,0 
 
ANALYTIC GEOMETRY 
 
•V><y^° 
 
ANALYTIC GEOMETRY 
 
 FOR 
 
 TECHNICAL SCHOOLS ANI) COLLEGES 
 
 P. A. LAMHERT, M.A. 
 
 INSTKUCTOli IX MATHEMATICS, LEHIGH UNIVERSITY 
 
 THE MACMILLAN COMPANY 
 
 LONDON: MACMILLAN & CO., Ltd. 
 
 1897 
 
 All rights reserved 
 
,5 
 
 QK^ 
 
 Copyright, 1897, 
 By the MACMILLAN COMPANY. 
 
 Norfaaati 19rfS3 
 
 .1. S. Cushins S; Co. Berwick & Smith 
 Norwood Mass. U..S.A. 
 
PREFACE 
 
 The object of this text-book is to furnish a natural 
 but thorough introduction to the principles and applica- 
 tions of Analytic Geometry for students who have a 
 fair knowledge of Elementary Geometry, Algebra, and 
 Trigonometry. 
 
 The presentation is descriptive rather than formal. 
 The numerous problems are mainly numerical, and are 
 intended to give familiarity with the method of Analytic 
 (ieometry, rather than to test the student's ingenuity in 
 guessing riddles. Answers are not given, as it is thought 
 better that the numerical results should be verified by 
 actual measurement of figures carefully drawn on cross- 
 section paper. 
 
 Attention is called to the applications of Analytic 
 Geometry in other branches of Mathematics and Physics. 
 The important engineering curves are thoroughly dis- 
 cussed. This is calculated to increase the interest of the 
 student, aroused by the beautiful application the Analytic 
 Geometry makes of his knowledge of Algebra. The 
 historical notes are intended to combat tlie notion that 
 a mathematical system in all its completeness issues 
 Minerva-like from the brain of an individual. 
 
 P. A. LAMBERT. 
 
 80()5,';4 
 
TABLE OF CONTENTS 
 
 ANALYTIC GEOMETRY OF TWO DIMENSIONS 
 
 CHAPTER I 
 
 Rectangi'lar Coordinates 
 uniriE i-AOK 
 
 1. Introduction 1 
 
 2. Coordinates 1 
 
 3. The Point in a Straight Line 2 
 
 4. The Point in a Plane 3 
 
 5. Distance between Two I'oints ....... 7 
 
 6. Systems of Points in the Phme 8 
 
 CHAPTER II 
 Equations of Geometric Figures 
 
 7. Tlie Straight Line 
 
 . 13 
 
 8. The Circle 
 
 . 15 
 
 0. Tlie Conic Sections 
 
 . 15 
 
 10. The Ellipse 
 
 . 18 
 
 11. The Hyperbola 
 
 . 21 
 
 12. The Parabola 
 
 . 24 
 
 CHAPTER HI 
 Plottixc; ou Ai.gki'.uaic Equations 
 
 13. General Theory 
 
 14. Locus of First Degree I'^piation 
 
 15. Straight Line through a Point 
 IG. Tangents .... 
 
CONTENTS 
 
 17. Points of Discontinuity . 
 
 18. Asymptotes 
 
 19. Miixiinum and Minimum Ordinates 
 
 20. Points of Inflection 
 
 21. Diametric Method of Plotting Equations 
 
 22. Summary of Properties of Loci . 
 
 PAGE 
 
 . 33 
 
 . 34 
 
 . 30 
 
 . 37 
 
 . 39 
 
 . 39 
 
 CHAPTER IV 
 PLOTTiNr, OF Transcendental Equations 
 
 Elementary Transcendental Functions 45 
 
 Exponential and Logaritlimic Functions 45 
 
 Circular and Inverse Circular Functions 47 
 
 Cycloids 54 
 
 Prolate and Curtate Cycloids 57 
 
 Epicycloids and Hypocycloids 58 
 
 Involute of Circle 59 
 
 CHAPTER V 
 
 Tkansfokmation of Coordinates 
 
 30. Transformation to Parallel Axes . 
 
 31. From Uectangular Axes to Rectangular 
 
 32. <)bli(iue Axes 
 
 33. From Rectangular Axes to Oblique 
 
 34. General Transformation 
 
 35. The Problem of Transformation . 
 
 CHAPTER VI 
 
 Polar Coordinates 
 
 30. Polar Coordinates of a Point . 
 
 37. Polar Equations of Geometric Figures 
 
 38. Polar Equation of Straiglit Line . 
 
 39. Polar Equation of Circle 
 
 40. Polar Equations of the Conic Sections 
 
 41. Plotting of Polar Eciuations . 
 
 42. Transformation from Rectangular to Polar Coordinates 
 
CONTENTS 
 
 CH AFTER VII 
 Propertiks of the Straight Line 
 
 AKTICI.E 
 
 43. Equations of the Straight Line . 
 
 41. Angle between Two Lines . 
 
 4;"). Distance from a Point to a Line . 
 
 4<;. E(iuations of Bisectors of Angles 
 
 47. Lines through Intersection of Given Lines 
 
 48. Three Points in a Straight Line . 
 40. Three Lines through a Point 
 50. Tangent to Curve of Second ( )rder 
 
 PAGE 
 
 81 
 84 
 85 
 86 
 87 
 88 
 89 
 91 
 
 CHAPTER VIII 
 Properties of the Circle 
 
 51. Equation of the Circle 93 
 
 52. Connnon Chord of Two Circles 94 
 
 53. Power of a Point 95 
 
 54. Coaxal Systems 97 
 
 55. Orthogonal Systems 98 
 
 56. Tangents to Circles 1*'^ 
 
 57. Poles and Polars • .102 
 
 58. Reciprocal Figures 194 
 
 59. Inversion 196 
 
 60. 
 61. 
 62. 
 63. 
 64. 
 65. 
 66. 
 67. 
 
 CHAPTER IX 
 
 Properties of the Conic Sections 
 
 General Equation HI 
 
 Tangents and Normals H"^ 
 
 Conjugate Diameters 119 
 
 Supplementary Chords 122 
 
 Parameters 1-'* 
 
 The Elliptic Compass 12<> 
 
 Area of the Ellipse 127 
 
 Eccentric Angle of Ellipse 128 
 
 Eccentric Angle of the Hyperbola 130 
 
CONTENTS 
 
 CHAPTER X 
 
 Second Degree Equation 
 
 ARTICLE PAGE 
 
 69. Locus of Second Degree Equation 133 
 
 70. Second Degree Equation in Oblique Coordinates . . . 138 
 
 71. Conic Section through Five Points 141 
 
 72. Conic Sections Tangent to Given Lines 142 
 
 73. Similar Conic Sections 144 
 
 74. Coufocal Conic Sections 146 
 
 CHAPTER XI 
 Line Coordinates 
 
 75. Coordinates of a Straight Line 149 
 
 76. Line Equations of the Conic Sections 151 
 
 77. Cross-ratio of Four Points 151 
 
 78. Second Degree Line Equations 152 
 
 79. Cross-ratio of a Pencil of Four Rays 153 
 
 80. Construction of Projective Ranges and Pencils .... 155 
 
 81. Conic Section through Five Points 157 
 
 CHAPTER XII 
 
 Analytic Geoaikti;y of thi; Complkx Yari 
 
 82. Graphic Rcpioscntation of the Cuniiilex Variable 
 
 83. Arithmetic Operations applied to Vectors . 
 
 84. Algebraic Functions of the Complex Variable . 
 
 85. Generalized Transcendental Functions 
 
 160 
 162 
 165 
 168 
 
 ANALYTIC GEOMETRY OF THREE DIMENSIONS 
 
 CH.VPTER XIII 
 
 Point, Line, and Plane in Space 
 
 86. Rectilinear Space Coordinates 171 
 
 87. Polar Space Coordinates 173 
 
 88. Distance between Two Points 174 
 
CONTENTS 
 
 89. Equations of Lines in Space 
 
 90. Equations of the Straight Line . 
 9L Angle between Two Straight Lines 
 
 92. The Plane 
 
 03. Distance from a Point to a Plane 
 
 94. Angle between Two Planes 
 
 CHAPTER XIV 
 
 Curved Sukfaces 
 
 95. 
 
 90. 
 
 97. 
 
 98. 
 
 99. 
 100. 
 101. 
 102. 
 
 103. 
 104. 
 105. 
 100. 
 107. 
 108. 
 100. 
 110. 
 111. 
 112, 
 113. 
 
 Cylindrical Surfaces . 
 
 Conical Surfaces 
 
 Surfaces of Revolution 
 
 The Ellipsoid 
 
 The Hyperboloids 
 
 The Paraboloids . 
 
 The Conoid 
 
 Equations in Three Variables 
 
 CHAPTl 
 
 Second Degree Equation in Three Variables 
 
 Transformation of Coordinates 
 
 Plane Section of a Quadric 
 
 Center of Quadric 
 
 Tangent Plane to Quadric . 
 
 Reduction of General Equation of Quadric 
 
 Surfaces of the First Class 
 
 Surfaces of the Second Class 
 
 Surfaces of the Third Class 
 
 Quadrics as Ruled Surfaces 
 
 Asymptotic Surfaces . 
 
 Orthogonal Systems of (Juadrics 
 
 R XV 
 
ANALYTIC GEOMETRY 
 
 CHAPTER I 
 
 EEOTANGULAK COORDINATES 
 
 Art. 1. — Introduction 
 
 The object of analytic* geometry is the study of geometric 
 figures by tlie processes of algebraic analysis. 
 
 The three fundamental problems of analytic geometry are: 
 
 To find the equation of a geometric figure or the e(;[uations 
 of its several parts from its geometric definition. 
 
 To construct the geometric figure represented by a given 
 equation. 
 
 To find the relations existing between the geometric prop- 
 erties of figures and the analytic properties of equations. 
 
 Art. 2. — Coordinates 
 
 Any scheme by means of which a geometric figure may be 
 represented by an equation is called a system of coordinates. 
 
 * The reasoning of pure geometry, the geometry of Euclid, is mainly 
 synthetic, that is, starting from something known we pass from conse- 
 (luence to consequence until something new results. The reasoning of 
 algebra is analytic, that is, assuming what is to be demonstrated we pass 
 from consequence to consequence until the relation between the unknown 
 and the known is found. The term "analytic geometry" is therefore 
 equivalent to algebraic geometry. The application of algebra to the de- 
 termination of the properties of geometric figures was invented by 
 Descartes (1596-1050), a French philosopher, and published in Leyden 
 in 1G37. 
 
2 ANALYTIC GEOMETRY 
 
 The coordinates of a point are the quantities which deter- 
 mine the position of the point. 
 
 Along the line of a railroad the position of a station is 
 determined by its distance and direction from a fixed station ; 
 on our maps the position of a town is determined by its lati- 
 tude and longitude, the distances and directions of the town 
 from two fixed lines of the map ; the position of a point in a 
 survey is determined by its distance and bearing from a fixed 
 station. 
 
 On these different methods of determining the position of 
 a point are based different systems of coordinates. 
 
 Akt. 3. — The Point in a Straight Line 
 
 On a straight line a single quantity or coordinate is sufficient 
 to determine the position of a point. Let be a fixed point 
 
 -8 -7 .-6 -5 -4 -3-2-10 1 2 3 4 5 6 7 8 
 Fio. 1. 
 
 in the line; adopt some length, such as 01, as the linear unit; 
 call distances measured from towards the right positive, dis- 
 tances measured from towards the left negative. Let a 
 point of the line be represented by the number which ex- 
 presses its distance and direction from the fixed point 0. 
 Then to every real number, positive or negative, rational or 
 irrational, there corresponds a definite point in the straight 
 line, and to every point in the line there corresponds a definite 
 real number. This fact is expressed by saying that there is 
 a " one-to-one correspondence " between the points of the line 
 and real numbers. 
 
 The algebra of a single real variable finds a geometric inter- 
 pretation in the straight line. Denoting by x the distance 
 and direction of a point in the straight line from 0, that is 
 letting X denote the coordinate of the point, the equation 
 cc^ — 2a; — 8 = locates the two points (4), (— 2), in the 
 straight line. 
 
UECT ANGULAR COORDINATES 3 
 
 Problems. — 1. Locate in the straight Hue the points 3; —2; 1^ ; 
 -2.5; -5; f. 
 
 2. Locate VG ; -VS; VlO ; VT. 
 
 Suggestion. — The numerical value of VS can be found 
 only approximately. The hypotenuse of a rit;ht triangle /^ 
 whose two sides about the right angle are 2 and 1, repre- '^/ 
 sents v'5 exactly. I 
 
 3. rind the point midway between xj and x^. Fig. 2. 
 
 4. Find the point dividing the line from Xi to X'2 internally into seg- 
 ments whose ratio is /•. 
 
 5. Find the puiiit dividing the lino from Xi to X2 externally into seg- 
 ments wiinsi' ratio is r. 
 
 6. Locatt! the roots of a;2 + 2 a; - 8 = 0. 
 
 7. Locate the roots of xr — i x — i = 0. 
 
 8. Locate the roots of x^ - x- + 11 x - = 0. 
 
 9. Find the points dividing into three equal parts the line from 2 
 to 14. 
 
 10. Find the points dividing into three ecjual parts the line from X] 
 to X.J. 
 
 11. Find the point dividing a line 8 feet long internally into segments 
 in the ratio 3 : 4. 
 
 12. A uniform bar 10 feet long has a weight of 15 pounds at one end, 
 of 25 pounds at the other end. Find the point of support for equilib- 
 rium. 
 
 Al^T. 4. — Thk Poikt in a Plane 
 
 To determine the position of a point in a plane, assume two 
 straight lines at right angles to each other to be fixed in the 
 plane. These lines are called the one the A'-axis, the other 
 the F-axis. The distance from a point, in llic plane to either 
 axis is moasiired on a line ]):ira]l('l to tlie other axis; the 
 direction of the point I'roui tlic axis is indicated by the alge- 
 braic sii^n prchxcd to the nniiilicr expressing,' tlie distance from 
 tlio axis. 
 
4 ANALYTIC GEOMETRY 
 
 Distances measured parallel to the X-axis to the right from 
 the l''-axis are called positive ; those measured to the left from 
 the F-axis are called negative. The distance and direction of 
 a point from the T-axis is called the abscissa of the point, and 
 is denoted by x. 
 
 
 
 
 
 + 
 
 Y 
 
 
 
 
 
 ( 
 
 r3,2) 
 
 
 
 
 
 
 
 G,2) 
 
 
 
 
 
 
 
 
 
 
 
 
 -X 
 
 
 
 
 
 A 
 
 
 
 
 + X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 ( 
 
 -3-2) 
 
 
 
 
 
 
 (3-2) 
 
 
 
 
 
 
 
 -Y 
 
 
 
 
 
 Distances measured parallel to the F-axis upward from the 
 X-axis are called positive; those measured downward from 
 the X-axis are called negative. The distance and direction of 
 a point from the X-axis is called the ordinate of the point, and 
 is denoted by y. 
 
 The axes of reference cut the plane into four parts. Calling 
 the part +X^+F the first angle, +F^4~X the second angle, 
 -XA~ Y the third angle, ~ F^l+X the fourth angle, it is seen 
 that in the first angle ordinate and abscissa are both positive ; 
 in the second angle the ordinate is positive, the abscissa nega- 
 tive ; in the third angle ordinate and abscissa are both negative ; 
 in the fourth angle the ordinate is negative, the abscissa posi- 
 tive. 
 
 The a1)scissa of a point determines a straight line parallel to 
 the l''-axis in which the point must lie. For, by elementary 
 geometry, the locus of all points on one side of a straight line 
 
RECTANG ULA li COORD IN A TES 
 
 ami equidistant from the straight line is a straight line parallel 
 to the given line. 
 
 The ordinate of a point determines a straight line parallel to 
 the X-axis in which the point must lie. 
 
 If both ordinate and abscissa of a point are known, the point 
 must lie in each of two straight lines at right angles to each 
 other, and must, therefore, be the intersection of these lines. 
 I lence ordinate and abscissa together determine a single point 
 in the plane. 
 
 Conversely, to a point in the plane there correspomls one 
 ordinate and one abscissa. For through the point only one 
 straight line parallel to the I'-axis can be drawn. This fact 
 determines a single value for the abscissa of the point. 
 Through the given point only one parallel to the X-axis can 
 be drawn. This determines a single value for the ordinate of 
 the i)oint. 
 
 The abscissa and ordinate of a point as defined are together 
 the rectangular* co- 
 ordinates of the point. 
 The point whose co- 
 ordinates are x and y 
 is spoken of as the 
 point (.r, ?/). There is 
 a " one-to-one corre- 
 spondence " between 
 the symbol (x, y) and 
 the points of the 
 Xl'-plane. 
 
 Problems. — 1. Locate the point (3, — 4). 
 
 Lay off ;] linear units on the X-axis to the right from the origin, and 
 thrru is found the straight line parallel to the 3'-axis, in whicli the point 
 must lie. On this line lay off 4 linear units downward from its intersec- 
 tion with tlie J-axis, and the point (3, — 4) is located. 
 
 * This method of representing a point in a plane was invented by Des- 
 cartes. Hence these coordinates are also called Cartesian coordinates. 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 3,-4) 
 
 
 
 
 
 
 
 
 
 1 
 
 
6 
 
 ANA L YTIC GEOMETli Y 
 
 2. Locate (-3,0); (0,4); (1, -1); (-1,-1); (-7,5); (10,-7); 
 (15, 20). 
 
 3. Locate (2i^3); (- 1, SJ); (% - 51)^(7.8, - 4.5). 
 Locate (V2, \/5); (-Va, Vl7); (V50, V75). 
 Construct the triangle whose vertices are (4, 5), ( — 2, 7), 
 Find the point midway between (4, 7), (0, 5). 
 Find the point midway between (x', y'), (a;", y"). 
 Find the area of tlie triangle whose vertices are (0, 0), (0, 8), (0,0). 
 Find the area ol the triangle whose vertices are (2, 1), (5, 4), (9, 2). 
 
 4. 
 
 1. 
 
 -3, -6). 
 
 
 Y 
 
 
 
 
 
 (5 
 
 4) 
 
 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 ^ 
 
 ::^ 
 
 (-97-2 
 
 
 1 
 
 > 1 "l 
 
 Z 
 
 L- 
 
 - 
 
 
 
 ■ 
 
 
 
 a' 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 (2:o) 
 
 
 (5 
 
 0) 
 
 
 
 (9i0) 
 
 Fio. 5. 
 Suggestion. — The area of the triangle is the area of the trapezoid 
 wliose vertices are (2, 1), (2, 0), (5, 4), (5, 0), plus the area of tlie 
 trapezoid whose vertices are (5,4), (5, 0), (9, 2), (9, 0), minus the area 
 of the trapezoid whose vertices are (2, 1), (2, 0), (9, 2), (9, 0). 
 
 10. Show that double the area of the triangle whose vertices are 
 (xi, 2/i), (X2, 2/2), (X3, 2/3) is 2/1 (a-3 - .r2)+ 2/2(a'i - a^3)+ 2/3(^2 - .^i). 
 
 11. Show that double the area of the quadrilateral whose vertices are 
 (.1-1, 2/1), (a^2, 2/2), (scs, 2/3), {Xi, 2/4) is yx{Xi - X2) + 2/2(xi - 0:3) + 2/3(0:2 - Xi) 
 + 2/4(*'3 - a^i). 
 
 12. Show that double the area of the pentagon whose vertices are 
 (a;i, 2/1), (3^2, 2/2), ('^3, 2/3), (X4, 2/4), (A-5, 2/5) is 2/1 (a-5 - 0:2) + 2/2(^1 - s's) 
 
 + 2/3(^2 - 3:4) + 2/4(«3 - X5) + y^ixi - Xi). 
 
 Notice that double the area of any polygon is the sum of the products 
 of the ordinate of each vertex by the difference of the abscissas of the 
 adjacent vertices, these differences being taken in the same direction, 
 anti-clockwise, around the entire polygon. 
 
 13. Find the area of the triangle whose vertices are (12, - 5), (- 8, 7), 
 (10, 15). 
 
 14. Find the" condition that (x, y) lie in the straight line through 
 
 (x', 2/'). (*"' y")- 
 
RECTANGULAR COORDINATES 7 
 
 15. Show that the points (1, 4), (3, 2,), (- 3, 8) lie in a straight line. 
 
 16. The vertices of a pentagon are (-J, 3), (— 5, 8), (11, — 4), (0, 12), 
 (14, 7). Plot the pentagon and find its area. 
 
 17. A piece of land is bounded by straight lines. From the survey 
 the rectangular coordinates of the stations at the corners referred to 
 a N. S. line and an E. W. line through station A are as follows, distances 
 measured in chains : 
 
 ^00 D 22.85 17.19 
 
 B 14.30 - 15.04 E 7.42 40.09 
 
 C 22.85 -4.18 F -8.29 29.80 
 
 Plot the survey and find the area of the piece of land. 
 
 18. Find tiie point which divides the line from (x', y') to (x", y") 
 internally into segments whose ratio is r. 
 
 19. Find the point which divides the line from (x', y') to (x", y") 
 externally into segments whose ratio is r. 
 
 20. Locate the points (2, — 9), (— fi, 5), and also the points dividing 
 the line joining them internally and externally in the ratio 2 : 3. 
 
 21. Show that the points (x, y), (x, — y) are symmetrical with respect 
 to the X-axis. 
 
 22. Show that the points (x, y), (— x,y) are symmetrical with respect 
 to the r-axis. 
 
 23. Show that the points (x, y), (— x, ~ y) are symmetrical with 
 respect to the origin. 
 
 Art. 5. — Distance between Two Points 
 
 The distance between the points (x', ?/'), (x", y") is the hypote- 
 nuse of the right triangle 
 whose two sides about the 
 riglit angle are («' — x") 
 and {y' — y"). Hence 
 
 d = ^{x' -x"y + (y'-y"f. 
 
 Problems. — 1. Find distance 
 between the points (4, 2) , (7, 5) ; 
 
 (-3,6), (4,-9); (0,8), (7,0); F,,;. r,. 
 
 (15, -17), (8,2); (-4, -7), (-12, -19). 
 
 2. Derive formula for distance from (x', y') to the origin. 
 
 3. Find distance from origin to (5, 9) ; (7,-4); (12,-15); (■ 
 
 
 Y 
 
 
 
 
 
 ^ 
 
 iKv 
 
 ') 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 (X," 
 
 i") 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 9, 14). 
 
8 ANALYTIC GEOMETRY 
 
 4. Find the lengths of the sides of the triangle whose vertices are 
 (-3, -2), (7,8), (-5,0). 
 
 5. Tlie vertices of a triangle are (0, 0), (4, —5), (—2, 8). Find 
 the lengths of the medians. 
 
 6. Find the distance between the middle points of tlie diagonals of 
 the quadrilateral whose vertices are (2, 3), (—4, 5), (6, - 3), (U, 7). 
 
 7. Show that the points (6, 0), (1^, 15), (- 3, - 12), (- 7^, - 3) 
 are the vertices of a parallelogram. 
 
 8. Find the center of the circle circumscribing the triangle whose ver- 
 tices are (2, 2), (7, - 3), (2, - 8). 
 
 9. Find the equation which expresses the condition that the point 
 (x, y) is equidistant from (4, - 5), (— 3, 7). 
 
 10. Find the equation which expresses the condition that the distance 
 from the point (x, y) to the point ( — 3, 2) is 5. 
 
 11. Find the equation which locates the point (x, y) in the circum- 
 ference of a circle whose radius is r, center (a, b). 
 
 Akt. 6. — Systems of Points in the 1'lane 
 
 If any two quantities, which may be called x and y, are so 
 related that for certain values of x, the corresponding values 
 of y are known, the different pairs of corresponding values of 
 X and y may be represented by points in the XF-plane. 
 
 Comparative statistics and experimental results can fre- 
 quently be more concisely and more forcibly presented graphi- 
 cally than by tabulating numerical values. In the diagram 
 the abscissas represent the years from 1878 to 1891, the corre- 
 sponding ordinates of the full and dotted lines the production 
 of steel in hundred thousand long tons in the United States and 
 Great Britain respectively.* The diagram exhibits graphically 
 the information contained in the adjacent table, condensed 
 from " Mineral Kesources," 1892. Observe that if the points are 
 
 * In the figure the linear unit on the X-axis is 5 times the linear unit 
 on the r-axis. It will be noticed that the essential feature of a system 
 of coordinates, the "one-to-one correspondence" of the symbol (x, y) 
 and the points of the A"l'-plane, is not disturbed by using different scales 
 for oidiuates and abscissas. 
 
RECTANGULAR COORDINATES 
 
 9 
 
 inaccurately located the diagram becomes not only worthless, 
 but misleading. 
 
 45i 
 
 
 1878 '79 
 
 '80 '81 
 
 '82 '83 
 
 'S-t 
 Fm. 
 
 '85 'SO '87 
 T. 
 
 '88 '89 
 
 '90 '91 
 
 
 U.S. 
 
 G. B. 
 
 
 
 U.S. 
 
 n. p.. 
 
 1878 
 
 7.3 
 
 10.6 
 
 
 1885 
 
 17.1 
 
 19.7 
 
 1879 
 
 9.3 
 
 10.9 
 
 
 1886 ■ 
 
 25.6 
 
 23.4 
 
 1880 
 
 12.5 
 
 13.7 
 
 
 1887 
 
 .33.4 
 
 31.5 
 
 1881 
 
 15.9 
 
 18.0 
 
 
 1888 
 
 29.0 
 
 34.0 
 
 1882 
 
 17.4 
 
 21.9 
 
 
 1889 
 
 33.8 
 
 36.7 
 
 1883 
 
 16.7 
 
 20.9 
 
 
 1890 
 
 42.8 
 
 36.8 
 
 1884 
 
 15.5 
 
 18.5 
 
 
 1801 
 
 39.0 
 
 32.5 
 
 The table furnishes a number of discrete points which in the 
 figure are connected by straight lines to assist the eye. 
 
 Problems. — Exhibit graphically the information contained in the fol- 
 lowins? tables : 
 
 1894 
 
 Cost of steel 
 
 rails per 
 
 long ton 
 
 in Penn.sylvania 
 
 mills f 
 
 rom 18(_ 
 
 . (Mineral Resources.) 
 
 
 
 
 
 
 1867 $166.00 
 
 1874 
 
 $04.25 
 
 1881 
 
 $61.13 
 
 1888 
 
 $29.83 
 
 1868 158.50 
 
 1875 
 
 68.75 
 
 1882 
 
 48.50 
 
 1889 
 
 29.25 
 
 1869 132.25 
 
 1876 
 
 59.25 
 
 1883 
 
 37.75 
 
 1890 
 
 31.75 
 
 1870 106.75 
 
 1877 
 
 45.50 
 
 1884 
 
 30.75 
 
 1891 
 
 29.92 
 
 1871 102.50 
 
 1878 
 
 42.25 
 
 1885 
 
 28.50 
 
 1892 
 
 30.00 
 
 1872 112.00 
 
 1879 
 
 48.25 
 
 1886 
 
 34.50 
 
 1893 
 
 28.12 
 
 1873 120.50 
 
 1880 
 
 67.50 
 
 1887 
 
 37.08 
 
 1894 
 
 24.00 
 
10 
 
 ANALYTIC GEOMETRY 
 
 2. Commercial value of one ounce gold in ounces silver from 1855 to 
 1894. (Report of Director of Mint.) 
 
 1855 
 
 15.38 
 
 1865 
 
 15.44 
 
 1875 
 
 16.59 
 
 1885 
 
 19.41 
 
 1850 
 
 15.38 
 
 1866 
 
 15.43 
 
 1876 
 
 17.88 
 
 1886 
 
 20.74 
 
 1857 
 
 15.27 
 
 1867 
 
 15.57 
 
 1877 
 
 17.22 
 
 1887 
 
 21.13 
 
 1858 
 
 15.38 
 
 1868 
 
 15.59 
 
 1878 
 
 17.94 
 
 1888 
 
 21.99 
 
 1859 
 
 15.19 
 
 1869 
 
 15.60 
 
 1879 
 
 18.40 
 
 1889 
 
 22.09 
 
 1860 
 
 15.29 
 
 1870 
 
 15.57 
 
 1880 
 
 18.05 
 
 1890 
 
 19.76 
 
 1861 
 
 15.50 
 
 1871 
 
 15.57 
 
 1881 
 
 18.16 
 
 1891 
 
 20.92 
 
 1862 
 
 15.35 
 
 1872 
 
 15.63 
 
 1882 
 
 18.19 
 
 1892 
 
 23.72 
 
 1863 
 
 15.37 
 
 1873 
 
 15.92 
 
 1883 
 
 18.64 
 
 1893 
 
 26.49 
 
 1864 
 
 15.37 
 
 1874 
 
 16.17 
 
 1884 
 
 18.57 
 
 1894 
 
 32.56 
 
 3. Expense of moving freight per ton mile on N.Y. C. & H.R. R.R. 
 from 1866 to 1894. (Poor's Railway Manual.) 
 
 1866 
 
 <?'2.16 
 
 1873 
 
 J? 1.03 
 
 1880 
 
 ^54 
 
 1887 
 
 fM 
 
 1867 
 
 1.95 
 
 1874 
 
 .98 
 
 1881 
 
 .56 
 
 1888 
 
 .59 
 
 1808 
 
 1.80 
 
 1875 
 
 .90 
 
 1882 
 
 .60 
 
 1889 
 
 .57 
 
 1869 
 
 1.40 
 
 1876 
 
 .71 
 
 1883 
 
 .68 
 
 1890 
 
 .54 
 
 1870 
 
 1.15 
 
 1877 
 
 .70 
 
 1884 
 
 .62 
 
 1891 
 
 .57 
 
 1871 
 
 1.01 
 
 1878 
 
 .60 
 
 1885 
 
 .54 
 
 1892 
 
 .54 
 
 1872 
 
 1.13 
 
 1879 
 
 .55 
 
 1886 
 
 .53 
 
 1893 
 1894 
 
 .54 
 .57 
 
 4. Pressure of saturated steam in pounds per square inch at intervals 
 of 9° from 32'^ to 428° Fahrenheit. (Based on Regnaulu's results.) 
 
 32° 
 
 .085 lbs. 
 
 131° 
 
 2.27 lbs. 
 
 230° 
 
 20.80 lbs. 
 
 329° 
 
 101.9 lbs 
 
 41 
 
 .122 
 
 140 
 
 2.88 
 
 239 
 
 24.54 
 
 338 
 
 115.1 
 
 50 
 
 .173 
 
 149 
 
 3.02 
 
 248 
 
 28.83 
 
 347 
 
 129.8 
 
 59 
 
 .241 
 
 158 
 
 4.51 
 
 257 
 
 33.71 
 
 356 
 
 145.8 
 
 68 
 
 .333 
 
 167 
 
 5.58 
 
 260 
 
 39.25 
 
 305 
 
 163.3 
 
 77 
 
 .456 
 
 176 
 
 6.87 
 
 275 
 
 .45.49 
 
 374 
 
 182.4 
 
 86 
 
 .607 
 
 185 
 
 8.38 
 
 284 
 
 52.52 
 
 383 
 
 203.3 
 
 95 
 
 .800 
 
 194 
 
 10.16 
 
 293 
 
 00.40 
 
 392 
 
 225.9 
 
 104 
 
 1.06 
 
 203 
 
 12.20 
 
 302 
 
 09.21 
 
 401 
 
 250.3 
 
 113 
 
 1.38 
 
 212 
 
 14.70 
 
 311 
 
 79.03 
 
 410 
 
 276.9 
 
 122 
 
 1.78 
 
 221 
 
 17.53 
 
 320 
 
 89.80 
 
 419 
 
 428 
 
 305.5 
 336.3 
 
RECTA NG ULA R COOR DIN A TES 
 
 11 
 
 In these problems it is evident that theoretically there corresponds 
 a determinate value of the ordinate to every value of the abscissa. Hence 
 the ordinate is called a function of the abscissa, even though it may be 
 impossible to express the relation between ordinate and abscissa by a 
 formula or analytic function. 
 
 5. Suppose a body falling freely under gravity down a vertical guide 
 wire to have a pencil attached in such a manner that the pencil traces 
 a line on a vertical sheet of paper moving 
 horizontally from right to left with a uni- 
 form velocity. To determine the relation 
 between the distance the body falls and the 
 time of falling.* 
 
 Take the vertical and horizontal lines 
 through the starting point as axes of refer- 
 ence, and let 01, 12, 23, •••, be the equal 
 distances through which the sheet of paper 
 moves per second, the spaces 05, 510, •■•, on 
 the vertical axis represent 5 feet. Then 
 the ordinate of any point of the line traced 
 by the pencil represents the distance the 
 body has fallen during the time represented 
 by the abscissa of the point. Careful meas- 
 urements show that the distance varies as 
 the square of the time. Calling the distance 
 .s', the time t, the distance the body falls 
 the first second \ g, where g is found by 
 experiment to be 32.16 feet, the relation 
 between ordinate and abscissa of the line 
 traced by the pencil is expressed by the 
 
 
 
 
 -1 
 
 r 
 
 
 
 
 
 
 
 1 
 
 
 I 3 
 
 4 
 
 
 
 
 5 
 
 \ 
 
 
 
 
 
 
 
 10 
 
 \ 
 
 
 
 
 
 
 
 15 
 
 ^ 
 
 . 
 
 
 
 
 
 
 20 
 
 
 \ 
 
 
 
 
 
 
 25 
 
 
 \ 
 
 
 
 
 
 
 30 
 
 
 \ 
 
 
 
 
 
 
 35 
 
 
 
 
 
 
 
 
 40 
 
 
 
 
 
 
 
 
 45 
 
 
 
 
 
 
 
 
 50 
 
 
 
 
 
 
 
 
 55 
 
 
 
 
 
 
 
 
 60 
 
 
 
 
 
 
 
 
 65 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 proportion 
 
 , which leads to the equa- 
 
 tion .<? = \ gt'. The curve and the equation express the same physical 
 law, tJK! one algebraically, the other geometrically. 
 
 In this problem the ordinate is an analytic function of the abscissa, for 
 the relation between the two is expressed by a formula. 
 
 The ordinate is a continuous function of the abscissa ; that is, the 
 difference between two ordinates can be made as small as we please by 
 sufficiently diminishing the difference between the corresponding 
 
 * This is the principle of Morin's apparatus for determining cxperi 
 mentally the law of falling bodies. 
 
12 ANALYTIC GEOMETRY 
 
 6. A body is thrown horizontally with a velocity of v feet per second. 
 The only force disturbing the motion of the body taken into account is 
 gravity. Find the position of the body t seconds after starting. 
 
 Calling the starting point the origin, the horizontal and vertical lines 
 through the origin the A'-axis and F-axis respectively, the coordinates of 
 the body t seconds after starting are x — vt, y = — ^ gfi. Eliminating t, 
 
 y = 2_ X-, an equation which expresses the relation existing between 
 
 2 v^ 
 the coordinates of all points in the path of the body. 
 
CHAPTER II 
 
 EQUATIONS OF GEOMETRIO EIGUEES 
 Art. 7. — The Straight Line 
 
 A point moving in a plane generates either a straight line 
 or a plane curve. Frequently the geometric law governing tlie 
 motion of the point can be directly expressed in the form of 
 an equation between the coordinates of the point. This equa- 
 tion is called the equation of the geometric figure generated 
 by the point. 
 
 Draw a straight line through the origin. By elementary 
 
 geometry -^" = -^ = ^^ = • • •. This succession of equal ratios 
 
 ..Irt Aa^ Aa.2 
 expresses a geometric property 
 which characterizes points in the 
 straight line ; for every point in 
 the line furnishes one of these 
 ratios, and no point not in the 
 straight line furnishes one of these 
 ratios. Calling the common value 
 of these ratios m, and letting x 
 and y denote the coordinates of 
 any point in the line, the equation 
 y = mx expresses the same geo- 
 metric property as the succession of equal ratios. Hence if tlie 
 point (x, y) is governed in its nurtion by the equation, it generates 
 a straight line through the origin. Uy trigonometry m is the 
 tangent of the angle through which the X-axis must be turned 
 anti-clockwise to bring it into coincidence with the straight line. 
 13 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 7 
 
 y 
 
 r 
 
 
 
 
 I 
 
 >/ 
 
 / 
 
 
 
 
 A 
 
 / 
 
 
 
 
 
 X 
 
 / 
 
 1 i" T 
 
 
 
14 
 
 ANALYTIC GEOMETRY 
 
 
 
 Y 
 
 
 
 
 /^ 
 
 
 
 
 
 
 / 
 
 /^ 
 
 
 
 
 
 
 // 
 
 / 
 
 ^-' 
 
 
 
 
 / 
 
 /^ 
 
 
 
 
 
 
 y 
 
 ■7- 
 
 n 
 
 y 
 
 
 
 
 
 X— 
 
 
 
 A 
 
 
 
 
 
 This angle is called the angle which the line makes with the 
 X-axis, and its tangent is called the slope of the line. 
 
 Give the straight line y=^mx a motion of translation parallel 
 to the y-axis upward through a distance n. The ordinate of 
 every point in the line in the new position is n greater than 
 the ordinate of the same point in the line through the origin. 
 
 Hence the equation of the 
 straight line, whose slope is m, 
 and which intersects the l''-axis 
 at a point n linear units above 
 the origin, is 2/ = '"'-^ -f ^- ''^ 
 is called the intercept of the 
 line on the Y-axis, x and y 
 are called the current coordi- 
 nates of the straight line, m 
 and n are called the parameters 
 of the straight line. To every 
 straight line there corresponds one pair of values of m and n ; 
 for a straight line makes only one angle with the X-axis, and 
 intersects the T-axis in only one point; conversely, to every 
 pair of values of m and n there corresponds only one straight 
 line. 
 
 Problems. — 1. Write the equation of the line parallel to the F-axis 
 at a distance of 5 linear units to the right of the F-axis. 
 
 2. Write the equation of the line parallel to tlie X-axis intersecting 
 the F-axis 6 below the origin. 
 
 3. Write the equation of the straight line through the origin making an 
 angle of 45° with the A'-axis. 
 
 4. Find the equation of the line making an angle of 135° with the 
 X-axis, intersecting the I'-axis 5 above the origin. 
 
 5. Write the equation of the line whose slope is 2, intercept on 
 F-axis — 5. 
 
 6. Find the equation of the path of a point moving in such a manner 
 that it is always equidistant from (3, — 5), (— 3, 5). 
 
 7. Find the equation of the path of a point moving in such a manner 
 that it is always equidistant from (4, 2), (- 3, 5). 
 
EQUATIONS OF GEOMETRIC FIGURES 15 
 
 8. Find the equation of the locus of the points equidistant from (7, 4), 
 (-3, -5). 
 
 9. Find the equation of the straight line bisecting the line joining 
 (2, — 5), (G, 3) at right angles. 
 
 AiiT. 8. — The Circle 
 
 According to the geometric definition of the circle the point 
 (x, y) describes the circumference of a circle with radius r, 
 center (a, b), if the point (x, y) moves in the XF-plane in such 
 a manner that its distance from (a, h) is always r. This con- 
 dition is expressed by the equation {x — a)- + (y — b)- = r, 
 which is therefore the equation of a circle. 
 
 Problems. — 1. Write the equation of the circle whose radius is 5, 
 center (2, - 3). 
 
 2. Find the equation of the circle with center at origin, radius r. 
 
 3. Find equation of circle radius 5, center (5, 0). 
 
 4. Find equation of circle radhis 5, center (5, 5). 
 
 5. Find equation of circle radius 5, center (—5, 5). 
 
 6. Find equation of circle radius 5, center (—5, — 5). 
 
 7. Find equation of circle radius 5, center (0, — 5). 
 
 8. Find equation of circle radius 5, center (0, 5). 
 
 Art. 9. — Thk Comc Sections 
 
 After studying the straight line and circle, the old Greek 
 mathematicians turned their attention to a new class of curves 
 which they called conic sections, because these curves Avere 
 originally obtained by intersecting a cone by a plane. Tt was 
 soon discovered that these curves may be defined thus : 
 
 A conic section i s a curve traced by a point muving in a 
 plane iiisucli_a manner tliatthe ratio of the distances from the 
 moving point to a fixed point and to a fixed line is constant. 
 
 This definition will be used to construct these curves, to 
 obtain their properties, and to find their equations. The fixed 
 point is called the focus, the fixed line the directrix of the 
 conic section. When the constant ratio, called the character- 
 istic ratio and denoted by e, is less than unity, the curve is 
 
16 
 
 ANALYTIC GEOMETRY 
 
 called au ellipse ; when greater than unity, an hyperbola ; 
 
 when equal to unity, a parabola.* 
 
 The following proposition is due to Quetelet (1796-1874), a 
 
 Belgian scholar : 
 
 If a right circular cone is cut by a plane, and two spheres 
 
 are inscribed in the cone tangent to the plane, the two points 
 
 of contact are the foci of the section of the cone by the plane ; 
 
 and the straight lines in which this plane is cut by the planes 
 
 of the circles of contact of spheres and cone are the directrices 
 
 corresponding to these two foci respectively. 
 
 Let the plane cut all the elements of one sheet of the cone. 
 
 F, F' are the points of contact of the spheres with the cutting 
 
 plane; F any point in the intersection of plane and surface of 
 cone ; T, T' the points of contact 
 of element of cone through P 
 with spheres. The plane of the 
 elements Sa, Sa' is perpendicular 
 to the cutting plane and the plane 
 of the circles of contact. Since 
 tangents from a point to a sphere 
 are equal, PF= FT, FF = FT'. 
 Hence 
 
 FF + FF' = FT+ FT' = TT', 
 
 Fig. 11. a constant. Through F draw 
 
 DD' perpendicular to the parallels HH', KK'. From the simi- 
 
 lar triangles FDT and FD'T', 
 
 FT 
 FT 
 
 PD 
 FD' 
 
 hence 
 
 r>y composition 
 
 FF PD 
 
 TT 
 
 -, by interchanging means, 
 1)1/ ^ * * ' FD 
 
 PF ^ PD 
 FF' PD'' 
 PF _ TV 
 DD'' 
 
 J^W" PD' P77" 
 
 a constant. Similarly, ^^- = -^, i-^ 
 ^'TT' DD'' PD' 
 
 TT 
 DD 
 
 • Call the points 
 
 * Cayley, in the article on Analytic Geometry in tlie Britannioa, niiiMi 
 edition, calls this definition of conic sections the definition of Apolloiiius. 
 ApoUonius, a Greek mathematician, about 203 b.c, wrote a treatise on 
 Conic Sections. 
 
EQUATIoys OF GKOMICTUHJ FKiUUES 
 
 17 
 
 of intersection of the straight line FF' with the section of 
 the cone V, V'. Since 
 
 VF + VF = FF' + 2 VF = TT 
 and VF + VF' - FF + 2 VF = TT', 
 
 VF=VF' 
 and T 'F + VF = VF' + 1 7^" = I ' V = 2'7^ 
 
 Hence the constant ratio 77^ = 7777, i^^ l*^ss than unity, and the 
 
 conic section is an ellipse. 
 
 It is seen that the ellipse may also be defined as the locus of 
 the points, the snni of whose distances from two fixed points, 
 the foci, is constant. 
 
 Let the plane cut both sheets of the cone. With the same 
 
 notation as before, PF= PT PF' = PT' ; lienee 
 
 PF- PF' = TV = a constant. 
 
 , PT PD 
 From the similar triangles PDT and PD'T , ^^,= 7777,; 
 
 PP PD PF PI) , ' ^ ^ ^ 
 
 hence — = -— . P.y division £±- = ±^- hence 
 PF' PD' ^ TV DD' 
 
 PF ^ TV 
 PD DD' 
 
 a constant. 
 
18 
 
 ANALYTIC GEOMETRY 
 
 Similarly, = 
 
 is greater than unity, and the 
 
 PD' DD'' DB' 
 
 conic section is an hyperbola. 
 
 The hyperbola may also be defined as the locus of points, the 
 difference of whose distances from the foci is constant. 
 
 Let the cutting plane and the element MN make the same 
 angle 6 with a plane perpendicular to the axis of the cone. 
 The intersections of planes through the element 3fN with the 
 cutting plane are perpendicular to the intersection of cutting 
 plane with plane of circle of contact. 
 
 FF=: FT— MN= PD, and the conic section is a parabola, 
 focus F, directrix HIl'. 
 
 Art. 10. — The Ellipse, e<l 
 Construction. — Let F be the focus, HH' the directrix. 
 Througli F draw i^Vr perpendicular to HH', &nd on the perpen- 
 dicular to FK through 
 F take the points P 
 and P' such that 
 PF ^P'F^ 
 FK FK ^' 
 Through K and P, and 
 through A" and P', draw 
 straight lines. Draw 
 any number of straight 
 lines parallel to HH', 
 intersecting KP and 
 KP' in r?„ n^, n^, v^, •••, 
 FK in ?)?j, m-j, vi^, •••. 
 With F as center and 
 mj?;, as radius describe 
 an arc intersecting niit, 
 in R. Then 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 r' 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 n 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 'i 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 "'"' 
 
 y 
 
 
 
 -~^ 
 
 ^ 
 
 
 
 
 T 
 
 / 
 
 
 .' . 
 
 ' 
 
 
 
 
 
 
 N 
 
 
 K 
 
 y 
 
 / 
 
 VF 
 
 />1 
 
 r, 
 
 ^ 
 
 
 A 
 
 
 
 [ 
 
 ' \ 
 
 v' 
 
 
 K 
 
 \ 
 
 V. 
 
 
 
 
 ni] 
 
 
 
 
 / 
 
 
 - 
 
 — 
 
 
 
 ,N 
 
 ■. 
 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 ^ 
 
 X 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 ,> 
 
 < 
 
 
 ', 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 \. 
 
 
 TTliK 
 
 A' FK ' 
 
EQUATIONS OF GEOMETRIC FIGURES l!» 
 
 and Pi is a point in tlie ellipse. Similarly, an infinite number 
 of points of the cnrve may be located. 
 
 Definitions. — The perpendicular through the focus to the 
 directrix is called the axis of the ellipse. The axis intersects 
 the curve in the points V and V', dividing FK internally and 
 externally into segments whose ratio is e. The points V and 
 F' are called the vertices, the point A midway between V 
 and V, the center of the ellipse. The finite line VV is the 
 transverse axis or major diameter, denoted by 2a; the line 
 PjPi perpendicular to VV at A and limited by the curve is 
 the conjugate axis or minor diameter, denoted by 26; the 
 finite line PP' is called the parameter of the ellipse, denoted 
 by 2 p. The lines KP and KP' are called focal tangents. 
 The ratio of the distance from the focus to the center to the 
 semi-major diameter is called the eccentricity of the ellipse. 
 
 Properties. — The foci F and F' are equidistant from the 
 center A. By the definition of the ellipse VF = e • VK, 
 V'F= e ' VK Subtracting, FF' = e- VV. Dividing by 2, 
 
 AF = e • AV Hence e = ^^—~, that is, in the ellipse the 
 
 o 
 eccentricity equals the characteristic ratio. 
 
 FP 
 
 By definition ? = e, and by construction 
 
 ^ AK ^ 
 
 FP,= An, 
 
 By definiti(m eccentricity e = =^ 
 ^ ^ AV a 
 
 From the figure, VF = AV— AF = a — ae = a (1 — e) ; 
 
 VF = A V + vlP = a + ae = a(l + e). 
 
 FPand VF' are called the focal distances. 
 
 VT+ V'T _VF+ V'F_ ^^ 
 
 2 2 
 
 Hence ^17r=^. 
 e 
 
 ^ . ., AF Va-- 
 a eccentricity e = = 
 
 -b' 
 
20 
 
 A NA L YTIC GEOMETli Y 
 
 15y clefinitiun ^ = e, hence FA^= "'^^~^^ ^^=e, 
 
 hence V'K^''-^^-^^- 
 e 
 
 From the fiLnire i^/f = ^lA" - .li^ = 'i - ae = "'^^~^'^ ; 
 
 F'A'= .1 A + AF' = 1^ + «e = li(l±i^. 
 
 By cletiiiitiou 
 
 --^= e, hence 
 FK ' 
 
 p — ail — e-)— a[ 1 -^ — = a— = — 
 
 \ a- J a^ a 
 
 Equation — Take the axis of the ellipse as X-axis, the 
 perpendicular to the axis through the center as F-axis. Let P 
 
 be any point of the curve, 
 its coordinates x and y. 
 Tlie problem is to express 
 the definition PF=e- PH 
 by means of an e(]uation 
 between x and y. The 
 definition is equivalent to 
 PF''=e^-PH\ which is 
 the same as 
 
 pff + (AD + AFf 
 = e\AK+ADy, 
 
 which becomes y'^ + (x + aey — e^[ - + a; ] , 
 
 reducincf to — + — — ^ = 1. 
 
 a' a\l - e') 
 
 Since the point (o, b) is in the curve, a\l — e^) = 6', and the 
 
 equation finally becomes - -f-^= 1, 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (0,?J 
 
 ) 
 
 p 
 
 
 
 
 H 
 
 — 
 
 — 
 
 ^ 
 
 ^ 
 
 ■^- 
 
 — 
 
 — 
 
 ~Z^ 
 
 
 ■\ 
 
 
 
 
 
 / 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 \ 
 
 
 K 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 («,n 
 
 
 
 \, 
 
 F 
 
 
 
 A 
 
 
 
 D 
 
 
 1 
 
 X 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 •^ 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 
 
EQUATIONS OF GEOMETUKJ FIGURES 21 
 
 Summary. — Collfrtiiig tlic results of the preceding pani- 
 
 graphs, the fuudanieutal properties of the ellipse -- -f • , = 1 are : 
 
 a- b'^ 
 
 Distance from focus to extremity of conjugate diameter a 
 
 Distance from center to directrix - 
 
 e 
 Distance from focus to center ae 
 
 Distance from focus to near vertex 
 Distance from focus to far vertex 
 
 Distance from directrix to near focus 
 Distance from directrix to far focus 
 Distance from directrix to near vertex 
 Distance from directrix to far vertex 
 
 Eccentricity 
 
 Square of semi-conjugate diameter 
 
 Semi-parameter 
 
 Art. 11. — The Hyperbola, e>l 
 
 Construction. — Draw FK through the focus F perpendicular 
 to the directrix ////'. On the perpendicular to FK through F 
 
 take the i)oints P and P' such that -— - = — - = e. Through K 
 
 ^ FA FK 
 
 and P, and through K and P' draw straight lines. Draw any 
 number of parallels to ////', and on these parallels locate points 
 of the curve exactly as was done in the ellipse. The hyperbola 
 consists of two infinite branches. The vertices Fand V divide 
 FK internally and externally into segments whose ratio is e. 
 The construction shows that the parallels to ////' l)etween V 
 and V do not contain points of the curve. The notation is the 
 same as for the ellipse. 
 
 a(l-e) 
 
 
 a{\ + e) 
 
 
 a{\ - e2) 
 
 
 e 
 
 
 a(l -1- e2) 
 
 
 e 
 
 
 a(l-e) 
 
 
 e 
 
 
 a{\+e) 
 
 
 e 
 
 
 e- («'- 
 
 h-^y^ 
 
 e — 
 
 O 
 
 
 62=«i(l - 
 
 -e^) 
 
 p = a{\-e 
 
 .) = ^ 
 a 
 
22 
 
 ANAL VriC GEOMETli Y 
 
 Properties. — From the definition of the hyi)erbohi VF— e • VK, 
 V'F^e ■ V'K. Adding FF'=e- VV] dividing by 2, jLF^e-AV. 
 eccentricity, that is in the hyperbola also the 
 
 Hence 
 
 AF 
 
 characteristic ratio equals the eccentricity. 
 
 
 \ 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 // 
 
 
 
 
 \ 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 H 
 
 
 
 n. 
 
 // 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 ^ 
 
 'R 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 % 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 N^ 
 
 
 
 P. 
 
 /; 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 s 
 
 \ 
 
 
 / 
 
 1/ 
 
 
 
 
 
 
 
 
 
 
 F' 
 
 v' 
 
 
 
 
 A 
 
 
 k\ 
 
 / 
 
 V. 
 
 /'F 
 
 vu, 
 
 m 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 / 
 
 / 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 / 
 
 
 
 p-^ 
 
 \ 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 s 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 Hi 
 
 
 
 \ 
 
 
 
 From the figure 
 
 VF=AF-AV= a{e - 1) ; V'F= AF -\- AV = a{e + 1). 
 
 By definition 
 
 YK==e, hence VK= ''^"-^^ ■, 1'^ = e, hence V'K= 
 e FA 
 
 a(e + 1) 
 
 VK 
 
 From the figure AK ^ AV — F/i = a — a 
 From the figure 
 
 e — 1 _ a^ 
 e 
 
 FK 
 
 ^AF-AK = ''^'-% F'K 
 
 AK-{-AF = 
 
 aO- + 1) 
 
 By definition pU= c, hence p = a{e- — 1). 
 
 Equation. — To find the equation of the hyperbola take the 
 axis of the curve as X-axis, the perpendicular to the. axis 
 
EQrATIO.XS OF GEOMKiniC FKJl'llKS 2o 
 
 tlii'oai;li llic (•('iittn- as I'-axis. Let /' he any ituiiit of the curve, 
 its coordinates x and //. The piubleiu is to express the delini- 
 tiou FF=e- PH by lueaus ot an e(iuation between x and y. 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 kp 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ''/ 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ' 
 
 1 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 f' 
 
 
 
 
 
 A 
 
 
 K 
 
 
 F 
 
 / 
 
 
 |D 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 — 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 _ 
 
 — 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 H' 
 
 
 
 
 \ 
 
 \ 
 
 
 The definition is equivalent to PF' — e' • PII , Avliich is tlie 
 same as PD' -it{AD — AFf = <i-{AD — AKf, which becomes 
 
 y- + (.c — ac)' — e-( x — -] , reducing to "— -J- = 1. 
 
 \ ej ' tt- a'-(e- — 1) 
 
 IMaciug <i'{('' — 1)= b-, the ecjuation takes the form ^ — -., = 1. 
 
 Since Ir = a-c^ — a-, it is seen from the figure that an arc 
 described from the vertex as a center witli a radius e(iual to 
 distance from focus to center intersects the F-axis at a distance 
 b from the center. 2 6 is called the conjugate or minor diameter 
 of the hyperbola. 
 
 Summary. — Collecting the results of the preceding para- 
 graphs, the fundamental properties of the hyperbola 
 
 ^.2 
 
 = 1 an 
 
24 
 
 A iVM L Y TIC GEOMETlt Y 
 
 Distance from vertex to extremity uf coiijujjate diameter ae 
 Distance from fucus to center 
 Distance from focus to near vertex 
 Distance from focus to far vertex 
 
 Distance from directrix to near vertex 
 
 Distance from directrix to far vertex 
 
 Distance from directrix to near focus 
 
 Distance from directrix to far focus 
 
 Eccentricity 
 
 Square of semi-conjugate diameter 
 
 Semi-parameter 
 
 ae 
 
 
 a{t 
 
 -1) 
 
 ait 
 
 + 1) 
 
 a(e 
 
 -1) 
 
 
 e 
 
 fl(e 
 
 + 1) 
 
 
 e 
 
 a(e 
 
 2-1) 
 
 
 e 
 
 a(e 
 
 2+1) 
 
 
 e 
 
 e = 
 
 (a2 + ^2)^ 
 
 a 
 
 62. 
 
 = rt2(e2_ 1) 
 
 P = 
 
 .,(e'-l)J-- 
 
 AuT. 12. 
 
 
 
 Y 
 
 
 
 
 
 
 /)t2 
 
 
 H 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 n 
 
 / 
 
 
 P= 
 
 
 
 
 
 
 / 
 
 ^> 
 
 ^ 
 
 
 
 
 
 
 ^ 
 
 <' 
 
 
 
 
 
 
 P 
 
 /- 
 
 
 / 
 
 
 
 
 
 
 / 
 
 
 '/ 
 
 
 
 
 
 K 
 
 / 
 
 V 
 
 /f 
 
 
 
 D 
 
 
 
 
 \ 
 
 \ 
 
 %s 
 
 
 
 
 
 X 
 
 
 
 \ 
 
 •• 
 
 \ 
 
 
 
 
 ~ 
 
 — 
 
 
 P' 
 
 \ 
 
 ^^ 
 
 ^ ^ 
 
 
 
 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 \. 
 
 — 
 
 ~K 
 
 
 
 
 fix 
 
 \ 
 
 \ 
 
 Pa 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^. 
 
 PiiE Parabola, e = l 
 
 Through F, the foctis, draw FK 
 perpendictilar to HH.\ the direc- 
 trix. On the perpendicular to FK 
 through F lay off FP = FF = FK. 
 Draw the focal tangents KP and 
 KP', draw a series of parallels to 
 HH', and locate points of the pa- 
 rabola as in the case of ellipse and 
 hyperbola. From the figure, it is 
 seen that the distance from vertex 
 to focus is ^ p, distance from vertex 
 to directrix is ^j), the parameter 
 being 2 j^- 
 
 To find the equation of the pa- 
 rabola, take the axis of the curve 
 as X-axis, the perpendicular to the 
 axis at the vertex as F-axis. Let 
 J*(x, ?/) be any point in the curve. 
 
EQUATIONS OF GEOMETIIW FIGURES 
 
 25 
 
 The problem is to express the deliiiitioii PF = I)K by means 
 of an eqnation between x and y. The definition may bo written 
 1^'^ = UK', which is the same as PD' + TTf = ( VI> + VKf, 
 wliich becomes 
 
 if + i^c-^pf^ix + llif, 
 
 re(bicing to y- = 2 jkv- 
 
 A parabola whose focus and directrix are known may be 
 generated mechanically as in- 
 dicated in the figure. 
 
 Problems. — 1. Construct the el- 
 lipse wliosc pariviiK'ter is G, eccentri- 
 city 2. 
 
 2. Construct tlie hyperbola whose 
 panuueter is S, eccentricity J. 
 
 3. Construct the ellipse whose 
 diameters are 10 and 8. Find tlie 
 equation of the ellipse, its eccentri- 
 city, and parameter. 
 
 4. Construct the hyperbola whose diameters are 8 and C>. Find Uw. 
 e(iuation of the liyperbola, its eccentricity, and parameter. 
 
 5. Construct the parabola whose parameter is 12 and find its e(iuation. 
 
 6. Find the equation of the ellipse whose eccentricity is j^, major 
 diameter 10. 
 
 7. The diameters of an hyperbola are 10 and 0. Find distances from 
 center to focus and directri.K. 
 
 8. The distances from focus to vertices of an hyperbola are 10 and 2. 
 Find diameters. 
 
 H 
 
 -==: 
 
 ■-^ 
 
 ^ 
 
 
 
 ^ 
 
 ^ 
 
 ^ 
 
 
 
 
 --, 
 
 
 P^ 
 
 >< 
 
 k 
 
 
 
 
 
 
 
 ^ 
 
 
 •-. 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 .... 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 K 
 
 
 \ 
 
 F 
 
 
 
 
 
 
 X 
 
 - 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 \, 
 
 
 
 
 
 9. The parameter of a parabola is 1-2. Find distance from focus to 
 point in curve who.se abscissa is 8. 
 
 10. Find diameters of the ellip.se whose parameter is 10, eccentricity I. 
 
 11. In an ellipse, the distance from vertex to dinctrix is 0, eccentri- 
 city J. Find diameters and construct ellipse. 
 
26 
 
 ANALYTIC GEOMETRY 
 
 12. In the ellipse — + ■'" = 1 show that the distances from the foci to 
 a- h- 
 the point (x, y) are r = a - ex, r' = a + ex. r and »•' are called the focal 
 radii of the point (;c, //). The sum of the focal radii of the ellipse is con- 
 stant and equal to 2 a. 
 
 x" 
 
 13. In the hyperbola 
 
 ^- = 1 show that the focal radii of the point 
 5- 
 
 The constant difference of the focal 
 
 (x, ?/) are r — ex - a, r = ex + a, 
 radii of the hyperbola is 2 u. 
 
 14. Find the equation of the ellipse directly from the definition: The 
 ellipse is the locus of the points the sum of whose distances from the foci 
 equals 2 a. 
 
 Take the line through the foci as A'-axis, the point midway between 
 the foci as origin. When the point (x, y) is on the F-axis its distances 
 from F and F' are each equal to a. 
 Call AF — AF' = c, the distance of 
 (x, y) when on the T-axis from the 
 origin b. Then ci^ - c- = b'\ The 
 geometric condition PF + PF' = 2 a 
 is expressed by the equation 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 ,P 
 
 x?/) 
 
 
 
 
 
 / 
 
 l/ 
 
 
 ,y 
 
 ^ 
 
 
 '\ 
 
 \ 
 
 
 
 / 
 
 ^ 
 
 >■ 
 
 ^ 
 
 
 
 l\ 
 
 \ 
 
 
 
 \ 
 
 F' 
 
 
 
 A 
 
 
 D 
 
 ' ) 
 
 X 
 
 
 \ 
 
 
 
 
 
 
 
 / 
 
 
 
 ^ 
 
 
 _ 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VyH (x-r)H V//-+(x + 0'-^ = 2 ff, 
 
 — which reduces to 
 
 = 1. 
 
 The definition used in this problem 
 
 suggests a very simple mechanical 
 
 construction of the ellipse whose foci 
 
 and major diameter are known. 
 
 Fasten the ends of an inextonsible string of constant length 2 a at the 
 
 foci F and F' . A pencil point guided in the plane by keeping the string 
 
 stretched traces the ellipse. 
 
 15. Find the equation of the hyperbola directly from the definition : 
 The hyperbola is the locus of the points the difference of whose distances 
 from the foci is 2 a. 
 
 Take the line through tlie foci as X-axis, the point midway between 
 the foci as origin. Call AF = AF' = c, c^ - a" = U^. 
 
 The condition PF' - PF=2a leads to the equation 
 
 Vy'^ + (X + c)2 
 
 which reduces to 
 
 
 \/y"- + (a 
 1. 
 
 c)-^ 
 
FA^UATIONS OF dKOMFTIUC FKiUIiES 27 
 
 The inechiuiical fonstnictidu of tho hyperbola is effected as indicated 
 in tlie figure. 
 
 "" 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 >^^ 
 
 ;;^- 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5^ 
 
 r 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 ^ 
 
 •^ 
 
 =-^ 
 
 ^^ 
 
 
 X 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 r^ 
 
 
 f-^^ 
 
 ^ 
 
 
 
 '" 
 
 /, 
 
 ifj) 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 r 
 
 
 < 
 
 --' 
 
 
 
 
 / 
 
 '/ 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 r 
 
 
 
 ^ 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 V 
 
 t^ 
 
 ^ 
 
 F'/ 
 
 
 
 
 
 A 
 
 
 
 
 \ 
 
 F 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fic. 21. 
 
 16. Two pins fixed in a ruler are constrained to move in grooves at 
 right anf,des to each other. Show that every point of the ruler describes 
 an ellipse whose semi-diameters are the distances from the point to tho 
 pins. Tliis device is called an elliptic compass. 
 
 Tlio folldwint;- statcMneiits may help to form an idea of flio 
 importance of the conic sections : 
 
 The planets and asteroids move in ellipses with the sun 
 at one focus. 
 
 The eccentricity of the earth's orl)it is about j.\^. 
 
 The eccentricity of the moon's elliplic }iath about the earth 
 is about J^. 
 
 Nearly all comets move in parabolas with the sun at the focus. 
 
 Tho caljlo of a suspension bridLfe, if the load is uniformly 
 distributed over the horizontal, takes the form of a ]iaral)ola. 
 
 A projectile, iinless projected vei-tically, moves in a ])arabola, 
 if the earth's attraction is the only distinl)ing forcu^ takcui into 
 account. 
 
CHAPTER III 
 
 PLOTTING or ALGEBEAIO EQUATIONS 
 AuT. 13. — General Theory 
 
 The locus of the points (x, y) whose coordinates are the pairs 
 of real values of x and y satisfying the equation f(;x, y) = is 
 called the graph or locus of the equation. 
 
 Constructing the graph of an equation is called plotting the 
 equation, or sketching the locus of the equation. 
 
 An equation f{x, y) = is an algebraic equation, and ?/ an 
 algebraic function of x, when only the operations addition, sub- 
 traction, multiplication, division, involution, and evolution 
 occur in the equation, and each of these only a finite number 
 of times. 
 
 When the equation has the form y =f(x), y is called an 
 explicit function of x; when the equation has the form 
 f(x, ?/) =0, y is called an implicit function of .a*. 
 
 The locus represented by an equation f(x, y) — depends 
 on the relative values of the coefficients of the equation. For 
 mf{x, y) = 0, wliere m is any constant, is satisfied by all the 
 pairs of values of x and y which satisfy f(x, ?/) = 0, and by no 
 others. 
 
 If the graphs of two equations /i(.)-, y) = 0, /.(.r, y) =0 an' 
 constructed, the coordinates of the points of intersection of 
 these graphs are the pairs of real values of x and y which 
 satisfy /i {x, y) = and f, {x, y)—0 simultaneously. 
 
 Occasionally it is possible to obtain the geometric definition 
 of a locus directly from its equation, and then construct the 
 locus mechanically. The equation x^ -^-y"^ = 25 is at once seen 
 28 
 
PLOTTING OF ALGEBRA W EQUATIONS 20 
 
 to represent a circle with center at oritijin, radius 5. In general 
 it is necessary to locate point after point of the locus by assign- 
 ing arbitrary values to one variable, and computing the corrc- 
 si)onding values of the other from the equation. 
 
 Akt. 14. — Locus OF First Degukk Equation 
 
 The locus of the general first degree equation Itotween two 
 variables x and y, Ax -\- B>/ + C = 0, is the locus of 
 
 ^ b"^ B 
 
 Moving the locus represented by this equation parallel to the 
 I'-axis upward through a distance -'^, increases each ordinate 
 
 fi ^ ... 
 
 l)y — . Hence the equation of the locus in the new position is 
 ?/ = — — .r, which represents a straight line through the origin, 
 
 since the ordinate is proportional to the abscissa. The equa- 
 tion Ax -\- By + C = () therefore represents a straight line whose 
 slope is — --, and whoso intercept on the F-axis is — — . The 
 intercept of this line on the X-axis, found by placing y equal 
 to zero in the equation and solving for x, is — -• 
 
 The straight line represented by a lirst degre(> c(pi:iii(>u may 
 be constructed by determining the point of intersection of th(( 
 line with the F-axis and the slope of the line, by dcici'mining 
 any point of the line and the slope of tlie line, by determining 
 the points of intersection of the line with tlie coordinate axes, 
 by locating any two points of the line. 
 
 Problems. — Construct by the different methods the linos repn^sentcd 
 by the equations. 
 
 1. 2x + 32/ = 6. 3. \x-\y = \. 5. •%;( = 1- 
 
 2. j/ = x-5. 4. ix-4?/ = 2. 6. ^+^-1. 
 
30 
 
 ANALYTIC GEOMETRY 
 
 7. Show that ?/- — 2 .r>j — 8 x- = represents two straight lines through 
 the origin. 
 
 8. Show that a homogeneous equation of the ?ith degree between x and 
 y represents n straight lines through the origin. 
 
 9. Construct the straight line ^ 4- ^ = 1 and the circle x- + 2/" = 25 
 
 and compute the coordinates of the points of intersection. Verify by 
 measurement. 
 
 AiiT. 15. — Stuaigiit Ltxe through a Point 
 
 Tliroiigh the fixed point (Xf,. 
 an an.ufle « with the X-axis. 
 
 yo) draw a straight line making 
 Let (x, y) be any point of this 
 line, d the distance of (;r, ?/) 
 from (.r,|, ?/(,). From the figure 
 X — a'o = d cos a, y — ?/o --= d sin a, 
 whence x = .r,, 4- d> cos a, 
 y = y^^ + d sin a, 
 an d y — ?/„ = tan a (x — .r„). That 
 is, if a i)oint (a-, y) is governed 
 in its motion by the ecpiation 
 Fro. 22. 1/ ~ 1/0 = tan a (x — x,|), 
 
 it generates a straight line 
 through (a'o, ?/o), making an angle a with the X-axis, and the 
 coordinates of the point in this line at a distance d from (.r,„ ?/„) 
 are x = Xo-\- d cos a, y = yQ-\- d sin a. The distance, d, is posi- 
 tive when measured from (xq, y^) in the direction of the side of 
 the angle a through (x„, 7/,,) ; negative Avhen measured in the 
 opposite direction. 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■^ 
 
 
 
 
 
 d 
 
 {X 
 
 }!> 
 
 
 
 
 
 
 (X 
 
 ,^ 
 
 < 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 — 
 
 A 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 siraiglit lino 
 terms of the 
 
 Problems. — 1. Express the coordinates of a point in tl 
 through (2, ".) making an angle of 30' with tlio A'-axis i 
 distance from (2, 3) to the point. 
 
 2. On the straight line through (o, — 2) making an angle of 00" with 
 the X-axis, find the coordinates of the points whose distances from 
 (?., - 2) are 10 and - 10. 
 
riOTTINa OF ALaEBRAW ^JQUAriONS 
 
 31 
 
 3. Write the oiiuation of the s(rai,i;ht line tlimu-li ( - 2, 5) and making 
 an angle of 45 with the A'-axis. 
 
 4. Write the equation of the straight line thniugh (4, - 1) wiiose 
 slope is J. 
 
 5. Find the distances from the point (2, ;5) to the points of intersection 
 of the line through this point, making an angle of 30° with the A'-axis 
 and the circle x^ + y'^ = 25. 
 
 The coordinates of any point of the given line are x = 2 + (?cos.OO\ 
 y = 3 + d sin 30°. These values of x and y substituted in the equation of 
 the circle x^ + ?/ = 25 give the equation d^ + (4 cos 30° + 6 sin 30°)rt = 12, 
 which determines the values of d for the points of intersection. 
 
 AiJT. 16. — Taxcexts 
 
 To plot a numerical algebraic equation involving; two vari- 
 ables, put it into the form y=f(x) if possible. Comi)uto the 
 values of y for different values of x, and locate the points whose 
 coordinates are the pairs of corresponding real values of x and 
 ?/. Connect the successive points by straight lines, and observe 
 the form towards which the broken line tends, as the nunilicr 
 of points locaiod is indofinitoly increased. This limit of tlu^ 
 broken line is tlic locus of tlie eqnatiim. 
 
 ExAiNrPLK. — Plot ?/- + a-- = 9. 
 Here y = ± V'.) — x-, a- = d 
 .T = — 4 —3 
 
 ji = ± V^^ 
 
 +1 
 
 ±3 • ±2V2 
 
 or extracting the roots 
 
 r = - ;; - 2 
 
 ,/ = ± 2.2;;7 ±2. SI'S 
 
 // has two iMimerically equal values for each value of x. 
 Hence the locus is symmetrical w^ith respect to the X-axis. 
 For a like reason the locus is symmetrical with respect to the 
 
 : V'.) - y-. 
 
 — 2 
 
 -1 
 
 
 ± VT) 
 
 ±2V2 
 
 
 -f 2 +3 
 ± Vr. 
 
 + 4 
 
 
 10 -f 1 
 
 js ± ;; ± 2.siis 
 
 + 2 
 
 ± 2.2;m 
 
 
 
32 
 
 ANAL YTIC GEOMETR Y 
 
 Y 
 
 I'-axis. For values of a- > + 3 and for values of a;< — 3, 
 y is imaginary. Hence the curve lies between the lines 
 a; = +3, a; = — 3. The curve also lies between the lines ?/= +3, 
 y=—o. Locating points of the 
 locus and connecting them by 
 straight lines, the figure formed 
 apjj roaches a circle more and 
 more closely as the number of 
 points located is increased. The 
 form of the equation shows at 
 once that the locus is a circle 
 whose radius is 3, center the 
 origin. 
 
 Through a point (.t,,, ?/„) of the 
 circle an infinite number of 
 straight lines may be drawn. The coordinates of any point of 
 the straight line y — ?/o = tan a{x — a;,,) through (.?(„ ?/(,), making 
 an angle a with the X-axis are .t = ccq + dcos, a, y — ?/„ + f' sin n. 
 The point (x, y) is a point of the circle x"^ -{-y^ = 9 wlien 
 
 (x„ + d cos ay + (,Vo + (I sin a)' = 9, 
 that is, when 
 
 (1) (.i-,2 + 7/,/ - 9) + 2 (cos a • a-o + sin a ■ y„)d + (1~ = 0. 
 Equation (1) determines two values of d, and to each of these 
 values of d there corresponds one point of intersection of line 
 and circle. Since the point 
 (xo, ?/o) is in the circle x^ -i- y'^ = 9, 
 the first term of equation (I) is 
 zero, hence the equation has two 
 roots equal to zero when 
 
 cos n • .T„ + sin a • v/,, = 0, 
 
 that is, when tan « = 
 
 .Vo 
 
 
 
 
 
 
 Y 
 
 \ 
 
 
 
 
 
 
 
 /^ 
 
 
 
 
 ..'■o 
 
 .'/o) 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 ^. 
 
 
 
 
 
 A 
 
 
 
 
 N 
 
 k^ 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 \. 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 To 
 
 d = there corresponds the point 
 (xo, ?/„), and when both roots of 
 
PLOTTING OF AIj; hlUlA KJ ICijUATIONS :;:j 
 
 eqiialiuu (1) ;iic! zero, Hit- two jH»iiil„s ol' iiitL'rsi'clitiii of the 
 straight line // — y,, = t;ui <« (,f — .r„) and the eircle x'- + j/- — \) 
 coincide at {j\„ //„), and the line is the tangent to the circle 
 (.?•„, _?/„). Hence the eqnation of the tangent to the circle x--\-y-=*J 
 at the point {x„, y^ is 
 
 which reduces to xxy + yy^ — 9. 
 
 A tangent to any curve is defined as a secant having two 
 points of intersection with the curve coincident.* \\y the 
 direction of the curve at any point is meant the direction of 
 the tangent to the curve at the point. 
 
 The circle x? 4- ?/- = i) at the point (.r,,, //,,) makes, with the 
 
 X-axis, tan~'( — " " ). At the points corresponding to .*; = U the 
 angles are tan-'( ^ )= loS^'oT' and tan-'f -^^- I — 41'' 2.';'. 
 
 Problems. — 1. Sliuw that -"'-" + ••'"=1 is tanj'ent to the ellipse 
 •|2 + p = l at(:r,„2,,). 
 
 2. Show that ^^-" - y'-'^ = 1 is tani^oiit to the hyperbola 
 
 orr «2 
 
 ^2-ft^=l at(.ro, yo). 
 
 3. Show that yun - p{x + re) is tangent to the parabola if- - '2 px at 
 (■'•0, 2/0) • 
 
 Ai;t. 17. — I'oix'is (JF Discontinuity 
 
 KXAMI'LK. — Plot >f = ^^-^^^■ 
 
 X — 'J 
 
 a; = -cc •••-;■> -4 -,S -2-1 (» +1 -f-lj +:; + -| ... -f ^ 
 
 y = 4- 1 ... 4-i +'j, +1 +\ -t -2 Tco +4 +l>i-... +1 
 
 * The secant definition of a tangent is due to Descartes and Fermat. 
 
 D 
 
34 
 
 ANAL YTIC GEO ME TR Y 
 
 From X = to x- = + 2, y is negative and iiu-reases iiuleli- 
 nitely in numerical valne as x approaches 2. From x — + 2 
 to x — + cTj, y is positive and diminishes from + qk> to + 1. 
 
 y is negative, and decreases 
 numerically from — i to 
 wliile X passes from to 
 — 1. y is positive and in- 
 creases to + 1 from x = —1 
 to a; := — Oj . The curve 
 meets each of the two 
 straight lines x = 2 and 
 II = 1 at two points infi- 
 nitely distant from the 
 origin. 
 
 The point corresponding 
 to X = 2 is a point of dis- 
 continnity of the curve. 
 J,',,;. 25. I'^^Ji" if two abscissas are 
 
 taken, one less than 2, 
 tlie other greater than two, the difference between the corre- 
 sponding ordinates approaches infinity Avhen the difference 
 between the abscissas is indefinitely diminished, while the 
 definition of continuity requires that the difference between 
 two ordinates may be made less than any assignable quantity 
 by sufficiently diminishing the difference between tlie corre- 
 sponding abscissas. 
 
 
 
 
 
 
 Y 
 
 
 \ 
 
 
 
 
 ~ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 — 
 
 — 
 
 — 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 ~ 
 
 - 
 
 — 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 AkT. 18. ASYMI'TOTES 
 
 Example. — Plot / — x"- = 4. Here // — ± V.ir + 4. 
 a; = -co 4 -3 -2 -1 0+1 +2 
 
 + 3 +4 ...+CO 
 Z/ = ±oo-.. ±4.47 ±3.r>l ±2.83 ±2.24 ±2 ±2.24 ±2.83 
 
 ±3.61 ±4.47"- ± ^ 
 
PLOTTli\G OF ALUKIIRAIC EQUATIONS 
 
 1/ lias two iiiiiiici'ically ('(pial iH'al values with opposiit; si^i's 
 fur every value of x. Tlie values of // iuereuse iudetiuitely 
 in numerical value as x in- 
 creases indeHnilely in iniuieri- 
 cal value. It now l)ec(Mnes 
 important to determine whether, 
 as was the case in Art. 17, a 
 strai.n'ht line can be drawn 
 whieh meets the eurve in two 
 points iniinitely distant from 
 the origin. The [xjints of in- 
 tt'rseetion of the straight line 
 // — iiix -f "■ iiiiil the locus of 
 y- — x- = 4 are found by making F"^- '-''• 
 
 these equations sinudtaneous. Eliminating y, there results the 
 equation in x, (?/r — 1) x' + 2 mnx + n- — 4 == 0. The problem 
 is so to determine m and n that this equation has two infinite 
 roots. An equation has two infinite roots when the coefficients 
 of the two highest powers of the unknown quantity are zero.* 
 Hence y = mx-\-n meets y"^ — x^ = 4: at two points infinitely 
 distant from the origin when nv — 1 = 0, 2 mn = 0, whence 
 7/1 = ± 1, n = 0. There are, therefore, two straight lines y = x 
 and ?/ = — X, each of which meets the locus of y- — x- = 4 at 
 two points infinitely distant from the origin. These lines are 
 called asymptotes to the curve. 
 
 \ 
 
 ^ 
 
 
 
 
 Y 
 
 
 
 / 
 
 ^ 
 
 
 \ 
 
 
 
 
 
 y 
 
 r/ 1 
 
 
 
 \ 
 
 V 
 
 
 
 y 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 \/ 
 
 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 
 X 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 
 
 
 ^ 
 
 
 
 \ 
 
 \ 
 
 ^ 
 
 L 
 
 
 ^ 
 
 
 
 
 
 
 \ 
 
 /^ 
 
 
 
 
 
 
 L_ 
 
 
 ^\l 
 
 Problem. — Show that y 
 
 ±-x are asymptotes to the liyperbola 
 X- y^ , 
 
 a- 
 
 62 
 
 * Place x = - in (1) ax" + hx"^ + cx""-^ + •■■ + h-x^ + Ix + m = 0. 
 
 There results (2) a + bz + cr^ + ... + kz" ^ + Iz" • + mz" = 0. Eiinatioii 
 (2) has two zero roots wlit'ii a = 0, b - 0. Hence L'(iualion (1) h;i,s two 
 infinite roots when a = 0, b = 0. 
 
36 
 
 ANA L VTIC GEOMETli Y 
 
 y = 
 
 AiiT. 19. — Maximum and Minimum Okdinatks 
 
 EXAMI'LK. l^lut // = X'' — 1 X -\- 7. 
 
 ^-y,... -4 -3 -2 -1 +1 +1\ +2 +3.--+^ 
 ..-29 +1 +13 +13 +7 +1 -i +1 +13. ..+00 
 
 For .T = + 1, _v = + 1 ; fol- 
 ic =+ 2, ?/ = + 1 ; for a; = 11 
 y= — \. Hence between x=l 
 and a: — 2 the curve passes 
 below the X-axis, turns and 
 again passes above the X-axis. 
 At the turning point the ordi- 
 nate has a niinimuni value ; 
 that is, a value less than the 
 ordinates of the points of the 
 curve just before reaching and 
 just after passing the turning 
 point. The point generating 
 the curve moves upward from 
 a; = to .T = — 1, but some- 
 where between x =— 1 and 
 X — — 2 the point turns and 
 starts moving towards the 
 X-axis. At this turning point the ordinate is a maximum; 
 that is, greater than the ordinates of the points next the turn- 
 ing point on either side. 
 
 To determine the exact position of the turning points, let x' 
 be the abscissa, ?/' the ordinate of the turning point. Let h 
 be a very small quantity, y, the value of y corresponding to 
 X = x' ± h. Then ?/i - ?/' must be positive when y' is a mini- 
 mum, negative when >/' is a maximum. Now 
 
 ,/, -y' = (3 x' - 7) ( ± //) + .'5 -v ( ± J'Y + ( ± /')'• 
 h may be taken so small that the lowest power of h deter- 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 : T 
 
 
 
 
 
 
 t \ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 T 
 
 1 r 
 
 
 
 
 
 J^ 
 
 f^ 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 27. 
 
PLOTTING OF ALCKHnAW Ei^UATlONti 87 
 
 mines the si!j,ii of //, — //'.* //, — //' (-an tlicrciore have the same 
 sign for ± h only when the coefficient of the lirst power of h 
 vanishes. This gives 3 x*^— 7 = 0, whence x = ± V^. x = + V|, 
 rendering y^ — y' positive for ± li, corresponds to a minimum 
 ordinate; a; = — V|, rendering //i — ,v' negative for ± //, corre- 
 sponds to a maximum ordinate. _ 
 
 ANhen x = + V|, .'/ = 7 - V" V^I = - .2 ; wlien x = - Vf, 
 y^7 +i^V21 = U.2.t 
 
 The values of a; which make // — are the roots of the ctpui- 
 tion ur' — 7 .f + 7 = 0. These values of x aiv. tlie abscissas of 
 the points wliere the locus of y = x^ — 7x + 7 intersects the 
 X-axis. X' — 7 .« + 7 = 0, therefore, has two roots between 
 -|- 1 and -f 'J, and a negative root between — ."> and — 4. 
 
 AUT. 20. I'olXTS OK IXFLECTIOX 
 
 ExAMi'LE. — Plot y + x-y — x = 0. Here // — 
 
 X ^-cc 3 - 2 - 1 +1 +2 + 3 ... -f X) 
 
 If (x, y) is a point of the locus, (— x,— y) is also a point of 
 the locus. Hence the origin is a center of symmetry of the 
 locus. A line may be drawn through the origin intersecting 
 the curve in the symmetrical points /-• and P'. If this line is 
 
 * Let s = ah^ + hh^ + civ' + (IW' + ••• be an infinite scries with finite 
 coefficients, and let li be greater numerically than the largest of the co- 
 efficients b, c, <1, ••-. Tlien hh + eh- + dh^ + ••• < /t — and 
 
 \ — h 
 
 s = h\a + hh + '•//■•! -h ,1h^ ...) = /(,3(rt i ^1), when A<h — • 
 
 1 — ft 
 
 When h is indefinitely diminished, h — '- — diminishes indefinitely. Con- 
 1 — h 
 
 sequently A becomes less than the finite quantity a, and s has the sign 
 
 of a/i*. 
 
 t This method of examining fur maxima and minima was invented by 
 
 Fermat (1590-l(iG;3). 
 
38 
 
 ANALYTIC G EOMETR Y 
 
 turned about xi until P coincides with A, F' must also coincide 
 with A. The line through A now becomes a tangent to the 
 curve, but this tangent intersects the curve. From the figure 
 it is seen that the coincidence of three points of intersection 
 
 Y_^ 
 
 at the point of tangency, and the consequent intersection of the 
 curve by the tangent, is caused by the fact that at the origin 
 the curve changes from concave up to convex up. Such a point 
 of the curve is called a point of inflection. 
 
 To find the analytic condition which determines a point of 
 inflection, let (a-,,, yo) be any point of // + x'-i/ — x = 0. The 
 coordinates of any point on a line through (.i-,„ ?/„) are 
 X — Xq + d cos a, !i = //„ + d sin a. The points of intersection 
 of line and curve correspond to the values of d satisfying 
 the equation 
 
 O/o + -V^/o — -^'o) + (sin « — cos « + 2 cos a ■ a;,?/,, + sin a • .r,,^) d 
 -f (cos^ a • ?/o^ + 2 sin a cos a • x^y) d- + cos" a sin a • d'^ — 0. 
 
 The first term of this equation vanishes by hypothesis, and if 
 the coefticients of d and d- also vanish, the straight line and 
 curve have three coincident points of intersection at (Xf,, ?/o)- 
 The simultaneous vanishing of the coefticients of d and d'^ 
 requires that the equations 
 
 sin a — cos a -\- 2 cos a • aVi,'/o + sin a • xj^ — 
 and cos- « • %- + 2 sin a cos a ■ .r,, = 
 
 determine the same value for tan «. This gives the equation 
 
 l-2av/„ . 
 
 ; reducing to ^/^) — o x^fi/^ + 2 .(\, = 0, which to- 
 
PLOTTING OF ALGEURAKJ EQUATIONS 
 
 81) 
 
 gcther with //„- + x^{y„ — a;o = determines the three puiuts of 
 iutiection (0, 0), (V3, ^V3), (- V3, - \Vo). 
 
 Art. 21. — Diametukj Mictuod of Tlotting Equations 
 ExAMi'LE. — riot 
 y- — 2 xy -\-'S.xr — Hj x — 0. 
 
 1 1 ere y — x ± Vl(J x — 2 x'. 
 
 Draw the strai.^•ht line y = x. 
 Adding to and subtracting from 
 the ordinate of this line, corre- 
 sponding to any abscissa x, the 
 cpiantity VlO x — 2 x^, the corre- 
 sponding ordinates of the re- 
 qnired locus are obtained. 
 
 This locus intersects the line 
 y = X when VlGx—2xr=0, that 
 is when x = and x = 8. y is 
 real only for values of x from 
 to 8. The curve intersects the 
 X-axis when x = VlG x — 2 a?, '''"^- '"'*• 
 
 that is Avhen x=^. Points of the curve are located by the 
 
 table, 
 
 x = +1 +2 +3 
 
 Vl6 x-2 x' = ±Vl4 ±2V6 ±V3() 
 
 + 4 -1-5 +6 +7 +8 
 
 ±4V2 ±V30 ±2VG ±VT4 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 /- 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 1 
 
 
 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 
 A 
 
 / 
 
 
 
 
 
 / 
 
 
 
 X 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 / 
 
 
 
 
 
 
 V 
 
 
 
 / 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 AUT. 22. SuMMAliV OK FliOl'KUTlKS OF LoOI 
 
 From the discussions in the i)receding articles, the following 
 conclusions are ol)tained : 
 
 1. If the absolute term of an eijuation is zero, the origin is 
 
 a point of the hicus of tlu' equation. 
 
40 ANALYTIC GEOMETIIY 
 
 2. To iind wlu've the locus of an equation intersects the 
 X-axis, pkice y = in the equation and solve for x ; to find 
 where the locus intersects the F-axis, place x= and solve 
 for y. 
 
 3. The abscissas of the points of intersection of the locus of 
 ij = f{x) with the X-axis are the real roots of the equation 
 
 4. If the equation contains only even powers of y, the locus 
 is symmetrical with respect to the X-axis ; if the equation con- 
 tains only even powers of x, the locus is symmetrical with 
 respect to the Y-axis. The origin is a center of symmetry of 
 the locus when (— x, — y) satisfies the equation, because {x, y) 
 does. 
 
 5. Tlie points of intersection of the straight line 
 
 2/ — ^u^ tan«(:« — a-o) 
 
 with the locus of J\x, ?/)= are the points {x, y) correspond- 
 ing to the values of d Avhich are the roots of the equation 
 obtained by substituting x = x^ + d cos a, y — y^-}- d sin a in 
 f{x, y)— 0. The number of points of intersection is equal to 
 the degree of the equation, and is called the order of the curve. 
 
 6. The distances from any point {xq, y^ to the points of in- 
 tersection of the straight line y — ?/„ = tan a(x. — x^ Avith the 
 locus of fix, y)—0 are the values of d which are the roots 
 of the equation obtained by substituting x = .Vo -f- d cos a, 
 y — ?/,, + d sin a in J\x, y) — 0. 
 
 7. The tangent to /(.r, y) = at (x*,,, ?/„) in the locus is 
 found by substituting x= .»■„ + d cos«, y = ?/„ + d sin a in 
 /(.t, _?/)= 0, equating to zero the coefficient of the first power 
 of d, and solving for tan a. This value of tan u makes 
 
 y-y, = timu(x- x^) 
 
 the equation of the tangent tof(x, _?/)= at (xq, y^). 
 
 8. If the curve f(x, y) = has infinite branches, the values 
 of 'tn. and u found by substituting mx + n for // in the equation 
 
PLOTTING OF ALGEIiUAW EQUATIONS 41 
 
 f{x,ij)=0, and tNiuating to zero the coetticients of tlio two 
 liighest powers of x in the resultini^ equation, deteruiine tlie 
 line // = mx -\- u which meets the curve at two points at in- 
 finity; that is, the asymptote. 
 
 9. To examine the locus of ?/=/(.»•) for maximum and mini- 
 mum ordinates form /(.r ± //) — /(.!•). Equate to zero the co- 
 efficient of the first power of h, and solve for x. The values of 
 X which make the coefficient of the second power of h positive 
 correspond to minimum, those which make this coefficient 
 negative correspond to maximum ordinates. 
 
 10. To determine the points of inflection of f(x, y)= 0, sul)- 
 stitute X — .To -f d cos a, y = yQ-\-d sin cc in f(x, y) = 0. In the 
 resulting equation place the coefficients of d and d'- ecpial to 
 zero, and equate the values of tan a obtained from these equa- 
 tions. The resulting equation, together with the equation of 
 the curve, determines the points of inflection. 
 
 Problems. — I'lot tlie numerical algebraic equations: 
 
 1. 2x + :],j~7. 5. (X- 2)0/ + 2)= 7. 10. yi = -\Ox. 
 
 2 ^^U=\ 6- ■'■^ + y'' = -^- 11- ^ ■'■'- + ^>J^ = 36. 
 
 '48' 7. a:-2-?/2 = 25. 12. 4x^-9>/ = m. 
 
 3- a-// = 4. 8. X- - ?/2 = - 25. 13. 4 x- - ?/2 = - 30. 
 
 4. (x-2)ij = [,. 9. if=z\Ox. 14. ?/-i = 10.C-X2. 
 
 15. 2/- = X- - 10 r. 26. ?/ = x- - 4 .c + 4. 
 
 16. x^ + 10 ./•// + >/i ^ 25. 27. 2/2 = (a: + 2) (X - 3). 
 
 17. .t2 + 10 x)/ + >/ + 25 = 0. 28. 2/2 = x2 - 2 x - 8. 
 
 18. 2/"^ = 8 x2 - x^ + 7. 29. 2/" = x2 - 4 x + 4. 
 
 19. x2 + 2 xy + y2 ^ 25. 30. 2/ = (x - 1 )(x - 2) (x - 3) . 
 
 20. .r2 + 10 .r.v + f = 0. 31. y^ = x'^ - G x"- + \\ x - G. 
 
 21. I/- = .rt - x2. 32. y = x* - 5 x:- + 4. 
 
 22. 2/2 = x2 - X*. 33. 2/- = x« - 5 x2 + 4. 
 
 23. 2/- = x^ - x\ 34. y = x» -I- 2x3 - 3x2 - 4 X -I- 4. 
 
 24. 2/ = (x + 2)(x-3). 
 
 35. y=_^L_ 
 2/ = x2-2x-8. -^ l-x'J 
 
42 
 
 ANALYTIC GEOMETRY 
 
 2x-l 
 3 X + 5' 
 
 y + 5 
 3 - x' 
 
 4-3x 
 5x — 6 
 
 39. ?/ - 2 x?/ - 2 = 0. 
 
 40. y- + 2 x^ + 3 x"^ - 4 X = 0. 
 
 41. m2 = x3-2x-^-8x. 
 
 36. y 
 
 37. y 
 
 38. 2/ 
 
 42. ?/ = x3-9x2 + 24x + 3. 
 
 43. ?/=(3x-5)(2x + 9). 
 
 44. 2/'i = x3 - 2 x2. 
 
 45. 2/2 + 2x2/- 3x2 + 4x = 0. 
 
 46. 2/^ -2x2/ + x2 + X = 0. 
 
 47. 2/^ + 4 X2/ + 4 x2 - 4 = 0. 
 
 48. 2/"-^ - 2 X2/ + 2 X'- - 2 X = 0. 
 
 49. 2/--2x2/ + 2x2+22/ + x + 3 = 0. 
 
 50. 2/" - 2 X2/ + x- - 4 2/ + X + 4 = 0. 
 
 1. 
 
 x2 + 2x- 15 = 0. 
 
 5. 
 
 x^ - 7 X + 7 = 0. 
 
 2. 
 
 x3-3x- 10 = 0. 
 
 6. 
 
 x'^ - 7 X - 7 = 0. 
 
 3. 
 
 x2-4x + 4 = 0. 
 
 7. 
 
 x3 + 7 X + 7 = 0. 
 
 4. 
 
 x2 - 5 X + 9 = 0. 
 
 8. 
 
 x3-5x + 2 = 0. 
 
 riot the real roots of the following equations : 
 
 9. X* - 5 x'- + 4 = 0. 
 
 10. xHa^'Hx-+x+l=0. 
 
 11. x^ + X- + X + 1 = 0. 
 
 12. x^ - X- + X + 1 = 0. 
 
 riot the real roots of the following pairs of simultaneous equations : 
 
 1. 2/^ = 10x, x^ + 2/2 = 25. Plot the.se equations to the same axes. 
 The coordinates of the points of intersection of the loci are the pairs of 
 real values of x and y which satisfy 
 each of the given equations. The 
 points of intersection are (2.07,4.42), 
 (2.07, -4.42). 
 
 By the angle of intersection be- 
 tween two curves is meant the angle 
 between the tangents to the curves at 
 their intersection. Hence the angle 
 between two curves is the differ- 
 ence between the angles the tangents 
 to the curves at their intersection 
 make with the X-axis. Calling the 
 angles the tangents to 2/^ = 10 x and 
 x2 + 2/2 = 25 at the point of intersec- 
 tion (xo, 2/0) make with tlie A'-axis 
 a' and a respectively, 
 
 tan a' 
 
 -y 
 
 
 ^" v^ 
 
 z / \ 
 
 y 7 V 
 
 t t \ 
 
 
 A X 
 
 V ^t ^ 
 
 \ \ L 
 
 ^ X z 
 
 ^- -< 
 
 
 ^\ 
 
PLOTTING OF ALGEllUAIC EQUATIONS 
 
 43 
 
 Evaluating for a-o = 2.07, |/n = 4.48, tan a' = 1.10, tan a = — .47 ; whence 
 a' = 48° 4', a = 154° 50', ami the anglo between the curves is 104° 52'. 
 
 2. 2 X + 3 y = 5, ?/ = i .'c + 3. 8. /- + ;/- = 25, ij- = 10 x - x^. 
 
 3. y = 3 X + 5, .T^ + !/-^ = 25. 9. 3 .c^ + 2 y-^ = 7, ?/ - 2 x = 0. 
 
 4. X- + J/'- = 9, */2 = 10 X — x^. 10. y- = 4 X, 2/ — X = 0. 
 
 5. 2/'^ = 10 X, 4 x2 - y^ = ;](!. 11. 2 x^ - r' = 14, x^ + 2/^ = 4. 
 
 6. 2 x- — IJ- = 14, X- + y'^ = !). 12. x- -f ?/- = 25, x'^ — )/- = 4. 
 
 7. >/ = x^ — 7 X + 7, ?/ — X = 0. 
 
 Solve the following equations graphically 
 
 1. x2 — X - (5 = 0. Plot // = X- and y = x + C> to the same axes. For 
 the points of intersection of the loci x- = x + d ; that is, x- — x — C = 0. 
 Hence the abscissas of the jioints of 
 intersection of y = x^ and y — x + 
 are the real roots of x- — x — = 0. 
 For all quadratic equations, 
 
 X- + ax + h = 0, 
 
 the curve ;/ = x- is the same, and the 
 roots are the abscissas of the points 
 of intersection of the straight line 
 y = — ax — h with this curve. 
 
 In like manner the real roots of 
 any trinomial equation x" + rtx4-/;=0 
 ai-e the abscissas of the points of in- 
 tersection of y=x" and y + ax-\-b=0. 
 
 2. x--3x + 2 = 0. 
 
 3. x"- + 5 X + -- 
 
 4. .r2 -4=0. 
 
 5. :,•;:_ Ox -10: 
 
 0. 
 
 0. 
 
 
 \ 
 
 
 
 
 Y 
 
 
 
 1 
 
 
 
 \ 
 
 
 
 
 
 
 
 I/. 
 
 — 
 
 
 
 I 
 
 
 
 
 
 A 
 
 
 
 \ 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 X 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 Fk! 
 
 .•?!. 
 
 
 
 
 
 6. .r2-4x- 15 = 0. 
 
 7. 3x2-12x + 2=0. 
 
 8. x'-i+5x + 10 = 0. 
 
 9. X-- 5x + 5 = 0. 
 
 10. 
 
 x3 - 7 X + 7 = 0. 
 
 11. 
 
 x^ + 7 X + 7 = 0. 
 
 12. 
 
 x^ + 7 X - 7 = 0. 
 
 13. 
 
 .x«-7x-7 = 0. 
 
 14. 
 
 x'»-10x+ 15 = 0. 
 
 15. 
 
 a:»-10x-15 = 0. 
 
44 
 
 ANALYTIC GEOMETRY 
 
 Sketch the following literal algebraic ecjuations : 
 
 ?/2 = a;3 
 
 1. ?/2 _ 3.3 _ (5 _ c)x^ — bcx. Here y — ± vx(x — 6) (x + c). Unless 
 numerical values are assigned to b and c, it is impossible to plot the equa- 
 tion by the location of points. How- 
 ever, the general nature of the locus may 
 be determined by discussing the equa- 
 tion. The A'-axis is an axis of sym- 
 metry, the origin a point of the locus. 
 For < X < 6, y ifi imaginary ; when 
 X = b, y = ; for x > b, y has two nu- 
 merically equal real values with op- 
 posite signs, increasing indefinitely in 
 numerical value with x. For > x > - r, 
 y has two numerically equal values with 
 opposite signs ; for x = — c, y = ; for 
 ^ < — <"» 2/ is imaginary. Sketching a 
 curve in accordance witli these condi- 
 tions, a locus of the nature shown in the 
 figure is obtained. 
 
 ili 
 
 = 1. 
 
 X2 
 
 Pig. 32. 
 = 1. 
 
 3. ?/ = ax. 6. 
 
 4. ?/ = ax + b. 7. 
 
 »/- = -J, px. 
 
 X- + y- — a-. 
 
 b-^ 
 
 b-^ 
 
 = 1. 
 
 13. 
 
 (x-rt)(2/- b) = m. 
 
 14. 
 
 y = (x-a)(x-b)ix-c). 
 
 15. 
 
 2/2 = (x-a)(x-6)(x-0. 
 
 16. 
 
 V2-(X a)-^^-^ 
 
 17. y-x =4 a- (2 a - x). 
 
 18. ?/- = (x - a) (x -I- /;) (x - c). 
 
 11. ?/ =(x - a)(x -f ?;). 
 
 12. ?/- =(x- ffl)(x-(- ft). 
 
 19. ^y-=x^, the semi-cubic parabola.* 
 
 20. rr// = x^, the cubic parabola. 
 ?/'- = (x — f'i)(x — P2)(x — Cs), ei, real, Ci, cs, conjugate imaginarios. 
 2/- = (x - Pi) (x - eo) (x - es), <'i, ^1;, '';!, real, ^ > r.2 > ^3- 
 y' = (x — (?i)(x — <'2)(x - P3), fii, ^25 Cti real, C] = r.:>, ei > pj. 
 ?/- = (x - ei)(x — r2)(x - Pz), P\, e-2, Cs, real, ^1 > ^2, ^2 = ea- 
 ?/2 =: (X - Pi) (x - r.,) (x - fs), Ci = e-2 = es- 
 
 21 
 
 25 
 
 * The rifling of a cannon, wlion the bore is rolled out on a plane, 
 technically " (Icvclopeil," is a srmi-cul)ic paralmla. 
 
CHAPTER IV 
 
 PLOTTING OF TRANSCENDENTAL EQUATIONS 
 Art. 23. — Elemkntauv Tiianscendkntal Fltnctions 
 
 Transoendental e(]u<ati()iis are e(iuat.it»ns involvinj;- ti'aiiscon- 
 dental functions. 
 
 The elementary transcendental functions are the exiioiion- 
 tial, logarithmic, circular or trigonometric, and inverse circular 
 functions. 
 
 The expression of ti-anscendental Cnnctioiis by means of tlu' 
 fundamental operations of algehra is possil)k^ only hy means 
 of infinite series. 
 
 AliT. 24. ExroXKXTIAL and LoOAUrTHMIC FirXCTIONS 
 
 The general type of the exponential function is y = h-a", 
 where a is called the base of the exponential function and is 
 always positive. 
 
 To plot the exponential func- 
 tion numerically, suppose /> = 1 , 
 c = 1, (( = U. Then y = 2'' and 
 
 a;= — CO 4 — 3 — 2 — 1 
 
 11= 0... i, I. \ I 
 
 1 2 3 4...r>D. 
 
 1 2 4 S 1 ('.... X. 
 Vov all values of a the locus 
 
 of y = a' contains the point (0, J) 
 and indefinitely approaches the 
 X-axis. Increasing the value of 
 a causes the locus to recede more 
 45 
 
 _ Y _ 
 
 I , ^ — 
 
46 ANALYTIC GEOMETRY 
 
 rapidly from the X-axis for x > 0, and to approach the X-axis 
 more rapidly for x<(). When « = !, the locus is a straight 
 line parallel to the X-axis. When a < 1, the locus approaches 
 the X-axis for x>l, and recedes from the X-axis for x < 0. 
 
 When c is not unity the function y = a" may be Avritten 
 y =(cfy, and the base of the exponential function becomes a". 
 When b differs from unity, each ordinate of y = h - a" is the 
 corresponding ordinate of ?/ = a" multiplied by b. 
 
 To plot the exponential function y = b • a" graphically, com- 
 pute ?/n and ?/i , the values of y corresponding to x = and 
 X = a'l, where x^ is any number not zero. Adopt the following 
 notation for corresponding values of x and y. 
 x= 4a-, -'Sxi -2x, -x, x, 2x, ?.x, 4.x,--- 
 
 y = — 2/- 4 Vz y-2 y-i ih Vx y-2 y-s yi---- 
 
 Then •lsl = l^ = '!h^'!h = yi=.h=... ««,. On two intersect- 
 
 y 2 y-i 2/o yi v-i .Vs 
 
 ing straight lines take OA — yo, OB = yi. Join .1 and B, 
 
 B 
 
 x./ \A. 
 
 O K A C E C. 
 
 Fin. .'54. 
 
 draw BC making angle OBC = angle OAB. Then draw CD, 
 DE, EF, •••, parallel to AB and BC alternately ; AH, II K, KL, 
 • ••, parallel to BC and AB alternately. From similar triangles 
 0K_ OII^ OA ^OB^OC^OD^ OE 
 ()L~ 0K~ OII~ OA OB OC 01) 
 Hence, if 0.4 = ?/o, OB = yi, it follows that OL = y_:„ OK 
 
PLOTTING OF TIlANSCENDENTAL EQUATIONS 47 
 
 = V .,. on = i/^x, OC — y.2, OD = II .^ ; that is, the ordi nates cor- 
 resi)onding iox = — ox^, — 2:Ci, — x^, 2a-,, Sifj become known 
 and the points of the curve can be located. 
 
 The logarithmic function ex = log„ {by) is equivalent to tlie 
 exponential function ?/ = a". When y = log a.- is plotted, tlie 
 
 logarithm of the product of any two numbers is the sum of 
 the ordinates of the abscissas which represent the nund)ers, 
 and the product itself is the aV)Scissa corresponding to this 
 sum of the ordinates. 
 
 The slide rule is based on this principle. In the slide rule 
 the ordinates of the logarithmic curve are laid off on a straight 
 line from a common point and the ends marked by the corre- 
 sponding abscissas. 
 
 Art. 25. — Circular and Inverse Circular Functions 
 
 Ty\ definition, am AOP 
 
 PD 
 OP 
 
 P'D' 
 OP' 
 
 angle ^lOP in circular measure is 
 
 Hence, if the radius OA' is the linear 
 unit, the line P'D' is a geometric rep- 
 resentative of sinvlOP, the arc A'P' 
 a geometric representative of the 
 angle AOP. The measure of the 
 angle ylOPis 1 when arc AP= OP; 
 that is, the unit of circular measure 
 is the angle at the center which in- 
 tercej^ts on the circumference an arc 
 equal to the radius. The unit of 
 circular measure is callcil the radi 
 of four riglit angles, or .".r.0°, is ' 
 
 and the value 
 arc A P ai 
 
 of the 
 c.I'P' 
 
 OP 
 
 OP 
 
 radian is ecpiivalent to 
 
 360'' 
 
 r°.3 -. 
 
48 
 
 ANALYTIC GEOMETRY 
 
 Calling angles generated by the anti-clockwise motion of 
 OA positive, angles generated by the clockwise motion of OA 
 negative, there corresponds to every value of the abstract num- 
 ber X a determinate angle. 
 
 Unless otherwise specified, angles are expressed in circular 
 measure. When an arc is spoken of without qualiiieation, an 
 arc to radius unity is always understood. 
 
 In tables of trigonometric functions angles are generally ex- 
 pressed in degrees. Hence, to plot y = sin x numerically, 
 assign arbitrary values to x, find the valne of the correspond- 
 ing angle in degrees, and take from the tables the numerical 
 value of sin x. 
 
 
 
 
 
 Y 
 
 
 
 
 \ 
 
 
 
 
 / 
 
 
 X 
 
 \ 
 
 \ 
 
 
 ^ ^^ 
 
 / 
 
 A 
 
 
 
 \^ 
 
 
 
 
 
 
 
 
 
 
 - 200° 32' 
 
 ?/= ••• 
 
 + .350 
 
 
 3. 
 
 - ,S5° 57 
 
 
 - .997 
 
 \ 
 
 1 
 
 28° 39' 
 
 57° 18' 
 
 .479 
 
 .841 
 
 3 
 
 171° 53' 
 
 .141 
 
 -4 
 
 - 143° 14' 
 
 - .598 
 
 -114° 35 
 - .909 
 
 -1 
 
 -57° 18' 
 - .841 
 
 -28° 39' 
 -.479 
 
 
 
 
 
 85° 57' 114° 35' 143° 14' 
 .997 .909 .598 
 
 3 .... 
 
 171' 
 
 .141 
 
 In itvactical problems the ecpiation frequently occurs in the 
 form y = sine (ttx). Were 
 
PLOTTING OF TRANSCENDENTAL EQUATIONS 4'J 
 
 ?/= Wli 1 W'2 -^V2 -1 
 
 jV2 1 jV2 -|V2 -1 -^-V2 
 
 ^Vii 
 
 Y 
 
 lAtbz:: 
 
 To plot ^ = sin X grapliically, draw a circle witli radius unity, 
 divide the circiunfereuce into any number of equal parts, and 
 placing the origin of arcs at the origin of coordinates, roll the 
 circle along the X-axis, marking on the A'-axis the points of 
 
 division of the circumference 0, 1, 2, .'5, 4, 5, (>, •••. Througli 
 the points of division of the circumference draw perpendicu- 
 lars to the diameter through the origin of arcs 00, 11, 22, «*>.'», 
 44, 55, GG, •■•. On the perpendiculars to the X-axis at the 
 points 0, 1, 2, o, 4, 5, G, •••, lay off the distances 00, 11, 22, ',y,\, 
 44, 55, GG, •••, respectivt'ly. lu this inauucr any nundjer of 
 points of y = sin x may be located. 
 
50 
 
 ANALYTIC GEOMETRY 
 
 On account of the perioilicity of sin.T, the locus of ?/ = sin x 
 consists of an infinite number of repetitions of the curve 
 obtained from a; = to a; = 2 tt. The locus has maximum or- 
 
 dinates y — -{-1 corresponding to x = (4n + 1)^, 
 
 minimum 
 
 ordinates y = — 1 corresponding to x = (4 m + 3)^, where n is 
 
 any integer. The locus crosses the a>axis when x = mr. 
 
 To plot y—o sin a.", it is only necessary 
 to multiply each ordinate oi y — sin x 
 by 3. This is effected graphically by 
 drawing a pair of concentric circles, one 
 with radius luiity, the other with radius 
 3. Since OP' is the linear unit, F'D' rep- 
 resents sin X, and PD represents 3 sin x, 
 *"^' '^^' while X is represented by the arc A'P'. 
 
 To plot v/=3 sin a-+sin (2 x), plot ?/i=3 sin a; and ?/2=sin (2 a-) 
 on the same axes. The ordinate of ?/ = 3 sin x + sin (2 .}•) cor- 
 
 
 
 
 
 N 
 
 
 
 ~ 
 
 
 'h 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 / 
 
 ■\; 
 
 
 
 
 
 ; 
 
 ■\ 
 
 \ 
 
 
 
 
 / 
 
 
 % 
 
 
 
 
 1/ 
 
 
 v> 
 
 
 
 
 
 
 \>, 
 
 
 
 
 / 
 
 
 \ 
 
 \ 
 
 
 
 '''^ 
 
 
 \ 
 
 / '• 
 
 \ 
 
 A 
 
 I' 
 
 \ 
 
 \ 
 
 v' 
 
 ^^ 
 
 
 I'/ 
 
 \ 
 
 X 
 
 
 \^ 
 
 /j 
 
 
 \^ 
 
 
 \ 
 
 
 /j 
 
 
 ' 
 
 
 \ 
 
 
 
 
 
 
 V 
 
 
 ij 
 
 
 
 
 \ 
 
 
 i 
 
 
 
 
 
 S 
 
 /) 
 
 
 
 
 \ 
 
 y 
 
 
 
 
 
 \ 
 
 \, 
 
 
 
 
 
 
 \ 
 
 '/ 
 
 
 
 
 
 
 
 
 
 
 
 V, 
 
 ' 
 
 
 
 
 
 v 
 
 
 
 
 responding to any value of x is the sum of t])o ordinates of 
 yi — 3 siiiiv and y., = sin (2.r) corresponding to the same value 
 of X. 
 
 When the sine-function occurs in the form y — a sin (wt + 6), 
 where w is uniform angular velocity in radians, t time in sec- 
 onds, a is called the amplitude, 6 the epoch angle. The periodic 
 
PLOTTING OF THANSCEN DENTAL EOUATIOXS hi 
 
 time is t = - — Tlie construction of the curve is indicated in 
 
 the iigure. The jtrojcction of unirorni motion in the circum- 
 I'ci'ent'e of a circle on a diamctei' is caUcd harmoiuc motion. 
 
 f=2 
 
 t = ] 
 
 1=0 
 
 !J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 r\ 
 
 
 
 
 
 
 
 / 
 
 ^-N 
 
 \ 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 
 
 \ 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 t 
 
 
 
 
 \ 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 
 
 
 
 \ 
 
 
 
 1 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 \ 
 
 
 1 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 \ 
 
 k/ 
 
 1 
 
 
 
 
 — 
 
 — 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 To add gi'apliically two sine-functions of equal periods 
 ?/, = «i sin (co^ + ^,), ?/^ = ao (sin w^ + ^o), draw a pair of con- 
 centric circles with rndii «, and a~y Let 1\0D and P.,OD be 
 oj^ + $1 and (ot -f d., corresponding to the same value of (. Then 
 
52 
 
 ANAL YTIC GEOMETU F 
 
 I\()D - P.,OD = ^1 - e.,. The ijarallelogram on 01\ and 01\ 
 for different values of t is the same j)arallelogram in different 
 positions. This parallelogram has the same angular motion 
 as OPi and 01\. Now y^ = PiAj 2/2 = P^D.,, hence 
 
 p/;-PiA + P.A = 2/i + 2/.. 
 
 and the sum of the sine-functions corresponding to the circular 
 motions of Pi and P., is the sine-function corresponding to the 
 circular motion of P. The resultant sine-function has the same 
 period as the component sine-functions, its amplitude is OP, 
 its epoch angle the angle POD corresponding to the position 
 of P for t = 0. The resultant sine-function is y = a s\n{wt-\-6), 
 where a^ = a^^ + a./ -\-2 ttia^ cos {61 — 62), 
 
 a I sin $1 + €(,2 sin O2 * 
 cij cos 61 + a.^ cos 62 
 
 (1) ij = sm~^x is equivalent to x 
 
 tan 6 . 
 
 is equivalent to x ■ 
 
 sin ^; (o) y = sin 
 o 
 
 sin y ; (2) // = 3 sin~' x 
 is equivalent to 
 
 a; = 3 sin:;; (4) /y = 3 sii 
 
 is equivalent to x — 2 sin 
 
 graphic interpretation of ccpuitions (1), (2), (3), (4) is shown 
 in figures (a), (b), (c), (d), which also indicate the nu^nncr of 
 plotting the equations graphically. 
 
 * A jointed paralk'lograin is used for conipoundiiig harmonic motions 
 of different periods in Lord Kelvin's tidal clock. 
 
PLOTTING OF TRANSCENDENTAL EQUATIONS 53 
 
 The reiiuiining circular and inverse circular fuuc^tions are 
 l)k)ttcd in a manner entirely analogous to that employed in 
 l)lotting y = s'lnx and y = sin"' x. 
 
 Problems. — riot 1. )j = 2^. 2. */ = lO-'. 3. y=(\y. 4. y=(.l)^ 
 
 5. i/ = 2-^. 6. .v = 5-2^. 7. y=3^\ 8. .v=:e^* Q. y = e-\ 10. ij = \(e^ + e-^). 
 This function is called the hyperbolic cosine, and is written y = cosh a;. 
 
 11. y — - (e^ + e-^^) or y = c cosh x. This is the equation of the catenary.t 
 
 the form assumed by a perfectly flexible, homogeneous chain whose ends 
 are fastened cat two points not in the same vertical. 
 
 12. y — I (e-^ — e""^). This is the hyperbolic sine, and is written sinh r. 
 
 13. y = K 
 
 14. y = logiox. 
 
 19. ;K-2 = logi„(y + 5). 
 
 20. r + 5 = logio (^ - 2). 
 
 1. y = logo X. 
 
 17. 2/ = ;]log.,x. 
 
 ;. , = iog^,^_x. 
 
 18. 2x = logio2/. 
 
 26. 
 
 »/ = .3sinx. 
 
 27. 
 
 )/ = sin (4x). 
 
 28. 
 
 y = sm(lw + ^x). 
 
 29. 
 
 rj — 3 + sin x. 
 
 30. 
 
 2/ = 3 + sin(7r + 2x). 
 
 31. 
 
 J/ = sinx + 2 sin -■ 
 
 22. ^ ^ = logio(y+l). 
 
 X + a 
 
 23 y = sin ^^' 
 
 ■ ^ 2 32. 2/ = 2 sin (2 x) + 3 sin (3 x) . 
 
 24. ?/ = .sin(2x). 33. ?/ = 3sin(2 + 2x) + fisin(l + 4x). 
 
 25. ;/ = sin (x + .[ tt). 34. y = cosx. 
 
 35. »/ = 5 cosx. 39. ?/ = secx. 43. y = versx. 
 
 36. y -2 cos (1 + Gx). 40. y = 2 secx. 44. y = covers x. 
 
 37. ?/ = tanx. 41. ?/ = cosecx. . , ,, 
 
 45. X = sm-i ^• 
 
 38. y = 2 + tan(l + x). 42. y = cosec (x - 1). 2 
 
 * e represents the base of the Napierian sy.stem of logarithms, a tran.scen- 
 dental number, whose value to nine places is 2.718281828. y — e" may 
 be plotted graphically by computing the values of y corresponding to any 
 two values of X ; numerically by writing the function in the form x = log,.»/ 
 and using a table of Napierian logarithms. 
 
 t The catenary was invented by Jolni and James nernouUi. The center 
 of gravity of the catenary is lower than for any other position of the 
 same chain with the same fixed points. 
 
54 ANALYTIC GEOMETRY 
 
 46. 2 M = cos-i X. -r b2. x — 2 — shr^ y. 
 
 50. w = 3 cos-i -• 
 
 47. 2/ = J taii-i x. 3 53. x+3 = siu-i (i/-2). 
 
 48. 2/ = sec-i(x-3). ^^ y = 5sin-i?. ^*- ^ = cos-i (?/ - 1). 
 
 49. ?/ = 2 + vers-i X. 4 55. 2/+2=:cos-i(a;-2). 
 56. y = sill (] nt). b1. y = sin (] ivt + \ v). 
 
 58. y — sill (.] wt + 2 t) + .siii(^ tt^ + v). 
 
 59. ;/ = sin (I nt) + sin (} irt + ^ tt). 
 
 60. y = sin (|: 7r( + 1 tt) + sin {\ -ret + i tt). 
 
 The elementary transcendental functions are of great impor- 
 tance in mathematical physics. For instance, if a steady 
 electric current /flows through a circuit, the strength i of the 
 current t seconds after the removal of the electromotive force 
 
 is given by the exponential function i= le ^, where R and L 
 are constants of the circuit. 
 
 The quantity of light that penetrates different thicknesses of 
 glass is a logarithmic function of the thickness. 
 
 The sine-function is the element by whose composition any 
 single-valued periodic function may be formed. Vibratory 
 motion and wave motion are periodic. The sine-function or, 
 as it is also called, the simple harmonic function, thus becomes 
 of fundamental importance in the mathematical treatment of 
 heat, light, sound, and electricity. 
 
 Art. 26. — Cycloids 
 
 A circle rolls along a fixed straight line. The curve traced 
 by a point fixed in the circumference of the circle is called a 
 cycloid.* The fixed line is called the base, the point whosr 
 distance from the base is the diameter of the generating circli^ 
 the vertex, the perpendicular from the vertex to the base th<' 
 
 * Curves generated by a point fixed in the plane of a curve which \\A[a 
 along some fixed curve are called by the general name "roulettes.'' 
 Cycloids are a special class of roulettes. 
 
a XI 
 in 
 
 P LOTTING OF TRANSCENDENTAL EQCATlONH .55 
 
 mS of tlicryi'loiil. The courtliiiates of any inuiitof the cycloid 
 ay be expressed as trausceudental i'uiictions cd' a variable 
 angle. 
 
 Take the base ol' the cyck)id as X-axis, the perpendicular to 
 the base where the cycloid meets the base as I'-axis, and call the 
 angle made by the radius of the generating circle to the tracing 
 point with the vertical diameter 6. l>y the nature of the 
 
 cycloid AK= arc PK = rO, y = PD = OK - OL ^ r - r cos d, 
 X = AD = AK — DK=^ i-e — r sin 9. Kence (1) x = rO - r sin 0, 
 y = )• — r cos 9 for every value of 9 determine a point of the 
 cycloid. The equation of the cycloid between x and y is 
 obtained either directly from the figure, 
 
 X = AK — DK — arc PK — PL = r vers"'* — Vli ni 
 
 r 
 
 or by eliminating 9 between ecpuitions (1), 
 
 T 
 
 V 
 
 . V -5 ry - y-, 
 
 hence 
 
 Vi/ 
 
 ry 
 
 Now r vers~^-^ has for the same value of y an infinite number 
 
 of values differing by 2tt, and V2ry — // is a two-valued func- 
 tion which is real only for values of y from to -f 2 r. Hence 
 the equation determines an infinite number of values of x for 
 every value of y between and + 2 r. This agrees with the 
 nature of the curve as determined by its generation. 
 
56 
 
 A NA L YTIC GEOMETR Y 
 
 By observing that tlie center of the generating circle is 
 always in the line parallel to the base at a distance equal to 
 the radius of the generating circle, the generating circle may 
 readily be placed in position for locating any point of the 
 cycloid. At the instant the point P is being located the 
 generating circle is revolving about K, hence the generating 
 point P tends at that instant to move in the circumference of 
 a circle whose center is A' and radius the chord KP. The tan- 
 gent to the cycloid at P is therefore the perpendicular to the 
 chord KP at P, that is the tangent is the chord PII of the 
 generating circle.* 
 
 The perpendicular to the tangent to a curve at the point of 
 tangency is called the normal to the 
 curve at that point. Hence the chord 
 KP is the normal to the cycloid at P. 
 
 Take the axis of the cycloid as X-axis, 
 the tangent at the vertex as F-axis. 
 By the nature of the cycloid 
 
 3//ir=arcP/i, 
 
 MN— semi-circumference KPH. 
 
 X =AD = HL=OH-OL = r-r cos 0, 
 
 7j = PD = LD+PL=(MN-Miq-\-PL 
 
 = arc IIP + PL= rO + r sin 0. 
 
 That is, 
 
 (1) X: 
 
 y 
 
 rd -\- r sin i 
 
 determine for every value of ^ a point of the cycloid. The 
 equation between x and y is found either directly from the 
 figure, 
 
 y =z LD + PL = r vers"' - + V2 rx — x- ; 
 
 This method of drawing a tangent to the cycloid is due to Descartes. 
 
V LOTTING OF THAN SiCEN DENTAL EQUATIONS 57 
 
 or by elimiiiatini;- 6 between eiiuations (1), 
 
 ^=:cos-/l --")=: vers-' ^, 
 
 .6^^V2' 
 
 hence 
 
 y=r vers"' - + V2 rx 
 r 
 
 Art. 27. — Prolate and Curtate Cycloids 
 
 When the genercating point instead of being on the eircuni- 
 ference is a point fixed in the pkme of the rolling circle, the 
 curve generated is called the prolate cycloid when the point is 
 within the circumference, the curtate cycloid when the point 
 is without the circumference. Let a be the distance from the 
 center to the generating point. From the figures the equations 
 of these curves are readily seen to be 
 
 x = r6 — a sin 0, y = r — a cos 6. 
 
 Fio. 40 
 
 * If the cycloid is concave up and the tanfjcnt at. the vertex horizontal, 
 the time required by a particle sliding down the cycloid, suiJjiosed friction- 
 less, to reach the vertex is independent of the starting point. On account 
 of this property, discovered by Huygens in 1G73, the cycloid is called the 
 tautochrone. The frictionless curve along which a body must slide to 
 pass from one point to another in the shortest time is a cycloid. On 
 account of this property, discovered by John Bernoulli in 1G9G, the cycloid 
 is called the brachistochrone. 
 
58 
 
 ANALYTIC GEOMETRY 
 
 Art. 
 
 Epicycloids and Hypocycloids 
 
 If a circle rolls along the circumference of a fixed circle, the 
 curve generated by a point fixed in 
 the circumference of the rolling 
 circle is called an epicycloid if the 
 circle rolls along the outside, an 
 hypocycloid if the circle rolls along 
 the inside of the circumference of 
 the fixed circle. By the nature of 
 Oi ~U D X the epicycloid arc HO = arc HP, 
 that is E-6^r- (f>. From the 
 figure X = AD = AL + DL 
 Fiu. 47. =(R + r)cos + r cos CPM. 
 
 ... Ji + r. „ _ 
 
 180°-: 
 
 Hence x = (R + r)cos 
 
 PD = CL 
 
 R + r 
 
 CM 
 
 = (R + r)sin^ — r sin 
 
 By the nature of the hypocycloid R • 6 
 
 = r • <^, hence ^ = ^-6. x = AD 
 r 
 
 X = AL - PM 
 
 = (R - r)cos e- 7- cos CPM. 
 
 NowCPJ/=lSO°-</, + ^ 
 R- 
 
 x=(R — r)cos 6 + r cos 
 
 180° 
 R- 
 
 Hence 
 
 PD = CL - CM = (R- r)s\n O-r sin 
 
 Epicycloids and hypocycloids are used in constructing gear teeth. 
 
PLOTTING OF TRANSCENDENTAL EQUATIONS f)!) 
 
 AiiT. 29. — Involutk of Cikcli': 
 
 A string whose length is the circumference of a circle is 
 wound about the circumference. One end is fastened at and 
 the string unwound. If the string is kept stretched, its free 
 
 end traces a curve which is called the involute of the circle. 
 From the nature of the involute, IIP is tangent to the fixed 
 circle and equals the arc HO, which equals Ji$. 
 
 X = AD = AL + KP = 11 cos + BO sin 6, 
 
 y=PD= IIL - IIK = n sin 6 - 116 cos 6* 
 
 * The invohite is also used in cnnstructing gear teeth. 
 
CHAPTER V 
 
 TEANSrOEMATION Or COOKDINATES 
 
 Art. 30. — Transformation to Parallel Axes 
 
 Let P be any point in the plane. Keferred to the axes 
 X, Y the point P is represented by {x, y) ; referred to the 
 parallel axes Xj, Yi, the point 
 P is represented by (x^, y-^. Let 
 the origin A^ be {m, n) when 
 referred to the axes X, Y. 
 — Xi Prom the figure x = m + x^, 
 y = n -{- yi. Since (x, y) and 
 — X (xi, ;Vi) represent the same point 
 P, if f{x, ?/) = is the equation 
 of a certain geometric figure 
 when interpreted with refer- 
 ence to the axes X, Y, f(m + Xi, n -|- ?/i) = is the equation of 
 the same geometric figure when interpreted with reference to 
 the axes Xj, Y^. 
 
 Example. — The eqiiation of the 
 circle whose radius is 5, center (2, 3) 
 is (1) (a; - 2)2 + (y - 3)^ = 25. Draw 
 a set of axes Xj, Yi parallel to X, 
 Y through (2, 3). Then x = 2 + x„ 
 y = ?y -\- ?/,. Substituting in equation 
 (1), there results (2) x^- + y,^ = 25. 
 Notice that the equation of a geo- 
 metric figure depends on the position 
 of the geometric figure with respect 
 to the axes. 
 GO 
 
TltANSFOIiMATION OF COOHDINATES 
 
 Gl 
 
 AUT. 31. — FUOM KeCTANGULAU AxKS to JiK( TAN(i[M.AIl 
 
 Let (x, y) represent any point in the plane referred to the 
 axes X, F; (x„ y^ the same point referred to the axes X,, \\, 
 where Xj, Yi are obtained by turning A', Y about A througli 
 the angle a. Now 
 
 x=AD = An-KD' 
 
 = x'l cos a — .'/i sin a, 
 y=. P1) = D'H+PK 
 
 — .Xisin a + yi cos a. 
 
 Since (.x*, .7) and (.i-,, yO rep- 
 resent the same point 7*, 
 /(x, ?/) = interpreted on the 
 A", Y axes and r><i. 52. 
 
 /(a'l cos a — ?/i sin «, a;, sin « + ?/, cos «) = 
 interpreted on the Xj, Fj axes represent the same geometric 
 figure. 
 
 Example. — y = a- + 4 is the equation of a straight line. To 
 find a set of rectangular axes, the origin remaining the same, to 
 which when this line is referred 
 its equation takes the form yi=n, 
 substitute in the given equation 
 
 X = Xi cos a — yi sin a, 
 y = Xi sin a + y^ cos a. 
 There results 
 
 Xy (sin a — cos«) 
 
 + ?/i (sin a + cos «) = 4, 
 
 and this equation takes the re- 
 quired form when 
 
 sin a — cos a = 0, 
 
 that is, when a = 45°. Substituting this value of a, the trans- 
 formed equation becomes ?/, = 2 V2. 
 
ANALYTIC GEOMETRY 
 
 Art. 32. — Oblique Axes 
 Hitherto the axes of reference have been perpendicular to 
 each other. The position of a point in the plane can be equally 
 well determined when the axes are oblique. The ordinate of the 
 
 point F referred to the 
 / / / oblique axes X, Yis the 
 distance and direction 
 of the point from the 
 X-axis, the distance 
 being measured on a 
 parallel to the I''-axis, 
 the side of the X-axis 
 on which the point lies 
 being indicated b}^ the 
 algebraic sign prefixed 
 " '"' ""' to the number express- 
 
 ing this distance. Similarly the abscissa of the point P is the 
 distance and direction of the point from the F-axis, the dis- 
 tance being measured on a parallel to the X-axis, the alge- 
 braic sign prefixed to this distance denoting on what side of 
 the F-axis the point lies. 
 
 Problems. — Tlie angle between the oblique axes being 45' : 
 
 1. Locate the points (3, - 2); (- 5, 4); (0, 8); (- 4, - 7); (2i, - 3); 
 
 Observe that the geometric 
 figure represented by an equa- 
 tion depends on the system of 
 coordinates used in plotting the 
 equation. 
 
 4. Find the equation of a 
 straight line referred to oblique 
 axes including an angle /3. The 
 method used to find the equation 
 of a straiafht lino referred to 
 
 ( 
 
 v/5, 
 2. 
 
 -V7); (-21, V 
 Plot 2/ = 3 X ; ?/ = 
 
 '10) 
 3x 
 
 + 5; 
 
 y = 
 
 -2x 
 
 
 3. 
 
 Plot x^ + y^^ IG 
 
 ; !/■ 
 
 = 4x 
 
 . x^ 
 ' 9 
 
 -! = 
 
TRAysFOnMAriON OF COOIiDrXATES 
 
 03 
 
 rcctancjular axes sliows tliat the equation of a strai,i,'lit line referred to 
 oblique axes is y = vix + «, where in — 
 
 of the line on the I'-axis. 
 
 sni (/S — a) 
 
 and n is the intercept 
 
 5. Show that V (x' - x")'^ + (y' - y")'^ + 2(3;' - x") {y' - y") cos /3 is 
 the distance between the points (z', ?/'), (x", y") when the angle between 
 the axes is /3. 
 
 6. Find the equation of the circle whose radius is Ji, center (m, 7i), 
 when the angle between the axes is /3. 
 
 7. Show that double the area of the triangle whose vertices are 
 
 (3^1, yO, (3-2, !/2), (a^s, 2/3) 
 is {yi(^3 - X2) + yo(xi - T:i) + y3(x-2 - xi)}sin /3. 
 
 AuT. 33. — From Rectangular Axes to Oi-.lique 
 
 It is sometimes desirable to find tlie equation of a £;eometric 
 figure referred to oblique axes when the equation of this figure 
 referred to rectangular axes is 
 known. This manner of ob- 
 taining the equation of a figure 
 referred to oblique axes is fre- 
 quently a simpler problem than 
 to obtain the equation directly. 
 To accomplish the transformation, 
 the rectangular coordinates of a 
 point must be expressed in terms 
 of the oblifjue coordinates of the Fio. 5g. 
 
 same point. From the figure 
 
 a: = AD = AII+ D'K= .r, cos a + ,v, cos a', 
 y = PD= D'II+ PK= .r, sin a + //, sin u'. 
 
 ExAMi'LE. — To find the equation of the h_viHTl)i)l;i rcfcvvcd 
 to its asymptotes from the common C(piatit)n of the hyperbola, 
 x^ _ jf _ 1 
 
64 
 
 ANALYTIC GEOMETin 
 
 b 
 
 The asymptotes of the hyperbola, y 
 of the rectangle on the axes. Hence 
 b 
 
 ± -X, are the diagonals 
 a 
 
 cos a — 
 
 Va- + b- 
 
 and the transformation for- 
 mulas become 
 
 " (.-^-i + z/O, 
 
 Va^ + b' 
 b 
 
 Va^ + 6- 
 Substituting in the common 
 equation of the hyperbola 
 
 and reducing, a\y^ = '-, 
 
 4 
 the equation of the hyperbola referred to its asymptotes. 
 
 The formulas for passing from oblique axes to rectangular, 
 'y the origin remaining the same, are 
 
 x = AD = AH-D'K 
 
 _ y, sin (/3 — a) _ ?/, cos ((3 — a) 
 sin/? sin/3 
 
 = PD = D'll + PK 
 
 _ Xi sin a ?/i cos n 
 sin /3 sin (3 
 
 Art. 34. — General Transformation 
 
 The general formulas for transforming from one set of recti- 
 linear axes to another set of rectilinear axes, the origin of the 
 second set when referred to the first set being (m, n), are 
 
TUANSF(>i;MAril)i\ OF I'OOUDl X ATKS 
 
 05 
 
 x = AD=An + A/r-\-J)'h', 
 
 _ .r I s i 11 (/3 - u) //i sill {13 -a') 
 
 -'"+ sm(3 ^ sm(3 ' 
 2/ = PL* = .l,7t* + D'T + I'K, 
 
 .r, sill u , Vi sill a' 
 
 From tliesc L;eiieral formulas all tlie ijrecediiiy formulas may 
 bo derived by substituting for ?/;, v, fS, a, a' tlieir values in 
 each special case. However, if it is observed that in every 
 case the figure used in deriving the transformation formulas 
 is constructed by drawing the coordinates of any ])oint P 
 referred to the original axes, and the coordinates of the same 
 point referred to the new axes, then through the foot of the 
 new ordinate parallels to the original axes, it is simpler to 
 derive these formulas directly from the figure, whenever they 
 are needed. 
 
 Art. 35. — Thk ruouLEM of Tiia\sfoi;i\i.\tion 
 
 An examination of the transformation formulas shows that 
 the values of the rectilinear coordinates of any point in terms 
 of any other rectilinear coordinates of the same ])oint are of 
 the first degree. Hence transformation from one set of recti- 
 
66 ANALYIUC GEOMETRY 
 
 linear coordinates to auotlier rectilinear set does not change 
 the degree of the equation of the geometric figure. 
 
 Two classes of problems are solved by the transformation of 
 coordinates : 
 
 I. Having given the equation of a geometric figure referred 
 to one set of axes, to find the equation of the same geometric 
 figure referred to another set of axes. 
 
 II. Having given the equation of a geometric figure referred 
 to one set of axes, to find a second set of axes to which when 
 the geometric figure is referred its equation takes a required 
 form. 
 
 Problems. —Transform to parallel axes, given the coordinates of the 
 new origin referred to the original axes. 
 
 1. 2/ - 2 = 0(x + 5), origin (- 5, 2). 
 
 2. (x - 3) (?/ - 4) = 5, origin (3, 4). 
 
 3. y = 2x + 5, origin (0, 5). 
 
 4. a;2 + ?/2 4- 2 X + 4 ?/ = 4, origin ( - 1, - 2). 
 
 5. a;2 + 2/2 + fi »/ = 7, origin (0, - 3). 
 
 6. X- + 2/2 - G X = 16, origin (3, 0). 
 
 7. 2/^ + 4 2/ - G X = 4, origin (0, - 2). 
 
 8. 25(2/ + 4)2 + lC(x - 5)2 = 400, origin (5, - 4). 
 
 9. ,;2 + f. ^ 25, origin ( - 5, 0). ^^ ^J + ?^ = 1, origin (0, - h). 
 
 10. x2 + 2/2 = 25, origin (0, - 5). "' ''' 
 
 11. x2 + 2/2 = 25, origin (-5, -5). '^- | + g = 1, origin (- a, - 6). 
 
 12. ^^t^ 1, origin (- a, 0). 15. ^-^- = 1, origin (a, 0). 
 a- 62 a^ h^ 
 
 Transform from one rectangular set to a second rectangular set, the 
 second set being obtained by turning the first about the origin tlirough 45'^. 
 
 16. x2 + 2/- = 4. 18. y + x = 5. 21. y'^-3ry + x«,= 0. 
 
 17. x2 - 2/2 = 4. 19. 2/2 ^ jy _ a:2 = G. 22. ?/" + 3 ry - .r' = 0. 
 
 20. if' + 4 xy + x2 = 8. 
 
 Notice that to plot equations 21 and 22 directly requires tlie sohition of 
 a cubic equation, whereas the transformed equations are plotted by the 
 solution of a quadratic equation. 
 
TliANSFORMATION OF COORDINATES 
 
 (37 
 
 In the following problems the first equation is the equation of a geo- 
 metric figure referred to rectangular axes. The origin of a parallel set of 
 axes is to be ftiuml to which when the geometric figure is ri'ferred its 
 equation is tlie second eiiuation given. 
 
 23. .'/ + 2 = 4(x - :J) ; y = -1 •^■- 26. if - X- - lU x = ; x- - ij- = 25. 
 
 24. (.<• + 1) (// + 5) = 4 ; xij ^ 1. 27. U' - 1U(..: -|- 5) = ; >f = 10 x. 
 
 25. f + x^ + lOx=0; x:- + y'=25. 28. if + x;^ + -i >j--2 x=l\ ; x- + f = YG. 
 
 In the following problems the first equation is tlu; equation of a geo- 
 metric figure referred to rectangular axes; find the inclination of a secoiul 
 set of rectangular axes to the given, origin remaining the same, to which 
 when the geometric figaire is referred its equation is the second eiiuation 
 given. 
 29. y = x + 4; y = 2V2. ^^ 
 
 „o ^ 2n ■ ,-2 4- y2 = 2:,. 32. ^ - 2g = 1 ; :r// 
 
 30. 
 
 y- 
 
 33. //■' - 3 axy + .';' = ; y^ 
 
 - y 
 3V2aa;' 
 
 d' 4- }i~ 
 
 2x3 
 
 2x -|-;j\/2a 
 
 Construct the locus of tlie first etiuation in the following problems by 
 drawing tlie axes A'l, I'l and plotting the second eipiation. 
 
 34. 11-2 = log(x -f- ;:i) ; ?/i = log Xi. 36. ?/ -f 3 == 2^ t •» ; y.^ = 2a. 
 
 35. 2/ = 3 sin(x + 5); 2/1 = 3 sin Xi. 37. y + 5 = tau(x - 3); yi = tan Xi. 
 
 1, obtain 
 
 3.2 ,,2 
 From the common eciuation of the hyperbola,-— — -- 
 
 the equation of the hyperbola referred to oblique axes through the center, 
 
 1,2 
 
 such that tan a tan a' = — 
 
 a- 
 
 The transformation formulas 
 
 are x = Xi cosa -f- 2/1 cosa', 
 
 y = Xisina -1- yi sin a' ; 
 
 the transformed equation 
 
 /cos^a sin-a\ , 
 
 [—2 ^p' 
 
 + 2f "°^°' 
 
 sm g sm g' 
 
 The condition tan a tan a 
 the equation of the hyperbola referred to the oblique axes becomes 
 
 Fic. f>i». 
 renders the coefficient of Xij/i zero, and 
 
68 
 
 ANALYTIC GEOMETRY 
 
 .siii-c 
 
 
 l.'/r = 1- 
 
 Since only values uf a and a' less than 180° need be considered, the con- 
 dition tan a tan a' = - shows that a and a' are either both less than 90° or 
 
 «" h b 
 
 both greater than 90", and that if tana <-, tana' >-. Since the equa- 
 tions of the asymptotes of the hyperbola are y = ±~x, it follows that if 
 the Xi-axis intersects the hyperbola, the Ti-axis cannot intersect it. 
 Calling the intercepts of the hyperbola on the Xi- and I'l-axis respectively 
 ai and biV— 1, the equation referred to the oblique axes becomes 
 
 39. From the common equation of the ellipse, ~ -|- - = 1, obtain the 
 
 62 
 
 62 
 
 P P 
 
 (m, n) is y=~(x + 7n), tano'=— . 
 
 n n 
 
 equation of the ellipse referred to oblique axes such that tan a tan a' 
 
 40. From the common equation of the parabola y^ = 2px obtain the 
 equation of the parabola referred to oblique axes, origin (m, n) on the 
 parabola, the AVaxis parallel to the axis of the parabola, the JVaxis 
 tangent to the parabola. 
 
 n- = 2 pm, a = 0, and, since the equation of the tangent to y"^ = 2px at 
 The transformation formulas be- 
 come X = m + Xi + ?/i cos a', ?/ = n + ?/i sin a', and the transformed equa- 
 
 rfl -|- /i'.2 
 
 tion reduces to yi^ = 2 .ti, or yi~ ■= 2(p + 2 m)xi. 
 
 41. To determine a set of oblique axes, with the origin at the center, to 
 which when the ellipse is referred, its equation has the same form as the 
 common equation of the ellipse -^ + ra = 1- 
 
 The substitution of 
 
 X = a:i cosa + ?/iC0Sa', 
 2/ = Xi sin a -1- yi sin a' 
 
 transforms the equation -2+72 = 1 
 
 into 
 
 /Cos2a 
 
 V rt2 
 
 + 
 
 ft2 
 
 
TRANSFOh'MATIO.Y OF COORDINATES G9 
 
 'I'lio problem requires thai the coefficient of Xij/i bo zero, hence 
 
 tan a tan a.' = '- 
 
 d- 
 
 Tlie problem is indeterminate, since the etiualion between a and a' admits 
 an infinite number of solutions. Let a and a' in the figure represent one 
 solution, then (^ + ^^^^^-3 + ^^' + ^li^^y,. ^ 1 is the equa- 
 tion of the ellipse referred to the axes A'l, I'l. Call the intercepts of the 
 ellipse on the axes Xi and I'l respectively ai and hi , and the equation 
 becomes ^ + f^ = 1. 
 
 When the equation of the ellipse referred to a pair of lines through the 
 center contains only the squares of the unknown quantities, these lines 
 are called conjugate diameters of the ellipse. The condition of conjugate 
 
 diameters of the ellipse is tan a tan a' = — — . 
 
 a' 
 
 42. Determine a set of oblique axes, with the origin at the center, to 
 which, when the hyperbola is referred, its equation takes the same form 
 as the common equation of the hyperbola. 
 
 The result, tanatana' = — , shows that the problem is indeterminate. 
 jfl «'- 
 
 tana tan a' = — is the condition of conjugate diameters of the hyperbola. 
 d^ 
 
 43. Determine a set of oblique axes, origin at center, to which, when 
 the hyperbola is referred, its equation takes the form xij — c. 
 
 44. Determine origin and direction of a set of oblique axes to which, 
 when the parabola is referred, its equation has the same form as the 
 common equation of the parabola. 
 
 45. Show that the equation of the parabola y'^ = 2pz when referred to 
 its focal tangents becomes x^ + y^ — a^, where a is the distance from the 
 new origin to the points of tangency. 
 
CHAPTER VI 
 
 POLAR OOOEDINATES 
 
 Art. 36. — Tolau Cooudinates of a Point 
 
 In the plane, suppose the point A and the straight line AX 
 through A fixed. A is called the pole, AX the polar axis, 
 p ,/ The angle which a line AP makes 
 
 ^'"''^^ with AX is denoted by 6. is 
 positive when the angle is con- 
 ceived to be generated by a line 
 X starting from coincidence with AX 
 turning about A anti-clockwise ; 9 
 is negative when generated by a 
 line turning about A clockwise. 
 When 6 is given, a line through A 
 is determined. On this line a point is determined by giving 
 the di-stance and direction of the point from A. . The direction 
 from A is indicated by calling distances measured from A in 
 the direction AP of the side of the angle positive, those 
 measured in the opposite direction negative. The point P is 
 denoted by the symbol (r, ^), the point P' by the symbol 
 (-r, e). The symbols (r, ^ + 27r?i), (-'', ^ +(2?i + l)7r), 
 where n is any integer, denote the same point. To every sym- 
 bol (r, &) there corresponds one point of the plane ; to every 
 point of the plane there corresponds an infinite number of 
 symbols (r, d). Under the restriction that r and 6 are positive, 
 and that the values of B can differ only by less than 2 tt, there 
 exists a one-to-one correspondence between the symbol (/•, Q) 
 70 
 
POLAR coon DIN ATES 71 
 
 aud the points of the plane, the pole only excepted, r and 
 are called the polar coordinates of the point. 
 
 1. Locate the points whose polar coordinates are (2, 0) ; 
 (-3,0); (3,l,r); (-2,7r); (4,f,r); (-4,l7r); (4,-|7r); (1,1); 
 (-2,1); (-1,0); (1,180°); (-4,45°); (4,225°); (-4,405°); 
 (0, 0) ; (0, 45°) ; (0, 225°). 
 
 2. Show that r'- + r"- - 2 r'r" cos (0' - 0") is the distance be- 
 tween (?•', 6'), (r", 0"). 
 
 3. Find the distances between the following pairs of points, 
 (4,l7r), (3,7r); (8,J-7r), (6,f,r); (2V2, -^7^), (l,i7r);(0,0), 
 (10, 45°); (5, 45°), (10, 90°) ; (-0, 120°), (- 8, 30°). 
 
 AuT. 37. — PoLAK Equations ok CiEoiiETiiio Fiuujies 
 
 The conditions to be satisfied by a moving point can some- 
 times be more readily expressed in polar coordinates than in 
 rectilinear coordinates. If a point moves in the XF-plane in 
 such a manner that its distance from the origin varies directly 
 as the angle included by the X-axis and the line from the 
 origin to the moving point, the rectangular equation of the 
 
 locus is Va.'- -f y- — a tan~^-,, the polar equation r = a9. 
 
 Desired information about a curve is often obtained more 
 directly from the polar equation than from the rectilinear 
 equation of the curve. This is especially the case when the 
 distances from a fixed point to various points of the curve are 
 required. Thus if the orbit of a comet is a parabola with the 
 sun at the focus, the comet's distance from the sun at any time 
 is obtained directly from the polar equation of the parabola. 
 
 AuT. 38. — Polar Equation of Stuakjut Link 
 
 A straight line is determined when tlie lengtli of tlic perpen- 
 dicular from the pole to the line and the angle included by this 
 perpendicular and the polar axis are given. Call the per- 
 
ANAL YTIC GEOMETR Y 
 
 peiuliciilar p, the angle a, and let (r, 6) be any point of the 
 line. The equation 
 
 Fig. C3. \ 
 
 ; for 6 = a. 
 
 cos {p — a) 
 expresses a relation satisfied by 
 the coordinates r, of every point 
 of the straight line and by the 
 coordinates of no other point; 
 that is, this is the equation of 
 the straight line. For 6 — 0, 
 
 from ^ = to ^ = 90° + «, r is 
 cos « ' 
 
 positive ; from = 00° + a to = 270° + a, r is negative ; 
 from 6 = 270° + a to 6 = 3G0°, r is again positive. For 
 6 = 90° + a and for 6 = 270° + «, r = ± oo . These results ob- 
 tained from the equation agree with facts observed from the 
 figure. 
 
 A straight line is also determined by its intercept on the 
 polar axis and the angle the line makes witii the polar axis. 
 
 Call the intercept b, the 
 v'^-' angle a, and let (r, 6) be any 
 
 point in the line. Then 
 
 6 sin a . ^, 
 
 r =—. — 7 ;rr IS thc cqua- 
 
 sin(« — ^) 
 
 tion of the line. For ^ = 0, 
 
 r = b; ?• is positive from 
 
 ^ = to 6 = a; negative 
 
 from ^ = a to 6 = 180° + a; 
 
 ^"'- ^- again positive from 6 = 180° 
 
 + atoe = 360°. ¥ov 0=a and 6 = 180° + «, r = ± oo. These 
 
 results may be obtained from the equation or from the figure. 
 
 Akt. 39. — Polar Equation of Circle 
 
 The equation of a circle whose radius is E when the pole is 
 at the center, the polar axis a diameter, is r = E. 
 
POLAR COOUDl NATES 
 
 73 
 
 AVheu tlic pole is on tlio circuinfeieuce, tlio polar axis a 
 diameter, the equation of the circle is r = 2 li cos 0. 
 
 r is positive from ^ = 0° to ^ = 90°, negative from $ = 90° to 
 6 = 270°, and again positive from 6 = 270° to ^ = 300°. The 
 entire circumference is traced from ^ = 0° to 6 = 180°, and 
 traced a second time from ^ = 180° to ^ = ,'>60°. 
 
 The polar equation of a circle radius R, center (>•', 6'), cur- 
 rent coordinates r, 6, is »-^ — 2 r'r cos (6 — 6')= lir — r'-, whence 
 r = r' cos (6 - $') ± -y/R- - r'- sm\e - J'), r is real and has 
 two unequal values when sin-(^ — ^')< -y,; that is, wIkmi 
 
 R ■ R ^ 
 
 <sin(^ — ^')< — ; these values of r become equal, and 
 
 r r jy 
 
 the radius vector tangent to the circle, when sin(^ — ^')= ± — ; 
 
 R- ^ 
 
 r is imaginary when sin^(^ — 6')> —- 
 
 ^1 
 
 Art. 40. — Polar Equations of the Conic Sectioxs 
 
 Take the focus as pole, the axis of the conic section as polar 
 axis. From the definition of a conic section 
 
 r = e- DE = e (DA + AE) =e['- + r cos d 
 
 Hence 
 
 r—j^ + ercofiO, r 
 
 1 — e cos 6 
 
74 
 
 ANALYTIC GEOMETRY 
 
 Since in the parabola e = 1, the polar equation of the parab- 
 
 Ola IS r = :; 1,- 
 
 In the ellipse and hyper- 
 bola the numerical value of 
 the semi-parameter p is 
 
 a(l-e^); 
 hence the polar equation of 
 ellipse and hyperbola is 
 
 1 — e cos 
 
 In the ellipse e is less than 
 unity, and r is therefore 
 
 Fig. 07. . -^ 
 
 always positive. For ^ = 0, 
 r — a(l + e), showing that the pole is at the left-hand focus. 
 In the hyperbola e is greater than unity, and r is positive 
 
 from 6 = to 6 = cos~^ - in the first quadrant, negative from 
 
 1 ^ 1 
 
 ,-1^ ;r, 4-1.0 fi,.of r.,no,iT.o,.f fr> fl — f-og-'- in the fourth 
 
 cos~' - in the first quadrant to 
 e 
 
 quadrant, again positive from 
 rant to ^ = 3G0' 
 
 ,^il 
 
 cos^^ - in the fourth quad- 
 e 1 
 
 r becomes infinite for 6 = cos ^ - ; hence lines 
 
 ' 1 
 
 through the focus making angles whose cosine is - with the 
 
 axis of the hyperbola are parallel to the asymptotes of the 
 hyperbola. 
 
 Problems. — 1. The length of the perpendicular from the pole to a 
 straight line is 5 ; this perpendicular makes with the polar axis an angle 
 of 45°. Find the equation of the line and discuss it. 
 
 2. Derive and discuss the polar equation of the straight line parallel 
 to the polar axis and 8 above it. 
 
 3. Derive and discuss the equation of the straight line at right angles 
 to the polar axis, and intersecting the polar axis 4 to the right of the pole. 
 
 4. Derive and discuss the equation of the circle, radius 5, center 
 
 (10, iTT). 
 
POLAR COORDINATES 
 
 75 
 
 5. Derive and discuss Uie equation of tlie circle, radius 10, center 
 (5, Iw). 
 
 6. Derive and discuss tlic equation of the circle, radius 8, center 
 (10, ]7r). 
 
 7. Derive and discuss the equation of the circle 
 (15, tt). 
 
 8. Derive and discuss the equation of the circle. 
 (10, ^tt). 
 
 9. Find the polar equation of the parabola whose parameter is 12. 
 Find the polar equation of the ellipse whose axes are 8 and 6. 
 Find the polar equation of the ellipse, parameter 10, eccentricity \ 
 Find the polar equation of the ellipse, transverse axis 10, eccen 
 
 radius 10, center 
 
 radius 10, center 
 
 10. 
 11. 
 12. 
 tricity 
 13. 
 14. 
 15. 
 
 4 c2,-2 co^l , 
 
 Find the polar equation of the hyperbola whose axes arc 8 and G. 
 
 Find polar equation of hyperbola, transverse axis 12, parameter 0. 
 
 Find polar equation of hyperbola, transverse axis 8, distance be- 
 tween foci 10. 
 
 16. Find the eijuation of the locus of a point moving in such a manner 
 that the product of the distances 
 of the point from two fixed 
 points is always the scjuare of 
 the half distance between the 
 fixed points. This curve is called 
 the lemniscate of Bernoulli. 
 
 By definition ViV^ = c^. From 
 the figure rx^ = r- + c^-2cr cos 6, 
 
 r^ = r2 -f- c2 + 2 cj- cos 9, hence rrro- — r* + 2 c^r'^ + <: 
 and r2 = 2 ^2 (2 cos2 ^ - 1), r^ = 2 c^ cos (2 0). 
 
 Corresponding pairs of values of Vi 
 and r2 may be found by drawing a 
 circle with radius r, to this circle a 
 tangent whose length is c. The dis- 
 tances from the end of the tangent 
 to the points of intersections of the 
 straight lines through the end of the 
 tangent with the circumference are 
 corresponding values of r^ and j-o, for 
 7\S ■ Tli = c'. The inter.sections of 
 arcs struck off from Fi and Fo as 
 centers with radii TS and Tli deter- 
 mine points of the lemniscate. 
 
76 
 
 ANALYTIC GEOMETRY 
 
 17. A bar turns around and slides on a fixed pin in such a manner 
 tliat a constant lengtli projects beyond a fixed straiglit line. Find the 
 equation of the curve traced by the end of the bar. This curve is called 
 the conchoid of Nicomedes. 
 
 
 ^^-^ 
 
 Y 
 
 -^^ 
 
 
 ^ ' 
 
 — a, 
 
 / 
 
 
 — 
 
 m 
 
 b 
 
 A 
 
 / 
 
 
 A 
 
 
 X 
 
 Take the fixed point A as focus, the line ^X, parallel to the fixed line 
 mn, as polar axis. Call the distance from the i^ole to the fixed line h, 
 the constant length projecting beyond the fixed line a. Then 
 
 The conchoid is used to trisect an angle graphically. Let GAH be the 
 angle. From any point B in one 
 side of the angle draw a perpen- 
 dicular mn to the other. With 
 vertex of angle as fixed point, 
 mn as fixed line, and FG = 2 BA 
 as constant distance, construct a 
 conchoid. At B erect perpen- 
 dicular BC to mn, and join its 
 A point of intersection with con- 
 
 "'■ ' ■ choid C and J. by a straight line. 
 
 GAC is \ CtAII, for, drawing through D, the middle point of BC, a parallel 
 to mn and joining B and E, the triangles ABE and BEC are isosceles. 
 Kence BAG = BE A = 2BCA = 2 GAC. 
 
 
 ^ 
 
 C G 
 
 
 
 -^ 
 
 
 ^^^ 
 
 ^^ 
 
 ^ 
 
 \ 
 
 D 
 \ 
 
 Ae 
 
 
 "^ 
 
 m 
 
 
 \ 
 
 
 
 n 
 
 
 
 E 
 
 
 F 
 
 
 Art. 41. — Plottinct of Polar Equations 
 
 
 Example. — Plot r = 10 cos 0. 
 
 
 e = () ^TT ItT fir TT fTT §77 | TT 
 
 2,r 
 
 r = l() nV2 -5V2 -10 -5VL> - SV^ 
 
 10 
 
POL A R COORBINA TES 
 
 77 
 
 If tlie iiuinl.or oi' 
 points located I'roiu ^ = 
 to ^ = 2 TT is indefinitely 
 increased, the polygon 
 formed by joining tlu? 
 successive points ap- 
 proaches the circumfer- 
 ence of a circle as its 
 limit. Tlie form of the 
 equation shows at once 
 that the locus is a circle 
 whose radius is 5. 
 
 Example. — Plot r = aO. 
 
 
 
 e = -4: -3 -2 
 
 -1 
 
 
 
 r = — 4a — 3 a — 2 a 
 
 — a 
 
 
 
 12 3 4 
 a 2 a 3 a 4 a 
 
 The curve is called 
 the spiral of Archimedes. 
 In rectangular coordi- 
 nates the equation of 
 this spiral is transcen- 
 dental. 
 
 Example. — Plot r = 
 
 $ = cos-> f 
 
 r = — 2 T 00 
 
 3 — T) cos 6 
 
 ±cx> 
 
78 
 
 ANALYTIC GEOMETIIY 
 
 From 6 = to 6 = cos~^ f , r varies continuovisly from — 2 to 
 
 — cc ; from ^ = cos~^f to 
 = TT, r decreases contimi- 
 ously from + co to -{- -h ; 
 from 6 = Tr to 6 = cos~^ | in 
 the fonrtli qxiadrant, r in- 
 creases continuously from ^j 
 to + CO ; from 6 = cos~^f in 
 the fourth quadrant to 
 6 = 2 TT, r increases from 
 
 — CO to — 2. r is discon- 
 tinuous for ^=cos"'|. This 
 equation represents an h}'- 
 perbola whose less focal 
 
 distance is i, greater focal distance 2, semi-parameter A, eccen- 
 tricity f 
 
 Example. — Plot /-^ = 8 cos (2 9). 
 = 0° 22.^° 45° 135° 157i° 180° 
 
 r = ± 2.828 ± 2.378 imaginary ± 2.37 
 
 ± 2.828 
 
 From 180° to 300° the 
 curve is traced a second 
 time. The pole is a cen- 
 ter of symmetry of the 
 curve. ' 
 
 Problems. — Plot 
 
 COS 
 
 cos (6 — J tt) 
 
 3. r = ncnsi{3 0). 
 
 4. )• = 2 cos 0. 
 
 5. r= asm (2 0). 
 
 6. r = acos(o0). 
 
 7. r = asin(Se). 
 
 8. »' = rtsin(4^). 
 
 9. r = a sin (5 6). 
 
 10. r = ^, the reciprocal spiral. 
 
 11. r — a", the logaritlimic spiral. 
 
POLAR COOIiT)TNATES 
 
 79 
 
 12. 
 
 
 , the lituus 
 
 13. 
 
 
 2 
 
 ' 1 
 
 - cos 
 
 11 
 
 
 5 
 
 
 2 
 
 - 3 cos » 
 
 15 
 
 
 4 
 
 
 ' 3 
 
 - 2 cos » 
 
 IB 
 
 
 10 
 
 1 + cos e 
 
 17. ?• = « (1 + cos e), the cardioid. 
 
 18. J- = 4(1 -COS0). 
 
 19. r = 5 + 2 sin 0. 
 
 20. 
 
 r — 2p cot ^ cosec ff. 
 
 21. 
 
 ^. _ 4 COS ^ 
 
 1 + 3 sin2 e 
 
 22. 
 
 4 COS e 
 
 1-5 sin- 
 
 23. 
 
 3 sin e cos 
 
 sin^ d + cos-'' ^ 
 
 24. 
 
 r = rt(sin2e + cos2<?). 
 
 25. 
 
 r- cos (2 e) - 4. 
 
 26. 
 
 )-2sin(2e)=8. 
 
 27. 
 
 r- cos 1 e = 2. 
 
 28. 
 
 )•- = 10 sin (2^). 
 
 Art. 42. — Transformation from Rectangular to Polar 
 Coordinates 
 
 If the rectangular equation of a geometric figure is given, 
 and the polar equation is desired, find the values of the rectan- 
 gular coordinates x and y of any point in terms of the polar 
 coordinates r and of the same point; substitute in the rec- 
 tangular equation f(x,y)=0, and the resulting equation 
 F(r, 6)=0 is the polar equation of the figure. 
 
 Let the pole A' referred to the rectangular coordinates be 
 (m, n), 6' the angle made by the polar axis with the X-axis. 
 Then 
 
 X = AD = m + r cos (6 + 6'), 
 
 y = PD = n + r sin (d + 6'). 
 
 When the pole is at the origin, 
 and the polar axis coincides with 
 the X-axis, these formulas be- 
 come X = r cos 6, y — r sin $. 
 
 If the polar equation of a 
 geometric figure is given and 
 the rectangular equation is desired, find the values of the 
 polar coordinates r and 6 of any point in terms of the rectan- 
 
80 ANALYTIC GEOMETRY 
 
 gular coordinates x and y of the same point ; substitute in the 
 polar equsition F(r, 6) = 0, and the resulting equation /(a;, ?/)=:0 
 is the rectangular equation of the curve. 
 From the figure 
 
 X — m 
 
 r = y/{x - my +{y- w)^ cos (6 + 6') = 
 sin(^ + ^')- •'~'" 
 
 ^{x - mf + (2/ - ?0' 
 
 V(.x'-m/+(y-n)^ 
 
 When the origin is at the pole, and the X-axis coincides with 
 the polar axis, these formulas become 
 
 r — V.V- + ]j\ cos Q = ^ sin 9 = • 
 
 VaT- + y- Var + y- 
 
 Problems. — Transform from rectangular coordinates to polar, pole at 
 origin, polar axis coinciding with A'-axis, and plot the locus from both 
 equations. 
 
 1. x2 + ?/^ = 25. _ 6. 2/2 = i(4.T-a;2). 
 
 2. X- + 2/2 - 10 X = 0. 7. ?/ = - 1 (4 X - x^). 
 
 3. y^ = 2j-)X. 8. if -Sx>j + x^ = 0. 
 
 4. a;2 - 2/2 = 25. 9. (x^ + 2/-)'^ = «" i^' - f')- 
 
 5. x2/ = 9. 
 
 Transform from polar coordinates to rectangular coordinates, X-axis 
 coinciding with polar axis. 
 
 10. r = «, origin at pole. ^^ ,. ^ 9 ^ p^j^ ^^ ^^^ ^^ 
 
 11. r = 10 cos e, origin at pole. 4 - 5 cos ^ 
 
 12. 9-2 = «2 cos (2 61) , origin at pole. le. r = , pole at (4, 0). 
 
 5 — 4cos& 
 
 13. )•- cos .', — 2, origin at pole. , cos f 2 0^ . • . , 
 
 - ' ° ^ 17. r- = '- ' , origm at pole. 
 
 „ cos* e 
 
 14. ,. = i^ , poleat(ip, 0). , .. n •• . , 
 
 1 - cos 18. r2cos'*^ = 1, origni at pole. 
 
CHAPTER VII 
 
 PROPERTIES OF THE STRAIGHT LINE 
 
 AuT. 43, — Equations of the Straight Line 
 
 The various conditions determining a straight line give rise 
 to different forms of the equation of a straight line. 
 
 I. The equation of the straight line determined by tlie two 
 l.oints (x', ?/'), (x", y"). 
 
 The similarity of the triangles 
 rP'D and P'P"D' is the geometric 
 condition which locates the point 
 P{x, y) on the straight line through 
 P'ix', y') and P"(.^•", ?/")• Tl>is 
 condition leads to the equation 
 
 y-y ^ '^, — ^ 
 x' — x" 
 
 tancrular coordinates 
 
 ^(x — o:'). In rec- 
 
 Fio. 77. 
 
 tan a, where 
 
 « IS 
 
 the an,<rl( 
 
 the 
 
 line 
 
 y -?/ _ 
 x' — x" 
 
 makes with the X-axis. In oblique coordinate! 
 sin a 
 
 -, where /3 is the angle between the axes, a 
 x' — x" sin (/3 — a) 
 
 the angle the line makes with the X-axis. 
 
 II. The equation of a straight line through a given point 
 (x', y') and making a given angle « with the X-axis is 
 y — y' = tan «(a; — x'). If the point {x', y') is the intersection 
 (0, n) of the line with the X-axis and tan « = m, the equation 
 becomes y = 7nx + n, the slope equation of a straight line. 
 
 On the straight line y — y' — tan a{x — x') the coordinates of 
 the point whose distance from (.«', ?/') is (7, are x = x' -f d cos a, 
 y = y' -\- d sin a. 
 
 a 81 
 
82 
 
 ANALYTIC GEOMETRY 
 
 III. The equation of the straight line whose intercepts on 
 the axes are a and 6. 
 
 Let {x, y) be any point in the 
 line. From the figure 
 
 a—x _y 
 
 a b 
 
 1, the 
 
 which reduces to - + - 
 a h 
 intercept equation of a straight 
 
 Fig. 78. Iji^g. 
 
 IV. When the length p and the inclination « to the X-axis 
 
 of the perpendicular from 
 the origin to the straight 
 line are given. 
 
 Let {x, y) be any point in 
 the straight line. From the 
 figure, AB+BC=p, hence 
 
 X cos (i + y sin « = p. 
 This is the normal equa- 
 tion of a straight line. 
 The different forms of 
 the equation of a straight line can be obtained from the general 
 first degree equation in two variables Ax -\- By -\- C = 0, which 
 always represents a straight line. 
 
 (a) Suppose the two points (x', y'), (x", y") -to lie in the line 
 represented by the equation ^x + i>?/ -|-C= 0. The elimina- 
 tion of A, B, C from (1) Ax+By+C=0, (2) Ax'+By'+C=0, 
 (3) Ax" -f By" + C = by subtracting (2) from (1) and (3) from 
 (1), and dividing the resulting equations gives 
 
 y-y = 
 
 y' — y' 
 
 x' - x' 
 
 (x-x'). 
 
 (b) Callin; 
 X-axis ((, on 
 
 : the intercept of the line Ax -\- By +C —0 on the 
 
 C 
 the F-axis b, for y = 0, x = - = a, for x — 0, 
 
 y — — b. Substituting in the equation Ax -f By + C = 0, 
 
 there results --f-?^=l. 
 a h 
 
I'liOriCliTIES OF THE STliAKniT LINE 83 
 
 (f) Tlie equation .l.i- + B;/ +C= U may be written 
 
 ^ b'^ b' 
 
 wliicli is of the form y = vix + n. 
 
 (d) Let Ax + Bij + C = and x cos a + y sin a =]> repre- 
 sent the same line. There must exist a constant factor m 
 such that VI Ax + vi By + i>iC = and x cos « + y sin « — p = 
 are identical. I'rom tliis identity mA = cos a, mB — sin a, 
 -,itC = —2>- The iirst two equations give nrA^ + vt^B'- — 1, 
 
 hence iii — — - That is, 
 
 Vvl- + B' 
 
 ^A' + B' VA' + B' y/A' + B' 
 is tlie normal form of the e(]uation of the straight line repre- 
 sented by Ax + By + C = 0. 
 
 The nature of the problem generally indicates what form of 
 the equation of the straight line it is expedient to use. 
 
 Problems. — 1. AVrite the tuiuation of the straight line through the 
 points (2,3), (-1, 4). 
 
 2. Write tlie eiiuation of the straight line througli (- 2, 3), (0, 4). 
 
 3. Write the intercept eijuation of tlie straight line through (4, 0), 
 (0, 3). 
 
 4. Write the equation of the straight line whose perpendicular dis- 
 tance from the origin is 5, this i)erpentlicular malting an angle of 30° with 
 the A'-axis. 
 
 5. Write the e(iuation ^ + | = 1 in the slope form. 
 
 6. Write the equation 2 x - 3 ?/ = 5 in the normal form. 
 
 7. Write the equation of the straight line through (4, -3), making 
 an angle of 135° with the A'-axis. 
 
 8. On the straight line through (- 2, 3), making an angle of 30° with 
 the A'-axi-s, find the coordinates of the point whose distance from 
 ( - 2, 3) is 0. 
 
 9. The vertices of a triangle are (3, 7), (5, - 1), (-3, 5). Write 
 equations of meilians. 
 
84 
 
 ANALYTIC GEOMETRY 
 
 Art. 44. — Angle between Two Lines 
 
 Let V be the angle between the straight lines y — mx + n, 
 y = m'x + n'. From the figure 
 V= u — a', hence 
 
 . Tr_ tan a — tan a' 
 1 + tan a tan a' 
 Since tan « = m, tan «' = 7/1' 
 
 — tanF= 
 
 When the 
 
 1 + ?«?u' 
 lines are parallel, V=0, which 
 requires that m — m'. When the lines are perpendicular, 
 
 F=00°, Avhich requires that 1 -\-mm' = 0. or m' = 
 
 m 
 If the e<|uations of the lines are written in the form 
 
 Ax + By+C=0, A'x + ]^y+a = 0, tanF^^^-||. 
 
 The lines are parallel when A'B — AB' = 0, perpendicular 
 when AA' + BB' = 0. 
 
 The equation of the straight line through (x', y') perpen- 
 dicular to y = mx + H is y — y' = (x — x'). 
 
 m 
 
 The equation of the straight line through (x', y') parallel to 
 y = mx + n is y — v' — ta (x — x'). 
 
 Let the straight line y — y' = tan a'(x — x') through the 
 point (x', ?/') make an angle 6 with 
 the line y — mx -f n. From the 
 figure, «' = ^ + «. Hence 
 , ,_ tan^+tan« _ ta,n6 + m 
 
 1 — tan ^ tan a 1—m tan 6' 
 since tan a = m. Therefore the 
 equation of the line through (x', y') 
 
 Fic. 81. 
 
 making an angle 6 with the line y — mx-\-n is 
 tan 6 + m 
 1 — 7)1 tan ( 
 
 y-y 
 
 .(x-x'). 
 
PEOPEPiTIES OF THE STliAiailT LINE 
 Problems. — 1. Find the angle the line 
 
 85 
 
 •^ = 1 makes with the 
 3 
 A'-axis. 
 
 2. Find the angle between the lines 2x + 3y = 1, lx+ lij = 1. 
 
 3. Find the e(iuation of the line through (4, -2) parallel to 
 5x-7i/ = 10. 
 
 4. Find the equation of the line through (1, 3) parallel to the line 
 through (2, 1), (-3, 2). 
 
 5. Find the equation of the line through the origin perpendicular to 
 3x-?/ = 5. 
 
 6. Find the equation of the line through (2, - 3) perpendicular to 
 |a;-.\y = l. 
 
 7. Find the ecpation of the line through (0, - 5) perpendicular to 
 the line through (4, 5), (2, 0). 
 
 8. The vertices of a triangle are (4, 0), (5, 7), (-0, 3). Find the 
 equations of the perpendiculars from the vertices to the opposite sides. 
 
 9. The vertices of a triangle are (3, 5), (7, 2), (- 5, - 4). Find the 
 equations of the perpendiculars to the sides at their middle points. 
 
 10. Write equation of line through (2, 5), making angle of 45' with 
 2x-3i/ = G. 
 
 Akt. 45. — Distance from a Point to a Line 
 
 Write the equation of the given line in the normal form 
 cccos« + ?/sin«-i:» = 0. Through the given point P{x', y') 
 draw a line parallel to the 
 given line. The normal 
 equation of this xiarallel 
 line is 
 
 X cos a + y sill (t = AP'. 
 
 Since (x', y') is in this line, 
 
 ■ x' cos a 4- y' sin « = AP'. ^^^ ^, 
 
 Subtracting p = AD', there 
 
 results x'cosa + y' sin a- p=PD; that is, the perpendicular 
 
 distance from the point {x',y') to the line x cos a + y sin a -p=0 
 
86 ANALYTIC GEOMETRY 
 
 is x' cos « + y' sin « — p. The manner of obtaining this result 
 shows that the perpendicular FD is positive when the point P 
 and the origin of coordinates lie on different sides of the given 
 line ; negative when the point P and the origin lie on the same 
 side of the given line. 
 
 The perpendicular distance from {x\ y') to Ax + J5^ + C'= 
 is found by writing this equation in the normal form 
 
 V^' + B" ^'A' + B" V^- + B- 
 
 j{x' 4- Bii' + C 
 and api)lving the former result to be PD = ' „ ' 3= — 
 i L ^ ^ V.l- + B' 
 
 This formula determines the length of the perpendicular; the 
 algebraic sign to be prefixed, which indicates the relative posi- 
 tions of origin, point, and line, must be determined as before. 
 
 Problems. — 1. Find distance from (-2,3) to 3 x + 5 y = 15. 
 
 2. Find distance from origin to | x — | ?/ = 7. 
 
 3. Find distance from (4, - 5) to line through (2, 1), (-3, 5). 
 
 4. Find distance from (3, 7) to ^^_^ = IjL=li^. 
 
 5. The vertices of a triangle ar6 (3,2), (-4,2), (5, -7). Find 
 lengths of perpendiculars from vertices to opposite sides. 
 
 6. The sides of a triangle are ?/ = 2 x + 5, 3 - ^ = !> 4 x - 7 y = 12. 
 Find lengths of perpendiculars from vertices to opposite sides. 
 
 7. The sides of a triangle are 2/ = 2x + 3, 2/ = -|x + 2, y -x-b. 
 Find area of triangle. 
 
 Art. 46. — Equations of Bisectors of Angles 
 
 Let the sides of the angles be Ax+By+C=0, A'x+Bhj^-G'=0. 
 The bisector ah is the locus of all points equidistant from the 
 given lines such that the points and the origin lie either on the 
 
I' HOP Eli TIES OF THE STRAIGHT LL\E 
 
 87 
 
 same side of each of the two ; 
 of each of the two given Hues. 
 In either case the perpendicu- 
 lars from any point {x, y) uf 
 the bisector to the given lines 
 have the same sign, and the 
 equation of the bisector is 
 Ax -f Bif + C _ A'x + B'n + C" 
 
 riven lines or on diiferent sides 
 
 V^- + B' VA" + B" 
 
 The bisector cd is the locus 
 
 of all points equidistant from ^'"- ^^• 
 
 the given lines and situated on the same side of one of the 
 
 given lines with the origin, while the other line lies between 
 
 the points of the bisector and the origin. The perpendiculars 
 
 from any point (x, ?/) of the bisector cd to the given lines are 
 
 therefore numerically equal but with opposite signs, and the 
 
 n ,, T ^ ,. Ax + By + C A'x + B'jf + C 
 equation of the bisector m is .' _ 
 
 VA' + B' 
 
 -VA" + B' 
 
 Problems. — 1. Find the bisectors of the angles whose sides are 
 3 .X + 4 (/ = 5, Hx - 1 >j = 2. 
 
 2. Find the bisectors of the angles whose sides are ^x — ly = 1, 
 2/ = 2x-3. 
 
 3. Find locus of all points cciuidistant from the lines 2x + 7 y = 10, 
 8 a; — 5y = 15. 
 
 4. The sides of a triangle are 5a: + 3 ?/ = 9, l x + I y = I, y = d x - 10. 
 Find the bisectors of the angles. 
 
 5. The sides of a triangle are 7x + 5i/ = 14, lOx — 15y = 21, y = 3x + 7. 
 Find the center of the inscribed circle. 
 
 AuT. 47. — Lines thuough Ixtekskctiox of Givk.v Lines 
 
 Let (1) Ax + B>/+C.= and (2) A'x + B'>/ + C = Ije the 
 given lines. Then (3) Ax + By + C -\-k (A'x + B'y + C) = 0, 
 where k is an arbitrary constant, represents a straight line 
 
88 ANALYTIC GEOMETRY 
 
 tlirougli the point of intersection of (1) and (2). For equation 
 (3) is of tlie first degree, hence it represents a straight line. 
 Equation (3) is satistied when (1) and (2) are satisfied simul- 
 taneously, hence the line represented by (3) contains the point 
 of intersection of the lines represented by equations (1) and (2). 
 If the line Ax + B>j+C + Jc (A'x + B'y + C") = is to contain 
 
 the point (x\ y'), k becomes - ^,^, _^ J^y _^ ^r Hence 
 
 is the equation of the line through (x', y'), and the intersection 
 of (1) and (2). 
 
 If the equations of the given lines are written in the normal 
 form, (1) X cos a + ?/ sin « — p = 0, (2) x cos «'+ ?/ sin «' — i''= '*? 
 the A; of the line through their point of intersection 
 
 (3) X cos u + y sin a — 2> + k i^' cos a' + y sin a' — 2>') = 
 
 -,. , ^ ■ ■ i. ^ i- 7 a;cos«+?/sin«— /> 
 
 has a direct geometric interpretation. A; = — ■■ — -. ; :, 
 
 a;cos« +?/sin« — ^> 
 
 that is, A; is the negative ratio of the distances from any point 
 (x, y) of the line (3) to the lines (1) and (2). 
 
 Problems. — 1. Find the equation of the line through the origin and 
 the point of intersection of 3x - 4?/ = 5 and 2 x + 5?/ = 8. 
 
 2. Find the equation of the locus of the points whose distances from 
 
 the lines i; x - 5 y + 2 = 0, - - ?^ = 1 are in the ratio of 2 to 3. 
 3 6 
 
 3. Find the equation of the line through (- 2, 3) and the intersection 
 of the lines 8 x - 5 ?/ = 15, 3 x + 10 ?/ = 8. 
 
 AiiT. 48. — TuuEE Points in a Straight Line 
 
 Let the three points (x', ?/'), (a-*", y"), {x"\ V'") lie in a straight 
 line. The equation of the straight line through the first two 
 
 points is y -y' = -[, ~-'„ {x-x'). By hypothesis the point 
 x — X 
 
PnOPEliTlES OF THE STRAIGUT LINE 8'J 
 
 ?/' — ?/" 
 (x'", y'") lies in tliis liue, hence y"' —y' = \, —^,(^"' ~ ^')- 
 
 Simplifying-, (1) x'y'" - x"y"' + x"y' - x"'y' + x'y" - x"'y" = 0. 
 When this equation is satisfied the three points lie in a 
 straight line, whether the coordinates are rectangular or 
 oblique. Notice that (1) expresses the condition that the 
 area of the triangle whose vertices are (x', y'), (x", y"), 
 (x'", y'") is zero. 
 
 Problenls. — 1. In a parallelogram each of the two sides through a vertex 
 is prolonged a distance equal to the length of the other side. Prove that 
 the opposite vertex of the parallelogram and the ends of the produced sides 
 lie in a straight line. 
 
 2. In a jointed parallelogram on two sides through a common vertex 
 two points are taken in a straight line with the opposite vertex. Show 
 that these three points are in a straight line however the parallelogram is 
 distorted. 
 
 Art. 49. — Tiiiiek Links through a Point 
 
 Let the three lines Ax + % + C'= 0, A'x + B'y + C" = 0, 
 A"x-j- B"y -\- C" = pass through a common iwint. IMako 
 the first two of these equations simultaneous, solve for x and y, 
 and substitute the values found in the third equation. There 
 results 
 
 AB'C" + A'B"C+A"BC' - A"B'C-A'BC" - AB"(J' = 0, 
 wliieh is the condition necessary for the intersection of the 
 given lines. 
 
 The three lines necessarily have aconiiudu ])oiut if constants 
 K,, K.,, K;, can be found such that ki(Ax -\- By + C)+ k2{A'x + 
 J^'ll + C")+ K^{A"x + B"y + 6'")= is identically satisfied. 
 For the values of x and y which satisfy Ax -\- By -\- C =0, and 
 A'x -}- B'y -f- C" = simultaneously must then also satisfy 
 A"x + B"y + C" = ; that is, the point of intersection of the 
 first two lines lies in the third line. 
 
 The second criterion is frequently more convenient of appli- 
 cation than the first. 
 
90 
 
 ANALYTIC GEOMETRY 
 
 Problems. — 1. The bisectors of the angles of a triangle pass through 
 a common point. 
 
 Let the normal equations of the three sides of the triangle be 
 a;cosa + 2/sina-i)i=0, xcosj3 + 2/sin/3-i)2=0, x cos 7 +2/ sin 7-^93=0. 
 Denote the left-hand members of these 
 \ equations by a, /3, 7. Then = 0, 
 
 |3 = 0, 7 = represent the sides of 
 the triangle, and a, /3, 7 evaluated for 
 the coordinates of any point (x, y) 
 are the distances from this point to 
 the sides of the triangle. Hence the 
 equations of the bisectors of the angles 
 are a- p =0, ^-7=0, 7-0 = 0. 
 The sum of the equations of the bi- 
 sectors is identically zero, therefore 
 the bisectors pass through a common 
 point. 
 
 2. The medians of a triangle pass through a common point. 
 
 For every point in the median through C, 
 
 hence 
 Simi- 
 
 sin B sin A 
 a sin A- ^sinB = is the equation of the median through C. 
 larly the equation of the median through B is found to be 
 
 7 sin C — a sin ^4 = 0; 
 of the median through A, /3 sin B -y sin C = 0. The sum of these equa- 
 tions vanishes identically. 
 
 3. The perpendiculars from the vertices of a triangle to the opposite 
 sides pass tlirough a common point. 
 
 The equation of the perpendicular through C is aros A — yScos J5 = ; 
 through B, 7 cos C - a cos .1 = 0; througli A, & cos 27 - 7 cos C = 0. 
 
riiOPERTIES OF THE STRAIGHT LINE 91 
 
 Akt. 50. — Tangent to Cukvk ok Skcond Okdkr 
 
 The general equation of the curve of the second order is 
 ax- + 2bxy + cy- -\-2dx + 2e>/ +/=(). Let (x„, y^ be any 
 point iu the curve. The equation y — ?/„ = tan a{x — Xq) repre- 
 sents any line through {xg, y^. The line cuts the curve of the 
 second order in two points and is a tangent when the two 
 points coincide. The coordinates of any point in the straight 
 line are x = Xq-{-1 cos a, y = y^ -\- 1 sin «. The points of inter- 
 section of straight line and curve of second order are the points 
 corresponding to the values of I satisfying the equation 
 
 {ax,; + 2 6.tv/o + cy,; + 2 dx, + 2 ey,, +/) 
 + (2 a.rii cos « + 2 hx.;^ sin « + 2 hy^ cos « 
 + 2 cyo sin « + 2 r/ cos « + 2 e sin a)l 
 + (a cos- « -f 2 6 cos « sin « + c sin- «)Z^ = 0. 
 
 Since (.i-q, .Vo) is in the curve, the absolute term of the equation 
 vanishes. If the coefficient of the first poAver of I also van- 
 ishes, the equation has two roots equal to zero, that is the two 
 points of intersection 6f y — ?/„ = tan a{x — a-,,) with the curve 
 coincide at (xq, ?/o) when 
 
 ax,^ cos a + hx^ sin « + hy^^ cos « + ry,, sin a -\- d cos a-\-c sin a = 0. 
 The equation of the tangent is found by eliminating cos a and 
 sin« from the three equations x = x^ -\- I cor a, ?/ = ?/„ + Z sin «, 
 (ixq cos « +■ bxo sin a + by„ cos a + cy^ sin a + fZ cos a-\- e sin « = 0. 
 This elimination is best effected by multiplying the third 
 equation by I, then substituting from the first two equations, 
 I cos a=x—Xo, I sin a^y—y^. The resulting equation reduces to 
 
 axxo + b(xy, + x„y) + ryy, + r/(.r + .^•,.) + K.'/ + ?/..) + /= 0. 
 The law of formation of the equation of the tangent from the 
 equation of the curve is manifest. 
 
 Problems. — 1. Write the equation of tlie tangent to x- + y"^ — r"^ at 
 (;^o, yo). 
 
92 ANALYTIC GEOMETRY 
 
 2. Write the equation of the tangent to — + f^ = 1 at (xo, ?/o). 
 
 d^ 0- 
 
 3. Write the equation of the tangent to •'- = 1 at {xq, yo). 
 
 a- b'^ 
 
 4. Write the equation of the tangent to y- = 2px at (xo, yo). 
 
 5. Find the equation of the tangents to 4 x^ + y- = 30 at the points 
 wliere x~\. 
 
 6. At what point of x^ — ?/2 = 1 nnist a tangent be drawn to make an 
 angle of 45° with the X-axis ? 
 
 7. Find the angle under which the line y = hx — b cuts the circle 
 x2 + ?/2 = 49. 
 
 8. Find tlie angle between the curves y'^ = C a-, 9 2/2 + 4 ^2 = 30. 
 
 9. Find the equations of the normals to the ellipse, hyperbola, and 
 parabola at the point {xo, yo) of the curve. 
 
 The normal to a curve at any point is the perpendicular through the 
 point to the tangent to tlie curve at the point. 
 
 10. Wliere must the normal to ^" + ^ = 1 be drawn to make an angle 
 of 135'" with the X-axis ? 
 
 11. Find the equation of the normal to ?/2 = 10 x at (10, 10). 
 
 12. Find equations of focal tangents to ellipse ^ + f- = 1. 
 
 a2 b^ 
 
CHAPTER VIII 
 
 PKOPERTIES or THE CIRCLE 
 AuT. 51. — Equation of tiik Cikclr 
 
 The equation of the circle referred to rectansjfuhir axes, 
 radius E, center (a, b), is (x — af +{y — h)- = R'\ This equa- 
 tion represents all circles in the Xl'-plane. The equation 
 expanded becomes x^ -\-y~ — 2ax — 2 by -f «- + ?>- — K' = 0. an 
 equation of the second degree lacking the term in .17/, and hav- 
 ing the coefficients of x^ and y^ equal. 
 
 Conversely, every second degree equation lacking the term 
 in xy, and having the coefficients of x^ and y- equal, represents 
 a circle when interpreted in rectangular coordinates. Such an 
 equation has the form x- -f ?/■ — 2 ax — 2 by + c = 0, whicli 
 when written in the form (x — ay -\-(y — b)' — (V + ^- — c, is 
 seen to represent a circle of radius ((t- + /r' — c)-, with center 
 at («, b). a, b,c are called the parameters of tlie circle, aiul 
 the circle is spoken of as the circle (a, b, 0). 
 
 When the center is at the origin, a = 0, b =^ 0, and the eipia- 
 tion of the circle becomes x^ + y^= R'- 
 
 When the X-axis is a diameter, the F-axis a tangent at the 
 end of this diameter, the circle lying on the ]>ositive side of 
 the F-axis, a = R, b = (), and the ecjuation of the circle be- 
 comes 2/^ = 2 Rx — x". 
 
 Problems. — Write tlie cciuafions of tlic fnllmvinn; circles: 
 1. Center (-2, 1), radius 5. 2. Center (- 5, 5), radins 5. 
 
 3. Center (- 10, 15), radius 5. 4. Center (0, 0), radius 5. 
 
 0:J 
 
94 ANALYTIC GEOMETBY 
 
 5. Find equation of circle througli (0, 0), (4, 0), (0, 4). 
 
 6. Find center and radius of circle through (2, — 1), (— 2, 1), (4, 5). 
 
 7. Find center and radius of circle x"^ -{■ y'^ -\- i x — Id y = 1 . 
 
 8. Find center and radius of circle x- -\- y- + l(i x = 11. 
 
 9. Does the line 3 x — 5 y = 12 intersect the circle 
 
 a;2 + j/2 - 8 X + 10 2/ = 50 ? 
 10. Find the points of intersection of the circles 
 
 x2 + 2/2-10x + 6y = 20, x2 + 2/2 + 4x- 15y = 25. 
 
 Art. 52. — Common Chord op Two Circles 
 
 The coordinates of the points of intersection of the circles 
 a? -{-y'^ -2ax -2hy -\- c = Q, x^ + y^ -2a'x -2 b'y -\-c' = 
 
 satisfy the equation 
 (a;2 + ?/2 - 2 ax - 2 by + c) - (x^ + y^-2 a'x - 2 b'y + c') = 0, 
 
 which reduces to 
 
 (a - a')x + (b - b')y +(<■• - r)= 0. 
 
 This is the equation of the straight line through the points of 
 intersection of the circles, that is the equation of the common 
 chord of the circles. 
 
 The intersections of two circles may be a pair of real points, 
 distinct or coincident, or a pair of conjugate imaginary points. 
 Since the equation of the straight line through the points of 
 intersection is in all cases real, it follows that the straight line 
 through a pair of conjugate imaginary points is real. 
 
 Problems. — Write the equations of the common chords of the pairs of 
 circles : 
 
 1. X- + y- -Gx + 4y = 12, x" + y^ - 'ix + 6y = 12. 
 
 2. x2 + ?y2 - lOx - 6?/= 15, x^ + ?/2 + lOx + 6?/ = 15. 
 
 3. x" + y- + lx + Sy = 20, x- -y y- + 4x - \0y = 18. 
 
PROI'EirriES OF THE CIRCLE 
 
 95 
 
 AkT. 53. POWKR OF A POIXT 
 
 Let (;c', y') be any point in the plane of the circle 
 
 The eqnation of any straight line throngh (x', y') is 
 
 y — y' = tan a (x — x'), 
 and on this line the point at a distance d from (x', y') has for 
 coordinates x = x' + d cos a, yz=y'-\-d sin «. The distances 
 from (x', y') to the points of intersection of line and circle are 
 the values of d found by solving the equation 
 
 |;(.^' _ ay + iy' - by - R'] + [2 (x' - a) cos a 
 + 2(y' - h) sin «] d + d- = 0. 
 
 Since the product of the roots of an equation equals numeri- 
 cally the absolute term of the equation, it follows that the 
 product of the distances from the point (x', ?/') to the points of 
 intersection oi y — y' = tan «(x — x') with the circle 
 
 (.^ _ ay + (y - by = E"^ is (x' - o)- + (y' - by - /?-. 
 
 This product is independent of a ; that is, it is the same for 
 all lines through (x', ?/'). This constant product is called the 
 power of the point (x', ?/') with respect to the circle. 
 
 The expression {x' — a)--f (?/' — by — Er is the square of the 
 distance from {x\ y') to the center (a, b) minus the square of the 
 radius. This difference, when the 
 point (x', ?/') is without the circle, 
 is the square of the tangent from 
 the point to the circle ; when the 
 point (.!•', )/') is within the circle, 
 this difference is the square of 
 half the least chord through the 
 point. 
 
 Let *S' represent the left-hand 
 member of the equation xr -\- y- — 2 a 
 S = is the equation of the circle, and S evaluated for the co- 
 
 Fir,. S". 
 
 2 by + c = 0. Then 
 
96 ANALYTIC GEOMETRY 
 
 ordinates of any point (x, y) is the power of that point with 
 respect to the circle. 
 
 Let /i5i = and So = represent two given circles. aS'i = S2 
 is the equation of the locus of the points whose powers with 
 respect to /S'l = and /S'2 = are equal. This equation, Avhich 
 may be written Si — S-, = 0, represents a straight line called 
 the radical axis of the two circles. The radical axis of two 
 circles is their common chord. 
 
 If three circles are given, Si — 0, S., = 0, S-^= 0, the radi- 
 cal axes of these circles taken two and two are Si — S^— 0, 
 S2 — Ss = 0, S3 — Si=: 0. The sum of these three equations is 
 identically zero, showing that the radical axes of three circles 
 taken two and two pass through a common point. This point 
 is called the radical center of the three circles. 
 
 Problems. — 1. Find the locus of the points from which tangents to 
 the circles x- + 7/- -i- 4x - 8y = b, x^ + 2/2 - 6x = 7 are equal. 
 
 2. Find the point from which tangents drawn to the three circles 
 a;2 + y2 _ 2 x = 8, x2 + 2/2 + 4 y - 12, x2 + ?/- + 4 x + 8 ?/ = 5 are equal. 
 
 3. Find the length of the tangent from (— 3, 2) to the circle 
 
 (x - 7)2 + (y- 10)2 = 9. 
 
 4. Find the length of tlie tangent from (10, 15) to the circle 
 
 x2-|- ?/2-4x + C?/- 12. 
 
 5. Find tlie length of the shortest chord of the circle 
 
 x2 + ?/2 - X + 4 ?/ = 3 
 through the point (—4, 3). 
 
 6. Find the equation of the radical axis of x2 + ?/2 + 5.r - 7 y = 15, 
 X- + 2/2 - 3 a: + 8 2/ = 10. 
 
 7. Find the radical center of x- + y- - ".x - 5, xr + y" - 4x + y - 8, 
 a;- + y^ + 7 2/ = 9. 
 
 8. Find the point of intersection of tlic tln-ee common chords of the 
 circles x2 + 2/" - 4x - 2 2/ = 0, x2 + ?/2 + 2x + 2 2/ = ll, x2-|- 2/"-6x + 42/ = 17 
 taken in pairs. 
 
I'JlorKin'lKS OF TIIK CIliCLIC 97 
 
 AuT. 54. — Coaxal Svstkms 
 
 Let Si — and iS.2 = re[)i't'sent two circles. Then 
 
 ,S', - kS, = 0, 
 
 for all values of the parameter A;, represents a circle tlirough 
 the intersections of Si = 0, aS^ = 0. The equation ^i — kS., — 0, 
 interpreted geometrically, gives the proposition, the locus of all 
 the points vi^hose powers with respect to two circles *Si = 0, 
 ^2 = are in a constant ratio is a circle through the points of 
 intersection of the given circles. 
 
 Si — kS^ = 0, by assigning to k all possible values, represents 
 the entire system of circles such that the radical axis of any 
 pair of circles of the system is the radical axis of *S, = and 
 S., = 0. 
 
 If tlie parameters of *S'i = and aS'2 = are a', h', r' and 
 a", b", c" respectively, the parameters of Si — kSo=^ are 
 
 a' — k(i" h' — kh" c' — kc" 
 1-/0 ' 1-k ' 1-k ' 
 
 Let aS' = represent a circle, L = a straight lino. Then 
 S —kL = represents the system of circles through the points 
 of intersection of circle and line. The commnii radical axis of 
 this system of circles is the line L = 0. 
 
 Circles having a common radical axis are called a coaxal 
 system of circles. 
 
 Problems. — 1. Write the equation of tlie system of circles tlirougli 
 the points of intensection of x^ + y- — 2 x + G ?/ = 10 and x- + y- — 4y = 8. 
 
 2. Find the equation of the circle through the points of intersection of 
 X- + y^-2x + Gy = 0, X- ■+ y"^ -4y = 8, and the point (4, - 2). 
 
 3. Find the equation of the circle through the points of intersection of 
 oc^ + 2/2 4. 10 7/ = 6, I X - 1 2/ = 3, and the point (4, 5). 
 
 4. Find the equation of the locus of all the points which have etiual 
 powers with respect to all circles of the coaxal system determined by the 
 circles x^ + j/2 - 3x + 7 y = 15 and x^ + y'^ + [>x - iy = 12. 
 
98 
 
 A NA L YTIC GEO MET R Y 
 
 Art. 55. — Okthogonal Systems 
 Two circles 
 a? + if — '^ a'x ~-2b'y + c' — 0, x^ + if — 2 a"x — 2 b"y + c" = 
 
 intersect at riglit angles when the square of the distance be- 
 tween their centers equals the 
 sum of the squares of their radii ; 
 that is, when 
 
 {a' — a")-+{b' — b"f 
 
 ^a"+b'--c' + a"'+b"''-c", 
 or 2a'a" + 2b'b" -c' -c" ^0. 
 
 If the circle (oj, bi, Cj) cuts each 
 of the two circles (a', b', c'), 
 (a", b", c") orthogonally, it cuts every one of the circles 
 
 ka" b' - kb" c' - kc 
 
 1-k 1-k 1-k 
 
 of the coaxal system orthogonally. For the hypothesis is 
 expressed by the equations 
 
 2a'a, + 2 b% - c'- c^ = 0, 2a"a, + 2b"bi -c" -c, = 0; 
 
 the conclusion by the equation 
 
 9 g' — ka" c b' — ^'^'\ 
 
 ''l-k^'l-k ' 
 
 c' - kc' 
 1-k 
 
 0, 
 
 which is a direct consequence of the equations of the hypothesis. 
 
 The condition that the circle (aj, &i, Cj) cuts the circles 
 (a', b', c') and («", b", c") orthogonally, is expressed by two 
 equations between the three parameters ttj, &„ Ci. These equa- 
 tions have an infinite number of solutions, showing that an 
 infinite number of circles can be drawn, cutting the given 
 circles orthogonally. 
 
 Let Oi, bi, Ci and a.^, bi, c^ be the parameters of any two circles 
 aSi = 0, aSj = cutting aS' = and S" = orthogonally. Then 
 
PliOPEliTIES OF THE CIRCLE 
 
 99 
 
 all circles of the coaxal system Si - kyS., = cut orthogonally 
 all circles of the coaxal system S' - k'S" = 0. For the equa- 
 tions 
 
 2a'((i + - f>'fJi — c' — c, = 0, 
 
 2 a"a, + '2 b"h, - c" -(\ = 0, 
 2a'a., + 2b'b2-c' -c., = 0, 
 2 a"a., + 2 b"b2 - c" - c.,= 0, 
 have as consequence 
 r,a' — k'a" a, — k,cu , r,b' — k'b" b^ — k^b.2 c' — k 
 
 (1) 
 (2) 
 (3) 
 (4) 
 
 Ci — kiC2 _ 
 
 1-k' 1-/h ■" 1-k' 1-ki 1-k' 1-ki 
 
 Subtracting (2) from (1), and (3) from (1), there results 
 
 2 (a' - a") ai + 2 (b' - b") b,-c' + c" = 0, (5) 
 
 2 (a, - a.) a' + 2 (6i - b.^ 6' - Ci + C2 = 0. (G) 
 
 Equation (5) shows that the centers of the orthogonal system 
 Si - kySo = lie in the 
 radical axis of the sys- 
 tem S'—k'S" = 0; equa- 
 tion (6) shows that the 
 centers of the system 
 S' - k'S" = lie in the 
 radical axis of the sys- 
 tem Si - kiS2 = 0. 
 
 Take for X-axis the 
 line of centers of the 
 system S' — k'S", for 
 y-axis the radical axis 
 of this system. Then 
 the equation of any 
 circle of the system 
 becomes 
 x-+y^-2a'x+c' = 0. (a) 
 
 Since by hypothesis 
 
 fiQ. 89. 
 
100 ANALYTIC GEOMETRY 
 
 tlie power of (0, ?/') is the same for all circles of the system, 
 c' must be a fixed constant. In like manner it is found that 
 the equations of the orthogonal system Si — k^S^ — have the 
 
 form 
 
 ar+^?/^-2 6i^ + Ci=0, {(3) 
 
 where Cj is a fixed constant. The condition for the orthogonal 
 intersection of two circles when applied to («) and (/3) becomes 
 Ci = — c'. Hence the equations of two orthogonal systems of 
 circles, when the radical axes of the systems are taken as 
 reference axes, are 
 
 X- + y^ — 2 a'x + c' = 0, x'^ + 11'— l>'u — c' = 0, 
 where «' and h' are parameters, c' a fixed constant. 
 
 The radii of the circles of the two orthogonal systems are 
 given by the equations r^ — a^'^ — c\ r'- = h'" -\- c' respectively. 
 When r and r' become zero the circles become points, called 
 the point circles of the system. In every case one of the or- 
 thogonal systems has a pair of real, the other a pair of imagi- 
 nary, point circles.* 
 
 Problems. — 1. Find the equation of the locus of the centers of the 
 circles which cut orthogonally the circles a;'- + y- — 4 .x + y = 15, 
 x2 -I- 1/2 -f 5 X - 8 2/ = 20. 
 
 2. Find the equation of the circle through the point (2, — 3) and cut- 
 ting orthogonally the circles x--\-y'^-]-^ x — 7 2/ = 18, x'^-l-y^ — 2 x—iy~\2. 
 
 3. Find the equation of the circle cutting orthogonally x'^+y'^ — \0 x = 9, 
 3.2 4. 2/2 = 25, x2 + y-i-8y = IG. 
 
 * Through every point of the plane there passes one circle of each of 
 the orthogonal systems. The point in the plane is determined by giving 
 the two circles on which it lies. This leads to a system of bicircular 
 coordinates. 
 
 If heat enters an infinite plane disc at one point at a uniform rate, 
 and leaves the disc at another point at the same uniform rate, when the 
 temperature conditions of the disc have become permanent, the lines of 
 equal temperature, the isothermal lines, and the lines of flow of heat are 
 systems of orthogonal circles. The points where the heat enters and 
 Jeaves the disc are the point circles of the isothermal system. 
 
rnOPEHTIKS OF THE CIliC'LlS' '^"^ " 11)1 
 
 4. Find the equation of the system of circles cutting orthogonally the 
 coaxal system detennined by x'^-\-if+ix + 6y-\5, x^ + y-+2 z-S y = l2. 
 
 5. Write the equation of the two orthogonal systems of circles whose 
 real point circles are (0, 4), (0, — 4). 
 
 Art. 56. — Takgents to Circles 
 
 The equation of a tangent to the circle x^ + -if = r'^ at the 
 point (a*o, ?/o) of the circumference is xxq + yy^ = r\ 
 
 Let (xi, yi) be any point in the plane of the circle x--\-y'=i~, 
 (x', y'), (x", y"), the points of contact of tangents from (x^, y^) 
 to the circle. Then (.r,, y{) must lie in each of the lines 
 xx' + yy' = 7", xx" + yy" = r-; that is, x^x' + y^y' = r, and 
 Xix" + yiy" = rl Hence the equation of the chord of contact 
 is xxi + yyi = r^. 
 
 The distance from the center of the circle to the chord of 
 
 contact is -, which is less than, equal to, or greater 
 
 (a.V + 2/i-)' 
 than r, according as the point (a-j, y^ lies without the circum- 
 ference, on the circumference, or within the circumference. In 
 the first case the points of contact of the tangents from (.Xi, r/i) 
 to the circle are real and distinct, in the second case real and 
 coincident, in the third case imaginary. In all cases the chord 
 of contact is real. 
 
 In the equation y = mx + n let m be a fixed constant, n a 
 parameter. The equation represents a system of parallel 
 straight lines. The value of n is to be determined so that 
 the line represented by y = mx + n is tangent to the circle 
 x^ + y~ = r^. The line is tangent to the circle when the per- 
 pendicular from the center of the circle to the line equals the 
 radius; that is, when = 7% ?i= ±rVl + m-. There- 
 
 fore, the equations of tangents to v? ■{- y'^ = r parallel to 
 y = mx -f n are y = mx ± r Vl + m'. 
 
102 ANALYTIC GEOMETRY 
 
 Problems. — 1. Find the equations of the tangents to x- -\- y- — 25 at 
 x = 3. 
 
 2. Find the chord of contact of tangents from (2, —3) to x- + y"^ = 1. 
 
 3. Find the points of contact of tangents from (5, 7) to x- + y- — 9. 
 
 4. Find the equations of tangents to x- + ?/- = 16, making angles of 
 45° with the A'"-axis. 
 
 5. Find the equations of tangents to x^ +?/'- = 25 parallel to ?/=3 x+5. 
 
 6. Find the equations of tangents to x^ + y'- = 25 perpendicular to 
 2/ = 3 X + 5. 
 
 7. Find the slopes of tangents to x- + ?/- = 9 through (4, 5). 
 
 8. Find the equations of the tangents to x^ + ?/2 = \Q through (5, 7). 
 
 9. The chord of contact of a pair of tangents to x' + ?/2 = 25 is 
 2 X + 3 // = 5. Find the intersection of the tangents. 
 
 10. Find equation of tangent to (x — a)^ + (y — b)- — f- at (x', y') of 
 circumference. 
 
 Art. 57. — Poles and Polars 
 
 Since it is awkward to speak of the chord of contact or the 
 point of intersection of a pair of imaginary tangents, the point 
 (xi, ?/i) is called the pole of the straight line xx^ + yi/i — i^ with 
 respect to the circle s? -\- y- = r, and xx^ + yy^ — r^ is called the 
 polar of the point (xj, y^). (x^, ?/,), 
 which may be any point of the 
 plane, determines uniquely the 
 line xxi + yyi — r- ; and conversely, 
 xxi -\- yyi = r, which may be any 
 straight line of the plane, deter- 
 mines uniquely the point (iCj, y^. 
 The relation between pole and polar 
 therefore establishes a one-to-one correspondence between the 
 points of the plane and the straight lines of the plane. 
 
 The polar of (.rj, ?/j) with respect to ar-|- ?/-=?- is xxi+yy^=r-, 
 
 the line through the center of the circle and {x^, ?/i) is ?/ = ' '.t". 
 
 Hence the line through the pole and the center is perpen- 
 
PliOrEllTIES OF THE (JIRCLK 
 
 103 
 
 dicular to the polar, and the angle included by lines fiom the 
 center to any two points equals the ant,de included l)y the 
 polars of the two points. 
 
 The distance from the center of the circle .r +//- = r to the 
 
 l)olar of (.1-,, ?/i) is 
 
 that is, the radius is the geomet- 
 
 ric mean between the distances of the center from pole and 
 polar. 
 
 The i)olar of (.).•„ y^), with respect to the circle x- + U' = t", is 
 constructed geometrically by draAving a perpendicular to the 
 line joining (.t„ ?/i) and the center of the circle at the point 
 whose distance from the center is the third proportional to the 
 distance from (a*,, ?/,) to the center and the radius of the circle. 
 The pole of any line, with respect to the circle x- -\- y- = r, is 
 constructed geometrically by laying off from the center on the 
 perpendicular from the center to the line the third propor- 
 tional to distance from center to line and the radius of the 
 circle. 
 
 The polar of (.r„ y{) Avith respect to the circle x- + y- = r- is 
 xxi + yyi = ?•-, the polar of (x.,, y.,) is xx^ + yy-j = r- The condi- 
 tion which causes (a'l,?/,) 
 to lie in the polar of 
 (X2,y2)is x^Xo+yiy2^r^\ 
 this is also the condi- 
 tion which causes the 
 polar of (.r„ //i) to con- 
 tain (x.,, y-_^. Hence 
 the polars of all points 
 in a straight line pass 
 through the pole of the 
 line, and the poles of 
 all lines through a 
 point lie in the polar 
 of that point. 
 
104 ANALYTIC GEOMETRY 
 
 Problems. — 1. Write the equation of the polar of (2, :]) with respect 
 to x~ + y- = 10. 
 
 2. Find the point whose polar with respect to x- + y- = d is S x + 7 y = 
 
 18. 
 
 3. Find distance from center of circle x- + y- = 26 to polar of (3, 4). 
 
 4. Find equation of polar of {x', y') with respect to circle (x - a)'^ 
 + (2/ - '0- = '•-• 
 
 5. Find polar of (0, 0) with respect to x- + y- = r^. 
 
 Art. 58. — Reciprocal Figures 
 
 If a geometric figure is generated by the continuous motion 
 of a point, the polar of the generating point takes consecutive 
 positions enveloping a geometric figure. To every point in the 
 first figure there corresponds a tangent to the second figure ; 
 to points of the first figure in a straight line there correspond 
 tangents to the second figure through a point; to a multiple 
 point of the first figure there corresponds a multiple tangent 
 in the second. If two points of intersection of a secant of 
 the first figure become coincident, in which case the secant 
 becomes a tangent, the pole of the secant at the same time 
 must become the point of intersection of two consecutive tan- 
 gents of the second figure, that is a point of the second figure. 
 Hence the first figure is also the envelope of the polars of the 
 points of the second figure. For this reason these figures are 
 called reciprocal figures. Reciprocation leads to the principle 
 of duality in geometry.* 
 
 Problems. — 1. To find the reciprocal of the circle C with respect to 
 the circle 0, x- + y- = r^. 
 
 * The principle of duality was developed by Poncelet (1822) and Ger- 
 gonne (1817-18) as a consequence of reciprocation, independently of 
 reciprocation by Mobius and Gergonne, 
 
pnoPEirriEs of the circle 
 
 105 
 
 The line nir{xxx + mn = f-) is tlie polar of the center C{xu 2/i) 
 with respect to the circle ; p (Xa, 2/2) is the pole of any tangent 
 
 PT{xx.z + ijih = r^) 
 to the circle C. Then 
 
 OC = (:'•!- + 2/1-)', 
 
 pK 
 
 CP 
 
 X1X2 + yiyo - r- 
 
 X\Xi + 
 
 Op^(x2-+y2~)l 
 Hence 
 
 OC-pK= CP- Op, 
 
 or ^ = ^'. 
 pK CP 
 
 oc 
 
 —^ is constant, and therefore p nmst generate a conic section whose focus 
 
 00 
 
 is O, directrix ////', eccentricity -— . This conic section is an ellipse 
 
 when O is within the circumference of the circle C, a parabola when is 
 on the circumference, an hyperbola when is without the circumference. 
 
 2. Find the reciprocal of a given triangle. 
 
 Call the vertices of the given triangle A{xx, y{), B{xn, 2/2), C(xz, yz)- 
 The polars of these vertices with respect to x^ + y~ = r'^ are 
 
 bc(xxx + 2/2/1 = '•^)' 
 
 ac{xx2 + 2/2/2 = »•-), 
 
 a5(a:a:3 + 2/2/3 = »•')• 
 Triangles such that the ver- 
 tices of the one are the 
 poles of the sides of the 
 other are called conjugate 
 triangles. The conjugate 
 triangle of the triangle cir- 
 cumscribed about a circle 
 with respect to that circle 
 is the triangle formed by 
 joining the points of con- 
 tact. 
 
106 ANALYTIC GEOMETRY 
 
 3. The straight lines joining the corresponding vertices of a pair of 
 conjugate triangles intersect in a common point. 
 
 The equations of the lines through the corresponding vertices are 
 Aa, (rciX3 + 2/12/3 - r'^) {xx^ + yy^ - r~) 
 
 - (a;iX2 + 2/12/2 - r'^){xxz + 2/2/3 - r-) = ; 
 Bh, {xiXo + 2/12/2 - r^){xxz + 2/2/3 - r-) 
 
 - {x^xz + 2/22/3 - r^) {xx^ + 2/^1 - V'-) = ; 
 Cc, {XiXs + yzys - r^) (xxi + 2/2/1 - r^) 
 
 -(X1X3 + 2/12/3 - r'^) {xx2 + 2/2/2 - »■") = 0. 
 The sum of these equations is identically zero, therefore the lines Aa, 
 Bb, Cc, pass through a common point. 
 
 4. Show that if a triangle is circumscribed about a circle the straight 
 lines joining the vertices with the points of contact of the opposite sides 
 pass through a common point. 
 
 5. Reciprocate problem 3. 
 
 The figure formed by the conjugate triangles ABC, ahc is its own 
 reciprocal. The poles of the lines joining the corresponding vertices of 
 ABC and abc are the points of intersection of the corresponding sides 
 of ABC and abc. Hence the reciprocal of problem 3 is, the points of 
 intersection of the corresponding sides of a pair of conjugate triangles lie 
 in a straight line. 
 
 6. Reciprocate problem 4. 
 
 The reciprocal of the circle is a conic section, the reciprocals of the 
 points of contact of the sides of the triangle are tangents of the conic sec- 
 tion, the reciprocals of the vertices of the triangle are the chords of the 
 conic section joining the points of tangency, hence the poles of the lines 
 from the vertices to the points of contact of the opposite sides in the given 
 figure are the points of intersection of the sides of the triangle inscribed 
 in the conic section with the tangents to the conic section at the opposite 
 vertices of the triangle. These three points of intersection must lie in a 
 straight line. 
 
 Art. 59. — Inversion* 
 
 Let P'(.T„ ?/i) be any point in the plane of the circle x- + y-=r", 
 P(x, 11) the intersection of the polar of P', (1) xx^ + ?///i = i~, and 
 
 * The value of inversion in geometric investigation was shown by 
 Pliicker in 1831. The value of inversion in the theory of potential was 
 shown by Lord Kelvin in 1845. 
 
PUOPERTIES OF THE CIRCLE 
 
 107 
 
 the diameter through P', (2) y = -Av. Then OP • OP' = r'-, that 
 
 ^Vheu r becomes 
 
 is (r - PA)(r + P'A)= r-, ^vheiice 
 
 J 1_^1 
 
 PA FA V 
 
 infinite the circle becomes a 
 straight line, PA = P'A, and P 
 and P' become symmetrical points 
 with respect to the line. P is said 
 to be obtained from P' by inver- 
 sion, by the transformation by 
 
 reciprocal radii vectors, or by symmetry with res})ect to the 
 circle. This transformation establishes a one-to-one corre- 
 spondence between the points within the circle and the points 
 without the circle. 
 
 The coordinates of P are obtained in terms of the coordinates 
 of P' by making (1) and (2) simultaneous and solving for x 
 
 and ?/. There results x = — — -'— y, y = -—-—• 
 xl + y,- Xi- -\- y^ 
 
 ilarly. 
 
 t^x 
 
 2/1 
 
 r'y 
 
 X' + y- X- + y- 
 
 If the point (.t„ ?/,) describes a circle 
 
 x,\-\-y:--2aj\-2hy, + c = 0, 
 the inverse point (.r, ?/) traces a curve whose equation is 
 r\-c- 4- r'^v" 2 arx 2 hr>i 
 
 (x- + iff x^ + y- X- + y- 
 which reduces to 
 
 + c = 0, 
 
 o , o sar- 
 
 ^■" + v ;« 
 
 c 
 
 2 hr 
 
 y + 
 
 the erpiation of a, circle. Hence iuvcrsiou Iransrorms llie circle 
 
 (a, b, c) into the circle 
 
 ("alliu''- the radius of tlie 
 
108 
 
 ANALYTIC GEOMETRY 
 
 given circle R, the radius of the transformed circle R', 
 
 R- = ^+^J^. 
 
 That is, 
 
 c. c- 
 
 R = -R. 
 
 When c = 0, R' = cc; that is, the transformed circle becomes 
 a straight line, c = is the condition which causes the center 
 of the inversion circle, which has been taken at the origin of co- 
 ordinates, to lie in the circumference of the given circle (a, 6, c). 
 The inverse of a geometric figure may be constructed mechan- 
 ically by means of an apparatus called Peaucellier's inversor. 
 
 The apparatus consists of six 
 rods, four of equal length h 
 forming a rhombus, and two 
 others of equal length a con- 
 necting diagonally opposite 
 vertices of the rhombus with 
 a lixed point 0. The rods 
 are fastened together by 
 
 Fig. 95. . i. n r 4. 
 
 pins so as to allow perfect 
 freedom of rotation about the pins. If P is made to follow a 
 given curve, P' traces the inverse, the center of inversion being 
 O and the radius of inversion (p? -f If)^. For 
 
 OP=a cos 6 — bcos 6', OP =^acos6 + b cos 6', a sin 0=1 sin 6>'. 
 
 Hence OP • OP' = a- cos- d - Ir cos^ 6', o? sin- 6 -h- sin- 6' = 0, 
 
 and by addition OP • OP = cr - Jr. 
 
 If the point P describes the circumference of a circle passing 
 through 0, P must move in a straight line. Therefore the 
 inversor transforms the circular motion of P about 0', mid- 
 way between and P, as center, into the rectilinear motion 
 of P'. 
 
rUorKllTlKS OF THE CIRCLE 
 
 109 
 
 The cosine of the angle between two circles {a, h, c), (a', h', c') 
 is found from the ecjuation 
 
 (rt _ a'f + {b - b'f = r- + r" - 2 rr' cos 6 
 
 . 2aa' + 2 hh' - c — d 
 
 to be 
 
 2Va^ +7>2 - c V^2 ^ in _ ^ 
 
 The circles obtained by inverting the given circles are 
 Calling their included angle $', 
 
 cos ^' 
 
 2ffa'>-'' 2 ^j?/r^ _ ?;; _ r^ 
 cc' cc' c c' 
 
 which reduces to 
 
 cos 9' = 
 
 2 «a' + 2 6/y - c - c' 
 
 2 Va^ + ?/ _ c V«'' + 6'2 - c' 
 
 Hence the angle between two circles is not altered by inversion. 
 For this reason inversion is called an equiangular or conformal 
 transformation. 
 
 If two orthogonal systems of circles are inverted, taking for 
 center of inversion one of the points of intersection of that 
 
110 ANALYTIC GEOMETRY 
 
 system of circles which has real points of intersection, one of 
 the systems of circles transforms into a system of straight lines 
 through a point. Hence the other system of circles must trans- 
 form into a system of concentric circles whose common center 
 is this point. 
 
CHAPTER IX 
 
 PROPERTIES OF THE CONIO SECTIONS 
 
 Akt. 60. — General, Equation 
 
 A point governed in its motion by the law — the ratio of 
 the distances from the moving point to a fixed point and 
 to a fixed line is constant — generates a conic section. To 
 express this definition by an eqnation between the coordinates 
 of the moving point, let the moving point be (x, y), the fixed 
 point F, the focus (in, n), the fixed line UH'. the directrix 
 a; cos a + y sin « — ^^ — 0. Calling 
 the constant ratio e, the defini- 
 tion is expressed by the equation 
 PF' = e^ • PD^, which becomes 
 
 {m - xf + (n - yf 
 
 = e- (x cos « 4- 2/ sin a — py. 
 
 a is the angle which the axis of 
 the conic section makes with the 
 X-axis, 1^ the distance from the 
 origin to the directrix. 
 
 By assigning to m, n, e, a, p their proper values in any 
 special case, this general eqnation becomes the equation of 
 any conic section in any position whatever in the XF-plane. 
 For example, to obtain the common equation of the ellipse, 
 which is the equation of the ellipse referred to its axes, make 
 
 m = ae, n = 0, a — 0, p = "^, 1 — e^ = ';,• The general equation 
 
 a -, 
 e 
 
112 
 
 A NA L YTIC GEOMETR Y 
 
 becomes (ae — .^•)- -\- y- = {ex — a)-. Expanding and collecting 
 terms, / + (1 — e-) X' = a- (1 — e-), or ^ + ^ = 1. 
 
 To obtain the equation of the hyperbola referred to its axis 
 and the tangent at the left-hand vertex, make m — a(l + e), 
 
 „ = 0, « = 0, p='^Sl+-^, l-e' = --- The general eqna- 
 
 e a- 
 
 tion becomes (a + ae — a-)- + f = (ex - a — ae)-. Expanding 
 and collecting terms, ?/- = (1 — e-) (2 ax — x"), or 
 
 ?/" 
 
 ^4(2rtx--x-). 
 
 Problems. — From the general equation of a conic section referred to 
 rectangular axes, obtain : 
 
 1. The common equation of the hyperbola. 
 
 2. The common equation of the parabola. 
 
 3. The equation of the ellipse referred to its axis and the tangent at 
 the left-hand vertex. 
 
 4. The equation of the ellipse referred to its axis and the tangent at 
 the right-hand vertex. 
 
 5. The equation of the hyperbola referred to its axis and the tangent 
 at the right-hand vertex. 
 
 6. The equation of the parabola referred to its axis and the perpen- 
 dicular to the axis through the focus. 
 
 7. The equation of the ellipse referred to its axis and the perpendicu- 
 lar to the axis through the focus. 
 
 8. The equation of the hyperbola referred to its axis and the directrix. 
 
PUOPEUTIES OF THE CONIC SECTIONS 
 
 113 
 
 9. 'I'hc c'liuation of tlir panibola referred to its axis and the dirrctrix. 
 
 10. Show that in the hyperbolas ^"-'^"=1, ■'/' _ ^ = i tlie traiis- 
 
 a- b'^ b- u- 
 
 verse axis of the first is the conjugate axis of tlie second, and vice versa. 
 Such hyperbolas are called a pair of conjugate hyperbolas. 
 
 11. Derive from the general equation of a conic section the equation 
 
 fc2 
 
 of the hyperbola conjugate to — 
 
 m = 0, n - be, a = 90^ p = '\ 1 - e^ = - ?^ 
 e b- 
 
 12. Show that the straight lines tj =±-x are the conunon asymptotes 
 
 a 
 
 of the pair of conjugate hyperbolas — — ''- = 1, — — •'' = — 1. 
 «- b- a- b- 
 
 13. Find the equation of the ellipse focus (—3,2), eccentricity |, 
 major axis 10, the axis of the ellipse making an angle of 45° with the 
 A'-axis. 
 
 14. Find the equation of the ellipse whose focal distances are 2 and 8, 
 center (5, 7), axes parallel to axes of reference. 
 
 15. Find the equation of the hyperbola whose axes are 10 and 8, cen- 
 ter (3, — 2), axis of curve parallel to X-axis. 
 
 16. Find the equation of the parabola whose parameter is 0, vertex 
 (2, — 3), axis of parabola parallel to A'-axis. 
 
 Art. 61. — Tangents and Nokmals 
 
 Using the common equations of ellipse, hyperbola, and parab- 
 ola, the equations of tangents to these curves at the point 
 
 (.Tu, y/o) of the curve are 
 
 tt- "^ V ' a- lr~ ' 
 yjhi=2){x-\-x^, respectively. 
 The slopes of these tangents 
 
 are for the ellipse 
 
 
 for the hyperbola — ■'-, for 
 
114 
 
 ANALVriC GEOMETRY 
 
 the parabola — Calling the intercepts of the tangent on the 
 
 1ft) 2 7 2 
 
 X-axis X, on the F-axis Y, for the ellipse X = —, Y=—, for 
 the hyperbola X = ^, Y= , for the parabola X — — Xo, 
 
 y= i_?/||. X and y may in each case be determined geometri- 
 cally, and the tangent drawn as indicated in the figure. 
 
 Suppose the point (,<■„ ?/,) to be any point in the plane of the 
 ellipse "-T, + •— = 1. Let {x', y'), (x", y") be the points of con- 
 tact of tangents from (a-j, y{) to the ellipse. Then must (xi, y^) 
 
 ... , „ , ,. xx' ?/?/' . xx" ?/w" ^ ,, , . ,, 
 
 he m each of the hues -^ -f ^4- = 1, -^ + '^ — 1 5 that is, the 
 a- ¥ a- 0- 
 
 XyV ?/,?/ 
 
 equations — 5- + ' ' 
 a- 
 
 b' 
 
 1 ^^ + -M! 
 ' a- b- 
 
 1 must be true. Hence 
 
 the points of contact lie in the line 
 
 +f=i, 
 
 diich 
 
 therefore the chord of contact. Similarly, it is found that the 
 
 X- "■' 
 points of contact from (.r'l, y/j) to the hyperbola -r, - 
 
 1 and 
 
 yih 
 
 to the parabola y' — lpx lie in the lines -—2^ — 72^ = 1 a,nd 
 ?///, = j>(;« -f- x^ respectively. The coordinates of the points of 
 contact of tangents through (.»■„ v/j) to a conic section are found 
 
prxOPEUTIES OF THE CONIC SECTIONS 115 
 
 by making the equations of the chord of contact and of the 
 conic section simultaneous and solving for x and //. 
 
 A theory of poles and polars with respect to any conic sec- 
 tion might be constructed entirely analogous to the theory of 
 poles and polars with respect to the circle. 
 
 The equation y = mx + n, where m is a fixed constant, n a 
 parameter, represents a system of parallel straight lines. For 
 any value of n, the abscissas of the points of intersection of 
 straight line and ellipse %,-\-% = l are found by solving the 
 equation (b' + aha-) x- + 2 a-mnx + a" (a- - b-) = 0. These ab- 
 scissas are equal, and the line ?/ = nix + n becomes a tangent 
 
 to the ellipse -;, + -'^ = l when u' = b'- + a-ni-. Therefore 
 u- b- 
 
 y = mx ± (b- + a-m-y are the two tangents to the ellipse whose 
 
 slope is m. In like manner it is found that the tangents to 
 
 the hyperbola whose slope is m are y = nix ±(a-iii' — b'-)- ; the 
 
 P 
 
 tangent to tlu; parabola whose slope is vi is y — rax + — — 
 
 The equations of the normals to ellipse, hyperbola, and parab- 
 ola at the i)oint (.»•„, ?a,) of the curves are y — 7j,, = -^(x — x^^, 
 y -?/„ = - "/"(.f - .f,), // - //„ = - -''^(x - .Vu) respectively. 
 
 Problems. — 1. Find the eiiuatioiis of taiip:cnts to the ellipse whose 
 axes arc S and at the points wliose distance from the T-axis is 1. 
 
 2. Find the eiiuatioiis of the focal tangents of ellipse, hyperbola, and 
 parabola. 
 
 3. From the point (fi, 8) tangents are drawn to tlu' ellipse ^-|-^=1. 
 F'ind the coordinates of the points of contact and the equations of the 
 tangents. 
 
 4. At what point of the parabola ?/- = 10x is the slope of the tan- 
 gent 1 h ? 
 
 5. On an elliptical track whose major axis is due east and west and 1 
 mile long, minur axis ! mile lonsr, in what direction is a man traveling 
 
116 ANALYTIC GEOMETRY 
 
 when walking from west to east and ] mile west of the north and south 
 line ? 
 
 6. Write the equations of tangents to ^ + ^ = 1 making an angle 45° 
 with the X-axis. 
 
 7. Write the equations of the tangents to — — ^ = 1 perpendicular to 
 2x-32/ = 4. ^ ^ 
 
 8. Write the equation of the tangent to y- = 8x parallel to ^ + -^ = ^• 
 
 9. Find the slopes of the tangents to — + ^- = 1 through the point 
 
 9 4 
 
 (4,5). ?/ = mx + (4 + 9m2)2 is tangent to— + ^= 1. Since (4, 5) is m 
 
 i 9 4 
 
 the tangent, 5 = 4 wi + (4 + 9 m^) 2. Solve for m. 
 
 10. Find the slopes of tangents to ^ - ^ = 1 through (2, 3). 
 
 11. Find the slopes of tangents to y' = Gx through (-5, 4). 
 
 12. Find the points of contact of tangents to y" = {Jx through ( -5, 4). 
 
 13. Find the intercepts of normals to ellipse, hyperbola, and parabola 
 on X-axis. 
 
 14. Find distances from focus to point of intersection of normal with 
 axis for each of the conic sections. 
 
 15. Prove that tangents to ellipse, hyperbola, or parabola at the ex- 
 tremities of chords through a fixed point intersect on a fixed straight line. 
 
 16. Prove that the chords of contact of tangents to a conic section 
 from points in a straight line pass through a common point. 
 
 17. Show that the tangent to the ellipse at any point bisects the angle 
 made by one focal radius to tlie point with the prolongation of the other 
 focal radius to the point. 
 
rUOVERTIKS OF THE CONIC SECTIONS 
 
 117 
 
 The ratio of the focul nulii is "^^ = - — ^- Since 
 PF' a + exo 
 
 AF= AF' = ae and AT- 
 
 Xo F'T~ a 
 
 (« - ex^) 
 
 — (« + ea;o) 
 
 Hence EJL = I1L^ and Pr bisects FPS. 
 F'T PF' 
 
 18. In the hyperbohx the tangent at any point bisects the angle in- 
 cluded by the focal radii to the point. 
 
 19. In the parabola the tangent at any point bisects the angle included 
 by the focal radius to and the diameter through the point.* 
 
 \D' 
 
 Fig. 104. 
 
 On problems 17, 18, 19 is based a simple method of drawing tangents 
 to the conic sections through a given point. With the given point as 
 center and radius equal to distance from given point to one focus strike 
 
 * Since it is true of rays of light, heat, and sound that the reflected ray 
 and the incident ray lie on different sides of the normal and make equal 
 angles with the normal, it follows that rays emitted from one focus of an 
 elliptic reflector are concentrated at the other focus ; that rays emitted 
 from one focus of an hyperbola reflector proceed after reflection as if 
 emitted from the other focus ; that rays emitted from the focus of a 
 parabolic reflector after reflection proceed in parallel lines. 
 
 It is this property of conic sections that suggested the term focus or 
 " burning point." 
 
118 
 
 A NA L YTIC GEOMETR Y 
 
 off an arc. In the parabola the parallels to the axis through the inter- 
 sections of this circle with the directrix determine the points of tan- 
 gency. For TF = TD, hence the triangles TPF, TPD are equal and PT 
 is tangent to the parabola. In ellipse and hyperbola strike off another 
 arc with the second focus as center and radius equal to transverse axis. 
 
 Lines joining the second focus with the points of intersection of the two 
 arcs determine the points of tangency. In the ellipse T'F' + T' F = 2 a, 
 and by construction T'F' + TD' = 2 a, hence T'F - T'D'. The trian- 
 gles T'PF, T'PD' are equal, and PT' is tangent to the ellipse. In 
 the hyperbola TF' - TF = 2 a, TF' - TD = 2a; hence TD = TF, the 
 triangles TPD, TPF are equal, and PT is tangent to the hyperbola. 
 
 20. Show that the locus of the foot of the perpendicular from the focus 
 
 of the ellipse \--l-=zl to the tangent is the circle described on the 
 
 a- b'^ 
 major axis as diameter. 
 
 The equation of the perpendicular from the focus {ae, 0) to the tan- 
 gents y = mx ± (b- + n-m^) •2 is my + x = ae. Make these equations 
 simultaneous and eliminate m by squaring both equations and adding. 
 There results x'^ + y^ = a'^. 
 
 21. Show that the locus of the foot of the perpendicular from the focus 
 
 of the hyperbola ^ - ^- = 1 to the tangent is the circle described on the 
 
 a^ b'^ 
 transverse axis as diameter. 
 
 22. Show that the locus of the foot of the perpendicular from the focus 
 of the parabola y^ = 2pxto the tangent is the F-axis. 
 
PUOI-KliTIKS or THE CONIC SECTIONS 
 
 119 
 
 Problems 20, 21, 22 may be used to construct the cuuic sections as 
 envelopes when the focus and the vertices are known. 
 
 23. Prove that for ellii)se and hyperbola the product of the perpen- 
 diculars from foci to tangent is constant and eipial to h'-. 
 
 24. Prove that in the parabola the locus of the point of intersection of 
 a line through the vertex perpendicular to a tangent with the ordinate 
 through the point of tangency is a semi-cubic parabola. 
 
 AkT. 62. (JONJUGATK DiAMETERS 
 
 Let {xo, ?/„) be the point of intersection of the diameter 
 
 w = tan^ • X with the ellipse — \-^^=l, and call the angle 
 a' Ir 
 
 made by the tangent to the ellipse at (.r,,, ij^^) with the X-axis $\ 
 
 Then 
 
 Y 
 
 tan d = •^, 
 
 tan^' = -^ 
 
 tan 6 tan 6' = - 
 
 Now let (a'l, ?/,) be the point 
 of intersection of 
 
 Fig. 107. 
 
 y = tan 9' - x 
 
 with the ellipse, and call the angle made by the tangent to the 
 ellipse at (a;,, ?/i) with the X-axis 0. Then 
 
 tan 
 
 ?/i 
 
 tan e=^- ^, tan 
 
 tan 6' = - 
 
 Ir 
 
 Hence the condition tan 6 tan & = 
 
 causes each of the 
 
 diameters of the ellipse y = tan • x, y = tan 0' ■ x to be par- 
 allel to the tangent at the extremity of the other. Such 
 diameters are called conjugate diameters of the ellipse. 
 
120 
 
 ANALYTIC GEOMETRY 
 
 The equation of tlie ellipse —-{---^^^l referred to a pair of 
 (r b'- 
 conjugate diameters and in terms of the semi-conjugate diam- 
 eters a' and b'is— + ^=l. (See Art. 35, Prob. 39.) This 
 
 equation shows that each of a pair of conjugate diameters 
 bisects all chords parallel to the other. The axes of the 
 ellipse are a pair of perpendicular conjugate diameters. 
 
 Let (xu, 2/o) be the point of intersection oi y — tan 6 • x with 
 the hyperbola — — ^ = 1, and call the angle made by the tan- 
 gent to the hyperbola at {xq, y^) with the X-axis 6'. Then 
 .Vo 4-„„ flf _ ^^•^•o tan 6* tan 6>' = — . Since y = ~x and 
 
 b 
 
 y — X are 
 
 a 
 
 tan 0^'^, tan 0' 
 x„ 
 
 b%, 
 
 tl 
 
 le common 
 
 asymptotes of the pair of con- 
 jugate hyperbolas 
 
 and — — - 
 
 cr b'' 
 l,it is evident 
 
 that the condition 
 
 tan d tan $' = — 
 
 causes y = tan 0' • x to inter- 
 sect --^=-1 if 
 
 tan I 
 
 intersects —-■£=!. Now suppose (x^, y^ to be the point of 
 
 a- y- 
 
 intersection of the line y = tan 6' • x with the conjugate hyper- 
 
 bola 
 
 1, and call the angle made by the tangent to 
 
 this hyperbola at (.Ti, ?/i) with the X-axis 6. Then tan & 
 
 tan 
 
 b-x, 
 
 tan e tan 6' = 
 
 Diameters of the hyperbola 
 b^ 
 
 satisfying the condition tan 6 tan 0' =-7, are called conjugate 
 diameters of the hyi^erbola. 
 
PliOPEliTIEtS OF THE CONIC SECTIONS 
 
 121 
 
 The ecjuation of the hyperbola '- 
 of conjugate diameters 
 diameters a' and b', is -^. — r^=l. (See Art 
 
 cr Ir 
 and in terms of tl 
 
 referred to a pair 
 
 le semi-conjngate 
 
 Prob. 38.) 
 
 This equation shows that chords of an hyperbola parallel to any 
 diameter are bisected by the conjugate diameter. The axes of 
 the hyperbola are perpendicular conjugate diameters. 
 
 The equation of the parabola referred to a diameter, and a 
 tangent at the extremity of 
 the diameter, is y^ — 2piX. 
 (See Art. 35, Prob. 40.) This 
 equation shows that any diam- 
 eter of the parabola bisects 
 all chords parallel to the tan- 
 gent at the extremity of the 
 diameter. The axis of the 
 parabola is that diameter 
 which bisects the system of 
 parallel chords at right angles. 
 
 It is now possible to deter- 
 mine geometrically the axes, focus, and directrix of a conic 
 section when the curve only is given. In the case of the 
 ellipse draw any pair of parallel chords. Their bisector is a 
 diameter of the ellipse. 
 With the center of the 
 ellipse as center strike 
 off a circle intersecting 
 the ellipse in four points. 
 The bisectors of the two 
 pairs of parallel chords 
 joining the points of in- 
 tersection are the axes 
 
 of the ellipse. An arc struck off with extremity of minor 
 axis as center, and radius equal to semi-major axis, inter- 
 
122 
 
 ANALYTIC GEOMETRY 
 
 sects the major axis in the foci. The directrix is perpendicular 
 to the line of foci where the focal tangents cross this line. 
 
 In the case of the hyperbola the directions of the axes are 
 found as for the ellipse. The focus is determined by drawing 
 a perpendicular to any tangent at the point of intersection of 
 this tangent with the circumference on the transverse axis. 
 Drawing the focal tangents determines the directrix. The 
 conjugate axis is limited by the arc struck off with vertex as 
 center and radius equal to distance from focus to center. 
 
 In the case of the parabola, 
 after determining a diameter 
 by bisecting any pair of paral- 
 lel chords, and the axis by 
 bisecting a pair of chords per- 
 pendicular to the diameter, 
 the focus is determined by 
 the property that the tangent 
 bisects the angle included by 
 diameter and focal radius to 
 point of tangency. 
 
 Art. 63. — Supplementary Chords 
 
 Chords from any point of ellipse or hyperbola-to the extrem- 
 ities of the transverse axis are called supplementary. Let 
 
 (x', y') be any point of the 
 
 The 
 
 ellipse ^„ + j-^ 
 
 equations of lines through 
 
 (a;', ?/'), (a, 0) and (.-»', ?/'), 
 ( — a, 0) are 
 
 Fig. 112. 
 
 •/ {X - a), 
 
 x' — a 
 
 , y' 
 
PliOPKRTlES OF THE CONIC SECTIONS 
 
 123 
 
 Calling the angles made l)y tlie supplementary eliords with 
 
 the X-axis </> and <^', tan <^ tan 4>' — 
 
 V 
 
 From the equa- 
 
 x'-). Hence tan </> tan (/>' = ;. 
 
 d- 
 
 — = 1 , tan <f) tan </> = — . 
 
 a- Ir a' 
 
 V are a pair of conjugate diam- 
 
 1 when tan ■ tan $' = — —. Hence 
 d- 
 
 tion of the ellipse, //'- = —(a' 
 
 In like manner for the hyperbola 
 y = tan 6 • x and ?/ = tan 6 
 
 eters of the ellipse '—4--^-- 
 d- h- 
 tan 6 • tan 6' = tan 4> • tan </>', 
 from which it follows 
 that if one of a pair of 
 supplementary chords is 
 parallel to a diameter the 
 other chord is parallel to 
 the conjugate diameter. 
 This proposition is dem- 
 onstrated for the hyper- ' 
 bola in the same manner. 
 
 On this proposition are based simple methods of drawing 
 tangents to ellipse or hyperbola, either through a point of the 
 curve or parallel to a given line. To draw a tangent to the 
 ellipse at any point P, dra^v a 
 diameter through P, a sup- 
 plementary chord parallel to 
 this diameter, and the line 
 through /'parallel to the other 
 supplementary chord is the 
 tangent. 
 
 To draw a tangent to the 
 hyperbola parallel to a given 
 straight line, draw one sup- 
 plementary chord parallel to the given line, and the diameter 
 parallel to the other supplementary chord determines the points 
 of tangency. 
 
124 ANALYTIC GEOMETEY 
 
 To draw a pair of conjugate diameters. of an ellipse, includ- 
 ing a given angle, construct on 
 ^^^^^^ /^^^ \ ^^^ major axis of the ellipse a 
 
 /^^ ^--V nh circular segment containing the 
 
 — x ^""^^^ / "^ ~^ ^ -J ^ given angle. From the point of 
 
 / V /' - -^^^^T"^-----/! intersection of the arc of the seg- 
 
 1 ^v^y^ 1^ ^^ ]\f) ment and the ellipse draw a pair 
 
 I "^ / \ of supplementary chords. The 
 
 PiQ 115 diameters parallel to these chords 
 
 are the required diameters. 
 
 Art. 64. — Parameters 
 Since — + ^ = 1 is the equation of an ellipse referred to any 
 
 pair of conjugate diameters, it is readily shown that the 
 squares of ordinates to any diameter of the ellipse are in the 
 ratio of the rectangles of the segments into which these ordi- 
 nates divide the diameter. The same proposition is true of 
 the hyperbola. 
 
 Taking the pair of perpendicular conjugate diameters of tlie 
 ellipse as reference axes and the points (cte, 'p), (0, IS), the 
 proposition leads to the proportion ^ = — ^^ — v^^^y whence 
 
 -^ — — - that is, the i:)arameter to the transverse axis of the 
 2& 2a' ' _ 
 
 ellipse is a fourth proportional to the transverse and conjugate 
 axes. Generalizing this result, the parameter to any diaineter 
 of ellipse or hyperbola is the fourth proportional to that diame- 
 ter and its conjxigate. 
 
 In the common equation of the parabola, y''- = 2i')X, the 
 parameter 2p is the fourth proportional to any abscissa and 
 its corresponding ordinate. Generalizing this definition, the 
 parameter to any diameter of the parabola is the fourth propor- 
 tional to any abscissa and its corresponding ordinate with 
 respect to this diameter. 
 
iniOPEIiTIES OF THE CONIC SECTIONS 
 
 125 
 
 ^Vhen (in, v) on the parabola y" = 2px- is taken as origin, the 
 diameter througli {m, n) as X-axis, the tangent at (?//., n) as 
 l''-axis, the e(i[uation of the pa- 
 rabola takes the form Y 
 
 yf = -J2h^i- 
 
 (See Art. 35, I'rob. 40.) 
 
 ^' sure 
 
 Hence 22h = 4(?ji + J- jj); that 
 is, the parameter to any di- 
 ameter of a parabola is four 
 
 times the focal radius of the vertex of that diameter. Calling 
 the focal radius /, the equation of the parabola becomes 
 
 r_ 
 
 1, find the equation of the 
 
 Problems. — 1. In the ellipse 
 rlianieter conjugate to y = x. 
 
 2. Find the angle between the supplementary chords of the ellipse 
 
 '■!.' ^111=1 at the extremity of the minor axis. 
 n- //- 
 
 3. Find the point of the ellipse ^ + ^'-1 at which supplementary 
 chords include an angle of 45°. 
 
 4. Show that the maximum angle between a pair of supplementary 
 x' , if , ..„ ..„_, 2 ah 
 
 h 
 
 chords of the ellipse 
 
 1 IS tan-' 
 
 «•' o~ ¥ — a- 
 
 5. Show that a pair of conjugate diameters of an hyperbola cannot 
 include an angle greater than 90°. 
 
 6. Construct the ellipse whose equation referred to a pair of conjugate 
 
 -f^ 
 
 Find focus and dircc- 
 
 diameters including an angle of 45° 
 trix of this ellipse. 
 
 7. Find the equation of the hyperbola whose axes arc 8 and 6 re- 
 ferred to a pair of conjugate diameters, of which one makes an angle of 
 45° with the axis of the hyperbola. Find lengths of the semi-conjugate 
 diameters. 
 
126 
 
 ANALYTIC GEOMETRY 
 
 8. Find equation of parabola whose parameter is 
 ter through (8, 8) and tangent at this point. 
 
 Find the locus of the centers of chords of 
 
 2x 
 
 referred to diame- 
 
 ^ = 1 parallel to 
 4 
 
 is y^=- 
 
 a parabola referred to 
 
 10. The equation of a pai-abola referred to oblique axes including an 
 angle of 60° isy- = lOx. Sketch the parabola and construct its focus and 
 directrix. 
 
 11. A body is projected from A in the direction AY with initial 
 velocity of v feet per second. Gravity is the only disturbing force. Find 
 the path of the body and its velocity at any instant. 
 
 Taking the line of projection as F-axis and the vertical through A as 
 
 X-axis, the coordinates of the body t seconds after projection are 
 
 x = I gfi, y = vt; the equation of the 
 
 path of the body, found by eliminating t, 
 
 (J 
 
 tangent and diameter through point of 
 tangency. Comparing this e<iuation with 
 2/2 = 4 /x, the equation of parabola re- 
 ferred to tangent and diameter, v'^=2 (jf ; 
 that is, the initial velocity is the velocity 
 acquired by a body falling freely from 
 the directrix of the parabola to the start- 
 ing point. 
 
 If the body is projected from any 
 point of the parabola along the tangent 
 to the parabola at that point, and with a velocity equal to the velocity of 
 the body projected from A wlien it reaches that point, the path of the 
 body is the path of the body projected from A. Hence it follows that 
 the velocity of the body at any point of the parabola is the velocity 
 acquired by a body freely falling from the directrix of the parabola to 
 that point. 
 
 Art. 65. — The Elliptic Compass 
 
 Let i^ -f -^ = 1 aud 3l + -^^ = 1 1)0, two ellipses ccmstructed 
 on the same major diameter.- Let ?/, and ?/. be ordinates cor- 
 respondintr to the same abscissa, then ■— = ^^; that is, if ellipses 
 
riiOPEIiTIES OF THE CONIC SECTIONS 
 
 127 
 
 are constructed on the same major diameter, corresponding 
 ordinates are to each other as the minor diameters. The circle 
 described on the major diameter of the ellipse is a variety of 
 the ellipse, hence the ordinate 
 of an ellipse is to the corre- 
 sponding ordinate of the cir- 
 cumscribed circle as the minor 
 diameter of the ellipse is to 
 the major diameter. 
 
 On this principle is based 
 a convenient instrument for 
 drawing an ellipse whose axes 
 are given. On a rigid bar 
 take PH=a, PK^h. Fix 
 pins at H and K which slide in grooves in the rulers X and "J 
 
 perpendicular to each other 
 Fo 
 
 
 
 lY P' 
 
 . 
 
 ^.''- 
 
 ' — 
 
 ^< 
 
 ^ ' 
 
 
 
 
 1 \ 
 
 
 
 
 1 \ 
 
 / 
 
 
 
 
 / 
 
 
 / / 
 
 1 \^ 
 
 x\ 
 
 
 ^/ ^ 
 
 D i^ 
 
 
 
 
 1 
 
 ■"^^^ 
 
 / 
 
 H ^y 
 
 
 ^~~^ 
 
 Y' 
 
 
 PI) PII 
 
 compass. 
 
 P traces the ellipse ^ -f ^ = 1 . 
 This instrument is called the elliptic 
 
 Art. 66. — Area of the Ellipse 
 
 Erect any number of perpendiculars to the major diameter 
 of the ellipse, and beginniug at the right draw through the 
 points of intersection of these 
 perpendiculars with the ellipse 
 and the circumscribed circle 
 parallels to the minor diam- 
 eter. There is thus inscribed 
 in the ellipse and in the circle 
 a series of rectangles. The 
 corresponding rectangles in 
 ellipse and circle have the 
 same base, and their altitudes 
 are in the ratio of h to a. 
 
128 
 
 A NAL YTIC GEOMETR Y 
 
 Hence the sum of the areas of the rectangles inscribed in 
 the ellipse bears to the sum of the rectangles inscribed in the 
 circle the ratio of h to a. By indefinitely increasing the 
 number of rectangles, the sum of the areas of the rectangles 
 inscribed in the ellipse approaches the area of the ellipse as its 
 limit, and at the same time the sum of the areas of the rec- 
 tangles inscribed in the circle approaches the area of the circle 
 
 as its limit. At the limit therefore ^^-^ r-^^ = -, hence 
 
 , area of circle a 
 
 area of ellipse = - • 7ra^ = irah. 
 a 
 
 Art. 67. — Eccentric Angle op Ellipse 
 
 At any point {x, y) of the ellipse ^, + 4, ■ 
 
 1 produce the 
 
 ordinate to the transverse axis to meet the circumscribed circle 
 and draw the radius of this circle to the point of meeting. 
 
 The angle <^ made by this 
 radius with the transverse 
 axis of the ellipse is called 
 the eccentric angle of the 
 point (.r, ?/). Erom the figure 
 x—.a • cos <)!>, 
 
 y = -. 1\D=^ - ■ asinc^ 
 (( a 
 
 = h • sin <^. 
 The coordinates of any point 
 {x, ?/) of the ellipse are thus 
 expressed in terms of the 
 single variable ^. 
 Let AP^ and AP^ be a pair of conjugate diameters of the 
 1, 6 and d' the angles these diameters make 
 
 ^'. Let 
 
 ellipse I + |: 
 
 Avith the axis of the ellipse. Then tan 6 tan 6' 
 
I'llOPKUTIES OF THE CONIC SECTIONS 
 
 129 
 
 (.I'l, ?/i) be the coordinates, ^i the eccentric angle of J\; {.i:,,y.,) 
 tlie coordinates, <f>.2 the eccentric angle of F2. Then 
 
 tan e 
 
 _?/, _ h sin 
 
 
 •i'l 
 
 a cos ^1 
 
 tai 
 
 X., 
 
 _ 6 sin <^o 
 
 <<- cos </)j 
 
 tan e tan 0' 
 
 
 
 h- sin 
 
 <^i sin <)!)2 
 
 
 a- cos 
 
 </>! cos ^2 
 
 
 
 </>itan^2 
 
 
 - ^'l 
 
 
 
 (r 
 
 
 Hence tan c^, tan <^, = - 1 and c/,, and <^. differ l.y 00°; that is, 
 the eccentric angles of the extremities of a pair of conjngate 
 diameters of the ellipse differ by 90°. 
 
 Call the lengths of the semi-conjugate diameters a, and />,. 
 
 Then a{ = .c,- + v/f = or cos'' <^i + Ir sin- ^i, 
 
 h{ = a- cos- (f)-. + b'' sin^ (f>2 = «^ sin^ ^1 + h- cos^ ^1, 
 
 since cj^. = 90° + <^i. T.y addition ac + b^' = (r + b-; that is, 
 the sum of the squares of any pair of conjugate diameters of 
 the ellipse equals the sum of the squares of the axes. 
 The conjugate diameters are of equal length when 
 
 a- cos- <i> -\-b- sin- cji = a- sin' <f) + h- cos^ cf> ; 
 
 that is, when 
 
 tan- <^ = 1, tan </> = ± 1, <^ = 45° or 13")°. 
 
 The cipiationsof the cc^ual conjugate diameters are y = ±~x, 
 
 and their length Vw(u' + b'-). 
 
 The area of the parallelogram circumscribed about the 
 
 K 
 
130 
 
 .1 NAL VTIC G EOMETll Y 
 
 ellipse with its sides parallel to a pair of conjugate diameters 
 
 is 4 6' • AN. The equation 
 of the tangent to the el- 
 lipse at (x', y') is 
 
 xx' ?///' _ ^ 
 
 The point (x', y') is the 
 same as (a cos ^j, h sin <^i), 
 and the tangent may be 
 written 
 
 X cos <^i y sin c^j 
 
 1. 
 
 The length of the perpendicular from the origin to this 
 tangent is AJS — 
 
 / cos- </>! sin-</)| \ - ^1 
 1^ a? ^ U^ ) 
 
 Hence 4 6' • AN= 4a6; that is, the area of the circumscribed 
 parallelogram equals the area of the rectangle on the axes. 
 
 Art. 68. — Eccentric Angle of the Hyperbola 
 
 On the transverse axis of the hyperbola describe a circle. 
 Through the foot of the ordinate of any point (x, y) of the 
 hyperbola draw a tangent to this 
 circle ; the angle made by the 
 radius to the point of tangency 
 and the axis of the hyperbola is 
 called the eccentric angle of the 
 point {x, y). From the hgure 
 x = a • sec ^ and, since 
 
 7.2 
 
 y- — _^ (a- — X-), y = h • tan 4>. 
 
 Let Al\ and AP-. be a pair of conjugate diameters of the 
 
I'llOl'EliTlES OF THE CONIC SECTIONS 
 
 131 
 
 hyperlxtla 
 
 1; (.i\,i/i) the coordinates, c/), the eeeeutric 
 
 angle of the point I\ ; 6 and 
 0' the angles included by the 
 conjugate diameters and the 
 axes of the hyperbola. Then 
 
 b tan <^, 
 
 tan 
 
 .Tj (( sec </>! 
 
 Since tan 6 tan $' = , 
 tane'= r 
 
 a sm (pi 
 
 Hence the equations of the 
 conjugate diameters are y = ^ 
 
 point of intersection of ?/ = 
 
 x,y = 
 
 a sm 
 
 X- 
 
 dtl 
 
 Po, th 
 -1, i 
 
 a sin ^1 cr 0- 
 
 (((tan^i, 6secc^i). Therefore APi — Ui' = a- sec"^ cl> + b- tmr 4>, 
 AFi = bi^ = a^ tan^ <^i + b^ sec- <^i. By subtraction a^- — bc 
 = a^ — b-; that is, the difference between the squares of any 
 pair of conjugate diameters of the hyperbola equals the differ- 
 ence of the squares of the axes. 
 
 The area of the parallelogram whose sides are tangents to a 
 pair of conjugate hyperbolas at the extremities of a pair of con- 
 jugate diameters is Aby AN. The equation of the tangent to 
 
 x^ y- i , , , 7 i. , \ • sec <f>. tan Aj , ^ mv,r> 
 
 — — ^=1 at (a sec cb^, b tan <ii) is ^.r --^y = 1. ihe 
 
 a^ b'^ a b 
 
 perpendicular from tlie origin to this tangent is 
 
 AN: 
 
 + 
 
 tan- </>, 
 
 •a- b'- 
 
 Hence the area of the parallelogram equals iab; that is, the 
 area of the rectangle on the axes. 
 
sec<^..^. 
 a 
 
 tan (^, 
 6 '' 
 
 - sec </,, 
 
 ..+-^.. 
 
 a 
 
 tanc^,_^ 
 a 
 
 sec <^i _ 
 b -^ 
 
 — tan <^i 
 a 
 
 ■x + '-^^^-^.y 
 
 132 ANALYTIC GEOMETRY 
 
 The equations of the sides of the parallelogram are 
 
 1, (1) 
 
 1, (2) 
 
 1, (3) 
 
 - 1. (4) 
 
 Making these equations simultaneous and combining by addi- 
 tion or subtraction, it is found that the vertices of the parallelo- 
 gram lie in the asymptotes y = ± -x. 
 
 Problems. — 1. Find the area of the ellipse whose axes are 8 and 6. 
 
 2. What are the eccentric angles of the vertices of the ellipse ? of the 
 ends of the focal ordinate to the transverse axis ? 
 
 3. The extremity of a diameter of the ellipse — -|-f-= 1 is (xi, yi), 
 
 a-' 0^ 
 
 the extremity of the conjugate diameter (x2, 2/2)- Find X2 and 2/2 in terms 
 of Xi and yi. 
 
 4. Solve the same problem for the hyperbola. 
 
 5. In the hyperbola whose axes are 10 and the length of a diameter is 
 15. Find the length of the conjugate diameter. 
 
 6. Find the lengths of the equal conjugate diameters of the ellipse 
 whose axes are 12 and 8. Also the equation of this ellipse referred to its 
 equal conjugate diameters. 
 
CHAPTER X 
 
 SECOND DEGKEE EQUATION 
 
 AuT. 69. — Locus OF Second Deouee Equation 
 
 Write the general second degree equation in two variables 
 in the form 
 
 (ur + 2 bx!f + qf + 2 dx + 2 ey + / = 0. (1) 
 
 The problem is to determine the geometric iigure represented 
 by this equation when interpreted with respect to the rectangu- 
 lar axes X, Y. The equation of this geometric figure when 
 referred to axes Xj, \\, parallel to X, Y, with origin at (a-o, y^, 
 becomes 
 aa;/ + 2 bx,y, + cy,- + 2 {ax, + hy^ + d)x, + 2 {bx, + ry, + e)y, 
 + ax,' + 2 bx„yo + cy,' + 2 dx, + 2 cy, +f= 0. (2) 
 
 The geometric figure is symmetrical with respect to the new 
 origin (a^o, y^ if the coefficients of the terms in the first powers 
 of the variables in equation (2) are zero. The coordinates of 
 the center of symmetry of the figure are therefore determined 
 by the equations ax, + by, + fZ = 0, bx, + c?/o + e = 0. Whence 
 ^ eb - cd^ ^ db - ae rj.^^ center is a determinate finite 
 ac — b- ac — ¥ 
 
 point only when ac — b' ^ 0. 
 
 Suppose ac — b^ =^ 0. The absolute term of ecpiation (2) be- 
 comes 
 
 ax,- + 2 6a-ov/o + c?/o' + 2 dx, -\-2eyo+f 
 
 = Xo(axo + by, + (Z) + 2/o (c?/o + ^.I'o + <')+ dx, + ey, + f 
 
 , , , ^ acf+2bde-ae--cd'-fb- 
 
 = dx, + c!/o + / = -"^^^^ TV, 
 
 ac — b' 
 
 133 
 
134 ANALYTIC GEOMETRY 
 
 Writing the last expression , equation (2) becomes 
 
 ac — b- 
 
 axf + 2 bx,y, + c?/f + —^^ = 0, (3) 
 
 ac — c>- 
 
 or axj- + 2 6a;i?/i + ci/f + A; = 0, (4) 
 
 where k — dx,^ + e^y,, + ./". 
 
 If A = 0, equation (3) becomes 
 
 ax{' + 2 bx,ii, + ci/f = 0, (5) 
 
 which determines two values real or imaginary for •— ; that is, 
 
 the equation resolves into two linear equations, and hence 
 represents two straight lines. An e(|uation which resolves into 
 lower degree equations is called reducible, and the function of 
 the coefficients, A, whose vanishing makes this resolution pos- 
 sible, is called the discriminant of the equation. 
 
 Turn the axes Xj, Yi about the origin (xq, ?/(,) through an 
 angle 6. Equation (4) becomes 
 
 (a cos^ e + c sin- ^ + 2 & sin ^ cos 6)x.f 
 
 + (a sin- 6 -j-c cos^ 6 — 2b sin 6 cos 6)yi 
 + 2 { (c - a) sin ^ cos ^ + 5(cos- 6 - sin- 0) I x.fli. + fc = 0. 
 Determine 6 by equating to zero the coefficient of x^^^ 
 
 whence tan 2 (9 = '^ ^ . Writing the res.ulting equation 
 
 a — c 
 3Ixi + Ny^^ + k = Q, it follows that 
 
 M+ N= a + c, il/- JV=(a - c)cos(2 ^)+ 2 6 sin(2^). 
 From tan (2^)=-^, sin (2^) = — -, 
 
 cos (2 6) = '^ — ^ -• 
 
 \^b'+{a-c)X' 
 
 Therefore, M-\-N=^a + c, 3/- iV^= ^6' +(« - c)-Ss and 
 MN= ac — b-. Now the equation 3fx.f + Ny-r + A' = repre- 
 sents an ellipse referred to its axes when M and N have like 
 signs, an hyperbola referred to its axes when 31 and N have 
 
SECOND DEGREE EQUATION 135 
 
 unlike signs. Hence the second degree equation represents an 
 ellipse when ac — b'^> 0, an hyperbola when ac — b^ < 0. 
 
 tan (2^)= ^ determines two values for 2 $, and the radi- 
 a — c 
 
 cal \-ih- +(a — c)-J ^ has the double sign. To resolve the ambi- 
 guity take 2 6 less than 180°, which makes sin (2 6) positive, and 
 requires that the sign of the radical be the same as the sign of b. 
 When a — c and the radical have the same sign, cos (2 9) is posi- 
 tive and 2 6 is less than 90° ; when a — c and the radical have 
 different signs, cos (2^) is negative and 2 6 is greater than 90°. 
 The ambiguity may be resolved and the squares of the semi- 
 axes calculated in this manner. The equation tan (2 6*) = -^^ — -, 
 
 ■written "^ = '^ , determines two values for tan 0. 
 
 1 — tan- a — c 
 Call these values tan 6i and tan Oo, and let 0^ locate the Xa-axis, 
 $2 the Fg-axis. In the equation axi^ + 2 bxiy^ + cj/f +k = 0, 
 substitute Xj = r cos 0, ?/i = r sin 0, and solve for ?-l There re- 
 sults r' = -1c 'i- + t&^^'0 Calling the values of r 
 
 a + 2 & tan 6^ -f c tan- 
 corresponding to tan Oi and tan 6.^ respectively r^^ and ?•2^ the 
 equation of the ellipse or hyperbola referred to the axes X2, Y^, 
 
 IS ^,-|---^;=l. 
 
 rr ?2- 
 
 When oc — b- = 0, the general ecjuation becomes 
 
 ax- -f 2 a)(^xy -f cf + 2dx + 2 e>j + / = 0, 
 which may be written {a^x+chjf+2 dx-\-2 e//+f=(). Trans- 
 form to rectangular axes with a-.c + c-y = for X-axis, the 
 origin unchanged. Then 
 
 tan 6=~— and sin 6 = ~ ^^' ^ , cos 6 = — ^— ^• 
 c' (a + cY (a + c)-^ 
 
 The transformation formulas become 
 
 (a + c)'^ (a + c)^ 
 
136 ANALYTIC GEOMETllY 
 
 The transformed equation is 
 
 2,0 ft-^ + c^e c) ft-e - ckl _ f 
 
 Vi + -^ T y^ — ' r ^1 I' 
 
 {a+cy {a + cY {a + cy 
 
 which may be written in the form 
 
 0/.-„y = 2 "'''~''y (.r,-m), 
 the equation of a paraboLa whose parameter is 2 — ^, and 
 
 (« + cy 
 
 whose vertex referred to the axes Xj, Y^, is {m, n). 
 
 The condition cfc — lr = causes the center (a'o, ?/o) of the 
 conic section to go to infinity. Hence the parabola may be 
 regarded as an ellipse or hyperbola with center at infinity. 
 When the discriminant A also equals zero, the parabola be- 
 comes two straight lines intersecting at infinity ; that is, two 
 parallel straight lines. 
 
 It is now seen that every second degree equation in two 
 variables interpreted in rectangular coordinates represents 
 some variety of conic section.* 
 
 Problems. — Determine the variety, magnitude, and position of the 
 conic sections represented by the following equations : 
 
 1. 14 x2 - 4 xy + 11 ?/2 - 44 X - 58 2/ + 71 = 0. 
 
 ac — h'=-\- 150, therefore the equation represents an ellipse. The 
 center is determined by the equations 
 
 14 xo - 2 2/0 - 22 = 0, - 2 xo + 11 yo - 29 = 0, 
 
 * The three varieties of curves of the second order are plane sections of 
 a right circular cone, which is for this reason called a cone of the second 
 order. When the conic section becomes two parallel straight lines, the 
 cone becomes a cylinder. 
 
 Newton (1642-1727) discovered that the curves of the third order arc 
 plane sections of five cones which have for bases the curves 21-25 on 
 page 44. Pliicker (1801-18G8) showed that curves of the third order 
 have 219 varieties. 
 
SECOND DEGUEE EQUATIOy 
 
 137 
 
 to be the point (2, 3). k = dxo + ei/o +/ 
 mined in direction by tan (2 0) = — j, 
 wlience 2 tan"- ^ - 3 tan 6-2 = 0, tan 
 = 2 or - J. M+N= 25, 3IN = 150. 
 If the X-axis corresponds to tan 6 = 2, 
 M — Xniust have the same sign as b. 
 Tiicrefore 
 
 M _ ,Y = _ 5, .1/ = 10, .V = 15. 
 The equation of tlie ellipse 
 
 10 a-2- -h 15 2/2- = 60, 
 
 The axes are deter- 
 
 2/2- 
 
 FiG. 125. 
 
 2. x:^ - 3 xij + 2/- + 10 X - 10 2/ + 21 = 0. 
 
 ac — b" = ~:l, therefore the equation represents an hyperbola. The 
 center, determined by the equations Xo — % ijo + 5 = 0, — ^ xo + 2/0 — 5=0, 
 is (— 2, 2). A; = fZ.ro + eyo+f=+ \. The axes are determined in direc- 
 tion by tan (2 0) =00, whence 0i=45°, 62 -IS^''- By substituting in 
 
 r^- = -k "^tJ^^ , ,,. = 2, r^ = -l 
 
 rt + 2 ?* tan + c tan- 
 
 The equation of the hyperbola referred to its own axes is I x- — ly- = I. 
 
 Fm. 127 
 
 3. 9 a:2 - 21 xij + 16 2/2 - 18 x - 101 y + 19 = 0. 
 
 rtc — 62 — 0, therefore the equation represents a parabola. Write the 
 equation in the form (3x-42/)2-18x-101 2/-M9=0. Take 3.x-42/ = 
 as X-axis of a rectangular system of coordinates, the origin unchanged. 
 Then tan = ?, sin0=i?, cos0 = v;, and the transformation formulas 
 
138 ANALYTIC GEOMETRY 
 
 become x = '^^^~^^S y = §_^lJiAll. The transformed equation is 
 
 5 5 
 
 25?/i2-75a-i-70?/o+19 = 0, wliicli may be written (?/i-|)2=3 (xi + g). 
 Hence the parameter of the parabola is 3, the vertex referred to the new 
 axes ( — I , I) . 
 
 4. 2/2 + 2 a;y + 3 x2 - 4 X = 0. 5. y^ + 2 xy - 3 oc^ - 4 x - 0. 
 
 6. ?/ -2xy + x^ + x = 0. 7. y^ -2 xij + 2 = 0. 
 
 8. ?/ + 4 x?/ + 4 x2 - 4 = 0. 9. 3 x2 + 2 xy + 3y'^= 8. 
 
 10. 4 x2 - 4 x?/ + 2/2 - 12 X + 6 ?/ + 9 = 0. 
 
 11. x^ — xy — 6 2/2 = 6. 
 
 12. x^ + xy + y-^ + x + y = 1. 
 
 13. 3 x2 + 4 xy + 2/2 - 3 X - 2 ?/ + 21 = 0. 
 
 14. 5 x2 + 4 X2/ + 2/"^ — 5 X — 3 2/ — 19 = 0. 
 
 15. 4 x2 + 4 X2/ + 2/- - 5 X - 2 2/ - 10 = 0. 
 
 Art. 70. — Second Degree Equation in Oblique 
 Coordinates 
 
 To determine the locus represented by 
 
 ax' + 2 hxy + c/ + 2 dx + 2 c?/ + /= 0, (1) 
 
 when interpreted in oblique axes including an angle ft, let 
 
 a'x" + 2 6 'x'y' + cY' + 2 rt'-^'' + 2e'y'+f' = (2) 
 
 be the result obtained by transforming the given equation to 
 rectangular axes, the origin unchanged. Since (x, y) repre- 
 sents any point P referred to the oblique axes,- and {x\ y') the 
 same point referred to rectangular axes, the expressions 
 
 ^ + ?/ + 2 xy cos /3 and x'- + y'^ 
 are each the square of the distance from P to the origin. 
 
 Hence x^ + ?/- + 2 xy cos ^ = x'^ + 7j'\ (3) 
 
 By hypothesis 
 
 ax' + 2 &.^7/ + c//- = «'•'«'- + 2 6'.);'.v' + c^/'-. (4) 
 
 Multiply the identity (3) by X and add the product to (4). 
 There results the identity 
 
 (a + X)x^ + 2(b + \ cos /3) xy + (c + A) ?/^ 
 
 = (a' + X) x'2 + 2 b'x'y' + (c' + X) y". 
 
SECOND DEGREE EQUATION 130 
 
 Now any value of A wliicli makes the left-hand member of 
 this identity a perfect square must also make tlie ri,L,^ht-han(l 
 memlier a perfect square. Tlic left-hand mcialjor is a perfect 
 
 , n> + X cos BV <■ + X. 
 
 S(iuare when / — ^ = — — , 
 
 \ a+X J a+X 
 
 , , , • , . .> , a +c — 2 6 cos 8 ^ , ac — b'- ^ 
 
 tliat IS, when X- -\ — — :-— ^^ X -\ ; = 0. 
 
 sin^ /3 sin- /3 
 
 The riL,dit-hand nieniher is a perfect square when 
 
 A- + 0-t' + b')X + a'c' - b'- = 0. 
 
 Since these equations determine the same values for X, 
 
 a'e'-b'-' = '-^^^^. 
 sin- /5 
 
 Therefore ac — b^ is greater than zero when a'c' — b'- is greater 
 than zero. When a'c' — b''^ > 0, equation (2) represents an 
 ellipse when interpreted in rectangular coordinates. Conse- 
 quently when ac — &- > equation (1) represents an ellipse 
 when interpreted in oblique coordinates. In like manner it 
 follows that equation (1) interpreted in oblique coordinates 
 represents an hyperbola when ac — Z^- < 0, a parabola when 
 ac - 62 = 0. 
 
 Problems. — 1. Two vertices of a trianc;le move along two intersecting 
 straight lines. Find the curve traced by the third vertex. 
 From the figure are obtained the pro- 
 
 portionsof y = ^'^Cg + °), /y 
 
 b sin oj 
 
 X _ sin (e + CO - p) 
 a sin w 
 
 whence 
 
140 
 
 ANALYTIC GEOMETRY 
 
 Substituting in sin^ d + cos- ^ = 1, there results 
 
 af _ 2 sin (a - ;3 + co) _^ tf ^ sin'-^ (a + ^ - w) ^ 
 a'^ " ah Ir sin- w 
 
 tlie equation of an ellipse. 
 
 2. Find the envelope of a straight line which moves in such a manner 
 that the sum of its intercepts on two 
 intersecting straight lines is constant. 
 
 Let - + - = 1 be the moving straight 
 a b 
 line, then must a -i- b = c, where c is a 
 constant. The equation of the straight 
 
 line becomes - -| — — 1, which may 
 
 a c — a 
 be written a^ -\-(y — x — c)a = ex. The 
 equation determines for every point 
 P(x, y) two values of a, to which cor- 
 respond two lines of the system inter- 
 secting at (x, ?/). When these two 
 values of a become equal, the point (;*•, y) becomes the intersection of 
 consecutive positions of the line; that is, a point of the envelope of 
 
 the line. Hence the point 
 (x, y) of the envelope must 
 satisfy the condition that 
 the equation in a has equal 
 roots. The equation of the 
 envelope is therefore 
 
 (y — X — c)2 + 4 c.^ = 0, 
 which reduces to 
 ?/2 '- 2 xy + xr — 2 cy + 2 ex 
 
 + C2 = 
 
 and represents a parabola. 
 This problem furnishes 
 method frequently used 
 to construct a parabola tan- 
 gent to two given straight lines at points equidistant from their intersec- 
 tion. Mark on the lines starting at their intersection the equidistant 
 points 
 1, 2, 3, 4, 5, G, 7, 8, ••■, -1, -2, -3, -4, -5, - G, -7, -8, •••. 
 
SECOND DEGREE EQUATION 
 
 141 
 
 If the given points are + 5 on one line and + 5 on the other, the straight 
 lines joining the points of the given lines the sum of whose marks is + 5 
 envelop the pai-abola required. 
 
 3. Through a fixed point a system of straight lines is drawn. Find 
 the locus of the middle points of the segments of tliese lines includid by 
 the axes of reference. 
 
 4. Find the envelope of a straight line of constant lungtli whose ex- 
 tremities slide in two fixed intersecting straight lines. 
 
 Art. 71. — Conic Section through Five Points 
 
 Let (a-i, ?/i), (.«,,, ?/,), (a;,,, ?/..j), {x^, y^) be four points of which no 
 three are in the same straight line. Let a = be the straight 
 line through (x^, y{), (x.,, yS)] b = the line through (.i\,, y.^, 
 (xs, 2/3) ; c = the line through (a%, y.), (x^, y^ ; d = the line 
 through (a-4, 2/4), O^'i, Z/i)- 
 The equation ac-\-'kbd=(), 
 where k is an arbitrary 
 constant, represents a 
 conic section through the 
 four points. For, since a, 
 b, c, d are linear, the equa- 
 tion ac + kbd = is of the 
 second degree, and must 
 therefore represent a conic 
 
 section. The equation is satisfied by a = and b = 0, condi- 
 tions which determine the point (ic^, 3/2) ; by a = and d — 0, 
 determining the point (xi, y^); by c = 0, b = 0, determining 
 (x.j, 1/3) ; by c = 0, d = 0, determining (a;^, y^). Since k is arbi- 
 trary, ac + kbd = represents any one of an infinite number 
 of conic sections through the four given points. 
 
 If the conic section is required to pass through a fifth point 
 (x'5, 2/5) not in the same straight line with any two of the four 
 points {xi, yy), (x.,, y^, {x^, y^), (x^, y^), the substitution of the 
 coordinates of (x^, 2/5) in ac + kbd = determines a single value 
 
142 
 
 ANALYTIC GEOMETRY 
 
 for Jc. Therefore five points of wliich no three lie in the same 
 straight line completely determine a conic section. 
 
 Problems. — Find the equations of conic sections tlirough the five 
 points. 
 
 1. (1,2), (3,5), (-1,4), (-3, -1), (-4, 3). 
 
 The equations of the sides of the quadrilateral whose vertices are 
 the first four points are a = Sx — 2y + 1 =0, b = x — 4tj + 17 = 0, 
 c = 5 X — 2 ?/ + 13 = 0, d = Sx — 4y + 5 = 0. The equation of a conic 
 section through these four points is therefore 
 
 (3x - 2?/ + l)(5x - 2?/ + 13)+ i-(a; - 4^ + 17)(3x - 4?/ + 5) = 0. 
 Substituting the coordinates of the fifth point (—4, 3), k = W- The 
 equation of the conic section through the five points is 
 
 79 x2 - 320 xy + 301 ?/2 + noi x - 1665 y + 1580 = 0. 
 
 2. (2, 3), (0, 4), (- 1, 5), (- 2, - 1), (1, - 2). 
 
 3. (1, 3), (4, - G), (0, 0), (9, - 9), (16, 12). 
 
 4. (- 4, - 2), (2, 1), (-6, 3), (0, 0), (2, - 1). 
 
 5. (- i, - i), (2, 1), (f, 2), (-J, - 3), (I,- I). 
 
 6. (3, V5), (-2, 0), (-4, - Vl2), (3, - V5) (2, 0). 
 
 7. (1,2), (2, 1), (3, -2), (0,4), (3,0). 
 
 8. (2,3), (-2,3), (4,1), (1,3), (0,0). 
 
 Art. 72. — Conic Sections Tangent to Given Lines 
 
 Let « = and b = represent two straight lines intersected 
 hy the straight line c = 0. The equation ab — kc' = repre- 
 sents a conic section tan- 
 gent to the lines a = 0, 
 6 = at the points of inter- 
 section of c — 0. For the 
 equation ab — kc' = is of 
 the second degree, and the 
 points of intersection of the 
 line a = with ab — kc^ = 
 ^'"^ ''^^' coincide at the point of in- 
 
 tersection of the lines a = 0, c = 0, which makes a = tangent 
 to the conic section. For a like reason b = is tangent to 
 
SECOND DEGliEK EQUATION 
 
 143 
 
 ab — kc- — 0. Since k is arbitrary, an infinite number of conic 
 sections can be drawn tangent to the given lines at the given 
 points. 
 
 The equation of a conic section tangent to the lines x = (), 
 y = lit the i)oints (a, 0), (0, h) is 
 
 a b 
 
 Kxy = 0. 
 
 (1) 
 
 The points of intersection of this conic section and the line 
 
 MK - + ^ = 1, lie in the 
 m 11 
 
 locus of the equation 
 
 i? + f_5_?'Y = A>,. (2) 
 a b m nj 
 
 This last equation is homo- 
 geneous of the second degree, 
 and hence represents two 
 straight lines from the origin 
 through the points of inter- 
 
 r a" , 2/ 1 A Fig. 138. 
 
 section of [--—1 = 
 
 VI n 
 
 and (- + - — 1) — Kxy = 0. The straight lines represented by 
 
 equation (2) coincide, and - + - — 1 = is tangent to 
 
 when 
 
 is a perfect square ; that is, whei 
 
 a b ' 
 
 I b m - ' ^ 
 
 \rt mj \b 71 J I \a mj \b nj 2 S 
 
144 ANALYTIC GEOMETRY 
 
 whence K= if^- lY/^i - ^Y (3) 
 
 \(i iiij \h nj 
 
 Similarly, — + - — 1 = is tangent to (1) when 
 
 Equations (3) and (4) determine the values of - and - in terms 
 
 of the arbitrary constant K, which shows that an infinite num- 
 ber of conic sections can be drawn tangent to four straight 
 lines no three of which pass through a common point. If 
 
 [- — = 1 is also tangent to the conic section represented by 
 
 equation (1), 
 
 A-=4fl^i)fl-l> (5) 
 
 Equations (3), (4), (5) determine -, -,aud A" uniquely, proving 
 
 a h 
 that only one conic section can be drawn tangent to five straight 
 lines no three of which pass through a common point. This 
 proposition is the reciprocal of the j)roposition of Art. 71 and 
 might have been demonstrated by the method of reciprocal 
 polars. 
 
 Problems. — 1 . Find the equation of the parabola tangent to two 
 straight lines including an angle of 60° at points whose distances from 
 their point of intersection are 2 and 4. 
 
 2. Find the equation of the conic section tangent to two straight lines 
 including an angle of 45° at (3, 0), (5, 0), and containing the point (7, 8), 
 the given straight lines being the axes of reference. 
 
 Art. 73, — Similar Coxic Sections 
 
 The points P{x, y) and P^iinx, my) lie in the same straight 
 line through the origin 0, and 0I\ = m • OP. The distance 
 between any two positions of P^, (mx', my'), (mx", my") is m 
 
SECOND DEGREE EQUATION 
 
 145 
 
 times tlie distance between tlie correspondiut,^ positions of 
 P,{x\y'),{x'\y"). For 
 
 { (m.f' - mx")- + {my' - myy\ ^ = m \ (^x' - x'J + (//' - y"f\ i 
 
 Representing the point P by {x, y), the point P^ ])y (X, Y), 
 when {x, y) traces a geometric figure, the point (A", Y) traces a 
 figure to scale m times as large. The effect of tlie substitution 
 
 X Y . 
 
 X——, ?/ = — is therefore simijly to change the scale of tlu; 
 
 drawing. Figures thus related are said to be similar. When 
 
 the two equations /(;r, y) = 0, fi~-, — j = are interpreted in 
 
 the same axes, their loci are similar and similarly placed ; 
 
 when interpreted in differ- ^ 
 
 ent axes but including the 
 
 same angle, the loci are 
 
 similar. Ellipses similar 
 
 to — + -i- = 1 are repre- 
 
 sented by Al + ^ 1. 
 
 All ellipses 
 
 of a 
 
 similar 
 
 system have 
 
 the same ec- 
 
 centricity, for 
 
 
 «-^ 
 
 r - 7;* 
 m-a^ 
 
 ^ 
 
 _ a- - 
 
 -b\ 
 
 
 Y2 
 
 In like manner, all hyperbolas of a similar system -^ • 
 
 ??i^a^ m'b- 
 have the same eccentricity. The parabolas similar to y^ = 2x)x 
 are represented by the equation Y^ = 22)mX. 
 Taking as corresponding points 
 
 J'(^, y), Pi(mx, my), !'.,(- mx, 
 
 'Z/), 
 
146 
 
 ANALYTIC GEOMETRY 
 
 the figure traced by Pi is similar to that traced by P, the figure 
 traced by P2 is symmetrical to that traced by P^. 
 
 The change of scale of a drawing may be effected mechani- 
 cally by means of an instrument called the pantograph, which 
 consists of four rods jointed together in such a manner as to 
 form a parallelogram ABOC with sides of constant length, 
 but whose angles may be changed with perfect freedom. On 
 the rods AB and AC fix two points P and Pj in a straight 
 line with 0. If the point is fixed in the plane, and the 
 point P is made to take any new position P', and the cor- 
 responding position of Pj is P/, the points P', 0, P/ in Fig. 
 134 are always in a straight line, the triangles Pi CO and 
 
 Pi'A'P' are similar and 
 ' P hence 
 
 qpi 
 
 OP' 
 
 P,'C 
 
 CA' 
 
 m 
 
 CA 
 
 a constant which may be 
 denoted by m. Taking 
 as origin of a system of 
 rectangular coordinates, 
 if Pis (a-, 2/), Pi is 
 
 (— mx, — my). 
 
 If the point P is fixed in 
 the plane and taken as origin of a system of rectangular coordi- 
 nates, if the point is (x, y), the point Pj is {mx, my). There- 
 fore, if the point is made to trace any locus, the point P, 
 traces a similar figure to a scale m times as large. 
 
 The equation 
 
 Art. 74. — Confocal Conic Sections 
 1, 
 
 If 
 
 (1) 
 
 a^ -f- A 6' -f A 
 
 where a? > W represents an ellipse when A > — 6-, an hyperbola 
 when — a^ < A < — &^, an imaginary locus when A < — al The 
 
SECOND DEGREE EQUATION 
 
 147 
 
 distance from focus to center of the ellii)se,s and liyperbolas 
 represented by eqiiation (1) is \(i' + \ — Ir — \\'- = {tC- — U')-. 
 Hence equation (1) when interpreted for different values of A. 
 in the same rectangular axes represents ellipses and hyperbolas 
 having common foci ; that is, a system of confocal conic sections. 
 Through every point {x', y') of the plane there passes one 
 ellipse and one hyperbola of the confocal system 
 
 a^ + A h' + X 
 For the conic sections passing through {x\ y') corresptmd to 
 the values of X satisfying the equation 
 
 .. 
 
 1. (2) 
 
 a- + X b- + X 
 This function of A, 
 
 .^1^ + ^ 1, 
 
 a- + A h- + X 
 
 is negative when A = + co, 
 positive just before A be- 
 comes — 6-, negative when 
 A is just less than — li' 
 and again positive when 
 A is just greater than — cr. 
 Hence equation (2) deter- 
 mines for A two values, one between + x and — h'-, the other 
 between — Jr and — a'. 
 
 The ellipse and hyperl)o]a of the confocal system 
 
 (t- -H A //- -f A 
 til rough the point (.c', //') intersect at right angles. 
 
 Let Ai and A. be the values of A satisfying the equation 
 
 X' 
 
 + 
 
 a- + A h- + X 
 
 = 1. Then 
 
 + 
 
 a- + Ai &' -f Ai 
 
 a- + X., U- + A. 
 
148 ANALYTIC GEOMETRY 
 
 represent ellipse and hyperbola tlirougli (x-', y'). The tan- 
 gents to this ellipse and hyperbola at (x', y') are 
 
 iiy 
 
 a" + Ai h- + Ai 
 
 xx' yy' 
 
 a^ + A2 b^ + A2 
 
 (1) 
 (2) 
 
 From the equations 
 
 + r7^^=l. -.-^^ + 
 
 a^ + Ai b- + \i a- + \2 b'' + X., 
 
 is obtained by subtraction 
 
 (cr + Ai) (a' + \,) {b' + Aj) (6^ + A.) 
 
 which is the condition of perpendicularity of tangents (1) 
 and (2). 
 
 Since through every point in the plane there passes one ellipse 
 and one hyperbola of the confocal system, the point of the 
 plane is determined by specifying the ellipse and hyperbola in 
 which the point lies. This leads to a system of elliptic coordi- 
 nates. 
 
 If heat flows into an infinite plane disc along a finite straight 
 line at a uniform rate, when the heat conditions have become 
 permanent, the isothermal lines are the ellipses, the lines of 
 flow of heat the hyperbolas of the confocal system. The same 
 is true if instead of heat any fluid flows over the disc, or if an 
 electric or magnetic disturbance enters along the straight line. 
 
CHAPTER XI 
 LINE OOOKDINATES 
 
 Art. 75. — Coordinates of a Straight Line 
 
 If the equation of a straight line is written in the form 
 ux -\-vy + 1 = 0, u and v are the negative reciprocals of the 
 intercepts of the line on the axes. To every pair of values of 
 H and V there corresponds one straight line, and conversely; that 
 is, there is a " one-to-one correspondence " between the symbol 
 («, v) and the straight lines of the plane, u and v are called 
 line coordinates.* 
 
 If {u, V) is fixed, the equation ux -{- vy -\- 1 = expresses the 
 condition that the point {x, y) lies in the straight line («, v). 
 The system of points on a straight line is called a range of 
 points. Hence a first degree point equation represents a range 
 of points and determines a straight line. 
 
 If (x, y) is fixed, ux + vy + 1 = expresses the condition that 
 the line (», v) passes through the point (x, y). The system of 
 lines through a point is called a x^encil of rays. Hence a first 
 degree line equation represents a pencil of rays and determines 
 a point. 
 
 The equations ?/.ri + vy^ -|- 1 = 0, nx^ + vy., + 1=0 determine 
 the points {x^, y,), {x^, y^ respectively. \ \\^' ^ '\+^) 
 represents for each value of X one point of the line through 
 (a^i, ?/i), (.i\,, 7/2). X is the ratio of the segments into Avhich the 
 point corresponding to X divides the finite line from (.r„ y,) to 
 
 * riiickcr in Germany and Cliasles in France developed the use of line 
 coordinates at about the same time (1829). 
 149 
 
150 ANALYTIC GEOMETUr 
 
 {x2, 2/2)- There is a " one-to-one correspondence " between A 
 and tlie points of the line throngh (a-j, y^, (.i-^, y^. 
 
 1-f-A 1 + A 
 
 which reduces to {ux^ + vy^ + 1) -|- A {ax^ + vy2 + 1) = 0, is the 
 line equation of the point A. Denoting m.Ti + vy^ -f 1 by L, 
 UX2 + v?/2 + 1 by 3f, L + \3I— represents the range of points 
 determined hy L = 0, 3£— 0. 
 
 The rays of the pencil determined by the lines 
 
 UiX + Viy + 1 = 0, ti.^x -\- v^y + 1 = 
 
 are represented by the equation 
 
 (u,x -f v,y + 1) + A (n.x + v-^y + 1) = 0, 
 
 which may be written '^!l±^x + !!l±J^ + 1 = 0. 
 •^ 1+A 1 + A 
 
 Hence (n, + Xv v,±M:: 
 
 V 1+A 1+A 
 are the lines of the pencil determined by (11^, Vi), (xi2, v^). There 
 is a " one-to-one correspondence " between A and the rays of 
 the pencil. Denoting UiX -[■ Viy -\-l by P, u^x + Vjy + 1 by Q, 
 P + AQ = represents the rays of the pencil determined by 
 P=0, Q = 0. ' 
 
 Problems. — 1. Construct the lines (4, 1); (- 2, 5); (- i, - J). 
 
 2. Construct the pencil represented by 3 ?i - 2 y ^- 1 = 0. 
 
 3. Construct the range represented by 2 .x — 3 ?/ -f 1 = 0. 
 
 4. Locate the point determined by 4 ?( -^ 5 v -|- 1 = 0. 
 
 5. Draw the line determined by 3x — 5 ?/ + 1 = 0. 
 
 6. Write the equation of the range of points determined by 
 
 2ti-Sv + l = 0, i?t-f^w-fl=0. 
 
 7. Write the equation of the pencil of rays determined by 
 
 2x-Sy + l=0, J a;-f i?/ + l = 0. 
 
LINE COORDINATES 151 
 
 Akt. 76. — Line Equatioxs of the Conic Sections 
 
 The equation of the tangent to the ellipse ^—■^•-L^—l at 
 
 a- b- 
 (.T,,, y/o) is ^' + "''•'^ = 1. Comparing the equation of the tangent 
 
 a- U- 
 with nx + (•>) + 1 = it is seen that the line coordinates of the 
 
 tangent are a = — ' ", i' = — ^", whence a-,, = — u-u, ?/„ = — Irv. 
 If the point of tangency (.Tn, ?/„) generates the ellipse "^ + ii — •'-> 
 the tangent {ii, v) envelopes the ellipse. Hence the line equa- 
 tion of the ellipse, when the reference axes are the axes of the 
 ellipse, is a-ir + b'-v^ — 1. 
 
 Problems. — 1. Show that the Hue equation of the circle x"^ + y- — r" 
 
 is iC- + y- = — 
 r- 
 
 2. Show that the Hue eiiuatiou of the hyperbola 
 
 1 is d-xi- — b-v- = 1. 
 
 a- h- 
 
 3. Show that the Hue equatiou of the parabola y"^ = 2px is pv'^ = 2 u. 
 Construct the euvelopes of the equations 
 
 4. - + - = - 5. G. ifi-\- v"^ = ' 8. 9 1*2 - 4 u2 = i. 
 
 U V 
 
 5. uv-\. 7. 9!t2 + 4tj2-i, 9. 8i;2-u = 0. 
 
 Art. 77. — Cross-ratio of Four Points 
 
 The double ratio -^--. — ^ is called the cross-ratio of the four 
 CB I)B 
 points A, B, C, D, and is denoted by the symbol (ABCB). If 
 the point A is denoted by 
 
 i = 0, the point B by J/= 0, > ^ »< ^ 
 
 the points C and D respec- ^'"- '"■ 
 
 tively by L + XiM=Q and L + XoM^O, it follows that 
 
 ^ = Ai, ^ = X,,, and (ABCD) = ^- Take any four points of 
 CB JJB Ao 
 
152 ANALYTIC GEOMETRY 
 
 tlie range L + X3/= corresponding to Aj, X.,, A3, A4, and repre- 
 sent L + Aji)/ by Li, L + LM by i»/i, whence /v + A,,.1/ is 
 
 represented by L,-^^^M„ L + A, J/ by L, - ^i^il/-i. 
 
 Ao — A3 Ao — A4 
 
 The four points corresponding to Aj, A., A3, A4 are represented 
 by the equations 
 
 L, = 0, 3/1 = 0, A - ^^^^^^3/, = 0, A - ^i-^^1A = 0, 
 A2 — A3 A2 — A4 
 
 and their cross-ratio is ^^ ~ ^ -~ ^ Since the four points 
 
 ^2 — X^Xi — A4 
 
 Ai, Ao, Ag, A4 can be arranged in 24 different ways, the cross- 
 ratio of four points takes 24 different forms, but these 24 
 different forms are seen to have only six different vaUies. 
 
 i-f A3/=0, L' + XM' = represent two ranges of points. 
 By making the point of one range determined by a value of 
 A correspond to the point of the other range determined by the 
 same value of A, a " one-to-one correspondence " is established 
 between the points of the two ranges, and the cross-ratio of 
 any four points of one range equals the cross-ratio of the corre- 
 sponding four points of the second range. Such ranges are 
 called projective. 
 
 Art. 78. — Second Degkke Line Equations 
 
 Remembering that each of the equations 
 
 L + \M=0, i'-fAJ/' = 
 
 for any value of A represents the entire pencil of rays through 
 the point of the range corresponding to A, it is evident that the 
 equation LM' — L'M= 0, obtained by eliminating A between 
 L + \M=^ 0, L' + AIT' = 0, represents the system of lines join- 
 ing the corresponding points of the two projective point ranges. 
 This equation is a second degree line equation, and it becomes 
 necessary to determine the locus enveloped by the lines repre- 
 sented by the equation. 
 
LINE COOliDINATES 153 
 
 Let nx + vy + 1 = represent any i^oint (x, y) of the plane. 
 Writing the values of L, M, L', 31' in full, the elimination of 
 u and V from the equations i(x + t'^ + 1 = 0, 
 
 iixi + viji + 1 + X(ux2 + vyo + 1) = 0? 
 ux' + vy' + 1 + X(ux" + vy" + 1) = 0, 
 
 determines a quadratic equation in X with coefficients of the 
 first degree in (x, ?/), GX^ + HX + K= 0. To the two values 
 of X which satisfy this equation there correspond the tangents 
 from (x, y) to the envelope of LM' — VM — 0. When these 
 tangents coincide, the point (.r, ?/) lies on the envelope. 
 
 4 IP - GK^ 
 
 causes the coincidence of the tangents, and is therefore the 
 point equation of the envelope. The point equation being of 
 the second degree, the envelope is a conic section. 
 
 The degree of a line equation denotes the number of tan- 
 gents that can be drawn from any point in the plane to the 
 curve represented by the equation, and is called the class of 
 the curve. 
 
 Art. 79. — Cross-ratio of a Pencil of Four Eays 
 
 Let a pencil of four rays, 
 
 P=0, Q = 0, P-|-X,Q = 0, P-fA,Q = 0, 
 
 be cut by any transversal in the four points A, B, C, D. j^ is 
 the common altitude of the triangles whose common vertex 
 is 0, and whose bases lie in the transversal. Then 
 
 i> • CA = OA - OC • sin COA, p • DA = OA ■ OD ■ sin DO A, 
 p -03=00 -OB- sin COB, p - DB = OD • OB ■ sin DOB, 
 
 and (ABCD) = ^"^ ^^^^ - ^"^^^^^ . This double sine ratio is 
 ^ ^ sin COB sin DOB 
 
154 ANALYTIC GEOMETRY 
 
 called the cross-ratio of the pencil of four rays. It is evident 
 
 that central projection does 
 not alter the cross-ratio of 
 four points in a straight 
 'PA \ "^^ line. 
 
 Writing the equation 
 P-|-XQ = in the com- 
 Q=0 plete form 
 u^x + i\y + 1 
 
 FiG' 138. and this in the form 
 
 the factor ^ll^ll+i^iX is seen to be the negative ratio of the 
 
 distances from any point of the line P + \Q = to the lines 
 
 P = 0, Q = 0. Hence 
 
 (7a_ sinC0^1 __. Da' ^ ^mPOA ^ ^ 
 
 Cb sin COB " Db' sin DOB " 
 
 ^ sinCO^^sin^DOA^Xi^^j^g cross-ratio of the four rays 
 sin COB sin X)0i5 Xg 
 
 P^O, Q = 0, P4-AiQ = 0, P + X,Q = 0. 
 
 Representing 
 
 P+X,Q by Pi, P + XoQ by Q„ P-f-XgQ 
 
 is represented by 
 
 P^ _ k^A^ Q„ p + X.Q by P. - ^^ Qi. 
 
 Xa — X3 Ao — A4 
 
 Hence the cross-ratio of the four points of the pencil P+XQ—0 
 
 , . , , - , 1 • Xi — Xi Xo — X) 
 
 corresponding to Xj, Xo, X;,, A4 is -^ — • 
 
 X2 — A.3 Aj^ • — A4 
 
LINE COORDINATES 155 
 
 By making the ray of F+kQ = determined by a value of 
 \ correspond to the ray of F' +XQ' = determined by the 
 same value of X, a " one-to-one correspondence " is established 
 between the rays of the two pencils, and the cross-ratio of any 
 four rays of one pencil equals the cross-ratio of the correspond- 
 ing four rays of the other pencil. Such pencils are called 
 projective pencils. 
 
 The equation of the locus of the points of intersection of the 
 corresponding rays of the two projective pencils F + XQ = 0, 
 F' -t- XQ' = is FQ' — F'Q = 0. This is a second degree point 
 equation and represents a conic section.* 
 
 Art. 80. — Construction" of Projective Ranges and 
 Pencils 
 
 If there exists a " one-to-one correspondence " between the 
 points of two ranges, between the rays of two pencils, or be- 
 tween the points of a range and the rays of a pencil, the ranges 
 and pencils are projective. 
 
 Let F=0, Q = 0, determining the range or pencil F-}-XQ=0, 
 correspond to F^ = 0, Qi = 0, determining the range or pencil 
 Pi+ A,Qi = 0, and let a "one-to-one correspondence" exist 
 between the elements X of the first system and the elements 
 Ai of the second system. This "one-to-one correspondence" 
 interpreted algebraically means that Xi is a linear function 
 
 of X; that is, Xj = ^Jhj±A. By hypothesis, Xi = when X = 0, 
 
 ' ' cX + d ^ ^^ ^ ' 
 
 and X, = CO when X = cc, hence b = 0, c = 0, and X, = - X. 
 
 Let X = / and X, = /, be a third ])air of corresponding ele- 
 
 * A complete projective treatment of conic sections is developed in 
 Steiner's Theorie der Kcgelschnitte, 1800, and in Chasles' G^om^trie 
 Sup^rieure, 1852, and in Cremona's Elements of Projective Geometry, 
 translated from the Italian. 
 
156 
 
 ANALYTIC GEOMETRY 
 
 ments;tlien - = -, Ai = -X, and the equations of tlie systems 
 
 P+XQ = 0, Pi + Ai(3i = become P + XQ = 0, IP^ + XI,Q,^0. 
 
 Now the elements of 
 ^ IF, + XIQ, = 
 
 are the elements of 
 
 P, + XQ, = 0, 
 hence the systems between 
 whose elements there exists a 
 '^ one-to-one correspondence " 
 are the projective systems 
 Fm. m P-{-XQ = 0, Pi + AQ, = 0. 
 
 This analysis also shows that the correspondence of three ele- 
 ments of one system to three elements of another makes the 
 systems projective. 
 
 Projective systems are constructed geometrically, as follows : 
 Let the points 1, 2, 3 on one straight line mm correspond to 
 
 the points 1, 2, 3, respec- 
 tively, on another straight 
 line nn. Place the two 
 lines with one pair of 
 corresponding points 2, 2 
 in coincidence. Join the 
 point of intersection of 
 the lines through 1, 1 and 
 3, 3 with 2. Take the 
 3 points of intersection of 
 lines through with mm 
 and 7in as corresponding 
 points, and a " one-to-one correspondence " is established between 
 the points of the ranges mm, nn, which are therefore projective. 
 In like manner, if three rays 1, 2, 3 of pencil m correspond 
 to the rays 1, 2, 3, respectively, of pencil w, by placing the cor- 
 responding rays 1, 1 in coincidence, and drawing the line 00 
 
LINE COORDINATES 
 
 157 
 
 through the points of intersection of the corresponding rays 
 2, 2 and 3, 3, and taking rays from m and 7i to any point of 00 
 as corresponding rays, a " one-to-one correspondence " is estalv 
 lished between the rays of the two pencils, and the pencils are 
 projective. 
 
 Art. 81. — Coxic Section through Five Points 
 
 It is now possible by the aid of the ruler only to construct a 
 conic section through five points or tangent to five lines. Take 
 two of the given points 1, 2 as the vertices of pencils, the pairs 
 of lines from 1 and 2 to the remaining three points 3, 4, 5, 
 respectively, as corresponding rays of projective pencils. The 
 
 
 2 
 
 /^ 
 
 7 
 
 1(5)/ 
 
 / 
 
 
 ). 
 
 y 
 
 (7) 
 
 (3^ 
 
 3 
 
 line 11 is a transversal of pencil 1, 22 of pencil 2. 0, the 
 intersection of 51 and 32, is the vertex of a pencil of which 
 11 and 22 are transversals. Hence the pencils 1 and 2 arc 
 projective, and corresponding rays are rays to the points of 
 intersection of the rays of pencil with 11 and 22. The 
 
158 
 
 ANALYTIC GEOMETRY 
 
 intersections of these corresponding rays are points of the 
 required conic section. 
 
 Take tAvo of the five given lines 11 and 22 as bearers of point 
 ranges on which the points of intersection of the other lines 
 
 33, 44, 55 respectively are corresponding points. The line 00 
 is a common transversal of the pencils (11), (22). Hence 
 corresponding points of the projective ranges 11 and 22 are 
 located by the intersection with 11 and 22 of lines connecting 
 (11) and (22) respectively with any point of 00. The straight 
 lines connecting corresponding points are tangents to the 
 required conic section. 
 
 Notice that the construction of the conic section tangent to 
 five straight lines is the exact reciprocal of the construction of 
 the conic section through five points. 
 
 The figure formed by joining by straight lines six arbitrary 
 points on a conic section in any order whatever is called a six- 
 
LINE COORDINATES 
 
 ir)9 
 
 7-f-\!(2)\4 
 
 .''5 
 
 point Taking 1 and 5 as vertices of pencils whose correspond- 
 ing rays are determined by the points 2, 3, 4, the points of 
 intersection of 16 with 11 and of 5iy with 55 must lie in the 
 same ray of the auxiliary pencil 0; that is, in any six-point of 
 a conic section the inter- ^i 
 
 section of the three pairs \ 1\ 
 
 of opposite sides are in 
 a straight line. This is 
 Pascal's theorem.* 
 
 Reciprocating Pascal's 
 theorem, Brianchon's theo- 
 rem is obtained. — In the 
 figure formed by drawing 
 tangents to a conic section 
 at six arbitrary points in fig. u3. 
 
 any order whatever (a six-side of a conic section), the straight 
 lines joining the three pairs of opposite vertices pass through 
 a common point.f 
 
 By Pascal's theorem any number of points on a conic section 
 through five points may be located by the aid of the ruler ; by 
 Brianchon's theorem any number of tangents to a conic section 
 tangent to five straight lines may be drawn by the aid of the 
 ruler. 
 
 * Discovered by Pascal, 1040. 
 t Discovered by Brianclion, 180G. 
 
CHAPTER XII 
 ANALYTIC GEOMETRY OP THE COMPLEX VAEIABLE 
 
 Art. 82. — Graphic Represextation op the Complex 
 Variable 
 
 The expression x + iij, where x and y are real variables and 
 i stands for V— 1, is called the complex variable, and is fre- 
 quently represented by z. Vx' + if is called the absolute value 
 of z and is denoted by 1 2; | or | a; + iy |. 
 
 If a; + iy is represented by the point (x, y), a " one-to-one 
 correspondence" is established be- 
 ^x-^iy tween the complex variable x + iy 
 
 j and the points of the XF-plane. 
 
 [ The X-axis is called the axis of 
 
 X \ reals, the F-axis the axis of imagi- 
 
 ^ naries. Denoting the polar coordi- 
 
 FiG. 144. nates of (x, y) by r and 6, x= r cos 6, 
 
 y = r sine, and 2=x-+ «/=?• (cos ^+r sin ^), where r=Vo^+f, 
 
 6 = tan-^^. r is the absolute value, and 6 is called the anipli- 
 
 X 
 
 tude of the complex variable x + iy. Hence to the complex 
 variable x + iy there corresponds a straight line determinate 
 in length and direction. A straight line determinate in length 
 and direction is called a vector. Hence there is a " one-to-one 
 correspondence" between the complex variable and plane 
 vectors. As geometric representative of the complex variable 
 may be taken either the point (x, y) or the vector which deter- 
 mines the position of that point with respect to the origin.* 
 
 * Argand (1806) was the first to represent the complex variable by 
 points in a plane. Gauss (1831) developed the same idea and secured 
 for it a permanent i>lace in mathematics. 
 160 
 
COMPLEX VAIIIABLE 
 
 161 
 
 Calling a liiu^ etiual in Icnii^'tli to the linear unit and laid olF 
 from the origin along the positive direction oi' the axis of reals 
 the uuit vector, the complex variable 
 
 z — X -\- i>i — r (cos 6 + I sin 6) 
 
 represents a vector obtained by multiplying the unit vector by 
 the absolute value, then turning the resulting line about its ex- 
 tremity at the origin through an angle equal to the amplitude of 
 the complex variable. When the 
 complex variable is written in 
 the form r(cos ^ + « sin(9), r is 
 the length of the vector, 
 
 cos 6 -}- i sin 9 
 
 the turning factor. In analytic 
 
 trigonometry it is proved that 
 
 cos 6 + i sin 6 = c'".* Hence the 
 
 complex variable r{vo9,e + i sin^)= r-c'^ whore the stretching 
 
 factor (tensor) and turning factor (versor) are neatly sc[)a,ratcd. 
 
 Problems. — 1. Locate the points rcpresmtud by 2 + t5; 3-i2; 
 
 - 1 + i2 ; i5 ; - i4 ; - 3 - i ; - t 7 ; + i7. 
 
 2. Draw the vectors represented by 3 + i2; 1— i3; — 2+i3; 
 
 - 1 -i4; - i5 ; 3 -/; 1 + /. 
 
 3. Show that e-'"^' = 1, when n is any integer. 
 
 4. Show that r • e'(9+-"'^' represents the same point for all integral 
 values of n. 
 
 2nni 
 
 5. Locate the different points represented by e"^ for integral values 
 of n. 
 
 I(g+2n7r) 
 
 6. Locate the different points represented by 5 • e •« for integral 
 values of n. 
 
 *This relation was discovered by Eulcr (1707-1783). 
 
162 ANALYTIC GEOMETRY 
 
 Akt. 83. — Arithmetic Operations applied to Vectors 
 
 The sum of two complex variables Xj + ii/i and X2 + iy2 is 
 (.I'l + X,) + / (^1 + //,) . H euce 
 
 |(a-i + iyi) + (•'^•2 + ^'^2)1 = y/{xi + x^y+ivi + 2/2)', 
 
 and. the amplitude of the sum is tan"' -'' A The graphic 
 
 Xj ~|~ ^2 
 
 representation shows that the vector corresponding to the sum 
 ^-rC^ + ii/) is found by constructing the vec- 
 ' "" / tor corresponding to Xi + i?/i ^iid. 
 / using the extremity of this vector 
 as origin of a set of new axes 
 w^ ^. parallel to the first axes to con- 
 struct the vector corresponding 
 
 ^ to X2 + ^2/2• The vector from the 
 F'G- ^■^6. origin to the end of the last vec- 
 
 tor is the vector sum. The vector sum is independent of the 
 order in which the component vectors are constructed. From 
 the figure it is evident that 
 
 I {^1 + Wi) + {^2 + iVi) I > l-^'i + m\ + \^2 + il/2\- 
 
 The difference between two vectors x^ + iyi and Xo + iy^ is 
 (x^ — x.^+ i{yi — y^. The graphic representation shows that 
 the vector corresponding to the difference is found by construct- 
 ing the vector corresponding to x-^ + iyi and adding to it the 
 vector corresponding to — x.2 — iy-,- It is seen that 
 
 I (x, + iy,) - {x. + iy^ | = V(a-i - x.y- + (?/i - y^, 
 
 the amplitude of the difference is tan-^ -^^' ~^^- , and that the 
 
 a-, - x. 
 
 equality of two complex variables requires the equality of 
 
 the coefficients of the real terms and the imaginary terms 
 
 separately. 
 
CO^fPLEX V Alii MILE 
 
 1G: 
 
 The product of two complex variables is most readily found 
 by writing these variables in the form r^ • e''^', r.^ • e'^^. The 
 product is i\r.2 • e''(«»+*2', showing that the absolute value of the 
 product is the product of the absolute values of the factors and 
 the amplitude of the product is the sum of the amplitudes of 
 the factors. Hence, writing the complex variables in the 
 form Ti (cos 6i + i sin ^i), n (cos 60 + i sin O.j), the product is 
 'V*2[cos (^1 + ^2) + ^ sill (^1 + ^2)]? which of course can be shown 
 directly. 
 
 Construct tlie vector corresponding to the multiplier r, • e'^* 
 and join its extremity Pj to the extremity of the unit vector 
 01. Construct the vector corre- 
 sponding to the midtiplicand 
 7-2 • e'^% and on this vector OP., as 
 a side homologous to 01 construct 
 a triangle OP^P similar to OPjl ; 
 then OP is the product vec- 
 tor. For, from the similar tri- 
 angles 0P= Ti • r^, and the angle 
 XOP=6i+62- The product vec- 
 tor is therefore formed from the 
 vector which is the multiplicand 
 
 in the same manner as the vector which is the multiplier is 
 formed from the unit vector. The product vector is indepen- 
 dent of the order of the vector factors and can be zero only 
 when one of the factors is zero. 
 
 The quotient of two complex variables i\ • e''^', ?*2' e'^- is 
 
 '"1 . e'(9i 62) . 
 
 that is, the absolute value of the quotient is the quotient of the 
 absolute values, and the amplitude of the quotient is the ampli- 
 tude of the dividend minus the amplitude of the divisor. 
 
 Construct the vectors OP, and OPo corresponding to dividend 
 and divisor respectively, and let 01 be the unit vector. On 
 
164 
 
 ANALYTIC GEOMETRY 
 
 OPi as a side homologous to 01\ construct the triangle OI\P 
 similar to OP^l, then OP is the quotient vector, for OP = -j 
 
 and the angle XOP is ^i — $2- The quotient vector is obtained 
 from the vector which is the dividend in the same manner as 
 the unit vector is obtained from the 
 vector which is the divisor. 
 Extracting the ??i root of 
 z = r • e'^ = r • e'f^+^'w) 
 
 there results z"^ = 9-™ . e ^ "» 
 Since n and m are integers, 
 
 Avhere q is an integer and r can have any value from to m — l. 
 
 <'^ + ^ 
 
 Hence 2;"' = r™-e~'" " ; that is, the m root of z has m 
 
 values which have the same absolute value and amplitudes 
 differing by — beginning with — .* 
 
 Problems. — 1. Add (2 + 1 5) , ( - 3 + 1 2) , (5 - i 3). 
 
 2. Find the value of (3 - i2) + (7 + i4) - (0 - i3). 
 
 3. Find absolute value and amplitude of 
 
 (4-j.3) + (2 + i5)-(-3 + i4). 
 
 4. Construct (2 - i 3) x (5 + i 2) h- (4 - i 5) . 
 
 5. Find absolute value and amplitude of (10 - i 7) x (4 - iS ). 
 
 6. Find absolute value and amplitude of (15 + f8) x (5 - i2). 
 
 7. Construct (2 + 1 3)3. 9. Construct (7 + i 4) ^ 
 
 8. Construct (8 - 1 5)^. 10. Construct (9 - i 7) t. 
 
 * In mechanics coplanar forces, translations, velocities, accelerations, 
 and the moments of couples are vector quantities ; that is, quantities 
 which are completely determined by direction and magnitude. Hence 
 the laws of vector combination are the foundation of a complete graphic 
 treatment of mechanics, 
 
COMPL EX VA R I A BLE 
 
 165 
 
 11. Construct the five fifth roots of unity. 
 
 12. Construct the roots of 2- — 3 2 + f) = 0. 
 
 Put 2 = a; + ill. There results (x^ _ 2/2 - 3 x + 5) + t (2 xy - 3 ?/) = 0. 
 Plot ofl - y- - 3 X + 5 = and 2xy - Sy = 0. The values of z deter- 
 nihied by the intersections of these curves are the roots of z'^— 3 2 + 5=0. 
 
 AiiT. 84. — Algkp.raic Functions of thk Complex Variable 
 
 The geonietric representative of the real variable is the point 
 system of the X-axis and the geometric representation of a 
 function of a real variable 7/ = f(x) is the line into which this 
 function transforms the X-axis. 
 
 The geometric representative of the complex variable is the 
 point system of the XF-plane, and the geometric representation 
 of a function of a complex variable u + iv=f(x-\- i;/) is the 
 system of lines into which this function transforms systems of 
 lines in the XF-plane. 
 
 ^ 
 
 
 : 
 
 
 1 
 
 ' 
 
 
 Y 
 
 1 
 
 
 
 
 I 
 
 fi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^a 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 rf 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 d' 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 ' 
 
 ' 
 
 
 
 2 
 
 V 
 
 1 
 
 
 : 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '—Ct 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \' 
 
 
 
 
 
 
 A' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 d' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fio. 149. 
 
 When the complex variable is written in the form x -f- ??/, it 
 is convenient to use the systems of parallels to the X-axis and 
 to the F-axis. Take the function iv = z-\- c, where iv stands 
 for u 4- iv, z for x -\- vj, c for a + ih, then 
 
 ?t + iv = (x -f a) + / (// + '>) and ii. = x-\- a, v = .y + h. 
 
166 
 
 ANALYTIC GEOMETRY 
 
 If in the XF-plane a point moves in a parallel to the F-axis, 
 X is constant, and consequently u is constant. Hence the func- 
 tion 10 — z + c transforms parallels to the F-axis into parallels 
 to the F-axis in the f/F-plane. In like manner it is shown 
 that lo — z + c transforms parallels to the X-axis into parallels 
 to the {/-axis. If the variables to and z are interpreted in the 
 same axes, the function io = z + c gives to every point of the 
 XF-plane a motion of translation equal to the translation 
 which carries A to c. 
 
 When the complex variable is written in the form r • e'^, it is 
 convenient to use a system of concentric circles and the system 
 of straisfht lines throudi their common center. Take the func- 
 
 tion lu — c-z, when to stands for R • e'®, z for r • e'^ and c for 
 r' • e'« ; then R • e'® = rr' ■ e'(«+e ^ and R = rr', © = 6 + 6'. If a 
 point in the XF-plane describes the circumference of a circle 
 center at origin, r is constant, and consequently R is constant, 
 and the corresponding point describes a circumference in the 
 C/F-plane, center at origin, and radius r' times the radius of 
 the corresponding circle in the XY-plane. If the point in the 
 XF-plane moves in a straight line through the origin, 6 is con- 
 stant, and consequently © is constant, and the corresponding 
 
coMrLEx V. 1 in A hle 
 
 107 
 
 p(iiiit in the CF-plane moves in a straight line through the 
 origin. If the variables w and z are interpreted in the same 
 axes X and Y, the function xo = c • z either stretches the XY- 
 plane outward from the origin, or shrinks it toward the origin, 
 according as r' is greater or less than unity, and then turns the 
 whole plane about the origin through the angle 0'. 
 
 R = 
 
 The function lo 
 1 ^ 
 
 may be written R • e'® : 
 
 ■'^', whence 
 
 z r 
 
 A circle in the XF-planc with center at the 
 
 origin is transformed into a circle in the (/F-plane with center 
 at the origin, the radius of one circle being the reciprocal of 
 the radius of the other. A straight line through the origin in 
 
 V 
 
 the XF-plane making an angle 6 with the X-axis, is trans- 
 formed into a straight line through the origin in the t/F-plane 
 making an angle — Avith the {7-axis. If iv and z are inter- 
 preted in the same axes, the function lo =- is equivalent to a 
 
 z 
 transformation by reciprocal radii vectors with respect to the 
 unit circle, and a transformation by symmetry with respect to 
 the axis of reals. 
 
168 
 
 ANALYTIC GEOMETRY 
 
 In the equation xv = z^, or R • e'® = ?-^ • e'''^, iv is a single 
 valued function of z, but 2 is a three-valued function of w. 
 
 Since r = i2% ^ = ® + ^^^, the absolute values of the three 
 
 3 3 
 values of z are the same, but their amplitudes differ by 120°. 
 The positive half of the (7-axis, ^ = 0, corresponds to the posi- 
 tive half of the X-axis, and the lines through the origin 
 making angles of 120° and 240° with the X-axis. The entire 
 C/F-plane is pictured by the function iv = z^ on each of the 
 three parts into which these lines divide the XF-plane. 
 
 Art. 85. — Generalized Transcendental Functions 
 
 Since z = x^ iy = r • 6'^^+""-', log z = log r + i{e + 2 mr). The 
 equation w = log 2 may be written u + iv = log ?• -f- i (^ -f- 2 mr). 
 
 Y 
 
 V 
 
 
 
 
 
 
 
 277 
 
 
 
 
 
 
 
 77r 
 i 
 
 
 
 
 
 
 
 37r 
 
 
 
 
 
 
 
 55 
 
 
 
 
 
 
 
 TT 
 
 
 
 
 
 
 
 ¥ 
 
 
 
 
 
 
 
 y2 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 U 
 
 
 
 
 1 
 
 f 
 
 ■'s 
 
 
 
 Hence u = log r, v = ^ + 2 htt. To the circle r = constant in 
 the XF-plane there corresponds in the C7F-plane a straight 
 line parallel to the F-axis ; to the straight line 6 = constant in 
 the XF-plane there corresponds in the f/F-plane a system of 
 parallels to the i7-axis at distances of 2 tt from one anotlier ; w - 
 is an infinite valued function of z, but 2 is a single valued func- 
 
COMPLEX VARIABLE 
 
 1G9 
 
 tion of IV. Tlie entire Xl'-plane is i)ictui'ed between any two 
 successive parallels to the C-axis at distances of 2 tt. 
 Writing the function ?o = sin (x + itj) in the form 
 u -\- IV = sin X cos iij + cos x sin iij, 
 and remembering that 
 
 cosh ?/ = 1 (e" + e *) = cos «//, sinh ?/ = i(e^' - 6-")= - i sin iy, 
 there results 
 
 M -j- iv = cosh ?/ sin .r — / sinh ?/ cos x, 
 whence 
 
 ?t = cosh ?/ sin x, v = — sinh ?/ cos ^'j and sin x = 
 
 cosh u 
 
 cos 
 
 a;= ^, cosh ?/=-r^, sinh^ = 
 
 sinh >j sm x cos x 
 
 sma; 
 
 Substituting in sin-.v+cos-a;=l and cosh-.y-sinli-// = l, there 
 results 
 
 w 
 
 V 
 
 1, — V = i- 
 
 cosh-?/ ' sinh'-^y sui^x cos-iK 
 
 
 Y 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 I/, TT 
 
 H^ 
 
 ,,,. 
 
 T 
 
 
 
 
 
 
 X 
 
 \V4^ \ 
 
 V 
 
 _3___ 
 2 
 
 //X 
 
 
 1 
 
 Uax 
 
 V^' / ffA\ 
 
 
 
 irn 1 " 
 
 
 
 Iv/ ] 
 
 / yyi 
 
 — 
 
 W- 
 
 -'^/jU 
 
 These equations when x and ?/ are respectively constant rep- 
 resent a system of confocal conic sections with the foci at 
 (4- 1, 0), (- 1, 0). The entire system of ellipses filling up the 
 C7F-plane is obtained by assigning to y values from + co to 
 — oc ; the entire system of hyperbolas filling up the C/F-plane 
 
170 ANALYTIC GEOMETRY 
 
 is obtained by assigning to x values from to 2 7r. Hence 
 v: = sin (x + iy) pictures that part of the XF-plane between 
 two parallels to the F-axis at a distance of 2 tt from each other 
 on the entire C/F-plane.* 
 
 Problems. — 1. Show that ■?« = - transforms the system of straight 
 z 
 
 lines through a + ih, and the system of circles concentric at this point 
 
 into systems of orthogonal circles. 
 
 2. Find what part of the AT-plane is transformed into the entire 
 {/ ^''-plane by the function w = z"^. 
 
 3. Into what systems of lines does tp = cos 2; transform the parallels to 
 the X-axis and to the F-axis ? 
 
 *The geometric treatment of functions of the complex variable has 
 been extensively developed by Riemann (1826-G6) and his school. 
 
ANALYTIC GEOMETRY OF THREE 
 DIMENSIONS 
 
 CHAPTER XIII 
 
 POINT, LINE, AND PLANE IN SPACE 
 
 Art. 
 
 — Rectilinear Space Coordinates 
 
 Through a point in space draw any three straight lines not 
 in the same plane. The point is called the origin of coordi- 
 nates, the lines the axes of coordinates, the planes determined 
 by the lines taken two and two, the coordinate planes. The 
 distance of any point P from 
 a coordinate plane is meas- 
 ured on a parallel to that axis 
 which does not lie in the plane, 
 and the direction of the point 
 from the plane is denoted by 
 the algebraic sign prefixed to ^^^ 
 the number expressing the dis- ___' 
 tance. The interpretation of 
 these signs is indicated in the 
 figure. If the distance and 
 direction of the point from the 
 yZ-plane is given, x — a, the ^'"- ''"* 
 
 point must lie in a determinate plane parallel to the J'Z-plane. 
 If the distance and direction of the point from the XZ-plane is 
 given, y = h, the point must lie in a determinate plane parallel 
 to the XZ-plane. If it is known that x = a and y = b, the 
 point must lie in each of two planes parallel, the one to the 
 171 
 
172 ANALYTIC GEOMETRY 
 
 FZ-plane, the other to the XZ-plane, and therefore the point 
 must lie iu a determinate straight line parallel to the Z-axis. 
 If the distance and direction of the point from the XF-plane 
 2; = c is also given, the point must lie in a determinate plane 
 parallel to the XF-plaue and in a determinate line parallel to 
 the Z-axis ; that is, the point is completely determined. 
 
 Conversely, to every point in space there corresponds one, 
 and only one, set of values of the distances and directions of 
 the point from the coordinate planes. For through the given 
 point only one plane can be passed parallel to a coordinate 
 plane, a fact which determines a single value for the distance 
 and direction of the point from that coordinate plane. 
 
 The point whose distances and directions from the coordi- 
 nate planes are represented by x, y, z is denoted by the symbol 
 {x, y, z), and x, y, z are called the rectilinear coordinates of the 
 point. There is seen to be a "one-to-one correspondence" 
 between the symbol {x, y, z) and the points of space. 
 
 Observe that x = a interpreted in the ZX-plane represents 
 a straight line parallel to the Z-axis ; interpreted in the 
 XF-plane a straight line parallel to the F-axis ; but when 
 interpreted in space it represents the plane parallel to the 
 FZ-plane containing these two lines. The equations x = a, 
 y = b interpreted in the XF-plane represents a point ; inter- 
 preted in space they represent a straight line through this 
 point parallel to the Z-axis. 
 
 If the axes are perpendicular to each other, the coordinates 
 are called rectangular, in all other cases oblique. 
 
 Problems. — 1. Write the equation of tlie plane parallel to the rZ-plane 
 cutting the X-axis 5 to the right of the origin. 
 
 2. What is the equation of the FZ-plane ? 
 
 3. What is the locus of the points at a distance 7 below the XF-plane ? 
 Write equation of locus. 
 
 4. Write the equations of the line parallel to the X-axis at a distance 
 -f 5 from the XF-plane and at a distance - 5 from the XZ-plane. 
 
POINT, LINE, AND PLANE IN SPACE 
 
 173 
 
 5. Write the equations of the origin. 
 
 6. What are the coordinates of the point on the Z-axis 10 below the 
 Xr-plane ? 
 
 7. What arc the equations of the Z-axis ? 
 
 8. What are the equations of a line parallel to the Z-axis ? 
 
 9. Explain the limitations of the position of a point imposed by 
 placing X = -I- 5, then y = — 5, then z — - 3. 
 
 10. Locate the points (2, - 3, 5); (- 2, 3, - 5). 
 
 11. Locate (0, 4, 5); (2, 0, - 3). 
 
 12. Locate (0, 0, - 5); (0, - 5, 0). 
 
 13. Show that (o, b, c), {-a, b, c) are symmetrical with respect to 
 the rZ-plane. 
 
 14. Show that («, b, c), (— «, — b, c) are symmetrical with respect to 
 the Z-axis. 
 
 15. Show that (a, b, c), (- a, - b, - c) are symmetrical with respect 
 to the origin. 
 
 Art. 87. — Polar Space Coordinates 
 
 Let (x, y, z) be the rectangular coordinates of any point F in 
 space. Call the distance from the origin to the point r, 
 the angle made by OP with its 
 projection OP' on the XF-plane 
 0, the angle made by the projec- 
 tion OP' with the X-axis <^. r, 
 (fi, 6 are the polar coordinates of 
 the point P. From the figure 
 
 OP' = r ■ cos e, 
 
 X = OP' ■ cos cf) = r cos 6 cos 4>, 
 y = OP' • sin ^ = r cos 6 sin (^, 
 z = r sin 6, 
 
 formulas which express the rec- /Y 
 
 tangular coordinates of any point ^"'' ^^' 
 
 in space in terms of the polar coordinates of the same point 
 
174 
 
 ANAL YTIC GEOMETll Y 
 
 From the figure are also obtained r ={x- + 9/ + z^)'^, sin^ =-, 
 tan 4> = '-, formulas which express the polar coordinates of any 
 point in space in terms of the rectangular coordinates of the 
 same point. 
 
 Problems. — 1. Locate the points whose polar coordinates are 5, 15°, 
 60" ; 8, 90°, 45^ 
 
 2. Find the polar coordinates of the point (3, 4, 5). 
 
 3. Find the rectangular coordinates of the point (10, 30°, 60°). 
 
 4. Find the distance from the origin to the point (4, 5, 7). 
 
 Art. 
 
 Distance between Two Points 
 
 Let the rectangular coordinates of the points be {x', y\ z'), 
 (x", y", z"). From the figure 
 
 If- = D" + (z' - z"y, D" = (x' - x"f + (?/' - y")\ 
 
 hence (1 ) D' = {xJ - x"f + (ij' - ?/")- + (2' - z"f. 
 
 x' — x" is the projection of D on the X-axis; y' — y" the pro- 
 
 Z 
 
 ^-v., 
 
POINT, LINE, AND PLANE IN SPACE 175 
 
 jection of D on the I'-axis; z' — z" the projection of D on the 
 ^axis.* Calling the angles which B makes with the coordi- 
 nate axes respectively X, Y, Z, 
 
 x' — x" = D cos X, y' — y" = D cos Y, z' — z" = D cos Z. 
 Substituting in (1), there results 
 
 D- cos- X + D- cos- Y+ D- cos- Z = D% 
 whence cos- X + cos' Y+ cos- Z=l; 
 that is, the sum of the squares of the cosines of the three 
 angles which a straight line in space makes with the rectangu- 
 lar coordinate axes is unity. 
 
 The distance from (x', y', z') to the origin is Va;'" + y'^ + z'^. 
 If the point {x, y, z) moves so that its distance from {x', y', z') 
 is always R, the locus of the point is the surface of a sphere 
 and (x - x'Y + {y- y'-) + {z-z'f = Er, which expresses the 
 geometric law governing the motion of the point, is the equa- 
 tion of the sphere whose center is (x', y\ z'), radius R. 
 
 Problems. —1. Find distance of (2, - 3, 5) from origin. 
 
 2. Find the angles which the line from (3, 4, 5) to the origin makes 
 with the coordinate axes. 
 
 3. Find distance between points ( - 2, 4, - 5), (3, - 4, 5), 
 
 4. Write equation of locus of points whose distance from (4, - 1, 3) 
 is 5. 
 
 5. Write equation of sphere center at origin, (2, 1,-3) on surface. 
 
 6. The locus of points equidistant from (x', y', z'), (x", y", z") is the 
 plane bisecting at right angles the line joining these points. Find the 
 equation of the plane. 
 
 7. Find the equation of the plane bisecting at right angles the line 
 joining (2, 1,3), (4,3, -2). 
 
 * The projection of one straight line in space on another is the part of 
 the second line included between planes through the extremities of the 
 first line peri^endicular to the second. The projection is given in direc- 
 tion and magnitude by tlie product of the line to be projected into tlie 
 cosine of the included angle. 
 
176 
 
 A NA L YTIC GEOMETR Y 
 
 8. Show that 
 
 y' + ir 
 
 + z< 
 
 is the point midway be- 
 
 tween (x', j/', z')^ {%", y", z"). 
 
 9. Find the point midway between (4, 5, 7), (2, — 1, 3). 
 
 10. Find the equation of the sphere wliich has the points (4, 5, 8), 
 (2, — 3, 4) at the extremities of a diameter. 
 
 11. Write the equation of the spliere with the origin on the surface, 
 center (5,0, 0). 
 
 12. Find angles which the line through (2, 3, - 5), (4, - 2, 3) makes 
 with the coordinate axes. 
 
 13. The length of the line from the origin to (x, ?/, z) is ?•, the line 
 makes with the axes the angles a, /8, 7. Show that x = rcosa, ?/ = rcos 3, 
 z = r cos 7. 
 
 Art. 
 
 Equations of Lines in Space 
 
 Suppose any line in space to be given. From every point of 
 the line draw a straiglit line perpendicular to the XZ-plane. 
 There is formed the surface which projects the line in space 
 on the XZ-plane. The values of x and z are the same for all 
 points in the straight line which projects a point of the line 
 
 in space on the XZ-plane. 
 Hence the equation of the 
 projection of the line in 
 space on the XZ-plane when 
 interpreted in space repre- 
 sents the projecting surface. 
 The projection of the line in 
 space on the XZ-plane deter- 
 mines one surface on which 
 /Y the line in space must lie. 
 
 The projection of the line 
 in space on the I''Z-plane determines a second surface on 
 which the line in space must lie. The equations of the pro- 
 jections of the line in space on the coordinate planes XZ and 
 
POLXT, LINE, AND PLANE IN SPACE 
 
 177 
 
 YZ therefore determine the line in space and are called the 
 equations of the line in space. By eliminating z from tlie 
 equations of the projections of the line on the planes XZ and 
 YZ, the equation of the projection of the line on the XF- 
 plane is found. 
 
 Art. 90. — E(juATroNs of the STUAKiiix Link 
 
 iglit line on the 
 h «, y = bz + (3. 
 
 The equations of the projections of the sti 
 coordinate planes XZ and YZ are x = az 
 The geometric meaning of 
 a, b, a, fi is indicated in 
 the figure. The elimina- 
 tion of z gives 
 
 y-(3 = l{x-a), 
 
 the equation of the pro- 
 jection of the line in the 
 XF-plane. 
 
 Two points, 
 
 (x', y', z'), {x", y", z"), 
 determine a straight line 
 m space. The projection 
 of the line through the 
 points {x\ y,' z'), (x", y", z") on the ZX-plane is determined by 
 the projections (a;', z'), (x", z") of the points on the ZX-plane, 
 likewise the projection of the line on the ZF-plane is deter- 
 mined by the points {z', y'), (z", y"). Hence the equations of 
 the straight lines through {x', y', z'), (x", y", z") are 
 
 Fm. 159. 
 
 ^-(z-z'), y-y' = 
 
 -(z-z'). 
 
 A straight line is also determined by one point and the direc- 
 tion of the line. Let {x', y', z') be one point of the line, «, /3, y 
 
 N 
 
178 
 
 ANALYTIC GEOMETRY 
 
 the angles which the line makes with the axes X, Y, Z respec- 
 tively. Let {x, y, z) be any point 
 of the line, d its distance from 
 {x', y', z'). Then 
 
 z-x' ^y-y' ^z-z' ^_^ 
 cos a cos 13 cos y 
 is the equation of the line. This 
 equation is equivalent to the equa- 
 tions x=x' + d cos a, y=y'+d cos (3, 
 z = z' + d cos y, which express the 
 coordinates of any point of the line 
 in terms of the single variable d. 
 
 If the straight line (1) contains 
 the point (x", y", z"), 
 x"-x' ^ y"-y' ^ z" -z' , 
 
 cos a cos /8 cos y 
 Eliminate cos a, cos (3, cos y from (1) and (2) by division, and 
 the equation of the straight line through two points is obtained 
 x — x' _ y — y' _ z — z' 
 x" — x' y" — y' z" — z' 
 as found before, a, (3, y are called the direction angles of the 
 straight line. 
 
 Problems. — 1. The projections of a straight line on the planes XZ 
 y z 
 
 Find the projection on the XY 
 - 5, ?/ = 2 2 — 3 with the coordinate 
 
 and YZ are 2 x + 3 
 plane. 
 
 2. Find the intersections of x 
 planes. 
 
 3. Write the equations of the straight line through (2, 3, 1), ( - 1, 3, 5) . 
 
 4. Write the equations of the straight line through the origin and the 
 point (4, - 1, 2). 
 
 5. Write the equations of the straight line through (3, 1, 2) whose 
 direction angles are (60°, 45°, G0°). 
 
 6. The direction angles of a straight line are (45°, 60°, 60°) ; (4, 5, 6) 
 is a point of the line. Find the coordinates of the point 10 from (4, 5, 6). 
 
POINT, LINE, AND PLANE IN SPACE 
 
 ITU 
 
 Akt. 91. 
 
 Let 
 
 X — a 
 cos « 
 
 Angle hetwkex Two Stuaigux Lines 
 
 X 
 
 y-b ^z~c 
 cos (i cos y 
 
 if 
 
 cos /8' 
 
 cos a' cos li' cos y 
 be the straight lines. The angle between the lines is by definition 
 the angle between parallels to 
 the lines through the origin. 
 Let OM' and OM" be these 
 parallels through the origin. 
 From any point P'{x\ y\ z') 
 of OM' draw a perpendicular 
 P'P" to OM". Then OP" is 
 the projection of OP' on OM", 
 and OP" is also the projection 
 of the broken line {x' + y' + z') 
 on OM".* Hence / 
 
 r' cos 6 = x' cos a' + y' cos ^' 
 
 + Z' cos y, 
 
 = '-cos a -|-— cos 
 r' y' 
 
 + -C0Sy 
 7 
 
 that 
 
 cos = cos a COS a' + cos /3 cos (3' + cos y cos y'. 
 
 (1) 
 
 * The sum of the projections of 
 the parts of a broken line on any 
 straight Hne is the part of the line 
 included between the projections of 
 tlie extremities of the broken line. 
 a!) is the projection of AB ; be is the 
 projection of BC; ac is the projec- 
 tion of AB + BC. 
 
 Fio. 162. 
 
180 ANALYTIC GEOMETRY 
 
 If the equations of the lines are written in the form 
 
 X = az + a, y — bz + (3] x= a'z + «',?/ = b'z + (3', 
 
 the equations of parallel lines through the origin are 
 
 X = az, y = bz ; x = a'z, y — b'z. 
 Let (x', y', z') be any point of the first line, its distance from 
 the origin r'. Then x' — az', y' — bz', r'? = x'- + y'^ + z'-, whence 
 
 cos a 
 
 x' 
 ~ r' 
 
 r' 
 
 = 
 
 
 a 
 
 
 
 VI 
 
 
 
 
 cos/8 
 cosy 
 
 b 
 
 + 
 
 b' 
 
 vr 
 
 + a- 
 1 
 
 T 
 
 1?' 
 
 
 
 
 
 Likewise if {x", y", z") is any point of the second line, r" its 
 distance from the origin. 
 
 Substituting in (1) 
 
 cos ii' 
 
 x" 
 r" 
 
 a' 
 
 
 
 V'l -f- «'- 
 
 ■■ + b'^ 
 
 cosfi' 
 
 r" 
 
 b' 
 
 
 VI + a"^ 
 
 '■ + b" 
 
 cosy' 
 
 _z" 
 r" 
 
 1 
 
 
 VI + a" 
 
 + b" 
 
 
 
 l+aa' + bb' 
 
 Vl + a^ + bWl + a" + b" 
 When the lines are perpendicular, cos ^ = 0, whence 
 
 1 + aa' + bb' = 0. 
 When the lines are parallel, cos^ = 1, whence 
 ^ _ 1 + aa,' + bb' 
 
 Vl + a- + bWl + a" + b" 
 which reduces to 
 
 (a' - ay + (b' - by + (ab' - a'by = 0. 
 
POINT, LINE, AND PLANE IN SPACE 181 
 
 This eciuation requires that a = a', b = b' ; that is, if two lines 
 are parallel, their projections on the coordinate planes are 
 parallel. 
 
 The equations of the straight line through (x', ?/', z') parallel 
 to x=az + a, y==bz-{- ^ are a; — x'= a {z — z'), y ~ y' = b{z — z'). 
 
 The straight line (1) x — x' = a'(z — z'), y — y' = b' {z — z') 
 through the point (x', y', z') is perpendicular to the straight line 
 (2) X = az + a, y = bz + ft when a' and b' satisfy the equation 
 1 + cm' + bb' = 0. This equation is satisfied by an infinite 
 number of pairs of values of a' and b'. This is as it ought to 
 be, for through the given point a plane can be passed perpen- 
 dicular to the given line, and every line in this plane is perpen- 
 dicular to the given line, and conversely. Hence if the straight 
 line (1) is governed in its motion by the equation l + aa'-j-6^' = 0, 
 it generates the plane through (x', y', z') perpendicular to the 
 straight line (2). 1 -f aa' -\- bb' = is the line equation of the 
 plane. 
 
 To find the relation between the constants in the equa- 
 tions of two straight lines x = az + a, y = bz + ^, x = a'z + a', 
 y — b'z + /3', which causes the lines to intersect, make these 
 equations simultaneous and solve the equations of the projec- 
 tions on the XZ-plane, also the equations of the projections on 
 
 the I'Z-plane, for z. The two values of z, and ^^ ^ 
 
 a — a' b — b' 
 
 must be equal if the lines intersect. Hence for intersection 
 the equation (a — a') (y8' — ft) — (b — b') {a' ~ a) = must be 
 satisfied, and the coordinates of the point of intersection are 
 
 aa' — a'a bB' — b'B «' — a -.^r, , -, 
 
 x = , y = -^ -, z = When a = a and 
 
 a — a' b — b' a — a' 
 
 I) = //, the point of intersection is at infinity, and the lines are 
 parallel, as found before. 
 
 Problems. — 1. Find the angle between the lines 
 
 x = Sz + \, y = ~ 22 + 5; x = z + 2, y = - z + i. 
 2. Find the angle between the lines through (1, 1, 2), (-3, - 2, 4) 
 and (2, 1, - 2), (3, 2, 1). 
 
182 
 
 ANALYTIC GEOMETRY 
 
 3. Find equations of line through (4, - 2, 3) parallel to a: = 4 2: + 1, 
 y = 2 z — b. 
 
 4. Find line through (1, - 2, 3) intersecting x = -2z-\-o, y=z + 5 
 at right angles. 
 
 5. Find distance from (2, 2, 2) to line x = 2 z -\- l, y = - 2 z + S. 
 
 6. Find equations of line intersecting each of the lines x = 3^ + 4, 
 y = -z + 2 and y = 2 z - 5, x - - z + 2 at right angles. 
 
 7. For what value of a do the lines x = Sz + a, y = 2z + 5 and 
 X = - 2 z - o, y = i z - d intersect ? 
 
 8. Find the equations of the straight line through the origin intersect- 
 ing at right angles the line through (4, 2, - 1), (1, 2, - 3). 
 
 9. Find distance of point of intersection of lines x = 2z-\-l, 
 y = 2z + 2 and x = z + 5, y-iz-6 from origin. 
 
 10. Find distance from origin to line x-iz-H, y = — 2z + 3. 
 
 Art. 
 
 The Plane 
 
 A plane is determined when 
 the length and direction of the 
 perpendicular from the origin 
 to the plane are given. Call 
 the length of the perpendicular 
 p, the direction angles of the per- 
 pendicular a, (3, y. Let P(.i-, ?/, z) 
 be any point in the plane. The 
 projection of the broken line 
 (;f + y + z) on the perpendicular 
 OP' equals p for all points in the 
 plane and for no others. Hence 
 xcosa + y cos /8 + 2: cos y = p is 
 the equation of the plane. This 
 is called the normal equation of 
 the plane. 
 Every first degree equation in three variables when inter- 
 preted in rectangular coordinates represents a plane. The 
 
POINT, LINE, AND PLANE LW SPACE 183 
 
 locus reprostMiiod by Ax + B>i + Cz + D = is the same as 
 the locus represeuted by u;cos a + ij (ios fi + z cos y - 2' = ^ i^' 
 
 Con 
 
 
 cos a 
 A 
 
 cos^ 
 B 
 
 _ cos y 
 
 c 
 
 D 
 
 
 ibiniiu 
 
 ; witli 
 
 c + cos- 13 + cos 
 
 r-y=\ 
 
 , cos a 
 
 
 A 
 
 
 V^i- 
 
 -\-B'+G' 
 
 cos 
 
 3 B 
 
 
 cos y = 
 
 
 C 
 
 "^ V^- +B' + C 
 
 V^l' + B' + C 
 
 
 P = - 
 
 
 D 
 
 
 
 
 V^l- 
 
 + B' + C 
 
 
 ice the 
 
 factor 
 
 ] 
 
 
 _ 
 
 
 Va:' + B-+ c- 
 
 transfonns Ax + Bi/ + Cz + D = into an e(iuatiou of the form 
 .1- cos a + y cos (3 + z cos y = p, which is the equation of a 
 plane. 
 
 •^ ^- •'/-)-?=: 1 is the equation of the plane whose intercepts 
 
 a b c 
 on the coordinate axes are a, b, c. This is the intercept 
 equation of the plane. 
 
 The plane represented by the equation Ax + By + Cz-{- D — 
 depends on the relative values of the coefficients. Hence the 
 equation of the plane has three parameters. To find the equa- 
 tion of the plane through three points {x\ y', z'), (_x", y", z"), 
 {x'", y'", z'"), substitute these coordinates for x, y, z in 
 (1) A'x + By + C'a; + 1 = 0, solve the resulting e(iuations for 
 A', B', C, and substitute in (1). 
 
 The intersections of a plane with the coordinate planes are 
 called the traces of the plane on the coordinate planes. The 
 equation of the trace of Ax + By +Cz + D = on the X^-plane 
 is found by making ?/ = in the equation of the plane. The 
 trace is therefore Ax + Cz + D = 0. The trace on YZ is 
 By + Cz + D = 0, on XY is Ax + By + D = 0. 
 
184 ANALYTIC GEOMETRY 
 
 For points in the intersection of the planes 
 Ax + By+Cz + D = and A'x + B'y + C'z-\-D' = 
 these equations are simultaneons. Eliminating >j, 
 
 {AB' - A'B)x + (CB' - C'B)z + {DB - D' D) = 0, 
 the equation of the projection of the intersection on the co- 
 ordinate plane XZ. In like manner the equations of the pro- 
 jections of the intersection on the planes YZ and XF are 
 obtained. 
 
 Problems. —1. Write the equation of the plane whose intercepts on the 
 axes are 2, — 4, — 3. 
 
 2. Find the equation of the plane through (2, - 3, 4) perpendicular 
 to the line joining this point to the origin. 
 
 3. Find the equation of the plane through (2, 5, 1), (3, 2, -5), 
 (1, -3,7). 
 
 4. Find the equations of the traces of3a;-?/ + 5z— 15 = 0. 
 
 5. Find the equations of the intersection of Sx + by ~ 7 z + 10 = 0, 
 bx -Utj + Sz - lb = 0. 
 
 6. Find the equation of the plane through (3, - 2, 5) perpendicular to 
 a: -1 _ ?/ + 2 _ g - 3 
 
 cos GO'^ cos 45"^ cos G0° 
 
 n. Find the direction angles of a perpendicular to the plane 
 
 2x-3?/+52 = 6. 
 8. Find the length of the perpendicular from the origin to 
 
 2x-3y + bz=^G. 
 
 Art. 93. — Distance from a Point to a Plane 
 
 Let (x', y\ z') be a given point, cc cos « + 2/ cos ^ + 2 cos y =i>, 
 a given plane. Through {x\ y', z') pass a plane parallel to the 
 given plane. The equation of this parallel plane is 
 a; cos « -h 2/ cos ^ + » cos y = OF". 
 
POINT, LINE, AND PLANE IN SPACE 
 
 m 
 
 The point (x', y', z') lies in this plane, therefore 
 x' cos a + y' cos ^ + 2' cos y = OP". 
 Subtracting OP from both 
 sides of this equation, 
 x' cos a + y' cos fi 
 
 -f- z' cos y — i^ = P-P" ; 
 that is, the perpendicular 
 distance from (x', y', z') to 
 xcosa + y cos (3 
 
 + 2 cos y — p = 
 is the left-hand member of 
 this equation evaluated for 
 (x', y', z'). The sign of the 
 perpendicular is plus when 
 the origin and the point 
 (x', y\ z') are on different 
 sides of the plane, minus 
 when the origin and the 
 
 , /,,,■. jl 1' 1(1. K«. 
 
 point (a; , y', z) are on the 
 same side of the plane. 
 
 The distance from {x', y', z') to the plane Ax-\-By+Cz-\-D=0 
 is found by transforming the equation of the plane into the 
 form X cos a -{- y cos 13 + z cos y — j? = to be 
 Ax' + By' + Cz' + D 
 
 Let .Tcos a-\-y cos /? + 2: cos y — p = and 
 
 X cos «' -f ?/ cos )8' -f 2 cos y' — ;y = 
 bo the faces of a diedral angle, 
 (.« cos «+?/ cos ^+2;cos y— ^))±(.); cos «'+?/ cos ^' + z cosy'— p')=0 
 
 is the equation of the locus of points equidistant from the 
 faces: that is, the eipiation oC the bisectors of the diedral 
 angle. 
 
186 ANALYTIC GEOMETRY 
 
 Problems. — 1. Find distance from origin to plane 
 Ux-13y+nz + 22 = 0. 
 
 2. Find distance from (3, - 2, 7) to 3 a; + 7 ?/ - 10 s + 5 = 0. 
 
 3. Write the equations of the bisectors of the diedral angles whose 
 faces are 2 X + 5 y — 7 z = 10, and ix-y + Gz — l!i = 0. 
 
 4. Find distance from (0, 5, 7) to - + | + ? = 1. 
 
 5. Find distance from origin to | ce — | ?/ -| | ^ = 1- 
 
 Art 94. — Angle between Two Planes 
 
 Let a; cos a -\- y cos ^+z cos y=2^, a; cos «'+?/cos^'+2:cosy'=jy 
 be two given planes, their included angle. The angle be- 
 tween the planes is the angle between the perpendiculars to 
 the planes from the origin. Hence 
 
 cos 6 = cos a cos a' + cos (3 cos ^' -\- cos y cos y'. 
 If the equations of the planes are in the form 
 
 Ax + B>j + Cz + D = 0, A'x + B'y + C'z + Z)' = 0, 
 
 cos a = ■ — > 
 
 V.4^ + B'+ C 
 
 COS «■ = ? 
 
 cosB — ^ > 
 
 cos IS' - ^' , 
 
 V^- + B'+ C 
 
 ^AJ' + B'-'+C" 
 
 
 
 pi 
 
 cos y' - 
 
 V^» + B' + C- 
 
 V^'- + B" + C" 
 
 r.no. .„...- AA' + BB'+CC . 
 
 VA' + B'+ CWA'^ + B'' + C"2 
 
 The planes are perpendicular when AA' + BB' + CC = ; 
 
 ,, , , . AA' + BB' + CC 1 . -, 
 
 parallel when 1 = — -'— — — ' r — =r — , which 
 
 V2- + B' + C-VA" + B'- + C- 
 reduces to {AB' - yl'i^)^ + {AC - A'Cf + (7JC" - B'Cf = 0, 
 1 A B C 
 
POINT, LINE, AND PLANE IN SPACE 1ST 
 
 The angle between the plane xcos a -\->/ con f3 + zcosy =p 
 y ~ •' =z ~ '^ is the conipUnnent of the 
 
 cos «' cos /8' cos y' 
 angle between the line and the perpendiculai- to the plane. 
 Hence sin 6 = cos a cos a' + cos ft cos /3' + cos y cos y'. If the 
 equations of line and plane are in the form x = az + «, 
 
 y = hz + p, and Ax + By -\- Cz-{- D = 0, 
 
 cos a = 
 
 Vyl- + B'-^ C 
 
 cos/8 = 
 
 B 
 
 V^l- + B' + C" 
 
 COSv = 
 
 C 
 
 a 
 
 coS|8' 
 
 COS y' 
 
 Vl + a2 + 62 
 
 Vl + a- 4- b^' 
 1 
 
 V^- + i3- + C - ' V 1 + a- + 62 
 
 Hence sin^ = ^a + m +C _. 
 
 VJ^M^- + C- Vl + a- + b^ 
 
 The line is parallel to the plane when ^hi -\- Bb -j- C=0; 
 perpendicular when 1= Aa + B b + C ^^^^^^^^ 
 
 VA' + B'+ (J- Vl + a' + b^ 
 reduces to {Ab - Baf -\- {A - Ccif + (B - Cbf = 0, whence 
 
 .1 , B 
 
 a = —, b = — 
 
 (f C 
 
 To find the intersection of the line x = az -{- a, y — bz -j- (S, 
 and the plane Ax + By + Cz + J9 = 0, make these equations 
 simultaneous, and solve for x, y, z. There results 
 
 ^ _ Au + B^ + D 
 
 Aa + Bb + C 
 
 If Aa -j- Bb + C—0, the point of intersection goes to infinity, 
 and the line and plane are parallel, as found before. If 
 Aa -I- B(3 + C also vanishes, z lieconies indeterminate, likewise 
 x and //, and the line lies wholly in the plane. 
 
188 ANALYTIC GEOMETRY 
 
 If the plane Ax-\- By + Cz-\- D = contains the point 
 (x', y', z') and the line x = az + a, y = bz-{-^, 
 
 Ax' + By' + Cz'+D = 0, Aa -\- Bb + C= 0, Aa + B(3-\-D = 0. 
 
 These equations determine the relative values of A, B, C, D, 
 hence the plane is determined. 
 
 The plane Ax + By -{- Cz + D = contains the two lines 
 X = az + a, y = bz + (^ and x = a'z + a', y = b'z + (3' when 
 Aa + Bb-^C^O, Aa + Bf3 + D^0, Aa' + Bb' + C^O, 
 Aa' + B/3' + D =0. These four equations are consistent only 
 
 when — — '^^'^ that is, when the lines intersect, and 
 
 b—b' /? — /3 
 
 then the relative values of A, B, C, D, which determine the 
 
 plane, are found by solving any three of the four equations. 
 
 Problems. — 1. Find angle between planes 10 x — 3 y + 4 £• + 12 = 0, 
 15 X + 11?/ -7 2 + 20 = 0. 
 
 2. Find angle between line x = 5 z + 7, y = S z — 2, and plane 
 2x- 15?/ + 200 + 18=0. 
 
 3. Find equation of plane through (4, — 2, 3) parallel to 
 3x-2y + z- 5 = 0. 
 
 4. Find equation of plane through (1, 2, — 1) containing the line 
 x = 2z — S, y = z + a. 
 
 5. Find equation of line through (4, 2, — 3) perpendicular to 
 x + Sy -2z + 4 = 0. 
 
 6. Find equation of plane containing the lines x = 2z + \, y = 2z + 2, 
 and x = z + 5, y = 4:Z — G. 
 
 7. Find angles which Ax + By + Cz + D = makes with the coordi- 
 nate axes. 
 
 8. Find angles which Ax + By + Cz + D ■= makes with the coordi- 
 nate planes. 
 
 9. Show that if two planes are parallel, their traces are parallel. 
 
 10. Show that if a line is perpendicular to a plane, the projections of 
 the line are perpendicular to the traces of the plane. 
 
POINT, LINE, AND PLANE IN SPACE 189 
 
 11. Show that ^ ~ ^' = ^''s:Jl!. = ?-IiA is perpendicular to 
 
 A B C 
 
 Ax + ny + a:: + D = 0. 
 
 12. Show that (x>-x"){x-x") + (7j' -y")(>/~y") + (z' -z")(:-z")=0 
 is a plane through (x", y", z") perpendicular to the line through 
 (x', y\ z') and (x", y", z"). 
 
 13. Find the equation of the plane tangent to the sphere x^ + y'^+z'^ — R- 
 at the point (x", y", z") of the surface. 
 
 14. Find the equation of the plane tangent to the sphere 
 
 (X - X')- + (2/ - y'y + {s- z'Y = R- 
 at the point (x", y" , s") of the surface. 
 
CHAPTER XIV 
 
 OUKVED SURPAOES 
 
 trix of a cylindrical surface 
 Z 
 
 Art. 95. — Cylindrical Surface.s 
 
 Let the straight line x = az + a, y = bz -\- (3 move in such a 
 manner that it always intersects the XF-plane in the curve 
 F(x, y) = 0, and remains parallel to its first position. The 
 straight line is the generatrix, the curve F{x, y)=0 the direc- 
 
 The generatrix pierces the 
 XF-plane in the point («, (3), 
 and therefore F(a, (3)^0. 
 This is the line equation of 
 the cylindrical surface, for 
 since a and b are constant, to 
 every pair of values of a and 
 13 there corresponds one posi- 
 tion of the generatrix, and 
 to all pairs of values of a 
 and 13 satisfying the equation 
 F(a, 13) =0 there corresponds 
 the generatrix in all positions 
 *''*"■ '^^^- while generating the cylindri- 
 
 cal surface. To obtain the equation of the cylindrical sur- 
 face in terras of the coordinates of any point (.r, y, z) of the 
 surface, substitute in F{(z, /8)=0 the values of a and ^ ob- 
 tained from the equations of the generatrix. There results 
 F(:x-az, y-bz)=0, the equation of the cylindrical surface 
 whose directrix is F(x, y) = 0, generatrix x=az-\-a, y=bz + (3. 
 190 
 
CURVED SURFACES 
 
 11)1 
 
 + ^ = 1. What does this equation become when elements are parallel 
 6- 
 
 ele- 
 
 Problems. — 1. Find the equation of the right circular cylinder whose 
 directrix is x- + y~ — f-, and axis the Z-axis. 
 
 2. TIic directrix of a cylinder is a circle in the A'l'-plane, center at 
 origin. The element of tlie cylinder in the ZA'-plane makes an angle of 
 45° with the A'-axis. Find equation of surface of cylinder. 
 
 3. Find general equation of surface of cylinder whose directrix is 
 
 X- , y^ 
 
 to Z-axis ? 
 
 4. Find equation of cylindrical surface directrix y- — \(ix 
 ments parallel to x = 22 + 5, ?/= — 3^ + 5. 
 
 5. Determine locus represented by 
 
 a; = a sin </>, y = a cos 0, z = r(p. 
 
 Since x- + y- = a-, the locus must lie on 
 the cylindrical surface whose axis is the 
 Z-axis, radius of base a. Points corre- 
 sponding to values of (p differing by 2 tt lie 
 in tlie same element of the cylindrical sur- 
 face. The distance between the successive 
 points of intersection of an element of the 
 cylindrical surface with the locus is 2 wc. 
 The locus is tlierefore the thread of a cylin- 
 drical screw with distance between threads 
 2 TTC. The curve is called the helix. 
 
 Art. 96. — Conical Surfaces 
 
 Let the straii^^ht lino x = az -f «, y = hz + (i move in such a 
 manner that it always intersects the XT-plane in the curve 
 F(x, ?/) = 0, and passes through the point {x\ ?/', z'). The 
 straight line generates a conical surface whose vertex is 
 (:r', ?/', z'), directrix F{x, y) = 0. The equations of the generatrix 
 are x — x' = a(z — z'), y — ?/' = h{z — z'), which may be written 
 x=az-{-{x'—az'), y=bz-{-(y'—hz'). This line pierces the XY- 
 
192 
 
 Analytic geometry 
 
 plane in {x' — az\ y' — bz'), and therefore F(x'—az', y' — bz')=0. 
 This is the line equation of the 
 conical surface, for to every pair 
 of values of a and b there corre- 
 sponds one position of the gen- 
 eratrix, and to all pairs of values 
 of a and b satisfying the equa- 
 tion F(x' — az', y'—bz) = there 
 corresponds the generatrix in all 
 positions while generating the 
 conical surface. To obtain the 
 equation of the conical surface 
 in terms of the coordinates of 
 any point (x, y, z) of the surface, 
 substitute in F(x' — az', y' — bz')= for a and b their values 
 obtained from the equations of the generatrix. There results 
 
 j^r x'z - xz' y'z-yz' \^ ^^^ equation of the conical surface 
 
 \ z — z' z~z' J 
 whose vertex is (x', y', z'), directrix F(x, y) = 0* 
 
 Problems. — 1. Find the equation of the surface of the right circular 
 cone whose axis coincides with the Z-axis, vertex at a distance c from tlie 
 origin. 
 
 2. Find the equation of the conical surface directrix ^ + l^= 1, ver- 
 tex (5, 2, 1). 
 
 3. Find the equation of the conical surface vertex (0, 0, 10), directrix 
 2/2 = 10 X - x~. 
 
 4. Find the equation of the conical surface vertex (0, 0, c), directrix 
 ^ + ?^'=1. 
 
 5. Find the equation of the conical surface vertex (0, 0, 10), directrix 
 a;'^ + 2/2^0. 
 
 * Surfaces which may be generated by a straight line are called ruled 
 surfaces. 
 
CURVED sun FACES 
 
 19a 
 
 Art. 97. — Sukkacks ok Kkvolution 
 
 Let 3fN be any line in the ZX-plane. When MN revolves 
 about the Z-axis, every point /* of JfiV deseribes the circumfer- 
 ence of a (drcle witli its center 
 on the Z-axis and whicli is pro- 
 jected on the XF-plane in an 
 equal circle. The equation of 
 the circle referred to a pair of (' 
 axes through its center parallel 
 to the axes X and Y is 
 or -j-y- = t^. 
 This is also the equation of the 
 })rojectiou of the circle on the 
 Xl''-})lane. The radius r is a 
 function of z whicli is given by 
 
 the equation of the generatrix r = F(z). Hence the equation 
 of the surface of revolution is obtained by eliminating /• from 
 the equations af' + y- — r' and r = F(z). 
 
 Problems. — 1. Find equation of surface of sphere, center at origin, 
 radius li. This sphere is generated by the revolution about the Z-axis of 
 a circle whoso e(iualion is r~ + z- = R-. Eliminate r from this equation 
 and x- -|- y- — ?•-, and the ecpiation of the sphere is found to be 
 
 X-' -f if -V Z'^:= R\ 
 
 2. Find equation of right circular cylinder vyliose axis is the Z-axis. 
 
 3. Find equation of right circular cone whose axis is Z-axis, vertex 
 (0, 0, c). 
 
 4. Find equation of right circular cone whose axis is Z-axis, vertex 
 (0, 0, 0). 
 
 5. Find equation of surface generated by revolution of ellipse about its 
 
 niajiir axis. This is the prolate spheroid. 
 
 6. Find ((juation of surface generated by revolution of ellipse about its 
 minor axis. This is the oblate spheroid. 
 
 o 
 
194 
 
 A NA L YTIC GEOMETK Y 
 
 1. Find equation of surface generated by revolution of hyperbola about 
 its conjugate axis. This is the hyperboloid of revolution of one sheet. 
 
 8. Find equation of surface generated by revolution of hyperbola about 
 its transverse axis. This is the hyperboloid of revolution of tv?o sheets. 
 
 9. Let PP' be perpendicular 
 to the JT-axis, but not in the 
 ZX-plane. Suppose PP' to re- 
 volve about the Z-axis. The 
 equation of the surface gener- 
 ated is to be found. 
 
 The equations of the projec- 
 tions of PP' on the planes ZX 
 and ZY are x = a.,y = bz. The 
 point P describes the circum- 
 ference of a circle whose equa- 
 tion is x2 + ^2 _ ,.2. The value 
 i of r depends on z, and from the 
 
 ^^«- 1'^"- figure r''= a?+ Vh'K Hence 
 
 the equation of the surface generated is x- + y'^ = &%- -f a'-. The surface 
 is thei'efore an hyperboloid of revolution of one sheet. 
 
 Akt. 
 
 The Ellipsoid 
 
 In the XF-plane there is the fixed ellipse ^-f--^ = 1, in the 
 .2 2 «' ^ 
 
 ZX-plane the fixed ellipse - + -" = 1. The figure generated 
 a- c- 
 by the ellipse which moves with 
 its center on the X-axis, the plane 
 of the ellipse perpendicular to the 
 X-axis, the axes of the ellipse in 
 any position the intersections of 
 the plane of the ellipse Avith the 
 fixed ellipses, is called the ellip- 
 soid. The equation of the ellipse 
 
 From the equations of the fixed ellipses 
 
C Uli \ 'ED S URFA CES 
 
 195 
 
 — +-- = 1, — + --=1, whence rs =lrll- \,rt—c-(l ]• 
 
 a- b- a- c- \ a-j \ a-j 
 
 Hence tlie equation of the generating ellipse in any position, 
 
 that is, the e(uiation of the ellipsoid, is — + 4, + — , = 1- When 
 
 (r b- & 
 a, b, c arc unequal, the figure is an ellipsoid with unequal 
 axes; when two of the axes are equal, the figure is an ellipsoid 
 of revolution, or spheroid; Avhen the three axes are equal, the 
 elli[)soid becomes the sphere. 
 
 Art. 
 
 The Hyperboloids 
 
 In the ZX-plane there is the fixed hyperbola — — r, = 1, in 
 the Zl'plane the fixed hyperbola ^ — ^, 
 
 cr 
 1. The figure gen- 
 
 Z 
 
 crated by the ellipse which 
 moves with its center on 
 the Z-axis, the plane of the 
 ellipse perpendicular to the 
 Z-axis, the axes of the el- 
 lipse in any position the 
 intersections of the plane 
 of the ellipse with the fixed 
 hyperbolas, is called the 
 hyperboloid of one sheet. 
 The equation of the ellipse Fig. i7t. 
 
 1. From the equations of the fixed hyperbolas 
 
 rs' rt 
 
 ri _?'=:!, ^-^=1, whence ^s'=a-(l + '^, 7r=b'(l+''\ 
 rr c- b' c- ' \ c-J \ c-J 
 
 Hence the equation of the generating ellipse in any posi- 
 tion, that is, the equation of the hyperboloid of one sheet, is 
 
 a- b- c- yS ^2 
 
 In the ZX-plane there is the fixed hyperbola — — = ^^ 
 
196 
 
 ANALYTIC GEOMETRY 
 
 in the XT-plane the fixed hyperbola — — ^-=^1. The figure 
 
 a^ h- 
 
 generated by the ellipse which moves with its center on the 
 
 X-axis, the plane of the el- 
 ^ '' lipse perpendicular to the 
 
 X-axis, the axes of the el- 
 lipse in any position the in- 
 tersections of the plane of 
 the ellipse with the fixed 
 hyperbolas, is called the 
 hyperboloid of two sheets. 
 The equation of the ellipse 
 
 is ^ + |r,= l- Fi'oi^ tlie 
 
 Fig. 172. '>'^ ^'^ 
 
 9 — 2 o -p 
 
 equations of the fixed hyperbolas — — -—-=1, — -^—^1 
 
 whence ri^h'{-^-\\, '^V^ (?(^^-\\. Hence the equation 
 
 of the generating ellipse in any position, that is, the equation 
 
 of the hyperboloid of two sheets, is — ■ 
 
 Art. 100. — The Paraboloids 
 
 In the XF-plane there is the fixed parabola -if' — 2 hx, in the 
 ZX-plane the fixed parabola z^=2 ex. The figure generated 
 by an ellipse which moves with its 
 center on the X-axis, its plane per- 
 pendicular to the X-axis, the axes 
 of the ellipse in any position the 
 intersections of the plane of the 
 
 X ellipse with the fixed parabola, is 
 
 called the elliptical paraboloid. The 
 equation of the ellipse is 
 
 rs rt 
 
CUR VED S UIIFA CES 
 
 11)7 
 
 From the equations of the fixed paraboUis rs' = 2 bx, rt = 2 ex. 
 Hence the equation of the generating ellipse in any position, 
 
 that is, the e(iuation of the elliptical paraboloid, is •—-{ — = 2x. 
 
 b c 
 
 In the ZX-plane there is the fixed parabola 2- = 2 ex, in 
 the Xl'-plane the fixed parabola y^ — — 2bx. The figure gener- 
 ated by an hyperbola which 
 moves with its center on the 
 X-axis, the plane of the hy- 
 ])erbola perpendicular to the 
 X-axis, the axes of the hy- 
 l)erbola the intersections of 
 the plane of the hyperbola 
 with the fixed parabolas, is 
 called the hyperbolic para- 
 boloid. The equation of the 
 hyperbola is 
 
 = 1. 
 
 z' _ y 
 
 rs rt' 
 From the equations of the 
 fixed parabolas rs = 2 ex, 
 ri'=—2bx. Hence the equa- 
 
 FiG. 174. 
 
 tion of the generating hyper 
 
 bola in any position, that is, the equation of the hyperbolic 
 
 paraboloid, is —=2x. 
 
 c b 
 
 Art. 101, —The Conoid 
 
 The center of an ellipse moves in a straight line perpendicu- 
 lar to the plane of the ellipse. The major axis is constant for 
 all positions of the ellipse, the minor axis diminishes directly 
 as the distance the ellipse has moved, becoming zero when the 
 
108 
 
 ANALYTIC GEOMETRY 
 
 ellipse has moved the distance c. The figure generated is called 
 the conoid with elliptical base. 
 The equation of the ellipse is 
 
 y- 
 
 where 
 
 4-^^ — 1 
 -:: ^ — 2 — -^' 
 s rt 
 
 a, and, from similar 
 
 z 
 
 triangles, — = -, whence rt 
 
 b c 
 
 -(c — z). The efjuation of the 
 c 
 
 generating ellipse in any posi- 
 tion, that is, the equation of the 
 
 conoid, is 
 
 lP{c 
 
 
 Art. 102. 
 
 Surfaces represented by Equations in 
 Three Variables 
 
 An equation ffi{x, ?/, z)=0, when intei-preted in rectangular 
 space coordinates, represents some surface. For when z is a 
 continuous function of x and y, if (x, y) takes consecutive posi- 
 tions in the XF-plane, the point {x, y, z) takes consecutive 
 positions in space. Hence the geometric representation of the 
 function </> (x, y, «) = is the surface into which this function 
 transforms the XF-plane. To determine the form and dimen- 
 sions of the surface represented by a given equation, the inter- 
 sections of this surface by planes parallel to the coordinate 
 planes are studied. 
 
 Problems. — Determine the form and dimensions of the surfaces rep- 
 resented by the following equations. 
 
 1. E--|-^-)- ^2— \ Tiie equation of the projection on the Xr-plane 
 9 4 
 of the intersection of the surface represented by this equation and a plane 
 
 g = c parallel to the AT-plane is ^ + ^ = 1 - c-. This equation repre- 
 
CURVED SURFACES 
 
 199 
 
 sents an ellipse whose dimensions are greatest when c = 0, diminish as the" 
 numerical value of c increases to 1, and are zero when c = ± 1. The 
 ellipse is imaginary when c is numerically greater than 1. 
 
 The equation of the projection on the ZA'-plane of the intersection of 
 
 the surface by a plane tj = b parallel to the ZX-plane is — \- z'^= I — — 
 
 9 4 
 
 which represents an ellipse whose dimensions are greatest when ft = 0, 
 diminish as b increases numerically to 2, are zero when ft = ± 2, and 
 become imaginary when ft is numerically greater than 2. 
 
 The equation of the projection on the TZ-plane of the intersection of 
 
 the surface by a plane x 
 
 a parallel to the FZ-plane is ^ + 2^ 
 4 
 
 a2 
 
 which represents an ellipse whose 
 dimensions are greatest when a = 0, 
 diminish as a increases numerically 
 to 3, are zero when a = ± 3, and be- 
 come imaginary when a is numeri- 
 cally greater than 3. 
 
 The sections of the surface made 
 by planes parallel to the coordinate 
 planes are all ellipses, the surface is 
 closed and limited by the faces of 
 the rectangular parallelopiped whose 
 faces are x — ±S, y = ±2, z = ±1. 
 From the equation it is seen that the origin is a center of symmetry, the 
 coordinate axes are axes of symmetry, the coordinate planes are planes of 
 symmetry of the surface. The figure is the ellipsoid with axes 3, 2, 1. 
 
 0. 
 
 10x = 0. 
 
 Fig. 176. 
 
 X- + 2/-^ - z^ 
 x"^ + >j- + z"^ 
 r/ + z^ ■ 
 
 X2 + J/2 . 
 
 10x = 0. 
 
 ^2:^1. 
 
 z' - 2 X + o !i 
 
 j2 + 2 X + 4 i' : 
 
 12. Show that the conoid is a 
 ruled surface. 
 
CHAPTER XV 
 
 SECOND DEGEEE EQUATION IN THKEE VAEIABLES 
 
 Art. 103. — Transformation of Coordinates 
 
 Take the point (a, b, c) referred to the axes X, Y, Z as the 
 origin of a set of axes X', Y', Z' parallel to X, Y, Z respec- 
 tively. Let (.r, y, z), {x', y', z') represent the same point referred 
 to the two sets of axes. From the figure x = x' + a, y = y'-\-b, 
 
 z = z' -\- c. 
 
 z' 
 
 Let X, F, Z be a set of rectangular axes, X\ Y\ Z any set 
 of rectilinear axes with the same origin. Denote the angles 
 made by A'' with A, F, Z by a, (3, y respectively, the angles 
 made by Y' with A, Y, Zhj a', y8', y', the angles made by Z' 
 with A, Y, Z by a", j8", y". If (a-, y, z) and (x', y', z') represent 
 the same point F, x is the projection of the broken line 
 200 
 
SECOND DEGREE EQUATION 201 
 
 (x' + If' + z') on tlio X-axis, y the projection of this broken line 
 on the i'-axis, z the projection of this broken line on the 
 Z-axis. Hence 
 
 X = x' cos a -\- ?/' cos a' + z' cos a", 
 
 y = x' cos f^ + ?/' cos /3' + z' cos /3", 
 Z = x' COS y -f- )/' COS y' + ;<;' cos y". 
 
 Since X, F, Z are rectangular axes, 
 
 cos^ a + cos^ /? + cos^ y = 1 , 
 
 COS-«'+ C0S-/5'+ COS-y' = 1, 
 
 cos^«" + cos-;8"-f cos^y" = 1. 
 If X', I"', Z' are also rectangular, 
 
 cos a cos a' -f cos ft cos /?' + cos y cos y' — 0, 
 cos « cos a" + cos /3 cos (3" + cos y cos y" = 0, 
 cos a' cos «" -f cos ft' cos /3" + cos y' cos y" = 0. 
 
 Problems. — 1. Transform x- + y^ + z- = 2G to parallel axes, origin 
 (-5,0,0). 
 
 2. Transform x- + 7j- + z- = 25 to parallel axes, origin ( — 5, - 5, — 5). 
 
 3. Transform ^ + ^ -f IT = 1 to parallel axes, origin ( - a, 0, 0). 
 
 a'^ b- c^ 
 
 4. Show that the first degi-ee equation in three variables interpreted 
 in oblique coordinates represents a plane. 
 
 5. Show that the equation of an elliptic cone, vertex at origin, and 
 
 3.2 y2 5-2 
 
 axis the Z-axis, is ~ + ^ — — = 0. 
 ' a^ 62 c-i 
 
 6. Derive the formulas for transformation from one rectangular sys- 
 tem to another rectangular system, the Z'-axis coinciding with the Z-axis, 
 the X'-axis making an angle d with the A'-axis. 
 
 Art. 104. — Plane Section of Quadric 
 
 Surfaces represented by the second degree equation in throe 
 variables 
 Ax"^ + By- + Cz- + 2 Dxy + 2Exz + 2 Fyz 
 
 * ■^2Gx + 2Hy + 2Kz + L = (1) 
 are known by the general name of quadrics. 
 
202 ANALYTIC GEOMETRY 
 
 To find the intersection of the surface represented by this 
 equation by any plane transform to a set of axes parallel to 
 the original set, having some point (a, b, c) in the cutting plane 
 as origin. The transformation formulas are 
 
 x = x' -\-a, y = y' -\- b, z = z' -\-c, 
 
 and the transformed equation is 
 
 Ax'' + By'' + Cz" + 2 D'x'tj' + 2 E'x'z' 
 
 + 2 F'y'z' + 2 G'x' + 2 H'y' + 2 K'z' + L' = 0, (2) 
 
 where G' = .la + Db + Ec + G, 
 
 H' = Bb + Da + Fc + H, K' = Cc + Ea + Fb + K, 
 
 L' = Aa' + Bb- + Cc- + 2 Dab + 2 Eac 
 
 + 2 F6c + 2 (^a + 2 /f & + 2 /ic + L. 
 
 Now turn the axes X', Y', Z' about the origin until the 
 X' F'-plane coincides with the cutting plane. This is accom- 
 plished by the transformation formulas 
 
 x' = Xi cos a + ?/i cos «' + Zi cos a", 
 y' = Xi cos y8 + v/i cos /5' + ^1 cos |8", 
 2' = .Tj cos y + ?/i cos y' + 2:1 cos y". 
 
 These formulas are linear, hence the equation of the quadric 
 in terms of {x^, y^, Zj) is of the form 
 
 A,x,' + B,y,' + C,z,' + 2 D,x,y, + 2 E,x,z, 
 
 + 2 jPj^^i^i + 2 (^^.Ti + 2 if,y, + 2 /r.^i + Xi = 0. (3) 
 
 Since the plane of the section is the Xj^Vplane, the equa- 
 tion of the intersection referred to rectangular axes in its own 
 plane is A.x^' -f JB,2/i' + 2 D.x.y^ + 2 G,x, + 2 H,y, + L,= 0, 
 which represents a conic section. Hence every plane section 
 of a quadric is a conic section. For this reason quadrics are 
 also called conicoids. 
 
SECOND DEGIIEE EQUATION 
 
 20'.] 
 
 Art. 105. — Ckxtku of Quadric 
 
 The surface represented by eq\iation (2) is symmetrical with 
 respect to the origin (a, b, c) wlien the coefficients of x', y', z' 
 are zero, for then if {x\ y', z') is a point of the surface, 
 
 (-•< -y', -2') 
 
 is also a point of the surface. Hence the center of the quadrie 
 (1) is found by solving the equations 
 
 Aa+ Db + Ec + G = 0, Bb + Da + Fc + 11 = 0, 
 
 and Cc + Ea + Fb + K= 0. 
 
 Problems. — 1. Find the center of the quadrie represented by 
 a:2 + ?/2 ^ 4 ^2 _ 8 a;5: + 2/ = 0. 
 
 2. Find the center of the quadrie represented by 
 
 .r2 - 2/2 4- 2;2 - 10 X + 8 2 + 15 = 0. 
 
 Art, 106. — T.vxgent Plane to Quadric 
 
 Let (.To, ?yo> 2:0) be any point of the quadric (1). The equa- 
 tions cc = .To + d cos (it.,y = ?/o + d cos p,z = Z(i-\- d cos y represent 
 all straight lines through (.t,„ ?yo> ^^- By substituting in (1) 
 
 = 
 
 + ^aV 
 
 + 2 oleosa 
 
 a'o 
 
 d+^lcos-« 
 
 + By,' 
 
 + 2Bcos^ 
 
 2/0 
 
 + 5cos^/8 
 
 + Cz,? 
 
 + 2 Coos y 
 
 2^0 
 
 + C'cos-y 
 
 + 2Z>.r,,Vo 
 
 + 2I>cos« 
 
 •Vo 
 
 + 2 Z> cos a cos /? 
 
 + 2Ex^, 
 
 + 2 Z) cos ^ 
 
 •Ty 
 
 + 2 £" cos a cos y 
 
 + 2FyoZo 
 
 + 2£oosy 
 
 .To 
 
 + 2i^C0S)8C0Sy 
 
 + 2Gx, 
 
 + 2£cosa 
 
 2^0 
 
 
 + 2 By, 
 
 + 2Fcosy 
 
 2/0 
 
 
 + 2Kz, 
 
 + 2FcoS|8 
 
 2^0 
 
 
 + L 
 
 + 2 (7 cos a 
 + 2//cos^ 
 + 2 A" cosy 
 
 
 
204 ANALYTIC GEOMETRY 
 
 an equation is found which determines the two values of d 
 corresponding to the points of intersection of straight line and 
 quadric. Since the point (a^o, yo, z^) lies in the quadric, the 
 term of this equation independent of d vanishes. If the co- 
 efficient of the first power of d also vanishes, the equation has 
 two roots equal to zero ; that is, every straight line through 
 the point (;Xq, ?/„, ^o), and whose direction cosines satisfy the 
 equation 
 
 A cos a • x^+B cos (3 • ?/o+ C'cos y • z^,-\-D cos a • ?/(, + Z) cos /5 • .Tq 
 -f^cosy • Xo+Ecosa ■ Z(t-\-Fcosy ■ y^+Fcos/B • Zq 
 -f GrCos« + HcosfS -f A'cos y = 
 
 is tangent to the quadric. To determine the surface repre- 
 sented by this equation multiply by d and substitute x — Xq for 
 d cos a, y — ?/o for dcos (3, z — z^ for d cos y. There results the 
 equation 
 
 AxX(, + Byyo -f Czz^ + D (;xy^, + x^y) + E (xz^ + x^) 
 
 + F(yZo + y,z) + G(x + x^) + H{y + y,) + K{z -f ^o) + ^ = 0, 
 
 which, since it is of the first degree in {x, y, z) represents a 
 plane. This plane, containing all the straight lines tangent to 
 the quadric at {xq, y^, Zq) is tangent to the quadric at (.Tu, ?/„, z^^. 
 Notice that the equation of the plane tangent to the quadric at 
 (xq, yo, Zq) is obtained by substituting in the equation of the 
 quadric xxq for x^, yy^ for y^, zz,, for z^, xy^ + x^y for 2 xy, 
 xZq + X(^ for 2 xz, yzo -f- y^z for 2yz, x -\- x^ for 2x, y -\- y^ for 2 y, 
 z + Zq for 2 z. 
 
 Let (x', y', z') be any point in space, (x^, yo, Zq) the point of 
 contact with the quadric (1) of any plane through (x', y', z') 
 tangent to the quadric. Then (Xq, y^, z^, (x', y', z') must satisfy 
 the equation 
 
 .4.r'.T„ + By'yo + Cz'z^ -f D (x'y, + ?/'a'„) -f ^(^'.^o -f .x-'^o) 
 
 -f F(z'y, + y'z,) + G (;«' + x^) + U{y' -{- y^) +K{z' + z,) +L = 0. 
 
SECOND DKCREK EQUATION 205 
 
 Hence the points of contact (.t,„ ?/„, z^■) must lie in a ]>lane, 
 and the locus of the points of contact is a conic section. 
 
 Problems. — 1. Write the equation of the phiiie tangent to 
 .T- + y- + z^ = B^ at (xo, 2/0, Zu). 
 
 2. Write the ciiuation of the plane tangent to 
 
 a:2 + y2 4. ^2 _ 10 a; + 25 = at (5, 0, 0). 
 
 3. Write the equation of the plane tangent to 
 
 ^ + ?^' + -'=lat (Xo, 2/0, 20). 
 a^ 62 c- 
 
 4. Write the equation of the plane tangent to 
 
 t^?l = 2xat (Xo, 2/0, 2o). 
 b c 
 
 5. Find equations of projections on planes ZX and ZY of locus 
 of points of contact of planes tangent to x- + y"^ + z"^ — 25 through 
 (7, - 10, 6). 
 
 6. Find equation of normal to— + ^ + — =1 at (x',ij',z'). The 
 
 a^ b'^ c^ 
 normal to a surface at any point is the line through that point perpen- 
 dicular to the tangent plane at that point. 
 
 7. Find the angle between the normal to — + ^^-{-^=1 at (x', y', z') 
 
 cfi h^ c'^ 
 and the line joining (x', y', z') and the center of the ellipsoid. 
 
 Art. 107. — Reduction of General Equation of Quadric 
 
 To determine the form and dimensions of the surfaces repre- 
 sented by the general second degree equation in three variables 
 when interpreted in rectangular space coordinates it is desirable 
 first to simplify the equation. This simplification is effected 
 by changing the position of the origin and the direction of the 
 axes. 
 
 The change of direction of rectangular axes is effected by 
 the formulas 
 
 X = x' cos a + y' cos «' + ^' cos a", 
 y = x' cos (3 ■{- y' cos /3' + 2' cos /3", 
 z = a;' cos y 4- y' cos y' -f- z' cos y", 
 
206 ANALYTIC GEOMETRY 
 
 where the nine cosines are subject to the six conditions 
 cos^ a + cos^ /3 + cos- y = 1 , 
 cos'^w' + cos^)8' + cos-y' = 1, 
 
 COS^«" + C0S-/3" + COS-y" = 1, 
 
 cos a cos «' + COS (3 cos (3' + cos y cos y' = 0, 
 cos a cos «" + cos (3 cos (3" + cos y cos y" = 0, 
 cos a' COS a" + cos yS' cos /3" + cos y' cos y" = 0. 
 Three arbitrary conditions may therefore be imposed on the 
 nine cosines. 
 
 Substituting for x, y, z in 
 
 Ax" + By- -^Cz--^2 Dxy + 2 Exz + 2Fyz + 2Gx 
 + 2Hy-^2 Kz + L = 0, 
 there results 
 
 Ax" + By" + Cz'- + 2 D'x'y' + 2 ^'a-'^' + 2 F'?/'^' + 2 G'x' 
 
 + 2H'y' + 2K'z' + L' = 0, 
 
 when D', E', F' are functions of the nine cosines. Equate 
 B', E', F' to zero and determine the directions of the rectangu- 
 har coordinates in space in accordance with these equations. 
 This transformation is always possible, hence 
 
 Ax' + Bf- + Cz' + 2 G'x + 2 IFy + 2 K'z + L' = 
 
 interpreted in rectangular coordinates represents all quadric 
 surfaces. 
 
 Now transform to parallel axes with the origin at (a, b, c). 
 The transformation formulas are 
 
 X = a -It x', y = h + y', z = c + z' 
 
 and the transformed equation 
 
 Ax"+By" + Cz" + 2(Aa + G')x' + 2(Bb +H')y'+2(Cc + K')z' 
 
 + (Aa' + BW + Cc- + 2 G\i + 2 //7> + 2 K'c + L') = 0. 
 
SECOND DEGREE EQUATION 207 
 
 Take advantage of the three arbitrary constants a, b, c to cause 
 the vanishing of the coefficients of x', y', z'. This gives 
 
 = -^ b = -— ' = -=^ 
 " A' B' '' C' 
 
 values whicli are admissible when .1 ^^ 0, B :^ 0, C ^ 0. The 
 
 resulting equation is of the form Lx' + 3Ii/ + iVV = P. 
 
 When A ^ 0, B ^ 0, C = 0, the transformation 
 
 x = a + x',y = h-\-y',z = c + z' 
 gives 
 
 Ax" + By" + 2(Aa+ G)x> + 2(Bb + H')y' + K'z' 
 
 + (Aa' + Bb' + 2 G'a + 2H'b + 2 K'c + L') = 0. 
 
 Equating to zero the coefficients of x', y' and the absolute term, 
 
 the values found for a, b, c are finite when A^O, -B =5^ 0, K' =^ 0. 
 
 The resulting equation is of the form Lx? + My- + N'z = 0. 
 
 When A^O, B^O, C = 0, K' = 0, the equation takes the 
 form LiT + My- + X'x + M'y + P = 0. 
 
 When url ^ 0, iJ = 0, C = 0, the equation takes the form 
 Mx"" + M'x + N'y + P = 0. 
 
 When A = 0, B — 0, C — 0, the equation is no longer of the 
 second degree. 
 
 Since x, y, z are similarly involved in 
 
 Ax" + By- + 0x^ + 2 G'x + 2 IVy + 2 K'z + L' = 0, 
 the vanishing of A and G' or of B and H' would lead to equa- 
 tions of the same form as the vanishing of C and 7i '. 
 
 Collecting results it is seen that the following equations 
 interpreted in rectangular coordinates represent all quadric 
 surfaces — 
 
 ^1:^0, B^Q, C4-(), Lx'+My- + Nz-=P I 
 
 A^^O, B^O, C=0, K'^0 Lx-+My''+N'z=0 II 
 
 A^O, B^O, C=0, K' = Lx^-^My^+M'y-{-L'x+P=0\ 
 A^O, B=0, C=0 Lx^+L'x-{-M'y+N'z + P=0\ 
 
 These equations are known as equations of the first, second, 
 and third class. 
 
208 ANALYTIC GEOMETRY 
 
 Art. 108. — Sukfaces of the First Class 
 
 The equation of the first class may take the forms 
 (a) ix- + 31 f + Nz^ = P, (b) Lx" + 3Iy- - Nz- = P, 
 (c) Lx^ + 3bf-Nz'^ = -P, 
 
 or similar forms with the coefficients of ar and z- or of 'if and z' 
 positive. 
 
 (a) The intersections of planes parallel to the coordinate 
 planes with Lx^ + 3Iy/ + Nz' = P are for 
 
 X = x', 3Iif + Nz~ = P- Lx'\ 
 an ellipse whose dimensions are greatest when x' = 0, diminish 
 as x' increases numerically, are zero for x' = ± \-jr, imaginary 
 
 — ' X/ 
 
 — ; 
 
 for y = ?/', Lx- + Nz- = P — My'-, 
 
 an ellipse whose dimensions are greatest when _?/' = 0, diminish 
 as y' increases numerically, are zero for ?/' = ±a/— , imaginary 
 when y' is numerically greater than -v/ — ; 
 
 for z = z', Lx- + 3ry- = P — Nz'-, 
 
 an ellipse whose dimensions are greatest for z' = 0, diminish 
 
 as z' increases numerically, are zero when z' = ±\^, imaginary 
 when z' is numerically greater than a / - • 
 
 Calling the semi-diameter on the X-axis a, on the I''-axis b, 
 
 on the .^-axis c, the equation becomes — -f^ + — = 1, the 
 ellipsoid. " ^ ^ 
 
 The figure represented by Lx^ -\- 3Df + Nz^ = — P is imagi- 
 nary. The equation Lx- + 3[y- + Nz- = represents the origin. 
 
 (b) Lx^ -\- 3fy^ — Nz^ = P. The intersections are for x = x', 
 
SECOND DKGRKE EQUATION 201) 
 
 3/)/- — Xz- = r — Lx'-, ;iu hyperbola whose real axis is parallel 
 to the I'-axis when — \/. <-<^'< +\-,> parallel to the Z-axis 
 
 when x' is numerically greater than -il -, and which becomes 
 
 two straight lines Avhen x' = ±-1/—; 
 ' L 
 
 for !/ = y', Lx"" - .V^- = P- My", 
 
 an liy}ierbola whose real axis is parallel to the X-axis when 
 
 parallel to the Z-axis when y' is numerically greater than 
 
 (/> jp 
 
 \/— , and which becomes two straight lines when ^' = ±\/— : 
 V M ^ M 
 
 for z-z', Lx' + My- = P + Nz'-, 
 
 an ellipse, always real, whose dimensions are least when z' = 0, 
 and increase indefinitely when z' increases indefinitely in nu- 
 merical value. Calling the intercepts of this surface on the 
 X-axis a, on the F-axis b, on the Z-axis cV— 1, the equation 
 
 becomes — -f -'^ — - = 1, the hyperboloid of one sheet. 
 a^ b' & 
 
 (c) Lx' -f My' -Nz' = - P. 
 
 The intersections are 
 
 for X = x', My' -Nz' = -P- Lx", 
 
 an hyperbola with its real axis parallel to the Z-axis, dimen- 
 sions least when a;' = 0, increasing indefinitely with the numeri- 
 cal value of x' ; 
 
 for y = y', Lx" -Nz' = - P- My", 
 
 an hyperbola with its real axis parallel to the Z-axis, dimen- 
 sions least when y' — 0, increasing indefinitely with the numeri- 
 cal value of y' ; 
 for z = z', Lx' + My- = Lz" - P, 
 
210 ANALYTIC GEOMETRY 
 
 an ellipse, imaginary when 
 
 —a/— < 2;' < -\-\y' dimensions 
 zero for z' — ±\j—, increasing indefinitely with the numerical 
 
 value of z'. 
 
 Calling the intercepts of this surface on the axes X, Y, Z 
 respectively, aV— 1, 6V— 1. c, the equation becomes 
 
 a? Ir c- 
 the hyperboloid of two sheets. 
 
 The surfaces of the first class are ellipsoids and hyperboloids. 
 
 Art. 109. — Surfaces of the Second Class 
 
 The equation of the second class may take the forms 
 (a) Lx' + Ml/ ± N'z = 0, (&) Lx" - Mxf ± N'z = 0. 
 
 (ct) Lx- + 3fy- = N'z. The intersections are 
 for x = x', Mf- = N'z - Lx'\ 
 
 a parabola Avhose parameter is constant, axis parallel to Z-axis, 
 and whose vertex continually recedes from the origin ; 
 for y = y', Lx^ = N'z - My", 
 
 a parabola whose parameter is constant, axis parallel to Z-axis, 
 and whose vertex continually recedes from the origin ; 
 for z = z', Lx^ + 3fy- = N'z', 
 
 an ellipse whose dimensions are zero for z' = and increase 
 indefinitely as z' increases from to + co, but are imaginary 
 for2;'<0. 
 
 This surface is the elliptic paraboloid. The equation 
 Lx^ + My^ = — N'z represents an elliptic paraboloid real for 
 negative values of z. 
 
 (b) Lx^ — My- = N'z. The intersections are 
 for x = x', My- = Lx'- - N'z, 
 
 a parabola of constant parameter whose axis is parallel to 
 
SECOyi) DEGREE EQUATION 
 
 211 
 
 the Z-axis and whose vertex recedes from the origin as x' 
 increases numerically ; 
 
 for y = ij', Lx^ = N'z + My'-, 
 
 a parabola of constant parameter whose axis is parallel to the 
 Z-axis and whose vertex recedes from the origin as y' increases 
 numerically ; 
 for z = z', Lxr — My- = N'z', 
 
 an hyperbola whose real axis is parallel to the X-axis when 
 z > 0, paralh'l to the I'-axis when z' < 0, and which becomes 
 two straight lines when z' = 0. 
 
 The surface is the hyperbolic paraboloid. 
 
 The surfaces of the second class are paraboloids. 
 
 Art. 110. — Surfaces of the Third Class 
 
 The equation Lx^ + 3fy^ + L'x + 31 'y + P = does not con- 
 tain z and therefore represents a cylindrical surface whose 
 elements are parallel to the 
 Z-axis. The directrix in 
 the XF-plane is an ellipse 
 Avhen L and 3f have like 
 signs, an hyperbola when 
 L and 31 have unlike signs. 
 
 The surface represented 
 by the e(piation 
 
 Lx- + L'x + 31' y 
 
 is intersected by the A'l"- 
 })lane in the parabola 
 Lx- + L'x + 3['y + P - 0, 
 by the ZX-plane in the 
 parabola 
 Lx- + L'x -\-N'z + P = 0, 
 
212 ANALYTIC GEOMETRY 
 
 by planes x = x' parallel to the I'Z-plane in parallel straight 
 
 lines 
 
 N'y + L'z + Mx'- + 3I'x' + P = 0. 
 
 Hence the surface is a parabolic cylinder with elements parallel 
 to the ZF-plane. 
 
 The surfaces of the third class are cylindrical surfaces with 
 elliptic, hyperbolic, or parabolic bases. 
 
 It is now seen that the second degree equation in three 
 variables represents ellipsoids, hyperboloids, paraboloids, and 
 cylindrical surfaces with conic sections as bases. Conical sur- 
 faces are varieties of hyperboloids. 
 
 Art. 111. — QuADRics as Ruled Surfaces 
 
 The equation of the hyperboloid of one sheet '—^ ^ = 1 — --^ 
 
 is satisfied by all values of x, y, z, which satisfy simultaneously 
 the pair of equations 
 
 l-l=t.(l-^, ^ + ?=.lfl+f\ (1) 
 
 or the pair 
 
 a c \ bj a c fx 
 
 - - ■^+A - + ^=lfl-?A (3) 
 
 a c \ oj a c fjL 
 
 c a'V b 
 
 when fx and fx' are parameters. For all values of /x equations 
 (1) represent two planes whose intersection must lie on the 
 hyperboloid. Likewise equations (2) for all values of fx' repre- 
 sent two planes whose intersection must lie on the hyperboloid. 
 There are therefore two systems of straight lines generating 
 the hyperboloid of one sheet. 
 
 Each straight line of one system is cut by every straight line 
 of the other system. For the four equations (1) and (2) made 
 simultaneous are equivalent to the three equations 
 
SECOND DEGREE EQUATION 21:J 
 
 from ^vlli{•]l 
 
 // M — /a' X 1 + llfx' Z _\ — fjifx.' 
 
 b /u. + /a' « /A + /a' c /a + /a' 
 Ko two straight lines of the same system intersect. "Write 
 the equations of lines of the first system corresponding to //j 
 
 and fx.0. INfaking the equations simultaneous (/jLi—fi^)! 1 — -- Wo, 
 
 /"l 1 \ / \ \ ^/ 
 
 and (- .Yl+?^]=0. Hence either n-i—fx-., or y = h and 
 
 II = — h. Since _?/ cannot be at once + h and — 6, //.i = yu,o ; that 
 is. two lines of the same system can intersect only if they 
 coincide. 
 
 Observing that the equation of the hyperbolic paraboloid 
 
 '^ = 2 .T is satisfied by the values of x, y, z, which satisfy 
 
 either of the pairsj, of equations 
 
 z y _ I X 
 Vc Vb /* ' 
 
 Vc Vb 
 
 0) 
 
 z y 2x 
 
 Vc Vb /' 
 
 Vc V6 
 
 (-0 
 
 it can be shown that this surface niaj^ be generated by two 
 systems of straight lines ; that each line of one system is in- 
 tersected by every line of the other, and that no two lines of 
 the same system intersect. 
 
 The equations of ellipsoid, hy})erl)()l()id of two sheets and 
 of elliptical paraboloid cannot be resolved into real factors of 
 the first degree, consequently these surfaces cannot be gener- 
 ated by systems of real straiglit lines. 
 
 Akt. 112. — AsvMi'i'oTic Sri;FA(;?:s 
 From the equaticm of the hypcrlxiloid of one sheet 
 
 b- 
 
214 ANALYTIC GEOMETRY 
 
 it is found that 
 
 
 ^x^y 
 
 the powers of a^'if + h-x^ in the denominators increasing in 
 the expansion by the binomial formula. Hence the z of the 
 
 hyperboloid -„ + ^ — - = 1, and the z of the cone 
 o} b^ c- 
 
 •ii + ^_ _ ^ = 
 
 ce h"- & 
 
 approach equality as x and y are indefinitely increased ; that 
 is, the conical surface is tangent to the hyperboloid at infinity. 
 
 ■^ — ^ = is shown to be asymp- 
 a- h^ <T 
 
 totic to the hyperboloid of two sheets '-, — ^ — ;, = 1- 
 
 a^ IP- & 
 
 Art. 113. — Orthogonal Systems of Quadrics 
 
 The equation (1 ) — ^ h tt^^ + -^r^— = 1' where a>h>c 
 
 and A is a parameter, represents an ellipsoid when co > A > — c^, 
 an hyperboloid of one sheet when — c^ > A > — Ir, an hyper- 
 boloid of two sheets when — 6^>A> — a', an imaginary sur- 
 face when A < — al 
 
 Through every point of space {x\ y\ z') there passes one 
 ellipsoid, one hyperboloid of one sheet, and one hyperboloid of 
 two sheets of the system of quadrics represented by equa- 
 tion (1). Por, if A is supposed to vary continuously from 
 + oo to — (X) through 0, the function of A, 
 
 a^ -I- A b- + X c- + A. 
 
 1, 
 
SECOND DEGREE EQUATION 215 
 
 is — when A = -f vd and + when A is just greater than — r, 
 
 — when A is just less than — c" and + when A is just greater 
 than — //-, — wlien A is just less than — b^ and again + when 
 A is just greater than — cr. Hence 
 
 must determine three real values for A; one between + oo and 
 
 — r, another between — c- and — b^, a third between — b' and 
 
 — ((-. 
 
 Let A„ Ao, A3 be the roots of equation (2) ; that is, let 
 
 g-'-' y'- z'- _ . -, 
 
 ^.'2 -,,»2 ^i-' 
 
 - 1, (4) 
 
 a- + A., b'^ + A, c- + A; 
 
 a- + A3 6- + A3 C- + A3 ■ ^'^ 
 
 The equations of tangent planes to the quadrics of system 
 (1) corresponding to Ai, As, A3 at the point of intersection 
 {x\ y\ z') are 
 
 XX 
 
 + 
 
 b' + A, 
 
 + - 
 
 22 
 
 r + A, 
 
 = 1, 
 
 xx' 
 «■ + A. 
 
 + 
 
 b' + \, 
 
 + 
 
 zz' 
 
 = 1, 
 
 xx' 
 
 + 
 
 
 + 
 
 zz' 
 
 = 1. 
 
 ((•- + A3 b' + A3 t- + A3 
 
 The condition of perpendicularity of the iii'st two [tlanes 
 
 ^ +__.'/"' .+ ^ 
 
 (a' + A,)(«- + A.) {b-' + X,){b-' + Ao) (c- + AO(r + A.) 
 
 is a conse(pience of (.'i) and (4). In like manner it is shown 
 that the three tangent planes are mutually perpendicular. 
 
216 ANALYTIC GEOMETRY 
 
 Hence equation (1) represents an orthogonal system of quad- 
 rics. 
 
 Since through every point of space there passes one ellipsoid, 
 one hyperboloid of one sheet, and one hyperboloid of two 
 sheets of the orthogonal system of quadrics, the point in 
 space is determined by specifying the quadrics of the orthogo- 
 nal system on which the point lies. This leads to elliptic 
 coordinates in space, developed by Jacobi and Lame in 1839, 
 by Jacobi for use in geometry, by Lame for use in the theory 
 of heat. 
 
 If a bar kept at a constant temperature is placed in a homo- 
 geneous medium, when the heat conditions of the medium 
 have become permanent the isothermal surfaces are the ellip- 
 soids, the surfaces along which the heat flows the hyperboloid s, 
 of the orthogonal system of quadrics. 
 
NEW AMERICAN EDITION OF 
 
 HALL AND KNIGHT'S ALGEBRA, 
 
 FOR COLLEGES AND SCHOOLS. 
 
 Revised and Enlarged for the Use of American Schools 
 
 and Colleges. 
 
 By FRANK L. SEVENOAK, A.M., 
 
 Assistant Principal of the Acadi-mic Di-partmcnt, Stevens 
 Institute of TiJinolosy. 
 
 Half leather. 12mo. $ 1 . lO. 
 
 JAMES LEE LOVE, Instructor of Mathematics, Harvard University, 
 Cambridge, Mass. : — I'mfessor Sevcnoak's rcvisimi of tlie F.lriiieiitnry AlgLbra 
 is WW L\i client l>i»ik. I wish I could persuade all the teachers titling boys for the 
 Laurence Sucntihc Sch.iol to use it. 
 
 VICTOR C. ALDERSON, Professor of Mathematics, Armour Institute, 
 
 Chicago, 111.: — We have Msea the lOn-lish Eililiou for the past two years in our 
 Scienlilic Academy The new edition is superior to the ohl, ami we shall certainly 
 use it. In my opiiiiou it is the best of all the elementary algebras. 
 
 AMERICAN EDITION OF 
 
 ALGEBRA FOR BEGINNERS. 
 
 By H. S. HALL, M.A., and S. R. KNIGHT. 
 
 NEVISED BY 
 
 FRANK L. SEVENOAK, A.M., 
 
 Assistant Principal of the Academic Department, Stevens 
 Institute of Technology. 
 
 16mo. Cloth. 60 cents. 
 An edition of this book containing additional chapters on Radicals and 
 the Binomial Theorem will be ready shortly. 
 JAMES S. LEWIS, Principal University School, Tacoma, Wash.: — 1 have 
 
 examined Hall and Knight's "Algebra for Beginners " as revised by Professor .Sev- 
 enoak, and consider it altogether the best book for the purpose intended that I 
 know of 
 
 MARY McCLUN, Principal Clay School, Fort Wayne, Indiana: — I have 
 
 examined the .Mgebra (piite carefully, and I fuul it the best 1 have ever .seen. Its 
 greatest value is found in the simple and clear language in which all its definitions 
 are expressed, and in the fact that each new step is so carefully explained. The ex- 
 amples in each chapter are well selected. I wish all teachers who leach Algebra 
 might be able to use the "Algebra for Beginners." 
 
 THE MACMILLAN COMPANY, 
 
 66 FIFTH AVENUE, NEW YORK. 
 
AMERICAN EDITION 
 
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 LOCK'S 
 
 TRIGONOMETRY FOR BEGINNERS, 
 
 WITH TABLES. 
 
 Revised for the Use of Schools and Colleges 
 By JOHN ANTHONY MILLER, A.M., 
 
 Professor of Mechanics and Astronomy at the Indiana University. 
 
 8vo. Cloth. $1.10 net. 
 
 IN PREPARATION. 
 
 AMERICAN EDITION 
 
 OF 
 
 HALL and KNIGHT'S 
 ELEMENTARY TRIGONOMETRY, 
 
 WITH TABLES. 
 
 By H. S. HALL, M.A., and S. R. KNIGHT, B.A. 
 
 Revised and Enlarged for the Use of American 
 Schools and Colleges 
 
 By FRANK L. SEVENOAK, A.M., 
 
 Assistant Principal of the Academic Department, Stevens 
 Institute of Technology. 
 
 THE MACMILLAN COMPANY, 
 
 66 FIFTH AVENUE. NEW YORK. 
 
ELEMENTARY SOLID GEOMETRY. 
 
 HENRY DALLAS THOMPSON, D.Sc, Ph.D. 
 
 Professor of Mathematics in Princeton University. 
 
 i2mo. Cloth. $i.io, net. 
 
 This is an elementary work on Geometry, brief and interesting, well 
 and well written. — School of Mines Quarterly. 
 
 THE ELEMENTS OF GEOMETRY. 
 
 By GEORGE CUNNINGHAM EDWARDS, 
 
 Associate Professor of Mathematics in the University of California. 
 
 i6mo. Cloth. $i.io, net. 
 
 PROF. JOHN F. DOWNEY, University of Minnesota : — There is a gain in 
 its being loss formal th;in many of the works on this siiljict The arrangement and 
 treatment ;ire such as lo develop in the student ability to do geometrical work. The 
 book would furnish the preparation necessary for admission to this University. 
 
 PRIN. F. 0. MOWER, Oak Normal School, Napa, Cal.:-Of the fifty or 
 more English and American editions of Geometry which I have on my shelves, I 
 consider this one of the best, if not the best, of them all. I shall give it a trial in my 
 next class beginning that subject. 
 
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 66 FIFTH AVENUE, NEW YORK. 
 
MATHEMATICAL TEXT-BOOKS 
 
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 ARITHMETIC FOR SCHOOLS. 
 
 By J. B. LOCK, 
 
 Author qf " Trigonometry for Beginners," "Elementary Trigonometry," etc. 
 
 Edited and Arranged for American Scliools 
 
 By CHARLOTTE ANGAS SCOTT, D.SC, 
 
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 1 6mo. Cloth. 75 cents. 
 
 " Evidently the work of a thoroughly good teacher. The elementary truth, that 
 arithmetic is common sense, is the principle which pervades the whole book, and no 
 process, however simple, is deemed unworthy of clear explanation. Where it seems 
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 ' Trigonometry ' and the present work are, to our mind, models of what mathematical 
 school books should be." — The Literary World. 
 
 FOR MORE ADVANCED CLASSES. 
 
 ARITHMETIC. 
 
 By CHARLES SMITH, M.A., 
 
 Author of " Elementary Algebra," "A Treatise on Algebra," 
 
 AND 
 
 CHARLES L. HARRINGTON, M.A., 
 
 Head Master of Dr. J. Sach's School for Boys, New York. 
 1 6mo. Cloth. 90 cents. 
 
 A thorough and comprehensive High School Arithmetic, containing many good 
 examples and clear, well-arranged explanations. 
 
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 than ordinary value, and there is also a useful collection of miscellaneous examples. 
 
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