THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES \j The RALPH D. REED LIBRARY DKPARTMKNT OF GEOLOGY UNIVER.S1TY of CALIFORNIA L08 ANGELES, CAUF. Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/courseinmathemat01woodiala A COURSE IN MATHEMATICS FOR STUDENTS OF ENGINEERING AND APPLIED SCIENCE BY FEEDERICK S. WOODS AND FREDERICK H. BAILEY Professors of Mathematics in the Massachusetts Institute of Technology Volume I ALGEBRAIC EQUATIONS FUNCTIONS OF ONE VARIABLE, ANALYTIC GEOMETRY DIFFERENTIAL CALCULUS GINN AND COMPANY BOSTON • NKW YORK • CHICAGO • LONDON ATLANTA • DAT.T.A8 • COLUMBU8 • SAN KKANCISCO Copyright, 1907, by Frederick S. Woods and Frederick H. Bailby ALL RIGHTS RESERVED 620.12 Cbe gtbenttum l^rtet GINN AND COMPANY • PRO- PRIETORS • BOSTON • U.S.A. Geology Library PKEFACE This book is the first volume of a course in mathematics designed to present in a consecutive and homogeneous manner an amount of material generally given in distinct courses under the various names of algebra, analytic geometry, differential and integral calculus, and differential equations. The entire course covers the work usually required of a student in his first two years in an engineering school, the first volume containing the work of the first year. In arranging the material, however, the traditional division of mathematics into distinct subjects is disregarded, and the principles of each subject are introduced as needed and the subjects developed together. The objects are to give the student a better grasp of mathematics as a whole, and 6f the interdependence of its various parts, and to accustom him to use, in later applications, the method best adapted to the problem in hand. At the same time a decided advantage is gained in the introduction of the principles of analytic geometry and calculus earlier than is usual. In this way these subjects are studied longer than is otherwise possible, thus leading to greater familiarity with their methods and greater freedom and skill in their application. In carrying out this plan in detail the subject-matter of this volume is arranged as follows: 1. An introductory chapter on elimination, including the use of determinants. This chapter may be postponed or omitted, if a teacher prefers, without seriously affecting the subsequent work. 2. Graphical representation. Here the student learns the use of a system of coordinates and the definition and plotting of a function. 3. The study of the algebraic polynomial. This includes the analytic geometry of the straight line, the more important t GS' iv PREFACE theorems of the theory of equations, and the definition of a derivative. Simple applications of the calculus to problems involving tangents, maxima and minima, etc., are given. In this way a student obtains an introduction to the piinciples of the calculus, free from the difficulties of algebraic computation. 4. The study of the algebraic function in general. The knowl- edge of analytic geometry and calculus is here much extended by new applications of the principles already learned. Simple applications of integration are also introduced. The study of the conies forms part of the work in this place, but other curves are also used and care is taken to avoid giving the impression that analytic geometry deals only with conic sections ; in fact, the chapters which deal especially with the conies may be omitted without affecting the subsequent work. 5. The study of the elementary transcendental functions. It has been thought best to assume the knowledge of elementary trigonometry, since that subject is often presented for admission to college, — a tendency which should be encouraged. Tlie chapter discusses the graphs, the differentiation of transcendental functions, and the solution of transcendental equations. 6. The work closes with chapters on the parametric represen- tation of curves, polar coordinates, and curvature. In the first of these chapters the solution of locus problems, which, from some standpoints, is the most important part of analytic geometry, finds its natural place ; for this problem involves, in general, the expression of the coordinates of a point on a locus in terms of an arbitrary parameter, and possibly the elimination of the parameter. As compared with the usual first course in analytic geometry, there will be found in this volume fewer of the properties of the conic sections, 'except as they appear in problems set for the student. On the other hand, a greater variety of curves are given, and it is believed that greater emphasis is placed on the essential principles. All work in three dimensions is postponed to the second year, and is to be taken up in the second volume in connection with functions of two or more variables, partial differentiation, and double and triple integration. PREFACE V This volume contains the matter usually given in a first course in differential calculus, with the exception of differentials, series, indeterminate forms, partial differentiation, envelopes, and some advanced applications to curves. These subjects will find their appropriate place in the further development of the course in the second volume. Integration has been sparingly used as the inverse operation of differentiation, and without employing the integral sign. Simple applications to areas and velocities are given. To do more would require the expenditure of too much time on the operation of integration, and the introduction of too many new ideas into one year's work. The integral, as a limit of a sum, with its many applications, will form an important part of the second year's work. In the preparation of the text the needs of a student who desires to use mathematics as a tool in engineering and scientific work have been primarily considered, but it is believed that the course is also adapted to the student who studies mathematics for its own sake. Abstract discussions are avoided and frequent applications and illustrations are given. Illustrations, however, which are beyond the range of a first-year student's knowledge of physical science are omitted. The proofs are made as rigorous as the maturity of the student will admit. It is to be remembered in this connection that the earlier chapters are to be studied by students w^ho have just entered college. In the preparation of the book the authors have had the advice and criticism of the mathematical department of the Massachu- setts Institute of Technology. In particular, they are indebted to the head of the department. Professor H. W. Tyler, at whose invitation the book has been written, and whose suggestions have been most valuable. Massachusetts Institute of Technology September, 1907 CONTENTS CHAPTER I — ELIMINATION Article Page 1, 2. Determinant notation 1 3. Properties of determinants . 6 4. Solution of n linear equations containing n unknown quantities, when the determinant of the coefficients of the unknown quantities is not zero 12 5. Systems of n linear equations containing more than n unknown quantities 15 6. Systems of n linear equations containing n unknown quantities, when the determinant of the coefficients of the unknown quantities is zero 17 7. Systems of linear equations in which the number of the equa- tions is greater than that of the unknown quantities ... 18 8. Linear homogeneous equations 21 9. Eliminants 23 Problems 25 CHAPTER II — GRAPHICAL REPRESENTATION 10. Real number 28 11. Zero and infinity 29 12. Complex numbers 31 13. Addition of segments of a straight line 32 14-15. Projection 34 16. Coordinate axes 35 17. Distance between two points 36 18-19. CoUinear points 38 20. Variable and function 40 21. Classes of functions 43 22. Functional notation 44 Problems 45 vii viii CONTENTS CHAPTER III — THE POLYNOMIAL OF THE FIRST DEGREE Article Page 23. Graphical representation 50 24-26. The general equation of the first degree 52 27. Slope 54 28. Angles 55 29. Problems on straight lines 58 30-31. Intersection of straight lines 61 32. Distance of a point from a straight line 63 33. Normal equation of a straight line 64 Problems 65 CHAPTER IV — THE POLYNOMIAL OF THE ^Yth DEGREE 34-36. Graph of the polynomial of the second degree 70 37. Discriminant of the quadratic equation 73 38. Graph of the polynomial of the nth degree 74 39. Solution of equations by factoring 77 40-41. Factors and roots 78 42-43. Number of roots of an equation 80 44-45. Conjugate complex roots 82 46. Graphs of products of real linear and quadratic factors ... 83 47. Location of roots 86 48. Descartes' rule of signs 87 49-51. Rational roots .89 52. Irrational roots 92 Problems 94 CHAPTER V — THE DERIVATIVE OF A POLYNOMIAL 53. Limits 97 54. Slope of a curve 99 55. Increment 100 56. Continuity 101 57. Derivative .102 58. Formulas of difFerentiation 103 59. Tangent line 104 60. Sign of the derivative 106 61. Maxima and minima 108 62. The second derivative 110 63. Newton's method of solving numerical equations 114 64. Multiple roots of an equation 116 Problems 118 CONTENTS ix CHAPTER VI — CERTAIN ALGEBRAIC FUNCTIONS AND THEIR GRAPHS Article Page 65-66. Square roots of polynomials 121 67. Functions defined by equations of the second degree in ?/ . . 127 68. Functions involving fractions . • 128 69. Special irrational functions 131 Problems 133 CHAPTER Vll — CERTAIN CURVES AND THEIR EQUATIONS 70-72. The circle 134 73-75. The ellipse 139 76-78. The hyperbola 142 79-80. The parabola 146 81. The conic 148 82. The witch 149 83. The cissoid 151 84. The strophoid 152 85. Examples 154 Problems 155 » CHAPTER Vni — INTERSECTION OF CURVES 86. General principle 161 87-89. /i(x,y) = and /2(x,.?/) = 161 90. fi(x,y)='0 iiTidf„(x,y) = 10(3 91. /„(z, ?/) = 0and/,(x, .y)=0 168 92-93. If^(x,y) + kf„(x,y)=0 171 Problems 175 CHAPTER IX — DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 94. Theorems on limits 178 95. Theorems on derivatives 179 96. Formulas 184 97. Derivative of u" 185 98. Higher derivatives . 187 99. Differentiation of implicit algebraic functions 188 100. Tangents 190 101. Normals • 191 102. Maxima and minima 192 103. Point of inflection 194 X COKTENTS. Akticle Page 104. Limit of ratio of arc to chord 195 105. The derivatives — and -^ . ; 196 (Is d,i 106. Velocity 198 107. Components of velocity 200 108. Acceleration and force 202 109. Other illustrations of the derivative 203 110. Integration 205 Problems 209 CHAPTER X — CHANGE OF COORDINATE AXES 111. Introduction 217 112-114. Change of origin without change of direction of axes . . 217 115. Change of direction of axes without change of origin . . 221 116. Oblique coordinates 223 117. Change from rectangular to oblique axes without change of origin 224 118. Degree of the transformed equation 225 Problems 225 CHAPTER XI — THE GENERAL EQUATION OF THE SECOND DEGREE 119. Introduction 229 120. Removal of the xy-tevm 229 121. The equation Ax^ + By'^ + 2 Gx + 2 Fy + C = . . . .231 122. The limiting cases 234 123. The determinant AB - H^ 235 124. The discriminant of the general equation 236 125. Classification of curves of the second degree 237 126-127. Center of a conic 238 128. Directions for handling numerical equations 240 129. Equation of a conic through five points 241 130. Oblique coordinates 244 Problems 244 CHAPTER XII — TANGENT, POLAR, AND DIAMETER FOR CURVES OF THE SECOND DEGREE 131. Equation of a tangent 246 132. Definition and equation of a polar 247 133. Fundamental theorem on polars 247 CONTENTS xi Article Page 134. Chord of contact 248 135. Construction of a polar 249 136. The harmonic property of polars 249 137. Reciprocal polars 251 138-140. Definition and equation of a diameter 252 141. Diameter of a parabola 254 142. Parabola referred to a diameter and a tangent as axes . . 255 143. Diameters of an ellipse and an hyperbola 256 144. Conjugate diameters 258 145. Ellipse and hyperbola referred to conjugate diameters as axes 259 146-147. Properties of conjugate diameters 260 Problems 262 CHAPTER XIII — ELEMENTARY TRANSCENDENTAL FUNCTIONS 148. Definition 266 149. Graphs of trigonometric functions 266 150. Graphs of inverse trigonometric functions 269 151. Limits of and '- — 270 h h 152. Differentiation of trigonometric functions 272 153. Differentiation of inverse trigonometric functions . . . .276 154. The exponential and the logarithmic functions 279 155. The number e 280 156. Limits of (1 + A)* and ^1^ 283 157-159. Differentiation of exponential and logarithmic functions . 284 160. Hyperbolic functions 288 161. Inverse hyperbolic functions 291 162. Transcendental equations 293 Problems 296 CHAPTER XIV — PARAMETRIC REPRESENTATION OF CURVES 163. Definition 302 164. The straight line 302 165. The circle 303 166. The ellipse 303 167. The cycloid 305 168. The trochoid 306 169. The epicycloid 307 170. The hypocycloid 309 xii CONTENTS Article Page 171. Epitrochoid and hypotrochoid , 309 172. The involute of the circle 311 173. Time as the arbitrary parameter 313 174. The derivatives 315 175-176. Applications to locus problems 316 Problems 323 CHAPTER XV — POLAR COORDINATES 177. The coordinate system 329 178. The spirals 331 179. The conchoid 334 180. The limacon 336 181. The ovals of Cassini 338 182. Relation between rectangular and polar coordinates . . . 341 183. The straight line 342 184. The circle 342 185. The conic, the focus being the pole 343 186. Examples 344 187. Direction of a curve 345 188. Derivatives with respect to the arc 347 189. Area 348 Problems 349 CHAPTER XVI — CURVATURE 190. Definition of curvature 353 191-192. Radius of curvature 354 193. Coordinates of center of curvature 356 194. Evolute and involute 357 195. Properties of evolute and involute . 359 196. Radius of curvature in parametric representation .... 360 197. Radius of curvature in polar coordinates 361 Problems 362 Answers 365 Index 381 A COURSE m MATHEMATICS CHAPTEE I ELIMINATION 1. Determinaiit notation. Elimination is the process of obtain- ing from a certain number of equations containing two or more unknown quantities one or more equations which do not contain all of these quantities. The quantities removed are said to have been eliminated. The solution of equations is essentially the elim- ination of all but one of the unknown quantities. The process of elimination leads to the formation of certain expressions in the coefficients, for which a special name and a corresponding notation have been invented. In this chapter we shall consider equations of the first degree, or linear equations. These are equations in which no term contains more than one unknown quantity, and that in the first degree. Ex.1. aix + 6i2/ + ci = 0, a2X + 62^ + C2 = 0. To eliminate y, multiply the first equation by 62, the second by — 61, and add. To eliminate x, multiply the first equation by — a^, the second by ai, and add. There results (ai62 - a26t) a; + (C162 - C261) = 0, /o\ (aib2 — a^bi) y + (aiC2 — OiCi) = 0. Unless 0162 — (tibi = 0, equations (2) give at once the solution of (1). If 0162 — a^bi = 0, the method used to eliminate y also eliminates x, and the equations need further discussion, to be given in § 6. 1 ELIMINATION Ex. 2. ai« + hy + ciz + di = 0, a^x + b^y + CiZ + d2 = 0, Osx + bsV + C3Z + dz = 0. (1) To eliminate y and 2, multiply the first equation by {baCs — ftgCa), the second by - {bid - bsci), the third by (61C2 - 62C1), and add. There results [ai(&2C8 - bsCn) -(h{biC3 - bgCi) + 03(6100 - ftgCi)]* + [di (62C8 - bsCi) - d^ (61C3 - 63C1) + da (61C2 - 62C1)] = 0, or (oibjCs + ajftsCi + OsftiCa — aihc^ — Oa^iCa — 036201) x + {dibuPs + d^ci + ds&iCa - ^16300 - ^06103 - dzb^Ci) = 0. (2) To eliminate z and «, multiply the first equation by — (0203 — a^c^), the second by (OiCg — agCi), the third by — (ciCj — 0201), and add. There results {ai62C8 + a263f-"i + 036102 — 016302 — O261C8 — 036201) y + (aid2C8 + a^fiaCi + asdiC2 — axd^c^ — a^diCs — 03^201) = 0. (3) To eliminate x and y, multiply the first equation by (0263 — 0362), the second by — (O163 — 0361), the thii-d by (0162 — O261), and add. There results (O1&2C8 + O263C1 + 036102 — O163C2 — 026103 — O362O1) z + (aibjds + 0263(^1 + 0361^2 — Oi63d2 — 0261^3 — 0362^1) = 0. (4) Equations (2), (3), and (4) give the solution of (1), unless Oi&aCs + OibzCi + 036102 — 016302 — O261C3 — O362C1 = 0. The exceptional case will be considered in § 6. The binomials which occur in the solution of Ex. 1 are called determinants of the second order. The symbol is used to denote the determinant afi^— afi^. Then equations (2) of Ex. 1 may be written «i ^1 » + = 0, «i ^1 'y + a, c, 2 1 "1 = 0. DETERMINANT NOTATION 3 The polynomials which occur in the solution of Ex. 2 are called determinants of the third order. The symbol «1 \ Ci ^2 h ^2 «3 h Cs is used to denote the determinant «AC8+ aA(^l+ «8*1^2— «^A^2— «2^lC8— CCsK^^l- The results of Ex. 2 may then be written a. \ Ci a^ h ^2 « + a. K Cs C?j ftj Cj d^ &2 Cg = 0, C?3 63 Cg «i h Ci «2 h ^2 2/ + «8 i^3 Cs «! d. «1 «2 d. ^2 ^8 d. Cz = 0, «1 h H a. h C2 2 + «8 h ^8 «1 \ d. ^2 K d. «8 h ds = 0. By the work of Ex. 2, a^ &j Cj ^2 ^2 «2 = a. ^2 «2 ^3 ^3 ^ ^1 -fa. 5„ c„ which may be taken as the definition of a determinant of the third order. Similarly a determinant of the fourth order is indicated by the symbol a, b, c, d. ^2 \ «2 d. (^S K ^8 d. a, h, c. d. and is defined as equal to &2 C2 d^ h ^3 ^3 ?>. c, d. I ^1 ^1 d^ -a\\ C3 <^3 + a. ^ c, d^ ?>2 ^2 C?2 -«4 h c, c?. ^ Cl < &2 ^^2 d. ^'3 ^3 d. 4 ELIMINATION If now each of these determinauts of the third order is expressed in terms of determinants of the second order, we shall have finally the determinant of the fourth order expressed as an algebraic polynomial of twenty-four terms. 2. In general a determinant of the nth order is an algebraic polynomial involving n^ quantities, called elements. The symbol of the determinant is obtained by writmg the elements in a square of n rows and n columns. If in such a symbol a row and a col- umn are omitted, there is left the symbol of a determinant of the next lower order. This new determinant is said to be a minor of the original determinant, and is said to correspond to the element which stands at the intersection of the omitted row and column. We shall now give as definition : A determinant is equal to the algebraic sum of the products obtained by multiplying each element of the first column by its corresponding minor, the signs of the products being alternately plv^ and minus. By repeated application of the same definition to the minors obtained, we eventually make the value of the determinant depend upon determinants of the second order, and thus obtain the poly- nomial indicated by the original symbol. Students who desire a more general definition and discussion of determinants are referred to treatises on the subject. We shall derive here, as simply as possible, only those properties which are of use in solving equations. Before doing so, however, we need to show that the word "column" may be changed to "row" in the above definition, thus : A determinant is also equal to the sum of the products obtained by multiplying each element of the first row by the corresponding minor, the signs of the products being alter- nately plus and minus. For a determinant of the third order the student may verify that '. = a. -&. -^-Cx DETERMIKANT NOTATION 6 The theorem thus shown to he true for a determinant of the third order may be proved for one of the fourth order as follows : = a. = «1 C2 ^2 C3 d, c, d^ -«2 h c, d, C3 d,^ c, d] + ^3 \ d, 2 ^2 '4 ^4 -a^ b^ c^ d, K c, d^ h C3 d^ (by definition) K C2 d^ K C3 d^ b, c, d^ -ah ""^ ^^ -Ci 63 £?3 ^4 d. + d. \ «3 1 ^4 ^4 J H ' c, d. -^1 K d, b, d. + d^ &2 C, 1 ^4 <^j -Ci ^2 ^' +d \ C2I ^3 ^3 J (as already proved) \ c^ d^ h C3 ^3 b, c, d^ H ' G, d. -«3 ^2 d-2 c, d. + a. ^2 ^2 "1 C3 d^ f r ^-i d„ . ^'^V^b\d\ -«3 K d„ \d. K d. 1 -.{ ' ^3 "C3 C4 -^3 \ «2 K C4 + ^4 h ^2 ^3 ^3 } (by a rearrangement) = a. h C2 d^ ^2 ^2 ^2 ^2 ^2 ^2 «2 ^2 ^2 h C3 ^3 -h ^3 C3 ^3 + c, «3 *3 ^3 -d. <^3 h h 64 C4 d. ^4 C4 ^4 ^4 &4 d^ ^4 &4 C4 (by definition) In a similar manner the theorem may be proved successively for determinants of the fifth, the sixth, and, eventually, any order. g ELIMINATION 3. Properties of determinants. 1. A determinant is unchanged in value if the rows and the columns are interchanged in such a manner that the first row becomes the first column, the second row the second column, and so on. The student may verify that «1 «2 «1 \ «i as h ^2 = *8 K ^3 a. a„ c, c„ This proves the theorem for determinants of the second and the third orders. To prove it for one of the fourth order, proceed as follows : a^ 6j Cj d^ ^2 h ^2 ^2 C'S h ^i ^3 ^4 ^4 ^4 ^4 \ c, d^ *1 Ci <^l 6, C, d. ^1 «i ^1 = «! h Cs ds -«2 ^3 ^'S ^8 + ^3 h C2 ^2 -a^ ^2 ^2 ^2 K C4 ^4 \ C, rf. \ c, <^, h Cs ^3 «! ^2 «8 «4 h b h b h h K ^ ^'3 ^'4 ^1 ^2 ^4 ^1 ^2 *3 t'l ''2 "s "4 C, C^ C3 c, d^ d^ d^ d^ = «i ^2 C3 C4 -^2 C, C3 c. + ^3 C, C^ C, -a. <^1 ^2 ^3 ^2 ^3 < ^1 < ^4 ^1 C?2 ^4 ^1 ^2 ^3 The expressions on the right of these equations are equal, and hence the determinants of the fourth order are equal. In the same manner the theorem may be proved for determinants of higher order. It follows from this theorem that any property which is true of the rows is also true of the columns, and vice versa. The following theorems are stated for both rows and columns, but are proved for the rows only. PROPERTIES OF DETERMINANTS 2. If two consecutive rows {or columns) of a determinant are interchanged, the sign of the determinant is changed. The student may verify that a^ b„ = = — a„ 5 «1 \ «i K h ^2 «i h ^1 a^ ^ ^i a^ \ ^'2 = «1 ^ ^1 > «2 h ^2 = — % &3 C3 ^3 h ^3 a 3 ^3 ^3 ^3 h ^^3 «2 K ^2 The theorem is then proved for determinants of the second and the third orders. To prove it for a determinant of the fourth order, consider \ c, d. h ^2 ^3 ^3 d, d. and I, c. d. h C3 By definition, «j h^ Cj d a, &2 ^2 do % &3 ^^3 ^3 «^ l^ c^ d^ a^ h^ Cj c?j «, &3 Cg 6?3 ^2 ^2 ^2 ^2 «4 ^4 C4 f?^ &2 C2 ^2 ^1 ^1 ^1 ^1 ^1 ^1 \ Cj d^ = «1 ^3 C3 f^3 -S ^3 C3 ^5 + «3 ^2 «2 <^2 — a^ b^ Cg d^ &4 C4 ^4 ^4 C4 f?4 I, c, d^ h C3 ^3 K C3 ^3 \ ^x dx ^1 ^1 ^1 &^ Cj c?^ = «i ^2 ^2 ^2 -^3 &2 Cg d^ + «2 63 C3 (^3 -«4 &3 C3 C?3 ^'4 C4 «^4 K H d\ ^4 C^ ^4 ^2 ^2 ^2 Comparing these two expressions, it will be noticed that the minors which multiply a^ and a^ (the elements of the unchanged rows) differ in the two expressioils by the interchange of two con- secutive rows, and that the minors which multiply a^ and a^ (the elements of the interchanged rows) are the same in the two expres- sions but are preceded by opposite signs. It is evident on reflection that these laws always hold ; and hence, if the theorem is true for determinants of any order, it is true for determinants of the 8 ELIMINATION next higher order. The theorem is known to be true for determi- nants of the third order ; hence it is universally true. 3. A determinant is equal to the algebraic sum of the products obtained by multiplying each element of any row {or column) by its corresponding minor, the sign of each product being ]}lus or minus according as the sum of the number of the row and the number of the column in which the element stands is even or odd. For the hth. row may be made the first row of a new determi- nant by ^ — 1 interchanges of two consecutive rows. By theorem 2, if k is odd, the new determinant is equal to the original one ; and if k is even, the new determinant is equal to minus the original one. The new determinant may now be expressed by definition as the algebraic sum of the elements of its first row multiplied by their mmors, which are the same as those of the ^th row of the original determinant. Hence the original determinant is equal to the alge- braic sum of the elements of its kth. row multiplied by their minors, the products being alternately plus and minus when h is odd, and alternately minus and plus when k is even. From this the law of signs as given in the theorem at once follows. Ex. a\ bi Cl di (h b. C2 d. as b. C8 ds a* 64 Ci di ai 61 Cl di 02 ft2 C2 d.2 a* h Ci d4 as 63 ca da 61 Cl di 04 62 C2 di 1 63 C8 da + 64 «x 61 Cl di Oi 64 Ci d* 02 62 C2 do as 63 C3 ds ai Cl di ^2 C2 d^ as C3 da -Ci at 64 C4 di ai 61 Cl di 02 62 C2 d2 as 63 C3 ds a\ 61 di 02 62 d2 03 63 d3 + di ai 61 Cl tta 62 C2 as 63 C3 When a determinant is thus expressed it is said to be expanded according to the elements of the kth row. We shall call the coefficient of an element the quantity which multiplies it in the expansion. Then the coefficient of an element is plus or minus the corre- sponding minor according as the nu7nber of the row added to the number of the column is eveii or odd. PROPERTIES OF DETERMINANTS 9 The coefficient of a^ shall be denoted by A^, that of h^ by B^, and so on. Then = b,B, + b,B,+ b,B, = c^C^ + CgCg + C3C3 = a^A^+\B^+c^C^ = a^A^+\B^+c^C^. 4. If any two rows {or columns) of a determinant are inter- changed, the sign of the determinant is changed. For suppose the determinant is expanded as in theorem 3, and that two rows other than that used in the expansion be inter- changed. A similar interchange takes place in the minors of the expansion. Hence, if the theorem is true for each of the minors, it is true for the determinant. In other words, if the theorem is true for determinants of any order, it is true for those of the next higher order. But the theorem is certainly true for determinants of the second order. Hence it is always true. 5. If two rows (or cohtmns) of a determinant are the same, the determinant is equal to zero. Let a determinant with two rows the same be expanded accord- ing to the elements of some other row. Each minor of the expan- sion has two rows the same. Hence, if the theorem is true for determinants of any order, it is true for determinants of the next higher order. But the theorem is certainly true for determinants of the second order, for afi^- ^1^1 ~ ^• Hence it is universally true. 10 ELIMINATION 6. Tlie sum of the products obtained by multiplying the elements of any row {or column) by the coefficients of the corresponding ele- ments of some other row {or column) is zero. Consider, for example, = a,A, + b,B^ +c^C,+ d,D^ If we replace a^, b^, c^, d^, on the right-hand side of this equation by a^, b^, c^, d^, the same substitution must be made on the left- hand side. Then we have = a^A^ + b^B^ + c^C^ + dj)^. «1 \ H d «2 K ^2 d, «S \ ^3 d, a. K ^4 d^ «i \ <^X d. «4 h ^4 < «3 h «3 d. a. K '^4 < But the determinant is zero, by theorem 5 ; therefore a^A^+b,B^+ cf^+ d^D^ = Q. It is evident that the proof is general and establishes the theorem. 7. If each element of any row {or column) is multiplied by the same quantity, the determinant is multiplied by the same quantity. This follows at once from theorem 3. For example, 8 h kc^ d. kc^ d. kc^ ds kc^ d. = kc^ C^ + kc^ C^ -f- kc^ C3 + kc^ (7^ = kic^c,+ c,c^^+ c,c,+ c^c;\ «1 K ^1 d. «2 b. c. d. «3 h ^3 d. «4 h ^4 d. PROPERTIES OF DETERMINANTS 11 8. If each of the elements of any row {or column) is increased hy the same multiple of the corresponding element of any other row {or column), the value of the determinant is unchanged. We wish to show, for example, that (1) r «1 \ h ^1 «2 h ^1 ^2 ^3 ^3 Cg d^ *4 ^4 ^4 ^4 a^ b^ + kd^ c, d^ a^ \+M^ ^2 ^2 «3 &3+K ^3 dz a^ h^+kd^ c, d. (2) Let the coefficients of the elements in the second column of (1) be B^, B^, B^, B^. It is evident that these are also the coeffi- cients of the elements of the second column of (2). Hence (2) is {\ + kd;) B^ + {b^ + kd^) B^ + (&3 + Mg) B^ + (&, + kd^ B^, which equals b^B^ + b^B^ + b^B^ + b,B, + k {d^B^ + c?2^2 + d^B^ + d^B^). The coefficient of k in this equation is zero, by theorem 6, and the remaining terms equal the determinant (1). Hence (2) = (1). It is evident that the proof is general. The following are special cases : If A; = 1, the elements of one row or column are added to the corresponding elements of another row or column ; if ^^ = — 1, the elements of one row or column are subtracted from those of another row or column. This theorem is often used in simplifying determinants. Ex. 1. Consider 1-2 1 2 3-5 3 5 - 1 2 3-4 3-5 2 5 (1) If the elements of the second cohimn are added to those of the fourth column this becomes 1-2 1 3-5 3 -1 2 3-2 8-5 2 (2) 12 ELIMINATION If twice the elements of the first column are added to those of the second column, (2) becomes 10 1 3 13 (3) -10 3-2- ^ ' 3 12 If the elements of the first column are subtracted from those of the third column, (3) becomes 10 3 10 -10 4-2 3 1-1 (4) Expressing (4) as the sum of the product of the elements of the first row and their coefficients, it becomes and this is equal to 1 4 - 1-1 14-21 - 1 = -2. Ex, 2. Consider X y 1 Xi yi 1 X2 2/2 1 By successive subtraction of the elements of one row from those of another we have X y I xi yi 1 = «2 2/2 1 x-Xi 2/ - 2/1 Xi ^1 1 ^2 2/2 1 X -Xi 2/ - 2/1 SCl -X2 yi - 2/2 X2 2/2 1 \x Xi 2/ 2/1 ^ijg |f^g{ transformation being made by X1-X2 2/1-2/2 ', „ * theorem 3. 4. Solution of n linear equations containing n unknown quan- tities, when the determinant of the coefficients of the unknown quantities is not zero. We are now prepared to show that the method used in § 1 to solve equations with two or three unknown quantities can be so generalized as to apply to any system of equations of the first degree in which the number of equations is equal to the number of the unknown quantities. For convenience we will take the case of four equations, but the student will readily see that the method is perfectly general. n EQUATIONS WITH n UNKNOWNS Cousider the equations a^x + bji/ + c^z + djW + e^ = 0, a^x + b^y + c^ + d^w + e^ = 0, a^x + h^y + C32 + d^w + e^ = 0, a^a; + &42/ + c^z + f^^w + e^ = 0. 13 (1) (2) (3) (4) «1 ^ ^1 ^1 a^ K ^2 ^2 «8 h C3 ^3 ^4 K ^'4 d. Let the determinant of the coefficients of the unknown quan- tities X, y, z, w be denoted by D, so that D = and let A^ denote the coefficient of a^, B^ the coefficient of &^, and so on. We assume i> ^ 0. If now we multiply (1) by A^, (2) by ^„ (3) by A^, (4) by A^, and add the results, we have, by theorems 3 and 6, § 3, Dx + c^A^ + ^2^2 + ^3^3 + ^4^4 = 0- (5) Similarly, by using B^, B^, B^, B^ as multipliers, we have By + e,B^ + e^B^ + ^3^3 + e^B^ = ; (6) by using C^, C^, C^, C^ as multipliers, we have Bz + e^C,+ e,C,+ e^C,+ e^C, = 0; (7) and by using D^, D^, D^, D^ as multipliers, we have Dw + e^D^ + ej)^ + e3Z>3 + ej)^ = 0. (8) Now it is clear that any values of x, y, z, w which satisfy (1), (2), (3), (4) satisfy also (5), (6), (7), (8). Conversely, any values which satisfy (5), (6), (7), (8) satisfy also (1), (2), (3), (4). For if we multiply (5) by a^, (6) by h^, (7) by c^ (8) by d^, and add, we obtain (1). Similarly (2), (3), (4) can be obtained from (5), (6), (7), (8). Hence (1), (2), (3), (4) and (5), (6), (7), (8) are equivalent equations. 14 ELIMINATION Now e^A, + e^A^ + e^A^ + e^A^ = e,B^^e^B^-^e,B^ + e,B,= e,C,+ e,C,+ e^C,+ e,C,= «, ^ <^1 d. ^. K ^2 d. «« h ^8 < «4 \ ^4 d. «1 H '^l d. ^2 ^2 Cg d. «8 «3 H d. a. «4 H d. tj 5j e^ d^ «3 ^3 ^8 ^8 *4 ^4 ^4 ^4 and 6,i),+ e,l>2+e3A+«4^4 = ^1 ^1 ^1 ^1 ^2 62 c^ e^ ^3 ^3 ^3 ^3 ^4 &4 C4 «^ Hence the solution of (5), (6), (7), and (8) is X — «1 \ \ ^. «« K «2 «!, «« h <^3 ^8 «4 h <^, < a, \ ^1 d. «2 K «2 d. «8 ■K ^3 < ^4 h C4 ^4 2/ = «1 «1 «1 ^X «2 «2 ^2 d. «3 «3 ^8 d. ^4 «4 «4 d. «1 ^ ^1 d. ^2 ^2 ^2 d„ a^ &3 C3 d. ^4 &4 ^4 d. a. \ «i ^X «2 K «2 «^2 «8 h «3 «^8 «4 K «4 < «t h ^'l d. «2 K ^2 d. «« K <^3 d. «4 h ^4 < «i ^ Ci «1 ^2 ^0 ^2 ^2 «3 ^3 «3 «3 «4 K C4 «4 «1 K Cl ^1 «2 K «2 «^2 «3 h «8 «58 «4 h C4 d. and this is the solution, and the only solution, of (1), (2), (3), (4). n EQUATIONS WITH MORE THAN n UNKNOWNS 15 Hence we may state the following important theorem : Any system of n linear equations containing n unknown quan- tities has one and only one solution when the determinant formed by the coejfficients of the unknown quantities is not zero. This solution may be written down at once, for each unknown quantity is equal to mmus a fraction, of which the denominator is the determinant of the coefficients and the numerator is a similar determinant formed by replacing the coefficients of that unknown quantity by the absolute terms. Ex. 1. Ex. 2. 3x + 52/-4 = 0, 2a; -32/ + 7 = 0. -4 5 3 -4 7 -3 23 2 7 29 3 6 " 19' ^^ 3 6 ~ 19 2 -3 2 2x-3y+ z-l = 0, 4x + 5y-22 + 2 = 0, x-2 7/ + 32-3 = 0. -3 1-3 1 2 5-2 3-2 3 2-3 1 4 5-2 1-2 3 = 0, y = - 2 -3 -1 4 5 2 1 -2 -3 2 -3 1 4 5 -2 1 -2 3 2-1 1 4 2-2 1-3 3 -3 1 5-2 -2 3 = 1. = 0, 5. Systems of n linear equations containing more than n un- known quantities. When in a set of linear equations the number of equations is less than the number of unknown quantities, the equations have usually an infinite number of solutions, but may have none. The general method of procedure in solving them is to pick out a number of the unknown quantities equal to the number of the equations and having the determinant of their coefficients IQ ELIMINATION not zero. These are solved by the method of § 4. We then have these unknown quantities expressed in terms of the others. Ex.1. 2x + Sy+ 2 + 4 = 0, x-2y + 3z + 2 = 0. I> = \' ?l=-7. If we choose x and y for the unknown quantities, we have 12 31 |l -2| Then, solving as in § 4, we have I z + 4 31 3z + 2 -2 11 a; = — 1 — - = z — ^, 2 3 7 ' |l -2| 12 z + 4| 1 3z + 2 5 « = — —. — ^ = -z : " 2 3 7 |l -2| and since z may be given any value whatever, the equations have an infinite number of solutions. Ex.2. 2x + 3y+ z + 4 = 0, 2x + 3y + 2z + 3 = 0. If we choose to solve for x and y, we have 12 3 2 3 I) = \l f| = 0. But if we choose to solve for y and z, we have n— ^ — ^ -^-13 2|-'^- The solutions are y = — ^x — ^, z = l. It is possible that no selection of the unknown quantities will lead to a determmaut of the coefficients which is not zero. In this case the equations may have no solution. The discussion is too complex for this book, but the student will probably have no diffi- culty with the cases likely to occur in practice. n EQUATIONS WITH n UNKNOWNS 17 Ex.3. 2x + Sy + z + 4 = 0, 2x + 3y + z + S = 0. The determinant of any pair of unknown equations is zero. By subtracting the second equation from the first we have 1 = 0, showing the equations to be contradictory. 6. Systems of n linear equations containing n unknown quan- tities, when the determinant of the coefficients of the unknown quantities is zero. Consider again equations (1), (2), (3), (4) of § 4, but with the assumption that Z> = 0. We may proceed exactly as in § 4, but equations (5), (6), (7), (8) do not now contain the unknown quantities. In fact, these equations are, in general, con- tradictory, and consequently equations (1), (2), (3), (4) have, in general, no solution, Ex. 1. X - 2/ + z + 3 = 0, 2x-{-y + Sz + 1 = 0, « + 2y + 2z + 4 = 0. Here B 1-11 2 13 1 2 2 0. Eliminating y and z by the method of § 4, we have x — 24 = 0, which is absurd. Hence the equations have no solution. It is, of course, possible that when D = each of the other determinants in (5), (6), (7), (8) Ib also zero. Each of these equa- tions is then simply = 0, and gives no direct information about the solutions of (1), (2), (3), (4). As a matter of fact, in this case, (1), (2), (3), (4) have, in general, an infinite number of solutions, but may, under special conditions, have no solutions. The general discussion is too complex to be given here. We shall simply state the following theorem : A set of linear equations containing n unknown quantities has, in general, no solution when the determinant of the coefficients of the unknoivn quMntities is zero, but may, under certain conditions, have an infinite number of solutions. 13 ELIMINATION In practice, one of the n equations may be temporarily set aside, and the other w — 1 equations, which contain 11 unknown quan- tities, may be examined by the method of § 5. If these equations can be solved, the solution can be tested in the equation which has been set aside. Ex. 2. 2x- 32/+ 2-1 = 0, X- 22/ + 3z + 4 = 0, 7a;-ll?/ + 6z+ 1 = 0. If the method of § 4 is used, the result is = 0. Solving the first two equa- tions for X and y, we have ic = 7 z + 14, y = 5 z + 9, and these results are found on trial to satisfy the last of the given equations. Since z may have any value, the equations have an infinite number of solutions. 7. Systems of linear equations in which the number of the equations is greater than that of the unknown quantities. If there are more equations of the first degree than there are unknown quantities, there will be, in general, no values of the imknown quantities which satisfy all equations. There may be such values, however, when certain relations exist among the coefficients of the equations. To obtain these relations we may pick out a num- ber of equations equal to the number of tlie unknown quantities and solve them. If the solution is substituted in the remainmg equations, there will result certain expressions in the coefficients which must be zero if the equations are to be satisfied. The most important case is that in which there are n-\-l equa- tions containing n unknown quantities. For example, consider a^x + l^y -f- Cj2 + d^ = Q, a^x + hju + c^ + rf., = 0, a^x -h &^ -h c„z + <3 = 0, a, A' -\-'^i^y + c^z + (l^ = 0. n + 1 EQUATIONS WITH n UNKNOWNS 19 The solution of the first three equations, if ^0, is (§4) x = d, ^ <^X d. (>. ^2 ds K ^3 «i \ '^l «2 K «2 «3 ^3 ^3 ^ ^'i ^1 *2 ^2 «!, *3 ^3 d. «1 K ^1 a^ ^2 .«2 «3 ^3 «3 «1 d. ^1 «2 d. <^2 «3 ds C3 «1 \ «1 «2 b. ^2 «3 ^ C3 «1 «1 ^. «2 ^2 d. «3 ^3 d. «1 ^ h «2 *2 Cg «3 ^ «3 «1 ^ ^, «2 62 ^2 «3 *3 «53 «1 ^'l «1 «2 K ^2 «3 *3 ^3 Substituting these values in the first member of the last equa- tion, we have — a. h ^1 + K \ d^ ". d^ — c. a^ \ d^ ^2 h ^2 + d. «3 ^3 ^3 b, c, \ ^2 «2 K ^2 which, by theorem 3, § 3, is the same as h «i ^^ K ^2 ^^2 \ C3 d. b. ^4 d. ^2 «2 20 ELIMINATION a, \ h rf «2 K ^2 d. ^8 h ^3 d. «4 h ^4 d Hence, in order that the last equation may be satisfied, we must have = 0. Extending this to any number of variables, we have the theorem : In order that a system of n + 1 linear equations containing n unknovm quantities shall have a solution, it is necessary that the determinant formed from the coefficients of the unknown quantities and the absolute terms shall he zero. Ex. 1. x+ 2/+ z-2 = 0, 2x+ y- z + 3 = 0, x-22/-3« + 4 = 0, 5x-3y-4« + l = 0. Here 111-2 2 1-1 3 1 _2 -3 4 6-3-4 1 = 0, showing that if the first three equations have a solution it will satisfy the fourth equation. In fact, the solution is x = 1, y = — 2, z = 3. It should be noted that the converse of the theorem stated is not necessarily true. All that has been proved is that if n of the equations have a solution, that solution satisfies the {n + l)st equa- tion when the determinant is zero. But the determinant may be zero when the equations are contradictory. Ex.2. 2x-3?/+ 2 + 1 = 0, 2x-3y + 52 + 2 = 0, 2x-3i/-6z-3 = 0, 2x-3?y + 2z-8 = 0. Here 2-311 2-362 2 -3 -6 -3 2-3 2-8 = 0, but any three of the equations may be seen to be contradictory by the method of §8. LINEAR HOMOGENEOUS EQUATIONS 21 8. Linear homogeneous equations. An equation is homoge- neous with respect to the unknown quantities when the sum of the exponents of the unknown quantities is the same in each term. In particular an equation of the first degree is homogeneous when each of the terms contains one of the unknown quantities; for example, where x^, x^, x^, x^ are the unknown quantities. Tliis equation is, of course, satisfied by placing x^ = 0, x^ = 0, x^ = 0, x^ = 0, but in practice tliis solution is generally unimportant. In such equations, in fact, it is usually the ratios of the unknown quantities wliich are important ; for if each unknown quantity is multiplied by the same number, the equation is unaltered. In fact, if we place X, X 1 "^2 "^3 the homogeneous equation just written becomes the non-homogene- ous equation a^ -H a^ + a^ + a^ = 0. In this manner a set of homogeneous equations containing n unknown quantities may be reduced to a set of non-homogeneous equations containmg ii — \ unknown quantities by dividing each equation by one of the unknown quantities. The methods of the previous articles may then be used. But tliis method of proced- ure is open to the objection that the unknown quantity by which the equations are divided may possibly be zero when the division is invalid. It is better, therefore, to handle the homogeneous equa- tions as they stand, slightly modifying the methods used for non- homogeneous equations in a manner which will be clear from the examples. Ex. 1, a^xx + a'^X'!, + 03X3 + ^4X4 = 0, 61X1 + 62X2 + 63X3 + &4X4 = 0, (1) C]X\ + C2X2 + CgXs + C4X4 = 0. '>') ELIMINATION We will handle this by the method of § 4, in that we temporarily look upon Xi, Xa, Xa as the unknown quantities. We have, in the first place, = 0, 61 02 62 63 Xi + O4X4 fl2 O3 64X4 62 &3 Cl C2 C3 C4X4 C2 Cs 61 02 62 03 63 X2 + Oj 04X4 03 6i 64X4 63 Cl C2 C3 Cl C4X4 C3 ai 61 O2 62 03 63 X8 + Oi C2 O4X4 61 62 64X4 Cl Ca cs Cl C2 C4X4 = 0, = 0, which may be written as Oi 02 03 61 fe2 &3 Xi + Cl C2 C3 Oi 02 as h 62 63 X2- Cl C2 C3 1 Ox O2 O3 61 62 63 X3 + Cl C2 Cs O2 as 04 62 hs 64 C2 Cs C4 fli 61 Oa &3 O4 64 Cl Cs C4 Oi 02 62 O4 hi Cl C2 Ci X4 = 0, X4 = 0, X4 = 0. (2) (3) (4) From these follow : Xi : X2 : Xs : X4 = 02 O3 04 62 63 64 : — C2 Cs C4 Oi 03 04 1 61 63 64 • 1 Cl Cs C4 1 Oi 02 O4 &1 62 h : — Cl C2 C4 ai 02 as 61 62 63 Cl C2 Cs (5) The result (5) holds even when one or more, but not all, of the determinants involved are equal to zero. Then the corresponding unknown quantities are equal to zero. For example, if Oi 03 04 61 h 64 = 0, Cl Cs C4 Ol 02 04 61 62 h Cl C2 C4 and the other determinants in (5) are not zero, (3) and (4) show that X2 = and Xs = 0, while (2) shows that the ratio of Xi and X4 are correctly given by (5). If all the determinants in (5) are zero, the values of the unknowns are not thereby determined. In this case, two of the equations (1) should be solved for two of the unknown quantities in terms of the others, and the results tested for the last equations. ELIM^ANTS 23 It should be noted that contradictory equations cannot occur. The student should compare the contradictory equations 2x-3y + 4 = 0, 2x-3y -2=0, with the homogeneous equations 2xi — 3iC2+4 Xs = 0, 2 »i — 3 a^ — 2 ajs = 0. By subtracting one equation from the other we have 6x3 = 0, whence X3 = and Xi : X2 = 3 : 2. Ex. 2. The four equations aiXi + aix<2. + 03X3 + 04X4 = 0, hyxx + 623:2 + 63X3 + 643:4 = 0, CiXi + C2X2 + C3X3 + C4X4 = 0, . d\Xx + ^23:2 + ^3X3 + ^4X4 = 0, have, of course, the common solutious, xi = 0, X2 = 0, X3 = 0, X4 = 0. In order that they may also be satisfied by the same ratios of the unknown quantities, it is necessary that a\ an = 0. a 6 c 6 c a c a b CHAPTEE II GRAPHICAL REPRESENTATION 10. Real number. The science of mathematics deals with vari- ous kinds of numbers, each of which has arisen through the desire to perform, without restriction, some one of the fundamental oper- ations. The simplest numbers are the positive integers, or whole numbers. If one restricts himself to the use of these, he may add or multiply together any two of them without obtaining a new kind of number ; but he may not divide one number by another not exactly contained in it, nor subtract a larger number from a smaller. In order that division may always be performed, the common frac- tions, which are the quotients of one integer divided by another, are necessary. In order that subtraction may always be possible, the idea of a negative number must be introduced. The integers and fractions, both positive and negative, together form the class of rational numbers. On these numbers the operations of addition, subtraction, multiplication, and division may always be performed without leading to a new kind of number. The operation of evolution, however, leads to two new kinds of numbers , — the irrational, exemplified by V2 ; and the complex, of wliich V— 2 is an example. The complex numbers will be noticed in § 12 ; we shall here speak only of the irrational numbers. An irrational number is defined as one wliich cannot be expressed exactly as an integer or a common fraction, but which may be so expressed approximately to any required degree of accuracy. The simplest examples are the roots of rational numbers ; for example, V? may be approximately expressed as f f^^ ro-Uh ^^^■' ^^^^ c^^" not be expressed exactly. There are also irrational numbers which are not the roots of numbers and cannot be expressed by means of radical signs. A familiar exj^mple is the number 7r = 3.14159---. An irrational number may be either positive or negative. The 28 ZEKO AND INFINITY 29 rational and the irrational numbers together form the class of real numbers. A rigorous investigation of the nature and properties of these numbers, especially of the irrational numbers, is too advanced for this book. An elementary discussion, however, is given in any course in algebra, and is here assumed as known. The real numbers may be represented graphically on a number scale, constructed as follows : On any straight line assume a — j — i — i — i — i — i — i — i-i-f •^ *^ -U -3 -2 -1 1 2 S U fixed point as the zero point, or origin, and lay off positive numbers in one direction and negative numbers in the other. If the line is horizontal, as in fig. 1, it is usual, but not necessary, to lay off the positive numbers to the right of and the negative numbers to the left. Then any point M on the scale represents a real number, namely, the number which measures the distance of M from ; positive if M is to the right of 0, and negative if M is to the left of 0. Conversely, any real number is represented by one and only one real point on the scale. 11. Zero and infinity. There are two mathematical concepts usually included in the number series, for which special rules of operation are needed. These are zero, represented by the symbol 0, and infinity, represented by the symbol oo. Zero arises in the first place by subtracting a quantity from an equal quantity ; thus, a — a = 0. It signifies in this sense the absence of quantity — nothing. It cannot, then, either operate upon a quantity or be operated upon; for all operations imply the existence of the quantities concerned. Literally, then, the n n expressions a x 0, - > - > are meaningless. However, it is possible to put into these symbols conventional meanings, as follows: Take the three expressions ax, -> -> and consider what hap- a X pens when x is taken smaller and smaller, constantly nearer to zero but never equal to it. It requires only elementary arith- metic to see that ax and - may each be made as small as we a 30 GRAPHICAL KEPllESENTATlOX please by taking a; sufficiently small, while - becomes indefinitely great as x decreases, and may be made larger than any quantity we may choose to name. We may express the first two results concisely by the formulas a X = 0, - = 0. a We can express the last result in a formula, however, only by introducing the concept infinity. Wlien the value of a quantity is indefinite, but the quantity is increasing or decreasing in such a way that its numerical value is greater than any assigned quan- tity, however great, it is said to become infinite. It is then denoted by the symbol oo, called infinity. We can accordingly express our third result by the formula a = "' which means that when the denominator of a fraction decreases, be- coming constantly nearer to zero, the value of the fraction increases and becomes greater than any quantity which can be named. The symbols a ^ a X 00, — > — 00 a are also literally meaningless. We can, however, give a conven- tional meaning to them by writing ax, -> -, and studying the X a effect of increasing x indefinitely. Elementary arithmetic leads to the results expressed by the formulas a X 00 = oo, — = 0, — = 00. CO a Two other forms also occur in practice, namely, - and — • These 00 arise when we have a fraction - in which the numerator and the denominator either approach zero together or increase indefi- nitely together. The value of the fraction cannot be determined unless we know a law to govern x and y. These fractions are consequently called indeterminate forms, and will be considered later in the course. COMPLEX NUMBERS 31 Neither zero nor infinity can be said to have an intrinsic alge- braic sign. In some cases a quantity may increase in value, remaining always positive. It is then said to be + co. At other times it may increase numerically, remaining always negative. It is then said to be — co. Often, however, the quantity is iadefi- nitely great in such a way that the sign is ambiguous. An example is tan 90°. If an acute angle is made nearer and nearer to 90°, its tangent increases indefinitely, remaining positive. But if an obtuse angle is made nearer and nearer to 90°, its tan- gent increases indefinitely, remaining negative. Hence we say tan 90° = 00, and no algebraic sign can be attached to it. Similar considerations hold for the sign of zero. 12. Complex numbers. If one restricts himself to the use of the real numbers, named in § 10, it is impossible to perform the operation of evolution without exception ; for the even root of a negative number is not a real number. It is therefore necessary, if the generality of all algebraic operations is to be maintained, to introduce a new kind of number, called a complex number. These numbers will be used very little in this volume, and the following resume of the matter usually contained in algebra is sufficient for our present purposes. A further discussion will be given in the second volume. The imaginary unit is V— 1, and is denoted by i. Then e=-i. By multiplying this equation successively by i, we find i^=— i, i* = 1, i^ = i, i^= — l, • • • ; and, in general, — — i'' = l, 'i" + ' = t, i"'^' = -l, i^^+'" = -i, where k is zero or any integer. If h is any real number, the product M is called a pure imagi- nary number. The square root of any negative number is pure imaginary ; thus, 32 GRAPHICAL REPRESENTATION If a and h are any two real numbers, the combination a + bi is caUed a complex imaginary number, or, more simply, a complex number. A complex number reduces to a pure imaginary number when a = 0, and to a real number when 5 = 0. If a = and 5 = 0, the complex number a + bi = 0; and conversely, if a + bi = 0, then a = and 5 = 0. All operations with complex numbers are carried out by using the ordinary laws of algebra and replacing all powers of i by their values just determined. Ex. 1. V-3 X V32 z= iVs X i V2 = ?:2 Ve = - Ve. 3j-V34_3 + 2i 2 + 2i_6 + 10i + 4i2 2 + lOi l+sV^ Ex. 2. 4 2-2i 2 + 2i 4-4 i^ Two complex numbers such as a + 5* and a — bi, where a and b have the same values in each, are called conjugate complex numbers. Their product is a real number ; thus, (a-{-bi)(a-bi) = a^-\.b\ It is clear that the complex numbers have no place on the num- ber scale of § 10. 13. Addition of segments of a straight line. Consider any straight line connecting two points A and B. In elementary geometry only the position and the length of the line are consid- ered, and consequently it is immaterial whether the line be called AB or BA ; but in the work to follow it is often important to con- sider the direction of the line as well. Accordingly, if the direction of the line is considered as from A to B, it is called ^j5; but if the direction is considered from B to A, it is called BA. It will ^e seen later that the distinction -4 B c between AB and BA is the same Fig. 2 ^^ t-hat between + a and — a in algebra. Consider now two segments AB and BC on the same straight line, the point B being the end of the first segment and the begin- ning of the second. The segment AC is called the sum of AB and BC, and is expressed by the equation AB+BC=AC. (1) SEGMENTS OF A STEAIGHT LINE 33 This is clearly true if the points are in the position of fig. 2, but it is equally true when the points are in the position of fig. 3, Here the line BC, being opposite in direction to AB, cancels part of it, j[ J ^ leaving ^C. F^,, 3 If, in the last figure, the point C is moved toward A, the sum AC becomes smaller, until finally when C coincides with A we have AB + BA = 0, or BA=-AB. (2) If the point C is at the left of A, as in fig. 4, we still have 4B + BC = AC, where AC = -CAhy (2). , It is evident that this addition — ^ '^ ^ is analogous to algebraic addition, ■pj^ ^ and that this sum may be an arith- metical difference. From (1) we may obtain by transposition a formula for sub- traction, namely, BC = AC-AB. (3) This is universally true since (1) is universally true. This result is particularly important when applied to segments of the number scale of § 10. For if x is any number corresponding to the point M, we may always place x = OM, since both x and OM are positive when M is at the right of 0, and both x and OM are negative when M is at the left of 0. Now let 31^ and M^ be any two points, and let x^ = OM^ and x^ = OM^. Then 3I^M^ = OM^ - OM^ = «2 - ^r On the other hand, M^M^ = OM^ — 031^ = x^—x,_= — M^M^. It is clear that the segment 31^31^ is positive when 31^ is at the right of Jtfp and is negative when M„ is at the left of 31^ Hence, the length and the sign of any segment of the number scale is found ly subtracting the value of the x corresponding to the beginning of the segment from the value of the x corresponding to the end of the ser^m^ent. 34 GRAPHICAL REPKESENTATION 14. Projection. Let AB and MN (figs. 5, 6) be any two straight lines in the same plane, the positive directions of which are respec- tively AB and MN. From A and B draw straight lines perpendicu- lar to 3IJSf, intersecting it at points A' and B' respectively. Then A' B' is the projection oi AB on MN, and is positive if it has the direction MN (fig. 5), and is negative if it has the direction NM (fig. 6). Denote the angle between MN and AB by ^, and draw A C par- allel to 3IN. Then in both cases, by trigonometry, AC=ABcos(f>. But AC=A'B', and therefore A'B' = AB cos with AB. COORDINATE AXES 35 The respective components of Fi and F2 are represented by A'B' and A'C, and the resultant component is represented by A'B' + A'C". But A'B'=Fi cos 0, and ^'(7'= F2 cos (j> ; hence, by substitution, tlie resultant component is F^ cos + jFs cos cj). It is to be noted that in fig. 8 Fi and F2 have opposite signs. 15. The projection of a broken line upon a straight line is defined as the algebraic sum of the projections of its segments. Let ABCDE (fig. 9) be a broken line, MN o. straight liue in the same plane, and AE the straight line joining the ends of the broken line. Draw AA', BB', CC, BD\ and EE' perpendicular to MN', then ^^_ A!B\ B'C, CD', J)'E\ and A'E' ^ are the respective projections on MNoi AB, BC, CD, DE, and AE. But A'B' + B'C + CD' + D'E' = A'E'. (^T § 13) Hence, the projection of a broken line upon a straight line is equal to the projection of the straight line joining its extremities. Ex. If ABCDE (fig. 9) represents a polygon of forces, we have the result: the component of the resultant in any direction is the sum of the components of the forces in that direction. X- M 16. Coordinate axes. Let X'X and Y' Y be two number scales at right angles to each other, with their zero points coincident at 0, as in fig. 10. Let P be any point in the plane, and through P draw straight Imes perpendicular to X'X and Y'Y respectively, intersectmg them at M and iV". If now, as in § 13, we place X = OM, and y — ON, it is clear that to any point P there corresponds one and only one pair of numbers x and y, and to any pair of numbers corresponds one and only one point P. Y' Fig. 10 X 36 GRAPHICAL REPRESENTATION If a point P is given, x and y may be found by drawing the two perpendiculars MP and NP as above, or by drawing only one per- pendicular as MP. Then MP = OiV^= y and 0M= x. On the other hand, if x and y are given, the point P may be located by finding the points M and N corresponding to the num- bers X and y on the two number scales, and drawing perpendiculars to X'X and Y' Y respectively through M and N. These perpen- diculars intersect at the required point P. Or, as is often more convenient, a point M corresponding to x may be located on its number scale, and a perpendicular to X'X may be drawn through M, and on this perpendicular the value of y laid off. In fig. 10, for example, M corresponding to x may be found on the scale X'X, and on the perpendicular to X'X at M, MP may be laid off equal to y. When the point is located in either of these ways it is said to be plotted. It is evident that plotting is most conveniently per- formed when the paper is ruled in squares, as in fig. 10. These numbers x and y are called respectively the abscissa and the ordinate of the point, and together they are called its coordi- nates. It is to be noted that the abscissa and the ordinate, as defined, are respectively equal to the distanpes from Y' Y and X'X to the point, the direction as weU as the magnitude of the distances being taken into account. Instead of designating a point by writing x = a and y = — b,itis customary to write P(a, — h), the abscissa always being written first in the parenthesis and separated from the ordinate by a comma. X'X and Y' Y are called the axes of coordinates, but are often referred to as the axes of x and y respectively. 17. Distance between two points. Let P^{x^, y^) and P,{x^,y^ be two points, and at first assume that P^P^ is parallel to one of the coordinate axes, as OX (fig. 11). Then y^^y^ Now M^M^, the projection of P^P^ on OX, is evidently equal to P^P^. But Y M^M^ = x^-x^{^ 13). Hence I\T^=i X^ Xy (1) In like manner, if x^ = x^ P^P^ is parallel to OY, and Fio.ii P.n=^y-yy (2) ^. R Ml M, ^ DISTANCE BETWEEN TWO POINTS 37 If x^ 4-- x^ and y^ ^ y^, P^P^ is not parallel to either axis. Let the points be situated as in fig. 12, and through ij and P^ draw straight lines parallel respectively to OX and OY. They will meet at a point R, the coordinates of which are readily seen to be {x„ y,). By (1) and (2), P^B = x^— x^, RP^ = ^2 - Vv But in the right triangle P^RIl, PP = whence, by substitution, we have Fig. 12 PxP. = ^(^2 -*'i)'+ (2/2-2/1)'- (3) It is to be noted that there is an ambiguity of algebraic sign on account of the radical sign. But since P^Il is parallel to neither coordinate axis, the only two directions in the plane the positive directions of which have been chosen, we are at liberty to choose either direction of P^P^ as the positive direction, the other becoming the negative. It is also to be noted that formulas (1) and (2) are particular cases of the more general formula (3). Ex. Find the coordinates of a point equally distant from the three points Pi(l, 2), P2(- 1, - 2), and Pz(2, - 5). Let P (x, y) be the required point. Then PiP= P^P and P2P = P3P. But PiP = V(x - 1)2 + {y- 2)2, P2P = V(a; + l)2 + (2/ + 2)2, P3P = V(x - 2)2 + (y+ 5)2. V(x - 1)2 + {y- 2)2 = V(x + 1)2 + (y + 2)2, V(a; + 1)2 + {y + 2f = V(x - 2)2 + (y + 5)2, ehence, by sohition, x = f and y - - ^. Therefore the required point is 38 GRAPHICAL EEPRESENTATION 18. CoUinear points. Let P{x, y) be a point on the straight line determined hy P^{x^, y^) and Bi{x^, y^, so situated that P^P = lil^P^). There are three cases to consider according to the position of the point P. If P is between the points P^ and P (fig. 13), the 3 X P/ R/ / M, M M, ' Fig. 13 Fig. 14 segments P^P and P^P^ have the same direction, and P^P P^P^ ; therefore / is a positive number greater than unity. Finally, if P is beyond P^ from P^ (fig. 15), PyP and P^P, have opposite directions, and / is a negative number, its numerical value ranging all the way from to oc. In the first case P is called a point of internal division, and in the last two cases it is called a point of external division. In all three figures draw P^M^, PM, and I^M^ perpendicular to OX. In each figure OM=OM^ + M^M; and since P^P = I (P^P,), M^M= 1{M^M^, by geometry. . • . OM = OM^ + / {M^M^), I Fig. 15 whence, by substitution, x = x^-\- l(x2—x^). (1) By drawing lines perpendicular to OY we can prove, in the same way, y = y.+ i{y.-yx)- (2) COLLINEAE POINTS 39 In particular, if P bisects the line P^P^, I = ^, and these formulas become 2-^2 X = Ex. 1. Find the coordinates of a point J of the distance from Pi (2, 3) to P2(3, -3). If the required point is P(x, y), X = 2 + f (8 - 2) = 22, 2/ = 3 + -?(-3-3) = f. Ex. 2. Prove analytically that the straight line dividing two sides of a tri- angle in the same ratio is parallel to the third side. Let one side of the triangle coincide with OX, one vertex being at O. Then the vertices of the triangle are 0(0, 0), A{xi, 0), B{X2, 2/2) (fig. 16). Let CD divide the sides OB and AB so that OC = 1{0B) and AD = 1{AB). If the coordinates of C are denoted by (X3, 1/3) and those of D by (X4, 2/4), then, by the above formulas, Xa = 1x2, 2/3 = ly2, and X4 = Xi + Z(xo — xi), 2/4 = l)/2- Since 7/3 = 2/4, CD is parallel to OA. Fig. 16 19. Let us now see what happens as different real values are assigned to I. Wlien I = 0, P coincides with P^ (fig. 17). As / increases in value, the point P moves along the line toward P^ till, when / = 1, it coincides with P,. As the value of I con- tinues to increase, the point P continues to move along the line away from ^ and in the same direc- tion as before. If negative values are assigned to /, in ascending order of numer- ical magnitude, the point P moves along the line, away from P^, in the opposite direction from P,. Fig. 17 40 GRAPHICAL REPRESENTATION It follows that ^1 + ^ («2 - ^i) and y^ +l{y^— 2/1) may be made to represent the coordinates of any point of the straight line determmed by the pomts P^ and i^ by assigning the appropriate value to I, the range of values for each segment of the line being indicated in fig. 17. Ex. Consider the straight line determined by the two points Pi(— 1, — 4) and P2(5, 6). Any other point P on tliis line has the coordinates x = -l + 6Z, ^ = -44-10^. When Z < 0, it is clear that a;< — 1, y < — 4k; hence P lies at the left of Pi. When < i < 1, it is clear that — l\, it is clear that a; > 5, y >Q; hence P lies at the right of P2. 20. Variable and function. A quantity which remains un- changed throughout a given problem or discussion is called a constant. A quantity which changes its value in the course of a problem or discussion is called a variable. If two quantities are so related that when the value of one 'is given the value of the other is determined, the second quantity is called a function of the first. Wlien the two quantities are variables the first is called the independent variable, and the function is sometimes called the dependent variable. As a matter of fact, when two related quantities occur in a problem it is usually a matter of choice which is called the independent variable and which the function. Thus, the area of a circle and its radius are two related quantities such that if one is given the other is deter- mined. We can say that the area is a function of the radius, and likewise that the radius is a fvmction of the area. The relation between the independent variable and the function can be graphically represented by the use of rectangular coordi- nates. For, if we represent the independent variable by x and the corresponding value of the function by y, x and y will determine a point in the plane, and a number of such points will outline a curve indicating the correspondence of values of variable and function. This curve is called the graph of the function. EXAMPLES OF FUNCTIONS 41 Ex. 1, An important use of the graph of a function is in statistical work. The following table shows the price of standard steel rails per ton in the respective years: 1895 $24.33 1896 28.00 1897 18.75 1898 17.62 1899 28.12 1900 $32.29 1901 27.33 1902 28.00 1903 28.00 1904 28.00 If we plot the years as abscissas, calling 1895 the first year, 1896 the second year, etc., and plot the price of rails as ordinates, making one unit of ordinates correspond to ten doUare, we shall locate the points Pi, P^, . . ., Pio in fig. 18. In order to study the variation in price, we join these points in succession by straight Fig. 18 lines. The resulting broken line serves merely to guide the eye from point to point, and no jwint of it except the vertices has any other meaning. It is to be noted that there is no law connecting the price of rails with the year. Also the nature of the function is such that it is defined only for isolated values of x. Ex. 2. As a second example we take the law that the postage on each ounce or fraction of an ounce of fii-st-class mail matter is two cents. The postage is then a known function of the weight. Denoting each ounce of weight by one unit of x, and each two y cents of postage by one unit of y, we have the series of straight lines (fig. 19) parallel to the axis of X, representing corresponding values of weight and postage. Here the function is defined by United States law for all positive values of x, but it cannot be expressed in elementary mathe- matical .symbols. A peculiarity of the graph is the series of breaks. The lines are not connected , but all points of each line represent correspond- ing values of x and y. Fig. 19 O 42 GKAPHICAL KEPRESENTATION Ex. 3. As a. third example, differing in type from eacli of the preceding, let us take the following. While it is known that there i.s some physical law con- necting the pressure of saturated steam with its temperature, so that to every temperature there is somo corresponding pressure, this law has not yet been formulated mathematically. ^'everthele.ss, knowing some corresponding values of temperature and prassure, we can construct a curve that is of considerable value. In the table* below, the temperatures are in degrees Centigrade and the pressures are in millimeters of mercury. Y J 7 r j 7 t zr 1 t ^ i .1 r 1 J r r t t. 1 _ t - J t _ - 4 J. J r ol 1 1 X Temperature Pressub 100 760 105 906 110 1074.7 115 1268.7 120 1490.5 125 1743.3 130 2029.8 135 2353.7 140 2717.9 146 3126.1 150 3581.9 Let 100 represent the zero point of tempera- ture, and let each unit of x represent 5 degrees of temperature ; also let each unit of y represent 100 millimeters of pressure of mercury, and locate the points representing the corresponding values of temperature and pre.ssure given in the above table. Through the points thus located draw a smooth curve (fig. 20) i.e. one which has no sudden changes of direction. AVhile only the eleven points located are exact, all other points are approxi- mately accurate, and the curve may be used for approximate computation as follows : A-ssume any temperature, and, laying it oif as an abscissa, measure the corresponding ordinate of the curve. While not exact, it will, nevertheless, give an approximate value of the corre- sponding pressure. Similarly, a pre.ssure may be assumed and the corresponding temperature determined. It may be added that the more closely together the tabulated values are taken, the better the approximation from the curve, but the curve can never be exact at all points. Fig. 20 »From C. H. Peabody's " Steam Tables," computed for sea level at a latitude of 45 degrees. CLASSES OF FUNCTIONS 43 Ex. 4. As a final example, we will take the law of Boyle and Mariotte for per- fect gases, namely, at a constant temperature the volume of a definite quantity of gas is inversely proportional to its pressure. It follows that if we repre- sent the pressure by x and the corre- k spending volume by y. then y = -, X where fc is a constant and x and y are positive variables. A curve (fig. 21) in the first quadrant, the coordinates of every point of which satisfy this equation, represents the comparative changes in pressure and volume, show- ing that as the pressure increases by a certain amount the volume is decreased more or less, according to the amount of pressure previously exerted. This example differs from the pre- ceding in that the law of the function is fully known and can be expressed in a mathematical formula. Consequently, we may find as many points on the curve as we please, and may therefore con- struct the curve to any required degree of accuracy. Fig. 21 21. Classes of functions. We shall consider in this book only those functions of one variable which can be expressed by means of elementary mathematical symbols. The simplest kind of such functions is the algebraic polyTiomial, expressed by ao«"-f ftj^t-"-^ + • • • + «„_!« + «„, where all the exponents are positive integers and the coefficients a^, «!, •••,«„_ J, «„ are real or complex numbers or zero. The number n is the degree of the polynomial. These functions are discussed in Chaps. Ill and IV. The quotient of two algebraic polynomials is a rational algebraic fraction, expressed by «„»" -f ajiz;"""^ +••• + «„_]''» + «„ Examples of functions of this kind are discussed in Chap. VL 44 GKAPHICAL REPRESENTATION If a function requires for its expression the use of radical signs combined with algebraic polynomials, it is an example of an irra- tional algebraic function ; for example. Ab + JIZI. Examples of such functions are found in Chap. VI. The general definition of an algebraic function is given in Chap. IX, and examples of non-algebraic, or transcendental func- tions, are given in Chap. XIII. 22. Functional notation. When y is a function of x it is cus- tomary to express this by the notation Then the particular value of the function obtained by giving x a definite value a is written f{a). For example, if tben /(2) = 2«+3-22-f-l=21, /(0)=0«+3-0^+l = l, /(-3) = (-3)H3(-3)^+l = l, /(a) = a'+3a2-f 1. If more than one function occurs in a problem, one may be expressed asf(x), another as F{x), another as (x), and so on. It is also often convenient in practice to represent different functions by the symbols f(x), f^(x), f^{x), etc. If /(a;) is any function, and we place y =/('«), we may, a^ already noted, construct a curve which is the graph of the function. The relation between this curve and the equation y =f{x) is such that all points the coordinates of which satisfy the equation lie on the curve ; and conversely, if a point lies on the curve, its coordinates satisfy the equation. PROBLEMS 45 The curve is said to be represented by the equation, and the equa- tion is called the equation of the curve. The curve is also called the locus of the equation. Its use is twofold, — on the one hand, we may study a function by means of the appearance and the properties of the curve, and, on the other hand, we may study the geometric properties of a curve by means of its equation. Both methods will be illustrated in the following pages. PROBLEMS 1. Find the perimeter of the triangle the vertices of which are (2, 3), (- 3, 3), (1, 1). 2. Prove that tlie triangle the vertices of which are (— 4, — 3), (2, 1), ( — 5, 5) is isosceles. , 3. Prove that (6, 2), (- 2, - 4), (5, - 5), (-1, 3) are points of a circle the center of which is (2, — 1). What is its radius ? 4. Prove that the quadrilateral of which the vertices are (2, 2), (4, 5), (—1, 4), (— 3, 1) is a parallelogram. 5. Find a point equidistant from the points (—3, 4), (5, 3), and (2, 0). 6. Find the center of a circle passing through the points (0, 0), (—3, 3), and (5, 4). 7. Find a point on the axis of x which is equidistant from (0, 4) and (-3,-3). 8. A point is equally distant from the points (1, 1) and (— 2, 3), and its distance from OY is twice its distance from OX. Find its coordinates. 9. Find the points which are 4 units distant from (2, 3) and 5 units distant from the axis of y. 10. A point of the straight line joining the points (—4, — 2) and (4, — 6) divides it into segments which are in the ratio 3 : 6. What are its coordinates ? 11. Find the coordinates of a point P on the straight line determined by Pi (2, - 1) and P2 (- 4, 5), when |^ = ^ • 12. On the straight line determined by the points Pi (2, 4) and P2( — 1, — 3) find the point three fourths of the distance from Pi to P2. 13. If P (x, y) is a point on the straight line determined by Pi (xi, 2/1) and ■P2(X2, 2/2), such that — ^ = -, prove PP2 ^2 hxi + hxi ?i2/2 + hVi x = 1 w = • h + I2 k + k 46 GKAPHICAL REPRESEKTATION 14. The middle point of a certain line is (1, 2) and one end is the point (—3, 5). Find the coordinates of the other end. 15. To what point must the line drawn from (1, —1) to (—4, 5) be extended in the same direction that its length may be trebled ? 16. One end of a line is at (2, — 6) and a point one fourth of the distance to the other end is (— 1, 4). Find the coordinates of the other end of the line. 17. Find the points of trisection of the line joining Pi(0, 3) and P2(6, — 3). 18. Find the lengths of the medians of the triangle (2, 1), (0, —3), (— 4, 0). 19. Given the three points A{- 3, 3), B{S, 1), and C{6, 0) upon a straight A.D AB line. Find a fourth point D such that = DC BC 20. Given four points Pj, P2, P3, P4. Find the point halfway between Pi and P2, then the point one third of the distance from this point to P3, and finally the point one fourth of the distance from this point to P4. Show that the order in which the points are taken does not affect the result. 21. Prove analytically that if in any triangle a median is drawn from the vertex to the base, the sum of the squares of the other two sides is equal to twice the square of half the base plus twice the square of the median. 22. Prove analytically that the straight line drawn between two sides of a triangle so as to cut off the same proportional parts measured from their com- mon vertex is the same proportional part of the third side. 23. Prove analytically that if two medians of a triangle are equal the tri- angle is iso.sceles. 24. Prove analytically that in any right triangle the straight line drawn from the vertex to the middle point of the hypotenuse is equal to one half the hjrpotenuse. 25. Prove analytically that the lines joining the middle points of the opposite sides of a quadrilateral bisect each other. 26. Show that the sum of the squares on the four sides of any quadrilateral is equal to the sum of the squares on the diagonals, together with four times the square on the lin6 joining the middle points of the diagonals. 27. Prove analytically that the diagonals of a parallelogram bisect each other. 28. Prove analytically that the line joining the middle points of the non- parallel sides of a trapezoid is one half the sum of the parallel sides. 29. OABC is a trapezoid of which the parallel sides OA and CB are per- pendicular to OC. D is the middle point of AB. Prove analytically that OD = CD. PROBLEMS 47 30. The following table gives the price of a bushel of wheat in the New York market from 1890 to 1904. Construct the graph. 1890 .983 1895 .669 1900 .804 1891 1.094 1896 .781 1901 .803 1892 .908 1897 .954 1902 .836 1893 .739 1898 .952 1903 .863 1894 .011 1899 .794 1904 1.107 31. The following table shows hourly barometric readings at a United States weather bureau station. Construct the graph. 1 A.M. 28.85 9 A.M. 29.04 6 P.M. 29.13 2 28.87 10 29.05 6 29.18 3 28.90 11 29.05 7 29.21 4 28.92 12 m 29.05 8 29.24 5 28.94 1 P.M. 29.05 9 29.25 6 28.97 2 29.06 10 29.29 7 28.98 3 29.08 11 29.29 8 29.02 4 29.10 12 29.29 32. The following table shows the number of inches of rainfall in Boston during the years 1880-1891. Construct the graph. 1880 38.89 1886 46.47 1881 49.22 1887 41.91 1882 48.42 1888 60.27 1883 35.56 1889 54.79 1884 53.86 1890 50.21 1885 44.07 1891 49.63 33. The following is a portion of a railway time-table. The letters indicate stations, and the adjacent number gives the distance fi'om A to each of the othei' stations. The second and the third columns give the times at which two trains running in opposite directions leave each of the stations. Make a graph showing the motion of each train and thus determine the time and place of their passing. A 10.45 AM. 2.00 F99 1.06 P.M. 10.48 B 21 1.30 G 126 9.53 C44 11.50 12.56 H 151 2.59 8.56 D64 12.11 P.M. 1177 7.48 E 84 11.30 A.M. K200 4.15 7.00 A.M. 48 GRAPHICAL REPRESE:N^TATI0X 34. The following table shows the amount of SI. 00 put at interest at 4% compounded annually. Construct the graph. 6yr. 1.217 30 yr. 3.242 10 1.480 35 3.946 16 1.801 40 4.801 20 2.191 45 5.841 25 2.666 50 7.116 35. Make a graph showing the relation between the side and the area of a square. 36. Make a graph showing the relation between the radius and the area of a circle. 37. Make a graph showing the relation between the radius and the volume of a sphere. 38. The space s through which a body falls from rest in t seconds is given by the formula s = I gt^. Assuming g = 32, construct the graph. 39. The velocity acquired by a body thrown towards the earth's surface with a velocity Vq is given at the end of t seconds by the formula v = Vo + gt. Construct the graph. 40. Two particles of mass mi and mz at a distance d from eacK other attract each other with a force F, given by the formula F = cP Assuming mi = 5 and m^ = 20, construct the graph of F. 41. Ohm's law for an electric current is ^ ^ Electromotive force Current = .. Resistance Assuming the electromotive force to be constant, plot the curve showing the relation between the resistance and the current. 42. n/(x) = X* - 3x2 + 7 X - 1, find/(3), /(O), f{a), f{a + h). 43. If /(x) = x8 + 1, show that/(2) - 4/(1) =/(0). 44. If /(x) = x< + 2x2 + 3, prove that/(- x) =/(x). 45. If /(x) = x6 + 3x8 - 7x, prove that/(-'x) = -/(x). 46. If /(x) = x2 - o2, prove that/(a) =/(- a). 4 7. If /i (x) = x2 + o2, and /2 (x) = 2 x, prove that /i (a) - af. (a) = 0. 48. If /(x) = (x - I) (x2 - 1) (.3 _ i^^ , prove that / (a) = -/(I) . PROBLEMS 49 49. If /(x) = ^-^, prove that /(a) • /(- a) = 1. 50. If fix) = ^' + 2a:8 + 2x + l ^ p^^^^ ^j^^^ ^ Jl\ ^ X2 \X/ 51. If/(x)=.^p±l,find/(3),/(0),/(-3),/(a),/Q). 52. If /(x) = |x, prove that (x + l)/(x) =/(x + 1). 53. If /i(x) =-v/- + \/|' and f^iz) = \/- - \/^' P^O'^'^ ^^^^^^ [/i(a=)?- [/2(a;)P =[/!(«)?• 54. If /(x) = ^^ , prove that /[/(x)] = x. CHAPTEE III THE POLYNOMIAL OF THE FIRST DEGREE 23. Graphical representation. An algebraic polynomial of the first degree is of the form mx + 1, where m and h are numbers, which may be positive or negative, integral or fractional, rational or irrational. We shall restrict the values of m and h, however, to real numbers. In particular cases h may be zero, when the poly- nomial becomes the monomial mx. To obtain the graph of the polynomial, we write y = mx + h, (1) and proceed as in the examples of the previous chapter. We assign to X any number of values assumed at pleasure, say x^, x^, x^, x^, etc. ; compute the corresponding values of y, namely. y^ = mx^ + h, y^ = m,x^+h, y^ = mx^+h, y^ = mx^+h, (2) and plot the points m^z> 2/3). P^i' Vi) (fig. 22). We then Fig. 22 draw the straight lines p,p„ nn, P,P„ each connecting two successive points, and shall prove that these lines form one and the same straight line. For that purpose draw no THE STllAIGHT LI:NE 51 through each point Imes parallel to the coordinate axes, forming the triangles shown in fig. 22. Then, by § 13, T[li.^ = x,^ x^, J^A^ = x^ x^, J^A^ = x^ x^, ^2^2 = ^2 - Vv ^^Z^l = ^3 - 2/2 . ^-^4^ = 2/4-2/3- By subtracting each equation in (2) from the one below it, we have 2/3-2/2 = ^'K-«3-^"2)' y4-2/3 = ^^('''4-'^3). Whence |Z|l ^ |^ ^ |Z| = ,,. (4) or, by (3), jpi, liR, IIR^ Hence the triangles of the figure are similar, and the angles R^P^P^, R,P,Ps, RiPPi are equal. Therefore the line P^P^P^P^ is a straight line. Again, let us take on this line any other point, such as ^, which has not been used in constructing the graph, and draw JIR^ and J?^^ parallel to OX and OY respectively. Then, since the triangles PJi^P^ and I^R^I^ are similar, R,P, _R.^J^, p,r,~i(k' that is, '.l^^ =. '!^~^ = m. (by (4)) 4 2 1 Therefore y^ = mx^— mx^ + y^, whence, by substituting the value of y^ given in (2), 2/5 = '^^5 + ^' Hence the coordinates of P-, satisfy the equation (1). We have now shown that all points the coordinates of which satisfy eqiiation (1) lie on a straight line, and that any point on 52 THE POLYNOMIAL OF THE FIRST DEGREE the line has coordinates which satisfy (1). We have accordingly proved the foUowijig proposition : The equation y = mx + h always represents a straight line. 24. The general equation of the first degree. The equation Ax + By + C={), where A, B, and C may be any numbers or zero, except that A and B cannot be zero at the same time, is called the general equation of the first degree. We shall prove : The general equa- tion of the first degree vnth real coefficients always represents a straight line. 1. Suppose A ^ {^ and B ^ 0. If any value of x is assumed, the value of y is determined. Therefore y is a, function of x, which may be expressed by solving the equation for y ; thus, A C y = X ^ B B This equation is of the form y = mx + h, and therefore repre- sents a straight line by §23. 2. Suppose yl = 0, 5 T^ 0. The equation is then By + C=0, or 2/=- 1' B All points the coordinates of which satisfy this equation lie on a straight line parallel to OX at a distance units from it ; B and, conversely, any point on tliis line has coordinates which satisfy the equation. Hence the equation represents this line. 3. Suppose A^O,B = 0. The equation is then Ax+C=0, or x = --, A and represents a straight line parallel to F at a distance — — units from it. -^ Therefore the equation Ax + By + C=0 always represents a straight line. THE STRAIGHT LINE 53 25. In order to plot a straight line it is, in general, convenient to find the points L and K (fig. 23), in which it cuts OX and OY respectively. If the coordinates of L are (a, 0) and those of K are (0, 5), these coordinates will satisfy the equation Ax + By+ C= 0. By substitution we find F C a = > A B Fig. 23 The quantities a and h, which are equal in magnitude and sign to OL and OK respectively, are called the intercepts of the straight line. It is evident that the h found here is the same as in y = tnx + h. If C=0, i.e. if the equation is Ax-\-By=Q, then a = and 5 = 0, and the straight line passes through the origin. To plot the line, we must find by trial the coordinates of another point which satisfy the equation, plot this point, and draw a straight line through it and the origin. Ex. 1. Plot the line 3a; — 52/ + 12 = 0. Placing y = 0, we find a = — 4. Placing X = 0, we find b = 2f . We lay off Oi = - 4, OK - 2f , and draw a straight line through L and K. Ex. 2. Plot the line 3x — 5 i/ = 0. Here a = and 6=0. If we place x = 1, we find 2/ = 4. The line is drawn through (0, 0) and (1, ^). 26. Any straight line may he represented hy an equation of the first degree. The proof consists in showing that the coefficients A, B, C, in the general equation of the first degree, may be so chosen that the equation may represent any straight line given in advance. Let (x^, y^) and (x^, y^) be any two points on a given straight line. The coordinates of these points will satisfy Ax-\-By + C=0, (1) provided A, B, C have such values that Ax, + By,+ C = 0, Ax^ + By^+C=0, 54 THE POLYNOMIAL OF THE FIRST DEGREE Solving these equations for the ratios oiA,B, C, we have (by § 8) A:B:C = |y, 1 X, 1 . ^'i yi \y. 1 X, 1 ^2 2/2 (2) If these values are used in (1), that equation represents a straight line which has two points in common with the given lin^, and therefore coincides with it throughout. Hence the theorem is proved. The result of substituting from (2) in (1) is = 0, which is the equation of a line through two given points. 27. Slope. Let P^{x^, y^) and P^{x^, y^) (figs. 24, 25) be two points upon a straight line. If we imagine that a point moves along the line from ij to i^, the change in x caused by this motion is measured in magnitude and sign by x^—x^, and the X y 1 ^1 Vx I «2 ^2 1 Fig. 24 Fig. 25 change in y is measured by y^—y^ We define the slope of the straight line as the ratio of the change in y to the change in x as a point moves along the line, and shall denote it by the letter m. "We have then, by definition, m = 2^2-^1 ANGLES 55 It appears from equations (4) (§23) that the letter m in the equation y = mx + J has the meaning just defined. It follows that if the equation of a straight line is in the form Ax+By + C = 0, its slope may he foimd by solving the equation for y and taking the coefficient of x, thus, AC A y = X — —•> whence m = "^ B B B A geometric interpretation of the slope is readily given. For if we draw through I^ a line parallel to OX, and through ^ a line parallel to Y, and call E the point in which these two lines inter- BB sect, then x^— x^ =I\B, and y^ — y^ —BB^ ; and hence m = — ^ • It is clear from the figures, as well as from equations (4) (§ 23), that the value of m is independent of the two points ij and B^ and depends only on the given line. We may therefore choose B^ and B^ (as in figs. 24 and 25) so that I^B is positive. There are then two essentially different cases, according as the line runs up or down toward the right hand. In the former case BB^ and m are positive (fig, 24) ; in the latter case BJF^ and m are negative (fig. 25). We may state this as foUows : The slope of a straight line is positive when an increase in x causes an increase in y, and is negative vjhen an increase in x causes a decrease in y. When the line is parallel to OX, y„ = y^, and consequently m = 0, as explained in § 11. If the line is parallel to OY, x^=. x^, and therefore m = oo in the sense of § 11. 28. Angles. The slope of a straight line enables us to solve many problems relating to angles, some of which we take up in this article. 1. The angle between the axis of x and a known line. Let a known line cut the axis of x at the point L. Then there are four angles formed. To avoid ambiguity, we shall agree to select that one of the four which is above the axis of x and to the right of 56 THE POLYNOMIAL OF THE FIRST DEGREE the line, and to consider LX as the initial line of this angle. We shall denote this angle by <^. Then if we take P any poiut on Fig. 26 Fig. 27 the terminal line of ^ and drop the perpendicular MP, we have, in the two cases represented by figs. 26 and 27, tan^ = MP LM MP But ^^^^^ is equal to the slope of the line. Therefore LM ^ ^ tan = m. If the straight line is parallel to OF, <^ = 90° and tan (f> = cc. If the line is parallel to OX, no angle ^ is formed ; but since 7n = 0, we may say tan <^ = 0, whence <^ = 0° or 180°. 2. Parallel lines. If two lines are parallel, they make equal angles with OX, and hence their slopes are equal. It follows that two equations which differ only in the absolute term, such as and Ax +By + C„_ = Q, represent parallel lines. More generally, two straight lines, and A„x + B„y + C, = 0, are parallel if ^1 J5„ = 0. ANGLES 57 3. Perpendicular lines. Let AB and CD (fig. 28) be two lines inter- secting at right angles. Through P draw PR parallel to OX and let RPD = <^^ and RPB = (f)^. Then tan ^ = m^ and tan ^^ = ^2. where m^ and ^/ig ^^"^ the slopes of the lines. But by hypothesis, (^,= ^ and RPD = (f}^. Then Fig. 29 and hence tan yS = tan {<^„ — j) _ tan <^„ — tan ^^ 1 -|- tan 0., tan <^^ 58 THE POLYN^OMIAL OF THE FIRST DEGREE But tan ^ = 7n^ and tan (f)^ = m^, where m^ is the slope of CD and m^ is the slope of AB. Therefore „ m„—m, tan p = 1 + m^mj If <^2 is always taken greater than ^^, tan yS will be positive or negative according as /3 is acute or obtuse. 29. Problems on straight lines. We shall solve in this article certain important problems which depend on the equation y = mx + h. The essential problem is, in every case, to determine m and h so that the line will fulfill certain conditions. Since two quantities are to be determined, two conditions are necessary and sufficient ; hence, in general, one and only one straight line can be found to satisfy two given conditions. 1. To find the equation of a straight line which has a known slope and passes through a known point. Let m^ be the known slope and P^{Xy, y^ be the known point. Tlie equation of the line will be of the form y = m^x + h, where h, however, is unknown. But the line contains the point Py Therefore y^ = 7n^x^+h, whence h = y^— m^Xy The required equation is, therefore, y = m^x + y^-my)c^\ or, more symmetrically, y-y^ = m^{x-x^). Ex. Find the equation of a straight line with the slope - § passing tlirough the point (5, 7). Fir^ method. We have y = — § x + 6 ; then 7 = -§(5) + &, whence 6 = s^L. Therefore the required equation is or, finally, 2x + 32/-31 = 0. PROBLEMS OX STRAIGHT LINES 59 Second method. By substituting in tlie formula we have y_7 = -|-(a;-5), whence 2x + 3y — 31 = 0, as before. 2. To find the equation of a straight line passing through a known jpoint and parallel to a known line. The slope of the required line is the same as that of the given line, which can be found by § 27. Hence the problem is the same as the preceding. Ex. Find the equation of a straight line passing through ( — 2, 3) and parallel to3x-5y + 6 = 0. First method. The slope of the given line is ^. Therefore the required line is 2/-3=f(x + 2), or 3x-5y + 21 = 0. Second method. As explained in § 28, 2, we know that the required equation is of the form Sx-Sy + O^O, where C is unknown. Since the line passes through (—2, 3), 3(- 2) -5(3) + = 0, whence C = 21. Therefore the required equation is 3x-5y + 21 = 0, 3. To find the equation of a straight line passing through a known point and perpendicular to a known line. The slope of the required line may be found from the slope of the given line, as in § 28, 3. The problem is then the same as problem 1. Ex. 1. Find a straight line through (5, 3) perpendicular to7x + 9y + l = 0. First method. The slope of the given line is — ^. Therefore the slope of the required line is |. By problem 1, the required line is 2/-3 = f(x-6), or 9x- 72/ -24 = 0. Second method. As shown in § 28, 3, we know that the equation of the required line is of the form 9x-7y+C = 0. Substituting (5, 3), we find C = — 24. Hence the required line is 9x — 7j/ — 24 = 0. 60 THE POLYNOMIAL OF THE FIEST DEGKEE Ex. 2. Find the equation of the perpendicular bisector of the line joining (0, 5) and (5, — 11). The point midway between the given points Ls (^, — 3), by § 18. The slope of the line joining the given points is — ^-, by § 27. Hence the required line passes through (§, — 3), with the slope ^^. Its equation is y + 3 = /5(x-|), or lOx - 32y - 121 = 0. 4. To find the equation of a straight line through two known points. This problem has already been solved in § 26, and the result given in the form which is the same as X y I ^1 2^1 1 a^2 ^2 1 X -•»! y -Vx X, — X 2 ^1-^21 = 0, = 0. (Ex. 2, § 3) Or, by § 27, the slope of the required line is x^ x^ Hence, by problem 1, the equation of the required line is Ex. Find a straight line through (1, 2) and (- 3, 6), By the formula, 5 — 2 y - ^ = _ 3 _ ^ (X - 1), or 3x + 4y-ll = 0. 5. To find the condition that three known points should lie on the same straight line. If the three points are (x^, y^), (x^, y^), and («3, y,), the condition that they should lie on the same straight line is - - as is evident from 4. x^ Vl 1 ^2 y^ 1 «8 yz 1 = 0, INTEKSECTION OF STRAIGHT LINES 61 30. Intersection of straight lines. Let A^x + B^y + C; = and A^x + B^y + Cg = (1) be the equations of two straight lines. It is required to find their point of intersection. Since the coordinates of any point on one of the lines satisfy the equation of that line, the coordinates of a point on both lines must satisfy both equations simultaneously. Hence the coordinates of the point of intersection of the lines is found by solving the two equations. There are three cases. 1. A A ^0. A ^2 The solutions are then X = c, c. ^1 A A ^1 ^2 y = A c, A ^2 A A A ^2 The two straight lines intersect in the corresponding point. ^1 B. = 0, but at least one of the determinants. B„ and not equal to zero. The equations are then contradictory and the straight lines do not intersect. In fact, § 28 shows that the straight Imes are parallel. This case may be brought into connection with case 1 as follows : In case 1 suppose that A^B^ — A„B^ is very small, but not zero. The values of x and y are then very large, assuming that the numerators are not small, and the point of intersection is then very remote. 62 THE POLYNOMIAL OF THE FIRST DEGREE Let now the lines be changed in such a manner that ^1^2—^2-^1 approaches zero. The values of x and y increase indefinitely, the point of intersection recedes indefinitely, and the lines approach parallelism. = 0, C„ B„ = 0, A.. C, = 0. The equations are then not independent but represent the same straight line. In this case the attempt to use the solutions as given in 1 leads to the indeterminate form ^ (§ 11). 31. If the three straight lines A^oi+B^y + C^=Q, A„x + B^y + 0.-,= 0, (1) (2) (3) pass through the same point, the three equations have a common solution, and therefore A A Cx A B, a A A cl = 0. (4) Also, if the three straight lines are parallel, the determinant (4) is zero. For if (1), (2), and (3) are parallel, A^B^ — A^B^=^0, 0, A^Bj^—A^Bg = 0, and therefore A^B,-A,B. Conversely, if A A Ci A A c. A A c. A A Cr A A c. A A C3 = 0. = 0, the Imes (1), (2), and (3) either pass through the same point or are parallel. For, by § 7, if two oi' the lines intersect, the coordinates of the point of intersection satisfy the other. DISTANCE FROM A STRAIGHT LINE 63 32. Distance of a point from a straight line. Take the equation of any straight line, written in the form y — mx — b = 0, (1) and consider the polynomial y — mx — h, (2) which stands upon the left-hand side. We may substitute in (2) the coordinates (x^, y^ of any point P^, and thus obtain a value of (2) which is zero when P^ lies on the line (1), but not other- wise. We wish now to obtain the meaning of y^— mx^ — h when P^ is not on (1). For that purpose, let LK (fig. 30) be the line (1), and let MP^, the ordinate j^,g 30 of ij, cut LK in Q. Then the abscissa of Q is x^ and its ordinate is MQ. From (1) MQ = mx^-\- h. Hence y^ — ma;^ — ^ = 2/i ~ (^^i + ^) = MPi-MQ = QP^. It is clear that y^ — inx^ — & is a positive quantity when {x^, y^ lies above the hne LK, and is a negative quantity when {x^, y^ lies below LK. It is also evident from the triangle P^QR, and from a like triangle in other cases, that the length of P^R is numeric- ally equal to P^Q cos <^. But tan <^ = m, and hence We have, then. cos<^ P^R mx^ — h ±ViT m 64 THE POLYNOMIAL OF THE FIRST DEGREE We may, if we wish, always choose the + sign in the denomi- nator. Then P^R is positive when ij is above y = mx + h, and negative when i^ is below. If the equation of the straight line is in the form Ax + By+C=Q, A C m= and & = Therefore B B and Ax^ + By^ +C=B{y^— mx^ — 5), Ax, + By,+ C y/A' + B^ It appears, then, that the polynomial Ax^ + By^ + C and the per- pendicular PyR are positive for all points on one side of the line Ax-\-By -\-C= 0, and negative for all points on the other side. To determine which side of the line corresponds to the positive sign, it is most convenient to test some one point, preferably the origin. 33. Normal equation of a straight line. Let LK (fig. 31) be any straight line and let OD be the normal (or perpendicular) drawn from the origin. Let the length of OD be p and let the angle XOD be a. Take P any point on LK. The projection of OP on OD is equal to the sum of the projections of OM and MP (§ 15). But the projection of OP on OD is p, since ODP is a right angle. The projection of OM on OD is xcoBa (§ 14), and that of 3IP is y cos (a — 90°) = 2/ sin a. Hence Fig. 31 or p = x cos a + y sin a, X cos a-\- y sin a — j? = 0. This equation, being true for the coordinates of any point on LK and for those of no other point, is the equation of LK, It is called the normal equation of a .straight line. NORMAL EQUATION 65 Since sin'^a; + cos^a = 1, it follows from § 32 that Xj^ cos a + 2/j sin a — p is numerically equal to the distance of {x^, y^ from X cos a + 2/ sin a; — ^ = 0. It is sometimes desirable to change an equation Ax +By + C = into X cos a + 2/sina — jt? = 0. For that purpose it is enough to notice that since any value of {x, y) which satisfies one equation must satisfy the other, the one is a multiple of the other. Hence J A = k cos a, B = k sin a, C = — ly, where k is an unknown factor. But from these last equations we have A' + B'' = k^. Therefore cos a = sma = p = ±^A' + B'' B -c ±y/A' + B- Since p is to be positive, the sign of the radical must be oppo- site to that of C. PROBLEMS Plot the graphs of the following equations : 1. 5a; -3?/ + 10 = 0. 3. a; + 3?/ - 7 = 0. 5. 3x + 52/ = 0. 2. 4x + 6?/ + 12 = 0. 4. 2x-9y = 0. 6. 4a; + 7 = 0. 7. .5y-8 = 0. 8. Two numbers are to be found such that one half of one plus one third of the other is eciual to unity. Show how one number may be graphically found when the other is known. 66 THE POLYNOMIAL OF THE FIRST DEGREE 9. A plane figure is in the form of a square, 3 ft. on one side, surmounted by a triangle constructed on one of its sides as a base. Express the area of the above figure in tenns of the altitude of the triangle, and plot the graph of the function. 10. Express the number of inches in any length as a function of the number of centimeters, and express the same as a graph. 11. A uniform elastic string of length I is subjected to a stretching force/. If V is the new length, l'=l(l + mf), where m is a constant. Plot the graph, showing the relation between I' and /. 12. K t represents the boiling point in degrees Centigrade at a height h in meters above sea level, then approximately h = 295 (100 — t). Plot the graph. 13. The pressure on a square unit of horizontal surface immersed in a liquid is equal to the weight of the column of liquid above it. Express the pressure at a depth x below the surface of a body of water, the density of the water being taken as unity. Express also the pressure x units below the surface of a body of water over which is a body of oil of density .9 and of depth 8 units. Plot the graphs. 14. A road starts at an elevation of 100 ft. above sea level and has a uniform up grade of 15 per cent ; i.e. it rises 15 ft. in every 100 ft. of horizontal length. Express the distance above sea level on the road as a function of the horizontal distance from the point of departure, and construct the graph. 15. A tank of water contains 100 gal. A tap is opened, causing the water to flow out at a uniform rate of § gal. per minute. Express the amount of water in the tank as a function of the time, and construct the graph. 16. Find the equation of the straight line of which the slope is 7 and the intercept on OY is — 3. 17. Find the equation of the straight line passing through the point (0, — 3) and making an angle of 135° with OX. 18. y ind the equation of a straight line making an angle of 60° with OX and cutting off an intercept — 5 on OY. 19. A straight line making a zero intercept on OY makes an angle of 120° with OX. Find its equation. 20. A straight line making a zero angle with OX cuts OY at a point 5 units from the origin. Find its equation. 21. Find the acute angle between the lines 2x-3y+5 = an^ x+2y + 2 = 0. 22. Find the acute angle between the lines 2x + Sy - = and 2x + y+l = 0. 23. Find the acute angle between the lines 4x + y — 2 = and 3x + oy + S = 0. 24. Show that 2x + 14i/ — 17 = bisects one of the angles between the lines 8z4-6y-ll = andSx -4y + 3 = 0. 25. Find the equation of the straight line through the point (- 4, 5) parallel to the line 6x — iy+l = 0. PROBLEMS 67 26. Find the equation of the straight line through (3, — 1) parallel to the line X - 2/ = 8. 27. Find the equation of the straight line through the point (2, — 11) per- pendicular to the line 9x — 8y + 6 = 0. 28. Find the equation of the straight line through the origin perpendicular to the line 6x + 5?/— 3 = 0. 29. Find the equation of the straight line through the points (— 2, — 3) and (0, 4). 30. Find the equation of the straight line through the points (2, — 1) and (3, 2). 31. Find the equation of the straight line through the points (—1, 3) and (—1, 5). 32. Find the angle between the straight lines drawn from the origin to the points of trisection of that part of the line 6x + iy = 24: which is included between the coordinate axes, 33. Find the equation of the perpendicular bisector of the line joining (-3, 6) and (-4, 1). 34. A straight line is perpendicular to the line joining the points (—4, — 2) and (2, — 6) at a point one third of the distance from the first to the second point. AVhat is its equation ? 35. Find the equation of the straight line through (3, 5) parallel to the straight line joining (2, 5) and (— 6, —2). 36. Find the equation of the straight line parallel to the line 2z — 3^ + 5 = and bisecting the straight line joining (— 1, 2) and (4, 5). 37. Find the equation of the straight line perpendicular to 3x — 5^ = 9 and bisecting that portion of it which is included between the coordinate axes. 38. What is the equation of a straight line the intercepts of which on the axes of X and y are 2 and — 5 respectively ? 39. What is the equation of the straight line the intercepts of which on the axes of X and y are — 4 and — 7 respectively ? 40. In the triangle A {- 2, - 2), B {1, - 8), C (0, - 7), a straight line is drawn bisecting the adjacent sides AB&nd BC. Prove that it is parallel to ^C and half as long. 41. Find the equation of a straight line through (4, ^) and the point of intersection of the lines 3x — 4y — 2 = and 12x — loy — 8 = 0. 42. Find the equation of the straight line passing through the point of inter- section of X — 22/ — 5=0 and 2x — 3y — 8 — and parallel to3x — 2?/ + 2 = 0. 43. Find the equation of the straight line through the point of intersection of6x — 2y — 11 = and 4x — Cy— 5 = and perpendicular to4x— 2/ + l = 0. 68 THE POLYNOMIAL OF THE FIRST DEGREE 44. Find the equation of the straight line joining the point of intersection of the lines 2x — y + o = and x+ y + 1 — and the point of intersection of the lines x — y — 7 = and 2x + y — o = 0. 45. Determine the value of m so that the line y = mx + 3 shall pass through the point of intersection of the lines y = 2x + 1 and y = x + o. 46. Find the vertices and the angles of the triangle formed by the lines x = 0,x-y + 2 = 0, and 2x + 32/-21 = 0. 47. Find the distance of (3, 5) from the line y = 4 x - 8. On which side of the line is the point ? 48. How far distant from the line 2x + Sy + 8 = qis the point (7, - 4), and on which side of the line is it ? 49. Find the distance from the point (6, — a) to the line - + - = 1 a b 50. The base of a triangle is the straight line joining the points (- 1, 3) and (5, - 1). How far is the third vertex (6, - 2) from the base ? 51. The vertex of a triangle is the point (6, - 2) and the base is the straight line joining (- 3, 2) and (4, 3). Find the lengths of the base and the altitude. 52. Find the distance between the two parallel lines 4x + Sy -10 = and 4x + 3 2/ - 8 = 0. ■ 53. A straight line is 7 units distant from the origin and its normal makes an angle of 30° with OX. What is its equation ? 54. The normal to a straight line which is 5 units distant from the origin makes the acute angle tan-i J with OX. What is the equation of the line ? 55. A straight line 4 units distant from the origin makes an angle of 45° with OX. What is its equation ? 56. The normal to a straight line makes an angle tan-i|with OX. The line passes through the origin. What is its equation ? 57. The normal to a straight line makes an angle of 90° with OX. The line is 7 units distant from the origin. What is its equation ? 58. Find a point on the line 4 x + 3 y = 12 equidistant from the point" (-1, -2) and (1,4). 59. Find the equation of the perpendicular bisector of the base of an isosceles triangle having its vertices at the points (3 2) (-2 -3^ and (2, - 5). V ' /' V . /, 60. A point is equally distant from (2, 1) and (- 4, 3), and the slope of the straight line joining it to the origin is §. Where is the point ? 61. A point is 7 unite distant from the origin, and the slope of the straight hue joining it to the origin is §. What are ite coordinates ? 62. Perpendiculars are let fall from the point (.5, 0) upon the sides of the triangle tlie vertices of which are at the points (4, 3), (- 4, 3) and (0 - 5) Show that the feet of the three perpendiculai-s lie on a straight line PEOBLEMS 69 63. Find a point on the line x + 2 y — 3 = 0, the distance of which from the axis of X equals its distance from tlie axis of y. 64. One diagonal of a parallelogram joins the points (4, — 2) and (— 4, — 4). One end of the other diagonal is (1, 2). Find its equation and length. 65. Find the equations of the straight lines through the point (—2, 0) making an angle tan-i | with the line 3x + 4y + 6=:0. 66. Find the equations of the straight lines through (2, 2) making an angle of 45° with the line 3 x - 2 ?/ = 0. 67. Find the equations of the straight lines through the point (2, 1) making an angle tan-i ^ with the line 2x — y — 3=0. 68. Derive the equation of the straight line making the intercepts a and h on the axes of x and y respectively. 69. Prove analytically that the locus of points equally distant from two points is the perpendicular bisector of the straight line joining them. 70. Prove analytically that the medians of a triangle meet in a point. 71. Prove analytically that the perpendicular bisectors of the sides of a tri- angle meet in a point. 72. Prove analytically that the perpendiculars from the vertices of a tri- angle to the opposite sides meet in a point. 73. Prove analytically that the perpendiculars from any two vertices of a triangle to the median from the third vertex are equal. 74. Prove analytically that the straight lines joining the middle points of the adjacent sides of any quadrilateral form a parallelogram. 75. Prove analytically that the straight lines drawn from a vertex of a paral- lelogram to the middle points of the opposite sides trisect a diagonal. CHAPTER IV THE POLYNOMIAL OF THE JNTth DEGREE 34. Graph of the polynomial of the second degree. The polynomial of the second degree is aa^+hx + c. Its graph may be plotted by equating it to y and proceeding as in §§ 20 and 23. Ex. 1. a;2 + 2x + 2. Place y = x^+2x + 2 and assume integral values of x. The corresponding values of x and y are given in tlie following table : y X y - 1 1 -2 2 -3 5 -4 10 -5 17 As in § 20, we plot these points (fig. 32), and are then to draw a smooth curve through them. But we notice that these points are nearer together in some places than in others. It follows that in some parts the curve would be more accurate than in others. To obviate this diffi- culty we assume such fractional values of x as will locate points between the more widely separated points already plotted. We thus form the table : x y 1.5 7.3 2.5 13.3 3.5 21.3 -2.5 3.3 X y -3.5 7.3 -4.5 13.3 -5.5 21.3 Fig. 32 Plotting these points also, and drawing the curve, we have (fig. 32) the graph of the given polynomial, a;^ + 2x + 2. The graph lies entirely above the axis of x, and recedes constantly from it as x increases numerically, since the polynomial is positive for all values of x, and increases in value as x increases. 70 POLYNOMIAL OF THE SECOND DEGREE 71 Ex. 2. 2x2 + x -6. Place y = 2x^ + X — Q and assume integral values of x Hence the table : y -6 -3 4 15 X - 1 -2 -3 y On plotting these points (fig. 33) we see that it is desirable to assume fractional values of x. Hence the table : X y 1.5 2.3 6.9 2.6 10.1 -1.5 -3 -2.5 4 X y -3.3 12.5 -3.7 17.7 - .5 -6 - .3 -6.1 - .7 -5.7 Fio. 33 In obtaining this new set of points we have assumed — .5, — .3, — .7 as values for x, with the aim of locating as closely as possible the turning point, or vertex, as it will be called, of the curve. Plotting these points also, we draw the curve (fig. 33). It is especially to be noted that the curve cuts the axis of X when x = — 2 and when x = 1.5. But these two values of x, since they make 2 x'^ + x — equal to zero, are the roots of the equation 2 x^ + x — 6 = 0. As the graph of the polynomial in Ex. 1 did not intersect the axis of x, we conclude that the equation formed by placing it equal to zero ha s no real roots. Solving that equation we find that, in fact, the roots are — 1 ± V— 1. 35. Let us now consider the general polynomial of the second degree, ax^+hx + c, of which the two polynomials just plotted are special cases. If we place y = ax^ + hx + c, we can write 7/ = a b cl 3y'+-x+-\ a aj h y c ^- + — + - 2 a/ a ¥— 4ac 4 a' + x-h h 72 THE POLYNOMIAL OF THE Nth DEGREE -4 4: a' The expression in brackets is the constant, -— ^ — > plus a function of x, {x + -^-]> which is always positive except for h V 2 a/ x = — ——, when it is zero. 2 a At first we shall regard a as positive. It follows that y has its least value when x = Therefore the lowest point, the vertex, 2 a of the curve will be ( > ) • As values greater and \ 2 a 4a / less than are assigned to x, x + - — increases numerically, 2 a 2 a y increases, and the corresponding point of the curve rises in the plane. Moreover, if x is assigned the values — 1- k and h A;, A; being any assumed constant value, the corresponding 2 a values of y are the same. Hence the curve is symmetrical with respect to the straight line x = — - — > which line passes through the vertex of the curve parallel to the axis of y. If a is negative, it can be proved in the same way that the curve has an axis of symmetry, a; = — - — > which passes through its vertex, which is in this case the highest point of the curve. 36. Now that we have proved that the graphs of all quadratic polynomials in x are alike, having a vertex and an axis of sym- metry passing through it, we can plot them more easily than was possible before, as is shown by the two following examples. Ex. 1. 4z2_4x + l. I y = 4a;2_4x + l = 4(x2_x + i) = 4(a;-^)2. I Therefore the vertex of the graph is (^, 0), and the / axis of symmetry is the line x= ^. Beginning with / the value ^, we assign to x values greater and less / than ^, thereby locating points on both sides of the axis of symmetry, and plot the graph which is repre- sented in fig. 34. V We see that the equation 4x2_4x + l = has two equal real roots, the graph being tangent (§ 37) to the Fig. 34 axis of x at the point x = \. DISCRIMINANT 73 Ex.2. -2x2 + 3x. y = -2x2 + 3x = -2(x2-|x) = -2[{x- 1)2-^9^]. Therefore the vertex of the graph is (|, |) and the axis of symmetry is the line x = |. The graph is represented in fig. 35. We see that it crosses the axis of z at two different points. Hence the equation — 2x2 + 3x=0 has two unequal real roots, which are found to be and f . Fid. 35 37. Discriminant of the quadratic equation. Turning now to the constant — in the equation 4a^ y = a h^—4ac 4 a' + x + 2a of § 35, we have three cases to consider. 1. If &'— 4 ac > 0, the vertex of the graph is below the axis of X when a > 0, and above the axis of x when a < 0, and accord- ingly the graph intersects the axis of x in two points. 2. If &'— 4 ac = 0, the vertex of the graph is on the axis of x, and hence the graph intersects the axis of a; in a single point. 3. If b'^—4:ac < 0, the vertex of the graph is above the axis of X when a > 0, and below the axis of x when a < 0, and the graph does not intersect the axis of x at all. Now let us suppose that different values are assigned to the constants a, h, and c, in such a way as to make &^— 4 ac decrease, beginning with a positive value. Then the vertex of the graph rises or falls in the plane until, when J^— 4 ac = 0, it lies on the axis of x. At the same time, the points in which the graph inter- sects the axis of x have been approacliing each other, and finally coincide, when the graph is said to be tangent to the axis of x. Eecalling that the abscissas of the points of the graph on the axis of X are the real roots of the equation formed by placing the expression equal to zero, we can tabulate the following results. 74 THE POLYNOMIAL OF THE iS^TH DEGKEE 1. If &^ — 4 ac > 0, the graph of a^^ +hx + c intersects the axis of X at two points, and the equation aa^ -\-bx-i- c = has two real roots, which are unequal. 2. If 6^— 4 ac = 0, the graph of aa^+ hx-\- c is tangent to the axis of X, and the equation aa?+hx + c = has two real roots, which are equaL 3. If &^— 4 ac < 0, the graph of ax^-\- hx + c is entirely on one side of the axis of x, and the equation ai? + Jx + c = has only imaginary roots. The expression J^ — 4 ac is called the discriminant of the quad- ratic equation, as its sign indicates the nature of the roots of the equation. Y 38. Graph of the polynomial of the nth degree. Let the polynomial be a^x"-\- a^af-'+ a^3(f-'^-\ 1- a^_^x+a^. In general this polynomial contains n + 1 terms. If any term is lacking, we may consider that its coefficient lias become zero. We will begin by plotting the graphs of some 5^ special numerical cases. Ex. 1. z3. Place y = x^ and assume values of x. Hence the table ; Fio. 36 X y 1 1 2 8 - 1 - 1 - 2 - 8 .5 .1 - .5 - .1 z y 1.5 3.4 - 1.5 - 3.4 2.3 12.2 - 2.3 - 12.2 2.7 19.7 - 2.7 - 19.7 Drawing a smooth curve through these points, we have the curve of fig. 36. It is called a cubical parabola. EXAMPLES OF GRAPHS 76 Ex. 2. X*. Place y = X* and assume values of x. Hence the table ; X y 1 1 2 16 -1 1 -2 16 .5 .1 .7 .2 .8 .4 .9 .7 1.1 1.6 1.3 2.9 1.6 6.1 X y 1.7 8.4 1.9 13.0 - .5 .1 - .7 .2 - .8 .4 - .9 .7 -1.1 1.6 -1.3 2.9 -1.5 5.1 -1.7 8.4 -1.9 13.0 The curve is represented in fig. 37. Ex. 3. x5. Place 7/ = x^ and assume values of x. Hence the table : Fig. 37 -X X y 1 1 2 32 - 1 - 1 -2 -32 .7 .2 .9 .6 1.2 2.6 1.4 5.4 1.6 10.6 1.7 14.2 X y 1.8 18.9 1.9 24.8 - .7 - .2 - .9 - .6 -1.2 - 2.5 -1.4 - 5.4 -1.6 -10.5 -1.7 - 14.2 - 1.8 - 18.9 - 1.9 -24.8 Fig. 38 The curve is represented in fig. 38. In each of the three examples above, the curve crossed the axis of x at the origin, and the corresponding equation had the root zero. 76 THE POLYNOMIAL OF THE Nth DEGREE Ex. 4. x^ -2x- + ox -6. Place y = x^ — 2 x^ + Sx — a and assume values of x. Hence the table : x y - 6 1 - 4 2 3 12 -1 -12 Fig. 39 - 2 i - 28 The curve is represented in fig. 39. This curve crosses the axis of x at the point x = 2, and hence the equation x^ — 2x2 + 3x-6 = has 2 for a real root . Its other roots are imaginary, i.e. ± V— 3. Ex. 5. 4x8 + 4x2- 9x - 9. tlace y = 4x' + 4x2 _ 9x — 9 and assume values of x. Hence the table : X y 1.5 - 2.6 2.5 4.6 2.7 7.2 -1.5 -18.4 -1.7 -21.8 X y - 9 1 -10 2 21 -1 -2 - 7 X y 1.6 1.3 -5.2 1.7 6.9 - .5 -4.0 -1.5 -1.3 .7 This curve is represented in fig. 40. It crosses the axis of X at three points, — when x = 1.5, when x = — 1.5, and when x = — l. Hence ±1.5 and — 1 are real roots of the equation 4x8 + 4x2 -9x -9 = 0. Without discussing any more numerical examples we can see that, in general, the abscissas of the points on the axis of x of the graph of the polynomial Fig. 40 are real roots of the equation a^of+ a^x"-'+a^oif-'+ ... + a„_,x+ a„ = 0. SOLUTION BY FACTORING 77 Conversely, the real roots of the equation a(,af + a^af~^+ a^o(f~^-\- • • • + a„_ia;+ «„ = are the abscissas of the points at which the graph of ftpaf + a^x''~^+ a^3(f*~^+ • • • + a„_jiz; + a„ intersects the axis of x, for they make y = 0. Moreover, if the graph of the polynomial does not intersect the axis of x, the corresponding equation has no real roots; and conversely, if the equation has no real roots, the graph of the polynomial does not intersect the axis of x. 39. Solution of equations by factoring. Let f{x) be a poly- nomial which can be separated into factors fy{x), fj^x), f^{x), • • • , each of which is necessarily of lower degree than /(a?). Then the equation /(^)=0 (1) may be written in the form A{^)-U^)-U^)'-- = ^- (2) It is evident that any value of x which makes one of the fac- tors f-^{x), f^(x), fJx), ■ • • zero, satisfies equation (2), and hence equation (1), i.e. is a root of equation (1). But such a value of x is evidently a root of some one of the equations Conversely, any root of equation (1) must satisfy equation (2), and hence must make some one of the factors /^{x), f^(x), fj^x), • • • zero ; for if no one of these factors is zero, their product cannot be zero. Hence the solution of the equation f{x) = is reduced to the solution of the separate equations /i(^)=0, /.(^)=o, /3(^)=0, In applying this method it is usually desirable to have no fac- tor of higher degree than the second ; but there is no advantage in carrying the factoring any further, as any quadratic equation can be readily solved. 78 THE POLYNOMIAL OF THE Ath DEGREE Ex. 1. Solve the equation x^ = 8. By transposition, x'^ — 8 = ; whence, by factoring, (x — 2) (x^ + 2 a; 4- 4) = 0. .-. x-2 = or x2 + 2x + 4 = 0; whence x = 2 or — 1 ± V— 3. Since the original equation might have been written x = '^S, we see that the three values of x which have been found are each a cube root of 8. In fact, every number has three cube roots, which may be found by solving the equation formed by placing x^ equal to the number. Ex. 2. Solve the equation x* + 9 = 0. This equation may be written (x* + 6x2 + 9) -6x2 = 0; whence, by factoring, (x2 + Vo x + 3) (x2 — V6 x + 3) = 0. .-. x2 + Vox + 3 = 0, or x2 - V6x + 3 = ; - V6 ± V3^ Ve + V^Te whence X — — = or —= - . 2 2 It is to be noted that every number has four fourth roots, which may be found by a method similar to that suggested above for linding its three cube roots. 40. Factors and roots. It follows immediately from the pre^ ceding article that if x — r is a factor of f{x), then r is a root of the equation f{x) = 0. Conversely, if r is a root of the equation f(x)=0, then the polynomial f {x) is divisible hy x — r. Let f{x) = a^x^^ a^af -' + • • • + a,^_^x + a„, and let r be a root of f{x) = 0. Then f{r) = a.,r^ + a^r'-i + . . . + a^_^r + a„ = 0. r,f{x)=f{x)~f(r) _ = (a^aj" + a^a;"-' + • • • + rr.,,,, a? + aj - («o^" + «i^"~' H + «„_ir + a„) = a„(af - r") + a^(a--' - r'-^) + • • • + a„_>(.t^ - r). As f{x) is expressed as a series of terms each of which, being the difference of the same positive integral powers of x and r, is divisible by x-r, it follows that /(a-) is divisible by a;-r. FACT0E8 AInD ROOTS 79 Ex. By inspection — 1 is a root of the equation a;* + x3 + 2x2 + 3x + l = 0. (1) Hence x + 1 is a factor of tlie left-hand member of tlie equation, which may accordingly be written (x + l)(x3 + 2x + l) = 0. (2) Additional roots of equation (1) may now be found by solving the equation x3 + 2x + l = 0by methods given in §§ 62 and 63. It is to be noted that the solution of the original equation has been simplified by making it depend upon the solution of a depressed equation, i.e. one of degree lower than the degree of the original equation. 41. By means of the second theorem we can form an equation which shall have any given quantities, r^, r.,, • • •, r^ as roots. For if r^, r^, • • • are the roots of the equation, its left-hand member must contain the factors x — r^, x — r^, • ■ • , the right-hand mem- ber being zero. Therefore the equation (x — r^) (x — r^) ■ ' • (x — rj = has the required quantities as roots. Moreover, this equation can have no other roots, since any other value of x will make no fac- tor equal to zero, and hence the product will not be zero. There- fore the required equation is (x — Vj) {x — r„)---(x — r„) = 0. Ex. 1. Form the equation having as roots 2 -f- 3 V — 1, 2 — 3 V— 1, — -J. The required equation is (a; _ 2 - 3 V^) (x - 2 + 3 V^) (x + -J) = 0, or [(X - 2)2 + 9] [3x + 1] = 0, or 3x3-11x2 4- 35x + 13 = 0. This method of forming an equation suggests a method of factor- ing a quadratic expression. For if r^ and r^ are the roots of the quadratic equation ax^+ hx+ e= 0, then aaf-i- hx + c i& divisible by x — r^ and x — r,, ; and hence ax^ + bx + c = a (x — r^) (x — r^). 80 THE POLYNOMIAL OF THE .Vth DEGREE Ex. 2. Factor 6 x^ + x - 1. The roots of the equation 6x2 + x — 1 = are _ ^ and ^. ,-. 6x2 + X - 1 = 6(x + ^) (X - ^) = 2(x + ^).3(x-i) = (2x + l)(3x-l). Ex. 3. Factor 4 x^ + 4 x - 2. The roots of the equation 4x2 + 4x — 2 = are ... 4x2 + 4x - 2 = 4(x - ^if^^) (x - :^if^) = (2x + 1 - V3) (2x + 1 + V3). Ex. 4. Factor x2 + 4 x + 6. The roots of the equation x2 + 4x + 6 = are — 2 ± V— 2. .-. x2 + 4x + 6 = (x + 2 - V3^) (x + 2 + V^). 42. Number of roots of an equation. The fundamental propo- sition concerning the roots of an equation is that every equation formed by placing a polynomial equal to zero has at least one root. The proof of this proposition, however, depends upon methods too advanced to be used here. We shall therefore assume it as proved, and proceed to prove, as a consequence of it, that every equation of the nth degree has n roots, and only n roots. Let the given equation a^af + ajOf-^H \.a^_^x+ a^ = be denoted by f{x) =0. (1) Since this equation must have at least one root, let r^ be that root. Then f{x) is divisible hy x~r^ (§ 40) and therefore f{x) = {x-r,)f,{x), (2) /j(ic) being the other factor, and necessarily of degree n — 1. Equation (1) can now be written {^-r,)f,{x)=0, (3) and any root of f^{x) = (4) is a root oif{x) = (§ 39). NUMBER OF ROOTS 81 But equation (4) must have at least one root ; and if we let r^ be that root, and reason as before, we may write U(x) = (x-T^f,{x), (5) f^ip^ being of degree n — 1. By substitution in (2) we shall have f{x) = {x-r^(x-T^f^{xy s (6) After separating n linear factors in this way, the last quotient will be (Xq. Therefore we shall have f{x) = aj^a - r^) {x-r^)---{x- rj, (7) the polynomial being expressed as the product of n linear factors. Then the equation f(x) = may be written af,{x — r^) {x — r^---{x — r„) = 0, (8) whence it is seen to have n roots (§ 39), i.e. r^, r^, • • •, r„. It can have no other roots ; for if we let x have any value other than r^ 1\- ■ -, ov r„, no factor of the first member of (8) is zero, and hence the product in the first member is not equal to zero. Therefore the equation of the nth. degree has n, and no more than n, roots, and the polynomial of the nth. degree can always be separated into n linear factors. In general, however, it is not possible to determine these factors where n > 4. It is to be noted that the roots may all be different, or some of them may occur more than once. In the latter case the equation is said to have multiple roots. 43. If now the left-hand member of equation (8) of § 42 is expanded, the equation appears in the original form a^af + a^af' ^ -j h a^,_iX + «„ = 0, and it is evident that (-r,) + (-r,) + (-r3)-|-...+(-r„)=% "■0 (1) and that (- r,) (- r,) (- r^) ■ - ■ (- O = ^ ' (2) 82 THE POLYNOMIAL OF THE Ntu DEGKEE Equations (1) and (2) express respectively the following theorems: 1. The sum of the roots of an equation with their signs changed is the coefficient of x^'^ divided hy that of «". 2. The product of the roots of an equation vnth their signs changed is the constant term, divided hy the coefficient of af. Other theorems of this type are given in works on the theory of equations, but only these two have been stated here, since they are of special service in finding the remaining root of an equation after all the others have been determined. Ex. 1. Three roots of the equation 2 x* + Tx^ + 8 a;2 + 2 a; - 4 = are - 2, _ 1 _ V— 1, and — 1 + V— 1. Find the fourth root. The sum of all the roots is — ^, and the sum of the three roots known is — 4. Therefore the fourth root is — | — (— 4), or ^. Ex. 2. Two roots of the equation 36 x^ — 7 x + 1 = are ^ and — J. Find the third root. Tlae sum of the two roots known is — ^, and the sum of all the roots is 0, since the coefficient of x^ is 0; therefore the third root is — (— ^), or ^. Or the product of the roots known is — ^, and the product of all the roots is — 3^5 ; therefore the third root is (— ^^) -;- (— ^), or ^. 44. Conjugate complex roots. Nothing was said in § 42 as to the nature of the roots r^, r„, • • • , r„. But if the coefficients «o» «i> • • • » ^n ^re all real, and if a + hi is one of the roots, then a — hiSs, also a root. For if a + Z>t is a root of f{:£) = 0, then f{a + hi) = 0. When f{a + hi) is expanded the terms can be separated into two sets, — those containing a alone or involving only even powers of hi as a factor, and those involving only odd powers of hi as a factor. By § 12 the terms of the first set are all real and their sum may be denoted by A ; and the terms of the second set contain i to the first power as a factor, and their sum may be denoted by Bi (B, of course, being real). Then f{a + hi) = may be written A + Bi = 0, whence (§ 12), ^ = and 5 = 0. If, in the above, we replace hi by — hi, it is evident that the terms in the first set are not affected, as they involve only even GRAPHS OF PRODUCTS 83 powers of li as a factor, and those in the second set, involving only odd powers of hi as a factor, are changed in algebraic sign only. Therefore we have f\a + (— hiy] = A — Bi. But we have seen that A = Q and J? = ; therefore f\ct + (— hi)] = 0. Since f\a + (— 11)]= f [a — hi), however, it follows that f{a — hi) = 0, and a — &i is a root of the given equation f{x) = 0. This fact is usually stated by saying that complex roots occur in pairs. It follows that an equation of even degree may not have any real roots, and that an equation of odd degree must have an odd number of real roots, and thus at least one real root. 45. It was proved in § 42 that every polynomial is equivalent to the product of n linear factors, i.e. «o(^ - ^i) {x-r^---{x- r„), where r^, 7\, • • • , r„ are the roots of the corresponding equation. Now if any one of these roots is complex, there will be a corre- sponding conjugate complex root. Let a + hi and a — hi be two such roots. Then the corresponding factors are (x — a — hi) and (x — a + hi), which combine into (x—a)^-i-h^, a real quadratic factor. Therefore every polynomial with real coefficients is equivalent to the product of real linear and quadratic factors. 46. Graphs of products of real linear and quadratic factors. 1. All the factors linear and none repeated, as a^{x - r,) {x-r^)---{x- r„). Placing y equal to this expression, we have y = a^{x - r,) {x-r,y--{x- r„). It is evident that the graph intersects the axis of a; at ?i dis- tinct points for which x =■ r^, x = r^, ■ • ■ , x = r^, and at no other points, as no other values of x make y zero. Now let the quan- tities T^, r^, ■ • • , r^ be arranged in the order of their magnitude, r^ being the least. Then if at first x < r^, all the factors are nega- tive ; and if x changes so that r^< x < r^, the first factor becomes positive while all the others remain negative. Therefore y changes 84 THE POLYNOMIAL OF THE Ath DEGREE sign when x changes from being less than r^ to being greater than rj, and the curve crosses the axis of x at the point x = r^. Again, if x changes so that at first 7\< x < i\ and then r^ 0, and the corresponding part of the curve lies above the axis of x. If — .5 < X < 2, the first two factors are positive and the third is nega- tive ; therefore y <0, and the corre- sponding part of the curve lies below the axis of x. Finally, if X > 2, all the factors are positive ; therefore y>0, and the correspond- ing part of the curve lies above the axis of X. 3. Assuming values of x and finding the corresponding values of 2/, we plot the curve, as repre- sented in fig. 41. Fig. 41 Fig. 42 Ex.2. y = .5(x + 2.5)(x-l)2. 1. If X = — 2.5 or 1, 2/ = 0, and there are two points of the curve on the axis of X. 2. If X < — 2.5, the first factor is negative and the second factor is posi- tive ; therefore y <0, and the corre- sponding part of the curve lies below the axis of x. If — 2.50, and the corresponding part of the curve lies above the axis of x. Finally, if « > 1, we have the same result as when — 2.6 — 3, the first factor is positive ; there- fore 2/ > 0, and the corresponding part of tlie curve is above the axis of X. 3. Assuming values of x, and finding the corresponding values of y, we plot the curve as repre- sented in fig. 43. 47. Location of roots. From the work of the last article it is evident that the real roots of the equation f(x)=0 determine points on the axis of X at which the graph of f{x) crosses or touches that axis. Moreover, if x^ and x^ (x^ < x^) are two values of x, such that /(Xj) and f{x^) are of opposite algebraic sign, the graph is on one side of the axis when x = x^, and on the other side when x = x^. Therefore (§56) it must have crossed the axis an odd number of times between the points x — x^ and x = x^. Of course it may- have touched the axis at any number of intermediate points. Since a point of crossing corresponds to an odd number of roots of an equation, and a point of touching corresponds to an even number of roots, it follows that the equation f(x) = has an odd number of real roots between x^^ and x^. The above gives a ready means of locating the real roots of an equation in the form f(x) = 0, for we have only to find two values of x, as x^ and x^, for which f{x) has different signs. We then know that the equation has an odd number of real roots between these values, and the nearer together x^^ and x^, the more Fig. 43 DESCARTES' RULE OF SIGNS 87 nearly do we know the values of the intermediate roots. In locat- ing the roots in this manner it is not necessary to construct the corresponding graph, though it may be helpful. 48. Descartes' rule of signs. When in a polynomial a term with one sign is immediately followed by one with the opposite sign, there is said to be a variation of sign. For example, in the polynomial 3 aj* + 2 a;^ — 3 ic^ + ic — 2 there are three variations. The variations of sign in the left-hand member of an equation are often of value in locating the real roots of the equation, for the number of positive roots of the equation f (x) = cannot exceed the number of variations of sign in its left-hand member. This rule is known as Descartes' rule of signs. For example, the equation 3x^-i-2x^ — 3x^ + x— 2 = cannot have more than three positive roots, as there are three variations of sign in its left-hand member. To determine the greatest possible number of negative roots, replace ic by — x'. The roots of the resulting equation will be those of the original equation with their signs changed. Accord- ingly the original equation can have no more negative roots than this new equation has positive roots. If, in the equation 3 x* -\- 2 x^ — 3 x'^ + x — 2 = 0, x is replaced by — x', tlie new equation is 3 x'* — 2 x'^ — 3 x'^ — x' — 2 = 0. As this equation cannot have more than one positive root, the original equation cannot have more than one negative root. Sometimes, l)y Descartes' rule, we can prove that an equation has imaginary roots. For example, the equation 3 a;^ -|- «^ -f- 2 = can have no positive root, and not more than one negative root. Being of odd degree, it has at least one real root (§ 44); therefore it has one negative root and two imaginary roots. In order to prove Descartes' rule we will first prove that if any polynomial f(x) is multiplied by x — r, ivhere r is a positive quan- tity, the product has at least one more variation than has f{x). Assuming the first term of f{x) to l)e positive, we will inclose all the terms preceding the first minus §ign in a parenthesis. In a second parenthesis we will inclose all the terms with a minus sign before a positive sign occurs again, and so on. Suppose, then, that the first minus sign appears in the term containing 88 THE POLYNOMIAL OF THE Nth DEGREE x""*, the next plus sign occurs in the term containing af-\ etc., and that all the terms after that containing x""" have the same sign as that term. We can now write — {a^^af-^ H \- a,_iaf -'+') ±K^"'" + ••• + ««)> (1) where all the terms within each parenthesis are of the same sign, i.e. plus. Therefore each parenthesis marks a variation. To multiply f(x) hj x — r we shall multiply first by x, then by — r, and add the partial products. The result is an equation of the following form : ' + (&^-' + i±...) ±(&„af-"' + ^±...)HFa„r, (2) where h^. = a^ + ra^._i, hi = ai + rai_^, etc., and accordingly are positive. The signs before each parenthesis of (2) are the same as in (1), but the signs within the parenthesis are not necessarily all plus. But however the signs may occur within any parenthesis, there is at least one variation between tlie first term of one parenthesis and the first term of the following parenthesis. Hence, if we con- sider the parentheses only, the number of variations in the prod- uct is not less than the number of variations in f{x). But, in addition, we have the last term of the product, i.e. :f a„r, the sign of which differs from the sign of the first term in the last parenthesis. Hence there is at least one more variation in {x — r) f{x) than in f{x), as we set out to prove. Now the equation having the roots r^, rg, • • • , r„ is (§ 41) {X- T^) {X- r^) ■ ■ ■ {X - t;)= {). In expanding the left-hand member every time we multiply by a factor corresponding to a positive root, we add at least one variation of sign. Hence the number of positive roots cannot exceed the number of variations, as stated in Descartes' rule. EATIONAL BOOTS 89 49. Rational roots. The real roots of any equation are either rational or irrational (§ 10), and the rational roots must be either integral or fractional. We will now derive methods of finding the rational roots, beginning with the integral roots. An easy method of determining the integral roots depends upon the following theorem : If the equation is written in the forni a^a^ + a^a""-^ H h «„_!«; + a„ = 0, (1) ivhere all the coefficients are integers, any integral root r must he a factor of a„. It has been proved in § 40 that the left-hand member of (1) is divisible hy x — r. Since the coefficient of x is unity, and all the coefficients in the dividend are integers, all the coefficients in the quotient are integers. But the last coefficient in the quotient multiplied by r must be «^„, since there is no remainder. Hence the theorem is proved. Accordingly, to find the integral roots of any equation with integral coefficients, we have merely to try the integral factors of a„. When an integral root has been found, we depress the degree of the equation as in § 40, and apply the process to the new equation. In this way all the integral roots may be found. In case no integral factor of a„ proves to be a root, it follows that the equation can have no integral root. Ex. Find the integral roots of the equation 4 X* - 4 x3 - 26x2 + X + 6 = 0. The integral roots of this equation must be factors of 6, so that we have to tiy ± 1, ± 2, ± 3, ± 6. By trial it is found that — 2 is a root, and the degree of the equation is depressed by dividing the left-hand member by x -1- 2, the depressed equation being 4 x^ — 12 x^ — x -H 3 = 0. The only possible values of integral roots of this equation are db 1, ± 3, and 3 is found to be a root. Dividing the left-hand member by x — 3, we have, as the depressed equation, 4x2— 1 = 0, the roots of which are ± ^. Therefore the roots of the original equation are — 2, 3, ± J. While all the integral roots of an equation may be found by tliis method, it is evident that it fails for fractional roots, as there is no way of determining what fractions ought to be tried. This difficulty is obviated by the two theorems in the next article. 90 THE POLYNOMIAL OF THE 3th DEGREE 50. If a^ is unity and all the other coejgHcients are integers, the equation cannot have a rational fraction in its lowest terms as a root. Let the equation be af + a^af-'^+a^af-^-\ ha„_^x + a„=0, p and, if possible, let the rational fraction - , wliich is in its lowest terms, be a root. Then (f)"-'(f)""-'(f)""--»-(f)---- Multiply through by q"'^, and transpose to the second member all terms but the first. Then ^ = - a^p"-'- aj)''-\ "n-ii??""' - «„2""'- By hypothesis p and q have no common factor, and therefore — is a rational fraction in its lowest terms, while the right-hand member of the equation is an integral expression. But two such p expressions cannot be equal, and hence — , the rational fraction in its lowest terms, cannot be a root of the equation. Moreover, every equation in the form ttoX" + ttjO^-' + a^cff"-- -\ h a^_^x + a„ = 0, in which a^ is not unity, can he tra-nsformed into an equation with integral coefficients in tvhich the coefficient of the highest power of the unknow7i quantity shall he unity. For, dividing tlirough by «„, we have a. a„ a , a ic"+-iiC»-i + -^af'-2-|-...-f--^^a;-}-- = 0. (1) If « is a root of this equation, let a? = — > 7n. being an integer, and substitute in (1). Then w" ffoWi a.m'^ «„ m a,, ^ ' RATIONAL ROOTS 91 Multiplying (2) by wi", we have x"'-\-[-^m x'"-' + {-^m^]x'"-' + In-l,l 2 2l^./n-2 We can now determine m by inspection in such, a way that all the coefficients of (3) shall be integers. The roots of this new equation are m times the roots of the original equation. Ex. Transform equation 12x3 + 16a;2 _ 6x — 3 = to an equation having integral coeflBcients, the coefficient of the highest power of x being unity. Dividing by 12, we have Multiplying the roots of this equation by an integer m, we insert in each term a power of to such that the sum of its exponent and that of x' shall be equal to the degree of the equation, thus obtaining X'8 + (J to) X'2 - (^\ to2) X' - (4 7n,8) = 0. For ^ m to be an integer, to must equal 3 k where k is an integer. Then ^'^ m2 becomes j''^ (9 k^), and this is an integer only when k = 21; i.e. in = 61, I being an integer. Finally, ^ to^, or ^ (6 Z)^, is an integer if 1 = 1, the least value of to being the one desired. Therefore we let to = 6, and our required equation is x'3 + 8x'2 - 15x' - 64 = 0, the roots of which are six times the roots of the original equation. The roots of this equation are found by the method of § 49 to be — 2, 3, and — 9. Hence the roots of the original equation are — ^, ^, and — ^. We are thus in a position to determine the rational fractional roots of any equation with rational coefficients. 51. We now see that to find all the rational roots of any equa- tion, we first find all its integral roots and then all its fractional roots, as indicated in the following example. 92 THE POLYNOMIAL OF THE .Vth DEGREE Ex. Find all the rational roots of the equation 2x*-5x3_2a:2-7x + 30 = 0. (1) By Descartes' rule of signs this equation cannot have more than two posi- tive roots, and not more than two negative roots. If any of the roots are inte- gral, they will be among the factors of 30, i.e. ± 1, ± 2, ± 3, ± 5, ± 6, ± 10, ± 16, ± 30. By trial we find + 2 to be a root, and the depressed equation is 2x8 -x2-4x- 15 = 0. (2) By trial we find that this new equation has no integral roots, no factor of 15 being a root. Accordingly we proceed to find fractional roots. Dividing equation (2) through by 2 and then multiplying the rpots by m, we have x'3 - (^ m) x'2 - (2 m^) z' - (V- rnS) = 0. (3) To make the coefficients of (3) integral we take wi = 2, and the equation becomes a;'8_x'2_8x'-60 = 0. (4) By trial we find an integral root of this equation to be 5, and the depressed equation is „ . ,„ „ ^ x2-|-4x + 12 = 0, (5) the roots of which are — 2 ± 2 V— 2. Theref ore the three roots of the transformed equation (4) are 5 and — 2 ± 2 V— 2, and the roots of the first depressed equation (2) are ^ and — 1 ± V— 2, so that the roots of the given equation are 2, ^, and — 1 ± V— 2. It is to be noted that in this example, after having found all the rational roots, we were able to find the remaining roots also, since the last depressed equation was of no higher degree than the second. 52. Irrational roots. It should be borne in mind that rational roots occur only for special values or systems of values of the coefficients. Hence, after removing the rational roots, if any, by the previous methods, we have, in general, to determine irrational roots in order to have all the real roots of the equation. But from the definition of an irrational quantity (§ 10) it is evident that we cannot find an irrational root exactly. We may, however, find an approximate value to any required degree of accuracy. There are various methods of approximation, one of which imme- diately follows. A more rapid method is given in § 63.* * A method of solving algebraic equations, known as Horner's method, is found in most treatises on the theory of equations. It is convenient in arrangement of work and speedy in the hands of an expert. It may therefore be recommended to one who has often to solve equations. On the other hand, the methods of §§ 52, 6.3 of this book have two advantages. They may be applied to other than algebraic equa- tions (see § 162), and depend upon principles which, if once mastered, are not easily forgotten. IRRATIONAL ROOTS 93 Fig. 44 Let the given equation 'bef(x)=0, and the graph of the left- hand member be as in fig. 44, where OM^ = x^ and OM^ = x^. Then M^F^ =f{x^) and M^P^=f{x^, and since f{x^ and f{x^ are of opposite sign, the curve crosses the axis of x between M^ and M„, and there is at least one real root of f{x)=Q between x^ and x^ (§ 47). Not only does the curve cross the axis of x at some point be- tween Jfj and M^, but it is evident from fig. 44 that the straight line P^P^ also intersects the axis of x at some point between M^ and M^, as M^. If the points M^ and M^ are near together, i.e. if x^ and x^ differ only by a small amount, the curve in most cases differs only slightly from the straight line P^P^. Hence, if we replace the curve by the straight line, the abscissa of the point at which P^P^ intersects the axis of x will be approximately the root of the equation. If OJ/3 is denoted by x^, it is evident (fig. 44) that there is a root of f(x)=0 between x^ and x^, a smaller interval than that between x^ and x^, in which the root was first located. If, however, the graph oi f{x) had been as in fig. 45, the root would have been between x^ and x^, an interval smaller, of course, than that between x^ and x^. If f{x^ has the same sign as f{x^, we have the first case (fig. 44) ; and if f{x^ has the same sign as f{x.^, we have the second case (fig. 45). In the first case, repeating the proc- ess, using iCg in place of x^, we can find an x^ between which and x^ the root must lie ; and in the second case, using x^ in place of x^, we can find an x^ between which and j\ the root must he. 94 THE POLYNOMIAL OF THE Nth DEGREE Moreover, it is evident that the successive values of x, ie. x^, «4) ^6> • • •> found in this way are each nearer to the true value of the root of f(x)=0 than the one preceding. Ex. Find the root of the equation x3 + 2x — 17 = between 2 and 3. Here Xi = 2 and X2 = 3 ; al8o/(2) = - 5 and/(3) = 16. The equation of the straight line determined by the points (2, - 5) and (3, 16) is (§ 29) , - 5 - 16 , y + 5= ^_3 (X - 2). Its intercept on OX, found by letting y = 0, is 2.2 +, and /(2.2) = — 1.952. Since /(2.2) has the same sign as/(2), the second straight line is determined by the points (2.2, - 1.952) and (3, 16). Its intercept on OX is 2.28 +,and /(2.28) = - 0.587648. Since /(2.28) and /(2.2) have the same sign, the third straight line is determined by the points (2.28, -0.587648) and (3, 16). Its intercept on OX is 2.3+, and /(2.3) = — 0.233. The fourth straight line is determined by the points (2.3, -0.233) and (3,16). Its intercept on OX is 2.31 +, and /(2.31) = - 0.053609. The fifth straight line is determined by (2.31, -0.053609) and (3, 16). Its intercept on OX is 2.312. Hence the irrational root of x^ + 2 x — 17 = 0, accurate to two places of decimals, is 2.31. By continuing this process we can find any desired number of decimal places of the root. It is to be noted that we are obliged to find one more decimal place than the number of decimal places to which the root is to be accurate. The approximation is more rapid if the first decimal place is found by the method of § 47. PROBLEMS Plot the graphs of the following quadratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corre- sponding equation : 1. 2x2 + 3x-2. 4. -3x2 + 5x. 2. 9x2-3x-2. - 5. -9x2 + 12x-7. 3. 4x2 + 4x + 3. 6. 4x2-4x-l. 7. For what values of a are the roots ofax2 + 3x + 7 = equal ? What are the roots ? 8. Prove that the roots of (ex + —J - 8 ax = are equal for all values of a and c, and find them. ^ ^ 9. Prove that there is no real value of m for which the roots of x2 + (mx + 3)2 - 16 = are equal. PKOBLEMS 95 For what values of k are the roots of the following quadratic equations (1) equal ? (2) real and unequal ? (3) imaginary ? 10. 2x2 + 3a; + 2 = A;. 11. x2 + (2 - A;)x + 1 = 0. 12. {k + l)x2 + (A; - l)x + (^ + 1) = 0. Plot the graphs of the following polynomials : 13. x3 - ax. (a > 0.) 19. x^ - 12 x + 3. 14. x3 - 4x2 + X + 1. 20. 2x* + x3 - 4x2 - lOx - 4. 15. x8- 3x2 + 1. 21. 4x* + 12x3 + 7x2-28x-6. 16. x3 + x2 + 2x + 5. ' 22. 3x* - 10x3 - 5x2 ^ 2x. 17. x3 - x2 + X - 4. 23. X* + 6 x8 + 10x2. 18. x3 + 6x-6. 24. 2x5 + 2x*-7x3-8x2-4x. Find all the roots of the following equations : 25. 8x3 = 27. 28. 5x6 + 27x2 = 2x«-64x*. 26. 8x6 - 63x8 - 8 = 0. 29. {2x - a)* - (3x + a)* = 0. 27. x6-5x3 + 12x = 2x3 + 3x. 30. x*- 2(a2+l)x2 + (a2 -1)2 = 0. Form the equations having the following values for their roots : 31.0,2,3. 32. a + Vb,a-Vb, -a. 33. 0, 0, 2 a ± 6, ± V2b. 34. Form a quadratic equation with real coefficients having 2 + 3i for one of its roots. Factor the following quadratic expressions : 35. 4x2 + 8x- 7. 38. x2 + 2ax- a + a2. 36. 4x2 + 12 X + 11, 39, ^2x2 + 2 a6x - a. 37. 4 a2x2 + 2 ax + 1. 40. a2x2 + 2abx + b + b'l If ri and r^ are the roots of the equation x2 + px + g = 0, find the values of the following expressions in terms of p and q without solving the equation : 41. r2 + r|. 42. rf + r.i'. 43.1 + 1. 44. 1 + i. 45.^ + ^. If ^ii ''25 ra are the roots of the equation x^ + _px2 + gx + r = 0, find the val- ues of the following expressions in terms of the coefficients without solving the equation : , . ' 46. (rl + r| + r.2) + 2 (nrz + rgrs + nn) + 3 rirgrs- 47. r{r2r3 + rhaTi + rhir^. 48. h 1 ■^ nrz r2r3 rsn 49. Show that if a + VS is a root of an equation with rational coefficients, then a — Vft is also a root. 96 THE POLYNOMIAL OF THE Nth DEGREE Plot the graphs of the following expressions, and find all the roots of the corresponding equations : 50. (X + 1) (a; - 2) (z - 4). 56. (2z + 5)(x2 + 2x + 3). 51. (x-2)(x-4)(2x + 3). 57. (x-5)(2x2 + 3x + 2). 52. (X - 4) (2x + 1) (3x + 5). 58. (x + 2) (x - 3) (x - 2)2. 53. (X + 3) (X - 1)2. 59. (X - 2) (X + 2) (x2 + 2). 54. (2x - l)(x - 3)2. 60. (X - 2)2(2x2 + 2x + 1). 55. (x-2)(2x + 3)2. 61. (x + l)(2x-l)(3x2 + 2x + 3). Find all the roots of the following equations : 62. x» - 4x2 - 2x + 6 = 0. 67. 8x3 - 28x2 + 30x - 9 = 0. 63. x8 - 3 x2 + 4 = 0. 68. 12x8 - 44 x2 + 5x + 7 = 0. 64. 3x3 _ 7x2 - 8x + 20 = 0. 69. 3x3 + 10x2 + iqx - 12 = 0. 65. 4x3-8x2-35x4-75 = 0. 70. 3x3 + 10x2 + 2x - 8 = 0. 66. x8 + 4x2 + 4x + 3 = 0. 71. 4x< + 8x3 + 3x2 - 2x - 1 = 0. 72. 6x* - 11x8 - 37x2 + 36X + 36 = 0. 73. 3x< - 17x3 4- 41x2 - 53x + 30 = 0. 74. 2x* - 9x3 - 9x2 + o7x - 20 = 0. 75. 18x< - 27x3 + 10x2 + I2x - 8 = 0. 76. 16x* + 16x8 - 72x2 - 20x + 25 = 0. 77. x8 - 2x< - 4x3 - 4x2 + 15x + 18 = 0. 78. 4x5 + 12x* + 11x3 + 5x2 - 3x - 2 = 0. 79. 12x5 + 44x* - 55x3 - 95x2 + 63x - 9 = 0. 80. 2x6 - 6x* - 13x3 + 13x2 + 5x - 2 = 0. Determine by Descartes' rule of signs the nature of the roots of the follow- ing equations : 81. x3 + 6x-7 = 0. 84. 3x* + 4x8 + 4x + 3 = 0. 82. x3 + 2x + 3 = 0. 85. X* + x2 - X - 6 = 0. 83. x8 + 2x2 + 5 = 0. 86. X* - 4x2 + 1 = 0. Find the real roots of the following equations, accurate to two decimal places : 87. a^ + 3x - 7 ^ 0. 89. X* - 12x + 7 = 0.. 88. i8 + X + 5 = 0. 90. X* - 3x3 + 3 = 0. 91. x3-x2-6x + l = 0. f I CHAPTEK V THE DERIVATIVE OF A POLYNOMIAL 53. Limits. A variable is said to approach a constant as a limit, when, under the law ivhich governs the change of value of the variable, the difference between the variable and the constant becomes and remains less than any quantity which can be narned, no matter how small. If the variable is independent, it may be made to approach a limit by assigning to it arbitrarily a succession of values follow- ing some known law. Thus, if x is given in succession the values 2" — 1 ■^1 ~" 2"' "^2 ~" 1» "^3 — t' » "^n ~ On and so on indefinitely, it approaches 1 as a limit. For we may make x differ from 1 by as little as we please by taking n suffi- ciently great ; and for all larger ^ ^ values of n the difference be- ? | f T ¥ / tween x and 1 is still smaller. * 234 This may be made evident graphically by marking off on a number scale the successive values of X (fig. 46), when it will be seen that the difference between x and 1 soon becomes and remains too minute to be represented. Similarly, if we assign to x the succession of values «i = J, ^2= — ^, «^3 = -4, ^4 = ~ 5' ■ ■ ■ ' ^n — \~^) n + \ ' X approaches as a limit (fig. 47). ~s~7 s e i 2 f -t— ) 1 H 1 + 1 Fig. 47 If the variable is not independent but is a function of x, the values which it assumes as it approaches a limit depend upon 97 98 THE DERIVATIVE OF A POLYNOMIAL For example, let y =f{x), the values arbitrarily assigned to x. and let x be given a set of values approaching a limit a. Let the corresponding values of y be Vv 2/2' 2/3. 2/4' •'•> Vny •••• Tlien if there exists a number A, such that the difference between y and A becomes and remains less than any assigned quantity, y is said to approach ^ as a limit as X approaches a in the man- ner indicated. This may be seen graphically in fig. 48, where the values of x approacliing a are seen on the axis of abscissas apd the values of y approaching A are seen on the axis of ordinates. The curve of the function is con- tinually nearer to the line y =A. In the most common cases, the limit of the function depends only upon the limit a of. the inde- pendent variable and not upon the particular succession of values that X assumes in approaching a. This is clearly the case if the graph of the function is as drawn in fig. 48. Ex. 1. Consider the function x2 + 3 X - 4 y = z ' X — 1 and let x approach 1 by passing through the succession of values x = l.l, x = 1.01, x = 1.001, x = 1.0001, ••.. Then y takes in succession the values 2/ = 5.1, 77=5.01, y = 5-001, 2/ = 5.0001. It appears as if y were approaching the limit 5. To verify this, we place x = 1 + A, where h is not zero. By substituting and dividing by h we find 2/ = 6 + A. From this it appears that y can be made as near 5 as we please by taking h sufficiently small, and that for smaller values of h, y is still nearer 5. Hence 6 is the limit of j/ as x approaches 1. Moreover, it appears that this limit is inde- pendent of the .succe.ssion of values which x assumes in approaching 1. SLOPE OF A CURVE 99 Ex. 2. Consider y = as X approaches zero. 1-vT-x Give X in succession the values .1, .01, .001, .0001, • • •. Then y takes the values 1.9487, 1.9950, 1.9995, 1.9999, •••, suggesting the limit 2. In fact, by multiplying both terms of VT by 1 + Vl — X we find y = 1 + vT^ X for all values of x except zero. Hence it appears that y approaches 2 as x approaches 0. We shall use the symbol = to mean " approaches as a limit." Then the expressions Lim x = a and X = a have the same significance. The expression \Am.f{x) = A is read " the limit oi f{x), as x approaches a, is A." 54. Slope of a curve. By means of the conception of a limit we may extend the definition of " slope," given in § 27 for a straight line, so that it may be applied to any curve. For let i^ and ^ be any two points upon a curve (fig. 49). If ij and P^ are connected by a straight line, the slope of this line is — ^ • If P„ and i^ are close enough together, the straight line PyP^ will differ only a little from the arc of the curve, and its slope may be taken as approximately the slope of the curve at the point P^ Now this approximation is closer, the nearer the point P, is to Py Hence we are led naturally to the following definition : The slope of a curve at a point P^ix^, y^) is the limit approached Fig. 49 by the fraction «//(> ^"~ W-i where x„ and y., are the coordinates of a second point P^ on the curve, and where the limit is taken as P^ moves toward P^ along the curve. 100 THE DERIVATIVE OF A POLYNOMIAL Ex. 1. Consider the curve y = x^ and the point (5, 25) upon it, and let Xi = 5, yi = 25. We take in succession various values for x^ and y^ corresponding to points on the curve which are nearer and nearer to (xi, j/i), and arrange our results in a table as follows : X2 2/2 Xo, - Xi 2/2 - 2/1 2/2-2/1 Xg-Xi 6 36 1 11 11 5.1 26.01 .1 1.01 10.1 6.01 25.1001- .01 .1001 10.01 5.001 25.010001 .001 .010001 10.001 The arithmetical work suggests the limit 10. To verify this, place X2 = 5 + /;. Vi — ^1 Then ya = 25 + 10 A + A^ Consequently = 10 + A, and as Xa approaches Xo — Xi Vn — Vi Xi, h approaches and approaches 10. Hence the slope of the curve X2 — Xi y = X* at the point (5, 25) is 10. Ex. 2. Find the slope of the curve y = - at the point (3, ^). We have here We place xi = 3, 2/1 Z2 = 3 + A, 2/2 = Then X2 — Xi = A, j/2 — 2/1 = -h 9 + 3A , and 3 + A X2 — Xi 9 + 3A As P2 approaches Pi along the curve, h approaches 0, and the limit of ^2 — 2/1 ■ 1 — - — IS — - ; hence the slope of the curve at the point (3, ^) is — J. In a similar manner we may find the slope of any curve the equation of which is not too complicated ; but when the equation is complicated there is need of a more powerful method for find- ing the limit of ^ _^ - This method is furnished by the opera- tion known as differentiation, the first principles of which are explained in the following articles. 55. Increment. When a variable changes its value the quan- tity which is added to its first value to obtain its last value is called its increment. Thus if x changes from 5 to ^\, its CONTINUITY 101 increment is ^. If it changes from 5 to 4|, the increment is — ^. So, in general, if x changes from x^ to x^, the increment «„— a.%. It is customary to denote an increment by the I is symbol A (Greek delta), so that Aa = x^— x^, and x^ = x^ + Ax. If 2/ is a function of x, any mcremeut added to x will cause a corresponding increment of y. Thus, let y =f(x), and let x change from x^ to x^. Then y changes from y^^ to y^, where 2/i =/K) and y„ =f{x.;). Hence Ay =f(x^) -f(x^). But, as shown above, x^ = x^ + Ax, so that Ay = f{x^ + Aa:;) — f{x^. 56. Continuity. ^ function y is called a continuous function of a variable x when the increment of y approaches zero as the increment of x approaches zero. It is clear that a continuous function cannot change its value by a sudden jump, since we can make the change in the function as small as we please by taking the increment of x sufficiently small. As a consequence of this, if a continuous function has a value A when x = a, and a value B when x = 'b, it will assume any value C, lying between A and B, for at least one value of x between a and h (fig. 50). In particular, if f{a) is posi- tive and f{h) is negative, f{x) = for at least one value of x between a and 5. An algebraic polynomial is a continuous function, but we shall omit the proof. The postage function (§20) is an example of a function which is discontinuous at certain points. Other examples are found in §§ 149, 154. x=a Fig. .50 102 THE DERIVATIVE OF A POLYNOMIAL When Ax and Ay approach zero together it usually happens that — approaches a limit. In this case y is said to have a Ax derivative, defined in the next article. 57. Derivative. When y is a continuous function ofx,the deriva- tive of y with respect to x is the limit of the ratio of the increment of y to the increment of x, as the increment of x approaches zero. dy The derivative is expressed by the symbol -~ \ or, if y is expressed by f{x), the derivative may be expressed by f'{x). Th.ViS,iiy=f{x), ^ =f'(x) = Lim ^ = Lim /(^ + ^/)-/(^) . dx Ax=oA£c Ax=o Ax The process of finding the derivative is called differentiation, and in carrying out the process we are said to differentiate y with respect to x. The process of differentiation involves, according to the defini- tion, the following four steps : 1. The assumption of an increment of x. 2. The computation of the corresponding increment of y. 3. The division of the increment of y by the increment of x. 4. The determination of the hmit approached by this quotient, as the increment of x approaches zero. Ex. 1. Find the derivative oi y = z^. (1) Assume Ax = h. (2) Compute Ay = {z + h)^ -x^ = Sx^h + Sxh^- + h^ (3) Find :^ = 3 x2 + 3 xA + K\ Ax (4) The limit is evidently 3x2. Hence — = 3x2. dz Ex. 2. Find the derivative of - • X (1) Place y = - and assume Ax = h. X (2) Compute Ay ^ ^ ^ X + h X x2 + xA (8)Find^ = I Ax x2 + xA (4) The limit is clearly , and therefore ^ = - i. x2 dx x2 I FOKMULAS OF DIFFERENTIATION 103 It appears that the operations of finding the derivative of f{x) are exactly those which are used in finding the slope of the curve y =zf{x). Hence the derivative is a function wliich gives the slope of the curve at each point of it. 58. Formulas of differentiation. The obtaining of a derivative by carrying out the operations of the last article is too tedious for practical use. It is more convenient to use the definition to obtain general formulas which may be used for certain classes of functions. In this article we shall derive all formulas necessary to differentiate a polynomial. 1. — = naaf'^, where n is a positive integer and a any dx constant. Let y = ax^. (1) Assume Ax = h. (2) Then Ay = a(x+hy— aaf = afnaf-'h + ""^"j ^K ''-'7i'+ • • ■+h\ (3) ^ = a(naf-'+ '^^''r'^K -'h + • • • + h"-'). ^ Ax \ [2 / (4) Taking the limit, we have -^ = naaf \ 2. — — - = a, where a is a constant. dx This is a special case of the preceding formula, n being here equal to 1. The student may prove it directly. dc 3. — = 0, where c is a constant. dx SiDce c is a constant, Ac is always 0, no matter what the value of X. Hence — =0, and consequently the limit -j- = ^^ Ax <^^ 104 THE DERIVATIVE OF A POLYNOMIAL 4, The derivative of a polynomial is found hy adding the derivatives of the terms in order. Let y = a^af'-\-a^af-'+--' + a^_,x + a„. (!) Assume Ax=h. (2) Then Ay = a,{x + A)"+ a^{x + /i)"-'+ • • • + a„_,{x + h)-{-a„ - [a^^af+ a^3f-^+ • • • + a„_iX + aj = h [naf^af-^ + {n — 1) a^af "^-j h a„_^] + ^[n(n-l)a,s(f-' + {n-l){n-2)a,af-'-h--' + a,^_,] H + h^'a^. (3) ^ = «aoic"-i + (?i - 1) a,af -2+ • • ■ + a„_, Aa; +!()+. ..+A n-l. "0* 2 (4) Taking the limit, we have -^ = waoaf-i + (w — l)a,af-^H h «„_!• Ex. Find the derivative of /(x) = 6x5 - 3a;* + 5x3 - 7 a;2 + 8x - 2. Applying formulas 1, 2, or 3 to each term in order, we have /'(x) = 30x* - 12x3 + 15x2 - 14x + 8. 59. Tangent line. A tangent to a curve is the straight line approached as a limit hy a secant line as two points of intersection of the secant and the curve are made to approach coincidence. It is immaterial in what manner the two points of intersection are made to approach coincidence. In § 37 this was done by considering the curve as moved in the plane. In § 88 the secant is considered as moving parallel to itself until it becomes a tangent. In this article we are especially interested in determin- ing a tangent at a known point of the curve. Let us call this TANGENT LINE 105 point JJ and a second point on the curve P,. Then if a secant is drawn through ij and i^ of a curve (fig. 51), and the point F^ is made to move along the curve toward i^, which is kept fixed in position, the secant will turn on ^ as a pivot, and wUl approach as a limit the tangent F^T. The point F^ is called the point of contact of the tangent. From the definition it follows that the slope of the tangent is the same as the slope of the curve at the point of contact; for the slope of the tangent is evidently the limit of the slope of the secant, and this limit is the slope of the curve, by § 54. The equation of the tangent is readily written by means of § 29, when the point of contact is known. For, let (x^, y^ be the point of contact, and let [-t-] denote the value of -7- when x = x^ \"'Vi ^^ /dy\ and y = y^. Then {x^, y^ is a point on the tangent and ( -p j is its slope. Therefore its equation is ^ '^ The equation of the tangent may also be written in terms of the abscissa of the j)oint of contact. Let a be the abscissa of the point of contact of a tangent to a curve y =f{x), and let f{x) represent as usual the derivative of f{x). Then the ordinate of the point of contact is f{a) and the slope of the tangent is f (a), in accordance with § 22. Hence the equation of the tangent is y--f(a) = (x-a)fia). (2) Ex. 1. Find the equation of the tangent to the curve ^^ = x^ at the point (Xi, Vi) on it. Using formula (1), we have y -yi = Sxl{x-Xi). But since (xi, j/i) is on tlie curve, we have 7/1 — xf. Tlierefore the equation can be written J/ = 3 xf X — 2 Xi^ 106 THE DERIVATIVE OF A POLYNOMIAL Fig. 52 Ex. 2. Find the equation of the tangent to y = x^ + Sx at the point the abscissa of which is 2. We will use equation (2). Then /(x) = z2 + 3x, /(x) = 2x + 3. /(2) = 10, /(2) = 7. Therefore the equation is y — 10 = 7(x — 2), or y = 7x — 4. If PT (fig. 52) is a tangent line and cf> the angle it m^kes with dy OX,'ita slope equals tan '^. The sign of the derivative enables us to determine whether a func- tion is increasing or decreasing in accordance with the following theorem : When the derivative of a func- tion is positive the function is in- creasing ; when the derivative is negative the function is decreasing. To prove this, consider y =/(«), and let us suppose that dy . . dy Av Av -r- is positive. Then, since ~ is the limit of -.^ , it follows that -r^ C'X cix Ax Ax SIGN OF THE DERIVATIVE 107 is positive for sufficiently small values of Ax; that is, if Ax is assumed positive, Ay is also positive, and the function is increas- ing. Similarly, if -r- is negative. Ay and Ax have opposite signs for sufficiently small values of Ax, and the function is decreasing by definition. dy ' Ex. 1. If y =: a;2 _ a; _ 6, — = 2« — 1, which is negative when x<^ and positive when x > ^. Hence the function is decreasing when x<^ and increas- ing when x>^, as is sliown in fig. 53. Ex.2. If y = |(x3-3x2-9x+27), Y dx ^ * = |(x + l)(x-3). Now — is positive wlien x < — 1, negative when — 1 < x < 3, and positive when X > 3. Hence tlie function is increasing wlien x < — 1, decreasing when X is between — 1 and 3, and increasing when x > 3 (fig. 54). It remains to examine the cases in which -^ = 0. ax Eefer- FiG. 54 ring to the two examples just given, we see that in each the values of x which make the derivative zero separate those for which the function is increasing from those for which the function is decreasing. The points on the graph which correspond to these zero values of the derivative can be described as turning points. Likewise, whenever f'{x) is a continuous function of x, the values of x for which the derivative is positive are separated from those for which it is negative by values of x for which it is zero (§ 56 ). Now in most cases which occur in elementary work f'{x) is a continuous function. Hence we may say. The values of x for which a function changes from an increas- ing to a decreasing function are, in general, values of x which make the derivative equal to zero. 108 THE DERIVATIVE OF A POLYNOMIAL The converse proposition is, however, not always true. A value of X for which the derivative is zero is not necessarily a value of X for which the function changes from increasing to decreasing or from decreasing to increasing. For consider \{x^-^^+21x-\% I Its derivative is ar^— 6 a? 4-9 = (a?— 3)^ / which is always positive. The func- / tion is therefore always increasing. ^ Wlien a; = 3 the derivative is zero and the corresponding shape of tlie graph is shown in fig. 55. 61. Maxima and minima. The X turning points of the graph of a function correspond to the maxi- mum and the minimum values of the function. These terms are more precisely defined as follows : /(a) %8 a maximum value of the function f [x) when f [a ±h) f{a) for all values of h sufficiently small. In passing through a maximum value the function changes from an increasing to a decreasing function, and in passing through a minimum value the 'function changes from a decreas- ing to an increasing function. From the work of the previous article we may accordingly frame the following rule for finding the maxima and the minima values of a function : Find the derivative of the function, place it equal to zero, and solve the resulting equation. Take each root thus found and see if the derivative has opposite signs as x is taken first a little smaller and then a little larger than the root. If the sign of the derivative changes from plus to mimis, the root substituted in the fwnetion gives a maximum value of the function. If the sign of the derivative changes from minus to plus, the root suhstituted in the function gives a minimum value of the function. MAXIMA AND MINIMA 109 This rule is most readily applied when the derivative can be factored. The change of sign is then determined as in § 46. In § 62 will be given a method of distinguishing between a maxi- mum and a minimum, which may be used when the factoring of the derivative is not convenient. In practical problems the ques- tion as to whether a value of x for which the derivative is zero corresponds to a maximum or a minimum can often be deter- mined by the nature of the problem. Ex. 1. Find the maximum and the minimum values of /(x) = xs _ 5x* -f- 5x3 + 10a;2 - 20a; + 5. We find /(x) = 5 x* - 20x3 + 15x2 -|- 20x - 20 = 5(x2-l)(x2-4x + 4) = 5(x + l)(x-l)(x-2)2. Tlie roots of /' (x) = are — 1, 1, and 2. As x passes through — 1, /'(x) changes from + to — . Hence x = — 1 gives /(x) a maximum value, namely 24. As X passes through + 1, /"(x) changes from — to +. Hence x = + 1 gives /(x) a minimum value, namely — 4. As x passes through 2, /' (x) does not change sign. Hence x = 2 gives /(x) neither a maximum nor a minimum value. Ex. 2. A rectangular box is to -be formed by cutting a square from each corner of a rectangular piece of cardboard and bending the resulting figure. The dimensions of the piece of cardboard being 20 by 30 inches, required the largest box which can be found. Let X be the side of the square cut out. Then if the cardboard is bent along the dotted lines of fig. 56, the dimensions of the box are 30 — 2 x, 20 — 2 x, z. Let y be the volume of the box. Then y = X (20 - 2 x) (30 - 2 x) = 600x- 100x2 + 4x3. dy dx = 600 - 200 x + 12x2. Equating this to zero, we have 3x2 _60x + 150 = 0, 25 ± 5 V? Hence dy dx 3.9 or 12.7. 12(x-3.9)(x-12.7). a- X [ 30-2X Ti ?! 1 Fig. 56 dx changes from + to — as x passes through 3.9. Hence x = 3.9 gives the maximum value 1056+ for the capacity of the box. x = 12.7 gives a mini- mum value of ?/, but this has no meaning in the problem for which x must lie between and 10. 110 THE DERIVATIVE OF A POLYNOMIAL Ex. 3. The deflection of a girder resting on three equally distant supports and loaded uniformly is given by the equation V = C (- l^ + Slx^ - 2x*), where C is a constant, I the distance between the supports, and x the distance from the end support. Required the point of maximum deflection. dz Equating this to zero, we have 8x3-9^2 + 18 = 0. It is clear that in the practical problem x 2, and has a point of inflec- tion when x = 2. When x = 0, — = fPy dx and -— < ; the corresponding value of y is therefore a maximum. When « = 4, -— = and " ^ > ; the corresponding value of y is therefore a minimum. EXAMPLES 113 3 x2 + o, Ex. 2. y = x^ + ax = x(x^ + a), dy dx dx2 The curve is concave downward when x < 0, is con- cave upward wlien x > 0, and has a point of inflection when X = 0. In addition we distinguish two cases : (1) a positive. dy dx is always positive, and the curve cuts OX only at the origin (fig. 62). (2) a negative. The curve has a maximum ordinate when X = I and has a mini- mum ordinate when H- y = + 2a Fig. 62 0, or -H V — a It cuts OX when x (fig. 63). Ex. 3. y = x^ + ax + b. The graph of this function may be obtained by moving the graph of Ex. 2 through the dis- tance b up or down, according to the sign of b. Our interest is especially with the intei'cepts on OX. The curve obtained from (1) of Ex. 2 cuts the axis of x in one and only one point. The curve obtained from (2) of Ex. 2 will intersect OX in three points, will intersect OX in one point and be tangent in another, or will intei'sect OX in one point only, accord- ing as the numerical value of b is less than, equal to, or greater than the distance of the tui'ning point of the curve from OX ; that is, according as b^ = 2a r^a\ Y\ 3/ This condition reduces to 62 Fig. 03 4 27 0. 114 THE DERIVATIVE OF A POLYNOMIAL {,2 qS It is to be noticed that when a > 0, - + — > 0. Hence we may cover all 4 27 cases by the statement : The eauation x^ + ax + b = has three unequal reed roots, two equal real roots 52 a' < and one other real root, or one real and two complex roots, according aa - + — = 0. 63. Newton's method of solving numerical equations. The results of this chapter may be applied to finding approximately the irrational roots of a numerical equation. We first find, by the method of § 47, two numbers x^ and x^, between which a root of /(a:)=0 is known to lie. It is necessary to take care that neither /'(a;) norf"{x) is zero for any value of x between x^^ and x^. Then f{x) is always increasing or decreasing between x^ and x^ and hence only one root of f{x) = lies between x^ and x^. Also N Xi D/ / C U (1) D X2 (S) N Fig. 64 the curve y= /(a?) is always concave upward or concave down- ward between aj^ and x^. Hence the curve has one of the four shapes of fig. 64. It appears that in each case a tangent at one of the points M ox N will intersect the axis of ic in a point C which lies between x^ and x^. In practice it is most convenient to sketch the curve with attention to the signs of the first and the second derivative, and to find the tangent at that end at which it lies between the curve and the ordinate of the point of contact. The intersection of the tangent with OX is then nearer to the J I NEWTON'S METHOD 115 intersection of the curve, i.e. to the required root of the equation, than is the abscissa of the point of contact. For example, in fig. 64, (1) and (4), the equation of the tangent is fix ) and its point of intersection with OX is x — •^ , ^ • Hence the root which was at first known to lie between x^ and x^ is now fix ) known to lie between x. and x„ — ^; ^' • It is well in practice to combine this method with the method of § 52. For, if we draw the secant MN, it will intersect the axis of ic in a point. D, and the root of the equation lies between C and D. But C and D are closer together than are x^ and x^, so that we have narrowed down the interval within which the root lies. Ex. 1. Find the root of a;^ — 6 x — 13 = 0, which lies between 3 and 4. Here /(a;) = x^ - 6 x - 13, /'(x) = 3x2-6, /"(x)=6x. When X = 3,/(x) = — 4 ; and when x = 4,/(x) = 27 ; while between x = 3 and X = 4, f'{x) and f"{x) are positive. Hence the graph is as in fig. 64, (1), where M is (3, - 4) and N is (4, 27). The tangent at N is y -21 = 42(x-4). Hence, for C, x = 4-2j = 3.36. The equation of MN is y - 27 = 31 (x - 4). Hence, for D, x = 4 - §^ = 3. 13. Therefore the root lies between 3.13 and 3.36. As this does not fix the first decimal figure of the root, it is advisable to apply § 47 again. We find /(3. 1) = - 1.809 and /(3.2) = + .568. Hence the root lies between 3.1 and 3.2. Accordingly, the point M is now (3.1, — 1.809), and the point N is (3.2, .568). The equation of the tangent at N is y- .568 = 24.72(x-3.2), and for the new point C x — 3.17702. The secant MN is y - .568 = 23.77 (x - 3.2) and for D x = 3.176. The root of the equation therefore lies between 3.176 and 3.177. This result is close enough for most practical purposes, but if the operations are carried out once more it is found that the root lies between 3.1768148 and 3.1768144. 116 THE DERIVATIVE OF A POLYNOMIAL Ex.2. In §61, Ex.3, the root of 8x^ -Olx"^ + 1^ = was found to lie between A21 and .43 Z, Placing /(a;) = 8x8-9ix2 + Z3, we have /'(x) = 24 a;- - 18 Ix, /"(x) = 48x-18Z, so that/'(x) is negative and /"(x) positive, when x is between A21 and AS I. Hence the curve has the shape of fig. 64, (3). The tangent at (.42 Z, .005104^3) meets OX where x = .42153 1. The chord connecting (.42 1, .005104 P) and (.43 1, - ,028044 1^) meets OX where x = .42154 Z. The root is therefore determined to four decimal places. 64. Multiple roots of an equation. If f{x) = aQ3if+a^xr-^+a^x"-^-] \- a„_^os^-{- a„_iX + a^, f'{x) = na^af'-'^ + {n - l)a^c(f'-'' + {n - 2)a^af-'^-\ + 2 a„_^x + a„_^, f\x) = n(n- l)a^ar-'+{n -l){n- 2)a^o^-^ + {n-2){n- 3) a^af -*+ • • • + 2 a„_j, f"'{x) = n{n-l){n-2) a^-" + {n-l){n,-2){n — S)a^af-*-] , and so on. Now let /(a), /'(a), f"{a), /'"{a), etc., denote the result of placing x = a va. these functions, and f{a + h) denote the result of placing x = a + hin f{x). One readily computes that f{a + ^) = f{a) + hf'(a) + ^ f"{a) + ^ f"'(a) + . . . + «^». (1) In (1) place h = x — a and it becomes f{x) =:f(a) + {x- a)f'{a) + ^^^V"(«) + ^^^/'»+--- + «o(^-<. (2) If now a is a double root oif{x) = Q,f{x) is divisible by {x—af, by § 42, and therefore, by (2), f{a) = 0, f{a) = 0. If a is a triple root of f{x) = 0, /(.>:) is divisible by (x— a)\ and therefore /(a) = 0, f'(a) = 0, /" (a) = 0. Similar statements may be made for multiple roots of higher order. MULTIPLE KOOTS 117 Conversely, if f{a) = and f'{ci) = 0, (2) shows that f(x) is certainly divisible by {x — aY and perhaps by a higher power of X — a. Therefore a is a multiple root of f(x) = 0. We have then the result: A multiple root of/{x) = Ois also a root off'{x)=0, and conversely. Hence we may find the multiple roots of f{x) = by equating to zero the highest common factor of f{x) and f'{x) and solving the resulting equation. The condition that an equation f{x) = should have multiple roots is the vanishing of the discriminant of the equation, which is the eliminant of the equations /{x) = and f'{x) = 0, and may be found by the method of § 9. Ex. 1. Find the discriminant of ax^ + 6x + c = 0. We have to find the condition that the two equations ax2 + 6x + c = and 2 ax + 6 = should have a common root. Multiplying the last equation by x, we have 2 ax2 + 6x = 0, and the determinant of the coefficients and the absolute terms of the three equations is a b c 2 a 6=0, 2a & 62 _ 4 ac = 0. Ex. 2. Find the discriminant of x^ + ox + 6 = 0. We must find the eliminant of this and 3x2 + a = 0. Multiplying the first equation by x, and the second by x and x^, we have the five equations X* + ax2 + 6x = 0, x3 + dx + 6 = 0, 3x* + ax2 =0, 3x8 +ax =0, 3x2 +a = 0, 1 a 6 1 a 6 3 a 0=0, 3 a 3 0a 4a3 + 2762 = 0. (See § 62, Ex. 3.) ■and their eliminant is 118 • THE DERIVATIVE OF A POLYNOMIAL PROBLEMS Find the respective slopes of the following curves at the points noted: (1) by an approximate numerical calculation, as in § 54 ; (2) by placing x equal to the abscissa of the given point, plus A, and allowing h to approach zero : 1. y = a;3 at (2, 8). 2. y = a;2 - 3 X at (0, 0). 3. y = a;3_3x + l at (1, -1). 4. Find the derivative of x^ — a; by using the definition but not the formulas. 5. Find the derivative of 3 x* + 2 x by using the definition but not the formulas. Find the derivative of each of the following expressions by the fonnulas : 6. |X6 -|X5 + X. 7. 4x8-6x2 + 5x-8. 8. 6x9-6x8 + 7x«-4x4-2x2 + 3x-9. 9. By expanding and differentiating show that the derivative of (3 x + 2)* isl2(3x + 2)8. 10. By expanding and differentiating show that the derivative of (x + a)" is n(x + a)»-i. 11. Find the equation of the tangent to the curve i/ = x* + 3 at the point the abscissa of which is — 2. 12. Show that the equation of the tangent to the curve y = x^ + ax + 6 at the point (Xi, i/i) is y = (3x,'' + a)x - 2xf + 6. 13. Show that the equation of the tangent to the curve y = ax- + 2 6x + c at the point (xi, yi) is y = 2(axi + b)x — axf + c. 14. Determine the point of intersection of the tangents to the curve y = x^ — 6x + 7 at the points the abscissas of which are — 2 and 3 respectively. 15. Find the angle between the tangents to the curve y = 2x2 — 3x + lat the points the abscissas of which are — 1 and 2 respectively. 16. Find the area of the triangle included between the coordinate axes and the tangent to the curve y = x^ at the point (2, 8). 17. Find the points on the curve y = x* — 3x + 7at which the tangents are parallel to the line y = dx + S. 18. How many tangents has the curve y = x8— 4x2 + x — 4 which are parallel to the line y + 4x + 7=0? Find their equations. 19. Find the points on the curve y = x^ + x^ — 6 at which it makes an angle of 45° with OX. PROBLEMS 119 Find the values of x for which the following expressions are respectively increasing and decreasing: 20. x2 + 4a;-7. ' 22. a;* + 8 x - 10. 21. x3-2x2 + 8. 23. X*- 2x2 + 6. 24. Find the lowest point of the curve y = 3x^ — 8x + l, 25. Find the turning points of the curve ?/ = i x* — 2 x2 + i. Find the maximum and the minimum values of the following expressions : 26. 3x8 _ 2x2 _ 5a; + 1. 27. 3x5 - 25x3 + 60x - 50. 28. Prove that the largest rectangle with a given perimeter is a square. 29. A rectangular piece of cardboard a in. long and b in. broad has a square cut out of each corner. Find the length of a side of this square when the box formed from the remainder has its greatest volume. 30. Find the dimensions of the greatest rectangle which can be inscribed in a given isosceles triangle with base b and altitude h. 3 1 . Find the right circular cylinder of greatest volume which can be inscribed in a sphere of radius a. 32. Find the right circular cylinder of greatest volume which can be cut from a given right circular cone. • 33, Find the point of the line 3x + y = Q such that the sum of the squares of its distances from the two points (5, 1) and (7, 3) may be a minimum. 34. Among all circular sectors with a given perimeter find the one which has the greatest area. 35. A rectangular box with a square base and open at the top is to be made out of a given amount of material. If no allowance is made for thickness of material or waste in construction, what are the dimensions of the largest box that can be made ? 36. A length I of wire is to be cut into two portions, which are to be bent into the forms of a circle and a square respectively. Show that the sum of the areas of these figures will be least when the wire is cut in the ratio ir : 4. 37. A piece of galvanized iron b ft. long and a ft. wide is to be bent into a U-shaped water pipe b ft. long. If we assume that the cross section of the pipe is exactly represented by a rectangle on top of a semicircle, what are the dimensions of the rectangle and the semicircle that the pipe may have the greatest capacity ; (1) when the pipe is closed on top ? (2) when it is open on top ? 38. A stream flowing with the velocity a strikes an undershot water wheel, giving it the velocity x. Assuming that the efficiency of the wheel is propor- tional to the velocity x of the wheel and the loss of velocity a — x of the water, what is the velocity of the wheel when it has its greatest eflBciency ? 120 THE DERIVATIVE OF A POLYXOMIAL 39. A gardener has a certain length of wire fencing w4th which to fence three sides of a rectangular plot of laud, the fourth side being made by a wall already constnicted. Required the dimensions of the plot which contains the maximum area. 40. For a continuous girder of uniform section, uniformly loaded, and con- sisting of three equal spans, the deflection in the middle span is given by the equation v — C {l^x — 6 l^x- + lOlx^ — 5x*), where C is constant, I the length of the span, and x the distance from a point of support. Find the greatest deflection. 41. If p is the density of water and t the temperature between 0° and 30° C, p = po(l + lt + mt^ + vi^), where po is the density when t = 0, and / = .000052939, m = - .0000065322, n = .00000001445. Show that the maximum density occure when « = 4.108°. 42. Show that the curve y = ax^ + bx + c is concave upward or downward according as a is positive or negative. 43. Show that the curve y = x^ + ax + b is concave upward when x is posi- tive and concave downward when x is negative. Determine the values of x for which the foUowhig curves are concave upward or downward : 44. y = x8-3x2-24. 45. y = a;5_5a; + 6. Find the points of inflection of the following curves : 46. Gy = x^-Gx^-\-Gx + 1. 47. 12y = x* - 6x^ + 12x^ - 2x+ 1. 48. y = 3x^- lOx* + 10x8 + 6x - 8. 49. y = 3x5 - 5x* 4- 20x8 - 60x2 + 20x - 5. 50. Prove that the curve y = ax^ + bx^ + ex + d always has one and only one point of inflection. Find the real roots of the following equations accurate to two decimal places : 51. x8 - x2 - 2x + 1 =0. 54. x^ - 3x2 - 2x + 5 = 0. 52. x8 + 3x2 + 4x + 5 = 0. 55. x* - x^ - x2 + x - 1 == 0. 53. x8-2x -5 = 0. Show that each of the following equations has equal roots and solve it : 56. x8-x2-8x+ 12 = 0. 57. x* - 2(l-a)x8 + (l_ 3a)x2 + a = 0. Find the condition that each of the following equations should have equal roots : 58. x8 + 3 ax2 + 6 = 0. 60. x* + 4 ax + 6 = 0. 59. X* + 4ax8 + 6 = 0. 61. aox3 + 3aix2 + Soox + as = 0. CHAPTEE VI CERTAIN ALGEBRAIC FUNCTIONS AND THEIR GRAPHS 65. Square roots of polynomials. In the previous chapters the discussion has been restricted to the polynomial. We will next study the square root of the polynomial. At first let us assume that the polynomial can be separated into 71 linear real factors, as in § 42. We have, then, y = ± Va, (X - r,) {x-r^)---{x- r„), (1) and the graph of this function can readily be constructed by con- sidering the graph of y = «o (^ - ^i) {-^ - ^^2) •••(•»- r„), (2) as given in § 46. In the first place, the graph of (1) will intersect the axis of ic in the same points as the graph of (2), i.e. in the points x = 1\, x = r^, • • •, as for these values of x the product under the radical sign is zero. In the second place, wherever the graph of (2) is below the axis of X, the expression under the radical sign in (1) is negative, tlie value of the radical is imaginary, and hence there is no cor- responding point of the graph. If, however, the graph of (2) is above the axis of x, there are two values of y in (1), equal in magnitude and opposite in sign, and correspondingly there are two points of the graph situated symmetrically with respect to OX. Therefore OX is an axis of symmetry. As the negative values of the expression under the radical sign are separated from the positive values by zero, it follows that the values of x which make the expression zero, i.e. r^,r„,---, r„, are of the utmost importance in plotting these graphs. In fact, the lines X = 1\, x=.r^,---,x = r^ divide the plane into sections bounded by straight lines parallel to F, in which there will be no part of the 121 122 CEKTAIN ALGEBRAIC FUNCTIONS graph if the corresponding values of x make the expression negative, and in which there will be a part of the graph if the corresponding values of x make the expression positive. Hence the first step in plotting the graph is the drawing of these lines and the determi- nation of which sections of the plane should be considered. Ex. \. y = ± V(x + 2) (X - 1) (X - 5). K X = — 2, 1, or 5, y = 0, and the graph intersects the axis of x at three points. The lines x = — 2, x = l, x=5 divide the plane (fig. 65) into four sections. If a; < — 2, all three factors of the product are negative ; hence the radical is imaginary and there can be no part of the graph in the corresponding section of the plane. Fig. 65 If - 2 < X < 1, the first factor is positive and the other two are negative ; hence the radical is real and there is a part of the graph in the corresponding section of the plane. If 1 < X < 5, the first two factors are positive and the third is negative ; hence the radical is imaginary and there can be no part of the graph in the corresponding section of the plane. GRAPHS 123 Finally, if x > 5, all three factors are positive ; hence the radical is real and there is a part of the graph in the corresponding section of the plane. Therefore the graph consists of two separate parts, and is seen (fig. 65) to consist of a closed loop and a branch of infinite length. Ex. 2. y = ± V(x + 4) (X + 2) (x - 1) (x - 4). If X = — 4, — 2, 1, or 4, y = 0, and the graph intersects the axis of x at four points. The lines x = — 4, x= — 2, x = l, and x = 4 divide the plane (fig. 66) into five sections. I -X Fig. 66 If X < - 4, all four factors are negative ; hence the radical is real and tlaere is a part of the graph in the first section. If - 4 < X < - 2, the first factor is positive and the others are negative ; hence the radical is imaginary and there can be no part of the graph in the second section. If - 2 < X < 1, the first two factors are positive and the other two are nega- tive ; hence the radical is real and there is a part of the graph in the third section. 124 CERTAIN ALGEBRAIC FUNCTIONS If 1 < X < 4, the fii-st three factors are positive and the last is negative ; hence the radical is imaginary and there can be no part of the graph in the fourth section. Finally, if x > 4, all the factors are positive ; hence the i-adical is real and there is a part of the graph in the fifth section. In this example we see that the graph consists of three separate parts, and is seen (fig. 66) to consist of a closed loop and two infinite branches. Ex.3, y = ± V- (X + 4) (X + 2) (X - 1) (X - 4), The plane is divided into five sections (fig. 67) by the lines x = X = 1, and X = 4. - 4, X = - 2, Fig. 67 Proceeding as in the previous two examples, we find y to be real if -4 — 2 and imaginary if x < — 2. Therefore there is a part of the graph to the right of the line x = — 2, but there can be no part of the graph to the left of that line unless x can have such value as to make the coeflicient of the radical zero ; and this coefficient is zero only when x equals unity. Hence all of the graph lies to the right of the line X = — 2, as shown in fig. 68. Comparing this example with Ex. 1 of § 65, we see that by changing the factor X — 5 to x — 1 we have joined the infinite branch and the loop, making a single curve crossing itself at the point Ex. 2. y = ± V(x + 2)2(x - 1) = ± (x + 2) Vx - 1. The line x = 1 divides the plane (fig. 69) into two sections. If X > 1, the radical is real and there is a part of the graph in the corresponding section of the plane. If x < 1, the radical is imaginary and there will be no points of the graph except for such values of x as make the coefficient of the radical zero. There is but one such value, i.e. - 2, and therefore there is but one point of the graph, i.e. (—2, 0), to the 126 CERTAIN ALGEBRAIC FUNCTIONS left of the line x = 1. The graph consists, then (fig. 69), of the isolated point A and the infinite branch. Comparing this example also with Ex. 1 of § 65, we see that by changing the factor x — 5 to x + 2 we have reduced the loop to a single point, leaving the infinite branch as such. Fig. 70 Ex.3, y = ± V_ (X + 4) (X + 2)2(x - 4) = ± (X + 2) V- (X + 4) (X - 4). The lines x = - 4 and x = 4 divide the plane (fig. 70) into three sections. If - 4 < X < 4, the radical is real and there is a part of the graph in the cor- responding portion of the plane. If x < - 4 or x > 4, the radical is imaginary ; and since in the corresponding sections there is no value of x which makes x + 2 zero, there can be no part of the graph in those sections.- It is represented in fig. 70. Comparing this example with Ex. 3 of § 65, we see that the changingof x - 1 to X + 2 has brought the two loops together, forming a single closed curve cross- ing itself at the point ( - 2, 0). FUNCTIONS DEFINED BY EQUATIONS 127 67. Functions defined by equations of the second degree in y. If we have given an algebraic equation involving both y and x, y is thereby defined as a function of x. For if x is assigned any value, the corresponding values of y are determined by means of the equation. In particular, if the equation involves no power of y higher than the second, it may be readily solved for y, and the work of finding the graph is similar to that already done. In many important cases the solution of the equation is of the form y = c± y/(x — Vj) (x — r^)--'. Comparing this case with the previous one, we see that y = c is an axis of symmetry instead of y — 0, and that in all other respects the work is similar. Ex. 2x2 + 2/2 + 3x -42/- 5 = 0. Solving for y, we have y = 2± V-2x2_ 3x + 9, or, after the expression under the radical sign has been factored, y = 2± V-2(x-|)(x + 3). The lines x = — 3 and x = | divide the plane (fig. 71) into three sections, and, proceeding as before, we find that the curve is entirely in the middle section, i.e. when — 3 < x < ^, and that the line 2/ = 2 is an axis of symmetry. Fig. 71 In case the given equation is of higher degree in y than the second, but of the first or the second degree in x, it is evident that we can solve for x in terms of y and proceed as above, working from the y axis instead of the x axis. It should be added that given any equation in x and y, since either may be regarded as the independent variable and the other as the function, we have perfect freedom of choice to solve for y in terms of x, or for x in terms of y, according to convenience. 128 CERTAIN ALGEBRAIC FUNCTIONS 68. Functions involving fractions. If the expression defining a function contains fractions, the function is not defined for a value of X which makes the denominator of any fraction zero (§ 11). But if a? = a is a value which makes the denominator zero, but not the numerator, and x is allowed to approach a as a limit, the value of the function increases indefinitely and is said to become infinite. The graph of a fimction then runs up or down indefinitely, approaching the line x = a indefinitely near, but never reaching it. We have thus a graphical representation of the discussion of infinity in § 11. When a fimction becomes infinite it is discontinuous (§ 56). In fact, this is the only kind of discontinuity which can occur in an algebraic function. Ex. 1. y Fig. 72 a;-2 It is evident that y is real for all values of x; also if a; < 2, y is negative, and if a; > 2, y is positive. Moreover, as x increases toward 2, y is negative and becomes indefinitely great ; while as x decreases toward 2, y is positive and becomes indefinitely great. We can accordingly assign all values to x except 2, that value being excluded by § 11. The curve is repre- sented in fig. 72. It is seen that the nearer to 2 the value assigned to x, the nearer the correspond- ing point of the curve to the line x = 2. In fact, we can make this distance as small as we please by choosing an appropriate value for x. At the same time the point recedes indefinitely from OX along the curve. Now when a straight line has such a position with respect to a curve that as the two are indefinitely prolonged the distance between them approaches zero as a limit, the straight line is called an, asymptote of the curve. FUNCTIONS INVOLVING FRACTIONS 129 It follows from the above definition that the line x = 2 and also the line y=0 are asymptotes of this curve. In this example it is to be noted that the asymptote a; = 2 is determined by the value of x which makes the function infinite. It is clear that all equations of the type 1 y = X — a represent curves of the same gen- eral shape as that plotted in fig. 72. Ex.2. y = + x + 2 If X = — 2 or if X = 2, y is infinite ; hence these two values may not be assigned to x, all other values, however, being possible. The curve is repre- sented in fig. 73. By a discussion similar to that of Ex. 1, it may be proved that the lines x = — 2 and x = 2, which correspond to the values of X which make the function infinite, and also the line y = 0, are asymptotes of the curve. This curve is a special case of that represented by y 1 4- X — a X — b and it is not difficult to see how the curve represented by 1 1 1 y + + + X — a X — b X — c will look for any number of terms. 1 Ex. 3. y = (X - 2)2 All values of x may be assumed except 2. The curve is represented in fig. 74. It is evident that the lines x = 2 and y = are asymptotes. This curve is a special case of that represented by _ 1 ^~(x-a)2' which is itself a special case of 1 1 Fig. 74 (X - a)2 (X - 6) + 130 CERTAIN ALGEBRAIC FUNCTIONS Ex. 4. y* = x-3 As in § 67, we solve for y, forming tlie equation ion y = ±yj^ The line X = 3 (fig. 75) divides the plane into two sections, and it is evident that there can be no part of the curve in that section for which a; < 3. Moreover, this Fig. 75 line X = 3 is an asymptote, as in the preceding examples. The curve, which is a special case of that represented by Is represented in fig. 75. It is to be noted that the axis of x also is an asymptote. Ex.5. y = ?i±i. X To plot this curve we write the equation in the equivalent form y = x + -. (1) It is evident that all values except may be assigned to x, that value being excluded as it makes y infinite. Let us also draw the line y = x, (2) a straight line passing through the origin and bisecting the first and the third quadrants. SPECIAL IRRATIONAL FUNCTIONS 13i Comparing equations (1) and (2), we see that if any value xj, is assigned to x, the corre- sponding ordinates of (1) and (2) are respectively xi -\ and Xi ■, Xi, and that they differ by — . Moreover, the numerical value of this difference decreases as greater numerical values are assigned to Xi, and it can be made less than any assigned quantity however small by tak- ing Xi sufficiently great. It follows that the line y = x isan asymptote of the curve. It is also evident that the line x = 0, determined by the value of x which makes the function infi- nite, is an asymptote. The curve is represented in fig. 76. (£)A Fig. 70 69. Special irrational functions. Ex. 1. 2/2 = x3. Writing this equation in the form y = ±x Vx, we see that y is an irrational function of x, and that its graph is symmetrical with respect to OX and lies entirely to the right of the axis y. It is represented in fig. 77, and is called the semicubical parabola. In general, if the equation expressing the _y function is of the form y = kx", the function is rational or irrational according as n is integral or fractional. In § 38 we have plotted the graphs of some of the rational func- tions of this type for the special case when k= 1 and n has the values 3, 4, and 5 respectively. Above we have just plotted the graph of one of the irrational functions, i.e. when n = ^. The grajDhs of the irrational functions y = x^, ■pio, 77 y = X*, and y = x^ may be obtained by assuming values for x and plotting as above, or by rewriting the equations in the forms x = y^, x = y*, and x = y^, when it is immediately evident that their graphs are respectively the same in shape as those of the 132 CEKTAIN ALCiEBRAIC FUNCTIONS Fig. 78 rational functions y = x^, y = x*, and y = x^ already plotted, the axes of X and y, however, being changed in position. It is to be noted that the graphs of all the functions expressed by the equation y = x" pass through the points (0, 0) and (1, 1). Ex. 2. a;' + 2/5 = a'. If y is defined as a function of x the equation x' + y^ = a\ it is evident that its graph will lie entirely in the first quadrant, since both x and y must be positive, and that its relative positions with respect to the two axes of coordinates are the same (fig. 78). The curve is a parabola (§ 79). If the equation is put in the form 2/ = (a* — x^)-, it is .seen that y is an irrational function of x. Ex. 3. xi + y^ = al Writing this equation in the form y = ± (a* — x^)-, we see that 2/ is an irrational function of x, and that its graph is symmetrical with respect to OX and bounded by the lines x = — o and X = a. In the same way we may show that the graph is symmetrical with respect to OY and bounded by the lines y = ~ a and y= a. It is repre- sented in fig. 79, and is a four-cu.sped hypocycloid. Ex. 4. x3 + y^ -3axy = 0. The graph of this equation, by which y is defined as an irrational function of x, is repre- sented in fig. 80, and is known as the Folium of Descartes. It is symmetrical with respect to the line y = x and has the line X + y + a = as an asymptote. While it may be plotted by assuming values for X and solving the corresponding cubic equations for y, it is more easily plotted when different axes of coordinates are chosen (see Ex. 38, Chap. X). PROBLEMS .00 PROBLEMS Plot the graphs of the following equations : 1. y2 = (X - 1) {x'^ - 4). 29. 2/3 = a;2(x + 2). 2. y2 ^ (X + 2) (8x - x2 _ 15). 30. {y + 2)3 = (x - 1) (x2 - 4). 3. 4 2/2 = (x + 3)(2x-3)2. 31. X2/ = 7. 4. 42/2 = x2{x + l). • 32. xy = -1. 5. 2/2 = (X- 3)2 (.5-2 X). __ 16 66. y = 6. ?/2 = (3x + 2)(9x2-4). ^-* 7. 2/2 = (X - 2)2(4x2 - 4x - 15). 34. ?/ = ^- L_. (X - 1)2 (X + 3)2 8. y2^(4x2-l){x2-4). 9. 2/2 = {2x+5)2(6 + x-x2). ^^- 22/ = 3x4--- 10. 2/2 = -x2(x + 3)2(x + l). 36. 2/-2 = 2(x-l)+ ^ . 11. 2/2 = x2{x-2)2(x-3). „„ 1 37. (2/ -2)2 = ^. 12. 2/2 = (1 - x2) (x2 - 9). a^ + 1 13. 2/2 = (2x-5)(x2 + 2). 38. y2 = 5i^±^. 14. 2/2=(x-2)2(x2-f-2). 15. 2/- = (x-2)(2x-3)2(x2+x+l). ^^ y^^^Zr^- 16. 10^2- 42-4 _a;6. 3 40. 2/2 = ? 17. x2-2/2-4x + 6y-l = 0. x2-6x + 8 18. 4x2 + 92/2+ 4x- 122/ -31=0. *^- a;22/2 + 36 = 4 2/2. 19. x2 - 2/3 + 32/2 + 2/ - 3 = 0. ^2- l^'^'^/' = 62a;2(a2 - 2ax). 20. x2-2/4(4 + 2/) = 0. 43. 2/2 = ^!^^^±^. 21. [x2 + .S(2/-l)][x2-3(2/-l)]=0. 44. 2/(x2 + a2) = a2(a-x). 22. (2/-l)2 = (x-l)2(x-4). 45. 2/2(x2 + a2)_a2a;2. 23. (2/ _ x)2 = 9 - x2. 46. a*2/2 + 62a;4 = a262x2. 24. (X + ?/)2=: 2/2(2/ + 1). ^ ^ •^Vi'^; 47. 2/2(a2 + x2) = x2(a2-x2). 25. x2 - 4 X2/ + 8 2/2 - 2/* = 0. 48. xy2 = 4a2(2a-x). ^^■^h^h'- 49.. = x= + i. 27. 2/3 = x4. J 28. 2/3 = X (x2 - 4). • ^ ^ ^ + ^" CHAPTEE VII CERTAIN CURVES AND THEIR EQUATIONS 70. The circle. Wheu a curve has been defined by a geometric property it is often possible to find the equation of the curve by expressing the definition in algebraic symbols. This equation serves, then, as a means for plotting the curve and also as a basis for examiniug its other properties. In this chapter we shall derive the equation of certain important elementary curves, beginning with the circle. A circle is the locus of a point at a constant distance from a fixed point. The fixed point is the center of the circle and the constant distance is the radius. Let {d, e) (fig. 81) be the coordi- nates of the center C, and r the radius of the circle. Then if P (x, y) is a point on the circle, x and y must satisfy the equation (x-df^{y-ef = r', (1) by § 17. Conversely, if x and y satisfy the equation (1), the point {x, y) is at a distance r from {d, e) and therefore lies on the circle. Therefore (1) is the equation of the circle (§ 22). Equation (1) expanded gives x-+f~ 2dx-2ey + d'+e^-r^=0; and if this is multiplied by any quantity A, it becomes Ax' + Af+2Gx+2Fy + C=0, where Fig. 81 (2) d=-- A e=--, d?Jre^-r' = -' A A 134 THE CIRCLE 135 Ex. The equation of a circle with the center (^, — J) and the radius § is (a;-J)2 + (y + J)2 = 4, which reduces to 12 a;^ + 12 ys _ 12 x + 8 y - 1 = 0. 71. Conversely, the equation where A^ 0, represents a circle, if it represents any curve at all. To prove this, we will transform the equation as follows : a?+2-x + y^ + 2-y=--, A ^ A"^ A ^,0^ , <^\ 2,0^ , ^' G^+F^-AC 4/ y A I A' There are then three possible cases : 1. G^ + F^-AOO. The equation is then of the type (1), § 70, 1, ^ ^ ^' -2 G^+F^~AC ,^, , where a = 7'^ = ,r=^ — , and therefore represents A A A^ . a circle with the center ( , | and the radius \ \ A AJ > A^ 2. G'' + F--AC=Q. The equation is then which can be satisfied by real values of x and y only when G F x = and y— Hence the equation represents the point — -> I . This may be called a circle of zero radius, regarding it as the limit of a circle as the radius approaches zero. 3. G^-\-F^ — AC<0. The equation can then be satisfied by no real values of x and y, since the sum of two positive quantities cannot be negative. Hence the equation represents no curve. 136 CERTAIN CURVES AND THEIR EQUATIONS Ex. 1. The equation x2 + j/2-2x + 4y4-l = may be written (X - 1)2 + (2/ + 2)2 = 4, and represents a circle with center (1, - 2) and radius 2. Kx. 2. The equation x^ + y- - 2 x + iy + o = may be written (x'- 1)2 + (y + 2)2 = 0, and is satisfied only by the point (1, - 2). Ex. 3. The equation x^ + y^ -2x + 4y + 7 = may be written (X - 1)2 + (y + 2)2 = - 2, and represents no curve. 72. To find the equation of a circle which will satisfy given conditions, it is necessary and sufficient to determine the three quantities d, e, r, or the ratios of the four quantities A,G,F, C. Each condition imposed upon the circle leads usually to an equa- tion involving these quantities. In order to determine the three quantities it is necessary and in general sufficient to have three equations. Hence, m general, three conditions are necessary and sufficient to determine a circle. It is not important to enumerate all possible conditions which may be imposed upon a circle, but the following three may be mentioned. 1. Let the condition be imposed upon the circle to pass through the known point (x^, y^. Then {x^, y^ must satisfy the equation of the circle ; therefore d, e, and r must satisfy the condition {x,-df+{y-ef = r\ 2. Let the condition be imposed upon the circle to be tangent to the known straight line Ax + By + C=Q. Then the distance from the center of the circle to this line must equal the radius ; therefore, by § 32, d, e, and r must satisfy the condition Ad + Be + C - = ± r. y/A' + B^ The sign will be ambiguous, unless from other conditions of the problem it is known on which side of the line the center lies. THE CIRCLE 137 3. Let it be required that the center of the circle should lie on the line Ax + i?y + C= 0, Then d and e must satisfy the condition Ad + Be + C=Q. Ex. 1. Find the equation of the circle through the three points (2, — 2), (7, 3), and (6, 0). The quantities d, e, and r must satisfy the three conditions (2-d)2 + (-2-e)2 = r2, (7 - d)2 + (3 - e)2 = r2, (6 - d)2 + (0 - e)2 = r2. Solving these we have d = 2, e = 3, and r = 5. Therefore the required equation is (x-2)2 + (y._3)2 = 25, or x2 + 2/2 - 4 a; - 6 y - 12 = 0. Ex. 2. Find the equation of the circle which passes through the points (2, —3) and (—4, —1) and has its center on the line Sy + x — IS = 0. The quantities d, e, and r must satisfy the conditions (2-d)2+ (-3-e)2 = r2, (_4_d)2 + (_l_ e)2 = r-2, 3 e + d - 18 = 0. Solving these equations we find d = |, e = y-, r2=-l-|-S.. Therefore the required equation is or a;2 + ^2 _ 3a; _ 11 2/ - 40 = 0. Ex. 3. Find the equation of a circle which is tangent to the lines 17 x + 2/ -35 = and 13x + II2/ + 50 = 0, and has its center on the line 88 x + 70 2/ + 15 = 0. The quantities d, e, and r must satisfy the conditions 17d + e-35 ■■±r. V290 - 13d -lie -50 = ±r, V290 88d + 70e + 15 = 0. 138 CERTAIN CURVES AND THEIR EQUATIONS These equations have the two solutions and d = 6, e = — ^^ , 6 ' 3V29O 20 Hence each of the two circles 3a;2 + 32/2+ 5x- 5j/-20 = and 40x2 + 402/2 _400x + 620y + 2429 = satisfies the conditions of the problem. Ex. 4. The equation of a circle through three given points is most readily found by means of the equation ^x2 + .42/2 + 2Gx + 2Fy-\-C=0. If (a^i, yi), {3^1 1/2)1 and (X3, 2/3) are the three given points, the quantities A, G, F, C must satisfy the equations ^x 2 + Ay^ + 2Gxi + 2Fyi + C=0, Ax^ + Ayl + 2 Gx2 + 2 Fyo + C = 0, Ax^ + Ay^ + 2 Gx3 + 2 Fys + C = 0. There are here four homogeneous equations in the unknowns .4, G, jP, C, and the result of eliminating the unknowns is, by § 9, x2 + ?/2 X y 1 ^X+Vl *! Vi 1 xi + yi X2 2/2 1 ^s+v! X3 Vs 1 0, (1) which is the required equation of the circle. It is to be noticed that the coefficient of x2 4- 2/2 in (1) is xi Vi 11 X2 2/2 1 1 . xs Vz 11 When this is zero, equation (1) is of the first degree and represents a straight line. But when Xi 2/1 1 X2 Vi 1 X8 2/8 1 = 0, the points (Xi, 2/1), (X2, 2/2), and (X3, 2/3) are on the same straight line (§ 29, 5) and cannot determine a circle. THE ELLIPSE 139 73. The ellipse. A71 ellipse is the locus of a point the sum of the distances of which from two fixed points is constant. The two fixed points are called the foci. Let them be denoted by F and F' (fig. 82) and let the axis of x be taken through them and the origin halfway between them. Then if P is any point on the ellipse and 2 a represents the constant sum of its distances from the foci, we have F'P + FP=2a. (1) From the triangle F'PF it follows that F'F<2a. Hence there is a point A on the axis of x and to the right of F which satisfies the definition. We have then F'A + FA=2a, or {F'O + OA) + {OA- OF) =2 a, whence OA = a. Let us now place OF OA = e, where e < 1. Then the coordinates of F and F' are {±ae, 0). Computing the values of F'P and FP by § 17, and substituting in (1), we have V(a?+ aeY+ y^ + V(^ — ae)'^ + y^ = 2> (2) By transposing the second radical to the right-hand side of the equation, squaring, and reducing, we have a — ex= ^{x—aeY+f = FP. (3) Similarly, by transposing the first radical in (2), we have a + ex = y/{x + aef +f = F'P. (4) 140 CERTAIN CURVES AND THEIR EQUATIONS Either (3) or (4) leads to the equation or -,+ ./ , =1. (6) Since e <1, the denominator of the second fraction is positive and we place „ , ,o thus obtaining ~2 + ^2 ~ -^- C^) We have now shown that any point which satisfies (1) has co- ordinates which satisfy (7). To show, conversely, that any point whose coordinates satisfy (7) is such as to satisfy (1), let us assume (7) as given. We can then obtain (6) and (5), and (5) may be put in each of the two forms ar^ + 2 aex + a^c^ + if = «- + 2 aex + ^o?, ar^ — 2 aex + cj^c^ + if = ar—1 aex + e^x?, the square roots of which are respectively F'F=±{a + ex), FP = ±{a — ex). These lead to one of the four following equations : F'P+FP=2a, F'P-FP=2a, -F'P + FP=2a, -F'P-FP = 2a. Of these, the last one is impossible, since the sum of two nega- tive numbers cannot be positive; and the second and third are impossible, since the difference between FP and F'P must be less than F'F, which is less than 2 a. Hence any point which satisfies (7) satisfies (1), and therefore (7) is the equation of the ellipse. THE ELLIPSE 141 74. Placing y = in (7), § 73, we find x = ± a. Placing x = 0, we find y = ±h. Hence the ellipse intersects OX in the two points A{a, 0) and A' {— a, 0), and intersects OF in two points B{Q, h) and -B'(0, — h). The points A and ^' are called the vertices of the ellipse. The line AA', which is equal to 2 a, is called the mayor axis, and the line ^^', which is equal to 2 h, is called the wtTior axis of the ellipse. Solving (7) first for y and then for x, we have M y = ±-^a'-ci» A' and a; = ± - V&-^ - y"^ B' Fig. 83 iV B' K These equations show (1) that the elHpse is symmetrical with respect to both OX and OY, (2) that x can have no value numer- ically greater than a, (3) that y can have no value numerically greater than b. If we construct the rectangle KLMN (fig. 83), which has for a center and sides equal to 2 a and 2 h respec- tively, the ellipse will lie entirely within it ; and if the curve is constructed in one quadrant, it can be found by symmetry in all quadrants. The form of the curve is shown in figs. 82 and 83. 75. Any equation of the form (7), § 73, in which a > b, represents an ellipse with the foci on OX. For if we place, as in §73, &"' = a'(I-e'), we find Vc^^' and may fix i^and F', which in § 73 were arbitrary in position, by the relation 0F = — OF' = ae. The foci may be found gi'aphically by placing the point of a com- pass on B and describing an arc with the radius a. This arc will intersect AA' in the foci ; for since OB = h and OF = Va^ — b\ BF=a. 142 CERTAIN CURVES AND THEIR EQUATIONS Similarly an equation of the form (7), § 73, in wliich h>a, represents an ellipse in which the foci lie on BB' at a distance VV^—a^ from 0. In this case BB' = 2 & is the major axis and AA' = 2 a is the minor axis. It may be noted that the nearer the foci are taken together, the smaDer is e and the more nearly h = a. Hence a circle may he considered as an ellipse with coincident foci and equal axes. 76. The hyperbola. An hyperhola is the locus of a point the difference of the distances of which from two fixed points is constant. Fig. 84 The two fixed points are called the foci. Let them be F and F' (fig. 84) and let FF' be taken as the axis Of x, the origin being lialfway between F and F'. Then if P is any point on the hyperbola and 2 a is the constant difference of its distances from F and F', we have either or F'P-FP = 2a, FP-F'P = 2a. (1) (2) Since in the triangle F'PF the difference of the two sides FP and F'P is less than F'F, it follows that F'F >2a. There is therefore at least one point A between O and F which satisfies the definition. THE HYPERBOLA 143 Then F'A—AF=2a, or {F'0 + 0A)-{0F-0A)=2a; whence OA = a. We may therefore place — - = e, where e > 1. OA Then the coordinates of F and F' are (± ae, 0) and equations (1) and (2) become V(aj + aef + / — V(a; — ae)"-^ +f = 2a (3) and V(a; — aef + 2/"^ — V(a; + aef + y- = 2 a. (4) By transposing one of the radicals to the right-hand side of these equations, squaring, and reducing, we obtain from (3) either + a= y/{x + aef + f = F'P, — a = y/{x — aef -\-f=FP', and from (4) we obtain either ex or ex ■ {ex + a) = y/{x + aef+y'' = F'P, or — {ex — a) = ■>J{x — aef + / = FP. Any one of the last four equations gives {l-e')x'+f = d\l-e\ (5) ^ , f a^ a\l-e^) ^.+ ..f .. =!• (6) But since e >l,a^{l — e'^) is a negative quantity and we may write a^{l — e^) = — b^, thus obtaining ^-C = l. (7) b 144 CERTAIN CURVES AND THEIR EQUATIONS Then any point wliich satisfies (1) or (2) satisfies (7). Conversely, by retracing our steps, we find that if the coordinates of a point P satisfy (7), then F'P = ±(ex + a) and FP = ±{ex — a). Hence we must have either F'P — FP = 2a, — F'P+FP = 2a, F'P + FP = 2a, or -F'P — FP = 2a. The equation F'P + FP = 2 a is impossible, for F'P + FP> F'F, and 2a < F'F. The equation — i^'P — i^P = 2 a is also clearly impossible. Hence any point which satisfies (7) satisfies either (1) or (2). Therefore (7) is the equation of the hyperbola. 77. If we place y = in (7), § 76, we have x = ±a. Hence the curve intersects OX in two points, A and A', called the vertices. It x= 0, y is imaginary. Hence the curve does not intersect OY. Solving (7), § 76, for y and x respectively, we have a and x=±- Vy+i^. These show (1) that the curve is symmetrical with respect to both OX and OY, (2) that x can have no value numerically less than a, and (3) that y can have all values. Moreover, the equation for y can be written THE HYPERBOLA 145 As X increases the term — decreases, approaching zero as a limit. Hence the more the hyperbola is prolonged, the nearer it comes to the straight lines y = ± - x. Therefore the straight lines y = ±- X are the asymptotes of the hyperbola. They are the diagonals of the rectangle constructed as in fig. 85, and are used F Fig. 85 , conveniently as guides in drawing the curve. The line AA' is called the transverse axis and the hne BB' the conjugate axis of the hyperbola. The shape of the curve is shown in figs. 84 and 85. 78. Any equation of the form (7), § 76, where a and h are any positive real values, represents an hyperbola with the foci on AA'. VaF+l? For if we place —h^ = a'^(l — e'), we find e = ■ and may 146 CERTAIN CURVES AND THEIR EQUATIONS find the position of the foci from the equations OF = — OF' = ae. Similarly any equation of the form represents an hyperbola mth the foci on BB'. \ih = a, the hyperbola is called an equilateral hyperhola and its equation is either ar^— z/^ = a^ or — ar^+ y^ = a^ 79. The parabola. A parabola is the locus of a point equally distant from a fixed point and a fixed straight line. The fixed point is called the focus and the fixed straight line the directrix. Let the line through the focus perpendicular to the directrix be taken as the axis of x, and let the origin be taken on this line halfway between the focus and the directrix. * ^ Let us denote the abscissa of the focus by p. In fig. 86 let i^ be the focus, BS the directrix intersecting OX at D, and let P be any point on the curve. Then the coordinates of -X F are (p, 0), those of D are {—p, 0), and the equation of BS is « = —p. Draw from P a line parallel to OX intersecting BS in N. If F is on the right of BS, P must also lie on the right of BS, and by the definition FP = NP. If, on the other hand, F is on the left of BS, P is also on the left of BS and FP = PN=-NP. In either case FP^ = NP^. But FP = (x-pY-\- f, and NP = x + p; hence {x~pf+f = (x + p)', which reduces to -f = i^px. (by § 17) (1) THE PARABOLA 147 Any point on the parabola then satisfies this equation. Conversely, it is easy to show that if a point satisfies this equation, it must so lie that FP = ± NP, and hence lies on the parabola. Equation (1) shows (1) that the curve is symmetrical with respect to OX, (2) that x must have the same sign as p, and (3) that y increases as x increases numerically. The position of the curve is as shown in fig. 86 when jp is positive. When p is neg- ative F lies at the left of and the curve extends toward the negative end of the axis of x. Similarly the equation a? = 4:py represents a parabola for which the focus lies on the axis of y, and which extends toward the positive or the negative end of the axis of y according as p is positive or negative. In all cases is called the vertex of the parabola and the line determined by and F is called its axis. 80. If Pi{x^, 2/i) and Po{x^, y„) are two points on the parabola y^ = 4:px (fig. 87), then yl = ^px^; hence Fig. 87 That is, tlie squares of the ordinates of a parabola are to each other as the abscissas. Conversely, if in any curve the squares of the ordinates are to each other as the abscissas, the curve is a parabola. For let i^ be a known point and P any point on the curve. Then, by hypothesis. which may be written y^ = ^x. But this is the same as y^ = 4:px, where p = 148 CERTAIN CURVES JlSD THEIR EQUATIONS 81. The conic. A conic is the locus of a point the distance of which from a Jixed point is in a constant ratio to its distance from a fixed straight line. The fixed point is called the focus, the fixed line the directrix, and the constant ratio the eccentricity. We shall take the directrix as the axis of y, and a line through the focus F as the axis of x, and shall caU the coordinates of the focus (c, 0), where c represents OF and is positive or negative according as F lies to the right or the left of O. Let P be any point on the conic; connect P and F, and draw PN per- pendicular to OY. Then by definition FP = ±eXP, (1) according as P is on the right or the left of OT. In both cases FP* = e'- XP\ But FP* = (a: — ef + y», by § 17, and NP = z. Therefore for any point on the conic r / / /•I / \ \ \ \ \ \ \ \ \ \ \ \ Fio. 88 {x-c)'+f = t^Ji*. (2) It is easy to show, conversely, that if the coordinates of P sat- isfy (2), P satisfies (1). Hence (2) is the equation of the conic. It is clear that the parabola is a special case of a conic, for the definition of the latter becomes that of the former when e = 1. It is also not difficult to show that the ellipse is a special case of a conic, where the eccentricity is « of § 73 and < 1. For if P (fig. 89) is a point on the ellipse -i + ^ = 1> we found in § 73 that a 6- FP = a- ex, F'P = a + ex, or FP = e(^-x\ F'P = e(- + x\ THE CONIC 149 If now we take the point D so that OD — - » and Z)' so that OD' = » draw the lines DS and Z>'*Sf' perpendicular to OX, the line N'FN perpendicular to I>S, and the ordinate MF, we have - — x = OI)— OM = MD = PN, e - + x = D'0-{- OM = D'M = N'P. Fig. 89 The ellipse has therefore two directrices at the distances ± - e from the center. When the ellipse is a circle, e = and the directrices are at infinity. In a similar manner we may show that the hyperbola is a special case of a conic where e > 1. In § 114, Ex. 3, we shall prove that the conic is always either an ellipse, a parabola, or an hyperbola. 82. The witch. Let OBA (fig. 90) be a circle, OA a diameter, and LK the tangent to the circle at A. From draw any line intersecting the circle at B and LK at C. From B draw a line parallel to LK and from C a line perpendicular to LK, and call the intersection of these two Hnes P. The locus of P is a curve called the witch. 150 CERTAIN CURVES AND THEIR EQUATIONS To obtain its equation we wiU take the origin at and the line OA as the axis of y. We will call the length of the diameter of the circle 2 a. Then by continuing CP until it meets OX at M, and calling {x, y) the coordinates of P, we have OM=x, MP = y, MP OB 0A=MC=2a. OBOC In the triangle OMC, — — = — - = ° MC OC OC If AB is drawn, OB A is a right angle and consequently 0B0C = 0A\ also OC' = OM^'+MC^ (1) Fig. 90 Therefore that is, and finally. JifP ^ OA ^C~ OM^ + MC^ y __ 4 a'^ . 2a y x'+4.a' (2) (3) (4) Conversely, if equation (4) is satisfied by any point, we can deduce equations (3), (2), and (1) in order, and hence show that the point is on the witch. Solving (4) for x, we have . ^ \2a-y THE CISSOID 151 This shows (1) that the curve is symmetrical with respect to OY, (2) that y cannot be negative nor greater than 2 a, and (3) that ?/ = is an asymptote. 83. The cissoid. Let ODA (fig. 91) be a circle with the diam- eter OA, and let LK be the tangent to the circle at A. Through draw any Ime intersecting the circle in D and LK in E. On OE lay off a distance OP equal to DE. Then the locus of P is a curve called the cissoid. To find its equation, we will take as the origin of coordinates and OA as the axis of x, and will call the diameter of the circle 2 a. Draw MP perpendicular to OA. Then if ^ and Z> are connected, a triangle ADE is formed similar to OMP ; whence OP _AE MP~ DE By hypothesis DE = OP. Therefore W^MPAE. (2) Also, in the similar triangles OAE and 0PM, AE _ MP . 0A~~ OM' whence, from (2), ^2 _ OA . MP' or X -\- y = whence y^ = OM 2af, 9.n.— This equation is satisfied by the coordinates of any point upon the cissoid. Fig. $)1 152 CERTAIN CURVES AND THEIR EQUATIONS Conversely, if we assume equation (6), we may deduce (5) and (4), aud then by aid of (1) and (3) we have OP = DE. Therefore (6) is the equation of the cissoid. It may be written y = ±x 1 X \2a-x From this it appears (1) that the curve is symmetrical with re- spect to OX, (2) that no value of x can be greater than 2 a or less than 0, and (3) that the line x = 2a is an asymptote. 84. The strophoid. Let ZA' and i?^' (fig. 92) be two straight lines intersecting at right A angles at 0, and let A be a fixed point on LK. Through A draw any straight line intersecting RS in. D, and lay off on AD in either direction a distance DP equal to OD. The locus of P is a curve called the strophoid. To find its equation, take LK as the axis of X and BS as the axis of y, and call the coordi- nates of A (a, 0). By the definition the point P may fall in any one of the four quadrants. THE STROPHOID 153 If we take the positive direction on AD as measured from A towards D, we have OD=PI> when F is in the first quadrant, OD = -PD when F is in the second quadrant, -OD = -FD when F is in the third quadrant, and -OD = FD when F is in the fourth quadrant. These four equations are equivalent to the single equation od'=fi>\ (1) From the similar triangles OAD and APM, OD MF y AD AF V(ic- ■cif+f FD MO OM AD AO OA X a f {x-af+y^ a" Hence ^^_af+f-d^ ^^ is an equation satisfied by any point on the curve. Conversely, if (2) is given, (1) may be deduced. Therefore (2) is the equation of the strophoid. It may be written = ±Xy^ \a — x ^ ^a-\- X This shows (1) that the curve is symmetrical with respect to OX, (2) that no value of x can be less than - a nor greater than + a, and (3) that « - - « is an asymptote. 154 CEKTAIN CURVES AND THEIR EQUATIONS 85. Examples. The use of the equation of a curve in solving problems connected with the curve will be constantly illustrated throughout the book. The following examples depend upon prin- ciples already given. Ex. 1. Prove that in the ellipse the squares of the ordinates of any two points are to each other as the products of the segments of the major axis made by the feet of these ordinates. We are to prove that (fig. 93) A'Mi ■ M2A ' Let the coordinates of Pi be X (p^ii Vi) ^-nd let those of P^ be (X2, 2/2)- Then (g + xi){a - xi) Fig. 93 y\ a'^ - x^ (« + ^^H'^ - ^) But yi = MiPi, a + Xi = A'O + OMi = A'Mi, a-Xi=OA - OMi = MiA, y-i = M2P2, a + X2 = A'Mz, a — X2 = M2A. Hence the proposition is proved. Ex. 2. If MiPi is the ordinate of a point Pi of the parabola, y2 _ 4 px^ and a straight line di-av?n through the middle point of ilfiPi parallel to the axis of x cuts the curve at Q; prove that the intercept of the line MiQ on the axis of y equals § MiPi. Let the coordinates of Pi (fig. 94) be y? (Xi, yi). Then xi = — from the equation of the parabola. 4p By construction, the ordinate of Q is Vi Since Q is on the parabola its abscissa is found by placing y = ~ in y'^ = 4px. The / 2 \ coordinates of Q are then ( — - , —)• The \16p 2/ cobrdinates of M^ are (xi, 0), which are the same as ( — » Ol. Hence the equation of MQ is, by § 29, Fig. 94 8px + 3yiy-2y^ = 0. The intercept of this line on OF is § yi = 2 ilfiPi, which was to be proved. PROBLEMS 155 PROBLEMS 1. Find the equation of the circle having the center (2, —4) and the radius 3. 2. Find the equation of the circle having the center (— §? ^) and the radius 6. 3. Find the equations of the circles having the line joining (2, 3) and ( — 3, 1) as a radius. 4. Find the equation of the circle having the line joining (a, — h) and ( — a, 6) as a diameter. 5. Find the equations of the circles of radius a which are tangent to the axis of y at the origin. 6. Find tiie equations of the circles of radius a which are tangent to both coordinate axes. 7. Find the equation of the circle having as a diameter that part of the line 2x — Sy + 6 — which is included between the coordinate axes. 8. Find the center and the radius of the circle x^ + ?/2 + 4 a; — 10 y — 36 = 0. 9. Find the center and the radius of the circle x^+y^-\-4x-Qy-\-l = 0. 10. Find the center and the radius of the circle 3x^ + 3y^ — 9x + 6y — 2 = 0. 1 1. Find the center and the radius of the circle 5x^+ 5y^ + 2x — 4ty + 1 = 0. 12. Prove that two circles are concentric if their equations differ only in the absolute term. 13. Show that the circles x^ + y^ + 2Gx + 2 Fy + C = and x^ + y^ + 2 G'x + 2 Yy + C" = are tangent to each other if V((?-G')2 + (F-i?")2 = Vg2 + F2 _ c- ± VG'2 + 2f"2 _ c". 14. Find the equation of the circle which passes through the points (0, 3), (3, 0), (0, 0). 15. Find the equation of the circle circumscribing the triangle with the vertices (0, 2), (- 1, 0), (0, - 2). 16. Find the equation of the circle circumscribed about the triangle the sides of which are x + y — 2 = 0, 9x + oj/ — 2 = 0, 2/ + 2x — 1 = 0. 17. Find the equation of the circle passing through the point (— 2, 4) and concentric with the circle x2 + ?/2— 5x + 4?/ — 1 = 0. 18. A circle which is tangent to both coordinate axes passes through (4, 2). Find its equation. 19. The center of a circle which is tangent to the axes of x and y is on the line 2x — 3y + 6 = 0. What is its equation ? 20. A circle of radius 5 passes through the points (2, — 1) and (3, — 2).' What is its equation ? 21. The center of a circle which passes through the points (1, — 2) and (- 2, 2) is on the line 8x-42/ + 9 = 0. What is its equation ? 156 CERTAIN CURVES AND THEIR EQUATIONS 22. A circle which is tangent to OX passes through (-3, 2) and (4, 9). What is its equation ? 23. The center of a circle which is tangent to the two parallel lines x — 3 = and a; - 7 = is on the line y = 2 x + 4. What is its equation ? 24. The center of a circle is on the line 2 x + y = 0. The circle passes through the point (4, 2) and is tangent to the line 4x-3?/-15 = 0. What is its equation ? 25. Find the equation of the circle circumscribing the isosceles triangle of which the altitude is 4 and the base is the line joining the points (— 3, 0) and (3, 0). 26. Find the equation of the ellipse the foci of which are (±3, 0) and the major axis of which is 8. 27. Find the equation of the ellipse the foci of which are (0, ± 2) and the major axis of which is 6. 28. Find the equation of an ellipse when the vertices are (±6, 0) and one focus is (4, 0). 29. Determine the semiaxes o and h in the ellipse — H = 1, so that it will pass through (1, 4) and (2, — 3). 30. If the vertices of an ellipse are (± 5, 0) and its foci are (±3, 0), find its equation. 31. The center of an ellipse is at the origin and its major axis lies along OX. If its major axis is 8 and its eccentricity is ^, find its equation. 32. Find the equation of an ellipse when its center is at the origin, one focus at the point (— 3, 0), and the minor axis equal to 8. 33. Find the equation of an ellipse the eccentricity of which is ^ and the foci of which are (0, ± 6). 34. Given the ellipse 9x2 + 16y2 = 144. Find its semiaxes, eccentricity, and foci. 35. Find the eccentricity and the equation of an ellipse, if the foci lie half- way between the center and the vertices, the major axis lying on OX. 36. Find the equation of an ellipse the eccentricity of which is f and the ordinate at the focus is 5, the center being at the origin and the major axis lying along OX. 37. Find the equation and the eccentricity of the ellipse if the ordinate at the focus is one fourth the minor axis. 38. Find the eccentricity of an ellipse if the line connecting the positive ends of the axes is parallel to the line joining the center to the upper end of the onlinate at the left-hand focus. 39. Find the equation of an ellipse when the foci are (±2,0) and the directrices are x = ± 5. 40. Given the ellipse 2x^-\-Zy^ = \. Find its semiaxes, foci, and directrices. PKOBLEMS 157 41. Find the equation of an hyperbola if the foci are (± 3, 0) and the trans- verse axis is 4. 42. Find the equation of an hyperbola if the foci are (0, ± 4) and the trans- verse axis is 4. 43. An hyperbola has its center at the origin and its transverse axis along OX. If its eccentricity is ^ and its transverse axis is 5, find its equation. 44. Find the equation of an hyperbola when the vertices are (± 4, 0) and the eccentricity is I. 45. Show that the eccentricity of an equilateral hyperbola is equal to the ratio of a diagonal of a square to its side. 46. Find the equation of an hyperbola the vertices of which are halfway between the center and the foci, the transverse axis lying on OX. 47. Find the equation of the hyperbola with eccentricity 3 which passes through the point (2, 4), its axes lying on OX and OY. 48. Find the equation of an equilateral hyperbola which passes through (5, - 2). 49. Find the equation of the hyperbola which has the points (0, ± f V2) for foci and passes through the point (2, — 1). 50. The sum of the semiaxes of an hyperbola is 17 and its eccentricity is \^. Find its equation, if its axes lie on OX and OY. 51. Find the equation of the hyperbola which has the asymptotes y = ± ^x and passes through the point (1, 1). 52. Express the angle between the asymptotes in terms of the eccentricity of the hyperbola. 53. If the vertex of an hyperbola lies two thirds of the distance from the center to the focus, find the slopes of the asymptotes. 54. Given the hyperbola 4x^ — 25]/^= 100. Find its eccentricity, foci, and asymptotes. 55. Find the equation of the hyperbola which has the lines y = ± § x for its asymptotes and the points ( ± 4, 0) for its foci. 56. Show that 1 = 1, where k is an arbitrary quantity, a^ — k^ 62 _ j(2 x2 v^ represents an ellipse confocal to — I- — = l,when ^2 <; 52 j and represents an a;2 y2 a^ t>'^ hyperbola confocal to — + ^= 1, when k'^ >62 but< a"^, a^ being considered greater than &2. «^ ^^ 57. Find the equation of an hyperbola when the foci are (± 7, 0) and the directrices are x = ± 4. X2 ?/2 58. Given the hyperbola — = 1. Find its eccentricity, foci, directrices, and asymptotes. 158 CERTAIN CURVES AND THEIR EQUATIONS 59. A perpendicular is drawn from a focus of an hyperbola to an asymptote. Show that its foot is at distances a and b from the center and the focus respectively. 60. Show that in an equilateral hyperbola the distance of a point from the center is a mean proportional "between its focal distances. 61. Determine p so that the parabola y'^ = ipx shall pass through the point (2, - 3). 62. An arch in the form of a parabolic curve is 29 ft, across the bottom and the highest point is 8 ft. above the horizontal. What is the length of a beam placed horizontally across the arch 4 ft. from the top ? 63. The cable of a suspension bridge hangs in the form of a parabola. The roadway, which is horizonUl and 240 ft. long, is supported by vertical wires attached to the cable, the longest being 80 ft. and the shortest being 30 ft. Find the length of a supporting wire attached to the roadway 50 ft. from the middle. 64. Find the equation of a circle through the vertex and the ends of the double ordinate through the focus of the parabola y^ = A px. 65. Find the equation of the circle through the vertex, the focus, and the upper end of the ordinate at the focus, of the parabola y^ + 12x = 0. 66. Find the equation of the locus of a point the distances of which from (8, - 2) and (-4, 1) are equal. 67. Find the equations of the locus of a point the distance of which from the axis of X equals five times the distance from the axis of y. 68. Find the equation of the locus of a point the distance of which from the axis of X is one third its disUnce from (0, 3). 69. Find the equation of the locus of a point the distance of which from the line x = 3 is equal to its distance from (4, - 2). 70. What is the locus of a point the distance of which from the line 3x + 4y — 6 = is twice its distance from (2, 1) ? 71. A point moves so that its distance from the axis of y equals its distance from the point (5, 0). Find the equation of its locus. 72. A point moves so that the square of its distance from the point (0, 2) equals the cube of its distance from the axis of y. Find its locus. 73. Find the locus of the points at a constant distance 6 from the line 4x + Sy -6 = 0. 74. Find the locus of points equally distant from the lines 2x + 3w-6 = and Sx — 2y + 1 = 0. 75. Show that the locus of a point which moves so that the sum of its dis- tances from two fixed straight lines is constant is a straight line. 76. Find the equations of the locus of a point equally distant from two fixed straight lines. PROBLEMS 159 77. A point moves so that its distances from two fixed points are in a con- stant ratio k. Sliow tliat tlie locus is a circle except when k — \. 78. A point moves so that the sum of the squares of its distances from the sides of an equilateral triangle is constant. Show that the locus is a circle and find its center. 79. A point moves so that the square of its distance from the base of an isosceles triangle is equal to the product of its distances from the other two sides. Show that the locus is a circle and an hyperbola which pass through the vertices of the two base angles. 80. A point moves so that the sum of the squares of its distances from the four sides of a square is constant. Find its locus. 81. A point moves so that the sum of the squares of its distances from any number of fixed points is constant. Find its locus. 82v Find the locus of a point the square of the distance of which from a fixed point is proportional to its distance from a fixed straight line. 83. Find the locus of a point such that the lengths of the tangents from it to two concentric circles are inversely as the radii of the circles. 84. A point moves so that the length of the tangent from it to a fixed circle is equal to its distance from a fixed point. Find its locus. 85. Find the equation of the locus of a point the tangents from wliich to two fixed circles are of equal length. 86. Straight lines are drawn through the points (— a, 0) and (a, 0) so that the difference of the angles they make with the axis of x is tan- 1 - . Find the locus of their point of intersection. 87. The slope of a straight line passing through (a, 0) is twice the slope of a straight line passing through (— a, 0). Find the locus of the point of inter- section of these lines. 88. A point moves so that the product of the slopes of the straight lines joining it to A {—a, 0) and B («, 0) is constant. Prove that the locus is an ellipse or an hyperbola. 89. If, in the triangle ABC, taw A tan^B = 2 and AB is fixed, show that the locus of C is a parabola with its vertex at A and focus at B. 90. Given the base 2 6 of a triangle and the sum s of the tangents of the angles at the base. Find the locus of the vertex. 91. Find the locus of the center of a circle which is tangent to a fixed circle and a fixed straight line. 92. Prove that the locus of the center of a circle which passes through a fixed point and is tangent to a fixed straight line is a parabola. 93. A point moves so that its shortest distance from a fixed circle is equal to it8 distance from a fixed diameter of that circle. Find its locus. 160 CERTAIN CURVES AXD THEIR EQUATIONS 94. is a fixed point and AB is a fixed straight line. A straight line is drawn from meeting AB at Q, and in OQ a point P is taken so that OPOQ = k^. Find the locus of P. 95. If a straight line is drawn from the origin to any point Q of the line y = a, and if a point P is taken on this line such that its ordinate is equal to the abscissa of Q, find the locus of P. 96. A OB and COD are two straight lines which bisect each other at right angles. Find the locus of a point P such that PA • PB = PC • PD. 97. AB and CD are perpendicular diameters of a circle and M is any point on the circle. Through Jf, AM and BM are drawn. AM intersects CD in N, and from J\r a line is drawn parallel to AB meeting BM in P. Find the locus of P. 98. Given a fixed line AB and a fixed point Q. From any point R in AB a perpendicular to AB is drawn equal in length to RQ. Find the locus of the end of this perpendicular. 99. Let OA be the diameter of a fixed circle. From J5, any point on the circle, draw a line perpendicular to OA and meeting it in D. Prolong the line Z>B to P, so that OD:DB=OA: DP. Find the locus of P. 100. Two straight lines are drawn through the vertex of a parabola at right angles to each other and meeting the curve at P and Q. Show that the line PQ cuts the axis of the parabola in a fixed point. 101. In the parabola y^ = ^px an equilateral triangle is so inscribed that one vertex is at the origin. What is the length of one of its sides ? 102. Prove that in the ellipse half of the minor axis is a mean proportional between AF and FA'. 103. Prove that in the ellipse or the hyperbola the ordinate at the focus is an harmonic mean between AF and AF'. 104. If from any point P of an hyperbola PK is drawn parallel to the transverse axis, cutting the asymptotes in Q and R, then PQ- PR- a^. If PK is drawn parallel to the conjugate axis, then PQ ■ PR = - b-. 105. Show that the focal disUnce of any point on the hyperbola is equal to the length of the straight line drawn through the point parallel to an asymptote to meet the corresponding directrix. 106. Prove that the product of the distances of any point of the hyperbola from the asymptotes is constant. 107. Prove that in the hyperbola the squares of the ordinates of any two points are to each other as the products of the segments of the transverse axis made by the feet of these ordinates. 108. Lines are drawn through a point of an ellipse from the two ends of the minor axis. Show that the product of their intercepts on OX is constant. 109. Pi is any point of the parabola y^ = 4px, and P^Q, which is perpen- dicular to OPi, intersects the axis of the parabola in Q. Prove that the pro- jection of PiQ on the axis of the parabola is always 4 p. CHAPTEE VIII INTERSECTION OF CURVES 86. General principle. If f„J(x, y) is an expression involving "^"'^- /..(»=. y) = o (1) is the equation of a curve containing all points the coordinates of which satisfy (1), and containing no other points. Similarly if /„(ie, y) is any second expression in x and y, /„(^,^) = (2) is the equation of a second curve. It follows that if we consider these two equations, any point common to the two corresponding curves will have coordinates satisfying both (1) and (2) ; and that, conversely, any values of x and y which satisfy both (1) and (2) are coordinates of a point common to the two curves. Hence, to find the joints of intersection of two curves, solve their equa- tions simultaneously. We have already discussed in § 30 the simplest case of this problem, i.e. the intersection of two straight lines. We shall now discuss some more complex cases. 87. /,(x, t/) = Oand/,(;c, (/) = 0. Let fM,y)=^ (1) be a linear equation, and f^ {x, y)=0 (2) be a quadratic equation. Since a linear equation always represents a straight line, this problem is to find the points of intersection of a straight Ime and a curve. Solving (1) for either x or y, and substitut- ing the result in (2), we obtain in general a quadratic equation, as, for example, aa^ + hx + c=:0, if (1) has been solved for y. We shall call this equation the resultant equation {§ 9). If the roots of this equation are denoted 161 162 INTERSECTION OF CURVES by x^ and x^, x^ and x^ are the abscissas of the required points of intersection. The corresponding ordinates are found by substi- tuting x^ and x^ in succession in (1). But according to § 37 there are three cases to be considered in the solution of the resultant equation. (1) The roots x^ and x^ may be real and unequal, in which case there are two points of intersection. (2) The roots x^ and x^ may be real and equal, in which case the corresponding ordinates are equal and the two points coincide. As in § 37, we may regard this case as a limit- ing case when the position of the curves is changed so as to make x^ and x^ approach each other, i.e. so as to make the points of inter- section of the straight line and the curve approach each other along •the curve. Accordingly, the straight line represented by equation (1) is tangent to the curve represented by equation (2). (3) Finally, the roots x^ and x^ may be imaginary, in which case no real points of intersection can be found, and the curves do not intersect. Ex. and 1. Find the points of intersection of 3x-2y-4 = r (1) (2) Fig. 96 Solving (1) for y and substituting tlie result in (2), we have x^ - 6x + 8 = 0, the roots of which are 2 and 4. Substituting these values of x in (1), we find the STRAIGHT LIKE AND COKIC 163 corresponding values of y to be 1 and 4. Therefore the points of intersection are (2, 1) and (4, 4) (fig. 95). Ex. 2. Find the points of intersection of and 6x-4y -9 = a;2 - 4 2/ = 0. (1) (2) Solving (1) for y and substi- tuting the result in (2), we have x2-6x + 9 = 0. The roots of this equation are equal, each being 3. Hence the straight line is tangent to the curve. Substi- tuting 3 for X in (1), we find 2/ = I ; hence the point of tan- gency is (3, f ) (fig. 96). Ex. 3. Find the points of intersection of 3x-2y-5 = (1) and x2 - 4 y = 0. (2) Proceeding as in the tw o pr evious examples, we obtain x^ — 6 x + 10 = 0, the roots of which are 3 ± V — 1. Hence the straight line does not intersect the curve (fig. 97). The correspond- ing values of y are 2 ± | V— 1. It is to be noted that the straight lines of these three examples all have the same direction, differing only in the intercept on the axis of y. 88. The work of the last article suggests a method '^ of finding the tangent to any curve represented by an equation of the second degree, the slope of the tangent being given. For if m of the required tan- gent is known, its equation may be written y = mx + b, where h is not known. According to the definition of a tangent, how- ever, b must have such value that the points of intersection of Fig. 97 164 INTERSECTION OF CURVES straight line and curve shall be coincident. This condition enables us to determine b, as shown in the following examples. Ex. 1. Find the equation of tlie tangent to the parabola 3 x^ + 2 7/ =0, the slope of the tangent being 2. Since the slope of the tangent is 2, its equation may be written y = 2x + b. Substituting this value of y in the equation of the parabola, •we have the equation 3a;2 + 4x + 26 = 0. Since the line is to intersect the curve in two coincident points, this equation must have equal roots. The condition for equal roots, by § .37, is (4)2 - 4 (.3) (2 b) = 0, whence we find 6 = |-. = 2x + |-, or 6x-Sy + 2 = (fig. Fig. 98 Therefore the required tangent is Ex. 2. Find the equation of the tangent to the ellipse x2 -I- 4 y2 = 4, the slope of the tangent being ^. The equation of the tangent is y = ^z + b. Substituting this value of y in the equation of the ellipse, we have x2 + 2 6a; + (2 62 - 2) = 0. Fig. 99 The condition that this equation shall have equal roots is (2 6)2 - 4 (2 62 - 2) = 0, whence 6 = ± V2. In this case there are two tangents having the required slope -^ (fig. 99), the equations of which are respectively j^ = i x + Vi and y = ^x-V2 or x-2y±2V2 = 0. By this same method the following formulas for a tangent with known slope m may be derived : 1. The tangent to the parabola y- = Apx is y = mx + EXCEPTIONAL CASES 165 2. The tangents to the ellipse — + ^ = 1 are = mx ± Va^m^ + b^. X IT 3. The tangents to the hyperbola — — 4 = 1 are a- ■ l)- y = mx ± ^ d^m^— If. 89. It was stated in § 87 that the result of the substitution is in general a quadratic equation. In exceptional cases, however, the resultant equation may be linear, as in the first of the following examples, or even impossible, as in the second example. Ex. 1. Consider 2a; -5?/ -10 = (1) and 4x2-252/2 = 100. (2) Fig. 100 Substituting in (2) tlie value of y from (1), we have tlie equation 40 x — 200 = 0, wlience x = 5. .-. y = 0, and tlie straight line and the curve intersect in a single point (o, 0) (tig. 100). E.\. 2. Consider and 2x-5y + 4 = 4x2 -25?/2 + i6x- 84 = 0. (1) (2) Fig. 101 Substituting in (2) the value of y from (1), we have — 100 = 0. But this equation is impossible. Hence the given equations are contradictory, and the straight line and the curve do not intersect (fig. 101). 166 INTERSECTION OF CURVES These exceptional cases, of which the above are illustrative examples, may be regarded as limiting cases as follows : If x^ and x^ are the roots of the resultant equation aaP + bx + c = 0, 2c X, = _5 + v^ - 4ac 2a -&-V&^- -4ac -6-V&--4ac 2c 2a —h -f Vft^— 4 ac Now as a == 0, the resultant equation approaches the linear equa- tion &« + c = 0. At the same time x.= and a;„ = cx) . There- l ^ fore, if a is made to approach zero by changing the position of either the straight line or the curve in the plane, the case in which only one solution of the linear and the quadratic equations is found is the limiting case of intersection of the straight line and the curve as one point of intersection recedes indefinitely from the origin. If a = and & = 0, both x^ and x^ increase indefinitely. Hence the case in which the linear and the quadratic equations are con- tradictory is the limitmg case of intersection, as both points of intersection recede indefinitely from the origin. 90- /i(^, 1/) = and /„(x, y) = 0. Let /i(-^> y) = (1) be a linear equation, and fj^x, y) = (2) be an equation of the wth degree where w > 2. The degree of a curve is defined as equal to the degree of its equation. Accord- ingly, this problem is to find the points of intersection of a straight line and a curve of the w,th degree where %->% and the method is the same as that of § 87. The resultant equation, after sub- stitution from the linear equation, is, in general, of the wth degree, and its solution is found by the methods of Chaps. IV and V. The number of points of intersection will be the same as the number of real roots of the resultant equation. Hence a straight STRAIGHT LINE AND CURVE OF A^th DEGREE 167 line can intersect a curve of the nth degree in n points at most. If the resultant equation has multiple roots, they correspond, in general, to points of tangency of the straight line and the curve, as in § 88 ; and if the resultant equation is of degree less than n, it can be shown that the straight line is the limiting position of one in which one or more points of intersection have been made to recede indefinitely. Ex. 1. Find the points of intersection of y = 2x (1) and y2 = x(x-3)2. (2) The resultant equation is a;[(x-3)2-4x] = 0, or a;[x2-10x + 9] = 0. Its roots (§ 39) are the roots of x = and x2-10x + 9 = 0, which are 0, 1, and 9. The corresponding values of y are found from (1) to be 0, 2, and 18. Therefore the points of intersection are (0, 0), (1, 2), and (9, 18) (fig. 102). Ex. 2. Find the points of intersection of y = 3x + 2{l) and y = x^. (2) Fig. 102 The resultant equa- tion isx^ — 3x — 2 = 0. One root is found (§ 49) to be 2, and the depressed equation isx2 + 2x + 1 = 0. Its roots are equal, both being - 1. The corresponding values of y, found from (1), are 8 and - 1. Therefore the points of intersection are (2, 8) and (- 1, - 1), the latter being a point of tangeucy (fig. 103). Fig. 103 168 INTERSECTION OF CURVES Ex. 3. Find the points of intersection of 2x + 2/-4 = (1) and y2 = x(x2-12). (2) The resultant equation is x^ - 4x2 + 4x - 16 = 0, or (x - 4) {x^ -f 4) = 0, the roots of which are 4 and ± 2 -Z^. The corresponding values of 2/, found Fig. 104 from (1), are — 4 and 4^4 V— 1. The only real solution of equations (1) and (2) being x = 4 and i/ = — 4, the straight line and the curve intersect in the single point (4, - 4) (fig. 104). 91. /„(x, i/)=Oand/„(;t, i/)=0. Let LM^y) = ^ (1) be an equation of the wth degree and /„(•«, 2/) =0 (2) be an equation of the nth degree, where m and n are both greater than unity. The method is the same as in the preceding cases, i.e. the elimination of either x or y, the solution of the resultant equa- tion, and the determination of the corresponding values of the unknown quantity eliminated. The equation resulting from the CURVES OF Mth AND Nth DEGREE 169 elimination is, in general, of degree mn, and the number of simul- taneous solutions of the original equations is mn. If all these solutions are real, the corresponding curves intersect at m7i points. If, however, any of these solutions are imaginary, or are alike, if real, the corresponding curves will intersect at a number of points less than 'tnn. Hence, two curves of degrees m and n respectively can intersect at mn points and no more. No attempt at a complete discussion will be made, on account of the unlimited number of cases which are possible. We shall merely solve a few illustrative examples, noting any interesting geometrical facts that may occur in the course of the sokition: Ex. 1. Find the points of intersection of y2_2x = (1) and x2 + 2/2_8 = o. (2) Subtracting (1) from (2), we elimi- nate y, thereby obtaining the resultant equation a;^ + 2 a; — 8 = 0, the roots of which are — 4 and 2. Substituting 2 and — 4 in either (1) or (2), we find the corresponding values of ?/ to be ± 2 and ± 2 V— 2. The real solutions of the equations are accordingly x = 2, y = ±2, and the corresponding curves intersect at the points (2, 2) and (2, - 2) (fig. 105). From the figure it is also evident that the value — 4 for x must make y imaginary, as both curves lie entirely to the right of the line x = — 4. Y ° Ex. 2. Find the points of inter- section of x2 - 3 2/ = (1) and 2/2_3x = 0. (2) Substituting in (2) the value of y from (1), we have x* — 27x = 0. This equation may be written x(x-3)(x2 + 3x + 0) = 0, the roots of which are 0, 3, and _ 3 4- 3 V — 3 — Substituting these values of x in (1). we find the corre- Fio. lOG spending values of y to be 0, 3, Fig. 105 -X 170 INTEKSECTION OF CUEVES and - 3^8^^ Therefore the real solutions of these equations are a; = 0, y = 0, and x = 3, y = 3. If we had substituted the values of x in (2), we should have at first seemed to find an additional real solution, y — — S when x = 3. But — 3 for 2/ makes x imaginary in (1), as no part of (1) is below the axis of X. Geometrically, the line x = 3 intersects the curves (1) and (2) in a common point and also intersects (2) in another point. Therefore the only real solutions of these equations are the ones noted above, and the corre- sponding curves intersect at the two points (0, 0) and (3, 3) (fig. 106). We 8ee, moreover, that any results found must be tested by substitution in both of the original equations. The re main ing two solutions of these equations found by letting x = _3±3Vr3 are imaginary. Ex. 3. Find the points of intersection of 2x2 + 32/2 = 35 and xy = 6. (1) (2) Since these equations are homogeneous quadratic equations we place y = mx (3) and substitute for y in both (1) and (2). The results are 2 x^ + 3 m'h? = 35 and mx2 = 6, whence x2 = 35 (4) 2 + 3m2 and x2 = 6 m (6) 35 6 5 m (6) 2 + 3 jft2 from which we find m = i or 4. Fig. 107 If m = ^, from (5) x = ± 2 ; and from (3) the corresponding values of y are ± 3. If m = |, in like manner we find X = ± '^ Vg and y — ±^ V6. Therefore the ellipse and the hyperbola intersect at the four points (2, 3), (- 2, - 3), (I V6, f V6), (- I V6, - I V6) (fig. 107). It should be noted that (3) is the equation of a straight line through the origin, so that when we solve (6) for m we determine the slopes of the straight lines passing through the origin and intersecting the two given curves at their common points. SYSTEMS OF CURVES 171 Ex. 4. Find the points of inter- section of 2 2/2 = x-2 (1) and x2 - 4 2/2 = 4. (2) Eliminating y, we have a;'^ 2a; = 0, Fig. 108 the roots of which are and 2. Wlien X = we fi nd from either (1) or (2) y = ± V— 1, and when X = 2 eitlier (1) or (2) reduces to y'^ = 0, whence y = 0. two curves are taijgent at the point (2, 0) (fig. 108). Therefore these Ex. 5. Find the points of intersec- uuxiux x2 = 2y (1) and x^ — 3 XT/ + 2/3 = 0. (2) Eliminating j/, we have x6- 4x3 = 0, which may be written x^ (x^ — 4) = 0. X The real roots of this equation are 4^ and 0, the latter being a triple root, and the t wo imaginary roots are 4*(-l±V-3) ^ ,. — i -• Corresponding 2 values of y a re fou nd to be 2^, 0, and 2-^(- 1 T ^^^^)- Therefore the Fig. 109 curves intersect at the two points (4*, 2i) and (0, 0) (fig. 109). At the point (0, 0) the parabola (1) is tangent to one part of (2) and passes through another part of (2), and for this reason the point is to be regarded as a triple point of intersection. ^^' Ifmi^t y)~^ kfn{x, i/)=0. If we have two expressions f^{x, y) and f,^{x, y), we have seen in § 86 that we can form the equations of two curves by placing each of them separately equal to zero, i.e. A(^,2/) = 0, (1) and /„(^,2/) = 0. (2) Let us now form the equation of a tliird curve by multiplying fjx, y) and fjx, y) by I and k respectively, where I and k are 172 INTERSECTION OF CURVES any two quantities which are independent of both x and y, and placing the sum of the products equal to zero, i.e. ifj<^>y)+¥n{^>y) = ^- (3) This third curve has the following two important properties : 1. It passes through all points common to curves (1) and (2). For the coordinates of any such points make /„,(a?, y) = ^ and fj^x, y) = 0, since they satisfy (1) and (2). Hence they will satisfy (3), ie. be coordinates of a point of curve (3). If either / or A; is placed equal to zero, (3) reduces to either (2) or (1) as a special case. 2. If neither I nor k is zero, it intersects curves (1) and (2) at no other points than their common points. For the coordi- nates of any point on (1), for example, but not on (2), make /^{x, y) = and /„(«, y) different from zero. Hence they will not satisfy (3), and the corresponding point cannot be a point of (3). It follows that if (1) and (2) have no points in common, (3) intersects neither (1) nor (2). If we treat (1) and (2) apart from possible geometrical interpretation, however, it is evident that the imaginary solutions of (1) and (2) are solutions of (3). By assigning different values to I and k we may make (3) satisfy another condition, as will be illustrated in the following examples : Ex. 1. Find the equation of the straight line passing through the point of intersection of the lines Y 2x + y-l = (1) and x + 2y -3 = (2) and the point (1, 2). l{2x+7j-l) + k{x + 27j-3) = (3) passes through the point of intersection of (1) and (2), and is the equation of a straight line, since it is an equation of the first degree. Since (3) is to pass through the point (1, 2), (1, 2) must satisfy (.3). Therefore i (2 + 2 - 1) + fc (1 + 4 - 3) = 0, or 3 ; + 2 /<: = 0. Therefore, if we substitute k = - ^lin (3) and simplify, we shall have the equation of the required line. It is found tobex-4y + 7 = (fig. 110). SYSTEMS OF CUKVES 173 Ex. 2. Find the equation of a straiglit line passing through the point of intersection of the lines and and parallel to the line x-2y + 1 = x-3y + 3 = 0, 2x + 3y + 8 = 0. As in Ex. 1, the required line is lix - 2y + 1) + k{x - 3y + 3) = 0, which may be written {I + k)x + (- 21 - Sk)y + {I + 3k) = 0. Since this line is to be parallel to (3), . l + k 2l-3k (1) (2) (3) (4) (§ 28, 2), whence 2 3 * = — I ^ Substituting this value of k in (4) and simplifying, we have as our required line 2a; + 3y-12 = (fig. 111). Fig. Ill Both of these examples could also have been solved by finding the point of intersection of the given lines and then, by the methods of Chap. Ill, passing the line through the point subject to the given condition. 93. In the two examples of the last article both equations were of the first degree. In this article we will solve some examples in which one or both equations are of the second degree. 174 INTERSECTION OF CURVES Ex. 1. Find the equation of a circle determined by the points of intersec- tion of the straiglit line 2x-y-6 = (1) and the circle x2 + y2-6x-6y -7 = (2) and the point (1, - 1) (fig. 112). The equation l{2x-y-6) +A;(x2+2/2-6x-6y-7) = (3) is the equation of a circle, since the coefficients of x^ and y^ are equal ; and since it passes through the points of intersection of (1) and (2), it only -piQ 112 remains to choose I and k so that it shall pass through the point (1, —1). 3 I Substituting (1, — 1) in (3), we have 3 Z + 5 fc = 0, whence k = Accordingly, the equation (3) of the required circle, in simplified form, is 3 x2 + 3 2/2 - 28 X - 13 ?/ + 9 = 0. Ex. 2. Find an equation representing the system of circles passing through the points of intersection of the circles x2 + 2/2 _ 9 = and x2 + j/2 - 4x - 2 2/ - 11 = (fig. 113). The equation Z(x2 + 2/2- 9) + A:(x2+2/2-4x-2 2/- 11)=0 is the required equation, for by its form it is the equation of a circle, and passes through the two points common to (1) and (2). By assigning different values to I and k we can make (3) represent any, and hence every, circle passing through the common points of (1) and (2). In other words, it represents the required system of circles. In particular, if I and k are assigned such values as to make the coefficients of x' and y^ vanish, i.e. k = — I in this example, the equation reduces to Fig. 113 2x + 2/ + l = 0. (4) But this is the equation of a straight line, and since it must, from the way in which it was formed, pass through the points common to the two circles, it must be the equation of their common chord. PROBLEMS 175 In general, if x^ + y^ + 2 GiX + 2Fiy + Ci = 0, (6) x2 + y2 ^. 2 Ggx + 2 F^y + Ca = 0, (6) are the equations of any two circles, we derive the third equation 2{Gi-G2)x + 2{Fi-F2)y + {Ci-C2) = ' (7) by assuming k = — I. If the circles intersect, (7) is the equation of their common chord ; but if they do not intersect, (7) is called their radical axis. It may easily be proved to be perpendicular to the line of centers and is the locus of points from which equal tangents, one to each circle, may be drawn. PROBLEMS Find where and how the straight line intersects the curve of the second degree in each of the following cases : 1. 2x + 3y = 5, 4x2 + 92/2 + 16x-182/-ll = 0. 2. X - y + 1 = 0, (X + 2)2 _ 4 2/ = 0. 3. x-22/ + 4 = 0, 2x2-?/2 + 8x + 22/ + 13 = 0. 4. 2/ - 2x = 0, x2 + ^2 _ a; + 3y = 0. 5. X - 2 2/ + 4 = 0, 5x2 - 4 2/2 + 20 = 0. 6. 2/ = 8x-5, 2x2 + X2/-3y2 + 6x4-42/ + 4 = 0. 7. 2x + 32/-6 = 0, x2 + 4 2/2-4 = 0. 8. X + 2/ - 4 = 0, x2 - 2 X2/ + 2/2 - 20 = 0. 9. Find the length of the chord of the circle x2 + 2/2 + 8x — 42/ + 10 = cut from the line 2x — Sy + S = 0. 10. Find the tangent to the curve x2 + 6x-22/ + 5 = with slope 2. 11. For what value of p will the parabola y^ = ipx be tangent to the line 2/-3x + l = 0? 12. Find the tangents to the ellipse 4 x2 + 9 7/2 = 36 which are parallel to the line joining the positive ends of the axes. 13. Find the tangent to the curve b^x^ + a^y^ + 2 a62x = perpendicular to the line ax + by = ab. 14. Prove that the line y = -mx + 2c Vm. is always tangent to the hyper- bola xy = c2, and that the point of contact is ( —^ ' c Vmj • 15. Find the point of contact of tlie tangent to the curve x2_4y2 + 2x2/-2x + 42/ = with slope ^. 16. Find the points of intersection of the line 82/ - 26x = and the curve x22/2 + 36 = 4 2/2. 17. Find the points of intersection of the line 2/ = 2x - 3 and the curve 42/2 = (x + 3)(2x- 3)2. 176 INTEKSECTION OF CURVES 18. Find the points of intersection of the line x — 2y + 2 — and the cissoid X (x2 + 2/2) = 4 2/2. 19. Find the points of intersection of the line x = 2y and the curve 16 2/2 = 4 X* - a*. 20. Find the points of intersection of the line y = 2 x — 2 and the cissoid a;(x2 + 2/2) = 4 2/2. 21. Find the points of intersection of the line y = mx and the cissoid x(x2 + 2/2) = 2a2/2. 22. Find the points of intersection of the line x — y — 1 = and the witch y x2 + 4 Find tlie points of intersection of the following pairs of curves : 23. 42/2 = x2(x + 1), 2/2 = x(x + 1)2. 24. 2/2 = 12 x, ^2 = (a; + 2) (X - 3)2. 25. x2 = y^{y + 2), x2 = (2/ - 1)2(2/ + 1). 26. Find the points of intersection of the parabolas 2/2 = 4 ax + 4 a2 and 2/2=-46x + 4 62. 27. Find the points of intersection of the parabola x2 = 4a2/ and the witch 8a3 y = X2 + 4 o2 28. Find the points of intersection of the cissoid y^ = and the parabola 2/2 = 4 ax. a — x 29. Find the poiaits of intersection of the cissoid y^ = and the circle x2 + 2/2-4ax = 0. 2a-x 30. Find the points of intersection of the hyperbola xy = 2aP- and the witch _ %d? ^~x2 + 4a2" 31. Find the points of intersection of the witch y = and the cissoid 4«3 *^ + 4a2 X2 = 5a — 42/ 32. Find the points of intersection of the circle x2 + 2/2=5 a2 and the witch x2 + 4 a2 33. Find the equation of a straight line through the point of intersection of 7x - 2/ - 18 = (1) and x - 3 2/ - 14 = (2) and the point (- 2, 1), without finding the point of interaection of (1) and (2). 34. Find the equation of a straight line through the point of intei-section of 2 X - y + 5 = (1) and x - 42/ + 13 = (2) and parallel to the line 2x + 62/ + 2 = 0, without finding the point of intersection of (1) and (2). PROBLEMS 177 35. Find the equation of a straight line through the point of intersection of 4x - 6?/ - 5 = (1) and G a; - 4?/ - 5 = (2) and perpendicular to the line X — Sy + 1 = 0, without finding the point of intersection of (1) and (2). 36. A circle passes through the origin of coordinates and the points of intersection of the circle x^ + y^ = 14 and the line 2x + Sy + b = 0. Find its equation. 37. Prove that (1, 1) is a point of the common chord of the two circles a;2 + 2/^ — 4 X = and x^ + |/2 _ 4 y = 0. 38. Find the circle passing through (1, — 3) and the points of intersection of the two circles x2 + 2/2 — 4x — 4^/ — 8 = and x^ + y^ + x + y — 4 = 0. 39. Find a curve of the second degree passing through (1, 1) and the points of intersection of the curves Sx^ + 5y'^ — 15 = and 2x^ — 3y^ — G = 0, and tell what kind of a curve it is. 40. Prove that a parabola can be passed through the points of intersection of the curves x^ - 2y^ + x + 2y + 1 = and 3x2 + 4 2/2_2x-2 = 0. 41. The center of a circle is at the vertex ^ of a parabola y^ = 4px, and its diameter is 3 J.F, i^ being the focus of the parabola. Prove that their common chord bisects AF. 42. Show that the circle described on any focal radius of a parabola as diameter is tangent to the tangent at the vertex of the parabola. 43. Show that the circle described on any focal chord of a parabola as a diameter is tangent to the directrix of the parabola. 44. If a circle is described from a focus of an hyperbola as center, with its radius equal to half the conjugate axis, prove that it will touch the asymptotes at the points where they intersect the corresponding directrix. CHAPTER IX DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 94. Theorems on limits. In operations with limits the follow- ing propositions are of importance. 1. The limit of the sum of a finite number of variables is equal to the sum of the limits of the variables. We will prove the theorem for three variables ; the proof is easily extended to any number of variables. Let X, Y, and Z be three variables, such that LimX = J, Lim Y = B, JAm Z = C. From the definition of Hmit (§ 53) we may write X = A+a, Y = B + b, Z=C + c, where a, b, and c are three quantities each of which becomes and remains numerically less than any assigned quantity as the variables approach their limits. Adding, we have X+Y-\-Z = A+B + C+a + b + c. Now if e is any assigned quantity, however small, we may make a, b, and c each numerically less than - > so that a + b + c is numerically less than e. Then the difference between X+Y+Z and A+B + C becomes and remains less than e, that is, lim {X+ Y + Z) = A + B+C= Lim X+ Lim Y+ Lim Z. 2. The limit of the product of a finite number of variables is equal to the product of the limits of the variables. Consider first two variables X and Y such that Lim X = A and Lim Y = B. As before, we have X = A + a and Y = B -i-b. Hence XY = AB + bA + aB + ab. 178 THEOREMS ON LIMITS 179 Now we may make a and h so small that hA, aB, and ah are e 3 Lim XY = AB = (Lim X) (Lim Y). each less than - . where e is any assigned quantity, no matter how small. Hence Consider now three variables X, Y, Z. Place XY= U. Then, as iust proved, UmUZ = {UmU){lAmZ); that is, lim XYZ = (Lim XY) (Lim Z) = (Lim X) (Lim Y) (Lim Z). Similarly the theorem may be proved for any finite number of variables. 3. The limit of a constant times a variable is equal to the con- stant times the limit of the variable. The proof is left for the student. 4. The limit of the quotient of two variables is equal to the quo- tient of the limits of the variables, provided the limit of the divisor is not zero. Let X and Y be two variables such that LimX = ^ and limY^B. Then, as before, X = A + a, Y = B-{-b. „ X A + a , X A A + a A aB — bA Hence — = > and = r- = — :; — ; — • Y B + b Y B B + b B B^ + bB Now the fraction on the right of this equation may be made less than any assigned quantity by taking a and b sufficiently small. TT ^. X A Lim X Hence Lim — = — = Y B LimF The proof assumes that B is not zero. 95. Theorems on derivatives. The definitions of increment, continuit} , and derivative given in Chap. V are perfectly general, although they are there applied only to algebraic polynomials. 180 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS In order to extend the process of differentiation to other func- tions, we shaU. need the following theorems : 1. The derivative of a function plus a constant is equal to the derivative of the function. Let w be a function of x which can be differentiated, let c be a constant, and place y — ^jl. c Then if x is increased by an increment Ax, u is increased by an increment Au, and c is unchanged. Hence the value of y becomes u + Au + c. Whence Ay = (w + Au + c) — {u + c) = Au. Therefore ^ = ^' Ax Ax and, taking the limit of each side of this equation, dy _ du dx dx Ex. 2/ = 4x3 + 3, ^ = 1(4x3) = 12x2. dx dx 2. The derivative of a constant times a function is equal to the constant times the derivative of the function. Let -JA be a function of x which can be differentiated, let c be a constant, and place _ ^^^ Give X an increment Ax, and let Au and Ay be the correspond- ing increments of u and y. Then Ay = c(u-{- Au) — cu = cAu. (by theorem 3, § 94) Hence Ay _ Au Ax Ax and , . Ay -r • Aw Lim — ^ = c Lim -— Ax Ax Therefore dy du -^ = c — > dx dx by the definition of a derivative. THEOREMS ON DERIVATIVES 181 Ex. 2/ = 5(x3 + 3x2 + l), dv d -^ = 5--(x3 + 3x2 + 1) = 5(3x2 + 6x) = lo(x2 + 2x). uX uX 3. The derivative of the sum of a fnite number of functions is equal to the sum of the derivatives of the functions. Let u, V, and w be three functions of x which can be differen- tiated, and let , , y = u -{- V + w. Give X an increment Ax, and let the corresponding increments of u, V, w, and y be Aw, Av, Aw, and Ay. Then Ay = [u + A'2<- + V + Av + w + A'Z^;) — [u + v + w) = Ait + A^ + Avj ; , Ay Aw Av Aw whence t^ = 7 — ^-r~ + -i;—' Ax Ax Ax Ax Now let Ax approach zero. By theorem 1, § 94, -r . Ay _ . Aw -r . A«7 -r • Aw . Lim — ^ = Lim f- Lim 1- Lim --— > Ax Ax Ax Ax that is, by the definition of a derivative, dy _ du dv dw (A/*A/ tl/t/j- (A/t/y Ct'X' The proof is evidently applicable to any finite number of functions. Ex. ?/ = x*- 3x3 + 2x2-7 X, ^ = 4x3-9x2 + 4x-7. dx 4. The derivative of the product of a. finite member of functions is equal to the sum of the products obtained by multiplying the derivative of each factor by all the other factors. Let u and v be two functions of x which can be differentiated, ^^^let y=^uv. 182 DIFFERE^^TIATION OF ALGEBRAIC FUNCTIONS Give X an increment Aa;, and let the corresponding increments of u, V, and y be Aw, Av, and Ay. Then Ly = (w + Aw) (v + At-) — wv = w Av + ■?; Aw + Azt • Av , Aw Av Aw . Aw . and —^ = w-— +?;—- + -— -Av. A;r ^x Ax Aa; If now Aic approaches zero, we have ,. Aw ^. Av . -r- Aw , -r- Aw _. . Lim — ^ = w Lim \- v Lim h Lim -— • Lim Av. A« Aa; o^x Ax ,(. q -^, T^ ^ T • Aw i^w ^ . Aw du T . Av dv , -r • a a But Lim —^ = -f-, Lim — = — - > Lim -— = -- > and Lim A?; = ; Ax dx Ax dx Ax dx ^, .„ dy dv , du thereiore -^ = w — — I- ^ -7- • dx dx dx Again, let y = uvw. Eegarding uv as one function and applying the result already obtained, we have dy dw d (uv) -^ = uv — — I- w —^ — - ax dx dx dw . = uv — — I- w dx [dv dul dx dxj dw dv du = uv— — f- uw -— + vw-—- dx dx dx The proof is clearly applicable to any finite number of factors. Ex. y = (3z-5)(x2 + l)ic3, ^ = (3x - 5) (x^ + D^-i^ + (3x - 6).3^(?!_±i) + (,. + i),3l(^^^ dx ^ '^ ' dx ^ ' dx ^ ' dx = (3x - 5)(x2 + l)(3x2) + (3x - 5)x3(2x) + (x^ + l)x3(3) = (18x3 - 25x2 + 12x - 15)x2. 5. The derivative of a fraction is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. THEOREMS ON DERIVATIVES 183 l^et y where u and v are two functions of x which can be v differentiated. Let Aic, Ai*, Av, and Ay be as usual. Then . _u-\- Au ti _v Ail — u Av V + Av V v^-{- V Av and Au Av Ay Ax Ax Ax v^+ V Av Now let Ax approach zero. By § 94, -r . Au -, . Av V Lim u Lim — Ay Ax Ax Lim-^— = 7-, T"- — 1 Ax V + V Lim Av dy du dv dx dx whence dx v' Ex. y — , " x2 + 1 dy (a;2 + l)(2x)- -(x2-l)2a; 1)2 4x dx (x2 + (X2 + 1)2 ^. If y is a function of x, then x is a function of y, and the derivative of x with respect to y is the reciprocal of the derivative of y with respect to x. Let Ax and Ay be corresponding increments of x and y. Then Ax 1 Ay^Ay' Ax whence lim -r— = Ax dx 1 that is, ~r = t" ' dy dy dx 7. If y is a function of u and ic is a function of x, then y is a fu7iction of x, and the derivative of y with respect to x is equal to the derivative of y with respect to u times the derivative of u with respect to x. 184 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS An increment Ax determines an increment Au, and this in turn determines an increment Ay. Then evidently Ay _ Ay Au Ax Au Ax whence Lim — ^ = lim — ^ • Ax Au Lim -— > Ax that is, dy _ dy du dx du dx Ex. y = = W2 + 3tt + 1 , where u = —, dy _ dx~ = (2 u + 3) '(- 2\ 2 + 3x2 2 _ X3/ ~ X2 X8 ~ 4 + 6x2 X5 The same result is obtained by substituting in tlie expression for y the value of u in terms of x, and then differentiating. 96. Formulas. The formulas proved in the previous article are : d(u + c) _du (2) (3) (4) (5) (6) (7) (8) dx dx d(cu) du dx dx d(u + v) du dv dx dx dx d{uv) dv 1 du dx dx dx , /u\ du dv \v/ dx dx dx ^ v^ dx 1 dy~ dy' dx dy _ dy du dx du dx dy dy_du dx~ dx' du brmula (8) is a combination of (6) and (7), DERIVATIVE OF U"" 185 97. Derivative of u". If tt is any function of x wliich can be differentiated and n is any real constant, then —- — - = nu — • To prove this formula we shall distinguish four cases : 1. When ri is a positive integer. d{u'') _ divJ') du dx du dx „_, du (by (7), § 96) m.»-^-. (by(l), §58) 2. When % is a positive rational fraction. P Let n = — where p and q are positive integers, and place p y = u\ By raising both sides of this equation to the g-th power we have ?/« = w". Here we have two functions of x which are equal for all values of X. If we give x an increment Aic, we have A(y'') = A(wa A(y^)_A(^0. Hx Ax and therefore - "^ — \ > dx dx whence QV^'^ — = pW'^ — - » dx dx since ^ and q are positive integers. Substituting the value of i) and dividing, we have dx q dx • Hence in this case also d hi") „ \du dx dx 186 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 3, When w is a negative rational number. Let n = — m, where tw is a positive number, and place y =z U "" = d{u'^) Then -^ = dx dx m 1 du dx — ^2m = -m-1 <^^' — mu -— dx Hence in this case also d{u^) _ dx n 1 du dx (by (5), § 96) (by 1 and 2) 4. When n is an irrational number. The formula is true in this case also, but the proof will not be given. As a particular case of this formula, it appears that 1, § 58, is true for all real values of n. Ex.1. y = (x3 + 4x2- 5a; + 7)3, -^ = 3(a;3 + 4x2 - 5a; + 7)2— (x^ + 4x2 - 5x + 7) dx dx = 3(3x2 + 8x _ 5)(x3 + 4x2 _ 5x + 7)2. (by § 58) # Ex. 2. y = ^/x2 + — = x^ + x-3, x3 ^ =!«■*- 3 x-" (by (3), §96) _^ _3 Ex. ^. y-{x + l)Vx2 + 1, = (a; + i)[i(x2 + i)-'.2x] + (x2 + i)i = ^(^ + (x2 + l)i _2x2 + x + l Vz2 + 1 HIGHER DERIVATIVES 187 dy_l / a; \- ^ d^ / x \ cto "" 3 \x8 + 1/ da; U* + 1/ 1 /x8 + i\3 1-2x8 _ 1-2x8 ~3x3(x3 + l)4' 98. Higher derivatives. It has been noted already (§ 62) that the derivative of the derivative of a function is called the second derivative of the function. Similarly the derivative of the second derivative is called the third derivative, and so on. The succes- sive derivatives are commonly indicated by the following notation. y =^f(x), the original function, dx -^ = f'{x), the first derivative, -— I — ) = — =^ =f"(x), the second derivative, ax \dxj dx — ( — 4 ) = -4 =f"'(x), the third derivative, \dx / dx'^ dx \dx dx' d"v •L = /^"^(a;), the wth derivative. It is noted in § 22 that /(a) denotes the value of f{x) when x — a. Similarly f{a), f"{a), f"'{a), are used to denote the values of f'{x), f"{x), f"'{x) respectively when x = a. It is to be empha- sized that the differentiation is to be carried out before the sub- stitution of the value of x. Ex. If/(x) = ^-^^, find/"(0). ,,, , -x2 + 2x + l. ^^''^- (x-^ + l)2 ' ^„, , 2x8-Gx2-6x + 2, ^^^^ = WT^' .-. /"(O) = 2. 188 DIFFERENTIATIOX OF ALGEBRAIC FUNCTIONS 99. Differentiation of implicit algebraic functions. Consider any equation of the form P^ + I>xf~'' + I>2t~'' + I>zy'"^+ ■ • • +Pn-iy+K = ^> (1) where w is a positive integer, and where some or all of the coeffi- cients Pffl Pi, • • • , jt>„, are polynomials in x. By means of this equa- tion, if a value of x is given, values of y are determined. For if a numerical value is given to x, the coefficients become numerical and the equation is of the kind discussed in Chap. IV, which has been shown always to have n roots. Hence (1) defines y as a function of x. This is the most general form of an algebraic function. When (1) is solved for y, so that y is expressed in terms of x, y is an explicit algebraic function. When (1) is not solved for y, y is an implicit algebraic function. For example, ^ a? — ^ xy + h y^ — ^ x + 1 y — ^ = ^, which may be written 5 2/' + (7 - 4 ic) y + (3 a;' - 6 a: - 8) = 0, defines y as an implicit function of x. If the equation is solved for y, giving -7+4a^±V209-}-64x-44ar^ ^ = lo ' y is expressed as an explicit function of x. It may be shown by advanced methods that y defined by (1) is a continuous function of x and has a derivative with respect to X. Assuming tliis, it is possible to find the derivative without solving (1), for we have in (1) a function of x which is always equal to zero. Hence its derivative is zero. The derivative may be found by use of the formulas of the previous article, as shown in the examples. Ex.1. Given x2 + y2 = 5. Then djz^ + y^) ^ dx ' dy that IS, 2x + 2y— =0', dx whence — = — . dx y IMPLICIT functio:n^s 189 The derivative may also be found by solving the equation for y. Then y = ±y/b- iC'^, dy _ ^ -X _ x dx V5 - x2 y Ex. 2. Given y^ -xy -1 = 0. Then d(2/3) d(xy)_^ dx dx Hence Sy^^/-xf--y = dx dx and dy y dx Sy^ — X The second derivative may be found by differentiating the result thus obtained. Ex. 3. If a;2 + y2 = 5, we have found ^ = - -. dx y Therefore ^ _ _ d^ /x\ c2 dx \y/ d?y dx^ dx \y, dy dx Ex. 4. If 7/3 — xy — 1 = 0, we have found y2 y^ + x^ yS dy y dx 3 2/2 _ a; ,dy di^y'^-x) {Sy^-x)-f-y ' ' V , _,, • d^y dx dx Then —^ = dx2 (3 2/2-x)2 (3 2/2 _ x)2 (3z/2_x)-^ yi^ 1) ^ \S 2/2 - X \3 y2 - a: / (3 2/2-a;)2 _ -2x2/ ■" (32/2 -x)3' 190 DIFFERENTIATION OF ALGEBKAIC FUNCTIONS 100. Tangents. It has been shown in § 59 that the tangent to a curve ?/ =/(a^) at a point (x^, y^ is where ( — | denotes the value of ^ at Ix, yX We wiU apply this \dxji dx to some of the curves of Chap. VII, obtaining results for future reference. Ex. 1. Consider the circle Ax"^ + Ay'^ ^r 2Gx + 2Fy + C = 0. Differentiating, we have 2Ax + 2Ay^ + 2G + 2F— = Q; dx dx dy_ Ax + G L6I1C6 dx Ay + F Hence the equation of the tangent . is it is, y- -yi = - Axi + G Ayi + F ■^yi -t r Axix - Ax^ + Ayiy - Ay^ + Gx - Gxi + Fy - Fyi = 0. This equation may be simplified by adding to it the identity Ax^ + Ay^ + 2 Gxi + 2 Fi/i + C = 0, which follows from the fact that (xi, y{) is on the circle. There results ^xix + Ayxy + G (x + Xi) + F{y + ^i) + C = 0. This result is easily remembered from its resemblance to the equation of the circle. The proofs of the next three examples are left to the student. Ex. 2. The tangent to the ellipse - + — = 1 is — + — = 1, Ex. 3. The tangent to the hyperbola ^-^ = lis?l?-^ = l. a2 62 (j2 ft2 Ex. 4. The tangent to the parabola j/2 = 4px is y^y = 2p(x + Xi), NORMALS Ex. 5. Consider the witch x'h/ + 4 a^y — 8 a^ = Differentiating, we have 191 ox dz Hence the equation of the tangent is y -Vi 2X1^1 ; (« - «i) ; that is, But Xi'' + 4 o2 zly + 4 a2y - 4 a^y^ + 2 Xi^ix - 3 xfyi = 0. XjVi + 4 o2yi - 8 a3 = 0. Hence the equation of the tangent may be written 2 xi^ix + (X 2 + 4 a2) 2/ + 8 a^yi - 24 a^ = 0. Ex. 6. In the same manner the tangent to the cissoid x^ + xy- — 2 ay^ = at the point (xi, y{) is found to be (3 x^ + y^) X + (2 xxyi - 4 ayi) y-2 ay^ = 0. 101. Normals. The normal to a curve at any point is the straight line perpendicular to the tangent at that point. To find its equation first find the slope of the tangent and then apply problem 3, § 29. X^ w2 Ex. 1. For the ellipse h — = 1 the slope of the tangent at (Xi, yi) is 62x, "' *' Hence the equation of the normal at (xi, yi) is a-'yi ah/i y-yi = ^{x- xi), which is a%ix - b^iy - (a2 - b!')xiyi = 0. If y = 0, a2 - 62 X = Xi = C'^Xi. a2 Hence in fig. 114 NF=OF-ON-ae- e'^xx, F'N = F'O + ON=ae + e^Xi. Then F'N _ a + exi _ F'P-i 'nF ~ a-exi ~ FPi ' (§73) Fig. 114 and therefore, by plane geometry, the angle F'PiF is bisected by NPi ; that is, in an ellipse the normal bisects the angle between the focal radii drawn to the point of contact. 192 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 102. Maxima and minima. The discussiou of § 61 applies here without change. Ex. 1. A lever with the fulcrum at one end A (fig. 115) is to be used to lift a jp weight w applied at a distance a from the fulcrum by means of a force applied at the other end B. The lever weighing n units per unit of length, required the length of the lever that the force required may be a minimum. Let X = AB, the length of the lever, 6 the angle it makes with the horizontal, and F the force applied at B. Then the weight of the lever is nx, and may be considered as applied at C, the middle point of AB. By the law of the lever. Fig. 115 Fx cos e = wa cos 6 + nxl-) cos 6, Then and Fz= wa X nx dF_ dx wa -I d^F 2wa dx2 X3 2ioa dF ^ , d^F . \l , — = and >0. \ n dx da;2 When X Therefore this is the required length. Ex. 2. Light travels from a point A in one medium to a point B in another, the two media being separated by a plane first medium is Vi and in the second v-z, required the path in order that the time of propagation from A to B shall be a minimum. It is evident that the path must lie in the plane through A and B perpendicular to the plane separating the two media, and that the path will be a straight line in each medium. We have, then, fig. 116, where j\fiV represents the intersection of the plane of the motion and the plane separating the two media, and ACB represents the path. If the velocity in the Fig. 110 MAXIMA AND MINIMA 193 Let MA = a, NB = b, MN = c, and MC = x. Then AC = VaF+1^^ and CB = V(c — a;)2 + 62, xhe time of propagation from J. to 5 is therefore _ Va2 + a-2 V(c - a;)2 + 62 , dt X c — X whence — = , ^ Vi Va2 + X2 1)2 V(C - X)2 + 62 d2< a2 62 and — = + dx^ t)i(a2 + x2) J »2 [(c - a;)2 + 6^]^ dH Since — is always positive, tlie time is a minimum wlien Vi VcC^ + X2 V2 V(C - X)2 + (1) This equation may be solved for x, but it is more instructive to proceed as follows : X MC Va2 + x2 --l^' = sin ^. c - X _ C'-ZV' _ : sin \j/. V(c - X)2 + 62 <^B sin Vi sin f V2 Then equation (1) is Now is the angle made by J. C with the normal at C and is called the angle of incidence, and xp is the angle made by CB with the normal at C and is called the angle of refraction. Hence the time of propagation is a minimum when the sine of the angle of incidence is to the sine of the angle of refraction as the velocity of the light in the first medium is to the velocity in the second medium. This is, in fact, the law according to which light is refracted. A case of a maximum or a minimum value sometimes occurs when the derivative is infinite and consequently discontinuous. Therefore the case is not included in the previous discussions. In practice the infinite values of the derivative may be examined by the rule of § 61. Ex. 3. 2/ = — ~, ~=- > 0. V3 dx.'^ Vs dx:^ 2a dx2 d^y n •* 2 a _|>0; If -— : ; and if x > 2, — ^ < 0. Hence the point for which x = 2 is a dx^ dx2 point of inflection, since on the left of that point the curve is concave upward and on the right of that point it is concave downward (fig. 118). The ordinate of this point isO. Fig. 118 104. Limit of ratio of arc to chord. The student is fa- miliar with the determination of the length of the circumference of a circle as the limit of the length of the perimeter of an inscribed regular polygon. So, in general, if the length of an arc of any curve is required, a broken line connecting the ends of the arc is constructed by drawing a series of chords to the curve as in fig. 119. Then the length of the curve is defined as the limit of the sum of the lengths of these chords as each approaches zero, and as their number there- fore increases without limit. The .B manner in which this limit is ob- tained is a question of the Integral Calculus, and will not be taken up here. We may use the definition, how- ever, to find the limit of the ratio of the length of an arc of any curve to the length of its chord, as the length of the arc approaches zero as a limit, i.e. as the ends of the arc approach each other along the curve. Accordingly, let P^ and P. (fig. 120) be any two points of a curve, i^T^ the chord joining them, and F^T and P^T the tangents to the curve at those points respectively. We assume that the arc P^I^ lies entirely on one side of the chord ^^, and is concave toward Fig. 119 196 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS the chord. These conditions can in general be met by taking the points J^ and i^ near enough together. Then it foUows from the definition that whence P^T+TP^ arc P^J^ PP > PP >1. Fig. 120 If TH is the perpendicular from T to P^P^, and if the angles P^P^T and -^i^T are denoted by a and yS respectively, then P^T = P^R sec a, and TP^ = RP^ sec ^ = (^^ - P^R) sec /S. .-. i^T + T^ = iji^ sec a; + (iji^-iji^) sec /3 = P^P^ sec yS + i^^ (sec a — sec /3), and P^T+TP^ _ P,P^ sec ^ + P^R {seca— sec /3) PP ~ PP 12 Pi? = sec yS + -'— (sec a — sec )Q). Now, as Pi and JP approach each other along the curve, a and ^ both approach zero as a limit, whence seca and secyS approach PR unity as a limit ; and since -^~ is always less than unity, it fol- -^■'2 PT+ IP lows that the limit of — r^ is unity. PP arc PP ^ Hence ^—^ lies between unity and a quantity approaching ?^. arc PP unity as a limit, and therefore the limit of — -, — ^ is unity, i.e. J\P, the limit of the ratio of an arc to its chord as the arc approaches zero as a limit is unity. 105. The derivatives -r- and -y- • On any given curve let the as as distance from some fixed initial point measured along the curve to any point P be denoted by s, where s is positive if P lies in one DIRECTION OF A CLTRVE 197 direction from the initial point, and negative if P lies in the opposite direction. The choice of the positive direction is purely arbitrary. We shall take as the posi- tive direction of the tangent that which shows the positive direction of the curve, and shall denote the angle between the positive direction of OX and the positive direction of the tan- gent by 4>. Now for a fixed curve and a fixed initial point, the position of a point P is determmed if s is given. Hence x and y, the coordinates of P, are func- tions of s, which in general are con- tinuous and may be differentiated. We will now show that dx ds = cos , arc PQ dx ds COS(f), dy . , -^ = sin 9. ds (1) 198 DIFFERENTIATIOK OF ALGEBRAIC FUNCTIONS From (1) we obtain by division dy ds dy tan

0, an increase of time corresponds to an increase of s ; while if v < 0, an increase of time causes a decrease of s. Consequently, the velocity is positive when the body moves in the direction in which s is measured, and negative if it moves in the opposite direction. Ex. 1. If a body is thrown up from tlie eartli with an initial velocity of 100 ft. per second, the space traversed, measured upward, is given by the equation s = 100t- IC) t'. Then • t> = ^ = 100 - 32«. , at When « < 3^, u >0 and when e > 3^, u <0. Hence the body rises for 3^ seconds, and then falls. The highest point reached is 100 (3^) - 16(3|)2 = 156^. Ex. 2. A man standing on a wharf 20 ft. above the water pulls in a rope attached to a boat at the uniform rate of 3 ft: per second. Required the velocity with which the boat approaches the wharf. 200 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS Let A (fig. 123) be the position of tlie man and C that of the boat. Let ^B = /i = 20, AC= s, and BC = x. dx We wisli to find dt Now therefore X = Vs'^ - 400 ; X s ds ds dt dt Vs-' - 400 '^t But, by hypothesis, s is decreasing at the rate of 3 ft. per second ; therefore = — 3, and tlie required expression for the velocity of the boat is 3s dx dt y/g2 _ 4y(J To express this in terms of the time we need to know the value of s when t = 0. Suppose this to be Sq ; then and s dx s,i — 3 1. 3s„ + 9« '^ Vs^j'-400-O.So« + 9«2 107. Components of velocity. When a body moves along its path, straight or curved, from P to ^ (fig. 124), where PQ = AS, X changes by an amount PR = A«, and y changes by an amount RQ = Ay. We now have Lim 'T~ — 'T-='^' = velocity of the body in its path. Ax dx Lim — - = -J- = v^ = component of ve- locity parallel to OA'. Lim — = J = ^y = component of ve- locity parallel to O Y. Fig. 124 Otherwise expressed, v represents the velocity of P, v^ the velocity of the projection of P upon OX, and v^ the velocity of the projection of P on Y. COMPONENTS OF VELOCITY 201 dx dt By (7), § 96, and (3), § 105, dx ds ds dt dsY dt)' V^ = V COS (f), dy _ dy ds dt ds dt dxV /dy'' "^ dt) \dt whence v^ = V sin ^, V- = V- + V' Ex. A man walks across the diameter of a circular courtyard at a uniform rate. A lamp, at one extremity of the diameter perpendicular to the one on which he walks, throws his shadow on the wall. Required the velocity of the shadow along the wall. In fig. 125 let L be the lamp, M the man, and S the shadow. Let a be the radius of the courtyard and c the uniform velocity of the man. Let the variable OM = Xi dX] where — = c. Then the equation dt of the line LS is ax — Xiij — axi = 0, and that of the circle is a;2 + ?/2 =: (• ". Solving these equations, we have, for the coordinates of S, 2a^x a-" Henc^- a2 + a;f y = a- + x{ a- — Xy and and dt dy di (a2 + x{f 2a^x? dx, ^ „ — = 2 a-c - ^ „. ., dt (a2 + x^y 4:a\ dxi ,, , -— = - 2a-c- 2axi v'i= { — ) + \dt (a^ + xf)- dt " ~ (a;- + x{r fc-» + 2 a2xf + x^ m-*^' (a2 + X,-)* 2 a-c The requii-ed velocity is ^ a -r X| The above solution can be simplified by the use of trigonometric functions. See Ex. 2, § 1<>3. 202 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 108. Acceleration and force. When the motion of a body is not uniform, the velocity at the end of an interval of time is not the same as at the beginning. Let v be the velocity at the begin- ning of the interval A<, and ?; + Ai; the velocity at the end. Then the limit of the ratio of the change in the velocity to the change in time, as the latter approaches zero as a limit, is called the acceleration in the path ; that is, if a denotes this acceleration, _dv _ d (ds\ _ d?s "'~dt~di\di/~df' When a is positive an increase of t corres'ponds to an increase of V. This happens when the body moves with increasing velocity in the direction in which s is measured, or with a decreasing velocity in the direction opposite to that in which s is measured. When a is negative an increase of t causes a decrease of v. This happens when the body moves with decreasing velocity in the direction in which s is measured, or with increasing velocity in the direction opposite to that in which s is measured. The force F which acts in the direction of the path of a moving body is measured by the product of the mass rii and the accelera- tiona. Thus ^^ ^^ . F — ma = m — = w — r • dt df From this it appears that a force is considered positive or nega- tive according as the acceleration it produces is positive or nega- tive. Hence a force is positive when it acts in the direction in which s is measured, and negative when it acts in the opposite direction. Ex. Let s = A + Bt+\ Ct\ Then v = B+Ct, a = C, and F= mC. If So and Vo denote the values of s and v when t = 0, -we have, from the last equations, So = ^, Vo = B, and the original equation may be vsritten S=so + vot+ ]^ai?. ILLUSTRATIONIS OF THE DEEIVATIVE 203 As a special case, suppose a body of mass m thrown vertically upward from a point h ft. above the surface of the earth with an initial velocity of Vq ft. per second. Then, if s is measured upward from the surface of the earth, we have 80 = A, F = — mg, a = — g, where g is the acceleration due to gravity. Then the expression becomes 109. Other illustrations of the derivative. 1. Bate of change. \i y = f{x), a change of Aa; units in the value of X causes a change of Ay units in the value of y. Then A?/ — ^ is the change in y per unit of change in x ; that is, the change in y which would be caused by the change of a unit in x, if A2/ were proportional to Aic. Passing to the limit, we have dv -— = rate of change of y with respect to x. For example, the velocity of a moving body is the rate of change of the space with respect to the time, and the acceleration is the rate of change of the velocity with respect to the time. 2. Momentum. The momentum of a moving body is the product of the mass and the velocity ; that is, if M is the momentum, M = mv. Now, from S 108, F = m -— = , ' = — — • ^ dt dt dt The force is therefore the derivative of the momentum with respect to the time, or, in other words, the rate of change of the momentum with respect to the time. 3. Kinetic energy. The kinetic energy of a moving body is equal to half the product of the mass into the square of the velocity; that is, if E is the kinetic energy, E = \ mv^. ^, dE d(lmv^) dv ds dv dv _, Then -^- = -^ ~ = mv-- = m — -— =m-- = F; ds ds ds dt as at that is, the force is the derivative of the kinetic energ}' with respect to the space traversed, or, in other words, the rate of change of the kinetic energy with respect to the space. 204 DIFFEKENTIATION OF ALGEBEAIC FUNCTIONS 4. Coefficient of expansion. Let a substance of volume ?? be at a temperature t. If the temperature is increased by A^, the pressure remaining constant, the vohnne is increased by A v. The change A?; per unit of volume is then — » and the ratio of this change per unit of volume to the change in the temperature is 1 Av V A^' The limit of this ratio is called the coefficient of expansion ; that is, the coefficient of expansion equals - -y- • In other words, the coefficient of expansion is the rate of change of a unit of volume with respect to the temperature. 5. Elasticity. Let a substance of volume v be under a pressure p. If the pressure is increased by A^, the volume is increased by Av — £^v. The change in volume per unit of volume is then V The ratio of this change per unit of volume to the change in the 1 Av pressure is — j and the limit of this' is called the compres- V Lp sibility ; that is, the compressibility is the rate of change of a unit volume with respect to the pressure. The reciprocal of the compressibihty is called the elasticity, wliich is therefore equal to ~ v — • dv Ex. For a perfect gas at constant temperature, P = -- Therefore the elasticity is dp I k\ k -v-- = -v(- ) = -=p; dv \ vV V that is, the elasticity of a perfect gas is equal to the pressure. G. Areas. Let2/=/(^)(fig.l26) Ite any curve, C a fixed point, and F{x, y) a variable point upon it. We shall assume that P lies at the right of C and that the por- tion of the curve between C and P lies above the axis of x. M ¥u.. 120 INTEGRATION 205 Draw the ordinates BC and MP and let A denote the area BMFC. Then ^ is a function of x, since it is determined when OM = a; is given. Give x an increment A^ = MN, and draw the ordinate NQ and the Unes BR and QS parallel to OX. Then BQ = Ai/, MNQP = A A, MNBP = MB • MN = y Ax, MNQS = NQ • MN = (y + Ay) Ax. But, from the figure, MNRB < MNQB < MNQS * ; that is, yAx which lies be- tween y and y + Ay, also approaches y ; that is, dA^ dx If the curve lies below the axis of x (fig. 127), and we place, as before, MNRB = y Ax and MNQS= {y + Ay) Ax, these areas are negative. We shall then have, as before, — dA dx = y, but the area is now considered as negative. 110. Integration. In many applications of the calculus the derivative is known, and the problem presents itself to find M N R Q Fio. 127 * If the curve runs down toward the right, the inequality signs will be reversed. 206 DIFFEEENTIATION OF ALGEBRAIC FUNCTI0:J^S the function which has that derivative. For example, it may be required to find a curve when its slope is known, or to find the space traversed by a particle with known velocity or acceleration, or to find the area bounded partly by a known curve, or to find a function which has a known rate of change. The process by which a function is found from its derivative is called iiitejration. Differentiation and integration are then inverse processes, as are addition and subtraction, multiplication and division, involution and evolution. The methods of integra- tion are in general complex and must be studied later in the integral calculus. At this time we shall give some simple exam- ples where the integration can be carried out by reversing the formulas of differentiation. In the first place, however, we must notice that the integration of a given function does not lead to a unique result. For, as we have seen already (§95), d {u + c) __ du dx dx where c is any constant whatever ; that is, two functions which differ hy an additive constant have the same derivative. Conversely, if two functions have the same derivative, they differ hy an additive constant. _- , , dv du For let -r = -r' dx dx Then dv du _ ^ dx dx or d{v-u) _Q dx Hence* V — M = c, where c = constant ; that is, V =ic + c. The constant c cannot be determined by integration, but must be fixed by the special conditions of the problem in which it occurs. * A proof of this conclusion will be given in the second volume. INTEGRATION 207 Ex. 1. Required the curve the slope of which at any point is twice the abscissa of the point. By hypothesis, dy dx Therefore y = x^ + c. (1) = 2x. Any curve whose equation can be derived from (1) by giving c a defi- nite value satisfies the condition of the problem. If it is required that the curve should pass through the point (2, 3), we have, from (1), 3 = 4 + c ; whence c = — 1, and therefore the equation of the curve is „, ^o . y = x^ — I. But if it is required that the curve should pass through (—3, 10), we have, from (1), 10 = 9 + c ; whence c — 1, and the equation is y = x^ + -i. Ex. 2. Required the space traversed by a particle if its velocity is equal to the square of the time. By hypothesis. Therefore V = — — t-. dt s- 1 1^ + c. The constant c can be determined if we Itnow the position of the particle at a given time. For instance, if when t — the particle is at the point from which 8 is measured, we must have c = 0. On the other hand, if wlien t = the particle is two units from the point at which s = 0, we have c = 2. Ex. 3. Required the space traversed by a body if the acceleration is propor- tional to the time. -nr 1- dv d^s ,. We have o = — = ■ — = kt, dt *2 where A; is a known constant. Then v = ds_l di~2 kt^ + ci. and S = -M^ + Cit + C2. The constants Ci and C2 can be determined if we know the position and the velocity of the body at a given time. If, for examole, we know that when < = 0, a = 0, and v = 4, we have cj = 0, ci = 4. 208 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS Ex. 4. Find the area bounded by the curve y = ^ (x^ — 3x'^ — Ox + 27), the axis of X, and the ordinates x = 4 and x = 5. If A is the area C DP M (fig. 129), where OC = 4 and OM = x, then (§ 109, 6) d A 1 — = -(x3-3x2-9x + 27). whence (1) Therefore 1 /x* 9 8\4 2 x) / 2 Fig. 129 If X = 4, MP coincides with CD and therefore J. = 0. Substituting in (1) the corresponding values x = 4, yl = 0, we find c = — |. 9 2' If X = 5, ^ = CDEF. Hence CZ>J?F = ^ (fi-l A _ 125 - 2f A + 135) - I = 2/j. Ex. 6. Find the area bounded by the axis ofx and tlie portion of the curve y = ^ (x^ — 3 x2 — 9 X + 27) between x = — 3 and x = 3. We now let A = the area 6?iVQ (fig. 129). Then, as before. dA_l dx ~8 (x3 -3x2- 9x + 27). 1 /x* 9 \ ^ = - I x3 - - x2 + 27 X ) + 8\4 2 / When X = - 3, ^ = ; therefore c = 2^^ , and A = -(-~x^-~x'^ + 21x] + ^^'^ 8\4 2 ' 32 Placing X = 3, we have area GQH — 13 J^. PROBLEMS 209 PROBLEMS dy Find -^ in each of the following cases : dx 1. 2/ = (3a; + l)(x2 + 2x + 1). 16. y 2. y = (3x2 + 6x + 1) (5x2 + lOx + 5). x^ + x2 + 1 x + a Vx2 + 1 4. y: 5. y 6. y x + a x3 + l X3-1 2x2- 4x + 3 3x2- 6x + 1 x^ — x- 2 + X- 1 18. 2/ = (2x -3)2(x + l)3. 19. 2/ = (3x - 5)2(x2 - 5x + 1). 20. 2/ = (x + l)Va;2- 1. 21. 1/ = (x2 - 4x + 3)'^(x3 + 1)3. 2.2. y = Vx + 1 + Vx - 1. a;'*-l 23. 2/ = x + Vx2 + 1. 2 7. 2/ = 2xi + 8x^-- + -.. 2L y=W^^Tl + -^. V3x2 + 1 8. z/ = 4x2-0x + ^-|. 25. 2/=^(|-J, 9. 7/ = V^-J-. „„ Vx2 + 1 Vx 26. y = — • ^ X X — 1 10. y = ^X2 — -^X + = =:• „„ (X2 + 1)5 Vx ^x2 27. y = ^^ ^, ^X vx (X3 + 1)* 11. y = (3x2 -ox + 6)2. ^ 12. 2/ = (x2 + l)3. 28. 2/ 13. ?/ = V4x2 + 5x-6. -Q 3/ — ^~ / 14. y = Vx2 + X - 1. ^02 - x2 15. 2/^ -J—. 30.2/ X + Vl + X2 X x2 + 1 * x + Va2 + x2 Find — from each of the following equations : dx 31. X* - 4x22/2 + 2/3 = 0. 34. x5 + 2/* - x' - 2/ = 0. 32. x5 - 2/5 - x3 + 2/ = 0. 35. (X + 2/)^ + (X - 2/)^ = a. 33. x«2/* + (X - yY = 0. 36. y"- = ^-^ • X y Find ^ and —^ from each of the following equations : dx dx2 37. 5 x2 + 2 2/2 = 10. 40. 2/' = ct^ (X + 2/). 38. x^ + y' = a^. 41_ 2/' + 2/ = x^. 39. t^ + ^ = 1 ?! 4. tL_ a2 62 "• 42. 2/' -2x3 + 4x2/ = 0. 210 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 43. Find the tangent and the normal to the parabola y^ — 4y — 6x — 9 = at a point the abscissa of which is — 2. 44. Find the equations of the tangent and the normal to the circle x^ + y^-iz + 6y-2i = at the point (1, 3). 45. Find the equation of the tangent.to the witch y = at the point for which x = 1. 46. Find the tangent to the curve x^ — y^ + x^ — y = at the point the abscissa of which is 1. 47. Find the tangent to the curve x^y + x^ — x^ + y = at the point the abscissa of which is 1. 48. Find the equation of the tangent to the curve y^ — xy — a = at the point {xi, Vi). 49. Find the equation of the tangent to the curve x = y^ + 1 &t the point (xi, yi). 50. Find the equation of the tangent to the curve y- = x^ at the point (xi, yi). 51. Find the equations of the tangent and the normal to the curve y — x -i — at the point (xi, yi). 52. Find the equation of the tangent to the curve Vx + Vy = Va at the point (Xi, yi). 53. Find the equation of the tangent to the curve x^ + y^ — a* at the point (xi, yi)- 54. Find the tangent and the normal to the ellipse 3x^ + 5y^ = lo at the upper end of the ordinate through the right-hand focus. 55. Find the equations of the tangent and the normal to the hyperbola 4 x^ — 2/2 — 12 at a point the abscissa of v/hich is equal to its ordinate. 56. Find in terms of x, y, and — the projections upon OX of the portions dx of the tangent and the normal between the point of contact and OX. These are called the suhtangent and the subnormal. dy 57. Find in terms of x, y, and — the lengths of the portions of the tangent dx included between the point of contact and the coordinate axes. 58. Prove that a normal to an hyjjerbola makes equal angles with the focal radii drawn to the point where the normal intersects the hyperbola. 59. Prove that a normal to a parabola makes equal angles with the axis of the parabola and the line drawn from the focus to the point where the normal intersects the parabola. 60. Show that for an ellipse the segments of the normal between the point of the curve at which the normal is drawn and the axes are in the ratio a^ : b"^. 61. Find the point at which the tangent to the curve x^ — xy — 1 = has the slope 2. PROBLEMS 211 62. Find the coordinates of a point on the ellipse 1- — = 1 such that the tangent there is parallel to the line joining the positive extremities of the major a,nd the minor axes. 63. Find a point on the ellipse -^ + ^ = 1 such that the tangent there is equally inclined to the two axes. 64. Prove that the portion of a tangent to an hyperbola included by the asymptotes is bisected by the point of tangency. 65. If any number of hyperbolas have the same transverse axis, show that tangents to the hyperbolas at points having the same abscissa all pass through the same point oh the transverse axis. 66. If a tangent to an hyperbola is intersected by the tangents at the verti- ces in the points Q and B, show that the circle described on QR as a diameter passes through the foci. 67. Prove that the ordinate of the point of intersection of two tangents to a parabola is the arithmetical mean between the ordinates of the points of con- tact of the tangents. 68. If P, Q, and R are three points on a parabola, the ordinates of which are in geometrical progression, show that the tangents at P and R meet on the ordinate of Q. 69. Show that the tangents at the extremities of the latus rectum* of a parabola are perpendicular to each other. 70. Prove that the tangents at the ends of the latus rectum of a parabola intersect on the directrix. 71. Prove analytically that if the normals at all points of an ellipse pass through the center, the ellipse is a circle. 72. Prove that the tangent at any point of the parabola y'^='ipx will meet the directrix and the latus rectum produced in two points equidistant from the focus. 73. Find the length of the perpendicular from the focus of the parabola y2 = 4px to the tangent at any point (xi, j/i), in terms of Xi and p. 74. If perpendiculars are let fall on any tangent to a parabola from two given points on the axis which are equidistant from the focus, prove that the difference of their squares is constant. 75. Show that the product of the perpendiculars from the foci of an ellipse upon any tangent equals the square of half the minor axis. 76. Find the equation and the length of the perpendicular from the center to any tangent to the ellipse 1- — = 1. 77. At what angles t do the loci 2/2-4a;-|-4=Oand2/-a; + l = intersect ? *The latus rectum of a conic is the chord through the focus perpendicular to the axis. fThe angle between two curves is the angle between their tangents at their point of intersection. 212 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 78. Find the angle between the straight line y — 2x — 2 and the cissoid X {x^ + y''^) — i y'^ at each of their points of intersection. 79. At what angle do the circles x^ -hy- -Q = 0, x^ + y^ — iix-6y + 9 = intersect ? 80. Prove that the center of each of the circles x^ + y^ = a2 and x^ + y^-2ax = is a point of the other, and find the angle at which they intersect. 81. At what angle do the circle x'^ + y^ = 21 and the parabola y'^ = ix inter- sect each other ? 82. Show that the curves \- — = 1 and ^ = 1 cut each other at right angles and are confocal. ' 83. Prove that an ellipse and an hyperbola with the same foci cut each other at right angles. 84. If two concentric equilateral hyperbolas are described, the axes of one being the asymptotes of the other, show that they intei-sect at right angles. 85. Find the angle between the parabolas y"^ — iax and x^ = 4ay at each of their points of intersection. 86. Find the angle between the parabola x" = iay and the witch y = — at each of their points of intersection. "^ 87. Prove that the cissoid y^ = and the parabola y^ = 4ax intersect at right angles at the origin. "" 88. Find the angles of intersection of the cissoid y- = and the circle x2 + y2_4ax = 0. 2a -X 89. Find the angle of intersection of the witch 8 a^ 4 7/8 y = — ^ and the -cissoid x^ — x'^ + ia^ 5 a — 4 y 90. Find the angles of intersection of the circle x^ + y- = a a^ and the witch 8a3 y «2 + 4 a^ 91. Find the angle between the strophoid y = ±x\ and the circle 0,09 \ a + X 92. Find the angles of intersection of the curves 2/2 = 2 ax and x^ + y^ — S axy = 0. 93. It is required to fence off a rectangular piece of ground to contain a given area, one side to be bounded by a wall already constructed. Required the dimensions of the rectangle which will require the least amount of fencing. 94. A man on one side of a river, the banks of which are assumed to be parallel straight lines i mi. apart, wishes to reach a point on the opposite side of the river and 3 mi. further along the bank. If he can walk 4 mi. an hour and swim 2 mi. an hour, find the route he should take to make the trip in the least time. PKOBLEMS 213 95. A rectangular piece of land to contain 96 sq. rd. is to be inclosed by a fence and divided into two equal parts by a fence parallel to one of the sides. What must be the dimensions of the rectangle that the least amount of fence may be required ? 96. What are the dimensions of the rectangular beam of greatest volume that can be cut from a log a ft. in diameter and b ft. long, assuming the log to be a circular cylinder ? 97. The hypotenuse of a right triangle is given. How shall the sides be chosen so that the area shall be a maximum ? 98. Two towns A and B are situated respectively 2 mi. and 3 mi. back from a straight river from which they are to get their water supply, both from the same pumping .station. At what point on the bank of the river should the station be placed, that the least amount of piping may be required, if the neai'est points of the river to A and B respectively are 10 mi. apart? 99. AB and CD are two parallel lines distant b units apart. A transversal BF is drawn, intersecting the transversal AD at E. For what position of F is the sum of the ai'eas of the two triangles AEB and FED a minimum ? 100. A right cone is generated by revolving an isosceles triangle of constant perimeter about its altitude. Show that the cone of greatest volume will be obtained when the length of the side of the triangle is three fourths the length of the base. 101. Into a full conical wine glass whose depth is a and angle at the base is 2 a there is carefully dropped a spherical ball of such size as to cause the greatest overflow. Show that the radius of the ball is a sin a sin a + cos 2 a 102. Two ships are sailing uniformly with velocities w, v along lines inclined at an- angle 6. Given that at a certain time the ships are distant respectively a and 6 from the point of intersection of their courses, show that the least dis- tance between the ships is (av — 6(t)sin (u2 + v"^ -2uv cos 6)^ 103. Find the least ellipse which can be described about a given rectangle, assuming that the area of an ellipse with semiaxes a and b is -n-ab. 104. Find what sector must be taken out of a given circle in order that it may form the curved surface of a cone of maximum volume. 105. The stiffness of a rectangular beam varies as the product of the breadth and the cube of the depth. Find the dimensions of the stiffest rectangular beam that can be cut from a circular cylindrical log of radius a in. 106. Tlie strength of a rectangular beam varies as the product of its breadth and the .s &, e = ■ ; and the foci of the ellipse a are at (x-' = ± ae, y' = 0), or, what is the same thing, (a; = ± ae + x^, y = y^). The directrices are x' = ± - > or x = x^ ::r -■ In a similar manner 220 CHANGE OF COORDINATE AXES is the equation of an hyperbola with its center at {x^^ y^ and its axes parallel to OAT and OY; and represents a parabola with its vertex at {x^, y^ and its axis parallel to OX. Any equation which can be reduced to a form similar to one of these can be discussed in a similar manner. A general treat- ment of such equations will be found in Chap. XI. We shall give here some examples. Ex.1. 16x2 + 252/2 + 64a; -1502/ -111 = 0. Rewriting, we have 16(a;2 + 4a;)+ 25(2/2 _ 6 y) = 111, whence 16(x2 + 4x + 4) + 25(2/2 - 62/ + 9) = 400, or (x + 2)2 (2/-3)2 ^^ 25 16 Placing now x = — 2 + x', y — Z -\- y', we have \-^— = \. 25 16 This is an ellipse with semiaxes 5 and 4, and eccentricity |. Its center is at (x' = 0, 2/' = 0), its foci are at (x' = ± 3, y' = 0), and its directrices are x' = ± '^^- = ±83- Hence the original equation represents an ellipse with semiaxes 5, 4, and eccentricity 3. Its center is at (— 2, 3), its foci are (—5, 3) and (1, 3), and its directrices are x = — 10^ and x = 6^. Ex.2. 5^2 _ 102/ - 4x -7 = 0. Rewriting, we have 6 (2/2 — 2 ?/) = 4 (x + |), 5(2/2_2y + l) = 4(x + |+|), or (2/ - 1)2 = 4 (X + 3). Placing now x = — 3 + x', 2/ = 1 + 2/', we have 7/2 = 4 x'. which represents a parabola with vertex (x'= 0, 2/'= 0). Its axis is along (YX'; its focus is (x' = ^, 2/' = 0), and its directrix is x' = - 1. Therefore the original equation represents a parabola with its vertex at (—3, 1) and its axis parallel to OX. Its focus is (—24, 1) and its directrix is a; = - 3.L CHANGE OF DIRECTION OF AXES 221 Ex. 3. (X - C)2 + 2/2 - e2a;-2. This is the equation of tlie conic, as found in § 81. We may write it as (1 - e2)a;2 _ 2 ex + 2/2 = - A Then if e ^i 1, we may proceed as follows: (1 - e2) (x^ - -l^x + ] + 2/2 ^ _ c2 + _£!_ , (1 - e2) (x —Y+ 2/2 = -^ , ^V l-e2/^^ l-e2 2/-* + — ^ — = 1. c2e2 (1 - e2)2 1 _ e2 (<2g2 We may now place = a^, (1 - e2)2 = a2(l-e2) = ±62^ 1 - e2 and 1 - e2 e the sign of 6 being ± 1 according as e < 1. The equation is then a2 62 The equation accordingly represents an ellipse or an hyperbola with center at 0,0). If e = 1, the equation (x — c)2 + y^ = ^x^ becomes y^ = 2cx - c* = 2 c (x 1 , which represents a parabola with the vertex at /- , 0| . 115. Change of direction of axes without change of origin. Case I. Rotation of axes. Let OX and OY (fig. 131) be the original axes, and OX' and Y' be the new axes, making Z with OX and Oy respectively. Then ZXOY' = 90° + , and ZYOX' ^90° -(f>. 222 CHANGE OF COOKDl^ATE AXE« Let P be any point in the plane, its coordinates being x and y with respect to OX and OY, and x' and y' with respect to OA"' and F'. Then by construction OM = x, 0N= y, OM' = x', and J/'P = y'. Draw OP. The projection of OP on OX is OM, and the projection of the broken line OM'P on OX is OM' cos(f> + i»/'P cos (90° + ) -X or OM' cos ^ — 3f'Psin^. .-. 0Jf=01f' cos (^-J/'P sin (^, (1) by § 15. In like manner the projection of OF on OY is ON, and the projection of the broken line OM'P on OF is Olf' cos (90° — <^) + J/'Pcos~ y' sin ^, y = a;' sin^ + y' cos^. Ex. 1. Transform the equation xy = 5 to new axes, having the same origin and making an angle of 45° with the original axes. x' Here

— y' sin 0, y = x' sin + y' cos 0, where is to be determined. Substituting in the equation and collecting like terms, we have (34 cos2 + 4 1 sin2 — 24 sin cos 0) x^ + (34 sin2 + 41 cos2 + 24 sin cos 0) t/2 + (24 sin2 + 14 sin cos - 24 cos2 0)xj/ = 100. By the conditions of the problem we are to choose so that 24 sin2 + 14 sin cos - 24 cos2 = 0. OBLIQUE COOKDINATES 223 One value of satisfying this equation is tan-i |. Accordingly we substitute sin^ = I and cos^ = |, when the equation reduces to x^ + 2y^ = 4, which is the equation of an ellipse. Case II. Interchange of axes. If the axes of x and y are simply interchanged, their directions are changed, and hence such a trans- formation is of the type imder consideration in this article. The formulas for such a transformation are evidently x = y', y = x'. Case III. Rotation and interchange of axes. Finally, if the axes are rotated through an angle <^ and then interchanged, the formulas, being merely a combination of the two already found, are x = y' cos — x' sin0, y = y' sin^ + x' cos^. A special case of some importance occurs when cf)= 270°. We have then x = x', y = — y'. Cases II and III, it should be added, occur much less frequently than Case I. In case both the origin and the direction of the axes are to be changed, the processes may evidently be performed successively, preferably in this order: (1) change of origin; (2) change of direction. 116. Oblique coordinates. Up to the present time we have always constructed the coordinate axes at right angles to each other. This is not necessary, however, and in some problems, indeed, it is of advantage to make the axes intersect at some other angle. Accordingly, in fig. 132, let OX and OY intersect at some angle &> other than 90°. We now define x for any point in the plane as the distance from OY to the point, measured parallel to OX; and y as the distance from OX to the point, measured parallel to OY. The algebraic signs are determined according to the same rules as were adopted in § 16. It is immediately evident that the rectangular coordinates are but a special case of this new type of coordinates, called oUique 224 CHANGE OF COORDINATE AXES coordinates, since the new definitions of x and y include those previously given. In fact, the term Cartesian or rectilinear co- ordinates includes both the rectangular and the oblique. Oblique coordinates are usually less convenient than the rectan- gular, and are very little used in this book. If necessary, the formulas obtained by using rectangular coordinates can be trans- formed into similar ones in oblique coordinates by the formulas of the following article. When no angle is specified the angle between the axes is understood to be a risht angle. 117. Change from rectangular to oblique axes without change of origin. Let OTand OY (fig. 133) be the original axes at right angles to each other, and OX^ and OY' the new axes, making angles <^ and (/>' , respectively with OX. n Then co = cf)' — ^. Let P be any point in the plane, X its rectangular coordinates being x and y, and its ob- lique coordinates being x' and y'. Draw PM parallel to OY, PM' parallel to OY', M'N parallel to OY, and RM'N' parallel to OX. Then Z RM'P = <\>'. But OM =0N + NM =0N+ M'N' = 031' cos ', MP = MN' + N'P = NM' + N'P = OM' sm <^ -F M'P sin <^'. .*• x= x' cos-{- y' cos(f>', y = x' sin + y' sin (f>'. Ex. Transfonn the hyperbola x-* a2 62 = 1 to its asymptotes as axes. Since the equations of the asymptotes are 2/ = ± - x, = tan- ' / — 1 , and (f/ = tan-i - , if we choose to have the hyperbola lie in the first and the third quadrants with respect to the new axes. The formulas of transformation become b Va-2 + 62 {X' + y'), y = Va- + 62 i-x' + y'). Substituting and simplifying, we have as the new equation xy Unless b — a, the axes are obli(iue and w = 2 taii^' - • a a2 + 62 PROBLEMS 225 118. Degree of the transformed equation. In reviewing this chapter we see that the expressions fur the original coordinates in terms of the new are all of the first degree. Hence the result of an}' transformation cannot be of higher degree than that of the origmal equation. On the other hand, the result cannot be of lower degree than that of the original equation ; for it is evident that if any equation is transformed to new axes and then back to the original axes, it must resume its original form exactly. Hence if the degree had been lowered by the first transformation, it must be increased to its original value by the second transformation. But this is impossible, as we have just noted. It follows that the degree of an equation is unchanged by any single transformation of coordinates, or by any number of succes- sive transformations. In particular, the proposition that any equa- tion of the first degree represents a straight line is true for oblique as for rectangular coordinates. PROBLEMS 1. What are the new coordinates of the points (2, 3), {— 4, 5), and (5, — 7) if the origin is transferred to the point (.3, — 2), the new axes being parallel to the old ? 2. Transform the equation x^ + Ay^ — 2x + 8y + l = 0to new axes parallel to the old axes and meeting at the point (1, — 1) with respect to the old axes. 3. Transform the equation ?/3 - 6?/2 + 3a;2 + 12 ?/ - 18x + 35 = to new axes parallel to the original axes and meeting at (2, — 3) with respect to the original axes. 4. Find the equation of the ellipse when the origin is taken at the lower extremity of the minor axis, and the minor axis is the axis of y. 5. Find the equation of the ellipse when the origin is at the left-hand vertex, the major axis lying along OX. 6. Find the equation of the hyperbola when the origin is at the left-hand vertex, the transverse axis lying along OX. 7. Find the equation of the strophoid when the origin is at A (fig. 92), the axes being parallel to those of § 84. 8. Find the equation of the strophoid when the asymptote is the axis of y, the axis of x being as in § 84. 9. Find the equation of the witch (fig. 90) when LK is the axis of x and OA the axis of y. 226 CHANGE OF COORDINATE AXES 10. Find the equation of tlie witch when the origin is taken at the center of the circle used in constructing it, the axes being parallel to those of § 82. 11. Find the equation of the cissoid when its asymptote is the axis of y and its axis is the axis of x. 12. Find the equation of the cissoid when the origin is at the center of the circle used in its definition, the direction of the axes being as in § 83. 13. Find the equation of the parabola when the origin is at t'.io fo -as and the axis of x is the axis of the curve. 14. Find the equation of the parabola when the axis of the curve and the directrix are taken as the axes of x and y respectively. 15. Transform ?/2 — 8x — 10 2/ + 1 = to new axes parallel to the old, so choosing the origin that the new equation shall contain only terms in y^ and x. 16. Transform the equation 12 x^ + 18 y^ _ 12 x + 12 ?/ — 31 = to new axes parallel to the old, so choosing the origin that there shall be no terms of the first degree in the new equation. 17. Show that any equation of the form xy + ax + 6j/ + c = can always be reduced to the form xy = k hj choosing new axes parallel to the old, and determine the value of A;. 18. Show that the equation ax^ + by^ + ex + dy + e = (a ?i 0, hjtiQi) can always be put in the form ax^ + hy'^ = fc by choosing new axes parallel to the old, and determine the value of k. 19. Show that the equation y^ + ay + bx + c = Q (^ ?^ 0) can always be reduced to the form y^ + 6x = by choosing new axes parallel to the given ones. 20. Find the equation of an ellipse if its axes are 6 and 2, its center is at (—3, 2), and its major axis is parallel to OX. 21. Find the equation of an ellip.se if its axes are a and ^, its center is at (— 2, — 3), and its major axis is parallel to OX. 22. Find the equation of an hyperbola if its transverse axis is 4, its conju- gate axis 2, its center at (1, — 2), and its transverse axis parallel to OX. 23. Find the equation of an hyperbola if its transverse axis is v2, its con- jugate axis V §, its center at (2, 3), and its transverse axis parallel to OX. 24. The vertex of a parabola is at (3, — 2) and its focus is at (5, — 2). Find its equation. 25. The vertex of a parabola is at (4, 5) and its focus is at (4, 1). Find its equation. 26. The center of an ellipse is at the point (2, 3), its eccentricity is ^, and the length of its major axis, which is parallel to the axis of x, is 10. What is the equation of the ellipse ? 27. Find the equation of an ellipse when the vertices are (—2, 0), (4, 0), and one focus is at the origin. PROBLEMS 227 28. The center of an hyperbola is at (- 1, - 2), its eccentricity is 1^, and its transverse axis, which is parallel to OX, is 4. Find its equation. 29. The vertex of a parabola is at the point (- 4, - 2), and it passes through the origin of coordinates. Find its equation, its axis being parallel to OX. 30. Given the ellipse 4x^ + 9y^ + 8x - S6y + i -0; find its eccentricity, center, vertices, foci, and directrices. 31. Given the ellipse 3a;2 + 5y2 + i8 x - 20y + 32 = ; find its eccentricity, center, vertices, foci, and directrices. 32. Given the hyperbola 9 x^ — 4 y^ — 6ix — 32 y — 19 = ; find its eccen- tricity, center, vertices, foci, directrices, and asymptotes. 33. Given the hyperbola 3x2 — 2 2/2 ^ 6x + 8y— 11 = 0; find its eccentricity, center, vertices, foci, directrices, and asymptotes. 34. Given the parabola 72x2 + 48x + 180y - 37 = 0; find its vertex, focus, axis, and directrix. 35. Given the parabola y^ — 5x + 6y — 1 — 0; find its vertex, focus, axis, and directrix. 36. What are the coordinates of the points (0, 1), (1, 0), (1, 1) if the axes are rotated through an angle of 60° ? 37. Transform the equation 3x2 + 3?/2 — lOxy + 8 = to a new set of axes by rotating the original axes through an angle of 45°, the origin not being changed. 38. Find the equation of the folium x^ + y^ — 3 axy = after the axes have been rotated through an angle of 46°. 39. By rotating the axes through an angle of 45° and changing the origin, prove that the curve x' + y^ = «^ is a parabola. 40. Transform 6x2 — 12xy + 10?/2 _ 14 = o to a new set of axes, making an angle tan-i | with the origiual'set. 41. Show that the equation x2 + ?/2 = a^ will be unchanged by transforma^ tion to any pair of rectangular ^.xes, if the origin is unchanged. 42. Transform the equation x"^ - y^ = 36 to new axes bisecting the angles between the original axes. 43. Transform the equation ix'^ - 3xy + 8y^ = I to one which has no xy-term, by rotating the axes through the proper angle. 44. By rotating the axes through the proper angle transform the equation 3 x2 + 2 Vs xy + ?/■- + 2 X - 2 V3 ?/ = to another which shall have no term in xy. 45. Transform the equation x2 _ 6 2/2 _ 6 V3 xy + [2 + 12 V3] X + [20 - 6 V3] y - 15 + 12 V3 = to a new set of rectangular axes making an angle of 60° with the original axes and intersecting at the point (-1, 2) with respect to the original axes. 228 CHANGE OF COORDINATE AXES 46. Transform the equation ix- + 9y^ = 36 from rectangular axes to oblique axes with the same origin, making angles tan-i^ and tan-i(— J) respectively with OX. 47. Find the equation of the hyperbola Sx^ — 4y^ = 12 referred to its asymp- totes as coordinate axes. 48. Show that the lines y = ±x intersect the strophoid at the origin only, and find the equation of the curve referred to these lines as axes. 49. Transform the equation 2x^ — Sy^ = Q from rectangular axes to oblique axes having the same origin and making the angles tan-' ^ and tan-' ^ respec- tively with OX. 50. Prove that the formulas for transposing from a set of rectangular axes to a set of oblique axes having the same origin and the same axis of x are x = x' + y' cos w, y = y' sin w, where w is the angle between the oblique axes. 51. By transforming the equation y = mx + 6 by the formulas of example 60, show that the equation of a straight line in oblique coordinates is sind> sin(« — ) where u is the angle between OX and OY, the angle between the line and OX, and c the intercept on OY. 52. Derive the result of example 51 directly by use of the trigonometric formulas connecting the sides and the angles of an oblique triangle. 53. By use of the transformation of example 60, prove that the equation of a circle in oblique coordinates is (X - d)2 + (y _ e)2 4- 2 (X - d) (y - e)cos u = r^, where w is the angle between the axes, and (d, e) is the center. 54. Obtain the result of example 53 directly by use of the trigonometric relations connecting the sides and the angles -of an oblique triangle. CHAPTEE XI THE GENERAL EQUATION OF THE SECOND DEGREE 119. Introduction. The most general equation of the second degree is of the form Ax'+ 2 Hxi/ + Bf-{-2Gx+2Fy+C=0, where the coefficients may have any values, including zero, except that A, B, and H cannot be zero together. We shall proceed to show that this equation always represents an ellipse, an hyperbola, a parabola, or a limiting case of one of these, if it represents any curve, and shall derive criteria by which the nature of tlie curve can be readily determined. 120. Removal of the xy-t&rm. Let us make a transformation of coordinates to new rectangular axes, making an angle ^ with the original ones, the origin being unchanged. The formulas of transformation are (§ 115) x = x' cos 4> — l/' sin (f), y = x' sincf) + y' cos (f>. Substituting, we have A'x''' + 2 H'x'y' + B'y'^ + 2 G'x' + 2 F'y' + C" = 0, where A' = A cos^ ^ + 2H sin (f> cos, H' = (B — A) sin (}> cos (f)-\-H (cos^<^ — sin^-(? sine/), and C'=C. 229 230 GENERAL EQUATION OF SECOND DEGREE We may now determine - sm^cfy) = 0. This equation is equivalent to 2 ^ cos 2 <^ + (5 - ^) sin 2 <^ = 0, 2ff whence tan 2(f> = A-B 2H or ^ = J tan ' A-B To compute the values of A^ and B\ we have A! =A cos^+ 2Hsia. +B sia^ .l+cos2<^ . 1— cos2<^ = A -+B:sin2(t>+B = ^[^+5 + (^-i?)cos2<^4- 2^sin20]. 2H But, since tan 2 ^ = A-B sin 2 <^ = ± "' > cos 2 <^ = ± and therefore A! ^Wa-^B± ( ^-^f+4^ 1 2L V(^-J5f+4^'J = 1 [^ + j5 ± V(^-^f+4^']. Sunilarly, 5' = ^ [^ 4- -S :f ^ {A-Bf^- A.H''\ From these results it follows that A'B' = AB-H^. Hence if AB — H^ is positive, A' and ^' have the same sign ; if AB — H^ is negative, ^' and B' have opposite signs ; if AB — ^^ is zero, either A' or B' is zero. EQUATION WITHOUT THE xy-TERM 231 The discussion of the general equation is then reduced to that of the simpler equation This equation we will consider in the next two articles, dropping the primes for convenience. 121. The equation Ax"" + By" -\- 2 Gx + 2 Fy-\- C = 0. We shall prove the theorem : The equation Aaf + Bif+2Gx+2Fy + C=0, where the coefficients are such that AF^' + BG^-ABC^^ 0, represents a conic, if it represents any curve at all. In particular, (1) when A and B have the same sign, it represents an ellipse * or no curve ; (2) when A and B have opposite signs, it represents an hyperhola ; (3) when either A or B is zero, it represents a parabola. Suppose first that neither ^ nor 5 is zero. Then the equation may be rearranged as follows : A(cc'+2~x\+B(f+2^y) = -C. We may then complete the squares of the expressions in the parentheses; thus, Aj Y BJ AB * The circle is considered a special case of an ellipse (see § 75). 232 GENERAL EQUATION OF SECOND DEGREE Since AF'^ + BG^ — ABC is not zero, we may divide by the right- hand member of the equation, obtaining M N where, for convenience, we place AF' + BG'^ — ABC M = N = A'B AF^ + BG--ABC AB- We may now transfer the origin of coordinates to the point (C F\ > — - b the new axes remaining parallel to the old, by the formulas ^ ^ A ' ^ B ^ The equation is then \-^ = \. ^ M N Now if A and B have the same sign, M and N wUl have the same sign. If this sign is positive, we may place M= a^, N= V^, and the equation is d' ^ h' ' which represents an ellipse. The axes of the ellipse are parallel to the original coordinate (C ft'X J — - I referred to the original axes, li A=B, the ellipse is a circle. If M and N are both negative, the equation ^ + ^' = 1 M N can be satisfied by no real values of x and y. EQUATION WITHOUT THE xy-TERM 233 If A and B have opposite signs, M and N have opposite signs, and we may place either M= a^, iV = — h'\ or J/ = — a~, JV = V^, thus obtaining either ,., ,., ^- _ /; ^ a' J)' ' -^ + 1^ = 1' either of which represents an hyperbola. The axes of the hyperbola are parallel to the original coordinate axes, and its center is at the point ( > ^ | referred to the • 1 V ^ ^/ origmai axes. ^ ^ The first and the second parts of the theorem are therefore proved. Consider now the case in wliich either tI or ^ is zero. If, for example, A = 0, B ^ 0, the equation is Bf+2Gx-\-2Fij + C=0, and the condition to be fulfilled by the coefficients is BG^ ^ 0, which is equivalent to G "^ 0, since B cannot be zero. We may arrange the equation as follows : 1^+2— y=— 2— X ^ B^ B B Completing the square, we have F\^ 2GI . C F^ y + -^)=--^{^ + ^^- Bj B\ 2G 2 GB If now we transform to a new origin by placing X— 1 \- x', y = — — + y> 2G 2GB ^ B ^' we have ■y'^ = x', which is the equation of a parabola. Similarly, if ^ = but A^ ?/= ■• ^ A "^ B Hence in this case the equation represents a point. This may be considered the limiting case of the ellipse. 2. A and B have opposite signs. We may put the equation in the form , ^2 / r^\2 ^(. + _') + 5(,+ _) = 0, or Ax"' + By"' = 0. THE DETERMINANT AB - H^ 235 Since A and B have opposite signs, we will consider A as posi- tive and B as negative. The equation can then be separated into two real factors {VAx' + y/'^y') {VAx' ~ V^y') = 0. Consequently the equation represents the two straight lines inter- sectmg in the point ic' = 0, y' = 0, or a? = > 2/ = This may be considered the limiting case of the hyperbola, 3. One of the coefficients ^ or -B is zero. For example, let ^ = 0, ^ ^ 0. Then the condition AF'' + BG^-ABC=0 becomes G = 0. Hence the equation is Bf+2Fi/-{-e=0. This may be factored into B{y-y^{y-y^) = ^, and accordingly represents either two parallel straight lines, two coincident straight lines, or no real locus, according as y^ and y^ are real and unequal, real and equal, or imaginary. This is considered a limiting case of the parabola. 123. The determinant AB — //'. Returning now to the gen- eral equation of the second degree, Ax^+2Hxy-{-Bf+2Gx+2Fy + C=0, and remembering that if it is reduced to the form ^ V + B'y"" + 2 G'x' + 2 F'y' + C' = Oy we have AB-H^'^A' B', we may state the foUo^snng theorem : The equation Ac(^+2JH'xy+By^+2Gx+2Fy-\-C=0 always represents a conic or one of the limiting cases, if it repre- sents any curve at all. 286 GENERAL EQUATION OF SECOND DEGREE 1. If AB—H' > 0, the equation represents an ellipse, a point, or no curiae. 2. If AB — H^ < 0, the equation represents an hyperbola or two intersecting straight lines. 3. If AB —H^ = 0, the equation represents a parabola, two par- allel lines, two coincident lines, or no curve. 124. The discriminant of the general equation. We have seen in § 122 that A'x''' + B'y'^ + 2 G'x' + 2 F'y' + C = (1) represents one of the limiting cases of the conic sections when A'F'^ + B'G''-A'B'C' = 0. It is useful to have this condition in terms of the coefhcients of the general equation Ax'-^2Hxy + By''^-2Gx + 2Fy-\-C=0. (2) This might be done by substituting for A', B', G' , F', and C the values given in § 120, but this method is tedious. We may obtam the result by noticing that the first member of (1) can be factored rationally in x and y when it represents a limiting case, and not otherwise. The same must be true of equation (2). We shall pro- ceed then to find the condition under which (2) can be factored. 1. Assume ^ ^ 0. (2) may now be considered as a quadratic equation in x, and factored by the method of § 41. Solving (2) for X, we have ^ -{Hy+ G) ±-^/JlP-AB)y'+ 2 y{HG-AF) + {G'-CA) It is necessary, however, that y should not appear under the radi- cal sign, and for this it is necessary and sufficient that the quantity under the radical sign must be a perfect square. The necessary and sufficient condition for this is (§ 37) {HG-AF)--{H''-AB){G''-CA)=Q, that is, ylBC+2FGH-AF^-BG^-Cff^ = 0. (3) CLASSIFICATION OF CONICS 237 2. Assume ^ = 0, but B ^^ 0. The equation may then be con- sidered as a quadratic equation in y, and handled in the same manner as before with the same result. 3. Assume A = (), B = 0. Then If cannot equal zero. The equation can consequently be written r* V c The factors of this, if they exist at all, are clearly of the form {x+a){y + h)^Q, whence a = — > 5 = — » ah = H H 2H The necessary and sufficient condition that two quantities a and h can be found satisfying these equations is But this is just what (3) becomes when ^ = 0, ^ = 0. Hence, the necessary and sicfflcient condition that Ax^+2Hxy + By^-\-2Gx+2Fy + C=Q represents a limiting case of a conic is ABC +2 FGH -AF^- B G^ - 6'//' = 0. The expression (3) is called tlie discriminant of (1) and is denoted by A. In determinant form A = 125. Classification of curves of the second degree. The results of the previous articles are exhibited in the table on the following page, which gives the simplest forms to which the general equation Ax}+ 2Hxy+Bf+ 2Gx+2Fy + C=0 can be reduced under the various hypotheses, where D = AB-H\ A =ABC+ 2FGH-AF^-BG--CIP. A H G H B F G F C 238 GENERAL EQUATION OF SECOND DEGREE At^O A = D>0 l2 7/2 or no curve a;2 7/2 D<0 x2 ?/2 Hyperbola- -^ = 1, X2 w2 -^2 + ^ = 1 Two Intersecting straight lines 02 62 D = Parabola 2/2 = 4 pa;, or x^ = 4py Two parallel straight lines {y - 2/i) {y - 2/2) = 0, or (x - a;i) (x - X2) = 0, or no locus 126. Center of a conic. It is frequently desirable to find the center of a conic represented by the general equation. Now, if the origin of coordinates is taken at the center of the curve, the equation can contain no terras of the first degree in x and y ; for if it is satisfied by any point (x^, y^, it must also be satisfied by the symmetrically placed point (— x^, — y^. We will accordingly take the center £is (ic^, y^ and make the transformation The general equation then becomes Ax'^ + 2 Hx'y' + By" + 2 {Ax, + Hy, + G)x'+2 {Hx, + By, + F) y' + ^<+ 2Hx^, + Byl+ 2 Gx,+ 2Fy,+ C=0, where, by the condition for the center, Ax, + ffy,+ G = 0, irx,-{-By, + F=0. By multiplying each of these by a properly chosen factor and add ing, we obtain the equivalent equations (1) (AB - H"") x^ = HF- BG, (AB - H') y^=HG- AF. (2) CENTEK OF A CONIC 239 Three cases then occur: 1. AB — H^'^O. Equations (2) have then a single solution and the curve has a center. This occurs for the ellipse, the hyperbola, and their limiting cases. 2. AB — IT^ = 0, but not each of the expressions HF—BG and HG—AF equal to zero. At least one of equations (2) ex- presses an absurdity, and hence equations (1) have no solution and the curve has no center. This occurs in the case of the parabola. 3. AB-IT'' = 0, JTF-BG=0, ITG-AF=0. Equations (2) are each = 0. Equations (1) are identical, and any point on the line expressed by each of them is a center of the curve. In this case one easily calculates that A = 0. The curve then consists of two parallel straight lines (§ 125), and the line of centers is the line halfway between the two parallel lines. 127. If for the equation Ax'+2 Hxy + Bf+1Gx+2Fy + C=0 the origin is transferred to the center of the curve, when such exists, the equation becomes Ax'^+2Hx'y' + By'''+C' = Q, where C = Ax^ + 2 Hx,y, + Byl +2Gx^+2 Fy^ + C. This quantity C may be expressed in terms of the original coeffi- • cients as follows. Take the equations (1) of § 126, multiply the first one by x^, the second by y^, and add them. There results Ax^ + 2 Hx^, + By^ + Gx, + Fy, = (i. Subtracting this from the value of C', as given above, we have C' = Gx, + Fy,+ C, whence, by substituting the values of x^ and y^, as given by (2) (§ 126), we have ABC+2FGH-AF^-BG^-Cff^ A C' = AB-H' D 240 GENERAL EQUATION OF SECOND DEGREE 128. Directions for handling numerical equations. In case it is necessary to reduce a uumerical equation to its simplest form, the procedure, based on the foregoing discussion, is as follows : First compute AB — H^ and determine the type of the curve (§ 123). A may also be computed if wished, but it is not necessary. If AB — H^ ^ 0, find the center, as in § 126, and transfer the origin to it. Then, as in § 120, turn the axes through an angle (p = w tan = tan A-B ■s/(^A-Bf+4:H''±{A-B) computing A' and B' by the formulas of § 120. The two values of tan^ are the slopes of the axes of the curve. If AB — H'^ = 0, write the equation in the form {y/~Ax + ^yf+2Gx + 2Fy + C=Q, Vb being taken with the same sign as H, and let , -y/Ax + V^y , Vbx — VAv y' = — 7=^' «' = — , ^ - ■■Va + b ^a + b Solve these equations for x and y and substitute in the given equation. The equation is now in the form y'^-i- 2 G'x' + 2F'y' + C" = 0, and the further reduction is made by the method of § 121. Ex.1. 8x2 _ 4xy + 52/'2-36x + 18?/ + 9 = 0. Here AB — 11^ = 30, and the curve is an ellipse or a limiting case of an ellipse. The center, found by § 126, is (2, — 1), and the equation transferred to the center as origin becomes 8 x'2 - 4 xY + 5 ?/'2 - 36 = 0. We now turn the axes through (p = ^ tan-i(— |) = tan-i2 or tan-i(— ^), and find, from § 120, ^' = 9 or 4, B' =4 or 9. The ambiguity is removed by noticing that if we take tan (f> — 2, the formulas of transformation (§ 115) are , z" — 2 ij" , 2 x" + y" 2/ = which give A' = 4, B' = 9. The simplest equation is then 4 x"2 + 9 ?/"2 = 36. The slopes of the axes are 2 and — J^, CONIC THROUGH FIVE POINTS 241 Ex. 2. 36a;2-48x2/ + Wy^ + 52x - 260y - 39 = 0. (6 X - 4 r/)2 4- 52 a; - 260 y - 39 = 0, 6x — 4?/ Sx — 2y Here We write and place y V52 Vl3 4x — Gy — 2x 8y V52 Vl3 Solving for x and y and substituting, we have y'S 4. VlSj/' + Vl3x' -3=0, or y''2 = — VTs x", Vl3 ^~ The curve is a parabola, the axis of which is y" = or 6 x where Vl3 y = ¥ 4 2/ + 13 = 0. 129. Equation of a conic through five points. The general equation of the second degree 1x^+2 Hxy + By^+2Gx + 2Fy + C=Q contains six constants, the ratios of which are alone essential. Five independent equations are sufficient to determine these ratios. Therefore a conic is, in general, determined by five conditions. The simplest conditions are that the conic should pass through the five points (a?„ y^, [x^, y^, (x^, y^), {x^, y^, and {x.^, y,). The five equa- tions to determine the ratios of A, If, B, G, F, and C are then Axl + 2 Hx^y^ + Byf + 2Gx^+ 2Fy^+ C = 0, Axl+2Hx^y^ + By^+ 2 Gx^+2Fy^+C = 0, Axl+ 2Hx,y^ + By^+ 2 Gx^+2Fy,+ C ={), Axl+2Hx^^ + Byl+ 2 Gx^+ 2Fy, + C=0, Ax^ + 2 Hx^y^ + Byl -^ 2Gx^+ 2Fy^+C = Q. Eliminating the coefficients between these and the general equa- tion, we have = 0, x-" xy f X y xl ^iVx yl ^1 2/1 ^! ^^y^ yl x^ y. ^^8 ^^z yl x^ 2/3 x! ^43/4 yl X, y^ < ^6^5 yl X, y. 242 GENERAL EQUATION OF SECOND DEGREE which is the required equation of a conic through five given points. The equation of a conic through five points may also be found in the following manner : Let us take any four of the given points and connect them by straight lines so as to form a quadrilateral (fig. 134). Let the equation of I^J^ be A^x + B^y + C^ = 0, or, more shortly, f^{x, y) — 0. Similarly, let the equation of P^l^ be fj^x, y) = 0, that of ^^ be f.J^x, y) = 0, and that of F^J^ be f^{x, y) = 0. Form now the equation Vi{x, y) ■ Ai-^, y) + ¥2(«. y) ■ A{^> y) = o. (i) where I and k are undetermined factors. This equation is of the second degree in x and y ; therefore it represents a conic section. More- over, this conic section passes through Il\ for the coordinates of P^ make f^{x, y) = and f^{x, y) = 0, and therefore satisfy equation (1). Simi- larly, this conic passes through ^, ^, and j^. If now we substitute in (I) the coordinates of j^, we de- termine values of I and k, which we must assume in order that the conic may pass through i^. We thus de- termine the equation of a conic through the five given points. Ex. Let it be required to pass a conic through the points Pi(2, 3), P^i — 1, 2), P3(-3, -1), P4(0, -4), P5(l, 1). The equation of P1P2 isa;-3y + 7 = 0, that of P2P3 is3x-22/ + 7=:0, that of PsP4 is X + y + i =: 0, and that of P4P1 is7a;-2y-8 = 0. We form the equation l{x -Sy + '!)(x + y + 4) + k(Sx - 2y + 'J){7 x - 2y - 8) = 0, and, substituting the coordinates of P5, find k = ^l. Hence the required conic is (X - 3 2/ + 7) (a; + 2/ + 4) + I (3 a; - 2 y + 7) (7 X - 2 y - 8) = 0, or 109x2-108x2/ + 82/2 + 169x -10?/ -168 = 0. If three of the points lie in a straight line, the method is appli- cable, but it is evident that the conic must be one of the limiting CONIC THKOUGH FIVE POINTS 243 cases, for it must consist of the straight line in which the three points lie, and the straight line connecting the other two points. If four or five of the points lie in a straight line, the method is not applicable. It is geometrically evident that in this case the problem is indeterminate ; for the conic may consist of the straight line in which the four points lie, together with any line through the fifth point, if that is not on the line with the four, or any line what- ever if the fifth point lies on a straight line with the four others. If it is required to determine a parabola, only four points are necessary. This follows from the fact that one relation connecting the coefficients is always given, namely, AB — H^ = 0. We form, as before, the equation ^/i(«'. y) ■ M^> y) + ^M^> y) ■ f^i-^^ y) = o- We form, then, the equation AB — H'^ = out of the coefficients of this equation. The result is a quadratic equation in -> and K hence we will have two, one, or no real parabolas, according as the values of j are real, equal, or imaginary. It should be noticed that fC in this connection " parabola " may mean two parallel straight lines. Ex. Let it be required to pass a parabola tlirough the points Pi(l, — 1), P2(2, 3), Ps(2, - 6), P4(5, 7). We find the equations of the following lines : P1P2, 4x — y — 6 = 0; P2P3, X - 2 = ; P3P4, 4x-2/-13 = 0; P4P1, 2a;-y-3 = 0. The equation of the conic is then Z(4x - 2/ - 5) (4x - 2/ - 13) + fc(a; - 2)(2x - 2/ - 3) = 0, or (16Z + 2A:)x2 + (-8Z- A:)x2/ + i2/2 + ( _ 72 J - 7 fc) X + (18 i + 2 i) 2/ 4- «5 Z + 6 fc = 0, and the condition AB — R"^ = d'xs (16Z + 2A:)Z-(4Z + ^A:)2 = 0, whence k-Oov—%1. There are accordingly two parabolas, 16x2 - 8 X2/ + 2/2 - 72 X + 18 2/ + 65 = 0, and 2/=^-16x + 2 2/ + 17 = 0. The first equation, however, represents a limiting case of a parabola, since it factors into 4x_2/-5 = and 4x-2/-13 = 0, which represent two parallel straight lines. 244 GENEEAL EQUATION OF SECOND DEGEEE 130. Oblique coordinates. We have assumed, thus far, that the general equation is referred to rectangular coordinates. If, how- ever, the equation Ax" + 2 Hxy + Bf+2Gx-j-2Fy + C=0 has reference to oblique coordinates, it may be transformed to any conveniently chosen pair of rectangular coordinates. Formulas for this purpose are given in § 117, and it has been proved in § 118 that such a transformation does not alter the degree of the equa- tion. Therefore the new equation is of the form A'x'^ + 2 H'x'y' 4- B'y"" + 2 G'x' + 2 F'y' + C'=0. This equation may now be investigated by the methods of this chapter. Hence we have the result : Any equation of the second degree, whether referred to rectangu- lar or to oblique coordinates, represents a conic. PROBLEMS Determine the nature and the position of the following conies : 1. 4xj/ + .3 2/2_8x + 16?/ + 19 = 0. 2. a;2 _ 6xy + 9 2/2 -280«- 20 = 0. 3. Ilx2_4xy + 14i/2_26x + 32y + 59 = 0. 4. 5x2 - 26xy + 51/2 + 10x-2Gy + 71 = 0. 5. 4:xy + 6x-8y + 1 = 0. 6. x2-2x2/ + y2 + 2x-22/ + l = 0. 7. 13x2 + 10 xy -}- 13 2/2 + 6x - 42y - 27 = 0. 8. x2 -4xy - 2 2/2- 14x + 4 2/ + 25 = 0. 9. 6x2 - 5x2/ - 62/2 -46x- 9?/ + 60 = 0. 10. 4 x2 - 8 x?/ + 4 2/2 + 6 X - 8 2/ + 1 = 0. 11. x2 + 6x2/ + 92/2 -6x -18 2/ + 5 = 0. 12. 41x2 - 24x2/ + 342/2 - 188x + 116?/ + 196 = 0. 13. 31 x2 - 24 X2/ + 21 2/2 + 48x - 84 2/ + 84 = 0. 14. Show that, if A and B in the general equation have opposite signs, the conic is an hyperbola. 15. Show that the conic represented by the general equation is an equilateral hyperbola when A = ~ B. PEOBLEMS 245 16. Prove that the necessary and sufficient conditions that the general equation should represent a circle are A = B, H — 0, provided the axes are rectangular. 17. Show that, if the general equation contains the term in xy and not more than one of the terms containing x^ or y^, the conic is an hyperbola. 18. Show that xy + ax + by + c = is the general equation of the hyperbola when the axes of coordinates are parallel to the asymptotes. ,_. ^ ... - 19. Prove that any homogeneous equation in x and y represents a system of straight lines passing through the origin. 20. Find the angle between the two straight lines represented by the equation Ax^ + 2 Hxy + By^ = 0. 21. Show that the asymptotes of the hyperbola are parallel to the two straight lines Ax^ + 2 Hxy + By^ = 0, 22. Show that, if the focus is taken upon the directrix, the conic becomes one of the limiting cases. Find the equations of the conies through the following points : 23. (3, 2), (- 2, - 3), (J, - 3), (2, - 2), (|, - f). 24. (1, 2), (6, 3), (3, 2), (2, 1), (9, 2). 25. (0, a), (a, 0), (0, -«),{- a, 0), (a, a). 26. (1, 1), (-1,6), (2,4), (0,3), (3,1). 27. Find the equation of a parabola through the four points (4, — 4), (9, 4), (6, - 1), (5, - 2). 28. A point moves so that the sum of the squares of its distances from two intersecting straight lines is constant. Prove that the locus is an ellipse, and find its eccentricity in terms of the angle between the lines. CHAPTER XII TANGENT, POLAR, AND DIAMETER FOR CURVES OF THE SECOND DEGREE 131. Equation of a tangent. It has been shown in § 59 that the tangent to a curve at a point {x^, y^ is where ( — ) denotes the value of -^ at (x., yX \dx/i ax Applying this theorem to the conic Aa?+2 Hxy + Bf+ 2Gx+2Fy + C = 0, we first find, by differentiation, 2Ax-h2Hy+2Hx^ + 2By^ + 2G+2F^=0, ax ax ax . dy Ax + Hy + G whence , = — -77 tt 7; ' dx Hx+By + F Therefore the equation of the tangent at the point {x^, y^ is Ax^ + Hy^+G. , y — y, = ^^ ix — X.), ^ ^' Bx^ + By^ + F^ ''' that is, Ax^x — Ax^ + Hxy^ + Hx^y — 2 Hx^y^ + ^ViV ~ ^Vl + Gx — Gx^ + Fy — Fy^ = 0. This equation may be simplified by adding to it the identity Ax*-^ 2Hx^y^ + Byl+ 2Gx^+2Fy^ + C= 0, which follows from the fact that {x^, y^ is on the conic. There results Ax^x + H{x^y + xy;) + By^y + G{x + x^) + F{y + y;)+ C =Q. This result is easily remembered from its resemblance to the equa- tion of the conic. 246 POLAR 247 132. Definition and equation of a polar. We have just seen in § 131 that the equation Ax^x + H(x,y + xy^) + By^y +G{x+x;) + F(y + y,) + C = (1) represents the tangent line to the conic As(^-\-2Hxy+By^+2Gx + 2Fy + C=0, (2) provided the point (x^, y^ is on the conic. But no matter what is the position of the point {x^, y^, (1), being of the first degree, repre- sents some straight line which from the form of the equation must in some way be related to the conic (2) and the point {x^, y^. This line is called the polar of the point {x^, y^ with respect to the conic, and the point is called the pole of the line. The tangent line now appears as only the special case of the polar which occurs when the pole is on the conic. Ex. 1. The polar of the point (3, — 2) with respect to the ellipse 4a;2 + 5 2/2_2x + 3y-l = is 12x-10y-(x + 3) + §(2/-2)-l = 0, or 22x-17y-14 = 0. Ex. 2. Find the pole of the line 2 x — 3^ + 6 = with respect to the hyper- bola 4x2-5T/2 + 4x-2i/ + 3 = 0. The polar of (Xj, yi) is 4xix - 5yiy + 2{x + Xi) - (2/ + 2/i) + 3 = 0, or (4xi + 2)x + (- 5yi -l)y + 2xi - yi + S = 0. This will be the same as the given line if 4xi + 2 ^ 5?/i + 1 _ 2xi - yi + 3 2 3 6 ' These reduce to the two equations for Xi and yi, 12x1-102/1 + 4 = 0, 2x1-11^1 + 1=0; whence • Xi = - ^|, ?/i = ^V- 133. Fundamental theorem on polars. When the equation Ax'+2Hxy+By^+2Gx+2Fy + C = (1) represents one of the limiting cases of the conies, the polar has little importance. We shall therefore assume that the conic is 248 TANGENT, POLAE, AND DIAMETER either an ellipse (including the circle), a parabola, or an hyperbola. The properties of its poles and polars are then conveniently found by use of the proposition : If P^is any 'point on the polar of another point ij, the polar of II passes through Py For the polar of Il(x^, y^ with respect to (1) is Ax^x + H{x^y + xy^) + By^y +G{x + x^) + F{y + y^) + C = 0, (2) and if P^ix^, y^ is on (2), we must have Ax^x^ + II{x^y^-\- x^y^ + By^y^_ + G (^,+ x^ + F{y^-^ y,) + C= 0. (3) Again, the polar of P^ with respect to (1) is Ax^x + H{x^y + xy^) + By^y j^G{x + x^} + F{y + 7/,) + C = 0, (4) and this passes through {x^, y^) because of (3). 134. Chord of contact. An inspection of the figures of the conies shows that a point not on a conic must lie so that in general either two tangents or no tangent can be drawn from it to the conic. In the former case the point is said to be outside the conic ; in the latter case, inside. Let us take now a point 7^ outside the conic, and let the two tangents drawn from it to the conic touch the conic in L and K (fig, 135). Now the polar of a point on a conic is the tangent to the conic at that point (§ 132). Hence I^L is the polar of L, and I^K is the polar of K. Therefore, by the fundamental theorem (§ 133), the polar of 7J must pass through L and K. Hence the polar is the straight line LK, which is called the chord of contact of tangents from 7J. Conversely, if a straight line intersects a conic, its pole is the point of intersection of the tangents at the points of intersection. The proof of this is left to the student. POLAR 249 The chord of contact may be used to find the equations of the tangents through a point not on the conic. Ex. Find the tangents to the conic x^ + 2xy + y^ + 2x + 6y + l = which pass tlirougli the point (4, — 2). Since this point is not on the conic, its coordinates not satisfying the equa- tion of the conic, we form the equation of its polar, i.e. 3x + 5?/ — 1 = 0, which will be the chord of contact of the tangents drawn from the point to the conic, provided any can be drawn. Solving the equations of the polar and the conic simultaneously, we find that they intersect at the points (7, — 4) and (2, — 1). Hence there are two tangents which are respectively 2x + 3y — 2 — and x + 2y = 0. •■■■'■ ■ ■•- ■■■ -■•■■'•': - , ■ 135. CottstiUction of a. ^lar. Whethfer a point lies inside or outside a conic, the polar may be obtained by the following con- struction. Draw through i^ (fig. 136) two straight lines, one intersecting the conic in L and K, and the other intersecting the / / ^y ^^^^s conic in M and N. Let the tangents at L and K intersect in H and the tangents at M and iV in- tersect in S. Then B is the pole of LK and S is the pole of UN, by § 134 Since i^ lies on both LK and 3IN, its polar passes through B and S by the fundamental theorem. Therefore ES is the required polar. This construction may also be used when ^ is outside the conic. 136. The harmonic property of polars. An important property of poles and polars is stated in the theorem : Any secant passing through I^ is divided har- monically hy the conic and the polar of P^. Let -^iNT (fig. 137) be any secant through P^, M and N be the points in which P^N cuts the conic, and Q the point in which it cuts the polar of P^. We are to prove that the line MN is divided Fig. 137 250 TANGENT, POLAR, AND DIAMETER harmonically, i.e. that it is divided externally and internally in the same ratio. We are to prove, then, that P^M _ MQ F^N~ Qjsr' whence, by placing MQ =F^Q—P^M, QN=P^N—P^Q, and solving for P^Q, we have 2 PM ■ PN ^^ P^M+P^N Let the point P^ be {x^, y^, the equation of the conic be Aa?-\-2Hxy + Bf+2Gx + 2Fy + C=Q, (1) and that of the polar of P^ be Ax^x + H{x^y + xy^ + By^y + G{x + x;) + F{y + ^/j) + C = 0. (2) Let {x, y) be a variable point on F^N, r the variable distance JJP, and d the angle made by F^N and OX. Then . ^-«^i . ^ y-Vx cos V = , sm a = , r r that is, x = r cos + x^, y = r sin. ■{■ y^ (3) Now if P coincides with either M or iV, the values of x and y given by (3) satisfy (1). Substitution gives r^ [A cos' 0+2II sin 0cos0+B sin' 0] + 2r [Ax^ cos + -^'(iCj sin + y^ cos ^) + %, sin ^ + G! cos ^ + i^ sin 6'] + C" = 0, where C = Ax'- + 2 Hx,y, + J5yf + 2 (?a;i + 2 Fy^ + C. The roots of this equation are P^M and P^N. Hence, by § 43, 2 [J«,cos ^ +//(«! sin ^ + yj cos0)-i-By^ sin ^ + (?cos^ ^-i^sin ^] F^M-F^N whence A cos' (9 +2 7/ sin 0cos0+B sin'^ ^ cos'^ + 2^sin ^ cos ^ + 5 sin^^ 2F,M-F,N FJf+P,N , (4) ^a;jCos^+-H'(ajiSin^ + 2/jCOS^)+%iSin^ + <9cos^+i^sin^ ^ EECIPROCAL POLAES 251 Also, if the point P coincides with Q, the values of x and y given by (3) satisfy (2). Substitution gives r [Ax^ cos 6 + H{x^ smd + y^ cos d)+By^ siD.d + Gco80 + Fsin. 6] + C" = 0. The root of this is F^Q. Therefore F^Q C (5) Ax^ cos 6-\-H(x^ sin 6 + y^ cos 6) + By^ mid+G cos 6 +i>''sin 6 Comparing (4) and (5), we have _ 2F,MF^N ^^~ f^m+f^n' which was to be proved. The theorem of this article is often made the basis of the definition of the polar. 137. Reciprocal polars. Consider a given conic and a rectilinear figure, such as the triangle ABC with sides a, b, c (fig. 138). Con- struct the lines a', b', c', the polars of A, B, C, respectively with respect to the conic. The lines a', b', c' form a new triangle A'B' C. The fundamental theorem shows that A', B', C' are the poles of a, b, c respectively. Hence the two triangles are so related that the vertices of one are the poles of the sides of the other. They are called reciprocal polars. A similar construction holds for any figure composed of straight lines. Consider next any curve K and a tangent line a (fig. 139). Let A be the pole of a with respect to a conic C. As the tangent rolls 252 TANGENT, POLAR, AND DIAMETER around the curve K, the point A describes another curve h Let a and h be two tangents to K, and M their point of intersection, and let A and B be the two corresponding points of k, and m the chord AB. Then, by the fundamental theorem, m is the polar of M. Now let a and h approach coincidence. Then M ap- proaches a point on K, B and A approach coinci- dence, and m approaclies a tangent to h. Hence tlie points of K are the poles of the tangents to k. We have then two curves such that the points of either are the poles of the tangents of the other. These curves are called reciprocal polars. The study of reciprocal polars forms an important part of geom- etry, but lies outside the limits of this work. 138. Definition and equation of a diameter. A diameter of a conic is the locus of the iniddle points of a system of parallel chords. I, i^t Asg'+2 Hxy+ Bf-^ 2 Gx + 2Fy + C=Q (1) be any conic (fig. 140), RS an^^ chord which makes the angle 6 with OX, and -?J(«i, y-^ the mid- dle point of this chord. Take P{x,y) any point on the chord, and let P^P = r, where r is posi- tive if iJP has the direction of as, and negative if I^P has the direction SB. Then for any position of P we have -X Fig. 140 = cos 6, r ' 2/ -2^1 sin^: whence X = x^+ r cos d, y = y^+ r sin 0. (2) DIAMETER 253 Now if P coincides with either B or S, the values of x and y in (2) satisfy (1). Substituting, we have r" [^ cos' (9 + 2 iT sin 6^ cos 6" + ^ sm='^] + 2 r [Ax^ cos 6 + Hx^ sin ^ + Hy^ cos 6 -\-By^&ind + Gco^d + F&md] + [Axl + 2 Hx^y^ + %,2 ^ 2 Gx^ + 2 i^'^/i + C] = 0, (3) the roots of which are P^S and P^R. But, by hypothesis, P^R = — -^*S^. Hence the roots of equation (3) are equal in magnitude and opposite in sign. Therefore the coefficient of r in (3) must be zero, that is, Ax^ cos 6 +Hx^ sin 6 + Hy^ cos ^ + %j sin 6* + G^ cos ^ + i<" sin ^ = 0. (4) If, for convenience, we assume that cos 6^0, and this will gen- erally be the case, we may divide by cos and replace tan by the usual symbol for the slope m, thus obtaining Ax^ + Hy^ +G + m {Hx^ + By^ + F) = 0. (5) If we allow RS to move parallel to itself, so that m remains fixed but P^ changes, (5) always holds true, and in fact shows that 7J is always a point of the straight line Ax + Hy + G + m{Hx + By+F)=Q. (6) Conversely, any point P^{Xy, y^) on line (6) makes the values of r in (3) equal in magnitude but opposite in sign, and if i^ lies so that these roots are real, it will be the middle point of a chord with slope m. ,-. g lo i?i.^s;r> iii" The straight Ime (6) is of infinite length, and it is customary to regard the entire line as the diameter, though it is evident that not all of its points correspond to chords of the system which intersect the conic in real points. i^w .-iuj ui.na 139. The last statement of the previous article may be explained as follows : The equation y = ??ix + 6 may be made to represent any line of slope m by assigning an appropriate value to b. For some values of b the corresponding line intersects the conic (1) of § 138 in real points, and is one of the chords bisected by the diameter (6). 254 TANGENT, POLAK, AND DIAMETER For other values of 6, however, the line does not intersect the conic in real points, the simultaneous values of x and y satisfying their equations being imaginary. But if these imaginary values of x and y are substituted for Xi, X2 and yi, yi respectively in the formulas x = — ^ , y = ^ — — of § 18, the resulting values of x and y are real, and furthermore they satisfy the equation of the diameter. This fact is sometimes expressed by saying that the line is a choi'd of the conic which intersects it in imaginary points, and that its middle point is a real point of the diameter. It is from this point of view that the entire line is regarded as the diameter, since every point of it is the middle point of some chord of the system. 140. If the conic has a center, every diameter passes through the center. For, by § 126, the center satisfies the equations Ax + Hi/ + G = 0, Hx+By + F=0, and hence satisfies (6) of § 138 for any and all values of m. In the parabola, hovjever, all diameters arc parallel to each other and to the axis ; for the slope of the diameter is, from (6), § 138, A. + Hm I But for the parabola H = ^ AB, so that the slope of ^ + ^^ A -J~A~ \l~ the diameter becomes --^= » which reduces to ;= • SAB + Bm y/B This is independent of m, and equal to the slope of the axis (§ 128). It is evident that the axes of a conic are diameters, for from the symmetry of the curves they contain the middle points of all chords which are perpendicular to them. In fact, they are the only diameters which are perpendicular to the chords which they bisect, as will be proved later on. 141. Diameter of a parabola. If the equation of the parabola is written in its simplest form, 'if = 4:px, the equation of the diameter becomes y = — • m From this equation it is evident that the only diameter perpen- dicular to the chords which it bisects is the axis of the parabola. Ex. 1. Find the equation of the diameter of the parabola 2y2^,3x = bisecting chords with slope 2. Since m = 2 and p = — ^, the equation of the diameter is, y = ^ » or 2 2/ + 3 = 0. ^ DIAMETER OF A PARABOLA 255 Ex. 2. A diameter of the parabola y^ = 2x passes through the point (2, — 1). What is its equation, and what is the slope of the chords bisected by it ? If m is the slope of the chords bisected, the equation of the diameter is y — —. But (2, — 1) is a point of this diameter. m 1 J .-. — 1 = — , whence rn = — 1 ; also the diameter is y = , ov y = — \. tn — 1 This equation of the diameter could have been written down immediately, for the diameter is parallel to OX, so that if one of its points is distant — 1 from OX, all its points are distant — 1 from OX, and its equation isy = — 1. If we solve the equations of the diameter and the parabola simultaneously, we find the coordinates of 0' (fig. 141), their point p 2p^ of intersection, to be The equation of the tangent at 0' is found to be 3/ = mx + whence it is seen that its slope is m. Calling 0' the end of the diameter, we express the above theo- rem as follows : The tangent at the end of a diameter is parallel to the chords bisected hy the diameter. If we consider the tangent as the limiting position of a chord which is moved, yet retains its original slope, the above theorem seems almost immediately evident. 142. Parabola referred to a diameter and a tangent as axes. Let O'X' (fig. 141) be a diameter of parabola f = ^px, (1) bisecting chords of slope m, and O'Y' be the tangent at 0'. Then p 1p^ ^m^ m and the slope of O'Y' is m. First transposing (1) to O'X' and O'Y", where O'Y" is parallel to OY, we have the formulas of transfor- mation the coordinates of 0' are x = ^, + x". m y = ^ + y". The new form of the equation is y"^+^y"=4:px". m Fig, 141 256 TANGENT, POLAR, AND DIAMETER Using now the 'formulas of transformation of § 117, which become a?" =x'+ ^ > y" = /^^ > Vl + mt: c:. . , ' -.:..: . . V 1 + tlV^ since <^ = and <^' = tan" ' m, we have, finally, ,, . ; : By § 1 7, however, FO' = P(^-^/'^) . Therefore if we denote FO' by p', after dropping the primes from « and 2/, the equation becomes y^ = 4 j?'x. ^^ ""It is to be noted that an equation in the form y"^ = 4:]px always represents a parabola, the x axis being a diameter, the y axis a tangent, and the distance of the focus from the origin being one fourth the coefficient of x. 143. Diameters of an ellipse and an hyperbola. If the equation + -. ~ 2 2 of the ellipse is written in its simplest form, — + ^ = 1, and the a^ If common slope of the chords is denoted by m^ the equation of the diameter becomes V V = 7, — X. If the slope of the diameter is denoted by m,, m^= — > whence m,w„ = 5 • ' . -If b =^ a^ m^ni^ csLnnot in general be —1, cmd the diameter of an ellipse cannot in general he perpendictdar to the chords which it bisects. The single exception is when the chords are parallel to either axis, in which case the diameter is the other axis and is perpendicular to the chords which it bisects, as noted above. If Z> = a, the ellipse becomes a circle, and m^ni^ is always equal to — 1. Hence the diameter of a circle is always perpendicidar to the chords which it bisects. DIAMETERS OF CENTRAL CONICS 257 62 jtti being 4 becomes — a^mi 9 mi Ex. 1. Find the equation of a diameter of the ellipse ix^ + 9y^ = 3(j bisoft- ing chords parallel to the line a;-|-2y + l = 0. Here a^ = 9, h"^ = 4, and in\ = — ^. .-. the diameter is y = — or 9 2/- 8x = 0. Ex. 2. 2y + 3x = 0isa diameter of the ellipse 4x2 + 9 ^2 = 3^ the slope of the chords which it bisects ? The slope of the diameter is — |, and by the formula is — the slope of the chords bisected. As a^ = 9 and 62 = 4, — 3 4, 8 .-. = , whence mi = — 2 9 77H 27 Ex.3. Find the diameter of the circle 4x2 + 4 2/2 + 43; — By — 11 = bisecting chords of slope 2. The center of the circle is (— -^, 1), so that the required diameter will be 2/ - 1 = - |(x + I), or 2 X + 4 ?/ - 3 = 0. Ex. 4. Find the diameter of the circle 4x2 + 42/2 + 4x — By — 11=0, which passes through the point (2, — 1). The center of the circle is (— ^, 1), and the straight line determined by the two points (2, — 1) and (— i, 1), i.e. 4x + oy — 3 = 0, is the required diameter. In the case of the hyperbola —, — —, 1 it is to be noticed that the parallel chords may be drawn in two ways. They may join points on the same branch of the hyperbola, or points of one branch to points of the other branch, as repre- sented in fig. 142. In whichever way the chords are drawn, if their common slope is denoted by m^, the equa- tion of the diameter is y = Fig. 142 This equation differs from that for the diameter of the ellipse only in the sign of the right-hand member. .^ If W2 is the slope of the diameter, m^m^ = — > and, as in the case of the ellipse, a diameter of an hyperbola cannot be perpendic- ular to the chords it bisects, except in the two special cases of the transverse axis and the conjugate axis. 258 TANGENT, POLAR, AND DIAMETER 144. Conjugate diameters. In § 143 we have seen that if the slope of the chords of the ellipse — + ^ = 1 is denoted by m^, and the slope of the diameter is denoted by m^, w„ = — am. whence m^m^ = — (1) Similarly, if the slope of the chords is m^, the slope of the diam- eter bisecting them must be — -^ — > which, by (1), must be m,. am\ Hence the proposition: If m^ and m^ are the slojjes of two diameters of an ellipse, and rthj)n„ = > then each diameter bisects all chords parallel to the other. Such diameters are called conjugate diameters. As the major and the minor axis each bisects chords parallel Fig. 143 to the other, they are conjugate diameters. It follows that : 1. The two axes are the only pair of conjugate diameters which are perpendicular to each other. 2. If one of two conjugate diameters of an ellipse makes an acute angle with the axis of x, the other makes an obtuse angle h^ with the axis of x. For if m^ > 0, m^< 0, since m^m^= r^- But a positive slope corresponds to an acute angle, and a negative slope to an obtuse angle. Hence the upper portions of conjugate diameters always lie on opposite sides of the minor axis, as OA^ and OB^ in fig. 143, A^A^ and B^B^ being conjugate diameters. In similar manner for the hyperbola — of two diameters m^ and m„ are such that m,m„ = — > y _ = l,if the slopes CONJUGATE DIAMETERS 259 the corresponding diameters are conjugate, and each bisects all chords parallel to the other. The transverse and the conjugate axes are conjugate diameters, each of which bisects chords parallel to the other. It follows that : 1. The two axes are the only pair of conjugate diameters that are perpendicular to each other. 2. Two conjugate diameters make either both acute or both obtuse angles with the transverse axis ; for m^m^ being always positive, vi^ and m^ have the same sign. 3. Two conjugate diameters lie on opposite sides of either asymp- tote ; for since m.m„ = — > if ?», < - > then m^> - > and the corre- ^ ^ a^ a 'a sponding conjugate diameters are on opposite sides of the asymptote 2/ = ^x(fig.l46). 145. Ellipse and hyperbola referred to conjugate diameters as axes. Let the conjugate diameters OA^ and OB^ of the ellipse ^ + 1 = 1 a'^ b' (1) (fig. 144) be chosen as new axes OX' and OY', and let them make angles (f> and ^' respectively with OX. Then the formulas of trans- formation are x = x' cos + y' cos ', y = x' sin 4> + y' sin ^', (2) where Le. tan <^ tan <^ I _ Fig. 144 b^ sin <^ sin ' cos <^ cos ' _ n b' + a' "' (3) since OX' and OY' are conjugate diameters. 260 TANGENT, POLAR, AND DIAMETER Substituting in (1) and collecting like terms, we have cos^(f> sin^^X ,2 , o/cos ^ cos (f>' sin (f> sin '\ , , v + ,'2^' + !«^V=l. x'y (4) But the coefficient of x'y' is zero, by virtue of (3) ; and if the intercepts on OX' and OY' are denoted by a' and b' respectively, i.e. OA^ = a' and OB^ = b', (4) becomes 4- =^ = 1 (5) where a' = and &' = Icos^ (f) sin^^, X V since — + — = 1, as noted above, a2 62 3. The area of the parallelogram formed by drawing tangents to an ellipse at the ends of conjugate diameters is constant and equal to 4 ab. Let Ti T^ T-s T4 (fig. 145) be a parallelogram formed by the tangents at the ends of the con- jugate diameters A1A2 and B1B2. Now the area of this parallelogram is evi- dently fo ur times th e area of the parallelogram ^lOlJiTi. But^iTi= Oi'i \a*v^ 4- 6*x^ iCiX = 6' = ~ i , from work above ; and since the equation of AiTi is — - a^ 22 , ViV _ 1 the perpendicular distance from to ^ 1 Ti is, by § 32, — ; - _ • 62 "^ / VaV + 6V /^a*y^ + ¥x^\/ a^b-^ \ , , ,, Hence the area of AiOBi Tj = ( \ ^- I , = a&, and the \ a& /\^a*yl + b*xl/ area of the large parallelogram is 4 ab, as was to be proved. 262 TANGENT, POLAR, AND DIAMETER 147. It was noted in § 144 that conjugate diameters of the hyperbola = 1 lie on opposite sides of the asymptotes, whence it follows that if one of two conjugate diameters intersects the hyperbola, the other cannot inter- sect it. In order, then, to state for the hyperbola propositions analogous to 2 and 3 of the last article, it is customary to consider, in connection with the above hyperbola, the hyperbola 1 = 1. These two hyperbolas are called con- jugate hyperbolas, either one being considered the primary and the other being called the conjugate. It may readily be proved that if the slopes of two diameters are such that 62 rmm^ = — , they are conjugate diameters of both the above hyperbolas. More- over it is evident (fig. 146) that if one diameter in- tersects one hyperbola, the other intersects the conju- gate hyperbola. Now if OAi and 0B\ are conjugate diameters, and OAx is called o', as in § 145, and we apply the same method as was ap- plied to the ellipse, we shall find OBi = 6' of § 145. With this value of 6', theorem 2, § 146, becomes for the hyperbola a'2 — 6'2 = a2 — 62, while theorem 3 is the same for the hyperbola as for the ellipse. The proofs of these last statements are left to the student, the work being exactly like that for the ellipse. Fig. 146 PROBLEMS Find the polars of each of the following points with respect to the given conic, and find the points in which the polar intersects the conic : 1. (1, 2), 23x2 -\\xy + iy^ + 36x - 9y + 9 = 0. 2. (-1, -2), 3x2-3xy + 4x + y-3 = 0. 3. (0, 0), 2x2-22/2-2x + 22/-l = 0. 4. (4, -2), 5^/2 + \%y 4-4x4. 5 = 0. Find the poles of each of the following polars with Vespect to the given conic : 5. 2x-y = 0, x2 + 8xy- 22/2 -12x4- 62/ -9 = 0. 6. x-32/4-2 = 0, x2-f 1/2 _ 2x4-42/ = 0. 7. x-f 2 2/-13 = 0, 3x2 + 82/2 -26x- 762/ -I- 231 = 0. 8. 3x-22/-9 = 0, 3x2-4j/24-6x-242/-45 = 0. PEOBLEMS 263 Find the equations of the tangents from each of the following points to the given conic : 9. (2, 3), 4x2 - 5X2/ + 2 2/2 + 3a; - 22/ = 0. 10. (0, 1), 3x2 - 42/2 + 12x = 0. 11. (1, -2), 2x2 -22/2- 6x- 62/ -1 = 0. 12. (2, 4), x2 + 2/2 - 6x - 2 2/ + 5 = 0. 13. (2, 0), 5 2/2 + 4x - 2 2/ - 8 = 0. 14. (-1, -1), 3x2 + 8 2/2- 8X-122/ + 4 = 0. 15. Prove that the polar of a given point with respect to any one of the circles x2 + 2/^ — 2 fcx + c2 = 0, when k is variable, always passes through a fixed point whatever the value of k. 16. T is the pole of a chord PQ of the parabola y^ = 4px. Prove that the perpendiculars from P, T, and Q upon any tangent to the parabola are in geometric progression. 17. If P is any point, LM its polar with respect to any central conic, C the center of the conic, R the point in which the perpendicular from C to LM meets LM, and S the point in which the perpendicular from P to LM meets the axis of the conic, prove CR ■ PS = b'^. 18. Prove that the perpendicular from any point (xi, 2/1) to its polar with respect to any central conic intersects the axis of the conic at a distance e2xi from the center of the conic. 19. Prove that if in any conic the pole of the normal at P lies on the normal at Q, then the pole of the normal at Q lies on the normal at P. 20. If Pi and P2 are any two points, and C the center of a conic, show that the perpendiculars from Pi and C to the polar of P2 are to each other as the perpendiculars from P2 and C to the polar of Pi. 21. If mi is the slope of the polar of a point Pi with respect to the ellipse x2 w2 1 = 1, and mg is the slope of the line joining Pi to the center, show that o* f>^ yi mirrii = Find the similar relation for the hyperbola. a2 22. Prove that the portion of the axis included between the polars of two points with respect to a parabola equals the projection on the axis of the line joining the points. 23. Show that for any conic section the polar of the focus is the directrix. 24. Where is the polar of the center of an ellipse or hyperbola with respect to that curve ? x2 ifl 25. In the ellipse 1- — = 1 find the equations of two conjugate diameters, o2 62 one of which bisects the chord deteimined by the upper end of the minor axis and the right-hand focus. 264 TANGENT, POLAR, AND DIAMETER 26. If Pi and P2 are the extremities of any two conjugate diameters of the ellipse 1- ~ = 1, prove that the sum of the squares of the perpendiculars drawn from Pi and P2 to the major axis of the ellipse is equal to 6^. 27. Show that there can be only one pair of equal conjugate diameters of the ellipse h ^ = 1, namely y — -x^y — x. a^ 6'^ a a 28. Show that the equation of any ellipse referred to its equal conjugate diameters as axes is x^ + y^ = 2 29. In any ellipse show that the diameters parallel to the lines joining the extremities of the axes are conjugate. 30. One diameter of the ellipse \- — = 1 passes through the upper end of the right-hand latus rectum. What is the slope of the conjugate diameter ? 31. What must be the relation between the semiaxes a and b of an ellipse when the diameters passing through the upper extremities of the left-hand latus rectum and the right-hand latus rectum are conjugate ? 32. Show that the polar of any point on a diameter of a central conic is parallel to the conjugate diameter. 33. Show that if an ellipse and an hyperbola have the same axes in magni- tude and position, then the asymptotes of the hyperbola coincide with the equal conjugate diameters of the ellipse. 34. Prove that tangents at the ends of conjugate diameters of an hyperbola intersect on the asymptotes. 35. Prove that the straight line joining the ends of a pair of conjugate diam- eters of an hyperbola is parallel to one asymptote and bisected by the other. 36. If an hyperbola has a pair of equal conjugate diameters, prove that it is an equilateral hyperbola. 37. Show that in an equilateral hyperbola conjugate diameters are equally inclined to the asymptotes. 38. Show that in an equilateral hyperbola all diameters at right angles to each other are equal. 39. Show that every diameter of an equilateral hyperbola is equal to its conjugate. 40. Prove that the tangents at the ends of any chord of a conic intersect on the diameter which bisects the chord. 41. The chords which join the ends of any diameter to any point of the curve are called supplemental chords. Prove that two diameters which are parallel to any pair of supplemental chords are conjugate. 42. If the tangent at the vertex A of an ellipse cuts any two conjugate diameters produced in T and t, show that AT ■ At = — tfi. PROBLEMS . 265 43. Show that if any tangent meets any two conjugate diameters, the prod- uct of its segments is equal to the square of the lialf of tlie parallel diameter. 44. If from the focus of an ellipse a pei-pendicular is drawn to a diameter, show that it will meet the conjugate diameter on the corresponding directrix. 45. The tangent at any point Pi of an ellipse cuts the equal conjugate diameters in T and T^. Show that the triangles TCP\ and TiCPi are in the ratio CT'^ : CTi\ 46. Show that the product of the focal distances of any point of a central conic is equal to the square of half the corresponding conjugate diameter. 47. Find where the tangents from the foot of the directrix will meet the hyperbola, and what angles they will make with the transverse axis. 48. Show that the perpendicular from the focus upon a polar with respect to an ellipse or an hyperbola meets the line drawn from the center to the pole on the corresponding directrix. CHAPTER XIII ELEMENTARY TRANSCENDENTAL FUNCTIONS 148. Definition. Any function of x which is not algebraic is called transcendental. The elementary transcendental functions are the trigonometric, the inverse trigonometric, the exponential, and the logarithmic functions, the definitions and the simplest properties of which are supposed to be known to the student. In this chapter we shall discuss the graphs and the derivatives of these functions. 149. Graphs of trigonometric functions. Ex. \. y = sinx. » ■ The values of y are found from a table of trigonometric functions. In plot- ting it is desirable to express x in circular measure ; e.g. for the angle 180° we lay off X = TT = 3.1416. When x is a multiple of tt, y = 0; when x is an odd TT multiple of — , y = ± 1 ; for other values of x, y is numerically less than 1. The graph consists of an indefinite number of congruent arches alternately above and below the axis of x, the width of each arch being tt and the height being 1 (fig. 147). Y Fig. 147 Tlie curve y = sin x may be con.structed without the use of tables by a method illustrated in fig. 148. Let Pi be any point on the circumference of a circle of radius 1 with its center at C, and let .40 be a diameter of the circle extended indefinitely. With a pair of dividers lay off on A produced a distance OiVj equal to the arc 0P\. This may be done by considering the arc OPi as composed of a number of straight lines each of which differs unappreciably from its arc. From ^i draw a line 266 TRIGONOMETRIC FUNCTIONS 267 perpendicular to A 0, and from Pi draw a line parallel to A 0. Let these lines intersect in Qi. Then iViQi = JfiPi = CPi sin OCPi. But CPi = 1, and the cir- cular measure of OCPi is OPi= ONi. If, then, we take ONi = x, NiQi= y. Fig. 148 Qi is a point of the curve y = sin x. By varying the position of the point Pi we may construct as many points of the curve as we wish. The figure shows the construction of another point Q2. Ex. 2. y = a sin bx. TT TT When X is a multiple oi ~, y = 0; when x is an odd multiple of — ? 2/ = ± o ; b 26 for all other values of x, y is numerically less than a. The curve is similar in its general shape to that of Ex. 1, but the width of each arch is now - , and its height is a. Fig. 149 shows the curve when a = 3 and 6 = 2. Fig. 149 Ex. 3. y = a sin {6x + c). c Place X = + x', y = y'. The equation then becomes y' = a sin 6x'. The graph is consequently the same as in Ex. 2, the effect of the term + c being merely to shift the origin. 268 ELEMENTARY TRANSCENDENTAL FUNCTIONS Ex. 4. y = a cos hx. This may be written y = asmlbx -\ — |, which is a cur\'e of Ex. 3. Hence the graph of the cosine function differs from that of the sine function only in its position. Ex. 5. y = sin X + ^ sin 2 x. The graph is found by adding the ordinates of the two curves y = sin x and y = ^ sin 2 X, as shown in fig. 150. ,-7/=^ sin 2x ^ -^y^sin x-t-jsin 231 -s y=sin X Fig. 150 Ex. 6. y = sin y = when - = Attt, i.e. when x = -, where k is any integer. Hence the X k " graph crosses the axis of x at the points 1, ^, ^, \, \, etc. Between any con- secutive two of these points y varies continuously from to ± 1 and back to Fig. 151 zero. It follows that as x approaches 0, the corresponding point on the graph oscillates an infinite number of times back and forth between the straight lines INVERSE TRIGONOMETRIC FUNCTIONS 269 y = ±1. It is therefore physically impossible to construct the graph in the neighborhood of the origin. This is shown in fig. 151 by the break in the cur%e. It should be borne in mind, however, that the value of y can be calculated for any value of x no matter how small. E.g. let x = — - ; then - = = 10 x 125 X 12 + — ir, and y = sin — = sin 75° = .9659. 12 ' 12 The value of 2/ is not defined for x = 0, and the function is discontinuous at that point. Ex. 7. 2/ = tanx. When X is a multiple of tt, y = 0; when x is an odd multiple of - , y is infinite, in the sense of §§ 11 and 68. The curve has therefore an iinlimited number of asymptotes perpendicular to OX, namely x = ± — » x = ± — » • • • , which divide the plane into an infinite number of sections, in each of which is a distinct branch of the curve, as shown in fig. 152. Fig. 162 150. Graphs of inverse trigonometric functions. The graphs of the inverse trigonometric functions are evidently the same as those of the direct functions, but differently placed with reference to the coordinate axes. It is to be noticed particularly that to any value of X corresponds an infinite number of values of y. '270 ELEMEIs^TARY TliANSCEXDENTAL FUoS'CTIONS Ex. 1. 2/ = sin-la;. From this x = sin y, and we may plot the graph by assuming values of y and computing those of x (fig. 153). Fig. 1o3 151. Limits of Fig. 154 Ex. 2. y = tan-ix. Then x = tan j/, and the graph is as in fig. 154. These curves show clearly that to any value of x corresponds an infinite number of values of y. sin^ and 1 — cos A In order to apply the h h methods of the differential calculus to the trigonometric functions, it is necessary to know the limits approached by — : — and as h approaches zero as a limit, it being assumed that h is expressed in circular measure. 1. Let AOB (fig. 155) be the angle h, r the radius of the arc AB described from as a center, a the ^ ' length of AB,p the length of the per- pendicular BC from B to OA, and t the length of the tangent drawn from B to meet OA produced in D. Fig. 155 ^^ p V' ^^^^^1^^ c ^1 V -^ 1 / ^^^ 1 / ■v^ / / ■^^ / / '^^v // ^v ,. / ^^. / B' CERTAIN LIMITS 271 Eevolve the figure on OA as an axis until B takes the position B'. Then BCB' =2p, BAB' = 2 a, B'D = BD. Evidently BD + DB' > BAB' > BCB', whence t> a> p. Dividing through by r, we have tap T r r that is, tan h> h> sin h. Dividing by sin h, we have cos h sin h or, by inverting, cos h < — - — < 1, Now as h approaches zero, cos h approaches 1. Hence > which lies between cos h and 1, must also approach 1 ; that is, -r . sin h . Lim —— = 1. 1 cos Jb 2. To find the limit of — — - — - » as h approaches 0, we proceed as follows : , ; 2sin^J sin'^^ , /sin^^ 1 — cos h ^ 2 ^ 2 ^ h 2 h h h ^\ Ik 2 \ 2 . h sm- Now as h approaches zero as a limit, — -^ approaches 1, as just shown, and therefore - ■ j approaches zero, by 2, § 94. Therefore Lim = 0. A±0 h 272 ELEMENTARY TRANSCENDENTAL FUNCTIONS 152. Differentiation of trigonometric functions. The formulas for the differentiation of trigonometric functions are as follows, where u represents any function of x which can be differentiated : d . du —- sm u = cos u—-} . (1) dx dx d . du ,-, — cos u= — sm u-—> (2) dx dx d . 2 <^^ /ov — tan -i^ = sec M — - > (3) dx dx d . ^ du ..^ -— COtW = — CSC M-—> (4) dx dx d du — sec u = sec u tan u-—> (5) dx dx d . du _^ -— cscw =— cscw cotM — -• (6) dx dx 1. By ( 0> S 96, -- sm % = -— sm w • -- • dx du dx To find — - sin u, we place y = sin u. du Then if u receives an increment Aw, y receives an increment A?/, where / * \ » . . , * s • o / Aw\ . Am Ay = sm {u + Azt) — sm u^ 1 cos i u + -— I sm — > the last reduction being made by the trigonometric formula . , „ a -\-h . a — b sm a — sm o = 2 cos — - — sm — - — 2 2 Then we have * a . Au . Aw V sm — — , sm -— Ay ^^ / Au\ 2 / , Au\ 2 — ^ = 2 cos I u + — — = cos u + — — Au \2/Aw V2/Aw 2 Let Au approach zero. By 2, § 94, . Aw sm— - Lim — = Lim cos ( w + — - ) Lim — r ' Aw \ 2 / Aw DIFFERENTIATION . 273 But Lini — ^ = -^ , Lim cos ( m + — - ) = cos u, and Lim Alt du \ 2 / = 1(§151). 2 Hence -— sin w = cos u du d . du and —- sm u = cos u - — dx dx 2. To find — cos u, we write dx ■ i-^ cos w = sin I — — t* Then -— cos w = -7- sin \— — ^t dx dx \l /tt \ d (it =°°%2-'7rf:;i2-''' <''y<'» "K \du du = — sm w ^- • dx 3. To find — tan u, we write sinw tan u = cos w ^, d ^ d smu Then -r tan -26 = -^ aa; dx cos -it cos u — sin u — sm u —- cos u ax ax (hvi^) ^^^) cos^u \"j \ /> / . K du (cos^u + siw'u) — cos^u (by (1) and (2)) du = sec w -7-- dx 274 ELEMENTARY TRANSCENDENTAL FUNCTIONS . 4. To find -T- cot u, we write ax cos u cot u — sin w ^, d ^ d cosu Then --cotu = : — ax dx sinw d d . sin u — cos u — cos ti — sin u = 3i^.^ (by (5). § 96) ^-sin^^-cos^.^ (by (1) and (2)) = — CSC M • — -• dx 5. To find — - sec u, we write dx sec u = = (cos u) \ cosu Then — sec u= — (cos ti)~ ^ -— cos u (by § 97) dx dx _ sin u du cos^w dx du = sec it tan u — — ■ (by (2)) 6. To find — CSC u, we write dx CSC w = -; = (sm u) . suiti Then — esc ?t = — (sin u)~^ — - sin u (by § 97) aa? dx = — CSC M cot w — • (by (1)) ax Ex. 1. y = tan 2 x — tan^ x = tan 2 x — (tan x)^. — = sec2 2 X (2 x) — 2 (tan x) — tan x dx dx dx = 2 sec2 2 X — 2 tan x sec^ x. DIFFERENTIATION 275 Ex. 2. A particle moves in a straight line so that s = A; sin bt, where t = time, s = space, and b and k are constants. Then velocity = w = — = bk cos bt, dt acceleration = a = — '- = — b^k sin bt = — 6%, dt^ ' force = F= ma = — mb^s. Let be the position of the particle when ^ = 0, and let OA = + k and OB = — k. Then it appears from the formulas for s and v that the particle oscillates forward and backward between B and A. It describes the distance _ 2 77" OA in the time — , and moves from B to A and back to B in the time 26 -6 The formula F = — mb^s shows that the particle is acted on by a force directed toward O and pro- portional to the distance of the particle from 0. The motion of the particle is called simple harmonic motion. Ex. 3. A wall is to be braced by means of a beam which must pass over a lower wall 6 units high and standing a units in front of the first wall. Required the shortest beam which can be used. Let AB = I (fig. 156) be the beam, and C the top of the lower wall. Draw the line CD parallel to OB and let EEC = 0. Then that is, when 1 = BC + CA = EC esc + DC sec = bcsc0 + a sec 9 — = — 6 CSC ^ cot ^ + a sec tf tan do _ asin^e - bco^^0 Si\vfi cos"^ 6 — = 0, when a sin^ 0=b cos^ 5, de bi tan = — • 270 ELEMENTARY TRAKSCENDEKTAL FUNCTIONS When e has a smaller value than this, a sin^d < h cos^^, and when d has a larger value, a sin^tf > b cos^tf. Hence Hs a minimum when tan d ■= — Then o* i = 6 CSC ^ + a sec tf 6 Vg^ + 6^ g VaJ + 6* g* = (a^ + b^)^. 153. Differentiation of inverse trigonometric functions. The formulas for the differentiation of the inverse trigonometric func- tions are as follows : 1. —&\n~^u= rr^ — > when sin"'w is in the first or the VI — w fourth quadrant; = —> when sin~'w is in the second or V 1 — w ^j^ third quadrant. 2. --T- cos~*w = — > when cos"'w is in the first or the ^ ~ ^ second quadrant ; > when cos~'w is in the third or the fourth quadrant. Vl — tt'^ dx ax 1 + u ax . d ^ , 1 du 4 -- cot-^u = — -— • ax 1 + u ax 5. — -sec"^^ = - — — > when sec" 'w is in the first or the dx . uVu' - 1 dx ^j^.^^ quadrant ; — > when sec~^ic is in the second wVit IX ^^ ^-^^ fourth quadrant. 6, — csc~'w= y^^z^ — > when esc ^w is in the first or dx uy/u' - 1 dx ^-^^ ^j^.^^^ quadrant ; — , when csc"'?^ is in the second or y/u^ — l dx uy/u the fourth quadrant. DIFFERENTIATION 277 The proofs of these formulas are as follows : 1. If y = sin"^w, then sin y = u. Hence ^^^^'^~'J~' (^"7 § 152) dy 1 du or • _£ = dx cosy dx But cos y = Vl — u^, when y is in the first or the fourth quad- rant, and cosy =— vl — w^ when y is in the second or the third quadrant. 2. If y = cos'^w, then cos y = u; dy du whence —siny-f- = or lA/Jb (a/Ju dy _ 1 du dx sin y dx But sin 3/ = + Vl— u^ when y is in the first or the second quadrant, and siuy = — Vl — ti"-^ when y is in the third or the fourth quadrant. 3. If y = tsinr^u, then tan y = u. Hence whence 4. If then Hence whence 2 dy du . dx dy _ 1 du dx 1+' u^ dx y = ■■ cot" 'u. coty = u. 2 <^V du dx dy _ 1 ( dx 1 + w' ( du 278 ELEMENTARY TRANSCENDENTAL FUNCTIONS 5. If y = sec"'tf, then sec y = u. Hence dy du secytany-f- = — , dx dx whence dy 1 du dx sec y tan y dx But sec y = u and tan y = + ^u^ — 1 when y is in the first or the third quadrant, and tan y = — ^li^ — 1 when y is in the second or the fourth quadrant. b. if y = csc"'w, CSC y = u. Hence dy du — CSC y cot y -T- = -rr-y dx dx whence dy 1 du dx CSC y cot y dx But CSC y = u and cot y = + v ?t^ — 1 when y is in the first or the third quadrant, and cot y = — vw^ — 1 when y is in the second or the fourth quadrant. Ex. 1. y = sin-i Vl — x^, where y is an acute angle. ^ Vl - (1 - a;2) y is the g-th root of the ^th power of a. If ic is a positive irrational num- ber, the approximate value of y may be obtained by expressing x approximately as a rational number. If a? = 0, 3/ = a" = 1. Finally, if x = — m, where m is any positive number, 3/ = a,""* = — • Practically, however, the value of a^ is most readily obtained by means of the inverse function, the logarithm ; for if then a; = log„y. When a = 10, tables of log- arithms are readily accessible. Suppose a is not 10, and let h be such a number that 10* = a, I.e. and h = log,oa. Then we have y = a^={lQy = W. Hence hx = log^^^y, ^ log,„y log,o2/ logio^ Fig. 167 Ex. 1. The graph oi y = log(i.6)X is shown in fig. 157. It is to be noticed that the curve has the negative portion of the y axis for an asymptote, and has no points corresponding to negative values of z. 280 ELEMENTARY TRAKSCENDEXTAL FUXCTIONS Ex. 2. The graph oi y = (1.5)* is shown in fig. 158. Fig. 158 155. The number e. In the theory and the use of the exponential and the logarith- mic functions, an important part is played by a certain irrational number, commonly denoted by the letter e. This number is defined by an X infinite series, thus : It will be shown in the second volume that this series converges; i.e. that the greater the number of terms taken the more nearly does their sum approach a certain number as a limit. Assuming this, we may compute e to seven decimal places by taking the first eleven terms. There results e= 2.7182818. ••. When y = e^,x is called the natural or Napierian logarithm of y. The student will discover as he proceeds with his study that the use of Napierian logarithms in theoretical work causes simpler formulas than would arise with the use of the common logarithm. Hence, in theoretical discussions, the expression logic usually means the Napierian logarithm. On the other hand, when the chief inter- est is in calculation of numerical values, as in the solution of tri- angles, logic usually means logjoX In this hook we shall use log x for log^x. Tables of values of log^a: and (f are found in many collections of tables, and may be used in finding the graphs. It is evident, however, that the graphs will not differ in general shape from those in Exs. 1 and 2 of § 154. We give the graphs of certain other functions which involve e and present other points of interest. GRAPHS 281 Ex. 1. y = e-=^. The curve (fig. 159) is symmetrical with respect to OY and is always above OX. When x = 0,y = 1. As x increases numerically y decreases, approaching zero. Hence OX is an asymptote. Y Fig. 159 Ex. 2. y -(ea + e »). This is the curve (fig. 160) made by a string held at the ends and allowed to hang freely. It is called the catenary. Ex. 3. y = e~°^sin6x. The values of y may be computed by multiplyi:ig the ordinates of the curve y _ e-ax by the value of sin bx for the corresponding abscissas. Since the values of sin6x oscillate between ± 1, the value of e-'"sin6x cannot exceed those of e-°^. Hence the graph lies in the portion of the plane between the curves y = e-«^ and ?/ = - e-"^. When x is a multiple of - , y is zero. The graph 282 ELEMENTAEY TIIANSCEKDE:N^TAL FUNCTIOI^S therefore crosses the axis of x an infinite number of times. Fig. 161 shows the graph when a = 1, 6 = 2 7r. 1 \ 1 \ ^\ /\^^ ~}^--t)^' K 1 Now as h approaches zero k approaches 0, and (1 + ^)* approaches e by the previous proof. Hence log(l + ky approaches log e, which is 1. Therefore e*— 1 Lim — - — = 1. »=o h 157. Differentiation of exponential and logarithmic functions. The formulas for the differentiation of the exponential and logarith- mic functions are as follows, where, as usual, it represents any func- tion which can be differentiated with respect to x, log means the Napierian logarithm, and a is any constant: dx dx d 1 I du -7- log W = ;- • (2) dx u dx — a-^a-loga^^. (3) dx "" dx ^ ' d , _ log„g du . dx u dx DIFFERENTIATION 285 dx du dx To find — e", place y = e". Then if w receives an increment Aw, du y receives an increment Ay, where Ay = e" + ''" — e" = e" (e^" — 1). Then Aw "" Aw Now let Aw approach zero. By § 94, Lim — ^ = 6" Lim — -^ Aw Aw But -..Ay ^y c? „ Lim -^ = -/■ = — e", Lu du du and Lim — - — = 1. Aw Therefore — e" = e", ait and d „ ^du 2. If y = logw, then e>' = u. Hence ydy _du dx dx whence dy 1 du _ldu dx e" dx u dx by 2, § 156 by(l) 3. Let y=a". Then it is always possible to find a quantity b such that a = e\ whence ^ = log a. 286 ELEMENTARY TRANSCENDENTAL FUNCTIONS Then y = {e")" = e"\ and dx dx dx = (log«)».|. 4 If y = log^-?^, then a« = u, and „ T dy du . a^ log a -^ = — - > dx dx whence dy 1 1 du dx log a u dx But if log a = h, a = e^, 1 whence (^ = e, and therefore \ = log.*. Hence ■ ^^\og^dv. dx u dx Ex.1, y = log(x2_4x + 6). iZj/ 2x — 4 by(i) by (3) dx x2 _ 4 x + 6 Ex. 2. y = e-=^. _ : ■ da; ■ Ex. 3. 2/ = e-'^cosftx. dy .d,, d,,, ,, ., -^ = cosox — (e-'«^) + e-"^ — (cosox) = — ae-°^cos6x — oe-»*sinc>x. dx dx dz , EXPONENTIAL FUNCTION 287 158. An important property of the exponential functions is expressed in the following theorem : If the rate of change of a function is proportional to the value of the function, the function is an exponential function. Let -^ = ay. Then l^ = a. y dx Hence log y = ax + c^, or y = g^^^+'^i = e'^'e'^ = ce'^. Ex. Let p be the atmospheric pressure at the distance h above the surface of the earth and p the density of the air. We will assume that the density is proportional to the pressure. Then if po and po are the density and the pressure respectively at the surface of the earth, Po Po' whence p = — • p. '^ Po Let now the height h be increased by a distance Ah. The pressure will be increased by an amount Ap, where — Ap is equal to the weight of a column of air standing on a base of unit area and having a height Ah. If p is the density at the height h and p — Ap the density at the height h + Ah, it is evident that the weight of this column of air lies between (p — Ap) Ah and pAh ; that is, (p — Ap) Ah< — Ap < pAh, Ap whence p — Ap < r < P- Ah Taking the limit, we have dp ... .^Ap PO ■— z=Limit— =- p = - ^p. dh Ah Po -Bih ... . Therefore p = ce po . .•..•■... _?o A ..... •Since when it = 0, e i'o =1 and p = po, it follows that c = p^.' "' • Hence p = poe I'o , , Po, Po /. = -log^. 288 ELEMENTARY TRANSCENDENTAL FUNCTIONS 159. Sometimes the work of differentiating a function is sim- plified by first taking the logarithm of the function and then applying the formulas of this article. Ex. 1. Let = Vr X2 + X2 - Hence and + X''' = ^l0g(l-x2)-^.l0g{l+«2). 1 dy _ X X y dx~ I - x^ 1 + X* -2x ~ (1 - x-2) (1 + x2) dy _ —2xy dx~ (1- cc2) (1 + a;2) -2x l-x2 x'^) \1 (1 - x2) (1 + x-i) \l + x3 -2x (l + x2)V'l-x< This method is especially useful for functions of the form u", where u and v are both functions of x. Such functions occur rarely in practice, and cannot be differentiated by any of the formulas so far given. By taking the logarithm of the function, however, a form is obtained which may be differentiated. Ex. 2. Let y = x«'°^. Then log y = log (x"" ^) = sin X • log X. Therefore — - = (sin x) - + cos x ■ log z, y dx X dy and -^ = x"° ^- J • sin X + x*'°^ cos x • log x. dx 160. Hyperbolic functions. Certain combinations of exponential functions are called hyperbolic functions. In their names and properties they are analogous to the trigonometric functions, but the reason for this cannot be shown at present. The fundamen- tal hyperbolic functions are the hyperbolic sine (sinh), the hyper- bolic cosine (cosh), the hyperbolic tangent (tanh), the hyperbolic HYPERBOLIC FUNCTIONS 289 cotangent (coth), the hyperbolic secant (sech), and the hyperbolic cosecant (cosech), defined by the equations sinha; = cosh X = tanh X = coth a; = sech X = cosech X = e^—e-^ 2 ' e^+e-^ 2 ' sinha; e^—e X cosh a: e^-f-e" X cosh X e^^+e- -X sinh X e* — e~ X 1 2 cosh a? e^'+e- X 1 2 sinh a? e^ — e" Fig. 164 Fio. 105 The graph of sinh x is given in fig. 164, that of cosh x in fig. 165, and that of tanh x in fig. 166. 290 ELEMENTARY TRANSCENDENTAL FUNCTIONS Relations between hyperbolic functions may be derived by expressing each in terms of the exponential functions. The student may in this way prove the following relations: cosh'^a; — sinh^a; = 1, tanh^ic + sech^a? = 1, coth'* X — cosech^ x = \, sinh (a? ± y) = sinha? coshy ± cosh a; sinhy, cosh {x ± y) = cosh X cosh y ± sinh x sinh y, ^ 1 / , X tanhaj±tanhy tanh(a;±y)— ^ Fig, 166 The derivatives of the hyperbolic functions are readily obtained by differentiating the equations which define them. We have in this way : , , •' a . , , an -T— smh u = cosh u-—> ax ax d , . , du -r- cosh u = smh u — - , dx dx d ^ , 1 2 du -— tanh u = sech w -— , dx dx d ^, , ^ du -r- coth u=— cosech u—-, dx ax -T- sech u =— sech u tanh u — - , dx dx —— cosech u = — cosech u coth u — -• dx dx INVERSE HYPERBOLIC FUNCTIONS 291 161. Inverse hyperbolic functions. If X = sinli y, then ' y = smh"^a;, called the inverse hyperbolic sine of x. This function may be expressed as a logarithm as follows : We have y = sinh~ ^ x, and X = sinh y = — Placing e~^= — and clearing of fractions, we have e^''—2xe''=l. Treating this as a quadratic equation in «", we have e^=x±Va^+l; but since we know that for any real value of y, e" is positive, we discard the minus sign before the radical and have y = sinh"' ic = log {x + vV+T). In the same manner, the student may prove the following : cosh~'a; = log {x ±Vic^— 1) = ±log tanh~'a; = | log = ± log (x + ^y? — 1)' coth"'a;= | log \—x x + 1 x — 1 , l±Vl-a;*-^ _^, l + Vl-ar' sech"' a? = log = ± log ^ X X cosech"^' ic = log • 292 ELEMENTARY TRANSCENDENTAL FUNCTIONS The derivative of the inverse hyperbolic functions can be obtained by differentiating the expressions just obtained, or by proceeding in the same manner as in § 153. In either way we find: , . , a . , , 1 du — sinh u =■—= — J dx V M^ -\-l dx d , _, , 1 du — cosh '■u = ±—: —-, dx ^u'-l dx d ^ , _, 1 du -— tanh ^u = - r-— , dx 1 — u dx d ^. _, 1 du --coth u=- --—, dx 1 — udx d , , 1 du — sech ^u = zf — — , dx u\l— u^ dx d , _, 1 du — cosech u = dx uV 1+ u^ dx Ex. Consider the motion of a particle of unit mass falling from rest, and impeded by a force proportional to the square of its velocity. The total force acting on the particle is then g — kv'^, where g is the acceleration due to gravity, and A; is a constant. Hence — = g-kv^; at 1 dv whence = 1, g - kv^ dt 1 1 dv , or = 1. ff ^ k . dt 9 To bring this under one of the known formulas of differentiation we will place whence dv _ jg du We have, therefore, — - — = 1 ; whence — — =tanh-i u z=t + c, Vkg i-u^ dt 1 {k tanh-i-%/-i' = < + f. TRANSCENDENTAL EQUATIONS 293 But since the body falls from rest, when t = 0, v = 0; therefore c = 0. The equation may be written that is, Hence V = ■%- tanh t Vkg^ ^ sinh t ^ 1 or a < — 1, there are no real solutions ; otherwise there are an infinite number of solutions. TT Let us call the smallest positive root Xj, where < xi < - if a is positive, and ] r -Ztt/ Nc'T / \7r 2w/ \ y \ „.» / - Fig. 167 TT < Xi < 2 TT if a is negative. The value of Xx must be found from a table or approximately from the graph. The next largest positive root is then tt — Xi when a is positive, and 3 tt — Xi when a is negative ; and all other roots, positive or negative, are found by adding or subtracting multiples of 2 tt. Hence the general solution is 2 A;7r + Xi and (2 A; + 1) rr - Xi, or, more compactly written, /c7r + (-l)'^^Xi, where k is any positive or negative integer or zero. Ex. 2. cosx = a. The general solution is 2 ^tt ± xi, where Xi is the smallest positive solution and k is an integer or zero. The proof is left to the student. Ex. 3. tanx = a. The general solution is kiz + Xi. The proof is left to the student. 294 ELEMENTARY TRANSCENDENTAL FUNCTIONS Ex.4, cos 2 a; = 2 cos X. When an equation involves two or more trigonometric functions it is well to write it in terms of one. The above equation may be written 2 cos^x — 1=2 cosx, which is a quadratic equation in cos x. Solving, we have in the first place cos x = -J- ± -^ V3, but the plus sign may be disregarded, since for real angles cos x is not greater tlian 1 numerically. The equation cos X = J - 1 Vs is now to be solved as in Ex. 2. There results x = 2 Arir i 1.946. Ex. 5. tanx = kx. The roots of this equation are the abscissas of the points of intersection of the curve y — tanx and the straight line y = kx (fig. 168). Fig. 168 The two intersect at the origin, but the other intersections depend upon the value of k. Since the slope of the curs'e y — tan x is 1 when x = 0, and > 1 when 01, 0 1, the smallest positive root lies between and — ; if < A; < 1, the smallest positive root lies between tt and — ; and if fc < 0, the smallest positive root lies between — and tt. We shall now find the smallest positive root in the special case tan X = 2 X. We must first locate the root (§ 47), either by the graph or by means of a table. If a table is used, it must be one in which angles are given in radians. We shall use the table on page 132 of Professor B. O. Peirce's "Short Table of Integrals." We find, by looking for a place in the tables where the tangent of an angle is approximately equal to twice the angle, that when x = 1.1G36 (G6° 40'), tanx = 2.3183, and when x = 1. 1665 (66° 50'), tan x = 2.3369. Consider now the curve y = tan x — 2 x. When xi = 1.1636, yi = - .0089, and when Xg = 1.1665, yz = .0039. Hence the curve intersects OX between xi and X2, and a root of the equation tan X — 2 X = is therefore located to two decimal places. To locate the root more closely we will use the method of § 63. We have ^ = sec2x - 2, dx and — ^ = 2 tan x sec^ x, both of which are positive when x is between Xi and Xg. Hence that portion of ^i^^onrye y = tanx-2x appears as in fig. 64, (1), and its intersection with OX lies between the tangent at (X2, 2/2) and the chord connecting {xi, yi) and {X2, 2/2)- The tangent at (X2, 2/2) is y - .0039 = 4.461 (x - 1. 1665) ; 0128 the chord is y -. 0039 = '-^— (x - 1 . 1665) , and the point of intersection of each with OX is found to be x = 1.1656 to four places of decimals. This is therefore the root of the equation to four decimal places. Ex.6. e^-4x2-2x + 3 = 0. The roots of this equation are the abscissas of the points of intersections of the curves y = e^ and j/ = 4x2 + 2x - 3, and may be found graphically or by means of tables to lie between - 1 and - 2 and between and 1. To determine the root between and 1, we place y = e^ - 4x2 - 2x + 3. When Xi = 0, 2/1 = 4, and when X2 = 1, 2/2 = — •282. 296 ELEMEXTAKY TKAXSCENDENTAL FUNCTIONS Also ^ = e^-8x-2, dx and — = e^ - 8, which are both negative when x is between and 1. Hence the portion of the curve in question has the shape of fig. 64, (4), and its intersection with OX lies between that of the tangent at {x^, y^) and that of the chord connecting (xi, yi) and (Xj, yo). The tangent is y + . 282 = - 7.282 (x-1), which intersects OX when x = .97 — . The chord is 2/ + .282 = - 4.282 (x-1), which intersects OX when x = .93 + . If we now place Xi = .93, yi = .2149, and if Xj = .97, 2/2 = — -0657, the tan- gent at (x^, y,) is ^ ^ ^.„ ^ _ ^^^gi (^ _ .97), which intersects OX where x = .9608— ; and the chord between (Xi, yi) and («2, Vi) is _ oaof? y + .0657 = (X - .97), .04 ^ ^' which intersects OX where x = .9606+. Hence a root of the equation lies between .9606 and .9608. PROBLEMS Plot the graphs of the following equations : 1. y = ctnx, 11. y = xsin-. x 2. y = secx. j _ 12. y = x^sin-- 6. y — cscx. x 4. y = versx. . 13. y = el^. 5. y = ^ sin 3 x. L-^x is ,10 14. y = xe ^ . 6. y = sm X + ^ sin 3 X. ^ 7. y = sin X + sin 2 x. 15. y = xe^. 8. y = 2 sin X - sin 2 X. 16. y = log (sin x). 9. y = cosx+^cos3x. 17. 2/ = tan- 1 (ox + 6). X — 1 10. y = 1 - A cosx - * cos 2 x. 18. y = log . ■* X + 1 19. Plot the graph of the equation y = - sin x, and determine what relation it has to the hyperbolas xy = ± 1. 20. Plot the graph of the equation y = sin x', and show that the distance between two consecutive intercepts on OX approaches zero as a limit. PROBLEMS 297 dt/ Find — in each of the following cases : dx 21. 2/ = sin (ox + 6) cos (ax — 6). 45. y = sin- 1 (2 x Vl — x^). 22. y = tan (ax + 6) ctn (ox + c). ._ , 2x — 1 46. y = csc-i 23. 8ec4x 24. ctn 2 X + 2 y — • csc2x 25. sec3x y = - — ^ — -^- 2 V^ X 47. y = sec- 1 V4 x2 + 4 X + 2. 48. y = CSC- • - I x2 + — I . 49. y = e^ + 2ar. lauox-t-i L2 _ a2 26. y = csc2x-ctn2x. 50. y = \ogy^^^ ^ ^^- 27. y = sec"'nxcsc»mx. ^ 28. 2/ = sec22x + tan2x. 51. y = a^i-=^e^i-^. 29. 1/= ctn4xcsc2x. 52. y = log(2x + 1 + 2 Vx2 + x). 30. y = sin(xcosx). 53. y = e»: -v^a. 31. y = (cos2x + f)sin8x. 54. y = o*""^. 32. 2/ = (2 sec^x + 3 sec2x) sinx. 55. y = x^ logx2. sin2x 56. y = otan^^ec^'^. 33. y = Vl — cos2x 57. y = tan 2 x • a"*= '- "'. 34. y = cos Vl — x2. 58. y = e(<' + *)''sinmx. __ sin2x cos2x 59. w = csc-i(sec2x). 35. y = _ sinx COSX CA / , V tan-iV^ SO. y — {x + a)e ^ <-. 36. y = sin-i :• , e"' - e-^ Vi + x2 61- 2' = ta"-'^^rqr^x- 37. y = tan-i-^=. g2 ,, - ip.. ^ ^^" '^ + ^ Vrr^ '*''• ^~ "^^ tanx + 3 38. y = cos-i:l-^- . 63. y = sec-i^ — 1 + x ^ (J 64. y = e^=<'««cos(xsina). 39. y = sec- 1 — --• 65. 2/ = log tan (x2 + a2). a-x 66. y = x cos->x - Vl - x2. 40. y = sin-i ^ " + * 67. 2/ = - log (sec ax + tan ox). 41. y = sec-il(|Vx + ^y gg ,^^(a; + oVr3^)e«--'-. 42. 2/ = CSC- 1 (x2 + 2 X). 69. y = log Vl + x2 + x ctn- 1 x. 43. 2/ = ctn-i^^-2tan-:l 70. y = ^^ - log VT^. ^ x2 - a2 a Vl-x2 .. 1 ,6 + acosx _. e<^(o sin mx — m cos mx) 44. y = — cos-^ ; 71. v = — ^ ; ;, " Va2 - 62 a + 6cosx " m^ + a^ 298 ELEMENTARY TRAJSTSCENDENTAL FUNCTIONS 72. y = a;2 ctn-i _ -|- a^ tan-i ax. X a -g , , * X sin-ix X X y = \og( ^ )- \1 + Vl-X2/ 74. y = log(x + Vx2 - a2) + cos-i 75. y = log(x + Vx2 - a2) _ csc-i-. 76. y = tan-i Vx^ - 2x - Ig ^jg^lj) . Vx2 — 2 X ,/ /a -6^ x\ =^ Un- 1( \ tan - I . b2 \\a + 6 2/ 77.2/ = -^ Va2 78. 2/ = log tan (2 x + 1) + esc (4 x + 2). mn r^ ', 1 "^2 OX — X2 79. y = v2 ax — x^ + a cos-i 80. 2/ = X Va2 + x2 + a21og(x + VoM^). o, , 2 v^ - 1 2V^sec-'2Vx 81. y = log ^ — \2Vx+l V4x-1 ort X — a ,/r r , a*^ . ,x — o 82. y = V2ax — x2-) sin-i 83. y = tan- 1 (x - Vx2 - 1) + log Vx* - 1 84. y = -M^±IL + csc-i V^^^T^. Ve2^+ 2e*- 1 85. y = ^ Vl - x2 _ A _ ?!\ tan-i Vl _ x2. 86. 2/ = -^ log , ^ + log(x + Vl + x2). 2V2 V2 + 2x2 + x 87. y = (sin V^)'*°^. 90. y = (x)««. 88. y = . Find — in each of the following cases : dx 93. x!' + secxy = 0. 97. e^ sin y - ei' cos x = 0. 94. ytan-ix-2/2 + x2 = 0. ^8. y sin x + x cos 2/ = x2^. 95. ysinx — cos (x — y) = 0, 99. 7/ logx = xsiny. ni> 100. X2/ = tan-i-. 96. ye"v = ox™. '^ y PROBLEMS 299 1-x , 1+x „. , dy d'hj dhj Find -^, -4, -4 m each of the following dz dx^ dx^ ^ 101. log (x2 + 2/2) - 2 tan-i ?^ = 0. 103. log ^— ^ - log i^t^ = 1. X l+y l-y 102. e- + ev = l. 104. x-2/ = log(a; + y). 105, &^+y = x«. 106. At what points is the curve j/ = sin x + sin 2 x parallel to the axis of x ? 107. What value must be assigned to m that the curve y = h tan-'(x+m) bmx may be parallel to OX at the point the abscissa of which is 1 ? 108. Find the angle of intersection of the curves 3/ = sinx and y = cosx. 109. Find the angle of intersection of the curves y = sin x and y = sin (x + a). 110. Find the angle of intersection of the curves y = sinx and y = sin2x. 111. Show that the portion of the tangent to the curve o, a + Va2 - x2 y = - log ===: — V a^ — x2 2 a - Va2 - x2 included between the point of contact and the axis of y is constant. (From this property the curve is called the tractrix.) 112. Find the points of inflection of the curve y = 2 sin x — ^ sin 2 x. 113. Find the points of inflection of the curve xy = a^log-- 114. Find the points of inflection of the curve y = e-^ 115. Prove that the curve y = ^x — ^sinx + j^ sin 2 x has an indefinite number of points of inflection, and that two of them lie between the points for which x = 6 and x = 10 respectively. 116. Plot the curve y = sin^x, finding maxima and minima, and points of inflection. 117. Plot the curve y = e-«^cos6z, and prove that it is tangent to the curve y = g-oa: wherever they have a point in common. Find maxima and minima and points of inflection of this curve when a = 6 = 1. 118. Plot the curve y = x"e-^ (n > 0), finding maxima and minima and points of inflection. 119. A body moves in a plane so that x = a cost + b, y = a sint + c, where t denotes time and a, 6, and c are constants. Find the path of the body, and show that its velocity is constant. 120. A rectilinear motion is expressed by the equation s = 5 — 2 cos' t. Show that the motion is a simple harmonic motion, and express the velocity and the acceleration at any point in terms of s. 300 ELEMENTARY TRANSCEKDE:N^TAL FUNCTIOXS 121. A, the center of one circle, is on a second circle witli center at B. A moving straiglit line, AMN, intersecting the two circles at M and N respec- tively, has constant angular velocity about A. Prove that BN has constant angular velocity about B. 122. Two particles are moving on the same straight line, and their dis- tances from the fixed point O on the line at any time t are respectively X = a cos ut and x' = a cos lut-] — j, w and a being constants. Find the greatest distance between them. ^ ' 123. A ladder b ft. long leans against a side of a house. Its foot is drawn away in the horizontal direction at the rate of a ft. per second. How fast is its center moving ? 124. If a particle moves so that s = e- 2 «^' (a sin ht + b cos hi), find expressions for the velocity and the acceleration. Hence show that the particle is acted on by two forces, one proportional to the distance from the origin and the other proportional to the velocity. Describe the motion of the particle. 125. If s = ae*"' -|- be-'-', show that the particle is acted on by a repulsive force which is proportional to the distance from the point from which s is measured. 126. BC is a rod a ft. long, connected with a piston rod at C, and at B with a crank AB b ft. long, revolving about A. Find Cs velocity in terms of AB's angular velocity. 127. A man walks along the diameter, 200 ft, in length, of a semicircular courtyai-d at a uniform rate of 5 ft. per second. How fast will his shadow move along the wall when the rays of the sun are at right angles to the diameter ? . 128. How fast is the shadow in the preceding problem moving if the sun's rays make an angle a with the diameter ? 129. Given that two sides and the included angle of a triangle have at a certain moment the values 6 ft., 10 ft., and 30° respectively, and that these quantities are changing at the rates of 3 ft., — 2 ft., and 10° per second respec- tively, what is the area of the triangle at the given moment, and how fast is it changing ? 130. One side of a triangle is I ft., and the opposite angle is a. Find the other angles of the triangle when its area is a maximum. 131. A tablet 8 ft. high is placed on a wall .so that the bottom of the tablet is 20 ft. from the ground. How far from the wall should a person stand in order that he may see the tablet at the best advantage, i.e. in order that the angle between the lines from the observer's standpoint to the top and the bottom of the tablet may be the greatest ? 132. A weight P is dragged along the ground by a force F. If the coefficient of friction is K, in what direction should the force be applied to produce the best result ? PROBLEMS 301 133. An open gutter is to be constructed of boards in such a way that the bottom and the sides, measured on the inside, are to be each 4 in. wide, and both sides are to have the same slope. How wide should the gutter be across the top in order that its capacity may be as great as possible ? 134. Above the center of a round table is a hanging lamp. What must be the ratio of the height of the lamp above the table to the radius of the table that the edge of the table may be most brilliantly lighted, given that the illumi- nation varies invereely as the square of the distance and directly as the cosine of the angle of incidence '? 135. A steel girder 27 ft. long is to be moved on rollers along a passageway and into a corridor 8 ft. in width at right angles to the passageway. If the hori- zontal width of the girder is neglected, how wide must the passageway be in order that the girder may go around the corner ? 136. Find the area of an arch of the curve tj = sin x. 137. Find the area bounded by the axis of y and the portion of the curves ?/ = sin X, y = cos x, lying between x = and x = tt. 138. Find the area bounded by the portions of the curves y =: ^ sin 2 x and ?/ = sin X + ^ sin 2 x that extend between x =^ and x — tt. 139. Find the area between the curve y = e^, the axis of x, and the ordinates X = and x = 1. 140. Find the area bounded by the axis of x, the catenary, and the ordinates X = ± a. 141. Find the area bounded by the axis of x, the curve y = -, and the ordinates x = 1 and x = 2. ^ 142. Find where the ordinate of the witch should be drawn in order that the area between that ordinate, the witch, the axis of y and the axis of x should be equal to the area of the circle used in the definition. 143. Show that for the catenary — = -(e" + e~«), and thence find an expression for the length of s. 144. Find the curve the slope of which at any point is k times the reciprocal of the abscissa of the point, and which passes through (2, 3). 145.. Find the curve the slope of which at any point is k times the ordinate of the point, and which passes through the point {a, h). 146. Find the space traversed by a moving body in the time t if its velocity is proportional to the distance traveled. Solve the following equations : 147. tanx = cosx. 152. tanx = x. 148. cos2x = Jcosx. 153. tanx=^x. 149. sin 2 tf cos 2 fl 4- 2 sin ^ = 0. 154. x - ^ .sin x = ■^^. 150. sin4x - 2sinxcos2x = 0. 155. e^=«2. 151. sin*x+3cos*x-4sin2xcos2x = 0. 156. logx = ^x. CHAPTEE XIV PARAMETRIC REPRESENTATION OF CURVES 163. Definition. Thus far we have considered a curve as represented by a single equation connecting x and y. Another useful method is to express x and y each as a function of a third independent variable ; thus : where t is an independent variable and f^{f) and f^{t) are continuous functions of t. As t varies, x and y also vary, and the point {x, y) traces out a curve. By eliminating t be- tween the two equations the curve may often be expressed by a single equation between X and y. 164. The straight line. Let Pi{x^, 2/1) (fig- 169) be a fixed point on a straight line and <^ be the angle which the line makes with a line P^R parallel to OX. Let P{x, y) be any point on the line, and r the distance from P^ to P, where r is positive when P is on the terminal line of ^, and negative when P^ is on the backward exten- sion of the terminal line. Then, for all possible positions of P Fig. 169 x — x^ y — y^ — -— = cos, y — d sin^, Fig. 170 the parametric equations of the circle with <^ as the arbitrary parameter. 166. The ellipse. Take the ellipse ^ + ^ = 1 (a,>h) and on its major axis as a diameter construct a circle. Take F{x, y) (fig. 171) any point on the ellipse, draw the ordinate MF 304 PARAMETRIC REPRESENTATION OF CURVES and prolong it imtil it meets the circle in Q. Call the coordinates of Q {x, y'). Then from the equation of the circle y '=V^^ and from the equation of the ellipse Fig. 171 ^ a ^ Hence y == — y'. Draw the line OQ, mak- ing the angle XOQ = , y = ^ sii^ ^> the parametric equations of the ellipse. ^ is called the eccentric angle of a point on the ellipse, and the circle x^+'i^=o?' is called the auxiliary circle. Ex. The parametric equations of an ellipse may be used to find its area. For if A is the area bounded by the ellipse, the axis of y, the axis of x, and any ordinate MP (fig. 171), then (6, § 109) dA — = V- dx dA dA dA _d^_ d dx dx — asinc^ But (1) and y = 6 sin <^, Therefore (1) is equivalent to dA d(j) Hence A = abf ah sin2 (p = ab sin 2 cos 2 (^ — 1 + c. THE CYCLOID 305 When <6 = - , ^ = ; hence c = . 2' 4 Therefore A = ab(^^ 2 4/ When (f> = 0, A is one fourth the area of the ellipse. Therefore the whole area of the ellipse equals Trab. 167. The cycloid. If a circle rolls upon a straight line each point of the circumference describes a curve called a cycloid. Let a circle of radius a roll upon the axis of x and let C (fig. 172) be its center at any time of its motion, N its point of contact with OX, and P the point on its circumference which describes the cycloid. Take as the origin of coordinates, 0, the point found by rolling the circle to the left until P meets OX. Then OJ:i= arc FK Draw 3fP and CJSF each perpendicular to OX, PR parallel to OX, and connect C and P. Let NOP = 0. Then x = OM=ON- MN = arc NP - PR = a(j) — a siu . y = MP = NC — RC = a — a cos (f). Hence the parametric representation of the cycloid is x= a(). 306 PARAMETRIC REPRESENTATION OF CURVES By eliminating the equation of the cycloid may be written x== a cos but this is less convenient than the parametric representation. At each point where the cycloid meets OX a sharp vertex called a cusp is formed. The distance between two consecutive cusps is evidently 2 rra. 168. The trochoid. When a circle rolls upon a straight Une, any poiat upon a radius, or upon a radius produced, describes a curve called a trochoid. M ^ Fig. 173 Let the circle roll upon the axis of x, and let C (figs, 173 and 174) be its center at any time, N its point of contact with the axis of X, F(x, y) the point which describes the trochoid, and Fig. 174 K the point in which the liae CP meets the circle. Take as the origin the point found by rolHng the circle toward the left until K is on the axis of x. Then ON^iixcNK. THE EPICYCLOID 307 Draw PM and CN perpendicular to OX, and through P a line parallel to OX, meeting CN or CN produced, in R. Let the radius of the circle be a, CP be h, and NCP be <^. Then x=^OM = ON-MN = sivc NK-PB = a^ — h sin . Then whence arc KN = h(f), arc NP = aO : We now have x = Oil = OL-h LM = OC cos KOC - CP cos SPC = (a -i-h) cos — a cos (^ + ^) = (a + 6) cos 9 — a cos 9. y = MP = LC-RC = OC sin KOC - CP sin ^PC = (a + &) sin ^ — a sin (^ + ^) / , 7,\ • ji . a + b = (a + 0) sm 9 — a sin 9. The curve consists of a number of congruent arches the first of which corresponds to values of between and 2 ir, tliat is, to- 2(177 values of ^ between and Similarly the Xth arch corre- sponds to values of between — — -— ^ — and — Hence b the curve is a closed curve when, and only when, for some value of k, ^^ — is a multiple of 2 tt. If a and b are incommensurable, CL 7) 10 this is impossible, but if - = — > where — is a rational fraction in b q q its lowest terms, the smallest value of Tc = q. The curve then con- sists of q arches and wiuds p times around the fixed circle. THE HYPOCYCLOID 309 170. The hypocycloid. When a circle rolls upon the inside of a fixed circle, each point of the rolling circle describes a curve called the hypocycloid. If the axes and the notation are as in the previous article, the equations of the hypocycloid are x = (b — a) cos ^-\- a cos rf>, a y = (b — a) sin . The proof is left to the student. The curve is shown in fig. 176. Fig. 176 171. Epitrochoid and hypotrochoid. The epitrochoid and hypotrochoid are generated by the motion of any point on the radius of a circle which rolls upon the outside or the 310 PARAMETRIC REPRESENTATION OF CURVES inside of a fixed circle. If h is the distance of the generating point from the center of the moving circle, and the notation is otherwise the same as in the previous articles, the equations of the epitrochoid are x = {a-\-h) cos (f> — h cos (f>, y = (a + b) sin.(f) — h sm c^i, a Fig. 177 and of the hypotrochoid are T x = (b~ a) cos , y = (h — a) sin d> — h sin 6. a THE EPITROCHOID 311 The proofs are left to the student. The curves are shown in figs. 177, 178, and 179, 180 respectively. Fig. 178 172. The involute of the circle. If a string, kept taut, is unwound from the circumference of a circle, its extremity describes a curve called the involute of the circle. Let (fig. 181), be the center of the circle, a its radius, and A the point at which the extremity of the string is on the circle. Take as the origin of coordinates and OA as the axis of x. Let P {x, y) be a point on the involute, PK the line drawn from P tangent to the circle at K, and ^ the angle XOK. Then PK represents a portion of the unwinding string, and hence KP = Q.VC AK = a<^. THE INVOLUTE OF THE CIRCLE 313 Now it is clear that for all positions of the point K, OK makes TT an angle — -^ with Y. Hence the projection of OK on OX is always OK cos^= a cos^, and its projection on OY is OK cos 7r\ . TT ^ — — I = « sin<^. Also KP always makes an angle ~ ^ with ( Fig. 181 OX and ir — (j> with Y. Hence the projection of KP on OX is KP cos(<^ — — I = a<^ sin^, and its projection on OF is KP cos (tt — (/>) = — a^ cos. The projection of OP on OX is x, and upon OY is 2/. Hence, by the law of projections, § 15, X = a cos^ + a sin(f>, y = a sin ^ — ci<^ cos ^. 173. Time as the arbitrary parameter. An important use of the parametric representation of curves occurs in mechanics in find- ing the path of a moving point acted on by known forces. Here the independent parameter is usually the time. 314 PARAMETRIC REPRESENTATION OF CURVES Ex. 1. A particle moves "in a circle with uniform velocity, k. Then, if s represents the arc traversed, and a the radius of the circle, s = kt and d> = - = — . a a Therefore the equations of the circle are (§ 165), U x = a cos — , a . kt w = a sin — . a This shows that the projections of P on the coordinate axes have simple harmonic motions of the same amplitude. Ex. 2. A particle Q moves with uniform velocity along the auxiliary circle of an ellipse (§ 166) ; required the motion of its accompanying point, P. kt As in Ex. 1, = — . Hence the equations of the path are kt X = a cos — , a ^ . kt y = sm — , a showing that the projections of P upon OX and OY have simple harmonic motion of amplitudes a and b respectively. Ex. 3. A projectile is shot with an initial velocity vq in an initial direction which makes an angle a with the horizontal direction. Then the initial com- ponent of velocity in the horizontal direction is vo cos a and in the vertical direction is Vq sin a. If the resistance of the air is neglected, the only force acting on the projectile is that of gravity. Hence if we take the origin at the initial position of the projectile, and the axis of x horizontal, we have ¥'="' d^y _ dt^ ^' which give X = Cit -r C2, But when t = = 0, we have X = 0, y = = 0. y = -yt^ + Cst + €4. dx , dy , — = Vo cos a, and -^ : dt dt Vo sm a. Hence the parametric equations of the path of the projectile are X = Vot cos a, y — Vot sin or — J gt^. Eliminating t from these equations, we have y = X tan a — 2 Vq cos2 a or 2 Vq y cos^ a = 2 Vo^x sin a cos a — gx^ which shows that the curve is a parabola. THE DERIVATIVES 315 174. The derivatives. When a curve is defined by the equations dy (1) dt Ex. For the cycloid q X = a{<(> — sin^), we have, by (8). §96. | = | y = a(l — cos^y, dy dy d4> a sin -cot*^ dx dx a(l— COS0) 2 d

i^ being perpendicular to PQ, is the normal. If it is required to find -r-^' we may proceed as follows : d (dy d?y d {dy\ _ dt \dx da? dx \dx/ dx di (2) Ex. For the cycloid dy _ dx~ cot-, 2 dx d(p ~ a(l — cos(^) = 2 sin2 - 2 d-^y — cosec2 - 2 2 2asin2- 2 1 dx2 4 a sin* 2 31G PARAMETRIC REPRESENTATION OF CURVES Formula (2) may be expanded as follows: d (dy d^y dx dj^x dy ~dt~~d¥'di dxV dt) dy\_d Idt \ _ df dt \dx) dt\dx\ \dil ^y dx dj^x dy cPy 'de'di~~dell dj? /dxV \dt} /dsV By multiplying equation (3), § 105, by ( — ) > we have \dt/ dsY_/dxY /dyV dt) ~ \dt/ \dt (3) (4) 175. Application to locus problems. In finding the Cartesian equation of the locus of a point which satisfies a given condition, it is often convenient to employ the principles of parametric rep- resentation ; for by fixing the attention upon a single point of the required locus, it is frequently possible to express its coordi- nates in terms of a single parameter. The required equation is then found by eliminating the parameter. Ex. 1. Locus of the point of inter- section of perpendicular tangents to a parabola. Let the parabola be y^ = i px (fig. 183), and let the equation of any tangent to it be written (§ 88) y = mx + P (1) If m is replaced by 110 y=( )x + _L. \ mj 1 , we have Fig. 183 X mm, m (2) as the equation of a tangent perpendicular to (1). Therefore, if P(x, y) is the point of intersection of (1) and (2), P is any point of the locus. LOCUS PROBLEMS 317 Solving (1) and (2), we find and X = — p (S) (4) which are the parametric representations of the locus, the parameter evidently being m. But for all points of the locus x = — p, and (3) is the Cartesian equation of the locus. It is to be noted that in this example the elimination of the param- eter is unnecessary, since one of the equations does not contain it. As (3) is the equation of the directrix, we have the proposition : Perpendicular tangents to a parabola meet on the directrix. Ex. 2. Locus of the point of intersection of perpendicular tan- gents to an ellipse. Let the ellipse be 1 — (fig. 184), and let the equation of any tangent to it be written (§ 88) y = mx ± Va^m'^ + 6^. Then the equation of a tangent perpendicular to (1) will be y = --± rn Ia2_ \m2 + 62, and P{x, y), the point of inter- section of (1) and (2), will be any point of the locus. Solving (1) and (2), we find Fig. 184 ± m r-v/^ + 62 _ Va2m2 + 62! y = »n2 + l \m^ -t + 62 + Va2»i- + 62 m2-M (8) (4) as the parametric representations of the locus in terms of the parameter m. To eliminate m, we square the respective values of x and y and add, the result being x2 + 2/2 = a^ + 62. (5) The locus is seen to be a circle concentric with the ellipse and having its radius equal to the chord joining the ends of the major and the minor axes of the ellipse. While (3) and (4) form the explicit parametric representation of the locus, x and y being expressed explicitly in terms of the parameter m, <{1) and (2) may be regarded as the implicit parametric representation of the locus, for x and y, the coordinates of any point of the locus, are expressed implicitly in terms of m. 318 PARAMETRIC REPRESENTATION OF CURVES From this point of view it is evident that we may eliminate m directly from (1) and (2) to find the Cartesian equation of the locus. Accordingly we write (1) and (2) in the forms y — mx = ± Va%i2 + 62, my + x — ± Va2 + b^m^, and square and add, the result being (1 + m2) (X2 + 2/2) = (1 + ^2) (o2 + 62)^ or (1 + m2) (x2 + ^2 _ a2 _ 52) = 0. As 1 + m' cannot be zero, since by hypothesis m must be real, we may cancel out this factor. The result, x2 + 2/2 _ a2 _ 62 = 0, is the same as that found by the previous method. Ex. 3. Lociis of the foot of the perpendicular from the focus of a parabola to any tangent. Let the parabola be 2/2 = 4px (fig. 185), and let P y = mx + (1) be any tangent. Then the perpendic- ular to the tangent from the focus is y = -l^{x-p). (2) Their point of intersection, P(x, y), is any point of the locus. Solving (1) and (2), we find Fig. 185 and x = y (8) (4) The locus is therefore x = 0, the tangent at the vertex of the parabola. If we proceed from the implicit parametric representation, we may elimi- nate the parameter m by substituting in (1) its value found from ('2). The result is x[2/2 + (p — x)2] = 0, which breaks up into two equations, i.e. X = 0, y^ + (x — p)2 =0. As the last equation represents a single pointj it is evident by the geometry of the problem that the required equation is x = 0, as was found by the other method. We see then that when we eliminate the parameter from the equations expressing x and y in terms of it, we must examine our result carefully to be sure that no extraneous factor is left in it. LOCUS PKOBLEMS 319 Ex. 4. Locus of the foot of the perpendicular from the vertex of a parabola to any tangent. Let the parabola be y'^ = 'ipx (fig. 186), and y = 7nx + (1) be any tangent. Then the perpen- dicular to (1) from the vertex is y = X. m Solving (1) and (2), we find -P y = m2 + l P (2) (3) (4) m (m2 + 1) as the explicit parametric represen- tation of the locus. The Cartesian equation of the locus is most readily- found by substituting in (1) the value of m from (2), and reducing. The result is x8 Fig. 186 y^ = - (6) p + x which is the equation of a cissoid (§ 83) situated on the negative axis of x. The last two loci are special examples of pedal curves, i.e. loci of the feet of perpendiculars drawn from any chosen fixed point to tangents to a given curve. 176. In the examples of the last article the parametric repre- sentation of the locus was in terms of a single parameter. In the examples of this article the parametric representation, whether implicit or explicit, is in terms of two parameters, which are not independent, however, since they are connected by a single equa- tion. The problem of finding the Cartesian equation of the locus is, then, the elimination of two parameters from three equations. Ex. 1. Through the vertex of a parabola a line is drawn perpendicular to any tangent. Required the locus of the intersection of this line and the ordinate through the point of contact of the tangent. Let Pi(xi, 2/i) be any point of the parabola y^ = ipx (fig. 187), PiT the tangent at Pi, and OT the perpendicular to PiT from the vertex 0. Then the equation of PiT is yiy = 2p{x + xi), (1) 2/1 and the equation of T is y = - —X. 2p The equation of the ordinate MiPi through Pi is X = Xi. (2) (8) 320 PARAMETRIC REPRESENTATION OF CURVES If P(x, y) is the point of intersection of (2) and (3), P is any point of the locus, and (2) and (3) form the implicit parametric representation of the locus in terms of the parameters Xi and yi. Since Pi (Xi, yi) is by hypothesis any point of the parabola, its coordinates satisfy the equation of the parabola, and the parameters Xi and yi satisfy the equation y^ = 4pxi. (4) Fig. 187 Solving (2) and (3) for Xi and yi and substituting their values in (4), we thereby eliminate them and have, as the Cartesian equation of the locus. 2/2 = _ x8. P (5) From the form of the equation tlie locus is seen to be a semicubical parabola. It may be added that the explicit parametric representation of the locus is - Vi^i 2p readily found to be x = Xj and y = , where y^ - 4pxi. LOCUS PROBLEMS 321 Ex. 2. Locus of the middle points of chords of an ellipse, drawn through one end of its major axis. Let the ellipse be 1 = 1 (fig. 188), and Pi(a;i, yi) be any point of the a^ b^ ellipse. Then APi is any chord through A, and P(x, y), its middle point, is any point of the required locus. Since the coordinates of A are (a, 0), by § 18 xi + a and y = y\ Then (1) and (2) are the explicit para- metric representations of the locus in terms of the parametere Xi and y^ which satisfy the equation (3) — + — = 1, 02 62 Fig, 188 since Pi is any point of the ellipse. To find the Cartesian equation of the locus, we substitute in (3) the values of Xi and yi from (1) and (2). The result is Accordingly the locus is an ellipse with its center at ( -, ) and its semiaxes equal respectively to - and - ■ Ex. 3. Locus of the point of intersection of tangents at the ends of conjugate diameters of an ellipse. "~^ ^/-"""T^:^-- ^ \(i) y^ ^icC"^ ^ ^ >^^r^\ 0- ^ J J \s,^^ Jir-——— __ -^ -^^2 ^y ^--^ X Fig. 189 X2 V^ Let the ellipse be 1-^ = 1 (fig. 189), and OAi and OBi be any two con- a2 62 / 6 \ jugate diameters. If Ax is (xi, yj), -Bi is ( ^, — ^-\ by ICx. 2, § 146. \ 6 a J 322 PARAMETRIC REPRESENTATION OF CURVES Then the tangents at Ai and Bx will be respectively and where Solving (1) and (2), we find z = y = 62 ab ^ ab~ ' a2 62 6xi — ayi bxi + ayi (1) (2) (3) (4) (6) as the explicit parametric representations of the locus. If we write (4) and (5) in forms bz — bxi — ayi and ay = bxi + ayi respec- tively, and square and add, we have 62a;2 + a22/2 = 2 (62x2 + a^^), or 62x2 + a2y2 = 2 a262, (6) by virtue of (3). y^ As (6) may be written + = 1, we see that the required locus is (a V2)2 (6 V2)2 an ellipse, concentric with the given ellipse and with the semiaxes a V2 and 6 V2. Ex. 4. P1P2 is any chord of an ellipse perpendicular to its major axis A\Ai. Find the locus of point of intersection of AiPi and A2P2. Fig. 190 x2 y^ Let the ellipse be 1- — = 1 (fig- 190), and the coordinates of Pi and P2 be a2 62 respectively (xi, yi) and (xi, — yi). Then the equation of AiPi and A2P2 are respectively „ y xi + a Vi a — xi {x + a), (z - a). (1) (2) PROBLEMS 323 which are accordingly the implicit parametric representation of the locus. The parameters Xi and yx satisfy the equation ^ + g = l. (8) Taking the product of (1) and (2), we have y^ = -lL^{x^-a\ (4) which may be written y"^ — — (x2 — dF), (6) a2 by virtue of (3). As (6) may hyperbola concentric with the ellipse and having the same semiaxes. As (6) may be written = 1, we see that the required locus is an a2 62 PROBLEMS 1. Show that X = pt^, y = 2pt are parametric equations of the parabola. 2. Find the equations of the tangent and the normal to the parabola when the equations of the parabola are as in problem 1. 3. Find the parametric equations of the parabola when the parameter is the slope of a line through the vertex. 4. Find the equations of the tangent and the normal to a parabola when the equations of the curve are as in problem 3. 5. Find the parametric equations of the ellipse when the parameter is the slope of a straight line through the center. 6. Find the parametric equations of the ellipse when the parameter is the slope of a straight line through the left-hand vertex. 7. Find the parametric equations of the cissoid when the parameter is the angle AOP (fig. 91). 8. Show that x = t, y — are parametric equations of the witch. a2 + <2 2 (I 2 (I 9. Show that x = , y — — are parametric equations of the \^t^ <(l + «2) ^ ^ cissoid. What is the geometric significance of i ? 10. Find the equation of the tangent to the cissoid if the equations of the curve are as in problem 9. 324 PARAMETRIC REPRESENTATION^ OF CURVES Find the Cartesian equations of each of the following curves : . , o <2 + 2 a t^ + 2t 11. X = - ■ , y = ~ ■ — . 2 P+\ 2 i^ + 1 12. X — , y = \-^t^ 1 + «3 ,o *:^ , kH l«j. x = a-\ -^, y = at + a(l + <2) a(l + «2) ,. l + 2« l + < 14. x = , y = 15. X = -, y t-1 t^-1 .„ e' + e-' e' — C-' 16, X = , y = . {e'-2e-'y^ {e'-2e-'f 17. x = t^ + St + 2^ y = t^-l. , o ct ct^ ^°- ^ = '. \ ; ;r' y (a + U) (1 + «2) " (a + 6t) (1 + «2) 19. Eliminate < from X _ cos < — sin < y _ cos < + sin ^ a e' a e' and prove that the curve represented is a logarithmic .spiral (§ 178). 20. Let be the center of a circle with radius a, ^ a fixed point, and B a moving point on the circle. If the tangent at B meets the tangent at A in C, and P is the middle point of BC, find the equations of the locus of P in para- metric form, using the angle AOB as the arbitrary parameter, OA as the axis of a;, and as the origin. Also find the Cartesian equation of the locus. 21. OBCD is a rectangle with OB = a and BC = c. Any line is drawn through C, meeting OB in E, and the triangle EPO is constructed so that the angles CEP and EPO are right angles. Find the parametric equations of the locus of P, using the angle DOP as the parameter, OB as the axis of x, and as the origin. Find also the Cartesian equation of the locus. 22. Let AB be a given line, a given point, a units from AB, and k a given constant. On any line through 0, meeting AB in M, take P so that OM ■ MP = A:2. Find the parametric equations of the locus of P, using as the origin, the perpendicular from O to ^Z? as the axis of x, and the angle between OX and OP as the parameter. Also find the Cartesian equation. 23. A and B are two points on the axis of ?/ at a distance — a and 4- a respectively from the origin. AH ifi any line through A meeting the axis of x at H. BK is the perpendicular from B on AH, meeting it at K. Tiirough K a line is drawn parallel to the axis of x and through H a line is drawn parallel to the axis of y. These lines meet in P. P"ind the parametric equations of the locua of P, using the angle BAK as the parameter. Also find the Cartesian equation. PROBLEMS 325 24. Let OA be the diameter of a fixed circle and LK tlie tangent at A. From draw any line intersecting the circle at B and LK at C, and let P be the middle point of BC. Find the parametric equations of the locus of P, using the angle A OP as the parameter, OA as the axis of y, and O as the origin. Find also the Cartesian equation. 25. Show that the tangent to the ellipse at any point and the tangent to the auxiliary cii'cle at the corresponding point pass through the same point of the major axis. 26. Prove that the eccentric angles of the ends of a pair of conjugate diameters of an ellipse differ by — • 27. Show that the perpendicular from either focus upon the tangent at any point of the auxiliary circle of an ellipse equals the focal distance of the corre- sponding point of the ellipse. 28. Q is the point on the auxiliary circle of the ellipse, corresponding to the point P of the ellipse. The straight line through P parallel to OQ meets OX at L and OY at M. Prove PL = b, and PM = a. 29. Find the equation of the tangent at any point of an ellipse in terms of the eccentric angle at that point. 30. What elevation must be given to a gun to obtain the maximum range on a horizontal line passing through the muzzle of the gun ? (In this and the following examples the resistance of the air and the effect of all forces except gravity are neglected.) 31. What elevation must be given to a gun to obtain a maximum range on an oblique line passing through the muzzle of the gun and making an angle ^ with the horizontal ? 32. What elevation must be given to a gun that the projectile should pass through a point in the horizontal line passing through the muzzle and 6 units from it ? 33. A gun stands on a cliff h units above the water. What elevation must be given to the gun that the projectile may strike a point Tn the water b units from the base of the cliff ? 34. Find the parametric equations of the curve described by any point in the connecting rod of a steam engine. 35. If a circle rolls on the inside of a fixed circle of twice its radius, what is the form of the curve generated by a point of the circumference of the rolling circle ? 36. Show that the hypocycloid generated when the rolling circle has J the radius of the fixed circle has the Cartesian equation x* -f- y^ = 6'. 37. If a wheel rolls with constant angular velocity on a straight line, required the velocity of any point on its circumference ; also of any point on one of the spokes. 326 PARAMETRIC REPRESENTATION OF CURVES 38. If a wheel rolls with constant angular velocity on the circumference of a fixed wheel, find the velocity of any point on its circumference and on its spoke. 39. Show that the highest point of a wheel rolling with constant velocity on a road moves twice as fast as each of the two points in the rim whose dis- tance from the ground is lialf the radius of the wheel. 40. If a string is unwound from a circle with constant velocity, find the velocity of the end in the path described. 41. AB and CD are perpendicular diameters of a circle of radius E. AM is a chord of the circle, rotating about A so that the angle BAM varies uniformly. AM is extended to N so that J!f^=the chord MB. Find the path of N, the velocity of N in its path, and the components of the velocity respectively parallel to AB and CD. 42. 0, CK, 0" are three points on a straight line and 0"(y = ^0(7. LK is drawn through C pei-pendicular to 00" ^ and any point M is taken on LK. From M a straight line is drawn perpendicular to 0"Jf, and through a straight line is drawn parallel to 0"M. These lines intersect in P. Required the locus of P. 43. is a fixed point and LK a fixed straight line. Any point M is taken on LK, and the line OM is drawn and prolonged to P so that OM ■ OP = k^, where A; is a constant. Find the locus of P. 44. Show that the locus of points symmetrical to the vertex of a parabola with respect to its tangent lines is a cissoid. 45. Let OA be the diameter of any circle and LK the tangent at A, Through draw any line intersecting the circle in D and LK in E. Lay off on OE produced the distance EP = OD, and find the locus of P. 46. Let a circle with center at intersect the axis of y at J. and the axis of X at C. Take two points G and E on the circle equidistant from A. If the ordinate of G intersects the line CE in P, prove that the locus of P is a cissoid. 47. From a point a units from the axis of x lines are drawn to OX, and from the point where each line meets the axis a line of tlie same length is drawn at right angles to the first line. Find the equation of the locus of the end of this last line. 48. OA is a diameter of a circle and LK the tangent at A. Through any line is drawn meeting the circle in B and LK in C. Through B a line is drawn perpendicular to OA and meeting it in M. Finally MB is prolonged to P so that MP = AC. Find the locus of P. 49. Find the path described by any point of a tangent line which rolls upon a circle without slipping. 50. CD is perpendicular to OX and distant a units from O. Through A , any point on CD, a straight line OA is drawn, and from A a perpendicular is drawn PEOBLEMS 327 to OA, intersecting OX at B. From B a straight line is drawn parallel to OY, intersecting OA at P. If m denotes the slope of OA, find the parametric and the Cartesian equations of the locus of P, 51. Prove that the pedal of a parabola with respect to any point is a cubic curve which passes through that point. 52. Prove that the pedal of the ellipse \- — = \ with respect to the center is the curve (x2 + y2)2 = a'^n^p. + 52^2. <^^ ^ 53. A line of constant length k moves with its extremities on the two axes of coordinates. Find the locus described by any point of the line. 54. A straight line has its extremities on the coordinate axes and passes through a fixed point. Find the locus of its middle point. 55. If the ordinate NP of an hyperbola be produced to Q, so that NQ, = FP, find the locus of Q. 56. Find the locus of the points of intersection of normals at corresponding points of the ellipse and the auxiliary circle. 57. P is any point of a parabola, A the vertex, and through A a straight line is drawn perpendicular to the tangent at P. Find the locus of the point of intersection of this line with the diameter through P, and also the locus of the point of intersection of this line with the ordinate through P. 58. Two equal parabolas have their axes parallel and a common tangent at their vertices, and straight lines are drawn parallel to the axes. Show that the locus of the middle points of the parts of the lines intercepted between the curves is an equal parabola. 59. Find the locus of the intersection of the ordinate, produced if necessary, of any point on an ellipse with the perpendicular from the center upon the tangent at that point. 60. Two parabolas have the same axis, and tangents are drawn from points on the first to the second. Prove that the middle points of the chords of con- tact with the second lie on a parabola. 61. Chords of an ellipse are passed through a fixed point. Find the locus of their middle points. 62. From a point P on an ellipse straight lines are drawn to the vertices A and A\ and from A and A' straight lines are drawn perpendicular to AP and A'P. Show that the locus of their point of intersection is an ellipse. 63. Show that the locus of the point of intersection of two tangents to a parabola, the ordinates of the points of contact of which are in a constant ratio, is a parabola. 64. If the tangent to the parabola y^ = 4px meets the axis at T and the tangent at the vertex A at J5, and the rectangle TABQ is completed, show that the locus of Q is the parabola y^ + px = 0. 328 PARAMETRIC REPRESENTATION OF CURVES 65. Find the locus of the feet of the perpendiculars from the focus to the normals of the parabola y^ = 4px. 66. Show that perpendicular normals to the pai'abola y- = 4px intersect on the curve y'^ = px — Sp^. 67. Find the locus of the intersection of a pair of perpendicular tangents to an hyperbola. 68. Two tangents to an ellipse are so drawn that the product of their slopes is constant. Show that the locus of their point of intersection is an ellipse or an hyperbola according as the product is negative or positive. 69. Prove that the locus of the point of intersection of two tangents to a parabola is a straight line if the product of their slopes is constant. 70. Find the locus of the foot of the perpendicular from either focus of an hyperbola to any tangent. 71. Let AB be the diameter of a circle and O its center. Let NQ be the ordinate of a point Q on the circle and P another point of the circle, so related to Q tjiat OP revolves uniformly from OA through a right angle in the same time that QN travels at a unifoi-m rate from A to 0. If OP and QN intersect in R, find the locus of R. 72. Find the equations of the cycloid when the tangent at its highest point is the axis of x, the normal at the vertex is the axis of y, and the angle is the angle through which the radius has rotated after passing through the highest point. 73. Prove that the area of an arch of the cycloid above the axis of x is three times the area of the rolling circle. 74. Prove that for a cycloid — = 2 a sin - , and thence find its length from . dd> 2 cusp to cusp. ^ 75. Show that for an epicycloid — = 2(a + 6)sin — , and thence find its length from cusp to cusp, ^ CHAPTEE XV POLAR COORDINATES 177. Coordinate system. So far we have determined the posi- tion of a point in the plane by two distances, x and y. We may, however, use a distance and direction, as follows : Let (fig. 191), called the origin or foh, be a fixed point, and OM, called the initial line, be a fixed line. Take P any point in the plane and draw OP. Denote OP by r and the angle MOP by 0. Then r and Q are called the 'polar coordinates of the point P(r, 6), and when given will completely determine P. Fig. 191 For example, the point (2, 15°) is plotted by laying off the angle MOP = 15° and measuring OP = 2. OP, or r, is called the raditis vector and the vectorial angle of P. These quantities may be either positive or negative. A negative value of is laid off in the direction of the motion of the hands of a clock, a positive angle in the opposite direction. After the angle has been constructed, positive values of r are measured from along the terminal line of 0, and negative values of r from O along the backward extension of the terminal line. It follows that the same point may have more than one pair of coordinates. 320 330 POLAR COORDINATES Thus (2, 195°), (2, - 165°), (- 2, 15°), and (- 2, - 345°) refer to the same point. In practice it is usually convenient to restrict 6 to positive values. Plotting in polar coordinates is facilitated by using paper ruled as in figs. 192 and 193. The angle 6 is determined from the num- bers at the ends of the straight lines, and the value of r is counted off on the concentric circles, either towards or away from the num- ber which indicates 6, according as r is positive or negative. When an equation is given in polar coordinates the correspond- ing curve may be plotted by giving to 6 convenient values, com- puting the corresponding values of r, plotting the resulting points, and drawing a curve through them. Ex. 1. r = a costf. a is a constant which may be given any convenient value. We may then find from a table of natural cosines the value of r which corresponds to any value of 0. 165) 180 195 By plotting the points corresponding to values of 6 from 0° to 90° we obtain the 9,TcABC0 (fig. 192). Values of 6 from 90° to 180° give the arc ODEA. Values of GRAPHS 331 e from 180° to 270° give again the arc ABCO, and those from 270° to 360° give the arc ODEA. Values of d greater than 300° can clearly give no points not already- found. The curve is a circle (§ 184). Ex. 2. r = o sin 3 6. As increases from 0° to 30°, r increases from to a ; as ^ increases from 30° to 60°, r decreases from a to 0; the point P{r, 6) traces out the loop OAO (fig. 193). As e increases from 60° to 90°, r is negative and decreases from to — a •, 5 if as 6 increases from 90° to 120°, r increases from — a to ; the point (r, 0) traces out the loop OBO. As 6 increases from 120° to 180°, the point (r, 0) traces out the loop OCO. Larger values of give points already found, since sin 3 (180° + 0) = — sin 3 $. The three loops are congruent because sin 3 (60° + 0) = — sin 3 0. This curve is called a rose of three leaves. 178. The spirals. Polar coordinates are particularly well adapted to represent certain curves called spirals, of which the more important follow. 332 POLAR COORDINATES Ex. 1. The spiral of Archimedes, r = ae. In plotting 6 is usually considered in circular measure. When $ =z 0, r = 0, and as 6 increases r increases, so that the curve winds infinitely often around Fig. Voi the origin while receding from it (fig. 194). In the figure the heavy line repre- sents the portion of the spiral corresponding to positive values of 0, and the dotted line the portion corresponding to negative values of 6. Ex. 2. The hyperbolic spiral, rd = a, a L- K Fig. 19.J As increases indefinitely r approaches zero. Hence the spiral winds infi- nitely often around the origin, continually approaching it but never reaching it THE SPIRALS 333 (fig. 195). As approaches zero r increases without limit. If P is a point on the spiral and NP the perpendicular to the initial line, ■»TT, • « sin^ NP = rsmd = a e Hence as 6 approaches zero as a limit, NP approaches a (§ 161). Therefore the curve comes constantly nearer to, but never reaches, the line LK, parallel to OM at a distance a units from it. This line is therefore an asymptote. In the figure the dotted portion of the curve corresponds to negative values of 0. Ex. 3. The logarithmic spiral, r = e"*. When 6 = 0, r = 1. As ^ increases r increases, and the ciirve winds around the origin at increasing distances from it (fig. 196). When is negative and increasing numerically without limit, r approaches zero. Hence the curve winds infinitely often around the origin, continually approaching it. The dotted line in the figure corresponds to negative values of 0. Fig. 196 A property of this spiral is that it cuts the radii vectors at a constant angle. The student may prove this after reading § 187. We shall now give examples of the derivation of the polar equa- tion of a curve from the definition of the curve. 334 POLAR COORDINATES 179. The conchoid. Take a fixed point (fig. 197) and a fixed straight line BC. Through draw any line OR intersecting BC in D, and on OB lay off a constant distance DP or DQ, measured from D in either direction. The locus of P and ^ is a curve called the conchoid. From the definition the conchoid consists of two parts, one generated by P, the other by Q. We may obtain the whole curve, Fig. 198 however, by allowing the line OR to revolve in the positive direc- tion through an angle of 360° and always laying off the distance h, measured from D in the direction of the terminal line of the angle AOR. Then if AOR is in the first quadrant, we obtaia the upper half of the curve described by P ; if ^ OR is in the second quad- rant, we have the lower half of the curve described by Q; HA OR is in the third quadrant, we have the upper half of the curve THE CONCHOID 335 described by Q ; and if A OB is in the fourth quadrant, we have the lower half of the curve described by P. To find its polar equation, take as the origin and the line OA perpendicular to BC as the initial line. Let OA = a and the con- stant distance DP = h. Call the coordinates of P (r, 6), where 6 =A OR. When ^ is in the first or the fourth quadrant, r = OD + DP = OD + h ; when 6 is in the second or the third quadrant, r = — OD -\-DQ = — OD + h. Fig. 199 But OD = a sec d when ^ is in the first or the fourth quadrant ; and OD = — a sec 6 when ^ is in the second or the third quadrant. Hence for all points on the conchoid r = a sec ^ + &. The conchoid has three shapes according as a > i (fig. 197), a = 6 (fig. 198), a, and lay off on this line a constant length measured from D in either direction. The locus of the points P and ^ thus found is a curve called the limagon. Take as the pole, the diameter OA as the initial line of a sys- tem of polar coordinates, and call the diameter of the circle a and Fig. 200 the constant length h. Then it is clear that the entire locus can be found by causing OD to revolve through an angle of 360° and laying off DP = h always in the direction of the terminal line of ADD. Let the coordinates of P be (r, 6), where 0=AOD. Then r = OD + DP when 6 is in the first or the fourth quadrant, and r = — OD + DP when 6 is in tlie second or the third quadrant. But it appears from the figure that OD = OA cos 6 when 6 is in the first or the fourth quadrant, and OD = — OA cos d THE LIMAgON 337 when 6 is in the second or the third quadrant. Hence for any point on the limagon r = a cos 6 + 1). In studying the shape of the curve there are three cases to be distinguished. O^cos\-i) Fig. 201 1. h>a. r is always positive and the curve appears as in fig. 200. 2. b < a. r is positive when cos d > ' negative when 7 h ^ cos 6 < ' and zero when cos = • The curve appears as a a ^'^ in fig. 201. 3. h = a. The equation now becomes r = a (cos ^ + 1) = 2 a cos^ - ■ r is positive except when 6 = 180°, when it is zero. The curve appears as in fig. 202 and is called the cardioid. 338 POLAE COORDmATES The cardioid is an epicycloid for wMch the radii of the fixed and the rolling circles are the same. The proof of this is left to the student. Fig. 202 181. The ovals of Cassini. If a point moves so that the product of its distances from two fixed points is constant, it generates a r Fig. 203 curve called an oval of Cassini. Let F^ and F^ (fig. 203) be the two fixed points, called the foci, and If the constant product of THE OVALS OF CASSINI 339 the distances of a point of the curve from F^ and F^. Take F^F^ as the initial line and the point .0, halfway between F^ and F^, as the pole of a system of polar coordinates, and let P be a point on the curve. Then, by definition, F^P . F^P = l\ (1) By trigonometry, :^' = OB'^j^OF^- 2 OP • OF^ cosF^OP = r'+ €0"+ 2 ra cos(9, where (r, 6) are the coordinates of P and 2a= F^F^. Also TJ^'' = op'^j^OF^- 2 OP • OF^ QosF^OP = r'+ a"- 2 ra cos^. Substituting in (1), we have (r2+ay_4aV'cos'(9 = J*, which is the same as /_ 2 aV cos 2 (9 + a*- 5* = 0. (2) To determine the form of the curve, it is convenient to solve (2) for r^, obtaining r" = a^cos2e± ^a* cos'2 -{a*-b*). (3) We have, then, three cases to consider 1. a^ < If. The quantity under the radical sign in (3) is posi- tive and greater than a* cos^ 2 6 for all values of 6. Therefore r^ in (3) has two real values, one positive and one negative. Conse- quently r has two, and only two, real values equal in magnitude and opposite in sign. The curve therefore consists of a single oval, symmetric with respect to the origin (fig. 203). a*— h* 2. a^ > If. When cos*^ 2 > — the quantity under the rad- ical sign in (3) is positive and less than a* cos^ 2 6. Hence for these values of 6 there are two real positive values of r^ and there- fore four real values of r, two positive and two negative. When a^—h* . . cos^ 2 < — the quantity under the radical sign in (3) is 340 POLAE COOKDINATES negative, and hence all values of r are imaginary. When cos*^ 2 Q a* — b* Va* — b* = — there are two real values of ?', namely r = ± The curve consists of two distinct ovals (fig. 204). -M Fig. 204 3. a^ = b^. Equation (2) then factors into the two equations r^ = and r^— 2 a^ cos 2 ^ = 0. But r^ = is satisfied only by the origin, which is also a point on the second equation. Fig. 20.3 Hence r' = 2 a' cos 2 d (4) is the full equation of the locus in this case. From (4) it appears that r has two real values equal in magnitude but opposite lq sign when 0<^<->or— <6'<— ,or— <6'<2 7r. Further, 4 4 4 4 r = when 6 = — , —^ > —7- , or — — ; and r is imaginary when 4 4 5 TT 4 4 4 4 4 4 The curve appears as in fig. 205 and is given the special name of the lemniscate. CHANCxE OF cooedi:n:ates 341 182. Relation between rectangular and polar coordinates. Let the pole O and the initial line OM of a system of polar coordinates be at the same time the origin and the axis of a; of a system of rectangular coordinates. Let P (fig. 206) be any point of the plane, {x, y) its rectangular coordinates, and {r, 6) its polar coordinates. Then, by the definition of the trigonometric functions, X cos 6 = -) sm a = - > r whence follows, on the one hanC, X = r cos 6, y = r sin Q, and, on the other hand. Fig. 206 r = -Vaf + y , sin ^ = y \^x^ + y^ cos tr = Va;^ r (2) By means of (1) a transformation can be made from rectangular to polar coordinates, and by means of (2) from polar to rectangular coordinates. Ex. 1. The equation of tlie cissoid (§ 83) is 2/2 = 2a — X Substituting from (1) and making simple reductions, we have the polar equation 2asin2 6i r = cos^ Ex. 2. The polar equation of the lemniscate is r2 = 2 a2 cos 2 e. Placing cos2^ = cos2 — sin^e and substituting from (2), we have the rec- tangular equation (x2 + ?/2)2=:2a2(x2-2/2). 342 POLAR COORDINATES 183. The straight line. Take the equation of the straight hne in the normal form x cos a-{-y sin a—p = and substitute the values of x and y from (1), § 182. There results r (cos cos a + sin sin a) — j? = 0; whence r cos {0 — a) = p. A reference to § 33 shows that {p, a) are the polar coordinates of the point in which the normal from the origin meets the straight line. If a = and p = a, we have the special equation r cos = a, or r = a sec 0, as found in § 179. If the straight liue passes through the origin, ^ = 0. The equa- tion of the line then becomes cos {6 — a)= 0, or simply = — + a, which is of the form = c. 184. The circle. If (d, e) are the rectangular coordinates of the center of the circle and a its radius, its equation is If (h, a) are the polar coordinates of the center and (r, 0) those of any point, the pole and the initial line of the polar coordinates being the origin and the axis of x, respectively, of the rectangular system, we have, by (1), § 182, x = r cos 0, y = rsm0, d = b cos a, e = h sin a. We obtain, by substitution, r'—2rb (cos cos a + sin ^ sin a)+h^— a^ = 0, or r^-2rbcos(0-a) + b^-a^==O. (1) P{r,e) THE CONIC 343 This result may also be directly obtained from fig. 207 by noticing CF"" ='0C^ +0F'' - 2 OP • OC cos Foa "When the origin is at the center of the circle, b = 0, and (1) becomes simply r = a. (2) When the origin is on the circle, b = a, and (1) becomes r—2a cos (^ — a) = 0, ^^ which may be written ^'^- -^^ r = aQ cos 6 + a^ sin 6, (3) where a^ and a^ are the intercepts on the lines ^ = and 6 = — respectively. Wlien the origin is on the circle and the initial line is a diameter, (3) becomes r = «() cos 6. (4) When the origin is on the circle and the initial line is tangent to the circle, (3) becomes r = a^ sin 6. (5) 185. The conic, the focus being the pole. From § 81 the equa- tion of a conic, when the axis of x is an axis of the conic and the axis of 3/ is a directrix, is {x — cf + y^ = e^x^. We may transfer to new axes having the origin as the focus and the axis of x as the axis of the conic by placing x = c + x', y = y', thus obtaining x'^-\- y'^ = e'^{x' + cf. If we now take a system of polar coordinates having the focus as the pole and the axis of the conic as the initial line, we have x' = r cos 6, y' — "f sin 6. 344 POLAR COORDINATES The equation then becomes r^ = e^ {r cos 6 + cf, which is equivalent to the two equations . r = 1 — e cos ce 1 + e cos 6 Either of these two equations alone will give the entire conic. To see this, place 6 = 6^ in the second equation, obtaining T, = — ce l-\- e cos 6^ Now place 6 = 'Tr -\- 6^ in the first equation, obtaining r = — r^. The points {9^, rj and (tt + 6^, — r^ are the same. Hence any point which can be found from the second equation can be found from the first. ce Therefore r = e cos d is the required polar equation. 186. Examples. We shall now give examples of the use of polar coordinates in solving problems. Ex. 1. Prove that if a secant is drawn through the focus of a conic, the sum of the reciprocals of the seg- ments made by the focus is constant. Let PiPo (fig- 208) be any secant through the focus F, and let FPi = ri and FPi = r^, and the angle MFP = 0. Then the polar coordinates of Pi are (ri, 0) and those of Pa are (ra, d + ir). From the polar equation of the conic we have Fig. 208 ri = r2 1 — ecos^ ce Hence 1- Ti r^ ce e cos {d + tt) ce 1 + e cos DIRECTION OF A CUEVE 345 Ex. 2. Find the locus of the middle points of a system of chords of a circle all of which pass through a fixed point. Take any circle with the center C (fig. 209) and let be any point in the plane. If O is taken for the pole and OC for the initial line of a system of polar coordinates, the equation of the ^^^^ p circle is r2-2r6cose + 62_a2 = 0. (1) Let P1P2 be any chord through and let OPi = ri, OP2 = rg. Then ri and r% are the two roots of equation (1) which correspond to the same value of d. Hence n + r2 = 2 6 cos 6. If Q is the middle point of P1P2, and we now place OQ = r, we have ri + r2 , . r = = ocos^. ,, „^^ 2 iiG. 209 But this is the polar equation of a circle through the points and C. 187. Direction of a curve. The direction of a curve expressed in polar coordinates is usually determined by means of the angle between the tangent and the radius vector. Let F(r, 6) (fig. 210) be any point on the curve, VQ /'■ PT the tangent at P, and i/r the angle made by PT and the radius vector OP. Give 6 an increment A6=P0Q, expressed in circular measure, thus fix- ing a second point of the curve Q(r + /^r, 6 + A6). To determine Ar describe a circle with center and radius OQ, intersecting OP produced in R. Then OB = OQ = r + Ar, PE = Ar, and SiTG PQ = As, Fig. 210 s being measured from some initial point A. 346 POLAR COORDINATES Draw also the chord PQ and the straight line QS perpendicular to OP and meeting it in S. Then ^^ = (r + Ar) sin A^, OS = {r + At) cos A6, SB = OE-OS = (r + Ar)(l— cosA^), and PS = PE-SB = Ar — {r + Ar) (1 — cos A^). As A0 approaches zero, the chord PQ approaches the limiting position PT and the angle EPQ approaches i/r. But in the triangle SPQ SQ tanEPQ = ^^ (r + Ar) sin A^ Ar — (r + A?') (1 — cos A^) sin A^ (r + Ar) A6> Ar , . , 1 — cos A^ -— — (r + Ar) - (1) A^ ' ' Ad Now as.:A^ approaches zero T- / A \ T- sinA^ . lam (r + A?-) = r, Lim — —r- = 1, A0 ,. Ar ch , ,. 1— cosA^ a/ciki\ Hence, by" taking the limit of (1), ■tan '«ir = 4^. (2) ar •To If it is desired to find the angle MNP = (f), it may be done by the evident relation DERIVATIVES WITH RESPECT TO THE ARC 347 188. Derivatives with respect to the arc. In the triangle FQS (fig. 210) SQ BmSPQ- chord P^ SQ QxcPQ arcFQ chord P^ _(r + Ar) sin A^ arc PQ As chord P^ ^ ,sinA(9 Ad arc PQ = (r + Ar) A0 As chord Pg , As A^ approaches zero, SPQ approaches yjr, Lim — —^ = 1, and Lim f^'^f^ = 1 (§ 104) ; hence chord P^ ^^ ^ sin '\lr = r—-- (1) as By dividing (1) just obtained by (2) of the previous article, dr ds cos ■\(r = — — (2) From (1) and (2) we obtain /dsY By multiplying (3) by | -r^ I we obtain d0j ^\ddi ' ^ ' /dsV and by multiplying (3) by ( ^r- ) we obtain CI drj \ dr 'J.yjr'^)'+l. (6) 348 POLAR COORDINATES 189. Area. Let C (fig. 211) be a fixed point and P {r,6) a variable point on the curve r =f(6), and let A denote the area of the figure OCF, bounded by the arc of the curve CP and the radii OC and OP. Then A is a function of 6, since the value of fixes the position of the point P. If 6 is increased by A^ = angle POQ, A is in- creased by A A = area POQ. From describe arcs of circles PS and QR with radii OP = r and OQ = r + Ar respectively. Yio. 211 Then in the figure aresi POS a straight line is drawn perpendicular to CD, intersecting OB at P. Find the locus of P. 350 POLAR COORDINATES Transform the following equations to polar coordinates : 33, y2 = 4px, 36. x2 + 2/2 - 8 ox - 8 ay = 0. 34. xy = 7. 37. x* + x2y2 _ a2y2 = q. 35 ^ + ?^ = 1 ^^- ^'^^ + ^')' = ''' (•^' ~ ^'>- ' a2 62 • 39. x^ + y3-3axy^0. 40. Find the polar equation of the cissoid when the pole is A and the initial line is OA (fig. 91). 41. Find the polar equation of the strophoid (1) when the pole is and the initial line OA (fig. 92); (2) when the pole is A and the initial line is OA. 42. In the strophoid (fig. 92) show that 112 AP.APi = a:^, and -t^ + -!— = -^, AP APi AN where AN is the projection of ^0 on AD. Transform the following equations to rectangular coordinates : 43. rcosM-- J + rcosM+-| = 12. 46. r = atan«. 44. r = asin«. 47. r2 = a2sin«. g 45. r = a(cos2^ + sin2«). 48. r2 = a2sin-. 49. Find the Cartesian equation of the rose of four petals »• = a sin 2 6. 50. Find the Cartesian equation of the cardioid r = a(l — cos 5). 51. Find the Cartesian equation of the ovals of Cassini r* - 2a2r2cos2^+ a* - 6* = 0. 52. Find the Cartesian equation of the limagon r = a cosO + b. 53. Find the Cartesian equation of the conchoid r = a sec 5 + 6. 54. Find the Cartesian equation of the logarithmic spiral r = e"^. 55. In a parabola prove that the length of a focal chord which makes an angle of 30° with the axis of the curve is four times the focal chord perpen- dicular to the axis. ^ , 56. A comet is moving in a parabolic orbit around the sun at the focus of ,the parabola. When the comet is 100,000,000 miles from the sun the radius . vector makes an angle of 60° with the axis of the orbit. What is the equation of the comet's orbit ? How near does it come to the sun ? 57. A comet moving in a parabolic orbit around the sun is observed at two ■ Boinfs of its path, its focal distances being 5 and 15 million miles and the angle between them being 90°. What is its distance from the sun when it is nearest it ? 58. If a straight line drawn through the focus of an hyperbola, parallel to an asymptote, meets the curve at P, prove that FP is one fourth the chord through the focus perpendicular to the transverse axis. PROBLEMS 351 59. The focal radii of a parabola are extended beyond the curve until their lengths are doubled. Find the equation of the locus of their extremities. 60. If Pi and P2 are the points of intersection of a straight line drawn from any point O to a circle, prove that OPi ■ OP2 is constant. 61. If Pi and P^ are the points of intersection of a straight line from any point O to a fixed circle, and Q is a point on the same straight line such that 00 = ^ ^^^ ' ^^^ , find the locus of Q. ^ OP1 + OP2 ^ 62. Secant lines of a circle are drawn from the same point on the circle, and on each secant a point is taken outside the circle at a distance equal to the portion of the secant included in the circle. Find the locus of these points. 63. From a point a straight line is drawn intersecting a fixed circle at P, and on this line a point Q is taken so that OP ■ OQ = k^. Find the locus of Q. 64. Find the polar equation of a conic if the pole is a vertex and the initial line an axis. 65. Find the locus of the middle points of the focal chords of a conic. 66. Find the locus of the middle points of the focal radii of a conic. 67. If P1PP2 and QiFQ-2 are two perpendicular focal chords of a conic, prove that 1 is constant. PiF-FPi. Q1F.FQ2 68. Prove that the angle between the normal and the radius vector to any point of the lemniscate is twice the angle made by the radius vector and the initial line. 69. Show that for any curve in polar coordinates the maximum and the minimum values of r occur in general when the radius vector is perpendicular to the tangent. 70. If a straight line drawn through the pole perpendicular to a radius t" vector OP meets the tangent in A and the normal in B, show that OA = — dr ^^ and OB = — v; de , de These are called the polar subtangent and the polar subnormal respectively. 71. If p is the pei-pendicular distance of a tangent from the pole, prove that p = nRI 72. When a point traverses the curve r=f{d) with a uniform angular velocity, find the rate at which r is changing and the rate of the point along the curve. 73. When a point moves along the curve r=/{S) at a uniform rate, find the rates at which r and 6 are changing. 352 POLAR COOEDINATES 74. Find the velocity of a point moving in a lima^on when B changes uniformly. 75. A point moves along the radius vector with a constant velocity a, while the radius vector revolves about with a constant velocity w. Find the path of the point. 76. Find the total area bounded by the curve r^ = a^ sin 0. 11. Find the area of a loop of the curve r^ = a^ sin 3 d. 78. Find the area swept over by the radius vector of the spiral of Archimedes as changes from to ir. 79. Find the area swept over by the radius vector of the logarithmic spiral as changes from to tt. Q 80. Find the area swept over by the I'adius v-ector of the curve r = asin- as changes from to 2 tt. 81. Find the area swept over by the radius vector of the curve r = atanfl TT as changes from to — • 4 82. Find the total area of the limagon (6 > a). 83. Find the total length of the cardioid. 84. Prove that the length of an arc of the logarithmic spiral is proportional to the difference of the radii vectores drawn to its ends. 85. Show that if the angle between the tangent to a curve and the radius vector to the point of contact is one half the vectorial angle, the curve is a cardioid. CHAPTER XVI CURVATURE 190. Definition of curvature. If a point describes a curve the change of direction of its motion may be measured by the change of the angle (§ 59). For example, in the curve of fig. 212, a AP^ = s and F^I^ = As, and if ^^ and (f)^ are the values of which as is called the curvature of the curve at the point Py Hence the curvature of a curve is the rate of change of the direction of the curve with respect to the length of the arc (§ 109). If -~ is constant, the curvature as is constant or uniform; other- Applying this definition to the Fig. 213 wise the curvature is variable. circle of fig. 213 of which the center is C and the radius 353 354 CURVATUKE Hence -J- = -, and the circle is a curve of constant ds a is a, we have A^ = -?^Ci^ ; and hence As = aA ds _}__ds_ ^^ d4~ d4>' ds (by (6), § 96) K the equation of the curve is in rectangular coordinates. and whence dx N \dxl (by § 105) \dx/ (by § 59) A d<^ dx^ dx ^Jdy^ \dx) ds ds dx ' ^^d4)~'df (by (8), § 96) dx Mm i d'y cb? RADIUS OF CURVATURE 355 In the above expression for p there is an apparent ambiguity of sign, on account of the radical sign. If only the numerical value of p is required, a negative sign may be disregarded. Ex, Find the radius of curvature of the ellipse h — = 1. Here dy_^_^ dx a?y and — ^ = dx2 a^y^ {a*y'^ + 6*x2)* •■• P = Another formula for p, i.e, r /dx^^-^^ (Tx df may be found by defining ^ as the angle between OY and the tangent, and interchanging x and y in the above derivation. ds 192. According to the definition, i.e. p = -tt' it is evident that d

increase at the same time, and is negative when one increases as the other decreases. For convenience we shall assume in the following work that s always increases from left to right * along the curve (figs. 214-217). Then ^ is always in the first or the fourth quadrant, and hence sec ^ is always positive. But sec = Vl+tan^^= \l 1 + ( -^^ ) • Therefore in the formula NIII . da? the sign of p is the same as the sign of -—• Hence p is positive when the curve is concave upward, and negative when the curve is concave downward. •The results and the proof are the same if s is measured from right to left along the curve ; hence the proof is left to the student. 356 CURVATURE 193. Coordinates of center of curvature. The center of the circle described in § 191 is called the center of curvature corre- sponding to the point. Let C{a, /3) (fig. 214) be the center of curvature corresponding to the point P{x, y) of the curve. Draw CL and FM parallel to OY, and NR through P parallel to OX. Then OL = 0M+ ML = OM+ PiV; Now and LC = LN+NC = MP + NC. ZBPC = + 90°, PC = p, Fig. 214 since p>0, the curve being concave upward. Therefore, by the definition of the trigonometric functions, PiV = PC cos i^PC = /? cos (<^ + 90°) = - /3 sin , NC = PC sin PPC = p sin(<^ + 90°) = p cos<^. .'. a = X — p sin, ^ = y i- pcos, /3 = y + /) cos , we may find the slope of the evolute at any point by assuming a, /3, x,y,p, and as functions of s, the length of arc along the involute. Then da _dx d^ • M^P ds ds ds ds = cos (f) — p COS <^ ( - ) — sin <^ = — sinrf)-f-- ds dfi dy . . d(b , , dp -7- = -/^— /^sm-^ + cos-p as ds ds ds dp ds = sbi(f) — p sin (f> {-)-{- cos ^ -~ \p/ ds = cos ; but -—- = -—> da da da ds ds by (8), § 96 ; and if tan^' is the slope of the evolute at the assumed dfi point, — = tan ', and hence tan ' and differ by 90°, and the tangent to the evolute at any point is perpendicular to the tangent to the involute at the corresponding point (fig. 218). If we square and add the above equations, we have /da\ \ds) daV (d^Y_(dp ds/ \ds/ \ds But if we denote the length of arc along the evolute by s', we have _= - i_|_/z^\ ; and if we regard s', a, ^, as expressed in da N \da) terms of s, the length of arc along the involute, we have 360 CURVATURE ds I \ds/ . da ~~ I /daV ' ds n| W whence Hence as ^\ds/ \ds/ c?s/ \ds/ and -y- = ± , as as ds' _ C?/3 6?S •'• s' = ±p + c. (by § 110) /if follows, then, that as the center of curvature moves along the evolute the radius of curvature increases or decreases hy exactly the distance traversed hy the center (fig. 218). From these two properties we see that an involute may be described by a pencil attached to the end of a string which is unwound from the evolute, the free portion being kept taut and tangent to the evolute. From any one evolute any number of involutes may be described by changing the length of the string. 196. Radius of curvature in parametric representation. If x and y are expressed in terms of any parameter t, the radius of curvature may be fovmd as follows: ds dt But and cix dt p = d(f) d- = tan-^-/ = tan-^--; dx ax (by (8), § 96) RADIUS OF CURVATURE 361 whence dt dx\ dj^y (dy\ d?x di/d^~\di)df ldy\ \dti (dxY \dtl dx d^y dy d^x Tt"df~^t'~df (dxV /dy \dt) \dt Therefore, by substitution, P = dx d^y dy d?x 'dt'~de~~dt"de Ex. Pind the radius of curvature of the cycloid Here the parameter is x = a — a sin 0, y = a- acos. dx d acoscft, d^x — = asinc^, dy -^ = asin^, d(p d"'y Hence, by substitution, p = a cos. _ [a2(l-cos0)2 + a2.sinV]^ a (1 — cos -.sin«. Substitutmg these values and simplifying, we have, as the required formula, \ddl d&' Ex. Find the radius of curvature of the cardioid r = a(l — cos^). Here — = a sin d and — — a cos 6. de dff^ _ [a2(l - cos^)2 + a? sin2fl]i ' a2(l- cos^)2 + 2a2sin2tf _ a(l - cose)acos9 [2a2(l-cos^)]3 2?a„ „,i = ^^ -^ = (1 - costf)*, 02(3-3 00861) 3 ^ p = f(2ar)i. PROBLEMS fj - _ - 1. Find the radius of curvature of tlie catenary y — ~ {e" + e "). 2. Find the radius of curvature of the cissoid y^ — 2a — X 3. Find the radius of curvature of the four-cusped hypocycloid x^ + y^ = a*. 4. Find the radii of curvature of the curve a*y^ = a^* — x^ at the points (0, 0) and (a, 0). 5. Find tlie radius of curvature of the curve (-) +(-) =1 at the point (0, b). ^"^ ^^^ PROBLEMS 363 6. Find the radii of curvature of tlie curve y^ = ax{x — Sa) at the points where it crosses the axis of x. 7. Find tlie radius of curvature of the curve e'^ = sin y at the point (xi, yi). 8. Find the slope and the radius of curvature of the curve y + log(l — x^) = at the origin of coordinates. 9. Show that the radius of curvature of the curve r = a (sin 6 + cos 0) is constant. 10. Find the radius of cui^ature of tlie curve r = a (2 cos^ — 1). 11. Find tlie radius of curvature of the curve r = a sin^ - . Find the greatest and the least values of the radius of curvature. 12. Find the radius of curvature of the lemniscate r^ = 2a^ cos 2 8. 13. Given the curve x = 2 cos t — cos 2t, y = 2 sint — sm2 1. Find the radius of curvature in terms of t, and show that it will be greatest when t = Tr. 14. Find the e volute of the parabola y^ = ipx. 15. Find the radius of curvature of the tractrix a, a + Va2-x2 _ y — -log — ■ Va2 - x2. 2 a _ Va2 - x2 16. Prove that the evolute of the tractrix is the catenary. 17. Prove that the evolute of a cycloid is an equal cycloid. 18. Find the evolute of the four-cusped hypocycloid x = a cos^4>, y = a sin^^. 19. Find the evolute of the ellipse from the parametric equations x = a cos^, y = bsm (f>. 20. Prove that the center of curvature of any point of the logarithmic spiral is the point of intersection of the normal with the perpendicular to the radius vector. 21. Find the circle of curvature of the curve y = er'^ when x = 0. 22. Show that the catenary ?/ = ^(e^ + e-^) and the parabola 2/ = 1 + ^x^ have the .same tangent and the same circle of curvature at their point of intersection. 23. Find the point of minimum curvature on the curve y — log x. 24. Find the points of greatest and of least curvature of the sine curve y = sinx. 25. Find the points on the ellipse for which the curvature is a maximum or a minimum. 26. Show that the curvature of the parabola y = ax? + 6x + c is a maximum at the vertex. 364 CURVATURE 27. Find the condition for a maximum or a minimum of the curvature k, dx2 where A; = Mm' 28. At what points on tlie curve y = log sin x is the radius of curvature unity, and in what direction from the point on tlie curve is the center of curvature ? 29. Show that the product of the radii of curvature of the curve y — ae~^ at the two points for which x = ±ais a'^(e + e-i)^. 30. If the angle between the radius vector to the point of contact and the straight line drawn from the pole perpendicular to the tangent is either a maxi- ANSWEKS (The answers to some problems are intentionally omitted.) CHAPTER I Page 25 1. 9. 3. x-x2. 5. 17. 7. 1. 2. x-y. 4. 4. 6. -18. 8. 2aJc 9. ahc + 2fgh-aP-hg'^-ch'^. 11. Ix-Qy-b. 10. a62 + hc^ + ca^ - ac^ - ba^ - cb^. 12. 3. 13. 2aia2CiC2 + 01616202 + 02616301 — a^c^ — aScf — ai6,Jci — 0261%. Page 26 2^ 13 zfcVei 33. a;= 2, 2/=-2, z = f. 2 ' 34. a; = - 5, 2/ = 0, z = 4. 25. 0, 6 ± V39. 35. X = - 1, 2/ = 0, z = 0. 26. 8x + 2/-13 = 0. 38. x = i(a-26 + c + d), 27. x2 + 2/2 - X - ?7 = 0. 7/ = i (a + 6 - 2 c + d), 28. x2 + 2/2-3x + i/- 4 = 0. z = i(« + ^ + c-2d), 29. x2 - (a + 6) X + tt6 - /i- = 0. lo = { (- 2 a + 6 + c + d). 30. x3 - (a + 6 + c) x2 + (a6 + 6c 39. Xi : X2 : X3 = 3 : - 5 : - 2. + ca — /2 — |; (3) A:<|. A; = 0or4; (2) fc<0orA;>4; 0 — |. |(-3±3 V-3). 26. 2, -i, -1±V^, {{1±- 27. 0, H±l±Vl3). 0, 6, ± V^ ± V^. 28. 29. 30. 31. 0, -2ffl, — (-l±5i). 32. x3 - ax2 - (a2 + 6)x + a^ - ab ±{a + 1), ± (a - 6x3- 13x2 + 6x = 0. = 0. 33. x6 _ 4aj;5 + {4a2 - 62 - 26)x* + 8 aSx^ - (8 a26 - 263)x2 - q. 34. x2 - 4 X + 13^ 0. 35. (2x + 2-Vll)(2x+2 +VlT). 36. (2 X + 3 + V^) (2 X + 3 - V- 2). 37. (2 ax + J +4 V- 3) (2 ax + J - ^ V^). 38. (X + a + Va) (x + a - Va). ANSWERS 367 89. {ax + b + Va + 62) (ox + 6 - Va + ^). 40. (ax + b + i Vb) (ax + b - i Vft). 41.p2_29. ^^ p^-2q 46. p2-3r. 42. S^jg — p3. ^2 ' 47. j)r. 43. -P. 46 ^'^-^g . 48.^. 9 Q r Page 96 62. 1, ^(3±v^). 66. -3, H-l±^^^)- 70--f,-l±'^. 63. 2,2, -1. 67. ^, §, f. 71. - 1, _ 1, ± i. 64- 2, 2, - |. 68. i, I, - f 72. 3, - 2, f , -j. 65. -3, f, f. _ 69. I, -2dhV-2. 73. 2, |, 1±V^, 74. 4, - f , 1(3 ± V5). 80. 1, -2, -^,2 ± Vs. 75. ±f, J(3±V^>. 87. 1.41. 76. i, -5, 1(1 J.V6). 88. -1.52. 77. 2, 3, - 1, - 1 ± vCr2. 89. 2.05, .59. 78. -2, ±J, |(_l±Vr3). 90. 1.18,2.87. 79. I, ± # , _ 2 ± Vs. 91. .16, 2.93, - 2.09. CHAPTER V Page 118 1. 12. 3. 0. 14. (4|, 55f). 16. lOf. 2. -3. 11. 32x + j/ + 45 = 0. 15. tan-i^V 17. (2, 9), (- 2, 5). 18. 4x + 2/ + 2 = 0, lOSx + 27y + 58 = 0. 19. (-1, -6), (1, -512). Page 119 20. Increasing if x > — 2 ; decreasing if x < — 2. 21. Increasing if x < or^ > ^ ; decreasing if — V2 ; decreasing if x < — V2. 23. Increasing ifx>lor— l 1; , 54. 1.20, 3.13, -1.38. downward if x < 1. 55. 1.51, -1.18. 45. Upward if x > 0, downward if x < 0. 56. 57. 2, 2, - 3. 1, 1, - a ± Va-^ - a. 46. (2, - h). 58. b{b + 4a^) = 0. 48. (0, - 8). 59. 62(27a< -6) = 0. 49. (1, - 27). 60. ¥ - 27 a* = 0. 51. 0.45, 1.80, - 1.25. 61. See Ex. 23, Chap. I. CHAPTER VII Page 155 5. x2 + 2/2 ± 2ax = 0. 6. x2 + 2/2 -t 2 ax ± 2 ay + a2 = 0. 7. x2 + 2/2 + 3 x^ 2 2/ = 0. 8. (- 2, 5); V6o. 9. (-2, 3); 2 V3. 10. (3., - 1) ; 1 Vlil. 11- {~hi)> 0. 14. x2 + 2/2-3x-32/ = 0. 15. X2 + 2/2- 3x -4 = 0. 16. x2 + 2/2 + 26 X + 16 2/ - 32 = 0. 17. x2 + 2/^ - 5x + 4 y - 46 = 0. 18. x2 + 2/2- 20x- 202/ + 100 = 0, x2 + 2/2-4x-42/ + 4 = 0. 19. x2 + 2/2 - 12x - 12 2/ + 30 = 0, 25x2+25 2/2 -l-60x-602/ + 3G = 0. 20. x2 + 2/2 + 2x + IO2/+ 1 = 0, a;2 + y2 _ 12 X - 4 2/ + 15 = 0. 21. 2x2 + 22/2 + 6x + 3 2/ -10=0. Page 156 22. x2 + 2/2 + 22 X - 34 2/ + 121 = 0, 30. x2 + 2/2-2x- 102/ + 1 = 0. 31. 23. X2 + 2/2- 10x-28 7/ + 217 = 0. 32. 24. x2 + 2/2 + 22 X - 44 y - 20 = 0, 33. x2 + 2/2 + 2 X - 4 2/ - 20 = 0. 34. 25. 4x2 + 42/2 ±7y- 36 = 0. 35. 26. 7x2+ I6y2_ii2 = 0. 36. 27. 9x2 + 5y2_ 45 = 0. 37. 28. 5x2 + 92/2-180 = 0. 38. 29. I V385, i V105. 39. 40. ^ V2, i V3; (± iAo) age 157 41. 5x2-42/2-20 = 0. 48. 42. 3 2/2 -x2- 12 = 0. 43. 28x2-36 2/2- 175 = 0. 44. 24x2-252/2-384 = 0. 46. 3x2 - 2/2 - 3a2 = 0. 47. 8x2-2/2-16 = 0, 8y2_a;2_ 124 = 0. 16 x2 + 25 2/2 - 400 = 0. 5x2 4.9y2_80 = 0. 16 x2 + 25^2 _ 400 = 0. 196x2 + 132 2/2 - 14553 = 0. 4,3; J V7; (±V7, 0). J; 3x2 + 4^2 _ 3^2= 0. 6x2 + 9^2 _ 405 = 0. a;2 + 4 2/2 _ a2 ^ ; }, V3. \ V2. 3x2 + 52/2-30 = 0. ; 2 X ± Ve = 0. 48. x2 - 2/2 - 21 = 0. 49. x2 -82/2 + 4 = 0. 50. 25x2 _ 144 2/2 _ 3500 = 0. 51. 252/2- 9x2- 16 = 52. cos- 53. ^ V5, i V5. ANSWERS 360 54. 58. Page 61. 62. 63. 71. 72. Page 80. 81. 82. Page 94. 95. i -v^; (±v^, 0); 2x±5y = 0. iVl3; (±VT3, 0); 13x±9 Vl3 158 55. 52x2 _ll7y2_ 576 = 0. 57. 3x2-41/2-84 = 0. 0; 2x±3y = 0. 2^? V2. 67. y ± 5x = 0. 68. x2 _ 82/2 _6y + 9 = 0. 69. 2/2 + 4?/-2x + 11 = 0. 64. x2 + y2 _ 5px = 0. 65. x2 + 2/2 + 3x-6y = 0. 66. 7x- 32/ + 2 = 0. 70. 91x2 + 842/2 - 24X2/ - 364x - lb2y + 464 = 2/2 -10x + 25=0. 73. 4x + 32/-31=0, 4x+ 32/ + 19 = (2/ - 2)2 ± x8 + x2 = 0. 74. 5 X + y - 5 = 0, X - 5 2/ + 7 = 0. 159 Circle. 83. Concentric circle. Circle. 84. Straight line. Circle. 85. Straight line. 91. Parabola. 86. Circle. 87. Two straight lines. 90. Parabola. 93. Two parabolas. 160 Circle. Parabola. 96. Hyperbola. 97. Parabola. 98. Hyperbola, 99. Witch. 101. 8i)V3. CHAPTER VIII Page 175 1. (1, 1), (- 2, 3). 2. (0, 1). 4. (0,0), (-1, -2). 5. (1, 21). 6. (1,3), (I, -1). 8. (2 ± VB, 2 1^ V6). 9. ^6, Vl3. Page 176 18. (2, 2). 19. (0,0), (±V2, ±i>^) 20. (2, 2), (f, - I). 21. (0, 0), 1^- 22. (2, 1). . + »l2 1 + ,„2 10. 2x-?/ + 2 = 0. 11. -3. 12. 2x + 32/±6V2 = 0._ 13. bx-ay + ab± ab V2 = 0. 16. (0, 0), (4, li). 16. (±lf, ±5), (±li, ±3J). 17. (I, 0), (1, - 1). 23. (0, 0), (-1, 0). 24. (1, ±2V3), (6, ±6V2). 25. (± I- V48 T 3 Vl3, Q )• 26. {b - a, ± 2 Va6). 27. (± 2a, a). 28. 29. 30. 31. Page 35. 36. (0,0), (fa, ±faV2). (2 a, a). (±2a, a). 177 3x + y -I = 0. 5 a;2 + 5 2/2 + 28 X + 42 2/ (0, 0), (- 2 a + 2 a A ± 2 a V2 Vi - 2) 32. (±2 a, a). 33. 5x + 42/ + 6 = 0. 34. 2x + 5y-13 = 0. 38. 3x2 +3^2 + 13a; +13?/ -4 = 0. 39. x2 + 82/2 -9 = 0. 370 ANSWERS CHAPTER IX Page 209 1. 9x2 + 14x + 6. 2. 20(x + l)(3x2 + 6x + 2). 2a (X + o)2 6x2 (X8 - 1)2 14 (1 - X) (3x2-6x + l)a' 1 ^^^^V|-^> (X + 1)2 3Vxi 8. 2(4x 9 3) + _(6_7x). x + 1 2xVx 10. l/A_JL_J_ + _^> A^x <^2 a;^x x^x2y 11. 2(3x2_5x+6)(6x- 5). 12. 6x(x2 + l)2. IS «* + ^ 14. 2 V4x2 + 6x-6 2x + 1 16. - 16. - 17. - 3 v^(x2 + X - 1)» 2x (X2 + 1)2 6(3x2 ^-2x) (X8 + X2 + 1)2 5x \^(X2 + 1)6 18. 5(2x-l)(2x-3)(x + l)2. 19. (3x- 6) (12x2 -55 X +31). 20 2^' + ^ + l 21. Vx2 + 1 3x*-10x3 + 6x2 + x-2 (x2-4x + 3)*(x3 + l)l 22. V_L=-f-i=V 2VVx + l Vx-1/ 23. 1 + ^ . Vx2 + 1 24. 2x/ 25. 1 1 30. 31. 32. 33. 34. 41. 42. .\^(3X2+1)2 \/ (3x2+1) 1 .) (X + l)Vx2-l X + 1 27. 28. 29. (X - 1)2 Vx2 + 1 X-X2 (X2 + l)i(x8 + 1)* 2x2 + 1 Vl + X2 -2x. a-' (a2 - x2)^ X — Va2 + x2 cfi Va2 + x2 4 X (2 y2 _ x2) 2/(3 2/ -8x2)' 5x^-3x2 52/* -1 3 [x2y* + (X - 2/)2] 3 (X - 2/)2 - 4 x^ys ' 3x2- 5 x< 4y8_i 35. 36. 37. 38. X + Vx2 - 2/2 y V X - 2/(x - 2/)2 _6x. 25 2 y ' 2 2/3 x6 6 a^x^ y 18 6* 39. ; - 40 6a*y 3 2/2 - a2 (a2 - 3 y^f 3x2 6x(l-32/2) 32/2 + r (1 + 32/2)3 ' 2(3x2-2 2/). 128x2/ 3 j/2 + 4 X (3 2/^ + 4 x)3 ANSWERS 371 Page 210 43. tan., 3x + y + 6 = 0; nor. ,x — 3y-\-b = Q: tan. , 3x — 2/4-9 = 0; nor. ,x + 3y — 7 = 0. 44. X - 6 y + 17 = 0, 6 X + 2/ - 9 = 0. 45. x + 2y - 2 = 0. 46. 4x-3?/-l = 0. 47. x + 2y -1 = 0. 48. (Sy^-xi)y-yiX-Xiyi-Sa=0. 49. X - 3 y^y + 2 Xi - 3 = 0. 50. 2yiy -3x^x + xl = 0. 51. (2 - xf)x + xfy - 3xi = 0. 52. xi~*x + yi~^?/ = a*. 53. Xi~^x + 2/i~i?/ = a^. 54. 52/ + Vl0x-6 VH = 0, lOy- 5V1OX + 4 V5 = 0. tan., 4x — 2/ — 6 = 0; nor., x + 4y — 10 = 0: tan., 4x — 2/ + 6 = 0; nor. , X + 4 2/ + 10 = 0. X 65 66 dy ax 61, dy dx (-0.57,2.08) Page 211 62. (±ia V2, ± (1 '' , lbV2] 76. 62 N^ a22/ix - 62xi2/ = ; o262 63. ^6^x2 TT . -, tan 4 73. V Va2 + 62 Vp(p + Xi). 2+62/ 77. + a*y{ age 212 78. 79. 80. 0, tan-i |. TT 2' TT 3' 86. -, tan-i|. 2 * 86. tan-13. 88. -, tan-i V2. 2 90. tan-i|. 91. 0, tan-isVs. 92. 0, -, tan-ijv^. 93. The length is twice 81. tan-1 5 Vs. 89. tan-1/5. the breadth. 94. He walks 2.86 mi. Page 213 95. 8rd., 12 rd. 96. Cross section is a square. 97. Of equal length. 98. 4 mi. from nearest point on bank to A. 99. jFD = (V2 -\)AB. 103. Area of ellipse is - area of rectangle. 104. Central angle of sector is |7r VO. 105. Breadth a, depth a Vs. 106. Breadth, f a V3; depth, fa V6. Page 214 107. Velocity in still water \a mi. per hour. 108. Radius of base equals alti- tude. 109. Altitude is | radius of sphere. 110. a- 6m Vn2 - m2 6n mi. on land, Vna" mi. in water. 372 ANSWERS 111. Altitude is J '^ radius of semicircle. 112. Altitude is jr distance between vertex of pai-abola and bound- ing straight line. 113. (&, a). 114. (±|V3, I). 115. maximum value when x = i, minimum value when x — \; points of inflection when x= — 1 1±V6 or 5 122. Stationary when < = 0, 4, or moving backward when 4 < 123. 20, 10 V5; (100,20). 116. minimum ordinate, x = — -\ V3 maximum oixlinate, x— — points of inflection (± a, 0). 117. (±i V 3, I). /± - V27 -3 V33, \ 6 V3 (0, 0), 119 ± :^ V5 V33 12 120.(±|V^,±^>^). 121. (1, 3), (5, - 5). 8 ; maximum velocity when t t<8. rii) = 1. Page 215 124. Whenx= i; parallel. 126. Velocity of top: velocity of bottom = distance of bottom from wall distance of top from ground, 126. 54 TT cu. ft. per hour. 132. 3 y = x» + 9z - 19. 127. 389 ft. per minute. 133. Sx-2xy -2 = 0. 128. 41.9 ft. per second. 134. 3y = 2x^ + U. 129. 15 ft. per second. 130. 0.2 hi. per .second. 131. 17.9 mi. per hour. 136. - -1 = A;(l-x). y 137. 90,000, 677iV- Page 216 138. 108. 140. hk n + 1 141. J «2. 142. 851. 143. K^ a^. 144. 10|. 145. 17 1 CHAPTER X. Page 225 1. {- 1, 5), (- 7, 7), (2, - 5). 2. a;2 + 4 2/2 - 4 = 0. 3. 2/^ - 15 ?/2 -f 3 x2 + 75 y - 6 X - 106 = 0. 4. 62a;2 ^ a2y2 _ 2 a^by = 0. 5. 62x2 4. a^yi _ 2 ab^-x = 0. 6. 62a;2 _ a2y2 - 2 ab^ = 0. X (a + x)2 7. 2/2 = - Page 226 2a + X 10. y = 4 a3 - ox" x2 + 4 a2 ' 11. y.^ (2a + x)8 8. 2/'' = 9. 2/ = (X - a)2(2 a-x) 2ax2 xa + 4 o2 12. 2/^ = (^±^. a-x 13. y^ = 4px -f 4p* 14. 2/2 = 4px — 4j)'. ANSWEES 373 16. 2x2 + 32/2-6 = 0. ■ 4a 4 6 n.ah-c. 20. x2 + 9^2 + 6a;_36y + 36 = 0. 21. 196 x2 + 900 2/2 + 784 X + 5400 y + 8875 = 0. 22. x2 - 42/2 _ 2x- 162/- 19 = 0. 25. x2 - 8x + I62/ - 64 = 0. 23. 2x2-62/2-8x + 36y-47=0. 26. 3x2 + 42/2- 12x - 242/ - 27=0. 24. 2/2 + 42/-8x + 28 = 0. 27. 8x2 + 92/2 - 16x - 64 = 0. Page 227 28. 5x2 - 42/2 + 10x- 162/ -31 = 0. 29. 2/2 + 4 2/ - X = 0. 30. i V5j (- 1, 2) ; (2, 2), (- 4, 2) ; (- 1 ± V5, 2) ; 6x + 5 ± 9 V5_= 0. " 31. I VlO; (-3, 2); (-3±V5, 2); (-3±V2, 2) ; 2x + 6 ± 5 -^2 =_0. 32. i Vl3; (3, -4); (5, -4), (1, -4); (3±vT3,-4); 13x-39±4 Vl3 = r, ; 3x-2y- 17 = 0, 3x + 22/- 1 = 0. 33. 1 VlO; (-1, 2)_; (-1 ±V2, 2); (- 1 +V5, 2); 5x + 5i:2 V5 = 0; V3(x + l)±V2(2/-2) = 0. 34. (-1, ^); (-^, -I); 3x + l=0; 82/-7=0. 35. {- 2, - 3) ; (- I, - 3) ; 2/ + 3 = ; 4x + 13 = 0. 36. (i V3, i), (i, - 1 V3), 40. x2 + 14 2/2 - 14 = 0. /I + V3 1 - V3\ *2. xy = - 18, or xy = 18. y ^' — 2 — ;■ ^^- 17x2 + 7 2/2-2 = 0, 37. x2- 42/2 -4=0. or 7x2+ 172/2-2 = 0. ,„ , x2(3aV2-2x) 44. x2±2/ = 0, or2/2±x = 0. ^^.y^ = -^ ^ : 45.2x2-2/2-1 = 0. 3 a V2 + 6x Page 228 46.5x2 + 8 2/2 = 40. 48. 1^ + VV ^ (^^^ - ^ + V 47. 4 X2/ = 7. ■ \x - 2/7 aV^ + x-y 49. 5x2-62/2 = 30. CHAPTER XI Page 244 1. Hyperbola; center, (—7, 2); slopes of axes, 2 and — \. 2. Parabola; slope of axis, \; vertex, (||, — 4if). 3. No curve. 4. Hyperbola ; center, (—1,0); slopes of axes, 1 and — 1. 5. Hyperbola; center, (2, — |); slopes of axes, 1 and — 1, 6. The line x — 2/ + 1 = taken twice. 7. Elliijse; center, (— 1, 2); slopes of axes, 1 and — 1. 8. A pair of straight lines intersecting at (3, — 2), and having the slopes - 1 ± i Ve. 9. A pair of straight lines intersecting at (3, — 2), and liaving tlie slopes § and — |. 10. Parabola; slope of axis, 1; vertex, (— 4^, — \\). 374 ANSWERS 11. The parallel straight lines x + 3y-5 = 0, x + 3y-l = 0. 12. Ellipse; center, (2, - 1); slopes of axes, f and - |. 13. Point, (0, 2). Page 245 20. tan-i : A + B 23. 2x2 + 3xy + 2/2 + 12x-13?/-50 = 0. 24. xy - 2r/2 _ 2x + 4y = 0. 25. x^ - xy + y"^ - a"^ = 0. 26. 6x2 + 5x2/ + 2/* - 29x - 13?/ + 30 = 0. 27. 9x2-12xy + 4y2_ii7x + 78y + 380 = 0, or 49x2 _ 66 xy + 16i/2 - 621x + 35 4y + 1964 = 0. V. -Jtan2^-1 1 - tan2^ if tan ^ < 1, if tan ^ > 1 (/3 the angle between 2 2 ^ 2 28 tan- the lines). 2 CHAPTER XII Page 262 1. 6x-y + 3 = 0; (0,3), (-1,-2). 5.(0,3). 2. X + y - 3 = ; (0, 3), (1, 2). 6. (|, - \). 3. X- 2/ + 1 = 0; (-1,1). 7.(2,3). 4. 2/-2x + 6 = 0;(l,-3),(2,-l). 8. (1, -2). Page 263 9. 8x-2y = 0,x-2/ + l=0. 14. x - 3?/- 2 = 0, 2x - 2/ + l = 0. 10. x = 0, x-2/ + l=0. 21 -. 11. 3x + 2/-l = 0. ■ a2 12. 2 X - 2/ = 0, X + 2 2/ - 10 = 0. 24. At infinity. 18. X + 2 2/ - 2 = 0, X - 3 y - 2 = 0. 25. 6ex - oj/ = 0, bx + aey = 0. Page 264 30. -e. 81- 0^^ = 2 62. Page 265 47. (ae, ± ^) ; tan-J(±e). CHAPTER XIII Page 297 21. a cos 2 ax. 22. o [sec* (ax + h) ctn (ax + c) - tan (ax + &) esc* (ox + c)]. 28. -8csc24x. 26. sec2x. 2(2ctn2x — 1) 27. mn sec™ nx esc" wx (tan ?ix — ctn mx). csc2x 28. 2sec2 2x(2tan2x + l). 3sec3x(tan3x-l) 29. - 2 csc2x(2 csc24x + ctn 4xctn 2x). (tan3x + l)2 30. cos(xcosx) (cosx — xsinx). ANSWERS 375 31. 5sin2xcos3x. 49. 2(x + l)e^ + 2»:. 32. 8 sec^x — 3 sec x. 2 a^ 33. — — cosx. V2 X* — a* 51. e^T^^^cF^^i-^^^ ^—\ V(l - X2)* Vl - xV 34. — == sin Vl-x2. V(l - x2)^ 36. secxtanx. Vx^Tx ^■^,- .,...VH(i-!5?). 37. . 64. a'ana^loga- sec^x. Vl-x2 55. 2x{l + 21ogx). gg 1 66. loga-sec2x(sec2x + 2tan2x)at''n3-sec»a:. (1 + x) Vx 67. 2sec2x(sec2x + loga- tan22x)a«e«''^ 1 68. [2{a + x)sinmx + m cosmx]e(« + =')^ 69. -2. 39 Va2 - x2 40. a + X \x ^ ^ ' -, 1 ox. 41. — • e2a: + e-2x (x + 1) vx 1 61. 62. 63. 42. - • 3 + 5sm2x (x2 + 2 X) Vx2 + 2 X - 1 ^^ 2 43. 0. 44. ^^ 64. e^<:os«cos(a + xsina). 65. 45. - • sm(2x2 + 2a2) 66. cos-^x. 46. 47. 48. - a + b cos X 2 Vl-X2 1 (2x-l) Vx2-x 1 2x2 + 2x + l 4x CO 4x 67. sec ax. 68. (a2 + l)e««i""*'". 69. ctn-ix. . cos-^x 70. V(l - X2)8 X* + 1 71. ef^sinmx. Page 298 72. 2xctn-i«. ^g_(x-l)log(x_D. * (x2-2x)3 sin-ix 1 — :;i 77. 78. a + b cos X 74. ^ _ 1 78. 4csc(4x + 2)[l-ctn(4x + 2)]. Vx2 - Ofi Va2 - X2 Yg ^ . 76 1 jx + a X \x — a' V2 ax - a;2 80. 2 Va2 + x2. 376 ANSWERS sec- 12 Vx 91. 2/x^(l + loga;). • Vx(4x-1)3' 92 yr tan-i(a + x) ^ log(a + x) -| 82- V2ax-x2. L « + ^ l+(a + x)d' 83. 2 - Vx2- 1 2/ (tan xy ) 2x(x2-l) ■ 93 84. - , log X — X tan xy y (e^ + 2) Ve2^+ 2e^-l ^^^fTxa (e2x +e-)log(e^ + 2 )^ 94. ^^^^^^_,^ V(e2 a: + 2e x_i)3 ^^ y C08X + sin (x - y) 85. X tan-i Vl — x^. ' sin {x — y) — sinx 86. ^^i±^'. ' 96. '"^ 2 + x2 {ny + \)x V r r nH e2' sin X + e^ sin w 87. -^(sec2 Vxlogsin Vx + l). 97. ^. 2 Vx ^ ° 01 cos X — e* cos y y a» y(l -cos x)- cosy 88. i;^(xctnx-logsinx). M- ^i^^ _ ^(i + ,5^^)' . 89. yx^fl + logx + (logx)2l. 99. ^^^^V -V L* J X log X — x2 cos y 90. ye^(l + logxV IQQ. Vd-^^-y^) \a^ / x(l+x2 + y2) Page 299 101 ^ + ^ , 2(x2+y2) 4(x + 2y)(x2 + y2) ■ X - y (X - y)3 (x - y)5 102. - e^-y, - e*-J'(H- e^-p), - e^-2'(l + e^-») (1 + 2e*-!'). 103 ^-y' 2(x + y)(y2-l) 6 (x + y)2(l - y^) ■ X2 - 1 ' (X2 - 1)2 ' (pfl - 1)3 104 ^ + y -^ 4 (X + 2/) 8(x + y)(l-2x-2y) ■ x + y + l' (x + y + l)3' (x + y + l)6 jQg y-g y(l+logx)-2x 2y[l + logx + (logx)2]-3x(l+logg) ■ x(l-logx)' [x(l-logx)]2 ' [x(l-logx)]3 - 1 ± V33 106. X 8 107. 1 or 2. ,,„ , o ; , '^ 112. X = «7r, X = 2 KTT ± - • 108. tan-i2V^. 3 113. (ae^ fae-^). 109. tan-i(2tan-sec-V \ 2 2/ 110. tan-4, tan-13. 114. (^^--*) 116. Maxima when x = (2 A; + 1)-, minima when x — kir] points of inflection -when X = (2fc + !)-• 117. Maxima when x =(2fc + |)7r, minima when x =(2*; + |)7r; points of in- flection when X = kit. ANSWERS 377 118. Maximum when x_= n, minimum wlien x = 0, (n = 2fc): points of inflection when x = n ± Vn, (n ?i 1) ; x = 2, (n = 1 ); x = 0, (n = 2 k + 1). 119. Circle. 120. 2 V(s - 3) (5 - s), 4(4-8). Page 300 122. a. 123. ab 128. 500 sin a ; , where x is 2 V62_a2t2• 126. -bsind 62 sin 5 cos ^ times Vl0,000-x2sin2a the distance from the center. 1 29 . 1 5 sq. ft. ; 9. 03 sq. ft. per second. Va2- 62 sin2 e 130. each. velocity of AB. 1 where 2 2 e = CAB. 131. 23.7. 500 132. At an angle tan-^fc with the 127. , wl lere x is the Vl0,000-x2 ground. distance from the center. Page 301 - 133. 134. Sin. 1:V2. 145. log- = ft(x-a). 6 135. 5V5ft. 146. s = ce^'. 136. 2. 147. A7r + (-l)«^.6661. 137. V2-I. 148. 2fc7r ± .567, 2kTr ± 2.206. 138. 2. 149. ktr. 139. e-1. 150. fcTT, (2A: + l)j, 2kir±1- 4 140. e 151. A;7r ±^, ^^'ri -• 4 141. log 2. 142. X = 2 a. 152. 4.4934. a j '^ *\ 163. 4.275. 143. - (ga _ e «) -f c. 154. 0.199. « 155. -0.7085. 144. 2/ - 3 = & log - . 2 166. 1.857, 4.54. CHAPTER XIV Page 323 2. tan., X - iy + p«2 = ; nor., tx + y -2pt- pt^ = 0. 4p 4p " 3. X = — ^» 2/ = <2 ' ^ i 4. tan., <2x _ 2 «y + 4p = ; nor., 2 t^x + t^y - ipt^ -Sp = 0. ab , aht 6. r.=z± —— y = J:— — y6-2 + a2«2 V62 + a2«2 _ a(62- OT2a2) _ 2a62m ^ ■ ^ - 62 + m2a2 ' ^ ~ 62 + m2a2 ' . „ 2asin3(^ 7. x = 2asm2<^, y = COS0 10. (l + 3«2)x-2i3y-2a=:0. 378 ANSWERS Page 324 ■ ^ 2x-a 3i2 + l 12. a;3 + 1/3 _ 3 oxy = 0. 16. (3 y - x)2 = 2 Vx2 - 2/2. 13. 2/2 (ax - a2) = x2 (a2 + A;2 - ax). 17. 9 y = (x - ?/)2 - 6 (x - 2/). 14. 3 2/2 - 4 xy + 2 2/ - 1 = 0. 18. (x2 + y^) (ox + 62^) = cxy. /— - — tan' ■ll 19. Vx2 + 2/2 = a V2 e^ o^ a/.„^^\. 2x + a ja-x 20. X = a cos2 - , 2/ = - I sin ^ + tan - ; ?/ = \ ~ 22\ 2/ 2\x 21. x = (a — c tan^) sin2^, 2/ = (acosfl — csin^)sin^; 2/(x2 + 2/2) = x(a2/ - ex). 22. X = - (a2 + A;2 cos2 ^), y = - {a^ tan » + fc2 gin cos fl) j a(x- a) (x2 + 2/2) = i2a;2. a{a2 — x2) 23. X — a tan ^, 2/ = a cos 2 ^ ; 2/ = -^; r-^ • a2 + x2 Page 325 24. X — a sin d (cos ^ + sec 6), y = a cos ^ (cos d + sec tf) ; 2/(x2 + 2/2) = a(x2 + 2 2/2). 29. 6x cos + a2/ sin <^ — a6 = 0. 30. !^. 31. !r + ^. 32. ^sin-i?-^ 4 4 2 2 «„^ 8,. ,,„.,'l±:^<±lMs:^\ gb 34. X = a cos ^ + i V62 _ a2 sin"^ S, y = {I — I) a sin ^, where the center of the driving wheel is the origin, a the length of the radius of the driv- ing wheel, b the length of the connecting rod, and lb the distance along the rod from the wheel to the point. 35. Straight line. 37. 2 aw sin - ; w Va^ — 2 aAcos + ^2^ where u is the constant angular velocity. Page 326 38. 2 au sin - , w v a'*' — 2 ah cos 6 + h^. ■ 2 40. adw, where a is the radius of the circle, ad the distance through which the point of the string in contact with the wheel has moved along the rim of the wheel, and w the constant angle of velocity. 41. X = a(cos2e + sin2tf), 2/ = a(l + sin2^ - cos2d) : 2 V2 aw, 2 a (cos 2 ^ — sin 2 ^) w, 2 a (sin 2 ^ + cos 2 ^) u. ^„ „ x2(3a + x) 48. The witch. 43. 1/2 =: ^ • a — x 49. X = a (cos 6 + 0sm 0), 43. Circle. y = a (sin — cos 0). 46. ?/ (x2 + 2/2) = 2 a (x2 + 2 2/2). 50. x = a(l 4- m2), ?/ = ma (1 + ?n2) ; 47. y = x — a. ay^ = x2(x — a). ANSWERS 379 Page 327 53. Ellipse. 54. Hyperbola. 55. Straight line. 56. Concentric circle. 57. z + 2p-0, py^ = afi. 59. Concentric ellipse. 61. Ellipse. Page 328 65. Parabola. 67. Concentric circle. 70. Concentric circle. TTX 71. y = xctn 2a 72. x = a{4) + sm), y = a{-l + cos^). 74. 8 a. 8a{a + b) b 75 Page 26. 27. 28. 349 CHAPTER XV (l 0353 a, ^y (l^-l)'(i^'T)- 29. (0,0),(a,±^),(a,±^) 80. Circle. 31. r = acos2^. a cos 2 32. r coa^9 Page 350 40. r^ sin2(? cos ^ + (2 a + r cos 6)^ = 0. 41. r cos 61 = a cos2 (?, (r^ + a^) cos^ + 2 ar = 49. (x2 + 2/2)3 _ 4 a-^x^y^ = 0. 60. (x2 + y^ + ax)2 - a2(x2 + y"-) = 0. 61. (a;2 4- 2/2)2 _ 2 a2 (x2 _ 2/2) + a* - 6* = 0. 82. {X2 + 2/2 _ ax)2 = 62(a;2 + 2/2). 63. (X - a)2 (x2 + 2/2) = &2x2. 54. log(x2 + y2)-2atan-i^- 56. 25,000,000. 57. 1,200,000, or 4,800,000. Page 59. 61. 62. 63. 2c 64. r = 351 r = - 1 — costf Straight line. Circle. Circle. 2 ce cos ^ 65. r = 66. 2r = ce2 cos d — eP'CO&^B ce 1 — e cos B 72. «/'(«), « Vt/(«)]2 + [/'(fl)]2. 1 - e2 cos2 ^ 78 ^r(e) V[/(e)]2 + [ne)f V[/(fl)]2 + [/'()s, a(6-4co8g)8 27p^ 9-6cos« i« a Va2 - .r^ 3 fl ^ 11. -asin2-; greatest value, - a ; 15. -U.0111--; t-iettLest value, -a ,o , .j ,i .-i 4 3^ '4 ' 18. (a; + 2/)J + (j-- ?/)1^ 2.0 least value, 0. 21. «2 ^ ^2 _ ^ _ q. 12. 2-^. 3r Page 364 23. Minimum curvature when x = 24. Maximum curvature when x = (2 A; + 1) — ; minimum curvature when x = krr. 25. Maximum curvature at ends of major axis; minimum curvature at ends of minor axis. L \dx/ J dx3 dx \dxV 28. (2fe + l)|. INDEX [The numbers refer to the pages.] Abscissa, 36 Acceleration, 202 Addition of segments of a straight line, 32 Algebraic functions, 43, 121 differentiation of, 178 implicit, 188 Angle between two lines, 57 between two curves, 211 eccentric, 304 vectorial, 329 Angles, 65 Arc length of, 195 limit of ratio to chord, 195 derivatives with respect to, 197, 347 Archimedes, spiral of, 332 Area, 204 of an ellipse, 304 in polar coordinates, 348 Asymptote, 128 of an hyperbola, 145 Auxiliary circle of ellipse, 304 Axes of an ellipse, 141 of an hyperbola, 145 Axis, of symmetry, 121 of a parabola, 147 radical, 175 Bisection of a line, 39 Cardioid, 337 Cassini, ovals of, 338 Catenary, 281 Center of a conic, 238 Change of origin without change of direction of axes, 217 of direction of axes without change of origin, 221 from rectangular to oblique axes without change of origin, 224 from rectangular to polar coordi- nates, 341 Chord of contact, 248 Circle, 134 through a known point, 136 tangent to a known line, 136 with center on known line, 137 through three known points, 138 parametric equation of, 303 involute of, 311 polar equation of, 342 of curvature, 354 Cissoid, 151 Classes of functions, 43 Coefficient of an element of a determi- nant, 8 Collinear points, 38 Complex numbers, 31 roots of an equation, 82 Components of velocity, 200 Concavity of a curve, 112 Conchoid, 334 Conic, l48, 229 classification, 237 through five points, 241 polar equation, 343 Conjugate complex numbers, 32 axis of an hyperbola, 145 diameters, 258 hyperbolas, 262 381 382 INDEX Constant, 40 of integration, 206 Contact, point of, 105 chord of, 248 Continuity, 101 Coordinate axes, 35 cliange of, 217 Coordinates rectangular, 35 transformation of, 217 oblique, 223 Cartesian, 224 polar, 329 relation between rectangular and polar, 341 Curvature, 353 radius of, 354, 360, 361 circle of, 354 center of, 356 Curve, Cartesian equation of, 44 slope of, 99 degree of, 166 parametric equations of, 302 polar equation of, 330 Curves, intersection of, 161 Curves of second degree, 229 Cycloid, 305 Degree of a curve, 166 Depressed equation, 79 Derivative, 102 of a polynomial, 97, 103 sign of, 106, 111 second, 110 higher. 111, 187 theorems on, 179 of w, 185 illustrations of, 203 ■with respect to an arc, 196, 347 Descartes' rule of signs, 87 folium of, 132 Determinants, 1 elements of, 4 minors of, 4 properties of, 6 expansion of, 8 Diameters, of a conic, 252 of a parabola, 254 of an ellipse, 256 of an hyperbola, 257 conjugate, 258 Differentiation, 102 of a polynomial, 103 of algebraic functions, 178 successive, 187 of implicit functions, 188 Differentiation, formulas of for a polynomial, 103 general, 184 for M», 185 for trigonometric functions, 272 for invei-se trigonometric func- tions, 276 for exponential functions, 284 for logarithmic functions, 284 for hyperbolic functions, 290 for inverse hyperbolic functions, 291 Direction of a curve, 197 in polar coordinates, 345 Directrix of a parabola, 146 of a conic, 148 of an ellipse, 149 of an hyperbola, 149 Discontinuity, 101 examples of, 41, 128, 268, 282 Discriminant, 117 of a quadratic equation, 73, 117 of a cubic equation, 114, 117 of the general equation of the sec- ond degree, 236 Distance between two points, 36 of a point from a straight line, 63 e, the number, 280 Eccentric angle of ellipse, 304 Eccentricity of a conic, 148 Elasticity, 204 Elements of a determinant, 4 Eliminants, 23 Elimination, 1 INDEX 383 Ellipse, 139 referred to conjugate diameters as axes, 269 parametric representation of, 303 Energy, kinetic, 203 Epicycloid, 307 Epitrochoid, 309 Equation in one variable solution by factoring, 77 with given roots, 79 depressed, 79 number of roots of, 80 sum and product of roots of, 82 complex roots of, 82 solution of, 89 Newton's method of solution of, 114 multiple roots of, 116 resultant, 161 Equation of a curve, 44 Equations in several variables, linear, 1 systems of, 12 homogeneous, 21 Equations in two variables of first degree, 52 of second degree, 229 Equations, transcendental, 293 Evolute, 357 Expansion of a determinant, 8 coefficient of, 204 Explicit algebraic function, 188 Exponential functions, 279 differentiation of, 284 Factoring, solution of equations by, 77 of quadratic expressions, 79 Factors and roots of an equation, 78 of a polynomial, 81, 83 Foci of an ellipse, 139 of an hyperbola, 142 Focus of a parabola, 146 of a conic, 148 Folium of Descartes, 132 Force, 202 Function, 40 Functional notation, 44 Functions, classes of, 43 algebraic, 43, 121 irrational, 44, 131 transcendental, 44, 266 defined by equations of the second degree, 127 involving fractions, 128 trigonometric, 266 inverse trigonometric, 269 exponential, 279 logarithmic, 279 hyperbolic, 288 inverse hyperbolic, 291 Graph, 40 Graphical representation, 28 Harmonic property of polars, 249 division of a line, 250 motion, 275 Homogeneous equations, 21 Horner's method, 92 Hyperbola, 142 equilateral, 146 referred to asymptotes as axes, 224 referred to conjugate diameters as axes, 259 conjugate, 262 Hyperbolic functions, 288 inverse, 291 differentiation of, 290, 292 Hyperbolic spiral, 332 Hypocycloid, 309 four-cusped, 132 Hypotrochoid, 309 Imaginary numbers (see Number, Complex) Implicit algebraic function, 188 Increment, 100 Infinity, 29, 128 Inflection, points of, 112, 194 Initial line, 329 Integration, 205 Intercepts, 53 384 INDEX Interchange of axes, 223 Intersection of curves, 161 number»of points of, 169 Involute, 357 of circle, 311 Irrational number, 28 algebraic functions, 44, 131 roots of an equation, 92, 114 Isolated point, 126 Kinetic energy, 203 Latus rectum, 211 Limit of ratio of arc to chord, 195 . sin ft , 1 — cos h _^- of and —, 270 h h of (1 + h)'' and 1 283 Limiting cases of a conic, 234 Limits, 97 theorems on, 178 Locus, 45 Locus problems, 316 Logarithm, Napierian, 280 Logarithmic function, 279 differentiation of, 284 spiral, 333 Lemniscate, 340 Lima^on, 336 Maxima and minima, 108, 112, 192 Minors of a determinant, 4 Momentum, 203 Motion, uniform, 199 harmonic, 275 Multiple roots of an equation, 116 Napierian logarithm, 280 Newton ■;S method of solving numerical equations, 114 Normal, 64, 191 Normal equation of straight line, 64 ' Number, real, 28 complex, 31 Oblique coordinates, 223 Ordinate, 36 Ovals of Cassini, 338 Parabola, 146 referred to tangent at ends of latus rectum, 132 referred to a diameter and a tan- gent as axes, 255 cubical, 74 semi-cubical, 131 Parallel lines, 56, 59 Parametric representation of curves, 302 Pedal curves, 319 Perpendicular lines, 57, 59 Plotting, 36, 329 Point of division, 38 Polar of a point, 247 Polar coordinates, 329 Polars, reciprocal, 251 Pole of a straight line, 247 of a system of polar coordinates, 329 Polynomial, 43 of first degree, 50 of second degree, 70 of nth degree, 74 factors of, 81, 83 derivative of, 97 \ square root of, 121 Problems on straight lines, 58 Products, graphs of, 83 Projection, 34 Radical axis, 175 Radius vector, 329 of curvature, 354, 360, 361 Rate of change, 203 Rational algebraic function, 43 number, 28 roots, 89 Reciprocal polars, 251 Resultant, 23 equation, 161 Roots of an equation and factors, 78 INDEX 385 Roots of an equation number of, 80 sum and product of, 82 complex, 82 location of, 86 rational, 89 irrational, 92, 114 multiple, 116 Rose of three leaves, 331 Rotation of axes, 221 Slope of a straight line, 64 of a curve, 99 Solution of simultaneous equations, 12 of algebraic equations, 77, 89, 114 of transcendental equations, 293 Spiral of Archimedes, 332 hyperbolic, 332 logarithmic, 333 Straight line, 50 satisfying two conditions, 58 normal equation of, 64 parametric equations of, 302 polar equation of, 342 Strophoid, 152 Subnormal, 210 polar, 351 Sub tangent, 210 polar, 351 Supplemental chords, 264 Sylvester's method of elimination, 24 Systems of curves with common points of intersection, 171 Tangent, 73, 84, 104, 190 to a conic with given slope, 163 to a conic at a given point, 246 Tractrix, 299 Transcendental functions, 44, 266 equations, 293 Transformation of coordinates, 217, 341 Transverse axis of hyperbola, 145 Trigonometric functions, 266 differentiation of, 272 inverse, 269 differentiation of, 276 Trochoid, 306 Turning points of a graph, 107 Variable, 40 Variation of sign, 87 Vector, radius, 329 Vectorial angle, 329 Velocity, 198 components of, 200 Vertex, 71 of a parabola, 147 Vertices of an ellipse, 141 of an hyperbola, 144 Witch, 149 Zero, 29 Date Due m PRINTED IN U. S. A. \ 000 584 802 3 The RAIPH D. IKED LIBRARf DEPARTMENT OF GEOLOGY UNIVKRSrrY of CALIFORNIA 1-08 ANGKLES. CALIF.