CD a Cvl o Irving Stringham -.^.^ ^ ijj^^asrsssraaasasasai A TREATISE ANALYTIC GEOMETRY, ESPECIALLY AS APPLIED TO THE PROPERTIES OF CONICS: INCLUDING THE MODERN METHODS OF ABRIDGED NOTATION. WRITTEN FOR THE MATHEMATICAL COURSE OF JOSEPH KAY, M.D., BY GEORGE H. HOWISON, M.A., PROFESSOR IN WASHINGTON UNIVERSITY. CINCINNATI: WILSON, HIISTKLE read sin w. " 103, " 21: for -f read = .' " 103, " 22 : dele the period at the close. " 221, 24: fork"=A:8inCTe&dk"=8inA:sinC. " 247, " 9 : put the 2 outside of the brace. " 269, " 28 : for of read to, and dele the period. CONTENTS. INTRODUCTION: THE NATURE, DIVISIONS, AND METHOD OF THE SCIENCE. PAGE. I. DETERMINATE GEOMETRY : Principles of Notation, 5 Examples, . . . . H 8 Principles of Construction, ....... 8 Examples, . . . . . . . . . .16 DETERMINATE PROBLEMS : In a given triangle, to inscribe a square, . . . .16 " " " " a rectangle with sides in given ratio, 17 To construct a common tangent to two given circles, . . 18 a rectangle, given area and difference of sides, . 21 Examples, . . . . . . . . .23 II. INDETERMINATE GEOMETRY : 1. Development of its Fundamental Principle: The Convention of Co-ordinates, .... 26 Distinction between Variables and Constants; defi- nition of a Function, ..... 28 Equations between co-ordinates : their geometric meaning, 29 The Locus denned and illustrated, . . . .33 11. Its Method outlined; in what sense it is Analytic: Manner of employing geometric equations to estab- lish properties, . " . . . . .35 Special analytic character of the Algebraic Calculus, 38 Elements of analysis added by the Convention of Co- ordinates, 39 III . Its Divisions and Subdivisions: Algebraic and Transcendental Geometry, ... 41 Orders of algebraic loci : Elementary and Higher Geometry, ,.>... . . . 42 Loci in a Plane and in Space : Geometry of Two and of Three Dimensions, 42 (v) vi CONTENTS. BOOK FIEST: PLANE CO-OEDINATES. PART I. ON THE REPRESENTATION OF FORM BY ANALYTIC SYMBOLS. CHAPTER FIEST. THE OLDER GEOMETRY : BILINEAR AND POLAR CO-ORDINATES. SECTION I. THE POINT. PAGE BILINEAR OR CARTESIAN SYSTEM OP CO-ORDINATES : Explanation in detail, 48 Expressions for Point on either Axis; for the Origin,. . 50 POLAR SYSTEM OP CO-ORDINATES :....... 52 Expression for the Pole; for Point on Initial Line, . . 53 Distance, in both systems, between any Two Points in a plane, . 55 Co-ordinates of Point cutting this distance in a given ratio, . . 56 TRANSFORMATION OF CO-ORDINATES : I. To change the Origin, Axes remaining parallel to their first position, 59 II. To change the Inclination of the Axes, Origin remaining the same, 59 Particular Cases: 1. From Rectangular Axes to Oblique, ... 60 2. From Oblique to Rectangular, 61 3. From Rectangular to Rect- angular, .... 61 III. To change System from Bilinears to Polars, and con- versely, 62 IV. To change the Origin, and make either previous Trans- formation at the same time, 63 GENERAL PRINCIPLES OF INTERPRETATION : I. Any single equation between co-ordinates represents a Locus, ......... 65 II. Any two simultaneous equations represent Determinate Points, 67 III. Any equation lacking absolute term, represents Locus passing through Origin, ..... 69 IV. Transformation of Co-ordinates does not affect Locus, nor change the Degree of its Equation, ... 69 SPECIAL INTERPRETATION OF EQUATIONS : Tracing their Loci by means of Points, 71 Definitions and illustrations, . . . . . . .72 Examples: Equations to some of the Higher Plane Curves, 75 CONTENTS. vii SECTION II. THE RIGHT LINE. A. THE BIGHT LINE UNDER GENERAL, CONDITIONS. I. Geometric Point of View : Equation to Right Line is always of the First Degree. PACK. Equation in terms of angle made by Line with axis X, and of its intercept on axis Y, 79 " " its intercepts on the two axes, ... 81 " " its perpendicular from the Origin, and angle of perp'r with axis X, 83 Polar equation, deduced geometrically, 84 II. Analytic Point of View : Every equation of First Degree in two variables represents a Right Line. Proof of the theorem by Algebraic Transformation of the general equation of First Degree, 87 Proof by means of the Trigonometric Function implied in the equation, 87 Proof by Transformation of Co-ordinates, ..... 89 Analytic deduction of the Three Forms of the equation, . . 92 Reduction of Ax + By + C= to the form x cos a + y cos /? /> = 0, . 95 Polar Equation obtained by Transformation of Co-ordinates, . . 97 B. THE RIGHT LINE UNDER SPECIAL CONDITIONS. Equation- to Eight Line passing through Two Fixed Points, . 98 Angle between two Right Lines : condition that they shall be parallel or perpendicular, 100 Equation to- Right Line parallel to given Line; perpendicular to given Line, 101 Equation to Right Line passing through given Point, and parallel to given Line, .......... 103 Equation to any Right Line through a Fixed Point, . . 103 Equation to Right Line through a given Point, and cutting a given line at given angle, 105 Equation to Perpendicular through a given Point, . . . 106 Length of Perpendicular from (x,y) on x cos a 4-ycos p = ; also on Ax + By + C= 0, 107 Equation to any Right Line through the intersection of two given ones, 109 Meaning of equation L + kL' = 0, 110 Equation to Bisector of angle between any two Right Lines, . . 113 Equation to the Right Line situated at Infinity, .... 116 Equations of Condition : Condition that Three Points shall lie on one Right Line, . . 117 " " Three Right Lines shall meet in One Point, . 118 " " Movable Right Line shall pass through a Fixed Point, 118 Vlll CONTENTS. C. EXAMPLES ON THE RIGHT LINE. Examples in Notation and Conditions, 120 Examples of Kectilinear Loci, 125 SECTION III. PAIRS OF EIGHT LINES. I. Geometric Point of View: Equation to a Pair of Right Lines is always of Second Degree. Formation of equations in the type of LMN . . . . = : their con- sequent meaning, ........ 130 Interpretation of equation LL'= 0, ...... 130 Equation to Pair of Eight Lines passing through a Fixed Point, . 131 Meaning of the equation Ax z + 2Hxy + By 2 =0, . . .132 Angle between the Pair Ax* + 2Hxy + Bif = 0, . . . -133 Condition that they shall cut at right angles, .... 134 Equation to Bisectors of angles between Ax" + 2Hxy + By 2 ^ 0, . 134 Case of Two Imaginary Lines having Real Bisectors of their angles, 134 II. Analytic Point of View: The Equation of Second Degree in two variables, upon a Determinate Condition, represents Two Right Lines. Proof of the theorem by the mode of forming LL' = 0, . . . 135 Condition on which Ax 1 + 2Hxy + By 1 - + 2Gx + 2Fy + C'= repre- sents Two Right Lines, 136 SECTION IV. THE CIRCLE. I. Geometric Point of View : Equation to Circle is always of Sec- ond Degree. Equation to the Circle, referred to any Rectangular Axes, deduced from geometric definition, . . 138 " " " " Oblique Axes, . . . .139 " " " " Rectangular Axes with Origin at Center, 139 " " " " Diameter and Tangent at its extremity, .... 140 Polar Equation to the Circle, 140 II. Analytic Point of View : The Equation of Second Degree in two variables, upon a Determinate Condition, represents a Circle. Proof of theorem by comparison of the General Equation with that to Circle, 142 CONTENTS. ix PAGE, Condition that Ax 2 + IHxj + /?/ + 2Gx + 2Fy + C'= Q shall rep- resent a Circle, 143 To determine Magnitude and Position of Circle, given its equa- tion, 144 Condition that a Circle shall touch the Axes, . . . . / . 145 Examples, 146 SECTION V. THE ELLIPSE. I. Geometric Point of View: Equation to Ellipse is always of Sec- ond Degree. Equation to the Ellipse deduced from geometric definitions, . . 149 Its general Form, referred to Axes of Curve and Focal Center, . 150 Center of a Curve defined : proof that Focal Center is center of the Ellipse, 151 Polar Equation to Ellipse, Center being Pole, 151 " " " Focus " " 153 II. Analytic Point of View: The Equation of Second Degree in two variables, upon a Determinate Condition, represents an Ellipse. Reduction of Ax 2 + 2Hxy + Bif + 2Gx + 2Fy+ C= to Center of its Locus, ........... 155 Condition that it represent an Ellipse is H z AB<0, . . .159 The Point, as intersection of Two Imaginary Right Lines, a partic- ular case of the Ellipse, 161 Examples, 165 SECTION VI. THE HYPERBOLA. I. Geometric Point of View: Equation to Hyperbola is always of Second Degree. Equation to the Hyperbola deduced from geometric definitions, . 169 Its general Form, referred to the Axes and Focal Center, . 170 Proof that the Focal Center is the center of the Hyperbola, . 172 Polar Equation to Hyperbola. Center being Pole, .... 172 " " " " Focus " " .... 174 II. Analytic Point of View : The Equation of Second Degree in two variables, upon a Determinate Condition, represents an Hyperbola. Equation to Hyperbola compared with Reduced Equation of Second Degree, . 175 Condition that the latter shall represent an Hyperbola is H*-AB>Q, 176 Two Right Lines intersecting, a particular case of the Hyperbola, . 177 x CONTENTS. PAGE. Examples on the Hyperbola, 178 SECTION VII. THE PARABOLA. I. Geometric Point of View: Equation to Parabola is always of Second Degree. Equation to Parabola deduced from geometric definitions, . . 181 Its general Form, referred to Axis and Directrix, . . . 182 Polar Equation to Parabola, Focus being Pole, .... 183 II. Analytic Point of View: The Equation of Second Degree in two variables, upon a Determinate Condition, represents a Parabola. Additional transformation of General Equation, under condition of non-centrality, 185 Condition that it represent a Parabola is H 2 AS= 0, . . . 188 The Right Line as Center of Two Parallels, . . . .189 " " " as Limit of Two Parallels, a particular case of the Parabola, 190 Examples, 191 SECTION VIII. Locus OF SECOND ORDER IN GENERAL. Summary of Conditions already imposed upon the General Equa- tion, 195 Proof that these exhaust the varieties of its Locus, . . . 196 Conies defined : Classification into Three Species, .... 197 Re'sume of argument for the theorem : Every Equation of the Second Degree in two variables, represents a Conic, .... 197 CHAPTER SECOND. THE MODERN OEOMETRY:-TRILINEAR AND TANGENTIAL CO-ORDINATES. SECTION I. TRILINEAR CO-ORDINATES. Trilinear Method of representing a Point, 199 Origin of the Method: the Abridged Notation, .... 200 Geometric meaning of the constant k in a + &/?= 0, . . . . 201 Interpretation of the equations a &/? = 0, a = 0, . . . 201 The Notation extended to equations in the form Ax + By + C= 0, . 202 Meaning of the equation la -t- mft +ny = : condition that it repre- sent any Right Line, ........ 203 Examples : Any line of Quadrilateral, in terms of any Three, . 206 CONTENTS. xi PAGE. The symbols a, /?, y may be considered as Co-ordinates, . . . 207 Peculiar Nature of Trilinear Co-ordinates : each a Determined Func- tion of the other two, 208 Equation expressing this Condition is oa + b/3 + cy = M, . . 209 General trilinear symbol for a Constant ; namely, 7c (a sin A + ~ /I sin B + -y sin C], 210 To render homogeneous any given equation in Trilinears, . . 210 Trilinear Equation to Right Line, 212 " " " " parallel to a given one, . . 212 " " " " situated at infinity, . . .212 Condition, in Trilinears, that two Right Lines shall be at right angles to each other, 213 Trilinear Equation to Right Line joining Two Fixed Points, . . 214 " " any Conic, referred to Inscribed Triangle, . 215 Circle, " " " . 216 Same for any Concentric Circle, . . 216 " " " Triangle of Reference having any sit- uation, 217 General Equation of Second Degree in Trilinears : *. e., Trilinear Equation to any Conic, Triangle of Reference having any sit- uation, ........... 218 Trilinear Equations to Chord and Tangent of any Conic, . . . 219 Examples of Trilinear Notation and Conditions, . . . 220 SECTION II. TANGENTIAL CO-ORDINATES. In Tangential system, Lines are represented by co-ordinates, and Points by equations, ........ 225 Cartesian Co-efficients are Tangential Co-ordinates : Tangential Equa- tions are Cartesian Equations of Condition namely, that a Line shall pass through two Consecutive Points on a given Curve, 225 Geometric interpretation of Tangentials : how they represent a LOCKS. Reason for Name, . . . 226 The Right Line in the Tangential System, ..... 227 ENVELOPES denned : Condition that a Right Line shall touch a Curve is the Tangential equation to the Curve ; or simply the Equa- tion to the Envelope of the Line, ...... 228 Development of the Tangential Equation to a Conic, referred to In- scribed Triangle, 228 Reciprocal relation between Points and Lines: the Principle of Duality, 235 Description of the Method of Reciprocal Polars : its relation to the Modern Geometry, ......... 238 Examples illustrating Tangentials, 242 Xll CONTENTS. PART II. ON THE PROPERTIES OF CONICS. CHAPTER FIKST. THE RIGHT LINE. PAQE Area of a Triangle in terms of the Co-ordinates of its vertices, . 246 " " " given the Right Lines which inclose it, . . 246 Compound ratio of segments of the Three Sides by any Trans- versal = - 1, .......... 248 Compound ratio of segments of the Three Sides by any three Con- vergents = +1, . . . . . . . . . 248 Various cases of Three Convergents occurring in any Triangle, solved by the Abridged Notation, ...... 249 Further application of Trilinears : The property of Homology ; Axis and Center of Homology, ...... 250 Quadrilaterals when Complete: Centers of their three Diagonals lie on one Right Line, ........ 251 Harmonic and Anharmonic Properties :...... 253 Constant ratio among Segments of Transversals to any Linear Pencil, .......... 253 Harmonic and Anharmonic Pencils, ...... 254 Anharmonic of a, (3, a + kfl, a + k'{3 = ^ , ..... 255 a, /?, a + &/?, a fc/?, form a Harmonic Pencil, .... 255 Anharmonio of any Pencil a + k/3, a + 10, a + mfi, a + n/3 (*-)(*-*)' ' Definition of Homographic Systems of lines, . . . . 257 Examples involving properties of the Right Line : Triangles, ........... 257 Harmonics of a Complete Quadrilateral, ..... 258 CHAPTER SECOND. THE CIRCLE. I. THE Axis OP X: Every Ordinate a mean proportional between the correspond- ing segments, 259 Every Right Line meets the Curve in Two Points, . . 260 Discrimination between Real, Coincident, and Imagi- nary points, 260 Chords defined. Equation to any Chord, . . . .262 II. DIAMETERS : Definition : Locus of middle points of Parallel Chords. Equation, 263 CONTENTS. xiii PAGE. Every diameter passes through Center, and is perpendicular to bisected Chords, ....... 264 CONJUGATE DIAMETERS defined each bisects Chords parallel to the other, 264 Conjugates of the Circle are at right angles, . . . 265 III. TANGENT : Definition : Chord meeting Curve in Two Coincident Points, 265 Equation, 266 Condition that a Right Line shall touch Circle. Auxiliary Angle, 267 Analytic Construction of Tangent through (x',y'): Two Tan- gents, real, coincident, or imaginary, .... 268 Length of Tangent from (x, y) = !/", 269 Subtangent its definition and value, .... 270, 271 IV. NORMAL: Definition of Normal. Equation, ...... 270 The Normal to Circle passes through Center. Length con- stant, 271 Subnormal its definition and value, 271 V. SUPPLEMENTAL CHORDS : Definition : Equation of Condition, .... 271, 272 In the Circle, they are always at right angles, . , . 272 VI. POLE AND POLAR: Development of the conception of the Polar, .... 273 I. Chord of Contact to Tangents from (x',y'), . . 273 II. Locus of- intersection of Tangents at extremities of convergent chords, ...... 273 III. Tangent brought under this conception, . . . 274 Construction of Polar from its definition, .... 276 Polar is perpendicular to Diameter through Pole: its dis- tance from Center, 277 Simplified geometric construction, ..... 277 Distances of any two points from Center are proportional to distance of each from Polar of the other, . . . 278 Conjugate and Self-conjugate Triangles defined: they are homologous, 278, 279 SYSTEMS OF CIRCLES. I. SYSTEM WITH COMMON RADICAL Axis : Radical Axis defined : its Equation, S S' = 0, . . .280 It is perpendicular to the line of the centers, . . . 281 Construction : Combine # '=- with y= and observe the foregoing, 281 The three Radical Axes belonging to any three Circles meet in one point: Radical Center, 281 xiv CONTENTS. PAGE. To construct the Radical Axis by means of the Radical Center, 281 Radical Axis of Point and Circle; of Two Points, . . 282 Definition of System of Circles with Common Radical Axis ; Their Centers lie on one Right Line. Their Equa- tion : x 2 + if 2kx ff = 0, 282 To trace the System from the equation, 283 Locus of Contact of Tangents from any point in the C. R. A. is Orthogonal Circle, 283 Geometric construction of the System : Limiting Points, 284 Analytic proof of the existence of the Limiting Points, . 285 II. Two CIRCLES WITH COMMON TANGENT : To determine the Chords of Contact, 287 The Tangents intersect on Line of Centers, and cut it in ratio of Radii, . . . " 289 Every Right Line through these Points of Section is cut sim- ilarly by the two Circles 289 The Centers of Similitude, 289 The three homologous Centers of Similitude belonging to any three Circles, lie on one Right Line. The Axis of Similitude, 290 THE CIRCLE IN THE ABRIDGED NOTATION. If a Triangle be inscribed in a Circle, and Perpendiculars be dropped from any point in the Circle upon the three sides, their feet will lie on one Right Line, 291 Angle between Tangent and Chord = angle inscribed under corre- sponding arc, 292 Trilinear Equation to the Tangent, referred to Inscribed Triangle, . 292 Tangents at Vertices of Inscribed Triangle cut Opposite Sides in points lying on one Right Line, ..... 292 Linea joining Vertices of Inscribed Triangle to those of Tri- angle formed by Tangents meet in one Point, . . 292 Radical Axis in Trilinears, 293 Examples on the Circle, 293 CHAPTER THIRD. THE ELLIPSE. I. THE CURVE REFERRED TO ITS AXES. THE AXES. THEOREM I. Focal Center bisects the Axes. Corresponding inter- X 2 II 2 pretation of -f z = 1, 297 CONTENTS. xv PAGE. THEOREM II. Foci fall within the Curve, 298 THEOREM III. Vertices of Curve equidistant from Foci, . . 298 THEOREM IV. Sum of Focal Radii = length of Transverse Axis. To construct Curve by Points, ..... 298 THEOREM V. Semi-conjugate Axis a geometric mean between Focal Segments of Transverse, .... 299 Cor. Distance from Focus to Vertex of Conjugate = Semi-transverse. To construct Foci, . . . 299 THEOREM VI. Squares of Ordinates to Axes are proportional to Rectangles of corresponding Segments, . . 300 Cor. The Latus Rectum, and its value, . . . 300 THEOREM VII. Squares of Axes are to each other as Rectangle of any two segments to Square of Ordinate, . . 301 THEOREM VIII. Ordinate of Ellipse : Corresponding Ordinate of Circumscribed Circle : : Semi-conjugate : Semi- transverse, ........ 301 Cor. I. Analogous relation to Inscribed Circle. Con- structions for the Curve, 302 a 2 I 2 Cor. 2. Interpretation of = e z : e defined as the Eccentricity, 303 THEOREM IX. The Focal Radius of the Curve is a Linear Function of corresponding Abscissa, ..... 304 LINEAR EQUATION to Ellipse : p = a ex, . .304 Verification of the Figure of the Curve by means of its equation, . 304 DIAMETERS. Diameters : Equation to Locus of middle points of Parallel Chords, 305 THEOREM X. Every Diameter is a Right Line passing through the Center, 306 Cor. Every Right Line through Center is a Diam- eter, 306 THEOREM XI. Every Diameter of an Ellipse cuts Curve in Two Real Points, .... . 306 Length of Diameter, in the Ellipse, 306 THEOREM XII. Transverse Axis the maximum, and Conjugate the minimum Diameter, ...... 306 THEOREM XIII. Diameters making supplemental angles with Axis Major are equal, ....... 307 Cor. Given the Curve, to construct the Axes, . . 307 THEOREM XIV. If a Diameter bisects Chords parallel to a second, second bisects Chords parallel to first, . . 307 CONJUGATE DIAMETERS defined : Ordinates to any Diameter, . . 308 To construct a pair of Conjugates, I 308 xvi CONTENTS. PACK. Equation of Condition to Conjugates, in the Ellipse, is tan 0. tan Q' = , . . . . . 308 or THEOREM XV. Conjugates in the Ellipse lie on opposite sides of Axis Minor, ....... 308 Equation to Diameter Conjugate to that through any given point, . 309 Cor. The Axes are a case of Conjugates, . . 309 (Jiven the co-ordinates to extremity of any Diameter, to find those to extremity of its Conjugate, ...... 309 THEOREM XVI. Abscissa to extremity of Diameter : Ordinate to extremity of its Conjugate : : Axis Major : Axis Minor, 310 THEOREM XVII. Sum of squares on Ordinates to extremities of Conjugates, constant and = 6 2 , . . 310 Length of any Diameter in terms of Abscissa to extremity of its Conjugate, 310 THEOREM XVIII. Square on any Semi-diameter = Rectangle Focal Radii to extremity of its Conjugate, . . 311 THEOREM XIX. Distance, measured on a Focal Chord, from ex- tremity of any Diameter to its Conjugate, is constant, and equals the Semi-Major, . . 311 THEOREM XX. Sum of squares on any two Conjugates is constant and = sum squares on Axes, . . . 312 Angle between any two Conjugates, . . . 312 THEOREM XXI. Parallelogram under any two Conjugates, constant and = Rectangular under Axes, . . . 313 Cor. 1. Curve has but one set of Rectangular Con- jugates, 313 Cor. 2. Inclination of Conjugates is maximum when a' = V, 314 THEOREM XXII. Equi-conjugates : they are the Diagonals of Cir- cumscribed Rectangle, 314 Cor. Curve has but one pair of Equi-conju- gates, 314 Anticipation of the Asymptotes in the Hyperbola, .... 315 THE TANGENT. Equation to the Tangent, referred to the Axes, .... 315 Condition that a Right Line touch an Ellipse. Eccentric Angle, . 316 Analytic construction of the Tangent through any point : Two, real, coincident, or imaginary, ...... 317 THEOREM XXIII. Tangent at extremity of any Diameter is parallel to its Conjugate, ...... 318 Cor. Tangents at extremities of any Diameter are parallel. Circumscribed Parallelogram, . 318 CONTENTS. xvii PAGE. THEOREM XXIV. Tangent bisects the External Angle between Focal Radii of Contact, . . . .319 Cor. 1. To construct a Tangent to the Ellipse at a given point, 319 Cor. 2. Derivation of the term Focus, . . 319 Intercept by Tangent on Axis Major: Constructions for Tangent by means of it, 320 The SUBTANGENT defined : Distinction between Subtangent of the Curve and of a Diameter, ....... 320 THEOREM XXV. Subtangent of Curve is Fourth Proportional to Abscissa of Contact and the corresponding segments of Axis Major, .... 321 Cor. Construction of Tangent by means of Cir- cumscr. Circle : Subtan. not function of &, . 321 THEOREM XXVI. Perpendicular from Center on Tangent is Fourth Proportional to the corresponding Semi- conjugate and the Semi-axes : p = - , . 322 Length of the Central Perp'r in terms of its angles with the Axes, . 322 THEOREM XXVII. Locus of Intersection of Tangents cutting at right angles is Concentric Circle, . . 323 Focal Perpendiculars upon Tangent: their length, .... 323 THEOREM XXVIII. Focal Perpendiculars on Tangent are propor- tional to adjacent Focal Radii, . . . 324 THEOREM XXIX. Rectangle under Focal Perpendiculars, constant and - 6 2 , 324 THEOREM XXX. Locus of foot of Focal Perpendicular is the Cir- cumscribed Circle, 324 Cor. Method of drawing the Tangent, common to all Conies, 325 THEOREM XXXI. If from any Point within a circle a Chord be drawn, and a perpendicular to it at the point of section, the Perpendicular is Tan- gent to an Ellipse, . . . . . 326 Cor. The Ellipse as Envelope, . . .326 THEOREM XXXII. Diameters through feet of Focal Perpendiculars, parallel to Focal Radii of Contact, . . 327 Cor. Diameters parallel to Focal Radii of Con- tact, meet Tangent at the feet of the Focal Perpendiculars, and are of the constant length = 2n, 327 THE NORMAL. Equation to the Normal, referred to the Axes, 328 THEOREM XXXIII. Normal bisects the Internal Angle between Focal Radii of Contact, 328 An. Ge. 2. xviil CONTENTS. PACK. Cor. 1. To construct a Normal at given point on the Ellipse, 329 Cor. 2. To construct a Normal through any point on Axis Minor, .... 329 Intercept of Normal on Axis Major : Constructions by means of it, 329 THEOREM XXXIV. Normal cuts distance between Foci in segments proportional to adjacent Focal Radii, . 330 The SUBNORMAL defined : Subnormal of the Curve its length, . 330 THEOREM XXXV. Normal cuts its Abscissa in constant ratio = a 2 -b 2 IT-' 33 Length of Normal from Point of Contact to either Axis, . . 330 THEOREM XXXVI. Rectangle under Segments of Normal by Axes = Square on Conjugate Semi-diameter, . 331 Cor. Equal, also, to Rectangle under corre- sponding Focal Radii, .... 331 THEOREM XXXVII. Rectangle under Normal and Central Perpen- dicular, constant and = a 2 , . . . 331 SUPPLEMENTAL AND FOCAL CHORDS. Equation of Condition to Supplemental Chords, .... 332 THEOREM XXXVIII. Diameters parallel to Supplemental Chords are Conjugate, 332 Cor. 1. To construct Conjugates at a given in- clination. Caution, 333 Cor. 2. To construct a Tangent parallel to given Right Line, 333 Cor. 3. To construct the Axes in the empty Curve, 333 Focal Chords Special Properties, 334 THEOREM XXXIX. Focal Chord parallel to any Diameter, a third proportional to Axis Major and the Di- ameter, 335 II. THE CURVE REFERRED TO ANY Two CONJUGATES. DIAMETRAL PROPERTIES. Equation to Ellipse, referred to Conjugates, 336 THEOREM XL. Squares on Ordinates to any Diameter, proportional to Rectangles under corresponding Segments, 337 THEOREM XLI. Square on Diameter : Square on Conjugate : : Rect- angle under Segments : Square on Ordinate, . 337 THEOREM XLII. Ordinate to Ellipse : Corresponding Ordinate to Circle on Diameter : : V : a', . . . 338 Cor. 1. Given a pair of Conjugates, to construct the Curve, . 338 CONTENTS. xix Cor. 2. General interp'n of x 2 + if = a" : Ellipse, referred to Equi-conjugates, . . . 339 Figure of the Ellipse with respect to any two Conjugates, . . 339 CONJUGATE PROPERTIES OF THE TANGENT. Equation to Tangent, referred to Conjugates, 330 THEOREM XLIII. Intercept of Tangent on any Diameter, third proportional to Abscissa of Contact and the Semi-diameter: x=, . . . . 340 THEOREM XLIV. Rectangle under Intercepts by Variable Tangent on Two Fixed Parallel Tangents, constant and = Square on parallel Semi-diameter, . 341 THEOREM XLV. Rectangle under Intercepts on Variable Tangent by Two Fixed Parallel Tangents, variable and = Square on Semi-diameter parallel to Variable, 341 THEOREM XL VI. Rectangle under Intercepts on Variable Tangent by any two Conjugates equals Square on Semi-diameter parallel to Tangent, . . 342 Cor. 1. Diameters through intersections of Va- riable Tangent with Two Fixed Parallel ones, are Conjugate, 342 Cor. 2. Given two Conjugates in position and magnitude, construct the Axes, . THE SUBTANGENT TO ANY DIAMETER : its length = 342 343 X Cor. Construction of Tangent by means of Auxiliary Circle, ...... 343 THEOREM XLVII. Rectangle under Subtangent and Abscissa of Contact : Square on Ordinate : : a' 2 : I' 2 , . 344 THEOREM XLVIII. Tangents at extremities of any Chord meet on its bisecting Diameter, .... 345 PARAMETERS. Parameter to any Diameter defined : Third proportional to Diam- eter and Conjugate, 345 Parameter of the Curve : identical with Latus Rectum, . . . 345 THEOREM XLIX. In the Ellipse, no Parameter except the Princi- pal is equal in value to the Focal Double Ordinate, 346 THE POLE AND THE POLAR. Development of the Equation to the Polar : Definition, . . 346349 Cor. Construction of Pole or Polar from its definition, 349 xx CONTENTS. PAGE. THEOREM L. Polar of any Point, parallel to Diameter Conjugate to the Point, 350 Special Properties : Polar of Center ; of any point on axis of x ; on Axis Major, ......... Cor. Second geometric construction for Polar, POLAR OF Focus : its distance from center, and its direction, . THEOREM LI. Ratio between Focal and Polar distances of any point on Ellipse, constant and = e, . . 352 Cor. 1. On the construction of the Ellipse according to this theorem, . . . . . .352 Cor. 2. Polar of Focus hence called the Directrix, . 353 Cor. 3. Second basis for the name Ell-ipxe, . . 353 THEOREM LII. Line from Focus to Pole of any Chord, bisects focal angle which the Chord subtends, . . . 354 Cor. Line from Focus to Pole of Focal Chord, per- pendicular to Chord, 354 III. THE CURVE REFERRED TO ITS Foci. Interpretation of the Polar Equations to the Ellipse, . . . 355 Development of the Polar Equation to a Tangent, . * . . . 355 Polar proof of Theorem XIX compared with former proof, and with that by pure Geometry, ........ 356 IV. AREA OF THE ELLIPSE. THEOREM LIII. Area of Ellipse = T times the Rectangle under its Semi-axes, . 358 V. EXAMPLES ON THE ELLIPSE. Loci, Transformations, and Properties, ...... 358 CHAPTER FOURTH. THE HYPERBOLA. I. THE CURVE REFERRED TO ITS AXES. THE AXES. THEOREM I. Focal Center bisects the Axes. Corresponding inter- x 2 if pretation of - J - == 1, 363 a b The Axis Conjugate, conventional : Equation to the Conjugate Hy- perbola, 364 THEOREM II. Foci full without the Curve, 365 THEOREM III. Vertices of Curve, equidistant from Foci, . . . 365 THEOREM IV. Difference Focal Radii = Transverse. The Curve by Points, 365 CONTENTS. xxi THEOREM V. Conventional Semi-conjugate, geometric mean be- tween Focal Segments of Transverse, Cor. Dist. from Center to Focus = dist. between ex- tremities of Axes. To construct Foci, THEOREM VI. Squares on Ordinates to Axes, proportional to Rect- angles under corresponding Segments, Cor. The Latus Rectum, and its value, THEOREM VII. Squares on Axes are as Rectangle under any two Segments to Square on their Ordinate, Analogy of Hyperbola to Ellipse, with respect to Circle on Trans- verse, defective. Circle replaced by the Equilateral Hyperbola, THEOREM VIII. Ordinate Hyperbola : Corresponding Ordinate of its Equilateral : : b ' a, Cor. Interpretation of ? z . e defined as the Eccentricity, ....... THEOREM IX. Focal Radius of Curve, a Linear Function of corre- sponding Abscissa, ...... LINEAR EQUATION to Hyperbola: p = exa, . Verification of Figure of Curve by means of its equation, DIAMETERS. 366 366 367 367 367 368 369 370 371 371 Diameters : Equation to Locus of middle points of Parallel Chords, 371 THEOREM X. Every Diameter a Right Line passing through the Center, .371 Cor. Every Right Line through center is a Diam- eter, 372 THEOREM XI. "Every Diameter cuts Curve in Two Real Points" untrue for Hyperbola, 372 Cor. 1. Limit of those diameters having real intersec- tions: 372 Cor. 2. All diameters cutting Hyperbola in Imagi- nary Points, cut its Conjugate in Two Real ones, 373 Length of Diameter, in the Hyperbola, THEOREM XII. Each Axis the minimum diameter for its own curve, . THEOREM XIII. Diameters making supplemental angles with Trans- verse are equal, ...... Cor. Given the Curve, to construct the Axes, . THEOREM XIV. If a Diameter bisects Chords parallel to a second, second bisects those parallel to first, CONJUGATE DIAMETERS : Ordinates, . ... To construct a pair of Conjugates, .... Equation of Condition to Conjugates, in Hyperbola : tan 0. tan 9' - 2 , 375 374 374 374 374 374 xxii CONTENTS. PAGE. THEOREM XV. Conjugates in the Hyperbola lie on same side of Conjugate Axis, 375 Equation to Diameter Conjugate to that through any given point, . 375 Cor, The Axes are a case of Conjugates, . . 376 Given co-ordinates to extremity of Diameter, to find those to ex- tremity of its Conjugate, 376 THEOREM XVI. Abscissa ext'y of any Diameter : Ordinate ext'y of its Conjugate : : Transverse : Conjugate, 376 THEOREM XVII. Diff. squares on Ordinates to extremities of Con- jugates, constant and = b 2 , . . . 376 Length of any Diameter in terms of Abscissa to extremity of its Conjugate, 377 THEOREM XVIII. Square on any Semi-diameter = Kectangle Focal Kadii to extremity of its Conjugate, . . 378 THEOREM XIX. Distance, measured on Focal Chord, from extrem- ity of any Diameter to its Conjugate, constant and equal to Semi-Transverse, . . . 378 THEOREM XX. Difference of squares on any two Conjugates, con- stant and = difference of squares on Axes, 378 Angle between any two Conjugates, . . . 379 THEOREM XXI. Parallelogram under any two Conjugates, constant and = Rectangular under Axes, . . . 379 Cor. 1. Curve has but one set of Rectangular Conjugates, 380 Cor. 2. Inclination of Conjugates diminishes with- out limit : the conception of Equi-conjugates replaced by that of Self -conjugates, . . 380 THEOREM XXII. The Self-conjugates in the Hyperbola are Diago- nals of the Inscribed Rectangle, . . . 381 Cor. Curve has two, and but two, Self-conjugates, 381 Analogy of the Self-conjugates to the Equi-conjugates of the Ellipse, 382 THE TANGENT. Equation to Tangent, referred to the Axes, ..... 382 Condition that a Right Line touch an Hyperbola. Eccentric Angle, . 383 Analytic construction of the Tangent through any point : Two, real, coincident, or imaginary, 384 THEOREM XXIII. Tangent at extremity of any Diameter, parallel to its Conjugate, 385 Cor. Tangents at extremities of any Diameter are parallel. To circumscribe Parallelogram, . 385 THEOREM XXIV. Tangent bisects the Internal Angle between Focal Radii of Contact, 386 Cor. 1. To construct Tangent to Hyperbola, at a given point, 386 Cor. 2. Derivation of the term Focus, . . .386 CONTENTS. XXlll PAGE. Intercept by Tangent on the Transverse Axis : Constructions by means of it, 387 The SUBTANGENT to the Hyperbola 387 THEOREM XXV. Subtangent of Curve a Fourth Proportional to Abscissa of Contact and corresponding seg- ments of Transverse, 387 Cor. 1. Defect supplied in the analogy between Hyperbola and Ellipse, respecting Circle on 2a, 388 Cor. 2. Construction of Tangent by means of Inscribed Circle, 388 THEOREM XXVI. Central Perpendicular on Tangent, a Fourth Pro- portional to corresponding Semi-conjugate and the Semi-axes : p = , . . . 389 o Length of Central Perpendicular in terms of its angles with Axes, . 389 THEOREM XXVII. Locus of Intersection of Tangents cutting at right angles is Concentric Circle, . . 390 Focal Perpendiculars on Tangent: their length, .... 390 THEOREM XXVIII. Focal Perpendiculars proportional to adjacent Focal Radii, 390 THEOREM XXIX. Rectangle under Focal Perpendiculars, constant and = W, . ... 391 THEOREM XXX. Locus of foot of Focal Perpendicular is In- scribed Circle, 391 Cor. To draw Tangent by the method common to all Conies, 391 THEOREM XXXI. If, from any point without a Circle, a Chord be drawn, and a perpendicular to it at the point of section, the Perpendicular is Tan- gent to an Hyperbola, .... 392 Cor. The Hyperbola as Envelope, . . . 392 THEOREM XXXII. Diameters through feet of Focal Perpendiculars are parallel to Focal Radii of Contact, . 393 Cor. Diameters parallel to Focal Radii of Con- tact, meet Tangent at the feet of Focal Perpendiculars, and are of the constant length = 2a, 393 THE NORMAL. Equation to the Normal, referred to the Axes, 393 THEOREM XXXIII. Normal bisects the External Angle between Focal Radii of Contact, . . . .394 Cor. 1. If Ellipse and Hyperbola are confocal, Normal to one is Tangent to other at inter- section, ....... 394 xxiv CONTENTS. PAGE. Cor. 2. To construct Normal at any point on Hyperbola, 394 Cor. 3. To construct Normal through any point on Conjugate Axis, . . . 394 Intercept of Normal on Transverse Axis : Constructions by means of it, 395 THEOREM XXXIV. Normal cuts distance (produced) between Foci in segments proport'l to Focal Radii, . 395 The SUBNORMAL : Subnormal of the Hyperbola its length, . . 396 THEOREM XXXV. Normal cuts its Abscissa in the constant ratio = , 396 Length of Normal from Point of Contact to either Axis, . . 396 THEOREM XXXVI. Rectangle under Segments of Normal by Axes = Square on Conjugate Semi-diam- eter, 396 Cor. Equal, also, to Rectangle under corre- sponding Focal Radii, .... 396 THEOREM XXXVII. Rectangle under Normal and Central Perp'r on Tangent, constant and = a 2 , . . 397 SUPPLEMENTAL AXD FOCAL CHORDS. Equation of Condition to Supplemental Chords, .... 397 THEOREM XXXVIII. Diameters parallel to Supplemental Chords are Conjugate, 397 Cor. 1. To construct Conjugates at a given inclination, 397 Cor. 2. To construct a Tangent parallel to given Right Line, 398 Cor. 3. To construct the Axes in the empty Curve, 398 Focal Chords Properties analogous to those for the Ellipse, . . 398 THEOREM XXXIX. Focal Chord parallel to any Diameter, a third proportional to the Transverse and the Diameter, 399 II. THE CURVE REFERRED TO ANY Two CONJUGATES. DIAMETRAL PROPERTIES. Equation to Hyperbola, referred to Conjugates. Conjugate and Equilateral Hyperbola, ........ 400 THEOREM XL. Squares on Ordinates to any Diameter, proportional to Rectangles under corresponding Segments, . 401 THEOREM XLI. Square on Diameter : Square on its Conjugate : : Rectangle under Segments : Square on their Ordinate, ... .401 CONTENTS. xxv PAGE. THEOREM XLII. Ordinate to Hyperbola : Corresponding Ordinate to Equilateral on Axis of x : : b f : a', . 401 Bern. Failure of analogy to Ellipse in respect to Diametral Circle, 401 Figure of the Hyperbola, with respect to any pair of Conjugates, . 402 CONJUGATE PROPERTIES OF TANGENT. Equation to Tangent, referred to Conjugates, 402 THEOREM XLIII. Intercept of Tangent on any Diameter, third pro- portional to Abscissa of Contact and the Senai-diameter : x - , . . . 402 THEOREM XLIV. Rectangle under Intercepts by Variable Tangent on Two Fixed Parallel Tangents, constant and Square on parallel Semi-diameter, . 403 THEOREM XLV. Rectangle under intercepts on Variable Tangent by Two Fixed Parallel Tangents, variable and = Square on Semi-diameter parallel to Variable, 403 THEOREM XLVI. Rectangle under Intercepts on Variable Tangent by any two Conjugates = Square on Semi- diameter parallel to Tangent, . . . 403 Cor. 1. Diameters through intersection of Vari- able Tangent with Two Fixed Parallel Tan- gents are Conjugate, 403 Cor. 2. Given two Conjugates in position and magnitude, to construct Axes, . . . 403 x n a n THE SUBTANGENT TO ANY DIAMETER : its length = , . . 404 Cor. Construction of Tangent by means of Aux- iliary Circle, 404 THEOREM XLVII. Rectangle under Subtangent and Abscissa of Contact : Square on Ordinate : : a' 2 : b' z , . 405 THEOREM XL VIII. Tangents at extremities of any Chord meet on its bisecting Diameter, 405 PARAMETERS. Definitions. Parameter of Hyperbola identical with its Latus Rectum, 406 THEOREM XLIX. In the Hyperbola, no Parameter except the Principal equal in value to the Focal Double Ordinate, 406 THE POLE AND THE POLAR. Development of the Equation to the Polar : Definition, . . 406 408 Cor. Construction of Pole or Polar from its definition, 409 An. Ge. 3. XXVI CONTENTS. THEOREM L. Polar of any Point, parallel to Diameter Conjugate to the Point, 409 Polar of Center; of any point on Axis of x ; on Transverse Axis, 410 Cor. Second geometric construction for Polar, . 410 POLAR OP Focus : its distance from center, and its direction, . . 410 THEOREM LI. Ratio between Focal and Polar distances of any point on Hyperbola, constant and = e, . . 411 Cor. 1. Curve described by continuous Motion, . 411 Cor. 2. Polar of Focus hence called the Directrix, 412 Cor. 3. Second basis for name Hyperbola, . .412 THEOREM LII. Line from Focus to Pole of any Chord, bisects focal angle which the Chord subtends, . . 413 Cor. Line from Focus to Pole of Focal Chord, per- pendicular to Chord, 413 III. THE CURVE REFERRED TO ITS Foci. Interpretation of the Polar Equations to the Hyperbola, . . . 413 Development of the Polar Equation to a Tangent, .... 414 IV. THE CURVE REFERRED TO ITS ASYMPTOTES. ASYMPTOTES denned: Derivation of the name, .... 414 THEOREM LIII. Self-conjugates of Hyperbola are Asymptotes, . 416 Angle between the Asymptotes; its value in the Eq. Hyperb., . 416 Equations to the Asymptotes, ........ 416 THEOREM LIV. Asymptotes par. to Diag'ls of Semi-conjugates, . 417 THEOREM LV. Asymptotes limits of Tangents, .... 418 THEOREM LVI. Perpendicular from Focus on Asymptote = Con- jugate Semi-axis, ...... 419 THEOREM LVII. Focal distance of any point on Hyperbola = dis- tance to Directrix on parallel to Asymptote, . 419 Equation to Hyperbola, referred to its Asymptotes, .... 420 Equation to Conjugate Hyperbola, . . . 420 THEOREM LVIII. Parallelogram under Asymptotic Co-ordinates, constant and = -Q-> ..... 421 THEOREM LIX. Right Lines joining two Fixed Points on Curve to a Variable one, make a constant intercept on Asymptote, 422 Equation to the Tangent, referred to Asymptotes, .... 422 " Diameter through any given point, .... 422 " " Conjugate to x'y' . Equations to the Axes, . 422 Co-ordinates of extremity of Diameter conjugate to x'y' , . . 423 THEOREM LX. Segment of Tangent by Asymptotes, bisected at Contact, 423 Cor. The Segment = Semi-diameter conjugate to point of contact, 423 CONTENTS. xxvn PAGE. THEOREM LXI. Rectangle intercepts by Tangent on Asymptotes, constant and = a 2 + b 2 , 423 THEOREM LXII. Triangle included between Tangent and Asymp- totes, constant and = ab, .... 424 THEOREM LXIII. Tangents at extremities of Conjugates meet on Asymptotes, ....... 424 THEOREM LXIV. Asymptotes bisect the Ordinates to any Diam- eter, 425 Cor. I. Intercepts on any Chord between Curve and Asymptotes are equal, .... 425 Cor. 2. Given Asymptotes and Point, to construct the Curve, 425 THEOREM LXV. Rectangle under Segments of Parallel Chords by Curve and Asymptote, constant and = b' 2 , . 426 V. AREA OP THE HYPERBOLA. THEOREM LXVI. Area of Hyperbolic Segment equals log. Abscissa extreme point, in system whose modulus = sin : or, A = sin 0. te', 428 VI. EXAMPLES ON THE HYPERBOLA. Loci, Transformations, and Properties, 428 CHAPTER FIFTH. THE PARABOLA. I. THE CURVE REFERRED TO .ITS Axis AND VERTEX. THE AXIS. THEOREM I. Vertex of Curve bisects distance from Focus to Directrix, 431 Interp'n of symbol p in y z = 4/>(a t />), . . . 431 THEOREM II. Focus falls within the Curve, 431 Transformation to ?/ 2 = 4px, ..... 432 Cor. To construct the Curve by points, . . 432 THEOREM III. Square on any Ordinate - Rectangle under Abscissa and four times Focal distance of Vertex, . . 432 Cor. Squares on Ordinates vary as the Abscissas, . 432 The Latits Rectum defined. Its value = 4p, 433 Relation between the Parabola and the Ellipse : proof that Ellipse becomes Parabola when a increases without limit, . . . 433 Cor. 1. Analogue, in Parabola, of Circumscribed Circle in Ellipse, 434 Cor. 2. Interpretation of I ) = l=e z : e de- V a- Ja = 00 fined as the Eccentricity, 435 xxviii CONTENTS. PAGE. THEOREM IV. Focal Radius of Curve, a linear function of corre- sponding Abscissa, ...... 436 LINEAR EQUATION to Parabola: p=p + x, . . 436 Verification of Figure of Curve by means of its equation, . . 437 Nature of its infinite branch as distinguished from that of Hyperbola, 437 DIAMETERS. Diameters: Equation to Locus of middle points Parallel Chords, . 438 THEOREM V. Every Diameter is a Right Line parallel to the Axis, 439 Cor. 1. All Diameters are parallel, . . . 439 Cor. 2. Every Right Line parallel to Axis, i. .... 443 Intercept by Tangent on Axis : its length ^ #' , . . . . . 444 THEOREM IX. Foot of Tangent and Point of Contact equally dis- tant from Focus, 444 Cor. To construct Tangent at any point on Curve, or from any on Axis, 444 SUBTANGENT TO THE CURVE : its length = 2x' , 445 THEOREM X. Subtangent to Curve is bisected in Vertex, . . 445 Cor. 1. To construct Tangent at any point on Curve, or from any on Axis, 445 Cor. 2. Envelope of lines in Isosceles Triangle, . 446 Focal Perpendicular on Tangent : to determine its length, . . 446 THEOREM XI. Focal perpendicular varies in subduplicate ratio to Focal distance of Contact, .... 447 Length of Focal Perpendicular in terms of its angle with Axis, . 447 CONTENTS. xxix PAGE. THEOREM XII. Locus of foot of Focal Perp'r is the Vertical Tangent, 447 Cor. 1. Construction of Tangent by general Conic Method, 448 Cor. 2. Circle to radius infinity is the Right Line, 448 THEOREM XIII. If from any Point a right line be drawn to a fixed Right Line, and a perpendicular to it through the point of section, the Perpendicular will touch a Parabola, 449 Cor. The Parabola as Envelope, . . . .449 THEOREM XIV. Locus of intersection of Tangent with Focal Chord a,t any fixed angle is Tangent of same inclina- tion to Axis, 450 THEOREM XV. The angle between any two Tangents to a Para- bola = half the focal angle subtended by their Chord of Contact, 451 THEOREM XVI. Locus of intersection of Tangents cutting at right angles is the Directrix, 451 Cor. New illustration of Right Line as Circle with infinite radius, 451 THE NORMAL. Equation to the Normal, referred to Axis and Vertex, . . . 451 THEOREM XVII. Normal bisects External Angle of Diameter and Focal Radius to its Vertex, . . . .452 Cor. To construct Normal at any point, . . 452 Intercept by Normal on Axis : its length in terms of the Abscissa of Contact, 452 Constructions for the Normal by means of its Intercept, . . . 453 THEOREM XVIII. Foot of Normal equidistant from Focus with Foot of Tangent and Point of Contact, . . 453 Cor. Corresp'g constructions for Normal or Tang., 453 SUBNORMAL TO THE PARABOLA, ........ 453 THEOREM XIX. Subnormal to the Parabola, constant and = 2p, . 453 Length of the Normal determined, 453 THEOREM XX. Normal double corresp'g Focal Perpendicular, . 454 II. THE CURVE IN TERMS OP ANY DIAMETER. DIAMETRAL PROPERTIES. Equation to Parabola, referred to any Diameter and Vert'l Tangent, 454 THEOREM XXI. Vertex of any Diameter bisects distance between Directrix and the point in which the Diam- eter is cut by its Focal Ordinate, . . . 455 THEOREM XXII. Focal distance Vertex of any Diameter = Focal distance Principal Vertex divided by the square of the Sine of Angle between Diam- eter and its Vertical Tangent, . . .456 XXX CONTENTS. THEOREM XXIII. Square on Ordinate to any Diameter = Rectangle under Abscissa and four times Focal distance of its Vertex, Cor. Squares on Ordinates to any Diameter vary as the corresponding Abscissas, . THEOREM XXIV. Focal Bi-ordinate to any Diameter = four times Focal distance of its Vertex, Rem. Analogy of this Double Ordinate to the Latus Rectum peculiar to Parabola, Figure of the Curve with reference to any Diameter, 456 456 457 457 457 GENERAL, DIAMETRAL PROPERTIES OF THE TANGENT. Equation to Tangent, referred to any Diameter and Vert'l Tangent, 457 Intercept by Tangent on any Diameter: its length =x r , . . . 458 THEOREM XXV. Subtangent to any Diameter is bisected in its Vertex, 458 Cor. 1. To construct a Tangent to a Parabola from any point whatever, .... 458 Cor. 2. To construct an Ordinate to any Diameter, 458 THEOREM XXVI. Tangents at extremities of any Chord meet on its bisecting Diameter, 459 THE POLE AND THE POLAR. Development of the Equation to the Polar, .... 459461 Cor. Construction of Pole and Polar from their definitions, 461 THEOREM XXVII. Polar of any Point, parallel to Ordinates of corre- sponding Diameter, 462 Polar of any point on Axis of x; on principal Axis, . . 462 POLAR OF Focus : its identity with the Directrix, .... 462 THEOREM XXVIII. Ratio between Focal and Polar distances of any point on Parabola, constant and = e, . . 463 Rem. 1. Vindication of original definition and construction of Curve, .... 463 Rem. 2. Second basis for the name Parabola, . 464 THEOREM XXIX. Line from Focus to Pole of any Chord bisects focal angle which Chord subtends, . . 464 Cor. Line from Focus to Pole of Focal Chord perpendicular to Chord, .... 464 Examples : Intercept on Axis between any two Polars = that be- tween perp'rs from their Poles, .... 465 Circle about Triangle of any three Tangents passes through Focus, ....... 465 PARAMETERS. Parameter defined as Third Prop'l to Abscissa and its Ordinate, 465 CONTENTS. xxxi PAGK. THEOREM XXX. Parameter of any Diameter = four times Focal distance of its Vertex, .... 466 Cor. Parameter of the Curve = four times Focal distance of the Vertex, .... 466 Rein. New interpretation of various Theorems and Equations, ...... 466 THEOREM XXXI. Parameter of any Diameter = its Focal Bi- ordinate, 466 Cor. Parameter of Curve Latus Rectum, . 466 Eem. The Theorem holds in the Parabola alone of all the Conies, . . . . . 466 Parameter of any Diameter in terms of Abscissa of its Vertex, . 467 " " " Principal, . . . .467 THEOREM XXXII. Parameter inversely proportional to sin 2 of Ver- tical Tangency, ...... 467 III. THE CURVE REFERRED TO ITS Focus. Interpretation of the Polar Equation to Parabola, .... 467 Development of the Polar Equation to Tangent, . . . 468 IV. AREA OP THE PARABOLA. THEOREM XXXIII. Area of any Parabolic Segment = Two-thirds the Circumscribing Rectangle, . . . 470 V. EXAMPLES ON THE PARABOLA. Loci, Transformations, and Properties, 470 CHAPTEE SIXTH. THE CONIC IN GENERAL. I. THE THREE CURVES AS SECTIONS OF THE CONE. Definitions, 473 Conditions of the several Sections, and their Geometric order, . 474 II. VARIOUS FORMS OF EQUATION TO THE CONIC IN GENERAL. General Equation in Rectangular Co-ordinates at the Vertex, . . 475 Equation to the Conic, in terms of the Focus and its Polar, . . 478 Linear Equation to the Conic, 479 Equation to the Conic, referred to any two Tangents, . . . 480 The Conic as Locus of the Second Order in General, . . . 482 III. THE CURVES IN SYSTEM AS SUCCESSIVE PHASES OF ONE FORMAL LAW. Order of the Curves, as given by Analytic Conditions, . . . 483 Classification of the Conies, 485 xxxii CONTENTS. PAGE. IV. DISCUSSION OP THE PROPERTIES OP THE CONIC IN GENERAL. The Polar Relation, 486 Diameters : Development of the Center, 495 Development of the conception of Conjugates and of the Axes, . 498 Development of the Asymptotes in general symbols, . . . 501 Similar Conies defined, ......... 506 V. THE CONIC IN THE ABRIDGED NOTATION. Fundamental Anharmonic Property of Conies, .... 507 Development of Pascal's and Brianchon's Theorems, . . . 508 BOOK SECOND: CO-ORDINATES IN SPACE. CHAPTER FIRST. THE POINT. Rectangular Co-ordinates in Space explained, .... 514 Expressions for Point on either Reference-plane; for Point on either Axis; for Origin, ...... 515 Polar Co-ordinates in Space, ........ 516 The doctrine of Projections, . . . . . . . .517 Distance between any Two Points in Space, ..... 520 Relation between the Direction-cosines of any Right Line, . 521 Co-ordinates of Point dividing this distance in Given Ratio, . . 522 Transformation of Co-ordinates : 522 I. To change Origin, Reference-planes remaining parallel to first position, ........ 522 II. To change Inclination of Reference-planes, . . . 522 III. To change System from Planars to Polars, and con- versely, 523 General Principles of Interpretation : 524 I. Single equation represents a Surface, 524 II. Two equations represent Line of Section, .... 524 III. Three equations represent mnp Points, .... 524 IV. Eq. wanting abs. term represents Surface through Origin, . 524 V. Transf 'n of Co-ordinates does not affect Space-Locus, . 524 CHAPTER SECOND. LOCUS OF FIRST ORDER IN SPACE. Equation of First Degree in Three Variables represents a Plane, . 525 General Form of Equation to any Plane, 526 Equation to Plane in terms of its Intercepts on Axes, . . . 527 CONTENTS. xxxin PAGE. Equation to Plane in terms of its Perpendicular from Origin and Direction-cosines, 527 Transformation of Ax + By + Cz + D = to the form last obtained, 528 THE PLANE UNDER SPECIAL CONDITIONS, 529 Equation to Plane through Three Fixed Points, . . . 529 Angle between two Planes : conditions that they be par- allel or perpendicular, 529 Equation to Plane parallel to given Plane; perpendicular to given Plane, 531 Length of Perp'r from (xyz) on x cos a -j- y cos /? + z cos y = 0, 532 " " " " on Ax + By + Cz + D = 0, . . 532 Equation to Plane through Common Section of two given ones, P+/cP' = 0, 532 Equation to Planar Bisector of angle between any two Planes, 533 Condition that Four Points lie on one Plane, .... 533 " " Three Planes pass through one Right Line, . 533 " " Four Planes meet in one Point, . . .534 QUADRIPLANAR CO-ORDINATES: Abridged Notation in Space, . . 534 la + m/3 + ny + r6 = represents a Plane in Quadriplanars, . 535 LINEAR Loci IN SPACE: solved as Common Sections of Surfaces, . 535 The Right Line in Space as common Section of Two Planes, . 535 Equation to Right Line in terms of Two Projections, . . 536 Symmetrical Equations to the Right Line in Space, . . 537 To find the Direction-cosines of a Right Line, . . . 537 Angle between Two Right Lines in Space, .... 538 Conditions as to Parallelism and Perpendicularity, . . . 539 Equation to Right Line perpendicular to given Plane, . . 539 Angle between a Right Line and a Plane, .... 540 Condition that a Right Line lie wholly in given Plane, . . 540 Condition that Two Right Lines in Space shall intersect, . . 541 Examples involving Equations of the First Degree, .... 541 CHAPTER THIED. LOCUS OF SECOND ORDER IN SPACE. General Equation of Second Degree in Three Variables: its gen- eral interpretation, ......... 542 SURFACES OF SECOND ORDER IN GENERAL: Criterion of the Form of any Surface furnished by its Sections with Plane, 543 Every Plane Section of Second Order Surface is a Curve of Sec- ond Order, 543 Properties common to all Quadrics, ..... 544 552 Classification of QUADRICS, or Surfaces of Second Order, . . 553 Summary of Analogies between Quadrics and Conies, ' . . 561 xxxiv CONTENTS. PAGE. SURFACES OP REVOLUTION OF THE SECOND ORDKR: . . . . 562 General Method of Revolutions explained, .... 563 Equation to the Cone, ........ 564 Demonstration that all Curves of Second Order are Conies, . 565 Equation to the Cylinder, 567 " " Sphere, 567 Equations to the Ellipsoids of Revolution, .... 568 " " Hyperboloids " 568 The Ellipse of the Gorge, and its Equation, . . 569 " to the Paraboloid, 569 The Tangent and Normal Planes to the Quadrics, . . . .570 EXAMPLES, 573 NOTE. The references in the present treatise are to RAT'S New Higher Algebra, and RAY'S Geometry and Trigonometry. INTRODUCTION: THE NATURE, DIVISIONS, AND METHOD OF ANALYTIC GEOMETRY. ANALYTIC GEOMETRY: ITS NATURE, DIVISIONS, AND METHOD. 1. BY Analytic Geometry is meant, speaking gen- erally, Geometry treated by means of algebra. That is to say, in this branch of mathematics, the properties of Figures, instead of being established by the aid of dia- grams, are investigated by means of the symbols and processes of algebra. In short, analytic is taken as equivalent to algebraic. 2. Accordingly, and within recent years especially, the science has sometimes been called Algebraic Geom- etry. It is preferable, however, for reasons which will appear farther on, to retain the older and more usual name. Why algebraic treatment should be considered analytic, in what precise sense geometry is called analytic if treated by algebra, when it is not called so if treated by the ordinary method, will appear as w r e proceed. But, for the present, the attention needs to be fixed upon the simple fact, that, in connection with geometry, analytic means algebraic. 3. The properties of Figures are of two principal classes : they either refer-to magnitude, or else to position and form. Thus, The areas of circles are to each other as the squares upon their radii, is a property of the first (3) ANAL YTIC GEOMETR Y. * kind ; of the second is, Through any three points, not in the same right line, one circle, and but one, can be passed. 4. Accordingly, geometric problems are either Prob- lems of Dimension or Problems of Form. 5. Corresponding to these two classes of problems, there are, in Analytic Geometry, two main divisions; namely, DETERMINATE GEOMETRY and INDETERMINATE GEOMETRY. The methods of these two divisions we now proceed to sketch. I. DETERMINATE GEOMETRY. 6. The geometry of Dimension is called DETERMINATE, because the conditions given in any problem in which a dimension is sought must be sufficient to determine the values of the required magnitudes ; or, to speak from the algebraic point of view, these conditions are always such as give rise to a group of independent equations, equal in number to the unknown quantities involved, and there- fore determinate.* 7. In completing a problem of Determinate Geometry, there are two distinct operations : the SOLUTION and the CONSTRUCTION. 8. The Solution for the required parts, consists in representing the known and unknown parts of the figure in question by proper algebraic symbols, and finding the roots of the equations which express the given relations of those parts. The Construction of the parts when found, consists in drawing, according to geometric principles, the geo- metric equivalents of the determined roots. * Algebra, Art. 159, compared with 168. INTRODUCTION. 5 The principles underlying each of these operations will now be developed. PRINCIPLES OF NOTATION. 9. These are all derived from the algebraic convention that a single letter, unaffected with exponent or index, shall stand for a single dimension ; or, as it is commonly put, that each of the literal factors in a term is called a dimension of the term. From this, it follows that the degree of a term is fixed by the number of its dimen- sions. Our principles therefore are : 1st. Any term of the first degree denotes a LINE, of de- terminate length. For, by the convention just stated, it denotes a quantity of one dimension. When applied to geometry, therefore, it must denote a magnitude of one dimension. But this is the definition of a line of fixed length. Accordingly, a, x denote lines whose lengths have the same ratio to their unit of measure that a and x respectively have to 1. 2d. Any term of the second degree denotes a SURFACE, of determinate area. For it denotes a magnitude of two dimensions, that is, a surface ; and since each of its dimensions denotes a line of fixed length, the term, as their product, must denote a surface of equal area with the rectangle under those lines. In fact, it is usually cited as their rectangle. Thus, ab denotes the rectangle under the lines whose lengths are a and b. Similarly, x 2 denotes the square upon a side whose length is x. 3d. Any term of the third degree denotes a SOLID, of determinate volume. For it is the product of the lengths of three lines, and hence denotes a volume equal to that of the right parallelepiped between those lines, and is so cited. Thus, abc is the right parallelepiped whose edges 6 ANALYTIC GEOMETRY. have severally the lengths a, b, c; and a? denotes the cube on the edge whose length is x. 4th. An abstract number, or any other term of the zero degree, denotes some TRIGONOMETRIC FUNCTION, to the radius 1. The general symbol for a term of the zero a degree may be written 7, since the number of dimen- sions in a quotient equals the number in the dividend less that in the divisor. If, now, we lay off any right line AB = b, describe a semicircle upon it, and, taking the chord BC = a, join AC: we shall have (Geom., Art. 225) the triangle ABC right-angled at C, Hence, (Trig., Art. 818,) - = sin A. o If the base, instead of the hypotenuse, were taken = b, we should have a v- = tan A. b If the hypotenuse were taken = a, and the base = b, we should have - = sec A ; and so on. 5th. A polynomial, in geometric use, is always HOMO- GENEOUS, and denotes (according to its degree) a length, an area, or a volume, equal to THE ALGEBRAIC SUM of the magnitudes denoted by its terms. By the ordinary convention of signs, it must denote the sum mentioned; it is therefore necessarily homogeneous, since the sum- INTRODUCTION. 1 mation of magnitudes of unlike orders is impossible. We can not add a length to an area, nor an area to a volume. Corollary. Hence, if a given polynomial be apparently not homogeneous, it is because one or more of the linear dimensions in certain of its terms are equal to the unit of measure, and consequently represented by the im- plicit factor 1. WJien, therefore, such a polynomial occurs, before constructing it, render its homogeneity apparent by supplying the suppressed factors. Thus, a 3 -f a?b c fg = a* + a 2 b c X 1 X 1 fg X 1 . Similarly, for we may write and so on. 6th. Terms of higher degrees than the third have no geometric equivalents. For no magnitude can have more than three dimensions. Corollary. If expressions apparently of such higher degrees occur, they are to be explained by assuming 1 as a suppressed divisor, and constructed accordingly. Thus, = ' and so on. Remark These six principles enable us to represent by proper symbols the several parts of any geometric problem, and to interpret the result of its solution, as indicating a line, a rectangle, a parallelopiped, etc. We then construct the magni- tude thus indicated, according to the principles to be explained in the next article. An. Ge. 4. 8 ANALYTIC GEOMETRY. EXAMPLES. 1. Render homogeneous a?b -\- c d 2 . 2. In what different Avays may the degree of la be reckoned ? Of 5xyl Of V 5 (# 2 -|-y 2 )? State their geometric meaning for each way. 3. Interpret geometrically i/2o6; 1/3; V^z; and \^6a. 4. Adapt a& to represent a line: also, i/ota. 5 . What does V d 1 + 6 2 represent ? What V m* + n z I' 2 r 2 ? 6. vd being given as denoting a surface, render its form con- sistent with its meaning. 7. Render c - ~ homogeneous of the second degree. 8. Render - homogeneous of the first degree. 9. Render - ^ homogeneous of the zero degree. 10. Adapt a 5 ^ 2 to represent a solid; a surface; a line. What is the geometric meaning of a 4 6~ 2 ? of a 5 6~ x ? of a~ 2 ? PRINCIPLES OF CONSTRUCTION. 1O. In these, we shall confine ourselves to construct- ing the roots of Simple Equations and Quadratics. I. The Root of the Simple Equation. This may assume certain forms, the construction of which can be generalized. The following are the most important: 1st. Let x = a =L b. Here, (Art. 9, 5) x denotes a line whose length is the algebraic sum of those denoted by a and b. Therefore, on any right line, take a point A as the starting-point, or Origin, and lay off (say to the right) the unit of measure till AB = a. Then, if b is pos- -^ i j +r~ idee, by laying off, in the same direction and on the same line as before, JBCb, INTRODUCTION. we obtain A as the required line ; for it is evidently the sum of the given lengths. But, if b is negative, its effect will be to diminish the departure from A ; hence, in that case, it must be laid off as BD = BC = b. We thus obtain AD for the required line ; and it is obviously equal to the difference of the given lengths. If b > a and negative, then x is wholly negative. Our construction answers to this condition. For then the extremity of b will fall to the left of the origin A, say at E, and the line x will therefore be represented by AE, and measured wholly to the left of A. This brings into view the important principle that the signs -f- and are the symbols of measurement in opposite directions. Hence, if we have a linear polynomial, its negative terms are to be constructed by retracing such a portion of the distance made from the origin, correspond- ing to its positive terms, as their length requires. If we have two monomials with contrary signs, they must be laid off in opposite directions from the origin. ab 2d. Let x = . In c this case, x denotes a line whose length is a fourth proportional to c, a, and b. Therefore, draw two right lines, AC and AE, making any angle with each other. On the one, lay off AB =c 9 and AC ' = a; on the other, AD = b. Join ED, and draw CE parallel to BD- then will AE be the line required. For, (Geom., 307) the triangles ABD, ACE being similar, we have AB : AC :: AD : AE; or, c : a :: b : AE. 10 ANALYTIC GEOMETRY. 3d. Let x = -j- . Putting -= = k, this may be written ko x = . 9 ab Therefore, construct k = -^-, as in the preceding case, and, with the line thus found, apply the same construc- tion to x. abed k'd abc kc in like manner, - 7 = - 7 , by putting; k = = fgh h fy 9 And, in the same way, we may construct any quotient of the first degree. II. The Roots of the Pure Quadratic. Three or four cases deserve attention : 1st. Let x = V ab. Here we have a line whose length is the geometric mean of a and b. There are several constructions, but the following is as elegant as any. On any right line, lay off AB = a, and BC = b. Upon AC = a -f 6, describe a semicircle. At B erect a perpendicular meeting the curve in D: BD is the line sought. For (Geom., 325) BD = I^ABXBC = Vdb = x. Striptly speaking, since the radical V ab has a double sign, there are two lines answering to x, equal in length, but measured in opposite directions. And for this, in fact, the construction provides: since there is a semi- circle below, as well as above AC, to which the per- pendicular dropped from B has the same length as but is drawn in a direction exactly opposite. ft.J.ef. 1NTR 01) UCTION. 1 1 2d. Let x = l/a 2 -|- ac. Writing this in the form we perceive that we have to construct the geometric mean of a and a -\- c. On any right line lay oiF A B = a, and B C= c : then A - a -f c. Describe a semicircle about AC, erect the perpendicular BD, and join AD: AD is the line required. (Geom., 324, 2.) In this case, to satisfy the negative value of x, we must lay off a and c from A to the left, and throw our semicircle below the (a -f- c) thus formed. The geometric equiv- alent of x is the chord joining A and the point where the perpendicular from the extremity of a meets this downward semicircle. And this chord is obviously drawn from A in exactly the opposite direction to AD. 3d. Let x = Va' 2j rb 2 . This gives us the side of a square whose area equals the sum of two given squares. Ac- cordingly, lay off AB = a ; at B erect a perpendicular, and upon it take BC = b; join A C: then (Geom., 408) AC is the line required. 4th. Let x = V / ~^ r b 2 . In this case, we are to con- struct the side of a square whose area equals the differ- ence of two given squares. Lay off AB = c, and erect upon it a semi- circle. From A as a center, with a radius A C = b, describe an arc cut- ting the semicircle in C. Join CB, arid (Geom., 409) BO will be the required line. We leave the discussion of the negative values of x, in the last two cases, to the student. 12 ANALYTIC GEOMETRY. III. Roots of the Complete Quadratic. Let us consider the Four Forms separately. (See Alg., 231.) 1st. The First Form of the complete Quadratic is Its roots are given in the formula x = p f -f f. To construct these : Lay off AB = q. At B erect the perpendicular BC = p. From G as center, with the radius BQ describe a circle. Join AC, and produce the line to meet the cir- cle in E: AD and EA are the required roots. For, by the con- construction, AC = V p 2 -f- > q-, the roots are real and unequal. The construcr tion also indicates this. For, so long as the hypotenuse CD=p is greater than the perpendicular CB = q, it will intersect AF in two real points. If we suppose p = q, the roots are real, but equal. This, too, is involved in the construction. For, when the radius CD=p becomes equal to CB = q, the circle touches AF; that is, its two points of section with AF become coincident in B, and AD = AB = AE. Fourth: If, in (5), (6), (7), (8), we suppose q > p, the roots become imaginary. With this, again, the construc- tion perfectly agrees. For, if q > p, the radius CD is less than the perpendicular CB, and the circle cuts AF in imaginary points. The supposition, moreover, re- quires us to construct a right triangle whose hypotenuse An. Ge. 5. 16 ANALYTIC GEOMETRY. shall be less than its perpendicular a geometric impos- sibility. This agrees with the well-known algebraic principle, that imaginary roots arise out of some incon- gruity in the conditions upon which the equation is founded. EXAMPLES. 1. Construct x = V I' 1 + m 2 n 2 , first, by placing m 2 n 2 = k' 2 ; secondly, by placing I'* -\- m* Tc 1 ; thirdly, by placing I' 2 n 2 = & 2 . Show that the three constructions give the same line. Imn -4- & 2 A 2. Construct x = . 3. Construct v'5, l/3afo, and >/a 2 -j-6 2 - Imn & 2 A * 4. Construct x = . 5. Construct o? = -\/ it-*/ ( 2 mn). \ n \ n^ DETERMINATE PROBLEMS. 11. The mode of applying the foregoing principles to the solution of these problems, may be best exhibited in a few examples. EXAMPLES. 1. In a given triangle, to inscribe a square. A triangle is given when its base and altitude are given; we are therefore here re- quired to find the side of the inscribed square in terms of the base and altitude of the given triangle. If we draw the annexed diagram, representing the problem as if solved, and designate the base of the triangle by b, its altitude by A, and the side of the inscribed square by x: then, since the triangles CAB, CEF are similar, we have (Geom., 310) * The student should pay strict attention to the geometric meaning of the signs + and , as explained above. He should also see that the given algebraic expressions are put into the most convenient forms, before constructing. INTRODUCTION. 17 AB : EF : : CD : Off; or, b : x : h : A x. bh .' . x = . That is, the side of the inscribed square is a fourth proportional to the base, the altitude, and their sum. We therefore construct it as in Art. 10, I. 2d. Or it may be more conveniently done as follows: Produce the base of the given tri- angle until BL A ; through L draw LM, parallel and equal to BC; join MA t and from JV, the point where MA cuts BC, drop a perpendicular upon AB: then is NO the side required. For, letting fall MP perpendicular to AL, we have, by the similar triangles MAL, NAB, AL : AB :: MP : NO; that is, b + h : b : : A : NO. bh \ Note. The student should consider what several positions the side of the square may assume according as the triangle is acute- angled, right-angled, or obtuse-angled. 2. In a given triangle, to inscribe a rectangle whose sides are in a given ratio. Let x and y represent the two sides, and r their con- stant ratio. Then we shall have x (1). And, as in the previous example, (the other symbols remaining the same,) we obtain the proportion b : y : : A : A x. .-. hy = bh bx (2). Eliminating y between (1) and (2), bh This value we construct in the same manner as the side of the inscribed square. In fact, as is obvious, the first problem is merely a particular case of the present one ; for a square is a rectangle, the ratio between the sides of which is equal to 1. The solution, too, 18 ANALYTIC GEOMETRY. shows this ; for the value of x in the present example becomes that obtained in the former, when r 1. Produce, then, the base of the given triangle until BL equals rh ; and complete the drawing exactly as in C _M the case of the inscribed square : the point -ZV, in which the diagonal AM cuts the side of the tri- angle, is a vertex of the required rectangle. Let the student prove this. 3. To draw a common tangent to two given circles. Here our data are the radii of the circles, and the distance between their centers. Let r denote the radius of the circle on the left, and r' that of the other. Let d = the distance between the centers. The problem may be otherwise stated : Required a point, from which, if a tangent be drawn to one of two given circles, it will also touch the other. From the method of constructing a tangent, (see eom., 230,) it follows that this point is some- where on the line join- ing the centers. Hence, drawing the diagram as annexed., it is evident that our unknown quan- tity is the intercept made by the tangent on this line : that is, we let x= CT. Now we have (Geom., 333) In like manner, M'T* = N'TXI/T; or, M'T*= (x d + r^) (x d -r') (2). Expanding, and dividing (1) by (2), MT* xt r* (3). But, by similar triangles, MT : :: r : 7' .-. MT< INTE OD UCTION. 19 Substituting in (3), and reducing, (r 2 r' 2 ) x 2 2r 2 dx + r*d z = : Before constructing this result, let us interpret it. We observe that our problem involves the solution of a quadratic, and that we thus obtain a double value for x = CT. The required tangent, therefore, cuts the line of the centers in two points; that is, there are two points from which a common tangent to the two circles may be drawn.* Let us now consider the two values of x more minutely. We shall find all the geometric facts of the problem perfectly repre- sented in them. First take the value numerically the greater, namely, rd X= 7=?'' Since this must be numerically greater than rf, and since the definition of a tangent renders it impossible that the point sought should fall within either of the circles, the point determined by this value of x is beyond both. If r ^> r*, x is positive, and the point T falls to the right of both circles, as in the diagram; if r=M'P, represents P; -a = OM', with l=M'P", represents P" ; + a = OM, with b = MP'", represents P'". INTRODUCTION. 27 The lines X'X, Y'Y are called axes; their intersec- tion is called the origin; OM, MP, etc., are called co-ordinates. If, now, instead of the particular lines OM, MP, etc., we take x and y as general symbols for the co-ordinates of a point, and denote by a and b the values they as- sume for any particular point, we obtain the algebraic expression for a determinate point in a given plane, namely, x = a in which a and b may have any value from to oo, and be either positive or negative. As already hinted, the foregoing is only one form of the convention upon which rests the whole structure of the geometry of Form. Several others are used, differing from the present in the nature of the assumed limits and of the means by which the point is referred to them : in some, as in the present one, the co-ordinates are linear; in others, one of them is angular. The present form, moreover, applies only to points in a given plane; forms suitable for representing a point any- where in space, are obtained by assuming for Fixed Limits planes instead of lines. But whether the forms of the convention apply to points in space or to points in a given plane, and however they may differ in their details, they all agree in this: that a point shall be de- termined by referring it to certain Fixed Limits by means of certain Elements of Reference, called Co- ordinates. Definition. The Co-ordinates of a point are it linear or angular distances from certain assumed limits. 28 ANALYTIC GEOMETRY. Corollary. By the Convention of Co-ordinates we therefore mean, The agreement that the algebraic symbols of magnitude shall denote the co-ordinates of a point. 20. This convention once established, the connection between an equation and a geometric form will readily become apparent. The discovery of this connection, in its universal bearing, and the first exhaustive applica- tion of it to the discussion of curves, was the work of the French philosopher DESCAETES. His method was first published in 1637, in his treatise De la Geometric. We shall now show that the connection alluded to really exists; but must first define certain conceptions on which it depends. 21. Variables and Constants. In analytic inves- tigations, the quantities considered are of two classes: variables and constants. Definition. A Variable is a quantity susceptible, in a given connection, of an infinite number of values. Definition, A Constant is a quantity susceptible of but one value in any given connection. Remark. In problems of analysis, constants impose the con- ditions; variables are subject to them. Constants are represented by the first letters of the alphabet; variables by the last. At times, both are designated by such Greek letters as may be con- venient. 22. Functions. In the investigations belonging to Indeterminate Geometry, the variables are so connected by the conditions of the problem in hand, that any change in the value of one produces a corresponding change in that of the others. Definition. A Function is a variable so connected with others, that its value, in every phase of its changes, INTE OD UCTION. 29 is derived from theirs in a uniform manner. Thus, in y = ax -f 6, y is a function of x; in z 2 =mx* -f- ny 2 -\- I, z is a function of x and ?/. Remark. Functions are classed, according to the number of the variables on which they depend, as functions of one variable, functions of two variables, etc. 23. With these definitions in view, we may state our Fundamental Principle with greater exactness, thus : Every equation betiveen variables that denote the co-ordi- nates of a pointy represents, in general, a geometric form. The proof of this now follows. 24. Equations between Co-ordinates: their Geometric Meaning. Every equation is the expres- sion of a constant relation between the variables which enter it. Further, if we solve any equation for one of its variables in terms of the others, it becomes ap- parent that such variable is a function of the rest. Accordingly, by varying either, we may cause all of them to vary together, by differences as great or as small as we please; but, so long as the constants that express the manner in which each is derived from the others remain unchanged, all the changes must comply with one uniform law. That is, whatever be the absolute value of either variable, its relative value, as compared with the others, is always the same. If either changes by infinitely small differences, the others must change by corresponding infinitesimals. If, then, we assume that the variables in an equation denote co-ordinates, the equation itself must represent a number of points, as many as we please, all of which have co-ordinates of the same relative values. Now, since these co-ordinates vary by differences as small as we please, the equation really represents an infinite 30 ANALYTIC GEOMETRY. number of points, lying infinitely near to each other, and thus forming a continuous series. This continuous series of positions, moreover, has a definable form, of the same nature in all its parts; since, from the definition of an equation, every point in the infinite succession must comply with a law of position, the same for all: a law expressed by the constants in the equa- tion, which subject the variable co-ordinates to an inflex- ible relation in value. Every equation between variables that denote co-ordinates must therefore, in general, represent a geometric form. Remark. It will represent a line or a surface, according as the co-ordinates are taken in a plane or in space. 25. A few illustrations will render the principle just proved still more apparent. For the sake of variety, we will take these from the converse point of view, from which it will appear that every attempt to state a law of form in algebraic symbols results in an equation between co-ordinates. To simplify, let us confine ourselves to rectilinear co-ordinates in a given plane, and (since the axes of reference may be any two intersecting right lines) suppose XX, Y'Y to intersect at right angles: the co-ordinates OM, MP will then be at right angles to each other. First: Let it be required to represent in algebraic symbols a right line parallel to the axis Y'Y. The law of this form -^ plainly permits the variable point P of the line to be at any IP distance above or below X'X, but restricts it to being at a constant distance from Y'Y: a condition imposed by TR OD UCTION. 31 assuming that y = MP varies without limit, while, at the same time, x = OM remains unchanged. Thus we see that, in a right line parallel to the axis Y'Y 9 the co-ordinate y has no determinate value, but the co- ordinate x has a fixed and unchangeable value. Hence the algebraic expression for such a line is the equation x = constant. Similarly, a right line parallel to the axis X'X is rep- resented by the equation y = constant. Second: Let it be required to represent algebraically a circle whose center is at the origin 0. The law of this form is, that the variable point P shall maintain a con- stant distance from 0. But it is obvious, upon inspecting the diagram, that the distance of any point from the origin is equal to V x 2 - -\- y 2 . Hence, the con- dition that the point shall be upon a circle whose center is at 0, gives us V x 2 + f -- constant ; and, squaring, we represent the circle by the equation X 2 _j_ yi __ cons tant = r-. 26. Let us now return to a more exact consideration of the Fundamental Principle. The student will have noticed, that, in both forms of stating it hitherto, the forms used in Arts. 23 and 24, we have been careful to say that it is true in general. This restriction is nec- essary, and of great importance ; for there are certain An. Ge. 6. 32 ANALYTIC GEOMETRY. equations which no real values of the variables will sat- isfy : and such, of course, can denote only imaginary, or impossible, forms. Others can only be satisfied by in- finite values of the variables, and consequently denote a series of points situated at infinity : a conception as im- possible, geometrically, as that corresponding to the previous class of equations. Others, again, can be sat- isfied by only one set of real values for the variables, and therefore represent a single point ; while others, which can be satisfied by a fixed, finite number of values, but by no others, represent a finite number of separate points. Others, still, are satisfied by distinct sets of values, each set being capable of an infinite number of values within itself, and having a distinct relation among the variables which belong to it ; and such equations represent a group of distinct, though related forms. All this makes it clear that, to hold universally, our Fundamental Principle must be stated in more abstract terms. We should be obliged to say, merely, that every equation between co-ordinates represents some conception relating to form or position, were it not that the happy expedient of a technical term saves us from this cum- brous circumlocution. It being established that every equation between co-ordinates has some equivalent in the province of geometry, it only remains to assign a name to that equivalent a name generic enough to include not only surfaces and lines, but all the exceptional cases, real, imaginary, or at infinity, that have been mentioned above. 27. Loci. To include all the cases that may arise under the conception that an equation has geometric meaning, the term locus is used. INTR OD UCTION. 33 Definition. A Locus is the series of positions, real or imaginary, to which a point is restricted by given con- ditions of form. Corollary. Since the locus is the geometric equiva- lent of the equation, we may state the Fundamental Principle of Indeterminate Geometry universally as fol- lows : Every equation between variables which denote the co-ordinates of a point., represents a locus. 28. The locus being the fundamental conception of purely analytic geometry, it is of the utmost importance that correct views of it be secured at the outset. The beginner is liable to conceive of it loosely, or else too narrowly. To guard against these errors, let us illus- trate what has been said or implied above somewhat more at length. I. Classification. Loci are either Geometric or merely Analytic : the former, when they can be repre- sented in a diagram ; the latter, when they can not. Creometric Loci include real surfaces, lines, points, and related groups of either. Merely Analytic Loci include imaginary loci and loci at infinity, and loci to be explained hereafter under the conception of a locus in general. The first have no existence whatever, except in the equations which sym- bolize them; the last two exist to abstract thought, but can neither be drawn nor imagined. The value of con- sidering these merely analytic loci, lies in their important bearings upon some of the higher problems of the science. II. Conformity to Law. It is essential to the con- ception of a locus, that it shall conform to some definite law. No form that comes within the scope of analytic geometry can be generated at hazard; no locus is the least capricious. For it is always the counterpart of an 34 ANALYTIC GEOMETRY. Y' equation ; and every equation, by means of its constants, maintains among its variables, throughout their infinite changes, a uniform relation in value. We must there- fore avoid the error of supposing that a broken, irregular, mixed figure, such as the line in the annexed diagram, or such as we dash off with the hand at a scribble, is a locus. Y On the contrary, a locus is, in a certain important sense, homo- geneous. That is, throughout its whole extent, it is so far alike x- as to be represented by one equation, and but one. No point in it can be found whose co-ordinates do not satisfy this one equation. HI. Variety of Meaning. The idea of the locus should be conceived broadly enough to include, in addi- tion to surfaces and lines, the various exceptional species enumerated in Art. 26. The attention has, perhaps, been sufficiently called to cases where it is a point, or a series of separate points, and where it is imaginary, or at infinity. But it is worth while to repeat that a locus is not neces- sarily a single figure. For example, the equation represents, as will be proved in the treatise which is to fol- low, two right lines, such as AA', BB f , bisecting the sup- plemental angles between the X ' axes. Again, as will also be- come evident upon a further acquaintance with the subject, the equation INTRODUCTION. 35 (x 2 + y 2 r-) (x 2 + ifr f represents three circles, such as My M' 7 and M" 9 having a common center at the origin. Examples of this kind might be greatly multi- plied, but these are perhaps enough to render the prin- ciple clear, and to fix it in the memory. II. THE METHOD OUTLINED ; IN WHAT SENSE IT IS ANALYTIC. 29. The following is a very simple example of the method by which Analytic Geometry investigates the properties of figures. The beginner, of course, must accept upon authority the meaning of the equations employed. To prove that the tangent to a circle is perpendicular to the radius drawn to the point of contact. Let the axes be rectangular, and the center of the circle at the origin. The equation to the circle is, in that case, /y-2 I 0.2 ~,2 ' U ~ The equation to the tangent at any point P, whose co- ordinates are x\ y' , is x'x -f- y'y r 2 36 ANALYTIC GEOMETRY, The equation to the .radius OP, referred to the same axes, is x f y y'x = (2). Now, it is known that when two equations of the first degree, referred to rectangular axes, interchange the co-efficients of x and ?/, at the same time changing the sign of one of them, they represent two right lines mutually perpendicular. Inspecting (1) and (2), we see that they answer to this condition. Hence, the lines which they represent are mutually perpendicular. 30. Generalizing from the foregoing illustration, we may sum up the method of our science as follows : I. Any locus, the subject of investigation, is repre- sented by its equation. II. This equation is then subjected to such trans- formations, or such combinations with the equations to other loci, as the conditions of the problem may require. III. The geometric meaning of these transformations and combinations, as derived from the convention of co-ordinates, is duly noted. In the same way, the form of the final result is interpreted. Thus the properties of the locus are deduced from the mere form of its equa- tion. 31. We have now reached a point from which to obtain a clear view of the reasons why geometry, when treated by means of algebra, should be called analytic. We must, in the first place, warn the beginner that the reason most obviously suggested by the method just described, is not among them. It is true, certainly, that this method assumes the equation to any locus to be the synthesis, or expression in a single formula, of all its 1NTR OD UCTION. 3? properties. It is true that these properties are drawn out from the equation by a process of real analysis namely, by solving or transforming the equation, thus causing it to give out the several conditions which its original form unites in one symbol. But this obvious fact, that we proceed from a com- plex unity to the elements that have vanished into it, by directly taking apart the unity itself, is not the dis- tinctive reason for calling the Geometry of the Equation analytic ; for it is a fact which does not distinguish it from the Geometry of the Diagram. In this last-named form of the science, the whole scheme of demonstration consists merely in developing what certain definitions and axioms imply : that is, the basis of the reasoning 1 all that gives it force and validity is analytic. And, in fact, the same is true in every department of math- ematics. 32. In short, the term analytic is applied to the Geometry of the Equation, not so much by way of con- trast, as of emphasis. It should not be taken as imply- ing that the Geometry of the Diagram is wholly synthetic, and the Geometry of the Equation wholly analytic, each to the exclusion of the other. Both are analytic, both synthetic ; and in both, the vital principle of the proofs is analysis. But to the Geometry of the Equation, the character of analysis belongs in a special sense, and in a higher degree; just as that of synthesis belongs, in a special sense and higher degree, to the Geometry of the Diagram. It involves, moreover, certain phases of an- alysis, of a higher and more subtle kind than ordinary geometry attains. 33. Now, this special analytic character, it owes to its use of the algebraic symbol. The question therefore 38 ANALYTIC GEOMETRY. naturally arises : How does the use of this symbol bring with it this special character ? The answer is In two ways : first, by giving scope to the analytic tendency inseparable from algebraic investigation ; secondly, by introducing the convention of co-ordinates, with its added elements of analysis. We will illustrate both of these ways somewhat in detail. 34. Special Analytic Character of the Alge- braic Calculus. This is due to two facts : the first, that operations with symbols necessarily thrust into prom- inence the analytic phase of the thinking they imply, while they obscure the synthetic ; the second, that the Theory of Equations the essence of the science of algebra, and the ground upon which all its investiga- tions are based is an application of analysis, pecu- liarly complex and subtle. That these are facts, will best be seen by considering them separately. I. To exhibit the first, let it be borne in mind that every demonstration involves both analysis and syn- thesis analysis, in thinking out the steps connecting the premise with the conclusion; synthesis, in arrang- ing those steps in their due order, and constructing the conclusion as their unity. Now, if we use ordinary language in making this array, clearness can not be secured without stating these steps one by one ; thus we seem to begin with parts, and to construct the conclusion as the whole which they compose. But if we employ algebraic language, our premise is written down in a formula, and, at the outset, our atten- tion is fixed upon it as a whole. The formula is the permanent object of our thought; the operations, the transformations it undergoes, seem transient and subor- dinate, and their results but dependent phases of its INTE OD UCTION. 39 original form. Derived from the formula by a partic- ular series of transformations, while a number of others are equally possible, the conclusion stands in the mind, not as a whole but rather as a part. It appears to us as but one of many elements involved in the original formula elements that may be made to show them- selves, if we apply other transformations. Thus the synthetic phase, though as real here as in using ordi- nary language, is lost to view, and we are only con- scious of the analytic. II. That the Theory of Equations involves a subtle and peculiar form of analysis, is obvious, and need not be enlarged upon. It is sufficient merely to recall its topics and their accessories, such as the Doctrine of Co-efficients, the Theory of Roots their Number, Form, Situation, and Limits, the. Discussion of Series, and the Binomial Theorem. But it is important to mention, that, so controlling a part does this Theory play in the whole science of algebra, and so emphat- ically does it embody a method peculiarly analytic, the science itself, from its earliest years, has been known by the name of Analysis. And it was mainly in allusion to the fact that the Geometry of the Equation brings the discussion of Form within the scope of this Theory, that the title analytic was originally applied to it. 35. Elements of Analysis added by the Con- vention of Co-ordinates. This convention has a twofold analytic meaning: I. First, it asserts that the conceptions of Position and Form are merely relative ones, always implying certain fixed limits to which they are referred the positions of points are their distances from these limits ; the forms of loci are the relative positions of their con- An. Ge. 7. 40 ANALYTIC GEOMETRY. stituent points. This assertion reaches the real essence of Position and Form, and gives to the science based upon it an element of analysis not attained in the Ge- ometry of the Diagram. II. Secondly, in referring the form of every locus to fixed limits by means of the co-ordinates of every point, the convention really determines that form by decom- posing it into infinitely small elements. It thus brings the form under the highest analytic conception known to mathematics, and prepares for its discussion by the various branches of the Infinitesimal Calculus. 3G. To recapitulate : The Geometry of the Equation is called Analytic, first, because its use of algebraic processes puts forward the analytic, and retires the synthetic phase of every demonstration, thus rendering us conscious of investigation rather than of proof; secondly, because its method consists in applying those special modes of analysis which mark the Theory of Equations; thirdly, because its convention of co-ordi- nates penetrates to the real nature of Position and Form, resolving them into their essential constituents Fixed Limits and Distance ; finally and most signifi- cantly, because it resolves all Forms into elements infinitely small, and thus brings the discussion of loci within the sphere of the Infinitesimal Calculus the highest expression of mathematical analysis. 37. These facts constitute a sufficient reason for pre- ferring to call the science Analytic Geometry, rather than Algebraic. The former title, more forcibly than the latter, calls up the characteristics of its method, as they have been detailed above. Moreover, the term algebraic is ambiguous; for it is generally used, in con- trast to transcendental, to characterize operations which INTR OD UCTION. 41 involve only addition, subtraction, multiplication, di- vision, or involution and evolution with constant indices. But Analytic Geometry considers all loci whatsoever, not only Algebraic but Transcendental, whether the latter be Exponential, Logarithmic, or Trigonometric. 38. The superiority of the Geometry of the Equation to that of the Diagram consists partly in its greater brevity and elegance, but mainly in its greater power of generalization. For since the equation to any locus is the complete synthesis of all its properties, our power of investigating and discovering these is limited only by our ability to transform the equation and to determine the form, limits, number, and situation of its roots. And since every equation denotes some locus, the equations of the several degrees may be discussed in their most general forms. Loci may thus be grouped into Orders, according to the degree of their equations, and proper- ties common to an entire Order may be discovered by absolute deduction properties which, if they could be established at all by ordinary geometry, would in- volve the most tedious processes of comparison and induction. III. THE SUBDIVISIONS OF THE SCIENCE. 39. The subdivisions of Indeterminate Geometry refer to the nature of the loci discussed in each ; and these are classified, primarily, according to the form of their equa- tions ; secondarily, according to their situation in a plane, or in space. 40. Hence, the first division of Indeterminate Geom- etry is into TRANSCENDENTAL and ALGEBRAIC. Transcendental Geometry discusses those loci whose equations involve transcendental functions ; that is, 42 ANALYTIC GEOMETRY. functions which depend on either a variable exponent, a logarithm, or one of the expressions sin, cos, tan, etc. Algebraic Geometry, those whose equations in- volve none but algebraic functions; that is, functions which imply only the operations of addition, subtraction, multiplication, division, or involution and evolution with constant indices. 41. Algebraic loci are classed into Orders, according to the degree of their equations referred to rectilinear axes. Thus, the locus whose equation is of the first degree, is called the locus of the First order; those whose equations are of the second degree are called loci of the Second order ; and so on. The loci of the First and Second orders, on account of their simplicity, symmetry, and limited number, are considered to form a class by themselves ; and those of all higher orders are grouped together as a second class. 42. Accordingly, the second division of our subject is that of Algebraic Geometry into ELEMENTARY and HIGHER. Elementary Geometry is the doctrine of loci of the First and Second orders. Higher Geometry is the doctrine of loci of higher orders than the Second. 43. Each of these divisions based upon the form of the equations considered, falls into the province of Plane or of Solid Geometry, according as the equa- tions are between plane co-ordinates, or co-ordinates in space; that is, according as the system of reference is two intersecting lines, giving rise to two co-ordinates for every point ; or three intersecting planes, giving rise to three co-ordinates. INTRODUCTION. 43 Hence, we have GEOMETRY OF Two DIMENSIONS, and GEOMETRY OF THREE DIMENSIONS. 44. The relations which the various divisions of An- alytic Geometry sustain to each other, will be best understood from the following SYNOPTICAL TABLE OF DIVISIONS. ' DETERMINATE. ANALYTIC GEOMETRY INDETERMINATE - f OF TWO DIMENSIONS. TRANSCENDENTAL J V. OF THREE DIMENSION& ALGEBRAIC - HIGHER ELEMENTARY OF TWO DIMENSIONS. OF THREE DIMENSIONS. OF TWO DIMENSIONS. OF THREE DIMENSIONS. 45. In the present treatise, we do not purpose more than an introduction into the wide domain which the foregoing scheme presents. We shall confine ourselves to Elementary Geometry, not entering upon the other departments any further than may prove necessary in order to present the subject in its true bearings. And, restricting ourselves to the discussion of loci of the First and Second orders, we can not within that compass give more than a sketch of the doctrine, methods, and resources of the science, in its present advanced condition. We shall, however, consider the loci of the 44 ANALYTIC GEOMETRY. first two orders, both in a plane and in space. Ac- cordingly, our treatise falls naturally into two Books: the first, upon Plane Co-ordinates; the second, upon Co-ordinates in Space. NOTE. As Greek characters are extensively used in all analytic investi- gations, and as we shall very frequently employ them in the following pages, we subjoin a list for the benefit of readers unacquainted with Greek. A a B/3 Ty A 6 E e Z C H r, alpha. beta. gamma. delta. epsilon. zeta. eta. theta. I i K K A A MyU N v Hf o iota. kappa. lambda. mu. nu. . omicron. pi. Pp 2 a T r T v rho. sigma. tau. upsilon. phi. chi. 2)si. omega. BOOK FIRST: PLANE CO-ORDINATES. PLANE CO-ORDINATES. PART I. THE REPRESENTATION OF FORM BY ANALYTIC SYMBOLS. 46. In applying to plane curves of the First and Second orders the method sketched in the foregoing pages, our work will naturally divide itself into two portions: we shall first have to determine the equations which represent the several lines to be discussed; and then deduce from these equations the various properties of the corresponding lines. Accordingly, our First Book falls into two parts : PART I. ON THE REPRESENTATION OF FORM BY AN- ALYTIC SYMBOLS. PART II. ON THE PROPERTIES OF CONICS. 47. We say Properties of Conies, because, as will be shown hereafter, all the lines of the First and Second orders may be formed by passing a plane through a right cone on a circular base. By varying the position of the cutting plane, its sections with the conic surface (47) 48 ANALYTIC GEOMETRY. will assume the forms of the several lines; and these may therefore be conveniently grouped under the gen- eral name of Conic Sections, or Coriics. It must be added, however, that this use of the term Conies is wider than ordinary. For, speaking strictly, we mean by the Conies the curves of the Second order alone ; namely, the Ellipse, the Hyperbola, and the Parabola; and ordinarily the line of the First order is not included in the term. 48. We shall therefore proceed to develop the modes of representing in algebraic language the Right Line, the Circle, the Ellipse, the Hyperbola, and the Parabola. And as these modes of representation are all derived from the conventions adopted for representing a point, we shall begin by explaining in full the principal forms of those conventions, which were merely sketched in the Introduction. We shall obtain, first, the formulae in ordinary use, or those which may be said to constitute the Older Geom- etry; and, afterward, those belonging to the Modern Geometry, based on what is called the Abridged No- tation. CHAPTER FIRST. THE OLDER GEOMETRY : BILINEAR AND POLAR CO-ORDINATES. SECTION I. THE POINT. BILINEAR OR CARTESIAN SYSTEM OF CO-ORDINATES. 4O. Resuming the topic and diagram of Art. 19, let us examine the Cartesian* system of co-ordinates * Cartesian, from Cartesius, the latinized form of Descartes' name. BILINEAR CO-ORDINATES. 49 more minutely, and develop its elements in complete detail. P being any point on a given plane, two right lines X'X and Y r Y are drawn in that plane, intersecting each other in 0. From P, a line PM is drawn parallel to Y'Y. The distances OM, MP being known, the position of P is de- termined. These distances are called the bilinear co- ordinates of the point P. They are also termed recti- linear, and sometimes parallel, co-ordinates. They are frequently cited as the Cartesian co-ordinates of the point. The co-ordinate MP, drawn parallel to Y'Y, is called the ordinate of the point P. The co-ordinate OM, which the former cuts off from X'X, is called the abscissa of the point. The abscissa of a point is represented by the symbol x; its ordinate, by the symbol y. The two lines X'X and Y'Y are called the axes of reference, or simply the axes. X'X, on which the ab- scissas are measured, is called the axis of abscissas; or, more briefly, the axis of x. Y' Y, parallel to which the ordinates are drawn, is called the axis of ordinates; or, for brevity, the axis of y. The point 0, in which the axis of x cuts the axis of y, is called the origin. The angle YOX is called. the inclination of the axes. It is designated by the Greek letter co, and may have any value from to 180. If w = 90, the axes and co-ordinates are said to be rectangular; if a> has any other value, they are said to be oblique. The two axes, being of infinite length, divide the 50 ANALYTIC GEOMETRY. whole planar space about into four angles. These are numbered to the left, beginning at the line OX. XOY is the first angle; YOX' is the second; XOY', the third; and Y'OX, the fourth. The signs + and are used in connection with the co-ordinates, and are taken to signify measurement in- opposite directions. Positive abscissas are measured to the right from 0, as OM; negative ones, to the left; as OM. Positive ordinates are measured upward from X X, as MP ; negative ones, downward; as MP'" . By attributing proper values to the co-ordinates x and y^ and taking account of their signs, we may represent any point in either of the four angles. Thus, > denotes a point in the first angle. ~ a \ " " " second " ~ a \ " " third " b ) x = y X = ~~ = ~~ ^ a \ " " " fourth " = o) Corollary 1. For any point on the axis of x, we shall evidently have while x, being susceptible of any value whatever, is in- determinate. Hence, the equation just written is the equation to the axis of x. Corollary 2. For any point on the axis of y, we shall have *=o, while y is indeterminate. Hence, the equation last writ- ten is the equation to the axis of y. BILINEAR CO-ORDINATES. 51 Corollary 3. For the origin, we shall obviously have jp = 0, = 0; and these expressions are therefore the symbol of the origin. Remark 1, For the sake of brevity, any point designated by Cartesian co-ordinates is written and cited as the point x y, the point a b, the point (3, 5), etc. These expressions are not to be confounded with alge- braic products. Remark 2, The symbols x and y are used for co- ordinates of a variable point, and are therefore general in their signification. But it is often convenient to repre- sent particular, or fixed, points by the variable symbols ; especially when their positions, though fixed, are arbi- trary. In such cases accents, or else inferiors, are used with the x and y. Thus x' y f , x" y", x l y } , x 2 y 2 , all rep- resent points which are to be considered as fixed, but fixed in positions chosen at pleasure. This distinction between the point x y as general, and points such as x' y', x 2 y 2 as particular, should be carefully remembered. Remark 3, A point is said to be given by its co- ordinates, when their values and that of the angle co are known. And when so given, the point may always be represented in position to the eye. For we have only to draw a pair of axes with the given inclination, and lay off by any scale of equal parts we please the given ab- scissa and ordinate. In practice, it is most convenient to lay off the co-ordinates on the axes, and draw through the points thus determined, lines parallel to the axes ; the intersection of the latter will be the point required. The truth of this will be apparent on inspecting the dia- gram at the head of this article. 52 ANALYTIC GEOMETRY. EXAMPLES. 1. Represent the point ( 5, 3) in rectangular co-ordinates. 2. Represent the point ( 3, 7), axes oblique and w = 60. 3. With the same axes as in Ex. 2, represent the points (1, 2), (-3,4), (-5, -6), (7, -8). 4. Represent the points corresponding to the co-ordinates given in Ex. 3, axes being rectangular. 5. Given w = 135, to represent the points (4, 1), (8, 2) (2, 8)- 6. Given = 90, represent the points (3, 4), (3, 4), ( 3, 4), (-3, -4). 7. With same axes, represent (6, 8) and (8, 6) ; also (6, 8) and (-8,6). 8. With a still = 90, represent the distance between (2, 3) and (4, 5). 9. With same axes, represent the distance between (4, 5) and (-3,2). 10. Axes rectangular, represent the distance between (0, 6) and (_ 5, 5) ; the distance between (0, 0) and (6, 0). Does the latter distance depend on the value of w, or not? POLAR SYSTEM OF CO-ORDINATES. 5O. A second method of representing the position of a point on a given plane, is founded on the fact that we naturally determine the position of any object by finding its direction and distance from our own. Hence, if we are given a fixed point and a fixed right F line OX passing through it, we > shall evidently know the posi- j tion of any point P, so soon as we have determined the angle p v - ..---'' ^ XOP and the distance OP. This method of representing a point is known as the method of Polar Co-ordinates. POLAR CO-ORDINATES. 53 The fixed point is called the pole; the fixed line OX, the initial line. The distance OP is termed the radius vector; the angle XOP, the vedorial angle. It is customary to represent the former by the letter p, and the latter by 6. In this system, accordingly, a point is cited as the point p 6, the point // 6', etc. By attributing proper values to p and #, we may repre- sent any point whatever in the plane PXO. Thus, ' > denotes the point P. 6 = XOP ) P' OF 1 (< p , 5' =AW j prrr '" = XOP" f The student will observe that all the angles 0, 0', 6", 6 f " are estimated from XO toward the left. Corollary 1. For the pole, we evidently have ,o = 0: which may therefore be considered as the equation to that point. Corollary 2. For any point on the initial line to the right of the pole, we have 6 = 2 n TT *. For any point on the same line to the left of the pole, we have = (2 n + 1) TT. * We shall frequently employ the symbol TT = the semi-circumference to radius 1, to denote the angle 180 ; on the principle that angles are measured by the arcs which subtend them. 54 ANALYTIC GEOMETRY. In these expressions, n may have any integral value from upward. Note It is customary to measure the angle 6 from XO toward the left; and the radius vector p, from O in such a direction as to bound the angle. When any distinctions of sign are admitted in polar co-ordinates, the directions just named are considered positive; while an angle measured from OX toward the right, and a radius vector measured from O in the direction opposite to that which bounds its angle, are considered negative. Thus, the point P is commonly denoted by the positive angle = XOP and the positive vector p= OP ; but it may also be represented by the negative angle XOP" = {^ 6} and the then negative vector p = OP. And, again, the same point may be represented by the positive angle 0" = 6 + TT = XOP" and the negative vector p = OP. The student will not fail to note that the signs -f and , as applied to a line revolving about a fixed point, have a signification quite different from that in connection with bilinear co-ordinates. They discriminate between radii vectores, not necessarily as measured in opposite directions, but in directions having opposite relations to the bounding of the vectorial angle. We may therefore define the positive direction of a radius vector to be that which extends from the pole along the front of the vectorial angle; and the negative, to be that ex- tending opposite. In practice, negative values of p and 6 are excluded from the ordinary formulas; but the distinction of sign just explained has an important bearing on the principles of the Modern Geometry. For this reason, it should be mastered at the outset. Remark. To represent any point given in polar co- ordinates, we have only to draw the initial line, and lay off at any point taken for the pole, an angle equal to the given angle 6 : then the distance p being measured from the pole, the required point is obtained. EXAMPLES. 1. Represent in polar co-ordinates the point (p = 8; = 7r). 2. Represent (p = 8 ; 6 = 0) and (p = 8 ; 6=ir). 3. Represent (p = 15 ; 6 =^\ and (p = .5 ; 6 = 3 ~} . DISTANCE BETWEEN TWO POINTS. 55 . Represent (p = 6; P == ^H fp = 6; 0= - - -g-Y and 5. Represent the distance between (p = 8; 6 = j) and * DISTANCE BETWEEN ANY TWO POINTS. 51. Any two points being given by their co-ordinates, the distance o between them is given. For, First: let the two points be x' y' and x" y". Taking P and /Y F P' to represent the points, we / ^^-^/ have P'P=3, / / By Trig., 865, P'P 2 = PQ 2 + P f Q 2 2PQ.P'Q cos PQP = PQ 2 + P'Q 2 + 2PQ.P'Q cos YOX. (Trig., 825). That is, 3 2 = (x" x'Y+(y" y') 2 + 2(x" x'} (y" y)cosa>. Corollary 1. If w = 90, then (Trig., 834) the last term in the foregoing expression vanishes, and we have, for the distance between two points in rectangular co- ordinates, 8> = (x"-x'y+(y"-y<)\ Corollary 2. For the distance of any point x y from the origin, we have (Art. 49, Cor. 3) $2 #2 _j_ y2 _|_ 2 xy cos co : which, in rectangular co-ordinates, becomes An. Ge. 8. 56 ANALYTIC GEOMETRY. Second: let the two points be p' 6' and p" 6". In the , for this case, OP=p',XOP=0'; diagram, for this case, Then, as before, o x ppt == OP 2 + P0 2 20P. PO cos POP; that is, o 2 = p' 2 -f p"' 2 2 ///>' ' cos (0" 6'). Corollary, For the distance of any point p 6 from the pole, we have (Art. 50, Cor. 1) $ 2 = p 2 or o = p : which agrees with our definition of the radius vector. Note In using the formulae of this article, be careful to observe the signs of x / y', x // y // . POINT DIVIDING IN A GIVEN RATIO THE DISTANCE BETWEEN TWO GIVEN POINTS. 52. Let the given points be x l y l9 x 2 y 2 ; and the given ratio, m : n. Denote the co-ordinates of the required point by x and y. By Geom., 313, we have in the / j^ p diagram annexed, where OM= x, MP=y; OM' = x l , M'P = yj 2} & 2) /O PR : RQ :: PP : PP" ; or, x Xt : x 2 x :: m : n. _ mx 2 -\- nx l m -f- n By like reasoning, we find y = DISTANCE DIVIDED IN GIVEN RATIO. 57 If the distance between two points x y l9 x 2 y 2 were cut externally in the given ratio, we should have x x l \ x x 2 : : m : n. mx?> nx, . . x = and, similarly, m n And this we should expect : for, if the point P fell beyond P", the segment PP" would be measured in the direction opposite to P'P, and n would have the negative sign. 53. As the student may have surmised from what he has already noticed, formulae referred to rectangular axes are generally simpler than those referred to oblique. For this reason, it is preferable to use rectangular axes when- ever it is practicable. Hereafter, then, the attention should be fixed chiefly upon those formulae which corre- spond to rectangular axes. In some cases, formulae are true for any value of co. Such are those deduced in the last article. In the examples given hereafter, the axes are supposed to be rectangular, unless the contrary is men- tioned. EXAMPLES. 1. Draw the triangle whose vertices are (2, 5), ( 4, 1), (-2, -6). 2. Find the lengths of the three sides of the same triangle. 3. Express algebraically the condition that x y is equidistant from (2, 3) and (4, 5). Ans. x + y 1. 4. Find the distance between ( 3, 0) and (2, 5). 5. Determine the co-ordinates of the point equidistant from the three points (1, 2), (0, 0), and (5, - 6). 58 ANALYTIC GEOMETRY. 6. Solve Ex. 2, supposing w successively and -r . o 4 7. Find the co-ordinates of the point bisecting the distance between xi y\ and x z y 2 8. The point x y is midway between (3, 4) and ( 5, 8) : find its distance from the origin, 9. x y divides externally the distance between (2, 8) and ( 5, 3) in the ratio 6 : 7. What is its distance from the point midway between (3, 4) and (6, 8) ? 10. Given the points (p = 5 ; 6= 30) and (p 6 ; = 225) to find the distance between them, and the polar co-ordinates of its middle point. TRANSFORMATION OF CO-ORDINATES. 54. A point being given by its co-ordinates, we can at pleasure change either the axes or the system to which they refer it. The process is called Transforma- tion of Co-ordinates. The position of the point is of course not affected by such a change. Its co-ordinates merely assume new values corresponding to the new axes or new system. The transformation is effected by substituting for the given co-ordinates their values in terms of the elements be- longing to the new limits. General formulae for these substi- tutions are easily obtained. In investigating them, it is convenient to consider the subject in four cases; namely, I. To change the Origin, the direction of the axes remaining the same. II. To change the Inclination of the Axes, the origin remaining the same. III. To change System from Bilinears to Polars, and conversely. IV. To change the Origin, at the same time trans- forming by II or III. TRANSFORMATION OF CO-ORDINATES. 59 55. Case First: To transform to parallel axes through a new origin. Let x and y be the co-ordinates of the point for the primitive axes ; and X and Y its co-ordinates for the proposed parallel axes. Let m, n be the co-ordinates of the new origin. Then, if OY, OX represent the primitive axes, and O'Y', O'X' the new : we shall have x = OM, y = MP; X=0'M', Y= M'P; and m = OS, n = SO'. Now, from the diagram, OM=OS+0'M> and that is, x = m -\- X, which are the required formulae of transformation. 5G. Case Second : To transform to new axes through the primitive origin, the inclination being changed. Let co represent the inclination of the primitive axes. Let a. -- the angle made by the new axis of x with the primitive ; and ft = that made by the new axis of y. Drawing the annexed dia- gram, we have x = OM, y -- MP; X = OM', Y = M'P; a=X f OX = M'OR, and ft -- : Y'OX = PM'S. Then, letting fall PQ and M'R perpendicular to OJT, and M'S perpendicular to PQ, we obtain (Trig., 858) MP sin PMQ = QP = RM r + SP. Q X 60 ANALYTIC GEOMETRY. But R M' + SP = OM' sin M OR -f M'P sin PMS. .-. MP sin PMQ = (W sin J^P 072 + M'P sin P.'; or, y sin o> = A r sin a -\- Y sin /9. By dropping perpendiculars from P and Jf' upon OJ 7 , and completing the diagram, we should obtain by the same principles x sin co = X sin (co a) -j- Y sin (a> ,9). The details of the proof are left to the student. We have, then, as the required formulae of transforma- tion, x sin co X sin(ft> ) + l^sin (w /9), ?/ sin (o = X sin -f- !F" sin /9 : which include all cases of bilinear transformation. The particular transformations which may arise under this general case, are various ; those of most importance are as follows : Corollary 1. To transform from rectangular axes to oblique, the origin remaining the same. Making co = 90 in our general formulae, we obtain (Trig., 834, 841) x = X cos a -\- Y cos /?, y X sin a -f- Ysin /9. Or we may obtain the formulae geometrically, as follows : Supposing the angle YOX to be a right angle, OM will obviously coincide with OQ, and MP with QP, and we shall have OM=OR + M'S = OM' cos M'OR -f M'P cos PM'S . . x = X cos a -\- Y cos /9 ; MP = RM + SP = OM sin MOR + M'P sin PM'S . . y -X" sin a -\- Y sin ft. TRANSFORMATION OF CO-ORDINATES. 61 Corollary 2. To transform from oblique axes to rectan- gular, the origin and the axis of x remaining the same. Here a = 0, and /3 = 90: hence, (Trig., 829, 834,) x sin a) = X sin co Y cos co, y sin co = Y. Geometrically as follows : OX, OY being the primitive axes, and OX, F the new : x=OM,y = MP; X= OM' Y=MP. By Trig., 859, OM=OM' M'M = OM f M'P cot PMM' ; or, x = X Ycot co. . ' . x sin co =X$m co Fcos co. Again, by Trig., 858, MPsmPMM=MP; or, y sin co = Y. Corollary 3. To revolve the rectangular axes through any angle 6. Here a = 6, [3 = 90 + 6, and co = 90. Substituting in the general formulae, we obtain x = X cos -f Y cos (90 + 6), y = X sin 6 + T sin (90 + 0) : expressions which, by Trig., 843, become x = X cos F sin 6; y = Xsin 6 + Fcos 6. Or we may deduce the formulae independently, from the diagram annexed. Here x = OM, y = MP; X=OM r , Y=M'P; 6 = M f OS = ROM=M'PQ. Drawing MS and M'Q perpendicular respect- ively to OS and PQ, we shall have O M SX \ 62 ANALYTIC GEOMETRY. OM= OSM'Q = OM' cos M'OSM'P sin M'PQ, MP = SM' J r QP= OM' sin M ' OS + M'P cos M'PQ. That is, M y=X sin# + Fcosfl. 57. Case Third: To transform from a bilinear to a polar system of co-ordinates, or conversely. Let OX, OF be the primitive rectangular axes, and OX f the initial line of the proposed polar system. Let a = the angle which the initial line makes with the axis of x : it will be positive or neg- ative according as OX' lies above or belotv OX. It is obvious from the diagram that we shall have x = p cos (6 + a), y = p sin (6 -f- a) : formulae by which we can either find x y in terms of p 0, or p 6 in terms of x y. In applying them, strict atten- tion must be paid to the* sign of , according to the con- vention named above. Note. We have confined the discussion of this case to the change from rectangular axes, as this alone is of very frequent occurrence in practice. It may be well, however, to give the formulas of the general case, in which is supposed to have any value whatever. Assuming, in the above diagram, the angle YOX to be oblique, we should have in the triangle OMP, (Trig., 867,) p: x :: sm u : sm {w (0 + a)} .'. x- fa , N p : y : : sm u : sm (0 + ) y= gn P sin ( e + These formulas evidently become those obtained above, when = 90. TRANSFORMATION OF CO-ORDINATES. 63 Corollary. If the axis of x is taken as the initial line, a ; and we have x = p cos 0, y = p sin : a set of formulae in very extensive use. 58. Case Fourth: To combine a change of origin with any other transformation. To effect this, we first apply the formulae of Art. 55, and thus pass to the new origin with a system of axes parallel to the primitive. That is, in effect, we remove the original system to the new position which the proposed origin requires. The formulae for the special transforma- tion in hand are then applied, and the whole change is accomplished. From the nature of the formulae in Art. 55, it is obvious that the present case is solved analytically by merely adding to the expressions for x and ?/, the co-ordinates m and n of the new origin. Thus, in general, by Art. 56, X sin (co a) 4- Y sin (to dl) x = m -f / / ^ , sin -f-coj . X sin a -f Y sin 8 ""; + ; -i5^r ' To pass from bilinears to polars when the pole is a dif- ferent point from the origin, we have x = m -f- p cos (0 -f a), y = n -\- p sin (6 -f- a). 59. It should be observed that in applying these various formulae, great care is to be exercised in respect to the signs of the constants involved. And in the ex- amples which follow, where given points are to be trans- formed, the same care should be taken with respect to all An. Go. 9. 64 ANALYTIC GEOMETRY. the known co-ordinates. It is recommended that the student reduce every problem to drawing, at least in the earlier stages of his studies. In no other way will he readily acquire the habit of bringing every analytic process to the test of geometric interpretation. EXAMPLES. 1. Given the point (5, 6) : what are its co-ordinates for parallel axes through the origin (2, 3) ? 2. Transform ( 3, 0) to parallel axes through (4, 5); to parallel axes through (5, 3); through (3, 5); through (-3,0). 3. Given in rectangular co-ordinates the points (1, 1), ( 1, 1), (2, 1), and (3, 3): find their polar co-ordinates, the origin being the pole, and the axis of x the initial line. 4. Transform the points in Ex. 3, supposing the pole to be at ( 4, 5), and the initial line to make with the axis of # an angle = _ 30. 5. Solve Ex. 4, on the supposition that a = 45. 6. Find the rectangular co-ordinates of (P = 3; 6 = 60) and (p = 3 ; = 60), the origin and axis of x coinciding respect- ively with the pole and the initial line. Find the same, supposing the origin at (2, 1), and the angle a = 30. 7. The co-ordinates of a point for a set of axes in which u = 60, satisfy the equation Sx -f- 4?/ 8 0: what will the equation become when transformed to w' = 45, a = 15? What, when in addition the origin is moved to ( 3, 2) ? 8. Transform x 2 -f y 1 r" 1 to parallel axes through (a, I). 9. It is evident that when we change from one set of rectangular axes to another having the same origin, x 1 -f~ y l must be equal to x n -j- 3/ 2 , since both express the square of the distance of the point from the origin. Verify this by squaring the expressions for x and y given in Art 56, Cor. 3, and adding the results. 10. Transform to rectangular co-ordinates the following equations in polar ; origin same as pole, and axis of x as initial line : c*; p 2 = c 2 cos 2 (9. INTERPRETATION IN GENERAL. 65 GENERAL PRINCIPLES OF INTERPRETATION. ' GO. In an important sense, the whole science of Analytic Geometry may be said to consist in knowing how to translate algebraic symbols into geometric facts. Supposing that we have solved any geometric problem by analytic methods, our result must be some algebraic expression. Hence, in the end the question is, How shall we interpret that expression into the geometric property which we are seeking ? The principles governing such an interpretation are to be mastered in their fullness, only through an exhaustive study of the whole field of Analytic Geometry. But there are a few of them, which, lying at the root of all the others, are of universal application, and must be de- termined at the outset. In the following articles, we will state and establish them. The student may notice that the illustrations and, in some cases, the phraseology refer to Cartesian co-ordinates ; but this does not affect the generality of the principles, since we can convert any geometric expression into one relating to the Cartesian system by transformation of co-ordinates. 61. A single equation betiveen plane co-ordinates repre- sents a plane IOCHS. This theorem follows directly from the corollary of Art. 27. It may be well, however, to add here some illustrations of the principle. It is plain that if we have any equation between two variables, as ax* -f bx*y -f cxf + dtf +/= 0, we may assign to x any value we please, and obtain a corresponding value for y. The unknown quantities in such an equation are therefore not determinate. On the contrary, there is a series of values, infinite in number, 66 ANALYTIC GEOMETRY. any of which will satisfy the equation. Taking these values as denoting the co-ordinates of a point, the equa- tion must represent an infinite number of points. But, though infinite in number, these points can not be taken at random; for the equation can not be satisfied by values arbitrary for y as well as x, but only by such values of y as its own conditions require in answer to the assigned values of x. Hence, the infinite series of points which the equation represents, conforms in all its members to the same law of position : a law ex- pressed in the uniform relation which the equation establishes between the values of its variables. Such a series of points must constitute a line. To illustrate by a diagram, we may suppose that in a given equation between two va- riables, x has the value Om. Corresponding to this there will be, let us say, three values of y, represented by mp, mq, mr. We thus determine three points, p, q, r. Again, supposing x = Om' , we determine three other points, p f , q f , /. And again, making x -- Om" , we obtain the points p", q", r". We may continue this process as long as we please, and determine any number of points, by assigning successive values to x. By taking these sufficiently near each other, and drawing a line through the points thus found, we may determine the figure of the locus which the equation represents. Remark. We must here carefully recall the proposition, stated in the Introduction, that to render the principle of this article uni- versally true we must take into account imaginary loci, loci at infinity, and cases where a locus degenerates into disconnected points or a single point. For example, the equation x* + y* + 1 = INTERPRETATION IN GENERAL. 67 can not be satisfied by any real values of x and y : consequently, in order to bring it within the terms of our principle, we must say that it denotes an imaginary locus. It should be borne in mind that this amounts to saying that it has no geometric locus. Again, the equation Ox + Oy + c = can be satisfied by none but infinite values of x and ?/. All the points on its locus are therefore at an infinite distance from the origin, and it can be brought within the terms of our principle only by saying that it denotes a locus at infinity. This, too, is only another way of saying that it has no geometric locus. Again, the equation (x - a) 2 + 7/ 2 = obviously can not be satisfied unless we have, at the same time, (x- a) 2 =0 and if = 0; that is, it admits of no values except x = a and y = 0. Accordingly, if we would bring it within our principle, we must say that it denotes a locus which has degenerated into a point situated on the axis of x, at a distance a from the origin. We shall learn here- after that this point is an infinitely small circle, having (a, 0) for its center. 62. Any two simultaneous equations between plane co- ordinates represent determinate points in a given plane. For, given two equations between two variables, we can determine the values of x and y by elimination. Moreover, the points which such a pair of simultaneous equations determines, are the points of intersection common to the two lines which the equations respectively repre- sent. For it is obvious that the values of x and y found by elimination, must satisfy both of the equations ; hence, the points which these values represent must lie on both of the lines represented by them : that is, they are the points common to those lines. 68 .ANALYTIC GEOMETRY. Corollary 1. Hence, To find the points of intersection of tivo lines given by their equations, solve the equations for x and y. Remark. T he geometric meaning of simultaneity and elimination may be made clearer by the accompanying diagram. Let the curve A be represented - / Q , ^^ % by the equation and the curve B by a second equation y = ?(*)* (2): then, supposing A to intersect B in the points p r and p ff , and in these points only, it is obvious that the x and y of equation (1) will become identical with the x and y of equation (2) when we substitute in both equations the co-ordinates of p' and p" ; for we shall then have, in both equations, x = Om' or Om", y = m'p' or m rf p". And it is equally plain that the x and y of the two equa- tions will not be the same, but different, so long as they represent the co-ordinates of any other point. Thus, if in (2) we make x = OM, we shall have y = MP : values which, it is manifest from the figure, the x and y of (1) can not have. Since, then, the variables in the equations to different curves will in general have different values ; and since, even in curves that intersect, the variables in their re- spective equations will become identical in value only at # Equations (1) and (2) are read "y = any function of x" and "y any other function of x." INTERPRETATION IN GENERAL. 69 the points of intersection : we learn the important principle, that to suppose two geometric equations simultaneous is to suppose that their loci intersect. In short, simultaneity means intersection ; and elimination determines the intersect- ing points. Corollary 2, Two equations, of the m th and n th degree respectively, represent mn points. For (Alg., 246), elimination between them involves the solution of an equation of the mn th degree ; and such an equation (Alg., 396, 397) will have mn roots. Two lines, therefore, of the m th and w th order respect- ively, intersect in mn points. Two lines of the first order, for example, have but one point of intersection ; two of the second, have four ; a line of the first order intersects one of the second, in two points ; a line of the second order cuts one of the third, in six; and so on. It should be observed that any number of these mn points may become coincident : a fact which will be indi- cated, of course, by the existence of a corresponding number of equal roots in the equation obtained by elimi- nation. Or, any number of them may become imaginary ; or, in certain cases, be situated at infinity : facts respect- ively indicated by the presence of imaginary and infinite roots. 63. An equation lacking the absolute term represents a line passing through the origin. For every such equation will be satisfied by the values x = 0, y = ; and these (Art. 49, Cor. 3) are the co- ordinates of the origin. 64. Transformation of co-ordinates does not alter the degree of an equation, nor affect the form of the locus^which it represents. 70 ANALYTIC GEOMETRY. For, supposing the equation to be originally of the ?i th degree, the term which tests that degree may be written Mx r y s ^ in which r -j- s = n. Now the most gen- eral case of transformation (compare Arts. 56, 58) will require us to substitute for this an expression of the form M(aX+ IY+ ey (a'X+ b'Y+ / 4- lx* - Sy = 0. SPECIAL INTERPRETATION. 71 SPECIAL INTERPRETATION OF PARTICULAR EQUATIONS. 65. The foregoing principles illustrate the doctrine that there is a general connection between an equation and the locus of a point. But every curve * has its own particular equation, and we may appropriately close our discussion of the Theory of Points by explaining briefly how to discover the form and situation of a curve from its equation. 66. The Special Interpretation of an Equation consists in tracing, by means of determined points, the curve which it represents. 67. In order to trace any curve from its equation, we solve the equation for either of its variables, say for y. We then assign to x various values at pleasure, and com- pute the corresponding values of y. Then, drawing the axes, we lay down the points corresponding to the co- ordinates thus found. A curve traced through these points will approximately represent the locus of the equation. Could we take the points infinitely near each other, we should obtain the exact curve. 68. Attention to certain characteristics of the given equation and of the values of the variable for which it is solved, will enable us to decide certain questions con- cerning the peculiar form of the corresponding curve. These algebraic characteristics, and their geometric meaning, AVC will now specify. 69. If the given equation is of a degree higher than the first, for every value assigned to x there will arise *It is customary to call any plane locus a curve, even though this in- volves the apparent harshness of saying that the right line or an isolated point is a curve. 72 ANALYTIC GEOMETRY. two or more values of y. The several points correspond- ing to the common abscissa are said to lie on different PORTIONS of the curve. Thus, in the figure, the points p, p', p" lie on one portion of the curve represented; the points q, q r , q" on another; and the points r, r', r" on a third. The LIMITS of a portion that is, the points where it merges into another portion are the points whose abscissas cause two values of the ordinate to become equal. Corollary. Hence, To test whether a curve consists of several portions, note whether its equation is of a degree higher than the first. To find the limits of the portions, observe what values of x give rise to equal roots for y. TO. If all the values assigned to x within the limits separating two portions of a curve, make the y'a of the two portions numerically equal but of opposite sign, the corresponding points of these portions will be equally distant from the axis of x. Similar conditions with respect to the axis of y will determine points equally distant from that axis. Two portions of a curve, whose points are thus situated with reference to either axis, are said to be SYMMETRICAL to that axis. A curve is symmetrical, when all its por- tions taken two and two are symmetrical. Corollary. Hence, To test for symmetry, note whether the values of either variable, corresponding to all values of the other between the limits of two portions of a curve, appear in pairs, numerically equal with contrary 71. If any value assigned to x gives rise to imaginary SPECIAL INTERPRETATION. 73 values for y, the corresponding point or points will be imaginary. That is, the curve is interrupted at such points. And if, between any two values of either vari- able, the corresponding values of the other are all imaginary, the curve does not exist between the corre- sponding limits. Thus, in the curve by solving for y we obtain y= -Vx 2 a 2 : a so that y is real for every value of x which lies beyond the limits x = a and x = a, but is imaginary for every value of x lying between them ; and the curve is interrupted in the latter region. When the extent of a curve is nowhere interrupted, and it suffers no abrupt changes in curvature, * it is said to be CONTINUOUS. A curve may be either continuous throughout or composed of continuous parts. Corollary, To test for continuity in extent, note whether the equation to a curve gives rise to limiting values of either variable, beyond or between which the values of the other are imaginary. Remark, To test for continuity in curvature, we employ the Differential Calculus. The continuous parts of a curve are called its BRANCHES. A branch should be distinguished from a portion of a curve : a branch may consist of several portions; or a portion, of several branches. * Cumaturc i. e. the rate at which a curve deviates from a riylit line. 74 ANALYTIC GEOMETRY. A branch of a curve may degenerate into isolated points, or a single point: such points are called CONJU- GATE POINTS. Corollary. The number and extent of the branches belonging to a curve may often be determined by exam- ining the limits beyond or between which its equation gives rise to imaginary values of the variables. Thus, if y will be real for all values of x lying between the limits x = a and x = a, but imaginary for all lying beyond. The curve therefore consists of a single branch, surround- ing a portion of the axis of x w r hose length = 2a. If _ _^_ b 2 a y is real for all values of x lying beyond the limits x = a and x = a, but imaginary for all lying between. Hence, the curve consists of two branches, separated by a portion of the axis of x whose length = 2a. If y = 2 Vpx, y is imaginary for all negative values of x, but real for all positive values. Hence, the curve consists of a single infinite branch, extending from the origin toward the right. Remark. Conjugate points belong to a class, known as SINGULAR POINTS, whose existence can not in general be tested without the aid of the Differential Calculus. If, however, a given equation is obviously satisfied by none but isolated values between certain limits, its locus between those limits will consist of conjugate points. SPECIAL INTERPRETATION. 75 73. We add a few examples, merely premising that it is often convenient first to find the situation of the axes to which the given equation is referred. This is done by making the x and y of the equation suc- cessively equal to zero : the resulting values of y and x (Art. 62, Cor. 1) are the intercepts made by the curve on the axis of y and of x respectively. We have given, for the sake of widening a little the student's view of the subject, a few equations to Higher Plane Curves, both Algebraic and Transcendental. These curves, of course, are beyond the province of the present work; and the reader who desires full information in regard to them is referred to SALMON'S Higher Plane Curves or to the writings of PLUCKER, PONCELET, and CHASLES. It will be most convenient, in tracing the curves of the following examples, to use paper ruled in small squares, whose constant side may be taken for the linear unit. Let the limits of the imaginary values, if such exist in any equation, be first found : then, within the sphere of real values, let the abscissas be taken near enough together to determine the figure of the curve. The axes are rectangular: as we shall always suppose them in examples, unless the contrary is indicated. EXAMPLES. 1. Represent the curve denoted by y 2x + 3. Making, successively, x = and y = 0, we obtain y = 3 and x = - . The curve therefore cuts the axis of y at a distance 3 above the origin, g and the axis of x at a distance - to the 1 and lay down the corresponding points. and the axis of x at a distance to the left of the origin. Draw the axes 76 ANALYTIC GEOMETRY. The equation being of the first degree, the curve consists of but one portion, y is obviously real for all real values of x : the curve is there- fore of infinite extent. Making x successively 3, 2, 1, 1, 2, 3, the corresponding values of y are 3, 1, 1, 5, 7, 9. Laying down the points (- 3, - 3), (- 2, - 1), (- 1, 1), (1, 5), (2, 7), (3, 9), we find that they all come upon the right line drawn through (0, 3) and * s therefore the curve represented by the given equation. if 1 / ' 2. Interpret ~ + |- = 1. Making x = 0, we obtain or the curve cuts the axis of y in two points: one at the distance 2 above the origin, and the other at the same distance below it. Making y = 0, we obtain whence the curve cuts the axis of x in two points equally distant from the origin, and on opposite sides of it. Since the equation is of the second degree, the curve consists of two portions; and as the values of y coincide and = when x. = 3, these portions are separated by the axis of x. Solving for y, we find o y = 3 1/9 - x 2 : hence, y will become imaginary when x > 3 or x < 3. The curve therefore has no point beyond its intersections with the axis of x. But for every value of x between the limits 3 and 3, y is real; that is, the curve consists of a single continuous branch. Making, now, x successively equal to 3, 2.5, 2, 1, 0, 1, 2, 2.5, 3, the corresponding values of y are 0, 1.2, 1.5, 1.9, 2, 1.9, 1.5, 1.2, 0. From these values, we see that the curve is sym- metrical to both axes ; and, laying down the sixteen points thus found, we determine the figure of the curve as annexed. It is an ellipse. 3. Interpret the equation y mx. x 2 - y 2 4. Interpret ^ ~ = 1. This curve is an hyperbola. 5. Interpret y* = 4#. This curve is a parabola. THE EIGHT LINE. 77 6. Interpret y = x 3 and y 2 = x z . The first of these curves is a cubic parabola; and the second, a semi-cubic parabola. 7. Interpret x z (a x) y 2 - = 0. This is known as the Cissoid of Diodes. 8. Interpret x*if = (a 2 y 2 } (b + y) 2 . This is the Conchoid of Nicomedes. 9. Interpret x = versin -1 y * 1/2 ry y' L . This is the Com- mon Cycloid. 10. Interpret y sin ar, the Ckrve o/ /Smes ; y = cos #, the o/" Cosines ; and y log x, the Logarithmic Curve. SECTION II. THE RIGHT LINE. THE RIGHT LINE UNDER GENERAL CONDITIONS. 74. In discussing the mode of representing the Right Line by analytic symbols, we shall in the first place have to determine the various forms of the equation which rep- resents any right line; that is, our problem will be to represent the Right Line under general conditions only. This accomplished, we shall then pass to the more partic- ular forms of the equation those which represent the Right Line under such special conditions as passing through two given points, passing in a given direction through one given point, etc. 75. In showing that a certain curve is represented by a certain equation, we may proceed in either of two ways. * This is the notation for an inverse triyonomct.-ic function, and is in this case read " the arc whose versed-sine is y." Similar expressions occur in terms of the other trigonometric functions : as, x sin 1 y;x = tan~ a y ; etc., read " x = the arc whose sine is y," " x= the arc whose tangent is y," and so on. 78 ANALYTIC GEOMETRY. First, we may begin by assuming some fundamental property of the curve, define the curve by means of it, and, with the help of a diagram which brings it into rela- tion with elementary geometric theorems, embody it in an equation between the co-ordinates of any point on the curve an equation from which all other prop- erties may be deduced by suitable transformations. Or, secondly, we may begin without any geometric assumptions except those on which the convention of co-ordinates is founded; may take an equation of any degree, in its most general form; and, by the purely analytic processes of algebraic, trigonometric, or co- ordinate transformation, reduce the equation to such simpler forms as will show us the species, figure, and properties of the corresponding curve. The latter method is the purely analytic one ; the former mingles the processes of geometry and analysis. TO. In the present Book, both of these methods will be applied in succession. It will be natural to set out from the geometric point of view: for in this way we shall secure simplicity and clearness, by constantly bringing the analytic formulae and operations to the test of interpretation by a diagram. After the char- acteristic forms of the equation to any locus have been obtained by the aid of geometry, and the beginner has become familiar with their geometric meaning, he may safely ascend to the higher analytic stand- point, and will be able to descend from it with some real appreciation of the scientific beauty which it brings to light. We proceed to apply to the Right Line the two methods mentioned, and shall follow the order which has just been indicated. EQUATION TO EIGHT LINE. 79 I. GEOMETRIC POINT OF VIEW : THE EQUATION TO THE RIGHT LINE IS ALWAYS OF THE FIRST DEGREE. TT. There are three principal forms of the equation to the Right Line, arising out of the three sets of data by which the position of the line is supposed to be deter- mined. Each of these will prove to be of the first degree. 78. Equation to the Right Line in terms of its angle wills the axis of .r and its intercept on the axis of ?y. It is obvious, on inspecting the diagram, that the position of the line DT is given when the angle DTX* and the intercept OD are given. Let x and y denote the co-ordi- nates OM, MP of any point P on the line. Let the angle DTX= a, and let OD = 6. Then, drawing MR parallel to DT, we shall have (Trig, 867) OR : OM : : sin OME : sin OEM. That is, b y : x : : sin a, : sin ( to). sin a. . 7 ' = srr=-5j* + J; or, putting m = -. sm a , sm (co a) y = mx -f b: which is the equation in the terms required. ~- :: ~ The student will not fail to notice that the angle which a lino makes with the axis of x is always measured, positively, from that axis toward the left; an angle measured from the axis toward the right, is negative. This principle holds true of the angle between any two lines. An. Ge. 10. 80 ANALYTIC GEOMETRY. Corollary 1, In the equation just obtained, the axes are supposed to have any inclination whatever. If the axes are rectangular, co = 90 and m = tan a. Hence, in y = mx -\- 5, when referred to rectangular axes, m denotes the tangent of the angle which the line makes with the axis of x; but when the equation is referred to oblique axes, m de- notes the ratio of the sines of the angles which the line makes with the two axes respectively. Remark. In interpreting an equation of the form y = mx -\- 6, and tracing the line corresponding to the values m and b have in it, account must be taken of the signs of those constants. >:< The constant m will be posi- tive or negative according as the angle a is less or greater than co. For m = ~ : which is positive or neg- sin (co ) ative upon the condition named, according to Trig., 829 ; since a is supposed not to exceed 180. The constant b (Art. 49) will be positive or negative according as the intercept on the axis of j falls above or below the, origin. Thus, in the case of the line in the diagram, m is negative, and b positive. If the axes are rectangular, the sign of m will be -f- or - according as a. is acute or obtuse. For, in that case, m = tan a ; and the variation of sign is determined by Trig., 825. Corollary 2. We can thus determine the position of a right line with respect to the angles about the axes, by merely inspecting the signs of its equation. * The quantities m and b are constants, since they can have but one value for any particular right line. But the equation is true for any right line, because we can assign to m and 6 any values we please. They are hence called arbitrary constants. EQUATION TO EIGHT LINE. 81 If m is negative, and b positive, 'the line crosses the axis of y above the origin, and makes with the axis of x an angle greater than CD : it therefore crosses the latter at some point to the right of the origin, and so lies across the first angle. If m and b are both positive, the line lies across the second angle. If m and b are both negative, the line lies across the third angle. If m is positive, and b negative, the line lies across the fourth angle. [The student may draw a diagram and verify the last three statements.] Corollary 3. If m = 0, we shall have sin a = . . a = 0, and the line will be parallel to the axis of x. Corollary 4. If m = oo, sin (to a) = . * . a = a, and the line will be parallel to the axis of y. If m = oo when the axes are rectangular, we shall have tan a. = oo . . a = 90, and the line will be per- pendicular to the axis of x : which is essentially the same result as before. Corollary 5, If b = 0, the line must pass through the origin. But, in that case, the equation becomes y = mx: which is therefore the equation to a right line passing through the origin . 79. Equation to the Right Line in terms of its intercepts on the two axes. The diagram shows that 82 ANALYTIC GEOMETRY. the position of DT- is given when OT and OD are given. Let OT=tf,and OD=b; rep- resent by x and y, as before, the co-ordinates OM, MP of any point P on the line. By similar triangles, we have OT: OD : : MT : MP ; that is, a : b : : a x : y. which is the symmetrical form of the required equation. Remark 1. When interpreting an equation of this form, the signs of the arbitrary constants a and b must be observed. By doing this, we can fix the position of the line with respect to the four angles, as in the preceding article. When a and b are both positive, the line lies in the first angle, as in the diagram. When a is negative, and b positive, the line lies in the second angle. When a and b are both negative, the line lies in the third angle. When a is positive, and b negative, the line lies in the fourth angle. Remark 2. This form of the equation to the Right Line is much used on account of its symmetry. It also deserves to be noticed on account of its resemblance to the analogous equations to the Conies, which we shall de- velop in due time. It is applicable, as is manifest from the investigation, to rectangular and oblique axes alike. EQUATION TO RIGHT LINE. 83 SO. Equation to the Right Line in terms of its perpendicular from the origin and the angle made by the perpendicular with the axis of x. By examining the diagram, it becomes evident that the po- sition and direction of DT are given, if the length of OR per- pendicular to Dl\ and the angle TOR which it makes with OX, are given. Let OR= p; and let /o the angle TOR = a, whence the angle DOR = (o . Then, a and b representing the intercepts OT and OD as before, we have (Trig., 859) a = - b = cos a cos (at a) Substituting these values of a and b in the equation of Art. 79, we obtain x cos ay cos (o) a) _ -, P P Clearing of fractions, x cos a -f- y cos (at a) = p: which is the equation sought. Remark. The co-efficients of x and y in the foregoing equation are called the direction-cosines of the line which the equation represents. In using this form of the equation, it is most convenient to suppose that the angle a may have any value from to 360, and that the perpendicular p is always positive that is, (Art. 50, Note,) measured from in such a direction as to bound the angle. This convention can not be too carefully remembered. 84 ANALYTIC GEOMETRY. Corollary. If the axes are rectangular, we shall have (Trig., 841) x cos a -f- y sin a = p : a form of the equation having the greatest importance, on account of its relations to the Abridged Notation. 81. The three forms of the equation to the Right Line are therefore as follows: y = mx -f b (1), x cos a -f- y cos (to a) = p (3). They are all of the first degree. Either of them may be derived from any other, by merely substituting for the constants in the latter their values in terms of those involved in the form sought. For (see diagram, Art. 80) we have b P A = -- ; a = ; o = P cos a cos ( A ' O u 4 13. In which angle does the line 5 x i/3 -y 21ie? 14. Trace the lines /> cos (0 45) = 8 and p sin = 6. What is the value of a in the second line, and on which side of the initial line is it measured ? 15. Find the intercepts of the line 5x + ly 9 = 0, and trace the line. EIGHT LINE IN STRICT ANALYSIS. 87 II. ANALYTIC POINT OF VIEW: EVERY EQUATION OF THE FIRST DEGREE REPRESENTS A RIGHT LINE. 85. The most general form of the equation of the first degree in two variables, is ^ + % + <7=o, in which A, B and C are arbitrary constants, and may have any value whatever. We propose to prove that this equation, no matter what the values of A, and C, always represents a right line. Solving the equation for y, we obtain A C y- ~S X ~B' That is, y is always equal to x multiplied by an arbitrary constant, plus an arbitrary constant. In other words, the equation is always reducible to the form and therefore (Art. 78) always represents a right line. 86. A second proof of the same proposition, by means of the trigonometric function which the equation implies, is as follows : Write the equation in the form (to which we have just shown that it is always reducible) y mx -f b : in which m and b are merely abbreviations for the arbitrary A C constants -^ and -^ . Now the equation, being true for every point of its locus, must be true for any three points x'y', x"y", x'"y" f , Hence, y ' = mx f +6 (1), y" = x" + ft (2), y'" = mx" t -}-'b (3). An. Ge. 11. 88 ANALYTIC GEOMETRY. Supposing, then, that the abscissas are taken in the order of magnitude, the equation y = mx -f b shows that the ordinates will also be in the order of magnitude ; that is, if we take x" greater than a/, and x f " greater than x" , we shall either have y" greater than y', and y" 1 greater than y h ', or else y" less than y', and y'" less than y". Accordingly, we subtract (1) from (2), and (2) from (3), and by comparison obtain y x" x' (4)- Since the form of the locus we are seeking, whatever it be, is (Art. 64) independent of the axes, let us for convenience refer the equation to rectangular axes, OX and OY. Draw the indefinite curve AB, to represent for the time being the unknown locus. Take P, P", P" as the three points x'y', x"y", x'"y'"; let fall the corresponding ordinates PM', P"M", P"M"'; draw the chords P'P", P"P"; and make PR, P'S parallel to OX. Then, from (4), we have P'R P"S that is, (Trig., 818,) tan P"P',K = tan P"'P"# .-. P"P f R=P'"P'S. Hence, the three points P r , P", P" lie on one right line. But P', P", P" are any three points of the locus. P" may therefore be anywhere on it between P' and P //r , and RIGHT LINE IN STRICT ANALYSIS. 89 is independent * of them. Hence, as we may take the points as near each other as we please, all the points of the locus lie on one right line ; that is, the locus itself is a right line. 87. A third proof of the same proposition is furnished by transformation of co-ordinates. being given for geometric interpretation, is of course referred to some bilinear system of co-ordinates. Sup- pose the original axes to be rectangular, and let us transform the equation to a new rectangular system having the origin at the point x'y r . To effect this, write (Art. 56, Cor. 3, cf. Art. 58) for x and y in the given equation x' -j- x cos y sin 6 and y' -j- x sin 6 -f- y cos 6. This gives us (A cos -f B sin 0} x (A sin 6 B cos B)y -f Ax' -f / -f C= as the equation to the unknown locus, referred to the new axes. Since a/, ?/, and are arbitrary constants, we may subject them to any conditions we please. Let us then suppose that x f and y' satisfy the relation and that the value of 6 is such that A cos 6 -f- B sin 6 i. e. tan = : ~ This is essential to the argument. For three points of a curve may lie on one right line, if the third is determined by the other two. Thus, in the annexed diagram, P , P' , P" are three points on the curve AB ; yet they all lie on the right line P P" . The reason is, that P" is determined by joining P and P". 90 ANALYTIC GEOMETRY. The first of these suppositions means that the new origin is taken somewhere on the unknown locus, since its co- ordinates satisfy the given equation; the second, that the new axis of x makes with the old, an angle whose A tangent is -- -^ Applying these suppositions to the transformed equa- tion, we obtain, after reductions, Hence (Art. 49, Cor. 1) the locus coincides with the new axis of x. And, in general, since the equation Ax -\- By -f 0=0 can, upon the suppositions above made, always be reduced to the form y = 0, we conclude that it represents a right line, which passes through the arbitrary point x'y', and makes with the primitive axis of x an angle whose tangent is found by taking the negative of the ratio between the co-efficients of x and y. That is to say, since these co- efficients are also arbitrary, Ax -j- By -f- C = is the Equation to any Right Line. 88. We have thus shown, by three independent demon- strations, that we can take the empty form of the General Equation of the First Degree, and, merely granting that it is to be interpreted according to the convention of co- ordinates, evoke from it the figure which it represents. It must not be supposed, however, that we ivere ignorant of the figure of the Right Line when we set out upon the foregoing transformations. On the contrary, each of the three demonstrations just given presupposes the figure of the Right Line, and certain of its properties. What we did not know is, that the equation Ax -f- By -)- C= represents, and always represents, that figure. EQUATION OF THE FIRST DEGREE. 91 It is important to call attention to this, because the significance of the result just obtained is sometimes over- estimated; and because the case of the Right Line is different in this respect from that of any higher locus. In the strictly analytic investigation of loci of higher orders than the First, not even the figure of the curves is presupposed, but is conceived as being learned for the first time from their equations. But the whole scheme of Analytic Geometry takes the figure and elementary properties of the Right Line for granted ; as is obvious from the nature of the convention of co-ordinates and of the theorems for transformation. 89. Starting, then, from Ax -\- By -\- C as the equation to the Right Line in its most general form, our next step will naturally be to determine the meaning of the constants A, _B, and C. This meaning will be found to vary according to the data by which we may suppose the position of a right line to be fixed. In dis- cussing the Right Line from the geometric point of view, we found that its equation assumed three forms, de- pending upon the three sets into which the data for its position naturally fall. We shall now see that the gen- eral equation will assume one or another of those forms, according as the constants in it are interpreted by one or another of the sets of data. OO. The first step toward a correct interpretation of these, is to observe that the arbitrary constants in our equation are but two: a proposition which we might infer from the fact that in an equation we are concerned only with the mutual ratios of the co-efficients. But 92 ANALYTIC GEOMETRY. its truth will be obvious, if we consider that an equa- tion may be divided by any constant without affecting the relation between its variables, and therefore without affecting the locus which it represents. Accordingly, if we divide Ax -f- By + C = by either of its constants, for example (7, we obtain A B a form in which there are only two arbitrary con- stants. Corollary. Hence, Two conditions determine a right line. Conversely, A right line may be made to satisfy any two conditions. This agrees with the fact that the data upon which the three forms of the equation to the Right Line were developed geometrically, are taken by twos. (See Arts. 78, 79, 80.) 91. We now proceed to the analytic deduction of those forms. In this process, the meaning of the ratios among the constants A, B, and C will duly appear. I. Let the data be the angle tvhich the line makes with the axis of x, and its intercept on the axis of y. We use the symbols , o, m, b to denote the same quantities as in Art. 79. If in Ax -f By + C = we make y = 0, we obtain / A f o o\ (~\ rn /-i \ x =. ~r (^fi.ri. ooj \j \ / If we make x = 0, we obtain y = -g= (Art. 83) OD = b (2). OD A Dividing (2) by (1), 7 yf,= ^>. OD A ~7o-^. rV / \ MEANING OF THE CONSTANTS. 93 Hence, (see Trig., 867,) 2 = rin(-a,) = ~ W (3) ' c From (2), we have B = -r ; and from (3), A mB ? Substituting these values in the original equation, b we obtain mC . . y = mx -f- b. A C Corollary 1. By (3), m = ^; and by (2), b=^. Hence, in the equation to a given right line the ratio between the co-efficients of x and y, taken with a con- trary sign, denotes the ratio between the sines of the angles which the line makes with the two axes: or, when the axes are rectangular, it denotes the tangent of the angle made with the axis of x ; and the ratio of the absolute term to the co-efficient of y, taken with a contrary sign, denotes the intercept of the line on the axis of y. Corollary 2, Hence, to reduce an equation in the form Ax -j- By + C to the form y = mx -j- 6, we merely solve the equation for y. II. Let the data be the intercepts of the line on the two axes. Here a and b have their usual signification. Making y = in Ax -f By -j- = 0, we find as before C ^ /^ ~~A~ ~ a (*) ^P Making x = 0, we obtain <7_ 94 ANALYTIC GEOMETRY. c c From (1), A = ; and from (2), JB = j . Substi- tuting for A and B in the equation, and reducing, x y --f f = 1. a ' 6 Corollary 1, From (1) and (2), we see that the ratios of the absolute term to the co-efficients of x and y re- spectively, taken with contrary signs, denote the intercepts of the line on the axis of x and the axis of y. Corollary 2. To reduce Ax -f By -f C = to the rm~ + T-=slj we divide it by its absolute term, and, if necessary, change its signs. III. Let the data be the perpendicular from the origin on the line, and the angle of that perpendicular with the axis of x. We use p and a in the same sense as in Art. 80. Making y and x successively equal to in the general equation, we obtain, as before, (1), _ = l>= (Trig., 859) (2) . Ccosa From (1), A= --- From (2), B = - C cos Substituting for A and B in the general equation, we have C cos a C cos ((o x cos -f~ ?/ cos (co ) p. CONSTANTS IN TERMS OF DIRECTION-COSINES. 95 Corollary, From (1) and (2) we learn that A cos a. -B cos (co a) ' or, the ratio between the co-efficients of x and y, denotes the ratio between the direction-cosines. Remark. The reduction of Ax -f By -f C = to the form x cos a+y cos (to a)=p is of such importance that we shall discuss its method in a separate article. 92. Reduction of Ax -f By -f C -= to the form o?cos-f ycos(w a) =/>. The problem may be more precisely stated: To find the values of cos , cos (w a), and p, w ferm* o/ A, B, and C. If, in the preceding article, we divide (1) by (2), we find 5 = ^co"'^ That is > ( Tri S'> 845 > IV ; 839 >) B i = cos co -\- sin co tan a. B . * . tan a= = sin co A sin co T/(l+tan 2 a) i/(^l 2 +^ 2 2 ^l^cos 01) jB cos a B sin w ~~ C cos a Csm co '' - A "V(A* +B 2 2 AB cos o>) ' Therefore, to make the required reduction, Multiply the .. n 7,7 _ sin co _ * equation throughout ly ^/ (A 2 + B 2 _ 2 AB cos co) ' * The following elegant solution of this problem is by SALMON : Conic Sections, p. 20 : " Suppose that the given equation when multiplied by a certain factor R is reduced to the required form, then RA = cos a, .K# = cos j3. But it can easily be proved that, if a and ft be any two angles whose sum is w, we shall have 9G ANALYTIC GEOMETRY. Remark. Since we have agreed always to consider p a positive quantity, it may be necessary to change the signs of the given equation, before multiplying, so that its abso- lute term when transposed to the second member may be positive. Corollary 1, If the axes are rectangular, we shall have A B .Bv c Accordingly, Ax -f- By -j- C = is reduced to the form x cos oi-\-y sin a = p, by dividing all its terms by V A z -\-B 2 , after the necessary changes of sign. Great importance pertains to the transformation ex- plained in this corollary. The general reduction to x cos a -f- y cos (to a)=p is of comparatively infre- quent use. Corollary 2. From the values of p above obtained, we learn that the length of the perpendicular from the origin upon the line Ax + By -|- C= is cos 2 a + cos 2 (3 2 cos a cos (3 cos u = sin 2 a). Hence, B 2 (A*+ B' 2 2 AB cos u) =sin 2 w; and the equation reduced to the required form is A sino JBsincj _ ^ ^ B z -2 AB cos u) J \ And we learn that B* - 2 ABcos u) ' y(A* + B 2 - 2 AB cos u) are respectively the cosines of the angles that the perpendicular from the origin on the line Ax-\-By-{- C=Q makes with the axes of x and y; and that * t- is the length of that P er P endicular -" POL A R EQ UA TION B Y ANAL YSIS. 97 C sin co ~ ]/ (A 2 + B 2 2 AB cos at) ' and that this length becomes C V(A* + B>) when the axes are rectangular. 93. Polar Equation to the Right Line, deduced analytically. The polar equation may be obtained from the Cartesian as follows : Let Ax -f- By -f C = 0, referred to rectangular axes, be reduced to the form x cos o.-\-y sin a=p. Transforming to polar co-ordinates, we have (Art. 57, Cor.) x = p cos $, and y = p sin : and the equation becomes /> (cos 6 cos -f sin d sin a) =_p ; that is, jO cos (6 a) = p, the equation of Art. 82. Corollary. Transforming the original equation to polars, we have p (A cos 6 + B sin 6) + 0= 0. Hence, an equation in the form p (A cos d + J5sin 0) =(7 may be reduced to the form p cos (0 a)=p by dividing each term by 94. Before advancing to the more particular forms of the equation to the Right Line, the student should make sure of having mastered the general ones which precede, and the principles which have been developed in the course of the discussions just closed. To this end, let the following exercises be performed. 98 ANALYTIC GEOMETRY. EXAMPLES. 1. Transform 3x 5y + 6 = to y = 0. What is the angle made by the new axis of x with the old ? 2. What angle does the line Sx + 12y -f 2 = make with the axis of x f What is the length of its intercept on the axis of y 1 In which of the four angles does it lie ? 3. Find the intercepts of the line x -f- 3?/ 3 = 0. 4. Find the ratio between the direction-cosines of the right line 2x + 3^ + 4 = 0, being = 60. 5. What is the tangent of the angle made with the axis of x by the perpendicular from the origin on 3x 1y 6 = 0? 6. Reduce all the equations in the previous examples to the form x cos a, -\-y sin a p, and determine a and p for each line. 7. Reduce 'Sx + 4y = 12 to the form x cos a + y cos (w a) p. What are the values of a and (w a), supposing a successively equal to 30, 45, 60, and sin" 1 f ? 8. Find the length of the perpendicular from the origin on 3x + 4y + 12 = 0, under the several values of o last supposed. 9. Find the length of the perpendicular from the origin on 3x 4y 12=0, axes being rectangular. 10. Reduce 2 p cos 6 3 p sin0 = 5 to the form p cos (6 a)=p. What are the values of a and p ? THE RIGHT LINE UNDER SPECIAL CONDITIONS. 95. Equation to the right line passing through Two Fixed Points. Let the two points be x f y f , x"y". Since they are points on a right line, their co-ordinates must satisfy the equation Ax + By + 0= (1), and we therefore have Ax? +By> +C=Q (2), Ax" + By" +0=0 (3). RIGHT LINE THROUGH TWO POINTS. 99 Subtracting (2) from (1) and (3) successively, we obtain A (x x f ) -j- B (y y f ) = 0, A(x"-x')+(y"- 1J ')=0, and thence y yf y't yf y_ & y y /A\ xx' ~ x" x f which is the equation required. For it is the equation to some right line, since it is of the first degree; and it is the equation to the line passing through the two given points, because it vanishes when either x' and y' or x" and y" are substituted in it for x and y. & tj Corollary 1. Equation (4) may evidently be written (y f y"} x (x f x") y + x'y" y'x" = : a form often useful, though the form (4) is more easily remembered. Corollary 2. If in the last equation we suppose x"= and y"= 0, we obtain the equation to the right line passing through a fixed point and the origin. Remark The same equation, (4), might have been obtained geometrically. For, since the triangles PRP', P'tSP" are similar, we have PR _P'S RP / ~ SP" ' that is, after changing the signs of the equation, y - y' y"-y' O M M' M" 100 ANALYTIC GEOMETRY. It is worth while to place the analytic and geometric proofs thus side by side, in order to make sure that the geometric meaning of all the symbols in the equations developed, shall be clearly under- stood. 96. Angle between two right lines given by their equations. All formulae for angles are greatly simplified by the use of rectangular axes. We therefore present the subject of this article first in the form which those axes determine. Let the two lines be y = mx + b and y = m'x -f- b f ; and let tp = the angle between them. From the dia- gram,

and . Hence (Trig, 838) '(A'' 2 + E ri 2 A'B' cos w) sin a = !/ (^4 /2 + /2 - 2 ^S 7 cos w) ' B A cos w Therefore (Trig, 845, HI and iv . ,_,_*' x _ (AB / + ^ x ^) cos c^ ~ Whence (Trig, 839) .d'.B)sino (AB / + ^ x ) cos w ' a formula which evidently becomes that of Art. 96, Cor. 1, if a 90. Corollary 1 The condition that the two lines shall be parallel, is AB' A'B = : from the identity of which with the condition of Art. 96, Cor. 2, we learn that the condition of parallelism is independent of the value of w. The same follows from the fact that the condition itself is not a function of w. Corollary 2. The condition that the lines shall be perpendicular to each other, is AA' + B& (AB' + A'B) cos w = 0. 98. Equation to a right line parallel to a given one. Let the given line be y = mx -\- b. The required equation will be of the form y m'x -\- b'. 102 ANALYTIC GEOMETRY. But in this, the condition of parallelism (Art. 96, Cor. 2 ; Art. 97, Cor. 1) gives us m f = m. The required equation is therefore y = mx -f- b'. Corollary, If the given line were Ax -\- By -f- C= 0, the equation would be of the form Ax -\- B'y -\- C' = 0. A C f In that case, m = j^ , and b f = IT, . The required BC' equation would therefore be, after writing C" for -/- , From this we infer that the equations to parallel right lines differ only in their constant terms. 99. Rectangular equation to a right line per- pendicular to a given one. The equation will be of the form y = m'x + 6', in which m' is determined by the condition (Art. 96, Cor. 3) 1 + mm' = 0. Hence, m! = ; and the equation is y = x + b f . y m Corollary, When the given line is Ax -f- By + 0= 0, the equation to its perpendicular is (Art. 91, I, Cor. 1) Ay Bx + Ci=Q. Hence, if two right lines are perpendicular to each other, their rectangular equations interchange the co-efficients of x and y, and change the sign of one of them. SYMMETRIC EQUATION TO RIGHT LINE. 103 100. Equation to a right line perpendicular to a given one, axes being oblique. From the condition of perpendicularity, (Art. 97, Cor. 2,) A' (A B cos w) + B' (B A cos w) = A' B A cos w Hence ' -W = A-Bw*> and the required equation (Art. 91, I, Cor. 1) is (A B cos w) y (B A cos ) x + C 2 = 0. 101. Equation to a right line parallel to a given one, passing through a Fixed Point. By Art. 98 we have the form of the equation, y = mx j r V (1). Calling the fixed point x'y', we therefore obtain y' = mx f + ^ : b f = y' mx f . Substituting for b f in (1), the equation sought is y y f =m(x x'}. Corollary 1. If in this equation we suppose m indeter- minate, the direction of the line is indeterminate ; and we have the equation to any right line passing through a fixed point. Corollary 2 Let sin a : sin to = k, and sin (w ) : sin co = h. Then k : h -\- sin a : sin (co a) =m ; and the equation may be written. k h or, if we denote either of these equal ratios by Z, y y' ___ x ~ x ' _ j . An. Ge. 12. 104 ANALYTIC GEOMETRY. a formula often convenient, and known as the Symmetrical Equation to the Right Line. Corollary 3. If the axes are rectangular, we have k : h = sin a : cos a ; and the equation may be more conveniently written y y f x x r where s and c are abbreviations for the sine and cosine of the angle which the line makes with the axis of x. 1O2. Geometric meaning of the ratio /. If, in the annexed diagram, P' denote the fixed point x'y' , and P any point of a right line passing through it, PR being /Y parallel to OX, \ / yy' PR sin co / \p 1 \ k smPP'R > / R/..\P P' R sin co / h But (Trig., Hence, sin P'PR / M M ' 871) PR sin co P'Rsmco T\X siuPP'R 1 1 sin P'PR ' y y' xx' k h l denotes the distance from a fixed point x'y' to any point xy of a right line passing through it. Remark 1. So long as the point xy is variable, I is of course indeterminate. But the formula enables us to find the distance from x'y' to any given point on the line, by merely substituting for x or y the abscissa or ordinate of such given point. LINE CUTTING ANOTHER AT GIVEN ANGLE. 105 Thus, supposing the given point of the line to be x"y ff , we should have (assuming the axes for convenience to be rectangular) y"-y f i zs ' x" - x' ~r =c - Squaring both sides of these equations, adding, and remembering (Trig., 838) that s 2 -f c 2 = 1, we obtain which agrees with the formula (Art. 51, I, Cor. 1) for the distance betiveen two given points. Remark 2. The signs -\- and in connection with I denote distances measured along the line in opposite di- rections from x r y f . Thus, if -f I were measured in the direction P'D, I would be measured in the direction P' T; and vice versa. Speaking with entire generality, + I must be laid off from x'y' in the positive direction of the line. (Aft. 50, Note), and I in the negative. 1O3. Rectangular equation to a right line passing through a Fixed Point, and cutting a given line at a Given Angle. Let the given line be y = mx -f 5, and let 6 = the given angle. The equation sought (Art. 101, Cor. 1) is of the form ~ AB f A'B as the co-ordinates of the common point. 1O7. Equation to a right line passing through the intersection of two given ones. If we multiply the equations to two given lines each by an arbitrary constant, and add the results, thus : I (Ax +By+CT) + m (A'x + B'y + C'} = 0, the new equation will represent a right line passing through the intersection of the lines Ax + By -f- 0= and A'x + B'y + (? = Q. For it manifestly denotes some right line, since it is of the first degree. Moreover, it is satisfied by any values of x and y that satisfy Ax + By -f- (7=0 and A'x -f B'y -f C f = simultaneously ; for its left member must vanish whenever the quantities Ax + By -j- C and 110 ANALYTIC GEOMETRY. A'x -f- B'y -j- C' are both equal to zero. That is, it passes through a point whose co-ordinates satisfy the equations of both the given lines. But such a point (Arts. 62, 106) is the intersection of the two lines. Corollary. By varying the values of the constants I and m, we can cause the above equation to represent as many different lines as we please, all passing through the intersection of the two given ones. Remark. The truth of the above equation is inde- pendent of a), and of the form of the equations to the given lines. This is manifest from the method by which it was obtained. It may therefore, when convenience requires, be written I {x cos o.-\-y cos (co a) p } A -\-m{x cos/3 + 2/cos (at ft) p'} ' or I (x cos a -j- y sin p) -\- m (x cos ft -f- y sin ft p'} = 0. 1O8. Meaning of tne equation L+kI/=O.If we put k = m : I, and represent by L and L' the quantities which are equated to zero in the equations of the two given lines, the equation of the preceding article becomes L + kL' = 0. We shall now prove that this is the equation to any right line passing through the intersection of two given ones. * * The beginner may suppose that this has been done already, in the preceding article. But we merely proved there, that the equation may represent an infinite number of lines answering to the given condition. Now, that an infinite number of lines is not the same as all the lines passing through the intersection of two others, is evident. For between two intersecting right lines there are two angles, supplemental to each other, in each of which there may be an infinite number of lines passing through the common point. LINE THROUGH INTERSECTION. HI Let 6 = the angle made with the axis of x by the line which the equation represents. Then (Art. 91, 1, Cor. 1) sin A + kA' sin (a* 0) B -f kB' ' Therefore (A + kA f ) sin CD tan u = (A cos a)B)+k (A' cos * B') Hence, as k may have any value from to oo, and be either positive or negative, tan 6 may have any value, either positive or negative, from to oo. That is, the equation is consistent with any value of 6 what- ever. Therefore, if L = and L' = are the equations to two right lines, L -f kL' = is the equation to any right line passing through the intersection of the two. Corollary 1. The equation to a particular line inter- secting two others in their common point, is formed from the above by assigning to k such a value, in terms of the conditions which the line must satisfy, as the relation L -f kU = implies. Thus, if the condition were that the intersecting line make with the axis of x a given angle 0, we should have, from the value of tan 6 above, A sin co (A cos co B ) tan 6 ~ Z'~sin > (A r "cos^i >~~ rF'Jtan^ ' or, in case the axes were rectangular, A -f- B tan ~A' + 't&u0' An. Ge. 13. 112 ANALYTIC GEOMETRY. If the condition were that the intersecting line pass through a fixed point x'y' , we should have (Ax f + By' + tf) rM (Ax> + B'y' + C r ) = : Ax? + 3^+0 A'x' 4- B'y 1 + 6" ' Corollary 2. The si0w of k has a most important geometric meaning. Obviously, in order to change its sign, a quantity must pass through the value or oc. Now if k 0, we have by Cor. 1 A A + B tan 6 = . . tan = -g- . And if k = GO, A! A' + jB' tan = . . tan == -g?-. That is, (Art. 91, I, Cor. 1,) a ^Ae instant when k changes sign, the line which passes through the intersection of two given ones coincides with one of them. Hence, of the lines L -j- ~kL' = and L kL' = 0, one lies in the angle be- tween Tj = Q and U = supplemental to that in which the other lies. It now remains to determine which of these supple- mental angles corresponds to -f- k, and which to k. If the two lines are x cos a -f- y cos (co ) p = and /?) / 0, we shall have x cos a -f i/ cos (w ) jy cos + cos w that is, (Art. 105,) one of the geometric meanings of k is, the negative of the ratio between the perpendiculars let fall on tivo given lines from any point of a line passing through their intersection. Hence, when k is positive, those per- pendiculars have unlike signs ; and Avhen k is negative, their signs are like. That is, (Art. 105, Cor. 3,) the per- pendiculars corresponding to + k fall one on the side of CRITERION OF ITS POSITION. 113 one line next the origin, and the other on the remote side of the other line ; while those corresponding to k fall both on the side of the lines next the origin or both on the side remote from it. In other words, the line corresponding to + k lies in the angle remote from the origin, e. g. P'L'; and the line corre- sponding to k lies in the same angle as the origin, e. g. PL. For convenience, we shall call the angle R'SQ f , remote from the origin, the external angle between the given lines ; and the angle RSQ, in which the origin lies, the internal angle. Hence, if L = and L f = are the equations to any two right lines, L 4- kL' = denotes a line passing through their intersection and lying in the external angle between them ; and L kL' == denotes one lying in the internal angle. 1O9. Equation to a right line bisecting the angle between two given ones. Any point on the bisector being equally distant from the two given lines, we have (Art. 105 cf. Cor. 2) Ax -f By + A'x + B'y + C f or co ft) p f }* (2), * The student may at first think that both members of (1) and (2) should have the double sign. But since an equation always implies the possibility of changing its signs, it is evident that we should write the expressions as above. 114 ANALYTIC GEOMETRY. according as the given lines are Ax + By -{- (7 and A'x 4- B'y -f- (?' = or a; cos a + y cos (w a) p == and a; cos /9 + # cos (w /?) p' 0. Expressions (1) and (2) are the principal forms of the required equation. Corollary 1. When the axes are rectangular, the equation becomes Ax + By+C A'x + B'y+C' l/(A*+&) V(A'* + B'*) or x cos a -f- y sin JP = (x cos /? + ?/ sin /? p') (4). Corollary 2. Expressions (1), (2), (3), (4) are evidently in the form L kL' = 0. Hence (Art. 108, Cor. 2) there are two bisectors : one lying in the external angle of the given lines, and the other in the internal. For the sake of brevity, we shall call the former the external bisector ; and the latter, the internal bisector. Corollary 3. If we put, as we conveniently may, a = x cos a -f- y cos (co ) p, fi = x cos ft -\- y cos (co ft) p f , expressions (2) and (4) will be included in the brief and striking form a = 0. Hence, if a 0, ft = are the equations to any two right lines, the line a -f- ft = bisects the external angle between them ; and the line -/3 = bisects their internal angle. BISECTOR OF ANGLE. 115 Caution. From the proposition just reached, the stu- dent is apt to rush to the conclusion that L L' - is the equation to the bisector of the angle between the lines L == and L r = 0, without regard to the values of L and U . This is a grave error. It assumes that the value of k, in the case of a bisector, is always 1. When the equations to two lines are in terms of the perpendicular from the origin and its angle with the axis of x ; that is, when L~x cos a -f- y cos (co a) p and L' x cos /9 -j- y cos (co /9) p f , k = 1. But from equation (1) we have (A* + B' 2 2 A'B f cos which is obviously not in general equal to 1. The condition that it shall have that value is A 2 -f B 2 2 AB cos co = A' 2 + B' 2 2 A'B' cos co or, when the axes are rectangular, Corollary 4. If we denote by r the particular value which k assumes in the case of a bisector, then L + rL' == represents the external, and L rL' = the internal bisector of the angle between the lines L= and L 1 = 0. 116 ANALYTIC GEOMETRY. In these expressions, r is to be determined from the relation I/ (A 2 + B 2 - 2 A B cos co) r = L T/(A' 2 + J?' 2 2 ^!'J3' cos w) : which, in case the axes are rectangular, becomes 110. Equation to a right line situated at in- finity. To assume that a right line is at an infinite distance from the origin is to assume that its intercepts on the axes are infinite. Hence, we have c c -r = GO and - - -5 oo. That is, supposing C to be finite, A = and B = 0. The required equation is therefore Oz + Oy + 0=0: in which C is finite. We shall cite it in the somewhat inaccurate but very convenient form 0=0. Remark. The student will of course remember that a line at infinity is not a geometric conception at all in fact does not exist, in any sense known to pure geometry. As an analytic conception, however, it has important bearings; and the equation just obtained is useful in some of the higher investigations of curves. 111. Equations of Condition. When elements of position and form sustain certain geometric relations to each other, the constants which enter their analytic equiv- alents must sustain corresponding relations. In other THREE POINTS ON ONE EIGHT LINE. 117 words, if the geometric relations exist, the constants satisfy certain equations. Such equations are called Equations of Condition. Thus we saw (Art. 97, Cor. 1) that if two right lines Ax + By -f C= and A'x + .B'y + C" = are parallel, the constants A, J9, A f , B' must satisfy the equation and that if they are mutually perpendicular, the constants must satisfy the equation AM + BB' (AB 1 + A'B) cos co === 0. 112. Condition that Three Points shall lie on One Right Line. Ifz } y } , x$ 2 , x^/ 3 lie on one right line, x$ 3 must satisfy the equation to the line which passes through x } yt and x$ 2 . Hence, (Art. 95,) the equation of condition is (y\ 2/2) %z fci x 2 ) ?/ 3 + x } y 2 y l x 2 = : which may for the sake of symmetry be written y, (x 2 x.) -f- y 2 (x., x } ) + y 3 (x, x 2 ) = 0. '/^~~^^ Xl ( jj N '* Remark. It is worth while to notice the order of the elements which enter into the latter form of this equation. In writing sym- metrical forms, the analogous symbols must be taken in a fixed order, which will be best understood by conceiving of the successive symbols as forming a circuit, about which we move according to the annexed diagram. Thus, as in the last equation, we pass from x\ to .r 2 , from x 2 to #3, and from x 3 to x\ ; and so round again : always going back to the FIRST element when the list has been completed, and then proceeding as before in numerical succession. The advantage of symmetrical forms is very decided, especially when we have to compare or combine analogous equations. But unless this order is observed, the methods of reasoning based upon it will of course fail ; and, in some cases, false conclusions may be drawn by com- bining equations according to rules which presuppose it. 118 ANALYTIC GEOMETRY. 113. Condition that Three Right Lines shall meet in One Point. If three lines Ax + By -f C= 0, A'x -f B'y + C' = Q, A"x -f "?/ -f <7" = pass through the same point, the co-ordinates of intersection for the first two must satisfy the equation to the third. Hence, (Art. 106,) the required condition is A" (BC'B r C) +B" ( CA' C'A] + C" (AB'A'B) = 0. 114. A condition often more convenient in practice, is derived from the principle of Arts. 107, 108. For if three right lines meet in one point, the equation to the third must be in the form of the equation to a right line passing through the intersection of the first and second. Therefore, supposing the three lines to be Z 0, L' = 0, L" = 0, we can always find some three constants, Z, m, and n, such that -nL" = lL+mL' 9 where n may be either a positive or a negative quan- tity. Hence, the condition is That is, three right lines meet in one point when their equations, upon being multiplied respectively by any three constants and added, vanish identically. Remark. For brevity, we shall often refer to three right lines passing through one point by the name of convergent*. 115. Condition that a Movable Right tine shall pass through a Fixed Point. Comparing Art. 101, Cor. 1 with Art. 91, I, Cor. 1, it is evident that the equation to a right line passing through a fixed point whose co-ordinates are I : n and m : n, may be written A (nx T) - B (ny m) = 0. MOVABLE LINE THROUGH FIXED POINT. 119 And comparing this with the general equation to a right line, we have (lA + mB)=nC. The required condition is therefore Hence, a movable right line passes through a fixed point, so long as the co-efficients in its equation suffer no change in- consistent with their vanishing when multiplied each by a fixed constant and added together. Corollary. The criterion of this article may be other- wise taken as the condition that any number of lines shall pass through one point. For every line whose co-efficients satisfy the equation IA -f- mB -\- nC = 0, must pass through the point I : n, m : n. 116. In this connection also, Art. 108 furnishes us with a second condition. For, since we may always regard a fixed point as the intersection of two given right lines, the most general expression for a right line passing through a fixed point is L + kL' -= 0. By writing L and L' in full, and collecting the terms, this becomes (A + kA r ) x + (B+ kB r ) y + (C + kC'} = 0. Now the condition that the line shall be movable is that k be indeterminate. Hence, a movable ric/ht line passes c/ J. through a fixed point whenever its equation involves an indeterminate quantity in the first degree. Corollary. The co-ordinates of the fixed point may be found by throwing the given equation into the form L -f kL' = 0, and solving L = and L' == for x and y. 120 ANALYTIC GEOMETRY. 117. A third condition may be obtained as follows : Suppose that we have an equation of the form (Ax f + By f +C)* + (^ #~+*&tf+C')Jl + (A"J + B'ty -f C") \ ~~ in which x f , y' are the co-ordinates of any point on the line MX' -{- Ny' -f P=0. By means of the latter relation, we can eliminate y' from the given equation, which will then contain the indeter- minate x' in the first degree. Therefore, a movable rigid line passes through a fixed point whenever its equation involves in the first degree the co-ordinates of a point which moves along a given right line. 118. If in any equation Ax+ By -{- C=Q, C : A or C : B is constant, the corresponding line (Art. 91, II, Cor. 1) makes a constant intercept on one of the axes. Hence, as a fourth condition, a movable right line passes through a fixed point whenever its equation satisfies the relation C : A constant or C : B constant. Corollary. A particular case of this is, that the line passes through a fixed point when (70. And, in fact, (Art. 63) every such line does pass through the origin. Remark It has seemed worth while to present the condition of this article separately, as it is often convenient in practice. It is obviously, however, only a particular case of Art. 116. EXAMPLES ON THE RIGHT LINE. I. NOTATION AND CONDITIONS. 119. In some of the exercises which follow, the student must use his judgment as to the selection of the axes. The labor of solving will be much lessened by a judicious choice. A few hints have been given where they seemed necessary. EXAMPLES ON THE EIGHT LINE. 121 1 . Form the equations to the sides of a triangle, the co-ordi- nates of whose vertices are (2, 1), (3, 2). ( 4, 1). 2. The equations to the sides of a triangle are x -{- y 2, x 3v/ = 4 and 3x + 5y -|- 7 = : find the co-ordinates of its vertices. 3. Form the equations to the lines joining the vertices of the triangle in Ex. 1 to the middle points of the opposite sides. 4. Form the equation to the line joining a/?/ to the point mid- way between ,r // y // and x /// 'i/ /// ; or, show that, in general, the equations to the lines from the vertices of a triangle to the middle points of the opposite sides are (y" +/"-2/ )x-(x" +x"'2x' )y+(x" y' -y" x' )+(x'"y r -y'"x' )=0, (y"'+y' -2y" )x-(x"'+x' -2x" )y+(x'"y"-y" f x")+(x t y" -y' x")=Q, (/ +y" -2y"'}x-(x' +x" -2x'")y+(x' y'"-y f x"'}+(x"y"'-y"y"> )=Q. 5. In the triangle of Ex. 1, form the equations to the perpen- diculars from each vertex to the opposite side. What inference as to the shape of the triangle ? 6. Prove that, in general, the equations to such perpendiculars are (x" -x"')x + (y" -y"')y + (x' x'" -\-y' y'")-(x> x" -f y' y" ) = Q, > +/' y> )-(x" x'"-\-y" y'"} = Q, " '""-x'"x' --'"> = 0. 7. Prove that the general equations to the perpendiculars through the middle points of the sides are 2 -*'" 2 ) -f '*-x> 2) 4- y 2 (y'"*-y' 2), (*' -x" ) x -f (y -y" )y=Y 2 (x' 2 -x" 2) -}- y, ( y > * -y" 2). 8. Find the angle between the lines x-\-y = \ and y x= 2, and determine their point of intersection. 9. Write the equation to any parallel of x cos a -f- y sin a p 0. Decide whether x sin a ?/ cos a=p / or x sin a -j- y cos a =p" may be parallel to it; and, if so, on what condition. 122 ANALYTIC GEOMETRY. 10. Taking for axes the sides a and b of any triangle, form the equation to the line which cuts off the m ih part of each, and show that it is parallel to the base. What condition follows from this? 11. Prove that y= constant is the equation to any parallel of the axis of x, and x = constant the equation to any parallel of the axis of y, whether the axes are rectangular or oblique. 12. Two lines AB, CD intersect in O; the lines AC, BD join their extremities and meet in E ; the lines AD, BC join their extremities and meet in F: required the condition that EF may be parallel to AB. 13. Form the equation to the line which passes through (2, 3) and makes with the line y=3x an angle = 60. 14. Form the equation to the line which passes through (2, 3) and makes with the line 3x -4y = an angle 6 = 45. 15. We have shown that, in rectangular axes, A : B = tan a, but that, in oblique axes, A : B - sin a : sin (w a). Prove that, in all cases, tan a = A sin : (A cos w B). 16. Axes being oblique, show that two lines will make with the axis of x angles equal but estimated in opposite directions (one above, the other below) upon the condition B B / 1 7. Find the length of the perpendicular from (3, 4) on the line 4x + 2y 7 = 0, when <*> = 60. On which side of the line is the given point? 18. Find the length of the perpendicular from the origin on the line a (x a) + b (y b) = 0. 19. Given the equations to two parallel lines: to find the distance between them. 20. What points on the axis of x are at the distance a from the 21. Form the equation to the bisectors of the angles between Zx -f- 4y 9=0 and \"2x -j- 5y 3 = ; - the equation to any right line passing through their intersection. EXAMPLES ON THE RIGHT LINE. 123 22. Prove that whether the axes be rectangular or oblique the lines x-\- y =-Q, x y = are at right angles to each other, and bisect the supplemental angles between the axes. Show analytically that all bisectors of supplemental angles are mutually perpendicular. 23. Find the equation to the line passing through the intersec- tion of 3x 5y -\- 6 = and 2x -f y 8 = 0, and striking the point (5, 6). 24. Find the equation to the line joining the origin to the inter- section of Ax + By + C^-- and A'x + B'y + C' = 0. 25. Show that the equation to the line passing through the inter- section of Ax + By + C= and A'x + B'y -f C= 0, and parallel to the axis of *, is (A B' A'B) y+ (A C/ A'C) = 0. Does this agree with the theorem of Ex. 11? 26. Find the equation to the line passing through the intersec- tion of x cos a -f- ysina =p, x cos ft -f- y sin /3 =p' and cutting at right angles the line x cos 7 + y sin y =p". 27. Given any three parallel right lines of different lengths ; join the adjacent extremities of the first and second, and produce the two lines thus formed until they meet; do the same with respect to the second and third, and the third and first: the three points of intersection lie on one right line. 28. Given the frustum of a triangle, with parallel bases: the intersection of its diagonals, the middle points of the bases, and the vertex of the triangle are on one right line. [Take vertex of triangle as origin, and sides for axes.] 29. On the sides of a right triangle squares are constructed ; from the acute angles diagonals are drawn, crossing the triangle to the vertices of these squares ; and from the right angle a perpen- dicular is let fall upon the hypotenuse : to prove that the diagonals and the perpendicular meet in one point. [Let the lengths of the sides be a and 6, and take them for axes] 30. Prove that the three lines which join the vertices of a tri- angle to the middle points of the opposite sides are convergents, taking for axes any two sides. [See also their equations above, Ex. 4.] 124 ANALYTIC GEOMETRY. 31. The three perpendiculars from the vertices on the sides, and the three that rise from the middle points of the sides, are each convergents. [See their equations, Exs. 6 and 7.] 32. The three bisectors of the angles in any triangle are con- vergen^s: for their equations are (x cos a -f- y sin a p ) (x cos ft + y sin /? p / ) = 0, (x cos j3 + y sin (3 p' ) (x cos y + y sin y p") = 0, (x cos 7 -f y sin y _p") (x cos a -f- y sin a p ) = 0, if we suppose the origin to be within the triangle. 33. Through what point do all the lines y = mx, Ax + By 0, 2x = 3y, x cos a + y sin a = 0, p cos = pass ? 34. Decide whether the lines 2x 3y -f 6 = 0, 4x -f 3y 6 = 0, 5.r 52/+10 = 0, 7x + 2y 4 = 0, * y-f 2 = pass through a fixed point. 35. Given the line 3x 5y-\-&=Q: form the equations to five lines passing through a fixed point, and determine the point. 36. Given three constants 2, 3, 5 : form the equations to five lines passing through a fixed point, and determine the point. 37. Given the vertical angle of a triangle, and the sum or differ- ence of the reciprocals of its sides: the base will move about a fixed point. 38. If a line be such that the sum of the perpendiculars, each multiplied by a constant, let fall upon it from n fixed points, is = 0, it will pass through a fixed point known as the Center of Mean Position to the given points. The conditions of the problem (Art. 105, Cor. 1) give us m' (x' cos a + y' sin a p) + m" (x" cos a + y" sin a p) | -f m" 1 (x'" cos a + y'" sin a p) -f or, putting Z(mx')as an arbitrary abbreviation for the sum of the mx's, 2 (TO?/') for the sum of the my's, and (i) for the sum of the m's, X (mx' ) cos a + 2 (my' ) sin a 2 (TO) p = 0. Solving for p, and substituting its value in xcos a + y sin a p = 0, the equation to the movable line becomes 2 (m) xZ (mx 1 ) + tan o { 2 (TO) y - 2 (my 1 ) } = : which (Art. 116) proves the proposition. [Solution by Salmon.] RECTILINEAR LOCI. 125 39. If the three vertices of a triangle move each on one of three convergents, and two of its sides pass through fixed points x'y', xf'y", the third will also pass through a fixed point. [Take the two exterior convergents for axes.] 40. If the vertex in which the two sides mentioned in Ex. 39 meet, does not move on a line convergent with the two on which the other vertices move, to find the condition that the third side may still pass through a fixed point. II. RECTILINEAR LOCI. 12O. The following examples illustrate the method of solving problems in which the path of a moving point is sought. When a point moves under given conditions, we have only to discover what relation between its co-ordinates is implied in those conditions : then by writing this relation in algebraic symbols we at once obtain the equation to its locus. As the investigation of loci is one of the principal uses of Analytic Geometry, it is important that the student should early acquire skill in thus writing down any geometric condition. The relation between the co-ordi- nates is sometimes so patent as to require no investiga- tion : for instance, if we were required to determine the locus of a point the sum of the squares on whose co-ordi- nates is constant, we should at once write the equation % 2 -f- y 2 r 2 and discover (Art. 25, II) that the locus is a circle. But as a general thing the relation must be developed from the conditions by means of the geometric or trigo- nometric properties which they imply. The process may be much simplified by a proper choice of axes. As a rule, the equations are rendered much shorter and easier to interpret by taking for axes two prominent lines of the figure to which the conditions give rise. Still, by taking 126 ANALYTIC GEOMETRY. axes distinct from the figure, we sometimes obtain equa- tions whose symmetry more than compensates for their loss of simplicity. For instance, in the equations of Exs. 4, 6, and 7, when the first is obtained, the second and third can be written out at once by analogy. 1. Given the base of a triangle and the difference between the squares on its sides : to find the locus of its vertex. Let us take for axes the base and a perpendicular at its middle point. Let the base == 2m, and the difference between the squares on the sides = ri 1 . Then, the co-ordinates of the vertex being x and y, we shall have RP~ = r/ 2 + (m + x) 2 and PQ 1 = f + (m x) 2 . That is, ?/ 2 + (m + x) 2 \y z + (m x) 2 } = n 2 ; or the equa- tion to the required locus is 4 mx = n 2 . Hence (Art. 25, I) the vertex moves upon a right line perpendicular to the base. 2. Find the locus of the vertex of a triangle, given the base and the sum of the cotangents of the base angles. Using the same axes as in Ex. 1, and putting cot R + cot Q n, we have from the diagram cot R and cot Q = . Hence, the equation to the required locus is and (Art. 25, I) the vertex moves along a line parallel to the base at a distance from it = 2m : n. 3. Given the base and that rcot-R=fc scotQ=p g, find the locus of the vertex. 4. Given the base and the sum of the sides, let the perpendicular to the base through the vertex be produced upward until its length equals that of one side : to find the locus of its extremity. 5. Given two fixed lines OM, ON; if any line MN parallel to a third fixed line OL in- tersect them : to find the locus of P where MN is cut in a given ratio, so that MP = nMN. CV RECTILINEAR LOCI. 127 Here it will be most convenient to take for axes the two fixed lines OM, OL. Then, since N is & point on the line ON, we shall have MN=mOM; and, substituting in the given equation, we find the equa- tion to the required locus, namely, y = mnx. Hence (Art. 78, Cor. 5) the point of proportional section moves on a right line passing through the intersection of the two given lines. 6. Find the locus of a point P, the sum of whose distances from O3/, OL is constant. 7. A series of triangles whose bases are given in magnitude and position, and whose areas have a constant sum, have a common vertex : to find its locus. 8. Given the vertical angle of a triangle, and the sum of its sides : to find the locus of the point P where the base is cut in a given ratio, so that mPX =nPY. TJ M x 9. Determine the locus of P in the annexed diagram under the following successive conditions; PQ being perpendicular to OQ, and PR to OR: i. OQ + OR = constant. N n. QR parallel to y =mx. in. QR cut in a given ratio by y=mx + l>. M Q 121. Hitherto we have found the equations to required loci by expressing the given conditions directly in terms of the variable co-ordinates. But it is often more con- venient to express them at first in terms of other lines, and then find some relation between these auxiliaries by which to eliminate the latter: the result of eliminating will be an equation between the co-ordinates of the point whose locus is sought. This process of forming an equation by eliminating an indeterminate auxiliary is extensively used, and is of especial advantage when investigating the intersections of movable lines. An. Ge. 14. 128 ANALYTIC GEOMETRY. 10. Given the fixed point A on the axis of re, and the fixed point B on the axis of y ; on the axis of x take any point A', and on the axis of y any point B', such that OA / -f OB' = OA + O : to find the locus of the intersection of AB / and A / B. If OA = a, and 0# = 6 . . OA' = a + (k indeterminate), and OB' = b k. Hence, (Art. 79,) by clearing of fractions and collecting the terms, the equations to AB' and A' B may be written bx + ay ab + k (a x) = 0, bx + ay ab + k (y b) = 0. Subtracting, we eliminate the indeterminate k; and the equation to the required locus is x + y a + b. # 11. In a given triangle, to find the locus of the middle point of the inscribed rectangle. 12. In a given parallelogram, whose adjacent sides are a and 6, to find the locus of the intersec- tion of AB' and A'B : the lines A A', BB / being any two paral- y A'^- lels to the respective sides. / --- ""7 /,> [The statement of this problem / /Sf will apparently involve two indeter- / /-'' / minate quantities j but both can be eliminated at one operation.] 13. A line is drawn parallel to the base of a triangle, and the points where it meets the two sides are joined transversely to the extremities of the base : to find the locus of the intersection of the joining lines. 14. Through any point in the base of a triangle is drawn a line of given length in a given direction : supposing it to be cut by the base in a given ratio, find the locus of the intersection of the lines joining its extremities to those of the base. 15. Given a point and two fixed right lines; through the point draw any two right lines, and join transversely the points where they meet the fixed ones : to find the locus of the intersection of the joining lines. *See Salmon's Conic Sections, p. 46. PAIRS OF RIGHT LINES. 129 When we have to determine the locus of the extremity of a line drawn through a fixed point under given conditions, it is generally convenient to employ polar co-ordinates. We make the fixed point the pole, and then the distance from it to the extremity of the moving line becomes the radius vector. 16. Through a fixed point O is drawn a line OP, perpendicular to a line QR which passes through the fixed point Q : to find the locus of its extremity P, if OP. OR = constant. Let the distance OQ = a, and let OP.OR=m*. From the diagram, OR = OQ cos ROQ. Hence, the equation to the locus of P is pa cos = m z ; Q and (Art. 82, Cor.) P moves on a right line perpendicular to OQ. 17. One vertex of an equilateral triangle is fixed, and the second moves along a fixed right line : to find the locus of the third. 18. In a right triangle whose two sides are in a constant ratio, one acute angle has a fixed vertex, and the vertex of the right angle moves on a given right line: to find the locus of the remaining vertex. 19. Given the angles of any triangle : if one vertex is fixed, and the second moves on a given right line, to find the locus of the third. [The student will readily perceive that Exs. 17 and 18 are particular cases of Ex. 19. He should trace this relation through the equations to the corresponding loci.] 20. Given the base of a triangle, and the sum of the sides; through either extremity of the base, a perpendicular to the adjacent side is drawn : to find the locus of its intersection with the bisector of the vertical angle. SECTION III. PAIRS OF RIGHT LINES. Since the equation of the first degree always represents a right line, there is but one locus whose equa- tion is of the first degree. But there are several loci 130 ANALYTIC GEOMETRY. whose equations are of the second degree ; and, in accord- ance with the principle of Art. 76, we shall consider these separately before discussing their relations to the locus of the Second order in general. In the case of each, we shall first obtain its equation, and then find the con- dition on which the general equation of the second degree will represent it. When this has been done for all of them, we can pass to the purely analytic ground, and show that the equation of the second degree always represents one of these lines. We begin with the cases in which it represents a pair of right lines. These have a special interest, as furnish- ing the title by which the Right Line takes its place in the order of Conies. I. GEOMETRIC POINT OF VIEW : THE EQUATION TO A PAIR OF RIGHT LINES IS OF THE SECOND DEGREE. 124. Any equation in the type of LMN. . . = will obviously be satisfied by supposing either of the factors L, Mj N) etc., equal to zero. If, then, L= 0, M= 0, N= 0, etc., are the equations to n different lines, their product will be satisfied by any values of x and y that satisfy either of them. That is, LMN. . . = will be satisfied by the co-ordinates of any point on either of the n lines: its locus is therefore the group of lines severally denoted by the separate factors. Hence, we form the equation to a group of lines by multiplying together the equations to its constituents. 125. Equation to a Pair of Right Lines. By the principle just established, the required equation is or, by writing the abbreviations in full, expanding, and collecting the terms, PAIR OF LINES THRO UGII FIXED POINT. 131 X+&QV +A"C' which is manifestly a particular case of the general equa- tion of the second degree in two variables, Ax 2 -f 2Hxy + By 2 -\-ZGx-\- %Fy -f C 0, * in which J., 5, (7, F, G, H are any six constants. 126. Equation to a Pair of Right Lines passing through a Fixed Point. The equations to two right lines passing through the point x'y' (Art. 101, Cor. 1) are A'(x - d)+B' (y - y 1 } = 0, A(x - x')-tB"(y-y') == 0. Hence, (Art. 124,) the required equation is or, since A' A", A'B"+A"B', B'B" are independent of each other, A(x - ^ + 2H(x - aO(y - y') + B(y - yj = 0. Corollary 1, The equation to a pair of right lines passing through the origin (Art. 49, Cor. 3) will be Ax 2 + 2Hxy + By 2 = 0. Corollary 2. Of the two equations last obtained, the former is evidently homogeneous with respect to (x x f ) and (?/ y'} ; and the latter, with respect to x and y. Hence, Every homogeneous quadratic in (x x') and '*" The equation of the second degree in two variables is usually written Ax 2 + Bxy + Cy + Dx + EI/ + F^ 0. I have followed Salmon in departing from this familiar expression. The sequel will show, however, that the new form imparts to the equations derived from it a simplicity and symmetry which far outweigh any incon- venience that may arise from its unfamiliar appearance. 132 ANALYTIC GEOMETRY. (j ~JO represents two right lines passing through the fixed point x'y' ; and, Every homogeneous quadratic in x and j represents two right lines passing through the origin. 127. The equation Ax 2 -f- 2Hxy -f- By 2 deserves a fuller interpretation, as it leads to conditions which have a most important bearing on the relations of the Right Line to the Conies. If we divide it by x 2 , it assumes the form of a complete quadratic in y : x, But (Arts. 124 ; 78, Cor. 5) it may also be written Hence, (Alg., 234, Prop. 2d,) its roots are the tangents of the angles made with the axis of x by the two lines which it represents. Now if we solve (1) for y : x, we obtain y__ HV(H 2 -AB) , x~ B that is, the roots are real and unequal when H 2 > AB ; real and equal when H 2 = AB ; and imaginary when H 2 0, the equation denotes two real right lines passing through the origin ; if H 2 AB = 0, two coincident ones ; but in case H 2 AB < 0, two imaginary ones. Corollary. The reasoning here employed is obviously applicable to the equation of Art. 126. The meaning of any homogeneous quadratic may therefore be deter- mined according to the following table of corresponding analytic and geometric conditions : ANGLE OF A PAIR OF LINES. 133 ff 2 - - AB > . . Two real right lines passing through a fixed point. H 2 AB . . Two coincident right lines passing through a fixed point. H 2 JJ?<0 .-. Two imaginary right lines passing through a fixed point. Note By thus admitting the conception of coincident and im- aginary lines as well as of real ones, we are enabled to assert that every quadratic which satisfies certain conditions represents two right lines. In fact, the result just obtained permits us to say that every equation between plane co-ordinates denotes a line (or lines), and to include in this statement such apparently exceptional equa- tions as Of this equation, we saw (Art. 61, Rem.) that the only geometric locus is the fixed point (a, 0). But it is evidently homogeneous in (x a\ (y 0) and fulfills the condition H' 1 AB = mm But we saw (Art. 127) that m and m! are the roots of the equation Ax 2 + 2 Hxy + By 2 = 0. Hence, m m = mm = B 134 ANALYTIC GEOMETRY. Therefore tan

0, H 2 AB = Q, H 2 AB < in the general equation to a pair of right lines. From (1), it follows that the roots of this equation may be written hence (taking account of the preceding corollary) they will be real and unequal, when H 2 > AB; will differ by a constant, when H 2 = AB ; and will be imaginary, when H 2 < AB. But these roots are the ordinates of the two lines represented by the equation of Art. 125 : hence, in any equation of the second degree whose co-efficients satisfy the condition (2), we shall have the following criteria : H 2 AB > /. Two real, intersecting right lines. H 2 AB = .'. Two parallel right lines. H 2 AB < /. Two imaginary, intersecting right lines. EXAMPLES ON PAIRS OF LINES. 137 Corollary. If H 2 AB = 0, and F 2 BC=Q at the same time, the two lines are coincident. Hence, A right line fanned by the coincidence of two others is the limit of two parallels. EXAMPLES. 1. Form the equation to the two lines passing through (2, 3), (4, 5) and (1, 6), (2, 5). 2. What locus is represented by xy = ? By x 2 y* = ? By x 1 5xy + 6?/ 2 = ? By ;e 2 2xy tan y a = 0? 3. What lines are represented by a? 6# 2 y -j- 11 xy 2 6^ = 0? 4. Show what loci are represented by the equations # 2 -f- y* = 0, x * + xy = 0, x 2 + 3/ 2 + 2 = 0, *y - a* = 0. 5. Interpret (* a) (?/ 6) = 0, (x a) 2 + (y ) 2 = 0, and Cr-y+a) 2 -fO + y-a) 2 :=0. 6. Show that (y 3# + 3) (3y + * 9) = represents two right lines cutting each other at right angles. 7. Find the angles between the lines in Ex. 2. 8. What is the angle between the lines x* -f- x y 67/ 2 =0? 9. Write the equation to the bisectors of the angles between the pair x 2 5xy + 6y 2 = 0. 10. Write the equations to the bisectors of the angles between the pairs x 1 y 2 = and x 1 a?y + y 2 = 0. 11. Show that the pair 6.r 2 + bxy 6?/ 2 = intersect at right angles. 12. Show that 6ar 2 + 5xy 6?/ 2 = bisect the angles between the pair 2x* + I2xy + ly 1 6. 13. Verify that a; 2 5xy + 4y 2 + x -f 2y 2 = represents two right lines, and find the lines. 14. Show that 9z 2 I2xy + 4y z 2x + y 3 = does not rep- resent right lines, and find what value must be assigned to the co- efficient of x in order that it may. 15. Show that 4z 2 I1xy+ 9y 2 4x + 6y 12 = represents two parallel right lines, and find the lines. 138 ANALYTIC GEOMETRY. 16. Show that 4x* I2xy + 9/ 4x + 6y + 1 = denotes two coincident right lines, and find the line which is their limit. 17. Show that 5z 2 \1xy + 9?/ 2 2x + &y + 10 = represents two imaginary right lines, and find them. 18. Show that if A, B, (7, L, N are any five constants, the equation to any pair of right lines may be written (Ax + By+ C)*--= (Lx + JV); that the lines are real, when (Lx + N) 2 is positive; imaginary, when (Lx + Ny is negative; parallel, when i = 0; and coinci- dent, when L and N are both = 0. 19. Form an equation in the type of Ax* + 2Hxy + By 2 + 2Gx + 2J?V+C=0 with numerical co-efficients, which shall represent two real right lines. 20. Form a similar equation representing two parallel lines, and one representing two coincident ones; also, one representing two imaginary lines. SECTION IV. THE CIRCLE. I. GEOMETRIC POINT OF VIEW : THE EQUATION TO THE CIRCLE IS OF THE SECOND DEGREE. 133. The Circle is distinguished by the following remarkable property : The variable point of the curve is at a constant distance from a fixed point, called the center. 134. Rectangular equation to the Circle. If x y represent any point on the curve, and r its constant distance from the center gf, the required equation (Art. 51, I, Cor. 1) will be ( x - g y + (y-fy = r\ After expansion and reduction, this assumes the form 2fy + (g 2 +f 2 r 2 ) = 0: EQUATION TO CIRCLE. 139 which is evidently a particular case of the general equa- tion Ax 2 + ZHxy + By 2 + 2Gx + ZFy + C= 0. Remark The annexed diagram will ren- der clearer the geometric meaning of the equation just obtained. In this, we have OM=x, MP=y; OM'=g, M'C=f; and CP = r. Now, drawing PQ parallel to OX, we obtain by the Pythagorean theo- rem PQ 2 + QC* = CP\ That is, 13|> Equation to tbe Circle, axes being oblique. _ To obtain this, we have simply to express, in terms proper for oblique axes, the fact that the distance from xy to gf is constant. Hence, (Art. 51, I,) the equation is or, in the expanded form, * 2 + 2xy cos a) +y 2 2 (g +/cos w)x 2 (f+gcosw)y+(gi+2gfcos,<>)+f' 2 r 2 )=0. Equation to the Circle, referred to rec- tangular axes witn tne Center as Origin. Since the equation of Art. 134 is true for any origin, we have only to suppose in it g = and / = 0, and the equation now sought is Remark The student will recognize this as the equation of Art. 25, II. Its great simplicity commends it to constant use. It may also be written a form analogous to that of the equation to a right line, 140 ANALYTIC GEOMETRY. 137. Equation to the Circle, referred to any Diameter and the Tangent at its Vertex. Since the diameter of a circle is perpendicular to its vertical tangent, in order to obtain the present equation we have merely to transform the last one to parallel axes through ( r, 0). Writing, then, x r for x in x 2 + 2/ 2 = r 2 , and reducing, we find # 2 + y 2 %rx = ; or, y 2 = 2rx x 2 . Kemark In the equation just obtained, we suppose the origin to be at the left-hand vertex of the diameter. It is customary to adopt this convention. If the origin were at the right-hand vertex, we should have to replace the x of x 2 + y' 2 = r 2 by x -f r, and the equation would be z 2 + if + 2 rx = 0; or. y 2 = (2 rx + x*}. The equation of this article is verified by the o diagram. For (Geom., 325) MF* = OM.MV; that is, y 1 s= x (2r x) = 2rx x 2 . The form of this equation (Art. 63) shows that the origin is on the curve : which agrees with our hypothesis. 138. Polar Equation to the Circle. The property that the distance from the center (d, ) to the variable point of the curve is constant, when expressed in polars, (Art. 51, II) gives us p 2 + d 2 2pd cos (0 a) == r 2 . Hence, the equation now sought is p 2 2 t od cos (0 a) = r 2 d 2 . Corollary 1. Making a = in the foregoing expres- sion, we obtain p 2 2pd cos 6 = r 2 d 2 : POLAR EQUATION TO CIRCLE. 141 the polar equation to a circle whose center is on the initial line. Corollary 2 Making d = 0, we obtain p 2 = r 2 ', or, p = constant : the polar equation to a circle whose center is the pole. Remark. We may verify these equations geometrically as follows : Let OX be the initial line, and C the center of the circle. and CP = r. But by Trig., 865, OP 2 + OC 2 - 2 OP . C cos COP = CP\ Hence, p 2 2pd cos (6 a) == r 2 d*. If the point C fell on OX, COP would become XOP; and we should have If C coincided with 0, OP would become CP ; and we should have p2_- ? .2. or ^ p = constant These equations all imply that for every value of 6 there will be two values of p : which is as it should be, since it is obvious from the diagram that the radius vector OP, corresponding to any angle XOP, cuts the curve in two points, P and P'. EXAMPLES. 1. Form the equation to the circle whose center is (3, 4), and whose radius = 2. 2. Lay down the center of the circle (a; 2) 2 -f (y 6) 2 = 25, and determine the length of its radius. 3. Do the same in the case of the circle (*+ 2) 2 + (y 5) 2 = 1; in the case of the circle x* + y 2 = 3. 4. Form the equation to the circle whose center is (5, 3) and whose radius =1/7, when =:60 . 142 ANALYTIC GEOMETRY. 5. Form the equation to the circle whose radius = 3, and whose center is (3, 0). Transform the equation to the opposite vertex of the diameter. 6. Form the equation to the circle whose radius = 6, and whose center lies, at a distance from the pole = 5, on a line which makes with the initial line an angle = 60. 7. Form the equation to the circle whose center is on the initial line at a distance of 16 inches from the pole, and whose radius = 1 foot. 8. Transform the equation of Ex. 6 to rectangular axes, the origin being coincident with the pole. 9. Form the equation to the circle whose center is at the pole, and whose radius = 3. Transform it to rectangular axes, origin same as pole. 10. Form the equation to an infinitely small circle whose center is (a, 0). [Compare the result with the Remark, Art. 61, and the Note, Art. 127.] II. ANALYTIC POINT OF VIEW: THE EQUATION OF THE SECOND DEGREE ON A DETERMINATE CONDITION REPRESENTS A CIRCLE. 189. The theorem is implied in the fact established in Arts. 134, 135, that the equation to the Circle is a particular case of the general equation of the second degree. To determine the condition on which the gen- eral equation will represent a circle, we have therefore merely to compare its co-efficients with those of the equation to the Circle written in its most general form. I4O. We saw (Art. 134) that the rectangular equation to the Circle is Since g and f may be either positive or negative, and r is not a function of either / or g, this may be written in the still more general form A (x 2 + if) + 2fo CIRCLE AND GENERAL EQUATION. 143 If, then, the equation Ax 1 + ZHxy + By* + 2Gx + 2Fy + C= represent a circle, it must assume the form just obtained. But in order to this, we must have H'= with A = B : which therefore constitutes the condition that the general equation of the second degree shall represent a circle. Corollary. It is obvious from the condition just deter- mined, that when the equation of the second degree represents a circle, it also fulfills the condition ff 2 AB<0. 141. From the result of Art, 135, it follows that H = A cos a with A = B is the condition in oblique axes that the equation of the second degree shall represent a circle. And it is evident that in this case, too, we have the condition 142. If we are given an equation in the general form we can at once determine the position and magnitude of the corresponding circle. For, by transposing (7, adding -7 to both mem- bers, and dividing through by A, the given equation may be thrown into the form 2 +FZ AC A 2 Comparing this with the equation of Art. 134, namely, 144 ANALYTIC GEOMETRY. we obtain, for determining the co-ordinates of the center, G F 9- -3' /= -3' and for determining the radius, r = A Corollary 1. From the expressions for g and /, which are independent of (7, we learn the important principle, that the equations to concentric circles differ only in their constant terms. Corollary 2. From the expression for r, we derive the followin conclusions : I. G 2 + F 2 AC>0 .'. the circle is real II. G 2 -f- F 2 AC= /. the circle is infinitely small. III. G 2 -\-F 2 AC' -f a/'), 2jF : ^L = - (y f + y"), and (7 : J. = a/o" = #y . Hence, (Art. 142,) for deter- mining the center, _x>_ + x" y' + y m CJ ~ 2 I" 2 and for determining the radius, r = \ V (x'*^ x" 2 } -{- (y 12 + y" 2 ). Corollary. Hence, the equation to any circle whose intercepts on the axis of x are given, is x ' 2 + y 2 (X + *") * 2 /y + v'x" = o ; the equation to any circle whose intercepts on the axis of y are given, is x-' +f- 2gx - (y' + */") y + y'y" = ; and the equation to the circle whose intercepts on both axes are given, is 2 (x^f] - 2 (x'+x") x-2 (y'+y") y+( x 'x"+y'y"} = 0. 144. Of the two equations the first (Alg., 237, 1) will have equal roots when G 2 -=AC; and the second, when JF 2 = AC. Hence, G 2 AC=Q is the condition that a circle shall touch the axis of x ; F 2 AC= is the condition that it shall touch the axis of j ; and is the condition that it shall touch both axes. 146 ANALYTIC GEOMETRY. EXAMPLES. I. NOTATION AND CONDITIONS. 1. Decide whether the following equations represent circles : 3.? 2 + 5xy ly 1 + 2x 4y + 8 = ; 5,-r 2 Sy 8 + 3x 2y + 7 = ; 5z 2 -1- 5y 2 IQx 30y + 15 = 0. 2. Determine the center and radius of each of the circles z 2 +7/ 2 + 42/-4a;-l=0, z 2 -f 7/ 2 + 6x 3y 1 = 0. 3. Form the equation to the circle which passes through the origin, and makes on the two axes respectively the intercepts -f- h and -f- k- 4. Find the points in which the circle x' 2 ^y' 2 = 3 intersects the lines x+y + 1= 0, x+y 1=0, and 2x + y 1/5=9. 5. What must be the inclination of the axes in order that each of the equations x z xy -f /' 2 Aa; hy = 0, x 2 -f- xy -{- y 2 A a; Ay = 0, may represent a circle ? Determine the magnitude and position of each circle. 6. Write the equations to any three circles concentric with 2(s 2 + y 2 ) + 6x 4y 12 = 0. 7. Form the equation to the circle which makes on the axis of x the intercepts (5, 12), and on the axis of y the intercepts (4, 15). Determine the center and radius of the same. 8. What is the equation to the circle which touches the axes at distances from the origin, each = a ? at distances = 5 and 6 respectively ? 9. ABC is an equilateral triangle: taking A as pole, and AB as initial line, form the polar equation to the circumscribed circle. Transform it to rectangular axes, origin same as pole, and axis of x as initial line. 10. If S and S / = are the equations to any two circles, what does the equation S P/S" = represent, k being arbitrary ? EXAMPLES ON THE CIRCLE. 147 II. CIRCULAR LOCI. 1. Given the base of a triangle and the sum of the squares on its sides : to find the locus of its vertex. Taking the base and a perpendicular through its middle point for axes, putting 2s 2 = the given sum of squares, and in other respects using the notation of Ex. 1, p. 126, we have PR 2 = 3/2 + (m + a-) 2 , PQ* = y 2 + (m - xp. Hence, the equation to the required locus is M Q and the vertex moves upon a circle whose center is the middle point of the base, and whose radius = V 2 m' z . 2. Given the base and the vertical angle of a triangle : to find the locus of the vertex. 3. Given the base and the vertical angle of a triangle : to find the locus of the center of the inscribed circle. 4. Find the locus of the middle points of chords drawn from the vertex of any diameter in a circle. 5. Given the base and the ratio of the sides of a triangle: to find the locus of the vertex. 6. Given the base and vertical angle: to find the locus of the intersection of the perpendiculars from the extremities of the base to the opposite sides. 7. When will the locus of a point be a circle, if the square of its distance from the base of a triangle bears a constant ratio to the product of its distances from the sides ? 8. When will the locus of a point be a circle, if the sum of the squares of its distances from the three sides of a triangle is con- stant ? 9. ABC is an equilateral triangle, and P is a point such that PA = PB + PC: find the locus of P. 10. A C B is the segment of a circle, and any chord A C is pro- duced to a point P such that AC nCP : to find the locus of P. 148 ANALYTIC GEOMETRY. 11. To find the locus of the middle point of any chord of a circle, when the chord passes through any fixed point. 12. On any circular radius vector OQ, OP is taken in a constant ratio to OQ : find the locus of P. 13. Find the locus of P, the square of whose distance from a fixed point O is proportional to its distance from a given right line. 14. O is a fixed point, and AS a fixed right line; a line is drawn from O to meet AB in Q, and on OQ a point P is taken so that OQ.OP= k* : to find the locus of P. 15. A right line is drawn from a fixed point O to meet a fixed circle in Q, and on OQ the point P is so taken that OQ.OP= k 2 : to find the locus of P. SECTION V. THE ELLIPSE. I. GEOMETRIC POINT OF VIEW I THE EQUATION TO THE ELLIPSE IS OF THE SECOND DEGREE. 145. The Ellipse may be defined by the following property : The sum of the distances from the variable point of the curve to two fixed points is constant. We may therefore trace the curve and discover its figure by the following process : Take any two points F' and F, and fasten in them the extremities of a thread whose length is greater than F'F. Place the point of a pencil P against the thread, and slide it so as to keep the thread constantly stretched: P in its motion will describe an ellipse. For, in every position of P, we shall have F'P+FP= constant, as the sum of these distances will always be equal to the fixed length of the thread. EQUATION TO ELLIPSE. 149 146. The two fixed points, F' and F, are called the foci; and the distances with a constant sum, F'P and FP, the focal radii of the curve. The right line drawn through the foci to meet the curve in A! and A, is called the transverse axis. The A , point 0, taken midway between F' and F, we may for the present call the focal center. The line B'B, drawn through at right angles to A' A, and terminated by the curve, is called the conjugate axis. 147. Equation to the Ellipse, referred to its Axes. Let %c = the constant distance between the foci, and 2a = the constant sum of the focal radii. Then, from the diagram above, F'P 2 = (x-\- c) 2 -f- ?/ 2 , and FP 2 = (x c) 2 -f- y z . Hence, the fundamental prop- erty of the Ellipse, expressed in algebraic symbols, will be Freeing this expression from radicals, we obtain the required equation, (a 2 c 2 ) x 2 + a 2 if = a 2 (a 2 - c 2 ) (1) . To abbreviate, put a 2 c 2 = b 2 , and this becomes b 2 x 2 + ay = a 2 b 2 (2): which may be more symmetrically written ^ . 2/! -, a 2~r 52 1. Eemark The student will observe the analogy between the last form and the equation to the Right Line in terms of its intercepts. 150 ANALYTIC GEOMETRY. 148. It is important that we should get a clear con- ception of the general form of the equation just obtained. Let us for a moment return to the form (1), (a 2 c 2 ) x 2 + ay = a 2 (a 2 c 2 ). In this, the definition of the Ellipse requires that a 2 c 2 shall have the same sign as a 2 ; for the sura of the dis- tances of a point from two fixed points can not be less than the distance between them : that is, a can not be numerically less than c, and consequently a 2 not less than c 2 . Therefore, in the equation to the Ellipse, the co-effi- cients of X L and y 1 must have like signs. Advancing now to the form (2), b 2 x 2 -f- a 2 y 2 a 2 b 2 , it is obvious that the constant term a 2 b 2 will have but one sign, whether the co-efficients of x 2 and y 2 be both positive or both negative. Supposing, then, that a 2 and b 2 are both positive, the equation after transposition would be b 2 x*-\- a 2 f a*b 2 = Q (3). Supposing them both negative, it would become, after transposition and the changing of its signs, b 2 x 2 + a 2 y 2 + a 2 b 2 = (4). Now, what is the meaning of the supposition that a 2 (and thence b 2 ) is negative ? Plainly (since in that case we shall have, instead of a, ai/--l) it signifies that the corresponding ellipse is imaginary* Hence, admitting into our conception of this curve the imaginary locus of (4), we learn that the equation to the Ellipse is of the general form A'x 2 + B'y 2 + C 1 = : * This interpretation is put beyond question by the form of equation (4) itself, which denotes an impossible relation. POLAR EQUATION TO ELLIPSE. 151 in which A! and B f are positive, and C r is either positive or negative according as the curve is imaginary or real. 149. We may at this point derive from the equation to the Ellipse a single property of the curve, as we shall need it in discussing the general equation of the second degree. Definition. A Center of a Curve is a point which bisects every right line drawn through it to meet the curve. Theorem, In any ellipse, the focal center is the center of the curve. The equation to any right line drawn through the focal center (Art. 78, Cor. 5) is y = mx. Comparing this with the equation to any ellipse, namely, x 2 y 2 ^ + ^ = :1 ' we see that if x f , y' satisfy both equations, #', y r will also satisfy both. In other words, the points in which the ellipse is cut by any right line drawn through the focal center may be represented by x r , y' and x f , y r . But these symbols (Art. 51, I, Cor. 2) necessarily denote two points equidistant from the focal center : which proves the proposition. 150. Polar Equation to the Ellipse, the Center being the Pole. Replacing (Art. 57, Cor.) the x and y of a 2 y 2 + b 2 x* = a 2 b 2 by p cos 6 and p sin 6, we find 97 ? 2 a~fr " a z sin 2 # -j- b 2 co& 2 6 ' that is, (Trig., 838,) 9 _ a 2 b 2 p " a 2 (a 2 b 2 ) cos 2 ' An. Ge. 16. 152 ANALYTIC GEOMETRY. Divide both terms of the second member of this expres- sion by a 2 , and, to abbreviate, put the result will be the form in which the required equation is usually written, namely, v P i - 9 - Ta ' 1 e 2 cos 2 d Remark. The equation indicates that for any value of there will be two radii vectores, numerically equal with contrary signs. The accom- panying diagram verifies this result; for the two points of the ellipse, P and P 1 ', evidently correspond to the same angle 0, and the point P' has the radius vector OP' = OP. 151. Special attention should be given to the two abbreviations used above, 2 12 ,1 ^V a? c 2 = b 2 and - = e - We shall find hereafter that they represent elements of great significance in the Conies. For the present, how- ever, they are to be regarded as abbreviations merely. It is evident that by combining them we obtain the relation c = ae. Corollary, Hence, the central polar equation to the Ellipse may be written n 2 _ ~ ~ a 2 (1 e 2 ) a form which will frequently prove more convenient than that of Art. 150. POLE AT FOCUS. 153 152. Polar Equation to the Ellipse, the Focus being the Pole. From the annexed diagram, we have F'P = p, and FP = y(p 2 + 4c 2 4/w cos 6). Hence, expressing the fundamental property of the Ellipse, a 2 c 2 p = - " c cos Replacing c by its equal ae, we obtain the usual form of the equation, " 1 g cos # Remark. The student should carefully discriminate between this equation and that of the corollary to Art. 151. From their striking similarity, the two are liable to be confounded. In using this equation, it is to be remembered that in it the left- hand focus is taken for the pole. In practice, this assumption is generally found to be the more convenient. The student may show, however, that when the right-hand focus is the pole the equation to the Ellipse is p = 1 -f e cos EXAMPLES. 1. Given the two points ( 3, 0) and (3, 0); the extremities of a thread whose length =10 are fastened in them: form the equation to the ellipse generated by pushing a pencil along the thread so as to keep it stretched. 2. In a given ellipse, half the sum of the focal radii = 3, and half the distance between the foci = 2 : write its equation. 3. Form the equation to the ellipse whose focus is 3 inches from its center, and whose focal radii have lengths whose constant sum = 1 foot. 4. In a given ellipse, the sum of the focal radii = 8, and the difference between the squares of half that sum and half the dis- tance between the foci = 9 : write its equation. 154 ANALYTIC GEOMETRY. 5. Show that in each of the ellipses hitherto given, the focal center bisects the lines y = 2#, y 3#, y = 5#, and any others whose equations are in the form y = mx. 6. Write the central polar equation to the ellipse in which the difference between the squares of half the sum of the focal radii and half the distance between the foci 9, and the ratio of the distance between the foci to the sum of the focal radii = 1:2. 7. "Write the central polar equation to the ellipse of Ex. 2. 8. Find, in the same ellipse, the ratio of the sum of the focal radii to the distance between the foci, and write the equation in the form corresponding to that in the corollary to Art. 151. 9. In a given ellipse, the sum of the focal radii = 12, and the ratio between that sum and the distance from the left-hand focus to the right-hand one = 3 : write its polar equation, the focus being the pole. 10. The focus being the pole, form the polar equation to the ellipse of Ex. 3. What would the equation be, if the right-hand focus were the pole ? II. ANALYTIC POINT OF VIEW: THE EQUATION OF THE SECOND DEGREE ON A DETERMINATE CONDITION REPRESENTS AN ELLIPSE. 153. We have seen (Art. 148) that the equation to the Ellipse may always be written in the form A'x 2 + B'y 2 + C r = 0, in which A' and B' are positive, and C r is indeterminate in sign. If, then, we can show that the general equation of the second degree is reducible to this form, and can find real conditions upon which the reduction may always be effected, we shall have established the theorem at the head of this article. ELLIPSE AND GENERAL EQUATION. 155 154. In order that the general equation of the second degree, Ax* + 2Hxy + By 2 + 2Gx + ZFy + C= 0, may assume the form A''* 2 + B f y 2 + C' = 0, the co-efficients of its x, y, and xy must all vanish. The question therefore is : Can we transform the equation to axes such as will cause these co-efficients to disappear ? 155. As we have seen, the equation A r x 2j rB'y 2 -\- C f =0 is referred to the center of the Ellipse, and to its axes, which by definition cut each other at right angles. As- suming, then, as we may, that the general equation of the second degree as above written is referred to rectan- gular axes, our first step will naturally be to determine, if possible, the center of the locus which it represents, and to reduce it to that center as origin. Let x r , y' be the co-ordinates of the center sought, and let us transform Ax 2 + 2ffxy -f By 2 + 2Gx -j- 2Fy + tf= (1) to parallel axes passing through x'y' . Replacing (Art. 55) the x and y of (1) by (x r -j- x) and (y f -f y), we obtain, after reductions, Ax 2 -f 2Hxy + By 2 \ -f Ax f2 -f ZHx'y' + By' 2 + 2Gx f + 2Fy r + C ) Now, since this equation is referred to the center as origin, it must (Art. 149) be satisfied equally by x, y and x,y. But in order to this, we must have 156 ANALYTIC GEOMETRY. Eliminating between these simultaneous equations, we find the co-ordinaies of the center, BGHF f AFffG ~ E*-AB ' y ~~- R 2 AB ' It is obvious that these values of x f and y' will be finite so long, and only so long, as H 2 AB is not equal to zero. Hence we conclude that the locus of (1) has a center, which is situated at a finite distance from the origin or at infinity, according as (1) does not or does fulfill the condition H 2 AB = Q. If, then, in the result of our first transformation above, we substitute these values of x' and y 1 ', we shall obtain an equation to the locus of (1), referred to its center. Now the only elements of that result which depend on xf and y', are the co-efficients of x and y, and the abso- lute term. Of these, the first two vanish, when the finite co-ordinates of the center are substituted in them ; the third may be thrown into the form {Axf+Htf+G) *'+ (By' + Hx'+F] y>+ (Gx'+Fy'+C} : and if in this we substitute the finite co-ordinates of the center, it becomes or, ABC+2FGHAF 2 -BG 2 -CH* H 2 AB Hence, putting C f to represent either (a) or (5), the equation of the second degree, reduced to the geometric center of its locus, is '=Q (2). ELLIPSE AND GENERAL EQUATION. 157 156. The transformation from (1) to (2) has destroyed the co-efficients of x and y, but the co-efficient of xy still remains. Reduction to the center as origin is therefore not sufficient to bring (1) into the form And, in fact, we might have anticipated as much; for the equation to the Ellipse, of which the required form is the type, is referred not to the center merely, but to the axes of the curve. To destroy the co-efficient of xy, then, we must resort to additional transformation ; and our next step will naturally be to determine, if possible, the axes of the locus represented by (2), and to revolve the reference-axes until they coincide with them. Let 6 = the angle made with the reference-axes of (2) by the possible axes of the locus. Replacing (Art. 56, Cor. 3) the x and y of (2) by x cos 6 y sin and x sin 6 -f y cos 0, we obtain, after reductions, (A cos 2 d-\-2H sin dcosd + B sin 2 0) x 2 ^ - 2 { (AB) sin 6 cos 6H(cos 2 6 sin 2 d)}xy I + C'=Q ; -|- (A sin 2 6 2#sin dcosd + B cos 2 6) y 2 J that is, (Trig., 847 : I, n, IV,) -\ L+C"=0. + J {(A+B) (A)cos202ffsm2d}y 2 ) If, in this expression, we equate to zero the co-efficient of xy, we shall have (A B] sin 20 2#cos 20 = : .-. tan 20 = J?) 158 ANALYTIC GEOMETRY. That is, since a tangent may have any value positive or negative from to GO, 20 (and therefore 6) is a real angle ; in other words, there do exist two real lines, at right angles to each other, which in virtue of their destroying the co- efficient of xy we may call axes of the curve to the locus of (2). Accordingly, if in the equation last obtained we substitute for the functions of 2# their values as implied in (c),* the resulting equation will represent the locus, referred to these so-called axes. cos 9 # " Substituting these values in our last equation, we find , ~ + J { (A + B) - i/(A whence, by writing = J {(A + B) + l/p-J?) 2 + (2#) 2 } (d), (e), the equation of the second degree, reduced to the axes of its locus, is 4V + y + C" = o (3). *By Trig., 836, sin A = -; cos A = - . But since c represents the hypotenuse, and a and b the sides, of a right triangle: c = Va* + b*~. a ~b Hence, when the tangent is given, e. g. tan A = j- , we at once derive the sine and cosine by writing CRITERION OF THE ELLIPSE. 159 From (3) it appears, that, setting aside the question of signs, the general equation of the second degree can be reduced to the required form ; provided it is not subject to the condition H 2 -AB = 0. 158. It remains, then, only to inquire what condition the general equation must fulfill in order that its reduced form (3) may have that combination of signs which (Art. 148) is characteristic of the Ellipse. If A! and B 1 are both positive, we shall have A'B' = positive ; or, by substituting for A' and B' from (d) and (e) above, and reducing, AB H 2 = positive ; that is, changing the signs of both sides of the expression, If 2 AB = negative. Hence, The equation of the second degree represents an ellipse whenever its co-efficients fulfill the condition H 2 AB<0. 1*>O. At the close of Art. 148 we saw that the sign of C r is plus or minus according as the ellipse is imag- inary or real. Let us then seek the conditions which the general equation must fulfill in order to distinguish between these two states of the curve. Applying the condition H 2 AB < to the value of G 1 as given in (b) of Art. 155, we see that, for C' to be negative, we must have ABC + 2FGH AF 2 BG 2 CH 2 < ; and, for C f to be positive, ABC + 2FGHAF 2 BO 2 OH 2 > 0. An. Ge. 17. 160 ANALYTIC GEOMETRY. In other words, we find that the same quantity which (Art. 131) by vanishing indicates a pair of right lines as the locus of the general equation, by changing sign indi- cates the transition from the real to the imaginary ellipse. This quantity is called, in modern algebra, the Discrimi- nant of the general equation ; and we may appropriately represent it by the Greek letter J. Adopting this nota- tion, we have H* AB<0 with J<0 as the condition that the equation of the second degree shall represent a real ellipse; and H 2 AB0 as the condition that it shall represent an imaginary one.* 16O. We can now see, at least in part, the real bearing of the conditions in terms of H 2 AB which we some time ago developed respecting Pairs of Right Lines. Comparing the results of Arts. 131, 132, we infer that H 2 AB<0 with J = (7) is the condition that the equation of the second degree shall represent two imaginary intersecting lines. But this condition evidently lies between the two criteria ff 2 AB < with J ' < (&), H 2 AB<0 with J>0 (m); so that we can not pass from (k) to (m) without passing through (7). We thus learn that two imaginary right lines intersecting each other, form the limit between the real and the imaginary ellipse. If we now revert to the equation (Art. 132) denoting * In testing any given equation by these criteria, we must see that its signs are so arranged that A (the co-efficient of x 2 ) may be positive. The conditions with respect to A, are derived on this assumption. POINT AND CIRCLE AS ELLIPSE. 161 two right lines, and take its two roots separately, we see that the two lines are / TT _, / JJT'Z /\ D \ I 7? I / Tjt i / T/i'> > /Y \ A ( XZ K 1 -/l-O ) iC ~+~ .Dl/ ]~ \Ju |/ M " /> L/ ) :::=: U. Eliminating between these equations, and recollecting [Art. 131, (1)1 that (HF-BGY we find, as the co-ordinates of intersection for the two lines, BG HF AFHG ~ ~ H 2 AB' That is, (Art. 155,) the lines intersect in the center of the locus of the general equation. But we have seen that this center is real, irrespective of the state of H 2 AB\ and is finite, so long as H 2 AB is not zero. Hence, when- ever the equation of the second degree represents two intersecting lines, their intersection is a finite real point, whether they be real or imaginary. Uniting the two conclusions thus reached, we obtain the following important theorem : The Point, as the inter- section of two imaginary right lines, is the limiting case of the Ellipse. Remark, This result is corroborated by the equation (Art. 148) to the Ellipse itself. For if, in the expression ' A'x 2 + BY + C" = 0, we suppose A = 0, then C" = , and the equation becomes which (Art. 126, Cor. 2 cf. Art. 127) denotes two imaginary lines passing through the origin ; that is, in this case, through the center. 161. The Point and the Pair of Imaginary Intersecting Lines have thus been brought within the order of Conies. We shall now show that the Circle likewise belongs there. 162 ANALYTIC GEOMETRY. The condition that the equation of the second degree shall represent a circle (Art. 140) is But, as we noticed in the corollary to Art. 140, this is merely a special form of the condition H 2 AB<0. Hence, the Circle is a particular case of the Ellipse. Resuming, then, the equation to the Ellipse, namely, (a 2 c 2 ) x 2 + a 2 ?/ 2 = a 2 (a 2 c 2 ), we notice that it already fulfills the condition ff=Q. Adding the condition A = B, necessary to make it rep- resent a circle, we obtain, as characteristic of the Circle, a 2 G 2 = a 2 .-. c = Q. We hence learn that the Circle is an ellipse whose two foci have become coincident at the center. Moreover, the Circle is real, vanishes, or is imaginary, on the same conditions as the Ellipse. For we saw (Art. 142, Cor. 2) that it assumes these several phases according as the quantity is positive, zero, or negative. Now, if we apply to the Discriminant A the conditions H= 0, A = B, we find, as true for the Circle, -A = A(G* + F* AC). And since we are always to suppose A positive, we have A < .*. a real circle. A = .. a point. A > .*. an imaginary circle. Remark. In allusion to the fact that its foci do not in general vanish in the center, the Ellipse may be called eccentric. THEOREMS OF TRANSFORMATION. 163 1G2. The following table exhibits the analytic con- ditions thus far imposed upon the equation of the second degree, with their geometric consequences : ( Real v A<0. II' 2 AB<0 ( ff== ' A ~ B = .'. Eccentric Curve . J Point . A=0. } (imag. . A>0. Ellipse ) (-Real . A<0. ^ff=,A- B = 0. -.Circle J Point . A=0. (imag. . A>0. 163. Theorems of Transformation. Before ad- vancing further, it will be well to collect from the foregoing discussion the theorems it implies respecting transformation of co-ordinates. They are often conve- nient in performing the reductions to which they relate. Theorem I. In transforming any equation of the second degree to parallel axes through a new origin: 1. The variable terms of the second degree retain their original co-efficients. 2. The variable terms of the first degree obtain new co-efficients, which are linear functions of the new origin. 3. The constant term is replaced by a new one, which is the result of substituting the co-ordinates of the new origin in the original equation. For, in applying this transformation to equation (1) of Art. 155, the co-efficients of x 2 , xy, and y 2 continued to be A, 2H, and B ; the co-efficients of x and y respectively replaced G and F by AaS + Htf + G, By' + Hx' + F; and, for the new constant term, we obtained Ax /2 + ZHx'y' + Ey n + 2Gx / + 2Fy' + C. Theorem II. In transforming any equation of the second degree from one set of rectangular axes to another, the quantities A -j- B, H 2 AB remain unaltered. For the equation near the middle of p. 157 is the result of this transformation ; and if we add together the co-efficients of x~ and y 2 in it, after representing them by A / and B' ', we obtain A / + B / = A + B. 164 ANALYTIC GEOMETRY. In likof manner, representing the co-efficient of xy by 2// / , and performing the necessary operations, we find Theorem III 7/* in the process of transforming an equation of the second degree, the co-efficients of x and j vanish, the new origin is the center of the locus. For, in that event, the new equation will be satisfied equally by x, y and x, y ; that is, all right lines drawn through the new origin to meet the curve will be bisected by that origin. Corollary, If only one of these co-efficients vanish, the new origin will lie on a right line passing through the center. For we must then have either that is, the co-ordinates of the new origin must satisfy one of the equations (Art. 155) by eliminating between which we determined the center. Theorem IV. If the co-efficient o/xy vanish, the new reference-axes, if rectangular, will be parallel to the axes of the locus. For when in Art. 156* this co-efficient vanished, with the center as origin, the new reference-axes coincided with the axes of the locus; hence, if the origin is at any other point, they must be parallel to them. * As the beginner is liable to misapprehend the argument of Art. 156, it may be well to restate it, in the form which the present connection suggests: When we revolved the reference-axes through the angle 9 7-7" e = y 2 tan- A-B> (which was found by equating the co-efficient of xy to zero} we produced an equation (3) identical in form with that previously obtained 'for the Ellipse by referring it to its axes. So far then as concerned the Ellipse, the new reference-axes were identical with the two lines which (Art. 146) we had described as the axes of that curve. But on account of their power to reduce the general equation to a fixed form, these two lines were properly assumed to have a fixed relation to its locus, analogous to that which they bore to the Ellipse ; and hence were called the " axes " of that locus. EXAMPLES ON THE ELLIPSE. 165 1O4. These theorems not only furnish criteria for selecting such transformations as will represent required geometric conditions, but they enable us to shorten the process of transformation. Thus, knowing Theorem I, we can henceforth write the result of transforming to parallel axes, without going through with the ordinary substitutions. Knowing Theorem II, we can immediately write the central equation of a second order curve from its general equation, by merely setting down the first three given terms and adding a new constant term found as in Theorem I, 3. By uniting Theorems II and IV, we may shorten the process of reduction to the axes. For, if such a reduc- tion is required, we shall have H f = ; and, therefore, A' + &=A + B with A'B' = AB H 2 : two equa- tions from which we can easily find A' and B r . C r is found as in Theorem I, 3. It is preferable, however, to write the reduced equation at once ; for its form is A'x 2 + B'y 2 + C' = Q: in which A', B r are found by formulae (d) and (e) Art. 156, and C r is obtained as before. When the given equation is already central, this reduction becomes very brief; since we do not then have to calculate C'. EXAMPLES. I. NOTATION AND CONDITIONS. 1. Determine by inspection the locus of each of the equations 2x* + 3?/ 2 = 12, 2. Transform 3z 2 + 4.ry + if 5x by 3 = to parallel axes through (2, 3). Is the curve an ellipse? 166 ANALYTIC GEOMETRY. 3. Reduce x*+ 2xy y 1 + Sx + 4y 8 = to the center. Does this represent an ellipse ? 4. If in a given equation of the second degree // 0, what condition must be fulfilled in order that the equation may represent an ellipse ? What, if A or _B equals zero ? 5. Transform Ux 2 4xy + lit/ 2 = 60 to the axes of the curve, by all three methods. What is the origin in the given equation? What is the locus, and is the same locus indicated by the reduced equation ? 6. Find the center of 5z 2 + 4xy -f if 5x 2y 19, and reduce the equation to it. Show that the curve is a real ellipse, both by the original equation and the reduced one. 7. Show that 14-r 2 4xy -f- lly 2 = denotes an infinitely small ellipse ; that is, an ellipse in the limiting case. 8. Show that 5;c 2 + 4zy -f if 5x 2y -f 19 = represents an imaginary ellipse, and verify by writing the equation as referred to the axes. 9. Find the equation to the Ellipse, the origin 3/y' being on the curve, and the reference-axes parallel to the axes of the curve. 10. Find the equation to an ellipse whose conjugate axis is equal to the distance between its foci, taking for axes the two lines that join the extremities of the conjugate to the left-hand focus. II. ELLIPTIC LOCI. 1. Find the locus of the vertex of a triangle, given the base and the product of the tangents of the base angles. Taking the base and a perpendicular through its middle point for axes, calling the given product I* : 2 , and in other respects retaining the notation of Ex. 1, p. 126, we shall have Q 7/ Hence, by the conditions of the problem, m -/_ ^ = 2 J anc * ^ Q equation to the required locus is Therefore, (Art. 147) the vertex moves on an ellipse whose center is the middle point of the base, and whose foci are on the base at a distance from the center m)/n 2 I 2 : n. ELLIPTIC LOCI. 167 2. Find the locus of the vertex, when the base and the sum of the sides are given. 3 Find the locus of the vertex, given the base and the ratio of the sides. 4. Given the base, and the product of the tangents of the halves of the base angles : to find the locus of the vertex. 5. Two vertices of a given triangle move along two fixed lines which are at right angles : to find the locus of the third. 6. A right line of given length moves so that its extremities always lie* one on each of two fixed lines at right angles to each other: to find the locus of a point which divides it in a given ratio. 7. In a triangle of constant base, the two lines drawn through the vertex at right angles to the sides make a constant intercept on the line of the base : find the locus of the vertex. 8. The ordinate of any circle a 2 + /3 2 = r 2 is moved about its foot so as to make an oblique angle with the corresponding diam- eter: find the locus of its extremity in its new position. 9. The ordinate of any circle x 2 -f- y 1 = r* is augmented by a line equal in length to the corresponding abscissa: find the locus of the point thus reached. 10. To the ordinate of any circle there is drawn a line, from the vertex of the corresponding diameter, equal in length to the ordinate: find the locus of the point of meeting. 1 1 . In any ellipse, find the locus of the middle point of a focal radius. 12. Find the locus of the extremity of an elliptic radius vector prolonged in a constant ratio. 13. A right line is drawn through a fixed point to meet an ellipse : find the locus of the middle point of the portion intercepted by the curve. 14. Through the focus of an ellipse, a line is drawn, bisecting the vectorial angle, and its length is a geometric mean of the radius vector and the distance from the focus to the center; find the locus of its extremity. 15. Through any point Q of an ellipse, a line is drawn parallel to the transverse axis, and upon it QP is taken equal to the corre- sponding focal radius : find the locus of P. 1G8 ANALYTIC GEOMETRY. SECTION VI. THE HYPERBOLA. I. GEOMETRIC POINT OF VIEW: THE EQUATION TO THE HYPERBOLA IS OF THE SECOND DEGREE. 165. The Hyperbola is characterized by the following property : The difference of the distances from the variable point of the curve to two fixed points is constant. Hence, we may trace the curve, and determine its figure, as follows : Take any two points, as F' and F. At F', pivot the corner of a ruler F'R; at F, fasten one end of a thread, whose length is less than that of the ruler. Then, having attached the other end to the ruler at R, stretch the thread close against the edge of the ruler with the point of a pencil P. Move the ruler on its pivot, and slide the pencil along its edge so as to keep the thread continually stretched : the path of the pencil- point will be an hyperbola. For, in every position of _P, we shall have F'P FP= (F'P + PR) (FP + PR) = F'RFPR. That is, the difference of the distances from the variable point P to the two fixed points F' and F will always be equal to the difference between the fixed lengths of the ruler and the thread ; or, we shall have F'P FP = constant. By pivoting the ruler at jP, and fastening the thread at F', we shall obtain a second figure similar in all respects to the former, except that it will face in the opposite direction. The complete curve therefore con- sists of two branches, as represented in the diagram. EQUATION TO HYPERBOLA. 169 166. The two fixed points, F f and F, are called the foci of the curve ; and the variable distances with a con- stant difference, F'P and FP, are termed its focal radii. The portion A' A of the right line drawn through the foci, is called the transverse axis. The point 0, taken midway between the foci, we shall for the present call the focal center. It is apparent from the diagram, that the right line Y f Y, drawn through the focal center at right angles to the transverse a.xis, does not meet the curve. We shall find, however, that a certain portion of it has a very significant relation to the Hyperbola, and is convention- ally known as the conjugate axis. For the present, we shall speak of the whole line under that name. 167. Equation to the Hyperbola, referred to its Axes. Putting 22 c 2 a 2 = b 2 and - -33- =*, a 2 may evidently be derived from the two used in connection with the Ellipse (Art. 151), by substituting b 2 for b 2 . By combining them, however, we still obtain the relation c =ae. Corollary, Hence, the central polar equation to the Hyperbola may be written a 2 (l-e 2 ) . a formula which we leave the student to distinguish from that given for the Ellipse in the corollary to Art. 151. 174 ANALYTIC GEOMETRY. 172. Polar Equation to the Hyperbola, the Focus being the Pole. From the annexed diagram, we have F f P = p, and FP = -|/( ( o 2 -f 4c 2 4pc cos 0). Hence, expressing the defining prop- erty of the Hyperbola, P I/Go 2 + 4c 2 4fj C cos 6)=2a: c cos 6 a Replacing c by its equal ae, we obtain the usual form of the equation, ; 1 ecosd Remark The apparent identity of this equation with that of Art. 152, we leave the student to explain; and he may show that when the right-hand focus is the pole, the equation will be _ a(g'-l) P 1 e cos 6 EXAMPLES. 1. Given the two points (3, 0) and (3, 0); on the first is pivoted a ruler whose length = 20, and in the second is fastened a thread whose length = 16: form the equation to the hyperbola generated by means of this ruler and thread as in Art. 165. 2. In a given hyperbola, half the difference of the focal radii = 2, and half the distance between the foci = 3 : write its equa- tion. Why can not this example be derived from Ex. 2, p. 153, by merely substituting "difference" for "sum"? 3. Form the equation to the hyperbola whose focus is 1 foot from its center, and whose focal radii have the constant difference of 3 inches. 4. In a given hyperbola, the difference of the focal radii = 8, and the difference between the squares of half that difference and half the distance between the foci = 9 : write its equation. HYPERBOLA AND GENERAL EQUATION. 175 5. Write the equations to the hyperbolas which are the conju- gates of those preceding. 6. Write the central polar equation to the hyperbola in which the squares of half the difference of the focal radii and half the distance between the foci differ by 9, and the ratio of the dis- tance between the foci to the difference of the focal radii =2:1. 7. Write the central polar equation to the hyperbola of Ex. 2. 8. Find, in the same hyperbola, the ratio which the difference of the focal radii bears to the distance between the foci, and write the equation in the form given in the corollary to Art. 171. 9. In a given hyperbola, the difference of the focal radii = 12, and the ratio of that difference to the distance between the foci = 1:3. Write its polar equation, the focus being the pole. 10. The focus being the pole, form the equation to the hyperbola of Ex. 3. What would this be, if the right-hand focus were the pole? II. ANALYTIC POINT OF VIEW: THE EQUATION OF THE SECOND DEGREE ON A DETERMINATE CONDITION REPRESENTS AN HYPERBOLA. 173. To establish this theorem, we must show (Art. 168) that the general equation of the second degree is reducible to the form in which A! is positive, and B r negative ; and we must be able to find real conditions upon which the reduction can always be effected. From (3) of Art. 156, we already know that, apart from the question of signs^ the general equation is re- ducible to the required form. It only remains, then, to determine the condition which must be fulfilled in order that the signs of A' and B' may be such as characterize the Hyperbola. An. Ge. 18. 176 ANA L YTIC G EG METE Y. 174. If A' is positive, and B' negative, the product A'B' must be negative. Therefore (see Art. 158) for the Hyperbola we shall have AB H 2 = negative ; or, after changing the signs throughout, H 2 AB = positive. Hence, The equation of the second degree represents an hyperbola whenever its co-efficients fulfill the condition H 2 -AB>0. 175. Let us now inquire what additional criteria the equation must satisfy, in order to distinguish between a primary hyperbola and its conjugate. For the reduced form of the equation, (Art. 168,) the curve is primary or conjugate according as C' is nega- tive or positive. But [Art. 155, (b)] we have J C'= in the case of the Hyperbola therefore, since H 2 AB is positive, C 1 will be negative when J is positive, and positive when J is negative. Hence, H 2 - AB > with J > is the condition that the equation of the second degree shall represent a primary hyperbola; and H 2 AB > with J < is the condition that it shall represent a conjugate one. RIGHT LINE AS HYPERBOLA. 177 17G. The limit separating the two conditions just determined, is evidently with J = 0. But (Arts. 131, 132; cf. 160) this is the condition that the equation of the second degree shall represent two real right lines intersecting in the center of its locus. Hence, Two real right lines, intersecting in its center, form the limiting case of the Hyperbola. Kemark The student may verify this by applying the condition A = to the equation A'x* - B'f + C' = 0. 177. In Art. 161, we found the Circle to be a partic- ular case of the Ellipse. We shall now see that the Hyperbola has an analogous case. One way of satisfying the criterion of the Hyperbola is, to have in the general equation the condition A + = (). For then H 2 AB will become H 2 + A 2 : which is necessarily positive. But if A -f- B = 0, then (Art. 163, Th. II) A' + B' = Q; and, after dividing through by A', the equation referred to the axes will become x 2 y 2 constant : an expression denoting an hyperbola, by the condition just established, and strongly resembling the equation to the Circle, x 2 -f- if = constant. Corollary. Suppose now we push this hyperbola to its limiting case. Its equation will of course continue to fulfill the condition A -}- B = Q, and the corresponding pair of right lines will therefore (Art. 128, Cor.) intersect at right angles. Accordingly, this curve is known as the Rectangular Hyperbola. 178 ANALYTIC GEOMETRY. 178. The following table presents, in their proper subordination, the several conditions by which we may distinguish the varieties of the Hyperbola in the equation of the second degree : ( Primary .* A > 0. ' A + B = .'. Oblique . . j Two Intersect. Lines . A=0. H*-AB>Q 'Conjugate . A < 0. Hyperbola /-Primary -.- A > 0. A + B=Q .'. Rectangular -j Two Perpendiculars . A = 0. '-Conjugate . A<0. By comparing this table with that of Art. 162, the student will see the truth of the Remark under Art. 167. EXAMPLES. I. NOTATION AND CONDITIONS. 1. Determine whether the following equations represent hyper- bolas : 3z 2 Sxy + 5?/ 2 6z + 4y 2 = 0, *'+ 2xy y' 2 + x- 3*/ + 7 = 0, 5z 2 \2.xy 7y 2 + Sx IQy + 3 = 0. 2. Show that 2z 2 I2xy + 5y 2 6x + 8y 9 = represents an oblique primary hyperbola. 3. Show that 3z 2 + 8xy 3y--\-6xlQy + 5 = Q represents a rectangular conjugate hyperbola. 4. Show that 3z 2 -f Sxy 3y 2 + 6x + lOy 5 = represents a rectangular primary hyperbola. 5. Verify the proposition of Ex. 2 by reducing the equation to the axes of the curve. 6. Verify the propositions of Ex. 3 and 4 in the same manner. 7. Given the hyperbola 5o; 2 6y 2 = 30 : form the equation to its conjugate, and find the quantities a, 6, c, and e. 8. Show that 2x 2 + xy 1 5?/ 2 x + 19y 6 = denotes an oblique hyperbola in its limiting case, and find the corresponding center. HYPERBOLIC LOCI. 179 9. Show that 3* 2 Sxy 3?/ 2 -f x + Yly 10 = denotes a rectangular hyperbola in its limiting case, and find the center. 10. Transform the two equations last given, to the centers of their respective curves. II. HYPERBOLIC LOCI. 1. Given the base of a triangle, and the difference of the angles at the base : to find the locus of the vertex. Since the difference of the base angles is given, the tangent of their difference is given. Let us call it = -^ . Then, using the axes and notation of Ex. 1, p. 126, we shall have II m x m + x 1 H 5^ 5 a. h'> or > m* - x*+y* h Hence, the equation to the required locus is ax 2 + 2hxy a.y i = am 2 ; and the vertex moves (Art. 177) on a rectangular hyperbola whose center [Art. 155, (2)] is the middle point of the base, and whose transverse axis h [Art. 156, (c)] is inclined to the base at an angle = % tan"" 1 ; that is, at an angle = half the complement of the given difference between the base angles. 2. Given the base of a triangle, and the difference between the tangents of the base angles : to find the locus of the vertex. 3. Find the locus of the vertex, given the base of a triangle, and that one base angle is double the other. 4. Find the locus of the vertex in an isosceles triangle, when the extremity of one equal side is fixed, and the other equal side passes through a fixed point. 5. Given the vertical angle of a triangle, and also its area: find the locus of the point where the base is cut in a given ratio. 6. Find the locus of a point so situated in a given angle, that, if perpendiculars be dropped from it upon the sides of the angle, the quadrilateral thus formed will be of constant area. 180 ANALYTIC GEOMETRY. M Q, X 7. Given a fixed point and a fixed right line : to find the locus of P, from which if there be drawn a right line to the fixed point and a perpendicular to the fixed line, they will make a constant intercept on the latter. 8. In the annexed diagram, QP is perpen- dicular to OQ, and RP to 072: find the locus of P, on the supposition that QR is constant. 9. Supposing that QR in the same diagram passes through a fixed point, find the locus of the intersection of two lines drawn through Q and R parallel respectively to OR and O Q. 10. QR is a line of variable length, revolving upon the fixed point a/3: find the locus of the center of the circle described about the triangle ORQ. 11. QR moves between OQ and 072 so that the area of the triangle ORQ is constant: find the locus of the center of the circumscribed circle. 12. A circle cuts a constant chord from each of two intersecting right lines : find the locus of its center. 13. Find the locus of the middle point of any hyperbolic focal radius. 14. From the extremity of any hyperbolic focal radius a line is drawn, parallel to the transverse axis and equal in length to the radius : find the locus of its extremity. 15. The ordinate of an hyperbola is prolonged so as to equal the corresponding focal radius: find the locus of the extremity of the prolongation. SECTION VII. THE PARABOLA. I. GEOMETRIC POINT OF VIEW: THE EQUATION TO THE PARABOLA IS OF THE SECOND DEGREE. 1TO. The Parabola may be defined by the following property: The distance of the variable point of the curve from a fixed point is equal to its distance from a fixed right line. EQUATION TO PARABOLA. 181 We may therefore trace the curve and find its figure as follows : Take any point F, and draw any right line D'D. Along the latter, fix the edge of a ruler; and in the former, fasten one end of a thread whose length is equal to that of a second ruler RD, which is right-angled at D. Then, having attached the other end to this ruler at JR, keep the thread stretched against the edge RD with the point P of a pencil, while the ruler is slid on its edge QD along D'D toward F: the path of P will be a parabola. For, in every position of P, we shall have FP= PD, as these distances will in all cases be formed by subtract- ing the same length RP from the equal lengths of the thread and ruler. 18. The fixed point F is called the focus of the par- abola, and the fixed line D'D its directrix. The line OF, drawn through the focus at right angles to the directrix, and extending to infinity, is called the axis of the curve. The point A, where the axis cuts the curve, is termed the vertex. We shall refer to the distance FP under the name of the focal radius. 181. Equation to the Parabola, referred to its Axis and Directrix. By putting 2p = the constant distance of the focus from the directrix, we shall have, in the above diagram, FP= ~]/{(x 2p) 2 -f y*} and PD = x. The algebraic expression for the defining property of the Parabola will therefore be 182 ANALYTIC GEOMETRY. Clearing of radicals, and reducing, we find the required equation, y* = p(x p). . Let us now investigate the general type to which this equation conforms. It may evidently be written f-px + 4p 2 = (1), and is therefore a particular case of the general equation +<7' = (2), in which B', G r , C r are any three constants whatever. Accordingly our real object is, to determine whether every equation in the type of (2) represents a parabola. We may settle this point as follows : Let us transform (1) to parallel axes whose origin is somewhere on the primitive axis of z, say at the distance x f from the given origin. To effect this, we merely re- place the x of (1) by x' -f- x, and thus obtain y 2 4px x'} (3). Now, since (1) represents a parabola, (3) also does. But in (3), since x' is arbitrary, the absolute term may have any ratio whatever to the co-efficient of x. More- over, by taking x' of the proper value, we can render the absolute term positive or negative at pleasure, and, by supposing 2j9 susceptible of the double sign, we shall accomplish the same with respect to the co-efficient of x. By carrying out these suppositions, and then multiplying the whole equation by some arbitrary constant, we can give it three co-efficients which will be entirely arbitrary, and may therefore write it POLAR EQUATION TO PARABOLA. 183 To show, then, that every equation in the type of (2) represents a parabola, we have only to prove that 2p may be either positive or negative, without affecting the form of the curve represented by (1). Now this supposition is clearly correct ; for (see dia- gram, Art. 179) a negative value of 2p merely indicates that the focus is taken on the left of D'D, instead of on the right: while, by using the thread and ruler on the left of the directrix, we can certainly describe a curve similar in all respects to PAL, except that it will face in the opposite direction. Hence we conclude that the equation to the Parabola may always be written in the form 5y + 2G'x+C' = 0, J9', 6r r , C f being any three constants whatever ; and, con- versely, that every equation of this form represents a parabola. 183. Polar Equation to the Parabola, the Focus being the Pole. From the annexed diagram, we have FP = p and OM= OF + FM=2p + p cos 0. Accordingly, the polar expression for the fundamental property of the Parabola will be p = 2p -+- p cos 0. o The required equation is therefore costf" Hemark This expression implies that the angle is measured from FX toward the left. The student may show that, supposing 9 to be estimated from FO toward the right, the equation to the Parabola will be P 1+cos An. Ge. 19. 184 ANALYTIC GEOMETRY. 184. To exhibit in part the analogy of the equation just obtained to those of the Ellipse and Hyperbola in Arts. 152, 172, let us agree to write as an abbreviation characteristic of the Parabola, and analogous to those adopted in Arts. 150, 170 for the other two curves. We may then write (see also Art. 627) " ~ 1 ecosd Corollary, Adopting the convention last suggested, we may arrange the abbreviations referred to, according to their numerical order, thus : Ellipse . . . . e < 1. Parabola . . . . e = l. Hyperbola . . . . e > 1. EXAMPLES. 1. Given the points (4, 0), (1, 0), (3, 0): write the rectangular equations to the three parabolas of which they are the foci. 2. Write the rectangular equation to the parabola whose focus is the point ( - 3, 0). 3. Transform the equations just found to parallel axes passing through the foci of their respective curves. 4. What are the positions of the foci with respect to the direc- trices, in the parabolas y l = 4 (a; 4), 4y 2 = 3 (4x + 3), and 5?/ 2 -6z -1-9 = 0? 5. Write the focal polar equation to the parabola whose focus is 2 feet distant from the directrix, and find the length of its radius vector when = 90. Also, find the polar equation to any parabola, the pole being at the intersection of the axis and directrix. PARABOLA AND GENERAL EQUATION. 185 II. ANALYTIC POINT OF VIEW : THE EQUATION OF THE SECOND DEGREE ON A DETERMINATE CONDITION REPRESENTS A PARABOLA. 185. To establish this theorem, we must show that there are real conditions upon which the general equation of the second degree may always be reduced to the form (Art. 182) B'y* + 2G'x + C' = 0. 186. In the investigation on which we are about to enter, we must confine our attention to those equations of the second degree whose co-efficients fulfill the condition For we have already proved (Arts. 158, 174) that every equation of the second degree in which H 2 AB is not equal to zero, represents either an ellipse or an hyperbola. 187. Further : The restriction just established carries with it the additional one, that, in the equations we are permitted to consider, the condition can not occur. For, reverting to the general value of the Discrimi- nant (Art. 159), we have J = ABC + 2FGH AF 2 BG 2 CH 2 . Whence, multiplying the second member by B : B, adding H 2 F 2 H 2 F 2 to the numerator of the result, and factoring, we may write _ (H 2 - AB) (F 2 -BC} (HFBG) 2 J- 186 ANALYTIC GEOMETRY, But, by the preceding article, we must assume H 2 AB = ; hence, for the purposes of the present inquiry, (HFBGY A = -3- Now the condition H 2 AB = Q obviously requires that A and B shall have like signs ; and we have agreed (see foot-note, p. 160) to write all our equations so that A shall be positive: therefore B, in the present inquiry, is positive. Whence it follows, that the fore- going expression for A is essentially negative; unless HF BG = 0, when it will vanish. Our proposition is therefore established. 188. The restrictions of the two preceding articles being accepted, our actual problem is, to determine whether the equation Ax* + ZHxy + Bf + 2Gx + 2Fy +0=0 (1), in which we suppose IP AB 0, can be reduced to the form that is, whether it can be subjected to such a transforma- tion of co-ordinates as will destroy the co-efficients of x\ xy, and y. 189. In the first place, then, we can certainly destroy the co-efficient of xy. For, to effect this, we need only revolve the axes through an angle 0, such that [Art. 156, (c)~\ we may have a condition compatible with any values of A, H, and B. PARABOLA AND GENERAL EQUATION. 187 Making this transformation, therefore, we shall get an equation of the form A'x 2 -f B'y 2 -f 2G'x -f 2F'y -f C = : in which (see the third equation in Art. 156) A' = % {( A + B) + (A B) cos 2d -f 2H sin 26}, B f = J {(J. + B) (A 5) cos 2# 2IZ"sin 20}, and (Art. 56, Cor. 3) G' = G cos/? -f.Fsin0, F' =F cos 6 G sin 6. Now, from the value of tan 20 above, (see foot-note, p. 158,) we know that sin 2d = VTn ms i /ouv2i . cos 2f) = or, by applying the condition H 2 = AB, and taking the radical as negative, that and therefore, by Trig., 847, iv, and by again applying the condition H 2 = AB, that H B smd = , frT2 _, TV. , cos 6 = rrnrr-jv, Substituting these values in the expressions for A f , B', G', and J 7 ', we obtain so that the proposed transformation has destroyed the co-efficient of z 2 as well as that of xy, and (1) becomes B'y* + 2G'x + 2F'y -{-0=0 (2). 188 ANALYTIC GEOMETRY. BOO. In the second place, we can certainly destroy the co-efficient of y. For our ability to do so depends on finding some new origin, to which if we transform (2) in parallel axes, the new co-efficient of y shall be zero ; and that we can find such an origin is easily proved. For, if the new origin be x'y', the new equation (Art. 163, Th. I) will assume the form B'f + 2G'x + 2 (B'y 1 + F 1 ) y + C' = : in which the co-efficient of y will vanish, if /_ F _ y w B' ' VH 2 + and y' being thus necessarily finite and real, while x' is indeterminate, there is an infinite number of points, lying on one right line, to any of which if we reduce (2) by parallel transformation, the co-efficient of y will disappear, and (2) will become _By + 20'a;+C" = (3). 191. We see, then, that we can reduce (1) to the required form; and that, too, without imposing any condition upon it other than the original one, that H 2 AB shall be equal to zero. Hence, The equation of the second degree will represent a parabola whenever its co-efficients fulfill the condition Corollary. It follows from this, that whenever the equation of the second degree denotes a parabola, its first three terms form a perfect square. PARALLELS AS PARABOLA. 189 From the restriction established in Art. 187, we conclude that the Parabola presents only two varieties of the condition just determined. They are, i. H 2 AB = with J < 0. ii. H 2 AB = with J = 0. By referring to Arts. 131, 132, it will be seen that the second of these is identical with the condition upon which the equation of the second degree represents two parallels. Hence, Two parallels constitute a particular case of the Parabola. 193. If we apply the criterion of the Parabola to the two lines at whose intersection (Art. 155) the center of the second order curve is found, we shall have A equal to the quotient of H 2 by B; and the two lines will become =Q (n) : which (Art. 98, Cor.) are evidently parallel. Hence, since we may always suppose that parallels intersect at infinity, the center of a parabola is in general situated at infinity. 194. To this general law, however, the case brought out in Art. 192 presents a striking exception. For, when the Parabola passes into two parallels, J vanishes ; and we obtain (Art. 187) 190 ANALYTIC GEOMETRY. so that, in this case, the lines in (n) become coincident, and the center is any point on the line We thus arrive at the conception of the Right Line as the Center of Tivo Parallels. Remark __ The result of this article is fully corroborated by the equations to the two parallels themselves. For (Art. 160) these are _______ Hx + By + F+ V F' 1 BC = Q, which obviously represent two lines equally distant from Hx+ By -f- JF=0. 195. Two parallels, considered as a variety of the Parabola, present three subordinate cases, each of which has its proper criterion. For, since the equations to the parallels are ' 1 - J*C--= 0, By + F VF 2 BO = 0, we shall evidently have the following series of conditions : I. F 2 BOO .'. Two parallels, separate and real. II. F 1 BO=Q .'. Two coincident parallels, in. F 2 BC<^0 /. Two parallels, separate but imaginary. Corollary. Hence, The Right Line, as the limit of two parallels, is the limiting case of the Parabola. Remark. It is noticeable that the limit into which the two parallels vanish when F 2 -BC=Q, is the line EXAMPLES ON THE PARABOLA. 191 which we have just shown to be the center of the rec- tilinear case of the Parabola. 196. The results of the foregoing articles, as fixing the varieties of the Parabola and their corresponding analytic conditions, may be summed up in the following table : H*AB=Q A<0 -'- Center atlnfinity. Parabola. {Two Real Parallels . F 2 BC>Q. Single Eight Line v F*-BC=Q. Twolmag. Parallels '.' EXAMPLES. I. NOTATION AND CONDITIONS. 1. Show why the following equations represent parabolas: 4x* -f 1 2xy + 9?/ 2 + 6x 10y + 5 = 0, (2x 5y) 2 = 3a:+4y 5, 5?/ 2 Qx + 1y 1 = 0. 2. Show that the preceding equations represent true parabolas, having their centers at infinity ; but that denotes two parallels, whose center is the line 2x = 3y. 3. Show that 4x* llxy + 9y 2 + Sx I2y -f 5 = denotes two imaginary parallels, whose center is the real line 2x oy 2 Q', and that 4x 2 12xy + 9y 2 Sa; + 12y + 4 = denotes the limit of these parallels. 4. Reduce 9z 2 24xy + 1 6?/' 2 + 4x 8y 1 = to the form 5. Show that when a parabola breaks up into two parallels, the line Hx -f- By + F= becomes the axis of the curve. 192 ANALYTIC GEOMETRY. II. PARABOLIC LOCI. 1. Given the base of a triangle, and the sum of the tangents of th.3 base angles : to find the locus of the vertex. Let us take the base for the axis of y, and a per- / pendicular through its middle point for the axis of x. Then, in the annexed diagram, OM will be the ordi- P T nate, and MP the abscissa of the variable vertex P. X' V Therefore, supposing the length of the base = 2m, \ and the given sum of tangents = n, we shall have \ my and the equation to the locus sought will be ny 2 2mx ?im 2 = 0. Hence, (see close of Art. 182,) the vertex moves on a parabola whose axis is the perpendicular through the middle of the base. By comparing this equation with (3) of Art. 182, it will be seen that the distances of the directrix and focus from the base are respectively 2. Given the base and altitude of a triangle : to find the locus of the intersection of perpendiculars drawn from the extremities of the base to the opposite sides. 3. Given a fixed line parallel to the axis of a:, and a movable line passing through the origin: to find the locus of a point on the latter, so taken that its ordinate is always equal to the portion of the former included between the axis of y and the moving line. 4. Lines are drawn, through the point where the axis of a par- abola meets the directrix, so as to intersect the curve in two points: to find the locus of the points midway between the intersections. 5. Through any point Q of a circle, OQ is drawn from the center O, and QR made a chord parallel to the diameter EOI and bisected in S: to find the locus of P, where OQ and ES intersect. 6 Find the locus of the center of a circle, which passes through a given point and touches a given right line. [Take given line for axis of y, and its perpendicular through given point for axis of x.~] 7. Given a right line and a circle : to find the locus of the center of a circle which touches both. [Take perpendicular to given line, through center of given circle, for axis of xJ] LOCUS OF SECOND ORDER. 193 S. OA is a fixed right line, whose length = a ; about 0, a second line POP' revolves in such a manner that the product of the areas PAO, P'AO = a 4 , and their quotient = cot 2 % POA : to find the locus of P or P'. [Polars.] 9. Given the base of a triangle, and that the tangent of one base angle is double the cotangent of half the other: to find the locus of the vertex. 10. Find the locus of the center of a circle inscribed in a sector of a given circle, one of the bounding radii of the sector being fixed. SECTION VIII. THE Locus OF THE SECOND ORDER IN GENERAL. 197. We have now seen that the equations to the Pair of Right Lines, the Circle, the Ellipse, the Hyper- bola, and the Parabola, are all of the second degree. We have proved, too, that the general equation of the second degree may be made to represent either of these loci, by giving it co-efficients which fulfill the proper con- ditions ; and, in the course of the argument, it has come to light that the Point, the Pair of Lines in their various states of intersection, parallelism, and coincidence, and the Circle, are phases of the three curves mentioned last. The latter result suggests the question, Is not the gen- eral equation, considered without reference to any of these conditions, the symbol of some locus still more generic than either the Ellipse, the Hyperbola, or the Parabola, of which these three curves are themselves successive phases ? It is the object of the present section, to show that this ques- tion, in a certain important sense, is to be answered in the affirmative ; and to aid the student in forming an exact conception of what is meant by the phrase Locus of the Second Order in General. 194 ^ ANALYTIC GEOMETRY. 198. We proceed, then, to show that such a locus exists ; and to explain the peculiar nature of the existence which belongs to it. In the first place, a moment's reflection upon the dis- cussions in the preceding pages will convince us that hitherto we have not regarded the equation Ax* + ZHxij + % 2 + 2fe + 2Fy + C= (1) in the strictly general aspect at all. For we have sup- posed its co-efficients to be subject to some one of the three conditions and therefore to be actual numbers, since it is only in actual numbers that the existence of such conditions can be tested. But, obviously, we can conceive of equation (1) as not yet subjected to any such conditions, the constants A,B,C, F,G,H not being actual numbers, but symbols of possible ones; and, in fact, we must so con- ceive of it, if we would take it up in pure generality. In the second place, not only does the equation await this purely general consideration, but when so considered it still has geometric meaning. For, though its co-efficients are indeterminate, its exponents are numerical and fixed : it therefore still holds its variables under a constant law, not so explicit as before, but certainly as real. It is still impossible to satisfy it by the co-ordinates of points taken at random ; it will accept only such as will combine to form an equation of the second degree. Since, then, we must consider (1) in its purely general aspect as well as under special conditions ; since, even in this aspect, it still expresses a law of form; and since this law, consisting as it does in the mere fact that the THE LOCUS AS PURE LA W. 195 equation is of the second degree, must pervade all the curves of the Second order: it follows that this law may be regarded as a generic locus, whose properties are shared alike by the Ellipse, the Hyperbola, and the Parabola. By the phrase Locus of the Second Order in General we therefore mean not a figure but an abstract law of form. It exists to abstract thought, but can not be drawn or imagined. To illustrate the nature of its existence by a more familiar case, we may compare it to that of the generic conception of a parallelogram. We can define a parallelogram ; but if we attempt to imagine or draw one, we invariably produce some particular phase of the conception either a rhomboid, a rhombus, a rect- angle, or a square. In the same way, the Locus of the Second Order exists so as to be defined ; but not other- wise, except in its special phases. In short, when speaking of it, we are dealing with a purely analytic conception ; and the beginner should avoid supposing that it is any thing else. 2OO. Further : This common law of form not only manifests itself in all the three curves we have been considering, but they may be regarded as successive phases of it, whose order is predetermined. For, as we have seen, they may be supposed to arise out of the general locus whenever the condition characteristic of each is imposed on the general equation. Now these conditions may be summed up as follows : 0, J = 0, A > 0. 196 ANALYTIC GEOMETRY. Hence, since lies between and +, while the condi- tions in A actually entered our investigations as subordi- nates of those in H 2 AB, and the conditions in F 2 BC as subordinates of J = 0, it follows that the three curves have a natural order corresponding to the analytic order of their criteria, and that their several varieties have a similar order corresponding to theirs. Accordingly, we should expect the curves to occur thus : Ellipse ; Parabola ; Hyperbola. In due time hereafter, this order will be verified geometrically. 2O1. Our three curves and their several varieties are thus shown to be species of the Locus of the Second Order : are there any others ? We shall now show that there are not, by proving that every equation of the second degree must represent one of the three curves already considered. We have just seen that all the conditions hitherto imposed on the general equation are subordinate to the three, and we have proved that any equation of the second degree fulfilling either of these must represent one of the three curves. But no equation of the second degree can exist without being subject to one of these conditions; for, whatever be the numerical values of A, .77, and B, we can always form the function H 2 AB, which can not but be less than, equal to, or greater than zero. Hence, the series of conditions already imposed on the general equation exhaust the possible varieties of its locus, and we have the proposition at the head of this article. 2O2. We mentioned in Art. 47, that the term Conic Section or Conic is used to describe a curve of the THE CONIC IN GENERAL. 197 Second order. From what has now been shown, we may define the Conic in General as the embodiment of that general law of form which is expressed by the uncondi- tioned equation of the second degree. It also follows that there are three species of the Conic, corresponding to the three leading conditions which have become so familiar. 2OB. Let us now recapitulate the argument by which we have thus gradually established the theorem : Every equation of the second degree represents a conic. I. We proved (Arts. 153 162) that every equation of the second degree whose co-efficients fulfill the condition H 2 AB < 0, repre- sents an ellipse, and showed that the Point, the Pair of Imaginary Lines, and the Circle, are particular cases of that curve. II. We proved (Arts. 173 178) that every equation of the second degree whose co-efficients fulfill the condition H 2 AB > 0, repre- sents an hyperbola, and showed that the Pair of Real Intersecting Lines are a case of that curve. III. We proved (Arts. 185 196) that every equation of the second degree whose co-efficients fulfill the condition H' L A B = 0, repre- sents a parabola, and showed that Two Parallels, whether separate, coincident, or imaginary, are a case of that curve. IV. We combined (Art. 200) the results of the three preceding steps, and inferred that, of the conditions previously imposed, the three H 2 AB<0, H*-AB--=0, H 2 were all that we needed to consider in testing the leading signifi- cation of the equation of the second degree, since all the others had proved to be subordinates of these. V. We showed (Art. 201) that these three conditions are of such a nature that every equation of the second degree must be subject to one of them; and thence inferred the theorem. 2O4. The existence of three conditions by which the signification of the general equation may be varied, indi- cates, as we have already noticed, three species of the Conic. But we must not overlook a previous subdivision 198 ANALYTIC GEOMETRY. of the locus. The three conditions are themselves subject to classification : two of them are finite, while the third is infinitely small. This classification, too, has its geo- metric counterpart : for the Ellipse and Hyperbola, in which H 2 AB is finite, have each a finite point as their center; while the Parabola, in which H 2 - AB is infi- nitely small, has no such center. In order, then, to include all the facts, we must say that the Conic consists of two families of curves, one central and the other non- central; and that these two families break up into the three species which we have already described. 2O5. The entire Locus of the Second Order, with its subdivisions arranged according to their 'mutual relation- ships as fixed by their analytic conditions, may be pre- sented as follows: ORDER. FAMILY. SPECIES. CONIC Central Ellipse Hyperbola VARIETY. Eccentric Circle Oblique CASE. Real. Point. Imaginary. Real. Point. Imaginary. Primary. Intersect. Lines. Conjugate. (Primary. Perpendiculars. Conjugate. f Center at Infinity. Non- \ Parabola f Real Parallels. r -raraooia -j Center a _. T>. I-XT- \ Single Line. Right Line _ [ Imag. Parallels. I Central TRILINEAR CO-ORDINATES. 199 CHAPTER SECOND. THE MODERN GEOMETRY: TRILINEAR AND TANGENTIAL CO-ORDINATES. SECTION I. TRILINEAK CO-ORDINATES. 2O6. Modern geometers frequently employ tHe fol- lowing method of representing a point : Any three right lines that form a triangle, as AB, BC, CA, are assumed as the Fixed Limits to which all posi- tions shall be referred. The position /A N B\ of any point P in the plane of the triangle is then determined by finding the lengths PL, PM, PN of three perpendiculars dropped from it upon the three fixed lines. The triangle whose sides are thus employed as limits, is called the triangle of reference. The three perpendic- ulars let fall upon its sides from any point, are termed the trilinear co-ordinates of the point, and are designated by the Greek letters , /?, f. 2OT. On a first glance, this system of co-ordinates seems redundant ; for, in the Cartesian system, we have seen that two co-ordinates are sufficient to determine a point; and it is obvious from the diagram that P is de- terminable by any two of the perpendiculars PL, PM, PN. The reader will therefore not be surprised to learn that the new method came into use as an unexpected con- sequence of abridging Cartesian equations. The process An. Ge. 20. 200 ANALYTIC GEOMETRY. by which the trilinear system thus grows out of the bilinear, will now be explained. 2OS. In Art. 108, we have already hinted at the Abridged Notation, which gives rise to the system of which we are speaking. We will now present the subject in detail. If the equation to any right line is written in terms of the direction-cosines of the line, namely, in the form x cos a -\- y sin p = 0, we may use a as a convenient abbreviation for the whole member equated to zero ; for it naturally recalls the ex- pression into which it enters as so prominent a constant. Similarly, in the equations x cos ft -j- y sin ft p r =0, x cos f + y sin Y p" = 0, we may represent the first members by ft and f. Thus the equations to any three right lines may be written = 0, = 0, 7* = 0. The brevity of these expressions is advantageous, even when they are taken separately ; but it is not until we combine them, that the chief value of the abridgment appears. We then find that it enables us to express, by simple and manageable symbols, any line of a given figure in terms of three others. 2O9. It is this last named fact, which constitutes the fundamental principle of trilinear co-ordinates. That we can so express a line, follows from our being able (see Art. 108) to write, in terms of the equations to two given right lines, the equation to any line passing through their intersection. ABRIDGED NOTATION. 201 21O. We may convince ourselves of this, by a few simple examples. If a = 0, ft = represent any two right lines, then (Art. 108) a + k@ = is the equation to any right line passing through their intersection, provided Jc is indeterminate. Now &, in this equation, (Art. 108, Cor. 2) is the negative of the ratio of the perpendiculars dropped from any point in the line a -f- kp = upon the two lines a = and ft = 0. Therefore, writing & so as to display its intrinsic sign, a -f &j0 = denotes a right line passing through the intersection of a = and /9 = 0, and lying in that angle of the two lines which is external to the origin ; but denotes one lying in the same angle as the origin. Moreover, when the perpendiculars mentioned are equal, the value of k = 1 ; and we have (Art. 109, Cor. 3) the equation to the bisector of the external angle between two given lines, and a fi = the equation to the bisector of the internal angle. Suppose, then, that we have a given triangle, whose sides are the three lines Granting, as we may, that the origin is within the triangle, the equations to the three bisectors of the angles will be 202 ANALYTIC GEOMETRY. The equations to the three lines which bisect the three external angles of the triangle, will be Thus, these six lines of the triangle are all expressed in terms of its three sides. . We can of course extend this system of abbre- viations to the case of lines whose equations are in the general form y+C= 0, by representing the member equated to zero by a single English letter, such as L or v. Thus, L -f kl/ = or v + Tcv' = denotes a line passing through the intersection of the lines It must be borne in mind, however, that in these equa- tions k does not denote the negative of the ratio of the perpendiculars mentioned above, and in consequence does not become 1 when those perpendiculars become equal. Hence, in these cases, the equations to the ex- ternal and internal bisectors of the angle between two given lines, are respectively (see Art. 109, Cor. 4) L + rL' = or lv+mv' = Q, L rL'=Q or lv mv f = Q, in which r, or m : I, is to be determined by the formula on p. 116. In this notation, if the three sides of a triangle are u=Q, v 0, w=Q, ABRIDGED NOTATION. 203 the three bisectors of its interior angles may be repre- sented by lu mv = 0, mv nw = 0, nw lu = ; and the three bisectors of its exterior angles, by lu + mv = 0, mv -f- nw = 0, ra# -f- lu = 0. Here, too, the six lines are all expressed in terms of the three sides. 212. Having thus learned how to interpret the equa- tions when a = 0, /? = 0, f=Q are given as forming a tri- angle, we next advance to the interpretation of la -j- TTi/9 -{- nf=Q : an equation of which the preceding six are evidently particular cases, and in which Z, m, n are any three constants whatever. On the surface, this looks like the condition (Art. 114) that the three lines a 0, ft = 0, y=Q shall meet in one point. But the terms of that condition are, that three lines will meet in one point whenever three con- stants can be found such that, if the equations to the lines be each multiplied by one of them, the sum of the products will be zero. Hence, the Z, m, and n of that condition are not absolutely arbitrary, but only arbitrary within the limits consistent with causing the function la -j- mj3 -j- nf to vanish identically. On the contrary, the ?, m, and n of the present equation are absolutely arbitrary. With this fact premised, let us now investigate the meaning of the equation. 204 ANALYTIC GEOMETRY. If we replace a, /?, and ? by the functions for which they stand, and reduce the equation on the sup- position that x and y have but one signification in all its three branches, we obtain (I cos a -j- m cos ft ~f n cos f) x\ + (7 sin a -f- m sin /9 -(- w sin y) ?/ V = : -f Zp -f ???/ -f- ?ip" which is evidently the equation to some right line, since it conforms to the type Ax + By + C = Q. Its full significance, however, will not appear until we discuss it under each of three hypotheses concerning the relative position of the three lines whose equations enter it. To this discussion, we devote the next three articles. 214. Let the three lines 0, /? = 0, f = meet in one point. When this is the case, the equation la + mft -f- wr = denotes a right line passing through the point of triple intersection. For the co-ordinates of this point will render , ft, and f equal to zero simultaneously, and will therefore satisfy the equation just written. 215. Let the three given lines be parallel. In this case, their equations (Art. 98, Cor.) may be written a = 0, a + c r = 0, a + c" = 0, and the equation we are discussing will thus become (I -j- m + n) a + (mc f + nc") = ; that is, it will assume the form a -f c = 0, a, ft, r , AS CO-ORDINATES. 205 and will therefore denote a right line parallel to the three given ones. 216. Let the three given lines form a triangle. In this case the preceding results are avoided ; hence, the equation of Art. 213 assumes the form Ax -{-By-}- 0=0 without imposing any restriction upon the values of A, B, and C. In this case, therefore, it denotes any right line whatever. This result is simply the extension of that obtained in Arts. 210, 211 ; and shows that we can (as was stated in Art. 208) express any line of a figure in terms of three given ones. 217. From the results of the last three articles, we conclude that la -f mft + 71? = is the equation to any right line ; provided, however, that the lines a = 0, = 0, r = are so situated as to form a triangle. This proviso, we can not too carefully remember. 218. The examination of an example somewhat more complex than any yet presented, will convince us that this abridged method of writing equations is in effect a new system of co-ordinates, applicable to lines of any order. Before commencing this example, it may be necessary to remind the student that the use of a Greek letter for an abbreviation, always implies that the equation to the line so represented is in the form xcosa -|-#sina p = 0, and the like ; and that, when the fundamental equations 206 ANALYTIC GEOMETEY. are in any other form, they will be abridged by means of English characters. For the sake of still greater brevity, the line a = is often cited as the line a, the line f) = as the line /9, etc. The point of intersection of two lines is frequently spoken of as the point a/9, etc. The last notation should be care- fully distinguished from that of the co-ordinates of the same point. 219. Example. Any line of a quadrilateral in terms of any three. Let ABEF be any quadrilateral, and let a = 0, = 0, 7=0 be the equations to its three sides J3C, CA, AB, We can now represent any other line in the figure, in terms of a, /?, A f and 7. Suppose the origin of bilinear co-ordinates to be within the tri- angle ABC, and, in fact, within the triangle EOB. The equation to AE, which passes through the intersection of p and 7, and lies in their internal angle, will be of the form mp ny=0 (AE). The equation to BF, which passes through the intersection of 7 and a, and lies in their internal angle, will be of the form ny la = Q (BF). The equation to EF, which joins the intersections of (a, AE) and (/?, BF), and lies in the external angle of the first two lines, but in the internal angle of the second pair, must be formed (Art. 108, Cor. 2) so as to equal either the sum of la and m/3 ny or the difference of m/3 and ny In. It is therefore la + mp ny = Q (EF). The equation to CD, which joins a/3 to the intersection of (7, EF), and lies in the external angle of both pairs of lines, must be the sum of either la and m/3, or ny and la -f- m(3 ny. Consequently, it is la + mp = Q (CD), , /9, }', AS CO-ORDINATES. 207 The equation to OC, which joins aft to the intersection of (AE> BF], and lies in the internal angle of a/3, but in the external angle of the other two lines, must be equal either to the difference of la and m{3 or the sum of mp ny and ny la. Accordingly, it is la- m p=Q (OC). Finally, the equation to OZ), which joins the intersection of (y, EF) to that of (AE } BF), and lies in the internal angle of both pairs of lines, must be formed so as to equal the difference of either la -f m{3 ny and ny, or mp ny and ny la. Therefore, it is We have thus expressed all the lines of the quadrilateral in terms of the three lines a, /3, and y. We can do more : we can solve problems involving the properties of the figure, by means of these equations, and test the relative positions of its lines without any direct reference to the x and y which the symbols a, /3, y conceal. Thus, the form of the equations la mp = 0, m(3 ny = 0, ny la = shows (Art. 114) that the three lines OC, AE, BF meet in one point, and the same relation between OZ>, AE, BF is shown in the form of their equations. 22O. We see, then, that by introducing this abridged notation we can replace the Cartesian equations in x and y by a set of equations in , /9, y. Moreover, it is notice- able that all these abridged equations to right lines are of the first degree with respect to , /9, and y. Since, therefore, we operate upon these symbols (as the last example shows) just as if they were variables, and since they combine in equations which satisfy the condition that an equation to a right line must be of the first degree, it appears that we may use a, /?, f as co-ordinates. Thus, we may say that the equation la + mp+nr = Q is the equation to any right line, and that , /9, f are the co-ordinates of any point in the line. An. Ge. 21. 208 ANALYTIC GEOMETRY. What now is the system of reference to which these co-ordinates belong? We have seen that, in order to the interpretation we have given of the last-named equation, the line , ft, 7* must form a triangle. We know, too, (Art. 105) that a is the length of a perpendicular let fall from any point to the line a ; that ft is the length of a perpendicular from the same point to the line ft; and that 7- is the length of a perpendicular from the same point to the line y. Hence, we see that if , ft, 7- are taken as co-ordinates, they are referred to the tri- angle formed by and that they signify the three perpendiculars dropped from any point in its plane upon its three sides. In short, we have come out upon the system of trilinear co-ordinates described in Art. 206. PECULIAR NATURE OF TRILINEAR CO-ORDINATES. 221. Before making any further application of the new system, it is important to notice that trilinear co- ordinates are in one respect essentially different from bilinear. In the Cartesian system, the x and y are independent of each other, unless connected by the equation to some locus. In the trilinear system, on the contrary, the , ft, and 7 are each of them determined by the other two ; that is, there is a certain equation between them, which holds true in all cases, whether the point which they represent be restricted to a locus or not. In the language of analysis, we express this peculiarity by saying that each of these co-ordinates is a determined function of the other two. That trilinear co-ordinates are subject to such a con- dition, is proved by the fact, that two co-ordinates com- TRILINEARS ALWAYS CONDITIONED. 209 pletely fix a point, and must therefore determine the value of any third. The equation expressing the exact limits of the condition, we now proceed to develop. 222. Let ABC be the triangle of reference, a = being the equation to the side BC, y3 = the equation to CA, and f = the equation to AB. Then, if P be any point in the plane of the triangle, the three perpendiculars PL, PM, N , PN will be represented by , /9, f. Supposing now that a, 6, and c denote the lengths of the three sides BC, CA, AB, we shall have aa = twice the area of BPC, bp = . CPA, cf = " " APE. But the sum of these areas is constant, being equal to the area of the triangle of reference ; therefore, repre- senting the double area of this triangle by M, we obtain aa -j- 6/9 -f cy M: which is the constant relation connecting the trilinear co-ordinates of any point. Remark When we say that the sum of the areas of APE, BPC, CPA is equal to the area of ABC, we of course mean their algebraic sum. For, if we take a point outside of the triangle of reference, such as P', we shall evidently have ABC= AP f B -f- BP'C CP'A that is, in such a case, one of the three areas becomes negative. And this is as it should be ; for, as we have agreed to suppose the Cartesian origin to be WITHIN the triangle of reference, the perpen- dicular P / M is negative, according to the third corollary of Art. 105. 210 ANALYTIC GEOMETRY. 223. This peculiarity of trilinear co-ordinates may be made to promote the advantages of the system. In the first place, it gives rise to a very symmetrical expression for an arbitrary constant. For if k be arbi- trary, kM will also be; and we shall have, for the symbol alluded to, k(aa + bp + c r ) (1). We can also modify this symbol, and render it still more useful. Let 1 : r = the ratio of the side a, in the triangle of reference, to the sine of the opposite angle A. Then, by Trig., 867, we shall have sin A sin B sin a b c Now, by the principle of Art. 222, r (aa -f 5/3 -j- cf) = constant ; , hence, substituting for ra, rb, re from the equal ratios above, a sin A -f- /3 sin B -f- f sin C = constant ; or, the symbol for an arbitrary constant may be written k (a sin A + /3 sin B -f- r sin 0) (2). 224. In the second place, the peculiarity of trilinears enables us to use homogeneous equations in all cases. For, if a given trilinear equation is not homogeneous, we can at once render it so by means of the relation in Art. 222. Thus, if the given equation were p=p, we might write 225. In the third place, instead of the actual trilinear co-ordinates of a point, we may employ any three quan- tities that are in the same ratio, without affecting the TRANSFORMATION TO BILISEARS. 211 equation to the locus of the point. For, since all tri- linear equations are homogeneous, the effect of replacing , /3, Y by />, /9/9, pf will only be to multiply the given equation by the constant p : which of course leaves it essentially unchanged. This principle will often prove of great convenience. Remark. In case it becomes desirable to find the actual tri- linear co-ordinates when they have been displaced in the manner described, we may proceed as follows : Let I, m, n be the three quantities in a constant ratio to a, /?, y: then, supposing the ratio to be 1 : r, a = rl, P = rm, y = rn. Hence, (Art. 222,) r (la -f- mb + nc} = M: from which r is readily found. 226. Finally, the property in question enables us to pass from trilinear co-ordinates to Cartesian. For, by means of the relation aa-\-bft-{-cf M, we can convert any trilinear equation into one which shall contain only ft and f. Then, supposing the side f of the triangle of reference to be the axis of x, and the side /9 the axis of ?/, we shall have ft = x sin A, f = y sin A. Corollary. If the triangle of reference should be right- angled at A, the reduction-formulae will become But, in general, to pass from trilinears to rectangulars when the side f is taken for the axis of x, we must use ft = xs'mA y cos A, f = y. The- student may draw a diagram, and verify the last formulae. 212 ANALYTIC GEOMETRY. The advantage of the trilinear system consists in this : Its equations may all be referred to three of the most prominent lines in the figure to which they belong, and hence become shorter and more expressive than those of the Cartesian system, which can go no farther in the process of simplification than the use of two prominent lines. The student will see a good illus- tration of this in the equations to the bisectors of the internal angles of a triangle. The trilinear equations are = 0, /9 r = 0, r~ = which are much simpler than the Cartesian ones given in Ex. 32, p. 124. TRILINEAR EQUATIONS IN DETAIL. 228. Equation to any Right Uiie. This we have already (Arts. 212217) found to be la + mfi + n r = Q. 229. Equation to a right line parallel to a given one. It is obvious geometrically, that the , /?, 7- of the parallel line will each differ from the a, /9, 7 of the given one by some constant. Hence, the given line being la -j- mf! 4- nf = 0, the required equation [Art. 223, (2)] will be la + mfi + ny + k (a sin A -f /? sin B -f 7 sin C) = 0. 230. Equation to a right line situated at in- finity. The Cartesian equation to this is (7=0, in which (Art. 110) C is a finite constant. Therefore (Art. 223) the trilinear equation is a sin A -f- /9 sin B + 7 sin C= 0. TEILINEAR EQUATIONS. 213 L. Condition that two right lines shall be mutually perpendicular. Let the two lines be la + mp + n r = 0, Va + m'ft -j- n' r = 0. By writing , /?, 7- in full, collecting the terms, and apply- ing (Art. 96, Cor. 3) the criterion AA r -f BB 1 = 0, we obtain, as the required condition, (mn f -f- m'n) cos (/9 7-) \ (nl r -f- n'Z) cos (7* a) > = 0. (?m ; -f- Z'm) cos ( /5) J Now, in this expression, , /?, 7* are the angles made with the axis of x by perpendiculars from the origin on the lines , /9, f. Supposing the latter to form the triangle of reference, and to inclose the origin, it is evident from the diagram, that /? 7- is the angle between the perpendiculars /? and f ; that f a is the angle between the perpendiculars f and a; and that a. between the perpendiculars a and /?. From the proper- ties of a quadrilateral it then follows, that /9 ^ is the supplement of A 9 f a of B, and a /9 of C. Hence, the required condition may be otherwise written X'" is the angle mm ((mn' + m'r. nn' < (nl f -f n' [ (lm r -f I'rt n) cos A 7) cos B I'm) cos (7 Corollary. The condition that la -f- mft -f be perpendicular to f = 0, is = 0. = shall n = m cos I cos 214 ANALYTIC GEOMETRY. 232 Length of a perpendicular from any point to a given right line. Let the point be 0/3^, and the line la -j- mft -f ^T = 0- Write the latter equa- tion in full, collect its terms, and apply the formula of Art. 105, Cor. 2. We thus find p _ _ l/(7 2 -|-m 2 -f-^ 2 2mn cosA 2nl cos.5 2lm cos (7) 233. [Equation to a right line passing through Two Fixed Points. The form of this will of course be la + mp -f w r = ; and our problem is, to determine the ratios l:m: n so that the line shall pass through the two points 1 /? 1 ?' 1 , Since the two points are to be on the line, we shall have Z! -f mfc -f wf, = 0, Ia 2 + mft 2 + nf 2 = 0. Solving for ? : n and m : n between these conditions, we find : m : n = The required equation is therefore far* r&) + ^ (n 2 - ir 2 ) -f r ( A A 2 ) = o. Corollary, Hence, in trilinears, the condition that three points shall lie on one right line is n&) -f A (Ti 2 ir 2 -f 3 i = o. Expanding, and re-collecting the terms, we may write this more symmetrically (see Art. 112), 2 (An rA) 4- TRIUNE AR EQUATIONS. 215 234. Theorem. Every trilinear equation of the second degree represents a conic. For, since , ft, f are linear functions of x and y, every such equation is reducible to an equation of the second degree in x and y. But the latter (Art. 203) will represent a conic. Remark. In passing now to the trilinear expressions for curves of the Second order, we shall at first suppose the triangle of reference to have a special position, such as will tend to simplify the resulting equations. The more general equations, corresponding to any position of the triangle, will be investigated afterward. 235. Equation to the Conic, referred to the Inscribed Triangle. To obtain this, we must form an equation of the second degree in , ft, f, such as the co-ordinates of the vertices of the reference-triangle will satisfy. Now, at the vertex A, we have ft = 0, Y = ; at the vertex B, Y = 0, a = ; and, at the vertex C, a 0, ft = 0. Hence, the required equation may be written -f naft = ; for this expression is obviously satisfied by either of the three suppositions Corollary. Dividing through by aftf, we may write the equation just found in the more symmetrical form m n 216 ANALYTIC GEOMETRY. 23G. Equation to the Circle, referred to the Inscribed Triangle. Our problem here is, to deter- mine I : m : n so that shall represent a circle. Write a, /9, 7- each in full, expand the equation, collect the terms with reference to x and ?/, and apply the cri- terion (Art. 140) H=Q, A B = 0. The conditions in order that (1) may represent a circle will thus be found to be I cos (j9 -\- Y) -\- m cos (f + ) -f- 7i cos (a -f- /?) = 0, Z sin (/3 + f) + w sin (7- -f- a) -f- w sin (a + /9) 0. Solving these for Z : n and m : w, and applying Trig., 845, in, we obtain I : m : n = sin (ft f) ' s i n (/ ) : s ^ n ( a $) But (see Art. 231) ft T = ISO A, r a=ISQB, a ft = 180 C. Hence, the equation sought is sin A sin B sin C n \ -Q ~T~ = ". Corollary. The equation to any circle concentric with the one circumscribed about the triangle of reference, will only differ from the foregoing (Art. 142, Cor. 1) by some constant. Hence, it may be written sin A .sin B sin C , , . n T> 1 r -\ A; (asm A -{- ft sm B -{- ? sm C). 237. General equation to the Circle. The equation of the preceding article applies only to the TRILINEAR EQUATIONS. 217 circle described about the triangle of reference. We are now to seek an equation which will represent any circle. The rectangular equation to a circle (Art. 140) may be put into the form X* _f- ^ + AX _|_ ty ^ C = Q In this, the only arbitrary constants are .A, B, C. Hence, rectangular equations to different circles will vary only in the linear part, and the equation to any circle may be formed from that of a given one by merely adding to the latter an arbitrary linear function. Now this property is as true of trilinear as of rectangular equations, since a trilinear equation is only a rectangu- lar one written in a peculiar way. We can therefore form the equation to any circle from that of the circle described about the triangle of reference, by adding to the latter, terms in the type of la -f- vnft -j- nf. By clearing of fractions, and replacing sin A, sin B, sin C by , 6, c, which are in the same ratio, the equation to the circumscribed circle becomes a ?r + b?a -f oafi = 0. Hence, the required equation is in which M is the fixed constant a a -}- 5/9 -f- + 2jffa0 == 0. Kemark. It is worthy of notice, that, if we suppose 7=1, this expression becomes Aa? -f 2Ha{3 + B(P + 2Ga + 2F/3 + C= 0, and is, in form, identical with Ax 1 + 2Hxy + By* + 2Gx + 2Fy+C= 0, the Cartesian equation to the Conic in General. This fact has led Salmon to the opinion that Cartesian co-ordinates are a case of triUnear.* 239. Problem, To determine the condition in order that the general triUnear equation of the second degree may represent a circle. From the equation of Art. 222, it is evident that we have the following relations : aa 2 = Ma 6 a/3 cya, bp = Mi3 c/ty ao/3, cf My aya bfiy. Substituting from these for a 2 , /3 2 , y 2 in the equation of Art. 238, we find that it may be written * See his Conic Sections, p. 67, 4th edition ; but compare the principles of Arts. 221, 222. TEILINEAE EQUATIONS. 219 Comparing this with the equation to the circle, a(3y + by a + cap + M (la + m{3 + ny ) = 0, we see that it will represent a circle, provided a b Hence, the required condition is Ca 2 - Ac 1 = 24O. Equations to the Chord and the Tangent of any Conic. For the sake of simplicity, we shall suppose the inscribed triangle to be the triangle of reference. I. The equation to the chord is the equation to the right line which joins any two points on the curve, as a'ft'f, a."fi n f. Now since these points are on the conic, we shall have (Art. 235, Cor.) I Tti n I m n a' j3 f f ' a 77 /9"~ ~~ p 7 ~ We can now easily see that the equation to the chord may be written la mfi n r ff For this is the equation to a right line, since it is of the first degree ; and to the line that passes through a'ft'f, a"fi"f, since it is satisfied by the co-ordinates of either point, inasmuch as they convert it into one of the con- ditions just found above. II. The point a"ft tr f f may be supposed to approach the point a'^f as closely as we please. At the moment of coincidence, the chord becomes the tangent at a' 220 ANALYTIC GEOMETRY. and we also have a" a', /3" = /9', f = f. By making the corresponding substitutions in the equation to the chord, we shall therefore obtain the equation to the tangent. It is la J. P -il nr n ^-H-^r-+pj- 241. There are many other equations, more or less symmetrical, representing the Conic or the Circle, but our limits forbid their presentation. Those already developed have the widest application, and are sufficient to illustrate the trilinear method. The student who wishes further information on the subject, may consult SALMON'S Conic Sections. EXAMPLES. 1. When will the locus of a point be a circle, if the product of its perpendiculars upon two sides of a triangle is in a constant ratio to the square of its perpendicular on the third? Take the triangle mentioned, as the triangle of reference ; and repre- sent the constant ratio spoken of, by k. The conditions then give us so that the locus is, in general, a conic of some form. In order that it may be a circle, the equation just found must satisfy the condition of Art. 239. Applying this, and observing the fact that in the present equation C= k, 2H= 1, and A,B,F,G each = 0, we find that the locus will be a circle if kb* = ka* = ab. From the first of these conditions, we obtain a= b; from the second and third, k = l. The required condition then is, that the triangle shall be isosceles, and the constant ratio unity. It is interesting to notice how clearly the equation above written expresses the position of the locus with respect to the triangle. To find where the side a cuts the conic, we simply make (Art. 62, Cor. 1) a^ in the equation a/? = ky*. The result is y 2 = ; that is, the equation whose roots are the co-ordinates of the points of intersection is a quadratic with equal roots. Hence, CONIC BY TWO TANGENTS AND CHORD. 221 (Art. 62, 3d paragraph of Cor. 2,) the line a meets the curve in two coin- cident points ; in other words, it touches the conic. Moreover, since the quadratic of intersection is y 2 = 0, the two points coincide on the line y. We thus learn that the conic touches the side a of the triangle at the point ya. If we make /?= in the equation, we again obtain y 2 = 0. Hence, the conic touches the side /? of the triangle at the point /?y. If we make y = 0, we get either a = or /3 = ; the side y therefore cuts the conic in two points : one on the side a, the other on B ; or, one the point ya, the other, the point /?y. We may sum up the whole result by saying that a/? = fcy 2 denotes a conic, to which two of the sides of the triangle of reference are tangents, while the third is a chord uniting their points of contact. 2. Form the trilinear equations to the right lines joining the vertices of a triangle to the middle points of the opposite sides. Take the triangle itself as the triangle of reference, and suppose the Cartesian origin within it. The required equations (Art. 108, Cor. 2) will be of the form a-fc/? = 0, /3-k'-y=Q, y-k"a=0: in which we are to determine k, k' , k" so that the lines shall bisect the sides of the triangle. We know that k is = the ratio of the perpendiculars dropped from the middle of y on a and /? respectively. But this ratio is evidently = sin B : sin A. Similarly, k' sin C : sin B ; and k" = A : sin (7. Hence, the equations sought are a sin A 13 sin B = 0, Bain B y sin C= 0, y sin C a sin A = 0. 3. Form the trilinear equations to the three perpendiculars which fall from the vertices of a triangle upon the opposite sides. We begin, as before, with the forms of the equations, namely, a-fc/?=0, p-k'y=Q, y-fc" a =0. Now the condition that the first line shall be perpendicular to y (Art. 231, Cor.) gives us cos B A; cos ^4 = 0. Hence, k cos B : cos A; and, similarly, k'= cos C:. cos B, k" = cos C: cos A. The required equations are therefore a cos A (3 cos B = 0, /? cos B y cos C= 0, y cos C a cos A = 0. 4. Form the trilinear equations to the three perpendiculars through the middle points of the sides of a triangle. The middle points of the sides may be regarded as the intersections of y, a, and with the lines of Ex. 2. Hence, the for m of the equations will be /?sin#+ny=0, sin B y sinC+la = 0, y sin C a sin A+ m/3 = 0. 222 ANALYTIC GEOMETRY. But the condition of Art. 231, Cor., gives us n = sin (AB), lsm(B C), m = sin ( C A). Therefore, the equations sought are a sin A sin B + y sin (A B) = 0, (3 sin B - y sin C + a sin ( B C) = Q, y sin C- a sin A + B sin ( C- A)-0. If the student will now compare the equations of the last three ex- amples with those of Exs. 4, 6, 7 on page 121, he will at once see how much simpler the trilinear expressions are than the Cartesian. 5. Show that the equation representing a perpendicular to the base of a triangle at its extremity, is a + ycos-B= 0. 6. Show that the lines o A-/? = 0, j3ka = Q are equally inclined to the bisector of the angle between a and (3. 1. Prove that the equation to the line joining the feet of two perpendiculars from the vertices of a triangle on the opposite sides is a cos A + P cos B y cos (7=0. Also, that the equation to a line passing through the middle points of two sides is a sin A + /3 sin B y sin C=Q. 8. Show that the equation to a line through the vertex of a triangle, parallel to the base, is a sin A -f (3 sin .8=0. 9. Find the equation to the line which joins the centers of the inscribed and circumscribed circles belonging to any triangle. By the principle of Art. 225, we may take for the co-ordinates of the first point 1, 1, 1 ; and, of the second, cos A, cos B, cos C. Hence, (Art. 233,) the equation is a (cos B cos C) + 0(cos C cos A) + y (cos A cos B) = 0. 10. What is the locus of a point, the sum of the squares of the perpendiculars from which on the sides of a triangle is constant? Show that when the locus is a circle, the triangle is equilateral. 11. Find, by a method similar to that of Art. 231, the tangent of the angle contained by the lines la -f m/3 + ny = 0, I' a -f m'/3 + n'y = 0. 12. Write the equation to the circle circumscribing the triangle whose sides are 3, 4, 5. 13. A conic section is described about a triangle ABC; lines bisecting the angles A, B, and C meet the conic in the points A / , B', and C / : form the equations to A'B, A f C, A'B'. INSCRIBED AND ESCRIBED CONICS. 223 14. Find an equation to the conic which touches the three sides of a triangle. If the equations to the three sides of the triangle are a = 0, /? 0, y 0, the required equation may be written Verify this, by clearing of radicals, and showing that y, a, and /? are alt tangents to the curve. (See Art. 240.) Note, The preceding equation is of great importance in some investi- gations, and it will be found upon expansion to involve four varieties of sign, in the terms containing /?y, ya, a/?. This agrees with the fact that there are four conies which touch the sides of a triangle, namely, one inscribed, and three escribed that is, tangent to one side externally, and to the prolongations of the other two internally. If we suppose a, 0, y all pos- itive, therefore, the equation will represent the inscribed conic ; thus, The equations to the escribed conies will be y^Ti + Vlntf + y^ = 0, Via + y^n~/3 + V^= 0, VTa + V^f3 + V^ = 0, since, in each, one set of perpendiculars must fall on the triangle exter- nally. 15 Find the equation to the circle inscribed in any triangle. We may derive this from Via + Vm\l + J^iy^O, by clearing of radicals on the assumption that a, /?, y are positive, and then taking I, m, n so as to satisfy the condition of Art. 239. Or, we may develop the equation from that of the circumscribed circle, as follows*: Let the sides of the triangle formed by joining the points of contact of the inscribed circle be a', 0', y'. Its equation (Art. 236) will then be sin A' sin B' sin C' ~~^~ ~&~ ~7~ =0 - But, with respect to the triangles AB'C', etc., -we have (Ex. 1) a' 2 =/?y, /?'2 = ya, y'2=a/?; and A'= 90 - Y 2 A, B' = 90 - % B, C'=W-%C. Substituting these values, and multiplying the resulting expression throughout by ^a/j/y, the required equation is found to be cos M A ^a + cos } B VJ+ cos K C f V'7= 0. The student may now investigate the equations to the three escribed circles. By the principle developed in Ex. 14, Note, these may be written down at once, from the equation just found. * Hart's solution, quoted by Salmon. An. Ge. 22. 224 ANALYTIC GEOMETRY. SECTION II. TANGENTIAL CO-ORDINATES. 242. We now come to a method of representing lines and points, which, in connection with the trilinear, plays a very important part in the Modern Geometry. It is known as the method of Tangential Co-ordinates, and was first employed by MOBIUS, in his Barycentrisclie Calcul, which appeared in 1827. We can here give only an outline of the system. For a full exhibition of it in its most important applications, the student is referred to the work just cited, to Pliicker's System der analytischen Geometric, and to Salmon's Conic. Sections. 218. The following considerations will bring into view the relations of the new system to the Cartesian and the trilinear. Let AX + py + v = be the equation to a right line. Since the position of the line is determined by fixing the values of /, //, v, it is evident that we may regard these co-efficients as the co-ordinates of a right line. Suppose then we take ^ /jt, v as variables, and connect them by any equation of the first degree, /-j-5/^-j-cv=0. Such an equation (Art. 115, Cor.) is the condition that the whole system of lines denoted by /# + p.y -f- v = shall pass through*a fixed point, whose co-ordinates are a : c, b : c. Since, then, the point becomes known whenever ah + b/jt -|- cv = has known co-efficients, we must regard the condition just written as the equation to a point. Accepting this result, and putting , /?, etc., as abbre- viations for the expressions equated to zero in the equa- tions to different points, we shall have, by the analogy of Art. 108, la -f mfi = as the equation to a point TANGENTIAL CO-OEDINATES. 225 dividing in a given ratio the distance between the points a = 0, = 0. Similarly, la mft = 0, mft ny 0, ny la = denote three points which lie on one right line. 244. From what has just been said, we can see that we may have a system of notation in which co-ordinates represent rigid lines, while points are represented ly equa- tions of the first degree between the co-ordinates. This is what we mean by a system of tangential co-ordinates. The tangential method is in a certain sense the recip- rocal of the Cartesian. It begins with the Right Line, and, by means of an infinite number of right lines all passing through the same point, determines the Point; the Cartesian method, on the contrary, begins with the Point, and determines the Kight Line as the assemblage of an infinite number of points. In the tangential sys- tem, accordingly, the Right Line fulfills the office assigned to the Point in the Cartesian : it is the determinant of all forms, which are conceived to be obtained by causing an infinite number of right lines to intersect each other in points infinitely close together. 345. Article 243 has shown us that Cartesian co- efficients are tangential co-ordinates, and that the tan- gential equation of the first degree is a Cartesian equation of condition. We shall presently see that tangential equations are always Cartesian conditions, and in fact signify that a right line passes through two consecutive points on a given curve; that is, through two points infinitely near to each other; and so, is a tangent to the curve. 246. The truth of this will appear from the following geometric interpretation of tangentials, which will bring 226 ANALYTIC GEOMETRY. the system into relation to curves of all orders, and show that we can represent any curve in this notation. Let AS, CD, EF be any three lines, each passing through two consecutive points of the curve LM, and therefore touching it, say at P, Q, R. It is plain from E ~j-/ the diagram, that the perimeter of the polygon formed by such tangents will approach the curve more and more closely as the number of the tangents is increased. Hence, when the number becomes infinite, the perimeter will coincide with the curve. Suppose then that ^, p, v being continuous variables we connect them by an equation expressing the condition that every line represented by Xx + py -f v = shall touch LM. We shall then have an infinite system of tangents to LM; that is, we shall have the curve itself. Hence, the equation of condition in /, /jt, v may be taken as the symbol of the curve, and is called its tangential equation for obvious reasons. 247. To form the tangential equation to any curve, we therefore have only to find the condition in ^, //, v which must be fulfilled in order that the line fa + py + v = 0, or la + p$ -f ^ = 0, may touch the curve. This may be done by elimi- nating one variable between the equation to the line and that of the curve, and then forming the condition that the resulting equation may have equal roots. For the roots of the resulting equation are the co-ordinates of the points in which the line cuts the curve (Art. 62, Cor. 1) ; and, if these roots are equal, the points of ENVELOPES. 227 section become coincident (that is, consecutive), and the line is a tangent. Instead of this method, it is often preferable to employ another, which will be illustrated a little farther on. 248. The Right Line in Tangentials. This is represented not by an equation, but by co-ordinates. The symbol of a fixed right line is therefore a' A + b' fi -f c' v = a"X + V'fji + c"u t= For, by solving between these simultaneous equations to two points, we can fully determine the ratios X : /j. : v ; that is, we can determine the line which joins the points represented by the given equations. This result is in harmony with the fact already no- ticed, that the Eight Line plays in the tangential system the same part that the Point does in the Cartesian. In the abridged notation, the simultaneous equations a = 0, /9 are the tangential symbol of a right line. For the sake of brevity, we shall generally speak of the line joining the points a and ft (that is, the points whose equations are a = 0, /3 = 0) as the line /9; just as, in the trilinear system, we call the point in which the lines a and ft intersect, the point aft. ENVELOPES. 249. We have called the geometric equivalent of a Cartesian or trilinear equation the locus of a point. Similarly, the geometric equivalent of a tangential equation is called the envelope of a right line. This term needs explanation. Every tangential equation takes the co-efficients of la. -f ftp -f- v? = as variables. It therefore implies the 228 ANALYTIC GEOMETRY. existence of an infinite series of right lines, the successive members of which have directions differing by less than any assignable quantity, and intersections lying infinitely near to each other. Such a series of consecutive direc- tions, blending at consecutive points, will of course form a curve., whose figure will depend on the equation con- necting the variable co-efficients /, /j, v. A curve thus defined by a right line whose co-ordinates vary continu- ously, is what we mean by an envelope. From the rea- soning of Art. 246, it is evident that a right line always touches its envelope. Definition. The Envelope of a right line is the series of consecutive directions to which it is restricted by given conditions of form. Or, The envelope of a right line is the curve which it always touches. Remark. We thus come upon an essential distinction between the tangential system and the Cartesian. In the latter, a curve is conceived to be the aggregate of an infinite number of positions; in the former, it is regarded as the complex of an infinite number of directions. It appears from this article, that, instead of calling the condition that a right line shall touch a curve the tangential equation to the curve, we may say it is the equation to the envelope of the line. 25O. Tangential equation to OBC Conic cir- cumscribed about a Triangle. We might form this by the method of Art. 247, but can proceed more rapidly as follows : The trilinear equation to the tangent of a conic, referred to the inscribed triangle, (Art. 240, II) is la mft n r m ^72-f^+p=- 1), the equation to the conic itself being THE CONIC AS ENVELOPE. 229 Hence, in order that the line Xa + ///? -\- vy = may- touch the curve, we must have ),a! 2 = pl, fjtp f2 = pm, vf' 2 = pn; that is, But a 1 ', /9', / are the co-ordinates of the point of contact, and must satisfy the equation to the curve. Therefore, after substituting in (2) the values just found, 1/0 -f- V'rnjjL -f l/m> = is the condition that / -f ^ -j- ^ = shall touch the conic. In other words, it is the tangential equation to the conic. Kemark. By clearing this equation of radicals, we shall find that it is of the second degree in A, ^ v. 251. The further investigation of tangentials requires that we now turn aside from our direct path, to examine the method of finding the envelope of a right line when the constants which enter its equation are subject to given conditions. By the terms of this problem, the equation to the line may be written so as to involve a single indeterminate quantity ; for instance, in the form m 2 a 2kmf -f &/9 = 0, where m is indeterminate, and k is fixed. For, by means of the given conditions, we can eliminate one of the arbi- trary constants which enter the original equation, and thus leave but one. Now, by the definition of an envelope (Art. 249), the right line m 2 a Zkm? + kfi = (1), 230 ANALYTIC GEOMETRY. is tangent to its envelope. In this particular instance, then, the envelope is such a curve that only two tangents can be drawn to it from a given point. For if we suppose the line (1) to pass through the given point oJfi'f, we shall have a'.m 2 Zkf.m + fc^ = (2), from which to find m, the indeterminate in virtue of which (1) represents a system of tangents to the envelope; and, since (2) is a quadratic in m, only two lines of the system can pass through a'fi'f. This property of the envelope is designated by calling it a curve of the Second class : the class of a curve being determined by the number of tangents that can be drawn to it from any one point. We learn, then, that if the equation to a right line involves an indeterminate quantity in the n th degree, the envelope is a curve of the n th class. * The question still remains, How shall we obtain the equation to this envelope? The definition of Art. 249 answers this ; for since the envelope is the series of consecutive directions of the tangent, we have only to form the condition that the n values of the indeterminate m may be consecutive, and the required equation is found. Now this condition is of course the same as that which requires the equation in m to have equal roots. For example, the equation to the envelope of (1) is aft = kf 2 . Hence, To find the envelope of a right line, throw its equation, by means of the given conditions, into a form involving a single indeterminate quantity in the n th degree : the condition that this equation in m shall have equal roots, is the equation to the envelope. * The student must not confound the class of a curve with its order. A conic belongs to the Second order, and also to the Second class; hut other curves do not in general show this agreement. TANGENTIAL EQUATION TO CONIC. 231 Example. Given the vertical angle and the sum of the sides of a triangle : to find the envelope of the base. Take the sides for axes ; then the equation to the base is in which o and b are subject to the condition a -f b c. The equation to the base may therefore be written 2 + (y * <0 a + ex = 0. Hence, the equation to the envelope is (y *-c) 2 = 4cx; or, x' 2 - Ixy + j/ 2 2cx 2cy + c 2 = 0. The required envelope is therefore (Art. 191) a parabola; which touches the sides x= and y 0, since the equations for deter- mining its intercepts on the axes are x 1 lex -f c 2 = 0, if Icy -f c 2 = 0. We may now resume the direct course of our investi- gation, and finish this part of it by establishing, in their proper order, the theorems necessary for interpreting tangential equations of the first and second degrees. 252. Tangential equation to any Conic. We shall here follow the method of Art. 247, and find the condition that Xa. -f- ///9 -f- wf = may touch the conic Aa? -f B^ -f Cf + 2Ffr + 2 GTOL -f ZHafi = 0. Eliminating f between the equation to the line and that of the conic, and collecting the terms of the result with reference to a : /9, we obtain ? 2 = 0. Forming the condition that this complete quadratic in a : /? may have equal roots, we get the required tangential equation, namely, Guv An. Ge. 23. 232 ANALYTIC GEOMETRY. which, after expansion and division by v 2 , becomes ~ Now it is remarkable that the co-efficients of (1) are the derived polynomials (see Alg., 411) of the Discriminant J, obtained by supposing the variable to be successively A, B, C, F, 6r, H. They may there- fore be aptly denoted by a, 6, y Via-}- l/mj3 + Vn-y = Q. 8. Find the equation to the envelope of the right line whose co-efficients fulfill the condition and the equation to the envelope of one whose co-efficients satisfy V A + l/m/T+ Vm> = 0. What is the meaning of the results ? 9. Prove that the envelope of the conic represented by in which I, m, n are subject to the condition I/A? -f V~pn+ Vvn = 0, is the right line Aa + ftp -f- vj = 0. 10. Prove that the envelope of the conic Vla-\- Vmfi + l/ny = 0, whose co-efficients satisfy the relation is the right line Aa + pft + V J = 0. 11. Find the envelope of a right line, the perpendiculars to which from two given points contain a constant rectangle. 12. The vertex of a given angle moves along a fixed right line while one side passes through a fixed point: to find the envelope of the other side. 13. A triangle is inscribed in a conic, and its two sides pass through fixed points : to find the envelope of its base. 14. Prove that if three conies have two points common to all, their three reciprocals will have two tangents common to all ; and conversely. 244 ANALYTIC GEOMETRY. 15. Establish the following reciprocal theorems, and determine the conditions for the cases noticed under them : If two vertices of a triangle move along fixed right lines while the three sides pass each through a fixed point, the locus of the third vertex is a conic. But if the points through which the sides pass lie on one right line, the locus will be a right line. In what other case will the locus be a right line? If two sides of a triangle pass through fixed points while the three vertices move each along a fixed right line, the envelope of the third side is a conic. But if the lines on which the vertices move meet in one point, the envelope will be a point. [Compare Ex. 39, p. 125.] In what other case will the envelope be a point? [Compare Ex. 40, p. 125.] PLANE CO-ORDINATES. PART II. THE PROPERTIES OF CONICS. 266. The investigations of Part I, have taught us the methods of representing geometric forms by analytic symbols ; and furnished us, in the resulting equations to curves of the First and Second orders, with the necessary instruments for discussing those curves. We now proceed to apply our instruments to the determination of the properties of the several conies. We shall adhere to the order of Part I, considering first the several vari- eties in succession, and then, by way of illustrating the method of investigating the common properties of a whole order of curves, determining those of the Conic in General. CHAPTER FIRST. THE RIGHT LINE. 267. Under this head, we only purpose developing a few properties, noticeable either on account of their usefulness or their relations to the new or to the higher geometry. (245) 246 ANALYTIC GEOMETRY. 268. Area of a Triangle in terms of its Ver- tices. Let the vertices be x \y\-> X 2t/2i x )z-> represented in the diagram by A, -B, C. It is obvious that for the re- quired area we have A BC=ALMB+BMNC CALN ; that is, or OTT 3T N X 2 7= y, (x-x ? ) + y 2 (x-x,) + 7/3 (x-x. 2 ). Remark. This expression for the double area of the triangle is identical with that which in Art. 112 is equated to zero as the condition that three points may lie on one right line. We thus discover the latter con- dition to be simply the algebraic statement, that, when three points lie on one right line, the triangle which they determine vanishes: which obviously accords with the fact. 269. Area of a Triangle in. terms of its inclos- ing lines. Let the three lines be Ax -j- By -)- C= 0, MX + B'y + C' = 0, A"x + B"y + C" = 0. Their in- tersections will form the vertices of the triangle ; hence, substituting for x } y } , x 2 y 2 , x z y^ in the preceding formula, according to Art. 106, we obtain 2T= - C A / - C / A Y C" B" B"C -B C' A B / A / B i i A / B" A // B / A"B A B" C / A" - C // A / ' \B"C -B C" B C' 7D/ /-^ A / B" -A "B' (A"B A B" A B / - A / B C"A Q A" i \B C' w C B / C" B // C / A"B A B"(A B / A / B AREA OF A TRIANGLE. 247 Now if we reduce each of the sets of fractions inside the braces to a common denominator, the three new numera- tors will be respectively E" I B( C"A' C / A ") +B'( CA"-C" A) +B"( C' ACA') \ , B \B(C''A'C / A"}+B'(CA"-C"A}+B"(C / A-CA'}\, B' lB(C / 'A / C / A // )+B'(CA"-C // A)+B"(C / ACA')l. Hence, the final expression for the required area may be written I B( C"A' CM") (AB'-A'B) (A'B"-A"B')(AB"A"B) Remark, The numerator of this expression may be otherwise written so that (Art. 113), if the three lines pass through the same point, 2 T vanishes. On the other hand, the expression for 2T becomes infinite (Art. 96, Cor. 2) whenever any two of the lines are parallel. In both respects, then, the formula accords with the facts. 27O. Ratio in which the instance between Two Points is divided by a Given Line. Let m : n be the ratio sought, and x\y^ x. 2 y 2 the given points. The co-ordinates of division (Art. 52) will then be ^ ___ mx 2 + nx l __ my, -H?|/, m + n m -f- n But these must of course satisfy the equation to the given line ; hence, A (mx. 2 -f nxj -f- B (my, -f- ny,} + C(m + ri) = 0. m_ Ax, + i + n~ An. Ge. 24. 248 ANALYTIC GEOMETRY. Corollary. If the given line passed through two fixed points x 3 y 3 and x 4 y 4 , we should have (Art. 95, Cor. 1) _ n~ (y-, y,) x 2 (x z x,) y 2 as the ratio in which the distance between two fixed points is divided by the line joining two others. TRANSVERSALS. 27 JU Definition, A Transversal of any system of lines is a line which crosses all the members of the system. Thus, LMN is a transversal of the three sides AB, BC, CA in the triangle ABC. . Theorem. In any triangle, the compound ratio of the segments cut off upon the three sides by any trans- versal is equal to 1. In the above diagram, let the vertices A, B, C be represented by #,?/,, x. 2 y 2 , x 3 y ?t , and the transversal LN by Ax + By -f- C= 0. Then (Art. 270) AL LB = - (Ax l +By l + C) : (Ax 2 +By 2 + C), BM:MC= CN : NA = Multiplying these equations member by member, we obtain the proposition. 273. Theorem. In any triangle, the compound ratio of the segments cut off upon the three sides by any three convergents that pass through the vertices is equal to -f- 1. For, if the point of convergency be 0, represented by x 4 y 4 while the vertices are denoted as in the preceding THREE CONVERGENTS. 249 article, we shall have (Art. 270, Cor.), after merely re- arranging the terms, PB _K(yi yi)+si(ys y^+^Cyi 2/2) and the product of these equations is the algebraic expression of the theorem. We shall next give some illustrations of the uses of abridged notation, as applied to rectilinear figures and to right lines in general. TRIPLE CONVERGENTS IN A TRIANGLE. 274. Theorem. The three bisectors of the internal angles of a triangle meet in one point. For their equations are a /3=0, ft f 0, f 0; and (Art. 114) these vanish identically when added together. 275. Theorem. The bisectors of any two external angles of a triangle, and the bisector of the remaining internal angle, meet in one point. For their equations are a-{-ft=Q 9 j3-\-r=Q, f a=0, etc. But (Art. 108) f a is a line passing through the intersection of a-f-/? and /9-f f. 27G. Theorem. The three lines which join the vertices of a triangle to the middle points of the opposite sides meet in one point. For (Ex. 2, p. 221) their equations are respectively asm A ft sin ,5 0, ftsinB 7- sin (7=0, fsin(7 asinyl=0, and therefore vanish identically when added together. 250 ANALYTIC GEOMETRY. 2*7*7. Theorem, The three perpendiculars let fall from the vertices of a triangle upon the opposite sides meet in one point. For (Ex. 3, p. 221) the corresponding equations are a cos A /9 cosB=Q, ft cosB ?- cos (7=0, y cos C a cosJ. 0. 278. Theorem, The three perpendiculars erected at the middle points of the sides of a triangle meet in one point. For (Ex. 4, p. 221) we have found their equations to be a sin A (3 sin B + 7 sin (A B) = 0, psinB ysin C-f a sin (B C] = 0, 7 sin (7 asin^L-f (3s'm(C A)=0; and, if we multiply these by sin 2 (7, shrJ., sin 2 _B respect- ively, we shall cause them to vanish identically. (See Trig., 850, Ex. 2.) HOMOLOGOUS TRIANGLES. 2*70. Definitions. Two triangles, the intersections of whose sides taken two and two lie on one right line, are said to be homologous. The line on which the three intersections lie is called the axis of homology. Any two sides that form one of the three intersec- tions are termed corre- sponding sides; and the angles opposite to them, corresponding angles. Thus, in the diagram, the triangles ABC, A f B'C f are homologous with respect to the axis LMN. AB, A'B'; BC, B'C' ; CA, C'A' are the corresponding sides; and A y A'; B) B'; (7, C", the corresponding angles. HOMOLOGY. 251 280. Theorem. In any two homologous triangles, the right lines joining the corresponding vertices meet in one point. Let a, /?, f be the sides of ABC, and take the latter for the triangle of reference ; the equation to the axis LN may then be written la -f- mft -j- ny = 0. Suppose the Cartesian origin to be somewhere between the tri- angles, and (Art. 108, Cor. 2) the equations to AB', B'C', C'A, which pass through the intersections of a, /?, f with the axis, will be (I Oa+m/?+n7=0, la+(mm / )/3+n-y=Q, la+mp+(n n')y=0. Subtracting the second of these from the first, the third from the second, and the first from the third, we get I' a m'p = 0, m f p n'r = 0, n'r Va = 0. But these equations (Art. 107) evidently denote the lines BB', CO', AA'; and they vanish identically, when added together. Remark. The point in which the lines joining the corresponding vertices meet, is called the center of homology. 281. Theorem. If the lines joining the corresponding vertices of two triangles meet in one point, the intersections of the corresponding sides lie on one right line. This theorem, the converse of the preceding, is ob- tained at once by merely interpreting the equations of the foregoing article in tangentials. We leave the student to carry out the details. COMPLETE QUADRILATERALS. S82. Definitions. A Complete Quadrilateral is the figure formed by any four right lines intersecting in six points. 252 ANALYTIC GEOMETRY. The three remaining right lines by which the six points of intersection can be joined two and two, are called the diagonals of the quadrilateral. Thus, ABCDEF is the complete quadrilateral of the four lines AB, BE, EF, FA, which meet in the six points B, E, F, A, C, D. AE, BF are the two inner diagonals, and CD is the outer one, some- times called the third. 383. Theorem. In any complete quadrilateral, the middle points of the three diagonals lie on one right line. Let a, /?, f, d be the equations to the four sides of the quadrilateral, and let the respective lengths of BE, EF, FA, AB equal a, b, c, d. Then L, M, N being the middle points of the three diagonals, we have the following equa- tions to the lines drawn from the vertices to the middle points of the bases of the triangles ABE, EFA, ABF, FEB: dd aa = (BL), bp c r = (FL) ; c r dd=Q (AM), aa - 6/9 = (EM). Hence, L and M both lie upon the line aa bp + cr dd = Q (1) , as this obviously passes through the intersection of (BL, FL), and of (AM, EM). If we now put Q = the double area of ABEF, we shall have and thence aa bp + crd8 = = 2 (aa + c r ) Q. AN HARMONICS. 253 Therefore (Art. 229) the line (1) is parallel to the two lines aa -j- cf and bp -j- dd, and midway between them. It accordingly bisects the distance between ;- (which is a point on the first) and fld (which is a point on the second). That is, N lies on (1) : which proves our proposition. * THE ANHARMONIC RATIO. 284. Definition. A Linear Pencil is a group of four right lines radiating from one point. Thus, OA, OQ, OB, OP constitute a linear pencil. 2S5 Theorem, If a linear pencil is cut by any transversal in four points A, Q, B, P, the ratio AP.QB : AQ.PB is constant. For, putting p = the perpendicular from upon the transversal, we have p.AP= A. OP sin AOP; p.QB OB.OQsmQOB; p.AQ = OA.OQsinAOQ; &ndp.PB = OB.OPsiuPOB: whence p\AP. QB = OA. OB. OP.OQsmAOPsm QOB, p\ AQ.PB = OA. OB. OP. OQsmAOQ sin POB. AP. QB _ sin A OP sin QOB ' AQ.PB ~~ sin AOQ sin POB' a value which is independent of the position of the transversal. Remark, The constant ratio just established is called the anharmonic ratio of the pencil. By reasoning sim- ilar to that just used, it may be shown that the ratios AP.QB : AB.QP&nd AB .QP : AQ.BP are also con- *See Salmon's Conic Sections, Ex. 3, p. 64. 254 ANALYTIC GEOMETRY. stant. To these, accordingly, the term anharmonic is at times applied ; but it is generally reserved for the par- ticular ratio to which we have assigned it, and we shall always intend that ratio when we use it hereafter. Note The Anharmonic Ratio has an important place in the Modern Geometry, especially in connection with the doctrine of the Conic. Its existence, however, has been known since the time of the Alexandrian geometer PAPPUS, who gives the property in his Mathematics Collectiones, Book VII, 129, and who probably belongs to the fourth century. The name anharmonic was given by CHASLES. But the bearing of the ratio upon the new geometry had been pre- viously investigated by MOBIUS, who called it the Ratio of Double Section (Doppelschnittsverhaltniss). 286. Definition. An Harmonic Proportion sub- sists between three quantities, when the first is to the third as the difference between the first and second is to the difference between the second and third. Thus, if the pencil in the above diagram cut the trans- versal so as to make AP:AQ :: APAB : AB AQ, the whole line AP would be in harmonic proportion with its segments AB and AQ. Corollary. A line is divided harmonically, when it is cut into three segments such that the whole is to either extreme as the other extreme is to the mean. For the. proportion given above may obviously be written AP:AQ::BP:BQ. 287. Definitions. An Harmonic Pencil is one which cuts its transversals harmonically. I. From the final equation of Art. 285, it is evident that when the anharmonic ratio of a pencil is numerically equal to 1, the pencil is harmonic. II. The four points in which a line is cut by an har- monic pencil, are called harmonic points. HARMONIC PENCILS. 255 III. Linear pencils are in general termed anharmonic, because they do not in general cut off harmonic segments from their transversals. 288. Theorem. The anharmonic of the pencil formed by the four lines a, ft a + &ft + W$ is equal to k : k'. Let OA be the position of #, OB of ft OP of a -f &ft and 0$ of a -f- &'ft Then, & being the ratio of the perpendiculars from OP upon 0J., 0.Z?, and Jc f the ratio of those from OQ on the same lines, we shall have &=sin^l0P: siuPOB, and k'=amAOQ : smQOB. Hence > k_ , siuAOPsiuQOB t k'~ smAOQsinPOB '' which (Art. 285) proves the proposition. Corollary. If k r = k, the anharmonic of the pencil becomes numerically equal to 1. Hence (Art. 287, I) the important property : The four lines ,/?,+ 7c/?, a 7c/9 form an harmonic pencil. Also, by the analogy of tangentials: The four points , /9, a + 7c/9, a 7c/9 are harmonic. . Theorem. The anharmonic of any four lines 9, a + Z/3, a -f- m/9, a -f nfi is equal to (n k) (ml) Let OJ., OB represent the lines a and ft and 07f, OJ^, OM 9 ON the four lines a-f ^A a-Hft a+wift a+nft. Then, if rs be any transversal, the anharmonic of the four given lines will be au.eo : ae.ou; or, what amounts to the same thing, it will be An. Ge. 25. 256 ANALYTIC GEOMETET. (ru ra) (ro re) : (re ra) (ru ro) . Now, taking the lengths of the perpendiculars from a, e, o, u upon a, both trigonometrically and from the equations to the given lines, we obtain ru = nf) cosec Ore, ra = - left cosec Ors, ro = m-ft cosec Ors, re Z/3 cosec Ors. Substituting these values in the ratio laslr written, and recollecting that, since the anharmonic is independent of the position of the transversal, we may take rs parallel to the line /?, and thus render the perpendicular /9 a constant, we find after reductions R ^ (n k) (m I) (n m) (I k) Remark. The student can easily convince himself that the harmonic and anharmonic properties above obtained are true for lines whose equations are of the more general form .L = 0, M= 0, L -{- kM= 0, etc. 2OO. Theorem. If there be two systems of right lines, each radiating from a fixed point, and if the several mem- bers of the one be similarly situated with those of the other in regard to any two lines of the respective groups, then ivill the anharmonic of any pencil in the first system be equal to that of the pencil formed by the four correspond- ing lines of the second. For, in such a case, the two reference-lines of the first group being L = 0, L' = 0, and those of the second M=Q, M f =, the corresponding pencils will be (Z/+&Z/, L -f W, L + mL', L + nL') and (M + kM', M+ IM', M-\- mM ', M + nM'). Hence, the theorem follows directly from the result of Art. 289. HOMOGRAPHIC LINES. 257 291. Definition. Systems of lines whose correspond- ing pencils have equal anharmonics are called homographic systems. Thus, in the diagram of Art. 289, the two systems (OA, OK, OL, OM, ON, OB) and (0>A, O'K', O'L', O'M', O'N', O'B) are intended to represent a particular case of homographics. 292. In the examples which follow, some are best adapted for solution by the old notation, and others by the abridged. We have room for only a few of the manifold properties of the Right Line. EXAM PL ES. 1. If from the vertices of any triangle any three convergents be drawn, and the points in which these meet the opposite sides be joined two and two by three right lines, the points in which the latter cut the sides again will lie on one right line. 2. Any side of a triangle is divided harmonically by one of the three convergents mentioned in Ex. 1, and the line joining the feet of the other two. 3. In the figure drawn for the two preceding examples, deter- mine by tangentials all the points that are harmonic. 4. In any triangle, the two sides, the line drawn from the vertex to the middle of the base, and the parallel to the base through the vertex, form an harmonic pencil. 5. The intersection of the three perpendiculars to the sides of a triangle, the intersection of the three lines drawn from the vertices to the middle points of the sides, and the center of the circum- scribed circle, lie on one right line. 6. APE, CQD are two parallels, and AP-.PB:: DQ: QC: to prove that the three right lines AD, PQ, BC are convergent. 7. From three points A, B, D, in a right line A BCD, three con- vergents are drawn to a point P; and through C is drawn a right line parallel to AP, meeting PB in E and PD in F: to prove that AD.BC : AB. CD :: EC: CF. 258 ANALYTIC GEOMETRY. 8. The six bisectors of the angles of any triangle intersect in only four points besides the vertices. 9. If through the vertices of any triangle lines be drawn parallel to the opposite sides, the right lines which join their intersections to the three given vertices will meet in one point. [Use both notations in succession.] 10. If through the vertices of any triangle there be drawn any three convergents whatever, to prove that these three lines and the three sides of the triangle may be respectively represented by the equations v w = 0, w u = Q, u v = 0, v-\- w = ?i, w -\-u = %, u-\-v = %. 11. \tOAA' A", OBE'R" are two right lines harmonically divided, the former in A and A', the latter in .B and B', the lines AB, A'B', A // B // either meet in one point or are parallel. 12. If on the three sides of a triangle, taken in turn as diagonals, there be constructed parallelograms whose sides are parallel to two fixed right lines, the three remaining diagonals of the parallelograms will meet in one point. 13. The three external bisectors of the angles of any triangle meet the opposite sides in three points which lie on one right line. 14. If three right lines drawn from the vertices of any triangle meet in one point, their respective parallels drawn through the middle of the opposite sides also meet in one point. 15. In every quadrilateral, the three lines which join the middle points of the opposite sides and the middle points of the diagonals, meet in one point. 16. If the four inner angles A,B,E,F of a complete quadrilateral (see diagram, Art. 219) are bisected by four right lines, the diago- nals of the quadrilateral formed by these bisectors will pass through the two outer vertices of the complete one, namely, one through C and the other through D. 17. Let the two inner diagonals of any complete quadrilateral (same diagram) intersect in O : the diagonals of the two quadri- laterals into which either CO or DO divides ABEF, intersect in two points which lie on one right line with O. 18. In any complete quadrilateral, any two opposite sides form an harmonic pencil with the outer diagonal and the line joining their intersection to that of the two inner diagonals. PROPERTIES OF THE CIRCLE. 259 19. Also, the two inner diagonals are harmonically conjugate to the two lines which join their intersection to the two outer vertices. 20. Also, two adjacent sides are harmonically conjugate to their inner diagonal and the line joining their intersection to that of the outer diagonal and the remaining inner one; etc. CHAPTER SECOND. THE CIRCLE. 203. Before attempting the discussion of the three Conies strictly so called, it will be advantageous to illustrate the analytic method by applying it to that case of the Ellipse with whose properties the reader is already familiar from his studies in pure geometry : we mean, of course, the Circle. As we proceed in this application, we shall be enabled to define those elements of curves in general, which constitute at once the leading objects and principal aids of geometric analysis. THE AXIS OF X. 294. The rectangular equation to the Circle referred to its center (Art. 136) is a; 2 + y 2 = r 2 . If we solve this for ?/, we obtain y = V(r+x) (r x). But, from the diagram, r -\- x = AM, and r x = MB ; and we have the well-known property of Geom., 325, The ordinate to any diameter of a circle is a mean proportional between the corresponding segments. 260 ANALYTIC GEOMETRY. 295. If we eliminate between the equation to a circle x 2 -f y 2 = r 2 and that of any right line y = mx -\- b by substituting for y in the former from the latter, we shall obtain, as determining the abscissas of intersection between a right line and a circle, the quadratic (1 -f- m 2 ) x 2 + 2mb. x + b 2 r 2 = 0. Now the roots of this quadratic are real and unequal, equal, or imaginary, according as (1 + m 2 ) r 2 is greater than, equal to, or less than & 2 . Hence, when the first of these conditions occurs, the right line will meet the circle in two real and different points ; when the second, in two coincident points ; when the third, in two imaginary points. Adhering, then, to the distinction between these three classes of points, we may assert, with full generality, Every right line meets a circle in two points, real, coin- cident, or imaginary. 296. It is so important that the distinction just alluded to shall be exactly understood in our future investigations, that we consider it worth while to illustrate it somewhat more at length. I. The conception of two real points, situated at a finite distance from each other, is of course already clear to the student. We therefore merely add, that such points are sometimes called discrete, or discontinuous points. II. The conception of coincident, or, as they are more significantly called, consecutive points, is peculiar to the analytic method. The most general definition of con- secutive points is, that they are points whose distance from each other is infinitely small. It may aid in rendering this definition clear, to think of two points which are drawing closer and closer together, which tend to meet but not CONSECUTIVE POINTS. 261 to pass each other, and whose mutual approach is never for an instant interrupted. The distance between two such points is evidently less than any assignable quantity ; for however small a distance we may assign as the true one, the points will have drawn nearer together in the very instant in which we assign it : so that their distance eludes all attempts at finite statement, and can only be represented by the phrase infinitely small. The geometric meaning of this analytic conception varies with its different applications. Thus, it may sig- nify exactly the same thing as the single point which, in the language of pure geometry, is common to a curve and its tangent. For since the distance between con- secutive points is infinitely small ; that is, so small that we can not assign a value too small for it; we may assign the value 0, and take the points as absolutely coincident. It is in this aspect, mainly, that we shall use the conception in our future inquiries. Hence, as from the infinite series of continuous values which the distance between two consecutive points must have, we thus select the one corresponding to the moment of coincidence, we have preferred to designate the concep- tion by the equivalent and for us more pertinent phrase coincident points. v The student should be sure that he always thinks of the distance between coincident points as a true infinit- esimal. The error into which the beginner almost always falls is, to think of a very small, instead of an infinitely small, distance. He thus confounds with two consecutive points, two discrete ones extremely close together, between which there is of course a finite distance. The conse- quence is, that he finds in such a distance, however small, an infinite number of points lying between his supposed consecutives, and fancies that all the arguments based 262 ANALYTIC GEOMETRY. on the conception of consecutiveness are fallacious. Whereas, if he excludes from his thoughts, as he should, all points separated by any finite distance however small, he will have points absolutely consecutive, in the only sense that mature reflection attaches to that term. III. The phrase imaginary points is also peculiar to analytic investigation. It really means, w r hen translated into the language of pure geometry, that the correspond- ing points not only do not exist, but are impossible. But, as we have mentioned once before, the expression, together with the accessories which serve to carry out its use, is found to be of real value in developing certain remote analogies in the properties of curves. We shall therefore retain it, only cautioning the student not to be misled by a false interpretation of it. 297. Definition. A Chord of any curve is any right line that meets it in two points. 298. Equation to a Circular Chord. Let the equation to the given circle be x 2 -j- y 2 = r 2 . Since the chord passes through two points of the curve, its equation (Art. 95) will be of the form y y' __ y" y' . x x'~~ x"x'' in which x r y r , x"y rf , since they both lie upon the circle, are subject to the condition 2/2 -f y n = r 2 = X ff2 _j_ y m f Hence, x r * x m = y" 2 y' 2 \ and we obtain y"y f _ ' CHORD AND DIAMETER. 263 Therefore the required equation to a chord is y y' x f + x" . x x' = ~ y' + y" ' in which x'y', x"y" are the points in which the chord cuts the circle. Corollary, By a course of analysis exactly similar to that just used, we find the equation to any chord of the circle (x g) 2 -f- (y f) 2 = r 2 , namely, y y' _ x f + x" 2g x x'~ ~ y' -f y" 2/ in which x'y', x"y" are the intersections of the circle with its chord, and gf is its center. DIAMETERS. 2OO. Definition, A Diameter of any curve is the locus of the middle points of parallel chords. 3OO. Equation to a Circular Diameter. To find this, we must form the equation to the locus of the middle points of parallel chords in a circle. Let x 9 y be the co-ordinates of any middle point : the formula for the length of the chord from xy to the point x'y' of the curve (Art. 102) gives us either x' =x-\- cl or y 1 = y -f- si. But since x'y' is a point on the circle x 2 -\- y 2 = r 2 , we have + c/) 2 + (?/ + s0 2 = r 2 ; or, remembering that s 2 -f c2 = 1> we get, for determining the distance I of the point xy from the circle, the quadratic I 2 + 2 (c-x + sy) I + (x 2 + f r 2 ) -0. 264 ANALYTIC GEOMETRY. Now, as xy is the middle point of a chord, the two values of I in this quadratic must be numerically equal with contrary signs. Hence, (Alg., 234, Prop. 3d,) the co- efficient of I must vanish, and we get ex -f sy = 0. But, in the present inquiry, s and c are the sine and cosine of the angle which a chord through xy makes with the axis of x\ and as this angle is the same for all parallel chords, the equation ex -f- sy = is a constant relation between the co-ordinates of the middle points of a series of parallel chords. That is, it is the required equation to any diameter of the circle. 301. If be the inclination of a series of parallel chords to the axis of x, the equation just obtained may be written y = xcott). Hence, (Arts. 63; 96, Cor. 3,) we have the familiar property, 4 < ,.* Every diameter of a circle passes through the center, and is perpendicular to the chords which it bisects. Corollary. From this we immediately obtain the im- portant principle : If a diameter bisects chords parallel to a second, the second bisects those parallel to the first. 302. Definition. By Conjugate Diameters of a curve, we mean two diameters so related that each bisects chords parallel to the other. 303. We have seen (Art. 301) that the equation to any diameter of a circle may be written y = x cot 0, in which expression 6 is the inclination of the chords CONJUGATE DIAMETERS. TANGENT. 265 which the diameter bisects. Hence, by the preceding definition, the equation to its conjugate will be y x tan 6 a). But, putting 0' == the inclination of the chords bisected by this conjugate, its equation will take the form y zcot 6' (2). Therefore, as the condition that two diameters of a circle may be conjugate, we have, since (1) and (2) are only different forms of the same equation, tan 6 tan 6'= I (3): in which 6 and 6' may be taken as the inclinations of the two diameters (since these are each perpendicular to the chords which they bisect), and we learn (Art. 96, Cor. 3) that The conjugate diameters of any circle are all at right angles to each other. THE TANGENT. 3O4. Definition. A Tangent of any curve is a chord which meets it in two coincident points. In applying this definition, the student must keep in mind the principles of Art. 296, II. The annexed diagram will aid him in apprehending the definition cor- rectly. Let PP" be any chord passing through the two distinct points P r and P n ', and let PT be a tangent parallel to P'P". Suppose 266 ANALYTIC GEOMETRY. P'P" to move parallel to itself until it coincides with PT. It is evident that as P'P" advances toward PT, the points P f and P" will move along the curve toward each other, and that when P'P" at length coincides with PT, they will become coincident in P, which is called the point of contact. We shall often allude to the position of the two coincident points through which a tangent passes, by the term contact alone. 305. Equation to a Tangent of a Circle. Let the given circle be represented by x 2 -J- y 2 = r 2 . Then, to obtain the required equation, we have only to suppose, in the equation to any chord (Art. 298), that the co-ordi- nates x' and x", y r and y", become identical. Making this supposition, reducing, and recollecting that x' 2 -j- y' 2 = r 2 , we obtain x'x + y'y = r 2 : in which x r y' is the point of contact. Corollary. To obtain the equation to a tangent of the circle (x #) 2 + (y f) 2 = r 2 -, which differs from x* -j- y 1 = r 2 only in having the origin removed to the point ( r 2 ; that is, when x'y f is outside the circle. Imaginary, when x' 2 -j- y' 2 < r 2 ; that is, when x'y' is within the circle. Coincident, when x 12 + y' 2 = r 2 ; that is, when x'y' is on the circle. LENGTH OF TANGENT. 269 3O9. Length of the Tangent from any point to a Circle. Let xy be any point in the plane of a circle whose center is the point gf. Then (Art. 51, I, Cor. 1) for the square of the distance between xy and gf, we have Putting t -- the required length of the tangent, we get (since the tangent is perpendicular to the radius at contact) t 2 = 8 2 r 2 . That is, t 2 =(x-g) 2 +(y-f) 2 -r 2 . From this we learn that if the co-ordinates x and j of any point be substituted in the equation to any circle, the result will be the square of the length of the tangent drawn from that point to the circle. Eemark. We have shown (Art. 142) that the general equation A (x 2 + if) is equivalent to (x g) 2 + (y /) 2 = r 2 , provided we take out A as a common factor. Hence, the property just proved applies to every equation denoting a circle, provided it be reduced to the form in question by the proper division. If, then, we write S as a convenient abbreviation for the left member of the equation to a circle, in which the common co-efficient of x 2 and y 2 is unity, we get t 2 = S. Corollary The square of the length of the tangent from the origin of the circle. A (x 2 + y 2 ) + 2Gx + 2% + # = 0, is equal to C: A: that is, to the quotient of the absolute term by the common co-efficient of x 2 and y 2 . 270 ANALYTIC GEOMETRY 310. Definition. The Subtangent of a curve, with respect to any axis of rr, is the portion of that axis inter- cepted between the foot * of the tangent and that of the ordinate of contact. Thus, if OT is consid- ered the axis of x, MT is the subtangent correspond- ing to PT of the inner curve, and to PT of the outer. 311. Subtangent of the Circle. To obtain the length of this, we find the intercept OT, cut off from the axis of x by the tangent, and subtract from it the abscissa of contact OM. The equation to the tangent being x'x + y'y = r\ to find the intercept on the axis of x, we make y 0, and take the corresponding value of x. Thus, r 2 x=-,=OT. x' Hence, for the subtangent MT, we have r 2 x' 2 subtan = - x' That is, Any subtangent of a circle is a fourth proportional to the abscissa of contact and the two segments into which the ordinate of contact divides the corresponding diameter. THE NORMAL. 312. Definition. The Normal of a curve is the right line perpendicular to a tangent at the point of contact. * The point in which a line meets the axis of x is termed the foot of the line. Similarly, the point where a line meets any other is sometimes called its foot. NORMAL AND SUBNORMAL. 271 313. Equation to a Xormal of a Circle. Since the normal is perpendicular to the line x'x-\-y'y = r 2 , and passes through the point of contact x'y', its equation (Art. 103, Cor. 1) is or, after reduction, y'x x'y = 0. The form of this expression (Art. 95, Cor. 2) shows that every normal of a circle passes through the center : a property which we might have gathered at once from the definition. 314. Definition. The portion of the normal included between the point of contact and the axis on which the cor- responding subtangent is measured, is called the length of the normal. For example, P f in the diagram of Art. 310. From the result of Art. 313, it follows that the length of the normal in any circle is constant, and equal to the radius. 315. Definition. The Subnormal of a curve, with respect to any axis of x, is the portion of that axis inter- cepted between the foot of the normal and that of the corresponding ordinate of contact. Thus, in the diagram of Art. 310, OM is the subnormal corresponding to the point P . Hence, for the Circle, we have subnor = x'. That is, Any subnormal of a circle is equal to the corre- sponding abscissa of contact. SUPPLEMENTAL CHORDS. 316. Definition. By Supplemental Chords of a circle, we mean two chords passing respectively through the extremities of a diameter, and intersecting on the curve. An. Ge. 26. 272 ANALYTIC GEOMETRY. Thus, AP, BP are supplemental chords of the circle whose center is and whose radius is OA. 317. Condition that Chords of a Circle be Supplemental. Take for the axis of x the diameter through whose extremities the chords pass, and for the axis of y a second diameter perpendicular to the first. Let tp - the inclination of one chord, say of AP, and f . Hence, at their point of intersection (Art. 62, Rem.) we shall have the condition y z (x 1 ?- 2 ) tan (p tan cp f ; or, since they intersect upon the circle x z -\-y 2 =r 2 , tan

(1), in which x'y' is the point from wliicli the tangents are drawn. The equation to the locus of the intersection of two tangents drawn at the extremities of any chord passing through a fixed point, is x'x + y f y=r* (2), in which x'y' is the point through which the movable chord is drawn. The equation to any tangent of the same circle to which the chord (1) and the locus (2) refer, is x'x + y'y = r* (3), in which x'y 1 is the point of contact. What, then, is the significance of this remarkable identity in the equations to these three lines ? It certainly means that there is some law of form common to the tangent, the chord of contact, and the locus mentioned. For (1), (2), (3) assert that the chord of contact is connected with the point from which the two corresponding tangents are drawn, and that the right line forming the locus of the inter- POLE AND POLAR. 275 section of two tangents at the extremities of a chord passing through a fixed point is connected with that point, in exactly the same way that the tangent is con- nected with its point of contact. Now a right line in the plane of a circle must be either a tangent, a chord of contact, or a locus corresponding to (2) : hence we learn that the Circle possesses the remarkable property of imparting to any right line in its plane the power of determining a point ; and reciprocally. This property is known as the principle of polar re- ciprocity ; or, as it is sometimes called, the principle of reciprocal polarity. It is fully expressed in the following twofold theorem : I. If from a fixed point chords be drawn to any circle, and tangents to the curve be formed at the extremities of each chord, the intersections of the several pairs of tangents will lie on one right line. II, If from different points lying on one right line pairs of tangents be drawn to any circle, their several chords of contact will meet in one point. The truth of (I) is evident from the equation of Art. 320 ; that of (II) appears as follows : Let Ax-{-By-\-C=Q be any right line. The chord of contact corresponding to any point x'y' of this line (Art. 319) is x f x-\- y f y = r 2 . Now the co-efficients of this equation are connected by the linear relation Ax' and the chord passes through a fixed point by Art. 117. We shall find, as we . go on, that the property just proved of the Circle is common to all conies. The reciprocal relation between a point and a right line is expressed by calling the point the pole of the line, and the line the polar of the point. 276 ANALYTIC GEOMETRY. 322. Definitions, The Polar of any point, with respect to a circle, is the right line which forms the locus of the intersection of the two tangents drawn at the ex- tremities of any chord passing through the point. Thus, LL'L" is the polar of P. The Pole of any right line, with respect to a circle, is the point in which all the chords of contact corre- sponding to different points on the line intersect each other. Thus, P is the pole of LL'L". These definitions enable us to con- struct the polar when the point is given, or the pole when we have the line. Thus, if P be the point, we draw any two chords through it, as MPN, OPQ, and the corre- sponding pairs of tan- gents, ML,NL;OL", QL". The line which joins the points L and L", in which the re- spective pairs of tangents intersect, is the polar of P. On the other hand, if LL" be the given line, we draw from any two of its points, as L and L ff , two pairs of tangents, LM, LN; L"0, L"Q, and the two correspond- ing chords of contact, MN, OQ: the point P in which the latter meet, is the required pole. We have given the construction in the form above be- cause it answers in all cases. It is evident, however, from the results of Art. 321, that when P is without the circle, its polar is the corresponding chord of contact QR; and CONSTRUCTION OF POLAR. 277 that when it is on the curve, its polar is the corresponding tangent TS. In these cases, then, our drawing may be modified in accordance with these facts. 323. From all that has now been shown, it follows that the equation to the polar of any point x'y', with respect to the circle x 2 -j- y 2 = r 2 , is x'x -j- y'y = r 2 . Now the equation to the line which joins x'y' to the center of the same circle (Art. 95, Cor. 2) is Hence, (Art. 99,) The polar of any point is perpendicular to the line which joins that point to the center of the corre- sponding circle. Corollary This property affords a method of con- structing the polar, simpler than that explained in the preceding article. For (Art. 92, Cor. 2) the distance of the polar from the center of its circle is r 2 in which (Art. 51, I, Cor. 2) l/V 2 -f y r * is the distance of the pole from the center ; hence, To construct the polar, join the pole to the center of the circle, and from the latter as origin lay off upon the resulting line a distance forming a third proportional to its ivhole length and the radius: the perpendicular to the first line, drawn through the point thus reached, ivill be the polar required. In the next four articles, we will present a few striking properties of polars. 278 ANALYTIC GEOMETRY. The condition that a point a/y shall lie upon the polar of x'y' is of course Now, obviously, this is also the condition that x f y / shall lie upon the polar of x"y" . Therefore, If a point lie upon the polar of a second, the second will lie upon the polar of the first. Corollary Hence, The intersection of two right lines is the pole of the line which joins their poles. Remark. We shall find hereafter that these properties are com- mon to all conies. 325. The distance of x'y' from the polar of a/'/' (Art. 105, Cor. 2) is P' = : and the distance of a/'y" from the polar of a/y is sV'+vV'-r 2 Hence, The distances of two points from each other s polars are pro- portional to their distances from the center of the corresponding circle. 326. Definitions Two triangles so situated with respect to any conic that the sides of the one are polars to the vertices of the other, are called conjugate triangles. Thus, in the diagram, AE C and abc are conjugate triangles with respect to the circle EQS. The corresponding sides of two conjugate triangles are those sides of the second which are opposite to the poles of the sides of the first, and A< *^ reciprocally. The corresponding an- gles lie opposite to the polars of the several vertices. Thus, AE and ab are corresponding sides; A and a, corresponding angles ; etc. A triangle whose sides are the polars of its own vertices is called self -con jugate. To draw a self-conjugate triangle, take any point P, and form its polar; on the latter, take any point T, and form its polar: this new polar (Art. 324) will pass through P, and will of CONJUGATE TRIANGLES. 279 course intersect the polar of P in some point Z; join P2 7 , and PTZ will be the required triangle. For TZ is the polar of P, and ZP of T, by construction; while PT is the polar of Z by the cor- ollar to Art. 324. Theorem. The three lines which join the corresponding vertices of two conjugate triangles meet in one point. Let the vertices of one triangle be xfy', x"y", x /// y /// ; the sides of the other will then be x'x + y'y = r\ y/'x + y"y = r\ x'"x -f y'"y = r*. For brevity, write these equations P x = 0, P xx = 0, P xxx ^= ; and let PI X/ , P/ x/ denote the results of substituting x'y* in P xx and P xxx ; P/ xx , P/, the results of substituting *"y" in P"' and P x ; and P 3 X , P 3 X/ , the results of substituting y/"y"' in P x and P /x . For the three lines joining the corresponding vertices (see diagram, Art. 326) we shall then have (Art. 108, Cor. 1) P/".P" P/ x .P /// = (Aa), p, .p't'pv'.p' = o P/ X .P X -P/ .P XX -O Now, writing the abbreviations in full, we get P l /// =P./; P/ x P/; P 2 XXX P/ x . Hence, the three equations just written vanish iden- tically when added, and the proposition is proved. Corollary By Art. 281, it follows that the intersections of the corresponding sides of two conjugate triangles lie on one right line. That is, conjugate triangles are homologous. SYSTEMS OF CIRCLES. I. SYSTEM WITH COMMON RADICAL AXIS. 328. Any two circles lying in the same plane give rise to a very remarkable line, which is called their radical axis. Its existence and its fundamental property will appear from the following analysis: An. Ge. 2T. 280 ANALYTIC GEOMETRY. Let S=Q, S'=Q be the equations to two circles, so written that in each the common co-efficient of x 2 and y 1 is unity. Then will the equation S + kS r =Q in general denote a circle passing through the points in which >S and S f intersect; for in it x 2 and y 1 will have the common co-efficient (1 -f- 7c), and obviously it will vanish when S and iS f vanish simultaneously. To this theorem, how- ever, there is one exception, namely, when Jc= 1. The resulting equation is then and, being necessarily of the first degree, denotes a right line. Moreover, this equation is satisfied by an infinite series of continuous values, whether any can be found to satisfy S and S f simultaneously or not. That is, The line S S' is real even when the two common points of the circles, through ivhich it passes, are imaginary. When the circles intersect in real points, the line is of course their common chord; and it might still be called by that name even when the points of intersection are imaginary, if we chose to extend the usage in regard to imaginary points and lines which has been so frequently employed. To avoid this apparent straining of language, however, the name radical axis has been generally adopted. The equation S=S f asserts (Art. 309, Rem.) that the tangents to S and /S", drawn from any point in its locus, are equal. Hence, The radical axis of two circles is a right line, from any point of which if tangents be drawn to both of them, the two tangents will be of equal length. 329. Writing S S' in full, and reducing, the equa- tion to the radical axis becomes (Art. 134) ' ! +f n ) } RADICAL AXIS. 281 Now the equation to the line joining the centers of the two circles (Art. 95, Cor. 1) may be written Therefore, (Art. 99, Cor.,) The radical axis of two circles is perpendicular to the line which joins their centers* Corollary. Hence, To construct the radical axis of two circles , find its intercept on the axis of x by making j = in the equation S S' 0, and through the extremity of the intercept draw a perpendicular to the line of the centers. Remark. This construction is applicable in all cases; but, when the circles intersect in real points, the axis is obtained at once by drawing the common chord. 33O. If S, S r , S" be any three circles, the equations to the three radical axes to which the group gives rise will be which evidently vanish identically when added. Hence, The three radical axes belonging to any three circles meet in one point, called the RADICAL CENTER. Corollary. We may therefore construct the radical axis as follows : Find the radical center of the two given circles witli respect to any third, and through it draw a perpendicular to the line of their centers. The annexed diagram will illus- trate the details of the pro- cess. In it, c and c' are the centers of the two given cir- cles, C the radical center, and CQC' the radical axis. 282 ANALYTIC GEOMETRY. 331. Two special cases of the radical axis deserve notice. First: We have seen that a point may be re- garded as an infinitely small circle. Hence, a point and a circle have a radical axis; that is, given any point and any circle, we can always find a right line, from any point of which if we draw a tangent to the circle and a line to the given point, the two will be of equal length. The axis is of course perpendicular to the line drawn from the given point to the center of the circle. It lies without the circle, whether the point be within or without ; for, as the radical axis always passes through the points common to its two circles, if it cut the given circle, the given point would form two consecutive points of that curve. From this it appears, that, when the given point is on the given circle, the axis is the tangent at the point. Second : If both circles to which a radical axis belongs become points, we have a line every point of which is equally distant from two given ones. Hence, the radical axis of two points is the perpendicular bisecting the dis- tance between them. We now proceed to the properties of the entire system of circles formed about a common radical axis. 332. Definition. By a System of Circles with a Common Radical Axis, we mean a system so sit- uated with respect to a fixed right line, that, if a tangent be drawn to each circle from any point in the line, all these tangents will be of equal length. The simplest case of such a system is that of the infinite series of circles which can be passed through two given points. The CIRCLES WITH COMMON AXIS. 283 diagram illustrates this case. Another is that of a series of circles touching each other at a common point. The appearance of the system when the two common points are imaginary, will be pre- sented farther on, in connection with the method of constructing the system. It follows directly from Art. 329, that all the centers of the system lie on one rig Jit line at right angles to their radical axis. 333* Equation to any member of the System. The equa- tion to any circle whose center lies on the axis of #, at a distance g from the origin, may be written (Art. 134) x* + f-2gx = r 2 g 1 ', so that the circle will cut the axis of y in real or imaginary points according as r 2 g 2 is positive or negative, the quantity Vr' 1 g' 1 representing half the intercept on the axis of y. Hence, if in the system of circles with a common radical axis, the common line of centers be taken for the axis of x, and the common radical axis for the axis of y : by putting k = the arbitrary distance of the center from the origin, and o 2 constant = r 2 & 2 , we may write the equa- tion to any member of the system and the corresponding system will cut the radical axis in real or in imaginary points according as o 2 is positive or negative. Corollary, Hence, To trace the system from the equation, assume different centers corresponding to arbitrary values of k, and from them, with radii in each case equal to y k* d= ^ 2 , describe circles. 334. The' Orthogonal Circle. From the definition of the system (Art. 332), it follows that the locus of the point of contact of the tangent drawn from any point in the common radical axis to any member of the system, is a circle. Since, then, the corre- sponding tangents of the system are all radii of this circle, tangents to this circle at the points where it cuts the several members of the system will be perpendicular to the respective tangents of the system. In other words, the circle in question cuts every member of the system at right angles,* and may therefore be called the orthogonal circle of the system. - ;: The angle between two curves is the angle contained by their re- spective tangents at the point of intersection. 284 ANALYTIC GEOMETRY. Since the center of such a circle is any point on the radical axis, there is an infinite series of orthogonal circles for every system with a common axis. But for the special purpose to which we are about to apply it, any one of these may be selected, of which we shall speak as the orthogonal circle. 335. Construction of tlic System. We can now construct the system geometrically, in all cases. The only case that needs illustration, however, is that of the system- passing through two imaginary common points. Since (Art. 334) the tangents of the orthogonal circle are all radii of the circles forming the system, we may draw any number of these circles as follows : Lay down any right line MN, and any perpendicular to it HQ. On the latter, take any point C as a center, and, with any radius (7Q, describe a circle cutting MN in the points m and n. At any points a, 6, c, d, e of this circle, draw tangents to meet MN in 1, 2, 3, 4, 5; and from these points as centers, with radii in each case equal to the corre- sponding tangent, describe circles. It is evident that the radii Ca, (76, Cfc, etc., of the fundamental circle, w'ill all be tangents to the respective circles last drawn. Hence HQ, is the common radical axis of all these circles, and the circle C-Qmn is orthogonal to them. 33G. Properties of the System: Limiting: Points From the nature of the foregoing construction and the resulting diagram, we obtain the following properties : PONCELETS LIMITING POINTS. 285 I. The circles of a system having two imaginary common points, in which the orthogonal circle cuts the line of centers in two real points m and w, exist in pairs : to every circle on the right of the radical axis, corresponds an equal one on the left, with an equally distant center. II. In a system of this character, the center of no circle can lie nearer to the radical axis than m or n. For the radius of the vari- able member of the system continually diminishes as the tangent of the orthogonal circle advances from Q toward m and n, and at m and n it vanishes. III. But the center of a member of the system may be as remote from the radical axis as we please. For the tangents of the orthog- onal circle at Q meet the line of centers at infinity. IV. Hence, the points m and n, and the axis RQ form the inferior and superior limits of the system; in short, are the corresponding limiting members of it : m and n being equal infinitesimal cirqles at equal distances from the axis, and the axis itself being the resultant of two coincident circles having equal infinite radii. V. The circle C-Qmn is drawn in the diagram to cut MN in real points ; but if the student will draw a new diagram, in which C-Qmn fails to cut MN, he will find that the circles of the resulting system all cut each other in two real points on the line RQ. Hence, a system of circles with a common radical axis intersect each other in two real or imaginary points, according as the limiting points m and n are imaginary or real. VI. The limiting points m and n are by construction equally distant from every point in RQ. Moreover, every orthogonal circle is described from some point in RQ, and cuts every member of the system at right angles. Hence every orthogonal circle passes through the limiting points. . That is, The orthogonal* of any system of circles with a common radical axis, form a complemental system, whose radical axis is the line joining the centers of the conjugate system. VII. Hence, if a system of circles intersect in two real points, the conjugate system of orthogonal circles will intersect in two imaginary ones; and reciprocally. 337 The Limiting Points by Analysis. If a system of circles cut its radical axis in two imaginary points, the equation to any member of the system (Art. 333) is z 2 + 6. Hence, the two points (y = 0, x = 6) and (y = 0. x (5) are the infinitesimal circles which we have called the limiting points of the system; for we have just shown that they have the property of Art. 336, II, and they are represented (Art. 61, Hem.) by the equation (a; =F 0)2 -f- f = ; that is, z 2 + f =F2 dx = g) cos 6 + (/'/) sin 6 = r r', (/-#) cos +(/'-/) sin = r + r' ; 288 ANALYTIC GEOMETRY. x ' q ?/ / f or, after replacing cos and sin by their values, and - (/-} (i), is the equation to the chord of contact for the direct tangents. Similarly, ($', _ r {cos (0 a) =h I/cos 1 ' (ft a) cos 2 a} sin Similarly, for the circle 8', we get r' { cos (0 a) l/cos a (0 a) cos' 2 a} P2 : sm a Therefore, p l : p 2 :: r : r 7 . Now these vectors of *S' and 8' are the segments formed by the two circles on any right line drawn through O or /. Hence, All right lines drawn through the intersection of the common tangents of two circles are cut similarly by the circles, namely, in the ratio of the radii. Remark On account of this property, the points in which the common tangents intersect are called centers of similitude. 290 ANALYTIC GEOMETRY. 343* Axis of Similitude. This name is given to a certain right line whose relation to three circles we will now develop. Let gf, g'f, g // f // be the centers of any three circles, and r, r', r" their radii. The co-ordinates of the external center of similitude for the first and second (Art. 341) will then be rg'-Sg y = r' those of the corresponding center for the second and third, (2); and those of the corresponding center for the third and first, //_ // ,,,/y r y// y" = (3). r" r Now, if we make the necessary sub- stitutions and reductions, we shall find that (1), (2), (3) satisfy the condition of Art. 112. Hence, Any three homolo- gous centers of similitude belonging to three circles lie on one right line, called the AXIS OF SIMILITUDE. Corollary. If two circles touch each other, one of their centers of similitude becomes the point of contact. Hence, If in a group of three circles the third touches the other two, the line joining the points of contact passes through a center of similitude of the two. Remark. The homologous centers of similitude are either all three external, as in the diagram, or else two internal and the third external. Corresponding to the latter case, there will of course be three different axes of similitude; making in all four such axes for every group of three circles. THE CIRCLE IN THE ABRIDGED NOTATION. 344. We have room for only a few examples of the uses to which this notation can be advantageously applied CIRCLE IN ABRIDGED NOTATION. 291 in the case of the Circle. The illustrations given will afford the beginner some further insight into the method, and the reader who desires fuller information must con- sult the larger works to which we have already referred in connection with this subject. 343. Since a, /?, y are the perpendiculars dropped from any point P to the three sides of a triangle, it is evident that the function fty sin A -f~ ya sin B -f- af3 sin C denotes (Trig., 874) the double area of the triangle formed by joining the feet of those perpendiculars; for the angle A, included between the sides ft and y, will be either the supplement of the angle between the perpendiculars ft and y, or else equal to it: and so, also, of the angles B and C. Now (Art. 236) if the point P be on the circumference of the circumscribed circle, we have ft-y sin A + ya sin B + aft sin C= ; /^2l>N that is, the triangle contained between the feet of the perpendiculars from P vanishes, and we obtain the following theorem : The feet of the perpendiculars dropped from any point in a circle upon the sides of an inscribed triangle lie on one right line. 346. The equation to the circle circumscribed about a triangle may, by factoring, be written y (a sin B + ft sin A) + aft sin C= : which shows that the line a sin B -f- ft sin A meets the circle on the line a, and also on the line ft; since, if a sin 5+ ft sin ^4 = in the above equation, we get a/3 = 0, a condition satisfied by either a = or ft = 0. But the only point in which either a or ft meets the circle is their intersection: hence a sin B + ft sin A is the tangent of the circle at aft. Now (Ex. 8, p. 222) a sin A +ft sin B is the parallel to the base of the triangle, passing through its vertex; and (Ex. 6, p. 222) this parallel, and therefore the base, has the same inclina- tion to a or ft as the tangent a sin B -f- ft sin A has to ft or a. Hence, the tangent of the circumscribed circle, at the vertex of a triangle, 292 ANALYTIC GEOMETRY. makes the same angle with either side as the base does with the other; or we have the well-known theorem: The angle contained by a tangent and chord of any circle is equal to that inscribed under the intercepted arc. 347. The equation obtained in the preceding article denotes the tangent at the vertex C of a triangle inscribed in a circle ; and analogous equations may at once be written for the tangents at the other two vertices A and B. The equation to the tangent at any point of a circle circumscribed about a given triangle (compare Arts. 236; 240, II) is a sin A j3 sin y sin C a'* = 0. 348. The equations to the tangents at the vertices of an inscribed triangle (Art. 346) may be written +-B- y + sinC^ sin (7 sin A Now (Art. 108) the line ft (1), (2), (3). a sin.4 sinB sinO passes through the intersection of (1) with 7, of (2) with a, and of (3) with /3. Hence, The tangents at the vertices of an inscribed triangle cut the opposite sides in points which lie on one right line. 349. Subtracting (2) from (3) above, (3) from (1), and (1) from (2), we get _ ___ - sin A sinj3 -4- s'mB -- sin (7 =0- ' sin C s'mA which (Art. 108) are the equations to the three lines which join the intersections of the tangents at the vertices to the intersections CIRCLE IN ABRIDGED NOTATION. 293 of the sides. Hence, (Art. 114,) The lines which join the vertices of a triangle to those of the triangle formed by drawing to its circum- scribed circle tangents at its vertices, meet in one point. Remark The theorems of the last two articles, which are illus- trated in the diagram, are evidently a particular case of homology (Art. 327) due to a pair of conjugate triangles. 35O. Radical Axis in Trilinears. The equations to any two circles, in the abridged notation, (Arts. 236, 237) are sin A sin E sin C 7ir/ , n + -j- + - + M (la + mfi + nf) = 0, sin A sin B sin C . 1/f - /7/ IQ , . A + g- -j- + M(l'a + m'p + n'r) = 0. Hence, their radical axis, S S', is denoted by la -f m p + n r = I' a + m'fi + n' r . Corollary. The radical axis of any circle and the circle circumscribed about the triangle of reference, is represented by la -\- mfi -{- nf = 0. EXAMPLES ON THE CIRCLE. 1. Find the intersections of the line 4.r+3y = 35c with the circle Also, the tangents from the origin to the circle a; 2 -f ?/ 2 6s 2^+8=0. I. Show that the equation to any chord of a circle may be written (X X ')(X- X") + (y - y>} (y - y") = ^ _|_ y t _ ^ the origin being at the center, and ary, x"y" being the extremities of the chord. ! Find the polar of (4, 5) with respect to z 2 + y 2 3x 4y = 8, and the pole of 2x + 3y = 6 with respect to (x l) 2 -f (y 2) 2 = 12! 294 ANALYTIC GEOMETRY. 4. Prove that the condition upon which Ax -f- By +(7=0 will touch the circle (x #) 2 + (y /) 2 = r' 2 is Ag + Bf+C = ~ 5. Find the length of the chord common to the two circles (x - ) 2 + (y ~ P)' 2 = r 2 , (x - W + (y - a) 2 - r 2 . Also, the equations to the right lines which touch or + y 2 = ?' 2 at the two points whose common abscissa is 1. 6. Find the equation to the circle of which y = 2x-{-3 is a tangent, the center being taken for the origin. 1. Prove that the bisectors of all angles inscribed in the same segment of a circle pass through a fixed point on the curve. 8. Given the hypotenuse of a right triangle: the locus of the center of the inscribed circle is the quadrant of which the given hypotenuse is the chord. 9. Given two sides and the included angle of a triangle: to find the equation to the circumscribed circle. 10. The locus of a point from which if lines be drawn to the vertices of a triangle, their perpendiculars through the vertices will meet in one point, is the circle circumscribed about the triangle. 11. If any chord be drawn through a fixed'point on the diameter of a circle, and its extremities joined to either end of the diameter, the joining lines will cut from the tangent at the other end, portions whose rectangle is constant. [See Art. 137.] 12. The locus of the intersection of tangents drawn to any circle at the extremities of a constant chord is a concentric circle. [See Art, 307.] 13. If a chord of constant length be inscribed in a given circle, it will always touch a concentric circle. 14. If through a fixed point O any chord of a circle be drawn, and OP be taken an harmonic mean between its segments OQ, OQf, the locus of P will be the polar of 0. 15. If through any point O of a circle, any three chords be drawn, and on each, as a diameter, a circle be described, the three circles which thus meet in O will meet in three other points, lying on one right line. EXAMPLES ON THE CIRCLE. 295 16. If several circles pass through two fixed points, their radical axes with a fixed circle will pass through a fixed point. [This example may be best solved by means of the Abridged Notation, but can be done very neatly without it.] 17. Form the equation to the system of circles which cuts at right angles any system with a common radical axis, and prove, by means of it, that every member of the former system passes through the limiting points of the latter. 18. If PQ be the diameter of a circle, the polar of P with respect to any circle that cuts the first at right angles, will pass through Q. 19. The square of the tangent drawn to any circle from any point on another is in a constant ratio to the perpendicular drawn from that point to their radical axis. 20. If a movable circle cut two fixed ones at constant angles, it will cut at constant angles all circles having the same radical axis as these two. [First prove that the angle at which two circles cut each other, is determined by the formula j)-i = jf?2 + r i _ ^Rr cos 0, in which R, r are the radii of the circles, and D the distance between their centers.] 21. Find the equations to the common tangents of the two circles What is the equation to their radical axis ? 22. If a movable circle cut three fixed ones, the intersections of the three radical axes will move along three fixed right lines which meet in one point. 23. The radical axis of any two circles that do not intersect, bisects the distances between the two points of contact correspond- ing to each of the four common tangents. 24. If through a center of similitude belonging to any two circles, we draw any two right lines meeting the first circle in the points R and Rf , S and S / respectively, and the second in r and r', s and s / : then will the chords US and rs, It'S'' and r's', be parallel ; while US and r's', JR'S' and r$, will each intersect on the radical axis. An. Ge. 28. 296 ANALYTIC GEOMETRY. 25. Find the trilinear equation to the circle passing through the middle points of the sides of any triangle, and prove that this circle passes through the feet of the three perpendiculars of the triangle, and bisects the distances from the vertices to the point in which the three perpendiculars meet. [This circle is celebrated in the history of geometry, and, on account of passing through the points just mentioned, is called the Nine Points Circle.^ Find, also, the radical axis of this and the circumscribed circle. CHAPTER THIRD. THE ELLIPSE, i. THE CURVE REFERRED TO ITS AXES. We may most conveniently begin the discussion of the Ellipse by means of the equation which we obtained in Art. 147, namely, of , f i a 2 + j* At a later point in our investigations, we shall refer the curve to lines which have a relation to it more generic than that of the two known as the axes (Art. 146), which give rise to the equation just written. THE AXES. 352. If in the above equation we make y = 0, we shall obtain, as the intercept of the curve upon the transverse axis, x = a (1); PROPERTIES OF THE ELLIPSE. 297 and, making x = 0, we get, for the intercept upon the conjugate axis, y=b (2). Comparing (1) and (2), we see that the curve cuts both axes in two points, and that in each case these two points are equally distant from the focal center, which (Art. 147) was taken for the origin. Hence we have Theorem I. The focal center of any ellipse bisects the transverse axis, and also the conjugate. Corollary. We must therefore from this time forward interpret the constants a and b in the equation as respectively denoting half the transverse axis and half the conjugate axis. Remark This theorem follows, of course, directly from that of Art. 149. We .have purposely developed it by a separate analysis, however, in order that the student may see the consistency of the analytic method. 353. If in the equation of Art. 147, which may be written b we suppose x > a or < a, the corresponding values of y are imaginary; so that no point of the curve is farther from the origin, either to the right or to the left, than the extremities of the transverse axis. Now (Art. 147), for the distance from the origin to either focus, we have 298 ANALYTIC GEOMETRY. Hence, c can not be greater than a, though it may approach infinitely near to the value of a, as b dimin- ishes toward zero. Therefore, Theorem II. The foci of any ellipse fall within the curve. 354. Moreover, a c measures the distance of either focus from the adjacent vertex ; while the distance of either from the remote vertex = a -f c. We accord- ingly get Theorem III. The vertices of the curve are equally distant from the foci. 355. From Art. 352, the length ' of the transverse axis = 2a. But (Art. 147) 2a = the constant sum of the focal radii of any point on the curve. That is, Theorem IV. The sum of the focal radii of any point on an ellipse is equal to the length of its transverse axis. Corollary. This property gives rise to the following construction of the curve by points : Divide the transverse axis at any point M between the foci F r and F. From F r as a center, with a radius equal to the segment MA', strike A ' two small arcs, one above the axis, and the other below it. Then from F, with the remaining segment MA as radius, strike two more arcs, intersecting the two former in P and P' : these points will be upon the required ellipse ; for F'P -f FP= AA' = F'P' -f FP'. By using the radius MA 1 from F, and MA from F', two more points, P" and P'", may be found ; so that every division of the transverse axis will determine four points CONSTRUCTION OF THE FOCI. 299 of the curve. Thus, the point of division N will give rise to the four points Q, Q f , Q", Q" f . When enough points have been found to mark the outline of the curve dis- tinctly, it may be drawn through them ; if necessary, with the help of a curve-ruler. It is evident that this construction implies that the transverse axis and the foci are given. 356. The abbreviation b' 2 ~a 2 c 1 adopted (Art. 147) for the Ellipse, gives us b=l(a + c ) (a c). Hence, attributing to , 6, c the meanings now known to belong to them, we have Theorem V, The conjugate semi-axis of any ellipse is a geometric mean between the segments formed upon the transverse axis by either focus. Corollary. Transposing in the abbreviation above, we have b 2 + c 2 = a 2 . But, from the diagram, b 2 -f- c 2 = F'B 2 =FB 2 . Therefor a or < a, the corresponding values of y are imaginary; and, if we suppose y^>b or < b, the corresponding values of x are imaginary. III. It is continuous in extent. For, between the limits # a and x a, all the values of y are real. IV. It is symmetric to both axes. For, corresponding to every value of x between the limits a and a, the two values of y are numerically equal with opposite signs; and the same is true of the values of x corresponding to any value of y between b and b. BISECTORS OF PARALLEL CHORDS. 305 DIAMETERS. 362. Equation to any Diameter. We are re- quired to find the equation to the locus of the middle points of any system of parallel chords in an ellipse. Let xy be the variable point of this locus, 6' the common inclination of the bisected chords, and x'y' the point in which any chord of the system cuts the curve. Then (Art. 101, Cor. 3) x' = x I cos 6', y 1 = y I sin 6'. But x'y' is a point on the curve ; hence (Art. 147) (x lcosd 1 ) 2 , (y Zsin^) 2 __ 1 a* V That is, to determine the distance I between xy and x'y', we get ' +b 2 xcosd')l Now, xy being the middle point of any chord, the two values of I must be numerically equal with opposite signs. Therefore (Alg., 234, Prop. 3d) the co-efficient of I vanishes, and we obtain, as the required equation, 72 y -- 2 x cot 0'. v Q> Corollary. Let = the inclination of any diameter to the transverse axis. The co-efficient of x in the equation just found is equal (Art. 78, Cor. 1) to tan 6. Hence, as the condition connecting the inclination of any diameter with that of the chords which it bisects, tan tan 0'= - . 306 ANALYTIC GEOMETRY. 363. The equation to any diameter conforms to the type y = mx. Therefore (Art. 78, Cor. 5) we have Theorem X. Every diameter of an ellipse is a right line passing through the center. Corollary. Since 6' in the foregoing condition is ar- bitrary, 6 is also arbitrary. Hence, the converse of this theorem is true ; that is, Every right line that passes through the center of an ellipse is a diameter. 364. If we eliminate between the equation to the Ellipse and that of any diameter, the roots of the result- ing equation will be ~ which, it is evident, are necessarily real. Hence, Theorem XI. Every diameter of an ellipse cuts the curve in two real points. 365. Length of any Diameter. This is of course double the radius vector given by the central polar equa- tion (Art. 150), namely, 1 e 2 cos 2 6 Hence, given the inclination #, the length of the corre- sponding diameter can at once be found. 366. From the preceding formula, it is evident that the diameter is longest when 6 = 0, and shortest when 6 = 90. That is, Theorem XII. In every ellipse, the transverse axis is the maximum, and the conjugate axis the minimum diameter. Remark. For this reason, the transverse axis is called the axis major, and the conjugate, the axis minor. CONJ UGA TE DIA METERS. 307 *$G7. Moreover, since 6 enters the foregoing formula by the square of its cosine, the value of ft is the same for and 7i 0. Hence, Theorem XIII. Diameters which make supplemental angles with the axis major of an ellipse are equal. Corollary. The converse of this is also given by the formula ; so that, having the curve, we can always construct the axes as follows: Draw any two pairs of parallel chords, and, by means of A '( them, two diameters DQ, D'C: their intersection C (Art. 363) will be the center. From C describe any circle cutting the curve in four points P, ft P, Q f . The diameters PP, QQ' will then be equal, and, by the converse of the theorem above, the two bisectors of the angles between them will be the axes. These bisectors may be drawn most readily by forming the chords PQ, QP f which subtend the angles : they will be perpendicular to each other (Art. 317, Cor.), and their parallels through C will be the bisecting axes required. 3GS. Let and 6' be the inclinations of any two diameters to the axis major. Then the condition that the first shall bisect chords parallel to the second (Art. 362, Cor.) is 7,2 But this is also the condition upon which the second would bisect chords parallel to the first. Hence, Theorem XIV. If one diameter of an ellipse bisects chords parallel to a second, the second bisects chords parallel to the first. 308 ANALYTIC GEOMETRY. 369. Two diameters of an ellipse which are thus related, are called conjugate diameters, as in the case of the Circle. The re-appearance of this relation in connection with the Ellipse, gives occasion to define what is meant by ordinates to a diameter. Definition. The Ordinates to a Diameter are the right lines drawn from the curve to such diameter, parallel to its conjugate; or, they are the halves of the chords which the diameter bisects. Corollary, Hence, To construct a pair of conjugate diameters, draw any diameter D'D, and any two chords MN, PQ par- allel to it. Join the middle points of the latter by the line S'S, which will be the required conjugate. When the center C is not given, the con- struction is effected by drawing S'S through the middle of D'D, parallel to the two chords (double ordinates to D'D] by the aid of which this first diameter must in such a case be determined. 370. Equation of Condition for Conjugate Di- ameters. This, as we have already seen (Art. 368), is 7,2 371. This condition, since it shows that the tangents of inclination belonging to any two conjugate diameters have opposite signs, indicates that one of two conjugates makes an acute angle with the axis major, and the other an obtuse. Now the axis minor makes a right angle with the axis major; hence, Theorem XV. Conjugate diameters of an ellipse lie on opposite sides of the axis minor. EXTREMITIES OF CONJUGATES. 309 372. Equation to a Diameter conjugate to a Fixed Point. For brevity, we shall say that a diam- eter is conjugate to a fixed point, when it is conjugate to the diameter drawn through such point. If, now, x'y' be any fixed point, the diameter drawn through it (Art. 95, Cor. 2) will be y'x-x'y = Q (I). The equation to the conjugate will be of the form y = x tan 6' (2), in which (Art. 370) tan 6 f is determined by the condition tan tan 0' = - . a 2 But (1), tan# =y' : x'. Hence, the equation sought is Corollary, The equation to the diameter conjugate to that which passes through the point (a, 0) is evidently # 0. But the diameter through (#, 0) is the axis major, while x=Q denotes the axis minor. Hence, The axes of an ellipse constitute a case of conjugate diameters. It is from this fact, that the axis minor derives its name of the conjugate axis. 373* Problem. Given the co-ordinates of the extremity of a diameter, to find those of the extremity of its conjugate. Let x'y r be the extremity of the given diameter. The required co-ordinates, found by eliminating between the equation of Art. 372 and that of the Ellipse, are . ay' bx f *< = T> *:T*T' 374. The expressions just obtained, transformed into the pro- portions x c : if : : a : b, xf : y c : : a : 6, give us 310 ANALYTIC GEOMETRY. Theorem XVI The abscissa of the extremity of any diameter is to the ordinate of the extremity of its conjugate, as tJie axis major is to the axis minor. 375. Squaring the second expression of Art. 373, adding y f2 7 and remembering that x'y' satisfies the equa- tion to the Ellipse, we find Hence, in ordinary language, we have Theorem XVII. The sum of the squares on the ordinates of the extremities of conjugate diameters is constant, and equal to the square on the semi-axis minor. Remark. The student may prove the analogous prop- erty; The sum of the squares on the abscissas of the extremities of conjugate diameters is constant^ and equal to the square on the semi-axis major. S76. Problem. To find the length of a diameter in terms of the abscissa of the extremity of its conjugate. Let a! be half the length required. Then, x'y 9 being the extremity of a 1 ', we have (Art. 51, 1, Cor. 2) a' 2 = x' 2 -f y' 2 . Substituting for x r and y' from Art. 373, we get But x c and y c satisfy the equation to the Ellipse ; hence, n n ( a t _ r 2\ i & r 2 ~2 a P 2 ^ ) + -2 ^c - -jr- * c Therefore (Art. 151), for determining the required length, we have a' 2 = a 2 e 2 x c 2 . By a precisely similar analysis, we should find b f2 a 2 e z x' 2 . INTERCEPT ON FOCAL RADIUS. 311 Comparing the expressions last found with the formula of Art. 360, we obtain Theorem XVIII. The square on any semi-diameter of an ellipse is equal to the rectangle under the focal radii drawn to the extremity of its conjugate. 378. The result of Art. 376 leads to a noticeable property of the Ellipse, which we may as well develop in passing. Let x / y / be the extremity D of any diameter DJy '. The equation to the conjugate diameter S'S (Art. 372) will be The equation to Z>F, which joins D to the focus, (Arts. 95, 151) may be written Eliminating y between (1) and (2), we obtain (a 2 / 2 + I V 2 I* ae x') x = a*e y' 2 ; or, since x / and y / satisfy the equation to the Ellipse, (a 2 /- 2 - V ae x'} x = V ae (a 2 a/ 2 ). Hence, for the co-ordinates of M, we have _ ' a-ex* ' ~ a-ex / The length of DM (Art. 51,1, Cor. 1) will therefore be found from f_~v ( a ~~ ex : Reducing the last expression, we obtain , 2 _ 2ae x Now x^+y / ' i =a /2 ^ (Art. 376) a 2 eV = (Art. 375, Rem.) a 2 e~ (a 2 x'' 2 ). Hence, 6 = DM=a: a relation which may be expressed by Theorem XIX The distance from the extremity of any diameter to its conjugate, measured upon the corresponding focal radius, is constant, and equal to the semi-axis major. 312 ANALYTIC GEOMETRY. 3719. Let I)' denote the length of the semi-diameter conjugate to that whose extremity is x f y f . Then (Art. 51, I, Cor. 2) we shall have V = x* + y.'= (Art. 147) jiM-jJ (a 2 -*;-). Reducing, and applying the abbreviation of Art. 150, we get b' 2 = b 2 +e 2 x 2 . Now (Art. 376) a' 2 = a 2 e 2 x c 2 . Adding this expression to the preceding, we find a' 2 -j- b' 2 = a 2 + b\ giving us the following important property : Theorem XX. The sum of the squares on any two con- jugate diameters of an ellipse is constant, and equal to the sum of the squares on the axes. 3SO. Angle between two Conjugates. Let

= d r 0; whence (Trig., 845, in) sin

J 381. The expression just found, by a single trans- formation gives the relation a'b f sin

. ECCENTRIC ANGLE. 317 It is evident from the diagram that CM== CQcosQCM, and MP = PR sin PRM= CQ (MP : MQ) sin QCJIf = (Art. 359) CB sin QCM. That is, if x'y' be any point P of an ellipse, x / = a cos 0, 3/ x = i sin 0. By means of this relation, we can always express a point on an ellipse in terms of the single variable . Thus, the equation to the tangent at x'y* becomes - COS0+ V sin 6 = I, a b 388. Problem. If a tangent to an ellipse passes through a fixed pointy to find the co-ordinates of contact. Let x'y' be the required point of contact, and x"y" the given point. Then, since x"y" must satisfy the equation to the tangent, and x'y' the equation to the curve, we shall have the two conditions x'x" y'y" _^ x* y* ~ r ~ - ~ ~ '- Solving these for x' and y', we find \At V *X .^L_ V*/MfV*V JVCU Cb X = ~ b 2 x m 4- aV /T ~ " ' Corollary. From these values it appears, that from any given point tivo tangents can be drawn to an ellipse : real when b 2 x" 2 + a?y n<2 > a 2 b 2 , that is, when the point is without the curve ; coincident when b 2 x ff2 -f- a 2 y" 2 = a 2 b 2 , that is, when the point is on the curve ; imaginary when b 2 x" 2 -f a?y" 2 < 2 6 2 , that is, when the point is within the curve. 318 ANALYTIC GEOMETRY. 389o The equation to the tangent at x'y f (Art. 385) is x'x y'lf -f ir = i C 1 ) ; a" or and the equation to the diameter conjugate to that whose extremity is x'y' (Art. 372) is x x i y y ' A /o\ ~tf ~ ~~ ^ '" Now (Art. 98, Cor.) the lines (1) and (2) are parallel. That is, Theorem XXIII. The tangent at the extremity of any diameter of an ellipse is parallel to the conjugate diameter. Corollary. If we replace x' and y' by x 1 and y f 9 equations (1) and (2) still satisfy the condition of paral- lelism. Hence, Tangents at the extremities of a diameter are parallel to each other. Remark. If the student will form the equation to the parallel of (2) passing through x'y' 9 he will find that it is (1). In other words, the converse of our theorem is also true, and we can always construct a tangent at any point JP 9 by drawing the diameter PD and its conjugate, and making LPM parallel to the latter. In this way we can form the circumscribed parallelogram corresponding to any two diameters PD, QD'; and we here find the promised justification of the statement (Art. 381, Rem.), that the parallelogram under two conjugates is circumscribed, since its sides must be parallel to the conjugates, and therefore be tangents to the curve at their extremities. DIRECTION OF TANGENT. 319 39O. Let PT be a tangent to an ellipse at any point P, and let FP, F'P be the focal radii of contact. The equations to these lines may be written (Arts. 385, 95) '^-*' P=ir : formulse which, in certain cases, are more useful than the preceding. * This locus may be obtained even more readily, as follows : The equations to the two tangents (Arts. 386, Cor. ; 99, Cor.) may be written y mx = r m 2 a 2 + 6 2 , my + x VHaMP + a' 2 . Squaring and adding these expressions, we eliminate m, and get a 2 + 52. 324 ANALYTIC GEOMETRY. SOT. Comparing the two results of Art. 396, we get p : p' : : p : p', a relation expressed in ordinary terms by Theorem XXVIII, The focal perpendiculars upon any tangent of an ellipse are proportional to the adjacent focal radii of contact. SOS. Multiplying together the values of p and p', and observing Art. 377, we obtain P p'=b\ which is the algebraic expression of Theorem XXIX. The rectangle under the focal perpen- diculars upon any tangent is constant, and equal to the square on the semi- axis minor. SOO. The equation to any tangent of an ellipse (Art. 386, Cor.) being y mx = v ma? -j- 6 2 , that of the focal perpendicular, which passes through (1/V - b'\ 0), may be written (Art. 103, Cor. 2) my -f- x V a" b 2 . Squaring these equations, and adding them together, we obtain as the equation to the locus of the point in which the focal perpendicular meets the tangent. Hence, (Art. 136,) Theorem XXX. The locus of the foot of the focal per- pendicular upon any tangent of an ellipse, is the circle circumscribed about the curve. Corollary. From this property, we obtain the follow- ing method of constructing a tangent to any ellipse, GENERIC CONSTR UCTION FOR TANGENT. 325 a method which deserves special attention, because it is applicable alike to all the Conies, and holds good whether the point through which the tangent is drawn be without the curve or upon it. To dratv a tangent to an ellipse through any given point. Join the given point P with either focus F, and upon PF as a diameter describe a semicircle. Then through , where this semicircle cuts the circumscribed circle, draw PQ: it will touch the ellipse at some point T. For the angle FQP is inscribed in a semicircle, and Q is therefore the foot of the focal perpendicular upon PQ. In case, as in the second dia- gram annexed, the point P is on the ellipse, the circle described on PF will be found to touch the circumscribed circle at Q (see Ex. 8, p. 359). The con- struction still holds, however ; for the point of contact Q must lie on the line joining the centers of the auxiliary and circumscribed circles, and may therefore be found at once by joining C with the middle point of PF, and producing the line thus formed until it meets the cir- cumscribed circle. Remark. It is obvious that the ordinary method of drawing a tangent to a circle through a given point (Geom., 230), is only a particular case of the method here described: the case, namely, where the two foci of the ellipse become coincident at (7, when of course the ellipse becomes identical with the circumscribed circle. We have seen (Art. 388, Cor.) that two tangents can, in general, be drawn to an ellipse from a given point P; and the construction evidently corroborates this, since the auxiliary circle PQF must in general cut the circumscribed circle in two points. 326 ANALYTIC GEOMETRY. 4OO. From what has been shown in the preceding article, it follows that every chord drawn through the focus of an ellipse to meet the circumscribed circle is a focal perpendicular to some tangent of the ellipse. Now it is evident, that, a circle being given, any point within it may be considered as the focus of an inscribed ellipse. We have, then, Theorem XXXI. If from any point within a circle a chord be drawn, and a perpendicular to it at its extremity, the perpendicular will be tangent to the inscribed ellipse of which the point is .a focus. Corollary This is of course equivalent to saying that the inscribed ellipse is the envelope of the perpendicular. Advantage may be taken of this principle, to construct an ellipse approximately by means of right lines; for it is evident that by taking the perpendiculars sufficiently near together, we can approach the line of the curve as closely as we please. The diagram presents an example of this method. 4Olo If the student will now form, by the method of Art. 108, Cor. 1, the equation to the diameter CQ, that is, the equation to the line join- ing the origin to the inter- section of the tangent with its focal perpendicular (Art. 103, Cor. 2) a 2 y f x b 2 x'y = a 2 cy', he will find that it may easily be reduced to the form y'x-(x' + c)y = (CQ). POINT OF CONTACT. NORMAL. 327 Now the equation to the focal radius of contact F'T, which passes through the focus ( c, 0) and the point of contact x'y 1 ', (Art. 95) may be written y'x (x> + c) y + cy' = (F* T). But (Art. 98, Cor.) these equations show that CQ and F'T are parallel; and, by like reasoning, the same may be proved of CQ' and FT. Hence, Theorem XXXII. The diameters which pass through the feet of the focal perpendiculars upon any tangent of an ellipse, are parallel to the alternate focal radii of contact. Corollary, The equations to CQ and F'T evidently involve the converse theorem, Diameters parallel to the focal radii of contact meet the tangent at the feet of its focal perpendiculars. Hence, if in drawing a tangent through a given point P it becomes desirable, after obtaining (Art. 399, Cor.) the foot Q of the focal per- pendicular, to find the point of contact, we can do so by merely drawing F'T parallel to CQ. By combining this property with Arts. 389, 399, we learn that the distance between the foot of the perpen- dicular drawn from either focus to a tangent, and the foot of the perpendicular drawn from the remaining focus to the parallel tangent, is constant, and equal to the length of the axis major. THE NORMAL. 4O2. Equation to the Normal. The expression for the perpendicular drawn through the point of contact to the tangent An. Ge. 31. 328 ANALYTIC GEOMETRY. according to Art. 103, Cor. 2, is Clearing of fractions, dividing through by x'y', and putting (Art. 151) c 2 for a 2 5 2 , we may write the equation sought 4O3. If we seek the angle tp made by the normal with the left-hand focal radius of contact F'T, whose equation is y'x (x 1 -f- c) y -f- cy f 0, we get (Art. 96, Cor. 1) (x f + g) Similarly, for the angle

, y -f- y' = (x -+- #') tan ^'. Hence, at the intersection P, we shall have the condition y 2 ?/ /2 = (a; 2 x r2 ) tan ^ tan ___ '_ f) - V _ / 1 ecos# 1-f-ecos^' the constant a is the semi-axis major of the given ellipse, and the constant e its eccentricity. Further : If we replace 1 er by its value (Art. 151) 6 2 : # 2 , we may write these equations ? = a b 2 1 ecostf a 1-j-ccos^ Now (Art. 428) b 2 : a is half the parameter of the curve. e cos the upper or lower sign being used according as the right-hand or left-hand focus is taken for the pole. 444. Polar Equation to the Tangent. We shall obtain this most readily by transforming the equation from rectangular to polar co-ordinates, at the same time removing the pole to the left-hand focus, whose co-ordi- nates are ae, 0. We have (Art. 58) a/ = //cos 6' ae* x = p cos 6 ae, y f = p r sin 6 f , y = p sin 6. Hence, the transformed equation is (f* f cos 6' ae) (p cos , p'p COS d COS 6* 1. But p'O' is on the curve : therefore, (Art. 443,) e 356 ANALYTIC GEOMETRY. Substituting these values in the first and second terms of the last equation respectively, and reducing, (cosd f e) (f) cosd ae)-\-p sin# sin d'=a (1 ecos# r ) (1). Therefore, (Trig., 845, iv,) the required equation is _ a (I-* 2 ) cos (6 6'}e cos 6 ' Corollary, Since the equation to the diameter conju- gate to x'y' (Art. 372), differs from that of the tangent at x'y' only in having for its constant term, we have, by putting instead of a (1 e cos 6') in (1), and reducing, ae (cos 6' e) P ~~ = cos (0 6'}e cos 6 ' as the polar equation to the, diameter conjugate to that which passes through p'6'. These polar equations afford a proof of Theorem XIX so much simpler than the one given in Art. 378, that we present it here expressly to invite comparison. Let p / / be the extremity D of any diameter DIY , The equation to its conjugate SS' is ae (cos 0* e) cos (# 6 / ) e cos In this, making d d*, we obtain S' Now, from the polar equation to the curve, as given in Art. 152, 1 e cos (? Hence, AREA OF THE ELLIPSE. 357 Remark. The geometric proof of this theorem is perhaps still simpler. Thus: The normal DN bisects the angle F'DF (Art. 403), and is perpendicular to SS' (Art. 389). Hence, the triangle MDL is isosceles, and I)M=DL. But, by drawing a parallel to LM through F, it becomes apparent, since CF / =CF^ that F'M FL. Therefore, DM + DL = F'D F'M + FD + FL =F'D + FD. That is, (Art. 355), 2DM=A'A .-. DM=a. We have given these three proofs of the same proposition for the purpose of illustrating the importance of a proper selection of methods, even in the elementary work of the beginner. In attempting to es- tablish theorems, several methods are often available to the student, and he should select that which will combine rigor, simplicity, and elegance, in the highest degree. While analytic processes are gen- erally able to satisfy this condition the best, it nevertheless sometimes happens, that the proof from pure geometry is superior. In such a case, of course, the latter is to be preferred. A' M iv. AREA OF THE ELLIPSE. 446. The area of any ellipse whose axes are given, may be determined by the following application of the geometric method of infinitesimals. A A being the axis major of the curve, describe a circle upon it, and divide it into any number of equal parts at Z, M, N, etc. Erect the ordinates ZP, M Q, NR, etc., cutting the ellipse in p, q, r, etc. Join PQ, pq : then, since Lp : LP= Mq : MQ = b : a (Art. 359), the trapezoids Lq, LQ are in the ratio b : a. Now the same must be true of any two corresponding trapezoids: therefore, the area of the polygon inscribed in the ellipse is to the area of the corresponding polygon inscribed in the circle as b is to a. Hence, as this pro- portion holds true, no matter how many sides the inscribed 358 ANALYTIC GEOMETRY. polygons may have, and as we can make the number of sides as great as we please by continually subdividing the axis major, in the limiting case where the polygons vanish into their respective curves, we shall have area of ellipse : area of circumscribed circle = b : a. Now (Geom., 500) the area of the circle no, 2 . Therefore, putting A = the required area, That is, Theorem LIII. The area of an ellipse is equal to n times the rectangle under Us semi-axes. Corollary. Since rtab = i/7ra 2 . ;r6 2 , we have the addi- tional property : The area of an ellipse is a geometric mean between the areas of its circumscribed and inscribed circles. EXAMPLES ON THE ELLIPSE. 1. Find the equations to the tangent and normal at the extremity of the latus rectum, and determine the eccentricity of the ellipse in which -the normal mentioned passes through the extremity of the axis minor. 2. Find the equations to the diameter passing through the ex- tremity of the latus rectum, and the chord joining the extremities of the axes ; and determine the eccentricity of the ellipse in which these lines are parallel. 3. A point P is so taken on the normal of an ellipse, that its distance from the foot of the normal is in a constant ratio to the length of the normal : find the locus of P, and prove that when P is the middle point of the normal, its locus is an ellipse whose eccen- tricity e' is connected with that of the given one by the condition (1 <5 /2 )(l+e 2 ) 2 =l e\ 4. Prove that two ellipses of equal eccentricity and parallel axes can have only two points in common. Also, show that if three such ellipses intersect, their three common chords will meet in one point. EXAMPLES ON THE ELLIPSE. 359 5. If two parallels bo drawn, one from an extremity A of the axis major, and the other from the adjacent focus, meeting the axis minor in M and JV, the circle described from N as a center, with a radius equal to MA, will either touch the ellipse or fall entirely outside of it. 6. A circle is inscribed in the triangle formed by the axis major and any two focal radii : find the locus of its center. 7. Find a point on an ellipse, such that the tangent there may be equally inclined to both axes. Also, a point such that the tangent may form upon the axes intercepts proportional to them. 8. Prove that the circle described on any focal radius of an ellipse touches the circumscribed circle. 9. From the vertex of an ellipse a chord is drawn to any point on the curve, and the parallel diameter is also drawn: the locus of the intersection of this diameter and the tangent at the extrem- ity of the chord, is the tangent at the opposite vertex. 10. From the center of an ellipse, two radii vectores are drawn at right angles to each other, and tangents to the curve are formed at their extremities : the tangents intersect on the ellipse 11. Two ellipses have a common center, and axes coincident in direction, while the sum of the squares on the axes is the same in both: find the equation to a common tangent. 12. The ordinate of any point P on an ellipse is produced to meet the circumscribed circle in Q : the focal perpendicular upon the tangent at Q is equal to the focal distance of P. 13. The lines which join transversely the foci and the feet of the focal perpendiculars on any tangent, intersect on the corresponding normal, and bisect it. 14. If a right line drawn from the focus of an ellipse meets the tangent at a constant angle 0, the locus of its foot is a circle, which touches the curve or falls entirely outside of it according as cos 6 is less or greater than e. 15. When the angle between a tangent and its focal radius of contact is least, the radius = a; and when the angle between a tangent and its central radius of contact is least, the radius = J V ' d l -f- P. 360 ANALYTIC GEOMETRY. 16. The locus of the foot of the central perpendicular upon any tangent to an ellipse, is the curve 17. The locus of the variable intersection of two circles described on two conjugate semi-diameters of an ellipse, is the curve 18. If lines drawn through any point of an ellipse to the extrem- ities of any diameter meet the conjugate CD in M and N, prove that CM.CN = CD-. 19. In an ellipse, the rectangle under the central perpendicular upon any tangent and the part of the corresponding normal inter- cepted between the axes, is constant, and equal to a 2 I'*. 20. The condition that two diameters of an ellipse may be conju- gate, referred to a pair of conjugates as axes of co-ordinates, is b n tan 6 tan & = -- ^ 21. Normals at P and D, the extremities of conjugate diameters, meet in Q: prove that the diameter CQ is perpendicular to PD, and find the locus of its intersection with the latter. 22. Given any two semi-diameters, if from the extremity of each an ordinate be drawn to the other, the triangles so formed will be equal in area. Also, if tangents be drawn at the extremity of each, the triangles so formed will be equal in area. 23. Find the locus of the intersection of the focal perpendicular upon any tangent with the radius vector from the center to the point of contact. Also, the locus of the intersection of the central perpen- dicular with the radius vector from the focus to the point of contact. 24. The equi-conjugates being taken for axes, find the equation to the normal at P, and prove that the normal bisects the line joining the feet of the perpendiculars dropped from P upon the equi-conjugates. 25. Find the locus of the intersection of tangents drawn through the extremities of conjugate diameters. 26. Putting p, p' to denote the focal radii of any point on an ellipse, and for its eccentric angle, prove that EXAMPLES ON THE ELLIPSE. 361 27. Express the lengths of two conjugate semi-diameters in terms of the eccentric angle, namely, by a /2 = a~ cos 2 9 + I 2 sin 2 0, b /2 = a 2 sin 2 -\- Ir cos 2 <^>. 28. The ordinate MP of an ellipse being produced to meet the circumscribed circle in Q, find the locus of the intersection of the radius CQ with the focal radius FP. 29. Normals to the ellipse and circumscribed circle pass through the points P and Q just mentioned, and intersect in R: find the locus of E. [First show that the equation to the normal of the ellipse is _f?L_Jy_ = c ,, cos (j) sin (f> being the eccentric angle of the point of contact.] 30. Prove that the area of any parallelogram circumscribed about an ellipse may be expressed by 4ab area = where 0, tf are the eccentric angles corresponding to the points of contact of the adjacent sides. Show that this area is least when the points of contact are the extremities of conjugates. 31. Upon the axis major of an ellipse, two supplemental chords are erected, and perpendiculars are drawn to them from the vertices : show that the locus of the intersection of these perpendiculars is another ellipse, and find its axes. 32. Let CP, CD be any two conjugate semi-diameters : the supplemental chords from P to the extremities of any diameter are parallel to those from D to the extremities of the conjugate. 33. The rectangle under the segments of any focal chord is to the whole chord in a constant ratio. 34. The sum of two focal chords drawn parallel to two conjugate diameters is constant. 35. The sum of the reciprocals of two focal chords at right angles to each other is constant. 36. To a series of confocal ellipses, tangents are drawn from a fixed point on the axis major : find the locus of the points of contact. 362 ANALYTIC GEOMETRY. 37. Tangents to two confocal ellipses are drawn to cut each other at right angles : the locus of their intersection is a circle concentric with the ellipses. 38. Find the sum of the focal perpendiculars upon the polar of x't/. 39. The intercept formed on any variable tangent by two fixed tangents, subtends a constant angle at the focus. Also, the line which joins the focus to the point in which any chord cuts the directrix, is the external bisector of the focal angle subtended by the chord. 40. One vertex of a circumscribed parallelogram moves along one directrix of an ellipse : prove that the opposite vertex moves along the other, and that the two remaining vertices move upon the circumscribed circle. CHAPTER FOURTH. THE HYPERBOLA, i. THE CURVE REFERRED TO ITS AXES. 447. In discussing the Hyperbola by means of its equation (Art. 167) f! 2! 1 a 2 b* we shall avoid the repetition of much that has already been said in connection with the Ellipse, by considering that the similarity of the equations to these two curves makes most of the arguments used in the foregoing pages at once applicable to the Hyperbola. We shall therefore avail ourselves of the principle developed in the corollary to Art. 167, and, for details, shall refer the student to the proper article in the preceding Chapter. For the sake of bringing out the antithesis between the Ellipse and Hyperbola, alluded to in the Remark PROPERTIES OF THE HYPERBOLA. 363 under Art. 167, the theorems of this Chapter are num- bered like the corresponding ones of the preceding. THE AXES. 448. Making y and x successively equal to zero in the equation of Art. 167, we get, for the intercepts of the Hyperbola upon the lines termed its axes, x=~-:a, y =-- b l/ ::r L Hence, the curve cuts the transverse axis in two rea points equally distant from the focal center, and the conjugate axis in tw r o imaginary points situated on opposite sides of that center at the distance bi/--I. Assuming, then, the conjugate axis to be measured by the imaginary unit 1/--1, we may infer Theorem I. The focal center of any hyperbola bisects the transverse axis, and also the conjugate. Corollary. In the light of the analysis leading to this theorem, we should therefore interpret the constants a and b in the equation * 2 f_, a 2 " b 2 ~ as respectively denoting half the transverse axis and half the modulus of the imaginary conjugate axis. 449. At the outset (see Art. 166), we arbitrarily used the phrase conjugate axis to denote the whole line drawn through the center C at right angles to the transverse axis A' A. We now see that the phrase in strictness means an imaginary portion of that line, of the length = 2&T/ :=: 1. But, as was promised in Art. 166, we shall now show 364 ANALYTIC GEOMETRY. that a certain real portion of this line has a most signifi- cant relation to the Hyperbola, on account of which it is by universal consent taken for the conjugate axis. This relation depends on the companion-curve called the conju- gate hyperbola, whose equation we developed in Art. 168. By referring to the close of Art. 168, it will be seen that the equations to two conjugate hyperbolas, when referred to their common center and axes, differ only in the sign of the constant term. The equation to the curve whose branches lie one above and the other below the line A 1 A in the diagram, may therefore be written - y -- _1 a" ~ b 2 ~ Now, if in this we make x = 0, we get Hence, the conjugate hyperbola has a real axis, identical in direction with the imaginary axis of the primary curve, whose length is the same multiple of 1 that the length of the imaginary is of V 1. Moreover, it is found that this real axis of the conjugate hyperbola, when used in- stead of the imaginary one of the primary curve, enables us to state the properties of the latter in complete analogy to those of the Ellipse. It is customary, therefore, to lay off CB, CB 1 each equal to 6, and to treat the resulting line B'B as the conjugate axis of the original hyperbola, though in fact it is only the transverse axis of the conju- gate curve. Adopting this convention, the statement in Theorem I is to be taken without reference to imaginary quantities, and the constants a and b in the equations b 2 CONSTRUCTION OF CURVE AND FOCI. 365 are henceforth to be interpreted as denoting the semi-axes of the curve. 45O. If in the equation of Art. 167, which may be itten written b a 2 , we suppose x #, the corresponding values of y are imaginary ; so that no point of the curve is nearer to the origin, either on the right or on the left, than the extremities of the transverse axis. But (Art. 171), for the distance from the origin to either focus, we have c 2 = a 2 + b 2 . Hence, c can not be less than &, though it may approach infinitely near to the value of a, as b diminishes toward zero. Therefore, Theorem II, The foci of any hyperbola fall without the curve. 451. Moreover, c a measures the distance of either focus from the adjacent vertex; while the distance of either from the remote vertex = c -f- a. Hence, Theorem III. The vertices of the curve are equally distant from the foci. From Art. 448, the length of the transverse axis = 2a. But (Art. 167) 2a = the constant difference of the focal radii of any point on the curve. That is, Theorem IV, The difference of the focal radii of any point on an hyperbola is equal to the length of its trans- verse axis. Corollary. We may therefore construct the curve by points as follows: From either focus, as F', lay off 366 ANALYTIC GEOMETRY. F'M equal to the transverse axis. Then from F' as a center, with any radius F 1 R greater than F'A, describe two small arcs, one above the axis, and the other below it. From the remaining focus F as a center, with a radius MR, describe two other arcs, inter- secting the former in P and P r : these points will be upon the required hyperbola; for F'PFP=F'R MR = A' A = F'P' FP 1 . By using the radius F'R from F, and MR from F f , two points, P" and P'", may be found upon the second branch of the curve. The operation must be repeated until the outline of the two branches is dis- tinctly marked, when the curve may be drawn through the points determined. The conjugate curve may be formed in the same way, at the same time, if desired. 453. The abbreviation b 2 =c 2 a 2 adopted (Art. 167) for the Hyperbola, gives us Hence, attributing to a, b, c the meanings now known to belong to them, we have Theorem V. The conjugate semi-axis of any hyperbola is a geometric mean between the segments formed upon the transverse axis by either focus. Corollary. Transposing in the abbreviation above, we get c 2 a 2 -f b 2 . But, from the diagram, d> + b 2 = AB*. Therefore, The distance from the center to either focus of an hyperbola is equal to the distance between the extremities of its axes. Hence, when the axes are given, LATUS RECTUM OF HYPERBOLA. 367 we may construct the foci as follows : From the center (7, with a radius equal to the diagonal of the rectangle under the semi-axes, describe an arc cutting the transverse axis produced in F and F f : the two points of intersection will be the foci sought. 454. By an analysis similar to that of Art. 357, the details of which the student must supply ,* we obtain Theorem VI. The squares on the ordinates drawn to either axis of an hyperbola are proportional to the rect- angles under the corresponding segments of that axis. Corollary. For the ordinate passing through either focus, we shall therefore have But c z a 2 =b 2 . Hence, doubling FP or F'P', 2b 2 latus rectum = = . a 2a That is, The latus rectum of any hyperbola is a third proportional to the transverse axis and the conjugate. 455. Throwing the equations to the Hyperbola and its conjugate into the forms y 2 _ 5 2 x 2 _ a 2 (x+a)(x-a) = If ' (y + b)(y-b) = ? ' we at once obtain Theorem VII. The squares on the axes of any hyperbola are to each other as the rectangle under any two segments of either is to the square on the ordinate which forms the segments. * When seeking properties of the conjugate axis, we must of course use the equation to the conjugate hyperbola. An. Ge. 34. 368 ANALYTIC GEOMETRY. 45f5o If we put the equation to the Hyperbola into the form and compare it with that of the circle described upon the transverse axis, namely, with we see that the ordinates of the two curves, correspond- ing to a common abscissa, have only an imaginary ratio. The analogy between the Hyperbola and the Ellipse, so far as concerns the circle mentioned, is therefore defective. If, however, we suppose b = a in (1), we get f = x-- b 2 . Hence, Theorem XI, The proposition that every diameter cuts the curve in two real points, is not true of the Hyperbola. Corollary 1. It is obvious, however, that the intersec- tions will be real, and at a finite distance from the center, so long as a 2 tan 2 < b' 2 . Hence, the diameters corre- sponding to a 2 tan 2 6 = b 2 , that is, the two diameters whose tangents of inclination are respectively b b tan o r= , tan u = , a a form the limits between those diameters which have real intersections with the curve and those which have not. But, from the values of their tangents of inclination, these two diameters are the diagonals of the rectangle contained by the axes. We learn, then, that diameters which cut the Hyperbola in real points must either make with the transverse axis an angle less than is made by the first of these diagonals, or greater than is made by the second. LENGTH OF DIAMETER. 373 It deserves notice, that the condition a 2 tan 2 = b 2 renders the abscissas of intersection, as expressed above, infinite. The two limiting diameters therefore meet the curve at infinity: and we have come upon the analogue of the equi-conjugates in the Ellipse. We shall soon find that these lines are the most remarkable elements of the Hyperbola, giving it a series of properties in which the other Conies do not share. Corollary 2. Eliminating between y = x tan 6 and the equation to the conjugate hyperbola, we get L ab ~ y (a 2 tan 2 # 6 2 ) ' Here, then, the condition of real intersection is # 2 tan 2 #>6 2 . Hence, Every diameter that cuts an hyperbola in two imag- inary points, cuts its conjugate in two real ones. 462. Length of any Diameter. This being double the central radius vector of the curve, may be determined (Art. 170) by J2 " ~ e 2 cos 2 6 1 ' or, if the diameter meets the conjugate curve instead of the primary, by 7 2 P = 1 e 2 'cos 2 6 ' an expression readily obtained by transforming to polar co-ordinates (Art. 57, Cor.) the equation to the conjugate hyperbola, found in Art. 449. 463. The first of the above expressions is least when = ; and the second, when = 90. Hence, Theorem XII. Each axis is the minimum diameter of its own curve. Remark. We see, then, that the terras major and minor are not applicable to the axes of an hyperbola. 374 ANALYTIC GEOMETRY. 464. By the same argument as in Art. 367, we obtain Theorem XIII, Diameters which make supplemental angles with the transverse axis of an hyperbola are equal. Corollary. It also follows, as in the corollary to Art. 367, that we can construct the axes when the curve is given. The diagram illustrates the process in the case of the Hyperbola, and the student may transfer the statements of Art. 367, Cor., to this figure, letter by letter. We deem it unnecessary to repeat them. 465. The inclinations of two diameters being repre- sented by 6 and 6', the argument of Art. 368 obviously applies to the Hyperbola, with respect to the condition (Art, 459, Cor.) tan/? tan 0'= - Hence, Theorem XIV. If one diameter of an hyperbola bisects chords parallel to a second, the second bisects chords par- allel to the first. 466. Two diameters of an hyperbola which are thus related, are called conjugate diameters, as in the case of the Ellipse. The phrase ordinates to any diameter is also used in connection with the Hyperbola, to signify the halves of the chords which the diameter bisects ; or, the right lines drawn from the diameter, parallel to its conjugate, to meet the curve. Corollary. The construction of a pair of conjugates in an hyperbola, may therefore be effected exactly in the manner CONJUGATE DIAMETERS. 375 described in the corollary to Art. 369. The details may be gathered by applying to the parts of the annexed diagram, the statements of that corollary. 467. Equation of Condition for Conjugates in the Hyperbola. The conjugate of any diameter being parallel to the chords which the diameter bisects, the in- clinations of two conjugates must be connected in the same way as those of a diameter and its ordinates. Hence, if 6 and 0' represent the inclinations, the re- quired condition (Art. 459, Cor.) is 52 tan tan 0'=-,. a 2 Corollary, Hence, if tan < b : a, tan 6 1 > b : a ; and if tan 6 > I : a, tan 0' < I : a. Therefore (Art. 461, Cors. 1, 2), If one of two conjugates meets an hyperbola, the other meets the conjugate curve. Remark. The condition of this article might have been obtained from that of Art. 370, by merely changing b 2 into 6 2 . 468. The preceding condition shows that the tangents of inclination have like signs. Hence, the angles made with the transverse axis by two conjugates are either both acute, or else both obtuse. That is, Theorem XV. Conjugate diameters of an hyperbola lie on the same side of the conjugate axis. 469. Equation to a Diameter conjugate to a Fixed Point. Making the requisite change of sign (Art. 167, Cor.) in the equation of Art. 372, we get the one now sought, namely, An. Ge. 35 376 ANALYTIC GEOMETRY. Corollary. The diameter conjugate to that which passes through (a, 0) is therefore x = 0, that is, the conjugate axis. Hence, The axes of an hyperbola con- stitute a case of conjugate diameters. 47O. Problem, Given the co-ordinates of the extrem- ity of a diameter, to find those of the extremity of its conjugate. By the extremities of the conjugate diameter, are meant the points in which the conjugate cuts the conju- gate hyperbola. The required co-ordinates are therefore found by eliminating between the equation of Art. 469 and They are Remark, By comparing these expressions with those of Art. 373, we notice that the abscissa and ordinate of the conjugate diameter in the Ellipse have opposite signs, but in the Hyperbola like signs. This agrees with the properties developed in Arts. 371, 468. 471. The equations of Art. 470, like those of Art. 373, give rise to Theorem XVI. The abscissa of the extremity of any diameter is to the ordinate of the extremity of its conjugate, as the transverse axis is to the conjugate axis. 472. By following, with respect to the second ex- pression of Art. 470, the steps indicated in Art. 375, excepting that we subtract the ?/ /2 , we arrive at Theorem XVII, The difference of the squares on the ordinates of the extremities of conjugate diameters is con- stant, and equal to the square on the conjugate semi-axis. LENGTH OF CONJUGATES. 377 Remark. We leave the student to prove the analogous property : The difference of the squares on the abscissas of the extremities of conjugate diameters is constant, and equal to the square on the transverse semi-axis. 473. Problem, To find the length of a diameter in terms of the abscissa of the extremity of its conjugate. Let x'y r be the extremity of any diameter, a' half its length, and b' half the length of its conjugate. Then a' 2 = x' 2 +y'\ and we get (Art. 470) since x c and y c must satisfy the equation to the conjugate hyperbola. Hence, (Art. 171,) a'' 2 = e 2 x 2 -f- a 2 . By performing similar operations with respect to b f , we should get b' 2 = e 2 x.' 2 a 2 . 474. Between these results and those of Art. 376, there is a striking difference ; and, as only the value of I' 2 equals (Art. 457) the rectangle of the focal radii drawn to x'y', it appears as if the property proved of the Ellipse in Art. 377 were only true of the Hyperbola with respect to those diameters which meet the conju- gate curve instead of the primary. But when we reflect that it is entirely arbitrary which of two conjugate hyper- bolas we consider the primary, it becomes evident that the property of Art. 377 is also true of the diameters which meet the curve hitherto called the primary, pro- vided we suppose the focal radii in question to be drawn from the foci of the conjugate curve. With this under- standing, then, we may state 378 ANALYTIC GEOMETRY. Theorem XVIII. The square on any semi-diameter of an hyperbola is equal to the rectangle under the focal radii drawn to the extremity of its conjugate. 47*5. It is evident on inspection, that the fourth formula in Art. 378 will not be altered by changing 6 2 into b 2 . Hence, in the Hyperbola as well as in the Ellipse, we have (a ex' in which x'y* is the extremity D of any diameter IYD. But x n + y /2 = a' 2 = (Art. 473) eV + a 2 = (Art. 472, Rem. ) &\x / ' 1 a 2 ) + a 2 . Hence, after substituting and reducing, or, in the Hyperbola as well as in the Ellipse, we have Theorem XIX. The distance from the extremity of any diameter to its conjugate, measured upon the corresponding focal radius, is constant, and equal to the transverse semi-axis. 476. Let a', V denote the lengths of any two conju- gate semi-diameters in an hyperbola. Then (Art. 473) ' 2 (1). Also, b' 2 = x* + y 2 = x c 2 + { b 2 (x 2 -f a 2 ) : a 2 } , since z, and y c satisfy the equation to the conjugate hyperbola. Hence, (Art. 171,) 6/2 = 6 2 X 2 + 2 ( 2 ). Subtracting (2) from (1), member by member, a 12 b f2 = a 2 b 2 . Hence, as the antithesis of Art. 379, Theorem XX. The difference of the squares on any two conjugate diameters of an hyperbola is constant , and equal to the difference of the squares on the axes. PARALLELOGRAM OF CONJUGATES. 379 477. Angle between two Conjugates. Using the same symbols as in Art. 380, it is plain that, in the Hyperbola also, we shall have Substituting for x c and y c from Art. 470, reducing, and remembering that b 2 x f2 a?y' 2 = a 2 b 2 , we get ab 478. Clearing this expression of fractions, we have a'b' sin

/ \^ we draw M Q tangent to the inscribed circle at Q, and join Q to the center C, QCM is called the eccentric angle of P. Now (Trig., 860) CM=CQ sec QCM. Also, from the equation to the Hyperbola, combined with this value of CM, MP 2 = ~( CM 2 - a 2 ) = & tan 2 Q CM. Hence, if we represent the arbitrary point P by vftf, y/ = a sec 0, y' = I tan 9. Substituting for x / and y' in Art. 481, we may write the equation to the tangent, in this notation, x y . sec V tan 1. a b The analogy of the angle QCJJ/, as formed in the case of the Hyperbola, to the similarly named angle in the Ellipse, may perhaps be obscure to the beginner; but it will become apparent when we reach the conception of a hyperbolic subtangent. 384 ANALYTIC GEOMETRY. 484. Problem. If a tangent to an hyperbola passes through a fixed point, to find the co-ordinates of contact. Let x"y" be the fixed point, and x'y' the required point of contact. Then, changing the sign of b 2 in the results of Art. 388, we get , _ a 2 b 2 x" g= aY Va 2 y" 2 - b 2 x" 2 -f- a 2 b 2 tf x n2 a 2ym ~ ' f _a 2 b 2 ifb 2 x n Va 2 y" 2 -b 2 x" 2 -}- a 2 b 2 y - Corollary 1. The form of these values indicates that from any given point two tangents can be drawn to an hyperbola : real when a?y" 2 b 2 x" 2 -f- a 2 b 2 > 0, that is, when the point is inside of the curve ; coincident when a 2 y" 2 b 2 x" 2 -f- a 2 b 2 = 0, that is, when the point is on the curve ; imaginary when a 2 y" b 2 x" 2 -\- a 2 b 2 < 0, that is, vdien the point is outside of the curve. Corollary 2. With regard to any two real tangents drawn from a given point, it is evident that their ab- scissas of contact will have like signs, if they both touch the same branch of the curve, and unlike signs, if the two touch different branches. But, if the two values of x' above have like signs, then, merely numerical relations being considered, a 2 b 2 x n > a 2 y" I/a 2 / f2 b 2 x" 2 + a 2 b 2 ; that is, after squaring, transposing, and reducing, Hence, as y = (b : a) x is the equation to the diagonal of the rectangle formed upon the axes (Art. 461, Cor. 1), POSITION OF POINT OF CONTACT. 385 the ordinate of the point from which two tangents can be drawn to the same branch of an hyperbola must be less than the corresponding ordinate of the diagonal; that is, the point itself must lie somewhere within the space in- cluded between the self-conjugates CL, OR (or CM, ON) and the adjacent branch of the curve. Hence, generally, The two tan- gents which can be drawn to an hyperbola from any point inside of the curve, will touch the same branch or different branches, according as the point is taken within or without the angle of the self -conjugates which incloses the two branches. 485. The argument of Art. 889 will be seen, on a moment's inspection, to hold good when hyperbolic equations are substituted for the elliptic. Therefore, Theorem XXIII. The tangent at the extremity of any diameter of an hyperbola is parallel to the conjugate diameter. Corollary. Tangents at the extremities of a diameter are parallel to each other. Remark. By drawing any diameter and its conjugate, and passing a parallel to the latter through the extremity of the former, we can readily form a tangent to a given hyperbola. If we construct tangents at the extremities of both diameters, we shall have an inscribed parallelo- gram. Thus the diagram of Art. 478 is verified; for, as only one parallel to a given line can be drawn through a given point, lines drawn through the extremi- ties of conjugate diameters so as to form their parallelo- gram must be tangents to the curve. 386 ANALYTIC GEOMETRY. 486. Let PT be a tangent to an hyperbola at any point P, and FP, F'P its focal radii of contact. From the equations ,8 Ftfx a 2 y'y = a 2 b 2 ( PT), y'(x-c)-(x'-c)y=Q (FP), tf(x+c)-W+c)y=Q (F'P), we readily find, by the same steps as in Art. 390, tan FPT = . , tan F'PT= , . cy' cy' Hence, FPT = F'PT-, or, we have Theorem XXIV. The tangent of an hyperbola bisects the internal angle between the focal radii drawn to the point of contact. Corollary 1. We therefore obtain the following solu- tion of the problem : To construct a tangent to an hyper- bola at a given point. Draw the focal radii FP, F'P to the given point P. On the longer, say F'P, lay off PQ = FP, and join QF. Through P draw SPT at right angles to QF: then will SPT be the tangent sought. For QPF is by construction an isosceles tri- angle; and SPT, the perpendicular from its vertex to its base, must therefore bisect the angle F'PF. Corollary 2. Hence, all rays emanating from F, and striking the curve, will be reflected in lines which, if traced backward, converge in F'; and reciprocally. Accordingly, to suggest the resemblance between these points and the corresponding ones of the Ellipse, they are called the foci, or burning points, of the Hyperbola. PRINCIPAL SUB TANGENT. 387 487. Let us suppose y = in the equation b 2 x f x a 2 y'y = a 2 b 2 . We shall thus find, as the value of the intercept which the tangent makes upon the transverse axis, x=CT= x' In the Hyperbola, then, as well as in the Ellipse, this intercept is a third proportional to the abscissa of contact and the transverse semi-axis, and we have the same constructions for the tangent at any point P of the curve, or from any point T of the transverse axis, as are described in Art. 391. 488. The Subtangent. For the length of the subtangent of the curve in the Hyperbola, we have MT=CMCT-, or, by the preceding article, sub tan = x' 2 a) x' x 1 But x' + a = AM, and x r a = MA. Hence, Theorem XXV. The subtangent of an hyperbola is a fourth proportional to the abscissa of contact and the tivo segments formed upon the transverse axis by the ordinate of contact. Corollary 1. Let x c r be the abscissa of contact for any tangent to the circle described on the transverse axis of an hyperbola, and x h f that of anytangent to the hyperbola itself. Then (Art. 311), subtan circ. = 388 ANALYTIC GEOMETRY. Suppose, now, that x c ' = a 2 : x h ' ; that is (Art. 487), that the abscissa of contact in the circle is the intercept of a tangent to the hyperbola. We at once get subtan circ. = - r = subtan hyp. a* We see, then, that if from the foot T of any tangent to an hyperbola an ordinate TQ be drawn to the in- scribed circle, the tangent to this circle at Q will pass through My the foot of the ordinate of contact in the hyperbola ; or, If tangents be drawn to an hyperbola and its inscribed circle from the head and foot of any ordinate to either, the resulting sub- tangents will be identical. We thus learn that the corresponding points of an hyperbola and its inscribed circle are those which have a common subtangent. And, in fact, by turning to the dia- gram of Art. 392, it will be seen that the corresponding points of an ellipse and its circumscribed circle may be defined in the same way. Hence, the defect in the anal- ogy between the two curves with respect to those circles, which came to light in Art. 456, can now be supplied. Corollary 2, Accordingly, we can construct the tan- gent by means of the inscribed circle as follows : When the point of contact P is given, draw the ordinate PM 9 and from its foot M make MQ tangent to the inscribed circle at Q. Let fall tire circular ordinate QT, and join its foot T with the given point P. PT will be the required tangent, by the property established. When T the foot of the tangent is given, erect the CENTRAL PERPENDICULAR ON TANGENT. 389 circular ordinate TQ, and draw the corresponding tan- gent QM. From M 9 the foot of this, erect the hyperbolic ordinate MP, and join its extremity P with the given point T. Remark. By comparing the diagrams of Arts. 387, 483 with those of Art. 392 and the present article, the complete analogy of the eccentric angles in the two curves will, as we stated in Art. 483, become apparent. The eccentric angle of any point on either curve, may be defined as the central angle determined by the corresponding point of the circle described upon the transverse axis, it being under- stood that the "corresponding" points are those which have a common subtangent. 489. Perpendicular from the Center to any Tangent. The length of the perpendicular from the origin upon the line b 2 x'x a?y'y = a 2 b 2 , (Art. 92, Cor. 2) must be a?b 2 ab ~ V (6V 2 -f ay 2 ) ~ i/(e 2 x' 2 a 2 ) ' Now (Art. 473) e 2 x' 2 a 2 =b' 2 . Therefore, ab P = > or, as in the Ellipse, we have Theorem XXVI. The central perpendicular upon any tangent of an hyperbola is a fourth proportional to the parallel semi-diameter and the semi-axes. 4OO. Central Perpendicular in terms of its inclination to tlie Transverse Axis. Changing the sign of & 2 in the formula of Art. 394, we get 390 ANALYTIC GEOMETRY. 491o Making the same change in the final equation of Art. 395, we obtain, as the equation to the locus of the intersection of tangents to an hyperbola which cut at right angles, From this (Art. 136) we at once get Theorem XXVII, The locus of the intersection of tangents to an hyperbola which cut each other at right angles, is the circle described from the center of the hyperbola, with a radius = Vv? b\ 492. Perpendiculars from the Foci to any Tangent. For the length of the perpendicular from the right-hand focus (ae, 0) upon b 2 x f x a 2 y'y = a 2 b 2 , we have (Art. 105, Cor. 2) b 2 x f ae a 2 b 2 b(ex f a] ~ y(b*x' L + ay 2 ) ~~ y (e*x' 2 a 2 ) ' or, since (Arts. 457, 473) ex' a = p, and e 2 x' 2 a 2 =b f2 , = bp P b' ' And, in like manner, for the perpendicular from the left-hand focus, Corollary. Since b' 2 pp' (Art. 474), we may also write VP ,2 b Y ~- 493. Upon dividing the value of p by that of p', we obtain Theorem XXVIII. The focal perpendiculars upon any tangent of an hyperbola are proportional to the adjacent focal radii of contact. FOCAL PERPENDICULARS ON TANGENT. 391 And if we multiply these values together, pp' 6 2 ; or, \ve have Theorem XXIX. The rectangle under the focal perpen- diculars upon any tangent is constant, and equal to the square on the conjugate semi-axis. 494. Changing the sign of b 2 in the first two equa- tions of Art. 399, we get y mx = 1/wiV - my -f- x = Yd 2 -f b'\ as the equations to any hyperbolic tangent and its focal perpendicular. Adding the squares of these together, we eliminate in, and obtain x 2 + y 2 = a 2 as the constant relation between the co-ordinates of intersection belonging to these lines. Hence, (Art. 136,) Theorem XXX. The locus of the foot of the focal per- pendicular upon any tangent of an hyperbola, is the circle inscribed within the curve. Corollary. We may therefore apply in the case of the Hyperbola, the construction given in the corollary to Art. 399, as follows: To draw a tangent to an hyperbola, through any given point : Join the given point P with either focus F, and upon PF describe a circle cutting the inscribed circle in Q and Q'. The line which joins P to either of these points, for example An. Ge. 36. 392 ANALYTIC GEOMETRY. the line PQ, will touch the hyperbola at some point T\ for the angles PQF, PQ'F being inscribed in a semi- circle, Q and Q' are the feet of focal perpendiculars. When P is on the curve, and PF consequently a focal radius, we can prove, as in Ex. 8, p. 359, that the circle described on PF will touch the inscribed circle. The foot of the focal perpendicular must then be found by joining the middle point of PF with the center (7, and noting the point in which the resulting line cuts the inscribed circle. 493. We see, then, that if an hyperbola is given, every chord drawn from the focus to meet the inscribed circle must be a focal perpendicular to some tangent of the hyperbola. On the other hand, it is obvious that any point outside of a given circle, may be considered the focus of some circumscribed hyperbola. Hence, Theorem XXXI. If from any point without a circle a chord be drawn, and a perpendicular to it at its extremity, the perpendicular luill be tangent to the circumscribed hyper- bola of which the point is a focus. Corollary Since this is equivalent to saying that the hyperbola is the envelope of the perpendicular, we may approximate the outline of an hyperbola, as is done in the an- nexed figure, by drawing chords to a circle from a fixed point P outside of it, and forming perpendiculars at their extremities. It should be no- ticed, that only the parts of these perpendiculars which lie on opposite sides of the chord that determines them, enter into the formation of the curve; in the Ellipse, on the contrary, the perpendiculars lie on the same side of the determining chords. When the chords assume the NORMAL OF THE HYPERBOLA. 393 limiting positions PL, P72, so as to touch the circle at L and J?, the corresponding perpendiculars LN, MR are the two lines which we have named the self-conjugates. 4OG. A little inspection of the equations in Art. 401, after the sign of b 2 has been changed in the first and second, will show that the reasoning of that article is entirely applicable to the Hyperbola. Hence, Theorem XXXII, The diameters which pass through the feet of the focal perpendiculars upon any tangent of an hyperbola, are parallel to the corresponding focal radii of contact. Corollary. Hence, also, as in the case of the Ellipse, we have the converse theorem, Diameters parallel to the focal radii of contact meet the tangent at the feet of its focal perpendiculars. Con- sequently, after finding the foot Q of the focal perpen- dicular, we can determine the point of contact T, if we wish to do so, by sim- ply drawing F'T parallel to CQ. It follows, also, that the distance between the foot of the perpendicular drawn from either focus to a tangent, and tJte foot of the perpendicular drawn from the remaining focus to the parallel tangent, is constant, and equal to the length of the transverse axis. THE NORMAL. 497. Equation to the Normal. From the equa- tion of Art. 402, by changing the sign of 6 2 , we have _. -~ 394 ANALYTIC GEOMETRY, 498. Let PN be the normal to an hyperbola at any point P" and F'P the corresponding focal radii. The equa- tions to the latter (Art. 95) are Combining the equation to the normal with each of these in succession, we get (Art. 96) Hence, FPN=l%Q F'PN= QPN; and we have Theorem XXXIII. The normal of an hyperbola bisects the external angle between the focal radii of contact. Corollary 1. Comparing Theorems XXIV, XXXIII of the Hyperbola with the same of the Ellipse, we at once infer : If an ellipse and an hyperbola are confocal, the normal of the one is the tangent of the other at their intersection. Corollary 2, To construct a normal at any point P of the curve, we draw the focal radii FP, F'P, produce one of them, as F'P, until PQ = FP, and join QF: then will PN, drawn through P at right angles to QF, be the required normal. For it will bisect the angle FPQ, accord- ing to the well-known properties of the isosceles triangle. Corollary 3. To draw a normal through any point R on the conjugate axis, we pass a circle RF'R'F through the given point and the foci, and join the point where CONSTRUCTION OF NORMAL. 395 this circle cuts the hyperbola with the given point by the line RPN: this line will bisect the angle FPQ, because R is the middle point of the arc F'RF. It is important to notice, however, that the auxiliary circle cuts each branch of the curve in two points, as P and P', and that only one of these (P, in the diagram) answers the conditions of the present construction. For the line joining R to the other, as RP r , will bisect the internal angle between the focal radii, instead of the external. We thus see that we can use this method for drawing a tangent from any point in the conjugate axis : a statement which applies to the Ellipse also, provided the point R is outside of the curve. 499. Intercept of the Normal. Making y = in the equation of Art. 497, we obtain R K We can therefore, as in the case of the Ellipse (Art. 404), construct a normal at any point P of the curve, or one from any point N of the transverse axis. 500. By an argument in all respects similar to that of Art. 405, we have F'N : FN= F'P : FP-, that is, Theorem XXXIV. The normal of an hyperbola cuts the distance between the foci in segments proportional to the adjacent focal radii of contact. 501. Length of the Subnormal. For the portion of the transverse axis included between the foot of the 396 ANALYTIC GEOMETRY. normal and that of the ordinate of contact, we have MN= ON CM = e 2 x' x' = (e 2 1) x'. Hence, subnor = x'. a 2 502. Comparing the results of Arts. 499 and 501, ON : MN = c 2 : b 2 . Or, since c 2 = a 2 + 6 2 , we have Theorem XXXV. The normal of an hyperbola cuts the abscissa of contact in the constant ratio (a 2 + b 2 ) : b 2 . 503. Length of the Normal. Changing the sign of b 2 in the first formula of Art. 408, and then applying the formula of Art. 473, we get W 5O4. Hence, PN r .PR = b' 2 ; and we have Theorem XXXVI. The rectangle under the segments formed by the tivo axes upon the normal is equal to the square on the semi-diameter conjugate to the point of contact. Corollary. Hence, too, (Art. 474) PN.PR = pp r ; or, The rectangle under the segments of the normal is equal to the rectangle imder the focal radii of contact. 5O5o Also (Art. 489), putting Q for the foot of the central perpendicular on the tangent at P, CQ.PR = a 2 , and OQ . PN = b 2 . That is, SUPPLEMENTAL CHORDS. 397 Theorem XXXVII. The rectangle under the normal and the central perpendicular upon the corresponding tangent is constant, and equal to the square on the semi-axis other than the one to which the normal is measured. SUPPLEMENTAL AND FOCAL CHORDS. 506. Condition that Chords of an Hyperbola be Supplemental. Let ^, (p f denote the inclina- tions of any two supple- mental chords DP, D'P. Then, from Art. 412, by / , the characteristic change of sign, the required condition will be b 2 tan

n t> ( ~TI\ ^ PQ = (-. But (Art. 479, Cor.) sin

, being (Art. 557) the abscissas of P / and P, the points in which it cuts the curve, will be found by eliminating be- tween this equation and xy= k 2 . But, as D is on the curve, x'y' = k 2 : whence, by combining with xy = 426 ANALYTIC GEOMETRY. Substituting this value in the equation to the chord, we get ^7 + = 2c .'. a: 2 2cx'x= a/ 2 as the quadratic determining the required abscissas. Hence, (P x ) x = x'(c+V^ =: \), (P) x = x'(cV~c T =~\). Now, $ being the angle between the asymptotes, and the in- clination of tyQ, the distance from Q to any other point on is determined (Arts. 101, Cor. 2; 102) by the formula _ x x l _ (x ;TI) sin h sin (, and I/ the semi-diameter conjugate to D. Hence, P / Q = 6 / (c+ l/c 2 1), QP = 6 x (c 1/c- 1). Therefore, P X Q . QP = b /2 ; and we have Theorem LXV. jTAe rectangle under the segments formed \ipnn parallel chords by either asymptote is constant, and equal to the square on the semi-diameter parallel to the chords. v. AREA OF THE HYPERBOLA. 559. The area of the segment ALMP, included between the curve, the asymptote, the ordi- nate of the vertex, and the ordinate of any given point P, is by general consent called the area of the hyperbola. Its value may be determined as follows : * Let x' = the abscissa CM of the point P to which the area is to be computed. Since the co-ordinates of the vertex are equal to each other, we shall have (from the equa- tion x = W CL k. *See Hymers' Conic Sections, p.. 121, 3d edition. AREA OF THE HYPERBOLA. 427 It is customary to take the quantity k as the unit in this computation. Adopting this convention, let the distance LMbe so subdivided at n points R, S 9 . . . , M, that the abscissas CL, CR, CS, . . . , CM may increase by geometric progression. Then, if OR = x, we shall have Thus x' = x n ; or, x = x f : so that, as n increases, x diminishes, and converges toward 1 as n converges to infinity. Now, at R, $, . . . , If, erect n ordinates, and form n corresponding parallelograms RA, Sa, . . . , Mo, situated as in the figure. Then area RA RL . LA sin = ( C72 CL) LA sin = (x 1 ) sin c&, " Sa = SR . Ra sin = (x 2 x) sin = (x 1) sin $, u Te=TS.Se sin 0= (x 3 x 1 ) ^ sin ^ = (.r 1) sin^, and so on for the whole series of n parallelograms. Hence, replacing x by its value a/"", and putting 2' = the sum of the n parallelograms, we get I = n (x''* 1) sin (p. But an inspection of the diagram shows that the greater the number of the parallelograms, the more nearly does the sum of their areas approach the area ALMP. And since x, the ratio of the successive abscissas, tends to 1 as n tends to op, we can make the number of the equal parallelograms as great as we please ; in other words, the true value of the area ALMP is the limit toward which 2' converges as n converges to oo. Hence, area ALMP = n (a/n 1) sin < = n [{ 1 + (x 7 1) }n 1] sin . I x'. But (Alg., 376) sin

= the inclination of a diameter passing through any point P, and tf = that of the polar of P, the transverse axis being the axis of ' oc: then will tan $ tan ' = 1. 13. The circle which passes through the center of an equilateral hyperbola and any two points A and B, passes also through the intersection of two lines drawn the one through A parallel to the polar of J5, and the other through B parallel to the polar of A. 430 ANALYTIC GEOMETRY. 14. The locus of a point such that the rectangle under the focal perpendiculars upon its polar with respect to a given ellipse shall be constant, is an ellipse or an hyperbola according as the foci are on the same side or on opposite sides of the polar. 15. A line is drawn at right angles to the transverse axis of ai5. hyperbola, meeting the curve and its conjugate in P and Q: show that the normals at P and Q intersect upon the transverse axis. Also, that the tangents at P and Q intersect on the curve 16. Given two unequal circles: the locus of the center of the circle which touches them both externally, is an hyperbola whose foci are the centers of the given circles. 17. Every chord of an hyperbola bisects the portion of either asymptote included between the tangents at its extremities. 18. If a pair of conjugate diameters of an ellipse be the asymp- totes of an hyperbola, to prove that the points of the hyperbola at which its tangents will also touch the ellipse, lie on an ellipse con- centric and of the same eccentricity with the given one. 19. A tangent is drawn at a point P of an hyperbola, cutting the asymptote CY in E; from E is drawn any right line EKH cutting one branch of the curve in A" //; and Kk, PM, Hh are drawn parallel to CY cutting the asymptote CK in k, M, h: to prove that 20. Three hyperbolas have parallel asymptotes: show that the three right lines which join two and two the intersections of the hyperbolas, meet in one point. CHAPTER FIFTH. THE PARABOLA, i. THE CURVE REFERRED TO ITS Axis AND VERTEX. 56O. In discussing the Parabola, we shall find it- most convenient to transform its equation, as found in PROPERTIES OF THE PARABOLA. 431 Art. 181, to a new set of reference-axes. Before doing so, however, we may deduce an important property, which will enable us to give the constant p, involved in that equation, a more significant interpretation. THE AXIS. 561. Making y = in the equation mentioned, namely, in we obtain x = OA=p. Now (Art. 181) 2p = OF: hence, OA = J OF', or, we have Theorem I. In any parabola, the vertex of the curve bisects the distance between the focus and the directrix. Corollary. Whenever, therefore, the constant p pre- sents itself in a parabolic formula, we may interpret it as denoting the distance from the focus to the vertex of the curve. Remark. We might also infer the theorem of this article directly from the definition of the curve in Art. 179. 562. Since AF is thus equal to p, the distance of the focus from the vertex w r ill converge to whenever %p = OF converges to that limit, but will remain finite as long as 2p remains so. In other words (since we may take the focus F as close to the directrix D'D as we please), the focus may approach infinitely near to the vertex, but can not pass beyond it. Hence, Theorem II. The focus of a parabola falls within the curve. 563. Let us now transform the equation u 2 = 4p (xp), by moving the axis of y parallel to itself along OF to 432 ANALYTIC GEOMETRY. the vertex A. This we accomplish (Art. 55) by putting x -f p for x, and thus get f = 4px. 564. This equation asserts that the ordinate of the Parabola is a geometric mean of the abscissa and four times the focal distance of the vertex. It therefore leads directly to the fol- lowing construction of the curve by ^;:'-l. points, when the focus and the vertex L --- -< are given : Through the focus F draw the axis M'A, and produce it until AB = 4AF. Through the vertex A draw AD r perpendicular to the axis. At any convenient points on the axis, erect perpendiculars of indefinite lengths, as MP, M'P*; and upon the distances BM, BM', etc., as diameters, describe circles BDM, BD'M', etc. From D, D', . . . , where these circles cut the per- pendicular AD ', draw parallels to the axis : the points P. P', . . . , in which these meet the perpendiculars MP, M'P 1 , . . . , will be points of the parabola required. For we shall have PM= AD = AB.AM = and a similar relation for P'M' and all other ordinates formed in the same way. 565. The property asserted by the equation y* = and involved in the foregoing construction, may be other- wise stated as Theorem III. The square on any ordinate of a para- bola is equal to four times the rectangle under the cor- responding abscissa and the focal distance of the vertex. Corollary. Since p is constant for any given parabola, y z will increase or diminish directly as x does. That is, PARABOLA LIMIT OF ELLIPSE. 433 The squares on the ordinates of any parabola vary as the corresponding abscissas: >66. The double ordinate L'L passing through the focus, in the Parabola also, is called the latus rectum. Making x = p in the equation to the curve, we obtain the value of FL) namely, y = 2p. Hence, latus rectum = 4p. Corollary. This result would lead us to state Theorem III as follows: The square on any ordinate is equal to the rectangle under the corresponding abscissa and the latus rectum. 567. It is now important to show that a certain relation in form exists between the Parabola and the Ellipse, such that we may consider a parabola as the limiting shape to which an ellipse approaches when we conceive its axis major to increase continually, while its focus and the adjacent vertex remain fixed. By estab- lishing this, we shall be enabled to bring the symbol e, arbitrarily written = 1 in Art. 184, under the conception which gives it meaning in the other two conies. The equation we are now using for the Parabola being referred to the vertex, we must refer the Ellipse to its vertex, if we desire to exhibit the relation mentioned. Transforming, then, the equation of Art. 147 by putting x a for x, we get, as the equation to the Ellipse referred to its vertex F, 434 ANALYTIC GEOMETRY. Putting p as an arbitrary symbol for the distance VF between the vertex and the adjacent focus, we have (Art. 151) p = a l/V 6 2 : whence and the above equation becomes Suppose, now, that the distance VF remains fixed, while the whole axis major VA increases to infinity : in the limit, where a = GO, we get if = 4px, the equation to the Parabola : which proves our propo- sition. Corollary 1. Since we may thus regard a parabola as an ellipse with an infinitely long axis, that is, with a center infinitely distant from its vertex and its focus, it deserves to be considered whether the Parabola has any element analogous to the so-called circumscribed circle of the Ellipse. To settle this point, we only need to consider, that, if a tangent be drawn to a circle, and the radius of the circle be then continually increased without changing the point of contact, the circle will tend more and more nearly to coincidence with its tangent the greater the radius becomes; so that, if we were to suppose the radius infinitely great, the circle would become straight by actually coinciding with the tangent. If, then, we draw at the vertex of an ellipse a common ECCENTRICITY. 435 tangent to the curve and its circumscribed circle, and subject the axis major to continual increase under the conditions which will cause the curve to assume the form of a parabola in the limit where a = GO, the circumscribed circle will continually approach the common tangent as its center recedes from the vertex, and, in the limit where the ellipse vanishes into a parabola, will coincide with the tangent. We learn, then, that the Parabola has an ana- logue of the circumscribed circle; that this analogue is in fact the tangent of the curve at its vertex ; and that it is also the line used as the axis of y in the equation ?/2 4^ since this line and the tangent are both per- pendicular to the axis of the curve at its vertex, and must therefore coincide. All these results will soon be confirmed by analysis. Corollary 2, But, as a point of greater importance, the relation established above enables us to assert that the symbol e = 1 denotes the eccentricity of the corre- sponding parabola. For, in the Ellipse, we have a 2 6 2 _ b 2 . e - 1 ~ > and if in this we make a = GO, or suppose the curve to become a parabola, we get e = l. That is, we may con- sider a parabola to be an ellipse in which the eccentricity has reached the limit 1. Moreover, in the view taken of this subject in the corollaries to Arts. 359, 456, the condition e = 1 corresponds to that ellipse which has so far deviated from the curvature of its circumscribed circle as to vanish into the right line that forms its axis major. Now, when we recollect (Art. 195, Cor.) that a right line is a particular case of the Parabola, and that, in reaching the limit 1, e has assumed a value fixed and the same 436 ANALYTIC GEOMETRY. for all parabolas, it becomes evident that we may con- sider the condition e = 1 as marking that stage of devi- ation from circularity which characterizes the Parabola, and therefore, in connection with this curve also, appro- priately call e the eccentricity. Hence, the name parabola (derived from the Greek xapafidttsw, to place side by side, to make equal] may be taken as signifying, that, in the curve which it denotes, the eccentricity is equal to unity. In closing > this article, we would again direct the student's attention to the fact, that all parabolas have the same eccentricity. Let p = FP, the focal distance of any point on a parabola. Then, by the definition of the curve (BD being the directrix), FP = PD. Also, from the diagram, PD = BM= BA -f AM. Now, AM = x, and (Art. 561) BA =p. Hence, p =p + x. That is, Theorem IV. The focal radius of any point on a par- abola is a linear function of the corresponding abscissa. Remark. This expression for p is similar to those found in the case of the Ellipse and of the Hyperbola (Arts. 360, 457), and is called the Linear Equation to the Parabola. 5159. The methods of drawing the curve, given in Arts. 179, 564, have already familiarized the reader with the figure of the Parabola, and suggested a tol- erably clear conception of its details. Let us now see how the equation FIGURE OF THE PARABOLA. 437 verifies and completes the impressions, made by the diagrams : I. No point of the curve lies on the left of the perpendicular to the axis, drawn through the vertex. For this line is the axis of y; and, if we make x negative in the equation, the resulting values of y are imaginary. II. But the curve extends to infinity on the right of the per- pendicular mentioned, both above and below the axis. For y is real for every possible positive value of x. III. The curve is symmetric to the axis. For there are two values of y, numerically equal but opposite in sign, corresponding to every value of x. 57O. The Parabola, then, differs from the Ellipse, and resembles the Hyperbola, in having infinite con- tinuity of extent. There is, however, a marked dis- tinction between its infinite branch and the infinite branches of the Hyperbola, which calls for a more minute examination. In the first place, the limiting forms to which the two curves tend are essentially different : that of the Hyper- bola (Art. 176) being two intersecting right lines, which pass through its center ; while that of the Parabola (Arts. 192; 195, Cor.) is two parallel right lines, or, in the extreme case, a single right line. This, of itself, indicates a difference in the nature of the curvature in these two conies. Secondly, the branches of the Hyperbola, in receding from the origin, tend to meet the two lines called the asymptotes in two coincident points at infinity (Arts. 539, 543). Now, if we seek the intersections of any right line with a parabola, by eliminating between the equations y = mx -f b and y 2 = 4px, we find that their abscissas are (2p ml) V / p- x = 438 ANALYTIC GEOMETRY. In order, then, that these intersections may be coinci- dent, we must have inb = p : in which case, for the abscissas, we get ?.* (1) ' and, for the equation to the given line, y mx -f- ; or, m 2 x my -\-p = Q (2) . If, now, the coincidence takes place at infinity, we shall have, from (1), jn = Q-, and (2) will assume the form (Art. 110) C=Q. That is, any right line that tends to meet a parabola in two coincident points at infinity, is situated altogether at infinity ; or, what is the same thing, no parabola has any tendency to approach a finite right line in the manner characteristic of the Hyperbola. DIAMETERS. 571. Equation to any Diameter. In a system of parallel chords in a parabola, let 6 be the common in- clination to the axis, xy the middle point of any member of the system, and x'y' the point in which the chord cuts the curve. We have (Art. 101, Cor. 3) x' = x Z cos #, y t= y Zsin#. Hence, as x'y f is on the parabola mentioned, (y I sin d) 2 = 4p(x l cos 6). That is, for determining Z, we get the quadratic Z 2 sin 2 2 (?/ sin 2p cos 6) l = 4px y\ DIAMETERS OF THE PARABOLA. 439 But as xij is the middle point of a chord, the co-efficient of I vanishes (Alg., 234, Prop. 3d), and the locus of the middle point is therefore represented by the equation y sin 2p cos = 0. Hence, the required equation to any diameter is y = 2p cot 0. 572. Since p is fixed for any given parabola, and for any given system of parallel chords, this equation is of the form y = constant. Now such an equation (Art. 25) denotes a parallel to the axis of x. Hence, Theorem V. Every diameter of a parabola is a right line parallel to its axis. Corollary 1. From this, we at once infer: All the diameters of a parabola are parallel to each other. Corollary 2. The constant value of y in the above equation, being dependent on the arbitrary angle 0, is itself arbi- trary. The converse of our theorem is therefore true, and we have : Every right line drawn parallel to the axis of a par- abola is a diameter. Remark. The directrix being perpendicular to the axis, we might define a diameter of a parabola as any right line drawn perpendicular to the directrix. The diagram illustrates diameters from either point of view. 573. If we write (as we may) the equation y = constant in the form y = Ox -f 6, An. Ge. 40. 440 ANALYTIC GEOMETRY. and then eliminate y between it and y 1 = tyx, we get, for determining the intersections of a parabola with its diameter, the relation Ox 2 -j- Ox -f b 2 = 4px ; or, Ox 2 4px -\-b 2 =Q. Now (Alg., 238) the roots of this quadratic are x GO : whence, observing the form of these roots, we have Theorem VI. Every diameter of a parabola meets the curve in two points, one finite, the other at infinity. Eemark. The meaning of this theorem, in ordinary geometric Ian future, is, of course, that a diameter only meets the curve in the one finite point whose abscissa = 6 2 : 4p. The argument and the phraseology adopted here are only used for the purpose of com- pleting analogies, and will seem less forced when we approach the subject from a more generic point of view. 574. From the last theorem, it follows that no chord of a parabola can be parallel to the axis, and, therefore, that no diameter can bisect a system of chords parallel to a second diameter. We thus learn, that, in the Par- abola, the conception of conjugate diameters vanishes in the parallelism of all diameters. THE TANGENT. 575. Equation to any Chord If x'y', x"y" be the extremities of any chord in a parabola, we shall have y'* = px' and y"* = 4px". Hence, y" 2 y' 2 = 4p (x"x'\ and we get y"-y' *P x"-* - y' + y"' TANGENT OF THE PARABOLA. 441 Substituting this value for the second member of, the equation in Art. 95, we obtain the equation now re- quired, namely, y y' . _4p_ x x' . : i/ f -f- y" ' 576. Equation to the Tangent. Making y" = y' in the preceding equation, reducing, and recollecting that y' 2 4px f , we get 577. Condition that a Right Line shall touch a Parabola. We have seen, in the second part of Art. 570, that y mx -f- b will meet the parabola y 2 4px in two coincident points, whenever mbp. The condition now required is therefore *=. m Corollary, Every right line, then, whose equation is of the form y = mx -f- m is a tangent to the parabola y 2 =4px. We have here another instance of the so-called Magical Equation to the Tangent. 578. Problem, If a tangent to a parabola passes through a fixed point, to find the co-ordinates of contact. Putting x'y' for the required point of contact, and x"y" for the fixed point through which the tangent passes, we have (Arts. 563, 576) 442 ANALYTIC GEOMETRY. Solving these conditions for x' and y', we get *'=& y' = y" Vy"' 2 - Corollary. These values indicate that from any given point two tangents can be drawn to a parabola : real when y" 2 4px ff > 0, that is, when the point is ivitliout the curve; coincident when y" 2 4jwc" = 0, that is, when the point is on the curve ; imaginary when y" 2 4pz"< 0, that is, when the point is within the curve. 579o Definition. The halves of the chords which any diameter of a parabola bisects, are called the ordinates of the diameter. The term is also applied at times to the entire chords, or to the right lines formed by pro- longing them indefinitely. 58Oo Let x'y f be the extremity of any parabolic di- ameter. Then, from Art. 571, ?/' 2pcot#; and we get, for determining the angle 6 which the ordinates of the diameter make with the axis, y' Now the equation to the tangent at x'y' (Art. 576) may be written Hence, by the principle of Art. 78, Cor. 1, Theorem VII. The tangent at the extremity of any diameter of a parabola is parallel to the corresponding ordinates. DIRECTION OF THE TANGENT. 443 Corollary. Hence, further, the tangent at the vertex of the curve is perpendicular to the axis ; and we confirm by analysis the result of Art. 567, Cor. 1, namely, The vertical tangent and the axis of y are identical. 581. The equations to any tangent of a parabola and its focal radius of contact (Arts. 576, 95) may be written y'y = 2p (x + x') (PT), y'x+(p-x')y=py< (FP). For the angle FPT between these lines, we therefore have (Art. 96, Cor. 1) y' But, by Art. 580, this is also the value of tan QPT. Hence FPT = QPT', and we get Theorem VIII. The tangent of a parabola bisects the internal angle between the diameter and focal radius drawn to the point of contact. Corollary 1. To draw, then, a tangent to a parabola at any point P, we form the focal radius PF, and the diameter RP. We next prolong RP until PQ equals PF, join QF, and draw PT perpendicular to QF: it will be the tangent required, by virtue of the theorem just proved, and the isosceles triangle FPQ. Corollary 2, Since SPR=QPT=FPT, all rays that strike the concave of the curve in lines parallel to the axis will be reflected to F, which is therefore called the focus, as in the Ellipse and the Hyperbola. 444 ANALYTIC GEOMETRY. 582. Making y = in the equation y'y = 2 we obtain, as the value of the intercept formed by the tangent upon the axis of a parabola, The length of the intercept is therefore equal to that of the abscissa of contact, the sign minus denoting that it is measured to the left of the vertex. If, then, we add AF = p to this length, we get But (Art. 568) p + x' = FP. Hence, FT=FP; and we have Theorem IX. In any parabola, the foot of the tangent and the point of contact are equally distant from the focus. Corollary. This property obviously leads to the fol- lowing constructions : I. To draw a tangent at any point P of a parabola. Join the given point P with the focus JF, and from the latter as a center, with a radius equal to PF, describe an arc cutting the axis in T: the required tangent may then be formed by joining PT. II. To draw a tangent to a parabola from any point T on the axis. From the focus F as a center, with the radius FT, describe an arc, and note the point P in which this cuts the curve: P will be the point of contact, and the corresponding tangent may be formed by joining TP. SUBTANGENT OF THE PARABOLA. 445 583. The Sufrtaiigent. For the length of the sub- tangent of the curve in the Parabola, or of that portion of the axis which is included between the foot of the tan- gent and that of the ordinate of con- tact, we have TM= TA-\- AM; or, subtan = x' -f x' = 2x f . 584. Thus TA = AM= J TM, and we get the im- portant property, Theorem X. The subtangent of a parabola is bisected in the vertex. Corollary 1. We can now construct the tangent at any point of the curve or from any point on the axis, as follows : When the point of contact P is given, draw the ordinate PM, and on the prolonged axis lay off AT=AM: then, by the theorem just proved, T will be the foot of the tangent at P, which is found by joining PT. When the foot T of the tangent is given, lay off upon the axis AM -- AT, and erect the ordinate MP. The point P in which this meets the curve, will be the point of contact sought; and the tangent is obtained by join- ing TP. Corollary 2. As the value of the subtangent is depend- ent upon the peculiar form of the equation to the Para- bola, the present theorem is peculiar to this curve, and hence leads to the following mode of constructing it, which is often used by mechanics and draughtsmen. Lay down two equal right lines AB, AC making any convenient angle with each other. Bisect them in E and F, join EF, BC, and draw AX perpendicular to the latter: Ic will bisect EF in V, and BC in X, by the well-known properties of the isosceles triangle. 446 ANALYTIC GEOMETRY. Now divide AE and its equal EB into the same number of equal parts, and the equals AF, FC in the same manner: the whole lines A 5, AC will thus be subdivided into equal parts at the points 1, 2, 3, E, 4, 5, 6 and 6, 5, 4, F, 3, 2, 1. Having numbered these points in reverse order upon the two A lines, as in the diagram, join those which have the same numeral : the resulting lines will envelope a parabola, which we can approximate as closely as we please, by continually diminishing the distances Al, 1 ... 2, etc. For, by the construction, V is the middle point of AX; and the curve touches all the lines AE, AC, 1 ... 1, 6. . . 6, etc. : hence, with respect to the lines AB, AC, which may be regarded as limiting cases of all the others, it is a curve that bisects its subtangent in the vertex ; that is, a parabola. From another point of view, the curve here formed is the envel- ope of a line EF, which moves within the fixed lines AB, AC in such a manner that the sum of the remaining sides of the triangle EAF is constant, being equal to AB. It is therefore a parabola, by the result of the Example solved in Art. 251. 585. Perpendicular from the Foeiis to any Tan- gent. For the length of the perpendicular from the focus (p, 0) upon the line y'y^Zp (x -f- #'), we have (Art. 105, Cor. 2) x f ) 2p (p + = But (Art. 568) p-\-x f =p, the focal distance of the point of contact. Hence, 5S6. In this expression, p being constant, it is evident that P 2 will change its value as p changes its value ; or, P will change with the square root of p : a property usually expressed by FOCAL PERPENDICULAR ON TANGENT. 447 Theorem XL The focal perpendicular upon the tangent of a parabola varies in the subduplicate ratio of the focal radius of contact. 587. Focal Perpendicular in terms of its inclination to the Axis The perpendicular from (p, 0) upon the line whose equation (Art. 577, Cor.) is m 2 x my -f- p = 0, will be, according to Art. 105, Cor. 2, p _ m*p + p _ p ^ l/(m*+m'' ! ) m Put = the inclination of P, measured from the axis toward the right: then m = cot 0, and we get P=^sec0. 588. The equation to the tangent being written (Art. 577, Cor.) that of its focal perpendicular, which passes through (p, 0), will be Combining these so as to eliminate m, we get as the equation to the locus of the point in which the focal perpendicular meets the tangent. Hence, (Art. 580, Cor.,) Theorem XII The locus of the foot of the focal per- pendicular upon any tangent of a parabola, is the tangent at the vertex of the curve. Corollary 1. By means of this property, we can solve in" its most general form the problem. An. Ge. 41. 448 ANALYTIC GEOMETRY. \ To draw a tangent to a parabola through any given point. Let P be the given point, and join it with the focus F. On PF as a diameter, de- scribe a circle cutting the ver- tical tangent in Q and Q' : these points will be the feet of focal perpendiculars, as the angles PQF, PQ'F are inscribed in semicircles. Hence, a line join- ing P to either of them, for instance the line PQ, will touch the parabola in some point T. If the point of contact is required, produce the axis to meet PQ in 72, and then apply the method of Art. 582, Cor. When the given point is on the curve, as at T, the auxiliary circle will touch the vertical tangent ; but the point Q can still be found, by dropping a perpendicular from the middle point of FT upon A Y. Corollary 2. The present theorem, and the resulting construction, completely justify the view, taken in Art. 567, Cor. 1, that the vertical tangent of the Parabola is the analogue of the circle circumscribed about the Ellipse. We thus arrive at the conclusion, often serviceable in anal- ysis, that the Right Line may be denned as the circle whose radius is infinite. 589. Since the vertical tangent is the locus of the foot of the focal perpendicular on any other tangent, that is, the line in which the foot of every such perpendicular is found, it follows that every right line drawn from the focus to the vertical tangent is a focal perpendicular to some other tangent of the curve. Besides, it is obvious that any given point and right line may be regarded as the focus and vertical tangent of some parabola. Hence, PARABOLA AS ENVELOPE. 449 Theorem XIII. If from any point a line be drawn to a fixed right line, and a perpendicular to it be formed at the intersection, the perpendicular will be tangent to the parabola of which the point and fixed line are the focus and vertical tangent. Corollary Thus we see that a parabola is the envelope of such a perpendicular, and we may therefore approximate the outline of one, by drawing oblique lines to a given right line from a fixed point P, and erecting perpendicu- lars at their extremities. If formed close enough together, these perpendiculars will define the curve with considerable distinctness, as the diagram shows. 59O. Let us now ascend from the theorem of Art. 588 to the general one, in which a line from the focus meets the tangent at any fixed angle. Calling this angle 0, and writing the equation to the tangent (Art. 577^ Cor.) y=*+ (i), we obtain, for the equation to the intersecting line from the focus (Art. 103), m + tan0 ;,_^ ^ From (2), by clearing of fractions, expanding, and collecting terms, y mx = (1 + 2 ) ^ tan 6 mp. Subtracting this result from (1) member by member, and then transposing, (1 -f m 2 ) x tan 6 = \ - + mp. Dividing through by (1 + m 2 ), P in x tan d= .-. m = 450 ANALYTIC GEOMETRY. Substituting this value of m in (1), we get the equation to the locus of the intersection of the tangent and the focal line, namely, y = x tan 6 + p cot 0. But (Art. 577, Cor.) this denotes a tangent of the same parabola, inclined to the axis at the angle 6. Hence, Theorem XIV The locus of the intersection of a tangent with the focal line which meets it at a fixed angle, is the tangent which meets the axis at the same angle. 591. Angle between any two Tangents. Let x'y', x"y" denote the two points of contact Q, Q / '. The equations to the corresponding tan- gents (Art. 576) will then be Hence, applying the formula of Art. 96, Cor. 1, we get tan 592. This expression leads to a noticeable property of the Parabola, as follows : PQ, PQf being any two intersecting tangents, the equations to their focal radii of contact FQ, FQ / (Art. 95) will be y / y _ y" x p x'p xp x"p Substituting for x' and re" their values from the equation to the Parabola, and clearing of fractions, we may write these equations, (FQ), (x-p) - 4/) y = 4py" (x - p) From them, by Art. 96, Cor. 1, we get /2 - V) - n o 2 ) 2 - 4p* (y" - 3/0 2 NORMAL OF THE PARABOLA. 451 Now, if we apply the formula for the tangent of a double angle (Trig., 847, in) to the angle QPQ', whose tangent we found in the preceding article, we shall get, for the tangent of 2QPQ', the ex- pression just obtained. Hence QFQ / = 2QPQ / ; and we have Theorem XV. The angle between any two tangents of a parabola is equal to half the focal angle subtended by their chord of contact. 593. The equations to any two tangents of a parabola that cut each other at right angles (Arts. 577, Cor. ; 96, Cor. 3) will be , P x y = mx + , y = mp. m m Subtracting the second of these from the first, we obtain x = p as the equation to the locus of the intersection. But this equation denotes a right line perpendicular to the axis at the distance p on the left of the vertex; in other words, the line called the directrix in our primary definitions. Hence, Theorem XVI. The locus of the intersection of tangents which cut each other at right angles, is the directrix of the curve. Corollary. Since this locus, in the case of the Ellipse, is a circle concentric with the curve (Art. 395), this theorem again shows us that the Circle converges to the form of the Right Line as its radius tends to infinity, and that we may therefore correctly regard the Right Line as a circle with an infinite radius. THE NORMAL. 594. Equation to the Normal. The equation to the perpendicular drawn through the point of contact x'y' to the tangent y'y = 2p(x + O, is found by Art. 103, Cor. 2, and is therefore 452 ANALYTIC GEOMETRY. 595. Let P be any point x'y' on a parabola, PN the corresponding normal, DP a diameter through P, and FP the focal radius of its vertex. The equation to FP (Art. 95) is y'x (x t p)y=py f . Comparing this with the equation to the normal, we get (Art. 96, Cor. 1) 1 FPN== - But, since every diameter is parallel to the axis, DPN= , and we have, from the equation to the normal, Hence, FPN= 180 DPN = QPN-, and we obtain Theorem XVII. The normal of a parabola bisects the external angle between the corresponding diameter and focal radius. Corollary. To construct a normal at any point P of the curve, we therefore draw the focal radius PF, and the diameter DPQ, laying off upon the latter PQ = PF, and joining QF: then will PN, drawn perpendicular to QF, bisect the angle FPQ, and for that reason be the normal required. 596. Intercept of the Normal. Making y = in the equation of Art. 594, we obtain x = AN=2p -f x f . Corollary. This result enables us to construct a normal at any point of the curve, or from any point on the axis. SUBNORMAL OF THE PARABOLA. 453 For, if the point P is given, we draw the corresponding ordinate PM, and lay off MN to the right of its foot, equal to 2AF: we thus find JV, the foot of the required normal. When N is given, we lay off NM 2AF to its left, erect the ordinate MP, and join NP. 597. Since FN=ANAF=(2p+x r )p=p + x!, we have (Arts. 568, 582) Theorem XVIII. The foot of the normal is at the same distance from the focus as the foot and the point of contact of the corresponding tangent. Corollary. This is the same as saying that the three points men- tioned are on the same circle, de- scribed from the focus as center. Hence, to construct either the tan- gent or the normal, or both, pass a circle from the center F through either of the three points P, T, N, as one or another is given, and join the points in which it cuts the axis with the point in which it cuts the parabola. 598. L,ength of the Subnormal. For this (see diagram, Art. 596), we have MN=ANAM. That is (Art. 596), subnor = 2p. In other words (Art. 561, Cor.), we have obtained Theorem XIX. The subnormal of a parabola is constant, and equal to twice the distance from the focus to the vertex. 599. Length of the Normal. For the distance between x'y r and (2p -f a/, 0), we have (Art. 51, I, Cor. 1) PN 2 = 4p 2 + y 12 = 454 ANALYTIC GEOMETRY. Now (Art. 568) p + d = p. Hence, PN 2 = 4pp. But (Art. 585) pp = the square of the focal perpendicu- lar on the tangent at P. Therefore, Theorem XX. The normal of a parabola is double the focal perpendicular on the corresponding tangent. ii. THE CURVE IN TERMS OF ANY DIAMETER. GOO. The equation which we have thus far employed is only a special form of a more general one, and many of the properties proved by means of it are but particu- lar cases of generic theorems which relate, not to the axis, but to any diameter whatever. The truth of this will appear as soon as we transform 7/ 2 1= 4p Xy which is referred to the axis AX and the vertical tangent AY. to O any diameter A'X' and its vertical tangent A'Y f . This transformation we c;m effect by means of the formulae in Art. 56, Cor. 1, observing that the new axis of x is parallel to the primitive, and the of the formulae therefore equal to zero. If in addition we call the angle Y'TX not /? but 0, and put x'y' to denote the new origin A 1 , these formulae of transformation will become (Art. 58) . , x = x f -f x -f- y cos 0, y = y' + y sin 0- 6O1. Equation to the Parabola, referred to any Diameter and its Vertical Tangent. Replacing the y and x of -if = 4px by their values as given in the pre- ceding formulae, and collecting the terms, we get if sin 2 d + Z(y r sin 6 2p cos 0) y -f- ?/' 2 4px f = 4px. PARABOLA REFERRED TO ANY DIAMETER. 455 But, as the new origin x'y 1 is on the curve, y' 2 4px' 0. Also, since the new axis of y is a tangent, tan0 = 2p : T/'; so that y' sin 6 2p cos 6 = 0. Hence, the transformed equation is in reality if sin 2 /? = lpx (1). Putting p : sin 2 6 = p', we may write the equation y2 _. 4^/3. This at once shows that y 2 = 4px is the form which (1) assumes when the diameter chosen for the axis of x is that whose vertical tangent is perpendicular to it, so that sin 2 0=1. 602. Before employing our new equation itself, we may extend the property of Theo- rem I by means of relations derived from y 2 = 4px : this will better pre- pare the way for the use of y 2 = 4p'x. Let ZM/ be any diameter, and A! T T its vertical tangent. Parallel to the latter, draw FF' through the focus. Then, in the parallelogram F'T, we shall have A'F' = FT = (Art. 582) FA'. But, from the definition of the curve, supposing IfD to be the directrix, FA' = AD. Hence, A'D = A'F' ; or, Theorem XXI. The vertex of any diameter bisects the distance from the directrix to the point in which the diameter is cut by its focal ordinate. 603. The factor p' =p : sin 2 which enters the second member of equation (2), Art. 601, may be expressed in terms of p and x', by the following process : 456 ANALYTIC GEOMETRY. . According to Art. 576, tan 6 2p : y'. Hence, sin = / We have, then, Now (Art. 568) p -f x' = FA f , the focal distance of the vertex of any diameter. Hence, Theorem XXIL The focal distance of the vertex of any diameter is equal to the focal distance of the principal vertex, divided by the square of the sine of the angle between the diameter and its vertical tangent. 6O4. The expression just obtained aids us to interpret the new equation y 2 = 4p'x. For, from what precedes, the equation may be written ?/ 2 4 (p -f x') x, and we learn that the symbol p' signifies the focal distance of the vertex taken for the new origin. We therefore read from the equation at once, the following extension of Theorem III : Theorem XXIII. The square on an ordinate to any diameter is equal to four times the rectangle under the corresponding abscissa and the focal distance of the vertex. Corollary. Hence, the squares on the ordinates to any diameter vary as the corresponding abscissas. FOCAL DOUBLE ORDINATES. 457 605. In Art. 602, we found A'F' = FA r ^= p + x'. Hence, making x =p -f- x' in the equa- tion of Art. 604, we get ,p bfr- x Now, since every diameter bisects the & chords parallel to its vertical tangent, PQ = 2F'P=4FA'-, and we have Theorem XXIV. The focal double ordinate to any diameter is equal to four times the focal distance of its vertex. Remark. The value of the latus rectum (Art. 566), furnishes a particular case of this theorem ; and, as 4p signifies the length of the focal double ordinate to the axis, so 4p', by what precedes, represents that of the focal double ordinate to any diameter. From Arts. 429, 522, we see that this uniform analogy among the focal double ordinates to all diameters, is peculiar to the Parabola. 6O6. The reader may now interpret the equation and show by means of it, that, with reference to any diameter, the Parabola consists of a single infinite branch, tending to two parallel right lines as its limiting form, and symmetric to the diameter. DIAMETRAL PROPERTIES OF THE TANGENT. 607. Equation to the Tangent, referred to any Diameter. From the identity of form in the equations y 2 = 4p.r, y 2 = 4p f x, we at once infer that this must be 458 ANALYTIC GEOMETRY. GO8. Making y = in this equation, we get, for the intercept of the tangent on any diameter, x = x'. This shows that the tangent cuts any diameter on the left of its vertex, at a distance equal to the abscissa of contact. Thus the vertex of any diameter is situated midway between the foot of the tangent and that of its ordinate of contact, and we have, as the extension of Theorem X, Theorem XXV. The subtangent to any diameter of a parabola is bisected in tlie corresponding vertex. Corollary 1. This property enables us to construct a tangent to a parabola from any external point whatever. For, if T' be such point, we have only to draw the diameter T f M r \ form the tangent A'T at its vertex (by dropping a perpendicular from A' upon the axis, and setting off AT = the abscissa thus determined), take A'M' = A'T', draw M'P parallel to the tangent A'T, and join PT'. Corollary 2. By the same property, we can construct an ordinate to any diameter. This is either done in the way M'P was formed above, or as follows: Take any point T" on the axis, make AM='AT", erect the per- pendicular MP, and join PT". Then, T 1 being the point where this tangent cuts the given' diameter, take A'M' = A'T', and join M'P. 6O9. Let PQ be any chord of a parabola. Then (Art. 607) the equations to the tangents at its opposite POLAR IN THE PARABOLA. 459 extremities P and $, by referring them to its bisecting diameter, will be 2p' (x + x') y'y = 0, 2p' (x + x f ) + y f y = 0. Subtracting the first of these from the second, we get as the equation to the locus of the intersection. Hence, Theorem XXVI. Tangents at the extremities of any chord of a parabola meet on the diameter which bisects that chord. POLE AND POLAR. 610. We shall now prove that the polar relation is a property of the Parabola, following the same steps as in the Ellipse and the Hyperbola. 611. Chord of Contact in the Parabola. Let x f y' denote the point from w r hich the two tangents that deter- mine the chord are drawn, and x^ l9 x$ 2 the extremities of the chord. Since x'y f is upon both tangents, we have i/y - 2p' (x> + x,), y,y' = 2/ (x' + x 2 ). That is, the two extremities of the chord are on the line whose equation is y'y = 2p f (x + xT). And this is therefore the equation to the chord itself. 612. Locus of the Intersection of Tangents to the Parabola. Let x'y' be the fixed point through which the chord of contact of the intersecting tangents is drawn. Then, if x^y^ be the intersection of the two tangents, since x'y f is always on their chord of contact, we shall have (Art. 611) y,y' = Zp'^ + xJ. 460 ANALYTIC GEOMETRY. And this being true, however x } y l may change its position as the chord of contact revolves about x'y', the co-ordi- nates of intersection must always satisfy the equation This, therefore, is the equation to the locus sought. 613. Tangent and Chord of Contact taken iip into the wider conception of the Polar. Here, too, as well as in the Ellipse and the Hyperbola, the two equations just found are identical in form with that of the tangent. By the same reasonmg, then, as in Arts. 433, 526, we learn that the tangent and chord of contact in the Parabola are particular cases of the locus just dis- cussed. Now, too, by its equation, this locus is a right line; and, if we suppose x'y' to be any point on a given right line, the co-efficients of the equation in Art. 611 will fulfill the condition Ax' -}- By' + C= 0, and thus (Art. 117) the chord of contact will pass through a fixed point. In the curve now before us, therefore, we have the twofold theorem : I. If from a fixed point chords be drawn to any para- bola, and tangents to the curve be formed at the extremities of each chord, the intersections of the several pairs of tan- gents will lie on one right line. II. If from different points lying on one right line pairs of tangents be drawn to any parabola, their several chords of contact will meet in one point. Thus the law that renders the locus of Art. 612 the generic form of which the tangent and chord of contact are special phases, is the law of polar reciprocity: whence the Parabola, in common with the other two Conies, im- parts to every point in its plane the power of determining a right line; and reciprocally. POLAR IN THE PARABOLA. 461 614. Equation to the Polar with respect to a Parabola. This, as we gather immediately from the preceding results, is if referred to any diameter ; or, if referred to the axis, 615. Definitions. The Polar of any point, with re- spect to a parabola, is the right line which forms the locus of the intersection of the two tangents drawn at the extremities of any chord passing through the point. The Pole of any right line, with respect to the same curve, is the point in which all the chords of contact corresponding to different points on the line intersect. We have, then, the following constructions: When the pole P is given, draw through it any two chords T'T, S'S, and form the corresponding pairs of tangents, T'L and TL, S'M and SM: the line LM which joins the intersection of M the first pair to that of the second, will be the polar of P. When the polar is given, take any two of its points, as L and M, draw a pair of tangents from each, and form the corresponding chords of contact, T'T, S'S: the point P in which these intersect, will be the pole of LM. When the pole is without the curve, as at M 9 the polar is the corresponding chord of contact $'S' 9 and when it is on the curve, as at T, the polar is the tangent at T. In these cases, the drawing may be made in accordance with the facts. 462 ANALYTIC GEOMETRY. 616o Direction of the Polar. By referring the polar of any point to the diameter drawn through the point, the y' of its equation will become = 0, and the equation itself (Art. 614) will assume the form x = x f . This denotes a parallel to the axis of y. Hence, Theorem XXVII. The polar of any point, with respect to a parabola, is parallel to the ordinates of the diameter which passes through the point. Corollary 1. The equation x = x r , more exactly interpreted, gives us : The polar of any point on a diameter is parallel to the ordinates of the diameter, and its distance from the vertex of the diameter is equal, in an opposite direction, to the distance of the point. And, in particular, The polar of any point on the axis is the per- pendicular which cuts the axis at the same distance from the vertex as the point itself, but on the opposite side. Corollary 2. To construct the polar, therefore, draw a diameter through the pole, take on the opposite side of its vertex a point equidistant with the pole, and draw through this a parallel to the corresponding ordinates. 617. Polar of the Focus. The equation to this is obtained by putting (p, 0) for x'y' in the second equation of Art. 614, and is The focal polar of the Parabola is therefore identical with the line which in Art. 180 we named the directrix, and we shall presently see that our ability to generate the curve by the means employed in Art. 179, is due to the polar relation of that line to the focus. POLAR OF THE FOCUS. 463 G18. Let D'D represent the polar of the focus F. Then, obviously, PD = RA + AM ; or, the distance of any point on the curve from the polar is equal to the distance of the polar from the vertex, increased by the abscissa of the point. That is, PD = + x. = PJ)' } or, Now (Art. 568) p + x = FP\ whence, since e = 1 in the Parabola, FP Here, then, the property of Arts. 439, 532 again appears, and we have Theorem XXVIII, The distance of any point on a parabola from the focus is in a constant ratio to its distance from the polar of the focus, the ratio being equal to the eccentricity of the curve. Remark 1, We thus complete the circuit of our anal- ysis, and, as stated above, return upon the property from which we set forth in Art. 179. The significance of our present result consists in the fact, that we have translated the apparently arbitrary definition of Art. 179 into the generic law of polarity. And we may say that we have vindicated our method of generating the curve ; because we can now see that it is the mechanical expression of the power to determine a conic, which the Point and the Right Line together possess : a power reciprocally in- volved, of course, in that of the Conic to bring these two Forms into the polar relation. An. Ge. 42. 464 ANALYTIC GEOMETRY. It may deserve mention, that the construction of Art. 179, like those of the corollaries to Arts. 439, 532, em- ploys the parts of a right triangle in order to embody the constant ratio of the focal and polar distances, the constant in the case of the Parabola being the ratio of the base to itself. Remark 2. The name parabola thus acquires a new meaning. We may henceforth regard it as signifying the conic in which the constant ratio between the focal and polar distances equals unity. G1O. Focal Angle subtemlcd by any Tangent. By an analysis similar to that of Art. 440, x being the abscissa of the point P from which the tangent is drawn, p its radius vector FP, and > the angle PFT, we can show that P< 62O. This expression, like those of Arts. 440, 533, is inde- pendent of the point of contact. Hence, the angle PFT = the angle PFT' ; and, with respect to the whole angle TFT', we get Theorem XXIX. The right line that joins the focus to the pole of any chord, bisects the focal angle which the chord subtends. Corollary. In particular, The line that joins the focus to the pole of any focal chord is perpendicular to the chord. Remark. By comparing this corollary with the theorem of Art. 593, and bearing in mind that the directrix, as the polar of the focus, is the line in which the pair of tangents drawn at the ex- tremities of any focal chord will intersect, we may state the follow- ing noticeable group of related properties : If tangents be drawn at the extremities of any focal chord of a parabola, I . The tangents will intersect on the directrix. 2. The tangents will meet each other at right angles. 3. The line that joins their intersection to the focus will be perpen- dicular to the focal chord. PARAMETER OF THE PARABOLA. 465 621. We shall in this article solve two examples, which will show the beginner IIOAV to take advantage of such results as we have lately obtained. I. Given two points P, Q, and their polars T'T, 8'iS: to deter- mine the relation between the intercept wm, cut off on the axis by the polars, and the intercept MN, cut off by perpendiculars from the points. Let x'y'y x"y" denote P and Q. Then Prt MN=x" -x'. But the equations to the polars T' T, S'S are y'y-= 2/>(* + '), y"y=2i>(x + *"); and if we make y in these, and take the difference of the results, Hence, The intercept on the axis between any tico 2^olars is equal to that between the perpendiculars from their 2>olcs. II. To prove that the circle which circumscribes the triangle formed by any three tangents, passes through the focus. Let L, Q, R be the intersections of the tangents, and F the focus. By Art. 592, the angle L QR is half the focal angle subtended by S'S. Also, by Art, 620, the angle LFR = LFT + TFR=18Q minus half this same focal angle. Hence, LQR + LFR = 180, and the quadrilateral LQRF is an inscribed quadrilateral. That is, F is on the circle which circumscribes the triangle LQR. PARAMETERS. 622. Definition. The Parameter of a parabola, with respect to any diameter, is a third proportional to any abscissa formed on the diameter, and the corre- sponding ordinate. Thus, ?/ 2 parameter = *~- . 3/ 623. From the equation to the Parabola, y' 2 : x f = 4p'. Also, from Art. 604, p f = the distance of the vertex of a diameter from the focus. Hence, 466 ANALYTIC GEOMETRY. Theorem XXX. The parameter of any diameter in a parabola is equal to four times the focal distance of its vertex. Corollary. The parameter of the axis, or, as it is usually called, the principal parameter, is therefore equal to four times the distance from the focus to the vertex of the curve ; that is, its value is 4p. Remark. In the equations y 2 = 4p'x, y 2 = 4px, we are henceforward to understand that p 1 is one-fourth the parameter of the reference-diameter, and p one- fourth the parameter of the axis. Or, with greater generality, in any equation of the form P, the constant co-efficient of x, is to be interpreted as the parameter of the corresponding parabola, taken with respect to the diameter to which the equation is referred. 624. In Art. 605, we proved that the focal double ordinale to any diameter is equal to four times the focal distance of its vertex. Hence, Theorem XXXI. The parameter of any diameter in a parabola is equal to the focal double ordinate of that diameter. Corollary. Accordingly, the parameter of the axis is equal to the latus rectum : as we might also infer from the fact that both are equal to 4p. Remark. From Arts. 429, 522, as already noticed in another connection, we see that the Parabola is the only conic in which the parameter of every diameter is equal to the corresponding focal double ordinate. PARABOLA REFERRED TO ITS FOCUS. 467 625. It is sometimes useful to express the parameter of any diameter in terms that refer to the axis of the curve as the axis of x. To effect this, we either use the relation p'=p + d (1), where x' is the abscissa of the vertex of the diameter whose parameter is sought, measured on the axis ; or else where 6 is the angle made with the axis by the tangent at the vertex of the diameter. 626. The latter of the foregoing relations may be interpreted as Theorem XXXII. The parameter of any diameter varies inversely as the square of the sine of the angle which the corresponding vertical tangent makes with the axis. in. THE CURVE REFERRED TO ITS Focus. 627. The polar equations to the Parabola (Art. 183 cf. Rem.) being P '' ~ 1 : cos 6 ' our present knowledge leads us to assign to p its proper meaning, and to describe the numerator in the value of p as half the parameter of the curve. Moreover, since e = 1 in the Parabola, we may write 2p P '' Z 1 e cos 6 ' 468 ANALYTIC GEOMETRY. thus completely exhibiting the analogy of these ex- pressions to the corresponding elliptic equations of Art. 443 : an analogy which we partially established in Art. 184, and which verifies the proposition of Art. 567, that a parabola may be regarded as an ellipse in which the eccentricity has passed to the limiting value = 1. G2S. IPolar Equation to the Tangent. In seeking this, we shall avail ourselves of the property just men- tioned. If in the equation of Art. 444 we replace a (1 e 2 ) by its value 2p, as found in Art. 443, we may write the polar equation to an elliptic tangent cos (0 V) e cos Making e = 1 in this, we get the equation to a parabolic tangent, namely, cos (0 0') cos 6 iv. AREA OF THE PARABOLA. 629. The area of any parabolic segment, included between the curve and any double ordinate to the axis, may be computed as follows : Supposing A - PMQ to be the segment whose area is sought, divide its abscissa AM into any number of equal parts at B, (7, D, . . . , erect ordinates BL, CN, DR, ... at the points of division, and through their extremities Z, N, R, . . . draw parallels to the axis, producing both the ordinates and the parallels until they meet as in the AREA OF THE PARABOLA. 469 figure. The curve divides the circumscribed rectangle UM into two segments : and, by the process just de- scribed, there will be formed in both of these a number of smaller rectangles, corresponding two and two ; as UR to EM, ON to ND, EL to LC, etc. Let x'y', x' f y'' be any two successive points thus formed upon the curve ; for instance, L, N. Then area LC = y' (x" x'} = area EL = x r (y" - y'} = To express, then, the ratio of any interior rectangle to its corresponding exterior one, we shall have an equation of the form LG_ _ y" + y' _ i | y"_ EL' y' f y' ' Hence, the limiting value to which this ratio tends as y" converges to y r , is evidently = 2 : and therefore the ratio borne by the sum of the interior rectangles to the sum of the exterior, also tends to the limit 2 as y fr con- verges to y'. Now the condition that y" may converge to y' is, that the subdivision of AM shall be continued ad infinitum: and if this takes place, the sum o'f the in- terior rectangles will converge to the area of the interior segment APM\ and the sum of the exterior, to the area of the exterior segment APU. Therefore, APM = 2APU; or, putting x = AM, y = MP, and A the area of the interior segment, 470 ANALYTIC GEOMETRY. 2 Similarly, the segment AQM=-xy: whence, o Theorem XXXIII. The area of a parabolic segment cut off by any double ordinate to the axis, is equal to two-thirds of the circumscribing rectangle. Corollary. Since the same reasoning is obviously applicable to the equation y 2 = 4p'rc, we may at once state the generic theorem : The area of a parabolic seg- ment cut off by a double ordinate to any diameter ', is equal to two-thirds of the circumscribing parallelogram. EXAMPLES ON THE PARABOLA. 1. The extremities of any chord of a parabola being x'y', x"y", and the abscissa of its intersection with the axis being Z, to prove that xV = x 1 , y'y" 4px. 2. Two tangents of a parabola meet the curve in x / y / and x"y" \ their point of intersection being xy, show that 3. The area of the triangle formed by three tangents of a para- bola is half that of the triangle formed by joining their points of contact. 4. To prove that the area of the triangle included between the tangents to the parabolas y 1 mx, y l nx at points whose common abscissa a, and the portion of the cor- responding ordinate intercepted between the two curves, is equal to 5. To prove that the three altitudes of any triangle circum- scribed about a parabola, meet in one point on the directrix. EXAMPLES ON THE PARABOLA. 471 6. The cotangents of the inclinations of three parabolic tangents are in arithmetical progression, the common difference being = J: prove that, in the triangle inclosed by these tangents, we shall have area = p' 2 <3 3 . 7. Given the outline of a parabola: to construct the axis r.r.d the focus. 8. Find the equation to the normal of a parabola, in terms ot' its inclination to the axis; and prove that the locus of the foot of the focal perpendicular upon the normal is a second parabola, whose vertex is the focus of the given one, and whose parameter is one- fourth as great as that of the given one. 9. Show that the locus of the intersection of parabolic normals which cut at right angles, is a parabola whose parameter is one- fourth that of the given one, and whose vertex is at a distance = 3p from the given vertex. 10. The centers of a series of circles which pass through the focus of a parabola, are situated on the curve: prove that each circle touches the directrix. 11. To find the area of the rectangle included by the tangent arid normal at any point P of a parabola, and their respective focal perpendiculars ; and to determine the position of P when the rect- angle is a square. 12. If two parabolic tangents are intersected by a third, parallel to their chord of contact, the distances from their point of intersec- tion to their respective points of contact are bisected by the third tangent. 13. Show that the locus of the vertex of a parabola which has a given focus, and touches a given right line, is a circle, of which the perpendicular from the given focus to the given line is a diameter. 14. Show that, if a parabola have a given vertex, and touch a given right line, its focus will move along another parabola, whose axis passes through the given vertex at right angles to the given line, and whose parameter = the distance from the given vertex to the given line. 15. TP and TQ are tangent to a given parabola at P and Q, and TIFjtfins their intersection to the focus: prove that FP. FQ = FT\ An. Ge. 43. 472 ANALYTIC GEOMETRY. 16. A right angle moves in such a manner that its sides are respectively tangent to two confocal parabolas whose axes are coincident: to find the locus of its vertex. 17. Prove that the pole of the normal which passes through one extremity of the latus rectum of a parabola, is situated on the diam- eter which passes through the other extremity, and find its exact position on that line. 18. The triangle included by two parabolic tangents and their chord of contact being of a given area = a 2 , prove that the locus of the pole is a parabola whose equation is 19. Prove that, if two parabolas whose axes are mutually per- pendicular intersect in four points, the four points lie on a circle. 20. If two parabolas, having a common vertex, and axes at right angles to each other, intersect in the point x'y': then, denoting the latus rectum of the one, and I' that of the other, I : ' : : / : V. [This property has a special interest, on account of its connection with the ancient problem of the Duplication of the Cube. It affords, as the reader will observe, a method of determining graphically two geometric means between two given lines ; and was proposed for this purpose by MENECHMUS, a geometer of the school of Plato, in connection with his attempt to solve the problem just mentioned. The graphic problem of " two means " received a variety of solutions at the hands of the Greek geometers, two .of the most celebrated being discovered by DIOCLES and NICOMEDES. They are effected, respectively, with the help of the curves called the Cissoid and the Conchoid.'} CHAPTER SIXTH. THE CONIC IN GENERAL. 63O. Having in the previous Chapters become familiar with the properties of the several conies considered as THE CONIC IN GENERAL. 473 separate curves, let us now ascend to the wholly generic point of view, from which we may comprehend them not as isolated Forms, but as members of a united System, and, in fact, as successive phases of a generic locus which may be called the CONIC, whose idea we sketched in the eighth Section of Part I. We may begin by showing in what sense this name is descriptive of the system ; or, how the curves may be grouped together as sections of a cone. THE THREE CURVES AS SECTIONS OF THE CONE. 631. Definitions. A Cone is a surface generated by moving a right line which is pivoted upon a fixed point, along the outline of any given curve whose plane does not contain the fixed point. The fixed point is called the vertex of the cone; the given curve, its directrix; and the moving right line, its generatrix. Since the generatrix extends indefinitely on both sides of the vertex, the cone will consist of two exactly similar portions, extending from the vertex in opposite directions to infinity. Of these, one is called the upper nappe of the cone; and the other, the lower nappe. Any single position of the generatrix is called an ele- ment of the cone. When the directrix is a circle, the cone is called cir- cular; and the right line drawn through the vertex and the center of the directrix, is termed the axis of the cone. 632. Definitions. A Right circular Cone is a cone whose directrix is a circle, and whose axis is perpendic- ular to the plane of its directrix. The section formed with any cone by a plane, is termed a base of the cone ; consequently, the directrix 474 ANALYTIC GEOMETRY. of a right circular cone may be called its base. For this reason, such a cone is often named a right cone on a circular base. The diagram of the next article presents an example of one. 633. We shall prove, in the proper place in Book Second, the following propositions, which we ask the student to take upon trust for the present, in order that we may use them in grouping the three curves according to their geometric order : I. Every section formed by passing a plane through a right circular cone is a curve of the Second order. II. If the angle which the secant plane makes with the base is less than that made by the generatrix, the section is an ellipse. III. If the angle which the secant plane makes with the base is equal to that made by the generatrix, the section is a parabola. IV. If the angle which the secant plane makes with the base is greater than that made by the generatrix, the section is an hyperbola. These three cases are represented in the diagram : that of the Ellipse, at AE'j that of the Parabola, at LPR\ and that of the Hyperbola, at HA-A'H'. It is manifest, how- ever, from the fact that the secant plane makes with the base an angle successively less than, equal to, and greater than the angle made by the generatrix (whose angle must have a fixed value for any given cone), that the three sections may be formed by a single SECTIONS OF A CONE. 475 plane, by simply revolving it on the line in which it cuts the base. Beginning with it in such a position that its inclination to the base is less than that of the generatrix (whose inclination is often called the incli- nation of the side of the cone), and revolving it upward toward the position of parallelism to the side, we shall cut out a series of ellipses of greater and greater eccentricity. When the secant plane becomes parallel to the side, the section will be a parabola. When it is pushed still farther upward, so that its angle with the base becomes greater than that of the side, it will reach across the space between the two nappes of the cone, and pierce the upper as well as the lower one, and the section will be an hyperbola, whose two branches will lie in the two nappes respectively. From this it appears, that, granting the proposition that the sections are the curves mentioned, the natural geometric order in which they occur is : Ellipse, Para- bola, Hyperbola. This is the same as their analytic order, as we found it in Art. 200. We shall be able to give a more explicit account of their appearance as suc- cessive phases of the Conic, so soon as we have presented a fuller view of the modes in which we can represent that generic locus by analytic symbols. VARIOUS FORMS OF THE EQUATION TO THE CONIC. 634. Equation in Rectangular Co-ordinates at the Vertex. We have not as yet referred the three Conies to the same axes and origin : let us now do so, by transforming the central equations of the Ellipse and the Hyperbola to such a vertex of each curve as will correspond to the vertex of the Parabola. 476 ANALYTIC GEOMETRY. This will require us to transform the equation 2/ 2 =^( 2 -* 2 ) (1) to the left-hand vertex of the Ellipse ; and the equation 2/ 2 =5(* 2 - 2 ) (2) to the right-hand vertex of the Hyperbola. Accordingly, putting x a for x in (1), and x + a for x in (2), and expanding, we get 2b 2 b 2 , 2b 2 , b 2 y 2 = x -- - z 2 if = x A -- , x 2 . a a 2 a a 2 Hence, (Arts. 428, 521,) the equations to the Ellipse, the Hyperbola, and the Parabola may be written Remembering, now, that b 2 : a 2 = (1 e 2 ), and that its value in the case of the Parabola must therefore be = 0, we learn that the equation to the Conic in General is in which P is the parameter of the curve, and R the ratio between the squares of the semi-axes; and we have the specific conditions R < . . Ellipse, E = .'. Parabola, R > . * . Hyperbola. Corollary. By the three equations from which y 2 = Px + Ex 2 was generalized, we see that in the Ellipse, NAMES OF THE SECTIONS. 477 the square on the ordinate is less than the rectangle under the abscissa and parameter; that in the Parabola, the square is equal to the rectangle; and that in the Hyperbola, the square is greater than the rectangle. Remark According to PAPPUS (Math. Coll., VII: c. A. D. 350), the names of the three curves were originally given to designate this property. But EUTOCIUS (A. D. 560) says that the names were derived from the fact, that, according to the ancient Greek geome- ters, the three sections were cut respectively from an acute-angled, a right-angled, and an. obtuse-angled cone, by means of a plane always passing at right angles to the side. Thus, if the angle under the vertex of the cone were acute, the sum of that and the right angle made by the secant plane with the side would be less than two right angles, and the name ellipse was given, either to indicate this deficiency, or to show that the curve would then fall short of the upper nappe of the cone. But if the angle under the vertex were right, the mentioned sum of angles would be equal to two right angles, and the plane of the curve consequently be par- allel to the side of the cone : to denote which facts, the name para- bola was given. Finally, if the angle under the vertex were obtuse, the mentioned sum of angles would be greater than two right angles, and the name hyperbola was given, either to suggest this excess, or to indicate that the curve would then reach over to the upper nappe of the cone. It is noticeable here, that the early geometers supposed the three sections to be peculiar respectively to an acute-angled, a right-angled, and an obtuse-angled cone. The improvement of forming them all from the same cone by merely changing the inclination of the secant plane, was introduced by APOLLONIUS OF PERGA, B. c. 250. Which of these etymologies should have the preference, is a question among critics of mathematical history. It is remarkable, however, that the names represent equally well all the distinguish- ing properties of the curves, whether geometric or analytic : as the reader may perhaps have already observed for himself. The name parameter, which simply means corresponding measure, or co-efficient, is given to the quantities '2lr : a and y'' z : a/, on account of the position they occupy in the equation one or the other of them being the firs*, co-efficient in the second 478 ANALYTIC GEOMETRY. member, according as the conic is central or non-central. More- over, as the dimensions of the curve depend mainly upon the value of P, the ratios which it symbolizes are naturally termed par excel- lence the measures of the Conic. On the other hand, the name parameter may be defined in each conic by the value which it denotes for each, as in the preceding Chapters. 6&5. Equation in terms of the Focus and its Polar. We have seen (Arts. 439, 532, 618) that in every conic the distance of any point on the curve from the focus is in a constant ratio to its distance from the polar of the focus, the ratio being equal to the eccen- tricity. Hence, calling the focal distance />, and the distance from the polar (or directrix) o, we may write, as the equation to the Conic, p = e.d: which will denote an ellipse, a parabola, or an hyperbola, according as e is less than, equal to, or greater than unity. G36. It follows from the property mentioned above, that the Conic may be defined as the locus of a point whose distance from a fixed point is in a constant ratio to its distance from a fixed right line. In fact, this definition has been made the basis of several treatises upon the Conies. Calling the fixed point x f y f 9 the fixed right line Ax-\- By -f- (7=0, and the variable point of the curve xy, the equation to the Conic is If we suppose the arbitrary axes of this equation to be changed so that the given line shall become the axis of y, and a perpendicular to it through the given point the LINEAR EQUATION TO THE CONIC. 470 axis of x, we shall have B = C = 0, y' = 0, and a new value of x' which may be called 2p : e. We then get This represents an ellipse when e < 1, a parabola when e = 1, an hyperbola when e > 1 ; and, as it can be written is evidently the generic relation of which the equation in Art. 181 is a particular case. Remark. The formulas of this article are only modi- fied expressions of the relation p = e.d', and it is obvious, on comparing this with (1) above, that the focal distance of any point on a conic can always be expressed as a rational function of the co-ordinates of the point, in the first degree. We leave the student to prove the converse theorem, that a curve must be a conic, if the distance of every point on it from a fixed point can be expressed as such a rational linear function. 637. Linear Equation to the Conic. It is evident from what has just been said, that this title would cor- rectly describe either the expression of Art. 635 or the modified form of it given in (1) of Art. 636. But the phrase is in fact reserved to designate a still further modification of the same expression, which we will now obtain. Suppose the origin of abscissas to be at the focus, and the axis of x to be the perpendicular drawn through the focus to its polar: the distance from this polar (or direc- trix) to the point on the curve, will then be equal to the 480 ANALYTIC GEOMETRY. distance between the directrix and the focus, increased by the abscissa of the point; or, we shall have and the equation ft = e.d will become p = 2p -f ex. Remark. The so-called Linear Equations to the Ellipse, the Parabola, and the Hyperbola, namely (Arts. 360, 457, 568), will all assume the form just found, if we shift their respective origins to the focus, by putting x ae for x in the first, x + ae for x in the second, and x-\-p for x in the third. We leave the actual transformation to the student, only reminding him, that, in the first two curves, a ( I e 2 ) - 2p ; and that, in the third, e = l. 638. A form of the preceding equation with which the reader may sometimes meet, is r = mx -f- n, and any equation of this form, in which m and n are any two constants whatever, will denote a conic, whose eccen- tricity will = m, while its semi-latus rectum will = n. O39. Equation referred to Two Tangents. A useful ex- pression for the Conic may be developed as follows : Let the equation to the curve, referred to any axes whatever, be Ax 1 + 2Hxy + By* + 2Gx + 2Fy + C = Q (1). To determine the intercepts of the curve on the axes, we get, by making y and x successively = 0, Ax* + 2Gx + C=Q, Bif + 2Fy + C= 0. But if the axes are tangents, the two intercepts on each will be equal, these quadratics will have equal roots, and we shall have CONIC EEFEREED TO T \VO TANGENTS. 481 Putting into (1) the values of A and B which these conditions ive, ZCHxy + FY + 2GCx -f 2FCy + C 2 = 0. Whence, by adding 2FGxy 2FGxy, and re-arranging the terms, (Gx + *V + C)* = 2 (FG - C//) ary. This is the equation we are seeking; and, as the co-efficient of xy in it is arbitrary with respect to G f , F, C] we may write it (Gx + Fy + C? = Mxy (2), where G, F, C, M are any four constants whatever. Corollary 1. Making y and x successively = in this equation, we get the distances of the two points of contact from the origin, namely, C C ~G> y ~ ~F' Calling the first of these distances a, and the second K, we have - Corollary 2. The special modification of this which represents a parabola, deserves a separate notice. In order that (3) may denote a parabola, we must have (Art. 191) F2__ T JL. La/c ^J aV : o. condition satisfied by either p. = or /z = 4 : a/c. If // = 0, the equation becomes and denotes the chord of contact of the tangent axes. If ft 4 : a/c, we get, by taking the square root of both members of (3), 482 ANALYTIC GEOMETRY. or, after transposing and again taking the square root, an equation which is sometimes written T//O; -|- Vay = 1/a/c. G4O. Polar Equation to the Conic. By compar- ing Arts. 443, 535, 627, it becomes evident that the Conic may be represented by the general equation 2^ QN 1 e cos 6 In this, as the reader will see by referring to the original investigations (Arts. 152, 172, 183), the pole is at the focus, and the vectorial angle 6 is reckoned from the remote vertex. A more useful expression, however, and the only one universally applicable in Astronomy, is P ~~ = I-}-ecosd Here, 6 is reckoned from the vertex nearest the focus selected for the pole, and I denotes the semi-latus rectum. The conic represented is an ellipse, a parabola, or an hyperbola, according as e < 1, e = 1, or e > 1. 641. The Conic as the tocus of the Second Order in General. All the Cartesian equations that precede, are only reduced forms of the general and unconditioned equation Ax 2 + ZHxy + Bf -f 2Gx -f 2Fy + C= 0, which may be converted into any one of them by a proper transformation of co-ordinates, and to whose type they all conform. SYSTEM OF THE CONICS. 483 THE CONICS IN SYSTEM, AS SUCCESSIVE PHASES OF ONE FORMAL LAW. 642* That the Conies are successive phases of some uniform law, we have already seen in Section VII of Part I. Of the nature of that law, however, we were there unable to give any better account than this : that it expressed itself in the unconditioned equation of the second degree, and became visible in a threefold series of curves, determined by the successive appearance, in that equation, of the three conditions But we have now reached a position which will enable us to state the law in geometric language, to exhibit the elements of form which it embodies, and to trace the steps by which those elements cause the three curves to appear in an unbroken series. And it deserves especial mention, that this geometric statement of the law is fur- nished by the polar relation, as expressed in the definition of Art. 636. G43. The generic law of form which is designated by the name of The Conic, may therefore be stated as follows : The distance of a variable point from a fixed point shall be in a constant ratio to its distance from a given right line. Expressing this law in the equation (Art. 637) P = 2p + ex (1), let us observe the development of the system of the three curves, member after member, as the given line advances nearer and nearer to the fixed point. We have (Art. 637), for the distance of the given line from the fixed point, d = *JL (2). 484 ANALYTIC GEOMETRY. Also, supposing a perpendicular to the given line to be drawn through the fixed point, the perpendicular to be called the axis, and the point in which it cuts the curve to be called the vertex, we get, for the distance of this vertex from the fixed point, by making x = p in equa- tion (1), Let the generation of the system begin with the given line at an infinite distance from the fixed point. In that case, from (2), we shall have e = Q. Under this suppo- sition, equation (1) becomes which (Art. 138, Cor. 2) denotes a circle, described from the fixed point as a center, with a radius = the semi-latus rectum of the Conic : a result confirmed by the fact, that, under the same supposition, equation (3) becomes p'=2p-, or, the distance of the vertex from the focus becomes equal to the radius. Now let the given line move parallel to itself along the axis, assuming successive finite distances from the fixed point, but with the condition that every distance shall be greater than 2p. Then, from (2), e < 1 ; and, from (3), p' < 2p and > p : so that a continuous series of ellipses will appear, of ever-increasing eccentricity, but with a constant latus rectum, their vertices all lying within a segment of the axis p, contained between the points reached by measuring from the fixed point distances =p and 2p. Next, let the given line have attained the distance = 2jt? from the fixed point. We shall then have, from (2), e = 1 ; and, from (3), p' p. That is, we shall have & parabola, described upon the constant latus rectum. GENERATION OF THE SYSTEM. 485 Finally, let the given line advance from its last posi- tion, and approach the fixed point indefinitely. Then, d being less than 2p, we shall have, from (2), e >> 1 ; and, from (3), f/1 .-. Oblique. . Semi-latus rectum > Distance of Focal Polar J .-.HYPERBOLA. U^vT.-. Rect'r. PROPERTIES OF THE CONIC IN GENERAL. 645. The views thus far taken of the Conic in the present Chapter, although generic, have nevertheless been obtained from a standpoint not strictly analytic. For our results have been derived from a comparison of the properties in which the three curves, after sepa- rate treatment by means of equations based upon certain assumed properties, have been found to agree. But we shall now, for a few pages, ascend to the strictly analytic 486 ANALYTIC GEOMETRY. point of view, and, beginning with the unconditioned equation Ax 2 + ZHxy + Bf + 2Gx -f ZFy + (7=0, shall show how the properties common to the System of the Conies, or peculiar to its several members, may be developed from this abstract symbol, without assum- ing a single one of them. THE POLAR RELATION. 646. Intersection of the Conic with the Right JLiiie. If we eliminate between and y = mx -f- &, we shall obviously get a quadratic in x to determine the abscissa of the point in which the Conic cuts any right line. Hence, Every right line meets the Conic in two points, real, coincident, or imag- inary. In particular, for the points in which the curve meets the axes of reference, we get, by making y and x in (1) successively = 0, the determining quadratics Ax 2 -f 2Gx -\- C= 0, By 2 -f 2Fy -j- C= (2). 647. The Chord of the Conic. If a right line meets the Conic in two real points, x'y' and x rf y n , we may write its equation A (xx f ) (xx ir ) + 2H(xx') (yy") -f B (yy r ) (yy") For this is the equation to some right line, since, upon expansion, its terms of the second degree destroy each TANGENT OF THE CONIC. 487 other ; and to a line that passes through the points x'y' , x"y" of the curve, because if x' and y 1 or x" and y" be substituted for x and y in it, we either get or else which are simply the conditions that x'y', x"y" may be on the curve. 648. The Tangent of the Conic. Making x" = x', and y" = y r , in the preceding equation to the chord, we get the equation to the tangent, yj = Ax 2 + IHxy + Ef + 2 Gx + 2Fy + C\ which, after expansion, assumes the form ZAx'x + 2H(x'y + y'x) + ZBy'y + 2Gx + 2Fy + C Adding 2Gx f + %Fij + (7 to both members of this, and remembering that the point of contact x'y 1 must satisfy the equation to the Conic, we get the usual form of the equation to the tangent, namely, */') + ^=0 (1). By expanding, and re-collecting the terms, this may be otherwise written JFy+<7=0 (2). An. Ge. 44. ANALYTIC GEOMETRY. These equations express the law which always con- nects the co-ordinates of any point on the tangent with those of the point of contact. Hence, if a point through which to draw a tangent were given, and the point of contact were required, equation (1) would still express the relation between the co-ordinates of these points, only x'y' would then denote the given point through which the tangent would pass, and xy the required point of contact. That is, the equation which when x'y' is on the curve represents the tangent at x'y', when x'y' is sit- uated elsewhere denotes a right line on which will be found the point of contact of the tangent drawn through x'y' . Now this line, in common with every other, meets the Conic in two points: hence, From any given point, there. can be drawn to the Conic two tangents, real, coin- cident, or imaginary. We thus learn that our curve is of the Second class as well as of the Second order. 649. Chord of Contact in the Conic. From what has just been stated, it follows that or its equivalent form (2) above, is the equation to the chord of contact of the two tangents drawn through x'y'. 65O. -ILociis of the Intersection of Tangents whose Chord of Contact revolves about a Fixed Point. Let x'y f be the fixed point, and #,?/, the intersection of the two tangents corresponding to the chord. Then, as x'y 1 is by supposition always on the revolving chord of con- tact, we shall have the condition H(x iy f i + y') + 0= 0, POLAR IN THE CONIC. 489 irrespective of the direction of the chord. In other words, the intersection of the tangents will always be found upon a right line whose equation is Ax'x + H(x f y -f y'x) + By'y + G(x + ) + 0= 0. 651. The Point and the Right Une, Reciprocals with respect to the Conic. -The result of the preceding article may be stated as follows : If through a fixed point chords be drawn to the Conic, and tangents be formed at the extremities of each chord, the intersections of the sev- eral pairs of tangents will lie on one right line. Also, we may write the equation to the chord of con- tact of two tangents drawn through x'y' (Art. 649) (Ax' + Hy' + G) x + (ffz' + By' + F)y so that (Art. 117), if we suppose x'y' to move along a given right line, the chord of contact will revolve about a fixed point. In other words : If from different points lying on one right line pairs of tangents be drawn to the Conic, their several chords of contact will meet in one point. Combining these two properties, we see that our curve imparts to every point in its plane, the power of deter- mining a right line ; and to every right line, the power of determining a point. That is, it renders the Point and the Right Line reciprocal forms. 652. The Polar ami its Equation. We perceive, then, that the relation between a tangent and its point of contact, and the relation between the chord of contact and the intersection of the corresponding tangents, are only particular cases of a general law which, with respect 490 ANALYTIC GEOMETRY. to the Conic, connects any fixed point with a correspond- ing right line. From the result of the last article, more- over, it appears that we shall fitly express this law by calling the line which corresponds to any point, the polar (i. e. the reciprocal) of the point, and the point itself the pole of the line. We are henceforth, then, to consider the equation Ax'x + H(x'y + y'x) + By'y as in general denoting the polar of x'y' ; and must regard the tangent at x'y' as the position assumed by the polar when x'y' is on the Conic. 653. The Conic referred to its Axis and Ver- tex. Before taking out any additional properties of the curve, it will be best to reduce the general equation Ax 2 + ZHxy + By 2 + 2Gx + 2Fy + (7=0 to a simpler form. Supposing the axes of reference to be rectangular, let us revolve them through an angle #, such that tan 26 = A B We shall thus (Art. 156) destroy the co-efficient of xy, and the equation will assume the form Ax 2 + B'y 2 + 2G'x + ZF'y + (7=0. If in addition we remove the origin to a point x'y' , we shall get (Art. 163, Th. I) A r x 2 + B'y 2 + 2 (A'x r + G')x + 2 (B'y' + F') y + (A'x 12 + B'y' 2 -}- ZG'x' + ZF'y' + C) = 0. VERTICAL EQUATION TO THE CONIC. 491 In order, then, that the constant term and the co-efficient of y may vanish together, we must have simultaneously A'x' 2 + B'y' 2 + 2 G'x' + ZF'y' + (7=0, B'y' + F = ; that is, we must take the new origin #y at the inter- section of the curve with the right line By-\-F=Q. Making this change of origin, our equation becomes which, by putting 20" : B' =- P, and A' : B' = - R, and transposing, may be written Here, for every value of #, there will be two values of y, numerically equal with opposite signs : the curve is therefore symmetric to the new axis of x, which for that reason shall be called an axis of the curve. If we seek the intersections of the curve with the new axis of y, by making x = in the equation, we get y ; so that the new axis of y meets the curve in two coincident points at the origin ; or, in other words, is tangent to the curve at the origin. Besides, the new axes are rectan- gular: hence, combining this fact with those just estab- lished, the new origin is the extreme point, or vertex, of the curve ; and we learn that our axes of reference are the principal axis of the curve and the tangent at its vertex. And, in fact, our new equation is identical with that obtained in Art. 634. 654. Focus of the Conic, and its Polar. Taking up our equation in its new form (1), 492 ANALYTIC GEOMETRY. let e be such a quantity that e 2 = I + R (2), and let that point whose co-ordinates are be called the focus of the Conic. The equation to the polar of any point, referred to our present axes, is at once found from the general equation of Art. 652, by putting for A, H, B, G, F, C their values as given by (1). It is therefore P(x + x') (4). Hence, substituting for x f and y' the values given in (3), we get, for the polar of the focus, or, after replacing R by its value e 2 1 from (2), x= - P (6) Equation (6) shows that the polar of the focus is per- * pendicular to the axis of the Conic, and cuts it on the opposite side of the vertex from the focus, at a distance an eih part of the distance of the focus. And we shall see,, in a moment, that the ratio thus found between the distances of the vertex from the focus and from its polar, subsists between the distances of any point on the curve from those two limits. From (3) we have (Art. 51, 1, Cor. 1), for the distance p of any point xy from the focus, THE FOCUS AND ITS POLAR. 493 since xy is on the Conic. Replacing R by its value e 2 1, and reducing, we get 2(1 + ,) Also, for the distance from xy to the polar of the focus, we have, from (6) by Art. 105, Cor. 2, , 2 (1 + e) ex + P 2(1+,), Hence, dividing (7) by (8), we get H (9). That is, The distance of any point on the Conic from the focus, is in a constant ratio to its distance from the polar of the focus. The ratio e, we will call the eccentricity of the Conic. 655. The Species of the Conic, and their Fig- ures. The preceding investigation leads directly to the resolution of the vague and general Conic into three specific curves, and the generic property just developed will enable us at once to determine the figures of these. For since e 2 = 1 -(- R, we shall evidently have according as R is negative, equal to zero, or positive. Therefore, by embodying the property of (9) in the mechanical contrivances described in the corollaries to Arts. 439, 532, 618, we can generate three distinct curves, depending on the value of e ; as follows : In the first, e will fall short of 1 : whence the curve may be called an ellipse. 494 ANAL YTIC GEOMETR Y. In the second, e will equal 1 : whence the curve may be called a parabola. In the third, e will exceed 1 : whence the curve may be called an hyperbola. The figures of these curves are therefore such as the methods of generation give, and need not be drawn here, as they are already familiar. If we put x=P : 2 (1 -f e) in equation (1) of the pre- ceding article, we get, for the ordinate erected at the focus, * = --: y - 2 whence, calling the double ordinate through the focus the latus rectum, latus rectum = P (1). We thus obtain a significant interpretation for the par- ameter P of our equation ; but we do more. For, by the generic property of the preceding article, the distance of the focus from its polar must equal an eth part of its distance from the extremity of the latus rectum, and therefore can now be expressed by Hence, when e < 1, the semi-latus rectum will be less than the distance of the focus from its polar ; when e = 1, it will be equal to that distance ; and when e > 1, it will be greater than that distance. In other words, the classification reached above, is identical with that of Art. 644 : as may be further shown by the fact, that, if R = 1, e = ; and if R = + 1, e = 1/2. DIAMETERS AND THE CENTER. 495 Also, when e 0, equation (7) of the preceding article gives That is, when e = 0, the curve is such that all its points are equally distant from the focus, or its figure is that of the Circle. Hence, as e increases from toward oo, the figure of the curve may be supposed to deviate more and more from the circular form, and we see the pro- priety of calling e the eccentricity. DIAMETERS AND THE CENTER. 656. A very significant question in regard to any curve is, What is the form of its diameters, that is, of the lines that bisect systems of parallel chords in it ? Let us, then, settle this question for the Conic. 657. Equation to any Diameter. If we suppose 6' to be the common inclination of any system of parallel chords, x'y' the intersection of any member of the system with the Conic, and xy its middle point, we shall have (Art. 102) x f = x Zcos#', y'y where I is the distance from xy to x'y'. But since x'y' is on the Conic, we get, by equation (1) of Art. 646, A(x C=0. Expanding, collecting terms, and putting S for the first member of the general equation of the second degree, we get (A cos 2 0" + 2Hcos tf sin 0' + B sin 2 P) I' 2 2[(Ax + H An. Ge. 45. 496 ANALYTIC GEOMETRY. Now xij being the middle point of a chord, the two values of I given by this quadratic must be numerically equal with opposite signs. Hence, (Alg., 234, Prop. 3d,) the co-efficient of I vanishes, and we obtain, as the equa- tion to any diameter, (Ax -f- Hy + G) + (Hx + By -f F) tan ' 0' = 0, in which 6 f is the inclination of the chords which the diameter bisects. 658. Form and Position of IMamcters. Com- paring the equation just obtained with that of Art. 108, we learn that every diameter of the Conic is a right line, and passes through the intersection of the two lines G = Q, Hx + By + F = ; that is, through the point whose co-ordinates (Art. 106) are BGHF _ AFHG m ~ H* - AB ' ~ H* Moreover, putting = the inclination of any diameter, we have (Art. 108) A -f H tan 0' so that, as 6' is arbitrary, a diameter may have any in- clination whatever to the axis of x; or, every right line that passes through the point (1) is a diameter. Hence, as (1) is in turn upon every diameter, it is the middle point of every chord drawn through it, and may there- fore be called the center of the Conic. For the form and position of conic diameters in gen- eral, we therefore have the two theorems : .Every diam- eter is a right line passing through the Center; and, Every right line that passes through the center is a diameter. DIAMETERS AND THE CENTER. 497 Hence, the lines Ax + ffi/+G = are both diameters ; and, by making 6 r in the final equation of Art. 657, we learn that the former bisects chords parallel to the axis of x ; while, by making 0'= 90, we see that the latter bisects chords parallel to the axis of?/. From (1) we see that the center of the Conic will be at a finite distance from the origin, so long as H 2 AB is not equal to zero ; but will recede to infinity, if H 2 =AB. Now, by putting for A, B, F, G, H the values they have .when the equation to the Conic takes the form y 2 = Px + Rx\ we get the co-ordinates of the center, referred to the prin- cipal axis and its vertical tangent, namely, * = -4 , = (3), which show that the center is situated on the principal axis, at a distance from the vertex = Pi 2R. This distance, then, will be finite if R is either positive or negative, but infinite if R = 0. Hence, (Art. 655,) the diameters of the two curves which we have named the Ellipse and the Hyperbola, meet in a finite point, and are inclined to each other; but the diameters of the curve called the Parabola meet only at infinity, or, in other words, are all parallel: a result corroborated by the fact, that, if in (2) we replace A by the value H 2 : B, which it will have if H 2 AB, we get tan0= -J (4), showing that all the diameters of any given parabola arc equally inclined to the axis of x. 498 ANALYTIC GEOMETRY. 659. Farther Classificatin of the Conic. It thus appears that the Ellipse and the Hyperbola may be classed together as central conies, while the Parabola may be styled the non-central conic. Adding this prior subdivision, the table of Art. 644 will appear thus : Semi-latus Rectum < Dist. of Foe. " ' .-. ELLIPSE. ( e Dist. of Foe. Pol. ( e > l ' Oblique. .-. HYPERBOLA. | e = 1/2".-. Rect'r. NON- f Semi-latus Rectum = Dist. of Foe. Pol. CENTRAL 1 .'. PARABOLA. CONJUGATE DIAMETERS AND THE AXES. 66O. Relative inclination of Diameters and their Ordinates. The halves of the chords which a diameter bisects may be called its ordinates. If, then, 6 = the inclination of any diameter, and 6' = that of its ordi- nates, we have, from (2) of Art. 658, tan tan 6' + JT(tan -f tan d') + A = Q as the relation always connecting the inclinations of a diameter and its ordinates. Now this may either be read as the condition that the diameter having the inclination 6 may bisect chords having the inclination 6 1 ', or vice versa. Hence,. If a diameter bisect chords parallel to a second, the second will bisect chords parallel to the first. This property is however restricted to the central conies ; for it is impossible that the ordinates of any parabolic diameter should be parallel to a second, since all parabolic diameters are parallel to each other. AXES OF THE CONIC. 499 661. Condition that two Diameters be Conju- gate. We indicate that two diameters of a central conic are in the relation above-mentioned, by calling them conjugate diameters. Since, then, the conjugate of any diameter is parallel to its ordinates, by interpreting 6 and 6 f as the inclinations of two diameters, B tan 6 tan O f + IT (tan 6 + tan 6') + A = becomes the condition that the two diameters may be conjugate. 662. The Axes, and their liquation. The condi- tion just established may be referred to the principal axis and vertex of the Conic by putting for A,B, ^T the values they have in the equation y 2 = Px + fix 2 . It thus becomes tan tan/?' = E (1). In this, if we suppose 6 = 0, but not otherwise, we get tan 6 f = GO, and learn that the conjugate of the principal axis is perpendicular to it. Hence. In a central conic there is one, and but one, pair of rectangular conjugates.* We will call these rectangular conjugates the axes of the conic. The one hitherto named the principal, shall now be termed the transverse axis ; and the other, the conjugate axis. Their respective equations, referred to the same system as the conic y 2 = Px -f- Jlx 2 , will be P=Q (2). # Unless the conic is a circle : when R will = 1, and the condition (1) will become 1 + tan 9 tan 0' = ; so that (Art. 96, Cor. 1) all the con- jugates will be at right angles. 500 ANALYTIC GEOMETRY. For the first equation represents the axis of x ; and the second, a perpendicular to it passing through the center. Hence, P)y = (3) is the equation to both axes, in the same system of reference. 663. Equation to the Conic, in its Simplest Forms. If in the equation (1), we suppose JR=Q . . e = l, the quantity P : 2 (1 -f e), which by (3) of Art. 654 denotes the distance of the focus from the vertex, will become = ^ P: showing that in the Parabola the latus rectum P is equal to four times the focal distance of the vertex. Putting this latter dis- tance JP, and giving to R in (1) its corresponding value 0, the equation to the Parabola will be (2). But if R be positive or negative, or (1) denote the Ellipse or the Hyperbola, this simplification is impossi- ble. If, however, we transform (1) to the center and axes, by putting [x (P : 27)] for x 9 we get or, after obvious reductions, 4E 2 x 2 4 2 = P 2 . Here, making y and x successively = 0, we obtain, for the lengths of the semi-axes, p P r~r X = -r- _ , 77 -f- \ I _ "212' y ~2V R- ASYMPTOTES OF THE CONIC. 501 Putting a to denote the first of these lengths, and b to denote the second, we get :, Z = - (A), and the central equation becomes the upper sign corresponding to R, and the lower to + R. Equations (2) and (3) are the simplest forms of the equation to the Conic. In them, we have reached the same forms with which we set out upon the separate in- vestigation of the Parabola, the Ellipse, and the Hyper- bola. Of course, then, we can now develop all the prop- erties derived from them in the preceding Chapters, and the reader will be convinced of the adequacy of the purely analytic method without proceeding farther. We will therefore present but a single topic more, whose treat- ment from the generic point of view has an especial interest. THE ASYMPTOTES. 664. We have shown (Art. 646) that every right line meets the Conic in two points, real, coincident, or imag- inary. A particular case of the real intersections deserves notice. The quadratic by which we determine the intersections of a right line with the Conic, may sometimes take the form of a simple equation, by reason of the absence of the co-efficient A or B in the equation to the Conic. Thus Hxy + Bif + 2Gx + 2Fy+C= (1) 502 ANALYTIC GEOMETRY. gives, on making y = 0, only the simple equation- 2Gx+C=Q (2) to determine the intersections of the curve with the axis of x, apparently indicating but a single intersection. In fact, it does indicate a single finite intersection; but it is a settled principle of analysis (Alg., 238) that an equation arising in the manner (2) does, shall be re- garded as a quadratic of the form Ox 2 -f 2Hx + 5 = 0, one of whose roots is finite, and the other infinite. Hence, the consistent interpretation of such an equation as (2), will be that the corresponding line meets the Conic in one finite point and in one point infinitely distant from the origin. 665. Transforming the general equation to polar co-ordinates, we obtain (A cos 2 6+2H cos 6 sin + S sin 2 6) f) 2 + 2(#cos0 + .Fsin0)/> + C=0 (1). The condition, then, that the radius vector may meet the Conic at infinity is A cos 2 -f 2JF/cos 6 sin + B sin 2 = (2) : a quadratic in 6, and therefore satisfied by two values of the vectorial angle, which evidently will be real, equal, or imaginary, according as If 2 AB is greater than, equal to, or less than zero. Hence, as the origin may be taken at any point, Through any given point there can be drawn two real, coincident, or imaginary lines which will meet the Conic at infinity. ASYMPTOTES OF THE CONIC. 503 Moreover, since a change of origin (Art. 163, Th. I) does not affect the co-efficients A, H, B, the directions of these lines for any given conic will in all cases be determined by the same quadratic (2). That is, All lines that meet the Conic at infinity are parallel. It is to be noted, that in general each of the radii vectores determined by (2) also meets the curve in one finite point, whose position is given by the finite terms of (1), namely, by 2 (G cos + F sin 0) p + C= (3). A convenient method of finding the equation to the two lines which pass through the origin and meet the curve at infinity, will be to multiply (2) throughout by ft\ and then put x for ft cos 6, and y for ft sin d. We thus obtain Ax 2 + ZHxy + By 2 = (4), the equation of Art. 127. The two lines, then, in case the conic is an ellipse, will be imaginary; in case it is a parabola, they will be coincident; and in case it is an hyperbola, they will be real. 666. If we now suppose the general equation to be transformed to the center, the co-efficients G and F (Art. 163, Th. Ill) will vanish. For that origin, then, the condition (2) of the preceding article will occur simultaneously with the disappearance of the co-efficient of ft in (1), and the roots of the latter equation will therefore be simultaneously infinite and equal. Hence, Through the center there can be drawn tivo lines, each of which will meet the Conic in two coincident points at infinity. These tangents at infinity may appropriately be called asymptotes; since the curve must converge to them as it 504 ANALYTIC GEOMETRY. recedes to infinity, but can not merge into them except at infinity. Since, then, all the lines that meet the curve at infinity are parallel, equation (4) of the preceding article will in general denote a pair of parallels to the asymptotes, passing through the origin. Hence, sup- posing the center to be the origin, we have, for the equation to the asymptotes, Ax* + 2Hxy + Btf = (1), since the co-efficients A, H, B remain the same for every origin. This is the same as saying, that, given any central equation to a conic, the asymptotes are found by equating to zero its terms of the second degree. The form of (1) shows that these lines are real in the Hyperbola, coincident in the Parabola, and imaginary in the Ellipse. If now, in the condition of'Art. 661, we make tan 6' ' = cot 0, the corresponding conjugates will be at right angles ; that is, they will be the axes. But then #tan 2 6 + (A B) tan H= 0. Multiply this by /> 2 , put x for p cos 0, and y for p sin : then Hx*-(A-B)xy-H>f = Q (2), the equation to the axes, if the center is origin. Now (2) is the equation of Art. 129, and therefore denotes two right lines bisecting the angles between the lines represented by (1). Hence, The axes bisect the angles between the asymptotes, and are real whether the asymptotes are real or imaginary. CONDITIONS DETERMINING A CONIC. The general equation of the second degree, Ax 2 -}- 2Hxy + By- CONDITIONS DETERMINING A CONIC. 505 may of course be divided through by any one of its co- efficients, and therefore contains five, and only five, arbitrary constants. Hence, Five conditions are neces- sary and sufficient to determine a conic. Thus, a conic may be made to pass through five given points ; or, to pass through four points and touch a given line ; or, to pass through three points and touch two given lines ; etc. And in case the equation to a conic contains less than five constants, we must understand that the curve has already been subjected to a series of condi- tions, equal in number to the difference between five and the number of constants in its equation. Thus, the conic y* = Px + llx 2 has already been subjected to three conditions; namely, passing through a given point (the vertex), touching a given line (the axis of ?/), and having the focus on a given line (the axis of x). The solution of two general problems which are often of use in connection with conies, may conveniently be presented here. I. To determine the relation between the parameters P and R in the vertical equation to the Conic. From the first of the equations at (A) in Art. 663, we have (1). Also, by combining both of the equations at (A), (2). II. To determine the axes and eccentricity of a conic given by the general equation. Comparing (3) of Art. 506 ANALYTIC GEOMETRY. 663 with (3), (:** -' OC.ODsmCOD OD.OAsmDOA . c d Substituting in (1), and reducing, sin A OB sin COD _ , a . c ,~\ = But (Art. 285) the first member of (2) is the anharmonic of the pencil 0-ABCD, and the second is constant. Hence, The anharmonic of a pencil radiating from any point of a conic to four fixed points of the curve is constant. 671. Definitions. In any hexagon, two vertices are said to be opposite, when they are separated by two others. Thus, if J., _#, (7, D, E, F are the six successive vertices, A and _D, B and E, C and F are opposite. Two sides are also called opposite when separated by 508 ANALYTIC GEOMETRY. two others. Thus, AB and DE, BC and EF, CD and FA are opposite sides. Opposite diagonals are those which join opposite ver- tices, and are therefore three in number ; namely, AD, BE, OF. 672. Pascal's Theorem. Let 0, /3, f, X, /*, v be the successive sides of a hexagon inscribed in any conic. Then, if d be the diagonal joining the opposite vertices va and y/, the equations a r fc/fcj -- 0, fa IfjLd = (1) will each represent the conic. We may therefore sup- pose the constants k and I to be so taken that 07- kfto is identically equal to fa l[w; that is, in such a manner that ar h = (kp l/Ji)d (2). Hence, all the conditions that will cause 07 fa to vanish identically, are included in 3 = 0, kfi lp = (3). Now the points v0, yA evidently satisfy a.f fa = 0, and these by hypothesis are on the line d = : so that the points ax, j-v, which also satisfy 07- fa = 0, but which by hypothesis are not on the line d 0, must lie upon the line &/? 1/2 = 0. But, by the form of its equation, this line contains the point p[j>. In short, 0.x, y9//, ^v, which are the intersections of the opposite sides of the hexagon, are all on the same line. Hence, The opposite sides of any hexagon inscribed in a conic intersect in three points which lie on one right line. This is known as Pascal's Theorem. From this single property, its discoverer BLAISE PASCAL is said to have developed the entire doctrine of the Conic, in a system THEOREMS OF PASCAL AND BRIANCHON. 509 of four hundred theorems, when he was but sixteen years old; but his treatise was never published, and has un- fortunately been lost. Leibnitz, however, has given a sketch of it, in a letter written in 1676 to Pascal's nephew Perier. By joining six points on a conic in every possible way, we can form sixty different figures, each of which may be called an inscribed hexagon, and in each of which the intersections of the opposite sides will lie on one right line. Consequently there are sixty such lines for every six points on the curve, which are called the Pascal lines, or simply the Pascals, of the corresponding conic. O73. Brianciaon's Theorem. If we take the sym- bols of the preceding article as tangentials, , /9, 7-, X, /;., v will be the vertices of a hexagon circumscribed about a conic, and 3 will denote the intersection of the opposite sides va, fL The equations at (1) will then be tangen- tial equations to the conic, and the relation (2) will show that the three lines aX, /?//, ?v intersect in the same point Jcft 1/2 = 0. That is, The three opposite diagonals of any hexagon circumscribed about a conic meet in one point. This is known as Brianchon's Theorem, having been discovered in the early part of the present century by BRIANCHON, a pupil of the Polytechnic School of Paris. It was one of the fruits of Poncelet's Method of Recip- rocal Polar s. By producing six tangents to a conic till they meet in every possible way, we can form sixty different figures, each of which may be called a circumscribed hexagon. Consequently, for every six points of a conic, there are sixty different Brianchon points, determined by the system of six tangents; just as there are sixty different Pascal lines, determined by the system of six chords. 510 ANALYTIC GEOMETRY. EXAMPLES ON THE CONIC IN GENERAL. 1. If two chords at right angles to each other be drawn through a fixed point to meet any come, to prove that = constant ' where $, s are the segments of one chord, and /S* 7 , s / the segments of the other. 2. If through a fixed point O there be drawn two chords to any conic, and if their extremities be joined both directly and trans- versely, to prove that the line PQ which joins the intersection of the direct lines of union to the intersection of the transverse ones is the polar of O. 3. Prove that any right line drawn through a given point to meet a conic, is cut harmonically by the point, the curve, and the polar of the point; also, that the chord through any given point, and the line which joins that point to the pole of the chord, are harmonically conjugate to the two tangents drawn from the point. 4. A conic touches two given right lines : to prove that the locus of its center is the right line which joins the intersection of the tangents with the middle point of their chord of contact. 5. Prove that in any quadrilateral inscribed in a conic, as ABCD, either of the three points E, F, O is the pole of the line which joins the other two. By means of this property, show how to draw a tangent to any conic from a given point outside, with the help of the ruler only. [This graphic problem is only one of a series resulting from the method of transversals and anharmonics, all of which are solvable with the ruler alone : for which reason > the doctrine of the solutions is sometimes called Lineal Geometry.'] 6. Prove that in any quadrilateral circumscribed about a conic, each diagonal is the polar of the intersection of the other two. BOOK SECOND: CO-ORDINATES IN SPACE. An. Ge. 46. (511) CO-ORDINATES IN SPACE. 674. In removing, at this point in our investigations, the restriction which has confined loci to a given plane, we shall only enter upon the consideration of the most elementary parts of the Geometry of Three Dimensions. That is, we shall only undertake to give the student a clear general outline of the principles by which we rep- resent and discuss the surfaces of the First and Second orders. In order to accomplish this, we must begin, as in the case of the Geometry of Two Dimensions, by explaining the conventions for representing a point in space. CHAPTER FIRST. THE POINT. 675. About a century after the publication of Des- cartes' method of representing and discussing plane curves, CLAIRAUT extended the method to lines and surfaces in space, by the following contrivance for representing the position of any conceivable point in space. (513) 514 ANALYTIC GEOMETRY. Let XY, YZ, ZX be three planes of indefinite extent, intersecting each other two and two in the lines X'X, Y'Y, Z'Z. [The point Y f is supposed to be concealed behind the plane ZX in the dia- gram.] Then, if P be any point whatever in the surrounding space, its position will be known with reference to the three planes so soon as we find the length of PM drawn parallel to OZ, and of ML, MN drawn parallel respectively to OF and OX-, or, which is obviously the same thing, so soon as we find the lengths of OL, LM, MP. In the diagram, the three planes are represented at right angles to each other : a restriction which has the advantage of simplifying the whole subject, and which can always be secured by a proper transfor- mation, if the planes are in fact inclined at any other angle. We shall therefore suppose, in our investigations, that these reference-planes are always rectangular, unless the contrary is stated. The distances OL, LM, MP, or their equals OL, ON, OS, are called the rectangular co-ordinates of P, and are respectively represented by x, y, z. The lines OX, OY, OZ, of indefinite extent, are termed the axes : OX is the axis of x, OY the axis of y, and OZ the axis of z. The point 0, in which the three axes intersect, and which is therefore common to the three reference-planes, is named the origin. The reference-planes evidently divide the surrounding space into eight solid angles, which are numbered as RECTANGULAR CO-ORDINATES IN SPACE. 515 follows: Z-XOY is the first angle; Z- YOX', the second; Z-X'OY', the third; and Z-Y'OX, the fourth. Similarly, Z f - XOY is the fifth angle; Z'-YOX', the 8ta/i ; Z'-X'OY', the semi/A; and Z'-Y'OX, the eighth. By affecting the co-ordinates a:, ?/, 2 with the proper sign, we represent a point in either of the eight angles. Thus, First angle : x -fa, y = + J, 2 = + 9 g = Seventh " x = a, t/ = 6, 2; c Eighth " x = -f a, y 6, z c. The student will observe that the positive x lies to the right of the first vertical plane YZ, and the negative x to the left of that plane ; the positive y, in front of the second vertical plane ZX, and the negative y in the rear of that plane ; the positive 2, above the horizontal plane JTF, and the negative z below that plane. Corollary 1. For any point in the plane YZ, we shall evidently have x = (1), while y and z are indeterminate. Equation (1) is there- fore the equation to the first vertical reference-plane. For any point in the plane ZX, we shall have y = o (2), while z and x are indeterminate. Hence, (2) is the equation to the second vertical reference-plane. 516 ANALYTIC GEOMETRY. Finally, for any point in the plane XY, we shall have z = (3), while x and y are indeterminate. Hence, (3) is the equation to the horizontal reference-plane. Corollary 2, If a point is on the axis X'X, we shall have y = 0, z = simultaneously, while x is indetermi- nate. If the point is on the axis Y' Y, z = 0, x = simultaneously, while y is indeterminate. If the point is on the axis Z r Z, x = Q 9 y = simultaneously, while z is indeterminate. Hence, the pairs 3r = OV z=Q\ x = z = 0/ ? *=0| y = are respectively the equations to the axis of x, the axis of y, awcZ the axis of z. Corollary 3. At the point O, where the axes intersect each other, we shall evidently have, simultaneously, x = y = z = 0, and these three equations are the symbol of the origin. POLAR CO-ORDINATES IN SPACE. 676. If MN be a fixed plane, OX a fixed line in it, and a fixed point in that line, then, if any point P in the surrounding space be joined with 0, and a plane be passed through OP per- pendicular to MN, so as to intersect the latter in the line OR, the distance OP, and the angles FOR, ROX, POLAR CO-ORDINATES IN SPACE. 517 are called the polar co-ordinates of the point P. The distance OP is called the radius vector, and is repre- sented by the letter p; the angles POR, ROX are termed the vectorial angles, and are designated respect- ively by

cos /9, z = p cos ^. Squaring and adding these equations, and observing that p 2 = x 2 -f- y 2 -f- z 2 , we get, for the relation mentioned, cos 2 a -f cos 2 ft + cos 2 f = 1. 522 ANALYTIC GEOMETRY. POINT DIVIDING THE DISTANCE BETWEEN TWO OTHERS IN A GIVEN RATIO. 683. By an investigation analogous to that of Art. 52, the details of which the student can easily supply, the co- ordinates of such a point are found to be mx 2 -j- m -j- n = J ~ #j + n y\ z = mz. m -\- n TRANSFORMATION OF CO-ORDINATES. 684. To transform to parallel reference -planes passing through a new origin. Let x f 9 y f j z' be the co-or- dinates of the new origin, #, o " " i/, z the primitive co-ordinates of any point P, and X, Y, Z its co-ordinates in the new system. Then, as is evident upon inspecting the diagram, the formulae of transformation will be 685. To transform from a given rectangular system to a system having its planes at any inclination. Let the direction-angles of the new axis of x be , ft, Y ; those of the new axis of T/, a', /9 r , f ; and those of the new axis of 2, a", /9", f. Then, if we suppose each of the new co-ordinates N'P, M'P, Q'P to be projected TRANSFORMATION OF CO-ORDINATES. 523 on one of the old axes, the sum of the three projections (Art. 680, Cor.) will in each case be equal to the projec- tion of the radius vector OP. But the projection of OP on OX will be equal to the old x of P; its projection on OF, to the old y\ and its projection on OZ, to the old z. Hence, (Art. 679,) x = X cos a -f Y cos a! -f- Z cos a!', y = A"cos/9 + Fcos/5' + Zcos/?", 2 = A 7 " cos f -}- F cos f' -f Z cos f", are the required formulae of transformation. Remark, It must be borne in mind, in using these formulae, that the direction-cosines of the new axes are subject to the conditions (Art. 682) cos 2 a -\- cos 2 /9 + cos 2 f = 1, cos 2 a' -j- cos 2 ft + cos 2 f = 1, cos 2 a" -j- cos 2 /3" + cos 2 f r = I. 6860 To transform from a planar to a polar system in space. Let the planar system be rectangular. Then, the co- ordinates of any point P in the two systems being related as in the diagram, it is evident that we shall have x = p cos y = p cos z = sin cos , sin 6, From these equations we can evidently also find />,^>, in terms of x, y, z. 524 ANALYTIC GEOMETRY. Remark. To combine a change of origin with this transformation or that of the preceding article, we have merely to add the co-ordinates a/, y' , z' of the new origin to the values found for x, y, and z. GENERAL PRINCIPLES OF INTERPRETATION. 687. These follow from the convention of co-ordinates in space in much the same manner as the principles of Plane Analytic Geometry followed from the convention of plane co-ordinates. We may therefore state them without further argument, as follows: I. Any single equation in space-coordinates represents a surface. To hold with full generality, this statement must be understood to include (in addition to surfaces in the ordinary sense) imaginary surfaces, surfaces at infinity, surfaces that have degenerated into lines or points, and surfaces combined in groups. II. Two simultaneous equations in space-coordinates represent a line of section between two surfaces. This principle is also to be taken with restrictions corresponding to those above stated. III. Three simultaneous equations in space-coordinates represent mnp determinate points. These are the points of intersection of three surfaces, supposed to be of the w th , w th , and p ih order respectively. IV. An equation which lacks the absolute term, repre- sents a surface passing through the origin. V. Transformation of co-ordinates in space does not alter the degree of a given equation, nor affect the form of its locus in any way. THE PLANE. 525 CHAPTER SECOND. LOCUS OF THE FIRST ORDER IN SPACE. 688. Form of the Locus. The general equation of the first degree in three variables, may be written Ax+By+Cz+D = Q (1), where A, J9, (7, D are any four constants whatever. Transforming (1) to parallel axes passing through a new origin x'y', we get (Art. 684) Ax + By + Cz + (Ax' + BiJ + Cz r + D) = 0. Hence, if we suppose the new origin to be any fixed point in the locus of (1), the new absolute term will vanish, and our equation will take the form Ax + JBy+Cz = Q (2). If we now change the directions of the reference- planes (Art. 685), we shall get, after expanding and collecting terms, (A cos a -{ B cos /? -f- C cos 7- ) x ^ 4- (A cos a f + B cos ft' +Ccos r ')y > = 0. + (A cos a!' + B cos /3" + (7 cos r "} z ) Hence, if we can take the new reference-planes so as to give the new axis of x and the new axis of y such directions that A cos a -J- B cos /9 -f- C cos y = == A cos a! + cos /9' + (7 cos / (Q), we shall reduce our equation to the simple form Z = (3). 526 ANALYTIC GEOMETRY. Now, obviously, the transformation from (1) to (2) is always possible ; and that we can always effect the transformation from (2) to (8) will readily appear. For Ave can leave the primitive vertical reference-planes unchanged, obtaining our new system by merely revolv- ing the primitive horizontal plane about the origin: in which case, we shall have = 90 r , = 0; a' = 0, /S'^OO r f ; and the conditions at (Q), upon which the transformation we are now considering depends, will become C A sin f -f C cos ? = i. e. tan ^ _ , J*. B sin/ -f C cos/ = i.e. tan/ = _^: JD suppositions compatible with any real values of J., , C. We conclude, then, that by a proper transformation of co-ordinates we can always reduce the general equa- tion of the first degree to the form 3 = 0. But this [Art. 675, Cor. 1, (3)] denotes the new refer- ence-plane XY. Hence, (Art. 687, V,) The locus of the First order in space is the Plane; or, as we may otherwise state our result, Every equation of the first degree in space represents a plane. THE PLANE UNDER GENERAL CONDITIONS. 689. General Form of ttoe Equation to the Plane. From what has just been shown, we learn that Ax -f By -f Cz + D = is the Equation to any Plane. PLANE UNDER GENERAL CONDITIONS. 527 o The Plane in terms of its Intercepts on the Axes. Let the plane ABC, making upon the co- ordinate axes the intercepts OA = a, OB b, OC=c, represent any plane Ax -f By + Cz + D = 0. Making ?/ and 0, 2- and x, x and ^ simultaneously = in succession, we obtain from this equation (Art. 675, Cor. 2) D x= a = r .-. A D y = t=-- .-. z = c = D Substituting these values of A 9 B, C in the general equation, we obtain the equation to the Plane in terms of its intercepts, namely, The Plane in terms of the Direction-cosines of its Perpendicular. Let the perpendicular from the origin upon any plane be = p, and let its direction-angles be a, /?, f. Then, a, 6, c being the intercepts of the plane, we shall have cos a. cos COSf 528 ANALYTIC GEOMETRY. Substituting these values in the equation of the preced- ing article, we obtain x cos o.-\-y cos /? -f- z cos fp, the equation to the Plane in the terms now required. 692. Reduction of the General Equation to the form last found. We may suppose the reduction to be effected by dividing the general equation Ax + By + Cz + D = throughout by some quantity Q. If so, we shall have A = Qcosa, .5=^()cos/3, C=Qcosf: whence Q 1 (cos 2 a + cos 2 /9 4- cos 2 r ) = A 2 + B 2 -j- <7 2 . Now (Art. 682), cos 2 a -f cos 2 ^ + cos 2 f = 1. Hence, 1- ^ + 6"; and we learn that cos == COS -=: (7 ~ and that, for the perpendicular from the origin upon a plane given by the general equation, we have D P= By always taking the radical Q with that sign which will render p positive, the resulting signs of cos , cos /?, PLANE UNDER SPECIAL CONDITIONS. 529 cos f will indicate whether the direction-angles of the perpendicular are acute or obtuse. THE PLANE UNDER SPECIAL CONDITIONS. 693. Equation to a Plane passing through Three Fixed Points. By a process exactly analogous to that of Art. 95, this is found to be C (y" '"-y"V )*' -j + (y"V -y' z'")x" (. + (y' *" -y" ' )*"', in which x'y'z', x n y"z", x trr y'"z" f are the three points which determine the plane. 694. Angle between two Planes. This is evidently equal, or else supplemental, to the angle between the perpendiculars thrown upon the planes from the origin. Now, if p, p' be the lengths of these perpendiculars, , /9, 7- and ', /3', f their direction-angles, and d the distance between the points xyz, x'y'z' in which they pierce their respective planes, we shall have (Art. 681) where

be the radius vector of any point xyz on the parallel, we shall have (Art. 681, Cor.) and, from the above equations of projection, mx nx 538 ANALYTIC GEOMETRY. Solving the last three equations for x, y, z, we obtain _ Iff mp = 2 * ' = np But, by the doctrine of projections, x = p cos , y = p cos /?, z = p cos T. Substituting, and dividing through by p, I n m cos a = - , cos Q = n Corollary, To find the direction-cosines of a line whose projections are given in any form whatever, throw its equations into the form x x f y y f z z' . I m n when the required functions will be Z, m, n, each divided by vV H- wi* + w 2 . TO8. Angle between two JLines in Space. The angle between two right lines in space is obviously equal to that between their respective parallels through the origin. Hence, by formula (A) of Art. 694, cos 6 = cos a cos a! -\- cos /9 cos ft' + cos f cos f (1) : which expresses the angle between two right lines in terms of their direction- cosines. PLANE ANGLES IN SPACE. 539 Substituting for cos , cos a', etc., from the preceding article, we get * _ II' -j- mm f -f- nn' ,y\ . ~V(J 2 + m 2 + n 2 ) (I 12 + w' 2 + w'*) which expresses the aftgrZe between two right lines in terms of their projections. Corollary 1, The condition that two right lines in space shall be parallel, derived from (1), is cos a cos a' -\- cos ft cos ft -f cos 7- cos f 1 (1), or, derived from (2) by steps analogous to those in the first corollary of Art. 694, V I t m f _ m . n' _ n ,<^ ^" = m' ~^'~~~~n' J~~~~1 Corollary 2. The condition that two right lines in space shall be perpendicular to each other, derived from (1), is cos a cos a! -f- cos ft cos ft' -f cos f cos f (1), or, derived from (2), W + mm' + nn' = Q (2). 709. Equation to a Right Line perpendicular to a given Plane. If Ax + By -f Cz + i> = be the given plane, the required equation may be written x a y b z c ~A~ ~1T ~~C~ For we may suppose the perpendicular to pass through any fixed point abc; and, by Art. 692, its direction- cosines must be proportional to A, B, C. 710. Angle contained between a Right Line and a Plane. This being the complement of the angle 540 ANALYTIC GEOMETRY. contained between the given line and a perpendicular to the plane, if the given line be z z' I we have, by comparing Arts. 708, 709, Al + Bm + On sin 6 = y (A 2 -f B* + C 2 ) (F -f- m 2 + n?) Corollary, The condition that a right line shall be parallel to a given plane, is Al + Bm 4- On = 0. 711. Condition that a Right Line shall lie wholly in a given Plane. If a right line lies wholly in a given plane, the z co-ordinate resulting from an elimination between the equation to the plane and those of the line must of course be indeterminate. Hence, if the plane be Ax -\- By -f Cz -j- D = 0, and the line (x = mz -\- a, y = nz -j- 6), so that we have by elimi- nation A (mz -f a) -f B (nz -f b) -f Cz + D = 0, or Aa + M + D ~ Am + En + 0" we must have, as the condition required, the simultaneous relations Am + Bn+ 0= Aa -f 56 + Z> = 0. Remark. This result is corroborated by the fact, that the vanishing of the numerator of z indicates that the point (a, 5, 0), in which the line pierces the horizontal reference-plane, is in the given plane ; while the vanish- ing of the denominator shows, by the corollary to the EIGHT LINES MEETING IN SPACE. 541 previous article, that the line is parallel to the given plane : two conditions which obviously place the line wholly in that plane. 712. Condition that two Right Lines in Space shall intersect. Two right lines in space will not in general intersect, because the four equations x = mz -+- 0, x = m'z -+- a', y = nz -j- y = n'z -j- being in general independent, are not compatible with simultaneous values of the three variables #, y, z. If, then, the two lines represented by these four equations do intersect, one of the equations must be derivable from the other three, and the condition of such a derivation will be the required condition of intersection. We form this condition, of course, by eliminating x,y,z from the four equations. To do this, solve the first and third, and also the second and fourth, for , and equate the two values thus found. The result is n n' b V EXAMPLES INVOLVING EQUATIONS OF THE FIRST DEGREE. 1. Show that, if L, M and N, R be the equations to two inter- secting right lines, they will be connected by some identical rela- tion IL + mM + nN + rR = 0, and that the plane of the two intersecting lines may be represented by either of the equations IL + mM^O, nN+rR=--Q. 2. Find the equation to the plane which passes through the lines x a _ y b _ z c x a _ y b _ z c ~~~ '~~~~ ~~~ ~ ~~~ 542 ANALYTIC GEOMETRY. 3. Find the equations to the traces of any given plane upon the three reference-planes, and prove that if a right line be perpen- dicular to a given plane, its projections will be perpendicular to the traces of the plane. 4. Find the equations to the three planes which pass through the traces of a given plane upon the reference-planes, and are each perpendicular to the plane. 5. Find the equation to the plane which passes through a given right line and makes a given angle with a given plane. 6. If (a', /?', /), (a", /?", 7 X/ ) be the direction-angles of two right lines, prove that the direction-cosines of the external bisector of the angle between them, are proportional to cos of -(- cos a", cos /5' + cos /?", cos / -f cos /', and that those of the internal bisector are proportional to cos a' cos a", cos j8 x cos /3 X/ , cos / cos y /x . 7. Three planes meet in one point, and through the common section of each pair a plane is drawn perpendicular to the third prove that in general the planes thus drawn pass through one right line. 8. Find the equation to a plane parallel to two given right lines, and thence determine the shortest distance between the lines. 9. A plane passes through the origin: find the bisector of the angle between its traces on two of the reference-planes. 10. Prove that the locus of the middle points of all right lines parallel to a given plane, and terminated by two fixed right lines which do not intersect, is a right line. CHAPTER THIRD. LOCUS OF THE SECOND ORDER IN SPACE. | 713. The general equation of the second degree in three variables, which is the symbol of the space-locus of the Second order, may be written Ax 2 -f ZHxy -f Bf + ZKyz + Ez* + ZLzx -f 2fe + 2Fy + 2Dz+C=Q (1), SPACE-LOCUS OF THE SECOND ORDER. 543 where A,B,E; H,K,L\ C,D,F,Gr are any ten con- stants whatever. Since we can divide this equation throughout by 6 Y , it appears that the number of independent constants is nine. Hence, nine conditions are necessary and sufficient to determine the locus. Thus we learn that, for example, the space-locus of the Second order is a surface of such a form that one, and but one, such surface can be passed through any nine points which do not lie in the same plane. 714. The most general and complete criterion of the form of any surface, is afforded by its curves of section with different planes. Let us apply this criterion to test the figure of the surface denoted by (1). If in (1) we make 2 = 0, that is, if we combine (1) with the equation to the horizontal reference-plane, we get Ax* + 2Hxy + Eif -f 2Gx + 2Fy + 0= 0, the general equation to the Conic. Hence, as we can transform the reference-plane of XY to any plane that we please, and as such a transformation will not affect the degree of the equation of section just found, Every plane section of a surface of the Second order is a conic. Moreover, if we combine (1) with z = /, that is (Art. 695, Cor.), if we intersect our locus by any plane parallel to the plane of XY, we get Ax 2 + ZHxy + Bif + 2G'x + ZF'y +C' = 0, where G r =G + kL, F f =F+kK, C' = Hence, (Art. 669,) since the co-efficients A,H,B remain unchanged whatever be the value of k, The sections of a surface of the Second order by parallel planes are similar conies. An. Ge. 49. 544 ANALYTIC GEOMETRY. To denote, then, that the surface of the Second order is represented by an equation of the second degree in space, and that all its plane sections are curves of the Second order, we shall henceforth call it the Quadric. THE QUADBIC IN GENERAL. 715. To increase the clearness of our conception of the quadric figure, we must now reduce the general equation (1) to its simplest forms. We can effect this reduction most rapidly, however, by taking out a few leading properties of the surface, partly from the general equation itself, and partly from the results of its first transformations. 71O. Let us transform (1) to parallel axes through a new origin x'y'z'. Since we merely have to write x -f- x' for x, y -f- y' for y, and z -f 2' for z, it is easily seen that the new equation will be Ax 2 + Zffxy -|- Ef -f ZKijz -f- Ez 2 -f Lzx 2D'z -f C' = (2), where C' , the new absolute term, is the result of substi- tuting x'y'z' in (1), and in the new co-efficients of x, y, z we have G r = Ax' -f- Hy' + Lz' -f G, F' = Hx f + By' + Kz' + F, D f == Lx r -f Ky' + Ez' -f- D, these quantities being planar functions of the new origin. It deserves especial notice, that G r (Alg., 411) is the # For the discussion of Quadrics in complete detail, the reader is referred to SALMON'S Geometry of Three Dimensions, from which the investigations of the following pages have in the main been reduced. DISCRIMINANT OF THE QUADRIC. 545 derived polynomial (or derivative, as we shall call it for brevity) of C 1 with respect to x ; that F 1 is the derivative of C' with respect to v/; and D 1 ', the derivative of C f with respect to 2. Hence, if we write the original equation (1) in the abbreviated form U = 0, we may use for the four co-efficients C f , G 1 ', jf 7 ', D r the convenient symbols Z7', UJ, U,', U.'. TIT 1 . As we shall also find it convenient to employ the so-called discriminant of the equation U=Q, and several of its derivatives, we will determine their values before advancing farther. The discriminant of any function may be defined as the result obtained by solving its several derivatives for its variables, and then substituting the values of these in the function itself. Accordingly, solving for .r, y, z in U x =^ Ax + Hy + Lz + G = Q, U y = Hx + By + Kz 4- F = 0, U z = Lx + Ky -f Ez + D = 0, and then substituting in U, we get ABCE + 2ADFK+ 2BDGL + 2CHKL -f 2EFGH This, then, is the discriminant of the given quadric U= 0, Snd may be appropriately represented by A. If we now denote the several derivatives of J, taken with reference to 6r, F, D, C in succession, by 2#, 2/, 2c?, c, we shall have g = BDL -f ^MT 5^(7 -h GK 2 DHK FKL, f = ADK+ EGH AEF + FL 2 - DHL GKL, d = ^^7T+ 5(y^ ABD + DH 2 FHL GHK, c = ^1^^ + 2fflK ^^T 2 BU EH 2 . 546 ANALYTIC GEOMETRY. It will be convenient next to determine the condition upon which the radius vector of the Quudric will be bisected in the origin. To find this, throw the general equation into the vectorial form, by writing f) cos a for x, p cos ft for y, and p cos f for 2, which we may evidently do if a, ft, f are the direction-angles of the radius vector. Equation (1) then becomes {A cos 2 a -f- 2-fiTcos a cos /? -f- B cos 2 /? + 2-fiT cos p cos 7 + E cos 2 y-\-2L cos y cos a) /J 2 + 2(cosa-f _Fcos/3+>cos}')/> + (7=0. If the origin bisects the radius vector, this equation will have its roots numerically equal with opposite signs. Hence, the required condition of bisection is G cos a -f jPcos ft -f- D cos f = ; or, after multiplying through by />, and replacing the corresponding x 9 y, 2, we learn that all radii vectores bisected in the origin must lie in the plane 719. If, then, in the equation to the Quadric we had 6r, F, D all = 0, the condition of bisection would be satisfied for all possible values of , ft, ? ; or, in other words, every right line drawn through the origin to meet the quadric would be bisected in the origin, and the origin would be a center of the quadric. 720. Resuming now our transformations of equation (1), let us suppose the new origin x'y'z' to which (2) is referred, to be a center. The new 6r, F, D will then vanish, and we learn (Art. 716) that the center lies at the intersection of the three planes QUADRICS CENTRAL AND NON-CENTRAL. 547 Solving these three equations, we obtain, as the co-ordi- nates of the center, x ^ v z , y -- , z --- , c c c where #, /, d, c have the values given in the table of Art. 717. The center, then, is a single determinate point, and will be a finite real one if c is not zero, but not other- wise. Hence, Quadrics are either central or non-central* and central quadrics have only one center. 721. By taking, then, the center for origin, the equa- tion to any central quadric may be written Ax 2 -\- ZHxy +% 2 -h ZKyz +Ez 2 + 2Lzx + C'= (3), where (Art. 716) by substituting the co-ordinates of the center in 7', we readily find Gg + Ff + Dd 0c A u = - = > e where (r, -JF, D, C are the co-efficients of the planar terms of (1), and #, /, d, , ABE + 2HKL - A K 2 - BL' 2 - Elf 2 = A'B'E' + IH'K'L' - A'K't-B'L't-E'H'' 2 ." By solving these three equations for cither A', B' , or E', we obtain the cubic in the text above, where u is merely a symbol for the unknown co -efficient. THE QUADRIGA CLASSIFIED. 553 Hence, as we now have c 0, one of the roots of this cubic, and therefore one of the new co-efficients A', B', E', must vanish whatever be the directions of the new rectangular axes. By taking these axes parallel to the principal planes, thus causing the new H 1 ', K', L' to disappear, we can therefore reduce the original equation to the form B'f -f E'# + 2G'x + 2F'y + Wz + C= 0, as this transformation (Art. 685) does not affect the absolute term. And now we can remove the origin to the point in which the line F r = 0, D f = pierces the Quadric, thus destroying the absolute term as well as the co-efficients of y and z. The equation to a non- central quadric will then take the form B'y* + or, as it may be more symmetrically written, if + Qz 2 = Px (5). Equations (4) and (5) are the simplest forms of the space-equation of the second degree. CLASSIFICATION OF QUADRICS. T2T. By means of equations (4) and (5), we can now ascertain the several varieties of quadric surfaces, and the peculiar figure of each. 728. We begin with the CENTRAL QUADRICS, repre- sented by the equation A'x 2 -f B'if -f E'z 2 + C 1 = 0. 554 ANALYTIC GEOMETRY. I. Let A', B', E' all be positive. Then, if C" is neg- ative, the equation can at once be put into the form where a, 6, c are the lengths of the intercepts cut off upon the axes of x, y, and z respectively. The sections of the surface with any planes of the form z = &, x = Z, y = m, are the ellipses ^ t ^_ p f z^ _P_ i 2 * 2 _-,_^ a*~*~b*~ c 2 ' 6*" 1 V~ a 2 ' e*~ r a 5i 6 2 These are real for every positive or negative value of k, I, m that is not greater respectively than a, 6, c', but are imaginary for all greater values. The quadric, therefore, lies wholly inside the rectangular parallelepiped formed by the six planes z = : c, x = a, y = b, but is continuous within those limits, and, having elliptic sec- tions with the reference-planes and all planes parallel to them, is properly called the Ellipsoid. Its semi-axes are of course respectively equal to a, b, c, and we suppose that the reference-planes are so taken that a is in general greater than ft, and b greater than c. But the following particular cases must be considered: 1. If b = c, the section with any plane x I becomes y 2 -j- z 2 = constant, and is therefore a circle. The surface may then be generated by revolving an ellipse upon its axis major, and is called an ellipsoid of major revolution; or, with greater exactness, the Prolate Spheroid. 2. If b = a, the section with any plane z = Jc becomes a circle, the surface may be generated by revolving an ellipse upon its minor axis, and is therefore called an ellipsoid of minor revolution ; or, the Oblate Spheroid. THE QUADEICS CLASSIFIED. 555 3. If a = b = c = r, all the plane sections of the surface are circles, and the equation becomes which is therefore the equation to the Sphere. Next, if 0' is zero, our general central equation, taking the form 'z 2 = 0, can only be satisfied by the simultaneous values x = 0, y = 0, z = 0, and thus denotes the Point, which may therefore be regarded as an infinitely small ellipsoid. Finally, if C f is positive, the general equation becomes x 2 i/ 2 z 1 a* + F + ? = 7 1 ' which having the ellipsoidal form in its first member, but involving an impossible relation, may be said to denote an imaginary ellipsoid. II. Let A' and B' be positive, but E' be negative. Then, if C' be also negative, we can write the equation, in terms of the intercepts, + _ - 1 a 2 1 6 2 c 2 Here, the sections formed by the planes x = I, y = are the hyperbolas f z 2 __ I 2 3? ^ \ ^L ~~~ = ~ 9 ~~~'~ ~ 556 ANALYTIC GEOMETRY. which are real for all values of I and ???, but whose branches cease to lie on the right and left of the center, and are found above and below it, when I > a and m > b. The section by any plane z = k is an ellipse 5>f- a" er -1- -3- which being real for every value of A", the surface is continuous to infinity. This being so, and the sections by the vertical reference-planes and all their parallels being hyperbolas, the surface is called the Hyperboloid of One Nappe. The quantities a, 6, c are called its semi-axes, though it is evident, by making x = y = in the equation, that the axis of z does not meet the surface. The real mean- ing of c will soon appear. In general, a is supposed greater than b. In case we have a = b, the sections parallel to the plane XY become circles, the surface may be generated by revolving an hyperbola upon its conjugate axis, and is called an hyperboloid of revolution of one nappe. Next, when C' = 0, the equation assumes the form The section made by the reference-plane z = 0, is the point ^L'o; 2 -f- -S'?/ 2 - 0, while that by any parallel plane z = k, is the ellipse A'x 2 -f- By 2 = E'k 2 . The sections formed by the planes x = 0, y = 0, are pairs of inter- secting right lines, B'y 1 JE f z' 2 = Q and A'x 2 E'z 1 = ^\ as also the section by any vertical plane y = mx, is the pair of lines (A' -f m 2 B'} x 2 E'z 2 = 0. The surface is therefore a Cone, whose vertex is the origin. If we have j * -I- T\ / J- T~r / J- THE QUAD RIGS CLASSIFIED. 557 the cone is said to be asymptotic to the preceding hyper- boloid. If a = 6, or A' = B', the section by the plane z = k is the circle x 2 -}- y 2 constant, and the cone is a circular one. Finally, when C f is positive, by changing the signs throughout we may write the equation z - __ - y = \ c 2 a 2 b 2 The plane z = k now evidently cuts the surface in imag- inary ellipses so long as k < c, but in real ones when k passes the limit c whether positively or negatively. The surface therefore consists of two portions, separated by a distance = 20, and extending to infinity in opposite directions. The sections by the planes x = l, y = m, are hyperbolas. The surface is therefore called the Hyperboloid of Two Nappes. By making x = y = 0, we find that the intercept of this surface on the axis of z is c ; while the intercepts upon the axes of x and y, found by putting y = z = Q and z = x = Q, are the imaginary quantities aV 1, hi/ 1. Moreover, the sections by the planes # 0, # 0, being ^ 1 JL i !i ? i 2 ~~ 2 ~ ' 2 ~~ 2 ~ are hyperbolas Conjugate to those in which the same planes cut the Hyperboloid of One Nappe. We there- fore perceive that the present hyperboloid is conjugate to the former, and that the real meaning of c in the equation to the former is, the semi-axis of its conjugate surface. When a = b, this hyperboloid also becomes one of revolution, and is called an hyperboloid of revolution of two nappes. 558 ANALYTIC GEOMETRY. III. Let A' be positive^ and B' and E' both negative. The general equation may then be written ^ f_ ^ __ -i ' I' v> --- * a" or eH when (7' is negative ; or __ _ b* " c 2 "" a 2 " - when C' is equal to zero ; or y" L - - - i fr-' " c 2 "" 2 ~ when C r is positive. The present hypothesis therefore presents no new forms, but merely puts those of the pre- ceding supposition into a different order. It is usual, however, to write the equation to the Hyperboloid of Two Nappes in the form tf z 2 a 2 b* ' c*~ rather than in that obtained by the final supposition of II. The advantage of doing so appears in connection with the hyperboloids of revolution ; for, if we make b = c in the above equation, we learn that an hyperboloid of revolution of two nappes may be generated by revolving an hyper- bola upon its transverse axis. Under the present hypoth- esis, therefore, the hyperboloids of two nappes and of one may be generated by revolving the same hyperbola, first upon its transverse, and then upon its conjugate axis. Under the hypothesis of II, on the contrary, the two surfaces would be generated by a pair of conjugate hyperbolas revolving upon the same axis. THE QUADRICS CLASSIFIED. 559 Let us, secondly, consider the NON-CENTRAL QUADRICS, represented by the equation if + Qz 2 = Px. I. Suppose Q to be positive. Any plane x == I cuts the surface in an ellipse y 2 + Qz 2 = Ply which, if P is positive, will be real only on condition that I is not negative; or, if P is negative, only on condition that I is not positive. The surface, then, consists of a single shell, extending to infinity on one side of the plane YZ, and in fact touched by that plane in the point y 2 -\-Qz 2 = Q, that is, in the origin. The sections of this shell by the planes y = 0, 2 = 0, are the parabolas z 2 = (P : Q) x, y* Px. Hence, all sections by planes parallel to these are also parabolas, and the surface is the Elliptic Para- boloid. 1. If Q = 1, the section by the plane x = I is the circle y 2J r z 2 = Pl^ and the surface may be generated by revolv- ing a parabola upon its principal axis. It is then called a paraboloid of revolution. 2. We have thus far not made P = 0, because the transformation to y 2 + Qz 2 = Px in general excludes that supposition, being made upon the assumption that we can not, in the equation (Art. 726) + E'z 2 + 2G'x + ZF'y + 2D'z +(7=0, destroy all the three co-efficients G f , F', D' together. But if G r were itself = 0, then any point on the line F' = 0, D' = would be a center of the quadric ; and as this line would thus pierce the surface only at infinity, we could not, by placing the origin at the piercing-point, cause the absolute term to disappear. However, by taking An. Ge. 50. 560 ANALYTIC GEOMETRY. the origin upon the line F' = 0, D r = 0, the equation would be reduced to the form f + Qz* = R. This equation, at first sight, appears to represent an ellipse in the plane YZ. But, obviously, it is true not only for points whose x = 0, but for points answering to any value of x that corresponds to a given y and z. It denotes, then, a cylinder whose base is the ellipse y 2 + Qz 2 R) an( i whose axis is the axis of x. Hence we learn that a particular case of an elliptic paraboloid is the Elliptic Cylinder. When R = 0, this cylinder breaks up into two imag- inary planes, whose common section, however, being projected in the real point y 2 -j- Qz 2 0, is a real right line, perpendicular to the plane YZ. II. Suppose Q = Q. The equation ?/ 2 + Qz 2 = Px then becomes and therefore denotes the Parabolic Cylinder. By shifting the origin along the axis of #, the equation to this cylinder takes the more general form f = Px + N. When, therefore, P= 0, this cylinder breaks up into the two parallel planes which are real, coincident, or imaginary, according as N is positive, equal to zero, or negative. III. Suppose Q to be negative. Then the surface ANALOGIES OF QUADRICS TO CONICS. 561 will meet all planes parallel to y = and z = in par- abolas ; but, being met by any plane x = l in the real hyperbola if + Qz* = PI, is called the Hyperbolic Par- aboloid. It evidently meets the plane x = in the two inter- secting lines y~-\- Qz 2 = Q, and extends to infinity on both sides of that plane. As a particular case, 1. Corresponding to the negative Q, we have the cylinder As this meets the plane x = in the hyperbola y 1 -f- Qz 2 = R, it is called the Hyperbolic Cylinder. When R = 0, this evidently breaks up into two inter- secting planes. *73O. The foregoing include all the varieties of the Quadric. By means of the several equations contained in the two preceding articles, we could now proceed to develop all the known properties of these surfaces. The student, however, can hardly have failed to observe the remarkable analogy, not only in respect to the preceding classification and its corresponding equations but in re- spect to such properties as have already been developed, which subsists between these surfaces and the several varieties of the Conic. He will therefore anticipate that by applying to the equations we have just obtained, the methods with which the discussion of conies has now thoroughly familiarized him, he can obtain the analogous properties of quadrics for himself. Accordingly, several of the more important ones have been presented in the examples at the close of this Chapter. It may deserve mention, however, that in these quadric analogies to conies, lines generally take the place of 562 ANALYTIC GEOMETRY. points, and surfaces the place of lines ; also, that where conic elements go by twos, the quadric elements gene- rally go by threes. Thus, the conception of two conju- gate diameters is replaced by that of three conjugate planes. Often, in connection with this principle of substitution, the result of the analogy is altogether unexpected. For instance, the quadric analogue of the conic focus, is itself a conic, known as the focal conic, which lies in either of the principal planes ; so that, in general, every quadric has three infinitudes of foci. In fact, the subject of Foci and Confocal Surfaces is one of the most recent as well as the most intricate in connection with quadrics. It was originally investigated by CHASLES and MACCULLAGH independently ; of whose discoveries Salmon has given a brilliant account in the Abridged Notation.* SURFACES OF REVOLUTION OF THE SECOND ORDER. 731. Any surface that can be generated by revolving any curve about a fixed right line is called a surface of revolution. The revolving curve is named the generatrix, and the fixed line around which it moves is termed the axis. We came upon the Quadrics of Revolution and their equations, in the preceding investigations. But we shall here give some account of them from another point of view, for the sake of putting the student in possession of the general method of revolutions. Let the shaded surface in the diagram represent *See his Geometry of Three Dimensions, p. 101. SURFACES OF REVOLUTION. 563 the surface generated by any curve, revolving about an axis 00. Let the equa- tions to the "projecting z/1 cylinders" of the gener- atrix be =./ y= = (x tan c) 2 , the equation to the curve of section NBL. Transform- ing this to its own plane, we shall have, in the formulas of Art. 685, a = 6, ft = 90 ; a! = 90, /?' = ; a" = 90, ft" = 90. We therefore replace x by x cos 0, and leave y unchanged, thus obtaining, as the equation to any plane section of a right circular cone, x 2 (tan 2 ^> tan 2 0) cos 2 + ?/ 2 tan 2

tan 2

y> 9 and c 0, so that the secant plane passes through the vertex, the section becomes x 2 (tan 2

2, every right line in the plane Gx + Fy -f Dz = is a tangent to the surface at the origin. Hence, the equation just written is the equation to the tangent plane at the origin. Supposing the origin not to be on the surface, by transforming to any point x'y'z' of the surface, the equation to the tangent plane at such point would be, after putting for the new G, F, D their values as given in Art. 716, which, by re-transformation to the original axes, gives (x - x') U x f + (y y*) UJ -f (z - z') U z f = 0, TANGENT AND NORMAL PLANES. 571 as the general equation to the tangent plane at any point x'yV. Remark. The generic interpretation of this equation, as of its analogue in the Conies, is to regard it as the symbol of the polar plane of the point x'y'z', which in this view is not restricted to being a point upon the surface. 743. Tangent Planes to the different Quad- rics. The equations to these, in their simplest forms, are found by deriving U x f , UJ, U x f in the equations of Arts. 728, 729, and substituting the results in the general equation last obtained. In this way, we get ^ + 2/1 _|_ !! = i, which represents, in terms of the semi-axes, the tangent plane to any ellipsoid; ?L? _L #V__^ 1 d 2 ' ' b' 2 " " c 2 ~ which represents the tangent plane to any Jiyperboloid of one nappe; x'x y'y z'z ^ ~o? ~ ' ~V ' ~ ~&~~ ' which represents the tangent plane to any hyperboloid of two nappes; which represents the tangent plane to any paraboloid. 744. Normal of a Qiiadric. The right line per- pendicular to any tangent plane at its point of contact 572 ANALYTIC GEOMETRY. with a quadric, is called the normal of the surface at that point. Hence, by comparing Arts. 709, 742, we learn that x x' y y r z z' are the general equations to the normal at any point x' y' z'. By deriving U x , UJ, UJ in the equations of Art. 728, we obtain the equations to the normals of the central quadrics, namely, a 2 (x x f ) = b 2 (y y') = c 2 (z z'} x' y' z 1 in which we must use the upper or lower signs in ac- cordance with the variety of the surface. If we derive U x f , Z7/, U z f in the equation of Art. 729, we obtain x x' y' y as the equations to the normal of a paraboloid. These can of course be thrown into other forms when con- venient. 745. Normal Planes. Any plane that passes through the normal of a quadric at any point, is called a normal plane to the quadric. Comparing Arts. 698 and 744, we learn that the general equation to a normal plane will be of the form l(x-x f ) m(y-y'} _ (I + m) (z-Q UJ UJ U z ' where the arbitrary k of Art. 698 is for the sake of symmetry replaced by the ratio m : I. By deriving U x ' 9 /"/, U a f from any specific equation to either of the quadrics, and substituting the results in EXAMPLES. 573 the preceding formula, we can obtain the equation to a normal plane for any given quadric in any given system of reference. It is noticeable that the above equation involves the indeterminate ratio m : I. This is as it should be ; for there is obviously an infinite number of normal planes corresponding to any point on a quadric. When the normal plane, however, satisfies such conditions as de- termine it, we can readily find the corresponding value of in : Z. EXAMPLES ON THE QUADRICS. 1. Determine, by means of their discriminating cubics (see Art. 726), whether the quadrics 7# 2 + 6?/ 2 + 5z 2 4yz 4xy = 6, Ix 1 13y 2 + G 2 2 + 24zy + I'lyz Uzx = 84, 2# 2 + 3y 2 + 4s 2 + 6xy + 4ijz + 8zx = 8, are ellipsoids, hyperboloids, or paraboloids. 2. In any central quadric, the sum of the squares on three conju- gate semi-diameters is constant. 3. The parallelepiped whose edges are three conjugate semi- diameters, is of constant volume. 4. Tangent planes at the extremities of a diameter, are parallel. 5. The length of the central perpendicular upon a tangent plane is given by the equation . p 2 a* b* c 4 ' 6. The length of the same perpendicular, in terms of its direction- cosines, is p* = a 2 cos 2 a -f & 2 cos 2 /? + c 2 cos 2 7. 7. The sum of the squares on the perpendiculars to any three tangent planes is constant. 574 ANALYTIC GEOMETRY. 8. The locus of the intersection of three tangent planes which are mutually perpendicular, is the sphere 9. Find the equation to a diametral plane conjugate to a fixed point x'y'z'; and prove that if two diameters are conjugate, their direction-cosines fulfill the condition cos a cos of cos /? cos ft' cosy cos/ _ a ~~ r ~ ~~~ 10 The locus of the intersection of three tangent planes at the extremities of three conjugate diameters is the central quadric UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. DEC 18 1947 MAY 131948 > mr j^ ^\Bv Q^^ SAugSHt LD 21-100m-9,'47(A5702sl6)476 FEB ? AN 2 7 1955 LU U 9Jan'58TSx EC'D LO EC 18 19S7 MAR 2 1S39 - LD JUN 1 YU ZZ370 800547 UNIVERSITY OF CALIFORNIA LIBRARY