, and + -; — » maximum. But a numerical maximum, if — , is properly considered a minimum ; and a negative numerical minimum, is properly a maximum. dy * ^or ^ = CD , 606 Calculus, p. 9t Ex. 1. Examine y ordinates. SINGULAK POINTS. — POINTS OF INFLEXION. 113 x^ — 9^2 _^ 24^ + 16 for maxima and minima Solution. dx 3a;2 — 18a; + 24 = 0. x = 4. and 2, — ^ dx^ ex — 18. For d^y X = 4:, — = 6 ; hence x = 4: corresponds to a minimum, which value is 32. For d^y X = 2, -— = — 6 ; hence x = 2 corresponds to a maximum, which value is 36. [Note. — The student should construct the locus, and notice the points. Also substitute values for X a little greater than 4: and a little less, and the same for the point x =2, observing in the results the maxima and minima values of y. ] Ex. 2. Find the location and value of maxima and minima ordi- nates in the following curves : [1), y = x'^ — 5x* + 5x^ + 1 ; {2), y = x^ — Sx^ — 24.x + 85 ; (3), y = 5{x — x^) ; (4), y = {2ax — x^)^ ; {5),y = x^ — 8x^ + 22x^~2^x-{-12',{6),y = b-\-{x — ay; (l),y = x'^{a — xy \ (8), in the logarithmic curve ; (9), in the curve of tan- gents ; (10), in the cycloid ; (11), in the parabola ; (12), in the lem- niscate of Bernouilli. POINTS OF INFLEXION, (a) BY KECTANGULAB, CO-OBDINATES. 170* I>EF. — A JPoint of Jftflexiou is a point where a curve changes direction of curvature for continuously increasing values of X or y. Such a point is also characterized by the fact that the tan- gent at the point cuts the curve in the point of tangency. Iiiii. — In passing from P' to P", the curve M N changes direction of curvature, being convex downward at P', and upward at P". The point P at which this change occurs is a point of inflexion. The student should not confound a point of inflexion with such a point as P in M ' N '. It is true that reckon- ing along the curve from M ' to N ' the curve changes direction of curvature with reference to the axis of x ; but not so in reckoning along AX. From D' to D the curve is both concave and convex towards the axis, and does not change at P, but is limited there. ISO, I*TOh, — To determine points of iriflexion. Solution. — If examined with respect to the axis of x, since, when the curve is d'^y d^y convex downward we have A — -^, and when concave downward -, at the rZx2 cZxs Fig. 122. 120 PEOPEETIES OF PLANE LOCI. point of inflexion — must change sign, and hence must = 0, or oo, .-.If there be a point of inflexion it is where ^ = or oo. Having determined this point, either construct the curve in the neighborhood of it, or, better, substitute in ^ a value of X a httle greater, and one a little less than the critical value, and observe '^^®^^^'' ^ '■^''^^^ ^""^^ ""^^"^^^ ^^g^ ^^ ^^e point under consideration. The precaution in the latter part of this solu- tion is necessary ; for, though a varying quan- tity cannot change sign without passing through or oo, it does not necessarily change sign upon passing through these values. Thus, let M N be a curve whose equation is y ^=f{x). Now, as X passes from the value A D to that of AD', ?/ passes through 0, hut does not change its sign. In like manner by referring to Fig. 109, it will be seen that in the curve there delineated, y passes through oo without changing its sign. Ex. 1. Examine y = 6+ (^ — ciY for points of inflexion. Solution. -— = Q[x — a) = 0, gives x^a, as a critical point, i. e., one which d^p dy may have the property sought. Now for x'y> a, ^ is + : and for o! << a, ^ dx;^ dx- is — . Therefore there is a point of inflexion at x=^a. For x = a, y ^^h ; hence the point of inflexion is (a, 6). Ex. 2. Examine the following for points of inflexion : a'^y=,x'i — cjc^ : y = x-\-SGx^ — 2x^ — £c-i ; y = since; y=taiix; a; = logy; the witch of Agnesi. (b) BY POLAE CO-OEDINATES. ISl, IPvoh, — To test polar curves for points of infiexion. Solution. — The equation being put into the formp =f{r), we have seen that dp dp for — -[-, the curve is concave towards the pole, and for ~ — , it is convex. dr dp dp Therefore — = 0, or oo, indicates a critical point. If upon examination -=- is dr dr found to change sign at this point, the point is one of inflexion, q. e. d Ex. 1. Test the lemniscate of Bernouilli for points of infiexion. Solution. — The equation is r'^ = a'^ cos 2d ; whence we have p ±a^ and SINGULAR POINTS — MULTIPLE POINTS. 121 ± a 2 ' Putting = 0, r = 0. If, therefore, there is a point of inflexion dr it is at r = 0, that is, at the pole. Finally, ^ = ^-^ da'^ cos 26 = ± 3 cos 26, which changes sign for consecutive, real values of r ; i. e., when 6 passes from 45° to 135° for which change r passes through 0. Ex 2. Examine the lituus (r = ~) for points of inflection. There is a point of inflection at r = aV^> ^ = ^8° 38' + . /7/93 Ex. 3. Examine r--^ r for points of inflexion. (7^ i Solution. -^7- =: P -, and f- k dr (4^4 _ I2ar3 -\- 13aV2 — 4.a^r) ' (6r^_13ar + 6a^)(-Q^V)^ _^ Whence r = 0, fa, and |a. If, therefore, (4,^4 _ I2ar3 j^ ISaY^ — 4:a^rV^ there is a point of inflexion, it must be where r passes through 0, |a, or |a. But / ^ changes sign only with the factor 6r2 — 13ar -f- Ga^ ; and this factor does not dr change sign when r passes through 0, but does at r = |a and fa. (To determine these facts, substitute r = Q-^h, and r = — 7i ; also r = fa + ft,, and r = fa — ft, etc., /t being treated as infinitesimal.) . * . There is a point of inflection at r = fa. Where r = |«, = V3, or about 99°.26. Where r = |a, is imaginary. MULTIPLE POINTS. 182, Def. — There are two species of Multiple I^oints^ viz., 1st, A point where two or more branches of a curve intersect ; 2nd, A point where two or more branches are tangent to each other. The latter are some- times called Points of Osculation. The an- nexed figures illustrate both species. The first curve has a triple point of the first species at P, and the second a double point of the second species at P. Fig. 124. 183. JProb. muUiple points. ■To examine a curve for Fig. 125. Solution. — Since two or more branches pass through a multiple point, for x = the abscissa of such a point, y has but one value, while at other points near it, y has two or more values for each value of x. In explicit functions, or in functions 122 PEOPEKTIES OF PLANE LOCI. of a comparatively simple form, such a point can generally be determined by inspection. Having found a value of x for wliich y lias but one value, and on both sides of which it has two or more, form -^, and observe whether it has equal or dx dv unequal values at this point. If -^ has unequal values the branches of the curve intersect at the point, since their tangents do, and the point is of the first species. If — has but one value for these values of a; and y, the tangents to the branches at dx the point coincide and the point is of the second species. When the critical points are not readily determined by inspection, put the equa- tion in the form of an imphcit function without radicals. Let it be u =f{x, y) = 0. du Form — = — — . Now, as the equation of the locus did not contain radicals, dx du dy dv and as differentiation does not introduce them, the only way in which -- can have du __ . dy dx du several values is by taking the form -. Hence we have -5- = — 7" =^ n' ^^ dT^^ dy and — = 0, from which to determine critical values of x and y, (that is, those dy du values which may correspond to multiple points). Solving the equations — = 0, and — = 0, for x and y, see which of the values found satisfy the equation of the dy locus. K, at any point thus determined, y has but one real value for the particular value of X, and on both sides of it y has two or more real values, this point is a multiple point. Its species can be determined, as before, by evaluating -^- = -, for the particular values of x and y which locate the point. Ex. 1. Test for multiple points y = {x — a) vx + b. SoiiUTiON. — Since \/x is both -{- and — , y has in general two values. But it is evident that for x = 0,y has but one value, namely, 6 ; also for x = a, y has but one value, 6. These are the critical values of x and y. Upon the point (0, h), we observe that the branches do not pass through it ; since for x negative y is imaginarj^ Hence (0, b) is not a multiple point. But upon the point (a, b) we observe that y has two real values on each side of it. This is therefore a -■ , n . i. XT ^?/ Sx — a ,., f Fig. 126. double pomt. Now -- = ± —, which for ^^^- -^^"• (^ 2k/x x = a gives — =: ± \/a. .• . The point is of the first species, and the tangents dx SINGULAR POINTS. 123 to the curve at the point make angles with the axis of x whose tangents are -f- \/o„ and — s/a. The form of the curve is given in the figure. Ex. 2. Examine y^ == x^ — x'^ for multiple points. Ex. 3. Examine x'^ -{- 2ax^y — ay^=0 for multiple points. Solution. — As it is not easy to discover by inspection all the points to be examined du dv dx m this case, we will proceed by the second method. We find -- = ~ -^ = — dx du dy 4-05^ ■ I - 4^0,^0/ - — — ^^. Whence ix^ -j- ^(^^V = , and 2ax^ — 3ay^ = 0. These equations give the following critical values \ \ ^ ; 1 ^ ^ \ and i f (j/ = 0'(2/ = — la \y =z — la. But of these only the first set satisfy the equation of the curve. The point (0, 0) is, therefore, to be examined. Since hone but even powers of x are involved, a change in its sign does not change the form of the function ; hence the form of curve is the same on both sides of the axis of y. As the equation is a cubic, there it at least one real root, and hence one branch at least passes through the origin in the plane of the axes. To determine whether the other roots are real or imagi- nary, and hence whether the other branches lie in the same plane with the axes we might solve the equation. But this is not necessary. We can more readily determine the facts by examining the tangents. Evaluating -- = '- 11^ '^— ^ dx 2«x-^ — 3«2/2 for a; = 0, y = 0, we find :^- = 0, + \/2 and — \/2. Therefore there are three tangents, and the point is a triple point of the first species. The curve is that given in Fxq. 124, i^l82). Ex. 4. Examine ay^ — x'^y — ax"^ = for multiple points. du du Sug's. —The values arising from — = 0, and — = 0, are a; = 0, w = 0, and x = dx dy ' ;/ ' a ^S, y = — a. But only the first satisfy the equation of the curve. Evaluating dy 3x-'y-{-3ax-^ . ^. . ^ -, / . . dv\ dx = 3at -x- = ^^" *^"'" ^^^^^^' ^" ^^^ ("^^^^ ^ ^^" dx> P^= 1> P^ - 1 =^ 0, or (p — l)(p2 -L p -f- 1) = 0. Whence p r= 1, or — ^ dr \\/~3. Hence we see that there is but one tangent in this plane, and therefore but one branch passing through the origin, and no multiple point. Ex. 5. Show x^ -{- .cc2?/2 — ^ax'^y -f a^y"^ = has a multiple point of the second species at the origin. 124 PEOPEBTIES OF PLANE LOCI. CUSPS. IS 4:. Def. — A Cusp is a variety of tlie second species of double point, in whicli the osculating branches terminate in the point. Cusps are of two kinds : 1st, When the branches lie on different sides of the tangent ; 2nd, When the branches He on the same side of the tangent. 18 S, JPvoh, — To examine a curve for cu^s. Fig. 12X. Solution. — The process is tlie same as for multiple points of the second species, the only difference being that the branches stop at the point instead of running through it ; and hence that the values of y are real on one side and imaginary on the other. To ascertain of which kind the cusp is, we may compare the ordinates of the curve in the vicinity of the point, with the corresponding ordinate of the tangent ; &, 3. the corresponding ordinate of the tangent ; and the other, as S'E=?) — h ,<^b. The cusp IS therefore of the first kind. 8INGULAR POINTS. — CONJUGATE POraTS. 125 Ex. 2. Show that y = a -{- a; -{• bx^ -{- cx-^ has a cusp of the second 5. kind, if the sign of ^^ be considered as ambiguous, and that the equation of the tangent at the cusp is y = x -]- a. SuG. — To determine tlie kind of cusp, we liave -r-^ = 1 26 d= ht-cx^, both of whicli values are -\- for infinitesimal positive values of x. Therefore both branches of the curve are convex downward in the vicinity of the point, and the cusp is of the second kind. The curve has the general form represented in the figure. There is a point of inflexion in the lower branch at £C = &2 225c2' , and it cuts Fig. 129. the tangent at ic == — . Ex. 3. Show that cy"^ = x^ has a cusp of the first kind at the origin. Ex. 4 Show that (y - — b — cx^y :=: (^ — a)^ has a cusp of the second kind at (a, 6 -f- c^"^)* CONJUGATE POINTS. 186, Def. — A Conjugate JPoint is an isolated point the co- ordinates of which satisfy the equation, while in the vicinity of the point, and on each side, real values of one co-ordinate give imaginary values to the other. III.— In the equation y = {a -\- x)\/x, if x is nega- tive, y is, in general, imaginary ; but for the particular value ic = — a, y = 0. Hence P is a point in the lo- cus ; and as there are no other points in this plane adjacent to it, P is an isolated or conjugate point. On P A "\ X the right of the origin any real value of x gives two real, numerically equal values to y, with opposite signs. The curve has therefore two infinite branches on this side, which are symmetrical with respect to the axis -p^ -ioq of X. 187* JPvop, — At a conjugate point some one or more of the differ- ^' 1 rr ■ ^ ^1 ^^J ^^1 ^*Y . . . entiat coefficients — , -^, -= — , —-, etc., is imaginary. Dem. — ^Let y =f{x) be the equation of a curve having a conjugate point at (x, y). Then letting h represent an infinitesimal increment or decrement of x, and y' the corresponding value of y, we have y'=^f(x ± li) == an imagmary quantity, from the definition (186). But h'2 (Py h^ y =A^ ^ ^ — da; 1 ^ ck'^ I . 2 da;3 1 . 2 . 3 -]-, etc. 126 PROPERTIES OF PLANE LOCI. Now as y and h are both real, to make y' imaginary, some one or more of the „ . , dv d-y d?y , , . . coefacients ^, -^, -7—, etc., must be miaginary. q. e. d. 18 S, J*VOp. — Let — -, y has two numerically equal, real values, affected with opposite signs. .*. The curve is symmetrical with respect to the axis of x, and has two infinite branches extending to the right. Again -- = ± . ' ' " — . which for x= — y, becomes dx s/a + hx dx db ex, and for ic = 0, dy dx rh s/a- . * . At ( , ) the curve cuts the axis of x perpendicularly, and at (0, 0) it cuts it in two directions, viz., at tan-i( -f- v^a), and tan-i( — \/a). This also shows that (0, 0) is a multiple point, a double point. Examining for direction of curvature, we have -^ = ±: -, which is ^' 4(a + hx)^ db between and — -, and ± between 0, and -\- cc, . • . At the left of the origin, the curve is concave towards the axis of x, and at the right, convex. "We have a maximum and a minimum ordinate at a; = — -^^f y 36' — -v^d«, as ct 1 ^hx appears by solving the equation ± — — ^— = 0. s/a -j- hx It only remains to examine the infinite branches for asymptotes. ^ dx ihx^ X =^x — y-r = ^dy = CO, for re = 00 ; and a -)- f 6x _ dy ^ ihv'^ . Y = y — x-f- = — ' = =P GO, for x = oo. ^^ s/a 4" bx Therefore there are no asymptotes. From this investigation the curve is readily conceived to have the form given in the figure, which is constructed assuming a = 36. Fig. 133. Ex. 2. Trace tlie curve y^ = a^x\ Results. The curve is symmetrical with respect to the axis of x ; extends only to the right ; is convex to the axis of x ; has two infinite branches ; has a cusp of the first kind at the origin, with the axis of x for the common tangent ; and has no asymptote. X Ex. 3. Trace the curve 2/ =^ i , ,- ^ 1 -\- x^ Results. The curve cuts the axes at the origin under an angle of \7t ; has one infinite branch extending to the right above the axis of x. TRACING CURVES. 131 and another extending to the left below this axis ; has a maxi- mum ordinate at ^" = + 1, and a minimum at j; = — 1 ; has the axis of X as an asymptote to both branches ; has points of in- flexion at (0, 0), and at ^ = \/3, and x = — v 3 ; between the latter points is concave towards the axis of x, and beyond them is convex. Ex. 4. Trace y^ =: a^ — x^. Ex. 5. Trace {y — x'^)" == x\ Ex. 6. Trace ay^ — x^ + bx'^ = 0. Results. The curve cuts the axis of x at right angles at (6, 0) ; has a conjugate point at the origin ; has points of inflexion at x= ^h ', is concave to the axis of x from x=ib iox==^b, and convex beyond ; has two infinite symmetrical branches with- out asymptotes. Ex. 7. Trace ay^ — .r^ + {b — c)x-^ + bcx = 0. The form of the curve is given in the figure. Observe that when c = this locus becomes identical with the preceding, which is sometimes called the campanulate (bell shaped) parabola. Ex. 8. Trace the FoHum of Des Car- tes, whose equation is y'^ — Saxy -f x^ = 0. Ex. 9. Trace y^ = 2ax'^ — x^ Fig. 134. Fig. 135. Ex. 10. Trace ?/ = x^ X -. Examine a the curve for asymptotes, for maxima and minima ordinate s, for cusps, for direction of curvature, and points of inflexion. Fig. 136. 132 PROPERTIES OF PLANE LOCI. •Ex. 11. Trace y' = x^ + x-^ Examine X — 1 the curve for asymptotes, for limits, and for maxima and minima ordinates. 194, J*TOb, — To trace a curve of the second order, that is, the locus of Ay^ -f- Bxy + Cx2 + Dy + Ex + E = 0, by direct inspection of its equation. Solution. — One method of solving this problem has been given on pages 46 — 49. The present method is given as a good algebraic exercise, and in illustration of the remark in the preceding schoHum upon equations which take the form y = (p[x) =t ipix). Solving the equation for y we have 1st. K we construct the straight line of which y = — - — (B.X -[- D) is the equation (let it be rep- resented by M N in the figure), any value of x (as AD) which locates a point (as P) in this line, locates, in general, two points (P', P") in the curve, on opposite sides of the line and equally distant from it, this distance being the radical part ■ 4:A G)x^ + 2(i?i> — 2AE)x + {D^ — 4.AF). of the value of y. Therefore 2/ = ■ — irii.^^ + -^)» Fig. 138. is a diameter of the locus. 2nd. For such value or values of x as render the radical 0, y has but one value, and at this point, or these points, the locus cuts its diameter. Hence {B'^ — 4.AG)x-^ + 2(£i) — 1AE)x + (X>^ — 4.AF) = deter- -(JBx + X»). In general, this 2^^ mines where the locus cuts the diameter y = gives two values of a*, indicating that the locus cuts its diameter in two points, as in the ellipse and hyperbola. But if B^ — 4.AC = 0, the equation becomes 2{BD — 2AE)x -f (1)2 — 4.AF) = 0, which gives only one point of intersection, as in the parabola, a result which agrees with the fact that B^—4AG=0 characterizes a parabola {62). Locating the point, or points, at which the curve cuts its diam- eter, we know, if there are two points, and the curve is an ellipse, that it Hes be- tween these hmits, or, if an hyperbola, beyond. These facts will readily appear by observing whether intermediate values of x give real or imaginary values to ?/. Thus the limits of the curve appear, 3rd. If the locus is an ellipse, the values of y midway between the two values of TRACING CURVES. 133 X which correspond to the extremities of the diameter, make known a diameter parallel to tangents at the extremities of the former, and hence determine the cir- cumscribed parallelogram. Thus the situation of the ellipse becomes known. dth. If the locus is an hyperbola, we can determine a few values of y corres- ponding to values of x without the limits, and thus locate the curve. It is often expedient to find the intersections with the axes. 5th. If the locus is a parabola, having determined its diameter and vertex, a few values of x will make known sufi&cient points to enable us to sketch the curve. The intersections with the axes may also be of service. Ex. 1. Trace the curve whose equation is y2 — 2a;y + 2^7* + 2?/ + a? + 3 = 0. Sug's. — Since B'^ — 4:AC <^ 0, the locus is an el- lipse. Solving for y, we have y = OS — 1 ± \/ — .r2 — Sx — 2 ; whence y = x — 1 is a diameter, which we construct. n/ — x- — dx — 2=0, gives x = — 1, and — 2, the limits of the curve. Between these limits y is real, and without them it is imaginary. For a; = — li, 2/ := — 2, and — 3. Thus we find the circumscribed parallelogram. Fig. 139. Ex. 2. Trace the curve whose equation is t/2 -|- 2xy — 2x^ — 4?/ — 07 + 10 = 0. Sug's. — As B^ — iAC >* the locus is an hy- perbola. ?/ = — ^ + 2drv/3(x2— iP — 2). y = — X -f- 2 locates N M . From 3(aj'^ — x — 2) = 0, we find P and P ", at .r =-- 2, and — 1. Between these values y is imaginary ; hence the locus lies beyond these points to the right and left. Put- ting 2/ = 0, we have — 2a;2 — a; -[- 10 = 0, whence a; = 2, and —21, and the curve cuts the axis of a; at C and B. For ic = 4, 2/ = 3 • 5 and — 7-5 nearly, and we have 1 and 2. In hke manner a§ many points as we wish may be found ; but with the diameter and intersections with the axis, little or nothing more is necessary in order to form a pretty definite idea of the situation of the curve. Fig. 140. Ex. 3. Trace the locus y^ — 2xy -\- x^ — 4y -f- a; -f 4 = 0. 134 PEOPEETIES OF PLANE LOCI. Sug's. — Since JB^ — 4:AC = 0, the locus is a parab- ola, y = x -\- 2 is the equation of a diameter. For X =1 0, y Tz=.2. For x negative, y is imaginary. For .r = 3, 2/ = 8, and 2. Ex's. 4 to 7. In like manner trace the fol- lowing : 2/2 _|_ ^xy + 3j^2 — 4^ = ; ?/« — ^xy + 2j72 — 2^ = ; y^ _|_ 4^1/ _|_ 4j;2 — y 4=0; and 2/" — ^^y + 2^2 _j_ 2i/ — 2^ + 3 = 0. Fig. 141. lOS, I*VOh, — To trace a locus of the second order hy means of transformation of co-ordinates. Solution. — "We will illustrate this method by an example. The method itself is altogether too tedious for practical purposes, but is highly important as giving a clear view of a process which we have occasion to use for other purposes. Let us trace the locus whose equation is y^ -\- 2xy -{- Sx^ — 4:X = 0. This is an ellipse, since B'^ — 4:A C'<^ 0. We will find its equation when referred to its own axes. This requires transformation from one rectangular system to another. The formulce for this transformation are a; = x,^ cos a — yi sin a -\- m, and y z=z Xi sin a -\- yi cos a -\- n. Substituting these in the equation, we have =0. (Eq. A.) As the required form of the equation is Ay'^ -\- Bx- -\- F= 0, we desire to elimi- nate the terms containing x^y^, and y^ and ccj. To find the direction of the new axes, I. e. to determine the value of a, and to find the position of the new origin, i.e., to determine the values of m and n, which will effect this reduction, we place the coefficients of the terms to be eliminated each equal to 0, and solve the result- ing equations. These equations are (1) 2 sin a cos a — 2 sin2 a -f- 2 cos"^ a. — 6 sin a cos a: = ; (2) 2n cos a -\- 2m cos a — 2n sin a — 6m sin a -}- 4: sin a = \ cos'^a ?/r 4-2 sin a cos a Viyi-\- sin^a .•Ci2-f-27icosa ?/i-}-2nsin a Xi-\- n^ ■2 sin a cos a — 2 sin -a -f-2sinacosa 4-27ncosa -f 2msin \/2, and for r > 1 from = 45° to = 90°, r rapidly increases, and becomes — go at = 90°. The branch in this octant is, therefore, infinite. To ascertain more fully the char- acter of this branch, we form the subtangent Subt z= —-— dr rHQ sin20(cos20 — sin20)2 X COS" tan20 sec 0(1 — 2 sin-^0)2 cos'^0 Now since between 45° and 90°, 1 — 3 sin20 — 2 sin-*0 1—3 sin20 — 2 sin40 sin20 is between ^ and 1, tan between 1 and oo, Bec20 between 2 and oo, and tan and sec increase much more rapidly than sin 0, it is easy to see that subt. con- stantly increases and becomes — oo at = 90°. . • . This is a parabolic branch and approaches to parallelism with A"1". Finally, since r :=/(sin 0, cos 0), and only even powers of cos are involved, the values of r will be repeated in the inverse order as passes from 90° to 180°. x^ + x^ Ex. 4. Trace the locus whose equation is y'^ to the polar equation. SuG. — The polar equation with the pole at the origin is r (See Ex, 11, 193,) X -, by passing cos 0(1 — 2 cos^ 0) ■^♦»- SUCTION' VL Eate of Curvature. 197 • ®EF. — The Curvature of a plane curve is its rate of deviation from a tangent, and is measured by the subtenses of indefi- nitely small but equal arcs. III. — Let M N and mn be any two circles, AT" and AT"' tangents, and AS and AS' infinitely small but equal arcs. Then will TS and T'S', drawn per- 188 PROPERTIES OF PLANE LOCI. pendicular to the tangents, be the subtenses which measure the curvature of the arcs of the respective circles ; and we shall have curvature o/" M N : curvature of mn : : T"S : "T'S'. That curve is said to have the greatest curvature which de- viates most rapidly from its tangent ; thus, in circles, the greater the radius the less the curvature ; i. e., the curvature and radius are inverse functions of each other. It is also evident that the circumference of the same circle has the same curvature at all points ; while in other curves, as the conic sections, the curva- ture varies at every successive point. In the ellipse the cui-vature varies from its maximum at the extrem- ities of the transverse axis to its minimum at the ex- tremities of the conjugate axis. In the parabola and Fig. 146. hyperbola the curvature is greatest at the vertex and diminishes as the point recedes, becoming at infinity. It is the object of this section to present a method of measuring curvature, and of comparing the rates of curvature of the same curve at different points, and to ascertain the law of variation. For this purpose a circle is used, called the oscu- latory circle. 198. Def. — An Osculatory Circle is a circle which has the same curvature as a given curve at a given point ; or, it may be de- fined as the circle which has the closest contact with a given curve at a given point. III. — ^Let BDEC be an ellipse. K with the centres upon DC various circumfer- ences be passed through D, it is evident that they will coin- cide in very different degrees with the elUpse. Some will fall within, and others without. Now the one which coincides most nearly, as in this case M N, is the osculatory circle of the ellipse at the point D. The arc of the osculatory cir- cle in this case is exterior to the ellipse. The osculatory circle at the vertex, as m"n" is within, and at any other point, as P, cuts the ellipse, as will be shown hereafter. 199, T>E¥.—The Madiits of Curvature is the radius of the osculatory circle ; The Centre of Curvature is the centre of the osculatory circle ; and the point of closest contact is the point of osculation. BATE OF CURVATUBE. 139 200, T>EF.— Contact. Let M N and M'N' be two curves whose equa- tions are respectively y =f(^x) and y' = (p(^x'). Suppose the curves to have a common point P, so that for X = x' :^= /KD,y = y' = PD- Now if X and x' take the infinitesimal in- crement D D ', which we will represent by h, designating the corresponding Fig 148. values of y and y', by Y and Y' (S D' and S'D')? we have dy d'^y h^ d^y h^ d^y h" dx c/ip2 2 ^ 2 Y ' ^ M/^ l^ A D D' X Y=/(^ + /i)=?/ + ^/i+t1 -^ + In? d^y "~3 "^^^ 2-3-4 and X'=^(p{x^K) = y'-\- ^fi d^y'h^ d^y' h^ diy' /i< +, etc. ; -f , etc. dx'" ' dx'^1 ' dxJ^2-Z ' ^j7'''2-8-4 Subtracting the second of these equations from the first, we have Y - T = (3, - y) + (j^-^> + ( J-^.)-2 + (Js-A^jan +• *• Now the contact of these curves will evidently be closer as Y — Y' (SS') is less. We may therefore notice the following degrees of con- formity : 1st. If in the case of any two loci whose equations y =f(^x) and y' = (p(^x'), there is no value of x = x' which renders y = y', there is no common point. 2nd. If for x = x', y = y' and the differential co-efficients are un- equal, the contact is the slightest, and is mere Intersection, du du 3rd. If in addition to y = y', -^ = ~, and the succeeding coeffi- cients are unequal, the contact is closer than before and is called Tangency. This is called contact of the First Order. dy^dy' ^^^d^y^d^ ^ ' dx dx" dx^ dx'''^' and the succeeding coefficients unequal, the contact is of the Second Order, etc. 201* ScH. — A geometrical elucidation of this" subject is obtained by con- sidering that "an infinitesimal element of the curve commencing from a given point, being straight, is coincident with the tangent line at that point ; and the next element of the curve, being inclined at an angle to the former one, deviates from the tangent. Now let the two consecutive elements be of equal lengths, and from the extremity of the second let a perpen- dicular be drawn to the tangent : as this perpendicular is longer or shorter, the curve will deviate more or less from the tangent, that is, b© more or less 4th. If we have at the same time y 140 PKOPERTIES OF PLANE LOCI. bent."* Again, in general a rectilinear tangent is considered as having hoo points in common with a curve, and a circle three, since through the three consecutive points one circumference, and only one, can be passed. 202, Def. — A JParameter^ as the term is used in this and similar discussions, is an arbitrary constant entering into an equation of a locus, but which is made variable by hypothesis. Thus in the equation y = ax ~{- b, a and b are constants as ordinarily considered, that is have the same values throughout the same discussion. Again, they are arbitrary constants, since they may have any values. Finally, we may consider how a straight line changes position when a and h vary continuously. In this case a and h are called parameters. 20S* JPvop, — If one curve be given in sjjecies, magnitude, and position, that is entirely given, and a second given only in species, in gen- eral the highest order of contact possible is equal to the number of para- meters in the equation of the second curve less one. III. — As this proposition usually seems to the learner quite abstract, we will give a familiar illustration of its meaning before proceeding to its demonstration. Let dy'^ -\- 4ic2 = 36 be the first locus. The species is ellipse ; the magnitude is determined by the value of the axes 6 and 4 : the form of the equation determines the position of the locus. Thus this curve is given in species, magnitude and position, or entirely given. Constructing it we have the elhpse in the figure. Let the second equation be that of a circle in its general form, viz. , {X — m)" -\-{y — w)2 = r-, in which m, n, and r, are arbitrary constants, which we propose to treat as variables, thus making them parameters. It is evident that the closeness of contact of these two curves will depend on two things, the value of the ra- dius, and the position of the centre ; but the position of the centre depends upon the values of m and n. Hence the closeness ^^' of contact depends upon the values of the three parameters in, n, and r. Thus if P be the common point, by locating the centre at C, and using CP as radius, it is evident that the contact is much closer than when C is the centre and CP the radius. There is therefore some position of the centre and some value of the radius which will give the circle closer contact than any other. Moreover it is evident that we have given the widest possible opportunity for varying the contact, by taking that form of the equation of the circle which has the three parameters m, n, andr. We will now give the demonstration. X>EM. — Let y ^=f(x), and y' = (p{x') be the equations of the loci. In order that we may make y =^y' for some value of x = x we must have hberty to impose one arbitrary condition (i. e., to vary the second locus in at least one respect), but this requires one parameter. If, in addition to this parameter, there is a second (i e., if we can vary the curve in another respect) we can impose another arbitrary condi- * Price's Infinitesimal Calculus. KATE OF CUKVATURE. 141 dv dv' tion, as y = -^, and so on for any number of parameters. Hence we see that CvvC \AJvb one parameter makes intersection possible ; two make tangency or contact of the first order possible ; three contact of the second order, etc. 204:, Cor. 1. — The right line can ham in general no higher order of contact than the first {tangency), since its equation y = ax + b has but two parameters 2i andh. 205 • Con. 2. — As the equation of the circle in its general form has but three parameters, it can in general have no higher order of contact than the second, 200* Cor. 3. — The parabola can have contact of the third order, and the ellipse and hyperbola of the fourth. 207* ScH. — This discussion assumes that y =f[x), which is given in all respects, is of such a character as to allow of any degree of contact. Of course the possibilities of contact are limited as much by one of the loci as by the other. Thus, if the first locus were a circle and the second an ellipse, the contact could not in general be above the second order, although the ellipse has a possible contact of the fourth order with other curves. Again, in this discussion we have said "in general," since exceptions occur at certain singular points. Some of these will be noticed hereafter. Thus far we have given the broader view of osculation, although for the practical purpose of the measurement of curvature we might limit our view to the circle, as we shall do in the following propositions. 208, JProb, — To produce the general differential formulcB for the value of radius of curvature and the co-ordinates of the centre of curva- ture of any plane curve, in terms of the co-ordinates of the given curve. Solution 1. — Let y=fi^x) be the equation of the given locus, and (x' — w)2 -f (y' — n)2 = r2 the equation of the circle. Now as the equation of the circle con- tains three arbitrary constants, m, n, and r, we may impose three conditions and find the values of these constants which fulfill them. The conditions requisite for the closest contact which a circle can have, are, tor x =^ x', y = v', — = — , and '^ ^ dx dx d^v d^v' ■T^ = -j^—^. These therefore are the conditions to be imposed, and from which the values of m, n, and r are to be obtained. In any given case it will be suffi- dt/ d'^v cient to find the values of y, j-, and j—, in the equation of the locus, and also ax ctX'^ dv' d^v' the values of y', —, and — ^-, in the general equation of the circle, and equating the corresponding values find from the three equations thus formed the values of m, n and r. But for practical purposes, general formulcB are more convenient. These are readily produced, as follows : 142 PROPERTIES OF PLANE LOCI. Differentiating the equation of the circle twice in succession we have (1) (a;'-m) + (2/'-n)^ = In these equations and the general equation of the circle (3) {X' — m)2 -j- {y' — n)2 = r% dy d-y we can now substitute the values of y, -p, and ^-^ as obtained from the equation dx^ of the given locus considering x = x', and have (4) (5) {X — m) + (2/ — w)-- = 0, i + S + c^'— ^^. = «' (6) {x -^ my -\-{y — nY = r'. In order to solve ttiese equations for r, m, and n, we get from (5) dy^ (7) (8) a; — m = (9) r = ± o+i:y cZx2 which substituted in (4) gives Substituting these values in (6) and reducing we have which is the formula for radius of curvature. The co-ordinates of the centre (m and n) are written at once from (8) and (7). They are , (10) m = a; — (11) n = y + (l + ^tW dx^/dx referred to A as its origin. This equation is satisfied for none but negative values of n, and gives m = 0, litr, 4.7tr, etc., for n =0 ; and also forn = — 2r, m = ;rr, 3;rr, etc., as it should. EVOLUTES AND INYOLUTES. 151 ScH. — The student will readily dis- cern the character of the evolute of the cycloid from the property that the radius of curvature is always twice the normal. Thus if the two circles C, and C roll along the bases AX and AX' at equal rates so as to keep their centres in the same vertical line P' will describe the evolute as P does the involute. 5k^ •^ ""^"^ / ' \\ y \ A V ^)\' X A' X' Fig. 159. 220, JPvojy* — A [produced) normal to an involute is tangent to the evolute, the point of tangency is the centre of curvature, and consequently the normal thus produced is the radius of curvature. Dem. — Let (m, n) be any point in the evolute of A IVI, from it draw a normal to AM, and let {X, y) be the point at which it is normal. The equation of this nor- mal is y — n dx or dy X — m -f- Y'jy — ?2) = (1) . Now as the point (m, n) changes position (cc, y) also changes, and to observe the law of change we differentiate (1) for X, y, m and n as variables. This gives Fig. 160. dx — dm -{- dy- — dndy dx + {y ^^df = «' or ^+1:+*^ d"y dm dx- dx dx dndy dx^ = 0. (2) But as (m, n) is in the evolute we have {208) y d^y d'y dx^ whence ^ + % + 'y n) dx-^ dm dx 0. Therefore dropping these terms, (2) becomes dn dy dx^ dx dy dx dy dn dm = 0, and {X — m), which is the equation of a normal to Hence the equation y — n - the involute at {x, y), may be written '^Z - n) = ^(^ - m\ which is the equation of a tangent to the evolute at (m, n). q. e. d. 227 » Cor. — The i^adius of curvature and the arc of the evolute vary by equal increments ; that is, the arc of the evolute between tivo centres of curvature equals the difference between the corresponding i^adii of curva- ture. 162 PBOPERTIES OF PLANE LOCI. Dem. — Since the radius of curvature is a tangent to the evolute it coincides with the arc between two con- secutive points. Thus P and P' being consecutive points on the involute, the radius at P is to be consid- ered as having the two consecutive points C and C common with the evolute to which it is tangent ; and as P passes to P', the radius of curvature so changes po- sition as to have the consecutive points C and C" common, and to coincide with the curve between them. Thus it appears that the radius of curvature and the arc of the evolute vary by equal increments. Fig. 16L 228. ScH. — From these relations it is easy to see how an involute may be described mechanically from its evolute. For example, to draw a para- bola, make a pattern of the form AOCM Fig. 161, the edge OCM being the arc of an evolute to the required parabola, and AO =p, Fasten a cord at M and, wrapping it around the edge of the pattern, fasten a pencil to the free end at A. Keeping the string tight, move the pencil along as from A to P, P', R, and it will de- scribe the parabola which is an involute to OM. In like manner any curve can be described by means of a pattern of its involute. The cycloid and ellipse are drawn with special facility by this method. Thus, for the ellipse, take a thin rectan- ^ gular board ABED, and w^on it fasten two pat- ^i^- 162. terns ACOD, and BC'OE, the edges CO and CO being the evolute. Then fastening at O, one end of a string whose length is AGO, the free end will describe the semi-ellipse as it is moved from A to B. Upon this principle attempts have been made to make a pendulum vibrate in the arc of a cycloid. 229, Cor. — Every carve has one and only one evolute ; hut every evolute has an infinite number of involutes, since every point in the string describes an involute as the string unwraps from the evolute. ^» » SECTION YIIL Envelopes to Plane Curves. 230. Def. — An JEnvelope is the locus of the intersection of consecutive lines, or curves, represented by a given equation, when one or more of its parameters are made variable. III.— Let {x — w)2 -j- j/^ — r2 — be the equation of the locus whose envelope ENVELOPES TO PLANE CURVES. 153 a-b c cl Fig. 163. N' is required. Let m be the (variable) parameter. Let r = Bl, so that Baa' shall be one position of the given locus, which in this case is a circle. Now sup- pose m to take an infinitesimal incre- ment dm, putting the centre at 2, and giving {x — (m -f- dm)}^ + 2/'^ — r^ = as the equation of the consecutive locus. The intersections of these loci, as a, a', are points in the envelope. Again, let m take another infinitesimal increment, as 2 3, then h and h' are points in the envel- ope. In like manner the intersections of 3 and 4, 4 and 5, etc. , etc. , give points in the envelope. The envelope in this case is evidently the two parallel right lines MN, M N'. Were we to make r vary at the same time as m, the form of the envelope would be clianged, and would depend upon the relative rates of change of r and m. Of course, the student will understand that the points of intersection a, h, c, d, etc. , are only in the envelope when 1 2, 2 3, 3 4, etc., are infinitesimal ; in other words, the envelope is the limit toward which these consecutive intersections approach as the increments 2 3, 3 4, etc. , diminish. 231, JP'TOb, — To find the equation of the envelope of a given locus. Solution. — Let F{x, y, m) =0 be the equation of the given locus. The consec- utive locus will be F^x, y, m -f dm) = 0, or F{x, y, m) + d^F^x, y, m) = 0. If we now combine the equation of the locus with this equation of its consecutive, elim- inating m, we shall determine the locus of the intersection, i. e., the envelope. But since Ft^x, y, m) = 0, the equation of the consecutive can always be reduced to d,nF(x, y, m) = 0. Hence in practice we simply combine the equation of the locus with its first differential equation eliminating the parameter, thus obtaining the envelope, q. e. d. ScH. — It is of course possible that the consecutive loci may not intersect ; as, for example, x'^ -^ y"^ =^ r^, when r is made variable. Ex. 1. Find the envelope of y" = m(x — m). Solution. or == xdm -Differentiating with reference to m, we have = dni{x — m) - 2mdm. Whence w = ^x. Combining this with ;{x, y)] = 0, and q){x, y) = m, this becomes — ' — \- CLvu dF{x, y, m) dy dy . dy -7-, whence -— dx dx locus. dF{x, y, m) dx dF{x, y, m) ~ dy ', the same as in the equation of the Ex. 4. Find the locus to which the hypot- y enuse of a right angled triangle of con- ^ stant area is always tangent. Solution, — Let the constant area ABC =; a, the parameter AC = m. Then AB = --, m 2a tan BCX = , and the equation of BC is y = -X -\ . The equation sought is xy = -, the equation of an hyperbola. Ex. 5. What is the envelope of an ellipse which retains its axes in the same right lines, but varies in eccentricity so that AB = a con- stant, 771 ? Sug's. — Since AB = m, the equation of the ellipse is A''y^ -\- m^x- = A"m'^ ; in which A is the parameter. The equation of the envelope is xy = Im, an equilat- eral hyperbola referred to its asymptotes. As will appear hereafter, the area of an ellipse is TtAB. Hence the area of the above locus is constant. Ex. 6. From every point in the circumference of a circle, pairs of tangents are drawn to another circle. Find the locus to which the chord connecting corresponding points of tangency is constantly tangent. Solution. — Letting C be the centre of the first and A of the second circle, it is evident that as P moves around the circle P'P" -will change its position. The envelope of P'P" is required. Let CP =r, and AP" = r'. Let P be designat- ed as (m, n), P' as (m', n'), and P" as {yn", n"). The equation of the locus P'P" whose envelope is i-equired is y - - n' ;= 156 PROPERTIES OF PLANE LOCI. — ; 77(0; — m') (1). But by reason of the tangents PP' and PP" we have nn' -f- '^^''n' = ^'- (2) ; and nn" -f- inm" = r'- (3). Subtracting (3) from (2) we have n{n' — n") -f- m{m' — m") = ; whence m — m m = (35- n —, and (1) becomes y — n' m'), wy 4" mx == r'2. Thus we Fig. 169. find the equation of the given locus P' P" to be ny -\- mx = r'2 (4). Again, if we let the distance between the centres of the circles AC be repre- sented by a, we have the relation between n and m in the equation ?i2 4- (m — ay = r2 (5). The problem then is to find the envelope of (4), when the relation between n and m is that given in (5). Differentiating (4) and (5) considering m and n as variables, we have y— — Ux = 0, dm dn X ^ dn , . nx , — — = , and n- \- m — a = 0. .*. j- am y am y m — a = 0, my — nx = ay (6). Finally eliminating m and n between (4) (5), and (6), and reducing, we have r-2y'2 _|_ (r2 — a-);c- -\- laf'^x = r'^, as the equation of the envelope. Hence the envelope is a conic section. When a = it is a circle ; when a <^r, an ellipse ; when a = r & parabola ; and when a >> r, an hyperbola. 233, J^TOb. — An infinite number' of parallel right lines meet a given curve on the same side ; and where each meets the curve a line is drawn making an angle with the parallel which is bisected by the normal at that point. Required the envelope of the line. Solution. — Let M N be the given curve, PO one of the parallels, PQ the normal, and PG the locus whose envel- ope is sought. Our first purpose is to find the equation of PG. Let y' = cp{x') be the equation of M N , and v the tan- gent of the constant angle PSX. Since P, whose co-ordinates are x',y', is a, point in PG, the equation has the form y — y' = a{x — x) in which a is the tan- gent of PDX. Now PDX = PQX — DPQ -- — (PSX - PQX) = 2PQX — PSX. tan2PQX — tan PSX 1 -j- tan 2PQX tan PSX' IG Fig. 170. PQX — QPS = PQX Therefore tan PDX = Again, as PQ is normal to y' =^ (p(,x'), tan PQX ^^' -u . ot-.^x, 2 tan PQX — TT-; ; whence tan 2 PQX = -^ dy ' ^ 1 — tan-^ PQX —2 — dy' 1 dx2 dy^t Substituting, and ENVELOPES TO PLANE CURVES. 157 introducing v for tan PSX, we have dx'2 dy'' V — V a = 2— dy' 1 — -T-.- dx''-^ dy"^ dx' dx 2-—V dy Putting —7 = p, for convenience, the equation of PG becomes dy y — y = p^v 2p [x — x'). 1 — p'^ — "Apv From this equation, its first difierential equation, and the equation of the curve y' = ^{x'), if x, and y' be ehminated, the resulting equation between a; and y will be the equation of the envelope sought. But the difficulties of elimination are often insurmountable. We give two cases which are readily solved. 234, CoK. 1.—^ O P is parallel to AX, f = 0, and the equation PG becomes 2p y — y p2 1 (X - X'). 23S. Cor. 2. Hon becomes -If OP is perpendicular to AX, v= oo, and the equ^ V' " " 2p 236 » ScH. — It is a well known prop- erty of light that its rays impinging upon a reflecting surface are thrown off so as to make the angle between the reflected ray and the normal, equal to that between the incident ray and the normal. In con- sequence of this law, when the rays of the sun, which are practically parallel, are re- flected from a curved surface, the inter- sections of the consecutive reflected rays produce a luminous curve, called a Caustic^ which is an example of the envelope dis- cussed in the problem. The annexed figure affords an illustration. Let NAN' be a section of a circular cylindrical mir- ror, made perpendicular to its axis. Let 1 to 11 be rays of light parallel to the axis of the mirror AC. The envelope of the reflected rays is the caustic curve NM. MN shows the lower branch of the caus- tic, the rays not being represented. This cnrs'e may be seen inside of a ring lying (X - X'). Fio. 171. 158 PROPERTIES OF PLAISE LOCI. on a table in the light. It is famihar to the milkman, as **the cow's foot in the milk," which is the caustic formed upon the smooth surface of the milk in a bright tin pail, by reflection of the Hght from the inside of the pail. Ex. 1. To produce the equation of the caustic when the incident rays are parallel to the axis of a parabolic reflector. Solution.— We have y'^ = 4mx', and -— ; = 7--, 4m being the parameter of the V ^ = 4mx , ana — - = 77—, ^ ay 2m 4m' parabola. Substituting this value of p, and for x', j-, in the equation {234), we have after reduction yy'2 — 4m2?/ -\~ Am"y' = ^my'x (1). Differentiating (1) with respect to the parameter y', gives y' = . Sub- stituting this in (1), and reducing we have X = m ± V — y^, as the equation of the caustic. This can only be satisfied for 2/ = 0, x T=m; whence we see that the caustic is a point, the focus. Ex. 2. To find the caustic to the circle "when the incident rays are parallel to the axis of x. dx' 11' Solution.— Equation of circle y'^ -\- x'^ = r^. .*. ;r-; = ;• Equation of etc/ X 2r) , 2x'w' reflected ray (234:) y—y'z:^ — —^{^ — a;'), becomes t/ — y' = : '—ri^ — ^)^ p- — 1 2/ — ^ " o^ ^ - «^' =- ^ l~y^ \ y' - y)' o^' ^-^' = Kl ~ ^')^^'~^^ ^^^- Differentiating (1) with respect to x', we have y' ^ y \, . . m ^\x' — 1 --:^V-.)-(|,-^,)f.- V £C'2 y"^ } ^ \x' y'/y' Whence -2 = _ ( ^^f + l^y^-y) _1 + _, r'^/'ii'-i \ x'^ - % + '- vS + 0^^- - y^' x"i-\-y'-2 r-Vv'- 4- •'^'^\. , X -u- V J- -J J 1. oi'^ -^ y'^ . or T-^- = —-{ — ){y — y) '> which divided by gives X'^ T^ 3 3 — = — {y' — y). From which we find y' =z r y . y ^ y'^ v' x' ^ To find x', substitute in (1) for y' — y its value '— ;j-, and we have ?/'2 X'2 2/'x'2 W'2 X'2 ,c X X — x' = = X . But X 2 = r2 — V 2 whence x — x' = 2y'x' r2 2r^ ^ _r2 4-2?/2 , , 2r?x _, ,^. I i . , ... , , 2r^r -X , or X = jr-7-. Putting r y for 2/ , this becomes x = -7 2^2 r^^2y'^ o . . ^^4.2/ BEOTIEICATION OP PLANE CURVES. 159 Finally, squaring these values of y', and x' and substituting in y"^ -|- x'^ = r^, we nave r y -| ^ — = r2, wMch is tlie equation of the caustic sought. [Note. — At this stage of the course the student will need to acquaint himself with the elements of the Integral Calculus, as given in Chap. III. of the second part of this volume.] ■♦♦»• SECTION IX, Rectification of Plane Curves. (a) BY MEANS OF KECTANGULAR CO-ORDINATES. ^57. Def. — To Rectify a Curve is to find its length. The term arises from the conception that a right line is to be found which has the same length as the curve. 238, I^rop, — The formula for the rectification of plane curves is dz = Vdx.^ + dy2 ; in which z represents the length of the curve, and x and j the general co- ordinates. Dem. — Let M N be any plane curve, AD = X, and P D =: y ^® ^^1 co-ordinates, and let D D' represent dx ; then will P'E repre- sent dy, and PP' , dz \ i, e., dx, dy, and dz will represent contemporaneous infinitesimal in- crements of the co-ordinates and the arc. xC^T From the right angled triangle PEP' we R have at once dz = \/dx^ -\- dy^. q. e. d. Fig. 172. 239, ScH. — To apply this formula to any particular curve, we have simply to find dv or dy from the equation of the proposed curve, substitute it in the formula, and then integrate between proper limits. Ex. 1. Rectify the semi-cubical parabola whose equation is y^==ax^. (See e volute of common parabola.) Dem. — Differentiatins; y- = ax^, we have dy = -^dx, whence dy- = -— — dx"^ = ^a^x^ , ' dx^ = iaxdx"^. Substituting this value in the formula for rectification, it 4ax3 becomes dz = (dx' ^ ^axdx^) =(!-}- l^^) ^^ Integrating we have z =* 160 PROPERTIES OF PLANE LOCI. 8 #• ^;:-(l -j- f aa;) -}- C. To determine C we may reckon the length of the curve from the origin A, whence for cc = 0, 8 8 z = 0, and we have = — — \- G, or G= — ^r=-. The cor- A iCL A id reded integral is therefore z = ^— — [(1 -\- ^ax) — 1]. Aid To illustrate this result consider A M the curve whose length is to be found, or, which is to be rectified. As this curve is infinite in extent, we can inquire only for the length of some specified portion of it. Let it be required to find the length of the curve between the origin and the point P whose abscissa we will call h. Substituting this value of X, we have arc A P = z 27a [(1 + ^ahy — 1]. Were Fig. 173. it required to find the length of some other arc, as PP', we should integrate between the limits .r = A D, and ic = AD'. Thus, let AD = 6 and A D' = c. 8 ^ Besuming the indefinite or general integral z = ^-{1 + ^ax)^ -\- C, substituting A (CL successively x = h, and x= c, and subtracting the former result from the latter, we get for the length of the arc P P', the definite integral Ex. 2. Bectify the common parabola. ifidifi 1 I Solution.— From y"^ = 2px we have dx'^ = - — ^ ; whence dz = -( pa _i_ y^) (fy. To integrate this apply formula <(^ of reduction, and we have y Vp^ + y^ p j dy 2p 2^ v/p2 + y^ But /: dy \/p2-|-2/ = = log[2/+v/p2-h2/^]+C. ^=^^±^VS-iog[^+v/p-q:F]+c. 2p P^ Estimating the arc from the vertex, which is the origin, we have C= — %^ogp ; and A the corrected integral is z = y\/p'^-\-y^ p, Vy -\- \/p^ -\- y'^ 2p +|i°gp+^ ;'+^' ]. ScH. — Instead of integrating as above, we may expand (p2 _^ ^^y by Maclaurin's or the Binomial theorem, and then integrate each term separate- 1 1 y3 1 1 1 1 3/5 1 1 3 1 1 1 ^7 ly,obtaimng.= (y + -. -.---.-.-.-.- + -.-. --2-. -.-.--etc.) + C. Ex. 3. Bectify the circle. Solution. From x:^ -\- xf^ = r^ we have dy^ = —dx^, hence dz = ( ' — ) y2 \ y y dx RECTIFICATION OF PLANE CURVES. 161 dz = ' 7, and z = rsin—^--\- C, But this is only a restatement of the s r (r2 — X") ' problem and is of no use in the solution. We shall have to integrate in some other way. We may wi'ite -2 / X'\-i -^ dz = r{r'~ — Q(fi) da; = ( 1 ;^ ) dec = r(l — x^-) 'dx^, putting - = Xi for convenience. Expanding by Maclaurin's or the Binomial the- orem, and integrating each term separately, we have Restoring x this becomes ofi 3^5 1 -3 . 5.r7 2 • 3r^ "^ ii . 4 . 5ri "^ 2 • 4 • 6 • Tr- To determine C, reckon the arc from the axis of ordi- nates (B), whence for a; = z = 0, and therefore (7=0. Then making x = r we have the length of the quadrant <; + BX, z = r( (^ + 2^3+2-:r75 + 2.4.b.7 + ^'^0 Fig. 174. Bepresenting the sum of the series in the parenthesis l>y i"^, we have z = a quadrant = irit, and the whole cir- cumference = 27rr. Letting D be the diameter, this be- comes Dit, whence it appears that it, the sum of the series above, is the ratio of the circumference of a circle to its diameter. By extending the terms in this series sufficiently, reducing each to a decimal fraction and adding, we find it = 3.1415926-f-. For practical purposes it is usually taken as 3,1416, and for still ruder approximations as 3|. 240, Cor. 1. — The circumferences of circles are to each other as their radii, or as their diameters. 24:1, ScH. — The quantity tt has not only fundamental importance in ge- ometry, but has great historic interest. Upon it depend both the method of obtaining the circumference of a circle, of a given radius, and the area of a circle, as well as many other problems. The ancients sought with much diligence to discover its value. Archimedes (287 B. C.) found it to be between 3^^^ and 3^. Metius (1640) gave a nearer approximation in the fraction ff^f. In 1853 Mr. Rutherford presented to the Royal Society of London a computation by Mr. W. Shanks of Houghton-le- Spring, extend- ing the decimal to 530 places. The following is its value to 50 places : 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 10. Ex. 4. Rectify the cycloid. Solution. — We have dx^ = — ^ — - — , whence zry — 2/^ fdz = (2r)^/(2r — y)~'% = — 2{2r)\2r — y)^ -f a Beckoning the arc from the origin C =. 4r, and the corrected integral is z = 162 PROPERTIES OF PLANE LOCI. .i i 2(2r)''(2r — 2/)" + ^^- Making y = 2r, 2 = 5 the cycloidal arc = 4r, whence the entire arc of the cycloid is seen to be 4 times the diameter of the generatrix. 242, CoE. 2. — Any arc of a cycloid, edimated franx ike vertex, is equal to twice the corresponding chord of the generatrix. * Dem. — Resuming the indefinite inte- i J^ gral z = — 2(2r) ^{2r — ?/)- + C, if we estimate the arc from B, where y = 2r, we have C = ; and the corrected in- tegral is z = — 2(2r) ~{2r — y)~. This is the length of any arc estimated from B, as BP, PD being y. But in the right angled triangle BEC, . • . arc B P = 2 times chord B E. q. e. d. ScH. — Both the fact that the length of the cycloid equals 4 times the diameter of the generatrix, and that any arc estimated from the vertex equals 2 times the corresponding chord of the generatrix, are readily ob- served from the manner in which the curve is described from its evolute. Thus the radius of curvature at the vertex is 2 times the diameter of the generatrix. But this is the arc of the evolute. So also as the string un- winds from the evolute, the radius of curvature is seen to be the arc of the evolute and equal to twice the corresponding chord of the generatrix. (See Fig. 159.) 2^2. Ex. 5. Bectify the hypocycloid, whose equation is x'^ -{- y'^ = a^. BE =^ v/BC X BG = v'^TXAr — y). (See Fig. 167.) Ex. 6. Bectify the ellipse. The length of the entire curve is Qa. Solution. — For this purpose the equation 2/- + (1 — e^)^:^ = ^2(1 _ gs) is most convenient. From this we have dy'^ = —(1 — e^)^cbfi', whence x^ t dz = ^ \dx^ + —(1 — e^-ydx^ = dxVy'^ + x\l — e-^)2 - ^ ^.^ 1( 1 — e^){A^ — x^) -Kl — e2)2a;2 (1 — e2)(^2 — a;2) - - -^J- — a;2 4- x2 — e2a;2 A^ — x^ = dx A'^ — e'^x^ Adx A^ v/^'2 e^x^ 1 "■ — li" ) ' ^® Adx ( ^ e"-x-^ e-'x* Se^x^ have dz = — — J 1 — v^lT^^i I 2A-2 2.4.^-* 2 . 4 . 6^« r , , /^ dx e2 /* x"dx e* r x'^dx 3e^ /• x"dx Jdz = A j- — etc. e'±'^\i whence v/^2 — ^c^* e2 r x"dx e* r x*dx 2>e^ r 2.4.6^yy^IZ^, RECTIFICATION OF PLANE CURVES. 163 — etc. Now integrating each of these terms separately we have z = ^sin— i • 3e6 SbA^Y^A^/A^ . X X ,- \ a;3 /- "l x^ r-. ) 2 . 4 . 6^5 — etc., + a K we estimate the arc of the ellipse from the extremity of the conjugate axis, we have for a; = 0, z = ; whence substituting, we find (7=0. It j Again, making x = A, and observing that sin— '1 =zs — , we have _7tA e^/AnA\ e* r3A^/A TtAy\ *'"~T~2"1V2"T/~2X43LTV2*~2"/J 2- 3e6 (5A^r3A^/A TtAyi) T6-i^h~LH2-^-JJi-"*^- Uniting and multiplying by 4, we have for the entire circumference of the ellipse, i A o A/'-i ^' ^^' 3.3.5e6 N 4., = 2;r^(l -^^- ,.2.4.4 - 2.2.4.4.6.6 " ''''} This series converges more or less rapidly as the eccentricity is greater or less, but is always converging. (6) RECTIFICATION BY MEANS OF POLAR CO-ORDINATES 243 • JProp. — The formula for rectifying polar curves is dz = (r2d 3p Sp {b' —J) = f(2p)V^— a^). Ex. 2. Eind the area oi y = x x\ Solution. — Substituting the value of y in the general formula, it becomes dA = xdx — x^dx . • . J. = 5.^2 — ix-^ -\- C. Sketching the curve, we observe that it will be natural to inquire for the area AmB, or Am'B'. Thus reckoning the area from the origin, x = 0, gives ^= 0, and consequently C= 0. The corrected integral is A = i-r^ — 40;^ Hence area A?nB, or Am'B' j, ^nn = i. To find the area of any other portion as BCD, we integrate between the limits a; = A B == 1, and a; = A D = 2. gives A = 2 — 4: — i = — 2^. Hence area BCD =2i. This Ex. 3. Find the area oi y = x^ — b'^x Area betiveen x = and x=b is \b^. 166 PBOPERTIES OF PLANE LOCI. Ex. 4. Find the area of y = x^ + ax^, constructing tlie curve, and observing the natural Hmits of integration. BesuU, Between x = 0, and x = a, A = y^a" ; and between ^ = and 00 = — a, dA = -^^a*. Ex. 5. Construct and find the area of a^-y"^ = x^{a'^ — x'^). Area = %a^. Ex. 6. Eequired the quadrature of xy^ = a^ between the hmits b — c y=h, and y = c. Area = 2a^—^. Ex. 7. Required the quadrature of the circle. 1 X. Solution.— From x^-\-y^ = r% we have y=[r^ — x'^y, hence dA = (r^ — x'^^dx. JL Applying formula )^ -{- -i-r-sin-'-. The significance of this is readily seen by «^= inspecting Fig. 174. AD = a, and j^a{r^ — a^) = area of APD. Arc BP = ci a r sin— 1— ; whence -ir^ sin— i- = area sector A B P. r r If the segment has but one base, a,= r, and we have / dA = |7rr2 — ^hir- — \fif — -^r^sin-'-. From Fig. 174, we see that krcr^ = | '• the area of the circle, |-r2sin— i- = area BPA, and ib(r^ — &2)2 __ q^q^^ APD. r In each case we have |- the segment. Ex. 8. Find the area of the ellipse. Sug's.— The area=-/ {A^ — x'^)''^ dx. But / (^2 — cc2)2'(;^x is i the area of a circle whose radius is A, which is vtA^. . * . Area of ellipse = itAB. 2S0» CoK. — The area of an ellipse is to the area of the circumscribed circle as the conjugate axis is to the transverse axis. The area of an ellipse i", to the area of the inscribed circle as the transverse axis is to the conjugate axis. The area of an ellipse is a mean proportional between the inscribed and circumscribed circles. Ex. 9. Find the area of the cycloid. Sitg's. — We have dx = ^-^- r ; whence dA = ~ ^ — L (2ry — yrf (2ry — y^)^ S. _JL y^{2r — y) ''^dy. Integrating by applying formula ^ twice, we obtain /*2r s. 2r 3. _1 (2r — y) *(i?/ = fr2Ters— '2 = f r^TT, since vers— '2 = tc. . • . The entire area is STtr"^, or three times the generating circle. A somewhat indirect but simple method of quadrature is as follows : 1st. Find the area of APCB, the Y B DD' c r 7" ^ \ ^ N A / Fig. 180. E X 168 PBOPEBTIES OF PLANE LOCI. element of whicli DPP'D' = dx{2r — y) — {2r — y) ^-^ — - — {^ry —-fydy. Now considering the circle A'pC, we observe that an element as pdd'p' = dy{2ry — ?/-)^. .•. APOB =^^ area of the generatrix, =^7tr-. But ABC A' = 2r X A A' z= 2r X 7tr =^ 'litr-. .-. ACA' = fTrr-, or the entire area of the cycloid := 37trK Observe that both these integrals are to be taken between the same limits, viz., y = 0, and y =^ 2r. Ex. 10. Find the area of the curve a^y~ = a-b^a:-^ — b'^x'^. Area = ^ab. (6) QUADKATUEE OF POLAil CUKVES. 2S1, JPvoj), — An elemerit of the area of a polar curve i^ -i-r^d^? and the formula for the quadrature is dA=^r^d.d. Dem.— In Fig. 176 PAP' = dA. But area PAP' = AP' X iPD = (?• -\- dr)jrdS = jrr-dQ, omitting irdrdB, and also remembering that P D = rd6. . • . dA = ^r^dQ. Q. e. d. Ex. 1. Eind the area of the spiral of Archimedes. Solution. ■ — The equation is r = ^r— ; whence dr = 27r — -cZ9, and d6 = 27tdr. Substituting in the formula, we have dA = Ttr^dr, and A = iTtr^ -f- O K we estimate the area from the pole, we have 4 = 0, when r = 0, and conse- quently C = 0. The corrected integral is, therefore, A = iTtr^. This is the general expression for the area passed over by the radius vector in its revolution from its starting at to any value, as r At the end of one revolution J' = AP = 1, and A = area of the first spire = AmP = i;r= i the area of a circle described with A P as a radius. At the end of the second revolution r =; 2, and the area traced by the radius vector, or Aj = ^tc. This evidently includes twice the first spire -\- the second. The area of the second spire is, therefore f tT — ^Tt = 27t. The area of the first two spires is ^tt. Ex. 2. Show that the area of the Napierian logarithnTic spiral is ^- the square described on the radius vector. Ex. 3. Find the area of the Lemniscate of Bernouilli. (r2= a-cos Id. ) SoLTJTioN. — From the formula we have dA = ^a^cosWdB. "Whence A = ia2 / cos 20 dS = ia2sin26 = la'^ . • . The entire area is a% i. e., the square on •the semi-axis. Ex. 4. Trace the curve r = a(cos 26 -f sin 2^), and find its area. QUADRATURE OF SURFACES OF REVOLUTION. 169 SUG*S. r'? = a2(cos2 29 + sin2 29 + 2 sin 26 cos 20) = a\l + sin 40) - . A == ri(j2(i _j_ sin40)d0 = ^a^{fdd 4-J"sin40 cZ0}. The entire area, which is com- prised between 9 = and = 27t, is Tta'^. ^♦» SUCTIOJSr XL Quadrature of Surfaces of Eevolution, 2,S2, Def. — A Surface of Mevolution is a surface gener- ated by a line (right or curved) revolving around a fixed right line as an axis, so that sections of the volume generated made by a plane perpendicular to the axis are circles. Ill's. — A right line parallel to and revolving around another right line, in the manner described in the definition, generates a cylindrical surface with a circular base. A semi-elhpse revolving around its transverse axis generates a prolate spheroid, around its conjugate axis an ohlate spheroid. These two are varieties of ellipsoids. A paraboloid is generated by the revolution of a parabola around its axis. The number of these surfaces is, of course, infinite ; and the specific char- acter of each depends upon the nature of tne generating curve. 2S3, I^TOp, — The differential element of a surface of revolution is dS= 27ry\/dy' + dx\ Dem. — Let M N be the generatrix, AX the axis of rev- olution and P P' two consecutive points on the generatrix. As M N revolves about AX, any point, as P, whose co- ordinates are x and y describes a circle whose circumference is 27ty, The circumference described by the consecutive point P' will be 27t{'i/ -f- dy). PP' describes the frustum of a cone whose surface is half the sum of the circumfer- jij^^^ 3^32. ences traced by P and P' multiplied by PP', or = y -\- ^^ + ^ y ,/cly2^clx^^ Whence omitting 2itdy, it being an infinitesimal of a higher degree than the other terms, we have dS= 27Cy\/dy'^-\-dx^. q. e. d. ScH. — To apply this formula, let y = '>• ~l ia;(a2 _ ezx^y + i_ sin-> - + C. . • . ^S == tt- a;(«2 — esa;^)^ + - sin-' - + (7 . Eeckoning ;S^ = for ic = 0, C = ; and then putting a; = a we have for half the X 1 Ttdh surface 7tdb[(l — e-)- -\ — sin— 'e], or nh^ -\ sin— le. 6 6 Ex. 6. Apply the above formula to find the area of the surface of the prolate spheroid whose generatrix is 25?/^ + IGor^ = 400. Area, 235.41 nearly. ^ ♦ » SECTION XIL , Oubature of Volumes of Eevolution. 2SS, J^TOp, — The differential element of volume of a solid of revo- lution is dV = ;ry2dx. Dem. — Using Fig. 182, with the same notation, the volume generated by the revolu- tion of DPP D' about D D', is a frustum of a cone, and is therefore equal to three cones having for their common altitude the altitude of the frustum {dx), and for bases the upper base (7ty^), the lower base l'!t(y-\-dyY'\, and a mean proportional between XI j^ , r / . 1 -, Tr -rrr It' y^ -\-y^ -\-1ydy -\- dy^ A-v'^ -\-ydy)dx the two bases \7ty{y + dy)\ Hence dV — -^^^^^^^-^ lyry-/ — _ o (omitting terms having infinitesimals of a higher order) ity'^dx, Q. E. d. - Ex. 1. To find the volume of a sphere. Solution, y^ z= r^ — x\ .- . dV = 7t{r'^ — x^)dx = Ttr'^dx — itx'^d^, V = I {Ttr^dx — Ttx'^dx) = f ^7*3 ; or, letting D = the diameter, and doubling for the entire volume F= knD'^ 172 PROPERTIES OF PLANE LOCI. 2d 6, Cor. 1. — Since ^tti^ = ^r x 4i7rY% the volume of a sphere == the surface X ^ the radius. 2S7» CoK. 2. — The volumes of spheres are to each other as the cubes of their radii or their diameters. 2SS. CoE. 3. — The portion of a sphere included between two parallel planes is called a segment. If a and b represent the distances of these planes from the centre^ the volume of the segment is Y = 7r[v^{h — a) — i(b3 — a^)]. 239, Cor. 4. f tti-^ = §of Ttv^ x2t = ^ of the circumscribed cylinder. See Scholium under Art. 2S4:, Ex. 2. Find the volume of the Prolate Spheroid. Also of the ob- late Spheroid. Show that each is |- of the circumscribed cylinder. Deduce from each the volume of the sphere. Ex. 3. Find the volume of the paraboloid. Volume = one half the circumscribed cylinder. Ex. 4. Find the volume of the cissoid revolved about its asymptote. Solution. —Let the curve AM revolve about BT, then will PDP' D' be an elemental section of the vol- ume if DD' = cZ?/. PD = 2a — x, and the circle traced by P D in its revolution is ;r(2a — xy. There- fore the element of volume or d V= tt (2a — x) ^dy. From the equation of the curve y^ = - — - — -, we have dy = X 3a.r2 — x^ , (3a — x){1ax — x'^f , „,, — -dx = dx. Whence sub- Fig. 184 i- i X a i stituting, dV= 7t{3a —x){2ax — x'^^dx, = Za7tx^(2a — xYdx — 7tx^ {2a — x)'^dx. These terms may be integrated by the use of formulas ^ and <^, and we find ^a7tx^{2a — x)'^dx— j 7tx\2a — xf dx = jt^a\ »/o <♦♦»■ m/. T Y / /^/ K D' \ / p/ \ D IJ' \ A E I X SECTION XIIL Equations of Curves deduced by tlie aid of the Calculus. 2(y0, Def. — Tlie Tractrix or Equitangential Curve is generated by the motion of a weight drawn upon a plane by a cord of constant length, the extremity of the cord moving along a straight line not in the direction of the cord itself. The portion of the tangent inter- EQUATIONS OF CUEVES DEDUCED BY THE AID OF THE CALCULUS. 173 cepted between the curve and the fixed hne, is a constant quantity (the length of the string), and hence the second appellation. ItiTjUS. — Let a weight be placed at B, Fvj, 185, with a string of the length A B attached. As the extrem- ity of the string, A, is carried along the line AX, B will trace BM, the iractrix. (This conception sup- poses friction to exist but not momentum.) "When the extremity of the string is at any point in the line, '~fi\ d~o* t^ as "T, the weight will be at P, and it is evident that Fig. 185. PT", the tangent, will be constant. . 23 1. I^TOh, — To find the equation of the Tractrix. Solution.— Let P and P' be consecutive points on the curve, PD = y, PE = — dy, (minus since y decreases as a; increases), AD = a; and D D' = dx. Let AB = PT = a. Then DT = v/«2 — yi, and PE : EP' :: PD : DT, or by the notation — dy : dx : : y : (a'^ — 2/'") • • * • i^ ~\ = 0- ClX s- (a2_ 2/2)2 Or this differential equation may be obtained from the general differential value of the tangent ; thus (l-\-~ yy = a, whence -^ = zh . The -J- sign indicates that the curve is generated by motion to the left, as then tan— i -- is + dx ' and the — sign is to be taken when the curve is generated as in the illustration above. In order to have the equation in finite terms it remains to integrate this differential between proper hmits. The inferior limit of x is evidently 0, to which y = a cor- responds. The superior limit must be left general, as the curve extends to infinity. J. (^2 ^2)2 /•■e Putting the equation in the form dx = — -^ — dy we have / dx = y Jo '-—1-Ldy = - ( 1- M =alog. ^ Ja ^i;ra2_w2^2 c«2_„2>>2^ y{a^ — y^Y {a^ — y^f^ ^ («2 — y2y. .'. The equation is a; = « log •— (a^ — y^y. 262, JPvoh, — To find the equation of a curve whose subnormal is constant. Solution. — The general differential value of the subnormal is t/--. .*. Letting dx p be the constant value of the subnormal, we have y-j- =p, or ydy = p dx. In- tegrating, ?/■- = 2px -f- G' This is the equation of the parabola, as it should be, since the constant subnormal characterizes that curve. G is not determined by the problem. If the condition "which passes through the origin," were added to the problem, C would be 0. 174 PROPERTIES OF PLANE LOCI. 203, JProb, — To find the equation of the curve whose normal is of constant length. ax'- J ting this equal to the constant r, we have y\\-X- -^ ) = r, or — = 1, or \ dxV dx^ j/"2 dx = ^ j^ = y{r^ — 2/^) *(^2/- Placing the origin on the curve, so that the inferior hmit of the integral shall he x = 0, y = and we have I dx = 2/(r2 — y^) ^dy, or a; = r — {r^ — y'^)'^ whence ?/2 :=:, ^rx — x^. This is the well known equation of the circle, as it should be, the normal of the circle being its radius, and hence constant : 204. I^voh, — To determine the equation of the curve whose subtan- gent is constant (m). m log y = X + C. The logarithmic curve. 2SS, I^TOb, — To determine the equation of the curve ivhose sub- normal varies as the square of the abscissa. y2 = 2.X3 _|_ c. The semicubical parabola. 206. ^TOh. — To determine the equation of a curve such that the area shall equal twice the product of its co-ordinates. SoLimoN. jydx = 2xy, or ydx = 2xdy -}- 2ydx or — = — i — . Integrating, y X logy = — \\ogx -{-C= — log.'j;^ -f- ^- •*• C=logxy. As log a;^2<' is constant x'^y must be constant ; therefore we may put x^y = m, or xy'^ = m'^ is the equa- tion sought. 207. J^rob. — Find the equation of the curve whose arc varies as the square root if the third power of the abscissa. Solution. J (dx^ + dy^)- ^^ ^x^, using ?7i for any constant factor. Eemovmg the sign of integration by differentiation and squaring both sides we have dx'^ -f- dy'^ 1 8 1 = fm2xda;2, or dy = (f m^x — 1 ) '^ dx , Whence integrating, y = EF. — A Subtangent is the portion of the axis scissas intercepted between the foot of the ordi- nate from the point of tangency, and the inter- section of the tangent with this axis ; or it may be defined as the projection of the correspond- ing portion of the tangent upon the axis of x. DT is the subtangent corresponding to the point p. of ab- 272. ^Toh, — To find the length of the sub- tangent. Fig. 188. Solution. — Letting a represent the angle which a tangent to the curve at the specified point makes with the axis of x, find tan a as in {268). Whence from the PD , ,,, , , ,. y triangle PDT", Fig. 188, we have tan a:, or D T (the subtangent) DT ^ "'tana: A slightly diff'erent method of solution is to produce the equation of the tangent to the curve, and then find where it intersects the axis of x. This intercept, as AX, Fig. 188, and the abscissa AD make known the subtangent, which is always their algebraic difference. Ex. 1. Eind the value of the subtangent of the common parabola at {x', y'). Solution. «/'2 2px' -We find in {268 Ex. 1) that tan a = ^,. Whence Suht y y tana V = -^- = 2a;'. 273, ScH. — ^From this example we learn that the subtangent in a para- bola is always equal to twice the abscissa of the point of tangency. Hence to draw a tangent to a parabola at any point as R, Fig. 188, let fall the ordi- nate PD, take AT == AD, and draw PT. 180 PBOPERTIES OF PLANE LOCI. Ex. 2. What is tlie subtangent to y^- := 10^, at .r = 6? At2/===8? At 2/ = 12 ? Draw these tangents according to the schohum. Ex. 3. Find the value of the subtangent to the ellipse at {x', y'). SoLTTTioN. — "We have tana A^y'-i A^ — .r'2 . ,hjEx.4. {268), yrhenceSuht. = ^— -A'^y tan a B-x x not a direction. neglecting the — sign as we are inquiring simply for a value, Fig. 189. 274:, ScH. 1. — From this example we learn that the subtangent in the ellipse does not depend upon the conjugate axis, but only on the transverse axis and the abscissa of the point of tan- C| gency. Hence if sevei^al ellipses he drawn on the same transverse axis the subtangenis corresponding to the same abscissa are equal. Thus in Fig. 189 let APB, AP'B, AP' B, and AP"B be several ellipses on the same transverse axis CB, then AD being any abscissa [x'), the subtangent corresponding to each point of tangency P, P', P", P'", is J^i x'- '- — . Hence the tangents at these several points cut the axis of x at x the same point T. 275, ScH. 2. — This property affords a convenient method of drawing a tangent to an ellipse at a given point in the curve. Thus let P be the point, Fig. 189. Draw the circle upon the transverse axis CB, draw the ordinate PD and produce it till it meets the circumference of the circle in P", and then draw a tangent to the circle at P'". T being the point at which the latter intersects the axis of x, is also the point at which au corresponding tangents to ellipses on the same axis intersect. Hence draw- ing PT it is the tangent sought. 276, CoK. — The expression for the subtangent being independent of B, the property is the same in the hyperbola as in the ellipse {00). How- x'2 — A2 ever as x ^ A in the hyperbola, we may write Subt. to have it positive. in order Ex. 4. I^rojy* — In an ellipse or hyperbola if from any point in the curve a tangent and an ordinate be drawn to the transverse axis, half this axis is a mean proportional between the distances of the intersections from the centre.. NOEMALS. 181 Dem. — Let P be any point in tlie curve and PD, PX the ordinate and tangent. Then AX :AB ::AB:AD. For we have the equation of a tangent at P {x', y'), A''yy' rt B'^xx = ± A^B\ Making ?/ = 0, we get x=—,. But a is AX and x is A D. Hence X X : A :: A -.x, or AX : A B : : A B : A D. The demonstration being the same in each case. Fig. 190. Fig. 191. 277* ScH 1. — To drav) a tangent to a hyperbola at a giveri point. From the given point of tangency P, F'ig. 191, let fall the ordinate PD ; and upon the transverse axis HB, and the abscissa AD, draw semi-circumferences. From their intersection let fall LT a perpendicular upon the axis of x. Draw a line through P and T and it is the tangent sought. Proof. Drawing AL and LD, we have AD {ov x') : AL (or Jl) : : AL (or A) : AT. Whence AT = — and is the intercept made by a tangent at p. 278, ScH. 2.— The pupil can scarcely fail to notice the close analogy between the forms of the equations of the conic sections and the equations of their tangents ; 6y simply dropjnng the accents the latter return to the former. Thus dropping the accents A^-yy' + B^xx' = A'^B'- becomes A^y'^ -f B'^x'^ = A^B% ^'^yy' — B-^xx' = — A^B^ becomes A^y^ — B^x"^ = — A^B% yy' = P{^ + ^') becomes y^ = 2px, and yy' -^ xx' =z E-2 becomes y^ -{- x^ = Rk NORMALS. 279, Dep. — A l^OTTYial to a plane curve is a perpendicular to a tangent at the point of tangency. 280, JPvoh, — To produce the equation of a normal to a plane curve. Dem. — Let PE be a normal to the curve M N at the point P, the co-ordinates of which are {x\ y'). The equation of a tangent at the point P is 2/ — y' = a{x — a;'), in which a is the tangent of the angle which the tangent at P makes with the axis of x, that is tan SXX ; and is to be determined from the equation of the curve as in the preceding part of this sec- tion. Now the equation of a line passing Fig. 192. 182 PKOPERTIES OP PLANE LOCI. tkrougli (x'j y') and perpendicular to the line y — y' = cl{x — x), is y — y' =^ ' {x — a;') {39), the coefficient being the negative reciprocal of the tan- gent of the angle which the tangent to the curve makes with the axis of x. 251, CoE. — The tangent of the angle luhich a normal to a curve at a particular point makes ivith the axis of x is the negative reciprocal of the mngent of the angle which is made by a tangent to the curve at the same point. Ex. 1. To find the equation of a normal to an ellipse. Solution. — The equation of the tangent to the ellipse is A-yy' -f- -B-xx' = A^B^, B^x' B- ov y = J7~r'^ H r- Now the equation of any Hne passing through {x, y') is A-y y y — y' = ci{x — x') ; but in order that this should be a normal to the ellipse we must have a = ,,^' , , the negative reciprocal of the tangent of the angle which the A'^y' tangent makes with the axis of x. Hence y — V' = -^^—X^ — ^') is the equation JD-X of a normal to the ellipse. Ex. 2. Show that the equation of a normal to an hyperbola is Ex. 3. Show that the equation of a normal to the parabola is I y' f i\ y — y' = ~ -{oc — x'). y' Ex. 4. Show that the equation of a normal to the circle is 2/ == —,oc^ and hence is the radius. Ex. 5. What is the equation of a normal to the elhpse whose axes are 8 and 4 at ^ = 1? At a: = — 1? At ^ = 3? One equation is y = ^ ^vlx qp f v 7. Ex. 6. What is the equation of a normal to the parabola whose parameter is 9, at a: = 4 ? At ^ = 9 ? At ^ = 5|- ? One equation is 1/ = qp 2^ db 27. 252, Cor. — The general expressions for the tangents of the angles which normals to the conic sections make with the axis of x are : A-y' A^y' For the ellipse -^7^, for the hyperbola — :^ — ,, y' Y For the circle — ^, for the parabola — — . SUBNOBMALS. 183 SUBNORMALS. 283, Def. — The Subnormal is the projection of the normal upon the axis of x ; or it is the distance from the foot of the ordinate let fall from the point in the curve to which the normal is drawn, to the intersection of the normal with the axis of ^, as D E, Fig. 193. Ex. 1. Show that the subnormal in the B'^x' ellipse is equal to A^' and has the same Whence numerical value in the hyperbola. Sug's. — In case of the elhpse, let PE be the normal at P and ED the subnormal. Now --— - = tan PE D, or -4 — = —^,. E D Subnor B'^x Subnor = — —. A'' Ex. 2. Show that in the parabola the subnormal is constant and equals the semi-parameter. Show how this property may be used to draw a tangent to a parabola when the focus is known. Fig. 193. 284. JPvop, — In the parabola a line joining the focus and the inter- section of a tangent with the axis of j {a tangent at the vertex), is per- pendicular to the tangent. Dem. — Let F be the focus, PT a tangent, and PE a normal. Join the intersection L with F. Then is FL perpendicular to PX. For, since AT = AD (^. 1, 272), TL = LP. Again TF = AX + AF = AD + ^p. Also FE = AD+DE — AF = AD+p — ip = AD + ip. "Whence as X P and X E are bisected by F L it is parallel to the normal P E and hence perpendicular to the tangent PX. Q. e. d. Fig. 194. 28d. Cor. PF=TF=FE. Also angle PT F = T P F, and FPE=FEP. ^^am PFX==2PTX. 286. ScH. 1. — Having given the curve and its axis, to find the focus draw a tangent to any point, as P, by (273), and then erect a perpendicular to it where it intersects the tangent at the vertex. The intersection of this perpendicular with the axis of x, will be the focus 287- ScH. 2. — Having given the axis and focus, to draw a tangent and a normal at P, take FX= FP= FE and draw PX and PE. 288, ScH. 3.— Having the axis and focus, a tangent may be drawn making 184 PROPERTIES OF PLANE LOCI. any given angle with the axis of x, by making P FX = twice the given angle, and drawing a tangent from the point where P F intersects the curve.. ^ » » EJECTION XV, Special Properties of tlie Oonio Sections. [Note. — The impoi-tance of the Conic Sections renders it necessary that their properties should be more fully developed than is found expedient in a compendious presentation of the subject of the General Geometry, and hence this section. Similar sections might be added on other curves, as of the cycloid, the catenary ; or sections discussing the loci embraced by equations of the 3rd degree, or the 4th degree, etc. But these subjects are not of sufficient importance to reqinre treatment in an elementary course, nor capable of being epitomized so as to be brought within proper hmits for such a course. Those who wish to pursiie the subject farther wUl find Salmon's Conic Sections and Higher Plane Curves in two volumes, or Price's Infinitesimal Calculus in four large volumes, the best English resources. Todhunter's four volumes, two on the Calculus, and two on the Co-ordinate Geometry, are also among the most valuable recent treatises. The author of this volume proposes to prepare a second volume on loci in space, and a more extended course in the Calculus, as soon as he is able.] (a) EADII VECTOEES AND THE ANGLES WHICH THEY MAKE WITH A TANGENT. 2S9, Def. — A Radius VectOi^ of a conic section is a line drawn from a focus to a point in the curve. 200, JPTop, — In an ellipse the sum of the radii vectores to any point in the curve is constant and equal to the transverse axis, and in the hyperbola the difference is constant and equal to the transverse axis. Dem. P being any point in the curve, M let PF = r and PF' = r' . We have V P D"'-f D f' = v/^HM^e" XV-' Fig. 195. also r'=V p D^-f D F''^=^Vy'-\-{Ae-\-xf. But 2/2 = (^2 _ a;2)(i _ e2) {S2), Sub- stituting this value of y-, we have r = \/[A^ — x^){i — e'^) -f {Ae — x)-^ = \/A^ — 2Aex -\- e^x^ = A — ex\ and r' = V{A^—x^}{l — e'^)-\-[Ae -f- 03)2 = v/^a _|_ ^Aex + e^'^ = A-\- ex. Adding, r' -\- r =. 2A, for the ellipse. In the hyperbola A — ex, is negative, since ex ^ A, and we write j-' = JL -f- ex, and r = ex — A. Whence, subtracting, r' — r = 2 J., q. e. d. 201, CoR. — The length of a radius vector drawn to the nearer focus t.s r = A — ex, and to the more remote x' = A -f ^x. SPECIAL PKOPERTIES OF THE CONIC SECTIONS. 185 N • 292, ScH. — The principles enunciated in tliis proposition afford very simple means for constructing the loci mechanically. For the ellipse take a string equal in length to the transverse axis, and fastening its ends at the foci, put a pencil against the string and move it around the perimeter of the cm-ve, keeping the string tense. Thus F'PF Fig. 195, represents the string, and P the pencil when the point P is located. To construct an hyperbola, take a ruler AB, and a string BPF ; mak- ing the string shorter than the ruler by the length of the transverse axis of the required hyperbola ; fasten one end of the string to one end of the ruler, as at B, fasten the other end of the string at one focus, as at F, and the other end of the ruler at the other focus, as at F'. Place a pencil against the string and bear it against the side of the ruler, as at P, and keeping the string tense move the pencil around the curve. It is evident that F'P — PF = 2 A in all positions of P. Hence P traces the curve. To trace the other branch the attachments have to be changed, so that the free end of the string shall be attached at F', and the end of the ruler at F. Fia. 196. 203, JPtop* — The radii vectores drawn to any point in an ellipse of hyperbola make equal angles with the tangent at that point. Dem. — Let PF and PF', be the radii vectores, and M T the tangent, Fig's. 195, 197. ThenFPT=F PM. For, AT = — {137, Bx. 1, ox 276, Ex. 4), and AF =:A F'==Ae. Hence, in the ellipse, F"r= A A ~{A — ea'), and F'X := —{A 4- ex)'. X X and in the hyperbola FT = — (ex — A)^ and F'T -(ex -f A), Wherefore in Fig. 197. either case we have FT : F'T : : r : r' {201). Now drawing F'M parallel to PF we have FT : FT : : PF : F'M, or r : r' : : r : F'M. .• . F'M=r' = F'P, and F MP = FPT= F'PM. q. e. d. 204:, CoE. — In the ellipse the normal bisects the angle included by the radii vectores to the same point ; and in the hyperbola it bisects the angle included by one radius vector and the other produced. 186 PKOPEKTIES OF PLANE LOCI. ivi; Fig. 198. 295, ScH.— This principle affords one of the most convenient methods Df drawing tangents to these curves. 1st. To draw a tangent through a giveji point in the curve. Let P, Fig's. 195, 197, be the point. Draw the radii vectores to the point and bisect the included angle for the hyperbola, or the angle included by one radius vector and the other produced in the case of the elHpse. 2nd, To draw a tangent frovx a point with- out the curve. Let P be the point. Join P with the nearer focus, and from P as a centre pass an arc of a circle through that focus. From the other focus, with a radius equal to the transverse axis, strike an arc cutting the former as at D and D'. Join D and D' with F', and T and T' are the points of tangency. To prove this for T, join D and F, and F and T. Now F'T + TF =: F'D, since each =2-4. Hence TD = TF. Moreover DP = PF. Hence "TP is perpendicular to DF, and angle DTP = MTF'=PTF. Whence we know that PM is tangent at T. In a similar manner PM' can be shown perpendicular to FD', and hence tan- gent at T'. [The student should com- plete the figure and give the demon- stration in the case of the hyperbola.] Fig. 199. 3rd. If a circle be described on PF and another on CB, the lines passing from P through their intersections are tangents to the curve, and this whether P is in or without the curve. [Why? The student should be able to answer after having read in this section through the next two propo- sitions.] This method, however, is imjDracticable when p is without the curve, as it does not indicate the precise point of tangency. 200, JProp, — In the 'parabola the radius vector drawn to the point of tangency makes the same angle with the tangent as a diameter through the same point, or as the tangent does with the axis of abscissas. Dem. — Let PF be the radius vector, PF' the diameter, and MT the tangent. Then FPT = F'PM. For, AT= X {140, ov 272, Ex. 1), and A F = -^p. Hence FT = x -f Ip. But PF = PD = BE= BA + AE=rJp-f-a;. .-.FT = FP, and angle FPT=: FTP=F'PM. q.e.d. Fig. 200. SPECIAL PBOPERTIES OF THE CONIC SECTIONS. 187 297* Cor. — In the parabola the normal bisects the angle included by the radius vector and a diameter at the same point in the curve. 298, Sen.— To draw a iangerit to the poiiit P in the parabola, draw the radius vector and the diameter to the point, produce one of them (as PD) and bisect the angle thus formed. To draw a tangent from a point without as P", join the point with the focus, from P" as a centre, pass an arc of a circle through the focus, and through its intersections with the directrix draw diameters. The vertices of these diameters are the points of tangency on the curve, as P and P'. To prove this, observe that as PF == PD, and P"F = P"D, P"P is per- pendicular to DP and angle FPP" = DPP"= MPF'. [Let the stu- dent give the proof for the point P'.] [Note.— The properties demonstrated in these propositions give elliptic, hyperbolic, and paraboUc reflectors their pecuUar properties. Thus, rays of light, sound, or heat diverging from one focus of an elliptic reflector aie converged at the other ; diverging from one focus of an hyperbolic reflector, they diverge after reflection as though they proceeded directly from the other focus ; and diverging from the focus of a parabolic reflector, they are reflected parallel. Conversely to the last, incident rays parallel to the axis are concentrated at the focus of a para- bolic reflector.] 200, JPvop, — The rectangle of the perpendiculars from the foci upon the tangent of the ellipse or hyperbola is constant, and equal to the square of the semi-conjugate axis. Dem. — Let L'X be a tangent at any point P, and FL a perpendicular from the focus upon it. Pro- duce FL till it meets F'P, produced if necessary, in D, and draw AL. Since AF = AF' and FL = LD, A L is parallel to F' D and equal to i F' D = A B. Hence the foot of a perpendicular from the focus upon a tangent lies in the circumference of a circle described on the transverse axis. Now let F'L' be the perpendicular from the second focus upon the tangent L T, we are to show that FL X F'L' = BK Join A and L' and produce the line till it meets LF produced in L". Then the triangles AL'F' and AL"F are equal, and AL" = AL', and L" is in the cir- cumference of the circle described on C B. Finally, FL X F'L' = FL X FL" == FB X FC (CB and LL" being chords in the same circle). But FB X FC == {A -f- Ae){A — Ae) = ^^(l — e^) = B^ {49, 3rd and 7th). .-. FL X F'L' = B\ Q. E. D. Fig. 202. 300. Cor. — The semi-conjugate axis is a mean proportional between Qui focal distances. 188 PKOPERTIES OF PLANE LOCI. (&) SUPPLEMENTAHY CHOKDS AND CONJUGATE DIAMETERS. 801. Def. — The term Ordinate, as used in connection with the conic sections, may mean any line drawn from a point in the curve to any diameter, and parallel to a tangent^ at the extremity of that diameter. 302. Def. — SupplemeTitary Chords are chords drawn from any point in the curve to the extremities of any diameter. 303, Def. — One diameter is said to be conjugate to another when it is parallel to a tangent at the extremity of the latter. 304:, JProp, — In an ellipse the rectangle of the tangents of the angles which a pair of supplementary chords make with the transverse axis is B2 equal to A2' Dem, — Let PC and PB be supplemen- taiy chords drawn to the axis. Let the angles PCX and PBX be represented by a and a', and tan a = a, and tan a = a'. Then aa' = — : -— , A and B being the semi- axes. The equation of PC isy=a{x-\-A), and of PB y = a'{x — A), disregarding the signs of a and a', since PC and PB are, as yet, any lines passing through C, and B. For the intersection of thest lines these equations are simultaneous, and we may have y"^ = aa! {pfl — A^^. Again, when the point of intersection is in the ellipse, this result is simultaneous with the equation of the curve, A-yl^ -\- B-x- = A'^^B'^, ox y^= —{pfi — A~) ; D Fig. 303. A^^ B-i A- whence combining the two, we have aa'=: — -— . q. e. d. 303, CoK. 1. — By a similar course of reasoning, or by simply changing the sign of B^*, we have for the hyper- B^ I? bola aa' 306, CoK. 2. — In the ellipse, if supplementary chords are drawn to the extremities of the conjugate axis aa'= B2 Fig. 204. -j— {the same as before) if the angles are measured from the axis of x, A^ hut aa' = — A^ — if they are measured from the axis of y. SPECIAL PROPERTIES OF THE COXIC SECTIONS. 189 Dem. — The equations of P'E and P'D, Fig. 203, are respectively y — B = ax and y -j-- B = a' X ; whence t/^ — B^ = aa'x'^. From A-y^ -j- B-x^ -.= A^B^ we have ^2 B^ 2/2 — B- = —x'^. . ' . aa' = ■— . If the angles are reckoned from the axis A^ A^ of y we observe that for a we shall have , and for , 1 a , -. a aa = 4! 307 • Cor. 3. — In the hyperbola, if supplementary chords are drawn from any point in the conjugate curve to the extremities of the conjugate axis aa' =. — if the angles are reckoned from the axis of x, hut aa' = A2 . B2 if they are reckoned from the axis of j. Dem. — [Let the student supply the demonstration, and also show that if the chords be drawn from a point in the x hyperbola to the extremities of the conju- gate axis, aa' is not constant. ] SOS* Cor. 4. — In the circle this relation becomes aa' = — 1, or 1 -}- aa' = ; which shows that the chords are perpendicular to each other, which is a well known property of the circle. In the equilateral hy- perbola the relation is aa' = 1, or a ^= — , signifying that the angles are a complementary. 309, Cor. 5. — ^ one of two supplementary chords to either axis is parallel to one of two drawn to the other axis the other two are parallel. . If a is the Dem.— In either Fig. 203 or 204 if P'E is parallel to PC, P'D is parallel to PB. For we have m case of each set of chords aa = -t- ~— . A'^ same in each, a' is also. 310. ScH. — The — sign in the formula indicates that a and a' have op- aa = A-^ posite signs in the ellipse. Thus P being the point from which the chords are drawn, PBX > 90°- and < 180° gives — a, PCX Substituting these values, there results in the case of the ellipse ^-sin a sin a' 4- -S- cos oi cos oc' =r: 0, and of the hyperbola A'^ sin a sin a — B- cos oc cos oc' = 0, formulcE which are sometimes referred to as the squations of condition of conjugate diameters. dth. From, the relation aa B^ — we may also solve the 2nd above. Thus, if (j be the given included angle, (S = a — oc, and tan /? = B^ A-^a 1 -j- aa supposed known.. whence a' can be determined, as all the other quantities are SPECIAL PROPERTIES OF THE CONIC SECTIONS. 193 318. JProb* — To investigate the relation between conjugate diameters and the axes. Dem. — The equation of the ellipse and hyperbola referred to the conjugate diameters 2^2 and 2^2 is A^^^y^ ± Bz^x^ = ± A-y^B^^ {127, jEx's. 10 and 11) ; the 4- sign applying to the ellipse and the — sign to the hyperbola. Transforming this equation so that the reference shall be to the axes, by means of the formulce 2/2 we have Az^y^cos'^a zb B-i'^y- cos- a y cos a. — a; sm a sm {a a) X2 — X sm a y cos a sin (a a) {127, ^cm), ± A2^B2^Bin^{a' — a). - 2^2 ^^y cos asina -\- Az^x^ sin^ a 2B.2'xy cos a'sin a' ± B^^x- sin2 a' Comparing this with A'^y'^ ± B^x:^ ^ ± A^B% we have (1) A2 2 cos2 a ± -B22 cos'^a' = J.2, (2) Az'-'sin^a ± B^^sin^a' = ± ^2, (3) Az'^ cos a sin a ± 5^2 cos a' sin a = 0, and (4) Az^Bz'^sm'^ {a — a) = A^B^. 1. Adding (1) and (2) and remembering that sin2 -j- cos2 = 1, we have A^^ =h B^^ =^ A^ ± B% that is (a) In the ellipse the sum of the squares of conjugate diameters is constant, and equal to the sum of the squares on the axes. In the hyperbola the difference of the squares is constant and equal to the difference of the squares on the axes. cos a sin a _ . , , . , ^ -h 1 ; but as ^2 and B^ ^^^ conju- 2. Making A2'^=B2'^ (3) gives cos a sm a gate tan a tan a' = smasma B^ ^^ ■,,.-, . ,, , „ ., ; = -t- -— . Multiplymg the members of these cos a cos a A^ equations and rejecting common factors we have sin2 ^ J52 sm a —-, or A'^ cos a' B A' the — sign characterizing the ellipse since a' is obtuse in the ellipse, and the -f- sign characterizing the hyperbola, as in it a and a' are acute. Hence B cos a sm a indicates the position cos a of equal conjugate diameters in the ellipse, and = —. in the hyperbola. From Fiq. 210 we see that sm a: A D and A F meet this condition in the ellipse ; for AC A , . . . HB AK = B sin a cos KAC — cos a'' Hence and AH sin H A B Fiq. 210. (b) In the ellipse the conjugate diameters which fall upon the diagonals of the rect-i angle on the axes are equal, and a and a' are supplementary. A B 3. In the hyperbola in general, the condition ; = -: — is met only when the cos a: sma two diameters, as G F and DE Fig. 212, coincide and fall on the asymptote. Hence, in the hyperbola, asymptotes are the analogues of the equal conjugate diameters of the ellipse. But from -^22 — B^^ = A'^ — B^, we observe that if A =:s B, A2 = Bz, independently of a and a. Henc6 194 PROPERTIES OF PLANE LOCI. (c) In the hyperbola, in general, there are no equal {finite) conjugate diameters ; hut, in the equilateral hyperbola, any pair of conjugate diameters are equal each to each. The conjugate diameters of the equilateral hyperbola find their analogues in the diameters of a circle. 4. Making A2 = B^ in A^^- -}- B^^ = A^ -{- B% we find that J^ (d) The length of one of the equal conjugate diameters of an ellipse is \/2 v/A^-j-B--^. .-. The semi-conjugate diameter: the semi-diagonal on the axes :: 1 : V^. 5. Extracting the square root of both members of (4), we have A^^Bz sin (a' — a) = AB ; .which signifies that (e) The parallelogram formed by tangents drawn through the vertices of any pair of conjugate dia- meters is constant, and equal to the rectangle on the axes. This will be more apparent from Fig's. 211, 212. D A F = (a' ~ a), and D A = ^5^, ; whence AO = -S2 sill (<^' — ^)- Hence J.2-B2 sin {a' — a) = area AFHD = iLKIH = AB = i the rectangle on the axes. . • . L K I H = the rectangle on the axes, Ex. 1. Write the equation of an el- lipse referred to a pair of conjugate diameters whose lengths are 8 and 6, and the included angle tan~^ ( — 2). Having written the equation con- struct it as in Chapter I., Sec. II. Ex. 2. Write the equation of an hyperbola referred to conjugate dia- ^ig* 212. meters whose lengths are 12 and 8, and whose included angle is tan~^2. Construct as in the last example. Ex. 3. In an ellipse whose axes are 8 and 6 what is the length of a diameter which makes an angle of 45° with the axis oi x? What is the length of its conjugate ? Stjg's. — ^From the relation aa' = j > we learn thart the conjugate diameter makes with the axis of x an angle of 150° 39' nearly. Hence A2B2 sin {a' — a) = AB, becomes ^12^2 sinl50° 39' = 12, or AoB. = ~J^. AlsoA^^ 4- -822 = ^2 + B^ .49014 2 r z I gives A2^-i-B2~ = 2^5. These two equations will give the values of A2 and Bz- Ex. 4. In an ellipse whose axes are 8 and 6, what are the sides of the circumscribed parallelogram whose sides are parallel to the equal conjugate diameters ? What is the altitude of this parallelogram ? Altitude, 6.79 nearly. SPECIAL PEOPERTIES OF THE CONIC SECTIONS. 195 (€) PROPEETIES OF ORDINATES. 310. JPvop, — The squares of ordinates to the transverse axis of an ellipse are to each other as the rectangles of the segments into ivhich they respectively divide the axis. Dem.— Let PD =y, FD' = ?/'. AD = jc, A D' = X', then A-y^' + B^x^= A'^B'^ and A-y'^ -f- B-^x''^ = A^B^ ; whence y^ = B^ B-^ -— {A^ — x2) and y'^ = -r;( A^ — x'^). Divid- A^ A~ ing and rejecting the common factor we have ^~ = = ^—-^, — -^^ -', or yi ^2 _ a;2 ^A-\- x){A — x) y2 : 2/'2 : : (^ + x){A —x):{A-\- x'){A — x') or :: CD X DB : CD' X D'B. ^•^•^- Fig. 213. 320, Cor. 1. — The square of any ordinate to the transverse axis of an ellipse is to the rectangle of the segments into which it divides that axis, as the square of the conjugate axis is to the square of the transverse. Dem.— In the above proportion if y' == GA=B, {A -\- x'){A — x') = A-, and we have 2/2 : -B2 : : C D X D B : ^2. ... y2 : CD X DB : : 4:B-i : 4:A^ q. e. d. 321, CoE. 2. — The latus rectuyn is a third proportional to the trans- verse and conjugate axes. Dem.— In the last proportion let y become the focal ordinate P"F, which call p, and CD X DB becomes CF X FB = {A-\- c){A — c), c being AF. Now {A + c){A — c) = ^2 — c2 = B^ hence p^ iB'^ :: B'^ : A% or 2A :2B ::2B : 2p. Q. E. D. 322, ScH.— The prop- erties demonstrated in this proposition, and in the 1st and 2nd corolla- ries, are equally true for the hyperbola, and can be proved in the same way. In the case of the hyperbola, however, the statement should be, The rectangles of the dis- tances frota the feet of the ordinates to the ver- tices, instead of "the ^^^' '^^'^' rectangles of the segments, etc.," as, in this case the ordinates do not 196 PBOPEKTIES OF PLANE LOCI. divide the axis, but fall upon its prolongation ; so that, in Fig. 214, we have y2 : y'2 ; ; qq x BD : CD' X BD'. 323, CoE. 3. — In the case of the circle Coe. 1st shows that the square of the ordinate equals the rectangle of the segments into which it divides the diameter — a well known property. 324:. Coe. 4. — This proposition and Coe, 1st may be asserted of ordinates to the conjugate axis. [Let the student give the proof and a figure to illustrate it.] 32S, Coe. 5. — This pj^oposition and Coe. 1st may also be asserted of ordinates to any diameter of an ellipse or an hyperbola. Dem. — The corollary can be proved in the same way as the proposition, by using the equation of the curves referred to conjugate diameters {127 f Ex's. 10 and 11), since these equations are of the same form as those used above. In the annexed figures, therefore, PD' : P D ' :: CD X DB : CD' X D'B. Also PD' : CD X DB ::^2 :^'2. Fig. 215. 326. Coe. Q^.—From the last relation, it fol- lows that chords paral- lel to any diameter are bisected by its conjugate, i e.PD = DH,P'D' = D'H', etc. ; and hence that these curves are symmetrical with re- spect to any diameter. 327* ScH. — These principles, together with others already known, enable us to find the centre, axes, and foci of the curves, geometrically, when the perimeters alone are given. Thus, in the case of the ellipse, let the curve NHIM be given, to find the centre, axes, and foci. Draw any two par- allel chords as DE and BO, bisect them at K and L, and draw FG ; it Avill be a diameter by Coe. 6. Bisect this diameter and A will l)e the centre. From A with Fig. 216. Fig. 217. SPECIAL PROPERTIES OF THE CONIC SECTIONS. 197 a radius sufficiently long to cut the curve, construct the circle HIMN, join two of the intersections, as I and H, and perpendicular to this chord pass a line through the centre ; it will be the axis. [The student can readily fin- ish the problem.] The construction is the same for the hy- perbola except in find- ing the conjugate axis when the conjugate hyperbola is not given. Fig. 218. For this purpose use the proposition in Cok. 1. In the figure, take SR X VR : HR^ : : AV' : AO'^ ; whence AO can be constructed. S2S, IProp. — In different ellipses upon the same transverse axis, the corresponding ordinates to the transverse axis are to each other as the con- jugate axes of the respective curves. Dem.— We have PG : CG X GB : :AD^ :AB' alsoP'G' : CG XGB: : AD ': AB' and P G^ : CG XGB:: AD'": ABl .• . PG : PG : P"G : : AD : AD': AD , etc. Q. E. D. 329, Co^.—Amj ordi- nate to the transverse axis of an ellipse is to the correspond- ing ordinate of the circum- scribed circle as the conjugate axis of the ellipse is to the transverse. [ y" Sf" . Z-^^^^Z— dJ[^ p>\ /^ -^^^^1_— D' / P^\^\ (^ D / P^^^^\\\ c // 1 A V G J -^x^ "■^^^-—^ H 1 ^^^-^^^y/ "^ ^^ B Fig. 219. Dem. — Let CD"'B be the circumscribed circle, then as it may be considered as an ellipse with equal axes, we have PG : P"'G :: AD : AD"'(= AB% Q. E. D. 198 PROPERTIES OP PLANE LOCI. 330, ScH. — An instrument called a Trammel is constructed upon the prin- ciple enunciated in this corollary. It consists of two grooved bars X ' X , Y Y ', fastened together at right angles, and an adjustable arm PH. H and I are pins which can be fastened anywhere on PH, and have heads on the under side which run in the grooves of the bars. Any point in the movable bar, as P, traces an ellipse as H and I slide back and forth in the grooves. To prove that P is a point in an ellipse of which PH is the semi-transverse axis and PI the semi-conjugate, draw AP"' and PH parallel to it, Fig. 219. Produce PG till it meets HE drawn paraUel to AB, in E. Then AP'" = ^ = PH. Again, P" G : PG : : P"'A : PI, or ordinate of circle : ordinate of ellipse : : A : PI. And as this is true for all positions, PI being made = B, and PH = ^, P is always in the curve. We may also demonstrate directly that the locus of P is an ellipse. Prom Fig. 220, using the common notation PI:PH ::PD:PE, gives B : A :'. y : >/ J.2 — x^, or, squaring, B^ : A^ : : y^ : A-2 — x^ ; whence A^y^ + B^x^ =^ A^B\ 331, I^TOp, — In different ellipses on the same conjugate axis, cor- responding ordinates to this axis are to each other as the transverse axes of the respective curves. Dem GE .: AB : AD DG ~ We have PG :DG X and p'Q- ; J/2 X G E : : A B' .-. PG : PG :: AB Q. E. D. AD^ : AB' 332, Cor. — Any ordi- nate to an ellipse is to the cor- resjionding ordinate of the in- scribed circle, as the transverse axis of the ellipse is to the con- jugate. [The student may make the deduction from the proposition.] SPECIAL PEOPEETIES OF THE CONIC SECTIONS. 199 333, I*rop* — The squares of ordinates to any diameter of a para-r hola are to each other as their corresponding abscissas. Dem. — Referred to any diameter, as AX or AjXi, the equation of the parabola is y^ = 2px {127 f ^x,. 12). Whence, letting y and y' represent any two ordinates, as PD and P'D, or PiDi and P^'D]', and .-r and ;c' the corresponding abscissas we have y~ = 2px and 2/ '2 ■= 2px'. Dividing, — - = -^. q. e. d, 334:, CoK. — All chords drawn parallel to a tangent at the extremity of a diameter of a par- ^ig 222 abola are bisected by that diameter. 335. ScH. — Having the curve to find the axis and focus of a parabola, we draw any pair of parallel chords, and bisect tliem by a right line. This line is a diameter. Draw two other parallel chords perpendicular to the diameter thus found, bisect these chords by a right line, and it will be the axis. Find the focus by {164, or 284). (d) ECCENTEIC ANGLE. 336, Def. — The JEccentric Angle in an ellipse is the angle formed with the axis of abscissas by a line drawn from the centre to a point in the circumference of the circumscribed circle where a pro- duced ordinate meets it, that is P"'A B Fig. 219. 337. I*rop. — The abscissa of any p>oint in the ellipse equals the semi-transverse axis into the cosine of the eccentric angle, and the corres- ponding ordinate equals the semi-conjugate axis into the sine of the same angle. That is, letting q) represent the eccentric angle, X = A cos qp, and y z= B sin q). Dem.— In Fig. 219, AG = a: = P"'A cos P " A B — Acos q). Also PG = y = P I sin P I G = 5 sin 9>. ScH. — The introduction of this angle is a recent device to facilitate the deduction of certain properties of the ellipse. It enables us to transform an equation in terms of rectangular co-ordinates {x, y) into one containing but one variable,^, which is sometimes of much advantage. We will give a few specimens of its use. 33 S. I^vop, — The equation of a tangent to the ellipse in terms of the eccentric angle is A sin cp ■ y -\- B cos cp ■ x = AB. Dem. — The equation of a tangent to an ellipse is A^y'y ■\- B-x'x = A-B\ As (x', y') is a point in the ellipse, we have x' == A cos cp, and y' = .Bsin q). Substi- tuting these values and dividing by AB, we have A sin cp- y -\- B cos

' = 90° -j- (p ; whence sincp' = cos cp, and P' A'^= A- — e2 • A-cos'2 (p ^= A^ — e-x^. .' . rr' = P'A^. q. e. d. 342* JPvop,—The sum of the squares of any pair of conjugate diameters is constant and equal to the sum of the squares of the axes. Dem. — In Fig. 223 we have P' A' =^ x^- -{- y^^ = B-sin^cp' -f- ^2cos2^' ; or since (p' =90° -|-^ sin q)' = cos cp, and cos cp'^= — sin cp, ■ — -, 2 P'A = A^ sin2 q) -^ B- cos^ qj ; and in like manner, P A' = ^- cos- q) 4- B- sin2 q). Adding P A' + P' A" = A"- -{- B^-. Multiplying by 4, 4PA' + 4P'A'=:4^2_|_4^e. Q. E. D. ScH. — This proposition has been demonstrated before {318 ^ a), but is inserted here as its demonstration affords an example of the utility of the eccentric angle. Ex. 1. What is tlie eccentric angle of the extremity of the trans- SPECIAL PBOPERTIES OF THE CONIC SECTIONS. 201 verse axis ? What of the extremity of the latus rectum ? What of the extremity of the conjugate axis ? Ans., (p = 0°, cp = cos~^ e = sin~^ — cp = 90®. Ex. 2. In an elHpse whose axes are 8 and 6, what is the eccentric angle at :r = 1 ? What are the co-ordinates of the point of which the eccentric angle is 60° ? 45° ? 30° ? Ex. 3. In an ellipse whose axes are 12 and 8 what is the length of the diameter from the point whose eccentric angle is 60° ? SuG.— Calling the semi-diameter A^ we have A2- = A'^ cos^ g) -\- B'^sin'^q)^ 36 X {ky -+- 16 X (iv'3)2 = 21, and A^ = v/21. 343, JProp, — The intercepts of a secant between the hyperbola and its asymptotes are equal. Dem. — Let DD' be any secant, and P and P', the points in which it cuts the curve, be designated respectively as {x, y') and (x", y"). Since DD' is a line pass- ing through the two points (x, y'), and {x", y"), we have for its equation y — y' = y y -{x — x). And since (.r', y') and X — cc" ' {X", y") are points in the curve x'y' = x"y" = m. If in the equation of D D ' we make y = 0, x = AD, and x — x' = CD. Hence we have CD = x — x' = X y — y X Fig. 224. _ x"y" — y'x" y" — y' = X" = y" — y' C'P'. Now as PCD and P'C'D' are equiangular and have CD = C'P', the triangles are equal, and PD = P' D'. Q. e. d. ScH. — This proposition afibrds an ele- gant and convenient method of construct- ing the hyperbola. If the axes are given, put them in position and draw the asymptotes, which are the diagonals of the rectangle on the axes. Then, through the extremities of the transverse axis, draw a convenient number of radiant lines, as aa' , 'hh' , cc\ dd\ and make the intercepts \a' , 2b', 3c', 4c?' respectively equal to Ba, Bb, Be, Be?. Then are 1, 2, 3, 4, points in the curve. If the asymptotes are given, or the Fia. 225. 202 PROPERTIES OF PLANE LOCI. angle included, and any point in the curve as P, the asymptotes can be drawn ; and then radiant hues through P will be secants whose intercepts will make known points in the curve. PARAMETER TO ANT DIAMETER. o44. A Parameter to any iPiameter of an Ellipse or Hyperbola, is a third proportional to that diameter and its conjugate. In the Parabola it is a third proportional to any abscissa and its cor- responding ordinate. 34:S, J^rop, — The distance from any point in a Parabola to the focus is one fourth the parameter to the diameter from that point. Dem.— Let A2 F =/ ; then is y2^ = ¥^z- From Ex. 12, page 88, we have 2/2 2 2p = ■ , X2 j and also 2n sin a' — 2r) cos a' 0. From the latter, n^ sin^o:' p2sin"a'; whence sin^a' p2 cos'-'a' = p2 — p2 n2-f p2 • 2p 2(7i2 4- p2) 2(2pm -f- p2) Hence sin-'a' p p 4(to -}- ip), since n^ = 2pm. But m -f- sP = TF = FA2 = /. Therefore 2/2^= ¥^2, or iCg : 2/2 • • 2/2 '■ ¥ '} and 4/ is the parameter to the diameter A. 2^2 Q. e. d. Fig. 226. 34:0, CoR. 1.— The parameter to any diameter of a Parabola is four times the distance from the vertex of that diameter to the directrix. 34:7* CoE. 2. — The double ordinate to any diameter of a Parabola, which {ordinate) passes through the focus, is the parameter to that dia- meter. Dem. — Let AgH =X2, and LH = 2/2 1 whence 2/2^= ^f^i- Now A^H =; Xi= TF = A2F =/. Wherefore ?/2 2 = 4/2 ; and 2/2 = 2/. But IL = 2LH = 2y.2. =^ 4f. . • . I L is the parameter to A2a;2' 34S, JProp, — Any chord ichich passes through the focus of an Ellipse is a third proportional to the transverse axis and a diameter par- allel to the chord. Dem.— Let PF=:r, PFB = a:, and P'F = r'. Then P _„,-, ... V -, and r' = _ , 1 — e cos a 1-j-ecosa: {107)', whence r 4-r'= PP' =: 86, 5i2 = 2p 1 — e- coti'^a But from JEx. 10, page AHl — e2) A'^ siii'^a -f- B^ cos- a e- cos- a P' Fig. 227. Ap 1 — e^ cos^a 2 SPECIAL PROPERTIES OF THE CONIC SECTIONS. ; by substituting A^(l — e^) for B^ and reducing. Therefore 203 pp_ , or 2 J. : BjCi : : BiCj : PP'. q. e. d. 349. ScH. — Tlie statement in [347) is not true in case of the ellipse, as will appear from this proposition. CHORD OF CURVATURE. [Note.— The following proposition is designed to be read by those who have taken the Dlfterential Calculus, and have studied Section VI, Chapter IV, or have some knowledge of the subject of radius of curvature.] 350. A Chord of Curvature is a chord of the Osculatory Circle, drawn from the point of contact. 351. I^rop, — In the parabola, the chord of curvature which passes through the focus is the parameter to the diameter passing through the point of contact. Dem. O being the centre of the os- culatory circle at 3P, in the parabola whose focus is "F, IE*^wC is the chord of curvature passing through the focus, and is the parameter to IP3D, the diameter through IE*. For, draw IFXj perpendic- ular to the tangent through P, and we have from the similar triangles Jr^JbdiiMC and 1P:E*'Lj, ■F:E<, : lE^IS^C : : IT : FI-., or. :!E*1^ = -p-p 2N^ But 1*1^ = — j », I[ being the normal, and p the semi-latusrectum of the curve {211); !FIj = ^!F*IEj = \]Sf {164: or 284), and IT = A/^^ + XJF *' = \/\n^-\-~N'^1 4pi pi 1 /2>2 4- ^2 7^2 g N'Y ^ / - — {143, Ex. 2), remembering that p^ + y^ = W\ Substituting these values, we have, IPIMI = — s- x -pr- 2i^3 ]^ 2p 2i\r2 p' 2 W' = = 4Jr'Jb' and hence is P the parameter to IE*ID {340). 204 GENEEAL SCHOLIUM. 352, CoK. — The chord l»s intercepted on the diameter I»ID is equal to the chord of curvature j^assing through the focus, since angle CI*ID = TI^OSJ:. EKD OF PART FIRST. GENERAL SCHOLIUM. Book Second, treating of Loci in Space, is reserved for a second volume. The present volume is deemed sufficient for the use of all students in our colleges, except such as pursue mathematical studies as a specialty. Yol- ume n. -^11 contain Loci in Space, and a more extended course in the Calculus. THE INFINITESIMAL CALCULUS. INTRODUCTION. [>ToTE. — The four following chapters on the Diflferential Calcixlus are to be read iniinediately after the first three chapters of the General Geometry, that is, the first 92 pages of this volume.] 1, Qtiatltity is the amount or extent of that which may be measured ; it comprehends number and magnitude. (See Akt. 4, General Geometry, and the two SchoHums under it on pages 1 and 2.) 2, NlilfYlbeT is quantity conceived as made up of parts, and answers to the question, "How many?" (See Art. 5, Illustration, General Geometry.) S* Number is of two kinds, DiscontiTlttOUS and CoTltin- uous. 4, Discofltiflttous JVtcmber is number conceived as made up of finite parts ; or it is number which passes from one state of aggregation to another by the successive additions of finite units, i, e., units of appreciable magnitude. S» ContiflUOUS JVttmber is number which is conceived as composed of infinitesimal parts ; or it is number which passes from one state of value to another by passing through all intermediate values, or states. Ill's. — The method of conceiving number with which the pupil has become familiar in arithmetic and algebra, characterizes discontinuous number. Thus the number 1 3 is conceived as produced from 5 by the successive ad- ditions of finite units, either integral or fractional. In either case we advance by successive steps oi finite length. If we say 5, 6, 7, etc., tiU we reach 13, we pass by one •^^^- ^^ kind of steps; and, if we say 5.1, 5.2, 5.3, etc., till we reach 13, we pass by another sort of steps {tenths), but as really hy finite ones. If, however, we call the hne A B, Fig. 1, x, and C D, x', and conceive AB to slide to the position CD, increasing in length as it moves so as to keep its extremities in the lines O M and B D '* B D Fig. 3. 2 INFINITESIMAL Cx\LCULUS. O N , it will pass by infinitesimal elements of growth from the value x, te the value x' ; or, it will pass from one value to the other by passing through all intermediate values, and thus becomes an illustration of continuous number. Again, if the line A B, Fig. 2, be considered as gen- ^ erated by a point moving from A to B, and we call AC B the portion generated when the point has reached C, • ^^^' 2- X, and the whole line x', x will pass to x' , by receiving" infinitesimal increments, or by passing through all states of value between x and x'. A surface may be considered as generated by the mo- tion of a Hue, and thus afford another illustration of continuous number. Thus let the parallelogram AF be conceived as generated by the right line A B moving from AB to EF. When AB has reached the po- sition CD, call the surface traced, namely A BCD, X, and the entire surface A B E F, x' ; then will x pass to x' by receiving infinites- imal increments, or by passing through all intermediate values. Finally, as volumes may be conceived as generated by the motion of planes, all geometrical magnitudes -afford illustrations of continuous number. We usually conceive of time as discontinuous number, as when we think of it as made up of hours, days, weeks, etc. But it is easy to see that such is not ttie way in which time actually grows: A period of one day does not grow to be a period of one week by taking on a whole day at a time, or a whole hour, or even a whole second. It grows by imperceptible increments (additions). These incon- ceivably small parts of which continuous number is made up are called Infinites- imals. Motion and force afford other illustrations of continuous number. In fact, the conception which regards number as continuous, vsdll be seen to be less artificial — more true to nature — than the conception of it as discontinuous. 6. Jin Infinite Quantity is a quantity conceived under such a form, or law, as to be necessarily greater than any assignable quan- tity. 7. A.n Infinitesimal is a quantity conceived under such a form, or law, as to be necessarily less than any assignable quantity. 8. ScH. — By an infinite quantity is not meant one larger than any other, or the largest possible quantity. It simply means a quantity larger than any assignable quantity ; i. e. , larger than any one which has limits. The mathematical notion concerns rather the manner of conceiving the quantity, than its absolute value. Thus, a series of Is, as 1 1 1, etc., repeated with- out stopping, represents an infinite quantity, because, from the method of conceiving the quantity, it is necessarily greater than any quantity which we can assign or mention. If we assign a row of 9s reaching around the world, though it is an inconceivably great number, it is not as great as a series of Is extending without limit. Moreover, one infinite may be larger than another ; for a series of 2s extending without limit, as 2 2 2 2, etc., is INTEODUCTION. 3 twice as large as a series of Is conceived in the same way. It is never of any use to try to comprehend the magnitude of an infinite quantity ; we cannot do it ; although we can compare infinites just as well as finites. Again, and- what is more to our purpose, an infinitesimal quantity is not a quantity so small that there can be no smaller. There would be but one such quantity and hence no comparison of infinitesimals. All that is meant by the term as used in mathematics is, a quantity which is to be treated in the argument as less than any assignable quantity. Whether we can or cannot comprehend its absolute magnitude is of no manner of con- sequence. Nor is absolute value usually of any importance in pure mathe- matical reasoning. Thus 2 times 5 is 10 whether 5 be mites or moun- tains. In order to free himself from needless embarrassment in the use of infinitesimals, the student needs to keep constantly in mind the fact that, In pure mathematics, it is the relation of quantities, rather than their absolute values, with which we are concerned. 9» JPfop, — Any finite quantity divided by an infinite is an infinites- imal ; and any finite quantity divided by an infinitesimal is an infinite. Dem. — Let a represent any finite quantity and x any infinite. Then - is an in- X finitesimal ; for the value of a fraction depends upon the relative values of its numerator and denominator, and is less as the ratio of numerator to denominator is less. Now, in this case, a is infinitely less than x, by the definition of an infinite. Hence - is an infinitesimal. Again - is infinite if x is infinitesimal, since a is in- X X . finitely greater than cc. 10, CoK. — The reciprocal of an infinite is infinitesimal, and the re- ciprocal of an infinitesimal is infinite. 11, The products of infinites by infinites, and of infinitesimals by infinitesimals are denominated Ot^CTS : thus, if x and y are in- finites, x% 7/2, and coy are infinites of the Second Order ; if x, y, and z are infinites, x^, z^, xyz, x^y, xy^, etc., are infinites of the Third Order, The corresponding expressions are used with reference to infinitesimals, the product of two infinitesimals being caUed an infin- itesimal of the second order, of three, the third, etc. 12, ScH. — An infinite of a lower order sustains a relation to the next higher similar to that which a finite sustains to an infinite. Thus if x and y are infinites, x-, xy, and y'^ are infinitely greater than x and y. On the other hand if x and y are infinitesimals, x'^, xy, and y^ are infinitely less, and sus- tain a relation to x and y, similar to that which infinitesimals sustain to finites. INFINITESIMAL CALCULUS. AXIOMS. 13* From expressions containing the sum or difference of finites and infinites, the finites may be dropped without affecting the ratio. 14:, From expressions containing the sum or difference of infin- ites of different orders, the terms containing the lower orders may be dropped without affecting the ratio. 15, The order of an infinite is not altered by multiplying or divid- ing it by a finite. 10» From expressions containing the sum or difference of finites and infinitesimals, the infinitesimal terms may be dropped without affecting the ratio. 17* From expressions containing the sum or difference of infini- tesimals of different orders the terms containing the higher orders may be dropped without affecting the ratio. 18, The order of an infinitesimal is not changed by multiplying or dividing it by a finite. Ill's. — Although the above are conceived to be axioms in the strictest sense, that is truths to which the mind at once assents as soon as the terms used are clearly comprehended, the true notion of infinites and intinitesimals is so removed from common thought that a familiar illustration or two may aid the comprehen- sion. Suppose, then, that the quantities under consideration were the masses of matter in the earth and in the sun. If a grain of sand were added to or subtracted from each or either it would not appreciably affect the ratio of these masses. But in this instance the grain of sand is by no means infinitesimal with reference to either mass ; it is & finite, though very small part, of either mass. Again, let x and y be two infinite quantities, and a and 6 two finite ones. There can be no difference between — ==^ and - : since to assume such a difierence y ±zh y would be to assign some values to a and h, as respects x and y. But by hypoth- esis, the former have no assignable values in relation to the latter. ^ ..„.,, ^ a±x a Once more, if a and h are finite quantities, and x and y infinitesimal, = -, since x and y have no assignable values as compared with a and h. So also, x and y still being infinitesimal, — = — = -, as x^ and y^ are infinitesimals, (have no y±y' y assignable values) with respect to x and y. INTKODUCTIGN. 5 ETALUATION OF EXPRESSIONS CONTAINING EINITES AND INFINITESIMALS, AND FINITES AND INFINITES, Ex. 1. What is the value of the fraction - — — - if x is infinite and dx -\- b ' a and 6 finite ? Solution. — Since a and h liave no assignable values in relation to x they must ^x 2 be dropped, and we bave ^. Now dividing both terms by x, we have - as the value of when re is infinite and a and & finite. 'dx -\- h Ex. 2. What is the value of the fraction in the last example if x is infinitesimal and a and h finite ? Solution. — As x is infinitesimal ^x and 3ic are also infinitesimal, and hence have no value in relation to a and h, and must be dropped. Hence the value of the fraction is — -. Ex. 3. What is the value of -— — when x is infinite ? When X is infinitesimal ? Ans., When x is infinite, 6 ; when infinitesimal, 3. ^x X Ex. 4. What is the value of y in the equation y = when x --\-x X is infinite ? When x is infinitesimal ? Atis.j When x is infinite, — 5 ; when infinitesimal, -. Ex. 5. What is the value of y in the expression y = — - — when x JL "j~ X is infinite ? When x is infinitesimal ? Ans., When x is infinite, ; when infinitesimal, 1. -^ ^ -^x, . . .-, , . ax^ -\- hx"^ -^ ex -\r d . ••/»..« Ex. 6. What IS the value of when x is mnmte? m^3 _|_ ifirjQ'i -\- px 4- q When X is infinitesimal ? Ans., WTien x is infinite, — ; when infinitesimal, -. m q 2x^ — 5m^x Ex. 7. What is the value of y m the expression y = — • ox • TTIX when X is infinite ? When x is infinitesimal ? Ans., When x is infinite, ; when infinitesimal, 5m, 6 INFINITESIMAL CALCULUS. • doc'^ A- 2 7*^ 1 Ex. 8. "What is the value of y in the equation y = - when X is infinite ? "When x is infinitesimal ? Ans., When x is infinite, y=Qo', when infinitesimal, y = ■ — \. ?)X Ex. 9. When x and v are infinitesimals what is the value of *-- ? ^/zs., We cannot tell ; as we know nothing about the relation be- tween X and y. 3x Ex. 10. What is the value of — when y^ = 9x and x and y are infinite ? Ani^., oo. Ex. 11. Same as Ex. 10, only x and y infinitesimal? Ans., 0. Ex. 12. What is the value of y in the equation y^ = —- , when X is infinitesimal ? Ans., 0. CONSTANTS AND TARIABLES. 19. A. CottStatlt quantity is one which maintains the same value throughout the same discussion, and is represented in the no- tation by one of the leading letters of the alphabet. 20* VuTiable quantities are such as may assume in the same discussion any value, within certain limits determined by the nature of the problem, and are represented by the final letters of the alphabet. 2JL, CoR. — Any exjjression containing a variable is, when taken as a whole, a variable. Thus the value of the entire exjjression (4a — 3^^ -[-5)2 varies if x varies ; so that taken as a whole it is a variable. [Note. — These notions should be already familiar from General Geometry, page 9, and are in- troduced here only to give completeness, and for review.] 22, Variables are distinguished as Independent and Dependent. 23. An Independent Variable is one to which we assign arbitrary values, or upon whose law of variation we make some arbi- trary hypothesis. 24:, A- Dependent Yariahle is one which varies in value in consequence of the variation of the independent variable or vari- ables. IiiL. — ^ThuB, in the equation of the parabola, y'^ = 2px, if we assign arbitrary INTRODUCTION. values to x and find tlie corresponding values of y, we make X the independent variable, and p the dependent variable. Again, and what is more to our present purpose, if we as- sume X to vary in some particular way, as by taking on equal increments, as DD', D'D", D"D"', etc., 2/ will evi- dently vary ia some other way, but still in a way depending upon the way in which x varies, and upon the nature of the curve, or, what is the same thing, upon the form of the equation of the curve. In this case also, x is the independ- ent and 7/ the dependent variable. Fig. 4. ScH. — This distinction is made simply for convenience, and is not founded in any difference in the nature of the variables ; either variable may be treated as the independent variable. 2S, A.n JSquicresceflt variable is one which is assumed to in- crease or decrease by equal increments or decrements, as x in the last illustration. 26. Contemporaneous Tncrefnents are such as are generated at the same time. iLii. — Thus let^=/(cc) represent the equation of AM in the figure. Suppose we contemplate the values of x and y at the point P'. Now if ic takes the increment D'D", y takes the contemporaneous increment P"E'. So also we see that DD', P'E, and PP' are contemporaneous incre- ments of the abscissa, ordinate, and arc, respectively. Fig. 5. FUNCTIONS AND THEIR FORMS, 27* JL Function is a quantity, or a mathematical expression, conceived as depending for its value upon some other quantity or quantities. III. — A man's wages for a given time is a function of the amount received per day ; or, in general, his wages is a function of both the time of service and the amount received per day. Again, in the expressions y = 2ax-, y = x^ — 26x + 5, 2/ = 2 log ax, y = a'', y is, a function of x ; since, the numbers 2, 5, a and & being considered constant, the value of y depends upon the value we assign to x. For a like reason \/a'^ — x"^, and 3aa;2 — 2\/a; may be spoken of as functions of x. Once more, the ordinate of a curve is a function of the abscissa. ScH. — There is a sense in which the dependent variable (or function) is a function of the constants as well as of the variable or variables which enter into its value. So also it is a function of the form of the expression, that is, its value depends in part upon the form of the expression as well as upon the value of the independent variable. Thus if we have y = a\ogx 8 INFINITESIMAL CALCULUS. -f- h, and y = x^ — ex, though in each case ?/ is a function of x, speaking according to the definition, nevertheless it is not the same function in both cases. Its value depends upon the value of x, upon the constants, and upon the form of the expression involving these quantities. But the con- ception expressed in the definition is the ordinary one. 28* Functions are classified by their forms as Algebraic and TraTiscendentalf and the latter are subdivided into Trigono- metrical and Circular^ Logarithmic and Exponential. 29, An Algebraic Function is one which involves only the elementary methods of combination, viz., addition, subtraction, mul- tiplication, division, involution and evolution. Thus in y= ax^ — 3^", y is an algebraic function of x. 30, A Trigonometrical Function is one which involves sines, cosines, tangents, cotangents, etc., as variables ; thus ?/=:sinar, y = sin X tan x, etc. 31, A Circular Function is one in which the concept is a variable arc (in the trigonometrical the concept is a right line). These are written thus : y = sin~^^, read " y equals the arc whose sine is a; "; y == isiii'^x, read " y equals the arc whose tangent is x." III. — Notice that in the expression 2/ = tan— i .-j;, it is the arc which we are to think of, while in the expression x = tan y it is the tangent, which is a right line. Trigonometrical functions are right lines ; circular functions are arcs. These functions are mutually convertible into each other ; thus y = sin—' x is equivalent to a; = sin y, the only difference being that in the former we think of the arc, the sine being given to tell what arc, and in the latter, we think of its sine, the arc being given to tell what sine. The circular functions y = sin~^^, y = cosr'^x^ y = sec~^x, etc., are often called Inverse Trigonometrical Functions. 32, A Logarithmic Function is one which involves loga- rithms of the variable ; as y = log x, log^ 2/ = 3 log ax, etc. 33, An Fxponential Function is one in which the vari- able occurs as an exponent ; as y = a"", z = x'-', etc. 34, Functions are further distinguished as Explicit and Im- plicit, 35, An Explicit^ Function is a variable whose value is ex- pressed in terms of another variable or other variables and constants. Thus in y = 2ax^ — 3^"^, y is an explicit function of x. * From explicitum, unfolded. The function is disentangled from the other quantities. INTRODUCTION. 9 36, An Implicit^ Fmiction is a variable involved in an equation which is not solved. Thus in x"^ — ^xy -f 2?/ = 16, ?/ is an implicit function of x, or x is an implicit function of y. When we can solve the equation, an implicit function may always be expressed as exphcit. 37* dotation. When we wish to write that y is an explicit function of x, and do not care to say precisely what the form of the function is, we write y =f(x), read "i/ = a function of x." If we wish to indicate several different forms of dependence in the same discussion, we use other letters, as 2/=/(^), y=F{x), y= cp{x), etc., or use subscripts or accents as y =f(x), y ==zf'(^x), etc. Such symbols are read "y = the/", large F, cp, f sub-one, f prime, etc., function of x" as the case may be. When we wish to write that x and y are functions of each other, or that y is an implicit function of x, or x an implicit function of y, without being more specific, we write F{x, y) = 0, or f{x, y) == 0, or (p{x, y) r= 0, etc. ; and read "function x and y = 0," the F func- tion X and 2/ = 0, etc. This form symbolizes any equation between two variables with all the terms transposed to the first member. 3S, Again, functions are distinguished as Incveasitig and Decreasing, ' 30* A.n Increasing Function is a function that increases as its variable increases, and decreases as its variable decreases. 4:0, Jl Decreasing Function is a function which decreases as its variable increases, and increases as its variable decreases. III,. — In the expressions t/- = 'Ipx, y = log 'X,, y t^ a^, y is an increasing function of X. In the expressions y = —, y"- -\- x^ = E-, y = log — , y is a decreasing func- tion of X. For what vahies of cc is 2/ an increasing function of its variable, and for what a decreasing, in the following : y^ = ax^ — x'^, y = sin a;, y =z cosa; ? 41, OOhe Infinitesimal Calculus treats of Continuous Number^ and is chiefly occupied in deducing the relations of the con- temporaneous infinitesimal elements of such number from given re- lations between finite values, and the converse process, and also in pointing out the nature of such infinitesimals and the methods of using them in mathematical investigation. * From implicituiti, infolded, entangled. 10 INFINITESIMAL CALCULUS. III. — Let y^ = 8x be the equation of the parabola in the figure. Here we have the relation between finite values of y and x expressed. Now suppose x takes an infinitesimal increment as. D D ' *, what increment does y take ? The cal- . 4 cuius shows us that the increment which y takes is - times y as large as the increment which x takes ; that is, it shows us the relation between the elements of the variables y and x, when we know the relation between finite values. This is the province of the Differential Calculus, The converse of Fig. 6. this problem is, What is the equation of the curve whose ordinate varies - times as fast as its abscissa ? that is, having given the relation between the infinitesimal elements of y and x, to find the relation between finite values. This is the prov- ince of The Integral Calculus. 4:2, There are two branches of the Calculus, yiz., XTie Differ- ential Calculus^ and The Integral Calculus, * Of course all such attempts to represent infinitesimals to tlie eye, are egregious exaggerations; nevertheless they are of great service to the roiad. THE INFINITESIMAL CALCULUS. CHAPTER I. THE JDIFFEBENTIAL CALCULUS. SECTION L DifFerentiation of Algebraic Functions. 4:8, The DiffereTitial Calculus is that branch of the Infin- itesimal Calculus which treats of the methods of deducing the relations between the contemporaneous infinitesimal elements of vari- ables, from given relations between finite values. 4:4:, A. Diffevential is the difference between two consecutive states of a function, or variable. It is the same as an infinitesimal. 45, Consecutive Values of a function or variable are values which differ from each other by less than any assignable quantity. Consecutive Points on a line are points nearer to each other than any assignable distance. III. — Suppose y = 2x^ — Sx. Now let x be supposed to increase infinitesimally, 2/ will also change infinitesimally. Call the new value of y, y\ Then y' =2x'^ — 3x'. In such a case a; and x' are consecutive values of the variable, and y and y' are consecutive values of the function. But by this we do not mean that x and x' {or J and y' ) are so nearly equal thai there can he no intermediate value, for this would be to make an infinitesimal mean a quantity so small that there can be no smaller, which is not its meaning as used in mathematics (7). AH that is meant by saying that y and y' are consecutive values is that they are to he reasoned upon as having no assignable difference. So also in speaking of consecutive points on a line, as D and D', or P and P', Mg. 6, we do not conceive them as actually in juxtaposition ; but we mean simply that we are to reason upon them as nearer each other than any assignable distance. 46, Wotatiofl, The differential of a variable (one of its infini- tesimal elements) is represented by writing the letter d before it. 12 THE DIFFEKENTIAL CALCULUS. Thus, doc, read, " differential x" Of course the letter d is not to be con- founded with a factor ; it is simply an abbreviation for differential. [Caution. — The student should be careful and not allow himself to read such expressions as dy, dx, etc. , by merely naming the letters as he would ay, ax, etc. The former should always be read " differential y," "differential x," etc.] RULES FOR DIFFERENTIATING ALGEBRAIC FUNCTIONS. 47. BULE 1. To DIFFERENTIATE A SINGLE YAEIABLE SIMPLY WHITE THE LETTER d BEFORE IT. Dem. — Let us take the function y =z x. The consecutive state of the variable is ic -|- dx. Now representing the change in y which is produced by this change in X by cZy {dx and dy being the contemporaneous increments of the variable and the function), we have 1st state of the function, y = x, 2nd, or consecutive state, y -\- dy ^x -\- d^. Subtracting the 1st from the 2nd, dy = dx, which being the difference between two consecutive states of the function is its differen- tial {4:4:). Q. E. D. ScH. — This rule is evidently only the same thing as the notation requires, and its formal demonstration would be unnecessary except for the purpose of uniformity in treating the several cases of differentiation. III. — Let M N be a line passing through the origin and making an angle of 450 with the axis of x. Its equation is ?/ = cc. Let P be any point in the line, AD = ic, and P D = ?/. Let a; take the infinitesimal increment D D ' {dx), then y becomes P' D'. Now the first state of the function is P D = A D, or 2/ = .T, The second or consecutive state is PD + P'E = AD+ DP", ""r y^dy = x-\-dx. Subtracting we have P' E ^ D D', or dy = dx. Now that the increment of y (or dy) is equal to the in- Y /jyi crement of x (or dx) in this case is readily seen from the figure ; for, as P'PE = 45°, P'E = PE, or DD'. dy = dx, then, means that the contemporaneous incre- ments of X and y are equal, or that x and y increase at the same rate. DDT X Fig. 7. 48, B ULE 2. — Constant factors or divisors appear in the differ- ential THE SAME AS IN THE FUNCTION. Dem. — Let us take the function y = ax, in which a is any constant, integral or fractional. Let .v take an infinitesimal increment and become x -\- dx ; and let dy be the contemporaneous increment of 2/, so that when x becomes x -\- dx, y be- comes y -|- dy. We then have DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 13 1st state of tlie function, y ■= ax\ 2nd, or consecutive state, y -\- dy = a'x -\- dx) Subtracting the 1st from the 2nd, a.x -j- adx: dy = adx, ■ ■which being the difference between two consecutive states of the function is its differential {44:) . Now the factor a appears in this differential just as it was in the function, q. e. d. Iljl. — Let y =z ax he the equation of the line MN. PD and P'D' representing consecutive ordinates, DD' represents dx, and P' E represents dy. Here it is evident that P'E =a X DD'; for from the triangle PP'E we have P'E = tanP'PE X PE. But tan P'PE = tan MAX = a. Th'e meaning in this case is, therefore, that the ordinate increases a times as fast as the abscissa. ^^^ °- If « :■= 1, or tan 45°, the ordinate and abscissa increase at equal rates ; if a <^ 1, i. e. , if the angle is less than 45° the ordinate increases more slowly than the abscissa ; if « ^ 1, the ordinate increases more rapidly than the abscissa. 40, RULE 3. — Constant terms disappear in differentiating: or THE DIFFERENTIAL OF A CONSTANT IS . Dem. — Let US take the function y = ax -±1 h, in which a and h are constants. Let x take an infinitesimal increment and become cc -\- dx ; and let dy be the con- temporaneous increment of y, so that when x becomes x -\- dx, y becomes y -\- dy. We then have 1st state of the function, y = ax ±h \ 2nd, or consecutive state, y -\- dy = a(x + (^^) rt &, or y -\- dy = ax -\- adx =fc h. Subtracting the 1st state from the 2nd, dy =^ adx, which being the difference between two consecutive states of the function is its differen- tial {44). Now from this differential the constant term ± 6 has disappeared. We may also say that as a constant retains the same value there is no difference between its consecutive states (properly it has no consecutive states). Hence the differential of a constant may be spoken of (though with some latitude) as 0. Q. E. D. III.— Let y==ax-\-hhe the equation of the line M N. Now the relative rates of increase of the abscissa and or- dinate, that is the relation of dy to dx, is evidently not affected by h which is A B ; for, if we were to draw a line through the origin parallel to M N , the contemporane- ous increments of its co-ordinates would be the same as those of M N . Again, we can see that the constant term does not affect the differential, i. e., the difference between the consecutive states of y, by observing that these two states are represented by PD and P' D', each of which contains the constant as a part of it, whence the difference between them is not affected by it. SO, Cor. — An infinite variety of functions differing from each other only in their constant terms still have the same differential. Fig. 9. 14 THE DIFFEEEKTIAL CALCULUS. 31, RULE 4. — To differentiate the algebraic sum of several va- riables, DIFFERENTIATE EACH TERM SEPARATELY AND CONNECT THE DIFFEREN- TIALS WITH THE SAME SIGNS AS THE TERMS. Dem. — Let u = x -{- y — z, u representing the algebraic sum of the variables X, y, and — z. Then is the differential of this sum or du=.dx -\- dy — dz. For let dx, dy, and dz be infinitesimal increments of x, y, and z ; and let du be the in- crement which u takes in consequence of the infinitesimal changes in x, y, and z. We then have 1st state of the function, m = £c + 2/ — ^ ■> 2nd, or consecutiye state, u -\- du = x -\- dx -\- y -\- dy — {z -\- dz), or u -^ du := X -}- dx -{- y -^ dy — z — dz. Subtracting the 1st state from the 2nd, du = dx -{- dy — dz. q. e. d. III. — "We may illustrate this by conceiving x and y to be forces acting to raise a weight, and z a force acting to prevent the raising, u being the aggregate effect of all, i. e. their algebraic sum (Complete Algebra, 65). Now if x, y, and z each re- ceive an infinitesimal increment, which we will call respectively dx, dy, and dz, it is evident that the increment of lifting force is dx -\- dy, and as the increment of the depressing force is dz, the combined effect of the change is dx -\- dy — dz, which is the change in u. Moreover, since this quantity dx -\- dy — dz is the ag- gregate of a finite number of infinitesimals, it must be itself infinitesimal. Hence the change in u is infinitesimal, or du. ScH. — It is important to notice that the above reasoning is entirely inde- pendent of the relative values of the infinitesimals dx, dy, and dz. These may be conceived as equal, or as sustaining any finite ratio whatever to each other, only so that they remain infinitesimal. 32, RULE 5. — The differential of the product of two variables 15 the differential of the first into the second, plus the differential OF THE SECOND INTO THE FIRST. Dem, — Letw =^ xy be the first state. The consecutive state is w -f- dw = {x -\- dx){y -\- dy) =xy -\- ydx -\- xdy -\- dxdy. Subtracting the 1st state from the 2nd, or consecutive state, we have dxL = ydx -f- ^^y + <^^' • ^V- Now ydx and xdy are infinitesimals of the 1st order, and dx • dy, being the product of two infinitesi- mals, is of the 2nd order and must be dropped (17). Therefore du = ydx -f- xdy. Q. E. D. III. — Let u represent the area of the rectangle A BC D, £C = ^ e C * A B, and y = *BC. Then u = xy. Let B& represent dx, and p - Oc",dy. Whence ShCc' = *ydx, DdCc"=*xdy, Cc'cc" ==* dx ' dy, and du = *BhCc' + DdOc" -\- Cc'cc". Now since cc' is infinitesimal and c'h is finite, Cc"cc" is infinitesimal with A B reference to B&Cc', as for a like reason it is with reference to Fig. 10. DdCc" ; hence it is to be omitted as having no assignable value with reference to them. Another view which may be taken of this is to consider that it is the rate at which * In such cases = signifies " is represented by," and is used for brevity. DIFFEBENTIATION OP ALGEBRAIC FUNCTIONS. 15 the rectangle is increasimj when a; = A B and 1/ = BC, not the amount of change in the area after x and y shall have increased more or less : in other words, we seek for the difference between consecutive values of the area. Now it is easy to see that the rale at which the rectangle A BCD starts to increase, depends upon the length of the side BC iy) and the rate at which it starts to move to the right, -j- the length of DC (x) and the rate at which it starts to move upward. Letting dx represent the rate at which A B starts to increase (by being the amount which it would increase in an infinitesimal of time), and dy represent in like manner the rate at which y starts to increase, we readily see that du = ydx -f- scdy is the rate at which the area starts to increase. Moreover, we see that this is equally true whether dy =^ dx, or whether one is any finite multiple of the other ; all that is necessary being that both be infinitesimals of the same order. ^3, B ULE 6. — The differential of the product of several varia- bles IS THE SUM OF THE PRODUCTS OF THE DIFFERENTIAL OF EACH INTO THE PRODUCT OF ALL THE OTHERS. Dem. — Let u := xyz ; then du = yzdx -f- xzdy -f- xydz. For the 1st state of function is u = xyz, 2nd, or consecutive state, u-\-duz= {x-\-dx){y-\-dy){z-\-dz), or u-{-du ■= xyz -\- yzdx -f- xzdy -\- xydz -f- xdydz -f- ydxdz -\- zdxdy -\- dxdydz. Subtracting and dropping infinitesimals of higher orders than the first we have du =- yzdx -f- xzdy -\- xydz. In a similar manner the rule can be demonstrated for any number of variables. Q. E. D. S4:, RULE 7. — The differential of a fraction having a variable NUMERATOR AND DENOMINATOR IS THE DIFFERENTIAL OF THE NUMERATOR MULTIPLIED BY THE DENOMINATOR, MINUS THE DIFFERENTIAL OF THE DENOMI- NATOR MULTIPLIED BY THE NUMERATOR, DIVIDED BY THE SQUARE OF THE DENOMINATOR. Dem. — Let u = - ; then is du = — ^ — - — -. For clearing of fractions yu = x. Differentiating this by Kul^ 5, udy -\- ydu = dx. Substituting for u its value, we have ' — : — f- ydu = dx. Finding the value of du, we have du = - — '■ — -. y y Q. E. D. SS» CoR. — The differential of a fraction hamng a constant numerator and a variable denominator is the product of the numerator with its sign changed into the differential of the denominator, divided by the square of ' the denominator. Dem. — Let u = -. Differentiating this by the rule and calling the differential of the constant (a), 0, we have du = = ~. o. e. d. 16 THE DIFFERENTIAL CALCULUS. ScH. — ^If the numerator is variable and the denominator constant it falls under Rule 2. S6. RULE 8. — The differential of a vaeiable affected with an EXPONENT IS THE CONTINUED PEODUCT OF THE EXPONENT, THE VARIABLE WITH ITS EXPONENT DIMINISHED BY 1, AND THE DIFFERENTIAL OF THE VARIABLE. Bem. — 1st. When the exponent is a positive integer. — Let y = x"^, m being a pos- itive integer ; then dy = mx^~^dx. For y z= x^ ::= x • x • x ' xio m factors. Now differentiating this by Bute 6, we have dy = {XXX to m — 1 factors) dx -f- {xxx to m — 1 factors) dx + etc., to m terms, or dy = ic"»-^dx + x'^—^dx -\- x"'—^dx -f- etc., to m terms. . • . dy =: mx'^—^dx. - m . 2nd. When the exponent is a positive fraction. — Let y =z xn, — being a positive TO !^ — 1 fraction ; then dy = — X" dx. For involving both members to the nth power n we have 2/" = ^C". Differentiating as just shown, ny"~^dy = mx^—^dx. Now m mn — m from y = a «, we have y"—^ = x~i . Substituting this in the last form, we have mn — m »j. mv — m ™ in nx n~dy = mx'^—^dx, or dv = — x^ Ti dx = — X"~ dx. ^ n n 3rd. When the exponent is negative. — Let y = x—", n being integral or fractional ; then dy= — nx-^—^dx. For i/ = x—" = — , which differentiated by jRwte 7, Cor., gives dy=: ;; — '— = — nxr-"—^dx. All three of which forms agree with the enunciation of the rule. q. e. d. S7* Cor. — The differential of the square root of a variable is the dif- ferential of the variable divided by twice the square root of the variable. Dem. — Let y = \/x = x . Differentiating by the rule we have dy =-ix^ dx=. 1 -2 , dx iX dX = =. Q. E. D. 2v/x ScH. — Special rules can be readily made for other roots, but it is un- necessary. The square root is of such frequent occurrence as to make the special process expedient. Of course the general rule can always be used, if desired. EXERCISES. [Note. — The following examples are designed to give practical skill in applying tlie rules for differentiating algebraic functions. The student should not advance beyond these, till he has the rules firmly fixed in memory, and can apply them with facility to all forms of algebraic func- tions.] Ex. 1 . Differentiate y = Qx — 4. dy = 6dx. QuEKT. — What three iiales apply? Be careful to repeat the rules in applying them to the solution of the examples, and thus render them familiar. DIFFEBENTIATION OF ALGEBRAIC FUNCTIONS. 17 -Ex. 2. Differentiate y = a'' + da'x-' + Sa'x^ + x^ Solution. — The differential of i/ is d^/- [Bepeat Rule 1.] To differentiate the second member we notice 1st, that it consists of several terms, and hence proceed to differentiate each term separately. [Repeat liule 4.] a^ being a constant term, disappears. [Eepeat Eule 3.] To differentiate Sa^x^, we notice l«t that the con- stant factor 3a4 will be a factor in the differential. [Repeat Eule 2.] The differ- ential of a;2 is 2xdx. [Repeat Bule 8.] Hence the differential of Sa'^x'^ is GalTc/a?. [In like manner proceed with the other terms, giviiig the reason for each step hy repeating the appropriate rule.^ Ex. 3 . Differentiate u = 2ax — 3^^ _|_ ahx^ — 5. Result, du = (2a — 6x -{- dabx^)dx. Ex. 4. Differentiate y = ^x^ — ■ 2x — Bin. Ex. 5. Differentiate u = ab — ■ 6x^ -{- 2ax. Ex. 6. Differentiate it = ax-y\ QuEBiES. — What is the most general feature of the function ax^^? What rula applies first? Rule 5. What other rule apphes ? Result, du = 2axy^dx -f dax^y^dy. 2 Ex. 7. Differentiate u = 6ax^y^. 1 2. Result, du = 4,ax~^y^dx -\- ISax^y^dy. Ex. 8. Differentiate y = 2bz-^ + Sax'^z'^. 6 2 1 'Sax'^dz Abdz Result, dy = 5ax^z^dx + r- • ^ 2^z ^' 11 -r. 7 xdy -f ydx Ex. 9. Differentiate u = x'^y'^. Besult, — ■ — —'- — . 2^-^i/^ P Ex. 10. From y^ = 2px find the value of dy. dy = -dx. Ex. 11. From A^y- -\- B^x^ = A^B^ find the value of dy. Ex. 12. From A^y^ — B^x^ = — A^B^ find the value of dy. dy = -— -a^. X Ex. 13. From ^24-2/2 = R2 find the value of dy. dy = dx. Ex. 14. From 2xy^ — ay^ £= x^ find the value of dy. ^j, = ^dx. 4:Xy — "lay 18 THE DIFFEKENTIAL CALCULUS. Ex. 15. Differentiate u = r-~. Result, du = ~ -, 32/3 Sy^ 1 dx Ex. 16. Differentiate v = -. dy = . ^ X x^ Ex. 17. Differentiate u = - — %r—. du = ^^ ^ b — 22/2* (6 — 2]/2)2* Ex. 18. Differentiate y = —zr-- dy = -- x 2xdx = -—dx. SiTG. — Do not treat this as a fraction under Kule 7. Ex. 19. Differentiate u == x-y^z. 2.^2 — 3 Ex. 20. Differentiate u = 4j7 4- X' Opebation. dK = -^(^^^ - 3)(^-^ + »^) - d(ix + x^)(2g_--_3) _ 4:Xdx{ 4:X + a;g) — (4da; + 2a;d.-r) (2a;3 — 3) _ {4x(4x-{-x'^) — (4 + 2a;) (2.^^ — 3)} da; (8a;^ + 6a; + 12)da; ~ (4x + a;2)2 * Stjg^s. — The first step is the application of the rule for fractions, since the func- tion is a fraction with a variable numerator and a variable denominator. The second step is to perform the difierentiation of 2,'r2 — 3, and 4.x -|- s;^. This step involves the rules for constant factors, variables affected with exponents, constant terms, and the sum of variables. The remainder of the work is reduction and addition of terms. ^ r.^ ^n^ .■ , 2a:^ 7 8<22^3 — 4075^ Ex. 21. Differentiate u = . du = — —dx. (a2 — x^y ^ ^ a, Ex. 22. Differentiate y = . dy = dx. /][ 2j7 x'^^dx Ex. 23. Differentiate y = ^ ' '^ . dy == rr-— r ^ t < -"> ^ (1 _|_ a;2) Ex. 24. Differentiate y = Ex. 25. Differentiate v = . ^ 1 — X Ex. 26. Differentiate y ==: Sx"^ — 4. dy == Smx'^-'dx. m 29712 ">—" Ex. 27. Differentiate y = 2mx\ dy == x « dx. ^ n a-' — - x^' a — X X 1 + X 1 + x^ 1 + 372 1 ^2* X DIFFERENTIATION OP ALGEBRAIC FUNCTIONS. 19 m 1 Ex. 28. Differentiate w = Sno;"!/". m — n 1 v% 1 — n du = 1mx~^y^doc + 2x''y~dy. 1 71 Ex. 29. Differentiate i/ = — . dy = zTl^^' Ex. 30. Differentiate y = ^ x:^ — a^. Opebation. — By the special rule for the square root (57), we have dy == d(iK3 — a:^ 3x-cZx Ex. 31. Differentiate y = V~ax ■\- Vc^. ad.v 3c-.r-dr , , i -i 3c -i-, , a^ -f 3ca7 ^2/ = — -^ + —7= = {W^ + 17^ )^^> o^ -=-dx. iVax ■ Wc:^x^ ^ 2va7 /- X Ex. 32. Differentiate y = av a; — -. o Ex. 33. Differentiate y = V ax -\- hx'^ -\- cx^. Ex. 34. Differentiate y = {ax- — x^y. Solution. — Kegarding' «x2 — x^ as a variable, it is affected with the exponent 4 ; hence we have dy = ^{ax'^ — x^)'^ X d^ax- — x^), the operation of differentiating the variable ax- — x^ being as yet unperformed. Performing this operation and reducing, we have dy = 4(aa;- — x^Y X (2ax — ^x'^)dx = 8ax^(a — xydx — 12a;8(a — xydx. Ex. 35. Differentiate y= {a + bx")^. dy==^^-i-{a f hx'^)'^bxdx. Ex. 36. Differentiate y == (a^ + ^2)3. dy = 6x{a''- + x'^ydx. a , 6a^ - Ex. 37. Differentiate y = r- dy = — - — ■ — -r-dx. •^ {ly^ + x^y^ {b- + ^2)'« Ex. 38. Differentiate 7/ = (1 + 2j72)(1 -}- 4^;^). Solution.— Regarding this function as the product of the two variables 1 -|- 2x^ and 1 + 4x^ we have dy = d^l + 2x^) X (1 + 4.x^) + d(l + 4.^3) X (1 + Sx^). Performing the operation of differentiating 1 + 2.1?^ and 1 + 4.x^, we have dy = 4a;(l 4- 4a;3)(2a; + 12x2(1 -f 2£c2)da; = 4a;(l + Zx + 10x3)dx. Ex. 39. Differentiate y = {x^ -\- a.){Sx^ + 6). dy = (1507^ + 35^2 + 6ax)dx. x^ , 3^2 4- ^3 Ex. 40. Differentiate y = 7-— -. dy = — — -^^. ^ (1 + :r)2 ^ (1 + ^)3 a , 3a^.r Ex. 41. Differentiate y == -t r-. dy (a — ar)3 (a — :r)^ 20 THE DLFFEBENTIAL CALCULUS. Ex. 42. Differentiate y = —z -. dy— — j^—- — r — • ^ {ab ^2)3 ^ (a6 — ^r2)4 "^x. 43. Differentiate, without first expanding, y = {1 -^ x)*{l -\- x^y. dy = 4(1 + xy{l -i- x^){l -}- X + 2x^)dx. Ex. 44. Differentiate y == x^ — \/i — x^ „ , dx'-dx av = 2xdx -\ ■ ^ ^ 2v/l~^- / : ( n ^ j dx Ex. 45. Differentiate u == v2ax — x"^. du = — - — . '^^2ax — xi Ex. 46. Differentiate u = Va^ + ^^ X Vb"^ + j/^. (^^4- y^)xdx + (a2 + x-^)ydy . \/a2 _j_ ^-2 X V 62 + ?/2 Ex. 47. Dinerentiate i/ = — - a^/ = v^a2 — ^2 v/(a2 — x^y Ex. 48. Differentiate j/ X \/l + J72 Sug's. 2/=a;(l+x2) I .-. d?/ = dx(l +3:2) ^ -f a;. d(l+a;2) ^==da;(l+a;2) ^ — £c2(l -j- a;2) *dx = — ^ = — -. Or, we may apply the rules (1+^2)2 (1+^2)2 „ - ,. ^ i XT, J dxVl 4- £c2 — a;dv/l + ic2 tor a fraction and a square root, thus du = ■ —. — = xdx dXs/l -f- iC2 £C v^l + ^^ dx(l -f- a;2) — x^dx dx 1 _l_ r2 3. Ji (1 — 3^)cZj; Ex. 49. Differentiate u= {l-\-x)vl — x. du 2v'l — X Ex. 50. Differentiate u = — . du = \/{i — x^y (1 — ^2)i* Ex. 51. Differentiate u = — . 2v a2^2 — ^4 — 04(^2 — 2x^)dx du= ^. 2j;2(a2 — 072)2 Ex. 52. Differentiate u = \^ x -\- \/l + x\ xdx Sug's. — Squaring u2 = a; -}- s/1 -f- x'^. 2iLdu == dx + — . du = v/l + a:-' dx -j- DIFFERENTIATION OP ALGEBRAIC FUNCTIONS. 21 xdx (a; -f y/l 4- x')dx a;2 Or, we may differentiate without squaring, thus du 2v- 1 4- a;' xdx dx -j y- 2 Va; 4- n/IT^^ 2 v/r+¥' V a; -f v/r+^'^ ^^^ + '^^ x^dx X Ex. 63. Differentiate u V a'^ -\- x-^ — X Sug's. — As the denominator is more involved in the differential of a fraction than the numerator, it is expedient to reduce the fraction to a form having as sim- ple a denominator as possible. Bationalizing this denominator, we have u = ^2 — a2 a-^ aVa;-^ + a^ «^ Ex. 54. Differentiate w = ' . V x'^ 4-14-^ du = 2 \ 2x g ^l±-L I dx, VX'' 4- 1 11 s/x Ex. 55. Differentiate u = 1 + \/: X „ , 1 — v/.'T Vl — s/x s/l — X . dx SUGS. U = K. z — : = — = . du ^ + ^a? \/l4-v/x l + v^a; 2(l4->/a;)ya; — a;2 X \" , nx''~^dx (X \ ) . du =2 1 -^ x/ Ex. 57. Differentiate u = (i 4- x)"-^'' \/l -i- X — VI — X 1 + v/l — ^^ . du = ; — dx. Ex. 58. Differentiate w = >/a; • V ^a; 4- 1 , 7a;^ 4- 4 , du = — dx. 22 THE DIFFERENTIAL CALCULUS. Ex. 59. Differentiate u = N/2:r— 1— V25;~l — \/2a7-^l— -, etc., to infinity. Sug's.— We liave u = v^'^x — 1 — m ; whence m^ = 2x — 1 ~ m, and u — si iv/8x — 3. . • . di* = ± v^Sx — 3 Ex. 60. Differentiate u = ,y du v/ + V (c2 — a;2 J? )']■ 36 4^7 — "^ ILLUSTRATIVE EliLOIPLES. [Note. — The following examples are designed to illustrate more fully the significance of the process of differentiation.] Ex. 1. In a parabola whose parameter is 12, which is increasing the faster at a; = 2, the ordinate or the abscissa, and how much ? At ^ = 3 ? At ^ = 8 ? At ^ = 24 ? How does the relative rate of change vary as we recede from the vertex? At what point are ordi- nate anid abscissa varying equally ? Solution. — The equation of this parabola is y- = 12x. Differentiating, -we have dy = -dx. Now, as differentiat- ^ ing is the process of finding the difference between two consecutive states of a function, dx represents one of the infinitesimal increments of a;, as DD', and dy the coniemporaneous, infinitesimal increment of y, as n P' E. We, therefore, learn from dy = -dx that in gen- eral dy is - times as great as dx ; or. in other words, that 2/ a — y changes _ times as fast as x. At P where a; = 2, y ■= \/24. Hence, at this y 6 point, dy = — ■z=dx = ^v/edx ; that is, y is increasing nearly li times as fast as x. >/2i At P" where re = 3, ?/ = 6, and dy = dx ; that is, x and y are increasing equally. In general, at the focus the ordinate and abscissa of a parabola are increasing equally, since at this point y =^ p. At P''^ where cc = 8, y is, increasing only about .G as fast as x. At P^' where cc = 24, y is increasing at the still slower rate of about .35 as fast as x. Finally, it is evident, fi'om a slight inspection of the figure, Fig. 11. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 23 that y increases less and less rapidly as x becomes larger, x continuing to increase at a uniform rate. At cc = co, y ceases to increase, i. e. the branches become par- allel to the axis of x.- Ex. 2. Examine the relative rates of change of the ordinate and abscissa in the eUipse. JB-x Solution.— Differentiating A-y^ + B-x"^ = A'^B'^, we find dy = — -jr-dx — -A- y p., :dx. On this we observe 1st, That the — sign shows that x and y AVA^ — a;2 are decreasing functions of each other ; that is, that as a; takes an increment y takes a decrement. This is evident from a consideration of the curve. 2nd, That Bx in general terms y diminishes times as fast as x increases. 3rd, At A\^A^ ~ a;2 X = 0, I e. at the extremity of the conjugate axis, y is not increasing or decreas- Bx ing, since here = 0, and dy = Q -dx — O. At the extremity of the As/A^ — x^ transverse axis dy = — oc • dx, i. e. y is decreasing infinitely faster than x increases. There are, therefore, all relative rates of change between x and y from to oc. Moreover as x begins to increase from 0, y commences to decrease (at first slowly, Bx as the fraction • — is small when x is small), and then more and more AVA^ — x'^ rapidly as x increases, till it reaches an infinitely rapid rate of decrease at x = A. Bx This, it is easy to see, is the law of change in the fraction — _ as x in- AvA'^ — x'^ creases. The same law is also rendered probable from an inspection of the curve. Finally, we may inquire at what point the relative rates of change sustain any given relation to each other, as, for example, when y decreases twice as fast as x in- creases, or just as fast, or 10 times as fast. Thus when y decreases twice as fast as Bx X increases, we must have dy= — Mx, i. e. ^rr = 2. From this we find 2A^ a; = i . — . - ; hence at these points, y is diminishing twice as fast as x is v/4^2 -f J52 increasing. Ex. 3. A boy is running on a horizontal plane directly toward the foot of a tower 100 feet in height. How much faster is he nearing the "foot than the top of the tower? How far is he from the foot of the tower when he is approaching the base twice as fast as he approaches the top ? How far off must he be to be approaching both base and top equally ? Where is he when he is not approaching the top at all, or is making infinitely more progress toward the base than towards the top? When he is at 200 feet from the base of the tower how much faster is he approaching the base than the top ? Sug's. — Let AB represent the tower, and AX the line in the plane of the base in which the boy is approaching the base. Suppose the boy at any point, as P 24 THE DIITEEENTLIL CALCULUS. and let AP =^ x, and PB = y. Then 2/2 — .-r^rrr 10000. WhencB dy = -dx. Hence we see that in general he is only approach- ing the top an -th part as fast as he is the base ; i. e., letting PP' represent an infin- ^^'^^ ^^' itesimal element of the distance to the foot of the tower, P F represents a contem- poraneous, infinitesimal element of the distance to the top ; and also, that P F is X an -th part of PP'. Secondly, when he is approaching the foot of the tower 1 X 1 twice as fast as he is the top ; we have dy = ^dx, or - = - ; whence y = 2cc. But ^ 2 2/2 100 2/- — x^=^ 10000 ; and, substituting, 3a;'- = 10000, or x = — - = uS nearly. Lastly, \/3 /J. 200 25 when he is at 200 feet from the base y = \/50000 = 224 nearly, and - = -— = — y 224 2o 25 25 nearly. Hence dy = ^-^^x, or he is approaching the top — as fast as he is the 28 2o base. [Let the pupil decide the other points himself. ] Ex. 4. A sliip is sailing northwest at 15 miles an hour. At what rate is she making north latitude ? An^., At 10.6054- niiles an hour. Sitg's. — Let 2/ represent any distance run in the northwest course, and x the cor- responding northing. Then as the course is northwest there is made in the same time X westing, and we have y^ = 2x-. From this dy = — dx, and the ship is run- 2x — 2x 3iing -- times as fast as she is making northing. But y = x\/% whence -— = ^x - - - — ^: = \/2, and dy = \^1dx, or dx = ^V'ldy ; i. e., she is making northing a:\/'2 , 707-f- as fast as she is running. Ex. 5. In the function y = 27x -|- 8^72, required the value of x when y is increasing 45 times as fast as :3t. Result, x = o. Ex. 6. What is the relative rate of variation of the side and alti- tude of an equilateral triangle? i. e., if the side takes an infinitesi- mal increment, what is the contemporaneous infinitesimal increment of the altitude ? When the side is increasing at the rate of 2 inches per second how rapidly is the altitude increasing ? Is the relative rate of increase constant or variable ; that is, does the altitude increase more or less rapidly in comparison with the side when the side is small than it does when it is large, or is the relative rate of increase always the same? Sug's. —Let y = the altitude and x one cf the sides of the tdangle. Then LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 25 3x Sx ~ ?/2 = |aj2 and di/ = ~-dx = ^-dx = i\/ddx. Hence we see that the infinitesi- 4?/ 2s/ '3x mal increment of y is always i\/3 times as much as the contemporaneous infini- tesimal increment of x. When x is increasing at 2 inches per second y is increas- ing 2>/3 times 2 inches, or \/3 inches per second. ScH. — The student should now be able to comprehend with considerable clearness the object of the Differential Calculus; viz., having given the relation between finite values of variables, to find the relation between the contemporaneous infinitesimal increments of those variables, or their rela- tive rate of cl^ange. Thus, in the last example, the relation between the altitude and one side of an equilateral triangle, 3/2 = lx~, is the relation between finite values of the variables, from which we find the relation be- tween the contemporaneous infinitesimal increments d^ and dx, by the Dif- ferential Calculus. ■4»»- SUCTIOJSr IL Differentiation of Logaritlmlio and Exponential Functions.* S8, The JI£odulus of a system of logarithms is a constant factor which depends upon the base of the system and characterizes the system. SO, I^vop. — The differential of the logarithm of a variable is the differential of the variable multiplied by the modulus of the system, di- vided by the variable ; or, in the Napierian system the modulus being 1, the differential of the logarithm is the differential of the variable divided by the variable. Dem. — Let 2/ = a*.", n being constant. Then log y = n log a;. Differentiating dy y = x", we have dy = nx^—^dx, orn = fr- = — - = -r-> since X"—^ = --. Again, ^ .^ x'^-Hx y ^ dx X ^ -dx — X X whatever the differentials of log y and log x are, we have d{log y) ■= n - dQiOgx), ox n = - \ . Placing these values of n equal to each other, we obtain d(logic) ° \ dy - , ' . = -— . Now let m be the factor by which -- must be multiplied to make a(logfl;) dx V X it equal to ^(log?/), then is (^(logic) = . * See 3^, 33, 26 THE DIFFERENTIAL CALCULUS. "We are now to show that m is a constant depending upon the base of the system. To do this take y ^z""', from which we find as before n = / °^ ^^ = ~. But m a{log z) dz z is the ratio of d{logy) to — ; hence d(logz) = . Thus we see that in any case y 2 *the same ratio exists between the differential of the log. of a number, and the differ- ential o! the number divid^-d by the number. Therefore m is a constant factor. To show that m depends il^)wu tue base of the system we have but to recur to the definition of a logaritlim to see that the only quantities involved are ihe number, its logarithm, and the base of the system. Of these the two former are variable, whence, as the base is the only constant in the scheme, m is a function of the base. * Finally, as m depends upon the base of the system, the base may be so taken that m := 1. The system of logarithms founded on this base is called the Napie- rian system, q. e. d. 00, JPtoj)- — The differential of an exponential function with a constant base is the function itself, into the logarithm of the hose, into the differential of the exponent, divided by the modulus. Dem. — Let 2/ = «'. Taking the logarithms of both members log 2/ = £c log a. Differentiating — - = log adx, or dy = ^^ , remembering that y = a^, and that log a is constant, q. e. d. Ql, Cor. 1. — The differential of an expjonentiol function with a con- stant base, taken with reference to the Napierian system, is the function itsef, into the logarithm of the base, into the differential of the exponent. Thus if J == a"", cly = p/ log adx. 02 » Cor. 2. — If the base of the exponential is the base of the system cf logarithms in reference to ivhich the differeyitiation is made, we have, in general, dy = , or in the Najjierian system dy = e'^dx, since the log- aritlim (f the base of a system, taken in that system, is 1, and in the Na- pierian system e is used to represent the base and m = 1. G3, JPvo2>o — The differential of an exponential with a variable base is best obtained by passing to logarithms, and then differentiating. III.. — Let w = 2/"". Passing to logarithms, log it = a; log y. Differentiating, we , . , mdu , , , mxdy , , ulogy dx , u x dy have m neneral, = lo!? y dx 4- , whence du = + — = -^ u y m ' y 11^ loc 11 dx ii^xdii ' ^^-^ — '■ 1- — — '-. If the logarithms are taken in the Napierian system, m = 1, * What tliis relation is, it does not concern us at present to know. It will be determined here- after. LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 27 and du = y* log y dx -{- y—^xdy. If in addition y = x, so that u = a;*, du == x*(loga; -f- l)dx. EXERCISES. [Note. — The following exercises are designed to familiarize the rules for differentiating loga- rithmic and exponential functions, and give the needed facility in applying them.] Ex. 1. Differentiate u = x log x. du = logxdx -\- mdx, or {logx + l)dx. Ex. 2. Differentiate u == log x^ du = 2m--, or . ° XX d X dx Ex. 3. Differentiate u = log^ x. du = 2m log x- — , or 2 log x — . X X Ex. 4 Differentiate u = x""'. du = af'x' ] log x{\og ^ 4- 1) + - r dx/ a}°^ -^ loff a , Ex. 5. Differentiate u = a}"^', du == ^-dx. X , xdx Ex. 6. Differentiate u = losf \/l — x^. du = — . ^ 1 — x^ # 6^+1-7 Ex. 7. Differentiate u = log (3x^ + x). du = ■ dx. ^ ^ ^ 6x^ + X Ex. 8. Differentiate u = log {x -\- \/l + x'^). du = yi +a;2 2 Ex. 9. What is the differential of w = a"" in the common system 2 2 when a is the base of the system ? du = — a"" xdx. m Ex. 10. Differentiate u = e^"^"", in the common system, e being the . , udx . mudx base of the Napierian system. du = log e == — — . SuG. — If the student has studied the subject of logarithms as usually presented in our higher Algebras, he has learned that the common logarithm of the Napier- ian base is the modulus of the common system ; L e. , in this example log e = m. This fact will also appear hereafter. V X'i -\- I X Ex. 11. Differentiate u = log • V X- 4- 1 + ^ SuG.— First rationalize the denominator of the fraction, obtaining u Mx log (\/x-2 4- 1 — ic)2 = 2 log( \/x2 -)- 1 — x), and then differentiate, du = — \/iC2-j-l * The student will observe for himself whether common or Napierian logarithms axe used. 28 THE DIFFERENTIAL CALCULUS. 64» ScH. — The differentiation of algebraic functions is often performed with greater faciUty by first passing to logarithms. Ex. 12. Differentiate u = . 1 — x^ SuG. — Passing to logarithms we have log u = log (1 -f- x^) — log (1 — ^-)- Biffer- du 2xclx — 2xdx 4:Xdx Axdx entiatmg, — = — - — = —— — — tj —- .' . clu X — ^^^- = — ^ — . This example illustrates the method (1 -\-x-^){l — x-^) 1 — x^ (1 — rC2)2 referred to in the scholium, although the student will find the direct method quite as expeditious. Ex. 13. Differentiate u = x{a^ + x^)va^ — x% by first passing to logarithms. du = — — dx. va^ — X- Ex. 14. Differentiate w = (a" + l')^. du = 2a"(a" + 1) logadx. a"" — 1 - , 2^"" log- adx Ex. 15. Differentiate u == —- -. du = -. a'' -^ 1 {a'' + 1)2 66. Cor. — The ordinary rule (SO) for differentiating a variable affected with an exponent applies when the exponent is imaginary. Dem — Let u = x^ *'— ^ Passing to logarithms, log u = a V — 1 log x. Differentiat- du , — -dx , . — -xidx , — - aVZl_i mg, — = av — 1 — . . • . dit = av — 1 = av — 1 x dx. q. e. d. ILLUSTRATIYE EXAMPLES. Ex. 1. Which increases the faster, a number or its logarithm ? Solution. — Let x represent any number and y its logarithm, so that y = log x. "We now wish to find the relation between the contemporaneous, infinitesimal in- crements of X and y ; i. e., if the number {x) changes how does the logarithm (y) change? Hence we differentiate, and have dy = -dx. Prom this we see that the increment of the logarithm {dy) is — times the increment of the number {dx). Therefore when ic << m the logarithm increases faster than the number ; when X > m t'le logarithm increases more slowly than the number ; and when x = m they incujabe equally. [Note. —The student should not fail to see in every such example the real object of the Dif- ferential Calculus {42). In the last example the relation between finite values of the A-ariables x and v is y = log. x. The relation between the contemporaneous, infinitesimal elements of these variables is found by differentiating, this being the object of the Differential Calculus.] Ex. 2. When the number is 2124 and is conceived as passing on to larger values by the law of growth of continuous number, i e. by LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 29 taking on infinitesimal increments, liow much faster is the number increasing than its common logarithm ? If this relative rate of change continued uniform (which it does not) while the number passed to 2125, i. e. increased by 1, how much would the logarithm have in- creased ? Solution. — Letting x be any number and y its logarithm, we have found that dy = —dx. But m, the moduhis of the common system = .434:29448. Hencef X . 43429448 when X = 2324, we have dy = '- — — — — dx = .000204dc, or the increment of the logarithm is .000204 part of the increment of the number. The number is, there- 1000000 fore increasing — — — — , or about 4902 times as fast as the logarithm. Secondly, If this relative rate of change continued the same while the number passed from 2124 to 2125, I. e. increased by 1, the logarithm would increase once .000204, or .000204. Hence the logarithm of 2125 would be .000204 larger than the logarithm of 2124. GEOMETEicAii Illustbation. — Let M N be the curve whose equation is y = log x. Take A D = 2124 ; then will P D represent its loga- rithm. Let D D ' represent dx ; then will p -. o P'E represent (Z?/.* Ex. 3. The common logarithm of 327 is 2.514548. What is the log- arithm of 327.12, on the hypothesis that the relative rate of change of the number and its logarithm continues uniformly the same from 327 to 327.12 that it is at 327? 43429448 SuG.— At 327 dy = '- — — — dx = .001328d'r. Now as the number 327 increases .12 to become 327.12 ; according to the hypothesis the logarithm increases .12 times .001328 or .000159. Hence the logarithm of 327.12 is 2.514707. ScH. — The hypothesis that the relative rate of change of a number and its logarithm continues constant for comparatively small changes in the num- ber, is sufficiently accurate for practical purposes, and is the assumption made in using the tabular difference in the table of logarithms, as explained in The Complete School Algebka {125), and in the introduction to the table of logarithms {14:) in the volume on Geometry and Trigonometry. Ex. 4. What should be the tabular difference in the table of loga- rithms for numbers between 2688 and 2689 ? Ans., .00016156+. QuEKT. — How is it that the tabular difference found in the table of logarithms for * The figure is necessarily out of proportion, as the true relation of y and x requires that A D be nearly 700 times as long as PD. 30 THE DIFFERENTIAL CALCULUS. numbers between 2688 and 2689, is 162? Sbow how the method of nsing this tabular difference makes the result agree substantially with the method of inter- polating now being presented. Ex. 5. According to the arrangement of our common tables, show that the tabular difference corresponding to 7487 is 58. ■#♦» SUCTION III Differentiation of Trigonometrical and Circular Functions. TEIGONOMETKICAL FUNCTIONS. SG, J^TOp, — The differential of the sine of an arc {or angle) is the cosine of the same arc into the differential of the arc. Dem. — Let X be any arc (or angle) and y its sine, i. e. let y = sin.r. If a; takes an infinitesimal increment (dx), let dy represent the contemporaneous infinitesi- mal increment of y. Then the consecutive state of the function is y -\- dy ^ sin {x -\- dx) = sin x cos dx -j- sin dx cos x. Now cos da; = 1, since as an angle grows less its cosine approaches the radius in value, and at the limit, is radius. Moreover, as an angle grows less the sine and the corresponding arc approach equality, and at the limit we have sin dx = dx.* The consecutive state may therefore be written y -{- dy = sin x -|- cos x dx. From this subtract y = sin x and we have dy = cos x dx, which, being the difference between two consecutive states of the function, is the differential, q. e. d. 67 » JPvop, — The differential of the cosine of an arc (or angle) has the opposite algebraic sign from the function, and is numerically equal to the sine of the same arc into the differential of the arc. Dem. — Let x be any arc (or angle) and y its cosine, so that y = cos a. Since cos X = sin (90° — x) we have y = sin (90° — x). Differentiating this by the pre- ceding proposition, we obtain dy = cos (90° — cc) X d{90° — x) = cos (90° — x){— dx) = — sin ic dx, since cZ 90° — x) = — dx, and cos (90° — x) = sin x. q. e. d. 68, ScH. — The opposition in the signs of the differential of the cosine, and of the corresponding arc, signifies that they are decreasing functions of each other [40) ; ^. e., if one takes an incremeyit the other suffers p de- crement. * The student m&Y be inclined to say that at the limit siu dx = 0. This is time, and no error would follow from the assumption ; but the statement in the demonstration is equally true, and we consider sin dx = dx instead of = 0, simply because we do not wish to have dx vanish from the formula, our object being to find the relation between dy and dx. TRIGONOMETEICAL FUNCTIONS. 31 69 • JProp* — The differential of the tangent of an arc (or angle) is the square of the secant of the same arc into the dfferential of the arc ; or for the square of the secant we may write the reciprocal of the square of the cosine. Dem. — Let 11 = tana;. Now tan x = '-, whence y = '-. Differentiatine ■^ cos X cos X Sin ^ this, observine that '- is a fraction with a variable numerator and denominator, cos a; and hence can be differentiated by the rule for fractions {54:), and the two propo- ^^ ^^. -, , cos cede sin a;') — sin.'rd(cosa;) coH-xdx-\-s\u'^xdx sitions (oo. 07), we have ay = = ■ ■^ ^ cos2a; cos- a; cos2 X 4- sin2 .r , 1 , = ■ ax = ax = sec'^xax. q. e. d. cos2 a; cos^ a; Another Demonstbation. — The consecutive state of the function y = tan x, , . . , , , -, V tan =r 4- tan di3 isnax 4- d.v ,, _,„ being y -\- dy = tan (a; + dx) = ; , = -,-> the difference 1 — tan ic tan da; 1 — tan a; aa; between the two states, i. e. the differential is dy = -^ — ^ tan x = 1 — tanxaa; ^ _j_ tan- X —dx =(14- tan2a;)di; = sec~xdx. [Let the student give the detailed 1 — tan xdx . explanation of the process. ] 70. JPvop, — The differential of the cotangent of an arc (or angle) has the opposite algebraic sign from the function, and is numerically equal to the square ofjhe cosecant of the same arc into the differential of the arc : or for the squafe cf the cosecant we may turite the reciprocal of the square of the sine. Dem. — Let y = cot x = tan (90° — a;). Differentiating by the last proposition dy = sec2 (90^ — x) X cZ(90o — a;) = cosec^ x{ — dx) = — cosec^ a;da;, or r—^dx. Q. E. D. cos X [Let the student demonstrate this rule by remembering that y = cot x = Binx and also by taking the difference between the consecutive states y = cot x, and y ~\- dy = cot (x -{- dx), developing and reducing as in the second demonstration under {69).} Query. — What is the significance of the opposition in signs? 71» JPvop, — The differential of the secant of an arc (or angle) is the tangent of the same arc into its secant into the differential of the arc. Dem. — Let y = sec a; = . Differentiating by {55, 67) "we have dy = sin a; da; sin a; 1 , , = X X dx = tan x sec a;aa;. q. e. d. cos2 X cos x cos X 82 THE DIFFERENTIAL CALCULUS. 72* JPvop. — The differential of the cosecant of an arc (or ang>) ],as the opposite algebraic sign from the function, and is numerically equal to the cotangent of the same arc into its cosecant into the di^erential of the arc. Dem. — Let y = coseccc = sec (90° — x). Differentiating by the last proposition, dy =z tan(90° — a;)sec(90° — x)d{'dO° — a;) = cot£ccosecjK( — dx) =^ — cotxcosecadx. Q. E. D. [Let the student demonstrate this proposition from the relation y ^ cosec x = sin x' Query. — What is the significance of the opposition in signs? 73, IPTOp,— The differential of the versed-sine of an arc (or angle) is the sine of the same arc into the differential of the arc. Dem.— Let y = versic =1 — cosic. Differentiating by {S7), dy = smxdx. Q. E. D, QuEEY. — Why should the differential of the versed-sine be numerically the same as the differential of the cosine, but have an opposite sign ? Illustrate geometri- cally. 74. J^vop, — The differential of the coversed-sine of an arc {or angle) has the opposite algebraic sign from the function, and is numerically equal to the cosine of the saine arc into the differential of the arc. Dem. — (Similar to the preceding.) QuEBY. — Why should the coversed-sine have the same differential as the sine, but with an opposite algebraic sign ? Illustrate geometrically. EXERCISES. Ex. 1. Differentiate u = sin x cos x. SuG. — Observe that we have here the product of two variables, viz., since and cos X. Hence du = cos .-c d(sin x) -f sin x d(cos x) = cos- xdx — sin^ xdx = (cos^a — sin2 x)dx, or (2 cos2 x — l)dx, or (1 — 2 sin^ x)dx, or cos 2 x dx. • Ex. 2. Differentiate u = cos^a;. SuG. — Observe that this is the cube of the variable cos re. Hence apply (56) and we have dii = 3 cos^ x d{cos x) = — 3 cos^ re sin a dx = 3(sin3 x — sin x)dx. Ex. 3. Differentiate u = tan 5x. SuG. du = sec2 5xd{5x) = 5 sec2 5x dx. Ex. 4. Differentiate u = cot^ x^-. du = — Qx^ cot x^ cosec^ x^ dx. Ex. 5. Differentiate w == sins x cos x. du = sin2 x(^ — 4 sin^ x)dx. TEIGONOMETEICAL FUNCTIONS. 33 Ex. 6. Differentiate w == 3 sin'' x. du = 12 sin^ cc cos x dx. Ex. 7. Differentiate u = cos mx. du = — m Binmxdx. Ex. 8. Differentiate u = sin 3^ cos 2^. du = (3 cos 3^7 cos 2x — 2 sin ^x sin 2x)dx. Ex. 9. Differentiate u = sec^ ^x. du = 10 sec^ 5a: tan 5a7 c?a:. Ex. 10. Differentiate u = tan" nx. du = 7i^ tan"~^ no; sec^ no: d'^. Ex. 11. Differentiate u = log sin x. Solution. — ^We have here a logarithm to differentiate, {. e. the logarithm of sin a;. Hence the differential is the differential of sin x, divided by sin x, in the Napier- ian system, or m times this, in the common system. Therefore du = — -^ • = sin X m cos X dx smx m cot X dx, or in the Napierian system, cot x dx. Ex. 12. Differentiate u = log cos x. du = — m tan x dx, or — tan x dx. Ex. 13. Differentiate u = log tan x. - sec2 X _ dx 2dx du = ■ dx = tan X sin x cos x sin 2x' Ex. 14. Differentiate u = log cot j;. Ex. 15. Differentiate u = log sec x. Ex. 16. Differentiate u = log cosec x. du = — cot x dx. Ex. 17. Differentiate u = e^'cos x, e being the Napierian base. Sug's. du = Cdicosx) + cos x die") = — e'sinxdx + e^cosxdx = e*(cos X — sin x)dx. Ex. 18. Differentiate u = cce*'*"'^ Sug's. du ■= e''°'"dx + xe''°^'d{cosx) = e^^'^dx — xe'^^'"' Bin x dx = grosx^l _ a; sin (c)c?a;. Ex. 19. Diffsj-entiate w = — ^ TXl du = e*^ sin x dx. Ex. 20. Differentiate u = log v sin ^ + log v cos a;. Sug's. w = 5 log sin a; + ^ log cos as. .•.(!« = J(cot x — tan x)dx = - — —. tan aX 34 THE DIFFEBENTIAL CALCULUS. ef^ ILLUSTRATIVE EXAMPLES. [Note. — The object of these examples is to still farther illustrate the meaning of the process of differentiation.] Ex. 1. Whicli changes the faster an arc or its sine ? "What is the relative rate of change ? When is the disparity greatest and when least? What is the relative rate of change when the arc is 60°? When 20° ? When 80° ? SoiiTTTiON. — From y = sin a;, we have by differen- RrT=SlP tiating, dy = cosxdx. The meaning of this is, that if the arc {x, AP) takes an infinitesimal increment {dx, Pp) the sine {y, P/) takes an infinitesimal incre- ment {dy, pE) which is cosjc times the increment of the arc. Now cos x is, in general, less than unity, so that the increment of the sine is, in general, less than the contemporaneous increment of the arc. But as x grows less cos x becomes greater and approaches unity as X approaches 0. So, also, cos a? approaches as x approaches 90°. Hence the disparity between the contemporaneous increments of an arc and its sine is less as the arc is less, disappears when the arc is 0, and becomes infinite when the arc is 90°. For x = we may, therefore , con- sider the arc and its sine to be increasing at equal rates. For x = 90°, the arc is increasing infinitely faster than its sine. When x = 60° cos. r = ?. Hence at 60° the sine is increasing just i as fast as the arc. In the figure, letting P'p' represent dx,p'E.' represents dy andp'E' = iP'p'. "When .a; = 20° cos a; = .94 nearly. Hence at 20° the sine is increasing . 94 as fast as the arc. At 80°, the sine is in- creasing only about .17 as fast as the arc. These facts are illustrated in the figure. Ex. 2. Assuming that the relative rate of increase remains con- stantly the same as at 40°, how much does the sine increase when the arc increases from 40° to 40° 10' ? What when the arc increases to 41°? 8 14159 StJG. — Since the arc of 10' = — ^ = .0029088 ; we find the increase of the 180 X 6 sine, on the above hypothesis, to be . 002228 , which is slightly in excess of the real increase, as will be found by examining a table of natural sines in which the de- cimals are extended to 7 places. The table gives ,0022156. At the same rate of increase the sine of 41° should be .01^9 above the sine of 40° ; whereas from a table the increase is found to be .0132714. [The student will observe that the cause of this disagreement is that the rela- tive rate of increase of the sine as compared with its arc, is greater at 40° than at any point between 40° and 40° 10', or at any point after 40°,] Ex. 3. The natural tangent of 27° 20' is .5168755. ' Assuming that the relative rate of increase of the tangent as compared with its arc TRIGONOMETRICAL FUNCTIONS. Zo remains the same as at this point, for the next 25" increase of the arc, what is the natural tangent of 27° 20' 25" ? Ans., .517029. Ex. 4. Which increases faster, the arc or its tangent ? "When is this difference greatest? When least? What is the value of the arc when the tangent is increasing just twice as fast as the arc? Ansiver to the last, 45°. Ex. 5. The natural cosine of 5° 31' is .995368. Assuming that the relative rate of change of the cosine as compared with the arc re- mains the same as at 5° 31', while the arc increases to 5° 32'_, what is the cosine of 5° 32'? Ans., .995340. Ex. 6. At 36° what is the relative rate of increase of the arc and the logarithm of its tangent? SuG. — From ii, = log tan x, we have du=m {teinx-\-cotx)dx. When x = 36° this becomes dit = .43429 X 2.102925da; ; or the logarithm of the tangent increases about .91 times as fast as the arc. Ex. 7. The logarithmic cosine of 67° 30' is 9.582840. Assuming that the relative rate of change of the logarithmic cosine and the arc remains the same as at this point while the arc passes to 67° 31', what is the logarithmic cosine of the latter arc ? Ans. , 9. 582535. Ex. 8. The log cot 58°21' = 9.789868. On the same assumption . as above, what is the decrease of this logarithm for 1 second increase in the arc? Ans., .OOOOO'ill. Ex. 9. The log cos 42° 14' = 9.869474. What is the corresponding tabular difference ? Ex. 10. At what rate relative to its velocity, is a point in the cir- cumference of a wheel revolving in a vertical plane, ascending, when it is 60° above the horizontal plane through the centre of motion? Ans., One half as fast. CIRCULAR FUNCTIONS. '^S, I^TOp* — The differential of an arc in terms of its sine is the differential of the sine difjided by the square root of 1 minus the square of the sine ; or the differential of the sine divided by the cosine. Dem. — Let y = sin— ^a;*, whence x = siny. Differentiating and finding the dx value of dy, we have dy = . But cos y = \/l — sin^y = s/l — x^. .'. dy i= dx , Q. E. D. \/l — a;2 * This notation is explained in the Trigonometry of this series. It means simply "y = the aro wh.066 Bine is «, and hence y = 6in""'x is equivalent to x = sin y." 36 THE DIFFERENTIAL CALCULUS. 70' ScH. — The student should not fail to observe the essential identity of this proposition with [06). Thus, when we differentiate u = sin a;, we get du = cos xdx, which expresses the differential of the sine (u) in terms of its arc (x). From this we have dx = = — , which expresses cosa; ^/l—u^ the differential of the arc (x) in terms of the sine [u). The one conception is the converse of the other. 77. JPvop* — The differential of an arv in terms of its cosine has the opposite sign from the function, and is nuvfierically equal to the differential of the cosine divided by the square root of 1 minus the square of the co- sine ; or the differential of the cosine divided by the sine. Dem. — Let y=cos—^ x, whence x = cos y. Differentiating, and finding the value f]l* cl'X of dy, we have dy = r^ — = . q. e. d. ^ ^ sm2/ v^l — X-' 7S, ScH — Compare this and the following propositions with their equiv- alents in Trigonometrical functions, as was done in the case of the preced- ing proposition. 79. JPvop* — The differential of an arc in term.s of its tangent is the differential of the tangent divided by 1 plus the square of the tangent. Dem. — Let y = tan— i cc, whence x = tan?/. Differentiating and finding the dx dx value of dy, we have dy = — ~ = q — ■. — -, since sec^w =1-1- tan^?/ = 1 -f x^ ^ sec-?/ 1 -[- x^ Q. E. D. ^0. JPvop, — The differential of an arc in terms of its cotangent has the 02:)posite sign from tlie function, and is numerically equal to the differ- ential of the cotangent divided by 1 idIus the square of the cotangent. Dem. — Let y = cot-i cc, whence x = cot y. Differentiating, and finding the value (It S.PO of dy, we have dy =^ ^r— = — :; — ; — ;;. Q- e. d, ^' ^ cosec2 y 1 j^ X' SI, Pvop, — The differential of an arc in terms of its secant is the differential of the secant divided by the secant into the square root of the square of the secant minus 1. Dem. — Let y = sec-i x, whence x = sec y. Differentiating and finding the value dx dx '>f dy we have dy = "- — — = ■ since tan y = s/sec^j/ — 1 = \/x--^ — 1. secy tan 2/ ^cVx^ — l Q. £. D. CIRCULAR FUNCTIONS. 37 S2, Prop. — The differential of an arc in terms of its cosecant has the opposite sign from the function, and is numerically equal to the differ- ential of the cosecant divided by the cosecant into the square root of the square of the cosecant, minus 1. Dem. — Let y = cosec-i a;, whence x = cosec?/. Differentiating, and finding tlie dx dx valtie of dy, we have dy = = . since coty = \/cosec^y — 1 cosecycot?/ ^s/x- 1 V X^ 1. Q. E. D. S3, I^TOp, — The dfferential of an arc in terms of its versed-sme is the differential of the versed-sine divided by the square root of twice the versed-sine minus the square of the versed-sine. Dem. — Let y = Yers—^x, whence x = \ersy. Differentiating and finding tho (Xx value of dy we have dy = . ' . But sin y = \/l — coss y = \/l — (1 — vers y)^ = dx \/l — (1 — xy^ = \/-Ax — a;2. Therefore, substituting, dy = — — "- . q. e. d. V2x — a;2 S4:, JPvop. — The diffey^ential of an arc in terms of its coversed-sine has the opposite sign from the function, and is numerically equal to the d,ifferential of the coversed-sine divided by the square root of twice the coversed-sine minus the square of the coversed-sine. Dem. — Let y = covers—' a;, whence x = covers y. Differentiating, and finding the , . , , , cZa; dx dx value of dy, we have ay= — = cos y ^i _ gij^sy yi _ ^1 _ covers s^)2 dx dx =zr. Q. E. D. v/1 — (1 — xy s/'tc — x^ EXERCISES. Ex. 1. Differentiate y == sin""'^- ; y = cos~^- ; y = tan~^- ; y = cot~^- ; y = sec~^— ; y = cosec""^- ; y = vers~^- ; y = covers"^-. StTG. — We have dy = — . -■ ' . by (75). Now since dl -) = — , we have dx r dx ^ ,., - ,/ a; . dx (X cos— ^ - ) = 38 THE DIFFERENTIAL CALCULUS. rdx rdx .. = .(t.n-.^-) = ^-^^; ., = .(eot-.5) = --^,; ., = .(sec- ?) = rdx / x\ rdx / x\ — - ; dy = d{ cosec-i - ) = :=:z ; dy = d[ vers-i - ) = _- zVx^—r^ ^ ^^ xVx'^ — r^ ^ rJ v/9 -. , ,/ £c\ dx and dy = a( covers—' - I = \ ^/ ^2ra; — x^ Geometeical IiiiiUSTBATioN. — Let O A = 1, and OA' = r. Let y = arc C A (to radius 1), and x= C B' the sine of the same number of degrees as y, but to a radius r. Now CB = C B ^, and we have y (or CA) = C B' V sin—' C B = sin—' = sin— ' -, the arc {y) being taken to radius 1 while the sine x is taken to the radius r. dx \/2rx — jc2 B A B' AT Fig. 15. Ex. 2. Differentiate ^ = sin-^- ; ^ = cos"^- ; ^ = tan"^- ; ^ = f IT IT y V TV L-1^ y _i^ y 1^ V 1^ -. V ^^ cot ^- ; - = sec ^-- ; - = cosec - ; - = vers" - : and - = covers -. f f T T TV TV V Besults ill order : dy r^dx rdx dy r- -\- x^ ; dy \/r2 — x^ r^dx ^; dy = rdx \/ri fi x2 -; dy. r'^dx rdx s/^rx — x'-^ ; and dy = XV X- — r^ o^dx ; dy = r^-dx XV X- j-2 r2 + 072 ' ; dy = v2rx — x^ Geometrical L^lustbation, — Li Fig. 15 let OA' = r, C'A' = y, and V X C'B' = X. Now if OA = 1, we have CA = -, and CB = -. Hence r r dy = d(sin— ' a;)to radius r = rdx v/r2 — a;2 r, etc. 85, ScH. — The results in the last example will be seen to correspond with those in Ex. 1, by noticing that in Ux. 1 y represents CA, whereas in Ex. 2 it represents C'A'. Now an increment of C'B' (which is x in both cases) which makes an increment in C A, will make r times as great an incre- ment in C'A'. Hence we have but to multiply the increments of CA (the c?y's) as found in Ex. 1, by r to get the corresponding increments in C'A', which are the c?y's in Ex. 2. X Ex. 3. Differentiate u = tan~^-. y du = Ex. 4. Differentiate u = BLvr^{2x\/l — x^). du = ydx — xdy 'Idx v/l — ar» CIECULAR FUNCTIONS. 39 Ex. 5. Differentiate u = cos~\a7\/l — x^). Sug's. — By the rule the differential of the arc u is negative, and numerically equal to the differential of its cosine, jc\/l — x^' divided by the square root of 1 minus the square of its cosine. The differential of ck \/l — x~ is dx\/l — x^ — and 1 — {x^l — a;2)2 = 1 — x2 ^ a;4 ... ciy, \/l — x\ (1 — 2x2)dx V{1 — x2 -f x4)(l — x^) X dx Ex. 6. Differentiate u = sin~' — • . du == 3^.37 Ex. 7. Differentiate u = sin""^(3a: — 4^3). du = -- V 1 ^2 Ex. 8. Differentiate u = vers"^?/ — \/2r2/ — y^, understanding that vers~^?/ is taken to radius r. SuG. dw = ^^1 _ rdy—ydy ^ ydy s/'lry — y- \/2ry — y~ \/2ry — y^ Ex. 9. Differentiate u = tan~^(\/l + x'^ — x). du = Ex. 10. Differentiate u = log a^/y— ^- 1- i- tan~^a;. dx 2(1 + ^=)' dr SuG. w = 4 log (1 + as) — 4 log (1 — ic) + ^ tan-» x. du = — £- fl;4 mdx Ex. 11. Differentiate v = sin~'ma;^ — - cZv = / — = ^ vl — m2^-5 Ex. 12. Differentiate y = e»'*'^~\ tZy = e*"~'" ^"^ 1 + ^■^' Ex. 13. Differentiate y = tan""^^ ^ — . dy 1 — x^' 1 + x^' Ex. 14. Differentiate ?/ = x^^''~\ 1 dy == x'^'' ^ :j — ^i— ^ ^ — Y dx. x{l — x^) GENERAL SCHOLIUM, 86, The preceding sections comprise the fundamental rules of the differ- ential calculus ; and it only remains to extend and apply them, in order to complete this portion of our subject. 40 THE DIFFERENTIAL CALCULUS. SUCTION IV, Successive Differentiation and Differential Coefficients. SUCCESSIVE DIFFEKENTIATIOK 87 » Bef. — Successive Differentials are differentials of dif- ferentials ; or a successive differential is the difference between two consecutive states of a differential. III. — Let M N Fig. 16, be a straight line whose equation is y ^=^ ax -\- b ; whence dy = adx. Now suppose X to be considered equicrescent, and let DD', D'D", D"D"',and D"'D''^ represent the successive equal increments. P'E, P"E', P"'E", and P'vE'" represent the contemporaneous incre- ments of y, i. e. the dy's. But in this case the dy's are all equal. Hence there being no difference be- tween two successive states of dy, as between P'E and P"E', there is no successive differential, or the differential of dy is 0, since dy is constant. This fact appears also from the rela- tion dy = adx, in which, if we conceive dx to be constant (i. e., x equicrescent), adx is constant ; whence dy, which equals adx, is constant. But consider in a similar manner the parabola in Fig. 17, Fig. 16. whose equation is y'^ 2px ; whence dy =^— . Still con- FiG. 17. sidering dx as constant, i. e. DD' = D'D" = D"D"' = D"'D'^, etc., it is evident that the dy's, which are represented by P'E, P"E', P"'E", etc., are not constant. Now the difference between any two successive values of dy, as between P'E and P"E', is a successive differ- ential, i. e. a differential of a differential. The fact that dy is a variable in this case when dx is constant is also Tfdx readily seen from its value dy =- — . In this pdx is constant, but y is variable. Hence dy varies inversely as y. 88. Def. — A Second Differential is a differential of a first differential, is represented by d^y, and read " Second differential y." A. Third Diff'erential is a differential of a second differential, is represented by d?y, and read " Third differential y." In like man- ner we have fourth, fifth, etc., differentials. ScH. — The student should be careful not to confound d^y with dy'^. The latter is the square of dy. Nor should the superior 2 in d'^y be mistaken for an exponent : it has no analogy to an exponent. Observe the significa- SUCCESSIVE DIFFERENTIATION. 41 tion of the several expressions d'^y, dy^ and d{y'^). The latter is equivalent to lydy. 89, JProp, — Second differentials are formed by differentiating first differentials, third differentials by differentiating second differentials, etc., according to the rules already given. This proposition is self-evident, since the differentials are expressed as algebraic, trigonometric, logarithmic, or exponential functions, the rules for differentiating which are those heretofore given. Ex. 1. Produce the several successive differentials of y == ax*. Solution. — Differentiating y = ax\ we have dy = 4:ax^dx. Differentiating this differential remembering that d{dy), i. e. the differential of dy is written d~y, and that dx is constant, we have d-y = 12ax'^dx dx, or 12ax^ dx-. In like manner dif- ferentiating d'^y = VlaxHx"^, we have d'^y = 24«a; dx? . And again d'*y = 2^adx*. Here the operation terminates, since d'^y being equal to 24«dx-* is constant. Ex. 2. Produce the several successive differentials oi y = Sx* — dx^ — 5x. ' dy = (32^3 — 9^i2 — h^dx. Results, - dHj = (96^2 — 18^)^^% dHj = (19207 — l%)dx\ d^y = ld2dx^ Ex. 3. Produce the first six successive differentials oi y = sin x. r dy = cos x dx, d^y = • — sin x dx^, Results, \ d^y = — cos x dx^, d^y == sin x dx*, [ d^y = cos X dx-', d^y = — sin x dx^. QuEBY. — Does the above process ever terminate ? Ex. 4. What is the 3rd differential oi y = af ? d^y = n{n — l)(n — ■1)x''~^dx^. Ex. 5. Produce the 4:th differential oi y = ax^. 15a dx* Ex. 6. Produce the first six successive differentials of y = cos^. Ex. 7. Produce the first four successive differentials oi y = logx, in the common system. ^ , , mdx ^ m dx^ , 2m xdx^ 2m dx^ _ Results, dy = , d^y = — , d^y = = ^, d*y = ' ^ X ^ x^ ^ X* x^ ^ 6??i dx* X* 42 THE DIFFERENTIAL CALCULUS. Ex. ^ Produce the first four successive differentials oi y =: log (1 + ^)> in the connnon systera. ^ , , mdx ^ m dx^ , ImilA-x) dx"^ 2m dx^ Results, dy = , d^y = — — , d^y = — y-—^ — '- = , ^ l^x ^ {1+xy ^ {1 + xy {l-{-xy , Qmdx* Ex. 9. Produce the fourth differential oi y = ef. d^y = e'dx*. Ex. 10. Produce the fourth differential oi y = a^, in the common - log^a , system. dni = a'^dx*. DIFFERENTIAL COEFFICIENTS. 90, Defs. — A First JDifferenticd Coejficient is the ratio of the differential of a function to the differential of its variable, and is dy represented thus, -7^, y being a function of the variable x. A Second Differential Coefficient is the ratio of the second differential of a function to the square of the differential of its variable, and is expressed thus, -r-^. A Third Differential Coefficient is the ratio of the third differential of the function to the cube of the differential of its vari- d^y able, and is represented thus, -7^. In Hke manner the nth differen- (XX d^y tial coefficient is -7—. dx" S/u civ 111. — Having y = ax\ we obtain -- = 5ax^. In strict propriety — is a symbol representing the general conception of the ratio of an infinitesimal increment of the function to the contemporaneous infinitesimal increment of its variable ; and 5ax* is, in this case, its value. But it is customary to speak of either as the dif- ferential coefScient. The appropriateness of the term differential coefficient arises from the fact that the Sax' is the coefficient by which the differential of the vari- able has to be multiplied in order to produce the difi'erential of the function. Strictly, therefore, the differential coefficient is the coefficient of the differential of the variable ; but it is customary to speak of it as the differential coefficient of* the function. * The "of" meaning, perliaps, "derived from," or "appertaining to." DIFFERElilTIAL COEFFICIENTS. 43 Ex. 1. Given y ■= ax^ — x^, to find the 1st, 2nd, and 3rd differential coefficients. Results, — = 3ax^ — 2x, — ^ = 6ax — 2, -— ^ = 6a. ax ax'^ dx^ 1 -\- X Ex. 2. Given v = z , to find the 5th differential coefficient. . 1 — X dni 240 dx'^ (1 — a;)« Ex. 3. Given y = fl^w^Z [Let the student give the proof.] tjc \~dw~\ Ex. 1. Given u = tan~^-, and y^ -{- x^ = r", to find -- . c TTT 1 r<^w"l <^w , du dy ^ ,..-,, du ^ du Solution. — VVe have -- == - — L _- -Ji. Remembenng that ^- and -- are LaxJ dx dy dx <= ^ ^y 50 THE DIFFEEENTIAL CALCULUS. dx partial differential coefacients, we have from u =^ tan-^-, du = bsbbee /acx dx 2/' ydx du y ^ ^.^ ^ ^du x , = -^ 3 wnence — = - . In like manner we find — = ^. Also from y^-f-a;? =r2, we find ^-^ = — -. Substituting these values, we obtain f--! = ^ 4- ^ — - Y— -\ dx y ^ \_dx_\ r2^\ rVV y). _ 2/ , jK^ _ y^ + x -2 _ r2 _ 1 1 r^ r ?/ ~ r2y ~ r-y ~ y ^^ ^^2 _ ,^' Ex. 2. Given w = iQjr^{xy), and y =: e* to form f— 1. V.dxi BesuU, r^l = f:(l+f2. LdxA 1 + ^2e'^ Ex. 3. Given u = z"- -\- y^ -{■ zy, and z = sin ^, y = ^> to form f— 1- Ldxj SxTG's.-We have [--1 = -- ^ + - - LdxJ fZ^/ dx dz dx = (32/2 _j_ 2)ex _^ (2z 4- 2/) (cos a;) = (3e2^ -[- sin .-rje^^ j-f- (2 sin a; -f e^)(cosa;) -= Se^^^ -j- 6^(sina; -f- cosic) + 2 sin jc cos a; = 3e3== -]- e^(sin ic -f- cos a) -|- sin 2x. Ex. 4. Given w == yz, and y = ^, z == ^-^ — Ax^ + 12j;2 — 24^ + 24, to find g]. Sesnlt, g^] = e-... Ex. 5. Given u = sin-^(^ — q), and^= Sx, q=:4:X^, to find f-^l. c ' xtr T. <^'^ 1 du — 1 dp buGs. — We have — = — ^ — = — . -£- = 3 «/> v/1 — {p — 5)2 Q'? v/1 — (p — g)2 t^^ and 1^ = 12^2. Whence Pill = ^ _ _ ^^:^^ "•^ LdxJ ^1 __ (p _ g)2 v/l _ (p _ ^j2 3 -^ 12.r2 3 v/1 — 9a;2 + 24x4 — 16x6 v^l — x-^ Ex. 6. Given li = — p- ^ + ^ and 2/ = log^, to find that r— 1 = a:3(log;r)2. Ex. 7. Given w = --^, where p = a sin a:, and q == cos a;, to find that I— 1 = e^sina;. [Note.— In such, examples as the above, it is of course possible to substitute in the ftinction u, the values of the several variables on wbicb it depends, in terms of the single variable upon FUNCTIONS OF SEVERAL VARIABLES. 61 which each of them depends, and then have m = a function of a single variable, which can be differentiated by the elementary processes. But it is the chief design of these examples to far miharize the important formulcB used, and render their meaning clear. Their precise practical value cannot be appreciated until the student has made further progress] IMPLICIT FUNCTIONS. du 102, I*VOp. — Having f (x, y) = = u, -r^ = — —- ; in which — , and — are the partial differential coefficients of the function taken with reference to x andj respectively. -r. -n, ^^x 1 rdvr\ du , du dy ^ , Dem. — From {99} we have I — = - — j_ — - _£. But as u remains constantly equal to 0, for all simultaneous values of x and y, when both x and y have taken on contemporaneous changes, du = 0. Therefore I -— = 0, and — 4- — - -^ = : ^ LdxJ dx dy dx du . dy dx whence -^ = — -r-. q. e. d. dx du dy ScH. — It is to be observed that though I y J = 0, it by no means follows du du ^ _ , , „ . „ , that -y-, or -- = 0. For example, y^ -\- x^ — r^ = is of the form f{x, y) ■= 0. Now if y changes and x does not, the function changes and is no longer := 0. So also if x changes and y does not, the function is not 0. Bub if both change together according to the law of their mutual depend- ence, i. e. if the changes are what we have called contemporaneous, the function remains equal to ; and its total differential is 0. [The student can observe the geometrical signification of these statements, by noticing that y- -\- x'^ — r2 r= is the equation of a circle, and that the function is when x and y vary together, according to their mutual dependence : but when one varies and the other does not, the function varies ; i. e. the point falls out of the circumference. Illustrate in like manner from y- — 2px ^ 0.] dy Ex. 1. Given x^ + ^ax^y — ay^ = 0, to form -r- npon the principle 0,00 just demonstrated. Solution. — Putting m = = cc^ -|- '2>ax'^y — ay^, we have dxU = 4.x"dx -|- 4:axydx, and dyii = 2ax^dy — Say^dy ; whence we have the partial differential coefficients du da; ' ^ dy dx du 2ax^ — Say* 52 THE DUTERENTIAL CALCULUS. Ex. 2. Given ax^ + ^'^V — ^V' = 0, to form — - as above. ax ^ ,, dy Sax^ 4- Sar^v Result, -f- = ^. ao; X-' — oay^ dy Ex. 3. Given y^ — 2axy -{- x^ — b"^ = 0, to form -^ as above. ax dy ay — x Result, v- = • ax y — ax dxj Ex. 4. Given y^ — Sy + ^ = 0, to form -f- as above. dx Result -^ == ' dx 3(1 — y^y dy Ex. 5. Given x^ — y" = 0, to form — ^ as above. dx jtesuu, ^ = y'-^y'^'gy dx x^ — xy log X V / = — dv Ex. 6. Given a^ + v secfo:?/) = 0, to form ~ as above. dx _, ^, dy yV BeQ.{xy)i?in(xy) 4- 20" yx^~^\o^a Result, -^ = '■ — . —^ ^-^- — . ^^ xV Beo, {xy) tan (xy) -\- 2a''''x^logalogx COMPOUND FUNCTIOISS. 103» Bef. — A Compound Function is a function of a function. Thus, if u = f{y), and y = (p{x), u = y"[^(^)],* and u is said to be a compound function of x. This relation is often indi- cated by saying that " u is a function of x through y," or that " u is indirectly a function of x through y." In case u = y(^, ?/)' ^^^ 2/ ^^= ^(^)j "^^ ^^J ^^^^ ^ ^^ directly a function of or, and also indirectly through y. 104. Frop.—If u = f(y) and y^ cp{x), ;^ = ^ ^- Dem. — From u = f{y), "we have du = ;7-cZ2/- ^^t from 2/ == ^(aj)> "we have dy = dy dy ^ _ , du dy ^ ^ du du dy —da;. Hence du = — -r-d^, and ^- = — ^- Q- e. d. ax dy a.^• dx dy dx ScH. — It will be seen that this is only a particular case of the preceding ; but it is of such frequent occurrence that it is thought best to give it prom- inence. * Read, "u eqiials the/ function of tbe (p function of x." SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES. 53 Ex. 1. Given u = a^, and y = 6', to form ~ on the principle just demonstrated. r, <:^u , . dy . . . du dudy . Solution. — = ay\o3,a, and -^ = &=^log&. .•.-—= — — - = a«'loga X dy ° dx dx dy dx &* log 6 = ayh^ log a log &, or a* iF log a log h. du Ex. 2. Given u == 6t/-», and y = ax^,to form — as above. CtJu du ^ _ dx du Ex. 3. Given u = log y, and y == log x to form — as above. ax du 1 1 1 dx y X xlogx 10S» JPvop, — Having given ?i=^(z) Sbndz=f(x, y) to differen- tiate u with respect to x and y without previously eliminating z. Dem. — Since w is a function of x and y, we have Now w is a function of x through, z (i. e. it is a compound function of a). Hence —- = —-—• and for a like reason -—==-- -^ (lOd). Therefore, substituting, dx dz dx dy dz dy du dz ^ , du dz , du = — --dx 4- -r- -T-dy. q. e. d. dz dx dz dy ScH. — The student should not fail to observe that all the truths developed in this section are but deductions from the proposition that the total differ- ential of a function of several variables equals the sum of the partial dif- ferentials. With this key in hand, he can readily unlock the mysteries of the whole subject. ■^♦» SUCTION VL Successive DifFerentiation of Functions of Two Ikdependent Variables, and of Implicit Functions. 106, JPvop, — In a function of two independent variables, both va- riables may be considered equicrescent ; i. e., their differentials may be regarded as constant. IrjL. — This proposition is an axiom, and it is only necessary that its import be clearly understood. Thus, if u =f{x, y) and x and y are independent, any change which X may undergo does not affect y, and any change which y may undergo 54 THE DIFFERENTIAL CALCULUS. does not affect x, as this is what is meant by their being independent. We may therefore conceive each of them to change according to any law we please ; and it is found convenient to conceive that x increases by equal infinitesimal increments, as heretofore, and that y also increases by equal infinitesimal increments. Thus dx and dy are constants ; but it does not follow that we are to regard dy = dx. In fact this would be to estabUsh a relation between x and y, and hence would be contrary to the hypothesis 107, Def. — When u = f{x, y), and x and y are independent of each other, dj,u and dyU are, in general, functions of x and y, and hence may be differentiated with respect to either variable, thus ob- taining a class of Second I^artial Differentials, In like manner these second partial differentials are in general functions of X and y, and may be differentiated with reference to either, giving rise to Third I*artial Differentials ; etc., etc. lOS, JS^otatton. — Havmg u = f\^x, y), -i—^' TIT ^ ^ ^^ dydx^ and -^—dy^ are the symbols for the second partial differen- dydx ^ dy^ d^u tials. The third partial differentials are indicated thus, -^dx^, d^u , , d'^u , , d'^u ^ , d^u ^ ^ ^ dHi -dx^dy or --t — dydx% dxdy^ or ■j----dy'dx, and -^dy\ dx"-dy dydx^ ^ ' dxdy^ dy'^dx ' dy^ In each case the form of the numerator indicates the number of dif- ferentiations, and the denominator the variable or variables with which the successive differentiations have been made, and the order. Thus ——-diMx signifies that u = f{x, y) has been differentiated dymx three times in succession, twice with reference to y and once with re- ference to x, and in this order. So, in general, ^ dx^-^dy"" sig- nifies that u =f{x, y) has been differentiated m — n times in succes- sion with reference to x, and then n times with reference to y. Ex. 1. Given u = x'^y'^ to form the several successive partial dif- ferentials. dii du d'-u Remits, —dx = 2y^xdx, —dy = 2x-'ydy ; t^/^^ = '^y^dx'', -dxdy =^ 4:xydxdy ; -f-dy^ = ^xmy"~ ; t-«^^ = 0, dxdy ^ ^ ^ ' dy' ^ dx' d^u ^ ^ .77 d^u . , A J J . d^^j , A j^d^dy = iydx'dy, j^^dxdy^ = ^^dxdy- ; -dy' = ; dhi -^-^-dx'dy^ r= ^dx^dy\ etc. SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES. 55 Sug's. — Having -^-dx = 2y^xdx, to differentiate it with, respect to jc, we notice in the first member that it will give • dx ; the dx being written in the de- nominator, and as a factor to designate with respect to which variable the differen- tiation is made, and also in accordance with the principle that the differential coefficient multiplied by the differential of the variable, is the differential of the function. Now, the differential coefficient being the differential of the function divided by the differential of the variable, we have for the differential coefficient du K^^"") Kd'x^'d of --dx taken with respect to x -—. • hence the differential is - dx, dx ax dx Finally, observing that dx is constant, this becomes ^ da;, or -jT^dx^. In like manner the student should analyze the other processes. It is of the utmost impor- tance that he fully comprehend the reasons for these processes ; if he do not he will become hopelessly entangled in the subsequent operations. Ex. 2. Produce the successive partial differentials of u = {x -{■ 2/)"* with respect to x ; also with respect to y. du Results, d^u or -^dx = m(^ + y)"'~^dx, —dx^, or dj,d^u = m{m — l)(.r + y)"'~^dx^, dHi -r-d^^ o^ d^d^d^u ^= m{m — l)(m — 2)(^ + yY~'^dx^, etc. ScH. — ^When u =/{x, y), the following forms are called Partial DiffeV" .. , ^ ^ > ^ d^u d-u d-u d?u d^u d^u d"'u enttal Coefficients : -5—, — -, - — --, — -, - — — , -—7—, ; — -, dx^ dy^ dydx dx^ dx^dy dxdy^ dx^^^'dy" etc. Ex. 3. Form the successive partial differential coefficients of w = sin (a; +2/) with respect to y. ^ ^^ du / , s d^u . , ^ d^u Besidts, —== Gos{x -^ y), — = —sm{x-j-y), — = —cos{x-\-y), d-^u . . . d°u , — = sm {x + y), — - = cos(^ + y), etc. Ex. 4. Form the successive partial differential coefficients of w = cos(^ . — y) with respect to x. ^ J, du d-'u , . d^u . , . EesultSy -5- = — smix — y), — - = — cosix — v), -7— == sm(a:— v), etc. Ex, 5. Form the successive partial differential coefficients of w = 56 THE DIFFERENTIAL CALCULUS. log {x + y) with respect to x, and also with respect to y in the com- mon system of logarithms. J, du m d^u m d^u 2m(x -\- y) dx X -\- y dx^ {x + y)' dx^ [x + ?/)4 , etc. The partial differential coefficients with respect to 2/ are altogether similar, 100. JPfop, — Jf u = f(x, y), in which x and j are independent, and several differentiations he performed^ m with reference to one variable and n with reference to the other, the result is the same whatever the order of the ope7^ations. ^ - , ,^ -, ., , d'^u , , d'^u , , Dem. — 1st., To snow that ■ dxdy = -r—r-dydx. acco/y aycix dn —dx =f{x -j- dx, y) —f{x, y), and ax d-u -dxdy = f{x -\- dx, y -\- dy) — f{x, y -\- dy) — [/(« + dx, y) —f{x, i/)] = dxdy f{x -\- dx,y -\- dy) —f{x, y + dy) —f{x + dx, y) + f{x, y). Again, -^-dy = f{x, y + dy) —fix, y) and d'^u dydx = f{x .\. dx,y -\- dy) — f{x + dx, y) — \_f{x, y + dy) — f{x, y)1 = dydx fix -{. dx,y + dy) —fix + dx, y) —fix, y -\- dy) -\- fix, y). These two results being identical, we have -——dxdy = dydx, (1). 2nd., To show that - — - dx'^dy = , , dydx^. dx^dy ^ dydx'^ ^ d^u d/hi For convenience of notation put dxdyU for y-r dxdy, and dyd^u for ^— ^ dydx. Then, as before shown dydxU = dxdyU = fix, y), and hence may be differentiated with reference to x or y. Differentiating with reference to x, we have, dxdydxU = d'^u dsdxdyU* But by (1) dxdyidxU) = dydxidxu) or dydxdxU. Whence j iydx^ = In Hke manner we may proceed to any extent desired. d^u d^i [Let the student show that t; — ir-dx^dy^ = - ; ' , dy-dx^"] *• dx^dy^ ^ dy^dx^ ■' Ex. 1. Given u == xlogy in which x and y are independent, to form the several second and third partial differentials, and to show that dydxU = dxdyU, and also that dydxdxU = dxdydxU = dxdxdyU.* -n T, d^u ^ . . , 7 , ^ dHi ^ ^ , . -, , X d^dy Results, -r—dx^ (i.e. dxdxU) = 0, - — —dxdy {i.e. dxdja) = , ' dx"^ ^ "" "" ^ ' dxdy ^ ^ ' ^ y * These opieratione are sometimes indicated thus : d„u= d^u, d.„ji= d^,„u= d_„u, etc. yx xy yx* xjf* dcx|f SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES. 57 T d"u dydx dHc , xdu'^ d^u , , and - — —dydx also = -^ , -7—dy^ = '—, - — -—dxdy'^ == dydx ^ y dy'i ^ y dxdy^ ^ dxdi/^ - d^u '—, and -; — 7— = 0. y" dydx^ Ex. 2. Given u = x^y + ay"^, to form the second, third, and fourth partial differential coefficients, and show the convertibility of the in- dependent differentiations. _, -, du .. du , _. d'^u ^ d^u _ d^u Hesults, -rr- = ox'^y, — - = ^3 _l 2ay: -— = bxy, — — = 2a, -; — — = dx^ dx ^ dy ^ dx'^ ^ dy^ dxdy _ d'^u ^ , d'^u ^ d^u ^ d^u ^ d-^u and -; — — = Sx" also ; - — = by, — — == 0, - — 7- = 6x = — — -— , dydx dx^ dy^ dxmy dyax^ d^u ^ d^u d*u ^ ^ = 3-—^- ; 3— = 0, etc., etc. dxdy'^ dy-'dx dx'^ Ex. 3 to 7. As above form and compare the successive partial dif- ^2 yi J^ ferential coefficients of the followine^ : u = : u = tan~^- : u = sin X cos y ; u = x^ ; and u = {x -\- y)'*. 110, JPfob, — To form the successive differentials of a function of two independent variables. Dem. —Let u =f{x, y), in which x and y are independent variables. The total differential being equal to the sum of the partials, we have Now, remembering that as x and y are independent and hence may be treated as equicrescent, dx and dy may be considered constant, and remembering also that du du — , and — are, in general, functions of x and y {107), we proceed to differentiate €vvO ^y (1) again. Thus and \dy)'='d^x^''-^ d^^^y^ which substituted give d'^u o^^w J J , d;^'^ J 1 , d^u. ^ d^u^ „ , „ d^u ^ ^ , d^u , d-'u = 7— <^^2 _^ ^^ydx -f- ——dxdy 4- 5— dys = -r-,dx^ 4-^-—rdxdy -\ dy^, dx-^ dxdy dydx ^ ^ dy^ ^ dx'^ ' dxdy dy^ ^ dht .J dH J . ^ - - ^^ '^^^^ Sy^^'^y = dyd^^y^"" ^^^^^' Again, differentiating this second differential, we have ^ _ du * To perform this operation observe that — is treated as a function of x and y, and hence its dx total differential is equal to the sum of its partial differentials. The partial differential with respect to aj is _2fdx, and with reference to 2/. — —dy. (ja;2 dxdy 58 THE DIFFERENTIAL CALCULUS. <^=o-+-o-^+^-> a( --. — ;- ) = -; — T-dx + ; — r-dy, \dxdy/ dx^dy dxdy^ and cZ( -— ) = dx A- -^—dy. Substituting, we have Kdy'^J dy^dx ' dy^ ^ ^ d^w , , d^u , ^ , ^ d'^u , ^ , ^ dht , , , d^u , ^ , d% , = -v-cZx3 _L -^ — -dx^dy 4- 2- — -dx-dy 4- 2 , , dxdy'^ 4- ——-dy^dx 4- -—dy^ dx^ ^ dxHy '^ ^ dx;^dy "^ ' dxdy'^ "^ ^ dy^dx ^ ^ dy^ ^ ^^^ 7 , . O <^^W 7 » 7 , r, ^^^* 7 „ 7 , <^^W 7 , = -r;-dx^ 4- 3- — -dx^dy 4- 3- — -dy^dx 4- —-dy\ dx^ ^ dxHy -^ ^ dyHx -^ ^ dy^ ^ . d^u , „, d-u _ , _ dhi , , „ ^ since -; — --dx-dy = ^— ; — — dxdydx = --^ — dydx% etc. dx^dy dxdydx ^ dydx^ ^ In like manner we may proceed to differentiate as often as desired. ScH. — A little observation will enable the student to write out any re- quired differential of it = /{x, y) by analogy from the above. He only- needs to notice that every distinct form of the partial differential of the required order is involved, and making x the leading letter insert the coef- ficients as in the binomial formula. Thus d^u = — 'dx''> + 5 dx^dy -j- dx^ dx^dy ^^ d'u , , , „ , ^„ d^u , „ , „ , _ d'm ^ , , dm , . 10- — —dx^dy'^ + 10 dx'^dy^ + 5 dxdy* 4 dy: dbfldy^ ^ dx'^-dy^ ^ dxdy^ ^ dy'^ ^ 111, ^TOh, — To form the successive differential coefficients of an implicit f miction of a single variable. Solution. — ^Let u =f{x, y) = 0, in which y is an implicit function. We are to . d% d^y ^ form -7^, — ^, etc. dx2' dic3' du dii dx- ' First we have -^ = by {102). (1). dy The form for differentiating this is /du\du /du\du d^y \dx/7ly \dy)dx . , . „ , , • ^ ^. •T-, = — , „ • — dx, since the second member is a fraction. \dy) To perform the operations thus indicated we have to remember that -r- and tt- dx dy are functions of x and y. Hence d( — ) = --dx 4- -r—-dy* \dx/ (Zx2 ^ dxdv^ SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES. 59' and d( ~~) = - — —dx 4- -r-dy. Dividing these values by dx and substituting:, \dy/ dydx dy- ° du/d-u d"u dy\ du/ d-u d'u dy\ d^j diKdx'^ dx dy dx / dxXdydx dy^dxj we have -^ = — ^ 7 . > ■ \dy) dii Finally, substituting in this the value of — as given in (1) we have du f d-u d'^u dx \ du I d^u I dy \ dx^ dx dy du | dx | dy dx dx^ /duy \dy) /du\^/d"u\ d^u du du d"u du du d^u/du\^ \dy/\dx-/ dxdy dx dy dydx dx 7ly dyAclx) \d4/) d'U/duY n ^"^ ^^ ^^ _t d^u/duY dxAdyJ dxdy dx dy dyAdx/ (- \d duy .dyJ In like manner the higher coefficients may be produced, but the forms are too complicated for elementary purposes. Ex. 1. Form the first and second differential coefficients of y as a function of x, when y^ — 2axy -f x^ — 62 = 0. du dy dx — 2aw 4-2^7 ay — x ^ Solution. -- = — •— = — —- = . For convemence of dx du 2?/ — 2ax y — ax dy notation put -- = = p, whence p is a function of x and y. Hence | - - r= dx y — ax f ^ ^ LdxJ dp* dp* dy dx dy dx' — (y — ax) 4- aiciy — ^) , dp* /ay — ^\ . ^ ^''iy — ^'^^ — («?/ — ^) I (99). But \f\ = % f = dri^^^) - dx X LdxJ dX' dx \y — ax/- and f = d./'^^IH^) ^ dy = 2< dw \y — ax/ Y2 (y — ax)^ dy \y — ax/ ' {y — ax) Reducing, -- = -, and -~ = '-. Substituting these values and dx {y — axy-i dy {y — ax)'-^ dv also the value of -- as at first found, we have, dx d^y _ (a^ — Vy (a^ — l)a; ay — x ( a^ — l)(y ^ — 2axy -f- x'^) <^^ {y — ^^y^ (2/ — ^^)^ y — «* iy — «^)^ Ex. 2. Form the first and second differential coefficients of ?/ as a function of x, when y^ -\- x^ — r^ = 0. dy X d-^y r^ \' dx y' dx^ T/3* * Beiuembar that these are partial differeutial coafficidnts. 60 THE DIFFERENTIAL CALCULUS. SuQ. — Be particular to use the method now being illustrated. Ex. 3. Given ^3 _}_ Saxy -f- i/3 = 0, to form -j-, and —^, by the method for differentiating implicit functions. _ , di/ x'^ -\- ay d y ^a?xy Results, -^ = dx V' -\' ^•^' ^'^■^ (Z/' + CLXf Ex. 4. Given y^ — %jcy -f o^- = 0, to form the first and second differ- ential coefficients of ?/ as a function of x, by substituting in (1) and ''2) of the preceding demonstration. ^ , du ^ du ^ „ d% d^u d^u dy d^u/du\^ ^dH( dii du d^u/du\~ 2w t^ d'v dxAdyJ "dxdy dx dy di/Adx/ ,•/ — ^ g^jj^ _^ __ — : . (22/ — 2 a;)2 — 2(— 2)(— 22/)(2y — 2x^ + 2(— 2.y)2 ^ _ —mj:y — x)-\-8y'^ ^ (2?/ — 2iCj3 (2?/ ~ 2a;/ y(y — 2a;) (y — x)3 * . dy ^ d'^y ... Ex. 5. Given cos {x + y) = 0, to form -^, and — by substituting as above. du . , , du • / , N <^'^^ , I N d^u — cos (a; + ?/), -^ = — cos (.r + 2/). Substituting, -- = — 1, and ~ = — cos (■'g + y) sin-^(.r -f y^ 4- 2 cos (a; -{- ;?/) sin^. x + y^ — cos(a; -|- ?/ ■ sin^ ;r + y^ _ ' — tsm^^^x 4-2/; These results are as might have been anticipated, since for cos ^.v -^ y) = 0, X + 2/ = 90° ; hence as one arc (x) increases, the other {y) decreases at the same rate. Therefore -/ = — 1, and, consequently, t^ = 0. dx a*'' Ex. 6. Solve Ex's 1 — 3 inclusive by substituting in the general/orm- ulce (1) and (2). DERITED EQUATIONS. 112. From w = =f{x, y), we have du ^ == _ — (1) dx du dy CHANGE OF THE INDEPENDENT VARIABLE. 61 d'^u dy\ du/ d"u d^u dy\ dxdii dx) dxSdiidx dy- dx^ du/d^u d^u dy\ du/ d-u d^u dy ^ d'^y dy\dx^ dxdy dx) dxKdiidx dy ^^ and— - = ' T , (2), dx-^ /du^ ^ " \d^) -r, ^x -. du dy du ^ /-, s , • ■, . ,, -, rw^^ From (1), we nave j~t^+3~^=0- (J-O' which is called TJie First Derived JEqiiation, or The Differential Equa- tion of the First Order, From (2) we obtain , du/d^u d^u dy\ du/ d^u d^u dy\ du d^y dy\dx^ dxdy dx/ dxxdydx dy^ dx/ dy dx^ du dy du d'^u d^u dy dx/ d'^u d^u dy\ dx^ dxdy dx du\dydx dy^ dx/ Ty dHi d-u dy dy/ d^u d^u dy\ dx^ dxdy dx dxXdydx dy^ dx/ dHL d'^u dy d/Hi/dyy dx^ dxdy dx dy^\dx/ ' Whence, transposing, we have, du d^y d^u dy dHi./dy\" d'^u dy dx^ dxdy dx dy'^dx) ' dx^ ' ^ ^^' which is called The Second Derived Fquation or The Differential Fquation of the Second Order, In a similar manner the Third Derived Equation is found to be du d^y f dm d^u dy ] d^y dH(./dy\^ d^u /dy\^ dy dx^ I dxdy dy^ dx ) dx^ dyAdx/ dxdy^dx/ ^ d^u dy dHi ax"ay dx dx'^ SECTION YIL Change of the Independent Variable. 113, In considering functions of a single variable, as ?/ = f{x), the hypothesis which we usually make that x is equicrescent, and hence that dx is constant, gives to all the differentials and differential coefficients of the function after the first, a different form from what they would have had if such hypothesis had not been made. Thus &2 THE DIFFERENTIAL CALCULUS. ^/dy\ d^ , . . ,■ d^ydx — d'^xdy d{-r-\-= —-, when x is equicrescent, but —— '-, when neither •variable is regarded as equicrescent (i. e. when dy and dx are both treated as variable). In the course of a discussion it sometimes be- comes important to change the conception and regard y as the equi- crescent, or independent variable, and x as the function. Or it may be desirable to introduce a new variable of which ^ is a function, and make it the equicrescent variable. Either of these changes can be readily effected by first giving to the expression under consideration the form which it would have had if neither variable had been treated as equicrescent. Then, to make y equicrescent, remember that all its differentials above the first are 0, and drop out the terms affected by them. To introduce a new inde- pendent equicrescent variable, as 6, of which j: is a function, simply substitute in the general form in which neither x nor y is equicres- cent, the values of x, dx, d-x, etc., in terms of the new equicrescent variable 0, dy d'Y d^y 114z. JProh, — To find the forms which ^, — -, —-, etc., take when -' ^ dx dx^ dx^ neither variable is considered equici^escent. Dem. — Since the hypothesis of au equicrescent variable has not modified the form of --, in it a; or v may be considered equicrescent, or neither, at pleasure. dx " Again — = — —— . Now differentiating the latter without regarding dx as dx' dx / dy\ d^y dx — d'^x dy EF. — Mciclaurin's Formula is a formula for devel- oping a function of a single variable in terms of the ascending powers of that variable and finite coefficients which depend upon the form of the function and upon its constants. 122, JProh, — To produce Maclaurin's Formula. Solution. — Let y = f{x) be the function to be developed. It is proposed to discover the law of the development, when the function can be developed in the form y =zf{x) = A -{- Bx -\- ac2 4- Dx^ -|_ ^x^ -f , etc., in which A, B, C, D, etc. , are independent of x and depend upon the form of the function, and its constants. Producing the successive differential coefScients, remembering that A, B, C, Z), etc., are constant, we have, ^^ = ^ -f 2ac + 3Z>.r2 -f 4.Ex^ -f , etc., dx I ' ' ;^ = 2C -f 2 . SDcc + 3 . 4^2 +, etc., ^ = 2 . 3Z> 4- 2 - 3 • 4^x -f , etc., ^ = 2 . 3 .4J5:4-, etc. Now as the coefficients A, B, C, B, etc., are independent of x, they are the same 68 APPLICATIONS OF THE DITFEBENTIAI. CALCULUS. for all values of it, and if we can find what they should be for any one value of x we shall have their values in all cases. Now, if a; = we have (?/) =/(0) = A, the expressions (?/), and /(O) signifying the value of the function when cc = 0. -- (I) = ^' (S) = ''■ (S) = - -• (i) = - 3 . .i., t.e ( ) si,n. fying in each case the value of the particular function when x = 0. Hence we find A = (y), B = (^\ C= C?^),-^, D = (^) ^. E=('^) -^-, etc. ^^^ \dxr \dxVl-2 Xdx^J 1.2.3' WW 1.2.3.4 Substituting these values, we have +, etc., • 2.3.4: which is the formula required. 123, ScH. 1. — The student should become perfectly familiar -with this important formula, and for this purpose it will be well to describe it thus : Maclaurin's Formula develops y = f[x) into a series of terms, the first of which is tlie value of the function when x = Q \ the second is the first dif- ferential coefficient of the function, x being made 0, into x ; the third, the x^ second differential coefficient, x being made 0, into '— , etc. A 124:, ScH. 2. — This formula may also be written y = f[x) = /(O) + /i(0)» AiO), etc., signify the same as {y), \-j), (^)' ®*^-' respectively. Ex. 1. To develop y = {a -{- x)\ by Maclaurin's Formula. dv d^xi SoiiTTTioN. — Differentiating successively, we have -^ = 7(a -\- xY, -^ = dx dx^ 6.7(«4-cc)o, ^ = 5.6.7(a+a:)4, ^ = 4t'5'6'7{a-\-x)^ ^ = 3-4.5. 6-7(a+a;)2, ^ = 2-3.4.5.6.7(a + x), — = 1-2. 3-4. 5-6-7. Here the differentiation termi- nates. Making x = 0, we have, (y) = {a -\- 0)7 = a^, (j-J = l(a -\- 0)c = 7aG, (^) = C.7a^(*^) = 5.0.7a., f^^) = 4.5.6.7«^ ^ = 3.4.5.6.7a^ ^^ = \dx2/ VdcV \di;4/ dx* da;^ 2-3.4.5.6. 7a, and TI^) = 1.2-3.4.5.6-7. Substituting in the formula, we obtain /y-i rjfi /p4 J/ = («r -f .'r)7 = «7 ^ 7«6a; -|- 6 - 7a"- + 5 . 6 . Ta^^ -f 4.5.6.7a3^-^ + qiyi q^Q ^qI or, reducing, y = {a -\- x^ = a'' -]~ la^x + 21^^x2 -f 35a'»ic3 -)- 35«^.'C^ + 21«9a;^ -f- 7ax^ 4" ^'^j 3, result identical with that given by actual multiplication, or by the Binomial Formula. DEVELOPMENT OF FUNCTIONS. 69 Ex. 2. To deduce the Binomial Formula from Maclaurin's Formula. . Solution. — Let t/= {a-\-x)'"\ in which m is either integral or fractional, positive or negative. Then differentiating successively, and taking the values for x = 0, we have (2,) = a", (|) = ma-'-K (g) = m(m - 1)«"-. (g) = m(m — l)(m — 2)a"»-3, (j^J = w(m — l)(m — 2)(m — 3)a"— * +, etc. Sub- stituting in Maclaurin's Formula, we obtain y = (a -{- x)'" = a"' -\- mct^—^x -f- m{m — l)a^-' — -\-m{m — l)(m — 2)a"»-^-— + "K'^ — l)(m — 2)(?n — 3)a™—' 2 ' ' '' ' 2-3 ' ' '" '" • li.3.4 -f-, etc., or we may write y = [a -\- x)™ = a'" + w^"*^^ -f T~^~' — ^'^ '^' + m(m — l)(m — 2) „ , m(m — l)(m — 2)(m — 3) , , , ■,.,.,■, _^ ^^ iam-zx^ ^ !^ '\ ^ -^""-^x* +, etc., which is the Binomial Formula. Ex. 3. Develop y = sin x. S.C.. (,) = 0, (I) = 1, (g) = 0, g = - 1. eu. .-. . = sin. = x^ , x^ x' , , X + , etc. 1.2.3^1.2.3.4.5 1.2.3.4.5.6.7^* Ex. 4. Develop y = cos x. X^ X* x^ Result, y = cosx = l-—-^:^-^-^^- ^^^^^^ + ; etc. 1-2 -3 4-5-6 7-8 125, ScH. — These formulce enable us to compute the natural sine and cosine of any arc directly. Thus, to obtain the natural sine of 10°, we have It .':p = yr; = .174533 nearly. This value substituted in the formulcF., will give the sin 10° = .17365, and cos 10° = .98481. The series converge so rapidly that but few terms are necessary. 1 Ex. 5. Develop y = {a^ -\- hx)^ hj Maclaurin's Formula. Sug's. y = (a-^ -\- bx)^, .-. (y) = a, dy dx . S, = (a= + 6x) =„ + ---_+_ _, etc. * This notation signifies " x being made equal to Q. 70 APPLICATIONS OF THE DIFFEEENTIAL CALCULUS. Ex. 6. Develop y == \/l -\- a^. ^ -, /:; - ^ 1 ^ x^ X* x^ bx^ Result, y=Vl + X"' = (1 + vr=)2 =l+__-+__— _+, etc. Ex. 7. To produce the logaritlimic series. Solution. — This series is the development of y = log (1 -)- x). Differentiating dv with reference to a system of logarithms whose modulus is m, we have, -r = dx m d^y m d^y 2m d^y 2 - 3m .^ iq:^'d^2-~(TT^'d^^'"a+^' d^4 = -(rqr^'^*''- Whence(t/) = ^-1 = °; (l) = - (S) = -- (S) = -' (5^) = ----- -• «^''- stituting in Maclaurin's Formula, we have y = log (1 -|- a;) = 7n(ic — gic^ + ix^ — ix* +, etc.), the law of the series being ajDparent. 120. Cor. 1. — Since in the Napierian system m = 1, we have y = log(l + x) = X — 1x2 + ^x3 — ix^ + ix5 — etc. 127* ScH. 1. — This formula is not adapted to the purpose of computing logarithms, since it is diverging for integral values of x. Thus, letting ic = 2, we have y = log3 = 2 — 2 + f — 4 + ^e^. — ^ etc., in which each term after the first two is greater than the preceding, and hence ex- tending the series does not approximate the value of log 3. From the series in the corollary, however, a converging series may be readily deduced. The following is a simple method : Substituting — x for x we have log (1 — x) = — X — \x^ — ^x"^ — ^x^ — \x^ — , etc. Subtracting this from the former, we obtain log (1 + ^) — log {l—x)= log ] \^-^ \ =2{x-^ i.^3 _}, la;-- + W +, etc.) (1 — X ) Now putting X = , whence — ^^- = -^~, we have log = ^ ^ 2z -i-1 1 — x z z log iz + 1) — log^ = 2^ — ^ \ 1 h . ^ \ Jt h, etc. ), ^^ ^ ^ ^ V22 + 1^3(22 + l)3^5(22+l)=^7(2^+lj7^' /' or log (2 + 1) = log2 + "li— 1 ^ \ ^ I ^ 1 — {-, etc. \ ^ ^ \22 + l ^3(22 + 1)3^5(22 + 1)^^ 7(2^ + 1)7^9(22 + 1)9^' / This series converges for all positive values of z, and more rapidly as 2 in- creases. To apply this formula in computing a table of Napierian logarithms, first let 2 = 1, whence log 2 = + 0/1,1,1,1,1, 1 , 1 1 ,^\ V3 ^ 3 • 33 ^ 5 . 3' ^ 7 -3' ^ 9 • 3^ ^ 11 -B-' ^ 13 . S'-'- 15 • 3'^ ^ / DEVELOPMENT OF FUNCTIONS. 71 The mimerical operations are conveniently performed as follows : 3 2.00000000 9 .66666667 9 .07407407 9 .00823045 9 .00091449 9 .00010161 9 .00001129 9 .00000125 .00000014 l0£ 1 3 5 7 9 11 13 15 2 = .66666667 .02469136 .00164609 .00013064 .00001129 .00000103 .00000009 .00000001 .69314718 Second. To find log 3, make z = 2, whence log 3 (- + - \5 ^ 3 . Compuiaiion. ^5 5 25 25 25 25 + 3-53 ' 5 2.00000000 + 7-5 log 2 + . _f -L 4- etc. ) .40000000 .01600000 .00064000 .00002560 .00000102 .40000000 .00533333 .00012800 .00000366 ■00000011 .40546510 .69314718 Third. Fourth. il 9'^3 To find log 4 find 1 To + 93 ' 5 • 9-i ' 7-9 Computation. 9 log 2 .■•. log 3 = 1.09861228 log4 = 21og2 = 2 X log 5. Let ^ = 4, 2.00000000 69314718 = 1.38629436. whence log 5 = log 4 -|- 81 81 81 .22222222 .00274348 .00003387 .00000042 log 4 .22222222 .00091449 .00000677 ■ 00000006 "722314354 1.38629436 .-. log 5 = 1.60943790 In like manner we may proceed to compute the logarithms of the prime numbers from the formula, and obtain those of the composite numbers, on the principle that the logarithm of the product equals the sum of the log- arithms of the factors. The Napierian logarithm of the base of the common system, 10, = log 5 + log 2 = 2.30258508. 12 S» Cor. 2. — The logarithms of the same number in different sys- tems are to each other as the moduli of those systems ; and the logarithm of a number in any system equals the Napierian logarithm of the same number multiplied by the modidus of the proposed system. 120a ScH. 2. — To find the modulus of the com,mon system of logaritlims. 72 APPLICATIONS OF THE DIFFERENTIAL CALCULUS. we have com. loff. x = m Nap. loe:. x, whence m = — — '-—^ — . Now hav- ing computed the Napierian logarithm of 10, by the formula above, and found it to be 2.302585, we have m = — — \ \,, = ,, ^^^ ^ - = Nap. log. 10 2.302585 .43429448+. ISO, ScH. 3. — To compute a table of common logarithms, first compute the Napierian logarithms and then multiply by the modulus of the common system, .43429448. Ex. 8. To ascertain the relation of the modulus of a system of log- arithms to its base. SoiiUTioN. — Developing y = a^, by Maclaurin's Formula, we have 1 1 05- 1 a;^ 1 rc^ !' = "' = 1 + m^ + »-^ 2- + m> a— + m. l^aTI + ^*°- ^^'^ Again, putting a =^1 -\-h, and developing by the Binomial Formula, we obtain ' . 2-3-4-5 Expanding and collecting the coefficients of the 1st power of x we find it to be ^-2-+3--4+5-6-+'^*"- Finally, since series (1) and (2) are equal the coefficients of like powers of x are 1 62 53 54 55 ^G equal, and — =6 — tt + t; -7--+-? ;; — hj etc. ; or restoring a and findmg m 23456 the value of m, we have 1 vn — — ^ (a — 1) — i(a — l)2 + i(a — 1)3 — i(a — l)4 + i(a — 1)^ — -L(a— l)6-f, etc. 131, ScH. — To find e, the base of the Napierian system. Since the log- arithms of the same number in different systems are to each other as the moduli of those systems, we have com. loge : Nap. log e(= 1) :: .43429448 : 1. .•. com. loge= .43429448, and e from the table of common logarithms, which we have shown how to compute, is 2.718281+. Ex. 9. To develop y = a"", i. e. to produce the exponential series. X X' x^ Result, 1/ = a- = 1 + log a J + log^ aj-^ + log^ a ^ ^ ^ + x^ X 132. ScH. — ^If a = e, the Napierian base, this becomes q/ = e' — l-{-- + x-^ , a;3 , X* , x^ , . \~, etc. 12^1..2-3^1-2-3.4^1-2.3.4-5^ If a: = 1, we have DEVELOPMENT OF FUNCTIONS.- 73 y==^_,_2 + - + -- + ^-g-- + 2.3.4.5 +' ^t«- ' a ^^^^^ for finding the Napierian base, although the series converges slowly. Ex. 10. Develop y = tan ^x. SoiiUnoN. — Differentiating -- z= - — ; , which by division becomes ~ =z\ — dx 1 + a;2 •' dx a;2 + jc* — icG -f a;« — x'" +, etc. Differentiating successively, -r-^ = — 2a; + 4a;'' — Gx^ + Sx^ — lOx^ 4-, etc. dx'-^ -!| = — 2 + 3.4a;2 — 5.6a;^ + 7-8a;6 — 9-10x8+, etc. -^ = 2.3-4rc — 4-5-6x3 4- 6-7.8x5 — B-Q-lOx^-f , etc. ^ = 2-3.4 — 3-4.5.6x2 + 5.6.7-8x''—7-8.9-10xe-h, etc. ^. = — 2-3.4.5-6X+4 5-6.7-8x3 — 6.7-8-9.10x5 -fete, ax" ' menee („ = tan-.<, = 0, (|) = X, (g) = 0, (^^) = - 2, (g) . 0, [y4) = 2-3-4, f-— jzizO, etc. Introducing these values into Maclaurin's Formula, we have 2/ = tan— 'a; =. x — \x'^ -\- \x^ — ^x"^ -f" 9^^ — A^" +> etc. ScH. — By means of this development we are enabled to find the value It Tt of It. Thus let y = 45° = j, whence x = 1, and we have y = — = * 11 1 1|1 1,1 1,1 1,1 1,. tan-l=l-- + -_- + --- + ----+--_+, etc. 133, Pvop, — Though Maclaurin's Formula is applicable to a very great variety of forms of functions of a single variable^ it will not develop ALL such functions. The truth of this theorem will be substantiated if we can present examples of functions of a single variable which the formula will not develop properly. This we proceed to do. 1 . Ex. 1. Show that y = x^ i^ not properly developed by Maclaurin*s Formula. Solution. — From y = x , we have -r- = — ;, -rp^ = -, etc. Hence (v) = 0, ax „ 5 ax2 I \s/ J f -^ j = - = 00, ( — j = — oc, etc . Substituting these values in the formula have y == X = -4- cox ^ — co'----f-, etc. Such results as these will be simply we 74 APPLICATIONS OP THE DIPPERENTIAL CALCULUS. unintelligible to the learner at first. But in this case it is easy to see that the de- yeloijment will consist of pairs of terms of the same general form as (x>x — oo — , 2t To ascertain just what is to be understood by this binomial, let us restore the values of oo as they were before x was made equal to 0, only using x to indicate the X that is 0. We then have '- '—, or ^-1-. Now as x = 0, 2ic'^ 4.x' '^ 8a;' ^ this becomes — ^ , which is oo for all values of x except 0, and indeterminate for that. In like manner, it may be shown that each succeeding pair of terms equals 00 . Hence we have the absurd result that ?/ = x^ = oo, for aU values of x, since the development should be true for all values of the variable. Ex. 2. Show that y = log x is not properly developed by Maclau- rin's Formula. Sug's. — The result is similar to the preceding except that the first term is oo in this case. Each succeeding binomial may be seen to be oo, as in the former case. Hence we have the absurd result that y = log a; = oo for all values of x. 1 Ex. 3. Show that y = cot x, and y=^ a" are not properly developed bv Maclaurin's Formula. 134, ScH. — The occasion of the inapplicability of Maclaurin's Formula, in such cases as just given, is the fact that the form of the function is such that the coefficients -f-, — ^, etc., or the function itself, or both, become a.v ax' infinite for x = 0, which is contrary to the hypothesis upon which the formula was produced. Whether, in such cases, the failure to develop cor- rectly by this formula is due to the fact that the particular function is in- capable of any development, or whether it is simply because it will not develop in the particular form assumed in this theorem, does not as yet ap- pear, and our limits forbid our entering upon the question. TATLO?.'S FORMULA, 135. Def. — Taylor's Formula is a formula for developing a function of the sum of two variables in terms of the ascending powers of one of the variables, and finite coefficients which depend upon the other variable, the form of the function, and its constants. 136. Lemina* — Jff^ u = f (x + y) the partial differential coefficients du _ du -- ana -z— are equal. dx dy ^ Dem.— Having u = f(x -f- y), if x take an increment, we have u -f dxu = f(x -^ dx -{- ij) =f[{x -\- y) -\- dx] ; whence d^u = f [{x -{- y) -{- dx] —,f{x -f- y). Again, if y take an increment, we have u -\-dyU =f{x-\-y-{-dy) =fl{x-'ry)-^dy]; DEVELOPMENT OF FUNCTIONS. 75 whence dyu =f{{x -}- y) -\- dy} —fix + y)' Now the form of the values of dxU. and dyU, as regards the way in which x and y are involved, is the same ; hence, if it were not for dx and dy, they would be absolutely equal. Passing to the differ- ential coefficients by dividing the first by dx and the second by dy, we have du /[(.r + y) -\-dx-]—f[x-\-y) du _ /[(x + ?y) + dy] — .Ax-f y ) = — , ana -J- — . x)Ut, dx dx dy dy in differentiating, the differential of the variable enters into every term ; hence f[{x + 2/) + dx] — fix -j- y), as it would appear in application, would have a dx in each term which would be cancelled by the dx in the denominator in the coef- ficient, and — would be independent of dx. In like manner — is independent of dy. dx (ly Hence, finally, as these values of the partial differential coefficients are simply func- tions of {x-\-y), of the same form, and not involving dx or dy, they are equal, q. e. d. ScH. — The substance of this demonstration is that the values of the dif- ferential coefficients depend upon the form of the function, and are inde- pendent of the increment of the variable. Therefore Avhen the form of the function is such as to give to the partial differential coefficients, the same form with respect to the variables, the coefficients are equal. But suppose . du f[{x-\-dx)if]—f{xy) f[xy + ydx) — f{xy ) ^ we have u =. fixy). -— = -^ ; = -^— ; dx dx _ dx and ^ =^^''^^ + "^^^ -■^^''^^ ^^^^'" + ""'^-'^ -A^y), In these coef- dy dy dy ficients we see that the form is not such as to involve x and y in the same way ; hence they are not necessarily equal. A few examples will render the truth of the lemma more clear, Ex. 1. Given u-= (^ + y)"* to show that the partial differential coefficients are equal. -,. -, du , V , -, du , . N„ , Besults, — = m{x + y) , and — = m{x + y) • Ex. 2. Given u = log {x -{- y) to shovr that the partial differential „ du 1 du 1 coefficients are equal. Results, -r- = — ■ — , and -7- = — ■ — :. •^ dx X -\- y dy X -\- y Ex. 3. Given u == tan~^(a; -\-y) to show that the partial differential coefficients are equaL ^ ^ du 1 ^ du 1 Besults, -- = r-, and 3- dx 1 + (^ + yY dy 1 + {^' + yY -j , show that the partial differential coefficients are not equal. ^ , du mx'^~^ ^ du 7??.r'"v"'~^ „ _™ , Results, ~ = — -— , and -- = "^ — = — rnx'^y-^'K ' dx 2/ dy ?/2"> ^ Ex. 5. Given u = log (xy)j show that the partial differential coef- ficients are not equal Results^ -— = - and -- = -. * ax X dy y 76 APPLICATIONS OF THE DIFFEKENTIAL CALCULUS. 137* I*VOh, — To produce Taylor's Formula. Solution. — Let u =f{x -\- y) be the function to be developed. It is proposed to discover the law of the development when the function can be developed in the form u =f{x ^y) = A + By+0!^ + Dy^^Mj* +, etc., (1), in which A, B, C, etc., are independent of y, and dependent upon x, the form of the function, and its constants. Differentiating with respect to y, remembering that as A is independent of y it will disappear, and that as the factors B, C, B, etc., are likewise independent of y, they are to be regarded constant, we have ^ = 5 + 2Q/ 4- 3i)2/2 + 4Z2/3 +, etc. (2). Again, differentiating with respect to jc, we have du dA* , dB dC' , dB ^ ^ -r = T- -\- -T-y + T-y- + T-y^ +» etc. (3). dx dx ^ dx^ ^ dx^ ^ dx^ ~' ^ ' Hence by (136) S + 2C, + ZDr- + W +, etc. = ^ + ^2, + ^-3 + ^^y, +, etc. Now, by the theory of development by indeterminate coefficients, the coefficients of like powers of y are equal, and we have B = ^A, 2a = ^, 3i> = 5^?. 4£'=^, etc. dx dx dx dx But as (1) is true for all values of y, we may make y =0, whence A =f(x) = u, du letting u represent the value of the function u when y = 0. Hence B ==-—, (X*C r —-i- —I ^^■^' ^ _ etc., which is the Binomial Formula. Ex. 2. Develop u = log {^x -\- y). Sug's. — This being a function of the sum of two variables, we apply Taylor's T, , , , du 1 dfiu 1 d?u' 2 Formula, w = logx, — — =-, - — = , = ~, etc. ax, X ax?- 's?' dx,^ 'x? Henceu = log(.r + 2/)=log^ + |-^|4-£-£+,etc. Ex. 3. Develop u = a='+^ Besult, u = a\l + log a y + -^y^2/^ + -|^2/' +> etc.). Ex. 4. Develop w = sin (^ + 2/)- „ , , . du d^u' . d^u SuG s. u = sm X, — — = cos a;, - — 7= — sm x, - — == — cos x, etc. Hence dx dx'^ dx^ V . y^ /j/3 y4 u = sm {X -\- y) = sm x -j- cos a;^ — sm x- — - — cos x '^ - -\- sin .r- — ' -f- ,in.,(l - ^-^ + 1-2^-4 - 1.2./4.5.6 +' "*"•> + "'''^^(2' - T^ + i.2.3'.4. "5 - l.2.3.4'.5-6-7 +' '''''•' = ^'"*'=<>«2' + <=os.rsiny, since the series in the parentheses are equal respectively to cosy and sin 2^ {124, Ex's 3, and 4). Ex. 5. Develop u = cos {x -\- y). yi yi yB ^esult,u=cos{x-^y)=Gosx{l—~-\- ^^^^ — ^^^^^^^ -\-,etG.) - ^^ whicli developed by Taylor's Formula gives S "" 2x2 "^ 3^ "~ i^ "^' ^*^' } "^ ^^^^^ *^® modu- lus of the system of logaritlims. ScH. — If h be considered infinitesimal with respect to x, so that we have Ji = dx, we may drop all the terms within the parenthesis except the first, and write y' = log re -j '-. This is the consecutive state of the function y = log X. Hence subtracting the latter from the former we have y' — y = dy = d log X = . This result is as it should be, in accordance with the X rule for differentiating a logarithm. Ex. 2. Given y = 3x — 2x^ — 5, to find y', which represents the value of the function after x has taken the increment h. Result, y =z 3;r — 2^3 _ 5 4_ (3 _ Qoc^)h — llx^ — 12^ = 3a; — 2^3 _ 5 _|- (3 _ 6a72)/i — 6a7/i2 — 2h\ ScH. — This result may be easily verified by direct substitution. Thus, y' = 3{,r + h) — 2(.r 4- hy — 5. Expanding, y' = Sx + 3h — 2x^ — Gx^h — Qxhi _ 2^3 _ 5 =, 3^ _ 2a73 — 5 + (3 — 6^2) 7j — 6x71^ - 2h\ 140, Prop. — Though Taylor's Formula gives the general form of the development of a function of the sum of two variables, there are sometime.^ particular values of one or the other of the variables for which the development is not true. We will illustrate this proposition with a few examples. JL Ex. 1. Develop u-=i {x -\- y — a)^ by Taylor's Formula, and show that the development is false when x =^ a. , , d du ,^ -h 1 d^u -f Solution, u = (a; — a) , ——== i{x — a) = j, ——^= — ^{x — a) dx ^ a dx^ 2{x — a) : , etc. Hence substituting in Taylor's Formula, f dx^ „,„ ^J 4(0: — a)^ ^{x—ay DEVELOPMENT OF FUNCTIONS. 79 i i y y^ ^v^ we have w = (a;4-2/ — «) = (* — <*) -\ 1 ^H ^ — .etc. 2{x — af 8(x — a)''^ m.x — af Now, no absurdity appears in this series for general values of a; , but for x = a the series becomes oc, while (x -^ y — a) = 2/ ? ^or the same value. But by hy- pothesis X and y are independent and the development should be true for any value of y irrespective of the value assigned to x. Hence the conclusion that for X = a, y^ = oc is contradictory to the hypothesis, and false. ScH. — It is evident that any form of function which, when developed by this formula, gives a factor of the form [x =F «)"* in the denominator of any term in the development, will afford an instance similar to the above, and the development will not be true for x = ± a, since for this value [x =+:«)"'=:= 0, and the terms in the denominators of which it occurs will reduce to oo. 7. Ex. 2. For what value of x is the development of w= (^x-\-y-{-b)''^ by Taylor's Formula, untrue ? Ans., x = — h. Ex. 3. Required the value of the function after x has taken an in- 3. crement h, when y = h -\- (a; + c)^ + (-^ — ^)^' For what value of x does the development fail ? Result, y' = h -^ (x ^ cy ■}- {x — a)^ + [2(^ + c) + ^{x — aY]h -f [2 + |(^ _ «)-i]|' _ !(:. -_ a)-t^ +, etc. y'= cc when x = a, and hence the development fails for this value. ScH. 1. — If h = dx the above development is true for all values of x, for 3 1 then we have y' = b -\- {x -\- c)2 -f (.r — a^ -{- [2{x + c) + f (.t — a) ]h, which is the same as would be obtained by substituting x -j- h for x in the first state of the function and developing, and then making h = dx, and dropping the higher powers of h. For x ^ a this becomes y'= Z> + (a -f c)^ -f- 2 (a -f c)h, which is as it should be, since for x -{- h = a -{- h, y' = h -\r {a -\- h-]- cY + [a -\- h — af = 5 + «2 + 2a^ + 2ac + 7^2 + Ihc + c^ + ]{' = (dropping higher powers of h) b -\- a^ -\- lah + 2«c + 2hc -f- c^ = 5 -j- (a2 + 2ac + c2) + {2ah + 2c^) = 6 + (a + c)2 + 2(a + c)h. ScH. 2. — ^It will be observed that when Maclaurin's Formula fails to give the true development of a function it fails for- all values of the variable ; but when Taylor's fails it is only for particular values, the general develop- ment being still true. GENEKAii Scholium. — There are many other important /brmM^cc for the development of functions, but the prescribed limits of this volume pre- clude their presentation. 80 APPLICATIONS OF THE DIFFERENTIAL CALCULUS. SECTION 11. Evaluation of Indeterminate Expressions, 14:X, The following forms are called The Indeterminate Forms, viz., 00 -, — X 00, 00 — 00, 0°, oo«, 1". U 00 Whenever an expression assumes any one of these forms, the impor- tant question to be determined is whether it is ideally indeterminate, for it often happens that the indetermination is only apparent. Of these forms, - is the fundamental one, to which all the others can be reduced. IiiL. — That - is an indeterminate form, is readily seen when we observe that the divisor, 0, multipUed by any finite number, produces the dividend, 0. We may show that each of the other forms can be reduced to the first, and hence that they are indeterminate forms. Thus, let a represent a finite quantity ; then a a 00^5 a O^^^oo. .^^ -xr iu = — . But - = - X - = zz- That — is an indeterminate form may also be a CO a a ao seen directly ; since one infinity may be any number of times another, and the symbols oo do not mean that numerator and denominator are the same infinity. Again OX '^ = - X t: = 7{, « being any finite quantity. Also oo — oo is inde- a terminate, since the difierence between two infinities may be any quantity what- ever. Taking 0" and passing to logarithms, we have log = 0( — oc) = — X oo, which has been shown equal to -. Finally, applying logarithms to ooo, and 1 ", the former becomes Ologoo z= X cc, and the latter oologl = oo x 0. ^ 142, The apparent indetermination often occurs from the intro- duction of some hypothesis which introduces a factor 0, into both terms of the fraction. ^3 jp3 III. — What is the value of when a; = a ? Making x = a reduces the a — X (j3 X^ . expression to - ; whence it would appear that is indeterminate for x = a. u a — X ^3 a;3 But that such is not the case is evident, since = a^ -{- ax -4- x'^ which = 3o2 a — X when x = a. This apparent indetermination arises from the fact that the hypoth- esis x = a introduces a factor into numerator and denominator. This factor being divided out, the true value is seen. But it is not always easy to discover i EVALUATION OF INDETEKMINATE EXPRESSIONS. 81 the factor which becomes 0, so as to be able to cancel it ; hence the necessity of some general method of procedure. f(x) 14:3. I^vob, — To evaluate y = , \ for x == a, when for this value of the variable the /unction assumes the form —. Solution. — Let y' be the function when x has taken an increment h, so that f(x -\- h) y' = — ■ — —, Developing /(aj -f- /*,) and wix 4- h) by Taylor's Formula, and- for cp{x -\- h) simplicity using f {x), f'{x), (p\x), (p"{<^), etc., for the coefficients, we have fix) + f{x)^ + /"(^)r^'-77 +, etc. I ex -^- ft I y fix + /n _' '•^^ -r J v-^^i -r J v-^^i . 2 ^ ^' ^ •'- "- , logx logy.=o=-Y- X sin'x 2 sin X cos x 2 sin X cos x - COSiC iccosa; cosx — xsina; cosx sin X sin2 X ? 2 sin X = 0. . • . y = 1. EVALUATION OF INDETERMINATE EXPRESSIONS. 87 , . . . 1 . , . loar sin x , . ^ ... Also y = (smir)*'"== gives logy = sm a; log sm ^ = —- (differentiating) COS6C *Cx= cot a; 1 . = 0. . • . 2/ = 1. cosec X cot X cosec x Ex. 3. Evaluate y = (cot^)^*'^* for x = 0. SuG. — Put this in tlie form — . Thus logv = sin x log cot a; = — ■ (dif- 00 ^ cosec iCa:=0 cosec2 a; - . . cot X cosec X sm cc . lerentiatme) = = = = - = 0. — cosec X cot X cot^ x cos^ x 1 Ex. 4. Evaluate y = (1 + nxy for x = 0, „ . log(l 4- nx) ^.^ 1. X- 1 5^ SuG. log ?/ = ^^ ■ = -. Differentiating, log 2/2=0 = - = n. X x = ^ •*• , • . y=e\ Ex. 5. Evaluate y = {Qosiax)}"""'"^'^^ for ^ = 0. SuG. y = {cos(ax)}<'°^«<=^^'=^) = 1", when x = 0. Passing to logarithms log cos («x) s (ax) = — ^ ^^ = -, when x = 0. Differentiat- sin2 (cxj — a tan (ax) a tan (ax) — a- sec2(ax) a- mg twice ogT/^^o 2c sin (ex) cos (cic) csin(2ca;) 2c2cos(2ca;) 2e^ « * 88 APPLICATIONS OF THE DIFFERENTIAL CALCULUS. SUCTION III. Maxima and Minima of Functions of One Variable, 14:8, Def. — A M^aociniutn value of a function of a single vari- able is a value which is greater than the immediately preceding and the immediately succeeding values ; i. e., the value when the variable takes an infinitesimal decrement, and the value when the variable takes an infinitesimal increment. Ill's. — Let y = sin x. When x = —, y \b b, maximum, since it is greater than the immediately preceding and the immediately succeeding values. If x takes an increment h, making y' = sinf^ f- ^ )> or a decrement, — h, so that y" = sinf — — ^^\y is evidently greater than y' and y" , as at 90° the sine is greater than it is at a little more or a little less than 90°. Again, constructing the equation y'^ = 603^ — x'^, we find the right hand branch to be as given in the figure. Here y =:f{x), and y is & maximum when a; = A D = 4, since for x infinitesimally less or greater than 4, y is less than for x = 4. The maximum value of y is, therefore, y = -^6-4^ — ^4^ = 3^ nearly. ^xq 19 Once more, let y = 8x — a;^. If x = 1, y = 7 ', if x = 2, 2/ = 12 ; if a; = 3, 2/ = 15 ; if x = 4, 2/ = 16 (a maximum) ; if cc = 5, ?/ = 15 ; if a; = 6, 2/ = 12 ; and if a; = 7, y = 7. Hence it appears that as x increases y in- creases till it has attained a certain value, when although x is made to continue its increase, y begins to diminish. The point at which the function ceases to in- crease and begins to decrease is its maximum. In this case it will be found that however little x varies from 4, either way, y becomes less than 16. Thus if x ^= 3.9, y = 15.99 ; and if a; = 4.1, y = 15.99. 149, I^EF. — A. ]\Hnil7lU7¥l value of a function of a single vari- able is a value which is less than the immediately preceding and the immediately succeeding values ; i. e. , the value when the variable takes an infinitesimal decrement, and the value when the variable takes an infinitesimal increment. tc Ill's. — Let 2/ = coseca;. As x approaches — y diminishes and approaches 1, reaching 1 at a; = - . "When x passes — , y begins to increase, so that 2/ = 1> is a it A minimum value of the function y = cosec x. Again, y = x" — Q>x -\- 10, has a minimum value for a; = 3, at which value MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE. 89 y = 1. By substituting values of « a little greater than 3, as 3.01, and a little less as 2.09, y will be found to be greater than 1 in both cases. The locus of the function is given in Mg. 20, where P D represents the minimum value of y. ISO, CoE. — The same function may have several maxima or several minima values, and these may he equal or unequal. Moreover, a maximum value may be equal to or even less than a minimum value of the same function. Ill's.— The function y := x^ — 8x3 _^ 22x2 — 24a; -}- 12, has minima values for x = 1, and x = 3, which values are both ?/ = 3 ; or two equal minima values, as illustrated by the ordi- nates at P and P" in the figure. For x =^2 y = i,, a maximum value, as illustrated by P'D'. Again, let y = /(.t?) be the equation of M N referred to AX and AY Fig. 22. Then PD, P"D", and P"'D'"^are maxima values of y ; and P'D', and P"'D"' are minima values. But the several maxima values are unequal and the minimum P' D' is greater than the maximum P^vQiv. Fig. 20. ADD' D" Fig. 21. 151, ScH. — It will be observed that the terms maximum and mini- mum, as here used, do not mean the greatest possible and least possible. Thus, if we ask for the maximum value of 3/ in 3/ = «;•* — Zax'^ — 5, we do not inquire, "what is the greatest possible value which y can have ? but simply, whether if x vary con- tinuously through all possible values, there is any point at which y will at- tain a greater value than it had immediately preceding that point, and than, it will have immediately after passing that point ; and, if there be such a value of y, what it is. Fig. 22. 1S2, JPfop. — In an explicit function of a single variable, y = f(x), dy the first diffey^ential coefficient, —jChanges sign from, -f to — ,for contin- uously increasing values of the variable, where the function is at a maxi- mum, and from — to -{- where the function is at a minimum. Hence for such values the first differential coefficient == or oo. Dem. — Let y =f^x) be the function. First, For x = x', suppose y becomes y' , a maximum. Then ?/' = f'^x') is at a maximum. Now the immediately preceding 90 APPLICATIONS OF THE DIFFERENTIAL CALCULUS. dx') —fix') flu* "fi % state of tlie function is fix' — dx"), and we have -4-, = — — ; tt- ''^ " dx {x'—dx) By hypothesis /(a' —dx) — f{x') is — *, and as (x' — dx') — x' is evidently — , we have dv' — -j— Again, the immediately succeeding state to y' =f{x') isf{x'-\-dx'); hence we have -r—, = dx dy' f{x'-\-dx')—f{x') \x' -j- dx) — x and as {x -\- dx) — x is evidently -\-, we have By hypothesis /(a' -\-dx') — /(.'r') is dy' dx Therefore where y' = dy' fyX ) is a maximum -— changes sign from to dx')-f{x') dx ) — x x' is evidentlv — is — , since by Again dv' fix' ■ Second. If y' = fix') is at a minimum -^^ = — — - dx (X hypothesis fix — dx') — fix') is -|-, and (cc' — dx') dy' f(x' + dx') —fix) . ^-u • ^- ' , ^ 'N ^/ 'N • I A d^' ^ ix + dx) — x' "^' "'^ ^ hypothesis /,a: + dx ) —fix ) is +, and {x -\- dx) — x is evidently -j-- Finally, since when a varying function changes sign it passes through or go, dy we have -^ = or oo for maxima and minima values of the function, o. e. r». dx Geometeical IiiLUSTEATioN. — T"S being tangent to the curve M N at P, P' being a consecutive point so that P E represents dx, and P'E dy, we observe that the angle P'PE = a, the angle which the line makes with the axis of abscissas. Hence tan a = tan P'PE = -=r-r=- = - - ; i. e. the first dif- PE dx Fig. 23. ferential coefficient of the ordinate regarded as a function of the abscissa, represents the tangent of the angle which a tangent to a plane curve makes with the axis of abscissas. Now, observing Fig. 22 we see that as x is increasing, and y approaching a maximum value as PD, the tangent to the curve makes an acute angle ; hence dy approaching P from the left -- is dx At P the tangent becomes parallel to the axis of X ; tan a d'd^ -- = 0. Immediately upon passing P, a becomes obtuse, and consequently tan a = -^is — . dx So also in approaching a minimum value as P' D' from the left it appears that a is obtuse, and hence -- ax passing P', a becomes acute and -- 4-. dx ; at this point, P', a = 0, and dy dy dx 0; and after * The hypothesis is that y' =J\x') is a maximum, i. e. is greater than either the immediately preceding or the immediately succeeding state of the function. But fix' — dx') is the immedi- ately preceding state, and/(a;' + dx') is the immediately succeeding state. Hence/(a;''— flfa;0< /(xO, and/(a3' + cfaK) /4a2ic2 — 'lax:^ = —r- ^r^r- = -, a maximum. \ y ^' 3\/3 Ex. 2. What values of x render y = x-^ — 9^=^ + 24.x — 16 a maxi- mum or a minimum, and what are the maxima values of y ? Results, X = 2 corresponds to a maximum, and x = 4: to a mini- mum. The maximum value is y = 4, and the minimum y = 0. Ex. 3. Examine y = x^ — ^x^- — 24.x + 85 for maxima and minima. Results, For ^ = 4, ?/ = 5, a minimum ; For X = — 2, y = 113, a maximum. , Ex. 4. Examine y = 5(x — ^r^) for maxima and minima. SuG. — Drop the 5. x = i, gives y ==. ^, a maximum. Ex. 5. Examine y = {2ax — x^)'^ for maxima and minima. SuG. — Use y' = 2ax — x^. x = a, gives y = a, a maximum, and — a, a min- imum. Ex. 6. Examine y = x* — Sx^ -\- 22a;2 — 24^ + 12 for maxima and minima. Sug's. ^f = 4x'^ — 24.'k2 4- 44a; — 24 =-. 0, or x^ — Gx2 + IL-r — G = 0. To ax find the roots of this equation, observe that the factors of the absolute term with its sign changed are 1, 2, and 3 (Complete School Algebra, 111). By trial these are found to be the values oi x, x = l gives ?/ = 3, a minimum ; .^• = 2 gives 2/ = 4, a maximum ; .r = 3 gives ?/ = 3, a minimum (see III. Fig. 21). Ex. 7. Examine y=x^ — 5x^ + 5x^ + 1 foi' maxima and minima. Results, The critical values of x are 0, 0, 1, 3. For x ■= 1, ij = 2, a maximum ; for x = d, y = — 26, a minimum, x = does not correspond to either a maximum or minimum value of y. SuG. — That x = does not correspond to either a maximum or a minimum is determined as follows : dy Having -- = 5x4 — 20x3 + 15x2, substitute — h and -\- h for x, and evaluatef^ dx dv the expression for k infinitesimal, thus determining whether -- changes sign or not^ cvX 94 APPLICATIONS OF THE DIFFERENTIAL CALCULUS, in passing through x = 0. Thus JJ = 5(0 — h)" — 20(0 — hy -f 15(0 — A)2 = 5;i4 + 20/13 + 157^2 = 15;i2, when h is infinitesimal. Again -- = 5/i4 — 20A3 ux- -f- 15^2 ^= 15^2, when h is infinitesimal. Therefore, as -- has like signs on both sides of a; = 0, and consecutive with it, it does not change sign in passing through a; = 0. Hence jb = does not correspond to either a maximum or a min- imum. Ex. 8. Examine y z=h -\- \x — a)^ for maxima and minima. dv Stjg's. — = 3(a; — aY = 0, gives x = a. Hence if there is any maximum or dv minimum it must be v = &, as no other value of x than x = a will render -^ = 0. dx d^V dv Again, since this value renders -r— = 0, we examine it by ascertaining whether -rr dx^ dx dv dv changes sign a.tx = a. -^ = 3(a — h — a)2 = 3h' is the value of -^ next preced- (XX ax dv ing X = a; and —■ = 3(a -f- ^ — a)^ = STi^ is the next succeeding value. There- fore, as -- does not change sign at a; = a, the function has no maximum nor mini- dx mum value. Ex. 9. Examine y = a{x — by -\- c for maxima and minima values. Sug's. -^ = 4:(x — hy = 0. .-. x = h. % = 4(& — h — hy = — 4.h^, and dx ^ ' dx dv' dv' -J— = 4(6 4" ^ — 6)3 = 47i3, are the values of — — immediately preceding and suc- (X*o CuC dy' ceedmg x = h ; hence, as — — changes sign from — to 4- ^t this point, x = 6 cor- dx responds to a minimum. .-. y = a(b — 6)^ -J- c = c is a minimum. Ex. 10. Examine y = {x — l)^{x -\-2y for maxima and minima. Sug's. -^ = A{x — iy{x + 2)3 -f- 3[x — iy{x -f 2)2 == {{x — l)3(x + 2)-'} {4(a;-f2)4-3(a; — 1)} =(« — l)3(a;+2)2(7a;-f5)=0. .-. .r— 1 = 0, a;+2 = 0, 7a; -|- 5 = 0, give x = 1, x := — 2, x = — fas the critical values of x. d^V -^=.3(a; — l)2(a;-|-2)2(7x + 5)4-2(x — l)3(x + 2)(7a; + 5)-l-7(a; — l)3(cc + 2)2 = (123) . 92 for X =^1, and x = — 2, but is — -^ — —- for x =^ — ^. The latter value, there- ' 74 ' 12 9' fore, corresponds to a maximum, and gives y =z ( — f — ly{ — ^ -|- 2)3 =: — — — , a maximum. To ascertain whether x = \ corresponds to a maximum or minimum, notice MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE. 95 that -^ = n—h~ 1)3(1 — h-^ 2)2(7 — 7h-{-5)= — h^S — /i)2(12 — Ih) is - ax and '^^ = a+h- 1)3(1 + /i + 2)2(7 + 7/i + 5) = h\3 + 70^(12 + 7h) is +. Hence at x = 1, -- changes sign from — to +, and there is a minimum at this value. dx dy dy_ dx This minimum is 2/ = 0. Finally, to test a; =— 2,^ = (— 2 — /i — 1)3(— 2 — /i + 1)\— 14 — 7/i + 5) = (— 3 — K)\-~ h)\— 9 — 7/i), which is -f . Again, ^ ^ (— 2 + /i — 1)^ (_ 2 4- /i + 2)2(— 14 -I- 7/i 4- 5) = (— 3 -f /i;3(+ )i)\— 9 + 7/i) is also +. Therefore a; = — 2 does not correspond either to a maximum or a minimum. 156. ScH. — It is usually easy to see, without going through with the de- tails of the substitution, whether -^ changes sign with h in such cases as dx dy the above ; that is, whether \i x= a is the critical value we are testing, -~ will have a different sign when we substitute a -\- h, iox x, from what it will when we substitute a — % for x. Ex. 11. Examine v = -^ — J, for maxima and minima. ^ (a; — 3)2 Stjg's. -- = - = 0, gives for the critical values re = — 2, x = 13. d« (a? — 3)3 dy _ (x + 2)2(c.-13) _ ^ _ 3 _ ^j^^^^^ ^ _ 3_ da; (X — 3)3 . 5 V ^ d^v In this case it is better not to form -^ as it is complicated, but test the critical dx"^ dv values by noticing whether -- changes sign or not for these points, x = — 2 does not correspond to either a maximum or a minimum, x = 13, gives :y = 33|, a minimum, x = 3, gives i/ = oo, a maximum. ^ ScH.— The first 10 examples give x = co iov -j = oo, and hence give rise to no critical values, as .r = oo cannot correspond to a maximum or mini- mum, there being no succeeding value of the function. {X 1)2 Ex. 12. Examine y == -, =^- for maxima and minima. {x-\-iy Sug's.— Putting-5- = 0, gives x = 1, and 5, as the critical values. Putting — == oc, gives X = — 1. When x = 1, 2/ = 0, a minimum. When x — 5, y = j^y, dx a maximum. When x =. — 1, y is neither a maximum nor a minimum. 96 APPLICATIONS OF THE DIFFERENTIA!, CALCULUS. 3 Ex. 13. Examine y = b -{- {x — aj"^ for maxima and minima. Result, The critical value oi x i^ x = a. But this does not corres- dy pond to either a maximum or a minimum, since -^ does not change sign at this value. m SuG. —In this example — ^ = rb ex for a; — a, and hence cannot be used to dis- criminate between maxima and minima. 4 Ex. 14. Examine y = h -{- {x — a)^ for maxima and minima. Result, ?/ = 6 is a minimum. Ex. 15. Examine y = i — {x — aj^ for maxima and minima. Result, 2/ = 6 is a maximum. Ex. 16. Show that y = x^ — Sx- + 6^+7 has neither a maximum nor a minimum value. X Ex. 17. Show that y = - is a maximum when x = cos a;.* 1 4- j; tana? 1- '' dv cos"^ X .__. dy SuG. — = ————— . When x <" cos a;, -- is + ; but when x >> cos a, dx (l + a;tanx)2 ^ ' dx ' ^ ^is dx Ex. 18. Show that y = sin^ x cos x is sl maximum when x = 60°. sin X Ex. 19. Show that y = is a maximum when x ^ 4:5°. ^ 1 + tan^ GEOMETRICAL PROBLEMS. Ex. 1. Eequired the altitude of the maximum cylinder which can be inscribed in a given right cone with a circular base. Solution. — Let SO = a be the altitude, and AO = 6 the radius of the base of the given cone. Let ac = xhe the altitude, and cO = of = y he the radius of the base of the required cylinder. The function which is to he a maximum is the volume of the cylinder. Calhng this V, we have V = Tty^x. In this form V is a function of two variables x and y. But these variables being dependent upon each other, we can find the value of one in terms of the other. Thus, S/ : S O : : «/ : AO ; or, in the notation, a — x : a : : y :h ; whence y =i -[a — x). Substi- * When X = cost, x = 42"* 21' nearlj'. GEOMETRICAL PEOBLEMS. 97 tutmg this value of y, we have V = — (a — x^x, which is to be a maximum. Dropping the con- stant factor — {153, 1st), we have V'= {a — x^x a^x — 2ax-^-j-x^. dV' dx = a" — 4aa; + 3x2 _ o ; whence a; = ia ; that is, the axis of the cyHnder is i of the axis of the cone. From this we readily find y, the radius of the base of the cylinder =|&. . • . volume of cylinder = -^^ita}fi. But volume of A cone =: \Tidb~ ; whence volume of cylinder = f volume of cone. Fig. 25. Ex. 2. To find the axis of the maximum cone which can be inscribed in a given sphere. 771 Sug's. — Let AmB be the semicircle which, re- volved upon A B, generates the sphere, and Aa& the triangle which generates the cone. Let AO=r, A&=::ic, anda&=:2/- Then V = inr2/2x=i7rx2(2r — a), —2 since a6 =2/^ = A6 X6B=£c(2r — x). .•.a; = |r, or the altitude of the cone is f of the diameter of the sphere. Volume of sphere = ^itr'^, volume of maximum cone = -/f X a^rr^ ; or the cone :==: -^^ of the sphere. ScH.— In attempting the solution of such problems, first notice wliat the function is which is to be a maximum. Thus, in Fx. 1, it is the volume of a cylinder ; in Ex. 2, it is the volume of a cone. Having obtained the equa- tion expressing the function in terms of the variable or variables on which it depends, if there are two dependent variables involved, find from the conditions of the problem the relation between these variables, and sub- stitute for one of them its value in terms of the other. Finally, we have a function of a single variable, which can be examined for maxima and minima values in the usual way. Ex. 3. Required the cylinder of greatest convex surface which can be inscribed in a given right cone with a circular base. SuG. — The function is the convex surface of a cylinder. Using the same notation as in Ex. 1, and letting S represent the function, we have S = 27tyx = (a — x)x. .' . X =-la, and S = -3— ; that is, the altitude of the cylinder is i that of the cone ; and the convex surface of the cylinder is to the convex surface of the cone as - ; s/a^ -j- ^^j o^ ^s i the altitude of the cone is to its slant height. Ex. 4. Required the maximum Cylinder which can be cut from a 98 APPLICATIONS OF THE DIFFERENTIAL CALCULUS. given fephcre. The axis of the cylinder = f \/3 times radius of sphere. The cylinder is to the sphere as 1 : y'S. Ex. 5. Eequired the area of the greatest rectangle which can be inscribed in a given circle. The rectangle is a square, and its area = 2r2. Ex. 6. "What is the altitude of the maxi- mum rectangle which can be inscribed in a given parabola ? Sug's. — Let ac = X, af = y, and AX = «. Let A be the function, the area of the rectangle. Then A = 2a'?/. From the equation of the parabola aj" = 2p X A/, or y^ = 2p{a — x) ; whence A = 2x\/2p^a — X). A' =^ ax^ — x^, and x = |a. Ex. 7. Bequired the axis of the cone of maximum convex surface which can be inscribed in a given sphere. The axis == ^ the radius of the sphere. Ex. 8. Required the altitude of the maximum cone which can be inscribed in a given paraboloid, the vertex of the cone being at the intersection of the axis of the paraboloid with the base. Sug's. — Let ABC be the parabola whose revo- lution about AS as an axis generates the parabo- loid. Let AS ==^ a the axis of the paraboloid, oS = X, the altitude of the cone, and ao = y the radius of the base of the cone. The result is X = ia. Ex. 9. Required the maximum para- bola which can be cut from a right cone With a circular base, knowing that the area of a parabola whose limiting co-ordi- nates are x and y is |-.r?/. Sug's. — LetSO=a, BO = &, AX=a;,and f X = t/. The function is A (the area) = ^xy. But aX = y = v^BX X XC ; and CX : Fig. 28. C B : AX : S B, or CX : 26 : : a; : v/a'^ -f 62 ; 2hx whence CX = — p, letting S = Va'^ + ^^ ^o^ brevity. Then BX CB — CX 25 26x Fig. 29. GEOMETRICAL PROBLEMS. 99 -> x). Finally, A = ^x m 86 -x{S — x) ■= --\/x\S — X), and A' = Sx^ — X*. The result is a; = |/S^, that is, the axis of the parabola is 4 the slant height of the cone. The area of the parabola = ibS\/3. Notice that CX = |CB. Ex. 10. From a given quantity of material a cylindrical vessel with circular base and open top is to be made, so as to contain the greatest amount. What must be its proportions ? Sug's. — Let X = the altitude, y the radius of the base, and V the volume. dV dy Then V = Tty'^x is to be a maximum. Hence -y- = S^/x-^ -\- y'^ z= Q, or y = — 2a;--. But iTtyx -f 7ty^ litydy = ; whence -- = ■ dx dx dx s, the surface. Differentiating ^itxdy -f- 27tydx -f- y Substituting, y = — -— . . • . y = x, that x + y_ ^' ^ x + y is, the altitude = the radius of the base. The altitude = — • Ex. 11. Of all right cones of a given convex surface to determine that whose solidity is the greatest. The altitude = \/2 into the radius of the base. Ex. 12. To find the maximum rectangle inscribed in a given ellipse. Sug's. A = 4xy. A' = xy. — r— r= ?/ + ^'y "^ ^' '^^y^ -f ^2x2 — ^2^2. dy dx dx B'^x A^y y = dx dy B^x'i X— = . dx A^y X '.y :: A: B. That is, the sides of the rectangle are to each other as the axes of the ellipse. The sides of the rectangle are A\/2, and B\/2. Fig. 30. Ex. 13. To find the maximum cylinder which can be inscribed in a given ellipsoid, generated by the revolution of an ellipse about its 2 The axis of the cylinder = — -=A. transverse axis. Ex. 14. A person being in a boat 3 miles from the nearest point of the beach, wishes to reach in the shortest time a place 6 miles from that point along the shore ; supposing he can walk 5 miles an hour, but pull only at the rate of 4 miles an hour, required the place where he must land. Sug's. — Let AX =;r, and f = the time required to reach A by rowing from B to X, and walking from 5 4 is to be a mini- mum. He must land at X, 1 mile from A. FlO. 31r 100 APPLICATIONS OF THE DIPFEEENTIAL CALCULUS. Ex. 15. Divide a into two factors the sum of which shall be a min- imum. Result, The factors are equal. Ex. 16. The difference between two numbers is a ; required that the square of the greater divided by the less shall be a minimum. Result, The greater = twice the less. Ex. 17. To find the number of equal parts into which a must be divided, so that their continued product shall be a maximum. Sug's, — Tlie function is u=zl-\. logw = a;(loga — \ogx). u ^= .^'log« — a; logic. — = log a — log a; — 1 ^= 0. x = -. Arithmetically the problem is possible only when — is integral. Ex. 18. Eind a number x such that its ^th root shall be a maxi- mum. X = e. Ex. 19. A privateer wishes to get to sea unobserved, but has to pass between two lights, A and B, on opposite headlands, the distance be- tween Avhich is a. The intensity, at a unit's distance, of A is h, and of B, c. At what point must the privateer cross the line joining the lights, so as to be as little in the light as possible ; it being under- stood that the intensity of a light at any point equals its intensity at a unit's distance divided by the square of the distance from the light. . 6 c SuG. — Letting x = the distance from A, the function is u = — -] — x-i ' (a — x)2 J. ao X = — — ^ 3 , 3 -\- C Ex. 20. The intensity of illumination from a given light varies as the sine of the angle under which the light strikes the illuminated surface, divided by the square of its distance from the surface. Re- quired the height of a light directly over the centre of a given circle, so that it shall illuminate the circumference as much as possible. Sug's. — Let /represent the illumination at P, which is to be a sin l_PO maximum ; PO = B ; and LO =: .r. 7 = — ; — . But LP' Bin LPO =—-=—-. .•.!=:=—,= -^ ; whence ^^ ^^ LP' ^R.J^x^Y dl (E->-fx2)- — 3a-2(/?2 + ar2)- f i . T,. = ^R:^ -I- a:^). ' = «. im-\-x^-f-Zx'^{R^+x^)l=6 Ji2 _^ .,.< _ 3.,;: ^ 0, and X = Bs/\. GEOMETRICAL PROBLEMS. 101 Ex. 21. To find in a line joining the centres of two spheres, the point from which the greatest portion of spherical surface is visible. Sug's. — The function is the sum of the two zones whose altitudes are, M D and md ; hence we must obtain an expression for the areas of these zones Let CO =^ R, co=: r, Oo == a, PO = X, and Po = x =a — x. ^~^ 1^ ^ ^ From the right angled triangle PCO, R- =z Fig. 33. ]^x 7?2 DO X aJ ; whence MD=-R — DO=^ — , the altitude of the zone seen X fV'rv* . rt*3 on this sphere. In like manner md = ; — . Now the area of a zone being to X the area of the surface of its sphere as the altitude of the zone is to the diameter of the sphere, letting Z and z be the zones, Z : 4:7tR'^ : : — — ; : 2E, . • . Z = „ ^Rx — R^ A 1 • Ti ^ rx' —r^ a — x — r zTtR . And m like manner z = znr -, = Aitr- . x X a — a; X Ji Hence, letting S represent the function, we have, S = 27tR'^ -f- ^ a — x — r ,„ ^ R3 , r3 dS' R^ r^ . , 27rr-2 , S' = R^ ^ r^ . -- = -; ^- = ; whence a — X X a — X dx x- {a — xy^ 9 3. a. x= " '' — ■■" a. 3. )2 r (r^ 4- i?^)2-| ; and the entire surface = 27r r^ -f- J^^ • r' +R' Since 27tr^ -f- ^jtR^ is the sum of the hemispheres, S is always less than this sujn except when a = oo. GENERAL SCHOLIUM. The student should now resume the study of G-eneral Geometry at Chap- ter IV. OHAPTEE IE. THE INTEGMAL CALCULUS. SECTION I. Definitions and Elementary Forms. 157. The Integral Calculus is that branch of the Infinites- imal Calculus which treats of the methods of deducing the relations between finite values of variables, from given relations between the contemporaneous infinitesimal elements of those variables. It is the inverse of the Differential Calculus. 158. The Tvttegval of a differential function is another func- tion which being differentiated produces the differential. 150, IlfltegvatiOTl is the process of deducing the integral func- tion from its differential. I(y0, The Sign of Integration is J", which is a form derived from the old, or long s. It is the initial of the word simi, and came into use from the conception that integration is a process of summing an infinite series of infinitesimals. Ill's. — Suppose we have given dy = -—^ — — . This is a differential function, and we have given in the equation the relation between dy and dx. The Integral Calculus proposes to find the relation between y and x from such a relation between their differentials ; or, in other words, to find the function which being differen- tiated produces the given differential. The function in this case is ?/ = = , as 1 — iC^ will be proved by differentiating. The latter is therefore called the integral of the // Axidx ^x^ dy== I ^xdx and read, "the integral of dy equals the integral of .^ ' ~-_^, which equals (1 — X2)2 1—X^ Ix (1 — "x2)2' 1—X^ The conception of integration as a process having for its object the summation of an infinite series of infinites- imals may be illustrated by considering the area of an ellipse as composed of an infinite number of infinitesi- mal segments, as represented in the figure. Let A rep- resent the area of the ellipse ; whence cZA represents one of the infinitesimal segments, or elements of the axea. DEFINITIONS AND ELEMENTABY FOKMS. 103 Now it is found that dA = -{a^ — x^^) dx. By integration it is found that the entire area is Ttah, h and 6 being the semi-axes. But, as the entire area is the sum of the infinitesimal segments, the process of integration may be considered as having for its object the summing, or adding together of all the infinitesimals which go to make up the entire area. 101. Important General Statement. — Strictly speaking, there is no such thing as a Process of Integration. Whenever a differential is proposed for integration, the first question is, Is this a Knoivn Form f that is. Can we see by inspection what function, being differentiated, pro- duces this ? If we cannot thus discern the integral by a simple inspec- tion, the only question remaining is, Can we transform the differential into an equivalent expi^ession the integral of luhich we can recognize? Thus, in any case, we pass from the differential to its integral by a simple inspection ; and the sufficient reason always is, This expression is the integral of that, because, being differentiated, it produces it. THREE ELEMEIVTART PROPOSITIONS, 102, JPvop. 1. — Constant factors or divisors appear in the integral the same as in the differential, and hence may be written before or after the sign of integration at 2)leasure. Dem. — This is a direct consequence of the fact that constant factors or divisors appear in the differential the same as in the function {48). lOS* ^vop, 2, — To integr'ate the algebraic sum of several differen- tials, integrate each term separately, and connect the integrals by the same signs as their differentials were connected. Dem. — This is a direct consequence of the rule for differentiating the algebraic sum of several variables (51). 104, JPvop, 3, — An indeterminate constant must always be added td the integral of a function. Dem. —Since, in difierentiating, constant terms disappear, in returning from the diflferential to the integral we have to represent any possible constant terms by an indeterminate constant. ScH. — The method of disposing of this constant term, which we usually represent by C, will be presented hereafter.* The fact that there may be such a term is all that the student is expected to see at this point. To illus- trate, suppose y =^ dax^ -\- 12b, dy = ^ax dx. Now, if the latter alone were given, we might see that y = dax'^ was its integral, since being differentiated it would produce dy = 6axdx. But so will y = dax^-\~ any constant, as 126, or, as we represent it, y = 3«.r2 -J- G. * Section VII., closing illustration. 104 THE INTEGRAL CALCULUS. TWO ELEMENTARY RULES. lOS, R ULE 1. — Whenever a differential can be separated or trans- formed into three factors ; viz., 1st. Its constant factors ; 2nd. A vari- able factor affected with any exponent except — 1 ; and 3rd. A differen- tiat factor which is the differential of the 27id factor without its exponent, its integral is The product of the second factor with its exponent increased by 1, INTO the 1st or constant factor divided by the new exponent.* Dem.— This rule is evident from {162), and the rule for differentiating a variable affected with an exponent {56.. Thus, if y = m[/(a;)]«, dy = mn[/(a;)]«-i d.f{x), or mn X [/(''''^)]"~^ X d.f[x) ; whence to pass from the latter to the former, we have to suppress the differential factor, d.f{x), increase the exponent n — 1 by 1 making it n, and divide the constant factor mn by this n. In the exceptional case the exponent by which we would be required to divide according to the rule would be 1 — 1=0, whence the result would be oo. Ex. 1. Integrate dy = dax^dx. Solution, dy = 3a Xx^Xdx; whence y = jSax^dx = — x^ -\-G= ax^ -f" G. O Ex. 2. Integrate dy = ax^dx. /CL axHx = -x4 _[- C. Ex. 3. Integrate dy = (a -\- 3x^y6xdx. Solution, dy = 1 X{ci-\- Sx^y X Qxdx, which corresponds to the requirements of the rule, since d{a + 3x^) = 6xdx. .'. y = J{a-\- Sx^y6xdx = i{a + 3x2j3 -|- C. Ex. 4. Integrate dy = (a + ^x^yxdx. Solution. — The differential of the quantity within the parenthesis being Qxdx, we write dy = \ {a -\- 3a;2)3 x Qxdx, which conforms to the requirements of the rule. .'. y — f\{a + 3.r2)36arc?a; = -^{a + ^x'^Y + G. Ex. 5. Integrate dy = a{ax -\- hx^y^dx -\- '2h{ax -\- bx^)^xdx. Sug's. y=fla{ax-{-hx^ydx + 2b{ax-\-bx-^yxdx'] = f{{ax-\-lx^)\a-}-2hx)dx'] = /[I X (ax + hx^f X (« + 2hx)dx'] = i{ax -}- bx^^Y -f- G. 100, RULE 2. — "Whenever a differential can be written in, or transformed into a fraction whose numerator is the exact differen- tial OF its denominator, the integral is the Napierian logarithm of the denominator.* * In giving such rules the constant term of the integral is not mentioned, as its addition is always implied. DEFINITIONS AND ELEMENT AllY FOllMS. 105 Dem. — This is a direct consequence of the rule that the differential of the Na- pierian logarithm of a number is the differential of the number divided by the number. [This will be seen to be the exceptional case under the preceding rule.] 167. ELEMENTARY FORMS. 1. V = Cx^'dx = -^"+^ + G. Same as Bule 1. ^ -^ • n + 1 /dx — = log X -\- G. Same as Bule 2. S. y= fa'dx = -^a^ + G. ^ log a 3i. y = fe'-dx = e' -\- G. 4.. y = fcos X dx = sin x -\- G. 5. y = r — sin X dx = cos x -j- G. 6. V = / > or fsec^ x dx = tan x -{- G. J cos^^ -^ 7. V = / — — > or f — cosec^^ dx = cot^ + G. J sin2^ -^ 8. 2/ = rtan x sec xdx = sec x -{- G. 9. y = f — cot X cosec xdx = cosec x -\- G, 10. ?/ = fsin X dx = vers j: -f (7. 11. y z= C — cos X dx = covers x -\- G. 12. y = I . =: sin~^^ + G. 13. 2/ = / ; = COS-^^ + U. dx 14. 1/ = fzr^- = tan-^j7 + a 15. y= f— r^- = cot-^o; + G. ^ J 1 -{- x^ 16. 2/ = f — 7=^= = sec-^^ 4- G. J XV X'^ ■ 1 17. y = I — = cosec~^a; + C, J XV x-^ 1 18. ?/ = / = vers~^^ + G. ^ Vix 072 /dx . = covers-^o; + G. V 2,r — J72 Converse of (60). « (61), (( (66). « (67), it (69), -a " (70), ^2a6a; — 62^2 o a * or = f—^=Mz= = vers-'- + C, when 6 = 1. r dx 1 _,bx 8 V = / = T covers h ^j */ \^2abx — b'x^ ^ ^ or = f ^^ = covers-' -, when 6=1. J Vlax — x^ ^ Dem.— These forms may be considered as the converse of Ex's. 1,2, pages 38, 39. They may also be established by differentiating the result and showing that /I . hx\ its differential is the given differential function. Thus, a^^ sm — ) "^ DEFINITIONS AND ELEMENTARY FORMS. 107, '\aj la 1 hdx dx 1 \aj la _ ^, [The student ^ r 6^x2 & ja* — 62x2 ^ v^o2_52a;2 y/a2_ 523.2* I 65a;2 6 | a« — j should verify all of them in this way. ] A direct way of obtaining these integrals, and one with which the student should not fail to become familiar, is the following : /dx — — , we observe that it has the general form of the s/a'^ — &2a;2 differential of an arc in terms of its sine, which is — - To transform our \/l — x'-^ expression into this form, we have first to make the first term under the radical 1. This can be readily done thus, / — — ,/ \/a2— &x2 since the constant divisor a appears in the same form in the integral as in the dif- ferential {102). Now to make the quantity under the sign of integration the differential of an arc in terms of its sine, the numerator ought to be the differen- tial of the square root of the second term in the denominator, which is the square of the sine. But d( —\ = -dx. "We, therefore, need to introduce - into the nu- \ a / a a merator. This can be done by putting - outside the sign of integration as they will neutralize each other (162). Hence a -dx — . The quantity now under the sign of integration is the exact &x &X differentiaLof sin—'—, since it is the differential of the sine, — , divided by the ti a a &X f square root of 1 — the square of the sine, — (75). Hence, finally, as J dy=y\ , r dx 1 . hx we have y = I — = - sm— 1 [- C. J \/a2 _ 12^2 0, [The student should produce all these subordinate integrals in this way, for the benefit of the exercise. We give the outline of two more, which should be ex- plained at length as above.] h y /dx __ / dx 1 / dx la / a at / 62x2 ab ab i ^ . 62a;2 ~ ^ ^^ a" "*" 108 THE INTEGRAL CALCULUS. y — / dx /^ dx ^ /* ^ da; _ dx 1 / a / a covers—* — f- C. a 169. LOGARITHMIC TRIGONOMETRICAL FORMS. y ^ dx 2 cos2 (ix) — : — . r '" = sm (ix) cos (|jc) / tdJi{ix) J t-d.n{ix) ^ _ r dx r ■, r dx rdiirt — x) 2. y = I or sec;rcZa; = / -: ; = — / -: = [by (1)J ^ J cosx -^ J sm(47r — x) J sin(i;r — x) »- ♦' ^ ^J — log tan {i7t — ix) + ^' r dx r . -, /"cos xdx rd{^\n a;) . 3. y = I or I cot xdx = / — : = I — ■ = log sm x 4- (J. I tanic -^ / HVOLX J sin a; /^ dx r , rsiuxdx /•(i.'cos.r) 4. V = / or tanjcox = / = — I = — log cos a; = / cotic ^ J cos a; / cos a; log = log sec X A- €, cos X 5. V = / —■ = I rr-T = I -■ — rr— = \yl (1)1 log tan .r -I- C. ^ J smxcosrc / sm (^2:c) / sm ^^2^:;) l j v /j o ^ ScH — The above 32 forms must be so thoroughly memorized as to be in- stantly recognized. There is no doing anything in the integral calculus without this. These forms are to integxation what the multiplication table is to arithmetical operations. Thus we say, 7 goes into 56 8 times, because 8 times 7 = 56. In like manner we say that cot~'.r -(- is the integral of -, because cot~^a; -{- C differeniiated = — :j — -' — -. 1 + a;2' ' •" 1 + a;2 Ex. 1. Integrate dy = 'iax^dx. SoiiUTioN.-— The integral of dy is y, since y differentiated = dy. To integrate Zax^dx, notice that 3<2 X ^^ X dx, conforms to {105). ,•. y = fSax^dx = Ex. 2. Integrate dy = (2a + c^hxydx, Sug's. y = J {2a + 3bx)^dx = J(8a^ + 36a^bx -f BAdb^x^-\- 27b^x^)dx = DEFINITIONS AND ELEMENTAKY FORMS. 109 fSa^dx -\- JSGa'^bxdx + J^4:ab^afidx + j21h^xHx = 8a^x + ISa^bx' + ISab'^x^ + Y63a;4 .^ G (163). This may also be integrated by {165). Thus y = / oT X (2<^ + 3&x)3 X 3&cte =/^^(2«+ = j^ (2a 4- 3&a;)4 + ^ 5 which is the same as the preceding. xdx \/a'^ + x^ Sug's. y = l /^^ - = f{a^ + x^)~^xdM = /i X (a^ + x^)"^ X SxcZa; =s 2/ = = %hx^ + a y = X-' + a y 5 — 1^^ + a y^ == ^mx^ + a y = ^ 1 Sx^ + a Ex. 3. Integrate dy == ' xdx (a + a;2)i + a Ex. 4. Integrate dy = bx^dx. Ex. 5. Integrate 6?y == 3a7~^c?a7. 2 Ex. 6. Integrate ^y = 2x^dx. Ex. 7. Integrate dy = — 5mx~^dx, Ex. 8. Integrate dy = — . X* x^dx 1 Ex. 9. Integrate dy = -. y = |.(a» + jp3)^ + G. Ex. 10. Integrate dy = dx. (Sax-^ — x^)'^ 2ax x^ — i * _i Sug's. jdx = — (Sax^ — x^) (2ax — x'^)dx = _ ^ x (3aa;2 — x^) {3ax- — x^) ^ /2ax x^ ^ : -dx = — i{dax^ — x^)'' + a (Sax^—x^)'' Ex. 11. Integrate dy = 12bx{4:bx^ — 2cx^)^dx — 9ca?«(46a7« — 2cx^)'^dx. Sug's. 126a;(46a;2 — 2cx3) da; — dcx^ibx^ — 2cx^) dx = (4&a2 — 2cx3)* (12&a; — 9cx^)dx. Now in order that the factor (126a; — 9cx^)dx should be the dif- ferential of 4bx^ — 2ca;3 we should have 8 instead of 12 and 6 instead of 9. Hence we write f (4&x2 — 2ca;3) {8bx — 6GX^)dx. ,-. y = %{4:hx^ — 2c£c3)* -f C. 170, ScH. — It is not always easy to determine just what constant factor is required in order to make the differential factor the differential of the quantity within the parenthesis ; nor can such a factor always be found. To determine whether there is such a factor or not ; and, if there is, to find it, we may proceed as in the following examples. 110 THE INTEGBAL CALCULUS. Ex. 12. Integrate dy = '■ jdx. (26 + dax^ — 5a;3)^ Stjg's. — jdx = (2& -|- 3cwr2 — s^s) ^ (2ax — 5x^)dx. Suppose A to (26 + 3aa;2 — 5a;3)^ /I -i —{2b-{-dax^ — 5x^) {2aAx — 5Ax^)dx. It is required that A should fulfill the condition d{2b -\- Zax^ — 6a^) = {2dAz—5Ax^)dXy or 6ax — ISx^ = 2aAx — 5 Ax-. Now, as this is to be true for all values of x, we have 6a = 2aA, or ^ = 3 ; and also 15 = 5A, or A = d. Hence 3 is the factor sought, and we have y == J (2& + 3aa;2 — 5a;3) ^ (2ax — bx^)dx = J i(26 -f- 3aa;2 — 5x3) "* {a?^)]. Ex. 26. Integrate dy = ^ --. 2/ = log 5:. ^""'^'^ (8a — 3a:^)^ Ex. 27. Which of the following can be integrated by the method used in the last 6 examples ; viz., dy = — -^ — -dx ; c??/ = 2ir» , , a: — ^2, , 3^'^c?^ , 2a — 10a;2 dx',dy = -dx ; 6^2/ = ^^- ^ ; ^^Z = h" — e:::,^?^? 1_^3 ' ^ 3 — a;3 ' ^ 2^='— 5 *" 2aa; — Sa:^ Ex. 28. Integrate dy =^ ^ ' nx"^ 5(3a; _ a»)4da; h/^lx*dx — 108a2a;''da;4-54:a4a;2(^ _ 12aSa^a;+a8da;'* nx^ n> -( Slajcto — 108a2dx + — r— + --- ). 1 Ex. 29. Integrate dy = (h — 572)3^2,^-^, , = i63.l_^*-^.HA^-^-,V^+a -n, «^ -r , , , 5(2a — ^«)3, Ex. 30. Integrate dy = — -dx. y = 5[- ^V ^' + 6a logx - ix'] + a dx Ex. 31. Integrate dy = Slog^o; — . Sug's. 2/ = fs X (loga;)2 X — = logs a; + C, since — = d logaf. dx Ex. 32. Integrate dy = 2 logs -j; — . 2/ == i log"* ^ -\- G, * In all these examples c represents the constant of integration. DEFINITIONS AND ELEMENTARY FORMS. 113 doc 7TL Ex. 33. Integrate dy == mlog"a^ — . y = -losf""*'^ x4- G. . X n + 1 Ex. 34. Integrate dy = a?"" log adx. Sug's. — In order to make this conform to {107 f 3), we should have d{^x) =. Idx as a lactor. Hence we write ?/ = \ oP''^ log adx = J^kct-"" log a • 2dx = ia^-^ -j- C\ [The pupil should differentiate, verify, and so fully consider the case as to see the reason for the introduction of the constant factor. ] Ex. 35. Which of the following can be integrated by {107 ^ 3) ; dy = a" log a 2dx, or dy = da""' log a x dx? Am., The latter, y = ^a'^ + G. Ex. 36. Integrate dy = e"dx. 2/ == ae" + G. Ex. 37. Integrate dy = Se^dx. Ex. 38. Integrate dy = hd^'dx. Sug's. y = h fa-'dx = — fa^'^ log a 3dx = — a^* + C. •^ 3 log a*^ 3 log a 771 Ex. 39. Integrate dy = me''''dx. 2/ = — e"* + <^. Ex. 40. Integrate cZ?/ = cos {2x)dx. Sug's. — In order to make this conform to {107 f 4), we should have 2dx, i. e. the differential of the arc 2a;, instead of dx. Hence y = Jcos 2xdx = if cos 2x • 2dx = ^ sin 2x + a Ex. 41. Which of the following forms can be integrated by {107 f 4) ; dy ■= cos^x-ldx, or dy = cosx^xdx? Ans., The latter, y = -^sinojs + C. Ex. 42. Integrate dy = sins ^ cos xdx. Sug's. y = J sin^ a; cos a;da5 = J 1 X (sin x)^ X cos xdx = 4 sin^ a; + ^> accord- ing to {105 and jer, 4). Ex. 43. Integrate cZ^/ = sin {3x)dx. y = — ^ cos (3^) + G, or J vers (3a:) + (7. ScH. i vers [Sx] + = i [1 — cos (3a;)] + = i — i cos (3a;) + G. Ex. 44. Integrate dy = sin2 {2x) cos (2^)c?a:. 2/ = -|-sin3(2^) + G. Ex. 45. Integrate dy = cos^ {Sx) sin (3a;)c?j:. 114: THE INTEGRAL CALCULUS. Ex. 46. Integrate c^?/ = sec^x^xdx. y=^ tan 372+ Q Ex. 47. Integrate dy = 5 sec^ x^ • x^dx, ' 2/ = f tan x^ + 6'. Ex. 48. Integrate dy == 6 sec (4a;) tan {4:x)dx. y = 3. sec (4a:) + C'. Ex. 49. Integrate dy = 2 sin (a + Sx)dx. y=^— |cos(a + 3d7) + C. Ex. 50. Integrate dy = ^ cosec^ v2x ■ x '^dx. Sug's. « = ff cosec^ \/2a; • ic dx = — = fcoseca v^ • ^>/2 • a; dr = ^ *^ v/2 — 2_ cot v^2x + a v/2 Ex. 51. Integrate dy = 2 cosec (na;) • cot {nx)dx. y = cosec (nx) + G. Ex. 52. Integrate dy == e"'" " cos ajtZa;. y = e""" + C'. Ex. 53. Integrate dy = — e"'""' sin xdx. y = 6"'"" + C. dx Ex. 54. Integrate dy = — 7TT~T' 2/ == 2 tan {^x) + G. cos^ (-^a^j xdx Ex. 55. Integrate c^v = . -^ on - 1/ == i cot (3a;2) -f a ° sm" (da72) Ex. 56. Integrate dy = sin {ax)dx. V = - versin (ax) 4- G, or cos (ax) -\- C. ^ a a Ex. 57. Integrate dy = — cos {^x^)xdx. y == covers {^x^) + G, ov — sin {^x^) + C". ScH.— In the last, C" = C+ 1. In the 56th Ex., G' = C + -, Or 2dx Ex. 58. Integrate dy = V/I — 4:Xi Sug's. — The form of the denominator suggests at once that this may be the dif- ferential of some arc in terms of its sine. Observing that the numerator is the differential of the square root of 4ic'', we are enabled to conclude that y = = sm-i 2x H- a /; \/l — 4a;2 xdx Ex. 59. Integrate dy = . y === ^sin""^ {x-) -f 0. V 1 — a:^ Ex. 60. Integrate dy DEFINITIONS AND ELEMENTARY FOEMS. 115 dx ^2 — 9^2 — -dx ^ , dx dx \/2 >/2 /• dx SUGS. v/2 — y.r^ v/2v/l— |a;^ ^ v/2>/l — fx"^ ^ >/2— 9x-J ^ >/2 / — -dx .1 / v/2 1 . 3cc , ^ Qcdx Sx. 61. Integrate dy = \/^ — 5x4 „ , /• — xcZx /* — .r Ja; /* — 2\''^xdx ScTGs. y = I — = / — = — - = / — = — - — =5 J v/2 — 5.r^ J v/2 yi — fx^ ^ 2v/f v/'2 v/1 — f^t^ 1 /-— 'l^l.xdx 1 , r/5\ A n , ^ — / — = — ^cos-i ( - ) a;H + C. /5^ v/1 — fx^ 2>/5 L\^/ J 2v/5 Ex. 62. Integrate d/y = Ex. 63. Integrate dy = Ex. 64. Integrate c??/ = Ex. 65. Integrate dy = Ex. 66. Integrate dy = '6dx 4 + 9^2* , , , Zx „ 2/ — ^tan-^y + C/. j;^c?x 2/ — ^sin-X2^/2-) _|_ c. >/2 — 4^3 xcfj^ 2/ — ^ sm ' — 4- a 26 a v/qj2 1)2^4 x"dx 1 4- x^' 2/ .. Jtan-Va^s 4- (7. ^xr'^dx Ni 2a-^ — 6^^ Sug's. — As far as the variable is concerned this conforms to the diflferential of au arc in terms of its versed sine. Thus, the numerator = d\] 6a;*, as far as the variable is concerned ; and x^ = (x^ )2, which is the relation between the functions of the variable in the denominator of the form referred to. Hence, if we can adjust the constant factors to this form, the integral will be apparent. To effect the latter, we proceed as follows : -I --2 -^ S.'T dx .- 8.r ^dx , /^ 2x ^dx = vo — - - - -- ■■ ■ ■■ =: 4v/6 \ 2x ' - 6a;' Nj 2 • 6x ' — (6x' y- N/ 2 • 6a; ' — {Gx'y in which 6x being regarded as the variable, the expression has the desired form. .-. 2/ = 4v/6 vers-i (6a;*) 4. C, - 116 THE INTEGRAL CALCULUS. Ex. 67. Integrate ay = — y=z=z . y = -7= sec~^ — r- 4- G. XV 3x^ — 5 V 5 5t Ex. 68. Integrate dy = \/l4:a;2 — 3 y==^^osec-^[{^fx] + a MISCELLANEOUS EXERCISES UPON THE ELEMENTARY FORMS. [NoTK. — The following exercises are given without the integrals, as it is of first importance in the Integral Calculus, that the pupil be able to discover in the differential the probable form of the integral.] ■^ ^ n- , ,7 (1 — smx)dx Ex. 1. Integrate ay = ; . ° ^ X -\- cos X Ex. 2. Integrate dy = (2x^ + x~'^)dx. -r-, « -r , , -, ^dx - - x^dx . Ex. 3. Integrate dy = -— ; also dy •= Ex. 4. Integrate dy = 1 + X*' ^ ^ 1 4- ar* xdx \/l — x^ qqz ^x A- 3 Ex. 5. Integrate dy = -^—^^^^^dx. -r^ « -r . . •, ^dX , , xdx Ex. 6. Integrate dy = ^ , ^ ; also dy = ^^ , .^ . ^ 2 + 5^72 ^ 2 + 6ar« Ex. 7. Integrate c?^/ = (1 + cos J7)<^a:. (a 4- vxydx Ex. 8. Integrate dy = 7= . V X (2a2 + 4j72)dci7 I Ex. 9. Integrate dy = — . V a2 + x^ Mx Ex. 10. Integrate dy = Ex. 11. Integrate (Zt/ = cos^ x sin a;{x) • { (.r zb a)^ + _H "» in which il){x) represents the product of the real factors. 176. I*VOp, d, — Whenever the denominator of a rational fraction, f(x)dx as , —, whose numerator is of lower dimensions than its denominator, is real and resolvable into n real and equal quadkatic factors, the fraction can be decomposed into n partial fractions of the form ( Ax+B)dx (Cx + D)dx _ , (Ex + F)dx (Mx+N)dx ' r(i:±a)2-fb2]«"^[(x±a)2H-b'^]"-^"^ [(xdba)2+b2]"-^ ' " " (xrta)2-f b^' and these fractions integrated separately. Dem. — [The first part of the demonstration, showing that the fraction can be * Called conjugate imaginary faetors. 120 THE INTEGRAL CALCULUS. separated into this form, is identical with that of the last proposition, and the student can supply it.] Having separated the fraction into partial fractions as proposed, it remains to be shown that these partial fractions can be integrated. The general form is — ~ ; — , in which n is an integer. To reduce this to known forms put i{x ±z ay^ + &2]«' s X dz a = z, whence x = z =p a, dx = dz, and (x ± a)^ = z"^. Substituting these values, we have Aa)dz r {Ax + B)dx _ r^Az ^ Aa -\- B)dz _ r Azdz r {B ^ Aa)di J A{z-^ + h-^)-«zdz + / ■^f:y^,, ill which A'= B =F Aa. By {165) we have f A{z^ + l^)--zdz = - ^^^ _ -^^^^, _^ j^,y,^ ,- *In a subsequent article {192, formula 5|» it will be shown that the 1 may be made to depend upon / f-7 -, which in turn may be (22 _|_ 62)n ^ ^ ^ J (22 -^ b-^)n-l r A^dz ^, . ^, ... ..-./* A„+idz made to depend upon / , , ^,^„_2 > thus, m the end giving either / ^^, _^ ^,^„_„ 177. ScH. — In case the factors of the denominator are not readily seen, put it equal to 0, and solve the equation for the variable. According to the theory of the composition of equations, as developed in Higher Algebra, the variable minus each of the several roots in turn will be the factors. Ex. 1. Integrate ay == —r -^p: — — -r.. Solution.— Putting x^ + Qx'^ -f llic + 6 = 0, we find cc = — 1, — 2, and — 3.+ .-. ic3 -f 6a;2 -f llo; -f 6 = (.r + l){x + 2){x + 3), and we assume '' + ' ^ +^,+ " - ^A _|_ o.'ra -]-llx+6~ic+l~a; + 2 .t + 3 A(x + 2)(cc + 3) J3(a'. + l)(x. + 3) C(a; + l)(x -f- 2^ ^ (a; 4- l)(a; 4- 2)ia; + 3; "^ (a; -f l)(x + 2)(x -j- 3) "^ (x + l)(a; + 2)(x'+ 3) ^x^ 4- hAx 4- 6^ 4- Bx^ + 4.B.r 4- 3^ + Cl-r^ + BC-g + 26 ' £C^ -}- 6.^2 4- llx 4- 6 Whence x2 + 1 = (A + 5 4- (7).x2 4- (5^ + 45 4" 3C)x -|- 6^ + 35 4- 2a These members being identical, A^B^Q=\ U); 5^1+45 4-3(7=0 (2); and 6^ 4- 35 + 2(7= 1 (3). From (1), (2), and (3) we find ^ = 1, 5 = — 5, and C'= 5. Hence we have * This reduction might be exhibited here, but as the formula referred to is better for practical puriio.es, it is thought best to give the process but once, t Complete School Algebba, Part II., {ill). RATIONAL FRACTIONS. 121 { cfa + l)(x + 3)5 ) 51og(a; + 2) +51og(a; + 3) 4-logc = log] ^ ^ 2^ \' adx Ex. 2. Integrate a?/ r adx _ 1 r dx 1 r = 0. ' da; == / dx 4- / — = / — x:x--\- 2)-2da; _ / £ — = X log (x2 + 2) -I / — — — rr-. The last term can J (x-^ + 2)2 ^ ^'^ ^ ''^2(a;2 + 2) J (0:24-2)2 be integrated by formula ^ {192, Ex. 10). dx - Ex. 14. Integrate dy = —7; r. Sug's —The mmple factors ar# a? + v/^2, fl? — >/ — 2,>n.d » — 1 ; and the RATIONAL FBACnONS. 123 _^x -\-JB form of the partial fractions is - ' ' . . A = — h B = — i. and (7= ^. x--f2 x — 1 /dx , /T xdx , /• diT . , /* dx 1. , ^ , n ' r dx 1 « — tan— ^ h (7 ; since / „ , ^ = tan— i — . Ex. 15. Integrate t^2/ = x^ + X^ -\- X + 1* Ex. 16. Integrate c?v = ^• SuG s. —Assume ; = — r— -\ -\ — - ; whence A = — *• \x-f-l/ 6 ^2 X^ -\- X Ex. 17. Integrate dy = ; -dx. Ex. 18. Intesrrate dy = -dx. ^ ^ x^ + x^ + X -}- 1 -p ia T . . ^ 9^2 + 9^-128 , Ex. 19. Integrate dy = —z --., — ^r ax. & ^ x^ — 5x^ i- dx -\- d x^ — 1 Ex. 20. Integrate dy = -^ jdx. !/ = f-' + I log (07 + 2) + I log (^ — 2) + a 17 S, ScH.— It will be observed that the foregoing processes of separa- ting rational fractions into partial fractions make their integration depend on one or more of the following forms : /^ r dx r dx r xdx r xdx r dx -^ ^ ^' ./ X ±:a J x^ -\- a2' J x^ + a^' J (.x-2 4. a^y^' J [x^ + a^)^' ^ All of these forms except the last are integrable by the elementary pro- cesses. The integration of the last is effected by formula i|> {192). 4-'^-^ A^ - CT$ 124: THE INTEGBAL CALCULUS. V SECTION IIL Eationalization. 170* When polynomial radicals occur in a difterential wliicli we desire to integrate, it is sometimes possible and expedient to rational- ize the expression by the substitution of a new yariable which is soiue definite function of the variable in the given differential. A few of the more important cases are given in this section. BINOMIAL DIFFERENTIALS. 180. I^TOp* 1. — Every binomial differential can he reduced to the form x™(a + bx°)Pdx, in which m and n are integral, and n positive. Dem. — 1st. If X occurs in both terms of the binomial, and the form is x^{(ix^ -\- hx*)Pdx, we can remove from the parenthesis the factor x", or a', which has the less exponent. Thus suppose s <^t, we can write x^^ax' -j- hx^) p dx = (x^\p a -\-h— ] dx = x^+p^{a -\- bx^-^)pdx. In this form t — s is positive, since t^ s, but it may be fractional, r -{- ps may be either positive or negative, integral f 6 or fractional. Now let r -\- ps = ± f, and t — s = -[- - ; whence we have /I / X \a+bx fydx. ±- +- 2nd. In the latter form put x = z^f ] whence x ^ = z^*^, x ^ = z+^\ and dx = ±- +- hfz^-f—^dz. Substituting these values, we have x '\a -j- ^^ ^^dx = z±c/(a 4- hz+^'')Phf z^f-'^dz = hf z±'^f+^f-'^{a -J- hz+^^)Pdz, in which the exponents of z are integral, since c, e, /, and h are integers, and eh is positive. Therefore putting ± cf-{-hf — 1 = m, and eh = n, we have hfz^{a-\- bz'')pdz. q. e. d. 181. JProp. 2. — A binomial differential oftheform'K^(a.-{-hx''ydx {any or all the exponents being fractions) may be rendered rational by . , m + 1 . . , , putting a + bx = z'^, when is integral. Dem. — Putting a -{- hx^ = z^^ (1) p we have (« + & iK")^ = ^, (2) Differentiating (1), nh x'^—^dx = qz^—^dz. (3) m — n+1 Also from (1), a;'»^+i = ( J T J " * ^^^ m — »+l Multiplying (2), (3), and (4), n&x'"(a -f &x")?dx = gz^+g-/ ^ T j " ^2» rr/-l-l ? n /z'' — fl\~t ' or .r"'(a + hx")\1x — -^z'' •^''- — ■ — ) nh \ b / d.z. RATIONALIZATION. 125 m -j— 1 Now by hjrpothesis p -{-q — 1 is integral ; hence, if ' — is integral, the expression is rational, q. e. d. 182, JPvop* 3, — A binomial differential of the form x'"(a4-bx")^dx (any or all the exponents being /inactions) may be rendered rational 6?) 11^ + 1 P . . . putting a + bx" = z'^x", when [- ~ is integral. (1) (2) Dem. — Putting a -\-hx'^ = z?x", e have a T*" — 22 — 6* 1 a" 1 f . n^(''1 -h) ». (z? _ ly (3) and jc" = a "(z? — 6) «. (4) Multipljdng (2) by Z) and adding a, we have IT ah , az^ _. a 4- occ" = 4- a = -, (5) whence (a + 6a;")* = a9(z3 — 6) «zp. (6) 9 - ---1 Differentiating (3), do; = — -a^zi — 6) « zs-'dz. (7) Multiplying together (4), (6), and (7) and putting A for the constant factor, there results a;"i(a _{_ 'bx^'Ydx = ^(z« — 6) ^ " « ^z? + «— 'dz. 772 1 ■ 1 Ti Now by hypothesis p -\- q — 1 is integral ; hence, if — — f- - is integral, the expression is rational, q. e. d. 183, ScH. — ^Although the rationalization can always be effected as stated in the last two propositions, it does not always facilitate the integration. When in the former case — — 1 is a positive integer or 0, or in the Wi I J. T) latter 1- - + 1 is a negative integer or 0, the binomial ^ — b will 71 q have a positive integral exponent and can be expanded into a series of a finite number of terms, or a exponent and will be equal to 1. Hence in any such case the rationalization will lead directly to the integration. But if — — 1 is a negative integer in the former, or — \- — -\-l is a n n q positive integer in the latter, the exponent of ^ — b will be negative, and the rationalization will not generally lead to the integration ; and in fact it is not usTially expedient to rationalize in such cases. 126 THE INTEGRAI. CALGDLUS. 184:, Cob. — Every differential of the form dy== Ax'"(a+bx)''dx can he rationalized and integrated when either m or p is a positive integer. Dem. — If p is a positive integer, a-j-bx can be expanded into a series of a finite number of terms, whicli multiplied by x'"dx will give a series of monomials ; and an algebraic monomial can always be integrated by {IGS or 1S6). . , . , , .^. m4-l , m-|-l , If p is fractional or negative and m integral and positive, 1 = — 1, will be a positive integer or 0, and Prop. % will effect a rationalization which will lead directly to the integral. 1 Ex. 1. Integrate dy = x^{2 + ^x'^^dx. Sug's.— Since m = 5, and n = 2, ^^^-^t i — 2, a positive integer, and n Prop. 2, will lead to the integration. To rationalize, put 2 + 3x- = z^j (1) i whence (2 + 3a;2)-=z, (2) Differentiating (1) xdx = \zdz, (3) ~3 — /' ^^^ Multiplying (2), (3), and (4) together dy = x-{2 + Sx^fdx = -M^' — '^)^^^dz = -^{z^dz — 4:Z^dz + Az^dz). ,-. y = ijfiz^dz — 4^'idz + 4z^dz) = 5^,/^ — -|- -f- -|- j + C; or, restoring the value of ., 2, = -A- 1 '^^^ - i^^4^^ + ^^4^* [ + a [ThU result may be expanded and reduced, if desired. ] 3. Ex. 2. Integrate dy = x^{a + bx^)^dx. Ex. 3. Integrate dy = x^{a — x^) ^dx. 2/ = — J(a — x')^2a + x') + G. 1 Ex. 4. Integrate dy === x^{a -^ x)^dx, y = ^2^(a + x)^{5x-^ — Aax+ f a^) + 0. dx Ex. 5. Integrate dy = 1' ^4(1 _^ a;2)2 "- J = J x-*{l + a-) ~(ir. Here m = :— 4, n = 2, and p = -X; whence !!!L±i+^4-l = ^i-±i _ ^ + X = ^ 1, and Prop. 3, will lead to the integratioii. BATIONALIZATION. 127 l^utting 1 + fl?2 =5 z«a;« (1) ; we have x^ = (2) ; x == (3) ; sr-< = (z^ — iy (4); l+.r2 = l+-^i-^=-^- (5) ; (l+a;2)""* = z-i(z2_l)^ (6); and differentiating (3), dx = — (z2 — 1) ^z cZ2 (7)^ Multiplying together (4), (6), and (7), there results 3x^ y = /^-4(1 +x2) *da; = — /(z2 — l)(2z = z — iz3 _[_ c = (2a;3 — l)(14-.r2) _^ ^^ Ex. 6. Integrate cZi/ = 3 . (1 + ^.Ni^^ . ^ ax Ex. 7. Integrate dy = a(l + ^2) ^dx. y == — -^ — — 4- G. (1 + x"")^ Ex. a Integrate dy = a7-*(l --= ^x'^)~'^dx. 1 _L 4j;2 1 IRRATIONAL FRACTIONS. 18 S, JPvop, 1, — When a fraction contains none but monomial surds, it can be rationalized by substituting a new variable toith an expo- nent which is a common multiple of all the denominators of the fractional indices in the given expression. Dem. — The general form of such a fraction is m p ax" -\- hx'^ -\- etc. dx. r t a'x' -\- b'x'' -f- etc. m p r In this put X == z"9'*"» ®*'=- ; whence a;" = z'"^su, etc.^ ^q -_. 2«iwttj etc-, x' = i^vm, etc.^ jpit __ ^nqrt, etc.^ ^nd dx = (ngsM, etc.) z"?*"; etc. —^dz. These values substituted in the given fraction, evidently render it rational. 180, Cor. — This method is equally applicable when the fraction in- m volves no surd except one of the form (a + bx) °, 6y treating a + bx as the variable. 187* I*rop, 2,— When a fraction contains no surd but one of the form v/a -j- bx ± x% U can be rationalized by putting v/a+bx-j-x* = z— x 128 '. THE INTEGllAL CALCULUS. when x' is -f- ; and when x^ is — , s/n + bx — x^ == \/(x — ^){p — x) = (x — a)z, in which a and fi are the roots of the equation a + bx — x^ = 0. Dem. — 1st. When x^ is +. Putting >/a -{- bx -\- x' = z — x, a -{- hx -{- x"^ = ■2zx-\-X' ; whence x = 2-' — a z'^ -{- hz -\- a. z2 — 2zx-\-X' ; whence x = - — —7, dx = ! -^ dz, and Va A-bx-\- x^=^ 'Az-\-b {'Az -\- by- ' ' -. Hence as x, V'a -i- bx -{- x\ and djx are expressed in rational terms of z, the transformed fraction \viii be rational. . 2nd. When x- is — . Assuming \/a-]-bx — x^ = V {x — a)(/i — x) = {x— a)z, and squaring we have (x — «)(/? — x) = (x — ay^z% or /3— x= (x — a)z^ ; whence az-i -\- /5 . 2(a — /3)zdz ^ . — — 1^22 + ^ ) x=.^-^,dx==-^—,^nd ^a + bx-x^== ]_il_«^.= '' .^ _■ ■■ -. Hence as x, \/a-\-bx — x^, and dx are expressed in rational terms of z, the transformed fraction will be rational, q. b. d. i 2 Ex. 1. Integrate dy = dx. 5x^ A 8 Stjg's.— Put x==z^; whence d^/ = Y^' dz — ^^z^ dz. y = -^^x — |x' + C 1 Sx^dx Ex. 2. Integrate dy == — -' 2072 _ a;! {5. 2 1 • \ /pb 07^ ^X 1 1 if -g- + -2 + -3- + 4a;^ + 16^"^— 32 log (2 — o;*) [ + C. Ex. 3. Integrate dy = Sitg's.— Putting 1 -\- x = z^,dy = -^ — ^ _J ; whence 3/ == 2 tan-^l +" «) -f C. Ex. 4. Integrate dy = (1 + 4a7)^ ,=^r(iiif)!_3(i+4.)4 — 3 ^__i_^n^^_ ^ "^ (l + 4r)^ 3(1 + 40?)^-' Ex. 5. Integrate dy = — 2/ = log /—^ = h C: .rv/l 4-07 \/l + ^ + 1 SuG. — jEr's 4 and 5 can be performed hy {184) or {181). In fact these methods are essentially identical when there is but one surd of the form (a -}- ftx)". Ex. 6. Integrate dy = EATIONALIZATION. • 129 dx xVl -{- X -{- x^ Solution. — Put \/l -j- ic + .r- = z — x; whence z = x -\- \/l -|- cc -j- x\ X = -, dx = ■ — ---^ , and VI -\- x -{- x^ = —■ — — -^ — . Substitut- ing these values /dx /'2(z24-z + l) 2z + l 2z+l , /* 2dz log g ^ + C: 2 + a; + 2^1 + a; 4- ;»^ CfcJ7 Ex. 7. Integrate dy = V X- — X — 1 y = log [c{2x — 1 4- ^V'x-^ — x — l)]. __ ^ _ , , _ dx\/2x 4- ^2 Ex. 8. Integrate dy= — x^ Sug's. — ^Putting \/2aHh^ = z — ic, there results, in the nsual way dy = z2_|-4z+4 , z^-^z . 4(z-f l)dz (Zz , 4dz . , , -.n = — ■ • — dz = — ^^ — ■ = — -— A . .'. y = log (24-1) — - 4-C= log (cc 4- 1 4-v/2.'c 4- ^=^)- — r— =rr= 4- 0. 2 -^4- \/2^' 4- x^ dx Ex. 9. Integrate dy = \/ 2 — :;; — x"' Solution. — Put \/2 — x — x" = \/\x -j- 2)(1 — x) = (x 4- 2)z ; whence x = 1 — 2z'2 , 6zdz , , 3z r dx 2-2 + 1 (z24.l;2' z24_l ^ J ^2 — x—a;i -/r 4^ = — 2tan-iz4.(7=— 2tan-i ?_-^ 4- C. dx Ex. 10. Integrate dy V^l 4- a; — x^ Suo's. — From 1 -{- x — a;^ = 0, we learn that the factors are x — (^-.-f- i>/&) an4 (h — i\/5) — X. As these roots are so cumbrous it will be economy to take x — a and ^ — Of as the factors, as in the general demonstration. The differential in terms of « is dy == — :j-¥^. .-. y = — 2tan-i2 4-(7==2eot-V ii:-i^?^Z^4-C'. 130 THE INTEGRAL CALCULUS. SUCTION IV. Integration by Parts. 188. The Formula for Integration by !Part8 is Ju dv = uv — fv du. This formula is deduced directly from the differential of a product. Thus d{uv) = vdu -\- udv ; whence uv = jv du -f- ju dv, and ju dv = uv — fv du, FORMULA] OF REDUCTION. jfiL, 50, without affecting the other. INTEGRATION BY PARTS. 131 1.00. JProh, — To produce a formula for increasing the exponent of X without the parenthesis by the exponent of x within, in the form dy=:x'"(a+ bx")Mx; i. e. to make the integration of this form depend upon the form /x'"+''(a + bx°)Pdx. Solution. — Clearing (jA^) of fractions, transposing the first member and the last term of the second member, and dividing by a(m — n -f- 1)> we have j a;'^"(a 4- bx')Pdx = —^ J—^ LJT i — . Putting m — n = m', whence m = m' -\- n, this becomes r ,, ,, , , a;"»'+i(o4-6ar»y+i — i5>(np+m'+n-|-l) ra;"»'+"(a4-!/iC")PcZa: y = j x« '(a -f hT^ydx = ^^ — ■ ■■ —^-^ ■ ■ — -^ -— L i • aim -j- 1) or, dropping the accents, y=jx^{a-{-bx^)pdx= — — 7-.. ' ^-^- — —' (2S.) 191, J^rob. — To produce a formula for diminishing the exponent of the parenthesis by 1, in the form dy = x"'(a + bx")Pdx ; i. e. to make the integration — t xszt>\ ^bonce o ssr j?' 4. l, this becomes 132 THE INTEGEAL CALCULUS. * -^ ^ ' — ani^y -f- 1). or, dropping tlie accents, ^ -^ ^ ' '^ — a7i(p + 1) ScH. — Binomial differentials of this form, or such as may be readily re- duced to it, are of such frequent occurrence, and the formulae S^, H^, , called Formulce of Reduction, are so frequently efficient in reducing them to known forms, that these /ormwZce should be carefully memorized. Ex. 1. Integrate dy == j-. /x^dx f ~\ — ^ = J a;2(a2 — a;2) ^x, a form which corresponds to fx^{a -j- 6x'')Pda;, by considering m = 2, n = 2, a = a^, b = — 1, and p = — ^. We now observe that if the exponent of x outside the parenthesis, or in the nu- merator in the given form, were 0, so that x^ = 1, the integral would be a cir- cular function. Now formula {^) will so reduce this exponent ; hence we apply /'v"dx c ~ 5 — : ^ = J a;2(a2 — x'^) dx (a2 — a;2)^ a;2-2+i(a2 _ x^)~^'^^ — 052(2 — 2 + l\r.'3;2-2(a2 _ x2)~^da; -l[2(-^)-i-2-hl] i — 2 " ' 2 a;(a2 — £c2)2 a2 ^ -i =-^a;(a2 _ a;2)^ + / ^ •/ (a2 — jc2) i . o^ . ^ . « Ex. 2. Integrate c??/ = '- r Ex. 3. Integrate dy (1 _ a;2)2 3/ = — ia;2(l — 5r2)i _ 1(1 _ ^,)i _|_ a (2 + a;2)f y = a;«(2 + a;^) "^ + 4(2 + a?') ^ + C'= "^^ "^ ^ . + C. (2 4- ^'^r INTEGRATION BY PARTS. 133 Ex. 4. Integrate dy = '■ — ~. (1 — a;«)2 SuG. — Apply jfikj twice in succession, and we have /I ,1-4 , l-2.4\ , Ex. 5. Integrate dy = (1_^2)2 /I 15 135\ /- 1-3.5. , dx Ex. 6. Integrate z + C = /tan^xdx -y^pi:, = fzHz - fdz +fjj^^, itan'x — tanar-fx-f C. INTEGRATION BY PARTS. 14:X Ex. 18. integrate dy = tan^ xdx, y = ^iQji*x — ^tan2j7 + -J- log sec a; + (7. 197. JProb, — To integrate the forms dy = x"sinxdx, and dy = x" cos X dx. Solution. — To integrate dy = a^'sinxdx, put dv = Binxdx, whence u = x"*, u = — cosic, and du = nX^-'^dx. .-. y = — jc^cosa; -f wj^x" -' cos .r dr. By repeating the process the integration will finally depend upon the form ^ fcosxdx* or A' jsmxdx. In like manner y = fx"cosxdx = a;" sin a; — nf x''-^ Bin xdx, and the final forms are the same as before. Ex. 1. Integrate dy == x^ cos xdx. y = x^sinx -{- Sx^ cos x — 6^ sin ^ — 6 cos :r + C. Ex. 2. IntegTate dy == x'^ sin x dx. y == — ^^coso; + 4a;3sina7 + Vlx'^cosx — 24^ sin j; — 24cos^ + G. 198. Prop, — When m and n are integers, the form dy == sin"" X cos" X dx may be integrated in simple terms of the sines and cosines of the multiple arcs. DEM.~The form sin^a; cos"a; may be expressed in simple terms of multiple arcs by the use of the following formulae : (1)* sin £c sin?/ = \ cos {x — y) —^ cos {x -f y), (2) cos X cos y = ^ cos {x — y) -\-^ cos {x -\- y), (3) sin X cos y = i sin {x + 2/) + 2 sin {x — y). The truth of this statement and the manner of applying the formulae, may be seen most readily from the solution of a few examples. Ex. 1. Integrate in simple terms of the sines or cosines of multiple arcs dy = sin^ x cos^ x dx. Solution, sin^ x cos^ x = sin x (sin a; cos xY z= sin x[^ sin 2.'r]2 [From (3), making y = ;v. ] = \ sin ir[sin2 2x'] -^'r= L sinic[^ cos — ^ cos4£c] [From (1), making x =3 y = 2a;.] = ^ sin xW — |- cos 4a?] = \ sin X — \, sin x cos Ax = \ sin X — \{^ sin 5a: — |- sin 3a;] [From (3), making X = X, and y = dx.] = "li sin x — -i\- sin 5x -f- iV sin 3x. Hence y = J sin^ x cos2 xdx = i fsin xdx — J-jJ'sin 5xdx-\- -^gj^ sin Sx dx = ^ -^ eos.r — -4^8 cos 3x -f -g'o cos5x + C. * These formulas are essentially those of Art. S9, Plane Trigonometry. To put the formultt of that article into this form simply change * into ^{x -\-y) and y into j^{x — y). 142 THE INTEGRAL CALCULUS. Ex. 2. Integrate in simple terms of the sines or cosines of mnltiple arcs, dy = sin^ a;cos3 x dx. Sug's. siii3 X cos^ a; = s sin^ 2x = \ sin 2a; sin2 2a; = § sin 2a;(^ — ^ cos 4,x) = -jig sin 2a; — -j^^ sin 2a; cos 4a; = ^^g sin 2a; — -/g(^sin6a; — ^ sin 2a;) = i^-^ sin 2a; — ■^ sin 6a; + ^ sin 2a; = ^^ sin 2a! — -^^ sin 6a;. . • . y — f sin^ X cos^ xdx = y\ y sin 2a; d!a; •— ^*^ y sin 6a; .2/ ^1 — X2' 4-\)xdx4-l Jcos— 'a/ i -f- ioj'-^ -{- -^ fcos— ' a; d(cos— ' cc) = ^[a;(l — £k2)^] cos-1 .r — i(cos-ia;)2 + ix'^ + C. Ex. 3. Integrate dy = -^- —r-^dx, ^ ^ 1 + ^•'^ ?/ = tan~'^ (a: — i tan""^ x) — log Vl -\- x* + ^• 200* JPvob, — To integrate the forms dy = e"'' sin" x dx, and dy = e"'' cos" x dx. Method of Solution. — Put e«*da; = dv: whence sin" a: = w, v r= -e"*, and a dtt = n sin"— ^ X cos a; dr. 1 . n r . .' . 11 = —&"' sm" X I e"^ sm"-! x cos a; c7a:. Applying the formula for integration by parts again, putting dy == C'dr, u = sin"— 'a; cos X, v = — e"^, du = (n — 1) sin"— 2 .r cos^ a; dx — sin".rcZ;r, and a we have ' .• . y = j e"^ sin" x dx = — e"* sin" x — n \1 . , n — 1 r . ,.lr .,) - i -e"'" sin«— 1 x cos x e"=^ sm"— 2 cos2 xdx -i — I e«^ Bin"x dx Y . a i a a -^ a-^ ) Transposing the last term, uniting it with the first, and dividing by —^ — , we have <* . n . ^ , nin — 1) r ^ o > y = — ; — e^^sin^x ; — e^'^sm"— ^xcosx-J I e^^'sm"— -a;cos^ xdx. If now the last term be transposed and united with the first member and we divide by the coefficient, we shall have made the integration to depend upon a form in which n is diminished by 2. By successive applications of the same process, the integration may be made to depend upon the form Aje'^^dx when n is even, or A'fe'^^ sin a;dx when n is odd. The former is an elementary form ; and the formula lor integration by parts being applied to the latter, it becomes an integral without further process, since the co- efficient of the unintegrated term contains a factor n — 1, as will be seen above. By a process altogether similar the form d'?/ = e<^ cos^ x dx, can be integrated. Ex. 1. Integrate dij == e"" cos xdx. N Solution. — Put dv = C'^x : whence u = cos a?, v == -e"'', and du == — sinxdai a 144 THE INTEGRAL CALCULUS. y = je"' COS xdx = -&"= cos x -4 — fe<^ sin x dx *^ a a-' 1 , 1 • 1 r = -e"" ees x A e«^ sm a; I e<^ cos x dx. a a- a^^ .'. y = -——-e'^ cos X 4- — — -— — e*^ sin a; + C. (a cos a; -}- sin x) -^ C. a^ + 1 Ex. 2. Integrate dy = ef" sin^ a; c?a;. 2/ = tV^(s^^^ ^ + 3cos3 x 4- 3 sin x — 6 cos a;) -f C. Ex. 3. Integrate sin a; , /• da; — -\- ci t — (a 4" ^ cos .x/'+ ' / (^a 4- 6cosa;)"+' rh"' — a"' A- 'la'. a 4- ^cos.r'i — [a -\- 6cos.r)2* , 6 sin a* (n 4- 1) / ^ ^ ■ ■ dx = ; h ^ ' J (a 4- 6.cosx;"+'- (a 4- 6cosic/'+i /» /7t '^ dor /* d'V + (n + l)f ,f ^. ^ ^ ^ / (a4-6cosa;)» Transposing and uniting similar integrals, r dx bsinjc -I_1^ /* < ?g (n4-l)(&2 — a2) / ^^_^^cosa;)"+2 ~ (a4-6cosa;)«+i ~ ^V ^+ ^ (a4-6cosa;)«+» -+- n I ^. ' / (« 4- 6cosa;)» Dividing by (n 4- 1)(&- — «-). and writing 7i — 2 for n, we obtain * By adding and sxibtracting 2a2 -f 2a6 cos «, 62 — b2 cos2 1 = &2 -j- 2ai -f 2a&cos » — 2a2 — 2 lb C03 X — b? C082 2; = b2 — a2 -(- 2a(a 4- b cos t) — (a -i- b cos x}^. '^h dx INTEGRATION BY PARTS. 6 sin a; a(2n — 3) 145 dx -f-6cosx)» (w— 1)1,6- — a-)(a-{-6cosxj"-i (n — 1)(6"^ — a^)l (a+6cosa;)"-i n — 2 /• dx + t^)J (( (n — l){b'^ — a'^)J {a -}-b cos iCj"-=*' By means of repeated applications of this formula, or the process by which it was produced, the integration may be made to depend upon the form 1 a -\-o cos X (It X x To integrate dv == "■ , we remember that cos x = cos^ - — sin=^ -, and a -)- 6 cos X 4 2 cos2 - -j- sin2 jr = 1. Hence .=/: dx a -{- b cos X A dx cos2 - -\- sin2 i) + < / dx X {a -{- b) cos2 5 + (a — &) sins - = /^ sec2 - dx «y . a 4- 6 (a — i»^ («-&) -d(tan|) X a -}- X . ,x cos2 - — sin2 - dx cos2 (« + &) + (« — ^) COS2- 2ci!(^tan | j X + -^tan.| «.-.,* tan , a — b ^ „x a -\- b a -{- b 2 -, J i^LZ^^^tan^ j. + C, when a>6. y When a <1 &, we have dec / a+6 cosa; / , sec2 da; flj (&+a) — (& — a)tan2- ib 4- a)' i(tan|) (6+a)^— (& — «rtan| /. V (£ 2cZi (tan I) (6-i-a) — (5 — a)tan2- _l-^(tan|) (6 + af (64-ar+(& — a)Han- (6— ard^tan|j (52_a2f/ (&-fa)^ — (6 — a)^tan^ {b'^—a^? I (& + «)^ + (5 — a)* tan ^ (& — a)^dAan^j T— ^ + (1*2 — a2) log 1 J £C (6+a)^ -f- (6 — a)- tan - ' + a ■146 THE INTEGEAL CALCULUS. ScH. — The four preceding sections comprise the greater part of what is known concerning abstract methods of passing from the differential to the exact integral function of a single variable ; but there are many other methods of great practical importance for determining the approximace value of the integral of a differential which cannot be integrated by tne^e methods. One of the most useful and simple is given in the next section. ■♦4» SUCTION V, Integration by Infinite Series. 202, It often occurs that a differential can be expanded into an infinite series, the terms of which can be integrated separately If the result is a conyerging series, the value of the integral may be de- termined with sufficient accuracy for practical purposes by summing a finite number of terms. It may also sometimes happen that the law of the series is such that its exact sum may be found, although the series itself is infinite. This method is not only a last resort when the methods ol exact integration fail, but it is sometimes serviceable' by being more smiplt than they, even M^hen they are applicable ; moreover, it affords a method of developing such a function. 11 Ex. 1. Integi-ate dy = a;^{l — x^)^dx. Solution. — Expanding (1 — a;^)^ by the binomial or Maclaurin's theorein, w:^ have (1 _ x^f = 1 _ ia;2 _ ia;4 _ ^^^s _ _A_a.8 _ etc. ^ \ 1 pi p 5. p a. r ^?- r ■^^' . •. ?/ = j x^{l—x'^ydx=J x'dx—^J X dx—^J x^dx--^j x^ dx—^j x''- ax— etc 3. 2. JLL 15. ill = fcc^ — V — ^4^ — Tioae ^ — T2^«« — etc. + G. This series is converging for a; or d^y = Cydx^. From this we obtain y = IC^x^ -{- C^x -\- C3. Ex. 4. Given -v— == — to find the inteeral y, dx^ x^ ^ ^ y = log^ + ^C^x^ + C^x -}- G^, Ex. 5. Integrate d^y = sin x cos* x dx^. Solution. — Putsina; = v; whence dv = cosxdx, cos^xdx'^ = dv% and d^y =' sin X cos2 x dx^ = v dv^. From d-y = v dv-, we obtain as before y = ^v^ + 6'i i? -j- Cj . . • . 2/ = i sin3 a; -j- ^1 sin x -\- Cg. Ex. 6. Integrate d^y == cos x sin^ x dxK y = ^ COS3 X -}- C^ cos X + Cg. ScH. 2. — In order to integrate successively d"y =/{x)dx'', according to the foregoing process, it is necessary that we be able to integTate exactly /(.r)r/r, fi[x)dx, f^{x)dx, etc. It is evident that this will be frequently impossible. A method of approximation which is often serviceable in such cases, is readily obtained by means of Maclaurin's Formula. * We might write GaCi as the coefficient of this term ; but as Cl represents any constant, it is unnecessary to retain the 6a. t This is simply a convenient form in which to write the operation ; the student should under- stand the process and be able to explain it, as in the preceding solution. SUCCESSIVE INTEGRATION. 149 204, JProp. — The nth integral of f(x)dx" may be developed into a series by the following formula : ^ ^x";;;^^ , ^^^^^^ x" ^ fdf(x)- -"+^ l-2.3---(n-l)+'^('')]l-2-3---n + L dx Jl • 2 - 3 - - - (n + 1) rd2f(x) -i x"+- rd3f(x)-| x"+' "^L dx-^ Jl.2-3---(n+2)"^L~dx3 Jl-2-3---(n + 3)+'^*'^' Dem. —1st. Developing y = fy{x)dx" by Maclaurin's formula, we have y = fy{x)dx^ =. ify{x)dx'^^ + [/""y(;r)dT«-^]*| + [/"■y(x)dx»-^]^l. •3---(n — 1)' 2nd. Comparing this development with the development of fy{x)dx'' as made in the solution Akt. 203, remembering that .'C = in the bracketed factors of the present series, and that these factors are therefore constant, we see that the present series is that of Abt. 203 reversed, extended and generalized ; that [fyix)dx"] is Cn, [j "~y(a5)dx"— 1] is Cn—i, etc., and that the former series begins with the term [/(cc)]r— •.—,•; of the latter. Hence using C,„ (7„_i, etc. for these constant i-A-o EF. — A Definite Integral is an integral taken be- tween limits. A definite integral and the limits between which it is taken is sym- bolized thus : y = I f{x)dx. This signifies that the indefinite inte- gral ot f{x)dx is to be obtained, and first satisfied by substituting h for X, then by substituting a for x ; and that, finally, the latter is to be subtracted from the former, x = a and x ==h are called the limits of the integral, the former being called the inferior and the latter the superior limit.* Ex. 1. Find the value oi y == I 5x^dx. Sug's. — The indefinite integral is y' = ^x^ -f- ^• Now this is true for all values of x, hence for x = 3. For this value of x, we have * It is assumed tliat x can liave such values as we assign it, and that the function is continuous between these values, i. e. that it does not become imaginary or infinite for any intermediate valua of X. DEFINITE INTEGKATION AND THE CONSTANTS OF INTEGRATION. 151 y" =45 -\- C. In like manner for a; == 1 we have y'= f -h ^- Subtracting the latter from the former, y" — y'" = 43:j. . •. y = y" — y'" = j 5x^dx = 431. Ex. 2. Find the definite integral of dy==nxdx between the limits a and 6. y := j nxdx=— — -. Ex. 3. Find the value oi y = j {x^dx — h'^x dx). An&., — \h*. 209, Disposing of the Constant of Integration, There are two principal methods of disposing of the constant of in- tegration, G : 1st. By integrating between limits, the constant is eliminated. This IS illustrated in the preceding examples. 2nd. "When there is anything in the nature of the problem under discussion, from which we can know the yalue of the function for some particular value of the variable, by substituting hese values in the indefinite integral, the value of the constant G jan be found. And. a& 6" is a constant, if we find its value for any particular value ot the variable, we have it for all values of the variable. Ex, 1. Find the corrected integral of the function dz = {dx^ + ^ax dx^Y, on the hypothesis that 2 = w^hen x = Q. 8 ^ SuG's.— The indefinite integral is s = x^ (1 + %axf + C. Now if s = when 8 8 a; = 0, we have O = ^^^r- + 0. ,'. G = — ^=— Substituting this value of G in 8 ^8 tie indefinite integral, we have, as the corrected integral, z = h;=-(1 -{- %axY — x=-. 27rv ^ Ex 2 What is the value of (7, when du = — ^(^2 _j_ y^ydyyii u = ^ when y = ? Ans., (7 == - — ^Ttp^. Ex. 3 Given du = (2'/')2(2r — • y)~^dy, what is the value of 0, if zi -= C when y =zO? What if w = when y = 2r7 Answers, Ar^ 0. Iiiii. — A differential is one of the infinitesimal elements of which a quantity is conceived as composed. Thus, let A represent the area of the surface lying be- tween AM and AX, Fig. 35; PD being any ordinate, and a6 the consecutive 152 THE INTEGRAL CALCULUS. §(2p)y + a ordinate, PDdb may be considered as representing an element of the area, or dA. This element in the case of the parabola is found to be \/2pa; dx ; hence dA = \/2px dx. This expression therefore represents any one of the infinitesimal elements of which the quantity A is composed. The Indefinite Integral is A This is indefinite in two respects. 1st, it is true for any value of X ; 2nd, it is indefinite, and in fact indetermi- nate, as regards C, the value of which may be anything. As KEGABDs THE CONSTANT, if wc choose to estimate the area from A, so that A = when x = 0, we have = 0-{- C. . • . C= 0. Hence the corrected integral 1 3. is ^ = |(2p)^x*. This represents the area, estimated from A to any value we may choose to give x. If x= AZ^', A = area A P' D'. Again, if we choose to estimate the area from the focal ordinate H F, calling A = 0, when x = ^p, we have |(2p)^(|2>)^ -\- C, or ip^ -^ C. . • . C = — ip"^, and the corrected integral is Fig. 35. A = i{2p)^x^ — ap-' This represents the area estimated from the focal ordinate H F to any value we choose to give x. Thus if .r = A D ' , this corrected integral represents the area H FP' D'. Finally, suppose we ask. Where must the area be conceived as beginning in order that G = — m ? To meet this case we have i. 5. 0= |(2p)^a;^ — m, whence x -